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b(means,)j(in)d(particular,)h(that)i (in)e(the)h(cen)m(ter)h(of)165 1207 y(a)f(large)f(b)s(o)m(x)g(w)m(e)h (will)c(t)m(ypically)i(encoun)m(ter)i(more)f(or)g(less)g(the)g(same)h (con\014gurations)e(no)h(matter)165 1320 y(what)37 b(b)s(oundary)e (conditions)h(outside)g(this)g(b)s(o)m(x)h(are)h(imp)s(osed.)59 b(That)37 b(is,)h(if)e(w)m(e)h(compare)h(t)m(w)m(o)165 1433 y(indep)s(enden)m(t)d(realizations)i(of)g(the)g(system)h(in)e(the) h(b)s(o)m(x)g(with)f(di\013eren)m(t)g(b)s(oundary)g(conditions)165 1546 y(outside)25 b(then,)i(still)c(at)j(high)e(temp)s(eratures,)j(the) e(di\013erence)g(b)s(et)m(w)m(een)h(the)g(b)s(oundary)e(conditions)165 1659 y(cannot)32 b(b)s(e)f(felt)h(b)m(y)f(the)h(spins)d(in)h(the)i(cen) m(ter)h(of)e(the)h(b)s(o)m(x;)g(sp)s(eci\014cally)-8 b(,)30 b(there)i(should)e(not)h(exist)165 1772 y(an)m(y)39 b(path)f(from)f(the)i(b)s(oundary)d(to)i(the)h(cen)m(tral)f(part)g(of)g (the)h(b)s(o)m(x)f(along)g(whic)m(h)f(the)h(spins)e(of)165 1885 y(the)29 b(t)m(w)m(o)h(realizations)e(disagree.)40 b(This)27 b(picture)g(is)h(rather)h(robust)e(and)h(can)h(for)g(example) f(also)h(b)s(e)165 1998 y(applied)g(when)g(the)h(in)m(teraction)h(is)e (random;)h(see)h(Sections)f(7)h(and)f(9.)306 2111 y(A)m(t)h(lo)m(w)e (temp)s(eratures,)g(or)g(large)h(densities)d(\(when)i(the)g(in)m (teraction)g(is)f(su\016cien)m(tly)g(strong\),)165 2223 y(the)g(picture)f(ab)s(o)m(v)m(e)i(no)e(longer)g(holds.)39 b(Rather,)28 b(the)g(sp)s(eci\014c)e(c)m(haracteristics)j(of)e(the)h (in)m(teraction)165 2336 y(will)23 b(come)j(in)m(to)g(pla)m(y)f(and)g (determine)f(the)i(sp)s(eci\014c)e(features)i(of)f(the)h(lo)m(w)f(temp) s(erature)g(phase.)39 b(In)165 2449 y(man)m(y)24 b(cases,)j(the)d(lo)m (w)g(temp)s(erature)f(b)s(eha)m(vior)g(can)h(b)s(e)g(describ)s(ed)e(as) i(a)g(random)f(p)s(erturbation)f(of)165 2562 y(a)k(ground)e(state,)k (i.e.,)e(of)g(a)f(\014xed)g(con\014guration)f(of)i(minimal)c(energy)-8 b(.)40 b(Then)24 b(w)m(e)h(can)h(exp)s(ect)g(that)165 2675 y(at)e(lo)m(w)f(temp)s(eratures,)h(and)f(sometimes)g(ev)m(en)h(up) e(to)i(the)f(critical)f(temp)s(erature,)j(the)e(equilibrium)165 2788 y(phases)37 b(are)h(realized)f(as)h(a)g(deterministic)d(ground)i (state)i(con\014guration,)g(p)s(erturb)s(ed)c(b)m(y)i(\014nite)165 2901 y(random)26 b(islands)d(on)j(whic)m(h)f(the)i(con\014guration)e (disagrees)h(with)f(the)h(ground)f(state.)41 b(This)24 b(means)165 3014 y(that)e(the)g(ground)e(state)j(pattern)e(can)h(p)s (ercolate)f(through)g(the)h(space)g(to)g(in\014nit)m(y)-8 b(.)35 b(One)21 b(prominen)m(t)165 3127 y(w)m(a)m(y)33 b(of)f(con\014rming)e(this)g(picture)h(is)f(pro)m(vided)g(b)m(y)i(the)g (so)f(called)g(Pirogo)m(v{Sinai)g(theory)g(whic)m(h)165 3240 y(is)d(describ)s(ed)e(in)h(detail)h(e.g.)i(in)d([230)q(].)41 b(In)27 b(Section)h(8)h(w)m(e)g(will)d(discuss)g(some)j(other)g(tec)m (hniques)f(of)165 3353 y(establishing)g(the)j(same)g(geometric)g (picture.)306 3465 y(It)j(is)e(eviden)m(t)h(from)f(the)i(ab)s(o)m(v)m (e)g(that)g(p)s(ercolation)e(theory)h(will)e(pla)m(y)h(an)h(imp)s (ortan)m(t)g(role)f(in)165 3578 y(this)38 b(text.)66 b(In)38 b(fact,)k(w)m(e)d(will)d(mainly)g(b)s(e)i(concerned)h(with)e (dep)s(enden)m(t)h(p)s(ercolation,)i(but)e(one)165 3691 y(can)32 b(sa)m(y)h(that)f(indep)s(enden)m(t)e(p)s(ercolation)h(stands) g(as)h(a)g(protot)m(yp)s(e)g(for)g(the)g(study)e(of)i(statistical)165 3804 y(equilibrium)j(prop)s(erties)j(in)h(geometric)i(terms.)69 b(In)39 b(indep)s(enden)m(t)e(p)s(ercolation,)42 b(the)d(mo)s(del)g(is) 165 3917 y(extremely)29 b(simple:)38 b(the)30 b(comp)s(onen)m(ts)f(are) g(binary-v)-5 b(alued)27 b(and)h(indep)s(enden)m(t)f(from)h(eac)m(h)j (other.)165 4030 y(What)44 b(is)d(hard)h(is)g(the)h(t)m(yp)s(e)f(of)h (question)f(one)h(asks,)j(namely)c(the)h(question)f(of)h(existence)g (of)165 4143 y(in\014nite)30 b(paths)i(of)g(1's)g(and)g(their)f (geometry)-8 b(.)47 b(W)-8 b(e)33 b(will)d(in)m(tro)s(duce)h(p)s (ercolation)g(b)s(elo)m(w)g(but)g(refer)165 4256 y(to)f(other)f (publications)d(\(suc)m(h)j(as)h(the)f(b)s(o)s(ok)f(b)m(y)h(Grimmett)g ([108)q(]\))h(for)f(a)g(systematic)g(accoun)m(t)i(of)165 4369 y(the)g(theory)-8 b(.)306 4482 y(P)m(ercolation)41 b(will)c(come)k(in)m(to)f(pla)m(y)g(here)g(on)g(v)-5 b(arious)39 b(lev)m(els.)69 b(Its)40 b(concepts)h(lik)m(e)f(clusters,) 165 4595 y(op)s(en)25 b(paths,)h(connectedness)g(etc.)g(will)d(b)s(e)i (useful)e(for)i(describing)e(certain)i(geometric)h(features)g(of)165 4708 y(equilibrium)20 b(phases,)26 b(allo)m(wing)d(c)m (haracterizations)j(of)e(phases)g(in)g(p)s(ercolation)f(terms.)39 b(Examples)165 4820 y(will)27 b(b)s(e)h(presen)m(ted)i(where)e(the)i (\(thermal\))f(phase)g(transition)e(go)s(es)j(hand)e(in)g(hand)g(with)g (a)h(phase)165 4933 y(transition)39 b(in)g(an)h(asso)s(ciated)g(p)s (ercolation)f(pro)s(cess.)70 b(Next,)43 b(p)s(ercolation)d(tec)m (hniques)f(can)i(b)s(e)165 5046 y(used)22 b(to)h(obtain)f(sp)s (eci\014c)g(information)f(ab)s(out)h(the)h(phase)f(diagram)g(of)h(the)g (system.)38 b(F)-8 b(or)23 b(example,)165 5159 y(equilibrium)g (correlation)k(functions)e(are)j(sometimes)f(dominated)f(b)m(y)h (connectivit)m(y)h(functions)d(in)165 5272 y(an)e(asso)s(ciated)h(p)s (ercolation)e(problem)f(whic)m(h)h(is)g(easier)h(to)h(in)m(v)m (estigate.)39 b(Finally)-8 b(,)23 b(represen)m(tations)165 5385 y(in)g(terms)g(of)h(p)s(ercolation)e(mo)s(dels)g(will)f(yield)h (explicit)g(relations)g(b)s(et)m(w)m(een)i(certain)g(observ)-5 b(ables)22 b(in)165 5498 y(equilibrium)c(mo)s(dels)j(and)g(some)i (corresp)s(onding)d(p)s(ercolation)h(quan)m(tities.)37 b(In)22 b(fact,)j(the)d(resulting)165 5611 y(p)s(ercolation)i(mo)s (dels,)h(lik)m(e)f(the)g(random-cluster)g(mo)s(del,)h(ha)m(v)m(e)g (some)g(in)m(terest)g(in)e(their)h(o)m(wn)h(righ)m(t)1867 5895 y(4)p eop %%Page: 5 5 5 4 bop 165 304 a Fw(and)30 b(will)e(also)i(b)s(e)g(studied)f(in)g (some)h(detail.)306 417 y(This)h(text)j(is)d(supp)s(osed)g(to)i(b)s(e)f (self-con)m(tained.)48 b(Therefore)32 b(w)m(e)h(need)f(to)i(in)m(tro)s (duce)d(v)-5 b(arious)165 530 y(concepts)40 b(and)e(tec)m(hniques)g (whic)m(h)g(are)h(w)m(ell-kno)m(wn)f(to)h(some)g(readers.)66 b(On)38 b(the)h(other)g(hand,)165 643 y(imp)s(ortan)m(t)29 b(related)g(issues)g(will)d(not)k(b)s(e)f(discussed)e(when)i(they)h (are)f(not)h(explicitly)d(needed.)40 b(F)-8 b(or)165 756 y(more)44 b(complete)g(discussions)d(on)j(the)g(in)m(tro)s(ductory) e(material)i(w)m(e)g(will)d(refer)i(the)h(in)m(terested)165 869 y(reader)35 b(to)g(other)g(sources.)53 b(More)35 b(seriously)-8 b(,)34 b(w)m(e)h(will)d(not)j(include)d(here)i(a)h (discussion)d(of)j(some)165 981 y(imp)s(ortan)m(t)29 b(geometric)h(concepts)f(dev)m(elop)s(ed)g(in)f(the)h(1980's)i(for)d (the)i(study)e(of)h(critical)f(b)s(eha)m(vior)165 1094 y(in)34 b(statistical)h(mec)m(hanical)g(systems,)h(namely)e(random)h(w) m(alk)f(expansions)g(or)h(random)f(curren)m(t)165 1207 y(represen)m(tations.)39 b(F)-8 b(ortunately)g(,)28 b(w)m(e)e(can)g (refer)f(to)i(an)e(excellen)m(t)h(b)s(o)s(ok)f([77)q(])h(where)f(the)h (in)m(terested)165 1320 y(reader)33 b(will)c(\014nd)i(all)g(the)i (relev)-5 b(an)m(t)33 b(results)e(and)h(references.)47 b(Imp)s(ortan)m(t)32 b(steps)g(in)f(this)g(con)m(text)165 1433 y(include)d([5)q(,)j(6,)g(7,)g(11)q(,)f(40)q(,)h(41)q(])f(and)g (the)g(references)h(therein.)306 1546 y(Finally)-8 b(,)35 b(to)h(a)m(v)m(oid)f(misunderstanding,)e(the)i(random)f(geometry)i(in)e (the)h(title)f(of)h(this)f(w)m(ork)165 1659 y(should)f(not)i(b)s(e)g (confused)f(with)g(sto)s(c)m(hastic)h(geometry)i(\(or)e(geometric)h (probabilit)m(y\))d(whic)m(h,)i(as)165 1772 y(a)k(branc)m(h)e(of)h(in)m (tegral)f(geometry)-8 b(,)42 b(pro)m(vides)37 b(a)h(v)m(ery)g(in)m (teresting)f(to)s(ol-b)s(o)m(x)h(for)g(the)g(discussion)165 1885 y(of)i(morphological)e(c)m(haracteristics)i(of)g(random)e (\014elds)g(app)s(earing)g(in)g(statistical)i(ph)m(ysics)e(and)165 1998 y(b)s(ey)m(ond,)30 b(see)h([145)r(,)f(172)r(,)g(2)q(].)165 2160 y Fx(Ac)m(kno)m(wledgemen)m(ts:)111 b Fw(It)36 b(is)f(a)h (pleasure)f(to)h(thank)g(J.)f(Leb)s(o)m(witz)h(for)g(suggesting)g (\(and)f(in-)165 2273 y(sisting\))i(that)i(w)m(e)f(should)e(write)h (this)g(rep)s(ort.)63 b(W)-8 b(e)39 b(are)g(also)f(grateful)g(to)g(L.)g (Cha)m(y)m(es,)j(A.C.D.)165 2386 y(v)-5 b(an)34 b(En)m(ter)f(and)g(J.)g (L})-45 b(orinczi)33 b(who)g(lo)s(ok)m(ed)g(at)h(parts)f(of)h(the)g (man)m(uscript)e(and)h(made)g(n)m(umerous)165 2499 y(suggestions,)e (and)f(to)h(Y.)f(Higuc)m(hi)g(for)g(discussions)e(on)i(Prop)s(osition)e (8.5.)1867 5895 y(5)p eop %%Page: 6 6 6 5 bop 165 304 a Fy(2)135 b(Equilibrium)41 b(phases)165 510 y FA(2.1)113 b(The)37 b(lattice)165 682 y Fw(Our)h(ob)5 b(ject)40 b(of)f(study)g(are)g(ph)m(ysical)f(systems)h(with)f(man)m(y)h (constituen)m(ts,)j(spins)37 b(or)i(particles,)165 795 y(whic)m(h)g(will)e(b)s(e)i(lo)s(cated)h(at)g(the)g(sites)g(of)f(a)i (crystal)e(lattice)h Fv(L)p Fw(.)69 b(The)39 b(standard)g(case)h(is)f (when)165 908 y Fv(L)f Fw(=)f Fx(Z)438 875 y Fu(d)478 908 y Fw(,)j(the)e Ft(d)p Fw(-dimensional)d(h)m(yp)s(ercubic)h (lattice.)63 b(In)37 b(general,)j(w)m(e)e(shall)e(assume)h(that)i Fv(L)e Fw(is)165 1021 y(the)c(v)m(ertex)g(set)g(of)f(a)g(coun)m(table)h (graph.)45 b(That)32 b(is,)g Fv(L)g Fw(is)f(an)h(at)h(most)g(coun)m (table)f(set)h(and)e(comes)165 1134 y(equipp)s(ed)25 b(with)i(a)h(\(symmetric\))f(adjacency)h(relation.)39 b(Namely)-8 b(,)29 b(w)m(e)f(write)f Ft(x)e Fv(\030)g Ft(y)30 b Fw(if)d(the)g(v)m(ertices)165 1246 y Ft(x;)15 b(y)36 b Fv(2)c(L)i Fw(are)h(adjacen)m(t,)j(and)c(this)f(is)h (visualized)e(b)m(y)j(dra)m(wing)f(an)g(edge)h(b)s(et)m(w)m(een)h Ft(x)e Fw(and)g Ft(y)s Fw(.)54 b(In)165 1359 y(this)39 b(case,)44 b Ft(x)c Fw(and)f Ft(y)k Fw(are)d(also)g(called)f(neigh)m(b) s(ors,)i(and)f(the)g(edge)g(\(or)h(b)s(ond\))d(b)s(et)m(w)m(een)j Ft(x)f Fw(and)165 1472 y Ft(y)h Fw(is)36 b(denoted)i(b)m(y)f Fv(h)p Ft(xy)s Fv(i)p Fw(.)63 b(W)-8 b(e)39 b(write)e Fv(B)j Fw(for)d(the)h(set)g(of)g(all)e(edges)i(\(b)s(onds\))f(in)f Fv(L)p Fw(.)62 b(A)38 b(complete)165 1585 y(description)29 b(of)h(the)h(graph)f(is)f(th)m(us)h(giv)m(en)g(b)m(y)h(the)f(pairs)f (\()p Fv(L)p Ft(;)15 b Fv(\030)p Fw(\))31 b(or)f(\()p Fv(L)p Ft(;)15 b Fv(B)s Fw(\).)306 1698 y(In)27 b(the)h(case)h Fv(L)c Fw(=)f Fx(Z)1008 1665 y Fu(d)1049 1698 y Fw(,)k(the)g(edges)g (will)d(usually)g(b)s(e)i(dra)m(wn)g(b)s(et)m(w)m(een)h(lattice)g (sites)f(of)g(distance)165 1811 y(one;)52 b(hence)44 b Ft(x)49 b Fv(\030)f Ft(y)f Fw(whenev)m(er)d Fv(j)p Ft(x)29 b Fv(\000)g Ft(y)s Fv(j)49 b Fw(=)f(1.)82 b(Here,)49 b Fv(j)29 b(\001)h(j)44 b Fw(stands)g(for)g(the)g(sum-norm,)j(i.e.)165 1924 y Fv(j)p Ft(x)p Fv(j)37 b Fw(=)411 1860 y Fs(P)499 1886 y Fu(d)499 1947 y(i)p Fr(=1)632 1924 y Fv(j)p Ft(x)709 1938 y Fu(i)738 1924 y Fv(j)g Fw(whenev)m(er)g Ft(x)f Fw(=)g(\()p Ft(x)1488 1938 y Fr(1)1528 1924 y Ft(;)15 b(:)g(:)g(:)h(;)f(x)1781 1939 y Fu(d)1822 1924 y Fw(\))37 b Fv(2)f Fx(Z)2055 1891 y Fu(d)2095 1924 y Fw(.)60 b(This)36 b(c)m(hoice)h(is)f(natural)g(b)s(ecause)h(then)165 2037 y Fv(j)p Ft(x)22 b Fv(\000)g Ft(y)s Fv(j)32 b Fw(coincides)g(with)f (the)h(graph-theoretical)h(\(or)g(lattice\))g(distance,)g(viz.)g(the)f (length)g(of)h(the)165 2150 y(shortest)38 b(path)f(\(consisting)g(of)g (consecutiv)m(e)i(edges\))f(connecting)g Ft(x)f Fw(and)g Ft(y)s Fw(.)62 b(F)-8 b(or)38 b(con)m(v)m(enience,)165 2263 y(w)m(e)i(sometimes)g(use)g(the)f(same)h(notation)g(in)f(the)h (case)g(of)g(a)g(general)g(graph.)68 b(On)39 b Fx(Z)3250 2230 y Fu(d)3330 2263 y Fw(w)m(e)h(will)165 2376 y(o)s(ccasionally)26 b(distinguish)e(b)s(et)m(w)m(een)j(the)g(standard)g(metrics)f Ft(d)2364 2390 y Fr(1)2404 2376 y Fw(\()p Ft(x;)15 b(y)s Fw(\))26 b(=)2736 2311 y Fs(P)2824 2398 y Fu(i)2867 2376 y Fv(j)p Ft(x)2944 2390 y Fu(i)2986 2376 y Fv(\000)13 b Ft(y)3115 2390 y Fu(i)3142 2376 y Fv(j)p Fw(,)29 b Ft(d)3268 2390 y Fr(2)3307 2376 y Fw(\()p Ft(x;)15 b(y)s Fw(\))26 b(=)165 2488 y([)190 2424 y Fs(P)278 2451 y Fu(d)278 2511 y(i)p Fr(=1)397 2488 y Fw(\()p Ft(x)484 2502 y Fu(i)534 2488 y Fv(\000)21 b Ft(y)671 2502 y Fu(i)699 2488 y Fw(\))734 2455 y Fr(2)774 2488 y Fw(\)])834 2455 y Fr(1)p Fu(=)p Fr(2)977 2488 y Fw(and)32 b Ft(d)1203 2502 y Fq(1)1278 2488 y Fw(\()p Ft(x;)15 b(y)s Fw(\))30 b(=)f(max)1787 2502 y Fu(i)1831 2488 y Fv(j)p Ft(x)1908 2502 y Fu(i)1958 2488 y Fv(\000)21 b Ft(y)2095 2502 y Fu(i)2123 2488 y Fv(j)p Fw(.)48 b(Giv)m(en)33 b(an)m(y)g(metric)f Ft(d)h Fw(on)g Fv(L)p Fw(,)g(w)m(e)g(write)165 2601 y Ft(d)p Fw(\(\003)p Ft(;)15 b Fw(\001\))27 b(=)e(inf)694 2615 y Fu(x)p Fq(2)p Fr(\003)p Fu(;y)r Fq(2)p Fr(\001)1012 2601 y Ft(d)p Fw(\()p Ft(x;)15 b(y)s Fw(\))32 b(for)e(the)g(distance)g (of)h(t)m(w)m(o)h(subsets)d(\003)p Ft(;)15 b Fw(\001)26 b Fv(\032)f(L)p Fw(.)306 2714 y(W)-8 b(e)29 b(will)24 b(alw)m(a)m(ys)j(assume)g(that)h(the)f(graph)f(\()p Fv(L)p Ft(;)15 b Fv(\030)p Fw(\))28 b(is)e Fp(lo)-5 b(c)g(al)5 b(ly)31 b(\014nite)p Fw(,)d(whic)m(h)e(means)g(that)i(eac)m(h)165 2827 y Ft(x)40 b Fv(2)g(L)e Fw(has)h(only)f(a)i(\014nite)e(n)m(um)m(b)s (er)g Ft(N)1577 2841 y Fu(x)1660 2827 y Fw(of)h(nearest)h(neigh)m(b)s (ors.)65 b(In)38 b(other)i(w)m(ords,)h Ft(N)3305 2841 y Fu(x)3388 2827 y Fw(is)d(the)165 2940 y(n)m(um)m(b)s(er)32 b(of)i(edges)g(emanating)f(from)g Ft(x)p Fw(.)49 b Ft(N)1716 2954 y Fu(x)1793 2940 y Fw(is)32 b(also)i(called)e(the)i Fp(de)-5 b(gr)g(e)g(e)34 b Fw(of)f(the)h(graph)f(at)h Ft(x)p Fw(.)49 b(In)165 3053 y(man)m(y)37 b(cases)h(w)m(e)g(will)c(ev)m (en)k(assume)f(that)g(\()p Fv(L)p Ft(;)15 b Fv(\030)p Fw(\))38 b(is)e(of)h(b)s(ounded)e(degree,)40 b(whic)m(h)c(means)h(that) 165 3166 y Ft(N)f Fw(=)25 b(sup)506 3188 y Fu(x)p Fq(2L)661 3166 y Ft(N)734 3180 y Fu(x)803 3166 y Ft(<)g Fv(1)p Fw(.)40 b(Common)27 b(examples)g(of)h(suc)m(h)f(graphs,)h(b)s(esides)e Fx(Z)2823 3133 y Fu(d)2863 3166 y Fw(,)j(are)f(the)g(triangular)165 3279 y(lattice)f(in)f(t)m(w)m(o)i(dimensions,)d(and)h(the)g(regular)g (tree)i Fx(T)2080 3294 y Fu(d)2146 3279 y Fw(\(also)f(kno)m(wn)f(as)h (the)g(Ca)m(yley)g(tree)g(or)g(the)165 3392 y(Bethe)34 b(lattice\),)f(whic)m(h)e(is)g(de\014ned)g(as)i(the)f(\(unique\))f (in\014nite)f(connected)j(graph)e(con)m(taining)h(no)165 3505 y(circuits)d(in)g(whic)m(h)g(ev)m(ery)j(v)m(ertex)f(has)f(exactly) h Ft(d)21 b Fw(+)f(1)31 b(nearest)f(neigh)m(b)s(ors.)306 3618 y(A)g(region)e(of)h(the)g(lattice,)h(that)g(is)e(a)h(subset)f (\003)e Fv(\032)f(L)p Fw(,)k(is)f(called)g(\014nite)g(if)g(its)g (cardinalit)m(y)g Fv(j)p Fw(\003)p Fv(j)h Fw(is)165 3730 y(\014nite.)40 b(W)-8 b(e)31 b(write)e Fv(E)37 b Fw(for)30 b(the)g(collection)f(of)h(all)e(\014nite)h(regions.)40 b(The)29 b(complemen)m(t)h(of)g(a)g(region)f(\003)165 3843 y(will)i(b)s(e)h(denoted)h(b)m(y)f(\003)1003 3810 y Fu(c)1067 3843 y Fw(=)d Fv(L)22 b(n)g Fw(\003.)48 b(The)33 b(b)s(oundary)d Ft(@)5 b Fw(\003)34 b(of)f(\003)f(is)g(the)h(set)h(of)f (all)e(sites)i(\(v)m(ertices\))165 3956 y(in)c(\003)334 3923 y Fu(c)400 3956 y Fw(whic)m(h)g(are)h(adjacen)m(t)i(to)f(some)g (site)f(of)h(\003.)306 4069 y(A)m(t)37 b(some)g(o)s(ccasions)e(w)m(e)i (will)c(need)j(the)g(notion)f(of)h(thermo)s(dynamic)e(\(or)i (in\014nite)e(v)m(olume\))165 4182 y(limit,)29 b(and)g(w)m(e)i(need)f (to)h(describ)s(e)d(in)h(what)h(sense)g(a)h(region)e(\003)d Fv(2)f(E)38 b Fw(gro)m(ws)31 b(to)f(the)h(full)d(lattice)i Fv(L)p Fw(.)165 4295 y(F)-8 b(or)27 b(our)f(purp)s(oses,)f(it)h(will)d (in)i(general)h(b)s(e)f(su\016cien)m(t)h(to)h(tak)m(e)g(an)f(arbitrary) f(increasing)g(sequence)165 4408 y(\(\003)263 4422 y Fu(n)311 4408 y Fw(\))35 b(with)592 4344 y Fs(S)661 4431 y Fu(n)p Fq(\025)p Fr(1)814 4408 y Fw(\003)877 4422 y Fu(n)956 4408 y Fw(=)d Fv(L)p Fw(.)54 b(In)33 b(the)i(case)h Fv(L)c Fw(=)g Fx(Z)1938 4375 y Fu(d)1979 4408 y Fw(,)k(w)m(e)f(will)d (often)j(mak)m(e)h(the)e(standard)g(c)m(hoice)165 4529 y(\003)228 4543 y Fu(n)301 4529 y Fw(=)25 b([)p Fv(\000)p Ft(n;)15 b(n)p Fw(])668 4496 y Fu(d)727 4529 y Fv(\\)k Fx(Z)871 4496 y Fu(d)911 4529 y Fw(,)31 b(the)f(lattice)g(cub)s(es)f (cen)m(tered)i(around)e(the)h(origin.)39 b(As)30 b Fv(E)38 b Fw(is)29 b(a)h(directed)g(set)165 4642 y(ordered)37 b(b)m(y)g(inclusion,)f(w)m(e)i(will)d(o)s(ccasionally)h(also)h (consider)g(the)g(limit)e(along)i Fv(E)8 b Fw(.)62 b(In)36 b(eac)m(h)j(of)165 4755 y(these)31 b(cases)g(w)m(e)g(will)d(use)i(the)h (notation)f(\003)c Fv(")f(L)p Fw(.)165 4999 y FA(2.2)113 b(Con\014gurations)165 5170 y Fw(The)25 b(constituen)m(ts)g(of)g(our)f (systems)h(are)h(the)f(spins)e(or)i(particles)e(at)j(the)f(lattice)h (sites.)38 b(So,)26 b(at)g(eac)m(h)165 5283 y(site)g Ft(x)f Fv(2)g(L)g Fw(w)m(e)h(ha)m(v)m(e)h(a)f(v)-5 b(ariable)24 b Ft(\033)s Fw(\()p Ft(x)p Fw(\))j(taking)e(v)-5 b(alues)25 b(in)g(a)h(non-empt)m(y)f(set)i Ft(S)5 b Fw(,)26 b(the)g Fp(state)j(sp)-5 b(ac)g(e)27 b Fw(or)165 5396 y Fp(single-spin)j(sp)-5 b(ac)g(e)p Fw(.)40 b(In)26 b(a)h(magnetic)g(set-up)g(\(to)h(whic)m(h)d (w)m(e)i(mostly)f(adhere)h(for)f(simplicit)m(y\),)f Ft(\033)s Fw(\()p Ft(x)p Fw(\))165 5509 y(is)k(in)m(terpreted)f(as)i(the)f(spin)f (of)h(an)g(elemen)m(tary)h(magnet)g(at)g Ft(x)p Fw(.)41 b(In)28 b(a)i(lattice)f(gas)h(in)m(terpretation,)165 5622 y(there)37 b(is)e(a)h(distinguished)d(v)-5 b(acuum)35 b(state)j(0)d Fv(2)g Ft(S)41 b Fw(represen)m(ting)35 b(the)h(absence)h(of)f(an)m(y)h(particle,)1867 5895 y(6)p eop %%Page: 7 7 7 6 bop 165 304 a Fw(and)27 b(the)h(remaining)e(elemen)m(ts)i(corresp)s (ond)e(to)i(the)g(t)m(yp)s(es)f(and/or)h(the)f(n)m(um)m(b)s(er)g(of)g (the)h(particles)165 417 y(at)g Ft(x)p Fw(.)39 b(Unless)25 b(stated)j(otherwise,)e(w)m(e)h(will)d(alw)m(a)m(ys)j(assume)f(that)h Ft(S)32 b Fw(is)25 b(\014nite.)39 b(Elemen)m(ts)26 b(of)g Ft(S)32 b Fw(will)165 530 y(t)m(ypically)e(b)s(e)f(denoted)i(b)m(y)f Ft(a;)15 b(b;)g(:)g(:)g(:)r Fw(.)306 643 y(A)34 b Fp(c)-5 b(on\014gur)g(ation)35 b Fw(is)e(a)g(function)f Ft(\033)i Fw(:)c Fv(L)g(!)g Ft(S)38 b Fw(whic)m(h)32 b(assigns)h(to)h(eac)m(h)g (v)m(ertex)h Ft(x)30 b Fv(2)f(L)k Fw(a)h(spin)165 756 y(v)-5 b(alue)34 b Ft(\033)s Fw(\()p Ft(x)p Fw(\))f Fv(2)f Ft(S)5 b Fw(.)53 b(In)34 b(other)h(w)m(ords,)g(a)g(con\014guration)f Ft(\033)k Fw(is)33 b(an)i(elemen)m(t)g(of)f(the)h(pro)s(duct)f(space) 165 869 y(\012)29 b(=)f Ft(S)420 836 y Fq(L)473 869 y Fw(.)47 b(\012)32 b(is)g(called)f(the)i(con\014guration)f(space)h(and)f (its)g(elemen)m(ts)h(are)g(in)e(general)h(written)g(as)165 981 y Ft(\033)n(;)15 b(\021)s(;)g(\030)t(;)g(:)g(:)g(:)s Fw(.)42 b(\(It)31 b(is)f(sometimes)h(useful)e(to)j(visualize)d Ft(a;)15 b(b)g(:)g(:)g(:)32 b Fw(as)f(colors.)42 b(A)31 b(con\014guration)g(is)e(then)165 1094 y(a)j(coloring)e(of)h(the)h (lattice.\))43 b(A)31 b(con\014guration)g Ft(\033)j Fw(is)c(constan)m (t)i(if)e(for)h(some)g Ft(a)c Fv(2)f Ft(S)5 b Fw(,)31 b Ft(\033)s Fw(\()p Ft(x)p Fw(\))c(=)f Ft(a)31 b Fw(for)165 1207 y(all)36 b Ft(x)f Fv(2)g(L)p Fw(.)59 b(Tw)m(o)36 b(con\014gurations)g Ft(\033)k Fw(and)35 b Ft(\021)40 b Fw(are)d(said)e(to)j(agree)f(on)g(a)g(region)f(\003)f Fv(\032)g(L)p Fw(,)j(written)165 1320 y(as)g(\\)p Ft(\033)i Fv(\021)d Ft(\021)j Fw(on)e(\003",)h(if)e Ft(\033)s Fw(\()p Ft(x)p Fw(\))g(=)g Ft(\021)s Fw(\()p Ft(x)p Fw(\))h(for)f(all)f Ft(x)h Fv(2)f Fw(\003.)62 b(Similarly)-8 b(,)36 b(w)m(e)i(write)e(\\)p Ft(\033)41 b Fv(\021)36 b Ft(\021)41 b Fw(o\013)c(\003")h(if)165 1433 y Ft(\033)s Fw(\()p Ft(x)p Fw(\))26 b(=)f Ft(\021)s Fw(\()p Ft(x)p Fw(\))32 b(for)e(all)f Ft(x)36 b(=)-56 b Fv(2)25 b Fw(\003.)306 1546 y(W)-8 b(e)34 b(also)e(consider)f (con\014gurations)g(in)f(regions)i(\003)c Fv(\032)g(L)p Fw(.)45 b(These)32 b(are)g(elemen)m(ts)g(of)g Ft(S)3291 1513 y Fr(\003)3344 1546 y Fw(,)h(again)165 1659 y(denoted)f(b)m(y)f (letters)g(lik)m(e)g Ft(\033)n(;)15 b(\021)s(;)g(\030)t(;)g(:)g(:)g(:)s Fw(.)43 b(Giv)m(en)31 b Ft(\033)n(;)15 b(\021)31 b Fv(2)26 b Fw(\012,)32 b(w)m(e)f(write)g Ft(\033)2582 1673 y Fr(\003)2635 1659 y Ft(\021)2680 1673 y Fr(\003)2729 1654 y Fo(c)2796 1659 y Fw(for)g(the)h(con\014guration)165 1772 y Ft(\030)h Fv(2)28 b Fw(\012)k(with)f Ft(\030)t Fw(\()p Ft(x)p Fw(\))e(=)f Ft(\033)s Fw(\()p Ft(x)p Fw(\))33 b(for)f Ft(x)d Fv(2)f Fw(\003)k(and)g Ft(\030)t Fw(\()p Ft(x)p Fw(\))d(=)f Ft(\021)s Fw(\()p Ft(x)p Fw(\))33 b(for)f Ft(x)d Fv(2)f Fw(\003)2594 1739 y Fu(c)2629 1772 y Fw(.)46 b(Then,)32 b(ob)m(viously)-8 b(,)32 b Ft(\030)h Fv(\021)28 b Ft(\033)165 1885 y Fw(on)j(\003.)40 b(The)30 b(cylinder)e(sets)1253 2089 y Fv(N)1328 2103 y Fr(\003)1381 2089 y Fw(\()p Ft(\033)s Fw(\))e(=)f Fv(f)p Ft(\030)30 b Fv(2)25 b Fw(\012)g(:)55 b Ft(\030)30 b Fv(\021)24 b Ft(\033)34 b Fw(on)c(\003)p Fv(g)15 b Ft(;)165 2293 y Fw(\003)47 b Fv(2)g(E)8 b Fw(,)47 b(form)c(a)h(coun)m(table)f(neigh)m(b)s(orho)s(o)s(d)e(basis)h(of)i Ft(\033)50 b Fv(2)d Fw(\012;)i(they)44 b(generate)h(the)e(pro)s(duct) 165 2406 y(top)s(ology)31 b(on)e(\012.)40 b(Hence,)31 b(t)m(w)m(o)g(con\014gurations)e(are)i(close)f(to)g(eac)m(h)h(other)f (if)f(they)h(agree)h(on)f(some)165 2519 y(large)e(\014nite)e(region,)i (and)f(a)g(diagonal-sequence)h(argumen)m(t)g(sho)m(ws)f(that)h(\012)e (is)h(a)h(compact)g(in)e(this)165 2632 y(top)s(ology)-8 b(.)306 2745 y(W)g(e)35 b(will)30 b(often)j(c)m(hange)i(a)e (con\014guration)f Ft(\033)h Fv(2)d Fw(\012)i(at)i(just)e(one)i(site)e Ft(x)e Fv(2)f(L)p Fw(.)49 b(Changing)31 b Ft(\033)s Fw(\()p Ft(x)p Fw(\))165 2858 y(in)m(to)d(a)g(prescrib)s(ed)d(v)-5 b(alue)27 b Ft(a)f Fv(2)f Ft(S)32 b Fw(w)m(e)c(obtain)g(a)g(new)f (con\014guration)g(written)g Ft(\033)2923 2825 y Fu(x;a)3024 2858 y Fw(.)40 b(In)27 b(particular,)165 2971 y(for)j Ft(S)h Fw(=)25 b Fv(f\000)p Fw(1)p Ft(;)15 b Fw(+1)p Fv(g)32 b Fw(w)m(e)e(write)1295 3240 y Ft(\033)1350 3203 y Fu(x)1394 3240 y Fw(\()p Ft(y)s Fw(\))25 b(=)1633 3096 y Fs(\()1742 3184 y Ft(\033)s Fw(\()p Ft(y)s Fw(\))158 b(for)31 b Ft(y)d Fv(6)p Fw(=)d Ft(x)1742 3297 y Fv(\000)p Ft(\033)s Fw(\()p Ft(x)p Fw(\))83 b(for)31 b Ft(y)d Fw(=)d Ft(x)165 3505 y Fw(for)30 b(the)h(con\014guration)f(resulting)e(from)i (\015ipping)d(the)k(spin)d(at)k Ft(x)p Fw(.)306 3618 y(W)-8 b(e)36 b(will)c(also)j(deal)f(with)f(automorphisms)g(of)i(the)g (underlying)c(lattice)k(\()p Fv(L)p Ft(;)15 b Fv(\030)p Fw(\).)54 b(Eac)m(h)35 b(suc)m(h)165 3731 y(automorphism)d(de\014nes)h (a)g(measurable)g(transformation)g(of)g(the)h(con\014guration)e(space)i (\012.)50 b(The)165 3844 y(most)33 b(in)m(teresting)f(automorphisms)e (are)j(the)f(translations)f(of)h(the)h(in)m(teger)g(lattice)f Fv(L)c Fw(=)g Fx(Z)3389 3811 y Fu(d)3430 3844 y Fw(;)33 b(the)165 3957 y(asso)s(ciated)39 b(translation)e(group)g(acting)i(on)e (\012)h(is)f(giv)m(en)h(b)m(y)g Ft(\022)2378 3971 y Fu(x)2421 3957 y Ft(\033)s Fw(\()p Ft(y)s Fw(\))h(=)f Ft(\033)s Fw(\()p Ft(x)25 b Fw(+)g Ft(y)s Fw(\),)41 b Ft(y)f Fv(2)e Fx(Z)3402 3924 y Fu(d)3442 3957 y Fw(.)64 b(In)165 4069 y(particular,)31 b(an)m(y)h(constan)m(t)i(con\014guration)d(is)g (translation)f(in)m(v)-5 b(arian)m(t.)45 b(Similarly)-8 b(,)29 b(w)m(e)j(can)g(sp)s(eak)165 4182 y(ab)s(out)c(p)s(erio)s(dic)e (con\014gurations)i(whic)m(h)f(are)i(in)m(v)-5 b(arian)m(t)27 b(under)g Ft(\022)2441 4196 y Fu(x)2513 4182 y Fw(with)g Ft(x)h Fw(in)f(some)i(sublattice)f(of)165 4295 y Fx(Z)229 4262 y Fu(d)270 4295 y Fw(.)306 4408 y(Later)g(on,)g(w)m(e)g(will)c (also)k(consider)e(con\014gurations)g(whic)m(h)g(refer)h(to)h(the)f (lattice)h(b)s(onds)d(rather)165 4521 y(than)k(the)g(v)m(ertices.)41 b(These)29 b(are)h(elemen)m(ts)f Ft(\021)j Fw(of)e(the)f(pro)s(duct)f (space)h Fv(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)2814 4488 y Fq(B)2869 4521 y Fw(,)29 b(and)g(a)g(b)s(ond)f Ft(b)d Fv(2)g(B)165 4634 y Fw(will)h(b)s(e)i(called)g(op)s(en)g(if)f Ft(\021)s Fw(\()p Ft(b)p Fw(\))f(=)f(1,)30 b(and)e(otherwise)g(closed.) 40 b(The)28 b(ab)s(o)m(v)m(e)h(notations)g(apply)e(to)i(this)165 4747 y(situation)h(as)g(w)m(ell.)165 4990 y FA(2.3)113 b(Observ)-6 b(ables)165 5162 y Fw(An)45 b(observ)-5 b(able)44 b(is)g(a)h(real)g(function)e(on)i(the)g(con\014guration)g(space)g(whic) m(h)f(ma)m(y)h(b)s(e)g(though)m(t)165 5275 y(of)40 b(as)g(the)f(n)m (umerical)g(outcome)h(of)g(some)g(ph)m(ysical)e(measuremen)m(t.)69 b(Mathematically)-8 b(,)42 b(it)d(is)g(a)165 5388 y(measurable)44 b(real)g(function)f(on)h(\012.)82 b(Here,)48 b(the)d(natural)e (underlying)e Ft(\033)s Fw(-\014eld)j(of)g(measurable)165 5501 y(ev)m(en)m(ts)37 b(in)d(\012)g(is)h(the)g(pro)s(duct)f Ft(\033)s Fw(-algebra)i Fv(F)42 b Fw(=)33 b(\()p Fv(F)1981 5515 y Fr(0)2021 5501 y Fw(\))2056 5468 y Fq(L)2109 5501 y Fw(,)k(where)d Fv(F)2503 5515 y Fr(0)2578 5501 y Fw(is)g(the)i(set)f (of)h(all)e(subsets)g(of)165 5614 y Ft(S)5 b Fw(.)41 b Fv(F)f Fw(is)29 b(de\014ned)h(as)g(the)h(smallest)e Ft(\033)s Fw(-algebra)i(on)g(\012)f(for)g(whic)m(h)f(all)h(pro)5 b(jections)30 b Ft(X)7 b Fw(\()p Ft(x)p Fw(\))26 b(:)g(\012)f Fv(!)g Ft(S)5 b Fw(,)1867 5895 y(7)p eop %%Page: 8 8 8 7 bop 165 304 a Ft(X)7 b Fw(\()p Ft(x)p Fw(\)\()p Ft(\033)s Fw(\))31 b(=)d Ft(\033)s Fw(\()p Ft(x)p Fw(\))33 b(with)e Ft(\033)g Fv(2)d Fw(\012)k(and)g Ft(x)c Fv(2)g(L)p Fw(,)33 b(are)f(measurable.)46 b(It)32 b(coincides,)g(in)f(fact,)j(with)d(the) 165 417 y(Borel)g Ft(\033)s Fw(-algebra)g(for)f(the)g(pro)s(duct)g(top) s(ology)g(on)h(\012.)306 530 y(W)-8 b(e)30 b(also)f(consider)f(ev)m(en) m(ts)i(and)e(observ)-5 b(ables)28 b(dep)s(ending)e(only)i(on)g(some)h (region)g(\003)c Fv(\032)g(L)p Fw(.)40 b(W)-8 b(e)165 643 y(let)38 b Fv(F)369 657 y Fr(\003)459 643 y Fw(denote)g(the)g (smallest)e(sub-)p Ft(\033)s Fw(-\014eld)g(of)i Fv(F)46 b Fw(con)m(taining)37 b(the)h(ev)m(en)m(ts)h Fv(N)2929 657 y Fr(\001)2991 643 y Fw(\()p Ft(\033)s Fw(\))f(for)g Ft(\033)i Fv(2)c Ft(S)3551 610 y Fr(\001)165 756 y Fw(and)29 b(\001)c Fv(2)g(E)37 b Fw(with)28 b(\001)d Fv(\032)g Fw(\003.)40 b(Equiv)-5 b(alen)m(tly)d(,)28 b Fv(F)1752 770 y Fr(\003)1835 756 y Fw(is)g(the)h Ft(\033)s Fw(-algebra)h (generated)g(b)m(y)f(the)h(pro)5 b(jections)165 869 y Ft(X)i Fw(\()p Ft(x)p Fw(\))32 b(with)d Ft(x)c Fv(2)g Fw(\003.)41 b Fv(F)965 883 y Fr(\003)1048 869 y Fw(is)30 b(the)g Ft(\033)s Fw(-algebra)h(of)g(ev)m(en)m(ts)h(o)s(ccurring)d(in)g (\003.)306 981 y(An)c(ev)m(en)m(t)h Ft(A)e Fw(is)g(called)f Fp(lo)-5 b(c)g(al)36 b Fw(if)24 b(it)g(o)s(ccurs)g(in)f(some)i (\014nite)e(region,)j(whic)m(h)d(means)h(that)h Ft(A)h Fv(2)f(F)3561 995 y Fr(\003)165 1094 y Fw(for)k(some)h(\003)25 b Fv(2)g(E)8 b Fw(.)41 b(Similarly)-8 b(,)26 b(an)j(observ)-5 b(able)29 b Ft(f)34 b Fw(:)26 b(\012)f Fv(!)g Fx(R)k Fw(is)g(called)f Fp(lo)-5 b(c)g(al)41 b Fw(if)28 b(it)g(dep)s(ends)g (on)h(only)165 1207 y(\014nitely)i(man)m(y)h(spins,)e(meaning)i(that)g Ft(f)42 b Fw(is)31 b(measurable)g(with)g(resp)s(ect)h(to)g Fv(F)2897 1221 y Fr(\003)2983 1207 y Fw(for)g(some)g(\003)c Fv(2)g(E)8 b Fw(.)165 1320 y(More)33 b(generally)-8 b(,)32 b(an)g(observ)-5 b(able)31 b Ft(f)41 b Fw(is)31 b(called)g Fp(quasilo)-5 b(c)g(al)44 b Fw(if)31 b(it)g(is)g(\(uniformly\))f(con)m (tin)m(uous,)i(i.e.,)165 1433 y(if)37 b(for)h(all)f Ft(\017)g(>)h Fw(0)g(there)g(is)f(some)h(\003)g Fv(2)f(E)46 b Fw(suc)m(h)37 b(that)i Fv(j)p Ft(f)10 b Fw(\()p Ft(\033)s Fw(\))26 b Fv(\000)e Ft(f)10 b Fw(\()p Ft(\030)t Fw(\))p Fv(j)38 b Ft(<)g(\017)f Fw(whenev)m(er)h Ft(\030)k Fv(\021)37 b Ft(\033)k Fw(on)165 1546 y(\003.)48 b(The)32 b(set)h Ft(C)7 b Fw(\(\012\))32 b(of)h(con)m(tin)m(uous)f(observ)-5 b(ables)32 b(is)g(a)h(Banac)m(h)h(space)f(for)f(the)h(suprem)m(um)e (norm)165 1659 y Fv(jj)p Ft(f)10 b Fv(jj)26 b Fw(=)f(sup)579 1681 y Fu(\033)641 1659 y Fv(j)p Ft(f)10 b Fw(\()p Ft(\033)s Fw(\))p Fv(j)15 b Ft(;)31 b Fw(and)f(the)h(lo)s(cal)e(observ)-5 b(ables)30 b(are)g(dense)g(in)f(it.)306 1772 y(The)37 b(lo)s(cal)f(ev)m(en)m(ts)j(and)d(observ)-5 b(ables)36 b(should)f(b)s(e)i(view)m(ed)f(as)i(microscopic)e(quan)m(tities.)60 b(On)165 1885 y(the)32 b(other)h(side)d(w)m(e)j(ha)m(v)m(e)g(the)f (macroscopic)h(quan)m(tities)e(whic)m(h)f(only)h(dep)s(end)g(on)g(the)i (collectiv)m(e)165 1998 y(b)s(eha)m(vior)e(of)h(all)f(spins,)f(but)h (not)h(on)g(the)f(v)-5 b(alues)32 b(of)f(an)m(y)h(\014nite)f(set)h(of)g (spins.)43 b(They)31 b(are)h(de\014ned)165 2111 y(in)j(terms)g(of)h (the)f Fp(tail)j Ft(\033)s Fp(-algebr)-5 b(a)44 b Fv(T)56 b Fw(=)1587 2046 y Fs(T)1656 2134 y Fr(\003)p Fq(2E)1814 2111 y Fv(F)1879 2125 y Fr(\003)1928 2106 y Fo(c)1965 2111 y Fw(,)37 b(whic)m(h)d(is)h(also)g(called)g(the)g Ft(\033)s Fw(-algebra)i(of)e(all)165 2223 y(ev)m(en)m(ts)41 b(at)f(in\014nit)m(y)-8 b(.)66 b(An)m(y)39 b(tail)g(ev)m(en)m(t)h Ft(A)h Fv(2)e(T)62 b Fw(and)39 b(an)m(y)h Fv(T)22 b Fw(-measurable)39 b(observ)-5 b(able)39 b(is)f(called)165 2336 y Fp(macr)-5 b(osc)g(opic)p Fw(.)306 2449 y(As)29 b(a)h(\014nal)d(piece)i(of)g (notation)h(w)m(e)f(in)m(tro)s(duce)f(the)h Fp(indic)-5 b(ator)33 b(function)j Ft(I)2879 2463 y Fu(A)2965 2449 y Fw(of)29 b(an)g(ev)m(en)m(t)i Ft(A)p Fw(;)f(it)165 2562 y(tak)m(es)i(the)f(v)-5 b(alue)30 b(1)g(if)g(the)g(ev)m(en)m(t)i (o)s(ccurs)e(\()p Ft(I)1701 2576 y Fu(A)1759 2562 y Fw(\()p Ft(\033)s Fw(\))c(=)f(1)30 b(if)g Ft(\033)e Fv(2)d Ft(A)p Fw(\))31 b(and)f(is)f(zero)i(otherwise.)165 2803 y FA(2.4)113 b(Random)38 b(\014elds)165 2975 y Fw(As)32 b(the)g(spins)d(of)j(the)g 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Fw(.)61 b(The)37 b(probabilit)m(y)e (of)165 3652 y(an)42 b(ev)m(en)m(t)h Ft(A)e Fw(is)g Ft(\026)p Fw(\()p Ft(A)p Fw(\))j(=)f Ft(\026)p Fw(\()p Ft(I)1247 3666 y Fu(A)1304 3652 y Fw(\))h(=)1498 3583 y Fs(R)1537 3680 y Fu(A)1609 3652 y Ft(d\026)p Fw(,)h(and)40 b(w)m(e)i(omit)g(the)f (set)h(braces)g(when)e Ft(A)i Fw(is)e(giv)m(en)165 3765 y(explicitly)-8 b(.)38 b(F)-8 b(or)27 b(example,)g(giv)m(en)g(an)m(y)g Ft(x)e Fv(2)g(L)h Fw(and)g Ft(a)f Fv(2)g Ft(S)32 b Fw(w)m(e)27 b(write)f Ft(\026)p Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))26 b(=)f Ft(a)p Fw(\))i(for)f Ft(\026)p Fw(\()p Ft(A)p Fw(\))h(with)165 3878 y Ft(A)f Fw(=)f Fv(f)p Ft(\033)k Fv(2)c Fw(\012)f(:)56 b Ft(\033)s Fw(\()p Ft(x)p Fw(\))26 b(=)f Ft(a)p Fv(g)p Fw(.)41 b(Co)m(v)-5 b(ariances)31 b(are)g(abbreviated)f(as)g Ft(\026)p Fw(\()p Ft(f)10 b Fw(;)15 b Ft(g)s Fw(\))26 b(=)f Ft(\026)p Fw(\()p Ft(f)10 b(g)s Fw(\))21 b Fv(\000)f Ft(\026)p Fw(\()p Ft(f)10 b Fw(\))p Ft(\026)p Fw(\()p Ft(g)s Fw(\).)306 3991 y(Whenev)m(er)29 b(w)m(e)g(need)f(a)h(top)s (ology)f(on)g(probabilit)m(y)e(measures)i(on)g(\012,)h(w)m(e)g(shall)d (tak)m(e)k(the)e Fp(we)-5 b(ak)165 4104 y(top)g(olo)g(gy)p Fw(.)42 b(In)25 b(this)f(\(metrizable\))i(top)s(ology)-8 b(,)27 b(a)f(sequence)g(of)f(probabilit)m(y)e(measures)i Ft(\026)3165 4118 y Fu(n)3237 4104 y Fw(con)m(v)m(erges)165 4217 y(to)39 b Ft(\026)p Fw(,)h(denoted)f(b)m(y)f Ft(\026)945 4231 y Fu(n)1030 4217 y Fv(!)g Ft(\026)p Fw(,)i(if)d Ft(\026)1425 4231 y Fu(n)1472 4217 y Fw(\()p Ft(A)p Fw(\))i Fv(!)f Ft(\026)p Fw(\()p Ft(A)p Fw(\))h(for)f(all)f(lo)s(cal)g(ev)m(en) m(ts)j Ft(A)e Fv(2)3002 4153 y Fs(S)3071 4240 y Fr(\003)p Fq(2E)3229 4217 y Fv(F)3294 4231 y Fr(\003)3348 4217 y Fw(.)64 b(This)165 4330 y(holds)32 b(if)g(and)g(only)g(if)g Ft(\026)1014 4344 y Fu(n)1061 4330 y Fw(\()p Ft(f)10 b Fw(\))29 b Fv(!)h Ft(\026)p Fw(\()p Ft(f)10 b Fw(\))32 b(for)h(all)f(lo)s(cal,)h(or)g(equiv)-5 b(alen)m(tly)d(,)33 b(all)e(quasilo)s(cal)h(functions)165 4443 y Ft(f)10 b Fw(.)50 b(In)33 b(applications,)f Ft(\026)1001 4457 y Fu(n)1082 4443 y Fw(will)e(often)k(b)s(e)f(an)h(equilibrium)28 b(state)35 b(in)e(a)g(\014nite)g(b)s(o)m(x)g(\003)3121 4457 y Fu(n)3202 4443 y Fw(tending)f(to)165 4556 y Fv(L)g Fw(as)h Ft(n)28 b Fv(!)g(1)p Fw(,)33 b(and)e(w)m(e)i(are)g(in)m (terested)f(in)f(whether)g(the)i(probabilities)c(of)j(ev)m(en)m(ts)i(o) s(ccurring)c(in)165 4669 y(some)c(\014xed)e(\014nite)g(v)m(olume)h(ha)m (v)m(e)h(a)f(w)m(ell-de\014ned)e(thermo)s(dynamic)h(\(or)h(bulk\))f (limit.)36 b(That)25 b(is,)h(w)m(e)165 4781 y(observ)m(e)32 b(what)g(happ)s(ens)d(around)i(the)g(origin)f(\(via)i(the)f(lo)s(cal)g (function)f Ft(f)10 b Fw(\))31 b(while)f(the)h(b)s(oundary)165 4894 y(of)38 b(the)f(b)s(o)m(x)g(in)f(whic)m(h)g(w)m(e)h(realize)g(the) h(equilibrium)32 b(state)38 b(receeds)g(to)g(in\014nit)m(y)-8 b(.)59 b(As)37 b(there)h(are)165 5007 y(only)31 b(coun)m(tably)h(man)m (y)g(lo)s(cal)f(ev)m(en)m(ts,)j(one)e(can)g(easily)f(see)h(b)m(y)g(a)g (diagonal-sequence)g(argumen)m(t)165 5120 y(that)f(the)g(set)g(of)f (all)g(probabilit)m(y)e(measures)i(on)g(\012)g(is)f(compact)j(in)d(the) i(w)m(eak)g(top)s(ology)-8 b(.)165 5361 y FA(2.5)113 b(The)37 b(Hamiltonian)165 5533 y Fw(W)-8 b(e)40 b(will)c(b)s(e)i (concerned)h(with)e(systems)i(of)g Fp(inter)-5 b(acting)47 b Fw(spins.)64 b(As)38 b(usual,)i(the)e(in)m(teraction)h(is)165 5646 y(describ)s(ed)25 b(b)m(y)h(a)h(Hamiltonian.)38 b(As)27 b(the)g(spins)d(are)j(lo)s(cated)g(at)h(the)f(sites)f(of)h(a)g (graph)f(\()p Fv(L)p Ft(;)15 b Fv(\030)p Fw(\),)28 b(it)e(is)1867 5895 y(8)p eop %%Page: 9 9 9 8 bop 165 304 a Fw(natural)27 b(to)g(consider)g(the)g(case)h(of)f (homogeneous)h(neigh)m(b)s(or)e(p)s(oten)m(tials.)39 b(\(W)-8 b(e)29 b(will)24 b(deviate)k(from)165 417 y(homogeneit)m(y)33 b(in)d(Section)i(9)g(when)f(considering)e(random)i(in)m(teractions.\)) 45 b(The)31 b(Hamiltonian)f Ft(H)165 530 y Fw(then)g(tak)m(es)i(the)f (form)1100 734 y Ft(H)7 b Fw(\()p Ft(\033)s Fw(\))26 b(=)1436 653 y Fs(X)1430 828 y Fu(x)p Fq(\030)p Fu(y)1577 734 y Ft(U)10 b Fw(\()p Ft(\033)s Fw(\()p Ft(x)p Fw(\))p Ft(;)15 b(\033)s Fw(\()p Ft(y)s Fw(\)\))23 b(+)2223 653 y Fs(X)2263 828 y Fu(x)2358 734 y Ft(V)d Fw(\()p Ft(\033)s Fw(\()p Ft(x)p Fw(\)\))820 b(\(1\))165 1017 y(with)42 b(a)h(symmetric)f(function)f Ft(U)55 b Fw(:)46 b Ft(S)34 b Fv(\002)28 b Ft(S)50 b Fv(!)c Fx(R)20 b Fv([)g(f1g)q Fw(,)46 b(the)d Fp(neighb)-5 b(or-inter)g(action)p Fw(,)48 b(and)42 b(a)165 1130 y Fp(self-ener)-5 b(gy)48 b Ft(V)62 b Fw(:)42 b Ft(S)k Fv(!)c Fx(R)p Fw(.)70 b(The)40 b(in\014nite)e(sums)h (are)h(formal;)45 b(the)40 b(summation)f(index)g Ft(x)i Fv(\030)g Ft(y)165 1242 y Fw(means)27 b(that)g(the)f(sum)g(extends)g(o) m(v)m(er)i(all)d(b)s(onds)g Fv(h)p Ft(xy)s Fv(i)g(2)g(B)k Fw(of)e(the)f(lattice.)40 b Ft(U)c Fw(th)m(us)26 b(describ)s(es)f(the) 165 1355 y(in)m(teraction)33 b(b)s(et)m(w)m(een)g(spins)d(at)j(neigh)m (b)s(oring)d(sites,)j(while)e Ft(V)52 b Fw(migh)m(t)32 b(come)i(from)e(the)g(action)h(of)165 1468 y(an)e(external)f(magnetic)h (\014eld.)40 b(In)30 b(a)h(lattice)g(gas)g(in)m(terpretation)f(when)f Ft(S)i Fw(=)25 b Fv(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)33 b Fw(\(the)e(v)-5 b(alue)30 b(1)165 1581 y(b)s(eing)h(assigned)g(to)h (sites)g(whic)m(h)e(are)j(o)s(ccupied)d(b)m(y)i(a)g(particle\),)g Ft(V)52 b Fw(corresp)s(onds)31 b(to)h(a)g(c)m(hemical)165 1694 y(p)s(oten)m(tial.)306 1807 y(T)-8 b(o)35 b(mak)m(e)g(sense)f(of)g (the)g(formal)g(sums)f(in)f(\(1\))j(w)m(e)g(compare)g(the)f (Hamiltonian)e(for)i(t)m(w)m(o)h(dif-)165 1920 y(feren)m(t)e (con\014gurations)f Ft(\033)n(;)15 b(\021)33 b Fv(2)28 b Fw(\012)k(whic)m(h)f Fp(di\013er)k(only)g(lo)-5 b(c)g(al)5 b(ly)35 b Fw(\(or)d(are)h(\\lo)s(cal)f(p)s(erturbations")f(or)165 2033 y(\\excitations")i(of)f(eac)m(h)g(other\),)h(in)d(that)i Ft(\033)f Fv(\021)c Ft(\021)34 b Fw(o\013)e(some)g(\003)27 b Fv(2)g(E)8 b Fw(.)44 b(F)-8 b(or)33 b(suc)m(h)e(con\014gurations)g(w) m(e)165 2146 y(can)g(de\014ne)f(the)g Fp(r)-5 b(elative)34 b(Hamiltonian)468 2350 y Ft(H)7 b Fw(\()p Ft(\033)s Fv(j)p Ft(\021)s Fw(\))27 b(=)878 2269 y Fs(X)872 2443 y Fu(x)p Fq(\030)p Fu(y)1034 2350 y Fw([)p Ft(U)10 b Fw(\()p Ft(\033)s Fw(\()p Ft(x)p Fw(\))p Ft(;)15 b(\033)s Fw(\()p Ft(y)s Fw(\)\))23 b Fv(\000)d Ft(U)10 b Fw(\()p Ft(\021)s Fw(\()p Ft(x)p Fw(\))p Ft(;)15 b(\021)s Fw(\()p Ft(y)s Fw(\)\)])24 b(+)2363 2269 y Fs(X)2403 2443 y Fu(x)2513 2350 y Fw([)p Ft(V)c Fw(\()p Ft(\033)s Fw(\()p Ft(x)p Fw(\)\))i Fv(\000)e Ft(V)g Fw(\()p Ft(\021)s Fw(\()p Ft(x)p Fw(\)\)])189 b(\(2\))165 2627 y(in)31 b(whic)m(h)g(the)h(sums)f(no)m(w)h(con)m(tain) g(only)f(\014nitely)f(man)i(non-zero)h(terms:)43 b(the)33 b(\014rst)e(part)h(is)f(o)m(v)m(er)165 2740 y(those)26 b(neigh)m(b)s(or)d(pairs)h Fv(h)p Ft(xy)s Fv(i)h Fw(for)g(whic)m(h)e (at)j(least)f(one)g(of)g(the)g(sites)g(b)s(elongs)f(to)h(\003,)i(and)d (the)h(second)165 2853 y(part)31 b(is)e(o)m(v)m(er)j(all)d Ft(x)c Fv(2)g Fw(\003.)165 3097 y FA(2.6)113 b(Gibbs)38 b(measures)165 3268 y Fw(Gibbs)d(measures)g(are)h(random)f(\014elds)g (whic)m(h)f(describ)s(e)g(our)i(ph)m(ysical)e(spin)g(system)i(when)f (it)g(is)165 3381 y(in)d Fp(macr)-5 b(osc)g(opic)38 b(e)-5 b(quilibrium)35 b(with)i(r)-5 b(esp)g(e)g(ct)37 b(to)e(the)h(given)f (micr)-5 b(osc)g(opic)37 b(inter)-5 b(action)42 b Fw(at)34 b(a)g(\014xed)165 3494 y(temp)s(erature.)49 b(Here,)34 b(macroscopic)g(equilibrium)28 b(means)33 b(that)h(all)e(parts)g(of)i (the)f(system)g(are)g(in)165 3607 y(equilibrium)c(with)j(their)g (exterior)h(relativ)m(e)h(to)g(the)f(prescrib)s(ed)e(in)m(teraction)i (and)g(temp)s(erature.)165 3720 y(So)e(it)f(is)f(natural)g(to)i (de\014ne)f(Gibbs)f(measures)h(in)f(terms)h(of)h(conditional)e (probabilities.)165 3908 y Fx(De\014nition)36 b(2.1)46 b Fp(A)27 b(pr)-5 b(ob)g(ability)31 b(me)-5 b(asur)g(e)29 b Ft(\026)f Fp(on)h(the)g(c)-5 b(on\014gur)g(ation)30 b(sp)-5 b(ac)g(e)30 b Fw(\012)d Fp(is)h(c)-5 b(al)5 b(le)-5 b(d)30 b(a)f Fx(Gibbs)165 4020 y(measure)g Fp(for)h(the)f(Hamiltonian)i Ft(H)37 b Fp(in)29 b(\(1\))h(or)g(\(2\))g(at)g(inverse)f(temp)-5 b(er)g(atur)g(e)32 b Ft(\014)f Fv(\030)25 b Fw(1)p Ft(=T)43 b Fp(if)29 b(for)h(al)5 b(l)165 4133 y Fw(\003)26 b Fv(2)f(E)40 b Fp(and)34 b(al)5 b(l)33 b Ft(\033)28 b Fv(2)d Fw(\012)p Fp(,)1112 4338 y Ft(\026)p Fw(\()p Ft(X)33 b Fv(\021)25 b Ft(\033)35 b Fw(on)c(\003)15 b Fv(j)g Ft(X)33 b Fv(\021)25 b Ft(\021)36 b Fw(o\013)31 b(\003\))25 b(=)g Ft(\026)2426 4293 y Fu(\021)2426 4366 y(\014)s(;)p Fr(\003)2542 4338 y Fw(\()p Ft(\033)s Fw(\))831 b(\(3\))165 4551 y Fp(for)33 b Ft(\026)p Fp(-almost)f(al)5 b(l)32 b Ft(\021)d Fv(2)c Fw(\012)p Fp(.)41 b(In)32 b(the)g(ab)-5 b(ove,)32 b Ft(\026)1706 4507 y Fu(\021)1706 4579 y(\014)s(;)p Fr(\003)1821 4551 y Fw(\()p Ft(\033)s Fw(\))h Fp(is)e(the)h Fx(Boltzmann{Gibbs)j (distribution)165 4664 y Fp(in)e Fw(\003)g Fp(for)g Ft(\014)k Fp(and)d Ft(H)7 b Fp(,)32 b(which)i(is)f(given)e(by)1123 4913 y Ft(\026)1178 4869 y Fu(\021)1178 4941 y(\014)s(;)p Fr(\003)1293 4913 y Fw(\()p Ft(\033)s Fw(\))c(=)1551 4843 y Ft(I)1591 4862 y Fq(f)p Fu(\033)r Fq(\021)p Fu(\021)36 b Fr(o\013)24 b(\003)p Fq(g)p 1551 4892 432 4 v 1601 4975 a Ft(Z)1663 4989 y Fr(\003)1716 4975 y Fw(\()p Ft(\014)5 b(;)15 b(\021)s Fw(\))2007 4913 y(exp[)p Fv(\000)p Ft(\014)5 b(H)i Fw(\()p Ft(\033)s Fv(j)p Ft(\021)s Fw(\)])27 b Ft(:)842 b Fw(\(4\))165 5170 y Fp(Her)-5 b(e,)34 b Ft(Z)474 5184 y Fr(\003)527 5170 y Fw(\()p Ft(\014)5 b(;)15 b(\021)s Fw(\))35 b Fp(is)e(a)h(normalization)j(c)-5 b(onstant)35 b(making)f Ft(\026)2270 5126 y Fu(\021)2270 5198 y(\014)s(;)p Fr(\003)2419 5170 y Fp(a)g(pr)-5 b(ob)g(ability)35 b(me)-5 b(asur)g(e,)35 b(and)g(the)165 5283 y(c)-5 b(onstr)g(aint)39 b(that)e Ft(\033)j Fp(has)d(to)g(c)-5 b(oincide)36 b(with)h Ft(\021)j Fp(outside)d Fw(\003)f Fp(is)g(adde)-5 b(d)38 b(b)-5 b(e)g(c)g(ause)37 b(we)g(want)g(to)g(r)-5 b(e)g(alize)165 5396 y(these)42 b(pr)-5 b(ob)g(ability)43 b(me)-5 b(asur)g(es)43 b(imme)-5 b(diately)43 b(on)e(the)h(in\014nite)f(lattic)-5 b(e.)68 b(Note)41 b(that)h Ft(\026)3191 5352 y Fu(\021)3191 5424 y(\014)s(;)p Fr(\003)3348 5396 y Fp(in)f(fact)165 5509 y(only)34 b(dep)-5 b(ends)34 b(on)f(the)g(r)-5 b(estriction)34 b(of)f Ft(\021)j Fp(to)d Fw(\003)1782 5476 y Fu(c)1817 5509 y Fp(.)1867 5895 y Fw(9)p eop %%Page: 10 10 10 9 bop 165 304 a Fw(So,)28 b Ft(\026)g Fw(is)e(a)i(Gibbs)e(measure)h (if)f(it)h(has)g(prescrib)s(ed)e(conditional)h(distributions)e Fp(inside)j Fw(some)h(\014nite)165 417 y(set)k(of)g(v)m(ertices,)h(giv) m(en)f(that)g(the)f(con\014guration)g(is)g(held)f(\014xed)h Fp(outside)p Fw(,)i(and)e(these)h(conditional)165 530 y(distributions)d(are)k(giv)m(en)g(b)m(y)f(the)h(usual)e (Boltzmann{Gibbs)g(formalism.)46 b(This)30 b(de\014nition)g(go)s(es)165 643 y(bac)m(k)40 b(to)g(the)f(w)m(ork)g(of)g(Dobrushin)d([64)q(])j(and) g(Lanford)f(and)g(Ruelle)f([151)r(])i(in)f(the)h(late)g(1960's,)165 756 y(whence)h(Gibbs)e(measures)i(are)h(often)f(called)f Fp(DLR-states)p Fw(.)70 b(By)40 b(this)f(w)m(ork,)k(equilibrium)35 b(sta-)165 869 y(tistical)f(ph)m(ysics)f(and)h(the)g(study)g(of)g (phase)g(transitions)f(made)i(\014rm)e(con)m(tact)k(with)c(probabilit)m (y)165 981 y(theory)28 b(and)g(the)g(study)f(of)h(random)f(\014elds.)38 b(A)28 b(thermo)s(dynamic)e(justi\014cation)h(of)h(this)e(de\014nition) 165 1094 y(can)37 b(b)s(e)e(giv)m(en)h(b)m(y)g(the)h(v)-5 b(ariational)35 b(principle,)f(whic)m(h)h(states)i(that)g(\(in)e(the)h (case)h Fv(L)e Fw(=)f Fx(Z)3376 1061 y Fu(d)3417 1094 y Fw(\))i(the)165 1207 y(translation)f(in)m(v)-5 b(arian)m(t)35 b(Gibbs)e(measures)j(are)g(precisely)e(those)i(translation)e(in)m(v)-5 b(arian)m(t)35 b(random)165 1320 y(\014elds)e(whic)m(h)g(minimize)e (the)j(free)g(energy)h(densit)m(y)-8 b(,)34 b(cf.)h([151)q(,)g(96)q(].) 51 b(F)-8 b(or)35 b(b)s(etter)f(distinction,)f(the)165 1433 y(Gibbs)27 b(distributions)e Ft(\026)1012 1389 y Fu(\021)1012 1461 y(\014)s(;)p Fr(\003)1156 1433 y Fw(are)k(often)g (called)f Fp(\014nite)j(volume)e Fw(Gibbs)e(distributions,)e(whereas)k (the)165 1546 y(Gibbs)g(measures)h(are)h(sometimes)f(sp)s(eci\014ed)f (as)i Fp(in\014nite)h(volume)f Fw(Gibbs)e(measures.)306 1659 y(W)-8 b(e)37 b(write)e Fv(G)5 b Fw(\()p Ft(\014)g(H)i Fw(\))36 b(for)g(the)f(set)h(of)g(all)e(Gibbs)g(measures)h(with)f(giv)m (en)i(Hamiltonian)d Ft(H)43 b Fw(and)165 1772 y(in)m(v)m(erse)36 b(temp)s(erature)g Ft(\014)5 b Fw(.)57 b(In)35 b(the)h(sp)s(ecial)f (case)i Ft(U)44 b Fv(\021)34 b Fw(0)j(of)f(no)f(in)m(teraction,)j (there)e(is)f(only)g(one)165 1885 y(Gibbs)44 b(measure,)k(namely)c(the) h(pro)s(duct)f(measure)h(with)e(one-site)j(marginals)d Ft(\026)p Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))51 b(=)e Ft(a)p Fw(\))165 1998 y(=)33 b Ft(e)311 1965 y Fq(\000)p Fu(\014)s(V)16 b Fr(\()p Fu(a)p Fr(\))563 1998 y Ft(=)623 1933 y Fs(P)711 2021 y Fu(b)p Fq(2)p Fu(S)854 1998 y Ft(e)896 1965 y Fq(\000)p Fu(\014)s(V)g Fr(\()p Fu(b)p Fr(\))1140 1998 y Fw(.)55 b(In)34 b(general,)j(sev)m(eral)e(Gibbs)f (measures)h(for)f(the)i(same)f(in)m(teraction)165 2111 y(and)26 b(temp)s(erature)g(can)h(co)s(exist.)39 b(This)25 b(is)g(the)i(fundamen)m(tal)e(phenomenon)g(of)i(non)m(uniqueness)d(of) 165 2223 y(phases)30 b(whic)m(h)f(is)h(one)g(of)h(our)f(main)f(sub)5 b(jects;)30 b(w)m(e)h(return)e(to)j(this)d(p)s(oin)m(t)g(in)g(Section)h (2.7)i(b)s(elo)m(w.)306 2336 y(First)i(w)m(e)h(w)m(an)m(t)g(to)g (emphasize)e(an)h(imp)s(ortan)m(t)g(consequence)h(of)f(our)g (assumption)e(that)j(the)165 2449 y(underlying)k(in)m(teraction)i Ft(U)52 b Fw(in)m(v)m(olv)m(es)41 b(only)g(neigh)m(b)s(or)f(spins.)72 b(Due)41 b(to)i(this)d(assumption,)j(the)165 2562 y(Gibbs)27 b(distribution)e Ft(\026)976 2518 y Fu(\021)976 2590 y(\014)s(;)p Fr(\003)1120 2562 y Fw(only)i(dep)s(ends)g(on)h(the)h (restriction)e Ft(\021)2424 2577 y Fu(@)t Fr(\003)2547 2562 y Fw(of)h Ft(\021)k Fw(to)d(the)g(b)s(oundary)d Ft(@)5 b Fw(\003)29 b(of)165 2675 y(\003,)j(and)e(this)h(implies)d (that)k(eac)m(h)g(Gibbs)d(measure)i Ft(\026)c Fv(2)f(G)5 b Fw(\()p Ft(\014)g(H)i Fw(\))32 b(is)e(a)i Fp(Markov)i(r)-5 b(andom)35 b(\014eld)p Fw(.)44 b(By)165 2788 y(de\014nition,)29 b(this)g(means)h(that)h(for)f(eac)m(h)i(\003)25 b Fv(2)g(E)39 b Fw(and)29 b Ft(\033)g Fv(2)c Ft(S)2256 2755 y Fr(\003)1002 2970 y Ft(\026)p Fw(\()p Ft(X)33 b Fv(\021)25 b Ft(\033)33 b Fw(on)e(\003)15 b Fv(j)g(F)1691 2984 y Fr(\003)1740 2965 y Fo(c)1777 2970 y Fw(\))25 b(=)g Ft(\026)p Fw(\()p Ft(X)33 b Fv(\021)25 b Ft(\033)33 b Fw(on)e(\003)15 b Fv(j)g(F)2622 2985 y Fu(@)t Fr(\003)2717 2970 y Fw(\))p Ft(;)721 b Fw(\(5\))165 3153 y Ft(\026)p Fv(\000)p Fw(almost)45 b(surely)-8 b(.)85 b(This)44 b(Mark)m(o)m(v)j(prop)s(ert)m(y)e(will)e (b)s(e)h(an)i(essen)m(tial)f(to)s(ol)g(in)f(the)i(geometric)165 3265 y(argumen)m(ts)f(to)f(b)s(e)g(discussed)e(in)h(this)g(review.)81 b(There)43 b(is)g(in)g(fact)i(an)f(equiv)-5 b(alence)43 b(b)s(et)m(w)m(een)165 3378 y(Mark)m(o)m(v)k(random)d(\014elds)g(and)g (Gibbs)g(measures)h(for)g(nearest)g(neigh)m(b)s(or)f(p)s(oten)m(tials,) k(see)e(e.g.)165 3491 y(Av)m(erin)m(tsev)31 b([17)q(],)g(Grimmett)f ([107)r(])g(or)h(Georgii)f([96)q(].)306 3604 y(As)c(an)g(aside,)h(let)e (us)h(commen)m(t)h(on)e(the)h(case)h(when)e(the)h(in)m(teraction)g(of)g (spins)e(is)h(not)h(nearest-)165 3717 y(neigh)m(b)s(or)37 b(but)g(only)h(deca)m(ys)h(su\016cien)m(tly)e(fast)h(with)f(their)g (distance.)64 b(The)37 b(Boltzmann-Gibbs)165 3830 y(distributions)32 b(in)i(\(4\),)k(and)d(therefore)h(also)f(Gibbs)f(measures,)j(can)f (then)f(still)e(b)s(e)i(de\014ned,)h(but)165 3943 y(the)e(Gibbs)d (measures)i(fail)f(to)h(p)s(ossess)g(the)g(Mark)m(o)m(v)i(prop)s(ert)m (y)d(\(5\).)50 b(Rather)33 b(their)f(lo)s(cal)h(condi-)165 4056 y(tional)27 b(distributions)c Ft(\026)1003 4012 y Fu(\021)1003 4084 y(\014)s(;)p Fr(\003)1146 4056 y Fw(satisfy)k(a)g(w)m(eak)m(ening)h(of)f(the)h(Mark)m(o)m(v)h(prop)s (ert)m(y)d(called)h Fp(quasilo)-5 b(c)g(ality)165 4178 y Fw(or)42 b Fp(almost-Markov)j(pr)-5 b(op)g(erty)8 b Fw(:)67 b(for)42 b(ev)m(ery)h(\003)i Fv(2)f(E)50 b Fw(and)42 b Ft(A)j Fv(2)f(F)2524 4192 y Fr(\003)2577 4178 y Fw(,)i Ft(\026)2703 4134 y Fu(\021)2703 4206 y(\014)s(;)p Fr(\003)2818 4178 y Fw(\()p Ft(A)p Fw(\))d(is)e(a)h(con)m(tin)m(uous)165 4291 y(function)33 b(of)g Ft(\021)s Fw(.)51 b(So,)34 b(in)e(this)h(case,)i(Gibbs)d(measures)h(ha)m(v)m(e)i(prescrib)s(ed)c (con)m(tin)m(uous)i(v)m(ersions)g(of)165 4404 y(their)c(lo)s(cal)f (conditional)g(probabilities.)37 b(T)-8 b(o)30 b(obtain)e(a)i (su\016cien)m(tly)e(general)h(de\014nition)e(of)j(Gibbs)165 4517 y(measures)35 b(including)d(this)h(and)i(other)g(cases,)i(one)e (in)m(tro)s(duces)f(the)h(concept)h(of)f(a)g(sp)s(eci\014cation)165 4630 y Ft(G)30 b Fw(=)f(\()p Ft(G)472 4644 y Fr(\003)526 4630 y Ft(;)49 b Fw(\003)29 b Fv(2)g(E)8 b Fw(\).)49 b(This)31 b(is)h(a)h(family)f(of)h(probabilit)m(y)d(k)m(ernels)i Ft(G)2560 4644 y Fr(\003)2647 4630 y Fw(from)g(\(\012)p Ft(;)15 b Fv(F)3070 4644 y Fr(\003)3119 4625 y Fo(c)3156 4630 y Fw(\))33 b(to)h(\(\012)p Ft(;)15 b Fv(F)9 b Fw(\).)165 4742 y Ft(G)236 4756 y Fr(\003)290 4742 y Fw(\()p Fv(\001)p Ft(;)15 b(\021)s Fw(\))28 b(stands)e(for)g(an)m(y)h(distribution)c(of)j (spins)f(with)g(\014xed)g(con\014guration)h Ft(\021)2910 4756 y Fr(\003)2959 4738 y Fo(c)3021 4742 y Fv(2)f Ft(S)3168 4709 y Fq(Ln)p Fr(\003)3331 4742 y Fw(outside)165 4855 y(\003;)i(the)d(standard)e(case)j(is)e(the)h(Gibbs)e(sp)s (eci\014cation)h Ft(G)2063 4869 y Fr(\003)2116 4855 y Fw(\()p Fv(\001)p Ft(;)15 b(\021)s Fw(\))27 b(=)e Ft(\026)2477 4811 y Fu(\021)2477 4883 y(\014)s(;)p Fr(\003)2593 4855 y Fw(.)38 b(A)24 b(Gibbs)e(measure)i(is)e(then)165 4968 y(a)37 b(probabilit)m(y)c(measure)j Ft(\026)g Fw(on)g(\012)g (satisfying)f Ft(\026)p Fw(\()p Ft(A)15 b Fv(j)g(F)2091 4982 y Fr(\003)2140 4963 y Fo(c)2177 4968 y Fw(\))35 b(=)f Ft(G)2423 4982 y Fr(\003)2477 4968 y Fw(\()p Ft(A;)15 b Fv(\001)p Fw(\))38 b Ft(\026)p Fw(-almost)e(surely)e(for)i(all)165 5081 y(\003)26 b Fv(2)f(E)34 b Fw(and)25 b Ft(A)h Fv(2)e(F)9 b Fw(;)29 b(this)c(prop)s(ert)m(y)g(can)h(b)s(e)g(expressed)f(in)g(a)i 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b(set)h(of)g(all)f(\(in\014nite)165 1659 y(v)m(olume\))31 b(Gibbs)e(measures)h(is)f(then)h(equal)g(to)1446 1863 y Fv(G)5 b Fw(\()p Ft(\014)g(H)i Fw(\))27 b(=)1860 1782 y Fs(\\)1837 1966 y Fr(\003)p Fq(2E)1991 1863 y Fv(G)2045 1877 y Fr(\003)2098 1863 y Fw(\()p Ft(\014)5 b(H)i Fw(\))p Ft(:)165 2141 y Fw(Equiv)-5 b(alen)m(tly)d(,)27 b(a)h(probabilit)m(y)d (measure)i Ft(\026)h Fw(on)f(\012)g(is)f(a)i(Gibbs)e(measure)h(for)h (the)f(Hamiltonian)f Ft(\014)5 b(H)165 2254 y Fw(if)30 b(it)g(b)s(elongs)f(to)i(the)g(closed)f(con)m(v)m(ex)i(h)m(ull)c(of)j (the)f(set)h(of)g(limit)d(p)s(oin)m(ts)h(of)h Ft(\026)2833 2210 y Fu(\021)2833 2282 y(\014)s(;)p Fr(\003)2979 2254 y Fw(as)h(\003)25 b Fv(")h(L)p Fw(.)306 2367 y(One)39 b(imp)s(ortan)m(t)e(consequence)j(is)d(that)j Fv(G)5 b Fw(\()p Ft(\014)g(H)i Fw(\))40 b Fv(6)p Fw(=)f Fv(;)p Fw(.)66 b(This)36 b(is)i(b)s(ecause)h(eac)m(h)g Fv(G)3251 2381 y Fr(\003)3305 2367 y Fw(\()p Ft(\014)5 b(H)i Fw(\))39 b(is)165 2480 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y(size)43 b(e\013ects)h(\(whic)m(h)e (are)h(alw)m(a)m(ys)g(presen)m(t)f(but)g(whic)m(h)g(are)h(not)g(alw)m (a)m(ys)g(relev)-5 b(an)m(t)43 b(for)f(certain)165 4061 y(phenomena\))34 b(and)g(it)f(is)g(only)h(in)e(the)j(thermo)s(dynamic)d (limit)g(of)i(in\014nite)e(v)m(olume)i(that)g(w)m(e)h(can)165 4174 y(get)27 b(a)e(clean)h(and)e(precise)h(picture)f(of)i(realistic)e (phenomena)g(suc)m(h)h(as)h(phase)f(transitions)e(or)i(phase)165 4287 y(co)s(existence.)39 b(This)19 b(is)h(a)i(consequence)f(of)h(the)f (general)g(probabilistic)c(principle)h(of)j(large)g(n)m(um)m(b)s(ers.) 165 4400 y(In)35 b(this)g(sense,)j(in\014nite)33 b(systems)j(serv)m(e)h (as)f(an)f(idealized)g(appro)m(ximation)f(to)j(v)m(ery)f(large)g (\014nite)165 4512 y(systems.)165 4756 y FA(2.7)113 b(Phase)38 b(transition)g(and)g(phases)165 4928 y Fw(As)22 b(p)s(oin)m(ted)f(out)h (ab)s(o)m(v)m(e,)j(in)20 b(general)i(there)g(ma)m(y)h(exist)e(sev)m (eral)h(solutions)f Ft(\026)g Fw(to)i(the)f(DLR-equation)165 5040 y(\(3\))36 b(for)e(giv)m(en)h Ft(U)10 b Fw(,)36 b Ft(V)54 b Fw(and)34 b Ft(\014)5 b Fw(,)36 b(whic)m(h)e(means)g(that)h (m)m(ultiple)e(Gibbs)g(measures)h(exist.)53 b(The)34 b(sys-)165 5153 y(tem)j(can)g(then)f(c)m(ho)s(ose)h(b)s(et)m(w)m(een)g (sev)m(eral)g(equilibrium)32 b(states.)60 b(\(In)36 b(a)h(dynamical)e (theory)h(this)165 5266 y(c)m(hoice)e(w)m(ould)e(dep)s(end)g(on)h(the)g (past;)i(but)d(here)h(w)m(e)h(are)g(in)d(a)j(pure)e(equilibrium)c (setting.\))50 b Fp(The)165 5379 y(phenomenon)c(of)d(non-uniqueness)g (ther)-5 b(efor)g(e)44 b(c)-5 b(orr)g(esp)g(onds)47 b(to)d(a)f(phase)i (tr)-5 b(ansition)p Fw(.)77 b(In)41 b(fact,)165 5492 y(it)g(is)e(then)i(p)s(ossible)d(to)k(construct)f(di\013eren)m(t)f (Gibbs)f(measures)i(as)g(in\014nite)d(v)m(olume)j(limits)d(of)165 5605 y(Gibbs)f(distributions)d(with)j(di\013eren)m(t)h(c)m(hoices)h(of) f(b)s(oundary)e(conditions)h([91)q(,)h(96)q(].)65 b(Since)37 b(an)m(y)1844 5895 y(11)p eop %%Page: 12 12 12 11 bop 165 304 a Fw(t)m(w)m(o)34 b(Gibbs)e(measures)g(can)h(b)s(e)f (distinguished)d(b)m(y)k(a)g(suitable)e(lo)s(cal)h(observ)-5 b(able,)33 b(a)h(phase)e(tran-)165 417 y(sition)39 b(can)h(b)s(e)f (detected)i(b)m(y)f(lo)s(oking)e(at)j(suc)m(h)e(a)h(lo)s(cal)f(observ) -5 b(able)39 b(whic)m(h)g(is)g(then)g(called)g(an)165 530 y Fp(or)-5 b(der)37 b(p)-5 b(ar)g(ameter)p Fw(.)52 b(V)-8 b(arying)33 b(the)g(external)g(parameters)h(suc)m(h)f(as)g(temp) s(erature)g(or)g(an)g(external)165 643 y(magnetic)27 b(\014eld)d(\(whic)m(h)h(can)h(b)s(e)f(tuned)h(b)m(y)f(the)h(exp)s (erimen)m(ter)f(via)h(some)g(heatbath)g(or)g(reserv)m(oir\))165 756 y(one)35 b(will)e(observ)m(e)i(di\013eren)m(t)f(scenarios;)j(these) e(are)h(collected)f(in)e(the)i(so)g(called)f Fp(phase)k(diagr)-5 b(am)165 869 y Fw(of)31 b(the)f(considered)g(system.)306 981 y(As)40 b(w)m(e)f(ha)m(v)m(e)i(indicated)d(in)f(the)j(in)m(tro)s (duction,)g(the)f(phase)g(transition)f(phenomenon)g(is)g(of)165 1094 y(cen)m(tral)30 b(in)m(terest)f(in)f(equilibrium)d(statistical)k 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(elemen)m(ts,)j(the)d(extremal)g(Gibbs)e(measures,)165 1998 y(ha)m(v)m(e)32 b(a)e(trivial)f(tail)g Ft(\033)s Fw(-\014eld)g Fv(T)53 b Fw(\(whic)m(h)29 b(means)h(that)h(all)e(ev)m (en)m(ts)j(in)d Fv(T)52 b Fw(ha)m(v)m(e)32 b(probabilit)m(y)27 b(0)k(or)f(1\).)165 2111 y(Equiv)-5 b(alen)m(tly)d(,)35 b(all)e(macroscopic)i(observ)-5 b(ables)34 b(are)h(almost)f(surely)g (constan)m(t.)54 b(In)34 b(addition,)g(the)165 2223 y(tail)25 b(trivialit)m(y)e(can)i(b)s(e)g(c)m(haracterized)h(b)m(y)f(an)g (asymptotic)g(indep)s(endence)e(\(or)j(mixing\))d(prop)s(ert)m(y)-8 b(.)165 2336 y(On)44 b(the)g(other)g(hand,)j(an)m(y)e(Gibbs)d(measure)i Ft(\026)g Fw(can)h(b)s(e)e(decomp)s(osed)h(in)m(to)g(extremal)h(Gibbs) 165 2449 y(measures;)30 b(therefore)f(ev)m(ery)g(con\014guration)f (whic)m(h)f(is)h(t)m(ypical)g(for)g Ft(\026)h Fw(is)e(in)h(fact)h(t)m (ypical)f(for)h(some)165 2562 y(extremal)k(Gibbs)d(measure.)46 b(This)30 b(sho)m(ws)i(that)h(the)f(extremal)g(Gibbs)f(measures)h (corresp)s(ond)e(to)165 2675 y(what)i(one)f(can)h(really)e(see)i(in)e (nature)h(as)h(far)f(as)h(large)f(systems)g(in)f(equilibrium)d(are)32 b(concerned.)165 2788 y(The)23 b(extremal)g(Gibbs)f(measures)h (therefore)h(corresp)s(ond)d(to)j(the)g(ph)m(ysical)e(macrostates,)k (whereas)165 2901 y(non-extremal)45 b(Gibbs)e(measures)h(only)g(pro)m (vide)g(a)h(limited)e(description)f(when)i(the)h(system's)165 3014 y(precise)28 b(state)i(is)e(unkno)m(wn.)39 b(F)-8 b(or)29 b(all)f(these)h(reasons,)g(the)g(extremal)f(Gibbs)f(measures)i (are)g(called)165 3127 y(\(equilibrium\))36 b Fp(phases)p Fw(.)69 b(The)39 b(cen)m(tral)h(sub)5 b(ject)39 b(of)h(this)e(review)g (is)h(the)h(geometric)g(analysis)e(of)165 3240 y(their)c(t)m(ypical)h (con\014gurations,)g(and)f(thereb)m(y)h(the)g(analysis)f(of)h(the)g (phase)f(diagram)g(giving)g(the)165 3353 y(v)-5 b(ariation)37 b(in)f(the)i(n)m(um)m(b)s(er)f(and)g(the)g(nature)h(of)g(the)f(phases)h (as)g(one)f(c)m(hanges)i(v)-5 b(arious)37 b(con)m(trol)165 3465 y(parameters)31 b(\(coupling,)e(temp)s(erature,)i(external)f (\014elds,)f(etc.\).)306 3578 y(Often)j(it)g(is)e(natural)h(to)i (consider)e(automorphisms)f(of)i(the)g(graph)g(\()p Fv(L)p Ft(;)15 b Fv(\030)p Fw(\).)45 b(F)-8 b(or)33 b(example,)f(if)165 3691 y Fv(L)g Fw(=)f Fx(Z)426 3658 y Fu(d)500 3691 y Fw(w)m(e)k(consider)e(the)h(translation)g(group)f(\()p Ft(\022)1960 3705 y Fu(x)2004 3691 y Fw(\))2039 3714 y Fu(x)p Fq(2)p Fn(Z)2176 3695 y Fo(d)2216 3691 y Fw(.)52 b(A)34 b(homogeneous)h(phase)f(is)f(then)h(an)165 3804 y(extremal)29 b(Gibbs)d(measure)j(whic)m(h)e(is)g(also)h(translation)g (in)m(v)-5 b(arian)m(t.)39 b(On)27 b(the)i(other)f(hand,)g(w)m(e)h(can) 165 3917 y(regard)d(the)g(extremal)g(p)s(oin)m(ts)e(of)i(the)g(con)m(v) m(ex)h(set)f(of)g(all)f(translation)f(in)m(v)-5 b(arian)m(t)25 b(Gibbs)f(measures.)165 4030 y(These)31 b(are)g(ergo)s(dic,)g(whic)m(h) f(means)g(that)i(they)f(cannot)g(b)s(e)g(decomp)s(osed)f(in)m(to)h (distinct)e(transla-)165 4143 y(tion)35 b(in)m(v)-5 b(arian)m(t)34 b(probabilit)m(y)e(measures,)k(and)e(are)h(trivial)e(on)i(the)g Ft(\033)s Fw(-algebra)g(of)g(all)f(translation)165 4256 y(in)m(v)-5 b(arian)m(t)37 b(ev)m(en)m(ts.)63 b(Ho)m(w)m(ev)m(er,)41 b(these)d(extremal)f(translation)f(in)m(v)-5 b(arian)m(t)37 b(Gibbs)e(measures)i(need)165 4369 y(not)e(b)s(e)g(homogeneous)g (phases;)i(they)e(are)g Fp(only)44 b Fw(ergo)s(dic.)53 b(Y)-8 b(et,)38 b(ergo)s(dic)c(measures)h Ft(\026)f Fw(satisfy)h(a)165 4482 y(la)m(w)30 b(of)h(large)g(n)m(um)m(b)s(ers:)39 b(for)30 b(an)m(y)h(observ)-5 b(able)29 b Ft(f)40 b Fw(and)30 b(an)m(y)g(sequence)h(of)g(increasing)e(cub)s(es)g(\003,)1044 4719 y(lim)1022 4786 y Fr(\003)p Fq(")p Fn(Z)1156 4768 y Fo(d)1251 4658 y Fw(1)p 1217 4698 114 4 v 1217 4781 a Fv(j)p Fw(\003)p Fv(j)1364 4638 y Fs(X)1356 4822 y Fu(x)p Fq(2)p Fr(\003)1507 4719 y Ft(f)g Fv(\016)21 b Ft(\022)1690 4733 y Fu(x)1759 4719 y Fw(=)k Ft(\026)p Fw(\()p Ft(f)10 b Fw(\))91 b Ft(\026)p Fw(-almost)30 b(surely)-8 b(.)165 4997 y(Hence,)31 b(ergo)s(dic)f(Gibbs)e(measures)i (are)g(suitable)e(for)i(mo)s(delling)d(macrostates)32 b(in)c(equilibrium)d(if)165 5110 y(one)k(limits)e(oneself)h(to)i (measuring)d(certain)i(bulk)e(observ)-5 b(ables)27 b(or)i(macroscopic)g (quan)m(tities)f(with)165 5223 y(additivit)m(y)e(prop)s(erties.)38 b(Notice,)29 b(ho)m(w)m(ev)m(er,)g(that)f(there)g(exists)e(a)i(certain) f(non-uniformit)m(y)e(in)h(the)165 5336 y(literature)35 b(concerning)g(the)h(nomenclature.)56 b(Sometimes)35 b(these)h(ergo)s(dic)f(Gibbs)f(measures)i(are)165 5449 y(called)26 b(\(pure\))f(phases.)39 b(It)26 b(is)f(then)h(argued)g (that)h(it)e(migh)m(t)h(happ)s(en)e(that)j(t)m(w)m(o)g(phases)f(\(as)g (de\014ned)165 5562 y(ab)s(o)m(v)m(e\))34 b(for)f(a)g(system)f(can)h(b) m(y)g(no)f(means)h(b)s(e)f(macroscopically)g(distinguished)c(\(for)33 b(example)f(if)1844 5895 y(12)p eop %%Page: 13 13 13 12 bop 165 304 a Fw(one)30 b(is)f(a)h(translation)f(of)h(the)g (other\).)41 b(W)-8 b(e)31 b(do)f(not)g(wish)e(to)j(en)m(ter)f(in)m(to) g(a)g(detailed)f(discussion)e(of)165 417 y(these)k(p)s(oin)m(ts.)1844 5895 y(13)p eop %%Page: 14 14 14 13 bop 165 304 a Fy(3)135 b(Some)44 b(mo)t(dels)165 507 y Fw(In)36 b(this)g(section)h(w)m(e)h(discuss)d(brie\015y)g(the)i (phase)f(transition)f(b)s(eha)m(viour)h(of)h(some)g(protot)m(ypical)165 620 y(examples)43 b(of)g(Gibbs)e(systems.)77 b(Although)42 b(these)h(examples)g(are)g(fairly)e(standard)h(and)g(w)m(ell-)165 733 y(kno)m(wn)31 b(to)g(most)g(of)g(our)g(readers,)g(w)m(e)g(need)g (to)g(include)e(them)i(here)f(to)i(set)f(up)f(the)h(stage.)44 b(They)165 846 y(will)22 b(b)s(e)i(studied)f(in)g(detail)g(in)g(the)i (later)f(sections.)39 b(An)24 b(accoun)m(t)i(of)f(phase)f(transition)f (phenomena)165 959 y(in)30 b(more)i(general)f(lattice)h(mo)s(dels)e (can)i(b)s(e)f(found)f(in)g(man)m(y)h(other)h(sources,)g(including)c ([147)q(,)k(96)q(,)165 1072 y(212)r(,)e(213)r(,)g(180)r(].)165 1312 y FA(3.1)113 b(The)37 b(ferromagnetic)i(Ising)e(mo)s(del)165 1483 y Fw(The)e(Ising)e(mo)s(del)h(w)m(as)i(in)m(tro)s(duced)d(in)g (the)j(1920's)g(b)m(y)f(Wilhelm)e(Lenz)i([158)q(])g(and)g(his)e(studen) m(t)165 1596 y(Ernst)40 b(Ising)f([137)r(])h(as)h(a)g(simple)d(mo)s (del)h(for)h(magnetism)h(and,)h(in)d(particular,)j(ferromagnetic)165 1709 y(phase)23 b(transitions.)37 b(Eac)m(h)24 b(site)e Ft(x)k Fv(2)e(L)f Fw(can)h(tak)m(e)g(either)f(of)g(t)m(w)m(o)h(spin)e (v)-5 b(alues,)24 b(+1)f(\(\\spin)f(up"\))h(and)165 1822 y Fv(\000)p Fw(1)i(\(\\spin)f(do)m(wn"\),)j(so)e(that)g(the)g(state)h (space)g(is)e(equal)g(to)i Ft(S)k Fw(=)25 b Fv(f\000)p Fw(1)p Ft(;)15 b Fw(+1)p Fv(g)p Fw(.)40 b(The)25 b(Hamiltonian)e(is)165 1935 y(giv)m(en)32 b(b)m(y)f(\(1\))h(with)e Ft(U)10 b Fw(\()p Ft(\033)s Fw(\()p Ft(x)p Fw(\))p Ft(;)15 b(\033)s Fw(\()p Ft(y)s Fw(\)\))30 b(=)c Fv(\000)p Ft(\033)s Fw(\()p Ft(x)p Fw(\))15 b Ft(\033)s Fw(\()p Ft(y)s Fw(\))33 b(and)d Ft(V)21 b Fw(\()p Ft(\033)s Fw(\()p Ft(x)p Fw(\)\))28 b(=)e Fv(\000)p Ft(h)15 b(\033)s Fw(\()p Ft(x)p Fw(\).)45 b(The)31 b(parameter)165 2048 y Ft(h)26 b Fv(2)f Fx(R)j Fw(describ)s(es)d(an)j(external)f(\014eld.)38 b(The)27 b(\014nite)g(v)m(olume)g(Gibbs)f(distribution)e(in)i(a)i(b)s(o)m(x)f (\003)g(with)165 2161 y(external)38 b(\014eld)f Ft(h)h Fw(at)g(in)m(v)m(erse)g(temp)s(erature)g Ft(\014)43 b(>)37 b Fw(0)h(with)f(b)s(oundary)f(condition)g Ft(\021)42 b Fw(is)36 b(th)m(us)i(the)165 2274 y(probabilit)m(y)30 b(measure)i Ft(\026)1043 2229 y Fu(\021)1043 2302 y(h;\014)s(;)p Fr(\003)1251 2274 y Fw(on)g(\012)d(=)f Fv(f\000)p Fw(1)p Ft(;)15 b Fw(+1)p Fv(g)1935 2241 y Fq(L)2021 2274 y Fw(whic)m(h)31 b(to)i(eac)m(h)h Ft(\033)d Fv(2)d Fw(\012)k(assigns)g(probabilit)m(y) 165 2387 y(prop)s(ortional)d(to)777 2631 y Ft(I)817 2649 y Fq(f)p Fu(\033)r Fq(\021)p Fu(\021)34 b Fr(on)24 b(\003)1165 2630 y Fo(c)1197 2649 y Fq(g)1282 2631 y Fw(exp)1436 2487 y Fs(")1509 2631 y Ft(\014)1581 2537 y Fs(\020)1750 2550 y(X)1752 2710 y Fo(x)p Fm(\030)p Fo(y)1655 2766 y(x)p Fm(2)p Fl(\003)11 b(or)g Fo(y)r Fm(2)p Fl(\003)1990 2631 y Ft(\033)s Fw(\()p Ft(x)p Fw(\))k Ft(\033)s Fw(\()p Ft(y)s Fw(\))22 b(+)e Ft(h)2543 2550 y Fs(X)2536 2734 y Fu(x)p Fq(2)p Fr(\003)2686 2631 y Ft(\033)s Fw(\()p Ft(x)p Fw(\))2863 2537 y Fs(\021)2914 2487 y(#)2977 2631 y Ft(:)165 2936 y Fw(F)-8 b(or)42 b Ft(\014)48 b Fw(=)42 b(0)f(\(\\in\014nite)e(temp)s(erature"\))i(the)g(spin)e(v)-5 b(ariables)39 b(are)j(indep)s(enden)m(t)c(under)h Ft(\026)3413 2892 y Fu(\021)3413 2964 y(h;\014)s(;)p Fr(\003)3589 2936 y Fw(,)165 3049 y(but)c(as)g(so)s(on)g(as)g Ft(\014)j(>)33 b Fw(0)j(the)f(probabilit)m(y)e(distribution)e(starts)36 b(to)f(fa)m(v)m(our)h(con\014gurations)e(with)165 3162 y(man)m(y)d(neigh)m(b)s(or)e(pairs)g(of)i(aligned)e(spins.)39 b(This)29 b(tendency)h(b)s(ecomes)h(stronger)f(and)g(stronger)h(as)165 3275 y Ft(\014)36 b Fw(increases.)306 3388 y(In)h(the)h(case)h Ft(h)e Fw(=)g(0)h(of)g(no)f(external)h(\014eld,)g(the)g(mo)s(del)e(is)h (symmetric)f(under)g(in)m(terc)m(hange)165 3500 y(of)h(the)f(spin)e(v) -5 b(alues)36 b Fv(\000)p Fw(1)g(and)g(+1,)h(so)g(that)g(there)f(is)f (an)h(equal)g(c)m(hance)h(of)g(ha)m(ving)e(man)m(y)h(pairs)165 3613 y(of)j(plus)e(spins)f(or)j(ha)m(ving)f(man)m(y)h(pairs)e(of)i(min) m(us)e(spins.)63 b(This)37 b(dic)m(hotom)m(y)i(giv)m(es)g(rise)f(to)h (the)165 3726 y(follo)m(wing)34 b(in)m(teresting)g(b)s(eha)m(vior.)54 b(Supp)s(ose)33 b(that)j Fv(L)d Fw(=)f Fx(Z)2253 3693 y Fu(d)2294 3726 y Fw(,)k Ft(d)d Fv(\025)g Fw(2.)55 b(If)35 b Ft(\014)40 b Fw(is)34 b(su\016cien)m(tly)g(small)165 3839 y(\(i.e.,)f(in)e(the)h(high)e(temp)s(erature)i(regime\),)g(the)g (in)m(teraction)g(is)f(not)h(strong)g(enough)g(to)g(pro)s(duce)165 3952 y(an)m(y)e(long)f(range)h(order,)g(so)f(that)i(the)e(b)s(oundary)f (conditions)g(b)s(ecome)i(irrelev)-5 b(an)m(t)28 b(in)h(the)g (in\014nite)165 4065 y(v)m(olume)j(limit)e(and)i(the)g(Gibbs)f(measure) h(is)g(uniquely)d(determined.)45 b(By)33 b(ergo)s(dicit)m(y)e(and)h (the)g Fv(\006)165 4178 y Fw(symmetry)-8 b(,)27 b(the)e(limiting)e (fraction)i(of)g(plus)e(spins)g(will)g(almost)i(surely)f(b)s(e)h(1)p Ft(=)p Fw(2)h(under)e(this)g(unique)165 4291 y(Gibbs)e(measure.)38 b(In)22 b(con)m(trast,)k(when)21 b Ft(\014)28 b Fw(is)22 b(su\016cien)m(tly)g(large)h(\(in)f(the)h(lo)m(w)f(temp)s(erature)h (regime\),)165 4404 y(the)38 b(in)m(teraction)g(b)s(ecomes)g(so)g (strong)g(that)g(a)g(long)g(range)g(order)f(app)s(ears:)55 b(the)38 b(bias)e(to)m(w)m(ards)165 4517 y(neigh)m(b)s(or)23 b(pairs)g(of)h(equal)g(spin)e(then)i(implies)d(that)k(Gibbs)e(measures) h(prefer)f(con\014gurations)g(with)165 4630 y(either)h(a)h(v)-5 b(ast)24 b(ma)5 b(jorit)m(y)25 b(of)f(plus)e(spins)g(or)i(a)h(v)-5 b(ast)25 b(ma)5 b(jorit)m(y)24 b(of)g(min)m(us)f(spins,)g(and)h(this)f (preference)165 4742 y(ev)m(en)j(surviv)m(es)d(in)g(the)i(in\014nite)e (v)m(olume)h(limit.)36 b(The)25 b(system)f(th)m(us)h(undergo)s(es)e(a)i (phase)g(transition)165 4855 y(whic)m(h)33 b(manifests)g(itself)g(in)f (a)j(non-uniqueness)c(of)j(Gibbs)e(measures.)51 b(Sp)s(eci\014cally)-8 b(,)33 b(there)h(exist)165 4968 y(t)m(w)m(o)47 b(particular)c(Gibbs)g (measures)h Ft(\026)1523 4935 y Fr(+)1627 4968 y Fw(and)g Ft(\026)1873 4935 y Fq(\000)1932 4968 y Fw(,)49 b(obtained)44 b(as)h(in\014nite)d(v)m(olume)j(limits)d(with)165 5081 y(resp)s(ectiv)m(e)25 b(b)s(oundary)d(conditions)g Ft(\021)29 b Fv(\021)c Fw(+1)f(and)g Ft(\021)k Fv(\021)d(\000)p Fw(1,)h(whic)m(h)d(can)h(b)s(e)g(distinguished)c(b)m(y)k(their)165 5194 y(o)m(v)m(erall)h(densit)m(y)e(of)i(+1's:)38 b(the)24 b(densit)m(y)g(is)f(greater)i(than)f(1)p Ft(=)p Fw(2)i(under)c Ft(\026)2613 5161 y Fr(+)2696 5194 y Fw(and)i(\(b)m(y)h(symmetry\))f (less)165 5307 y(than)29 b(1)p Ft(=)p Fw(2)g(under)e Ft(\026)851 5274 y Fq(\000)910 5307 y Fw(.)40 b(This)27 b(is)g(the)i(sp)s(on)m(taneous)f(magnetization)h(phenomenon)e(that)i (Lenz)g(and)165 5420 y(Ising)23 b(w)m(ere)i(lo)s(oking)d(for)i(but)f(w) m(ere)i(discouraged)e(b)m(y)h(not)g(\014nding)e(it)h(in)g(one)h (dimension.)36 b(In)23 b(higher)165 5533 y(dimensions,)31 b(the)h(uniqueness)e(regime)i(and)f(the)i(phase)f(transition)e(regime)i (are)h(separated)g(b)m(y)f(a)165 5646 y(sharp)26 b(critical)h(v)-5 b(alue)26 b Ft(\014)993 5660 y Fu(c)1029 5646 y Fw(,)i(as)f(is)f (summarized)g(in)g(the)i(follo)m(wing)d(classical)i(theorem)g([187)r(,) g(63)q(,)h(65)q(]:)1844 5895 y(14)p eop %%Page: 15 15 15 14 bop 165 304 a Fx(Theorem)34 b(3.1)46 b Fp(F)-7 b(or)29 b(the)f(ferr)-5 b(omagnetic)29 b(Ising)e(mo)-5 b(del)29 b(on)f(the)g(inte)-5 b(ger)28 b(lattic)-5 b(e)28 b Fx(Z)3040 271 y Fu(d)3108 304 y Fp(of)g(dimension)165 417 y Ft(d)47 b Fv(\025)f Fw(2)f Fp(at)g(zer)-5 b(o)45 b(external)h(\014eld,)h(ther)-5 b(e)45 b(exists)g(a)g(critic)-5 b(al)45 b(inverse)f(temp)-5 b(er)g(atur)g(e)47 b Ft(\014)3178 431 y Fu(c)3260 417 y Fv(2)f Fw(\(0)p Ft(;)15 b Fv(1)p Fw(\))165 530 y Fp(\(dep)-5 b(ending)35 b(on)f Ft(d)p Fp(\))g(such)f(that)i(for)f Ft(\014)e(<)26 b(\014)1646 544 y Fu(c)1714 530 y Fp(the)34 b(mo)-5 b(del)35 b(has)f(a)g(unique)e (Gibbs)i(me)-5 b(asur)g(e)34 b(while)g(for)165 643 y Ft(\014)d(>)25 b(\014)394 657 y Fu(c)461 643 y Fp(ther)-5 b(e)34 b(ar)-5 b(e)33 b(multiple)h(Gibbs)e(me)-5 b(asur)g(es.)165 830 y Fw(A)29 b(sto)s(c)m(hastic-geometric)i(pro)s(of)d(of)h(this)f (result)g(will)e(b)s(e)i(giv)m(en)h(in)f(Section)g(6.)41 b(In)28 b(fact,)i(the)f(result)165 943 y(\(as)40 b(w)m(ell)e(as)h(its)g (pro)s(of)7 b(\))39 b(holds)e(for)i(an)m(y)h(graph)e(\()p Fv(L)p Ft(;)15 b Fv(\030)p Fw(\))40 b(in)d(place)j(of)f Fx(Z)2716 910 y Fu(d)2756 943 y Fw(,)j(except)e(that)f Ft(\014)3373 957 y Fu(c)3448 943 y Fw(ma)m(y)165 1056 y(then)34 b(tak)m(e)h(the)f(v)-5 b(alues)33 b(0)h(or)g Fv(1)p Fw(.)50 b(F)-8 b(or)34 b(instance,)h(on)e(the)h(one-dimensional) e(lattice)i Fx(Z)3225 1023 y Fr(1)3298 1056 y Fw(w)m(e)g(ha)m(v)m(e)165 1169 y Ft(\014)216 1183 y Fu(c)294 1169 y Fw(=)41 b Fv(1)p Fw(,)j(whic)m(h)39 b(means)h(that)h(there)g(is)e(a)i(unique)e(Gibbs)g (measure)h(for)h(all)e Ft(\014)5 b Fw(.)71 b(F)-8 b(or)41 b Fx(Z)3380 1136 y Fr(2)3420 1169 y Fw(,)i(the)165 1282 y(critical)27 b(v)-5 b(alue)28 b(has)f(b)s(een)h(found)e(to)j(b)s(e)e Ft(\014)1604 1296 y Fu(c)1665 1282 y Fw(=)1771 1246 y Fr(1)p 1771 1261 36 4 v 1771 1313 a(2)1831 1282 y Fw(log)q(\(1)17 b(+)2131 1207 y Fv(p)p 2206 1207 46 4 v 2206 1282 a Fw(2)q(\).)40 b(This)26 b(calculation)h(is)g(a)i(remark)-5 b(able)165 1395 y(ac)m(hiev)m(emen)m(t)31 b(whic)m(h)d(b)s(egan)h(with)f(Onsager)i ([186)q(])g(in)e(1944.)42 b(An)29 b(accoun)m(t)h(of)g(v)-5 b(arious)28 b(\(algebraic)165 1508 y(and/or)f(com)m(binatorial\))g (metho)s(ds)f(can)i(b)s(e)e(found)g(e.g.)40 b(in)26 b([220)r(,)h(171)q (].)40 b(Let)28 b(us)e(also)h(men)m(tion)g(the)165 1621 y(w)m(ork)f(done)g(in)e(1973)k(b)m(y)e(Abraham)f(and)g(Martin-L\177)-45 b(of)26 b([1])g(relating)f(these)h(exact)i(computations)d(to)165 1734 y(the)32 b(real)g(magnetization)g(in)f(the)h(appropriate)f(Gibbs)f (measures;)i(it)g(also)g(giv)m(es)g(the)g(result)f(that)165 1847 y(there)h(is)f(a)h(unique)d(Gibbs)h(measure)i Fp(at)g Fw(the)g(critical)f(v)-5 b(alue)31 b Ft(\014)h Fw(=)27 b Ft(\014)2525 1861 y Fu(c)2560 1847 y Fw(.)45 b(A)31 b(rigorous)g(calculation)g(of)165 1959 y(the)g(critical)e(v)-5 b(alue)30 b(in)f(higher)g(dimensions)e(is)j(b)s(ey)m(ond)f(curren)m(t)h (kno)m(wledge.)41 b(It)30 b(is)g(b)s(eliev)m(ed)f(that)165 2072 y(uniqueness)e(holds)h(at)i(criticalit)m(y)e(in)g(all)g (dimensions)f Ft(d)e Fv(\025)g Fw(2,)30 b(but)f(so)g(far)g(this)f(is)g (only)h(kno)m(wn)f(for)165 2185 y Ft(d)e Fw(=)f(2)31 b(and)e Ft(d)d Fv(\025)f Fw(4)31 b([11)q(].)306 2298 y(The)38 b(case)i(of)e(a)h(nonzero)g(external)f(\014eld)f Ft(h)i Fv(6)p Fw(=)f(0)h(is)e(less)h(in)m(teresting,)h(in)f(that)g(one) h(\014nds)e(a)165 2411 y(unique)27 b(Gibbs)g(measure)h(for)h(all)e Ft(\014)34 b Fw(and)28 b Ft(d)p Fw(.)40 b(The)28 b(in)m(tuitiv)m(e)g (explanation)f(is)h(that)h(for)f Ft(h)e Fv(6)p Fw(=)f(0)k(there)165 2524 y(is)37 b(no)g Fv(\006)g Fw(symmetry)g(whic)m(h)f(could)h(b)s(e)f (brok)m(en;)41 b(dep)s(ending)35 b(on)j(the)f(sign)g(of)g Ft(h)p Fw(,)j(the)e(system)f(is)165 2637 y(forced)d(to)g(prefer)f (either)f(+1's)i(or)f Fv(\000)p Fw(1's.)50 b(This)32 b(comes)i(from)f(the)h(fact)g(that)g(the)g(magnetic)g(\014eld)165 2750 y(acts)g(on)f(the)g(whole)g(v)m(olume,)g(whereas)g(the)g (in\015uence)e(of)j(a)f(b)s(oundary)e(condition)g(is)h(of)i(smaller)165 2863 y(order)39 b(as)h(the)g(v)m(olume)f(increases.)69 b(In)38 b(con)m(trast,)44 b(a)c(phase)f(transition)f(for)i Ft(h)h Fv(6)p Fw(=)f(0)g(do)s(es)f(o)s(ccur)165 2976 y(when)32 b Fx(Z)469 2943 y Fu(d)542 2976 y Fw(is)g(replaced)g(b)m(y)h (certain)f(nonamenable)g(graph)g(structures)g(for)h(whic)m(h)e(the)i(b) s(oundary)165 3089 y(of)f(a)f(v)m(olume)g(is)g(of)g(the)h(same)f(order) g(of)g(magnitude)g(as)g(the)h(v)m(olume)f(itself)f(\(whic)m(h)g(mak)m (es)i(them)165 3201 y(ph)m(ysically)j(p)s(erhaps)h(less)h(realistic\))f ({)h(an)h(example)f(is)f(the)h(regular)g(tree)h Fx(T)2917 3216 y Fu(d)2994 3201 y Fw(with)e Ft(d)h Fv(\025)f Fw(2;)42 b(w)m(e)165 3314 y(refer)35 b(to)g([215)r(,)g(96)q(,)g(140)q(].)54 b(A)35 b(phase)f(transition)g(can)h(also)f(o)s(ccur)h(for)f(a)i (non-zero)f(external)g(\014eld)165 3427 y(for)28 b(the)h(Ising)e(mo)s (del)g(on)i(a)f(half-space)h(where)e(it)h(is)g(due)f(to)j(the)e (so-called)g(Basuev)h(phenomenon)165 3540 y([19)r(,)h(20)q(].)306 3653 y(Because)39 b(of)f(the)f(simplicit)m(y)e(of)i(its)g(mo)s(del)f (assumptions,)h(the)h(standard)e(Ising)g(mo)s(del)g(has)165 3766 y(inspired)18 b(a)i(v)-5 b(ariet)m(y)21 b(of)g(tec)m(hniques)f (for)g(analyzing)g(in)m(teracting)g(random)f(\014elds.)36 b(Its)21 b(ferromagnetic)165 3879 y(structure)h(suggests)i(v)-5 b(arious)21 b(monotonicit)m(y)i(prop)s(erties)e(whic)m(h)g(can)i(b)s(e) f(c)m(hec)m(k)m(ed)j(b)m(y)d(the)h(coupling)165 3992 y(metho)s(ds)i(to)i(b)s(e)e(describ)s(ed)f(in)g(Section)h(4,)j(and)d (the)h(assumption)e(of)i(neigb)s(or)e(in)m(teraction)i(implies)165 4105 y(the)c(spatial)f(Mark)m(o)m(v)j(prop)s(ert)m(y)d(\(5\))i(whic)m (h)e(pla)m(ys)g(a)i(fundamen)m(tal)d(role)i(in)f(the)h(geometric)h (analysis)165 4218 y(of)j(t)m(ypical)g(con\014gurations.)38 b(Man)m(y)27 b(tec)m(hniques)e(whic)m(h)g(w)m(ere)h(dev)m(elop)s(ed)g (on)f(this)g(testing)h(ground)165 4331 y(turned)k(out)g(to)h(b)s(e)f (fruitful)d(also)k(in)e(more)h(general)h(cases.)165 4574 y FA(3.2)113 b(The)37 b(an)m(tiferromagnetic)i(Ising)f(mo)s(del)165 4746 y Fw(The)d(Ising)g(an)m(tiferromagnet)h(is)f(de\014ned)f(quite)h (similarly)e(to)j(the)g(ferromagnetic)g(case,)i(except)165 4859 y(that)32 b Ft(U)10 b Fw(\()p Ft(\033)s Fw(\()p Ft(x)p Fw(\))p Ft(;)15 b(\033)s Fw(\()p Ft(y)s Fw(\)\))33 b(is)d(tak)m(en)h(to)h(b)s(e)e(+)p Ft(\033)s Fw(\()p Ft(x)p Fw(\))15 b Ft(\033)s Fw(\()p Ft(y)s Fw(\))32 b(rather)f(than)f Fv(\000)p Ft(\033)s Fw(\()p Ft(x)p Fw(\))15 b Ft(\033)s Fw(\()p Ft(y)s Fw(\).)43 b(This)29 b(means)i(that)165 4971 y(neigh)m(b)s(oring)e(sites)h(no)m(w)g(prefer)g(to)h(tak)m(e)h Fp(opp)-5 b(osite)32 b Fw(spins.)306 5084 y(Supp)s(ose)h(that)h Ft(h)d Fw(=)g(0)j(and)g(that)g(the)g(underlying)d(graph)i(is)g (bipartite;)i(this)d(means)i(that)g Fv(L)165 5197 y Fw(can)28 b(b)s(e)f(partitioned)g(in)m(to)g(t)m(w)m(o)i(sets)f Fv(L)1506 5211 y Fu(ev)r(en)1683 5197 y Fw(and)f Fv(L)1920 5212 y Fu(odd)2058 5197 y Fw(suc)m(h)g(that)i(sites)e(in)f Fv(L)2821 5211 y Fu(ev)r(en)2998 5197 y Fw(only)h(ha)m(v)m(e)i(edges) 165 5310 y(to)j(sites)f(in)f Fv(L)651 5325 y Fu(odd)762 5310 y Fw(,)i(and)f(vice)g(v)m(ersa.)44 b(Clearly)-8 b(,)31 b Fx(Z)1852 5277 y Fu(d)1924 5310 y Fw(is)f(an)h(example)g(of)h (a)g(bipartite)e(graph.)43 b(In)30 b(this)165 5423 y(situation,)d(w)m (e)f(can)h(reduce)f(the)g(an)m(tiferromagnetic)h(Ising)e(mo)s(del)g(to) i(the)f(ferromagnetic)h(case)g(b)m(y)1844 5895 y(15)p eop %%Page: 16 16 16 15 bop 165 304 a Fw(a)31 b(simple)d(spin-\015ipping)e(tric)m(k:)41 b(The)30 b(bijection)f Ft(\033)f Fv($)i Fw(~)-49 b Ft(\033)33 b Fw(of)e(\012)f(de\014ned)f(b)m(y)1250 569 y(~)-50 b Ft(\033)s Fw(\()p Ft(x)p Fw(\))26 b(=)1544 425 y Fs(\()1653 512 y Ft(\033)s Fw(\()p Ft(x)p Fw(\))154 b(if)30 b Ft(x)25 b Fv(2)g(L)2294 526 y Fu(ev)r(en)2458 512 y Ft(;)1653 625 y Fv(\000)p Ft(\033)s Fw(\()p Ft(x)p Fw(\))83 b(if)30 b Ft(x)25 b Fv(2)g(L)2294 640 y Fu(odd)3498 569 y Fw(\(6\))165 833 y(maps)41 b(an)m(y)h(Gibbs)e(measure)h(for)g(the)h(an)m (tiferromagnetic)g(Ising)e(mo)s(del)g(to)j(a)e(Gibbs)f(measure)165 946 y(for)j(the)g(ferromagnetic)h(Ising)d(mo)s(del)h(with)g(the)h(same) g(parameters,)k(and)c(vice)g(v)m(ersa.)79 b(As)43 b(a)165 1059 y(consequence,)30 b(a)e(phase)g(transition)f(in)g(the)h(an)m (tiferromagnetic)g(mo)s(del)f(is)h(equiv)-5 b(alen)m(t)27 b(to)i(a)f(phase)165 1172 y(transition)k(in)f(the)j(ferromagnetic)f(mo) s(del)f(with)f(the)j(same)f(parameters.)49 b(Hence,)34 b(Theorem)f(3.1)165 1285 y(immediately)c(carries)h(o)m(v)m(er)i(to)f (the)f(an)m(tiferromagnetic)h(case.)306 1398 y(The)23 b(mo)s(del)f(b)s(ecomes)h(more)g(in)m(teresting)f(\(or,)j(at)f(least,)h (more)e(gen)m(uinely)e(an)m(tiferromagnetic\))165 1511 y(if)32 b(either)f Ft(h)e Fv(6)p Fw(=)f(0)33 b(or)f(the)h(graph)f(is)f (tak)m(en)i(to)h(b)s(e)d(non-bipartite.)45 b(Supp)s(ose)31 b(\014rst)g(that)i Ft(h)c Fv(6)p Fw(=)f(0)33 b(but)165 1624 y(still)k Fv(L)h Fw(=)h Fx(Z)626 1591 y Fu(d)666 1624 y Fw(.)65 b(If)38 b Fv(j)p Ft(h)p Fv(j)h Fw(is)f(small)f(and)h Ft(\014)43 b Fw(su\016cien)m(tly)37 b(large,)k(w)m(e)e(ha)m(v)m(e)h (the)e(same)h(picture)f(as)g(in)165 1736 y(the)h(case)h Ft(h)f Fw(=)g(0:)57 b(there)39 b(exist)f(t)m(w)m(o)i(distinct)d (phases,)j(one)f(ha)m(ving)f(a)h(ma)5 b(jorit)m(y)39 b(of)g(plus)d(spins)165 1849 y(on)e(the)h(ev)m(en)f(sublattice)g(and)f (a)i(ma)5 b(jorit)m(y)34 b(of)g(min)m(us)f(spins)f(on)i(the)g(o)s(dd)f (sublattice,)i(the)f(other)165 1962 y(one)43 b(ha)m(ving)f(a)g(ma)5 b(jorit)m(y)43 b(of)f(plus)e(spins)h(on)h(the)g(o)s(dd)f(sublattice)h (and)f(a)i(ma)5 b(jorit)m(y)42 b(of)h(min)m(us)165 2075 y(spins)32 b(on)i(the)g(ev)m(en)g(sublattice.)50 b(W)-8 b(e)35 b(will)c(sho)m(w)j(this)f(in)f(Section)i(8.5,)i(Example)d(8.17;) k(see)d(also)165 2188 y([65)r(,)j(96)q(].)62 b(Note)39 b(that)f(this)f(phase)g(transition)f(is)g(somewhat)i(di\013eren)m(t)f (in)f(\015a)m(v)m(or)j(compared)e(to)165 2301 y(that)45 b(in)d(the)i(ferromagnetic)h(Ising)d(mo)s(del:)67 b(whereas)44 b(in)e(the)i(Ising)f(ferromagnet)h(the)g(phase)165 2414 y(transition)34 b(pro)s(duces)g(a)i(breaking)e(of)i(a)f(state-space)j (symmetry)-8 b(,)37 b(the)e(phase)g(transition)f(in)g(the)165 2527 y(Ising)29 b(an)m(tiferromagnet)i(instead)e(breaks)h(the)g (translation)f(symmetry)h(b)s(et)m(w)m(een)h(the)f(sublattices)165 2640 y Fv(L)228 2654 y Fu(ev)r(en)408 2640 y Fw(and)f Fv(L)647 2655 y Fu(odd)758 2640 y Fw(.)306 2753 y(T)-8 b(o)29 b(see)f(what)f(happ)s(ens)f(in)g(the)i(case)h(of)e(a)h (non-bipartite)e(graph)h(w)m(e)h(consider)f(the)h(triangular)165 2866 y(lattice)h(whic)m(h)e(can)i(b)s(e)f(obtained)g(b)m(y)g(taking)g (the)h(usual)e(square)h(lattice)h Fx(Z)2775 2833 y Fr(2)2842 2866 y Fw(and)f(adding)f(an)h(edge)165 2978 y(b)s(et)m(w)m(een)33 b(eac)m(h)g(v)m(ertex)g Ft(x)e Fw(and)h(its)f(north-east)h(neigh)m(b)s (or)f Ft(x)21 b Fw(+)g(\(1)p Ft(;)15 b Fw(1\).)46 b(In)32 b(this)e(case,)k(one)e(exp)s(ects)165 3091 y(uniqueness)24 b(when)i Ft(h)f Fw(=)g(0,)j(and)e(existence)h(of)f(three)h(distinct)d (phases)i(when)g Fv(j)p Ft(h)p Fv(j)g(6)p Fw(=)f(0)h(is)g(small)f(and) 165 3204 y Ft(\014)36 b Fw(is)29 b(large.)41 b(Phase)31 b(transitions)d(in)h(these)i(mo)s(dels)e(w)m(ere)i(studied)e(in)g([65)q (,)i(129)q(],)g(for)f(example.)165 3448 y FA(3.3)113 b(The)37 b(P)m(otts)g(mo)s(del)165 3619 y Fw(A)c(natural)e (generalization)i(of)f(the)h(ferromagnetic)g(Ising)e(mo)s(del)g(is)h (the)g(\(ferromagnetic\))i(P)m(otts)165 3732 y(mo)s(del)29 b([194)q(],)h(in)e(whic)m(h)g(eac)m(h)j(spin)c(ma)m(y)j(tak)m(e)h Ft(q)e Fv(\025)24 b Fw(2)30 b(\(rather)g(than)f(only)g(t)m(w)m(o\))i (di\013eren)m(t)e(v)-5 b(alues.)165 3845 y(The)30 b(state)i(space)f(is) e(then)h Ft(S)h Fw(=)25 b Fv(f)p Fw(1)p Ft(;)15 b Fw(2)p Ft(;)g(:)g(:)g(:)j(;)d(q)s Fv(g)p Fw(,)31 b(and)f(the)h(pair)e(in)m (teraction)h(is)f(giv)m(en)i(b)m(y)1227 4049 y Ft(U)10 b Fw(\()p Ft(\033)s Fw(\()p Ft(x)p Fw(\))p Ft(;)15 b(\033)s Fw(\()p Ft(y)s Fw(\)\))28 b(=)d(1)c Fv(\000)f Fw(2)p Ft(I)2125 4068 y Fq(f)p Fu(\033)r Fr(\()p Fu(x)p Fr(\)=)p Fu(\033)r Fr(\()p Fu(y)r Fr(\))p Fq(g)2527 4049 y Ft(:)165 4254 y Fw(W)-8 b(e)24 b(con\014ne)e(ourselv)m(es)g(to)h(the)f(case)i (of)e(zero)h(external)f(\014eld,)h(so)f(that)h Ft(V)e Fw(\()p Ft(\033)s Fw(\()p Ft(x)p Fw(\)\))26 b Fv(\021)f Fw(0.)39 b(T)-8 b(aking)22 b Ft(q)28 b Fw(=)d(2)165 4367 y(and)f(iden)m(tifying)f(the)i(state)h(space)f Fv(f)p Fw(1)p Ft(;)15 b Fw(2)p Fv(g)27 b Fw(with)c Fv(f\000)p Fw(1)p Ft(;)15 b Fw(+1)p Fv(g)26 b Fw(w)m(e)f(reobtain)f(the)h (ferromagnetic)g(Ising)165 4479 y(mo)s(del)31 b(with)f(zero)i(external) g(\014eld.)42 b(Just)31 b(as)h(in)e(the)i(latter)g(case,)h(the)e(P)m (otts)i(in)m(teraction)f(fa)m(v)m(ours)165 4592 y(con\014gurations)g (where)g(man)m(y)h(neigh)m(b)s(or)e(pairs)g(agree,)k(and)d(Theorem)g (3.1)i(can)f(b)s(e)f(extended)g(to)165 4705 y(the)f(P)m(otts)h(mo)s (del)d(as)h(follo)m(ws,)g(as)h(w)m(e)g(will)c(sho)m(w)k(in)e(Section)h (6.3.)165 4893 y Fx(Theorem)k(3.2)46 b Fp(F)-7 b(or)34 b(the)f Ft(q)s Fp(-state)g(Potts)h(mo)-5 b(del)34 b(on)f Fx(Z)2097 4860 y Fu(d)2138 4893 y Fp(,)f Ft(d)26 b Fv(\025)g Fw(2)p Fp(,)33 b(ther)-5 b(e)33 b(exists)g(a)h(critic)-5 b(al)33 b(inverse)165 5006 y(temp)-5 b(er)g(atur)g(e)38 b Ft(\014)716 5020 y Fu(c)782 5006 y Fv(2)30 b Fw(\(0)p Ft(;)15 b Fv(1)p Fw(\))37 b Fp(\(dep)-5 b(ending)37 b(on)e Ft(d)h Fp(and)h Ft(q)s Fp(\))e(such)g(that)i(for)f Ft(\014)g(<)30 b(\014)2915 5020 y Fu(c)2985 5006 y Fp(the)36 b(mo)-5 b(del)37 b(has)g(a)165 5119 y(unique)32 b(Gibbs)h(me)-5 b(asur)g(e)34 b(while)f(for)g Ft(\014)e(>)25 b(\014)1668 5133 y Fu(c)1735 5119 y Fp(ther)-5 b(e)34 b(exist)e Ft(q)k Fp(mutual)5 b(ly)33 b(singular)h(Gibbs)e(me)-5 b(asur)g(es.)165 5306 y Fw(In)35 b(the)g(same)h(w)m(a)m(y)h(as)e(Theorem)g(3.1,)j(this)c (theorem)i(also)f(holds)f(on)h(general)h(graphs)e(pro)m(vided)165 5419 y(w)m(e)j(allo)m(w)f Ft(\014)597 5433 y Fu(c)668 5419 y Fw(to)h(b)s(e)e(0)i(or)f Fv(1)p Fw(.)58 b(The)35 b(P)m(otts)j(mo)s(del)d(di\013ers)f(from)i(the)g(Ising)f(mo)s(del)g(in) g(that,)j(for)165 5532 y Ft(q)k Fw(large)d(enough,)i(there)e(are)h(m)m (ultiple)c(Gibbs)i(measures)h(also)g(at)g(the)g(critical)f(v)-5 b(alue)39 b Ft(\014)44 b Fw(=)c Ft(\014)3554 5546 y Fu(c)3589 5532 y Fw(,)165 5645 y(as)e(demonstrated)g(b)m(y)g(Kotec)m(k)q(\023)-47 b(y)40 b(and)d(Shlosman)f([148)r(];)41 b(an)d(outline)e(of)i(a)g(pro)s (of)f(will)e(b)s(e)i(giv)m(en)1844 5895 y(16)p eop %%Page: 17 17 17 16 bop 165 304 a Fw(in)39 b(Example)h(8.21.)72 b(The)40 b(Onsager)g(critical)f(v)-5 b(alue)40 b(for)g(the)h(t)m(w)m (o-dimensional)e(Ising)g(mo)s(del)g(is)165 417 y(b)s(eliev)m(ed)25 b(to)i(extend)g(to)g(the)g(P)m(otts)g(mo)s(del)f(on)g Fx(Z)1861 384 y Fr(2)1926 417 y Fw(through)g(the)h(form)m(ula)e Ft(\014)2799 431 y Fu(c)2834 417 y Fw(\()p Ft(q)s Fw(\))h(=)3080 381 y Fr(1)p 3080 396 36 4 v 3080 448 a(2)3141 417 y Fw(log)q(\(1)12 b(+)3433 360 y Fv(p)p 3510 360 44 4 v 3510 417 a Ft(q)s Fw(\);)165 530 y(see)27 b(W)-8 b(elsh)25 b([221)r(],)i(for)e(example.)39 b(This)24 b(has)h(so)h(far)f(only)g(b)s (een)g(established)f(when)g Ft(q)29 b Fw(is)24 b(su\016cien)m(tly)165 643 y(large)31 b([149)q(].)165 879 y FA(3.4)113 b(The)37 b(hard-core)i(lattice)e(gas)h(mo)s(del)165 1051 y Fw(The)21 b(hard-core)g(lattice)g(gas)g(mo)s(del)f(\(or)h(hard-core)g(mo)s(del)e (for)i(short\))g(describ)s(es)e(a)i(gas)g(of)g(particles)165 1164 y(whic)m(h)37 b(can)i(only)e(sit)h(on)g(the)g(lattice)g(sites)g (but)g(are)g(so)h(large)f(that)h(adjacen)m(t)g(sites)f(cannot)h(b)s(e) 165 1277 y(o)s(ccupied)30 b(sim)m(ultaneously)-8 b(.)39 b(The)29 b(state)j(space)f(is)f Ft(S)g Fw(=)25 b Fv(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)p Fw(,)32 b(the)f(pair)e(in)m(teraction) 1033 1503 y Ft(U)10 b Fw(\()p Ft(\033)s Fw(\()p Ft(x)p Fw(\))p Ft(;)15 b(\033)s Fw(\()p Ft(y)s Fw(\)\))28 b(=)1689 1359 y Fs(\()1798 1447 y Fv(1)83 b Fw(if)29 b Ft(\033)s Fw(\()p Ft(x)p Fw(\))d(=)f Ft(\033)s Fw(\()p Ft(y)s Fw(\))h(=)f(1)1798 1559 y(0)129 b(otherwise,)165 1730 y(describ)s(es)47 b(the)i(hard)e(core)j(of)e(the)h(particles,)k(and)47 b(the)i(c)m(hemical)g(p)s(oten)m(tial)f(is)f Ft(V)20 b Fw(\()p Ft(\033)s Fw(\()p Ft(x)p Fw(\)\))57 b(=)165 1842 y Fv(\000)p Fw(\(log)17 b Ft(\025)p Fw(\))38 b Ft(\033)s Fw(\()p Ft(x)p Fw(\).)63 b(Here)38 b Ft(\025)g(>)f Fw(0)h(is)f(the)h (so-called)f(activit)m(y)h(parameter.)63 b(The)37 b(hard-core)h(mo)s (del)165 1955 y(sho)m(ws)i(some)g(similarities)c(to)41 b(the)f(Ising)e(an)m(tiferromagnet)j(in)d(an)i(external)f(\014eld)g (and)g(can,)j(in)165 2068 y(fact,)e(obtained)c(from)h(it)f(b)m(y)h(a)g (limiting)d(pro)s(cedure)h(\()p Ft(\014)42 b Fv(!)36 b(1)p Fw(,)i Ft(h)f Fv(!)f Fw(2)p Ft(d)p Fw(,)j Ft(\014)5 b Fw(\()p Ft(h)26 b Fv(\000)e Fw(2)p Ft(d)p Fw(\))37 b(=)f(const,)165 2181 y([68)r(]\).)58 b(Since)36 b Ft(U)46 b Fw(is)35 b(either)h(0)g(or)h Fv(1)p Fw(,)g(the)g(in)m(v)m(erse)f (temp)s(erature)g Ft(\014)42 b Fw(is)35 b(irrelev)-5 b(an)m(t)35 b(and)h(will)d(th)m(us)165 2294 y(b)s(e)c(\014xed)g(as)g (1,)h(and)f(w)m(e)h(can)f(v)-5 b(ary)29 b(only)g(the)g(parameter)h Ft(\025)p Fw(.)40 b(Finite)28 b(v)m(olume)i(Gibbs)d(distribitions)165 2407 y(can)34 b(then)g(b)s(e)e(though)m(t)j(of)e(as)h(\014rst)f (letting)g(all)g(spins)e(b)s(e)i(indep)s(enden)m(t,)f(taking)i(v)-5 b(alues)33 b(0)h(and)f(1)165 2520 y(with)d(resp)s(ectiv)m(e)i (probabilities)1377 2484 y Fr(1)p 1329 2499 132 4 v 1329 2551 a(1+)p Fu(\025)1501 2520 y Fw(and)1734 2484 y Fu(\025)p 1689 2499 V 1689 2551 a Fr(1+)p Fu(\025)1831 2520 y Fw(,)f(and)g(then)g (conditioning)e(on)j(the)f(ev)m(en)m(t)i(that)f(no)165 2633 y(t)m(w)m(o)g(1's)f(sit)f(next)g(to)h(eac)m(h)h(other)f(an)m (ywhere)f(on)g(the)h(lattice.)306 2746 y(The)36 b(phase)g(transition)e (b)s(eha)m(vior)h(of)h(the)h(hard-core)f(mo)s(del)e(on)i Fx(Z)2708 2713 y Fu(d)2749 2746 y Fw(,)h Ft(d)e Fv(\025)f Fw(2,)k(is)e(as)g(follo)m(ws.)165 2859 y(F)-8 b(or)33 b Ft(\025)e Fw(su\016cien)m(tly)g(close)g(to)i(0,)f(the)g(particles)f (are)h(spread)e(out)i(rather)g(sparsely)e(on)i(the)f(lattice,)165 2972 y(and)i(w)m(e)g(get)i(a)e(unique)e(Gibbs)h(measure,)h(just)g(as)g (in)f(the)h(Ising)f(an)m(tiferromagnet)i(at)g(high)d(tem-)165 3084 y(p)s(eratures.)40 b(When)29 b Ft(\025)h Fw(increases,)f(the)h (particle)f(densit)m(y)g(also)g(increases,)h(and)f(the)h(system)f (\014nally)165 3197 y(starts)34 b(lo)s(oking)e(for)i(optimal)e(pac)m (kings)h(of)h(particles.)49 b(There)33 b(are)h(t)m(w)m(o)h(suc)m(h)e (optimal)f(pac)m(kings,)165 3310 y(one)24 b(where)f(all)g(sites)g(in)f Fv(L)1059 3324 y Fu(ev)r(en)1232 3310 y Fw(are)i(o)s(ccupied)e(and)h (those)h(in)f Fv(L)2310 3325 y Fu(odd)2444 3310 y Fw(are)h(empt)m(y)-8 b(,)26 b(and)d(one)h(vice)f(v)m(ersa;)165 3423 y(w)m(e)32 b(denote)g(these)g(con\014gurations)f(b)m(y)g Ft(\021)1588 3437 y Fu(ev)r(en)1769 3423 y Fw(and)g Ft(\021)1992 3438 y Fu(odd)2103 3423 y Fw(,)g(resp)s(ectiv)m(ely)-8 b(.)44 b(\(These)32 b(c)m(hessb)s(oard)f(con-)165 3536 y(\014gurations)d(lo)s (ok)g(similar)e(to)j(those)g(fa)m(v)m(oured)g(in)f(the)g(Ising)g(an)m (tiferromagnet.\))41 b(F)-8 b(or)29 b(su\016cien)m(tly)165 3649 y(large)36 b Ft(\025)p Fw(,)h(the)f(in\014nite)e(v)m(olume)h (construction)g(of)h(Gibbs)e(measures)i(with)e(these)i(t)m(w)m(o)i(c)m (hoices)e(of)165 3762 y(b)s(oundary)i(condition)h(pro)s(duces)g (di\013eren)m(t)h(Gibbs)e(measures,)43 b(so)d(w)m(e)h(get)g(a)g(phase)f (transition)165 3875 y([65)r(].)165 4026 y Fx(Theorem)34 b(3.3)46 b Fp(F)-7 b(or)42 b(the)e(har)-5 b(d-c)g(or)g(e)43 b(mo)-5 b(del)42 b(on)f Fx(Z)1999 3993 y Fu(d)2039 4026 y Fp(,)h Ft(d)e Fv(\025)e Fw(2)p Fp(,)43 b(ther)-5 b(e)41 b(exist)f(two)i(c)-5 b(onstants)42 b Fw(0)d Ft(<)165 4139 y(\025)218 4153 y Fu(c)294 4139 y Fv(\024)h Ft(\025)458 4106 y Fq(0)458 4162 y Fu(c)534 4139 y Ft(<)g Fv(1)h Fp(\(dep)-5 b(ending)43 b(on)e Ft(d)p Fp(\))h(such)f(that)h(for)g Ft(\025)f(<)f(\025)2331 4153 y Fu(c)2407 4139 y Fp(the)i(mo)-5 b(del)42 b(has)g(a)g(unique)e(Gibbs)165 4252 y(me)-5 b(asur)g(e)34 b(while)g(for)f Ft(\025)25 b(>)g(\025)1122 4219 y Fq(0)1122 4275 y Fu(c)1189 4252 y Fp(ther)-5 b(e)34 b(ar)-5 b(e)33 b(multiple)h(Gibbs)e(me)-5 b(asur)g(es.)165 4404 y Fw(This)25 b(result)f(will)g(b)s(e)h(pro)m(v)m(ed)h(in)f (Section)h(6.7.)40 b(F)-8 b(rom)27 b(a)f(computer-assisted)g(pro)s(of)f ([201)r(])h(w)m(e)g(kno)m(w)165 4517 y(that)31 b Ft(\025)415 4531 y Fu(c)475 4517 y Fv(\025)25 b Fw(1)p Ft(:)p Fw(50762.)44 b(It)30 b(is)f(widely)f(b)s(eliev)m(ed)h(that)i(one)f(should)f(b)s(e)g (able)h(to)h(tak)m(e)h Ft(\025)3088 4531 y Fu(c)3148 4517 y Fw(=)24 b Ft(\025)3296 4484 y Fq(0)3296 4539 y Fu(c)3361 4517 y Fw(in)29 b(this)165 4630 y(result,)23 b(whic)m(h)f(w)m(ould)f(mean)h(that)h(the)g(o)s(ccurrence)f(of)h(phase) f(transition)e(is)i(increasing)f(in)g Ft(\025)p Fw(.)38 b(Suc)m(h)165 4742 y(a)d(result,)f(ho)m(w)m(ev)m(er,)i(w)m(ould)d (\(unlik)m(e)f(Theorems)i(3.1)h(and)e(3.2\))j Fp(not)e Fw(extend)g(to)h(arbitrary)e(graph)165 4855 y(structures;)27 b(some)f(coun)m(terexamples)g(w)m(ere)f(recen)m(tly)h(pro)m(vided)e(b)m (y)i(Brigh)m(t)m(w)m(ell,)g(H\177)-45 b(aggstr\177)g(om)27 b(and)165 4968 y(Winkler)i([38)q(].)306 5081 y(The)e(hard-core)g(mo)s (del)f(analogue)h(of)g(in)m(tro)s(ducing)e(an)i(external)f(\014eld)g (in)g(the)h(Ising)e(mo)s(del)h(on)165 5194 y Fx(Z)229 5161 y Fu(d)305 5194 y Fw(is)34 b(obtained)g(b)m(y)h(replacing)f(the)i (single)d(activit)m(y)j(parameter)f Ft(\025)h Fw(b)m(y)f(t)m(w)m(o)h (di\013eren)m(t)f(activities)165 5307 y Ft(\025)218 5321 y Fu(ev)r(en)392 5307 y Fw(and)23 b Ft(\025)615 5322 y Fu(odd)726 5307 y Fw(,)j(one)e(for)g(sites)g(in)e Fv(L)1429 5321 y Fu(ev)r(en)1602 5307 y Fw(and)i(the)g(other)h(for)e(sites)h(in)f Fv(L)2647 5322 y Fu(odd)2758 5307 y Fw(.)38 b(By)25 b(analogy)f(with)f (the)165 5420 y(Ising)j(mo)s(del,)h(one)h(w)m(ould)e(exp)s(ect)i(to)g (ha)m(v)m(e)h(a)e(unique)f(Gibbs)f(measure)i(as)h(so)s(on)f(as)h Ft(\025)3155 5434 y Fu(ev)r(en)3329 5420 y Fv(6)p Fw(=)d Ft(\025)3478 5435 y Fu(odd)3589 5420 y Fw(;)165 5533 y(this)32 b(w)m(as)h(conjectured)h(b)m(y)e(V)-8 b(an)34 b(den)e(Berg)i(and)e(Steif)g([27)q(])i(and)e(pro)m(v)m(ed)h(for)g(the)g (case)h Ft(d)c Fw(=)f(2)k(b)m(y)165 5646 y(H\177)-45 b(aggstr\177)g(om)32 b([117)r(].)1844 5895 y(17)p eop %%Page: 18 18 18 17 bop 165 304 a FA(3.5)113 b(The)37 b(Widom{Ro)m(wlinson)i(lattice) e(mo)s(del)165 476 y Fw(The)22 b(Widom{Ro)m(wlinson)d(mo)s(del)i(is)g (another)h(lattice)g(gas)g(mo)s(del,)h(where)e(this)f(time)i(there)g (are)g(t)m(w)m(o)165 589 y(t)m(yp)s(es)31 b(of)g(particles,)g(and)f(t)m (w)m(o)i(particles)e(are)h(allo)m(w)m(ed)g(to)h(sit)e(on)h(neigh)m(b)s (oring)e(sites)h(only)g(if)g(they)165 701 y(are)h(of)g(the)f(same)h(t)m (yp)s(e.)41 b(Actually)-8 b(,)31 b(Widom)e(and)h(Ro)m(wlinson)f([223)r (])h(originally)e(in)m(tro)s(duced)h(it)h(as)165 814 y(a)e(con)m(tin)m(uum)f(mo)s(del)f(of)i(particles)e(living)f(in)h Fx(R)1842 781 y Fu(d)1883 814 y Fw(;)j(see)f(Section)f(10.2)i(b)s(elo)m (w.)39 b(The)27 b(lattice)h(v)-5 b(arian)m(t)165 927 y(describ)s(ed)34 b(here)i(w)m(as)g(\014rst)f(studied)f(b)m(y)i(Leb)s (o)m(witz)f(and)h(Galla)m(v)m(otti)g([153)r(].)57 b(The)36 b(state)h(space)f(is)165 1040 y Ft(S)k Fw(=)35 b Fv(f\000)p Fw(1)p Ft(;)15 b Fw(0)p Ft(;)g Fw(+1)p Fv(g)p Fw(,)40 b(where)c Fv(\000)p Fw(1)g(and)f(+1)i(are)f(the)h(t)m(w)m(o)g(particle) e(t)m(yp)s(es,)j(and)e(0's)g(corresp)s(ond)f(to)165 1153 y(empt)m(y)c(sites.)41 b(The)29 b(pair)h(in)m(teraction)g(is)f(giv)m (en)i(b)m(y)1031 1418 y Ft(U)10 b Fw(\()p Ft(\033)s Fw(\()p Ft(x)p Fw(\))p Ft(;)15 b(\033)s Fw(\()p Ft(y)s Fw(\)\))28 b(=)1687 1274 y Fs(\()1795 1361 y Fv(1)83 b Fw(if)30 b Ft(\033)s Fw(\()p Ft(x)p Fw(\))15 b Ft(\033)s Fw(\()p Ft(y)s Fw(\))27 b(=)e Fv(\000)p Fw(1)15 b Ft(;)1818 1474 y Fw(0)106 b(otherwise,)165 1682 y(and)30 b(the)h(c)m(hemical)f(p)s (oten)m(tial)g(b)m(y)1082 1997 y Ft(V)20 b Fw(\()p Ft(\033)s Fw(\()p Ft(x)p Fw(\)\))27 b(=)1525 1800 y Fs(8)1525 1874 y(>)1525 1899 y(<)1525 2049 y(>)1525 2074 y(:)1641 1884 y Fv(\000)15 b Fw(log)h Ft(\025)1912 1898 y Fq(\000)2054 1884 y Fw(if)82 b Ft(\033)s Fw(\()p Ft(x)p Fw(\))26 b(=)f Fv(\000)p Fw(1)15 b Ft(;)1783 1996 y Fw(0)226 b(if)82 b Ft(\033)s Fw(\()p Ft(x)p Fw(\))26 b(=)f(0)15 b Ft(;)1641 2109 y Fv(\000)g Fw(log)h Ft(\025)1912 2123 y Fr(+)2054 2109 y Fw(if)82 b Ft(\033)s Fw(\()p Ft(x)p Fw(\))26 b(=)f(+1)15 b Ft(:)165 2311 y Fw(Here)40 b Ft(\025)442 2325 y Fq(\000)501 2311 y Ft(;)31 b(\025)610 2325 y Fr(+)709 2311 y Ft(>)40 b Fw(0)g(are)f(the)h(activit)m(y)g(parameters)f(for)h(the)f(t)m(w)m(o)i (particle)d(t)m(yp)s(es)i Fv(\000)p Fw(1)f(and)g(+1.)165 2424 y(As)h(in)e(the)h(hard-core)h(mo)s(del,)g(w)m(e)g(\014x)f(the)h (in)m(v)m(erse)f(tempreature)g Ft(\014)46 b Fw(=)40 b(1)f(and)g(only)g (v)-5 b(ary)39 b(the)165 2537 y(activit)m(y)d(parameters.)57 b(Gibbs)34 b(measures)i(can)g(then)f(b)s(e)g(though)m(t)i(of)e(as)h (\014rst)f(pic)m(king)g(all)f(spins)165 2650 y(indep)s(enden)m(tly)-8 b(,)29 b(taking)i(v)-5 b(alues)31 b Fv(\000)p Fw(1,)g(0)g(or)g(+1)h (with)d(probabilities)f(prop)s(ortional)h(to)i Ft(\025)3250 2664 y Fq(\000)3310 2650 y Fw(,)g(1,)h(and)165 2763 y Ft(\025)218 2777 y Fr(+)277 2763 y Fw(,)27 b(and)d(then)h(conditioning) e(on)i(the)g(ev)m(en)m(t)i(that)e(no)g(t)m(w)m(o)i(particles)d(of)h (di\013eren)m(t)f(t)m(yp)s(e)i(sit)e(next)h(to)165 2876 y(eac)m(h)k(other)e(in)f(the)i(lattice.)40 b(W)-8 b(e)28 b(are)g(mainly)e(in)m(terested)h(in)f(the)h(symmetric)g(case)h Ft(\025)3122 2890 y Fq(\000)3207 2876 y Fw(=)d Ft(\025)3356 2890 y Fr(+)3440 2876 y Fw(=)g Ft(\025)p Fw(,)165 2988 y(where)33 b(the)h(phase)f(transition)f(b)s(eha)m(vior)g(on)i Fx(Z)1833 2955 y Fu(d)1873 2988 y Fw(,)g Ft(d)d Fv(\025)f Fw(2)k(is)e(similar)f(to)j(the)g(Ising)e(mo)s(del:)46 b(F)-8 b(or)34 b Ft(\025)165 3101 y Fw(small,)e(there)g(is)f(a)h (unique)f(Gibbs)f(measure)i(in)f(whic)m(h)f(the)j(o)m(v)m(erall)f (densit)m(y)f(of)i(plus-particles)c(is)165 3214 y(almost)k(surely)f (equal)g(to)i(that)f(of)h(the)f(min)m(us-particles.)46 b(F)-8 b(or)33 b Ft(\025)g Fw(su\016cien)m(tly)f(large,)i(the)f(system) 165 3327 y(w)m(an)m(ts)28 b(to)f(pac)m(k)h(the)f(particles)e(so)i (densely)f(that)h(the)g Fv(\006)p Fw(1)g(symmetry)f(is)g(brok)m(en.)39 b(As)27 b(for)f(the)h(Ising)165 3440 y(mo)s(del,)39 b(one)g(can)f (construct)g(t)m(w)m(o)h(particular)e(Gibbs)f(measures)i Ft(\026)2557 3407 y Fr(+)2654 3440 y Fw(and)f Ft(\026)2893 3407 y Fq(\000)2990 3440 y Fw(using)f(b)s(oundary)165 3553 y(conditions)f Ft(\021)j Fv(\021)c Fw(+1)j(and)e Ft(\021)j Fv(\021)d(\000)p Fw(1;)k(for)d(small)e Ft(\025)i Fw(w)m(e)h(get)g Ft(\026)2327 3567 y Fr(+)2421 3553 y Fw(=)d Ft(\026)2581 3567 y Fq(\000)2676 3553 y Fw(whereas)i(for)g (large)g Ft(\025)g Fw(the)165 3666 y(t)m(w)m(o)27 b(measures)f(are)f (di\013eren)m(t)g(\(and)h(distinguishable)21 b(through)k(the)h (densities)e(of)h(the)h(t)m(w)m(o)h(particle)165 3779 y(t)m(yp)s(es\),)k(pro)s(ducing)d(a)j(phase)f(transition.)165 3966 y Fx(Theorem)k(3.4)46 b Fp(F)-7 b(or)41 b(the)g(Widom{R)-5 b(ow)5 b(linson)43 b(mo)-5 b(del)42 b(on)e Fx(Z)2375 3933 y Fu(d)2415 3966 y Fp(,)i Ft(d)d Fv(\025)f Fw(2)p Fp(,)k(with)f(activities)f Ft(\025)3446 3980 y Fq(\000)3543 3966 y Fw(=)165 4079 y Ft(\025)218 4093 y Fr(+)305 4079 y Fw(=)27 b Ft(\025)p Fp(,)34 b(ther)-5 b(e)35 b(exist)f Fw(0)28 b Ft(<)f(\025)1183 4093 y Fu(c)1245 4079 y Fv(\024)g Ft(\025)1396 4046 y Fq(0)1396 4102 y Fu(c)1459 4079 y Ft(<)g Fv(1)34 b Fp(\(dep)-5 b(ending)35 b(on)f Ft(d)p Fp(\))g(such)g(that)i(for)e Ft(\025)28 b(<)f(\025)3166 4093 y Fu(c)3234 4079 y Fp(the)34 b(mo)-5 b(del)165 4192 y(has)34 b(a)f(unique)f(Gibbs)g(me)-5 b(asur)g(e)34 b(while)g(for)f Ft(\025)25 b(>)g(\025)1908 4159 y Fq(0)1908 4215 y Fu(c)1975 4192 y Fp(ther)-5 b(e)34 b(ar)-5 b(e)33 b(multiple)h(Gibbs)e(me)-5 b(asur)g(es.)165 4380 y Fw(As)39 b(in)f(the)i(hard-core)f(mo)s(del,)i (w)m(e)e(exp)s(ect)h(that)g(one)f(should)e(b)s(e)i(able)g(to)g(tak)m(e) i Ft(\025)3138 4394 y Fu(c)3213 4380 y Fw(=)f Ft(\025)3377 4347 y Fq(0)3377 4402 y Fu(c)3411 4380 y Fw(,)i(but)165 4493 y(suc)m(h)f(a)g(monotonicit)m(y)f(is)g(not)h(kno)m(wn.)71 b(Examples)39 b(of)i(graph)f(structures)g(where)g(the)h(desired)165 4606 y(monotonicit)m(y)31 b(fails)e(can)h(b)s(e)g(found)f(in)g([38)q (].)306 4719 y(W)-8 b(e)36 b(furthermore)d(exp)s(ect)h(that)h(the)f (asymmetric)f(Widom{Ro)m(wlinson)g(mo)s(del)f(on)i Fx(Z)3363 4686 y Fu(d)3437 4719 y Fw(with)165 4832 y Ft(\025)218 4846 y Fq(\000)324 4832 y Fv(6)p Fw(=)47 b Ft(\025)495 4846 y Fr(+)597 4832 y Fw(alw)m(a)m(ys)d(has)g(a)f(unique)f(Gibbs)g (measure)h(\(similarly)d(to)45 b(the)e(Ising)f(mo)s(del)h(with)f(a)165 4944 y(nonzero)31 b(external)f(\014eld\),)g(but)f(this)h(also)g(is)g (not)g(rigorously)f(kno)m(wn.)1844 5895 y(18)p eop %%Page: 19 19 19 18 bop 165 304 a Fy(4)135 b(Coupling)43 b(and)i(sto)t(c)l(hastic)g (domination)165 507 y Fw(Geometry)23 b(alone)e(will)d(not)j(b)s(e)g (su\016cien)m(t)f(for)h(our)g(analysis)e(of)j(equilibrium)16 b(phases.)37 b(W)-8 b(e)22 b(also)f(need)165 620 y(some)26 b(probabilistic)c(to)s(ols)j(whic)m(h)e(allo)m(w)i(us)g(to)g(compare)h (di\013eren)m(t)f(con\014gurations)f(and)h(di\013eren)m(t)165 733 y(probabilit)m(y)39 b(measures.)72 b(So)41 b(w)m(e)g(need)g(to)h (include)c(another)k(preparatory)e(section)h(describing)165 846 y(these)31 b(to)s(ols)f(and)g(their)g(basic)f(applications)g(to)i (our)f(setting.)306 959 y Fp(Coupling)c Fw(is)f(a)g(probabilistic)d (tec)m(hnique)j(whic)m(h)f(has)h(turned)f(out)i(to)g(b)s(e)f(immensely) e(useful)g(in)165 1072 y(virtually)31 b(all)g(areas)j(of)e(probabilit)m (y)f(theory)-8 b(,)34 b(and)e(esp)s(ecially)f(in)g(its)h(applications)f (to)i(statistical)165 1184 y(mec)m(hanics.)68 b(The)38 b(basic)h(idea)g(is)f(to)i(de\014ne)f(t)m(w)m(o)h(\(or)g(more\))g(sto)s (c)m(hastic)g(pro)s(cesses)f(join)m(tly)f(on)165 1297 y(the)d(same)g(probabilit)m(y)d(space)j(so)g(that)g(they)g(can)g(b)s(e) f(compared)h(realization)m(wise.)52 b(This)33 b(direct)165 1410 y(comparison)23 b(often)h(leads)f(to)h(conclusions)e(whic)m(h)g(w) m(ould)g(not)i(b)s(e)f(easily)g(a)m(v)-5 b(ailable)23 b(b)m(y)g(considering)165 1523 y(the)29 b(pro)s(cesses)f(separately)-8 b(.)40 b(Although)28 b(an)g(indep)s(enden)m(t)e(coupling)g(is)i (sometimes)g(already)g(quite)165 1636 y(useful)22 b(\(as)i(w)m(e)h (will)c(see)j(in)e(Section)i(7.3,)i(for)d(example\),)j(it)d(is)f (usually)g(more)i(e\016cien)m(t)g(to)g(in)m(tro)s(duce)165 1749 y(a)29 b(dep)s(endence)f(whic)m(h)f(relates)h(the)h(t)m(w)m(o)h (pro)s(cesses)e(in)f(an)h(e\016cien)m(t)h(w)m(a)m(y)-8 b(.)42 b(One)27 b(suc)m(h)i(particularly)165 1862 y(nice)g (relationship)e(is)h(that)i(one)f(pro)s(cess)g(is)f(p)s(oin)m(t)m(wise) g(smaller)g(than)h(the)g(other)h(in)e(some)h(partial)165 1975 y(order.)38 b(This)22 b(case)j(is)d(related)i(to)g(the)g(cen)m (tral)g(concept)h(of)e Fp(sto)-5 b(chastic)29 b(domination)p Fw(,)e(via)d(Strassen's)165 2088 y(Theorem)40 b(\(Theorem)f(4.6\))j(b)s (elo)m(w.)67 b(W)-8 b(e)41 b(will)c(con\014ne)j(ourselv)m(es)f(to)h (those)h(parts)e(of)h(coupling)165 2201 y(theory)29 b(that)g(are)g (needed)e(for)i(our)e(applications;)h(a)g(more)h(general)f(accoun)m(t)i (can)f(for)f(example)g(b)s(e)165 2314 y(found)h(in)g(the)i(monograph)f (b)m(y)g(Lindv)-5 b(all)28 b([162)q(].)165 2557 y FA(4.1)113 b(The)37 b(coupling)i(inequalit)m(y)165 2729 y Fw(In)f(this)g(and)g (the)g(next)h(subsection)f(of)h(general)f(c)m(haracter,)43 b Fv(L)38 b Fw(will)e(b)s(e)i(an)g(arbitrary)g(\014nite)f(or)165 2841 y(coun)m(tably)32 b(in\014nite)d(set.)45 b(As)32 b(the)f(notation)h(indicates,)f(w)m(e)h(think)e(of)i(the)g(standard)f (case)h(that)g Fv(L)165 2954 y Fw(is)e(the)g(lattice)h(in)m(tro)s (duced)e(in)g(Section)h(2.1,)i(but)d(the)i(follo)m(wing)d(results)i (will)d(also)k(b)s(e)e(applied)f(to)165 3067 y(the)h(case)g(when)e Fv(L)h Fw(is)f(replaced)g(b)m(y)i(its)e(set)i Fv(B)h Fw(of)f(b)s(onds.)38 b(W)-8 b(e)29 b(consider)e(again)h(the)h(pro)s (duct)e(space)165 3180 y(\012)e(=)g Ft(S)413 3147 y Fq(L)466 3180 y Fw(,)30 b(where)g(for)g(the)h(momen)m(t)g Ft(S)k Fw(is)30 b(an)g(arbitrary)f(measurable)h(space.)306 3293 y(Supp)s(ose)h Ft(X)40 b Fw(and)32 b Ft(X)1041 3260 y Fq(0)1097 3293 y Fw(are)g(random)g(elemen)m(ts)h(of)f(\012,)h(and)e (let)i Ft(\026)f Fw(and)g Ft(\026)2822 3260 y Fq(0)2877 3293 y Fw(b)s(e)g(their)f(resp)s(ectiv)m(e)165 3406 y(distributions.)46 b(W)-8 b(e)35 b(de\014ne)e(the)g(\(half)7 b(\))33 b Fp(total)k (variation)g(distanc)-5 b(e)35 b Fv(k)p Ft(\026)22 b Fv(\000)g Ft(\026)2812 3373 y Fq(0)2835 3406 y Fv(k)34 b Fw(b)s(et)m(w)m(een)g Ft(\026)f Fw(and)g Ft(\026)3591 3373 y Fq(0)165 3519 y Fw(b)m(y)1289 3632 y Fv(k)p Ft(\026)20 b Fv(\000)g Ft(\026)1555 3594 y Fq(0)1578 3632 y Fv(k)26 b Fw(=)36 b(sup)1745 3709 y Fu(A)p Fq(\032)p Fr(\012)1919 3632 y Fv(j)p Ft(\026)p Fw(\()p Ft(A)p Fw(\))21 b Fv(\000)f Ft(\026)2304 3594 y Fq(0)2327 3632 y Fw(\()p Ft(A)p Fw(\))p Fv(j)1008 b Fw(\(7\))165 3847 y(where)22 b Ft(A)g Fw(ranges)h(o)m(v)m (er)g(all)e(measurable)h(subsets)f(of)h(\012.)38 b(The)22 b(coupling)e(inequalit)m(y)h(b)s(elo)m(w)g(pro)m(vides)165 3959 y(us)32 b(with)g(a)h(con)m(v)m(enien)m(t)h(upp)s(er)c(b)s(ound)h (on)h(this)g(distance.)47 b(T)-8 b(o)34 b(state)g(it)e(w)m(e)h(\014rst) f(need)g(to)i(de\014ne)165 4072 y(what)d(w)m(e)f(mean)h(b)m(y)f(a)h (coupling)e(of)h Ft(X)38 b Fw(and)30 b Ft(X)1810 4039 y Fq(0)1833 4072 y Fw(.)165 4260 y Fx(De\014nition)36 b(4.1)46 b Fp(A)27 b Fx(coupling)i Ft(P)40 b Fp(of)28 b(two)g Fw(\012)p Fp(-value)-5 b(d)28 b(r)-5 b(andom)30 b(variables)f Ft(X)35 b Fp(and)28 b Ft(X)3135 4227 y Fq(0)3159 4260 y Fp(,)g(or)h(of)e(their)165 4373 y(distributions)32 b Ft(\026)f Fp(and)h Ft(\026)1003 4340 y Fq(0)1026 4373 y Fp(,)f(is)g(a)g(pr)-5 b(ob)g(ability)33 b(me)-5 b(asur)g(e)32 b(on)g Fw(\012)16 b Fv(\002)g Fw(\012)30 b Fp(having)i(mar)-5 b(ginals)33 b Ft(\026)d Fp(and)i Ft(\026)3453 4340 y Fq(0)3476 4373 y Fp(,)f(in)165 4486 y(that)j(for)f(every)g(event)f Ft(A)26 b Fv(\032)f Fw(\012)1393 4690 y Ft(P)13 b Fw(\(\()p Ft(\030)t(;)i(\030)1662 4652 y Fq(0)1687 4690 y Fw(\))25 b(:)41 b Ft(\030)29 b Fv(2)c Ft(A)p Fw(\))h(=)f Ft(\026)p Fw(\()p Ft(A)p Fw(\))1112 b(\(8\))165 4894 y Fp(and)1345 5007 y Ft(P)13 b Fw(\(\()p Ft(\030)t(;)i(\030)1614 4969 y Fq(0)1638 5007 y Fw(\))26 b(:)40 b Ft(\030)1808 4969 y Fq(0)1857 5007 y Fv(2)25 b Ft(A)p Fw(\))g(=)g Ft(\026)2222 4969 y Fq(0)2245 5007 y Fw(\()p Ft(A)p Fw(\))i Ft(:)1063 b Fw(\(9\))165 5194 y(W)-8 b(e)37 b(think)d(of)i(a)g(coupling)d(as)j(a) g(rede\014nition)d(of)j(the)g(random)e(v)-5 b(ariables)34 b Ft(X)43 b Fw(and)35 b Ft(X)3186 5161 y Fq(0)3245 5194 y Fw(on)h(a)g(new)165 5307 y(common)26 b(probabilit)m(y)d(space)i(suc)m (h)g(that)h(their)e(distributions)e(are)j(preserv)m(ed.)39 b(Sometimes)25 b(it)g(will)165 5420 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Ft(\021)s Fw(\))43 b Ft(>)g Fw(0)p Fv(g)e Fw(is)165 3248 y(connected)27 b(in)e(the)h(sense)g(that)h(an)m(y)g(elemen)m(t)f(of)g (\012)g(with)f(p)s(ositiv)m(e)g Ft(\026)p Fw(-probabilit)m(y)f(can)i(b) s(e)f(reac)m(hed)165 3361 y(from)e(an)m(y)h(other)f(via)g(successiv)m (e)h(co)s(ordinate)f(c)m(hanges)h(without)e(passing)g(through)h(elemen) m(ts)g(with)165 3474 y(zero)32 b Ft(\026)p Fw(-probabilit)m(y)-8 b(.)165 3661 y Fx(Theorem)34 b(4.8)i(\(Holley\))45 b Fp(L)-5 b(et)36 b Fv(L)f Fp(b)-5 b(e)36 b(\014nite,)g(and)g(let)g Ft(S)41 b Fp(b)-5 b(e)35 b(a)h(\014nite)g(subset)g(of)g Fx(R)p Fp(.)51 b(L)-5 b(et)36 b Ft(\026)f Fp(and)165 3774 y Ft(\026)220 3741 y Fq(0)288 3774 y Fp(b)-5 b(e)44 b(pr)-5 b(ob)g(ability)47 b(me)-5 b(asur)g(es)46 b(on)e Fw(\012)p Fp(.)77 b(Assume)44 b(that)i Ft(\026)2178 3741 y Fq(0)2245 3774 y Fp(is)f(irr)-5 b(e)g(ducible)45 b(and)g(assigns)g(p) -5 b(ositive)165 3887 y(pr)g(ob)g(ability)35 b(to)e(the)g(maximal)i (element)e(of)g Fw(\012)f Fp(\(with)h(r)-5 b(esp)g(e)g(ct)35 b(to)e Fv(\026)p Fp(\).)41 b(Supp)-5 b(ose)35 b(further)e(that)799 4091 y Ft(\026)p Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))26 b Fv(\025)f Ft(a)15 b Fv(j)g Ft(X)34 b Fw(=)25 b Ft(\030)36 b Fw(o\013)31 b Ft(x)p Fw(\))26 b Fv(\024)f Ft(\026)1992 4054 y Fq(0)2015 4091 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))26 b Fv(\025)f Ft(a)15 b Fv(j)g Ft(X)34 b Fw(=)25 b Ft(\021)35 b Fw(o\013)c Ft(x)p Fw(\))473 b(\(11\))165 4296 y Fp(whenever)34 b Ft(x)27 b Fv(2)f(L)p Fp(,)33 b Ft(a)26 b Fv(2)h Ft(S)5 b Fp(,)33 b(and)h Ft(\030)t(;)15 b(\021)30 b Fv(2)d Ft(S)1621 4263 y Fq(Lnf)p Fu(x)p Fq(g)1852 4296 y Fp(ar)-5 b(e)34 b(such)g(that)g Ft(\030)d Fv(\026)26 b Ft(\021)s Fp(,)34 b Ft(\026)p Fw(\()p Ft(X)g Fw(=)26 b Ft(\030)37 b Fw(o\013)30 b Ft(x)p Fw(\))d Ft(>)f Fw(0)34 b Fp(and)165 4409 y Ft(\026)220 4376 y Fq(0)243 4409 y Fw(\()p Ft(X)f Fw(=)25 b Ft(\021)36 b Fw(o\013)31 b Ft(x)p Fw(\))26 b Ft(>)f Fw(0)p Fp(.)42 b(Then)33 b Ft(\026)25 b Fv(\026)1404 4423 y Fq(D)1489 4409 y Ft(\026)1544 4376 y Fq(0)1568 4409 y Fp(.)165 4596 y Fx(Pro)s(of:)37 b Fw(Consider)20 b(a)h(Mark)m(o)m(v)j(c)m(hain)c(\()p Ft(X)1584 4611 y Fu(k)1628 4596 y Fw(\))1663 4563 y Fq(1)1663 4623 y Fu(k)r Fr(=0)1817 4596 y Fw(with)g(state)j(space)f(\012)f(and)g (transition)f(probabilities)165 4709 y(de\014ned)32 b(as)h(follo)m(ws.) 48 b(A)m(t)34 b(eac)m(h)g(in)m(teger)f(time)g Ft(k)g Fv(\025)c Fw(1,)34 b(pic)m(k)f(a)g(random)f(site)h Ft(x)d Fv(2)f(L)j Fw(according)h(to)165 4822 y(the)c(uniform)c(distribution.) 37 b(Let)28 b Ft(X)1424 4837 y Fu(k)1492 4822 y Fw(=)d Ft(X)1663 4837 y Fu(k)r Fq(\000)p Fr(1)1824 4822 y Fw(on)j Fv(L)15 b(n)h(f)p Ft(x)p Fv(g)p Fw(,)30 b(and)d(select)i Ft(X)2779 4837 y Fu(k)2822 4822 y Fw(\()p Ft(x)p Fw(\))f(according)g (to)h(the)165 4935 y(single-site)34 b(conditional)g(distribution)d (prescrib)s(ed)i(b)m(y)i Ft(\026)p Fw(.)55 b(This)33 b(is)h(a)i(so-called)f(Gibbs)e(sampler)165 5048 y(for)f Ft(\026)p Fw(,)f(and)g(it)g(is)g(immediate)f(that)i(if)f(the)g(initial) e(con\014guration)i Ft(X)2556 5062 y Fr(0)2627 5048 y Fw(is)g(c)m(hosen)h(according)f(to)h Ft(\026)p Fw(,)165 5161 y(then)h Ft(X)450 5176 y Fu(k)527 5161 y Fw(has)g(distribution)c Ft(\026)k Fw(for)h(eac)m(h)g Ft(k)s Fw(.)50 b(De\014ne)33 b(a)h(similar)d(Mark)m(o)m(v)k(c)m(hain)e(\()p Ft(X)3112 5128 y Fq(0)3105 5187 y Fu(k)3148 5161 y Fw(\))3183 5128 y Fq(1)3183 5187 y Fu(k)r Fr(=0)3349 5161 y Fw(with)f Ft(\026)165 5274 y Fw(replaced)e(b)m(y)g Ft(\026)705 5241 y Fq(0)729 5274 y Fw(.)306 5387 y(Next,)42 b(de\014ne)c(a)h (coupling)e(of)i(\()p Ft(X)1513 5402 y Fu(k)1556 5387 y Fw(\))1591 5354 y Fq(1)1591 5413 y Fu(k)r Fr(=0)1763 5387 y Fw(and)f(\()p Ft(X)2065 5354 y Fq(0)2058 5413 y Fu(k)2101 5387 y Fw(\))2136 5354 y Fq(1)2136 5413 y Fu(k)r Fr(=0)2308 5387 y Fw(as)h(follo)m(ws.)65 b(First)38 b(pic)m(k)g(the)h(initial)165 5500 y(v)-5 b(alues)31 b(\()p Ft(X)547 5514 y Fr(0)587 5500 y Ft(;)15 b(X)709 5467 y Fq(0)702 5523 y Fr(0)742 5500 y Fw(\))31 b(according)g(to)g(the) g(pro)s(duct)f(measure)h Ft(\026)20 b Fv(\002)g Ft(\026)2404 5467 y Fq(0)2427 5500 y Fw(.)42 b(Then,)31 b(for)f(eac)m(h)i Ft(k)s Fw(,)g(pic)m(k)e(a)h(site)165 5612 y Ft(x)26 b Fv(2)e(L)30 b Fw(at)g(random)f(and)g(let)g Ft(U)1232 5627 y Fu(k)1304 5612 y Fw(b)s(e)g(an)h(indep)s(enden)m(t)d(random)i(v) -5 b(ariable,)29 b(uniformly)d(distributed)1844 5895 y(21)p eop %%Page: 22 22 22 21 bop 165 304 a Fw(on)33 b(the)h(in)m(terv)-5 b(al)32 b([0)p Ft(;)15 b Fw(1].)51 b(Let)33 b Ft(X)1281 319 y Fu(k)1324 304 y Fw(\()p Ft(y)s Fw(\))e(=)e Ft(X)1648 319 y Fu(k)r Fq(\000)p Fr(1)1782 304 y Fw(\()p Ft(y)s Fw(\))k(and)g Ft(X)2195 271 y Fq(0)2188 330 y Fu(k)2231 304 y Fw(\()p Ft(y)s Fw(\))d(=)g Ft(X)2562 271 y Fq(0)2555 330 y Fu(k)r Fq(\000)p Fr(1)2688 304 y Fw(\()p Ft(y)s Fw(\))k(for)e(eac)m(h)j(site)e Ft(y)f Fv(6)p Fw(=)e Ft(x)p Fw(,)165 417 y(and)g(up)s(date)g(the)g(v)-5 b(alues)30 b(at)h(site)f Ft(x)g Fw(b)m(y)h(letting)811 621 y Ft(X)886 636 y Fu(k)929 621 y Fw(\()p Ft(x)p Fw(\))26 b(=)f(max)p Fv(f)p Ft(a)h Fv(2)e Ft(S)31 b Fw(:)25 b Ft(\026)p Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))27 b Fv(\025)e Ft(a)15 b Fv(j)g Ft(X)33 b Fw(=)25 b Ft(\030)34 b Fw(o\013)d Ft(x)p Fw(\))26 b Fv(\025)f Ft(U)2881 636 y Fu(k)2923 621 y Fv(g)165 825 y Fw(and)774 938 y Ft(X)856 901 y Fq(0)849 961 y Fu(k)892 938 y Fw(\()p Ft(x)p Fw(\))h(=)f(max)p Fv(f)p Ft(a)g Fv(2)g Ft(S)31 b Fw(:)25 b Ft(\026)1701 901 y Fq(0)1724 938 y Fw(\()p Ft(X)1841 901 y Fq(0)1865 938 y Fw(\()p Ft(x)p Fw(\))h Fv(\025)f Ft(a)15 b Fv(j)g Ft(X)2294 901 y Fq(0)2344 938 y Fw(=)25 b Ft(\021)33 b Fw(o\013)e Ft(x)p Fw(\))26 b Fv(\025)f Ft(U)2918 953 y Fu(k)2960 938 y Fv(g)165 1105 y Fw(where)i Ft(\030)i Fw(=)c Ft(X)665 1120 y Fu(k)r Fq(\000)p Fr(1)798 1105 y Fw(\()p Fv(L)14 b(n)g(f)p Ft(x)p Fv(g)p Fw(\))28 b(and)f Ft(\021)h Fw(=)d Ft(X)1599 1072 y Fq(0)1592 1132 y Fu(k)r Fq(\000)p Fr(1)1725 1105 y Fw(\()p Fv(L)14 b(n)g(f)p Ft(x)p Fv(g)p Fw(\).)41 b(It)27 b(is)f(clear)h(that)h(this)e (construction)h(giv)m(es)165 1218 y(the)34 b(correct)h(marginal)e(b)s (eha)m(viors)g(of)h(\()p Ft(X)1643 1233 y Fu(k)1686 1218 y Fw(\))1721 1185 y Fq(1)1721 1245 y Fu(k)r Fr(=0)1888 1218 y Fw(and)f(\()p Ft(X)2185 1185 y Fq(0)2178 1245 y Fu(k)2222 1218 y Fw(\))2257 1185 y Fq(1)2257 1245 y Fu(k)r Fr(=0)2390 1218 y Fw(.)51 b(The)33 b(assumption)f(\(11\))k (implies)165 1331 y(that)f Ft(X)441 1346 y Fu(k)514 1331 y Fv(\026)c Ft(X)698 1298 y Fq(0)691 1357 y Fu(k)768 1331 y Fw(whenev)m(er)i Ft(X)1245 1346 y Fu(k)r Fq(\000)p Fr(1)1409 1331 y Fv(\026)e Ft(X)1593 1298 y Fq(0)1586 1357 y Fu(k)r Fq(\000)p Fr(1)1719 1331 y Fw(.)51 b(By)34 b(the)g(irreducibilit)m(y)29 b(of)34 b Ft(\026)2807 1298 y Fq(0)2830 1331 y Fw(,)g(the)g(c)m(hain)g(\()p Ft(X)3410 1298 y Fq(0)3403 1357 y Fu(k)3446 1331 y Fw(\))3481 1298 y Fq(1)3481 1357 y Fu(k)r Fr(=0)165 1444 y Fw(will)e(almost)i(surely)f (hit)g(the)h(maximal)f(state)j(of)e(\012)g(at)g(some)h(\014nite)e (\(random\))h(time,)h(and)f(from)165 1557 y(this)26 b(time)h(on)g(w)m (e)h(will)c(th)m(us)j(ha)m(v)m(e)h Ft(X)1443 1572 y Fu(k)1512 1557 y Fv(\026)d Ft(X)1690 1524 y Fq(0)1683 1583 y Fu(k)1726 1557 y Fw(.)39 b(Since)26 b(the)i(coupled)e(c)m(hain)g(\()p Ft(X)2854 1572 y Fu(k)2898 1557 y Ft(;)15 b(X)3020 1524 y Fq(0)3013 1583 y Fu(k)3056 1557 y Fw(\))3091 1524 y Fq(1)3091 1583 y Fu(k)r Fr(=0)3251 1557 y Fw(is)27 b(a)g(\014nite)165 1670 y(state)34 b(ap)s(erio)s(dic)c(Mark)m(o)m(v)k(c)m(hain,)f(\()p Ft(X)1497 1685 y Fu(k)1540 1670 y Ft(;)15 b(X)1662 1637 y Fq(0)1655 1696 y Fu(k)1698 1670 y Fw(\))33 b(has)f(a)h(limiting)c (distribution)g(as)j Ft(k)g Fv(!)d(1)p Fw(.)46 b(Pic)m(king)165 1783 y(\()p Ft(X)r(;)15 b(X)399 1750 y Fq(0)424 1783 y Fw(\))33 b(according)f(to)i(this)d(limiting)f(distribution)e(giv)m (es)33 b(a)g(coupling)e(of)h Ft(X)40 b Fw(and)32 b Ft(X)3184 1750 y Fq(0)3240 1783 y Fw(suc)m(h)h(that)165 1896 y Ft(X)g Fv(\026)25 b Ft(X)451 1863 y Fq(0)505 1896 y Fw(almost)30 b(surely)-8 b(,)30 b(whence)g Ft(\026)25 b Fv(\026)1551 1910 y Fq(D)1637 1896 y Ft(\026)1692 1863 y Fq(0)1745 1896 y Fw(b)m(y)31 b(Theorem)f(4.6.)1150 b Fk(2)165 2058 y Fw(A)42 b(non-dynamical)f(pro)s(of)g(of)h(Holley's)g(inequalit)m(y)e (b)m(y)i(induction)e(on)i Fv(jLj)p Fw(,)j(together)e(with)e(an)165 2171 y(extension)35 b(to)h(non-\014nite)e Ft(S)5 b Fw(,)37 b(w)m(as)f(giv)m(en)f(b)m(y)g(Preston)h([197)q(];)i(the)e(simplest)d (induction)g(pro)s(of)i(of)165 2284 y(an)c(ev)m(en)g(more)f(general)h (result)e(can)i(b)s(e)e(found)g(in)g([21)r(].)306 2397 y(As)39 b(a)f(consequence)h(of)g(Holley's)f(inequalit)m(y)e(w)m(e)j (obtain)f(the)g(celebrated)h(FK)m(G)g(inequalit)m(y)165 2510 y(\(Theorem)45 b(4.11)i(b)s(elo)m(w\))d(of)i(F)-8 b(ortuin,)48 b(Kasteleyn)d(and)f(Ginibre)f([83)q(],)49 b(who)44 b(stated)i(it)f(under)165 2623 y(sligh)m(tly)25 b(di\013eren)m(t)i(conditions.)38 b(It)26 b(concerns)h(the)g (correlation)f(structure)h(in)e(a)i(single)f(probabilit)m(y)165 2736 y(measure)31 b(rather)f(than)g(a)h(comparison)e(b)s(et)m(w)m(een)i (t)m(w)m(o)h(probabilit)m(y)c(measures.)165 2923 y Fx(De\014nition)36 b(4.9)46 b Fp(A)32 b(pr)-5 b(ob)g(ability)35 b(me)-5 b(asur)g(e)34 b Ft(\026)e Fp(on)h Fw(\012)f Fp(is)h(c)-5 b(al)5 b(le)-5 b(d)34 b Fx(monotone)e Fp(if)811 3128 y Ft(\026)p Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))26 b Fv(\025)f Ft(a)15 b Fv(j)g Ft(X)33 b Fw(=)25 b Ft(\030)37 b Fw(o\013)31 b Ft(x)p Fw(\))25 b Fv(\024)g Ft(\026)p Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))27 b Fv(\025)e Ft(a)15 b Fv(j)g Ft(X)33 b Fw(=)25 b Ft(\021)36 b Fw(o\013)31 b Ft(x)p Fw(\))484 b(\(12\))165 3332 y Fp(whenever)34 b Ft(x)27 b Fv(2)f(L)p Fp(,)33 b Ft(a)26 b Fv(2)h Ft(S)5 b Fp(,)33 b(and)h Ft(\030)t(;)15 b(\021)30 b Fv(2)d Ft(S)1621 3299 y Fq(Lnf)p Fu(x)p Fq(g)1852 3332 y Fp(ar)-5 b(e)34 b(such)g(that)g Ft(\030)d Fv(\026)26 b Ft(\021)s Fp(,)34 b Ft(\026)p Fw(\()p Ft(X)g Fw(=)26 b Ft(\030)37 b Fw(o\013)30 b Ft(x)p Fw(\))d Ft(>)f Fw(0)34 b Fp(and)165 3445 y Ft(\026)p Fw(\()p Ft(X)f Fw(=)25 b Ft(\021)36 b Fw(o\013)31 b Ft(x)p Fw(\))25 b Ft(>)g Fw(0)p Fp(.)165 3632 y Fw(In)m(tuitiv)m(ely)-8 b(,)34 b Ft(\026)g Fw(is)f(monotone)i(if)e(the)h(spin)e(at)j(a)f(site)g Ft(x)g Fw(prefers)f(to)h(tak)m(e)i(large)e(v)-5 b(alues)33 b(whenev)m(er)165 3745 y(its)d(surrounding)d(sites)j(do.)165 3933 y Fx(De\014nition)36 b(4.10)46 b Fp(A)32 b(pr)-5 b(ob)g(ability)34 b(me)-5 b(asur)g(e)34 b Ft(\026)d Fp(on)i Fw(\012)f Fp(is)g(said)h(to)g(have)g Fx(p)s(ositiv)m(e)h(correlations) 165 4046 y Fp(if)f(for)g(al)5 b(l)33 b(b)-5 b(ounde)g(d)34 b(incr)-5 b(e)g(asing)34 b(functions)f Ft(f)5 b(;)15 b(g)28 b Fw(:)41 b(\012)25 b Fv(!)g Fx(R)33 b Fp(we)g(have)1512 4250 y Ft(\026)p Fw(\()p Ft(f)10 b(g)s Fw(\))26 b Fv(\025)f Ft(\026)p Fw(\()p Ft(f)10 b Fw(\))15 b Ft(\026)p Fw(\()p Ft(g)s Fw(\))g Ft(:)165 4454 y Fw(Since)37 b(the)i(preceding)e (inequalit)m(y)f(is)h(preserv)m(ed)h(under)e(rescaling)h(and)h (addition)e(of)i(constan)m(ts)165 4567 y(to)d Ft(f)43 b Fw(and)33 b Ft(g)s Fw(,)i Ft(\026)f Fw(has)f(p)s(ositiv)m(e)g (correlations)g(whenev)m(er)h Ft(\026)c Fv(\026)2301 4581 y Fq(D)2393 4567 y Ft(\026)2448 4534 y Fq(0)2504 4567 y Fw(for)k(an)m(y)g(probabilit)m(y)d(measure)165 4680 y Ft(\026)220 4647 y Fq(0)279 4680 y Fw(with)k(b)s(ounded)f (increasing)g(Radon{Nik)m(o)s(dym)i(densit)m(y)f(relativ)m(e)h(to)h Ft(\026)p Fw(.)57 b(Theorem)35 b(4.6)i(th)m(us)165 4793 y(sho)m(ws)g(that)g Ft(\026)f Fw(has)g(p)s(ositiv)m(e)g(correlations)g (whenev)m(er)g Ft(\026)p Fw(\()p Ft(f)10 b(g)s Fw(\))36 b Fv(\025)f Ft(\026)p Fw(\()p Ft(f)10 b Fw(\))15 b Ft(\026)p Fw(\()p Ft(g)s Fw(\))37 b(for)g(all)e Fp(c)-5 b(ontinuous)165 4906 y Fw(b)s(ounded)34 b(increasing)g(functions)g Ft(f)44 b Fw(and)35 b Ft(g)s Fw(.)56 b(Hence,)38 b(the)d(prop)s(ert)m(y)g(of)h (p)s(ositiv)m(e)e(correlations)h(is)165 5019 y(also)c(preserv)m(ed)f (under)f(w)m(eak)i(limits.)165 5206 y Fx(Theorem)j(4.11)i(\(The)e(FK)m (G)h(inequalit)m(y\))45 b Fp(L)-5 b(et)37 b Fv(L)f Fp(b)-5 b(e)36 b(\014nite,)h Ft(S)42 b Fp(a)36 b(\014nite)h(subset)f(of)h Fx(R)p Fp(,)g(and)165 5319 y Ft(\026)32 b Fp(a)g(pr)-5 b(ob)g(ability)34 b(me)-5 b(asur)g(e)33 b(on)g Fw(\012)e Fp(which)i(is)f(irr)-5 b(e)g(ducible)32 b(and)h(assigns)f(p)-5 b(ositive)33 b(pr)-5 b(ob)g(ability)34 b(to)f(the)165 5432 y(maximal)c(element)d(of)h Fw(\012)f Fp(\(r)-5 b(elative)27 b(to)h Fv(\026)p Fp(\).)39 b(If)26 b Ft(\026)g Fp(is)h(monotone,)i(it)e (also)g(has)h(p)-5 b(ositive)27 b(c)-5 b(orr)g(elations.)1844 5895 y Fw(22)p eop %%Page: 23 23 23 22 bop 165 304 a Fx(Pro)s(of:)54 b Fw(Supp)s(ose)34 b Ft(\026)918 271 y Fq(0)978 304 y Fw(is)h(a)i(second)f(probabilit)m(y) e(measure)j(on)f(\012)g(suc)m(h)g(that)h Ft(\026)2991 271 y Fq(0)3014 304 y Fw(\()p Ft(\021)s Fw(\))g(=)e Ft(\026)p Fw(\()p Ft(\021)s Fw(\))p Ft(g)s Fw(\()p Ft(\021)s Fw(\))165 417 y(for)c(all)e Ft(\021)g Fv(2)d Fw(\012)k(and)g(some)h(p)s(ositiv)m (e)f(increasing)f(function)g Ft(g)s Fw(.)42 b(F)-8 b(or)32 b Ft(x)25 b Fv(2)h(L)p Fw(,)k Ft(a)c Fv(2)f Ft(S)36 b Fw(and)30 b Ft(\030)g Fv(2)25 b Ft(S)3416 384 y Fq(Lnf)p Fu(x)p Fq(g)165 530 y Fw(suc)m(h)32 b(that)h Ft(\026)p Fw(\()p Ft(X)i Fw(=)28 b Ft(\030)34 b Fw(o\013)d Ft(x)p Fw(\))d Ft(>)f Fw(0)33 b(w)m(e)f(write)f Ft(q)1774 544 y Fu(x)1818 530 y Fw(\()p Ft(a;)15 b(\030)t Fw(\))29 b(=)e Ft(\026)p Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))29 b Fv(\025)f Ft(a)15 b Fv(j)g Ft(X)36 b Fw(=)27 b Ft(\030)35 b Fw(o\013)30 b Ft(x)p Fw(\))j(and)e(de\014ne)165 643 y Ft(q)209 610 y Fq(0)206 665 y Fu(x)250 643 y Fw(\()p Ft(a;)15 b(\030)t Fw(\))31 b(similarly)c(in)i(terms)i(of)f Ft(\026)1373 610 y Fq(0)1396 643 y Fw(.)41 b(Then)611 861 y Ft(q)655 824 y Fq(0)652 884 y Fu(x)695 861 y Fw(\()p Ft(a;)15 b(\030)t Fw(\))h(/)g(\(1)21 b Fv(\000)f Ft(q)1210 824 y Fq(0)1207 884 y Fu(x)1250 861 y Fw(\()p Ft(a;)15 b(\030)t Fw(\)\))85 b(=)1728 780 y Fs(X)1726 963 y Fu(s)p Fq(\025)p Fu(a)1866 861 y Ft(\026)p Fw(\()p Ft(\030)2000 824 y Fu(x;s)2096 861 y Fw(\))15 b Ft(g)s Fw(\()p Ft(\030)2271 824 y Fu(x;s)2369 861 y Fw(\))2404 742 y Fs(\036)2493 780 y(X)2491 956 y Fu(s)f(r)s Fw(\))1484 3381 y(=)1649 3300 y Fs(X)1638 3484 y Fu(r)r Fq(2)p Fn(Q)1795 3262 y Fs(\022)1856 3381 y Ft(P)13 b Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))27 b Fv(\024)e Ft(r)s Fw(\))20 b Fv(\000)g Ft(P)13 b Fw(\()p Ft(X)2667 3343 y Fq(0)2691 3381 y Fw(\()p Ft(x)p Fw(\))26 b Fv(\024)f Ft(r)s Fw(\))3014 3262 y Fs(\023)1484 3597 y Fw(=)83 b(0)p Ft(:)165 3775 y Fw(Summing)28 b(o)m(v)m(er)k(all)d Ft(x)d Fv(2)e(L)30 b Fw(w)m(e)h(get)h Ft(P)13 b Fw(\()p Ft(X)33 b Fv(6)p Fw(=)25 b Ft(X)1824 3742 y Fq(0)1848 3775 y Fw(\))g(=)g(0,)31 b(whence)f Ft(\026)25 b Fw(=)g Ft(\026)2654 3742 y Fq(0)2708 3775 y Fw(b)m(y)30 b(\(10\).)527 b Fk(2)165 4014 y FA(4.3)113 b(Applications)38 b(to)f(the)g(Ising)h(mo)s(del)165 4186 y Fw(W)-8 b(e)43 b(will)38 b(no)m(w)k(apply)e(the)h(results)f(of)i(the) f(previous)f(subsection)g(to)i(the)g(ferromagnetic)f(Ising)165 4299 y(mo)s(del.)47 b(Let)33 b(\()p Fv(L)p Ft(;)15 b Fv(\030)p Fw(\))34 b(b)s(e)e(an)m(y)h(in\014nite)d(lo)s(cally)i (\014nite)f(graph.)48 b(F)-8 b(or)33 b(de\014niteness,)f(one)h(ma)m(y)h (think)165 4412 y(of)i(the)f(case)i Fv(L)c Fw(=)g Fx(Z)897 4379 y Fu(d)937 4412 y Fw(;)38 b(the)d(argumen)m(ts)h(are,)h(ho)m(w)m (ev)m(er,)h(indep)s(enden)m(t)33 b(of)j(the)f(particular)f(graph)165 4525 y(structure.)306 4638 y(As)40 b(in)f(Section)h(3.1,)j(w)m(e)d (write)f Ft(\026)1517 4593 y Fu(\021)1517 4666 y(h;\014)s(;)p Fr(\003)1733 4638 y Fw(for)h(the)g(Gibbs)e(distribution)e(in)j(a)h (\014nite)f(region)g(\003)165 4750 y(with)34 b(b)s(oundary)g(condition) g Ft(\021)j Fv(2)c Fw(\012)i(and)g(external)h(\014eld)e Ft(h)g Fv(2)f Fx(R)j Fw(at)g(in)m(v)m(erse)f(temp)s(erature)h Ft(\014)i(>)165 4863 y Fw(0.)78 b(Our)42 b(\014rst)g(application)f(of)i (Holley's)f(theorem)h(asserts)g(that)g(if)f(one)h(b)s(oundary)d (condition)165 4976 y(dominates)26 b(another,)i(then)e(w)m(e)h(also)f (ha)m(v)m(e)i(sto)s(c)m(hastic)f(domination)e(b)s(et)m(w)m(een)i(the)f (corresp)s(onding)165 5089 y(\014nite)k(v)m(olume)g(Gibbs)f (distributions.)165 5255 y Fx(Lemma)k(4.13)46 b Fp(If)33 b(the)g(b)-5 b(oundary)34 b(c)-5 b(onditions)35 b Ft(\030)t(;)15 b(\021)29 b Fv(2)c Fw(\012)32 b Fp(satisfy)h Ft(\030)c Fv(\026)c Ft(\021)s Fp(,)33 b(then)1548 5433 y Ft(\026)1603 5389 y Fu(\030)1603 5461 y(h;\014)s(;)p Fr(\003)1804 5433 y Fv(\026)1875 5447 y Fq(D)1961 5433 y Ft(\026)2016 5389 y Fu(\021)2016 5461 y(h;\014)s(;)p Fr(\003)2206 5433 y Ft(:)165 5638 y Fp(A)n(lso,)g(e)-5 b(ach)33 b Ft(\026)657 5594 y Fu(\030)657 5666 y(h;\014)s(;)p Fr(\003)866 5638 y Fp(has)g(p)-5 b(ositive)34 b(c)-5 b(orr)g(elations.)1844 5895 y Fw(23)p eop %%Page: 24 24 24 23 bop 165 304 a Fx(Pro)s(of:)55 b Fw(The)37 b(conditional)f (probabilit)m(y)e(of)k(ha)m(ving)f(a)g(plus)e(spin)h(at)i(a)f(giv)m(en) h(site)f Ft(x)g Fw(giv)m(en)g(the)165 417 y(con\014guration)g Ft(\030)k Fw(ev)m(erywhere)d(else)f(is)f(equal)h(to)h(\(1)26 b(+)e(exp[)p Fv(\000)p Fw(2)p Ft(\014)5 b Fw(\()p Ft(h)26 b Fw(+)2672 353 y Fs(P)2759 440 y Fu(y)r Fr(:)p Fu(y)r Fq(\030)p Fu(x)2967 417 y Ft(\030)t Fw(\()p Ft(y)s Fw(\)\)]\))3224 384 y Fq(\000)p Fr(1)3320 417 y Fw(,)39 b(whic)m(h)165 530 y(is)e(an)h(increasing)e(function)g(of)i Ft(\030)t Fw(.)62 b(Theorem)38 b(4.8)g(and)f(Theorem)h(4.11)h(th)m(us)e(imply)e (sto)s(c)m(hastic)165 643 y(domination)e(b)s(et)m(w)m(een)h(the)h(pro)5 b(jections)33 b(of)h(the)g(Gibbs)f(distributions)d(to)35 b Ft(S)2881 610 y Fr(\003)2968 643 y Fw(and)e(the)h(p)s(ositiv)m(e)165 756 y(correlations)e(prop)s(ert)m(y)-8 b(.)45 b(As)33 b(their)e(b)s(eha)m(vior)g(outside)g(\003)h(is)f(deterministic,)g(the)h (lemma)g(follo)m(ws.)165 869 y Fk(2)165 1026 y Fw(W)-8 b(e)46 b(write)e Ft(\026)638 988 y Fr(+)638 1054 y Fu(h;\014)s(;)p Fr(\003)859 1026 y Fw(and)g Ft(\026)1105 988 y Fq(\000)1105 1054 y Fu(h;\014)s(;)p Fr(\003)1326 1026 y Fw(for)g(the)h(\014nite)f(v) m(olume)h(Gibbs)e(distributions)e(obtained)j(with)165 1139 y(resp)s(ectiv)m(e)31 b(b)s(oundary)d(conditions)h Ft(\021)g Fv(\021)c Fw(+1)30 b(and)g Ft(\021)f Fv(\021)24 b(\000)p Fw(1.)41 b(Lemma)31 b(4.13)h(then)e(sho)m(ws)g(that)1362 1328 y Ft(\026)1417 1290 y Fq(\000)1417 1357 y Fu(h;\014)s(;)p Fr(\003)1618 1328 y Fv(\026)1689 1342 y Fq(D)1774 1328 y Ft(\026)1829 1284 y Fu(\021)1829 1357 y(h;\014)s(;)p Fr(\003)2030 1328 y Fv(\026)2101 1342 y Fq(D)2187 1328 y Ft(\026)2242 1290 y Fr(+)2242 1357 y Fu(h;\014)s(;)p Fr(\003)3453 1328 y Fw(\(13\))165 1518 y(for)41 b(an)m(y)g Ft(\021)k Fv(2)d Fw(\012.)72 b(This)38 b(sandwic)m(h)i(inequalit)m(y)e (rev)m(eals)j(the)g(sp)s(ecial)f(role)g(pla)m(y)m(ed)g(b)m(y)h(the)g (\\all)165 1631 y(plus")g(and)h(\\all)g(min)m(us")f(b)s(oundary)f (conditions.)75 b(Next)43 b(w)m(e)g(establish)e(the)i(existence)f(of)h (the)165 1744 y(limiting)33 b(\\plus)g(measure")j(discussed)d(in)h (Section)h(3.1.)56 b(W)-8 b(e)37 b(will)32 b(sa)m(y)k(that)g(a)f (measure)h Ft(\026)e Fw(on)i(\012)165 1856 y(is)c Fp(homo)-5 b(gene)g(ous)36 b Fw(if)31 b(it)i(is)f(in)m(v)-5 b(arian)m(t)32 b(under)f(all)h(graph)h(automorphisms)e(of)i(\()p Fv(L)p Ft(;)15 b Fv(\030)p Fw(\).)49 b(In)32 b(the)h(case)165 1969 y Fv(L)26 b Fw(=)h Fx(Z)416 1936 y Fu(d)456 1969 y Fw(,)k(a)h(homogeneous)f(measure)g(is)f(th)m(us)h(in)m(v)-5 b(arian)m(t)30 b(under)g(translations,)g(lattice)i(rotations,)165 2082 y(and)g(re\015ections)g(in)f(the)i(axes.)47 b(W)-8 b(e)34 b(write)e(\003)c Fv(")i(L)i Fw(for)g(the)g(limit)f(along)h(an)g (arbitrary)f(increasing)165 2195 y(sequence)g(of)g(\014nite)e(regions)h (whic)m(h)f(exhaust)h(the)h(full)d(graph)i Fv(L)p Fw(.)165 2393 y Fx(Prop)s(osition)37 b(4.14)46 b Fp(The)33 b(limiting)g(pr)-5 b(ob)g(ability)35 b(me)-5 b(asur)g(e)1559 2582 y Ft(\026)1614 2544 y Fr(+)1614 2610 y Fu(h;\014)1746 2582 y Fw(=)28 b(lim)1842 2642 y Fr(\003)p Fq("L)1990 2582 y Ft(\026)2045 2544 y Fr(+)2045 2610 y Fu(h;\014)s(;)p Fr(\003)165 2826 y Fp(exists.)41 b Ft(\026)505 2788 y Fr(+)505 2854 y Fu(h;\014)640 2826 y Fp(is)28 b(a)h(homo)-5 b(gene)g(ous)30 b(Gibbs)e(me)-5 b(asur)g(e)30 b(for)f(the)f(Ising)g(mo)-5 b(del)30 b(on)e Fw(\()p Fv(L)p Ft(;)15 b Fv(\030)p Fw(\))29 b Fp(with)g(external)165 2939 y(\014eld)34 b Ft(h)e Fp(and)i(inverse)e (temp)-5 b(er)g(atur)g(e)36 b Ft(\014)h Fp(and)d(has)g(p)-5 b(ositive)33 b(c)-5 b(orr)g(elations.)165 3137 y Fx(Pro)s(of:)55 b Fw(By)38 b(the)f(general)g(theory)h(in)d(Section)i(2.6,)k(the)c (limit)e(is)h(a)i(Gibbs)d(measure)i(whenev)m(er)165 3249 y(it)c(exists.)47 b(Also,)33 b(b)m(y)g(Lemma)f(4.13)j(the)d(limit)f(m)m (ust)h(ha)m(v)m(e)i(p)s(ositiv)m(e)e(correlations.)47 b(T)-8 b(o)33 b(sho)m(w)g(the)165 3362 y(existence)e(of)g(the)f(limit)e (w)m(e)j(note)g(that)1148 3552 y Ft(\026)1203 3513 y Fr(+)1203 3580 y Fu(h;\014)s(;)p Fr(\003)1404 3552 y Fv(\027)1475 3566 y Fq(D)1560 3552 y Ft(\026)1615 3513 y Fr(+)1615 3580 y Fu(h;\014)s(;)p Fr(\001)1922 3552 y Fw(whenev)m(er)f(\003)c Fv(\032)f Fw(\001)g Ft(:)821 b Fw(\(14\))165 3756 y(This)33 b(follo)m(ws)g(from)h(Lemma)g(4.13)i(b)s (ecause)f Ft(\026)1818 3717 y Fr(+)1818 3784 y Fu(h;\014)s(;)p Fr(\003)2027 3756 y Fw(is)f(obtained)f(from)h Ft(\026)2774 3717 y Fr(+)2774 3784 y Fu(h;\014)s(;)p Fr(\001)2994 3756 y Fw(b)m(y)g(conditioning)165 3869 y(on)d(the)f(increasing)f(ev)m (en)m(t)j(that)f Ft(X)i Fv(\021)25 b Fw(+1)30 b(on)g(\001)20 b Fv(n)h Fw(\003.)306 3981 y(No)m(w,)46 b(for)c(an)m(y)h(\014nite)d Ft(A)45 b Fv(\032)g Fw(\003,)g(if)c(\003)h(increases,)j(then,)g(b)m(y)d (\(14\),)k Ft(\026)2784 3943 y Fr(+)2784 4010 y Fu(h;\014)s(;)p Fr(\003)2960 3981 y Fw(\()p Ft(X)53 b Fv(\021)44 b Fw(+1)30 b(on)h Ft(A)p Fw(\))165 4109 y(decreases)c(and)d(therefore)i(con)m(v)m (erges)i(to)51 b(inf)1749 4123 y Fr(\001)1827 4109 y Ft(\026)1882 4071 y Fr(+)1882 4137 y Fu(h;\014)s(;)p Fr(\001)2067 4109 y Fw(\()p Ft(X)33 b Fv(\021)25 b Fw(+1)31 b(on)f Ft(A)p Fw(\))15 b(.)40 b(Note)27 b(that)f(this)e(limit)f(is)165 4222 y(ob)m(viously)h(in)m(v)-5 b(arian)m(t)24 b(under)f(an)m(y)i (automorphism)e(of)i(\()p Fv(L)p Ft(;)15 b Fv(\030)p Fw(\).)39 b(By)25 b(inclusion-exclusion)c(it)j(follo)m(ws)165 4335 y(that,)k(for)e(an)m(y)g(lo)s(cal)f(observ)-5 b(able)25 b Ft(f)10 b Fw(,)26 b Ft(\026)1495 4297 y Fr(+)1495 4363 y Fu(h;\014)s(;)p Fr(\003)1671 4335 y Fw(\()p Ft(f)10 b Fw(\))26 b(con)m(v)m(erges)i(to)e(an)g(automorphism)e(in)m(v)-5 b(arian)m(t)25 b(limit)165 4463 y(as)e(\003)i Fv(")h(L)p Fw(.)37 b(These)22 b(limits)e(determine)h(a)i(unique)d(homogeneous)j (probabilit)m(y)d(measure)i Ft(\026)3230 4424 y Fr(+)3230 4491 y Fu(h;\014)3359 4463 y Fw(whic)m(h,)165 4576 y(as)30 b(a)g(w)m(eak)g(limit)d(of)j(\014nite)e(v)m(olume)h(Gibbs)f (distributions,)e(is)i(a)i(Gibbs)e(measure.)40 b(The)29 b(lemma)g(is)165 4688 y(th)m(us)h(pro)m(v)m(ed.)2891 b Fk(2)165 4846 y Fw(Ob)m(viously)-8 b(,)40 b(replacing)e(the)h(\\all)f (plus")f(b)s(oundary)g(condition)h(b)m(y)g(the)h(\\all)g(min)m(us")e(b) s(oundary)165 4959 y(condition,)27 b(w)m(e)i(obtain)e(in)f(the)i(same)g (w)m(a)m(y)h(an)f(automorphism)e(in)m(v)-5 b(arian)m(t)27 b(in\014nite)e(v)m(olume)j(Gibbs)165 5072 y(measure)44 b Ft(\026)588 5034 y Fq(\000)588 5100 y Fu(h;\014)739 5072 y Fw(with)e(p)s(ositiv)m(e)g(correlations.)80 b(In)43 b(the)h(same)g(w)m(a)m(y)-8 b(,)48 b(Lemma)c(4.13)h(sho)m(ws)f(that)165 5194 y Fp(any)35 b(extr)-5 b(emal)36 b(Gibbs)e(me)-5 b(asur)g(e)35 b(has)g(p)-5 b(ositive)36 b(c)-5 b(orr)g(elations)42 b Fw(\(since)32 b(it)f(is)g(a)i(w)m(eak)f(limit)e(of)i Ft(\026)3438 5150 y Fu(\021)3438 5222 y(h;\014)s(;)p Fr(\003)165 5307 y Fw(for)39 b(suitable)f Ft(\021)s Fw(\).)67 b(Ho)m(w)m(ev)m(er,)44 b(p)s(ositiv)m(e)38 b(correlations)h(ma)m(y)g (fail)f(for)h(suitable)f(\()p Fv(L)p Ft(;)15 b Fv(\030)p Fw(\))39 b(and)g(some)165 5420 y(particular)20 b(non-extremal)i(Gibbs)d (measures.)38 b(F)-8 b(or)22 b(instance,)h(when)e Fv(L)k Fw(=)g Fx(Z)2801 5387 y Fr(3)2861 5420 y Fw(and)c Ft(\014)27 b Fw(is)20 b(su\016cien)m(tly)165 5533 y(large,)39 b(one)e(can)g(tak)m (e)i(a)e(con)m(v)m(ex)h(com)m(bination)e(of)h(t)m(w)m(o)h(di\013eren)m (t)f(so-called)f(Dobrushin)f(states;)165 5646 y(see)c([66)q(,)g(22)q (].)1844 5895 y(24)p eop %%Page: 25 25 25 24 bop 306 304 a Fw(W)-8 b(e)29 b(no)m(w)e(tak)m(e)i(the)f(limit)d (in)h(the)h(sandwic)m(h)f(inequalit)m(y)g(\(13\).)41 b(Let)28 b Ft(\026)f Fw(b)s(e)g(an)m(y)g(Gibbs)f(measure)165 417 y(for)39 b(the)h(Ising)e(mo)s(del)g(with)g(parameters)h Ft(h)h Fw(and)f Ft(\014)5 b Fw(.)67 b(T)-8 b(aking)39 b(the)g(mean)2837 348 y Fs(R)2907 417 y Ft(\026)p Fw(\()p Ft(d\021)s Fw(\))h(in)e(\(13\),)44 b(w)m(e)165 530 y(obtain)31 b(that)g Ft(\026)701 492 y Fq(\000)701 558 y Fu(h;\014)s(;)p Fr(\003)903 530 y Fv(\026)974 544 y Fq(D)1061 530 y Ft(\026)26 b Fv(\026)1213 544 y Fq(D)1300 530 y Ft(\026)1355 492 y Fr(+)1355 558 y Fu(h;\014)s(;)p Fr(\003)1531 530 y Fw(,)31 b(and)f(since)h(sto)s(c)m(hastic)h(domination)d(is)h(preserv)m (ed)h(under)165 643 y(w)m(eak)h(limits,)c(w)m(e)j(end)e(up)h(with)1518 756 y Ft(\026)1573 717 y Fq(\000)1573 784 y Fu(h;\014)1706 756 y Fv(\026)1777 770 y Fq(D)1862 756 y Ft(\026)25 b Fv(\026)2013 770 y Fq(D)2099 756 y Ft(\026)2154 717 y Fr(+)2154 784 y Fu(h;\014)3453 756 y Fw(\(15\))165 923 y(when)33 b Ft(\026)g Fw(is)g(an)m(y)g(Ising-mo)s(del)f(Gibbs)g (measure)h(for)g Ft(\014)5 b(;)31 b(h)p Fw(.)50 b(On)33 b(the)h(one)g(hand,)f(this)f(sho)m(ws)i(that)165 1035 y Ft(\026)220 997 y Fq(\000)220 1064 y Fu(h;\014)355 1035 y Fw(and)28 b Ft(\026)585 997 y Fr(+)585 1064 y Fu(h;\014)720 1035 y Fw(are)g(extremal,)h(and)e(th)m(us)g(equilibrium)c (phases)28 b(in)e(the)i(sense)g(of)g(Section)g(2.7.)41 b(On)165 1148 y(the)35 b(other)g(hand,)g(w)m(e)g(obtain)f(an)g (e\016cien)m(t)h(criterion)f(for)g(the)h(existence)g(of)f(a)h(phase)g (transition)165 1261 y(whic)m(h)29 b(w)m(as)i(\014rst)f(observ)m(ed)g (b)m(y)h(Leb)s(o)m(witz)f(and)g(Martin-L\177)-45 b(of)30 b([155)q(])h(and)f(Ruelle)f([205)q(].)165 1449 y Fx(Theorem)34 b(4.15)47 b Fp(F)-7 b(or)37 b(the)g(Ising)f(mo)-5 b(del)38 b(on)e(an)h(in\014nite)f(lo)-5 b(c)g(al)5 b(ly)39 b(\014nite)d(gr)-5 b(aph)38 b Fw(\()p Fv(L)p Ft(;)15 b Fv(\030)p Fw(\))37 b Fp(with)g(ex-)165 1562 y(ternal)d(\014eld)f Ft(h)26 b Fv(2)f Fx(R)33 b Fp(and)g(inverse)g(temp)-5 b(er)g(atur)g(e)35 b Ft(\014)5 b Fp(,)32 b(the)h(fol)5 b(lowing)34 b(statements)g(ar)-5 b(e)34 b(e)-5 b(quivalent.)211 1749 y Fw(\(i\))32 b Fp(Ther)-5 b(e)34 b(is)e(a)h(unique)f(in\014nite)h(volume)g(Gibbs)f(me)-5 b(asur)g(e.)211 1937 y Fw(\(ii\))32 b Ft(\026)420 1899 y Fq(\000)420 1965 y Fu(h;\014)552 1937 y Fw(=)25 b Ft(\026)703 1899 y Fr(+)703 1965 y Fu(h;\014)211 2139 y Fw(\(iii\))31 b Ft(\026)445 2101 y Fq(\000)445 2168 y Fu(h;\014)552 2139 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))27 b(=)e(+1\))g(=)g Ft(\026)1241 2101 y Fr(+)1241 2168 y Fu(h;\014)1349 2139 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))26 b(=)f(+1\))33 b Fp(for)h(al)5 b(l)33 b Ft(x)25 b Fv(2)g(L)p Fp(.)165 2327 y Fx(Pro)s(of:)48 b Fw(The)33 b(implications)d(\(i\))j Fv(\))h Fw(\(ii\))e Fv(\))h Fw(\(iii\))f(are)i(immediate.)49 b(\(iii\))31 b Fv(\))i Fw(\(ii\))g(follo)m(ws)f(directly)165 2440 y(from)e(\(15\))i(and)e(Prop)s(osition)e(4.12,)33 b(and)c(\(ii\))h Fv(\))g Fw(\(i\))g(from)g(\(15\).)1115 b Fk(2)165 2603 y Fx(Remarks:)57 b Fw(\(a\))40 b(In)e(the)g(case)i Ft(h)f Fw(=)g(0)g(of)g(no)f(external)h(\014eld,)g(assertion)g(\(iii\))e (is)g(equiv)-5 b(alen)m(t)38 b(to)165 2716 y Ft(\026)220 2677 y Fr(+)220 2744 y Fu(h;\014)328 2716 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))29 b(=)f(+1\))h(=)f(1)p Ft(=)p Fw(2)33 b(for)f(all)f Ft(x)e Fv(2)e(L)p Fw(,)33 b(b)m(y)f(the)g Fv(\006)g Fw(symmetry)g(of)g(the)h(mo)s(del.)45 b(An)31 b(extension)165 2829 y(of)g(Theorem)f(4.15)i(in)d(this)g(case)j(to)f (the)f Ft(q)s Fw(-state)i(P)m(otts)g(mo)s(del)d(will)f(b)s(e)i(giv)m (en)g(in)f(Theorem)h(6.10.)306 2941 y(\(b\))40 b(If)g(the)g(graph)f (automorphisms)f(act)j(transitiv)m(ely)e(on)g(\()p Fv(L)p Ft(;)15 b Fv(\030)p Fw(\))41 b(then,)h(b)m(y)d(homogeneit)m(y)-8 b(,)165 3054 y(assertion)42 b(\(iii\))e(is)h(equiv)-5 b(alen)m(t)42 b(to)g(ha)m(ving)g(the)g(equation)g(only)f(for)h Fp(some)g Ft(x)j Fv(2)f(L)p Fw(.)75 b(Using)41 b(for)165 3167 y(example)30 b(the)g(random-cluster)f(metho)s(ds)g(of)h(Section)g (6)g(one)h(can)f(obtain)f(the)h(same)h(equiv)-5 b(alence)165 3280 y(also)31 b(in)e(the)h(general)h(case,)g(assuming)e(only)h(that)h Fv(L)f Fw(is)f(connected.)306 3393 y(\(c\))42 b(If)d Fv(L)j Fw(=)f Fx(Z)839 3360 y Fu(d)879 3393 y Fw(,)i(\(iii\))38 b(is)i(equiv)-5 b(alen)m(t)39 b(to)i(the)f(condition)f(that)i(the)f (free)g(energy)h(densit)m(y)e(is)165 3506 y(di\013eren)m(tiable)30 b(with)f(resp)s(ect)i(to)h Ft(h)f Fw(at)h(the)f(giv)m(en)g(v)-5 b(alues)30 b(of)h Ft(h)g Fw(and)f Ft(\014)36 b Fw([155)r(].)43 b(By)31 b(the)g(celebrated)165 3619 y(Lee-Y)-8 b(ang)35 b(circle)d(theorem)h([203)r(],)g(this)f(is)g(the)h(case)g(whenev)m(er)g Ft(h)c Fv(6)p Fw(=)g(0.)48 b(Alternativ)m(ely)-8 b(,)34 b(one)f(can)165 3732 y(use)g(the)g(so-called)g(GHS)g(inequalit)m(y)e (to)j(establish)d(\(iii\))g(for)i Ft(h)d Fv(6)p Fw(=)f(0)k([196)r(].)48 b(Hence,)35 b(for)e(non-zero)165 3845 y(external)e(\014eld)e(the)h (Ising)f(mo)s(del)h(on)g Fx(Z)1551 3812 y Fu(d)1621 3845 y Fw(do)s(es)h(not)f(exhibit)f(a)h(phase)g(transition.)165 4007 y(W)-8 b(e)39 b(conclude)d(this)g(subsection)h(comparing)f(the)h (Ising-mo)s(del)f(\\plus")g(measures)h(for)g(di\013eren)m(t)165 4120 y(v)-5 b(alues)30 b(of)h(the)f(parameters.)41 b(\(F)-8 b(or)32 b(similar)27 b(results)j(in)f(a)h(lattice)h(gas)g(setting)g (see)g([156)q(].\))165 4308 y Fx(Prop)s(osition)37 b(4.16)46 b Fp(Consider)25 b(the)g(Ising)e(mo)-5 b(del)26 b(on)f(an)f(arbitr)-5 b(ary)26 b(gr)-5 b(aph)26 b Fw(\()p Fv(L)p Ft(;)15 b Fv(\030)p Fw(\))25 b Fp(at)f(two)h(inverse)165 4421 y(temp)-5 b(er)g(atur)g(es)44 b Ft(\014)759 4435 y Fr(1)799 4421 y Ft(;)30 b(\014)905 4435 y Fr(2)986 4421 y Fp(and)42 b(two)g(external)g(\014elds)g Ft(h)1998 4435 y Fr(1)2038 4421 y Ft(;)30 b(h)2145 4435 y Fr(2)2185 4421 y Fp(.)67 b(Supp)-5 b(ose)42 b(that)h(either)e Ft(\014)3147 4435 y Fr(1)3227 4421 y Fw(=)f Ft(\014)3389 4435 y Fr(2)3470 4421 y Fp(and)165 4534 y Ft(h)217 4548 y Fr(1)282 4534 y Fv(\024)25 b Ft(h)430 4548 y Fr(2)470 4534 y Fp(,)32 b(or)h Fw(\()p Fv(L)p Ft(;)15 b Fv(\030)p Fw(\))33 b Fp(is)f(of)h(b)-5 b(ounde)g(d)34 b(de)-5 b(gr)g(e)g(e)33 b Ft(N)i Fw(=)25 b(sup)2085 4556 y Fu(x)p Fq(2L)2240 4534 y Ft(N)2313 4548 y Fu(x)2389 4534 y Fp(and)33 b Ft(\014)2616 4548 y Fr(2)2656 4534 y Ft(h)2708 4548 y Fr(2)2773 4534 y Fv(\025)25 b Ft(\014)2920 4548 y Fr(1)2960 4534 y Ft(h)3012 4548 y Fr(1)3071 4534 y Fw(+)19 b Ft(N)10 b Fv(j)p Ft(\014)3320 4548 y Fr(1)3380 4534 y Fv(\000)19 b Ft(\014)3521 4548 y Fr(2)3561 4534 y Fv(j)p Fp(.)165 4647 y(Then)1551 4760 y Ft(\026)1606 4721 y Fr(+)1606 4788 y Fu(h)1647 4797 y Fl(1)1681 4788 y Fu(;\014)1741 4797 y Fl(1)1804 4760 y Fv(\026)1875 4774 y Fq(D)1961 4760 y Ft(\026)2016 4721 y Fr(+)2016 4788 y Fu(h)2057 4797 y Fl(2)2091 4788 y Fu(;\014)2151 4797 y Fl(2)2204 4760 y Ft(:)165 4974 y Fx(Pro)s(of:)71 b Fw(The)45 b(stated)h (conditions)e(imply)f(that)j Ft(\026)1995 4930 y Fu(\030)1995 5006 y(h)2036 5015 y Fl(1)2070 5006 y Fu(;\014)2130 5015 y Fl(1)2164 5006 y Fu(;)p Fq(f)p Fu(x)p Fq(g)2298 4974 y Fw(\(+1\))51 b Fv(\024)e Ft(\026)2710 4930 y Fu(\021)2710 5006 y(h)2751 5015 y Fl(2)2785 5006 y Fu(;\014)2845 5015 y Fl(2)2879 5006 y Fu(;)p Fq(f)p Fu(x)p Fq(g)3013 4974 y Fw(\(+1\))d(whenev)m(er)165 5109 y Ft(\030)30 b Fv(\026)25 b Ft(\021)s Fw(.)40 b(Hence,)31 b(b)m(y)d(Theorem)h(4.8,)i Ft(\026)1472 5071 y Fr(+)1472 5137 y Fu(h)1513 5146 y Fl(1)1547 5137 y Fu(;\014)s(;)p Fr(\003)1707 5109 y Fv(\026)1778 5123 y Fq(D)1864 5109 y Ft(\026)1919 5071 y Fr(+)1919 5137 y Fu(h)1960 5146 y Fl(2)1994 5137 y Fu(;\014)s(;)p Fr(\003)2158 5109 y Fw(for)e(all)f(\003,)i(and)e(the)h(prop)s(osition)e (follo)m(ws)165 5222 y(b)m(y)k(letting)e(\003)d Fv(")g(L)p Fw(.)2719 b Fk(2)165 5385 y Fw(If)27 b Fv(L)f Fw(has)h(b)s(ounded)e (degree)i Ft(N)10 b Fw(,)28 b(w)m(e)f(obtain)g(in)e(particular)g(a)j (comparison)e(of)h(Ising)e(and)i(Bernoulli)165 5498 y(measures.)77 b(Let)43 b Ft( )861 5512 y Fu(p)943 5498 y Fw(denote)g(the)g(Bernoulli) d(measure)i(on)h Fv(f\000)p Fw(1)p Ft(;)15 b Fw(1)p Fv(g)2618 5465 y Fq(L)2715 5498 y Fw(with)41 b(densit)m(y)g Ft(p)p Fw(.)77 b(Then)1844 5895 y(25)p eop %%Page: 26 26 26 25 bop 165 304 a Ft(\026)220 266 y Fr(+)220 332 y Fu(h;\014)354 304 y Fv(!)27 b Ft( )531 318 y Fu(p)602 304 y Fw(as)32 b Ft(\014)g Fv(!)27 b Fw(0)k(and)g Ft(\014)5 b(h)27 b Fv(!)1433 268 y Fr(1)p 1433 283 36 4 v 1433 335 a(2)1494 304 y Fw(log)1681 263 y Fu(p)p 1636 283 126 4 v 1636 335 a Fr(1)p Fq(\000)p Fu(p)1772 304 y Fw(.)43 b(The)31 b(preceding)f(prop)s(osition)f(and)i(Corollary)f(4.7)165 417 y(th)m(us)g(sho)m(w)h(that)1124 545 y Ft(\026)1179 507 y Fr(+)1179 573 y Fu(h;\014)1311 545 y Fv(\027)1382 559 y Fq(D)1468 545 y Ft( )1527 559 y Fu(p)1627 545 y Fw(for)61 b Ft(h)25 b Fv(\025)2008 484 y Fw(1)p 1980 524 102 4 v 1980 608 a(2)p Ft(\014)2107 545 y Fw(log)2328 484 y Ft(p)p 2250 524 203 4 v 2250 608 a Fw(1)20 b Fv(\000)g Ft(p)2482 545 y Fw(+)g Ft(N)165 751 y Fw(and)1098 879 y Ft(\026)1153 841 y Fr(+)1153 907 y Fu(h;\014)1286 879 y Fv(\026)1357 893 y Fq(D)1442 879 y Ft( )1501 893 y Fu(p)1602 879 y Fw(for)60 b Ft(h)26 b Fv(\024)1983 818 y Fw(1)p 1955 858 102 4 v 1955 941 a(2)p Ft(\014)2082 879 y Fw(log)2303 818 y Ft(p)p 2224 858 203 4 v 2224 941 a Fw(1)21 b Fv(\000)f Ft(p)2457 879 y Fv(\000)g Ft(N)35 b(:)165 1164 y FA(4.4)113 b(Application)38 b(to)f(other)g(mo)s(dels)165 1335 y Fw(Do)43 b(the)e(argumen)m(ts)h(of)g(the)f(previous)f (subsection)h(extend)g(to)h(the)g(other)g(mo)s(dels)e(of)h(Section)165 1448 y(3?)76 b(The)42 b(answ)m(er)g(to)g(this)f(question)h(is)f (di\013eren)m(t)g(for)h(the)g(di\013eren)m(t)g(mo)s(dels.)74 b(F)-8 b(or)43 b(the)f Fp(Potts)165 1561 y(mo)-5 b(del)46 b Fw(with)32 b Ft(q)j Fv(\025)c Fw(3,)36 b(Lemma)e(4.13)i(fails)c(b)s (ecause)i(the)h(conditional)d(probabilit)m(y)g(that)j(the)f(spin)165 1674 y(at)i(a)e(site)h Ft(x)f Fw(tak)m(es)i(a)f(large)g(v)-5 b(alue)34 b(is)f(not)i(increasing)e(in)g(the)i(surrounding)c(spin)h (con\014guration.)165 1787 y(Nev)m(ertheless,)e(the)e(P)m(otts)h(mo)s (del)e(admits)g(some)i(analogues)f(of)g(Prop)s(osition)e(4.14)k(and)d (Theorem)165 1900 y(4.15.)45 b(These)30 b(results)g(are)i(deep)s(er)e (than)h(their)f(Ising)g(coun)m(terparts,)i(and)e(will)f(b)s(e)h (demonstrated)165 2013 y(b)m(y)h(random-cluster)e(argumen)m(ts)i(in)e (Section)h(6.)306 2126 y(The)20 b Fp(Widom{R)-5 b(ow)5 b(linson)28 b(mo)-5 b(del)45 b Fw(exhibits)19 b(the)h(same)h (monotonicit)m(y)g(prop)s(erties)e(as)i(the)f(Ising)165 2239 y(mo)s(del.)55 b(W)-8 b(e)36 b(th)m(us)f(obtain)g(Widom{Ro)m (wlinson)e(analogues)j(of)f(Lemma)h(4.13)g(and)f(Prop)s(osition)165 2352 y(4.14.)67 b(Fixing)37 b(an)m(y)i(t)m(w)m(o)h(activit)m(y)e (parameters)h Ft(\025)1939 2366 y Fr(+)1998 2352 y Ft(;)31 b(\025)2107 2366 y Fq(\000)2205 2352 y Ft(>)38 b Fw(0)h(and)f(writing)e Ft(\026)2959 2319 y Fr(+)3056 2352 y Fw(\(resp.)j Ft(\026)3373 2319 y Fq(\000)3432 2352 y Fw(\))f(for)165 2465 y(the)i(asso)s(ciated)g (limiting)c(Gibbs)i(measures)h(with)g(\\all)f(plus")g(\(resp.)i(\\all)f (min)m(us"\))f(b)s(oundary)165 2578 y(conditions,)24 b(w)m(e)g(\014nd)e(that)i(an)m(y)f(other)h(Gibbs)e(measure)h(for)g(the) h(same)g(parameters)f(is)g(sandwic)m(hed)165 2690 y(\(in)d(the)g(sense) h(of)f(\(15\)\))j(b)s(et)m(w)m(een)e Ft(\026)1374 2657 y Fr(+)1453 2690 y Fw(and)f Ft(\026)1675 2657 y Fq(\000)1733 2690 y Fw(.)38 b(The)20 b(analogue)g(of)h(Theorem)f(4.15)i(reads)e(as)h (follo)m(ws.)165 2878 y Fx(Theorem)34 b(4.17)47 b Fp(F)-7 b(or)46 b(the)g(Widom{R)-5 b(ow)5 b(linson)50 b(mo)-5 b(del)47 b(on)f(an)g(in\014nite)g(lo)-5 b(c)g(al)5 b(ly)48 b(\014nite)d(gr)-5 b(aph)165 2991 y Fw(\()p Fv(L)p Ft(;)15 b Fv(\030)p Fw(\))33 b Fp(with)h(activity)f(p)-5 b(ar)g(ameters)35 b Ft(\025)1480 3005 y Fr(+)1540 2991 y Ft(;)30 b(\025)1648 3005 y Fq(\000)1732 2991 y Ft(>)25 b Fw(0)p Fp(,)33 b(the)g(fol)5 b(lowing)34 b(statements)g(ar)-5 b(e)33 b(e)-5 b(quivalent.)211 3179 y Fw(\(i\))32 b Fp(Ther)-5 b(e)34 b(is)e(a)h(unique)f(in\014nite)h (volume)g(Gibbs)f(me)-5 b(asur)g(e.)211 3366 y Fw(\(ii\))32 b Ft(\026)420 3333 y Fr(+)504 3366 y Fw(=)25 b Ft(\026)655 3333 y Fq(\000)211 3554 y Fw(\(iii\))31 b Ft(\026)445 3521 y Fr(+)504 3554 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))26 b(=)f(+1\))h(=)f Ft(\026)1193 3521 y Fq(\000)1252 3554 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))27 b(=)e(+1\))33 b Fp(for)g(al)5 b(l)33 b Ft(x)25 b Fv(2)g(L)p Fp(.)165 3741 y Fw(The)31 b Fp(Ising)j(antiferr)-5 b(omagnet)67 b Fw(and)31 b(the)h Fp(har)-5 b(d-c)g(or)g(e)36 b(lattic)-5 b(e)35 b(gas)f(mo)-5 b(del)66 b Fw(are)32 b(far)f(from)g(satisfying)165 3854 y(the)38 b(monotonicit)m(y)f(prop)s(erties)f(needed)h(for)g(the)h (argumen)m(ts)g(in)e(Section)h(4.3;)42 b(the)c(conditional)165 3967 y(probabilit)m(y)27 b(that)i(a)g(site)f Ft(x)h Fw(tak)m(es)h(the)f (v)-5 b(alue)28 b(+1)h(is)f Fp(de)-5 b(cr)g(e)g(asing)p Fw(,)31 b(rather)d(than)h(increasing,)f(in)f(the)165 4080 y(surrounding)k(con\014guration.)51 b(Ho)m(w)m(ev)m(er,)38 b(when)33 b(\()p Fv(L)p Ft(;)15 b Fv(\030)p Fw(\))34 b(is)g(bipartite)e(\(as)j(in)e(the)h(case)h Fv(L)c Fw(=)g Fx(Z)3538 4047 y Fu(d)3579 4080 y Fw(\))165 4193 y(w)m(e)43 b(can)f(use)f(again)h(the)g(tric)m(k)g(of)g(\(6\))h(to)g(\015ip)d(all)h (spins)f(on)h(the)i(o)s(dd)d(sublattice.)75 b(The)41 b(Ising)165 4306 y(an)m(tiferromagnet)36 b(is)d(then)h(mapp)s(ed)f(on)m (to)j(the)e(Ising)f(ferromagnet)j(with)d(a)h(staggered)i(external)165 4419 y(\014eld)42 b(\(ha)m(ving)h(alternating)g(signs)f(on)h(the)g(ev)m (en)h(and)e(the)h(o)s(dd)f(sublattices\).)79 b(Similarly)-8 b(,)43 b(the)165 4532 y(hard-core)26 b(mo)s(del)f(is)f(mapp)s(ed)g(in)m (to)i(a)g(mo)s(del)f(whic)m(h)f(also)h(exhibits)f(the)i(necessary)g (monotonicit)m(y)165 4645 y(prop)s(erties.)40 b(W)-8 b(e)32 b(th)m(us)e(obtain)g(analogous)h(results)e(whic)m(h)g(w)m(e)i (sp)s(ell)d(out)j(only)f(for)g(the)h(hard-core)165 4758 y(mo)s(del)f(with)g(activit)m(y)i Ft(\025)27 b(>)f Fw(0)15 b(:)44 b(There)30 b(exist)h(t)m(w)m(o)i(particular)d(Gibbs)f(measures)i Ft(\026)3065 4725 y Fu(ev)r(en)3065 4784 y(\025)3246 4758 y Fw(and)f Ft(\026)3478 4725 y Fu(odd)3478 4784 y(\025)3589 4758 y Fw(,)165 4871 y(obtained)k(as)h(in\014nite)d(v)m (olume)j(limits)d(of)i(\014nite)g(v)m(olume)g(Gibbs)f(distributions)e (with)i(resp)s(ectiv)m(e)165 4983 y(b)s(oundary)26 b(conditions)g Ft(\021)1049 4997 y Fu(ev)r(en)1226 4983 y Fw(and)h Ft(\021)1445 4998 y Fu(odd)1556 4983 y Fw(,)h(de\014ned)f(in)f(Section)i(3.4.)40 b(In)27 b(terms)h(of)g(these)g(t)m(w)m(o)h(Gibbs)165 5096 y(measures,)i(the)f(existence)h(of)g(a)f(phase)h(transition)d(can) j(b)s(e)f(c)m(haracterized)h(as)g(follo)m(ws.)1844 5895 y(26)p eop %%Page: 27 27 27 26 bop 165 304 a Fx(Theorem)34 b(4.18)47 b Fp(F)-7 b(or)47 b(the)g(har)-5 b(d-c)g(or)g(e)49 b(mo)-5 b(del)47 b(on)g(an)g(in\014nite)f(lo)-5 b(c)g(al)5 b(ly)48 b(\014nite)f(bip)-5 b(artite)47 b(gr)-5 b(aph)165 417 y Fw(\()p Fv(L)p Ft(;)15 b Fv(\030)p Fw(\))33 b Fp(with)h(activity)f(p)-5 b(ar)g(ameter)35 b Ft(\025)p Fp(,)e(the)g(fol)5 b(lowing)33 b(ar)-5 b(e)34 b(e)-5 b(quivalent.)211 605 y Fw(\(i\))32 b Fp(Ther)-5 b(e)34 b(is)e(a)h(unique)f(in\014nite)h(volume)g(Gibbs)f(me)-5 b(asur)g(e.)211 792 y Fw(\(ii\))32 b Ft(\026)420 759 y Fu(ev)r(en)420 819 y(\025)594 792 y Fw(=)25 b Ft(\026)745 759 y Fu(odd)745 819 y(\025)211 980 y Fw(\(iii\))31 b Ft(\026)445 947 y Fu(ev)r(en)445 1006 y(\025)594 980 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))26 b(=)f(1\))i(=)d Ft(\026)1212 947 y Fu(odd)1212 1006 y(\025)1323 980 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))27 b(=)e(1\))33 b Fp(for)g(al)5 b(l)33 b Ft(x)26 b Fv(2)f(L)p Fp(.)1844 5895 y Fw(27)p eop %%Page: 28 28 28 27 bop 165 304 a Fy(5)135 b(P)l(ercolation)165 507 y Fw(W)-8 b(e)33 b(will)c(no)m(w)j(in)m(tro)s(duce)f(the)h(ideas)f(of)h (random)f(geometry)i(w)m(e)f(referred)f(to)i(in)d(the)i(title)f(of)h (this)165 620 y(pap)s(er.)40 b(As)29 b(these)h(w)m(ere)h(dev)m(elop)s (ed)e(\014rst)f(in)h(the)h(framew)m(ork)f(of)h(p)s(ercolation)f(theory) -8 b(,)31 b(w)m(e)f(dev)m(ote)165 733 y(this)39 b(section)h(to)g(a)g (description)e(of)i(this)e(sub)5 b(ject.)69 b(W)-8 b(e)41 b(will)c(start)j(with)e(the)i(classical)f(case)i(of)165 846 y(indep)s(enden)m(t,)30 b(or)i(Bernoulli,)e(p)s(ercolation,)h(and)g (will)e(then)j(pro)s(ceed)f(to)i(the)e(case)i(of)f(dep)s(enden)m(t)165 959 y(p)s(ercolation.)47 b(In)32 b(the)h(subsequen)m(t)f(sections)h(w)m (e)g(will)d(see)j(ho)m(w)g(these)g(results)f(and)g(ideas)g(can)h(b)s(e) 165 1072 y(used)d(for)g(the)h(geometric)g(analysis)e(of)h(equilibrium)c (phases.)165 1315 y FA(5.1)113 b(Bernoulli)38 b(p)s(ercolation)165 1487 y Fw(Bernoulli)32 b(p)s(ercolation)i(w)m(as)h(in)m(tro)s(duced)e (in)g(the)i(1950's)h(in)d(pap)s(ers)h(b)m(y)g(Broadb)s(en)m(t)h(and)e (Ham-)165 1599 y(mersley)f([39)q(,)h(124)q(,)g(125)q(])g(as)g(a)f(mo)s (del)g(for)g(the)h(passage)g(of)g(a)g(\015uid)d(through)i(a)g(p)s (orous)g(medium.)165 1712 y(In)f(fact,)j(the)e(mo)s(del)f(has)g(app)s (eared)g(\014rst)g(in)g([79)q(])h(in)f(the)h(con)m(text)h(of)f(p)s (olymerization.)44 b(W)-8 b(e)33 b(giv)m(e)165 1825 y(here)28 b(only)f(a)h(brief)e(in)m(tro)s(duction;)h(a)i(thorough)e(treatmen)m(t) i(can)g(b)s(e)e(found)f(in)h(the)h(the)g(b)s(o)s(oks)f(and)165 1938 y(lectures)j(b)m(y)h(Grimmett)f([108)q(,)h(110)q(],)g(Cha)m(y)m (es)g(and)f(Cha)m(y)m(es)h([52)q(],)g(and)f(Kesten)g([143)r(].)306 2051 y(The)c(p)s(orous)g(medium)e(is)i(mo)s(delled)e(b)m(y)i(a)h(graph) f(\()p Fv(L)p Ft(;)15 b Fv(\030)p Fw(\),)28 b(and)d(either)h(the)h (sites)f(or)g(the)h(b)s(onds)165 2164 y(of)35 b(this)e(graph)g(are)i (considered)e(to)h(b)s(e)g(randomly)e(op)s(en)i(or)g(closed)g(\(blo)s (c)m(k)m(ed\).)52 b(W)-8 b(e)36 b(b)s(egin)c(with)165 2277 y(the)j(case)g(of)f(site)g(p)s(ercolation;)h(the)f(alternativ)m(e) h(case)g(of)f(random)f(b)s(onds)f(will)g(b)s(e)h(discussed)f(at)165 2390 y(the)f(end)f(of)g(this)f(subsection.)306 2503 y(The)44 b(basic)g(question)g(of)g(p)s(ercolation)f(theory)i(is)e(ho)m(w)i(a)f (\015uid)e(can)j(spread)e(through)h(the)165 2616 y(medium.)82 b(This)42 b(in)m(v)m(olv)m(es)j(the)f(connectivit)m(y)h(prop)s(erties)e (of)i(the)f(set)h(of)g(op)s(en)f(v)m(ertices.)84 b(T)-8 b(o)165 2729 y(describ)s(e)32 b(this)g(w)m(e)i(in)m(tro)s(duce)e(some)i (terminology)-8 b(.)49 b(A)34 b(\014nite)e Fp(p)-5 b(ath)35 b Fw(is)d(a)i(sequence)g(\()p Ft(v)3181 2743 y Fr(0)3221 2729 y Ft(;)15 b(e)3303 2743 y Fr(1)3343 2729 y Ft(;)g(v)3427 2743 y Fr(1)3467 2729 y Ft(;)g(e)3549 2743 y Fr(2)3589 2729 y Ft(;)165 2841 y(:)g(:)g(:)i(;)e(e)369 2856 y Fu(k)412 2841 y Ft(;)g(v)496 2856 y Fu(k)539 2841 y Fw(\),)40 b(where)c Ft(v)952 2855 y Fr(0)992 2841 y Ft(;)15 b(:)g(:)g(:)i(;)e(v) 1238 2856 y Fu(k)1317 2841 y Fv(2)37 b(L)g Fw(are)g(pairwise)f (distinct)f(v)m(ertices)j(and)f Ft(e)2932 2855 y Fr(1)2972 2841 y Ft(;)15 b(:)g(:)g(:)h(;)f(e)3215 2856 y Fu(k)3296 2841 y Fv(2)36 b(B)k Fw(are)165 2954 y(pairwise)19 b(distinct)g(edges)i (suc)m(h)f(that,)k(for)c(eac)m(h)i Ft(i)j Fv(2)g(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)j(;)d(k)s Fv(g)p Fw(,)23 b(the)e(edge)g Ft(e)2760 2968 y Fu(i)2809 2954 y Fw(connects)h(the)e(v)m(ertices)165 3067 y Ft(v)209 3081 y Fu(i)p Fq(\000)p Fr(1)359 3067 y Fw(and)31 b Ft(v)581 3081 y Fu(i)609 3067 y Fw(.)43 b(Ob)m(viously)-8 b(,)30 b(a)i(path)f(is)g(equiv)-5 b(alen)m(tly)30 b(describ)s(ed)f(b)m(y)i(its)g(sequence)g(\()p Ft(v)3126 3081 y Fr(0)3166 3067 y Ft(;)15 b(v)3250 3081 y Fr(1)3290 3067 y Ft(;)g(:)g(:)g(:)i(;)e(v)3536 3082 y Fu(k)3579 3067 y Fw(\))165 3180 y(of)36 b(v)m(ertices)h(or)f(its)f(sequence)i(\() p Ft(e)1318 3194 y Fr(1)1358 3180 y Ft(;)15 b(e)1440 3194 y Fr(2)1480 3180 y Ft(;)g(:)g(:)g(:)i(;)e(e)1724 3195 y Fu(k)1767 3180 y Fw(\))36 b(of)g(edges.)58 b(The)36 b(n)m(um)m(b)s(er)e Ft(k)39 b Fw(is)c(called)g(the)i Fp(length)165 3293 y Fw(of)31 b(the)f(path.)41 b(In)30 b(the)g(same)h(w)m(a)m(y)-8 b(,)32 b(w)m(e)f(can)g(also)f(sp)s(eak)g (of)g(in\014nite)e(paths)i(\()p Ft(v)2847 3307 y Fr(0)2887 3293 y Ft(;)15 b(e)2969 3307 y Fr(1)3010 3293 y Ft(;)g(v)3094 3307 y Fr(1)3133 3293 y Ft(;)g(e)3215 3307 y Fr(2)3256 3293 y Ft(;)g(:)g(:)g(:)q Fw(\))31 b(and)165 3406 y(doubly)h (in\014nite)f(paths)j(\()p Ft(:)15 b(:)g(:)h(;)f(v)1272 3420 y Fq(\000)p Fr(1)1367 3406 y Ft(;)g(e)1449 3420 y Fr(0)1489 3406 y Ft(;)g(v)1573 3420 y Fr(0)1613 3406 y Ft(;)g(e)1695 3420 y Fr(1)1735 3406 y Ft(;)g(v)1819 3420 y Fr(1)1859 3406 y Ft(;)g(:)g(:)g(:)r Fw(\).)50 b(A)34 b(region)f Ft(C)k Fv(\032)30 b(L)k Fw(is)e(called)h Fp(c)-5 b(onne)g(cte)g(d)35 b Fw(if)165 3519 y(for)24 b(an)m(y)h Ft(x;)15 b(y)28 b Fv(2)d Ft(C)31 b Fw(there)24 b(exists)g(a)g(path)g(whic)m(h)f(starts)h(at)h Ft(x)p Fw(,)h(ends)d(at)i Ft(y)s Fw(,)g(and)f(whic)m(h)f(only)g(con)m(tains) 165 3632 y(v)m(ertices)31 b(in)e Ft(C)7 b Fw(.)306 3745 y(An)28 b Fp(op)-5 b(en)31 b(p)-5 b(ath)30 b Fw(is)d(a)h(path)g(on)g (whic)m(h)e(all)h(v)m(ertices)i(are)f(op)s(en.)39 b(An)28 b Fp(op)-5 b(en)31 b(cluster)d Fw(is)f(a)h(maximal)165 3858 y(connected)38 b(set)f Ft(C)43 b Fw(in)36 b(whic)m(h)f(all)h(v)m (ertices)h(are)h(op)s(en;)h(here)e(maximal)e(means)i(that)g(there)g(is) f(no)165 3971 y(larger)f(region)g Ft(C)778 3938 y Fq(0)833 3971 y Fv(\033)e Ft(C)42 b Fw(whic)m(h)33 b(is)h(connected)i(and)f (only)f(con)m(tains)h(op)s(en)f(v)m(ertices.)56 b(An)34 b(in\014nite)165 4083 y(op)s(en)29 b(cluster)f(\(or)h(in\014nite)e (cluster,)i(for)g(short\))g(is)f(an)h(op)s(en)f(cluster)g(con)m (taining)h(in\014nitely)d(man)m(y)165 4196 y(v)m(ertices.)41 b(Using)28 b(these)h(terms,)h(w)m(e)f(ma)m(y)g(sa)m(y)g(that)h(the)f (existence)g(of)g(an)g(in\014nite)d(op)s(en)i(cluster)g(is)165 4309 y(equiv)-5 b(alen)m(t)30 b(to)h(the)g(fact)g(that)g(a)g(\015uid)d (can)j(w)m(et)g(a)g(macroscopic)f(part)h(of)f(the)h(medium.)306 4422 y(W)-8 b(e)27 b(no)m(w)f(turn)e(to)i(the)g(classical)f(case)h(of)g (Bernoulli)d(site)i(p)s(ercolation)g(with)f(reten)m(tion)i(param-)165 4535 y(eter)32 b Ft(p)27 b Fv(2)f Fw([0)p Ft(;)15 b Fw(1].)45 b(In)30 b(this)g(case,)j(eac)m(h)f(v)m(ertex)g(of)g Fv(L)p Fw(,)f Fp(indep)-5 b(endently)35 b(of)f(al)5 b(l)34 b(others)p Fw(,)f(is)d(declared)g(to)165 4648 y(b)s(e)i(op)s(en)g(\(and)g (represen)m(ted)g(b)m(y)h(the)f(v)-5 b(alue)32 b(1\))h(with)e (probabilit)m(y)f Ft(p)i Fw(and)g(closed)g(\(with)g(v)-5 b(alue)31 b(0\))165 4761 y(with)f(probabilit)m(y)f(1)21 b Fv(\000)f Ft(p)p Fw(.)42 b(W)-8 b(e)33 b(write)d Ft( )1559 4775 y Fu(p)1630 4761 y Fw(for)h(the)g(asso)s(ciated)g(\(Bernoulli\))f (probabilit)m(y)e(measure)165 4874 y(on)j(the)f(con\014guration)g (space)h Fv(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)1462 4841 y Fq(L)1517 4874 y Fw(.)306 4987 y(The)39 b(\014rst)g(question)g(to)h (b)s(e)e(ask)m(ed)i(is)f(whether)f(or)i(not)f(in\014nite)e(clusters)i (can)g(exist.)68 b(This)165 5100 y(dep)s(ends,)22 b(of)g(course,)i(on)d (b)s(oth)g(the)h(graph)f(\()p Fv(L)p Ft(;)15 b Fv(\030)p Fw(\))23 b(and)e(the)g(parameter)i Ft(p)p Fw(.)37 b(The)21 b(basic)g(observ)-5 b(ation)165 5213 y(is)30 b(the)g(follo)m(wing.)165 5400 y Fx(Prop)s(osition)37 b(5.1)46 b Fp(F)-7 b(or)56 b(Bernoul)5 b(li)56 b(site)f(p)-5 b(er)g(c)g(olation)59 b(on)d(an)f(in\014nite)h(lo)-5 b(c)g(al)5 b(ly)57 b(\014nite)e(gr)-5 b(aph)1844 5895 y Fw(28)p eop %%Page: 29 29 29 28 bop 165 304 a Fw(\()p Fv(L)p Ft(;)15 b Fv(\030)p Fw(\))p Fp(,)33 b(ther)-5 b(e)34 b(exists)f(a)g(critic)-5 b(al)33 b(value)g Ft(p)1611 318 y Fu(c)1671 304 y Fv(2)24 b Fw([0)p Ft(;)15 b Fw(1])35 b Fp(such)e(that)919 574 y Ft( )978 588 y Fu(p)1018 574 y Fw(\()p Fv(9)g Fw(an)d(in\014nite)e (op)s(en)i(cluster)o(\))c(=)2211 430 y Fs(\()2319 517 y Fw(0)84 b Fp(if)32 b Ft(p)25 b(<)g(p)2749 531 y Fu(c)2319 630 y Fw(1)84 b Fp(if)32 b Ft(p)25 b(>)g(p)2749 644 y Fu(c)2783 630 y Ft(:)165 838 y Fp(A)n(t)i(the)h(critic)-5 b(al)29 b(value)f Ft(p)d Fw(=)f Ft(p)1177 852 y Fu(c)1212 838 y Fp(,)k(the)g Ft( )1473 852 y Fu(p)1513 838 y Fp(-pr)-5 b(ob)g(ability)30 b(of)e(having)g(an)g(in\014nite)f(op)-5 b(en)29 b(cluster)f(is)g(either)165 951 y Fw(0)33 b Fp(or)h Fw(1)p Fp(.)165 1139 y Fx(Pro)s(of:)50 b Fw(A)34 b(momen)m(t's)h (though)m(t)g(rev)m(eals)f(that)h(the)g(existence)g(of)f(in\014nite)e (clusters)i(is)f(in)m(v)-5 b(arian)m(t)165 1252 y(under)31 b(a)i(c)m(hange)h(of)f(the)g(status)f(of)h(\014nitely)e(man)m(y)i(v)m (ertices.)48 b(By)32 b(Kolmogoro)m(v's)i(zero-one)g(la)m(w,)165 1365 y(the)k Ft( )388 1379 y Fu(p)428 1365 y Fw(-probabilit)m(y)e(of)h (ha)m(ving)h(in\014nite)d(clusters)i(is)f(therefore)j(either)e(0)h(or)f (1.)63 b(It)38 b(remains)e(to)165 1477 y(sho)m(w)28 b(that)h(this)d (probabilit)m(y)g(is)h(increasing)f(in)h Ft(p)p Fw(.)39 b(The)28 b(existence)g(of)g(an)g(in\014nite)e(op)s(en)h(cluster)g(is) 165 1590 y(ob)m(viously)i(an)i(increasing)e(ev)m(en)m(t,)j(so)e(w)m(e)h (are)g(done)f(if)g(w)m(e)g(can)h(sho)m(w)f(that)1277 1795 y Ft( )1336 1809 y Fu(p)1372 1818 y Fl(1)1436 1795 y Fv(\026)1507 1809 y Fq(D)1592 1795 y Ft( )1651 1809 y Fu(p)1687 1818 y Fl(2)1786 1795 y Fw(whenev)m(er)h Ft(p)2232 1809 y Fr(1)2296 1795 y Fv(\024)25 b Ft(p)2438 1809 y Fr(2)2477 1795 y Ft(:)951 b Fw(\(16\))165 1999 y(This)42 b(is)h(in)m(tuitiv)m(ely)e(ob)m(vious)i(and)g(can)h(b)s(e)f (pro)m(v)m(ed)h(b)m(y)g(the)g(follo)m(wing)e(elemen)m(tary)i(coupling) 165 2112 y(argumen)m(t.)78 b(Let)43 b Ft(Y)65 b Fw(=)45 b(\()p Ft(Y)20 b Fw(\()p Ft(x)p Fw(\)\))1318 2126 y Fu(x)p Fq(2L)1501 2112 y Fw(b)s(e)42 b(a)h(family)e(of)h(i.i.d.)f(random)h(v) -5 b(ariables)41 b(with)g(uniform)165 2225 y(distribution)35 b(on)k([0)p Ft(;)15 b Fw(1],)43 b(and)38 b(for)g Ft(p)h Fv(2)g Fw([0)p Ft(;)15 b Fw(1])40 b(let)f Ft(X)2004 2239 y Fu(p)2083 2225 y Fw(=)f(\()p Ft(X)2302 2239 y Fu(p)2343 2225 y Fw(\()p Ft(x)p Fw(\)\))2500 2239 y Fu(x)p Fq(2L)2679 2225 y Fw(b)s(e)g(de\014ned)f(b)m(y)i Ft(X)3342 2239 y Fu(p)3382 2225 y Fw(\()p Ft(x)p Fw(\))g(=)165 2338 y Ft(I)205 2356 y Fq(f)p Fu(Y)16 b Fr(\()p Fu(x)p Fr(\))p Fq(\024)p Fu(p)p Fq(g)522 2338 y Fw(.)39 b(It)25 b(is)g(then)g(clear)h (that)g Ft(X)1448 2352 y Fu(p)1513 2338 y Fw(has)f(distribution)d Ft( )2220 2352 y Fu(p)2285 2338 y Fw(and)j Ft(X)2532 2352 y Fu(p)2568 2361 y Fl(1)2632 2338 y Fv(\026)g Ft(X)2803 2352 y Fu(p)2839 2361 y Fl(2)2903 2338 y Fw(whenev)m(er)g Ft(p)3343 2352 y Fr(1)3408 2338 y Fv(\024)g Ft(p)3550 2352 y Fr(2)3589 2338 y Fw(.)165 2450 y(This)40 b(implies)f(\(16\))k(b) m(y)f(\(the)g(trivial)e(part)i(of)7 b(\))42 b(Theorem)f(4.6.)76 b(\(Note)43 b(that)g(w)m(e)f(ha)m(v)m(e)h(in)d(fact)165 2563 y(constructed)f(a)g(sim)m(ultaneous)e(coupling)g(of)h(all)g Ft( )1977 2577 y Fu(p)2016 2563 y Fw('s,)j(and)d(that)h(this)e (construction)i(w)m(ould)e(b)s(e)165 2676 y(used)30 b(in)f(Mon)m(te)j (Carlo)e(sim)m(ulations)e(of)i(Bernoulli)e(p)s(ercolation.\))1059 b Fk(2)165 2839 y Fw(W)-8 b(e)41 b(write)d Fv(f)p Ft(x)j Fv($)f(1g)f Fw(for)h(the)f(ev)m(en)m(t)i(that)f Ft(x)g Fv(2)f(L)g Fw(b)s(elongs)g(to)h(an)f(in\014nite)e(cluster,)k(and)e(set) 165 2952 y Ft(\022)208 2966 y Fu(x)252 2952 y Fw(\()p Ft(p)p Fw(\))32 b(=)g Ft( )562 2966 y Fu(p)601 2952 y Fw(\()p Ft(x)h Fv($)e(1)p Fw(\).)53 b(F)-8 b(or)35 b(homogenous)f (graphs)g(suc)m(h)g(as)g Fv(L)e Fw(=)f Fx(Z)2616 2919 y Fu(d)2657 2952 y Fw(,)k Ft(\022)2760 2966 y Fu(x)2804 2952 y Fw(\()p Ft(p)p Fw(\))f(do)s(es)g(not)h(dep)s(end)165 3065 y(on)29 b Ft(x)p Fw(,)g(and)g(then)f(w)m(e)i(write)e(simply)e Ft(\022)s Fw(\()p Ft(p)p Fw(\).)40 b(Equation)28 b(\(16\))i(sho)m(ws)f (that)g Ft(\022)2736 3079 y Fu(x)2780 3065 y Fw(\()p Ft(p)p Fw(\))g(is)f(increasing)g(in)f Ft(p)p Fw(.)165 3178 y(W)-8 b(e)32 b(also)e(mak)m(e)i(the)e(follo)m(wing)f(observ)-5 b(ation.)165 3365 y Fx(Prop)s(osition)37 b(5.2)46 b Fp(F)-7 b(or)33 b(any)g(in\014nite)f(lo)-5 b(c)g(al)5 b(ly)34 b(\014nite)e(c)-5 b(onne)g(cte)g(d)34 b(gr)-5 b(aph)34 b Fw(\()p Fv(L)p Ft(;)15 b Fv(\030)p Fw(\))p Fp(,)32 b(any)h Ft(x)25 b Fv(2)g(L)32 b Fp(and)165 3478 y(any)i Ft(p)25 b Fv(2)f Fw(\(0)p Ft(;)15 b Fw(1\))p Fp(,)35 b(we)e(have)g Ft(\022)1146 3492 y Fu(x)1189 3478 y Fw(\()p Ft(p)p Fw(\))26 b Ft(>)f Fw(0)33 b Fp(if)f(and)i(only)f(if)f Ft( )2114 3492 y Fu(p)2154 3478 y Fw(\()p Fv(9)h Fw(an)d(in\014nite)e (op)s(en)i(cluster)o(\))c(=)f(1)p Fp(.)165 3666 y Fx(Pro)s(of:)41 b Fw(F)-8 b(rom)30 b(Prop)s(osition)d(5.1)k(w)m(e)e(kno)m(w)h(that)f (an)h(in\014nite)d(cluster)h(exists)h(with)f(probabilit)m(y)f(0)165 3779 y(or)k(1.)42 b(The)30 b(implication)e(\\only)i(if)7 b(")30 b(is)g(therefore)h(immediate.)41 b(F)-8 b(or)31 b(the)g(\\if)7 b(")30 b(part,)h(w)m(e)g(note)h(that)165 3892 y(if)38 b(an)g(in\014nite)f(cluster)g(exists)i(with)e(p)s(ositiv)m (e)g(probabilit)m(y)-8 b(,)39 b(then)g(there)f(is)g(some)h Ft(N)48 b Fw(suc)m(h)39 b(that)165 4005 y Ft( )224 4019 y Fu(p)264 4005 y Fw(\()p Ft(A)367 4019 y Fu(N)435 4005 y Fw(\))26 b Ft(>)f Fw(0,)j(where)f Ft(A)1018 4019 y Fu(N)1113 4005 y Fw(is)g(the)h(ev)m(en)m(t)h(that)f(some)g(v)m(ertex)h (within)c(distance)i Ft(N)38 b Fw(from)27 b Ft(x)g Fw(b)s(elongs)165 4118 y(to)34 b(an)e(in\014nite)e(cluster.)46 b(On)32 b(the)h(other)f(hand)g(w)m(e)h(ha)m(v)m(e)g Ft( )2240 4132 y Fu(p)2280 4118 y Fw(\()p Ft(B)2384 4132 y Fu(N)2452 4118 y Fw(\))c Ft(>)f Fw(0,)34 b(where)e Ft(B)3053 4132 y Fu(N)3152 4118 y Fw(is)g(the)h(ev)m(en)m(t)165 4230 y(that)22 b(all)f(v)m(ertices)h(within)c(distance)j Ft(N)12 b Fv(\000)r Fw(1)22 b(from)f Ft(x)g Fw(are)h(op)s(en.)37 b(The)21 b(ev)m(en)m(t)i Ft(A)2785 4244 y Fu(N)2854 4230 y Fv(\\)r Ft(B)2986 4244 y Fu(N)3075 4230 y Fw(implies)18 b(that)k Ft(x)165 4343 y Fw(b)s(elongs)i(to)h(an)f(in\014nite)f (cluster.)38 b(But)24 b Ft(A)1583 4357 y Fu(N)1675 4343 y Fw(and)g Ft(B)1915 4357 y Fu(N)2007 4343 y Fw(are)h(increasing)e(ev)m (en)m(ts)j(\(see)f(Section)f(4.2\),)k(so)165 4456 y(that)e(w)m(e)g(can) g(apply)e(Theorem)h(4.11)i(to)g(obtain)d Ft(\022)1886 4470 y Fu(x)1930 4456 y Fw(\()p Ft(p)p Fw(\))i Fv(\025)f Ft( )2227 4470 y Fu(p)2267 4456 y Fw(\()p Ft(A)2370 4470 y Fu(N)2448 4456 y Fv(\\)10 b Ft(B)2588 4470 y Fu(N)2655 4456 y Fw(\))25 b Fv(\025)g Ft( )2870 4470 y Fu(p)2910 4456 y Fw(\()p Ft(A)3013 4470 y Fu(N)3081 4456 y Fw(\))p Ft( )3175 4470 y Fu(p)3215 4456 y Fw(\()p Ft(B)3319 4470 y Fu(N)3387 4456 y Fw(\))h Ft(>)e Fw(0.)165 4569 y Fk(2)165 4732 y Fw(Note)g(that)g(the)f(pro)s(of)f(ab)s(o)m(v)m(e)i(applies)d(to) j(the)f(m)m(uc)m(h)f(broader)h(class)f(of)h(all)f(measures)h(with)e(p)s (ositiv)m(e)165 4845 y(correlations)30 b(\(recall)g(De\014nition)f (4.10\),)k(rather)d(than)g(only)g(the)g(Bernoulli)e(measures.)306 4958 y(Next)40 b(w)m(e)f(ask)g(whether)f(b)s(oth)f(p)s(ossibilities)e (in)i(Prop)s(osition)f(5.2)k(really)d(o)s(ccur,)k(that)e(is,)h(if)165 5071 y(0)34 b Ft(<)f(p)394 5085 y Fu(c)462 5071 y Ft(<)g Fw(1.)56 b(F)-8 b(or,)38 b(only)c(in)g(this)g(case)i(w)m(e)g(really)e (ha)m(v)m(e)j(a)e Fp(nontrivial)i Fw(critical)d(phenomenon)g(at)165 5184 y Ft(p)211 5198 y Fu(c)246 5184 y Fw(.)40 b(The)27 b(answ)m(er)h(dep)s(ends)e(on)i(the)g(graph.)40 b(F)-8 b(or)29 b Fv(L)24 b Fw(=)h Fx(Z)2122 5151 y Fu(d)2191 5184 y Fw(with)h(dimension)g Ft(d)f Fv(\025)g Fw(2,)30 b(the)e(threshold)165 5296 y Ft(p)211 5310 y Fu(c)279 5296 y Fw(is)k(indeed)g(non)m(trivial,)h(as)g(is)g(stated)h(in)e(the)h (theorem)h(b)s(elo)m(w.)49 b(This)32 b(non)m(trivialit)m(y)f(of)j Ft(p)3407 5310 y Fu(c)3474 5296 y Fw(is)f(a)165 5409 y(fundamen)m(tal)f(ingredien)m(t)g(of)h(man)m(y)h(of)f(the)g(sto)s(c)m (hastic-geometric)i(argumen)m(ts)f(emplo)m(y)m(ed)f(later)165 5522 y(on.)41 b(On)29 b(the)i(other)g(hand,)e(it)h(is)g(easy)h(to)g (see)g(that)g Ft(p)2009 5536 y Fu(c)2068 5522 y Fw(=)25 b(1)31 b(for)f Fv(L)25 b Fw(=)g Fx(Z)2627 5489 y Fr(1)2666 5522 y Fw(.)1844 5895 y(29)p eop %%Page: 30 30 30 29 bop 165 304 a Fx(Theorem)34 b(5.3)46 b Fp(The)c(critic)-5 b(al)42 b(value)f Ft(p)1586 318 y Fu(c)1661 304 y Fw(=)g Ft(p)1819 318 y Fu(c)1853 304 y Fw(\()p Ft(d)p Fw(\))i Fp(for)e(site)h(p)-5 b(er)g(c)g(olation)44 b(on)e Fv(L)e Fw(=)h Fx(Z)3230 271 y Fu(d)3270 304 y Fp(,)i Ft(d)e Fv(\025)g Fw(2)p Fp(,)165 417 y(satis\014es)34 b(the)f(ine)-5 b(qualities)1660 501 y Fw(1)p 1558 542 250 4 v 1558 625 a(2)p Ft(d)21 b Fv(\000)f Fw(1)1843 563 y Fv(\024)25 b Ft(p)1985 577 y Fu(c)2045 563 y Fv(\024)2150 501 y Fw(6)p 2150 542 46 4 v 2150 625 a(7)2206 563 y Ft(:)1222 b Fw(\(17\))165 779 y Fx(Pro)s(of:)53 b Fw(W)-8 b(e)38 b(b)s(egin)c(with)h(the)h(lo)m(w)m(er)h(b)s(ound)d(on)i Ft(p)1996 793 y Fu(c)2030 779 y Fw(.)58 b(F)-8 b(or)37 b Ft(k)h Fw(=)d(1)p Ft(;)15 b Fw(2)p Ft(;)g(:)g(:)g(:)s Fw(,)37 b(w)m(e)g(write)e Ft(N)3264 794 y Fu(k)3343 779 y Fw(for)h(the)165 892 y(random)25 b(n)m(um)m(b)s(er)g(of)h(op)s(en)f (paths)g(of)h(length)f Ft(k)k Fw(starting)d(at)g(0.)40 b(On)25 b(the)h(ev)m(en)m(t)h Fv(f)p Fw(0)f Fv($)g(1g)g Fw(w)m(e)g(ha)m(v)m(e)165 1005 y Ft(N)238 1020 y Fu(k)306 1005 y Fv(\025)f Fw(1)31 b(for)f(eac)m(h)i Ft(k)s Fw(,)e(whence)1585 1118 y Ft(\022)s Fw(\()p Ft(p)p Fw(\))25 b Fv(\024)g Ft( )1927 1132 y Fu(p)1967 1118 y Fw(\()p Ft(N)2075 1133 y Fu(k)2118 1118 y Fw(\))15 b Ft(:)1260 b Fw(\(18\))165 1285 y(The)35 b(n)m(um)m(b)s(er)f(of)i(all)e(paths)h(of)h(length)e Ft(k)39 b Fw(starting)c(at)h(0)g(is)e(at)i(most)g(2)p Ft(d)p Fw(\(2)p Ft(d)25 b Fv(\000)f Fw(1\))3063 1252 y Fu(k)r Fq(\000)p Fr(1)3196 1285 y Fw(,)37 b(and)e(eac)m(h)165 1398 y(path)30 b(is)g(op)s(en)g(with)f(probabilit)m(y)f Ft(p)1407 1365 y Fu(k)1449 1398 y Fw(.)41 b(Hence)1369 1602 y Ft( )1428 1616 y Fu(p)1468 1602 y Fw(\()p Ft(N)1576 1617 y Fu(k)1619 1602 y Fw(\))26 b Fv(\024)f Fw(2)p Ft(d)p Fw(\(2)p Ft(d)d Fv(\000)e Fw(1\))2188 1564 y Fu(k)r Fq(\000)p Fr(1)2322 1602 y Ft(p)2368 1564 y Fu(k)165 1806 y Fw(whic)m(h)35 b(tends)g(to)i(0)f(as)h Ft(k)g Fv(!)e(1)g Fw(whenev)m(er)h Ft(p)e(<)h Fw(1)p Ft(=)p Fw(\(2)p Ft(d)22 b Fv(\000)e Fw(1\).)58 b(In)35 b(com)m(bination)g(with)g(\(18\))i(this)165 1919 y(implies)28 b(that)j Ft(\022)s Fw(\()p Ft(p)p Fw(\))25 b(=)g(0)31 b(for)f Ft(p)25 b(<)g Fw(1)p Ft(=)p Fw(\(2)p Ft(d)d Fv(\000)e Fw(1\),)31 b(and)f(the)g(\014rst)g(half)f(of)i(\(17\)) h(is)d(established.)306 2032 y(The)f(second)h(half)e(of)i(\(17\))h (only)d(needs)h(to)h(b)s(e)f(pro)m(v)m(ed)h(for)f Ft(d)e Fw(=)f(2;)k(this)f(is)f(b)s(ecause)h Fx(Z)3288 1999 y Fr(2)3356 2032 y Fw(can)h(b)s(e)165 2145 y(em)m(b)s(edded)c(in)m(to)h Fx(Z)834 2112 y Fu(d)900 2145 y Fw(for)g(an)m(y)g Ft(d)f Fv(\025)g Fw(2,)j(so)e(that)g Ft(p)1813 2159 y Fu(c)1848 2145 y Fw(\()p Ft(d)p Fw(\))g Fv(\024)f Ft(p)2133 2159 y Fu(c)2167 2145 y Fw(\(2\).)40 b(So)26 b(let)g Ft(d)g Fw(=)e(2.)40 b(W)-8 b(e)27 b(\014rst)e(need)h(some)165 2258 y(additional)i(terminology)-8 b(.)40 b(A)30 b Fv(\003)p Fw(-path)g(in)e Fx(Z)1678 2225 y Fr(2)1747 2258 y Fw(is)g(a)i(sequence) g(\()p Ft(v)2367 2272 y Fr(0)2407 2258 y Ft(;)15 b(v)2491 2272 y Fr(1)2531 2258 y Ft(;)g(:)g(:)g(:)i(;)e(v)2777 2273 y Fu(k)2820 2258 y Fw(\))30 b(of)f(distinct)f(v)m(ertices)165 2371 y(suc)m(h)34 b(that)i Ft(d)623 2385 y Fq(1)698 2371 y Fw(\()p Ft(v)777 2385 y Fu(j)t Fq(\000)p Fr(1)904 2371 y Ft(;)15 b(v)988 2385 y Fu(j)1025 2371 y Fw(\))32 b(=)g(1)j(for)f Ft(j)k Fw(=)31 b(1)p Ft(;)15 b(:)g(:)g(:)i(k)s Fw(.)53 b(Note)36 b(that)f(t)m(w)m(o)h(consecutiv)m(e)f(v)m(ertices)g(in)f(a)g Fv(\003)p Fw(-)165 2484 y(path)26 b(need)g(not)h(b)s(e)e(nearest-neigh) m(b)s(ors;)j(they)e(ma)m(y)h(also)f(b)s(e)g(\\diagonal)g(neigh)m(b)s (ors".)38 b(A)26 b Fv(\003)p Fp(-cir)-5 b(cuit)165 2596 y Fw(is)25 b(a)h(sequence)g(\()p Ft(v)774 2610 y Fr(0)814 2596 y Ft(;)15 b(v)898 2610 y Fr(1)938 2596 y Ft(;)g(:)g(:)g(:)h(;)26 b(v)1194 2611 y Fu(k)1237 2596 y Ft(;)15 b(v)1321 2610 y Fr(0)1361 2596 y Fw(\))25 b(suc)m(h)h(that)g(\()p Ft(v)1893 2610 y Fr(0)1933 2596 y Ft(;)15 b(v)2017 2610 y Fr(1)2057 2596 y Ft(;)g(:)g(:)g(:)h(;)f(v)2302 2611 y Fu(k)2345 2596 y Fw(\))26 b(is)f(a)h Fv(\003)p Fw(-path)g(and)f Ft(d)3066 2610 y Fq(1)3141 2596 y Fw(\()p Ft(v)3220 2611 y Fu(k)3263 2596 y Ft(;)15 b(v)3347 2610 y Fr(0)3387 2596 y Fw(\))26 b(=)e(1.)165 2709 y(Informally)-8 b(,)32 b(a)h Fv(\003)p Fw(-circuit)e(is)h(a)h Fv(\003)p Fw(-path)g(whic)m(h)e (ends)g(where)h(it)g(starts.)47 b Fv(\003)p Fw(-circuits)32 b(with)f(the)i(same)165 2822 y(set)27 b(of)f(sites)f(are)h(iden)m (ti\014ed.)38 b(A)26 b Fp(close)-5 b(d)29 b Fv(\003)p Fp(-cir)-5 b(cuit)26 b Fw(is)f(a)h Fv(\003)p Fw(-circuit)g(in)e(whic)m (h)h(all)f(v)m(ertices)j(are)f(closed.)306 2935 y(No)m(w)39 b(let)e Ft(M)47 b Fw(b)s(e)37 b(the)g(n)m(um)m(b)s(er)f(of)i(closed)f Fv(\003)p Fw(-circuits)g(that)h(surround)c(the)k(origin)e(0.)62 b(As)37 b Fx(Z)3575 2902 y Fr(2)165 3048 y Fw(is)32 b(a)h(planar)e (graph,)i(the)g(\\outer)g(b)s(oundary")e(of)i(a)g(\014nite)f(op)s(en)g (cluster)g(con)m(taining)g(0)h(de\014nes)f(a)165 3161 y(closed)f Fv(\003)p Fw(-circuit)e(around)h(0.)41 b(Hence,)31 b(the)g(ev)m(en)m(t)h Fv(f)p Fw(0)26 b Fv($)f(1g)31 b Fw(o)s(ccurs)f(if)f(and)h(only)f(if)h Ft(M)35 b Fw(=)25 b(0.)306 3274 y(The)h(n)m(um)m(b)s(er)g(of)g Fp(al)5 b(l)37 b Fw(\(not)27 b(necessarily)e(closed\))i Fv(\003)p Fw(-circuits)f(of)g(a)h(giv)m(en)g(length)f Ft(k)j Fw(surrounding)165 3387 y(0)37 b(allo)m(ws)e(the)h(follo)m(wing)e(crude)h(estimate.)58 b(Consider)34 b(the)i(leftmost)g(crossing)f(of)h(the)g Ft(x)p Fw(-axis)g(of)165 3500 y(suc)m(h)c(a)h(circuit;)f(the)h(lo)s (cation)e(of)i(suc)m(h)f(a)g(crossing)g(is)f(at)i(distance)f(at)h(most) g Ft(k)i Fw(from)d(the)g(origin,)165 3613 y(so)e(there)g(are)g(at)g (most)g Ft(k)j Fw(suc)m(h)c(lo)s(cations)g(to)i(c)m(ho)s(ose)f(from.)40 b(Starting)29 b(at)h(this)f(lo)s(cation,)g(w)m(e)i(ma)m(y)165 3726 y(trace)f(the)f Fv(\003)p Fw(-circuit)f(clo)s(c)m(kwise)h(\(sa)m (y\),)h(and)e(at)i(eac)m(h)g(step)e(w)m(e)h(ha)m(v)m(e)h(at)g(most)f(7) g Ft(d)3020 3740 y Fq(1)3095 3726 y Fw(-neigh)m(b)s(ors)e(to)165 3838 y(c)m(ho)s(ose)37 b(from.)56 b(Hence,)37 b(the)f(n)m(um)m(b)s(er)e (of)i Fv(\003)p Fw(-circuits)f(of)h(length)f Ft(k)j Fw(around)d(0)h(is) e(at)i(most)g Ft(k)19 b Fw(7)3456 3805 y Fu(k)r Fq(\000)p Fr(1)3589 3838 y Fw(.)165 3951 y(Eac)m(h)31 b(one)g(is)e(closed)i(with) e(probabilit)m(y)f(\(1)21 b Fv(\000)f Ft(p)p Fw(\))1863 3918 y Fu(k)1905 3951 y Fw(,)31 b(so)1331 4216 y Ft( )1390 4230 y Fu(p)1430 4216 y Fw(\()p Ft(M)10 b Fw(\))26 b Fv(\024)1749 4111 y Fq(1)1724 4136 y Fs(X)1720 4320 y Fu(k)r Fr(=1)1864 4216 y Ft(k)18 b Fw(7)1974 4179 y Fu(k)r Fq(\000)p Fr(1)2107 4216 y Fw(\(1)k Fv(\000)d Ft(p)p Fw(\))2380 4179 y Fu(k)2423 4216 y Ft(:)165 4496 y Fw(The)37 b(last)g(sum)g(is)f(\014nite)g(for)h Ft(p)g(>)f Fw(6)p Ft(=)p Fw(7.)63 b(Hence,)40 b(for)d(suc)m(h)g(a)h Ft(p)f Fw(and)f Ft(n)h Fw(large)h(enough,)h(there)e(is)165 4609 y(a)f(p)s(ositiv)m(e)e(probabilit)m(y)f(for)i(ha)m(ving)g(no)h(closed)f Fv(\003)p Fw(-circuit)f(around)h(0)g(whic)m(h)g(con)m(tains)g(a)h(site) f(of)165 4722 y(distance)d(at)h(least)f Ft(n)g Fw(from)f(0.)46 b(By)33 b(the)f(argumen)m(t)g(in)f(the)h(pro)s(of)f(of)i(Prop)s (osition)d(5.2,)j(it)f(follo)m(ws)165 4835 y(that)d Ft( )419 4849 y Fu(p)459 4835 y Fw(\()p Ft(M)35 b Fw(=)25 b(0\))h Ft(>)f Fw(0)k(for)e(suc)m(h)h Ft(p)p Fw(.)39 b(Hence)29 b Ft(\022)s Fw(\()p Ft(p)p Fw(\))c Ft(>)g Fw(0)j(for)g Ft(p)d(>)g Fw(6)p Ft(=)p Fw(7,)30 b(and)d(the)h(second)g(half)f(of)h (\(17\))165 4948 y(follo)m(ws.)3087 b Fk(2)165 5110 y Fw(The)36 b(exact)h(v)-5 b(alue)35 b(of)h(the)g(p)s(ercolation)e (threshold)g Ft(p)2042 5124 y Fu(c)2077 5110 y Fw(\()p Ft(d)p Fw(\))i(of)g Fx(Z)2403 5077 y Fu(d)2479 5110 y Fw(is)f(not)h(kno)m(wn)f(for)g(an)m(y)h Ft(d)f Fv(\025)e Fw(2.)165 5223 y(The)d(b)s(est)g(rigorous)g(b)s(ounds)e(for)i Ft(d)c Fw(=)e(2)31 b(are)g(presen)m(tly)1463 5428 y(0)p Ft(:)p Fw(556)c Ft(<)e(p)1837 5442 y Fu(c)1872 5428 y Fw(\(2\))h Ft(<)f Fw(0)p Ft(:)p Fw(680)1139 b(\(19\))1844 5895 y(30)p eop %%Page: 31 31 31 30 bop 165 304 a Fw(where)32 b(the)g(\014rst)g(inequalit)m(y)e(is)i (due)f(to)i(v)-5 b(an)32 b(den)f(Berg)i(and)f(Ermak)m(o)m(v)h([26)q(])g (\(inspired)c(b)m(y)j([176)r(]\))165 417 y(and)e(the)h(second)f(to)h (Wierman)f([224)r(].)40 b(F)-8 b(or)32 b(high)d(dimensions,)f(it)h(is)h (kno)m(wn)g(that)1562 619 y(lim)1537 679 y Fu(d)p Fq(!1)1729 619 y Fw(2)p Ft(d)16 b(p)1883 633 y Fu(c)1918 619 y Fw(\()p Ft(d)p Fw(\))26 b(=)f(1)15 b Ft(;)1211 b Fw(\(20\))165 853 y(see)29 b(Kesten)g([144)q(],)h(Gordon)e([104)q(])h(and)f(Hara)h (and)f(Slade)f([127)r(])h(for)g(this)g(and)f(\014ner)h(asymptotics.)165 966 y(On)36 b(some)h(particular)d(graphs,)k Ft(p)1355 980 y Fu(c)1426 966 y Fw(can)f(b)s(e)e(determined)g(exactly)-8 b(.)60 b(F)-8 b(or)38 b(example,)f(for)g Fv(L)e Fw(=)g Fx(T)3549 981 y Fu(d)3589 966 y Fw(,)165 1079 y(the)25 b(regular)f(\(Ca)m(yley)g(or)h(Bethe\))g(tree,)i(branc)m(hing-pro)s (cess)c(argumen)m(ts)i(immediately)d(sho)m(w)j(that)165 1191 y Ft(p)211 1205 y Fu(c)246 1191 y Fw(\()p Fx(T)354 1206 y Fu(d)394 1191 y Fw(\))h(=)f(1)p Ft(=d)p Fw(,)k(and)e(for)g(the)h (triangular)d(lattice)j(it)f(follo)m(ws)f(from)h(planar)f(dualit)m(y)g (that)i Ft(p)3322 1205 y Fu(c)3382 1191 y Fw(=)d(1)p Ft(=)p Fw(2)165 1304 y([143)r(].)306 1417 y(The)36 b(preceding)e (considerations)g(do)i(not)g(tell)e(us)h(what)g(happ)s(ens)f Fp(at)i Fw(the)g(critical)e(v)-5 b(alue)35 b Ft(p)3554 1431 y Fu(c)3589 1417 y Fw(.)165 1530 y(It)g(is)f(b)s(eliev)m(ed)g (that,)j(for)d(the)h(in)m(teger)h(lattice)f Fx(Z)1904 1497 y Fu(d)1979 1530 y Fw(of)g(an)m(y)h(dimension)c Ft(d)h Fv(\025)f Fw(2,)37 b(there)e(is)f Ft( )3380 1544 y Fu(p)3416 1552 y Fo(c)3452 1530 y Fw(-a.s.)165 1643 y(no)39 b(in\014nite)e(cluster,)j(whic)m(h)e(means)h(that)g Ft(\022)s Fw(\()p Ft(p)1834 1657 y Fu(c)1868 1643 y Fw(\))h(=)f(0;)44 b(so)39 b(far)f(this)g(is)g(only)g(kno)m(wn)g(for)h Ft(d)h Fw(=)f(2)165 1756 y(and)31 b Ft(d)26 b Fv(\025)h Fw(19,)32 b(see)g(Russo)e([209)r(])h(and)f(Hara)i(and)e(Slade)h([126)q(].)43 b(One)31 b(can)g(sho)m(w)g(that)h(the)f(relation)165 1869 y Ft(\022)s Fw(\()p Ft(p)292 1883 y Fu(c)326 1869 y Fw(\))50 b(=)e(0)d(implies)d(con)m(tin)m(uit)m(y)j(of)f Ft(\022)s Fw(\()p Ft(p)p Fw(\))h(at)g Ft(p)k Fw(=)f Ft(p)2096 1883 y Fu(c)2131 1869 y Fw(,)g(so)d(in)e(com)m(bination)h(with)f(the)i (trivial)165 1982 y(con)m(tin)m(uit)m(y)40 b(of)g Ft(\022)s Fw(\()p Ft(p)p Fw(\))f(in)f(the)i(sub)s(critical)d(regime)i(and)g(the)h (follo)m(wing)e(more)h(in)m(teresting)g(result)165 2095 y(\(whic)m(h)29 b(can)g(b)s(e)g(found)f(e.g.)i(in)e([108)r(]\))i(w)m(e) g(get)g(con)m(tin)m(uit)m(y)f(of)h Ft(\022)s Fw(\()p Ft(p)p Fw(\))e(throughout)h([0)p Ft(;)15 b Fw(1])31 b(as)f(so)s(on)f (as)165 2208 y(absence)i(of)g(in\014nite)d(clusters)h(at)i(criticalit)m (y)f(is)f(established.)165 2393 y Fx(Theorem)34 b(5.4)46 b Fp(F)-7 b(or)32 b(Bernoul)5 b(li)31 b(site)g(p)-5 b(er)g(c)g(olation) 34 b(on)d Fx(Z)2173 2360 y Fu(d)2213 2393 y Fp(,)g Ft(d)26 b Fv(\025)f Fw(2)p Fp(,)31 b(the)g(function)g Ft(\022)s Fw(\()p Ft(p)p Fw(\))f Fp(is)h(c)-5 b(ontin-)165 2506 y(uous)33 b(thr)-5 b(oughout)35 b(the)e(sup)-5 b(er)g(critic)g(al)35 b(r)-5 b(e)g(gime)33 b Fw(\()p Ft(p)1861 2520 y Fu(c)1896 2506 y Ft(;)15 b Fw(1])p Fp(.)306 2692 y Fw(So)42 b(far)g(w)m(e)g(w)m (ere)h(in)m(terested)f(in)e(the)i(existence)h(of)f(in\014nite)d (clusters.)75 b(In)41 b(the)h(sub)s(critical)165 2805 y(regime)f Ft(p)i(<)g(p)718 2819 y Fu(c)793 2805 y Fw(when)d(no)h (in\014nite)e(cluster)h(exists,)k(one)d(ma)m(y)h(ask)f(for)g(the)g (size)g(of)g(a)g(t)m(ypical)165 2918 y(cluster.)50 b(Let)34 b Fv(j)p Ft(C)760 2932 y Fr(0)799 2918 y Fv(j)g Fw(b)s(e)f(the)h (random)e(n)m(um)m(b)s(er)h(of)g(v)m(ertices)h(in)f(the)g(op)s(en)g (cluster)g(con)m(taining)g(the)165 3031 y(origin;)28 b(w)m(e)h(set)g Fv(j)p Ft(C)809 3045 y Fr(0)848 3031 y Fv(j)d Fw(=)f(0)k(if)e(the)h(origin)f(is)h(closed.)40 b(By)28 b(the)h(de\014nition)d(of)i Ft(p)2797 3045 y Fu(c)2832 3031 y Fw(,)h Ft( )2945 3045 y Fu(p)2985 3031 y Fw(\()p Fv(j)p Ft(C)3110 3045 y Fr(0)3150 3031 y Fv(j)c Fw(=)g Fv(1)p Fw(\))h(=)e(0.)165 3144 y(F)-8 b(or)40 b Fv(L)e Fw(=)g Fx(Z)611 3111 y Fu(d)651 3144 y Fw(,)j(w)m(e)e(ev)m(en) g(ha)m(v)m(e)h(the)e(stronger)h(statemen)m(t)h(that)f(the)g(exp)s (ected)f(v)-5 b(alue)38 b(of)h Fv(j)p Ft(C)3450 3158 y Fr(0)3489 3144 y Fv(j)g Fw(is)165 3257 y(\014nite.)165 3442 y Fx(Theorem)34 b(5.5)46 b Fp(F)-7 b(or)40 b(Bernoul)5 b(li)38 b(site)h(p)-5 b(er)g(c)g(olation)41 b(on)e Fx(Z)2211 3409 y Fu(d)2290 3442 y Fp(with)g(r)-5 b(etention)39 b(p)-5 b(ar)g(ameter)41 b Ft(p)36 b(<)f(p)3552 3456 y Fu(c)3586 3442 y Fp(,)165 3555 y(we)e(have)g Ft( )568 3569 y Fu(p)608 3555 y Fw(\()p Fv(j)p Ft(C)733 3569 y Fr(0)773 3555 y Fv(j)p Fw(\))26 b Ft(<)f Fv(1)p Fp(.)165 3741 y Fw(This)33 b(w)m(as)i(pro)m(v)m(ed)g(indep)s(enden)m(tly)d(b)m (y)i(Menshik)m(o)m(v)h([174)r(])g(and)f(b)m(y)g(Aizenman)g(and)g (Barsky)h([6)q(].)165 3854 y(The)30 b(pro)s(ofs)f(are)h(rather)g(in)m (v)m(olv)m(ed,)g(so)g(w)m(e)g(refer)g(the)g(reader)g(to)g(the)g (original)e(articles)i(and)f([108)r(].)165 3967 y(It)39 b(is)e(w)m(orth)i(noting)f(that)h(Theorem)f(5.5)h(fails)e(in)g(the)i (setting)f(of)h(general)f(graphs;)k(a)d(striking)165 4080 y(coun)m(terexample)31 b(is)f(the)g(\\three-one-tree")j(discussed) c(on)h(p.)g(936)i(of)e(Ly)m(ons)h([164)q(].)306 4193 y(Menshik)m(o)m(v)h(ev)m(en)g(sho)m(w)m(ed)f(that)h(the)g(distribution) 27 b(of)32 b(the)f(radius)e(of)j(the)f(op)s(en)g(cluster)g(con-)165 4306 y(taining)h(the)i(origin)e(deca)m(ys)i(exp)s(onen)m(tially)-8 b(.)48 b(The)33 b(follo)m(wing)f(ev)m(en)i(stronger)g(result)e(states)j (that)165 4419 y(the)29 b(same)h(is)d(true)i(for)f(the)h(distribution)d (of)j Fv(j)p Ft(C)1806 4433 y Fr(0)1845 4419 y Fv(j)p Fw(;)h(see)f(Grimmett)g([108)r(])f(for)h(a)g(pro)s(of)f(\(in)g(the)h (case)165 4531 y(of)i(b)s(ond)e(p)s(ercolation\).)165 4717 y Fx(Theorem)34 b(5.6)46 b Fp(F)-7 b(or)40 b(Bernoul)5 b(li)38 b(site)h(p)-5 b(er)g(c)g(olation)41 b(on)e Fx(Z)2211 4684 y Fu(d)2290 4717 y Fp(with)g(r)-5 b(etention)39 b(p)-5 b(ar)g(ameter)41 b Ft(p)36 b(<)f(p)3552 4731 y Fu(c)3586 4717 y Fp(,)165 4830 y(ther)-5 b(e)34 b(exists)f(a)g(c)-5 b(onstant)34 b Ft(c)f Fp(\(dep)-5 b(ending)34 b(on)f Ft(p)p Fp(\))g(such)f(that)1486 5032 y Ft( )1545 5046 y Fu(p)1585 5032 y Fw(\()p Fv(j)p Ft(C)1710 5046 y Fr(0)1750 5032 y Fv(j)25 b(\025)g Ft(n)p Fw(\))g Fv(\024)g Ft(e)2149 4995 y Fq(\000)p Fu(c)12 b(n)165 5234 y Fp(for)34 b(al)5 b(l)33 b Ft(n)p Fp(.)165 5420 y Fw(Lo)s(oking)41 b(at)h(a)f(\014xed)g (path)g(of)g(length)g Ft(n)f Fw(starting)h(at)h(0,)j(w)m(e)c (immediately)f(obtain)g(the)i(lo)m(w)m(er)165 5533 y(b)s(ound)31 b Ft( )507 5547 y Fu(p)547 5533 y Fw(\()p Fv(j)p Ft(C)672 5547 y Fr(0)712 5533 y Fv(j)f(\025)f Ft(n)p Fw(\))g Fv(\025)h Ft(p)1133 5500 y Fu(n)1179 5533 y Fw(.)48 b(So)33 b(the)g(preceding)f (upp)s(er)f(b)s(ound)g(is)h(b)s(est)g(p)s(ossible,)g(except)i(that)165 5646 y(the)d(optimal)e(constan)m(t)j Ft(c)f Fw(is)e(unkno)m(wn.)1844 5895 y(31)p eop %%Page: 32 32 32 31 bop 306 304 a Fw(W)-8 b(e)42 b(conclude)d(this)g(section)h(with)f (some)h(remarks)g(on)g Fp(Bernoul)5 b(li)42 b(b)-5 b(ond)43 b(p)-5 b(er)g(c)g(olation)p Fw(.)73 b(The)165 417 y(mo)s(del)35 b(is)g(similar,)g(except)h(that)h(no)m(w)e(the)h(edges)h(rather)e(than) h(the)g(v)m(ertices)g(in)f(\()p Fv(L)p Ft(;)15 b Fv(\030)p Fw(\))36 b(are)g(in-)165 530 y(dep)s(enden)m(tly)f(op)s(en)h(\(describ) s(ed)f(b)m(y)i(the)g(v)-5 b(alue)36 b(1\))i(or)e(closed)h(\(with)f(v)-5 b(alue)36 b(0\))i(with)d(resp)s(ectiv)m(e)165 643 y(probabilities)28 b Ft(p)i Fw(and)g(1)21 b Fv(\000)f Ft(p)p Fw(.)41 b(The)30 b(asso)s(ciated)h(con\014guration)f(space)h(is)e(th)m(us)i Fv(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)3135 610 y Fq(B)3189 643 y Fw(.)41 b(W)-8 b(e)32 b(write)165 756 y Ft(\036)219 770 y Fu(p)287 756 y Fw(for)27 b(the)h(asso)s(ciated)g(\(Bernoulli\))d (probabilit)m(y)g(measure)j(on)f Fv(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)2624 723 y Fq(B)2679 756 y Fw(.)39 b(In)27 b(the)h(presen)m(t)f(con)m(text)165 869 y(of)36 b(b)s(ond)e(p)s (ercolation,)i(an)f Fp(op)-5 b(en)38 b(p)-5 b(ath)37 b Fw(is)e(a)h(path)f(in)f(whic)m(h)g(all)g(edges)i(are)g(op)s(en,)g (and)f(an)g Fp(op)-5 b(en)165 981 y(cluster)31 b Fw(is)e(a)i(maximal)e (region)h Ft(C)i Fv(\032)25 b(L)k Fw(whic)m(h)g(is)h(connected,)h(in)e (that)i(for)f(an)m(y)h Ft(x;)15 b(y)28 b Fv(2)d Ft(C)37 b Fw(there)30 b(is)165 1094 y(an)h(op)s(en)e(path)h(in)f Ft(C)37 b Fw(from)30 b Ft(x)g Fw(to)i Ft(y)s Fw(.)306 1207 y(All)g(results)g(for)h(site)g(p)s(ercolation)g(discussed)e(so)j (far)f(extend)g(to)h(the)f(b)s(ond)f(p)s(ercolation)g(set-)165 1320 y(up.)62 b(This)36 b(is)g(no)i(surprise)d(b)s(ecause)i(b)s(ond)f (p)s(ercolation)h(is)g(equiv)-5 b(alen)m(t)37 b(to)h(site)g(p)s (ercolation)e(on)165 1433 y(the)46 b(so-called)f(co)m(v)m(ering)h (graph)e(for)h(whic)m(h)f Fv(B)k Fw(is)c(tak)m(en)i(as)g(set)f(of)h(v)m (ertices,)j(and)c(edges)h(are)165 1546 y(dra)m(wn)37 b(b)s(et)m(w)m(een)i(an)m(y)f(t)m(w)m(o)h(coinciden)m(t)e(elemen)m(ts)i (of)f Fv(B)s Fw(.)62 b(In)37 b(particular,)h(there)g(exists)g(again)g (a)165 1659 y(critical)d(v)-5 b(alue)35 b Ft(p)760 1673 y Fu(c)830 1659 y Fw(for)h(the)g(o)s(ccurrence)f(of)h(in\014nite)e(op)s (en)h(clusters,)h(and)f(Prop)s(ositions)f(5.1)j(and)165 1772 y(5.2,)j(Theorems)35 b(5.3,)40 b(5.5,)f(5.6)e(and)f(5.4,)j(and)d (the)g(asymptotic)h(form)m(ula)e(\(20\))j(are)f(still)d(true)i(in)165 1885 y(the)41 b(case)g(of)f(b)s(ond)f(p)s(ercolation.)70 b(What)40 b(is)g(generally)f(di\013eren)m(t,)k(are)d(the)h(critical)e (v)-5 b(alues)39 b(for)165 1998 y(site)34 b(and)g(b)s(ond)f(Bernoulli)f (p)s(ercolation)h(on)i(a)f(giv)m(en)h(graph.)52 b(One)34 b(remark)-5 b(able)33 b(case)j(is)d(that)i(of)165 2111 y(b)s(ond)j(p)s(ercolation)g(on)h Fx(Z)1084 2078 y Fr(2)1123 2111 y Fw(,)j(where)c(\(again)i(b)m(y)f(planar)f(dualit)m(y\))g Ft(p)2573 2125 y Fu(c)2647 2111 y Fw(=)i(1)p Ft(=)p Fw(2;)45 b(this)38 b(is)g(a)h(famous)165 2223 y(result)31 b(of)h(Kesten)f([142)r (].)45 b(In)31 b(the)g(sp)s(eci\014c)g(case)h(of)g(trees,)h(ho)m(w)m (ev)m(er,)h(site)d(and)g(b)s(ond)f(p)s(ercolation)165 2336 y(are)40 b(equiv)-5 b(alen)m(t.)67 b(In)38 b(particular,)i(for)f Fv(L)h Fw(=)g Fx(T)1833 2351 y Fu(d)1912 2336 y Fw(w)m(e)g(ha)m(v)m(e)g Ft(p)2319 2350 y Fu(c)2393 2336 y Fw(=)g(1)p Ft(=d)h Fw(for)e(b)s(oth)f(site)h(and)g(b)s(ond)165 2449 y(p)s(ercolation;)30 b(for)g(more)h(general)f(trees)h(a)g(form)m(ula)e(for)i Ft(p)2156 2463 y Fu(c)2220 2449 y Fw(w)m(as)g(giv)m(en)f(b)m(y)h(Ly)m (ons)f([164)r(].)165 2693 y FA(5.2)113 b(Dep)s(enden)m(t)38 b(p)s(ercolation:)51 b(the)37 b(role)h(of)f(the)h(densit)m(y)165 2864 y Fw(Our)23 b(main)f(sub)5 b(ject)23 b(is)g(the)h(analysis)e(of)i (equilibrium)18 b(phases)23 b(b)m(y)h(means)f(of)h(p)s(ercolation)f (metho)s(ds.)165 2977 y(In)38 b(this)g(case,)k(a)d(site)f(will)e(b)s(e) i(considered)f(as)i(op)s(en)f(if,)i(for)e(example,)j(the)d (con\014guration)g(in)g(a)165 3090 y(neigh)m(b)s(orho)s(o)s(d)24 b(of)j(this)e(site)h(sho)m(ws)g(a)h(sp)s(eci\014ed)e(pattern,)i(and)f (the)g(ev)m(en)m(ts)i(\\site)f Ft(x)f Fw(is)f(op)s(en")h(with)165 3203 y Ft(x)36 b Fv(2)g(L)g Fw(are)h(then)g(far)f(from)h(b)s(eing)e (indep)s(enden)m(t.)58 b(This)35 b(leads)h(us)g(to)h(considering)e(the) i(case)h(of)165 3316 y(dep)s(enden)m(t)30 b(p)s(ercolation.)40 b(In)29 b(this)g(subsection)h(w)m(e)h(do)f(some)h(\014rst)e(steps)i(in) e(this)g(direction.)306 3429 y(Our)36 b(starting)g(p)s(oin)m(t)g(is)g (the)h(follo)m(wing)e(question.)59 b(In)36 b(the)h(case)h(of)e (Bernoulli)f(p)s(ercolation,)165 3542 y(there)41 b(is)f(a)h(unique)d (parameter,)44 b(the)d(o)s(ccupation)f(probabilit)m(y)e(or)j(densit)m (y)f Ft(p)p Fw(,)j(whic)m(h)c(go)m(v)m(erns)165 3655 y(the)47 b(phase)f(diagram)f(and)h(allo)m(ws)g(to)h(distinguish)42 b(b)s(et)m(w)m(een)47 b(sub)s(critical)d(\(\\no)j(in\014nite)d(clus-) 165 3768 y(ter"\))c(and)e(sup)s(ercritical)e(\(\\at)k(least)f(one)g (in\014nite)d(cluster"\))j(b)s(eha)m(vior.)64 b(Do)s(es)39 b(this)f(also)g(hold)165 3881 y(in)f(general?)62 b(Sp)s(eci\014cally)-8 b(,)37 b(is)g(it)g(true)g(that)h(for)g(an)m(y)g(translation)e(in)m(v)-5 b(arian)m(t)37 b(probabilit)m(y)e(mea-)165 3993 y(sure)j Ft(\026)h Fw(on)f Fv(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)814 3960 y Fn(Z)864 3937 y Fo(d)907 3993 y Fw(,)41 b(the)e(o)s(ccurrence)g (of)f(an)h(in\014nite)e(cluster)h(only)f(dep)s(ends)g(on)i(the)g (densit)m(y)165 4106 y Ft(p)p Fw(\()p Ft(\026)p Fw(\))27 b(=)g Ft(\026)p Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))28 b(=)f(1\))32 b(of)g(op)s(en)e(sites)h Ft(x)c Fv(2)g Fx(Z)1753 4073 y Fu(d)1793 4106 y Fw(?)43 b(In)31 b(general,)h(the)f(answ)m(er)h (is)e(ob)m(viously)g(\\no",)j(as)165 4219 y(w)m(e)d(will)d(no)m(w)i (sho)m(w)g(b)m(y)g(t)m(w)m(o)i(simple)c(examples:)39 b(there)30 b(exist)f(translation)f(in)m(v)-5 b(arian)m(t)28 b(measures)h Ft(\026)165 4332 y Fw(on)24 b Fv(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)505 4299 y Fn(Z)555 4276 y Fo(d)622 4332 y Fw(with)23 b(arbitrarily)e(small)i(densities)g(suc)m(h)g(that)i (in\014nite)d(clusters)i(exist)g(almost)g(surely)-8 b(,)165 4445 y(and)36 b(also)g(translation)f(in)m(v)-5 b(arian)m(t)36 b(measures)g(with)e(densities)h(arbitrarily)e(close)k(to)g(1)f(for)g (whic)m(h)165 4558 y(no)31 b(in\014nite)d(clusters)h(exist)h(with)f (probabilit)m(y)f(1.)165 4746 y Fx(Example)34 b(5.7)46 b Fw(F)-8 b(or)30 b Ft(q)e Fv(2)d Fw(\(0)p Ft(;)15 b Fw(1\),)32 b(let)d(\()p Ft(Y)20 b Fw(\()p Ft(x)p Fw(\))p Ft(;)31 b(x)26 b Fv(2)e Fx(Z)p Fw(\))30 b(b)s(e)e(i.i.d.)g(random)h(v) -5 b(ariables)28 b(taking)h(v)-5 b(alues)165 4859 y(0)34 b(and)f(1)h(with)e(probabilit)m(y)f(1)23 b Fv(\000)f Ft(q)36 b Fw(resp.)d Ft(q)s Fw(.)50 b(W)-8 b(e)34 b(de\014ne)f(a)h (translation)e(in)m(v)-5 b(arian)m(t)33 b(random)g(\014eld)165 4971 y(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))p Ft(;)32 b(x)40 b Fv(2)g Fx(Z)718 4938 y Fu(d)758 4971 y Fw(\),)i Ft(d)e Fv(\025)g Fw(2,)i(b)m(y)d(setting)g Ft(X)7 b Fw(\()p Ft(x)p Fw(\))41 b(=)e Ft(X)7 b Fw(\()p Ft(x)2136 4985 y Fr(1)2177 4971 y Ft(;)15 b(:)g(:)g(:)h(;)f(x)2430 4986 y Fu(d)2471 4971 y Fw(\))40 b(=)g Ft(Y)20 b Fw(\()p Ft(x)2817 4985 y Fr(1)2857 4971 y Fw(\))39 b(for)g(eac)m(h)h Ft(x)g Fv(2)g Fx(Z)3549 4938 y Fu(d)3589 4971 y Fw(.)165 5084 y(W)-8 b(riting)45 b Ft(\026)g Fw(for)g(the)h(distribution)41 b(of)46 b(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))p Ft(;)32 b(x)50 b Fv(2)g Fx(Z)2142 5051 y Fu(d)2182 5084 y Fw(\),)g(w)m(e)45 b(ha)m(v)m(e)i Ft(p)p Fw(\()p Ft(\026)p Fw(\))j(=)g Ft(q)s Fw(,)f(but)c(with)f Ft(\026)p Fw(-)165 5197 y(probabilit)m(y)28 b(1)j(there)g(exist)f(in\014nitely)d(man)m(y)k(in\014nite)d(clusters,)h (ev)m(en)i(if)f Ft(q)j Fw(is)c(arbitrarily)f(small.)165 5410 y Fx(Example)34 b(5.8)46 b Fw(Again)33 b(let)f Ft(q)g Fv(2)d Fw(\(0)p Ft(;)15 b Fw(1\),)35 b Ft(d)30 b Fv(\025)e Fw(2,)34 b(and)e(\()p Ft(Y)21 b Fw(\()p Ft(x;)15 b(i)p Fw(\))p Ft(;)32 b(x)d Fv(2)f Fx(Z)p Ft(;)j(i)e Fv(2)g(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)i(;)e(d)p Fv(g)p Fw(\))34 b(b)s(e)e(i.i.d.)165 5523 y(random)39 b(v)-5 b(ariables)37 b(taking)i(v)-5 b(alues)38 b(0)i(and)e(1)i(with)d(probabilities)f(1)26 b Fv(\000)g Ft(q)42 b Fw(and)c Ft(q)s Fw(.)66 b(W)-8 b(e)40 b(de\014ne)f(a)1844 5895 y(32)p eop %%Page: 33 33 33 32 bop 165 304 a Fw(translation)30 b(in)m(v)-5 b(arian)m(t)29 b(random)h(\014eld)f(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))p Ft(;)32 b(x)25 b Fv(2)g Fx(Z)2061 271 y Fu(d)2101 304 y Fw(\))31 b(b)m(y)f(setting)1171 586 y Ft(X)7 b Fw(\()p Ft(x)p Fw(\))27 b(=)e Ft(X)7 b Fw(\()p Ft(x)1667 600 y Fr(1)1707 586 y Ft(;)15 b(:)g(:)g(:)i(;)e(x)1961 601 y Fu(d)2001 586 y Fw(\))26 b(=)2197 480 y Fu(d)2162 505 y Fs(Y)2158 687 y Fu(i)p Fr(=1)2287 586 y Ft(Y)20 b Fw(\()p Ft(x)2447 600 y Fu(i)2476 586 y Ft(;)15 b(i)p Fw(\))p Ft(:)165 857 y Fw(A)31 b(momen)m(t's)g(though)m(t)h(rev)m(eals)f(that)g Ft(\026)p Fw(-a.s.)g(there)g(exist)g(no)f(in\014nite)e(clusters,)j (despite)e(the)i(fact)165 970 y(that)g Ft(p)p Fw(\()p Ft(\026)p Fw(\))26 b(=)f Ft(q)699 937 y Fu(d)769 970 y Fw(ma)m(y)31 b(b)s(e)f(arbitrarily)e(close)i(to)h(1.)165 1182 y(These)25 b(examples)g(suggest)h(to)g(lo)s(ok)f(for)g(additional) f(assumptions)f(under)h(whic)m(h)g(high)g(\(resp.)h(lo)m(w\))165 1295 y(densit)m(y)35 b(guaran)m(tees)i(existence)f(\(resp.)f (nonexistence\))h(of)g(in\014nite)d(clusters.)55 b(P)m(ositiv)m(e)36 b(correla-)165 1408 y(tions)i(\(in)f(the)i(sense)f(of)h(De\014nition)e (4.10\))j(do)s(es)e(not)h(su\016ce,)h(b)s(ecause)e(b)s(oth)g(examples)g (ab)s(o)m(v)m(e)165 1520 y(ob)m(viously)29 b(ha)m(v)m(e)j(p)s(ositiv)m (e)d(correlations.)306 1633 y(An)43 b(alternativ)m(e)g(migh)m(t)f(b)s (e)g(to)h(assume)f Ft(R)q Fw(-indep)s(endence)e(in)i(the)g(sense)h (that)g Ft(X)7 b Fw(\(\003\))44 b(and)165 1746 y Ft(X)7 b Fw(\(\001\))32 b(are)e(indep)s(enden)m(t)e(for)j(an)m(y)f(t)m(w)m(o)i (\014nite)d(regions)h(\003)p Ft(;)15 b Fw(\001)26 b Fv(\032)f(L)30 b Fw(suc)m(h)g(that)1580 1950 y(min)1507 2009 y Fu(x)p Fq(2)p Fr(\003)p Fu(;y)r Fq(2)p Fr(\001)1820 1950 y Fv(j)p Ft(x)21 b Fv(\000)f Ft(y)s Fv(j)25 b Ft(>)g(R)165 2208 y Fw(for)33 b(some)h(giv)m(en)f Ft(R)q Fw(.)48 b(F)-8 b(or)34 b Fx(Z)1151 2175 y Fu(d)1191 2208 y Ft(;)15 b(d)30 b Fv(\025)g Fw(2,)k(this)e(giv)m(es)h(non)m(trivial)e(thresholds)g(0)g Ft(<)e(p)2995 2222 y Fr(1)3064 2208 y Ft(<)g(p)3210 2222 y Fr(2)3279 2208 y Ft(<)g Fw(1)34 b(\(de-)165 2321 y(p)s(ending)23 b(on)h Ft(R)q Fw(\))h(suc)m(h)f(that)h(existence)g(\(resp.)g (non-existence\))g(of)f(in\014nite)f(clusters)g(is)h(guaran)m(teed)165 2434 y(as)35 b(long)f(as)h Ft(p)p Fw(\()p Ft(\026)p Fw(\))d Ft(>)g(p)950 2448 y Fr(2)1023 2434 y Fw(\(resp.)j Ft(p)p Fw(\()p Ft(\026)p Fw(\))d Ft(<)g(p)1633 2448 y Fr(1)1672 2434 y Fw(\);)37 b(see)e(e.g.)h(Liggett,)g(Sc)m(honmann)e(and)g(Stacey) h([161)r(].)165 2547 y(Ho)m(w)m(ev)m(er,)d Ft(R)q Fw(-indep)s(endence) 27 b(rarely)h(holds)f(in)h(Gibbs)f(mo)s(dels.)39 b(F)-8 b(or)30 b(instance,)f(for)f(plus)f(phase)i(of)165 2660 y(the)j(Ising)e(mo)s(del)g(with)g(v)-5 b(anishing)29 b(external)j(\014eld)e(and)g(in)m(v)m(erse)i(temp)s(erature)f Ft(\014)h(>)26 b Fw(0,)33 b(the)e(spins)165 2772 y(at)40 b(an)m(y)f(t)m(w)m(o)i(v)m(ertices)e(are)h(alw)m(a)m(ys)f(strictly)f(p) s(ositiv)m(ely)f(correlated)i(no)g(matter)h(ho)m(w)f(far)g(apart)165 2885 y(they)31 b(are)g(\(although)f(the)g(correlation)g(do)s(es)g(tend) h(to)g(0)f(in)f(the)i(distance\).)306 2998 y(Ho)m(w)m(ev)m(er,)50 b(in)43 b(con)m(trast)j(to)f(what)f(w)m(e)g(just)g(sa)m(w)h(in)d(the)j (case)g(of)f(the)h(cubic)e(lattices,)48 b(the)165 3111 y(densit)m(y)28 b(do)s(es)h(pla)m(y)g(a)g(signi\014can)m(t)f(role)g (for)h(the)g(regular)f(trees)h Fx(T)2446 3126 y Fu(d)2487 3111 y Fw(.)40 b(T)-8 b(o)29 b(sho)m(w)g(this)f(w)m(e)h(consider)f(a) 165 3224 y(probabilit)m(y)d(measure)j Ft(\026)f Fw(on)g Fv(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)1404 3191 y Fq(L)1459 3224 y Fw(,)28 b(where)f(no)m(w)h Fv(L)d Fw(=)g Fx(T)2216 3239 y Fu(d)2283 3224 y Fw(with)i Ft(d)e Fv(\025)g Fw(2.)40 b(The)27 b(natural)g(analogue)165 3337 y(of)33 b(translation)e(in)m(v) -5 b(ariance)32 b(in)f(this)g(setting)h(is)g Fp(automorphism)38 b(invarianc)-5 b(e)33 b Fw(of)f Ft(\026)p Fw(,)h(whic)m(h)e(means)165 3450 y(that)e Ft(\026)e Fw(inherits)e(all)i(the)h(symmetries)e(of)i Fx(T)1695 3465 y Fu(d)1735 3450 y Fw(.)40 b(In)27 b(particular,)g(this) f(implies)f(that)j Ft(\026)p Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))27 b(=)e(1\))j(is)165 3563 y(indep)s(enden)m(t)j(of)i Ft(x)p Fw(,)h(so)f(that)g(the)g(densit)m(y)f Ft(p)p Fw(\()p Ft(\026)p Fw(\))h(is)f(w)m(ell-de\014ned.)46 b(As)33 b(opp)s(osed)f(to)i(the)f Fx(Z)3354 3530 y Fu(d)3427 3563 y Fw(case,)165 3676 y(ha)m(ving)25 b Ft(p)p Fw(\()p Ft(\026)p Fw(\))h(su\016cien)m(tly)e(close)i(to)g(1)g(no)m(w)f(do)s(es) g(guaran)m(tee)i(that)f(an)f(in\014nite)f(cluster)g(exists)h(with)165 3789 y(p)s(ositiv)m(e)f(probabilit)m(y)-8 b(.)37 b(This)23 b(is)g(also)i(true)f(in)g(the)h(b)s(ond)e(p)s(ercolation)h(case,)j (where)d Ft(p)p Fw(\()p Ft(\026)p Fw(\))h(is)e(de\014ned)165 3902 y(as)32 b(the)g(probabilit)m(y)d(that)j(a)g(giv)m(en)g(edge)g(is)f (op)s(en.)44 b(The)31 b(follo)m(wing)f(result)h(is)f(due)h(to)i(H\177) -45 b(aggstr\177)g(om)165 4014 y([118)r(].)165 4201 y Fx(Theorem)34 b(5.9)46 b Fp(F)-7 b(or)42 b(any)e(automorphism)k (invariant)d(site)f(p)-5 b(er)g(c)g(olation)44 b(mo)-5 b(del)41 b Ft(\026)f Fp(on)h Fx(T)3369 4216 y Fu(d)3449 4201 y Fp(with)165 4314 y(density)f Ft(p)p Fw(\()p Ft(\026)p Fw(\))c Fv(\025)807 4278 y Fu(d)p Fr(+1)p 807 4293 127 4 v 835 4346 a(2)p Fu(d)944 4314 y Fp(,)j(we)g(have)g Ft(\026)p Fw(\()p Fv(9)33 b Fw(an)d(in\014nite)e(op)s(en)i(cluster)o (\))37 b Ft(>)f Fw(0)p Fp(.)60 b(The)39 b(same)g(is)g(true)f(for)165 4427 y(b)-5 b(ond)34 b(p)-5 b(er)g(c)g(olation)36 b(on)d Fx(T)1041 4442 y Fu(d)1114 4427 y Fp(with)g(density)h Ft(p)p Fw(\()p Ft(\026)p Fw(\))25 b Fv(\025)1971 4391 y Fr(2)p 1925 4406 V 1925 4458 a Fu(d)p Fr(+1)2061 4427 y Fp(.)165 4630 y Fw(These)g(b)s(ounds)d(are)j(in)e(fact)i(sharp,)g(in) e(that)i(for)f(an)m(y)h Ft(p)g(<)2188 4594 y Fu(d)p Fr(+1)p 2188 4609 V 2216 4661 a(2)p Fu(d)2349 4630 y Fw(there)g(exists)f(some)h (automorphism)165 4742 y(in)m(v)-5 b(arian)m(t)21 b(probabilit)m(y)e (measure)i(on)g Fv(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)1676 4709 y Fn(T)1733 4721 y Fo(d)1797 4742 y Fw(with)20 b(densit)m(y)g Ft(p)p Fw(,)k(whic)m(h)c(do)s(es)h(not)g(allo)m(w)g(an)g(in\014nite)165 4855 y(cluster)33 b(with)f(probabilit)m(y)e(1,)35 b(and)d(similarly)e (for)j(the)g(case)i(of)e(b)s(ond)e(p)s(ercolation;)j(see)g([118)r(].)49 b(It)165 4968 y(follo)m(ws)23 b(from)h(Example)f(5.8)i(that)g(the)f (corresp)s(onding)e(threshold)g(for)i Fx(Z)2686 4935 y Fu(d)2750 4968 y Fw(is)f(trivial:)36 b(only)23 b(densit)m(y)165 5081 y(1)31 b(is)e(enough)h(to)h(rule)e(out)i(the)f(nonexistence)g(of)h (in\014nite)d(clusters.)39 b(The)30 b(in)m(tuitiv)m(e)f(reason)i(is)e (the)165 5194 y(follo)m(wing.)54 b(On)34 b Fx(Z)819 5161 y Fu(d)860 5194 y Fw(,)i(one)g(can)f(\014nd)f(\014nite)g(regions)h (\003)e Fv(\032)g(L)i Fw(with)f(arbitrarily)e(small)i(surface-to-)165 5307 y(v)m(olume)41 b(ratio,)k(whic)m(h)39 b(means)j(that)f(a)h(v)-5 b(ast)42 b(ma)5 b(jorit)m(y)41 b(of)g(sites)g(is)g(not)g(adjacen)m(t)h (to)g(a)g(v)m(ertex)165 5420 y(outside)35 b(\003;)j(w)m(e)e(can)f (simply)e(tak)m(e)k(\003)d(=)f(\003)1682 5434 y Fu(n)1763 5420 y Fw(=)g([)p Fv(\000)p Ft(n;)15 b(n)p Fw(])2138 5387 y Fu(d)2201 5420 y Fv(\\)24 b Fx(Z)2350 5387 y Fu(d)2425 5420 y Fw(with)34 b(large)i Ft(n)p Fw(;)h(this)d(prop)s(ert)m(y)h(of) 165 5533 y Fx(Z)229 5500 y Fu(d)306 5533 y Fw(is)h(kno)m(wn)h(as)g Fp(amenability)p Fw(.)61 b(Hence,)39 b(a)f(relativ)m(ely)e(small)f(n)m (um)m(b)s(er)g(of)i(closed)g(v)m(ertices)h(ma)m(y)165 5646 y(easily)h(\\surround")f(a)h(large)h(n)m(um)m(b)s(er)e(of)i(op)s (en)e(sites.)68 b(In)38 b(con)m(trast,)44 b(ev)m(ery)c(region)f(in)f Fx(T)3403 5661 y Fu(d)3482 5646 y Fw(has)1844 5895 y(33)p eop %%Page: 34 34 34 33 bop 165 304 a Fw(a)39 b(surface)f(of)g(the)g(same)g(order)g(of)g (magnitude)f(as)h(its)g(v)m(olume;)k(this)36 b(mak)m(es)j(it)f(imp)s (ossible)c(for)165 417 y(a)i(small)d(minorit)m(y)h(of)h(closed)g(v)m (ertices)h(to)g(surround)c(a)k(large)f(n)m(um)m(b)s(er)f(of)h(op)s(en)f (v)m(ertices.)56 b(This)165 530 y(in)m(tuition)24 b(can)i(b)s(e)f (turned)g(in)m(to)g(a)h(pro)s(of)f(using)g(the)h(so)g(called)f (mass-transp)s(ort)g(metho)s(d)g(sk)m(etc)m(hed)165 643 y(b)s(elo)m(w.)40 b(Benjamini,)29 b(Ly)m(ons,)h(P)m(eres)g(and)g(Sc)m (hramm)f([23)q(])h(ha)m(v)m(e)i(recen)m(tly)e(extended)g(this)f(metho)s (d)165 756 y(to)40 b(deriv)m(e)e(a)h(similar)d(dic)m(hotom)m(y)k(for)e (a)h(large)g(class)f(of)h(graphs,)i(including)35 b(Ca)m(yley)k(graphs)f (of)165 869 y(\014nitely)29 b(generated)i(groups.)165 1031 y Fx(Sk)m(etc)m(h)d(pro)s(of)f(of)h(Theorem)e(5.9:)37 b Fw(F)-8 b(or)25 b(simplicit)m(y)c(w)m(e)j(con\014ne)f(ourselv)m(es)g (to)i(the)e(case)i(of)f(b)s(ond)165 1144 y(p)s(ercolation)32 b(on)g Fx(T)841 1158 y Fr(2)880 1144 y Fw(.)46 b(W)-8 b(e)34 b(w)m(an)m(t)f(to)g(sho)m(w)f(that)h(if)e Ft(p)p Fw(\()p Ft(\026)p Fw(\))e Fv(\025)f Fw(2)p Ft(=)p Fw(3,)34 b(then)e(an)g(in\014nite)e(cluster)h(exists)165 1257 y(with)25 b(p)s(ositiv)m(e)g Ft(\026)p Fw(-probabilit)m(y)-8 b(.)37 b(Imagine)26 b(the)g(follo)m(wing)f(allo)s(cation)g(of)h(mass)g (to)h(the)f(edges)h(of)f Fx(T)3550 1271 y Fr(2)3589 1257 y Fw(.)165 1370 y(Originally)21 b(ev)m(ery)j(edge)g(receiv)m(es)h(mass) e(1.)39 b(Then)23 b(the)h(mass)f(is)g(redistributed,)f(or)i(transp)s (orted,)g(as)165 1483 y(follo)m(ws.)39 b(If)27 b(an)h(edge)g Ft(e)g Fw(is)f(op)s(en)g(and)g(is)g(con)m(tained)h(in)e(a)i(\014nite)e (op)s(en)h(cluster,)h(then)g(it)f(distributes)165 1596 y(all)41 b(its)f(mass)i(equally)e(among)i(those)g(closed)f(edges)h (that)g(are)g(adjacen)m(t)g(to)g(the)g(op)s(en)f(cluster)165 1709 y(con)m(taining)c Ft(e)p Fw(.)60 b(If)37 b Ft(e)g Fw(is)f(op)s(en)g(and)h(con)m(tained)g(in)e(an)i(in\014nite)e(op)s(en)h (cluster,)i(then)f(it)f(k)m(eeps)i(its)165 1822 y(mass.)70 b(Closed)39 b(edges,)44 b(\014nally)-8 b(,)41 b(k)m(eep)g(their)e(o)m (wn)h(mass)g(and)g(happily)d(accept)42 b(an)m(y)e(mass)h(that)165 1935 y(op)s(en)35 b(edges)g(decide)g(to)g(send)g(them.)54 b(The)35 b(exp)s(ected)g(mass)g(at)g(eac)m(h)i(edge)e(b)s(efore)g (transp)s(ort)f(is)165 2047 y(ob)m(viously)25 b(1,)j(and)e(one)g(can)h (sho)m(w)f(|)g(this)f(is)g(an)h(instance)g(of)h(the)f(mass-transp)s (ort)g(principle)d([23)q(])165 2160 y(|)33 b(that)g(the)g(exp)s(ected)g (mass)g(at)g(a)g(giv)m(en)g(edge)g(is)f(1)h(also)f(after)h(the)g (transp)s(ort.)47 b(Supp)s(ose)31 b(no)m(w,)165 2273 y(for)i(con)m(tradiction,)h(that)g Ft(p)p Fw(\()p Ft(\026)p Fw(\))c Fv(\025)g Fw(2)p Ft(=)p Fw(3)k(and)e(that)i(all)e(op)s(en)h (clusters)f(are)h(\014nite)g Ft(\026)p Fw(-a.s.)49 b(Then)32 b(all)165 2386 y(op)s(en)23 b(edges)h(ha)m(v)m(e)g(mass)f(0)h(after)f (transp)s(ort.)38 b(F)-8 b(urthermore,)25 b(since)d(eac)m(h)j(op)s(en)d (cluster)h(con)m(taining)165 2499 y(exactly)29 b Ft(n)e Fw(edges)h(has)g(exactly)g Ft(n)15 b Fw(+)g(3)27 b(adjacen)m(t)i (closed)f(edges)g(\(as)g(is)f(easily)g(sho)m(wn)g(b)m(y)g(induction)165 2612 y(|)33 b(it)e(is)h(here)g(that)h(the)f(tree)h(structure)f(is)g (used\),)g(the)h(mass)f(after)h(transp)s(ort)e(at)j(a)e(closed)g(edge) 165 2725 y(adjacen)m(t)g(to)f(t)m(w)m(o)h(op)s(en)e(clusters)f(of)i (sizes)f Ft(n)1728 2739 y Fr(1)1797 2725 y Fw(and)g Ft(n)2029 2739 y Fr(2)2098 2725 y Fw(has)g(mass)1402 2943 y(1)21 b(+)1647 2881 y Ft(n)1702 2895 y Fr(1)p 1569 2922 251 4 v 1569 3005 a Ft(n)1624 3019 y Fr(1)1683 3005 y Fw(+)f(3)1849 2943 y(+)2029 2881 y Ft(n)2084 2895 y Fr(2)p 1950 2922 V 1950 3005 a Ft(n)2005 3019 y Fr(2)2064 3005 y Fw(+)g(3)2236 2943 y Ft(<)25 b Fw(3)165 3182 y(Hence)32 b(the)e(exp)s(ected)h(mass)f (after)h(transp)s(ort)f(at)h(a)f(giv)m(en)h(edge)g Ft(e)g Fw(is)e(strictly)g(less)h(than)1237 3386 y(3)15 b Ft(\026)p Fw(\()p Ft(X)7 b Fw(\()p Ft(e)p Fw(\))28 b(=)d(0\))h(=)f(3\(1)c Fv(\000)f Ft(p)p Fw(\()p Ft(\026)p Fw(\)\))26 b Fv(\024)f Fw(1)p Ft(;)165 3590 y Fw(con)m(tradicting)31 b(the)f(mass-transp)s (ort)g(principle.)1673 b Fk(2)165 3834 y FA(5.3)113 b(Examples)38 b(of)g(dep)s(enden)m(t)h(p)s(ercolation)165 4005 y Fw(F)-8 b(rom)43 b(the)g(previous)e(subsection)h(the)g(reader)h(migh)m(t)f(get) i(a)e(rather)h(p)s(essimistic)d(view)h(of)i(the)165 4118 y(p)s(ossibilities)34 b(of)39 b(establishing)d(existence)k(\(or)f (non-existence\))g(of)g(in\014nite)d(clusters)i(for)g(dep)s(en-)165 4231 y(den)m(t)c(p)s(ercolation)f(mo)s(dels)g(on)g Fx(Z)1352 4198 y Fu(d)1393 4231 y Fw(.)50 b(This)32 b(is)h(certainly)g Fp(not)43 b Fw(the)34 b(case,)i(and)d(a)h(lot)g(can)g(b)s(e)f(done.)165 4344 y(One)e(standard)f(w)m(a)m(y)i(of)f(determining)e(the)i(p)s (ercolation)f(b)s(eha)m(vior)g(of)h(a)g(dep)s(enden)m(t)f(mo)s(del)g (is)g(b)m(y)165 4457 y Fp(sto)-5 b(chastic)35 b(c)-5 b(omp)g(arison)37 b(with)d(a)f(suitable)g(Bernoul)5 b(li)34 b(p)-5 b(er)g(c)g(olation)36 b(mo)-5 b(del)9 b Fw(:)44 b(F)-8 b(or)31 b(the)h(existence)f(of)165 4570 y(in\014nite)25 b(clusters,)j(it)e(is)h(su\016cien)m(t)f(to)i(sho)m(w)f(that)h(the)g (giv)m(en)f(dep)s(enden)m(t)f(mo)s(del)g(is)g(sto)s(c)m(hastically)165 4683 y(larger)31 b(than)f(the)h(Bernoulli)d(mo)s(del)i(for)g(some)i (parameter)f Ft(p)25 b(>)h(p)2474 4697 y Fu(c)2508 4683 y Fw(,)31 b(and)f(the)h(absence)h(of)e(in\014nite)165 4796 y(clusters)37 b(will)d(follo)m(w)i(if)g(the)h(mo)s(del)f(at)i (hand)e(is)g(sto)s(c)m(hastically)g(dominated)g(b)m(y)h(the)h (Bernoulli)165 4908 y(mo)s(del)30 b(for)g(some)g Ft(p)25 b(<)g(p)1015 4922 y Fu(c)1050 4908 y Fw(.)41 b(Let)30 b(us)g(demonstrate)h(this)e(tec)m(hnique)h(for)h(the)f(Ising)f(mo)s (del)h(on)g Fx(Z)3479 4875 y Fu(d)3519 4908 y Fw(.)306 5021 y(Consider)h(p)s(ercolation)g(of)i(plus)d(spins)g(in)h(the)i(plus) d(measure)i Ft(\026)2564 4983 y Fr(+)2564 5049 y Fu(h;\014)2672 5021 y Fw(,)h(de\014ned)e(in)g(Section)h(4.3.)165 5149 y(If)c(w)m(e)h(k)m(eep)g Ft(\014)k Fw(\014xed)28 b(then)f(Prop)s (osition)g(4.16)i(tells)f(us)f(that)i Ft(\026)2329 5111 y Fr(+)2329 5177 y Fu(h;\014)2464 5149 y Fw(is)f(sto)s(c)m(hastically)g (increasing)e(in)165 5262 y Ft(h)p Fw(.)41 b(Consequen)m(tly)-8 b(,)28 b(b)s(oth)g(the)h(probabilit)m(y)c(of)k(ha)m(ving)f(an)g (in\014nite)e(cluster)i(of)g(plus)f(spins,)g(as)h(w)m(ell)165 5375 y(as)k(the)f(probabilit)m(y)e(that)j(a)f(giv)m(en)h(v)m(ertex)g (is)e(in)g(suc)m(h)h(an)g(in\014nite)e(cluster,)i(are)h(increasing)e (in)g Ft(h)p Fw(.)165 5488 y(F)-8 b(urthermore,)37 b(as)f Fx(Z)905 5455 y Fu(d)981 5488 y Fw(is)e(of)i(b)s(ounded)d(degree)j Ft(N)44 b Fw(=)33 b(2)p Ft(d)p Fw(,)38 b(the)d(remarks)g(after)h(the)g (same)g(prop)s(o-)165 5601 y(sition)c(imply)e(that,)k(for)f(an)m(y)g (giv)m(en)g Ft(p)c Fv(2)g Fw(\(0)p Ft(;)15 b Fw(1\))34 b(and)e Ft(\014)5 b Fw(,)34 b(the)f(Ising)f(measure)g Ft(\026)2940 5562 y Fr(+)2940 5629 y Fu(h;\014)3080 5601 y Fw(sto)s(c)m(hastically)1844 5895 y(34)p eop %%Page: 35 35 35 34 bop 165 304 a Fw(dominates)25 b(the)h(Bernoulli)d(measure)i Ft( )1543 318 y Fu(p)1608 304 y Fw(when)g Ft(h)h Fw(is)e(large)i (enough,)g(and)f(is)f(dominated)h(b)m(y)g Ft( )3440 318 y Fu(p)3505 304 y Fw(for)165 417 y Ft(h)j Fw(b)s(elo)m(w)f(some)i(b)s (ound.)38 b(W)-8 b(e)29 b(ma)m(y)f(com)m(bine)f(this)g(observ)-5 b(ation)28 b(with)e(Prop)s(osition)g(5.1)j(to)f(deduce)165 530 y(the)j(follo)m(wing)e(critical)g(phenomenon:)165 694 y Fx(Theorem)34 b(5.10)47 b Fp(F)-7 b(or)29 b(the)f(Ising)g(mo)-5 b(del)30 b(on)e Fx(Z)1818 661 y Fu(d)1859 694 y Fp(,)h Ft(d)c Fv(\025)g Fw(2)p Fp(,)k(at)g(a)f(\014xe)-5 b(d)29 b(temp)-5 b(er)g(atur)g(e)31 b Ft(\014)5 b Fp(,)29 b(ther)-5 b(e)29 b(exists)165 807 y(a)k(critic)-5 b(al)34 b(value)e Ft(h)836 821 y Fu(c)897 807 y Fv(2)25 b Fx(R)33 b Fp(\(dep)-5 b(ending)34 b(on)f Ft(d)g Fp(and)g Ft(\014)5 b Fp(\))33 b(for)g(the)g(external)h(\014eld,)f(such)g(that)724 1048 y Ft(\026)779 1010 y Fr(+)779 1076 y Fu(h;\014)886 1048 y Fw(\()p Fv(9)f Fw(an)f(in\014nite)d(cluster)h(of)i(plus)d(spins)n(\)) e(=)2383 904 y Fs(\()2491 992 y Fw(0)84 b(if)29 b Ft(h)d(<)f(h)2929 1006 y Fu(c)2491 1105 y Fw(1)84 b(if)29 b Ft(h)d(>)f(h)2929 1119 y Fu(c)2979 1105 y Ft(:)165 1285 y Fw(As)32 b(w)m(e)g(shall)d(see) j(later)g(in)e(Theorem)h(8.2,)i(w)m(e)f(ha)m(v)m(e)h Ft(h)2074 1299 y Fu(c)2136 1285 y Fw(=)26 b(0)32 b(when)f Ft(d)c Fv(\025)g Fw(2)k(and)g Ft(\014)h(>)27 b(\014)3207 1299 y Fu(c)3242 1285 y Fw(.)44 b(Higuc)m(hi)165 1398 y([133)r(])32 b(has)f(sho)m(wn)g(that)h(the)g(p)s(ercolation)f (transition)f(at)i Ft(h)2234 1412 y Fu(c)2301 1398 y Fw(is)f(sharp,)g(in)f(that)i(the)g(connectivit)m(y)165 1511 y(function)41 b(deca)m(ys)j(exp)s(onen)m(tially)c(when)h Ft(h)46 b(<)e(h)1920 1525 y Fu(c)1956 1511 y Fw(,)h(and)d(that)g(the)h (p)s(ercolation)e(probabilit)m(y)f(is)165 1623 y(con)m(tin)m(uous)21 b(in)f(\()p Ft(\014)5 b(;)15 b(h)p Fw(\),)25 b(except)e(on)e(the)h (critical)e(half-line)f Ft(h)25 b Fw(=)g(0)p Ft(;)31 b(\014)g(>)25 b(\014)2657 1637 y Fu(c)2692 1623 y Fw(.)38 b(In)20 b(Section)h(6)h(b)s(elo)m(w,)h(w)m(e)165 1736 y(will)d(mak)m(e)j(a)g(similar)c(use)j(of)g(sto)s(c)m(hastic)h (comparison)f(argumen)m(ts)g(for)g(random-cluster)f(measures,)165 1849 y(cf.)32 b(Prop)s(osition)d(6.11.)44 b(The)31 b(sto)s(c)m(hastic)h (domination)d(approac)m(h)j(w)m(orks)f(also)g(in)f(the)h(framew)m(ork) 165 1962 y(of)g(lattice)g(gases)g(with)e(attractiv)m(e)j(p)s(oten)m (tial;)f(see)g(Leb)s(o)m(witz)f(and)f(P)m(enrose)i([156)r(].)306 2116 y(In)39 b(the)h(rest)g(of)g(this)f(subsection)f(w)m(e)j(shall)d (giv)m(e)i(some)g(examples)f(of)h(strongly)f(dep)s(enden)m(t)165 2228 y(systems)28 b(where)f(other)h(approac)m(hes)h(to)f(the)g (question)f(of)h(p)s(ercolation)f(are)h(needed.)40 b(T)m(ypically)25 b(in)165 2341 y(these)j(examples,)g(the)f(probabilit)m(y)e(that)j(all)e (v)m(ertices)i(in)d(a)j(\014nite)e(region)h(\003)g(are)h(op)s(en)e (\(or)i(closed\))165 2454 y(fails)g(to)h(deca)m(y)h(exp)s(onen)m (tially)d(in)h(the)h(v)m(olume)f(of)h(\003,)h(and)e(as)h(a)g (consequence,)h(the)f(random)g(\014eld)165 2567 y(neither)h(dominates)g (nor)f(is)h(sto)s(c)m(hastically)g(dominated)f(b)m(y)i(an)m(y)f(non)m (trivial)f(Bernoulli)f(mo)s(del.)306 2680 y(The)h(geometry)i(of)f(lev)m (el)f(heigh)m(ts)g(of)h(a)g(random)e(\014eld)g(forms)h(an)g(imp)s (ortan)m(t)g(ob)5 b(ject)30 b(of)g(study)165 2793 y(b)s(oth)24 b(from)g(the)g(theoretical)h(and)f(the)g(applied)e(side.)38 b(F)-8 b(or)25 b(example,)h(it)d(relates)i(to)g(the)g(presence)f(of)165 2906 y(hills)h(and)j(v)-5 b(alleys)27 b(on)h(a)g(rough)g(surface,)g(or) g(to)h(the)f(random)g(lo)s(cation)f(of)h(p)s(oten)m(tial)g(barriers)e (in)h(a)165 3019 y(dop)s(ed)h(semi-conductor.)40 b(T)-8 b(o)29 b(\014x)f(the)h(ideas)f(w)m(e)h(consider)f(a)h(random)f(\014eld) f Ft(X)33 b Fw(=)25 b(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))p Ft(;)15 b(x)27 b Fv(2)d Fx(Z)3538 2986 y Fu(d)3579 3019 y Fw(\))165 3132 y(with)36 b(v)-5 b(alues)36 b Ft(X)7 b Fw(\()p Ft(x)p Fw(\))38 b Fv(2)e Ft(S)41 b Fv(\032)36 b Fx(R)h Fw(whic)m(h)f(are)i(not)f(necessarily)f(discrete.)60 b(It)37 b(is)f(often)i(in)m(teresting)165 3245 y(to)44 b(divide)c Ft(S)48 b Fw(in)m(to)42 b(t)m(w)m(o)i(parts)f Ft(S)1357 3259 y Fr(1)1438 3245 y Fw(and)g Ft(S)1684 3259 y Fr(0)1765 3245 y Fw(and)f(to)i(de\014ne)e(a)h(new)f(discrete)g (random)g(\014eld)f Ft(Y)165 3358 y Fw(via)36 b Ft(Y)20 b Fw(\()p Ft(x)p Fw(\))35 b(=)e Ft(I)694 3376 y Fq(f)p Fu(X)5 b Fr(\()p Fu(x)p Fr(\))p Fq(2)p Fu(S)976 3385 y Fl(1)1012 3376 y Fq(g)1052 3358 y Fw(.)56 b(F)-8 b(or)37 b Ft(S)i Fw(=)34 b Fx(R)i Fw(one)g(t)m(ypically)f(considers)f Ft(S)2616 3372 y Fr(1)2689 3358 y Fw(=)g([)p Ft(`;)15 b Fv(1)p Fw(\))37 b(for)e(some)h(lev)m(el)165 3470 y Ft(`)g Fv(2)g Fx(R)p Fw(.)60 b(In)36 b(this)f(w)m(a)m(y)j(w)m(e)f (obtain)f(a)i(coarse-grained)f(description)e(of)i(a)g(system)g(of)f (con)m(tin)m(uous)165 3583 y(spins.)58 b(One)36 b(question)g(is)g(to)h (whic)m(h)f(exten)m(t)i(one)f(can)g(reconstruct)g(the)g(complete)g (image)g(from)165 3696 y(this)f(information.)58 b(W)-8 b(e)38 b(consider)e(here)g(a)i(di\013eren)m(t)e(question:)52 b(what)37 b(is)f(the)h(geometry)h(of)f(the)165 3809 y(random)f(set)i Fv(f)p Ft(x)e Fv(2)f Fx(Z)947 3776 y Fu(d)1023 3809 y Fw(:)i Ft(Y)20 b Fw(\()p Ft(x)p Fw(\))36 b(=)g(1)p Fv(g)p Fw(?)60 b(This)35 b(set)i(is)f(called)g(the)h(excursion)e(or)i (exceedance)i(set)165 3922 y(when)30 b(it)f(corresp)s(onds,)h(as)g(in)f (the)i(example)f(ab)s(o)m(v)m(e,)h(to)g(the)g(set)g(on)f(whic)m(h)f (the)h(original)f(random)165 4035 y(\014eld)g(exceeds)j(a)e(giv)m(en)h (lev)m(el.)40 b(F)-8 b(or)31 b(a)g(recen)m(t)g(review)f(of)h(this)e (sub)5 b(ject)30 b(w)m(e)h(refer)f(to)h([2)q(].)306 4148 y(W)-8 b(e)30 b(no)m(w)f(giv)m(e)g(four)f(examples)h(of)f(equilibrium)c (systems)29 b(with)f(con)m(tin)m(uous)g(spins)f(where)h(one)165 4261 y(can)34 b(sho)m(w)g(\(the)g(absence)g(of)7 b(\))34 b(p)s(ercolation)e(of)i(an)g(excursion)e(\(exceedance\))k(set.)51 b(Here)34 b(w)m(e)g(only)165 4374 y(state)c(the)f(results.)39 b(Some)29 b(hin)m(ts)e(on)i(the)g(pro)s(ofs)e(will)f(b)s(e)i(giv)m(en)h (later)f(in)g(Section)g(8)h(via)f(Theorem)165 4487 y(8.1.)42 b(Details)30 b(can)h(b)s(e)f(found)f(in)g(the)i(pap)s(er)e(b)m(y)h (Bricmon)m(t,)h(Leb)s(o)m(witz)f(and)g(Maes)h([36)q(].)165 4651 y Fx(Example)j(5.11)46 b Fw(Consider)30 b(a)h(general)g(mo)s(del)f (of)i(real-v)-5 b(alued)30 b(spins)f(\()p Ft(\033)s Fw(\()p Ft(x)p Fw(\))p Ft(;)j(x)26 b Fv(2)g Fx(Z)3165 4618 y Fu(d)3206 4651 y Fw(\))31 b(with)f(fer-)165 4764 y(romagnetic)h (nearest-neigh)m(b)s(or)f(in)m(teraction.)41 b(The)30 b(formal)f(Hamiltonian)g(is)g(giv)m(en)i(b)m(y)1412 4940 y Ft(H)7 b Fw(\()p Ft(\033)s Fw(\))26 b(=)f Fv(\000)1834 4859 y Fs(X)1828 5033 y Fu(x)p Fq(\030)p Fu(y)1975 4940 y Ft(\033)s Fw(\()p Ft(x)p Fw(\))p Ft(\033)s Fw(\()p Ft(y)s Fw(\))15 b Ft(:)1088 b Fw(\(21\))165 5194 y(The)28 b(reference)h(\(or)g(single-spin\))d(measure)i Ft(\025)d Fv(6)p Fw(=)g Ft(\016)1920 5208 y Fr(0)1989 5194 y Fw(on)j Fx(R)h Fw(is)e(assumed)h(to)h(b)s(e)f(ev)m(en)h(and)f(to)h(deca)m(y)165 5307 y(fast)39 b(enough)f(at)h Fv(\0061)f Fw(so)h(that)g(the)f(mo)s (del)f(is)h(w)m(ell)f(de\014ned.)63 b(Then,)40 b(for)e(an)m(y)g(Gibbs)f (measure)165 5420 y Ft(\026)g Fw(relativ)m(e)g(to)h(\(21\))g(with)e Ft(\026)p Fw(\(sgn\()p Ft(\033)s Fw(\(0\)\)\))j Ft(>)d Fw(0,)j(there)e(will)d(b)s(e)j(p)s(ercolation)f(of)h(all)f(sites)h Ft(x)f Fw(with)165 5533 y Ft(\033)s Fw(\()p Ft(x)p Fw(\))45 b Fv(\025)e Fw(0.)74 b(Suc)m(h)40 b(Gibbs)g(measures)h(alw)m(a)m(ys)h (exist)f(at)h(su\016cien)m(tly)e(lo)m(w)h(temp)s(eratures)g(when)165 5646 y Ft(d)26 b Fv(\025)f Fw(2.)1844 5895 y(35)p eop %%Page: 36 36 36 35 bop 165 304 a Fx(Example)34 b(5.12)46 b Fw(Consider)29 b(again)i(a)h(spin)d(system)i(\()p Ft(\033)s Fw(\()p Ft(x)p Fw(\))p Ft(;)h(x)26 b Fv(2)g Fx(Z)2521 271 y Fu(d)2561 304 y Fw(\),)32 b(where)e(no)m(w)h(the)g(Hamilto-)165 417 y(nian)e(has)h(the)h(`massless')f(form)1310 621 y Ft(H)7 b Fw(\()p Ft(\033)s Fw(\))26 b(=)f Fv(\000)1732 540 y Fs(X)1726 715 y Fu(x)p Fq(\030)p Fu(y)1873 621 y Ft( )s Fw(\()p Ft(\033)s Fw(\()p Ft(x)p Fw(\))d Fv(\000)e Ft(\033)s Fw(\()p Ft(y)s Fw(\)\))165 904 y(with)26 b Ft(\033)s Fw(\()p Ft(x)p Fw(\))h Fv(2)d Fx(R)k Fw(or)g Fx(Z)f Fw(and)g Ft( )j Fw(an)e(ev)m(en)g(con)m(v)m(ex)h(function.)38 b(The)27 b(single-spin)e(measure)i Ft(\025)g Fw(is)g(either)165 1017 y(Leb)s(esgue)33 b(measure)h(on)f Fx(R)h Fw(or)f(coun)m(ting)g (measure)g(on)h Fx(Z)p Fw(.)49 b(The)33 b(case)h Ft(\033)s Fw(\()p Ft(x)p Fw(\))d Fv(2)f Fx(Z)j Fw(and)g Ft( )s Fw(\()p Ft(t)p Fw(\))e(=)f Fv(j)p Ft(t)p Fv(j)165 1130 y Fw(corresp)s(onds)37 b(to)h(the)g(so-called)f(solid-on-solid)e (\(SOS\))i(mo)s(del)f(of)i(a)g Ft(d)p Fw(-dimensional)d(surface)i(in) 165 1242 y Fx(Z)229 1209 y Fu(d)p Fr(+1)360 1242 y Fw(;)32 b(the)f(c)m(hoice)h Ft(\033)s Fw(\()p Ft(x)p Fw(\))27 b Fv(2)f Fx(R)32 b Fw(and)e Ft( )s Fw(\()p Ft(t)p Fw(\))d(=)g Ft(t)1746 1209 y Fr(2)1816 1242 y Fw(giv)m(es)k(the)h(harmonic)e (crystal.)42 b(Let)32 b Ft(\026)f Fw(b)s(e)f(a)i(Gibbs)165 1355 y(measure)f(whic)m(h)f(is)f(obtained)i(as)g(in\014nite)d(v)m (olume)j(limit)d(of)j(\014nite)f(v)m(olume)h(Gibbs)e(distributions)165 1468 y(with)g(zero)h(b)s(oundary)d(condition.)39 b(\(In)30 b(the)f(con)m(tin)m(uous-spin)f(case,)j(suc)m(h)e(Gibbs)f(measures)h (exist)165 1581 y(for)34 b(an)m(y)g(temp)s(erature)g(when)e Ft(d)g Fv(\025)e Fw(3)k(and)g Ft( )s Fw(\()p Ft(t)p Fw(\))e(=)e Ft(\013t)2074 1548 y Fr(2)2136 1581 y Fw(+)23 b Ft(\036)p Fw(\()p Ft(t)p Fw(\),)35 b(where)e Ft(\013)f(>)e Fw(0)35 b(and)e Ft(\036)h Fw(is)e(con)m(v)m(ex)165 1694 y([33)r(].\))41 b(Then,)29 b(for)i(an)m(y)f Ft(`)25 b(<)g Fw(0,)31 b(there)g(is)e(p)s (ercolation)h(of)g(the)h(sites)f Ft(x)25 b Fv(2)g Fx(Z)2728 1661 y Fu(d)2799 1694 y Fw(with)k Ft(\033)s Fw(\()p Ft(x)p Fw(\))d Fv(\025)f Ft(`)30 b Fw(.)165 1907 y Fx(Example)k(5.13)46 b Fw(Consider)38 b(next)h(a)h(mo)s(del)e(of)h(t)m(w)m(o-comp)s(onen)m (t)j(spins)37 b Ft(\033)s Fw(\()p Ft(x)p Fw(\))k Fv(2)e Fx(R)3185 1874 y Fr(2)3225 1907 y Fw(,)j Ft(x)e Fv(2)g Fx(Z)3549 1874 y Fu(d)3589 1907 y Fw(,)165 2020 y Ft(\033)s Fw(\()p Ft(x)p Fw(\))26 b(=)f(\()p Ft(r)540 2034 y Fu(x)600 2020 y Fw(cos)16 b Ft(\036)791 2034 y Fu(x)835 2020 y Ft(;)f(r)916 2034 y Fu(x)975 2020 y Fw(sin)f Ft(\036)1156 2034 y Fu(x)1200 2020 y Fw(\),)31 b(with)e(formal)h(Hamiltonian)1400 2224 y Ft(H)7 b Fw(\()p Ft(\033)s Fw(\))26 b(=)f Fv(\000)1822 2143 y Fs(X)1816 2317 y Fu(x)p Fq(\030)p Fu(y)1963 2224 y Ft(\033)s Fw(\()p Ft(x)p Fw(\))c Fv(\001)f Ft(\033)s Fw(\()p Ft(y)s Fw(\))165 2514 y(and)34 b(some)h(rotation-in)m(v)-5 b(arian)m(t)35 b(and)f(suitably)f(deca)m(ying)i(reference)g(measure)f Ft(\025)h Fw(on)g Fx(R)3264 2481 y Fr(2)3303 2514 y Fw(.)54 b(Then,)165 2627 y(for)37 b(an)m(y)g(Gibbs)d(measure)j Ft(\026)f Fw(with)f Ft(\026)p Fw(\(cos)16 b Ft(\036)1704 2641 y Fr(0)1744 2627 y Fw(\))36 b Ft(>)f Fw(0,)k(there)d(is)g(p)s (ercolation)g(of)g(the)h(sites)f Ft(x)g Fv(2)f Fx(Z)3574 2594 y Fu(d)165 2740 y Fw(with)29 b(cos)16 b Ft(\036)563 2754 y Fu(x)633 2740 y Fv(\025)25 b Fw(0.)41 b(Suc)m(h)29 b(Gibbs)g(measures)h(exist)g(at)i(lo)m(w)e(temp)s(eratures)g(if)f Ft(d)d Fv(\025)f Fw(3.)165 2952 y Fx(Example)34 b(5.14)46 b Fw(Consider)35 b(again)h(the)g(massless)f(harmonic)g(crystal)h(of)g (Example)f(5.12)j(ab)s(o)m(v)m(e)165 3065 y(\(with)29 b Ft( )s Fw(\()p Ft(t)p Fw(\))d(=)f Ft(t)727 3032 y Fr(2)766 3065 y Fw(\))30 b(in)e Ft(d)d Fw(=)g(3)30 b(dimensions.)37 b(There)29 b(exists)g(a)h(v)-5 b(alue)28 b Ft(`)2537 3079 y Fu(c)2597 3065 y Ft(<)d Fv(1)k Fw(so)h(that)f(for)g(all)f Ft(`)e Fv(\025)f Ft(`)3580 3079 y Fu(c)165 3178 y Fw(there)31 b(is)e Fp(no)i Fw(p)s(ercolation)f(of)g(sites)g Ft(x)25 b Fv(2)g Fx(Z)1623 3145 y Fu(d)1694 3178 y Fw(with)k Ft(\033)s Fw(\()p Ft(x)p Fw(\))d Fv(\025)f Ft(`)p Fw(.)306 3391 y(Finally)-8 b(,)26 b(w)m(e)h(giv)m(e)f(an)g(example)g(of)h(a)f (strongly)g(correlated)g(system,)h(sharing)e(some)i(prop)s(erties)165 3504 y(with)40 b(the)i(harmonic)e(crystal)h(of)g(Example)f(5.14,)46 b(where)40 b(at)i(presen)m(t)f(there)h(is)e(no)h(pro)s(of)f(of)i(a)165 3616 y(p)s(ercolation)e(transition.)71 b(The)41 b(mo)s(del)e(is)h(one)i (of)f(the)g(simplest)e(examples)h(of)h(an)g(in)m(teracting)165 3729 y(particle)j(system.)84 b(What)45 b(mak)m(es)g(the)g(problem)e (di\016cult)g(is)g(that)i(the)g(random)f(\014eld)f(is)h(not)165 3842 y(Mark)m(o)m(v)36 b(\(not)f(ev)m(en)g(Gibbsian\))d(and)i(not)h (explicitly)c(describ)s(ed)i(in)f(terms)j(of)f(a)h(family)d(of)j(lo)s (cal)165 3955 y(conditional)29 b(distributions.)165 4168 y Fx(Example)34 b(5.15)46 b Fw(The)27 b(v)m(oter)i(mo)s(del)d(is)g(a)i (sto)s(c)m(hastic)g(dynamics)e(in)g(whic)m(h)g(individuals)c(\(v)m (oters\))165 4281 y(sitting)k(at)i(the)f(v)m(ertices)g(of)g(a)g(graph)g (up)s(date)f(their)g(p)s(osition)e(\(y)m(es/no\))29 b(b)m(y)e(randomly) e(selecting)i(a)165 4393 y(neigh)m(b)s(oring)19 b(v)m(ertex)k(and)e (adopting)f(its)h(p)s(osition,)h(see)f(Liggett)i([160)q(])f(for)f(an)g (in)m(tro)s(duction.)36 b(Using)165 4506 y(spin)g(language)i(and)f (putting)g(ourselv)m(es)h(on)f Fx(Z)1856 4473 y Fr(3)1896 4506 y Fw(,)i(the)f(time)g(ev)m(olution)f(of)h(this)f(v)m(oter)h(mo)s (del)f(is)165 4619 y(sp)s(eci\014ed)21 b(b)m(y)i(giving)e(the)i(rate)g Ft(c)p Fw(\()p Ft(x;)15 b(\033)s Fw(\))24 b(for)e(a)h(spin)e(\015ip)f (at)k(the)e(site)h Ft(x)f Fw(when)f(the)i(spin)e(con\014guration)165 4732 y(is)30 b Ft(\033)e Fv(2)d(f)p Fw(+1)p Ft(;)15 b Fv(\000)p Fw(1)p Fv(g)785 4699 y Fn(Z)835 4676 y Fl(3)875 4732 y Ft(;)1258 4883 y(c)p Fw(\()p Ft(x;)g(\033)s Fw(\))27 b(=)1646 4822 y(1)p 1646 4862 46 4 v 1646 4946 a(6)1723 4803 y Fs(X)1717 4977 y Fu(y)r Fq(\030)p Fu(x)1864 4789 y Fs(\020)1914 4883 y Fw(1)20 b Fv(\000)g Ft(\033)s Fw(\()p Ft(x)p Fw(\))p Ft(\033)s Fw(\()p Ft(y)s Fw(\))2420 4789 y Fs(\021)2497 4883 y Ft(:)165 5123 y Fw(There)i(is)f(a)h (one-parameter)h(family)e(of)h(extremal)g(in)m(v)-5 b(arian)m(t)21 b(measures)h Ft(\026)2706 5137 y Fu(p)2767 5123 y Fw(eac)m(h)h (obtained)e(asymp-)165 5236 y(totically)33 b(\(in)g(time\))g(from)g (taking)h(the)f(Bernoulli)e(measure)j Ft( )2388 5250 y Fu(p)2461 5236 y Fw(with)e(densit)m(y)h Ft(p)g Fw(as)h(initial)d (con-)165 5349 y(dition.)49 b(These)33 b(stationary)h(states)h Ft(\026)1492 5363 y Fu(p)1565 5349 y Fw(are)f(strongly)f(correlating.) 50 b(The)33 b(spin-spin)e(correlations)165 5462 y(deca)m(y)36 b(as)e(the)h(in)m(v)m(erse)f(1)p Ft(=r)k Fw(of)c(the)h(spin-distance)d Ft(r)37 b Fw(on)d Fx(Z)2262 5429 y Fr(3)2301 5462 y Fw(.)53 b(It)34 b(is)f(an)i(op)s(en)e(question)h(whether)165 5575 y(for)29 b Ft(p)f Fw(su\016cien)m(tly)g(close)h(to)g(1)h(the)f (plus)d(spins)h(p)s(ercolate,)j(and)e(whether)g(for)h(su\016cien)m(tly) e(small)g Ft(p)1844 5895 y Fw(36)p eop %%Page: 37 37 37 36 bop 165 304 a Fw(there)29 b(is)e(no)i(p)s(ercolation.)39 b(Sim)m(ulations)26 b(b)m(y)i(Leb)s(o)m(witz)g(and)g(Saleur)f([157)r(]) h(indicate)g(that)g(there)h(is)165 417 y(indeed)g(a)i(non-triv)-5 b(al)29 b(p)s(ercolation)g(transition)g(with)g(critical)h(v)-5 b(alue)29 b Ft(p)2624 431 y Fu(c)2684 417 y Fv(\031)c Fw(0)p Ft(:)p Fw(16.)306 530 y(The)i(same)h(problem)e(ma)m(y)i(b)s(e)e (considered)h(for)g Ft(d)e Fv(\025)g Fw(4.)40 b(F)-8 b(or)28 b Ft(d)e Fw(=)f(1)p Ft(;)31 b Fw(2,)e(ho)m(w)m(ev)m(er,)g(the)f (problem)165 643 y(is)38 b(not)i(in)m(teresting)e(b)s(ecause)h(in)f (these)h(cases)h Ft(\026)1885 657 y Fu(p)1964 643 y Fw(is)e(kno)m(wn)g (to)i(put)e(mass)h Ft(p)g Fw(on)g(the)g(\\all)g(+1")165 756 y(con\014guration)g(and)g(mass)g(1)27 b Fv(\000)f Ft(p)39 b Fw(on)g(the)g(\\all)g Fv(\000)p Fw(1")h(con\014guration.)67 b(Alternativ)m(ely)-8 b(,)42 b(one)e(ma)m(y)165 869 y(consider)31 b(the)h(same)g(mo)s(del)e(with)g Fx(Z)1451 836 y Fr(3)1522 869 y Fw(replaced)h(b)m(y)g Fx(T)2082 884 y Fu(d)2123 869 y Fw(;)h(Theorem)f(5.9)i(can)f(then)f(b)s(e)g(applied)e(to)165 981 y(sho)m(w)i(that)g(the)f(plus)f(spins)f(do)i(p)s(ercolate)h(for)f Ft(p)25 b Fv(\025)1999 946 y Fu(d)p Fr(+1)p 1999 961 127 4 v 2026 1013 a(2)p Fu(d)2135 981 y Fw(.)165 1225 y FA(5.4)113 b(The)37 b(n)m(um)m(b)s(er)i(of)e(in\014nite)i(clusters) 165 1397 y Fw(Once)c(in\014nite)d(clusters)h(ha)m(v)m(e)j(b)s(een)d (sho)m(wn)h(to)h(exist)f(with)f(p)s(ositiv)m(e)g(probabilit)m(y)f(in)h (some)i(p)s(er-)165 1509 y(colation)h(mo)s(del,)f(the)g(next)g(natural) g(question)f(is:)49 b Fp(How)38 b(many)f(in\014nite)g(clusters)h(c)-5 b(an)37 b(exist)h(si-)165 1622 y(multane)-5 b(ously?)71 b Fw(F)-8 b(or)41 b(Bernoulli)c(site)i(or)h(b)s(ond)e(p)s(ercolation)h (on)h Fx(Z)2584 1589 y Fu(d)2624 1622 y Fw(,)i(Aizenman,)g(Kesten)e (and)165 1735 y(Newman)34 b([12)q(])f(obtained)g(the)h(follo)m(wing,)f (no)m(w)g(classical)g(uniqueness)e(result:)45 b(with)33 b(probabilit)m(y)165 1848 y(1,)28 b(there)f(exists)g(at)g(most)g(one)g (in\014nite)e(cluster.)38 b(Simpler)24 b(pro)s(ofs)i(w)m(ere)h(found)f (later)g(b)m(y)h(Gandol\014,)165 1961 y(Grimmett)i(and)f(Russo)g([89)q (])h(and)f(b)m(y)h(Burton)f(and)g(Keane)i([42)q(].)40 b(The)28 b(argumen)m(t)h(of)g(Burton)g(and)165 2074 y(Keane)e(is)e(not) h(only)g(the)g(shortest)g(\(and,)h(arguably)-8 b(,)27 b(the)f(most)h(elegan)m(t\))h(so)e(far.)39 b(Also,)27 b(it)f(requires)165 2187 y(m)m(uc)m(h)f(w)m(eak)m(er)i(assumptions)c (on)i(the)h(p)s(ercolation)e(mo)s(del,)h(namely:)37 b(translation)24 b(in)m(v)-5 b(ariance)25 b(and)165 2300 y(the)30 b(\014nite-energy)f (condition)g(b)s(elo)m(w,)g(whic)m(h)f(is)h(a)h(strong)g(w)m(a)m(y)g (of)g(stating)g(that)g(all)f(lo)s(cal)g(con\014g-)165 2413 y(urations)j(are)g(really)f(p)s(ossible;)h(its)f(signi\014cance)h (for)g(p)s(ercolation)f(theory)i(had)e(b)s(een)h(disco)m(v)m(ered)165 2526 y(b)s(efore)e(b)m(y)h(Newman)f(and)g(Sc)m(h)m(ulman)f([183)q(].) 165 2713 y Fx(De\014nition)36 b(5.16)46 b Fp(A)30 b(pr)-5 b(ob)g(ability)34 b(me)-5 b(asur)g(e)32 b Ft(\026)f Fp(on)h Fv(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)2205 2680 y Fq(L)2260 2713 y Fp(,)31 b(with)h Fv(L)e Fp(a)i(c)-5 b(ountable)32 b(set,)f(is)h(said)g(to)165 2826 y(have)h Fx(\014nite)i(energy)e Fp(if,)f(for)h(every)g(\014nite)f(r)-5 b(e)g(gion)34 b Fw(\003)26 b Fv(\032)e(L)p Fp(,)1243 3030 y Ft(\026)p Fw(\()p Ft(X)33 b Fv(\021)25 b Ft(\021)36 b Fw(on)30 b(\003)15 b Fv(j)g Ft(X)33 b Fv(\021)25 b Ft(\030)37 b Fw(o\013)30 b(\003\))c Ft(>)f Fw(0)165 3235 y Fp(for)34 b(al)5 b(l)33 b Ft(\021)28 b Fv(2)d(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)820 3202 y Fr(\003)908 3235 y Fp(and)34 b Ft(\026)p Fp(-a.e.)d Ft(\030)f Fv(2)24 b(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)1723 3202 y Fr(\003)1772 3178 y Fo(c)1811 3235 y Fp(.)165 3447 y Fx(Theorem)34 b(5.17)i(\(The)e (Burton{Keane)i(uniqueness)f(theorem\))44 b Fp(L)-5 b(et)32 b Ft(\026)g Fp(b)-5 b(e)32 b(a)g(pr)-5 b(ob)g(ability)165 3569 y(me)g(asur)g(e)34 b(on)g Fv(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)868 3536 y Fn(Z)918 3513 y Fo(d)992 3569 y Fp(which)34 b(is)f(tr)-5 b(anslation)35 b(invariant)f(and)g(has)g (\014nite)e(ener)-5 b(gy.)43 b(Then,)33 b Ft(\026)p Fp(-a.s.,)165 3682 y(ther)-5 b(e)34 b(exists)f(at)g(most)h(one)f(in\014nite)f(op)-5 b(en)34 b(cluster.)165 3895 y Fx(Sk)m(etc)m(h)e(pro)s(of:)39 b Fw(Without)26 b(loss)g(of)h(generalit)m(y)g(w)m(e)g(can)g(assume)f (that)h Ft(\026)g Fw(is)f(ergo)s(dic)g(with)f(resp)s(ect)165 4008 y(to)g(translations.)37 b(F)-8 b(or,)26 b(one)e(can)g(easily)e (sho)m(w)i(that)g(the)g(measures)f(in)g(the)g(ergo)s(dic)g(decomp)s (osition)165 4121 y(of)e Ft(\026)g Fw(admit)f(the)h(same)h(conditional) d(probabilities,)h(and)g(th)m(us)h(inherit)d(the)k(\014nite-energy)e (prop)s(ert)m(y)-8 b(.)165 4233 y(Since)26 b(the)g(n)m(um)m(b)s(er)f Ft(N)36 b Fw(of)27 b(in\014nite)d(clusters)h(is)g(ob)m(viously)g(in)m (v)-5 b(arian)m(t)26 b(under)e(translations,)j(it)e(then)165 4346 y(follo)m(ws)42 b(that)h Ft(N)53 b Fw(is)42 b(almost)h(surely)e (equal)h(to)i(some)f(constan)m(t)h Ft(k)49 b Fv(2)c(f)p Fw(0)p Ft(;)15 b Fw(1)p Ft(;)g(:)g(:)g(:)k(;)c Fv(1g)p Fw(.)78 b(In)42 b(fact,)165 4459 y Ft(k)36 b Fv(2)c(f)p Fw(0)p Ft(;)15 b Fw(1)p Ft(;)g Fv(1g)p Fw(.)57 b(Otherwise,)35 b(with)e(p)s(ositiv)m(e)h(probabilit)m(y)e(eac)m(h)k(of)f(the)g Ft(k)j Fw(clusters)c(w)m(ould)f(meet)165 4572 y(a)38 b(su\016cien)m(tly)e(large)i(cub)s(e)e(\003;)41 b(b)m(y)d(the)f (\014nite-energy)g(prop)s(ert)m(y)-8 b(,)39 b(this)d(w)m(ould)g(imply)f (that)j(with)165 4685 y(p)s(ositiv)m(e)32 b(probabilit)m(y)e(all)h (these)j(clusters)d(are)i(connected)h(within)c(\003,)j(so)g(that)g(in)e (fact)j Ft(k)e Fw(=)d(1,)k(in)165 4798 y(con)m(tradiction)e(to)g(the)f (h)m(yp)s(othesis.)40 b(\(This)29 b(part)h(of)g(the)h(argumen)m(t)g(go) s(es)g(bac)m(k)g(to)g([183)r(].\))306 4911 y(W)-8 b(e)33 b(th)m(us)d(only)g(need)h(to)h(exclude)e(the)i(case)g Ft(k)d Fw(=)d Fv(1)p Fw(.)43 b(In)30 b(this)g(case,)i Ft(\026)p Fw(\()p Ft(N)37 b Fv(\025)26 b Fw(3\))h(=)f(1,)32 b(and)e(the)165 5024 y(\014nite-energy)c(prop)s(ert)m(y)g(implies)d (again)k(that)g Ft(\026)p Fw(\()p Ft(A)1946 5038 y Fu(x)1990 5024 y Fw(\))e(=)g Ft(\016)30 b(>)24 b Fw(0,)k(where)e Ft(A)2736 5038 y Fu(x)2806 5024 y Fw(is)g(the)g(ev)m(en)m(t)i(that)f Ft(x)f Fw(is)165 5137 y(a)33 b(triple)e(p)s(oin)m(t,)i(in)e(that)i (there)g(exist)f(three)h(disjoin)m(t)e(in\014nite)f(op)s(en)i(paths)g (with)g(starting)g(p)s(oin)m(t)165 5250 y Ft(x)p Fw(.)41 b(By)31 b(the)f(\(norm-\))h(ergo)s(dic)f(theorem,)h(for)f(an)m(y)h (su\016cien)m(tly)e(large)h(cubic)g(b)s(o)m(x)g(\003)g(w)m(e)h(ha)m(v)m (e)1281 5469 y Ft(\026)1336 5375 y Fs(\020)1385 5469 y Fv(j)p Fw(\003)p Fv(j)1498 5432 y Fq(\000)p Fr(1)1616 5389 y Fs(X)1608 5572 y Fu(x)p Fq(2)p Fr(\003)1759 5469 y Ft(I)1799 5483 y Fu(A)1852 5491 y Fo(x)1920 5469 y Fv(\025)25 b Ft(\016)s(=)p Fw(2)2149 5375 y Fs(\021)2226 5469 y Fv(\025)g Fw(1)p Ft(=)p Fw(2)15 b Ft(:)956 b Fw(\(22\))1844 5895 y(37)p eop %%Page: 38 38 38 37 bop 165 304 a Fw(On)24 b(the)g(other)g(hand,)h(for)f(geometrical) h(reasons)f(\(whic)m(h)f(are)i(in)m(tuitiv)m(ely)d(ob)m(vious)h(but)h (need)g(some)165 417 y(w)m(ork)36 b(when)e(made)h(precise\),)h(there)f (cannot)h(b)s(e)f(more)g(triple)e(p)s(oin)m(ts)h(in)g(\003)h(than)g(p)s (oin)m(ts)f(in)g(the)165 530 y(b)s(oundary)i Ft(@)5 b Fw(\003)39 b(of)f(\003.)63 b(Indeed,)39 b(eac)m(h)h(of)e(the)g(three)g (paths)g(lea)m(ving)f(a)i(triple)d(p)s(oin)m(t)h(meets)h Ft(@)5 b Fw(\003,)165 643 y(whic)m(h)28 b(giv)m(es)h(three)g(b)s (oundary)e(p)s(oin)m(ts)g(asso)s(ciated)i(to)h(eac)m(h)g(triple)d(p)s (oin)m(t)h(in)f(\003.)40 b(If)29 b(one)g(iden)m(ti\014es)165 756 y(these)i(b)s(oundary)c(p)s(oin)m(ts)i(successiv)m(ely)g(for)h(one) g(triple)e(p)s(oin)m(t)h(after)h(the)g(other)g(one)g(sees)h(that,)f(at) 165 869 y(eac)m(h)38 b(step,)h(at)e(least)g(one)g(of)g(the)g(b)s (oundary)d(p)s(oin)m(ts)i(m)m(ust)g(b)s(e)g(di\013eren)m(t)g(from)h (those)g(obtained)165 981 y(b)s(efore.)k(Hence,)1259 1094 y Fv(j)p Fw(\003)p Fv(j)1372 1057 y Fq(\000)p Fr(1)1490 1014 y Fs(X)1482 1197 y Fu(x)p Fq(2)p Fr(\003)1633 1094 y Ft(I)1673 1108 y Fu(A)1726 1116 y Fo(x)1794 1094 y Fv(\024)25 b(j)p Fw(\003)p Fv(j)2003 1057 y Fq(\000)p Fr(1)2098 1094 y Fv(j)p Ft(@)5 b Fw(\003)p Fv(j)26 b Ft(<)f(\016)s(=)p Fw(2)165 1340 y(when)32 b(\003)h(is)e(large)i (enough.)47 b(Inserting)32 b(this)f(in)m(to)i(\(22\))h(w)m(e)f(arriv)m (e)g(at)g(the)g(con)m(tradiction)f Ft(\026)p Fw(\()p Fv(;)p Fw(\))e Fv(\025)165 1453 y Fw(1)p Ft(=)p Fw(2,)i(and)e(the)g (theorem)h(is)e(pro)m(v)m(ed.)41 b(F)-8 b(or)31 b(more)f(details)g(w)m (e)g(refer)g(to)h(the)g(original)d(pap)s(er)h([42)q(].)41 b Fk(2)165 1616 y Fw(W)-8 b(e)38 b(stress)f(that)h(the)f(last)f (argumen)m(t)i(relies)d(essen)m(tially)h(on)h(the)g(amenabilit)m(y)f (prop)s(ert)m(y)g(of)h Fx(Z)3574 1583 y Fu(d)165 1729 y Fw(discussed)c(in)f(Section)i(5.2.)54 b(The)33 b(\014nite-energy)h (condition)f(is)g(also)h(indisp)s(ensable:)44 b(In)34 b(another)165 1842 y(pap)s(er)24 b([43)r(],)i(Burton)f(and)f(Keane)i (construct,)h(for)e(an)m(y)g Ft(k)j Fv(2)d(f)p Fw(2)p Ft(;)15 b Fw(3)p Ft(;)g(:)g(:)g(:)k(;)c Fv(1g)p Fw(,)27 b(translation)d(in)m(v)-5 b(arian)m(t)165 1955 y(p)s(ercolation)28 b(mo)s(dels)g(on)h Fx(Z)1130 1922 y Fr(2)1198 1955 y Fw(for)g(whic)m(h)f(\014nite)g(energy)h(fails)e(and)i(whic)m(h)e(ha)m (v)m(e)k(exactly)e Ft(k)j Fw(in\014nite)165 2068 y(op)s(en)k(clusters.) 58 b(F)-8 b(or)37 b(example,)h(w)m(e)e(ha)m(v)m(e)i Ft(k)g Fw(=)d Fv(1)h Fw(in)f(Example)h(6.1.)59 b(F)-8 b(ortunately)g(,)39 b(the)e(\014nite-)165 2180 y(energy)c(condition)d(holds)h(in)g(most)h (of)g(the)h(dep)s(enden)m(t)e(p)s(ercolation)g(mo)s(dels)g(whic)m(h)f (sho)m(w)i(up)f(in)165 2293 y(sto)s(c)m(hastic-geometric)i(studies)c (of)i(Gibbs)d(measures.)306 2406 y(The)i(situation)f(b)s(ecomes)h (radically)e(di\013eren)m(t)h(when)g Fx(Z)2263 2373 y Fu(d)2333 2406 y Fw(is)g(replaced)g(b)m(y)h(the)g(non-amenable)165 2519 y(tree)g Fx(T)419 2534 y Fu(d)459 2519 y Fw(.)40 b(Instead)28 b(of)h(ha)m(ving)f(a)h(unique)d(in\014nite)g(cluster,)j (sup)s(ercritical)c(p)s(ercolation)j(mo)s(dels)f(on)165 2632 y Fx(T)238 2647 y Fu(d)307 2632 y Fw(tend)i(to)g(pro)s(duce)f (in\014nitely)e(man)m(y)j(in\014nite)d(clusters.)40 b(It)28 b(is)g(not)h(hard)f(to)i(v)m(erify)e(that)h(this)f(is)165 2745 y(indeed)e(the)h(case)h(for)f(sup)s(ercritical)d(Bernoulli)h(site) i(or)g(b)s(ond)e(p)s(ercolation)i(\(except)h(in)e(the)h(trivial)165 2858 y(case)d(when)f(the)g(reten)m(tion)g(probabilit)m(y)e Ft(p)h Fw(is)g(1\),)k(and)c(a)i(corresp)s(onding)d(result)h(for)h (automorphism)165 2971 y(in)m(v)-5 b(arian)m(t)26 b(p)s(ercolation)f (on)i Fx(T)1207 2986 y Fu(d)1273 2971 y Fw(can)g(b)s(e)e(found)h(in)f ([118)q(].)40 b(On)25 b(more)i(general)f(nonamenable)g(graph)165 3084 y(structures,)38 b(the)f(uniqueness)e(of)i(the)f(in\014nite)f (cluster)h(prop)s(ert)m(y)g(can)h(fail)e(in)h(more)h(in)m(teresting)165 3197 y(w)m(a)m(ys)j(than)e(on)g(trees;)44 b(see)39 b(e.g.)h(Grimmett)e (and)g(Newman)g([111)r(])h(and)f(H\177)-45 b(aggstr\177)g(om)40 b(and)e(P)m(eres)165 3310 y([122)r(].)306 3422 y(Let)32 b(us)d(next)i(consider)e(the)i(particular)e(case)j(of)e(\(p)s(ossibly)e (dep)s(enden)m(t\))i(site)h(p)s(ercolation)e(on)165 3535 y Fx(Z)229 3502 y Fr(2)269 3535 y Fw(.)40 b(W)-8 b(e)30 b(kno)m(w)g(from)e(Theorem)h(5.17)i(that)f(under)d(fairly)h(general)h (assumptions)e(there)j(is)e(almost)165 3648 y(surely)36 b(at)h(most)h(one)f(in\014nite)d(op)s(en)j(cluster.)59 b(Under)36 b(the)h(same)h(asumptions)d(there)i(is)f(almost)165 3761 y(surely)25 b(at)i(most)f(one)g(in\014nite)e Fp(close)-5 b(d)27 b Fw(cluster)e(\(i.e.,)j(at)f(most)f(one)g(in\014nite)e (connected)j(comp)s(onen)m(t)165 3874 y(of)37 b(closed)e(v)m (ertices\).)59 b(In)35 b(fact,)k(the)d(pro)s(of)f(of)h(Theorem)g(5.17)i (ev)m(en)e(sho)m(ws)g(that)h(almost)f(surely)165 3987 y(there)31 b(is)f(at)h(most)g(one)g(in\014nite)d(closed)j Fv(\003)p Fw(-cluster.)41 b(\(Here,)32 b(a)f(closed)g Fv(\003)p Fw(-cluster)f(is)g(a)h(maximal)f(set)165 4100 y Ft(C)45 b Fw(of)39 b(closed)f(sites)g(whic)m(h)f(is)g Fv(\003)p Fw(-connected,)42 b(in)c(that)g(an)m(y)h(t)m(w)m(o)h Ft(x;)15 b(y)42 b Fv(2)c Ft(C)45 b Fw(are)38 b(connected)h(b)m(y)g(a) 165 4213 y Fv(\003)p Fw(-path)34 b(in)e Ft(C)7 b Fw(;)34 b Fv(\003)p Fw(-paths)f(w)m(ere)g(in)m(tro)s(duced)f(in)g(the)h(pro)s (of)f(of)h(Theorem)g(5.3.)50 b(An)m(y)33 b(closed)g(cluster)165 4326 y(is)h(part)h(of)g(some)g(closed)f Fv(\003)p Fw(-cluster.\))55 b(But)34 b(p)s(erhaps)f(an)i(in\014nite)e(op)s(en)h(cluster)g(and)g(an) g(in\014nite)165 4439 y(closed)28 b Fv(\003)p Fw(-cluster)f(can)h(co)s (exist?)39 b(Theorem)27 b(5.18)i(b)s(elo)m(w)e(asserts)h(that)f(under)f (reasonably)h(general)165 4552 y(circumstances)39 b(this)f(cannot)i (happ)s(en.)66 b(Under)38 b(sligh)m(tly)f(di\013eren)m(t)i(conditions)f (\(replacing)g(the)165 4664 y(\014nite-energy)24 b(assumption)e(b)m(y)h (separate)i(ergo)s(dicit)m(y)e(under)f(translations)g(in)h(the)g(t)m(w) m(o)i(co)s(ordinate)165 4777 y(directions\),)30 b(it)g(w)m(as)h(pro)m (v)m(ed)f(b)m(y)g(Gandol\014,)g(Keane)h(and)f(Russo)g([90)q(].)165 4965 y Fx(Theorem)k(5.18)47 b Fp(L)-5 b(et)39 b Ft(\026)f Fp(b)-5 b(e)39 b(an)g(automorphism)k(invariant)d(and)f(er)-5 b(go)g(dic)41 b(pr)-5 b(ob)g(ability)41 b(me)-5 b(asur)g(e)165 5078 y(on)33 b Fv(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)515 5045 y Fn(Z)565 5021 y Fl(2)639 5078 y Fp(with)33 b(\014nite)g(ener)-5 b(gy)33 b(and)g(p)-5 b(ositive)34 b(c)-5 b(orr)g(elations.)45 b(Then)759 5282 y Ft(\026)p Fw(\()p Fv(9)33 b Fw(in\014nite)28 b(op)s(en)h(cluster,)h Fv(9)i Fw(in\014nite)c(closed)j Fv(\003)p Fw(-cluster\))25 b(=)g(0)15 b Ft(:)165 5486 y Fw(Note)24 b(that)f(automorphism)e(in)m(v)-5 b(ariance)21 b(in)h(the)g Fx(Z)1892 5453 y Fr(2)1931 5486 y Fw(-case)i(means)e (that,)j(in)c(addition)g(to)i(translation)165 5599 y(in)m(v)-5 b(ariance,)23 b Ft(\026)f Fw(is)f(also)h(in)m(v)-5 b(arian)m(t)21 b(under)g(re\015ection)h(in)e(and)i(exc)m(hange)h(of)g(co)s(ordinate)e (axes.)39 b(Under)1844 5895 y(38)p eop %%Page: 39 39 39 38 bop 165 304 a Fw(the)27 b(conditions)e(of)i(the)f(theorem,)i(w)m (e)f(ha)m(v)m(e)h(in)d(fact)j(some)f(information)d(on)j(the)f (geometric)i(shap)s(e)165 417 y(of)37 b(in\014nite)c(clusters:)51 b(If)36 b(an)g(in\014nite)e(op)s(en)h(cluster)g(exists)h(and)f(th)m(us) h(all)f(closed)h Fv(\003)p Fw(-clusters)g(are)165 530 y(\014nite,)31 b(eac)m(h)i(\014nite)e(b)s(o)m(x)g(of)h Fx(Z)1208 497 y Fr(2)1278 530 y Fw(is)f(surrounded)e(b)m(y)i(an)h(op)s (en)f(circuit,)f(and)h(all)g(these)h(circuits)e(are)165 643 y(part)h(of)g(the)g(\(necessarily)f(unique\))f(in\014nite)g(op)s (en)h(cluster.)41 b(Hence)31 b(the)g(in\014nite)e(op)s(en)h(cluster)g (is)165 756 y(a)35 b(sea,)i(in)c(the)i(sense)g(that)g(all)e(\\islands") g(\(i.e.,)k(the)d Fv(\003)p Fw(-clusters)h(of)g(its)f(complemen)m(t\))h (are)g(\014nite.)165 869 y(Similarly)-8 b(,)41 b(if)e(a)i(closed)g Fv(\003)p Fw(-cluster)g(in)e Fx(Z)1601 836 y Fr(2)1681 869 y Fw(exists,)44 b(it)c(is)g(necessarily)f(a)i(sea)h(\(and)e(in)g (particular)165 981 y(unique\).)306 1094 y(The)22 b(corresp)s(onding)d (result)h(is)h(false)g(in)f(higher)h(dimensions.)35 b(T)-8 b(o)22 b(see)g(this,)g(consider)f(Bernoulli)165 1207 y(site)30 b(p)s(ercolation)f(on)g Fx(Z)993 1174 y Fr(3)1033 1207 y Fw(.)40 b(The)29 b(critical)g(v)-5 b(alue)29 b Ft(p)1867 1221 y Fu(c)1931 1207 y Fw(for)h(this)f(mo)s(del)f(is)h (strictly)g(less)g(than)g(1)p Ft(=)p Fw(2)i(\(see)165 1320 y(Campanino)41 b(and)g(Russo)h([47)q(]\),)k(whence)41 b(for)h Ft(p)j Fw(=)f(1)p Ft(=)p Fw(2)f(there)g(exist)e(almost)h (surely)f(b)s(oth)g(an)165 1433 y(in\014nite)28 b(op)s(en)i(cluster)g (and)g(an)g(in\014nite)e(closed)i(cluster.)306 1546 y(The)37 b(pro)s(of)e(of)i(Theorem)f(5.18)i(b)s(elo)m(w)e(is)g(based)g(on)g(a)h 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Fw(\))25 b Ft(>)g Fw(0.)41 b(Then,)30 b(b)m(y)g(ergo)s(dicit)m(y)-8 b(,)31 b Ft(\026)p Fw(\()p Ft(A)20 b Fv(\\)g Ft(B)5 b Fw(\))25 b(=)g(1.)41 b(\(This)29 b(is)g(the)i(only)e(use)h(of)165 2380 y(ergo)s(dicit)m(y)k(w)m(e)g(mak)m(e,)h(and)e(ergo)s(dicit)m(y)g (could)g(clearly)g(b)s(e)g(replaced)g(b)m(y)h(tail)e(trivialit)m(y)g (or)i(some)165 2493 y(other)d(mixing)d(condition.\))40 b(Next)31 b(w)m(e)g(pic)m(k)f Ft(n)g Fw(so)h(large)f(that)1027 2677 y Ft(\026)p Fw(\()p Ft(A)1185 2691 y Fu(n)1232 2677 y Fw(\))c Ft(>)e Fw(1)d Fv(\000)f Fw(10)1635 2639 y Fq(\000)p Fr(3)1791 2677 y Fw(and)60 b Ft(\026)p Fw(\()p Ft(B)2157 2691 y Fu(n)2204 2677 y Fw(\))26 b Ft(>)f Fw(1)20 b Fv(\000)g Fw(10)2607 2639 y Fq(\000)p Fr(3)2728 2677 y Ft(;)165 2861 y Fw(where)38 b Ft(A)504 2875 y Fu(n)590 2861 y Fw(\(resp.)g Ft(B)920 2875 y Fu(n)967 2861 y Fw(\))h(is)e(the)i(ev)m (en)m(t)h(that)f(some)g(in\014nite)d(op)s(en)h(cluster)h(\(resp.)h (some)f(in\014nite)165 2974 y(closed)33 b Fv(\003)p Fw(-cluster\))f(in) m(tersects)h(\003)1310 2988 y Fu(n)1386 2974 y Fw(=)28 b([)p Fv(\000)p Ft(n;)15 b(n)p Fw(])1756 2941 y Fr(2)1817 2974 y Fv(\\)21 b Fx(Z)1963 2941 y Fr(2)2003 2974 y Fw(.)46 b(Let)33 b Ft(A)2307 2941 y Fu(L)2307 2997 y(n)2392 2974 y Fw(\(resp.)f Ft(A)2715 2941 y Fu(R)2715 2997 y(n)2773 2974 y Fw(,)h Ft(A)2899 2941 y Fu(T)2899 2997 y(n)2987 2974 y Fw(and)f Ft(A)3234 2941 y Fu(B)3234 2997 y(n)3294 2974 y Fw(\))h(b)s(e)f(the)165 3087 y(ev)m(en)m(t)25 b(that)e(some)h(v)m(ertex)g(in)d(the)i(left)g(\(resp.)g(righ)m(t,)h (top)f(and)g(b)s(ottom\))g(side)f(of)h(the)g(square-shap)s(ed)165 3200 y(v)m(ertex)30 b(set)g(\003)643 3214 y Fu(n)707 3200 y Fv(n)18 b Fw(\003)833 3214 y Fu(n)p Fq(\000)p Fr(1)999 3200 y Fw(b)s(elongs)27 b(to)j(some)f(in\014nite)e(op)s(en)h (path)h(whic)m(h)e(con)m(tains)i(no)g(other)g(v)m(ertex)165 3313 y(of)i(\003)332 3327 y Fu(n)379 3313 y Fw(,)g(and)e(de\014ne)h Ft(B)948 3280 y Fu(L)943 3336 y(n)1000 3313 y Fw(,)g Ft(B)1129 3280 y Fu(R)1124 3336 y(n)1186 3313 y Fw(,)h Ft(B)1316 3280 y Fu(T)1311 3336 y(n)1401 3313 y Fw(and)f Ft(B)1652 3280 y Fu(B)1647 3336 y(n)1742 3313 y Fw(analogously)-8 b(.)41 b(Then)1351 3498 y Ft(A)1419 3512 y Fu(n)1491 3498 y Fw(=)25 b Ft(A)1655 3460 y Fu(L)1655 3520 y(n)1728 3498 y Fv([)19 b Ft(A)1876 3460 y Fu(R)1876 3520 y(n)1955 3498 y Fv([)g Ft(A)2103 3460 y Fu(T)2103 3520 y(n)2179 3498 y Fv([)h Ft(A)2328 3460 y Fu(B)2328 3520 y(n)2404 3498 y Ft(:)165 3682 y Fw(Since)j(all)g(four)h(ev)m(en)m(ts)h(in)e(the) h(righ)m(t)f(hand)g(side)g(are)i(increasing)d(and)i Ft(\026)g Fw(has)f(p)s(ositiv)m(e)g(correlations,)926 3867 y Ft(\026)p Fw(\()p Ft(A)1084 3881 y Fu(n)1131 3867 y Fw(\))84 b(=)f Ft(\026)p Fw(\()p Ft(A)1562 3829 y Fu(L)1562 3889 y(n)1634 3867 y Fv([)20 b Ft(A)1783 3829 y Fu(R)1783 3889 y(n)1861 3867 y Fv([)g Ft(A)2010 3829 y Fu(T)2010 3889 y(n)2085 3867 y Fv([)g Ft(A)2234 3829 y Fu(B)2234 3889 y(n)2295 3867 y Fw(\))1250 4004 y(=)83 b(1)20 b Fv(\000)g Ft(\026)p Fw(\()p Fv(:)p Ft(A)1779 3967 y Fu(L)1779 4027 y(n)1851 4004 y Fv(\\)g(:)p Ft(A)2061 3967 y Fu(R)2061 4027 y(n)2139 4004 y Fv(\\)g(:)p Ft(A)2349 3967 y Fu(T)2349 4027 y(n)2424 4004 y Fv(\\)g(:)p Ft(A)2634 3967 y Fu(B)2634 4027 y(n)2694 4004 y Fw(\))1250 4142 y Fv(\024)83 b Fw(1)20 b Fv(\000)g Ft(\026)p Fw(\()p Fv(:)p Ft(A)1779 4105 y Fu(L)1779 4165 y(n)1831 4142 y Fw(\))p Ft(\026)p Fw(\()p Fv(:)p Ft(A)2085 4105 y Fu(R)2085 4165 y(n)2143 4142 y Fw(\))p Ft(\026)p Fw(\()p Fv(:)p Ft(A)2397 4105 y Fu(T)2397 4165 y(n)2453 4142 y Fw(\))p Ft(\026)p Fw(\()p Fv(:)p Ft(A)2707 4105 y Fu(B)2707 4165 y(n)2767 4142 y Fw(\))26 b Ft(;)165 4327 y Fw(where)36 b Fv(:)g Fw(indicates)f(the)h(complemen)m(t)g(of)g (a)h(set)g(\(for)f(t)m(yp)s(ographical)f(reasons\).)58 b(By)37 b(the)f(auto-)165 4440 y(morphism)j(in)m(v)-5 b(ariance)41 b(of)g Ft(\026)p Fw(,)j Ft(A)1348 4407 y Fu(L)1348 4462 y(n)1401 4440 y Fw(,)g Ft(A)1538 4407 y Fu(R)1538 4462 y(n)1596 4440 y Fw(,)g Ft(A)1733 4407 y Fu(T)1733 4462 y(n)1830 4440 y Fw(and)c Ft(A)2085 4407 y Fu(B)2085 4462 y(n)2187 4440 y Fw(all)h(ha)m(v)m(e)h(the)g(same)f Ft(\026)p Fw(-probabilit)m(y)-8 b(,)43 b(so)165 4553 y(that)1387 4665 y Ft(\026)p Fw(\()p Fv(:)p Ft(A)1606 4628 y Fu(L)1606 4688 y(n)1658 4665 y Fw(\))26 b Fv(\024)f Fw(\(1)c Fv(\000)f Ft(\026)p Fw(\()p Ft(A)2165 4679 y Fu(n)2212 4665 y Fw(\)\))2282 4628 y Fr(1)p Fu(=)p Fr(4)165 4822 y Fw(and)30 b(therefore)719 5007 y Ft(\026)p Fw(\()p Ft(A)877 4969 y Fu(L)877 5029 y(n)930 5007 y Fw(\))25 b(=)g Ft(\026)p Fw(\()p Ft(A)1244 4969 y Fu(R)1244 5029 y(n)1302 5007 y Fw(\))h Fv(\025)f Fw(1)c Fv(\000)e Fw(\(1)i Fv(\000)f Ft(\026)p Fw(\()p Ft(A)1965 5021 y Fu(n)2013 5007 y Fw(\)\))2083 4969 y Fr(1)p Fu(=)p Fr(4)2219 5007 y Fw(=)25 b(1)20 b Fv(\000)g Fw(10)2561 4969 y Fq(\000)p Fr(3)p Fu(=)p Fr(4)2752 5007 y Ft(>)25 b Fw(0)p Ft(:)p Fw(82)i Ft(:)393 b Fw(\(23\))165 5191 y(In)30 b(the)h(same)f(w)m(a)m(y) -8 b(,)32 b(w)m(e)f(get)1406 5304 y Ft(\026)p Fw(\()p Ft(B)1570 5267 y Fu(T)1565 5327 y(n)1624 5304 y Fw(\))26 b(=)f Ft(\026)p Fw(\()p Ft(B)1945 5267 y Fu(B)1940 5327 y(n)2005 5304 y Fw(\))h Ft(>)f Fw(0)p Ft(:)p Fw(82)i Ft(:)1079 b Fw(\(24\))165 5461 y(No)m(w)31 b(de\014ne)f(the)h(ev)m(en)m (t)h Ft(D)c Fw(=)d Ft(A)1299 5428 y Fu(L)1299 5484 y(n)1371 5461 y Fv(\\)20 b Ft(A)1520 5428 y Fu(R)1520 5484 y(n)1598 5461 y Fv(\\)g Ft(B)1753 5428 y Fu(T)1748 5484 y(n)1828 5461 y Fv(\\)f Ft(B)1982 5428 y Fu(B)1977 5484 y(n)2042 5461 y Fw(.)41 b(F)-8 b(rom)31 b(\(23\))h(and)e(\(24\))i(w)m(e)e (obtain)1182 5646 y Ft(\026)p Fw(\()p Ft(D)s Fw(\))c Fv(\025)f Fw(1)20 b Fv(\000)g Fw(4\(1)i Fv(\000)d Fw(0)p Ft(:)p Fw(82\))28 b(=)d(0)p Ft(:)p Fw(28)i Ft(>)e Fw(0)g Ft(:)1844 5895 y Fw(39)p eop %%Page: 40 40 40 39 bop 165 304 a Fw(When)39 b Ft(D)j Fw(o)s(ccurs,)g(b)s(oth)c(the)i (left)f(side)f(and)g(the)i(righ)m(t)e(side)h(of)g(\003)2568 318 y Fu(n)2654 304 y Fw(are)h(in)m(tersected)g(b)m(y)f(some)165 417 y(in\014nite)c(op)s(en)h(cluster.)60 b(By)37 b(Theorem)g(5.17,)j (these)e(in\014nite)c(op)s(en)j(clusters)f(are)h(iden)m(tical)f(and)165 530 y(separate)d(their)d(\(common\))i(complemen)m(t)f(in)m(to)h(\(at)g (least\))g(t)m(w)m(o)g(pieces,)g(prev)m(en)m(ting)f(the)g(in\014nite) 165 643 y(closed)h Fv(\003)p Fw(-clusters)g(in)m(tersecting)g(the)g (top)g(and)g(b)s(ottom)g(sides)f(of)h(\003)2542 657 y Fu(n)2621 643 y Fw(from)g(reac)m(hing)g(eac)m(h)h(other)165 756 y(\(see)41 b(the)g(picture)e(on)h(p.)g(196)h(of)f([108)r(]\).)70 b(Consequen)m(tly)-8 b(,)43 b(there)d(exist)g(t)m(w)m(o)i(in\014nite)37 b(closed)j Fv(\003)p Fw(-)165 869 y(clusters,)30 b(in)f(con)m (tradiction)h(to)i(Theorem)e(5.17.)1686 b Fk(2)165 1031 y Fw(Theorem)27 b(5.18)i(admits)e(some)g(v)-5 b(arian)m(ts.)40 b(First,)28 b(the)f(assumption)f(of)h(ergo)s(dicit)m(y)g(can)h(b)s(e)f (a)m(v)m(oided)165 1144 y(if)42 b(the)g(assumption)f(of)i(p)s(ositiv)m (e)f(correlations)g(is)f(strengh)m(tened)i(to)g(the)g(condition)e(that) i Ft(\026)f Fw(is)165 1257 y(monotone)f(in)e(the)i(sense)f(of)g (De\014nition)f(4.9.)72 b(\(This)38 b(is)i(b)s(ecause)g(monotonicit)m (y)g(is)f(preserv)m(ed)165 1370 y(under)i(ergo)s(dic)h(decomp)s (osition,)i(so)e(that)h(Theorem)f(4.11)i(implies)39 b(p)s(ositiv)m(e)j (correlations)f(for)165 1483 y(eac)m(h)d(ergo)s(dic)e(comp)s(onen)m (t.\))60 b(Moreo)m(v)m(er,)41 b(as)c(the)g(preceding)e(pro)s(of)h(sho)m (ws,)i(ergo)s(dicit)m(y)e(is)g(only)165 1596 y(needed)26 b(to)g(sho)m(w)g(that)g(in\014nite)d(clusters)i(exist)g(with)g (probabilit)m(y)e(either)i(0)h(or)g(1,)h(and)e(translation)165 1709 y(in)m(v)-5 b(ariance)24 b(and)g(\014nite)f(energy)i(are)g(only)f (used)f(for)i(the)f(uniqueness)f(of)h(in\014nite)f(clusters.)37 b(W)-8 b(e)26 b(also)165 1822 y(need)35 b(only)e(the)i(in)m(v)-5 b(ariance)33 b(under)g(lattice)i(rotations)f(rather)h(than)f(all)f (re\015ections,)j(and)d(closed)165 1935 y Fv(\003)p Fw(-clusters)e(can) f(b)s(e)g(replaced)g(b)m(y)g(closed)g(clusters.)40 b(W)-8 b(e)32 b(ma)m(y)f(th)m(us)f(state)i(the)e(follo)m(wing)f(result.)165 2122 y Fx(Prop)s(osition)37 b(5.19)46 b Fp(Ther)-5 b(e)28 b(exists)g(no)g(pr)-5 b(ob)g(ability)29 b(me)-5 b(asur)g(e)29 b Ft(\026)e Fp(on)h Fv(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)2793 2089 y Fn(Z)2843 2066 y Fl(2)2911 2122 y Fp(which)29 b(has)f(p)-5 b(ositive)165 2235 y(c)g(orr)g(elations,)51 b(is)45 b(invariant)h(under)g(lattic)-5 b(e)46 b(r)-5 b(otations)47 b(and)f(the)g(inter)-5 b(change)46 b(of)f(the)h(states)g (1)165 2348 y(\(\\op)-5 b(en)c("\))34 b(and)f(0)g(\(\\close)-5 b(d"\),)34 b(and)f(satis\014es)1088 2552 y Ft(\026)p Fw(\()15 b Fv(9)32 b Fw(a)f(unique)e(in\014nite)f(op)s(en)h(cluster\))c (=)g(1)h Ft(:)165 2756 y Fw(This)j(prop)s(osition)f(will)g(b)s(e)h (applied)g(to)i(the)f(ferromagnetic)h(Ising)e(mo)s(del)g(in)g(Section)i (8.2.)1844 5895 y(40)p eop %%Page: 41 41 41 40 bop 165 304 a Fy(6)135 b(Random-cluster)44 b(represen)l(tations) 165 507 y Fw(In)23 b(the)h(previous)e(section)i(w)m(e)g(sa)m(w)g(a)g(n) m(um)m(b)s(er)e(of)i(dep)s(enden)m(t)f(p)s(ercolation)f(mo)s(dels.)38 b(Here)24 b(w)m(e)g(shall)165 620 y(fo)s(cus)39 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Fv(g)1583 4693 y Fq(B)1677 4726 y Fw(and)39 b(summing)f(o)m(v)m(er)j Ft(\033)j Fv(2)d(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)j(;)d(q)s Fv(g)3046 4693 y Fq(L)3099 4726 y Fw(.)69 b(Note)41 b(that,)165 4839 y(giv)m(en)32 b Ft(\021)s Fw(,)h(there)f(are)g(exactly)h Ft(q)1256 4806 y Fu(k)r Fr(\()p Fu(\021)r Fr(\))1423 4839 y Fw(spin)d (con\014gurations)h Ft(\033)k Fw(that)d(are)g(allo)m(w)m(ed,)h(in)d (that)j(an)m(y)f(t)m(w)m(o)165 4952 y(neigh)m(b)s(oring)d(v)m(ertices)i Ft(x)25 b Fv(\030)g Ft(y)33 b Fw(with)c Ft(\021)s Fw(\()p Fv(h)p Ft(xy)s Fv(i)p Fw(\))e(=)e(1)31 b(ha)m(v)m(e)g(the)g(same)f (spin.)39 b(W)-8 b(e)32 b(get)1021 5123 y Ft(P)1092 5086 y Fu(G;edg)r(e)1079 5146 y(p;q)1309 5123 y Fw(\()p Ft(\021)s Fw(\))84 b(=)1792 5043 y Fs(X)1665 5234 y Fu(\033)r Fq(2f)p Fr(1)p Fu(;:::)o(;q)r Fq(g)1992 5215 y Fm(L)2054 5123 y Ft(P)2125 5086 y Fu(G)2112 5146 y(p;q)2205 5123 y Fw(\()p Ft(\033)n(;)15 b(\021)s Fw(\))1511 5412 y(=)83 b Ft(q)1709 5375 y Fu(k)r Fr(\()p Fu(\021)r Fr(\))1865 5351 y Fw(1)p 1854 5391 V 1854 5474 a Ft(Z)1958 5331 y Fs(Y)1947 5515 y Fu(e)p Fq(2B)2090 5412 y Ft(p)2136 5375 y Fu(\021)r Fr(\()p Fu(e)p Fr(\))2266 5412 y Fw(\(1)21 b Fv(\000)e Ft(p)p Fw(\))2538 5375 y Fr(1)p Fq(\000)p Fu(\021)r Fr(\()p Fu(e)p Fr(\))1511 5644 y Fw(=)83 b Ft(\026)1720 5606 y Fu(G)1720 5666 y(p;q)1813 5644 y Fw(\()p Ft(\021)s Fw(\))26 b Ft(;)1844 5895 y Fw(42)p eop %%Page: 43 43 43 42 bop 165 304 a Fw(again)31 b(b)m(y)f(normalization.)2440 b Fk(2)165 464 y Fw(The)25 b(Edw)m(ards{Sok)-5 b(al)24 b(coupling)g Ft(P)1400 431 y Fu(G)1387 487 y(p;q)1506 464 y Fw(of)h Ft(\026)1659 431 y Fu(G)1659 491 y(\014)s(;q)1785 464 y Fw(and)g Ft(\036)2011 431 y Fu(G)2011 487 y(p;q)2130 464 y Fw(is)f(the)i(k)m(ey)g(to)g(using)e(the)h(random-cluster)165 577 y(mo)s(del)35 b(in)f(analyzing)h(the)h(P)m(otts)h(mo)s(del.)55 b(The)35 b(follo)m(wing)f(t)m(w)m(o)j(results)d(are)i(eac)m(h)h (other's)f(dual,)165 690 y(and)30 b(are)h(immediate)e(consequences)i (of)g(Theorem)f(6.2)h(and)f(the)h(de\014nition)d(of)i Ft(P)3033 657 y Fu(G)3020 713 y(p;q)3114 690 y Fw(.)165 882 y Fx(Corollary)35 b(6.3)46 b Fp(L)-5 b(et)43 b Ft(p)e Fw(=)h(1)28 b Fv(\000)e Ft(e)1388 849 y Fq(\000)p Fr(2)p Fu(\014)1526 882 y Fp(,)44 b(and)f(supp)-5 b(ose)43 b(we)f(pick)g(a)g (r)-5 b(andom)44 b(spin)f(c)-5 b(on\014gur)g(ation)165 995 y Ft(X)33 b Fv(2)25 b(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)i(;)e(q)s Fv(g)740 962 y Fq(L)826 995 y Fp(as)34 b(fol)5 b(lows:)273 1177 y(1.)46 b(Pick)41 b(a)h(r)-5 b(andom)44 b(e)-5 b(dge)42 b(c)-5 b(on\014gur)g(ation)44 b Ft(Y)62 b Fv(2)42 b(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)2237 1144 y Fq(B)2333 1177 y Fp(ac)-5 b(c)g(or)g(ding)44 b(to)e(the)h(r)-5 b(andom-cluster)393 1290 y(me)g(asur)g(e)34 b Ft(\036)799 1257 y Fu(G)799 1312 y(p;q)892 1290 y Fp(.)273 1475 y(2.)46 b(F)-7 b(or)38 b(e)-5 b(ach)38 b(c)-5 b(onne)g(cte)g(d)39 b(c)-5 b(omp)g(onent)40 b Ft(C)k Fp(of)37 b Ft(Y)20 b Fp(,)39 b(pick)e(a)g(spin)h(at)g(r)-5 b(andom)40 b(\(uniformly\))f(fr)-5 b(om)393 1588 y Fv(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)i(;)e(q)s Fv(g)p Fp(,)44 b(assign)e(this)g (spin)g(to)g(every)f(vertex)g(of)h Ft(C)7 b Fp(,)43 b(and)f(do)g(this)g (indep)-5 b(endently)43 b(for)393 1701 y(di\013er)-5 b(ent)33 b(c)-5 b(onne)g(cte)g(d)34 b(c)-5 b(omp)g(onents.)165 1883 y(Then)33 b Ft(X)40 b Fp(is)33 b(distribute)-5 b(d)34 b(ac)-5 b(c)g(or)g(ding)35 b(to)e(the)g(Gibbs)f(me)-5 b(asur)g(e)34 b Ft(\026)2379 1850 y Fu(G)2379 1909 y(\014)s(;q)2480 1883 y Fp(.)165 2102 y Fx(Corollary)h(6.4)46 b Fp(L)-5 b(et)42 b Ft(p)g Fw(=)f(1)27 b Fv(\000)f Ft(e)1386 2069 y Fq(\000)p Fr(2)p Fu(\014)1524 2102 y Fp(,)43 b(and)g(supp)-5 b(ose)43 b(we)f(pick)f(a)h(r)-5 b(andom)44 b(e)-5 b(dge)42 b(c)-5 b(on\014gur)g(ation)165 2215 y Ft(Y)46 b Fv(2)24 b(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)569 2182 y Fq(B)656 2215 y Fp(as)33 b(fol)5 b(lows:)273 2397 y(1.)46 b(Pick)29 b(a)h(r)-5 b(andom)32 b(spin)f(c)-5 b(on\014gur)g(ation)32 b Ft(X)g Fv(2)25 b(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)j(;)d(q)s Fv(g)2310 2364 y Fq(L)2393 2397 y Fp(ac)-5 b(c)g(or)g(ding)32 b(to)f(the)f(Gibbs)g(me)-5 b(asur)g(e)393 2510 y Ft(\026)448 2477 y Fu(G)448 2536 y(\014)s(;q)548 2510 y Fp(.)273 2695 y(2.)46 b(Given)32 b Ft(X)7 b Fp(,)33 b(assign)g(e)-5 b(ach)34 b(e)-5 b(dge)32 b Ft(e)26 b Fw(=)f Fv(h)p Ft(xy)s Fv(i)33 b Fp(indep)-5 b(endently)34 b(value)f Fw(1)g Fp(with)h(pr)-5 b(ob)g(ability)1513 2814 y Fs(\()1622 2902 y Ft(p)82 b Fp(if)33 b Ft(X)7 b Fw(\()p Ft(x)p Fw(\))26 b(=)f Ft(X)7 b Fw(\()p Ft(y)s Fw(\))1622 3014 y(0)83 b Fp(if)33 b Ft(X)7 b Fw(\()p Ft(x)p Fw(\))26 b Fv(6)p Fw(=)f Ft(X)7 b Fw(\()p Ft(y)s Fw(\))49 b Fp(,)393 3216 y(and)33 b(value)g(0)g(otherwise.)165 3398 y(Then)g Ft(Y)53 b Fp(is)32 b(distribute)-5 b(d)34 b(ac)-5 b(c)g(or)g(ding)35 b(to)e(the)g(r)-5 b(andom-cluster)35 b(me)-5 b(asur)g(e)34 b Ft(\036)2742 3365 y Fu(G)2742 3420 y(p;q)2835 3398 y Fp(.)165 3603 y Fw(As)22 b(a)f(w)m(arm-up)g(for)g(the)g(phase)g (transition)f(considerations)g(in)g(Section)h(6.3,)k(w)m(e)c(giv)m(e)h (the)g(follo)m(wing)165 3716 y(result)30 b(as)g(a)h(t)m(ypical)f (application)f(of)h(the)h(random-cluster)e(represen)m(tation.)165 3898 y Fx(Corollary)35 b(6.5)46 b Fp(If)41 b(we)f(pick)g(a)g(r)-5 b(andom)43 b(spin)e(c)-5 b(on\014gur)g(ation)42 b Ft(X)47 b Fv(2)38 b(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)j(;)d(q)s Fv(g)3036 3865 y Fq(L)3129 3898 y Fp(ac)-5 b(c)g(or)g(ding)43 b(to)165 4011 y(the)30 b(Gibbs)g(me)-5 b(asur)g(e)30 b Ft(\026)965 3978 y Fu(G)965 4038 y(\014)s(;q)1066 4011 y Fp(,)g(then)g(for)g Ft(i)25 b Fv(2)g(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)j(;)d(q)s Fv(g)30 b Fp(and)g(two)h(vertic)-5 b(es)29 b Ft(x;)15 b(y)29 b Fv(2)24 b(L)p Fp(,)30 b(the)g(two)g(events) 165 4124 y Fv(f)p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))27 b(=)e Ft(i)p Fv(g)33 b Fp(and)h Fv(f)p Ft(X)7 b Fw(\()p Ft(y)s Fw(\))27 b(=)e Ft(i)p Fv(g)33 b Fp(ar)-5 b(e)33 b(p)-5 b(ositively)34 b(c)-5 b(orr)g(elate)g(d,)36 b(i.e.)731 4321 y Ft(\026)786 4284 y Fu(G)786 4344 y(\014)s(;q)886 4321 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))27 b(=)e Ft(i;)15 b(X)7 b Fw(\()p Ft(y)s Fw(\))27 b(=)e Ft(i)p Fw(\))h Fv(\025)f Ft(\026)1885 4284 y Fu(G)1885 4344 y(\014)s(;q)1985 4321 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))27 b(=)e Ft(i)p Fw(\))15 b Ft(\026)2483 4284 y Fu(G)2483 4344 y(\014)s(;q)2584 4321 y Fw(\()p Ft(X)7 b Fw(\()p Ft(y)s Fw(\))27 b(=)e Ft(i)p Fw(\))15 b Ft(:)165 4528 y Fx(Pro)s(of:)46 b Fw(The)33 b(measure)f Ft(\026)1094 4495 y Fu(G)1094 4554 y(\014)s(;q)1227 4528 y Fw(is)g(in)m(v)-5 b(arian)m(t)32 b(under)f(p)s(erm)m(utation)h(of)h(the)g(spin)e(set)i Fv(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)j(;)d(q)s Fv(g)p Fw(,)34 b(so)165 4641 y(that)1134 4769 y Ft(\026)1189 4732 y Fu(G)1189 4792 y(\014)s(;q)1289 4769 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))27 b(=)e Ft(i)p Fw(\))h(=)f Ft(\026)1894 4732 y Fu(G)1894 4792 y(\014)s(;q)1994 4769 y Fw(\()p Ft(X)7 b Fw(\()p Ft(y)s Fw(\))27 b(=)e Ft(i)p Fw(\))h(=)2550 4708 y(1)p 2550 4748 46 4 v 2551 4832 a Ft(q)2620 4769 y(:)165 4971 y Fw(W)-8 b(e)32 b(therefore)f(need)f(to)h(sho)m(w)f(that) 1268 5184 y Ft(\026)1323 5146 y Fu(G)1323 5207 y(\014)s(;q)1424 5184 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))27 b(=)e Ft(i;)15 b(X)7 b Fw(\()p Ft(y)s Fw(\))27 b(=)e Ft(i)p Fw(\))g Fv(\025)2396 5123 y Fw(1)p 2377 5163 84 4 v 2377 5246 a Ft(q)2421 5220 y Fr(2)2486 5184 y Ft(:)165 5420 y Fw(W)-8 b(e)46 b(ma)m(y)g(no)m(w)e(think)g(of)h Ft(X)52 b Fw(as)45 b(b)s(eing)e(obtained)h(as)h(in)f(Corollary)f(6.3)j(b)m(y)f (\014rst)f(pic)m(king)f(an)165 5533 y(edge)32 b(con\014guration)f Ft(Y)47 b Fv(2)26 b(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)1332 5500 y Fq(B)1418 5533 y Fw(according)31 b(to)h(the)g(random-cluster)e (measure)h Ft(\036)3135 5500 y Fu(G)3135 5555 y(p;q)3260 5533 y Fw(and)f(then)165 5646 y(assigning)k(i.i.d.)g(uniform)g(spins)f (to)j(the)g(connected)g(comp)s(onen)m(ts.)56 b(Giv)m(en)35 b Ft(Y)20 b Fw(,)37 b(the)f(conditional)1844 5895 y(43)p eop %%Page: 44 44 44 43 bop 165 304 a Fw(probabilit)m(y)21 b(that)j Ft(X)7 b Fw(\()p Ft(x)p Fw(\))26 b(=)f Ft(X)7 b Fw(\()p Ft(y)s Fw(\))26 b(=)f Ft(i)f Fw(is)e(1)p Ft(=q)27 b Fw(if)22 b Ft(u)h Fw(and)g Ft(v)j Fw(are)e(in)e(the)h(same)h(connected)g(comp)s (onen)m(t)165 417 y(of)i Ft(Y)20 b Fw(,)26 b(and)e(1)p Ft(=q)693 384 y Fr(2)759 417 y Fw(if)g(they)h(are)g(in)f(di\013eren)m (t)h(connected)h(comp)s(onen)m(ts.)39 b(Hence,)27 b(for)e(some)h Ft(\013)f Fv(2)g Fw([0)p Ft(;)15 b Fw(1],)896 659 y Ft(\026)951 622 y Fu(G)951 682 y(\014)s(;q)1051 659 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))27 b(=)e Ft(i;)15 b(X)7 b Fw(\()p Ft(y)s Fw(\))27 b(=)e Ft(i)p Fw(\))h(=)f Ft(\013)2063 598 y Fw(1)p 2063 638 46 4 v 2064 722 a Ft(q)2139 659 y Fw(+)20 b(\(1)h Fv(\000)f Ft(\013)p Fw(\))2544 598 y(1)p 2525 638 84 4 v 2525 722 a Ft(q)2569 695 y Fr(2)2644 659 y Fv(\025)2769 598 y Fw(1)p 2750 638 V 2750 722 a Ft(q)2794 695 y Fr(2)2858 659 y Ft(:)3546 879 y Fk(2)165 1042 y Fw(An)32 b(easy)h(mo)s(di\014cation)d(of)i(the)h(ab)s(o)m(v)m(e) g(pro)s(of)e(sho)m(ws)h(that)g(if)f Ft(G)i Fw(is)e(connected)i(and)e Ft(\014)j(>)27 b Fw(0,)33 b(then)165 1155 y(the)e(correlation)f(b)s(et) m(w)m(een)h Ft(I)1168 1174 y Fq(f)p Fu(X)5 b Fr(\()p Fu(x)p Fr(\)=)p Fu(i)p Fq(g)1510 1155 y Fw(and)29 b Ft(I)1726 1174 y Fq(f)p Fu(X)5 b Fr(\()p Fu(y)r Fr(\)=)p Fu(i)p Fq(g)2066 1155 y Fw(is)29 b(in)g(fact)j Fp(strictly)f Fw(p)s(ositiv)m(e.)306 1268 y(Note)k(that)e(the)g(relation)f(b)s(et)m (w)m(een)i(the)f(random-cluster)e(mo)s(del)h(and)g(the)h(P)m(otts)h(mo) s(del)e(de-)165 1381 y(p)s(ends)h(crucially)f(on)j(the)f(fact)h(that)g (all)f(spins)e(in)h Fv(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)i(;)e(q)s Fv(g)35 b Fw(are)g Fp(a)i(priori)e Fw(equiv)-5 b(alen)m(t.)53 b(This)32 b(is)165 1494 y(no)g(longer)f(the)h(case)h(when)d(a)i (nonzero)g(external)g(\014eld)e(is)h(presen)m(t)g(in)g(the)g(Ising)g (mo)s(del.)43 b(Sev)m(eral)165 1607 y(attempts)29 b(to)g(\014nd)e (useful)f(random-cluster)h(represen)m(tations)h(of)g(the)h(Ising)e(mo)s (del)f(with)h(external)165 1720 y(\014eld)36 b(ha)m(v)m(e)j(b)s(een)d (made,)k(but)c(progress)h(has)g(b)s(een)g(limited.)59 b(P)m(erhaps)37 b(the)g(recen)m(t)i(duplication)165 1833 y(idea)30 b(of)h(Cha)m(y)m(es,)g(Mac)m(h)m(ta)i(and)d(Redner)f([60)r(]) h(represen)m(ts)g(a)h(breakthrough)f(on)g(this)f(problem.)165 2076 y FA(6.2)113 b(In\014nite-v)m(olume)39 b(limits)165 2248 y Fw(In)34 b(this)g(subsection)g(w)m(e)h(will)d(exploit)i(some)h (sto)s(c)m(hastic)h(monotonicit)m(y)f(prop)s(erties)e(of)i(random-)165 2360 y(cluster)27 b(distributions)c(on)k(\014nite)g(subgraphs)e(of)i Fx(Z)1928 2328 y Fu(d)1969 2360 y Fw(.)39 b(This)26 b(will)e(giv)m(e)k (us)f(the)g(existence)h(of)f(certain)165 2473 y(limiting)32 b(random-cluster)i(distributions,)e(and)i(also)h(the)g(existence)g(of)g (certain)g(limiting)d(Gibbs)165 2586 y(measures)f(for)f(the)g(P)m(otts) i(mo)s(del.)306 2699 y(The)44 b(basic)e(observ)-5 b(ation)44 b(is)e(stated)j(in)d(the)i(lemma)f(b)s(elo)m(w)f(whic)m(h)h(follo)m(ws) f(directly)h(from)165 2812 y(de\014nitions.)165 3000 y Fx(Lemma)33 b(6.6)46 b Fp(Consider)40 b(the)g(r)-5 b(andom-cluster)41 b(mo)-5 b(del)41 b(with)f(p)-5 b(ar)g(ameters)42 b Ft(p)d Fp(and)h Ft(q)i Fp(on)e(a)f(\014nite)165 3113 y(gr)-5 b(aph)46 b Ft(G)f Fp(with)g(e)-5 b(dge)45 b(set)f Fv(B)s Fp(.)77 b(F)-7 b(or)45 b(any)g(e)-5 b(dge)44 b Ft(e)j Fw(=)f Fv(h)p Ft(xy)s Fv(i)h(2)f(B)s Fp(,)h(and)e(any)g(c)-5 b(on\014gur)g(ation)47 b Ft(\021)j Fv(2)165 3226 y(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)385 3193 y Fq(B)r(nf)p Fu(e)p Fq(g)578 3226 y Fp(,)33 b(we)f(have)h(that)244 3495 y Ft(\036)298 3458 y Fu(G)298 3518 y(p;q)392 3495 y Fw(\()p Ft(e)g Fw(is)c(op)s(en)15 b Fv(j)g Ft(\021)s Fw(\))26 b(=)1043 3351 y Fs(\()1152 3431 y Ft(p)361 b Fw(if)30 b Ft(x)g Fw(and)g Ft(y)j Fw(are)e(connected)g(via)f(op)s(en)g(edges)h (in)e Ft(\021)1296 3503 y Fu(p)p 1162 3523 305 4 v 1162 3576 a(p)p Fr(+\(1)p Fq(\000)p Fu(p)p Fr(\))p Fu(q)1559 3544 y Fw(otherwise.)3453 3495 y(\(27\))165 3815 y(F)-8 b(or)37 b Ft(q)h Fv(\025)d Fw(1,)j(Lemma)f(6.6)g(means)f(in)f (particular)g(that)i(the)f(conditional)f(probabilit)m(y)e(in)i(\(27\))j (is)165 3928 y(increasing)33 b(in)g Ft(\021)38 b Fw(\(and)33 b(also)i(in)d Ft(p)p Fw(\).)53 b(This)32 b(allo)m(ws)h(us)h(to)h(use)f (Holley's)f(Theorem)h(and)g(the)g(FK)m(G)165 4040 y(inequalit)m(y)j(to) i(pro)m(v)m(e)g(the)f(follo)m(wing)f(v)m(ery)h(useful)f(result.)63 b(W)-8 b(e)39 b(write)f Ft(\036)2770 4007 y Fu(G)2770 4063 y(p)2867 4040 y Fw(for)g(Bernoulli)e(b)s(ond)165 4153 y(p)s(ercolation)30 b(on)g Ft(G)h Fw(with)e(parameter)i Ft(p)p Fw(.)165 4341 y Fx(Corollary)k(6.7)46 b Fp(F)-7 b(or)30 b(a)e(\014nite)g(gr)-5 b(aph)30 b Ft(G)e Fp(and)h(the)g(r)-5 b(andom-cluster)30 b(me)-5 b(asur)g(e)29 b Ft(\036)2961 4308 y Fu(G)2961 4363 y(p;q)3083 4341 y Fp(with)g Ft(p)24 b Fv(2)h Fw([0)p Ft(;)15 b Fw(1])165 4454 y Fp(and)34 b Ft(q)28 b Fv(\025)d Fw(1)p Fp(,)33 b(we)g(have)211 4642 y Fw(\(a\))64 b Ft(\036)444 4609 y Fu(G)444 4664 y(p;q)570 4642 y Fp(is)32 b(monotone,)j(and)e(ther)-5 b(efor)g(e)35 b(it)d(has)i(p)-5 b(ositive)33 b(c)-5 b(orr)g(elations,) 211 4838 y Fw(\(b\))63 b Ft(\036)449 4805 y Fu(G)449 4861 y(p;q)567 4838 y Fv(\026)638 4852 y Fq(D)724 4838 y Ft(\036)778 4805 y Fu(G)778 4861 y(p)853 4838 y Fp(,)211 5035 y Fw(\(c\))h Ft(\036)439 5002 y Fu(G)439 5058 y(p;q)557 5035 y Fv(\027)628 5049 y Fq(D)714 5035 y Ft(\036)768 5002 y Fu(G)896 5037 y Fo(p)p 778 5053 268 4 v 778 5094 a(p)p Fl(+\(1)p Fm(\000)p Fo(p)p Fl(\))p Fo(q)1107 5035 y Fp(.)165 5253 y(F)-7 b(urthermor)i(e,)35 b(for)f Fw(0)25 b Fv(\024)g Ft(p)1076 5267 y Fr(1)1141 5253 y Fv(\024)g Ft(p)1283 5267 y Fr(2)1347 5253 y Fv(\024)g Fw(1)33 b Fp(and)h Ft(q)28 b Fv(\025)d Fw(1)p Fp(,)32 b(we)h(have)211 5440 y Fw(\(d\))63 b Ft(\036)449 5407 y Fu(G)449 5463 y(p)485 5472 y Fl(1)519 5463 y Fu(;q)602 5440 y Fv(\026)673 5454 y Fq(D)759 5440 y Ft(\036)813 5407 y Fu(G)813 5463 y(p)849 5472 y Fl(2)883 5463 y Fu(;q)956 5440 y Fp(.)1844 5895 y Fw(44)p eop %%Page: 45 45 45 44 bop 165 304 a Fx(Pro)s(of:)40 b Fw(The)27 b(monotonicit)m(y)h(in) e(\(a\))j(is)d(just)h(the)h(observ)-5 b(ation)27 b(that)h(the)g (conditional)e(probabilit)m(y)165 417 y(in)f(\(27\))i(is)d(increasing)g (in)h Ft(p)g Fw(and)g(in)f Ft(\021)s Fw(.)39 b(P)m(ositiv)m(e)26 b(correlations)f(then)g(follo)m(ws)g(from)g(Theorem)g(4.11.)165 530 y(Next,)32 b(note)f(that)g(\(27\))h(implies)27 b(that)1385 680 y Ft(p)p 1171 720 474 4 v 1171 804 a(p)20 b Fw(+)g(\(1)h Fv(\000)f Ft(p)p Fw(\))p Ft(q)1680 741 y Fv(\024)25 b Ft(\036)1830 704 y Fu(G)1830 764 y(p;q)1923 741 y Fw(\()p Ft(e)31 b Fw(is)f(op)s(en)14 b Fv(j)h Ft(\021)s Fw(\))27 b Fv(\024)e Ft(p)834 b Fw(\(28\))165 983 y(for)39 b(all)f Ft(\021)43 b Fw(as)c(in)f(Lemma)h(6.6.)68 b(Theorem)39 b(4.8)h(in)e(conjunction)g(with)g(the)h(second)g(\(resp.)g(\014rst\)) 165 1096 y(inequalit)m(y)30 b(in)h(\(28\))i(implies)c(\(b\))j(\(resp.)f (\(a\)\).)46 b(Finally)-8 b(,)31 b(\(d\))h(is)e(just)h(another)h (application)e(of)i(\(d\))165 1209 y(and)e(Theorem)g(4.8.)2680 b Fk(2)165 1368 y Fw(Consider)23 b(no)m(w)i(the)f(in)m(teger)h(lattice) g Fx(Z)1504 1335 y Fu(d)1569 1368 y Fw(\(for)f(de\014niteness)f(and)h (simplicit)m(y\))f(with)g(its)h(usual)f(graph)165 1480 y(structure.)79 b(W)-8 b(e)44 b(asso)s(ciate)h(with)d(an)m(y)h (\014nite)f(region)h(\003)k Fv(\032)f Fx(Z)2420 1447 y Fu(d)2504 1480 y Fw(t)m(w)m(o)e(sp)s(eci\014c)e(random-cluster)165 1593 y(distributions)d(whic)m(h)i(corresp)s(ond)g(to)i(t)m(w)m(o)g (di\013eren)m(t)f(c)m(hoices)h(of)f(b)s(oundary)f(condition.)75 b(The)165 1706 y(latter)33 b(will)d(w)m(e)k(distinguished)29 b(b)m(y)j(a)h(parameter)g Ft(b)d Fv(2)e(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)p Fw(.)50 b(Let)33 b Fv(B)j Fw(b)s(e)c(the)h(set)g(of)g (all)e(nearest-)165 1819 y(neigh)m(b)s(or)e(b)s(onds)g(in)g Fx(Z)975 1786 y Fu(d)1016 1819 y Fw(,)h Fv(B)1134 1786 y Fr(0)1131 1845 y(\003)1214 1819 y Fw(the)h(set)g(of)g(all)e(edges)i (of)g Fv(B)h Fw(that)f(are)g(con)m(tained)g(in)e(\003,)i(and)f Fv(B)3408 1786 y Fr(1)3405 1845 y(\003)3488 1819 y Fw(the)165 1932 y(set)e(of)g(edges)g(with)e(at)i(least)g(one)g(endp)s(oin)m(t)e (in)g(\003.)40 b(\(The)27 b(di\013erence)g Fv(B)2624 1899 y Fr(1)2621 1958 y(\003)2688 1932 y Fv(n)14 b(B)2810 1899 y Fr(0)2807 1958 y(\003)2888 1932 y Fw(th)m(us)28 b(consists)f(of)g(all)165 2045 y(edges)j(leading)f(from)g(a)h(p)s(oin)m (t)e(of)i(\003)f(to)i(a)f(p)s(oin)m(t)e(of)i(\003)2028 2012 y Fu(c)2063 2045 y Fw(.\))40 b(W)-8 b(e)31 b(then)e(let)h Ft(\036)2710 2012 y Fu(b)2710 2071 y(p;q)r(;)p Fr(\003)2901 2045 y Fw(b)s(e)f(the)h(probabilit)m(y)165 2170 y(measure)h(on)f Fv(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)866 2137 y Fq(B)951 2170 y Fw(in)29 b(whic)m(h)g(eac)m(h)i Ft(\021)e Fv(2)c(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)1901 2137 y Fq(B)1985 2170 y Fw(is)30 b(assigned)f(probabilit)m(y)f(prop)s(ortional)g(to)1001 2404 y Ft(I)1041 2430 y Fq(f)p Fu(\021)r Fq(\021)p Fu(b)j Fr(o\013)24 b Fq(B)1377 2407 y Fo(b)1375 2454 y Fl(\003)1421 2430 y Fq(g)1461 2285 y Fs(\032)1571 2323 y(Y)1538 2519 y Fu(e)p Fq(2B)1666 2496 y Fo(b)1664 2543 y Fl(\003)1725 2404 y Ft(p)1771 2366 y Fu(\021)r Fr(\()p Fu(e)p Fr(\))1900 2404 y Fw(\(1)d Fv(\000)f Ft(p)p Fw(\))2173 2366 y Fr(1)p Fq(\000)p Fu(\021)r Fr(\()p Fu(e)p Fr(\))2393 2285 y Fs(\033)2480 2404 y Ft(q)2524 2366 y Fu(k)r Fr(\()p Fu(\021)r(;)p Fr(\003\))2753 2404 y Ft(;)165 2719 y Fw(where)k Ft(k)s Fw(\()p Ft(\021)s(;)15 b Fw(\003\))26 b(is)e(the)g(n)m(um)m(b)s(er)f (of)i(all)e Ft(\021)s Fw(-op)s(en)i(clusters)e(meeting)i(\003.)39 b(W)-8 b(e)25 b(call)f Ft(\036)2944 2686 y Fu(b)2944 2744 y(p;q)r(;)p Fr(\003)3130 2719 y Fw(the)h(random-)165 2832 y(cluster)38 b(distribution)c(in)j(\003)i(with)e(parameters)h Ft(p)g Fw(and)g Ft(q)j Fw(and)c(b)s(oundary)f(condition)h Ft(b)p Fw(.)64 b(In)38 b(the)165 2944 y(case)i Ft(b)f Fw(=)f(0,)j Ft(k)s Fw(\()p Fv(\001)p Ft(;)15 b Fw(\003\))40 b(is)e(simply)e(the)i(n)m(um)m(b)s(er)f(of)i(all)e(clusters)h(that)h (are)g(con)m(tained)f(in)f(\003;)43 b(this)165 3057 y(corresp)s(onds)27 b(to)h(forgetting)g(all)f(sites)g(in)g(\003)1675 3024 y Fu(c)1737 3057 y Fw(and)g(is)g(therefore)h(referred)f(to)i(as)f(the)g Fp(fr)-5 b(e)g(e)30 b(b)-5 b(oundary)165 3170 y(c)g(ondition)p Fw(.)58 b(On)34 b(the)i(other)g(hand,)g(supp)s(ose)e(that)i(\003)f(has) g(no)h(holes,)g(in)e(the)i(sense)f(that)h(\003)3412 3137 y Fu(c)3482 3170 y Fw(has)165 3283 y(no)31 b(\014nite)f(connected)i (comp)s(onen)m(ts;)g(since)e(w)m(e)i(can)f(alw)m(a)m(ys)h(assume)e (without)g(loss)h(of)g(generalit)m(y)165 3396 y(that)c(\003)f(is)g (connected,)i(w)m(e)f(call)e(suc)m(h)h(a)g(\003)h Fp(simply)j(c)-5 b(onne)g(cte)g(d)p Fw(.)41 b(Then,)26 b(in)f(the)h(case)h Ft(b)f Fw(=)f(1,)i(all)e(sites)165 3509 y(of)30 b(\003)331 3476 y Fu(c)396 3509 y Fw(ma)m(y)g(b)s(e)g(though)m(t)g(of)g(as)g(b)s (eing)e(\014rmly)g(wired)h(together,)i(whence)f(this)f(is)f(called)i (the)g Fp(wir)-5 b(e)g(d)165 3622 y(b)g(oundary)35 b(c)-5 b(ondition)p Fw(.)306 3735 y(Supp)s(ose)29 b(no)m(w)h(that)h(\003)26 b Fv(\032)f Fw(\001)30 b(are)h(t)m(w)m(o)g(\014nite)f(regions)g(in)f Fx(Z)2375 3702 y Fu(d)2415 3735 y Fw(.)41 b(Then)29 b Ft(\036)2772 3702 y Fu(b)2772 3760 y(p;q)r(;)p Fr(\003)2964 3735 y Fw(is)h(obtained)g(from)165 3861 y Ft(\036)219 3828 y Fu(b)219 3886 y(p;q)r(;)p Fr(\001)426 3861 y Fw(b)m(y)36 b(conditioning)d(on)i(the)h(ev)m(en)m(t)g Fv(f)p Ft(\021)i Fv(\021)33 b Ft(b)d Fw(on)h Fv(B)2111 3828 y Fu(b)2108 3886 y Fr(\001)2194 3861 y Fv(n)24 b(B)2326 3828 y Fu(b)2323 3886 y Fr(\003)2376 3861 y Fv(g)35 b Fw(whic)m(h)f(is)h(increasing)f (for)h Ft(b)e Fw(=)h(1)165 3974 y(and)c(decreasing)g(for)g Ft(b)c Fw(=)f(0.)41 b(Hence,)31 b(if)f Ft(q)e Fv(\025)d Fw(1)30 b(then)g(Corollary)f(6.7)j(\(a\))f(implies)d(that)874 4166 y Ft(\036)928 4129 y Fr(0)928 4189 y Fu(p;q)r(;)p Fr(\003)1115 4166 y Fv(\026)1186 4180 y Fq(D)1271 4166 y Ft(\036)1325 4129 y Fr(0)1325 4189 y Fu(p;q)r(;)p Fr(\001)1527 4166 y Fw(and)i Ft(\036)1758 4129 y Fr(1)1758 4189 y Fu(p;q)r(;)p Fr(\003)1945 4166 y Fv(\027)2016 4180 y Fq(D)2102 4166 y Ft(\036)2156 4129 y Fr(1)2156 4189 y Fu(p;q)r(;)p Fr(\001)2358 4166 y Fw(when)f(\003)d Fv(\032)f Fw(\001)g Ft(;)547 b Fw(\(29\))165 4359 y(in)27 b(complete)i(analogy)f (to)h(\(14\).)41 b(Moreo)m(v)m(er,)31 b(w)m(e)e(obtain)e(the)h(follo)m (wing)f(coun)m(terpart)h(of)g(Prop)s(osi-)165 4472 y(tion)h(4.14)h(on)f (the)g(existence)g(of)g(in\014nite-v)m(olume)e(limits.)38 b(W)-8 b(e)30 b(write)e(\003)d Fv(")h Fx(Z)2825 4439 y Fu(d)2894 4472 y Fw(for)i(the)h(limit)e(along)165 4585 y(some)32 b(\(an)m(y\))h(increasing)d(sequence)i(of)g(\014nite)e (simply)f(connected)k(subsets)e(of)g Fx(Z)2982 4552 y Fu(d)3023 4585 y Fw(,)h(con)m(v)m(erging)g(to)165 4698 y Fx(Z)229 4665 y Fu(d)300 4698 y Fw(in)d(the)i(usual)e(w)m(a)m(y)-8 b(.)165 4876 y Fx(Lemma)33 b(6.8)46 b Fp(F)-7 b(or)34 b Ft(p)25 b Fv(2)g Fw([0)p Ft(;)15 b Fw(1])34 b Fp(and)g Ft(q)28 b Fv(\025)d Fw(1)p Fp(,)32 b(the)h(limiting)g(me)-5 b(asur)g(es)1269 5069 y Ft(\036)1323 5031 y Fu(b)1323 5091 y(p;q)1442 5069 y Fw(=)47 b(lim)1538 5136 y Fr(\003)p Fq(")p Fn(Z)1672 5117 y Fo(d)1723 5069 y Ft(\036)1777 5031 y Fu(b)1777 5091 y(p;q)r(;)p Fr(\003)1964 5069 y Ft(;)109 b(b)25 b Fv(2)g(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)g Ft(;)165 5309 y Fp(exist)33 b(and)h(ar)-5 b(e)33 b(tr)-5 b(anslation)36 b(invariant.)165 5533 y Fw(This)26 b(con)m(v)m(ergence)k(result)c(has)h(consequences)i(for)e(the)g(con)m (v)m(ergence)j(of)e(Gibbs)e(distributions)d(for)165 5646 y(the)36 b(P)m(otts)h(mo)s(del,)f(as)f(w)m(e)h(will)d(sho)m(w)j(next.) 56 b(Let)36 b Ft(q)g Fv(2)e(f)p Fw(2)p Ft(;)15 b Fw(3)p Ft(;)g(:)g(:)g(:)s Fv(g)p Fw(,)37 b(and)e(for)h Ft(i)e Fv(2)f(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)j(;)d(q)s Fv(g)36 b Fw(and)1844 5895 y(45)p eop %%Page: 46 46 46 45 bop 165 304 a Fw(an)m(y)40 b(\014nite)d(region)i(\003)g(in)f Fx(Z)1149 271 y Fu(d)1228 304 y Fw(let)h Ft(\026)1423 271 y Fu(i)1423 330 y(\014)s(;q)r(;)p Fr(\003)1630 304 y Fw(denote)h(the)f(Gibbs)e(distribution)e(in)j(\003)h(for)g(the)g(P)m (otts)165 417 y(mo)s(del)34 b(at)i(in)m(v)m(erse)g(temp)s(erature)f Ft(\014)40 b Fw(with)34 b(b)s(oundary)f(condition)h Ft(\021)j Fv(\021)c Ft(i)j Fw(on)f(\003)2954 384 y Fu(c)2989 417 y Fw(.)55 b(F)-8 b(or)36 b Ft(i)e Fw(=)f(0,)k(let)165 530 y Ft(\026)220 497 y Fr(0)220 556 y Fu(\014)s(;q)r(;)p Fr(\003)423 530 y Fw(b)s(e)32 b(the)i(corresp)s(onding)d(Gibbs)h (distribution)e(with)i(free)i(b)s(oundary)d(condition,)i(whic)m(h)f(is) 165 653 y(de\014ned)26 b(b)m(y)g(letting)h Fv(L)e Fw(=)g(\003)h(in)g (\(4\),)i(i.e.,)g(b)m(y)f(ignoring)e(all)g(sites)i(outside)f(\003;)i(w) m(e)f(think)e(of)i Ft(\026)3337 620 y Fr(0)3337 680 y Fu(\014)s(;q)r(;)p Fr(\003)3533 653 y Fw(as)165 792 y(a)h(probabilit)m (y)e(measure)h(on)h(the)g(full)d(con\014guration)j(space)g Fv(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)i(;)e(q)s Fv(g)2656 759 y Fn(Z)2706 735 y Fo(d)2775 792 y Fw(b)m(y)27 b(using)g(an)g (arbitrary)165 905 y(extension.)306 1018 y(Still)33 b(for)i Ft(i)e Fw(=)f(0,)37 b(Theorem)d(6.2)i(sho)m(ws)f(that)h Ft(\026)1983 985 y Fr(0)1983 1044 y Fu(\014)s(;q)r(;)p Fr(\003)2186 1018 y Fw(and)f Ft(\036)2422 985 y Fr(0)2422 1043 y Fu(p;q)r(;)p Fr(\003)2618 1018 y Fw(admit)f(an)h(Edw)m(ards{Sok) -5 b(al)165 1144 y(coupling)32 b(\(on)i(\003\))g(when)f Ft(p)d Fw(=)g(1)23 b Fv(\000)f Ft(e)1450 1111 y Fq(\000)p Fr(2)p Fu(\014)1588 1144 y Fw(.)50 b(A)34 b(similar)d(Edw)m(ards{Sok)-5 b(al)33 b(coupling)f(is)g(p)s(ossible)f(for)165 1257 y Ft(i)26 b Fv(2)f(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)i(;)e(q)s Fv(g)27 b Fw(when)e(\003)h(is)e(simply)g(connected.)40 b(Indeed,)26 b(let)g Ft(P)2384 1224 y Fu(i)2371 1283 y(p;q)r(;)p Fr(\003)2559 1257 y Fw(b)s(e)f(the)h(probabilit)m(y)d (measure)165 1383 y(on)k Fv(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)j(;)d(q)s Fv(g)670 1350 y Fq(L)737 1383 y Fv(\002)f(f)p Fw(0)p Ft(;)h Fw(1)p Fv(g)1042 1350 y Fq(B)1122 1383 y Fw(corresp)s(onding)25 b(to)j(pic)m(king)e(a)h(random)f(site-and-b)s(ond)g(con\014guration)165 1496 y(according)31 b(to)g(the)f(follo)m(wing)f(pro)s(cedure.)276 1683 y(1.)47 b(Assign)29 b(to)i(eac)m(h)h(v)m(ertex)f(of)g(\003)1442 1650 y Fu(c)1507 1683 y Fw(v)-5 b(alue)30 b Ft(i)p Fw(,)h(and)e(to)i (all)f(edges)h(of)f Fv(B)23 b(n)e(B)2801 1650 y Fr(1)2798 1709 y(\003)2881 1683 y Fw(v)-5 b(alue)30 b(1.)276 1871 y(2.)47 b(Assign)21 b(to)i(eac)m(h)g(v)m(ertex)h(in)d(\003)h(a)h(spin)d (v)-5 b(alue)22 b(c)m(hosen)h(from)f Fv(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)i(;)e(q)s Fv(g)23 b Fw(according)f(to)h(uniform)393 1984 y(distribution,)i(assign)i(to)h(eac)m(h)i(edge)e(in)f Fv(B)1860 1951 y Fr(1)1857 2009 y(\003)1937 1984 y Fw(v)-5 b(alue)28 b(1)g(or)g(0)g(with)f(resp)s(ectiv)m(e)h(probabilities)c Ft(p)393 2097 y Fw(and)29 b(1)21 b Fv(\000)f Ft(p)p Fw(,)30 b(and)g(do)g(this)g(indep)s(enden)m(tly)d(for)j(all)f(v)m(ertices)i (and)f(edges.)276 2284 y(3.)47 b(Condition)23 b(on)j(the)g(ev)m(en)m(t) i(that)e(no)g(t)m(w)m(o)h(v)m(ertices)f(with)f(di\013eren)m(t)g(spins)f (ha)m(v)m(e)j(an)f(edge)g(with)393 2397 y(v)-5 b(alue)29 b(1)i(connecting)g(them.)165 2585 y(It)26 b(is)f(no)m(w)h(a)h(simple)c (mo)s(di\014cation)i(of)h(the)g(pro)s(of)f(of)h(Theorem)g(6.2)h(to)g(c) m(hec)m(k)g(that)g(the)f(v)m(ertex)h(and)165 2698 y(edge)38 b(marginals)e(of)i Ft(P)983 2665 y Fu(i)970 2723 y(p;q)r(;)p Fr(\003)1169 2698 y Fw(are)g Ft(\026)1383 2665 y Fu(i)1383 2724 y(\014)s(;q)r(;)p Fr(\003)1590 2698 y Fw(and)e Ft(\036)1827 2665 y Fr(1)1827 2723 y Fu(p;q)r(;)p Fr(\003)1989 2698 y Fw(,)k(resp)s(ectiv)m(ely)-8 b(.)61 b(\(Note)39 b(that)f(b)m(y)g(the) f(simple)165 2811 y(connectedness)f(of)f(\003)g(there)g(is)f(alw)m(a)m (ys)h(a)g(unique)e(comp)s(onen)m(t)i(con)m(taining)g(\003)2950 2778 y Fu(c)2985 2811 y Fw(.\))54 b(Analogues)35 b(of)165 2924 y(Corollaries)j(6.3)j(and)e(6.4)i(follo)m(w)e(easily)-8 b(.)68 b(This)38 b(leads)h(us)g(to)h(the)g(follo)m(wing)e(result)h (extending)165 3036 y(Prop)s(osition)29 b(4.14)j(to)f(the)f(P)m(otts)i (mo)s(del.)165 3224 y Fx(Prop)s(osition)37 b(6.9)46 b Fp(F)-7 b(or)33 b(any)h Ft(i)25 b Fv(2)g(f)p Fw(0)p Ft(;)15 b Fw(1)p Ft(;)g(:)g(:)g(:)j(;)d(q)s Fv(g)p Fp(,)33 b(the)g(limiting)g (pr)-5 b(ob)g(ability)35 b(me)-5 b(asur)g(e)1547 3428 y Ft(\026)1602 3391 y Fu(i)1602 3451 y(\014)s(;q)1728 3428 y Fw(=)46 b(lim)1823 3496 y Fr(\003)p Fq(")p Fn(Z)1957 3477 y Fo(d)2009 3428 y Ft(\026)2064 3391 y Fu(i)2064 3451 y(\014)s(;q)r(;)p Fr(\003)165 3708 y Fp(on)30 b Fv(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)j(;)d(q)s Fv(g)674 3675 y Fn(Z)724 3651 y Fo(d)794 3708 y Fp(exists)30 b(and)h(is)e(a)h (tr)-5 b(anslation)33 b(invariant)e(Gibbs)e(me)-5 b(asur)g(e)31 b(for)g(the)f Ft(q)s Fp(-state)f(Potts)165 3820 y(mo)-5 b(del)35 b(on)e Fx(Z)620 3787 y Fu(d)693 3820 y Fp(at)g(inverse)f(temp) -5 b(er)g(atur)g(e)35 b Ft(\014)5 b Fp(.)165 4008 y Fx(Pro)s(of:)37 b Fw(In)20 b(view)h(of)g(the)g(general)g(facts)h(rep)s(orted)e(in)g (Section)g(2.6,)25 b(the)c(limits)e(are)i(Gibbs)e(measures)165 4121 y(whenev)m(er)30 b(they)g(exist.)41 b(W)-8 b(e)31 b(th)m(us)e(need)h(to)h(sho)m(w)f(that)g Ft(\026)2173 4088 y Fu(i)2173 4147 y(\014)s(;q)r(;)p Fr(\003)2342 4121 y Fw(\()p Ft(f)10 b Fw(\))30 b(con)m(v)m(erges)i(as)e(\003)c Fv(")f Fx(Z)3238 4088 y Fu(d)3279 4121 y Fw(,)30 b(for)g(an)m(y)165 4234 y(lo)s(cal)39 b(observ)-5 b(able)39 b Ft(f)10 b Fw(.)67 b(F)-8 b(or)40 b(de\014niteness,)g(w)m(e)g(do)f(this)g(for)g Ft(i)h Fv(2)g(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)j(;)d(q)s Fv(g)p Fw(;)45 b(the)39 b(case)i Ft(i)f Fw(=)g(0)g(is)165 4347 y(completely)30 b(similar.)306 4460 y(Fix)37 b(an)h Ft(f)46 b Fw(as)38 b(ab)s(o)m(v)m(e,)j(and)c(let)g(\001)g Fv(\032)f Fx(Z)1718 4427 y Fu(d)1796 4460 y Fw(b)s(e)h(the)h(\014nite)e (region)h(on)g(whic)m(h)f Ft(f)47 b Fw(dep)s(ends.)60 b(As)165 4573 y(sho)m(wn)32 b(ab)s(o)m(v)m(e,)i(for)e(a)h(simply)d (connected)j(\003)f(w)m(e)h(ma)m(y)g(think)e(of)h(a)h Fv(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)j(;)d(q)s Fv(g)2903 4540 y Fn(Z)2953 4516 y Fo(d)2993 4573 y Fw(-v)-5 b(alued)32 b(random)165 4686 y(elemen)m(t)44 b Ft(X)51 b Fw(with)42 b(distribution)e Ft(\026)1424 4653 y Fu(i)1424 4712 y(\014)s(;q)r(;)p Fr(\003)1636 4686 y Fw(as)k(arising)e(b)m(y)h(\014rst)g(pic)m(king)f (an)i(edge)g(con\014guration)165 4812 y Ft(Y)i Fv(2)24 b(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)569 4779 y Fr(\003)653 4812 y Fw(according)28 b(to)g Ft(\036)1222 4779 y Fr(1)1222 4838 y Fu(p;q)r(;)p Fr(\003)1412 4812 y Fw(\(with)f Ft(p)e Fw(=)g(1)16 b Fv(\000)g Ft(e)2009 4779 y Fq(\000)p Fr(2)p Fu(\014)2146 4812 y Fw(\))28 b(and)g(then)g(assigning)e(random)h(spins) f(to)165 4925 y(the)31 b(connected)g(comp)s(onen)m(ts,)g(forcing)f (spin)e Ft(i)j Fw(to)g(the)g(\(unique\))e(in\014nite)f(cluster.)40 b(F)-8 b(or)32 b Ft(x;)15 b(y)28 b Fv(2)d Fw(\001,)165 5038 y(w)m(e)33 b(write)f Fv(f)p Ft(x)e Fv($)f Ft(y)s Fv(g)j Fw(for)h(the)g(ev)m(en)m(t)h(that)f Ft(x)f Fw(and)g Ft(y)k Fw(are)d(in)e(the)i(same)g(connected)h(comp)s(onen)m(t)e(in)165 5151 y Ft(Y)20 b Fw(,)33 b(and)e Fv(f)p Ft(x)d Fv($)g(1g)33 b Fw(for)e(the)h(ev)m(en)m(t)i(that)e Ft(x)g Fw(is)f(in)g(an)h (in\014nite)d(cluster.)45 b(Clearly)-8 b(,)32 b(the)g(conditional)165 5264 y(distribution)j(of)k Ft(f)48 b Fw(giv)m(en)38 b Ft(Y)59 b Fw(dep)s(ends)36 b(only)i(on)h(the)f(indicator)g(functions)f (\()p Ft(I)2965 5282 y Fq(f)p Fu(x)p Fq($)p Fu(y)r Fq(g)3187 5264 y Fw(\))3222 5278 y Fu(x;y)r Fq(2)p Fr(\001)3468 5264 y Fw(and)165 5377 y(\()p Ft(I)240 5395 y Fq(f)p Fu(x)p Fq($1g)496 5377 y Fw(\))531 5391 y Fu(x)p Fq(2)p Fr(\001)681 5377 y Fw(,)30 b(since)f(the)h(conditional)e(distribution)d (of)30 b Ft(X)37 b Fw(on)29 b(\001)h(is)e(uniform)g(o)m(v)m(er)i(all)f (elemen)m(ts)165 5499 y(of)k Fv(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)j(;)d(q) s Fv(g)653 5466 y Fr(\001)750 5499 y Fw(suc)m(h)32 b(that)i(\014rstly)d Ft(X)7 b Fw(\()p Ft(x)p Fw(\))31 b(=)e Ft(X)7 b Fw(\()p Ft(y)s Fw(\))34 b(whenev)m(er)f Ft(x)c Fv($)h Ft(y)s Fw(,)j(and)g(secondly)f Ft(X)7 b Fw(\()p Ft(x)p Fw(\))31 b(=)e Ft(i)165 5612 y Fw(whenev)m(er)35 b Ft(x)d Fv($)h(1)p Fw(.)53 b(Hence,)37 b(the)e(desired)e(con)m(v)m(ergence)k(of)e Ft(\026)2391 5579 y Fu(i)2391 5638 y(\014)s(;q)r(;)p Fr(\003)2560 5612 y Fw(\()p Ft(f)10 b Fw(\))35 b(follo)m(ws)e(if)h(w)m (e)h(can)g(sho)m(w)1844 5895 y(46)p eop %%Page: 47 47 47 46 bop 165 304 a Fw(that)23 b(the)g(join)m(t)f(distribution)d(of)j (\()p Ft(I)1365 323 y Fq(f)p Fu(x)p Fq($)p Fu(y)r Fq(g)1588 304 y Fw(\))1623 318 y Fu(x;y)r Fq(2)p Fr(\001)1852 304 y Fw(and)g(\()p Ft(I)2096 323 y Fq(f)p Fu(x)p Fq($1g)2352 304 y Fw(\))2387 318 y Fu(x)p Fq(2)p Fr(\001)2559 304 y Fw(con)m(v)m(erges)i(as)f Ft(n)i Fv(!)g(1)p Fw(.)38 b(This,)165 417 y(ho)m(w)m(ev)m(er,)30 b(follo)m(ws)c(from)h(Lemma)g (6.8)i(up)s(on)d(noting)g(that)i(\()p Ft(I)2284 435 y Fq(f)p Fu(x)p Fq($)p Fu(y)r Fq(g)2507 417 y Fw(\))2542 431 y Fu(x;y)r Fq(2)p Fr(\001)2776 417 y Fw(and)f(\()p Ft(I)3025 435 y Fq(f)p Fu(x)p Fq($1g)3280 417 y Fw(\))3315 431 y Fu(x)p Fq(2)p Fr(\001)3493 417 y Fw(are)165 530 y(increasing)i(functions.)2568 b Fk(2)165 773 y FA(6.3)113 b(Phase)38 b(transition)g(in)f(the)h(P)m(otts)e(mo)s(del)165 945 y Fw(As)25 b(promised,)f(this)f(subsection)h(is)g(dev)m(oted)h(to)g (pro)m(ving)f(Theorems)g(3.1)i(and)e(3.2,)j(using)c(random-)165 1058 y(cluster)29 b(argumen)m(ts.)41 b(The)29 b(original)f(source)i (for)f(the)h(material)f(in)g(this)f(subsection)h(is)f(Aizenman,)165 1171 y(Cha)m(y)m(es,)k(Cha)m(y)m(es)f(and)f(Newman)g([9)q(];)g(see)h (also)g([119)q(])g(for)f(a)h(sligh)m(tly)d(di\013eren)m(t)i(presen)m (tation.)306 1284 y(W)-8 b(e)41 b(consider)e(the)g(P)m(otts)i(mo)s(del) e(on)g Fx(Z)1731 1251 y Fu(d)1771 1284 y Fw(,)j Ft(d)f Fv(\025)g Fw(2.)68 b(All)38 b(the)i(argumen)m(ts)g(to)g(b)s(e)f(used)g (here,)165 1396 y(except)33 b(those)e(sho)m(wing)f(that)i(the)g (critical)e(in)m(v)m(erse)h(temp)s(erature)g Ft(\014)2569 1410 y Fu(c)2635 1396 y Fw(is)f(strictly)g(b)s(et)m(w)m(een)i(0)g(and) 165 1509 y Fv(1)p Fw(,)44 b(go)f(through)d(on)h(arbitrary)f(in\014nite) f(graphs;)47 b(w)m(e)42 b(stic)m(k)f(to)h(the)g Fx(Z)2724 1476 y Fu(d)2805 1509 y Fw(case)h(for)e(de\014niteness)165 1622 y(and)34 b(simplicit)m(y)e(of)j(notation.)54 b(W)-8 b(e)36 b(consider)d(the)i(limiting)d(Gibbs)h(measures)h Ft(\026)3025 1589 y Fu(i)3025 1649 y(\014)s(;q)3160 1622 y Fw(obtained)g(in)165 1735 y(Prop)s(osition)e(6.9.)50 b(F)-8 b(or)34 b Ft(i)c Fv(2)g(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)i(;)e(q)s Fv(g)p Fw(,)35 b(these)e(pla)m(y)g(a)h(role)f(similar)d(to)k(that)g(of) f(the)h(\\plus")e(and)165 1848 y(\\min)m(us")d(measures)h Ft(\026)966 1810 y Fr(+)966 1876 y Fu(\014)1054 1848 y Fw(and)f Ft(\026)1285 1810 y Fq(\000)1285 1876 y Fu(\014)1374 1848 y Fw(for)g(the)h(Ising)f(mo)s(del.)39 b(In)29 b(fact,)i(w)m(e)f (ha)m(v)m(e)h(the)f(follo)m(wing)e(result)165 1961 y(whic)m(h)38 b(extends)g(Theorem)g(4.15)j(to)e(the)g(P)m(otts)g(mo)s(del)f(and)g (also)g(giv)m(es)h(a)g(c)m(haracterization)h(of)165 2074 y(phase)30 b(transition)f(in)g(terms)i(of)f(p)s(ercolation)g(in)f(the)h (random-cluster)g(mo)s(del.)165 2261 y Fx(Theorem)k(6.10)47 b Fp(L)-5 b(et)35 b Ft(\014)g(>)29 b Fw(0)35 b Fp(and)h Ft(p)30 b Fw(=)f(1)22 b Fv(\000)g Ft(e)1823 2228 y Fq(\000)p Fr(2)p Fu(\014)1961 2261 y Fp(.)49 b(F)-7 b(or)36 b(any)f Ft(x)30 b Fv(2)f Fx(Z)2622 2228 y Fu(d)2697 2261 y Fp(and)36 b(any)g Ft(i)30 b Fv(2)f(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)j(;)d(q)s Fv(g)p Fp(,)165 2374 y(the)33 b(fol)5 b(lowing)34 b(statements)g(ar)-5 b(e)34 b(e)-5 b(quivalent.)211 2562 y Fw(\(i\))45 b Fp(Ther)-5 b(e)47 b(is)e(a)h(unique)f(Gibbs)h(me)-5 b(asur)g(e)47 b(for)f(the)g Ft(q)s Fp(-state)g(Potts)g(mo)-5 b(del)47 b(on)f Fx(Z)3126 2529 y Fu(d)3212 2562 y Fp(at)g(inverse)393 2675 y(temp)-5 b(er)g(atur)g(e)35 b Ft(\014)5 b Fp(.)211 2862 y Fw(\(ii\))32 b Ft(\026)420 2829 y Fu(i)420 2889 y(\014)s(;q)520 2862 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))27 b(=)e Ft(i)p Fw(\))h(=)f(1)p Ft(=q)s Fp(.)211 3060 y Fw(\(iii\))31 b Ft(\036)444 3027 y Fr(1)444 3083 y Fu(p;q)537 3060 y Fw(\()p Ft(x)26 b Fv($)f(1)p Fw(\))g(=)g(0)p Fp(.)165 3297 y Fw(As)j(w)m(e)g(will)d(see)k(in)d(a)i(momen)m(t,)h(it)f (is)e(the)i(p)s(ercolation)f(criterion)g(\(iii\))f(whic)m(h)g(is)h (most)h(con)m(v)m(enien)m(t)165 3410 y(to)j(apply)-8 b(.)40 b(In)30 b(this)f(con)m(text)j(w)m(e)f(note)g(that)744 3614 y Ft(\036)798 3577 y Fr(1)798 3637 y Fu(p;q)891 3614 y Fw(\()p Ft(x)26 b Fv($)f(1)p Fw(\))h(=)33 b(inf)1368 3673 y Fr(\003)p Fu(;)p Fr(\001)1510 3614 y Ft(\036)1564 3577 y Fr(1)1564 3637 y Fu(p;q)r(;)p Fr(\003)1726 3614 y Fw(\()p Ft(x)26 b Fv($)f Fw(\001)2031 3577 y Fu(c)2065 3614 y Fw(\))h(=)47 b(lim)2222 3681 y Fr(\003)p Fq(")p Fn(Z)2356 3663 y Fo(d)2407 3614 y Ft(\036)2461 3577 y Fr(1)2461 3637 y Fu(p;q)r(;)p Fr(\003)2623 3614 y Fw(\()p Ft(x)26 b Fv($)f Fw(\003)2915 3577 y Fu(c)2950 3614 y Fw(\))g Ft(;)418 b Fw(\(30\))165 3870 y(where)36 b Fv(f)p Ft(x)e Fv($)g Fw(\001)766 3837 y Fu(c)800 3870 y Fv(g)j Fw(stands)e(for)g(the)h(ev)m(en)m(t)h(that)f(there)g(exists)g(an)f(op)s (en)g(path)h(from)f Ft(x)g Fw(to)i(some)165 3983 y(site)d(in)e(\001)521 3950 y Fu(c)555 3983 y Fw(.)50 b(This)32 b(follo)m(ws)h(from)g(\(29\))i (and)d(the)i(fact)h(that)f Fv(f)p Ft(x)c Fv($)h Fw(\001)2603 3950 y Fu(c)2637 3983 y Fv(g)j Fw(decreases)g(to)h Fv(f)p Ft(x)30 b Fv($)h(1g)165 4096 y Fw(as)g(\001)25 b Fv(")h Fx(Z)513 4063 y Fu(d)553 4096 y Fw(.)306 4209 y(The)i(usefulness)d(of)j (the)g(p)s(ercolation)f(criterion)f(is)h(demonstrated)h(b)m(y)f(the)h (next)g(result)e(whic)m(h)165 4322 y(extends)35 b(the)g(scenario)g(for) g(Bernoulli)e(p)s(ercolation)h(to)h(the)h(random-cluster)e(mo)s(del.)53 b(T)-8 b(ogether)165 4435 y(with)29 b(Theorem)i(6.10,)h(this)d(giv)m (es)i(Theorem)f(3.2)h(with)e Ft(\014)2162 4449 y Fu(c)2223 4435 y Fw(=)2329 4399 y Fr(1)p 2329 4414 36 4 v 2329 4466 a(2)2389 4435 y Fw(log)q(\(1)21 b Fv(\000)f Ft(p)2744 4449 y Fu(c)2779 4435 y Fw(\).)165 4622 y Fx(Prop)s(osition)37 b(6.11)46 b Fp(F)-7 b(or)38 b(the)f(r)-5 b(andom-cluster)40 b(mo)-5 b(del)38 b(on)g Fx(Z)2384 4589 y Fu(d)2424 4622 y Fp(,)g Ft(d)c Fv(\025)f Fw(2)p Fp(,)38 b(and)g(any)g(\014xe)-5 b(d)38 b Ft(q)e Fv(\025)d Fw(1)p Fp(,)165 4735 y(ther)-5 b(e)34 b(exists)f(a)g(p)-5 b(er)g(c)g(olation)36 b(thr)-5 b(eshold)35 b Ft(p)1609 4749 y Fu(c)1669 4735 y Fv(2)25 b Fw(\(0)p Ft(;)15 b Fw(1\))34 b Fp(\(dep)-5 b(ending)34 b(on)f Ft(d)g Fp(and)h Ft(q)s Fp(\))e(such)h(that)1217 5004 y Ft(\036)1271 4967 y Fr(1)1271 5027 y Fu(p;q)1365 5004 y Fw(\()p Ft(x)25 b Fv($)g(1)p Fw(\))1734 4860 y Fs(\()1843 4948 y Fw(=)g(0)84 b Fp(for)33 b Ft(p)25 b(<)g(p)2426 4962 y Fu(c)2485 4948 y Ft(;)1843 5061 y(>)g Fw(0)84 b Fp(for)33 b Ft(p)25 b(>)g(p)2426 5075 y Fu(c)2485 5061 y Ft(:)165 5269 y Fx(Pro)s(of:)42 b Fw(The)30 b(statemen)m(t)i(of)e (the)h(prop)s(osition)d(consists)i(of)g(the)h(follo)m(wing)e(three)h (parts:)211 5456 y(\(i\))g Ft(\036)391 5423 y Fr(1)391 5479 y Fu(p;q)484 5456 y Fw(\()p Ft(x)c Fv($)f(1)p Fw(\))h(=)f(0)30 b(for)h Ft(p)f Fw(su\016cien)m(tly)f(small,)211 5644 y(\(ii\))g Ft(\036)416 5611 y Fr(1)416 5666 y Fu(p;q)510 5644 y Fw(\()p Ft(x)c Fv($)g(1)p Fw(\))h Ft(>)f Fw(0)31 b(for)f Ft(p)g Fw(su\016cien)m(tly)f(close)i(to)g(1,)g(and)1844 5895 y(47)p eop %%Page: 48 48 48 47 bop 211 304 a Fw(\(iii\))29 b Ft(\036)442 271 y Fr(1)442 326 y Fu(p;q)535 304 y Fw(\()p Ft(x)c Fv($)h(1)p Fw(\))k(is)g(increasing)f(in)g Ft(p)p Fw(.)165 502 y(W)-8 b(e)44 b(\014rst)d(pro)m(v)m(e)i(\(i\).)77 b(Supp)s(ose)40 b Ft(p)45 b(<)g(p)1609 516 y Fu(c)1643 502 y Fw(\()p Fx(Z)1742 469 y Fu(d)1783 502 y Ft(;)15 b Fw(b)s(ond)o(\),)45 b(the)e(critical)e(v)-5 b(alue)42 b(for)g(Bernoulli)e(b)s(ond)165 614 y(p)s(ercolation)30 b(on)g Fx(Z)828 581 y Fu(d)868 614 y Fw(.)41 b(F)-8 b(or)31 b Ft(")26 b(>)f Fw(0,)31 b(w)m(e)g(can)f(then)g(pic)m(k)g(\001)g(large)h(enough)f(so)h(that)1557 819 y Ft(\036)1611 833 y Fu(p)1651 819 y Fw(\(0)26 b Fv($)f Fw(\001)1949 781 y Fu(c)1983 819 y Fw(\))h Fv(\024)f Ft(")15 b(:)165 1034 y Fw(By)29 b(Corollary)e(6.7)j(\(b\),)f(w)m(e)g (ha)m(v)m(e)h(that)f(the)g(pro)5 b(jection)28 b(of)h Ft(\036)2302 1048 y Fu(p)2370 1034 y Fw(on)g Fv(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)2715 1000 y Fq(B)2763 977 y Fl(1)2761 1024 y(\001)2850 1034 y Fw(sto)s(c)m(hastically)28 b(dom-)165 1158 y(inates)i(the)h(pro)5 b(jection)30 b(of)h Ft(\036)1175 1125 y Fr(1)1175 1184 y Fu(p;q)r(;)p Fr(\003)1367 1158 y Fw(on)f Fv(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)1713 1124 y Fq(B)1761 1101 y Fl(1)1759 1148 y(\001)1850 1158 y Fw(for)30 b(an)m(y)h(\003)25 b Fv(\033)g Fw(\001,)30 b(so)h(that)1173 1377 y Ft(\036)1227 1339 y Fr(1)1227 1399 y Fu(p;q)r(;)p Fr(\003)1389 1377 y Fw(\(0)26 b Fv($)f(1)p Fw(\))h Fv(\024)f Ft(\036)1913 1339 y Fr(1)1913 1399 y Fu(p;q)r(;)p Fr(\003)2075 1377 y Fw(\(0)h Fv($)f Fw(\001)2373 1339 y Fu(c)2407 1377 y Fw(\))h Fv(\024)f Ft(")165 1581 y Fw(for)30 b(an)m(y)h(\003)26 b Fv(\033)f Fw(\001.)40 b(Since)29 b Ft(")i Fw(w)m(as)g(arbitrary)-8 b(,)30 b(w)m(e)g(\014nd) 1454 1785 y(lim)1432 1852 y Fr(\003)p Fq(")p Fn(Z)1566 1834 y Fo(d)1617 1785 y Ft(\036)1671 1748 y Fr(1)1671 1808 y Fu(p;q)r(;)p Fr(\003)1833 1785 y Fw(\(0)c Fv($)f(1)p Fw(\))g(=)g(0)165 2042 y(whic)m(h)k(in)h(conjunction)f(with)g(\(30\))j (implies)c(\(i\).)306 2154 y(Next,)51 b(\(ii\))46 b(can)g(b)s(e)f (established)f(b)m(y)i(a)h(similar)c(argumen)m(t:)72 b(Let)47 b Ft(p)e Fw(b)s(e)h(suc)m(h)f(that)i Ft(p)3453 2121 y Fq(\003)3543 2154 y Fw(=)165 2267 y Ft(p=)p Fw([)p Ft(p)21 b Fw(+)f(\(1)h Fv(\000)f Ft(p)p Fw(\))p Ft(q)s Fw(])38 b Ft(>)g(p)974 2281 y Fu(c)1008 2267 y Fw(\()p Fx(Z)1107 2234 y Fu(d)1148 2267 y Ft(;)15 b Fw(b)s(ond)o(\).)64 b(Corollary)37 b(6.7)i(\(c\))g(then)f(sho)m(ws)g(that)h Ft(\036)2968 2234 y Fr(1)2968 2293 y Fu(p;q)r(;)p Fr(\003)3168 2267 y Fv(\027)3239 2281 y Fq(D)3337 2267 y Ft(\036)3391 2281 y Fu(p)3427 2262 y Fm(\003)3505 2267 y Fw(for)165 2380 y(ev)m(ery)32 b(\003,)e(so)h(that)1441 2493 y(lim)1419 2561 y Fr(\003)p Fq(")p Fn(Z)1553 2542 y Fo(d)1604 2493 y Ft(\036)1658 2456 y Fr(1)1658 2516 y Fu(p;q)r(;)p Fr(\003)1820 2493 y Fw(\(0)26 b Fv($)f(1)p Fw(\))h Ft(>)f Fw(0)p Ft(;)165 2712 y Fw(pro)m(ving)30 b(\(ii\).)306 2825 y(T)-8 b(o)31 b(c)m(hec)m(k)h(\(iii\))d(w)m(e)i(note)g(that)g(Corollary)e(6.7)i (\(d\))g(implies)c(that,)32 b(for)e(an)m(y)h(\003,)1175 3029 y Ft(\036)1229 2992 y Fr(1)1229 3052 y Fu(p)1265 3061 y Fl(1)1299 3052 y Fu(;q)r(;)p Fr(\003)1451 3029 y Fv(\026)1522 3043 y Fq(D)1608 3029 y Ft(\036)1662 2992 y Fr(1)1662 3052 y Fu(p)1698 3061 y Fl(2)1732 3052 y Fu(;q)r(;)p Fr(\003)1888 3029 y Fw(whenev)m(er)g Ft(p)2334 3043 y Fr(1)2398 3029 y Fv(\024)25 b Ft(p)2540 3043 y Fr(2)2579 3029 y Fw(.)849 b(\(31\))165 3234 y(This)29 b(pro)m(v)m(es)i(\(iii\))e(and)h(thereb)m(y)g(the)h(prop)s(osition.) 1578 b Fk(2)165 3396 y Fw(Before)35 b(the)f(pro)s(of)e(of)i(Theorem)f (6.10)j(w)m(e)e(need)f(another)h(de\014nition)d(and)i(a)h(couple)f(of)g (lemmas.)165 3509 y(F)-8 b(or)31 b(a)g(\014nite)f(b)s(o)m(x)g(\003)g (in)f Fx(Z)1074 3476 y Fu(d)1145 3509 y Fw(and)h(a)g(spin)f (con\014guration)h Ft(\030)f Fv(2)c(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)i(;)e(q)s Fv(g)2677 3476 y Fu(@)t Fr(\003)2772 3509 y Fw(,)31 b(let)1386 3713 y Ft(A)1454 3676 y Fu(i)1454 3736 y(\030)1517 3713 y Fw(=)25 b Fv(f)p Ft(x)h Fv(2)f Ft(@)5 b Fw(\003)25 b(:)41 b Ft(\030)t Fw(\()p Ft(x)p Fw(\))26 b(=)f Ft(i)p Fv(g)165 3939 y Fw(for)j Ft(i)e Fw(=)f(1)p Ft(;)15 b(:)g(:)g(:)i(;)e(q)s Fw(.)40 b(W)-8 b(e)29 b(no)m(w)f(de\014ne)f(the)h Fp(r)-5 b(andom-cluster)33 b(distribution)f Ft(\036)2728 3895 y Fu(\030)2728 3966 y(p;q)r(;)p Fr(\003)2920 3939 y Fp(for)f Fw(\003)f Fp(with)i(b)-5 b(ound-)165 4077 y(ary)44 b(c)-5 b(ondition)45 b Ft(\030)t Fw(,)g(as)d(the)g(probabilit)m(y)d(measure)j(on)f Fv(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)2349 4043 y Fq(B)2397 4019 y Fl(1)2395 4067 y(\003)2489 4077 y Fw(whic)m(h)40 b(to)j(eac)m(h)g Ft(\021)k Fv(2)d(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)3516 4043 y Fq(B)3564 4019 y Fl(1)3562 4067 y(\003)165 4189 y Fw(assigns)30 b(probabilit)m(y)e(prop)s(ortional)g(to)1147 4429 y Ft(I)1187 4448 y Fu(D)r Fr(\()p Fu(\030)s(;\021)r Fr(\))1412 4310 y Fs(\032)1522 4348 y(Y)1489 4542 y Fu(e)p Fq(2B)1617 4519 y Fl(1)1615 4566 y(\003)1676 4429 y Ft(p)1722 4392 y Fu(\021)r Fr(\()p Fu(e)p Fr(\))1851 4429 y Fw(\(1)21 b Fv(\000)f Ft(p)p Fw(\))2124 4392 y Fr(1)p Fq(\000)p Fu(\021)r Fr(\()p Fu(e)p Fr(\))2344 4310 y Fs(\033)2421 4429 y Ft(q)2465 4392 y Fu(k)2502 4404 y Fo(\030)2536 4392 y Fr(\()p Fu(\021)r Fr(\))165 4742 y Fw(where)29 b Ft(k)474 4757 y Fu(\030)513 4742 y Fw(\()p Ft(\021)s Fw(\))h(is)f(the)h(n)m(um)m(b)s(er)e(of)i(connected)g(comp)s(onen)m(ts) g(in)f Ft(\021)j Fw(that)f(do)e(not)h(in)m(tersect)g Ft(@)5 b Fw(\003,)31 b(and)165 4855 y Ft(D)s Fw(\()p Ft(\030)t(;)15 b(\021)s Fw(\))35 b(is)d(the)h(ev)m(en)m(t)i(that)e (there)h(is)e(no)h(op)s(en)f(path)h(in)f Ft(\021)k Fw(connecting)e(an)m (y)f(t)m(w)m(o)i(v)m(ertices)e(in)f Ft(A)3576 4822 y Fu(i)3576 4882 y(\030)165 4991 y Fw(and)e Ft(A)410 4947 y Fu(j)410 5019 y(\030)478 4991 y Fw(for)h(an)m(y)f Ft(i)c Fv(6)p Fw(=)f Ft(j)5 b Fw(.)165 5194 y Fx(Lemma)33 b(6.12)46 b Fp(L)-5 b(et)35 b Ft(p)29 b Fw(=)f(1)23 b Fv(\000)e Ft(e)1302 5161 y Fq(\000)p Fr(2)p Fu(\014)1440 5194 y Fp(,)35 b(let)f Fw(\003)h Fp(b)-5 b(e)34 b(a)h(\014nite)g(r)-5 b(e)g(gion)36 b(in)e Fx(Z)2611 5161 y Fu(d)2652 5194 y Fp(,)h(and)g(\014x)g(some)g(b)-5 b(oundary)165 5307 y(c)g(ondition)45 b Ft(\030)i Fv(2)c(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)j(;)d(q)s Fv(g)1146 5274 y Fu(@)t Fr(\003)1240 5307 y Fp(.)72 b(Supp)-5 b(ose)44 b(that)g(we)f(pick)g(a)g(r)-5 b(andom)45 b(spin)e(c)-5 b(on\014gur)g(ation)45 b Ft(X)51 b Fv(2)165 5420 y(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)j(;)d(q)s Fv(g)547 5387 y Fr(\003)633 5420 y Fp(as)33 b(fol)5 b(lows.)273 5616 y(1.)46 b(Pick)31 b Ft(Y)46 b Fv(2)24 b(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)1002 5582 y Fq(B)1050 5559 y Fl(1)1048 5606 y(\003)1133 5616 y Fp(ac)-5 b(c)g(or)g(ding)35 b(to)e Ft(\036)1697 5572 y Fu(\030)1697 5643 y(p;q)r(;)p Fr(\003)1859 5616 y Fp(.)1844 5895 y Fw(48)p eop %%Page: 49 49 49 48 bop 273 304 a Fp(2.)46 b(F)-7 b(or)42 b(e)-5 b(ach)42 b Ft(i)f Fv(2)g(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)i(;)e(q)s Fv(g)42 b Fp(and)g(e)-5 b(ach)42 b(c)-5 b(onne)g(cte)g(d)43 b(c)-5 b(omp)g(onent)43 b Ft(C)48 b Fp(in)41 b Ft(\021)k Fp(interse)-5 b(cting)41 b Ft(A)3548 271 y Fu(i)3548 330 y(\030)3586 304 y Fp(,)393 417 y(assign)33 b(spin)g Ft(i)g Fp(to)g(every)f(vertex)h(in)g Ft(C)7 b Fp(.)273 604 y(3.)46 b(F)-7 b(or)39 b(al)5 b(l)39 b(other)g(c)-5 b(onne)g(cte)g(d)40 b(c)-5 b(omp)g(onents)41 b Ft(C)j Fp(in)38 b Ft(Y)20 b Fp(,)40 b(pick)e(a)g(spin)h(at)g(r)-5 b(andom)41 b(\(uniformly\))393 717 y(fr)-5 b(om)37 b Fv(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)j(;)d(q)s Fv(g)p Fp(,)37 b(assign)g(this)g(spin)f(to)h(every)f(vertex)g(of)h Ft(C)7 b Fp(,)36 b(and)h(do)g(this)g(indep)-5 b(endently)393 830 y(for)33 b(di\013er)-5 b(ent)33 b(c)-5 b(onne)g(cte)g(d)34 b(c)-5 b(omp)g(onents.)165 1017 y(Then)34 b Ft(X)40 b Fp(is)33 b(distribute)-5 b(d)35 b(ac)-5 b(c)g(or)g(ding)35 b(to)e(the)h(Gibbs)f(distribution)h Ft(\026)2518 973 y Fu(\030)2518 1045 y(\014)s(;q)r(;)p Fr(\003)2720 1017 y Fp(for)g(the)f(Potts)h(mo)-5 b(del)35 b(on)165 1130 y Fw(\003)e Fp(with)g(b)-5 b(oundary)35 b(c)-5 b(ondition)34 b Ft(\030)t Fp(.)165 1317 y Fx(Pro)s(of:)39 b Fw(This)24 b(is)g(a)i(straigh)m(tforw)m(ard)f(generalization)g(of)h(the)f(pro)s (ofs)g(of)g(Theorem)g(6.2)i(and)d(Corol-)165 1430 y(lary)30 b(6.3.)3056 b Fk(2)165 1617 y Fx(Lemma)33 b(6.13)46 b Fp(With)g(notation)h(as)e(ab)-5 b(ove,)49 b(we)c(have,)k(for)c(any)h Ft(\030)52 b Fv(2)47 b(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)j(;)d(q)s Fv(g)3127 1584 y Fu(@)t Fr(\003)3221 1617 y Fp(,)48 b(that)f(the)165 1738 y(pr)-5 b(oje)g(ction)35 b(of)e Ft(\036)748 1705 y Fr(1)748 1764 y Fu(p;q)r(;)p Fr(\003)942 1738 y Fp(on)g Fv(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)1292 1705 y Fq(B)1340 1681 y Fl(1)1338 1728 y(\003)1423 1738 y Fp(sto)-5 b(chastic)g(al)5 b(ly)36 b(dominates)e Ft(\036)2462 1694 y Fu(\030)2462 1766 y(p;q)r(;)p Fr(\003)2624 1738 y Fp(.)165 1951 y Fx(Pro)s(of:)40 b Fw(Just)26 b(write)h(do)m(wn)f(single-edge)h (conditional)e(distributions)f(for)j Ft(\036)2790 1907 y Fu(\030)2790 1978 y(p;q)r(;)p Fr(\003)2978 1951 y Fw(and)g Ft(\036)3206 1918 y Fr(1)3206 1977 y Fu(p;q)r(;)p Fr(\003)3395 1951 y Fw(\(as)g(in)165 2064 y(Lemma)k(6.6\))h(and)d(in)m(v)m(ok)m(e)j (Theorem)e(4.8.)1893 b Fk(2)165 2227 y Fw(W)-8 b(e)32 b(are)f(\014nally)d(ready)i(for)g(the)h(pro)s(of)f(of)g(Theorem)g (6.10.)165 2389 y Fx(Pro)s(of)f(of)g(Theorem)e(6.10:)39 b Fw(W)-8 b(e)25 b(b)s(egin)f(with)f(the)i(implication)d(\(i\))i Fv(\))h Fw(\(ii\).)37 b(If)25 b(there)f(is)g(a)h(unique)165 2502 y(Gibbs)e(measure)g(for)h(the)g Ft(q)s Fw(-state)h(P)m(otts)g(mo)s (del)e(on)h Fx(Z)2026 2469 y Fu(d)2090 2502 y Fw(at)g(in)m(v)m(erse)g (temp)s(erature)g Ft(\014)5 b Fw(,)25 b(then)f(w)m(e)g(ha)m(v)m(e)165 2615 y(in)29 b(particular)g(that)1540 2728 y Ft(\026)1595 2690 y Fr(1)1595 2750 y Fu(\014)s(;q)1721 2728 y Fw(=)c Ft(:)15 b(:)g(:)26 b Fw(=)f Ft(\026)2099 2683 y Fu(q)2099 2756 y(\014)s(;q)2214 2728 y Ft(:)165 2919 y Fw(But)35 b(since)f(b)m(y)g(symmetry)g Ft(\026)1191 2886 y Fu(i)1191 2945 y(\014)s(;q)1291 2919 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))34 b(=)d Ft(i)p Fw(\))i(=)f Ft(\026)1923 2874 y Fu(j)1923 2947 y(\014)s(;q)2023 2919 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))33 b(=)f Ft(j)5 b Fw(\))35 b(for)f(an)m(y)h Ft(i;)15 b(j)38 b Fv(2)32 b(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)i(;)e(q)s Fv(g)p Fw(,)37 b(w)m(e)165 3045 y(m)m(ust)31 b(ha)m(v)m(e)g Ft(\026)655 3012 y Fu(i)655 3071 y(\014)s(;q)756 3045 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))26 b(=)f Ft(i)p Fw(\))h(=)f(1)p Ft(=q)s Fw(,)32 b(and)d(\(ii\))h(is)f(established.)306 3158 y(Next)f(w)m(e)f (turn)f(to)h(the)g(implication)d(\(ii\))i Fv(\))g Fw(\(iii\).)38 b(By)27 b(the)g(Edw)m(ards{Sok)-5 b(al)26 b(coupling)f(of)h(edge)165 3271 y(and)k(site)g(pro)s(cesses)g(in)m(tro)s(duced)f(b)s(efore)h(Prop) s(osition)f(6.9,)i(w)m(e)g(ha)m(v)m(e)878 3474 y Ft(\026)933 3437 y Fu(i)933 3497 y(\014)s(;q)1034 3474 y Fw(\()p Ft(X)7 b Fw(\(0\))27 b(=)e Ft(i)p Fw(\))84 b(=)104 b(lim)1692 3541 y Fr(\003)p Fq(")p Fn(Z)1826 3523 y Fo(d)1878 3474 y Ft(\026)1933 3437 y Fu(i)1933 3497 y(\014)s(;q)r(;)p Fr(\003)2102 3474 y Fw(\()p Ft(X)7 b Fw(\(0\))27 b(=)e Ft(i)p Fw(\))1539 3716 y(=)1702 3654 y(1)p 1702 3695 46 4 v 1703 3778 a Ft(q)1778 3716 y Fw(+)1879 3654 y Ft(q)e Fv(\000)d Fw(1)p 1879 3695 201 4 v 1957 3778 a Ft(q)2126 3716 y Fw(lim)2104 3783 y Fr(\003)p Fq(")p Fn(Z)2238 3764 y Fo(d)2290 3716 y Ft(\036)2344 3678 y Fr(1)2344 3738 y Fu(p;q)r(;)p Fr(\003)2505 3716 y Fw(\(0)26 b Fv($)g Fw(\003)2791 3678 y Fu(c)2825 3716 y Fw(\))15 b Ft(:)553 b Fw(\(32\))165 3972 y(T)-8 b(ogether)32 b(with)d(\(30\),)j (the)f(result)e(follo)m(ws.)306 4084 y(Most)37 b(of)e(the)h(w)m(ork)g (is)e(needed)h(for)g(the)h(implication)d(\(iii\))h Fv(\))h Fw(\(i\).)56 b(Roughly)34 b(sp)s(eaking,)i(the)165 4197 y(absence)24 b(of)g(p)s(ercolation)e(in)h(the)g(random-cluster)g(mo)s (del)f(implies)f(that)j(ev)m(ery)g(\014nite)e(region)h(is)g(cut)165 4310 y(o\013)31 b(from)e(in\014nit)m(y)f(b)m(y)i(a)g(set)g(of)g(closed) g(edges.)41 b(Th)m(us,)29 b(indep)s(enden)m(tly)e(of)j(what)g(happ)s (ens)e(macro-)165 4423 y(scopically)-8 b(,)33 b(the)g(lo)s(cal)f(spins) f(feel)h(as)h(if)f(they)h(are)g(in)e(a)i(system)g(with)f(free)h(b)s (oundary)d(condition.)165 4536 y(This)f(mak)m(es)i(a)g(phase)f (transition)f(imp)s(ossible.)306 4649 y(T)-8 b(o)30 b(mak)m(e)g(this)d (in)m(tuition)g(precise)h(w)m(e)h(let)g Ft(\026)g Fw(b)s(e)f(an)h (arbitrary)e(Gibbs)g(measure)i(for)g(the)g(P)m(otts)165 4762 y(mo)s(del)d(at)j(in)m(v)m(erse)e(temp)s(erature)g Ft(\014)5 b Fw(.)40 b(W)-8 b(e)28 b(will)d(sho)m(w)i(that)h Ft(\026)d Fw(=)g Ft(\026)2444 4729 y Fr(0)2444 4788 y Fu(\014)s(;q)2544 4762 y Fw(,)j(the)g(limiting)c(measure)j(with)165 4875 y(free)h(b)s(oundary)e(condition)h(obtained)g(in)g(Prop)s(osition) f(6.9.)41 b(W)-8 b(e)29 b(\014x)e(an)m(y)h(lo)s(cal)f(observ)-5 b(able)28 b Ft(f)37 b Fw(and)165 4988 y(some)31 b Ft(")26 b(>)f Fw(0.)41 b(W)-8 b(e)31 b(then)f(can)h(\014nd)e(a)i(\014nite)e(b)s (o)m(x)h(\001)25 b Fv(\032)g Fx(Z)2123 4955 y Fu(d)2194 4988 y Fw(suc)m(h)30 b(that)961 5094 y Fs(\014)961 5144 y(\014)961 5193 y(\014)1014 5191 y Ft(\026)1069 5154 y Fr(0)1069 5214 y Fu(\014)s(;q)r(;)p Fr(\000)1233 5191 y Fw(\()p Ft(f)10 b Fw(\))20 b Fv(\000)g Ft(\026)1524 5154 y Fr(0)1524 5214 y Fu(\014)s(;q)1625 5191 y Fw(\()p Ft(f)10 b Fw(\))1765 5094 y Fs(\014)1765 5144 y(\014)1765 5193 y(\014)1818 5191 y Ft(<)25 b(")61 b Fw(for)30 b(all)f(\014nite)g (\000)c Fv(\033)g Fw(\001)g Ft(:)635 b Fw(\(33\))165 5418 y(In)28 b(view)f(of)h(\(30\))h(w)m(e)g(can)f(also)g(c)m(ho)s(ose)h (a)f(\014nite)f(b)s(o)m(x)h(\003)d Fv(\033)g Fw(\001)i(satisfying)g Ft(\036)2755 5385 y Fr(1)2755 5443 y Fu(p;q)r(;)p Fr(\003)2917 5418 y Fw(\()p Ft(x)e Fv($)h Fw(\003)3209 5385 y Fu(c)3243 5418 y Fw(\))g Ft(<)f("=)p Fv(j)p Fw(\001)p Fv(j)165 5531 y Fw(for)30 b(all)g Ft(x)25 b Fv(2)g Fw(\001,)30 b(and)g(th)m(us)1482 5644 y Ft(\036)1536 5606 y Fr(1)1536 5666 y Fu(p;q)r(;)p Fr(\003)1698 5644 y Fw(\(\001)25 b Fv($)g Fw(\003)2013 5606 y Fu(c)2048 5644 y Fw(\))h Ft(<)f(")g(:)1844 5895 y Fw(49)p eop %%Page: 50 50 50 49 bop 165 304 a Fw(Here,)32 b Fv(f)p Fw(\001)25 b Fv($)g Fw(\003)731 271 y Fu(c)766 304 y Fv(g)31 b Fw(is)e(the)i(ev)m (en)m(t)h(that)f(there)f(exists)g(an)g(op)s(en)g(path)g(from)g(\001)g (to)h(\003)3063 271 y Fu(c)3098 304 y Fw(.)306 417 y(Consider)37 b(the)i(complemen)m(tary)f(ev)m(en)m(t)i Ft(C)45 b Fw(=)39 b Fv(f)p Fw(\001)f Fv($)h Fw(\003)2323 384 y Fu(c)2358 417 y Fv(g)2403 384 y Fu(c)2438 417 y Fw(.)65 b(F)-8 b(or)39 b(an)m(y)g(edge)g(con\014guration)165 530 y Ft(\021)34 b Fv(2)29 b Ft(C)7 b Fw(,)33 b(\001)g(is)f(cut)i(o\013)f(from)g(\003) 1239 497 y Fu(c)1307 530 y Fw(b)m(y)g(a)g(set)h(of)f(closed)g(edges.)50 b(Indeed,)33 b(let)g(\000)g(b)s(e)f(the)i(union)d(of)i(all)165 643 y Ft(\021)s Fw(-op)s(en)e(clusters)e(meeting)i(\001.)40 b(Then)211 825 y(\(a\))31 b(\001)25 b Fv(\032)g Fw(\000)g Fv(\032)g Fw(\003,)31 b(and)211 1010 y(\(b\))f Ft(\021)s Fw(\()p Ft(e)p Fw(\))d(=)e(0)31 b(for)f(all)f(edges)i(from)f(\000)g(to) h(\000)1699 977 y Fu(c)1734 1010 y Fw(,)f(i.e.,)h Ft(\021)e Fv(\021)c Fw(0)30 b(on)h Fv(B)2396 977 y Fr(1)2393 1036 y(\000)2461 1010 y Fv(n)20 b(B)2589 977 y Fr(0)2586 1036 y(\000)2634 1010 y Fw(.)165 1192 y(Since)42 b(these)h(prop)s(erties)e (are)i(stable)f(under)f(\014nite)h(unions,)h(there)g(exists)f(a)h (largest)g(set)g(\000\()p Ft(\021)s Fw(\))165 1305 y(satisfying)31 b(\(a\))i(and)e(\(b\).)45 b(The)31 b(maximalit)m(y)g(implies)e(that,)k (for)f(eac)m(h)h(\014xed)e(region)g(\000,)i(the)f(ev)m(en)m(t)165 1417 y Fv(f)p Ft(\021)d Fw(:)d(\000\()p Ft(\021)s Fw(\))g(=)f(\000)p Fv(g)31 b Fw(only)e(dep)s(ends)g(on)h(the)h(status)f(of)h(the)f(edges)h (o\013)g Fv(B)2562 1384 y Fr(0)2559 1443 y(\000)2607 1417 y Fw(.)306 1530 y(Coming)37 b(to)g(the)h(core)g(of)f(the)g (argumen)m(t,)j(w)m(e)d(\014x)g(an)m(y)g(b)s(oundary)e(spin)h (con\014guration)g Ft(\030)41 b Fv(2)165 1643 y(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)j(;)d(q)s Fv(g)547 1610 y Fu(@)t Fr(\003)677 1643 y Fw(and)35 b(consider)g(the)h(Edw)m(ards{Sok)-5 b(al)34 b(coupling)g Ft(P)47 b Fw(=)33 b Ft(P)2663 1599 y Fu(\030)2650 1671 y(p;q)r(;)p Fr(\003)2848 1643 y Fw(of)j Ft(\026)3012 1599 y Fu(\030)3012 1671 y(\014)s(;q)r(;)p Fr(\003)3216 1643 y Fw(and)f Ft(\036)3452 1599 y Fu(\030)3452 1671 y(p;q)r(;)p Fr(\003)165 1784 y Fw(in)m(tro)s(duced)20 b(in)f(Lemma)i(6.12.)39 b(W)-8 b(e)23 b(write)d Ft(X)28 b Fw(for)21 b(the)g(random)f(spin)f(con\014guration)h(in)g Fv(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)i(;)e(q)s Fv(g)3523 1751 y Fn(Z)3573 1727 y Fo(d)165 1905 y Fw(and)30 b Ft(Y)50 b Fw(for)31 b(the)f(random)g(edge)h(con\014guration)f(in)f Fv(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)2159 1872 y Fq(B)2213 1905 y Fw(.)41 b(Then)29 b Ft(P)13 b Fw(\()p Ft(f)30 b Fv(\016)20 b Ft(X)7 b Fw(\))26 b(=)f Ft(\026)3056 1861 y Fu(\030)3056 1933 y(\014)s(;q)r(;)p Fr(\003)3225 1905 y Fw(\()p Ft(f)10 b Fw(\),)31 b(and)1052 2129 y Ft(P)13 b Fw(\()p Ft(Y)45 b Fv(2)25 b Ft(C)7 b Fw(\))25 b(=)g Ft(\036)1624 2085 y Fu(\030)1624 2156 y(p;q)r(;)p Fr(\003)1786 2129 y Fw(\()p Ft(C)7 b Fw(\))26 b Fv(\025)f Ft(\036)2104 2091 y Fr(1)2104 2151 y Fu(p;q)r(;)p Fr(\003)2265 2129 y Fw(\()p Ft(C)7 b Fw(\))26 b Ft(>)f Fw(1)20 b Fv(\000)g Ft(")165 2326 y Fw(b)m(y)29 b(Lemma)h(6.13)h(and)d(the)i(c)m(hoice)g (of)f(\003.)40 b(Assuming)28 b(without)g(loss)h(of)g(generalit)m(y)g (that)h Fv(k)p Ft(f)10 b Fv(k)26 b(\024)e Fw(1,)165 2439 y(w)m(e)31 b(can)g(therefore)g(conclude)e(that)1165 2539 y Fs(\014)1165 2589 y(\014)1165 2639 y(\014)1192 2636 y Ft(P)13 b Fw(\()q Ft(f)29 b Fv(\016)21 b Ft(X)h Fv(j)16 b Ft(Y)45 b Fv(2)25 b Ft(C)7 b Fw(\))20 b Fv(\000)g Ft(\026)2034 2592 y Fu(\030)2034 2665 y(\014)s(;q)r(;)p Fr(\003)2203 2636 y Fw(\()p Ft(f)10 b Fw(\))2328 2539 y Fs(\014)2328 2589 y(\014)2328 2639 y(\014)2381 2636 y Ft(<)25 b Fw(2)p Ft(")h(:)838 b Fw(\(34\))165 2844 y(Ho)m(w)m(ev)m(er,)43 b(the)c(conditional)e(exp)s(ectation)h(on)h(the)f(left)h(is)e(an)h(a)m (v)m(erage)j(of)e(the)g(conditional)d(ex-)165 2957 y(p)s(ectations)j Ft(P)13 b Fw(\()p Ft(f)35 b Fv(\016)26 b Ft(X)d Fv(j)15 b Fw(\000\()p Ft(Y)21 b Fw(\))39 b(=)f(\000\))h(with)e(\001)i Fv(\032)g Fw(\000)g Fv(\032)f Fw(\003,)j(and)d(these)h(in)e(turn)h(are) h(a)m(v)m(erages)i(of)165 3070 y(conditional)29 b(exp)s(ectations)i(of) f(the)h(form)1033 3267 y Ft(P)13 b Fw(\()p Ft(f)30 b Fv(\016)21 b Ft(X)32 b Fv(j)26 b Ft(Y)45 b Fw(=)25 b Ft(\021)34 b Fw(o\013)d Fv(B)1903 3230 y Fr(1)1900 3290 y(\000)1947 3267 y Ft(;)41 b(Y)k Fv(\021)25 b Fw(0)31 b(on)f Fv(B)2472 3230 y Fr(1)2469 3290 y(\000)2537 3267 y Fv(n)21 b(B)2666 3230 y Fr(0)2663 3290 y(\000)2711 3267 y Fw(\))165 3465 y(whic)m(h,)37 b(b)m(y)f(construction)f(of)i Ft(P)13 b Fw(,)37 b(are)g(equal)e(to)i Ft(\026)1934 3432 y Fr(0)1934 3491 y Fu(\014)s(;q)r(;)p Fr(\000)2098 3465 y Fw(\()p Ft(f)10 b Fw(\).)58 b(T)-8 b(ogether)37 b(with)e(\(33\))j (and)d(\(34\),)40 b(w)m(e)165 3577 y(conclude)30 b(that)1342 3593 y Fs(\014)1342 3643 y(\014)1342 3693 y(\014)1395 3690 y Ft(\026)1450 3646 y Fu(\030)1450 3718 y(\014)s(;q)r(;)p Fr(\003)1619 3690 y Fw(\()p Ft(f)10 b Fw(\))20 b Fv(\000)g Ft(\026)1910 3653 y Fr(0)1910 3713 y Fu(\014)s(;q)2010 3690 y Fw(\()p Ft(f)10 b Fw(\))2150 3593 y Fs(\014)2150 3643 y(\014)2150 3693 y(\014)2203 3690 y Ft(<)25 b Fw(3)p Ft(")h(:)165 3877 y Fw(T)-8 b(aking)35 b(the)f Ft(\026)p Fw(-a)m(v)m(erage)k(o)m(v)m(er)e Ft(\030)i Fw(and)c(letting)g Ft(")f Fv(!)f Fw(0)j(w)m(e)g(\014nally)e(get)j 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280 y(^)2339 304 y Ft(\036)2393 271 y Fu(B)s(;\030)2393 326 y(p;q)2546 304 y Fw(with)37 b Ft(\030)44 b Fv(\021)39 b Fw(1.)67 b(W)-8 b(e)40 b(can)f(omit)165 427 y(the)f(hat)h(when)e Ft(G)h Fw(=)f Fx(Z)1025 394 y Fu(d)1066 427 y Fw(,)j(b)s(ecause)e (there)g(is)f(alw)m(a)m(ys)h(exactly)h(one)f(in\014nite)e(cluster)h (regardless)165 540 y(of)k(the)g(con\014guration)f(on)h Ft(B)k Fw(\(this)40 b(is)f(related)i(to)g(Prop)s(osition)e(6.19)j(b)s (elo)m(w\).)71 b(On)40 b(the)h(other)165 653 y(hand,)26 b(the)g(t)m(w)m(o)h(di\013eren)m(t)e(w)m(a)m(ys)h(of)g(coun)m(ting)f (clusters)g(with)f(wired)g(b)s(oundary)g(condition)g(are)i(not)165 766 y(equiv)-5 b(alen)m(t)37 b(for)g(all)g(graph)g(structures;)j(a)e (simple)e(coun)m(terexample)h(is)g Ft(G)g Fw(=)g Fx(T)3030 781 y Fu(d)3108 766 y Fw(in)f(whic)m(h)g(the)165 878 y(wired)29 b(b)s(oundary)f(condition)h(giv)m(es)i(rise)f(to)h(sev)m (eral)f(in\014nite)e(clusters.)306 991 y(Still)36 b(in)h(the)h(con)m (text)h(of)f(general)g(in\014nite)e(graphs,)j(w)m(e)g(write)e Ft(B)42 b Fv(")c(B)j Fw(for)c(the)h(limit)e(along)165 1104 y(some)27 b(\(an)m(y\))g(sequence)g(of)f(\014nite)f(simply)f (connected)j(regions)e(increasing)g(to)i Fv(B)i Fw(in)24 b(the)j(usual)d(w)m(a)m(y)-8 b(.)165 1217 y(In)28 b(complete)i(analogy) f(to)h(\(29\))g(and)e(Lemma)h(6.8)h(w)m(e)g(then)e(obtain)h(the)g (follo)m(wing)e(monotonicit)m(y)165 1330 y(and)j(con)m(v)m(ergence)j (result.)165 1498 y Fx(Lemma)g(6.16)46 b Fp(F)-7 b(or)34 b Ft(p)25 b Fv(2)g Fw([0)p Ft(;)15 b Fw(1])p Fp(,)34 b Ft(q)28 b Fv(\025)d Fw(1)p Fp(,)33 b(and)h(any)f(two)g(\014nite)g(b) -5 b(ond)34 b(sets)f Ft(B)2824 1512 y Fr(1)2888 1498 y Fv(\032)25 b Ft(B)3053 1512 y Fr(2)3093 1498 y Fp(,)32 b(we)h(have)211 1666 y Fw(\(i\))63 b Ft(\036)424 1633 y Fu(B)477 1642 y Fl(1)512 1633 y Fu(;)p Fr(0)424 1688 y Fu(p;q)596 1666 y Fv(\026)667 1680 y Fq(D)753 1666 y Ft(\036)807 1633 y Fu(B)860 1642 y Fl(2)895 1633 y Fu(;)p Fr(0)807 1688 y Fu(p;q)954 1666 y Fp(,)32 b(so)h(that)h(the)f (limit)g Ft(\036)1737 1633 y Fr(0)1737 1688 y Fu(p;q)1856 1666 y Fw(=)e(lim)1952 1726 y Fu(B)s Fq("B)2106 1666 y Ft(\036)2160 1633 y Fu(B)s(;)p Fr(0)2160 1688 y Fu(p;q)2309 1666 y Fp(exists;)h(and)211 1909 y Fw(\(ii\))376 1885 y(^)365 1909 y Ft(\036)419 1876 y Fu(B)472 1885 y Fl(1)507 1876 y Fu(;)p Fr(1)419 1932 y Fu(p;q)591 1909 y Fv(\027)662 1923 y Fq(D)759 1885 y Fw(^)747 1909 y Ft(\036)801 1876 y Fu(B)854 1885 y Fl(2)889 1876 y Fu(;)p Fr(1)801 1932 y Fu(p;q)948 1909 y Fp(,)h(so)g(that)h(the)f(limit)1690 1885 y Fw(^)1678 1909 y Ft(\036)1732 1876 y Fr(1)1732 1932 y Fu(p;q)1851 1909 y Fw(=)e(lim)1947 1969 y Fu(B)s Fq("B)2113 1885 y Fw(^)2101 1909 y Ft(\036)2155 1876 y Fu(B)s(;)p Fr(1)2155 1932 y Fu(p;q)2304 1909 y Fp(exists.)165 2159 y Fw(No)m(w)g(let)f Ft(\036)g Fw(b)s(e)g(an)m(y)g(random-cluster)f (measure)h(of)g(either)g(t)m(yp)s(e,)g(compacti\014ed)g(or)h(not,)f (with)f(the)165 2272 y(giv)m(en)d(parameters)f Ft(p)g Fw(and)f Ft(q)s Fw(.)39 b(F)-8 b(urther)25 b(application)f(of)h (Theorem)g(4.8)h(\(or)f(Corollary)f(6.7\))j(implies)165 2385 y(that)1513 2498 y Ft(\036)1567 2460 y Fr(0)1567 2520 y Fu(p;q)1686 2498 y Fv(\026)1757 2512 y Fq(D)1842 2498 y Ft(\036)f Fv(\026)1993 2512 y Fq(D)2090 2474 y Fw(^)2079 2498 y Ft(\036)2133 2460 y Fr(1)2133 2520 y Fu(p;q)2241 2498 y Ft(:)165 2653 y Fw(This)j(is)g(analogous)i(to)g(the) g(sandwic)m(hing)d(relation)h(\(15\))j(for)e(the)h(Ising)e(mo)s(del.) 306 2766 y(F)-8 b(urthermore,)26 b(the)f(argumen)m(ts)g(of)f(Section)h (6.3)g(go)h(through)d(to)j(sho)m(w)e(that)h(the)g Ft(q)s Fw(-state)h(P)m(otts)165 2878 y(mo)s(del)37 b(on)g Ft(G)h Fw(at)g(in)m(v)m(erse)f(temp)s(erature)g Ft(\014)43 b Fw(has)37 b(a)h(unique)e(Gibbs)f(measure)j(if)e(and)h(only)g(if)f(the) 177 2967 y(^)165 2991 y Ft(\036)219 2958 y Fr(1)219 3014 y Fu(p;q)313 2991 y Fw(-probabilit)m(y)28 b(of)i(ha)m(ving)g(an)h (in\014nite)d(cluster)h(is)h(0,)h(where)f(as)g(usual)f Ft(p)c Fw(=)g(1)c Fv(\000)f Ft(e)3103 2958 y Fq(\000)p Fr(2)p Fu(\014)3240 2991 y Fw(.)306 3104 y(F)-8 b(or)42 b(De\014nitions)d(6.14)j(and)e(6.15)i(to)f(b)s(e)f(of)h(in)m(terest,)j (w)m(e)d(ha)m(v)m(e)g(to)h(establish)d(at)i(least)g(the)165 3217 y(existence)33 b(of)g(random-cluster)e(measures)i(of)f(the)h(t)m (w)m(o)h(t)m(yp)s(es.)47 b(The)32 b(follo)m(wing)f(theorem)i(tells)f (us)165 3330 y(that)42 b(at)g(least)g(for)f Ft(q)46 b Fv(\025)d Fw(1,)i(suc)m(h)c(measures)g(do)g(exist.)73 b(\(The)41 b(existence)h(problem)d(for)i Ft(q)47 b(<)c Fw(1)165 3443 y(remains)31 b(op)s(en)g(in)f(the)i(setting)g(of)f (general)h(graphs,)f(although)g(existence)i(has)e(b)s(een)g (established)165 3556 y(for)f Fx(Z)368 3523 y Fu(d)439 3556 y Fw(and)g Fx(T)689 3571 y Fu(d)729 3556 y Fw(;)h(see)g([109)q(])g (and)f([114)q(],)h(resp)s(ectiv)m(ely)-8 b(.\))165 3724 y Fx(Theorem)34 b(6.17)47 b Fp(F)-7 b(or)43 b Ft(p)e Fv(2)h Fw([0)p Ft(;)15 b Fw(1])44 b Fp(and)f Ft(q)i Fv(\025)c Fw(1)p Fp(,)k(we)d(have)g(that)h(\(i\))f Ft(\036)2691 3691 y Fr(0)2691 3746 y Fu(p;q)2827 3724 y Fp(is)f(a)i(r)-5 b(andom-cluster)165 3855 y(me)g(asur)g(e,)34 b(and)g(\(ii\))896 3831 y Fw(^)884 3855 y Ft(\036)938 3822 y Fr(1)938 3877 y Fu(p;q)1064 3855 y Fp(is)f(a)g(C-r)-5 b(andom-cluster)34 b(me)-5 b(asur)g(e.)165 4022 y Fw(F)d(or)30 b(the)f(pro)s(of)f(w)m(e)h (use)f(the)h(follo)m(wing)e(lemma)h(whic)m(h)g(c)m(haracterizes)i (random-cluster)d(measures)165 4135 y(in)k(terms)g(of)h(single-edge)g (conditional)e(probabilities.)41 b(F)-8 b(or)33 b Ft(e)28 b Fw(=)f Fv(h)p Ft(xy)s Fv(i)h(2)f(B)34 b Fw(and)d Ft(\030)h Fv(2)27 b(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)3396 4102 y Fq(B)r(nf)p Fu(e)p Fq(g)3589 4135 y Fw(,)165 4248 y(w)m(e)28 b(write)e(as)h(usual)e Fv(f)p Ft(x)h Fv($)f Ft(y)s Fv(g)i Fw(for)g(the)g(ev)m(en)m(t)i(that)e(there)g(exists)g(an)g(op)s(en)f (path)h(in)e Ft(\030)31 b Fw(from)c Ft(x)f Fw(to)i Ft(y)s Fw(.)165 4383 y(W)-8 b(e)34 b(also)f(write)e Fv(f)p Ft(x)926 4333 y Fu(C)870 4383 y Fv( )-15 b(!)28 b Ft(y)s Fv(g)33 b Fw(for)f(the)h(ev)m(en)m(t)h(that)f(there)g(either)f(exists)g(an)h (op)s(en)f(path)g(in)f Ft(\030)37 b Fw(from)165 4496 y Ft(x)g Fw(to)g Ft(y)s Fw(,)h(or)f Ft(x)f Fw(and)g Ft(y)j Fw(are)e(b)s(oth)f(in)f(in\014nite)f(clusters)i(of)h Ft(\030)t Fw(.)59 b(W)-8 b(e)38 b(think)d(of)h(this)g(C-connectivit)m (y)165 4626 y(notion)30 b Ft(x)581 4576 y Fu(C)525 4626 y Fv( )-15 b(!)25 b Ft(y)33 b Fw(as)e(allo)m(wing)e(paths)h(b)s(et)m(w) m(een)h Ft(x)f Fw(and)g Ft(y)j Fw(to)e(go)h(\\via)e(in\014nit)m(y".)165 4793 y Fx(Lemma)j(6.18)46 b Fp(Fix)36 b Ft(p)31 b Fv(2)g Fw([0)p Ft(;)15 b Fw(1])38 b Fp(and)e Ft(q)e(>)d Fw(0)p Fp(,)37 b(and)g(let)f Ft(\036)f Fp(b)-5 b(e)36 b(a)g(pr)-5 b(ob)g(ability)39 b(me)-5 b(asur)g(e)37 b(on)f Fv(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)3532 4760 y Fq(B)3586 4793 y Fp(.)165 4906 y(Then)37 b Ft(\036)g Fp(is)f(a)h(r)-5 b(andom-cluster)38 b(me)-5 b(asur)g(e)38 b(for)f Ft(p)f Fp(and)h Ft(q)i Fp(if)e(and)g(only)g(if)f(for)h(e)-5 b(ach)37 b Ft(e)c Fw(=)e Fv(h)p Ft(xy)s Fv(i)i(2)f(B)165 5019 y Fp(and)i Ft(\036)p Fp(-a.e.)e Ft(\030)d Fv(2)c(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)980 4986 y Fq(B)r(nf)p Fu(e)p Fq(g)1206 5019 y Fp(we)32 b(have)994 5265 y Ft(\036)p Fw(\()p Ft(e)i Fw(is)29 b(op)s(en)h Fv(j)15 b Ft(\030)t Fw(\))26 b(=)1711 5121 y Fs(\()1819 5201 y Ft(p)362 b Fw(if)29 b Ft(x)c Fv($)h Ft(y)35 b Fw(in)29 b Ft(\030)1964 5273 y Fu(p)p 1829 5293 305 4 v 1829 5346 a(p)p Fr(+\(1)p Fq(\000)p Fu(p)p Fr(\))p Fu(q)2227 5314 y Fw(otherwise,)3453 5265 y(\(36\))165 5511 y Fp(Similarly,)40 b Ft(\036)d Fp(is)g(a)g(C-r)-5 b(andom-cluster)39 b(me)-5 b(asur)g(e)38 b(for)g Ft(p)e Fp(and)i Ft(q)i Fp(if)c(and)i(only)g(if)e(\(36\))j (holds)f(with)165 5646 y Ft(x)298 5596 y Fu(C)243 5646 y Fv( )-15 b(!)24 b Ft(y)36 b Fp(inste)-5 b(ad)34 b(of)f Ft(x)25 b Fv($)g Ft(y)s Fp(.)1844 5895 y Fw(52)p eop %%Page: 53 53 53 52 bop 165 304 a Fx(Pro)s(of:)48 b Fw(W)-8 b(e)35 b(consider)d(only)h(the)h(\014rst)e(statemen)m(t,)37 b(as)d(the)f(C-case)i(is)e(completely)g(similar.)47 b(F)-8 b(or)165 417 y(the)41 b(\\only)e(if)7 b(")40 b(part)g(w)m(e)g(only)g (need)f(to)i(note)g(that)g(the)f(righ)m(t-hand)f(side)g(of)h(\(36\))i (is)d(equal)g(to)165 546 y Ft(\036)219 498 y Fq(f)p Fu(e)p Fq(g)p Fu(;\030)219 556 y(p;q)380 546 y Fw(\()p Ft(e)31 b Fw(is)f(op)s(en)f(\).)71 b(P)m(assing)40 b(to)h(the)f(\\if)7 b(")41 b(part,)h(w)m(e)f(ma)m(y)g(restrict)f(ourselv)m(es)g(to)h(the)g (case)g(of)165 659 y Ft(p)25 b Fv(2)g Fw(\(0)p Ft(;)15 b Fw(1\).)42 b(Assume)28 b(that)g Ft(\036)h Fw(satis\014es)e(\(36\))j (for)e(eac)m(h)h Ft(e)d Fw(=)e Fv(h)p Ft(xy)s Fv(i)i(2)f(B)31 b Fw(and)c Ft(\036)p Fw(-a.e.)j Ft(\030)f Fv(2)c(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)3396 626 y Fq(B)r(nf)p Fu(e)p Fq(g)3589 659 y Fw(.)165 772 y(Let)33 b Ft(B)g Fv(\032)28 b(B)35 b Fw(b)s(e)c(some)i(\014nite)e(edge)i(set.)47 b(W)-8 b(e)34 b(need)e(to)h(sho)m(w)f(that)h(the)f(conditional)f (distribution)165 884 y Ft(\036)p Fw(\()p Fv(\001j)p Ft(\030)t Fw(\))i(of)e Ft(\036)h Fw(giv)m(en)f(the)h(con\014guration)e Ft(\030)36 b Fw(on)31 b Ft(B)1830 851 y Fu(c)1895 884 y Fw(equals)g Ft(\036)2226 851 y Fu(B)s(;\030)2226 907 y(p;q)2371 884 y Fw(for)g Ft(\036)p Fw(-a.e.)i Ft(\030)t Fw(.)44 b(F)-8 b(or)32 b(this,)f(it)f(su\016ces)165 1006 y(to)37 b(c)m(hec)m(k)h(that)f(for)f(an)m(y)h(t)m(w)m(o)h (con\014gurations)d Ft(\021)s(;)15 b(\021)1965 973 y Fq(0)2025 1006 y Fv(2)35 b(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)2341 973 y Fq(B)2431 1006 y Fw(whic)m(h)35 b(agree)j(with)d Ft(\030)40 b Fw(on)c Ft(B)3440 973 y Fu(c)3511 1006 y Fw(w)m(e)165 1119 y(ha)m(v)m(e)1025 1210 y Ft(\036)p Fw(\()p Ft(\021)19 b Fv(j)c Ft(\030)t Fw(\))p 1013 1251 297 4 v 1013 1334 a Ft(\036)p Fw(\()p Ft(\021)1150 1308 y Fq(0)1190 1334 y Fv(j)g Ft(\030)t Fw(\))1345 1272 y(=)1484 1210 y Fv(f)1529 1146 y Fs(Q)1608 1233 y Fu(e)p Fq(2)p Fu(B)1764 1210 y Ft(p)1810 1178 y Fu(\021)r Fr(\()p Fu(e)p Fr(\))1939 1210 y Fw(\(1)21 b Fv(\000)f Ft(p)p Fw(\))2212 1178 y Fr(1)p Fq(\000)p Fu(\021)r Fr(\()p Fu(e)p Fr(\))2432 1210 y Fv(g)p Ft(q)2521 1178 y Fu(k)r Fr(\()p Fu(\021)r(;B)s Fr(\))p 1451 1251 1316 4 v 1451 1337 a Fv(f)1496 1273 y Fs(Q)1575 1360 y Fu(e)p Fq(2)p Fu(B)1730 1337 y Ft(p)1776 1311 y Fu(\021)1813 1292 y Fm(0)1836 1311 y Fr(\()p Fu(e)p Fr(\))1928 1337 y Fw(\(1)h Fv(\000)f Ft(p)p Fw(\))2201 1311 y Fr(1)p Fq(\000)p Fu(\021)2328 1292 y Fm(0)2351 1311 y Fr(\()p Fu(e)p Fr(\))2443 1337 y Fv(g)p Ft(q)2532 1311 y Fu(k)r Fr(\()p Fu(\021)2635 1292 y Fm(0)2658 1311 y Fu(;B)s Fr(\))3453 1272 y Fw(\(37\))165 1494 y(with)39 b Ft(k)j Fw(de\014ned)d(as)h(in)e(\(35\).)70 b(If)39 b Ft(\021)k Fw(and)c Ft(\021)1707 1461 y Fq(0)1771 1494 y Fw(di\013er)f(only)h(at)h(a)g(single)f(edge)h Ft(e)p Fw(,)j(then)c Ft(k)s Fw(\()p Ft(\021)s(;)15 b(B)5 b Fw(\))27 b Fv(\000)165 1607 y Ft(k)s Fw(\()p Ft(\021)298 1574 y Fq(0)323 1607 y Ft(;)15 b(B)5 b Fw(\))32 b(=)f Ft(k)s Fw(\()p Ft(\021)s(;)15 b(e)p Fw(\))25 b Fv(\000)e Ft(k)s Fw(\()p Ft(\021)1108 1574 y Fq(0)1132 1607 y Ft(;)15 b(e)p Fw(\),)37 b(whence)d(\(37\))i(is)d(immediate)h(from)g(\(36\).)54 b(In)34 b(the)g(general)h(case,)165 1720 y(w)m(e)g(in)m(terp)s(olate)f Ft(\021)j Fw(and)d Ft(\021)1078 1687 y Fq(0)1135 1720 y Fw(b)m(y)g(a)h(sequence)f(of)h(con\014gurations)e(whic)m(h)g (successiv)m(ely)h(di\013er)f(in)f(at)165 1833 y(most)f(one)g(edge,)g (and)f(use)g(a)h(telescoping)f(argumen)m(t.)1483 b Fk(2)165 1994 y Fx(Pro)s(of)41 b(of)f(Theorem)f(6.17:)50 b Fw(W)-8 b(e)36 b(pro)m(v)m(e)g(\(ii\))e(only)-8 b(,)36 b(as)f(\(i\))f(follo)m (ws)g(from)g(a)i(similar)c(argumen)m(t)165 2107 y(and)c(is)g(also)g(b)s (etter)h(kno)m(wn,)f(see)h(e.g.)h(Borgs)f(and)f(Cha)m(y)m(es)h([32)q (].)41 b(Fix)28 b Ft(e)d Fw(=)g Fv(h)p Ft(xy)s Fv(i)h(2)f(B)s Fw(.)39 b(By)29 b(Lemma)165 2242 y(6.18,)39 b(it)c(is)g(su\016cien)m(t) g(to)i(establish)d(\(36\))j(with)d(the)i(C-connectivit)m(y)g(relation) 3046 2192 y Fu(C)2990 2242 y Fv( )-15 b(!)35 b Fw(in)f(place)i(of)165 2355 y(the)31 b(standard)e(connectivit)m(y)i(relation)f Fv($)p Fw(.)306 2467 y(Let)38 b Ft(B)545 2481 y Fr(1)584 2467 y Ft(;)15 b(B)693 2481 y Fr(2)733 2467 y Ft(;)g(:)g(:)g(:)38 b Fw(b)s(e)e(an)g(increasing)f(sequence)i(of)g(\014nite)e(edge)j(sets)f (con)m(taining)f Ft(e)h Fw(and)f(con-)165 2580 y(v)m(erging)d(to)g Fv(B)i Fw(in)c(the)h(usual)f(sense.)47 b(W)-8 b(e)34 b(write,)e(with)f(sligh)m(t)h(abuse)g(of)g(notation,)i Ft(\030)t Fw(\()p Ft(B)3251 2594 y Fu(i)3279 2580 y Fw(\))f(for)f(the) 165 2693 y(restriction)e(of)g Ft(\030)35 b Fw(to)c Ft(B)959 2707 y Fu(i)1007 2693 y Fv(n)21 b(f)p Ft(e)p Fv(g)p Fw(.)41 b(W)-8 b(e)32 b(recall)e(from)g(the)g(martingale)g(con)m(v)m(ergence)j (theorem)e(that)973 2869 y(^)961 2893 y Ft(\036)1015 2855 y Fr(1)1015 2916 y Fu(p;q)1108 2893 y Fw(\()p Ft(e)g Fw(is)e(op)s(en)h Fv(j)15 b Ft(\030)t Fw(\))26 b(=)49 b(lim)1768 2951 y Fu(j)t Fq(!1)1969 2869 y Fw(^)1957 2893 y Ft(\036)2011 2855 y Fr(1)2011 2916 y Fu(p;q)2104 2893 y Fw(\()p Ft(e)31 b Fw(is)e(op)s(en)h Fv(j)15 b Ft(\030)t Fw(\()p Ft(B)2711 2907 y Fu(j)2748 2893 y Fw(\)\))635 b(\(38\))165 3154 y(for)316 3130 y(^)304 3154 y Ft(\036)358 3121 y Fr(1)358 3177 y Fu(p;q)452 3154 y Fw(-a.e.)32 b Ft(\030)d Fv(2)c(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)1024 3121 y Fq(B)r(nf)p Fu(e)p Fq(g)1217 3154 y Fw(.)306 3302 y(W)-8 b(e)38 b(supp)s(ose)d(\014rst)g(that)i Ft(x)1356 3252 y Fu(C)1300 3302 y Fv( )-15 b(!)35 b Ft(y)k Fw(fails)c(in)g Ft(\030)t Fw(.)58 b(Then)36 b(at)h(least)f(one)h(of)f(the)h(v)m (ertices)g Ft(x)f Fw(and)165 3414 y Ft(y)k Fw(is)35 b(in)h(a)h (\014nite)e(cluster)h(of)h Ft(\030)t Fw(,)i(and)d(consequen)m(tly)g (there)h(is)f(some)h Ft(m)f Fw(\(dep)s(ending)f(on)i Ft(\030)t Fw(\))f(suc)m(h)165 3549 y(that)29 b Fv(:)p Fw(\()p Ft(x)589 3499 y Fu(C)533 3549 y Fv( )-15 b(!)25 b Ft(y)s Fw(\))i(can)h(b)s(e)f(v)m(eri\014ed)g(b)m(y)g(just)g(lo)s (oking)g(at)h Ft(\030)t Fw(\()p Ft(B)2308 3563 y Fu(m)2375 3549 y Fw(\).)40 b(\(This)26 b(is)h(a)h(consequence)g(of)g(the)165 3662 y(sp)s(ecial)h(concept)j(of)e(C-connectivit)m(y)-8 b(.\))42 b(F)-8 b(or)31 b(an)m(y)g Ft(n)25 b Fv(\025)g Ft(j)30 b Fv(\025)25 b Ft(m)31 b Fw(w)m(e)f(then)g(ha)m(v)m(e)1132 3857 y(^)1120 3881 y Ft(\036)1174 3843 y Fu(B)1227 3851 y Fo(n)1270 3843 y Fu(;)p Fr(1)1174 3903 y Fu(p;q)1329 3881 y Fw(\()p Ft(e)h Fw(is)f(op)s(en)f Fv(j)15 b Ft(\030)t Fw(\()p Ft(B)1936 3895 y Fu(j)1974 3881 y Fw(\)\))26 b(=)2389 3819 y Ft(p)p 2175 3859 474 4 v 2175 3943 a(p)20 b Fw(+)g(\(1)h Fv(\000)f Ft(p)p Fw(\))p Ft(q)165 4148 y Fw(so)31 b(that)g(b)m(y)f(the)h(de\014nition)d(of)1276 4124 y(^)1264 4148 y Ft(\036)1318 4115 y Fr(1)1318 4170 y Fu(p;q)1442 4148 y Fw(w)m(e)j(get)1163 4355 y(^)1151 4379 y Ft(\036)1205 4342 y Fr(1)1205 4402 y Fu(p;q)1299 4379 y Fw(\()p Ft(e)g Fw(is)e(op)s(en)h Fv(j)15 b Ft(\030)t Fw(\()p Ft(B)1906 4393 y Fu(j)1943 4379 y Fw(\)\))26 b(=)2359 4318 y Ft(p)p 2145 4358 V 2145 4441 a(p)20 b Fw(+)g(\(1)h Fv(\000)e Ft(p)p Fw(\))p Ft(q)165 4627 y Fw(b)m(y)31 b(letting)f Ft(n)25 b Fv(!)g(1)p Fw(.)41 b(Then)29 b(w)m(e)i(let)f Ft(j)h Fv(!)26 b(1)k Fw(and)g(use)g(\(38\))i (to)f(deduce)f(the)h(C-v)m(ersion)f(of)g(\(36\))i(in)165 4762 y(the)f(case)g Ft(\030)40 b(=)-55 b Fv(2)24 b(f)p Ft(x)848 4712 y Fu(C)792 4762 y Fv( )-15 b(!)25 b Ft(y)s Fv(g)p Fw(.)306 4897 y(W)-8 b(e)32 b(go)f(on)g(to)g(the)f(case)i Ft(\030)d Fv(2)c(f)p Ft(x)1504 4847 y Fu(C)1448 4897 y Fv( )-15 b(!)25 b Ft(y)s Fv(g)p Fw(.)41 b(In)30 b(analogy)h(to)g (\(38\),)h(w)m(e)f(ha)m(v)m(e)1330 5118 y(lim)1307 5176 y Fu(j)t Fq(!1)1507 5094 y Fw(^)1495 5118 y Ft(\036)1549 5081 y Fr(1)1549 5141 y Fu(p;q)1643 5118 y Fw(\()p Ft(x)1811 5068 y Fu(C)1755 5118 y Fv( )-15 b(!)25 b Ft(y)18 b Fv(j)d Ft(\030)t Fw(\()p Ft(B)2198 5132 y Fu(j)2235 5118 y Fw(\)\))26 b(=)f(1)165 5396 y(for)311 5372 y(^)299 5396 y Ft(\036)353 5363 y Fr(1)353 5418 y Fu(p;q)446 5396 y Fw(-a.e.)i Ft(\030)i Fv(2)c(f)p Ft(x)971 5346 y Fu(C)915 5396 y Fv( )-15 b(!)25 b Ft(y)s Fv(g)p Fw(.)39 b(F)-8 b(or)26 b(suc)m(h)e Ft(\030)29 b Fw(and)24 b(an)m(y)h Ft(")h(>)f Fw(0,)h(w)m(e)f(can)h(th)m(us)e (\014nd)f(an)i Ft(m)f Fw(\(dep)s(ending)165 5509 y(on)31 b Ft(\030)t Fw(\))f(suc)m(h)g(that)1329 5620 y(^)1317 5644 y Ft(\036)1371 5606 y Fr(1)1371 5666 y Fu(p;q)1464 5644 y Fw(\()p Ft(x)1632 5594 y Fu(C)1577 5644 y Fv( )-15 b(!)25 b Ft(y)18 b Fv(j)d Ft(\030)t Fw(\()p Ft(B)2020 5658 y Fu(j)2057 5644 y Fw(\)\))26 b Fv(\025)f Fw(1)20 b Fv(\000)g Ft(")1006 b Fw(\(39\))1844 5895 y(53)p eop %%Page: 54 54 54 53 bop 165 304 a Fw(for)28 b(an)m(y)g Ft(j)j Fv(\025)25 b Ft(m)p Fw(.)39 b(Next)29 b(w)m(e)f(use)g(the)g(de\014nition)d(of)1953 280 y(^)1941 304 y Ft(\036)1995 271 y Fr(1)1995 326 y Fu(p;q)2088 304 y Fw(.)40 b(F)-8 b(or)29 b(an)m(y)f Ft(n)d Fw(=)g(1)p Ft(;)15 b Fw(2)p Ft(;)g(:)g(:)g(:)r Fw(,)29 b(let)e Ft(Y)48 b Fw(and)27 b Ft(Y)3446 318 y Fu(n)3521 304 y Fw(b)s(e)165 435 y Fv(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)385 402 y Fq(B)440 435 y Fw(-v)-5 b(alued)32 b(random)g(edge)i (con\014gurations)f(with)e(distributions)2650 411 y(^)2638 435 y Ft(\036)2692 402 y Fr(1)2692 457 y Fu(p;q)2818 435 y Fw(and)3009 411 y(^)2997 435 y Ft(\036)3051 402 y Fu(B)3104 410 y Fo(n)3147 402 y Fu(;)p Fr(1)3051 457 y Fu(p;q)3239 435 y Fw(satisfying)165 548 y Ft(Y)218 562 y Fu(n)308 548 y Fv(\027)43 b Ft(Y)20 b Fw(;)47 b(this)39 b(is)i(p)s(ossible)d(b)m(y)j(Lemma)g(6.16.)75 b(W)-8 b(e)42 b(write)e Ft(P)2409 562 y Fu(n)2498 548 y Fw(for)g(the)i (probabilit)m(y)c(measure)165 661 y(underlying)28 b(this)h(coupling.)39 b(By)31 b(the)f(same)h(lemma)f(and)g(the)g(order)g(relation)g Ft(Y)2972 675 y Fu(n)3044 661 y Fv(\027)25 b Ft(Y)20 b Fw(,)30 b(w)m(e)h(ha)m(v)m(e)1304 865 y(lim)1276 915 y Fu(n)p Fq(!1)1475 865 y Ft(P)1533 879 y Fu(n)1580 865 y Fw(\()p Ft(Y)1668 879 y Fu(n)1715 865 y Fw(\()p Ft(B)1819 879 y Fu(j)1856 865 y Fw(\))25 b Fv(6)p Fw(=)g Ft(Y)20 b Fw(\()p Ft(B)2189 879 y Fu(j)2226 865 y Fw(\)\))26 b(=)f(0)15 b Ft(:)950 b Fw(\(40\))165 1126 y(Since)30 b Ft(Y)456 1140 y Fu(n)528 1126 y Fv(2)25 b(f)p Ft(x)792 1076 y Fu(C)736 1126 y Fv( )-15 b(!)25 b Ft(y)s Fv(g)31 b Fw(whenev)m(er)f Ft(Y)45 b Fv(2)25 b(f)p Ft(x)1814 1076 y Fu(C)1758 1126 y Fv( )-15 b(!)25 b Ft(y)s Fv(g)p Fw(,)31 b(w)m(e)f(can)h(write)391 1255 y Fs(\014)391 1305 y(\014)391 1354 y(\014)456 1328 y Fw(^)444 1352 y Ft(\036)498 1315 y Fu(B)551 1323 y Fo(n)594 1315 y Fu(;)p Fr(1)498 1375 y Fu(p;q)653 1352 y Fw(\()p Ft(x)821 1302 y Fu(C)766 1352 y Fv( )-15 b(!)24 b Ft(y)18 b(;)31 b(\030)t Fw(\()p Ft(B)1224 1366 y Fu(j)1261 1352 y Fw(\)\))21 b Fv(\000)1455 1328 y Fw(^)1443 1352 y Ft(\036)1497 1315 y Fr(1)1497 1375 y Fu(p;q)1590 1352 y Fw(\()p Ft(x)1758 1302 y Fu(C)1702 1352 y Fv( )-15 b(!)25 b Ft(y)18 b(;)31 b(\030)t Fw(\()p Ft(B)2161 1366 y Fu(j)2198 1352 y Fw(\)\))2294 1255 y Fs(\014)2294 1305 y(\014)2294 1354 y(\014)394 1580 y Fv(\024)83 b Ft(P)606 1594 y Fu(n)653 1486 y Fs(\020)703 1580 y Fv(f)p Ft(x)881 1530 y Fu(C)825 1580 y Fv( )-15 b(!)25 b Ft(y)33 b Fw(in)c Ft(Y)1254 1594 y Fu(n)1301 1580 y Ft(;)41 b(Y)1420 1594 y Fu(n)1467 1580 y Fw(\()p Ft(B)1571 1594 y Fu(j)1607 1580 y Fw(\))26 b(=)f Ft(\030)t Fw(\()p Ft(B)1912 1594 y Fu(j)1949 1580 y Fw(\))p Fv(g)h(4)f(f)p Ft(x)2339 1530 y Fu(C)2283 1580 y Fv( )-15 b(!)25 b Ft(y)34 b Fw(in)29 b Ft(Y)5 b(;)40 b(Y)20 b Fw(\()p Ft(B)2960 1594 y Fu(j)2997 1580 y Fw(\))25 b(=)g Ft(\030)t Fw(\()p Ft(B)3301 1594 y Fu(j)3338 1580 y Fw(\))p Fv(g)3419 1486 y Fs(\021)394 1808 y Fv(\024)548 1714 y Fs(\020)609 1784 y Fw(^)597 1808 y Ft(\036)651 1771 y Fu(B)704 1779 y Fo(n)747 1771 y Fu(;)p Fr(1)651 1831 y Fu(p;q)806 1808 y Fw(\()p Ft(x)975 1758 y Fu(C)919 1808 y Fv( )-15 b(!)25 b Ft(y)s Fw(\))20 b Fv(\000)1317 1784 y Fw(^)1305 1808 y Ft(\036)1359 1771 y Fr(1)1359 1831 y Fu(p;q)1452 1808 y Fw(\()p Ft(x)1621 1758 y Fu(C)1565 1808 y Fv( )-15 b(!)25 b Ft(y)s Fw(\))1840 1714 y Fs(\021)1910 1808 y Fw(+)20 b Ft(P)2059 1822 y Fu(n)2106 1808 y Fw(\()p Ft(Y)2194 1822 y Fu(n)2241 1808 y Fw(\()p Ft(B)2345 1822 y Fu(j)2382 1808 y Fw(\))26 b Fv(6)p Fw(=)e Ft(Y)c Fw(\()p Ft(B)2715 1822 y Fu(j)2752 1808 y Fw(\)\))26 b Ft(;)165 2064 y Fw(where)33 b Fv(4)g Fw(denotes)h(symmetric)e(di\013erence.)49 b(Since)33 b Fv(f)p Ft(x)2197 2014 y Fu(C)2141 2064 y Fv( )-15 b(!)30 b Ft(y)s Fv(g)j Fw(is)g(the)g(decreasing)g(limit)e(of)j (the)165 2176 y(lo)s(cal)c(ev)m(en)m(ts)i Fv(f)p Ft(x)26 b Fv($)f Ft(y)33 b Fw(in)c(\001)p Fv(g)21 b([)f(f)p Ft(x)25 b Fv($)h Fw(\001)1617 2143 y Fu(c)1651 2176 y Ft(;)31 b(y)d Fv($)d Fw(\001)1972 2143 y Fu(c)2007 2176 y Fv(g)31 b Fw(as)f(\001)c Fv(")f(L)p Fw(,)31 b(an)f(analogue)h(of)g(\(30\))h (together)165 2289 y(with)d(\(40\))j(sho)m(ws)e(that)h(the)g(last)f (expression)f(tends)h(to)h(zero)h(as)e Ft(n)25 b Fv(!)g(1)p Fw(.)41 b(It)30 b(follo)m(ws)g(that)917 2515 y(lim)888 2565 y Fu(n)p Fq(!1)1099 2491 y Fw(^)1087 2515 y Ft(\036)1141 2478 y Fu(B)1194 2486 y Fo(n)1237 2478 y Fu(;)p Fr(1)1141 2538 y Fu(p;q)1296 2515 y Fw(\()p Ft(x)1465 2465 y Fu(C)1409 2515 y Fv( )-15 b(!)25 b Ft(y)18 b Fv(j)d Ft(\030)t Fw(\()p Ft(B)1852 2529 y Fu(j)1889 2515 y Fw(\)\))26 b(=)2093 2491 y(^)2081 2515 y Ft(\036)2135 2478 y Fr(1)2135 2538 y Fu(p;q)2228 2515 y Fw(\()p Ft(x)2396 2465 y Fu(C)2341 2515 y Fv( )-15 b(!)25 b Ft(y)18 b Fv(j)d Ft(\030)t Fw(\()p Ft(B)2784 2529 y Fu(j)2821 2515 y Fw(\)\))165 2760 y(whic)m(h)35 b(is)g(at)i(least)f(1)25 b Fv(\000)f Ft(")36 b Fw(b)m(y)g(\(39\).)59 b(But)37 b(since)1909 2736 y(^)1897 2760 y Ft(\036)1951 2727 y Fu(B)2004 2735 y Fo(n)2047 2727 y Fu(;)p Fr(1)1951 2783 y Fu(p;q)2106 2760 y Fw(\()p Ft(e)31 b Fw(is)e(op)s(en)h Fv(j)15 b Ft(\030)2609 2727 y Fq(0)2633 2760 y Fw(\))35 b(=)f Ft(p)i Fw(for)g(eac)m(h)h Ft(n)f Fw(and)f(all)165 2908 y Ft(\030)209 2875 y Fq(0)258 2908 y Fv(2)25 b(f)p Ft(x)522 2858 y Fu(C)466 2908 y Fv( )-15 b(!)25 b Ft(y)s Fv(g)p Fw(,)31 b(w)m(e)g(get)1064 3112 y Ft(p)20 b Fv(\000)g Ft(")26 b Fv(\024)54 b Fw(lim)1385 3162 y Fu(n)p Fq(!1)1596 3088 y Fw(^)1584 3112 y Ft(\036)1638 3074 y Fu(B)1691 3082 y Fo(n)1734 3074 y Fu(;)p Fr(1)1638 3134 y Fu(p;q)1793 3112 y Fw(\()p Ft(e)31 b Fw(is)e(op)s(en)h Fv(j)15 b Ft(\030)t Fw(\()p Ft(B)2400 3126 y Fu(j)2437 3112 y Fw(\)\))26 b Fv(\024)f Ft(p)15 b(:)165 3332 y Fw(Hence,)1195 3445 y Ft(p)20 b Fv(\000)g Ft(")25 b Fv(\024)1527 3421 y Fw(^)1515 3445 y Ft(\036)1569 3407 y Fr(1)1569 3467 y Fu(p;q)1662 3445 y Fw(\()p Ft(e)31 b Fw(is)f(op)s(en)g Fv(j)15 b Ft(\030)t Fw(\()p Ft(B)2270 3459 y Fu(j)2307 3445 y Fw(\)\))26 b Fv(\024)f Ft(p)15 b(;)165 3611 y Fw(and)34 b(since)g Ft(")h Fw(w)m(as)g(arbitrary)e(w)m(e)i(can)g(use)f(\(38\))i(to)f (deduce)f(the)h(C-v)m(ersion)f(of)g(\(36\))i(in)d(the)i(case)165 3746 y Ft(\030)30 b Fv(2)24 b(f)p Ft(x)499 3696 y Fu(C)443 3746 y Fv( )-15 b(!)25 b Ft(y)s Fv(g)p Fw(.)2793 b Fk(2)165 3909 y Fw(Let)23 b(us)f(no)m(w)g(brie\015y)f(address)h(the)g(issue)f (of)i(whether)f(the)g(t)m(w)m(o)i(t)m(yp)s(es)e(of)h(random-cluster)e (measures)165 4022 y(are)31 b(an)m(y)g(di\013eren)m(t.)40 b(The)30 b(follo)m(wing)f(result)g(sa)m(ys)i(that)g(v)m(ery)g(often)f (they)h(are)g(the)f(same.)165 4209 y Fx(Prop)s(osition)37 b(6.19)46 b Fp(L)-5 b(et)33 b Ft(\036)f Fp(b)-5 b(e)33 b(a)g(pr)-5 b(ob)g(ability)35 b(me)-5 b(asur)g(e)34 b(on)f Fv(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)2536 4176 y Fq(B)2623 4209 y Fp(with)1036 4414 y Ft(\036)p Fw(\()p Fv(9)33 b Fw(at)e(most)g(one)f(in\014nite)f(op)s(en)g(cluster\))c(=)g(1)15 b Ft(:)165 4618 y Fp(Then,)38 b(for)g(any)f Ft(p)32 b Fv(2)h Fw([0)p Ft(;)15 b Fw(1])38 b Fp(and)g Ft(q)d(>)d Fw(0)p Fp(,)38 b Ft(\036)f Fp(is)g(a)g(r)-5 b(andom-cluster)39 b(me)-5 b(asur)g(e)38 b(for)f Ft(p)g Fp(and)g Ft(q)j Fp(if)c(and)165 4731 y(only)e(if)e(it)h(is)f(a)h(C-r)-5 b(andom-cluster)35 b(me)-5 b(asur)g(e)34 b(for)f Ft(p)f Fp(and)i Ft(q)s Fp(.)165 4918 y Fw(This)d(means)h(that)h(whenev)m(er)f (\\uniqueness)e(of)i(the)h(in\014nite)d(cluster")h(can)i(b)s(e)e(v)m (eri\014ed,)h(the)h(t)m(w)m(o)165 5031 y(t)m(yp)s(es)24 b(of)h(random-cluster)e(measures)h(coincide.)37 b(An)24 b(example)g(is)f(obtained)g(if)h(w)m(e)g(consider)f(trans-)165 5144 y(lation)39 b(in)m(v)-5 b(arian)m(t)38 b(random-cluster)g (measures)h(for)g Fx(Z)2069 5111 y Fu(d)2109 5144 y Fw(,)j(since)c(the) i(Burton{Keane)f(uniqueness)165 5257 y(theorem)d(\(Theorem)f(5.17\))j (applies)33 b(in)h(this)g(situation.)55 b(F)-8 b(or)36 b Fx(Z)2444 5224 y Fu(d)2484 5257 y Fw(,)h(the)e(measures)h Ft(\036)3157 5224 y Fr(0)3157 5280 y Fu(p;q)3285 5257 y Fw(and)3479 5233 y(^)3467 5257 y Ft(\036)3521 5224 y Fr(1)3521 5280 y Fu(p;q)165 5370 y Fw(are)31 b(translation)f(in)m(v) -5 b(arian)m(t,)30 b(b)m(y)g(Lemma)h(6.8.)43 b(On)29 b(the)i(other)g(hand,)f(uniqueness)e(of)j(the)f(in\014nite)165 5483 y(cluster)36 b(t)m(ypically)g(fails)f(on)h(trees,)j(leading)d(to)h (v)m(ery)g(di\013eren)m(t)f(b)s(eha)m(vior)g(for)g(the)h(t)m(w)m(o)h(t) m(yp)s(es)f(of)165 5596 y(random-cluster)30 b(measures;)g(see)h([114)r (,)f(115)r(])g(for)g(a)h(discussion.)1844 5895 y(54)p eop %%Page: 55 55 55 54 bop 165 304 a Fx(Pro)s(of)42 b(of)f(Prop)s(osition)h(6.19:)52 b Fw(F)-8 b(or)36 b Ft(p)e Fw(=)g(0)i(or)f(1)h(the)g(result)e(is)h (trivial,)g(so)h(w)m(e)g(ma)m(y)g(assume)165 417 y(that)f Ft(p)30 b Fv(2)h Fw(\(0)p Ft(;)15 b Fw(1\).)53 b(The)33 b(conditional)f(probabilities)e(in)j(\(36\))i(and)e(its)g(C-coun)m (terpart)i(di\013er)d(only)165 530 y(on)f(the)f(ev)m(en)m(t)1321 643 y Ft(A)1389 657 y Fu(xy)1496 643 y Fw(=)25 b Fv(f)p Ft(x)1770 593 y Fu(C)1715 643 y Fv( )-15 b(!)24 b Ft(y)s Fv(g)d(n)g(f)p Ft(x)k Fv($)g Ft(y)s Fv(g)15 b Ft(:)165 810 y Fw(Hence)38 b(if)d Ft(\036)i Fw(is)e(a)i(random-cluster)f (measure)g(but)g(not)h(a)f(C-random-cluster)g(measure)g(\(or)h(vice)165 923 y(v)m(ersa\),)h(then)d Ft(A)745 937 y Fu(xy)862 923 y Fw(has)g(to)h(ha)m(v)m(e)g(p)s(ositiv)m(e)e Ft(\036)p Fw(-probabilit)m(y)g(for)h(some)g(edge)h Ft(e)e Fw(=)f Fv(h)p Ft(xy)s Fv(i)h(2)f(B)s Fw(.)55 b(But)165 1035 y(then)24 b(the)h(ev)m(en)m(t)h Ft(A)819 1049 y Fu(xy)908 1035 y Fv(\\)8 b(f)p Ft(e)31 b Fw(is)f(closed)p Fv(g)24 b Fw(has)h(p)s(ositiv)m(e)e Ft(\036)p Fw(-probabilit)m(y)-8 b(,)24 b(and)g(since)g(this)f(ev)m(en)m(t)j(implies)165 1148 y(the)31 b(existence)g(of)f(at)h(least)g(t)m(w)m(o)h(in\014nite)c (clusters,)i(w)m(e)g(are)h(done.)1078 b Fk(2)165 1311 y Fw(Muc)m(h)36 b(of)g(the)g(study)f(of)h(in\014nite)e(v)m(olume)i (random-cluster)e(measures)i(that)g(has)g(b)s(een)f(done)h(so)165 1424 y(far)i(concerns)f(the)h(issue)e(of)i(uniqueness)d(\(or)j (non-uniqueness\))d(of)j(random-cluster)e(measures.)165 1537 y(A)41 b(discussion)e(of)i(this)f(issue)g(w)m(ould,)j(ho)m(w)m(ev) m(er,)i(lead)c(us)f(to)s(o)i(far,)h(so)f(instead)e(w)m(e)h(advise)g (the)165 1650 y(reader)28 b(to)h(consult)e(Grimmett)g([109)r(],)h (H\177)-45 b(aggstr\177)g(om)30 b([114)q(])e(and)f(Jonasson)h([139)r(]) f(to)i(\014nd)d(out)i(what)165 1763 y(is)i(kno)m(wn)g(and)f(what)i(is)e (conjectured)i(in)e(this)g(\014eld.)165 2006 y FA(6.5)113 b(An)37 b(application)i(to)e(p)s(ercolation)h(in)g(the)f(Ising)h(mo)s (del)165 2178 y Fw(In)c(Theorem)h(5.10)h(w)m(e)g(ha)m(v)m(e)g(seen)f (that)g(the)g(probabilit)m(y)e(of)i(p)s(ercolation)f(of)h(plus)e(spins) f(in)i(the)165 2291 y(Ising)g(mo)s(del)f(is)g(an)h(increasing)f (function)g(of)i(the)g(external)f(\014eld.)51 b(A)35 b(m)m(uc)m(h)f(harder)f(question)h(is)165 2404 y(to)g(determine)f (monotonicit)m(y)g(prop)s(erties)f(of)h(p)s(ercolation)f(probabilities) e(as)k Ft(\014)k Fw(\(rather)33 b(than)g Ft(h)p Fw(\))165 2517 y(is)k(v)-5 b(aried.)62 b(An)38 b(in)m(teresting)f(op)s(en)g (problem)f(is)h(to)i(decide)e(whether)g(for)h Ft(G)g Fw(=)f Fx(Z)3079 2484 y Fu(d)3119 2517 y Fw(,)j Ft(d)e Fv(\025)g Fw(2,)i(the)165 2629 y(probabilit)m(y)1617 2754 y Ft(\026)1672 2716 y Fr(+)1672 2782 y Fu(\014)1731 2754 y Fw(\()p Ft(x)1900 2704 y Fr(+)1844 2754 y Fv( )-15 b(!)25 b(1)p Fw(\))165 2935 y(is)35 b(increasing)g(in)f Ft(\014)5 b Fw(.)57 b(Here)36 b(w)m(e)h(write)e Ft(\026)1595 2897 y Fr(+)1595 2964 y Fu(\014)1689 2935 y Fw(for)h(the)g(plus)e (phase)h(in)f(the)i(Ising)f(mo)s(del)g(at)h(in)m(v)m(erse)165 3083 y(temp)s(erature)31 b Ft(\014)37 b Fw(with)30 b(external)h (\014eld)f Ft(h)d Fw(=)g(0,)32 b(and)e Fv(f)p Ft(x)2166 3033 y Fr(+)2111 3083 y Fv( )-15 b(!)26 b(1g)32 b Fw(is)e(the)h(ev)m (en)m(t)i(that)f(there)f(exists)165 3196 y(an)36 b(in\014nite)e(path)i (of)g(plus)e(spins)g(starting)h(at)i Ft(x)p Fw(.)58 b(A)m(t)37 b(\014rst)e(sigh)m(t,)i(one)g(migh)m(t)e(b)s(e)h(seduced)f(in)m(to)165 3309 y(thinking)24 b(that)i(this)f(w)m(ould)f(b)s(e)h(a)h(consequence)h (of)f(the)g(connection)g(b)s(et)m(w)m(een)g(Ising)e(and)i(random-)165 3422 y(cluster)g(mo)s(dels,)h(and)f(the)h(sto)s(c)m(hastic)g (monotonicit)m(y)g(of)g(random-cluster)f(measures)g(as)h Ft(p)f Fw(v)-5 b(aries;)165 3535 y(see)41 b(\(31\).)73 b(Ho)m(w)m(ev)m(er,)45 b(suc)m(h)40 b(a)h(conclusion)e(is)h(un)m(w)m (arran)m(ted.)71 b(F)-8 b(or)41 b(example,)i(in)c(the)i(coupling)165 3648 y(of)35 b(Theorem)f(6.2)i(the)f(existence)g(of)f(an)h(op)s(en)e (path)i(b)s(et)m(w)m(een)g Ft(x)f Fw(and)g Ft(y)j Fw(in)d(the)g (random-cluster)165 3761 y(represen)m(tation)e(is)f(a)h(su\016cien)m(t) f Fp(but)j(not)g(ne)-5 b(c)g(essary)33 b Fw(condition)d(for)i Ft(x)f Fw(and)h Ft(y)i Fw(to)e(b)s(e)f(in)g(the)h(same)165 3873 y(spin)k(cluster.)61 b(In)36 b(fact,)41 b(H\177)-45 b(aggstr\177)g(om)39 b([116)q(])f(sho)m(w)m(ed,)i(b)m(y)d(means)g(of)h (a)f(simple)f(coun)m(terexample)165 3986 y(and)29 b(in)e(resp)s(onse)h (to)h(a)h(question)e(of)h(Cammarota)g([45)r(],)g(that)g(the)g (probabilit)m(y)e(that)i Ft(x)g Fw(and)f Ft(y)k Fw(are)165 4099 y(in)27 b(the)i(same)f(spin)f(cluster)g(need)h(not)h(b)s(e)f (increasing)e(in)h Ft(\014)5 b Fw(,)30 b(and)d(similarly)e(for)j(the)h (exp)s(ected)f(size)165 4212 y(of)j(the)f(spin)f(cluster)h(con)m (taining)f Ft(x)p Fw(.)306 4325 y(Ho)m(w)m(ev)m(er,)k(when)c(the)i (underlying)c(graph)j(is)f(a)i(tree,)h(monotonicit)m(y)e(in)f Ft(\014)36 b Fw(of)30 b(the)h(probabilit)m(y)165 4438 y(of)g(plus)d(p)s(ercolation)i(can)g(b)s(e)g(established:)165 4626 y Fx(Theorem)k(6.20)47 b Fp(F)-7 b(or)31 b(the)g(Ising)g(mo)-5 b(del)32 b(on)g(the)f(r)-5 b(e)g(gular)32 b(tr)-5 b(e)g(e)31 b Fx(T)2463 4641 y Fu(d)2503 4626 y Fp(,)g Ft(d)26 b Fv(\025)f Fw(2)p Fp(,)31 b(with)h(a)f(distinguishe)-5 b(d)165 4750 y(vertex)33 b Ft(x)p Fp(,)f(the)h(p)-5 b(er)g(c)g(olation) 36 b(pr)-5 b(ob)g(ability)35 b Ft(\026)1649 4712 y Fr(+)1649 4778 y Fu(\014)1708 4750 y Fw(\()p Ft(x)1876 4700 y Fr(+)1821 4750 y Fv( )-15 b(!)24 b(1)p Fw(\))33 b Fp(is)g(incr)-5 b(e)g(asing)33 b(in)g Ft(\014)5 b Fp(.)165 4938 y Fw(An)35 b(in)m(teresting)g(asp)s(ect)h(of)g(this)e(result)h(is)f(that)j(its)d (pro)s(of,)j(unlik)m(e)d(those)i(of)f(the)h(monotonicit)m(y)165 5051 y(results)28 b(men)m(tioned)g(earlier)f(in)g(this)h(section,)h(is) e Fp(not)j Fw(based)e(on)g(sto)s(c)m(hastic)i(domination)c(b)s(et)m(w)m (een)165 5163 y(the)34 b(probabilit)m(y)c(measures)j(in)f(question.)47 b(In)33 b(fact,)h(sto)s(c)m(hastic)g(domination)e(fails,)g(i.e.)h(it)g (is)f(not)165 5276 y(alw)m(a)m(ys)f(the)g(case)g(\(in)f(the)g(setting)h (of)f(Theorem)g(6.20\))j(that)1666 5481 y Ft(\026)1721 5442 y Fr(+)1721 5509 y Fu(\014)1761 5518 y Fl(1)1824 5481 y Fv(\026)1895 5495 y Fq(D)1981 5481 y Ft(\026)2036 5442 y Fr(+)2036 5509 y Fu(\014)2076 5518 y Fl(2)3453 5481 y Fw(\(41\))1844 5895 y(55)p eop %%Page: 56 56 56 55 bop 165 304 a Fw(when)41 b Ft(\014)465 318 y Fr(1)549 304 y Fv(\024)i Ft(\014)714 318 y Fr(2)754 304 y Fw(.)74 b(An)42 b(easy)g(w)m(a)m(y)g(to)h(see)f(this)e(is)h(as)h(follo)m(ws.)73 b(Just)41 b(as)h(in)e(Theorem)h(3.1,)46 b(let)165 417 y Ft(\014)216 431 y Fu(c)285 417 y Fw(b)s(e)33 b(the)g(critical)g(in)m (v)m(erse)g(temp)s(erature)h(for)f(non-uniqueness)e(of)i(the)h(Gibbs)e (measure.)50 b(\(It)34 b(is)165 530 y(straigh)m(tforw)m(ard)i(to)h(sho) m(w,)h(using)d(either)g(the)i(random-cluster)e(approac)m(h)h(or)h(the)f (metho)s(ds)f(in)165 643 y(Section)c(7,)g(that)g Ft(\014)833 657 y Fu(c)894 643 y Ft(>)25 b Fw(0)31 b(for)g Fv(L)25 b Fw(=)h Fx(T)1464 658 y Fu(d)1504 643 y Fw(.\))41 b(Pic)m(k)31 b Ft(\014)1860 657 y Fr(1)1925 643 y Ft(<)26 b(\014)2073 657 y Fr(2)2143 643 y Fw(in)k(\(0)p Ft(;)15 b(\014)2421 657 y Fu(c)2457 643 y Fw(\).)42 b(By)31 b(Theorem)f(4.15,)i(w)m(e)f (then)165 756 y(ha)m(v)m(e)37 b Ft(\026)435 717 y Fr(+)435 784 y Fu(\014)475 793 y Fl(1)513 756 y Fw(\()p Ft(\030)g Fw(:)49 b Ft(\030)t Fw(\()p Ft(y)s Fw(\))34 b(=)e(+1\))i(=)f Ft(\026)1342 717 y Fr(+)1342 784 y Fu(\014)1382 793 y Fl(2)1420 756 y Fw(\()p Ft(\030)38 b Fw(:)48 b Ft(\030)t Fw(\()p Ft(y)s Fw(\))34 b(=)f(+1\))h(=)e(1)p Ft(=)p Fw(2)37 b(for)e(ev)m(ery)h(v)m(ertex)g Ft(y)s Fw(.)55 b(If)34 b(no)m(w)i(\(41\))165 883 y(w)m(as)h(true)f(w)m(e)g(w)m(ould)f(ha)m(v)m (e)i Ft(\026)1224 845 y Fr(+)1224 911 y Fu(\014)1264 920 y Fl(1)1337 883 y Fw(=)e Ft(\026)1498 845 y Fr(+)1498 911 y Fu(\014)1538 920 y Fl(2)1612 883 y Fw(b)m(y)h(Prop)s(osition)e (4.12.)59 b(This,)36 b(ho)m(w)m(ev)m(er,)j(is)d(imp)s(ossible)165 996 y(b)s(ecause)31 b(the)f(t)m(w)m(o)i(measures)e(ha)m(v)m(e)i (di\013eren)m(t)e(conditional)e(distributions)f(on)j(\014nite)f (regions.)306 1109 y(Theorem)f(6.20)h(can)g(b)s(e)e(pro)m(v)m(ed)h (using)e(the)j(exact)g(calculations)e(for)h(the)g(Ising)e(mo)s(del)h (on)h Fx(T)3549 1124 y Fu(d)3589 1109 y Fw(,)165 1222 y(whic)m(h)36 b(can)h(b)s(e)g(found)e(in)h(e.g.)i(Spitzer)e([215)r(])h (and)g(Georgii)f([96)q(].)61 b(Here)38 b(w)m(e)f(presen)m(t)g(a)h (simpler)165 1335 y(pro)s(of)32 b(whic)m(h)f(do)s(es)h(not)g(require)f (an)m(y)i(exact)h(calculation,)e(but)g(whic)m(h)f(exploits)g (random-cluster)165 1448 y(metho)s(ds.)165 1611 y Fx(Pro)s(of)38 b(of)f(Theorem)g(6.20:)45 b Fw(As)32 b(usual)f(w)m(e)h(write)g Fv(L)g Fw(and)f Fv(B)k Fw(for)d(the)g(v)m(ertex)i(and)d(edge)i(sets)g (of)165 1724 y Fx(T)238 1739 y Fu(d)279 1724 y Fw(.)40 b(Since)1186 1845 y(lim)1182 1905 y Fr(\003)p Fq("L)1330 1845 y Ft(\026)1385 1807 y Fr(+)1385 1873 y Fu(\014)s(;)p Fr(\003)1501 1845 y Fw(\()p Ft(x)1669 1795 y Fr(+)1613 1845 y Fv( )-15 b(!)25 b(1)p Fw(\))h(=)f Ft(\026)2108 1807 y Fr(+)2108 1873 y Fu(\014)2166 1845 y Fw(\()p Ft(x)2335 1795 y Fr(+)2279 1845 y Fv( )-15 b(!)25 b(1)p Fw(\))165 2057 y(for)30 b(an)m(y)h Ft(\014)36 b Fw(in)29 b(analogy)i(to)g (\(30\),)h(it)e(su\016ces)g(to)h(sho)m(w)f(that)h(for)f Ft(\014)2456 2071 y Fr(1)2522 2057 y Fv(\024)24 b Ft(\014)2668 2071 y Fr(2)2739 2057 y Fw(and)29 b(an)m(y)i(\003,)g(w)m(e)g(ha)m(v)m (e)1177 2277 y Ft(\026)1232 2239 y Fr(+)1232 2305 y Fu(\014)1272 2314 y Fl(1)1306 2305 y Fu(;)p Fr(\003)1378 2277 y Fw(\()p Ft(x)1547 2227 y Fr(+)1491 2277 y Fv( )-15 b(!)25 b(1)p Fw(\))g Fv(\024)g Ft(\026)1985 2239 y Fr(+)1985 2305 y Fu(\014)2025 2314 y Fl(2)2059 2305 y Fu(;)p Fr(\003)2132 2277 y Fw(\()p Ft(x)2300 2227 y Fr(+)2244 2277 y Fv( )-15 b(!)25 b(1)p Fw(\))15 b Ft(:)851 b Fw(\(42\))165 2496 y(This)29 b(w)m(e)i(will)c(do)k(b)m(y)f(constructing)g(a)h(coupling)d Ft(P)44 b Fw(of)30 b(t)m(w)m(o)i Fv(f\000)p Fw(1)p Ft(;)15 b Fw(+1)p Fv(g)2630 2463 y Fq(L)2684 2496 y Fw(-v)-5 b(alued)30 b(random)f(ob)5 b(jects)165 2626 y Ft(X)240 2640 y Fr(1)307 2626 y Fw(and)26 b Ft(X)555 2640 y Fr(2)622 2626 y Fw(with)f(resp)s(ectiv)m(e)i(distributions)c Ft(\026)1827 2588 y Fr(+)1827 2654 y Fu(\014)1867 2663 y Fl(1)1901 2654 y Fu(;)p Fr(\003)2001 2626 y Fw(and)j Ft(\026)2229 2588 y Fr(+)2229 2654 y Fu(\014)2269 2663 y Fl(2)2303 2654 y Fu(;)p Fr(\003)2402 2626 y Fw(and)h(the)g(prop)s(ert)m(y)f(that) h(if)f Ft(x)3503 2576 y Fr(+)3448 2626 y Fv( )-15 b(!)165 2739 y(1)31 b Fw(in)e Ft(X)468 2753 y Fr(1)507 2739 y Fw(,)i(then)f(the)h(same)f(thing)g(happ)s(ens)e(in)h Ft(X)1929 2753 y Fr(2)1969 2739 y Fw(.)306 2852 y(Recall)g(the)g(Edw)m (ards{Sok)-5 b(al)28 b(coupling)f(of)i(spin)d(and)j(edge)g (con\014gurations)f(describ)s(ed)f(ahead)165 2965 y(of)49 b(Prop)s(osition)d(6.9.)96 b(In)48 b(the)g(presen)m(t)h(case)g(of)g (the)f(tree)h Fx(T)2450 2980 y Fu(d)2490 2965 y Fw(,)k(this)48 b(construction)g(requires)165 3077 y(the)42 b(C-v)m(ersion)g(of)g(coun) m(ting)g(clusters,)i(whic)m(h)c(corresp)s(onds)h(to)i(making)e(the)h (complemen)m(t)g(of)165 3190 y(\003)h(connected.)78 b(Therefore)42 b(w)m(e)h(will)d(w)m(ork)i(with)f(the)i(C-random-cluster)e (distributions.)74 b(Let)165 3303 y Ft(p)211 3317 y Fr(1)276 3303 y Fw(=)25 b(1)11 b Fv(\000)g Ft(e)552 3270 y Fq(\000)p Fr(2)p Fu(\014)682 3279 y Fl(1)746 3303 y Fw(and)25 b Ft(p)964 3317 y Fr(2)1029 3303 y Fw(=)f(1)11 b Fv(\000)g Ft(e)1304 3270 y Fq(\000)p Fr(2)p Fu(\014)1434 3279 y Fl(2)1473 3303 y Fw(,)27 b(and)e(let)h Ft(B)j Fw(=)c Fv(B)2081 3270 y Fr(1)2078 3329 y(\003)2156 3303 y Fv(\032)g(B)j Fw(b)s(e)d(the)h(set)h(of)e(edges)i(inciden)m(t)d(to)i(at)165 3416 y(least)j(one)g(v)m(ertex)h(in)d(\003.)40 b(W)-8 b(e)30 b(\014rst)e(let)h Ft(Y)1571 3430 y Fr(1)1638 3416 y Fw(and)f Ft(Y)1866 3430 y Fr(2)1934 3416 y Fw(b)s(e)g(t)m(w)m(o)i Fv(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)2446 3383 y Fq(B)2500 3416 y Fw(-v)-5 b(alued)28 b(random)g(edge)i(con\014g-)165 3529 y(urations)25 b(distributed)f(according)i(to)g(the)h (random-cluster)e(measures)2662 3505 y(^)2650 3529 y Ft(\036)2704 3485 y Fu(B)s(;)p Fr(1)2704 3554 y Fu(p)2740 3563 y Fl(1)2774 3554 y Fu(;)p Fr(2)2859 3529 y Fw(and)3043 3505 y(^)3031 3529 y Ft(\036)3085 3485 y Fu(B)s(;)p Fr(1)3085 3554 y Fu(p)3121 3563 y Fl(2)3156 3554 y Fu(;)p Fr(2)3215 3529 y Fw(,)i(and)e(suc)m(h)165 3665 y(that)30 b Ft(P)13 b Fw(\()p Ft(Y)520 3679 y Fr(1)585 3665 y Fv(\026)25 b Ft(Y)734 3679 y Fr(2)773 3665 y Fw(\))h(=)f(1;)30 b(this)e(is)h(p)s (ossible)e(b)m(y)i(the)1930 3641 y(^)1918 3665 y Ft(\036)1972 3620 y Fu(B)s(;)p Fr(1)1972 3689 y Fu(p;)p Fr(2)2088 3665 y Fw(-analogue)h(of)g(\(31\).)42 b Ft(X)2901 3679 y Fr(1)2970 3665 y Fw(and)28 b Ft(X)3220 3679 y Fr(2)3289 3665 y Fw(can)i(no)m(w)165 3778 y(b)s(e)h(obtained)g(b)m(y)h(assigning) e(spins)g(to)i(the)g(connected)g(comp)s(onen)m(ts)g(of)g Ft(Y)2769 3792 y Fr(1)2840 3778 y Fw(and)f Ft(Y)3071 3792 y Fr(2)3141 3778 y Fw(in)g(the)h(usual)165 3891 y(w)m(a)m(y;)f(these)f(spin)d(assignmen)m(ts)i(are)h(coupled)e(as)h (follo)m(ws.)40 b(First)28 b(w)m(e)i(m)m(ust)f(assign)g(spin)e(+1)i(to) h(all)165 4003 y(in\014nite)g(clusters)i(in)e Ft(Y)967 4017 y Fr(1)1039 4003 y Fw(and)h Ft(Y)1270 4017 y Fr(2)1309 4003 y Fw(.)46 b(Then)32 b(w)m(e)g(let)g(\()p Ft(Z)7 b Fw(\()p Ft(y)s Fw(\)\))2145 4017 y Fu(y)r Fq(2)p Fr(\003)2316 4003 y Fw(b)s(e)31 b(i.i.d.)g(random)g(v)-5 b(ariables)31 b(taking)165 4116 y(v)-5 b(alues)33 b(+1)h(and)g Fv(\000)p Fw(1)f(with)g(probabilit)m(y)e(1)p Ft(=)p Fw(2)k(eac)m(h,)h(and)d (assign)h(to)g(eac)m(h)h(\014nite)e(cluster)g Ft(C)40 b Fw(of)34 b Ft(Y)3575 4130 y Fr(1)165 4229 y Fw(and)29 b Ft(Y)394 4243 y Fr(2)462 4229 y Fw(the)g(v)-5 b(alue)28 b Ft(Z)7 b Fw(\()p Ft(y)s Fw(\),)30 b(where)e Ft(y)k Fw(is)c(the)h(\(unique\))f(v)m(ertex)i(of)f Ft(C)35 b Fw(that)30 b(minimizes)c(the)j(distance)165 4342 y(to)f Ft(x)p Fw(.)40 b(This)25 b(de\014ned)h Ft(X)979 4356 y Fr(1)1046 4342 y Fw(and)g Ft(X)1294 4356 y Fr(2)1334 4342 y Fw(.)40 b(A)27 b(momen)m(t's)h(though)m(t)f(rev)m(eals)h(that)f (the)h(set)f(of)h(v)m(ertices)g(that)165 4455 y(can)33 b(b)s(e)e(reac)m(hed)h(from)g Ft(x)g Fw(via)f(spins)f(in)g Ft(X)1654 4469 y Fr(1)1726 4455 y Fw(is)h(almost)h(surely)e(con)m (tained)i(in)f(the)h(corresp)s(onding)165 4568 y(set)f(for)f Ft(X)521 4582 y Fr(2)561 4568 y Fw(.)41 b(Hence)31 b(\(42\))h(is)d (established,)g(and)h(w)m(e)h(are)g(done.)1200 b Fk(2)165 4731 y Fw(Note)38 b(that)e(this)f(pro)s(of)h(did)e(not)i(use)g(an)m(y)g (prop)s(ert)m(y)g(of)g Fx(T)2249 4746 y Fu(d)2325 4731 y Fw(except)h(for)f(the)g(tree)h(structure,)g(so)165 4844 y(Theorem)30 b(6.20)i(can)f(immediately)e(b)s(e)g(extended)i(to)g (the)f(setting)h(of)g(arbitrary)e(trees.)165 5087 y FA(6.6)113 b(Cluster)37 b(algorithms)h(for)g(computer)g(sim)m(ulation)165 5259 y Fw(An)31 b(issue)f(of)h(great)h(imp)s(ortance)f(in)e (statistical)i(mec)m(hanics)g(whic)m(h)f(w)m(e)h(ha)m(v)m(e)i(not)e (touc)m(hed)h(up)s(on)165 5372 y(so)40 b(far)g(is)f(the)h(abilit)m(y)e (to)j(p)s(erform)d(computer)i(sim)m(ulations)e(of)i(large)g(Gibbs)e (systems.)69 b(Man)m(y)165 5484 y(\(most?\))40 b(questions)23 b(ab)s(out)h(phase)g(transition)e(b)s(eha)m(vior)h(etc.)j(can)e(with)f (curren)m(t)h(kno)m(wledge)g(only)165 5597 y(b)s(e)h(answ)m(ered)h (partially)d(\(or)j(not)g(at)g(all\))f(using)f(rigorous)h(mathematical) h(argumen)m(ts.)39 b(Computer)1844 5895 y(56)p eop %%Page: 57 57 57 56 bop 165 304 a Fw(sim)m(ulations)26 b(are)i(then)g(imp)s(ortan)m (t)f(for)g(supp)s(orting)e(\(or)j(rejecting\))h(heuristic)c(argumen)m (ts,)k(or)f(\(in)165 417 y(case)e(not)f(ev)m(en)g(a)g(go)s(o)s(d)f (heuristic)f(can)h(b)s(e)g(found\))g(to)h(pro)m(vide)f(ideas)f(for)i (what)f(a)h(go)s(o)s(d)f(conjecture)165 530 y(migh)m(t)g(b)s(e.)38 b(This)22 b(topic)i(is)f(somewhat)h(b)s(eside)e(the)i(main)f(issue)g (of)h(our)f(surv)m(ey)-8 b(,)26 b(but)d(since)h(random-)165 643 y(cluster)36 b(represen)m(tations)f(ha)m(v)m(e)i(pla)m(y)m(ed)f(a)g (k)m(ey)h(role)f(in)e(sim)m(ulation)g(algorithms)h(for)g(more)h(than) 165 756 y(a)h(decade)h(w)m(e)f(feel)f(that)h(it)g(is)e(appropriate)h (to)h(describ)s(e)f(some)h(of)f(these)i(algorithms.)58 b(In)36 b(fact,)165 869 y(it)e(w)m(as)g(the)g(need)f(of)h(e\016cien)m (t)g(sim)m(ulation)e(whic)m(h,)i(in)e(the)i(late)g(1980's,)j(spark)m (ed)c(the)h(reviv)-5 b(al)33 b(of)165 981 y(the)26 b(random-cluster)f (mo)s(del)g(\(Sw)m(endsen)g(and)g(W)-8 b(ang)27 b([219)r(]\))f(whic)m (h)f(up)g(to)h(then)g(had)f(raised)g(only)165 1094 y(little)30 b(in)m(terest)g(since)g(its)g(in)m(tro)s(duction)e(b)m(y)i(F)-8 b(ortuin)30 b(and)g(Kasteleyn)g(in)f(the)i(early)f(1970's.)306 1207 y(Consider)c(for)i(instance)f(the)h(Ising)f(mo)s(del)g(with)f (free)i(b)s(oundary)e(condition)g(on)i(a)g(large)g(cubic)165 1320 y(region)j(\003)26 b Fv(\032)g Fx(Z)689 1287 y Fu(d)729 1320 y Fw(.)42 b(Direct)31 b(sampling)e(from)i(the)g(Gibbs)e (distribution)e Ft(\026)2646 1335 y Fu(h;\014)s(;)p Fr(\003)2853 1320 y Fw(with)i(free)i(b)s(oundary)165 1433 y(condition)42 b(is)f(not)i(feasible,)i(due)d(to)i(the)f(h)m(uge)g(cardinalit)m(y)e (of)i(the)g(state)h(space)f(\012,)j(and)c(the)165 1546 y(\(related\))36 b(in)m(tractabilit)m(y)e(of)h(computing)f(the)h (normalizing)e(constan)m(t)k(for)d(the)i(Gibbs)d(measure.)165 1659 y(The)38 b(most)g(widely)d(used)i(w)m(a)m(y)i(to)f(handle)e(this)h (problem)f(is)h(the)h Fp(Markov)h(chain)h(Monte)g(Carlo)165 1772 y Fw(metho)s(d,)h(whic)m(h)c(dates)i(bac)m(k)g(to)h(the)e(1953)j (pap)s(er)c(b)m(y)i(Metrop)s(olis)e(et)i(al.)g([178)q(].)66 b(The)38 b(idea)g(is)165 1885 y(to)d(de\014ne)e(an)g(ergo)s(dic)g(Mark) m(o)m(v)j(c)m(hain)d(ha)m(ving)g(as)h(unique)e(stationary)i (distribution)c(the)k(target)165 1998 y(distribution)40 b Ft(\026)729 2013 y Fu(h;\014)s(;)p Fr(\003)905 1998 y Fw(.)80 b(Starting)43 b(the)g(c)m(hain)g(in)f(an)i(arbitrary)e(state) i(and)f(running)e(the)j(c)m(hain)165 2111 y(for)d(long)g(enough)f(will) f(then)i(pro)s(duce)e(an)i(output)g(with)f(a)h(distribution)c(close)42 b(to)f(the)h(target)165 2223 y(distribution.)h(An)32 b(example)g(of)g(suc)m(h)g(a)h(c)m(hain)f(is)f(the)h(single-site)f (heat)i(bath)f(algorithm,)g(whose)165 2336 y(ev)m(olution)g(is)g(as)g (follo)m(ws.)46 b(A)m(t)34 b(eac)m(h)f(in)m(teger)g(time,)g(a)g(v)m (ertex)g Ft(x)c Fv(2)f Fw(\003)33 b(is)e(c)m(hosen)i(at)g(random,)g (and)165 2449 y(the)26 b(spin)e(at)i Ft(x)f Fw(is)g(replaced)g(b)m(y)g (a)h(new)f(v)-5 b(alue)25 b(according)h(to)g(the)g(conditional)d (distribution)f(\(under)165 2562 y Ft(\026)220 2577 y Fu(h;\014)s(;)p Fr(\003)396 2562 y Fw(\))33 b(of)g(the)g(spin)e(at)j Ft(x)f Fw(giv)m(en)g(the)g(spins)d(at)k(its)e(neigh)m(b)s(ors.)47 b(It)33 b(is)f(immediate)g(that)h Ft(\026)3344 2577 y Fu(h;\014)s(;)p Fr(\003)3553 2562 y Fw(is)165 2675 y(stationary)j(for)e (this)g(c)m(hain,)i(and)f(ergo)s(dicit)m(y)f(of)h(the)h(c)m(hain)e (follo)m(ws)g(from)h(elemen)m(tary)g(Mark)m(o)m(v)165 2788 y(c)m(hain)30 b(theory)g(up)s(on)f(c)m(hec)m(king)i(that)g(it)e (is)g(ap)s(erio)s(dic)f(and)i(irreducible.)37 b(The)30 b(problem)e(with)h(this)165 2901 y(approac)m(h)38 b(is)e(that)i(the)g (time)f(tak)m(en)h(to)g(come)g(close)g(to)g(equilibrium)32 b(ma)m(y)38 b(b)s(e)f(v)m(ery)g(long.)61 b(F)-8 b(or)165 3014 y(example,)38 b(let)f Ft(h)e Fw(=)g(0.)60 b(Then,)37 b(for)f Ft(\014)41 b(<)35 b(\014)1679 3028 y Fu(c)1750 3014 y Fw(\(with)h Ft(\014)2050 3028 y Fu(c)2121 3014 y Fw(de\014ned)f(as)i(in)e(Theorem)h(3.1\),)k(the)c(time)165 3127 y(tak)m(en)43 b(to)f(come)g(within)d(a)i(\014xed)g(small)f(v)-5 b(ariational)40 b(distance)h(from)g(the)g(target)i(distribution)165 3240 y(gro)m(ws)32 b(only)e(lik)m(e)g Ft(n)15 b Fw(log)h Ft(n)31 b Fw(in)e(the)i(size)g(of)g(the)g(system)g(\(here)g Ft(n)g Fw(is)f(the)h(n)m(um)m(b)s(er)e(of)j(v)m(ertices)f(in)f(\003\)) 165 3353 y(whereas)23 b(in)f(con)m(trast)j(the)e(time)g(gro)m(ws)g (\(stretc)m(hed\))i(exp)s(onen)m(tially)c(in)h(the)h(size)g(of)h(the)f (system)g(for)165 3465 y Ft(\014)31 b(>)25 b(\014)394 3479 y Fu(c)429 3465 y Fw(;)i(see)g(e.g.)g([169)r(,)f(168)q(].)40 b(This)24 b(means)h(that)i(sim)m(ulation)d(using)g(this)h(heat)i(bath)e (algorithm)g(is)165 3578 y(computationally)j(feasible)g(ev)m(en)h(for)g (fairly)e(large)i(systems)g(pro)m(vided)f(that)h Ft(\014)i(<)25 b(\014)3091 3592 y Fu(c)3126 3578 y Fw(,)k(but)f(not)h(for)165 3691 y Ft(\014)i(>)25 b(\014)394 3705 y Fu(c)429 3691 y Fw(.)38 b(What)24 b(happ)s(ens)d(for)i Ft(\014)30 b(>)25 b(\014)1449 3705 y Fu(c)1507 3691 y Fw(is)d(that)i(if)e(the)h(c)m(hain) f(starts)i(in)d(a)j(con\014guration)e(dominated)165 3804 y(b)m(y)32 b(plus)d(spins,)h(then)h(the)g(plus)e(spins)h(con)m(tin)m (ue)h(to)h(dominate)f(for)g(an)g(astronomical)g(amoun)m(t)h(of)165 3917 y(time,)39 b(and)e(similarly)c(for)k(starting)g(con\014gurations)f (dominated)h(b)m(y)g(min)m(us)e(spins.)59 b(The)36 b(set)i(of)165 4030 y(con\014gurations)d(where)f(the)h(fraction)g(of)g(plus)e(spins)g (is)i(around)f(1)p Ft(=)p Fw(2)i(\(rather)f(than)g(around)f(the)165 4143 y(fractions)28 b(predicted)f(b)m(y)h(the)g(magnetization)h(in)e (the)h(in\014nite-v)m(olume)e(Gibbs)h(measures)h Ft(\026)3381 4105 y Fr(+)3381 4171 y Fu(\014)3468 4143 y Fw(and)165 4271 y Ft(\026)220 4232 y Fq(\000)220 4299 y Fu(\014)279 4271 y Fw(\))37 b(has)g(small)e(probabilit)m(y)f(and)j(th)m(us)f(can)h (b)s(e)f(seen)h(as)g(a)g(\\b)s(ottlenec)m(k")i(in)c(the)i(state)h (space,)165 4384 y(slo)m(wing)30 b(do)m(wn)f(the)i(con)m(v)m(ergence)i (rate.)306 4496 y(A)44 b(w)m(a)m(y)h(to)f(tac)m(kle)h(the)f (exceedingly)f(slo)m(w)g(con)m(v)m(ergence)j(rate)f(in)d(the)i(phase)f (co)s(existence)165 4609 y(regime)36 b(is)f(to)h(use)f(the)h(heat)h (bath)e(algorithm)g(for)g(the)h(corresp)s(onding)e(random-cluster)g(mo) s(del)165 4722 y(rather)h(than)g(for)f(the)h(Ising)f(mo)s(del)g (itself,)h(and)f(only)g(in)g(the)h(end)f(go)i(o)m(v)m(er)g(to)g(the)f (Ising)e(mo)s(del)165 4835 y(b)m(y)40 b(the)g(random)f(mapping)f (describ)s(ed)g(in)h(Corollary)f(6.3.)70 b(This)38 b(has)i(the)g (disadv)-5 b(an)m(tage)41 b(that)165 4948 y(the)c(calculation)e(of)i (single-site)e(\(or,)k(rather,)f(single-edge\))e(conditional)f (probabilities)e(b)s(ecome)165 5061 y(computationally)c(more)h (complicated)g(due)g(to)g(the)h(p)s(ossible)c(dep)s(endence)i(on)h (edges)h(arbitrarily)165 5174 y(far)g(a)m(w)m(a)m(y)i(\(see)f(Lemma)e (6.6\).)44 b(This)29 b(disadv)-5 b(an)m(tage,)32 b(ho)m(w)m(ev)m(er,)h (seems)e(to)g(b)s(e)f(b)m(y)h(far)g(out)m(w)m(eighed)165 5287 y(b)m(y)j(the)h(fact)f(that)h(the)f(con)m(v)m(ergence)j(rate)e(of) f(the)g(Mark)m(o)m(v)i(c)m(hain)d(\(for)i Ft(\014)h(>)31 b(\014)2961 5301 y Fu(c)2996 5287 y Fw(\))k(app)s(ears)e(to)i(b)s(e)165 5400 y(v)m(ery)41 b(m)m(uc)m(h)f(faster)g(than)g(for)g(the)g(heat)h (bath)f(applied)e(directly)h(to)i(the)f(spin)e(v)-5 b(ariables.)69 b(The)165 5513 y(reason)31 b(for)f(this)g(phenomenon)f(is)h(that)h(the) g(random-cluster)e(represen)m(tation)i(\\do)s(esn't)g(see)g(an)m(y)165 5626 y(di\013erence")37 b(b)s(et)m(w)m(een)h(the)f(plus)e(state)j(and)f (the)g(min)m(us)e(state.)62 b(This)35 b(approac)m(h)j(can)f(of)g (course)1844 5895 y(57)p eop %%Page: 58 58 58 57 bop 165 304 a Fw(b)s(e)32 b(used)g(also)h(for)f(the)h Ft(q)e Fv(\025)e Fw(3)k(P)m(otts)h(mo)s(del,)e(and)g(is)f(due)h(to)h (Sw)m(een)m(y)g([218)r(].)47 b(Later,)34 b(Propp)e(and)165 417 y(Wilson)23 b([200)r(])i(built)d(on)i(this)g(approac)m(h)g(b)m(y)h (coupling)d(sev)m(eral)j(suc)m(h)f(Mark)m(o)m(v)i(c)m(hains)e(\(i.e.)h (running)165 530 y(them)j(in)f(parallel\))g(in)g(an)h(ingenious)e(w)m (a)m(y)-8 b(,)30 b(pro)s(ducing)c(an)i(algorithm)f(whic)m(h)g(runs)f (for)i(a)h(random)165 643 y(amoun)m(t)38 b(of)g(time)f(\(determined)f (b)m(y)h(the)h(algorithm)e(itself)7 b(\))37 b(and)g(then)g(outputs)g(a) h(state)g(whic)m(h)165 756 y(has)29 b Fp(exactly)h Fw(the)f(target)i (distribution.)36 b(The)29 b(running)d(time)j(of)g(this)f(algorithm)g (turns)g(out)h(\(from)165 869 y(exp)s(erimen)m(ts\))35 b(to)g(b)s(e)g(mo)s(derate)g(except)h(for)f(the)g(case)h(of)f(large)g Ft(q)j Fw(and)c Ft(\014)40 b Fw(close)35 b(to)h(the)f(critical)165 981 y(v)-5 b(alue.)40 b(The)27 b(Propp{Wilson)g(approac)m(h,)i(kno)m (wn)e(as)h(exact)i(or)e(p)s(erfect)g(sim)m(ulation,)f(has)h(receiv)m (ed)165 1094 y(a)36 b(v)-5 b(ast)37 b(amoun)m(t)f(of)f(atten)m(tion)i (among)f(statisticians)f(during)e(the)j(last)g(few)f(y)m(ears)h(\(see)h (e.g.)g(the)165 1207 y(annotated)i(bibliograph)m(y)c([225)r(]\))j(and)g (w)m(e)g(b)s(eliev)m(e)f(that)i(it)e(has)h(in)m(teresting)f(p)s(oten)m (tial)h(also)f(in)165 1320 y(ph)m(ysics.)306 1433 y(There)43 b(is,)i(ho)m(w)m(ev)m(er,)i(another)c(Mark)m(o)m(v)i(c)m(hain)d(whic)m (h)g(app)s(ears)g(to)h(con)m(v)m(erge)i(ev)m(en)f(faster)165 1546 y(than)36 b(those)g(of)h(Sw)m(een)m(y)-8 b(,)38 b(Propp)d(and)g(Wilson.)56 b(W)-8 b(e)37 b(are)g(talking)e(ab)s(out)g (the)i(Sw)m(endsen{W)-8 b(ang)165 1659 y([219)r(])44 b(algorithm,)k(whic)m(h)43 b(runs)f(as)j(follo)m(ws)e(for)h(Ising)f (and)h(P)m(otts)i(mo)s(dels)d(on)h(a)h(graph)e(with)165 1772 y(v)m(ertex)d(set)f Fv(L)f Fw(and)g(edge)h(set)g Fv(B)s Fw(:)57 b(Starting)38 b(with)f(a)i(spin)d(con\014guration)i Ft(X)2890 1786 y Fr(0)2969 1772 y Fv(2)g(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)j(;)d(q)s Fv(g)3450 1739 y Fq(L)3503 1772 y Fw(,)41 b(a)165 1885 y(b)s(ond)29 b(con\014guration)i Ft(Y)1000 1899 y Fr(0)1065 1885 y Fv(2)25 b(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)1371 1852 y Fq(B)1456 1885 y Fw(is)30 b(c)m(hosen)h(according)g(to)g(the)g(random)f(mapping)f(de\014ned)g(in) 165 1998 y(Corollary)36 b(6.4.)62 b(Then)36 b(another)i(spin)d (con\014guration)h Ft(X)2192 2012 y Fr(1)2269 1998 y Fw(is)g(pro)s(duced)g(from)g Ft(Y)3045 2012 y Fr(0)3122 1998 y Fw(b)m(y)h(assigning)165 2111 y(random)e(spins)e(to)i(the)h (connected)g(comp)s(onen)m(ts,)g(i.e.)f(b)m(y)g(the)h(random)e(mapping) f(of)i(Corollary)165 2223 y(6.3.)42 b(This)28 b(pro)s(cedure)h(is)g (then)h(iterated,)h(pro)s(ducing)c(a)k(new)e(edge)i(con\014guration)f Ft(Y)3137 2237 y Fr(1)3206 2223 y Fw(and)f(a)i(new)165 2336 y(spin)24 b(con\014guration)h Ft(X)974 2350 y Fr(2)1014 2336 y Fw(,)h(etc.)40 b(By)26 b(com)m(bining)e(the)i(t)m(w)m(o)h (corollaries,)e(w)m(e)h(see)g(that)g(if)f Ft(X)3202 2350 y Fr(0)3267 2336 y Fw(is)g(c)m(hosen)165 2449 y(according)32 b(to)h(the)f(target)h(distribution,)c(then)j(the)g(same)g(holds)e(for)i Ft(X)2690 2463 y Fr(1)2730 2449 y Fw(,)g(and)f(consequen)m(tly)h(for) 165 2562 y Ft(X)240 2576 y Fr(2)280 2562 y Ft(;)15 b(X)395 2576 y Fr(3)435 2562 y Ft(;)g(:)g(:)g(:)r Fw(.)40 b(In)28 b(other)h(w)m(ords,)g(the)g(target)h(distribution)25 b(is)j(stationary)h(for)g(the)g(c)m(hain)f Fv(f)p Ft(X)3367 2577 y Fu(k)3411 2562 y Fv(g)3456 2529 y Fq(1)3456 2589 y Fu(k)r Fr(=0)3589 2562 y Fw(,)165 2675 y(and)g(b)m(y)g(the)g (\(easily)g(v)m(eri\014ed\))f(ergo)s(dicit)m(y)h(of)g(the)g(c)m(hain)g (w)m(e)g(ha)m(v)m(e)i(a)e(v)-5 b(alid)26 b(Mark)m(o)m(v)k(c)m(hain)e (Mon)m(te)165 2788 y(Carlo)i(algorithm.)41 b(Although)30 b(it)g(is)g(not)h(exact)h(in)e(the)h(sense)f(of)h(the)g(Propp{Wilson)e (algorithm,)165 2901 y(it)e(app)s(ears)g(to)i(con)m(v)m(erge)g(m)m(uc)m (h)f(faster,)h(th)m(us)e(in)f(practice)i(allo)m(wing)e(sim)m(ulation)g (of)i(systems)g(that)165 3014 y(are)35 b(orders)f(of)g(magnitude)g (larger.)52 b(Heuristically)-8 b(,)34 b(the)h(reason)f(for)g(this)g (faster)g(con)m(v)m(ergence)j(is)165 3127 y(that)27 b(large)g(c)m(h)m (unks)f(of)g(spins)e(ma)m(y)j(\015ip)e(sim)m(ultaneously)-8 b(,)26 b(allo)m(wing)f(the)h(c)m(hain)g(to)h(tunnel)e(through)165 3240 y(an)m(y)34 b(b)s(ottlenec)m(ks)f(in)f(the)h(target)i (distribution.)45 b(Ho)m(w)m(ev)m(er,)36 b(rigorous)c(upp)s(er)f(and)i (lo)m(w)m(er)g(b)s(ounds)165 3353 y(on)26 b(the)g(time)f(tak)m(en)h(to) h(come)f(close)g(to)g(equilibrium)21 b(are)26 b(to)g(a)g(large)g(exten) m(t)h(lac)m(king,)f(although)f(Li)165 3465 y(and)f(Sok)-5 b(al)24 b([159)q(])h(ha)m(v)m(e)g(pro)m(vided)e(a)i(lo)m(w)m(er)g(b)s (ound)d(demonstrating)i(the)g(phenomenon)f(of)i(\\critical)165 3578 y(slo)m(wing)30 b(do)m(wn")g(as)h Ft(\014)k Fw(approac)m(hes)c Ft(\014)1489 3592 y Fu(c)1524 3578 y Fw(.)306 3691 y(The)25 b(Sw)m(endsen{W)-8 b(ang)25 b(algorithm)f(has,)i(since)e(its)g(in)m (tro)s(duction)f(in)h(1987,)k(b)s(ecome)d(the)g(stan-)165 3804 y(dard)34 b(approac)m(h)h(to)g(sim)m(ulating)e(Ising)g(and)h(P)m (otts)i(mo)s(dels.)52 b(In)m(teresting)35 b(v)-5 b(arian)m(ts)34 b(and)g(mo)s(di\014-)165 3917 y(cations)i(of)f(this)f(algorithm)h(ha)m (v)m(e)h(b)s(een)f(dev)m(elop)s(ed)f(b)m(y)i(W)-8 b(ol\013)35 b([226)r(])h(and)e(Mac)m(h)m(ta)k(et)e(al.)f([165)r(];)165 4030 y(the)26 b(last)g(pap)s(er)f(is)g(an)g(in)m(teresting)g(attempt)i (at)g(com)m(bining)d(the)i(original)e(approac)m(h)i(of)g(Sw)m(endsen) 165 4143 y(and)31 b(W)-8 b(ang)33 b(with)d(ideas)h(from)g(so)g(called)g Fp(invasion)j(p)-5 b(er)g(c)g(olation)35 b Fw(\(see)d([55)q(]\))g(to)g (get)h(an)e(algorithm)165 4256 y(sp)s(eci\014cally)39 b(aimed)g(at)j(sampling)c(from)i(a)h(Gibbs)e(distribution)e(at)k(the)g (critical)e(in)m(v)m(erse)h(tem-)165 4369 y(p)s(erature)f Ft(\014)592 4383 y Fu(c)628 4369 y Fw(,)j(i.e.)e(where)f(the)h(use)g (of)g(other)g(algorithms)f(ha)m(v)m(e)i(pro)m(v)m(ed)f(to)g(b)s(e)g (most)g(di\016cult.)165 4482 y(Generalizations)35 b(of)h(the)f(Sw)m (endsen{W)-8 b(ang)36 b(algorithm)f(for)g(v)-5 b(arious)34 b(mo)s(dels)g(other)i(than)f(Ising)165 4595 y(and)d(P)m(otts)i(mo)s (dels)d(ha)m(v)m(e)j(also)e(b)s(een)g(obtained,)g(see)h(e.g.)h(Campb)s (ell)c(and)i(Cha)m(y)m(es)h([48)q(],)g(Cha)m(y)m(es)165 4708 y(and)d(Mac)m(h)m(ta)j([58)q(,)e(59)q(],)f(and)g(H\177)-45 b(aggstr\177)g(om)32 b(et)f(al.)f([121)r(].)165 4951 y FA(6.7)113 b(Random-cluster)38 b(represen)m(tation)g(of)g(the)f (Widom{Ro)m(wlinson)i(mo)s(del)165 5123 y Fw(The)27 b(random-cluster)f (mo)s(del)g(can)i(b)s(e)e(seen)i(as)f(a)h(p)s(erturbation)d(of)i (Bernoulli)e(b)s(ond)h(p)s(ercolation,)165 5235 y(where)e(the)h (probabilit)m(y)d(measure)j(is)e(c)m(hanged)j(in)d(fa)m(v)m(our)i(of)g (con\014gurations)f(with)f(man)m(y)i(\(for)f Ft(q)k(>)165 5348 y Fw(1\))c(or)g(few)f(\(for)h Ft(q)k(<)d Fw(1\))f(connected)g (comp)s(onen)m(ts.)39 b(A)23 b(fairly)f(natural)g(question)h(is)f(what) i(happ)s(ens)d(if)165 5461 y(w)m(e)29 b(p)s(erturb)c(Bernoulli)h(site)h (p)s(ercolation)g(in)g(the)h(same)g(w)m(a)m(y)-8 b(.)41 b(F)-8 b(or)29 b(lac)m(k)f(of)g(an)g(established)e(name,)165 5574 y(w)m(e)i(call)f(the)g(resulting)f(mo)s(del)g(the)h (site-random-cluster)g(mo)s(del.)38 b(Let)28 b Ft(G)g Fw(b)s(e)f(a)g(\014nite)g(graph)f(with)1844 5895 y(58)p eop %%Page: 59 59 59 58 bop 165 304 a Fw(v)m(ertex)32 b(set)f Fv(L)g Fw(and)f(edge)h(set) g Fv(B)s Fw(.)41 b(F)-8 b(or)32 b(a)f(site)f(con\014guration)g Ft(\021)f Fv(2)d(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)2671 271 y Fq(L)2726 304 y Fw(,)31 b(w)m(e)g(write)f Ft(k)s Fw(\()p Ft(\021)s Fw(\))h(for)g(the)165 417 y(n)m(um)m(b)s(er)26 b(of)i(connected)g(comp)s(onen)m(ts)f(in)f(the)h(subgraph)f(of)h Ft(G)h Fw(obtained)e(b)m(y)h(deleting)f(all)g(v)m(ertices)165 530 y Ft(x)31 b Fw(with)e Ft(\021)s Fw(\()p Ft(x)p Fw(\))d(=)f(0)31 b(and)e(their)h(inciden)m(t)f(edges.)165 713 y Fx(De\014nition)36 b(6.21)46 b Fp(The)31 b Fx(site-random-cluster)i(measure)d Ft( )2456 680 y Fu(G)2453 735 y(p;q)2577 713 y Fp(for)i Ft(G)f Fp(with)h(p)-5 b(ar)g(ameters)35 b Ft(p)25 b Fv(2)165 835 y Fw([0)p Ft(;)15 b Fw(1])36 b Fp(and)f Ft(q)30 b(>)d Fw(0)34 b Fp(is)g(the)g(pr)-5 b(ob)g(ability)36 b(me)-5 b(asur)g(e)35 b(on)g Fv(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)2202 802 y Fq(L)2290 835 y Fp(which)35 b(to)f(e)-5 b(ach)35 b Ft(\021)c Fv(2)c(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)3247 802 y Fq(L)3335 835 y Fp(assigns)165 948 y(pr)-5 b(ob)g(ability)1005 1121 y Ft( )1067 1083 y Fu(G)1064 1143 y(p;q)1158 1121 y Fw(\()p Ft(\021)s Fw(\))26 b(=)1463 1059 y(1)p 1408 1100 156 4 v 1408 1183 a Ft(Z)1477 1157 y Fu(G)1470 1206 y(p;q)1588 977 y Fs(\()1670 1040 y(Y)1655 1224 y Fu(x)p Fq(2L)1806 1121 y Ft(p)1852 1083 y Fu(\021)r Fr(\()p Fu(x)p Fr(\))1988 1121 y Fw(\(1)21 b Fv(\000)f Ft(p)p Fw(\))2261 1083 y Fr(1)p Fq(\000)p Fu(\021)r Fr(\()p Fu(x)p Fr(\))2488 977 y Fs(\))2570 1121 y Ft(q)2614 1083 y Fu(k)r Fr(\()p Fu(\021)r Fr(\))2749 1121 y Ft(;)165 1374 y Fp(wher)-5 b(e)34 b Ft(Z)491 1341 y Fu(G)484 1396 y(p;q)610 1374 y Fp(is)e(a)h(normalizing)i(c)-5 b(onstant.)165 1557 y Fw(Analogously)40 b(to)i(the)e(usual)g(random-cluster)f(mo)s (del)h(living)e(on)i(b)s(onds,)i(taking)f Ft(q)k Fw(=)d(1)f(giv)m(es) 165 1670 y(the)32 b(ordinary)f(Bernoulli)e(site)j(p)s(ercolation)f Ft( )1788 1684 y Fu(p)1827 1670 y Fw(,)i(while)d(other)i(c)m(hoices)g (of)g Ft(q)j Fw(lead)d(to)g(dep)s(endence)165 1782 y(b)s(et)m(w)m(een)f (v)m(ertices.)306 1895 y(T)-8 b(aking)40 b Ft(q)k Fw(=)d(2)f(is)f(of)i (particular)d(in)m(terest)i(b)s(ecause)g(it)f(leads)h(to)g(a)h (represen)m(tation)f(of)g(the)165 2008 y(Widom{Ro)m(wlinson)g(mo)s(del) g(whic)m(h)h(is)f(similar)f(to)j(\(and)g(in)e(fact)i(sligh)m(tly)e (simpler)f(than\))j(the)165 2121 y(usual)31 b(random-cluster)h (represen)m(tation)g(of)h(the)f(Ising)f(mo)s(del.)46 b(Let)33 b Ft(\026)2649 2088 y Fu(G)2649 2148 y(\025)2740 2121 y Fw(b)s(e)f(the)g(Gibbs)f(measure)165 2234 y(for)24 b(the)f(Widom{Ro)m(wlinson)f(mo)s(del)h(with)f(activit)m(y)i Ft(\025)f Fw(on)h Ft(G)p Fw(,)i(i.e.)d Ft(\026)2515 2201 y Fu(G)2515 2260 y(\025)2598 2234 y Fw(is)f(the)i(probabilit)m(y)d (measure)165 2347 y(on)31 b Fv(f\000)p Fw(1)p Ft(;)15 b Fw(0)p Ft(;)g Fw(+1)p Fv(g)739 2314 y Fq(L)824 2347 y Fw(whic)m(h)29 b(to)i(eac)m(h)h Ft(\030)d Fv(2)c(f\000)p Fw(1)p Ft(;)15 b Fw(0)p Ft(;)g Fw(+1)p Fv(g)2002 2314 y Fq(L)2087 2347 y Fw(assigns)30 b(probabilit)m(y)e(prop)s(ortional)g (to)1357 2464 y Fs(Y)1297 2653 y Fq(h)p Fu(xy)r Fq(i2B)1539 2545 y Ft(I)1579 2564 y Fq(f)p Fu(\030)s Fr(\()p Fu(x)p Fr(\))p Fu(\030)s Fr(\()p Fu(y)r Fr(\))p Fq(6)p Fr(=)p Fq(\000)p Fr(1)p Fq(g)2082 2464 y Fs(Y)2067 2648 y Fu(x)p Fq(2L)2218 2545 y Ft(\025)2271 2508 y Fq(j)p Fu(\030)s Fr(\()p Fu(x)p Fr(\))p Fq(j)2458 2545 y Ft(:)165 2834 y Fw(The)i(follo)m(wing)f(analogues)i(of)f(Corollaries)f(6.3)i(and)f (6.4)h(are)g(trivial)e(to)i(c)m(hec)m(k.)165 3017 y Fx(Prop)s(osition) 37 b(6.22)46 b Fp(L)-5 b(et)46 b Ft(p)i Fw(=)1400 2981 y Fu(\025)p 1355 2996 132 4 v 1355 3049 a Fr(1+)p Fu(\025)1496 3017 y Fp(,)h(and)d(supp)-5 b(ose)47 b(we)f(pick)f(a)h(r)-5 b(andom)48 b(spin)e(c)-5 b(on\014gur)g(ation)165 3142 y Ft(X)33 b Fv(2)25 b(f\000)p Fw(1)p Ft(;)15 b Fw(0)p Ft(;)g Fw(+1)p Fv(g)806 3109 y Fq(L)894 3142 y Fp(as)33 b(fol)5 b(lows.)273 3325 y(1.)46 b(Pick)31 b Ft(Y)46 b Fv(2)24 b(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)1002 3292 y Fq(L)1090 3325 y Fp(ac)-5 b(c)g(or)g(ding)35 b(to)e Ft( )1662 3292 y Fu(G)1659 3348 y(p;)p Fr(2)1754 3325 y Fp(.)273 3511 y(2.)46 b(Set)32 b Ft(X)7 b Fw(\()p Ft(x)p Fw(\))27 b(=)e(0)33 b Fp(for)g(e)-5 b(ach)33 b Ft(x)25 b Fv(2)g(L)33 b Fp(such)f(that)i Ft(Y)20 b Fw(\()p Ft(x)p Fw(\))26 b(=)f(0)p Fp(.)273 3696 y(3.)46 b(F)-7 b(or)37 b(e)-5 b(ach)37 b(op)-5 b(en)37 b(cluster)f Ft(C)43 b Fp(of)36 b Ft(Y)20 b Fp(,)37 b(\015ip)g(a)f(fair)h(c)-5 b(oin)36 b(to)h(de)-5 b(cide)37 b(whether)g(to)g(give)e(spin)i Fw(+1)393 3809 y Fp(or)c Fv(\000)p Fw(1)g Fp(in)f Ft(X)40 b Fp(to)33 b(al)5 b(l)33 b(vertic)-5 b(es)33 b(of)g Ft(C)7 b Fp(.)165 3992 y(Then)33 b Ft(X)40 b Fp(is)33 b(distribute)-5 b(d)34 b(ac)-5 b(c)g(or)g(ding)35 b(to)e(the)g(Widom{R)-5 b(ow)5 b(linson)36 b(Gibbs)c(me)-5 b(asur)g(e)34 b Ft(\026)3154 3959 y Fu(G)3154 4018 y(\025)3213 3992 y Fp(.)165 4198 y Fx(Prop)s(osition)j(6.23)46 b Fp(L)-5 b(et)46 b Ft(p)i Fw(=)1400 4163 y Fu(\025)p 1355 4178 V 1355 4230 a Fr(1+)p Fu(\025)1496 4198 y Fp(,)h(and)d(supp)-5 b(ose)47 b(we)f(pick)f(a)h(r) -5 b(andom)48 b(spin)e(c)-5 b(on\014gur)g(ation)165 4323 y Ft(Y)46 b Fv(2)24 b(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)569 4290 y Fq(L)657 4323 y Fp(as)33 b(fol)5 b(lows.)273 4506 y(1.)46 b(Pick)31 b Ft(X)i Fv(2)25 b(f\000)p Fw(1)p Ft(;)15 b Fw(0)p Ft(;)g Fw(+1)p Fv(g)1239 4473 y Fq(L)1327 4506 y Fp(ac)-5 b(c)g(or)g(ding)34 b(to)g Ft(\026)1892 4473 y Fu(G)1892 4533 y(\025)1950 4506 y Fp(.)273 4692 y(2.)46 b(Set)32 b Ft(Y)20 b Fw(\()p Ft(x)p Fw(\))26 b(=)f Fv(j)p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))p Fv(j)34 b Fp(for)f(e)-5 b(ach)34 b Ft(x)25 b Fv(2)g(L)p Fp(.)165 4875 y(Then)33 b Ft(Y)53 b Fp(is)32 b(distribute)-5 b(d)34 b(ac)-5 b(c)g(or)g(ding)35 b(to)e(the)g(site-r)-5 b(andom-cluster)35 b(me)-5 b(asur)g(e)34 b Ft( )2920 4842 y Fu(G)2917 4898 y(p;)p Fr(2)3012 4875 y Fp(.)165 5081 y Fw(W)-8 b(e)22 b(remark)e(that)h(for)f Ft(q)28 b Fv(2)d(f)p Fw(3)p Ft(;)15 b Fw(4)p Ft(;)g(:)g(:)g(:)s Fv(g)p Fw(,)23 b(these)e(results)e(extend)i(in)e(the)h(ob)m(vious)g(w)m (a)m(y)h(to)g(a)g(connection)165 5194 y(b)s(et)m(w)m(een)37 b Ft( )584 5161 y Fu(G)581 5217 y(p;q)709 5194 y Fw(and)e(the)h (generalized)f(Widom{Ro)m(wlinson)f(mo)s(del)h(whith)e Ft(q)39 b Fw(t)m(yp)s(es)c(of)h(particles)165 5307 y(rather)31 b(than)f(just)g(2)g(\(and)g(strict)h(repulsion)c(b)s(et)m(w)m(een)k (all)e(particles)h(of)g(di\013eren)m(t)g(t)m(yp)s(e\).)306 5420 y(Man)m(y)f(of)e(the)h(argumen)m(ts)f(applied)f(to)i(Ising)e(and)h (P)m(otts)i(mo)s(dels)d(in)g(Section)h(6.3)i(can)e(no)m(w)h(b)s(e)165 5533 y(applied)33 b(to)j(the)f(Widom{Ro)m(wlinson)e(mo)s(del)h(in)g(a)h (similar)d(manner.)54 b(T)-8 b(o)35 b(apply)f(Theorem)h(4.8,)165 5646 y(w)m(e)27 b(need)f(to)h(calculate)g(the)g(conditional)e (probabilit)m(y)e(in)j(the)g(site-random-cluster)g(mo)s(del)f(that)i(a) 1844 5895 y(59)p eop %%Page: 60 60 60 59 bop 165 304 a Fw(giv)m(en)25 b(v)m(ertex)i(is)d(op)s(en)g(giv)m (en)h(the)g(status)h(of)f(all)f(other)h(v)m(ertices.)40 b(F)-8 b(or)26 b Ft(x)f Fv(2)g(L)f Fw(and)h Ft(\021)k Fv(2)24 b(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)3388 271 y Fq(Lnf)p Fu(x)p Fq(g)3589 304 y Fw(,)165 417 y(w)m(e)31 b(get)1114 587 y Ft( )1176 550 y Fu(G)1173 610 y(p;q)1266 587 y Fw(\()p Ft(x)g Fw(is)e(op)s(en)h Fv(j)15 b Ft(\021)s Fw(\))27 b(=)2107 526 y Ft(p)15 b(q)2212 493 y Fr(1)p Fq(\000)p Fu(\024)p Fr(\()p Fu(x;\021)r Fr(\))p 1950 566 706 4 v 1950 652 a Ft(p)g(q)2055 626 y Fr(1)p Fq(\000)p Fu(\024)p Fr(\()p Fu(x;\021)r Fr(\))2362 652 y Fw(+)20 b(1)h Fv(\000)f Ft(p)3453 587 y Fw(\(43\))165 800 y(where)43 b Ft(\024)p Fw(\()p Ft(x;)15 b(\021)s Fw(\))45 b(is)d(the)i(n)m(um)m(b) s(er)e(of)h(op)s(en)f(clusters)h(of)g Ft(\021)k Fw(whic)m(h)42 b(in)m(tersect)h Ft(x)p Fw('s)h(neigh)m(b)s(orho)s(o)s(d)165 913 y Fv(f)p Ft(y)51 b Fv(2)c(L)g Fw(:)63 b Ft(y)50 b Fv(\030)d Ft(x)p Fv(g)p Fw(.)81 b(If)43 b(the)h(degree)h(of)f(the)f(v)m (ertices)i(in)d Ft(G)i Fw(is)f(b)s(ounded)e(b)m(y)j Ft(N)10 b Fw(,)47 b(sa)m(y)-8 b(,)48 b(then)165 1026 y(0)35 b Fv(\024)g Ft(\024)p Fw(\()p Ft(x;)15 b(\021)s Fw(\))36 b Fv(\024)e Ft(N)46 b Fw(for)36 b(an)m(y)g Ft(x)f Fv(2)f(L)h Fw(and)h Ft(\021)i Fv(2)c(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)2056 993 y Fq(Lnf)p Fu(x)p Fq(g)2256 1026 y Fw(.)58 b(F)-8 b(or)36 b(\014xed)g Ft(q)i Fw(and)e(an)m(y)g Ft(p)3218 993 y Fq(\003)3292 1026 y Fv(2)e Fw(\(0)p Ft(;)15 b Fw(1\),)165 1139 y(w)m(e)39 b(can)f(th)m(us)f(apply)f(Theorem)i(4.8)g(to)h(sho)m(w) e(that)i Ft( )2108 1106 y Fu(G)2105 1162 y(p;q)2236 1139 y Fw(sto)s(c)m(hastically)e(dominates)g Ft( )3308 1153 y Fu(p)3344 1134 y Fm(\003)3422 1139 y Fw(for)g Ft(p)165 1252 y Fw(su\016cien)m(tly)32 b(close)i(to)g(1,)h(and)d(is)h(dominated) f(b)m(y)h Ft( )1977 1266 y Fu(p)2013 1247 y Fm(\003)2086 1252 y Fw(for)g Ft(p)g Fw(small)f(enough.)50 b(The)32 b(argumen)m(ts)i(of)165 1365 y(Section)c(6.3)i(leading)d(to)h(a)h(pro)s (of)e(of)i(Theorem)f(3.1,)h(with)e(the)i(random-cluster)e(mo)s(del)g (replaced)165 1478 y(b)m(y)i(the)f(site-random-cluster)g(mo)s(del,)f (therefore)i(go)g(through)f(to)h(sho)m(w)f(Theorem)g(3.4.)306 1591 y(One)c(thing)e(that)i(do)s(es)g Fp(not)g Fw(go)g(through)f(in)g (this)f(con)m(text,)29 b(ho)m(w)m(ev)m(er,)f(is)d(the)h(analogue)g(of)f (\(31\).)165 1704 y(The)31 b(reason)h(for)f(this)f(is)g(that,)i(in)e (con)m(trast)j(to)f(\(27\),)h(the)f(conditional)d(probabilit)m(y)g(in)h (\(43\))j(fails)165 1817 y(to)40 b(b)s(e)e(increasing)f(in)g Ft(\021)s Fw(,)k(so)e(that)g(Theorem)g(4.8)g(is)f(not)h(applicable)d (for)j(comparison)e(b)s(et)m(w)m(een)165 1930 y(site-random-cluster)f (measures)h(with)f(di\013eren)m(t)g(v)-5 b(alues)37 b(of)g Ft(p)p Fw(.)60 b(In)36 b(fact,)k(the)d(analogue)g(of)h(\(31\))165 2042 y(for)27 b(site-random-cluster)f(measures)h(sometimes)f(fails,)h (and)f(moreo)m(v)m(er)i(the)f(o)s(ccurrence)g(of)g(phase)165 2155 y(transition)33 b(for)h(the)g(Widom{Ro)m(wlinson)e(mo)s(del)h(on)h (certain)g(graphs)g(fails)e(to)j(b)s(e)f(increasing)e(in)165 2268 y Ft(\025)p Fw(,)f(as)g(demonstrated)f(b)m(y)g(Brigh)m(t)m(w)m (ell,)g(H\177)-45 b(aggstr\177)g(om)32 b(and)e(Winkler)f([38)q(].)306 2381 y(Another)36 b(consequence)g(of)f(the)h(failure)d(of)j(the)f (conditional)f(probabilit)m(y)f(in)h(\(43\))i(to)g(b)s(e)f(in-)165 2494 y(creasing)i(is)g(that)g(the)h(FK)m(G)g(inequalit)m(y)d(\(Theorem) i(4.11\))i(cannot)f(b)s(e)e(applied)f(to)j Ft( )3251 2461 y Fu(G)3248 2517 y(p;q)3342 2494 y Fw(.)61 b(As)37 b(a)165 2607 y(consequence,)30 b(the)d(pro)s(of)g(of)h(Theorem)f(6.10)j (cannot)e(b)s(e)f(adapted)h(to)g(the)g(case)h(of)e(the)h(m)m(ultit)m (yp)s(e)165 2720 y(\()p Ft(q)i Fv(\025)d Fw(3\))32 b(Widom{Ro)m (wlinson)d(mo)s(del.)43 b(In)30 b(fact,)j(suc)m(h)e(a)h(Widom{Ro)m (wlinson)d(analogue)j(of)f(The-)165 2833 y(orem)i(6.10)h(is)e(kno)m(wn) g(to)i(b)s(e)e Fp(false)p Fw(,)i(as)f(sho)m(wn)f(b)m(y)h(Runnels)d(and) i(Leb)s(o)m(witz)h([207)q(];)i(see)e(also)g([57)q(])165 2946 y(and)d([184)r(].)1844 5895 y(60)p eop %%Page: 61 61 61 60 bop 165 304 a Fy(7)135 b(Uniqueness)45 b(and)f(exp)t(onen)l(tial) h(mixing)367 453 y(from)f(non-p)t(ercolation)165 656 y Fw(In)f(the)g(previous)e(section)i(w)m(e)h(sa)m(w)f(examples)f(where) h(phase)f(transition)g(in)f(one)j(system)f(w)m(as)165 769 y(equiv)-5 b(alen)m(t)34 b(to)g(the)g(existence)h(of)f(in\014nite)e (clusters)h(in)f(another,)k(suitably)c(de\014ned,)h(system.)52 b(In)165 882 y(this)44 b(section)h(w)m(e)h(shall)d(discuss)g(v)-5 b(arious)44 b(approac)m(hes)h(where)f(conclusions)g(ab)s(out)g(the)h (phase)165 995 y(transition)35 b(b)s(eha)m(vior)g(can)h(only)e(b)s(e)i (dra)m(wn)e(from)i(nonexistence)g(\(and)f(not)h(from)f(existence\))i (of)165 1108 y(in\014nite)29 b(clusters.)42 b(On)30 b(the)h(other)g (hand,)f(these)i(approac)m(hes)f(t)m(ypically)f(apply)g(to)h(a)h(m)m (uc)m(h)f(wider)165 1221 y(range)h(of)f(mo)s(dels.)42 b(W)-8 b(e)33 b(address)d(t)m(w)m(o)j(problems:)40 b(the)32 b(uniqueness)d(of)i(the)g(Gibbs)f(measure,)i(and)165 1334 y(the)44 b(deca)m(y)g(of)f(correlations)f(for)h(a)h(giv)m(en)f (Gibbs)e(measure.)79 b(In)42 b(fact,)47 b(the)d(general)f(theme)g(of) 165 1447 y(this)35 b(section)i(can)f(b)s(e)g(stated)h(as)f(follo)m(ws:) 51 b(T)-8 b(o)37 b(whic)m(h)e(exten)m(t)i(can)g(a)f(giv)m(en)g(spin)f (b)s(e)g(in\015uenced)165 1560 y(b)m(y)41 b(a)g(con\014guration)f(far)h (a)m(w)m(a)m(y?)73 b(If)41 b(suc)m(h)f(an)h(in\015uence)e(disapp)s (ears)g(in)g(the)i(limit)d(of)j(in\014nite)165 1673 y(distance,)24 b(it)e(follo)m(ws)f(\(dep)s(ending)f(on)h(the)i(setting\))f(that)h (either)e(there)h(is)f(no)h(long-range)h(in\015uence)165 1785 y(of)39 b(b)s(oundary)d(conditions)g(at)j(all)e(\(implying)e (uniqueness)h(of)i(the)g(Gibbs)f(measure\),)j(or)e(that)h(a)165 1898 y(sp)s(eci\014c)i(lo)m(w)h(temp)s(erature)g(phase)g(exhibits)e (some)j(mixing)d(prop)s(erties.)75 b(In)41 b(b)s(oth)h(cases,)k(the)165 2011 y(decreasing)34 b(in\015uence)e(comes)i(from)f(the)h(absence)g(of) g(in\014nite)d(clusters)i(of)h(suitable)e(t)m(yp)s(e)h(whic)m(h)165 2124 y(could)44 b(transp)s(ort)f(a)i(dep)s(endence)e(b)s(et)m(w)m(een)i (spins.)80 b(So,)47 b(b)s(oth)d(uniqueness)e(and)h(mixing)g(will)165 2237 y(app)s(ear)30 b(here)g(as)h(a)g(consequence)g(of)f(non-p)s (ercolation.)306 2350 y(In)d(a)h(\014rst)f(part,)h(w)m(e)g(will)d (address)i(the)g(problem)f(of)i(uniqueness.)38 b(In)26 b(fact,)k(w)m(e)e(will)d(encoun)m(ter)165 2463 y(conditions)37 b(whic)m(h)h(not)h(only)e(imply)g(the)h(uniqueness)f(of)h(the)h(Gibbs)e (measure,)k(but)d(also)h(lead)165 2576 y(us)32 b(in)m(to)g(a)g(regime)g (where)f(`all)g(go)s(o)s(d)h(things')f(happ)s(en,)g(i.e.,)i(where)f (the)g(unique)e(Gibbs)g(measure)165 2689 y(exhibits)38 b(nice)i(exp)s(onen)m(tial)f(mixing)f(prop)s(erties)g(and)h(the)h(free) g(energy)h(dep)s(ends)d(analytically)165 2802 y(on)33 b(all)e(relev)-5 b(an)m(t)32 b(parameters.)47 b(\(In)32 b(general,)h(the)g(uniqueness)d(of)i(the)h(Gibbs)e(measure)h(do)s(es)g (not)165 2915 y(imply)40 b(the)i(absence)h(of)f(other)g(critical)f (phenomena,)j(whic)m(h)d(migh)m(t)h(manifest)f(themselv)m(es)h(as)165 3028 y(singularities)c(of)k(the)f(free)g(energy)g(or)g(other)h(thermo)s (dynamic)d(quan)m(tities.)72 b(F)-8 b(or)42 b(example,)h(in)165 3140 y(Section)29 b(9)g(w)m(e)f(will)e(see)j(that)g(in)f(the)g (so-called)h(Gri\016ths')e(regime)h(of)h(a)g(disordered)d(system)j (there)165 3253 y(is)h(a)h(unique)d(Gibbs)h(measure,)h(but)g(the)h (free)f(energy)h(is)e(not)i(analytic.\))306 3366 y(The)42 b(`nice)h(regime')f(ab)s(o)m(v)m(e)i(is)d(usually)g(referred)g(to)j(as) e(the)h(high)e(temp)s(erature,)46 b(or)c(w)m(eak)165 3479 y(coupling,)22 b(lo)m(w)g(densit)m(y)-8 b(,)23 b(or)f(also)f (analytic)h(regime,)h(and)e(is)g(usually)e(studied)h(b)m(y)i(high)e (temp)s(erature)165 3592 y(cluster)27 b(expansions.)39 b(Dobrushin)25 b(and)i(Shlosman)f([70)q(,)i(71)q(])g(dev)m(elop)s(ed)f (a)h(b)s(eautiful)d(and)i(general)165 3705 y(theory)39 b(describing)d(a)j(regime)g(of)g(`complete)g(analyticit)m(y')f(b)m(y)h (v)-5 b(arious)37 b(equiv)-5 b(alen)m(t)38 b(prop)s(erties.)165 3818 y(One)k(of)g(these)g(ranks)f(at)i(the)f(top)g(of)g(a)g(hierarc)m (h)m(y)g(of)g(mixing)e(prop)s(erties.)73 b(While)41 b(complete)165 3931 y(analyticit)m(y)46 b(mak)m(es)g(precise)f(what)g(actually)h(the)f (`nice)h(regime')f(is,)k(and)c(applies)f(mainly)f(to)165 4044 y(high)32 b(temp)s(eratures)i(or)f(large)h(external)f(\014elds,)g (it)g(is)g(not)g(limited)e(to)k(this)d(case)j(only)d([75)q(].)51 b(The)165 4157 y(relationships)26 b(b)s(et)m(w)m(een)j(this)f(and)g (related)g(notions)g(and)g(also)h(with)e(dynamical)g(prop)s(erties)g (ha)m(v)m(e)165 4270 y(b)s(een)f(studied)f(in)g(man)m(y)i(pap)s(ers.)38 b(Although)26 b(some)h(of)g(these)g(ha)m(v)m(e)h(an)e(explicit)f (geometric)j(\015a)m(v)m(or,)165 4382 y(w)m(e)39 b(do)g(not)g(discuss)d (them)j(here)f(b)s(ecause)h(of)f(limitations)f(of)h(space.)66 b(W)-8 b(e)40 b(rather)e(refer)h(to)g(the)165 4495 y(sources)31 b([70)q(,)f(71)q(,)h(217)q(,)g(168)q(,)g(170)q(])g(and)e(also)i(to)g (the)f(references)h(follo)m(wing)e(condition)g(\(68\).)306 4608 y(In)i(Section)h(7.3)h(w)m(e)f(shall)d(discuss)h(an)i(application) e(of)i(the)f(p)s(ercolation)g(metho)s(d)g(to)i(the)e(lo)m(w)165 4721 y(temp)s(erature)f(regime,)h(and)e(see)i(ho)m(w)g(p)s(ercolation)e (estimates)i(for)f(the)g(co)m(v)-5 b(ariance)32 b(b)s(et)m(w)m(een)f(t) m(w)m(o)165 4834 y(distan)m(t)k(observ)-5 b(ables,)35 b(com)m(bined)f(with)f(con)m(tour)j(estimates,)h(giv)m(e)e(rise)f(to)h (exp)s(onen)m(tial)f(mixing)165 4947 y(prop)s(erties.)165 5190 y FA(7.1)113 b(Disagreemen)m(t)38 b(paths)165 5362 y Fw(Let)29 b(\()p Fv(L)p Ft(;)15 b Fv(\030)p Fw(\))29 b(b)s(e)f(an)h(arbitrary)e(lo)s(cally)g(\014nite)h(graph,)g(and)g(supp) s(ose)f(w)m(e)i(are)g(giv)m(en)g(a)g(neigh)m(b)s(or)e(in-)165 5475 y(teraction)g Ft(U)35 b Fw(:)25 b Ft(S)16 b Fv(\002)10 b Ft(S)30 b Fv(!)25 b Fx(R)h Fw(and)f(a)h(self-p)s(oten)m(tial)f Ft(V)45 b Fw(:)26 b Ft(S)k Fv(!)25 b Fx(R)p Fw(.)39 b(Consider)24 b(the)i(asso)s(ciated)g(Gibbs)165 5588 y(distributions)36 b Ft(\026)761 5544 y Fu(\021)761 5616 y(\014)s(;)p Fr(\003)916 5588 y Fw(in)m(tro)s(duced)i(in)g(\(4\).)69 b(More)40 b(generally)-8 b(,)42 b(w)m(e)e(could)f(consider)f(an)h(arbitrary)1844 5895 y(61)p eop %%Page: 62 62 62 61 bop 165 304 a Fw(Mark)m(o)m(v)37 b(sp)s(eci\014cation)d(\()p Ft(G)1130 318 y Fr(\003)1184 304 y Fw(\))1219 318 y Fr(\003)p Fq(2E)1398 304 y Fw(in)f(the)j(sense)f(of)g(Section)g(2.6.)56 b(Suc)m(h)35 b(sp)s(eci\014cations)e(app)s(ear,)165 417 y(in)k(particular,)h(if)e(w)m(e)i(ha)m(v)m(e)h(an)f(in)m(teraction)f (of)h(\014nite)e(range)i Ft(R)q Fw(,)i(sa)m(y)e(on)f Fx(Z)2880 384 y Fu(d)2921 417 y Fw(,)i(and)e(dra)m(w)h(edges)165 530 y(b)s(et)m(w)m(een)i(all)e(sites)g(of)h(distance)g(at)h(most)f Ft(R)q Fw(.)66 b(Ho)m(w)m(ev)m(er,)43 b(for)c(de\014niteness)e(and)h (simplicit)m(y)f(w)m(e)165 643 y(stic)m(k)27 b(to)f(the)g(setting)g (describ)s(ed)e(b)m(y)i(the)g(Hamiltonian)e(\(1\).)41 b(W)-8 b(e)27 b(will)c(often)j(consider)f(the)h(in)m(v)m(erse)165 756 y(temp)s(erature)h Ft(\014)33 b Fw(as)27 b(\014xed)g(and)f(then)h (simply)e(write)i Ft(\026)2039 711 y Fu(\021)2039 783 y Fr(\003)2119 756 y Fw(instead)f(of)i Ft(\026)2585 711 y Fu(\021)2585 784 y(\014)s(;)p Fr(\003)2700 756 y Fw(.)40 b(If)26 b(\003)i(is)e(a)i(singleton,)f(w)m(e)165 869 y(use)j(the)h(shorthand)e Ft(x)h Fw(for)g Fv(f)p Ft(x)p Fv(g)p Fw(.)306 981 y(W)-8 b(e)39 b(lo)s(ok)e(for)h(a)g(condition)e (implying)f(that)j(there)g(is)e(only)h(one)h(Gibbs)e(measure)h Ft(\026)h Fw(for)f(the)165 1094 y(Hamiltonian)29 b(\(1\),)j(i.e.,)f(a)f (unique)f(probabilit)m(y)f(measure)i(on)g(\012)25 b(=)g Ft(S)2596 1061 y Fq(L)2679 1094 y Fw(satisfying)907 1284 y Ft(\026)p Fw(\()15 b Fv(\001)g(j)g Ft(X)34 b Fv(\021)25 b Ft(\021)33 b Fw(o\013)e(\003\))26 b(=)f Ft(\026)1779 1240 y Fu(\021)1779 1311 y Fr(\003)1953 1284 y Fw(for)30 b Ft(\026)p Fw(-almost)h(all)e Ft(\021)g Fv(2)24 b Fw(\012)h Ft(:)165 1474 y Fw(Since)31 b(this)g(prop)s(ert)m(y)g(needs)h(only)f (to)i(b)s(e)e(c)m(hec)m(k)m(ed)j(for)d(singletons)g(\003)d(=)g Fv(f)p Ft(x)p Fv(g)k Fw(\(cf.)h(Theorem)f(1.33)165 1586 y(of)i([96)q(]\),)h(it)e(is)g(su\016cien)m(t)g(to)h(lo)s(ok)f(for)g (conditions)f(on)i(the)f(single-spin)e(Gibbs)h(distributions)e Ft(\026)3571 1554 y Fu(\021)3571 1609 y(x)165 1699 y Fw(with)e Ft(x)e Fv(2)e(L)p Fw(.)40 b(In)m(tuitiv)m(ely)-8 b(,)29 b(w)m(e)h(w)m(an)m(t)g(to)g(express)f(that)h Ft(\026)2152 1666 y Fu(\021)2152 1722 y(x)2196 1699 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))26 b(=)f Ft(a)p Fw(\))30 b(dep)s(ends)d(only)i (w)m(eakly)g(on)165 1812 y Ft(\021)38 b Fw(\(whic)m(h)c(can)h(b)s(e)f (exp)s(ected)h(to)g(hold)e(for)i(small)e Ft(\014)5 b Fw(\).)54 b(This)33 b(dep)s(endence)g(can)i(b)s(e)f(measured)g(b)m(y) 165 1925 y(the)d(maximal)e(v)-5 b(ariation)1412 2038 y Ft(p)1458 2052 y Fu(x)1527 2038 y Fw(=)48 b(max)1623 2099 y Fu(\021)r(;\021)1717 2080 y Fm(0)1740 2099 y Fq(2)p Fr(\012)1853 2038 y Fv(k)p Ft(\026)1953 2001 y Fu(\021)1953 2061 y(x)2018 2038 y Fv(\000)20 b Ft(\026)2164 2001 y Fu(\021)2201 1977 y Fm(0)2164 2061 y Fu(x)2228 2038 y Fv(k)2273 2052 y Fu(x)2342 2038 y Ft(;)1086 b Fw(\(44\))165 2238 y(where)1497 2351 y Fv(k)p Ft(\027)6 b Fv(k)1638 2365 y Fr(\001)1727 2351 y Fw(=)59 b(sup)1823 2428 y Fu(A)p Fq(2F)1973 2439 y Fl(\001)2042 2351 y Fv(j)p Ft(\027)6 b Fw(\()p Ft(A)p Fw(\))p Fv(j)1172 b Fw(\(45\))165 2565 y(is)26 b(the)h(total)g(v)-5 b(ariation)26 b(norm)g(on)h(the)g(sub-)p Ft(\033)s Fw(-algebra)f Fv(F)2139 2579 y Fr(\001)2229 2565 y Fw(of)h(ev)m(en)m(ts)h(whic)m(h)e(dep)s(end)f(only)g(on)i(the) 165 2678 y(spins)i(in)g(\001.)40 b(W)-8 b(e)32 b(write)d Fx(p)i Fw(as)f(a)h(shorthand)e(for)h(the)h(family)e(\()p Ft(p)2391 2692 y Fu(x)2435 2678 y Fw(\))2470 2692 y Fu(x)p Fq(2L)2609 2678 y Fw(.)306 2791 y(Giv)m(en)e(t)m(w)m(o)h (con\014gurations)d Ft(\030)t(;)41 b(\030)1470 2758 y Fq(0)1519 2791 y Fv(2)24 b Fw(\012,)j(a)g(path)g(in)e Fv(L)h Fw(will)e(b)s(e)i(called)f(a)i Fp(p)-5 b(ath)31 b(of)e(disagr)-5 b(e)g(ement)165 2904 y Fw(\(for)30 b Ft(\030)k Fw(and)29 b Ft(\030)633 2871 y Fq(0)657 2904 y Fw(\))h(if)f Ft(\030)t Fw(\()p Ft(x)p Fw(\))c Fv(6)p Fw(=)g Ft(\030)1136 2871 y Fq(0)1160 2904 y Fw(\()p Ft(x)p Fw(\))30 b(for)g(all)e(its)i(v)m(ertices)g Ft(x)p Fw(.)41 b(F)-8 b(or)30 b(eac)m(h)h(\014nite)e(region)h(\003)25 b Fv(\032)g(L)30 b Fw(and)f(an)m(y)165 3030 y(t)m(w)m(o)k (con\014gurations)e Ft(\021)s(;)g(\021)1078 2997 y Fq(0)1133 3030 y Fw(on)g(\003)1323 2997 y Fu(c)1390 3030 y Fw(w)m(e)g(will)e (construct)j(a)g(coupling)e Ft(P)44 b Fw(of)32 b Ft(\026)2802 2986 y Fu(\021)2802 3058 y Fr(\003)2886 3030 y Fw(and)f Ft(\026)3119 2986 y Fu(\021)3156 2963 y Fm(0)3119 3058 y Fr(\003)3214 3030 y Fw(describing)165 3143 y(the)40 b(di\013erence)f(of)g(these)h(measures)f(in)f(terms)h(of)h(paths)e(of)i (disagreemen)m(t)g(running)c(from)j(the)165 3256 y(b)s(oundary)29 b Ft(@)5 b Fw(\003)31 b(in)m(to)f(the)h(in)m(terior)f(of)h(\003.)41 b(In)m(tuitiv)m(ely)-8 b(,)30 b(these)h(paths)f(of)h(disagreemen)m(t)g (then)f(sho)m(w)165 3369 y(ho)m(w)39 b(deep)f(inside)e(the)j (in\015uence)d(of)j(the)g(b)s(oundary)d(conditions)h(can)i(still)d(b)s (e)i(felt.)64 b(W)-8 b(e)40 b(write)165 3505 y Fv(f)p Fw(\001)377 3449 y Fq(6)p Fr(=)321 3505 y Fv( )-15 b(!)35 b Ft(@)5 b Fw(\003)p Fv(g)37 b Fw(for)f(the)g(ev)m(en)m(t)i(in)d Ft(S)1447 3472 y Fr(\003)1524 3505 y Fv(\002)23 b Ft(S)1679 3472 y Fr(\003)1769 3505 y Fw(that)36 b(there)h(exists)e(a)i(path)f(of) g(disagreemen)m(t)h(from)165 3618 y(some)31 b(p)s(oin)m(t)f(of)g(a)h (set)g(\001)25 b Fv(\032)g Fw(\003)30 b(to)h(some)g(p)s(oin)m(t)e(of)i Ft(@)5 b Fw(\003.)306 3731 y(Although)33 b(the)g(coupling)e Ft(P)46 b Fw(to)34 b(b)s(e)e(constructed)i(is)e(not)h(b)s(est)g(suited) f(for)h(direct)f(use,)i(it)e(has)165 3844 y(a)h(useful)e(sp)s(ecial)h (feature:)45 b(its)32 b(disagreemen)m(t)h(distribution)c(is)j(sto)s(c)m (hastically)g(dominated)g(b)m(y)h(a)165 3957 y(Bernoulli)h(measure.)56 b(This)34 b(will)f(allo)m(w)i(us)g(to)h(conclude)f(that)h(absence)g(of) g(p)s(ercolation)f(for)g(the)165 4069 y(latter)c(implies)d(uniqueness)g (of)i(the)h(Gibbs)e(measure)h(for)g(the)h(Hamiltonian)d(\(1\).)306 4182 y(W)-8 b(e)36 b(write)d Ft( )762 4196 y Fn(p)846 4182 y Fw(for)h(the)g(Bernoulli)e(measure)i(on)g Fv(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)2254 4149 y Fq(L)2343 4182 y Fw(with)33 b Ft( )2613 4196 y Fn(p)2663 4182 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))33 b(=)e(1\))h(=)g Ft(p)3298 4196 y Fu(x)3375 4182 y Fw(for)i(all)165 4295 y Ft(x)k Fv(2)e(L)p Fw(,)k(and)c Ft( )722 4309 y Fn(p)p Fu(;)p Fr(\003)878 4295 y Fw(for)h(the)h(analogous)g(pro)s(duct)e(measure)i (on)f Fv(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)2678 4262 y Fr(\003)2733 4295 y Fw(.)63 b(As)37 b(in)f(Section)i(4,)i(w)m(e)165 4408 y(use)35 b(the)f(notation)h Ft(X)7 b Fw(\()p Ft(x)p Fw(\))35 b(and)f Ft(X)1357 4375 y Fq(0)1381 4408 y Fw(\()p Ft(x)p Fw(\))h(for)f(the)h(pro)5 b(jections)34 b(from)g(\012)23 b Fv(\002)f Fw(\012)34 b(to)i Ft(S)5 b Fw(.)53 b(The)34 b(follo)m(wing)165 4521 y(theorem)d(is)f(due)f(to)i(v)-5 b(an)31 b(den)e(Berg)i(and)f(Maes)i([27)q(].)165 4697 y Fx(Theorem)i(7.1)46 b Fp(F)-7 b(or)40 b(e)-5 b(ach)39 b(\014nite)g Fw(\003)d Fv(\032)f(L)j Fp(and)i(e)-5 b(ach)39 b(p)-5 b(air)40 b Ft(\021)s(;)31 b(\021)2459 4664 y Fq(0)2519 4697 y Fv(2)k Fw(\012)j Fp(ther)-5 b(e)40 b(exists)e(a)h(c)-5 b(oupling)165 4818 y Ft(P)39 b Fw(=)25 b Ft(P)416 4833 y Fr(\003)p Fu(;\021)r(;\021)579 4814 y Fm(0)639 4818 y Fp(of)33 b Ft(\026)801 4774 y Fu(\021)801 4845 y Fr(\003)886 4818 y Fp(and)h Ft(\026)1118 4774 y Fu(\021)1155 4750 y Fm(0)1118 4845 y Fr(\003)1214 4818 y Fp(having)f(the)g(fol)5 b(lowing)34 b(pr)-5 b(op)g(erties:)211 5027 y Fw(\(i\))32 b Fp(F)-7 b(or)34 b(e)-5 b(ach)33 b Ft(x)26 b Fv(2)e Fw(\003)p Fp(,)33 b Fv(f)p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))27 b Fv(6)p Fw(=)e Ft(X)1454 4994 y Fq(0)1478 5027 y Fw(\()p Ft(x)p Fw(\))p Fv(g)h Fw(=)f Fv(f)p Ft(x)1945 4971 y Fq(6)p Fr(=)1890 5027 y Fv( )-15 b(!)24 b Ft(@)5 b Fw(\003)p Fv(g)34 b Ft(P)13 b Fp(-a.s.)211 5218 y Fw(\(ii\))32 b Fp(F)-7 b(or)33 b(the)g(distribution)h Ft(P)1243 5174 y Fq(6)p Fr(=)1230 5245 y(\003)1335 5218 y Fp(of)e Fw(\()p Ft(I)1516 5237 y Fq(f)p Fr(\()p Fu(X)5 b Fr(\()p Fu(x)p Fr(\))p Fq(6)p Fr(=)p Fu(X)1853 5218 y Fm(0)1878 5237 y Fr(\()p Fu(x)p Fr(\))p Fq(g)2012 5218 y Fw(\))2047 5232 y Fu(x)p Fq(2)p Fr(\003)2220 5218 y Fp(under)33 b Ft(P)13 b Fp(,)32 b(we)h(have)g Ft(P)3025 5174 y Fq(6)p Fr(=)3012 5245 y(\003)3109 5218 y Fv(\026)3180 5232 y Fq(D)3266 5218 y Ft( )3325 5232 y Fn(p)p Fu(;)p Fr(\003)3468 5218 y Ft(:)211 5401 y Fw(\(iii\))e Fp(F)-7 b(or)33 b(e)-5 b(ach)34 b Fw(\001)25 b Fv(\032)g Fw(\003)p Fp(,)1089 5618 y Fv(k)p Ft(\026)1189 5574 y Fu(\021)1189 5646 y Fr(\003)1263 5618 y Fv(\000)20 b Ft(\026)1409 5574 y Fu(\021)1446 5551 y Fm(0)1409 5646 y Fr(\003)1473 5618 y Fv(k)1518 5632 y Fr(\001)1606 5618 y Fv(\024)25 b Ft(P)13 b Fw(\(\001)1965 5563 y Fq(6)p Fr(=)1910 5618 y Fv( )-15 b(!)25 b Ft(@)5 b Fw(\003\))26 b Fv(\024)f Ft( )2434 5632 y Fn(p)2483 5618 y Fw(\(\001)15 b Fv($)g Ft(@)5 b Fw(\003\))26 b Ft(:)536 b Fw(\(46\))1844 5895 y(62)p eop %%Page: 63 63 63 62 bop 165 308 a Fx(Pro)s(of:)40 b Fw(W)-8 b(e)29 b(construct)f(a)g(coupling)e(\()p Ft(X)r(;)15 b(X)1706 275 y Fq(0)1731 308 y Fw(\))28 b(of)g Ft(\026)1950 264 y Fu(\021)1950 335 y Fr(\003)2030 308 y Fw(and)f Ft(\026)2259 264 y Fu(\021)2296 240 y Fm(0)2259 335 y Fr(\003)2351 308 y Fw(b)m(y)g(the)h(follo)m(wing)e(algorithm.)39 b(In)27 b(a)165 421 y(preparatory)g(step)g(w)m(e)g(in)m(tro)s(duce)f(an)h (arbitrary)f(linear)f(ordering)h(on)g(\003)h(,)h(set)g(\001)d(=)g (\003,)j(and)e(de\014ne)165 534 y Ft(X)7 b Fw(\()p Ft(x)p Fw(\))27 b(=)e Ft(\021)s Fw(\()p Ft(x)p Fw(\))p Ft(;)15 b(X)784 501 y Fq(0)809 534 y Fw(\()p Ft(x)p Fw(\))26 b(=)f Ft(\021)1101 501 y Fq(0)1124 534 y Fw(\()p Ft(x)p Fw(\))31 b(for)g Ft(x)25 b Fv(2)g Fw(\001)1656 501 y Fu(c)1690 534 y Fw(.)306 647 y(F)-8 b(or)38 b(\014xing)d(the)i(main)e (iteration)i(step,)h(supp)s(ose)d(that)i(\()p Ft(X)r(;)15 b(X)2514 614 y Fq(0)2539 647 y Fw(\))37 b(is)f(already)g(de\014ned)f (on)i(the)165 760 y(complemen)m(t)i(of)g(a)g(non-empt)m(y)g(set)g(\001) g Fv(\032)g Fw(\003)f(and)g(is)g(realized)g(as)h(a)g(pair)e(\()p Ft(\030)t(;)15 b(\030)3005 727 y Fq(0)3029 760 y Fw(\))39 b(o\013)g(\001,)i(where)165 873 y(\()p Ft(\030)t(;)15 b(\030)328 840 y Fq(0)352 873 y Fw(\))26 b Fv(\021)f Fw(\()p Ft(\021)s(;)15 b(\021)680 840 y Fq(0)705 873 y Fw(\))29 b(o\013)h(\003.)40 b(Conditional)27 b(on)i(the)g(ev)m(en)m (t)h(that)g(\()p Ft(X)r(;)15 b(X)2467 840 y Fq(0)2492 873 y Fw(\))26 b Fv(\021)f Fw(\()p Ft(\030)t(;)15 b(\030)2812 840 y Fq(0)2836 873 y Fw(\))29 b(o\013)g(\001,)h(w)m(e)f(consider)165 999 y(the)41 b(Gibbs)e(distributions)e Ft(\026)1203 955 y Fu(\030)1203 1026 y Fr(\001)1306 999 y Fw(and)j Ft(\026)1548 955 y Fu(\030)1582 931 y Fm(0)1548 1026 y Fr(\001)1651 999 y Fw(obtained)g(b)m(y)g(conditioning)f Ft(\026)2757 955 y Fu(\021)2757 1026 y Fr(\003)2850 999 y Fw(and)h Ft(\026)3092 955 y Fu(\021)3129 931 y Fm(0)3092 1026 y Fr(\003)3196 999 y Fw(on)h Ft(X)49 b Fv(\021)42 b Ft(\030)165 1112 y Fw(resp.)34 b Ft(\030)431 1079 y Fq(0)488 1112 y Fw(o\013)g(\001,)h(and)e(w)m(e)h(pic)m(k)f(the)h(smallest)f(v)m (ertex)i Ft(x)c Fw(=)g Ft(x)p Fw(\()p Ft(\030)t(;)15 b(\030)2460 1079 y Fq(0)2484 1112 y Fw(\))31 b Fv(2)g Fw(\001)i(for)h(whic)m(h)e(there)i(exists)165 1225 y(some)41 b(v)m(ertex)g Ft(y)k Fv(2)c Fw(\001)956 1192 y Fu(c)1031 1225 y Fw(with)e Ft(y)45 b Fv(\030)c Ft(x)f Fw(and)g Ft(\030)t Fw(\()p Ft(y)s Fw(\))i Fv(6)p Fw(=)g Ft(\030)2090 1192 y Fq(0)2113 1225 y Fw(\()p Ft(y)s Fw(\).)71 b(If)40 b(suc)m(h)g(an)g Ft(x)g Fw(do)s(es)g(not)h(exist,)i(w)m(e)165 1351 y(ha)m(v)m(e)30 b Ft(\026)428 1307 y Fu(\030)428 1379 y Fr(\001)516 1351 y Fw(=)25 b Ft(\026)667 1307 y Fu(\030)701 1283 y Fm(0)667 1379 y Fr(\001)758 1351 y Fw(on)j Fv(F)947 1365 y Fr(\001)1039 1351 y Fw(b)m(y)g(the)g(Mark)m (o)m(v)i(prop)s(ert)m(y)-8 b(,)29 b(so)g(that)g(w)m(e)f(can)h(tak)m(e)h (the)e(ob)m(vious)g(optimal)165 1464 y(coupling)f(for)h(whic)m(h)f Ft(X)33 b Fv(\021)24 b Ft(X)1207 1431 y Fq(0)1259 1464 y Fw(on)29 b(\001,)f(and)g(w)m(e)g(are)h(done.)40 b(If)28 b(suc)m(h)g(an)g Ft(x)g Fw(do)s(es)g(exist,)g(w)m(e)h(consider)165 1585 y(the)40 b(single)f(v)m(ertex)i(distributions)36 b Ft(\026)1475 1541 y Fu(\030)1475 1613 y Fr(\001)p Fu(;x)1638 1585 y Fw(=)41 b Ft(\026)1805 1541 y Fu(\030)1805 1613 y Fr(\001)1867 1585 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))43 b(=)d Fv(\001)15 b Fw(\))41 b(and)e Ft(\026)2617 1541 y Fu(\030)2651 1518 y Fm(0)2617 1613 y Fr(\001)p Fu(;x)2780 1585 y Fw(=)i Ft(\026)2947 1541 y Fu(\030)2981 1518 y Fm(0)2947 1613 y Fr(\001)3010 1585 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))42 b(=)f Fv(\001)15 b Fw(\))40 b(on)165 1698 y Ft(S)5 b Fw(.)58 b(Conditionally)33 b(on)j(\()p Ft(X)r(;)15 b(X)1249 1665 y Fq(0)1274 1698 y Fw(\))35 b Fv(\021)f Fw(\()p Ft(\030)t(;)15 b(\030)1612 1665 y Fq(0)1636 1698 y Fw(\))36 b(o\013)h(\001,)g(w)m(e)f(then)g(let)g(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))p Ft(;)15 b(X)2830 1665 y Fq(0)2856 1698 y Fw(\()p Ft(x)p Fw(\)\))37 b(b)s(e)e(distributed)165 1825 y(according)30 b(to)h(an)f(optimal)f(coupling)g(\(as)i(in)d (De\014nition)h(4.3\))j(of)e Ft(\026)2523 1780 y Fu(\030)2523 1852 y Fr(\001)p Fu(;x)2675 1825 y Fw(and)g Ft(\026)2907 1780 y Fu(\030)2941 1757 y Fm(0)2907 1852 y Fr(\001)p Fu(;x)3029 1825 y Fw(.)40 b(The)30 b(coupling)165 1938 y(\()p Ft(X)r(;)15 b(X)399 1905 y Fq(0)424 1938 y Fw(\))29 b(is)e(then)h(de\014ned)f(on)h(the)h(set)g Ft(x)16 b Fv([)f Fw(\001)1732 1905 y Fu(c)1767 1938 y Fw(,)29 b(so)f(that)h(w)m (e)g(can)f(replace)h(\001)f(b)m(y)g(\001)16 b Fv(n)g Ft(x)28 b Fw(and)g(rep)s(eat)165 2050 y(the)j(preceding)e(iteration)h (step.)306 2163 y(It)25 b(is)e(clear)h(that)h(the)f(algorithm)g(ab)s(o) m(v)m(e)h(stops)f(after)h(\014nitely)d(man)m(y)j(iterations)e(and)h (giv)m(es)g(us)g(a)165 2285 y(coupling)d(of)i Ft(\026)672 2240 y Fu(\021)672 2312 y Fr(\003)747 2285 y Fw(and)f Ft(\026)971 2240 y Fu(\021)1008 2217 y Fm(0)971 2312 y Fr(\003)1035 2285 y Fw(.)38 b(Prop)s(ert)m(y)22 b(\(i\))h(is)e (eviden)m(t)h(from)g(the)h(construction,)h(since)e(disagreemen)m(t)165 2398 y(at)44 b(a)f(v)m(ertex)h(is)d(only)h(p)s(ossible)f(if)g(a)i(path) g(of)g(disagreemen)m(t)g(leads)f(from)g(this)g(v)m(ertex)h(to)h(the)165 2519 y(b)s(oundary)-8 b(.)84 b(F)-8 b(or)46 b(\(ii\),)j(w)m(e)c(note)h (that)g(the)g(measures)f Ft(\026)2230 2475 y Fu(\030)2230 2546 y Fr(\001)p Fu(;x)2397 2519 y Fw(and)g Ft(\026)2644 2475 y Fu(\030)2678 2451 y Fm(0)2644 2546 y Fr(\001)p Fu(;x)2811 2519 y Fw(are)h(mixtures)d(of)j(the)165 2632 y(Gibbs)23 b(distributions)d Ft(\026)1003 2599 y Fu(\033)1003 2654 y(x)1073 2632 y Fw(with)i(suitable)h(b)s(oundary)f(conditions)g Ft(\033)s Fw(,)j(b)m(y)f(the)g(consistency)g(of)g(Gibbs)165 2745 y(distributions.)37 b(Hence)1459 2858 y Fv(k)p Ft(\026)1559 2813 y Fu(\030)1559 2885 y Fr(\001)p Fu(;x)1702 2858 y Fv(\000)20 b Ft(\026)1848 2813 y Fu(\030)1882 2790 y Fm(0)1848 2885 y Fr(\001)p Fu(;x)1970 2858 y Fv(k)2015 2872 y Fu(x)2084 2858 y Fv(\024)25 b Ft(p)2226 2872 y Fu(x)2295 2858 y Ft(:)165 3011 y Fw(By)31 b(construction,)f(this)g (means)g(that)h(in)e(eac)m(h)i(iteration)f(of)h(the)f(main)g(step)g(w)m (e)h(ha)m(v)m(e)962 3187 y Ft(P)13 b Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))27 b Fv(6)p Fw(=)e Ft(X)1477 3150 y Fq(0)1500 3187 y Fw(\()p Ft(x)p Fw(\))h Fv(j)g Fw(\()p Ft(X)r(;)15 b(X)1933 3150 y Fq(0)1958 3187 y Fw(\))25 b Fv(\021)g Fw(\()p Ft(\030)t(;)15 b(\030)2277 3150 y Fq(0)2301 3187 y Fw(\))31 b(o\013)g(\001\))25 b Fv(\024)g Ft(p)2774 3201 y Fu(x)165 3364 y Fw(for)33 b Ft(x)28 b Fw(=)g Ft(x)p Fw(\()p Ft(\030)t(;)15 b(\030)701 3331 y Fq(0)725 3364 y Fw(\),)34 b(so)e(that)h(\(ii\))f(follo)m(ws)g(b)m(y)g (induction.)44 b(Finally)-8 b(,)32 b(\(iii\))f(follo)m(ws)g(directly)g (from)h(\(i\))165 3477 y(and)e(\(ii\))g(b)s(ecause)g(for)g(eac)m(h)i (\001)25 b Fv(\032)g Fw(\003)917 3667 y Fv(k)p Ft(\026)1017 3622 y Fu(\021)1017 3694 y Fr(\003)1090 3667 y Fv(\000)20 b Ft(\026)1236 3622 y Fu(\021)1273 3599 y Fm(0)1236 3694 y Fr(\003)1300 3667 y Fv(k)1345 3681 y Fr(\001)1434 3667 y Fv(\024)25 b Ft(P)13 b Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))27 b Fv(6)p Fw(=)e Ft(X)2045 3629 y Fq(0)2069 3667 y Fw(\()p Ft(x)p Fw(\))30 b(for)h(some)f Ft(x)c Fv(2)e Fw(\001\))165 3843 y(b)m(y)31 b(the)f(coupling)f(inequalit)m(y)g (\(10\).)42 b(The)30 b(pro)s(of)f(is)h(therefore)h(complete.)809 b Fk(2)165 3997 y Fw(Although)32 b(the)g(algorithm)f(in)g(the)h(pro)s (of)f(ab)s(o)m(v)m(e)j(is)d(quite)g(explicit,)g(it)h(is)f(not)h(easy)h (to)g(deal)e(with)165 4109 y(directly)-8 b(.)41 b(In)30 b(particular,)f(it)h(is)g(not)h(clear)f(in)g(whic)m(h)f(w)m(a)m(y)j (the)f(coupling)d(dep)s(ends)h(on)i(the)g(c)m(hosen)165 4222 y(ordering,)24 b(b)s(ecause)g(the)g(site)g Ft(x)f Fw(to)i(b)s(e)e(selected)h(in)f(eac)m(h)h(step)g(dep)s(ends)e(on)i(\()p Ft(\030)t(;)15 b(\030)2925 4189 y Fq(0)2949 4222 y Fw(\))24 b(and)f(is)g(therefore)165 4335 y(random.)37 b(Nev)m(ertheless,)24 b(if)c(the)h(Gibbs)f(distributions)d(are)22 b(monotone)g(\(in)e(the)h (sense)g(of)g(De\014nition)165 4448 y(4.9\),)32 b(w)m(e)f(get)h(some)f (extra)g(prop)s(erties.)165 4602 y Fx(Remark:)k Fw(Supp)s(ose)19 b Ft(S)26 b Fw(is)20 b(linearly)e(ordered)i(and)g(the)h(conditional)e (distributions)e Ft(\026)3089 4569 y Fu(\030)3089 4624 y(x)3153 4602 y Fw(are)k(sto)s(c)m(has-)165 4715 y(tically)34 b(increasing)f(in)g Ft(\030)t Fw(.)52 b(Then,)35 b(if)e Ft(\021)j Fv(\026)31 b Ft(\021)1691 4682 y Fq(0)1715 4715 y Fw(,)36 b(the)e(coupling)f Ft(P)47 b Fw(of)35 b(Theorem)f(7.1)h(can)g(b)s(e)f(c)m(hosen)165 4827 y(in)f(suc)m(h)h(a)h (w)m(a)m(y)g(that,)h(in)d(addition)g(to)i(prop)s(erties)e(\(i\))h(to)h (\(iii\),)f Ft(X)39 b Fv(\026)31 b Ft(X)2749 4794 y Fq(0)2807 4827 y Ft(P)13 b Fw({a.s.)36 b(and,)f(for)f(eac)m(h)165 4940 y Ft(x)26 b Fv(2)e Fw(\003,)830 5076 y Fv(k)p Ft(\026)930 5032 y Fu(\021)930 5104 y Fr(\003)1003 5076 y Fv(\000)c Ft(\026)1149 5032 y Fu(\021)1186 5009 y Fm(0)1149 5104 y Fr(\003)1213 5076 y Fv(k)1258 5090 y Fu(x)1328 5076 y Fv(\024)25 b Ft(P)13 b Fw(\()p Ft(x)1663 5021 y Fq(6)p Fr(=)1607 5076 y Fv( )-15 b(!)25 b Ft(@)5 b Fw(\003\))26 b Fv(\024)f Fw(\()p Fv(j)p Ft(S)5 b Fv(j)21 b(\000)f Fw(1\))15 b Fv(k)p Ft(\026)2525 5032 y Fu(\021)2525 5104 y Fr(\003)2600 5076 y Fv(\000)20 b Ft(\026)2746 5032 y Fu(\021)2783 5009 y Fm(0)2746 5104 y Fr(\003)2810 5076 y Fv(k)2855 5090 y Fu(x)2924 5076 y Ft(:)504 b Fw(\(47\))165 5229 y(This)29 b(is)g(b)s(ecause)h(in)e(eac)m(h)j(step)f(of)g(the)h (algorithm)e(pro)m(ving)g(Theorem)g(7.1)j(w)m(e)e(can)g(ac)m(hiev)m(e)i (that)165 5342 y Ft(X)7 b Fw(\()p Ft(x)p Fw(\))27 b Fv(\024)e Ft(X)574 5309 y Fq(0)597 5342 y Fw(\()p Ft(x)p Fw(\),)32 b(and)d(for)i(the)f(second)h(inequalit)m(y)d(in)h(\(47\))j(it)e(is)g (su\016cien)m(t)f(to)j(note)f(that)373 5547 y Ft(P)13 b Fw(\()p Ft(x)613 5491 y Fq(6)p Fr(=)557 5547 y Fv( )-15 b(!)25 b Ft(@)5 b Fw(\003\))26 b(=)f Ft(P)13 b Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))26 b Ft(<)f(X)1536 5509 y Fq(0)1560 5547 y Fw(\()p Ft(x)p Fw(\)\))h Fv(\024)1929 5466 y Fs(X)1839 5654 y Fu(a)p Fq(2)p Fu(S)t Fq(nf)p Fu(m)p Fq(g)2139 5547 y Fw([)p Ft(P)13 b Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))27 b Fv(\024)e Ft(a)p Fw(\))20 b Fv(\000)g Ft(P)13 b Fw(\()p Ft(X)2979 5509 y Fq(0)3003 5547 y Fw(\()p Ft(x)p Fw(\))26 b Fv(\024)f Ft(a)p Fw(\)])h Ft(;)1844 5895 y Fw(63)p eop %%Page: 64 64 64 63 bop 165 304 a Fw(where)36 b Ft(m)f Fw(is)g(the)h(maximal)f (elemen)m(t)h(of)g Ft(S)5 b Fw(.)57 b(F)-8 b(or)36 b(details)f(w)m(e)h (refer)g(to)g([27)q(].)58 b(In)35 b(particular,)g(for)165 417 y Fv(j)p Ft(S)5 b Fv(j)26 b Fw(=)f(2)31 b(w)m(e)g(ha)m(v)m(e)g (equalit)m(y)f(in)f(\(47\).)165 577 y(Let)f(us)d(apply)h(this)f(remark) i(to)g(the)g(ferromagnetic)g(Ising)e(mo)s(del)h(with)f(external)i (\014eld)e Ft(h)h Fw(=)f(0)i(and)165 690 y(an)m(y)35 b(in)m(v)m(erse)g(temp)s(erature)f Ft(\014)5 b Fw(,)36 b(with)d(b)s(oundary)g(conditions)g Ft(\021)j Fv(\021)c Fw(+1)i(and)g Ft(\021)2913 657 y Fq(0)2969 690 y Fv(\021)e(\000)p Fw(1)j(outside)f(of)165 802 y(some)28 b(\014nite)e(region)h(\003)f Fv(2)e(E)8 b Fw(.)40 b(Then,)27 b(b)m(y)g(the)h(spin)d(\015ip)h (symmetry)h(and)f(sto)s(c)m(hastic)i(monotonicit)m(y)-8 b(,)583 998 y Ft(\026)638 959 y Fr(+)638 1026 y Fu(\014)s(;)p Fr(\003)754 998 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\)\))27 b(=)e Ft(\026)1206 959 y Fr(+)1206 1026 y Fu(\014)s(;)p Fr(\003)1321 998 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))27 b(=)e(1\))c Fv(\000)f Ft(\026)1930 959 y Fq(\000)1930 1026 y Fu(\014)s(;)p Fr(\003)2046 998 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))26 b(=)f(1\))h(=)f Fv(k)p Ft(\026)2709 959 y Fr(+)2709 1026 y Fu(\014)s(;)p Fr(\003)2845 998 y Fv(\000)20 b Ft(\026)2991 959 y Fq(\000)2991 1026 y Fu(\014)s(;)p Fr(\003)3107 998 y Fv(k)3152 1012 y Fu(x)165 1193 y Fw(and)30 b(therefore,)h(b)m(y)f(\(47\),)1317 1334 y Ft(\026)1372 1296 y Fr(+)1372 1362 y Fu(\014)s(;)p Fr(\003)1488 1334 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\)\))27 b(=)e Ft(P)13 b Fw(\()p Ft(x)2124 1278 y Fq(6)p Fr(=)2068 1334 y Fv( )-15 b(!)25 b Ft(@)5 b Fw(\003\))26 b Ft(:)165 1496 y Fw(W)-8 b(e)47 b(emphasize)d(that)i(this)e(relation)g(is)g (completely)h(similar)d(to)k(what)f(w)m(e)h(obtained)e(for)h(the)165 1609 y(random-cluster)30 b(represen)m(tation,)g(viz.)1317 1804 y Ft(\026)1372 1766 y Fr(+)1372 1833 y Fu(\014)s(;)p Fr(\003)1488 1804 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\)\))27 b(=)e Ft(\036)1939 1767 y Fr(1)1939 1827 y Fu(p;)p Fr(2)p Fu(;)p Fr(\003)2102 1804 y Fw(\()p Ft(x)15 b Fv($)h Ft(@)5 b Fw(\003\))165 2015 y(for)42 b Ft(p)i Fw(=)g(1)28 b Fv(\000)f Ft(e)734 1982 y Fq(\000)p Fr(2)p Fu(\014)872 2015 y Fw(;)48 b(cf.)42 b(equation)f(\(32\).)77 b(The)41 b(coupling)f Ft(P)13 b Fw(,)45 b(ho)m(w)m(ev)m(er,)h(is)41 b(less)g(explicit,)i(and)165 2128 y(the)c(geometric)g(ev)m(en)m(t)g(in) m(v)m(olv)m(es)f(site)g(p)s(ercolation)f(rather)h(than)g(b)s(ond)f(p)s (ercolation)g(as)h(for)g(the)165 2241 y(random{cluster)32 b(measure,)i(but)e(the)g(exact)j(corresp)s(ondence)d(b)s(et)m(w)m(een)h (the)g(magnetization)g(for)165 2354 y(the)e(spin)d(system)j(and)f(the)g (p)s(ercolation)g(probabilit)m(y)e(of)i(the)h(geometric)g(system)g(is)e (the)i(same.)306 2467 y(Let)h(us)e(no)m(w)g(turn)g(to)h(the)g(main)f (result)g(of)g(this)g(subsection,)g(the)h(uniqueness)d(theorem.)43 b(Let)165 2580 y Ft(\026;)31 b(\026)331 2547 y Fq(0)385 2580 y Fw(b)s(e)g(an)m(y)h(t)m(w)m(o)h(Gibbs)d(measures)i(for)f(the)h (Hamiltonian)e(\(1\))i(at)h(some)f(in)m(v)m(erse)f(temp)s(erature)165 2693 y Ft(\014)5 b Fw(.)41 b(Inequalit)m(y)30 b(\(46\))h(then)g(sho)m (ws)f(that)962 2901 y Fv(k)p Ft(\026)20 b Fv(\000)g Ft(\026)1228 2864 y Fq(0)1251 2901 y Fv(k)1296 2915 y Fr(\001)1385 2901 y Fv(\024)64 b Fw(sup)1481 2980 y Fu(\021)r(;\021)1575 2961 y Fm(0)1599 2980 y Fq(2)p Fr(\012)1712 2901 y Fv(k)p Ft(\026)1812 2857 y Fu(\021)1812 2929 y Fr(\003)1885 2901 y Fv(\000)20 b Ft(\026)2031 2857 y Fu(\021)2068 2834 y Fm(0)2031 2929 y Fr(\003)2095 2901 y Fv(k)2140 2915 y Fr(\001)2229 2901 y Fv(\024)25 b Ft( )2384 2915 y Fn(p)2433 2901 y Fw(\(\001)15 b Fv($)h Ft(@)5 b Fw(\003\))165 3155 y(whenev)m(er)31 b(\001)25 b Fv(\032)g Fw(\003)g Fv(2)g(E)8 b Fw(.)41 b(Letting)30 b(\003)25 b Fv(")h(L)k Fw(w)m(e)h(\014nd)1397 3350 y Fv(k)p Ft(\026)20 b Fv(\000)g Ft(\026)1663 3312 y Fq(0)1686 3350 y Fv(k)1731 3364 y Fr(\001)1820 3350 y Fv(\024)25 b Ft( )1975 3364 y Fn(p)2024 3350 y Fw(\(\001)15 b Fv($)h(1)p Fw(\))165 3545 y(whic)m(h)29 b(giv)m(es)i(the)g(follo)m(wing)e(uniqueness)f(result.)165 3749 y Fx(Theorem)34 b(7.2)46 b Fp(If)39 b Ft( )949 3763 y Fn(p)999 3749 y Fw(\()15 b Fv(9)32 b Fw(an)f(in\014nite)d(op)s(en)h (cluster)h(\))c(=)f(0)39 b Fp(then)24 b(the)h(set)f Fv(G)5 b Fw(\()p Ft(\014)g(H)i Fw(\))25 b Fp(of)f(Gibbs)g(me)-5 b(a-)165 3861 y(sur)g(es)28 b(for)h(the)f(Hamiltonian)h(\(1\))f(at)g (inverse)g(temp)-5 b(er)g(atur)g(e)30 b Ft(\014)j Fp(is)27 b(a)h(singleton.)41 b(In)28 b(p)-5 b(articular,)30 b(this)165 3974 y(holds)35 b(if)d Fw(sup)623 3996 y Fu(x)682 3974 y Ft(p)728 3988 y Fu(x)797 3974 y Ft(<)25 b(p)939 3988 y Fu(c)973 3974 y Fp(,)33 b(the)g(critic)-5 b(al)33 b(density)g(for)h (Bernoul)5 b(li)33 b(site)f(p)-5 b(er)g(c)g(olation)36 b(on)d Fw(\()p Fv(L)p Ft(;)15 b Fv(\030)p Fw(\))p Fp(.)165 4178 y Fw(A)38 b(w)m(eak)m(er)i(v)m(ersion)d(of)h(Theorem)f(7.2)i(w)m (as)g(obtained)e(\014rst)g(in)f([24)r(])i(using)e(a)i(pro)s(duct)f (coupling)165 4291 y(instead)f(of)h(Theorem)f(7.1;)41 b(see)c(Prop)s(osition)e(7.10)j(b)s(elo)m(w)d(and)h(also)h([28)q(].)59 b(In)36 b(some)h(cases,)i(the)165 4404 y(simple)29 b(pro)s(duct)g (coupling)f(nev)m(ertheless)j(giv)m(es)g(equiv)-5 b(alen)m(t)29 b(results;)h(cf.)g(the)h(discussion)d(in)h([27)q(].)306 4517 y(F)-8 b(or)35 b(a)f(large)f(class)h(of)f(regular)g(graphs)g(suc)m (h)g(as)h Fx(Z)2096 4484 y Fu(d)2136 4517 y Fw(,)h(the)e(assumption)f (of)i(Theorem)f(7.2)i(not)165 4630 y(only)g(implies)d(the)k(uniqueness) d(of)i(the)g(Gibbs)f(measure)h(but)g(ev)m(en)h(yields)d(certain)i(exp)s (onen)m(tial)165 4742 y(mixing)c(prop)s(erties.)47 b(This)31 b(can)i(b)s(e)f(seen)h(almost)g(immediately)e(b)m(y)i(com)m(bining)e (inequalit)m(y)h(\(46\))165 4855 y(with)e(Theorem)i(5.6)g(on)g(the)f (exp)s(onen)m(tial)g(tail)f(of)i(the)g(distribution)27 b(of)32 b(the)g(cluster)e(diameter)h(in)165 4968 y(sub-critical)h (Bernoulli)g(p)s(ercolation.)50 b(W)-8 b(e)35 b(will)c(use)i(similar)e (argumen)m(ts)j(in)f(Section)h(9.2)g(in)f(the)165 5081 y(con)m(text)g(of)d(random)g(in)m(teractions.)306 5194 y(Let)i(us)f(discuss)f(no)m(w)h(some)h(sp)s(ecial)e(cases.)45 b(Clearly)-8 b(,)31 b(the)g(conditions)f(of)i(Theorem)f(7.2)h(hold)165 5307 y(when)h Fv(L)c Fw(=)h Fx(Z)j Fw(with)f(the)h(usual)f(graph)g (structure,)i(since)e(then)h Ft(p)2466 5321 y Fu(c)2530 5307 y Fw(=)d(1.)50 b(This)31 b(giv)m(es)i(uniqueness)165 5420 y(of)g(the)f(Gibbs)f(measure)h(for)h(one-dimensional)d (nearest-neigh)m(b)s(or)i(systems.)46 b(Next)33 b(w)m(e)g(consider)165 5533 y(the)i(case)g Fv(L)d Fw(=)g Fx(Z)784 5500 y Fu(d)824 5533 y Fw(,)k Ft(d)c Fv(\025)f Fw(2.)54 b(Recall)34 b(the)g(b)s(ound)e (\(19\))k(for)f(the)f(p)s(ercolation)g(threshold)e Ft(p)3338 5547 y Fu(c)3407 5533 y Fw(when)165 5646 y Ft(d)26 b Fw(=)f(2,)31 b(and)f(the)g(large{dimensions)f(asymptotics)h(of)h Ft(p)2125 5660 y Fu(c)2189 5646 y Fw(in)e(\(20\).)1844 5895 y(64)p eop %%Page: 65 65 65 64 bop 165 304 a Fx(Example)34 b(7.3)46 b Fp(The)33 b(Ising)f(ferr)-5 b(omagnet.)43 b Fw(Let)31 b Ft(\014)f(>)25 b Fw(0)31 b(b)s(e)e(an)m(y)i(in)m(v)m(erse)f(temp)s(erature)f(and)h Ft(h)g Fw(an)165 417 y(external)h(\014eld.)39 b(Then,)30 b(for)g(an)m(y)g Ft(x)p Fw(,)h(w)m(e)g(obtain)f(from)g(\(13\))h(b)m(y)g (a)f(short)h(computation)597 621 y Ft(p)643 635 y Fu(x)712 621 y Fw(=)25 b Fv(k)p Ft(\026)908 583 y Fr(+)908 649 y Fu(h;\014)s(;x)1095 621 y Fv(\000)20 b Ft(\026)1241 583 y Fq(\000)1241 649 y Fu(h;\014)s(;x)1408 621 y Fv(k)1453 635 y Fu(x)1522 621 y Fw(=)25 b([tanh\()p Ft(\014)5 b Fw(\()p Ft(h)22 b Fw(+)e(2)p Ft(d)p Fw(\)\))h Fv(\000)f Fw(tanh\()p Ft(\014)5 b Fw(\()p Ft(h)21 b Fv(\000)f Fw(2)p Ft(d)p Fw(\)\)])p Ft(=)p Fw(2)15 b Ft(:)165 825 y Fw(Hence,)35 b(the)f(Gibbs)d(measure)i(is)g(unique)e(when)h Ft(h)f Fw(=)e(0)34 b(and)e(tanh\(2)p Ft(d\014)5 b Fw(\))32 b Ft(<)e(p)2900 839 y Fu(c)2934 825 y Fw(,)k(or)f(if)g Fv(j)p Ft(h)p Fv(j)d Ft(>)g Fw(2)p Ft(d)k Fw(is)165 938 y(so)d(large)f(that)h(2)p Ft(d)26 b(<)f(p)956 952 y Fu(c)1021 938 y Fw(cosh)1193 901 y Fr(2)1233 938 y Fw(\()p Ft(\014)5 b Fw(\()p Fv(j)p Ft(h)p Fv(j)22 b(\000)e Fw(2)p Ft(d)p Fw(\)\),)32 b(for)e(example.)165 1151 y Fx(Example)k(7.4)46 b Fp(The)32 b(har)-5 b(d-c)g(or)g(e)35 b(lattic)-5 b(e)33 b(gas.)42 b Fw(Setting)30 b Ft(\014)g Fw(=)25 b(1,)30 b(w)m(e)h(see)f(that)g Ft(p)2934 1165 y Fu(x)3003 1151 y Fw(=)25 b Ft(\025=)p Fw(\(1)c(+)d Ft(\025)p Fw(\))30 b(for)165 1264 y(an)m(y)j Ft(x)p Fw(,)f(so)h(that)f(uniqueness)e(of)j (the)f(Gibbs)e(measure)i(follo)m(ws)g(for)f Ft(\025)e(<)e(p)2772 1278 y Fu(c)2807 1264 y Ft(=)p Fw(\(1)22 b Fv(\000)g Ft(p)3093 1278 y Fu(c)3127 1264 y Fw(\).)46 b(\(This)31 b(can)165 1377 y(also)g(b)s(e)e(obtained)h(b)m(y)g(using)f(the)i(pro)s (duct)e(coupling)g(men)m(tioned)h(ab)s(o)m(v)m(e,)h(cf.)g([28)q(].\)) 165 1589 y Fx(Example)j(7.5)46 b Fp(The)35 b(Widom{R)-5 b(ow)5 b(linson)38 b(lattic)-5 b(e)36 b(gas.)48 b Fw(W)-8 b(e)34 b(tak)m(e)g(again)f Ft(\014)h Fw(=)28 b(1)33 b(and)f(set)h Ft(\025)3455 1603 y Fr(+)3543 1589 y Fw(=)165 1702 y Ft(\025)218 1716 y Fq(\000)305 1702 y Fw(=)27 b Ft(\025)p Fw(.)45 b(It)32 b(turns)f(out)h(that)g(the)g(maxim)m(um)e(in)h (equation)h(\(44\))h(is)e(attained)h(for)f(the)h(b)s(oundary)165 1815 y(conditions)f Ft(\021)g Fv(\021)c Fw(0)32 b(and)f Ft(\021)1079 1782 y Fq(0)1135 1815 y Fw(equal)g(to)i(+1)e(and)h Fv(\000)p Fw(1)f(on)h(\(at)h(least\))f(t)m(w)m(o)i(di\013eren)m(t)d (neigh)m(b)s(ors)f(of)i Ft(x)p Fw(,)165 1928 y(whence)g Ft(p)531 1942 y Fu(x)602 1928 y Fw(=)c(2)p Ft(\025=)p Fw(\(1)23 b(+)e(2)p Ft(\025)p Fw(\))32 b(for)g(an)m(y)g Ft(x)p Fw(.)45 b(It)32 b(follo)m(ws)f(that)h(the)g(Gibbs)f(measure)g (is)g(unique)f(when)165 2041 y Ft(\025)c(<)f(p)386 2055 y Fu(c)420 2041 y Ft(=)p Fw(\(2\(1)d Fv(\000)e Ft(p)784 2055 y Fu(c)819 2041 y Fw(\)\).)165 2253 y(It)25 b(is)e(in)m(teresting) h(to)h(compare)f(the)h(uniqueness)d(condition)h(of)h(Theorem)g(7.2)h (with)e(the)i(celebrated)165 2366 y(Dobrushin)e(uniqueness)g (condition,)j(cf.)f([96)q(])h(and)f(the)g(original)f(pap)s(ers)g([65)q (,)i(66)q(].)39 b(This)23 b(condition)165 2479 y(reads)1251 2592 y(sup)1299 2660 y Fu(x)1403 2511 y Fs(X)1444 2686 y Fu(y)1613 2592 y Fw(max)1538 2653 y Fu(\021)r Fq(\021)p Fu(\021)1667 2634 y Fm(0)1721 2653 y Fr(o\013)h Fu(y)1873 2592 y Fv(k)p Ft(\026)1973 2555 y Fu(\021)1973 2615 y(x)2038 2592 y Fv(\000)c Ft(\026)2184 2555 y Fu(\021)2221 2531 y Fm(0)2184 2615 y Fu(x)2248 2592 y Fv(k)2293 2606 y Fu(x)2362 2592 y Ft(<)25 b Fw(1)p Ft(:)925 b Fw(\(48\))165 2839 y(The)23 b(constrain)m(t)f(\\)p Ft(\021)30 b Fv(\021)25 b Ft(\021)1028 2806 y Fq(0)1082 2839 y Fw(o\013)30 b Ft(y)s Fw(")23 b(means)g(that)g(the)g(con\014gurations)f Ft(\021)s(;)15 b(\021)2650 2806 y Fq(0)2697 2839 y Fw(di\013er)22 b(only)g(at)h(the)g(v)m(ertex)165 2952 y Ft(y)s Fw(.)40 b(F)-8 b(or)30 b(systems)g(with)d(hard-core)j(exclusion)d(or)j(in)d (certain)j(an)m(tiferromagnetic)f(mo)s(dels)f(it)h(often)165 3065 y(happ)s(ens)k(that,)k(for)e(ev)m(ery)h Ft(y)g Fv(2)c Ft(@)5 b(x)p Fw(,)37 b(the)e(maxim)m(um)f(in)f(\(48\))k(is)d(actually)g (the)h(same)h(as)f(that)g(in)165 3178 y(\(44\),)i(see)e([27)q(].)51 b(Dobrushin's)32 b(uniqueness)g(condition)g(then)i(tak)m(es)h(the)f (form)g(sup)3091 3200 y Fu(x)3150 3178 y Fv(j)p Ft(@)5 b(x)p Fv(j)15 b Ft(p)3366 3192 y Fu(x)3442 3178 y Ft(<)30 b Fw(1.)165 3291 y(F)-8 b(or)45 b Fv(L)i Fw(=)h Fx(Z)635 3258 y Fu(d)719 3291 y Fw(and)43 b Ft(p)955 3305 y Fu(x)1046 3291 y Fw(=)k Ft(p)1210 3305 y Fr(0)1293 3291 y Fw(indep)s(enden)m(tly) 41 b(of)j Ft(x)p Fw(,)j(this)c(means)h(that)g Ft(p)2875 3305 y Fr(0)2962 3291 y Ft(<)j Fw(1)p Ft(=)p Fw(\(2)p Ft(d)p Fw(\),)j(while)165 3404 y(Theorem)38 b(7.2)g(only)f(requires)f Ft(p)1312 3418 y Fr(0)1388 3404 y Ft(<)h(p)1542 3418 y Fu(c)1577 3404 y Fw(,)i(and)e(it)g(is)g(kno)m(wn)g(that)h Ft(p)2567 3418 y Fu(c)2639 3404 y Ft(>)f Fw(1)p Ft(=)p Fw(\(2)p Ft(d)27 b Fv(\000)d Fw(1\))39 b(for)e Ft(d)h(>)e Fw(1.)165 3516 y(Ho)m(w)m(ev)m(er,)41 b(if)35 b(the)h(constrained)g (maxim)m(um)f(in)g(Dobrushin's)g(condition)g(is)g(m)m(uc)m(h)h(smaller) f(than)165 3629 y(the)i(unconstrained)e(maxim)m(um)g(in)g(\(44\),)40 b(then)c(Dobrushin's)f(condition)g(will)f(b)s(e)h(w)m(eak)m(er)j(than) 165 3742 y(that)33 b(of)f(Theorem)g(7.2.)47 b(F)-8 b(or)33 b(example,)f(for)g(the)h(Ising)e(ferromagnet)h(on)g Fx(Z)2813 3709 y Fu(d)2886 3742 y Fw(with)f(external)h(\014eld)165 3855 y Ft(h)43 b Fw(=)f(0,)i(Dobrushin's)39 b(condition)g(requires)g (that)j(2)p Ft(d)15 b Fw(tanh)h Ft(\014)47 b(<)42 b Fw(1)f(whic)m(h,)i (in)c(view)h(of)h(\(20\),)k(is)165 3968 y(less)37 b(restrictiv)m(e)g (than)h(the)f(condition)f(obtained)h(in)f(Example)h(7.3.)63 b(Th)m(us,)38 b(roughly)e(sp)s(eaking,)165 4081 y(Theorem)j(7.2)h(w)m (orks)e(b)s(est)h(for)f(\\constrained")h(systems)g(with)e(strong)i (repulsiv)m(e)e(in)m(teractions)165 4194 y(and)g(lo)m(w-dimensional)d (lattices)k(\(or)f(graphs)f(with)g(small)g Fv(j)p Ft(@)5 b(x)p Fv(j)p Fw('s\))38 b(for)f(whic)m(h)f(reasonable)h(lo)m(w)m(er)165 4307 y(b)s(ounds)c(of)h(the)h(critical)f(probabilit)m(y)e Ft(p)1577 4321 y Fu(c)1646 4307 y Fw(are)j(a)m(v)-5 b(ailable.)53 b(Examples)33 b(are)i(the)g(hard-core)g(lattice)165 4420 y(gas)c(and)f(the)h(Widom{Ro)m(wlinson)d(lattice)j(gas)g(on)f Fx(Z)2055 4387 y Fr(2)2125 4420 y Fw(considered)f(ab)s(o)m(v)m(e.)306 4533 y(There)37 b(is)e(also)i(another)f(reason)h(wh)m(y)f(Theorem)g (7.2)i(is)e(useful.)57 b(Namely)-8 b(,)39 b(its)d(condition)f(of)165 4646 y(non-p)s(ercolation)h(is)g(a)i(global)e(condition:)52 b(the)37 b(absence)h(of)f(p)s(ercolation)f(do)s(es)h(not)g(dep)s(end)e (on)165 4758 y(the)h(v)-5 b(alue)34 b(of)h Ft(p)720 4772 y Fu(x)799 4758 y Fw(at)h(an)m(y)f(single)f(site)h Ft(x)p Fw(.)54 b(In)34 b(particular,)h Ft(p)2267 4772 y Fu(x)2346 4758 y Fw(could)f(b)s(e)g(large)h(or)g(ev)m(en)h(b)s(e)f(equal)165 4871 y(to)g(1)f(for)f(all)g Ft(x)p Fw('s)g(in)g(an)g(in\014nite)f (subset)h(\(sa)m(y)-8 b(,)36 b(a)e(p)s(erio)s(dic)d(sublattice\))i(of)h Fv(L)p Fw(;)h(once)g(the)e Ft(p)3354 4885 y Fu(x)3398 4871 y Fw('s)h(are)165 4984 y(su\016cien)m(tly)c(small)f(on)i(the)g (complemen)m(tary)g(set,)h(there)f(is)f(still)f(no)i(in\014nite)d(op)s (en)i(cluster.)42 b(This)165 5097 y(can)21 b(b)s(e)g(applied)d(to)k (non-translation)e(in)m(v)-5 b(arian)m(t)20 b(in)m(teractions)g(where,) j(in)c(general,)k(it)e(is)f(imp)s(ossible)165 5210 y(to)37 b(obtain)e(uniform)f(small)g(b)s(ounds)g(on)h(the)h Ft(p)1821 5224 y Fu(x)1865 5210 y Fw('s)g(\(or)g(on)g(the)g(strength)g(of)f(the)h (in)m(teraction,)i(as)165 5323 y(w)m(ould)33 b(b)s(e)f(required)g(b)m (y)h(the)h(Dobrushin)d(condition)h(or)i(for)f(some)h(standard)f (cluster-expansion)165 5436 y(argumen)m(t\).)42 b(W)-8 b(e)31 b(will)d(come)j(bac)m(k)h(to)f(this)e(p)s(oin)m(t)g(in)g (Section)i(7.3.)1844 5895 y(65)p eop %%Page: 66 66 66 65 bop 165 304 a FA(7.2)113 b(Sto)s(c)m(hastic)37 b(domination)i(b)m(y)f(random-cluster)h(measures)165 476 y Fw(Recen)m(tly)-8 b(,)37 b(Alexander)c(and)g(Cha)m(y)m(es)i([14)q (])f(in)m(tro)s(duced)f(a)h(v)-5 b(arian)m(t)34 b(of)h(the)f (random-cluster)f(tec)m(h-)165 589 y(nique)g(that)h(applies)f(to)h(a)h (substan)m(tially)d(greater)j(class)e(of)i(systems)e(than)h(those)h (considered)d(in)165 701 y(Section)39 b(6.)65 b(This)36 b(approac)m(h)j(in)m(v)m(olv)m(es)f(a)h(so-called)f Fp(gr)-5 b(aphic)g(al)43 b(r)-5 b(epr)g(esentation)41 b Fw(of)d(the)h(original) 165 814 y(system.)50 b(The)33 b(graphical)f(represen)m(tation)h(is)g (sto)s(c)m(hastically)g(dominated)f(b)m(y)h(a)h(random-cluster)165 927 y(mo)s(del,)29 b(and)f(absence)h(of)g(in\014nite)e(clusters)h(in)g (this)g(random-cluster)f(mo)s(del)h(implies)e(uniqueness)165 1040 y(of)40 b(the)f(Gibbs)f(measure)h(for)g(the)g(original)f(spin)f (system.)68 b(The)38 b(price)h(to)h(pa)m(y)f(for)g(the)h(greater)165 1153 y(generalit)m(y)30 b(is)f(that)h(the)g(implication)d(go)s(es)j (only)f(one)h(w)m(a)m(y:)41 b(p)s(ercolation)29 b(in)f(the)i (random-cluster)165 1266 y(mo)s(del)g(do)s(es)g(not,)g(in)g(general,)g (imply)e(non-uniqueness)g(of)j(Gibbs)d(measures.)306 1379 y(W)-8 b(e)33 b(assume)e(that)h(the)f(state)i(space)e Ft(S)37 b Fw(is)30 b(a)h Fp(\014nite)j(gr)-5 b(oup)32 b Fw(with)e(unit)g(elemen)m(t)i(1;)g(the)f(in)m(v)m(erse)165 1492 y(elemen)m(t)d(of)f Ft(a)f Fv(2)f Ft(S)32 b Fw(is)26 b(denoted)h(b)m(y)g Ft(a)1445 1459 y Fq(\000)p Fr(1)1540 1492 y Fw(,)h(so)f(that)h Ft(a)1943 1459 y Fq(\000)p Fr(1)2037 1492 y Ft(a)e Fw(=)f Ft(aa)2303 1459 y Fq(\000)p Fr(1)2422 1492 y Fw(=)g(1.)40 b(F)-8 b(or)28 b(simplicit)m(y)d(w)m(e)i (assume)165 1605 y(that)g(the)g(underlying)c(graph)j(is)f Fv(L)g Fw(=)g Fx(Z)1548 1572 y Fu(d)1615 1605 y Fw(\(although)h(this)f (will)f(not)j(really)e(matter\).)41 b(W)-8 b(e)27 b(consider)165 1718 y(the)33 b(Hamiltonian)e(\(1\))i(for)g(a)f(pair)g(p)s(oten)m(tial) g Ft(U)42 b Fw(and)32 b(with)f(no)h(self-energy)-8 b(,)34 b Ft(V)49 b Fw(=)28 b(0.)48 b(By)32 b(adding)165 1831 y(some)i(constan)m(t)h(to)f Ft(U)44 b Fw(\(whic)m(h)33 b(do)s(es)g(not)h(c)m(hange)g(the)g(relativ)m(e)g(Hamiltonian\))e(w)m (e)i(can)g(arrange)165 1943 y(that)d Ft(U)36 b Fv(\024)25 b Fw(0.)41 b(The)29 b(basic)h(assumption)f(is)g(that)i Ft(U)41 b Fw(is)29 b Fp(left-invariant)p Fw(,)j(so)e(that)1542 2123 y Ft(U)10 b Fw(\()p Ft(a;)15 b(b)p Fw(\))26 b(=)f Ft(u)p Fw(\()p Ft(a)2068 2086 y Fq(\000)p Fr(1)2163 2123 y Ft(b)p Fw(\))1216 b(\(49\))165 2303 y(for)34 b(all)f Ft(a;)15 b(b)32 b Fv(2)f Ft(S)38 b Fw(and)c(the)g(ev)m(en)h(function)d Ft(u)g Fw(=)f Ft(U)10 b Fw(\(1)p Ft(;)15 b Fv(\001)p Fw(\))33 b Fv(\024)e Fw(0.)52 b(Note)35 b(that)g(this)d(setting)i (includes)165 2416 y(the)g Ft(q)s Fw(-state)h(P)m(otts)g(mo)s(del)e (for)g(whic)m(h)g Ft(S)j Fw(=)30 b Fx(Z)1807 2430 y Fu(q)1879 2416 y Fw(and)j Ft(u)e Fw(=)f Fv(\000)p Fw(2)15 b Ft(I)2414 2434 y Fq(f)p Fr(0)p Fq(g)2524 2416 y Fw(.)51 b(F)-8 b(or)35 b(an)m(y)f(\014nite)e(\003)f Fv(\032)g Fx(Z)3437 2383 y Fu(d)3511 2416 y Fw(w)m(e)165 2528 y(consider)f(the)g(Gibbs)f (distribution)797 2735 y Ft(\026)852 2691 y Fu(\021)852 2763 y(\014)s(;)p Fr(\003)967 2735 y Fw(\()p Ft(\033)s Fw(\))d(=)1224 2666 y Ft(I)1264 2684 y Fq(f)p Fu(\033)r Fq(\021)p Fu(\021)34 b Fr(o\013)24 b(\003)p Fq(g)p 1224 2714 430 4 v 1274 2797 a Ft(Z)1336 2811 y Fr(\003)1389 2797 y Fw(\()p Ft(\014)5 b(;)15 b(\021)s Fw(\))1704 2735 y(exp)1858 2616 y Fs(\024)1922 2735 y Fv(\000)20 b Ft(\014)2160 2654 y Fs(X)2084 2842 y Fq(h)p Fu(xy)r Fq(i2B)2308 2853 y Fl(\003)2370 2735 y Ft(u)p Fw(\()p Ft(\033)s Fw(\()p Ft(x)p Fw(\))2634 2698 y Fq(\000)p Fr(1)2730 2735 y Ft(\033)s Fw(\()p Ft(y)s Fw(\)\))2938 2616 y Fs(\025)165 3005 y Fw(at)39 b(in)m(v)m(erse)f(temp)s(erature)f Ft(\014)43 b Fw(with)37 b(b)s(oundary)e(condition)i Ft(\021)k Fv(2)c Fw(\012.)62 b(Here)39 b(w)m(e)f(write)f Fv(B)3251 3019 y Fr(\003)3341 3005 y Fw(for)h(the)165 3118 y(set)f(of)f(all)f(b)s (onds)g Ft(b)g Fv(2)f(B)39 b Fw(with)c(at)i(least)f(one)h(endp)s(oin)m (t)d(in)h(\003.)58 b(The)36 b(graphical)f(represen)m(tation)165 3231 y(of)j Ft(\026)331 3187 y Fu(\021)331 3259 y(\014)s(;)p Fr(\003)483 3231 y Fw(will)d(b)s(e)h(based)h(on)g(b)s(ond)e (con\014gurations)i Ft(!)i Fv(2)d(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)2428 3198 y Fq(B)2474 3209 y Fl(\003)2526 3231 y Fw(.)61 b(Eac)m(h)38 b(suc)m(h)e Ft(!)k Fw(will)35 b(also)i(b)s(e)165 3344 y(view)m(ed)31 b(as)g(a)h(subset)f(of)g Fv(B)1097 3358 y Fr(\003)1150 3344 y Fw(,)g(and)g(the)g(b)s(onds)e(in)h Ft(!)k Fw(will)29 b(b)s(e)h(called)g(op)s(en.)43 b(The)30 b(k)m(ey)i(idea)f(of)g(this)165 3457 y(represen)m(tation)g(is)g(tak)m (en)h(from)e(the)i(classical)e(high)f(temp)s(erature)i(expansion.)42 b(F)-8 b(or)32 b(\014xed)e Ft(\014)i(>)26 b Fw(0)165 3570 y(and)k(an)m(y)h Ft(a)25 b Fv(2)g Ft(S)35 b Fw(w)m(e)c(in)m(tro)s (duce)e(the)i(di\013erence)1447 3750 y Ft(R)1516 3764 y Fu(a)1583 3750 y Fw(=)25 b Ft(e)1721 3712 y Fq(\000)p Fu(\014)15 b(u)p Fr(\()p Fu(a)p Fr(\))1989 3750 y Fv(\000)20 b Fw(1)25 b Fv(\025)g Fw(0)15 b Ft(:)1122 b Fw(\(50\))165 3929 y(With)30 b(this)f(notation)i(w)m(e)g(can)g(write)852 4136 y Ft(\026)907 4092 y Fu(\021)907 4164 y(\014)s(;)p Fr(\003)1022 4136 y Fw(\()p Ft(\033)s Fw(\))84 b(=)1395 4066 y Ft(I)1435 4085 y Fq(f)p Fu(\033)r Fq(\021)p Fu(\021)34 b Fr(o\013)24 b(\003)p Fq(g)p 1395 4115 V 1444 4198 a Ft(Z)1506 4212 y Fr(\003)1559 4198 y Fw(\()p Ft(\014)5 b(;)15 b(\021)s Fw(\))1932 4055 y Fs(Y)1849 4243 y Fq(h)p Fu(xy)r Fq(i2B)2073 4254 y Fl(\003)2120 4136 y Fw(\(1)21 b(+)f Ft(R)2381 4155 y Fu(\033)r Fr(\()p Fu(x)p Fr(\))2517 4137 y Fm(\000)p Fl(1)2601 4155 y Fu(\033)r Fr(\()p Fu(y)r Fr(\))2740 4136 y Fw(\))1231 4433 y(=)1395 4363 y Ft(I)1435 4382 y Fq(f)p Fu(\033)r Fq(\021)p Fu(\021)34 b Fr(o\013)24 b(\003)p Fq(g)p 1395 4412 V 1444 4495 a Ft(Z)1506 4509 y Fr(\003)1559 4495 y Fw(\()p Ft(\014)5 b(;)15 b(\021)s Fw(\))1962 4352 y Fs(X)1849 4548 y Fu(!)r Fq(2f)p Fr(0)p Fu(;)p Fr(1)p Fq(g)2102 4524 y Fm(B)2143 4541 y Fl(\003)2268 4352 y Fs(Y)2209 4540 y Fq(h)p Fu(xy)r Fq(i2)p Fu(!)2449 4433 y Ft(R)2518 4452 y Fu(\033)r Fr(\()p Fu(x)p Fr(\))2654 4434 y Fm(\000)p Fl(1)2738 4452 y Fu(\033)r Fr(\()p Fu(y)r Fr(\))2902 4433 y Ft(:)165 4721 y Fw(This)27 b(sho)m(ws)h(that)h Ft(\026)880 4677 y Fu(\021)880 4749 y(\014)s(;)p Fr(\003)1024 4721 y Fw(is)f(the)g(\014rst)g(marginal)f(distribution)e(of)k(a)f (probabilit)m(y)e(measure)j Ft(P)3387 4677 y Fu(\021)3374 4749 y(\014)s(;)p Fr(\003)3518 4721 y Fw(on)165 4849 y(\012)20 b Fv(\002)g(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)562 4816 y Fq(B)608 4827 y Fl(\003)660 4849 y Fw(,)31 b(namely)1040 5078 y Ft(P)1111 5034 y Fu(\021)1098 5106 y(\014)s(;)p Fr(\003)1214 5078 y Fw(\()p Ft(\033)n(;)15 b(!)s Fw(\))26 b(=)1566 5009 y Ft(I)1606 5027 y Fq(f)p Fu(\033)r Fq(\021)p Fu(\021)34 b Fr(o\013)24 b(\003)p Fq(g)p 1566 5057 V 1616 5140 a Ft(Z)1678 5154 y Fr(\003)1731 5140 y Fw(\()p Ft(\014)5 b(;)15 b(\021)s Fw(\))2080 4997 y Fs(Y)2021 5185 y Fq(h)p Fu(xy)r Fq(i2)p Fu(!)2261 5078 y Ft(R)2330 5098 y Fu(\033)r Fr(\()p Fu(x)p Fr(\))2466 5079 y Fm(\000)p Fl(1)2550 5098 y Fu(\033)r Fr(\()p Fu(y)r Fr(\))2714 5078 y Ft(;)165 5364 y(\033)29 b Fv(2)c Fw(\012,)30 b Ft(!)e Fv(2)d(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)844 5331 y Fq(B)890 5342 y Fl(\003)942 5364 y Fw(.)41 b(The)29 b(second)i(marginal)e(distribution)e(of)j Ft(P)2537 5319 y Fu(\021)2524 5392 y(\014)s(;)p Fr(\003)2671 5364 y Fw(is)f(equal)h(to)1282 5602 y Ft(\015)1334 5558 y Fu(\021)1329 5630 y(\014)s(;)p Fr(\003)1445 5602 y Fw(\()p Ft(!)s Fw(\))c(=)f Ft(W)1796 5558 y Fu(\021)1783 5630 y(\014)s(;)p Fr(\003)1898 5602 y Fw(\()p Ft(!)s Fw(\))2028 5483 y Fs(\036)2116 5602 y Ft(Z)2178 5616 y Fr(\003)2231 5602 y Fw(\()p Ft(\014)5 b(;)15 b(\021)s Fw(\))27 b Ft(;)1844 5895 y Fw(66)p eop %%Page: 67 67 67 66 bop 165 304 a Fw(where)1164 417 y Ft(W)1263 373 y Fu(\021)1250 445 y(\014)s(;)p Fr(\003)1366 417 y Fw(\()p Ft(!)s Fw(\))25 b(=)1715 336 y Fs(X)1617 521 y Fu(\033)r Fq(\021)p Fu(\021)34 b Fr(o\013)24 b(\003)2007 336 y Fs(Y)1947 524 y Fq(h)p Fu(xy)r Fq(i2)p Fu(!)2188 417 y Ft(R)2257 437 y Fu(\033)r Fr(\()p Fu(x)p Fr(\))2393 418 y Fm(\000)p Fl(1)2476 437 y Fu(\033)r Fr(\()p Fu(y)r Fr(\))3453 417 y Fw(\(51\))165 690 y(is)i(the)g(\\graphical)g(w)m(eigh) m(t")h(of)g(an)m(y)f Ft(!)j Fv(2)24 b(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)1828 657 y Fq(B)1874 668 y Fl(\003)1926 690 y Fw(.)40 b(The)26 b(probabilit)m(y)e(measure)i Ft(\015)3037 646 y Fu(\021)3032 718 y(\014)s(;)p Fr(\003)3174 690 y Fw(on)g Fv(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)3516 657 y Fq(B)3562 668 y Fl(\003)165 803 y Fw(is)28 b(called)g(the)h Fp(gr)-5 b(aphic)g(al)34 b(distribution)j Fw(or)29 b(the)g Fp(gr)-5 b(ey)31 b(me)-5 b(asur)g(e)37 b Fw(\(since)29 b(it)f(ignores)h(the)g(spins)d(whic)m(h)165 916 y(are)f(considered)e (as)h(colors\).)39 b(The)23 b(graphical)g(represen)m(tation)h(of)g Ft(\026)2490 871 y Fu(\021)2490 944 y(\014)s(;)p Fr(\003)2630 916 y Fw(th)m(us)f(obtained)h(is)f(analogu-)165 1029 y(ous)28 b(to)h(the)f(random-cluster)g(represen)m(tation)g(of)g(the)g (P)m(otts)i(mo)s(del)d(and)g(can)i(b)s(e)e(summarized)g(as)165 1141 y(follo)m(ws.)165 1329 y Fx(Lemma)33 b(7.6)46 b Fp(In)37 b(the)g(set-up)g(describ)-5 b(e)g(d)38 b(ab)-5 b(ove,)39 b(the)e(Gibbs)g(distribution)h Ft(\026)2895 1285 y Fu(\021)2895 1357 y(\014)s(;)p Fr(\003)3047 1329 y Fp(c)-5 b(an)37 b(b)-5 b(e)37 b(derive)-5 b(d)165 1451 y(fr)g(om)34 b(the)f(gr)-5 b(aphic)g(al)35 b(distribution)f Ft(\015)1452 1407 y Fu(\021)1447 1479 y(\014)s(;)p Fr(\003)1595 1451 y Fp(by)f(me)-5 b(ans)34 b(of)f(the)g(c)-5 b(onditional)35 b(pr)-5 b(ob)g(abilities)1021 1681 y Ft(P)1092 1637 y Fu(\021)1079 1709 y(\014)s(;)p Fr(\003)1196 1681 y Fw(\()p Ft(\033)s Fv(j)p Ft(!)s Fw(\))26 b(=)f Ft(W)1627 1637 y Fu(\021)1614 1709 y(\014)s(;)p Fr(\003)1729 1681 y Fw(\()p Ft(!)s Fw(\))1860 1635 y Fq(\000)p Fr(1)2056 1600 y Fs(Y)1954 1789 y Fu(b)p Fr(=)p Fq(h)p Fu(xy)r Fq(i2)p Fu(!)2280 1681 y Ft(R)2349 1701 y Fu(\033)r Fr(\()p Fu(x)p Fr(\))2485 1682 y Fm(\000)p Fl(1)2569 1701 y Fu(\033)r Fr(\()p Fu(y)r Fr(\))2733 1681 y Ft(:)165 1976 y Fp(That)34 b(is,)1129 2089 y Ft(\026)1184 2045 y Fu(\021)1184 2117 y(\014)s(;)p Fr(\003)1300 2089 y Fw(\()p Ft(\033)s Fw(\))26 b(=)1659 2008 y Fs(X)1547 2205 y Fu(!)r Fq(2f)p Fr(0)p Fu(;)p Fr(1)p Fq(g)1800 2181 y Fm(B)1841 2198 y Fl(\003)1906 2089 y Ft(\015)1958 2045 y Fu(\021)1953 2117 y(\014)s(;)p Fr(\003)2069 2089 y Fw(\()p Ft(!)s Fw(\))15 b Ft(P)2285 2045 y Fu(\021)2272 2117 y(\014)s(;)p Fr(\003)2389 2089 y Fw(\()p Ft(\033)s Fv(j)p Ft(!)s Fw(\))26 b Ft(:)165 2376 y Fw(F)-8 b(or)33 b(the)f(P)m(otts)g(in)m(teraction)g Ft(u)27 b Fw(=)h Fv(\000)p Fw(2)15 b Ft(I)1538 2394 y Fq(f)p Fr(0)p Fq(g)1679 2376 y Fw(with)31 b(state)i(space)f Ft(S)g Fw(=)27 b Fx(Z)2606 2390 y Fu(q)2644 2376 y Fw(,)32 b(the)g(graphical)f(represen-)165 2489 y(tation)36 b(ab)s(o)m(v)m(e)g (is)e(easily)g(seen)h(to)h(coincide)e(with)g(the)h(random-cluster)f (represen)m(tation)h(studied)165 2602 y(in)30 b(Section)h(6.)43 b(One)30 b(imp)s(ortan)m(t)g(feature)i(is)e(that)h(the)h(graphical)d(w) m(eigh)m(ts)j(factorize)g(in)m(to)f(cluster)165 2714 y(terms.)47 b(Indeed,)33 b(eac)m(h)g(b)s(ond)e(con\014guration)h Ft(!)k Fw(divides)30 b(\003)i(in)m(to)h(connected)g(comp)s(onen)m(ts)g (called)165 2827 y(op)s(en)27 b(clusters)g(\(whic)m(h)g(ma)m(y)i(p)s (ossibly)c(consist)i(of)h(isolated)f(sites\).)40 b(The)27 b(set)i(of)f(b)s(onds)e(b)s(elonging)165 2940 y(to)31 b(an)e(op)s(en)g(cluster)g Ft(C)36 b Fw(is)28 b(denoted)i(b)m(y)g Ft(!)1630 2954 y Fu(C)1688 2940 y Fw(.)41 b(W)-8 b(riting)28 b Fv(C)5 b Fw(\()p Ft(!)s Fw(\))31 b(for)e(the)h(set)g(of)g(all)e(op)s (en)h(clusters)g(w)m(e)165 3053 y(then)h(obtain)g(that)1286 3166 y Ft(W)1385 3122 y Fu(\021)1372 3194 y(\014)s(;)p Fr(\003)1488 3166 y Fw(\()p Ft(!)s Fw(\))25 b(=)1808 3085 y Fs(Y)1739 3273 y Fu(C)5 b Fq(2C)t Fr(\()p Fu(!)r Fr(\))2025 3143 y Fw(\026)1999 3166 y Ft(W)2098 3122 y Fu(\021)2085 3194 y(\014)s(;)p Fr(\003)2200 3166 y Fw(\()p Ft(C)q(;)15 b(!)2398 3180 y Fu(C)2458 3166 y Fw(\))960 b(\(52\))165 3424 y(with)849 3514 y(\026)822 3537 y Ft(W)921 3492 y Fu(\021)908 3565 y(\014)s(;)p Fr(\003)1024 3537 y Fw(\()p Ft(C)q(;)15 b(!)1222 3551 y Fu(C)1282 3537 y Fw(\))25 b(=)1716 3456 y Fs(X)1438 3647 y Fu(\033)r Fq(2)p Fu(S)1574 3628 y Fo(C)1626 3647 y Fr(:)12 b Fu(\033)r Fq(\021)p Fu(\021)33 b Fr(on)24 b Fu(C)5 b Fq(\\)p Fu(@)t Fr(\003)2273 3456 y Fs(Y)2189 3644 y Fq(h)p Fu(xy)r Fq(i2)p Fu(!)2411 3655 y Fo(C)2479 3537 y Ft(R)2548 3556 y Fu(\033)r Fr(\()p Fu(x)p Fr(\))2684 3537 y Fm(\000)p Fl(1)2768 3556 y Fu(\033)r Fr(\()p Fu(y)r Fr(\))2932 3537 y Ft(:)165 3806 y Fw(\(W)-8 b(e)25 b(mak)m(e)g(the)f (usual)e(con)m(v)m(en)m(tion)j(that)f(the)g(empt)m(y)g(pro)s(duct)e(is) h(equal)g(to)h(1;)j(hence)3146 3783 y(\026)3119 3806 y Ft(W)3218 3762 y Fu(\021)3205 3834 y(\014)s(;)p Fr(\003)3321 3806 y Fw(\()p Ft(C)q(;)15 b(!)3519 3820 y Fu(C)3579 3806 y Fw(\))165 3919 y(=)26 b Fv(j)p Ft(S)5 b Fv(j)31 b Fw(if)e Ft(C)37 b Fw(is)30 b(an)g(isolated)g(site.\))42 b(T)-8 b(ogether)32 b(with)d(Lemma)i(7.6,)h(equation)e(\(52\))i(sho)m (ws)f(that)g(the)165 4032 y(spins)i(b)s(elonging)g(to)i(disjoin)m(t)e (op)s(en)h(clusters)g(are)h(conditionally)d(indep)s(enden)m(t.)52 b(In)34 b(particular,)165 4145 y(w)m(e)f(can)g(sim)m(ulate)f(the)g (spin)f(system)h(b)m(y)h(\014rst)f(dra)m(wing)f(a)i(b)s(ond)e (con\014guration)g Ft(!)36 b Fw(with)31 b(w)m(eigh)m(ts)165 4257 y(\(51\))h(and)e(then)g(obtain)g(in)f(eac)m(h)j(op)s(en)d(cluster) h(a)h(spin)d(con\014guration)i(according)g(to)h Ft(P)3258 4213 y Fu(\021)3245 4286 y(\014)s(;)p Fr(\003)3361 4257 y Fw(\()p Ft(\033)s Fv(j)p Ft(!)s Fw(\).)306 4370 y(Supp)s(ose)38 b(w)m(e)i(knew)f(that)h(there)f(is)g(no)g(p)s(ercolation)g(in)f(the)h (graphical)f(represen)m(tation,)k(in)165 4483 y(the)e(sense)f(that)h (max)949 4497 y Fu(\021)1006 4483 y Ft(\015)1058 4439 y Fu(\021)1053 4511 y(\014)s(;)p Fr(\003)1168 4483 y Fw(\(0)h Fv($)g Ft(@)5 b Fw(\003\))41 b Fv(!)f Fw(0)g(as)f(\003)i Fv(")g(L)p Fw(.)67 b(The)39 b(conditional)f(indep)s(endence)f(of)165 4596 y(spins)24 b(in)h(di\013eren)m(t)h(op)s(en)f(clusters)g(w)m(ould)g (then)h(suggest)h(that)g(there)f(is)f(only)g(one)i(Gibbs)d(measure)165 4709 y(for)36 b(the)g(spin)d(system.)57 b(Unfortunately)-8 b(,)37 b(this)e(is)f(not)i(kno)m(wn)g(\(though)f(w)m(eak)m(er)i (statemen)m(ts)h(are)165 4822 y(established)c(in)g([58)q(]\).)57 b(Ho)m(w)m(ev)m(er,)39 b(one)c(can)h(mak)m(e)g(a)g(sto)s(c)m(hastic)h (comparison)d(of)i(the)f(graphical)165 4935 y(distributions)29 b(with)i(wired)f(random-cluster)i(distributions)c(\(Lemma)33 b(7.7)g(b)s(elo)m(w\),)g(and)e(the)i(ab-)165 5048 y(sence)f(of)f(p)s (ercolation)e(in)h(the)h(dominating)e(random-cluster)h(distribution)d (will)h(then)j(guaran)m(tee)165 5161 y(that)39 b(the)e(original)f (system)i(has)f(a)h(unique)e(Gibbs)g(measure.)63 b(This)36 b(will)f(b)s(e)i(ac)m(hiev)m(ed)h(in)e(The-)165 5274 y(orem)k(7.8)h(allo)m(wing)e(to)i(b)s(ound)c(the)k(dep)s(endence)d(on)i (b)s(oundary)e(conditions)g(in)h(terms)h(of)g(the)165 5387 y(connectivit)m(y)31 b(probabilit)m(y)d(in)h(a)i(random-cluster)e (mo)s(del.)1844 5895 y(67)p eop %%Page: 68 68 68 67 bop 306 304 a Fw(T)-8 b(o)39 b(this)d(end)i(w)m(e)g(also)g(need)f (to)i(consider)d(Gibbs)h(distributions)d Ft(\026)2708 260 y Fu(f)2708 332 y(\014)s(;)p Fr(\003)2861 304 y Fw(with)i Fp(fr)-5 b(e)g(e)40 b(b)-5 b(oundary)165 444 y(c)g(ondition)p Fw(.)42 b(These)27 b(admit)g(similar)e(graphical)h(represen)m(tations)h Ft(\015)2481 399 y Fu(f)2476 472 y(\014)s(;)p Fr(\003)2620 444 y Fw(based)g(on)g(b)s(ond)f(con\014gura-)165 557 y(tions)33 b(inside)e(\003;)k(that)f(is,)f(the)h(b)s(onds)e(leading)g (from)g(\003)i(to)g(\003)2331 524 y Fu(c)2399 557 y Fw(are)g(remo)m(v)m (ed.)50 b(In)33 b(the)g(follo)m(wing,)165 669 y(the)e(sup)s(erscript)d Ft(f)39 b Fw(will)28 b(refer)i(to)h(this)e(case.)306 782 y(The)f(sto)s(c)m(hastic)i(comparison)d(with)g(random-cluster)h (distributions)c(will)i(b)s(e)i(form)m(ulated)g(us-)165 895 y(ing)546 1041 y Ft(R)616 1004 y Fq(\003)680 1041 y Fw(=)d(max)795 1101 y Fu(a)p Fq(2)p Fu(S)961 1041 y Ft(R)1030 1055 y Fu(a)1097 1041 y Ft(;)1248 1018 y Fw(\026)1228 1041 y Ft(R)h Fw(=)1462 980 y(1)p 1429 1020 112 4 v 1429 1104 a Fv(j)p Ft(S)5 b Fv(j)1571 961 y Fs(X)1565 1144 y Fu(a)p Fq(2)p Fu(S)1712 1041 y Ft(R)1781 1055 y Fu(a)1848 1041 y Ft(;)106 b(p)25 b Fw(=)g Ft(R)2216 1004 y Fq(\003)2255 1041 y Ft(=)p Fw(\(1)d(+)e Ft(R)2563 1004 y Fq(\003)2602 1041 y Fw(\))25 b Ft(;)107 b(q)28 b Fw(=)d Ft(R)3029 1004 y Fq(\003)3068 1041 y Ft(=)3133 1018 y Fw(\026)3113 1041 y Ft(R)h(:)220 b Fw(\(53\))165 1283 y(Note)41 b(that)g(these)f (quan)m(tities)f(dep)s(end)f(on)i Ft(\014)45 b Fw(since)39 b(the)h Ft(R)2296 1297 y Fu(a)2377 1283 y Fw(in)f(\(50\))i(do.)69 b(In)39 b(the)h(case)h(of)f(the)165 1396 y Ft(r)s Fw(-state)i(P)m(otts) f(mo)s(del)e(when)h Ft(u)i Fw(=)f Fv(\000)p Fw(2)15 b Ft(I)1628 1415 y Fq(f)p Fr(0)p Fq(g)1738 1396 y Fw(,)43 b(w)m(e)e(ha)m(v)m(e)g Ft(R)2239 1363 y Fq(\003)2320 1396 y Fw(=)h(1)27 b Fv(\000)f Ft(e)2644 1363 y Fq(\000)p Fr(2)p Fu(\014)2822 1396 y Fw(and)40 b Ft(q)45 b Fw(=)c Ft(r)s Fw(;)k(that)c(is,)165 1509 y(in)d(this)g(case)i(the)f (parameters)h Ft(p)f Fw(and)f Ft(q)k Fw(are)d(nothing)f(but)h(the)g (standard)f(parameters)i(of)f(the)165 1622 y(random-cluster)31 b(represen)m(tation.)47 b(F)-8 b(or)33 b Ft(p)f Fw(and)f Ft(q)k Fw(as)e(ab)s(o)m(v)m(e)g(w)m(e)g(consider)e(no)m(w)h(the)h (wired)d(\(resp.)165 1735 y(free\))h(random-cluster)e(distribution)e Ft(\036)1552 1702 y Fr(1)1552 1760 y Fu(p;q)r(;)p Fr(\003)1744 1735 y Fw(\(resp.)k Ft(\036)2052 1702 y Fr(0)2052 1760 y Fu(p;q)r(;)p Fr(\003)2214 1735 y Fw(\))f(in)f(\003)i(as)f(in)m(tro)s (duced)f(in)g(Section)h(6.2.)165 1940 y Fx(Lemma)j(7.7)46 b Fp(F)-7 b(or)26 b(any)f Fw(\003)g Fv(2)g(E)8 b Fp(,)26 b Ft(\014)k(>)25 b Fw(0)g Fp(and)h Ft(p;)15 b(q)27 b Fp(as)e(ab)-5 b(ove,)26 b Ft(\015)2324 1896 y Fu(\021)2319 1968 y(\014)s(;)p Fr(\003)2460 1940 y Fv(\026)2531 1954 y Fq(D)2617 1940 y Ft(\036)2671 1907 y Fr(1)2671 1966 y Fu(p;q)r(;)p Fr(\003)2857 1940 y Fp(and)g Ft(\015)3078 1896 y Fu(f)3073 1968 y(\014)s(;)p Fr(\003)3214 1940 y Fv(\026)3285 1954 y Fq(D)3370 1940 y Ft(\036)3424 1907 y Fr(0)3424 1966 y Fu(p;q)r(;)p Fr(\003)3586 1940 y Fp(.)165 2121 y Fx(Pro)s(of:)130 b Fw(W)-8 b(e)41 b(only)e(pro)m(v)m(e)i(the)f (\014rst)g(statemen)m(t)i(since)d(the)h(second)g(is)f(similar)f(and)h (simpler.)165 2234 y(According)30 b(to)g(Section)g(6.1,)h(the)f(w)m (eigh)m(ts)g(of)g(the)g(random-cluster)f(distribution)d Ft(\036)3110 2201 y Fr(1)3110 2259 y Fu(p;q)r(;)p Fr(\003)3301 2234 y Fw(are)k(pro-)165 2346 y(p)s(ortional)f(to)1537 2399 y Fs(\022)1686 2457 y Ft(p)p 1608 2497 203 4 v 1608 2581 a Fw(1)21 b Fv(\000)f Ft(p)1820 2399 y Fs(\023)1882 2422 y Fq(j)p Fu(!)r Fq(j)1986 2518 y Ft(q)2030 2481 y Fu(k)r Fr(\()p Fu(!)r(;)p Fr(\003\))165 2725 y Fw(with)35 b Fv(j)p Ft(!)s Fv(j)h Fw(the)h(n)m(um)m(b)s(er)d(of)j(op)s(en)e(b)s (onds)f(and)i Ft(k)s Fw(\()p Ft(!)s(;)15 b Fw(\003\))37 b(the)f(n)m(um)m(b)s(er)f(of)h(op)s(en)g(clusters)f(meeting)165 2838 y(\003)k(\(where)g(all)e(clusters)h(touc)m(hing)h Ft(@)5 b Fw(\003)39 b(are)g(wired)e(together)j(in)m(to)f(a)g(single)e (cluster\).)66 b(Up)38 b(to)i(a)165 2950 y(constan)m(t)32 b(factor,)g(the)e(Radon{Nik)m(o)s(dym)g(densit)m(y)g(of)g Ft(\015)2139 2906 y Fu(\021)2134 2979 y(\014)s(;)p Fr(\003)2280 2950 y Fw(relativ)m(e)h(to)g Ft(\036)2769 2917 y Fr(1)2769 2976 y Fu(p;q)r(;)p Fr(\003)2961 2950 y Fw(is)e(th)m(us)h(giv)m(en)h(b) m(y)1105 3205 y Ft(F)13 b Fw(\()p Ft(!)s Fw(\))26 b(=)f Ft(W)1527 3161 y Fu(\021)1514 3233 y(\014)s(;)p Fr(\003)1630 3205 y Fw(\()p Ft(!)s Fw(\))1760 3086 y Fs(\036)1847 3205 y Fw(\()p Ft(R)1952 3168 y Fq(\003)1991 3205 y Fw(\))2027 3163 y Fq(j)p Fu(!)r Fq(j)2116 3205 y Fw(\()p Ft(R)2221 3168 y Fq(\003)2261 3205 y Ft(=)2326 3182 y Fw(\026)2306 3205 y Ft(R)q Fw(\))2411 3168 y Fu(k)r Fr(\()p Fu(!)r(;)p Fr(\003\))2649 3205 y Ft(:)165 3449 y Fw(Since)30 b Ft(\036)457 3416 y Fr(1)457 3475 y Fu(p;q)r(;)p Fr(\003)650 3449 y Fw(has)h(p)s(ositiv)m(e)f(correlations,)h(the)g(lemma)f(will)f (therefore)i(b)s(e)f(pro)m(v)m(ed)i(once)f(w)m(e)h(ha)m(v)m(e)165 3562 y(sho)m(wn)37 b(that)g Ft(F)50 b Fw(is)36 b(a)h(decreasing)g (function)e(of)i Ft(!)s Fw(.)60 b(T)-8 b(o)37 b(this)f(end)g(w)m(e)i (let)e Ft(!)j Fv(\026)d Ft(!)3046 3529 y Fq(0)3106 3562 y Fw(b)s(e)g(suc)m(h)h(that)165 3675 y Ft(!)225 3642 y Fq(0)274 3675 y Fw(=)25 b Ft(!)e Fv([)d(f)p Ft(b)p Fv(g)31 b Fw(for)f(a)h(b)s(ond)d Ft(b)e Fv(2)f(B)1346 3689 y Fr(\003)1419 3675 y Fv(n)20 b Ft(!)s Fw(.)306 3788 y(W)-8 b(e)30 b(\014rst)e(consider)g(the)h(case)g(when)f Ft(b)d Fw(=)g Fv(h)p Ft(xy)s Fv(i)k Fw(is)f(not)h(connected)g(to)g Ft(@)5 b Fw(\003)29 b(and)f(joins)g(t)m(w)m(o)i(op)s(en)165 3901 y(clusters)c Ft(C)556 3915 y Fu(x)600 3901 y Ft(;)31 b(C)721 3915 y Fu(y)787 3901 y Fv(2)25 b(C)5 b Fw(\()p Ft(!)s Fw(\).)40 b(F)-8 b(or)28 b(eac)m(h)g(op)s(en)e(cluster)g Ft(C)33 b Fw(let)2240 3878 y(\026)2214 3901 y Ft(W)12 b Fw(\()p Ft(C)7 b Fw(\))25 b(=)2602 3878 y(\026)2575 3901 y Ft(W)2674 3857 y Fu(\021)2661 3929 y(\014)s(;)p Fr(\003)2777 3901 y Fw(\()p Ft(C)q(;)15 b(!)2975 3915 y Fu(C)3035 3901 y Fw(\))27 b(b)s(e)f(as)h(in)e(\(52\).)165 4014 y(Supp)s(ose)33 b(w)m(e)i(stipulate)e(that)j(the)e(spin)f Ft(\033)s Fw(\()p Ft(z)t Fw(\))j(at)f(an)m(y)g(site)f Ft(z)i Fv(2)c Ft(C)41 b Fw(is)34 b(equal)g(to)h(some)g Ft(a)d Fv(2)g Ft(S)5 b Fw(.)53 b(It)165 4127 y(is)32 b(then)f(easy)i(to)g(see)g(that)g(the)f(remaining)e(sum)h(in)g(the)i (de\014nition)c(of)2712 4104 y(\026)2686 4127 y Ft(W)12 b Fw(\()p Ft(C)7 b Fw(\))33 b(do)s(es)e(not)i(dep)s(end)165 4240 y(on)e Ft(a)f Fw(and)g(th)m(us)g(has)g(the)h(v)-5 b(alue)1327 4217 y(\026)1301 4240 y Ft(W)12 b Fw(\()p Ft(C)7 b Fw(\))p Ft(=)p Fv(j)p Ft(S)e Fv(j)p Fw(.)42 b(Prescribing)28 b(the)j(v)-5 b(alues)29 b(of)i Ft(\033)s Fw(\()p Ft(x)p Fw(\))g(and)f Ft(\033)s Fw(\()p Ft(y)s Fw(\))h(in)e(this)165 4353 y(w)m(a)m(y)j(w)m(e)f(th)m(us)f(\014nd)e (that)750 4526 y(\026)723 4549 y Ft(W)12 b Fw(\()p Ft(C)921 4563 y Fu(x)986 4549 y Fv([)20 b Ft(C)1132 4563 y Fu(y)1193 4549 y Fv([)g Ft(b)p Fw(\))26 b(=)1496 4526 y(\026)1470 4549 y Ft(W)12 b Fw(\()p Ft(C)1668 4563 y Fu(x)1712 4549 y Fw(\))1789 4526 y(\026)1762 4549 y Ft(W)h Fw(\()p Ft(C)1961 4563 y Fu(y)2003 4549 y Fw(\))26 b Fv(j)p Ft(S)5 b Fv(j)2175 4511 y Fq(\000)p Fr(2)2376 4468 y Fs(X)2285 4656 y Fu(\033)r Fr(\()p Fu(x)p Fr(\))p Fu(;)12 b(\033)r Fr(\()p Fu(y)r Fr(\))2603 4549 y Ft(R)2672 4568 y Fu(\033)r Fr(\()p Fu(x)p Fr(\))2808 4549 y Fm(\000)p Fl(1)2892 4568 y Fu(\033)r Fr(\()p Fu(y)r Fr(\))3031 4549 y Ft(;)165 4847 y Fw(and)34 b(therefore)g Ft(W)830 4802 y Fu(\021)817 4875 y(\014)s(;)p Fr(\003)932 4847 y Fw(\()p Ft(!)1027 4814 y Fq(0)1051 4847 y Fw(\))d(=)1238 4824 y(\026)1219 4847 y Ft(R)15 b(W)1402 4802 y Fu(\021)1389 4875 y(\014)s(;)p Fr(\003)1505 4847 y Fw(\()p Ft(!)s Fw(\).)51 b(Since)33 b Ft(k)s Fw(\()p Ft(!)2097 4814 y Fq(0)2121 4847 y Ft(;)15 b Fw(\003\))32 b(=)e Ft(k)s Fw(\()p Ft(!)s(;)15 b Fw(\003\))24 b Fv(\000)e Fw(1)34 b(and)f Fv(j)p Ft(!)3136 4814 y Fq(0)3160 4847 y Fv(j)e Fw(=)f Fv(j)p Ft(!)s Fv(j)23 b Fw(+)f(1,)165 4960 y(it)29 b(follo)m(ws)g(that)h Ft(F)13 b Fw(\()p Ft(!)915 4927 y Fq(0)939 4960 y Fw(\))25 b(=)g Ft(F)13 b Fw(\()p Ft(!)s Fw(\),)31 b(pro)m(ving)d(the)i(claim)e(in)g(the)i (\014rst)f(case.)41 b(If)29 b Ft(b)h Fw(links)d(some)j(cluster)165 5073 y(to)g(the)f(b)s(oundary)e(whic)m(h)h(otherwise)g(w)m(as)i (separated)f(from)g(the)g(b)s(oundary)-8 b(,)28 b(then)g(the)i(argumen) m(t)165 5185 y(ab)s(o)m(v)m(e)i(sho)m(ws)e(again)g(that)h Ft(F)13 b Fw(\()p Ft(!)1289 5152 y Fq(0)1313 5185 y Fw(\))26 b(=)f Ft(F)13 b Fw(\()p Ft(!)s Fw(\).)306 5298 y(Next)28 b(w)m(e)e(consider)f(the)i(case)g(when)e Ft(b)h Fw(=)f Fv(h)p Ft(xy)s Fv(i)h Fw(closes)h(a)f(lo)s(op)g(in)f Ft(!)k Fw(but)c(is)h(still)e(not)i(connected)165 5411 y(to)31 b(the)g(b)s(oundary)-8 b(.)39 b(Since)30 b(clearly)1327 5607 y Ft(R)1396 5627 y Fu(\033)r Fr(\()p Fu(x)p Fr(\))1532 5608 y Fm(\000)p Fl(1)1616 5627 y Fu(\033)r Fr(\()p Fu(y)r Fr(\))1780 5607 y Fv(\024)25 b Fw(max)1895 5666 y Fu(a)p Fq(2)p Fu(S)2061 5607 y Ft(R)2130 5621 y Fu(a)2197 5607 y Fw(=)g Ft(R)2363 5570 y Fq(\003)2427 5607 y Ft(;)1844 5895 y Fw(68)p eop %%Page: 69 69 69 68 bop 165 304 a Fw(w)m(e)36 b(\014nd)d(that)i Ft(W)790 260 y Fu(\021)777 332 y(\014)s(;)p Fr(\003)893 304 y Fw(\()p Ft(!)988 271 y Fq(0)1011 304 y Fw(\))e Fv(\024)g Ft(W)1282 260 y Fu(\021)1269 332 y(\014)s(;)p Fr(\003)1384 304 y Fw(\()p Ft(!)s Fw(\))15 b Ft(R)1599 271 y Fq(\003)1639 304 y Fw(.)54 b(On)34 b(the)h(other)g(hand,)h(in)d(this)h(case)i(w)m(e) f(ha)m(v)m(e)h Fv(j)p Ft(!)3462 271 y Fq(0)3486 304 y Fv(j)c Fw(=)165 417 y Fv(j)p Ft(!)s Fv(j)21 b Fw(+)g(1)31 b(and)g Ft(k)s Fw(\()p Ft(!)787 384 y Fq(0)810 417 y Ft(;)15 b Fw(\003\))28 b(=)e Ft(k)s Fw(\()p Ft(!)s(;)15 b Fw(\003\),)32 b(so)g(that)f Ft(F)13 b Fw(\()p Ft(!)1889 384 y Fq(0)1913 417 y Fw(\))27 b Fv(\024)f Ft(F)13 b Fw(\()p Ft(!)s Fw(\).)43 b(As)31 b(w)m(e)h(are)f(considering)e(the)j (wired)165 530 y(random-cluster)42 b(measure,)k(this)c(argumen)m(t)h (remains)e(v)-5 b(alid)42 b(if)f Ft(b)i Fw(joins)e(t)m(w)m(o)k (clusters)d(already)165 643 y(attac)m(hed)33 b(to)e(the)f(b)s(oundary) -8 b(.)2344 b Fk(2)165 805 y Fw(W)-8 b(e)40 b(are)f(no)m(w)g(in)e(a)i (p)s(osition)d(to)k(state)g(the)e(main)g(result)f(of)i(Alexander)f(and) g(Cha)m(y)m(es)h([14)q(],)i(an)165 918 y(estimate)26 b(on)f(the)g(dep)s(endence)f(of)h(Gibbs)e(distributions)e(on)k(their)f (b)s(oundary)f(condition)h(in)f(terms)165 1031 y(of)k(p)s(ercolation)e (in)g(the)h(wired)f(random-cluster)g(distribution.)36 b(Recall)25 b(the)i(notation)f(\(45\))i(for)e(the)165 1144 y(total)32 b(v)-5 b(ariation)29 b(norm)h(on)g(the)h(sub-)p Ft(\033)s Fw(-algebra)f Fv(F)1921 1158 y Fr(\001)2014 1144 y Fw(of)h(ev)m(en)m(ts)h(in)d(some)h(\001.)165 1332 y Fx(Theorem)k(7.8)46 b Fp(Consider)h(the)f(spin)g(system)g(with)g(p)-5 b(air)47 b(inter)-5 b(action)47 b(\(49\))g(at)f(some)g(inverse)165 1445 y(temp)-5 b(er)g(atur)g(e)40 b Ft(\014)e(>)33 b Fw(0)p Fp(,)39 b(and)f(let)f Ft(p;)30 b(q)40 b Fp(b)-5 b(e)37 b(given)f(by)h(\(53\).)56 b(Then,)38 b(for)g(any)g Fw(\001)32 b Fv(\032)h Fw(\003)h Fv(2)e(E)46 b Fp(and)37 b(any)165 1558 y(p)-5 b(air)34 b(of)f(b)-5 b(oundary)35 b(c)-5 b(onditions)34 b Ft(\021)s(;)15 b(\021)1421 1525 y Fq(0)1471 1558 y Fv(2)25 b Fw(\012)p Fp(,)1196 1770 y Fv(k)p Ft(\026)1296 1726 y Fu(\021)1296 1798 y(\014)s(;)p Fr(\003)1432 1770 y Fv(\000)20 b Ft(\026)1578 1726 y Fu(\021)1615 1703 y Fm(0)1578 1798 y Fu(\014)s(;)p Fr(\003)1693 1770 y Fv(k)1738 1784 y Fr(\001)1827 1770 y Fv(\024)25 b Ft(\036)1977 1733 y Fr(1)1977 1793 y Fu(p;q)r(;)p Fr(\003)2139 1770 y Fw(\(\001)g Fv($)g Ft(@)5 b Fw(\003\))15 b Ft(:)165 1975 y Fx(Pro)s(of:)40 b Fw(\(This)26 b(pro)s(of)g(is)h(di\013eren)m(t) f(from)h(the)g(one)h(that)g(app)s(eared)e(in)g([14)q(].\))40 b(Let)28 b Ft(A)f Fw(b)s(e)g(an)m(y)h(ev)m(en)m(t)165 2087 y(in)h Fv(F)336 2101 y Fr(\001)400 2087 y Fw(.)40 b(F)-8 b(rom)31 b(Lemma)g(7.6)g(w)m(e)g(kno)m(w)f(that)1233 2292 y Ft(\026)1288 2247 y Fu(\021)1288 2320 y(\014)s(;)p Fr(\003)1404 2292 y Fw(\()p Ft(A)p Fw(\))c(=)1664 2211 y Fs(X)1701 2385 y Fu(!)1799 2292 y Ft(\015)1851 2247 y Fu(\021)1846 2320 y(\014)s(;)p Fr(\003)1962 2292 y Fw(\()p Ft(!)s Fw(\))15 b Ft(P)2178 2247 y Fu(\021)2165 2320 y(\014)s(;)p Fr(\003)2282 2292 y Fw(\()p Ft(A)p Fv(j)p Ft(!)s Fw(\))g Ft(:)165 2556 y Fw(T)-8 b(o)30 b(con)m(trol)f(the)h Ft(\021)s Fw(-dep)s(endence)f(of)g(this)f (probabilit)m(y)e(w)m(e)k(will)c(pro)s(ceed)j(in)f(analogy)i(to)g(the)f (argu-)165 2669 y(men)m(t)j(for)f(the)h(implication)c(\(ii\))j Fv(\))g Fw(\(iii\))e(of)j(Theorem)f(6.10.)45 b(If)31 b Ft(!)e Fv(2)e(f)p Fw(\001)g Fv(6$)f Ft(@)5 b Fw(\003)p Fv(g)32 b Fw(then)f(equation)165 2782 y(\(52\))39 b(sho)m(ws)d(that)i (the)f(conditional)e(distribution)e Ft(P)2047 2738 y Fu(\021)2034 2810 y(\014)s(;)p Fr(\003)2150 2782 y Fw(\()p Ft(A)p Fv(j)p Ft(!)s Fw(\))38 b(do)s(es)e(not)h(dep)s(end)e(on)i Ft(\021)s Fw(.)60 b(So)37 b(w)m(e)165 2904 y(need)30 b(to)i(con)m(trol)e(the)h Ft(\021)s Fw(-dep)s(endence)f(of)h Ft(\015)1673 2860 y Fu(\021)1668 2932 y(\014)s(;)p Fr(\003)1783 2904 y Fw(\(\001)26 b Fv(6$)f Ft(@)5 b Fw(\003\).)41 b(This,)29 b(ho)m(w)m(ev)m(er,)j(do)s(es)e(not)h(seem)g(p)s(os-)165 3029 y(sible)j(directly)-8 b(.)54 b(So)35 b(w)m(e)h(will)d(replace)i Ft(\015)1563 2985 y Fu(\021)1558 3057 y(\014)s(;)p Fr(\003)1709 3029 y Fw(b)m(y)g(the)g Ft(\021)s Fw(-indep)s(enden)m(t)f Ft(\036)2654 2996 y Fr(1)2654 3055 y Fu(p;q)r(;)p Fr(\003)2851 3029 y Fw(b)m(y)h(using)f(a)i(suitable)165 3142 y(coupling)29 b(tric)m(k.)306 3255 y(By)d(Lemma)f(7.7)i(and)d(Strassen's)h(theorem)h (\(Theorem)f(4.6\))i(there)e(exists)g(a)g(coupling)f(\()3383 3232 y(~)3369 3255 y Ft(Y)c(;)3496 3232 y Fw(~)3482 3255 y Ft(Y)3556 3222 y Fq(0)3579 3255 y Fw(\))165 3368 y(of)35 b Ft(\015)325 3324 y Fu(\021)320 3396 y(\014)s(;)p Fr(\003)471 3368 y Fw(and)f Ft(\036)706 3335 y Fr(1)706 3394 y Fu(p;q)r(;)p Fr(\003)903 3368 y Fw(suc)m(h)g(that)1327 3345 y(~)1314 3368 y Ft(Y)52 b Fv(\026)1536 3345 y Fw(~)1522 3368 y Ft(Y)1595 3335 y Fq(0)1654 3368 y Fw(almost)34 b(surely)-8 b(.)53 b(If)2367 3345 y(~)2354 3368 y Ft(Y)2427 3335 y Fq(0)2483 3368 y Fv(2)32 b(f)p Fw(\001)h Fv(6$)f Ft(@)5 b Fw(\003)p Fv(g)p Fw(,)37 b(there)e(exists)g(a)165 3501 y(largest)c(\(random\))f(set)h(\000)25 b(=)g(\000\()1289 3478 y(~)1275 3501 y Ft(Y)1348 3468 y Fq(0)1372 3501 y Fw(\))30 b(suc)m(h)g(that)211 3689 y(\(a\))h(\001)25 b Fv(\032)g Fw(\000)g Fv(\032)g Fw(\003,)31 b(and)211 3876 y(\(b\))376 3853 y(~)362 3876 y Ft(Y)435 3843 y Fq(0)459 3876 y Fw(\()p Ft(b)p Fw(\))26 b(=)f(0)30 b(for)g(all)g(b)s (onds)e(connecting)j(\000)f(with)f(\000)2102 3843 y Fu(c)2137 3876 y Fw(.)165 4064 y(F)-8 b(or)336 4041 y(~)322 4064 y Ft(Y)395 4031 y Fq(0)444 4064 y Fv(2)25 b(f)p Fw(\001)g Fv($)g Ft(@)5 b Fw(\003)p Fv(g)26 b Fw(w)m(e)e(set)h(\000)g(=)g Fv(;)p Fw(.)39 b(Conditional)22 b(on)j(\000,)g(Lemma)g(7.7)g(and)f (Strassen's)g(theorem)165 4185 y(pro)m(vide)32 b(us)h(further)e(with)h (a)h(coupling)f(\()1623 4162 y(~)1609 4185 y Ft(Y)1662 4199 y Fr(\000)1710 4185 y Ft(;)1764 4162 y Fw(~)1750 4185 y Ft(Y)1823 4152 y Fq(0)1803 4211 y Fr(\000)1851 4185 y Fw(\))h(of)h Ft(\015)2078 4141 y Fu(f)2073 4213 y(\014)s(;)p Fr(\000)2216 4185 y Fw(and)f Ft(\036)2450 4152 y Fr(0)2450 4211 y Fu(p;q)r(;)p Fr(\000)2640 4185 y Fw(suc)m(h)f(that)3061 4162 y(~)3047 4185 y Ft(Y)3100 4199 y Fr(\000)3177 4185 y Fv(\026)3291 4162 y Fw(~)3278 4185 y Ft(Y)3351 4152 y Fq(0)3331 4211 y Fr(\000)3379 4185 y Fw(.)48 b(It)33 b(is)165 4298 y(then)d(easy)h(to)h(see)f(that)f (the)h(pair)e(of)i(random)e(v)-5 b(ariables)29 b(\()p Ft(Y)5 b(;)15 b(Y)2392 4265 y Fq(0)2415 4298 y Fw(\))31 b(de\014ned)e(b)m(y)887 4568 y(\()p Ft(Y)5 b(;)15 b(Y)1093 4530 y Fq(0)1117 4568 y Fw(\)\()p Ft(b)p Fw(\))26 b(=)1383 4424 y Fs(\()1491 4511 y Fw(\()1540 4488 y(~)1526 4511 y Ft(Y)1579 4525 y Fr(\000)1628 4511 y Ft(;)1682 4488 y Fw(~)1668 4511 y Ft(Y)1741 4478 y Fq(0)1721 4537 y Fr(\000)1769 4511 y Fw(\)\()p Ft(b)p Fw(\))84 b(if)30 b Ft(b)g Fw(is)f(con)m(tained)i(in)e(\000,)1508 4629 y(\()1557 4606 y(~)1543 4629 y Ft(Y)20 b(;)1670 4606 y Fw(~)1656 4629 y Ft(Y)1730 4596 y Fq(0)1753 4629 y Fw(\)\()p Ft(b)p Fw(\))100 b(otherwise)165 4845 y(is)32 b(still)e(a)j(coupling)e(of)h Ft(\015)1039 4801 y Fu(\021)1034 4873 y(\014)s(;)p Fr(\003)1182 4845 y Fw(and)g Ft(\036)1415 4812 y Fr(1)1415 4871 y Fu(p;q)r(;)p Fr(\003)1609 4845 y Fw(suc)m(h)g(that)h Ft(Y)49 b Fv(\026)28 b Ft(Y)2289 4812 y Fq(0)2345 4845 y Fw(almost)33 b(surely)-8 b(.)45 b(\(Notice)34 b(that)f(also)179 4955 y(~)165 4978 y Ft(Y)20 b Fw(\()p Ft(b)p Fw(\))26 b(=)f(0)j(for)f(all)f(b)s(onds)g(from)h(\000) g(to)h(\000)1525 4945 y Fu(c)1559 4978 y Fw(.\))40 b(W)-8 b(e)29 b(denote)f(the)f(underlying)d(probabilit)m(y)h(measure)i(b)m(y) 165 5091 y Ft(Q)237 5058 y Fu(\021)279 5091 y Fw(.)41 b(No)m(w)31 b(w)m(e)g(can)f(write)625 5331 y Ft(\026)680 5286 y Fu(\021)680 5359 y(\014)s(;)p Fr(\003)795 5331 y Fw(\()p Ft(A)p Fw(\))84 b(=)f Ft(Q)1243 5293 y Fu(\021)1285 5212 y Fs(\022)1346 5331 y Ft(P)1417 5286 y Fu(\021)1404 5359 y(\014)s(;)p Fr(\003)1520 5331 y Fw(\()p Ft(A)p Fv(j)p Ft(Y)20 b Fw(\))1756 5212 y Fs(\023)1017 5563 y Fw(=)83 b Ft(Q)1243 5526 y Fu(\021)1285 5444 y Fs(\022)1346 5563 y Ft(P)1417 5519 y Fu(\021)1404 5591 y(\014)s(;)p Fr(\003)1520 5563 y Fw(\()p Ft(A)p Fv(j)p Ft(Y)20 b Fw(\))15 b Ft(I)1811 5582 y Fq(f)p Fr(\000=)p Fq(;g)2021 5444 y Fs(\023)2102 5563 y Fw(+)20 b Ft(Q)2265 5526 y Fu(\021)2307 5444 y Fs(\022)2368 5563 y Ft(P)2439 5519 y Fu(\021)2426 5591 y(\014)s(;)p Fr(\003)2542 5563 y Fw(\()p Ft(A)p Fv(j)p Ft(Y)h Fw(\))15 b Ft(I)2834 5582 y Fq(f)p Fr(\000)p Fq(6)p Fr(=)p Fq(;g)3043 5444 y Fs(\023)3130 5563 y Ft(:)1844 5895 y Fw(69)p eop %%Page: 70 70 70 69 bop 165 304 a Fw(The)31 b(\014rst)g(term)g(in)f(the)i(last)f(sum) f(is)h(at)h(most)f Ft(Q)1891 271 y Fu(\021)1933 304 y Fw(\(\000)c(=)g Fv(;)p Fw(\))g(=)g Ft(\036)2409 271 y Fr(1)2409 330 y Fu(p;q)r(;)p Fr(\003)2571 304 y Fw(\(\001)g Fv($)f Ft(@)5 b Fw(\003\).)45 b(W)-8 b(e)32 b(claim)f(that)165 417 y(the)36 b(second)g(term)g(do)s(es)g(not)g(dep)s(end)e(on)h Ft(\021)s Fw(.)58 b(Indeed,)36 b(if)f(\000)f Fv(6)p Fw(=)g Fv(;)i Fw(then,)h(b)m(y)f(\(52\),)j Ft(P)3169 373 y Fu(\021)3156 445 y(\014)s(;)p Fr(\003)3272 417 y Fw(\()p Ft(A)p Fv(j)p Ft(Y)21 b Fw(\))34 b(=)165 557 y Ft(P)236 512 y Fu(f)223 585 y(\014)s(;)p Fr(\000)335 557 y Fw(\()p Ft(A)p Fv(j)p Ft(Y)516 571 y Fr(\000)564 557 y Fw(\))g(only)f(dep)s(ends)f(on)i(the)g (restriction)f Ft(Y)1973 571 y Fr(\000)2054 557 y Fw(of)h Ft(Y)54 b Fw(to)34 b(the)g(set)h(of)e(b)s(onds)g(inside)e(\000.)51 b(The)165 669 y(second)31 b(term)f(can)h(th)m(us)f(b)s(e)g(written)f (explicitly)f(as)1042 787 y Fs(X)1030 975 y Fu(G)p Fq(6)p Fr(=)p Fq(;)1190 868 y Ft(\036)1244 830 y Fr(1)1244 890 y Fu(p;q)r(;)p Fr(\003)1406 868 y Fw(\(\000)d(=)g Ft(G)p Fw(\))1788 787 y Fs(X)1741 971 y Fu(!)33 b Fr(in)24 b Fu(G)1970 868 y Ft(\015)2022 824 y Fu(f)2017 896 y(\014)s(;G)2139 868 y Fw(\()p Ft(!)s Fw(\))i Ft(P)2366 824 y Fu(f)2353 896 y(\014)s(;G)2475 868 y Fw(\()p Ft(A)p Fv(j)p Ft(!)s Fw(\))g Ft(;)165 1153 y Fw(whic)m(h)j(is)h(ob)m(viously)f(indep)s (enden)m(t)f(of)j Ft(\021)s Fw(.)41 b(The)30 b(theorem)g(no)m(w)h (follo)m(ws)e(immediately)-8 b(.)360 b Fk(2)165 1314 y Fw(T)-8 b(o)33 b(apply)d(the)i(theorem)h(w)m(e)f(consider)f(the)h (limiting)d(random-cluster)i(measure)g Ft(\036)3093 1281 y Fr(1)3093 1336 y Fu(p;q)3218 1314 y Fw(with)g(arbi-)165 1427 y(trary)e(parameters)g Ft(p)c Fv(2)p Fw(]0)p Ft(;)15 b Fw(1[)30 b(and)e Ft(q)h Fv(\025)c Fw(1)k(and)f(wired)f(b)s(oundary)g (condition;)h(recall)g(from)g(Section)165 1540 y(6.2)33 b(that)f(this)e(limiting)e(measure)j(exists.)43 b(By)32 b(Corollary)d(6.7\(d\),)k(it)e(mak)m(es)h(sense)f(to)h(de\014ne)f(the) 165 1653 y(p)s(ercolation)f(threshold)1226 1766 y Ft(p)1272 1780 y Fu(c)1307 1766 y Fw(\()p Ft(q)s Fw(\))c(=)e(inf)6 b Fv(f)p Ft(p)26 b Fw(:)f Ft(\036)1874 1728 y Fr(1)1874 1788 y Fu(p;q)1967 1766 y Fw(\(0)h Fv($)g(1)p Fw(\))f Ft(>)g Fw(0)p Fv(g)p Ft(:)165 1929 y Fw(W)-8 b(e)37 b(also)e(consider)e (the)j(threshold)d Ft(w)1503 1943 y Fu(c)1538 1929 y Fw(\()p Ft(q)s Fw(\))i(for)g(exp)s(onen)m(tial)f(deca)m(y)i(of)g (connectivities,)f(whic)m(h)f(is)165 2042 y(de\014ned)29 b(as)i(the)g(suprem)m(um)d(of)j(all)e Ft(p)p Fw('s)h(for)g(whic)m(h) 1343 2241 y Ft(\036)1397 2203 y Fr(1)1397 2263 y Fu(p;q)1490 2241 y Fw(\(0)c Fv($)f Ft(@)5 b Fw(\003\))26 b Fv(\024)f Ft(C)7 b(e)2099 2203 y Fq(\000)p Fu(c)k(d)p Fr(\(0)p Fu(;@)t Fr(\003\))165 2439 y Fw(uniformly)22 b(in)h(\003)h(\(or,)i(at)f (least,)h(for)e(\003)g(in)f(a)i(prescrib)s(ed)d(sequence)i(increasing)f (to)i Fv(L)p Fw(\))g(with)e(suitable)165 2552 y(constan)m(ts)36 b Ft(c)d(>)g Fw(0)i(and)f Ft(C)39 b(<)32 b Fv(1)p Fw(.)54 b(It)35 b(is)f(eviden)m(t)h(that)g Ft(w)2174 2566 y Fu(c)2209 2552 y Fw(\()p Ft(q)s Fw(\))e Fv(\024)f Ft(p)2505 2566 y Fu(c)2540 2552 y Fw(\()p Ft(q)s Fw(\);)37 b(for)e(large)g Ft(q)i Fw(it)e(is)f(kno)m(wn)165 2665 y(that)j Ft(w)433 2679 y Fu(c)467 2665 y Fw(\()p Ft(q)s Fw(\))e(=)f Ft(p)767 2679 y Fu(c)801 2665 y Fw(\()p Ft(q)s Fw(\))i([75)q(].)57 b(Theorem)35 b(7.8)i(then)e(giv)m(es)h(us)f(the)h(follo)m(wing)e (conditions)g(for)h(high-)165 2778 y(temp)s(erature)c(b)s(eha)m(vior;)e (compare)i(with)e(Theorem)h(7.2.)165 2984 y Fx(Corollary)35 b(7.9)46 b Fp(Whenever)30 b Ft(\014)k Fp(is)29 b(so)h(smal)5 b(l)30 b(that)g Ft(p)25 b(<)g(p)2158 2998 y Fu(c)2193 2984 y Fw(\()p Ft(q)s Fw(\))p Fp(,)30 b(ther)-5 b(e)30 b(is)f(a)g(unique)g(Gibbs)g(me)-5 b(asur)g(e)165 3097 y(for)40 b(the)g(Hamiltonian)g Ft(H)46 b Fp(with)40 b(p)-5 b(air)41 b(inter)-5 b(action)40 b(\(49\).)63 b(F)-7 b(urthermor)i(e,)43 b(if)c(in)g(fact)g Ft(p)e(<)g(w)3465 3111 y Fu(c)3500 3097 y Fw(\()p Ft(q)s Fw(\))165 3210 y Fp(then)i(the)g(spin)g(system)g (is)f(exp)-5 b(onential)5 b(ly)41 b(we)-5 b(ak-mixing)39 b(in)f(the)h(sense)f(that)i(ther)-5 b(e)39 b(ar)-5 b(e)39 b(p)-5 b(ositive)165 3323 y(c)g(onstants)46 b Ft(C)k(<)45 b Fv(1)p Ft(;)15 b(c)45 b(>)f Fw(0)f Fp(such)h(that)g(for)g(al)5 b(l)43 b Fw(\001)i Fv(\032)f Fw(\003)g Fv(2)g(E)51 b Fp(and)45 b(al)5 b(l)43 b(b)-5 b(oundary)45 b(c)-5 b(onditions)165 3436 y Ft(\021)s(;)15 b(\021)301 3403 y Fq(0)351 3436 y Fv(2)25 b Fw(\012)1168 3557 y Fv(k)p Ft(\026)1268 3513 y Fu(\021)1268 3585 y(\014)s(;)p Fr(\003)1404 3557 y Fv(\000)20 b Ft(\026)1550 3513 y Fu(\021)1587 3490 y Fm(0)1550 3585 y Fu(\014)s(;)p Fr(\003)1666 3557 y Fv(k)1711 3571 y Fr(\001)1799 3557 y Fv(\024)25 b Ft(C)d Fv(j)p Ft(@)5 b Fw(\001)p Fv(j)15 b Ft(e)2218 3520 y Fq(\000)p Fu(c)d(d)p Fr(\(\001)p Fu(;)p Fr(\003)2507 3496 y Fo(c)2539 3520 y Fr(\))2586 3557 y Ft(:)165 3764 y Fw(In)35 b(fact,)j(Alexander)d (and)g(Cha)m(y)m(es)h([14)q(])g(go)g(a)g(bit)f(further)f(in)g(their)h (exploration)f(of)i(`nice')f(high)165 3877 y(temp)s(erature)i(b)s(eha)m (vior,)g(sho)m(wing)f(that)h(for)f Ft(p)f(<)g(p)2028 3891 y Fu(c)2063 3877 y Fw(\(1\))i(the)g(unique)d(Gibbs)h(measure)i (satis\014es)165 3990 y(the)31 b(condition)e(of)h(`complete)i (analyticit)m(y')e(\(in)m(v)m(estigated)h(in)e([71)q(],)i(for)f (example\).)165 4232 y FA(7.3)113 b(Exp)s(onen)m(tial)38 b(mixing)h(at)e(lo)m(w)g(temp)s(eratures)165 4404 y Fw(In)44 b(the)g(previous)e(subsections)h(w)m(e)h(ha)m(v)m(e)h(seen)f(ho)m(w)g (sto)s(c)m(hastic-geometric)i(metho)s(ds)e(can)g(b)s(e)165 4517 y(used)e(to)h(analyze)g(the)g(high)e(temp)s(erature)h(b)s(eha)m (vior)g(of)g(a)h(spin)e(system)h(and,)j(in)d(particular,)165 4630 y(for)36 b(establishing)e(exp)s(onen)m(tial)h(deca)m(y)i(of)f (correlations.)58 b(Here)37 b(w)m(e)f(w)m(an)m(t)h(to)g(demonstrate)g (that)165 4742 y(similar)d(p)s(ercolation)i(tec)m(hniques)g(can)h(also) g(b)s(e)f(used)g(in)f(the)i(lo)m(w)g(temp)s(erature)f(regime)h(in)e (the)165 4855 y(presence)30 b(of)g(phase)g(transition.)39 b(W)-8 b(e)31 b(will)c(presen)m(t)j(a)g(metho)s(d)g(to)g(sho)m(w)g (that,)h(for)e(a)i(giv)m(en)e(phase)165 4968 y Ft(\026)p Fw(,)g(the)g(co)m(v)-5 b(ariance)29 b Ft(\026)p Fw(\()p Ft(f)10 b Fw(;)15 b Ft(g)s Fw(\))30 b(of)e(an)m(y)h(t)m(w)m(o)h(lo)s (cal)e(observ)-5 b(ables)27 b Ft(f)38 b Fw(and)28 b Ft(g)k Fw(deca)m(ys)d(exp)s(onen)m(tially)e(fast)165 5081 y(with)32 b(the)h(distance)f(b)s(et)m(w)m(een)i(their)e(dep)s(endence)f(sets.)49 b(\(The)32 b(problem)g(of)h(phase)f(transition)f(at)165 5194 y(lo)m(w)f(temp)s(eratures)h(will)c(b)s(e)j(addressed)f(in)g (Section)i(8.\))306 5307 y(As)i(a)g(matter)g(of)g(fact,)h(the)f (problem)d(of)j(exp)s(onen)m(tial)e(deca)m(y)j(of)f(co)m(v)-5 b(ariances)33 b(\(or)g(truncated)165 5420 y(correlation)g(functions\))f (arises)h(in)f(man)m(y)h(ph)m(ysical)f(situations.)48 b(Correlation)32 b(functions)g(are)i(re-)165 5533 y(lated)i(to)h(in)m (teresting)e(resp)s(onse)h(functions)e(or)i(to)h(\015uctuations)e(of)h (sp)s(eci\014c)f(order)h(parameters.)165 5646 y(Exp)s(onen)m(tial)j (deca)m(y)j(of)e(co)m(v)-5 b(ariances)42 b(also)e(pro)m(vides)f (estimates)i(on)g(higher)e(order)g(correlation)1844 5895 y(70)p eop %%Page: 71 71 71 70 bop 165 304 a Fw(functions,)34 b(ev)m(en)m(tually)g(pro)m(viding) d(in\014nite)h(di\013eren)m(tiabilit)m(y)f(of)j(the)g(free)g(energy)g (with)e(resp)s(ect)165 417 y(to)c(an)f(external)g(\014eld)f([72)q(].)40 b(Motiv)-5 b(ated)28 b(b)m(y)e(these)i(in)m(terests,)g(a)f(v)-5 b(ariet)m(y)28 b(of)f(tec)m(hniques)g(ha)m(v)m(e)h(b)s(een)165 530 y(dev)m(elop)s(ed.)40 b(The)27 b(most)h(familiar)e(approac)m(h)i (are)g(cluster)f(expansions)g(whic)m(h)f(apply)h(equally)f(w)m(ell)165 643 y(to)j(b)s(oth)d(the)i(high)e(temp)s(erature)h(\(or)h(lo)m(w)f (densit)m(y\))g(regime)h(and)f(the)g(lo)m(w)g(temp)s(erature)h(\(or)f (high)165 756 y(densit)m(y\))40 b(regime.)68 b(Although)39 b(they)g(often)h(emplo)m(y)g(geometric)h(concepts,)i(it)c(seems)h (useful)d(to)165 869 y(com)m(bine)42 b(them)f(with)f(ideas)h(of)h(p)s (ercolation)f(theory)g(to)i(mak)m(e)f(geometry)h(more)f(visible.)71 b(An)165 981 y(example)37 b(of)f(this)g(is)f(the)i(metho)s(d)f(to)h(b)s (e)f(describ)s(ed)e(b)s(elo)m(w)i(whic)m(h)f(is)g(tak)m(en)j(from)e(a)h (pap)s(er)e(b)m(y)165 1094 y(Burton)29 b(and)f(Steif)g([37)q(],)i (where)e(it)g(w)m(as)h(used)f(to)i(sho)m(w)e(that)i(certain)e(Gibbs)f (measures)i(exhibit)e(a)165 1207 y(p)s(o)m(w)m(erful)i(mixing)g(prop)s (ert)m(y)h(called)f(`quite)h(w)m(eak)h(Bernoulli)d(with)h(exp)s(onen)m (tial)h(rate'.)306 1320 y(W)-8 b(e)35 b(consider)d(a)h(spin)e(system)i (on)g(an)g(arbitrary)f(graph)g(\()p Fv(L)p Ft(;)15 b Fv(\030)p Fw(\))33 b(with)f(Hamiltonian)f(\(1\).)50 b(As)165 1433 y(b)s(efore,)31 b(the)f(essen)m(tial)g(feature)h(of)g(this)e (Hamiltonian)g(is)g(that)i(only)f(adjacen)m(t)h(spins)d(in)m(teract,)k (so)165 1546 y(that)38 b(the)g(Gibbs)e(distributions)d Ft(\026)1397 1502 y Fu(\021)1397 1574 y(\014)s(;)p Fr(\003)1550 1546 y Fw(in)j(\014nite)g(regions)h(\003)h(ha)m(v)m(e)g(the)g(Mark)m(o) m(v)h(prop)s(ert)m(y)-8 b(.)62 b(The)165 1659 y(in)m(v)m(erse)30 b(temp)s(erature)g Ft(\014)g(>)25 b Fw(0)31 b(do)s(es)e(not)h(pla)m(y)g (an)m(y)g(role)g(for)f(the)i(momen)m(t,)f(so)h(w)m(e)f(set)g(it)g (equal)f(to)165 1772 y(1)i(and)f(drop)f(it)h(from)g(our)g(notation.)306 1885 y(Our)37 b(starting)g(p)s(oin)m(t)g(is)g(the)g(follo)m(wing)g (estimate)h(on)f(the)h Ft(\021)s Fw(-dep)s(endence)g(in)e(terms)h(of)h (dis-)165 1998 y(agreemen)m(t)27 b(paths)d(for)h(t)m(w)m(o)h Fp(indep)-5 b(endent)34 b Fw(copies)25 b(of)g Ft(\026)2059 1953 y Fu(\021)2059 2025 y Fr(\003)2112 1998 y Fw(.)38 b(This)23 b(result)h(is)g(a)h(w)m(eak)g(v)m(ersion)g(\(and,)g(in)165 2111 y(fact,)i(a)e(forerunner)d([24)q(]\))j(of)g(Theorem)f(7.1.)40 b(It)24 b(is)f(a)i(pleasan)m(t)f(surprise)e(that)j(although)f(dev)m (elop)s(ed)165 2223 y(with)30 b(high)g(temp)s(erature)h(situations)f (in)g(mind,)f(it)i(also)g(pro)m(vides)f(a)i(useful)d(alternativ)m(e)j (to)g(some)165 2336 y(asp)s(ects)f(of)g(the)f(standard)g(lo)m(w)g(temp) s(erature)g(expansions.)165 2524 y Fx(Prop)s(osition)37 b(7.10)46 b Fp(F)-7 b(or)34 b(any)f Fw(\001)25 b Fv(\032)g Fw(\003)g Fv(2)g(E)41 b Fp(and)33 b Ft(\021)s(;)15 b(\021)2078 2491 y Fq(0)2128 2524 y Fv(2)25 b Fw(\012)p Fp(,)1149 2751 y Fv(k)p Ft(\026)1249 2707 y Fu(\021)1249 2779 y Fr(\003)1322 2751 y Fv(\000)20 b Ft(\026)1468 2707 y Fu(\021)1505 2684 y Fm(0)1468 2779 y Fr(\003)1532 2751 y Fv(k)1577 2765 y Fr(\001)1666 2751 y Fv(\024)25 b Ft(\026)1817 2707 y Fu(\021)1817 2779 y Fr(\003)1890 2751 y Fv(\002)20 b Ft(\026)2036 2707 y Fu(\021)2073 2684 y Fm(0)2036 2779 y Fr(\003)2100 2751 y Fw(\(\001)2292 2696 y Fq(6)p Fr(=)2236 2751 y Fv( )-15 b(!)25 b Ft(@)5 b Fw(\003\))26 b Ft(:)165 2974 y Fx(Pro)s(of:)110 b Fw(F)-8 b(or)37 b(brevit)m(y)e(let)g Ft(P)47 b Fw(=)34 b Ft(\026)1443 2929 y Fu(\021)1443 3001 y Fr(\003)1519 2974 y Fv(\002)24 b Ft(\026)1669 2929 y Fu(\021)1706 2906 y Fm(0)1669 3001 y Fr(\003)1733 2974 y Fw(,)37 b(and)e(write)f Ft(X)r(;)15 b(X)2413 2941 y Fq(0)2474 2974 y Fw(for)35 b(the)h(t)m(w)m(o)g(pro)5 b(jections)36 b(from)165 3086 y(\012)20 b Fv(\002)g Fw(\012)30 b(to)h(\012.)41 b(Then)29 b(for)h(an)m(y)h Ft(A)25 b Fv(2)g(F)1473 3100 y Fr(\001)1567 3086 y Fw(w)m(e)30 b(ha)m(v)m(e)1025 3299 y Ft(\026)1080 3255 y Fu(\021)1080 3326 y Fr(\003)1133 3299 y Fw(\()p Ft(A)p Fw(\))21 b Fv(\000)f Ft(\026)1438 3255 y Fu(\021)1475 3231 y Fm(0)1438 3326 y Fr(\003)1502 3299 y Fw(\()p Ft(A)p Fw(\))26 b(=)f Ft(P)13 b Fw(\()p Ft(X)33 b Fv(2)25 b Ft(A)p Fw(\))c Fv(\000)f Ft(P)13 b Fw(\()p Ft(X)2465 3262 y Fq(0)2514 3299 y Fv(2)25 b Ft(A)p Fw(\))h Ft(:)165 3503 y Fw(W)-8 b(e)37 b(decomp)s(ose)f(the)g(probabilities)c(on)j(the)h(righ)m(t-hand) e(side)h(in)m(to)g(the)h(t)m(w)m(o)h(con)m(tributions)d(ac-)165 3639 y(cording)27 b(to)h(whether)f(the)g(ev)m(en)m(t)i Fv(f)p Fw(\001)1532 3584 y Fq(6)p Fr(=)1476 3639 y Fv( )-15 b(!)25 b Ft(@)5 b Fw(\003)p Fv(g)29 b Fw(o)s(ccurs)e(or)g(not.)40 b(In)26 b(the)i(latter)g(case,)h(there)e(exists)165 3752 y(a)j(random)e(set)i(\000)25 b Fv(\032)g Fw(\003)k(con)m(taining)f (\001)h(suc)m(h)g(that)g Ft(X)k Fv(\021)25 b Ft(X)2213 3719 y Fq(0)2266 3752 y Fw(on)j Ft(@)5 b Fw(\000.)41 b(\(The)29 b(union)e(of)i(all)f(disagree-)165 3865 y(men)m(t)35 b(clusters)e(in)g(\003)h(meeting)g(\001)f(is)g(suc)m(h)h(a)g(set.\))53 b(Let)34 b(\000)g(b)s(e)f(the)i(maximal)d(random)i(subset)f(of)165 3978 y(\003)f(with)e(this)h(prop)s(ert)m(y)-8 b(.)44 b(Then)31 b(for)g(eac)m(h)i Ft(G)f Fw(the)g(ev)m(en)m(t)h Fv(f)p Fw(\000)27 b(=)g Ft(G)p Fv(g)33 b Fw(only)e(dep)s(ends)e(on)j (the)g(con\014g-)165 4091 y(uration)j(outside)f Ft(G)p Fw(,)j(and)e Ft(X)40 b Fv(\021)33 b Ft(X)1423 4058 y Fq(0)1482 4091 y Fw(on)i Ft(@)5 b(G)p Fw(.)56 b(The)35 b(Mark)m(o)m(v)i(prop)s(ert)m(y)d(therefore)i(implies)c(that,)165 4204 y(conditionally)26 b(on)i Fv(f)p Fw(\000)d(=)g Ft(G)p Fv(g)k Fw(and)e(\()p Ft(X)1482 4218 y Fu(G)1537 4199 y Fo(c)1574 4204 y Ft(;)15 b(X)1696 4171 y Fq(0)1689 4229 y Fu(G)1744 4211 y Fo(c)1781 4204 y Fw(\),)29 b Ft(X)1945 4218 y Fu(G)2032 4204 y Fw(and)e Ft(X)2288 4171 y Fq(0)2281 4229 y Fu(G)2369 4204 y Fw(are)h(indep)s(enden)m(t)e (and)h(iden)m(tically)165 4363 y(distributed,)38 b(and)f(this)g(sho)m (ws)h(that)h Ft(P)13 b Fw(\()p Ft(X)46 b Fv(2)38 b Ft(A;)31 b Fw(\001)2095 4281 y Fm(6)p Fl(=)2073 4363 y Fv(6$)38 b Ft(@)5 b Fw(\003\))39 b(=)f Ft(P)13 b Fw(\()p Ft(X)2689 4330 y Fq(0)2751 4363 y Fv(2)38 b Ft(A;)31 b Fw(\001)3110 4281 y Fm(6)p Fl(=)3088 4363 y Fv(6$)38 b Ft(@)5 b Fw(\003\).)65 b(The)165 4476 y(prop)s(osition)28 b(no)m(w)j(follo)m(ws)e(immediately) -8 b(.)1909 b Fk(2)165 4638 y Fw(What)36 b(is)d(gained)h(with)g(the)g (disagreemen)m(t)h(estimate)h(ab)s(o)m(v)m(e?)54 b(First,)35 b(let)g(us)f(observ)m(e)h(that)g(this)165 4751 y(estimate)42 b(pro)m(vides)e(b)s(ounds)f(on)h(co)m(v)-5 b(ariances)42 b(of)g(lo)s(cal)e(functions)f(in)h(terms)h(of)g(disagreemen)m(t)165 4864 y(p)s(ercolation.)165 5052 y Fx(Corollary)35 b(7.11)47 b Fp(Fix)40 b(any)g Fw(\003)f Fv(2)f(E)48 b Fp(and)40 b Ft(\021)i Fv(2)c Fw(\012)p Fp(.)63 b(L)-5 b(et)40 b Ft(f)49 b Fp(and)41 b Ft(g)i Fp(b)-5 b(e)40 b(any)g(two)h(lo)-5 b(c)g(al)42 b(functions)165 5165 y(dep)-5 b(ending)34 b(on)f(the)g(spins)h(in)e(two)i(disjoint)f(subsets)g Fw(\001)f Fp(r)-5 b(esp.)34 b Fw(\001)2421 5132 y Fq(0)2476 5165 y Fp(of)f Fw(\003)p Fp(.)42 b(Then)948 5392 y Fv(j)p Ft(\026)1028 5348 y Fu(\021)1028 5419 y Fr(\003)1081 5392 y Fw(\()p Ft(f)10 b Fw(;)15 b Ft(g)s Fw(\))p Fv(j)26 b(\024)f Ft(\016)s Fw(\()p Ft(f)10 b Fw(\))15 b Ft(\016)s Fw(\()p Ft(g)s Fw(\))29 b Ft(\026)1865 5348 y Fu(\021)1865 5419 y Fr(\003)1938 5392 y Fv(\002)20 b Ft(\026)2084 5348 y Fu(\021)2084 5419 y Fr(\003)2137 5392 y Fw(\(\001)2329 5336 y Fq(6)p Fr(=)2273 5392 y Fv( )-15 b(!)25 b Fw(\001)2541 5354 y Fq(0)2597 5392 y Fp(in)32 b Fw(\003\))165 5596 y Fp(wher)-5 b(e)34 b Ft(\016)s Fw(\()p Ft(f)10 b Fw(\))26 b(=)f(max)881 5611 y Fu(\030)934 5596 y Ft(f)10 b Fw(\()p Ft(\030)t Fw(\))20 b Fv(\000)g Fw(min)1366 5611 y Fu(\030)1419 5596 y Ft(f)10 b Fw(\()p Ft(\030)t Fw(\))32 b Fp(is)h(the)g(total)h (oscil)5 b(lation)34 b(of)f Ft(f)10 b Fp(.)1844 5895 y Fw(71)p eop %%Page: 72 72 72 71 bop 165 304 a Fx(Pro)s(of:)87 b Fw(By)29 b(rescaling)f(and)g (addition)f(of)i(suitable)f(constan)m(ts)i(w)m(e)f(can)g(assume)g(that) h(0)25 b Fv(\024)g Ft(f)35 b Fv(\024)25 b Fw(1)165 417 y(and)30 b(0)c Fv(\024)f Ft(g)k Fv(\024)c Fw(1.)41 b(Prop)s(osition)28 b(7.10)k(then)e(sho)m(ws)g(that)721 646 y Fv(j)p Ft(\026)801 602 y Fu(\021)801 674 y Fr(\003)854 646 y Fw(\()p Ft(f)10 b Fw(;)15 b Ft(g)s Fw(\))p Fv(j)27 b(\024)1213 531 y Fs(Z)1311 646 y Ft(\026)1366 602 y Fu(\021)1366 674 y Fr(\003)1419 646 y Fw(\()p Ft(d\030)t Fw(\))1595 531 y Fs(Z)1694 646 y Ft(\026)1749 602 y Fu(\021)1749 674 y Fr(\003)1802 646 y Fw(\()p Ft(d\030)1928 609 y Fq(0)1952 646 y Fw(\))f Ft(g)s Fw(\()p Ft(\030)2138 609 y Fq(0)2162 646 y Fw(\))2212 524 y Fs(\014)2212 574 y(\014)2212 624 y(\014)2212 673 y(\014)2240 646 y Ft(\026)2295 602 y Fu(\030)2295 678 y Fr(\003)p Fq(n)p Fr(\001)2438 659 y Fm(0)2464 646 y Fw(\()p Ft(f)10 b Fw(\))20 b Fv(\000)g Ft(\026)2755 602 y Fu(\030)2789 579 y Fm(0)2755 678 y Fr(\003)p Fq(n)p Fr(\001)2898 659 y Fm(0)2925 646 y Fw(\()p Ft(f)10 b Fw(\))3050 524 y Fs(\014)3050 574 y(\014)3050 624 y(\014)3050 673 y(\014)721 871 y Fv(\024)25 b Ft(\016)s Fw(\()p Ft(f)10 b Fw(\))15 b Ft(\016)s Fw(\()p Ft(g)s Fw(\))1203 756 y Fs(Z)1301 871 y Ft(\026)1356 827 y Fu(\021)1356 899 y Fr(\003)1409 871 y Fw(\()p Ft(d\030)t Fw(\))1585 756 y Fs(Z)1684 871 y Ft(\026)1739 827 y Fu(\021)1739 899 y Fr(\003)1792 871 y Fw(\()p Ft(d\030)1918 834 y Fq(0)1942 871 y Fw(\))25 b Ft(\026)2057 827 y Fu(\030)2057 903 y Fr(\003)p Fq(n)p Fr(\001)2200 884 y Fm(0)2247 871 y Fv(\002)20 b Ft(\026)2393 827 y Fu(\030)2427 804 y Fm(0)2393 903 y Fr(\003)p Fq(n)p Fr(\001)2536 884 y Fm(0)2562 871 y Fw(\(\001)2754 816 y Fq(6)p Fr(=)2699 871 y Fv( )-15 b(!)24 b Fw(\001)2966 834 y Fq(0)3020 871 y Fw(in)29 b(\003\))165 1100 y(b)s(ecause)h Ft(\030)f Fv(\021)c Ft(\030)710 1067 y Fq(0)759 1100 y Fv(\021)g Ft(\021)32 b Fw(on)e Ft(@)5 b Fw(\003.)41 b(By)30 b(carrying)e(out)i(the)g(last)f (in)m(tegration)h(w)m(e)g(obtain)f(the)h(result.)39 b Fk(2)165 1259 y Fw(The)30 b(b)s(ounds)e(ab)s(o)m(v)m(e)j(lea)m(v)m(e)h (us)d(with)g(the)h(task)h(of)f(estimating)g(the)g(probabilit)m(y)d(of)k (disagreemen)m(t)165 1372 y(paths)c(in)f(a)i(duplicated)d(system.)40 b(In)26 b(con)m(trast)j(to)f(the)g(situation)e(in)g(Section)h(7.1,)i(w) m(e)f(are)f(lo)s(oking)165 1485 y(no)m(w)34 b(for)f(estimates)h(v)-5 b(alid)32 b(at)i(lo)m(w)f(temp)s(eratures.)50 b(If)33 b(a)h(cluster)e(expansion)h(w)m(orks,)h(there)g(is)e(no)165 1598 y(need)22 b(to)g(lo)s(ok)f(an)m(y)h(further.)37 b(F)-8 b(or)22 b(instance,)h(a)g(lo)m(w)e(temp)s(erature)g(analysis)g (and)g(estimates)h(of)g(semi-)165 1711 y(in)m(v)-5 b(arian)m(ts)30 b(for)h(the)h(Ising)d(mo)s(del)h(can)i(b)s(e)e(obtained)g(using)g (standard)g(con)m(tour)i(represen)m(tations;)165 1824 y(see)h([67)q(].)47 b(It)32 b(needs)g(to)h(b)s(e)f(emphasized,)f(ho)m (w)m(ev)m(er,)k(that)e(the)f(main)f(step)h(of)h(cluster)e(expansions) 165 1937 y(consists)44 b(in)e(expanding)g(the)i(logarithm)f(of)h(the)g (partition)f(function.)80 b(Only)42 b(afterw)m(ards,)47 b(b)m(y)165 2050 y(taking)42 b(ratios)f(of)h(partition)e(functions,)j (do)s(es)f(one)g(obtain)f(expressions)f(for)h(co)m(v)-5 b(ariances)43 b(and)165 2163 y(higher)33 b(order)h(correlation)g (functions.)50 b(Therefore,)36 b(a)e(p)s(oin)m(t)f(to)i(appreciate)g (is)e(that)i(the)f(b)s(ound)165 2276 y(of)39 b(Corollary)f(7.11)i(pro)m (vides)d(a)j(direct)e(geometric)h(b)s(ound)e(on)i(co)m(v)-5 b(ariances)39 b(whic)m(h)f(a)m(v)m(oids)h(the)165 2389 y(mac)m(hinery)e(of)g(cluster)f(expansions)g(and,)i(in)e(particular,)h (the)h(problems)d(coming)i(from)f(taking)165 2501 y(logarithms.)42 b(As)31 b(a)g(consequence,)h(this)e(estimate)i(also)e(applies)g(to)h (some)h(cases)f(where)g(standard)165 2614 y(cluster)f(expansions)f(are) i(do)s(omed)f(to)h(fail.)306 2727 y(Let)36 b(us)e(illustrate)f(this)h (for)g(the)h(case)h(of)f(the)g(Ising)f(ferromagnet.)54 b(This)33 b(migh)m(t)i(not)g(b)s(e)f(the)165 2840 y(b)s(est)c(example)g (b)s(ecause)g(other)g(metho)s(ds)f(can)h(also)g(b)s(e)g(applied)d(to)k (it;)f(nev)m(ertheless,)g(it)g(is)f(useful)165 2953 y(to)41 b(demonstrate)f(the)g(tec)m(hnique)g(in)e(this)h(simple)e(case.)70 b(Afterw)m(ards)40 b(w)m(e)g(will)d(discuss)h(a)i(case)165 3066 y(where)30 b(cluster)g(expansion)f(tec)m(hniques)h(cannot)h(b)s(e) f(used)f(equally)h(easily)-8 b(.)306 3179 y(Consider)36 b(the)h(lo)m(w)g(temp)s(erature)f(plus)f(phase)i Ft(\026)2065 3141 y Fr(+)2065 3207 y Fu(\014)2161 3179 y Fw(of)g(the)g (ferromagnetic)h(Ising)d(mo)s(del)h(on)165 3304 y(the)31 b(square)f(lattice)h Fv(L)25 b Fw(=)h Fx(Z)1135 3271 y Fr(2)1204 3304 y Fw(with)k(zero)h(magnetic)g(\014eld.)40 b(By)31 b(taking)f(the)h(in\014nite)d(v)m(olume)i(limit)165 3417 y(in)f(Corollary)g(7.11)j(with)d Ft(\021)g Fv(\021)c Fw(+1)31 b(w)m(e)f(obtain)g(the)h(estimate)1055 3634 y Fv(j)p Ft(\026)1135 3596 y Fr(+)1135 3662 y Fu(\014)1194 3634 y Fw(\()p Ft(f)10 b Fw(;)15 b Ft(g)s Fw(\))p Fv(j)27 b(\024)e Ft(\016)s Fw(\()p Ft(f)10 b Fw(\))15 b Ft(\016)s Fw(\()p Ft(g)s Fw(\))28 b Ft(\026)1978 3596 y Fr(+)1978 3662 y Fu(\014)2057 3634 y Fv(\002)20 b Ft(\026)2203 3596 y Fr(+)2203 3662 y Fu(\014)2262 3634 y Fw(\(\001)2454 3578 y Fq(6)p Fr(=)2398 3634 y Fv( )-15 b(!)25 b Fw(\001)2666 3596 y Fq(0)2689 3634 y Fw(\))729 b(\(54\))165 3828 y(for)35 b(the)g(co)m(v)-5 b(ariance)36 b(of)f(an)m(y)g(t)m(w)m(o)h(lo)s(cal)e (functions)f Ft(f)5 b(;)15 b(g)38 b Fw(with)c(disjoin)m(t)f(dep)s (endence)h(sets)h(\001)p Ft(;)15 b Fw(\001)3566 3795 y Fq(0)3589 3828 y Fw(.)165 3964 y(Next)29 b(w)m(e)g(observ)m(e)f(that) h(the)f(ev)m(en)m(t)i Fv(f)p Fw(\001)1627 3908 y Fq(6)p Fr(=)1571 3964 y Fv( )-15 b(!)25 b Fw(\001)1839 3931 y Fq(0)1862 3964 y Fv(g)j Fw(is)f(clearly)h(con)m(tained)g(in)f(the)h (ev)m(en)m(t)h(that)g(there)165 4077 y(exists)39 b(a)h(path)f(from)g (\001)g(to)h(\001)1266 4044 y Fq(0)1328 4077 y Fw(along)g(whic)m(h)e (\()p Ft(X)r(;)15 b(X)2083 4044 y Fq(0)2107 4077 y Fw(\))41 b Fv(6)p Fw(=)f(\(+)p Ft(;)15 b Fw(+\).)68 b(The)39 b(latter)g(ev)m(en) m(t)i(will)c(b)s(e)165 4208 y(denoted)h(b)m(y)f Fv(f)p Fw(\001)806 4153 y Fm(6)p Fl(=)18 b(\(+)p Fo(;)10 b Fl(+\))851 4208 y Fv( )-15 b(!)81 b Fw(\001)1175 4175 y Fq(0)1198 4208 y Fv(g)p Fw(.)61 b(No)m(w,)40 b(Burton)d(and)f(Steif)h([37)q(])g (ha)m(v)m(e)i(sho)m(wn)d(ho)m(w)h(to)h(estimate)165 4321 y(the)f(probabilit)m(y)d(of)j(this)e(ev)m(en)m(t)j(to)g(o)s(ccur)e(in)f (the)i(duplicated)d(plus)h(phase.)59 b(The)36 b(result)f(is)h(the)165 4434 y(follo)m(wing.)165 4613 y Fx(Theorem)e(7.12)47 b Fp(F)-7 b(or)46 b(the)g(Ising)g(ferr)-5 b(omagnet)47 b(on)f Fx(Z)2152 4580 y Fr(2)2237 4613 y Fp(at)g(su\016ciently)f(lar)-5 b(ge)47 b Ft(\014)5 b Fp(,)49 b(ther)-5 b(e)46 b(exist)165 4726 y(c)-5 b(onstants)47 b Ft(c)h(>)f Fw(0)e Fp(and)h Ft(C)53 b(<)47 b Fv(1)e Fp(\(dep)-5 b(ending)46 b(on)f Ft(\014)5 b Fp(\))46 b(such)e(that)j(for)e(any)g(two)h(disjoint)g(sets) 165 4839 y Fw(\001)p Ft(;)15 b Fw(\001)357 4806 y Fq(0)406 4839 y Fv(2)25 b(E)8 b Fp(,)760 5047 y Ft(\026)815 5009 y Fr(+)815 5075 y Fu(\014)894 5047 y Fv(\002)20 b Ft(\026)1040 5009 y Fr(+)1040 5075 y Fu(\014)1099 5047 y Fw(\(\001)1235 4992 y Fm(6)p Fl(=)d(\(+)p Fo(;)11 b Fl(+\))1269 5047 y Fv( )-15 b(!)59 b Fw(\001)1571 5009 y Fq(0)1594 5047 y Fw(\))26 b Fv(\024)f Ft(C)d Fw(min)n Fv(fj)p Ft(@)2107 5061 y Fu(i)2136 5047 y Fw(\001)p Fv(j)p Ft(;)15 b Fv(j)p Ft(@)2350 5061 y Fu(i)2379 5047 y Fw(\001)2455 5009 y Fq(0)2478 5047 y Fv(jg)26 b Ft(e)2616 5009 y Fq(\000)p Fu(c)12 b(d)p Fr(\(\001)p Fu(;)p Fr(\001)2915 4986 y Fm(0)2937 5009 y Fr(\))2994 5047 y Ft(;)165 5241 y Fp(wher)-5 b(e)34 b Ft(@)470 5255 y Fu(i)499 5241 y Fw(\001)25 b(=)f Ft(@)5 b Fw(\(\001)859 5208 y Fu(c)895 5241 y Fw(\))32 b Fp(is)h(the)g(inner)g(b)-5 b(oundary)34 b(of)f Fw(\001)p Fp(.)165 5420 y Fw(Com)m(bining)f(this)h(theorem)h(with)e(\(54\))k(w)m (e)e(obtain)f(an)h(exp)s(onen)m(tial)f(b)s(ound)e(for)j(the)g(co)m(v)-5 b(ariance)165 5533 y(of)31 b(an)m(y)f(t)m(w)m(o)i(lo)s(cal)d(observ)-5 b(ables)29 b(in)g(the)i(lo)m(w)f(temp)s(erature)g(Ising)f(plus)f (phase.)40 b(While)29 b(this)g(result)165 5646 y(is)h(w)m(ell-kno)m (wn,)f(its)h(pro)s(of)g(b)s(elo)m(w)f(sho)m(ws)h(ho)m(w)h(one)f(can)h (pro)s(ceed)f(in)f(more)i(general)f(cases.)1844 5895 y(72)p eop %%Page: 73 73 73 72 bop 165 304 a Fx(Sk)m(etc)m(h)51 b(pro)s(of)f(of)h(Theorem)e (7.12:)67 b Fw(It)44 b(is)e(su\016cien)m(t)h(to)i(pro)m(v)m(e)f(the)g (statemen)m(t)h(with)d Ft(\026)3555 266 y Fr(+)3555 332 y Fu(\014)165 432 y Fw(replaced)j(b)m(y)f Ft(\026)734 393 y Fr(+)734 460 y Fu(\014)s(;)p Fr(\003)850 432 y Fw(,)49 b(where)44 b(\003)h(is)f(a)h(su\016cien)m(tly)e(large)i(square) g(b)s(o)m(x)g(con)m(taining)f(\001)p Ft(;)15 b Fw(\001)3349 399 y Fq(0)3372 432 y Fw(.)84 b(F)-8 b(or)165 559 y(brevit)m(y)g(,)37 b(let)f Ft(P)695 573 y Fr(\003)782 559 y Fw(=)e Ft(\026)942 521 y Fr(+)942 587 y Fu(\014)s(;)p Fr(\003)1081 559 y Fv(\002)24 b Ft(\026)1231 521 y Fr(+)1231 587 y Fu(\014)s(;)p Fr(\003)1346 559 y Fw(.)57 b(Supp)s(ose)33 b(that)k Fv(j)p Ft(@)5 b Fw(\001)2146 526 y Fq(0)2169 559 y Fv(j)35 b(\024)e(j)p Ft(@)5 b Fw(\001)p Fv(j)p Fw(,)38 b(\014x)d(an)g(arbitrary)g Ft(x)f Fv(2)f Ft(@)3461 573 y Fu(i)3490 559 y Fw(\001)3566 526 y Fq(0)3589 559 y Fw(,)165 709 y(and)g(supp)s(ose)g(that)h(the)g (ev)m(en)m(t)h Fv(f)p Ft(x)1422 654 y Fm(6)p Fl(=)17 b(\(+)p Fo(;)10 b Fl(+\))1456 709 y Fv( )-15 b(!)64 b Fw(\001)p Fv(g)34 b Fw(o)s(ccurs.)50 b(Let)34 b(\000)2391 723 y Fr(0)2461 709 y Fw(=)c(\000)2619 723 y Fr(0)2659 709 y Fw(\()p Ft(X)r(;)15 b(X)2893 676 y Fq(0)2918 709 y Fw(\))33 b(b)s(e)g(the)h(maximal)165 822 y(connected)h(set)e(con)m (taining)g Ft(x)g Fw(on)h(whic)m(h)e(\()p Ft(X)r(;)15 b(X)1894 789 y Fq(0)1918 822 y Fw(\))31 b Fv(6)p Fw(=)f(\(+)p Ft(;)15 b Fw(+\).)50 b(Also,)34 b(let)f(\000)g(b)s(e)g(the)g(union)e (of)j(\000)3575 836 y Fr(0)165 935 y Fw(and)j(all)f(\014nite)g(comp)s (onen)m(ts)i(of)f(\000)1396 949 y Fr(0)1435 899 y Fu(c)1470 935 y Fw(.)61 b(\000)37 b(is)f(enclosing)h(in)f(the)h(sense)g(that)h(b) s(oth)e(\000)h(and)g(\000)3421 902 y Fu(c)3493 935 y Fw(are)165 1048 y(connected.)43 b(In)30 b(fact,)i(\000)f(can)g(b)s(e)f (iden)m(ti\014ed)f(with)g(a)i(con)m(tour,)h(the)f(brok)m(en)g(line)e (whic)m(h)g(separates)165 1161 y(\000)h(from)g(its)g(complemen)m(t.)306 1274 y(Consider)35 b(the)h(set)h Fv(C)k Fw(of)36 b(all)f(enclosing)g (sets)i(con)m(taining)e Ft(x)h Fw(and)g(con)m(tained)g(in)f(\003.)58 b(F)-8 b(or)37 b(an)m(y)165 1387 y(in)m(teger)i Ft(`)f Fv(\025)f Fw(2,)k(let)d Fv(C)956 1402 y Fu(`)1026 1387 y Fw(b)s(e)g(the)g(set)g(of)h(all)e Ft(C)44 b Fv(2)37 b(C)44 b Fw(suc)m(h)37 b(that)i Fv(j)p Ft(@)2506 1401 y Fu(i)2535 1387 y Ft(C)7 b Fv(j)37 b Fw(=)h Ft(`)p Fw(.)63 b(Since)37 b(the)h(n)m(um)m(b)s(er)165 1500 y(of)33 b(con)m(tours)g (with)f(length)g Ft(`)g Fw(surrounding)d(a)k(giv)m(en)g(site)f(on)h (the)g(square)f(lattice)h(is)f(b)s(ounded)e(b)m(y)165 1613 y Ft(`)p Fw(3)248 1580 y Fu(`)315 1613 y Fw(and)i(since)g(eac)m(h) j(enclosing)d(set)h(with)f Fv(j)p Ft(@)1746 1627 y Fu(i)1775 1613 y Ft(C)7 b Fv(j)29 b Fw(=)h Ft(`)i Fw(uniquely)f(de\014nes)h(a)h (con)m(tour)h(with)e(length)165 1726 y(b)s(et)m(w)m(een)k Ft(`)f Fw(and)g(4)p Ft(`)p Fw(,)i(w)m(e)e(ha)m(v)m(e)i(that)f Fv(jC)1549 1741 y Fu(`)1582 1726 y Fv(j)d(\024)g Fw(3\(4)p Ft(`)25 b Fw(+)e(1\))p Ft(=)p Fw(2\(81\))2356 1693 y Fu(`)2428 1726 y Fw(gro)m(wing)35 b(exp)s(onen)m(tially)e(with)h Ft(`)p Fw(.)165 1838 y(No)m(w)d(w)m(e)g(can)g(write)656 2035 y Ft(P)714 2049 y Fr(\003)768 2035 y Fw(\()p Ft(x)880 1980 y Fm(6)p Fl(=)18 b(\(+)p Fo(;)10 b Fl(+\))915 2035 y Fv( )-15 b(!)59 b Fw(\001\))25 b(=)g Ft(P)1431 2049 y Fr(\003)1484 2035 y Fw(\(\000)c Fv(\\)f Fw(\001)25 b Fv(6)p Fw(=)f Fv(;)p Fw(\))j Fv(\024)2184 1954 y Fs(X)2076 2142 y Fu(`)p Fq(\025)p Fu(d)p Fr(\(\001)p Fu(;)p Fr(\001)2361 2123 y Fm(0)2384 2142 y Fr(\))2481 1954 y Fs(X)2456 2138 y Fu(C)5 b Fq(2C)2595 2150 y Fo(`)2641 2035 y Ft(P)2699 2049 y Fr(\003)2752 2035 y Fw(\(\000)26 b(=)f Ft(C)7 b Fw(\))25 b Ft(:)165 2322 y Fw(F)-8 b(urthermore,)31 b(for)f Ft(C)i Fv(2)25 b(C)1088 2337 y Fu(`)1151 2322 y Fw(w)m(e)31 b(ha)m(v)m(e)463 2518 y Ft(P)521 2532 y Fr(\003)574 2518 y Fw(\(\000)25 b(=)g Ft(C)7 b Fw(\))83 b Fv(\024)1188 2437 y Fs(X)1131 2622 y Fu(D)r Fq(\032)p Fu(@)1283 2632 y Fo(i)1309 2622 y Fu(C)1379 2518 y Ft(\026)1434 2480 y Fr(+)1434 2546 y Fu(\014)s(;)p Fr(\003)1550 2518 y Fw(\()p Ft(X)33 b Fv(\021)25 b(\000)p Fw(1)30 b(on)h Ft(D)s(;)f(X)j Fv(\021)25 b Fw(+1)30 b(on)g Ft(@)5 b(C)i Fw(\))1495 2752 y Fv(\002)15 b Ft(\026)1636 2713 y Fr(+)1636 2780 y Fu(\014)s(;)p Fr(\003)1751 2752 y Fw(\()p Ft(X)33 b Fv(\021)25 b(\000)p Fw(1)30 b(on)h Ft(@)2311 2766 y Fu(i)2339 2752 y Ft(C)c Fv(n)21 b Ft(D)s(;)30 b(X)j Fv(\021)25 b Fw(+1)30 b(on)g Ft(@)5 b(C)i Fw(\))26 b Ft(:)136 b Fw(\(55\))165 2948 y(No)m(w,)32 b(a)e(standard)g(P)m(eierls)f(estimate) j(\(see)f(e.g.)42 b([213)q(]\))31 b(sho)m(ws)f(that)829 3144 y Ft(\026)884 3106 y Fr(+)884 3172 y Fu(\014)s(;)p Fr(\003)999 3144 y Fw(\()p Ft(X)j Fv(\021)25 b(\000)p Fw(1)30 b(on)h Ft(D)s(;)15 b(X)33 b Fv(\021)25 b Fw(+1)30 b(on)g Ft(@)5 b(C)i Fw(\))26 b Fv(\024)f Ft(C)7 b Fw(\()p Ft(\014)e Fw(\))25 b Ft(e)2652 3107 y Fq(\000)p Fu(c)p Fr(\()p Fu(\014)s Fr(\))13 b Fq(j)p Fu(D)r Fq(j)3453 3144 y Fw(\(56\))165 3340 y(with)36 b(constan)m(ts)h Ft(c)p Fw(\()p Ft(\014)5 b Fw(\))p Ft(;)15 b(C)7 b Fw(\()p Ft(\014)e Fw(\))39 b(indep)s(enden)m(t)34 b(of)i(\003)h(satisfying)e Ft(c)p Fw(\()p Ft(\014)5 b Fw(\))37 b Fv(!)e(1)h Fw(as)h Ft(\014)k Fv(!)35 b(1)p Fw(.)59 b(Substi-)165 3453 y(tuting)31 b(\(56\))i(in)m(to)f(\(55\))h(w)m(e)f(obtain)f(the)h(theorem)g(b)m(y)f (simple)f(com)m(binatorics)h(and)g(summations.)165 3566 y Fk(2)165 3726 y Fw(W)-8 b(e)29 b(emphasize)d(that)i(the)f(sp)s (eci\014c)f(prop)s(erties)f(of)j(the)f(plus)e(phase)i Ft(\026)2582 3688 y Fr(+)2582 3754 y Fu(\014)2667 3726 y Fw(are)h(used)e(only)g(in)g(the)h(last)165 3839 y(step,)i(the)f(P)m (eierls)f(estimate)i(\(56\).)42 b(Before,)29 b(w)m(e)g(needed)f(only)f (the)h(Mark)m(o)m(v)i(prop)s(ert)m(y)-8 b(.)39 b(Therefore)165 3952 y(it)27 b(is)e(useful)g(to)j(note)f(that)g(the)g(P)m(eierls)f (estimate)h(is)f(not)h(limited)d(to)k(the)f(Ising)e(mo)s(del;)i(it)f (remains)165 4065 y(v)-5 b(alid)29 b(under)h(the)h(conditions)e(of)i (the)g(standard)f(Pirogo)m(v{Sinai)f(theory)i([213)r(].)42 b(In)30 b(particular,)g(it)165 4178 y(follo)m(ws)k(that)g(the)h (results)e(of)h(Burton)g(and)g(Steif)f([37)q(])i(on)f(the)g(ergo)s(dic) g(prop)s(erties)f(of)h(the)g(Ising)165 4291 y(mo)s(del)c(carry)g(o)m(v) m(er)i(to)f(more)f(general)h(Mark)m(o)m(vian)g(mo)s(dels)e(of)h(Pirogo) m(v{Sinai)g(t)m(yp)s(e.)306 4404 y(Let)43 b(us)f(\014nally)e(discuss)h (a)i(case)g(in)e(whic)m(h)g(a)i(P)m(eierls)e(estimate)j(of)e(the)h (form)f(\(56\))h(is)f(not)165 4517 y(a)m(v)-5 b(ailable.)40 b(Namely)-8 b(,)31 b(w)m(e)f(ask)g(for)g(co)m(v)-5 b(ariance)31 b(estimates)f(of)g(lo)s(cal)f(functions,)g(still)e(for)j(the)g(ferro-) 165 4630 y(magnetic)i(Ising)f(mo)s(del)f(in)h(a)h(large)f(square)h (\003,)g(but)f(no)m(w)h(for)f(some)h(b)s(oundary)e(condition)g Ft(\021)35 b Fw(not)165 4742 y(iden)m(tically)26 b(equal)g(to)i(+1.)40 b(This)26 b(question)g(arises,)i(for)f(example,)h(in)e(the)h(con)m (text)i(of)f(correlations)165 4855 y(atop)37 b(of)e(a)h(disordered)d (surface,)k(or)f(in)e(the)i(problem)e(of)h(establishing)e(a)j(Gibbsian) d(description)165 4968 y(of)i(non-Gibbsian)d(measures.)52 b(In)34 b(fact,)j(in)c([167)q(])i(the)g(metho)s(d)f(to)h(b)s(e)e (describ)s(ed)g(b)s(elo)m(w)g(is)h(used)165 5081 y(to)40 b(pro)m(v)m(e)g(that)f(the)h(pro)5 b(jection)38 b(to)i(a)f(line)e(of)j (the)f(lo)m(w-temp)s(erature)g(plus)e(phase)h(of)h(the)g(t)m(w)m(o-)165 5194 y(dimensional)28 b(Ising)h(mo)s(del)g(is)h(w)m(eakly)g(Gibbsian.) 306 5307 y(T)-8 b(o)31 b(b)s(e)f(sp)s(eci\014c,)f(supp)s(ose)g(that)i (the)f(b)s(oundary)f(condition)g Ft(\021)k Fw(on)d Ft(@)5 b Fw(\003)31 b(is)e(not)i(iden)m(tically)d(plus)165 5420 y(but)f(con)m(tains)g(a)h(large)f(prop)s(ortion)f(of)h(plus)e(spins;)i (w)m(e)g(stipulate)g(that)g Ft(\021)i Fv(\021)c Fw(+1)i(on)g(three)h (sides)e(of)165 5533 y(\003)g(while)d(on)j(the)g(remaining)d(side)i (only)f(a)i(large)g(fraction)f(of)h(the)f(spins)f(is)g(plus.)38 b(In)24 b(this)h(case,)j(\(56\))165 5646 y(cannot)h(b)s(e)e(true)h(b)s (ecause)g Ft(D)j Fw(can)d(b)s(e)f(small)g(and)g(close)h(to)h(the)f(b)s (oundary)e(of)i(\003.)40 b(F)-8 b(or)29 b(example,)f(if)1844 5895 y(73)p eop %%Page: 74 74 74 73 bop 165 304 a Ft(E)31 b Fw(=)25 b Ft(@)5 b(D)22 b Fv(\\)c Ft(@)5 b Fw(\003)25 b Fv(6)p Fw(=)g Fv(;)30 b Fw(and)f Ft(\021)k Fw(happ)s(ens)28 b(to)i(b)s(e)f(min)m(us)f(on)h Ft(E)35 b Fw(then)30 b(it)f(is)f(not)i(v)m(ery)g(unlik)m(ely)d(that)j (all)165 417 y(spins)h(in)g Ft(D)36 b Fw(are)d(min)m(us,)f(and)g (\(56\))i(will)c(not)j(hold.)47 b(Ho)m(w)m(ev)m(er,)35 b(one)e(can)g(tak)m(e)h(adv)-5 b(an)m(tage)35 b(of)e(the)165 530 y(fact)e(that)e(for)g(small)f Ft(D)k Fw(its)d(complemen)m(t)g(in)f Ft(@)1806 544 y Fu(i)1835 530 y Ft(C)35 b Fw(is)28 b(large,)i(and)f (vice)g(v)m(ersa.)41 b(In)29 b(other)g(w)m(ords,)g(to)165 643 y(estimate)e(the)g(righ)m(t-hand)e(side)g(of)h(\(55\))i(one)e (should)e(not)j(apply)e(\(56\))i(separately)g(to)g(eac)m(h)g(factor,) 165 756 y(but)j(rather)g(one)h(can)g(hop)s(e)e(to)i(estimate)h(their)d (pro)s(duct.)306 869 y(T)-8 b(o)26 b(mak)m(e)h(these)f(general)f (remarks)g(precise)g(w)m(e)h(consider)e(the)h(Ising)g(mo)s(del)f(on)h (the)h(half-plane)165 981 y Fx(Z)11 b Fv(\002)g Fx(Z)386 995 y Fr(+)446 981 y Fw(.)39 b(F)-8 b(or)27 b(an)m(y)f Ft(n)g Fw(w)m(e)g(consider)f(the)h(square)g(\003)1893 995 y Fu(n)1965 981 y Fw(=)f Fv(f)p Fw(\()p Ft(x)2193 995 y Fr(1)2234 981 y Ft(;)15 b(x)2326 995 y Fr(2)2365 981 y Fw(\))26 b Fv(2)f Fx(Z)2576 948 y Fr(2)2640 981 y Fw(:)h Fv(\000)p Ft(n)e Fv(\024)h Ft(x)2989 995 y Fr(1)3054 981 y Fv(\024)g Ft(n;)30 b Fw(0)c Ft(<)f(x)3479 995 y Fr(2)3543 981 y Fv(\024)165 1094 y Ft(n)p Fv(g)c Fw(touc)m(hing)g(the)g (b)s(oundary)e(line)g Fx(Z)q Fv(\002)q(f)p Fw(0)p Fv(g)p Fw(.)39 b(On)20 b(this)g(line)f(w)m(e)i(\014x)g(a)g(con\014guration)f Ft(\030)30 b Fv(2)24 b(f)p Fw(+1)p Ft(;)15 b Fv(\000)p Fw(1)p Fv(g)3533 1061 y Fn(Z)3589 1094 y Fw(,)165 1207 y(thereb)m(y)29 b(de\014ning)d(a)i(b)s(oundary)e(condition)h(on)h(one)g (part)g(of)h Ft(@)5 b Fw(\003.)40 b(On)27 b(the)h(remaining)f(part)g (of)i Ft(@)5 b Fw(\003)165 1320 y(w)m(e)27 b(imp)s(ose)e(plus)g(b)s (oundary)f(condition.)38 b(That)27 b(is,)f(w)m(e)h(c)m(ho)s(ose)g(the)g (b)s(oundary)d(condition)h Ft(\021)k Fv(\021)c Fw(+1)165 1433 y(on)k Ft(@)5 b Fw(\003)406 1447 y Fu(n)469 1433 y Fv(n)16 b Fw(\()p Fx(Z)h Fv(\002)e(f)p Fw(0)p Fv(g)p Fw(\))30 b(and)e Ft(\021)g Fv(\021)d Ft(\030)33 b Fw(on)28 b Fx(Z)16 b Fv(\002)f(f)p Fw(0)p Fv(g)p Fw(.)42 b(W)-8 b(e)29 b(ask)f(for)g(the)h(correlation)f(of)g(the)h(spins)d(at)j(the) 165 1546 y(sites)h Ft(x)c Fw(=)f(\(0)p Ft(;)15 b Fw(1\))32 b(and)e Ft(y)e Fw(=)d(\()p Ft(k)s(;)15 b Fw(1\))32 b(with)d(0)d Ft(<)e Fv(j)p Ft(k)s Fv(j)i Ft(<)f(n)p Fw(.)165 1727 y Fx(Theorem)34 b(7.13)47 b Fp(In)32 b(the)h(situation)h(just)f (describ)-5 b(e)g(d,)33 b(supp)-5 b(ose)34 b Ft(\030)j Fp(is)c(such)f(that)961 1887 y Fu(m)932 1912 y Fs(X)931 2095 y Fu(j)t Fr(=0)1068 1993 y Ft(\030)t Fw(\()p Ft(j;)15 b Fw(0\))28 b Fv(\025)d Fw(8)p Ft(m=)p Fw(9)126 b Fp(and)2111 1887 y Fq(\000)p Fr(1)2096 1912 y Fs(X)2054 2095 y Fu(j)t Fr(=)p Fq(\000)p Fu(m)2274 1993 y Ft(\030)t Fw(\()p Ft(j;)15 b Fw(0\))27 b Fv(\025)e Fw(8)p Ft(m=)p Fw(9)165 2270 y Fp(for)36 b(su\016ciently)g(lar)-5 b(ge)36 b Ft(m)p Fp(,)g(and)g(let)g Ft(\021)j Fp(b)-5 b(e)35 b(de\014ne)-5 b(d)36 b(as)g(ab)-5 b(ove.)51 b(Then)36 b(ther)-5 b(e)36 b(ar)-5 b(e)37 b(c)-5 b(onstants)37 b Ft(c)31 b(>)f Fw(0)165 2383 y Fp(and)k Ft(C)e(<)25 b Fv(1)32 b Fp(\(not)i(dep)-5 b(ending)33 b(on)g Ft(n)p Fp(\))g(such)f(that)1201 2580 y Fv(j)p Ft(\026)1281 2535 y Fu(\021)1281 2608 y(\014)s(;)p Fr(\003)1393 2616 y Fo(n)1439 2580 y Fw(\()p Ft(X)7 b Fw(\(0)p Ft(;)15 b Fw(1\);)g Ft(X)7 b Fw(\()p Ft(k)s(;)15 b Fw(1\)\))p Fv(j)32 b(\024)24 b Ft(C)e(e)2399 2542 y Fq(\000)p Fu(c)12 b Fq(j)p Fu(k)r Fq(j)165 2776 y Fp(whenever)33 b Ft(\014)38 b Fp(and)c Fv(j)p Ft(k)s Fv(j)f Fp(ar)-5 b(e)33 b(su\016ciently)g(lar)-5 b(ge)34 b(and)f Ft(n)25 b(>)g Fv(j)p Ft(k)s Fv(j)p Fp(.)165 2957 y Fx(Sk)m(etc)m(h)46 b(pro)s(of:)127 b Fw(W)-8 b(e)41 b(pro)s(ceed)e(as)h(in)e(the)h(pro)s (of)g(of)h(Theorem)f(7.12.)70 b(In)38 b(dealing)h(with)f(the)165 3070 y(righ)m(t-hand)i(side)g(of)i(\(55\))g(w)m(e)g(m)m(ust)f(tak)m(e)i (in)m(to)e(accoun)m(t)h(that)g(p)s(ossibly)c Ft(@)5 b(C)34 b Fv(\\)27 b Ft(@)5 b Fw(\003)44 b Fv(6)p Fw(=)f Fv(;)p Fw(.)73 b(W)-8 b(e)165 3182 y(therefore)27 b(replace)f Ft(@)5 b(C)33 b Fw(b)m(y)26 b Ft(@)5 b(C)19 b Fv(n)12 b Ft(@)5 b Fw(\003)26 b(in)f(the)h(pro)s(duct)f(term)i(and)e(also)h (estimate)h(the)g(probabilities)165 3295 y(of)k(in)m(tersections)f(b)m (y)g(conditional)f(probabilities,)e(yielding)h(the)j(upp)s(er)d(b)s (ound)959 3492 y Ft(\026)1014 3447 y Fr(+)p Fu(;\021)1014 3520 y(\014)s(;C)1135 3492 y Fw(\()p Ft(X)33 b Fv(\021)25 b(\000)p Fw(1)31 b(on)f Ft(D)s Fw(\))c Ft(\026)1841 3447 y Fr(+)p Fu(;\021)1841 3520 y(\014)s(;C)1962 3492 y Fw(\()p Ft(X)33 b Fv(\021)25 b(\000)p Fw(1)31 b(on)f Ft(@)2522 3506 y Fu(i)2550 3492 y Ft(C)d Fv(n)21 b Ft(D)s Fw(\))165 3709 y(for)43 b(the)g(summands)d(on)j(the)g(righ)m(t-hand)e(side)h(of)h (\(55\).)79 b(Here,)46 b Ft(\026)2600 3664 y Fr(+)p Fu(;\021)2600 3737 y(\014)s(;C)2764 3709 y Fw(stands)c(for)h(the)g(Gibbs)165 3821 y(distribution)32 b(in)i Ft(C)42 b Fw(with)35 b(b)s(oundary)e (condition)h(equal)h(to)i(+1)e(on)h Ft(@)5 b(C)30 b Fv(\\)23 b Fw(\003)36 b(and)f(equal)g(to)h Ft(\021)j Fw(on)165 3934 y Ft(@)5 b(C)14 b Fv(\\)8 b Ft(@)d Fw(\003.)40 b(T)-8 b(o)24 b(deriv)m(e)g(the)h(theorem)f(w)m(e)h(need)f(to)h(replace)f(the) h(P)m(eierls)e(estimate)i(\(56\))h(b)m(y)e(a)h(similar)165 4047 y(b)s(ound)h(on)i(the)h(last)f(pro)s(duct.)38 b(The)28 b(exp)s(onen)m(tial)f(deca)m(y)i(of)f(correlations)g(then)g(again)g (follo)m(ws)f(b)m(y)165 4160 y(simple)i(com)m(binatorics)h(and)f (summations.)306 4273 y(T)-8 b(o)41 b(mak)m(e)g(the)f(in\015uence)e(of) i(the)g(b)s(oundary)d(condition)i Ft(\021)k Fw(explicit)38 b(w)m(e)i(exploit)f(a)h(con)m(tour)165 4386 y(represen)m(tation)31 b(leading)e(to)i(the)g(estimate)921 4582 y Ft(\026)976 4538 y Fr(+)p Fu(;\021)976 4610 y(\014)s(;C)1098 4582 y Fw(\()p Ft(X)i Fv(\021)25 b(\000)p Fw(1)30 b(on)g Ft(D)s Fw(\))c Ft(\026)1803 4538 y Fr(+)p Fu(;\021)1803 4610 y(\014)s(;C)1924 4582 y Fw(\()p Ft(X)33 b Fv(\021)25 b(\000)p Fw(1)31 b(on)f Ft(@)2484 4596 y Fu(i)2513 4582 y Ft(C)c Fv(n)21 b Ft(D)s Fw(\))1285 4785 y Fv(\024)1692 4704 y Fs(X)1572 4884 y Fl(\000)p Fo(;)p Fl(\000)1667 4870 y Fm(0)1709 4884 y Fl(inside)h Fo(C)1471 4938 y Fl(\000)e(compatible)i(with)f Fo(D)1391 5000 y Fl(\000)1429 4986 y Fm(0)1471 5000 y Fl(compatible)h(with)f Fo(@)1956 5016 y(i)1982 5000 y(C)s Fm(n)p Fo(D)2151 4704 y Fs(Y)2138 4888 y Fu(\015)t Fq(2)p Fr(\000)2284 4785 y Ft(w)2349 4799 y Fu(\021)2391 4785 y Fw(\()p Ft(\015)5 b Fw(\))2564 4704 y Fs(Y)2529 4889 y Fu(\015)2569 4870 y Fm(0)2592 4889 y Fq(2)p Fr(\000)2683 4870 y Fm(0)2721 4785 y Ft(w)2786 4799 y Fu(\021)2828 4785 y Fw(\()p Ft(\015)2915 4747 y Fq(0)2938 4785 y Fw(\))26 b Ft(:)429 b Fw(\(57\))165 5194 y(The)26 b(righ)m(t-hand)f(side)h(is)f(de\014ned)g(as)i(follo)m (ws.)39 b(F)-8 b(or)27 b(an)m(y)f(con\014guration)g Ft(\033)j Fv(2)c(f)p Fw(+1)p Ft(;)15 b Fv(\000)p Fw(1)p Fv(g)3209 5161 y Fr(\003)3290 5194 y Fw(w)m(e)26 b(dra)m(w)165 5307 y(horizon)m(tal)h(resp.)g(v)m(ertical)g(lines)f(of)h(unit)f (length)g(b)s(et)m(w)m(een)i(neigh)m(b)s(oring)d(sites)i(of)g(opp)s (osite)g(spins,)165 5420 y(doing)32 b(as)h(if)f(the)h(b)s(oundary)e (spins)g(w)m(ere)i(all)f(plus;)h(w)m(e)g(then)f(obtain)h(a)g(disjoin)m (t)e(union)g(of)i(closed)165 5533 y(non-self-in)m(tersecting)g(p)s (olygonal)f(curv)m(es.)51 b(Eac)m(h)34 b(of)g(these)g(curv)m(es)f(is)g (called)g(a)h(con)m(tour)g Ft(\015)5 b Fw(,)35 b(and)165 5646 y(a)d(set)f(\000)f(of)h(con)m(tours)h(arising)d(in)g(this)h(w)m(a) m(y)i(is)d(called)h(compatible.)42 b(W)-8 b(e)31 b(th)m(us)g(ha)m(v)m (e)h(a)f(one-to-one)1844 5895 y(74)p eop %%Page: 75 75 75 74 bop 165 304 a Fw(corresp)s(ondence)40 b(b)s(et)m(w)m(een)h(spin)d (con\014gurations)h Ft(\033)k Fw(and)d(compatible)f(sets)h(\000)g(of)g (con)m(tours.)71 b(If)165 417 y Ft(\033)29 b Fv(\021)c(\000)p Fw(1)h(on)g Ft(D)s Fw(,)h(then)f(eac)m(h)i(comp)s(onen)m(t)f(of)f Ft(D)j Fw(is)c(surrounded)f(b)m(y)i(some)h(con)m(tour)g Ft(\015)k Fw(\(i.e.,)d(b)s(elongs)165 530 y(to)34 b(the)e(in)m(terior)g (In)m(t)15 b Ft(\015)37 b Fw(of)c Ft(\015)5 b Fw(\);)34 b(the)f(smallest)e(con)m(tours)i(surrounding)d(the)i(comp)s(onen)m(ts)h (of)g Ft(D)i Fw(are)165 643 y(collected)25 b(in)m(to)g(a)g(set)g(\000)f (of)h(con)m(tours.)39 b(Eac)m(h)26 b(set)f(\000)f(arising)f(in)g(this)h (w)m(a)m(y)h(is)f(called)g Fp(c)-5 b(omp)g(atible)29 b(with)165 756 y Ft(D)s Fw(.)40 b(The)27 b(probabilit)m(y)d(that)k(a)g (giv)m(en)f(set)h(\000)f(of)g(con)m(tours)h(o)s(ccurs)f(is)f(not)i (larger)f(than)3130 691 y Fs(Q)3208 779 y Fu(\015)t Fq(2)p Fr(\000)3359 756 y Ft(w)3424 770 y Fu(\021)3466 756 y Fw(\()p Ft(\015)5 b Fw(\),)165 869 y(where)840 999 y Ft(w)905 1013 y Fu(\021)947 999 y Fw(\()p Ft(\015)g Fw(\))26 b(=)f(exp)1345 880 y Fs(\024)1409 999 y Fv(\000)20 b Fw(2)p Ft(\014)5 b Fv(j)p Ft(\015)g Fv(j)21 b Fw(+)f(2)p Ft(\014)1986 918 y Fs(X)1932 1102 y Fu(x)p Fq(2)p Fr(In)n(t)13 b Fu(\015)2279 918 y Fs(X)2176 1103 y Fu(y)r Fq(2)p Fu(@)t Fr(\003:)p Fu(y)r Fq(\030)p Fu(x)2502 999 y Fw(\(1)21 b Fv(\000)f Ft(\021)s Fw(\()p Ft(y)s Fw(\)\))2895 880 y Fs(\025)165 1255 y Fw(and)h Fv(j)p Ft(\015)5 b Fv(j)21 b Fw(is)g(the)g(length)f(of)h Ft(\015)5 b Fw(;)25 b(this)20 b(can)h(b)s(e)g(seen)g(b)m(y)g(comparing)f(the)h(probabilit)m(y)e(of)i (a)h(con\014guration)165 1367 y(con)m(taining)30 b(\000)f(with)g(the)g (probabilit)m(y)f(of)i(the)f(con\014guration)h(obtained)f(b)m(y)g (\015ipping)e(the)j(spins)d(in)165 1416 y Fs(S)235 1503 y Fu(\015)t Fq(2)p Fr(\000)385 1480 y Fw(In)m(t)16 b Ft(\015)5 b Fw(.)40 b(These)30 b(observ)-5 b(ations)30 b(establish)f(the)i(inequalit)m(y)d(\(57\).)306 1593 y(Note)34 b(that)g(the)f(w)m(eigh)m(t)g Ft(w)1242 1607 y Fu(\021)1284 1593 y Fw(\()p Ft(\015)5 b Fw(\))33 b(of)g(a)g(con)m (tour)g Ft(\015)38 b Fw(dep)s(ends)31 b(on)h(the)h(b)s(oundary)e (con\014guration)165 1706 y Ft(\021)s Fw(;)42 b(this)36 b(is)g(b)s(ecause)i(w)m(e)f(ha)m(v)m(e)i(c)m(hosen)f(to)g(dra)m(w)f (the)g(con)m(tours)h(for)f(plus)e(b)s(oundary)h(conditions)165 1819 y(rather)f(than)f Ft(\021)s Fw(.)53 b(It)35 b(follo)m(ws)e(that)i Ft(w)1456 1833 y Fu(\021)1498 1819 y Fw(\()p Ft(\015)5 b Fw(\))35 b(do)s(es)g(not)f(necessarily)g(tend)g(to)h(zero)g(when)f Fv(j)p Ft(\015)5 b Fv(j)35 b Fw(gro)m(ws)165 1932 y(to)k(in\014nit)m (y;)h(this)d(is)h(in)e(con)m(trast)k(with)d(the)h(case)i Ft(\021)h Fv(\021)d Fw(+1.)64 b(The)38 b(standard)f(lo)m(w)h(temp)s (erature)165 2045 y(expansion)31 b(w)m(ould)h(therefore)g(b)s(ecome)h (m)m(uc)m(h)f(more)h(complicated.)46 b(Ho)m(w)m(ev)m(er,)35 b(if)c(the)h(densit)m(y)g(of)165 2158 y(plus)26 b(spins)g(in)g Ft(\021)31 b Fw(is)26 b(su\016cien)m(tly)h(large,)h(or)g(if)e Ft(@)5 b Fw(In)m(t)16 b Ft(\015)k Fv(\\)15 b Ft(@)5 b Fw(\003)27 b(is)g(rather)g(small,)g(the)h(standard)f(w)m(eigh)m(t)165 2271 y(exp[)p Fv(\000)p Fw(2)p Ft(\014)5 b Fv(j)p Ft(\015)g Fv(j)p Fw(])32 b(of)e(the)g(Ising)e(con)m(tours)j(will)c(dominate,)j (and)f(the)h(righ)m(t-hand)e(side)h(of)h(\(57\))i(can)e(b)s(e)165 2384 y(estimated,)h(as)g(w)m(e)g(will)c(sho)m(w)k(no)m(w.)306 2497 y(W)-8 b(e)29 b(unite)e(the)g(sets)h(\000)p Ft(;)15 b Fw(\000)1172 2464 y Fq(0)1223 2497 y Fw(in)26 b(\(57\))k(in)m(to)d(a) h(single)e(set)i(of)g(con)m(tours)2630 2474 y(~)2624 2497 y(\000)d(=)g(\000)15 b Fv([)g Fw(\000)3007 2464 y Fq(0)3029 2497 y Fw(.)40 b(The)27 b(con)m(tours)165 2609 y(in)290 2586 y(~)284 2609 y(\000)43 b(can)g(o)m(v)m(erlap,)k(but) c(a)g(site)g(of)g Ft(@)1532 2623 y Fu(i)1561 2609 y Ft(C)49 b Fw(can)44 b(only)e(b)s(elong)g(to)i(the)f(in)m(terior)f(of)h(at)h (most)g(t)m(w)m(o)165 2722 y(con)m(tours.)d(On)29 b(the)h(other)g (hand,)f(ev)m(ery)i(site)e(of)h Ft(@)1920 2736 y Fu(i)1948 2722 y Ft(C)37 b Fw(is)28 b(in)h(the)h(in)m(terior)e(of)i(at)g(least)g (one)g(con)m(tour)165 2835 y(of)277 2812 y(~)271 2835 y(\000,)k(and)e Fv(j)p Ft(@)639 2849 y Fu(i)668 2835 y Ft(C)7 b Fv(j)29 b(\025)h Ft(k)s Fw(,)k(the)f(distance)g(of)g(the)g (t)m(w)m(o)h(spins)d(considered.)47 b(These)33 b(ingredien)m(ts)f(allo) m(w)165 2948 y(us)i(to)i(con)m(trol)f(the)g(sum)f(on)h(the)g(righ)m (t-hand)f(side)f(of)i(\(57\).)56 b(If)35 b Ft(k)i Fw(is)d(so)h(large)g (that)h(the)f(densit)m(y)165 3061 y(of)i(plus)d(spins)g(in)h Ft(\030)40 b Fw(b)s(et)m(w)m(een)d(0)g(and)e(\()p Ft(k)s(;)15 b Fw(0\))38 b(exceeds)g(8)p Ft(=)p Fw(9)f(then)f(w)m(e)h(\014nd)d(for)i (an)m(y)h(collection)f(of)165 3174 y(con)m(tours)538 3151 y(~)532 3174 y(\000)25 b(=)g(\000)20 b Fv([)g Fw(\000)925 3141 y Fq(0)979 3174 y Fw(as)30 b(ab)s(o)m(v)m(e)1011 3297 y Fs(X)1008 3497 y Fr(~)-38 b Fu(\015)5 b Fq(2)1097 3480 y Fr(~)1093 3497 y(\000)1237 3297 y Fs(X)1182 3481 y Fu(x)p Fq(2)p Fr(In)n(t)15 b(~)-37 b Fu(\015)1559 3297 y Fs(X)1457 3482 y Fu(y)r Fq(2)p Fu(@)t Fr(\003:)p Fu(y)r Fq(\030)p Fu(x)1782 3378 y Fw(\(1)21 b Fv(\000)f Ft(\021)s Fw(\()p Ft(y)s Fw(\)\))27 b Fv(\024)e Fw(5)p Ft(=)p Fw(9)2480 3297 y Fs(X)2477 3497 y Fr(~)-37 b Fu(\015)t Fq(2)2566 3480 y Fr(~)2562 3497 y(\000)2621 3378 y Fv(j)s Fw(~)-48 b Ft(\015)5 b Fv(j)26 b Ft(:)165 3681 y Fw(This)j(yields)625 3804 y Fs(Y)612 3988 y Fu(\015)t Fq(2)p Fr(\000)759 3885 y Ft(w)824 3899 y Fu(\021)866 3885 y Fw(\()p Ft(\015)5 b Fw(\))1039 3804 y Fs(Y)1004 3989 y Fu(\015)1044 3971 y Fm(0)1066 3989 y Fq(2)p Fr(\000)1157 3971 y Fm(0)1195 3885 y Ft(w)1260 3899 y Fu(\021)1302 3885 y Fw(\()p Ft(\015)1389 3848 y Fq(0)1413 3885 y Fw(\))25 b Fv(\024)1582 3804 y Fs(Y)1569 3988 y Fu(\015)t Fq(2)p Fr(\000)1716 3885 y Fw(exp[)p Fv(\000)p Fw(8)p Ft(=)p Fw(9)15 b Ft(\014)22 b Fv(j)p Ft(\015)5 b Fv(j)p Fw(])2352 3804 y Fs(Y)2317 3989 y Fu(\015)2357 3971 y Fm(0)2379 3989 y Fq(2)p Fr(\000)2470 3971 y Fm(0)2508 3885 y Fw(exp[)p Fv(\000)p Fw(8)p Ft(=)p Fw(9)15 b Ft(\014)22 b Fv(j)p Ft(\015)3043 3848 y Fq(0)3066 3885 y Fv(j)p Fw(])k Ft(:)165 4178 y Fw(A)m(t)35 b(this)d(p)s(oin)m(t)g (the)h(standard)g(argumen)m(ts)g(tak)m(e)i(o)m(v)m(er)g(\(with)d Ft(\014)38 b Fw(replaced)33 b(b)m(y)g(4)p Ft(\014)5 b(=)p Fw(9\),)36 b(leading)c(to)165 4291 y(an)39 b(exp)s(onen)m(tial)e (estimate)j(of)f(\(57\).)67 b(F)-8 b(or)39 b(example,)i(one)e(can)g (conclude)f(the)g(pro)s(of)g(along)h(the)165 4404 y(lines)29 b(of)h(Lemma)h(2.5)g(of)g([37)q(].)2329 b Fk(2)1844 5895 y Fw(75)p eop %%Page: 76 76 76 75 bop 165 304 a Fy(8)135 b(Phase)45 b(transition)g(and)f(p)t (ercolation)165 507 y Fw(T)m(ypically)-8 b(,)24 b(t)m(w)m(o)h(ends)d (of)i(the)g(phase)f(diagram)f(are)i(amenable)f(to)h(mathematical)g (analysis.)37 b(One)23 b(is)165 620 y(the)h(high)f(temp)s(erature,)i (or)f(lo)m(w)g(densit)m(y)-8 b(,)25 b(regime)f(whic)m(h)e(w)m(as)j (discussed)d(in)g(the)j(previous)d(section)165 733 y(and)30 b(in)e(whic)m(h)h(the)h(system)g(can)h(b)s(e)e(view)m(ed)g(as)i(a)f (small)e(p)s(erturbation)g(of)i(an)g(indep)s(enden)m(t)e(spin)165 846 y(system.)49 b(The)32 b(other)i(end)e(is)g(the)h(lo)m(w)g(temp)s (erature)g(regime)f(whic)m(h)g(w)m(e)h(will)e(no)m(w)i(consider.)47 b(A)m(t)165 959 y(lo)m(w)25 b(temp)s(eratures,)h(the)e(energy)h (dominates)g(o)m(v)m(er)h(the)e(en)m(trop)m(y)i(whic)m(h)d(comes)j (from)e(the)h(thermal)165 1072 y(\015uctuations)30 b(of)h(the)g(spins.) 39 b(One)30 b(therefore)h(exp)s(ects)g(that)g(the)g(spin)e (con\014guration)h(is)f(t)m(ypically)165 1184 y(similar)19 b(to)k(some)e(frozen)h(zero)g(temp)s(erature)g(state,)j(whic)m(h)20 b(is)h(a)g(con\014guration)g(of)h(minimal)d(energy)165 1297 y(and)27 b(th)m(us)f(called)g(a)h Fp(gr)-5 b(ound)31 b(state)p Fw(.)40 b(The)27 b(similarit)m(y)d(of)j(a)g(lo)m(w)g(temp)s (erature)f(state)i(with)e(a)h(ground)165 1410 y(state)36 b(is)d(con)m(v)m(enien)m(tly)h(describ)s(ed)e(in)h(geometric)i(terms:) 48 b(one)35 b(imagines)d(that)j(the)f(spins)e(whic)m(h)165 1523 y(agree)h(with)d(the)i(giv)m(en)f(ground)f(state)j(form)e(an)g (in\014nitely)d(extended)k(sea,)g(whereas)f(those)h(spins)165 1636 y(whic)m(h)g(ha)m(v)m(e)h(c)m(hosen)h(to)f(deviate)g(from)f(the)h (ground)e(state)j(are)f(con\014ned)f(to)h(in)m(tersp)s(ersed)e (\014nite)165 1749 y(islands.)57 b(This)34 b(is,)k(of)e(course,)j(a)d (picture)g(of)g(p)s(ercolation)g(theory:)53 b(spins)34 b(that)j(agree)h(with)d(the)165 1862 y(ground)25 b(state)j(form)d(a)h (unique)e(in\014nite)g(cluster.)39 b(W)-8 b(e)27 b(are)f(th)m(us)g(led) f(to)i(the)f(concept)h(of)f Fp(agr)-5 b(e)g(ement)165 1975 y(p)g(er)g(c)g(olation)p Fw(,)34 b(whic)m(h)29 b(will)f(b)s(e)i (discussed)e(in)h(the)i(\014rst)e(part)i(of)f(this)g(section.)306 2088 y(In)e(fact,)h(agreemen)m(t)h(p)s(ercolation)c(is)h(in)m(timately) g(related)h(to)g(the)g(existence)g(of)g(a)h(phase)e(tran-)165 2201 y(sition.)40 b(If)29 b(sev)m(eral)i(distinct)d(ground)i(states)h (exist,)f(w)m(e)h(ma)m(y)g(hop)s(e)e(to)i(\014nd)e(at)h(lo)m(w)g(temp)s (eratures)165 2314 y(also)40 b(sev)m(eral)f(equilibrium)c(phases)j (whic)m(h)g(can)i(b)s(e)e(distinguished)e(b)m(y)j(agreemen)m(t)i(p)s (ercolation)165 2426 y(with)g(resp)s(ect)i(to)g(the)g(di\013eren)m(t)e (ground)h(states.)78 b(One)42 b(ma)m(y)h(ask)f(further)f(whether)h(the) h(geo-)165 2539 y(metric)d(picture)g(that)h(applies)d(to)j(lo)m(w)f (temp)s(eratures)h(remains)e(v)-5 b(alid)38 b(throughout)i(the)h(whole) 165 2652 y(non-uniqueness)c(region.)69 b(Ph)m(ysically)-8 b(,)40 b(this)f(is)g(a)h(matter)g(of)g(stabilit)m(y)e(of)i(the)g (ground)f(states.)165 2765 y(Mathematically)-8 b(,)38 b(it)d(means)h(to)g(lo)s(ok)f(for)g(conditions)f(under)g(whic)m(h)g (distinct)g(Gibbs)g(measures)165 2878 y(allo)m(w)c(distinct)f(sto)s(c)m (hastic-geometric)k(c)m(haracterizations.)306 2991 y(W)-8 b(e)37 b(will)32 b(approac)m(h)j(this)f(question)h(from)f(t)m(w)m(o)j (di\013eren)m(t)d(sides.)54 b(In)34 b(Sections)h(8.1)h(to)g(8.4)g(w)m (e)165 3104 y(in)m(v)m(estigate)23 b(whether)e(\\phase)h(transition)e (implies)e(p)s(ercolation".)38 b(W)-8 b(e)22 b(study)f(a)h(\014xed)f (equilibrium)165 3217 y(phase)38 b Ft(\026)f Fw(in)f(the)i (non-uniqueness)d(region)i(whic)m(h,)i(b)m(y)e(its)g(v)m(ery)h (construction,)i(can)e(b)s(e)e(view)m(ed)165 3330 y(as)42 b(a)g(random)e(p)s(erturbation)g(of)h(some)h(ground)f(state)h Ft(\021)s Fw(.)74 b(W)-8 b(e)43 b(then)e(will)e(see)j(that,)j(in)40 b(man)m(y)165 3443 y(cases,)j(spins)36 b(that)k(agree)g(with)e Ft(\021)k Fw(do)d(p)s(ercolate.)67 b(After)39 b(a)g(general)g (discussion)d(of)j(agreemen)m(t)165 3556 y(p)s(ercolation)h(in)e (Section)i(8.1,)k(w)m(e)d(in)m(v)m(estigate)g(this)e(concept)i(in)e (the)h(subsequen)m(t)f(subsections)165 3668 y(for)c(some)f(sp)s (eci\014c)g(mo)s(dels)f(including)e(the)j(Ising)f(ferromagnet)j(and)d (the)i(P)m(otts)h(mo)s(del.)51 b(In)34 b(the)165 3781 y(case)29 b(of)e(the)h(planar)e(lattice)i Fx(Z)1228 3748 y Fr(2)1294 3781 y Fw(with)e(its)h(limited)e(geometric)k(p)s (ossibilities)23 b(w)m(e)k(will)e(also)j(see)g(that)165 3894 y(con)m(v)m(ersely)-8 b(,)29 b(the)d(absence)h(of)f(phase)g (transition)e(sometimes)j(implies)c(an)j(absence)g(of)h(p)s (ercolation,)165 4007 y(and)33 b(that)h(in)f(the)g(case)i(of)f(phase)f (transition)f(one)i(has)f(restrictions)f(on)i(the)f(n)m(um)m(b)s(er)g (of)g(phases.)165 4120 y(\(Metho)s(dologically)-8 b(,)40 b(these)e(results)f(still)f(run)g(under)g(the)i(heading)f(\\phase)h (transition)e(implies)165 4233 y(p)s(ercolation".\))49 b(In)32 b(the)h(last)g(Section)g(8.5)h(the)f(con)m(v)m(erse)i(will)30 b(b)s(e)i(treated)i(more)g(systematically)165 4346 y(and)25 b(under)f(a)i(di\013eren)m(t)f(asp)s(ect:)39 b(w)m(e)26 b(will)d(sho)m(w)j(that)g(at)g(lo)m(w)g(temp)s(eratures)f(one)h(has)f (p)s(ercolation)165 4459 y(of)39 b(b)s(onds)e(along)i(whic)m(h)e(the)i (in)m(teraction)f(energy)h(is)f(minimal,)g(and)g(w)m(e)h(will)d(see)j (that)g(suc)m(h)g(a)165 4572 y Fp(gr)-5 b(ound-ener)g(gy)35 b(b)-5 b(ond)35 b(p)-5 b(er)g(c)g(olation)35 b Fw(often)d(implies)d(a)k (phase)e(transition.)43 b(T)-8 b(ak)m(en)33 b(together,)h(these)165 4685 y(results)25 b(will)f(sho)m(w)h(that)i(in)e(v)-5 b(arious)25 b(mo)s(dels)g(phase)g(transition)g(comes)i(along)f(with)e (the)j(existence)165 4798 y(of)33 b(a)g(ground-state)g(sea)g(with)e (\014nite)g(islands)f(\(deviation)i(islands\))e(on)i(whic)m(h)f(the)i (spins)d(deviate)165 4910 y(from)g(the)h(ground)e(state,)j(and)e(vice)g (v)m(ersa.)306 5023 y(A)f(theory)g(dev)m(eloping)f(this)f(picture)g(in) h(m)m(uc)m(h)g(more)h(detail)f(is)f(the)i(Pirogo)m(v{Sinai)e(theory)i (of)165 5136 y(phase)d(transition)e(whic)m(h)h(deals)g(with)f(the)i(lo) m(w)g(temp)s(erature)f(phase)h(diagram)f(in)f(the)i(presence)g(of)165 5249 y(sev)m(eral)g(stable)f(ground)f(states.)40 b(One)25 b(basic)g(idea)g(of)g(this)f(theory)i(is)e(to)i(treat)g(the)g(\014nite) e(deviation)165 5362 y(islands)f(of)i(a)g(lo)m(w)g(temp)s(erature)g (system)g(as)g(the)g(constituen)m(ts)g(of)g(a)g(lo)m(w)g(densit)m(y)f (gas)i(of)f(hard-core)165 5475 y(particles.)62 b(While)36 b(the)i(Pirogo)m(v{Sinai)e(theory)i(is)f(in)m(timately)f(related)i(to)g (the)g(sub)5 b(ject)37 b(of)h(this)165 5588 y(section,)43 b(it)d(is)f(m)m(uc)m(h)h(to)s(o)g(in)m(v)m(olv)m(ed)g(to)h(b)s(e)e(dev) m(elop)s(ed)g(here.)70 b(There)39 b(is,)j(ho)m(w)m(ev)m(er,)i(a)c(n)m (um)m(b)s(er)1844 5895 y(76)p eop %%Page: 77 77 77 76 bop 165 304 a Fw(of)32 b(exp)s(ositions)f(whic)m(h)f(ma)m(y)j (serv)m(e)g(as)f(general)g(in)m(tro)s(ductions)e(and)h(in)f(whic)m(h)h (man)m(y)h(additional)165 417 y(references)c(can)f(b)s(e)g(found.)38 b(W)-8 b(e)28 b(men)m(tion)f(only)f([37)r(,)h(69)q(,)g(76)q(,)g(213)r (,)g(214)q(])h(and)e([228)r(])h(to)h([230)r(].)40 b(Here)165 530 y(w)m(e)31 b(will)c(concen)m(trate)32 b(on)e(more)g(sp)s(eci\014c)f (results)g(whic)m(h)f(are)j(partly)e(b)s(ey)m(ond)g(the)h(Pirogo)m (v{Sinai)165 643 y(theory)-8 b(,)40 b(in)c(that)h(they)g(are)h(not)f (limited)e(to)i(lo)m(w)g(temp)s(eratures)g(but)f(rather)h(apply)e(to)j (the)f(full)165 756 y(non-uniqueness)27 b(region,)j(and)f(commen)m(t)h (o)s(ccasionally)f(on)g(some)h(relationships.)38 b(In)29 b(particular,)165 869 y(the)i(results)e(of)i(Section)f(8.5)h(are)g (similar)d(in)h(spirit)f(to)j(this)e(theory)-8 b(.)165 1106 y FA(8.1)113 b(Agreemen)m(t)37 b(p)s(ercolation)h(from)g(phase)h (co)s(existence)165 1278 y Fw(W)-8 b(e)33 b(consider)e(again)h(the)g (general)g(setting)g(of)g(Section)g(2.)46 b(\()p Fv(L)p Ft(;)15 b Fv(\030)p Fw(\))32 b(is)f(an)h(arbitrary)f(lo)s(cally)f (\014nite)165 1391 y(graph,)37 b Ft(S)j Fw(is)34 b(a)i(\014nite)e(set,) j(and)e(\012)e(=)g Ft(S)1584 1358 y Fq(L)1637 1391 y Fw(.)55 b(Supp)s(ose)34 b Ft(\026)g Fw(is)h(a)g(random)g(\014eld)f(and) g Ft(\021)j Fv(2)c Fw(\012)i(a)h(\014xed)165 1504 y(con\014guration.)68 b(W)-8 b(e)41 b(consider)d(the)i(ev)m(en)m(t)h Fv(f)p Ft(x)1925 1448 y Fu(\021)1861 1504 y Fv( )-15 b(!)40 b(1g)g Fw(that)g Ft(x)g Fv(2)g(L)f Fw(b)s(elongs)g(to)h(an)f (in\014nite)165 1617 y(cluster)c(of)h(the)g(random)f(set)h Ft(R)q Fw(\()p Ft(\021)s Fw(\))f(=)f Fv(f)p Ft(y)j Fv(2)d(L)f Fw(:)i Ft(X)7 b Fw(\()p Ft(y)s Fw(\))35 b(=)e Ft(\021)s Fw(\()p Ft(y)s Fw(\))p Fv(g)p Fw(,)39 b(and)c(w)m(e)h(sa)m(y)g(that)h Ft(\026)g Fp(exhibits)165 1740 y(agr)-5 b(e)g(ement)36 b(p)-5 b(er)g(c)g(olation)38 b(for)e Ft(\021)g Fw(if)31 b Ft(\026)p Fw(\()p Ft(x)1609 1685 y Fu(\021)1544 1740 y Fv( )-15 b(!)29 b(1)p Fw(\))g Ft(>)g Fw(0)k(for)f(some)h Ft(x)c Fv(2)g(L)p Fw(.)47 b(In)32 b(short,)h(w)m(e)g(will)d(then)165 1853 y(simply)d(sp)s(eak)h(of)h Ft(\021)s Fw(-p)s(ercolation.)40 b(T)-8 b(o)29 b(visualize)e(suc)m(h)h(an)g(agreemen)m(t,)j(it)d(ma)m(y) i(b)s(e)e(con)m(v)m(enien)m(t)h(to)165 1966 y(think)23 b(of)h(a)g(reduced)f(description)f(of)h Ft(\026)h Fw(in)e(terms)i(of)g (its)f(image)h(under)e(the)i(map)g Ft(s)2990 1980 y Fu(\021)3056 1966 y Fw(:)i(\012)f Fv(!)g(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)3534 1933 y Fq(L)3589 1966 y Fw(,)165 2079 y(whic)m(h)29 b(describ)s(es)g(lo)s(cal)h(agreemen)m(t)i(and)e(disagreemen)m(t)g (with)f Ft(\021)s Fw(,)i(and)f(is)g(de\014ned)f(b)m(y)1194 2307 y(\()p Ft(s)1272 2321 y Fu(\021)1314 2307 y Fw(\()p Ft(\033)s Fw(\)\)\()p Ft(x)p Fw(\))e(=)1719 2163 y Fs(\()1827 2250 y Fw(1)84 b(if)29 b Ft(\033)s Fw(\()p Ft(x)p Fw(\))d(=)f Ft(\021)s Fw(\()p Ft(x)p Fw(\))p Ft(;)1827 2363 y Fw(0)84 b(otherwise.)3453 2307 y(\(58\))165 2565 y(With)30 b(this)f(mapping,)g (w)m(e)i(can)g(write)f Fv(f)p Ft(x)1697 2509 y Fu(\021)1632 2565 y Fv( )-15 b(!)25 b(1g)h Fw(=)f Ft(s)2125 2532 y Fq(\000)p Fr(1)2125 2587 y Fu(\021)2219 2565 y Fv(f)p Ft(x)p Fv($1g)p Fw(.)306 2678 y(W)-8 b(e)34 b(are)f(in)m(terested)g (here)f(in)g(the)h(case)g(when)f Ft(\026)g Fw(is)g(a)h(Gibbs)e(measure) i(for)f(the)h(Hamiltonian)165 2790 y(\(1\),)e(and)d Ft(\021)k Fw(is)c(an)h(asso)s(ciated)g(ground)f(state.)41 b(W)-8 b(e)30 b(sa)m(y)g(that)f(a)h(con\014guration)e Ft(\021)g Fv(2)d Fw(\012)k(is)f(a)h Fp(gr)-5 b(ound)165 2903 y(state)33 b Fw(or,)f(more)g(explicitly)-8 b(,)30 b(a)i(ground)f(state)i (con\014guration)e(for)g(the)h(relativ)m(e)g(Hamiltonian)e(\(2\),)165 3016 y(if)j Ft(H)7 b Fw(\()p Ft(\033)s Fv(j)p Ft(\021)s Fw(\))32 b Fv(\025)e Fw(0)j(for)h(an)m(y)f(lo)s(cal)g(mo)s (di\014cation)f(\(or)i(\\excitation"\))h Ft(\033)h Fw(of)e Ft(\021)s Fw(.)50 b(In)33 b(other)g(w)m(ords,)h Ft(\021)j Fw(is)165 3129 y(a)c(ground)e(state)j(if,)e(for)g(an)m(y)h(region)f (\003)c Fv(2)g(E)8 b Fw(,)33 b(the)g(con\014guration)f Ft(\021)j Fw(minimizes)30 b(the)i(energy)h(in)e(\003)165 3242 y(when)i Ft(\021)451 3256 y Fr(\003)500 3237 y Fo(c)570 3242 y Fw(is)g(\014xed.)51 b(One)34 b(should)e(note)i(in)f(this)g(con)m (text)i(that,)h(in)d(the)h(lo)m(w)g(temp)s(erature)f(limit)165 3355 y Ft(\014)e Fv(")26 b(1)p Fw(,)j(the)g(\014nite-v)m(olume)f(Gibbs) g(distribution)d Ft(\026)1974 3311 y Fu(\021)1974 3383 y(\014)s(;)p Fr(\003)2119 3355 y Fw(from)k(\(4\))h(tends)e(to)i(the)g (equidistribution)165 3468 y(on)f(the)h(set)f(of)g(all)f (con\014gurations)h Ft(\033)j Fw(of)d(minimal)d(energy)k Ft(H)7 b Fw(\()p Ft(\033)s Fv(j)p Ft(\021)s Fw(\).)41 b(This)28 b(fact)h(suggests)h(that,)g(at)165 3581 y(least)i(in)d(some)j (cases,)g(the)g(lo)m(w)e(temp)s(erature)h(phase)g(diagram)g(is)f(only)g (a)h(sligh)m(t)f(deformation)h(of)165 3694 y(the)i(zero-temp)s(erature) g(phase)f(diagram)f(describing)f(the)j(structure)f(of)g(ground)f (states.)48 b(This)30 b(is)165 3807 y(precisely)22 b(the)h(sub)5 b(ject)22 b(of)h(Pirogo)m(v{Sinai)e(theory)i(whic)m(h)f(pro)m(vides)f (su\016cien)m(t)i(conditions)e(for)h(this)165 3920 y(to)32 b(hold,)d(prop)s(oses)h(a)h(construction)g(of)g(lo)m(w)f(temp)s (erature)h(phases)f(as)h(p)s(erturbations)d(of)j(ground)165 4032 y(states,)e(and)d(also)g(sho)m(ws)g(that)h(the)g(size)f (distribution)d(of)j(the)h(deviation)e(islands)f(has)i(exp)s(onen)m (tial)165 4145 y(deca)m(y)-8 b(.)306 4258 y(Supp)s(ose)24 b(next)h(that)g(the)g(Gibbs)e(measure)i Ft(\026)g Fw(is)f(related)h(to) g(the)g(ground)f(state)i Ft(\021)j Fw(in)23 b(some)i(w)m(a)m(y)-8 b(.)165 4371 y(F)g(or)26 b(example,)g Ft(\026)e Fw(migh)m(t)g(b)s(e)h (obtained)f(as)h(the)f(in\014nite)f(v)m(olume)h(limit)f(of)i(the)g (\014nite)e(v)m(olume)i(Gibbs)165 4484 y(distributions)c Ft(\026)746 4440 y Fu(\021)746 4512 y(\014)s(;)p Fr(\003)886 4484 y Fw(with)j(b)s(oundary)e(condition)i Ft(\021)s Fw(,)i(p)s(ossibly)c(along)i(some)i(subsequence.)38 b(\(Under)165 4597 y(the)32 b(conditions)d(of)i(the)h(Pirogo)m(v{Sinai)e(theory)h (suc)m(h)g(a)g(limit)e(alw)m(a)m(ys)j(exists.\))43 b(In)30 b(the)h(case)h(of)g(a)165 4710 y(phase)e(transition,)e(when)h(other)h (phases)g(than)f Ft(\026)h Fw(exist)f(and)g(one)h(is)f(in)m(terested)h (in)e(c)m(haracteristic)165 4823 y(prop)s(erties)f(of)i Ft(\026)p Fw(,)f(one)h(exp)s(ects)g(that)g(the)f(relationship)e(b)s(et) m(w)m(een)j Ft(\026)f Fw(and)g Ft(\021)j Fw(b)s(ecomes)e(manifest)e(in) 165 4936 y(a)35 b(macroscopic)f(pattern)g(of)g(the)g(t)m(ypical)f (con\014gurations,)i(in)d(that)j Ft(\026)e Fw(sho)m(ws)h Ft(\021)s Fw(-p)s(ercolation.)51 b(In)165 5049 y(short,)31 b(w)m(e)g(ask)f(for)g(the)h(v)-5 b(alidit)m(y)29 b(of)h(the)h(h)m(yp)s (othesis)511 5217 y Fv(jG)5 b Fw(\()p Ft(\014)g(H)i Fw(\))p Fv(j)27 b Ft(>)e Fw(1)p Ft(;)31 b(\026)f Fw(is)f(extremal)i(in)e Fv(G)5 b Fw(\()p Ft(\014)g(H)i Fw(\))32 b(and)d(related)i(to)g(a)g (ground)e(state)j Ft(\021)c Fv(2)d Fw(\012)511 5365 y(=)-15 b Fv(\))55 b Ft(\026)p Fw(\()p Ft(x)945 5309 y Fu(\021)880 5365 y Fv( )-15 b(!)25 b(1)p Fw(\))h Ft(>)f Fw(0)91 b Fv(8)15 b Ft(x)25 b Fv(2)g(L)g Ft(:)1655 b Fw(\(59\))165 5533 y(In)30 b(the)h(sp)s(eci\014c)f(cases)i(considered)e(b)s(elo)m(w)g (it)g(will)e(alw)m(a)m(ys)k(b)s(e)e(clear)h(in)e(what)i(sense)g Ft(\026)f Fw(and)h Ft(\021)j Fw(are)165 5646 y(related;)j(t)m(ypically) -8 b(,)35 b Ft(\026)f Fw(will)e(b)s(e)h(a)i(limiting)c(Gibbs)i(measure) h(with)g(b)s(oundary)e(condition)h Ft(\021)s Fw(.)53 b(W)-8 b(e)1844 5895 y(77)p eop %%Page: 78 78 78 77 bop 165 304 a Fw(emphasize)38 b(that)i(\(59\))g(do)s(es)e(not)h (hold)e(in)h(general;)43 b(a)c(coun)m(ter-example)g(can)g(b)s(e)f (constructed)165 417 y(b)m(y)f(com)m(bining)e(man)m(y)h(indep)s(enden)m (t)e(copies)i(of)h(the)g(Ising)e(ferromagnet)i(to)g(a)g(la)m(y)m(ered)g (system,)165 530 y(see)43 b(the)f(discussion)e(after)i(Prop)s(osition)e (8.3)j(b)s(elo)m(w.)75 b(Also,)45 b(ev)m(en)e(when)e(\(59\))i(holds,)h (it)d(do)s(es)165 643 y(not)h(necessarily)f(imply)e(that)j(the)g(phase) f Ft(\026)g Fw(is)g(uniquely)e(c)m(haracterized)k(b)m(y)e(the)h(prop)s (ert)m(y)f(of)165 756 y Ft(\021)s Fw(-p)s(ercolation.)306 869 y(Ho)m(w)34 b(can)g(one)f(establish)e(\(59\)?)50 b(In)33 b(the)g(con)m(text)i(of)e(the)g(Ising)f(mo)s(del,)h(Coniglio)e (et)j(al.)f([62)q(])165 981 y(and)f(Russo)h([208)q(])g(dev)m(elop)s(ed) f(a)h(con)m(v)m(enien)m(t)h(citerion)e(whic)m(h)g(is)f(based)i(on)f(a)h (m)m(ultidimensional)165 1094 y(analog)28 b(of)g(the)f(strong)h(Mark)m (o)m(v)h(prop)s(ert)m(y)e(and)f(th)m(us)h(can)h(b)s(e)f(used)f(for)h (general)h(Mark)m(o)m(v)h(random)165 1207 y(\014elds)g([36)q(,)i(100)q (].)41 b(One)30 b(v)m(ersion)g(is)f(as)i(follo)m(ws.)165 1362 y Fx(Theorem)j(8.1)46 b Fp(L)-5 b(et)38 b Fw(\()p Fv(L)p Ft(;)15 b Fv(\030)p Fw(\))39 b Fp(b)-5 b(e)37 b(a)i(lo)-5 b(c)g(al)5 b(ly)39 b(\014nite)f(gr)-5 b(aph,)41 b Ft(\026)d Fp(a)g(Markov)g(\014eld)g(on)h Fw(\012)34 b(=)h Ft(S)3351 1329 y Fq(L)3403 1362 y Fp(,)k(and)165 1475 y Ft(\021)k Fv(2)c Fw(\012)g Fp(any)i(c)-5 b(on\014gur)g(ation.)67 b(Supp)-5 b(ose)42 b(ther)-5 b(e)41 b(exist)g(a)f(c)-5 b(onstant)42 b Ft(c)e Fv(2)e Fx(R)j Fp(and)g(a)g(lo)-5 b(c)g(al)42 b(function)165 1588 y Ft(f)35 b Fw(:)25 b(\012)g Fv(!)g Fx(R)k Fp(dep)-5 b(ending)29 b(only)g(on)g(the)f(c)-5 b(on\014gur)g(ation)30 b(in)e(a)h(c)-5 b(onne)g(cte)g(d)30 b(set)e Fw(\001)p Fp(,)g(such)h(that)g Ft(\026)p Fw(\()p Ft(f)10 b Fw(\))25 b Ft(>)g(c)165 1701 y Fp(but)1434 1813 y Ft(\026)p Fw(\()p Ft(f)f Fv(j)15 b Ft(X)33 b Fv(\021)25 b Ft(\030)37 b Fw(on)30 b Ft(@)5 b Fw(\000\))26 b Fv(\024)f Ft(c)1107 b Fw(\(60\))165 1960 y Fp(for)49 b(al)5 b(l)48 b(\014nite)g(c)-5 b(onne)g(cte)g(d)49 b(sets)f Fw(\000)53 b Fv(\033)f Fw(\001)c Fp(and)g(al)5 b(l)49 b Ft(\030)56 b Fv(2)d Fw(\012)47 b Fp(with)i Ft(s)2614 1974 y Fu(\021)2655 1960 y Fw(\()p Ft(\030)t Fw(\))54 b Fv(\021)e Fw(0)c Fp(on)g Ft(@)5 b Fw(\000)p Fp(.)88 b(Then)165 2088 y Ft(\026)p Fw(\(\001)421 2032 y Fu(\021)356 2088 y Fv( )-15 b(!)25 b(1)p Fw(\))h Ft(>)f Fw(0)p Fp(,)33 b(i.e.,)f Ft(\026)g Fp(exhibits)h(agr)-5 b(e)g(ement)34 b(p)-5 b(er)g(c)g(olation)36 b(for)d Ft(\021)s Fp(.)165 2253 y Fx(Pro)s(of:)46 b Fw(Supp)s(ose)31 b(b)m(y)i(con)m(trap)s(osition)f (that)h Ft(\026)p Fw(\(\001)2039 2197 y Fu(\021)1974 2253 y Fv( )-15 b(!)29 b(1)p Fw(\))g(=)g(0.)48 b(F)-8 b(or)34 b(an)m(y)f Ft(")c(>)g Fw(0)k(w)m(e)g(can)g(then)165 2376 y(c)m(ho)s(ose)39 b(some)f(\014nite)f(\003)g Fv(\033)h Fw(\001)f(suc)m(h)g(that)i Ft(\026)p Fw(\(\001)1942 2321 y Fu(\021)1878 2376 y Fv( )-15 b(!)37 b Fw(\003)2145 2343 y Fu(c)2180 2376 y Fw(\))h Ft(<)f(")p Fw(.)63 b(F)-8 b(or)39 b Ft(\030)51 b(=)-55 b Fv(2)37 b(f)p Fw(\001)3064 2321 y Fu(\021)3000 2376 y Fv( )-15 b(!)37 b Fw(\003)3267 2343 y Fu(c)3302 2376 y Fv(g)p Fw(,)j(there)165 2489 y(exists)35 b(a)h(connected)g(set)f(\000)g(suc)m(h)g(that)h(\001)c Fv(\032)h Fw(\000)g Fv(\032)g Fw(\003)i(and)g Ft(s)2310 2503 y Fu(\021)2351 2489 y Fw(\()p Ft(\030)t Fw(\))f Fv(\021)f Fw(0)i(on)g Ft(@)5 b Fw(\000;)38 b(w)m(e)e(simply)c(let)j (\000)165 2602 y(b)s(e)c(the)g(union)e(of)j(\001)e(and)h(all)f Ft(\021)s Fw(-clusters)h(meeting)g Ft(@)5 b Fw(\001.)43 b(As)31 b(in)f(the)h(pro)s(of)f(of)i(Theorem)f(6.10,)i(w)m(e)165 2715 y(let)f(\000\()p Ft(\030)t Fw(\))h(b)s(e)e(the)i(largest)f(suc)m (h)g(set.)46 b(F)-8 b(or)33 b Ft(\030)f Fv(2)c(f)p Fw(\001)2010 2659 y Fu(\021)1945 2715 y Fv( )-15 b(!)28 b Fw(\003)2203 2682 y Fu(c)2238 2715 y Fv(g)33 b Fw(w)m(e)f(put)g(\000\()p Ft(\030)t Fw(\))c(=)g Fv(;)p Fw(.)46 b(Then,)32 b(for)g(eac)m(h)165 2828 y(\014nite)f(connected)i(set)f(\000)c Fv(6)p Fw(=)f Fv(;)p Fw(,)33 b(the)f(ev)m(en)m(t)h Fv(f)p Ft(\030)f Fw(:)c(\000\()p Ft(\030)t Fw(\))g(=)f(\000)p Fv(g)32 b Fw(dep)s(ends)e(only)h(on)h(the)g(con\014guration)165 2941 y(in)26 b(\003)13 b Fv(n)g Fw(\000,)27 b(whence)g(b)m(y)f(the)h (Mark)m(o)m(v)i(prop)s(ert)m(y)d Ft(\026)p Fw(\()p Ft(f)e Fv(j)15 b Fw(\000\()p Fv(\001)p Fw(\))27 b(=)e(\000\))h(is)g(an)h(a)m (v)m(erage)i(of)e(the)g(conditional)165 3053 y(probabilities)g(that)k (app)s(ear)f(in)f(assumption)g(\(60\).)42 b(>F)-8 b(rom)31 b(this)e(w)m(e)i(obtain)887 3218 y Ft(\026)p Fw(\()p Ft(f)10 b Fw(\))25 b Fv(\024)g Ft(c)15 b(\026)p Fw(\(\000\()p Fv(\001)p Fw(\))27 b Fv(6)p Fw(=)e Fv(;)p Fw(\))c(+)f Ft(\026)p Fw(\()p Fv(j)p Ft(f)10 b Fv(j)15 b Ft(I)2049 3236 y Fq(f)p Fr(\000\()p Fq(\001)p Fr(\)=)p Fq(;g)2332 3218 y Fw(\))26 b Ft(<)f(c)20 b Fw(+)g Ft(")15 b Fv(k)p Ft(f)10 b Fv(k)26 b Ft(:)165 3382 y Fw(Letting)31 b Ft(")25 b Fv(!)h Fw(0)k(w)m(e)h(\014nd)e Ft(\026)p Fw(\()p Ft(f)10 b Fw(\))25 b Fv(\024)g Ft(c)p Fw(,)31 b(con)m(tradicting)f(our)g (assumption.)898 b Fk(2)165 3532 y Fw(In)38 b(most)g(applications)e(w)m (e)j(will)c(ha)m(v)m(e)k(a)g(natural)e(candidate)h(for)g(the)g (function)f Ft(f)10 b Fw(.)62 b(Whenev)m(er)165 3645 y(distinct)40 b(phases)h(do)h(exist,)i(they)e(can)g(b)s(e)f (distinguished)d(b)m(y)j(some)h(order)f(parameter,)k(viz.)d(a)165 3758 y(lo)s(cal)31 b(function)g Ft(f)41 b Fw(ha)m(ving)31 b(di\013eren)m(t)g(exp)s(ectations)h(for)g(the)g(t)m(w)m(o)h(phases.)44 b(If,)32 b(in)e(addition,)h(some)165 3871 y(sto)s(c)m(hastic)40 b(monotonicit)m(y)f(is)f(a)m(v)-5 b(ailable)38 b(then)g(w)m(e)i(can)f (hop)s(e)f(to)h(establish)e(\(60\).)68 b(In)38 b(fact,)k(the)165 3983 y(p)s(ercolation)34 b(phenomena)h(stated)h(in)d(Examples)h(5.11)j (to)e(5.15)i(can)e(b)s(e)f(deduced)g(from)h(a)g(sligh)m(t)165 4096 y(mo)s(di\014cation)e(of)h(Theorem)g(8.1.)52 b(W)-8 b(e)36 b(will)31 b(not)j(go)h(in)m(to)f(the)g(details)f(of)h(these)h (examples)e(whic)m(h)165 4209 y(are)e(treated)h(in)d([36)q(],)i(but)e (rather)h(apply)f(Theorem)i(8.1)g(to)g(our)f(standard)g(examples.)165 4446 y FA(8.2)113 b(Plus-clusters)38 b(for)f(the)g(Ising)h(ferromagnet) 165 4618 y Fw(The)c(idea)f(of)h(agreemen)m(t)i(p)s(ercolation)d(w)m(as) h(\014rst)f(dev)m(elop)s(ed)g(in)g(the)h(con)m(text)h(of)f(the)g (ferromag-)165 4731 y(netic)g(Ising)e(mo)s(del)g([62)q(,)h(208)r(].)49 b(Let)34 b(us)f(apply)f(Theorem)h(8.1)h(to)g(this)e(standard)h(case.)50 b(W)-8 b(e)35 b(only)165 4844 y(consider)g(the)g(case)i(of)e(no)g (external)h(\014eld,)f(i.e.,)i(w)m(e)f(set)g Ft(h)e Fw(=)f(0,)k(so)f (the)f(only)g(parameter)h(is)e(the)165 4957 y(in)m(v)m(erse)27 b(temp)s(erature)g Ft(\014)k(>)25 b Fw(0.)40 b(W)-8 b(e)28 b(are)g(in)m(terested)f(in)f(agreemen)m(t)j(p)s(ercolation)d(for)h(the) h(constan)m(t)165 5070 y(con\014gurations)34 b Ft(\021)h Fv(\021)c Fw(+1)k(resp.)f Ft(\021)h Fv(\021)c(\000)p Fw(1,)36 b(whic)m(h)d(are)h(the)h(only)e(p)s(erio)s(dic)f(ground)h (states)i(of)g(the)165 5194 y(mo)s(del.)k(W)-8 b(e)28 b(write)908 5144 y Fr(+)852 5194 y Fv( )-15 b(!)26 b Fw(resp.)1315 5144 y Fq(\000)1259 5194 y Fv( )-15 b(!)27 b Fw(for)f(the)h(corresp)s(onding)e(connectedness)j(relation.)38 b(Our)26 b(\014rst)165 5307 y(result)35 b(sho)m(ws)h(that)g(if)f(there) h(is)f(a)h(phase)g(transition)e(then)i(there)g(is)f(plus-p)s (ercolation)e(for)j(eac)m(h)165 5420 y(Gibbs)h(measure)h(except)h(the)g (min)m(us-phase)e Ft(\026)1839 5382 y Fq(\000)1839 5448 y Fu(\014)1897 5420 y Fw(;)43 b(that)c(is,)g(assertion)f(\(59\))i (holds)d(for)h Ft(\021)k Fv(\021)c Fw(+1.)165 5533 y(This)28 b(result)h(\(due)h(to)g([208)r(]\))g(is)f(v)-5 b(alid)28 b(for)i(an)f(arbitrary)g(lo)s(cally)f(\014nite)h(graph)g(\()p Fv(L)p Ft(;)15 b Fv(\030)p Fw(\))31 b(with)d(\014nite)165 5646 y(critical)i(in)m(v)m(erse)g(temp)s(erature)g Ft(\014)1339 5660 y Fu(c)1374 5646 y Fw(.)1844 5895 y(78)p eop %%Page: 79 79 79 78 bop 165 304 a Fx(Theorem)34 b(8.2)46 b Fp(L)-5 b(et)36 b Ft(\026)f Fp(b)-5 b(e)36 b(an)f(arbitr)-5 b(ary)38 b(Gibbs)d(me)-5 b(asur)g(e)37 b(for)f(the)g(ferr)-5 b(omagnetic)37 b(Ising)e(mo)-5 b(del)165 428 y(with)41 b(p)-5 b(ar)g(ameters)42 b Ft(\014)h(>)37 b Fw(0)p Fp(,)42 b Ft(h)c Fw(=)f(0)p Fp(.)63 b(If)39 b Ft(\026)f Fv(6)p Fw(=)f Ft(\026)1850 390 y Fq(\000)1850 456 y Fu(\014)1948 428 y Fp(then)j Ft(\026)p Fw(\()p Ft(x)2393 378 y Fr(+)2337 428 y Fv( )-15 b(!)37 b(1)p Fw(\))h Ft(>)f Fw(0)j Fp(for)g(al)5 b(l)40 b Ft(x)e Fv(2)f(L)p Fp(.)62 b(In)165 576 y(p)-5 b(articular,)35 b(if)d Ft(\014)f(>)25 b(\014)926 590 y Fu(c)993 576 y Fp(then)34 b Ft(\026)1251 538 y Fr(+)1251 604 y Fu(\014)1309 576 y Fw(\()p Ft(x)1478 526 y Fr(+)1422 576 y Fv( )-15 b(!)25 b(1)p Fw(\))g Ft(>)g Fw(0)33 b Fp(for)h(al)5 b(l)33 b Ft(x)25 b Fv(2)g(L)p Fp(.)165 731 y Fx(Pro)s(of:)108 b Fw(By)36 b(the)f(sandwic)m(hing)e(inequalit)m(y)g(\(15\))k(and)e (Prop)s(osition)e(4.12,)38 b(there)d(exists)g(a)g(site)165 844 y Ft(x)26 b Fv(2)e(L)29 b Fw(suc)m(h)f(that)h Ft(\026)p Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))26 b(=)f(1\))h Ft(>)f(c)h Fv(\021)f Ft(\026)1652 805 y Fq(\000)1652 872 y Fu(\014)1711 844 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))26 b(=)f(1\).)41 b(On)28 b(the)h(other)f(hand,)g(the)h(analogue)g (of)165 957 y(inequalit)m(y)g(\(14\))j(for)e(the)h(min)m(us)d(b)s (oundary)h(condition)g(sho)m(ws)h(that)875 1121 y Ft(\026)p Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))27 b(=)e(1)15 b Fv(j)g Ft(X)33 b Fv(\021)25 b(\000)p Fw(1)31 b(on)f Ft(@)5 b Fw(\000\))26 b(=)f Ft(\026)2191 1083 y Fq(\000)2191 1149 y Fu(\014)s(;)p Fr(\000)2301 1121 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))27 b(=)e(1\))h Fv(\024)f Ft(c)165 1286 y Fw(for)32 b(ev)m(ery)h(\014nite)e(\000)d Fv(3)f Ft(x)p Fw(.)46 b(Theorem)32 b(8.1)h(th)m(us)f(giv)m(es)g(the)g(result)f (for)h(the)g Ft(x)g Fw(at)h(hand.)44 b(In)32 b(view)f(of)165 1399 y(the)g(\014nite)e(energy)i(prop)s(ert)m(y)f(of)g Ft(\026)p Fw(,)g(this)g(extends)g(easily)g(to)h(all)e(other)i Ft(x)25 b Fv(2)g(L)p Fw(.)630 b Fk(2)165 1548 y Fw(Let)36 b(us)f(rephrase)g(the)g(last)g(statemen)m(t)i(of)f(Theorem)f(8.2)i(as)e (follo)m(ws:)50 b(b)s(elo)m(w)35 b(the)g(critical)f(tem-)165 1661 y(p)s(erature)d(the)h(plus)d(spins)g(p)s(ercolate)j(in)e(the)h (plus)f(phase)h(and,)g(b)m(y)g(symmetry)-8 b(,)32 b(the)g(min)m(us)d (spins)165 1774 y(p)s(ercolate)h(in)d(the)i(min)m(us)e(phase)i Ft(\026)1386 1736 y Fq(\000)1386 1802 y Fu(\014)1445 1774 y Fw(.)40 b(In)28 b(the)h(case)h(of)f(graphs)f(with)g(symmetry)h (axes,)h(this)d(state-)165 1887 y(men)m(t)k(allo)m(ws)f(an)g(in)m (teresting)g(re\014nemen)m(t.)165 2042 y Fx(Prop)s(osition)37 b(8.3)46 b Fp(Supp)-5 b(ose)46 b Fw(\()p Fv(L)p Ft(;)15 b Fv(\030)p Fw(\))46 b Fp(admits)g(an)g(involutive)f(gr)-5 b(aph)47 b(automorphism)i Ft(r)e Fp(which)165 2155 y(maps)32 b(a)f(subset)f Fv(H)c(\032)f(L)30 b Fp(onto)i(its)e(c)-5 b(omplement)32 b Fv(H)1941 2122 y Fu(c)1976 2155 y Fp(,)e(and)i(that)f (for)g Ft(x)26 b Fv(2)e(H)q Fp(,)31 b Ft(y)d Fv(2)d(H)3072 2122 y Fu(c)3136 2155 y Fp(either)31 b Ft(x)25 b Fv(6\030)g Ft(y)165 2267 y Fp(or)30 b Ft(y)e Fw(=)d Ft(r)s(x)p Fp(.)40 b(F)-7 b(or)31 b Ft(x)25 b Fv(2)g(H)30 b Fp(let)f Fv(f)p Ft(x)1317 2218 y Fl(+)p Fo(r)r Fl(+)1296 2267 y Fv( )-15 b(!)25 b(1)32 b Fw(in)d Fv(H)q(g)g Fp(b)-5 b(e)29 b(the)h(event)f(that) i(ther)-5 b(e)30 b(exists)f(an)h(in\014nite)f(p)-5 b(ath)165 2380 y Ft(\015)39 b Fp(in)33 b Fv(H)i Fp(starting)f(fr)-5 b(om)35 b Ft(x)f Fp(such)f(that)i(al)5 b(l)34 b(spins)h(along)f(b)-5 b(oth)35 b Ft(\015)k Fp(and)34 b(its)g(r)-5 b(e\015e)g(ction)35 b(image)f Ft(r)s(\015)k Fp(ar)-5 b(e)165 2493 y(p)g(ositive.)43 b(If)33 b Ft(\026)681 2455 y Fr(+)681 2521 y Fu(\014)765 2493 y Fv(6)p Fw(=)25 b Ft(\026)916 2455 y Fq(\000)916 2521 y Fu(\014)1007 2493 y Fp(then)33 b Ft(\026)1264 2455 y Fr(+)1264 2521 y Fu(\014)1323 2493 y Fw(\()p Ft(x)1457 2443 y Fl(+)p Fo(r)r Fl(+)1435 2493 y Fv( )-15 b(!)25 b(1)33 b Fw(in)c Fv(H)q Fw(\))c Ft(>)g Fw(0)33 b Fp(for)g(al)5 b(l)33 b Ft(x)26 b Fv(2)e(H)q Fp(.)165 2663 y Fw(One)34 b(natural)e(case)j(to)g(think)d(of)i(is)f(when)f Fv(L)f Fw(=)f Fx(Z)1937 2630 y Fu(d)2012 2663 y Fw(for)j Ft(d)e Fv(\025)g Fw(2,)k Fv(H)f Fw(a)g(halfspace)g(with)e(b)s(oundary)165 2776 y(orthogonal)27 b(to)f(an)g(axis,)g(and)f Ft(r)j Fw(the)e(asso)s(ciated)g(re\015ection.)39 b(The)26 b(prop)s(osition)d (then)i(asserts)i(that)165 2889 y Ft(\026)220 2851 y Fr(+)220 2917 y Fu(\014)279 2889 y Fw(-almost)32 b(surely)f(there)h (exists)f(an)h(in\014nite)d(connected)k(mirror-symmetric)d(set)i(of)g (plus)e(spins.)165 3002 y(Another)k(in)m(teresting)g(case)h(is)e(when)f Fv(L)i Fw(consists)g(of)g(t)m(w)m(o)h(disjoin)m(t)d(copies)i(of)g(a)g (graph)g Fv(H)g Fw(whic)m(h)165 3115 y(are)40 b(not)f(connected)h(to)g (eac)m(h)g(other)f(b)m(y)g(an)m(y)h(b)s(ond.)65 b(In)38 b(this)g(case,)k Ft(\026)2706 3077 y Fr(+)2706 3143 y Fu(\014)2805 3115 y Fw(=)d Ft(\026)2970 3077 y Fr(+)2970 3143 y Fu(\014)s(;)p Fq(H)3123 3115 y Fv(\002)26 b Ft(\026)3275 3077 y Fr(+)3275 3143 y Fu(\014)s(;)p Fq(H)3401 3115 y Fw(,)42 b(and)165 3243 y(the)36 b(statemen)m(t)h(is)e(that)h(t)m(w)m (o)h(indep)s(enden)m(t)c(realizations)i(of)h Ft(\026)2400 3205 y Fr(+)2400 3271 y Fu(\014)s(;)p Fq(H)2562 3243 y Fw(exhibit)e(sim)m(ultaneous)g(plus-)165 3356 y(p)s(ercolation;)44 b(in)38 b(this)h(case,)k(the)d(preceding)f(prop)s(osition)e(w)m(as)j (observ)m(ed)g(b)m(y)g(Giacomin)f(et)h(al.)165 3469 y([100)r(].)j(It)31 b(is,)f(ho)m(w)m(ev)m(er,)j(not)e(p)s(ossible)d(to)k(tak)m(e)h(an)d (arbitrarily)f(large)i(n)m(um)m(b)s(er)e Ft(k)34 b Fw(of)d(indep)s (enden)m(t)165 3581 y(realizations)k Ft(X)726 3595 y Fr(1)766 3581 y Ft(;)15 b(:)g(:)g(:)h(;)f(X)1042 3596 y Fu(k)1121 3581 y Fw(of)35 b Ft(\026)1284 3543 y Fr(+)1284 3610 y Fu(\014)s(;)p Fq(H)1411 3581 y Fw(,)h(at)g(least)g(when)e Fv(H)i Fw(has)f(b)s(ounded)e(degree)j Ft(N)10 b Fw(.)55 b(F)-8 b(or,)37 b(if)d Ft(p)3483 3595 y Fu(c)3553 3581 y Fw(is)165 3694 y(the)d(Bernoulli)d(site)i(p)s(ercolation)f(threshold) g(of)i Fv(H)g Fw(and)f Ft(k)j Fw(is)c(so)i(large)g(that)1354 3859 y(sup)1349 3936 y Fu(x)p Fq(2H)1511 3859 y Ft(\026)1566 3821 y Fr(+)1566 3891 y Fu(\014)s(;)p Fq(f)p Fu(x)p Fq(g)1743 3859 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))27 b(=)e(1\))2185 3821 y Fu(k)2254 3859 y Ft(<)g(p)2396 3873 y Fu(c)165 4077 y Fw(then)31 b(the)h(set)g Fv(f)p Ft(x)27 b Fv(2)g(H)g Fw(:)h Ft(X)1118 4091 y Fr(1)1157 4077 y Fw(\()p Ft(x)p Fw(\))g(=)f Fv(\001)15 b(\001)g(\001)28 b Fw(=)e Ft(X)1710 4092 y Fu(k)1753 4077 y Fw(\()p Ft(x)p Fw(\))i(=)f(1)p Fv(g)32 b Fw(do)s(es)f(not)g(p)s(ercolate.)44 b(This)30 b(follo)m(ws)h(from)165 4190 y(a)e(standard)e(domination)g(argumen)m (t.)40 b(Since)27 b(the)h(la)m(y)m(ered)h(system)f(consisting)f(of)h Ft(k)j Fw(indep)s(enden)m(t)165 4302 y(copies)f(of)g(the)h(Ising)d(mo)s (del)h(with)g Ft(\014)h(>)25 b(\014)1617 4316 y Fu(c)1682 4302 y Fw(certainly)30 b(exhibits)e(a)i(phase)g(transition,)e(w)m(e)j (see)g(that)165 4415 y(h)m(yp)s(othesis)e(\(59\))j(do)s(es)e(not)h (hold)e(in)g(general.)165 4565 y Fx(Pro)s(of)47 b(of)e(Prop)s(osition)i (8.3:)126 b Fw(W)-8 b(e)41 b(iden)m(tify)c(eac)m(h)k Ft(\030)j Fv(2)c Fw(\012)f(with)f(\()p Ft(\030)t Fw(\()p Ft(x)p Fw(\))p Ft(;)15 b(\030)t Fw(\()p Ft(r)s(x)p Fw(\)\))3169 4579 y Fu(x)p Fq(2H)3363 4565 y Fv(2)40 b Ft(S)3525 4532 y Fq(H)3589 4565 y Fw(,)165 4678 y(where)32 b Ft(S)h Fw(=)c Fv(f\000)p Fw(1)p Ft(;)15 b Fw(1)p Fv(g)910 4645 y Fr(2)951 4678 y Fw(.)46 b(The)32 b(ev)m(en)m(t)i(under)c (consideration)i(then)f(corresp)s(onds)g(to)i Ft(\021)s Fw(-p)s(ercolation)165 4791 y(for)g(the)f(con\014guration)g Ft(\021)g Fv(2)c Ft(S)1245 4758 y Fq(H)1342 4791 y Fw(with)j Ft(\021)s Fw(\()p Ft(x)p Fw(\))f(=)e(\(1)p Ft(;)15 b Fw(1\))35 b(for)d(all)f Ft(x)e Fv(2)f(H)q Fw(.)47 b(Let)33 b Ft(f)38 b Fw(=)28 b Ft(X)7 b Fw(\()p Ft(x)p Fw(\))23 b(+)e Ft(X)7 b Fw(\()p Ft(r)s(x)p Fw(\).)165 4904 y(Then)25 b Ft(\026)453 4865 y Fr(+)453 4932 y Fu(\014)512 4904 y Fw(\()p Ft(f)10 b Fw(\))25 b(=)g(2)15 b Ft(\026)873 4865 y Fr(+)873 4932 y Fu(\014)933 4904 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\)\))27 b Ft(>)e Fw(0)h(b)m(y)g(the)g Ft(r)s Fw(-in)m(v)-5 b(ariance)25 b(of)h Ft(\026)2328 4865 y Fr(+)2328 4932 y Fu(\014)2386 4904 y Fw(.)40 b(On)25 b(the)h(other)g(hand,)g(let)f(\000)h Fv(\032)f(H)165 5037 y Fw(b)s(e)33 b(a)g(\014nite)f(set)h(con)m(taining)g Ft(x)p Fw(,)g(and)1489 5014 y(~)1484 5037 y(\000)c(=)g(\000)22 b Fv([)f Ft(r)s Fw(\000.)48 b(If)32 b(\()p Ft(\030)t(;)15 b(\030)2261 5004 y Fq(0)2285 5037 y Fw(\))30 b Fv(2)f Ft(S)2501 5004 y Fq(H)2595 5037 y Fw(=)g(\012)k(with)e Ft(s)3046 5051 y Fu(\021)3088 5037 y Fw(\()p Ft(\030)t(;)15 b(\030)3251 5004 y Fq(0)3275 5037 y Fw(\))30 b Fv(\021)f Fw(0)k(on)165 5150 y Ft(@)213 5164 y Fq(H)278 5150 y Fw(\000)25 b(=)g Ft(@)5 b Fw(\000)20 b Fv(\\)f(H)q Fw(,)30 b(then)g Ft(\030)1050 5117 y Fq(0)1099 5150 y Fv(\026)25 b(\000)p Ft(\030)33 b Fw(on)d Ft(@)1513 5164 y Fq(H)1578 5150 y Fw(\000,)g(and)g(therefore)g(\()p Ft(\030)t(;)15 b(\030)2411 5117 y Fq(0)2435 5150 y Fw(\))26 b Fv(\026)f Fw(\()p Ft(\030)t(;)15 b Fv(\000)p Ft(\030)t Fw(\))31 b(\(as)f(elemen)m(ts)h(of)f(\012\))165 5262 y(on)h Ft(@)351 5239 y Fw(~)345 5262 y(\000)25 b(=)g Ft(@)571 5276 y Fq(H)636 5262 y Fw(\000)20 b Fv([)f Ft(r)f(@)900 5276 y Fq(H)965 5262 y Fw(\000.)40 b(W)-8 b(e)32 b(can)e(th)m(us)g(write)450 5436 y Ft(\026)505 5397 y Fr(+)505 5464 y Fu(\014)563 5436 y Fw(\()p Ft(f)25 b Fv(j)15 b Fw(\()p Ft(X)r(;)g(X)942 5398 y Fq(0)968 5436 y Fw(\))25 b Fv(\021)g Fw(\()p Ft(\030)t(;)15 b(\030)1287 5398 y Fq(0)1311 5436 y Fw(\))31 b(on)f Ft(@)5 b Fw(\000\))84 b(=)e Ft(\026)1940 5388 y Fr(\()p Fu(\030)s(;\030)2055 5364 y Fm(0)2077 5388 y Fr(\))1940 5479 y Fu(\014)s(;)2007 5462 y Fr(~)2003 5479 y(\000)2108 5436 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\)\))22 b(+)e Ft(\026)2550 5388 y Fr(\()p Fu(\030)s(;\030)2665 5364 y Fm(0)2687 5388 y Fr(\))2550 5479 y Fu(\014)s(;)2617 5462 y Fr(~)2613 5479 y(\000)2718 5436 y Fw(\()p Ft(X)7 b Fw(\()p Ft(r)s(x)p Fw(\)\))1732 5623 y Fv(\024)82 b Ft(\026)1940 5575 y Fr(\()p Fu(\030)s(;)p Fq(\000)p Fu(\030)s Fr(\))1940 5666 y Fu(\014)s(;)2007 5649 y Fr(~)2003 5666 y(\000)2141 5623 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\)\))22 b(+)e Ft(\026)2583 5575 y Fr(\()p Fu(\030)s(;)p Fq(\000)p Fu(\030)s Fr(\))2583 5666 y Fu(\014)s(;)2650 5649 y Fr(~)2646 5666 y(\000)2783 5623 y Fw(\()p Ft(X)7 b Fw(\()p Ft(r)s(x)p Fw(\)\))57 b(=)e(0)1844 5895 y(79)p eop %%Page: 80 80 80 79 bop 165 304 a Fw(b)m(y)29 b(Lemma)f(4.13)i(and)e(the)h(symmetry)f (under)f Ft(r)k Fw(and)d(sim)m(ultaneous)f(spin)f(\015ip.)38 b(The)28 b(prop)s(osition)165 417 y(th)m(us)i(follo)m(ws)g(from)g (Theorem)g(8.1.)2143 b Fk(2)165 575 y Fw(In)45 b(the)g(remaining)e (part)i(of)g(this)f(subsection)g(w)m(e)h(consider)f(the)i(case)g(of)f (the)g(square)g(lattice)165 688 y Fv(L)25 b Fw(=)g Fx(Z)413 655 y Fr(2)453 688 y Fw(,)i(in)f(whic)m(h)f(w)m(e)i(can)h(obtain)e(m)m (uc)m(h)h(stronger)g(conclusions.)37 b(The)27 b(follo)m(wing)e(result)h (giv)m(es)h(a)165 801 y(complete)f(c)m(haracterization)h(of)e(the)h (non-uniqueness)d(regime)i(of)h(the)f(parameter)h(space)g(in)e(terms) 165 914 y(of)34 b(p)s(ercolation)e(of)i(plus)e(spins)f(in)h(the)i (Gibbs)d(measure)j Ft(\026)2229 876 y Fr(+)2229 942 y Fu(\014)2288 914 y Fw(.)49 b(It)34 b(is)e(due)h(to)h(Coniglio)e(et)i (al.)g([62)q(];)165 1027 y(see)d(also)g([131)q(].)165 1203 y Fx(Corollary)k(8.4)46 b Fp(F)-7 b(or)30 b(the)e(Ising)g(ferr)-5 b(omagnet)30 b(on)f(the)f(squar)-5 b(e)29 b(lattic)-5 b(e)29 b Fx(Z)2723 1170 y Fr(2)2790 1203 y Fp(with)g(no)f(external)i (\014eld)165 1316 y(and)37 b(inverse)e(temp)-5 b(er)g(atur)g(e)38 b Ft(\014)5 b Fp(,)36 b(the)f Ft(\026)1488 1278 y Fr(+)1488 1344 y Fu(\014)1547 1316 y Fp(-pr)-5 b(ob)g(ability)37 b(of)f(having)f(an)h(in\014nite)g(plus-cluster)f(is)h Fw(0)f Fp(in)165 1429 y(the)e(uniqueness)f(r)-5 b(e)g(gime)34 b Ft(\014)c Fv(\024)25 b Ft(\014)1294 1443 y Fu(c)1329 1429 y Fp(,)33 b(and)g Fw(1)g Fp(in)g(the)g(non-uniqueness)f(r)-5 b(e)g(gime)33 b Ft(\014)e(>)25 b(\014)3066 1443 y Fu(c)3101 1429 y Fp(.)165 1605 y Fx(Pro)s(of:)137 b Fw(The)42 b(existence)g(of)g (an)g(in\014nite)d(cluster)i(of)h(plus)e(spins)g(is)g(a)j(tail)e(ev)m (en)m(t)i(and)e(th)m(us,)165 1718 y(b)m(y)h(the)f(extremalit)m(y)h(of)f Ft(\026)1133 1680 y Fr(+)1133 1746 y Fu(\014)1192 1718 y Fw(,)j(has)d(probabilit)m(y)e(0)j(or)f(1.)74 b(The)41 b(case)h Ft(\014)49 b(>)43 b(\014)2929 1732 y Fu(c)3005 1718 y Fw(is)e(th)m(us)f(co)m(v)m(ered)165 1846 y(b)m(y)d(Theorem)f (8.2.)61 b(F)-8 b(or)37 b Ft(\014)k Fv(\024)35 b Ft(\014)1307 1860 y Fu(c)1343 1846 y Fw(,)j Ft(\026)1461 1808 y Fr(+)1461 1874 y Fu(\014)1556 1846 y Fw(coincides)e(with)f Ft(\026)2215 1808 y Fq(\000)2215 1874 y Fu(\014)2274 1846 y Fw(.)60 b(Th)m(us,)37 b(if)f(an)g(in\014nite)e(plus-cluster)165 1959 y(existed)25 b(with)g(probabilit)m(y)e(1)i(then,)i(b)m(y)e (symmetry)-8 b(,)27 b(an)e(in\014nite)e(min)m(us-cluster)h(w)m(ould)g (also)h(exist,)165 2072 y(in)43 b(con)m(tradiction)h(to)h(Theorem)f (5.18;)53 b(the)44 b(assumptions)f(of)h(this)f(theorem)h(are)h (satis\014ed)e(b)m(y)165 2185 y(Prop)s(osition)29 b(4.14.)2706 b Fk(2)165 2343 y Fw(Com)m(bining)29 b(the)i(corollary)f(ab)s(o)m(v)m (e)h(with)f(Prop)s(osition)e(4.16,)33 b(w)m(e)e(can)g(also)f(obtain)g (some)h(b)s(ounds)165 2456 y(for)f(the)h(p)s(ercolativ)m(e)f(region)g (of)h(the)g(Ising)e(mo)s(del)g(for)h Ft(h)c Fv(6)p Fw(=)f(0;)31 b(see)g([8])g(for)f(a)h(detailed)e(discussion.)306 2569 y(The)c(equiv)-5 b(alence)25 b(of)g(non-uniqueness)e(and)i(p)s (ercolation)f(just)g(observ)m(ed)i(for)f(the)g(Ising)f(mo)s(del)165 2681 y(on)37 b Fx(Z)362 2648 y Fr(2)438 2681 y Fw(cannot)g(b)s(e)f(exp) s(ected)h(to)g(hold)e(for)h(non-planar)f(graphs.)59 b(Consider,)37 b(for)f(example,)i(the)165 2794 y(Ising)d(mo)s(del)g(on)h(the)h(cubic)e (lattice)i Fx(Z)1553 2761 y Fr(3)1592 2794 y Fw(.)58 b(F)-8 b(or)37 b Ft(\014)j Fw(=)35 b(0)i(uniqueness)c(certainly)j (holds,)g(and)g(plus-)165 2907 y(p)s(ercolation)28 b(is)g(equiv)-5 b(alen)m(t)28 b(to)i(Bernoulli)c(site)i(p)s(ercolation)g(on)h Fx(Z)2482 2874 y Fr(3)2550 2907 y Fw(with)e(parameter)i(1)p Ft(=)p Fw(2.)42 b(But)29 b(a)165 3020 y(result)g(of)h([47)r(])g(states) h(that)g Ft(p)1194 3034 y Fu(c)1228 3020 y Fw(\()p Fx(Z)1327 2987 y Fr(3)1367 3020 y Fw(\))25 b Ft(<)g Fw(1)p Ft(=)p Fw(2.)42 b(The)30 b(plus)e(spins)g(th)m(us)h(p)s(ercolate)h(at)h Ft(\014)g Fw(=)25 b(0.)41 b(In)29 b(view)165 3133 y(of)f(Prop)s (osition)d(4.16,)30 b(this)c(is)g(still)f(the)i(case)i(for)e (su\016cien)m(tly)e(small)h Ft(\014)5 b Fw(,)28 b(so)g(that)g(plus-p)s (ercolation)165 3246 y(do)s(es)i(o)s(ccur)h(in)e(a)h(non-trivial)f (part)h(of)g(the)h(uniqueness)d(region.)306 3359 y(F)-8 b(or)25 b(the)e(planar)f(graph)h Fx(Z)1203 3326 y Fr(2)1242 3359 y Fw(,)i(ho)m(w)m(ev)m(er,)h(Theorem)d(5.18)i(do)s(es)e(not)h (only)e(imply)f(the)j(equiv)-5 b(alence)165 3472 y(of)33 b(phase)e(transition)g(and)h(p)s(ercolation,)g(but)f(also)h(giv)m(es)h (some)f(information)f(on)h(the)g(n)m(um)m(b)s(er)f(of)165 3585 y(phases)41 b(in)g(the)g(non-uniqueness)e(region.)74 b(As)42 b(a)g(w)m(arm-up)e(let)i(us)f(sho)m(w)g(that,)k(for)d(the)f (Ising)165 3698 y(ferromagnet)36 b(on)f Fx(Z)873 3665 y Fr(2)948 3698 y Fw(at)h(in)m(v)m(erse)f(temp)s(erature)g Ft(\014)k(>)33 b(\014)2138 3712 y Fu(c)2173 3698 y Fw(,)k(there)e(are)h (no)f(other)g(translation)g(and)165 3811 y(rotation)42 b(in)m(v)-5 b(arian)m(t)40 b(extremal)h(Gibbs)f(measures)h(than)g Ft(\026)2258 3772 y Fr(+)2258 3839 y Fu(\014)2357 3811 y Fw(and)g Ft(\026)2600 3772 y Fq(\000)2600 3839 y Fu(\014)2659 3811 y Fw(.)72 b(F)-8 b(or,)45 b(supp)s(ose)40 b(another)165 3923 y(suc)m(h)28 b(phase)f Ft(\026)g Fw(existed.)40 b(By)28 b(Theorem)f(8.2)i(and)e(the)h(Burton-Keane)g(uniqueness)d (theorem)j(5.17,)165 4036 y(there)d(exist)e(unique)f(in\014nite)g (plus-)h(and)g(min)m(us-clusters)f(with)g Ft(\026)p Fw(-probabilit)m(y) g(1.)39 b(As)24 b(an)g(extremal)165 4149 y(Gibbs)f(measure,)j Ft(\026)e Fw(has)g(p)s(ositiv)m(e)g(correlations;)i(recall)e(the)g (paragraph)g(b)s(elo)m(w)g(Prop)s(osition)e(4.14.)165 4262 y(Prop)s(osition)29 b(5.19)j(th)m(us)e(sho)m(ws)g(that)h Ft(\026)f Fw(cannot)h(exist.)306 4375 y(The)21 b(statemen)m(t)h(just)e (sho)m(wn)g(is)g(a)h(w)m(eak)h(v)m(ersion)e(of)h(the)f(follo)m(wing)g (result)f(whic)m(h)h(c)m(haracterizes)165 4488 y(all)33 b(translation)f(in)m(v)-5 b(arian)m(t)32 b(Gibbs)g(measures.)49 b(In)33 b(fact,)i(it)e(is)f(su\016cien)m(t)h(to)h(assume)f(p)s(erio)s (dicit)m(y)-8 b(,)165 4601 y(whic)m(h)29 b(means)i(in)m(v)-5 b(ariance)29 b(under)g(the)i(translation)e(subgroup)g(\()p Ft(\022)2481 4615 y Fu(x)2525 4601 y Fw(\))2560 4621 y Fu(x)p Fq(2)p Fu(p)p Fn(Z)2733 4602 y Fl(2)2801 4601 y Fw(for)h(some)h Ft(p)25 b(>)g Fw(1.)165 4777 y Fx(Prop)s(osition)37 b(8.5)46 b Fp(A)n(ny)34 b(p)-5 b(erio)g(dic)37 b(Gibbs)e(me)-5 b(asur)g(e)37 b Ft(\026)e Fp(for)g(the)h(Ising)f(ferr)-5 b(omagnet)37 b(on)f Fv(L)29 b Fw(=)h Fx(Z)3575 4744 y Fr(2)165 4890 y Fp(with)h(no)f(external)h(\014eld)g(and)g(inverse)e (temp)-5 b(er)g(atur)g(e)33 b Ft(\014)d(>)25 b(\014)2231 4904 y Fu(c)2296 4890 y Fp(is)30 b(a)g(mixtur)-5 b(e)31 b(of)f(the)g(two)h(phases)g Ft(\026)3555 4852 y Fr(+)3555 4918 y Fu(\014)165 5018 y Fp(and)j Ft(\026)397 4980 y Fq(\000)397 5046 y Fu(\014)456 5018 y Fp(.)165 5194 y Fw(Under)44 b(the)g(condition)f(of)h(translation)f(in)m(v)-5 b(ariance,)47 b(this)c(prop)s(osition)e(w)m(as)k(\014rst)e(deriv)m(ed)g (for)165 5307 y(large)35 b Ft(\014)41 b Fw(b)m(y)34 b(Galla)m(v)m(otti) i(and)f(Miracle-Sole)f([88)q(],)j(and)d(later)h(for)g(all)f Ft(\014)k(>)33 b(\014)2870 5321 y Fu(c)2940 5307 y Fw(b)m(y)h(Messager) j(and)165 5420 y(Miracle-Sole)f([177)q(])g(using)e(some)i(sp)s (eci\014c)f(correlation)g(inequalities;)h(it)f(follo)m(ws)g(also)h (from)f(the)165 5533 y(Onsager-form)m(ula)26 b(for)h(the)g(free)f (energy)h(densit)m(y)f(and)g(a)h(result)f(of)g(Leb)s(o)m(witz)h([152)q (].)40 b(W)-8 b(e)28 b(will)c(giv)m(e)165 5646 y(a)31 b(geometric)h(pro)s(of)d(b)s(elo)m(w.)1844 5895 y(80)p eop %%Page: 81 81 81 80 bop 306 304 a Fw(Remark)-5 b(ably)38 b(enough,)j(one)d(can)h(go)g (one)g(step)g(further:)55 b(eac)m(h)40 b(\(not)f(necessarily)e(p)s (erio)s(dic\))165 417 y(Gibbs)30 b(measure)h(for)g(the)h(Ising)e(mo)s (del)g(on)h(the)h(square)f(lattice)h(is)e(a)i(mixture)e(of)h(the)h (plus-phase)165 530 y(and)i(the)g(min)m(us-phase,)g(and)f(th)m(us)h (automatically)g(automorphism)e(in)m(v)-5 b(arian)m(t.)51 b(This)33 b(b)s(eautiful)165 643 y(result)42 b(w)m(as)h(obtained)f (indep)s(enden)m(tly)d(b)m(y)k(Aizenman)f([3)q(])g(and)g(Higuc)m(hi)g ([130)q(])h(based)f(on)h(the)165 756 y(w)m(ork)c(of)h(Russo)e([208)r (].)67 b(F)-8 b(or)39 b(more)h(general)f(t)m(w)m(o-dimensional)f (systems)h(the)g(absence)g(of)g(non-)165 869 y(translation-in)m(v)-5 b(arian)m(t)43 b(Gibbs)f(measures)i(at)h(su\016cien)m(tly)d(lo)m(w)i (temp)s(eratures)g(w)m(as)g(pro)m(v)m(ed)g(in)165 981 y([69)r(].)75 b(In)41 b(three)h(or)g(more)g(dimensions,)g(ho)m(w)m(ev)m (er,)k(non-translation)41 b(in)m(v)-5 b(arian)m(t)41 b(phases)g(of)h(the)165 1094 y(Ising)31 b(mo)s(del)f(do)h(exist;)h (this)f(is)f(a)i(famous)f(result)g(of)g(Dobrushin)f([66)q(],)i(see)g (also)f([22)r(])g(for)h(a)f(short)165 1207 y(pro)s(of.)165 1392 y Fx(Theorem)j(8.6)46 b(\(Aizenman{Higuc)m(hi\))38 b Fp(F)-7 b(or)38 b(the)f(Ising)h(ferr)-5 b(omagnet)39 b(on)e Fv(L)d Fw(=)f Fx(Z)3238 1359 y Fr(2)3315 1392 y Fp(with)38 b(no)165 1505 y(external)33 b(\014eld)g(and)f(inverse)g (temp)-5 b(er)g(atur)g(e)34 b Ft(\014)d(>)25 b(\014)1919 1519 y Fu(c)1954 1505 y Fp(,)32 b Ft(\026)2069 1467 y Fr(+)2069 1533 y Fu(\014)2159 1505 y Fp(and)h Ft(\026)2390 1467 y Fq(\000)2390 1533 y Fu(\014)2481 1505 y Fp(ar)-5 b(e)32 b(the)h(only)f(phases,)i(and)e(any)165 1618 y(other)i(Gibbs)f (me)-5 b(asur)g(e)34 b(is)e(a)h(mixtur)-5 b(e)34 b(of)f(these)g(two.) 165 1803 y Fw(The)44 b(pro)s(of)e(is)h(a)h(masterpiece)g(of)g (random-geometric)g(analysis)f(of)h(equilibrium)38 b(phases)44 b(and)165 1916 y(con)m(tains)35 b(v)-5 b(arious)34 b(ingenious)f (ideas,)j(but)e(unfortunately)f(it)h(is)g(to)s(o)i(long)e(to)i(b)s(e)e (sk)m(etc)m(hed)i(here.)165 2029 y(F)-8 b(or)36 b(the)f(full)e(result)h (w)m(e)h(th)m(us)g(need)g(to)h(refer)e(to)i(the)f(original)e(pap)s(ers) h(cited)h(ab)s(o)m(v)m(e,)j(as)d(w)m(ell)f(as)165 2142 y(to)f(the)f(surv)m(ey)g([4)q(].)46 b(Ho)m(w)m(ev)m(er,)35 b(to)e(pro)m(vide)e(an)h(idea)g(of)g(some)g(of)h(the)f(geometric)h (ideas)e(in)m(v)m(olv)m(ed)165 2254 y(w)m(e)i(will)d(no)m(w)i(giv)m(e)h (a)g(\(new\))g(geometric)g(pro)s(of)f(of)h(Prop)s(osition)d(8.5.)48 b(This)31 b(pro)s(of)g(resulted)h(from)165 2367 y(discussions)c(of)j (the)f(\014rst)g(author)g(with)f(Y.)i(Higuc)m(hi.)306 2480 y(In)e(this)g(pro)s(of)f(w)m(e)i(need)g(to)g(consider)e (in\014nite)g(clusters)g(in)h(halfplanes.)38 b(Here,)30 b(w)m(e)g(sa)m(y)h(that)f(a)165 2593 y(set)25 b Fv(H)h(\032)f Fx(Z)564 2560 y Fr(2)628 2593 y Fw(is)e(a)i(halfplane)d(if)i Fv(H)h Fw(is)e(a)i(translate)f(of)h(either)f(the)g(upp)s(er)e (halfplane)h Fv(f)p Ft(x)i Fw(=)g(\()p Ft(x)3321 2607 y Fr(1)3361 2593 y Ft(;)15 b(x)3453 2607 y Fr(2)3493 2593 y Fw(\))26 b Fv(2)165 2706 y Fx(Z)229 2673 y Fr(2)294 2706 y Fw(:)g Ft(x)397 2720 y Fr(2)462 2706 y Fv(\025)f Fw(0)p Fv(g)31 b Fw(or)f(its)g(complemen)m(t,)h(the)g(lo)m(w)m(er)g (halfplane,)d(or)j(a)g(translate)f(of)h(the)f(righ)m(t)h(and)e(left)165 2819 y(halfplanes)23 b(whic)m(h)g(are)h(similarly)d(de\014ned.)38 b(The)24 b(next)g(lemma)g(pro)m(vides)f(a)i(\014rst)f(step)g(in)f(the)i (pro)s(of)165 2932 y(of)31 b(Prop)s(osition)d(8.5.)165 3117 y Fx(Lemma)33 b(8.7)46 b Fp(Consider)35 b(the)e(Ising)h(ferr)-5 b(omagnet)35 b(on)f Fx(Z)2183 3084 y Fr(2)2222 3117 y Fp(,)g(and)g(let)g Ft(D)i Fp(b)-5 b(e)33 b(the)h(event)f(that)i(for)f (at)165 3230 y(le)-5 b(ast)30 b(one)g(halfplane)h Fv(H)e Fp(in)g Fx(Z)1197 3197 y Fr(2)1236 3230 y Fp(,)h(b)-5 b(oth)30 b Fv(H)f Fp(and)h Fv(H)1840 3197 y Fu(c)1903 3230 y Fp(c)-5 b(ontain)31 b(an)e(in\014nite)g(cluster)g(of)g(the)h (same)f(sign.)165 3343 y(Then)k Ft(\026)p Fw(\()p Ft(D)s Fw(\))26 b(=)f(1)33 b Fp(for)g(al)5 b(l)33 b Ft(\026)25 b Fv(2)g(G)5 b Fw(\()p Ft(\014)g(H)i Fw(\))34 b Fp(and)f Ft(\014)e(>)25 b(\014)1953 3357 y Fu(c)1988 3343 y Fp(.)165 3528 y Fx(Pro)s(of:)101 b Fw(Since)32 b(eac)m(h)j(Gibbs)c(measure)i(is) g(a)g(mixture)g(of)g(extremal)g(Gibbs)f(measures,)i(w)m(e)g(only)165 3640 y(need)d(to)i(sho)m(w)e(that)h Ft(\026)p Fw(\()p Ft(D)s Fw(\))27 b(=)g(1)k(for)g(an)m(y)h(extremal)g Ft(\026)p Fw(.)43 b(Supp)s(ose)30 b(the)h(con)m(trary)-8 b(.)45 b(Since)30 b Ft(D)k Fw(is)d(tail)165 3753 y(measurable,)f(it)f(then)h (follo)m(ws)f(that)h Ft(\026)p Fw(\()p Ft(D)s Fw(\))c(=)e(0)31 b(for)e(some)i(extremal)f Ft(\026)p Fw(.)40 b(W)-8 b(e)31 b(will)c(sho)m(w)j(that)g(this)165 3866 y(is)g(imp)s(ossible.)306 3979 y Fp(Step)35 b(1:)44 b Fw(Let)32 b Fv(H)g Fw(b)s(e)f(an)m(y)g (halfplane,)f Ft(r)k Fw(the)e(re\015ection)f(of)h Fx(Z)2413 3946 y Fr(2)2484 3979 y Fw(whic)m(h)e(maps)h Fv(H)h Fw(on)m(to)g Fv(H)3376 3946 y Fu(c)3411 3979 y Fw(,)g(and)165 4092 y Ft(\034)41 b Fw(:)31 b Ft(\033)j Fv(!)d(\000)p Ft(\033)36 b Fw(the)e(spin)e(\015ip)g(on)h(\012.)51 b(W)-8 b(e)35 b(sho)m(w)e(that)h Ft(\026)d Fw(=)g Ft(\026)22 b Fv(\016)h Ft(r)i Fv(\016)d Ft(\034)10 b Fw(.)51 b(Since)33 b Ft(\026)p Fw(\()p Ft(D)s Fw(\))e(=)f(0,)35 b(at)g(least)165 4205 y(one)28 b(of)h(the)f(halfplanes)d Fv(H)k Fw(and)e Fv(H)1367 4172 y Fu(c)1429 4205 y Fw(con)m(tains)h(no)g(in\014nite)d(min)m (us-cluster,)i(and)g(this)g(or)h(the)g(other)165 4318 y(halfplane)37 b(con)m(tains)j(no)e(in\014nite)f(plus-cluster.)64 b(In)38 b(view)h(of)g(the)g(tail)f(trivialit)m(y)f(of)i Ft(\026)p Fw(,)i(w)m(e)f(can)165 4431 y(assume)30 b(that)g Fv(H)h Fw(con)m(tains)f(no)g(in\014nite)d(min)m(us-cluster)h Ft(\026)p Fw(-almost)i(surely)-8 b(.)39 b(Hence,)31 b(for)f(an)m(y)g (giv)m(en)165 4544 y(\001)25 b Fv(2)g(E)36 b Fw(and)26 b Ft(\026)p Fw(-almost)i(ev)m(ery)g Ft(\030)h Fv(2)c Fw(\012,)j(there)f(exists)g(an)g Ft(r)s Fw(-symmetric)g(region)g (\000\()p Ft(\030)t Fw(\))e Fv(2)g(E)36 b Fw(suc)m(h)27 b(that)165 4657 y(\000\()p Ft(\030)t Fw(\))f Fv(\033)f Fw(\001)i(and)f Ft(\030)j Fv(\021)c Fw(1)j(on)f Ft(@)5 b Fw(\000\()p Ft(\030)t Fw(\))13 b Fv(\\)g(H)q Fw(.)40 b(The)27 b(last)f(prop)s(ert)m(y)h(implies)d(that)k Ft(\030)h Fv(\027)c Ft(r)16 b Fv(\016)d Ft(\034)d Fw(\()p Ft(\030)t Fw(\))28 b(on)f Ft(@)5 b Fw(\000\()p Ft(\030)t Fw(\),)165 4770 y(and)30 b(using)f(Lemma)h(4.13)i(and)e(the)h(\015ip-re\015ection) e(symmetry)h(of)g Ft(H)37 b Fw(w)m(e)31 b(\014nd)e(that)1056 4981 y Ft(\026)1111 4937 y Fu(\030)1111 5013 y(\014)s(;)p Fr(\000\()p Fu(\030)s Fr(\))1335 4981 y Fv(\027)1406 4995 y Fq(D)1492 4981 y Ft(\026)1547 4933 y Fu(r)r Fq(\016)p Fu(\034)8 b Fr(\()p Fu(\030)s Fr(\))1547 5013 y Fu(\014)s(;)p Fr(\000\()p Fu(\030)s Fr(\))1773 4981 y Fw(=)25 b Ft(\026)1924 4937 y Fu(\030)1924 5013 y(\014)s(;)p Fr(\000\()p Fu(\030)s Fr(\))2143 4981 y Fv(\016)c Ft(r)h Fv(\016)f Ft(\034)40 b Fw(on)31 b Fv(F)2610 4995 y Fr(\001)2698 4981 y Ft(:)165 5194 y Fw(Assuming)22 b(that)i(\000\()p Ft(\030)t Fw(\))g(is)e(maximal) g(in)g(a)i(large)g(b)s(o)m(x)f(\003)i Fv(\033)g Fw(\001,)g(w)m(e)f(can) f(apply)f(the)i(Mark)m(o)m(v)h(prop)s(ert)m(y)165 5307 y(of)35 b Ft(\026)e Fw(in)g(the)h(same)h(w)m(a)m(y)g(as)f(in)e(the)j (pro)s(of)e(of)h(Theorem)g(8.1.)53 b(This)32 b(yields)g(that)j Ft(\026)c Fv(\027)3193 5321 y Fq(D)3284 5307 y Ft(\026)23 b Fv(\016)g Ft(r)i Fv(\016)e Ft(\034)165 5420 y Fw(on)37 b Fv(F)363 5434 y Fr(\001)462 5420 y Fw(for)g(an)m(y)f(\001,)i(and)e (th)m(us)g Ft(\026)f Fv(\027)1474 5434 y Fq(D)1570 5420 y Ft(\026)24 b Fv(\016)h Ft(r)i Fv(\016)e Ft(\034)10 b Fw(.)59 b(\(The)36 b(preceding)f(argumen)m(t)i(is)f(a)h(v)-5 b(arian)m(t)36 b(of)165 5533 y(an)d(idea)f(of)h(Russo)f([208)r(].\))48 b(Using)32 b(the)h(absence)g(of)g(in\014nite)d(plus-clusters)g(in)i Fv(H)h Fw(or)g Fv(H)3260 5500 y Fu(c)3327 5533 y Fw(w)m(e)g(\014nd)165 5646 y(analogously)d(that)h Ft(\026)25 b Fv(\026)1006 5660 y Fq(D)1092 5646 y Ft(\026)20 b Fv(\016)h Ft(r)h Fv(\016)f Ft(\034)10 b Fw(.)41 b(Hence)31 b Ft(\026)25 b Fw(=)g Ft(\026)20 b Fv(\016)g Ft(r)j Fv(\016)e Ft(\034)40 b Fw(as)31 b(claimed.)1844 5895 y(81)p eop %%Page: 82 82 82 81 bop 306 304 a Fp(Step)41 b(2:)114 b Fw(Here)39 b(w)m(e)g(use)f(a)h(v)-5 b(arian)m(t)38 b(of)h(Zhang's)f(argumen)m(t)h (whic)m(h)e(w)m(as)i(explained)e(in)g(the)165 417 y(pro)s(of)29 b(of)h(Theorem)g(5.18.)42 b(T)-8 b(o)30 b(b)s(egin,)f(w)m(e)h(observ)m (e)g(that)h(the)f(comp)s(osition)e(of)i(t)m(w)m(o)h(re\015ections)e(in) 165 530 y(parallel)g(axes)j(is)d(a)i(translation.)41 b(Step)31 b(1)g(therefore)g(implies)d(that)j Ft(\026)g Fw(is)e(p)s(erio)s(dic)g(with)g(p)s(erio)s(d)f(2.)165 643 y(The)g(\015ip-re\015ection)e(symmetry)i(of)g Ft(\026)f Fw(implies)f(further)g(that)j Ft(\026)e Fw(is)g(di\013eren)m(t)h(from)f Ft(\026)3104 605 y Fr(+)3104 671 y Fu(\014)3191 643 y Fw(and)g Ft(\026)3420 605 y Fq(\000)3420 671 y Fu(\014)3479 643 y Fw(,)i(so)165 756 y(that)36 b(\(b)m(y)g(Theorem)f(8.2\))i(there)e (exist)g(b)s(oth)g(an)g(in\014nite)e(plus-)h(and)h(an)g(in\014nite)e (min)m(us-cluster)165 869 y Ft(\026)p Fw(-almost)e(surely)-8 b(.)41 b(By)31 b(the)f(Burton{Keane)i(uniqueness)c(theorem)j(5.17,)i (these)e(in\014nite)d(clusters)165 981 y(are)k(almost)g(surely)e (unique.)43 b(W)-8 b(e)32 b(no)m(w)g(c)m(ho)s(ose)h(a)e(square)h(\003) 27 b(=)g([)p Fv(\000)p Ft(n;)15 b(n)21 b Fv(\000)f Fw(1])2833 948 y Fr(2)2895 981 y Fv(\\)g Fx(Z)3040 948 y Fr(2)3111 981 y Fw(so)32 b(large)g(that)165 1111 y Ft(\026)p Fw(\(\003)417 1061 y Fr(+)361 1111 y Fv( )-15 b(!)42 b(1)p Fw(\))h Ft(>)f Fw(1)27 b Fv(\000)g Fw(10)1112 1078 y Fq(\000)p Fr(3)1207 1111 y Fw(.)72 b(Since)40 b Ft(\026)g Fw(is)g(extremal,)j Ft(\026)e Fw(has)f(p)s(ositiv)m(e)g(correlations.)71 b(By)41 b(the)165 1240 y(argumen)m(t)d(leading)e(to)i(\(23\))h(w)m(e)f (th)m(us)f(obtain)f(that)i Ft(\026)p Fw(\()p Ft(@)2195 1255 y Fu(k)2238 1240 y Fw(\003)2394 1190 y Fr(+)2338 1240 y Fv( )-15 b(!)36 b(1)p Fw(\))h Ft(>)g Fw(1)25 b Fv(\000)g Fw(10)3068 1207 y Fq(\000)p Fr(3)p Fu(=)p Fr(4)3271 1240 y Fw(for)37 b(some)165 1353 y Ft(k)32 b Fv(2)c(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)i(;)e Fw(4)p Fv(g)p Fw(,)34 b(where)e Ft(@)1087 1368 y Fu(k)1130 1353 y Fw(\003)g(is)g(the)g(in)m (tersection)g(of)g Ft(@)5 b Fw(\003)33 b(with)e(the)h Ft(k)s Fw('th)h(quadran)m(t)f(\(relativ)m(e)g(to)165 1466 y(the)h(axes)g Fv(f)p Ft(x)623 1480 y Fr(2)691 1466 y Fw(=)28 b Fv(\000)p Fw(1)p Ft(=)p Fw(2)p Fv(g)34 b Fw(and)e Fv(f)p Ft(x)1351 1480 y Fr(1)1419 1466 y Fw(=)c Fv(\000)p Fw(1)p Ft(=)p Fw(2)p Fv(g)p Fw(\).)49 b(F)-8 b(or)33 b(de\014niteness,)e(w)m(e)i(assume)f(that)h Ft(k)f Fw(=)c(1.)47 b(By)165 1579 y(the)31 b(\015ip-re\015ection)e(symmetry)-8 b(,)30 b(it)g(follo)m(ws)g(that)h(the)f(in)m(tersection)820 1791 y Fv(f)p Ft(@)913 1805 y Fr(1)954 1791 y Fw(\003)1098 1741 y Fr(+)1042 1791 y Fv( )-15 b(!)25 b(1)p Ft(;)45 b(@)1443 1805 y Fr(2)1483 1791 y Fw(\003)1627 1741 y Fq(\000)1572 1791 y Fv( )-15 b(!)25 b(1)p Ft(;)45 b(@)1973 1805 y Fr(3)2013 1791 y Fw(\003)2157 1741 y Fr(+)2101 1791 y Fv( )-15 b(!)25 b(1)p Ft(;)46 b(@)2503 1805 y Fr(4)2542 1791 y Fw(\003)2687 1741 y Fq(\000)2631 1791 y Fv( )-15 b(!)25 b(1g)165 1992 y Fw(has)i(probabilit)m(y)d(at)k(least) f(1)13 b Fv(\000)g Fw(4)g Fv(\001)g Fw(10)1430 1959 y Fq(\000)p Fr(3)p Fu(=)p Fr(4)1623 1992 y Ft(>)24 b Fw(0,)29 b(whic)m(h)c(is)h(imp)s(ossible)e(b)s(ecause)i(of)h(the)g(uniqueness) 165 2105 y(of)k(the)f(in\014nite)f(clusters.)39 b(This)29 b(con)m(tradiction)h(concludes)g(the)g(pro)s(of)g(of)h(the)f(lemma.)335 b Fk(2)165 2266 y Fx(Pro)s(of)29 b(of)f(Prop)s(osition)i(8.5:)81 b Fw(Let)24 b Ft(\026)g Fw(b)s(e)g(an)m(y)h(Gibbs)e(measure)h(in)m(v)-5 b(arian)m(t)23 b(under)g(\()p Ft(\022)3191 2280 y Fu(x)3235 2266 y Fw(\))3270 2286 y Fu(x)p Fq(2)p Fu(p)p Fn(Z)3443 2267 y Fl(2)3505 2266 y Fw(for)165 2379 y(some)33 b Ft(p)27 b(>)h Fw(1.)46 b(Using)31 b(the)h(ergo)s(dic)g(decomp)s(osition,)f(w)m (e)i(can)f(assume)g(that)g Ft(\026)g Fw(is)f(in)g(fact)i(ergo)s(dic)165 2492 y(with)27 b(resp)s(ect)g(to)h(this)f(group)g(of)g(translations.)39 b(By)28 b(Lemma)f(8.7,)j(there)d(exists)g(a)h(pair)e(\()p Fv(H)q Ft(;)15 b Fv(H)3443 2459 y Fu(c)3478 2492 y Fw(\))28 b(of)165 2605 y(halfplanes)g(suc)m(h)i(that,)h(with)e(p)s(ositiv)m(e)h (probabilit)m(y)-8 b(,)28 b(b)s(oth)i Fv(H)g Fw(and)g Fv(H)2622 2572 y Fu(c)2686 2605 y Fw(con)m(tain)h(in\014nite)d (clusters)165 2718 y(of)k(spins)e(of)i(the)f(same)h(constan)m(t)h (sign.)44 b(F)-8 b(or)32 b(de\014niteness,)f(supp)s(ose)f Fv(H)j Fw(is)d(the)i(upp)s(er)e(halfplane,)165 2831 y(and)39 b(the)h(sign)f(is)f(plus.)67 b(In)38 b(view)h(of)h(the)g(\014nite)e (energy)i(prop)s(ert)m(y)-8 b(,)42 b(it)d(then)g(follo)m(ws)g(that)h (also)165 2944 y Ft(\026)p Fw(\()p Ft(A)323 2958 y Fr(0)363 2944 y Fw(\))26 b Ft(>)f Fw(0,)31 b(where)f(for)g Ft(k)e Fv(2)d Fx(Z)1072 3161 y Ft(A)1140 3176 y Fu(k)1208 3161 y Fw(=)g Fv(f)p Fw(\()p Ft(k)s(;)15 b Fw(0\))1637 3111 y Fr(+)1581 3161 y Fv( )-15 b(!)25 b(1)30 b Fw(b)s(oth)g(in)f Fv(H)i Fw(and)f Fv(H)2578 3128 y Fu(c)2612 3161 y Fv(g)c Ft(:)165 3362 y Fw(Let)k Ft(A)g Fw(b)s(e)f(the)g(ev)m(en)m(t)i(that)f Ft(A)1206 3377 y Fu(k)1278 3362 y Fw(o)s(ccurs)f(for)g(in\014nitely)e (man)m(y)i Ft(k)g(<)c Fw(0)k(and)g(in\014nitely)e(man)m(y)i Ft(k)g(>)24 b Fw(0.)165 3475 y(The)30 b(horizon)m(tal)g(p)s(erio)s (dicit)m(y)d(and)j(P)m(oincar)m(\023)-43 b(e's)31 b(recurrence)f (theorem)h(\(or)f(the)h(ergo)s(dic)e(theorem\))165 3588 y(then)h(sho)m(w)h(that)g Ft(\026)p Fw(\()p Ft(A)953 3602 y Fr(0)1013 3588 y Fv(n)20 b Ft(A)p Fw(\))26 b(=)f(0,)31 b(and)f(therefore)h Ft(\026)p Fw(\()p Ft(A)p Fw(\))26 b Ft(>)e Fw(0.)306 3701 y(Next,)34 b(let)e Ft(B)37 b Fw(b)s(e)32 b(the)g(ev)m(en)m(t)i(that)e(there)h(exists)f(an)g (in\014nite)e(min)m(us-cluster.)44 b(W)-8 b(e)33 b(claim)f(that)165 3814 y Ft(\026)p Fw(\()p Ft(A)5 b Fv(\\)g Ft(B)g Fw(\))25 b(=)g(0.)39 b(Indeed,)23 b(supp)s(ose)f Ft(\026)p Fw(\()p Ft(A)5 b Fv(\\)g Ft(B)g Fw(\))24 b Ft(>)h Fw(0.)39 b(Since)22 b Ft(A)h Fw(is)e(tail)h(measurable)g(and)g(horizon)m(tally)165 3926 y(p)s(erio)s(dic,)27 b(w)m(e)j(can)f(use)f(the)h(\014nite)f (energy)i(prop)s(ert)m(y)e(and)g(horizon)m(tal)h(p)s(erio)s(dicit)m(y)d (of)j Ft(\026)g Fw(as)g(ab)s(o)m(v)m(e)165 4039 y(to)i(sho)m(w)g(that)g (the)f(ev)m(en)m(t)372 4240 y Ft(C)i Fw(=)25 b Ft(A)20 b Fv(\\)g(f)p Fw(\()p Ft(k)s(;)15 b Fw(0\))1067 4190 y Fq(\000)1011 4240 y Fv( )-15 b(!)25 b(1)31 b Fw(for)f(in\014nitely)d (man)m(y)j Ft(k)f(<)c Fw(0)31 b(and)e(in\014nitely)f(man)m(y)i Ft(k)f(>)c Fw(0)p Fv(g)165 4441 y Fw(has)33 b(p)s(ositiv)m(e)e (probabilit)m(y)-8 b(.)45 b(But)33 b(on)g Ft(C)38 b Fw(there)33 b(exist)g(in\014nitely)c(man)m(y)k(min)m(us-clusters,)e(whic)m(h)g(is) 165 4554 y(imp)s(ossible)c(b)m(y)k(the)f(Burton{Keane)h(theorem.)306 4667 y(T)-8 b(o)41 b(complete)f(the)f(pro)s(of,)j(w)m(e)e(note)g(that)g Ft(\026)p Fw(\()p Ft(B)5 b Fw(\))41 b Fv(\024)f Ft(\026)p Fw(\()p Ft(A)2352 4634 y Fu(c)2387 4667 y Fw(\))h Ft(<)g Fw(1,)h(and)d(th)m(us)g Ft(\026)p Fw(\()p Ft(B)5 b Fw(\))41 b(=)f(0)g(b)m(y)165 4780 y(ergo)s(dicit)m(y)-8 b(.)57 b(In)34 b(view)h(of)h(Theorem)f(8.2,)j(this)c(means)i(that)g Ft(\026)e Fw(=)f Ft(\026)2556 4741 y Fr(+)2556 4808 y Fu(\014)2615 4780 y Fw(.)56 b(In)35 b(the)g(case)i(considered,)165 4893 y(the)30 b(prop)s(osition)d(is)h(th)m(us)h(pro)m(v)m(ed.)40 b(The)29 b(other)h(cases)g(are)f(similar;)e(in)h(particular,)g(in)g (the)i(case)g(of)165 5005 y(negativ)m(e)i(sign)d(w)m(e)i(\014nd)e(that) i Ft(\026)25 b Fw(=)g Ft(\026)1455 4967 y Fq(\000)1455 5034 y Fu(\014)1514 5005 y Fw(.)2007 b Fk(2)165 5248 y FA(8.3)113 b(Constan)m(t-spin)39 b(clusters)e(in)h(the)f(P)m(otts)f (mo)s(del)165 5420 y Fw(Consider)42 b(the)i Ft(q)s Fw(-state)h(P)m (otts)g(mo)s(del)e(on)g(the)h(lattice)g Fv(L)k Fw(=)f Fx(Z)2470 5387 y Fu(d)2554 5420 y Fw(in)m(tro)s(duced)42 b(in)g(Section)i(3.3,)165 5533 y Ft(q)s(;)15 b(d)42 b Fv(\025)e Fw(2,)i(and)d(recall)g(the)h(results)f(of)g(Section)h(6.3)g (on)g(the)g(phase)f(transition)f(in)h(this)f(mo)s(del.)165 5646 y(The)j(p)s(erio)s(dic)e(ground)i(states)i(are)e(the)h(constan)m (t)h(con\014gurations)e Ft(\021)2664 5660 y Fu(i)2736 5646 y Fv(\021)j Ft(i)p Fw(,)g(1)h Fv(\024)e Ft(i)h Fv(\024)g Ft(q)s Fw(.)74 b(W)-8 b(e)1844 5895 y(82)p eop %%Page: 83 83 83 82 bop 165 315 a Fw(write)476 265 y Fu(i)404 315 y Fv( )-15 b(!)37 b Fw(for)f(the)h(agreemen)m(t)i(connectivit)m(y)e (relation)f(relativ)m(e)h(to)h Ft(\021)2712 329 y Fu(i)2740 315 y Fw(,)h(and)d(w)m(e)h(consider)f(the)165 428 y(limiting)28 b(Gibbs)i(measure)h Ft(\026)1179 395 y Fu(i)1179 454 y(\014)s(;q)1310 428 y Fw(at)h(in)m(v)m(erse)f(temp)s(erature)g Ft(\014)36 b Fw(asso)s(ciated)c(to)g Ft(\021)2921 442 y Fu(i)2949 428 y Fw(,)g(whic)m(h)d(exists)i(b)m(y)165 552 y(Prop)s(osition)25 b(6.9.)41 b(As)26 b(a)h(further)e(illustration) f(of)j(assertion)f(\(59\),)j(w)m(e)e(sho)m(w)g(that)g Ft(\026)3088 519 y Fu(i)3088 579 y(\014)s(;q)3215 552 y Fw(exhibits)e Ft(i)p Fw(-)165 665 y(p)s(ercolation)d(whenev)m(er)g (there)g(is)f(a)i(phase)f(transition.)36 b(This)21 b(is)g(a)i(P)m (otts-coun)m(terpart)h(of)e(Theorem)165 778 y(8.2.)55 b(F)-8 b(or)35 b(its)f(pro)s(of,)h(w)m(e)g(use)f(the)h(random-cluster)e (represen)m(tation)i(rather)f(than)h(Theorem)f(8.1)165 891 y(b)s(ecause)25 b(for)g Ft(q)j(>)d Fw(2)h(there)f(is)f(no)h(sto)s (c)m(hastic)h(monotonicit)m(y)g(a)m(v)-5 b(ailable)24 b(in)g(the)h(spin)e(con\014guration.)165 1072 y Fx(Theorem)34 b(8.8)46 b Fp(F)-7 b(or)29 b(the)g(Potts)f(mo)-5 b(del)30 b(on)e Fx(Z)1778 1039 y Fu(d)1847 1072 y Fp(at)g(any)h(inverse)f(temp) -5 b(er)g(atur)g(e)30 b Ft(\014)j Fp(with)c Fv(jG)5 b Fw(\()p Ft(\014)g(H)i Fw(\))p Fv(j)27 b Ft(>)165 1205 y Fw(1)p Fp(,)33 b(we)g(have)g Ft(\026)670 1172 y Fu(i)670 1232 y(\014)s(;q)770 1205 y Fw(\()p Ft(x)954 1156 y Fu(i)883 1205 y Fv( )-15 b(!)25 b(1)p Fw(\))g Ft(>)g Fw(0)33 b Fp(for)h(al)5 b(l)33 b Ft(x)25 b Fv(2)g Fx(Z)1903 1172 y Fu(d)1976 1205 y Fp(and)33 b Ft(i)26 b Fv(2)f(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)i(;)e(q)s Fv(g)p Fp(.)165 1387 y Fx(Pro)s(of:)86 b Fw(By)28 b(translation)g(in)m(v)-5 b(ariance)27 b(w)m(e)i(can)f(c)m(ho)s(ose)h Ft(x)c Fw(=)g(0.)41 b(In)27 b(Theorem)h(6.10)i(w)m(e)f(ha)m(v)m(e)g(seen)165 1500 y(that)g Ft(\036)414 1467 y Fr(1)414 1522 y Fu(p;q)508 1500 y Fw(\(0)d Fv($)f(1)p Fw(\))g(=)g Ft(c)h(>)f Fw(0)j(for)g Ft(\014)j(>)25 b(\014)1577 1514 y Fu(c)1612 1500 y Fw(,)k(where)f Ft(p)d Fw(=)g(1)16 b Fv(\000)g Ft(e)2284 1467 y Fq(\000)p Fr(2)p Fu(\014)2449 1500 y Fw(as)29 b(usual.)38 b(In)28 b(view)f(of)i(\(30\),)h(this)165 1613 y(means)f(that)h Ft(\036)692 1580 y Fr(1)692 1638 y Fu(p;q)r(;)p Fr(\003)854 1613 y Fw(\(0)c Fv($)f Fw(\003)1139 1580 y Fu(c)1174 1613 y Fw(\))h Fv(\025)f Ft(c)k Fw(for)g(all)f(\003)e Fv(3)e Fw(0.)41 b(But)29 b(for)g(the)h(Edw)m(ards-Sok)-5 b(al)28 b(coupling)f Ft(P)3465 1580 y Fu(i)3452 1638 y(p;q)r(;)p Fr(\003)165 1762 y Fw(of)35 b Ft(\026)328 1729 y Fu(i)328 1788 y(\014)s(;q)r(;)p Fr(\003)532 1762 y Fw(and)e Ft(\036)766 1729 y Fr(1)766 1787 y Fu(p;q)r(;)p Fr(\003)963 1762 y Fw(\(de\014ned)h(b)s(efore)g(Prop)s(osition)e(6.9\)) k(w)m(e)f(ha)m(v)m(e)h Fv(f)p Fw(0)d Fv($)f Fw(\003)2933 1729 y Fu(c)2968 1762 y Fv(g)h(\032)f(f)p Fw(0)3343 1712 y Fu(i)3272 1762 y Fv( )-15 b(!)32 b Fw(\003)3534 1729 y Fu(c)3569 1762 y Fv(g)165 1911 y Fw(almost)44 b(surely)-8 b(,)45 b(so)f(that)g Ft(\026)1161 1878 y Fu(i)1161 1938 y(\014)s(;q)r(;)p Fr(\003)1330 1911 y Fw(\(0)1529 1861 y Fu(i)1457 1911 y Fv( )-15 b(!)47 b Fw(\003)1734 1878 y Fu(c)1769 1911 y Fw(\))g Fv(\025)f Ft(c)p Fw(.)80 b(In)43 b(particular,)i Ft(\026)2755 1878 y Fu(i)2755 1938 y(\014)s(;q)r(;)p Fr(\003)2924 1911 y Fw(\(0)3123 1861 y Fu(i)3051 1911 y Fv( )-15 b(!)47 b Fw(\001)3341 1878 y Fu(c)3375 1911 y Fw(\))g Fv(\025)g Ft(c)165 2061 y Fw(whenev)m(er)35 b(0)f Fv(2)f Fw(\001)g Fv(\032)g Fw(\003.)56 b(Letting)35 b(\014rst)f(\003)g Fv(")f Fx(Z)1854 2028 y Fu(d)1930 2061 y Fw(and)i(then)f(\001)g Fv(")f Fx(Z)2575 2028 y Fu(d)2651 2061 y Fw(w)m(e)i(\014nd)f(that)i Ft(\026)3233 2028 y Fu(i)3233 2088 y(\014)s(;q)3333 2061 y Fw(\(0)3519 2011 y Fu(i)3448 2061 y Fv( )-15 b(!)165 2174 y(1)p Fw(\))26 b Fv(\025)f Ft(c)p Fw(,)31 b(and)e(the)i(theorem)g(follo)m(ws.)2057 b Fk(2)165 2334 y Fw(Next)22 b(w)m(e)f(ask)g(for)f(a)i(con)m(v)m(erse)g (stating)f(that)g(\\agreemen)m(t)i(p)s(ercolation)d(implies)e(phase)i (transition".)165 2447 y(As)30 b(w)m(e)f(already)g(noticed)g(in)f(the)i (case)g(of)g(the)f(Ising)f(mo)s(del,)h(this)f(can)i(b)s(e)e(exp)s (ected)i(to)g(hold)e(only)165 2560 y(in)c(the)i(case)g(of)f(a)h(planar) d(lattice.)40 b(But)25 b(then)g(a)h(coun)m(terpart)f(of)h(Corollary)d (8.4)k(do)s(es)e(indeed)e(hold,)165 2673 y(as)31 b(w)m(as)g(sho)m(wn)e (b)m(y)i(L.)f(Cha)m(y)m(es)h([56)q(].)165 2854 y Fx(Theorem)j(8.9)46 b Fp(F)-7 b(or)44 b(the)f(unique)f(Gibbs)h(me)-5 b(asur)g(e)44 b Ft(\026)2111 2869 y Fu(\014)s(;q)2254 2854 y Fp(of)f(the)g Ft(q)s Fp(-state)g(Potts)g(mo)-5 b(del)44 b(on)g(the)165 2967 y(squar)-5 b(e)27 b(lattic)-5 b(e)26 b Fx(Z)767 2934 y Fr(2)832 2967 y Fp(at)h(inverse)e(temp)-5 b(er)g(atur)g(e)28 b Ft(\014)j(<)25 b(\014)1959 2981 y Fu(c)1994 2967 y Fp(,)i(we)f(have)g Ft(\026)2434 2982 y Fu(\014)s(;q)2534 2967 y Fw(\()p Fv(9)33 b Fw(an)d(in\014nite)e Ft(i)p Fw(-cluster\))e(=)f(0)165 3080 y Fp(for)34 b(al)5 b(l)33 b Ft(i)25 b Fv(2)g(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)j(;)d(q)s Fv(g)p Fp(.)165 3261 y Fw(The)31 b(strategy)i(of)e(pro)m(ving)g(this)f (theorem)i(is)e(the)i(same)g(as)f(that)h(in)e(the)i(pro)s(of)e(of)i (Corollary)e(8.4.)165 3374 y(Supp)s(ose)38 b(the)h Ft(i)p Fw(-spins)e(p)s(ercolate)j(in)d Ft(\026)1564 3389 y Fu(\014)s(;q)1704 3374 y Fw(for)i(some)g Ft(i)p Fw(.)67 b(Then,)40 b(b)m(y)f(symmetry)-8 b(,)42 b(this)c(holds)f(for)165 3487 y(all)d Ft(i)p Fw(,)i(so)e(that)i (in)d(particular)f(the)j(1-spins)e Fp(and)j Fw(the)e(other)h(spins)d(p) s(ercolate.)54 b(Hence,)36 b(Theorem)165 3600 y(5.18)43 b(leads)d(to)i(a)f(con)m(tradiction,)j(pro)m(vided)39 b(w)m(e)i(can)g(sho)m(w)g(that)h(the)f(set)g(of)g(1's)h(has)e(p)s (ositiv)m(e)165 3713 y(correlations.)h(Theorem)30 b(8.9)h(th)m(us)f (follo)m(ws)g(from)g(the)g(follo)m(wing)f(lemma.)165 3894 y Fx(Lemma)k(8.10)46 b Fp(Consider)38 b(the)f(phase)g Ft(\026)1622 3861 y Fu(i)1622 3921 y(\014)s(;q)1759 3894 y Fp(of)f(the)h Ft(q)s Fp(-state)g(Potts)f(mo)-5 b(del)38 b(at)f(any)g(inverse)f(tem-)165 4007 y(p)-5 b(er)g(atur)g(e)39 b Ft(\014)f(>)32 b Fw(0)p Fp(,)38 b(and)g(let)f(the)g(mapping)h Ft(s)1705 4021 y Fu(i)1769 4007 y Fp(b)-5 b(e)37 b(de\014ne)-5 b(d)38 b(by)e(\(58\))i(with)g Ft(\021)e Fw(=)c Ft(\021)2953 4021 y Fu(i)2981 4007 y Fp(,)38 b Ft(i)33 b Fv(2)f(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)j(;)d(q)s Fv(g)p Fp(.)165 4120 y(Then)33 b(the)g(me)-5 b(asur)g(e)34 b Ft(\027)950 4135 y Fu(\014)s(;q)1076 4120 y Fw(=)25 b Ft(\026)1227 4087 y Fu(i)1227 4146 y(\014)s(;q)1348 4120 y Fv(\016)20 b Ft(s)1456 4082 y Fq(\000)p Fr(1)1456 4146 y Fu(i)1583 4120 y Fp(has)33 b(p)-5 b(ositive)34 b(c)-5 b(orr)g(elations.)165 4301 y Fx(Sk)m(etc)m(h)40 b(of)g(pro)s(of:)104 b Fw(By)34 b(symmetry)-8 b(,)36 b Ft(\027)1641 4316 y Fu(\014)s(;q)1776 4301 y Fw(do)s(es)d(not)i(dep)s(end)d(on)j Ft(i)p Fw(.)52 b(F)-8 b(or)35 b(de\014niteness)e(w)m(e)i(set)165 4414 y Ft(i)i Fw(=)g(1)g(in)f(the)h(follo)m(wing.)60 b(Since)36 b(the)i(prop)s(ert)m(y)e(of)i(p)s(ositiv)m(e)e(correlations)h(is)f (preserv)m(ed)h(under)165 4527 y(w)m(eak)26 b(limits,)e(it)h(is)f (su\016cien)m(t)g(to)i(consider)d(the)i(\014nite)f(v)m(olume)h(Gibbs)e (distribution)f Ft(\026)3153 4494 y Fr(1)3153 4553 y Fu(\014)s(;q)r(;)p Fr(\003)3346 4527 y Fw(and)j(its)165 4656 y(image)h Ft(\027)468 4671 y Fu(\014)s(;q)r(;)p Fr(\003)662 4656 y Fw(=)f Ft(\026)813 4623 y Fr(1)813 4682 y Fu(\014)s(;q)r(;)p Fr(\003)992 4656 y Fv(\016)10 b Ft(s)1090 4618 y Fq(\000)p Fr(1)1090 4681 y(1)1185 4656 y Fw(.)39 b(By)25 b(the)h(FK)m(G)g(inequalit)m(y)-8 b(,)25 b(Theorem)g(4.11,)j(it)d(is)g(further)f(su\016cien)m(t)165 4779 y(to)36 b(sho)m(w)e(that)h Ft(\027)756 4794 y Fu(\014)s(;q)r(;)p Fr(\003)960 4779 y Fw(is)f(monotone.)54 b(In)34 b(terms)g(of)h Ft(\026)2061 4746 y Fr(1)2061 4806 y Fu(\014)s(;q)r(;)p Fr(\003)2265 4779 y Fw(and)f(the)g(random)g(\014eld)f Ft(Y)53 b Fw(=)32 b Ft(s)3397 4793 y Fr(1)3436 4779 y Fw(\()p Ft(X)7 b Fw(\),)165 4892 y(this)30 b(means)g(that)h(the)f (conditional)f(probabilit)m(y)1149 5089 y Ft(q)1190 5103 y Fu(x)1233 5089 y Fw(\()p Ft(\030)t Fw(\))d(=)f Ft(\026)1524 5051 y Fr(1)1524 5111 y Fu(\014)s(;q)r(;)p Fr(\003)1693 5089 y Fw(\()p Ft(Y)20 b Fw(\()p Ft(x)p Fw(\))26 b(=)f(1)15 b Fv(j)g Ft(Y)46 b Fv(\021)25 b Ft(\030)35 b Fw(o\013)30 b Ft(x)p Fw(\))165 5307 y(is)g(increasing)f(in)h Ft(\030)g Fv(2)25 b(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)1165 5274 y Fn(Z)1215 5251 y Fo(d)1288 5307 y Fw(for)31 b(an)m(y)g Ft(x)25 b Fv(2)h Fw(\003.)42 b(Since)29 b(the)i(b)s(oundary)e (condition)g(is)h(\014xed)g(to)i(b)s(e)165 5420 y(equal)26 b(to)g(1)g(o\013)g(\003,)h(w)m(e)f(can)g(assume)g(that)g Ft(\030)k Fw(is)25 b(equal)g(to)h(1)g(o\013)h(\003,)g(and)e(it)g(is)g (su\016cien)m(t)g(to)h(pro)m(v)m(e)h(the)165 5533 y(inequalit)m(y)e Ft(q)624 5547 y Fu(x)667 5533 y Fw(\()p Ft(\030)t Fw(\))h Fv(\024)f Ft(q)944 5547 y Fu(x)987 5533 y Fw(\()p Ft(\030)1066 5500 y Fq(0)1090 5533 y Fw(\))h(for)g(an)m(y)g(t)m(w)m(o)h(suc)m(h)f Ft(\030)t(;)k(\030)1964 5500 y Fq(0)2013 5533 y Fw(that)c(di\013er)f (only)g(at)i(a)f(single)e(site)i Ft(y)i Fv(2)d Fw(\003)h(and)165 5646 y(are)h(suc)m(h)f(that)h Ft(\030)t Fw(\()p Ft(y)s Fw(\))e(=)g(0)i(and)e Ft(\030)1278 5613 y Fq(0)1302 5646 y Fw(\()p Ft(y)s Fw(\))g(=)g(1.)40 b(F)-8 b(or)27 b(suc)m(h)f Ft(\030)t(;)15 b(\030)2139 5613 y Fq(0)2163 5646 y Fw(,)27 b(the)f(inequalit)m(y)f Ft(q)2826 5660 y Fu(x)2869 5646 y Fw(\()p Ft(\030)t Fw(\))h Fv(\024)f Ft(q)3146 5660 y Fu(x)3189 5646 y Fw(\()p Ft(\030)3268 5613 y Fq(0)3292 5646 y Fw(\))i(simply)1844 5895 y(83)p eop %%Page: 84 84 84 83 bop 165 304 a Fw(means)32 b(that)g Ft(Y)20 b Fw(\()p Ft(x)p Fw(\))32 b(and)f Ft(Y)20 b Fw(\()p Ft(y)s Fw(\))32 b(are)g(p)s(ositiv)m(ely)e(correlated)i(under)e(the)i(conditional)e (distribution)165 417 y Ft(\026)220 435 y Fu(x;y)r Fq(j)p Fu(\030)404 417 y Fw(of)h Ft(\026)563 384 y Fr(1)563 443 y Fu(\014)s(;q)r(;)p Fr(\003)762 417 y Fw(giv)m(en)g(that)g Ft(Y)45 b Fv(\021)25 b Ft(\030)34 b Fw(o\013)d Fv(f)p Ft(x;)15 b(y)s Fv(g)p Fw(.)306 530 y(T)-8 b(o)45 b(sho)m(w)g(this)e(w)m (e)i(\014x)f Ft(x;)30 b(y)s(;)g(\030)t Fw(.)83 b(F)-8 b(or)45 b Ft(\026)1770 548 y Fu(x;y)r Fq(j)p Fu(\030)1924 530 y Fw(,)j(w)m(e)d(ha)m(v)m(e)g Ft(X)56 b Fv(\021)49 b Fw(1)c(on)f(the)g(complemen)m(t)h(of)165 643 y(\001)33 b(=)g Fv(f)p Ft(x;)15 b(y)s Fv(g)25 b([)e(f)p Ft(v)37 b Fv(2)c Fw(\003)24 b Fv(n)f(f)p Ft(x;)15 b(y)s Fv(g)35 b Fw(:)49 b Ft(\030)t Fw(\()p Ft(v)s Fw(\))34 b(=)f(0)p Fv(g)p Fw(.)56 b(W)-8 b(e)36 b(th)m(us)f(consider)f(the)i(graph)e Ft(G)i Fw(with)e(v)m(ertex)165 756 y(set)c(\001)f(and)f(edge)i(set)g Fv(B)s Fw(\(\001\))f(consisting)e(of)j(all)e(edges)h(of)g Fv(B)j Fw(with)c(b)s(oth)g(endp)s(oin)m(ts)f(lying)h(in)g(\001.)40 b(If)165 869 y(w)m(e)31 b(knew)e(that)i Ft(Y)20 b Fw(\()p Ft(x)p Fw(\))25 b(=)g Ft(Y)20 b Fw(\()p Ft(y)s Fw(\))26 b(=)f(0,)31 b(then)e Ft(\026)1721 887 y Fu(x;y)r Fq(j)p Fu(\030)1905 869 y Fw(w)m(ould)f(b)s(e)i(the)g(distribution)c(of)k(a)g (\()p Ft(q)23 b Fv(\000)c Fw(1\)-state)165 981 y(P)m(otts)39 b(mo)s(del)e(on)g Ft(G)i Fw(with)d(state)j(space)f Fv(f)p Fw(2)p Ft(;)15 b Fw(3)p Ft(;)g(:)g(:)g(:)k(;)c(q)s Fv(g)p Fw(.)63 b(No)m(w)39 b(that)f(w)m(e)g(don't)g(kno)m(w)f Ft(Y)20 b Fw(\()p Ft(x)p Fw(\))39 b(and)165 1094 y Ft(Y)20 b Fw(\()p Ft(y)s Fw(\),)37 b Ft(\026)473 1113 y Fu(x;y)r Fq(j)p Fu(\030)662 1094 y Fw(is)d(still)f(a)j(mo)s(di\014cation)e(of)h (this)f(\()p Ft(q)26 b Fv(\000)e Fw(1\)-state)37 b(P)m(otts)g(mo)s (del,)e(in)f(whic)m(h)g Ft(x)h Fw(and)f Ft(y)165 1207 y Fw(are)d(allo)m(w)m(ed)f(to)h(ha)m(v)m(e)h(the)f Ft(q)s Fw('th)f(spin)e(v)-5 b(alue)30 b(1.)306 1320 y(T)-8 b(o)36 b(describ)s(e)e(this)g(mo)s(di\014cation)g(w)m(e)i(supp)s(ose)e (\014rst)g(that)i Ft(x)f Fw(and)g Ft(y)j Fw(are)e(not)g(adjacen)m(t.)57 b(Let)165 1433 y Ft(n)220 1447 y Fu(x)299 1433 y Fw(b)s(e)35 b(the)h(n)m(um)m(b)s(er)e(of)i(neigh)m(b)s(ors)e Ft(v)39 b Fw(of)d Ft(x)g Fw(with)e Ft(\030)t Fw(\()p Ft(v)s Fw(\))h(=)f(1,)j (and)e(de\014ne)g Ft(n)2855 1447 y Fu(y)2932 1433 y Fw(accordingly)-8 b(.)56 b(The)165 1546 y(probabilit)m(y)28 b(w)m(eigh)m(t)j(of)g Ft(\026)1077 1565 y Fu(x;y)r Fq(j)p Fu(\030)1261 1546 y Fw(then)f(con)m(tains)g(the)h(additional)d(biasing)h(factor)1195 1750 y(exp[2)p Ft(\014)5 b Fw(\()p Ft(n)1550 1764 y Fu(x)1594 1750 y Ft(I)1634 1769 y Fq(f)p Fu(X)g Fr(\()p Fu(x)p Fr(\)=1)p Fq(g)1977 1750 y Fw(+)20 b Ft(n)2123 1764 y Fu(y)2164 1750 y Ft(I)2204 1769 y Fq(f)p Fu(X)5 b Fr(\()p Fu(y)r Fr(\)=1)p Fq(g)2524 1750 y Fw(\)])165 1954 y(whic)m(h)38 b(acts)i(lik)m(e)e(an)h(external)g(\014eld)f(at)i Ft(x)e Fw(and)h Ft(y)s Fw(.)66 b(F)-8 b(or)40 b(this)e(mo)s(di\014ed)f(P)m (otts)j(mo)s(del,)g(w)m(e)g(can)165 2067 y(still)24 b(de\014ne)i(a)g (mo)s(di\014ed)e(random-cluster)i(represen)m(tation)g(whic)m(h)f(giv)m (es)h(an)m(y)h(edge)g(con\014guration)165 2180 y Ft(\020)32 b Fv(2)25 b(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)543 2147 y Fq(B)r Fr(\(\001\))741 2180 y Fw(a)31 b(probabilit)m(y)d(prop)s (ortional)g(to)442 2385 y(\()p Ft(q)c Fv(\000)c Fw(1\))713 2347 y Fu(k)r Fr(\()p Fu(\020)5 b Fr(\))846 2385 y Fw(\()p Ft(q)24 b Fv(\000)c Fw(1)g(+)g Ft(e)1235 2347 y Fr(2)p Fu(\014)s(n)1356 2355 y Fo(x)1400 2385 y Fw(\))1435 2347 y Fu(k)1472 2355 y Fo(x)1511 2347 y Fr(\()p Fu(\020)5 b Fr(\))1605 2385 y Fw(\()p Ft(q)24 b Fv(\000)c Fw(1)g(+)g Ft(e)1994 2347 y Fr(2)p Fu(\014)s(n)2115 2355 y Fo(y)2157 2385 y Fw(\))2192 2347 y Fu(k)2229 2355 y Fo(y)2267 2347 y Fr(\()p Fu(\020)5 b Fr(\))2444 2304 y Fs(Y)2377 2492 y Fu(e)p Fq(2B)r Fr(\(\001\))2633 2385 y Ft(p)2679 2347 y Fu(\020)g Fr(\()p Fu(e)p Fr(\))2806 2385 y Fw(\(1)21 b Fv(\000)f Ft(p)p Fw(\))3079 2347 y Fr(1)p Fq(\000)p Fu(\020)5 b Fr(\()p Fu(e)p Fr(\))3312 2385 y Ft(:)165 2696 y Fw(Here)30 b Ft(p)25 b Fw(=)g(1)17 b Fv(\000)f Ft(e)737 2663 y Fq(\000)p Fr(2)p Fu(\014)875 2696 y Fw(,)29 b Ft(k)s Fw(\()p Ft(\020)7 b Fw(\))29 b(is)f(the)h(n)m(um)m(b)s(er)e (of)i(connected)h(comp)s(onen)m(ts)e Fp(excluding)k(singletons)g(at)165 2808 y Ft(x)g Fp(or)g Ft(y)s Fw(,)d(and)f Ft(k)689 2822 y Fu(x)734 2808 y Fw(\()p Ft(\020)7 b Fw(\))29 b(and)f Ft(k)1102 2822 y Fu(y)1144 2808 y Fw(\()p Ft(\020)7 b Fw(\))29 b(are)h(the)f(indicator)e(functions)h(of)h(ha)m(ving)g(a)g (singleton)f(connected)165 2921 y(comp)s(onen)m(t)33 b(at)h Ft(x)e Fw(resp.)g Ft(y)s Fw(.)48 b(A)32 b(spin)f (con\014guration)h(with)f(distribution)e Ft(\026)2781 2940 y Fu(x;y)r Fq(j)p Fu(\030)2968 2921 y Fw(is)i(then)i(obtained)165 3034 y(from)41 b(the)g(edge)h(con\014guration)f(b)m(y)g(assigning)f (spins)f(at)j(random)e(uniformly)e(from)j Fv(f)p Fw(2)p Ft(;)15 b(:)g(:)g(:)i(;)e(q)s Fv(g)165 3147 y Fw(to)42 b(connected)f(comp)s(onen)m(ts,)j(except)e(for)e(a)h(singleton)f(at)h Ft(x)p Fw(,)i(where)e(the)f(spin)f(is)h(tak)m(en)i(from)165 3260 y Fv(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)j(;)d(q)s Fv(g)41 b Fw(with)e(probabilities)e(prop)s(ortional)h(to)j(\()p Ft(e)2067 3227 y Fr(2)p Fu(\014)s(n)2188 3235 y Fo(x)2232 3260 y Ft(;)15 b Fw(1)p Ft(;)g(:)g(:)g(:)i(;)e Fw(1\),)45 b(and)39 b(similarly)e(for)k(a)f(sin-)165 3373 y(gleton)k(at)g Ft(y)s Fw(.)79 b(Just)42 b(as)i(in)e(Corollary)f(6.5,)48 b(this)42 b(represen)m(tation)i(giv)m(es)f(the)g(desired)f(p)s(ositiv)m (e)165 3486 y(correlation)37 b(of)h Ft(Y)20 b Fw(\()p Ft(x)p Fw(\))38 b(and)f Ft(Y)20 b Fw(\()p Ft(y)s Fw(\))38 b(under)e Ft(\026)1704 3504 y Fu(x;y)r Fq(j)p Fu(\030)1858 3486 y Fw(,)j(pro)m(vided)d(w)m(e)i(can)g(sho)m(w)f(that)h Ft(k)3101 3500 y Fu(x)3183 3486 y Fw(and)f Ft(k)3414 3500 y Fu(y)3493 3486 y Fw(are)165 3599 y(p)s(ositiv)m(ely)30 b(correlated)i(in)e(the)i(mo)s(di\014ed)d(random-cluster)i(mo)s(del.)43 b(Since)30 b(these)i(indicator)f(v)-5 b(ari-)165 3712 y(ables)38 b(are)g(decreasing,)i(it)e(su\016ces)f(to)i(c)m(hec)m(k)h (that)e(the)h(mo)s(di\014ed)d(random-cluster)g(mo)s(del)h(has)165 3825 y(p)s(ositiv)m(e)26 b(correlations,)h(whic)m(h)e(follo)m(ws)g (from)h(Theorem)g(4.11)i(b)m(y)f(v)m(erifying)e(that)i(it)f(is)f (monotone;)165 3938 y(this,)30 b(ho)m(w)m(ev)m(er,)i(is)d(similar)f(to) j(Lemma)g(6.6.)306 4050 y(The)k(case)h(when)e Ft(x)g Fw(and)h Ft(y)i Fw(are)e(neigh)m(b)s(ors)f(is)g(handled)f(analogously;) j(in)e(fact,)j(the)e(p)s(ositiv)m(e)165 4163 y(correlation)30 b(can)h(only)f(b)s(ecome)g(stronger)h(when)e Ft(x)i Fw(and)e Ft(y)k Fw(ha)m(v)m(e)f(an)e(edge)h(in)e(common.)298 b Fk(2)165 4407 y FA(8.4)113 b(F)-9 b(urther)37 b(examples)i(of)f (agreemen)m(t)g(p)s(ercolation)165 4578 y Fw(Here)j(w)m(e)g(treat)g (the)g(Ising)e(an)m(tiferromagnet,)k(the)e(hard-core)f(lattice)h(gas,)i (and)d(the)g(Widom{)165 4691 y(Ro)m(wlinson)29 b(lattice)i(mo)s(del,)e (and)h(shortly)f(men)m(tion)h(the)h(Ashkin{T)-8 b(eller)28 b(mo)s(del.)165 4854 y Fp(The)41 b(Ising)g(antiferr)-5 b(omagnet)p Fw(.)70 b(Consider)37 b(the)j(setting)f(of)g(Section)h (3.2.)68 b(W)-8 b(e)40 b(need)f(to)h(assume)165 4967 y(that)34 b(the)e(underlying)e(lattice)j Fv(L)f Fw(is)g(bipartite,)g (and)g(th)m(us)g(splits)f(o\013)i(in)m(to)f(t)m(w)m(o)i(parts,)f Fv(L)3286 4981 y Fu(ev)r(en)3468 4967 y Fw(and)165 5080 y Fv(L)228 5095 y Fu(odd)339 5080 y Fw(.)39 b(If)26 b Fv(j)p Ft(h)p Fv(j)g Ft(<)f Fw(2)p Ft(d)p Fw(,)j(there)e(exist)g(t)m(w) m(o)h(p)s(erio)s(dic)d(ground)h(states,)j Ft(\021)2439 5094 y Fu(ev)r(en)2615 5080 y Fw(and)d Ft(\021)2832 5095 y Fu(odd)2968 5080 y Fw(=)g Fv(\000)p Ft(\021)3180 5094 y Fu(ev)r(en)3329 5080 y Fw(,)j(where)165 5193 y Ft(\021)210 5207 y Fu(ev)r(en)385 5193 y Fv(\021)d Fw(1)i(on)g Fv(L)739 5207 y Fu(ev)r(en)915 5193 y Fw(and)g Ft(\021)1134 5207 y Fu(ev)r(en)1309 5193 y Fv(\021)e(\000)p Fw(1)i(on)g Fv(L)1734 5208 y Fu(odd)1844 5193 y Fw(.)40 b(\(There)27 b(are)h(no)f(other)g(p)s(erio)s(dic)e(ground)h(states,)165 5306 y(see)33 b(for)g(example)f([68)q(].\))48 b(The)32 b(phase)g(transition)f(in)g(this)g(mo)s(del)h(has)g(b)s(een)g(studied)f (in)g([65)q(])i(and)165 5419 y([129)r(].)73 b(Because)42 b(of)f(the)h(bipartite)d(structure,)44 b(w)m(e)d(can)g(\015ip)f(all)g (spins)f(on)h(a)i(sublattice)e(as)h(in)165 5532 y(\(6\),)d(whic)m(h)c (turns)g(the)h(mo)s(del)f(in)m(to)h(an)g(Ising)f(ferromagnet)i(in)e(a)h (staggered)i(magnetic)e(\014eld)f(of)165 5644 y(alternating)c(sign)g (on)g Fv(L)1007 5658 y Fu(ev)r(en)1186 5644 y Fw(and)g Fv(L)1426 5659 y Fu(odd)1537 5644 y Fw(.)40 b(The)30 b(latter)h(mo)s(del)e(still)f(satis\014es)i(the)h(FK)m(G)g(inequalit)m (y)-8 b(.)1844 5895 y(84)p eop %%Page: 85 85 85 84 bop 165 304 a Fw(As)24 b(p)s(oin)m(ted)g(out)g(in)f(Section)h (3.2,)i(for)e Ft(h)i Fw(=)f(0)g(there)f(is)f(a)i(one-to-one)h(corresp)s (ondence)d(b)s(et)m(w)m(een)i(all)165 417 y(Gibbs)j(measures)i(for)f (the)h(Ising)e(ferromagnet)i(and)f(the)h(Ising)e(an)m(tiferromagnet.)42 b(In)29 b(particular,)165 530 y(b)s(oth)22 b(mo)s(dels)f(then)g(ha)m(v) m(e)j(the)e(same)h(critical)e(in)m(v)m(erse)h(temp)s(erature)g Ft(\014)2587 544 y Fu(c)2622 530 y Fw(.)38 b(F)-8 b(or)23 b(general)f Fv(j)p Ft(h)p Fv(j)k Ft(<)f Fw(2)p Ft(d)p Fw(,)g(w)m(e)165 643 y(still)c(ha)m(v)m(e)i(t)m(w)m(o)h(limiting)c (Gibbs)g(measures)j Ft(\026)1724 599 y Fu(\021)1759 607 y Fo(ev)r(en)1724 671 y Fu(\014)1919 643 y Fw(and)f Ft(\026)2143 599 y Fu(\021)2178 611 y Fo(odd)2143 671 y Fu(\014)2281 643 y Fw(,)i(and)e(these)h(measures)f(ha)m(v)m(e)i(p)s(ositiv)m(e)165 756 y(correlations)k(relativ)m(e)h(to)g(the)f(\\staggered")j(ordering)c Ft(\033)i Fv(\026)c Ft(\033)2322 723 y Fq(0)2373 756 y Fw(i\013)j Ft(\033)s Fw(\()p Ft(x)p Fw(\))p Ft(\021)2702 770 y Fu(ev)r(en)2852 756 y Fw(\()p Ft(x)p Fw(\))e Fv(\024)f Ft(\033)3151 723 y Fq(0)3175 756 y Fw(\()p Ft(x)p Fw(\))p Ft(\021)3342 770 y Fu(ev)r(en)3492 756 y Fw(\()p Ft(x)p Fw(\))165 869 y(for)31 b(all)e Ft(x)d Fv(2)f(L)p Fw(.)40 b(Relativ)m(e)31 b(to)h(this)d(ordering,)h(an)g(analogue)h(of)g(the)f (sandwic)m(hing)f(inequalit)m(y)g(\(15\))165 981 y(holds;)e(for)f(more) h(details)f(see)h(Section)f(9)h(of)g([198)q(].)40 b(Here)27 b(is)f(a)h(v)m(ersion)f(of)g(statemen)m(t)j(\(59\))f(for)e(this)165 1094 y(mo)s(del.)165 1267 y Fx(Theorem)34 b(8.11)47 b Fp(Consider)38 b(the)g(Ising)f(antiferr)-5 b(omagnet)40 b(on)d(a)h(bip)-5 b(artite)39 b(gr)-5 b(aph)39 b Fw(\()p Fv(L)p Ft(;)15 b Fv(\030)p Fw(\))37 b Fp(in)h(an)165 1380 y(external)c(\014eld)f Ft(h)g Fp(at)g(any)h(inverse)e(temp)-5 b(er)g(atur)g(e)35 b Ft(\014)c(>)25 b Fw(0)p Fp(.)42 b(If)32 b Fv(jG)5 b Fw(\()p Ft(\014)g(H)i Fw(\))p Fv(j)27 b Ft(>)e Fw(1)p Fp(,)33 b(we)g(have)1475 1578 y Ft(\026)1530 1533 y Fu(\021)1565 1541 y Fo(ev)r(en)1530 1606 y Fu(\014)1704 1578 y Fw(\()p Ft(x)1816 1522 y Fu(\021)1851 1530 y Fo(ev)r(en)1818 1578 y Fv( )-15 b(!)26 b(1)p Fw(\))g Ft(>)f Fw(0)165 1764 y Fp(for)34 b(al)5 b(l)33 b Ft(x)25 b Fv(2)g(L)p Fp(.)165 1937 y Fw(This)40 b(follo)m(ws)g(from)h(Theorem)g(8.1)h(in)e (the)i(same)f(w)m(a)m(y)h(as)g(Theorem)f(8.2.)74 b(F)-8 b(or)42 b Fv(L)h Fw(=)g Fx(Z)3379 1904 y Fr(2)3419 1937 y Fw(,)h(the)165 2050 y(ob)m(vious)26 b(coun)m(terparts)h(of)g (Corollary)e(8.4)j(and)e(Prop)s(osition)e(8.5)k(are)f(also)f(v)-5 b(alid)25 b(since)h(the)h(pro)s(ofs)165 2163 y(of)k(these)g(results)e (carry)h(o)m(v)m(er)i(to)f(the)g(case)g(of)g(a)f(staggered)i(external)e (\014eld.)165 2320 y Fp(The)35 b(har)-5 b(d-c)g(or)g(e)38 b(lattic)-5 b(e)35 b(gas.)96 b Fw(As)32 b(w)m(e)h(ha)m(v)m(e)h(seen)f (in)e(Section)i(3.4,)h(this)e(mo)s(del)f(has)i(state)h(space)165 2433 y Ft(S)47 b Fw(=)41 b Fv(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)42 b Fw(and)d(corresp)s(onds)g(to)i(setting)f Ft(U)10 b Fw(\()p Ft(a;)15 b(b)p Fw(\))42 b(=)f Fv(1)p Ft(I)2320 2451 y Fq(f)p Fu(a)p Fr(=)p Fu(b)p Fr(=1)p Fq(g)2647 2433 y Fw(and)f Ft(V)20 b Fw(\()p Ft(a)p Fw(\))42 b(=)f Fv(\000)p Ft(a)15 b Fw(log)h Ft(\025)40 b Fw(in)165 2546 y(\(1\),)d Ft(a;)15 b(b)33 b Fv(2)f Ft(S)5 b Fw(;)36 b Ft(\025)c(>)g Fw(0)j(is)f(an)g(activit)m(y)h(parameter.)54 b(The)34 b(hard-core)g(mo)s(del)g(is)f(the)i(limit)d(of)j(the)165 2659 y(Ising)h(an)m(tiferromagnet)i(for)f Ft(\014)k Fv(!)c(1)g Fw(and)f Ft(h)h Fv(!)f Fw(2)p Ft(d)i Fw(along)f Ft(\014)5 b Fw(\(2)p Ft(d)26 b Fv(\000)e Ft(h)p Fw(\))37 b(=)2878 2623 y Fr(1)p 2878 2638 36 4 v 2878 2690 a(2)2939 2659 y Fw(log)16 b Ft(\025)p Fw(,)39 b(pro)m(vided)d(a)165 2772 y(con\014guration)26 b Ft(\033)j Fv(2)24 b(f\000)p Fw(1)p Ft(;)15 b Fw(+1)p Fv(g)1240 2739 y Fq(L)1321 2772 y Fw(is)26 b(mapp)s(ed)e(to)k(\(1)12 b Fv(\000)g Ft(\033)s Fw(\))p Ft(=)p Fw(2)27 b Fv(2)e(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)2546 2739 y Fq(L)2601 2772 y Fw(;)28 b(see)f([68)q(])g(for) f(details.)38 b(\(The)165 2885 y(phase)30 b(diagram)f(p)s(oin)m(t)g Ft(h)d Fw(=)f(2)p Ft(d;)15 b(\014)31 b Fw(=)25 b(+)p Fv(1)30 b Fw(of)g(the)g(Ising)e(an)m(tiferromagnet)j(is)e(highly)f (degenerate)165 2998 y(since)38 b(there)h(are)h(in\014nitely)35 b(man)m(y)-8 b(,)42 b(in)37 b(general)i(nonp)s(erio)s(dic,)f(ground)g (states.\))67 b(F)-8 b(or)40 b Fv(L)f Fw(=)g Fx(Z)3549 2965 y Fu(d)3589 2998 y Fw(,)165 3110 y(the)30 b(hard-core)g(lattice)h (gas)f(can)h(b)s(e)e(seen)h(as)g(a)g(gas)h(of)f(hard)f(\(i.e.,)h(non-o) m(v)m(erlapping\))g(diamonds.)165 3223 y(In)37 b(general,)i(w)m(e)e (still)e(assume)i(that)h Fv(L)f Fw(is)f(bipartite.)60 b(F)-8 b(or)38 b Ft(\025)e(>)g Fw(1,)k(the)d(hard-core)g(mo)s(del)f (then)165 3336 y(has)30 b(t)m(w)m(o)h(p)s(erio)s(dic)c(ground)i(states) i(of)f(c)m(hessb)s(oard)f(t)m(yp)s(e,)h(namely)f Ft(\021)2564 3350 y Fu(ev)r(en)2743 3336 y Fw(whic)m(h)g(is)f(equal)i(to)g(1)g(on) 165 3449 y Fv(L)228 3463 y Fu(ev)r(en)410 3449 y Fw(and)i(0)h (otherwise,)g(and)f Ft(\021)1324 3464 y Fu(odd)1465 3449 y Fw(=)d(1)22 b Fv(\000)f Ft(\021)1769 3463 y Fu(ev)r(en)1919 3449 y Fw(.)48 b(As)32 b(noticed)h(in)e(Section)i(4.4,)i(the)e(asso)s (ciated)165 3562 y(limiting)i(Gibbs)h(states)i Ft(\026)1102 3529 y Fu(ev)r(en)1102 3588 y(\025)1289 3562 y Fw(and)f Ft(\026)1528 3529 y Fu(odd)1528 3588 y(\025)1676 3562 y Fw(exist.)62 b(So,)39 b(follo)m(wing)d(the)i(program)f(stated)h(in)e (\(59\),)165 3675 y(w)m(e)e(ma)m(y)f(ask)g(whether)f(these)h(Gibbs)e (measures)h(exibit)g(agreemen)m(t)i(p)s(ercolation)e(in)f(the)i(case)h (of)165 3788 y(phase)c(transition.)40 b(The)30 b(answ)m(er)g(is)f (again)i(p)s(ositiv)m(e:)165 3961 y Fx(Theorem)j(8.12)47 b Fp(F)-7 b(or)28 b(the)g(har)-5 b(d-c)g(or)g(e)30 b(mo)-5 b(del)29 b(on)e(a)h(bip)-5 b(artite)29 b(gr)-5 b(aph)29 b Fv(L)e Fp(we)g(have)h(for)g(any)g(activity)165 4074 y Ft(\025)e(>)f Fw(0)p Fp(:)41 b(If)31 b Ft(\026)603 4041 y Fu(ev)r(en)603 4100 y(\025)777 4074 y Fv(6)p Fw(=)25 b Ft(\026)928 4041 y Fu(odd)928 4100 y(\025)1070 4074 y Fp(then)32 b Ft(\026)1326 4041 y Fu(ev)r(en)1326 4100 y(\025)1475 4074 y Fw(\()15 b Ft(x)1602 4018 y Fu(\021)1637 4026 y Fo(ev)r(en)1604 4074 y Fv( )-15 b(!)26 b(1)p Fw(\))g Ft(>)f Fw(0)31 b Fp(for)h(al)5 b(l)32 b Ft(x)25 b Fv(2)g(L)p Fp(,)31 b(and)h(similarly)h(with)f(`o)-5 b(dd')165 4187 y(in)33 b(plac)-5 b(e)34 b(of)e(`even)-9 b('.)165 4360 y Fw(This)32 b(result)h(is)g(completely)h(analogous)g(to)h(Theorem)e (8.11,)k(and)c(w)m(as)h(conjectured)g(b)m(y)g(Hu)g(and)165 4473 y(Mak)k([135)r(,)f(136)q(])g(from)g(computer)g(sim)m(ulations.)59 b(In)36 b(these)h(pap)s(ers,)h(the)f(authors)g(also)g(discuss)165 4586 y(the)c(case)h(of)e(hard-core)h(particles)e(on)i(a)g(triangular)d (lattice,)k(the)f(hard)e(hexagon)i(mo)s(del.)46 b(While)165 4699 y(Theorem)30 b(8.12)h(do)s(es)e(apply)g(to)h(the)g(hard)e (triangle)h(mo)s(del)g(on)g(the)h(hexagonal)g(lattice)g(\(whic)m(h)f (is)165 4811 y(bipartite\),)f(the)h(non-bipartite)d(triangular)h (lattice)i(with)d(nearest-neigh)m(b)s(or)i(b)s(onds)f(is)g(excluded.) 165 4924 y(The)g(results)g(of)g([135)r(,)h(136)q(])g(suggest)g(that)g (Theorem)f(8.12)i(still)d(holds)g(for)h(the)h(triangular)d(lattice.)165 5037 y(A)31 b(geometric)g(pro)s(of)f(of)g(this)g(conjecture)h(w)m(ould) e(b)s(e)h(of)g(particular)f(in)m(terest.)306 5150 y(The)i(hard-core)g (mo)s(del)f(on)h(the)g(square)g(lattice)g Fx(Z)2083 5117 y Fr(2)2153 5150 y Fw(admits)f(an)h(analogue)g(to)h(Corollary)e(8.4,) 165 5263 y(in)35 b(that)i(non)m(uniqueness)d(of)i(the)g(Gibbs)f (measure)h(is)f(equiv)-5 b(alen)m(t)35 b(to)i Ft(\021)2685 5277 y Fu(ev)r(en)2835 5263 y Fw(-p)s(ercolation)e(for)h(the)165 5376 y(Gibbs)29 b(measure)h Ft(\026)838 5343 y Fu(ev)r(en)838 5402 y(\025)987 5376 y Fw(;)h(see)g([100)r(])f(or)g([117)r(])h(for)f (more)g(details.)165 5533 y Fp(The)37 b(Widom{R)-5 b(ow)5 b(linson)39 b(lattic)-5 b(e)38 b(mo)-5 b(del.)106 b Fw(Consider)32 b(the)i(set-up)h(of)f(Section)g(3.5,)j(with)c(equal)165 5646 y(activities)i Ft(\025)610 5660 y Fr(+)701 5646 y Fw(=)d Ft(\025)857 5660 y Fq(\000)949 5646 y Fw(=)g Ft(\025)g(>)g Fw(0)j(for)g(the)f(plus)f(and)h(min)m(us)f(particles.)53 b(F)-8 b(or)35 b Ft(\025)e(>)f Fw(1)j(w)m(e)g(ha)m(v)m(e)h(t)m(w)m(o) 1844 5895 y(85)p eop %%Page: 86 86 86 85 bop 165 304 a Fw(distinct)40 b(p)s(erio)s(dic)e(ground)h(states)j Ft(\021)1496 318 y Fr(+)1598 304 y Fv(\021)g Fw(+1)f(and)f Ft(\021)2100 318 y Fq(\000)2202 304 y Fv(\021)i(\000)p Fw(1.)72 b(F)-8 b(rom)41 b(Section)g(4.4)h(w)m(e)f(kno)m(w)165 417 y(that)30 b(the)g(asso)s(ciated)g(limiting)d(Gibbs)h(measures)h Ft(\026)1994 379 y Fr(+)1994 445 y Fu(\025)2078 417 y Fw(=)c(lim)2300 432 y Fr(\003)p Fq("L)2452 417 y Ft(\026)2507 370 y Fu(\021)2542 379 y Fl(+)2507 445 y Fu(\025;)p Fr(\003)2650 417 y Fw(and)k Ft(\026)2881 379 y Fq(\000)2881 445 y Fu(\025)2970 417 y Fw(exist.)40 b(Moreo)m(v)m(er,)165 530 y(Theorem)h(4.17)h(asserts)f(that)h(a)f(phase)f(transition)g(o)s (ccurs)g(for)h(some)g(activit)m(y)g Ft(\025)g Fw(if)f(and)g(only)165 643 y(if)k Ft(\026)318 605 y Fr(+)318 671 y Fu(\025)377 643 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))51 b(=)f(1\))h Ft(>)e(\026)1094 605 y Fr(+)1094 671 y Fu(\025)1153 643 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))51 b(=)f Fv(\000)p Fw(1\))45 b(for)g(some)h Ft(x)j Fv(2)h(L)p Fw(.)84 b(No)m(w,)50 b(it)45 b(turns)e(out)j(that)f(in)165 756 y(this)39 b(mo)s(del)f(not)i (only)f(h)m(yp)s(othesis)f(\(59\))j(holds,)f(but)f(that)h(the)g(non)m (uniqueness)d(of)j(the)g(Gibbs)165 869 y(measure)34 b(is)g(in)e(fact)j (equiv)-5 b(alen)m(t)34 b(to)h(agreemen)m(t)h(p)s(ercolation,)e(not)g (only)g(for)g(the)g(square)g(lattice)165 981 y(but)29 b Fp(for)k(any)g(gr)-5 b(aph)p Fw(.)42 b(This)28 b(comes)i(from)f(the)h (nature)g(of)g(the)g(random-cluster)e(represen)m(tation)i(of)165 1094 y(Section)j(6.7,)j(whic)m(h)31 b(is)i(related)g(to)h(the)f(sites)g (rather)g(than)g(the)g(b)s(onds)f(of)h(the)g(lattice,)i(and)d(is)h(a) 165 1207 y(curious)26 b(exception)h(from)g(the)g(fact)i(that,)f(on)f (the)h(whole,)f(the)g(Widom{Ro)m(wlinson)f(mo)s(del)g(is)g(less)165 1320 y(amenable)33 b(to)g(sharp)f(results)g(than)g(the)h(Ising)f(mo)s (del.)46 b(Ho)m(w)m(ev)m(er,)36 b(b)m(y)d(the)g(reasons)g(discussed)d (in)165 1433 y(Section)j(6.7,)i(this)d(result)g(do)s(es)g Fp(not)i Fw(carry)f(o)m(v)m(er)h(to)g(the)f(m)m(ultit)m(yp)s(e)f (Widom{Ro)m(wlinson)f(lattice)165 1546 y(mo)s(del)f(with)f Ft(q)f Fv(\025)d Fw(3)30 b(t)m(yp)s(es)h(of)f(particles.)165 1734 y Fx(Theorem)k(8.13)47 b Fp(Consider)c(the)f(Widom{R)-5 b(ow)5 b(linson)46 b(lattic)-5 b(e)43 b(mo)-5 b(del)44 b(on)e(an)h(arbitr)-5 b(ary)44 b(gr)-5 b(aph)165 1847 y Fw(\()p Fv(L)p Ft(;)15 b Fv(\030)p Fw(\))33 b Fp(for)h(any)f (activity)g Ft(\025)25 b(>)g Fw(0)p Fp(.)42 b(Then)33 b(the)g(fol)5 b(lowing)34 b(statements)g(ar)-5 b(e)33 b(e)-5 b(quivalent.)211 2034 y Fw(\(i\))32 b Fp(The)h(Gibbs)g(me)-5 b(asur)g(e)34 b(for)f(the)g(p)-5 b(ar)g(ameter)35 b Ft(\025)e Fp(is)f(non-unique.)211 2235 y Fw(\(ii\))g Ft(\026)420 2197 y Fr(+)420 2263 y Fu(\025)478 2235 y Fw(\()p Ft(x)631 2177 y Fu(\021)666 2186 y Fl(+)591 2235 y Fv( )-15 b(!)25 b(1)p Fw(\))g Ft(>)g Fw(0)33 b Fp(for)g(some,)h(and)f(thus)g(al)5 b(l)34 b Ft(x)25 b Fv(2)g(L)p Fp(.)165 2423 y Fx(Sk)m(etc)m(h)38 b(of)f(Pro)s(of:)95 b Fw(Consider)30 b Ft(\026)1437 2376 y Fu(\021)1472 2385 y Fl(+)1437 2451 y Fu(\025;)p Fr(\003)1582 2423 y Fw(for)i(some)h(\014nite)d(\003.)46 b(In)31 b(the)i(same)f(w)m (a)m(y)h(as)f(the)h(random-)165 2536 y(cluster)c(represen)m(tation)g (of)h(Section)f(6.1)i(w)m(as)e(mo)s(di\014ed)e(in)i(Section)g(6.2)h(to) g(deal)f(with)f(b)s(oundary)165 2649 y(conditions,)22 b(w)m(e)h(can)e(mo)s(dify)f(the)i(site-random-cluster)f(represen)m (tation)g(of)h(Section)g(6.7)g(to)h(obtain)165 2762 y(a)k(coupling)d (of)j Ft(\026)751 2714 y Fu(\021)786 2723 y Fl(+)751 2790 y Fu(\025;)p Fr(\003)890 2762 y Fw(and)f(a)g(wired)f(site-random)g (cluster)h(distribution)c Ft( )2717 2729 y Fr(1)2714 2787 y Fu(p;)p Fr(2)p Fu(;)p Fr(\003)2878 2762 y Fw(,)27 b(so)f(that)h(analogues)165 2874 y(of)d(Prop)s(ositions)d(6.22)k(and)e (6.23)i(hold.)37 b(As)23 b(a)h(coun)m(terpart)g(to)g(equation)f(\(32\)) i(and)e(b)m(y)g(the)g(sp)s(eci\014c)165 2987 y(nature)30 b(of)h(the)g(site-random-cluster)e(represen)m(tation,)i(w)m(e)g(then)f (\014nd)e(that)453 3200 y Ft(\026)508 3153 y Fu(\021)543 3162 y Fl(+)508 3228 y Fu(\025;)p Fr(\003)621 3200 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))27 b(=)e(1\))c Fv(\000)f Ft(\026)1230 3153 y Fu(\021)1265 3162 y Fl(+)1230 3228 y Fu(\025;)p Fr(\003)1344 3200 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))26 b(=)f Fv(\000)p Fw(1\))h(=)f Ft( )2040 3162 y Fr(1)2037 3222 y Fu(p;)p Fr(2)p Fu(;)p Fr(\003)2200 3200 y Fw(\()p Ft(x)h Fv($)f Ft(@)5 b Fw(\003\))26 b(=)f Ft(\026)2757 3153 y Fu(\021)2792 3162 y Fl(+)2757 3228 y Fu(\025;)p Fr(\003)2871 3200 y Fw(\()p Ft(x)3023 3142 y Fu(\021)3058 3151 y Fl(+)2983 3200 y Fv( )-15 b(!)25 b Ft(@)5 b Fw(\003\))165 3404 y(for)30 b(all)g Ft(x)25 b Fv(2)g(L)p Fw(.)40 b(In)30 b(the)g(limit)f(\003)c Fv(")h(L)k Fw(w)m(e)g(obtain)g(b)m(y)h(an)f(analogue)h(to)g(\(30\))884 3622 y Ft(\026)939 3584 y Fr(+)939 3650 y Fu(\025)998 3622 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))27 b(=)e(1\))c Fv(\000)f Ft(\026)1607 3584 y Fr(+)1607 3650 y Fu(\025)1666 3622 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))26 b(=)f Fv(\000)p Fw(1\))h(=)f Ft(\026)2355 3584 y Fr(+)2355 3650 y Fu(\025)2414 3622 y Fw(\()p Ft(x)2567 3563 y Fu(\021)2602 3572 y Fl(+)2526 3622 y Fv( )-15 b(!)25 b(1)p Fw(\))h Ft(;)165 3826 y Fw(and)k(the)h(theorem)f(follo)m(ws)g(immediately)-8 b(.)1889 b Fk(2)165 3989 y Fw(T)-8 b(o)36 b(conclude)e(this)g (subsection,)h(w)m(e)h(note)f(that)h(h)m(yp)s(othesis)d(\(59\))k(also)e (holds)e(in)h(other)h(mo)s(dels.)165 4102 y(W)-8 b(e)30 b(men)m(tion)f(here)f(only)g(the)h Fp(Ashkin{T)-7 b(el)5 b(ler)32 b(mo)-5 b(del)30 b Fw([16)q(],)g(a)f(4-state)i(mo)s(del)c (whic)m(h)h(in)m(terp)s(olates)165 4215 y(in)k(an)h(in)m(teresting)g(w) m(a)m(y)h(b)s(et)m(w)m(een)g(the)g(4-state)h(P)m(otts)g(and)d(the)i(so) f(called)g(4-state)i(clo)s(c)m(k)f(mo)s(del,)165 4327 y(whic)m(h)29 b(is)h(also)g(accessible)g(to)h(random-cluster)f(metho)s (ds;)g(w)m(e)g(refer)h(to)g([58)q(,)f(61)q(,)h(192)q(,)g(210)q(].)165 4571 y FA(8.5)113 b(P)m(ercolation)37 b(of)h(ground-energy)g(b)s(onds) 165 4742 y Fw(So)23 b(far)f(in)f(this)h(section)g(w)m(e)h(considered)f (a)h(n)m(um)m(b)s(er)e(of)h(mo)s(dels)g(whic)m(h)f(are)i(kno)m(wn)f(to) h(sho)m(w)g(a)f(phase)165 4855 y(transition,)31 b(and)g(ask)m(ed)h (whether)f(this)g(phase)g(transition)f(go)s(es)j(hand)d(in)g(hand)h (with)f(agreemen)m(t)165 4968 y(p)s(ercolation.)38 b(These)23 b(results)f(run)g(under)g(the)h(heading)f(\\phase)i(transition)e (implies)e(p)s(ercolation",)165 5081 y(ev)m(en)33 b(though)e(for)h(the) g(square)f(lattice)h(w)m(e)h(established)d(results)g(of)i(con)m(v)m (erse)h(t)m(yp)s(e)f(coming)g(from)165 5194 y(the)f(imp)s(ossibilit)m (y)26 b(of)k(sim)m(ultaneous)f(o)s(ccupied)g(and)h(v)-5 b(acan)m(t)32 b(p)s(ercolation)e(on)g Fx(Z)3007 5161 y Fr(2)3046 5194 y Fw(.)306 5307 y(W)-8 b(e)29 b(no)m(w)f(tak)m(e)h (the)e(opp)s(osite)g(p)s(oin)m(t)f(of)i(view)f(and)f(ask)i(whether)f (\\p)s(ercolation)g(implies)d(phase)165 5420 y(transition".)37 b(More)23 b(precisely)-8 b(,)22 b(w)m(e)h(w)m(an)m(t)f(to)h(deduce)e (the)h(existence)h(of)f(a)g(phase)f(transition)f(\(at)j(lo)m(w)165 5533 y(temp)s(eratures)g(or)g(high)e(densities\))h(from)g(a)h(p)s (ercolation)f(result.)37 b(In)22 b(fact,)k(suc)m(h)d(an)f(idea)h(is)f (already)165 5646 y(implicit)40 b(in)h(P)m(eierls')h([187)r(])h(and)f (Dobrushin's)e([63)q(])j(pro)s(of)f(of)h(phase)f(transition)f(in)g(the) i(Ising)1844 5895 y(86)p eop %%Page: 87 87 87 86 bop 165 304 a Fw(mo)s(del,)35 b(and)e(is)h(an)g(in)m(tegral)g (part)g(of)h(the)f(Pirogo)m(v{Sinai)f(theory)-8 b(.)53 b(F)-8 b(or)35 b(mo)s(dels)e(with)g(neigh)m(b)s(or)165 417 y(in)m(teraction)21 b(as)g(in)f(the)h(Hamiltonian)e(\(1\),)24 b(the)d(underlying)d(principle)f(can)k(b)s(e)g(sk)m(etc)m(hed)h(as)f (follo)m(ws.)165 530 y(A)m(t)33 b(lo)m(w)e(temp)s(eratures)g(\(or)h (high)e(densities\),)h(eac)m(h)h(pair)e(of)i(adjacen)m(t)h(spins)c (\(or)j(particles\))f(tries)165 643 y(to)f(minimize)c(its)i(pair)g(in)m (teraction)g(energy)-8 b(.)41 b(Note)30 b(that)g(this)d(minimization)f (in)m(v)m(olv)m(es)j(the)g(b)s(onds)165 756 y(rather)g(than)g(the)g (sites)f(of)h(the)g(lattice.)41 b(So,)29 b(one)g(exp)s(ects)g(that)h(b) s(onds)d(of)i(minimal)d(energy)j({)g(the)165 869 y Fp(gr)-5 b(ound-ener)g(gy)34 b(b)-5 b(onds)32 b Fw({)f(prev)-5 b(ail,)29 b(forming)h(regions)g(separated)h(b)m(y)f(b)s(oundaries)f (that)i(consist)f(of)165 981 y(b)s(onds)f(of)h(higher)f(energy)-8 b(.)42 b(Suc)m(h)29 b(b)s(oundaries,)f(whic)m(h)h(are)i(kno)m(wn)f(as)g Fp(c)-5 b(ontours)p Fw(,)32 b(cost)g(an)e(energy)165 1094 y(prop)s(ortional)23 b(to)i(their)e(size,)j(and)e(are)g(therefore) h(t)m(ypically)e(small)g(when)h Ft(\014)29 b Fw(is)23 b(large.)39 b(This)23 b(implies)165 1207 y(that)i(the)g(ground-energy)f (b)s(onds)e(should)h(p)s(ercolate.)39 b(No)m(w,)26 b(the)f(p)s(oin)m(t) e(is)g(that)i(if)e(the)i(spins)d(along)165 1320 y(a)33 b(b)s(ond)e(can)h(c)m(ho)s(ose)i(b)s(et)m(w)m(een)e(di\013eren)m(t)g (states)i(of)e(minimal)d(energy)k(then)f(this)f(am)m(biguit)m(y)h(can) 165 1433 y(b)s(e)h(transmitted)h(to)g(the)g(macroscopic)g(lev)m(el)g(b) m(y)f(an)h(in\014nite)d(ground-energy)j(cluster,)g(and)f(this)165 1546 y(giv)m(es)k(rise)f(to)i(phase)e(transition.)59 b(In)36 b(other)h(w)m(ords,)h(the)f(classical)f(con)m(tour)h(argumen)m (t)g(for)g(the)165 1659 y(existence)j(of)g(phase)f(transition)f(can)i (b)s(e)f(summarized)f(in)g(the)h(phrase:)59 b(ground-energy)39 b(b)s(ond)165 1772 y(p)s(ercolation)27 b(together)i(with)e(a)h (\(clear-cut\))h(non-uniqueness)c(of)j(the)g(lo)s(cal)f(ground)g(state) i(implies)165 1885 y(non-uniqueness)f(of)j(Gibbs)e(measures.)40 b(W)-8 b(e)32 b(will)27 b(no)m(w)k(describ)s(e)d(this)i(picture)f(in)g (detail.)306 1998 y(W)-8 b(e)44 b(consider)e(the)h(cubic)f(lattice)h Fv(L)i Fw(=)h Fx(Z)1840 1965 y Fu(d)1922 1998 y Fw(of)d(dimension)d Ft(d)46 b Fv(\025)g Fw(2)d(with)e(its)h(usual)g(graph)165 2111 y(structure.)76 b(F)-8 b(or)43 b(de\014niteness)d(w)m(e)j (consider)e(the)i(Hamiltonian)d(\(1\))j(for)f(some)h(pair)e(p)s(oten)m (tial)165 2223 y Ft(U)49 b Fw(:)38 b Ft(S)30 b Fv(\002)c Ft(S)43 b Fv(!)38 b Fx(R)p Fw(.)64 b(W)-8 b(e)40 b(can)e(and)g(will)d (assume)j(that)h(the)f(self-p)s(oten)m(tial)g Ft(V)58 b Fw(v)-5 b(anishes;)41 b(this)c(is)165 2336 y(b)s(ecause)31 b(otherwise)f(w)m(e)g(can)h(replace)f Ft(U)41 b Fw(b)m(y)921 2572 y Ft(U)993 2535 y Fq(0)1017 2572 y Fw(\()p Ft(a;)15 b(b)p Fw(\))26 b(=)f Ft(U)10 b Fw(\()p Ft(a;)15 b(b)p Fw(\))21 b(+)1751 2511 y(1)p 1727 2551 93 4 v 1727 2635 a(2)p Ft(d)1845 2572 y Fw([)p Ft(V)g Fw(\()p Ft(a)p Fw(\))g(+)f Ft(V)g Fw(\()p Ft(b)p Fw(\)])26 b Ft(;)106 b(a;)15 b(b)26 b Fv(2)f Ft(S;)595 b Fw(\(61\))165 2803 y(whic)m(h,)30 b(together)i(with)d(the)h(self-p)s(oten)m(tial)g Ft(V)1790 2770 y Fq(0)1838 2803 y Fv(\021)25 b Fw(0,)31 b(leads)f(to)h(the)g (same)g(Hamiltonian.)39 b(Let)1556 3005 y Ft(m)25 b Fw(=)40 b(min)1757 3065 y Fu(a;b)p Fq(2)p Fu(S)1953 3005 y Ft(U)10 b Fw(\()p Ft(a;)15 b(b)p Fw(\))1231 b(\(62\))165 3248 y(b)s(e)30 b(the)h(minimal)c(v)-5 b(alue)30 b(of)h Ft(U)10 b Fw(.)306 3360 y(Giv)m(en)37 b(an)f(arbitrary)g(con\014guration)g Ft(\033)i Fv(2)e Fw(\012,)i(w)m(e)f(will)d(sa)m(y)j(that)g(an)f(edge)i Ft(e)e Fw(=)f Fv(f)p Ft(x;)15 b(y)s Fv(g)37 b(2)e(B)165 3473 y Fw(is)e(a)h Fp(gr)-5 b(ound-ener)g(gy)37 b(b)-5 b(ond)35 b Fw(for)e Ft(\033)k Fw(if)32 b Ft(U)10 b Fw(\()p Ft(\033)s Fw(\()p Ft(x)p Fw(\))p Ft(;)15 b(\033)s Fw(\()p Ft(y)s Fw(\)\))34 b(=)c Ft(m)p Fw(.)51 b(The)33 b(subgraph)f(of)i Fx(Z)3044 3440 y Fu(d)3117 3473 y Fw(consisting)f(of)165 3586 y(all)28 b(v)m(ertices)i(of)f Fx(Z)784 3553 y Fu(d)854 3586 y Fw(and)f(only)g(the)i(ground-energy)e(b)s(onds)g(for)h Ft(\033)j Fw(splits)27 b(then)i(o\013)g(in)m(to)g(connected)165 3699 y(comp)s(onen)m(ts)i(whic)m(h)e(will)f(b)s(e)i(called)f Fp(gr)-5 b(ound-ener)g(gy)34 b(clusters)d Fw(for)f Ft(\033)s Fw(.)41 b(W)-8 b(e)32 b(are)e(in)m(terested)h(in)e(the)165 3812 y(existence)36 b(of)f(in\014nite)e(ground-energy)i(clusters,)h (and)e(w)m(e)i(also)f(need)g(to)h(iden)m(tify)d(sp)s(eci\014c)h(suc)m (h)165 3925 y(clusters.)k(Unfortunately)-8 b(,)24 b(the)e(Burton{Keane) i(uniqueness)c(theorem)j(5.17)h(do)s(es)e(not)h(apply)e(here)165 4038 y(b)s(ecause,)28 b(for)e(an)m(y)h(Gibbs)e(measure,)i(the)g (distribution)c(of)j(the)h(set)g(of)g(ground-energy)f(b)s(onds)f(fails) 165 4151 y(to)40 b(ha)m(v)m(e)f(the)g(\014nite-energy)f(prop)s(ert)m(y) -8 b(.)65 b(W)-8 b(e)39 b(therefore)g(resort)g(to)g(considering)d (ground-energy)165 4264 y(clusters)45 b(in)f(an)m(y)i(\014xed)e(t)m(w)m (o-dimensional)g(la)m(y)m(er)i(of)g Fx(Z)2157 4231 y Fu(d)2197 4264 y Fw(;)53 b(the)46 b(uniqueness)d(of)i(planar)f (in\014nite)165 4377 y(clusters)28 b(can)i(b)s(e)e(sho)m(wn)g(in)g(our) g(case.)41 b(\(An)29 b(alternativ)m(e)h(argumen)m(t)f(a)m(v)m(oiding)g (the)g(use)f(of)h(planar)165 4490 y(la)m(y)m(ers)e(but)f(requiring)e (stronger)j(conditions)e(on)h(the)h(temp)s(erature)g(has)f(b)s(een)g (suggested)h(in)e([87)q(].\))165 4602 y(In)30 b(fact,)i(w)m(e)e(ha)m(v) m(e)i(the)f(follo)m(wing)d(result.)165 4789 y Fx(Theorem)34 b(8.14)47 b Fp(Consider)f(the)h(Hamiltonian)g(\(1\))f(on)g(the)g (lattic)-5 b(e)47 b Fv(L)i Fw(=)f Fx(Z)2994 4756 y Fu(d)3035 4789 y Fp(,)g Ft(d)h Fv(\025)g Fw(2)p Fp(,)g(with)165 4902 y(neighb)-5 b(or)38 b(inter)-5 b(action)38 b Ft(U)46 b Fp(and)38 b(no)f(self-p)-5 b(otential,)39 b(and)e(let)g Fv(P)44 b Fp(b)-5 b(e)37 b(any)g(planar)h(layer)g(in)e Fv(L)p Fp(.)53 b(\(So)165 5015 y Fv(P)34 b Fw(=)25 b Fv(L)33 b Fp(for)h Ft(d)26 b Fw(=)f(2)p Fp(.\))43 b(If)33 b Ft(\014)38 b Fp(is)33 b(lar)-5 b(ge)34 b(enough,)f(ther)-5 b(e)34 b(exists)f(a)h(Gibbs)f(me)-5 b(asur)g(e)34 b Ft(\026)26 b Fv(2)f(G)5 b Fw(\()p Ft(\014)g(H)i Fw(\))34 b Fp(which)165 5128 y(is)f(invariant)h(under)f(al)5 b(l)33 b(automorphisms)j(of)d Fv(L)f Fp(and)i(al)5 b(l)33 b(symmetries)g(of)g Ft(U)43 b Fp(such)33 b(that)803 5330 y Ft(\026)p Fw(\()p Fv(9)f Fw(a)e(unique)f(in\014nite)f(ground-energy)i(cluster)g(in)f Fv(P)7 b Fw(\))26 b(=)f(1)h Ft(:)165 5533 y Fp(In)31 b(the)f(ab)-5 b(ove,)31 b(a)g(symmetry)g(of)g Ft(U)40 b Fp(is)30 b(a)g(tr)-5 b(ansformation)34 b Ft(\034)40 b Fp(of)31 b Ft(S)k Fp(such)30 b(that)i Ft(U)10 b Fw(\()p Ft(\034)g(a;)15 b(\034)10 b(b)p Fw(\))26 b(=)f Ft(U)10 b Fw(\()p Ft(a;)15 b(b)p Fw(\))165 5646 y Fp(for)34 b(al)5 b(l)33 b Ft(a;)15 b(b)25 b Fv(2)g Ft(S)5 b Fp(;)33 b(such)f(a)h Ft(\034)43 b Fp(acts)33 b(c)-5 b(o)g(or)g(dinatewise)36 b(on)d(c)-5 b(on\014gur)g(ations.)1844 5895 y Fw(87)p eop %%Page: 88 88 88 87 bop 165 304 a Fw(Theorem)28 b(8.14)i(is)d(a)i(particular)d(case)j (of)g(a)f(result)f(\014rst)h(deriv)m(ed)f(in)g([92)q(])h(and)g(presen)m (ted)g(in)f(detail)165 417 y(in)k(Chapter)g(18)h(of)g([96)q(].)45 b(W)-8 b(e)33 b(will)c(sk)m(etc)m(h)k(its)e(pro)s(of)g(b)s(elo)m(w.)44 b(The)32 b(remark)-5 b(able)31 b(fact)h(is)f(that)h(this)165 530 y(t)m(yp)s(e)37 b(of)h(p)s(ercolation)e(often)h(implies)d(that)k Ft(\026)e Fw(has)h(a)g(non-trivial)e(extremal)i(decomp)s(osition,)h(so) 165 643 y(that)31 b(there)g(m)m(ust)f(b)s(e)g(a)h(phase)f(transition.) 39 b(In)30 b(fact,)h(this)e(happ)s(ens)g(whenev)m(er)h(the)h(set)1174 815 y Ft(G)1245 829 y Fu(U)1330 815 y Fw(=)24 b Fv(f)p Fw(\()p Ft(a;)15 b(b)p Fw(\))27 b Fv(2)e Ft(S)g Fv(\002)20 b Ft(S)31 b Fw(:)25 b Ft(U)10 b Fw(\()p Ft(a;)15 b(b)p Fw(\))26 b(=)f Ft(m)p Fv(g)848 b Fw(\(63\))165 988 y(of)35 b(b)s(ond)d(ground)h(states)i(splits)d(in)m(to)i(su\016cien)m(tly)f (disjoin)m(t)f(parts.)51 b(T)-8 b(o)35 b(explain)d(the)i(underlying)165 1101 y(mec)m(hanism)g(\(whic)m(h)f(ma)m(y)h(b)s(e)f(view)m(ed)h(as)g (the)g(core)h(of)f(the)g(classical)f(P)m(eierls)g(argumen)m(t,)i(and)f (a)165 1214 y(rudimen)m(tary)22 b(v)m(ersion)g(of)i(Pirogo)m(v{Sinai)e (theory\))h(w)m(e)h(consider)e(\014rst)g(the)i(standard)e(Ising)g(mo)s (del.)165 1375 y Fx(Example)34 b(8.15)46 b Fp(The)31 b(Ising)g(ferr)-5 b(omagnet)32 b(at)f(zer)-5 b(o)32 b(external)f (\014eld.)42 b Fw(In)27 b(this)g(mo)s(del,)h(w)m(e)h(ha)m(v)m(e)g(as) 165 1488 y(usual)k Ft(S)38 b Fw(=)32 b Fv(f\000)p Fw(1)p Ft(;)15 b Fw(1)p Fv(g)p Fw(,)38 b Ft(U)10 b Fw(\()p Ft(a;)15 b(b)p Fw(\))33 b(=)f Fv(\000)p Ft(ab)i Fw(for)h Ft(a;)15 b(b)32 b Fv(2)g Ft(S)5 b Fw(,)36 b Ft(m)c Fw(=)g Fv(\000)p Fw(1,)k(and)e Ft(G)2717 1502 y Fu(U)2809 1488 y Fw(=)e Fv(f)p Fw(\()p Fv(\000)p Fw(1)p Ft(;)15 b Fv(\000)p Fw(1\))p Ft(;)g Fw(\(1)p Ft(;)g Fw(1\))p Fv(g)p Fw(.)165 1601 y(Hence,)40 b(either)c(all)g(spins)e(of)j(a)g(ground-energy)g(cluster)f (are)h(negativ)m(e,)j(or)d(else)f(all)g(these)h(spins)165 1714 y(are)d(p)s(ositiv)m(e.)50 b(In)33 b(other)h(w)m(ords,)g(eac)m(h)h (ground-energy)e(cluster)g(is)f(either)i(a)g(min)m(us-cluster)d(or)j(a) 165 1827 y(plus-cluster.)39 b(This)28 b(implies)g(that)666 1999 y Fv(f9)i Fw(a)h(unique)d(in\014nite)h(ground-energy)g(cluster)h (in)f Fv(P)8 b(g)26 b(\032)f Ft(A)2776 2013 y Fq(\000)2855 1999 y Fv([)20 b Ft(A)3004 2013 y Fr(+)3088 1999 y Ft(;)165 2172 y Fw(where)32 b Ft(A)498 2186 y Fq(\000)589 2172 y Fw(and)g Ft(A)836 2186 y Fr(+)927 2172 y Fw(are)g(the)h(ev)m(en)m(ts) g(that)g(there)f(exists)g(an)g(in\014nite)e(cluster)h(of)i(negativ)m (e,)h(resp.)165 2285 y(p)s(ositiv)m(e,)h(spins)d(in)h Fv(P)7 b Fw(.)54 b(F)-8 b(or)35 b(the)g(Gibbs)e(measure)h Ft(\026)g Fw(of)h(Theorem)f(8.14)i(w)m(e)f(th)m(us)f(ha)m(v)m(e)h Ft(\026)p Fw(\()p Ft(A)3471 2299 y Fq(\000)3554 2285 y Fv([)165 2398 y Ft(A)233 2412 y Fr(+)293 2398 y Fw(\))41 b(=)g(1)g(and,)h(b)m(y)e(the)g(spin-\015ip)d(symmetry)j(of)g Ft(U)50 b Fw(and)39 b(th)m(us)h Ft(\026)p Fw(,)j Ft(\026)p Fw(\()p Ft(A)2779 2412 y Fq(\000)2838 2398 y Fw(\))f(=)f Ft(\026)p Fw(\()p Ft(A)3185 2412 y Fr(+)3244 2398 y Fw(\).)70 b(Hence)165 2511 y Ft(\026)p Fw(\()p Ft(A)323 2525 y Fq(\000)383 2511 y Fw(\))38 b Ft(>)h Fw(0)f(and)g Ft(\026)p Fw(\()p Ft(A)992 2525 y Fr(+)1051 2511 y Fw(\))h Ft(>)f Fw(0,)j(so)e(that)f(the)h(measures)f Ft(\026)2287 2478 y Fq(\000)2384 2511 y Fw(=)g Ft(\026)p Fw(\()p Fv(\001)15 b(j)p Ft(A)2716 2525 y Fq(\000)2776 2511 y Fw(\))39 b(and)e Ft(\026)3089 2478 y Fr(+)3187 2511 y Fw(=)h Ft(\026)p Fw(\()p Fv(\001)15 b(j)p Ft(A)3519 2525 y Fr(+)3579 2511 y Fw(\))165 2624 y(are)33 b(w)m(ell-de\014ned.)45 b(Since)31 b Ft(A)1165 2638 y Fq(\000)1224 2624 y Fw(,)i Ft(A)1350 2638 y Fr(+)1442 2624 y Fw(are)g(tail)e(ev)m(en)m(ts,)k(it)d(follo)m (ws)f(immediately)g(that)i Ft(\026)3230 2591 y Fq(\000)3289 2624 y Fw(,)g Ft(\026)3402 2591 y Fr(+)3493 2624 y Fw(are)165 2736 y(Gibbs)i(measures)h(for)h Ft(\014)5 b(H)i Fw(.)59 b(Also,)37 b Ft(A)1505 2750 y Fq(\000)1589 2736 y Fv(\\)24 b Ft(A)1742 2750 y Fr(+)1837 2736 y Fw(is)36 b(con)m(tained)h(in)e(the) h(ev)m(en)m(t)i(that)f(there)g(are)g(t)m(w)m(o)165 2849 y(distinct)28 b(ground-energy)g(clusters)g(in)g Fv(P)7 b Fw(,)30 b(and)f(therefore)g(has)g Ft(\026)p Fw(-measure)g(0.)40 b(Hence)30 b Ft(\026)3237 2816 y Fq(\000)3325 2849 y Fw(and)e Ft(\026)3555 2816 y Fr(+)165 2962 y Fw(are)j(m)m(utually)e (singular,)f(whence)j Fv(jG)5 b Fw(\()p Ft(\014)g(H)i Fw(\))p Fv(j)27 b Ft(>)e Fw(1.)165 3124 y(The)k(same)h(argumen)m(t)g (as)g(in)e(the)i(preceding)e(example)i(yields)d(the)j(follo)m(wing)e (theorem)i(on)f(phase)165 3236 y(transition)i(b)m(y)i(symmetry)f (breaking.)47 b(A)33 b(detailed)e(pro)s(of)h(\(in)g(a)h(sligh)m(tly)e (di\013eren)m(t)h(setting\))h(can)165 3349 y(b)s(e)d(found)f(in)g (Section)h(18.2)i(of)f([96)q(].)165 3511 y Fx(Theorem)j(8.16)47 b Fp(Under)30 b(the)h(c)-5 b(onditions)33 b(of)e(The)-5 b(or)g(em)32 b(8.14)g(supp)-5 b(ose)32 b(that)g(the)f(set)g Ft(G)3250 3525 y Fu(U)3340 3511 y Fp(de\014ne)-5 b(d)165 3623 y(by)33 b(\(63\))h(admits)g(a)f(de)-5 b(c)g(omp)g(osition)36 b Ft(G)1512 3637 y Fu(U)1596 3623 y Fw(=)25 b Ft(G)1763 3637 y Fr(1)1824 3623 y Fv([)19 b Ft(:)c(:)g(:)22 b Fv([)d Ft(G)2182 3637 y Fu(N)2283 3623 y Fp(such)32 b(that)273 3785 y(1.)46 b(the)23 b(sets)h Ft(G)775 3799 y Fu(n)822 3785 y Fp(,)h Fw(1)h Fv(\024)f Ft(n)g Fv(\024)g Ft(N)10 b Fp(,)25 b(have)f(disjoint)g(pr)-5 b(oje)g(ctions,)27 b(i.e.,)e(if)e Fw(\()p Ft(a;)15 b(b)p Fw(\))26 b Fv(2)f Ft(G)2989 3799 y Fu(n)3037 3785 y Fp(,)g Fw(\()p Ft(a)3173 3752 y Fq(0)3196 3785 y Ft(;)15 b(b)3275 3752 y Fq(0)3299 3785 y Fw(\))26 b Fv(2)e Ft(G)3516 3800 y Fu(n)3559 3781 y Fm(0)3586 3785 y Fp(,)393 3898 y(and)33 b Ft(n)25 b Fv(6)p Fw(=)g Ft(n)800 3865 y Fq(0)823 3898 y Fp(,)32 b(then)h Ft(a)26 b Fv(6)p Fw(=)f Ft(a)1303 3865 y Fq(0)1326 3898 y Ft(;)15 b(b)26 b Fv(6)p Fw(=)f Ft(b)1566 3865 y Fq(0)1589 3898 y Fp(,)32 b(and)273 4075 y(2.)46 b(for)30 b(any)h(two)g(indic)-5 b(es)30 b Ft(n;)15 b(n)1323 4042 y Fq(0)1371 4075 y Fv(2)25 b(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)i(;)e(N)10 b Fv(g)31 b Fp(we)f(have)35 b Fw(\026)-50 b Ft(\034)10 b Fw(\()p Ft(G)2402 4089 y Fu(n)2450 4075 y Fw(\))26 b(=)f Ft(G)2678 4090 y Fu(n)2721 4071 y Fm(0)2777 4075 y Fp(for)31 b(some)g(tr)-5 b(ansforma-)393 4188 y(tion)37 b Fw(\026)-50 b Ft(\034)41 b Fp(of)32 b Ft(S)23 b Fv(\002)18 b Ft(S)37 b Fp(which)32 b(is)g(either)g(the)g(r)-5 b(e\015e)g(ction,)33 b(or)g(the)f(c)-5 b(o)g(or)g(dinatewise)34 b(applic)-5 b(ation)35 b(of)393 4301 y(some)e(symmetry)h(of)f Ft(U)10 b Fp(,)32 b(or)i(a)f(c)-5 b(omp)g(osition)35 b(of)e(b)-5 b(oth.)165 4462 y(Then,)27 b(if)e Ft(\014)31 b Fp(is)25 b(su\016ciently)g(lar)-5 b(ge,)28 b(ther)-5 b(e)26 b(exist)f Ft(N)35 b Fp(mutual)5 b(ly)26 b(singular)g(Gibbs)g(me)-5 b(asur)g(es)26 b Ft(\026)3291 4429 y Fr(1)3331 4462 y Ft(;)15 b(:)g(:)g(:)h(\026)3547 4429 y Fu(N)165 4575 y Fv(2)25 b(G)5 b Fw(\()p Ft(\014)g(H)i Fw(\))p Fp(,)34 b(invariant)g(under)f(al)5 b(l)33 b(even)f(automorphisms)k(of)d Fx(Z)2377 4542 y Fu(d)2450 4575 y Fp(and)g(such)g(that)1184 4747 y Ft(\026)1239 4710 y Fu(n)1286 4747 y Fw(\()p Fv(9)f Fw(an)e(in\014nite)e Ft(n)p Fw(-cluster)i(in)f Fv(P)8 b Fw(\))25 b(=)g(1)165 4920 y Fp(for)34 b(al)5 b(l)33 b Fw(1)25 b Fv(\024)g Ft(n)g Fv(\024)g Ft(N)10 b Fp(.)42 b(In)32 b(p)-5 b(articular,)35 b(ther)-5 b(e)34 b(exist)e Ft(N)43 b Fp(distinct)33 b(phases)h(for)f Ft(\014)5 b(H)i Fp(.)165 5081 y Fw(In)24 b(the)h(statemen)m(t)i(ab)s(o)m(v)m(e,)g(an)e (in\014nite)d Ft(n)p Fw(-cluster)i(for)h(a)g(con\014guration)f Ft(\033)k Fw(is)c(an)g(in\014nite)f(cluster)h(of)165 5194 y(the)29 b(subgraph)e(of)i Fx(Z)879 5161 y Fu(d)948 5194 y Fw(obtained)f(b)m(y)h(k)m(eeping)f(only)g(those)h(edges)h Ft(e)25 b Fv(2)g(B)31 b Fw(with)d(\()p Ft(\033)s Fw(\()p Ft(x)p Fw(\))p Ft(;)15 b(\033)s Fw(\()p Ft(y)s Fw(\)\))28 b Fv(2)c Ft(G)3541 5208 y Fu(n)3589 5194 y Fw(,)165 5307 y(where)40 b Ft(x)f Fw(is)g(the)h(endp)s(oin)m(t)e(of)h Ft(e)h Fw(in)f(the)h(ev)m(en)g(sublattice)f Fv(L)2363 5321 y Fu(ev)r(en)2551 5307 y Fw(and)g Ft(y)44 b Fv(2)c(L)2990 5322 y Fu(odd)3140 5307 y Fw(is)f(the)h(other)165 5420 y(endp)s(oin)m(t)d(of)h Ft(e)p Fw(.)64 b(Also,)39 b(an)f(ev)m(en)h (automorphism)d(of)i Fx(Z)2166 5387 y Fu(d)2244 5420 y Fw(is)f(an)h(automorphism)f(lea)m(ving)g Fv(L)3465 5434 y Fu(ev)r(en)165 5533 y Fw(in)m(v)-5 b(arian)m(t.)306 5646 y(W)d(e)32 b(illustrate)d(this)g(theorem)i(b)m(y)f(applying)e(it)i (to)h(our)f(other)h(standard)e(examples.)1844 5895 y(88)p eop %%Page: 89 89 89 88 bop 165 304 a Fx(Example)34 b(8.17)46 b Fp(The)c(Ising)g (antiferr)-5 b(omagnet)44 b(in)d(an)h(external)h(\014eld.)69 b Fw(W)-8 b(e)41 b(ha)m(v)m(e)g(again)f Ft(S)46 b Fw(=)165 417 y Fv(f\000)p Fw(1)p Ft(;)15 b Fw(1)p Fv(g)p Fw(,)37 b(but)32 b(the)i(in)m(teraction)f(is)f(no)m(w)i Ft(U)10 b Fw(\()p Ft(a;)15 b(b)p Fw(\))31 b(=)f Ft(ab)22 b Fv(\000)2222 381 y Fu(h)p 2207 396 72 4 v 2207 448 a Fr(2)p Fu(d)2289 417 y Fw(\()p Ft(a)g Fw(+)g Ft(b)p Fw(\))34 b(for)f(some)h(constan)m(t) g Ft(h)d Fv(2)f Fx(R)p Fw(.)165 530 y(\(Here)46 b(w)m(e)g(applied)c (the)k(transformation)e(\(61\).\))86 b(If)45 b Fv(j)p Ft(h)p Fv(j)50 b Ft(<)g Fw(2)p Ft(d)45 b Fw(then)g Ft(m)50 b Fw(=)f Fv(\000)p Fw(1)c(and)f Ft(G)3434 544 y Fu(U)3543 530 y Fw(=)165 643 y Fv(f)p Fw(\()p Fv(\000)p Fw(1)p Ft(;)15 b Fw(1\))p Ft(;)g Fw(\(1)p Ft(;)g Fv(\000)p Fw(1\))p Fv(g)p Fw(.)45 b Ft(G)978 657 y Fu(U)1067 643 y Fw(splits)27 b(up)i(in)m(to)g(the)g(singletons)f Ft(G)2263 657 y Fr(1)2329 643 y Fw(=)d Fv(f)p Fw(\(1)p Ft(;)15 b Fv(\000)p Fw(1\))p Fv(g)31 b Fw(and)e Ft(G)3064 657 y Fr(2)3129 643 y Fw(=)c Fv(f)p Fw(\()p Fv(\000)p Fw(1)p Ft(;)15 b Fw(1\))p Fv(g)p Fw(.)165 756 y(This)31 b(decomp)s(osition)f(meets)j(the)f(conditions)e (of)j(the)f(theorem;)h(in)e(particular,)g Ft(G)3100 770 y Fr(1)3172 756 y Fw(and)g Ft(G)3421 770 y Fr(2)3493 756 y Fw(are)165 869 y(related)d(to)h(eac)m(h)g(other)f(b)m(y)f(the)h (re\015ection)g(of)g Ft(S)20 b Fv(\002)15 b Ft(S)5 b Fw(.)40 b(Consequen)m(tly)-8 b(,)28 b(there)g(exist)f(t)m(w)m(o)j(m)m (utually)165 981 y(singular)d(Gibbs)g(measures)i Ft(\026)1214 948 y Fr(1)1282 981 y Fw(and)f Ft(\026)1512 948 y Fr(2)1581 981 y Fw(whic)m(h)f(are)i(in)m(v)-5 b(arian)m(t)29 b(under)e(ev)m(en)j (automorphisms)d(and)165 1094 y(ha)m(v)m(e)37 b(an)f(in\014nite)e (cluster)h(of)h(c)m(hessb)s(oard)f(t)m(yp)s(e,)j(either)d(with)g(plus)f (spins)f(on)j(the)g(ev)m(en)h(cluster)165 1207 y(sites)30 b(and)g(min)m(us)f(spins)f(at)j(the)g(o)s(dd)e(cluster)h(sites,)g(or)g (vice)h(v)m(ersa.)165 1420 y Fx(Example)j(8.18)46 b Fp(The)37 b(Potts)h(mo)-5 b(del.)55 b Fw(In)35 b(this)e(case,)38 b Ft(S)f Fw(=)c Fv(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)i(;)e(q)s Fv(g)36 b Fw(for)e(some)i(in)m(teger)f Ft(q)g Fv(\025)e Fw(2)165 1533 y(and)h Ft(U)10 b Fw(\()p Ft(a;)15 b(b)p Fw(\))32 b(=)g(1)23 b Fv(\000)f Fw(2)p Ft(I)996 1551 y Fq(f)p Fu(a)p Fr(=)p Fu(b)p Fq(g)1194 1533 y Fw(.)51 b(Again)34 b Ft(m)e Fw(=)f Fv(\000)p Fw(1,)k(and)e Ft(G)2180 1547 y Fu(U)2271 1533 y Fw(=)e Fv(f)p Fw(\()p Ft(n;)15 b(n)p Fw(\))33 b(:)e(1)h Fv(\024)f Ft(n)g Fv(\024)g Ft(q)s Fv(g)p Fw(.)52 b(Theorem)165 1646 y(8.16)30 b(is)e(ob)m(viously)f (applicable,)g(and)h(w)m(e)h(reco)m(v)m(er)i(the)d(result)g(that)h(for) f(su\016cien)m(tly)g(large)g Ft(\014)34 b Fw(there)165 1759 y(exist)k Ft(q)i Fw(m)m(utually)c(singular,)i(automorphism)e(in)m (v)-5 b(arian)m(t)36 b(Gibbs)g(measures,)k(the)e Ft(n)p Fw(th)f(of)g(whic)m(h)165 1871 y(has)30 b(an)h(in\014nite)d(cluster)i (of)g(spins)e(with)h(v)-5 b(alue)30 b Ft(n)p Fw(.)165 2084 y Fx(Example)k(8.19)46 b Fp(The)36 b(har)-5 b(d-c)g(or)g(e)39 b(lattic)-5 b(e)37 b(gas.)51 b Fw(This)32 b(mo)s(del)h(has)g(state)j (space)e Ft(S)i Fw(=)31 b Fv(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)36 b Fw(and)165 2197 y(neigh)m(b)s(or)25 b(in)m(teraction)h Ft(U)36 b Fw(of)27 b(the)f(form)g Ft(U)10 b Fw(\()p Ft(a;)15 b(b)p Fw(\))26 b(=)f Fv(1)h Fw(if)f Ft(a)h Fw(=)f Ft(b)g Fw(=)g(1,)i(and)f Ft(U)10 b Fw(\()p Ft(a;)15 b(b)p Fw(\))26 b(=)f Fv(\000)3208 2156 y Fr(log)14 b Fu(\025)p 3208 2176 145 4 v 3244 2228 a Fr(2)p Fu(d)3362 2197 y Fw(\()p Ft(a)e Fw(+)g Ft(b)p Fw(\))165 2310 y(for)33 b(all)f(other)h(\()p Ft(a;)15 b(b)p Fw(\))30 b Fv(2)f Ft(S)1054 2277 y Fr(2)1094 2310 y Fw(.)48 b(Here)33 b Ft(\025)d(>)f Fw(0)k(is)f(an)h(activit)m(y)g (parameter,)h(and)f(w)m(e)g(ha)m(v)m(e)h(again)f(used)165 2423 y(the)d(transformation)f(\(61\).)42 b(F)-8 b(or)30 b Ft(\025)c(>)f Fw(1)30 b(w)m(e)g(ha)m(v)m(e)g Ft(G)1991 2437 y Fu(U)2076 2423 y Fw(=)25 b Fv(f)p Fw(\(0)p Ft(;)15 b Fw(1\))p Ft(;)g Fw(\(1)p Ft(;)g Fw(0\))p Fv(g)p Fw(,)35 b(so)30 b(that)g(Theorem)g(8.16)165 2536 y(applies.)59 b(Since)35 b(m)m(ultiplying)f Ft(U)46 b Fw(with)36 b(a)h(factor)h Ft(\014)k Fw(amoun)m(ts)37 b(to)h(c)m(hanging)e Ft(\025)p Fw(,)j(w)m(e)e(obtain)g(that)165 2649 y(for)c(su\016cien)m(tly)e(large) i Ft(\025)f Fw(there)h(exist)f(t)m(w)m(o)i(distinct)d(Gibbs)g(measures) i(with)e(in\014nite)f(clusters)i(of)165 2761 y(c)m(hessb)s(oard)e(t)m (yp)s(e,)h(just)f(as)g(for)h(the)f(Ising)f(an)m(tiferromagnet)i(at)h (lo)m(w)e(temp)s(eratures.)165 2974 y Fx(Example)k(8.20)46 b Fp(The)28 b(Widom{R)-5 b(ow)5 b(linson)31 b(lattic)-5 b(e)29 b(mo)-5 b(del.)42 b Fw(Here)25 b(w)m(e)h(ha)m(v)m(e)g Ft(S)k Fw(=)25 b Fv(f\000)p Fw(1)p Ft(;)15 b Fw(0)p Ft(;)g Fw(1)p Fv(g)28 b Fw(and)165 3087 y Ft(U)10 b Fw(\()p Ft(a;)15 b(b)p Fw(\))35 b(=)d Fv(1)j Fw(if)f Ft(ab)f Fw(=)g Fv(\000)p Fw(1,)k Ft(U)10 b Fw(\()p Ft(a;)15 b(b)p Fw(\))34 b(=)f Fv(\000)1676 3046 y Fr(log)13 b Fu(\025)p 1676 3066 V 1712 3118 a Fr(2)p Fu(d)1829 3087 y Fw(\()p Fv(j)p Ft(a)p Fv(j)25 b Fw(+)e Fv(j)p Ft(b)p Fv(j)p Fw(\))36 b(otherwise,)g Ft(a;)15 b(b)33 b Fv(2)g Ft(S)5 b Fw(.)54 b(If)35 b Ft(\025)e(>)g Fw(1)i(then)165 3200 y Ft(G)236 3214 y Fu(U)334 3200 y Fw(=)j Fv(f)p Fw(\()p Fv(\000)p Fw(1)p Ft(;)15 b Fv(\000)p Fw(1\))p Ft(;)g Fw(\(1)p Ft(;)g Fw(1\))p Fv(g)p Fw(.)68 b(Theorem)38 b(8.16)i(th)m(us)d(sho)m(ws)h (that)h(for)f(su\016cien)m(tly)f(large)h Ft(\025)g Fw(there)165 3313 y(exist)k(t)m(w)m(o)g(translation)f(in)m(v)-5 b(arian)m(t)40 b(Gibbs)g(measures)h(ha)m(ving)g(in\014nite)f(clusters)g(of)i(plus-)e (resp.)165 3426 y(min)m(us-particles.)165 3638 y(Although)i(the)h (results)f(in)f(the)i(examples)f(ab)s(o)m(v)m(e)i(are)f(w)m(eak)m(er)h (than)f(those)g(obtained)f(b)m(y)h(the)165 3751 y(random-cluster)27 b(metho)s(ds)h(of)g(Section)g(6)g(\(when)g(these)g(apply\),)g(the)g (ideas)g(presen)m(ted)g(here)g(ha)m(v)m(e)165 3864 y(the)i(adv)-5 b(an)m(tage)30 b(of)g(pro)m(viding)d(a)i(general)g(picture)f(of)h(the)g (geometric)h(mec)m(hanisms)e(that)i(imply)d(a)165 3977 y(phase)e(transition,)f(and)h(Theorem)f(8.16)i(can)f(quite)g(easily)f (b)s(e)g(applied.)36 b(Moreo)m(v)m(er,)29 b(the)c(ideas)f(can)165 4090 y(b)s(e)e(extended)g(immediately)e(to)j(systems)f(with)f (arbitrary)g(state)i(space)g(and)e(suitable)g(in)m(teractions.)165 4203 y(In)38 b(this)e(w)m(a)m(y)j(one)f(obtains)g(phase)f(transitions)f (in)h(anisotropic)g(plane)g(rotor)h(mo)s(dels,)h(classical)165 4316 y(Heisen)m(b)s(erg)23 b(ferromagnets)h(or)f(an)m(tiferromagnets,)j (and)c(related)h Ft(N)10 b Fw(-v)m(ector)25 b(mo)s(dels;)g(see)f (Chapter)165 4429 y(16)37 b(of)g([96)q(].)58 b(One)35 b(can)i(also)f(consider)f(next-nearest)i(neigh)m(b)s(or)e(in)m (teractions,)i(and)f(th)m(us)g(obtain)165 4541 y(v)-5 b(arious)33 b(other)h(in)m(teresting)f(examples;)i(for)e(this)f(one)i (has)g(to)g(consider)e(p)s(ercolation)h(of)h(ground-)165 4654 y(energy)e(plaquettes)f(rather)g(than)f(ground-energy)h(b)s(onds,) f(whic)m(h)f(is)i(in)e(fact)j(the)f(set-up)g(of)h([96)q(].)165 4767 y(Last)c(but)f(not)h(least,)h(the)f(symmetry)f(assumption)f(of)i (Theorem)f(8.16)i(can)f(often)g(b)s(e)f(replaced)g(b)m(y)165 4880 y(either)37 b(some)g(direct)f(argumen)m(t,)k(or)d(a)g(P)m(eierls)f (condition)g(in)f(the)i(spirit)e(of)i(the)g(Pirogo)m(v{Sinai)165 4993 y(theory;)31 b(see)g(Chapter)f(19)h(of)g([96)q(].)41 b(One)30 b(suc)m(h)g(extension)g(will)d(b)s(e)j(used)g(in)f(our)h(next) g(example.)165 5181 y Fx(Example)k(8.21)46 b Fp(First-or)-5 b(der)45 b(phase)f(tr)-5 b(ansition)44 b(in)e(the)h(Potts)g(mo)-5 b(del.)73 b Fw(Consider)39 b(again)i(the)165 5294 y(P)m(otts)f(mo)s (del)e(of)h(Example)e(8.18,)43 b(and)38 b(supp)s(ose)f(for)i(simplicit) m(y)d(that)j Ft(d)g Fw(=)g(2.)66 b(An)m(y)39 b(translate)165 5407 y(in)44 b Fx(Z)350 5374 y Fr(2)434 5407 y Fw(of)h(the)g(quadratic) f(cell)g Fv(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)1540 5374 y Fr(2)1626 5407 y Fw(is)44 b(called)g(a)h(plaquette.)84 b(F)-8 b(or)46 b(a)f(giv)m(en)g(con\014guration)165 5519 y Ft(\033)i Fv(2)42 b Fw(\012,)i(a)d(plaquette)h Ft(P)54 b Fw(is)40 b(called)g Fp(or)-5 b(der)g(e)g(d)53 b Fw(if)40 b(all)g(spins)f(in)h Ft(P)54 b Fw(agree,)45 b Fp(disor)-5 b(der)g(e)g(d)54 b Fw(if)40 b(no)h(t)m(w)m(o)165 5632 y(adjacen)m(t)28 b(spins)d(in)g Ft(P)40 b Fw(agree,)29 b(and)d Fp(pur)-5 b(e)35 b Fw(if)25 b(one)i(of)g(these)g(t)m(w)m(o)i (cases)e(o)s(ccurs.)39 b(If)27 b Ft(q)i Fw(\(the)f(n)m(um)m(b)s(er)d (of)1844 5895 y(89)p eop %%Page: 90 90 90 89 bop 165 304 a Fw(distinct)25 b(spin)e(v)-5 b(alues\))26 b(is)f(large)h(enough)f(then,)i(for)f(arbitrary)e Ft(\014)5 b Fw(,)27 b(there)g(exists)e(an)h(automorphism)165 417 y(in)m(v)-5 b(arian)m(t)27 b(Gibbs)g(measure)h Ft(\026)f Fw(supp)s(orted)f(on)i(con\014gurations)f(with)g(a)h(unique)e (in\014nite)g(cluster)h(of)165 530 y(pure)33 b(plaquettes.)49 b(This)32 b(v)-5 b(arian)m(t)33 b(of)h(Theorem)f(8.14)i(is)d(due)h(to)h (Kotec)m(k)q(\023)-47 b(y)36 b(and)c(Shlosman)g([148)r(],)165 643 y(see)39 b(also)g(Section)f(19.3.2)j(of)d([96)q(].)65 b(Clearly)-8 b(,)40 b(eac)m(h)f(cluster)f(of)h(pure)e(plaquettes)h (only)g(con)m(tains)165 756 y(plaquettes)j(of)f(the)g(same)h(t)m(yp)s (e,)i(either)d(ordered)f(or)h(disordered.)69 b(F)-8 b(or)41 b(some)f(sp)s(eci\014c)f(critical)165 869 y(v)-5 b(alue)40 b Ft(\014)461 883 y Fu(c)496 869 y Fw(\()p Ft(q)s Fw(\))h(b)s(oth)f(p)s (ossibilities)c(m)m(ust)k(o)s(ccur)g(with)f(p)s(ositiv)m(e)h (probabilit)m(y;)j(this)c(follo)m(ws)h(from)165 981 y(thermo)s(dynamic) 28 b(considerations,)h(namely)g(b)m(y)h(con)m(v)m(exit)m(y)h(of)f(the)g (free)g(energy)g(as)g(a)g(function)e(of)165 1094 y Ft(\014)33 b Fw([148)r(,)27 b(96)q(].)40 b(Conditioning)25 b(on)i(eac)m(h)i(of)f (these)g(t)m(w)m(o)g(cases)h(yields)d(t)m(w)m(o)i(m)m(utually)e (singular)g(Gibbs)165 1207 y(measures)c(with)e(an)i(in\014nite)d (cluster)i(of)h(ordered)f(resp.)g(disordered)f(plaquettes.)38 b(F)-8 b(urthermore,)23 b(all)165 1320 y(spins)k(of)h(a)h(cluster)e(of) i(ordered)e(plaquettes)i(m)m(ust)f(ha)m(v)m(e)h(the)g(same)f(v)-5 b(alue,)29 b(so)f(that)h(b)m(y)f(symmetry)165 1433 y(the)36 b(\\ordered")g(Gibbs)e(measure)h(can)h(b)s(e)e(decomp)s(osed)i(further) e(in)m(to)h Ft(q)j Fw(Gibbs)c(measures)h(with)165 1546 y(in\014nite)i(clusters)h(of)h(constan)m(t)h(spin)d(v)-5 b(alue.)66 b(As)39 b(a)g(result,)h(for)f(large)g Ft(q)i Fw(and)e Ft(\014)44 b Fw(=)39 b Ft(\014)3223 1560 y Fu(c)3259 1546 y Fw(\()p Ft(q)s Fw(\))g(there)165 1659 y(exist)30 b Ft(q)23 b Fw(+)c(1)31 b(m)m(utually)d(singular)g(Gibbs)h(measures)h (whic)m(h)f(b)s(eha)m(v)m(e)i(qualitativ)m(ely)e(similar)e(to)k(the)165 1772 y(disordered)k(phase)h(for)g Ft(\014)41 b(<)35 b(\014)1265 1786 y Fu(c)1300 1772 y Fw(\()p Ft(q)s Fw(\))i(resp.)g(the)f Ft(q)k Fw(ordered)c(phases)g(for)g Ft(\014)41 b(>)35 b(\014)2943 1786 y Fu(c)2978 1772 y Fw(\()p Ft(q)s Fw(\).)60 b(This)34 b(is)i(the)165 1885 y(\014rst-order)f(phase)g(transition)f (in)g(the)i(P)m(otts)g(mo)s(del)e(for)i(large)f Ft(q)s Fw(.)56 b(F)-8 b(or)36 b(further)e(discussions)f(w)m(e)165 1998 y(refer)d(to)i([227)q(,)f(148)q(,)g(150)q(,)g(149)q(])f(and)g(the) h(references)g(therein.)165 2185 y(W)-8 b(e)32 b(no)m(w)e(giv)m(e)h(an) f(outline)f(of)i(the)g(pro)s(of)e(of)i(Theorem)f(8.14.)165 2348 y Fx(Sk)m(etc)m(h)49 b(pro)s(of)f(of)g(Theorem)f(8.14:)63 b Fw(F)-8 b(or)43 b(simplicit)m(y)38 b(w)m(e)k(stic)m(k)g(to)h(the)e (case)i Ft(d)h Fw(=)g(2.)74 b(F)-8 b(or)165 2461 y(an)m(y)38 b(in)m(v)m(erse)e(temp)s(erature)h Ft(\014)42 b(>)35 b Fw(0)j(and)e(an)m(y)h(square)g(b)s(o)m(x)g(\003)2356 2475 y Fu(n)2439 2461 y Fw(=)f([)p Fv(\000)p Ft(n;)15 b(n)24 b Fv(\000)g Fw(1])2981 2428 y Fr(2)3046 2461 y Fv(\\)g Fx(Z)3195 2428 y Fr(2)3272 2461 y Fw(w)m(e)37 b(write)165 2574 y Ft(\026)220 2530 y Fu(per)220 2602 y(\014)s(;n)366 2574 y Fw(for)f(the)g(Gibbs)f(distribution)e(relativ)m (e)j(to)h Ft(\014)5 b(H)44 b Fw(in)35 b(the)h(b)s(o)m(x)g(\003)2584 2588 y Fu(n)2667 2574 y Fw(with)f(p)s(erio)s(dic)f(b)s(oundary)165 2687 y(condition.)51 b(The)34 b(latter)h(means)f(that)h(\003)1600 2701 y Fu(n)1681 2687 y Fw(is)f(view)m(ed)g(as)g(a)h(torus,)g(so)g (that)f(\()p Ft(i;)15 b(n)24 b Fv(\000)e Fw(1\))33 b Fv(\030)e Fw(\()p Ft(i;)15 b Fv(\000)p Ft(n)p Fw(\))165 2800 y(and)25 b(\()p Ft(n)11 b Fv(\000)g Fw(1)p Ft(;)k(i)p Fw(\))27 b Fv(\030)e Fw(\()p Fv(\000)p Ft(n;)15 b(i)p Fw(\))26 b(for)f Ft(i)h Fv(2)f Fw([)p Fv(\000)p Ft(n;)15 b(n)c Fv(\000)g Fw(1])g Fv(\\)g Fx(Z)p Fw(;)26 b(the)g(Hamiltonian)e Ft(H)2724 2767 y Fu(per)2717 2822 y(n)2856 2800 y Fw(in)h(\003)3021 2814 y Fu(n)3093 2800 y Fw(with)g(p)s(erio)s(dic)165 2913 y(b)s(oundary)32 b(condition)h(is)g(then)h(de\014ned)e(in)h(the)h (natural)f(w)m(a)m(y)-8 b(.)53 b(Let)35 b Ft(\026)2640 2880 y Fu(per)2779 2913 y Fw(b)s(e)f(an)g(arbitrary)e(limit)165 3025 y(p)s(oin)m(t)f(of)h(the)g(sequence)g(\()p Ft(\026)1135 2981 y Fu(per)1135 3054 y(\014)s(;n)1244 3025 y Fw(\))1279 3039 y Fu(n)p Fq(\025)p Fr(1)1417 3025 y Fw(.)45 b(Eviden)m(tly)-8 b(,)31 b Ft(\026)1974 2992 y Fu(per)2111 3025 y Fw(has)h(the)g (symmetry)f(prop)s(erties)f(required)165 3138 y(of)h Ft(\026)f Fw(in)f(Theorem)h(8.14,)i(and)e Ft(\026)1293 3105 y Fu(per)1424 3138 y Fv(2)25 b(G)5 b Fw(\()p Ft(\014)g(H)i Fw(\).)306 3251 y(T)-8 b(o)28 b(establish)d(p)s(ercolation)g(of)i (ground-energy)f(b)s(onds)f(w)m(e)i(\014x)g(some)g Ft(\013)e(<)g Fw(1)i(and)f(consider)g(the)165 3364 y(w)m(edge)35 b Fv(W)j Fw(=)30 b Fv(f)p Ft(x)i Fw(=)e(\()p Ft(x)985 3378 y Fr(1)1025 3364 y Ft(;)15 b(x)1117 3378 y Fr(2)1157 3364 y Fw(\))31 b Fv(2)f(L)h Fw(:)g Ft(x)1516 3378 y Fr(1)1586 3364 y Fv(\025)g Fw(0)p Ft(;)g Fv(j)p Ft(x)1866 3378 y Fr(2)1905 3364 y Fv(j)g(\024)g Ft(\013x)2173 3378 y Fr(1)2213 3364 y Fv(g)p Fw(.)51 b(Let)34 b Ft(A)2568 3378 y Fq(W)2681 3364 y Fw(b)s(e)f(the)h(ev)m(en)m(t)i(that)e(there)165 3477 y(exists)c(an)g(in\014nite)e(path)i(of)g(ground-energy)g(b)s(onds) f(in)f Fv(W)38 b Fw(starting)30 b(from)f(the)i(origin.)39 b(W)-8 b(e)31 b(w)m(an)m(t)165 3590 y(to)36 b(sho)m(w)e(that)i Ft(\026)767 3557 y Fu(per)873 3590 y Fw(\()p Ft(A)976 3604 y Fq(W)1056 3590 y Fw(\))c Ft(>)h Fw(3)p Ft(=)p Fw(4)j(when)d Ft(\014)40 b Fw(is)34 b(large)g(enough.)54 b(Supp)s(ose)33 b Ft(\030)46 b(=)-55 b Fv(2)32 b Ft(A)3012 3604 y Fq(W)3092 3590 y Fw(.)53 b(Then)34 b(there)165 3703 y(exists)h(a)g(con)m(tour)h(crossing)f Fv(W)7 b Fw(,)36 b(i.e.,)h(a)e(path)g Ft(\015)40 b Fw(in)34 b(the)h(dual)f (lattice)h Fv(L)2731 3670 y Fq(\003)2803 3703 y Fw(=)e Fx(Z)2971 3670 y Fr(2)3034 3703 y Fw(+)23 b(\()3173 3667 y Fr(1)p 3173 3682 36 4 v 3173 3734 a(2)3218 3703 y Ft(;)3269 3667 y Fr(1)p 3269 3682 V 3269 3734 a(2)3314 3703 y Fw(\))35 b(whic)m(h)165 3816 y(crosses)d(no)f(ground-energy)g(b)s(ond)e(for)i Ft(\030)k Fw(and)c(connects)h(the)f(t)m(w)m(o)i(half-lines)28 b(b)s(ordering)h Fv(W)7 b Fw(.)43 b(F)-8 b(or)165 3929 y(eac)m(h)32 b(suc)m(h)e(path)g Ft(\015)35 b Fw(w)m(e)c(will)d (establish)h(the)h(con)m(tour)h(estimate)1196 4133 y Ft(\026)1251 4095 y Fu(per)1357 4133 y Fw(\()p Ft(\015)36 b Fw(is)29 b(a)i(con)m(tour)q(\))25 b Fv(\024)g Fw(\()p Fv(j)p Ft(S)5 b Fv(j)p Ft(e)2287 4095 y Fq(\000)p Fu(\014)s(\016)2424 4133 y Fw(\))2459 4095 y Fq(j)p Fu(\015)t Fq(j)2558 4133 y Ft(;)870 b Fw(\(64\))165 4337 y(where)32 b Fv(j)p Ft(\015)5 b Fv(j)33 b Fw(is)e(the)i(length)f(\(the)g(n)m(um)m(b)s(er)f(of)i(v)m (ertices\))g(of)g Ft(\015)5 b Fw(,)33 b(and)f Ft(\016)g(>)c Fw(0)33 b(is)e(suc)m(h)h(that)h Ft(m)21 b Fw(+)h(2)p Ft(\016)36 b Fw(is)165 4450 y(the)31 b(second)f(lo)m(w)m(est)i(v)-5 b(alue)29 b(of)i Ft(U)10 b Fw(.)306 4563 y(Assuming)31 b(\(64\))i(w)m(e)g(obtain)e(the)h(theorem)g(as)h(follo)m(ws.)44 b(The)31 b(n)m(um)m(b)s(er)g(of)h(paths)g(of)g(length)f Ft(k)165 4676 y Fw(crossing)f Fv(W)38 b Fw(is)29 b(at)i(most)g Ft(ck)19 b Fw(3)1213 4643 y Fu(k)1286 4676 y Fw(for)30 b(some)h Ft(c)25 b(<)g Fv(1)31 b Fw(dep)s(ending)d(on)i Ft(\013)p Fw(.)41 b(Hence)1190 4913 y Ft(\026)1245 4876 y Fu(per)1351 4913 y Fw(\()p Ft(A)1454 4876 y Fu(c)1454 4936 y Fq(W)1534 4913 y Fw(\))26 b Fv(\024)f Ft(c)1750 4833 y Fs(X)1745 5017 y Fu(k)r Fq(\025)p Fr(1)1889 4913 y Ft(k)19 b Fw(\(3)p Fv(j)p Ft(S)5 b Fv(j)p Ft(e)2188 4876 y Fq(\000)p Fu(\014)s(\016)2325 4913 y Fw(\))2360 4876 y Fu(k)2428 4913 y Ft(<)2534 4852 y Fw(1)p 2534 4892 46 4 v 2534 4976 a(4)165 5204 y(for)30 b(su\016cien)m(tly)f(large) h Ft(\014)5 b Fw(.)41 b(By)30 b(the)g(rotation)g(in)m(v)-5 b(ariance)29 b(of)i Ft(\026)2337 5171 y Fu(per)2443 5204 y Fw(,)f(it)f(follo)m(ws)g(that)i Ft(\026)3138 5171 y Fu(per)3244 5204 y Fw(\()p Ft(A)3347 5218 y Fr(0)3387 5204 y Fw(\))26 b Ft(>)e Fw(0,)165 5317 y(where)45 b Ft(A)511 5331 y Fr(0)595 5317 y Fw(is)f(the)h(in)m(tersection)f(of)h Ft(A)1560 5331 y Fq(W)1685 5317 y Fw(with)e(its)i(three)g(coun)m (terparts)g(obtained)f(b)m(y)h(lattice)165 5430 y(rotations.)71 b(Roughly)39 b(sp)s(eaking,)j Ft(A)1467 5444 y Fr(0)1547 5430 y Fw(is)d(the)i(ev)m(en)m(t)g(that)g(the)g(origin)d(b)s(elongs)h (to)i(t)m(w)m(o)h(doubly)165 5543 y(in\014nite)36 b(ground-energy)i (paths,)h(one)f(b)s(eing)f(quasi-horizon)m(tal)g(and)g(the)h(other)g (quasi-v)m(ertical.)1844 5895 y(90)p eop %%Page: 91 91 91 90 bop 165 304 a Fw(Since)21 b Ft(\026)449 271 y Fu(per)576 304 y Fw(is)f(in)m(v)-5 b(arian)m(t)21 b(under)e(horizon)m(tal)i(and)g (v)m(ertical)g(translations,)h(the)g(P)m(oincar)m(\023)-43 b(e)22 b(recurrence)165 417 y(theorem)31 b(\(or)g(the)f(ergo)s(dic)g (theorem\))h(implies)d(that)j(the)g(ev)m(en)m(t)394 589 y Ft(A)462 603 y Fq(1)563 589 y Fw(=)25 b Fv(f)p Ft(\030)k Fv(2)c Fw(\012)g(:)h Ft(\022)1044 603 y Fu(x)1087 589 y Ft(\030)j Fv(2)c Ft(A)1310 603 y Fr(0)1380 589 y Fw(for)30 b(in\014nitely)e(man)m(y)i Ft(x)g Fw(in)f(eac)m(h)j(of)e(the)h(four)f (half-axes)p Fv(g)165 762 y Fw(has)h(also)f(p)s(ositiv)m(e)g Ft(\026)902 729 y Fu(per)1008 762 y Fw(-probabilit)m(y)-8 b(.)39 b(Eac)m(h)31 b(con\014guration)f(in)f Ft(A)2482 776 y Fq(1)2588 762 y Fw(has)h(in\014nitely)d(man)m(y)k(quasi-)165 875 y(horizon)m(tal)i(and)g(quasi-v)m(ertical)g(ground-energy)f(paths)h (in)f(eac)m(h)i(of)g(the)f(four)g(directions)f(of)h(the)165 987 y(compass,)k(and)e(b)m(y)g(planarit)m(y)f(all)g(paths)h(of)g (di\013eren)m(t)g(orien)m(tation)g(m)m(ust)g(in)m(tersect.)55 b(Therefore)165 1100 y(all)32 b(these)h(paths)g(b)s(elong)e(to)j(the)f (same)g(in\014nite)e(ground-energy)h(cluster)g(whic)m(h)g(has)g(only)g (\014nite)165 1213 y(holes,)f(and)g(is)f(therefore)h(unique.)41 b(Hence)32 b Ft(A)1743 1227 y Fq(1)1849 1213 y Fw(is)e(con)m(tained)h (in)f(the)h(ev)m(en)m(t)i Ft(B)j Fw(that)31 b(there)g(exists)165 1326 y(a)40 b(ground-energy)e(cluster)h(surrounding)c(eac)m(h)40 b(\014nite)e(set)i(of)f Fv(L)p Fw(,)i(and)e Ft(\026)2761 1293 y Fu(per)2867 1326 y Fw(\()p Ft(B)5 b Fw(\))39 b Ft(>)h Fw(0.)67 b(As)39 b Ft(B)k Fw(is)165 1439 y(a)35 b(tail-ev)m(en)m(t)g(and)f(in)m(v)-5 b(arian)m(t)34 b(under)e(all)i (automorphisms)e(of)j Fv(L)f Fw(and)f(all)h(symmetries)f(of)h Ft(U)10 b Fw(,)36 b(the)165 1552 y(theorem)31 b(follo)m(ws)f(b)m(y)g (setting)g Ft(\026)25 b Fw(=)g Ft(\026)1473 1519 y Fu(per)1579 1552 y Fw(\()15 b Fv(\001)g(j)p Ft(B)5 b Fw(\).)306 1665 y(It)37 b(remains)e(to)i(establish)e(the)h(con)m(tour)h(estimate)g (\(64\).)60 b(F)-8 b(or)37 b(this)e(it)h(is)f(su\016cien)m(t)h(to)h (sho)m(w)165 1778 y(that)1214 1891 y Ft(\026)1269 1847 y Fu(per)1269 1919 y(\014)s(;n)1379 1891 y Fw(\()p Ft(\015)f Fw(is)29 b(a)i(con)m(tour)q(\))25 b Fv(\024)g Fw(\()p Fv(j)p Ft(S)5 b Fv(j)p Ft(e)2309 1853 y Fq(\000)p Fu(\014)s(\016)2446 1891 y Fw(\))2481 1853 y Fq(j)p Fu(\015)t Fq(j)3453 1891 y Fw(\(65\))165 2042 y(when)24 b Ft(n)h Fw(is)e(so)j(large)e(that)i Ft(\015)k Fw(is)24 b(con)m(tained)h(in)e(\003)1809 2056 y Fu(n)1856 2042 y Fw(.)39 b(This)23 b(b)s(ound)g(is)h(based)h(on)f (re\015ection)h(p)s(ositivit)m(y)165 2155 y(and)35 b(the)g(c)m(hessb)s (oard)f(estimate,)k(whic)m(h)33 b(are)j(treated)g(at)f(length)g(in)e (Chapter)i(17)h(of)f([96)q(].)55 b(Here)165 2268 y(w)m(e)32 b(giv)m(e)g(only)e(the)i(principal)c(ideas.)43 b(The)31 b(basic)f(observ)-5 b(ation)31 b(is)g(the)g(follo)m(wing)f(consequence) i(of)165 2380 y(the)d(toroidal)f(symmetry)g(of)g Ft(\026)1236 2336 y Fu(per)1236 2409 y(\014)s(;n)1345 2380 y Fw(:)40 b(for)28 b(an)m(y)h Ft(i)c Fv(2)g(f)p Fw(0)p Ft(;)15 b(:)g(:)g(:)j(;)d(n)h Fv(\000)g Fw(1)p Fv(g)p Fw(,)30 b(the)e(con\014gurations)g(on)g(the)h(t)m(w)m(o)165 2508 y(parts)k(\003)464 2470 y Fr(+)464 2534 y Fu(n;i)585 2508 y Fw(=)d Fv(f)p Ft(x)h Fv(2)f Fw(\003)968 2522 y Fu(n)1045 2508 y Fw(:)g Ft(x)1152 2522 y Fr(1)1222 2508 y Fv(\025)g Ft(i)g Fw(or)h Ft(x)1548 2522 y Fr(1)1617 2508 y Fv(\024)f Ft(i)23 b Fv(\000)f Ft(n)p Fv(g)33 b Fw(and)g(\003)2241 2470 y Fq(\000)2241 2534 y Fu(n;i)2361 2508 y Fw(=)d Fv(f)p Ft(x)h Fv(2)f Fw(\003)2744 2522 y Fu(n)2821 2508 y Fw(:)g Ft(i)23 b Fv(\000)f Ft(n)30 b Fv(\024)f Ft(x)3260 2522 y Fr(1)3330 2508 y Fv(\024)h Ft(i)p Fv(g)k Fw(of)165 2621 y(\003)228 2635 y Fu(n)303 2621 y Fw(are)27 b(conditionally)e(indep)s(enden)m(t)g(and,)i(up)f(to)i (re\015ection,)g(iden)m(tically)e(distributed)e(giv)m(en)j(the)165 2734 y(spin)e(v)-5 b(alues)25 b(on)i(the)f(t)m(w)m(o)i(separating)e (lines)f Fv(f)p Ft(x)1800 2748 y Fr(1)1865 2734 y Fw(=)g Ft(i)p Fv(g)i Fw(and)f Fv(f)p Ft(x)2334 2748 y Fr(1)2399 2734 y Fw(=)f Ft(i)12 b Fv(\000)g Ft(n)p Fv(g)p Fw(.)40 b(Hence,)28 b(if)d Ft(f)5 b(;)15 b(g)30 b Fw(are)c(real)165 2847 y(functions)33 b(on)g Ft(S)751 2814 y Fr(\003)800 2822 y Fo(n)881 2847 y Fw(dep)s(ending)e(only)i(on)g(the)h (con\014guration)f(in)g(\003)2540 2809 y Fr(+)2540 2873 y Fu(n;i)2630 2847 y Fw(,)i(and)e Ft(g)2916 2814 y Fr(\()p Fu(i)p Fr(\))3034 2847 y Fw(is)f(the)i(function)165 2960 y(obtained)c(from)g Ft(g)35 b Fw(b)m(y)30 b(re\015ection)g(in)g(these)h (t)m(w)m(o)h(separating)e(lines)f(\(and)h(th)m(us)g(dep)s(ending)e(on)j (the)165 3073 y(con\014guration)25 b(in)f(\003)875 3034 y Fq(\000)875 3099 y Fu(n;i)966 3073 y Fw(\),)j(then)e(the)g(bilinear)d (form)j(\()p Ft(f)5 b(;)15 b(g)s Fw(\))27 b Fv(!)e Ft(\026)2343 3028 y Fu(per)2343 3101 y(\014)s(;n)2452 3073 y Fw(\()p Ft(f)10 b(g)2588 3040 y Fr(\()p Fu(i)p Fr(\))2671 3073 y Fw(\))26 b(is)e(p)s(ositiv)m(e)h(de\014nite)f(and)165 3186 y(th)m(us)h(satis\014es)f(the)i(Cauc)m(h)m(y{Sc)m(h)m(w)m(arz)g (inequalit)m(y)-8 b(.)38 b(Similar)22 b(Cauc)m(h)m(y{Sc)m(h)m(w)m(arz) 27 b(inequalities)22 b(hold)165 3298 y(for)30 b(v)m(ertical)h (re\015ections.)40 b(The)29 b(c)m(hessb)s(oard)h(inequalit)m(y)f(is)g (obtained)g(b)m(y)h(suitable)f(com)m(binations)165 3411 y(of)i(all)e(these,)i(as)g(w)m(e)g(will)c(illustrate)i(next.)306 3524 y(Let)j(us)e(mark)h(the)g(plaquettes)g(around)f(the)h(v)m(ertices) h(of)f Ft(\015)36 b Fw(with)29 b(a)j Fv(\017)p Fw(;)f(this)f(giv)m(es)i Fv(j)p Ft(\015)5 b Fv(j)31 b Fw(mark)m(ed)165 3637 y(plaquettes.)39 b(Marking)22 b(a)i(plaquette)f(indicates)f(that)i(at)f(least)h(one)f (of)g(its)g(b)s(onds)e(has)i(non-minimal)165 3750 y(energy;)35 b(lea)m(ving)d(it)h(unmark)m(ed)f(do)s(es)g(not)h(sa)m(y)h(that)f(it)f (consists)h(of)f(ground-energy)h(b)s(onds,)f(but)165 3863 y(that)g(w)m(e)f(don't)g(need)f(an)m(y)h(information)e(on)i(its)f (spins)e(.)42 b(In)30 b(the)h(case)h Ft(n)25 b Fw(=)h(2,)31 b(this)f(migh)m(t)g(lead)g(to)165 3976 y(the)h(picture)p 1667 4129 4 79 v 1755 4129 V 1843 4129 V 1931 4129 V 1669 4132 352 4 v 1667 4211 4 79 v 1755 4211 V 1843 4211 V 1871 4187 a Fq(\017)p 1931 4211 V 53 w(\017)p 1669 4214 352 4 v 1667 4293 4 79 v 1755 4293 V 1843 4293 V 1931 4293 V 1959 4270 a(\017)p 1669 4297 352 4 v 1667 4376 4 79 v 1755 4376 V 1843 4376 V 1931 4376 V 1669 4379 352 4 v 2091 4379 a Fr(.)165 4540 y Fw(T)-8 b(o)27 b(estimate)f(its)f(probabilit)m(y)e(w)m(e)k(use)e(rep)s(eatedly)g(the)h (Cauc)m(h)m(y{Sc)m(h)m(w)m(arz)h(inequalit)m(y)e(relativ)m(e)g(to)165 4653 y(suitable)32 b(pairs)f(of)j(re\015ection)e(lines.)47 b(Indicating)31 b(eac)m(h)k(time)d(only)g(the)h(pair)f(of)h(lines)f (used)g(next,)165 4766 y(and)24 b(omitting)h(the)f(ev)m(en)m(t)j(that)e (no)g(plaquette)f(is)g(mark)m(ed)h(\(whic)m(h)f(has)g(probabilit)m(y)f (1\),)j(w)m(e)g(obtain)490 5077 y Ft(\026)545 5033 y Fu(per)545 5105 y(\014)s(;n)670 4908 y Fs(0)670 5057 y(@)p 766 4981 4 79 v 942 4981 V 766 5060 V 942 5060 V 970 5037 a Fq(\017)53 b(\017)p 766 5139 V 942 5139 V 1058 5115 a(\017)p 766 5218 V 942 5218 V 1134 4908 a Fs(1)1134 5057 y(A)1263 5077 y Fv(\024)85 b Ft(\026)1474 5033 y Fu(per)1474 5105 y(\014)s(;n)1599 4883 y Fs(0)1599 5029 y(B)1599 5082 y(@)p 1697 4981 352 4 v 1723 5037 a Fq(\017)53 b(\017)g(\017)f(\017)1723 5115 y(\017)228 b(\017)p 1697 5142 V 2063 4883 a Fs(1)2063 5029 y(C)2063 5082 y(A)2136 4906 y Fr(1)p Fu(=)p Fr(2)2301 5077 y Fv(\024)86 b Ft(\026)2513 5033 y Fu(per)2513 5105 y(\014)s(;n)2637 4883 y Fs(0)2637 5029 y(B)2637 5082 y(@)2762 4954 y Fq(\017)53 b(\017)f(\017)h(\017)2762 5033 y(\017)g(\017)f(\017)h(\017)p 2735 5060 V 2762 5115 a(\017)228 b(\017)2762 5194 y(\017)g(\017)p 2735 5221 V 3102 4883 a Fs(1)3102 5029 y(C)3102 5082 y(A)3175 4906 y Fr(1)p Fu(=)p Fr(4)521 5538 y Fv(\024)85 b Ft(\026)732 5494 y Fu(per)732 5566 y(\014)s(;n)857 5369 y Fs(0)857 5518 y(@)971 5419 y Fq(\017)53 b(\017)g(\017)f(\017)971 5498 y(\017)h(\017)g(\017)f(\017)971 5576 y(\017)h(\017)g(\017)f(\017) 971 5655 y(\017)h(\017)g(\017)f(\017)1311 5369 y Fs(1)1311 5518 y(A)1384 5390 y Fr(1)p Fu(=)p Fr(8)1509 5538 y Ft(\026)1564 5494 y Fu(per)1564 5566 y(\014)s(;n)1688 5369 y Fs(0)1688 5518 y(@)1803 5419 y Fq(\017)p 1862 5442 4 79 v 2038 5442 V 228 w(\017)1803 5498 y(\017)p 1862 5521 V 2038 5521 V 228 w(\017)1803 5576 y(\017)p 1862 5600 V 2038 5600 V 228 w(\017)1803 5655 y(\017)p 1862 5679 V 2038 5679 V 228 w(\017)2143 5369 y Fs(1)2143 5518 y(A)2216 5390 y Fr(1)p Fu(=)p Fr(8)2381 5538 y Fv(\024)86 b Ft(\026)2593 5494 y Fu(per)2593 5566 y(\014)s(;n)2717 5369 y Fs(0)2717 5518 y(@)2831 5419 y Fq(\017)53 b(\017)g(\017)g(\017)2831 5498 y(\017)g(\017)g(\017)g(\017)2831 5576 y(\017)g(\017)g(\017)g(\017) 2831 5655 y(\017)g(\017)g(\017)g(\017)3172 5369 y Fs(1)3172 5518 y(A)3244 5390 y Fr(3)p Fu(=)p Fr(16)3430 5538 y Ft(:)1844 5895 y Fw(91)p eop %%Page: 92 92 92 91 bop 165 304 a Fw(In)30 b(general,)h(w)m(e)f(obtain)g(in)f(this)h (w)m(a)m(y)1113 508 y Ft(\026)1168 464 y Fu(per)1168 536 y(\014)s(;n)1277 508 y Fw(\()p Ft(\015)36 b Fw(is)29 b(a)i(con)m(tour)q(\))26 b Fv(\024)f Ft(\026)2075 464 y Fu(per)2075 536 y(\014)s(;n)2184 508 y Fw(\()p Ft(C)2284 522 y Fu(n)2331 508 y Fw(\))2366 471 y Fq(j)p Fu(\015)t Fq(j)p Fu(=)p Fq(j)p Fr(\003)2550 479 y Fo(n)2592 471 y Fq(j)2641 508 y Ft(;)165 712 y Fw(where)f Ft(C)487 726 y Fu(n)559 712 y Fw(is)f(the)i(ev)m(en)m(t)g(that)g(all)f (plaquettes)g(in)f(\003)1934 726 y Fu(n)2006 712 y Fw(con)m(tain)h(at)h (least)g(one)g(b)s(ond)e(of)h(non-minimal)165 825 y(energy)-8 b(.)51 b(But)34 b(if)e Ft(C)828 839 y Fu(n)908 825 y Fw(o)s(ccurs)h(then)h(at)g(least)f Fv(j)p Fw(\003)1820 839 y Fu(n)1868 825 y Fv(j)p Ft(=)p Fw(2)h(of)g(the)g(2)p Fv(j)p Fw(\003)2417 839 y Fu(n)2464 825 y Fv(j)g Fw(edges)g(in)e(\003) 2941 839 y Fu(n)3022 825 y Fw(are)i(no)f(ground-)165 938 y(energy)f(b)s(onds.)41 b(The)31 b(Hamiltonian)e Ft(H)1554 905 y Fu(per)1547 961 y(n)1691 938 y Fw(with)h(p)s(erio)s (dic)f(b)s(oundary)g(condition)h(is)g(therefore)h(at)165 1051 y(least)h(\(2)p Ft(m)22 b Fw(+)f Ft(\016)s Fw(\))p Fv(j)p Fw(\003)819 1065 y Fu(n)868 1051 y Fv(j)32 b Fw(on)f Ft(C)1117 1065 y Fu(n)1164 1051 y Fw(.)45 b(Since)31 b(there)h(is)f(at)h(least)g(one)g Ft(\033)f Fv(2)c Ft(S)2526 1018 y Fr(\003)2575 1026 y Fo(n)2654 1051 y Fw(with)j Ft(H)2945 1018 y Fu(per)2938 1074 y(n)3051 1051 y Fw(\()p Ft(\033)s Fw(\))f(=)e(2)p Fv(j)p Fw(\003)3436 1065 y Fu(n)3484 1051 y Fv(j)p Ft(m)p Fw(,)165 1164 y(it)j(follo)m(ws)g(that) 742 1368 y Ft(\026)797 1324 y Fu(per)797 1396 y(\014)s(;n)907 1368 y Fw(\()p Ft(C)1007 1382 y Fu(n)1054 1368 y Fw(\))c Fv(\024)1237 1288 y Fs(X)1210 1472 y Fu(\030)s Fq(2)p Fu(C)1341 1480 y Fo(n)1399 1368 y Ft(e)1441 1331 y Fq(\000)p Fu(\014)s Fr(\(2)p Fu(m)p Fr(+)p Fu(\016)r Fr(\))p Fq(j)p Fr(\003)1847 1339 y Fo(n)1892 1331 y Fq(j)1915 1274 y Fs(.)1983 1368 y Ft(e)2025 1331 y Fq(\000)p Fu(\014)s Fr(2)p Fu(m)p Fq(j)p Fr(\003)2289 1339 y Fo(n)2332 1331 y Fq(j)2381 1368 y Fv(\024)f Fw(\()p Fv(j)p Ft(S)5 b Fv(j)15 b Ft(e)2680 1331 y Fq(\000)p Fu(\014)s(\016)2817 1368 y Fw(\))2852 1331 y Fq(j)p Fr(\003)2921 1339 y Fo(n)2963 1331 y Fq(j)3012 1368 y Ft(:)165 1661 y Fw(This)29 b(giv)m(es)i (estimate)g(\(65\))h(and)d(completes)i(the)g(pro)s(of)e(of)i(Theorem)f (8.14.)731 b Fk(2)1844 5895 y Fw(92)p eop %%Page: 93 93 93 92 bop 165 304 a Fy(9)135 b(Random)44 b(in)l(teractions)165 507 y Fw(So)24 b(far)g(in)f(this)f(review,)j(the)f(spin)e(systems)i (considered)f(had)g(an)h(in)m(teraction)g(whic)m(h)f(w)m(as)h(in)m(v)-5 b(arian)m(t)165 620 y(under)20 b(all)g(automorphisms)g(of)h(the)h (underlying)c(graph)i Fv(L)p Fw(.)38 b(Here)22 b(w)m(e)f(will)e(assume) i(for)g(con)m(v)m(enience)165 733 y(that)37 b Fv(L)f Fw(is)f(the)i(cubic)e(lattice)i(\()p Fx(Z)1353 700 y Fu(d)1394 733 y Ft(;)15 b Fv(B)s Fw(\),)38 b Ft(d)d Fv(\025)g Fw(2,)j(but)e(the)g(in)m(teraction)h(b)s(et)m(w)m(een)g(adjacen)m(t)g (spins)165 846 y(will)28 b(no)i(longer)g(b)s(e)f(translation)g(in)m(v) -5 b(arian)m(t.)40 b(That)30 b(is,)g(instead)f(of)h(the)h(Hamiltonian)d (\(1\))j(w)m(e)g(no)m(w)165 959 y(consider)f(a)g(mo)s(di\014ed)f (Hamiltonian)f(of)j(the)f(form)860 1163 y Ft(H)7 b Fw(\()p Ft(\033)s Fw(\))26 b(=)1286 1082 y Fs(X)1190 1270 y Fu(b)p Fr(=)p Fq(h)p Fu(xy)r Fq(i2B)1517 1163 y Ft(J)1567 1178 y Fu(b)1627 1163 y Ft(U)10 b Fw(\()p Ft(\033)s Fw(\()p Ft(x)p Fw(\))p Ft(;)15 b(\033)s Fw(\()p Ft(y)s Fw(\)\))23 b(+)2299 1082 y Fs(X)2273 1274 y Fu(x)p Fq(2)p Fn(Z)2410 1255 y Fo(d)2461 1163 y Ft(h)2513 1177 y Fu(x)2572 1163 y Ft(V)e Fw(\()p Ft(\033)s Fw(\()p Ft(x)p Fw(\)\))560 b(\(66\))165 1458 y(where)25 b(the)h Ft(J)625 1473 y Fu(b)685 1458 y Fw(and)e(the)i Ft(h)1060 1472 y Fu(x)1129 1458 y Fw(ma)m(y)g(v)-5 b(ary)25 b(from)g(b)s(ond)f(to)i(b)s(ond,)f (resp.)g(from)g(site)g(to)h(site.)39 b(In)24 b(fact,)k(w)m(e)165 1571 y(are)36 b(in)m(terested)g(in)e(the)h(case)i(where)e(these)g (coupling)f(co)s(e\016cien)m(ts)i(sho)m(w)f(no)h(regular)e(structure,) 165 1684 y(and)j(th)m(us)f(assume)h(that)h(they)f(are)g Fp(r)-5 b(andom)p Fw(.)63 b(Suc)m(h)36 b(systems)h(of)h(spins)c(in)m (teracting)j(di\013eren)m(tly)165 1797 y(dep)s(ending)30 b(on)i(their)g(p)s(osition)e(and)i(in)f(a)h(w)m(a)m(y)i(go)m(v)m(erned) f(b)m(y)f(c)m(hance)i(are)f(kno)m(wn)e(as)i Fp(disor)-5 b(der)g(e)g(d)165 1909 y(systems)p Fw(.)41 b(W)-8 b(e)29 b(will)d(not)i(elab)s(orate)g(on)g(the)g(ph)m(ysical)f(origins)f(of)i (suc)m(h)g(random)f(in)m(teractions.)40 b(W)-8 b(e)165 2022 y(merely)32 b(men)m(tion)g(that)g(they)h(can)f(b)s(e)g(related)g (to)h(the)f(presence)g(of)h(impurities)28 b(or)33 b(defects)f(in)f(an) 165 2135 y(originally)24 b(homogeneous)k(system,)f(and)g(are)g(used)e (to)j(mo)s(del)d(quenc)m(hed)h(allo)m(ys)h(of)g(magnetic)g(and)165 2248 y(nonmagnetic)k(materials)e(lik)m(e)h Fj(F)-8 b(eAu)p Fw(.)42 b(F)-8 b(or)31 b(details)e(w)m(e)i(refer)f(to)h([84)q(,)g(30)q (,)f(78)q(].)306 2361 y(W)-8 b(e)41 b(assume)f(that)g(the)g(family)e Fx(J)j Fw(=)g(\()p Ft(J)1747 2376 y Fu(b)1782 2361 y Fw(\))1817 2376 y Fu(b)p Fq(2B)1986 2361 y Fw(of)f(coupling)e(co)s (e\016cien)m(ts)i(and)f(the)h(external)165 2474 y(\014elds)d Fx(h)h Fw(=)g(\()p Ft(h)698 2488 y Fu(x)743 2474 y Fw(\))778 2496 y Fu(x)p Fq(2)p Fn(Z)915 2477 y Fo(d)993 2474 y Fw(are)h(indep)s(enden)m(t,)f(and)f(eac)m(h)j(collection)d(constitutes) i(a)f(family)f(of)h(m)m(u-)165 2587 y(tually)33 b(indep)s(enden)m(t)e (and)i(iden)m(tically)f(distributed)f(real)i(random)g(v)-5 b(ariables.)49 b(Hence,)36 b(while)31 b(no)165 2700 y(realization)g(of) h(the)f(coupling)f(co)s(e\016cien)m(ts)i(is)e(translation)g(in)m(v)-5 b(arian)m(t,)31 b(w)m(e)h(still)e(ha)m(v)m(e)i(translation)165 2813 y(in)m(v)-5 b(ariance)34 b(in)e(a)j(statistical)e(sense.)52 b(W)-8 b(e)35 b(will)d(not)i(sp)s(ecify)f(the)h(underlying)d (probabilit)m(y)g(space,)165 2926 y(except)k(that)g(the)g(letter)f Ft(P)47 b Fw(will)32 b(b)s(e)h(used)h(to)g(denote)h(the)g(probabilit)m (y)c(measure)j(and)g(the)g(asso-)165 3039 y(ciated)e(exp)s(ectation.)44 b(The)30 b(random)h(families)e Fx(J)i Fw(and)g Fx(h)g Fw(are)h(often)f(referred)g(to)h(as)f(the)h Fp(disor)-5 b(der)p Fw(.)165 3151 y(The)34 b(disorder)e(is)h(called)g(b)s(ounded)f (if)h Ft(P)13 b Fw(\()p Fv(j)p Ft(J)1708 3166 y Fu(b)1744 3151 y Fv(j)31 b Ft(>)g(c)p Fw(\))h(=)g(0)i(for)g(some)g(\014nite)f Ft(c)p Fw(.)52 b(Ph)m(ysically)-8 b(,)34 b(this)f(is)165 3264 y(the)e(most)g(relev)-5 b(an)m(t)30 b(case.)306 3377 y(\(In)22 b(some)h(ph)m(ysical)d(applications)h(it)g(is)g(natural) g(to)i(assume)f(that)h(the)f Ft(J)2776 3392 y Fu(b)2833 3377 y Fw(are)g(not)g(indep)s(enden)m(t)165 3490 y(but)g(rather)h(ha)m (v)m(e)h(some)f(\014nite-range)g(dep)s(endence)e(structure,)j(but)f(w)m (e)g(will)d(not)j(include)d(this)i(case)165 3603 y(here.)41 b(W)-8 b(e)31 b(also)f(assumed)g(for)g(simplicit)m(y)d(that)k(the)f (disorder)e(is)i(real)f(v)-5 b(alued,)30 b(although)f(some)i(of)165 3716 y(the)i(follo)m(wing)e(also)h(applies)f(to)i(the)g(case)h(when)d Ft(J)1956 3731 y Fu(b)2023 3716 y Fw(or)i Ft(h)2189 3730 y Fu(x)2265 3716 y Fw(are)g(allo)m(w)m(ed)g(to)g(tak)m(e)h(the)f(v)-5 b(alue)32 b(+)p Fv(1)165 3829 y Fw(with)d(p)s(ositiv)m(e)h(probabilit)m (y)-8 b(.\))306 3942 y(In)35 b(Section)g(9.1)i(w)m(e)f(will)d(discuss)g Fp(dilute)-5 b(d)39 b(ferr)-5 b(omagnets)p Fw(.)58 b(A)35 b(b)s(ond-diluted)d(Ising)i(or)i(P)m(otts)165 4055 y(ferromagnet)g(on)e Fx(Z)872 4022 y Fu(d)947 4055 y Fw(can)h(b)s(e)f(view)m(ed)h(as)g(an)f (Ising)g(or)g(P)m(otts)i(mo)s(del)e(on)h(the)f(op)s(en)g(clusters)g (for)165 4168 y(Bernoulli)29 b(b)s(ond)h(p)s(ercolation)g(on)h Fx(Z)1454 4135 y Fu(d)1494 4168 y Fw(.)43 b(As)31 b(observ)m(ed)g(in)f ([9)q(],)h(these)h(mo)s(dels)e(can)h(quite)g(easily)f(b)s(e)165 4281 y(understo)s(o)s(d)j(using)f(their)i(random-cluster)f(represen)m (tation.)52 b(They)33 b(form)h(just)g(ab)s(out)g(the)g(only)165 4393 y(class)24 b(of)g(disordered)e(systems)i(where)f(the)h(phase)f (transition)g(can)h(b)s(e)f(in)m(v)m(estigated)i(in)d(suc)m(h)i (detail.)306 4506 y(Then,)31 b(in)e(Section)h(9.2,)j(w)m(e)e(study)e (the)i(so-called)g Fp(Gri\016ths)j(r)-5 b(e)g(gime)32 b Fw(whic)m(h)d(is)h(the)h(only)e(non-)165 4619 y(trivial)j(regime)h (for)g(disordered)f(systems)h(where)g(b)m(y)h(no)m(w)f(quite)g(general) g(results)g(are)g(a)m(v)-5 b(ailable.)165 4732 y(It)32 b(o)s(ccurs)g(at)g(in)m(termediate)f(temp)s(eratures)h(if)e(the)i (disorder)e(is)h(b)s(ounded,)f(or)i(at)g(arbitrary)e(high)165 4845 y(temp)s(eratures)39 b(if)f(the)h(disorder)e(is)g(un)m(b)s (ounded,)i(and)f(is)g(c)m(haracterized)i(b)m(y)f(the)g(fact)g(that)h (the)165 4958 y(Gibbs)35 b(measure)h(is)f(still)f(unique)g(but)h(fails) f(to)j(ha)m(v)m(e)g(a)g(nice)e(high)g(temp)s(erature)h(b)s(eha)m(vior)f (uni-)165 5071 y(formly)h(in)g(the)h(disorder.)59 b(The)36 b(study)g(of)h(random)g(Gibbs)e(measures)i(in)f(the)h(Gri\016ths)e (regime)165 5184 y(has)d(started)g(in)f(the)h(early)g(1980's)h(and)f (has)f(reac)m(hed)i(a)f(satisfactory)h(stage)g(only)e(recen)m(tly)-8 b(.)47 b(The)165 5297 y(simplest)30 b(and)h(also)g(most)g(p)s(o)m(w)m (erful)f(metho)s(ds)h(use)g(sto)s(c)m(hastic-geometric)i(represen)m (tations)f(and)165 5410 y(will)c(b)s(e)i(presen)m(ted)g(here.)306 5523 y(As)j(represen)m(tativ)m(e)h(for)f(the)g(large)g(literature)e(on) i(the)g(sub)5 b(ject)33 b(w)m(e)g(refer)g(to)g([13)q(,)g(18)q(,)g(29)q (,)g(46)q(,)165 5636 y(72)q(,)i(84)q(,)f(85)q(,)h(86)q(,)g(102)q(,)g (106)q(,)g(189)q(,)g(190)q(].)53 b(Dynamical)34 b(problems)f(\(whic)m (h)g(are)i(not)g(touc)m(hed)g(up)s(on)1844 5895 y(93)p eop %%Page: 94 94 94 93 bop 165 304 a Fw(here\))31 b(are)g(treated)g(e.g.)h(in)d([49)q(,) i(50,)g(51)q(,)f(15)q(,)h(103)q(,)g(112)q(,)g(101)q(].)165 541 y FA(9.1)113 b(Diluted)37 b(and)i(random)f(Ising)g(and)h(P)m(otts)d (ferromagnets)165 713 y Fw(The)k(random)g(P)m(otts)h(mo)s(del)e(is)g (de\014ned)g(as)i(follo)m(ws.)69 b(Spins)38 b(tak)m(e)k(v)-5 b(alues)40 b(in)f(the)h(state)i(space)165 826 y Ft(S)31 b Fw(=)25 b Fv(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)i(;)e(q)s Fv(g)p Fw(,)31 b(and)f(the)h(in)m(teraction)f(is)f(giv)m(en)i(b)m(y)f (the)h(Hamiltonian)d(\(66\))k(with)1311 991 y Ft(U)10 b Fw(\()p Ft(\033)s Fw(\()p Ft(x)p Fw(\))p Ft(;)15 b(\033)s Fw(\()p Ft(y)s Fw(\)\))28 b(=)d(2)15 b Ft(I)2067 1009 y Fq(f)p Fu(\033)r Fr(\()p Fu(x)p Fr(\))p Fq(6)p Fr(=)p Fu(\033)r Fr(\()p Fu(y)r Fr(\))p Fq(g)165 1156 y Fw(and)26 b Ft(V)46 b Fv(\021)25 b Fw(0.)40 b(Note)28 b(that)f(this)f(c)m(hoice)h (of)g Ft(U)37 b Fw(and)26 b Ft(V)47 b Fw(coincides)26 b(with)f(that)j(used)e(in)f(Section)h(6.1)i(for)165 1269 y(the)k(standard)e(P)m(otts)j(mo)s(del)d(and)h(di\013ers)e(from)i(that) h(in)e(Section)h(3.3)h(only)f(b)m(y)g(constan)m(ts)h(whic)m(h)165 1382 y(cancel)22 b(in)f(the)g(de\014nition)f(of)h(Gibbs)f (distributions.)34 b(The)21 b(Ising)g(mo)s(del)f(corresp)s(onds)g(to)j (the)e(c)m(hoice)165 1495 y Ft(q)28 b Fw(=)d(2.)41 b(As)28 b(for)h(the)f(disorder,)f(w)m(e)i(mak)m(e)h(the)e(essen)m(tial)h (assumption)d(that)k(the)e(random)g(coupling)165 1608 y(co)s(e\016cien)m(ts)i Ft(J)674 1623 y Fu(b)739 1608 y Fw(are)f(nonnegativ)m(e,)i(so)e(that)h(the)g(in)m(teraction)f(is)g (still)e(ferromagnetic.)41 b(Of)29 b(course,)165 1720 y(w)m(e)35 b(also)e(mak)m(e)i(the)f(general)g(assumption)e(of)i(this)f (section)h(that)g(the)g Ft(J)2695 1735 y Fu(b)2764 1720 y Fw(are)g(indep)s(enden)m(t)d(with)165 1833 y(the)g(same)g (distribution,)c(sa)m(y)32 b Ft(\031)s Fw(.)41 b(A)31 b(particular)e(case)i(of)g(sp)s(ecial)e(in)m(terest)i(is)f(that)h(of)g Fp(dilution)p Fw(,)g(in)165 1946 y(whic)m(h)e(the)g Ft(J)630 1961 y Fu(b)694 1946 y Fw(tak)m(e)i(the)f(v)-5 b(alues)29 b(1)h(and)e(0)i(with)e(probabilities)e Ft(p)k Fw(and)e(1)19 b Fv(\000)f Ft(p)p Fw(,)30 b(resp)s(ectiv)m(ely)-8 b(,)29 b(whic)m(h)165 2059 y(means)38 b(that)h Ft(\031)h Fw(=)e Ft(p\016)943 2073 y Fr(1)1008 2059 y Fw(+)25 b(\(1)h Fv(\000)f Ft(p)p Fw(\))p Ft(\016)1427 2073 y Fr(0)1467 2059 y Fw(.)63 b(F)-8 b(or)38 b Ft(p)g Fw(=)f(1)i(w)m(e)f(then)g(reco)m (v)m(er)h(the)f(homogeneous)h(P)m(otts)165 2172 y(mo)s(del)30 b(of)g(Section)g(3.3.)306 2285 y(In)g(the)h(follo)m(wing,)e(the)h (distribution)d Ft(\031)33 b Fw(of)e(the)g Ft(J)2034 2300 y Fu(b)2099 2285 y Fw(will)c(en)m(ter)k(only)f(through)g(the)g (quan)m(tities)685 2504 y(\026)-53 b Ft(p)p Fw(\()p Ft(\014)5 b(;)15 b(\031)s Fw(\))27 b(=)e Ft(P)1138 2385 y Fs(\022)1199 2504 y Fw(1)20 b Fv(\000)g Ft(e)1397 2466 y Fq(\000)p Fr(2)p Fu(\014)s(J)1569 2478 y Fo(b)1605 2385 y Fs(\023)1691 2504 y Ft(;)107 b(p)p 1823 2535 46 4 v -1 w Fw(\()p Ft(\014)5 b(;)15 b(\031)s Fw(\))27 b(=)e Ft(P)2283 2385 y Fs(\022)2490 2442 y Fw(1)20 b Fv(\000)g Ft(e)2688 2409 y Fq(\000)p Fr(2)p Fu(\014)s(J)2860 2421 y Fo(b)p 2354 2483 678 4 v 2354 2566 a Fw(1)h(+)f(\()p Ft(q)j Fv(\000)d Fw(1\))p Ft(e)2823 2540 y Fq(\000)p Fr(2)p Fu(\014)s(J)2995 2552 y Fo(b)3041 2385 y Fs(\023)165 2717 y Fw(for)32 b Ft(\014)g(>)27 b Fw(0.)45 b(Note)33 b(that)f(they)g(do)f(not)h(dep)s(end)e(on)h Ft(b)d Fv(2)e(B)s Fw(,)32 b(and)f(that)h(0)c Fv(\024)f Ft(p)p 2782 2748 46 4 v -1 w Fw(\()p Ft(\014)5 b(;)15 b(\031)s Fw(\))29 b Fv(\024)35 b Fw(\026)-53 b Ft(p)p Fw(\()p Ft(\014)5 b(;)15 b(\031)s Fw(\))28 b Fv(\024)f Ft(p)165 2830 y Fw(with)i Ft(p)c Fw(=)g Ft(P)13 b Fw(\()p Ft(J)695 2845 y Fu(b)756 2830 y Ft(>)25 b Fw(0\).)306 2943 y(F)-8 b(or)24 b(a)g(giv)m(en)f(realization)g Fx(J)i Fw(=)g(\()p Ft(J)1459 2958 y Fu(b)1494 2943 y Fw(\))1529 2958 y Fu(b)p Fq(2B)1682 2943 y Fw(of)e(the)h(disorder)d(and)i(in)m(v)m (erse)g(temp)s(erature)g Ft(\014)5 b Fw(,)25 b(w)m(e)f(can)165 3056 y(in)m(tro)s(duce)i(the)h(Gibbs)e(measure)h Ft(\026)1384 3023 y Fu(i)1384 3083 y(\014)s Fn(J)p Fu(;q)1553 3056 y Fw(obtained)g(from)g(the)h(Gibbs)e(distributions)e(with)i(constan)m (t)165 3169 y(b)s(oundary)30 b(condition)g Ft(i)e Fv(2)f(f)p Fw(1)p Ft(;)15 b(:)g(:)g(:)i(;)e(q)s Fv(g)32 b Fw(in)f(the)g (in\014nite)f(v)m(olume)h(limit.)42 b(This)30 b(limit)f(exists)i(b)m(y) h(the)165 3282 y(argumen)m(ts)h(of)f(Prop)s(osition)e(6.9,)k(since)d (these)i(use)f(only)f(the)h(sto)s(c)m(hastic)h(monotonicit)m(y)f (coming)165 3395 y(from)d(Corollary)e(6.7)j(\(a\))f(as)g(w)m(ell)f(as)h (the)g(random-cluster)e(represen)m(tation,)j(whic)m(h)d(b)s(oth)h (remain)165 3508 y(v)-5 b(alid)29 b(in)g(the)i(non-homogeneous)g(case.) 306 3621 y(The)h(k)m(ey)g(quan)m(tit)m(y)g(for)g(phase)f(transition,)g (the)h(order)f(parameter,)i(is)e(the)g(\\quenc)m(hed)h(mag-)165 3734 y(netization")1036 3864 y Ft(m)p Fw(\()p Ft(\014)5 b(;)15 b(\031)s Fw(\))26 b(=)1547 3803 y Ft(q)p 1469 3843 201 4 v 1469 3927 a(q)d Fv(\000)d Fw(1)1705 3864 y Ft(P)1776 3745 y Fs(\022)1837 3864 y Ft(\026)1892 3827 y Fu(i)1892 3887 y(\014)s Fn(J)p Fu(;q)2034 3864 y Fw(\()p Ft(X)7 b Fw(\(0\))27 b(=)e Ft(i)p Fw(\))c Fv(\000)2577 3803 y Fw(1)p 2577 3843 46 4 v 2578 3927 a Ft(q)2632 3745 y Fs(\023)2719 3864 y Ft(:)165 4055 y Fw(Indeed,)38 b(an)f(inhomogeneous)g(v)m(ersion)f(of)i(equation)f(\(32\))h(sho)m(ws)f (that)h Ft(m)p Fw(\()p Ft(\014)5 b(;)15 b(\031)s Fw(\))37 b(=)g Ft(P)13 b Fw(\()p Ft(\022)3335 4069 y Fu(q)3373 4055 y Fw(\()p Ft(\014)5 b Fx(J)p Fw(\)\),)165 4168 y(where)45 b Ft(\022)486 4182 y Fu(q)524 4168 y Fw(\()p Ft(\014)5 b Fx(J)p Fw(\))50 b(=)g Ft(\036)929 4135 y Fr(1)929 4191 y Fn(p)p Fu(;q)1032 4168 y Fw(\(0)g Fv($)g(1)p Fw(\))45 b(is)f(the)i(p)s(ercolation)e(probabilit)m(y)f(for)h(the)i(wired)d (in\014nite-)165 4291 y(v)m(olume)c(random-cluster)e(measure)i(with)e (b)s(ond)h(probabilities)d Ft(p)2514 4306 y Fu(b)2587 4291 y Fw(=)k(1)26 b Fv(\000)g Ft(e)2907 4258 y Fq(\000)p Fr(2)p Fu(\014)s(J)3079 4270 y Fo(b)3114 4291 y Fw(.)66 b(Hence,)42 b(w)m(e)165 4404 y(ha)m(v)m(e)31 b Ft(m)p Fw(\()p Ft(\014)5 b(;)15 b(\031)s Fw(\))27 b Ft(>)e Fw(0)30 b(if)e(and)h(only)g(if)f Ft(\022)1455 4418 y Fu(q)1493 4404 y Fw(\()p Ft(\014)5 b Fx(J)p Fw(\))26 b Ft(>)f Fw(0)30 b(with)e(p)s(ositiv)m(e)h Ft(P)13 b Fw(-probabilit)m(y)-8 b(.)38 b(But)30 b(whether)f(or)165 4517 y(not)c Ft(\022)364 4531 y Fu(q)402 4517 y Fw(\()p Ft(\014)5 b Fx(J)p Fw(\))26 b Ft(>)f Fw(0)g(do)s(es)g(not)g(dep)s(end)e(on)h(the)h(v)-5 b(alue)25 b(of)g Ft(J)2089 4532 y Fu(b)2148 4517 y Fw(for)f(a)i(single) d(b)s(ond)g Ft(b)p Fw(.)39 b(So,)26 b(Kolmogoro)m(v's)165 4630 y(zero{one)37 b(la)m(w)d(implies)e(that)j Ft(m)p Fw(\()p Ft(\014)5 b(;)15 b(\031)s Fw(\))33 b Ft(>)f Fw(0)j(if)f(and)g (only)g(if)f Ft(\022)2346 4644 y Fu(q)2384 4630 y Fw(\()p Ft(\014)5 b Fx(J)p Fw(\))33 b Ft(>)f Fw(0)j Ft(P)13 b Fw(-almost)35 b(surely)-8 b(,)35 b(and)165 4742 y(b)m(y)40 b(an)f(inhomogeneous)f(v)m(ersion)h(of)h(Theorem)f(6.10)i(the)e(latter) h(means)f(that)h(m)m(ultiple)d(Gibbs)165 4855 y(measures)i(for)g Ft(\014)5 b Fx(J)39 b Fw(exist)g(with)f Ft(P)13 b Fw(-probabilit)m(y)36 b(1.)68 b(Moreo)m(v)m(er,)43 b(an)c(inhomogeneous)f(v)m(ersion)h(of)165 4968 y(relation)30 b(\(31\))i(sho)m(ws)e(that)h Ft(\022)1192 4982 y Fu(q)1229 4968 y Fw(\()p Ft(\014)5 b Fx(J)p Fw(\))32 b(is)d(an)h(increasing)f(function)g(of)h Ft(\014)5 b Fx(J)p Fw(.)41 b(It)31 b(follo)m(ws)e(that)i Ft(m)p Fw(\()p Ft(\014)5 b(;)15 b(\031)s Fw(\))165 5081 y(is)28 b(increasing)f(in)h Ft(\014)33 b Fw(and)28 b(also)h(in)e Ft(\031)32 b Fw(\(relativ)m(e)d (to)g Fv(\026)1948 5095 y Fq(D)2009 5081 y Fw(\).)40 b(In)28 b(particular,)g(for)g(eac)m(h)i Ft(\031)i Fw(there)c(exists)h (a)165 5194 y(critical)d(in)m(v)m(erse)h(temp)s(erature)g Ft(\014)1329 5208 y Fu(c)1364 5194 y Fw(\()p Ft(\031)s Fw(\),)h(p)s(ossibly)c(=)h(+)p Fv(1)p Fw(,)j(suc)m(h)e(that)i Ft(m)p Fw(\()p Ft(\014)5 b(;)15 b(\031)s Fw(\))26 b Ft(>)f Fw(0)i(for)g Ft(\014)j(>)25 b(\014)3453 5208 y Fu(c)3488 5194 y Fw(\()p Ft(\031)s Fw(\))165 5307 y(and)30 b Ft(m)p Fw(\()p Ft(\014)5 b(;)15 b(\031)s Fw(\))27 b(=)e(0)30 b(for)g Ft(\014)h(<)25 b(\014)1209 5321 y Fu(c)1244 5307 y Fw(\()p Ft(\031)s Fw(\).)306 5420 y(It)i(remains)d(to)j(in)m(v)m (estigate)g(the)f(quenc)m(hed)g(magnetization)g Ft(m)p Fw(\()p Ft(\014)5 b(;)15 b(\031)s Fw(\).)41 b(The)25 b(follo)m(wing)g(lemma)165 5533 y(sho)m(ws)31 b(ho)m(w)g Ft(m)p Fw(\()p Ft(\014)5 b(;)15 b(\031)s Fw(\))32 b(can)g(b)s(e)e (estimated)h(in)f(terms)h(of)g(Bernoulli)e(b)s(ond)g(p)s(ercolation;)i (recall)f(the)165 5646 y(end)g(of)h(Section)f(5.1.)1844 5895 y(94)p eop %%Page: 95 95 95 94 bop 165 304 a Fx(Lemma)33 b(9.1)46 b Fp(L)-5 b(et)30 b Ft(\022)s Fw(\()p Ft(p)p Fw(\))25 b(=)g Ft(\036)1206 318 y Fu(p)1246 304 y Fw(\(0)h Fv($)f(1)p Fw(\))30 b Fp(b)-5 b(e)30 b(the)g(Bernoul)5 b(li)30 b(b)-5 b(ond)31 b(p)-5 b(er)g(c)g(olation)33 b(pr)-5 b(ob)g(ability)33 b(on)d Fx(Z)3574 271 y Fu(d)165 417 y Fp(with)k(p)-5 b(ar)g(ameter)35 b Ft(p)p Fp(.)42 b(Then)1213 600 y Ft(\022)s Fw(\()p Ft(p)p 1294 631 46 4 v -1 w Fw(\()p Ft(\014)5 b(;)15 b(\031)s Fw(\)\))27 b Fv(\024)e Ft(m)p Fw(\()p Ft(\014)5 b(;)15 b(\031)s Fw(\))27 b Fv(\024)e Ft(\022)s Fw(\()8 b(\026)-53 b Ft(p)o Fw(\()p Ft(\014)5 b(;)15 b(\031)s Fw(\)\))g Ft(:)165 782 y Fx(Pro)s(of:)51 b Fw(W)-8 b(e)36 b(will)c(use)i(an)h(inhomogeneous)f(limiting)e(v)m(ersion)i(of)h (the)g(domination)e(b)s(ounds)g(\(b\))165 895 y(and)d(\(c\))h(of)f (Corollary)f(6.7.)41 b(Although)29 b(they)i(w)m(ere)f(stated)h(only)e (in)g(the)h(case)h(of)f(a)g(homogeneous)165 1008 y(in)m(teraction,)38 b(they)e(do)g(extend)h(also)f(to)h(the)f(inhomogeneous)g(case.)59 b(De\014ne)36 b(t)m(w)m(o)h(families)e Fx(p)f Fw(=)165 1121 y(\()p Ft(p)246 1136 y Fu(b)281 1121 y Fw(\))316 1136 y Fu(b)p Fq(2B)485 1121 y Fw(and)39 b Fx(p)729 1088 y Fq(0)794 1121 y Fw(=)h(\()p Ft(p)986 1088 y Fq(0)986 1147 y Fu(b)1021 1121 y Fw(\))1056 1136 y Fu(b)p Fq(2B)1225 1121 y Fw(in)f(terms)g(of)h(a)g(realization)f Fx(J)h Fw(b)m(y)g Ft(p)2531 1136 y Fu(b)2606 1121 y Fw(=)g(1)27 b Fv(\000)g Ft(e)2929 1088 y Fq(\000)p Fr(2)p Fu(\014)s(J)3101 1100 y Fo(b)3136 1121 y Fw(,)43 b Ft(p)3250 1088 y Fq(0)3250 1147 y Fu(b)3325 1121 y Fw(=)d(\(1)27 b Fv(\000)165 1234 y Ft(e)207 1201 y Fq(\000)p Fr(2)p Fu(\014)s(J)379 1213 y Fo(b)415 1234 y Fw(\))p Ft(=)p Fw(\(1)h(+)f(\()p Ft(q)j Fv(\000)c Fw(1\))p Ft(e)1026 1201 y Fq(\000)p Fr(2)p Fu(\014)s(J)1198 1213 y Fo(b)1235 1234 y Fw(\))42 b(=)f Ft(p)1470 1249 y Fu(b)1504 1234 y Ft(=)p Fw(\()p Ft(p)1630 1249 y Fu(b)1692 1234 y Fw(+)26 b Ft(q)s Fw(\(1)i Fv(\000)e Ft(p)2084 1249 y Fu(b)2118 1234 y Fw(\)\).)71 b(Let)41 b Ft(\036)2511 1201 y Fr(1)2511 1256 y Fn(p)p Fu(;q)2654 1234 y Fw(b)s(e)f(the)g(asso)s(ciated)h(wired)165 1356 y(random-cluster)35 b(measure,)h(and)f Ft(\036)1419 1370 y Fn(p)1469 1356 y Fw(,)h Ft(\036)1584 1371 y Fn(p)1629 1352 y Fm(0)1691 1356 y Fw(the)g(corresp)s(onding)d(pro)s(duct)h (measures)h(on)h Fv(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)3535 1323 y Fq(B)3589 1356 y Fw(.)165 1469 y(An)30 b(inhomogeneous)g(v)m (ersion)g(of)h(Corollary)e(6.7)i(then)f(sho)m(ws)g(that)1027 1652 y Ft(\036)1081 1667 y Fn(p)1126 1648 y Fm(0)1153 1652 y Fw(\(0)c Fv($)f(1)p Fw(\))h Fv(\024)f Ft(\036)1677 1614 y Fr(1)1677 1674 y Fn(p)p Fu(;q)1780 1652 y Fw(\(0)h Fv($)f(1)p Fw(\))h Fv(\024)f Ft(\036)2304 1666 y Fn(p)2353 1652 y Fw(\(0)h Fv($)f(1)p Fw(\))h Ft(:)165 1834 y Fw(W)-8 b(e)32 b(no)m(w)f(tak)m(e)h(the)e(exp)s(ectation)h(with)f(resp)s(ect)g (to)h Ft(P)13 b Fw(.)41 b(In)30 b(view)g(of)g(the)h(preceding)f (remarks,)g(the)165 1947 y(middle)g(term)i(has)g Ft(P)13 b Fw(-exp)s(ectation)33 b Ft(m)p Fw(\()p Ft(\014)5 b(;)15 b(\031)s Fw(\),)34 b(while)c(the)j Ft(P)13 b Fw(-in)m(tegration)32 b(of)g(the)h(Bernoulli)c(mea-)165 2060 y(sures)f Ft(\036)446 2075 y Fn(p)491 2057 y Fm(0)545 2060 y Fw(and)g Ft(\036)774 2074 y Fn(p)851 2060 y Fw(again)h(leads)e(to)i(Bernoulli)d(measures,)i (namely)g(the)g(homogeneous)h(Bernoulli)165 2173 y(measures)i Ft(\036)610 2192 y Fu(p)p 610 2218 36 4 v -1 w Fr(\()p Fu(\014)s(;\031)r Fr(\))840 2173 y Fw(and)f Ft(\036)1077 2192 y Fr(\026)-41 b Fu(p)p Fr(\()p Fu(\014)s(;\031)r Fr(\))1271 2173 y Fw(.)41 b(The)30 b(lemma)g(follo)m(ws)f(immediately) -8 b(.)925 b Fk(2)165 2329 y Fw(Com)m(bining)35 b(the)i(lemma)f(with)f (the)i(discussion)d(b)s(efore)j(w)m(e)g(arriv)m(e)f(at)i(the)e(follo)m (wing)g(result)f(on)165 2442 y(phase)30 b(transition)f(in)g(the)i (random)e(P)m(otts)j(mo)s(del.)165 2611 y Fx(Theorem)i(9.2)46 b Fp(Consider)38 b(the)e(r)-5 b(andom)39 b(Potts)e(mo)-5 b(del)38 b(on)f Fx(Z)2372 2578 y Fu(d)2448 2611 y Fp(at)g(inverse)f (temp)-5 b(er)g(atur)g(e)39 b Ft(\014)e(>)32 b Fw(0)165 2724 y Fp(with)45 b(c)-5 b(oupling)45 b(distribution)g Ft(\031)s Fp(.)75 b(Set)44 b Ft(p)p Fw(\()p Ft(\031)s Fw(\))j(=)e Ft(\031)s Fw(\(]0)p Ft(;)15 b Fv(1)p Fw([\))48 b(=)d Ft(P)13 b Fw(\()p Ft(J)2569 2739 y Fu(b)2650 2724 y Ft(>)45 b Fw(0\))p Fp(,)j(and)d(let)f Ft(p)3295 2738 y Fu(c)3373 2724 y Fp(b)-5 b(e)44 b(the)165 2837 y(Bernoul)5 b(li)33 b(b)-5 b(ond)34 b(p)-5 b(er)g(c)g(olation)36 b(thr)-5 b(eshold)35 b(of)e Fx(Z)1795 2804 y Fu(d)1836 2837 y Fp(;)f(so)h Ft(p)2058 2851 y Fu(c)2118 2837 y Fw(=)25 b(1)p Ft(=)p Fw(2)34 b Fp(when)f Ft(d)26 b Fw(=)f(2)p Fp(.)251 3007 y Fw(\(i\))46 b Fp(If)39 b Fw(\026)-53 b Ft(p)p Fw(\()p Ft(\014)5 b(;)15 b(\031)s Fw(\))27 b Ft(<)e(p)923 3021 y Fu(c)989 3007 y Fp(then)32 b(with)h Ft(P)13 b Fp(-pr)-5 b(ob)g(ability)34 b Fw(1)e Fp(ther)-5 b(e)33 b(exists)f(only)h(one)f(Gibbs)g(me)-5 b(asur)g(e)33 b(with)393 3120 y(inter)-5 b(action)34 b Ft(\014)5 b Fx(J)p Fp(.)42 b(In)32 b(p)-5 b(articular,)35 b(this)e(holds)i(when)e Ft(p)p Fw(\()p Ft(\031)s Fw(\))26 b Ft(<)f(p)2574 3134 y Fu(c)2641 3120 y Fp(or)33 b Ft(\014)38 b Fp(is)32 b(smal)5 b(l)34 b(enough.)226 3300 y Fw(\(ii\))45 b Fp(If)37 b Ft(p)p 493 3331 46 4 v Fw(\()p Ft(\014)5 b(;)15 b(\031)s Fw(\))35 b Ft(>)f(p)946 3314 y Fu(c)1018 3300 y Fp(then)k Ft(m)p Fw(\()p Ft(\014)5 b(;)15 b(\031)s Fw(\))36 b Ft(>)e Fw(0)p Fp(,)39 b(and)f(with)h Ft(P)13 b Fp(-pr)-5 b(ob)g(ability)39 b Fw(1)f Fp(ther)-5 b(e)38 b(exist)g Ft(q)i Fp(distinct)393 3413 y(phases)d(for)f(the)g(inter)-5 b(action)38 b Ft(\014)5 b Fx(J)p Fp(.)51 b(In)36 b(p)-5 b(articular,)38 b(this)f(holds)g(when)f Ft(p)p Fw(\()p Ft(\031)s Fw(\))c Ft(>)e(p)3207 3427 y Fu(c)3278 3413 y Fp(and)36 b Ft(\014)41 b Fp(is)393 3526 y(lar)-5 b(ge)33 b(enough.)165 3696 y Fw(Another)f(w)m(a)m(y)h(of)f (stating)g(this)f(result)f(is)h(the)h(follo)m(wing.)44 b(Supp)s(ose)30 b Ft(\031)h Fw(=)c(\(1)22 b Fv(\000)f Ft(p)p Fw(\))p Ft(\016)3081 3710 y Fr(0)3142 3696 y Fw(+)g Ft(p)15 b(\031)3347 3710 y Fr(+)3437 3696 y Fw(with)165 3809 y Ft(\031)217 3823 y Fr(+)321 3809 y Fw(=)44 b Ft(\031)s Fw(\()p Fv(\001)15 b(j)p Fw(]0)p Ft(;)g Fv(1)p Fw([\),)48 b(and)42 b(let)g Ft(L)1319 3823 y Fr(+)1378 3809 y Fw(\()p Ft(\014)5 b Fw(\))45 b(=)1665 3740 y Fs(R)1720 3766 y Fq(1)1704 3836 y Fr(0)1810 3809 y Ft(e)1852 3776 y Fq(\000)p Fu(\014)s(t)1980 3809 y Ft(\031)2032 3823 y Fr(+)2091 3809 y Fw(\()p Ft(dt)p Fw(\))e(b)s(e)e(the)h(Laplace)g(transform)g(of)g Ft(\031)3530 3823 y Fr(+)3589 3809 y Fw(.)165 3921 y(\(Note)36 b(that)f(then)42 b(\026)-53 b Ft(p)p Fw(\()p Ft(\014)5 b(;)15 b(\031)s Fw(\))33 b(=)f Ft(p)p Fw(\(1)23 b Fv(\000)f Ft(L)1543 3935 y Fr(+)1602 3921 y Fw(\(2)p Ft(\014)5 b Fw(\)\))36 b(and)e Ft(p)p 2025 3952 V Fw(\()p Ft(\014)5 b(;)15 b(\031)s Fw(\))33 b Fv(\025)e Ft(p)p Fw(\(1)24 b Fv(\000)e Ft(q)c(L)2791 3935 y Fr(+)2850 3921 y Fw(\(2)p Ft(\014)5 b Fw(\)\).\))55 b(Then)33 b(there)165 4034 y(is)j(no)g(phase)g(transition)e(for)i Ft(p)f(<)g(p)1452 4048 y Fu(c)1486 4034 y Fw(,)j(whereas)e(for)g Ft(p)f(>)g(p)2277 4048 y Fu(c)2348 4034 y Fw(the)h(critical)f(in)m(v)m(erse)h(temp)s (erature)165 4147 y Ft(\014)216 4161 y Fu(c)251 4147 y Fw(\()p Ft(p;)15 b(\031)424 4161 y Fr(+)484 4147 y Fw(\))25 b Fv(\021)g Ft(\014)691 4161 y Fu(c)727 4147 y Fw(\()p Ft(\031)s Fw(\))31 b(is)e(\014nite)g(\(and)h(decreasing)h(in) e Ft(p)p Fw(\))h(and)g(sati\014es)g(the)g(b)s(ounds)1193 4301 y Ft(p)20 b Fv(\000)g Ft(p)1396 4315 y Fu(c)p 1193 4342 238 4 v 1267 4425 a Ft(pq)1466 4363 y Fv(\024)k Ft(L)1623 4377 y Fr(+)1682 4363 y Fw(\(2)15 b Ft(\014)1828 4377 y Fu(c)1864 4363 y Fw(\()p Ft(p;)g(\031)2037 4377 y Fr(+)2097 4363 y Fw(\)\))26 b Fv(\024)2299 4301 y Ft(p)20 b Fv(\000)g Ft(p)2502 4315 y Fu(c)p 2299 4342 V 2395 4425 a Ft(p)2571 4363 y(:)165 4589 y Fw(If)i Ft(\022)s Fw(\()p Ft(p)375 4603 y Fu(c)409 4589 y Fw(\))k(=)f(0)d(\(whic)m(h)g (is)f(kno)m(wn)h(to)h(hold)e(for)h Ft(d)j Fw(=)g(2,)g(and)c(is)h(exp)s (ected)g(to)h(hold)e(for)h(all)f(dimensions\))165 4702 y(then)42 b(uniqueness)d(holds)h(when)h Ft(p)j Fw(=)f Ft(p)1601 4716 y Fu(c)1677 4702 y Fw(or)f Ft(\014)49 b Fw(=)44 b Ft(\014)2066 4716 y Fu(c)2101 4702 y Fw(\()p Ft(p;)15 b(\031)2274 4716 y Fr(+)2333 4702 y Fw(\).)75 b(In)41 b(ph)m(ysical)f(terminology)-8 b(,)45 b(the)165 4815 y(preceding)30 b(b)s(ounds)e(on)i Ft(\014)1068 4829 y Fu(c)1103 4815 y Fw(\()p Ft(p;)15 b(\031)1276 4829 y Fr(+)1336 4815 y Fw(\))30 b(imply)e(that)j(the)g(so-called)f(crosso)m (v)m(er)i(exp)s(onen)m(t)e(is)g(1.)165 4984 y Fx(Example)k(9.3)46 b Fp(The)35 b(c)-5 b(ase)36 b(of)f(dilution.)49 b Fw(If)32 b Ft(J)1792 4999 y Fu(b)1860 4984 y Fw(is)g(1)h(or)f(0)i(with)d (probabilit)m(y)f Ft(p)j Fw(resp.)f(1)22 b Fv(\000)g Ft(p)32 b Fw(then)165 5097 y Ft(L)227 5111 y Fr(+)286 5097 y Fw(\()p Ft(\014)5 b Fw(\))26 b(=)f Ft(e)576 5064 y Fq(\000)p Fu(\014)679 5097 y Fw(.)38 b(Hence,)26 b(for)d Ft(p)i(>)g(p)1377 5111 y Fu(c)1435 5097 y Fw(the)f(critical)e(in)m(v)m (erse)i(temp)s(erature)f(satis\014es)g(the)g(logarithmic)165 5210 y(b)s(ounds)1118 5333 y Fv(\000)15 b Fw(ln)1304 5272 y Ft(p)20 b Fv(\000)g Ft(p)1507 5286 y Fu(c)p 1304 5312 V 1400 5395 a Ft(p)1577 5333 y Fv(\024)25 b Fw(2)15 b Ft(\014)1784 5347 y Fu(c)1820 5333 y Fw(\()p Ft(p;)g(\016)1981 5347 y Fr(1)2021 5333 y Fw(\))25 b Fv(\024)g(\000)15 b Fw(ln)2364 5272 y Ft(p)20 b Fv(\000)g Ft(p)2567 5286 y Fu(c)p 2364 5312 V 2438 5395 a Ft(pq)2637 5333 y(:)165 5533 y Fw(F)-8 b(or)44 b Ft(q)k Fw(=)e(2,)g(the)d(diluted)d(Ising)h(mo) s(del,)k(assertion)d(\(ii\))g(of)h(Theorem)f(9.2)i(giv)m(es)f(the)g (sligh)m(tly)165 5646 y(sharp)s(er)29 b(upp)s(er)f(b)s(ound)h Ft(\014)1084 5660 y Fu(c)1119 5646 y Fw(\()p Ft(p;)15 b(\016)1280 5660 y Fr(1)1320 5646 y Fw(\))26 b Fv(\024)f Fw(tanh)1659 5609 y Fq(\000)p Fr(1)1753 5646 y Fw(\()p Ft(p)1834 5660 y Fu(c)1869 5646 y Ft(=p)p Fw(\).)1844 5895 y(95)p eop %%Page: 96 96 96 95 bop 165 304 a Fx(Example)34 b(9.4)46 b Fp(The)32 b(c)-5 b(ase)33 b(of)f(p)-5 b(ower)34 b(law)f(singularities.)42 b Fw(Supp)s(ose)28 b Ft(\031)2642 318 y Fr(+)2701 304 y Fw(\()p Ft(dt)p Fw(\))e(=)f(\000\()p Ft(a)p Fw(\))3148 271 y Fq(\000)p Fr(1)3243 304 y Ft(t)3276 271 y Fu(a)p Fq(\000)p Fr(1)3407 304 y Ft(e)3449 271 y Fq(\000)p Fu(t)3534 304 y Ft(dt)165 417 y Fw(is)j(the)h(Gamma)h(distribution)25 b(with)j(parameter)h Ft(a)d(>)f Fw(0.)40 b(Then)28 b Ft(L)2464 431 y Fr(+)2523 417 y Fw(\()p Ft(\014)5 b Fw(\))26 b(=)f(\(1)18 b(+)f Ft(\014)5 b Fw(\))3048 384 y Fq(\000)p Fu(a)3145 417 y Fw(,)30 b(so)f(that)g(for)165 530 y Ft(p)c(>)g(p)378 544 y Fu(c)443 530 y Fw(the)31 b(critical)e(in)m(v)m(erse)h(temp)s (erature)g(satis\014es)g(a)h(p)s(o)m(w)m(er)f(la)m(w)h(with)e(exp)s (onen)m(t)h Fv(\000)p Fw(1)p Ft(=a)p Fw(:)906 678 y Fs(\022)1055 736 y Ft(p)1101 750 y Fu(c)p 977 776 238 4 v 977 860 a Ft(p)20 b Fv(\000)f Ft(p)1179 874 y Fu(c)1224 678 y Fs(\023)1285 701 y Fr(1)p Fu(=a)1417 798 y Fv(\000)h Fw(1)56 b Fv(\024)g Fw(2)15 b Ft(\014)1847 812 y Fu(c)1882 798 y Fw(\()p Ft(p;)g(\031)2055 812 y Fr(+)2115 798 y Fw(\))56 b Fv(\024)2332 678 y Fs(\022)2459 736 y Ft(p)2505 750 y Fu(c)2540 736 y Ft(q)p 2403 776 V 2403 860 a(p)20 b Fv(\000)g Ft(p)2606 874 y Fu(c)2650 678 y Fs(\023)2711 701 y Fr(1)p Fu(=a)2849 798 y Ft(:)165 1053 y Fw(Examples)30 b(with)f(other)h(kinds)f(of)h(singularities)e(can)i(easily)g(b)s(e)g (pro)s(duced)e([94)q(].)306 1166 y(Theorem)33 b(9.2)h(is)e(due)g(to)i ([9)q(].)48 b(Earlier,)33 b(a)g(generalized)f(P)m(eierls)g(argumen)m(t) i(w)m(as)f(used)f(in)g([93)q(])165 1279 y(to)e(sho)m(w)f(that)g(for)g (the)g(diluted)e(Ising)h(mo)s(del)g(\()p Ft(q)g Fw(=)d(2\))30 b(in)d Ft(d)f Fw(=)f(2)k(dimensions)e(a)i(phase)g(transition)165 1392 y(o)s(ccurs)f(almost)f(surely)f(when)h Ft(p)e(>)g(p)1445 1406 y Fu(c)1504 1392 y Fw(=)g(1)p Ft(=)p Fw(2)k(and)e Ft(\014)33 b Fw(is)26 b(large)i(enough.)39 b(In)27 b(fact,)i(this)e (pap)s(er)f(dealt)165 1505 y(mainly)37 b(with)f(the)j(case)g(of)f(site) g(dilution,)f(in)g(whic)m(h)f(sites)i(rather)g(than)g(b)s(onds)e(are)j (randomly)165 1618 y(remo)m(v)m(ed)44 b(from)e(the)h(lattice,)j(and)c (whic)m(h)f(in)g(the)i(presen)m(t)f(framew)m(ork)h(can)g(b)s(e)f (describ)s(ed)e(b)m(y)165 1731 y(setting)34 b Ft(J)517 1749 y Fq(h)p Fu(xy)r Fq(i)685 1731 y Fw(=)c Ft(\030)t Fw(\()p Ft(x)p Fw(\))p Ft(\030)t Fw(\()p Ft(y)s Fw(\))35 b(for)f(a)g(family)e(\()p Ft(\030)t Fw(\()p Ft(x)p Fw(\)\))1888 1753 y Fu(x)p Fq(2)p Fn(Z)2025 1734 y Fo(d)2100 1731 y Fw(of)i(Bernoulli)d(v)-5 b(ariables;)35 b(the)f Ft(J)3221 1746 y Fu(b)3289 1731 y Fw(are)g(th)m(us)165 1843 y(1-dep)s(enden)m(t.) 57 b(This)34 b(w)m(as)i(con)m(tin)m(ued)f(in)f([94)r(,)h(95)q(])h(for)g (a)g(class)f(of)h(random)f(in)m(teraction)g(mo)s(dels)165 1956 y(including)25 b(the)j(random-b)s(ond)f(Ising)g(mo)s(del)g(as)h (considered)f(here,)i(obtaining)e(impro)m(v)m(ed)g(b)s(ounds)165 2069 y(on)33 b Ft(\014)345 2083 y Fu(c)380 2069 y Fw(\()p Ft(p;)15 b(\031)553 2083 y Fr(+)612 2069 y Fw(\))33 b(for)f Ft(d)c Fw(=)h(2)j(as)h Ft(p)28 b Fv(#)h Fw(1)p Ft(=)p Fw(2.)48 b(Extensions,)32 b(in)f(particular)g(to)i Ft(d)28 b Fv(\025)h Fw(3,)k(w)m(ere)g(obtained)e(in)165 2182 y([53)r(].)44 b(The)31 b(diluted)f(Ising)g(mo)s(del)h(with)f(a)i (non-random)f(external)g(\014eld)g Ft(h)d Fv(6)p Fw(=)f(0)32 b(do)s(es)f(not)h(exhibit)165 2295 y(a)37 b(phase)e(transition;)j(this) d(w)m(as)h(sho)m(wn)f(in)g([93)q(])h(for)g Fv(L)e Fw(=)g Fx(Z)2298 2262 y Fu(d)2374 2295 y Fw(and)i(recen)m(tly)g(extended)g(to) h(quite)165 2408 y(general)31 b(graphs)e(in)g([123)r(].)306 2521 y(F)-8 b(or)34 b(the)e(diluted)e(Ising)h(mo)s(del)h(there)g(is)g (also)g(a)h(dynamical)e(phase)h(transition)e(at)k(the)e(p)s(oin)m(t)165 2634 y Ft(p)37 b Fw(=)g Ft(p)402 2648 y Fu(c)436 2634 y Fw(.)62 b(F)-8 b(or)38 b Ft(p)f(>)g(p)930 2648 y Fu(c)1002 2634 y Fw(and)g Ft(\014)1237 2648 y Fu(c)1272 2634 y Fw(\(1)p Ft(;)15 b(\016)1432 2648 y Fr(1)1473 2634 y Fw(\))37 b Ft(<)g(\014)42 b(<)37 b(\014)1905 2648 y Fu(c)1940 2634 y Fw(\()p Ft(p;)15 b(\016)2101 2648 y Fr(1)2141 2634 y Fw(\))38 b(the)g(relaxation)f(to)h(equilibrium)33 b(is)j(no)165 2747 y(longer)i(exp)s(onen)m(tially)f(fast)h([15)q(].)65 b(This)37 b(illustrates)f(that)j(uniqueness)d(of)i(the)h(Gibbs)d (measure)165 2860 y(do)s(es)f(not)f(in)g(itself)f(imply)f(the)j (absence)g(of)g(a)g(critical)f(phenomenon.)52 b(Beside)34 b(suc)m(h)h(dynamical)165 2973 y(phenomena,)c(there)g(are)h(also)e (some)i(static)f(e\013ects)h(of)g(the)f(disorder)e(in)g(the)i (uniqueness)e(regime,)165 3085 y(alb)s(eit)g(these)i(are)g(p)s(erhaps)e (less)g(remark)-5 b(able.)40 b(These)31 b(are)f(the)h(sub)5 b(ject)30 b(of)h(the)f(next)h(subsection.)165 3329 y FA(9.2)113 b(Mixing)38 b(prop)s(erties)g(in)g(the)f(Gri\016ths)h (regime)165 3500 y Fw(As)27 b(w)m(e)g(ha)m(v)m(e)h(seen)f(ab)s(o)m(v)m (e,)i(the)d(diluted)f(Ising)g(ferromagnet)j(sho)m(ws)e(sp)s(on)m (taneous)h(magnetization)165 3613 y(when)41 b Ft(p)i(>)h(p)664 3627 y Fu(c)740 3613 y Fw(and)d Ft(\014)49 b(>)44 b(\014)1194 3627 y Fu(c)1229 3613 y Fw(\()p Ft(p)p Fw(\))g Fv(\021)g Ft(\014)1555 3627 y Fu(c)1590 3613 y Fw(\()p Ft(p;)15 b(\016)1751 3627 y Fr(1)1791 3613 y Fw(\),)45 b(and)c(m)m(ultiple)e (Gibbs)h(measures)h(for)h Ft(\014)5 b Fx(J)41 b Fw(exist)165 3726 y(almost)d(surely)-8 b(.)63 b(In)37 b(the)h(uniqueness)e(region)h (when)g(still)f Ft(p)i(>)f(p)2482 3740 y Fu(c)2555 3726 y Fw(but)g Ft(\014)43 b(<)38 b(\014)2983 3740 y Fu(c)3018 3726 y Fw(\()p Ft(p)p Fw(\))g(w)m(e)h(need)e(to)165 3839 y(distinguish)18 b(b)s(et)m(w)m(een)j(t)m(w)m(o)h(di\013eren)m(t)f (regimes.)37 b(A)m(t)22 b(high)e(temp)s(eratures)g(when)g(actually)h Ft(\014)30 b(<)25 b(\014)3483 3853 y Fu(c)3543 3839 y Fv(\021)165 3952 y Ft(\014)216 3966 y Fu(c)251 3952 y Fw(\(1\),)35 b(the)e(critical)e(in)m(v)m(erse)h(temp)s(erature)h(of)f (the)h(undiluted)c(system,)34 b(w)m(e)f(are)f(in)g(the)g(so-called)165 4065 y(paramagnetic)j(case.)52 b(This)32 b(is)h(comparable)g(to)i(the)f (usual)e(uniqueness)g(regime)i(for)f(translation)165 4178 y(in)m(v)-5 b(arian)m(t)36 b(Ising)f(mo)s(dels.)57 b(A)m(t)38 b(in)m(termediate)e(temp)s(eratures,)i(namely)d(when)h Ft(\014)3013 4192 y Fu(c)3083 4178 y Ft(<)f(\014)40 b(<)35 b(\014)3437 4192 y Fu(c)3472 4178 y Fw(\()p Ft(p)p Fw(\),)165 4291 y(w)m(e)d(encoun)m(ter)g(di\013eren)m(t)f(b)s(eha)m(vior)f (arising)g(from)h(the)g(fact)h(that)g(the)g(system)f(starts)h(to)g (feel)f(the)165 4404 y(disorder.)38 b(This)25 b(regime)i(is)f(called)g (the)i Fp(Gri\016ths)j(r)-5 b(e)g(gime)p Fw(,)28 b(since)f(it)f(w)m(as) i(he)e([105)r(])h(who)g(disco)m(v)m(ered)165 4517 y(in)f(this)h (parameter)h(region)f(the)g(phenomenon)g(no)m(w)g(called)g(Gri\016ths') f(singularities.)37 b(He)28 b(studied)165 4630 y(site-diluted)21 b(ferromagnets)j(,)h(but)d(the)i(argumen)m(ts)f(remain)f(v)-5 b(alid)22 b(also)h(in)f(the)h(b)s(ond-diluted)d(case.)165 4742 y(The)25 b(basic)g(fact)h(is)e(the)i(follo)m(wing:)36 b(adding)24 b(a)i(complex)f(magnetic)g(\014eld)f Ft(h)i Fw(to)g(the)f(Hamiltonian)f(of)165 4855 y(the)k(diluted)d(Ising)h(mo)s (del)g(w)m(e)i(\014nd)e(that)i(the)g(partition)e(function)g(in)g(a)h(b) s(o)m(x)h(with)e(plus)f(b)s(oundary)165 4968 y(conditions,)35 b(as)g(a)g(function)e(of)i Ft(h)p Fw(,)h(can)f(tak)m(e)h(v)-5 b(alues)34 b(arbitrarily)e(close)j(to)g(zero.)54 b(The)34 b(reason)h(is)165 5081 y(that)j(t)m(ypically)e(a)h(large)g(part)g(of)g (the)g(b)s(o)m(x)g(is)f(\014lled)f(b)m(y)i(a)g(h)m(uge)g(cluster)g(of)g (in)m(teracting)g(b)s(onds,)165 5194 y(giving)22 b(a)h(con)m(tribution) e(corresp)s(onding)g(to)i(an)g(Ising)e(partition)h(function)f(in)g(the) i(phase)g(transition)165 5307 y(region.)41 b(The)30 b(radius)e(of)j (analyticit)m(y)f(of)h(the)f(free)h(energy)g(around)e Ft(h)d Fw(=)f(0)31 b(is)e(th)m(us)h(zero.)42 b(In)30 b(other)165 5420 y(w)m(ords,)42 b(the)e(magnetization)h Ft(m)p Fw(\()p Ft(\014)5 b(;)15 b(p;)g(h)p Fw(\))41 b(cannot)g(b)s(e)e (con)m(tin)m(ued)g(analytically)g(from)g Ft(h)i(>)g Fw(0)f(to)165 5533 y Ft(h)28 b(<)g Fw(0)k(through)f Ft(h)d Fw(=)f(0)33 b(when)d Ft(p)e(>)f(p)1478 5547 y Fu(c)1544 5533 y Fw(and)32 b Ft(\014)g(>)c(\014)1956 5547 y Fu(c)1991 5533 y Fw(.)45 b(So,)32 b(the)g(presence)g(of)g(macrosopic)g(clusters)165 5646 y(of)e(strongly)g(in)m(teracting)f(spins)f(\(on)i(whic)m(h)f(the)h (spins)d(sho)m(w)j(the)g(lo)m(w)f(temp)s(erature)h(b)s(eha)m(vior)f(of) 1844 5895 y(96)p eop %%Page: 97 97 97 96 bop 165 304 a Fw(the)28 b(corresp)s(onding)d(translation)i(in)m (v)-5 b(arian)m(t)26 b(system\))i(giv)m(es)g(rise)f(to)h(singular)d(b)s (eha)m(vior.)39 b(Related)165 417 y(phenomena)31 b(sho)m(w)g(up)f(in)f (a)j(large)f(v)-5 b(ariet)m(y)31 b(of)h(other)f(random)f(mo)s(dels,)g (though)h(not)g(necessarily)165 530 y(in)26 b(the)i(form)f(of)g (non-analyticit)m(y)g(in)f(the)h(uniqueness)e(regime;)k(in)d(general)h (it)g(ma)m(y)g(b)s(e)g(di\016cult)e(to)165 643 y(pinp)s(oin)m(t)d (their)h(precise)g(nature.)39 b(Nev)m(ertheless,)26 b(w)m(e)e(will)e (sp)s(eak)h(of)i(Gri\016ths')d(phase)i(or)g(Gri\016ths')165 756 y(regime)41 b(whenev)m(er)f(suc)m(h)h(singularities)c(are)k(exp)s (ected)h(to)f(o)s(ccur,)i(ev)m(en)f(when)d(a)j(pro)s(of)d(is)h(still) 165 869 y(lac)m(king.)46 b(These)31 b(terms)h(then)g(simply)e(indicate) h(that)h(the)h(usual)d(high)h(temp)s(erature)h(tec)m(hniques)165 981 y(cannot)f(b)s(e)f(applied)e(as)j(suc)m(h.)306 1094 y(As)45 b(another)f(illustration)e(w)m(e)j(consider)e(a)i(random)e (Ising)h(mo)s(del)f(with)g(un)m(b)s(ounded,)i(sa)m(y)165 1207 y(Gaussian)21 b(coupling)e(v)-5 b(ariables)20 b Ft(J)1319 1222 y Fu(b)1353 1207 y Fw(.)38 b(Then)20 b Ft(\014)5 b(J)1750 1222 y Fu(b)1806 1207 y Fw(is)20 b(also)h(un)m(b)s (ounded,)f(ev)m(en)i(for)f(arbitrarily)d(small)i Ft(\014)5 b Fw(,)165 1320 y(and)25 b(with)g(high)f(probabilit)m(y)f(a)j(large)g (b)s(o)m(x)f(con)m(tains)h(a)g(p)s(ositiv)m(e)f(fraction)g(of)h (strongly)f(in)m(teracting)165 1433 y(spins.)39 b(In)30 b(particular,)f(there)h(is)g(no)g(paramagnetic)h(regime,)f(and)g(the)h (whole)e(uniqueness)f(region)165 1546 y(b)s(elongs)k(to)h(the)f (Gri\016ths)g(phase.)46 b(F)-8 b(or)33 b(this)f(reason,)h(it)f(is)f(a)i (non-trivial)e(problem)f(to)k(sho)m(w)e(the)165 1659 y(uniqueness)39 b(of)h(the)h(Gibbs)e(measure.)71 b(F)-8 b(or)42 b(example,)h(the)e(standard)e(Dobrushin)f(uniqueness)165 1772 y(condition)30 b(encoun)m(tered)i(in)e(\(48\))j(\(cf.)f([65)q(,)g (70)q(]\))f(is)g(useless)f(in)g(this)g(case;)k(similarly)-8 b(,)28 b(a)k(naiv)m(e)f(use)165 1885 y(of)f(standard)g(cluster)f (expansion)g(tec)m(hniques)g(fails.)39 b(In)30 b(fact,)h(these)f(metho) s(ds)f(are)i(b)s(ound)d(to)i(fail)165 1998 y(since)h(they)g(also)g (imply)d(analyticit)m(y)j(whic)m(h)e(is)h(probably)f(to)s(o)j(m)m(uc)m (h)f(to)h(hop)s(e)e(for)g(\(ev)m(en)i(though)165 2111 y(w)m(e)f(cannot)g(dispro)m(v)m(e)f(it\).)306 2223 y(In)j(the)h(follo)m (wing)e(w)m(e)i(will)d(not)j(deal)f(with)f(the)i(singular)d(b)s(eha)m (vior)i(in)f(the)i(Gri\016ths)e(phase.)165 2336 y(Instead,)43 b(w)m(e)d(address)f(the)h(problem)e(of)i(sho)m(wing)e(nice)i(b)s(eha)m (vior,)h(whic)m(h)e(w)m(e)h(sp)s(ecify)e(here)i(as)165 2449 y(go)s(o)s(d)i(mixing)d(prop)s(erties)h(of)i(the)f(system.)74 b(W)-8 b(e)43 b(shall)d(presen)m(t)h(t)m(w)m(o)i(tec)m(hniques:)62 b(the)42 b(use)f(of)165 2562 y(random-cluster)30 b(represen)m(tations,) g(and)g(the)h(use)f(of)g(disagreemen)m(t)h(p)s(ercolation.)165 2723 y Fp(Applic)-5 b(ation)35 b(of)d(r)-5 b(andom-cluster)35 b(r)-5 b(epr)g(esentations.)85 b Fw(Consider)29 b(a)h(random)g(Ising)f (mo)s(del.)40 b(Spins)165 2835 y(tak)m(e)32 b(v)-5 b(alues)30 b Ft(\033)s Fw(\()p Ft(x)p Fw(\))c(=)f Fv(\006)p Fw(1,)31 b(and)f(the)g(formal)g(Hamiltonian)f(is)1262 3032 y Ft(H)7 b Fw(\()p Ft(\033)s Fw(\))26 b(=)f Fv(\000)1774 2952 y Fs(X)1678 3140 y Fu(b)p Fr(=)p Fq(h)p Fu(xy)r Fq(i2B)2005 3032 y Ft(J)2055 3047 y Fu(b)2115 3032 y Ft(\033)s Fw(\()p Ft(x)p Fw(\))p Ft(\033)s Fw(\()p Ft(y)s Fw(\))i Ft(:)936 b Fw(\(67\))165 3330 y(W)-8 b(e)35 b(set)g Ft(\014)h Fw(=)30 b(1.)52 b(Let)34 b Ft(\026)1002 3286 y Fu(\021)1002 3358 y Fn(J)p Fu(;)p Fr(\003)1150 3330 y Fw(b)s(e)f(the)h(asso)s (ciated)g(Gibbs)f(distribution)d(in)i(\003)f Fv(2)g(E)42 b Fw(with)32 b(b)s(oundary)165 3443 y(condition)f Ft(\021)h Fv(2)c Fw(\012.)46 b(F)-8 b(or)33 b(man)m(y)g(applications)d(it)i(is)g (imp)s(ortan)m(t)f(to)i(ha)m(v)m(e)h(go)s(o)s(d)e(estimates)h(on)f(the) 165 3556 y(v)-5 b(ariational)23 b(distance)h Fv(k)7 b(\001)g(k)1088 3570 y Fr(\001)1176 3556 y Fw(on)23 b(\001)i Fv(\032)g Fw(\003)f(\(see)h(\(45\)\))h(of)e(these)g(measures)f(with)g(di\013eren) m(t)g(b)s(oundary)165 3669 y(conditions.)165 3874 y Fx(De\014nition)36 b(9.5)46 b Fp(The)27 b(r)-5 b(andom)31 b(spin)d(system)g(ab)-5 b(ove)29 b(is)e(c)-5 b(al)5 b(le)-5 b(d)29 b Fx(exp)s(onen)m(tially)g (w)m(eak-mixing)165 3987 y Fp(with)34 b(r)-5 b(ate)34 b Ft(m)25 b(>)g Fw(0)33 b Fp(if)f(for)h(some)g Ft(C)f(<)25 b Fv(1)32 b Fp(and)i(al)5 b(l)33 b Fw(\003)26 b Fv(2)e(E)41 b Fp(and)34 b Fw(\001)25 b Fv(\032)g Fw(\003)963 4219 y Ft(P)1034 4100 y Fs(\022)1133 4219 y Fw(max)1110 4280 y Fu(\021)r(;\021)1204 4261 y Fm(0)1228 4280 y Fq(2)p Fr(\012)1341 4219 y Fv(k)p Ft(\026)1441 4175 y Fu(\021)1441 4247 y Fn(J)p Fu(;)p Fr(\003)1576 4219 y Fv(\000)20 b Ft(\026)1722 4175 y Fu(\021)1759 4152 y Fm(0)1722 4247 y Fn(J)p Fu(;)p Fr(\003)1836 4219 y Fv(k)1881 4233 y Fr(\001)1945 4100 y Fs(\023)2031 4219 y Fv(\024)25 b Ft(C)7 b Fv(j)p Fw(\001)p Fv(j)15 b Ft(e)2382 4182 y Fq(\000)p Fu(m)d(d)p Fr(\(\001)p Fu(;)p Fr(\003)2702 4158 y Fo(c)2734 4182 y Fr(\))2791 4219 y Ft(;)637 b Fw(\(68\))165 4462 y Fp(wher)-5 b(e)34 b Ft(d)p Fw(\(\001)p Ft(;)15 b Fw(\003)683 4429 y Fu(c)719 4462 y Fw(\))33 b Fp(is)f(the)h(Euclide)-5 b(an)33 b(distanc)-5 b(e)34 b(of)f Fw(\001)f Fp(and)i Fw(\003)2258 4429 y Fu(c)2293 4462 y Fp(.)165 4667 y Fw(V)-8 b(arious)33 b(other)g(mixing)d (conditions)i(can)h(also)f(b)s(e)g(considered.)47 b(A)32 b(stronger)h(condition)f(requires)165 4780 y(that)g(the)f(v)-5 b(ariational)30 b(distance)g(in)g(\(68\))i(is)e(exp)s(onen)m(tially)g (small)f(in)h(the)h(distance)f(b)s(et)m(w)m(een)i(the)165 4893 y(set)i(\001)f(and)g(the)g(region)g(where)g(the)g(b)s(oundary)e (conditions)h Ft(\021)37 b Fw(and)32 b Ft(\021)2618 4860 y Fq(0)2675 4893 y Fw(really)g(di\013er.)48 b(One)33 b(could)165 5006 y(also)e(restrict)f(\003)g(and/or)h(\001)f(to)h (regular)e(b)s(o)m(xes.)41 b(See)31 b([170)q(,)g(217)q(,)g(70)q(,)f(71) q(,)h(25)q(,)f(49)q(,)h(50)q(,)f(15)q(].)306 5119 y(Let)d(us)d(commen)m (t)j(on)e(the)h(signi\014cance)f(of)g(the)h(exp)s(onen)m(tial)f(w)m (eak-mixing)f(condition)h(ab)s(o)m(v)m(e.)165 5279 y Fx(Remarks:)67 b Fw(\(1\))44 b(Supp)s(ose)e(condition)g(\(68\))j (holds.)79 b(A)43 b(straigh)m(tforw)m(ard)g(application)f(of)i(the)165 5392 y(Borel-Can)m(telli)30 b(lemma)g(then)g(sho)m(ws)g(that)h(for)f (an)m(y)h Ft(m)2097 5359 y Fq(0)2145 5392 y Ft(<)25 b(m)1077 5597 y Fw(max)1054 5658 y Fu(\021)r(;\021)1148 5639 y Fm(0)1171 5658 y Fq(2)p Fr(\012)1285 5597 y Fv(k)p Ft(\026)1385 5553 y Fu(\021)1385 5625 y Fn(J)p Fu(;)p Fr(\003)1520 5597 y Fv(\000)20 b Ft(\026)1666 5553 y Fu(\021)1703 5530 y Fm(0)1666 5625 y Fn(J)p Fu(;)p Fr(\003)1780 5597 y Fv(k)1825 5611 y Fr(\001)1914 5597 y Fv(\024)25 b Ft(C)2075 5611 y Fn(J)2136 5597 y Fv(j)p Fw(\001)p Fv(j)15 b Ft(e)2319 5560 y Fq(\000)p Fu(m)2436 5536 y Fm(0)2471 5560 y Fu(d)p Fr(\(\001)p Fu(;)p Fr(\003)2662 5536 y Fo(c)2694 5560 y Fr(\))1844 5895 y Fw(97)p eop %%Page: 98 98 98 97 bop 165 304 a Fw(with)32 b(some)i(realization-dep)s(enden)m(t)d Ft(C)1557 318 y Fn(J)1633 304 y Ft(<)e Fv(1)k Ft(P)13 b Fw(-almost)34 b(surely)-8 b(.)47 b(In)m(tegrating)34 b(o)m(v)m(er)g Ft(\021)3274 271 y Fq(0)3331 304 y Fw(for)f(an)m(y)165 417 y(Gibbs)d(measure)h Ft(\026)840 431 y Fn(J)917 417 y Fw(w)m(e)g(\014nd)f(in)f(particular)h(that)i Ft(\026)2014 431 y Fn(J)2086 417 y Fw(=)26 b(lim)2309 439 y Fr(\003)p Fq(")p Fn(Z)2443 420 y Fo(d)2499 417 y Ft(\026)2554 373 y Fu(\021)2554 444 y Fn(J)p Fu(;)p Fr(\003)2699 417 y Fw(for)31 b(all)f Ft(\021)s Fw(,)i(implying)c(that)165 530 y Ft(\026)220 544 y Fn(J)300 530 y Fw(is)k(the)i(only)f(Gibbs)f (measure)i(\(and)f(dep)s(ends)f(measurably)g(on)i Fx(J)p Fw(\).)51 b(Moreo)m(v)m(er,)36 b(noting)e(that)165 643 y Ft(\026)220 657 y Fn(J)266 643 y Fw(\()p Ft(A)p Fv(j)p Ft(B)5 b Fw(\))31 b(=)635 574 y Fs(R)705 643 y Ft(\026)760 599 y Fu(\021)760 670 y Fn(J)p Fu(;)p Fr(\003)874 643 y Fw(\()p Ft(A)p Fw(\))15 b Ft(\026)1082 657 y Fn(J)1129 643 y Fw(\()p Ft(d\021)s Fv(j)p Ft(B)5 b Fw(\))35 b(for)e Ft(A)d Fv(2)f(F)1823 657 y Fr(\001)1920 643 y Fw(and)j Ft(B)j Fv(2)30 b(F)2359 657 y Fr(\003)2408 638 y Fo(c)2444 643 y Fw(,)k(w)m(e)g(see)g(that)f(this)f(realization-)165 756 y(dep)s(enden)m(t)e(Gibbs)f(measure)h Ft(\026)1278 770 y Fn(J)1354 756 y Fw(satis\014es)g(the)g(exp)s(onen)m(tial)g(w)m (eak-mixing)g(condition)1000 942 y(sup)677 1023 y Fu(A)p Fq(2F)827 1034 y Fl(\001)882 1023 y Fu(;)11 b(B)s Fq(2F)1066 1036 y Fl(\003)1107 1022 y Fo(c)1144 1023 y Fu(;)h(\026)1218 1034 y Fi(J)1258 1023 y Fr(\()p Fu(B)s Fr(\))p Fu(>)p Fr(0)1475 942 y Fv(j)p Ft(\026)1555 956 y Fn(J)1601 942 y Fw(\()p Ft(A)p Fv(j)p Ft(B)5 b Fw(\))20 b Fv(\000)g Ft(\026)2004 956 y Fn(J)2050 942 y Fw(\()p Ft(A)p Fw(\))p Fv(j)27 b(\024)e Ft(C)2401 956 y Fn(J)2462 942 y Fv(j)p Fw(\001)p Fv(j)15 b Ft(e)2645 905 y Fq(\000)p Fu(m)2762 881 y Fm(0)2797 905 y Fu(d)p Fr(\(\001)p Fu(;)p Fr(\003)2988 881 y Fo(c)3020 905 y Fr(\))3077 942 y Ft(:)351 b Fw(\(69\))306 1198 y(\(2\))39 b(Condition)c(\(69\))k(ab)s(o)m(v)m(e)g(also)e(implies) e(an)i(exp)s(onen)m(tial)g(deca)m(y)h(of)g(co)m(v)-5 b(ariances.)63 b(Let)38 b Ft(f)165 1311 y Fw(b)s(e)33 b(an)m(y)h(lo)s(cal)f(observ)-5 b(able)33 b(with)g(dep)s(endence)f(set) i(\001)d Fv(2)f(E)42 b Fw(and)33 b Ft(g)k Fw(b)s(e)c(an)m(y)h(b)s (ounded)e(observ)-5 b(able)165 1424 y(dep)s(ending)25 b(only)h(on)i(the)f(spins)e(o\013)j(\003,)g(where)f(\001)e Fv(\032)g Fw(\003.)39 b(Also,)28 b(let)f Ft(\016)s Fw(\()p Ft(f)10 b Fw(\))26 b(=)f(max)2939 1439 y Fu(\033)n(;\033)3039 1421 y Fm(0)3081 1424 y Fv(j)p Ft(f)10 b Fw(\()p Ft(\033)s Fw(\))k Fv(\000)g Ft(f)c Fw(\()p Ft(\033)3530 1391 y Fq(0)3554 1424 y Fw(\))p Fv(j)165 1537 y Fw(b)s(e)28 b(the)h(total)g(oscillation)d(of)j Ft(f)37 b Fw(and)28 b Ft(\016)s Fw(\()p Ft(g)s Fw(\))i(that)f(of)g Ft(g)s Fw(.)40 b(The)28 b(co)m(v)-5 b(ariance)30 b Ft(\026)2718 1551 y Fn(J)2764 1537 y Fw(\()p Ft(f)10 b Fw(;)15 b Ft(g)s Fw(\))29 b(of)f Ft(f)38 b Fw(and)28 b Ft(g)j Fw(then)165 1650 y(satis\014es)f Ft(P)13 b Fw(-almost)31 b(surely)e(the)h (inequalit)m(y)1039 1836 y Fv(j)p Ft(\026)1119 1850 y Fn(J)1165 1836 y Fw(\()p Ft(f)10 b Fw(;)15 b Ft(g)s Fw(\))p Fv(j)26 b(\024)f Ft(C)1588 1850 y Fn(J)1649 1836 y Fv(j)p Fw(\001)p Fv(j)15 b Ft(\016)s Fw(\()p Ft(f)10 b Fw(\))15 b Ft(\016)s Fw(\()p Ft(g)s Fw(\))g Ft(e)2189 1799 y Fq(\000)p Fu(m)2306 1775 y Fm(0)2345 1799 y Fu(d)p Fr(\(\001)p Fu(;)p Fr(\003)2536 1775 y Fo(c)2568 1799 y Fr(\))2600 1836 y Ft(=)p Fw(2)26 b Ft(:)165 2023 y Fw(Indeed,)44 b(a)e(short)f(computation)h(sho)m(ws)f(that)h Fv(j)p Ft(\026)1933 2038 y Fn(J)p Fu(;)p Fn(h)2044 2023 y Fw(\()p Ft(f)10 b Fw(;)15 b Ft(g)s Fw(\))p Fv(j)43 b Fw(is)d(not)i(larger)f (than)g(the)h(left-hand)165 2136 y(side)c(of)h(inequalit)m(y)e(\(69\))k (times)d Ft(\016)s Fw(\()p Ft(f)10 b Fw(\))15 b Ft(\016)s Fw(\()p Ft(g)s Fw(\))p Ft(=)p Fw(2,)46 b(cf.)39 b(inequalit)m(y)e (\(8.33\))k(in)d([96)q(].)66 b(If)39 b Ft(g)j Fw(is)c(lo)s(cal,)j(a)165 2249 y(similar)30 b(inequalit)m(y)h(holds)g(for)i(co)m(v)-5 b(ariances)34 b(relativ)m(e)e(to)i(\014nite)d(v)m(olume)i(Gibbs)e (distributions)d(in)165 2362 y(su\016cien)m(tly)h(large)i(regions)f (with)f(arbitrary)g(b)s(oundary)f(conditions.)165 2518 y(W)-8 b(e)35 b(will)c(no)m(w)i(in)m(v)m(estigate)i(under)d(whic)m(h)g 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Fw(\))i(=)e(\()p Ft(p)1529 3258 y Fu(b)1563 3243 y Fw(\))1598 3258 y Fu(b)p Fq(2B)1756 3243 y Fw(giv)m(en)j(b)m(y)h Ft(p)2162 3258 y Fu(b)2221 3243 y Fw(=)c(1)16 b Fv(\000)g Fw(exp[)p Fv(\000)p Fw(2)p Fv(j)p Ft(J)2820 3258 y Fu(b)2855 3243 y Fv(j)p Fw(].)41 b(Next)29 b(w)m(e)g(can)f(use)165 3356 y(a)j(recen)m(t)h(conca)m(vit)m(y)g(result)d(of)h([15)r(]:)165 3529 y Fx(Lemma)j(9.6)46 b Fp(L)-5 b(et)45 b Fx(K)31 b Fw(=)g(\()p Ft(K)1211 3544 y Fu(b)1246 3529 y Fw(\))1281 3544 y Fu(b)p Fq(2B)1447 3529 y Fp(b)-5 b(e)36 b(a)g(c)-5 b(ol)5 b(le)-5 b(ction)38 b(of)e(p)-5 b(ositive)37 b(r)-5 b(e)g(al)37 b(numb)-5 b(ers)37 b(and)g Fx(p)31 b Fw(=)g Fx(p)p Fw(\()p Fx(K)p Fw(\))165 3642 y Fp(denote)40 b(the)e(c)-5 b(ol)5 b(le)-5 b(ction)40 b(of)f(densities)f Ft(p)1564 3657 y Fu(b)1634 3642 y Fw(=)e(1)25 b Fv(\000)f Ft(e)1948 3609 y Fq(\000)p Fr(2)p Fu(K)2098 3621 y Fo(b)2133 3642 y Fp(.)59 b(F)-7 b(or)40 b(any)f(incr)-5 b(e)g(asing)39 b(function)g Ft(f)10 b Fp(,)39 b(the)165 3755 y(exp)-5 b(e)g(ctation)35 b Ft(\036)689 3722 y Fr(1)689 3781 y Fn(p)p Fu(;)p Fr(2)p Fu(;)p Fr(\003)862 3755 y Fw(\()p Ft(f)10 b Fw(\))32 b Fp(is)h(then)g(a)g(c)-5 b(onc)g(ave)34 b(function)e(of)h(e)-5 b(ach)34 b Ft(K)2479 3770 y Fu(b)2513 3755 y Fp(.)165 3938 y Fx(Pro)s(of:)46 b Fw(F)-8 b(or)34 b(brevit)m(y)e(w)m(e)h(set)h Ft(\036)1307 3905 y Fr(1)1307 3963 y Fn(p)p Fu(;)p Fr(2)p Fu(;)p Fr(\003)1509 3938 y Fw(=)29 b Ft(\036)p Fw(.)47 b(F)-8 b(or)34 b(an)m(y)f(\014xed)f(b)s (ond)f Ft(b)i Fw(w)m(e)g(consider)f(the)h(functions)165 4063 y Ft(F)13 b Fw(\()p Ft(p)317 4078 y Fu(b)352 4063 y Fw(\))30 b(=)f Ft(\036)p Fw(\()p Ft(f)10 b Fw(\))33 b(and)f Ft(G)p Fw(\()p Ft(K)1091 4078 y Fu(b)1126 4063 y Fw(\))e(=)f Ft(F)13 b Fw(\(1)23 b Fv(\000)e Ft(e)1599 4030 y Fq(\000)p Fr(2)p Fu(K)1749 4042 y Fo(b)1785 4063 y Fw(\).)48 b(Using)32 b(equation)h(\(27\))h(of)f(Lemma)g(6.6)h(w)m(e)f (then)165 4176 y(\014nd)c(that)1280 4289 y Ft(F)1351 4251 y Fq(0)1374 4289 y Fw(\()p Ft(p)1455 4304 y Fu(b)1490 4289 y Fw(\))c(=)g Ft(\036)p Fw(\()p Ft(f)g(g)1848 4303 y Fu(p)1884 4315 y Fo(b)1920 4289 y Fw(\))20 b Fv(\000)g Ft(\036)p Fw(\()p Ft(f)10 b Fw(\))15 b Ft(\036)p Fw(\()p Ft(g)2392 4303 y Fu(p)2428 4315 y Fo(b)2464 4289 y Fw(\))165 4447 y(and)1371 4559 y Ft(F)1442 4522 y Fq(00)1485 4559 y Fw(\()p Ft(p)1566 4574 y Fu(b)1600 4559 y Fw(\))26 b(=)f Fv(\000)p Fw(2)p Ft(F)1944 4522 y Fq(0)1967 4559 y Fw(\()p Ft(p)2048 4574 y Fu(b)2083 4559 y Fw(\))p Ft(\036)p Fw(\()p Ft(g)2250 4573 y Fu(p)2286 4585 y Fo(b)2322 4559 y Fw(\))h Ft(;)165 4717 y Fw(where)k(for)g Ft(\021)612 4732 y Fu(b)672 4717 y Fv(2)25 b(f)p Fw(0)p Ft(;)15 b Fw(1)p Fv(g)1369 4869 y Ft(g)1412 4883 y Fu(p)1448 4895 y Fo(b)1483 4869 y Fw(\()p Ft(\021)1563 4884 y Fu(b)1598 4869 y Fw(\))26 b(=)1917 4807 y(2)p Ft(\021)2007 4822 y Fu(b)2062 4807 y Fv(\000)20 b Fw(1)p 1765 4848 586 4 v 1765 4931 a Ft(p)1811 4887 y Fu(\021)1846 4899 y Fo(b)1811 4959 y Fu(b)1882 4931 y Fw(\(1)h Fv(\000)e Ft(p)2119 4946 y Fu(b)2154 4931 y Fw(\))2189 4905 y Fr(1)p 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Fu(p)1309 5672 y Fo(b)1345 5646 y Fw(\))26 b(=)f(2)p Ft(\036)p Fw(\()p Ft(\021)1681 5661 y Fu(b)1741 5646 y Fw(=)g(1\))p Ft(=p)2008 5661 y Fu(b)2064 5646 y Fv(\000)20 b Fw(1)25 b Fv(\025)g Fw(2)p Ft(=)p Fw(\(2)d Fv(\000)e Ft(p)2650 5661 y Fu(b)2684 5646 y Fw(\))h Fv(\000)f Fw(1)25 b Fv(\025)g Fw(0)p Ft(;)1844 5895 y Fw(98)p eop %%Page: 99 99 99 98 bop 165 304 a Fw(where)40 b(the)h(\014rst)e(inequalit)m(y)g(uses) h(the)g(fact)h(that)g(the)g(random-cluster)e(distribution)e(for)j Ft(q)k Fw(=)165 417 y(2)f(dominates)g(an)f(indep)s(enden)m(t)f(p)s (ercolation)g(mo)s(del)h(with)f(densities)g Ft(p)2781 384 y Fq(0)2781 443 y Fu(b)2861 417 y Fw(=)k Ft(p)3023 432 y Fu(b)3057 417 y Ft(=)p Fw(\(2)30 b Fv(\000)e Ft(p)3357 432 y Fu(b)3391 417 y Fw(\),)47 b(see)165 530 y(Corollary)29 b(6.7\(c\).)43 b(It)31 b(follo)m(ws)e(that)i Ft(G)1525 497 y Fq(00)1593 530 y Fv(\024)25 b Fw(0,)31 b(pro)m(ving)f(the)g (claimed)g(conca)m(vit)m(y)-8 b(.)548 b Fk(2)165 691 y Fw(The)30 b(preceding)g(lemma)f(implies)f(that)565 928 y Ft(P)636 809 y Fs(\022)697 928 y Ft(\036)751 890 y Fr(1)751 953 y Fn(p)p Fr(\()p Fq(j)p Fn(J)p Fq(j)p Fr(\))p Fu(;)p Fr(2)p Fu(;)p Fr(\003)1060 928 y Fw(\(\001)e Fv($)f Ft(@)5 b Fw(\003\))1464 809 y Fs(\023)1551 928 y Fv(\024)25 b Ft(\036)1701 890 y Fr(1)1701 950 y Fu(p;)p Fr(2)p Fu(;)p Fr(\003)1864 928 y Fw(\(\001)g Fv($)h Ft(@)5 b Fw(\003\))26 b Fv(\024)2423 847 y Fs(X)2390 1031 y Fu(x)p Fq(2)p Fu(@)t Fr(\001)2591 928 y Ft(\036)2645 890 y Fr(1)2645 950 y Fu(p;)p Fr(2)p Fu(;)p Fr(\003)2809 928 y Fw(\()p Ft(x)f Fv($)g Ft(@)5 b Fw(\003\))p Ft(;)165 1208 y Fw(where)38 b Ft(p)f Fw(=)g(1)25 b Fv(\000)g Fw(exp[)p Fv(\000)p Fw(2)15 b Ft(P)e Fw(\()p Fv(j)p Ft(J)1269 1223 y Fu(b)1305 1208 y Fv(j)p Fw(].)63 b(The)37 b(w)m(eak)h(mixing)e(prop)s (ert)m(y)h(\(68\))i(th)m(us)f(follo)m(ws)e(pro)m(vided)165 1321 y(w)m(e)h(can)f(sho)m(w)f(an)h(exp)s(onen)m(tial)f(deca)m(y)i(of)f (connectivit)m(y)g(in)e(the)i(wired)e(random-cluster)h(distri-)165 1434 y(bution)e Ft(\036)511 1401 y Fr(1)511 1460 y Fu(p;)p Fr(2)p Fu(;)p Fr(\003)674 1434 y Fw(.)50 b(The)34 b(latter)g(certainly) f(holds)f(when)h Ft(p)d(<)h(p)2277 1448 y Fu(c)2345 1434 y Fw(b)s(ecause)i Ft(\036)2738 1401 y Fr(1)2738 1460 y Fu(p;)p Fr(2)p Fu(;)p Fr(\003)2935 1434 y Fw(is)g(dominated)g(\(on) 165 1547 y(\003\))k(b)m(y)f(the)h(Bernoulli)c(measure)k Ft(\036)1408 1561 y Fu(p)1484 1547 y Fw(\(cf.)g(Corollary)e (6.7\(b\)\),)k(and)d(the)g(connectivit)m(y)h(of)f(a)h(sub-)165 1660 y(critical)28 b(Bernoulli)f(mo)s(del)h(deca)m(ys)i(exp)s(onen)m (tially)d(fast)j(\(recall)f(Theorem)f(5.6\).)42 b(So)29 b(w)m(e)h(arriv)m(e)e(in)165 1773 y(particular)h(at)i(the)g(follo)m (wing)e(result.)165 1957 y Fx(Theorem)34 b(9.7)46 b Fp(Consider)55 b(a)f(r)-5 b(andom)57 b(Ising)c(mo)-5 b(del)56 b(on)e Fx(Z)2392 1924 y Fu(d)2486 1957 y Fp(with)g(Hamiltonian)i(\(67\).)106 b(If)165 2070 y Fw(2)15 b Ft(P)e Fw(\()p Fv(j)p Ft(J)406 2085 y Fu(b)442 2070 y Fv(j)p Fw(\))32 b Ft(<)g Fv(\000)15 b Fw(ln)n(\(1)24 b Fv(\000)e Ft(p)1041 2084 y Fu(c)1076 2070 y Fw(\))36 b Fp(for)h(the)f(Bernoul)5 b(li)37 b(b)-5 b(ond)37 b(p)-5 b(er)g(c)g(olation)39 b(thr)-5 b(eshold)39 b Ft(p)2969 2084 y Fu(c)3039 2070 y Fp(of)e Fx(Z)3214 2037 y Fu(d)3290 2070 y Fp(then)g(the)165 2183 y(system)d(is)e(exp)-5 b(onential)5 b(ly)35 b(we)-5 b(ak-mixing.)165 2368 y Fw(As)30 b(the)g(pro)s(of)f(ab)s(o)m(v)m(e)i(sho)m(ws,)e(the)h(factor)h Fv(j)p Fw(\001)p Fv(j)f Fw(on)f(the)h(righ)m(t-hand)f(side)f(of)i (\(68\))i(can)d(actually)h(b)s(e)165 2481 y(replaced)g(b)m(y)g Fv(j)p Ft(@)5 b Fw(\001)p Fv(j)p Fw(.)306 2594 y(In)25 b(the)g(diluted)d(Ising)i(mo)s(del)g(at)h(in)m(v)m(erse)g(temp)s (erature)g Ft(\014)30 b Fw(\(see)c(Example)e(9.3\),)j(the)e(condition) 165 2707 y(of)41 b(the)g(preceding)e(theorem)i(reads)g(2)p Ft(\014)5 b(p)43 b(<)f Fv(\000)15 b Fw(ln)n(\(1)28 b Fv(\000)f Ft(p)2194 2721 y Fu(c)2228 2707 y Fw(\).)72 b(Therefore,)43 b(if)c Ft(d)i Fw(is)f(so)h(large)g(that)165 2820 y(2)p Ft(\014)261 2834 y Fu(c)312 2820 y Ft(p)358 2834 y Fu(c)419 2820 y Ft(<)27 b Fv(\000)15 b Fw(ln)o(\(1)21 b Fv(\000)g Ft(p)918 2834 y Fu(c)952 2820 y Fw(\))32 b(then)f(the)h(theorem)f(co)m(v)m(ers)i(part)f(of)f(the)h(Gri\016ths)e (regime.)43 b(This)30 b(fact)i(is)165 2933 y(eviden)m(t)c(in)e(the)h (case)h(of)g(an)f(un)m(b)s(ounded,)f(sa)m(y)i(Gaussian,)f(disorder)f(b) s(ecause)h(then)g(\(as)h(explained)165 3045 y(ab)s(o)m(v)m(e\))k(there) f(is)e(no)i(paramagnetic)f(phase.)306 3158 y(Exp)s(onen)m(tial)i(w)m (eak-mixing)h(for)g(random)f(Ising)g(mo)s(dels)g(can)h(also)g(b)s(e)g (sho)m(wn)f(b)m(y)h(other)h(ap-)165 3271 y(plications)24 b(of)i(random-cluster)e(domination.)38 b(Let)26 b(us)f(sk)m(etc)m(h)i (suc)m(h)e(an)h(alternativ)m(e)g(route.)39 b(Using)165 3384 y(the)23 b(random-cluster)e(represen)m(tation,)k(Newman)d([182)r (])g(sho)m(w)m(ed)h(that)g(\(p)s(oin)m(t)m(wise)e(in)g(the)i(disorder) 165 3497 y Fx(J)p Fw(\))1070 3623 y(max)1095 3684 y Fu(\021)r(;\021) 1189 3665 y Fm(0)1254 3623 y Fv(k)p Ft(\026)1354 3579 y Fu(\021)1354 3651 y Fn(J)p Fu(;)p Fr(\003)1489 3623 y Fv(\000)d Ft(\026)1635 3579 y Fu(\021)1672 3556 y Fm(0)1635 3651 y Fn(J)p Fu(;)p Fr(\003)1749 3623 y Fv(k)1794 3637 y Fr(\001)1883 3623 y Fv(\024)25 b Fw(2)2052 3543 y Fs(X)2040 3726 y Fu(x)p Fq(2)p Fr(\001)2200 3623 y Ft(\026)2255 3585 y Fr(+)2255 3655 y Fq(j)p Fn(J)p Fq(j)p Fu(;)p Fr(\003)2409 3623 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\)\))p Ft(;)745 b Fw(\(70\))165 3878 y(where)25 b Ft(\026)478 3840 y Fr(+)478 3910 y Fq(j)p Fn(J)p Fq(j)p Fu(;)p Fr(\003)657 3878 y Fw(is)g(the)g(Gibbs)f(distribution)e(in)i(\003)h(with)f(plus)g (b)s(oundary)f(conditions)h(for)h(the)h(Hamil-)165 4003 y(tonian)37 b(\(67\))j(with)c Ft(J)919 4018 y Fu(b)991 4003 y Fw(replaced)h(b)m(y)h Fv(j)p Ft(J)1566 4018 y Fu(b)1601 4003 y Fv(j)p Fw(.)62 b(On)37 b(the)h(other)f(hand,)i(Higuc)m (hi)e([132)q(])h(obtained)f(the)165 4116 y(estimate)1018 4228 y Ft(\026)1073 4190 y Fr(+)1073 4260 y Fq(j)p Fn(J)p Fq(j)p Fu(;)p Fr(\003)1226 4228 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\)\))28 b Fv(\024)1630 4148 y Fs(X)1623 4336 y Fu(y)10 b(=)-43 b Fq(2)q Fr(\003)1857 4148 y Fs(X)1772 4331 y Fu(z)s Fq(2)p Fr(\003:)11 b Fu(z)s Fq(\030)p Fu(y)2078 4228 y Ft(\026)2133 4184 y Fu(f)2133 4260 y Fq(j)p Fn(J)p Fq(j)p Fu(;)p Fr(\003)2286 4228 y Fw(\()p Ft(X)c Fw(\()p Ft(x)p Fw(\))p Ft(X)g Fw(\()p Ft(z)t Fw(\)\))165 4486 y(\(the)31 b(sup)s(erscript)c(`)p Ft(f)10 b Fw(')29 b(referring)g(to)h (the)g(free)g(b)s(oundary)e(condition\),)h(while)f(Olivieri,)f(P)m (erez)k(and)165 4599 y(Goulart)g(Rosa)g([185)q(])g(pro)m(v)m(ed)f(that) 1071 4840 y Ft(P)1142 4721 y Fs(\022)1203 4840 y Ft(\026)1258 4796 y Fu(f)1258 4872 y Fq(j)p Fn(J)p Fq(j)p Fu(;)p Fr(\003)1411 4840 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))p Ft(X)g Fw(\()p Ft(z)t Fw(\)\))1883 4721 y Fs(\023)1973 4840 y Fv(\024)25 b Ft(\026)2124 4796 y Fu(f)2124 4867 y(J)n(;)p Fr(\003)2233 4840 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))p Ft(X)g Fw(\()p Ft(z)t Fw(\)\))748 b(\(71\))165 5081 y(with)29 b Ft(J)34 b Fw(=)25 b Ft(P)13 b Fw(\()p Fv(j)p Ft(J)733 5096 y Fu(b)769 5081 y Fv(j)p Fw(\).)41 b(W)-8 b(e)31 b(are)g(th)m(us)e(bac)m(k)i(to)g(the)f(standard)f(Ising)g(Gibbs)f (distribution)f(in)h(\003)i(with)165 5194 y(zero)h(external)f(\014eld,) f(free)h(b)s(oundary)e(condition)g(and)h(coupling)g(constan)m(t)i Ft(J)9 b Fw(.)40 b(No)m(w)31 b(w)m(e)f(can)h(tak)m(e)165 5307 y(adv)-5 b(an)m(tage)26 b(of)e(the)g(second)h(Gri\016ths)d (inequalit)m(y)g(\(whic)m(h)h(w)m(e)i(did)d(not)i(discuss)e(so)j(far)e (in)g(this)g(text\))165 5420 y(stating)28 b(that)h(correlation)f (functions)e(suc)m(h)i(as)g(on)g(the)g(righ)m(t-hand)e(side)h(ab)s(o)m (v)m(e)i(are)g(monotone)f(in)165 5533 y(the)k(coupling)e(co)s (e\016cien)m(ts,)j(see)f(e.g.)h([160)r(].)45 b(This)30 b(implies)f(that)j(the)g(righ)m(t-hand)e(side)h(of)h(\(71\))h(is)165 5646 y(an)25 b(increasing)f(function)g(of)h(\003)g(and)f(th)m(us)h(b)s (ounded)e(ab)s(o)m(v)m(e)j(b)m(y)f(its)f(in\014nite)f(v)m(olume)h (limit.)37 b(But)25 b(for)1844 5895 y(99)p eop %%Page: 100 100 100 99 bop 165 304 a Ft(J)33 b Fw(less)24 b(than)f Ft(J)665 318 y Fu(c)701 304 y Fw(,)i(the)f(critical)f(coupling,)h(the)g(Gibbs)e (measure)i(is)f(unique)f(and)i(has)f(an)h(exp)s(onen)m(tial)165 417 y(deca)m(y)34 b(of)f(correlations,)g(see)g([7)q(].)48 b(W)-8 b(e)33 b(th)m(us)g(\014nd)e(that)i(for)f Ft(J)38 b(<)29 b(J)2486 431 y Fu(c)2554 417 y Fw(the)k(righ)m(t-hand)e(side)h (of)h(\(71\))165 530 y(has)j(an)f(exp)s(onen)m(tial)f(upp)s(er)g(b)s (ound)f Ft(C)22 b Fw(exp[)p Fv(\000)p Ft(m)p Fv(j)p Ft(x)h Fv(\000)h Ft(z)t Fv(j)p Fw(])36 b(for)f(suitable)f(constan)m(ts)j Ft(C)j(<)33 b Fv(1)j Fw(and)165 643 y Ft(m)i(>)g Fw(0.)64 b(T)-8 b(ogether)39 b(with)e(the)h(previous)e(estimates,)41 b(w)m(e)d(conclude)g(that)g Fp(under)i(the)g(c)-5 b(ondition)165 756 y Ft(P)13 b Fw(\()p Fv(j)p Ft(J)346 771 y Fu(b)382 756 y Fv(j)p Fw(\))42 b Ft(<)f(J)646 770 y Fu(c)681 756 y Fp(,)j(the)e(r)-5 b(andom)44 b(Ising)e(mo)-5 b(del)43 b(on)f Fx(Z)1960 723 y Fu(d)2042 756 y Fp(with)g(Hamiltonian)i(\(67\))e (is)g(exp)-5 b(onential)5 b(ly)165 869 y(we)-5 b(ak-mixing)p Fw(.)58 b(Again,)37 b(this)e(condition)f(co)m(v)m(ers)k(part)e(of)g (the)g(Gri\016ths)e(regime)i(for)f(the)h(diluted)165 981 y(Ising)30 b(ferromagnet.)306 1094 y(As)f(an)g(alternativ)m(e)g(to) h(the)f(use)f(of)h(Gri\016ths')f(inequalities)e(ab)s(o)m(v)m(e)k(w)m(e) g(can)f(also)f(apply)g(Theo-)165 1207 y(rem)i(6.2)i(and)e(Corollary)f (6.7\(b\),)j(giving)1189 1452 y Ft(P)1260 1333 y Fs(\022)1321 1452 y Ft(\026)1376 1408 y Fu(f)1376 1484 y Fq(j)p Fn(J)p Fq(j)p Fu(;)p Fr(\003)1529 1452 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\))p Ft(X)g Fw(\()p Ft(z)t Fw(\)\))2001 1333 y Fs(\023)2091 1452 y Fv(\024)25 b Ft(\036)2241 1466 y Fu(p)2281 1452 y Fw(\()p Ft(x)g Fv($)g Ft(z)t Fw(\))165 1697 y(where)45 b Ft(\036)497 1711 y Fu(p)581 1697 y Fw(is)f(the)h(b)s(ond)e(Bernoulli)g(measure)h(with)g(densit)m(y)g Ft(p)49 b Fw(=)g Ft(P)13 b Fw(\(1)31 b Fv(\000)e Fw(exp[)p Fv(\000)p Fw(2)p Fv(j)p Ft(J)3280 1712 y Fu(b)3315 1697 y Fv(j)p Fw(]\).)85 b(As)165 1810 y(men)m(tioned)32 b(ab)s(o)m(v)m(e,)i (its)e(connectivit)m(y)g(function)f(deca)m(ys)i(exp)s(onen)m(tially)e (fast)h(when)f Ft(p)d(<)g(p)3382 1824 y Fu(c)3417 1810 y Fw(.)46 b(W)-8 b(e)165 1923 y(therefore)26 b(conclude)f(that)h Fp(the)j(r)-5 b(andom)31 b(Ising)d(mo)-5 b(del)30 b(on)e Fx(Z)2243 1890 y Fu(d)2312 1923 y Fp(is)g(also)h(exp)-5 b(onential)5 b(ly)31 b(we)-5 b(ak-mixing)165 2035 y(when)34 b Ft(P)13 b Fw(\(1)21 b Fv(\000)f Fw(exp)o([)p Fv(\000)p Fw(2)p Fv(j)p Ft(J)1015 2050 y Fu(b)1051 2035 y Fv(j)p Fw(]\))26 b Ft(<)f(p)1304 2049 y Fu(c)1338 2035 y Fw(.)306 2148 y(The)j(ab)s(o)m(v)m(e)g(estimate)h(\(70\))g(do)s(es)e(not)h(hold) e(when)h(w)m(e)h(add)f(a)h(random)e(magnetic)i(\014eld)f(to)h(the)165 2261 y(Hamiltonian)h(\(67\),)986 2465 y Ft(H)7 b Fw(\()p Ft(\033)s Fw(\))26 b(=)f Fv(\000)1498 2385 y Fs(X)1402 2573 y Fu(b)p Fr(=)p Fq(h)p Fu(xy)r Fq(i2B)1729 2465 y Ft(J)1779 2480 y Fu(b)1829 2465 y Ft(\033)s Fw(\()p Ft(x)p Fw(\))p Ft(\033)s Fw(\()p Ft(y)s Fw(\))d Fv(\000)2318 2385 y Fs(X)2291 2577 y Fu(x)p Fq(2)p Fn(Z)2428 2558 y Fo(d)2479 2465 y Ft(h)2531 2479 y Fu(x)2591 2465 y Ft(\033)s Fw(\()p Ft(x)p Fw(\))p Ft(;)660 b Fw(\(72\))165 2761 y(with)36 b(i.i.d.)g(real)h(random)f(v)-5 b(ariables)36 b Ft(h)1553 2775 y Fu(x)1635 2761 y Fw(indep)s(enden)m(t)f(from)i(the)g Ft(J)2592 2776 y Fu(b)2627 2761 y Fw(.)61 b(Ho)m(w)m(ev)m(er,)41 b(if)36 b Ft(J)3253 2776 y Fu(b)3288 2761 y Ft(;)15 b(h)3380 2775 y Fu(x)3461 2761 y Fv(\025)37 b Fw(0)165 2873 y(then)30 b(w)m(e)h(can)g(tak)m(e)h(adv)-5 b(an)m(tage)32 b(of)f(Section)f(2)h (of)f([132)r(])g(to)h(replace)g(\(70\))h(with)997 3091 y(max)1023 3152 y Fu(\021)r(;\021)1117 3133 y Fm(0)1181 3091 y Fv(k)p Ft(\026)1281 3047 y Fu(\021)1281 3119 y Fn(J)p Fu(;)p Fn(h)p Fu(;)p Fr(\003)1481 3091 y Fv(\000)20 b Ft(\026)1627 3047 y Fu(\021)1664 3023 y Fm(0)1627 3119 y Fn(J)p Fu(;)p Fn(h)p Fu(;)p Fr(\003)1806 3091 y Fv(k)1851 3105 y Fr(\001)1940 3091 y Fv(\024)25 b Fw(2)2109 3010 y Fs(X)2096 3194 y Fu(x)p Fq(2)p Fr(\001)2257 3091 y Ft(\026)2312 3053 y Fr(+)2312 3119 y Fn(J)p Fu(;)p Fr(0)p Fu(;)p Fr(\003)2482 3091 y Fw(\()p Ft(X)7 b Fw(\()p Ft(x)p Fw(\)\))p Ft(;)165 3369 y Fw(and)30 b(w)m(e)h(can)g(con)m(tin)m(ue)f (as)h(ab)s(o)m(v)m(e.)165 3532 y Fp(Applic)-5 b(ation)39 b(of)e(disagr)-5 b(e)g(ement)39 b(p)-5 b(er)g(c)g(olation.)112 b Fw(As)35 b(w)m(e)h(ha)m(v)m(e)g(indicated)e(at)h(the)h(end)e(of)h (Section)165 3645 y(7.1,)i(the)e(idea)f(of)g(disagreemen)m(t)h(p)s (ercolation)f(can)h(b)s(e)e(applied)g(to)i(study)f(the)g(Gri\016ths)f (regime)165 3758 y(for)k(rather)f(general)g(random-in)m(teraction)g (systems.)59 b(T)-8 b(o)37 b(b)s(e)f(sp)s(eci\014c)g(w)m(e)g(consider)g (Ising)f(spins)165 3871 y(with)c(the)h(Hamiltonian)f(\(72\).)47 b(W)-8 b(e)33 b(consider)e(the)h(\014nite)f(v)m(olume)h(Gibbs)e (distribution)e Ft(\026)3327 3826 y Fu(\021)3327 3899 y Fn(J)p Fu(;)p Fn(h)p Fu(;)p Fr(\003)3538 3871 y Fw(in)165 3984 y(a)i(b)s(o)m(x)f(\003)g(with)f(b)s(oundary)f(condition)h Ft(\021)19 b Fw(;)29 b(as)h(b)s(efore,)f(the)g(subscripts)e Fx(J)p Ft(;)15 b Fx(h)30 b Fw(describ)s(e)e(the)h(random)165 4097 y(in)m(teraction.)41 b(W)-8 b(e)32 b(are)f(going)f(to)h(apply)e (Theorems)h(7.1)i(and)e(7.2)h(p)s(oin)m(t)m(wise)e(in)g(the)i (disorder.)39 b(As)165 4209 y(in)29 b(\(44\))j(w)m(e)f(th)m(us)f(ha)m (v)m(e)i(to)f(consider)e(the)i(v)-5 b(ariational)29 b(single-spin)e (oscillations)1288 4427 y Ft(p)1334 4390 y Fn(J)p Fu(;)p Fn(h)1334 4450 y Fu(x)1469 4427 y Fw(=)49 b(max)1565 4488 y Fu(\021)r(;\021)1659 4469 y Fm(0)1683 4488 y Fq(2)p Fr(\012)1796 4427 y Fv(k)p Ft(\026)1896 4383 y Fu(\021)1896 4455 y Fn(J)p Fu(;)p Fn(h)p Fu(;x)2087 4427 y Fv(\000)20 b Ft(\026)2233 4383 y Fu(\021)2270 4359 y Fm(0)2233 4455 y Fn(J)p Fu(;)p Fn(h)p Fu(;x)2403 4427 y Fv(k)2448 4441 y Fu(x)3453 4427 y Fw(\(73\))165 4688 y(whic)m(h)41 b(dep)s(end)f(on)h (the)h(disorder)e Fx(J)p Ft(;)15 b Fx(h)p Fw(.)75 b(Under)41 b(the)h(presen)m(t)f(assumptions,)i(\()p Ft(p)3101 4655 y Fn(J)p Fu(;)p Fn(h)3101 4710 y Fu(x)3212 4688 y Fw(\))3247 4710 y Fu(x)p Fq(2)p Fn(Z)3384 4691 y Fo(d)3466 4688 y Fw(is)e(a)165 4800 y(1-dep)s(enden)m(t)28 b(random)g(\014eld;)f(this) h(is)f(the)h(only)g(prop)s(ert)m(y)f(of)h(the)h(disorder)d(needed)i(b)s (elo)m(w.)39 b(No)m(w,)165 4913 y(from)f(Theorems)g(7.1)i(and)e(7.2)h (w)m(e)g(can)g(conclude)f(that)h(if)e(with)g Ft(P)13 b Fw(-probabilit)m(y)36 b(1)j(there)g(is)e(no)165 5026 y(Bernoulli)29 b(site-p)s(ercolation)h(with)f(densities)g(\(73\))k (then,)e(again)f(with)g Ft(P)13 b Fw(-probabilit)m(y)29 b(1,)i(there)g(is)165 5139 y(a)38 b(unique)e(Gibbs)g(measure)i(for)f Fx(J)p Ft(;)15 b Fx(h)p Fw(.)63 b(\(In)37 b(fact,)k(similarly)34 b(to)39 b(the)e(results)g(of)h(Section)f(7.2)i(the)165 5252 y(random)32 b(system)h(can)g(b)s(e)e(dominated)h(b)m(y)g(a)h (random)f(p)s(ercolation)g(system)g(whic)m(h)g(more)g(or)h(less)165 5365 y(coincides)g(with)f(a)i(sto)s(c)m(hastic-geometric)i(represen)m (tation)e(of)f(the)h(diluted)d(ferromagnet)k([101)q(].\))165 5478 y(The)30 b(follo)m(wing)f(theorem)i(is)e(an)i(immediate)e (consequence)i(of)g(the)f(results)f(of)i(Section)f(7.1.)1822 5895 y(100)p eop %%Page: 101 101 101 100 bop 165 304 a Fx(Theorem)34 b(9.8)46 b Fp(A)n(n)32 b(Ising)h(system)g(with)h(r)-5 b(andom)35 b(Hamiltonian)f(\(72\))g (satisfying)1490 546 y Ft(P)13 b Fw(\()p Ft(p)1642 509 y Fn(J)p Fu(;)p Fn(h)1642 569 y Fu(x)1753 546 y Fw(\))26 b Ft(<)2077 485 y Fw(1)p 1920 525 360 4 v 1920 609 a(\(2)p Ft(d)21 b Fv(\000)f Fw(1\))2239 583 y Fr(2)165 794 y Fp(is)33 b(exp)-5 b(onential)5 b(ly)35 b(we)-5 b(ak-mixing.)165 982 y Fw(Of)28 b(course,)i(w)m(e)f(no)m(w)g(ha)m(v)m(e)h(to)f(add)f(in) g(\(68\))i(the)f(subscript)d Fx(h)j Fw(referring)e(to)i(the)g(random)f (external)165 1095 y(\014eld.)165 1258 y Fx(Pro)s(of:)41 b Fw(Consider)28 b(Theorem)h(7.1.)41 b(Eviden)m(tly)-8 b(,)29 b(the)g(righ)m(t)g(hand)g(side)f(of)h(\(46\))i(can)f(only)e (increase)165 1370 y(if)33 b(the)h(densit)m(y)f(for)g(an)h(op)s(en)f (site)h(is)e(set)j(to)f(b)s(e)f(1)h(on)g(the)g(o)s(dd)e(sublattice)h Fv(L)2908 1385 y Fu(odd)3052 1370 y Fw(of)h Fx(Z)3223 1337 y Fu(d)3264 1370 y Fw(.)50 b(Due)34 b(to)165 1483 y(the)c(1-dep)s(endence)e(of)i(the)f(random)f(\014eld)g Fx(p)d Fw(=)g(\()p Ft(p)1928 1450 y Fn(J)p Fu(;)p Fn(h)1928 1506 y Fu(x)2039 1483 y Fw(\))2074 1506 y Fu(x)p Fq(2)p Fn(Z)2211 1487 y Fo(d)2280 1483 y Fw(noticed)k(ab)s(o)m(v)m(e,)i(the)e (remaining)f Ft(p)3504 1450 y Fn(J)p Fu(;)p Fn(h)3504 1506 y Fu(x)165 1596 y Fw(with)35 b Ft(x)h Fv(2)f(L)625 1610 y Fu(ev)r(en)810 1596 y Fw(are)h(m)m(utually)f(indep)s(enden)m(t.) 57 b(T)-8 b(aking)36 b(the)g Ft(P)13 b Fw(-exp)s(ectation)38 b(in)d(\(46\))j(w)m(e)e(th)m(us)165 1709 y(obtain)j(on)h(the)f(righ)m (t-hand)g(side)f(the)i(Bernoulli)d(p)s(ercolation)h(probabilit)m(y)g Ft( )2989 1723 y Fu(p;)11 b(ev)r(en)3205 1709 y Fw(\(\001)k Fv($)g Ft(@)5 b Fw(\003\),)165 1822 y(where)30 b Ft( )487 1836 y Fu(p;)11 b(ev)r(en)733 1822 y Fw(is)29 b(the)h(Bernoulli)d (measure)j(with)f(densit)m(y)g Ft(p)c Fw(=)g Ft(P)13 b Fw(\()p Ft(p)2565 1789 y Fn(J)p Fu(;)p Fn(h)2565 1845 y Fu(x)2676 1822 y Fw(\))30 b(on)g(the)g(ev)m(en)h(sublattice)165 1935 y Fv(L)228 1949 y Fu(ev)r(en)407 1935 y Fw(and)e(densit)m(y)g(1)h (on)f Fv(L)1159 1950 y Fu(odd)1270 1935 y Fw(.)40 b(The)30 b(exp)s(onen)m(tial)e(w)m(eak-mixing)h(prop)s(ert)m(y)g(no)m(w)h(follo) m(ws)f(b)m(y)g(an)165 2048 y(argumen)m(t)j(similar)c(to)j(that)g(follo) m(wing)f(\(18\);)i(note)g(that,)g(relativ)m(e)e(to)i Ft( )2702 2062 y Fu(p;)11 b(ev)r(en)2918 2048 y Fw(,)31 b(a)g(path)g(of)g(length)165 2161 y Ft(k)j Fw(is)29 b(op)s(en)h(with)f (probabilit)m(y)f(at)j(most)g Ft(p)1609 2128 y Fq(b)p Fu(k)r(=)p Fr(2)p Fq(c)1784 2161 y Fw(.)1737 b Fk(2)165 2324 y Fx(Remarks:)40 b Fw(\(1\))30 b(Supp)s(ose)d(w)m(e)j(add)e(a)h (uniform)e(magnetic)j(\014eld)d Ft(h)j Fw(to)f(the)h(random)e (Hamiltonian)165 2436 y(\(72\).)73 b(Under)40 b(the)h(conditions)e(of)i (Theorem)f(9.8)i(it)e(is)g(then)g(not)h(to)s(o)g(di\016cult)e(to)i(sho) m(w)g(that)165 2549 y(the)33 b(disorder-a)m(v)m(eraged)f(exp)s (ectation)h Ft(P)13 b Fw(\()p Ft(\026)1704 2564 y Fn(J)p Fu(;)p Fn(h)p Fr(+)p Fu(h)1911 2549 y Fw(\()p Ft(f)d Fw(\)\))32 b(of)g(an)m(y)h(lo)s(cal)e(function)g Ft(f)41 b Fw(is)31 b(an)h(in\014nitely)165 2662 y(di\013eren)m(tiable)d (function)g(of)i Ft(h)p Fw(,)g(see)g(e.g.)g([72)q(].)306 2775 y(\(2\))c(F)-8 b(ollo)m(wing)25 b(Theorem)h(7.2,)i(Theorem)d(9.8)i (requires)d(that)j(the)f(single-p)s(oin)m(t)e(densities)g(\(73\))165 2888 y(are)g(globally)f(small)f(enough)h(to)i(prev)m(en)m(t)f(p)s (ercolation)f(in)f(the)i(asso)s(ciated)g(Bernoulli)e(mo)s(del.)37 b(This)165 3001 y(condition)j(can)h(b)s(e)f(extended)h(to)h(so-called)f (constructiv)m(e)g(conditions)f(in)m(v)m(olving)f(\014nite)h(b)s(o)m (xes)165 3114 y(rather)31 b(than)f(single)f(sites)h([25)q(].)306 3227 y(\(3\))g(Another)e(v)m(ery)h(p)s(o)m(w)m(erful)e(and)h (transparen)m(t)g(treatmen)m(t)i(of)e(the)h(Gri\016ths)e(regime)h (\(based)165 3340 y(on)43 b(v)m(ery)g(similar)d(p)s(ercolation)i (ideas\))g(has)g(b)s(een)g(dev)m(elop)s(ed)g(in)g([72)q(].)78 b(This)40 b(pap)s(er)i(prop)s(oses)165 3453 y(a)36 b(tec)m(hnique)e (similar)e(to)k([18)q(])f(and)f([85)q(])h(of)g(a)h(resummation)d(in)h (the)h(high)e(temp)s(erature)i(cluster)165 3566 y(expansion.)i(The)22 b(b)s(ounds)f(then)h(allo)m(w)g(a)g(probabilistic)d(in)m(terpretation)j (of)h(the)g(expansion)e(linking)165 3678 y(it)30 b(with)f(a)i(b)s(ond)e (p)s(ercolation)g(pro)s(cess.)1822 5895 y(101)p eop %%Page: 102 102 102 101 bop 165 304 a Fy(10)135 b(Con)l(tin)l(uum)44 b(mo)t(dels)165 507 y Fw(All)23 b(mo)s(dels)g(considered)h(so)g(far)h (liv)m(ed)e(on)h(a)h(lattice.)39 b(Ph)m(ysical)23 b(systems)i(lik)m(e)f (real)g(gases,)i(ho)m(w)m(ev)m(er,)165 620 y(are)35 b(more)e (realistically)f(mo)s(delled)g(b)m(y)h(particles)g(living)f(in)g(con)m (tin)m(uous)i(space.)51 b(This)32 b(section)i(is)165 733 y(an)29 b(outline)e(of)i(ho)m(w)f(some)h(of)g(the)f(sto)s(c)m (hastic-geometric)j(ideas)d(dev)m(elop)s(ed)g(in)f(previous)g(sections) 165 846 y(can)34 b(b)s(e)f(applied)e(to)j(a)g(con)m(tin)m(uum)e (setting.)50 b(First,)33 b(in)f(Section)i(10.1,)h(w)m(e)f(consider)f (the)g(natural)165 959 y(con)m(tin)m(uum)g(analogues,)i(based)e(on)g(P) m(oisson)g(pro)s(cesses,)i(of)e(the)h(Bernoulli)d(p)s(ercolation)h(mo)s (dels)165 1072 y(in)m(tro)s(duced)39 b(in)h(Section)g(5.)72 b(Then,)43 b(in)c(Section)i(10.2,)k(w)m(e)c(consider)e(a)j(con)m(tin)m (uum)d(v)-5 b(arian)m(t)41 b(of)165 1184 y(the)29 b(Widom{Ro)m(wlinson) d(mo)s(del)h(in)m(tro)s(duced)g(in)g(Section)h(3.5)h(and)f(discuss)e (its)i(phase)g(transition)165 1297 y(b)s(eha)m(vior.)54 b(\(As)36 b(men)m(tioned)f(in)e(Section)i(10.2,)j(this)c(con)m(tin)m (uum)h(v)-5 b(arian)m(t)35 b(is)f(the)h(one)h(originally)165 1410 y(considered)30 b(b)m(y)g(Widom)g(and)f(Ro)m(wlinson)g([223)r(],)h (so)h(it)f(predates)g(the)h(lattice)g(mo)s(del.\))165 1654 y FA(10.1)113 b(Con)m(tin)m(uum)38 b(p)s(ercolation)165 1825 y Fw(Here)29 b(w)m(e)g(consider)e(the)h(basic)g(mo)s(dels)e(of)j (con)m(tin)m(uum)e(p)s(ercolation.)39 b(F)-8 b(or)29 b(a)g(thorough)f(treatmen)m(t)165 1938 y(of)e(the)f(mathematical)h (theory)f(of)h(con)m(tin)m(uum)e(p)s(ercolation)g(w)m(e)i(refer)f(to)h (Meester)g(and)f(Ro)m(y)h([173)r(].)306 2051 y(W)-8 b(e)31 b(\014rst)e(need)g(to)h(in)m(tro)s(duce)e(the)h(P)m(oisson)g(pro)s (cess)g(on)h Fx(R)2375 2018 y Fu(d)2445 2051 y Fw(and)e(its)h(subsets.) 40 b(Heuristically)-8 b(,)165 2164 y(a)34 b(P)m(oisson)f(pro)s(cess)g (with)f(in)m(tensit)m(y)h Ft(\025)d(>)g Fw(0)j(on)g Fx(R)1949 2131 y Fu(d)2024 2164 y Fw(is)f(a)i(random)e(set)i Ft(X)41 b Fw(of)33 b(p)s(oin)m(ts)f(in)g Fx(R)3363 2131 y Fu(d)3437 2164 y Fw(with)165 2277 y(the)f(prop)s(erties)e(that)221 2465 y(\(i\))76 b(for)28 b(an)m(y)g(b)s(ounded)e(Borel)j(set)f(\003)h (of)f Fx(R)1718 2432 y Fu(d)1787 2465 y Fw(with)f(v)m(olume)h Fv(j)p Fw(\003)p Fv(j)p Fw(,)i(the)e(n)m(um)m(b)s(er)f(of)h(p)s(oin)m (ts)f(of)i Ft(X)35 b Fw(in)393 2577 y(\003)30 b(is)f(P)m(oisson)h (distributed)d(with)h(mean)i Ft(\025)p Fv(j)p Fw(\003)p Fv(j)p Fw(,)h(i.e.,)g(for)f Ft(n)25 b Fw(=)g(0)p Ft(;)15 b Fw(1)p Ft(;)g Fw(2)p Ft(;)g(:)g(:)g(:)33 b Fw(the)d(probabilit)m(y)e (of)393 2690 y(seeing)i(exactly)h Ft(n)f Fw(p)s(oin)m(ts)f(in)g(\003)i (equals)e(exp\()p Fv(\000)p Ft(\025)p Fv(j)p Fw(\003)p Fv(j)p Fw(\)\()p Ft(\025)p Fv(j)p Fw(\003)p Fv(j)p Fw(\))2489 2657 y Fu(n)2539 2690 y Ft(=n)p Fw(!)15 b(;)196 2878 y(\(ii\))75 b(for)26 b(an)m(y)h(t)m(w)m(o)h(disjoin)m(t)d(suc)m(h)h (subsets)g(\003)1759 2892 y Fr(1)1825 2878 y Fw(and)g(\003)2061 2892 y Fr(2)2101 2878 y Fw(,)h(the)g(n)m(um)m(b)s(ers)e(of)i(p)s(oin)m (ts)f(observ)m(ed)g(in)g(\003)3575 2892 y Fr(1)393 2991 y Fw(and)j(in)g(\003)738 3005 y Fr(2)808 2991 y Fw(are)i(indep)s(enden) m(t.)165 3179 y(F)-8 b(or)28 b(a)f(construction)f(of)h(suc)m(h)g(a)g (pro)s(cess,)g(w)m(e)h(\014rst)e(consider)f(a)j(b)s(ounded)c(Borel)j (set)g(\003)g(of)g Fx(R)3376 3146 y Fu(d)3417 3179 y Fw(.)40 b(Let)165 3291 y(\012)231 3305 y Fr(\003)315 3291 y Fw(b)s(e)30 b(the)i(set)f(of)g(all)f(\014nite)g(subsets)h(of)g (\003.)43 b(A)31 b(P)m(oisson)f(pro)s(cess)h(on)g(\003)g(with)f(in)m (tensit)m(y)g Ft(\025)d(>)f Fw(0)31 b(is)165 3404 y(then)f(giv)m(en)h (b)m(y)f(a)h(random)e(elemen)m(t)i(of)g(\012)1651 3418 y Fr(\003)1734 3404 y Fw(ha)m(ving)f(distribution)c Ft(\031)2574 3419 y Fu(\025;)p Fr(\003)2688 3404 y Fw(,)31 b(where)743 3674 y Ft(\031)795 3689 y Fu(\025;)p Fr(\003)909 3674 y Fw(\()p Ft(F)13 b Fw(\))26 b(=)f Ft(e)1214 3637 y Fq(\000)p Fu(\025)p Fq(j)p Fr(\003)p Fq(j)1474 3569 y(1)1449 3594 y Fs(X)1443 3775 y Fu(n)p Fr(=0)1616 3613 y Ft(\025)1669 3580 y Fu(n)p 1616 3653 100 4 v 1626 3737 a Ft(n)p Fw(!)1741 3559 y Fs(Z)1839 3674 y Fv(\001)15 b(\001)g(\001)1961 3559 y Fs(Z)2059 3674 y Ft(I)2099 3693 y Fq(ff)p Fu(x)2209 3702 y Fl(1)2244 3693 y Fu(;:::)n(;x)2382 3701 y Fo(n)2424 3693 y Fq(g2)p Fu(F)10 b Fq(g)2615 3674 y Ft(dx)2714 3688 y Fr(1)2769 3674 y Fv(\001)15 b(\001)g(\001)h Ft(dx)2989 3688 y Fu(n)165 3946 y Fw(for)35 b(all)f Ft(F)47 b Fv(2)33 b(F)704 3960 y Fr(\003)757 3946 y Fw(,)k(the)e(smallest)g Ft(\033)s Fw(-\014eld)f(whic)m(h)g(allo)m(ws)g(us)h(to)h(coun)m(t)g (the)f(n)m(um)m(b)s(er)f(of)h(p)s(oin)m(ts)f(in)165 4059 y(eac)m(h)e(Borel)e(subset)g(of)h(\003.)306 4172 y(Next,)36 b(let)d(\012)g(b)s(e)f(the)i(set)g(of)f(all)f(lo)s(cally)g(\014nite)g (p)s(oin)m(t)h(con\014gurations)f(on)i Fx(R)3021 4139 y Fu(d)3061 4172 y Fw(;)h(lo)s(cally)d(\014nite)165 4284 y(means)k(that)g(an)m(y)f(b)s(ounded)e(set)j(con)m(tains)g(only)e (\014nitely)g(man)m(y)h(p)s(oin)m(ts.)55 b(The)35 b(P)m(oisson)g(pro)s (cess)165 4397 y(on)30 b Fx(R)369 4364 y Fu(d)440 4397 y Fw(with)f(in)m(tensit)m(y)h Ft(\025)g Fw(is)f(a)i(random)e(p)s(oin)m (t)g(con\014guration)h Ft(X)38 b Fw(distributed)27 b(according)j(to)h (the)165 4510 y(unique)h(probabilit)m(y)e(measure)k Ft(\031)1341 4525 y Fu(\025)1419 4510 y Fw(on)f(\012)g(whic)m(h,)g(when)f(pro)5 b(jected)34 b(on)f(an)m(y)g(b)s(ounded)e(Borel)j(set)165 4623 y(\003)26 b Fv(\032)f Fx(R)428 4590 y Fu(d)469 4623 y Fw(,)30 b(yields)e Ft(\031)831 4638 y Fu(\025;)p Fr(\003)944 4623 y Fw(.)41 b(Prop)s(erties)29 b(\(i\))g(and)h(\(ii\))f(ab)s(o)m(v)m (e)i(are)f(easily)f(v)m(eri\014ed,)g(and)h(mak)m(e)g(P)m(oisson)165 4736 y(pro)s(cesses)h(the)f(natural)g(analogues)g(of)h(Bernoulli)d (measures)i(for)g(lattice)h(mo)s(dels.)306 4849 y(T)-8 b(o)25 b(study)f(p)s(ercolation)f(prop)s(erties)f(of)j(the)f(P)m (oisson)g(pro)s(cess)g Ft(X)7 b Fw(,)26 b(w)m(e)f(need)f(to)h(in)m(tro) s(duce)e(some)165 4962 y(notion)32 b(of)g(connectivit)m(y)-8 b(.)47 b(A)32 b(natural)g(w)m(a)m(y)h(is)e(to)i(imagine)e(a)i(closed)f (Euclidean)e(ball)h Ft(B)5 b Fw(\()p Ft(x;)15 b(R)q Fw(\))32 b(of)165 5075 y(\014xed)25 b(radius)f Ft(R)j Fw(around)e(eac)m(h)h(p)s (oin)m(t)f Ft(x)h Fw(of)g(the)g(P)m(oisson)f(pro)s(cess,)i(whic)m(h)d (leads)h(us)g(to)i(considering)165 5188 y(the)f(random)e(subset)944 5165 y(\026)918 5188 y Ft(X)33 b Fw(=)1122 5124 y Fs(S)1191 5211 y Fu(x)p Fq(2)p Fu(X)1360 5188 y Ft(B)5 b Fw(\()p Ft(x;)15 b(R)q Fw(\))25 b(of)h Fx(R)1868 5155 y Fu(d)1908 5188 y Fw(.)39 b(Suc)m(h)25 b(random)f(subsets)g(are)h(widely)e(kno)m (wn)i(as)165 5301 y Fp(Bo)-5 b(ole)g(an)27 b(mo)-5 b(dels)p Fw(.)40 b(\(More)22 b(generally)-8 b(,)23 b Ft(B)5 b Fw(\()p Ft(x;)15 b(R)q Fw(\))22 b(could)f(b)s(e)g(replaced)f(b)m(y)i(a) g(closed)f(compact)i(random)165 5414 y(set)37 b(cen)m(tered)g(at)f Ft(x)p Fw(.\))58 b(This)34 b(particular)h(Bo)s(olean)h(mo)s(del)f(is)g (often)h(referred)f(to)i(as)f(the)g Fp(Poisson)165 5527 y(blob)g(mo)-5 b(del)36 b Fw(or)e Fp(lily)i(p)-5 b(ond)38 b(mo)-5 b(del)p Fw(.)53 b(Tw)m(o)34 b(p)s(oin)m(ts)f Ft(x;)d(y)k Fv(2)d Ft(X)41 b Fw(are)35 b(then)e(considered)g(as)h (connected)165 5639 y(to)f(eac)m(h)h(other)f(if)e(they)i(are)f (connected)i(in)1731 5616 y(\026)1705 5639 y Ft(X)7 b Fw(,)33 b(meaning)f(that)h(there)f(exists)h(a)f(con)m(tin)m(uous)g (path)1822 5895 y(102)p eop %%Page: 103 103 103 102 bop 165 304 a Fw(from)32 b Ft(x)f Fw(to)h Ft(y)j Fw(in)790 281 y(\026)764 304 y Ft(X)8 b Fw(.)44 b(By)32 b(scaling,)f(there)h(is)f(no)g(loss)g(of)h(generalit)m(y)g(in)e (setting)i Ft(R)c Fw(=)f(1)p Ft(=)p Fw(2,)34 b(so)e(that)165 417 y(t)m(w)m(o)39 b(balls)d(cen)m(tered)j(at)f Ft(x)g Fw(and)f Ft(y)j Fw(in)m(tersect)e(if)f(and)g(only)f(if)h Fv(j)p Ft(x)25 b Fv(\000)g Ft(y)s Fv(j)37 b(\024)g Fw(1,)j(where)d Fv(j)26 b(\001)f(j)38 b Fw(denotes)165 530 y(Euclidean)33 b(distance.)52 b(The)33 b(basic)h(result)f(on)h(Bo)s(olean)h(con)m(tin) m(uum)e(p)s(ercolation,)i(analogous)f(to)165 643 y(Theorem)c(5.3)i(for) e(ordinary)f(site)h(p)s(ercolation,)g(is)f(the)i(follo)m(wing.)165 830 y Fx(Theorem)j(10.1)47 b Fp(Pick)37 b(a)i(Poisson)g(pr)-5 b(o)g(c)g(ess)40 b Ft(X)46 b Fp(on)38 b Fx(R)2130 797 y Fu(d)2171 830 y Fp(,)i Ft(d)c Fv(\025)f Fw(2)p Fp(,)40 b(with)f(intensity)f Ft(\025)p Fp(.)59 b(L)-5 b(et)3452 807 y Fw(\026)3426 830 y Ft(X)42 b Fw(=)165 879 y Fs(S)235 966 y Fu(x)p Fq(2)p Fu(X)404 943 y Ft(B)5 b Fw(\()p Ft(x;)15 b Fw(1)p Ft(=)p Fw(2\))30 b Fp(b)-5 b(e)28 b(the)h(asso)-5 b(ciate)g(d)30 b(Bo)-5 b(ole)g(an)31 b(mo)-5 b(del,)30 b(and)f(let)g Ft(\022)s Fw(\()p Ft(\025)p Fw(\))f Fp(denote)h(the)f(pr) -5 b(ob)g(ability)31 b(that)165 1056 y(the)44 b(origin)f(b)-5 b(elongs)43 b(to)h(an)f(unb)-5 b(ounde)g(d)44 b(c)-5 b(onne)g(cte)g(d)45 b(c)-5 b(omp)g(onent)45 b(of)2668 1033 y Fw(\026)2642 1056 y Ft(X)7 b Fp(.)73 b(Then)43 b(ther)-5 b(e)44 b(exists)f(a)165 1169 y(critic)-5 b(al)34 b(value)f Ft(\025)759 1183 y Fu(c)818 1169 y Fw(=)25 b Ft(\025)967 1183 y Fu(c)1002 1169 y Fw(\()p Ft(d)p Fw(\))h Fv(2)f Fw(\(0)p Ft(;)15 b Fv(1)p Fw(\))34 b Fp(such)f(that)h Ft(\022)s Fw(\()p Ft(\025)p Fw(\))25 b(=)g(0)33 b Fp(if)f Ft(\025)26 b(<)f(\025)2588 1183 y Fu(c)2655 1169 y Fp(and)33 b Ft(\022)s Fw(\()p Ft(\025)p Fw(\))26 b Ft(>)f Fw(0)33 b Fp(if)f Ft(\025)25 b(>)g(\025)3515 1183 y Fu(c)3550 1169 y Fp(.)165 1357 y Fw(The)40 b(standard)f(pro)s(of)g(of)h(this)f (result)g(is)g(based)h(on)f(a)i(partitioning)c(of)k Fx(R)2847 1324 y Fu(d)2927 1357 y Fw(in)m(to)f(small)f(cub)s(es,)165 1470 y(reducing)29 b(the)i(problem)e(to)i(its)f(lattice)g(analogue,)h (Theorem)f(5.3;)i(see)f([173)r(])f(or)h([108)q(].)306 1583 y(Another)f(con)m(tin)m(uum)f(p)s(ercolation)g(mo)s(del)g(is)f (the)i(so)g(called)f Fp(r)-5 b(andom)35 b(c)-5 b(onne)g(ction)33 b(mo)-5 b(del)p Fw(,)32 b(or)165 1695 y Fp(Poisson)25 b(r)-5 b(andom)26 b(e)-5 b(dge)24 b(mo)-5 b(del)p Fw(,)25 b(whic)m(h)19 b(w)m(as)i(in)m(tro)s(duced)e(b)m(y)i(M.)g(P)m(enrose)h ([188)q(].)38 b(Let)21 b Ft(g)29 b Fw(:)41 b([0)p Ft(;)15 b Fv(1)p Fw(\))26 b Fv(!)165 1808 y Fw([0)p Ft(;)15 b Fw(1])34 b(b)s(e)d(a)h(decreasing)f(function)f(with)h(b)s(ounded)e (supp)s(ort)h(\(that)j(is)d Ft(g)s Fw(\()p Ft(x)p Fw(\))f(=)e(0)32 b(when)f Ft(x)g Fw(exceeds)165 1921 y(some)23 b Ft(R)j(<)f Fv(1)p Fw(\).)38 b(The)22 b(random)f(connection)i(mo)s(del)e(with)g(in) m(tensit)m(y)g Ft(\025)h Fw(and)g(connectivit)m(y)g(function)165 2034 y Ft(g)27 b Fw(arises)22 b(b)m(y)g(taking)h(a)g(P)m(oisson)f(pro)s (cess)g Ft(X)31 b Fw(in)21 b Fx(R)1846 2001 y Fu(d)1909 2034 y Fw(with)h(in)m(tensit)m(y)g Ft(\025)g Fw(and)h(indep)s(enden)m (tly)c(dra)m(wing)165 2147 y(an)k(edge)g(b)s(et)m(w)m(een)g(eac)m(h)h (pair)e(of)g(p)s(oin)m(ts)g Ft(x)h Fw(and)f Ft(y)j Fw(of)e Ft(X)30 b Fw(with)21 b(probabilit)m(y)f Ft(g)s Fw(\()p Fv(j)p Ft(x)5 b Fv(\000)g Ft(y)s Fv(j)p Fw(\).)39 b(This)21 b(setting)165 2260 y(includes)31 b(the)j(Bo)s(olean)f(mo)s(del,)h(whic) m(h)e(corresp)s(onds)f(to)j(the)g(c)m(hoice)g Ft(g)g Fw(=)c Ft(I)2858 2279 y Fr([0)p Fu(;)p Fr(1])2991 2260 y Fw(.)50 b(Theorem)33 b(10.1)165 2373 y(extends)c(to)h(this)e(mo)s (del:)39 b(F)-8 b(or)29 b Ft(g)k Fw(as)c(ab)s(o)m(v)m(e)h(with)1895 2304 y Fs(R)1950 2330 y Fq(1)1934 2400 y Fr(0)2040 2373 y Ft(g)s Fw(\()p Ft(x)p Fw(\))p Ft(dx)d(>)e Fw(0)k(and)f(dimensions)f Ft(d)e Fv(\025)g Fw(2,)30 b(there)165 2486 y(is)25 b(a)i(critical)d(v) -5 b(alue)26 b Ft(\025)906 2500 y Fu(c)966 2486 y Fw(=)f Ft(\025)1115 2500 y Fu(c)1149 2486 y Fw(\()p Ft(d;)15 b(g)s Fw(\))28 b(suc)m(h)e(that)g(in\014nite)e(connected)j(comp)s(onen) m(ts)f(a.s.)g(o)s(ccur)g(\(resp.)165 2599 y(do)37 b(not)f(o)s(ccur\))g (whenev)m(er)h Ft(\025)e(>)g(\025)1404 2613 y Fu(c)1475 2599 y Fw(\(resp.)h Ft(\025)f(<)g(\025)1981 2613 y Fu(c)2015 2599 y Fw(\))i(in)e(the)h(random)g(connection)g(mo)s(del)f(with)165 2712 y(in)m(tensit)m(y)30 b Ft(\025)g Fw(and)g(connectivit)m(y)h (function)e Ft(g)s Fw(.)306 2825 y(Muc)m(h)39 b(of)f(the)g(theory)g(of) g(standard)f(\(lattice\))i(p)s(ercolation)e(has)g(analogues)i(for)e (these)i(con-)165 2937 y(tin)m(uum)c(mo)s(dels.)55 b(An)36 b(example)g(is)e(the)j(uniqueness)c(of)j(the)g(in\014nite)e(cluster)h (\(Theorem)h(5.17\),)165 3050 y(whic)m(h)31 b(go)s(es)i(through)f(for)g (the)g(Bo)s(olean)h(and)f(random)f(connection)i(mo)s(dels.)45 b(See)32 b([173)r(])g(for)g(this)165 3163 y(and)e(m)m(uc)m(h)h(more.) 165 3407 y FA(10.2)113 b(The)37 b(con)m(tin)m(uum)i(Widom{Ro)m(wlinson) f(mo)s(del)165 3578 y Fw(The)23 b(con)m(tin)m(uum)e(Widom{Ro)m(wlinson) g(mo)s(del)h(is)g(a)h(mark)m(ed)g(p)s(oin)m(t)f(pro)s(cess)g(where)g (the)h(p)s(oin)m(ts)f(are)165 3691 y(of)k(t)m(w)m(o)g(t)m(yp)s(es:)38 b(w)m(e)26 b(call)e(them)h(plus-p)s(oin)m(ts)d(and)j(min)m(us-p)s(oin)m (ts.)36 b(\(F)-8 b(rom)26 b(no)m(w)f(on)g(and)f(throughout)165 3804 y(this)e(section,)i(w)m(e)f(drop)f(the)g(term)h(\\con)m(tin)m (uum")f(when)g(refering)f(to)j(this)d(mo)s(del,)i(and)f(instead)g(add) 165 3917 y(the)28 b(w)m(ord)e(\\lattice")j(when)d(talking)g(ab)s(out)h (the)h(mo)s(del)e(of)h(Section)g(3.5.\))41 b(F)-8 b(or)28 b(the)f(mo)s(del)f(de\014ned)165 4030 y(on)33 b(a)f(region)g(\003)d Fv(\022)f Fx(R)915 3997 y Fu(d)988 4030 y Fw(realizations)k(tak)m(e)h (v)-5 b(alues)32 b(in)f(\012)2116 4044 y Fr(\003)2190 4030 y Fv(\002)21 b Fw(\012)2348 4044 y Fr(\003)2401 4030 y Fw(;)34 b(the)e(\014rst)g(co)s(ordinate)g(describ)s(es)165 4143 y(the)h(lo)s(cations)f(of)h(the)f(plus-p)s(oin)m(ts,)f(and)h(the)h (second)f(co)s(ordinate)h(the)f(min)m(us-p)s(oin)m(ts.)45 b(There)32 b(is)165 4256 y(a)d(hard)d(sphere)i(in)m(teraction)f(prev)m (en)m(ting)h(t)m(w)m(o)h(p)s(oin)m(ts)e(from)h(coming)f(within)e (Euclidean)h(distance)165 4369 y Ft(R)40 b Fw(from)e(eac)m(h)i(other;)j (again,)f(w)m(e)d(set)g Ft(R)h Fw(=)f(1)g(without)f(loss)g(of)h (generalit)m(y)-8 b(.)66 b(This)37 b(in)m(teraction)165 4482 y(corresp)s(onds)29 b(to)i(the)g(Hamiltonian)1264 4686 y Ft(H)7 b Fw(\()p Fx(x)p Ft(;)15 b Fx(y)q Fw(\))27 b(=)1770 4605 y Fs(X)1691 4781 y Fu(x)p Fq(2)p Fn(x)p Fu(;y)r Fq(2)p Fn(y)1983 4686 y Fv(1)15 b Ft(I)2129 4704 y Fq(fj)p Fu(x)p Fq(\000)p Fu(y)r Fq(j\024)p Fr(1)p Fq(g)2491 4686 y Ft(;)165 4964 y Fx(x)p Ft(;)g Fx(y)28 b Fv(2)d Fw(\012)495 4978 y Fr(\003)563 4964 y Fw(.)306 5077 y(When)h(\003)g(is) f(b)s(ounded,)g(the)h(Widom{Ro)m(wlinson)e(mo)s(del)h(on)h(\003)g(with) f(in)m(tensit)m(y)g Ft(\025)h Fw(is)f(obtained)165 5190 y(b)m(y)31 b(conditioning)e(the)i(P)m(oisson)g(pro)s(duct)f(measure)h Ft(\031)2048 5205 y Fu(\025;)p Fr(\003)2182 5190 y Fv(\002)21 b Ft(\031)2326 5205 y Fu(\025;)p Fr(\003)2470 5190 y Fw(on)31 b(the)h(ev)m(en)m(t)g(that)g(there)f(is)f(no)165 5302 y(pair)39 b(of)g(p)s(oin)m(ts)f(of)i(opp)s(osite)f(t)m(yp)s(e)g (within)e(unit)h(distance)h(from)g(eac)m(h)h(other.)69 b(The)38 b(extension)165 5415 y(to)33 b Fx(R)356 5382 y Fu(d)429 5415 y Fw(is)e(done)g(in)g(the)h(usual)f(DLR)h(fashion:)42 b(a)32 b(probabilit)m(y)e(measure)i Ft(\026)f Fw(on)h(\012)21 b Fv(\002)g Fw(\012)31 b(is)g(a)i(Gibbs)165 5528 y(measure)25 b(for)g(the)g(Widom{Ro)m(wlinson)e(mo)s(del)h(at)i(in)m(tensit)m(y)e Ft(\025)h Fw(if,)g(for)g(an)m(y)g(b)s(ounded)e(Borel)i(set)h(\003,)165 5641 y(the)d(conditional)e(distribution)e(of)k(the)g(p)s(oin)m(t)e (con\014guration)h(on)h(\003)g(giv)m(en)f(the)h(p)s(oin)m(t)f (con\014guration)1822 5895 y(103)p eop %%Page: 104 104 104 103 bop 165 304 a Fw(on)26 b Fx(R)365 271 y Fu(d)417 304 y Fv(n)11 b Fw(\003)27 b(is)e(that)h(of)g(t)m(w)m(o)h(indep)s (enden)m(t)d(P)m(oisson)h(pro)s(cesses)h(conditioned)f(on)g(the)h(ev)m (en)m(t)i(that)e(no)165 417 y(p)s(oin)m(t)g(in)g(\003)h(is)e(placed)i (within)d(unit)h(distance)i(from)f(a)h(p)s(oin)m(t)f(of)h(the)g(opp)s (osite)f(t)m(yp)s(e,)i(either)e(inside)165 530 y(or)j(outside)f(\003.) 40 b(The)28 b(resem)m(blence)h(with)e(the)i(lattice)g(Widom{Ro)m (wlinson)e(mo)s(del)h(of)h(Section)f(3.5)165 643 y(is)i(eviden)m(t.)40 b(W)-8 b(e)32 b(ha)m(v)m(e)f(the)g(follo)m(wing)e(analogue)i(of)f (Theorem)g(3.4.)165 830 y Fx(Theorem)k(10.2)47 b Fp(F)-7 b(or)34 b(the)f(Widom{R)-5 b(ow)5 b(linson)37 b(mo)-5 b(del)34 b(on)g Fx(R)2407 797 y Fu(d)2448 830 y Fp(,)f Ft(d)26 b Fv(\025)g Fw(2)p Fp(,)33 b(with)h(activity)f Ft(\025)p Fp(,)g(ther)-5 b(e)165 943 y(exist)35 b(c)-5 b(onstants)36 b Fw(0)28 b Ft(<)g(\025)1005 910 y Fq(0)1005 966 y Fu(c)1068 943 y Fv(\024)g Ft(\025)1220 910 y Fq(00)1220 966 y Fu(c)1290 943 y Ft(<)g Fv(1)34 b Fp(\(dep)-5 b(ending)35 b(on)g Ft(d)p Fp(\))g(such)f(that)h(for)g Ft(\025)28 b(<)g(\025)3001 910 y Fq(0)3001 966 y Fu(c)3070 943 y Fp(the)34 b(mo)-5 b(del)36 b(has)165 1056 y(a)d(unique)f(Gibbs)h(me)-5 b(asur)g(e,)33 b(while)h(for)f Ft(\025)25 b(>)g(\025)1773 1023 y Fq(00)1773 1079 y Fu(c)1848 1056 y Fp(ther)-5 b(e)34 b(ar)-5 b(e)33 b(multiple)h(Gibbs)e(me)-5 b(asur)g(es.)165 1244 y Fw(The)35 b(pro)s(of)g(of)g(this)f(result)h(splits)e(naturally)g (in)m(to)j(t)m(w)m(o)g(parts:)51 b(\014rst,)36 b(w)m(e)f(need)g(to)i (demonstrate)165 1357 y(uniqueness)20 b(of)j(Gibbs)e(measures)h(for)g Ft(\025)h Fw(su\016cien)m(tly)e(small,)h(and)g(secondly)g(w)m(e)h(need) f(to)h(sho)m(w)f(non-)165 1470 y(uniqueness)j(for)h Ft(\025)h Fw(su\016cien)m(tly)e(large.)40 b(The)26 b(\014rst)g(half)g(can)h(b)s (e)f(done)h(b)m(y)f(a)h(v)-5 b(ariet)m(y)27 b(of)g(tec)m(hniques.)165 1583 y(F)-8 b(or)27 b(instance,)f(one)g(can)g(partition)e Fx(R)1475 1550 y Fu(d)1542 1583 y Fw(in)m(to)i(cub)s(es)e(of)i(unit)e (sidelength)g(and)h(apply)g(disagreemen)m(t)165 1695 y(p)s(ercolation)36 b(\(Theorem)g(7.1\).)61 b(Tw)m(o)36 b(observ)-5 b(ations)36 b(are)h(crucial)e(in)g(order)h(to)h(mak)m(e)g (this)f(w)m(ork:)165 1808 y(that)23 b(the)f(conditional)e(distribution) f(of)j(the)g(con\014guration)f(in)g(suc)m(h)h(a)g(cub)s(e)f(giv)m(en)h (ev)m(erything)g(else)165 1921 y(only)34 b(dep)s(ends)f(on)i(the)g (con\014gurations)f(in)g(its)g(neigh)m(b)s(oring)f(cub)s(es,)i(and)f (that)i(the)f(conditional)165 2034 y(probabilit)m(y)g(of)i(seeing)f(no) h(p)s(oin)m(t)e(at)j(all)e(in)f(a)i(cub)s(e)f(tends)g(to)i(1)f(as)g Ft(\025)f Fv(!)g Fw(0,)j(uniformly)33 b(in)j(the)165 2147 y(neigh)m(b)s(ors')29 b(con\014gurations.)306 2260 y(The)h(more)h(di\016cult)d(part,)i(the)g(non)m(uniqueness)e(for)i (large)g Ft(\025)p Fw(,)h(w)m(as)f(\014rst)g(obtained)f(b)m(y)h(Ruelle) 165 2373 y([204)r(])24 b(using)e(a)i(P)m(eierls-t)m(yp)s(e)f(argumen)m (t.)39 b(Here)24 b(w)m(e)h(shall)c(sk)m(etc)m(h)k(a)g(mo)s(dern)d(sto)s (c)m(hastic-geometric)165 2486 y(approac)m(h)g(using)e(a)h (random-cluster)g(represen)m(tation.)37 b(This)20 b(approac)m(h)i(is)e (due)h(mainly)e(to)j(Cha)m(y)m(es,)165 2599 y(Cha)m(y)m(es)47 b(and)d(Kotec)m(k)q(\023)-47 b(y)49 b([54)q(])c(\(but)h(see)g(also)f ([100)r(]\),)50 b(and)45 b(w)m(orks)g(in)f(sho)m(wing)h(b)s(oth)g (parts)g(of)165 2712 y(Theorem)30 b(3.4.)42 b(The)30 b(so)h(called)e Fp(c)-5 b(ontinuum)34 b(r)-5 b(andom-cluster)35 b(mo)-5 b(del)32 b Fw(is)d(de\014ned)g(as)i(follo)m(ws.)165 2924 y Fx(De\014nition)36 b(10.3)46 b Fp(The)31 b Fx(con)m(tin)m(uum)j (random-cluster)f(distribution)e Ft(\036)2931 2939 y Fu(\025;)p Fr(\003)3076 2924 y Fp(with)i(intensity)165 3037 y Ft(\025)g Fp(for)g(the)g(c)-5 b(omp)g(act)35 b(r)-5 b(e)g(gion)34 b Fw(\003)25 b Fv(\032)g Fx(R)1431 3004 y Fu(d)1504 3037 y Fp(is)33 b(the)g(pr)-5 b(ob)g(ability)35 b(me)-5 b(asur)g(e)34 b(on)f Fw(\012)2739 3051 y Fr(\003)2824 3037 y Fp(with)h(density)1314 3274 y Ft(f)10 b Fw(\()p Fx(x)p Fw(\))25 b(=)1690 3213 y(1)p 1625 3253 176 4 v 1625 3337 a Ft(Z)1687 3352 y Fu(\025;)p Fr(\003)1836 3274 y Fw(2)1881 3237 y Fu(k)r Fr(\()p Fn(x)p Fr(\))2047 3274 y Ft(;)109 b Fx(x)25 b Fv(2)g Fw(\012)2413 3288 y Fr(\003)3453 3274 y Fw(\(74\))165 3527 y Fp(with)34 b(r)-5 b(esp)g(e)g(ct)35 b(to)f(the)f(Poisson)h(pr)-5 b(o)g(c)g(ess)35 b Ft(\031)1622 3542 y Fu(\025;)p Fr(\003)1769 3527 y Fp(of)e(intensity)h Ft(\025)p Fp(;)f(her)-5 b(e)34 b Ft(Z)2624 3542 y Fu(\025;)p Fr(\003)2771 3527 y Fp(is)f(a)g (normalizing)i(c)-5 b(on-)165 3640 y(stant)34 b(and)g Ft(k)s Fw(\()p Fx(x)p Fw(\))f Fp(is)g(the)g(numb)-5 b(er)33 b(of)g(c)-5 b(onne)g(cte)g(d)34 b(c)-5 b(omp)g(onents)35 b(of)e(the)g(set)2764 3639 y Fw(\026)2759 3640 y Fx(x)25 b Fw(=)2935 3576 y Fs(S)3004 3663 y Fu(x)p Fq(2)p Fn(x)3153 3640 y Ft(B)5 b Fw(\()p Ft(x;)15 b Fw(1)p Ft(=)p Fw(2\))p Fp(.)165 3853 y Fw(In)29 b(analogy)i(to)f(the)g(corresp)s(ondence)f(b)s (et)m(w)m(een)i(the)f(lattice)g(Widom{Ro)m(wlinson)e(mo)s(del)h(and)g (its)165 3966 y(random-cluster)i(represen)m(tation)h(in)f(Prop)s (ositions)f(6.22)k(and)d(6.23,)k(w)m(e)d(obtain)g(the)g(con)m(tin)m (uum)165 4079 y(random-cluster)37 b(mo)s(del)g(b)m(y)g(simply)f (disregarding)f(the)j(t)m(yp)s(es)g(of)g(the)g(p)s(oin)m(ts)f(in)f(the) i(Widom{)165 4192 y(Ro)m(wlinson)c(mo)s(del,)h(with)f(the)h(same)h(c)m (hoice)g(of)f(the)g(parameter)h Ft(\025)p Fw(.)55 b(Con)m(v)m(ersely)-8 b(,)37 b(the)e(Widom{)165 4304 y(Ro)m(wlinson)21 b(mo)s(del)g(is)g (obtained)h(when)f(the)h(connected)i(comp)s(onen)m(ts)e(in)f(the)h(con) m(tin)m(uum)g(random-)165 4417 y(cluster)29 b(mo)s(del)f(are)h (assigned)g(indep)s(enden)m(t)d(t)m(yp)s(es,)k(plus)d(or)i(min)m(us)f (with)g(probabilit)m(y)e(1)p Ft(=)p Fw(2)31 b(eac)m(h.)165 4530 y(T)-8 b(o)37 b(see)f(wh)m(y)f(this)g(is)g(true,)i(note)g(that)f (for)f(an)m(y)i Fx(x)d Fv(2)g Fw(\012)2140 4544 y Fr(\003)2228 4530 y Fw(there)i(are)h(exactly)f(2)2987 4497 y Fu(k)r Fr(\()p Fn(x)p Fr(\))3164 4530 y Fw(elemen)m(ts)g(of)165 4643 y(\012)231 4657 y Fr(\003)306 4643 y Fv(\002)22 b Fw(\012)465 4657 y Fr(\003)551 4643 y Fw(whic)m(h)32 b(do)h(not)g(con)m(tradict)h(the)f(hard)f(sphere)h(condition)e(of)j (the)f(Widom{Ro)m(wlinson)165 4756 y(mo)s(del)d(and)f(whic)m(h)g(map)h (in)m(to)h Fx(x)f Fw(when)g(w)m(e)g(disregard)f(the)i(t)m(yp)s(es)f(of) h(the)g(p)s(oin)m(ts.)306 4869 y(Besides)d(the)h(random-cluster)e (represen)m(tation,)i(w)m(e)f(can)h(also)f(tak)m(e)i(adv)-5 b(an)m(tage)30 b(of)e(sto)s(c)m(hastic)165 4982 y(monotonicit)m(y)j (prop)s(erties.)39 b(Let)31 b(us)f(de\014ne)f(a)i(partial)e(order)h Fv(\026)g Fw(on)g(\012)20 b Fv(\002)g Fw(\012)30 b(b)m(y)g(setting)1077 5186 y(\()p Fx(x)p Ft(;)15 b Fx(y)q Fw(\))27 b Fv(\026)e Fw(\()p Fx(x)1511 5149 y Fq(0)1535 5186 y Ft(;)15 b Fx(y)1631 5149 y Fq(0)1655 5186 y Fw(\))95 b(if)e Fx(x)26 b Fv(\022)f Fx(x)2164 5149 y Fq(0)2218 5186 y Fw(and)k Fx(y)e Fv(\023)e Fx(y)2628 5149 y Fq(0)2677 5186 y Ft(;)751 b Fw(\(75\))165 5390 y(so)22 b(that)f(in)f(other)h(w)m(ords)g(a)g(con\014guration)g (increases)g(with)e(resp)s(ect)i(to)h Fv(\026)f Fw(if)f(plus-p)s(oin)m (ts)e(are)k(added)165 5503 y(and)28 b(min)m(us-p)s(oin)m(ts)e(are)i (deleted.)40 b(A)29 b(straigh)m(tforw)m(ard)f(extension)g(of)g(Theorem) g(10.4)i(b)s(elo)m(w)d(then)165 5616 y(implies)41 b(that)k(the)f(Gibbs) e(distributions)e(for)k(the)g(Widom{Ro)m(wlinson)e(mo)s(del)h(ha)m(v)m (e)i(p)s(ositiv)m(e)1822 5895 y(104)p eop %%Page: 105 105 105 104 bop 165 304 a Fw(correlations)35 b(relativ)m(e)h(to)g(this)e (order.)55 b(The)35 b(metho)s(ds)f(of)i(Sections)f(4.1)h(and)f(4.3)h (can)g(therefore)165 417 y(b)s(e)27 b(adapted)h(to)g(sho)m(w)g(that)g (the)g(Widom{Ro)m(wlinson)d(mo)s(del)i(on)g Fx(R)2551 384 y Fu(d)2619 417 y Fw(at)i(in)m(tensit)m(y)e Ft(\025)g Fw(admits)g(t)m(w)m(o)165 530 y(particular)j(phases)h Ft(\026)931 492 y Fr(+)931 558 y Fu(\025)1022 530 y Fw(and)g Ft(\026)1255 492 y Fq(\000)1255 558 y Fu(\025)1314 530 y Fw(,)h(where)f Ft(\026)1690 492 y Fr(+)1690 558 y Fu(\025)1780 530 y Fw(is)g(obtained)g(as)h(a)g(w)m(eak)g(limit)e(of)h(Gibbs)f (measures)165 643 y(on)i(compact)i(sets)f(\(tending)e(to)i Fx(R)1395 610 y Fu(d)1436 643 y Fw(\))f(with)f(b)s(oundary)f(condition) h(consisting)g(of)i(a)f(dense)g(cro)m(wd)165 756 y(of)f(plus-p)s(oin)m (ts,)d(and)i Ft(\026)992 717 y Fq(\000)992 784 y Fu(\025)1081 756 y Fw(is)f(obtained)h(similarly)-8 b(.)38 b(W)-8 b(e)31 b(also)f(ha)m(v)m(e)i(the)f(sandwic)m(hing)d(relation)1567 960 y Ft(\026)1622 922 y Fq(\000)1622 988 y Fu(\025)1706 960 y Fv(\026)1777 974 y Fq(D)1862 960 y Ft(\026)d Fv(\026)2013 974 y Fq(D)2099 960 y Ft(\026)2154 922 y Fr(+)2154 988 y Fu(\025)3453 960 y Fw(\(76\))165 1164 y(for)35 b(an)m(y)g(Gibbs)e (measure)h Ft(\026)h Fw(for)f(the)h(in)m(tensit)m(y)f Ft(\025)g Fw(Widom{Ro)m(wlinson)f(mo)s(del)g(on)i Fx(R)3230 1131 y Fu(d)3271 1164 y Fw(,)h(so)f(that)165 1277 y(uniqueness)28 b(of)j(Gibbs)e(measures)h(is)f(equiv)-5 b(alen)m(t)30 b(to)h(ha)m(ving)f Ft(\026)2363 1239 y Fq(\000)2363 1305 y Fu(\025)2447 1277 y Fw(=)25 b Ft(\026)2598 1239 y Fr(+)2598 1305 y Fu(\025)2657 1277 y Fw(.)306 1390 y(The)37 b(Gibbs)e(measure)i (for)g(the)g(Widom{Ro)m(wlinson)e(mo)s(del)h(on)h(a)g(b)s(o)m(x)g(\003) g(with)f(\\plus")g(\(or)165 1503 y(\\min)m(us"\))21 b(b)s(oundary)e (condition)h(corresp)s(onds)g(to)i(the)f(wired)f(con)m(tin)m(uum)g (random-cluster)g(mo)s(del)165 1616 y Ft(\036)219 1583 y Fr(1)219 1642 y Fu(\025;)p Fr(\003)357 1616 y Fw(on)k(\003)f(where)g (all)g(connected)i(comp)s(onen)m(ts)e(within)e(distance)j(1)p Ft(=)p Fw(2)h(from)e(the)h(b)s(oundary)d(coun)m(t)165 1729 y(as)35 b(a)g(single)e(comp)s(onen)m(t.)54 b(Arguing)34 b(as)g(in)g(Sections)g(6.3)i(and)e(6.7)h(w)m(e)g(\014nd)e(that)j (uniqueness)c(of)165 1842 y(Gibbs)22 b(measures)h(for)g(the)h(Widom{Ro) m(wlinson)e(mo)s(del)g(is)g(equiv)-5 b(alen)m(t)23 b(to)h(not)g(ha)m (ving)f(an)m(y)g(in\014nite)165 1954 y(connected)g(comp)s(onen)m(ts)f (in)e(the)i(con)m(tin)m(uum)f(random-cluster)g(mo)s(del.)37 b(Let)22 b Ft(\025)2888 1968 y Fu(c)2945 1954 y Fw(b)s(e)f(as)h(in)e (Theorem)165 2067 y(10.1.)57 b(Theorem)35 b(10.2)i(follo)m(ws)d(if)g(w) m(e)i(can)g(sho)m(w)f(that)g(the)h(con)m(tin)m(uum)e(random-cluster)g (mo)s(del)165 2180 y Ft(\036)219 2147 y Fr(1)219 2207 y Fu(\025;)p Fr(\003)363 2180 y Fw(with)28 b(su\016cien)m(tly)f(large)j (in)m(tensit)m(y)e Ft(\025)i Fw(sto)s(c)m(hastically)f(dominates)f Ft(\031)2745 2195 y Fu(\025)2786 2204 y Fl(1)2821 2195 y Fu(;)p Fr(\003)2923 2180 y Fw(for)h(some)g Ft(\025)3340 2194 y Fr(1)3405 2180 y Ft(>)c(\025)3554 2194 y Fu(c)3589 2180 y Fw(,)165 2304 y(whereas)31 b Ft(\036)564 2271 y Fr(1)564 2330 y Fu(\025;)p Fr(\003)703 2304 y Fv(\026)774 2318 y Fq(D)859 2304 y Ft(\031)911 2319 y Fu(\025)952 2328 y Fl(2)987 2319 y Fu(;)p Fr(\003)1090 2304 y Fw(for)f(some)h Ft(\025)1510 2318 y Fr(2)1575 2304 y Ft(<)25 b(\025)1724 2318 y Fu(c)1789 2304 y Fw(when)k Ft(\025)h Fw(is)g(su\016cien)m(tly)f (small.)306 2417 y(T)-8 b(o)39 b(this)d(end)h(w)m(e)h(need)f(a)h(p)s (oin)m(t)f(pro)s(cess)g(analogue)h(of)g(Theorem)f(4.8,)k(whic)m(h)36 b(is)h(based)g(on)165 2530 y(the)32 b(concept)g(of)g Fp(Pap)-5 b(angelou)35 b(\(c)-5 b(onditional\))36 b(intensities)c Fw(for)f(p)s(oin)m(t)f(pro)s(cesses.)44 b(Supp)s(ose)29 b Ft(\026)i Fw(is)g(a)165 2642 y(probabilit)m(y)c(measure)h(on)h(\012) 1172 2656 y Fr(\003)1253 2642 y Fw(whic)m(h)f(is)f(absolutely)h(con)m (tin)m(uous)g(with)g(densit)m(y)g Ft(f)10 b Fw(\()p Fx(x)p Fw(\))29 b(relativ)m(e)f(to)165 2755 y(the)36 b(unit)f(in)m(tensit)m(y) g(P)m(oisson)g(pro)s(cess)h Ft(\031)1610 2769 y Fr(1)p Fu(;)p Fr(\003)1718 2755 y Fw(.)57 b(F)-8 b(or)36 b Ft(x)f Fv(2)f Fw(\003)h(and)h(a)g(p)s(oin)m(t)f(con\014guration)g Fx(x)f Fv(2)g Fw(\012)3561 2769 y Fr(\003)165 2868 y Fw(not)d(con)m(taining)f Ft(x)p Fw(,)g(the)h(P)m(apangelou)g(in)m (tensit)m(y)e(of)i Ft(\026)f Fw(at)h Ft(x)f Fw(giv)m(en)h Fx(x)f Fw(is,)g(if)f(it)h(exists,)1454 3115 y Ft(\025)p Fw(\()p Ft(x)p Fv(j)p Fx(x)p Fw(\))c(=)1841 3054 y Ft(f)10 b Fw(\()p Fx(x)20 b Fv([)g(f)p Ft(x)p Fv(g)p Fw(\))p 1841 3094 425 4 v 1963 3178 a Ft(f)10 b Fw(\()p Fx(x)p Fw(\))2300 3115 y Ft(:)1128 b Fw(\(77\))165 3368 y(Heuristically)-8 b(,)26 b Ft(\025)p Fw(\()p Ft(x)p Fv(j)p Fx(x)p Fw(\))p Ft(dx)i Fw(can)e(b)s(e)g(in)m(terpreted)g(as)h(the)f(probabilit)m(y)e (of)j(\014nding)d(a)j(p)s(oin)m(t)e(inside)f(an)165 3481 y(in\014nitesimal)j(region)j Ft(dx)h Fw(around)f Ft(x)p Fw(,)g(giv)m(en)h(that)g(the)g(p)s(oin)m(t)e(con\014guration)h(outside) g(this)f(region)165 3594 y(is)34 b Fx(x)p Fw(.)53 b(Alternativ)m(ely)-8 b(,)35 b Ft(\025)p Fw(\()p Fv(\001)15 b(j)g(\001)p Fw(\))36 b(can)f(b)s(e)e(c)m(haracterized)j(as)f(the)f(Radon{Nik)m(o)s(dym)g (densit)m(y)f(of)i(the)165 3707 y(measure)519 3638 y Fs(R)589 3707 y Ft(\026)p Fw(\()p Ft(d)p Fx(x)p Fw(\))831 3643 y Fs(P)920 3730 y Fu(x)p Fq(2)p Fn(x)1069 3707 y Ft(\016)1109 3725 y Fr(\()p Fu(x;)p Fn(x)p Fq(nf)p Fu(x)p Fq(g)p Fr(\))1446 3707 y Fw(on)29 b(\003)19 b Fv(\002)f Fw(\012)1808 3721 y Fr(\003)1860 3707 y Fw(,)30 b(the)g(so-called)f (reduced)g(Campb)s(ell)e(measure)i(of)165 3820 y Ft(\026)p Fw(,)34 b(relativ)m(e)e(to)i(the)f(Leb)s(esgue)f(measure)h(times)f Ft(\026)g Fw([98)r(].)47 b(It)33 b(is)f(easily)g(c)m(hec)m(k)m(ed)i (that)g(the)e(P)m(oisson)165 3933 y(pro)s(cess)e Ft(\031)534 3948 y Fu(\025;)p Fr(\003)678 3933 y Fw(has)g(P)m(apangelou)h(in)m (tensit)m(y)f Ft(\025)p Fw(\()p Ft(x)p Fv(j)p Fx(x)p Fw(\))c(=)f Ft(\025)p Fw(.)306 4046 y(The)35 b(follo)m(wing)e(p)s(oin)m (t)g(pro)s(cess)h(analogue)h(of)g(Theorem)g(4.8)g(w)m(as)g(pro)m(v)m (ed)g(b)m(y)g(Preston)f([199)r(])165 4159 y(under)h(an)i(additional)e (tec)m(hnical)h(assumption,)h(using)e(a)i(coupling)e(of)i(so)g(called)f (spatial)g(birth-)165 4271 y(and-death)g(pro)s(cesses)g(similar)d(to)j (the)g(coupling)e(used)h(in)f(the)i(pro)s(of)f(of)h(Theorem)f(4.8.)58 b(Later,)165 4384 y(the)31 b(full)d(result)h(w)m(as)i(pro)m(v)m(ed)g (in)e(Georgii)h(and)g(K)s(\177)-48 b(uneth)29 b([98)q(])i(b)m(y)f(a)h (discretization)e(argumen)m(t.)165 4572 y Fx(Theorem)34 b(10.4)47 b Fp(Supp)-5 b(ose)36 b Ft(\026)e Fp(and)43 b Fw(~)-52 b Ft(\026)34 b Fp(ar)-5 b(e)35 b(pr)-5 b(ob)g(ability)37 b(me)-5 b(asur)g(es)37 b(on)e Fw(\012)2732 4586 y Fr(\003)2819 4572 y Fp(with)g(Pap)-5 b(angelou)37 b(in-)165 4685 y(tensities)c Ft(\025)p Fw(\()p Fv(\001)15 b(j)g(\001)p Fw(\))34 b Fp(and)966 4661 y Fw(~)962 4685 y Ft(\025)p Fw(\()p Fv(\001)15 b(j)g(\001)p Fw(\))34 b Fp(satisfying)1573 4889 y Ft(\025)p Fw(\()p Ft(x)p Fv(j)p Fx(x)p Fw(\))26 b Fv(\024)1954 4865 y Fw(~)1950 4889 y Ft(\025)p Fw(\()p Ft(x)p Fv(j)2120 4888 y Fw(~)2115 4889 y Fx(x)q Fw(\))165 5093 y Fp(whenever)34 b Ft(x)27 b Fv(2)g Fw(\003)33 b Fp(and)i Fx(x)p Ft(;)1103 5092 y Fw(~)1098 5093 y Fx(x)27 b Fv(2)g Fw(\012)1334 5107 y Fr(\003)1420 5093 y Fp(ar)-5 b(e)34 b(such)g(that)h Fx(x)27 b Fv(\022)2154 5092 y Fw(~)2149 5093 y Fx(x)q Fp(.)44 b(Then)34 b Ft(\026)27 b Fv(\026)2668 5107 y Fq(D)2762 5093 y Fw(~)-52 b Ft(\026)p Fp(,)33 b(in)h(that)g(ther)-5 b(e)35 b(exists)165 5206 y(a)e(c)-5 b(oupling)34 b Fw(\()p Ft(X)r(;)779 5183 y Fw(~)752 5206 y Ft(X)8 b Fw(\))33 b Fp(of)g Ft(\026)f Fp(and)41 b Fw(~)-52 b Ft(\026)32 b Fp(such)h(that)h Ft(X)f Fv(\022)1984 5183 y Fw(~)1958 5206 y Ft(X)40 b Fp(a.s.)165 5394 y Fw(Plugging)35 b(\(74\))j(in)m(to)e (\(77\))i(w)m(e)f(\014nd)d(that)j(the)g(con)m(tin)m(uum)e (random-cluster)g(measure)h Ft(\036)3332 5409 y Fu(\025;)p Fr(\003)3482 5394 y Fw(has)165 5507 y(P)m(apangelou)31 b(in)m(tensit)m(y)1473 5620 y Ft(\025)p Fw(\()p Ft(x)p Fv(j)p Fx(x)p Fw(\))26 b(=)f Ft(\025)15 b Fw(2)1963 5582 y Fr(1)p Fq(\000)p Fu(\024)p Fr(\()p Fu(x;)p Fn(x)p Fr(\))2281 5620 y Ft(;)1147 b Fw(\(78\))1822 5895 y(105)p eop %%Page: 106 106 106 105 bop 165 304 a Fw(where)40 b Ft(\024)p Fw(\()p Ft(x;)15 b Fx(x)p Fw(\))42 b(is)e(the)h(n)m(um)m(b)s(er)e(of)i (connected)g(comp)s(onen)m(ts)g(of)2531 240 y Fs(S)2601 327 y Fu(y)r Fq(2)p Fn(x)2747 304 y Ft(B)5 b Fw(\()p Ft(y)s(;)15 b Fw(1)p Ft(=)p Fw(2\))42 b(in)m(tersecting)165 417 y Ft(B)5 b Fw(\()p Ft(x;)15 b Fw(1)p Ft(=)p Fw(2\).)40 b(It)22 b(is)e(a)i(simple)e(geometric)j(fact)f(that)g(there)g(exists)g (a)g(constan)m(t)h Ft(\024)2823 431 y Fu(max)2992 417 y Fw(=)i Ft(\024)3140 431 y Fu(max)3284 417 y Fw(\()p Ft(d)p Fw(\))h Ft(<)f Fv(1)165 530 y Fw(suc)m(h)34 b(that)g Ft(\024)p Fw(\()p Ft(x;)15 b Fx(x)p Fw(\))32 b Fv(\024)e Ft(\024)1028 544 y Fu(max)1206 530 y Fw(for)j(all)f Ft(x)i Fw(and)f Fx(x)p Fw(;)i(for)f Ft(d)c Fw(=)h(2)j(w)m(e)g(ma)m(y)g(tak)m (e)h Ft(\024)2848 544 y Fu(max)3023 530 y Fw(=)30 b(5.)50 b(It)34 b(follo)m(ws)165 643 y(that)1403 756 y Ft(\025)15 b Fw(2)1516 718 y Fr(1)p Fq(\000)p Fu(\024)1647 726 y Fo(max)1804 756 y Fv(\024)25 b Ft(\025)p Fw(\()p Ft(x)p Fv(j)p Fx(x)p Fw(\))i Fv(\024)e Fw(2)p Ft(\025)1077 b Fw(\(79\))165 923 y(for)34 b(all)f Ft(x)g Fw(and)h Fx(x)p Fw(.)51 b(Hence,)36 b(applying)31 b(Theorems)j(10.4)h(and)e(10.1)j(w)m (e)e(\014nd)e(that)j Ft(\036)3081 890 y Fr(1)3081 949 y Fu(\025;)p Fr(\003)3226 923 y Fv(\026)3297 937 y Fq(D)3388 923 y Ft(\031)3440 938 y Fr(2)p Fu(\025;)p Fr(\003)3589 923 y Fw(,)165 1035 y(so)40 b(that)g(for)f Ft(\025)i(<)f(\025)898 1049 y Fu(c)933 1035 y Ft(=)p Fw(2)g(w)m(e)g(obtain)f(the)h(absence)g (of)f(un)m(b)s(ounded)e(connected)j(comp)s(onen)m(ts)g(of)165 1084 y Fs(S)235 1171 y Fu(x)p Fq(2)p Fn(x)384 1148 y Ft(B)5 b Fw(\()p Ft(x;)15 b Fw(1)p Ft(=)p Fw(2\))24 b(in)d(the)i(limit) d(\003)25 b Fv(")h Fx(R)1472 1115 y Fu(d)1535 1148 y Fw(of)d(con)m(tin)m(uum)e(random-cluster)h(measures.)38 b(On)21 b(the)i(other)165 1261 y(hand,)36 b(taking)e Ft(\025)f(>)g(\025)951 1275 y Fu(c)1001 1261 y Fw(2)1046 1228 y Fu(\024)1087 1236 y Fo(max)1214 1228 y Fq(\000)p Fr(1)1343 1261 y Fw(yields)g(the)i(presence)g(of)g(un)m(b)s(ounded)d (connected)k(comp)s(onen)m(ts)165 1374 y(in)29 b(the)i(same)g(limit.)38 b(Theorem)30 b(10.2)i(follo)m(ws)e(immediately)-8 b(.)306 1487 y(It)26 b(is)d(imp)s(ortan)m(t)i(to)g(note)h(that)f(this)f (approac)m(h)h(do)s(es)g(not)g(allo)m(w)g(us)f(to)h(sho)m(w)g(that)h (non)m(unique-)165 1600 y(ness)31 b(of)g(Gibbs)f(measures)g(dep)s(ends) g(monotonically)g(on)h Ft(\025)p Fw(.)42 b(The)31 b(reason)g(is)f (similar)f(to)j(what)f(w)m(e)165 1713 y(sa)m(w)g(for)f(the)g(lattice)h (Widom{Ro)m(wlinson)e(mo)s(del)g(in)g(Section)h(6.7:)41 b(the)31 b(righ)m(t)f(hand)f(side)g(of)i(\(78\))165 1826 y(fails)g(to)j(b)s(e)e(increasing)f(in)g Fx(x)p Fw(.)47 b(It)33 b(th)m(us)f(remains)g(an)g(op)s(en)g(problem)f(whether)h(one)h (can)g(actually)165 1939 y(tak)m(e)f Ft(\025)415 1906 y Fq(0)415 1961 y Fu(c)475 1939 y Fw(=)25 b Ft(\025)624 1906 y Fq(00)624 1961 y Fu(c)697 1939 y Fw(in)k(Theorem)h(10.2.)306 2052 y(There)36 b(are)f(sev)m(eral)h(in)m(teresting)f(generalizations)g (of)h(the)g(Widom{Ro)m(wlinson)e(mo)s(del.)55 b(Let)165 2165 y(us)37 b(men)m(tion)h(one)g(of)g(them,)i(in)c(whic)m(h)g(neigh)m (b)s(oring)g(pairs)g(of)i(particles)f(of)h(the)g(opp)s(osite)f(t)m(yp)s (e)165 2277 y(are)c(not)f(forbidden,)e(but)i(merely)f(discouraged.)45 b(Let)32 b Ft(h)d Fw(:)f([0)p Ft(;)15 b Fv(1)p Fw(\))29 b Fv(!)f Fw([0)p Ft(;)15 b Fv(1)p Fw(])34 b(b)s(e)d(an)h(\\in)m(tersp)s (ecies)165 2390 y(repulsion)24 b(function")h(whic)m(h)f(is)h (decreasing)h(and)f(has)h(b)s(ounded)e(supp)s(ort.)37 b(F)-8 b(or)27 b(\003)e Fv(\032)g Fx(R)3214 2357 y Fu(d)3281 2390 y Fw(compact)165 2503 y(and)30 b Ft(\025)25 b(>)g Fw(0,)31 b(the)g(asso)s(ciated)g(Gibbs)e(distribution)d Ft(\026)2021 2518 y Fu(h;\025;)p Fr(\003)2225 2503 y Fw(on)31 b(\012)2418 2517 y Fr(\003)2491 2503 y Fv(\002)20 b Fw(\012)2648 2517 y Fr(\003)2731 2503 y Fw(is)29 b(giv)m(en)h(b)m(y)h (its)e(densit)m(y)990 2753 y Ft(f)10 b Fw(\()p Fx(x)p Ft(;)15 b Fx(y)q Fw(\))26 b(=)1493 2691 y(1)p 1398 2732 237 4 v 1398 2815 a Ft(Z)1460 2830 y Fu(h;\025;)p Fr(\003)1685 2753 y Fw(exp)1839 2634 y Fs(\022)1920 2753 y Fv(\000)2090 2672 y Fs(X)2011 2848 y Fu(x)p Fq(2)p Fn(x)p Fu(;y)r Fq(2)p Fn(y)2304 2753 y Ft(h)p Fw(\()p Fv(j)p Ft(x)21 b Fv(\000)f Ft(y)s Fv(j)p Fw(\))2688 2634 y Fs(\023)2765 2753 y Ft(:)165 3032 y Fw(relativ)m(e)27 b(to)g Ft(\031)644 3047 y Fu(\025;)p Fr(\003)770 3032 y Fv(\002)12 b Ft(\031)905 3047 y Fu(\025;)p Fr(\003)1019 3032 y Fw(.)39 b(In\014nite)25 b(v)m(olume)h(Gibbs)f(measures)h(on)h(\012)12 b Fv(\002)g Fw(\012)26 b(are)g(then)h(de\014ned)e(in)g(the)165 3145 y(usual)31 b(w)m(a)m(y)-8 b(.)47 b(Leb)s(o)m(witz)32 b(and)g(Lieb)f([154)r(])h(pro)m(v)m(ed)g(non)m(uniqueness)e(of)j(Gibbs) d(measures)i(for)g(large)165 3258 y Ft(\025)j Fw(when)e Ft(h)p Fw(\()p Ft(x)p Fw(\))i(is)e(large)h(enough)g(in)f(a)i(neigh)m(b) s(orho)s(o)s(d)d(of)i(the)g(origin.)51 b(Georgii)34 b(and)g(H\177)-45 b(aggstr\177)g(om)165 3371 y([97)r(])35 b(later)h(established)e(the)i (same)g(b)s(eha)m(vior)f(without)g(this)f(condition,)i(and)f(for)h(a)g (larger)f(class)165 3484 y(of)j(systems,)i(using)d(the)g (random-cluster)g(approac)m(h.)63 b(This)36 b(in)m(v)m(olv)m(es)i(a)g (generalization)g(of)g(the)165 3596 y(con)m(tin)m(uum)27 b(random-cluster)g(mo)s(del,)h(whic)m(h)f(arises)g(b)m(y)h(taking)g (the)g(random)f(connection)h(mo)s(del)165 3709 y(of)45 b(Section)f(10.1)h(with)e(connectivit)m(y)i(function)e Ft(g)s Fw(\()p Ft(x)p Fw(\))49 b(=)f(1)30 b Fv(\000)f Ft(e)2491 3676 y Fq(\000)p Fu(h)p Fr(\()p Fu(x)p Fr(\))2730 3709 y Fw(and)44 b(biasing)e(it)i(with)f(a)165 3822 y(factor)32 b(2)471 3789 y Fu(k)r Fr(\()p Fn(z)p Fr(\))605 3822 y Fw(,)f(where)g Ft(k)s Fw(\()p Fx(z)p Fw(\))h(is)e(the)h(n)m(um)m(b)s (er)e(of)i(connected)h(comp)s(onen)m(ts)e(of)h(a)h(con\014guration)e Fx(z)h Fw(of)165 3935 y(p)s(oin)m(ts)d(and)h(edges.)41 b(T)-8 b(o)30 b(establish)d(the)j(phase)f(transition)e(b)s(eha)m(vior)i (of)g(this)f(\\soft-core)j(Widom{)165 4048 y(Ro)m(wlinson)k(mo)s(del")g (\(i.e.,)j(uniqueness)c(of)i(Gibbs)e(measures)i(for)g(small)e Ft(\025)i Fw(and)f(non)m(uniqueness)165 4161 y(for)25 b(large)h Ft(\025)p Fw(\))f(one)h(can)g(basically)d(use)i(the)h(same)g (argumen)m(ts)f(as)h(the)f(ones)h(sk)m(etc)m(hed)g(ab)s(o)m(v)m(e)h (for)e(the)165 4274 y(standard)g(Widom{Ro)m(wlinson)f(mo)s(del.)38 b(Ho)m(w)m(ev)m(er,)29 b(due)d(to)g(the)g(extra)h(randomness)e(of)h (the)g(edges)165 4387 y(some)34 b(parts)f(of)h(the)f(argumen)m(t)h(b)s (ecome)g(more)g(in)m(v)m(olv)m(ed.)49 b(In)33 b(particular,)f(there)i (is)e(no)i(longer)f(a)165 4500 y(deterministic)j(b)s(ound)h(\(corresp)s (onding)f(to)j Ft(\024)1806 4514 y Fu(max)1950 4500 y Fw(\))g(on)f(ho)m(w)g(m)m(uc)m(h)g(the)h(n)m(um)m(b)s(er)e(of)h (connected)165 4613 y(comp)s(onen)m(ts)f(can)h(decrease)g(when)d(a)j(p) s(oin)m(t)e(is)g(added)g(to)i(the)f(random-cluster)e(con\014guration;) 165 4726 y(th)m(us)30 b(more)h(w)m(ork)f(is)g(needed)g(to)h(obtain)f (an)g(analogue)h(of)f(the)h(\014rst)f(inequalit)m(y)e(in)h(\(79\).)306 4838 y(T)-8 b(o)46 b(conclude,)i(w)m(e)e(note)f(that)h(the)f(Widom{Ro)m (wlinson)e(mo)s(del)h(on)g Fx(R)2912 4805 y Fu(d)2998 4838 y Fw(has)h(an)g(ob)m(vious)165 4951 y(m)m(ultit)m(yp)s(e)34 b(analogue)g(with)g Ft(q)g Fv(\025)e Fw(3)j(di\013eren)m(t)f(t)m(yp)s (es)g(of)h(particles.)52 b(This)33 b(m)m(ultit)m(yp)s(e)g(mo)s(del)g (still)165 5064 y(admits)e(a)i(random-cluster)e(represen)m(tation)h (from)f(whic)m(h)g(the)h(existence)g(of)h(a)f(phase)g(transition)165 5177 y(can)45 b(b)s(e)e(deriv)m(ed)g([97)q(].)82 b(There)43 b(is,)k(ho)m(w)m(ev)m(er,)i(no)43 b(partial)g(ordering)g(lik)m(e)g (\(75\))j(giving)c(rise)h(to)165 5290 y(sto)s(c)m(hastic)32 b(monotonicit)m(y)e(or)g(an)h(analogue)f(of)h(\(76\).)1822 5895 y(106)p eop %%Page: 107 107 107 106 bop 165 304 a Fy(References)265 494 y Fh([1])45 b(Abraham,)32 b(D.B.)h(and)f(Martin-L\177)-42 b(of,)32 b(A.)h(\(1973\))e(The)i(transfer)f(matrix)e(for)i(a)g(pure)g(phase)g (in)g(the)398 593 y(t)n(w)n(o-dimensional)22 b(Ising)k(mo)r(del,)g Fg(Comm)n(un.)f(Math.)j(Ph)n(ys.)e Ff(32)p Fh(,)i(245{268.)265 767 y([2])45 b(Adler,)h(R.J.)d(\(1998\))e(On)i(excursion)e(sets,)47 b(tub)r(e)d(form)n(ulae,)g(and)f(maxima)c(of)k(random)e(\014elds,)398 867 y(preprin)n(t,)26 b(P)-7 b(AS)27 b(1060.)265 1041 y([3])45 b(Aizenman,)32 b(M.)h(\(1980\))f(T)-7 b(ranslation)29 b(in)n(v)-5 b(ariance)30 b(and)i(instabilit)n(y)d(of)k(phase)f(co)r (existence)g(in)g(the)398 1141 y(t)n(w)n(o-dimensional)22 b(Ising)k(system,)g Fg(Comm)n(un.)g(Math.)h(Ph)n(ys.)g Ff(73)p Fh(,)g(83{94.)265 1315 y([4])45 b(Aizenman,)21 b(M.)h(\(1980\))f(Instabilit)n(y)e(of)j(phase)g(co)r(existence)e(and)i (translation)d(in)n(v)-5 b(ariance)19 b(in)i(t)n(w)n(o)g(di-)398 1415 y(mensions,)h(in)h(K.)g(Osterw)n(alder)e(\(ed.\),)k Fg(Mathematical)c(problems)g(in)i(theoretical)e(ph)n(ysics)p Fh(,)i(Lecture)398 1514 y(Notes)k(in)g(Ph)n(ysics)e(116,)h(143{147,)f (Springer,)g(Berlin,)h(Heidelb)r(erg,)f(New)j(Y)-7 b(ork.)265 1688 y([5])45 b(Aizenman,)33 b(M.)g(\(1982\))f(Geometric)f(analysis)f (of)j(\010)2100 1658 y Fe(4)2171 1688 y Fh(\014elds)f(and)h(Ising)f(mo) r(dels.)g(P)n(arts)f(I)j(and)f(I)r(I,)398 1788 y Fg(Comm)n(un.)25 b(Math.)j(Ph)n(ys.)f Ff(86)p Fh(,)g(1{48.)265 1962 y([6])45 b(Aizenman,)30 b(M.)h(and)g(Barsky)-7 b(,)30 b(D.J.)i(\(1987\))e (Sharpness)g(of)h(the)g(phase)g(transition)d(in)i(p)r(ercolation)398 2062 y(mo)r(dels,)25 b Fg(Comm)n(un.)h(Math.)i(Ph)n(ys.)e Ff(108)p Fh(,)h(489{526.)265 2236 y([7])45 b(Aizenman,)27 b(M.,)j(Barsky)-7 b(,)28 b(D.J.)h(and)g(F)-7 b(ernandez,)28 b(R.)i(\(1987\))d(The)i(phase)g(transition)d(in)i(a)h(general)398 2335 y(class)c(of)j(Ising-t)n(yp)r(e)e(mo)r(dels)f(is)i(sharp,)g Fg(J.)g(Stat.)h(Ph)n(ys.)54 b Ff(47)p Fh(,)27 b(343{374.)265 2510 y([8])45 b(Aizenman,)27 b(M.,)i(Bricmon)n(t,)d(J.)i(and)g(Leb)r(o) n(witz,)g(J.L.)g(\(1987\))f(P)n(ercolation)d(of)29 b(the)g(minorit)n(y) 24 b(spins)398 2609 y(in)j(high-dimensional)21 b(Ising)26 b(mo)r(dels,)g Fg(J.)h(Stat.)h(Ph)n(ys.)f Ff(49)p Fh(,)g(859{865.)265 2783 y([9])45 b(Aizenman,)25 b(M.,)i(Cha)n(y)n(es,)f(J.T.,)h(Cha)n(y)n (es,)f(L.)h(and)f(Newman,)h(C.M.)g(\(1987\))f(The)h(phase)f(b)r (oundary)398 2883 y(in)h(dilute)f(and)h(random)f(Ising)g(and)h(P)n (otts)g(ferromagnets,)e Fg(J.)i(Ph)n(ys.)g(A)h Ff(20)p Fh(,)f(L313{L318.)223 3057 y([10])45 b(Aizenman,)27 b(M.,)j(Cha)n(y)n (es,)e(J.T.,)h(Cha)n(y)n(es,)f(L.)i(and)e(Newman,)h(C.M.)g(\(1988\))f (Discon)n(tin)n(uit)n(y)e(of)j(the)398 3157 y(magnetization)f(in)j (one-dimensional)c(1)p Fd(=)p Fc(j)p Fd(x)21 b Fc(\000)g Fd(y)s Fc(j)1985 3127 y Fe(2)2054 3157 y Fh(Ising)30 b(and)i(P)n(otts)f(mo)r(dels,)g Fg(J.)h(Stat.)h(Ph)n(ys.)e Ff(50)p Fh(,)398 3256 y(1{40.)223 3431 y([11])45 b(Aizenman,)24 b(M.)i(and)g(F)-7 b(ernandez,)26 b(R.)g(\(1986\))f(On)h(the)g(critical) d(b)r(eha)n(vior)h(of)h(the)i(magnetization)22 b(in)398 3530 y(high-dimensional)f(Ising)26 b(mo)r(dels,)g Fg(J.)h(Stat.)h(Ph)n (ys.)f Ff(44)p Fh(,)g(393{454.)223 3704 y([12])45 b(Aizenman,)33 b(M.,)i(Kesten)e(H.)g(and)h(Newman,)g(C.M.)f(\(1987\))f(Uniqueness)g (of)i(the)g(in\014nite)e(cluster)398 3804 y(and)27 b(con)n(tin)n(uit)n (y)e(of)j(connectivit)n(y)d(functions)i(for)g(short-)f(and)i (long-range)c(p)r(ercolation,)g Fg(Comm)n(un.)398 3904 y(Math.)j(Ph)n(ys.)g Ff(111)p Fh(,)g(505{532.)223 4078 y([13])45 b(Aizenman,)27 b(N.,)i(Klein,)e(A.)i(and)f(Newman,)g(C.)h(M.) f(\(1992\))g(P)n(ercolation)c(metho)r(ds)k(for)g(disordered)398 4177 y(quan)n(tum)g(Ising)f(mo)r(dels.)g(In:)40 b(Kotec)n(k)q(\023)-43 b(y,)28 b(R.)h(\(ed.\),)i Fb(Phase)h(T)-6 b(r)l(ansitions:)42 b(Mathematics,)33 b(Physics,)398 4277 y(Biolo)l(gy,)f(...)p Fh(.)d(129{137,)c(W)-7 b(orld)26 b(Scien)n(ti\014c,)g(New)i(Jersey)-7 b(,)26 b(London.)223 4451 y([14])45 b(Alexander,)23 b(K.S.)i(and)f(Cha) n(y)n(es)g(L.)g(\(1997\))g(Non-p)r(erturbativ)n(e)e(criteria)g(for)i (Gibbsian)e(uniqueness,)398 4551 y Fg(Comm)n(un.)j(Math.)j(Ph)n(ys.)f Ff(189)p Fh(,)g(447{464.)223 4725 y([15])45 b(Alexander,)25 b(K.S.,)h(Cesi,)g(F.,)h(Cha)n(y)n(es,)e(L.,)i(Maes,)f(C.)h(and)f (Martinelli,)c(F.)27 b(\(1998\))e(Con)n(v)n(ergence)f(to)398 4824 y(equilibrium)e(of)28 b(random)e(Ising)g(mo)r(dels)f(in)i(the)h (Gri\016ths')f(phase,)g Fg(J.)g(Stat.)h(Ph)n(ys)f Ff(92)p Fh(,)g(337{351.)223 4999 y([16])45 b(Ashkin,)32 b(J.)f(and)h(T)-7 b(eller,)30 b(E.)i(\(1943\))e(Statistics)g(of)i(t)n(w)n(o-dimensional) 26 b(lattices)j(with)j(four)f(comp)r(o-)398 5098 y(nen)n(ts,)c Fg(Ph)n(ys.)g(Rev.)g Ff(64)p Fh(,)h(178{184.)223 5272 y([17])45 b(Av)n(erin)n(tsev,)18 b(M.B.)h(\(1975\))e(Gibbs)h (description)f(of)h(random)f(\014elds)h(whose)g(conditional)d (probabilities)398 5372 y(ma)n(y)26 b(v)-5 b(anish)26 b Fg(Probl.)f(Inform.)i(T)-7 b(ransmission)23 b Ff(11)p Fh(,)k(326{334.)223 5546 y([18])45 b(Bassalygo,)34 b(L.A.)i(and)f (Dobrushin,)i(R.L.)f(\(1986\))e(Uniqueness)g(of)i(a)f(Gibbs)g(\014eld)g (with)g(random)398 5646 y(p)r(oten)n(tial)25 b({)i(an)h(elemen)n(tary)c (approac)n(h,)i Fg(Theory)h(Prob.)f(Appl.)h Ff(31)p Fh(,)g(572{589.) 1822 5895 y Fw(107)p eop %%Page: 108 108 108 107 bop 223 304 a Fh([19])45 b(Basuev,)24 b(A.G.)h(\(1984\))e (Complete)g(phase)h(diagrams)c(with)k(resp)r(ect)g(to)h(external)d (\014elds)i(at)g(lo)n(w)f(tem-)398 404 y(p)r(eratures)h(for)g(mo)r (dels)f(with)h(nearest-neigh)n(b)r(or)e(in)n(teraction)g(in)i(the)i (case)e(of)h(a)f(\014nite)h(or)f(coun)n(table)398 503 y(n)n(um)n(b)r(er)i(of)i(ground)e(states,)h Fg(Theor.)g(Math.)h(Ph)n (ys.)e Ff(58)p Fh(,)i(171{182.)223 678 y([20])45 b(Basuev,)24 b(A.G.)h(\(1987\))e(Hamiltonian)d(of)k(the)h(phase)f(separation)d(b)r (order)j(and)g(phase)g(transitions)d(of)398 777 y(the)28 b(\014rst)f(kind.I)r(I.)g(The)h(simplest)d(disordered)g(phases,)i Fg(Theor.)f(Math.)i(Ph)n(ys.)f Ff(72)p Fh(,)g(861{871.)223 952 y([21])45 b(Batt)n(y)-7 b(,)41 b(C.J.K.)d(and)h(Bollmann,)f(H.W.)h (\(1980\))f(Generalized)e(Holley{Preston)e(inequalities)h(on)398 1051 y(measure)d(spaces)i(and)h(their)f(pro)r(ducts.)h Fg(Z.)f(W)-7 b(ahrsc)n(heinlic)n(hk)n(eitstheori)o(e)29 b(V)-7 b(erw.)35 b(Geb)p Fh(.)g Ff(53)p Fh(,)h(157{)398 1151 y(174.)223 1325 y([22])45 b(v)-5 b(an)29 b(Beijeren,)f(H.)i (\(1975\))e(In)n(terface)h(sharpness)f(in)g(Ising)g(systems,)h Fg(Comm)n(un.)e(Math.)i(Ph)n(ys.)g Ff(40)p Fh(,)398 1425 y(1{6.)223 1599 y([23])45 b(Benjamini,)24 b(I.,)k(Ly)n(ons,)f(R.,)g(P)n (eres,)f(Y.)i(and)g(Sc)n(hramm,)c(O.)k(\(1999\))e(Group-in)n(v)-5 b(arian)n(t)23 b(p)r(ercolation)398 1699 y(on)k(graphs,)f Fg(Geom.)h(F)-7 b(unct.)28 b(Analysis)d Ff(9)p Fh(,)j(29{66.)223 1873 y([24])45 b(v)-5 b(an)29 b(den)g(Berg,)g(J.)g(\(1993\))e(A)j (uniqueness)e(condition)f(for)h(Gibbs)h(measures,)e(with)i(application) c(to)398 1973 y(the)j(2{dimensional)22 b(Ising)k(an)n(tiferromagnet)e Fg(Comm)n(un.)h(Math.)j(Ph)n(ys.)e Ff(152)p Fh(,)h(161{166.)223 2147 y([25])45 b(v)-5 b(an)33 b(den)h(Berg,)g(J.)f(\(1997\))f(A)i (constructiv)n(e)d(mixing)g(condition)g(for)i(2{D)g(Gibbs)g(measures)e (with)398 2247 y(random)25 b(in)n(teractions,)g Fg(Ann.)j(Probab.)e Ff(25)p Fh(,)h(1316{1333.)223 2421 y([26])45 b(v)-5 b(an)28 b(den)g(Berg,)f(J.)h(and)f(Ermak)n(o)n(v,)e(A.)j(\(1996\))f(A)h(new)g (lo)n(w)n(er)e(b)r(ound)i(for)g(the)g(critical)c(probabilit)n(y)398 2521 y(of)j(site)g(p)r(ercolation)d(on)k(the)g(square)e(lattice,)g Fg(Random)g(Structures)h(Algorithms)d Ff(8)p Fh(,)k(199{212.)223 2695 y([27])45 b(v)-5 b(an)30 b(den)g(Berg,)f(J.)h(and)g(Maes,)g(C.)g (\(1994\))f(Disagreemen)n(t)e(p)r(ercolation)g(in)i(the)h(study)h(of)f (Mark)n(o)n(v)398 2795 y(\014elds,)c Fg(Ann.)j(Probab.)d Ff(22)p Fh(,)h(749{763.)223 2969 y([28])45 b(v)-5 b(an)35 b(den)g(Berg,)g(J.)g(and)g(Steif,)h(J.)f(\(1994\))f(P)n(ercolation)c (and)35 b(the)g(hard)f(core)g(lattice)f(gas)h(mo)r(del,)398 3069 y Fg(Sto)r(c)n(h.)27 b(Pro)r(c.)g(Appl.)g Ff(49)p Fh(,)g(179{197.)223 3243 y([29])45 b(Berretti,)26 b(A.)i(\(1985\))e (Some)g(prop)r(erties)g(of)i(random)d(Ising)h(mo)r(dels,)g Fg(J.)h(Stat.)h(Ph)n(ys.)f Ff(38)p Fh(,)g(483{496.)223 3417 y([30])45 b(Binder,)24 b(K.)g(and)g(Y)-7 b(oung,)26 b(A.P)-7 b(.)24 b(\(1988\))g(Spin)g(glasses:)32 b(exp)r(erimen)n(tal)22 b(facts,)j(theoretical)d(concepts,)398 3517 y(and)27 b(op)r(en)h(questions,)e Fg(Rev.)h(Mo)r(d.)h(Ph)n(ys.)f Ff(58)p Fh(,)g(801{976.)223 3691 y([31])45 b(Biskup,)19 b(M.,)i(Borgs,)e(C.,)i(Cha)n(y)n(es,)e(J.T.)g(and)g(Kotec)n(k)q(\023) -43 b(y,)19 b(R.)h(\(1999\))d(The)i(P)n(otts)f(mo)r(del)g(with)g (external)398 3791 y(\014elds:)34 b(Gibbs)25 b(states)g(of)g(Edw)n (ards{Sok)-5 b(al,)22 b(random)h(cluster)h(and)h(spin)f(represen)n (tations,)f(preprin)n(t.)223 3965 y([32])45 b(Borgs,)27 b(C.)i(and)g(Cha)n(y)n(es,)e(J.T.)i(\(1996\))e(The)i(co)n(v)-5 b(ariance)26 b(matrix)g(of)j(the)g(P)n(otts)f(mo)r(del:)38 b(a)28 b(random)398 4065 y(cluster)e(analysis,)e Fg(J.)k(Stat.)g(Ph)n (ys.)e Ff(82)p Fh(,)h(1235{1297.)223 4239 y([33])45 b(Brascamp,)k(H.J.) f(and)f(Lieb,)k(E.H.)c(\(1976\))f(On)i(extensions)d(of)i(the)h (Brunn-Mink)n(o)n(wski)43 b(and)398 4339 y(Pr)n(\023)-39 b(ek)n(opa-Leindler)34 b(theorems,)42 b(including)37 b(inequalities)g(for)i(log)g(conca)n(v)n(e)g(functions,)k(and)d(with) 398 4439 y(an)27 b(application)d(to)j(the)h(di\013usion)e(equation,)g Fg(J.)i(F)-7 b(unct.)28 b(Anal.)f Ff(22)p Fh(,)g(366{389.)223 4613 y([34])45 b(Bricmon)n(t,)20 b(J)j(and)f(Kupiainen,)f(A)i(\(1997\)) e(In\014nite)h(dimensional)c(SRB)23 b(measures,)e Fg(Ph)n(ysica)f(D)j Ff(103)p Fh(,)398 4712 y(18{33.)223 4887 y([35])45 b(Bricmon)n(t,)28 b(J)i(and)h(Kupiainen,)e(A)i(\(1996\))e(High)g(temp)r(erature)h (expansions)e(and)i(dynamical)d(sys-)398 4986 y(tems,)g Fg(Comm)n(un.)e(Math.)j(Ph)n(ys.)e Ff(178)p Fh(,)h(703{732.)223 5161 y([36])45 b(Bricmon)n(t,)26 b(J.,)j(Leb)r(o)n(witz,)f(J.L.)g(and)h (Maes,)f(C.)h(\(1987\))e(P)n(ercolation)e(in)j(strongly)e(correlated)g (sys-)398 5260 y(tems:)36 b(the)28 b(massless)c(Gaussian)i(\014eld,)h Fg(J.)g(Stat.)h(Ph)n(ys.)f Ff(46)p Fh(,)g(1249{1268.)223 5435 y([37])45 b(Bricmon)n(t,)28 b(J.)i(and)g(Sla)n(wn)n(y)f(J.)h (\(1989\))f(Phase)g(transitions)f(in)h(systems)g(with)h(a)g(\014nite)g (n)n(um)n(b)r(er)f(of)398 5534 y(dominan)n(t)c(ground)i(states,)g Fg(J.)g(Stat.)h(Ph)n(ys.)f Ff(54)p Fh(,)g(89{161.)1822 5895 y Fw(108)p eop %%Page: 109 109 109 108 bop 223 304 a Fh([38])45 b(Brigh)n(t)n(w)n(ell,)20 b(G.R.,)25 b(H\177)-42 b(aggstr\177)g(om,)22 b(O.)i(and)g(Winkler,)e(P) -7 b(.)24 b(\(1998\))f(Nonmonotonic)e(b)r(eha)n(vior)h(in)h(hard-)398 404 y(core)j(and)i(Widom{Ro)n(wlinson)22 b(mo)r(dels,)j Fg(J.)j(Stat.)g(Ph)n(ys.)p Fh(,)f(to)g(app)r(ear.)223 575 y([39])45 b(Broadb)r(en)n(t,)h(S.R.)d(and)g(Hammersley)-7 b(,)43 b(J.M.)g(\(1957\))f(P)n(ercolation)c(pro)r(cesses)k(I.)h (Crystals)d(and)398 674 y(mazes,)26 b Fg(Pro)r(c.)g(Cam)n(bridge)e (Phil.)i(So)r(c.)h Ff(53)p Fh(,)h(629{641.)223 845 y([40])45 b(Brydges,)25 b(D.,)j(F)-7 b(r\177)-42 b(ohlic)n(h)24 b(J.)j(and)g(Sok)-5 b(al,)25 b(A.D.)j(\(1983\))d(The)i(random)e(w)n (alk)g(represen)n(tation)f(of)j(clas-)398 945 y(sical)35 b(spin-systems)g(and)j(correlation)c(inequalities.)g(I)r(I.)39 b(The)f(sk)n(eleton)e(inequalities,)g Fg(Comm)n(un.)398 1045 y(Math.)27 b(Ph)n(ys.)g Ff(91)p Fh(.)g(117{139.)223 1216 y([41])45 b(Brydges,)e(D.,)j(F)-7 b(r\177)-42 b(ohlic)n(h)39 b(J.)i(and)g(Sp)r(encer,)k(T.)d(\(1982\))e(The)h(random)f(w)n(alk)g (represen)n(tation)e(of)398 1315 y(classical)23 b(spin-systems)i(and)i (correlation)d(inequalities,)f Fg(Comm)n(un.)j(Math.)h(Ph)n(ys.)g Ff(83)p Fh(.)g(123{150.)223 1486 y([42])45 b(Burton,)22 b(R.)h(and)f(Keane,)g(M.)h(\(1989\))e(Densit)n(y)g(and)h(uniqueness)f (in)h(p)r(ercolation,)e Fg(Comm)n(un.)g(Math.)398 1586 y(Ph)n(ys.)26 b Ff(121)p Fh(,)h(501{505.)223 1757 y([43])45 b(Burton,)25 b(R.)h(and)f(Keane,)g(M.)h(\(1991\))e(T)-7 b(op)r(ological)21 b(and)k(metric)f(prop)r(erties)f(of)j(in\014nite)e (clusters)g(in)398 1857 y(stationary)h(t)n(w)n(o-dimensional)d(site)k (p)r(ercolation,)f Fg(Israel)g(J.)j(Math.)f Ff(76)p Fh(,)h(299{316.)223 2028 y([44])45 b(Burton,)21 b(R.)f(and)f(Steif,)i(J.)f(\(1995\))e (Quite)h(w)n(eak)g(Bernoulli)d(with)j(exp)r(onen)n(tial)f(rate)h(and)g (p)r(ercolation)398 2127 y(for)27 b(random)e(\014elds,)i Fg(Sto)r(c)n(h.)h(Pro)r(c.)e(Appl.)i Ff(58)p Fh(,)f(35{55.)223 2298 y([45])45 b(Cammarota,)22 b(C.)j(\(1993\))f(Sto)r(c)n(hastic)g (order)g(and)h(monotonicit)n(y)d(in)i(temp)r(erature)g(for)h(Gibbs)g (mea-)398 2398 y(sures,)h Fg(Lett.)j(Math.)e(Ph)n(ys.)g Ff(29)p Fh(,)g(287{295.)223 2569 y([46])45 b(Campanino,)20 b(M.)h(and)h(Klein,)e(A.)i(\(1991\))e(Deca)n(y)h(of)g(t)n(w)n(o{p)r (oin)n(t)e(functions)i(for)g(\()p Fd(d)6 b Fh(+)g(1\){dimensional)398 2669 y(p)r(ercolation,)28 b(Ising)h(and)i(P)n(otts)e(mo)r(del)g(with)h Fd(d)p Fh({dimensional)c(disorder,)j Fg(Comm)n(un.)f(Math.)j(Ph)n(ys.) 398 2768 y Ff(135)p Fh(,)c(499{515.)223 2939 y([47])45 b(Campanino,)24 b(M.)j(and)g(Russo,)g(L.)g(\(1985\))e(An)j(upp)r(er)f (b)r(ound)h(for)e(the)h(critical)d(p)r(ercolation)g(proba-)398 3039 y(bilit)n(y)g(for)k(the)g(three-dimensional)22 b(cubic)27 b(lattice,)f Fg(Ann.)i(Probab.)e Ff(13)p Fh(,)h(478{491.)223 3210 y([48])45 b(Campb)r(ell,)24 b(M.)i(and)h(Cha)n(y)n(es,)e(L.)h (\(1998\))f(The)i(isotropic)c(O\(3\))j(mo)r(del)e(and)j(the)f(W)-7 b(ol\013)26 b(represen)n(ta-)398 3309 y(tion,)g Fg(J.)i(Ph)n(ys.)e(A)p Fh(,)i Ff(31)p Fh(,)g(L255{L259.)223 3481 y([49])45 b(Cesi,)39 b(F.,)i(Maes,)g(C.)d(and)g(Martinelli,)f(F.)h(\(1997\))f(Relaxation)e (of)j(disordered)e(magnets)g(in)i(the)398 3580 y(Gri\016ths')26 b(regime,)f Fg(Comm)n(un.)h(Math.)i(Ph)n(ys.)e Ff(188)p Fh(,)h(135{173.)223 3751 y([50])45 b(Cesi,)21 b(F.,)j(Maes,)e(C.)g(and) g(Martinelli,)c(F.)23 b(\(1997\))d(Relaxation)f(to)j(equilibrium)16 b(for)22 b(t)n(w)n(o)f(dimensional)398 3851 y(disordered)k(Ising)h (systems)g(in)h(the)h(Gri\016ths')f(phase,)g Fg(Comm)n(un.)e(Math.)j (Ph)n(ys.)f(189)p Fh(,)f(323{335.)223 4022 y([51])45 b(Cesi,)22 b(F.,)j(Maes,)e(C.)g(and)g(Martinelli,)d(F.)j(\(1997\))f (Relaxation)e(of)j(random)e(magnets)h(:)34 b(exact)23 b(results)398 4121 y(for)k(the)h(Gri\016ths')f(regime,)d Fg(Mark)n(o)n(v)i(Pro)r(c.)g(Rel.)h(Fields)f Ff(3)p Fh(,)h(465{474.)223 4292 y([52])45 b(Cha)n(y)n(es,)26 b(J.T.)h(and)h(Cha)n(y)n(es,)e(L.)i (\(1988\))e(Les)h(Houc)n(hes)g(Lectures.)223 4464 y([53])45 b(Cha)n(y)n(es,)27 b(J.T.,)h(Cha)n(y)n(es,)f(L.)h(and)g(F)-7 b(r\177)-42 b(ohlic)n(h,)26 b(J.)i(\(1985\))e(The)j(Lo)n(w-T)-7 b(emp)r(erature)25 b(Beha)n(vior)g(of)j(Dis-)398 4563 y(ordered)e(Magnets,)h Fg(Comm)n(un.)e(Math.)j(Ph)n(ys.)f Ff(100)p Fh(,)g(399{437.)223 4734 y([54])45 b(Cha)n(y)n(es,)29 b(J.T.,)i(Cha)n(y)n(es,)e(L.)i(and)f(Kotec)n(k)q(\023)-43 b(y,)29 b(R.)i(\(1995\))d(The)j(analysis)c(of)j(the)g(Widom-Ro)n (wlinson)398 4834 y(mo)r(del)25 b(b)n(y)j(sto)r(c)n(hastic)d(geometric) g(metho)r(ds,)i Fg(Comm)n(un.)e(Math.)j(Ph)n(ys.)f Ff(172)p Fh(,)g(551{569.)223 5005 y([55])45 b(Cha)n(y)n(es,)26 b(J.T.,)i(Cha)n(y)n(es,)e(L.)i(and)f(Newman,)g(C.M.)h(\(1985\))e(The)i (sto)r(c)n(hastic)e(geometry)f(of)j(in)n(v)-5 b(asion)398 5104 y(p)r(ercolation,)24 b Fg(Comm)n(un.)i(Math.)i(Ph)n(ys.)e Ff(101)p Fh(,)h(383{407.)223 5275 y([56])45 b(Cha)n(y)n(es,)26 b(L.)h(\(1996\))f(P)n(ercolation)d(and)k(ferromagnetism)c(on)k Ff(Z)2425 5245 y Fe(2)2463 5275 y Fh(:)37 b(the)27 b Fd(q)s Fh(-state)g(P)n(otts)g(cases,)f Fg(Sto)r(c)n(h.)398 5375 y(Pro)r(c.)g(Appl.)h Ff(65)p Fh(,)h(209{216.)223 5546 y([57])45 b(Cha)n(y)n(es,)28 b(L.,)i(Kotec)n(k)q(\023)-43 b(y,)28 b(R.)i(and)f(Shlosman,)e(S.B.)j(\(1997\))e(Staggered)g(phases)g (in)h(diluted)f(systems)398 5646 y(with)f(con)n(tin)n(uous)e(spins,)i Fg(Comm)n(un.)e(Math.)j(Ph)n(ys.)f Ff(189)p Fh(,)g(631{640.)1822 5895 y Fw(109)p eop %%Page: 110 110 110 109 bop 223 304 a Fh([58])45 b(Cha)n(y)n(es,)39 b(L.)f(and)g(Mac)n (h)n(ta,)h(J.)f(\(1997\))f(Graphical)e(represen)n(tations)g(and)j (cluster)f(algorithms)c(I.)398 404 y(Discrete)26 b(spin)h(systems,)f Fg(Ph)n(ysica)f(A)j Ff(239)p Fh(,)f(542{601.)223 573 y([59])45 b(Cha)n(y)n(es,)25 b(L.)i(and)f(Mac)n(h)n(ta,)g(J.)g (\(1998\))g(Graphical)d(represen)n(tations)h(and)i(cluster)f (algorithms.)e(P)n(art)398 673 y(I)r(I,)28 b Fg(Ph)n(ysica)d(A)j Ff(254)p Fh(,)f(477{516.)223 842 y([60])45 b(Cha)n(y)n(es,)23 b(L.,)h(Mac)n(h)n(ta,)g(J.)f(and)h(Redner,)g(O.)g(\(1998\))e(Graphical) f(represen)n(tations)f(for)k(Ising)e(systems)398 942 y(in)27 b(external)e(\014elds,)i Fg(J.)g(Stat.)h(Ph)n(ys.)f Ff(93)p Fh(,)g(17{32.)223 1111 y([61])45 b(Cha)n(y)n(es,)27 b(L.,)i(McKellar,)c(D.)k(and)f(Winn,)g(B.)h(\(1998\))e(P)n(ercolation)d (and)k(Gibbs)g(states)g(m)n(ultiplicit)n(y)398 1211 y(for)f (ferromagnetic)d(Ashkin{T)-7 b(eller)24 b(mo)r(dels)h(on)i Ff(Z)2030 1181 y Fe(2)2068 1211 y Fh(,)h Fg(J.)f(Ph)n(ys.)g(A)h Ff(31)p Fh(,)f(9055{9063.)223 1380 y([62])45 b(Coniglio,)24 b(A.,)30 b(Nappi,)e(C.R.,)h(P)n(eruggi,)c(F.)k(and)g(Russo,)f(L.)g (\(1976\))g(P)n(ercolation)c(and)k(phase)g(tran-)398 1480 y(sitions)d(in)i(the)h(Ising)e(mo)r(del,)g Fg(Comm)n(un.)f(Math.)j (Ph)n(ys)e Ff(51)p Fh(,)i(315{323.)223 1649 y([63])45 b(Dobrushin,)22 b(R.L.)h(\(1965\))f(Existence)f(of)i(a)f(phase)g (transition)e(in)i(t)n(w)n(o)g(and)h(three)f(dimensional)d(Ising)398 1749 y(mo)r(dels,)25 b Fg(Th.)j(Probab.)e(Appl.)h Ff(10)p Fh(,)h(193{213.)223 1919 y([64])45 b(Dobrushin,)26 b(R.L.)i(\(1968\))e (Gibbsian)f(random)h(\014elds)g(for)h(lattice)e(systems)h(with)h (pairwise)e(in)n(terac-)398 2018 y(tions,)h Fg(F)-7 b(unct.)29 b(Anal.)d(Appl.)i Ff(2)p Fh(,)f(292{301.)223 2188 y([65])45 b(Dobrushin,)32 b(R.L.)g(\(1968\))f(The)h(problem)e(of)i(uniqueness)f (of)h(a)f(Gibbsian)f(random)h(\014eld)g(and)h(the)398 2287 y(problem)25 b(of)i(phase)g(transition,)e Fg(F)-7 b(unct.)29 b(Anal.)e(Appl.)g Ff(2)p Fh(,)g(302{312.)223 2457 y([66])45 b(Dobrushin,)60 b(R.L.)55 b(\(1972\))e(Gibbs)h(state)h (describing)d(co)r(existence)h(of)h(phases)g(for)g(a)h(three-)398 2556 y(dimensional)23 b(Ising)j(mo)r(del,)g Fg(Th.)i(Probab.)e(Appl.)h Ff(17)p Fh(,)g(582{601.)223 2726 y([67])45 b(Dobrushin,)26 b(R.L.)i(\(1996\))e(Estimates)f(of)i(semi-in)n(v)-5 b(arian)n(ts)22 b(for)k(the)i(Ising)e(mo)r(del)g(at)h(lo)n(w)f(temp)r(era-)398 2825 y(tures,)h Fg(Amer.)g(Math.)g(So)r(c.)h(T)-7 b(ransl.)26 b Ff(177)p Fh(,)h(59{81.)223 2995 y([68])45 b(Dobrushin,)j(R.L.,)i (Kolafa,)d(J.)e(and)g(Shlosman,)i(S.B.)e(\(1985\))f(Phase)g(diagram)e (of)j(the)h(t)n(w)n(o{)398 3095 y(dimensional)14 b(Ising)k(an)n (tiferromagnet)d(\(computer{assisted)g(pro)r(of)6 b(\),)21 b Fg(Comm)n(un.)c(Math.)i(Ph)n(ys.)f Ff(102)p Fh(,)398 3194 y(89{103.)223 3364 y([69])45 b(Dobrushin,)21 b(R.L.)g(and)f (Shlosman)f(S.B.)i(\(1985\))e(The)i(problem)d(of)j(translation)d(in)n (v)-5 b(ariance)17 b(of)k(Gibbs)398 3463 y(states)36 b(at)h(lo)n(w)e(temp)r(eratures,)j(in)e Fg(Mathematical)e(Ph)n(ysics)g (Reviews,)k(So)n(viet)e(Science)g(Reviews)398 3563 y(Section)26 b(C)p Fh(,)i(V)-7 b(ol.)27 b(5,)g(S.P)-7 b(.)27 b(No)n(vik)n(o)n(v,)e (ed.)j(\(Gordon)f(and)g(Breac)n(h,)f(New)i(Y)-7 b(ork\).)223 3732 y([70])45 b(Dobrushin,)23 b(R.L.)i(and)f(Shlosman,)e(S.B.)i (\(1985\))f(Constructiv)n(e)f(criterion)f(for)i(the)i(uniqueness)e(of)h (a)398 3832 y(Gibbs)k(\014eld.)g(In:)40 b(F)-7 b(ritz,)29 b(J.,)g(Ja\013e,)g(A.,)g(Szasz,)g(D.)h(\(eds.\),)f Fb(Statistic)l(al)j (me)l(chanics)g(and)f(dynamic)l(al)398 3932 y(systems.)d Fh(347{370,)c(Birkhauser,)h(Boston.)223 4101 y([71])45 b(Dobrushin,)26 b(R.L.)i(and)g(Shlosman,)d(S.B.)j(\(1987\))f (Completely)d(analytical)g(in)n(teraction:)34 b(construc-)398 4201 y(tiv)n(e)26 b(description,)f Fg(J.)j(Stat.)g(Ph)n(ys.)e Ff(46)p Fh(,)h(983{1014.)223 4370 y([72])45 b(v)n(on)28 b(Dreifus,)g(H.,)h(Klein,)e(A.)i(and)f(P)n(erez,)g(J.F.)g(\(1995\))g(T) -7 b(aming)26 b(Gri\016ths')i(singularities:)33 b(in\014nite)398 4470 y(di\013eren)n(tiabilit)n(y)22 b(of)28 b(quenc)n(hed)f (correlation)d(functions,)j Fg(Comm.)e(Math.)j(Ph)n(ys.)f Ff(170)p Fh(,)f(21{39.)223 4639 y([73])45 b(Edw)n(ards,)60 b(R.G.)c(and)f(Sok)-5 b(al,)60 b(A.D.)c(\(1988\))e(Generalization)d(of) k(the)h(F)-7 b(ortuin{Kasteleyn{-)398 4739 y(Sw)n(endsen{W)g(ang)21 b(represen)n(tation)e(and)j(Mon)n(te)f(Carlo)f(algorithm,)f Fg(Ph)n(ys.)i(Rev.)h Fh(D)h Ff(38)p Fh(,)g(2009{2012.)223 4908 y([74])45 b(Ellis,)34 b(R.S.)j(\(1985\))d Fg(En)n(trop)n(y)-7 b(,)37 b(Large)d(Deviations)g(and)i(Statistical)d(Mec)n(hanics)p Fh(,)j(Springer,)g(New)398 5008 y(Y)-7 b(ork.)223 5177 y([75])45 b(v)-5 b(an)27 b(En)n(ter,)f(A.C.D.,)j(F)-7 b(ernandez,)26 b(R.,)i(Sc)n(honmann,)e(R.H)i(and)f(Shlosman,)e(S.B.)i (\(1997\))f(On)h(com-)398 5277 y(plete)g(analyticit)n(y)c(of)28 b(the)g(2D)g(P)n(otts)e(mo)r(del,)g Fg(Comm)n(un.)g(Math.)h(Ph)n(ys.)g Ff(189)p Fh(,)223 5447 y([76])45 b(v)-5 b(an)44 b(En)n(ter,)k(A.C.D.,)i (F)-7 b(ernandez,)49 b(R.)c(and)f(Sok)-5 b(al)43 b(A.D.)j(\(1993\))d (Regularit)n(y)f(prop)r(erties)g(and)398 5546 y(pathologies)31 b(of)36 b(p)r(osition-space)c(renormalization-)o(gr)o(oup)e (transformations:)48 b(scop)r(e)35 b(and)g(limita-)398 5646 y(tions)26 b(of)i(Gibbsian)d(theory)-7 b(,)27 b Fg(J.)h(Stat.)g(Ph)n(ys.)e Ff(72)p Fh(,)h(879{1167.)1822 5895 y Fw(110)p eop %%Page: 111 111 111 110 bop 223 304 a Fh([77])45 b(F)-7 b(ernandez,)32 b(R.,)g(F)-7 b(r\177)-42 b(ohlic)n(h)29 b(J.)j(and)f(Sok)-5 b(al,)31 b(A.D.)h(\(1992\))e Fg(Random)h(W)-7 b(alks,)31 b(Critical)d(phenomena,)398 404 y(and)f(T)-7 b(rivialit)n(y)23 b(in)k(Quan)n(tum)f(Field)g(Theory)p Fh(,)g(Springer-V)-7 b(erlag,)24 b(Berlin.)223 575 y([78])45 b(Fisher,)20 b(K.H.)h(and)f(Hertz,)i(J.A.)e(\(1991\))f Fg(Spin)h(glasses)p Fh(,)g(Cam)n(bridge)d(Univ)n(ersit)n(y)g(Press,)k(Cam)n(bridge,)398 674 y(U.K.)223 845 y([79])45 b(Flory)-7 b(,)27 b(P)-7 b(.J.)28 b(\(1941\))g(Molecular)d(size)j(distribution)e(in)i (three-dimensional)23 b(gelation)j(I{I)r(I)r(I,)j Fg(J.)g(Am.)398 945 y(Chem.)e(So)r(c.)p Ff(63)p Fh(,)g(3083{3100.)223 1116 y([80])45 b(F)-7 b(ortuin,)31 b(C.M.)g(\(1972\))f(On)h(the)h (random-cluster)27 b(mo)r(del.)j(I)r(I.)h(The)g(p)r(ercolation)e(mo)r (del,)h Fg(Ph)n(ysica)398 1216 y Ff(58)p Fh(,)d(393{418.)223 1387 y([81])45 b(F)-7 b(ortuin,)27 b(C.M.)i(\(1972\))e(On)h(the)h (random-cluster)c(mo)r(del.)h(I)r(I)r(I.)k(The)e(simple)e (random-cluster)e(pro-)398 1486 y(cess,)j Fg(Ph)n(ysica)e Ff(59)p Fh(,)i(545{570.)223 1657 y([82])45 b(F)-7 b(ortuin,)26 b(C.M.)h(and)f(Kasteleyn,)f(P)-7 b(.W.)27 b(\(1972\))e(On)i(the)g (random-cluster)c(mo)r(del.)i(I.)i(In)n(tro)r(duction)398 1757 y(and)g(relation)e(to)i(other)g(mo)r(dels,)f Fg(Ph)n(ysica)f Ff(57)p Fh(,)i(536{564.)223 1928 y([83])45 b(F)-7 b(ortuin,)30 b(C.M.,)i(Kasteleyn,)e(P)-7 b(.W.)31 b(and)g(Ginibre,)f(J.)g(\(1971\))g (Correlation)d(inequalities)f(on)31 b(some)398 2028 y(partially)23 b(ordered)k(sets,)g Fg(Comm)n(un.)f(Math.)h(Ph)n(ys.)g Ff(22)p Fh(,)g(89{103.)223 2199 y([84])45 b(F)-7 b(r\177)-42 b(ohlic)n(h,)33 b(J.)h(\(1984\))f(Mathematical)e(asp)r(ects)j(of)g(the) h(ph)n(ysics)e(of)h(disordered)e(systems.)h(In:)50 b(Os-)398 2298 y(terw)n(alder,)25 b(K.,)j(Stora,)e(R.)i(\(eds.\),)h Fg(Critical)24 b(phenomena,)i(random)g(systems)g(and)i(gauge)e (theories)p Fh(.)398 2398 y(725{894,)e(North{Holland,)h(Amsterdam.)223 2569 y([85])45 b(F)-7 b(r\177)-42 b(ohlic)n(h,)17 b(J.)i(and)f(Im)n (brie,)g(J.)h(\(1984\))e(Impro)n(v)n(ed)f(p)r(erturbation)h(expansion)g (for)h(disordered)e(systems:)398 2669 y(b)r(eating)26 b(Gri\016ths')h(singularities,)22 b Fg(Comm)n(un.)k(Math.)i(Ph)n(ys.)e Ff(96)p Fh(,)i(145{180.)223 2840 y([86])45 b(F)-7 b(r\177)-42 b(ohlic)n(h,)27 b(J.)i(and)g(Zegarlinski,)c(B.)30 b(\(1987\))e(The)h (high)f(temp)r(erature)g(phase)h(of)h(a)e(long)g(range)g(spin)398 2939 y(glass,)d Fg(Comm)n(un.)h(Math.)h(Ph)n(ys.)g Ff(110)p Fh(,)g(121{155.)223 3110 y([87])45 b(F)-7 b(ukuy)n(ama,)34 b(M.)h(\(1998\))e(Discrete)h(symmetry)e(breaking)g(for)i(certain)f (short-range)f(in)n(teractions,)398 3210 y(preprin)n(t,)26 b(Graduate)h(Sc)n(ho)r(ol)e(of)j(Mathematical)c(Science,)j(Univ)n (ersit)n(y)d(of)k(T)-7 b(oky)n(o.)223 3381 y([88])45 b(Galla)n(v)n(otti,)31 b(G.)i(and)h(Miracle-Sole,)c(S.)k(\(1972\))e (Equilibrium)d(states)k(of)g(the)h(Ising)e(mo)r(del)g(in)h(the)398 3481 y(t)n(w)n(o-phase)26 b(region.)f Fg(Ph)n(ys.)i(Rev.)g(B)55 b Ff(5)p Fh(,)28 b(2555{2559.)223 3652 y([89])45 b(Gandol\014,)33 b(A.,)h(Grimmett,)e(G.)h(R.,)i(and)d(Russo,)i(L.)f(\(1988\))e(On)i(the) g(uniqueness)e(of)i(the)g(in\014nite)398 3751 y(op)r(en)27 b(cluster)g(in)f(the)i(p)r(ercolation)d(mo)r(del,)h Fg(Comm)n(un.)f (Math.)j(Ph)n(ys.)f Ff(114)p Fh(,)g(549{552.)223 3922 y([90])45 b(Gandol\014,)26 b(A.,)h(Keane,)g(M.)g(and)g(Russo,)g(L.)g (\(1988\))f(On)h(the)h(uniqueness)e(of)h(the)g(in\014nite)f(o)r (ccupied)398 4022 y(cluster)g(in)h(dep)r(enden)n(t)h(t)n(w)n (o-dimensional)22 b(site)27 b(p)r(ercolation,)d Fg(Ann.)k(Probab.)e Ff(16)p Fh(,)i(1147{1157.)223 4193 y([91])45 b(Georgii,)28 b(H.-O.)j(\(1973\))f(Tw)n(o)g(remarks)e(on)j(extremal)d(equilibrium)e (states,)31 b Fg(Comm.)e(Math)i(Ph)n(ys.)398 4292 y Ff(32)p Fh(,)c(107{118.)223 4464 y([92])45 b(Georgii,)22 b(H.-O.)i(\(1981\))g (P)n(ercolation)c(for)k(lo)n(w)f(energy)h(clusters)f(and)h(discrete)f (symmetry)g(breaking)398 4563 y(in)k(classical)c(spin)j(systems,)h Fg(Comm)n(un.)e(Math.)j(Ph)n(ys.)f Ff(81)p Fh(,)g(455{473.)223 4734 y([93])45 b(Georgii,)38 b(H.-O.)g(\(1981\))f(Sp)r(on)n(taneous)h (magnetization)d(of)j(randomly)e(dilute)h(ferromagnets,)h Fg(J.)398 4834 y(Stat.)28 b(Ph)n(ys.)e Ff(25)p Fh(,)h(369{396.)223 5005 y([94])45 b(Georgii,)33 b(H.-O.)h(\(1984\))g(On)g(the)h (ferromagnetic)c(and)k(the)g(p)r(ercolativ)n(e)c(region)i(of)h(random)f (spin)398 5104 y(systems,)26 b Fg(Adv.)i(Appl.)f(Prob.)f Ff(16)p Fh(,)i(732{765.)223 5275 y([95])45 b(Georgii,)17 b(H.-O.)i(\(1985\))f(On)g(the)i(ferromagnetic)15 b(and)k(p)r(ercolativ) n(e)d(region)h(of)h(random)g(spin)g(systems,)398 5375 y Fg(Con)n(temp.)26 b(Math.)i Ff(41)p Fh(,)f(135{147.)223 5546 y([96])45 b(Georgii,)34 b(H.-O.)i(\(1988\))e Fg(Gibbs)h(Measures)f (and)i(Phase)e(T)-7 b(ransitions)p Fh(,)35 b(de)h(Gruyter,)h(Berlin)c (New)398 5646 y(Y)-7 b(ork.)1822 5895 y Fw(111)p eop %%Page: 112 112 112 111 bop 223 304 a Fh([97])45 b(Georgii,)32 b(H.-O.)h(and)h(H\177) -42 b(aggstr\177)g(om,)32 b(O.)h(\(1996\))g(Phase)f(transition)f(in)i (con)n(tin)n(uum)f(P)n(otts)g(mo)r(dels,)398 404 y Fg(Comm)n(un.)25 b(Math.)j(Ph)n(ys.)f Ff(181)p Fh(,)g(507{528.)223 578 y([98])45 b(Georgii,)31 b(H.-O.)h(and)h(K)r(\177)-44 b(uneth,)34 b(T.)f(\(1997\))f(Sto)r(c)n(hastic)f(comparison)e(of)k(p)r (oin)n(t)f(random)f(\014elds,)i Fg(J.)398 678 y(Appl.)27 b(Probab.)f Ff(34)p Fh(,)h(868{881.)223 852 y([99])45 b(Gertsik,)39 b(V.M.)f(\(1976\))e(Conditions)g(for)h(the)h (non-uniqueness)e(of)h(the)h(Gibbs)f(state)h(for)f(lattice)398 952 y(mo)r(dels)25 b(ha)n(ving)h(\014nite)h(in)n(teraction)e(p)r(oten)n (tials,)g Fg(Math.)j(USSR)g(Izv)n(estija)e Ff(10)p Fh(,)h(429{443.)182 1126 y([100])44 b(Giacomin,)39 b(G.,)44 b(Leb)r(o)n(witz,)d(J.L.)f(and) g(Maes,)i(C.)e(\(1995\))f(Agreemen)n(t)f(p)r(ercolation)f(and)j(phase) 398 1226 y(co)r(existence)26 b(in)h(some)f(Gibbs)g(systems,)h Fg(J.)g(Stat.)h(Ph)n(ys.)f Ff(80)p Fh(,)g(1379{1403.)182 1400 y([101])44 b(Gielis,)36 b(G.)h(\(1998\))f(Spin)g(systems)g(with)g (random)f(in)n(teractions:)52 b(the)37 b(Gri\016ths')g(regime,)f(Ph.D.) 398 1500 y(thesis,)26 b(K.U.)i(Leuv)n(en.)182 1674 y([102])44 b(Gielis,)32 b(G.)i(and)g(Maes,)g(C.)g(\(1995\))e(The)i(uniqueness)f (regime)d(of)k(Gibbs)f(\014elds)g(with)g(un)n(b)r(ounded)398 1774 y(disorder,)25 b Fg(J.)i(Stat.)h(Ph)n(ys.)f Ff(81)p Fh(,)g(829{835.)182 1948 y([103])44 b(Gielis,)26 b(G.,)j(Maes,)g(C.)g (and)f(V)-7 b(ande)29 b(V)-7 b(elde,)28 b(K.)h(\(1999\))e(Randomly)g (in)n(teracting)e(particle)i(systems:)398 2047 y(the)h(uniqueness)e (regime,)f Fg(Ann.)j(Inst.)g(H.)g(P)n(oincar)n(\023)-39 b(e,)23 b(Ph)n(ys.)k(Theor.)p Fh(,)g(to)g(app)r(ear)g(\(Ma)n(y)g (1999\).)182 2222 y([104])44 b(Gordon,)26 b(D.M.)h(\(1991\))f(P)n (ercolation)c(in)k(high)g(dimensions,)d Fg(J.)k(London)f(Math.)h(So)r (c.)g(\(2\))g Ff(44)p Fh(,)f(373{)398 2321 y(384.)182 2496 y([105])44 b(Gri\016ths,)35 b(R.)g(\(1969\))e(Non{analytic)f(b)r (eha)n(viour)g(ab)r(o)n(v)n(e)h(the)i(critical)c(p)r(oin)n(t)j(in)g(a)g (random)f(Ising)398 2595 y(ferromagnet,)25 b Fg(Ph)n(ys.)h(Rev.)i (Lett.)g Ff(23)p Fh(,)f(17{19.)182 2770 y([106])44 b(Gri\016ths,)28 b(R.)h(and)g(Leb)r(o)n(witz,)f(J.L.)g(\(1972\))g(Random)f(spin)h (systems:)38 b(rigorous)26 b(results,)h Fg(J.)i(Math.)398 2869 y(Ph)n(ys.)d Ff(9)p Fh(,)i(1284{1292.)182 3044 y([107])44 b(Grimmett,)20 b(G.R.)i(\(1973\))e(A)i(theorem)d(ab)r(out)j(random)d (\014elds,)j Fg(Bull)d(London)i(Math.)h(So)r(c.)f Ff(5)p Fh(,)i(81{84.)182 3218 y([108])44 b(Grimmett,)25 b(G.R.)j(\(1999\))e Fg(P)n(ercolation)p Fh(,)e(2.)j(edition,)f(Springer,)f(New)j(Y)-7 b(ork.)182 3392 y([109])44 b(Grimmett,)39 b(G.R.)g(\(1995\))f(The)h (sto)r(c)n(hastic)d(random-cluster)g(pro)r(cess,)k(and)f(the)g (uniqueness)e(of)398 3492 y(random-cluster)24 b(measures,)h Fg(Ann.)j(Probab.)e Ff(23)p Fh(,)h(1461{1510.)182 3666 y([110])44 b(Grimmett,)22 b(G.R.)i(\(1997\))e(P)n(ercolation)d(and)k (disordered)e(systems,)i Fg(Lectures)g(on)g(Probabilit)n(y)c(The-)398 3766 y(ory)30 b(and)g(Statistics)f(\(Sain)n(t-Flour)f(1996\))p Fh(,)j(153{300,)d Fg(Lecture)j(Notes)f(in)g(Math.)h Ff(1665)p Fh(,)g(Springer,)398 3866 y(Berlin.)182 4040 y([111])44 b(Grimmett,)32 b(G.R.)h(and)g(Newman,)g(C.M.)g(\(1990\))f(P)n (ercolation)c(in)k Fc(1)22 b Fh(+)g(1)32 b(dimensions,)f Fg(Disorder)398 4140 y(in)e(Ph)n(ysical)d(Systems)i Fh(\(G.)i(R.)g (Grimmett)d(and)j(D.)g(J.)f(A.)h(W)-7 b(elsh,)30 b(editors\),)f (219{240,)d(Clarendon)398 4239 y(Press,)g(Oxford.)182 4414 y([112])44 b(Guionnet,)25 b(A.)h(and)g(Zegarlinski,)21 b(B.)26 b(\(1997\))f(Deca)n(y)g(to)g(equilibrium)c(in)k(random)f(spin)h (systems)g(on)398 4513 y(a)i(lattice)f(I)r(I,)i Fg(J.)f(Stat.)h(Ph)n (ys.)f Ff(86)p Fh(,)g(899{904.)182 4688 y([113])44 b(H\177)-42 b(aggstr\177)g(om,)20 b(O.)i(\(1995\))f(Random-cluster)d(measures)i (and)i(uniform)e(spanning)h(trees,)h Fg(Sto)r(c)n(h.)g(Pro)r(c.)398 4787 y(Appl.)27 b Ff(59)p Fh(,)g(267{275.)182 4962 y([114])44 b(H\177)-42 b(aggstr\177)g(om,)33 b(O.)i(\(1996\))e(The)i (random-cluster)c(mo)r(del)h(on)j(a)f(homogeneous)e(tree,)k Fg(Probab.)d(Th.)398 5061 y(Rel.)26 b(Fields)g Ff(104)p Fh(,)h(231{253.)182 5236 y([115])44 b(H\177)-42 b(aggstr\177)g(om,)28 b(O.)i(\(1996\))f(Almost)f(sure)i(quasilo)r(calit)n(y)24 b(fails)k(for)i(the)h(random-cluster)26 b(mo)r(del)j(on)h(a)398 5335 y(tree,)d Fg(J.)g(Stat.)h(Ph)n(ys.)f Ff(84)p Fh(,)g(1351{1361.)182 5509 y([116])44 b(H\177)-42 b(aggstr\177)g(om,)24 b(O.)j(\(1996\))f(A)h (note)h(on)e(\(non-\)monotonicit)n(y)e(in)i(temp)r(erature)g(for)h(the) g(Ising)f(mo)r(del,)398 5609 y Fg(Mark)n(o)n(v)f(Pro)r(c.)h(Rel.)h (Fields)f Ff(2)p Fh(,)i(529{537.)1822 5895 y Fw(112)p eop %%Page: 113 113 113 112 bop 182 304 a Fh([117])44 b(H\177)-42 b(aggstr\177)g(om,)34 b(O.)h(\(1997\))f(Ergo)r(dicit)n(y)d(of)k(the)h(hard)e(core)g(mo)r(del) g(on)g Ff(Z)2726 274 y Fe(2)2799 304 y Fh(with)h(parit)n(y-dep)r(enden) n(t)398 404 y(activities,)24 b Fg(Ark.)j(Mat.)h Ff(35)p Fh(,)f(171{184.)182 578 y([118])44 b(H\177)-42 b(aggstr\177)g(om,)26 b(O.)i(\(1997\))f(In\014nite)h(clusters)f(in)g(dep)r(enden)n(t)j (automorphism)24 b(in)n(v)-5 b(arian)n(t)25 b(p)r(ercolation)398 678 y(on)i(trees,)g Fg(Ann.)h(Probab.)e Ff(25)p Fh(,)i(1423{1436.)182 852 y([119])44 b(H\177)-42 b(aggstr\177)g(om,)27 b(O.)i(\(1998\))f (Random-cluster)e(represen)n(tations)h(in)h(the)i(study)g(of)f(phase)g (transitions,)398 952 y Fg(Mark)n(o)n(v)c(Pro)r(c.)h(Rel.)h(Fields)f Ff(4)p Fh(,)i(275{321.)182 1126 y([120])44 b(H\177)-42 b(aggstr\177)g(om,)30 b(O.)i(and)f(Jonasson,)h(J.)f(\(1998\))g(Phase)g (transition)e(in)j(the)g(random)e(triangle)f(mo)r(del,)398 1226 y Fg(J.)e(Appl.)g(Probab.)p Fh(,)f(to)i(app)r(ear.)182 1400 y([121])44 b(H\177)-42 b(aggstr\177)g(om,)23 b(O.,)j(v)-5 b(an)26 b(Lieshout,)f(M.N.M.)h(and)g(M\034ller,)d(J.)j(\(1997\))f (Characterisation)c(results)j(and)398 1500 y(Mark)n(o)n(v)j(c)n(hain)i (Mon)n(te)g(Carlo)e(algorithms)f(including)h(exact)i(sim)n(ulation)c (for)30 b(some)e(spatial)f(p)r(oin)n(t)398 1599 y(pro)r(cesses,)f Fg(Bernoulli)p Fh(,)e(to)j(app)r(ear.)182 1774 y([122])44 b(H\177)-42 b(aggstr\177)g(om,)20 b(O.)i(and)g(P)n(eres,)f(Y.)i (\(1999\))d(Monotonicit)n(y)f(of)j(uniqueness)f(for)h(p)r(ercolation)d (on)i(Ca)n(yley)398 1873 y(graphs:)31 b(all)17 b(in\014nite)h(clusters) g(are)g(b)r(orn)h(sim)n(ultaneously)-7 b(,)17 b Fg(Probab.)g(Th.)j (Rel.)f(Fields)e Ff(113)p Fh(,)k(273{285.)182 2047 y([123])44 b(H\177)-42 b(aggstr\177)g(om,)18 b(O.,)j(Sc)n(honmann,)f(R.H.)g(and)g (Steif,)h(J.E.)e(\(1999\))g(The)h(Ising)e(mo)r(del)g(on)h(diluted)g (graphs)398 2147 y(and)27 b(strong)f(amenabilit)n(y)-7 b(,)24 b(preprin)n(t.)182 2321 y([124])44 b(Hammersley)-7 b(,)20 b(J.M.)k(\(1957\))e(P)n(ercolation)d(pro)r(cesses.)j(Lo)n(w)n (er)g(b)r(ounds)h(for)g(the)h(critical)c(probabilit)n(y)-7 b(,)398 2421 y Fg(Ann.)28 b(Math.)g(Statist.)f Ff(28)p Fh(,)g(790{795.)182 2595 y([125])44 b(Hammersley)-7 b(,)25 b(J.M.)k(\(1957\))e(P)n(ercolation)d(pro)r(cesses)j(I)r(I.)i(The)g (connectiv)n(e)e(constan)n(t,)h Fg(Pro)r(c.)g(Cam-)398 2695 y(bridge)e(Phil.)f(So)r(c.)j Ff(53)p Fh(,)f(642{645.)182 2869 y([126])44 b(Hara,)24 b(T.)h(and)g(Slade,)g(G.)g(\(1994\))f(Mean)h (\014eld)f(b)r(eha)n(vior)f(and)i(the)g(lace)f(expansion,)g(in)g Fg(Probabilit)n(y)398 2969 y(and)g(Phase)g(T)-7 b(ransition)22 b Fh(\(ed.)j(G.)h(Grimmett\),)d(87{122,)g(Pro)r(ceedings)f(of)j(the)g (NA)-7 b(TO)25 b(ASI)g(meeting)398 3069 y(in)i(Cam)n(bridge)d(1993,)i (Klu)n(w)n(er.)182 3243 y([127])44 b(Hara,)33 b(T.)f(and)h(Slade,)g(G.) g(\(1995\))e(The)i(self-a)n(v)n(oiding)27 b(w)n(alk)k(and)i(p)r (ercolation)c(critical)g(p)r(oin)n(ts)j(in)398 3343 y(high)26 b(dimensions,)e Fg(Com)n(bin.)i(Probab.)g(Comput.)h Ff(4)p Fh(,)g(197{215.)182 3517 y([128])44 b(Harris,)37 b(T.E.)f(\(1960\))g(A) h(lo)n(w)n(er)d(b)r(ound)k(on)e(the)h(critical)d(probabilit)n(y)e(in)37 b(a)f(certain)f(p)r(ercolation)398 3617 y(pro)r(cess,)26 b Fg(Pro)r(c.)h(Cam)n(bridge)d(Phil.)h(So)r(c.)p Ff(56)p Fh(,)i(13{20.)182 3791 y([129])44 b(Heilmann,)32 b(O.J.)h(\(1974\))g (The)g(use)h(of)g(re\015ection)e(as)h(symmetry)e(op)r(eration)h(in)h (connection)f(with)398 3891 y(P)n(eierls')24 b(argumen)n(t,)h Fg(Comm.)h(Math.)i(Ph)n(ys.)e Ff(36)p Fh(,)i(91{114.)182 4065 y([130])44 b(Higuc)n(hi,)25 b(Y.)k(\(1981\))d(On)i(the)g(absence)f (of)h(non-translation)c(in)n(v)-5 b(arian)n(t)24 b(Gibbs)k(states)f (for)g(the)h(t)n(w)n(o-)398 4165 y(dimensional)23 b(Ising)j(mo)r(del,)f (in)i(:)37 b(J.)27 b(F)-7 b(ritz,)27 b(J.L.)g(Leb)r(o)n(witz)f(and)h (D.)h(Sz\023)-42 b(asz)27 b(\(eds.\),)h Fg(Random)e(\014elds)p Fh(,)398 4264 y(Esztergom)e(\(Hungary\))k(1979.)d(Amsterdam:)35 b(North-Holland,)25 b(V)-7 b(ol.)26 b(I,)i(517{534.)182 4439 y([131])44 b(Higuc)n(hi,)23 b(Y.)j(\(1985\))f(A)g(w)n(eak)g(v)n (ersion)e(of)i(RSW)h(theorem)e(for)h(the)h(t)n(w)n(o-dimensional)20 b(Ising)k(mo)r(del,)398 4538 y(in:)32 b(R.)21 b(Durrett)g(\(ed.\),)i(P) n(article)18 b(systems,)i(random)f(media,)h(and)h(large)d(deviations.)g Fg(Con)n(temp)r(orary)398 4638 y(Math.)27 b Ff(41)p Fh(,)h(Amer.)e (Math.)i(So)r(c.,)g(207{214.)182 4812 y([132])44 b(Higuc)n(hi,)23 b(Y.)j(\(1993\))e(Co)r(existence)f(of)j(in\014nite)d(\(*\){clusters)h (I)r(I:{Ising)f(p)r(ercolation)f(in)j(t)n(w)n(o)f(dimen-)398 4912 y(sion,)i Fg(Prob.)g(Th.)i(Rel.)f(Fields)f Ff(97)p Fh(,)h(1{33.)182 5086 y([133])44 b(Higuc)n(hi,)34 b(Y.)h(\(1993\))f(A)h (sharp)f(transition)e(for)i(the)h(t)n(w)n(o-dimensional)30 b(Ising)j(p)r(ercolation,)g Fg(Prob.)398 5186 y(Th.)27 b(Rel.)g(Fields)f Ff(97)p Fh(,)h(489{514.)182 5360 y([134])44 b(Holley)-7 b(,)24 b(R.)j(\(1974\))e(Remarks)f(on)i(the)g(FK)n(G)g (inequalities,)d Fg(Comm)n(un.)h(Math.)i(Ph)n(ys.)f Ff(36)p Fh(,)i(227{231.)182 5534 y([135])44 b(Hu,)25 b(C.-K.)f(and)g(Mak,)h (K.-S.)f(\(1989\))f(P)n(ercolation)d(and)k(phase)g(transitions)e(of)i (hard{core)e(particles)398 5634 y(on)27 b(lattices)e(:)37 b(Mon)n(te)28 b(Carlo)d(approac)n(h,)h Fg(Ph)n(ys.)g(Rev.)i(B)f Ff(39)p Fh(,)h(2948{2951.)1822 5895 y Fw(113)p eop %%Page: 114 114 114 113 bop 182 304 a Fh([136])44 b(Hu,)25 b(C.-K.)f(and)g(Mak,)h (K.-S.)f(\(1990\))f(P)n(ercolation)d(and)k(phase)g(transitions)e(of)i (hard{core)e(particles)398 404 y(on)27 b(lattices)e(with)j(pair)d(in)n (teractions,)g Fg(Ph)n(ys.)h(Rev.)i(B)g Ff(42)p Fh(,)f(965{968.)182 576 y([137])44 b(Ising,)26 b(E.)h(\(1925\))f(Beitrag)g(zur)h(Theorie)e (der)j(F)-7 b(erromagnetism)n(us,)22 b Fg(Z.)28 b(Ph)n(ys.)e Ff(31)p Fh(,)i(253{258.)182 749 y([138])44 b(Israel,)30 b(R.B.)h(\(1979\))f(Con)n(v)n(exit)n(y)f(in)h(the)i(theory)f(of)g (lattice)e(gases,)i(Princeton)e(Univ)n(ersit)n(y)g(Press,)398 848 y(Princeton.)182 1021 y([139])44 b(Jonasson,)24 b(J.)i(\(1999\))f (The)h(random-cluster)c(mo)r(del)i(on)i(a)f(general)f(graph)h(and)h(a)f (phase)h(transition)398 1120 y(c)n(haracterization)d(of)k (nonamenabilit)n(y)-7 b(,)23 b Fg(Sto)r(c)n(h.)28 b(Pro)r(c.)e(Appl.)i Ff(79)p Fh(,)f(335{354.)182 1293 y([140])44 b(Jonasson,)25 b(J.)j(and)f(Steif,)g(J.)h(\(1997\))e(Amenabilit)n(y)e(and)j(phase)g (transition)e(in)i(the)h(Ising)e(mo)r(del,)f Fg(J.)398 1392 y(Theor.)h(Probab.)p Fh(,)h(to)g(app)r(ear.)182 1565 y([141])44 b(Kamae,)39 b(T.,)i(Krengel,)e(U.)g(and)f(O'Brien,)i (G.L.)f(\(1977\))e(Sto)r(c)n(hastic)g(inequalities)d(on)k(partially)398 1664 y(ordered)26 b(spaces,)h Fg(Ann.)h(Probab.)e Ff(5)p Fh(,)h(899{912.)182 1837 y([142])44 b(Kesten,)23 b(H.)g(\(1980\))f(The) h(critical)c(probabilit)n(y)g(of)k(b)r(ond)g(p)r(ercolation)d(on)i(the) i(square)d(lattice)g(equals)408 1904 y Fe(1)p 408 1918 34 4 v 408 1965 a(2)451 1936 y Fh(,)27 b Fg(Comm)n(un.)f(Math.)i(Ph)n (ys.)e Ff(74)p Fh(,)i(41{59.)182 2109 y([143])44 b(Kesten,)27 b(H.)h(\(1982\))e Fg(P)n(ercolation)d(theory)k(for)g(mathematicians)p Fh(.)d(Boston)i(MA,)i(Birkhauser.)182 2281 y([144])44 b(Kesten,)33 b(H.)f(\(1990\))f(Asymptotics)f(in)h(high)g(dimensions)e (for)j(p)r(ercolation,)e Fg(Disorder)g(in)h(ph)n(ysical)398 2381 y(systems)26 b Fh(\(eds)i(G.)f(Grimmett)f(and)h(D.)h(W)-7 b(elsh\),)27 b(219{240,)e(Oxford)h(Univ)n(ersit)n(y)f(Press.)182 2553 y([145])44 b(Klain,)35 b(D.A.)i(and)f(Rota,)i(G.-C.)e(\(1997\))e Fg(In)n(tro)r(duction)h(to)h(geometric)c(probabilit)n(y)p Fh(,)j(Cam)n(bridge)398 2653 y(Univ)n(ersit)n(y)24 b(Press,)i(Cam)n (bridge.)182 2825 y([146])44 b(Kotec)n(k)q(\023)-43 b(y,)27 b(R.)j(\(1994\))d(Geometric)f(represen)n(tation)g(of)j(lattice)e(mo)r (dels)g(and)i(large)d(v)n(olume)g(asymp-)398 2925 y(totics,)d Fg(Probabilit)n(y)d(and)j(Phase)g(T)-7 b(ransition)21 b Fh(\(ed.)k(G.)f(Grimmett\),)f(Klu)n(w)n(er,)f(Dordrec)n(h)n(t,)i (153{176.)182 3098 y([147])44 b(Kotec)n(k)q(\023)-43 b(y,)26 b(R.)i(\(1995\))e(Phase)h(transitions)d(of)k(lattice)d(mo)r (dels.)h(Univ)n(ersit)n(\023)-39 b(e)24 b(de)k(Rennes)f(I)h(lectures.) 182 3270 y([148])44 b(Kotec)n(k)q(\023)-43 b(y,)23 b(R.)g(and)h (Shlosman,)d(S.)j(\(1982\))e(First-order)f(phase)i(transitions)d(in)j (large)e(en)n(trop)n(y)h(lattice)398 3370 y(systems,)k Fg(Comm)n(un.)f(Math.)j(Ph)n(ys.)f Ff(83)p Fh(,)g(493{550.)182 3542 y([149])44 b(Laanait,)22 b(L.,)j(Messager,)e(A.,)i(Miracle-Sole,) 20 b(S.,)25 b(Ruiz,)f(J.)g(and)f(Shlosman,)f(S.)j(\(1991\))d(In)n (terfaces)h(in)398 3642 y(the)32 b(P)n(otts)e(mo)r(del)g(I:)h(Pirogo)n (v{Sinai)25 b(theory)31 b(of)g(the)h(F)-7 b(ortuin{Kasteleyn)28 b(represen)n(tation,)i Fg(Com-)398 3741 y(m)n(un.)d(Math.)g(Ph)n(ys.)g Ff(140)p Fh(,)g(81{91.)182 3914 y([150])44 b(Laanait,)25 b(L.,)j(Messager,)e(A.)i(and)g(Ruiz,)f(J.)g(\(1986\))g(Phase)f(co)r (existence)g(and)i(surface)f(tensions)f(for)398 4013 y(the)i(P)n(otts)e(mo)r(del,)g Fg(Comm)n(un.)g(Math.)h(Ph)n(ys.)g Ff(105)p Fh(,)g(527{545.)182 4186 y([151])44 b(Lanford,)36 b(O.)g(and)f(Ruelle,)h(D.)g(\(1969\))f(Observ)-5 b(ables)33 b(at)i(in\014nit)n(y)f(and)h(states)h(with)f(short)g(range)398 4285 y(correlations)23 b(in)k(statistical)d(mec)n(hanics,)h Fg(Comm)n(un.)g(Math.)j(Ph)n(ys.)f Ff(13)p Fh(,)g(194{215.)182 4458 y([152])44 b(Leb)r(o)n(witz,)26 b(J.L.)h(\(1972\))f(Bounds)h(on)g (the)g(correlations)c(and)k(analyticit)n(y)d(prop)r(erties)h(of)i (Ising)f(spin)398 4557 y(systems,)g Fg(Comm)n(un.)f(Math.)j(Ph)n(ys.)f Ff(28)p Fh(,)g(313{321.)182 4730 y([153])44 b(Leb)r(o)n(witz,)20 b(J.L.)g(and)g(Galla)n(v)n(otti,)d(G.)j(\(1971\))f(Phase)g(transitions) e(in)j(binary)e(lattice)g(gases,)i Fg(J.)g(Math.)398 4830 y(Ph)n(ys.)26 b Ff(12)p Fh(,)i(1129{1133.)182 5002 y([154])44 b(Leb)r(o)n(witz,)34 b(J.L.)h(and)f(Lieb,)h(E.H.)f(\(1972\)) f(Phase)g(transition)f(in)h(a)h(con)n(tin)n(uum)f(classical)c(system) 398 5102 y(with)e(\014nite)g(in)n(teractions,)d Fg(Ph)n(ys.)j(Lett.)h Ff(39A)p Fh(,)g(98{100.)182 5274 y([155])44 b(Leb)r(o)n(witz,)27 b(J.L.)g(and)h(Martin-L\177)-42 b(of,)27 b(A.)h(\(1972\))f(On)h(the)g (uniqueness)f(of)h(the)h(equilibrium)23 b(state)28 b(for)398 5374 y(Ising)e(spin)h(systems,)f Fg(Comm)n(un.)f(Math.)j(Ph)n(ys.)f Ff(25)p Fh(,)g(276{282.)182 5546 y([156])44 b(Leb)r(o)n(witz,)c(J.L.)f (and)f(P)n(enrose,)i(O.)e(\(1977\))g(Cluster)f(and)i(p)r(ercolation)c (inequalities)g(for)j(lattice)398 5646 y(systems)26 b(with)h(in)n (teractions,)d Fg(J.)k(Stat.)g(Ph)n(ys.)e Ff(16)p Fh(,)i(321{337.)1822 5895 y Fw(114)p eop %%Page: 115 115 115 114 bop 182 304 a Fh([157])44 b(Leb)r(o)n(witz,)27 b(J.L.)h(and)g(Saleur,)e(H.)j(\(1986\))e(P)n(ercolation)c(in)28 b(strongly)e(correlated)f(systems,)i Fg(Ph)n(ysica)398 404 y Ff(138)p Fh(A,)g(194{205.)182 577 y([158])44 b(Lenz,)k(W.)d (\(1920\))e(Beitrag)e(zum)j(v)n(erst\177)-42 b(andnis)42 b(der)i(magnetisc)n(hen)d(Ersc)n(hein)n(ungen)g(in)j(festen)398 677 y(K\177)-42 b(orp)r(ern,)26 b Fg(Ph)n(ys.)h(Zeitsc)n(hr.)e Ff(21)p Fh(,)j(613{615.)182 851 y([159])44 b(Li,)24 b(X.-J.)g(and)g (Sok)-5 b(al,)24 b(A.)g(\(1989\))g(Rigorous)d(lo)n(w)n(er)h(b)r(ounds)i (on)g(the)h(dynamical)c(critical)g(exp)r(onen)n(ts)398 950 y(of)27 b(the)h(Sw)n(endsen{W)-7 b(ang)27 b(algorithm,)c Fg(Ph)n(ys.)k(Rev.)g(Lett.)h Ff(63)p Fh(,)g(827{830.)182 1124 y([160])44 b(Liggett,)26 b(T.M.)i(\(1985\))e Fg(In)n(teracting)g (P)n(article)e(Systems)p Fh(,)i(Springer,)g(New)i(Y)-7 b(ork.)182 1298 y([161])44 b(Liggett,)30 b(T.M.,)i(Sc)n(honmann,)e (R.H.)h(and)f(Stacey)-7 b(,)32 b(A.M.)f(\(1997\))e(Domination)f(b)n(y)i (pro)r(duct)h(mea-)398 1397 y(sures,)26 b Fg(Ann.)j(Probab.)d Ff(25)p Fh(,)h(71{96.)182 1571 y([162])44 b(Lindv)-5 b(all,)24 b(T.)k(\(1992\))e Fg(Lectures)h(on)h(the)g(Coupling)d(Metho)r (d)p Fh(,)j(Wiley)-7 b(,)26 b(New)i(Y)-7 b(ork.)182 1745 y([163])44 b(L\177)-42 b(orinczi,)33 b(J.,)j(Maes,)g(C.)e(and)g(V)-7 b(ande)35 b(V)-7 b(elde)34 b(K.)g(\(1998\))f(T)-7 b(ransformations)30 b(of)35 b(Gibbs)e(measures,)398 1844 y Fg(Prob.)26 b(Th.)i(Rel.)f (Fields)e Ff(112)p Fh(,)i(121{147.)182 2018 y([164])44 b(Ly)n(ons,)26 b(R.)i(\(1990\))e(Random)h(w)n(alks)e(and)i(p)r (ercolation)e(on)i(trees,)g Fg(Ann.)i(Probab.)d Ff(18)p Fh(,)h(931{958.)182 2192 y([165])44 b(Mac)n(h)n(ta,)32 b(J.,)g(Choi,)g(Y.S.,)i(Luc)n(k)n(e,)e(A.,)h(Sc)n(h)n(w)n(eizer,)e(T.)g (and)h(Cha)n(y)n(es,)g(L.)g(\(1995\))e(In)n(v)-5 b(aded)32 b(cluster)398 2292 y(algorithm)23 b(for)k(equilibrium)c(critical)h(p)r (oin)n(ts,)i Fg(Ph)n(ys.)h(Rev.)h(Lett.)g Ff(75)p Fh(,)f(2792{2795.)182 2465 y([166])44 b(Maes,)27 b(C.)g(\(1993\))g(Coupling)e(in)n(teracting) f(particle)h(systems,)i Fg(Rev.)g(Math.)h(Ph)n(ys.)f Ff(5)p Fh(,)g(457{475.)182 2639 y([167])44 b(Maes,)24 b(C.)h(and)f(K.)g(V)-7 b(ande)24 b(V)-7 b(elde)24 b(\(1997\))f(Relativ) n(e)f(energies)g(for)i(non-Gibbsian)e(states,)i Fg(Comm)n(un.)398 2739 y(Math.)j(Ph)n(ys.)g Ff(189)p Fh(,)g(277{286.)182 2912 y([168])44 b(Martinelli,)22 b(F.)k(\(1997\))f(Lectures)h(on)g (Glaub)r(er)f(dynamics)e(for)j(discrete)f(spin)g(mo)r(dels.)f(Sain)n (t-Flour)398 3012 y(Lecture)j(Notes.)182 3186 y([169])44 b(Martinelli,)31 b(F.)k(and)f(Olivieri,)d(E.)j(\(1994\))f(Approac)n(h)g (to)h(equilibrium)29 b(of)34 b(Glaub)r(er)f(dynamics)f(in)398 3285 y(the)c(one)f(phase)g(region.)e(I.)j(The)g(attractiv)n(e)d(case.)i Fg(Comm)n(un.)e(Math.)j(Ph)n(ys.)f Ff(161)p Fh(,)g(447{486.)182 3459 y([170])44 b(Martinelli,)31 b(F.)k(and)f(Olivieri,)d(E.)j (\(1994\))f(Approac)n(h)g(to)h(equilibrium)29 b(of)34 b(Glaub)r(er)f(dynamics)f(in)398 3559 y(the)c(one)f(phase)g(region)e(I) r(I:)j(The)g(general)d(case,)i Fg(Comm)n(un.)e(Math.)j(Ph)n(ys.)f Ff(161)p Fh(,)g(487.)182 3732 y([171])44 b(McCo)n(y)-7 b(,)42 b(B.M.)d(and)h(W)-7 b(u,)43 b(T.T.)d(\(1973\))f Fg(The)g(t)n(w)n(o-dimensional)34 b(Ising)k(mo)r(del)p Fh(,)j(Harv)-5 b(ard)39 b(Univ.)398 3832 y(Press,)26 b(Cam)n(bridge,)e(Mass.)182 4006 y([172])44 b(Mec)n(k)n(e,)26 b(K.R.)i(\(1998\))e(In)n(tegral)f(geometry)h(and)h(statistical)d(ph)n (ysics,)i(preprin)n(t.)182 4179 y([173])44 b(Meester,)27 b(R.)h(and)f(Ro)n(y)-7 b(,)27 b(R.)h(\(1996\))e Fg(Con)n(tin)n(uum)g(P) n(ercolation)p Fh(,)d(Cam)n(bridge)h(Univ)n(ersit)n(y)h(Press.)182 4353 y([174])44 b(Menshik)n(o)n(v,)19 b(M.V.)i(\(1986\))e(Coincidence)f (of)j(critical)16 b(p)r(oin)n(ts)k(in)f(p)r(ercolation)f(mo)r(dels,)i Fg(So)n(viet)e(Math-)398 4453 y(ematics)25 b(Doklady)h Ff(33)p Fh(,)h(856{859.)182 4626 y([175])44 b(Menshik)n(o)n(v,)33 b(M.V.,)k(Molc)n(hano)n(v,)c(S.A.)i(and)f(Sido)n(v)n(enk)n(o,)f(S.A.)i (\(1986\))e(P)n(ercolation)d(theory)j(and)398 4726 y(some)k (applications,)i Fg(Itogi)f(Nauki)g(i)g(T)-7 b(ec)n(hniki)37 b Fh(\(Series)h(of)h(Probabilit)n(y)c(theory)-7 b(,)42 b(Mathematical)398 4826 y(statistics)25 b(and)i(Theoretical)e(cyb)r (ernetics\))h Ff(24)p Fh(,)i(53{110.)182 4999 y([176])44 b(Menshik)n(o)n(v,)27 b(M.V.)i(and)g(P)n(elikh,)d(K.D.)j(\(1989\))f(P)n (ercolation)d(with)j(sev)n(eral)e(defect)k(t)n(yp)r(es,)f(an)g(esti-) 398 5099 y(mate)21 b(of)h(critical)c(probabilit)n(y)g(for)j(a)h(square) f(lattice,)g Fg(Math.)h(Notes)g(Acad.)g(Sc.)g(USSR)h Ff(46)p Fh(,)g(778{785.)182 5273 y([177])44 b(Messager,)24 b(A.)j(and)g(Miracle-Sole,)22 b(S.)k(\(1975\))g(Equilibrium)21 b(states)26 b(of)g(the)h(t)n(w)n(o-dimensional)21 b(Ising)398 5372 y(mo)r(del)k(in)i(the)h(t)n(w)n(o-phase)e(region,)g Fg(Comm)n(un.)f(Math.)j(Ph)n(ys.)e Ff(40)p Fh(,)i(187{196.)182 5546 y([178])44 b(Metrop)r(olis,)27 b(N.,)32 b(Rosen)n(bluth,)d(A.W.,)i (Rosen)n(bluth,)f(M.N.,)h(T)-7 b(eller,)28 b(A.H.)j(and)f(T)-7 b(eller,)28 b(E.)i(\(1953\))398 5646 y(Equations)18 b(of)i(state)g (calculations)c(b)n(y)k(fast)g(computing)e(mac)n(hine,)h Fg(J.)h(Chem.)g(Ph)n(ys.)f Ff(21)p Fh(,)j(1087{1091.)1822 5895 y Fw(115)p eop %%Page: 116 116 116 115 bop 182 304 a Fh([179])44 b(Minlos)25 b(R.A.)j(\(1967\))e (Gibbs')h(limit)e(distribution,)f Fg(F)-7 b(unct.)29 b(Anal.)e(Appl.)g Ff(1)p Fh(,)g(60{73.)182 478 y([180])44 b(S.)22 b(Miracle)d(Sole)i(\(1994\))g(On)g(the)i(microscopic)17 b(theory)22 b(of)g(phase)f(transitions,)f(in)i Fg(Pro)r(c.)f(XI)r(I)r (I)i(Sitges)398 578 y(Conf.)k Fh(\\25)g(y)n(ears)e(of)j(statistical)c (mec)n(hanics.")182 752 y([181])44 b(Molc)n(hano)n(v,)23 b(S.A.)j(and)f(Stepano)n(v,)h(A.K.)f(\(1983\))f(P)n(ercolation)e(in)i (random)g(\014elds.)h(I,)g Fg(T)-7 b(eoret.)25 b(Mat.)398 852 y(Fiz.)i Ff(55)p Fh(,)g(246{256)d(&)j(419{430.)182 1026 y([182])44 b(Newman,)19 b(C.M.)g(\(1994\))f(Disordered)f(Ising)g (systems)g(and)i(random)e(cluster)g(represen)n(tations,)h Fg(Prob-)398 1126 y(abilit)n(y)37 b(and)j(Phase)f(T)-7 b(ransition)37 b Fh(\(ed.)k(G.)f(Grimmett\),)i(NA)-7 b(TO)40 b(ASI)h(series)d(v)n(ol.)h(120,)j(Klu)n(w)n(er,)398 1226 y(Dordrec)n(h)n(t,)26 b(247{260.)182 1400 y([183])44 b(Newman,)23 b(C.M.)i(and)e(Sc)n(h)n(ulman,)g(L.S.)h(\(1981\))f (In\014nite)g(clusters)f(in)i(p)r(ercolation)d(mo)r(dels,)h Fg(J.)i(Stat.)398 1500 y(Ph)n(ys.)i Ff(26)p Fh(,)i(613{628.)182 1674 y([184])44 b(Nielaba,)26 b(P)-7 b(.)28 b(and)g(Leb)r(o)n(witz,)f (J.L.)h(\(1997\))f(Phase)g(transitions)f(in)h(the)i(m)n(ulticomp)r (onen)n(t)24 b(Widom{)398 1774 y(Ro)n(wlinson)g(mo)r(del)i(and)h(in)g (hard)g(cub)r(es)h(on)f(the)h(BCC-lattice,)d(preprin)n(t.)182 1948 y([185])44 b(Olivieri,)21 b(E.,)k(P)n(erez,)f(F.)i(and)f(Goulart)e (Rosa)h(Jr.,)h(S.)h(\(1983\))e(Some)g(rigorous)e(results)h(on)i(the)h (phase)398 2047 y(diagram)e(of)j(the)h(dilute)f(Ising)f(mo)r(del,)f Fg(Ph)n(ys.)i(Lett.)h Ff(94A)p Fh(,)g(309.)182 2222 y([186])44 b(Onsager,)25 b(L.)j(\(1944\))e(Crystal)g(statistics,)f(I.)j(A)g(t)n(w) n(o-dimensional)22 b(mo)r(del)k(with)h(an)g(order-disorder)398 2321 y(transformation,)d Fg(Ph)n(ys.)i(Rev.)i Ff(65)p Fh(,)f(117{149.)182 2496 y([187])44 b(P)n(eierls,)31 b(R.)i(\(1936\))f(On)h(Ising's)e(mo)r(del)h(of)h(ferromagnetism,)d Fg(Pro)r(c.)i(Cam)n(bridge)d(Phil.)j(So)r(c.)h Ff(32)p Fh(,)398 2595 y(477{481.)182 2770 y([188])44 b(P)n(enrose,)26 b(M.D.)i(\(1991\):)36 b(On)28 b(a)f(con)n(tin)n(uum)f(p)r(ercolation)f (mo)r(del,)h Fg(Adv.)i(Appl.)f(Probab.)g Ff(23)p Fh(,)g(536{)398 2869 y(556.)182 3044 y([189])44 b(P)n(erez,)21 b(J.F.)h(\(1993\))f(Con) n(trolling)c(the)23 b(e\013ect)f(of)g(Gri\016ths')f(singularities)c(in) k(random)f(ferromagnets,)398 3143 y Fg(Braz.)26 b(J.)i(Ph)n(ys.)e Ff(23)p Fh(,)h(356-362.)182 3318 y([190])44 b(P)n(etritis,)28 b(D.)i(\(1996\))f(Equilibrium)c(statistical)i(mec)n(hanics)g(of)j (frustrated)g(spin)f(glasses:)40 b(a)30 b(surv)n(ey)398 3417 y(of)d(mathematical)c(results,)k Fg(Ann.)h(Inst.)g(H.)g(P)n (oincar)n(\023)-39 b(e)23 b(Ph)n(ys.)k(Theorique)e Ff(64)p Fh(,)i(255{288.)182 3592 y([191])44 b(P\014ster,)23 b(C.-E.)f(and)h(V) -7 b(ande)23 b(V)-7 b(elde,)24 b(K.)f(\(1995\))e(Almost)h(sure)g (quasilo)r(calit)n(y)c(in)k(the)i(random)d(cluster)398 3691 y(mo)r(del,)k Fg(J.)j(Statist.)f(Ph)n(ys.)g Ff(79)p Fh(,)g(765{774.)182 3866 y([192])44 b(P\014ster,)25 b(C.-E.)g(and)g(V) -7 b(elenik,)24 b(Y.)i(\(1997\))e(Random-cluster)e(represen)n(tation)h (of)i(the)h(Ashkin{T)-7 b(eller)398 3965 y(mo)r(del,)25 b Fg(J.)j(Stat.)g(Ph)n(ys.)e Ff(88)p Fh(,)i(1295{1331.)182 4140 y([193])44 b(Pirogo)n(v,)23 b(S.A.)j(and)h(Sinai,)d(Y.G.)j (\(1976\))e(Phase)g(diagrams)e(for)j(classical)c(lattice)i(systems,)h Fg(Theor.)398 4239 y(Math.)i(Ph)n(ys.)g Ff(25)p Fh(,)g(1185{1192)182 4414 y([194])44 b(P)n(otts,)d(R.B.)e(\(1952\))f(Some)f(generalized)f (order-disorder)f(transformations,)j Fg(Pro)r(c.)g(Cam)n(bridge)398 4513 y(Phil.)25 b(So)r(c.)j Ff(48)p Fh(,)f(106{109.)182 4688 y([195])44 b(Preston,)30 b(C.J.)g(\(1974\))g Fg(Gibbs)g(states)g (on)g(coun)n(table)f(sets)p Fh(,)j(Cam)n(bridge)27 b(Univ)n(ersit)n(y)h (Press,)i(Cam-)398 4787 y(bridge.)182 4962 y([196])44 b(Preston,)27 b(C.J.)h(\(1974\))f(An)i(application)c(of)j(the)h(GHS)g (inequalit)n(y)24 b(to)29 b(sho)n(w)e(the)i(absence)e(of)i(phase)398 5061 y(transitions)24 b(for)j(Ising)f(spin)h(systems,)f Fg(Comm)n(un.)g(Math.)i(Ph)n(ys)p Fh(.)e Ff(35)p Fh(,)i(253{255.)182 5236 y([197])44 b(Preston,)23 b(C.J.)g(\(1974\))f(A)i(generalization)19 b(of)k(the)h(FK)n(G)f(inequalities,)d Fg(Comm)n(un.)h(Math.)j(Ph)n(ys)p Fh(.)f Ff(36)p Fh(,)398 5335 y(233{242.)182 5509 y([198])44 b(Preston,)26 b(C.J.)h(\(1976\))g Fg(Random)f(Fields)p Fh(,)g(Lect.)h(Notes)h(in)f(Math.)g Ff(534)p Fh(,)g(Springer{V)-7 b(erlag,)23 b(Berlin.)1822 5895 y Fw(116)p eop %%Page: 117 117 117 116 bop 182 304 a Fh([199])44 b(Preston,)d(C.J.)e(\(1977\))f (Spatial)f(birth-and-death)g(pro)r(cesses,)k Fg(Bull.)d(In)n(t.)h (Statist.)g(Inst.)h Ff(46)p Fh(\(2\),)398 404 y(371{391.)182 578 y([200])k(Propp,)33 b(J.G.)h(and)f(Wilson,)g(D.B.)g(\(1996\))g (Exact)f(sampling)e(with)j(coupled)f(Mark)n(o)n(v)f(c)n(hains)g(and)398 678 y(applications)23 b(to)28 b(statistical)c(mec)n(hanics,)h Fg(Random)h(Structures)h(Algorithms)d Ff(9)p Fh(,)k(223{252.)182 852 y([201])44 b(Radulescu,)32 b(D.C.)g(\(1997\))f Fg(A)i (computer-assisted)28 b(pro)r(of)k(of)g(uniqueness)f(of)h(phase)f(for)h (the)g(hard-)398 952 y(square)47 b(lattice)g(gas)g(mo)r(del)g(in)h(t)n (w)n(o)f(dimensions)p Fh(,)k(Rutgers)d(Univ)n(ersit)n(y)d(Ph.D.)k (thesis)f(\(New)398 1051 y(Brunswic)n(k,)25 b(New)j(Jersey\).)182 1226 y([202])44 b(Ruelle,)26 b(D.)i(\(1969\))e Fg(Statistical)f(Mec)n (hanics.)g(Rigorous)g(results)p Fh(,)h(Benjamin,)g(New)i(Y)-7 b(ork.)182 1400 y([203])44 b(Ruelle,)39 b(D.)f(\(1971\))f(An)h (extension)f(of)h(the)g(Lee{Y)-7 b(ang)37 b(circle)e(theorem,)k Fg(Ph)n(ys.)e(Rev.)h(Lett.)h Ff(26)p Fh(,)398 1500 y(303{304.)182 1674 y([204])44 b(Ruelle,)27 b(D.)j(\(1971\))e(Existence)f(of)i(a)g (phase)g(transition)d(in)i(a)h(con)n(tin)n(uous)e(classical)e(system,)j Fg(Ph)n(ys.)398 1774 y(Rev.)f(Lett.)h Ff(27)p Fh(,)g(1040{1041.)182 1948 y([205])44 b(Ruelle,)f(D.)f(\(1972\))e(On)h(the)g(use)g(of)h (\\small)37 b(external)j(\014elds")f(in)i(the)h(problem)c(of)k (symmetry)398 2047 y(breakdo)n(wn)26 b(in)g(statistical)f(mec)n (hanics,)g Fg(Ann.)j(Ph)n(ys.)e Ff(69)p Fh(,)i(364{374.)182 2222 y([206])44 b(Ruelle,)26 b(D.)i(\(1978\))e Fg(Thermo)r(dynamic)e(F) -7 b(ormalism)p Fh(,)23 b(Addison-W)-7 b(esley)g(,)25 b(Reading.)182 2396 y([207])44 b(Runnels,)25 b(L.K.)g(and)g(Leb)r(o)n (witz,)g(J.L.)g(\(1974\))f(Phase)g(transitions)f(of)i(a)g(m)n(ulticomp) r(onen)n(t)c(Widom{)398 2496 y(Ro)n(wlinson)j(mo)r(del,)i Fg(J.)h(Math.)h(Ph)n(ys.)f Ff(15)p Fh(,)g(1712{1717.)182 2670 y([208])44 b(Russo,)31 b(L.)g(\(1979\))f(The)i(in\014nite)e (cluster)f(metho)r(d)i(in)g(the)g(t)n(w)n(o-dimensional)26 b(Ising)j(mo)r(del,)i Fg(Com-)398 2770 y(m)n(un.)c(Math.)g(Ph)n(ys.)g Ff(67)p Fh(,)g(251{266.)182 2944 y([209])44 b(Russo,)25 b(L.)h(\(1981\))e(On)i(the)g(critical)c(p)r(ercolation)h (probabilities,)e Fg(Z.)26 b(W)-7 b(ahrsc)n(h.)25 b(V)-7 b(erw.)25 b(Gebiete)h Ff(56)p Fh(,)398 3044 y(229{237.)182 3218 y([210])44 b(Salas,)37 b(J.)g(and)g(Sok)-5 b(al,)38 b(A.D.)g(\(1996\))e(Dynamic)f(critical)f(b)r(eha)n(vior)h(of)i(a)g(Sw)n (endsen{W)-7 b(ang-t)n(yp)r(e)398 3318 y(algorithm)23 b(for)k(the)h(Ashkin{T)-7 b(eller)24 b(mo)r(del,)i Fg(J.)h(Stat.)h(Ph)n (ys.)f Ff(85)p Fh(,)g(297{361.)182 3492 y([211])44 b(Sepp\177)-42 b(al\177)g(ainen,)27 b(T.)j(\(1998\))e(En)n(trop)n(y)g(for)h (translation-in)n(v)-5 b(ari)o(an)m(t)24 b(random-cluster)i(measures,)i Fg(Ann.)398 3592 y(Probab.)e Ff(26)p Fh(,)h(1139{1178.)182 3766 y([212])44 b(Simon,)29 b(B.)i(\(1993\))e Fg(The)i(Statistical)d (Mec)n(hanics)g(of)j(Lattice)f(Gases,)g(V)-7 b(olume)29 b(1)p Fh(,)i(Princeton)e(Uni-)398 3866 y(v)n(ersit)n(y)c(Press,)h (Princeton.)182 4040 y([213])44 b(Sinai,)26 b(Y.G.)i(\(1982\))f Fg(Theory)g(of)h(Phase)e(T)-7 b(ransitions:)35 b(Rigorous)25 b(Results)p Fh(,)i(P)n(ergamon)e(Press,)h(Ox-)398 4140 y(ford.)182 4314 y([214])44 b(Sla)n(wn)n(y)-7 b(,)30 b(J.)i(\(1986\))e(Lo)n(w)g(temp)r(erature)h(prop)r(erties)e(of)j (classical)27 b(lattice)i(systems:)43 b(phase)31 b(transi-)398 4414 y(tions)e(and)i(phase)f(diagrams,)d Fg(Phase)j(T)-7 b(ransitions)27 b(and)k(Critical)c(Phemomena)g Ff(5)k Fh(\(C.)g(Dom)n(b)e(and)398 4513 y(J.L.)e(Leb)r(o)n(witz,)f(eds.\),)i (Academic)e(Press,)g(London)h(and)g(New)h(Y)-7 b(ork.)182 4688 y([215])44 b(Spitzer,)26 b(F.)i(\(1975\))f(Mark)n(o)n(v)e(random)h (\014elds)g(on)h(an)h(in\014nite)e(tree,)h Fg(Ann.)i(Probab.)d Ff(3)p Fh(,)h(387{398.)182 4862 y([216])44 b(Strassen,)38 b(V.)g(\(1965\))d(The)i(existence)f(of)h(probabilit)n(y)c(measures)i (with)i(giv)n(en)e(marginals,)g Fg(Ann.)398 4962 y(Math.)27 b(Statist.)g Ff(36)p Fh(,)h(423{439.)182 5136 y([217])44 b(Stro)r(o)r(c)n(k,)25 b(D.W.)i(and)f(Zegarlinski,)c(B.)k(\(1992\))e (The)i(logarithmic)21 b(Sob)r(olev)k(inequalit)n(y)d(for)k(discrete)398 5236 y(systems)g(on)h(a)g(lattice,)f Fg(Comm)n(un.)g(Math.)h(Ph)n(ys.)g Ff(149)p Fh(,)g(175{193.)182 5410 y([218])44 b(Sw)n(een)n(y)-7 b(,)37 b(M.)g(\(1983\))e(Mon)n(te)h(Carlo)e(study)i(of)g(w)n(eigh)n (ted)f(p)r(ercolation)e(clusters)h(relev)-5 b(an)n(t)35 b(to)h(the)398 5509 y(P)n(otts)26 b(mo)r(del,)g Fb(Phys.)31 b(R)l(ev.)d Ff(27)p Fh(,)f(4445{4455.)1822 5895 y Fw(117)p eop %%Page: 118 118 118 117 bop 182 304 a Fh([219])44 b(Sw)n(endsen,)33 b(R.H.)g(and)f(W)-7 b(ang,)34 b(J.-S.)e(\(1987\))f(Non)n(univ)n(ersal)d(critical)h (dynamics)h(in)i(Mon)n(te)g(Carlo)398 404 y(sim)n(ulations.)23 b Fg(Ph)n(ys.)j(Rev.)i(Lett.)g Ff(58)p Fh(,)f(86{88.)182 578 y([220])44 b(Thompson,)58 b(C.J.)52 b(\(1972\))g Fg(Mathematical)e(Statistical)g(Mec)n(hanics)p Fh(,)57 b(McMillan,)g(New)c(Y)-7 b(ork,)398 678 y(reprin)n(ted)26 b(b)n(y)h(Princeton)e(Univ.)j(Press.)182 852 y([221])44 b(W)-7 b(elsh,)36 b(D.J.A.)g(\(1993\))d(P)n(ercolation)e(in)j(the)i (random)d(cluster)h(pro)r(cess)g(and)h(the)g Fd(Q)p Fh(-state)f(P)n (otts)398 952 y(mo)r(del,)25 b Fg(J.)j(Ph)n(ys.)f(A)p Fh(,)h Ff(26)p Fh(,)f(2471{2483.)182 1126 y([222])44 b(W)-7 b(elsh,)27 b(D.J.A.)h(\(1994\))e(The)i(random)d(cluster)i(pro)r (cess,)f Fg(Discr.)h(Math.)g Ff(136)p Fh(,)g(373{390.)182 1300 y([223])44 b(Widom,)34 b(B.)g(and)g(Ro)n(wlinson,)g(J.S.)g (\(1970\))f(New)h(mo)r(del)f(for)h(the)g(study)h(of)f(liquid-v)-5 b(ap)r(or)31 b(phase)398 1400 y(transition,)25 b Fg(J.)i(Chem.)g(Ph)n (ys.)g Ff(52)p Fh(,)g(1670{1684.)182 1574 y([224])44 b(Wierman,)c(J.C.)g(\(1995\))e(Substitution)h(metho)r(d)g(critical)d (probabilit)n(y)g(b)r(ounds)k(for)f(the)i(square)398 1674 y(lattice)25 b(site)i(p)r(ercolation)e(mo)r(del,)g Fg(Com)n(bin.)h(Probab.)g(Comput.)h Ff(4)p Fh(,)g(181{188.)182 1848 y([225])44 b(Wilson,)36 b(D.)h(\(1998\))f(Annotated)g(bibliograph) n(y)c(of)k(p)r(erfectly)g(random)f(sampling)e(with)j(Mark)n(o)n(v)398 1948 y(c)n(hains,)25 b Fg(Microsurv)n(eys)f(in)j(Discrete)f(Probabilit) n(y)c Fh(\(D.)29 b(Aldous)d(and)h(J.)g(Propp,)f(eds\),)i(pp)g(209{220,) 398 2047 y(American)d(Mathematical)f(So)r(ciet)n(y)-7 b(.)26 b(See)i(also)398 2147 y Fa(http://dimacs.r)o(utg)o(er)o(s.e)o (du)o(/~d)o(bw)o(il)o(son)o(/e)o(xac)o(t/)21 b Fh(.)182 2321 y([226])44 b(W)-7 b(ol\013,)27 b(U.)i(\(1989\))e(Collectiv)n(e)d (Mon)n(te)k(Carlo)d(up)r(dating)j(for)f(spin)g(systems,)g Fg(Ph)n(ys.)g(Rev.)h(Lett.)h Ff(62)p Fh(,)398 2421 y(361{364.)182 2595 y([227])44 b(W)-7 b(u,)28 b(F.Y.)g(\(1982\))e(The)i(P)n(otts)f(mo) r(del,)f Fg(Rev.)h(Mo)r(d.)h(Ph)n(ys.)f Ff(54)p Fh(,)g(235{)f(268.)182 2770 y([228])44 b(Zahradn)-9 b(\023)-32 b(\020k,)17 b(M.)j(\(1984\))e (An)i(alternate)e(v)n(ersion)f(of)j(Pirogo)n(v{Sinai)13 b(theory)-7 b(,)20 b Fg(Comm)n(un.)e(Math.)h(Ph)n(ys.)398 2869 y Ff(93)p Fh(,)27 b(559{581.)182 3044 y([229])44 b(Zahradn)-9 b(\023)-32 b(\020k,)21 b(M.)j(\(1987\))f(Analyticit)n(y)e (of)i(lo)n(w-temp)r(erature)e(phase)i(diagrams)d(of)k(lattice)e(spin)h (mo)r(d-)398 3143 y(els,)j Fg(J.)i(Stat.)f(Ph)n(ys.)g Ff(47)p Fh(,)g(725{755.)182 3318 y([230])44 b(Zahradn)-9 b(\023)-32 b(\020k,)24 b(M.)k(\(1996\))e(A)i(short)f(course)f(on)i(the) g(Pirogo)n(v{Sinai)21 b(theory)-7 b(,)27 b(Rome)f(I)r(I)i(lectures.) 1822 5895 y Fw(118)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9905050702473--