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Fq(\()p Fp(y)s Fq(\))15 b Fp(:)1190 b Fq(\(2)p Fp(:)p Fq(3\))0 1802 y(F)-8 b(or)34 b Fp(\033)n(;)15 b(\034)40 b Fm(2)29 b Fq(\012)j(w)m(e)i(also)f(let)f Fp(H)1062 1769 y Fo(\034)1055 1827 y(V)1118 1802 y Fq(\()p Fp(\033)s Fq(\))f(=)e Fp(H)1450 1816 y Fo(V)1512 1802 y Fq(\()p Fp(\033)1599 1816 y Fo(V)1663 1802 y Fp(\034)1703 1816 y Fo(V)1761 1797 y Fi(c)1800 1802 y Fq(\))k(and)f Fp(\034)43 b Fq(is)32 b(called)g(the)h Fk(b)-5 b(oundary)37 b(c)-5 b(ondition)p Fq(.)50 b(Sometimes)0 1929 y(w)m(e)26 b(tak)m(e)h Fp(\034)36 b Fm(2)493 1906 y Fq(\026)482 1929 y(\012)25 b(=)g Fm(f)p Fq(0)p Fp(;)15 b Fm(\000)p Fq(1)p Fp(;)g Fq(+1)p Fm(g)1116 1896 y Fe(Z)1167 1868 y Fi(d)1229 1929 y Fq(and)26 b(w)m(e)g(call)e(it)i(a)g Fk(gener)-5 b(alize)g(d)29 b(b)-5 b(oundary)31 b(c)-5 b(ondition)p Fq(.)40 b(As)26 b(a)g(particular)e(case)0 2056 y(w)m(e)33 b(ha)m(v)m(e)g(the)f Fk(fr)-5 b(e)g(e)35 b(b)-5 b(oundary)36 b(c)-5 b(onditions)p Fq(,)35 b(giv)m(en)d(b)m(y)g Fp(\034)39 b Fq(=)28 b(0.)46 b(The)32 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Fp(\034)10 b Fq(\()p Fp(x)p Fq(\))31 b(for)f(all)f Fp(x)c Fm(2)g Fp(V)3244 2832 y Fo(c)1079 3010 y Fq(0)1033 b(otherwise.)3713 2942 y(\(2)p Fp(:)p Fq(5\))0 3292 y(where)38 b Fp(Z)340 3248 y Fo(\014)s(;\034)333 3318 y(V)489 3292 y Fq(is)f(the)i(prop)s(er) e(normalization)g(factor)i(called)f(partition)f(function.)63 b(W)-8 b(e)39 b(will)d(often)j(omit)f(the)0 3419 y(sup)s(erscript)g Fp(\014)45 b Fq(in)39 b(all)g(quan)m(tities,)j(since)e(in)e(all)i(our)f (results)g(the)i(v)-5 b(alue)39 b(of)i Fp(\014)k Fq(can)40 b(b)s(e)g(considered)f(a)i(\014xed)0 3546 y(parameter.)49 b(Giv)m(en)33 b(a)g(measurable)f(b)s(ounded)f(function)g Fp(f)42 b Fq(on)33 b(\012,)h Fp(\026)2428 3560 y Fo(V)2491 3546 y Fp(f)42 b Fq(denotes)33 b(the)g(function)f Fp(\033)g Fm(7!)e Fp(\026)3687 3513 y Fo(\033)3687 3571 y(V)3750 3546 y Fq(\()p Fp(f)10 b Fq(\).)0 3673 y(Analogously)-8 b(,)40 b(if)c Fp(X)45 b Fm(2)37 b(F)9 b Fq(,)40 b Fp(\026)1045 3687 y Fo(V)1108 3673 y Fq(\()p Fp(X)7 b Fq(\))39 b(=)e Fp(\026)1462 3687 y Fo(V)1525 3673 y Fq(1)-25 b(I)1578 3687 y Fo(X)1647 3673 y Fq(,)40 b(where)d(1)-25 b(I)2035 3687 y Fo(X)2142 3673 y Fq(is)37 b(the)h(c)m(haracteristic)g(function)e (on)i Fp(X)7 b Fq(.)63 b Fp(\026)p Fq(\()p Fp(f)5 b(;)15 b(g)s Fq(\))0 3800 y(stands)36 b(for)g(the)g(co)m(v)-5 b(ariance)38 b(\(with)d(resp)s(ect)h(to)h Fp(\026)p Fq(\))g(of)f Fp(f)45 b Fq(and)36 b Fp(g)s Fq(.)59 b(The)36 b(set)h(of)f(measures)g (\(2.5\))i(satis\014es)e(the)0 3927 y(DLR)30 b(compatibilit)m(y)f (conditions)855 4153 y Fp(\026)910 4167 y Ft(\003)964 4153 y Fq(\()p Fp(\026)1054 4167 y Fo(V)1117 4153 y Fq(\()p Fp(X)7 b Fq(\)\))27 b(=)e Fp(\026)1482 4167 y Ft(\003)1536 4153 y Fq(\()p Fp(X)7 b Fq(\))183 b Fm(8)15 b Fp(X)32 b Fm(2)25 b(F)191 b(8)15 b Fp(V)46 b Fm(\032)25 b Fq(\003)g Fm(\032\032)g Ff(Z)2967 4115 y Fo(d)3020 4153 y Fp(:)668 b Fq(\(2)p Fp(:)p Fq(6\))0 4378 y(A)30 b(probabilit)m(y)e(measure)j Fp(\026)f Fq(on)g(\(\012)p Fp(;)15 b Fm(F)9 b Fq(\))31 b(is)f(called)f(a)i Fk(Gibbs)i(me)-5 b(asur)g(e)38 b Fq(if)1066 4604 y Fp(\026)p Fq(\()p Fp(\026)1211 4618 y Fo(V)1274 4604 y Fq(\()p Fp(X)7 b Fq(\)\))26 b(=)f Fp(\026)p Fq(\()p Fp(X)7 b Fq(\))183 b Fm(8)15 b Fp(X)32 b Fm(2)25 b(F)191 b(8)15 b Fp(V)46 b Fm(2)24 b Ff(F)c Fp(:)885 b Fq(\(2)p Fp(:)p Fq(7\))0 4829 y(It)35 b(is)f(w)m(ell)g(kno)m (wn)h(that,)i(for)e(the)g(in)m(teraction)g(\(2.3\),)k(the)c(sequences)g Fp(\026)2558 4785 y Fo(\014)s(;)p Ft(+)2558 4856 y Fo(Q)2615 4865 y Fi(L)2720 4829 y Fq(and)f Fp(\026)2956 4785 y Fo(\014)s(;)p Fn(\000)2956 4856 y Fo(Q)3013 4865 y Fi(L)3118 4829 y Fq(con)m(v)m(erge)k(w)m(eakly)-8 b(,)37 b(as)0 4956 y Fp(L)25 b Fm(!)g(1)p Fq(,)j(to)g(the)f(Gibbs)e(measures)i Fp(\026)1310 4923 y Fo(\014)s(;)p Ft(+)1463 4956 y Fq(and)f Fp(\026)1691 4923 y Fo(\014)s(;)p Fn(\000)1845 4956 y Fq(resp)s(ectiv)m(ely)-8 b(.)39 b(W)-8 b(e)29 b(call)d Fk(sp)-5 b(ontane)g(ous)32 b(magnetization)k Fq(the)0 5083 y(function)29 b Fp(m)436 5050 y Fn(\003)503 5083 y Fq(:)c Ff(R)613 5097 y Ft(+)705 5083 y Fm(7!)g Fq([0)p Fp(;)15 b Fq(1],)32 b(giv)m(en)f(b)m(y)1156 5309 y Fp(m)1236 5271 y Fn(\003)1278 5309 y Fq(\()p Fp(\014)5 b Fq(\))26 b(=)f Fp(\026)1581 5271 y Fo(\014)s(;)p Ft(+)1707 5309 y Fq(\()p Fp(\033)s Fq(\(0\)\))i(=)60 b(lim)2070 5367 y Fo(L)p Fn(!1)2280 5309 y Fp(\026)2335 5265 y Fo(\014)s(;)p Ft(+)2335 5335 y Fo(Q)2392 5344 y Fi(L)2461 5309 y Fq(\()p Fp(\033)s Fq(\(0\)\))15 b Fp(:)972 b Fq(\(2)p Fp(:)p Fq(8\))0 5558 y Fh(15)p Fi(=apr)r(ile=)p Fh(1999)30 b Ft([6])1353 b Fq(2.2)p eop %%Page: 7 7 7 6 bop 0 91 a Fq(The)37 b(critical)g(in)m(v)m(erse)h(temp)s(erature)f Fp(\014)1390 105 y Fo(c)1466 91 y Fq(is)f(de\014ned)h(as)h(the)g (suprem)m(um)e(of)i(all)f Fp(\014)5 b Fq('s)38 b(suc)m(h)f(that)i Fp(m)3515 58 y Fn(\003)3556 91 y Fq(\()p Fp(\014)5 b Fq(\))39 b(=)e(0.)0 218 y(When)k Fp(d)j Fq(=)f(2)f(it)f(is)f(w)m(ell)g (kno)m(wn)h(that)h Fp(\014)1525 232 y Fo(c)1606 218 y Fq(=)h(\(1)p Fp(=)p Fq(2\))15 b(log)t(\(1)28 b(+)2266 143 y Fm(p)p 2342 143 46 4 v 75 x Fq(2\).)74 b(W)-8 b(e)43 b(in)m(tro)s(duce)d(the)h Fk(c)-5 b(anonic)g(al)45 b(Gibbs)0 345 y(me)-5 b(asur)g(es)39 b Fq(on)31 b(\(\012)p Fp(;)15 b Fm(F)9 b Fq(\))31 b(de\014ned)e(as)1044 580 y Fp(\027)1095 543 y Fo(\034)1089 603 y Ft(\003)p Fo(;N)1254 580 y Fq(=)c Fp(\026)1405 543 y Fo(\034)1405 603 y Ft(\003)1459 580 y Fq(\()p Fm(\001)16 b(j)g Fp(N)1649 594 y Ft(\003)1728 580 y Fq(=)25 b Fp(N)10 b Fq(\))182 b Fp(N)35 b Fm(2)25 b(f)p Fq(0)p Fp(;)15 b Fq(1)p Fp(;)g(:)g(:)g(:)k(;)c Fm(j)p Fq(\003)p Fm(jg)858 b Fq(\(2)p Fp(:)p Fq(9\))0 815 y(where)30 b Fp(N)336 829 y Ft(\003)421 815 y Fq(is)f(the)h(n)m(um) m(b)s(er)g(of)g(particles)f(\()p Fk(i.e.)k Fq(spins)28 b(equal)i(to)h(+1\))g(in)e(\003.)0 997 y Fk(The)f(dynamics.)40 b Fq(W)-8 b(e)27 b(consider)e(the)g(so{called)h(Ka)m(w)m(asaki)g (dynamics)e(in)g(whic)m(h)h(particles)f(\(spins)g(with)g Fp(\033)s Fq(\()p Fp(x)p Fq(\))i(=)0 1125 y(+1\))43 b(can)h(jump)d(to)j (nearest)g(neigh)m(b)s(or)d(empt)m(y)j(\()p Fp(\033)s Fq(\()p Fp(x)p Fq(\))j(=)g Fm(\000)p Fq(1\))c(lo)s(cations,)j(k)m (eeping)d(the)g(total)h(n)m(um)m(b)s(er)e(of)0 1252 y(particles)37 b(constan)m(t.)66 b(F)-8 b(or)39 b Fp(\033)i Fm(2)d Fq(\012,)i(let)e Fp(\033)1486 1219 y Fo(xy)1610 1252 y Fq(b)s(e)f(the)i(con\014guration) e(obtained)h(from)g Fp(\033)j Fq(b)m(y)d(exc)m(hanging)h(the)0 1379 y(spins)30 b Fp(\033)s Fq(\()p Fp(x)p Fq(\))i(and)f Fp(\033)s Fq(\()p Fp(y)s Fq(\).)46 b(Let)32 b Fp(t)1058 1393 y Fo(xy)1143 1379 y Fp(\033)f Fq(=)c Fp(\033)1379 1346 y Fo(xy)1496 1379 y Fq(and)k(de\014ne)g(\()p Fp(T)2026 1393 y Fo(xy)2112 1379 y Fp(f)10 b Fq(\)\()p Fp(\033)s Fq(\))28 b(=)f Fp(f)10 b Fq(\()p Fp(t)2576 1393 y Fo(xy)2661 1379 y Fp(\033)s Fq(\))32 b(The)f(sto)s(c)m(hastic)i(dynamics)d(w)m(e)0 1506 y(w)m(an)m(t)h(to)g(study)f(is)f(determined)g(b)m(y)i(the)f(Mark)m (o)m(v)i(generators)g Fp(L)2267 1520 y Fo(V)2330 1506 y Fq(,)e Fp(V)46 b Fm(\032\032)24 b Ff(Z)2715 1473 y Fo(d)2753 1506 y Fq(,)31 b(de\014ned)e(b)m(y)731 1741 y(\()p Fp(L)828 1755 y Fo(V)891 1741 y Fp(f)10 b Fq(\)\()p Fp(\033)s Fq(\))26 b(=)1305 1655 y Fg(X)1228 1854 y Ft([)p Fo(x;y)r Ft(])p Fn(2E)1460 1863 y Fi(V)1529 1741 y Fp(c)1568 1755 y Fo(xy)1653 1741 y Fq(\()p Fp(\033)s Fq(\))15 b(\()p Fm(r)1904 1755 y Fo(xy)1991 1741 y Fp(f)10 b Fq(\)\()p Fp(\033)s Fq(\))182 b Fp(\033)28 b Fm(2)d Fq(\012)15 b Fp(;)106 b(f)35 b Fq(:)25 b(\012)g Fm(7!)h Ff(R)507 b Fq(\(2)p Fp(:)p Fq(10\))0 2091 y(where)41 b Fm(r)350 2105 y Fo(xy)478 2091 y Fq(=)i Fp(T)645 2105 y Fo(xy)758 2091 y Fm(\000)27 b Fq(1)-25 b(I.)73 b(The)41 b(nonnegativ)m(e)h(real)f (quan)m(tities)f Fp(c)2378 2105 y Fo(xy)2464 2091 y Fq(\()p Fp(\033)s Fq(\))i(are)f(the)h Fk(tr)-5 b(ansition)45 b(r)-5 b(ates)50 b Fq(for)41 b(the)0 2219 y(pro)s(cess.)0 2374 y(The)30 b(general)g(assumptions)f(on)h(the)h(transition)e(rates)i (are)66 2502 y(\(1\))46 b Fk(Ne)-5 b(ar)g(est)31 b(neighb)-5 b(or)30 b(inter)-5 b(actions.)41 b Fp(c)1483 2516 y Fo(xy)1569 2502 y Fq(\()p Fp(\033)s Fq(\))28 b(dep)s(ends)d(only)h(on)h(the)g (spins)e(that)j(are)g(nearest)f(neigh)m(b)s(ors)f(of)227 2629 y(either)k Fp(x)g Fq(or)h Fp(y)66 2757 y Fq(\(2\))46 b Fk(Detaile)-5 b(d)34 b(b)-5 b(alanc)g(e.)41 b Fq(F)-8 b(or)32 b(all)d Fp(\033)f Fm(2)d Fq(\012)30 b(and)g([)p Fp(x;)15 b(y)s Fq(])26 b Fm(2)f(E)2012 2776 y Fe(Z)2060 2758 y Fi(d)929 2992 y Fq(exp)1068 2919 y Fg(\002)1105 2992 y Fm(\000)p Fp(\014)c(H)1324 3009 y Fn(f)p Fo(x;y)r Fn(g)1504 2992 y Fq(\()p Fp(\033)s Fq(\))1629 2919 y Fg(\003)1668 2992 y Fp(c)1707 3006 y Fo(xy)1792 2992 y Fq(\()p Fp(\033)s Fq(\))26 b(=)f(exp)2178 2919 y Fg(\002)2216 2992 y Fm(\000)p Fp(\014)20 b(H)2434 3009 y Fn(f)p Fo(x;y)r Fn(g)2615 2992 y Fq(\()p Fp(\033)2705 2955 y Fo(xy)2790 2992 y Fq(\))2825 2919 y Fg(\003)2863 2992 y Fp(c)2902 3006 y Fo(xy)2988 2992 y Fq(\()p Fp(\033)3078 2955 y Fo(xy)3163 2992 y Fq(\))470 b(\(2)p Fp(:)p Fq(11\))66 3228 y(\(3\))46 b Fk(Positivity)29 b(and)h(b)-5 b(ounde)g(dness.)41 b Fq(F)-8 b(or)27 b(all)e Fp(\014)31 b(>)25 b Fq(0)i(there)f(exist)g(p) s(ositiv)m(e)g(real)g(n)m(um)m(b)s(ers)e Fp(c)3229 3242 y Fo(m)3299 3228 y Fq(\()p Fp(\014)5 b Fq(\))27 b Fp(c)3491 3242 y Fo(M)3572 3228 y Fq(\()p Fp(\014)5 b Fq(\))27 b(suc)m(h)227 3355 y(that)1246 3483 y Fp(c)1285 3497 y Fo(m)1379 3483 y Fm(\024)e Fp(c)1514 3497 y Fo(xy)1600 3483 y Fq(\()p Fp(\033)s Fq(\))h Fm(\024)f Fp(c)1886 3497 y Fo(M)2149 3483 y Fm(8)p Fp(x;)15 b(y)28 b Fm(2)d Ff(Z)2516 3445 y Fo(d)2569 3483 y Fp(;)15 b(\033)29 b Fm(2)24 b Fq(\012)15 b Fp(:)787 b Fq(\(2)p Fp(:)p Fq(12\))0 3701 y(W)-8 b(e)41 b(denote)f(b)m(y)f Fp(L)665 3668 y Fo(\034)665 3726 y(V)t(;N)842 3701 y Fq(the)h(op)s(erator)g Fp(L)1446 3715 y Fo(V)1548 3701 y Fq(acting)g(on)g Fp(L)2028 3668 y Ft(2)2068 3701 y Fq(\(\012)p Fp(;)15 b(\027)2260 3668 y Fo(\034)2254 3726 y(V)t(;N)2392 3701 y Fq(\))40 b(\(this)f(amoun)m(ts)h(to)g(c)m(ho)s(osing)f Fp(\034)50 b Fq(as)40 b(the)0 3828 y(b)s(oundary)34 b(condition)h(and)h Fp(N)46 b Fq(as)37 b(the)f(n)m(um)m(b)s(er)f(of)i(particles\).)57 b(Assumptions)35 b(\(1\),)k(\(2\))e(and)f(\(3\))h(guaran)m(tee)0 3955 y(that)25 b(there)f(exists)f(a)i(unique)d(Mark)m(o)m(v)j(pro)s (cess)f(whose)f(generator)j(is)d Fp(L)2462 3922 y Fo(\034)2462 3980 y(V)t(;N)2599 3955 y Fq(,)i(and)f(whose)f(semigroup)g(w)m(e)h (denote)0 4083 y(b)m(y)d(\()p Fp(T)218 4039 y Fo(V)t(;N)s(;\034)205 4105 y(t)414 4083 y Fq(\))449 4097 y Fo(t)p Fn(\025)p Ft(0)574 4083 y Fq(.)38 b Fp(L)699 4050 y Fo(\034)699 4108 y(V)t(;N)857 4083 y Fq(is)20 b(a)h(b)s(ounded)e(op)s(erator)i(on)g Fp(L)1903 4050 y Ft(2)1943 4083 y Fq(\(\012)p Fp(;)15 b(\027)2135 4050 y Fo(\034)2129 4108 y(V)t(;N)2267 4083 y Fq(\))21 b(and)f Fp(\027)2541 4050 y Fo(\034)2535 4108 y(V)t(;N)2694 4083 y Fq(is)f(its)i(unique)e(in)m(v)-5 b(arian)m(t)20 b(measure.)0 4210 y(Moreo)m(v)m(er)32 b Fp(\027)450 4177 y Fo(\034)444 4235 y(V)t(;N)612 4210 y Fq(is)e Fk(r)-5 b(eversible)30 b Fq(with)f(resp)s(ect)i(to)g(the)f (pro)s(cess,)g Fk(i.e.)i Fp(L)2451 4177 y Fo(\034)2451 4235 y(V)t(;N)2619 4210 y Fq(is)d(self{adjoin)m(t)h(on)h Fp(L)3382 4177 y Ft(2)3422 4210 y Fq(\(\012)p Fp(;)15 b(\027)3614 4177 y Fo(\034)3608 4235 y(V)t(;N)3746 4210 y Fq(\).)0 4365 y(A)42 b(fundamen)m(tal)g(quan)m(tit)m(y)g(asso)s (ciated)h(with)e(the)h(dynamics)f(of)i(a)f(rev)m(ersible)f(system)i(is) e(the)i(gap)f(of)h(the)0 4492 y(generator,)32 b Fk(i.e.)1267 4620 y Fq(gap\()p Fp(L)1505 4582 y Fo(\034)1505 4642 y(V)t(;N)1643 4620 y Fq(\))25 b(=)g(inf)c(sp)s(ec)15 b(\()p Fm(\000)p Fp(L)2278 4582 y Fo(\034)2278 4642 y(V)t(;N)2430 4620 y Fd(\026)g Fq(1)-25 b(I)2537 4580 y Fn(?)2598 4620 y Fq(\))0 4810 y(where)37 b(1)-25 b(I)323 4770 y Fn(?)422 4810 y Fq(is)36 b(the)i(subspace)f(of)h Fp(L)1244 4777 y Ft(2)1284 4810 y Fq(\(\012)p Fp(;)15 b(\027)1476 4777 y Fo(\034)1470 4835 y(V)t(;N)1608 4810 y Fq(\))38 b(orthogonal)g(to)g (the)g(constan)m(t)h(functions.)60 b(W)-8 b(e)39 b(let)e Fm(E)46 b Fq(b)s(e)37 b(the)0 4937 y(Diric)m(hlet)29 b(form)h(asso)s(ciated)h(with)e(the)i(generator)g Fp(L)1853 4904 y Fo(\034)1853 4962 y(V)t(;N)1990 4937 y Fq(,)614 5238 y Fm(E)670 5201 y Fo(\034)662 5261 y(V)t(;N)799 5238 y Fq(\()p Fp(f)5 b(;)15 b(f)10 b Fq(\))25 b(=)g Fm(h)p Fp(f)5 b(;)30 b Fm(\000)p Fp(L)1408 5201 y Fo(\034)1408 5261 y(V)t(;N)1545 5238 y Fp(f)10 b Fm(i)1635 5255 y Fo(L)1685 5237 y Fh(2)1720 5255 y Ft(\(\012)p Fo(;\027)1862 5233 y Fi(\034)1858 5279 y(V)s(;N)1979 5255 y Ft(\))2037 5238 y Fq(=)2144 5177 y(1)p 2144 5218 V 2144 5301 a(2)2293 5152 y Fg(X)2215 5351 y Ft([)p Fo(x;y)r Ft(])p Fn(2E)2447 5360 y Fi(V)2517 5238 y Fp(\027)2568 5201 y Fo(\034)2562 5261 y(V)t(;N)2714 5165 y Fg(\002)2767 5238 y Fp(c)2806 5252 y Fo(xy)2906 5238 y Fq(\()p Fm(r)3017 5252 y Fo(xy)3103 5238 y Fp(f)g Fq(\))3193 5201 y Ft(2)3248 5165 y Fg(\003)3668 5238 y Fq(\(2)p Fp(:)p Fq(13\))0 5558 y Fh(15)p Fi(=apr)r(ile=)p Fh(1999)30 b Ft([7])1353 b Fq(2.3)p eop %%Page: 8 8 8 7 bop 0 91 a Fq(and)32 b(V)-8 b(ar)320 51 y Fo(\034)320 113 y(V)t(;N)490 91 y Fq(is)31 b(the)i(v)-5 b(ariance)32 b(relativ)m(e)g(to)h(the)g(probabilit)m(y)d(measure)i Fp(\027)2571 58 y Fo(\034)2565 116 y(V)t(;N)2702 91 y Fq(.)47 b(The)31 b(gap)i(can)g(also)f(b)s(e)f(c)m(harac-)0 218 y(terized)f(as)967 396 y(gap\()p Fp(L)1205 358 y Fo(\034)1205 418 y(V)t(;N)1343 396 y Fq(\))c(=)430 b(inf)1500 467 y Fo(f)7 b Fn(2)p Fo(L)1641 440 y Fh(2)1677 467 y Ft(\(\012)p Fo(;\027)1819 440 y Fi(\034)1815 486 y(V)s(;N)1935 467 y Ft(\))p Fo(;)12 b Ft(V)-6 b(ar)2110 440 y Fi(\034)2110 486 y(V)s(;N)2230 467 y Ft(\()p Fo(f)7 b Ft(\))p Fn(6)p Ft(=0)2480 323 y Fm(E)2536 290 y Fo(\034)2528 348 y(V)t(;N)2665 323 y Fq(\()p Fp(f)e(;)15 b(f)10 b Fq(\))p 2478 375 405 4 v 2478 458 a(V)-8 b(ar)2619 418 y Fo(\034)2619 480 y(V)t(;N)2757 458 y Fq(\()p Fp(f)10 b Fq(\))2908 396 y Fp(:)735 b Fq(\(2)p Fp(:)p Fq(14\))0 667 y(where)0 849 y(Our)28 b(main)g(results)f(are)j(the)f(follo)m(wing)f(upp)s(er)e (and)j(lo)m(w)m(er)g(b)s(ounds)e(on)i(the)g(sp)s(ectral)f(gap)i(of)f (the)g(generator)h(in)0 977 y(a)h(\014nite)e(v)m(olume.)0 1159 y Fr(Theorem)34 b(2.1.)66 1288 y Fc(\(a\))46 b(F)-8 b(or)36 b(all)e(in)m(tegers)h Fp(d)f Fm(\025)f Fq(2)i Fc(there)g(exists)g Fp(\013)1683 1302 y Ft(1)1724 1288 y Fq(\()p Fp(\014)5 b Fq(\))34 b Fp(>)f Fq(0)j Fc(suc)m(h)e(that,)j (for)e(all)f Fp(\014)39 b(>)32 b Fq(0)p Fc(,)37 b(for)e(all)f Fp(\034)43 b Fm(2)33 b Fq(\012)p Fc(,)j(for)f(all)227 1415 y(large)c(enough)f Fp(L)g Fc(and)g(for)g(all)f Fp(N)35 b Fm(2)25 b(f)p Fq(0)p Fp(;)15 b(:)g(:)g(:)j(;)d Fm(j)p Fp(Q)1881 1429 y Fo(L)1936 1415 y Fm(jg)p Fc(,)31 b(w)m(e)g(ha)m(v)m(e) 1591 1651 y Fq(gap\()p Fp(L)1829 1614 y Fo(\034)1829 1674 y(Q)1886 1683 y Fi(L)1935 1674 y Fo(;N)2025 1651 y Fq(\))26 b Fm(\025)f Fp(e)2224 1614 y Fn(\000)p Fo(\013)2328 1623 y Fh(1)2364 1614 y Fo(L)2414 1586 y Fi(d)p Fj(\000)p Fh(1)3668 1651 y Fq(\(2)p Fp(:)p Fq(15\))61 1889 y Fc(\(b\))45 b(Let)37 b Fp(d)e Fq(=)f(2)p Fc(,)k Fp(\014)i(>)34 b(\014)938 1903 y Fo(c)976 1889 y Fc(,)j(and)f Fp(m)e Fm(2)h Fq(\()p Fm(\000)p Fp(m)1617 1856 y Fn(\003)1658 1889 y Fq(\()p Fp(\014)5 b Fq(\))p Fp(;)15 b(m)1904 1856 y Fn(\003)1947 1889 y Fq(\()p Fp(\014)5 b Fq(\)\))p Fc(.)59 b(Then)35 b(there)h(exists)g Fp(\013)2988 1903 y Ft(2)3029 1889 y Fq(\()p Fp(\014)5 b(;)15 b(m)p Fq(\))36 b Fp(>)e Fq(0)i Fc(suc)m(h)g(that,)227 2016 y(if)f Fm(N)404 1983 y Fo(m)391 2041 y(L)506 2016 y Fq(=)f Fm(b)p Fq(\(1)24 b(+)g Fp(m)p Fq(\))p Fm(j)p Fp(Q)1062 2030 y Fo(L)1116 2016 y Fm(j)p Fp(=)p Fq(2)p Fm(c)37 b Fc(\()p Fk(i.e.)32 b Fm(N)1589 1983 y Fo(m)1576 2041 y(L)1694 2016 y Fc(is)i(the)i(n)m(um)m(b)s(er)e (of)h(particles)g(whic)m(h)f(giv)m(es)i(a)f(magnetization)227 2143 y(appro)m(ximately)30 b(equal)g(to)h Fp(m)p Fc(\),)g(then,)f(for)g (large)h(enough)f Fp(L)1617 2379 y Fq(gap)q(\()p Fp(L)1856 2342 y Fn(;)1856 2403 y Fo(Q)1913 2412 y Fi(L)1961 2403 y Fo(;)p Fn(N)2053 2382 y Fi(m)2043 2427 y(L)2117 2379 y Fq(\))c Fm(\024)e Fp(e)2315 2342 y Fn(\000)p Fo(\013)2419 2351 y Fh(2)2455 2342 y Fo(L)3668 2379 y Fq(\(2)p Fp(:)p Fq(16\))0 2644 y Fk(R)-5 b(emarks.)55 b Fq(While)34 b(the)h(lo)m(w)m (er)g(b)s(ound)d(is)i(a)h(quite)f(general)h(statemen)m(t,)j(the)d(upp)s (er)e(b)s(ound)g(\(2.16\))k(is)c(sp)s(ecial)0 2771 y(to)j(free)f(b)s (oundary)e(conditions,)i Fp(\014)k(>)33 b(\014)1420 2785 y Fo(c)1492 2771 y Fq(and)i Fm(j)p Fp(m)p Fm(j)e Fp(<)g(m)2021 2738 y Fn(\003)2063 2771 y Fq(.)55 b(In)34 b([L)-8 b(Y])36 b(it)f(has)f(b)s(een)h(sho)m(wn)f(that)i(if)e(one)i(has)f(a)0 2899 y(\\complete)h(analyticit)m(y")f(t)m(yp)s(e)g(of)g(condition)e(at) j(a)f(certain)g(in)m(v)m(erse)g(temp)s(erature)g Fp(\014)40 b Fq(and)34 b Fk(for)j(al)5 b(l)38 b(magnetic)0 3026 y(\014elds)44 b Fp(h)p Fq(,)h(then)c(gap\()p Fp(L)821 2982 y Fo(\014)s(;)p Fn(;)821 3053 y Fo(Q)878 3062 y Fi(L)927 3053 y Fo(;)p Fn(N)1019 3031 y Fi(m)1009 3076 y(L)1083 3026 y Fq(\))j Fm(\030)g Fp(L)1339 2993 y Fn(\000)p Ft(2)1478 3026 y Fq(follo)m(ws.)73 b(In)41 b(the)h(case)h(of)f(the)f (2-dimensional)f(Ising)g(mo)s(del,)k(this)0 3153 y(condition)35 b(is)h(kno)m(wn)g(to)h(b)s(e)f(true)g(when)f Fp(\014)41 b(<)35 b(\014)1734 3167 y Fo(c)1772 3153 y Fq(.)58 b(One)36 b(migh)m(t)h(exp)s(ect)g(a)f(p)s(olynomial)e(la)m(w)j(if)e Fp(\014)41 b(>)35 b(\014)3680 3167 y Fo(c)3754 3153 y Fq(and)0 3280 y Fm(j)p Fp(m)p Fm(j)26 b Fp(>)f(m)332 3247 y Fn(\003)403 3280 y Fq(\(this)30 b(is)f(certainly)h(true)g(if)g (e.g)h(there)g(is)e(only)g(one)i(particle)f(!\).)0 3463 y Fk(The)41 b(Glaub)-5 b(er)43 b(dynamics)p Fq(.)70 b(Ev)m(en)40 b(though)f(our)g(results)g(deal)g(with)g(the)h(gap)g(asso)s(ciated)g (to)g(the)g(Ka)m(w)m(asaki)0 3590 y(dynamics,)23 b(w)m(e)h(sometimes)f (need)g(to)h(use)f(results)e(for)i(the)h(Glaub)s(er)d(dynamics,)j (de\014ned)d(in)h(terms)h(of)g(transition)0 3718 y(rates)29 b Fp(c)260 3732 y Fo(x)306 3718 y Fq(\()p Fp(\033)s Fq(\))f(whic)m(h)f (is)g(the)h(rate)h(of)f(the)g(transition)e Fp(\033)j Fm(!)c Fp(\033)2064 3685 y Fo(x)2110 3718 y Fq(,)k(where)e Fp(\033)2479 3685 y Fo(x)2553 3718 y Fq(is)g(the)h(con\014guration)f (obtained)g(from)0 3845 y Fp(\033)34 b Fq(b)m(y)c(\015ipping)d(the)k (spin)d(at)j Fp(x)p Fq(.)41 b(The)30 b(generators)h(of)g(these)f(pro)s (cesses)h(are)f(denoted)h(b)m(y)f Fm(L)3159 3859 y Fo(V)3222 3845 y Fq(,)g(de\014ned)g(as)1120 4081 y(\()p Fm(L)1218 4095 y Fo(V)1282 4081 y Fp(f)10 b Fq(\)\()p Fp(\033)s Fq(\))25 b(=)1627 3995 y Fg(X)1618 4190 y Fo(x)p Fn(2)p Fo(V)1782 4081 y Fp(c)1821 4095 y Fo(x)1867 4081 y Fq(\()p Fp(\033)s Fq(\)\()p Fm(r)2103 4095 y Fo(x)2150 4081 y Fp(f)10 b Fq(\)\()p Fp(\033)s Fq(\))183 b Fp(\033)28 b Fm(2)d Fq(\012)888 b(\(2)p Fp(:)p Fq(17\))0 5558 y Fh(15)p Fi(=apr)r(ile=)p Fh(1999)30 b Ft([8])1353 b Fq(2.4)p eop %%Page: 9 9 9 8 bop 0 91 a Fr(3.)47 b(Preliminary)34 b(results)0 241 y Fq(W)-8 b(e)32 b(collect)e(in)f(this)g(section)i(some)g (preliminary)26 b(results)k(that)g(will)e(b)s(e)i(used)f(in)g(the)i (future.)40 b(The)29 b(reader)i(can)0 368 y(skip)e(them)h(during)e(a)j (\014rst)f(reading)f(and)h(come)h(bac)m(k)g(to)h(this)d(section)h(when) g(these)h(results)e(are)h(needed.)0 541 y Fr(Prop)s(osition)41 b(3.1.)53 b Fc(Let)35 b Fq(\003)e Fm(\032\032)e Ff(Z)1286 508 y Fo(d)1324 541 y Fc(.)54 b(Then,)34 b(for)h(all)e Fp(\014)38 b(>)32 b Fq(0)p Fc(,)k Fp(\034)42 b Fm(2)32 b Fq(\012)i Fc(and)g(for)h(all)e Fp(k)j Fm(2)c(f)p Fq(0)p Fp(;)15 b(:)g(:)g(:)i(;)e Fm(j)p Fq(\003)p Fm(j)24 b(\000)f Fq(1)p Fm(g)35 b Fc(w)m(e)0 667 y(ha)m(v)m(e)967 772 y Fm(j)p Fq(\003)p Fm(j)21 b(\000)f Fp(k)p 967 812 275 4 v 1002 896 a(k)j Fq(+)d(1)1268 833 y Fp(e)1310 796 y Fn(\000)p Ft(4)p Fo(d\014)1516 833 y Fm(\024)1623 772 y Fp(\026)1678 739 y Fo(\034)1678 797 y Ft(\003)1732 772 y Fm(f)p Fp(N)1850 786 y Ft(\003)1930 772 y Fq(=)25 b Fp(k)e Fq(+)d(1)p Fm(g)p 1623 812 656 4 v 1701 896 a Fp(\026)1756 864 y Fo(\034)1756 922 y Ft(\003)1810 896 y Fm(f)p Fp(N)1928 910 y Ft(\003)2008 896 y Fq(=)25 b Fp(k)s Fm(g)2314 833 y(\024)2421 772 y(j)p Fq(\003)p Fm(j)c(\000)f Fp(k)p 2421 812 275 4 v 2455 896 a(k)k Fq(+)c(1)2722 833 y Fp(e)2764 796 y Ft(+4)p Fo(d\014)0 1072 y Fk(Pr)-5 b(o)g(of.)32 b Fq(De\014ne)e Fp(\021)s Fq(\()p Fp(x)p Fq(\))d(=)e(\(1)c(+)e Fp(\033)s Fq(\()p Fp(x)p Fq(\)\))p Fp(=)p Fq(2)33 b(,)d Fp(N)1498 1086 y Ft(\003)1553 1072 y Fq(\()p Fp(\033)s Fq(\))c(=)1800 1004 y Fg(P)1896 1100 y Fo(x)p Fn(2)p Ft(\003)2040 1072 y Fq(\(1)21 b(+)f Fp(\033)s Fq(\()p Fp(x)p Fq(\)\))p Fp(=)p Fq(2,)33 b(then)356 1365 y Fp(\026)411 1328 y Fo(\034)411 1388 y Ft(\003)466 1365 y Fm(f)p Fp(N)584 1379 y Ft(\003)664 1365 y Fq(=)25 b Fp(k)e Fq(+)d(1)p Fm(g)26 b Fq(=)1144 1218 y Fg(P)1240 1314 y Fo(\033)r Ft(:)p Fo(N)1363 1323 y Fh(\003)1408 1314 y Ft(\()p Fo(\033)r Ft(\)=)p Fo(k)r Ft(+1)1717 1287 y Fp(e)1759 1254 y Fn(\000)p Fo(\014)s(H)1925 1226 y Fi(\034)1920 1272 y Fh(\003)1966 1254 y Ft(\()p Fo(\033)r Ft(\))2087 1218 y Fg(P)2183 1314 y Fo(x)p Fn(2)p Ft(\003)2343 1287 y Fp(\021)s Fq(\()p Fp(x)p Fq(\))p 1144 1344 1371 4 v 1632 1428 a(\()p Fp(k)e Fq(+)c(1\))p Fp(Z)1978 1396 y Fo(\034)1971 1454 y Ft(\003)2550 1365 y Fq(=)491 1648 y(=)678 1586 y(1)p 598 1627 207 4 v 598 1710 a Fp(k)j Fq(+)d(1)835 1561 y Fg(X)831 1756 y Fo(x)p Fn(2)p Ft(\003)986 1648 y Fp(\026)1041 1610 y Fo(\034)1041 1670 y Ft(\003)1110 1547 y Fg(\020)1164 1648 y Fp(e)1206 1610 y Fn(\000)p Fo(\014)s Ft(\()p Fn(r)1396 1619 y Fi(x)1437 1610 y Fo(H)1502 1583 y Fi(\034)1497 1628 y Fh(\003)1543 1610 y Ft(\)\()p Fo(\033)r Ft(\))1677 1648 y Fq(\(1)h Fm(\000)f Fp(\021)s Fq(\()p Fp(x)p Fq(\)\)1)-25 b(I)2128 1664 y Fn(f)p Fo(N)2223 1673 y Fh(\003)2269 1664 y Ft(=)p Fo(k)r Fn(g)2407 1547 y Fg(\021)2487 1648 y Fm(\024)2593 1586 y(j)p Fq(\003)p Fm(j)21 b(\000)f Fp(k)p 2593 1627 275 4 v 2628 1710 a(k)j Fq(+)d(1)2879 1648 y Fp(e)2921 1610 y Ft(4)p Fo(d\014)3045 1648 y Fp(\026)3100 1610 y Fo(\034)3100 1670 y Ft(\003)3154 1648 y Fm(f)p Fp(N)3272 1662 y Ft(\003)3352 1648 y Fq(=)25 b Fp(k)s Fm(g)0 1952 y Fq(where)d(w)m(e)h(used)f(the)g(c)m(hange)i(of)f(v)-5 b(ariable)21 b Fp(\033)28 b Fm(7!)e Fp(\033)1703 1919 y Fo(x)1771 1952 y Fq(to)d(obtain)f(the)h(second)f(equalit)m(y)h(and)e Fm(k)p Fp(e)3177 1919 y Fn(\000)p Fo(\014)s Fn(r)3339 1928 y Fi(x)3381 1919 y Fo(H)3446 1892 y Fi(\034)3441 1938 y Fh(\003)3491 1952 y Fm(k)3536 1966 y Fn(1)3639 1952 y Fm(\024)k Fp(e)3777 1919 y Ft(4)p Fo(d\014)0 2079 y Fq(for)30 b(the)h(last)f(inequalit)m(y)-8 b(.)39 b(The)30 b(lo)m(w)m(er)h(b)s(ound)d(is)i(analogous.)91 b Fd(\003)0 2252 y Fr(Prop)s(osition)36 b(3.2.)53 b Fc(Let)31 b Fp(\032)f Fc(b)s(e)g(a)h(probabilit)m(y)d(measure)i(on)g Fq(\012)25 b(=)g Fm(f)p Fq(0)p Fp(;)15 b Fq(1)p Fp(;)g(:)g(:)g(:)k(;)c(N)10 b Fm(g)p Fc(,)31 b(and)f(assume)g(that)1623 2445 y Fp(\032)p Fq(\()p Fp(i)p Fq(\))c(=)f Fp(\032)p Fq(\()p Fp(N)31 b Fm(\000)20 b Fp(i)p Fq(\))15 b Fp(:)1437 b Fq(\(3)p Fp(:)p Fq(1\))0 2637 y Fc(Then,)30 b(for)g(all)f(functions)g Fp(f)40 b Fc(on)30 b Fq(\012)g Fc(w)m(e)h(ha)m(v)m(e)952 2923 y Fq(V)-8 b(ar)q(\()p Fp(f)10 b Fq(\))25 b Fm(\024)g Fp(C)1405 2937 y Fo(\032)1511 2810 y(N)1478 2837 y Fg(X)1485 3031 y Fo(i)p Ft(=1)1609 2923 y Fq(\()p Fp(\032)p Fq(\()p Fp(i)p Fq(\))d Fm(^)e Fp(\032)p Fq(\()p Fp(i)g Fm(\000)g Fq(1\)\))2249 2850 y Fg(\002)2289 2923 y Fp(f)10 b Fq(\()p Fp(i)p Fq(\))20 b Fm(\000)g Fp(f)10 b Fq(\()p Fp(i)20 b Fm(\000)g Fq(1\))2868 2850 y Fg(\003)2907 2869 y Ft(2)0 3205 y Fc(where)1260 3419 y Fp(C)1325 3433 y Fo(\032)1393 3419 y Fq(=)25 b(4)15 b(\()p Fp(N)31 b Fq(+)20 b(1\))1859 3382 y Ft(2)1931 3264 y Fg(")2098 3419 y Fq(sup)1984 3500 y Fo(i)p Fn(\024)p Fo(N)q(=)p Ft(2)p Fo(;)11 b(j)t Fn(\024)p Fo(i)2375 3358 y Fp(\032)p Fq(\()p Fp(j)5 b Fq(\))p 2375 3398 161 4 v 2381 3482 a Fp(\032)p Fq(\()p Fp(i)p Fq(\))2546 3264 y Fg(#)2599 3283 y Ft(2)0 3678 y Fk(Pr)-5 b(o)g(of.)32 b Fq(W)-8 b(e)31 b(consider)f(a)g(con)m(tin)m (uous)g(time)g(Mark)m(o)m(v)i(c)m(hain)e(with)f(transition)g(rates)1304 3930 y Fp(c)p Fq(\()p Fp(i;)15 b(j)5 b Fq(\))27 b(=)1649 3801 y Fg(\032)1744 3849 y Fo(\032)p Ft(\()p Fo(j)t Ft(\))p 1744 3872 129 4 v 1748 3924 a Fo(\032)p Ft(\()p Fo(i)p Ft(\))1903 3893 y Fm(^)20 b Fq(1)91 b(if)29 b Fp(j)i Fq(=)25 b Fp(i)c Fm(\006)f Fq(1)1733 4012 y(0)342 b(otherwise.)3713 3930 y(\(3)p Fp(:)p Fq(2\))0 4181 y(Since)29 b(the)i(rates)g(satisfy)f (the)g(detailed)g(balance)g(condition)1498 4374 y Fp(\032)p Fq(\()p Fp(i)p Fq(\))15 b Fp(c)p Fq(\()p Fp(i;)g(j)5 b Fq(\))29 b(=)24 b Fp(\032)p Fq(\()p Fp(j)5 b Fq(\))15 b Fp(c)p Fq(\()p Fp(j;)g(i)p Fq(\))1315 b(\(3)p Fp(:)p Fq(3\))0 4567 y(the)32 b(probabilit)m(y)d(measure)i Fp(\032)g Fq(is)g(rev)m(ersible)f(with)g(resp)s(ect)h(to)i(the)e(c)m(hain.)44 b(The)31 b(asso)s(ciated)h(Diric)m(hlet)e(form)h(is)0 4694 y(giv)m(en)f(b)m(y)982 4914 y Fm(E)1030 4928 y Fo(\032)1072 4914 y Fq(\()p Fp(f)5 b(;)15 b(f)10 b Fq(\))25 b(=)1441 4800 y Fo(N)1408 4827 y Fg(X)1415 5021 y Fo(i)p Ft(=1)1554 4914 y Fp(\032)p Fq(\()p Fp(i)p Fq(\))p Fp(c)p Fq(\()p Fp(i;)15 b(i)23 b Fm(\000)d Fq(1\))15 b([)r Fp(f)10 b Fq(\()p Fp(i)20 b Fm(\000)g Fq(1\))h Fm(\000)f Fp(f)10 b Fq(\()p Fp(i)p Fq(\)])2719 4868 y Ft(2)2785 4914 y Fq(=)1312 5260 y(=)1441 5146 y Fo(N)1408 5174 y Fg(X)1415 5367 y Fo(i)p Ft(=1)1539 5260 y Fq(\()p Fp(\032)p Fq(\()p Fp(i)p Fq(\))22 b Fm(^)e Fp(\032)p Fq(\()p Fp(i)h Fm(\000)f Fq(1\)\))2180 5186 y Fg(\002)2219 5260 y Fp(f)10 b Fq(\()p Fp(i)p Fq(\))20 b Fm(\000)g Fp(f)10 b Fq(\()p Fp(i)20 b Fm(\000)g Fq(1\))2798 5186 y Fg(\003)2837 5205 y Ft(2)2893 5260 y Fp(:)0 5558 y Fh(15)p Fi(=apr)r(ile=)p Fh(1999)30 b Ft([9])1353 b Fq(3.1)p eop %%Page: 10 10 10 9 bop 0 91 a Fq(If)30 b(w)m(e)h(denote)g(b)m(y)f Fp(\025)g Fq(the)h(sp)s(ectral)f(gap)g(of)h(the)f(generator)i(of)e(the)h(c)m (hain,)f(w)m(e)h(ha)m(v)m(e)669 424 y(V)-8 b(ar)811 438 y Fo(\032)853 424 y Fq(\()p Fp(f)10 b Fq(\))25 b Fm(\024)1114 363 y Fq(1)p 1110 403 54 4 v 1110 487 a Fp(\025)1205 424 y Fm(E)1253 438 y Fo(\032)1295 424 y Fq(\()p Fp(f)5 b(;)15 b(f)10 b Fq(\))25 b(=)1645 363 y(1)p 1642 403 V 1642 487 a Fp(\025)1754 311 y Fo(N)1721 338 y Fg(X)1727 531 y Fo(i)p Ft(=1)1852 424 y Fq(\()p Fp(\032)p Fq(\()p Fp(i)p Fq(\))d Fm(^)d Fp(\032)p Fq(\()p Fp(i)i Fm(\000)f Fq(1\)\))2492 351 y Fg(\002)2531 424 y Fp(f)10 b Fq(\()p Fp(i)p Fq(\))21 b Fm(\000)f Fp(f)10 b Fq(\()p Fp(i)20 b Fm(\000)g Fq(1\))3111 351 y Fg(\003)3150 369 y Ft(2)3205 424 y Fp(:)0 758 y Fq(T)-8 b(o)28 b(conclude)f(the)h(pro)s(of)f(w)m(e)h (need)g(a)g(lo)m(w)m(er)g(b)s(ound)e(for)h Fp(\025)p Fq(.)40 b(Cheeger's)28 b(inequalit)m(y)e(\(see)j(Theorem)f(2.1)g(in)f ([LS]\))0 886 y(states)32 b(that)1780 1061 y Fp(\025)25 b Fm(\025)1993 1000 y Fp(I)2040 967 y Ft(2)p 1965 1040 144 4 v 1965 1124 a Fq(8)p Fp(M)0 1296 y Fq(where)30 b Fp(M)35 b Fq(=)25 b(sup)619 1318 y Fo(i)680 1296 y Fq(\()p Fp(c)p Fq(\()p Fp(i;)15 b(i)22 b Fq(+)e(1\))h(+)f Fp(c)p Fq(\()p Fp(i;)15 b(i)22 b Fm(\000)e Fq(1\)\))31 b(and)1310 1615 y Fp(I)h Fq(=)f(min)1478 1673 y Fo(A)p Fn(\032)p Ft(\012)1667 1468 y Fg(P)1763 1563 y Ft(\()p Fo(j;k)r Ft(\))p Fn(2)p Fo(A)p Fn(\002)p Fo(A)2128 1545 y Fi(c)2181 1536 y Fp(c)p Fq(\()p Fp(j;)15 b(k)s Fq(\))p Fp(\032)p Fq(\()p Fp(j)5 b Fq(\))p 1667 1594 912 4 v 1816 1677 a Fp(\032)p Fq(\()p Fp(A)p Fq(\))15 b(\()r(1)20 b Fm(\000)g Fp(\032)p Fq(\()p Fp(A)p Fq(\)\))0 1918 y(With)27 b(the)h(c)m(hoice)h(\(3.2\))g Fp(M)36 b Fm(\024)25 b Fq(2.)40 b(As)28 b(the)g(state)h(space)f(\012)g(is)f(coun)m(table)h (and)f(connected,)i(b)m(y)f(Corollary)e(4.4)j(in)0 2045 y([LS],)i(the)g(minim)m(um)e(can)i(b)s(e)f(tak)m(en)i(o)m(v)m(er)h(all) d(subsets)g Fp(A)d Fm(\032)f Fq(\012)k(suc)m(h)h(that)h Fp(A)f Fq(and)f Fp(A)2992 2012 y Fo(c)3061 2045 y Fq(are)h(connected.) 44 b(Using)0 2172 y(the)32 b(symmetry)f(b)s(et)m(w)m(een)h Fp(A)g Fq(and)f Fp(A)1282 2139 y Fo(c)1351 2172 y Fq(w)m(e)h(can)g (also)g(imp)s(ose)e Fp(\032)p Fq(\()p Fp(A)p Fq(\))e Fm(\024)f Fq(1)p Fp(=)p Fq(2.)46 b(W)-8 b(e)33 b(can)f(th)m(us)f (write,)h(using)e(\(3.1\),)0 2300 y(\(3.3\))i(and)e(\(3.2\),)833 2445 y(1)p 832 2485 48 4 v 832 2568 a Fp(I)915 2506 y Fm(\024)101 b Fq(sup)1011 2600 y Fo(i)p Fn(\024)1094 2607 y Fq([)1130 2571 y Fi(N)5 b Fj(\000)p Fh(1)p 1130 2586 135 4 v 1182 2622 a(2)1275 2607 y Fq(])1408 2363 y Fg(P)1504 2458 y Fo(j)t Fn(\024)p Fo(i)1640 2431 y Fp(\032)p Fq(\()p Fp(j)g Fq(\))p 1326 2485 556 4 v 1326 2568 a Fp(\032)p Fq(\()p Fp(i)p Fq(\))22 b Fm(^)d Fp(\032)p Fq(\()p Fp(i)i Fq(+)f(1\))1918 2506 y Fm(\024)2014 2378 y Fg(\024)2073 2445 y Fp(N)30 b Fq(+)20 b(1)p 2073 2485 240 4 v 2170 2568 a(2)2323 2378 y Fg(\025)2545 2506 y Fq(sup)2386 2600 y Fo(i)p Fn(\024)2469 2607 y Fq([)2505 2571 y Fi(N)5 b Fh(+1)p 2505 2586 132 4 v 2555 2622 a(2)2648 2607 y Fq(])2673 2600 y Fo(;)30 b(j)t Fn(\024)p Fo(i)2867 2445 y Fp(\032)p Fq(\()p Fp(j)5 b Fq(\))p 2867 2485 161 4 v 2873 2568 a Fp(\032)p Fq(\()p Fp(i)p Fq(\))3054 2506 y Fp(:)634 b Fq(\(3)p Fp(:)p Fq(4\))0 2822 y(As,)31 b(b)m(y)f(equation) g(\(3.1\),)63 b Fp(\032)p Fq(\()1023 2786 y Fo(N)7 b Ft(+1)p 1023 2801 157 4 v 1083 2853 a(2)1190 2822 y Fq(\))26 b(=)f Fp(\032)p Fq(\()1440 2786 y Fo(N)7 b Fn(\000)p Ft(1)p 1440 2801 158 4 v 1501 2853 a(2)1609 2822 y Fq(\))30 b(,)h(in)e(\(3.4\))j(w)m(e)f(can)g(tak)m(e)h Fp(i)25 b Fm(\024)g Fp(N)r(=)p Fq(2.)42 b(W)-8 b(e)32 b(th)m(us)e(obtain)1118 3071 y(1)p 1114 3112 54 4 v 1114 3195 a Fp(\025)1203 3133 y Fm(\024)25 b Fp(C)1364 3147 y Fo(\032)1432 3133 y Fq(=)g(4)30 b(\()p Fp(N)h Fq(+)20 b(1\))1913 3095 y Ft(2)1985 3032 y Fg(h)2122 3133 y Fq(sup)2028 3221 y Fo(i)p Fn(\024)2121 3196 y Fi(N)p 2121 3206 V 2133 3243 a Fh(2)2186 3221 y Fo(;)30 b(j)t Fn(\024)p Fo(i)2380 3071 y Fp(\032)p Fq(\()p Fp(j)5 b Fq(\))p 2380 3112 161 4 v 2386 3195 a Fp(\032)p Fq(\()p Fp(i)p Fq(\))2551 3032 y Fg(i)2594 3050 y Ft(2)2726 3133 y Fd(\003)0 3495 y Fq(In)30 b(the)g(follo)m(wing)f(prop)s(osition)f(w)m(e)j(need)f(to)h (in)m(tro)s(duce)e(a)i Fk(magnetic)i(\014eld)9 b Fq(,)32 b(so)e(w)m(e)h(also)g(de\014ne)1344 3736 y Fp(H)1427 3698 y Fo(h)1420 3758 y(V)1483 3736 y Fq(\()p Fp(\033)s Fq(\))26 b(=)f Fp(H)1806 3750 y Fo(V)1869 3736 y Fq(\()p Fp(\033)s Fq(\))c Fm(\000)f Fp(h)2182 3649 y Fg(X)2173 3844 y Fo(x)p Fn(2)p Fo(V)2337 3736 y Fp(\033)s Fq(\()p Fp(x)p Fq(\))15 b Fp(:)1159 b Fq(\(3)p Fp(:)p Fq(5\))0 4096 y(The)30 b(corresp)s(onding)e(Gibbs)h(measure)h(is)g(denoted)g(b)m (y)g Fp(\026)2005 4052 y Fo(h;\034)2005 4122 y Ft(\003)2115 4096 y Fq(.)0 4282 y Fr(Prop)s(osition)43 b(3.3.)53 b Fc(Let)37 b Fq(\003)f Fm(\032\032)f Ff(Z)1297 4249 y Fo(d)1335 4282 y Fc(,)k Fp(n)c Fm(2)g(f)p Fq(1)p Fp(;)15 b(:)g(:)g(:)j(;)d Fm(j)p Fq(\003)p Fm(j)25 b(\000)f Fq(1)p Fm(g)p Fc(,)39 b(and)d(let)h Fp(\014)5 b(;)15 b(h)37 b Fc(b)s(e)f(suc)m(h)h(that)g Fp(\026)3371 4239 y Fo(h;\034)3371 4309 y Ft(\003)3480 4282 y Fq(\()p Fp(N)3588 4296 y Ft(\003)3643 4282 y Fq(\))f(=)f Fp(n)p Fc(.)0 4410 y(Consider)43 b(a)i(Glaub)s(er{t) m(yp)s(e)f(dynamics)f(with)h(transition)f(rates)i Fp(c)2413 4424 y Fo(x)2459 4410 y Fq(\()p Fp(\033)s Fq(\))p Fc(,)50 b(generator)c Fm(L)3141 4366 y Fo(h;\034)3141 4436 y Ft(\003)3250 4410 y Fc(,)i(and)c(let)h Fp(c)3699 4366 y Fo(g)3699 4436 y(M)3829 4410 y Fq(=)0 4537 y(sup)137 4559 y Fo(x;\033)264 4537 y Fp(c)303 4551 y Fo(x)349 4537 y Fq(\()p Fp(\033)s Fq(\))p Fc(.)c(Then)834 4880 y Fp(\026)889 4837 y Fo(h;\034)889 4907 y Ft(\003)998 4880 y Fm(f)p Fp(N)1116 4894 y Ft(\003)1196 4880 y Fq(=)25 b Fp(n)p Fm(g)h(\025)1514 4725 y Fg( )1597 4818 y Fq(gap\()p Fm(L)1836 4774 y Fo(h;\034)1836 4845 y Ft(\003)1946 4818 y Fq(\))p 1597 4859 385 4 v 1664 4943 a Fm(j)p Fq(\003)p Fm(j)15 b Fp(c)1831 4899 y Fo(g)1831 4969 y(M)2012 4880 y Fm(^)20 b Fq(1)2138 4725 y Fg(!)2631 4819 y Fq(1)p 2251 4859 804 4 v 2251 4943 a(\()p Fp(n)h Fq(+)e(2\))c(\()p Fm(j)p Fq(\003)p Fm(j)23 b(\000)d Fp(n)f Fq(+)h(1\))0 5209 y Fk(Pr)-5 b(o)g(of.)32 b Fq(Consider)c(trial)h(function)1524 5340 y Fp(f)1569 5354 y Ft(\003)p Fo(;n)1690 5340 y Fq(\()p Fp(\033)s Fq(\))d(=)f(1)-25 b(I)1990 5356 y Fn(f)p Fo(N)2085 5365 y Fh(\003)2131 5356 y Ft(\()p Fo(\033)r Ft(\))p Fo(>n)p Fn(g)0 5558 y Fh(15)p Fi(=apr)r(ile=)p Fh(1999)30 b Ft([10])1317 b Fq(3.2)p eop %%Page: 11 11 11 10 bop 0 91 a Fq(where,)30 b(as)h(usual,)e Fp(N)736 105 y Ft(\003)820 91 y Fq(is)h(the)g(n)m(um)m(b)s(er)f(of)h(particles)g (in)f(\003.)40 b(The)30 b(Diric)m(hlet)f(form)h(asso)s(ciated)h(to)g (the)f(Glaub)s(er)0 218 y(dynamics)f(is)g(b)s(ounded)g(b)m(y)1288 345 y Fm(E)1336 359 y Ft(\003)p Fo(;h)1454 345 y Fq(\()p Fp(f)5 b(;)15 b(f)10 b Fq(\))25 b Fm(\024)g(j)p Fq(\003)p Fm(j)15 b Fp(c)1957 302 y Fo(g)1957 372 y(M)2054 345 y Fp(\026)2109 302 y Fo(h;\034)2109 372 y Ft(\003)2218 345 y Fm(f)p Fp(N)2336 359 y Ft(\003)2416 345 y Fq(=)25 b Fp(n)p Fm(g)0 531 y Fq(while)j(the)j(v)-5 b(ariance)30 b(is)g(giv)m(en)g(b)m(y)1131 755 y(V)-8 b(ar)1273 769 y Ft(\003)p Fo(;h)1391 755 y Fq(\()p Fp(f)10 b Fq(\))25 b(=)g Fp(\026)1692 711 y Fo(h;\034)1692 782 y Ft(\003)1802 755 y Fm(f)p Fp(N)1920 769 y Ft(\003)2000 755 y Fm(\024)g Fp(n)p Fm(g)15 b Fp(\026)2266 711 y Fo(h;\034)2266 782 y Ft(\003)2375 755 y Fm(f)p Fp(N)2493 769 y Ft(\003)2573 755 y Fp(>)25 b(n)p Fm(g)0 979 y Fq(so)31 b(that,)g(b)m(y)f(the)h (analogous)f(of)h(\(2.14\))i(for)d(the)g(Glaub)s(er)f(dynamics)g(w)m(e) i(ha)m(v)m(e)688 1295 y Fp(\026)743 1251 y Fo(h;\034)743 1322 y Ft(\003)853 1295 y Fm(f)p Fp(N)971 1309 y Ft(\003)1051 1295 y Fq(=)25 b Fp(n)p Fm(g)g(\025)1368 1139 y Fg( )1451 1233 y Fq(gap\()p Fm(L)1690 1189 y Fo(h;\034)1690 1259 y Ft(\003)1800 1233 y Fq(\))p 1451 1274 385 4 v 1519 1357 a Fm(j)p Fq(\003)p Fm(j)15 b Fp(c)1686 1314 y Fo(g)1686 1384 y(M)1846 1139 y Fg(!)1933 1221 y(\002)1986 1295 y Fp(\026)2041 1251 y Fo(h;\034)2041 1322 y Ft(\003)2151 1295 y Fm(f)p Fp(N)2269 1309 y Ft(\003)2349 1295 y Fm(\024)25 b Fp(n)p Fm(g)15 b Fp(\026)2615 1251 y Fo(h;\034)2615 1322 y Ft(\003)2724 1295 y Fm(f)p Fp(N)2842 1309 y Ft(\003)2922 1295 y Fp(>)25 b(n)p Fm(g)3133 1221 y Fg(\003)3186 1295 y Fp(:)502 b Fq(\(3)p Fp(:)p Fq(6\))0 1610 y(W)-8 b(e)32 b(no)m(w)e(b)s(ound)e(the)j(t)m(w)m(o)h(probabilities)27 b(in)i(the)h(RHS)g(of)h(\(3.6\).)42 b(F)-8 b(or)31 b(the)g(\014rst)e (one)i(w)m(e)g(ha)m(v)m(e)564 1871 y Fp(n)25 b Fq(=)g Fp(\026)795 1827 y Fo(h;\034)795 1898 y Ft(\003)904 1871 y Fq(\()p Fp(N)1012 1885 y Ft(\003)1067 1871 y Fq(\))h Fm(\025)f Fq(\()p Fp(n)20 b Fq(+)g(1\))31 b Fp(\026)1591 1827 y Fo(h;\034)1591 1898 y Ft(\003)1700 1871 y Fm(f)p Fp(N)1818 1885 y Ft(\003)1898 1871 y Fp(>)25 b(n)p Fm(g)g Fq(=)g(\()p Fp(n)20 b Fq(+)g(1\))2511 1770 y Fg(\020)2567 1871 y Fq(1)g Fm(\000)g Fp(\026)2778 1827 y Fo(h;\034)2778 1898 y Ft(\003)2888 1871 y Fm(f)p Fp(N)3006 1885 y Ft(\003)3086 1871 y Fm(\024)25 b Fp(n)p Fm(g)3282 1770 y Fg(\021)0 2095 y Fq(th)m(us)1494 2223 y Fp(\026)1549 2179 y Fo(h;\034)1549 2249 y Ft(\003)1658 2223 y Fm(f)p Fp(N)1776 2237 y Ft(\003)1856 2223 y Fm(\024)g Fp(n)p Fm(g)g(\025)2267 2161 y Fq(1)p 2184 2202 212 4 v 2184 2285 a Fp(n)20 b Fq(+)g(1)0 2450 y(Analogously)30 b(w)m(e)g(obtain)1346 2634 y Fp(\026)1401 2590 y Fo(h;\034)1401 2661 y Ft(\003)1510 2634 y Fm(f)p Fp(N)1628 2648 y Ft(\003)1708 2634 y Fp(<)25 b(n)p Fm(g)h(\024)2150 2573 y(j)p Fq(\003)p Fm(j)21 b(\000)f Fp(n)p 2036 2613 507 4 v 2036 2697 a Fm(j)p Fq(\003)p Fm(j)h(\000)f Fq(\()p Fp(n)g Fm(\000)g Fq(1\))0 2877 y(so)31 b(that)g(for)f(the)g(second)h (one)1011 3101 y Fp(\026)1066 3058 y Fo(h;\034)1066 3128 y Ft(\003)1176 3101 y Fm(f)p Fp(N)1294 3115 y Ft(\003)1374 3101 y Fp(>)25 b(n)p Fm(g)g(\025)1932 3040 y Fq(1)p 1702 3081 V 1702 3164 a Fm(j)p Fq(\003)p Fm(j)c(\000)f Fq(\()p Fp(n)g Fm(\000)g Fq(1\))2239 3101 y Fm(\000)g Fp(\026)2385 3058 y Fo(h;\034)2385 3128 y Ft(\003)2495 3101 y Fm(f)p Fp(N)2613 3115 y Ft(\003)2693 3101 y Fq(=)25 b Fp(n)p Fm(g)0 3383 y Fq(and)30 b(the)g(result)g(follo)m(ws.)90 b Fd(\003)0 3563 y Fq(F)-8 b(or)31 b Fp(x;)15 b(z)30 b Fm(2)25 b Ff(Z)478 3530 y Fo(d)516 3563 y Fq(,)30 b(w)m(e)h(de\014ne) f(the)g(ev)m(en)m(ts)1218 3788 y Fp(E)1285 3802 y Fo(xz)1393 3788 y Fq(=)25 b Fm(f)p Fp(\033)k Fm(2)c Fq(\012)g(:)g Fp(\033)s Fq(\()p Fp(x)p Fq(\))h(=)f(1)p Fp(;)47 b(\033)s Fq(\()p Fp(z)t Fq(\))26 b(=)f(0)p Fm(g)15 b Fp(:)1032 b Fq(\(3)p Fp(:)p Fq(7\))0 4013 y Fr(Prop)s(osition)34 b(3.4.)52 b Fc(Let)29 b Fq(\003)c Fm(\032\032)g Ff(Z)1258 3980 y Fo(d)1296 4013 y Fc(,)k Fq(\003)c(=)g Fp(V)36 b Fm([)15 b Fp(W)41 b Fc(with)27 b Fp(V)35 b Fm(\\)16 b Fp(W)37 b Fq(=)25 b Fm(;)p Fc(.)41 b(Let)28 b Fp(\027)j Fq(=)25 b Fp(\027)2909 3969 y Fo(\014)s(;\034)2903 4040 y Ft(\003)p Fo(;N)3071 4013 y Fc(and)j Fp(\032)3293 4027 y Fo(i)3348 4013 y Fq(=)d Fp(\027)6 b Fm(f)p Fp(N)3613 4027 y Fo(V)3702 4013 y Fq(=)25 b Fp(i)p Fm(g)p Fc(.)0 4140 y(Let)31 b(also)f Fp(c)384 4154 y Fo(i)440 4140 y Fq(=)25 b Fp(i)p Fq(\()p Fm(j)p Fp(W)13 b Fm(j)21 b(\000)f Fp(N)30 b Fq(+)20 b Fp(i)p Fq(\))31 b Fc(and)f Fp(c)1370 4107 y Fn(0)1370 4164 y Fo(i)1425 4140 y Fq(=)25 b Fp(i)p Fq(\()p Fm(j)p Fp(V)d Fm(j)e(\000)g Fp(N)30 b Fq(+)20 b Fp(i)p Fq(\))p Fc(.)41 b(Then,)30 b(for)g(all)f(functions)g Fp(f)40 b Fc(on)30 b Fq(\012)g Fc(w)m(e)h(ha)m(v)m(e)54 4375 y Fp(\027)6 b Fq(\()p Fp(f)25 b Fm(j)15 b Fp(N)323 4389 y Fo(V)412 4375 y Fq(=)24 b Fp(i)p Fq(\))d Fm(\000)f Fp(\027)6 b Fq(\()p Fp(f)25 b Fm(j)15 b Fp(N)954 4389 y Fo(V)1043 4375 y Fq(=)25 b Fp(i)20 b Fm(\000)g Fq(1\))26 b(=)160 4590 y Fm(\000)274 4529 y Fq(1)p 262 4569 70 4 v 262 4652 a Fp(c)301 4666 y Fo(i)385 4504 y Fg(X)377 4699 y Fo(x)p Fn(2)p Fo(V)369 4758 y(z)s Fn(2)p Fo(W)559 4590 y Fp(\027)21 b Fq([\()p Fm(r)761 4604 y Fo(xz)845 4590 y Fp(f)10 b Fq(\)1)-25 b(I)987 4604 y Fo(E)1040 4613 y Fi(xz)1134 4590 y Fm(j)15 b Fp(N)1247 4604 y Fo(V)1336 4590 y Fq(=)25 b Fp(i)p Fq(])c(+)1622 4529 y(1)p 1610 4569 V 1610 4652 a Fp(c)1649 4666 y Fo(i)1702 4529 y Fp(\032)1749 4543 y Fo(i)p Fn(\000)p Ft(1)p 1702 4569 171 4 v 1748 4652 a Fp(\032)1795 4666 y Fo(i)1926 4504 y Fg(X)1917 4699 y Fo(x)p Fn(2)p Fo(V)1909 4758 y(z)s Fn(2)p Fo(W)2100 4590 y Fp(\027)2165 4516 y Fg(\002)2203 4590 y Fp(e)2245 4553 y Fn(\000)p Fo(\014)s Fn(r)2407 4562 y Fi(xz)2481 4553 y Fo(H)2541 4562 y Fh(\003)2591 4590 y Fq(1)-25 b(I)2644 4604 y Fo(E)2697 4613 y Fi(z)r(x)2776 4590 y Fp(;)15 b(f)25 b Fm(j)15 b Fp(N)2999 4604 y Fo(V)3087 4590 y Fq(=)25 b Fp(i)c Fm(\000)f Fq(1)3371 4516 y Fg(\003)3434 4590 y Fq(=)165 4956 y(=)391 4894 y(1)p 272 4935 283 4 v 272 5018 a Fp(c)311 4987 y Fn(0)311 5045 y Fo(N)7 b Fn(\000)p Fo(i)p Ft(+1)609 4869 y Fg(X)600 5064 y Fo(x)p Fn(2)p Fo(V)592 5123 y(z)s Fn(2)p Fo(W)783 4956 y Fp(\027)21 b Fq([\()p Fm(r)985 4970 y Fo(xz)1068 4956 y Fp(f)10 b Fq(\)1)-25 b(I)1211 4970 y Fo(E)1264 4979 y Fi(z)r(x)1357 4956 y Fm(j)16 b Fp(N)1471 4970 y Fo(V)1559 4956 y Fq(=)25 b Fp(i)c Fm(\000)f Fq(1])g Fm(\000)2109 4894 y Fq(1)p 1990 4935 V 1990 5018 a Fp(c)2029 4987 y Fn(0)2029 5045 y Fo(N)7 b Fn(\000)p Fo(i)p Ft(+1)2342 4894 y Fp(\032)2389 4908 y Fo(i)p 2295 4935 171 4 v 2295 5018 a Fp(\032)2342 5032 y Fo(i)p Fn(\000)p Ft(1)2519 4869 y Fg(X)2510 5064 y Fo(x)p Fn(2)p Fo(V)2502 5123 y(z)s Fn(2)p Fo(W)2693 4956 y Fp(\027)2759 4882 y Fg(\002)2797 4956 y Fp(e)2839 4918 y Fn(\000)p Fo(\014)s Fn(r)3001 4927 y Fi(xz)3074 4918 y Fo(H)3134 4927 y Fh(\003)3184 4956 y Fq(1)-25 b(I)3238 4970 y Fo(E)3291 4979 y Fi(xz)3369 4956 y Fp(;)15 b(f)25 b Fm(j)15 b Fp(N)3592 4970 y Fo(V)3681 4956 y Fq(=)25 b Fp(i)3808 4882 y Fg(\003)0 5340 y Fk(R)-5 b(emark)p Fq(.)42 b(A)31 b(similar)c(statemen)m(t)33 b(is)c(con)m(tained)i(in)e (Lemma)h(3.1)i(in)d([L)-8 b(Y].)0 5558 y Fh(15)p Fi(=apr)r(ile=)p Fh(1999)30 b Ft([11])1317 b Fq(3.3)p eop %%Page: 12 12 12 11 bop 0 91 a Fk(Pr)-5 b(o)g(of.)32 b Fq(F)-8 b(or)31 b Fp(\033)d Fm(2)d Fq(\012,)30 b(let)h Fp(\031)907 58 y Fn(0)904 113 y Fo(x)949 91 y Fq(\()p Fp(\033)s Fq(\))26 b(=)f(\()p Fp(\033)s Fq(\()p Fp(x)p Fq(\))d(+)e(1\))p Fp(=)p Fq(2.)42 b(Adding)29 b(and)h(subtracting)f Fp(T)2789 105 y Fo(xz)2872 91 y Fp(f)40 b Fq(w)m(e)31 b(can)f(write)255 336 y Fp(\027)6 b Fq(\()p Fp(f)24 b Fm(j)16 b Fp(N)524 350 y Fo(V)612 336 y Fq(=)25 b Fp(i)p Fq(\))h(=)278 490 y(1)p 266 531 70 4 v 266 614 a Fp(c)305 628 y Fo(i)389 466 y Fg(X)380 661 y Fo(x)p Fn(2)p Fo(V)372 720 y(z)s Fn(2)p Fo(W)563 552 y Fp(\027)21 b Fq([\()p Fp(f)30 b Fm(\000)20 b Fp(T)908 566 y Fo(xz)991 552 y Fp(f)10 b Fq(\))p Fp(\031)1136 514 y Fn(0)1133 574 y Fo(x)1178 552 y Fq(\(1)21 b Fm(\000)f Fp(\031)1425 514 y Fn(0)1422 574 y Fo(z)1464 552 y Fq(\))15 b Fm(j)h Fp(N)1628 566 y Fo(V)1716 552 y Fq(=)25 b Fp(i)p Fq(])51 b(+)2033 490 y(1)p 2021 531 V 2021 614 a Fp(c)2060 628 y Fo(i)2144 466 y Fg(X)2135 661 y Fo(x)p Fn(2)p Fo(V)2127 720 y(z)s Fn(2)p Fo(W)2318 552 y Fp(\027)21 b Fq([\()p Fp(T)2497 566 y Fo(xz)2581 552 y Fp(f)10 b Fq(\))p Fp(\031)2726 514 y Fn(0)2723 574 y Fo(x)2768 552 y Fq(\(1)21 b Fm(\000)f Fp(\031)3015 514 y Fn(0)3012 574 y Fo(z)3054 552 y Fq(\))15 b Fm(j)g Fp(N)3217 566 y Fo(V)3306 552 y Fq(=)25 b Fp(i)p Fq(])3713 519 y(\(3)p Fp(:)p Fq(8\))0 947 y(After)32 b(the)g(c)m(hange)h(of)f(v)-5 b(ariable)30 b Fp(\021)h Fm(7!)d Fp(\021)1389 914 y Fo(xz)1472 947 y Fq(,)33 b(using)d(the)i (equalit)m(y)f Fp(\027)6 b Fq(\()p Fp(f)k(g)s Fq(\))28 b(=)g Fp(\027)6 b Fq(\()p Fp(f)f(;)15 b(g)s Fq(\))22 b(+)e Fp(\027)6 b Fq(\()p Fp(f)k Fq(\))p Fp(\027)c Fq(\()p Fp(g)s Fq(\),)33 b(w)m(e)g(can)f(write)0 1075 y(the)f(second)f(term)g (in)f(\(3.8\))k(as)313 1311 y(1)p 301 1352 V 301 1435 a Fp(c)340 1449 y Fo(i)423 1311 y Fp(\027)6 b Fm(f)p Fp(N)592 1325 y Fo(V)680 1311 y Fq(=)25 b Fp(i)c Fm(\000)f Fq(1)p Fm(g)p 423 1352 587 4 v 501 1435 a Fp(\027)6 b Fm(f)p Fp(N)670 1449 y Fo(V)759 1435 y Fq(=)24 b Fp(i)p Fm(g)1063 1286 y Fg(X)1054 1481 y Fo(x)p Fn(2)p Fo(V)1046 1540 y(z)s Fn(2)p Fo(W)1237 1372 y Fp(\027)1303 1299 y Fg(\002)1341 1372 y Fp(e)1383 1335 y Fn(\000)p Fo(\014)s Fn(r)1545 1344 y Fi(xz)1619 1335 y Fo(H)1679 1344 y Fh(\003)1729 1372 y Fq(\(1)d Fm(\000)f Fp(\031)1976 1335 y Fn(0)1973 1395 y Fo(x)2018 1372 y Fq(\))p Fp(\031)2108 1335 y Fn(0)2105 1395 y Fo(z)2147 1372 y Fp(;)15 b(f)25 b Fm(j)16 b Fp(N)2371 1386 y Fo(V)2459 1372 y Fq(=)25 b Fp(i)20 b Fm(\000)g Fq(1)2742 1299 y Fg(\003)2826 1372 y Fq(+)313 1686 y(1)p 301 1727 70 4 v 301 1810 a Fp(c)340 1824 y Fo(i)423 1686 y Fp(\027)6 b Fm(f)p Fp(N)592 1700 y Fo(V)680 1686 y Fq(=)25 b Fp(i)c Fm(\000)f Fq(1)p Fm(g)p 423 1727 587 4 v 501 1810 a Fp(\027)6 b Fm(f)p Fp(N)670 1824 y Fo(V)759 1810 y Fq(=)24 b Fp(i)p Fm(g)1063 1661 y Fg(X)1054 1856 y Fo(x)p Fn(2)p Fo(V)1046 1915 y(z)s Fn(2)p Fo(W)1237 1748 y Fp(\027)1303 1674 y Fg(\002)1341 1748 y Fp(e)1383 1710 y Fn(\000)p Fo(\014)s Fn(r)1545 1719 y Fi(xz)1619 1710 y Fo(H)1679 1719 y Fh(\003)1729 1748 y Fq(\(1)d Fm(\000)f Fp(\031)1976 1710 y Fn(0)1973 1770 y Fo(x)2018 1748 y Fq(\))p Fp(\031)2108 1710 y Fn(0)2105 1770 y Fo(z)2162 1748 y Fm(j)c Fp(N)2276 1762 y Fo(V)2364 1748 y Fq(=)25 b Fp(i)c Fm(\000)f Fq(1)2648 1674 y Fg(\003)2731 1748 y Fp(\027)h Fq(\()q Fp(f)j Fm(j)15 b Fp(N)3015 1762 y Fo(V)3104 1748 y Fq(=)25 b Fp(i)20 b Fm(\000)g Fq(1\))3713 1609 y(\(3)p Fp(:)p Fq(9\))0 2143 y(T)-8 b(aking)34 b Fp(f)42 b Fq(=)32 b(1)j(in)e(equation)h(\(3.8\))j(w)m(e)e(obtain)f (that)h(the)f(term)h(m)m(ultiplying)c Fp(\027)21 b Fq(\()p Fp(f)k Fm(j)15 b Fp(N)3065 2157 y Fo(V)3154 2143 y Fq(=)25 b Fp(i)20 b Fm(\000)g Fq(1\))35 b(in)f(\(3.9\))i(is)0 2270 y(equal)30 b(to)h(one)g(and)e(the)i(result)e(is)h(obtained.)0 2426 y(The)g(second)g(equalit)m(y)g(follo)m(ws)g(from)g(the)g(\014rst)g (b)m(y)g(in)m(terc)m(hanging)g Fp(V)51 b Fq(and)29 b Fp(W)104 b Fd(\003)0 5558 y Fh(15)p Fi(=apr)r(ile=)p Fh(1999)30 b Ft([12])1317 b Fq(3.4)p eop %%Page: 13 13 13 12 bop 0 91 a Fr(4.)47 b(Pro)s(of)36 b(of)f(the)f(lo)m(w)m(er)h(b)s (ound)0 244 y Fq(Here)h(w)m(e)f(pro)m(v)m(e)h(the)f(lo)m(w)m(er)g(b)s (ound)e(\(2.15\))38 b(of)d(the)g(main)f(Theorem)h(2.1.)56 b(In)34 b(the)h(\014rst)f(subsection)h(w)m(e)g(sho)m(w)0 371 y(ho)m(w,)29 b(at)g(least)g(when)e(the)h(b)s(oundary)f(condition)f (if)i Fk(fr)-5 b(e)g(e)35 b Fq(and)28 b(the)g(side)g Fp(L)g Fq(of)g(the)h(cub)s(e)e Fp(Q)3123 385 y Fo(L)3206 371 y Fq(is)g(a)i(p)s(o)m(w)m(er)f(of)g(t)m(w)m(o,)0 498 y(the)j(lo)m(w)m(er)g(b)s(ound)d(is)i(a)h(consequence)g(of)g(a)g(k) m(ey)g(inequalit)m(y)e(stated)j(in)d(Lemma)i(4.1.)42 b(The)30 b(second)h(subsection)0 625 y(is)e(where)g(our)g(hands)g(get)i (dirt)m(y)d(and)i(w)m(e)g(pro)m(v)m(e)g(of)g(Lemma)g(4.1.)42 b(Next)30 b(w)m(e)h(sho)m(w)e(ho)m(w)h(to)g(extend)g(the)g(results)0 752 y(to)g(an)g(arbitrary)e(b)s(oundary)g(condition)g(and)h(all)f(v)-5 b(alues)29 b(of)h Fp(L)f Fq(\(this)g(is)g(v)m(ery)h(simple\).)39 b(Finally)27 b(w)m(e)j(sho)m(w)g(ho)m(w,)0 879 y(with)f(our)h(approac)m (h,)h(w)m(e)g(reco)m(v)m(er)h(the)e(w)m(ell)g(kno)m(wn)g(result)f(for)h Fp(\014)g Fq(=)25 b(0.)0 1108 y Fu(4.1)35 b(Reduction)e(to)h(the)f(key) h(inequality)f(\(4.5\))0 1261 y Fq(The)26 b(core)i(of)e(the)h(pro)s(of) f(is)g(to)h(sho)m(w)g(that)g(gap)q(\(2\003\))f Fm(\025)f Fq(gap)q(\(\003\))15 b(exp\()p Fm(\000)p Fp(k)s(L)2557 1228 y Fo(d)p Fn(\000)p Ft(1)2693 1261 y Fq(\).)40 b(T)-8 b(o)27 b(b)s(e)f(more)h(precise,)g(for)f(ev)m(en)0 1388 y Fp(L)p Fq(,)37 b(consider)d(the)i(parallelepip)s(eds)31 b Fp(Q)1341 1344 y Fo(j)1341 1415 y(L)1431 1388 y Fq(for)k Fp(j)k Fq(=)34 b(0)p Fp(;)15 b Fq(1)p Fp(;)g(:)g(:)g(:)j(;)d(d)p Fq(,)37 b(de\014ned)d(as)i(the)g(set)g(of)f(all)g Fp(x)e Fm(2)h Fp(Q)3434 1402 y Fo(L)3523 1388 y Fq(suc)m(h)h(that)0 1515 y(0)c Fm(\024)g Fp(x)230 1529 y Fo(i)291 1515 y Fm(\024)g Fp(L=)p Fq(2)23 b Fm(\000)g Fq(1)34 b(for)g Fp(i)d Fq(=)g Fp(j)d Fq(+)22 b(1)p Fp(;)15 b(:)g(:)g(:)i(;)e(d)p Fq(.)52 b(Th)m(us)33 b Fp(Q)1883 1482 y Fo(d)1883 1540 y(L)1971 1515 y Fq(coincides)f(with)h Fp(Q)2641 1529 y Fo(L)2695 1515 y Fq(,)i(while)d Fp(Q)3068 1482 y Ft(0)3068 1540 y Fo(L)3153 1515 y Fq(=)f Fp(Q)3327 1531 y Fo(L=)p Ft(2)3455 1515 y Fq(.)51 b(Moreo)m(v)m(er)0 1642 y Fp(Q)72 1598 y Fo(j)t Ft(+1)72 1669 y Fo(L)233 1642 y Fq(is)29 b(the)i(disjoin)m(t)d(union)h(of)h Fp(Q)1232 1598 y Fo(j)1232 1669 y(L)1317 1642 y Fq(and)g(a)g(translate)h(of)g Fp(Q)2125 1598 y Fo(j)2125 1669 y(L)2179 1642 y Fq(.)40 b(Let)31 b(no)m(w)1302 1860 y Fp(a)p Fq(\(\003\))26 b(=)169 b(sup)1605 1940 y Fo(N)7 b Fn(2f)p Ft(0)p Fo(;:::)o(;)p Fn(j)p Ft(\003)p Fn(jg)2045 1860 y Fq(gap)q(\()p Fp(L)2284 1822 y Fn(;)2284 1884 y Ft(\003)p Fo(;N)2424 1860 y Fq(\))2459 1822 y Fn(\000)p Ft(1)2573 1860 y Fp(:)0 2156 y Fq(What)31 b(w)m(e)g(w)m(an)m (t)g(to)g(sho)m(w)f(is)g(that)h(for)f(some)h Fp(k)i Fq(whic)m(h)c(dep)s (ends)g(only)g(on)i Fp(\014)k Fq(w)m(e)c(ha)m(v)m(e)1027 2374 y Fp(a)p Fq(\()p Fp(Q)1182 2330 y Fo(j)t Ft(+1)1182 2400 y Fo(L)1313 2374 y Fq(\))25 b Fm(\024)g 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Fq(and)23 b Fp(W)37 b Fq(ha)m(v)m(e)25 b(b)s(een)e(turned)0 3460 y(o\013,)31 b Fk(i.e.)h Fq(w)m(e)f(de\014ne)e (an)i(in)m(teraction)1018 3716 y Fp(J)1068 3730 y Fo(xy)1179 3716 y Fq(=)1275 3615 y Fg(n)1351 3673 y Fq(0)91 b(if)29 b Fp(x)d Fm(2)e Fp(V)51 b Fq(and)30 b Fp(y)e Fm(2)d Fp(W)42 b Fq(\(or)31 b(vicev)m(ersa\))1351 3782 y(1)91 b(otherwise.)3713 3716 y(\(4)p Fp(:)p Fq(3\))0 3972 y(Then)29 b(w)m(e)i(de\014ne)f(the)g (follo)m(wing)f(\\dotted")j(quan)m(tities)e(for)g(the)g(\\decoupled")h (system)1043 4190 y(_)-41 b Fp(\027)31 b Fq(=)25 b Fp(\027)1250 4146 y Fo(J)n(;)p Fn(;)1244 4217 y Ft(\003)p Fo(;N)1400 4190 y Fp(;)1552 4167 y Fq(_)1531 4190 y Fp(L)g Fq(=)g Fp(L)1776 4146 y Fo(J)n(;)p Fn(;)1776 4217 y Ft(\003)p Fo(;N)1931 4190 y Fp(;)2088 4167 y Fq(_)2063 4190 y Fm(E)8 b Fq(\()p Fp(f)d(;)15 b(f)10 b Fq(\))25 b(=)40 b(_)-40 b Fp(\027)5 b Fq(\()p Fm(\000)p Fp(f)2702 4167 y Fq(_)2681 4190 y Fp(Lf)10 b Fq(\))15 b Fp(:)840 b Fq(\(4)p Fp(:)p Fq(4\))0 4408 y(The)30 b(main)f(ingredien)m(t)g(for)h(pro)m(ving)g (\(4.1\))i(is)d(the)i(follo)m(wing)0 4586 y Fr(Lemma)e(4.1.)53 b Fc(With)26 b(reference)h(to)h(the)f(notation)g(in)m(tro)s(duced)f(in) f(\(4.2\),)30 b(\(4.3\),)g(\(4.4\),)f(if)d Fp(W)39 b Fc(is)26 b(a)i(translate)f(of)0 4713 y Fp(V)48 b Fc(and)27 b Fp(d)322 4727 y Ft(1)363 4713 y Fq(\()p Fp(V)5 b(;)15 b(W)e Fq(\))26 b(=)f(1)i Fc(\(there)h(is)f(at)h(least)g(one)g(edge)g (connecting)g(the)f(t)m(w)m(o)i(sets\),)g(then)e(there)h(exists)f Fp(k)s Fq(\()p Fp(\014)5 b Fq(\))27 b Fp(>)e Fq(0)0 4841 y Fc(suc)m(h)30 b(that)716 4968 y Fq(V)-8 b(ar)869 4982 y Ft(_)-33 b Fo(\027)918 4968 y Fq(\()15 b(_)-40 b Fp(\027)6 b Fq(\()p Fp(f)25 b Fm(j)15 b Fp(N)1222 4982 y Fo(V)1285 4968 y Fq(\)\))26 b Fm(\024)f Fq(exp\()p Fp(k)s Fm(j)p Fq(\003)p Fm(j)1825 4901 y Fi(d)p Fj(\000)p Fh(1)p 1826 4916 114 4 v 1866 4952 a Fi(d)1955 4968 y Fq(\))2005 4867 y Fg(h)2089 4945 y Fq(_)2064 4968 y Fm(E)8 b Fq(\()p Fp(f)d(;)15 b(f)10 b Fq(\))20 b(+)35 b(_)-40 b Fp(\027)21 b 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Fp(\027)2790 48 y Fo(J)n(;)p Fn(;)2784 118 y Ft(\003)p Fo(;N)2925 91 y Fq(.)63 b(Then)37 b(for)h(all)e(functions)0 218 y Fp(f)f Fq(:)25 b(\012)196 232 y Ft(\003)275 218 y Fm(7!)g Ff(R)39 b Fq(w)m(e)31 b(ha)m(v)m(e)1053 345 y(V)-8 b(ar)1207 359 y Ft(_)-33 b Fo(\027)1240 345 y Fq(\()p Fp(f)10 b Fq(\))25 b(=)40 b(_)-40 b Fp(\027)6 b Fq(\(V)-8 b(ar)1726 359 y Ft(_)-33 b Fo(\027)1759 345 y Fq(\()p Fp(f)25 b Fm(j)15 b Fp(N)1977 359 y Fo(V)2040 345 y Fq(\)\))21 b(+)f(V)-8 b(ar)2376 359 y Ft(_)-33 b Fo(\027)2409 345 y Fq(\()16 b(_)-41 b Fp(\027)6 b Fq(\()p Fp(f)25 b Fm(j)15 b Fp(N)2713 359 y Fo(V)2776 345 y Fq(\)\))867 b(\(4)p Fp(:)p Fq(6\))0 516 y(The)30 b(k)m(ey)h(fact)g(no)m(w)g(is)e(that,)j (since)d(there)i(is)e(no)i(in)m(teraction)f(b)s(et)m(w)m(een)h Fp(V)51 b Fq(and)30 b Fp(W)13 b Fq(,)30 b(the)g(conditional)f(measure) 15 643 y(_)-40 b Fp(\027)6 b Fq(\()p Fm(\001)15 b(j)h Fp(N)240 657 y Fo(V)328 643 y Fq(=)25 b Fp(k)s Fq(\))31 b(is)f(a)g Fk(pr)-5 b(o)g(duct)35 b(me)-5 b(asur)g(e)7 b Fq(,)32 b Fk(i.e.)1346 835 y Fq(_)-41 b Fp(\027)6 b Fq(\()p Fm(\001)16 b(j)f Fp(N)1570 849 y Fo(V)1659 835 y Fq(=)25 b Fp(k)s Fq(\))g(=)g Fp(\027)2012 797 y Fn(;)2006 859 y Fo(V)t(;k)2140 835 y Fm(\002)19 b Fp(\027)2281 797 y Fn(;)2275 859 y Fo(W)n(;N)7 b Fn(\000)p Fo(k)2544 835 y Fp(:)0 1027 y Fq(Moreo)m(v)m(er)50 b(_)-40 b Fp(\027)6 b Fq(\()p Fm(\001)15 b(j)h Fp(N)642 1041 y Fo(V)734 1027 y Fq(=)29 b Fp(k)s Fq(\))34 b(is)e(the)h(rev)m(ersible)e(measure)i(of)g (a)g(Mark)m(o)m(v)i(pro)s(cess)d(whose)h(generator)h(is)e(the)h(sum)0 1154 y(of)e(the)f(t)m(w)m(o)i(comm)m(uting)e(generators)h Fp(L)1406 1119 y Fn(;)1406 1181 y Fo(V)t(;k)1540 1154 y Fq(+)20 b Fp(L)1693 1119 y Fn(;)1693 1181 y Fo(W)n(;N)7 b Fn(\000)p Fo(k)1946 1154 y Fq(.)41 b(By)31 b(consequence)g(w)m(e)g (ha)m(v)m(e)g(that)88 1372 y(V)-8 b(ar)242 1386 y Ft(_)-33 b Fo(\027)275 1372 y Fq(\()p Fp(f)25 b Fm(j)15 b Fp(N)493 1386 y Fo(V)581 1372 y Fq(=)25 b Fp(k)s Fq(\))h Fm(\024)98 1527 y(\024)f Fq(max)363 1453 y Fg(\010)431 1527 y Fq(gap)q(\()p Fp(L)670 1489 y Fn(;)670 1551 y Fo(V)t(;k)783 1527 y Fq(\))818 1489 y Fn(\000)p Ft(1)931 1527 y Fp(;)31 b Fq(gap\()p Fp(L)1225 1489 y Fn(;)1225 1551 y Fo(W)n(;N)7 b Fn(\000)p Fo(k)1479 1527 y Fq(\))1514 1489 y Fn(\000)p Ft(1)1627 1453 y Fg(\011)16 b(\002)1764 1527 y Fq(_)-40 b Fp(\027)5 b Fq(\()p Fm(\000)p Fp(f)10 b(L)2022 1489 y Fn(;)2022 1551 y Fo(V)t(;k)2135 1527 y Fp(f)24 b Fm(j)16 b Fp(N)2318 1541 y Fo(V)2406 1527 y Fq(=)25 b Fp(k)s Fq(\))c(+)35 b(_)-40 b Fp(\027)5 b Fq(\()p Fm(\000)p Fp(f)10 b(L)2972 1489 y Fn(;)2972 1551 y Fo(W)n(;N)d Fn(\000)p Fo(k)3226 1527 y Fp(f)24 b Fm(j)15 b Fp(N)3408 1541 y Fo(V)3497 1527 y Fq(=)25 b Fp(k)s Fq(\))3693 1453 y Fg(\003)3757 1527 y Fm(\024)98 1681 y(\024)g Fp(a)p Fq(\()p Fp(Q)349 1638 y Fo(j)349 1708 y(L)404 1681 y Fq(\))454 1608 y Fg(\002)522 1681 y Fq(_)-40 b Fp(\027)6 b Fq(\()p Fm(\000)p Fp(f)k(L)781 1644 y Fn(;)781 1706 y Fo(V)t(;k)893 1681 y Fp(f)25 b Fm(j)15 b Fp(N)1076 1695 y Fo(V)1165 1681 y Fq(=)24 b Fp(k)s 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y Fi(d)p Fj(\000)p Fh(1)1396 2373 y Fq([)15 b(1)21 b(+)f(2)p Fp(a)p Fq(\()p Fp(Q)1793 2329 y Fo(j)1793 2400 y(L)1849 2373 y Fq(\))15 b(])1965 2350 y(_)1939 2373 y Fm(E)9 b Fq(\()p Fp(f)c(;)15 b(f)10 b Fq(\))24 b Fm(\024)h Fp(e)2373 2335 y Fo(k)2412 2344 y Fh(1)2448 2335 y Fo(L)2498 2308 y Fi(d)p Fj(\000)p Fh(1)2621 2373 y Fp(a)p Fq(\()p Fp(Q)2776 2329 y Fo(j)2776 2400 y(L)2830 2373 y Fq(\))2906 2350 y(_)2880 2373 y Fm(E)9 b Fq(\()p Fp(f)c(;)15 b(f)10 b Fq(\))15 b Fp(:)521 b Fq(\(4)p Fp(:)p Fq(8\))0 2565 y(All)33 b(w)m(e)h(need)g(to)g(do)g(at) h(this)e(p)s(oin)m(t)g(is)g(to)i(\\remo)m(v)m(e)h(the)e(dot",)i Fk(i.e.)c Fq(restore)i(the)g(original)e(in)m(teraction)i(where)0 2692 y Fp(J)50 2706 y Fo(xy)161 2692 y Fq(=)25 b(1)k(for)g(all)g Fp(x)g Fq(and)g Fp(y)s Fq(.)40 b(This)27 b(is)i(v)m(ery)g(simple)f (since,)h(b)m(y)g(a)h(straigh)m(tforw)m(ard)f(computations,)h(w)m(e)g (\014nd)d(that)0 2818 y(there)k(is)e(a)i(constan)m(t)g Fp(k)813 2832 y Ft(2)885 2818 y Fq(suc)m(h)f(that)1030 3063 y 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Fn(\000)p Fo(i)p Ft(\))2720 1379 y Fq(\()p Fp(N)31 b Fm(\000)20 b Fp(i)p Fq(\))3042 1303 y Fp(:)601 b Fq(\(4)p Fp(:)p Fq(13\))0 1607 y(On)30 b(the)g(other)h(side,)e (thanks)h(to)h(Prop)s(osition)e(3.3)i(\(recall)f(\(4.12\)\),)j(w)m(e)e (kno)m(w)g(that)934 1936 y Fp(q)979 1882 y Ft(~)978 1899 y Fo(h)o Ft(\()p Fo(i)p Ft(\))1108 1936 y Fq(\()p Fp(i)p Fq(\))26 b Fm(\025)1331 1781 y Fg( )1414 1874 y Fq(gap\()p Fm(L)1654 1810 y Ft(~)1653 1827 y Fo(h)p Ft(\()p Fo(i)p Ft(\))p Fo(;)p Fn(;)1653 1901 y Fo(V)1842 1874 y Fq(\))p 1414 1915 464 4 v 1516 1998 a Fm(j)p Fp(V)20 b Fm(j)15 b Fp(c)1693 1955 y Fo(g)1693 2025 y(M)1908 1936 y Fm(^)20 b Fq(1)2034 1781 y Fg(!)2508 1875 y Fq(1)p 2147 1915 768 4 v 2147 1998 a(\()p Fp(i)h Fq(+)f(1\))15 b(\()p Fm(j)p Fp(V)22 b Fm(j)f(\000)e Fp(i)i Fq(+)f(1\))2941 1936 y Fp(:)702 b Fq(\(4)p Fp(:)p Fq(14\))0 2250 y(W)-8 b(e)33 b(need)f(at)g(this)f(p)s(oin)m(t)g(a)h(lo)m(w)m(er)g(b)s(ound)e (on)h(the)h(gap)g(of)g(the)g(generator)h(of)f(a)g(suitable)f(Glaub)s 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Fm(r)1939 4288 y Fo(xz)2022 4274 y Fp(f)10 b Fq(\)1)-25 b(I)2165 4288 y Fo(E)2218 4297 y Fi(xz)2311 4274 y Fm(j)16 b Fp(N)2425 4288 y Fo(V)2513 4274 y Fq(=)25 b Fp(i)15 b Fq(])848 4595 y Fp(A)916 4609 y Fn(\000)978 4595 y Fq(\()p Fp(i)p Fq(\))26 b(=)1331 4534 y(1)p 1212 4574 283 4 v 1212 4658 a Fp(c)1251 4626 y Fn(0)1251 4684 y Fo(N)7 b Fn(\000)p Fo(i)p Ft(+1)1655 4509 y Fg(X)1521 4708 y Ft(\()p Fo(x;z)s Ft(\))p Fn(2)p Fo(V)16 b Fn(\002)p Fo(W)1951 4595 y Fq(_)-40 b Fp(\027)21 b Fq([)15 b(\()p Fm(r)2153 4609 y Fo(xz)2236 4595 y Fp(f)10 b Fq(\)1)-25 b(I)2379 4609 y Fo(E)2432 4618 y Fi(z)r(x)2525 4595 y Fm(j)16 b Fp(N)2639 4609 y Fo(V)2727 4595 y Fq(=)25 b Fp(i)c Fm(\000)f Fq(1)15 b(])848 4917 y Fp(B)917 4931 y Ft(+)978 4917 y Fq(\()p Fp(i)p Fq(\))26 b(=)1224 4855 y(1)p 1212 4896 70 4 v 1212 4979 a Fp(c)1251 4993 y Fo(i)1303 4855 y Fp(\032)1350 4869 y Fo(i)p Fn(\000)p Ft(1)p 1303 4896 171 4 v 1350 4979 a Fp(\032)1397 4993 y Fo(i)1634 4831 y Fg(X)1500 5030 y Ft(\()p Fo(x;z)s Ft(\))p Fn(2)p Fo(V)15 b Fn(\002)p Fo(W)1930 4917 y Fq(_)-41 b Fp(\027)21 b Fq([)16 b Fp(g)2064 4931 y Fo(z)s(x)2162 4917 y Fp(;)31 b(f)24 b Fm(j)16 b Fp(N)2401 4931 y Fo(V)2489 4917 y Fq(=)25 b Fp(i)c Fm(\000)e Fq(1)c(])848 5238 y Fp(B)917 5252 y Fn(\000)979 5238 y Fq(\()p Fp(i)p Fq(\))26 b(=)1332 5177 y(1)p 1213 5218 283 4 v 1213 5301 a Fp(c)1252 5270 y Fn(0)1252 5328 y Fo(N)7 b Fn(\000)p Fo(i)p Ft(+1)1564 5177 y Fp(\032)1611 5191 y Fo(i)p 1518 5218 171 4 v 1518 5301 a Fp(\032)1565 5315 y Fo(i)p Fn(\000)p Ft(1)1848 5152 y Fg(X)1714 5351 y Ft(\()p Fo(x;z)s Ft(\))p Fn(2)p Fo(V)16 b Fn(\002)p Fo(W)2144 5238 y Fq(_)-40 b Fp(\027)21 b Fq([)15 b Fp(g)2278 5252 y Fo(xz)2377 5238 y Fp(;)30 b(f)25 b Fm(j)15 b Fp(N)2615 5252 y Fo(V)2703 5238 y Fq(=)25 b Fp(i)15 b Fq(])31 b Fp(:)3668 4788 y Fq(\(4)p Fp(:)p Fq(19\))0 5558 y Fh(15)p Fi(=apr)r(ile=)p Fh(1999)f Ft([17])1317 b Fq(4.5)p eop %%Page: 18 18 18 17 bop 0 91 a Fk(Estimate)33 b(of)g(the)g(term)g Fp(B)k Fk(in)c(\(4.17\))p Fq(.)42 b(In)30 b(order)g(to)h(estimate)g Fp(B)2278 105 y Ft(+)2338 91 y Fq(\()p Fp(i)p Fq(\))h(w)m(e)e(write)520 333 y(\()16 b(_)-40 b Fp(\027)20 b Fq([)c Fp(g)705 347 y Fo(z)s(x)803 333 y Fp(;)31 b(f)24 b Fm(j)16 b Fp(N)1042 347 y Fo(V)1130 333 y Fq(=)25 b Fp(i)c Fm(\000)f Fq(1)15 b(])q(\))1490 287 y Ft(2)1556 333 y Fm(\024)25 b Fq(V)-8 b(ar)1806 347 y Ft(_)-33 b Fo(\027)1839 333 y Fq(\()p Fp(f)25 b Fm(j)15 b Fp(N)2057 347 y Fo(V)2145 333 y Fq(=)25 b Fp(i)c Fm(\000)f Fq(1\))31 b(V)-8 b(ar)2649 347 y Ft(_)-33 b Fo(\027)2682 333 y Fq(\()p Fp(g)2760 347 y Fo(z)s(x)2859 333 y Fm(j)15 b Fp(N)2972 347 y Fo(V)3061 333 y Fq(=)25 b Fp(i)20 b Fm(\000)g Fq(1\))289 b(\(4)p Fp(:)p Fq(20\))0 575 y(and)30 b(w)m(e)h(notice)f(that)465 817 y(V)-8 b(ar)618 831 y Ft(_)-33 b Fo(\027)652 817 y Fq(\()p Fp(g)730 831 y Fo(z)s(x)829 817 y Fm(j)15 b Fp(N)942 831 y Fo(V)1031 817 y Fq(=)24 b Fp(i)d Fm(\000)f Fq(1\))26 b Fm(\024)f(k)p Fp(g)1559 831 y Fo(z)s(x)1643 817 y Fm(k)1688 780 y Ft(2)1688 840 y Fn(1)1791 817 y Fm(\024)g(k)p Fp(e)1974 780 y Fn(\000)p Ft(2)p Fo(\014)s Fn(r)2172 789 y Fi(xz)2246 780 y Fo(H)2306 789 y Fh(\003)2356 817 y Fm(k)2401 831 y Fn(1)2504 817 y Fm(\024)g Fp(e)2642 780 y Ft(2)p Fo(\014)s Fn(kr)2820 789 y Fi(xz)2894 780 y Fo(H)2954 789 y Fh(\003)3000 780 y Fn(k)3037 789 y Fj(1)3133 817 y Fm(\024)g Fp(e)3271 780 y Ft(8)p Fo(d\014)3410 817 y Fp(:)233 b Fq(\(4)p Fp(:)p Fq(21\))0 1059 y(Moreo)m(v)m(er,)33 b(recalling)c(\(4.11\))j (and)e(using)f(Prop)s(osition)f(3.1,)k(w)m(e)f(\014nd)931 1301 y Fp(\032)978 1315 y Fo(i)p Fn(\000)p Ft(1)p 931 1342 171 4 v 977 1425 a Fp(\032)1024 1439 y Fo(i)1137 1363 y Fm(\024)1440 1301 y Fp(i)p 1244 1342 423 4 v 1244 1425 a Fm(j)p Fp(V)20 b Fm(j)h(\000)f Fp(i)g Fq(+)g(1)1704 1301 y Fm(j)p Fp(W)13 b Fm(j)20 b(\000)g Fp(N)30 b Fq(+)20 b Fp(i)p 1704 1342 486 4 v 1755 1425 a(N)31 b Fm(\000)19 b Fp(i)i Fq(+)f(1)2200 1363 y Fp(e)2242 1325 y Ft(8)p Fo(d\014)2391 1363 y Fm(\024)25 b(j)p Fp(V)20 b Fm(j)15 b(j)p Fp(W)e Fm(j)p Fp(e)2816 1325 y Ft(8)p Fo(d\014)2955 1363 y Fp(:)0 1662 y Fq(A)30 b(straigh)m(tforw)m(ard)h(calculation)e (giv)m(es)280 1975 y Fp(c)319 1989 y Fo(u)392 1975 y Fq(=)499 1908 y(\()p Fp(N)i Fq(+)19 b(1\))c(\()p Fm(j)p Fp(V)22 b Fm(j)f Fq(+)f(1\))1190 1834 y Fg(\002)1244 1908 y Fq(\()p Fm(j)p Fp(W)13 b Fm(j)20 b(\000)g Fp(N)10 b Fq(\))15 b(\()p Fm(j)p Fp(V)21 b Fm(j)g Fq(+)f(1)g(+)g Fm(j)p Fp(W)13 b Fm(j)20 b Fq(+)g(1\))h(+)f(\()p Fp(N)31 b Fq(+)20 b(1\))15 b(\()p Fm(j)p Fp(V)22 b Fm(j)e Fq(+)g(1\))3242 1834 y Fg(\003)p 499 1954 2782 4 v 1699 2037 a Fq(\()p Fm(j)p Fq(\003)p Fm(j)h Fq(+)f(2\))2039 2011 y Ft(2)3317 1975 y Fq(=)392 2202 y(=)25 b(\()p Fp(N)31 b Fq(+)20 b(1\))15 b(\()p Fm(j)p Fp(V)21 b Fm(j)g Fq(+)f(1\))15 b(\()p Fm(j)p Fp(W)e Fm(j)21 b Fq(+)f(1\))15 b(\()p Fm(j)p Fq(\003)p Fm(j)22 b(\000)e Fp(N)30 b Fq(+)20 b(1\))15 b(\()p Fm(j)p Fq(\003)p Fm(j)22 b Fq(+)e(2\))2461 2165 y Fn(\000)p Ft(2)2575 2202 y Fp(:)3668 2055 y Fq(\(4)p Fp(:)p Fq(22\))0 2453 y(Moreo)m(v)m(er,)33 b Fp(c)464 2467 y Fo(i)525 2453 y Fq(is)c(increasing)g(in)g Fp(i)p Fq(.)41 b(So,)31 b(using)e(the)h(Sc)m(h)m(w)m(arz)h(inequalit)m(y)-8 b(,)30 b(w)m(e)h(obtain,)f(for)g Fp(i)25 b Fm(\025)g(b)p Fp(u)p Fm(c)c Fq(+)f(1,)431 2727 y Fp(B)505 2690 y Ft(2)500 2750 y(+)561 2727 y Fq(\()p Fp(i)p Fq(\))26 b Fm(\024)f Fp(c)823 2690 y Fn(\000)p Ft(2)823 2750 y Fo(u)936 2653 y Fg(\000)978 2727 y Fm(j)p Fp(V)20 b Fm(j)15 b(j)p Fp(W)e Fm(j)i Fp(e)1322 2690 y Ft(8)p Fo(d\014)1446 2653 y Fg(\001)1488 2668 y Ft(2)1544 2727 y Fq(\()p Fm(j)p Fp(V)20 b Fm(j)15 b(j)p Fp(W)e Fm(j)p Fq(\))2051 2641 y Fg(X)1916 2840 y Ft(\()p Fo(x;z)s Ft(\))p Fn(2)p Fo(V)k Fn(\002)p Fo(W)2332 2727 y Fq(\()31 b(_)-40 b Fp(\027)20 b Fq([)c Fp(g)2532 2741 y Fo(z)s(x)2630 2727 y Fp(;)31 b(f)24 b Fm(j)16 b Fp(N)2869 2741 y Fo(V)2957 2727 y Fq(=)25 b Fp(i)c Fm(\000)f Fq(1)15 b(])h(\))3332 2681 y Ft(2)3398 2727 y Fm(\024)688 3010 y(\024)25 b Fp(c)823 2973 y Fn(\000)p Ft(2)823 3033 y Fo(u)936 3010 y Fq(\()p Fm(j)p Fp(V)c Fm(j)15 b(j)p Fp(W)e Fm(j)p Fq(\))1294 2973 y Ft(4)1350 3010 y Fp(e)1392 2973 y Ft(32)p Fo(d\014)1582 3010 y Fq(V)-8 b(ar)1736 3024 y Ft(_)-33 b Fo(\027)1769 3010 y Fq(\()p Fp(f)25 b Fm(j)15 b Fp(N)1987 3024 y Fo(V)2076 3010 y Fq(=)25 b Fp(i)20 b Fm(\000)g Fq(1\))26 b Fm(\024)688 3241 y(\024)868 3180 y Fq(\()p Fm(j)p Fp(V)20 b Fm(j)15 b(j)p Fp(W)e Fm(j)p Fq(\))1225 3147 y Ft(2)1282 3180 y Fq(\()p Fm(j)p Fq(\003)p Fm(j)21 b Fq(+)f(2\))1622 3147 y Ft(4)p 795 3220 942 4 v 795 3304 a Fq(\()p Fp(N)30 b Fq(+)20 b(1\))1104 3277 y Ft(2)1161 3304 y Fq(\()p Fm(j)p Fq(\003)p Fm(j)h(\000)f Fp(N)30 b Fq(+)20 b(1\))1695 3277 y Ft(2)1763 3241 y Fp(e)1805 3204 y Ft(32)p Fo(d\014)1995 3241 y Fq(V)-8 b(ar)2148 3255 y Ft(_)-33 b Fo(\027)2182 3241 y Fq(\()p Fp(f)24 b Fm(j)16 b Fp(N)2400 3255 y Fo(V)2488 3241 y Fq(=)25 b Fp(i)c Fm(\000)f Fq(1\))15 b Fp(:)0 3545 y Fq(The)30 b(term)g Fp(B)473 3559 y Fn(\000)535 3545 y Fq(\()p Fp(i)p Fq(\))h(can)g(b)s(e)e(estimated)i(in)e(the)i (same)f(w)m(a)m(y)-8 b(.)43 b(So,)30 b(w)m(e)h(ha)m(v)m(e)884 3855 y Fp(B)f Fm(\024)25 b Fp(C)1144 3869 y Ft(1)1184 3855 y Fq(\()p Fp(\014)5 b Fq(\))1716 3793 y Fm(j)p Fq(\003)p Fm(j)1829 3760 y Ft(8)p 1322 3834 V 1322 3917 a Fq(\()p Fp(N)31 b Fq(+)20 b(1\))1632 3891 y Ft(2)1688 3917 y Fq(\()p Fm(j)p Fq(\003)p Fm(j)i(\000)d Fp(N)31 b Fq(+)19 b(1\))2222 3891 y Ft(2)2305 3855 y Fq(_)-40 b Fp(\027)21 b Fq(\()15 b(V)-8 b(ar)2560 3869 y Ft(_)-33 b Fo(\027)2593 3855 y Fq(\()p Fp(f)25 b Fm(j)16 b Fp(N)2812 3869 y Fo(V)2875 3855 y Fq(\))f(\))31 b Fp(:)652 b Fq(\(4)p Fp(:)p Fq(23\))0 4159 y Fk(Estimate)38 b(of)f(the)g(term)h Fp(A)f Fk(in)g(\(4.17\))p Fq(.)56 b(Before)36 b(starting)f(the)g(estimate)h(of)f(the)h(term)f Fp(A)g Fq(in)f(\(4.17\),)39 b(w)m(e)c(need)0 4286 y(the)c(follo)m(wing) d(de\014nition.)0 4475 y Fr(De\014nition)42 b(4.4.)53 b Fq(Giv)m(en)37 b(a)g(\014nite)e(connected)i(subset)f(\003)h(of)g Ff(Z)2290 4442 y Fo(d)2364 4475 y Fq(a)g Fk(p)-5 b(ath)40 b(choic)-5 b(e)39 b(in)k Fq(\003)37 b(is)f(a)h(collection)f Fp(\025)f Fq(=)0 4602 y Fm(f)p Fp(\025)98 4616 y Fo(xz)207 4602 y Fq(:)25 b(\()p Fp(x;)15 b(z)t Fq(\))27 b Fm(2)e Fq(\003)20 b Fm(\002)g Fq(\003)p Fm(g)31 b Fq(suc)m(h)f(that)h Fp(\025)1346 4616 y Fo(xz)1459 4602 y Fq(is)f(a)h(self{a)m(v)m(oiding)e (path)i(from)e Fp(x)i Fq(to)g Fp(z)j Fq(inside)28 b(\003.)0 4820 y(Giv)m(en)i(a)h(path)f(c)m(hoice)h Fp(\025)g Fq(in)e(\003,)h(and) g Fp(V)46 b Fm(\032)25 b Fq(\003)30 b(w)m(e)h(let)f Fp(W)38 b Fq(=)25 b(\003)p Fm(n)p Fp(V)51 b Fq(and)1112 5071 y Fm(G)1166 5085 y Fo(V)1229 5071 y Fq(\()p Fp(\025)p Fq(\))26 b(=)f(max)1475 5129 y Fo(e)p Fn(2E)1598 5138 y Fh(\003)1659 5071 y Fq(#)p Fm(f)p Fq(\()p Fp(x;)15 b(z)t Fq(\))26 b Fm(2)f Fp(V)41 b Fm(\002)20 b Fp(W)37 b Fq(:)26 b Fp(\025)2512 5085 y Fo(xz)2620 5071 y Fd(A)f Fp(e)p Fm(g)1096 5288 y(D)1166 5302 y Fo(V)1229 5288 y Fq(\()p Fp(\025)p Fq(\))h(=)140 b(max)1474 5351 y Ft(\()p Fo(x;z)s Ft(\))p Fn(2)p Fo(V)16 b Fn(\002)p Fo(W)1889 5288 y Fm(j)p Fp(\025)1967 5302 y Fo(xz)2050 5288 y Fm(j)0 5558 y Fh(15)p Fi(=apr)r(ile=)p Fh(1999)30 b Ft([18])1317 b Fq(4.6)p eop %%Page: 19 19 19 18 bop 0 91 a Fq(No)m(w)30 b(w)m(e)h(claim)e(that)h(for)f(some)i (constan)m(t)g Fp(C)1575 105 y Ft(2)1615 91 y Fq(\()p Fp(\014)5 b Fq(\),)31 b(the)f(quan)m(tit)m(y)g Fp(A)g Fq(de\014ned)f(in)f(\(4.18\))k(satis\014es,)e(for)g(all)e(path)0 218 y(c)m(hoices,)229 495 y Fp(A)e Fm(\024)f Fp(C)484 509 y Ft(2)525 495 y Fq(\()p Fp(\014)5 b Fq(\))1409 433 y Fm(j)p Fq(\003)p Fm(j)1522 400 y Ft(2)p 678 474 1618 4 v 678 557 a Fq(\()p Fp(N)30 b Fq(+)20 b(1\))15 b(\()p Fm(j)p Fp(V)22 b Fm(j)e Fq(+)g(1\))15 b(\()p Fm(j)p Fp(W)e Fm(j)22 b Fq(+)e(1\))15 b(\()p Fm(j)p Fq(\003)p Fm(j)22 b(\000)e Fp(N)30 b Fq(+)20 b(1\))2321 495 y Fm(D)2391 509 y Fo(V)2455 495 y Fq(\()p Fp(\025)p Fq(\))15 b Fm(G)2647 509 y Fo(V)2711 495 y Fq(\()p Fp(\025)p Fq(\))2887 408 y Fg(X)2865 603 y Fo(e)p Fb(@)p Fn(E)2996 612 y Fh(\003)3072 495 y Fq(_)-40 b Fp(\027)5 b Fq([\()p Fm(r)3243 509 y Fo(e)3283 495 y Fp(f)10 b Fq(\))3373 457 y Ft(2)3413 495 y Fq(])230 b(\(4)p Fp(:)p Fq(24\))0 813 y(Once)28 b(w)m(e)h(pro)m(v)m(e)g(inequalit)m(y)e(\(4.24\),)k(the)e(pro)s(of)e (of)i(Prop)s(osition)d(4.3)k(\(and)e(so)g(of)h(Lemma)f(4.1\))i(easily)e (follo)m(ws:)0 940 y(F)-8 b(or)34 b(statemen)m(t)i(\(1\))f(w)m(e)f (just)f(observ)m(e)i(that)f(the)g(big)f(fraction)g(in)g(\(4.24\))j(is)c (less)h(than)h(1,)h(and)e(that)i(trivially)0 1067 y Fm(G)54 1081 y Fo(V)117 1067 y Fq(\()p Fp(\025)p Fq(\))26 b Fm(\024)f(j)p Fp(V)c Fm(j)15 b(j)p Fp(W)e Fm(j)p Fq(,)27 b Fm(D)772 1081 y Fo(V)835 1067 y Fq(\()p Fp(\025)p Fq(\))f Fm(\024)f(j)p Fq(\003)p Fm(j)p Fq(,)i(for)f(an)m(y)g(path)f(c)m(hoice.)40 b(Statemen)m(t)27 b(\(2\))g(is)d(a)i(bit)f(more)h(complicated.)39 b(If)25 b(the)0 1194 y(geometry)34 b(is)d(lik)m(e)h(in)f(statemen)m(t)k (\(2\))e(w)m(e)g(then)f(use)g(the)h(follo)m(wing)e(path)h(c)m(hoice.)48 b(Giv)m(en)33 b Fp(x)28 b Fm(2)h Fp(V)52 b Fq(and)32 b Fp(z)h Fm(2)c Fp(W)0 1321 y Fq(start)f(increasing)e(\(or)h (decreasing\))h(the)f(\014rst)f(co)s(ordinate)h(of)h Fp(x)f 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5266 y Fg(\003)1699 5340 y Fm(\024)40 b Fq(_)-40 b Fp(\027)1861 5266 y Fg(\002)1929 5340 y Fq(_)g Fp(\027)5 b Fq(\(\()p Fm(r)2110 5354 y Fo(xz)2194 5340 y Fp(f)10 b Fq(\))2284 5302 y Ft(2)2340 5340 y Fm(j)15 b(B)2440 5354 y Fo(xz)2523 5340 y Fq(\))2573 5266 y Fg(\003)2637 5340 y Fq(=)40 b(_)-41 b Fp(\027)2798 5266 y Fg(\002)2851 5340 y Fq(\()p Fm(r)2962 5354 y Fo(xz)3046 5340 y Fp(f)10 b Fq(\))3136 5302 y Ft(2)3191 5266 y Fg(\003)3668 5340 y Fq(\(4)p Fp(:)p Fq(27\))0 5558 y Fh(15)p Fi(=apr)r(ile=)p Fh(1999)30 b Ft([19])1317 b Fq(4.7)p eop %%Page: 20 20 20 19 bop 0 91 a Fq(Let)31 b Fp(t)196 105 y Fo(x)p Fn(!)p Fo(z)377 91 y Fq(:)25 b(\012)g Fm(7!)g Fq(\012)30 b(b)s(e)g(the)g (trasformation)h(that)g(mo)m(v)m(es)g(a)g(particle)f(from)f Fp(x)i Fq(to)g Fp(z)j Fq(if)c(this)f(is)g(p)s(ossible,)g Fk(i.e.)1358 367 y Fp(t)1391 381 y Fo(x)p Fn(!)p Fo(z)1547 367 y Fp(\033)f Fq(=)1723 239 y Fg(\032)1807 312 y Fp(t)1840 326 y Fo(xy)1924 312 y Fp(\033)94 b Fq(if)30 b Fp(\033)e Fm(2)d Fp(E)2387 326 y Fo(xz)1807 421 y Fp(\033)211 b Fq(otherwise)30 b(,)0 653 y(and)g(let)g Fp(T)361 667 y Fo(x)p Fn(!)p Fo(z)517 653 y Fp(f)k Fq(=)25 b Fp(f)30 b Fm(\016)20 b Fp(t)865 667 y Fo(x)p Fn(!)p Fo(z)1021 653 y Fq(.)41 b(Then)29 b(w)m(e)i(ha)m(v)m(e)g(\()p Fm(r)1778 667 y Fo(xz)1862 653 y Fp(f)10 b Fq(\))1952 620 y Ft(2)2017 653 y Fq(=)25 b(\()p Fp(T)2201 667 y Fo(x)p Fn(!)p Fo(z)2357 653 y Fp(f)30 b Fm(\000)20 b Fp(f)10 b Fq(\))2613 620 y Ft(2)2673 653 y Fq(+)20 b(\()p Fp(T)2852 667 y Fo(z)s Fn(!)p Fo(x)3008 653 y Fp(f)30 b Fm(\000)20 b Fp(f)10 b Fq(\))3264 620 y Ft(2)3304 653 y Fq(,)31 b(th)m(us)1026 870 y Fp(Y)1079 884 y Fo(xz)1187 870 y Fq(=)41 b(_)-41 b Fp(\027)1349 797 y Fg(\002)1402 870 y Fq(\()p Fp(T)1490 884 y Fo(x)p Fn(!)p Fo(z)1646 870 y Fp(f)30 b Fm(\000)20 b Fp(f)10 b Fq(\))1902 833 y Ft(2)1957 797 y Fg(\003)2015 870 y Fq(+)36 b(_)-41 b Fp(\027)2172 797 y Fg(\002)2225 870 y Fq(\()p Fp(T)2313 884 y Fo(z)s Fn(!)p Fo(x)2469 870 y Fp(f)30 b Fm(\000)20 b Fp(f)10 b Fq(\))2725 833 y Ft(2)2780 797 y Fg(\003)2848 870 y Fp(:)0 1088 y Fq(Let)35 b(no)m(w)f Fp(\025)h Fq(b)s(e)e(an)m(y)i(path)f(c)m(hoice.)54 b(Thanks)33 b(to)i(Lemma)g(4.3)g(in)e([Y])i(w)m(e)g(get)h(that)f(there) f(exists)g Fp(C)3523 1102 y Ft(3)3564 1088 y Fq(\()p Fp(\014)5 b Fq(\))35 b(suc)m(h)0 1215 y(that)1014 1342 y(_)-40 b Fp(\027)1065 1268 y Fg(\002)1118 1342 y Fq(\()p Fp(T)1206 1356 y Fo(x)p Fn(!)p Fo(z)1362 1342 y Fp(f)29 b Fm(\000)20 b Fp(f)10 b Fq(\))1617 1304 y Ft(2)1673 1268 y Fg(\003)1736 1342 y Fm(\024)25 b Fp(C)1897 1356 y Ft(3)1937 1342 y Fq(\()p Fp(\014)5 b Fq(\))15 b Fm(j)p Fp(\025)2156 1356 y Fo(xz)2241 1342 y Fm(j)2335 1255 y Fg(X)2296 1451 y Fo(e)p Fb(@)p Fo(\025)2431 1460 y Fi(xz)2535 1342 y Fq(_)-40 b Fp(\027)5 b Fq([\()p Fm(r)2706 1356 y Fo(e)2746 1342 y Fp(f)10 b Fq(\))2836 1304 y Ft(2)2876 1342 y Fq(])0 1629 y(whic)m(h,)27 b(together)i(with)d(\(4.26\),)31 b(\(4.22\))f(and)d(the)g(de\014nition)e(of)j Fm(G)2272 1643 y Fo(V)2335 1629 y Fq(\()p Fp(\025)p Fq(\))g(and)f Fm(D)2730 1643 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829 2537 312 4 v 829 2630 a Fp(\027)880 2596 y Fn(;)874 2657 y Ft(\003)p Fo(;N)1014 2630 y Fq(\()p Fp(\033)s Fq(\))1176 2558 y(=)1283 2496 y Fp(\026)1338 2463 y Fo(\034)1338 2521 y Ft(\003)1392 2496 y Fq(\()p Fp(\033)s Fq(\))p 1283 2537 235 4 v 1283 2630 a Fp(\026)1338 2596 y Fn(;)1338 2657 y Ft(\003)1392 2630 y Fq(\()p Fp(\033)s Fq(\))1554 2558 y(=)1688 2496 y Fp(h)1740 2463 y Fo(\034)1786 2496 y Fq(\()p Fp(\033)s Fq(\))p 1661 2537 278 4 v 1661 2630 a Fp(\026)1716 2596 y Fn(;)1716 2657 y Ft(\003)1770 2630 y Fq(\()p Fp(h)1857 2604 y Fo(\034)1903 2630 y Fq(\))1975 2558 y Fm(\024)25 b Fp(e)2113 2520 y Ft(2)p Fo(\014)16 b Fn(k)p Fo(H)2308 2493 y Fi(\034)2303 2539 y Fh(\003)2349 2520 y Fn(\000)p Fo(H)2471 2493 y Fj(;)2466 2541 y Fh(\003)2511 2520 y Fn(k)2548 2529 y Fj(1)2645 2558 y Fm(\024)25 b Fp(e)2783 2520 y Ft(4)p Fo(d\014)s Fn(j)p Fo(@)t Ft(\003)p Fn(j)3057 2558 y Fp(:)0 2859 y Fq(Analogously)30 b(w)m(e)g(\014nd)1555 2981 y Fp(\027)1606 2948 y Fo(\034)1600 3006 y Ft(\003)p Fo(;N)1740 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b(\026)-60 b Fp(')q Fq(\()p Fp(v)s Fq(\))31 b(due)f(to)h(a)g(misprin)m(t\))d(and)i(the)g (result)f(is)1520 1095 y(\026)-59 b Fp(')p Fq(\()p Fp(v)s Fq(\))26 b(=)1815 1034 y(1)p 1815 1074 46 4 v 1815 1158 a(2)1872 1095 y Fp(w)r Fq(\()1974 1025 y Fm(p)p 2050 1025 48 4 v 70 x Fp(v)e Fm(^)2199 1032 y(p)p 2274 1032 85 4 v 2274 1095 a Fp(v)2318 1109 y Ft(0)2359 1095 y Fq(\))1319 b(\(5)p Fp(:)p Fq(2\))0 1364 y(where)31 b(the)g(constan)m(t) h Fp(w)i Fq(is)c(de\014ned)f(as)j Fp(w)d Fq(=)d Fp(W)13 b Fq(\()p Fp(\015)1777 1378 y Fo(w)1835 1364 y Fq(\))32 b(and)e Fp(\015)2126 1378 y Fo(w)2216 1364 y Fq(is)g(the)h Fk(Wul\013)i(curve)k Fq(\(see)32 b(e.g.)44 b([Sh])30 b(or)h([DKS]\))0 1491 y(whic)m(h)24 b(dep)s(ends)g(on)h Fp(\014)30 b Fq(and)25 b(it)g(is)f(c)m(haracterized)j(b)m(y)e(the)h (fact)g(that)g(it)f(is)f(the)i(unique)d(solution)h(to)i(the)f(follo)m (wing)0 1618 y(v)-5 b(ariational)29 b(problem)560 1847 y Fp(w)f Fq(=)d Fp(W)13 b Fq(\()p Fp(\015)930 1861 y Fo(w)989 1847 y Fq(\))25 b(=)g(min)o Fm(f)15 b Fp(W)e Fq(\()p Fp(\015)5 b Fq(\))26 b(:)f Fp(\015)36 b Fq(is)29 b(a)i(closed)f(curv)m(e)h(enclosing)e(a)i(unit)e(area)16 b Fm(g)f Fp(:)374 b Fq(\(5)p Fp(:)p Fq(3\))0 2075 y(The)30 b(singularit)m(y)e(p)s(oin)m(t)h Fp(v)918 2089 y Ft(0)990 2075 y Fq(satis\014es)g(the)i(equation)1552 2289 y(1)p 1552 2330 46 4 v 1552 2413 a(2)1608 2351 y Fp(w)1675 2288 y Fm(p)p 1751 2288 85 4 v 63 x Fp(v)1795 2365 y Ft(0)1861 2351 y Fq(=)25 b Fp(\034)1997 2365 y Fo(\014)2046 2351 y Fq(\(\(1)p Fp(;)15 b Fq(0\)\))g Fp(:)1357 b Fq(\(5)p Fp(:)p Fq(4\))0 2646 y(It)43 b(is)f(fairly)g(easy)h(to)h(sho)m(w)f (that)h(the)f(in\014m)m(um)e(in)g(\(5.1\))k(is)d(attained)h(when)g Fp(\015)p 2862 2679 53 4 v 48 w Fq(is)f(a)h(single)f(curv)m(e)h(whic)m (h)0 2774 y(corresp)s(onds)29 b(to)66 2929 y(\(1\))46 b(a)31 b(quarter)f(of)h(a)g(W)-8 b(ul\013)29 b(shap)s(e)h(cen)m(tered)h (at)h(one)e(of)h(the)f(four)g(corners)g(of)h Fp(Q)f Fq(if)f Fp(v)g Fm(\024)c Fp(v)3209 2943 y Ft(0)66 3057 y Fq(\(2\))46 b(a)31 b(unit)e(\(horizon)m(tal)h(or)h(v)m(ertical\))g(segmen)m(t)g(if) e Fp(v)g Fm(\025)c Fp(v)2075 3071 y Ft(0)2116 3057 y Fq(.)0 3212 y(W)-8 b(e)32 b(set)e(then)1085 3339 y(\000)1142 3353 y Fo(w)1200 3339 y Fq(\()p Fp(v)s Fq(\))d(=)e Fm(f)p Fp(\015)31 b Fm(2)24 b(D)k Fq(:)e Ff(V)p Fq(\()p Fp(\015)p 1893 3371 V 10 w Fq(\))g(=)e Fp(v)s(;)31 b(W)13 b Fq(\()p Fp(\015)5 b Fq(\))26 b(=)39 b(\026)-59 b Fp(')p Fq(\()p Fp(v)s Fq(\))p Fm(g)15 b Fp(:)900 b Fq(\(5)p Fp(:)p Fq(5\))0 3526 y(In)35 b(the)h(follo)m(wing)e(prop)s(osition)f(w)m(e)j(pro)m(v)m (e)h(a)f(stabilit)m(y)e(prop)s(ert)m(y)h(of)h(the)g(v)-5 b(ariational)35 b(problem)f(\(5.1\))j(whic)m(h)0 3653 y(will)28 b(b)s(e)h(used)h(later.)0 3834 y Fr(Prop)s(osition)42 b(5.1.)53 b Fc(Let)35 b Fp(v)i Fm(2)c Fq(\(0)p Fp(;)15 b Fq(1)p Fp(=)p Fq(2])p Fc(.)57 b(F)-8 b(or)36 b(all)e Fp(")g(>)f Fq(0)i Fc(there)h(is)e(a)h Fp(\016)j Fq(=)32 b Fp(\016)s Fq(\()p Fp(";)15 b(v)s Fq(\))36 b Fp(>)d Fq(0)j Fc(suc)m(h)f(that)g(if)f Fp(\015)p 3606 3867 V 39 w Fm(2)f(D)3859 3801 y Fn(\003)0 3962 y Fc(satis\014es)28 b Ff(V)p Fq(\()p Fp(\015)p 429 3994 V 10 w Fq(\))e(=)f Fp(v)32 b Fc(and)d Fp(W)13 b Fq(\()p Fp(\015)p 1029 3994 V 4 w Fq(\))26 b Fm(\024)40 b Fq(\026)-60 b Fp(')p Fq(\()p Fp(v)s Fq(\))19 b(+)e Fp(\016)s Fc(,)30 b(then)f(there)g(exists)f Fp(v)2351 3929 y Fn(0)2402 3962 y Fm(2)d Fq(\(0)p Fp(;)15 b Fq(1)p Fp(=)p Fq(2])32 b Fc(with)27 b Fm(j)p Fp(v)21 b Fm(\000)c Fp(v)3230 3929 y Fn(0)3256 3962 y Fm(j)25 b(\024)g Fp(")30 b Fc(and)e Fp(\015)3696 3976 y Ft(1)3762 3962 y Fm(2)d Fp(\015)p 3848 3994 V 0 4089 a Fc(suc)m(h)30 b(that)1077 4216 y Fq(inf)979 4279 y Fo(\015)t Fn(2)p Ft(\000)1115 4288 y Fi(w)1166 4279 y Ft(\()p Fo(v)1232 4261 y Fj(0)1258 4279 y Ft(\))1302 4216 y Fp(d)1349 4230 y Fo(H)1418 4216 y Fq(\()p Fp(\015)1500 4230 y Ft(1)1541 4216 y Fp(;)15 b(\015)5 b Fq(\))26 b Fm(\024)f Fp(")91 b Fq(and)2279 4130 y Fg(X)2176 4325 y Fo(\015)t Fn(2)p Fo(\015)p 2267 4352 43 4 v 4 w Ft(:)12 b Fo(\015)t Fn(6)p Ft(=)p Fo(\015)2478 4334 y Fh(1)2529 4216 y Ff(V)p Fq(\()p Fp(\015)5 b Fq(\))31 b Fm(\024)25 b Fp(")15 b(:)0 4526 y Fk(Pr)-5 b(o)g(of.)32 b Fq(The)e(pro)s(of)f(relies)g(on)i(the)f (follo)m(wing)f(preliminary)e(results,)i(Lemma)i(5.2)g(and)f(Lemma)g (5.3)0 4707 y Fr(Lemma)j(5.2.)53 b Fc(Let)30 b Fp(v)792 4721 y Fo(i)848 4707 y Fm(2)25 b Fq(\(0)p Fp(;)15 b Fq(1)p Fp(=)p Fq(2])33 b Fc(for)d Fp(i)c Fq(=)f(1)p Fp(;)15 b(:)g(:)g(:)i(;)e(n)p Fc(,)30 b(and)g(let)g Fp(v)f Fq(=)2373 4638 y Fg(P)2469 4661 y Fo(n)2469 4734 y(i)p Ft(=1)2606 4707 y Fp(v)2650 4721 y Fo(i)2681 4707 y Fc(.)40 b(Then)1460 4908 y Fo(n)1417 4935 y Fg(X)1424 5129 y Fo(i)p Ft(=1)1578 5022 y Fq(\026)-60 b Fp(')q Fq(\()p Fp(v)1702 5036 y Fo(i)1732 5022 y Fq(\))26 b Fm(\025)40 b Fq(\026)-60 b Fp(')p Fq(\()p Fp(v)s Fq(\))2080 4895 y Fg(r)p 2172 4895 311 4 v 2304 4960 a Fp(v)p 2183 5001 289 4 v 2183 5084 a Fq(max)2353 5098 y Fo(i)2398 5084 y Fp(v)2442 5098 y Fo(i)0 5340 y Fk(Pr)-5 b(o)g(of.)32 b Fq(W)-8 b(e)31 b(can)g(assume)f Fp(v)951 5354 y Ft(1)1017 5340 y Fm(\025)25 b Fp(v)1157 5354 y Ft(2)1223 5340 y Fm(\025)g Fp(:)15 b(:)g(:)27 b Fm(\025)d Fp(v)1590 5354 y Fo(n)1640 5340 y Fq(.)0 5558 y Fh(15)p Fi(=apr)r(ile=)p Fh(1999)30 b Ft([24])1317 b Fq(5.2)p eop %%Page: 25 25 25 24 bop 0 91 a Fk(Case)33 b(1:)42 b Fp(v)383 105 y Ft(1)449 91 y Fm(\024)25 b Fp(v)589 105 y Ft(0)630 91 y Fq(.)41 b(In)29 b(this)h(case)h(w)m(e)g(can)g(write)1035 290 y Fo(n)992 317 y Fg(X)999 511 y Fo(i)p Ft(=1)1153 404 y Fq(\026)-59 b Fp(')p Fq(\()p Fp(v)1277 418 y Fo(i)1308 404 y Fq(\))25 b(=)1475 342 y(1)p 1475 383 46 4 v 1475 466 a(2)1532 404 y Fp(w)1658 290 y Fo(n)1614 317 y Fg(X)1621 511 y Fo(i)p Ft(=1)1810 342 y Fp(v)1854 356 y Fo(i)p 1772 383 151 4 v 1772 408 a Fm(p)p 1847 408 75 4 v 1847 466 a Fp(v)1891 480 y Fo(i)1958 404 y Fm(\025)2065 342 y Fq(1)p 2065 383 46 4 v 2065 466 a(2)2121 404 y Fp(w)2256 342 y(v)p 2199 383 161 4 v 2199 408 a Fm(p)p 2275 408 85 4 v 58 x Fp(v)2319 480 y Ft(1)2396 404 y Fm(\025)40 b Fq(\026)-60 b Fp(')q Fq(\()p Fp(v)s Fq(\))2669 278 y Fg(r)p 2761 278 107 4 v 2790 342 a Fp(v)p 2772 383 85 4 v 2772 466 a(v)2816 480 y Ft(1)2883 404 y Fp(:)0 763 y Fk(Case)33 b(2:)42 b Fp(v)383 777 y Ft(1)449 763 y Fp(>)25 b(v)589 777 y Ft(0)630 763 y Fq(.)41 b(Let)31 b Fp(s)f Fq(b)s(e)f(suc)m(h)h(that)h Fp(v)1501 777 y Ft(1)1567 763 y Fm(\025)25 b Fp(:)15 b(:)g(:)27 b Fm(\025)e Fp(v)1935 777 y Fo(s)1999 763 y Fm(\025)g Fp(v)2139 777 y Ft(0)2205 763 y Fp(>)g(v)2345 777 y Fo(s)p Ft(+1)2501 763 y Fm(\025)g Fp(:)15 b(:)g(:)26 b Fm(\025)f Fp(v)2868 777 y Fo(n)2918 763 y Fq(.)40 b(Then)269 970 y Fo(n)225 997 y Fg(X)232 1191 y Fo(i)p Ft(=1)386 1084 y Fq(\026)-59 b Fp(')p Fq(\()p Fp(v)510 1098 y Fo(i)541 1084 y Fq(\))26 b(=)709 1022 y(1)p 709 1063 46 4 v 709 1146 a(2)765 1084 y Fp(w)832 983 y Fg(h)939 970 y Fo(s)891 997 y Fg(X)897 1191 y Fo(i)p Ft(=1)1037 1021 y Fm(p)p 1113 1021 85 4 v 63 x Fp(v)1157 1098 y Ft(0)1218 1084 y Fq(+)1390 970 y Fo(n)1347 997 y Fg(X)1309 1191 y Fo(i)p Ft(=)p Fo(s)p Ft(+1)1532 1021 y Fm(p)p 1607 1021 75 4 v 1607 1084 a Fp(v)1651 1098 y Fo(i)1697 983 y Fg(i)1765 1084 y Fm(\025)1872 1022 y Fq(1)p 1872 1063 46 4 v 1872 1146 a(2)1928 1084 y Fp(w)1995 1021 y Fm(p)p 2072 1021 85 4 v 2072 1084 a Fp(v)2116 1098 y Ft(0)2215 970 y Fo(n)2172 997 y Fg(X)2178 1191 y Fo(i)p Ft(=1)2318 957 y Fg(r)p 2409 957 107 4 v 2425 1022 a Fp(v)2469 1036 y Fo(i)p 2420 1063 85 4 v 2420 1146 a Fp(v)2464 1160 y Ft(1)2541 1084 y Fm(\025)2648 1022 y Fq(1)p 2648 1063 46 4 v 2648 1146 a(2)2704 1084 y Fp(w)2771 957 y Fg(r)p 2863 957 154 4 v 2874 1022 a Fp(v)2918 1036 y Ft(0)2958 1022 y Fp(v)p 2874 1063 133 4 v 2897 1146 a(v)2941 1160 y Ft(1)3042 1084 y Fq(=)39 b(\026)-59 b Fp(')p Fq(\()p Fp(v)s Fq(\))3314 957 y Fg(r)p 3406 957 107 4 v 3436 1022 a Fp(v)p 3417 1063 85 4 v 3417 1146 a(v)3461 1160 y Ft(1)3604 1084 y Fd(\003)0 1427 y Fr(Lemma)37 b(5.3.)53 b Fc(F)-8 b(or)35 b(all)e Fp(v)h Fm(2)d Fq(\(0)p Fp(;)15 b Fq(1)p Fp(=)p Fq(2])p Fc(,)38 b(all)33 b Fp(")f(>)f Fq(0)j Fc(there)g(exists)g Fp(\016)2319 1394 y Fn(0)2345 1427 y Fq(\()p Fp(";)15 b(v)s Fq(\))34 b Fp(>)d Fq(0)j Fc(suc)m(h)g(that)g(for)g(all)f Fp(\015)k Fm(2)31 b(D)36 b Fc(with)0 1555 y Ff(V)p Fq(\()p Fp(\015)5 b Fq(\))31 b(=)25 b Fp(v)33 b Fc(and)d Fp(W)13 b Fq(\()p Fp(\015)5 b Fq(\))26 b Fm(\024)39 b Fq(\026)-59 b Fp(')p Fq(\()p Fp(v)s Fq(\))21 b(+)f Fp(\016)1238 1522 y Fn(0)1295 1555 y Fc(w)m(e)31 b(ha)m(v)m(e)1607 1795 y Fq(inf)1508 1858 y Fo(\015)1550 1840 y Fj(0)1575 1858 y Fn(2)p Ft(\000)1669 1867 y Fi(w)1720 1858 y Ft(\()p Fo(v)r Ft(\))1831 1795 y Fp(d)1878 1809 y Fo(H)1947 1795 y Fq(\()p Fp(\015)5 b(;)15 b(\015)2126 1757 y Fn(0)2153 1795 y Fq(\))25 b Fm(\024)g Fp(")15 b(:)0 2097 y Fk(Pr)-5 b(o)g(of.)30 b Fq(This)c(statemen)m(t)31 b(follo)m(ws)c(froma)i(simple) d(compactness)j(argumen)m(t.)41 b(F)-8 b(or)29 b(a)g(pro)s(of)f(see)h (Lemma)f(12.4)i(in)0 2225 y([PV].)0 2411 y(W)-8 b(e)32 b(can)e(\014nally)e(pro)m(v)m(e)k(5.1.)42 b(Let)30 b Fp(\015)p 1188 2443 53 4 v 31 w Fq(=)25 b Fm(f)p Fp(\015)1454 2425 y Ft(1)1495 2411 y Fp(;)15 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Fq(\026)-60 b Fp(')q Fq(\()p Fp(v)s Fq(\))24 b(+)f Fp(\016)1825 5354 y Ft(1)1866 5340 y Fq(,)37 b(Thanks)d(to)h(Prop)s (osition)e(5.1)j(w)m(e)g(kno)m(w)f(that)h(there)f(is)0 5558 y Fh(15)p Fi(=apr)r(ile=)p Fh(1999)30 b Ft([26])1317 b Fq(5.4)p eop %%Page: 27 27 27 26 bop 0 91 a Fp(\015)47 105 y Ft(1)115 91 y Fm(2)26 b Fp(\015)p 202 123 53 4 v 5 w Fq(,)32 b Fp(v)358 58 y Fn(0)411 91 y Fm(2)26 b Fq(\()p Fp(v)f Fm(\000)c Fp(")736 58 y Fn(0)761 91 y Fp(;)15 b(v)25 b Fq(+)c Fp(")1004 58 y Fn(0)1030 91 y Fq(\))31 b(and)g Fp(\015)1321 105 y Fo(w)1407 91 y Fm(2)26 b Fq(\000)1551 105 y Fo(w)1610 91 y Fq(\()p Fp(v)1692 58 y Fn(0)1718 91 y Fq(\))32 b(suc)m(h)f(that)h Fp(d)2236 105 y Fo(H)2305 91 y Fq(\()p Fp(\015)2387 105 y Ft(1)2428 91 y Fp(;)15 b(\015)2515 105 y Fo(w)2575 91 y Fq(\))27 b Fp(<)f(")2776 58 y Fn(0)2834 91 y Fq(and)3011 23 y Fg(P)3107 118 y Fo(\015)t Fn(6)p Ft(=)p Fo(\015)3243 127 y Fh(1)3299 91 y Ff(V)p Fq(\()p Fp(\015)5 b Fq(\))32 b Fm(\024)27 b Fp(")p Fq(.)44 b(This)0 217 y(implies)1436 344 y Fm(j)p Ff(V)1522 358 y Fo(U)1588 344 y Fq(\()p Fp(\015)1670 358 y Ft(1)1711 344 y Fq(\))21 b Fm(\000)f Ff(V)1919 358 y Fo(U)1984 344 y Fq(\()p Fp(\015)2066 358 y Fo(w)2126 344 y Fq(\))p Fm(j)26 b(\024)f Fp(k)2355 358 y Ft(1)2396 344 y Fp(")2438 306 y Fn(0)0 498 y Fq(and)1169 624 y Fm(j)p Ff(V)1255 638 y Fo(U)1320 624 y Fq(\()p Fp(\015)p 1355 657 V 6 w Fq(\))20 b Fm(\000)g Ff(V)1615 638 y Fo(U)1681 624 y Fq(\()p Fp(\015)1763 638 y Ft(1)1804 624 y Fq(\))p Fm(j)26 b(\024)2089 538 y Fg(X)1986 734 y Fo(\015)t Fn(2)p Fo(\015)p 2077 761 43 4 v 4 w Ft(:)12 b Fo(\015)t Fn(6)p Ft(=)p Fo(\015)2288 743 y Fh(1)2339 624 y Ff(V)p Fq(\()p Fp(\015)5 b Fq(\))31 b Fm(\024)25 b Fp(")15 b(:)0 901 y Fq(Th)m(us)29 b(w)m(e)i(get)1178 1027 y Fm(j)p Ff(V)1264 1041 y Fo(U)1330 1027 y Fq(\()p Fp(\015)p 1365 1059 53 4 v 5 w Fq(\))21 b Fm(\000)f Ff(V)1625 1041 y Fo(U)1690 1027 y Fq(\()p Fp(\015)1772 1041 y Fo(w)1832 1027 y Fq(\))p Fm(j)25 b(\024)g Fq(\(1)c(+)f Fp(k)2252 1041 y Ft(1)2293 1027 y Fq(\))p Fp(")2370 990 y Fn(0)2422 1027 y Fm(\024)25 b Fp(")91 b Fd(\003)0 1219 y Fq(W)-8 b(e)32 b(conclude)d(this)h(section)g(with)f(t)m(w)m(o)j(results)d(whic) m(h)g(will)f(b)s(e)i(used)f(in)g(the)i(pro)s(of)e(of)i(Lemma)f(6.6.)0 1383 y Fr(Prop)s(osition)36 b(5.6.)53 b Fc(F)-8 b(or)31 b(all)f Fp(v)e Fm(2)d Fq(\(0)p Fp(;)15 b Fq(1)p Fp(=)p Fq(2])p Fc(,)33 b Fp(")26 b(>)f Fq(0)31 b Fc(there)f(exists)g Fp(\016)f(>)c Fq(0)31 b Fc(suc)m(h)f(that)h(if)e Fp(W)13 b Fq(\()p Fp(\015)p 3149 1415 V 5 w Fq(\))26 b Fp(<)3377 1359 y Fq(\026)3358 1383 y Fp( )s Fq(\()p Fp(v)s Fq(\))c Fm(\000)e Fp(")30 b Fc(then)1339 1538 y Fm(j)p Ff(V)p Fq(\()p Fp(\015)p 1460 1571 V 10 w Fq(\))21 b Fm(\000)e Fp(v)s Fm(j)i Fq(+)f Fm(j)p Ff(V)1933 1552 y Fo(U)1999 1538 y Fq(\()p Fp(\015)p 2034 1571 V 5 w Fq(\))h Fm(\000)f Fp(v)s(=)p Fq(2)p Fm(j)27 b Fp(>)e(\016)0 1695 y Fk(Sketch)35 b(of)f(the)h(pr)-5 b(o)g(of)p Fq(.)49 b(Assume)31 b(the)i(statemen)m(t) h(to)f(b)s(e)f(false.)46 b(Then)31 b(there)h(exists)g(some)h Fp(v)f Fm(2)c Fq(\(0)p Fp(;)15 b Fq(1)p Fp(=)p Fq(2],)36 b Fp(")28 b(>)h Fq(0)0 1821 y(suc)m(h)h(that)h(for)f(all)f Fp(\016)h(>)24 b Fq(0)31 b(there)g(is)e Fp(\015)p 1232 1853 V 31 w Fm(2)24 b(D)1468 1788 y Fn(\003)1540 1821 y Fq(suc)m(h)30 b(that)1339 1976 y Fm(j)p Ff(V)p Fq(\()p Fp(\015)p 1460 2008 V 10 w Fq(\))21 b Fm(\000)e Fp(v)s Fm(j)i Fq(+)f Fm(j)p Ff(V)1933 1990 y Fo(U)1999 1976 y Fq(\()p Fp(\015)p 2034 2008 V 5 w Fq(\))h Fm(\000)f Fp(v)s(=)p Fq(2)p Fm(j)27 b(\024)e Fp(\016)0 2131 y Fq(But)40 b(if)e(this)g(is)g(true)i(w)m(e)f(can)h(easily)e(construct)i(a)g(new)f Fp(\015)p 2037 2164 V 2089 2098 a Fn(0)2114 2131 y Fq(,)j(b)m(y)d (adding)f(to)i Fp(\015)p 2743 2164 V 45 w Fq(t)m(w)m(o)g(small)e (closed)h(curv)m(es)h(of)0 2258 y(appropiate)30 b(areas,)h(suc)m(h)f (that)66 2404 y(\(1\))46 b Ff(V)p Fq(\()p Fp(\015)p 323 2437 V 380 2371 a Fn(0)406 2404 y Fq(\))26 b(=)e Fp(v)34 b Fq(and)c Ff(V)878 2418 y Fo(U)943 2404 y Fq(\()p Fp(\015)p 978 2437 V 1031 2371 a Fn(0)1056 2404 y Fq(\))c(=)f Fp(v)s(=)p Fq(2)66 2532 y(\(2\))46 b Fp(W)13 b Fq(\()p Fp(\015)p 361 2564 V 413 2499 a Fn(0)439 2532 y Fq(\))25 b Fm(\024)g Fp(W)13 b Fq(\()p Fp(\015)p 729 2564 V 5 w Fq(\))21 b(+)f Fp(C)1000 2454 y Fm(p)p 1075 2454 44 4 v 1075 2532 a Fp(\016)0 2677 y Fq(So)30 b(if)g Fp(\016)k Fq(is)29 b(small)g(enough)h (w)m(e)h(use)f(the)h(de\014nition)d(of)1900 2653 y(\026)1881 2677 y Fp( )t Fq(\()p Fp(v)s Fq(\))j(and)f(w)m(e)h(get)g(a)g(con)m (tradiction.)91 b Fd(\003)0 2841 y Fq(F)-8 b(or)31 b Fp(v)s(;)15 b(w)29 b Fm(2)c Fq([0)p Fp(;)15 b Fq(1)p Fp(=)p Fq(2])33 b(w)m(e)d(de\014ne)g(the)h(quan)m(tities)1454 3006 y(^)-63 b Fp(m)1516 2969 y Ft(+)1577 3006 y Fq(\()p Fp(v)s Fq(\))26 b(=)f Fp(m)1896 2969 y Fn(\003)1937 3006 y Fq(\(1)c Fm(\000)f Fq(2)p Fp(v)s Fq(\))1453 3161 y(^)-63 b Fp(m)1515 3123 y Fn(\000)1577 3161 y Fq(\()p Fp(v)s Fq(\))26 b(=)f Fm(\000)p Fp(m)1967 3123 y Fn(\003)2008 3161 y Fq(\(1)c Fm(\000)f Fq(2)p Fp(v)s Fq(\))1433 3316 y(^)-62 b Fp(m)1496 3277 y Ft(+)1496 3342 y Fo(U)1556 3316 y Fq(\()p Fp(w)r Fq(\))27 b(=)e(2)p Fp(m)1941 3278 y Fn(\003)1983 3316 y Fq(\(1)p Fp(=)p Fq(2)d Fm(\000)e Fq(2)p Fp(w)r Fq(\))1432 3470 y(^)-62 b Fp(m)1495 3431 y Fn(\000)1495 3497 y Fo(U)1556 3470 y Fq(\()p Fp(w)r Fq(\))27 b(=)e Fm(\000)p Fq(2)p Fp(m)2012 3433 y Fn(\003)2054 3470 y Fq(\(1)p Fp(=)p Fq(2)d Fm(\000)d Fq(2)p Fp(w)r Fq(\))0 3635 y(T)-8 b(o)33 b(understand)d(wh)m(y)i(w)m(e)h(in)m(tro)s (duce)e(these)i(quan)m(tities,)g(let)f Fp(v)g Fq(=)c Ff(V)p Fq(\()p Fp(\015)p 2436 3667 53 4 v 10 w Fq(\))33 b(and)f(let)g Fp(w)f Fq(=)e Ff(V)3130 3649 y Fo(U)3195 3635 y Fq(\()p Fp(\015)p 3230 3667 V 6 w Fq(\).)47 b(If)32 b(the)g(set)h Fp(A)3853 3649 y Fo(\015)p 3853 3676 47 4 v 0 3762 a Fq(had)d(a)g(uniform)f(\\magnetization)i(densit)m(y")f Fm(\000)p Fp(m)1740 3729 y Fn(\003)1811 3762 y Fq(and)g(the)h(set)f Fp(B)2355 3776 y Fo(\015)p 2355 3802 V 2432 3762 a Fq(had)g(a)h (uniform)d(magnetization)j(densit)m(y)0 3888 y(+)p Fp(m)151 3855 y Fn(\003)192 3888 y Fq(,)25 b(then)41 b(^)-62 b Fp(m)523 3855 y Ft(+)583 3888 y Fq(\()p Fp(v)s Fq(\))25 b(w)m(ould)d(b)s(e)h(the)h(magnetization)g(in)e Fp(Q)i Fq(and)40 b(^)-62 b Fp(m)2279 3849 y Ft(+)2279 3915 y Fo(U)2363 3888 y Fq(the)24 b(magnetization)g(in)e Fp(U)10 b Fq(.)39 b(If)23 b(vicev)m(ersa)h(w)m(e)0 4014 y(ha)m(v)m(e)30 b(+)p Fp(m)359 3981 y Fn(\003)429 4014 y Fq(on)e Fp(A)621 4028 y Fo(\015)p 621 4055 V 696 4014 a Fq(and)g Fm(\000)p Fp(m)1022 3981 y Fn(\003)1092 4014 y Fq(on)h Fp(B)1286 4028 y Fo(\015)p 1286 4055 V 1361 4014 a Fq(then)f(w)m(e)h(get)47 b(^)-62 b Fp(m)1928 3981 y Fn(\000)1989 4014 y Fq(\()p Fp(v)s Fq(\))29 b(and)46 b(^)-63 b Fp(m)2390 3976 y Fn(\000)2390 4041 y Fo(U)2452 4014 y Fq(\()p Fp(w)r Fq(\))29 b(as)g(the)g(resp)s (ectiv)m(e)g(magnetizations)0 4141 y(on)38 b Fp(Q)g Fq(and)f Fp(U)10 b Fq(.)64 b(F)-8 b(or)38 b(an)m(y)h Fp(m)e Fm(2)h Fq([0)p Fp(;)15 b(m)1345 4108 y Fn(\003)1388 4141 y Fq(])38 b(w)m(e)g(let)g Fp(')p Fq(\()p Fp(m)p Fq(\))h(=)53 b(\026)-60 b Fp(')p Fq(\()p Fp(v)s Fq(\))39 b(and)f Fp( )s Fq(\()p Fp(m)p Fq(\))h(=)2866 4117 y(\026)2847 4141 y Fp( )t Fq(\()p Fp(v)s Fq(\))g(where)e Fp(v)42 b Fq(is)37 b(suc)m(h)g(that)0 4267 y Fp(m)25 b Fq(=)42 b(^)-62 b Fp(m)281 4234 y Ft(+)341 4267 y Fq(\()p Fp(v)s Fq(\).)0 4431 y Fr(Prop)s(osition)36 b(5.7.)53 b Fc(Let)31 b Fp(v)e Fm(2)24 b Fq(\(0)p Fp(;)15 b Fq(1)p Fp(=)p Fq(2])p Fc(,)34 b Fp(\016)29 b(>)c Fq(0)p Fc(,)31 b(and)f(let)g Fp(m)25 b Fq(=)42 b(^)-62 b Fp(m)2295 4398 y Ft(+)2355 4431 y Fq(\()p Fp(v)s Fq(\))p Fc(.)42 b(If)30 b(either)1276 4586 y Fm(j)p Fp(m)21 b Fm(\000)37 b Fq(^)-62 b Fp(m)1573 4549 y Ft(+)1633 4586 y Fq(\()p Fp(v)1715 4549 y Fn(0)1741 4586 y Fq(\))p Fm(j)21 b Fq(+)f Fm(j)p Fp(m)g Fm(\000)37 b Fq(^)-62 b Fp(m)2209 4547 y Ft(+)2209 4613 y Fo(U)2270 4586 y Fq(\()p Fp(w)2372 4549 y Fn(0)2398 4586 y Fq(\))p Fm(j)26 b(\024)f Fp(\016)0 4742 y Fc(or)1276 4868 y Fm(j)p Fp(m)20 b Fm(\000)37 b Fq(^)-62 b Fp(m)1572 4830 y Fn(\000)1633 4868 y Fq(\()p Fp(v)1715 4830 y Fn(0)1741 4868 y Fq(\))p Fm(j)21 b Fq(+)f Fm(j)p Fp(m)g Fm(\000)37 b Fq(^)-62 b Fp(m)2209 4829 y Fn(\000)2209 4895 y Fo(U)2270 4868 y Fq(\()p Fp(w)2372 4830 y Fn(0)2399 4868 y Fq(\))p Fm(j)26 b(\024)e Fp(\016)0 5022 y Fc(then)30 b(w)m(e)h(ha)m(v)m(e)1425 5149 y Fm(j)p Fp(v)24 b Fm(\000)19 b Fp(v)1655 5111 y Fn(0)1681 5149 y Fm(j)i Fq(+)f Fm(j)p Fp(v)s(=)p Fq(2)h Fm(\000)f Fp(w)2159 5111 y Fn(0)2185 5149 y Fm(j)26 b(\024)2359 5087 y Fq(2)p Fp(\016)p 2343 5128 122 4 v 2343 5211 a(m)2423 5185 y Fn(\003)0 5322 y Fq(W)-8 b(e)32 b(omit)e(the)g(elemen)m(tary)h(pro)s (of.)0 5558 y Fh(15)p Fi(=apr)r(ile=)p Fh(1999)f Ft([27])1317 b Fq(5.5)p eop %%Page: 28 28 28 27 bop 0 91 a Fr(6.)47 b(Pro)s(of)36 b(of)f(the)f(upp)s(er)h(b)s (ound)0 249 y Fq(In)f(this)g(section)i(w)m(e)f(pro)m(v)m(e)h(the)f(upp) s(er)e(b)s(ound)g(\(2.16\))38 b(whic)m(h)33 b(app)s(ears)i(in)e(part)i (\(b\))h(of)f(Theorem)g(2.1.)55 b(The)0 377 y(pro)s(of)42 b(pro)s(ceeds)g(along)g(the)h(follo)m(wing)d(steps:)65 b(\014rst)42 b(w)m(e)h(sho)m(w)f(that,)k(roughly)-8 b(,)45 b(\(see)e(\(6.16\))i(for)d(a)h(precise)0 504 y(statemen)m(t\))811 734 y(gap\()p Fp(L)1049 696 y Fn(;)1049 758 y Fo(Q)1106 767 y Fi(L)1154 758 y Fo(;)p Fn(N)1246 736 y Fi(m)1236 782 y(L)1310 734 y Fq(\))26 b Fm(\024)f Fp(C)c(L)1641 659 y(\026)1696 616 y Fo(\014)s(;)p Fn(;)1696 686 y Fo(Q)1753 695 y Fi(L)1806 586 y Fg(\010)1874 659 y Fp(m)1954 673 y Fo(Q)2011 682 y Fi(L)2063 659 y Fq(\()p Fp(\033)s Fq(\))27 b Fm(\030)e Fp(m;)45 b(m)2541 673 y Fo(U)2591 682 y Fi(L)2643 659 y Fq(\()p Fp(\033)s Fq(\))26 b Fm(\030)f Fp(m)2985 586 y Fg(\011)p 1641 713 1397 4 v 1931 815 a Fp(\026)1986 772 y Fo(\014)s(;)p Fn(;)1986 842 y Fo(Q)2043 851 y Fi(L)2095 742 y Fg(\010)2164 815 y Fp(m)2244 829 y Fo(Q)2301 838 y Fi(L)2353 815 y Fq(\()p Fp(\033)s Fq(\))h Fm(\030)f Fp(m)2695 742 y Fg(\011)3064 734 y Fp(:)624 b Fq(\(6)p Fp(:)p Fq(1\))0 1022 y(where)25 b Fp(U)320 1036 y Fo(L)400 1022 y Fq(is)g(the)h(rescaled)f(v)m(ersion)h(of)f(the)h(set)g Fp(U)36 b Fq(de\014ned)24 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Fp(:)1327 b Fq(\(6)p Fp(:)p Fq(5\))0 5085 y(Decomp)s(osition)29 b(\(6.4\))i(is)d(in)m(tuitiv)m(ely) f(obtained)i(b)m(y)g(cutting)g(all)f(three)i(and)e(four)h(edges)h (meetings,)f(along)h(the)0 5213 y(south{w)m(est)f(to)g(north{east)f (direction)f(\(see)i([CGMS])f(for)g(details\).)39 b(Fig.)28 b(2)h(sho)m(ws)f(a)g(set)g(of)h(edges)f(on)g(the)g(left)0 5340 y(and)i(the)g(corresp)s(onding)f(collection)h(of)g(con)m(tours)h (on)f(the)h(righ)m(t.)0 5558 y Fh(15)p Fi(=apr)r(ile=)p Fh(1999)f Ft([28])1317 b Fq(6.1)p eop %%Page: 29 29 29 28 bop 41 1145 a @beginspecial @setspecial %%BeginDocument: fig3.psc /inch {72 mul} def /black{ 0 setgray } def /gray2 {0.5 setgray} def /gray1 {0.8 setgray} def /gray0 {1 setgray} def /d 16 def /rad d 3 div def /trx 0 def /try 0 def /box { newpath moveto 1 inch 0 rlineto 0 1 inch rlineto -1 inch 0 rlineto closepath } def /fillbox { setgray fill } def /crs{ decx x y moveto ll uu decy decy x y moveto uu ll} def /crsround{ decx x y moveto lu uu decy decy x y moveto ul ll} def /crs3up{ decx x y moveto ll decy x y moveto uu ll} def /crs3upround{ decx x y moveto ll decy x y moveto ul ll} def /crs3left{ uu decy decy x y moveto uu ll} def /crs3leftround{ uu decy decy x y moveto ul ll} def /crs3down{ decx x y moveto ll uu decy x y moveto ll} def /crs3downround{ decx x y moveto lu uu decy x y moveto ll} def /crs3right{ decx x y moveto ll uu decy decy x y moveto uu} def /crs3rightround{ decx x y moveto lu uu decy decy x y moveto uu } def /incx {/x x d add def} def /incy {/y y d add def} def /decx {/x x d sub def} def /decy {/y y d sub def} def /xyl {x y lineto} def /uu {incy xyl} def /blankuu{ x y rad add moveto incy xyl} def /uublank{ incy x y rad sub lineto x y moveto} def /ll {incx xyl} def /blankll{ x rad add y moveto incx xyl} def /llblank{ incx x rad sub y lineto x y moveto} def /dd {decy xyl} def /blankdd{ x y rad sub moveto decy xyl} def /ddblank{ decy x y rad add lineto x y moveto} def /rr {decx xyl} def /blankrr{ x rad sub y moveto decx xyl} def /rrblank{ decx x rad add y lineto x y moveto} def /ul {incy x y x d add y rad arcto} def /ur {incy x y x d sub y rad arcto} def /dl {decy x y x d add y rad arcto} def /dr {decy x y x d sub y rad arcto} def /lu {incx x y x y d add rad arcto} def /ld {incx x y x y d sub rad arcto} def /ru {decx x y x y d add rad arcto} def /rd {decx x y x y d sub rad arcto} def /rect {/leny exch def /lenx exch def lenx {ll} repeat leny {uu} repeat lenx {rr} repeat leny {dd} repeat } def /pos {/py exch def /px exch def /x stx px d mul add def /y sty py d mul add def x y moveto} def /fillst{ gsave gray1 fill grestore stroke } def /lambda{ gsave newpath 0 0 pos 16 10 rect gray1 stroke grestore} def %-------- splitting rules example % %---- 1 0.65 0.65 scale trx try translate /stx 0 def /sty 0 def newpath 0 0 pos 40 14 rect stroke /stx 2 d mul def /sty 2 d mul def lambda newpath 0 0 pos ll ll uu uu ll dd ll ll uu uu uu ll dd ll ll dd dd ll dd rr rr uu rr rr dd 0 3 pos ll dd ll uu uu ll ll ll uu uu ll ll dd dd 9 {ll} repeat uu uu % rr rr rr dd rr rr rr rr uu uu ll uu uu ll ll dd ll uu uu ll ll ll dd rr rr dd rr dd ll uu ll ll dd dd 11 6 pos rr uu ll dd 9 7 pos rr uu rr uu ll ll dd dd 0 6 pos ll ll ll ll uu ll uu rr uu uu 2 10 pos dd ll dd dd rr uu rr rr uu uu ll ll 11 0 pos uu uu ll ll ll ll ll uu uu 11 2 pos uu rr rr dd ll ll stroke %--- 2 % /stx 22 d mul def /sty 2 d mul def lambda newpath 0 0 pos ll ll uu ul ll dd ll lu uu uu ul ll dd ll ll dd dd ll dd rr rr uu rr rd dd 0 3 pos ll dd lu uu uu ll ll lu uu uu ll ll dd dd 8 {ll} repeat lu uu uublank % 15 6 pos rr rr dd rr rr rr rr uu ul lu uu uu ll ll dd lu uu uu ll ll ll dd rr rr dd rd dd ll uu ll ll dd dr rr 11 6 pos rr ul ll dd 8 7 pos uu rr uu ll ll dd dr rr 0 6 pos ll ll ll ll uu ll uu rr uu uu 2 10 pos dd ll dd dd rr uu rr rr uu uu ll ll 11 0 pos uu ul ll ll ll ll ll uu uublank 10 2 pos lu uu rr rr dd ll stroke showpage %%EndDocument @endspecial 413 1300 a Fq(Fig.)40 b(2.)i(A)30 b(collection)g(of)h 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b Fq(when)27 b Fp(\015)34 b Fq(is)28 b(though)m(t)h(of)g(as)g(a)g(set)g(of)g(edges.)41 b(Th)m(us)27 b Fp(\016)s(\015)35 b Fq(can)29 b(either)f(b)s(e)g(empt)m (y)h(or)g(consist)f(of)h(pair)e(of)i(sites.)0 2013 y(Giv)m(en)h(a)h (collection)f(of)h(con)m(tours)g Fp(\015)p 1214 2045 53 4 v 30 w Fq(=)25 b Fm(f)p Fp(\015)1479 2027 y Ft(1)1520 2013 y Fp(;)15 b(:)g(:)g(:)i(;)e(\015)1769 2027 y Fo(n)1819 2013 y Fm(g)p Fq(,)31 b(its)e Fk(b)-5 b(oundary)33 b Fp(\016)s(\015)p 2479 2045 V 36 w Fq(is)c(de\014ned)h(as)1694 2327 y Fp(\016)s(\015)p 1737 2359 V 31 w Fq(=)1948 2214 y Fo(n)1920 2241 y Fg([)1911 2434 y Fo(i)p Ft(=1)2044 2327 y Fp(\016)s(\015)2134 2341 y Fo(i)2181 2327 y Fp(:)0 2635 y Fq(A)h(notion)g(of)h Fk(c)-5 b(omp)g(atibility)42 b Fq(b)s(et)m(w)m(een)32 b(con)m(tours)g(can)f(b)s(e)g(in)m(tro)s (duced)f(in)g(suc)m(h)h(a)h(w)m(a)m(y)g(that,)h(for)e(eac)m(h)h(simply) 0 2762 y(connected)e(\014nite)f(v)m(olume)g(\003)h(and)f(for)g(eac)m(h) i Fp(\034)k Fm(2)1749 2739 y Fq(\026)1739 2762 y(\012,)30 b Fp(\034)35 b Fm(6)p Fq(=)25 b(0,)30 b(the)g(mapping)e Fm(G)2719 2729 y Fo(\034)2714 2787 y Ft(\003)2798 2762 y Fq(is)g(one{to{one)k(from)d(\012)3642 2776 y Ft(\003)3726 2762 y Fq(on)m(to)0 2890 y(the)38 b(set)h(of)f(all)f(compatible)g (collections)g(of)h(con)m(tours)h(with)d(a)j(giv)m(en)f(b)s(oundary)e (determined)g(b)m(y)i Fp(\034)10 b Fq(.)63 b(If,)40 b(on)0 3017 y(the)d(other)g(side,)g(one)h(considers)d(free)i(b)s(oundary)e (conditions)g(then)h(the)h(mapping)e Fm(G)3045 2982 y Fn(;)3040 3043 y Ft(\003)3131 3017 y Fq(is)h(t)m(w)m(o{to{one)k(from)0 3144 y(\012)66 3158 y Ft(\003)151 3144 y Fq(on)m(to)32 b(the)f(set)h(of)f(all)f(compatible)g(collections)g(of)h(con)m(tours.) 44 b(W)-8 b(e)32 b(let)f Fp(C)2662 3111 y Fn(\003)2655 3169 y Ft(\003)p Fo(;\034)2802 3144 y Fq(b)s(e)g(the)g(image)g(of)g Fm(G)3510 3111 y Fo(\034)3505 3169 y Ft(\003)3559 3144 y Fq(,)h Fk(i.e.)g Fq(the)0 3271 y(set)f(of)g(all)f(compatible)f 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Fo(s)1761 5323 y Fq(\()p Fp(\015)p 1796 5356 53 4 v 5 w Fq(\))91 b Fp(B)2048 5286 y Fn(\016)2043 5346 y Fo(\015)p 2043 5373 47 4 v 2115 5323 a Fq(=)25 b Fp(B)2280 5337 y Fo(\015)p 2280 5364 V 2327 5323 a Fm(n)p Fq(\001)2448 5286 y Fo(s)2487 5323 y Fq(\()p Fp(\015)p 2522 5356 53 4 v 5 w Fq(\))15 b Fp(:)1064 b Fq(\(6)p Fp(:)p Fq(7\))0 5558 y Fh(15)p Fi(=apr)r(ile=)p Fh(1999)30 b Ft([29])1317 b Fq(6.2)p eop %%Page: 30 30 30 29 bop 0 91 a Fq(In)31 b(other)g(w)m(ords,)h(kno)m(wing)e(that)i Fp(\015)p 1198 123 53 4 v 37 w Fq(is)e(a)i(subset)f(of)h(the)f(set)h (of)g(all)e(con)m(tours)i(is)f(equiv)-5 b(alen)m(t)31 b(to)h(kno)m(wing)e(that)0 218 y Fp(\033)j Fq(=)c(+1)j(on)h(one)g (\\side")g(of)g Fp(\015)p 1013 250 V 38 w Fq(and)f Fp(\033)g Fq(=)d Fm(\000)p Fq(1)k(on)g(the)g(opp)s(osite)f(side,)h(or)f(vicev)m (ersa.)49 b(A)33 b(con)m(tour)h Fp(\015)k Fq(is)31 b(said)h(to)0 345 y(b)s(e)e Fp(s)p Fk({lar)-5 b(ge)38 b Fq(if)29 b(diam)o(\()p Fp(\015)5 b Fq(\))26 b Fp(>)f(s)p Fq(.)41 b(F)-8 b(or)31 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3067 y Fp(s)p 1080 3108 46 4 v 1080 3191 a Fq(2)1162 3128 y Fm(\024)25 b(j)p Fp(u)1335 3142 y Fo(i)p Ft(+1)1477 3128 y Fm(\000)20 b Fp(u)1620 3142 y Fo(i)1651 3128 y Fm(j)1676 3142 y Fn(1)1778 3128 y Fm(\024)25 b Fq(2)p Fp(s)182 b Fm(8)25 b Fp(i)g Fq(=)g(1)p Fp(;)15 b(:)g(:)g(:)i(;)e(n)21 b Fm(\000)e Fq(1)0 3397 y(Giv)m(en)37 b(no)m(w)g(a)g(con)m(tour)h Fp(\015)k Fq(w)m(e)c(sa)m(y)f (that)h Fp(\015)k Fq(is)36 b Fk(c)-5 b(onsistent)38 b Fq(with)e(an)h Fp(s)p Fq({sk)m(eleton)h Fp(P)49 b Fq(=)36 b([)p Fp(u)3179 3411 y Ft(1)3220 3397 y Fp(;)15 b(:)g(:)g(:)i(;)e(u) 3474 3411 y Fo(n)3524 3397 y Fq(],)39 b(and)d(w)m(e)0 3524 y(write)30 b Fp(\015)g Fm(\030)25 b Fp(P)13 b Fq(,)30 b(if)80 3680 y(\()p Fp(i)p Fq(\))46 b(All)29 b(v)m(ertices)i(of)g Fp(P)43 b Fq(lie)29 b(on)i Fp(\015)48 3808 y Fq(\()p Fp(ii)p Fq(\))47 b Fp(\016)s(\015)31 b Fq(=)25 b Fp(\016)s(P)17 3936 y Fq(\()p Fp(iii)p Fq(\))47 b Fp(d)274 3950 y Fo(H)344 3936 y Fq(\()p Fp(\015)5 b(;)15 b(P)e Fq(\))26 b Fm(\024)f Fp(s)0 4144 y Fq(It)36 b(is)e(easy)i(to)g(c)m(hec)m(k)i(that)e(for)f (eac)m(h)i(con)m(tour)f Fp(\015)k Fq(with)34 b(diam)o(\()p Fp(\015)5 b Fq(\))35 b Fm(\025)f Fp(s)p Fq(,)i(there)g(alw)m(a)m(ys)g (exists)f(a)h Fp(s)p Fq({sk)m(eleton)g Fp(S)0 4272 y Fq(consisten)m(t)31 b(with)f Fp(\015)5 b Fq(.)42 b(In)30 b(particular,)g(it)g(is)g(alw)m(a)m(ys)h(p)s(ossible)e(to)i(asso)s (ciate)h(a)f(particular)f Fp(s)p Fq({sk)m(eleton)h Fp(S)36 b Fq(to)c(an)m(y)0 4399 y Fp(s)p Fq({large)37 b(con)m(tour)g Fp(\015)5 b Fq(.)58 b(W)-8 b(e)38 b(assume)e(that)h(a)g(de\014nite)e(c) m(hoice)i(has)g(b)s(een)e(made)i(once)g(and)f(for)g(all,)h(and)3698 4376 y(^)3683 4399 y Fp(S)3739 4413 y Fo(s)3777 4399 y Fq(\()p Fp(\015)5 b Fq(\))0 4526 y(denotes)31 b Fk(the)g Fq(sk)m(eleton)g(of)f Fp(\015)5 b Fq(.)41 b(W)-8 b(e)31 b(extend)g(the)f(mapping)2046 4503 y(^)2030 4526 y Fp(S)2086 4540 y Fo(s)2155 4526 y Fq(to)h(collections)f(of)h(con)m(tours)g(b)m(y) 1342 4732 y(^)1326 4755 y Fp(S)1382 4769 y Fo(s)1421 4755 y Fq(\()p Fp(\015)p 1456 4787 53 4 v 5 w Fq(\))26 b(=)f Fm(f)1741 4732 y Fq(^)1725 4755 y Fp(S)1781 4769 y Fo(s)1820 4755 y Fq(\()p Fp(\015)5 b Fq(\))26 b(:)g Fp(\015)35 b Fq(is)30 b Fp(s)p Fq({large)15 b Fm(g)g Fp(:)1095 b Fq(\(6)p Fp(:)p Fq(11\))0 4984 y(Giv)m(en)29 b(\003)c Fm(2)g Ff(F)39 b Fq(and)28 b Fp(\021)h Fm(2)865 4961 y Fq(\026)855 4984 y(\012)f(w)m(e)h(also)g(\(improp)s(erly\))d (de\014ne,)j(for)f Fp(\033)h Fm(2)c Fq(\012)2453 4998 y Ft(\003)2507 4984 y Fq(,)2576 4961 y(^)2561 4984 y Fp(S)2617 4998 y Fo(s)2655 4984 y Fq(\()p Fp(\033)s Fq(\))i(=)2918 4961 y(^)2903 4984 y Fp(S)2959 4998 y Fo(s)2997 4984 y Fq(\()p Fm(G)3091 4940 y Fo(\021)3086 5011 y Ft(\003)3141 4984 y Fq(\()p Fp(\033)s Fq(\)\).)41 b(Finally)-8 b(,)28 b(giv)m(en)0 5111 y(a)j(set)g(of)f(sk)m(eletons)h Fm(S)h Fq(=)25 b Fm(f)p Fp(S)990 5125 y Ft(1)1031 5111 y Fp(;)15 b(:)g(:)g(:)i(;)e(S)1289 5125 y Fo(r)1329 5111 y Fm(g)31 b Fq(w)m(e)g(let)1342 5340 y Fp(E)1409 5354 y Fo(s)1448 5340 y Fq(\()p Fm(S)7 b Fq(\))26 b(=)f Fm(f)p Fp(\033)k Fm(2)24 b Fq(\012)1979 5354 y Ft(\003)2059 5340 y Fq(:)2124 5317 y(^)2109 5340 y Fp(S)2165 5354 y Fo(s)2204 5340 y Fq(\()p Fp(\033)s Fq(\))i(=)f Fm(S)7 b(g)0 5558 y Fh(15)p Fi(=apr)r(ile=)p Fh(1999)30 b Ft([30])1317 b Fq(6.3)p eop %%Page: 31 31 31 30 bop 0 91 a Fq(W)-8 b(e)32 b(also)e(set)1573 284 y Fp(W)13 b Fq(\()p Fm(S)7 b Fq(\))25 b(=)1973 171 y Fo(r)1925 198 y Fg(X)1932 392 y Fo(i)p Ft(=1)2072 284 y Fp(W)13 b Fq(\()p Fp(S)2262 298 y Fo(i)2292 284 y Fq(\))0 576 y(where,)30 b(for)g(an)m(y)h Fp(s)p Fq({sk)m(eleton)g Fp(S)f Fq(=)25 b([)p Fp(x)1295 590 y Ft(1)1336 576 y Fp(;)15 b(:)g(:)g(:)i(;)e(x)1590 590 y Fo(k)1635 576 y Fq(],)1437 918 y Fp(W)e Fq(\()p Fp(S)5 b Fq(\))25 b Fm(\021)1834 804 y Fo(k)1788 832 y Fg(X)1795 1025 y Fo(i)p Ft(=2)1935 918 y Fp(\034)1975 932 y Fo(\014)2024 918 y Fq(\()p Fp(x)2111 932 y Fo(i)2161 918 y Fm(\000)20 b Fp(x)2304 932 y Fo(i)p Fn(\000)p Ft(1)2428 918 y Fq(\))0 1312 y(In)30 b(the)g(follo)m(wing)f(t)m(w)m(o)j(prop)s(ositions)c(w)m 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Fq(+)k Fp(\026)2457 174 y Fo(\014)s(;)p Fn(;)2457 245 y Fo(Q)2514 254 y Fi(L)2567 218 y Fq(\()p Fp(K)2679 232 y Fo(\016)2719 218 y Fq(\))914 b(\(6)p Fp(:)p Fq(17\))0 402 y(In)30 b(order)g(to)h(estimate)g(the)f(second)h(term)f(w)m(e)h (can)g(use)f(Lemma)g(5.2)i(in)d([CGMS])h(whic)m(h)g(sa)m(ys)g(that)1245 675 y(lim)15 b(inf)1273 733 y Fo(L)p Fn(!1)1512 675 y Fm(\000)1602 613 y Fq(1)p 1594 654 62 4 v 1594 737 a Fp(L)1681 675 y Fq(log)i Fp(\026)1869 631 y Fo(\014)s(;)p Fn(;)1869 701 y Fo(Q)1926 710 y Fi(L)1978 675 y Fq(\()p Fp(K)2090 689 y Fo(\016)2131 675 y Fq(\))25 b Fm(\025)g Fp( )s Fq(\()p Fp(m)p Fq(\))d Fm(\000)e Fp(\016)0 936 y Fq(Next)31 b(w)m(e)g(w)m(an)m(t)g(to)g(pro)m(v)m(e)h(that)1218 1203 y(lim)15 b(inf)1247 1262 y Fo(L)p Fn(!1)1485 1203 y Fm(\000)1575 1142 y Fq(1)p 1567 1182 V 1567 1266 a Fp(L)1655 1203 y Fq(log)i Fp(\026)1843 1160 y Fo(\014)s(;)p Fn(;)1843 1230 y Fo(Q)1900 1239 y Fi(L)1952 1203 y Fq(\()p Fm(M)2096 1166 y Fo(m)2096 1226 y(L)2181 1203 y Fm(j)e Fp(K)2305 1166 y Fo(c)2298 1226 y(\016)2357 1130 y Fg(\001)2424 1203 y Fq(=)25 b(+)p Fm(1)986 b Fq(\(6)p Fp(:)p Fq(18\))0 1469 y(W)-8 b(e)47 b(denote)f(b)m(y)f Fp(R)690 1483 y Fo(\016)n(;L)843 1469 y Fq(the)h(set)g(of)f(all)g(compatible)f (collections)h(of)h(\(free)g(b)s(oundary)d(conditions\))i Fp(s)p Fq({large)0 1596 y(con)m(tours)31 b Fp(\015)p 367 1629 53 4 v 35 w Fq(in)e Fp(Q)627 1610 y Fo(L)682 1596 y Fq(,)h(suc)m(h)g(that)h Fp(W)13 b Fq(\()1288 1573 y(^)1273 1596 y Fp(S)1329 1610 y Fo(s)1368 1596 y Fq(\()p Fp(\015)p 1403 1629 V 5 w Fq(\)\))26 b Fp(<)f Fq(\()p Fp( )s Fq(\()p Fp(m)p Fq(\))d Fm(\000)e Fp(\016)s Fq(\))p Fp(L)p Fq(.)41 b(W)-8 b(e)32 b(also)e(let)764 1817 y Fp(E)831 1831 y Fo(\015)p 831 1858 47 4 v 903 1817 a Fq(=)25 b Fm(f)g Fp(\033)k Fm(2)c Fq(\012)1302 1831 y Fo(Q)1359 1840 y Fi(L)1436 1817 y Fq(:)h Fp(\015)p 1487 1849 53 4 v 35 w Fq(is)k(the)g(set)h(of)g(all)e Fp(s)p Fq({large)h(con)m(tours)h(for)f Fp(\033)k Fm(g)0 2038 y Fq(W)-8 b(e)32 b(can)e(then)g(write)1174 2165 y Fp(\026)1229 2121 y Fo(\014)s(;)p Fn(;)1229 2192 y Fo(Q)1286 2201 y Fi(L)1339 2165 y Fq(\()p Fm(M)1483 2127 y Fo(m)1483 2187 y(L)1567 2165 y Fm(j)16 b Fp(K)1692 2127 y Fo(c)1685 2187 y(\016)1729 2165 y Fq(\))25 b Fm(\024)78 b Fq(sup)1885 2241 y Fo(\015)p 1885 2268 43 4 v 4 w Fn(2)p Fo(R)2031 2251 y Fi(\016)n(;L)2143 2165 y Fp(\026)2198 2121 y Fo(\014)s(;)p Fn(;)2198 2192 y Fo(Q)2255 2201 y Fi(L)2308 2165 y Fq(\()p Fm(M)2452 2127 y Fo(m)2452 2187 y(L)2536 2165 y Fm(j)16 b Fp(E)2644 2179 y Fo(\015)p 2644 2206 47 4 v 2690 2165 a Fq(\))943 b(\(6)p Fp(:)p Fq(19\))0 2494 y Fk(Bound)33 b(on)g(\(6.19\))i(when)e Fp(\015)p 926 2527 53 4 v 30 w Fm(6)p Fq(=)25 b Fm(;)p Fq(.)0 2648 y(Assume)30 b Fp(\015)p 337 2680 V 30 w Fm(6)p Fq(=)25 b Fm(;)p Fq(.)0 2827 y Fr(De\014nition)35 b(6.5.)53 b Fq(Giv)m(en)29 b(a)i(p)s(ositiv)m(e)e (in)m(teger)h Fp(L)g Fq(and)f Fp(\016)g(>)c Fq(0)31 b(and)e(a)h (collection)g(of)g(large)g(con)m(tours)g Fp(\015)p 3532 2859 V 6 w Fq(,)g(w)m(e)g(sa)m(y)0 2954 y(that)h Fp(\033)d Fm(2)d Fp(E)430 2968 y Fo(\015)p 430 2995 47 4 v 507 2954 a Fq(pro)s(duces)k(a)i Fp(\016)s Fk({natur)-5 b(al)36 b(magnetization)e(p)-5 b(attern)40 b Fq(\(in)29 b Fp(Q)2484 2968 y Fo(L)2538 2954 y Fq(\))i(if)e(either)h(\(remem)m(b)s(er)g (\(2.2\)\))66 3109 y Fc(\(1\))227 3031 y Fg(\014)227 3086 y(\014)273 3109 y Fp(M)361 3123 y Fo(A)416 3104 y Fj(\016)416 3140 y Fi(\015)p 416 3164 42 4 v 471 3109 a Fq(+)9 b Fp(m)631 3076 y Fn(\003)672 3109 y Fm(j)p Fp(A)765 3076 y Fn(\016)765 3131 y Fo(\015)p 765 3158 47 4 v 813 3109 a Fm(j)853 3031 y Fg(\014)853 3086 y(\014)892 3109 y Fq(+)972 3031 y Fg(\014)972 3086 y(\014)1018 3109 y Fp(M)1106 3123 y Fo(A)1161 3104 y Fj(\016)1161 3140 y Fi(\015)p 1161 3164 42 4 v 1202 3123 a Fn(\\)p Fo(U)1301 3132 y Fi(L)1362 3109 y Fq(+)g Fp(m)1522 3076 y Fn(\003)1564 3109 y Fm(j)p Fp(A)1657 3076 y Fn(\016)1657 3131 y Fo(\015)p 1657 3158 47 4 v 1713 3109 a Fm(\\)g Fp(U)1845 3123 y Fo(L)1899 3109 y Fm(j)1939 3031 y Fg(\014)1939 3086 y(\014)1979 3109 y Fq(+)2059 3031 y Fg(\014)2059 3086 y(\014)2105 3109 y Fp(M)2193 3123 y Fo(B)2251 3104 y Fj(\016)2248 3140 y Fi(\015)p 2248 3164 42 4 v 2303 3109 a Fm(\000)g Fp(m)2463 3076 y Fn(\003)2505 3109 y Fm(j)p Fp(B)2604 3076 y Fn(\016)2599 3131 y Fo(\015)p 2599 3158 47 4 v 2645 3109 a Fm(j)2685 3031 y Fg(\014)2685 3086 y(\014)2725 3109 y Fq(+)2805 3031 y Fg(\014)2805 3086 y(\014)2851 3109 y Fp(M)2939 3123 y Fo(B)2997 3104 y Fj(\016)2994 3140 y Fi(\015)p 2994 3164 42 4 v 3035 3123 a Fn(\\)p Fo(U)3134 3132 y Fi(L)3196 3109 y Fm(\000)g Fp(m)3356 3076 y Fn(\003)3397 3109 y Fm(j)p Fp(B)3496 3076 y Fn(\016)3491 3131 y Fo(\015)p 3491 3158 47 4 v 3547 3109 a Fm(\\)g Fp(U)3679 3123 y Fo(L)3733 3109 y Fm(j)3773 3031 y Fg(\014)3773 3086 y(\014)3829 3109 y Fm(\024)227 3278 y Fp(\016)s(L)332 3245 y Ft(2)374 3278 y Fc(,)30 b(or)66 3406 y(\(2\))227 3329 y Fg(\014)227 3383 y(\014)273 3406 y Fp(M)361 3420 y Fo(A)416 3401 y Fj(\016)416 3438 y Fi(\015)p 416 3461 42 4 v 471 3406 a Fm(\000)9 b Fp(m)631 3373 y Fn(\003)672 3406 y Fm(j)p Fp(A)765 3373 y Fn(\016)765 3428 y Fo(\015)p 765 3455 47 4 v 813 3406 a Fm(j)853 3329 y Fg(\014)853 3383 y(\014)892 3406 y Fq(+)972 3329 y Fg(\014)972 3383 y(\014)1018 3406 y Fp(M)1106 3420 y Fo(A)1161 3401 y Fj(\016)1161 3438 y Fi(\015)p 1161 3461 42 4 v 1202 3420 a Fn(\\)p Fo(U)1301 3429 y Fi(L)1362 3406 y Fm(\000)g Fp(m)1522 3373 y Fn(\003)1564 3406 y Fm(j)p Fp(A)1657 3373 y Fn(\016)1657 3428 y Fo(\015)p 1657 3455 47 4 v 1713 3406 a Fm(\\)g Fp(U)1845 3420 y Fo(L)1899 3406 y Fm(j)1939 3329 y Fg(\014)1939 3383 y(\014)1979 3406 y Fq(+)2059 3329 y Fg(\014)2059 3383 y(\014)2105 3406 y Fp(M)2193 3420 y Fo(B)2251 3401 y Fj(\016)2248 3438 y Fi(\015)p 2248 3461 42 4 v 2303 3406 a Fq(+)g Fp(m)2463 3373 y Fn(\003)2505 3406 y Fm(j)p Fp(B)2604 3373 y Fn(\016)2599 3428 y Fo(\015)p 2599 3455 47 4 v 2645 3406 a Fm(j)2685 3329 y Fg(\014)2685 3383 y(\014)2725 3406 y Fq(+)2805 3329 y Fg(\014)2805 3383 y(\014)2851 3406 y Fp(M)2939 3420 y Fo(B)2997 3401 y Fj(\016)2994 3438 y Fi(\015)p 2994 3461 42 4 v 3035 3420 a Fn(\\)p Fo(U)3134 3429 y Fi(L)3196 3406 y Fq(+)g Fp(m)3356 3373 y Fn(\003)3397 3406 y Fm(j)p Fp(B)3496 3373 y Fn(\016)3491 3428 y Fo(\015)p 3491 3455 47 4 v 3547 3406 a Fm(\\)g Fp(U)3679 3420 y Fo(L)3733 3406 y Fm(j)3773 3329 y Fg(\014)3773 3383 y(\014)3829 3406 y Fm(\024)227 3575 y Fp(\016)s(L)332 3542 y Ft(2)0 3754 y Fq(W)-8 b(e)30 b(let)f Fp(E)357 3721 y Fo(\016)r Fn(\000)p Fo(nat)352 3777 y(\015)p 352 3804 V 595 3754 a Fq(the)g(subset)f(of)h(all)f(the)h(con\014gurations)f(in)g Fp(E)2166 3768 y Fo(\015)p 2166 3795 V 2241 3754 a Fq(whic)m(h)g (corresp)s(ond)f(to)j(a)f Fp(\016)s Fq({natural)h(magneti-)0 3881 y(zation)h(pattern.)0 4060 y(The)d(follo)m(wing)f(t)m(w)m(o)j (results,)e(Lemma)h(6.6)g(and)f(Lemma)h(6.7)g(are)g(no)m(w)g (su\016cien)m(t)f(for)g(\014nding)e(an)j(appropriate)0 4187 y(upp)s(er)f(b)s(ound)h(on)h(the)g(RHS)g(of)h(\(6.19\).)0 4367 y Fr(Lemma)j(6.6.)53 b Fc(F)-8 b(or)33 b(all)d Fp(\016)i(>)27 b Fq(0)p Fc(,)33 b Fp(m)27 b Fm(2)h Fq([0)p Fp(;)15 b(m)1538 4334 y Fn(\003)1580 4367 y Fq(\()p Fp(\014)5 b Fq(\)\))33 b Fc(there)f(exist)g Fp(\016)2268 4334 y Fn(0)2322 4367 y Fq(=)27 b Fp(\016)2463 4334 y Fn(0)2490 4367 y Fq(\()p Fp(\014)5 b(;)15 b(m;)g(\016)s Fq(\))34 b Fc(and)d Fp(L)3093 4381 y Ft(0)3161 4367 y Fq(=)c Fp(L)3321 4381 y Ft(0)3362 4367 y Fq(\()p Fp(\014)5 b(;)15 b(m;)g(\016)s Fq(\))34 b Fc(suc)m(h)0 4494 y(that)d(for)f(all)f Fp(L)c Fm(\025)g Fp(L)707 4508 y Ft(0)778 4494 y Fc(and)30 b(for)g(all)g Fp(\015)p 1221 4526 53 4 v 30 w Fm(2)25 b Fp(R)1453 4508 y Fo(\016)n(;L)1560 4494 y Fc(,)31 b(w)m(e)g(ha)m(v)m(e)1486 4714 y Fm(M)1595 4677 y Fo(m)1595 4737 y(L)1685 4714 y Fm(\\)20 b Fp(E)1833 4728 y Fo(\015)p 1833 4755 47 4 v 1905 4714 a Fm(\032)k Fq(\()p Fp(E)2107 4677 y Fo(\016)2142 4649 y Fj(0)2168 4677 y Fn(\000)p Fo(nat)2102 4737 y(\015)p 2102 4764 V 2341 4714 a Fq(\))2376 4677 y Fo(c)0 4936 y Fr(Lemma)39 b(6.7.)52 b Fc(F)-8 b(or)37 b(all)d Fp(\016)j(>)d Fq(0)p Fc(,)j Fp(m)d Fm(2)f Fq([0)p Fp(;)15 b(m)1578 4903 y Fn(\003)1621 4936 y Fq(\()p Fp(\014)5 b Fq(\)\))36 b Fc(there)g(exist)f Fp(c)f Fq(=)g Fp(c)p Fq(\()p Fp(\014)5 b(;)15 b(m;)g(\016)s Fq(\))36 b Fp(>)e Fq(0)h Fc(and)g Fp(L)3287 4950 y Ft(0)3362 4936 y Fq(=)e Fp(L)3528 4950 y Ft(0)3569 4936 y Fq(\()p Fp(\014)5 b(;)15 b(m;)g(\016)s Fq(\))0 5063 y Fc(suc)m(h)30 b(that)h(for)f(all)f Fp(L)d Fm(\025)f Fp(L)913 5077 y Ft(0)983 5063 y Fc(and)30 b(for)g(all)g Fp(\015)p 1426 5095 53 4 v 30 w Fm(2)25 b Fp(R)1658 5077 y Fo(\016)n(;L)1255 5323 y Fp(\026)1310 5280 y Fo(\014)s(;)p Fn(;)1310 5350 y Fo(Q)1367 5359 y Fi(L)1420 5323 y Fq(\(\()p Fp(E)1562 5286 y Fo(\016)r Fn(\000)p Fo(nat)1557 5346 y(\015)p 1557 5373 47 4 v 1771 5323 a Fq(\))1806 5286 y Fo(c)1859 5323 y Fm(j)15 b Fp(E)1966 5337 y Fo(\015)p 1966 5364 V 2013 5323 a Fq(\))25 b Fm(\024)g Fp(e)2211 5286 y Fn(\000)p Fo(cL)2351 5258 y Fh(2)2386 5286 y Fo(=s)p Ft(\()p Fo(L)p Ft(\))2563 5258 y Fh(2)2620 5323 y Fp(:)1023 b Fq(\(6)p Fp(:)p Fq(20\))0 5558 y Fh(15)p Fi(=apr)r(ile=)p Fh(1999)30 b Ft([34])1317 b Fq(6.7)p eop %%Page: 35 35 35 34 bop 0 91 a Fk(Pr)-5 b(o)g(of)36 b(of)e(L)-5 b(emma)36 b(6.6)p Fq(.)47 b(Assume)32 b Fp(\015)p 1209 123 53 4 v 33 w Fm(2)c Fp(R)1447 105 y Fo(\016)n(;L)1555 91 y Fq(,)33 b Fp(\015)p 1613 123 V 33 w Fm(6)p Fq(=)28 b Fm(;)33 b Fq(and)f(let)g Fm(S)j Fq(=)2387 68 y(^)2371 91 y Fp(S)2427 105 y Fo(s)2466 91 y Fq(\()p Fp(\015)p 2501 123 V 5 w Fq(\).)47 b(Since)31 b Fp(W)13 b Fq(\()p Fm(S)7 b Fq(\))29 b Fm(\024)f Fq(\()p Fp( )s Fq(\()p Fp(m)p Fq(\))23 b Fm(\000)e Fp(d)p Fq(\))p Fp(L)32 b Fq(w)m(e)0 218 y(ha)m(v)m(e)27 b(that)e(the)h(length)f(of)g Fm(S)32 b Fq(cannot)26 b(exceed)h Fp(C)1650 232 y Ft(1)1690 218 y Fp(L)e Fq(\()p Fp(C)1877 232 y Ft(1)1944 218 y Fq(dep)s(ends)e(on)i Fp(\014)31 b Fq(and)25 b Fp(m)p Fq(\).)39 b(But)25 b(since)g Fp(d)3284 232 y Fo(H)3353 218 y Fq(\()p Fm(S)7 b Fp(;)15 b(\015)p 3490 250 V 6 w Fq(\))26 b Fm(\024)f Fp(s)p Fq(\()p Fp(L)p Fq(\),)0 345 y(w)m(e)31 b(get)g(that)g(the)g(length)f(of)g Fp(\015)p 1020 377 V 36 w Fq(cannot)h(exceed)g Fp(C)1756 359 y Ft(2)1797 345 y Fp(s)p Fq(\()p Fp(L)p Fq(\))p Fp(L)p Fq(,)f(so)1574 564 y Fm(j)p Fq(\001)1675 526 y Fo(s)1714 564 y Fq(\()p Fp(\015)p 1749 596 V 5 w Fq(\))p Fm(j)c(\024)f Fp(C)2048 578 y Ft(3)2089 564 y Fp(s)p Fq(\()p Fp(L)p Fq(\))p Fp(L)1342 b Fq(\(6)p Fp(:)p Fq(21\))0 783 y(Assume)27 b(no)m(w)h(to)h(ha)m(v)m(e)g(a)f Fp(\016)953 750 y Fn(0)979 783 y Fq({natural)f(magnetization)i(pattern)f(for)f(some)h Fp(\016)2654 750 y Fn(0)2706 783 y Fp(>)d Fq(0.)40 b(F)-8 b(or)29 b(simplicit)m(y)c(w)m(e)j(assume)0 910 y(that)j(statemen)m(t)h (\(1\))g(of)e(De\014nition)f(6.5)j(holds.)39 b(The)30 b(case)h(when)f(\(2\))h(holds)e(is)g(analogous.)41 b(Let)31 b(then)1153 1181 y Fp(v)e Fq(=)1332 1119 y(1)p 1332 1160 46 4 v 1332 1243 a(2)1404 1080 y Fg(\020)1458 1181 y Fq(1)21 b Fm(\000)1647 1119 y Fp(m)p 1626 1160 122 4 v 1626 1243 a(m)1706 1217 y Fn(\003)1758 1080 y Fg(\021)1919 1181 y Fq(so)30 b(that)92 b Fp(m)25 b Fq(=)42 b(^)-62 b Fp(m)2569 1143 y Ft(+)2629 1181 y Fq(\()p Fp(v)s Fq(\))922 b(\(6)p Fp(:)p Fq(22\))0 1400 y(and)307 1619 y Fp(v)351 1633 y Ft(1)417 1619 y Fq(=)25 b Fm(j)p Fp(A)606 1581 y Fn(\016)606 1641 y Fo(\015)p 606 1668 47 4 v 653 1619 a Fm(j)p Fp(=L)785 1581 y Ft(2)826 1619 y Fp(;)107 b(w)1023 1633 y Ft(1)1089 1619 y Fq(=)25 b Fm(j)p Fp(A)1278 1581 y Fn(\016)1278 1641 y Fo(\015)p 1278 1668 V 1345 1619 a Fm(\\)20 b Fp(U)1488 1633 y Fo(L)1542 1619 y Fm(j)p Fp(=L)1674 1581 y Ft(2)1716 1619 y Fp(;)197 b(v)1982 1633 y Ft(2)2048 1619 y Fq(=)25 b Fm(j)p Fp(A)2237 1633 y Fn(S)2291 1619 y Fm(j)p Fp(=L)2423 1581 y Ft(2)2464 1619 y Fp(;)107 b(w)2661 1633 y Ft(2)2727 1619 y Fq(=)25 b Fm(j)p Fp(A)2916 1633 y Fn(S)2990 1619 y Fm(\\)20 b Fp(U)3133 1633 y Fo(L)3188 1619 y Fm(j)p Fp(=L)3320 1581 y Ft(2)3668 1619 y Fq(\(6)p Fp(:)p Fq(23\))0 1837 y(Then)29 b(w)m(e)i(ha)m(v)m(e)670 2043 y Fp(M)758 2057 y Fo(Q)815 2066 y Fi(L)893 2043 y Fq(=)25 b Fp(M)1077 2057 y Fo(A)1132 2038 y Fj(\016)1132 2075 y Fi(\015)p 1132 2098 42 4 v 1198 2043 a Fq(+)20 b Fp(M)1377 2057 y Fo(B)1435 2038 y Fj(\016)1432 2075 y Fi(\015)p 1432 2098 V 1499 2043 a Fq(+)g Fp(M)1678 2059 y Ft(\001)1738 2041 y Fi(s)1773 2059 y Ft(\()p Fo(\015)p 1801 2086 43 4 v 4 w Ft(\))1901 2043 y Fq(=)25 b Fp(M)2085 2057 y Fo(A)2140 2066 y Fj(S)2212 2043 y Fq(+)20 b Fp(M)2391 2057 y Fo(B)2446 2066 y Fj(S)678 2243 y Fp(M)766 2257 y Fo(U)816 2266 y Fi(L)893 2243 y Fq(=)25 b Fp(M)1077 2257 y Fo(A)1132 2238 y Fj(\016)1132 2274 y Fi(\015)p 1132 2298 42 4 v 1174 2257 a Fn(\\)p Fo(U)1273 2266 y Fi(L)1345 2243 y Fq(+)20 b Fp(M)1524 2257 y Fo(B)1582 2238 y Fj(\016)1579 2274 y Fi(\015)p 1579 2298 V 1621 2257 a Fn(\\)p Fo(U)1720 2266 y Fi(L)1792 2243 y Fq(+)g Fp(M)1971 2259 y Ft(\001)2031 2241 y Fi(s)2066 2259 y Ft(\()p Fo(\015)p 2094 2286 43 4 v 4 w Ft(\))p Fn(\\)p Fo(U)2263 2268 y Fi(L)2341 2243 y Fq(=)25 b Fp(M)2525 2257 y Fo(A)2580 2266 y Fj(S)2627 2257 y Fn(\\)p Fo(U)2726 2266 y Fi(L)2798 2243 y Fq(+)20 b Fp(M)2977 2257 y Fo(B)3032 2266 y Fj(S)3079 2257 y Fn(\\)p Fo(U)3178 2266 y Fi(L)3668 2156 y Fq(\(6)p Fp(:)p Fq(24\))0 2498 y(F)-8 b(rom)39 b(\(6.23\),)j(\(6.24\),)h(and)38 b(from)f(the)i(assumption)d(of)j(ha)m (ving)f(a)g Fp(\016)2441 2465 y Fn(0)2467 2498 y Fq({natural)g (magnetization)h(pattern,)h(w)m(e)0 2625 y(obtain)896 2752 y Fm(j)p Fp(m)1001 2766 y Fo(Q)1058 2775 y Fi(L)1131 2752 y Fm(\000)d Fq(^)-62 b Fp(m)1302 2715 y Ft(+)1362 2752 y Fq(\()p Fp(v)1441 2766 y Ft(1)1482 2752 y Fq(\))p Fm(j)21 b Fq(+)f Fm(j)p Fp(m)1759 2766 y Fo(U)1809 2775 y Fi(L)1881 2752 y Fm(\000)37 b Fq(^)-62 b Fp(m)2052 2713 y Ft(+)2052 2779 y Fo(U)2113 2752 y Fq(\()p Fp(w)2213 2766 y Ft(1)2254 2752 y Fq(\))p Fm(j)26 b(\024)f Fp(\016)2479 2715 y Fn(0)2525 2752 y Fq(+)20 b Fp(C)2681 2766 y Ft(3)2722 2752 y Fp(s)p Fq(\()p Fp(L)p Fq(\))p Fp(=L)0 2935 y Fq(If)38 b Fm(j)p Fp(m)204 2949 y Fo(Q)261 2958 y Fi(L)339 2935 y Fm(\000)26 b Fp(m)p Fm(j)38 b(\024)h Fq(10)p Fp(=L)886 2902 y Ft(2)966 2935 y Fq(and)f Fm(j)p Fp(m)1256 2949 y Fo(U)1306 2958 y Fi(L)1384 2935 y Fm(\000)25 b Fp(m)p Fm(j)39 b(\024)f Fq(10)p Fp(=L)1930 2902 y Ft(2)2011 2935 y Fq(and)f Fp(L)i Fq(is)e(large)i(enough,)h(the)f(previous)e (inequalit)m(y)0 3062 y(b)s(ecomes)1228 3189 y Fm(j)p Fp(m)21 b Fm(\000)37 b Fq(^)-62 b Fp(m)1525 3152 y Ft(+)1585 3189 y Fq(\()p Fp(v)1664 3203 y Ft(1)1705 3189 y Fq(\))p Fm(j)21 b Fq(+)f Fm(j)p Fp(m)g Fm(\000)37 b Fq(^)-62 b Fp(m)2173 3150 y Ft(+)2173 3216 y Fo(U)2234 3189 y Fq(\()p Fp(w)2334 3203 y Ft(1)2375 3189 y Fq(\))p Fm(j)26 b(\024)f Fq(2)p Fp(\016)2645 3152 y Fn(0)3668 3189 y Fq(\(6)p Fp(:)p Fq(25\))0 3372 y(F)-8 b(rom)31 b(Prop)s(osition)d(5.7)k (w)m(e)f(\014nd)d(no)m(w)1412 3648 y Fm(j)p Fp(v)c Fm(\000)c Fp(v)1640 3662 y Ft(1)1681 3648 y Fm(j)g Fq(+)g Fm(j)p Fp(v)s(=)p Fq(2)i Fm(\000)e Fp(w)2157 3662 y Ft(1)2198 3648 y Fm(j)25 b(\024)2359 3586 y Fq(4)p Fp(\016)2447 3553 y Fn(0)p 2355 3627 122 4 v 2355 3710 a Fp(m)2435 3684 y Fn(\003)3668 3648 y Fq(\(6)p Fp(:)p Fq(26\))0 3914 y(On)31 b(the)g(other)h(side,)f(since)g Fp(W)13 b Fq(\()p Fm(S)7 b Fq(\))28 b Fp(<)1357 3890 y Fq(\026)1338 3914 y Fp( )t Fq(\()p Fp(m)p Fq(\))21 b Fm(\000)g Fp(\016)s Fq(,)33 b(b)m(y)f(rescaling)e(and)h(b)m(y)h(Prop)s(osition)d(5.6)k(w)m (e)f(get)h(that)f(there)0 4041 y(exists)e Fp(\016)294 4008 y Fn(00)367 4041 y Fq(=)25 b Fp(\016)506 4008 y Fn(00)553 4041 y Fq(\()p Fp(m;)15 b(\016)s Fq(\))32 b(suc)m(h)e(that) 1439 4168 y Fm(j)p Fp(v)24 b Fm(\000)c Fp(v)1667 4182 y Ft(2)1707 4168 y Fm(j)h Fq(+)f Fm(j)p Fp(v)s(=)p Fq(2)i Fm(\000)d Fp(w)2183 4182 y Ft(2)2224 4168 y Fm(j)26 b Fp(>)f(\016)2414 4131 y Fn(00)3668 4168 y Fq(\(6)p Fp(:)p Fq(27\))0 4351 y(Finally)j(from)i(the)h(de\014nition)d(of)i(sk)m (eletons)h(it)f(follo)m(ws)g(\(see)h(for)f(instance)g(Lemma)h(5.13)h (of)e([DKS]\))h(that)1010 4632 y Fm(j)p Fp(v)1079 4646 y Ft(1)1140 4632 y Fm(\000)20 b Fp(v)1275 4646 y Ft(2)1316 4632 y Fm(j)25 b(\024)g Fp(C)1527 4646 y Ft(4)1579 4570 y Fp(s)p Fq(\()p Fp(L)p Fq(\))p 1579 4611 176 4 v 1635 4694 a Fp(L)1856 4632 y Fq(and)90 b Fm(j)p Fp(w)2183 4646 y Ft(1)2244 4632 y Fm(\000)20 b Fp(w)2400 4646 y Ft(2)2441 4632 y Fm(j)26 b(\024)f Fp(C)2653 4646 y Ft(4)2704 4570 y Fp(s)p Fq(\()p Fp(L)p Fq(\))p 2704 4611 V 2761 4694 a Fp(L)3668 4632 y Fq(\(6)p Fp(:)p Fq(28\))0 4890 y(for)20 b(some)h(univ)m(ersal)e(constan)m(t)j Fp(C)1144 4904 y Ft(4)1185 4890 y Fq(.)37 b(Inequalities)19 b(\(6.26\),)25 b(\(6.27\))e(and)d(\(6.28\))j(clearly)d(generate)i(a)f(con)m (tradiction)0 5018 y(if,)30 b(sa)m(y)-8 b(,)31 b(4)p Fp(\016)371 4985 y Fn(0)398 5018 y Fp(=m)523 4985 y Fn(\003)590 5018 y Fm(\024)25 b Fp(\016)729 4985 y Fn(00)777 5018 y Fp(=)p Fq(10)31 b(and)f Fp(C)1185 5032 y Ft(4)1226 5018 y Fp(s)p Fq(\()p Fp(L)p Fq(\))p Fp(=L)c Fm(\024)f Fp(\016)1673 4985 y Fn(00)1720 5018 y Fp(=)p Fq(10.)92 b Fd(\003)0 5196 y Fk(Pr)-5 b(o)g(of)34 b(of)f(L)-5 b(emma)34 b(6.7)p Fq(.)41 b(W)-8 b(e)32 b(can)e(write)1624 5323 y Fp(E)1696 5286 y Fo(\016)n(;nat)1691 5346 y(\015)p 1691 5373 47 4 v 1891 5323 a Fq(=)25 b Fp(Y)2040 5337 y Ft(1)2101 5323 y Fm([)20 b Fp(Y)2235 5337 y Ft(2)0 5558 y Fh(15)p Fi(=apr)r(ile=)p Fh(1999)30 b Ft([35])1317 b Fq(6.8)p eop %%Page: 36 36 36 35 bop 0 91 a Fq(where)30 b Fp(Y)316 105 y Fo(i)376 91 y Fq(is)g(the)g(ev)m(en)m(t)i(that)f(statemen)m(t)h(\(i\))f(of)f (the)h(De\014nition)e(6.5)i(holds.)39 b(Also,)31 b(w)m(e)f(ha)m(v)m(e) 1644 309 y(\()p Fp(Y)1732 323 y Fo(i)1762 309 y Fq(\))1797 271 y Fo(c)1860 309 y Fm(\032)25 b([)2017 271 y Ft(4)2017 331 y Fo(j)t Ft(=1)2147 309 y Fp(G)2219 271 y Fo(i)2219 331 y(j)0 527 y Fq(where)30 b Fp(G)335 494 y Ft(1)335 549 y(1)400 527 y Fq(=)25 b Fm(f)15 b(j)p Fp(M)669 541 y Fo(A)724 522 y Fj(\016)724 559 y Fi(\015)p 724 582 42 4 v 791 527 a Fq(+)20 b Fp(m)962 494 y Fn(\003)1004 527 y Fm(j)p Fp(A)1097 494 y Fn(\016)1097 549 y Fo(\015)p 1097 576 47 4 v 1144 527 a Fm(j)15 b(j)26 b(\025)f Fp(\016)s(L)1436 494 y Ft(2)1477 527 y Fp(=)p Fq(4)15 b Fm(g)32 b Fq(and)e(so)h(on.)40 b(Th)m(us)1032 795 y(\()p Fp(E)1139 757 y Fo(\016)n(;nat)1134 817 y(\015)p 1134 844 V 1309 795 a Fq(\))1344 757 y Fo(c)1407 795 y Fq(=)25 b(\()p Fp(Y)1591 809 y Ft(1)1632 795 y Fq(\))1667 757 y Fo(c)1725 795 y Fm(\\)19 b Fq(\()p Fp(Y)1893 809 y Ft(2)1934 795 y Fq(\))1969 757 y Fo(c)2032 795 y Fq(=)25 b Fm([)2189 757 y Ft(4)2189 817 y Fo(i)p Ft(=1)2331 795 y Fm([)2392 757 y Ft(4)2392 817 y Fo(j)t Ft(=1)2542 795 y Fp(G)2614 757 y Ft(1)2614 817 y Fo(i)2675 795 y Fm(\\)19 b Fp(G)2827 757 y Ft(2)2827 817 y Fo(j)0 1013 y Fq(So,)28 b(in)e(order)h(to)h(estimate)g(the)f(LHS)g(of)g(\(6.20\))j (it)d(is)f(enough)h(to)h(\014nd)e(an)h(appropriate)f(upp)s(er)g(b)s (ound)f(for)i(eac)m(h)0 1140 y(term)1585 1267 y Fp(\026)1640 1223 y Fo(\014)s(;)p Fn(;)1640 1294 y Fo(Q)1697 1303 y Fi(L)1750 1267 y Fq(\()p Fp(G)1857 1229 y Ft(1)1857 1289 y Fo(j)1917 1267 y Fm(\\)20 b Fp(G)2070 1229 y Ft(2)2070 1289 y Fo(j)2126 1267 y Fm(j)15 b Fp(E)2233 1281 y Fo(\015)p 2233 1307 V 2280 1267 a Fq(\))0 1449 y(W)-8 b(e)30 b(sho)m(w)f(no)m(w)f (ho)m(w)h(to)h(deal)e(with)f(one)i(of)g(them,)g(sa)m(y)h(the)f Fp(G)2165 1416 y Ft(1)2165 1472 y(1)2222 1449 y Fm(\\)17 b Fp(G)2372 1416 y Ft(2)2372 1472 y(4)2441 1449 y Fq(term,)29 b(since)f(the)h(others)g(can)g(b)s(e)f(treated)0 1576 y(analogoulsy)-8 b(.)41 b(Using)29 b(Prop)s(osition)g(6.1)i(and)f(the)g (spin)f(\015ip)f(symmetry)-8 b(,)31 b(w)m(e)g(get)36 1813 y Fp(\026)91 1769 y Fo(\014)s(;)p Fn(;)91 1840 y Fo(Q)148 1849 y Fi(L)201 1813 y Fq(\()p Fp(G)308 1775 y Ft(1)308 1835 y(1)369 1813 y Fm(\\)19 b Fp(G)521 1775 y Ft(2)521 1835 y(4)577 1813 y Fm(j)c Fp(E)684 1827 y Fo(\015)p 684 1853 V 731 1813 a Fq(\))26 b(=)61 2025 y(=)f Fp(\026)212 1982 y Fo(\014)s(;)p Fn(;)212 2052 y Fo(Q)269 2061 y Fi(L)322 1924 y Fg(\020)391 1952 y(\010)460 2025 y Fm(j)p Fp(M)573 2039 y Fo(A)628 2021 y Fj(\016)628 2057 y Fi(\015)p 628 2080 42 4 v 694 2025 a Fq(+)20 b Fp(m)865 1988 y Fn(\003)907 2025 y Fm(j)p Fp(A)1000 1988 y Fn(\016)1000 2048 y Fo(\015)p 1000 2075 47 4 v 1047 2025 a Fm(j)15 b(j)26 b(\025)f Fp(\016)s(L)1339 1988 y Ft(2)1380 2025 y Fp(=)p Fq(4)1485 1952 y Fg(\011)1559 2025 y Fm(\\)1640 1952 y Fg(\010)1708 2025 y Fm(j)p Fp(M)1821 2039 y Fo(B)1879 2021 y Fj(\016)1876 2057 y Fi(\015)p 1876 2080 42 4 v 1918 2039 a Fn(\\)p Fo(U)2017 2048 y Fi(L)2090 2025 y Fq(+)20 b Fp(m)2261 1988 y Fn(\003)2302 2025 y Fm(j)p Fp(B)2401 1988 y Fn(\016)2396 2048 y Fo(\015)p 2396 2075 47 4 v 2463 2025 a Fm(\\)g Fp(U)2606 2039 y Fo(L)2660 2025 y Fm(j)15 b(j)2740 1952 y Fg(\011)2820 2025 y Fm(\025)25 b Fp(\016)s(L)3021 1988 y Ft(2)3062 2025 y Fp(=)p Fq(4)3167 1921 y Fg(\014)3167 1975 y(\014)3167 2030 y(\014)3214 2025 y Fp(E)3281 2039 y Fo(\015)p 3281 2066 V 3342 1924 a Fg(\021)3422 2025 y Fq(=)61 2268 y(=)168 2207 y(1)p 168 2247 46 4 v 168 2330 a(2)240 2268 y Fp(\026)295 2224 y Fo(\014)s(;)p Fn(;)398 2197 y Fh(+)p Fi(;Q)515 2212 y(L)295 2295 y Fo(A)350 2277 y Fj(\016)350 2313 y Fi(\015)p 350 2336 42 4 v 391 2295 a Fo(;s)572 2167 y Fg(\020)641 2268 y Fm(j)p Fp(M)754 2282 y Fo(A)809 2263 y Fj(\016)809 2300 y Fi(\015)p 809 2323 V 876 2268 a Fq(+)20 b Fp(m)1047 2231 y Fn(\003)1088 2268 y Fm(j)p Fp(A)1181 2231 y Fn(\016)1181 2291 y Fo(\015)p 1181 2317 47 4 v 1228 2268 a Fm(j)15 b(j)26 b(\025)f Fp(\016)s(L)1520 2231 y Ft(2)1562 2268 y Fp(=)p Fq(4)1667 2167 y Fg(\021)1737 2268 y Fp(\026)1792 2224 y Fo(\014)s(;)p Fn(;)1895 2197 y Fj(\000)p Fi(;Q)2014 2212 y(L)1792 2295 y Fo(B)1850 2277 y Fj(\016)1847 2313 y Fi(\015)p 1847 2336 42 4 v 1889 2295 a Fo(;s)2072 2167 y Fg(\020)2141 2268 y Fm(j)p Fp(M)2254 2282 y Fo(B)2312 2263 y Fj(\016)2309 2300 y Fi(\015)p 2309 2323 V 2351 2282 a Fn(\\)p Fo(U)2450 2291 y Fi(L)2522 2268 y Fq(+)20 b Fp(m)2693 2231 y Fn(\003)2735 2268 y Fm(j)p Fp(B)2834 2231 y Fn(\016)2829 2291 y Fo(\015)p 2829 2317 47 4 v 2896 2268 a Fm(\\)g Fp(U)3039 2282 y Fo(L)3093 2268 y Fm(j)15 b(j)26 b(\025)f Fp(\016)s(L)3385 2231 y Ft(2)3427 2268 y Fp(=)p Fq(4)3532 2167 y Fg(\021)3587 2268 y Fq(+)56 2524 y(+)158 2462 y(1)p 158 2503 46 4 v 158 2586 a(2)230 2524 y Fp(\026)285 2480 y Fo(\014)s(;)p Fn(;)388 2453 y Fj(\000)p Fi(;Q)507 2468 y(L)285 2550 y Fo(A)340 2532 y Fj(\016)340 2569 y Fi(\015)p 340 2592 42 4 v 381 2550 a Fo(;s)564 2423 y Fg(\020)634 2524 y Fm(j)p Fp(M)747 2538 y Fo(A)802 2519 y Fj(\016)802 2555 y Fi(\015)p 802 2579 V 868 2524 a Fq(+)20 b Fp(m)1039 2486 y Fn(\003)1081 2524 y Fm(j)p Fp(A)1174 2486 y Fn(\016)1174 2546 y Fo(\015)p 1174 2573 47 4 v 1221 2524 a Fm(j)15 b(j)26 b(\025)f Fp(\016)s(L)1513 2486 y Ft(2)1554 2524 y Fp(=)p Fq(4)1659 2423 y Fg(\021)1730 2524 y Fp(\026)1785 2480 y Fo(\014)s(;)p Fn(;)1888 2453 y Fh(+)p Fi(;Q)2005 2468 y(L)1785 2550 y Fo(B)1843 2532 y Fj(\016)1840 2569 y Fi(\015)p 1840 2592 42 4 v 1881 2550 a Fo(;s)2062 2423 y Fg(\020)2131 2524 y Fm(j)p Fp(M)2244 2538 y Fo(B)2302 2519 y Fj(\016)2299 2555 y Fi(\015)p 2299 2579 V 2341 2538 a Fn(\\)p Fo(U)2440 2547 y Fi(L)2512 2524 y Fq(+)20 b Fp(m)2683 2486 y Fn(\003)2725 2524 y Fm(j)p Fp(B)2824 2486 y Fn(\016)2819 2546 y Fo(\015)p 2819 2573 47 4 v 2886 2524 a Fm(\\)g Fp(U)3029 2538 y Fo(L)3083 2524 y Fm(j)15 b(j)26 b(\025)f Fp(\016)s(L)3375 2486 y Ft(2)3416 2524 y Fp(=)p Fq(4)3521 2423 y Fg(\021)3602 2524 y Fm(\024)61 2779 y(\024)168 2718 y Fq(1)p 168 2758 46 4 v 168 2842 a(2)240 2779 y Fp(\026)295 2736 y Fo(\014)s(;)p Fn(;)398 2708 y Fj(\000)p Fi(;Q)517 2724 y(L)295 2806 y Fo(B)353 2788 y Fj(\016)350 2824 y Fi(\015)p 350 2848 42 4 v 391 2806 a Fo(;s)574 2678 y Fg(\020)644 2779 y Fm(j)p Fp(M)757 2793 y Fo(B)815 2775 y Fj(\016)812 2811 y Fi(\015)p 812 2835 V 854 2793 a Fn(\\)p Fo(U)953 2802 y Fi(L)1025 2779 y Fq(+)20 b Fp(m)1196 2742 y Fn(\003)1238 2779 y Fm(j)p Fp(B)1337 2742 y Fn(\016)1332 2802 y Fo(\015)p 1332 2829 47 4 v 1399 2779 a Fm(\\)f Fp(U)1541 2793 y Fo(L)1596 2779 y Fm(j)c(j)26 b(\025)f Fp(\016)s(L)1888 2742 y Ft(2)1929 2779 y Fp(=)p Fq(4)2034 2678 y Fg(\021)2110 2779 y Fq(+)2212 2718 y(1)p 2212 2758 46 4 v 2212 2842 a(2)2283 2779 y Fp(\026)2338 2736 y Fo(\014)s(;)p Fn(;)2441 2708 y Fj(\000)p Fi(;Q)2560 2724 y(L)2338 2806 y Fo(A)2393 2788 y Fj(\016)2393 2824 y Fi(\015)p 2393 2848 42 4 v 2434 2806 a Fo(;s)2618 2678 y Fg(\020)2687 2779 y Fm(j)p 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1887 a Fq(2)1599 1825 y Fm(g)g(\024)163 b Fq(max)1766 1883 y Fo(W)10 b Ft(:)p Fo(V)1923 1865 y Fj(\016)1961 1883 y Fn(\032)p Fo(W)g Fn(\032)p Fo(V)2227 1825 y Fp(\026)2282 1781 y Fo(\014)s(;)p Ft(+)2282 1851 y Fo(W)n(;)i(s)2421 1825 y Fm(fj)p Fp(m)2571 1839 y Fo(V)2630 1820 y Fj(\016)2672 1825 y Fq(\()p Fp(\033)s Fq(\))21 b Fm(\000)f Fp(m)2989 1787 y Fn(\003)3031 1825 y Fm(j)25 b(\025)3190 1763 y Fp(")p 3188 1804 V 3188 1887 a Fq(2)3245 1825 y Fm(g)378 b Fq(\(6)p Fp(:)p Fq(33\))0 2085 y(where)40 b(in)f(the)h(second)g (inequalit)m(y)f(w)m(e)h(used)g(Prop)s(osition)e(6.2.)71 b(By)41 b(construction)e(the)i(set)f Fp(W)53 b Fq(where)40 b(the)0 2212 y(maxim)m(um)f(is)f(attained)i(satis\014es)f Fm(j)p Fp(W)13 b Fm(j)41 b(\025)1559 2176 y Ft(3)p Fo(\013)p 1559 2191 84 4 v 1582 2243 a Ft(4)1653 2212 y Fp(L)1715 2179 y Ft(2)1795 2212 y Fq(and)f Fm(j)p Fp(W)f Fm(n)27 b Fp(V)2277 2179 y Fn(\016)2319 2212 y Fm(j)41 b(\024)f(j)p Fp(@)2569 2226 y Fo(s)2609 2212 y Fp(V)20 b Fm(j)41 b(\024)g Fp(c)2899 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3196 V 1780 3279 a Fq(4)1836 3216 y Fm(g)c Fq(+)2057 3150 y Fp(\026)2112 3106 y Fo(\014)s(;)p Ft(+)2112 3176 y Fo(W)2238 3150 y Fm(f)p Fp(m)2363 3164 y Fo(W)2446 3150 y Fq(\()p Fp(\033)s Fq(\))26 b Fm(\025)f Fp(m)2773 3117 y Fn(\003)2835 3150 y Fq(+)2938 3114 y Fo(")p 2937 3129 37 4 v 2937 3181 a Ft(4)2984 3150 y Fm(g)p 2004 3196 1080 4 v 2004 3294 a Fp(\026)2059 3251 y Fo(\014)s(;)p Ft(+)2059 3321 y Fo(W)2185 3294 y Fm(f)p Fq(all)15 b(con)m(tours)g(are) g(small)o Fm(g)3109 3216 y Fp(:)3668 3101 y Fq(\(6)p Fp(:)p Fq(34\))0 3514 y(Using)30 b(Lemma)g(3.1)i(of)e([I],)h(w)m(e)g (ha)m(v)m(e)g(that)g(there)g(exists)f(a)h(constan)m(t)g Fp(c)2474 3481 y Fn(0)2525 3514 y Fq(=)25 b Fp(c)2660 3481 y Fn(0)2686 3514 y Fq(\()p Fp(\013;)15 b(")p Fq(\))28 b Fp(>)c Fq(0)31 b(suc)m(h)f(that)1063 3797 y Fp(\026)1118 3753 y Fo(\014)s(;)p Ft(+)1118 3824 y Fo(W)n(;)12 b(s)1257 3797 y Fm(f)p Fp(m)1382 3811 y Fo(W)1465 3797 y Fq(\()p Fp(\033)s Fq(\))26 b Fm(\024)f Fp(m)1792 3760 y Fn(\003)1854 3797 y 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926 615 53 4 v 30 w Fq(=)25 b Fm(;)p Fq(.)41 b(In)30 b(this)f(case)j(w)m(e)f (write)703 855 y Fp(\026)758 812 y Fo(\014)s(;)p Fn(;)758 882 y Fo(Q)815 891 y Fi(L)868 855 y Fq(\()p Fm(M)1012 818 y Fo(m)1012 878 y(L)1097 855 y Fm(j)15 b Fp(E)1204 872 y Fn(;)1261 855 y Fq(\))26 b(=)f Fp(\026)1473 812 y Fo(\014)s(;)p Fn(;)1473 882 y Fo(Q)1530 891 y Fi(L)1578 882 y Fo(;s)1638 855 y Fq(\()p Fm(M)1782 818 y Fo(m)1782 878 y(L)1852 855 y Fq(\))g Fm(\024)g Fp(\026)2063 812 y Fo(\014)s(;)p Fn(;)2063 882 y Fo(Q)2120 891 y Fi(L)2168 882 y Fo(;s)2229 782 y Fg(\010)2297 855 y Fm(j)p Fp(m)2402 869 y Fo(Q)2459 878 y Fi(L)2532 855 y Fm(\000)20 b Fp(m)p Fm(j)25 b(\024)g Fq(10)p Fp(=L)3046 818 y Ft(2)3103 782 y Fg(\011)3171 855 y Fp(:)472 b Fq(\(6)p Fp(:)p Fq(38\))0 1092 y(Let)31 b Fp(Q)235 1059 y Fn(\016)235 1117 y Fo(L)314 1092 y Fq(=)25 b Fp(Q)482 1106 y Fo(L)537 1092 y Fm(n)p Fp(@)630 1106 y Ft(2)p Fo(s)705 1092 y Fp(Q)777 1106 y Fo(L)862 1092 y Fq(Then,)30 b(is)f Fp(L)h Fq(is)g(large)g(enough,)h 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Fm(\000)20 b Fp(m)p Fm(j)25 b(\024)g Fq(10)p Fp(=L)2055 2545 y Ft(2)2112 2509 y Fg(\011)2190 2582 y Fm(\024)g Fp(e)2328 2545 y Fn(\000)p Fo(cL)2468 2517 y Fh(2)2503 2545 y Fo(=s)p Ft(\()p Fo(L)p Ft(\))2680 2517 y Fh(2)2812 2582 y Fd(\003)0 5558 y Fh(15)p Fi(=apr)r(ile=)p Fh(1999)30 b Ft([38])1294 b Fq(6.11)p eop %%Page: 39 39 39 38 bop 0 91 a Fr(7.)47 b(A)35 b(lo)s(cal)g(large)g(deviation)g (result)0 246 y Fq(Here)d(w)m(e)f(pro)m(v)m(e)h(Theorem)f(6.3)i(follo)m (wing)c([IS].)i(W)-8 b(e)33 b(start)f(with)d(a)j(result)e(on)h(the)h (deca)m(y)g(of)f(the)h(connectivit)m(y)0 373 y(function)d(for)h(the)h Fp(d)25 b Fq(=)g(2,)31 b Fp(q)d Fq(=)d(2)31 b(FK)f(random)g(cluster)g (mo)s(del,)f(whic)m(h)g(impro)m(v)m(es)h(the)h(results)e(in)g([CCS].)0 608 y Fu(7.1)35 b(Deca)-6 b(y)34 b(of)f(connectivity)g(f)n(or)h(the)f (FK)h(measures)e(with)i(wired)g(bound)n(ar)-6 b(y)33 b(conditions)0 763 y Fq(W)-8 b(e)29 b(consider)e(here)h(the)g(FK)h (\\random)e(cluster")h(mo)s(del.)39 b(W)-8 b(e)29 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b(K)q(;\014)1496 2563 y Fj(\003)1533 2591 y Fo(;w)1312 2661 y Fn(G)1360 2643 y Fj(\003)1614 2634 y Fq(\()p Fp(!)1709 2597 y Fn(\003)1751 2634 y Fq(\))91 b(and)f Fp(\026)2169 2591 y Fo(F)10 b(K)q(;\014)s(;w)2169 2661 y Fn(G)2433 2634 y Fq(\()p Fp(!)s Fq(\))26 b(=)f Fp(\026)2740 2591 y Fo(F)10 b(K)q(;\014)2924 2563 y Fj(\003)2961 2591 y Fo(;)p Fn(;)2740 2661 y(G)2788 2643 y Fj(\003)3024 2634 y Fq(\()p Fp(!)3119 2597 y Fn(\003)3161 2634 y Fq(\))517 b(\(7)p Fp(:)p Fq(1\))0 2867 y(where)30 b(the)g(dual)f(in)m(v)m(erse)i (temp)s(erature)f Fp(\014)1497 2834 y Fn(\003)1569 2867 y Fq(is)g(giv)m(en)g(b)m(y)g(the)h(Krammer{W)-8 b(annier)29 b(equation)1658 3100 y Fp(e)1700 3062 y Ft(2)p Fo(\014)1811 3100 y Fq(=)c(tanh)15 b Fp(\014)2160 3062 y Fn(\003)2217 3100 y Fp(:)0 3333 y Fq(Notice)29 b(that)g Fp(\014)534 3300 y Fn(\003)601 3333 y Fp(<)c(\014)748 3347 y Fo(c)814 3333 y Fq(as)k(long)e(as)i Fp(\014)h(>)25 b(\014)1456 3347 y Fo(c)1494 3333 y Fq(.)40 b(W)-8 b(e)29 b(will)d(also)i(use)g (the)h(follo)m(wing)d(monotonicit)m(y)j(prop)s(ert)m(y)e(of)i(the)0 3460 y(FK)34 b(measures:)48 b(let)34 b Fp(U)42 b Fm(\032)31 b Fp(V)51 b Fm(\032\032)31 b Ff(Z)1280 3427 y Fo(d)1318 3460 y Fq(,)k(and)f(let)g Fp(f)43 b Fq(b)s(e)34 b(a)g(nondecreasing)g (function)e(on)i(\012)3135 3427 y Fo(F)10 b(K)3135 3485 y Fn(E)3174 3494 y Fi(U)3261 3460 y Fq(.)52 b(Then)33 b(\(see,)j(for)0 3587 y(instance)30 b([Pi])g(for)g(a)h(pro)s(of)7 b(\))1010 3820 y Fp(\026)1065 3776 y Fo(F)j(K)q(;)p Fn(;)1065 3847 y Fo(U)1245 3820 y Fq(\()p Fp(f)g Fq(\))25 b Fm(\024)g Fp(\026)1546 3776 y Fo(F)10 b(K)q(;)p Fn(;)1546 3847 y Fo(V)1727 3820 y Fq(\()p Fp(f)g Fq(\))25 b Fm(\024)g Fp(\026)2028 3776 y Fo(F)10 b(K)q(;w)2028 3847 y(V)2226 3820 y Fq(\()p Fp(f)g Fq(\))25 b Fm(\024)g Fp(\026)2527 3776 y Fo(F)10 b(K)q(;w)2527 3847 y(U)2725 3820 y Fq(\()p Fp(f)g Fq(\))15 b Fp(:)823 b Fq(\(7)p Fp(:)p Fq(2\))0 4093 y(F)-8 b(or)38 b(\003)f Fm(\032\032)e Ff(Z)512 4060 y Fo(d)588 4093 y Fq(and)h Fp(x;)15 b(y)40 b Fm(2)c Fq(\003)h(w)m(e)h (de\014ne)e Fm(f)p Fp(x)1711 4041 y Ft(\003)1652 4093 y Fm( )-15 b(!)p Fp(y)s Fm(g)38 b Fq(as)f(the)g(ev)m(en)m(t)i(that)f Fp(x)f Fq(is)f(connected)i(to)g Fp(y)i Fq(b)m(y)d(a)g(path)0 4220 y(of)f(b)s(onds)f(in)f Fm(E)540 4234 y Ft(\003)631 4220 y Fq(whic)m(h)g(are)j(FK{o)s(ccupied.)57 b(What)37 b(w)m(e)f(w)m(an)m(t)h(to)g(pro)m(v)m(e)g(is)e(the)h(exp)s(onen)m(tial) f(deca)m(y)j(of)e(the)0 4347 y(connectivit)m(y)31 b(function)1338 4475 y Fp(\034)1388 4431 y Fo(\014)s(;w)1378 4501 y Ft(\003)1513 4475 y Fq(\()p Fp(x;)15 b(y)s Fq(\))26 b(=)f Fp(\026)1900 4431 y Fo(F)10 b(K)q(;\014)s(;w)1900 4501 y Ft(\003)2164 4475 y Fm(f)p Fp(x)2320 4423 y Ft(\003)2261 4475 y Fm( )-15 b(!)p Fp(y)s Fm(g)15 b Fp(:)0 4704 y Fq(Notice)45 b(that)f Fm(f)p Fp(x)665 4652 y Ft(\003)606 4704 y Fm( )-15 b(!)p Fp(y)s Fm(g)44 b Fq(means)f(that)i(there)e(is)g(a)h(connection)g(whic)m (h)e(mak)m(es)j(no)e(use)h(of)g(the)f(\\b)s(oundary)0 4831 y(cluster",)30 b(otherwise)g(the)g(result)f(w)m(ould)g(clearly)g (b)s(e)h(false.)40 b(In)29 b([CCS])h(it)f(has)h(b)s(een)f(pro)m(v)m(en) i(that)g(the)f Fk(fr)-5 b(e)g(e)33 b(b.c.)0 4958 y Fq(connectivit)m(y) 39 b(function,)g(b)s(eing)f(equal)g(to)h(the)g(t)m(w)m(o{p)s(oin)m(t)g (correlation)f(function)g(of)g(the)h(Ising)e(mo)s(del,)j(has)0 5086 y(exp)s(onen)m(tial)34 b(deca)m(y)i(for)f(all)f Fp(\014)39 b(<)33 b(\014)1269 5100 y Fo(c)1306 5086 y Fq(.)55 b(The)35 b(wired)e(case)j(needs)f(ho)m(w)m(ev)m(er)i(a)e (di\013eren)m(t)g(treatmen)m(t)h(\(the)g(FK)m(G)0 5213 y(inequalit)m(y)f(sa)m(ys)j(that)f Fp(\034)881 5180 y Fo(\014)s(;w)1042 5213 y Fm(\025)f Fp(\034)1199 5180 y Fo(\014)s(;)p Fn(;)1307 5213 y Fq(\).)61 b(W)-8 b(e)38 b(also)f(p)s(oin)m(t)f(out)h(that)h(the)f(follo)m(wing)e(result)h(has)h (no)g(\\ob)m(vious")0 5340 y(extension)30 b(to)h(the)g(case)g Fp(d)26 b Fm(\025)f Fq(3.)0 5558 y Fh(15)p Fi(=apr)r(ile=)p Fh(1999)30 b Ft([39])1317 b Fq(7.1)p eop %%Page: 40 40 40 39 bop 0 91 a Fr(Theorem)34 b(7.1.)53 b Fc(F)-8 b(or)31 b(all)e Fp(\014)i(<)25 b(\014)1177 105 y Fo(c)1244 91 y Fc(there)31 b(exist)f Fp(C)7 b Fq(\()p Fp(\014)e Fq(\))p Fc(,)31 b Fp(m)p Fq(\()p Fp(\014)5 b Fq(\))26 b Fp(>)f Fq(0)31 b Fc(suc)m(h)f(that)h(for)f(all)f Fp(L)1147 348 y(\034)1197 305 y Fo(\014)s(;w)1187 375 y(Q)1244 384 y Fi(L)1322 348 y Fq(\()p Fp(x;)15 b(y)s Fq(\))27 b Fm(\024)e Fp(C)c(e)1783 311 y Fn(\000)p Fo(m)12 b Fn(j)p Fo(x)p Fn(\000)p Fo(y)r Fn(j)2284 348 y Fm(8)p Fp(x;)j(y)28 b Fm(2)d Fp(Q)2658 362 y Fo(L)2727 348 y Fp(:)0 613 y Fk(Pr)-5 b(o)g(of.)36 b Fq(W)-8 b(e)36 b(can)g(assume)e(that)i Fp(L)f Fq(is)f(greater)i(than)f(some)g Fp(L)2144 627 y Ft(0)2185 613 y Fq(\()p Fp(\014)5 b Fq(\).)55 b(F)-8 b(or)36 b Fp(l)e(<)f(L)p Fq(,)j(denote)g(b)m(y)f Fp(A)3344 627 y Fo(l)3372 613 y Fq(\()p Fp(x)p Fq(\))g(the)g(ev)m(en)m(t)0 741 y(that)44 b(the)f(top)g(side)f(of)h(the)h(square)f Fp(Q)1405 755 y Fo(l)1432 741 y Fq(\()p Fp(x)p Fq(\))h(is)e (FK{connected)j(to)e(the)h(lo)m(w)m(er)f(half)f(of)h Fp(Q)3238 755 y Fo(l)3266 741 y Fq(,)j Fk(i.e.)32 b Fq(to)44 b(the)f(set)0 869 y Fm(f)p Fp(y)36 b Fm(2)c Fp(Q)291 883 y Fo(l)318 869 y Fq(\()p Fp(x)p Fq(\))h(:)g Fp(y)576 883 y Ft(2)649 869 y Fm(\024)f Fp(x)804 883 y Ft(2)867 869 y Fq(+)23 b Fm(b)1012 834 y Ft(1)p 1012 849 37 4 v 1012 901 a(2)1060 869 y Fp(l)r Fm(cg)p Fq(,)36 b(b)m(y)f(a)g(path)f (of)h(FK{o)s(ccupied)f(b)s(onds)f(en)m(tirely)h(con)m(tained)h(in)e Fm(E)3503 886 y Fo(Q)3560 896 y Fi(l)3586 886 y Ft(\()p Fo(x)p Ft(\))3689 869 y Fq(.)53 b(Let)0 997 y(also)206 974 y(^)182 997 y Fp(A)250 1011 y Fo(l)278 997 y Fq(\()p Fp(x)p Fq(\))30 b(b)s(e)f(the)h(ev)m(en)m(t)h(that)g(the)e Fp(A)1368 1011 y Fo(l)1397 997 y Fq(\()p Fp(x)p Fq(\))h(o)s(ccurs)f(mo) s(dulo)f(a)i(rotation)g(b)m(y)g Fp(k)2764 961 y Fo(\031)p 2764 976 45 4 v 2768 1029 a Ft(2)2820 997 y Fq(,)g Fp(k)e Fq(=)d(0)p Fp(;)15 b Fq(1)p Fp(;)g Fq(2)p Fp(;)g Fq(3,)33 b Fk(i.e.)f Fq(that)f(an)m(y)0 1125 y(side)i(of)g Fp(Q)364 1139 y Fo(l)392 1125 y Fq(\()p Fp(x)p Fq(\))h(is)f(connected)i(to)f (the)g(\\furthest")g(half)e(of)i(the)g(square.)50 b(Let)34 b Fp(n)d Fq(=)f Fm(bj)p Fp(x)23 b Fm(\000)f Fp(y)s Fm(j)p Fp(=)p Fq(\(2)p Fp(l)r Fq(\))p Fm(c)p Fq(.)53 b(It)34 b(is)e(then)0 1293 y(easy)e(to)h(see)g(that)f(the)g(ev)m(en)m(t)h Fm(f)p Fp(x)1177 1237 y Fo(Q)1234 1246 y Fi(L)1145 1293 y Fm( )-15 b(!)p Fp(y)s Fm(g)31 b Fq(is)d(con)m(tained)j(in)d(the)i(ev) m(en)m(t)h Fm(A)2511 1307 y Fo(n)2561 1293 y Fq(\()p Fp(l)r Fq(\))f(de\014ned)f(as)h(follo)m(ws:)39 b(there)30 b(exists)0 1421 y Fp(j)h Fm(\025)25 b Fp(n)30 b Fq(and)f(a)i(sequence)g (of)g(disjoin)m(t)d(squares)i Fm(f)p Fp(Q)1744 1435 y Fo(l)1772 1421 y Fq(\()p Fp(x)1859 1388 y Fo(i)1890 1421 y Fq(\))p Fm(g)1970 1377 y Fo(j)1970 1446 y(i)p Ft(=1)2123 1421 y Fq(inside)f Fp(Q)2454 1435 y Fo(L)2538 1421 y Fq(suc)m(h)h(that)66 1590 y(\(a\))46 b Fp(x)25 b Fm(2)g Fp(Q)462 1604 y Fo(l)490 1590 y Fq(\()p Fp(x)577 1557 y Ft(1)618 1590 y Fq(\))31 b(and)f Fp(y)e Fm(2)d Fp(Q)1092 1604 y Fo(l)1119 1590 y Fq(\()p Fp(x)1206 1557 y Fo(j)1245 1590 y Fq(\))61 1726 y(\(b\))45 b(3)26 b Fp(<)f(d)441 1652 y Fg(\000)483 1726 y Fp(Q)555 1740 y Fo(l)583 1726 y Fq(\()p Fp(x)670 1693 y Fo(i)700 1726 y Fq(\))p Fp(;)15 b(Q)847 1740 y Fo(l)876 1726 y Fq(\()p Fp(x)963 1693 y Fo(i)p Ft(+1)1085 1726 y Fq(\))1120 1652 y Fg(\001)1188 1726 y Fm(\024)25 b Fp(l)92 b Fm(8)15 b Fp(i)26 b Fq(=)f(1)15 b Fp(:)g(:)g(:)h(j)26 b Fm(\000)20 b Fq(1)71 1862 y(\(c\))251 1839 y(^)227 1862 y Fp(A)295 1876 y Fo(l)323 1862 y Fq(\()p Fp(x)410 1829 y Fo(i)441 1862 y Fq(\))31 b(o)s(ccurs)f(for)g(eac)m(h)h Fp(i)26 b Fq(=)f(1)15 b Fp(:)g(:)g(:)i(j)5 b Fq(.)0 2031 y(Th)m(us)29 b(b)m(y)i(an)f(elemen)m(tary)h(P)m(eierl's)e(argumen)m(t)i (w)m(e)g(get)1094 2365 y Fp(\034)1144 2321 y Fo(\014)s(;w)1134 2392 y(Q)1191 2401 y Fi(L)1268 2365 y Fq(\()p Fp(x;)15 b(y)s Fq(\))27 b Fm(\024)1632 2251 y Fn(1)1602 2279 y Fg(X)1601 2472 y Fo(j)t Ft(=)p Fo(n)1735 2365 y Fq(\(3)p Fp(l)r Fq(\))1879 2327 y Ft(2)p Fo(j)1955 2365 y Fp(\026)2010 2321 y Fo(F)10 b(K)q(;\014)s(;w)2010 2392 y(Q)2067 2401 y Fi(L)2274 2365 y Fq(\()15 b Fm(\\)2385 2321 y Fo(j)2385 2390 y(i)p Ft(=1)2531 2342 y Fq(^)2507 2365 y Fp(A)2575 2379 y Fo(l)2603 2365 y Fq(\()p Fp(x)2690 2327 y Fo(i)2721 2365 y Fq(\))g(\))907 b(\(7)p Fp(:)p Fq(3\))0 2728 y(and,)30 b(using)f(\(7.2\))542 3085 y Fp(\026)597 3042 y Fo(F)10 b(K)q(;\014)s(;w)597 3112 y(Q)654 3121 y Fi(L)860 3085 y Fq(\()15 b Fm(\\)971 3042 y Fo(j)971 3111 y(i)p Ft(=1)1118 3062 y Fq(^)1094 3085 y Fp(A)1162 3099 y Fo(l)1190 3085 y Fq(\()p Fp(x)1277 3048 y Fo(i)1307 3085 y Fq(\))g(\))27 b Fm(\024)e Fp(\026)1570 3042 y Fo(F)10 b(K)q(;\014)s(;w)1570 3117 y Fn([)1619 3126 y Fi(i)1647 3117 y Fo(Q)1704 3127 y Fi(l)1731 3117 y Ft(\()p Fo(x)1800 3099 y Fi(i)1829 3117 y Ft(\))1862 3085 y Fq(\()15 b Fm(\\)1973 3042 y Fo(j)1973 3111 y(i)p Ft(=1)2120 3062 y Fq(^)2096 3085 y Fp(A)2164 3099 y Fo(l)2192 3085 y Fq(\()p Fp(x)2279 3048 y Fo(i)2309 3085 y Fq(\))g(\))26 b(=)2558 2967 y Fo(j)2517 2999 y Fg(Y)2516 3193 y Fo(i)p Ft(=1)2649 3085 y Fp(\026)2704 3042 y Fo(F)10 b(K)q(;\014)s(;w)2704 3117 y(Q)2761 3127 y Fi(l)2788 3117 y Ft(\()p Fo(x)2857 3099 y Fi(i)2886 3117 y Ft(\))2968 3085 y Fq(\()3043 3062 y(^)3018 3085 y Fp(A)3086 3099 y Fo(l)3115 3085 y Fq(\()p Fp(x)3202 3048 y Fo(i)3232 3085 y Fq(\))15 b(\))g Fp(:)356 b Fq(\(7)p Fp(:)p Fq(4\))0 3436 y(Thanks)29 b(to)i(Lemma)g(7.2)g(giv)m (en)g(b)s(elo)m(w)e(w)m(e)i(can)g(no)m(w)f(c)m(ho)s(ose)i Fp(l)27 b Fq(=)e Fp(l)2359 3450 y Ft(0)2400 3436 y Fq(\()p Fp(\014)5 b Fq(\))31 b(suc)m(h)f(that)1372 3693 y(\(3)p Fp(l)r Fq(\))1516 3656 y Ft(2)1558 3693 y Fp(\026)1613 3649 y Fo(F)10 b(K)q(;\014)s(;w)1613 3725 y(Q)1670 3735 y Fi(l)1696 3725 y Ft(\()p Fo(x)1765 3707 y Fi(i)1795 3725 y Ft(\))1877 3693 y Fq(\()1951 3670 y(^)1927 3693 y Fp(A)1995 3707 y Fo(l)2023 3693 y Fq(\()p Fp(x)2110 3656 y Fo(i)2141 3693 y Fq(\))15 b(\))26 b Fm(\024)f Fp(e)2390 3656 y Fn(\000)p Ft(1)2503 3693 y Fp(:)1185 b Fq(\(7)p Fp(:)p Fq(5\))0 3951 y(In)30 b(this)f(w)m(a)m(y)i(w)m(e)g (obtain)941 4284 y Fp(\034)991 4240 y Fo(\014)s(;w)981 4311 y(Q)1038 4320 y Fi(L)1116 4284 y Fq(\()p Fp(x;)15 b(y)s Fq(\))26 b Fm(\024)1479 4171 y Fn(1)1450 4198 y Fg(X)1448 4391 y Fo(j)t Ft(=)p Fo(n)1598 4284 y Fp(e)1640 4247 y Fn(\000)p Fo(j)1761 4284 y Fm(\024)f Fq(2)p Fp(e)1944 4247 y Fn(\000)p Fo(n)2076 4284 y Fq(=)g(2)p Fp(e)2259 4247 y Fn(\000bj)p Fo(x)p Fn(\000)p Fo(y)r Fn(j)p Fo(=)p Ft(\(2)p Fo(l)2654 4256 y Fh(0)2691 4247 y Ft(\))p Fn(c)2772 4284 y Fp(:)91 b Fd(\003)0 4650 y Fr(Lemma)33 b(7.2.)53 b Fc(Let)30 b Fp(\014)h(<)25 b(\014)977 4664 y Fo(c)1014 4650 y Fc(.)41 b(Then,)30 b(for)g(an)m(y)h(p)s(ositiv)m(e)e Fp(n)1494 4907 y Fq(lim)1473 4966 y Fo(l)p Fn(!1)1656 4907 y Fp(l)1685 4869 y Fo(n)1735 4907 y Fp(\026)1790 4863 y Fo(F)10 b(K)q(;\014)s(;w)1790 4934 y(Q)1847 4944 y Fi(l)2053 4907 y Fq(\()2113 4884 y(^)2088 4907 y Fp(A)2156 4921 y Fo(l)2185 4907 y Fq(\))25 b(=)g(0)15 b Fp(:)0 5212 y Fk(Pr)-5 b(o)g(of.)40 b Fq(The)e(pro)s(of)g(is,)h(in)f(turn,)h (based)g(on)f(the)h(follo)m(wing)e(lemma)h(quite)g(similar)e(to)j(an)f (analogous)h(result)0 5340 y(pro)m(v)m(ed)31 b(in)e([MOS])h(for)g (general)h(\014nite)e(range)i(t)m(w)m(o)g(dimensional)d(lattice)j(spin) d(systems.)0 5558 y Fh(15)p Fi(=apr)r(ile=)p Fh(1999)i Ft([40])1317 b Fq(7.2)p eop %%Page: 41 41 41 40 bop 0 91 a Fr(Lemma)33 b(7.3.)53 b Fc(Let)1230 218 y Fp(R)1299 232 y Fo(l)1352 218 y Fq(=)25 b Fm(f)p Fp(x)h Fq(=)f(\()p Fp(x)1754 232 y Ft(1)1795 218 y Fp(;)15 b(x)1887 232 y Ft(2)1928 218 y Fq(\))26 b Fm(2)f Fp(Q)2147 232 y Fo(l)2200 218 y Fq(:)40 b Fp(x)2317 232 y Ft(2)2383 218 y Fm(\024)2479 136 y(p)p 2555 136 29 4 v 82 x Fp(l)r Fm(g)15 b Fp(:)0 385 y Fc(Consider)38 b(the)j(ev)m(en)m(t)g Fp(B)874 399 y Fo(l)942 385 y Fc(that)g(there)f(is)f(a)i(connection)f (inside)e Fp(R)2379 399 y Fo(l)2447 385 y Fc(b)s(et)m(w)m(een)i(the)h (top)f(side)f(of)h Fp(R)3519 399 y Fo(l)3587 385 y Fc(and)g(the)0 512 y(b)s(ottom)31 b(half)e(of)i Fp(R)673 526 y Fo(l)700 512 y Fc(,)g(de\014ned)e(as)1253 696 y Fm(f)p Fp(x)d Fq(=)f(\()p Fp(x)1559 710 y Ft(1)1600 696 y Fp(;)15 b(x)1692 710 y Ft(2)1733 696 y Fq(\))26 b Fm(2)f Fp(Q)1952 710 y Fo(l)2005 696 y Fq(:)40 b Fp(x)2122 710 y Ft(2)2188 696 y Fm(\024)25 b(b)2324 614 y(p)p 2401 614 V 2401 696 a Fp(l)q(=)p Fq(2)p Fm(cg)15 b Fp(:)0 881 y Fc(If)30 b Fp(\014)g(<)25 b(\014)319 895 y Fo(c)387 881 y Fc(there)31 b(exist)f(p)s(ositiv)m(e)f(constan)m(ts)j Fp(")e Fc(and)g Fp(l)1850 895 y Ft(0)1921 881 y Fc(suc)m(h)g(that)1289 1065 y Fp(\026)1344 1021 y Fo(F)10 b(K)q(;\014)s(;w)1344 1092 y(R)1399 1102 y Fi(l)1608 1065 y Fq(\()p Fp(B)1712 1079 y Fo(l)1740 1065 y Fq(\))25 b Fm(\024)g Fq(1)c Fm(\000)f Fp(")182 b Fm(8)25 b Fp(l)i Fm(\025)e Fp(l)2530 1079 y Ft(0)2586 1065 y Fp(:)0 1250 y Fq(W)-8 b(e)29 b(p)s(ostp)s(one)d(the) h(pro)s(of)g(and)g(complete)g(instead)g(the)h(pro)s(of)e(of)i(Lemma)f (7.2.)41 b(Assume)27 b Fp(l)g Fm(\025)e Fq(9)i(and)g(denote)h(b)m(y)0 1377 y Fp(R)70 1344 y Fo(i)69 1403 y(l)131 1377 y Fq(the)k(v)m(ertical) f(translate)h(of)f Fp(R)1167 1391 y Fo(l)1227 1377 y Fq(b)m(y)g(the)g(amoun)m(t)h Fm(b)p Fq(2)p Fp(i)1959 1299 y Fm(p)p 2036 1299 V 2036 1377 a Fp(l)r Fm(c)p Fq(,)g Fp(i)27 b Fq(=)g(0)p Fp(;)15 b Fq(1)g Fp(:)g(:)g(:)s Fq(.)43 b(Notice)32 b(that)g(the)g(rectangles)g Fp(R)3777 1338 y Fo(i)p Fn(\000)p Ft(1)3776 1404 y Fo(l)0 1504 y Fq(and)h Fp(R)250 1471 y Fo(i)249 1529 y(l)314 1504 y Fq(are)h(separated)g(b)m(y)g(a)g(strip)e(of)i(width)e(larger)h(than)h (2.)51 b(Then)33 b(clearly)g(the)h(o)s(ccurrence)g(of)f(the)h(ev)m(en)m (t)0 1630 y Fp(A)68 1644 y Fo(l)127 1630 y Fq(implies)29 b(the)j(o)s(ccurence)g(of)f(the)h(ev)m(en)m(t)h Fp(B)1585 1644 y Fo(l)1644 1630 y Fq(for)e(eac)m(h)i(rectangle)f Fp(R)2450 1597 y Fo(i)2449 1656 y(l)2511 1630 y Fq(with)e Fp(i)e Fq(=)e(0)p Fp(;)15 b Fq(1)g Fp(:)g(:)g(:)j Fm(b)3194 1595 y Ft(1)p 3194 1610 37 4 v 3194 1662 a(3)3241 1553 y Fm(p)p 3317 1553 29 4 v 77 x Fp(l)r Fm(c)p Fq(.)44 b(Th)m(us,)31 b(if)f(w)m(e)0 1757 y(use)g(Lemma)h(7.3,)g(w)m(e)g(get)h (that)975 1941 y Fp(\026)1030 1898 y Fo(F)10 b(K)q(;\014)s(;w)1030 1968 y(Q)1087 1978 y Fi(l)1293 1941 y Fq(\()p Fp(A)1396 1955 y Fo(l)1425 1941 y Fq(\))25 b Fm(\024)g Fp(\026)1636 1898 y Fo(F)10 b(K)q(;\014)s(;w)1636 1968 y(R)1691 1978 y Fi(l)1900 1941 y Fq(\()p Fp(B)2004 1955 y Fo(l)2032 1941 y Fq(\))2078 1879 y Fh(1)p 2079 1889 31 4 v 2079 1926 a(3)2120 1849 y Fn(p)p 2180 1849 24 4 v 55 x Fo(l)q Fn(\000)p Ft(1)2326 1941 y Fm(\024)25 b Fq(\(1)c Fm(\000)f Fp(")p Fq(\))2702 1879 y Fh(1)p 2703 1889 31 4 v 2703 1926 a(3)2745 1849 y Fn(p)p 2805 1849 24 4 v 55 x Fo(l)p Fn(\000)p Ft(1)0 2162 y Fq(for)30 b(an)m(y)g Fp(l)d Fm(\025)e Fp(l)487 2176 y Ft(0)548 2162 y Fm(^)19 b Fq(9,)31 b(where)e Fp(l)1018 2176 y Ft(0)1089 2162 y Fq(and)h Fp(")g Fq(are)g(the)h (constan)m(ts)g(giv)m(en)f(in)e(Lemma)j(7.3.)41 b(Th)m(us)29 b Fp(\026)3181 2119 y Fo(F)10 b(K)q(;\014)s(;w)3181 2189 y(Q)3238 2199 y Fi(l)3445 2162 y Fq(\()p Fp(A)3548 2176 y Fo(l)3577 2162 y Fq(\))30 b(deca)m(ys)0 2289 y(at)37 b(least)g(as)g(a)f(negativ)m(e)i(exp)s(onen)m(tial)e(of)g(the)h(square) f(ro)s(ot)h(of)g(the)f(side)g Fp(l)i Fq(and)e(the)g(prop)s(osition)f (follo)m(ws)g(at)0 2416 y(once)c(from)f(the)h(trivial)d(inequalit)m(y)h Fp(\026)1326 2372 y Fo(F)10 b(K)q(;\014)s(;w)1326 2443 y(Q)1383 2453 y Fi(l)1590 2416 y Fq(\()1649 2393 y(^)1625 2416 y Fp(A)1693 2430 y Fo(l)1721 2416 y Fq(\))26 b Fm(\024)f Fq(4)p Fp(\026)1978 2372 y Fo(F)10 b(K)q(;\014)s(;w)1978 2443 y(Q)2035 2453 y Fi(l)2242 2416 y Fq(\()p Fp(A)2345 2430 y Fo(l)2373 2416 y Fq(\).)92 b Fd(\003)0 2630 y Fk(Pr)-5 b(o)g(of)41 b(of)f(L)-5 b(emma)41 b(7.3.)e Fq(Let)f Fp(M)49 b Fq(=)38 b Fm(f)p Fp(x)g Fm(2)g Fp(R)1600 2644 y Fo(l)1666 2630 y Fq(:)77 b Fp(x)1820 2644 y Ft(2)1898 2630 y Fq(=)38 b Fm(b)2058 2594 y Ft(1)p 2059 2609 37 4 v 2059 2661 a(2)2106 2552 y Fm(p)p 2181 2552 29 4 v 2181 2630 a Fp(l)r Fm(cg)h Fq(b)s(e)f(the)g(horizon)m(tal)g(segmen)m(t) h(whic)m(h)e(splits)0 2756 y Fp(R)69 2770 y Fo(l)129 2756 y Fq(in)m(to)c(roughly)e(t)m(w)m(o)i(equal)f(part)h(and,)f(for)g (an)m(y)h(giv)m(en)g(p)s(ositiv)m(e)e(in)m(teger)i Fp(\016)f(<)d(l)r(=) p Fq(2,)k(let)g(us)e(divide)g Fp(M)42 b Fq(in)m(to)33 b(a)0 2883 y(middle)28 b(part)i Fp(M)583 2897 y Ft(middle)832 2883 y Fq(and)g(a)g(lateral)g(part)h Fp(M)1653 2897 y Ft(lat)1773 2883 y Fq(as)f(follo)m(ws)1336 3072 y Fp(M)1424 3086 y Ft(lat)1540 3072 y Fq(=)25 b Fp(M)30 b Fm(n)21 b Fp(M)1908 3086 y Ft(middle)1208 3227 y Fp(M)1296 3241 y Ft(middle)1540 3227 y Fq(=)k Fm(f)p Fp(x)g Fm(2)g Fp(M)35 b Fq(:)51 b Fp(\016)29 b Fm(\024)c Fp(x)2260 3241 y Ft(1)2326 3227 y Fm(\024)g Fp(l)d Fm(\000)e Fp(\016)s Fm(g)15 b Fp(:)0 3416 y Fq(Accordingly)26 b(w)m(e)h(can)h(write)e(the)h(ev)m(en)m (t)h Fp(B)k Fq(as)27 b Fp(B)j Fq(=)25 b Fp(B)1887 3430 y Ft(lat)1990 3416 y Fm([)13 b Fp(B)2133 3430 y Ft(middle)2378 3416 y Fq(where)26 b Fp(B)2706 3430 y Ft(lat)2823 3416 y Fq(is)g(the)h(ev)m(en)m(t)h(that)g(the)f(lateral)0 3542 y(part)i(of)h Fp(M)10 b Fq(,)30 b Fp(M)540 3556 y Ft(lat)630 3542 y Fq(,)g(is)e(FK{connected)j(to)f(the)g(top)f(side)g (of)g Fp(R)2153 3556 y Fo(l)2210 3542 y Fq(and)g(similarily)d(for)j (the)g(ev)m(en)m(t)i Fp(B)3384 3556 y Ft(middle)3602 3542 y Fq(.)41 b(Then,)0 3669 y(thanks)30 b(to)h(the)g(FK)m(G)g (inequalit)m(y)-8 b(,)29 b(w)m(e)i(get)243 3853 y Fp(\026)298 3810 y Fo(F)10 b(K)q(;\014)s(;w)298 3880 y(R)353 3890 y Fi(l)562 3853 y Fq(\()p Fp(B)5 b Fq(\))26 b(=)f(1)20 b Fm(\000)g Fp(\026)1039 3810 y Fo(F)10 b(K)q(;\014)s(;w)1039 3880 y(R)1094 3890 y Fi(l)1303 3780 y Fg(\000)1345 3853 y Fp(B)1419 3816 y Fo(c)1414 3876 y Ft(lat)1524 3853 y Fm(\\)19 b Fp(B)1678 3816 y Fo(c)1673 3876 y Ft(middle)1891 3853 y Fq(\))26 b Fm(\024)f Fq(1)c Fm(\000)f Fp(\026)2260 3810 y Fo(F)10 b(K)q(;\014)s(;w)2260 3880 y(R)2315 3890 y Fi(l)2523 3853 y Fq(\()p Fp(B)2632 3816 y Fo(c)2627 3876 y Ft(lat)2717 3853 y Fq(\))p Fp(\026)2807 3810 y Fo(F)g(K)q(;\014)s(;w)2807 3880 y(R)2862 3890 y Fi(l)3072 3853 y Fq(\()p Fp(B)3181 3816 y Fo(c)3176 3876 y Ft(middle)3394 3853 y Fq(\))15 b Fp(:)244 b Fq(\(7)p Fp(:)p Fq(6\))0 4038 y(W)-8 b(e)32 b(no)m(w)e(\014nd)f(lo)m(w)m(er)h(b)s(ounds)e(for)i (b)s(oth)g(factors)h(in)e(the)i(RHS)f(of)g(\(7.6\).)43 b(The)29 b(term)i Fp(\026)3025 3994 y Fo(F)10 b(K)q(;\014)s(;w)3025 4064 y(R)3080 4074 y Fi(l)3289 4038 y Fq(\()p Fp(B)3398 4005 y Fo(c)3393 4063 y Ft(lat)3483 4038 y Fq(\))30 b(is)g(clearly)0 4164 y(b)s(ounded)36 b(from)i(b)s(elo)m(w)g(b)m(y)g(the)h(probabilit)m (y)d(that)j(all)e(v)m(ertical)i(b)s(onds)e(with)g(one)i(v)m(ertex)g(on) f Fp(M)3480 4178 y Ft(lat)3609 4164 y Fq(are)h(not)0 4291 y(FK{o)s(ccupied,)30 b(that)h(is)1467 4418 y Fp(\026)1522 4374 y Fo(F)10 b(K)q(;\014)s(;w)1522 4445 y(R)1577 4455 y Fi(l)1786 4418 y Fq(\()p Fp(B)1895 4380 y Fo(c)1890 4440 y Ft(lat)1980 4418 y Fq(\))25 b Fm(\025)g Fp(e)2178 4380 y Fn(\000)p Fo(C)t Ft(\()p Fo(\014)s Ft(\))p Fo(\016)0 4585 y Fq(for)j(a)h(suitable)d(constan)m(t)k Fp(C)7 b Fq(\()p Fp(\014)e Fq(\).)40 b(Using)27 b(the)i(exp)s(onen)m(tial)e 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Fo(M)1532 5368 y Fh(middle)1736 5249 y Fp(e)1778 5211 y Fn(\000)p Fo(m)p Ft(\()p Fo(\014)s Ft(\))p Fo(d)p Ft(\()p Fo(x;R)2184 5184 y Fi(c)2184 5230 y(l)2219 5211 y Ft(\))2277 5249 y Fm(\025)2384 5187 y Fq(1)p 2384 5228 46 4 v 2384 5311 a(2)3713 5132 y(\(7)p Fp(:)p Fq(7\))0 5558 y Fh(15)p Fi(=apr)r(ile=)p Fh(1999)30 b Ft([41])1317 b Fq(7.3)p eop %%Page: 42 42 42 41 bop 0 91 a Fq(pro)m(vided)27 b(that)i(b)s(oth)f Fp(\016)k Fq(and)c Fp(l)j Fq(are)e(tak)m(en)g(large)g(enough)f(dep)s (ending)e(on)j Fp(\014)5 b Fq(.)40 b(In)28 b(conclusion)f(the)h(RHS)g (of)h(\(7.6\))0 219 y(is)36 b(larger)i(than)f(1)25 b Fm(\000)760 183 y Ft(1)p 760 198 37 4 v 760 250 a(2)807 219 y Fp(e)849 186 y Fn(\000)p Fo(C)t Ft(\()p Fo(\014)s Ft(\))p Fo(\016)1141 219 y Fq(for)37 b(an)m(y)h Fp(l)h Fq(and)e Fp(\016)k Fq(large)d(enough.)61 b(The)37 b(pro)s(of)g(of)g (the)h(lemma,)h(and)e(th)m(us)g(of)0 347 y(Theorem)30 b(7.1,)i(is)d(complete.)92 b Fd(\003)0 620 y Fu(7.2)35 b(Pr)n(oof)f(of)f(Theorem)h(6.3)0 794 y Fq(F)-8 b(or)27 b Fp(\014)k(>)25 b(\014)388 808 y Fo(c)425 794 y Fq(,)j(let)f Fp(m)686 808 y Ft(0)726 794 y Fq(\()p Fp(\014)5 b Fq(\))26 b(=)f Fp(m)1054 761 y Fn(\003)1096 794 y Fq(\()p Fp(\014)5 b Fq(\)\(1)13 b Fm(\000)g Fq(2)p Fp(v)1488 808 y Ft(0)1531 794 y Fq(\),)28 b(where)e Fp(v)1922 808 y Ft(0)1989 794 y Fq(is)g(de\014ned)g(in)f(\(5.4\).)41 b(In)26 b(Prop)s(osition)f(2.3)j (of)f([CGMS])0 922 y(\(and)h(in)g(the)h(follo)m(wing)e(remark\))h(it)h (w)m(as)g(already)f(pro)m(v)m(ed)h(that,)h(under)d(the)h(same)h (assumptions)e(of)i(Theorem)0 1050 y(6.3)i(w)m(e)g(ha)m(v)m(e)h (\(remem)m(b)s(er)e(that)h Fm(N)1235 1017 y Fo(m)1222 1075 y(L)1329 1050 y Fq(=)25 b Fm(b)p Fq(\(1)d(+)e Fp(m)p Fq(\))p Fm(j)p Fp(Q)1870 1064 y Fo(L)1924 1050 y Fm(j)p Fp(=)p Fq(2)p Fm(c)p Fq(\),)1166 1367 y(lim)1132 1425 y Fo(L)p Fn(!1)1357 1367 y Fm(\000)1447 1305 y Fq(1)p 1439 1346 62 4 v 1439 1429 a Fp(L)1542 1367 y Fq(log)32 b Fp(\026)1745 1323 y Fo(\014)s(;)p Fn(;)1745 1394 y Fo(Q)1802 1403 y Fi(L)1854 1293 y Fg(\010)1922 1367 y Fp(N)1995 1381 y Fo(Q)2052 1390 y Fi(L)2131 1367 y Fq(=)25 b Fm(N)2315 1329 y Fo(m)2302 1389 y(L)2384 1293 y Fg(\011)2462 1367 y Fm(\025)g Fp(')p Fq(\()p Fp(m)p Fq(\))0 1675 y(with)i(equalit)m (y)g(if)g Fp(m)e Fm(\024)g Fp(m)910 1689 y Ft(0)951 1675 y Fq(.)40 b(Th)m(us)27 b(w)m(e)h(only)f(ha)m(v)m(e)i(to)g(consider)e (the)h(case)h Fp(m)c Fm(2)g Fq(\()p Fp(m)2890 1689 y Ft(0)2931 1675 y Fp(;)15 b(m)3051 1642 y Fn(\003)3093 1675 y Fq(\))28 b(and)g(w)m(e)g(m)m(ust)g(pro)m(v)m(e)0 1803 y(a)j(lo)m(w)m(er)f(b)s(ound)f(of)h(the)h(form)1235 1957 y Fp(\026)1290 1914 y Fo(\014)s(;)p Fn(;)1290 1984 y Fo(Q)1347 1993 y Fi(L)1400 1884 y Fg(\010)1468 1957 y Fp(N)1541 1971 y Fo(Q)1598 1980 y Fi(L)1676 1957 y Fq(=)25 b Fm(N)1860 1920 y Fo(m)1847 1980 y(L)1929 1884 y Fg(\011)2007 1957 y Fm(\025)g Fp(e)2145 1920 y Fn(\000)p Fo(L)p Ft([)p Fo(')p Ft(\()p Fo(m)p Ft(\)+)p Fo(\016)r Ft(\()p Fo(L)p Ft(\)])3713 1957 y Fq(\(7)p Fp(:)p Fq(8\))0 2175 y(with)33 b(lim)337 2189 y Fo(L)p Fn(!1)551 2175 y Fp(\016)s Fq(\()p Fp(L)p Fq(\))g(=)e(0.)52 b(F)-8 b(or)35 b(this)d(purp)s(ose)h(w)m(e)h(c)m(ho)s(ose)h(some)f Fp(m)d Fm(2)g Fq(\()p Fp(m)2654 2189 y Ft(0)2695 2175 y Fp(;)15 b(m)2815 2142 y Fn(\003)2857 2175 y Fq(\))34 b(once)h(and)e(for)h(all,) g(and)f(w)m(e)0 2304 y(establish)h(some)h(useful)f(notation.)56 b(W)-8 b(e)36 b(set)1621 2281 y(\026)1600 2304 y Fp(Q)1672 2318 y Fo(L)1760 2304 y Fq(=)d Fm(f)p Fp(x)h Fm(2)g Ff(R)2149 2271 y Ft(2)2230 2304 y Fq(:)f Fp(d)2335 2318 y Fn(1)2413 2304 y Fq(\()p Fp(x;)15 b(Q)2612 2318 y Fo(L)2667 2304 y Fq(\))34 b Fm(\024)f Fq(1)p Fp(=)p Fq(2)p Fm(g)i Fq(=)e([0)p Fp(;)15 b(L)p Fq(])3356 2271 y Ft(2)3422 2304 y Fm(\000)23 b Fq(\(1)p Fp(=)p Fq(2)p Fp(;)15 b Fq(1)p Fp(=)p Fq(2\))0 2432 y(and)29 b(w)m(e)h(de\014ne)f Fm(D)644 2399 y Fo(lb)641 2457 y(L)734 2432 y Fq(as)h(the)g(set)g(of)g(all)e(recti\014able)h (curv)m(es)h Fp(\015)h Fm(2)2251 2409 y Fq(\026)2230 2432 y Fp(Q)2302 2446 y Fo(L)2386 2432 y Fq(suc)m(h)f(that)g Fp(\015)35 b Fq(is)29 b(op)s(en)g(with)f(one)j(endp)s(oin)m(t)0 2560 y(on)d(the)g(left)f(side)g(of)736 2537 y(\026)715 2560 y Fp(Q)787 2574 y Fo(L)870 2560 y Fq(and)g(the)h(other)g(one)g(on) g(the)g(b)s(ottom)g(side)f(of)2495 2537 y(\026)2474 2560 y Fp(Q)2546 2574 y Fo(L)2601 2560 y Fq(.)39 b(Let)29 b Fm(B)2889 2527 y Fo(lb)2886 2585 y(L)2976 2560 y Fq(b)s(e)e(the)h (set)h(of)f(all)f(subsets)0 2688 y Fp(\025)32 b Fq(of)h Fm(E)239 2702 y Fo(Q)296 2711 y Fi(L)381 2688 y Fq(suc)m(h)f(that)h Fp(\025)f Fq(connects)h(the)g(left)f(side)f(of)h Fp(Q)1924 2702 y Fo(L)2011 2688 y Fq(with)f(its)h(b)s(ottom)g(side.)46 b(Similarly)-8 b(,)29 b(let)k Fm(B)3498 2644 y Fn(\003)p Fo(;lb)3495 2715 y(L)3648 2688 y Fq(b)s(e)f(the)0 2816 y(set)37 b(of)g(all)f(subsets)g Fp(\025)g Fq(of)924 2793 y Fg(e)911 2816 y Fm(E)967 2830 y Fo(Q)1024 2839 y Fi(L)1114 2816 y Fq(suc)m(h)g(that)h Fp(\025)g Fq(connects)g(the)g(left)g(side)e (of)2641 2793 y(\026)2620 2816 y Fp(Q)2692 2830 y Fo(L)2783 2816 y Fq(with)g(its)h(b)s(ottom)h(side.)59 b(Since)0 2944 y Fp(m)36 b Fm(2)h Fq(\()p Fp(m)329 2958 y Ft(0)370 2944 y Fp(;)15 b(m)490 2911 y Fn(\003)532 2944 y Fq(\),)39 b(the)f(set)f(\000)1000 2958 y Fo(w)1096 2944 y Fq(de\014ned)f(in)g (\(5.5\))j(consists)e(of)g(4)h(curv)m(es)f(and)g(eac)m(h)h(of)g(them)f (is)f(a)i(quarter)f(of)0 3072 y(W)-8 b(ul\013)28 b(shap)s(e)f(cen)m (tered)i(at)g(one)f(of)g(the)g(four)g(corners)g(of)g Fp(Q)p Fq(.)40 b(One)27 b(of)h(these)h(curv)m(es)f(suitably)e(rescaled) i(b)s(elongs)0 3201 y(to)j Fm(D)184 3168 y Fo(lb)181 3226 y(L)274 3201 y Fq(and)f(w)m(e)h(denote)f(it)g(b)m(y)h Fp(\015)1147 3168 y Fo(w)1142 3226 y(L)1206 3201 y Fq(.)40 b(W)-8 b(e)32 b(then)e(let)g Fp(w)1831 3215 y Fo(L)1911 3201 y Fq(=)25 b Fp(L)2069 3168 y Ft(1)p Fo(=)p Ft(4)2183 3132 y Fm(p)p 2259 3132 195 4 v 69 x Fq(log)16 b Fp(L)31 b Fq(and)e(de\014ne)h(the)h(sets)1290 3463 y Fg(e)1275 3486 y Fp(T)1341 3500 y Fo(L)1420 3486 y Fq(=)25 b Fm(f)p Fp(x)h Fm(2)1745 3463 y Fq(\026)1725 3486 y Fp(Q)1797 3500 y Fo(L)1876 3486 y Fq(:)g Fp(d)p Fq(\()p Fp(x;)15 b(\015)2153 3449 y Fo(w)2148 3509 y(L)2213 3486 y Fq(\))25 b Fm(\024)g Fp(w)2434 3500 y Fo(L)2489 3486 y Fp(=)p Fq(2)p Fm(g)1287 3641 y Fp(T)1340 3655 y Fo(L)1420 3641 y Fq(=)g Fm(f)p Fp(x)h Fm(2)1745 3618 y Fq(\026)1725 3641 y Fp(Q)1797 3655 y Fo(L)1876 3641 y Fq(:)g Fp(d)p Fq(\()p Fp(x;)15 b(\015)2153 3603 y Fo(w)2148 3663 y(L)2213 3641 y Fq(\))25 b Fm(\024)g Fp(w)2434 3655 y Fo(L)2489 3641 y Fm(g)15 b Fp(:)3713 3558 y Fq(\(7)p Fp(:)p Fq(9\))0 3933 y(The)32 b(di\013erence)g Fp(T)651 3947 y Fo(L)706 3933 y Fm(n)767 3910 y Fg(e)751 3933 y Fp(T)817 3947 y Fo(L)904 3933 y Fq(consists)g(of)h(t)m(w)m(o)h(disjoin)m(t)d (\\corridors",)i(whic)m(h)f(w)m(e)h(denote)g(b)m(y)g Fp(T)3235 3900 y Fo(i)3222 3958 y(L)3309 3933 y Fq(\(the)g(\\in)m (ternal")0 4061 y(one,)h Fk(i.e.)e Fq(closer)h(to)g(the)g(origin\))f (and)g Fp(T)1422 4028 y Fo(e)1409 4086 y(L)1496 4061 y Fq(\(the)i(\\external")f(one\).)49 b(With)32 b(this)g(notation)h(w)m (e)g(then)g(de\014ne)f(the)0 4189 y(k)m(ey)f(ev)m(en)m(t)h Fp(E)476 4156 y Fo(k)471 4214 y(L)556 4189 y Fq(as)e(the)h(set)g(of)g (all)e Fp(\033)f Fm(2)d Fq(\012)1428 4203 y Fo(Q)1485 4212 y Fi(L)1568 4189 y Fq(suc)m(h)30 b(that)3 4373 y(\()p Fp(E)105 4387 y Ft(1)146 4373 y Fq(\))46 b Fm(G)286 4338 y Fn(;)281 4400 y Fo(Q)338 4409 y Fi(L)391 4373 y Fq(\()p Fp(\033)s Fq(\))36 b(\(remem)m(b)s(er)f(\(6.5\)\))i(con)m(tains)e (exactly)h(one)f(op)s(en)f(con)m(tour)i Fp(\015)k Fq(suc)m(h)35 b(that)g Fp(\015)k Fm(2)32 b(B)3424 4329 y Fn(\003)p Fo(;lb)3421 4400 y(L)3542 4373 y Fq(,)37 b Fp(\015)h Fm(\032)32 b Fp(T)3845 4387 y Fo(L)227 4501 y Fq(and)e Fm(j)p Fp(\015)5 b Fm(j)26 b(\024)f Fp(k)s(L)15 b Fq(log)i Fp(L)3 4639 y Fq(\()p Fp(E)105 4653 y Ft(2)146 4639 y Fq(\))46 b(and)30 b(all)f(the)i(other)f(con)m(tours)h(ha)m(v)m(e)h (length)e(smaller)f(than)h Fp(k)18 b Fq(log)f Fp(L)p Fq(.)0 4813 y(Then,)30 b(follo)m(wing)e([IS],)j(w)m(e)g(write)879 5076 y Fp(\026)934 5033 y Fo(\014)s(;)p Fn(;)934 5103 y Fo(Q)991 5112 y Fi(L)1044 5003 y Fg(\010)1112 5076 y Fp(N)1185 5090 y Fo(Q)1242 5099 y Fi(L)1320 5076 y Fq(=)25 b Fm(N)1504 5039 y Fo(m)1491 5099 y(L)1573 5003 y Fg(\011)1651 5076 y Fm(\025)g Fp(\026)1802 5033 y Fo(\014)s(;)p Fn(;)1802 5103 y Fo(Q)1859 5112 y Fi(L)1912 5076 y Fq(\()p Fp(E)2019 5039 y Fo(k)2014 5099 y(L)2069 5076 y Fq(\))15 b Fp(\026)2174 5033 y Fo(\014)s(;)p Fn(;)2174 5103 y Fo(Q)2231 5112 y Fi(L)2284 5003 y Fg(\000)2341 5076 y Fp(N)2414 5090 y Fo(Q)2471 5099 y Fi(L)2549 5076 y Fq(=)25 b Fm(N)2733 5039 y Fo(m)2720 5099 y(L)2817 5076 y Fm(j)15 b Fp(E)2929 5039 y Fo(k)2924 5099 y(L)2979 5003 y Fg(\001)3668 5076 y Fq(\(7)p Fp(:)p Fq(10\))0 5340 y(and)30 b(w)m(e)h(estimate)g (separately)f(the)h(t)m(w)m(o)h(factors)f(in)e(the)h(RHS)g(of)h (\(7.10\).)0 5558 y Fh(15)p Fi(=apr)r(ile=)p Fh(1999)f Ft([42])1317 b Fq(7.4)p eop %%Page: 43 43 43 42 bop 0 91 a Fr(Prop)s(osition)34 b(7.4.)53 b Fc(Let)29 b Fp(\014)i(>)25 b(\014)1168 105 y Fo(c)1205 91 y Fc(.)40 b(Then)28 b(w)m(e)h(can)g(c)m(ho)s(ose)h(the)f(constan)m(t)h Fp(k)i Fc(in)27 b(suc)m(h)h(a)h(w)m(a)m(y)h(that)f(there)g(exists)0 218 y(a)i(p)s(ositiv)m(e)e(constan)m(t)j Fp(c)p Fq(\()p Fp(\014)5 b Fq(\))31 b Fc(suc)m(h)f(that,)i(for)e(an)m(y)g Fp(L)h Fc(large)f(enough)1295 445 y Fp(\026)1350 401 y Fo(\014)s(;)p Fn(;)1350 471 y Fo(Q)1407 480 y Fi(L)1460 445 y Fq(\()p Fp(E)1567 407 y Fo(k)1562 467 y(L)1616 445 y Fq(\))c Fm(\025)f Fp(e)1815 407 y Fn(\000)p Fo(L')p Ft(\()p Fo(m)p Ft(\))p Fn(\000)p Fo(c)p Ft(\()p Fo(\014)s Ft(\))2281 353 y Fn(p)p 2341 353 50 4 v 54 x Fo(L)13 b Ft(log)g Fo(L)2580 445 y Fp(:)0 672 y Fk(Pr)-5 b(o)g(of.)35 b Fq(The)e(pro)s(of)f(follo)m(ws)h(w)m(ord)g(b)m(y)g(w)m(ord)g(the)h (pro)s(of)e(of)i(the)g(analogous)f(statemen)m(t)j(discussed)31 b(in)h(Section)0 799 y(3.4)c(of)f([IS],)f(with)g(some)h(mo)s (di\014cations)d(due)i(to)i(the)f(fact)g(that)h(our)e(geometry)i(and)e (our)g(b)s(oundary)e(conditions)0 926 y(are)37 b(di\013eren)m(t)f(from) h(theirs.)58 b(Ho)m(w)m(ev)m(er,)41 b(in)36 b(our)g(con)m(text,)k(only) c(t)m(w)m(o)i(k)m(ey)g(p)s(oin)m(ts)d(of)i(the)g(pro)s(of)f(of)h([IS])g (ha)m(v)m(e)0 1053 y(to)g(b)s(e)f(repro)m(v)m(ed,)i(namely)e(Lemma)h (7.5)g(and)f(Lemma)h(7.6)g(whic)m(h)e(app)s(ear)h(b)s(elo)m(w.)58 b(Once)36 b(these)h(results)f(are)0 1181 y(a)m(v)-5 b(ailable,)36 b(the)f(rest)g(of)g(the)g(pro)s(of)f(is)g(an)h(almost)g(literally)e (transp)s(osition)f(to)k(our)f(case)h(of)f(their)f(argumen)m(ts)0 1308 y(and,)c(in)f(order)h(to)h(a)m(v)m(oid)g(useless)e(rep)s (etitions,)g(it)h(will)e(b)s(e)i(skipp)s(ed.)0 1462 y(The)g(strategy)i (en)m(visaged)e(in)f([IS])h(to)i(pro)m(v)m(e)f(Prop)s(osition)d(7.4)k (can)e(b)s(e)g(explained)f(in)g(simple)f(terms:)66 1617 y(\(1\))46 b(One)29 b(\014rst)g(sho)m(ws)g(\(see)h(lemma)f(7.5)h(b)s (elo)m(w\))f(that)h(with)e(probabilit)m(y)e(larger)j(than)g Fp(e)3150 1584 y Fn(\000)p Fo(L')p Ft(\()p Fo(m)p Ft(\))p Fn(\000)p Fo(c)p Ft(\()p Fo(\014)s Ft(\))3616 1530 y Fn(p)p 3677 1530 V 3677 1584 a Fo(L)12 b Ft(log)i Fo(L)227 1744 y Fq(there)31 b(exists)f Fp(\025)25 b Fm(2)g(B)938 1701 y Fn(\003)p Fo(;lb)935 1771 y(L)1086 1744 y Fq(suc)m(h)31 b(that)g Fp(\025)25 b Fm(\032)1678 1721 y Fg(e)1663 1744 y Fp(T)1729 1758 y Fo(L)1813 1744 y Fq(and)30 b Fp(\025)2043 1711 y Fn(\003)2115 1744 y Fq(is)g(FK{empt)m(y)-8 b(.)66 1872 y(\(2\))46 b(Then)36 b(one)h(pro)m(v)m(es)g(that)g(with)e(large)i (probabilit)m(y)d(there)j(exist)f(t)m(w)m(o)i(FK-clusters)e Fp(\025)3234 1886 y Fo(i)3264 1872 y Fp(;)15 b(\025)3357 1886 y Fo(e)3432 1872 y Fm(2)36 b Fp(B)3603 1839 y Fo(lb)3598 1897 y(L)3662 1872 y Fq(,)i(suc)m(h)227 2000 y(that)31 b Fp(\025)477 2014 y Fo(i)533 2000 y Fm(2)25 b Fp(T)685 1967 y Fo(i)672 2025 y(L)756 2000 y Fq(and)30 b Fp(\025)986 2014 y Fo(e)1051 2000 y Fm(2)24 b Fp(T)1202 1967 y Fo(e)1189 2025 y(L)1244 2000 y Fq(.)0 2154 y(When)42 b(b)s(oth)g(ev)m(en)m(ts)h (in)f(\(1\))h(and)f(\(2\))h(happ)s(en)e(then)h Fp(\025)2020 2168 y Fo(i)2092 2154 y Fq(and)g Fp(\025)2334 2168 y Fo(e)2415 2154 y Fq(b)s(elong)g(to)h(di\013eren)m(t)f Fp(F)13 b(K)7 b Fq({clusters)42 b(and)0 2281 y(therefore)28 b(the)f(asso)s(ciated)h(Ising)e(spins)f(will)f(ha)m(v)m(e)k(opp)s (osite)f(sign)f(with)g(probabilit)m(y)e(1)p Fp(=)p Fq(2.)41 b(Th)m(us,)27 b(conditioned)0 2408 y(to)j(the)f(existence)g(of)h(a)f (set)h Fp(\025)f Fq(as)g(in)e(\(1\),)k(with)d(large)h(probabilit)m(y)d (there)k(will)c(b)s(e)i(a)i(P)m(eierls)e(con)m(tour)i Fp(\015)g Fm(2)25 b Fp(B)3782 2364 y Fn(\003)p Fo(;lb)3777 2435 y(L)0 2535 y Fq(with)k Fp(\015)h Fm(\032)25 b Fp(T)433 2549 y Fo(L)488 2535 y Fq(.)41 b(W)-8 b(e)31 b(are)g(no)m(w)f(ready)h (to)g(state)g(our)f(\014rst)g(result.)40 b(Let)30 b Fm(C)2484 2549 y Fo(L)2569 2535 y Fq(b)s(e)g(the)g(ev)m(en)m(t)758 2762 y Fm(C)806 2776 y Fo(L)885 2762 y Fq(=)25 b Fm(f)31 b(9)p Fp(\025)25 b Fm(2)g(B)1335 2718 y Fn(\003)p Fo(;lb)1332 2789 y(L)1483 2762 y Fq(suc)m(h)30 b(that)h Fp(\025)25 b Fm(\032)2075 2739 y Fg(e)2059 2762 y Fp(T)2125 2776 y Fo(L)2210 2762 y Fq(and)30 b Fp(\025)2440 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Fq(de\014ne)g(the)g(three)h(mappings)d Fp(#)2057 3752 y Ft(v)n(ert)2187 3738 y Fp(;)31 b(#)2297 3752 y Ft(hor)2431 3738 y Fq(and)25 b Fp(#)2657 3752 y Ft(diag)2821 3738 y Fq(to)h(b)s(e)e(the)i(re\015ections)f(around)0 3865 y(the)34 b(lines)e(in)g Ff(R)540 3832 y Ft(2)620 3865 y Fq(giv)m(en)i(b)m(y)f Fp(x)e Fq(=)f Fm(\000)1256 3829 y Ft(1)p 1256 3844 37 4 v 1256 3896 a(2)1303 3865 y Fq(,)35 b Fp(y)e Fq(=)e Fm(\000)1625 3829 y Ft(1)p 1625 3844 V 1625 3896 a(2)1705 3865 y Fq(and)i Fp(y)h Fq(=)c Fm(\000)p Fp(x)22 b Fm(\000)g Fq(1)35 b(resp)s(ectiv)m(ely)-8 b(.)50 b(F)-8 b(or)34 b(simplicit)m(y)d(w)m(e)j(denote)0 3992 y(with)29 b(the)i(same)g(sym)m(b)s(ol)e(the)i(induced)e(mappings)g(on)h (the)h(con\014guration)f(space)h(\012)2952 3959 y Fo(F)10 b(K)3103 3992 y Fq(=)26 b Fm(f)p Fq(0)p Fp(;)15 b Fq(1)p Fm(g)3420 3959 y Fn(E)3459 3980 y Fa(Z)3488 3967 y Fh(2)3566 3992 y Fq(giv)m(en)31 b(b)m(y)0 4119 y Fp(#)54 4133 y Ft(v)n(ert)184 4119 y Fq(\()p Fp(!)s Fq(\)\([)p Fp(x;)15 b(y)s Fq(]\))27 b(=)e Fp(!)s Fq(\([)p Fp(#)871 4133 y Ft(v)n(ert)1001 4119 y Fq(\()p Fp(x)p Fq(\))p Fp(;)15 b(#)1217 4133 y Ft(v)n(ert)1348 4119 y Fq(\()p Fp(y)s Fq(\)]\))31 b(and)f(similarily)c(for)k(the)h(others.)227 4247 y(Let)h(no)m(w)f Fp(Q)653 4203 y Ft(#)653 4274 y Fo(L)744 4247 y Fm(\021)26 b Fp(#)895 4261 y Ft(#)959 4247 y Fq(\()p Fp(Q)1066 4261 y Fo(L)1121 4247 y Fq(\))32 b(where)e(#)d(=)f(v)m(ert)q Fp(;)15 b Fq(hor)p Fp(;)g Fq(diag)q(,)32 b(let)f(\003)c(=)f Fp(Q)2636 4261 y Fo(L)2711 4247 y Fm([)20 b Fp(Q)2864 4214 y Ft(v)n(ert)2864 4272 y Fo(L)3015 4247 y Fm([)g Fp(Q)3168 4214 y Ft(hor)3168 4272 y Fo(L)3298 4247 y Fm([)h Fp(Q)3452 4203 y Ft(diag)3452 4274 y Fo(L)3621 4247 y Fq(and)31 b(let)0 4374 y Fm(G)k Fq(=)29 b Fm(E)237 4388 y Fo(Q)294 4397 y Fi(L)369 4374 y Fm([)21 b(E)499 4393 y Fo(Q)556 4371 y Fh(v)n(ert)556 4417 y Fi(L)695 4374 y Fm([)h(E)826 4396 y Fo(Q)883 4375 y Fh(hor)883 4420 y Fi(L)1003 4374 y Fm([)g(E)1134 4405 y Fo(Q)1191 4375 y Fh(diag)1191 4429 y Fi(L)1315 4374 y Fq(.)48 b(Notice)34 b(that)g Fm(G)k Fq(con)m(tains)33 b(all)f(b)s(onds)f(in)g Fm(E)2880 4388 y Ft(\003)2967 4374 y Fq(min)m(us)g(those)j(connecting)0 4553 y(di\013eren)m(t)28 b Fp(Q)429 4510 y Ft(#)429 4580 y Fo(L)493 4553 y Fq('s.)41 b(So,)29 b(if)f Fp(A)h Fq(is)f(a)h(negativ)m(e)h(ev)m(en)m(t)g(in)e Fm(f)p Fq(0)p Fp(;)15 b Fq(1)p Fm(g)2033 4519 y Fn(E)2072 4528 y Fi(Q)2121 4543 y(L)2181 4553 y Fq(,)29 b(thanks)f(to)i(the)f (monotonicit)m(y)g(prop)s(ert)m(y)f(\(7.2\))0 4680 y(and)i(the)g (symmetry)g(of)h(the)f(FK{measures)h(w.r.t.)41 b(rotations)30 b(of)h(m)m(ultiples)d(of)i Fp(\031)s(=)p Fq(2,)i(w)m(e)f(ha)m(v)m(e)727 4941 y Fp(\026)782 4898 y Fo(F)10 b(K)q(;\014)s(;)p Fn(;)782 4968 y Fo(Q)839 4977 y Fi(L)1029 4941 y Fq(\()p Fp(A)p Fq(\))26 b(=)f Fp(\026)1344 4898 y Fo(F)10 b(K)q(;\014)s(;)p Fn(;)1344 4968 y(G)1591 4868 y Fg(\000)1632 4941 y Fp(A)21 b Fm(\\)f Fp(#)1856 4955 y Ft(v)n(ert)1985 4941 y Fq(\()p Fp(A)p Fq(\))h Fm(\\)f Fp(#)2279 4955 y Ft(hor)2388 4941 y Fq(\()p Fp(A)p Fq(\))h 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Ft([43])1317 b Fq(7.5)p eop %%Page: 44 44 44 43 bop 0 91 a Fq(W)-8 b(e)40 b(no)m(w)e(apply)f(the)i(ab)s(o)m(v)m (e)h(result)d(in)h(order)g(to)h(get)h(a)f(lo)m(w)m(er)f(b)s(ound)f(on)h Fp(\026)2759 48 y Fo(F)10 b(K)q(;\014)s(;)p Fn(;)2759 118 y Fo(Q)2816 127 y Fi(L)3006 91 y Fq(\()p Fm(C)3089 105 y Fo(L)3144 91 y Fq(\).)65 b(Remem)m(b)s(er)38 b(that)0 218 y(the)f(unique)e(W)-8 b(ul\013)36 b(curv)m(e)h Fp(\015)1017 185 y Fo(w)1012 243 y(L)1112 218 y Fm(2)e(D)1281 185 y Fo(lb)1278 243 y(L)1378 218 y Fq(is)g(a)i(quarter)g(of)g(the)g(W)-8 b(ul\013)36 b(curv)m(e)k(^)-48 b Fp(\015)2710 185 y Fo(w)2705 243 y(L)2806 218 y Fq(in)35 b(\003,)k(cen)m(tered)f(at)f(the)g(p)s(oin) m(t)0 345 y(\()p Fm(\000)117 309 y Ft(1)p 117 324 37 4 v 117 377 a(2)164 345 y Fp(;)15 b Fm(\000)286 309 y Ft(1)p 286 324 V 286 377 a(2)333 345 y Fq(\))31 b(and)f(enclosing)f(an) h(area)i(equal)d(to)j(2)p Fp(L)1747 312 y Ft(2)1788 345 y Fq(\()p Fp(m)1903 312 y Fn(\003)1965 345 y Fm(\000)20 b Fp(m)p Fq(\))p Fp(=m)2296 312 y Fn(\003)2338 345 y Fq(.)41 b(Therefore,)30 b(letting)1314 524 y Fg(e)1299 547 y Fp(T)1365 561 y Ft(\003)1444 547 y Fq(=)25 b Fm(f)p Fp(x)h Fm(2)f Fq(\003)g(:)41 b Fp(d)p Fq(\()p Fp(x;)19 b Fq(^)-49 b Fp(\015)2129 509 y Fo(w)2124 569 y(L)2189 547 y Fq(\))25 b Fm(\024)g Fp(w)2410 561 y Fo(L)2465 547 y Fp(=)p Fq(2)p Fm(g)0 749 y Fq(w)m(e)31 b(ha)m(v)m(e)g(that)g(the) g(ev)m(en)m(t)52 951 y Fm(C)100 965 y Ft(\003)180 951 y Fq(=)25 b Fm(f)15 b(9)30 b Fq(a)h(lo)s(op)e Fp(\025)i Fq(of)f(dual)f(b)s(onds,)g(inside)1645 928 y Fg(e)1630 951 y Fp(T)1696 965 y Ft(\003)1780 951 y Fq(around)h(the)g(origin,)f (suc)m(h)i(that)f Fp(\025)2983 918 y Fn(\003)3056 951 y Fq(is)f(FK-empt)m(y)16 b Fm(g)53 b Fq(\(7)p Fp(:)p Fq(13\))0 1152 y(is)30 b(clearly)f(con)m(tained)i(in)e(the)i(ev)m(en)m (t)h Fm(C)1345 1166 y Fo(L)1420 1152 y Fm(\\)20 b Fp(#)1555 1166 y Ft(v)n(ert)1684 1152 y Fq(\()p Fm(C)1767 1166 y Fo(L)1822 1152 y Fq(\))h Fm(\\)f Fp(#)2013 1166 y Ft(hor)2122 1152 y Fq(\()p Fm(C)2205 1166 y Fo(L)2259 1152 y Fq(\))h Fm(\\)f Fp(#)2450 1166 y Ft(diag)2588 1152 y Fq(\()p Fm(C)2671 1166 y Fo(L)2726 1152 y Fq(\).)42 b(Th)m(us,)29 b(thanks)h(to)i(\(7.12\))h(and)0 1279 y(using)c(Lemma)h(3.3.1)j(of)d ([IS],)g(w)m(e)h(get)893 1481 y Fp(\026)948 1437 y Fo(F)10 b(K)q(;\014)s(;)p Fn(;)948 1508 y Fo(Q)1005 1517 y Fi(L)1194 1481 y Fq(\()p Fm(C)1277 1495 y Fo(L)1332 1481 y Fq(\))26 b Fm(\025)f Fp(\026)1544 1437 y Fo(F)10 b(K)q(;\014)s(;w)1544 1508 y Ft(\003)1807 1481 y Fq(\()p Fm(C)1890 1495 y Ft(\003)1945 1481 y Fq(\))1980 1444 y Ft(1)p Fo(=)p Ft(4)2120 1481 y Fm(\025)25 b Fp(e)2258 1444 y Fn(\000)p Fo(L')p Ft(\()p Fo(m)p Ft(\))p Fn(\000)p Fo(c)p Ft(\()p Fo(\014)s Ft(\))2724 1389 y Fn(p)p 2784 1389 50 4 v 55 x Fo(L)13 b Ft(log)g Fo(L)3668 1481 y Fq(\(7)p Fp(:)p Fq(14\))0 1683 y(for)30 b(a)h(suitable)e(constan)m(t)i Fp(c)p Fq(\()p Fp(\014)5 b Fq(\))32 b(and)e(an)m(y)h Fp(L)f Fq(large)g(enough.)91 b Fd(\003)0 1905 y Fq(The)29 b(second)g(estimate)i(that)f(w)m(e)g(need) f(is)f(a)i(b)s(ound)e(on)h(the)h(dual)e(FK)h(connectivit)m(y)h (function)e(and)h(it)g(is)f(a)i(k)m(ey)0 2032 y(ingredien)m(t)37 b(to)i(carry)f(out)g(the)g(second)h(step)f(of)g(the)g(argumen)m(t)h(of) f([IS])g(\(see)h(the)f(estimate)h(ab)s(o)m(v)m(e)h(form)m(ula)0 2158 y(3.4.5)32 b(there\).)41 b(Consider)29 b(the)i(ev)m(en)m(ts)488 2360 y Fp(F)559 2323 y Fn(\017)546 2383 y Fo(L)626 2360 y Fq(=)25 b Fm(f)31 b(9)p Fp(\025)25 b Fm(2)g(B)1076 2327 y Fo(lb)1073 2385 y(L)1165 2360 y Fq(suc)m(h)30 b(that)h Fp(\025)26 b Fm(\032)f Fp(T)1808 2327 y Fn(\017)1795 2385 y Fo(L)1879 2360 y Fq(and)30 b Fp(\025)g Fq(is)g(FK{o)s(ccupied)g Fm(g)182 b(\017)26 b(2)f(f)p Fp(i;)15 b(e)p Fm(g)g Fp(:)0 2562 y Fq(Then)29 b(w)m(e)i(ha)m(v)m(e)h(the)e(follo)m(wing)f(result:)0 2737 y Fr(Lemma)k(7.6.)53 b Fc(Let)30 b Fp(\014)h(>)25 b(\014)977 2751 y Fo(c)1014 2737 y Fc(.)41 b(Then)930 2939 y Fq(lim)896 2997 y Fo(L)p Fn(!1)1106 2939 y Fp(\026)1161 2895 y Fo(F)10 b(K)q(;\014)s(;)p Fn(;)1161 2965 y Fo(Q)1218 2974 y Fi(L)1408 2939 y Fq(\()p Fp(F)1514 2901 y Fo(i)1501 2961 y(L)1556 2939 y Fq(\))26 b(=)f(1)91 b Fc(and)140 b Fq(lim)2102 2997 y Fo(L)p Fn(!1)2311 2939 y Fp(\026)2366 2895 y Fo(F)10 b(K)q(;\014)s(;)p Fn(;)2366 2965 y Fo(Q)2423 2974 y Fi(L)2613 2939 y Fq(\()p Fp(F)2719 2901 y Fo(e)2706 2961 y(L)2761 2939 y Fq(\))26 b(=)f(1)15 b Fp(:)0 3194 y Fk(Pr)-5 b(o)g(of.)29 b Fq(W)-8 b(e)29 b(consider)e(the)h(ev)m(en)m (t)h Fp(T)1230 3161 y Fo(i)1217 3219 y(L)1272 3194 y Fq(,)f(since)f(the)h(pro)s(of)f(for)h Fp(T)2137 3161 y Fo(e)2124 3219 y(L)2206 3194 y Fq(is)f(iden)m(tical.)38 b(Let)29 b Fp(Q)2928 3161 y Fn(\003)2928 3219 y Fo(L)p Fn(\000)p Ft(1)3100 3194 y Fq(=)c Fp(Q)3268 3208 y Fo(L)p Fn(\000)p Ft(1)3430 3194 y Fq(+)15 b(\(1)p Fp(=)p Fq(2)p Fp(;)g Fq(1)p Fp(=)p Fq(2\))0 3321 y(in)25 b(suc)m(h)h(a)h(w)m(a)m(y)g (that)g(\()p Fm(E)831 3335 y Fo(Q)888 3344 y Fi(L)941 3321 y Fq(\))976 3288 y Fn(\003)1044 3321 y Fq(=)e Fm(E)1188 3335 y Fo(Q)1245 3312 y Fj(\003)1245 3358 y Fi(L)p Fj(\000)p Fh(1)1378 3321 y Fq(.)39 b(Thanks)25 b(to)i(the)g(dualit)m(y)e (relationships)e(\(7.1\),)30 b(it)25 b(is)h(enough)g(to)h(pro)m(v)m(e)0 3447 y(that)1416 3574 y(lim)1381 3633 y Fo(L)p Fn(!1)1591 3574 y Fp(\026)1646 3531 y Fo(F)10 b(K)q(;\014)1830 3503 y Fj(\003)1868 3531 y Fo(;w)1646 3601 y(Q)1703 3579 y Fj(\003)1703 3625 y Fi(L)p Fj(\000)p Fh(1)1948 3574 y Fq(\(\(\()p Fp(F)2124 3537 y Fo(i)2111 3597 y(L)2167 3574 y Fq(\))2202 3537 y Fo(c)2239 3574 y Fq(\))2274 3537 y Fn(\003)2317 3574 y Fq(\))25 b(=)g(0)0 3802 y(where)20 b(the)g(ev)m(en)m(t)i(\(\()p Fp(F)770 3769 y Fo(i)757 3827 y(L)813 3802 y Fq(\))848 3769 y Fo(c)885 3802 y Fq(\))920 3769 y Fn(\003)982 3802 y Fq(is)e(the)g(dual)f(of)i(the)f (complemen)m(t)h(of)f Fp(F)2304 3769 y Fo(i)2291 3827 y(L)2346 3802 y Fq(,)i(and)e(it)g(ob)m(viously)f(implies)e(the)k(o)s (ccurrence)0 3929 y(of)31 b(the)f(ev)m(en)m(t)i Fp(D)575 3943 y Fo(L)660 3929 y Fq(giv)m(en)e(b)m(y)739 4200 y Fp(D)814 4214 y Fo(L)894 4200 y Fq(=)990 4072 y Fg(\032)1084 4139 y Fq(there)h(exists)f Fp(x;)15 b(y)28 b Fm(2)d Fp(Q)1891 4106 y Fn(\003)1891 4164 y Fo(L)p Fn(\000)p Ft(1)2069 4139 y Fq(suc)m(h)30 b(that)h Fp(d)p Fq(\()p Fp(x;)15 b(y)s Fq(\))26 b Fm(\025)f Fp(L)2912 4106 y Ft(1)p Fo(=)p Ft(4)1083 4262 y Fq(and)k(there)i(is)e(an)i(FK{connection)g(b)s(et)m(w) m(een)g Fp(x)f Fq(and)g Fp(y)3037 4072 y Fg(\033)3136 4200 y Fp(:)0 4461 y Fq(Thanks)f(to)i(Theorem)f(7.1)i(w)m(e)f(get)1365 4663 y Fp(\026)1420 4619 y Fo(F)10 b(K)q(;\014)1604 4592 y Fj(\003)1641 4619 y Fo(;w)1420 4689 y(Q)1477 4667 y Fj(\003)1477 4713 y Fi(L)p Fj(\000)p Fh(1)1722 4663 y Fq(\()p Fp(D)1832 4677 y Fo(L)1887 4663 y Fq(\))25 b Fm(\024)g Fp(L)2105 4625 y Ft(2)2146 4663 y Fp(C)7 b(e)2260 4625 y Fn(\000)p Fo(mL)2432 4598 y Fh(1)p Fi(=)p Fh(4)0 4864 y Fq(whic)m(h)29 b(go)s(es)i(to)g(zero)g(as)g Fp(L)25 b Fm(!)g(1)p Fq(.)91 b Fd(\003)0 5086 y Fq(W)-8 b(e)39 b(are)e(left)h(with)e(the)h(problem)f(of)i(pro)m(ving)e(a)i(lo)m(w)m (er)g(b)s(ound)d(on)j(the)f(last)h(factor)g(in)e(the)i(RHS)f(of)g (\(7.10\),)0 5213 y Fp(\026)55 5169 y Fo(\014)s(;)p Fn(;)55 5240 y Fo(Q)112 5249 y Fi(L)165 5139 y Fg(\010)233 5213 y Fp(N)306 5227 y Fo(Q)363 5236 y Fi(L)441 5213 y Fq(=)25 b Fm(N)625 5180 y Fo(m)612 5238 y(L)709 5213 y Fm(j)15 b Fp(E)821 5180 y Fo(k)816 5238 y(L)871 5139 y Fg(\011)924 5213 y Fq(,)30 b(whic)m(h)f(do)s(es)h(not)g(sp)s(oil)e(the)i(go)s(o)s (d)g(lo)m(w)m(er)h(b)s(ound)d(w)m(e)i(already)g(ha)m(v)m(e)h(on)f Fp(\026)3560 5169 y Fo(\014)s(;)p Fn(;)3560 5240 y Fo(Q)3617 5249 y Fi(L)3670 5139 y Fg(\000)3712 5213 y Fp(E)3784 5180 y Fo(k)3779 5238 y(L)3833 5139 y Fg(\001)3875 5213 y Fq(.)0 5340 y(As)g(in)f([IS])i(w)m(e)f(ha)m(v)m(e)0 5558 y Fh(15)p Fi(=apr)r(ile=)p Fh(1999)g Ft([44])1317 b Fq(7.6)p eop %%Page: 45 45 45 44 bop 0 91 a Fr(Prop)s(osition)36 b(7.7.)53 b Fc(Let)31 b Fp(\014)f(>)25 b(\014)1171 105 y Fo(c)1239 91 y Fc(and)30 b Fp(k)j Fc(b)s(e)d(\014xed.)40 b(Then)30 b(there)g(exists)g(a)h (constan)m(t)h Fp(c)p Fq(\()p Fp(\014)5 b(;)15 b(k)s Fq(\))32 b Fc(suc)m(h)e(that)1103 324 y Fp(\026)1158 280 y Fo(\014)s(;)p Fn(;)1158 350 y Fo(Q)1215 359 y Fi(L)1267 324 y Fq(\()15 b Fp(N)1390 338 y Fo(Q)1447 347 y Fi(L)1526 324 y Fq(=)25 b Fm(N)1710 286 y Fo(m)1697 346 y(L)1794 324 y Fm(j)15 b Fp(E)1906 286 y Fo(k)1901 346 y(L)1971 324 y Fq(\))26 b Fm(\025)f Fp(e)2170 286 y Fn(\000)p Fo(c)p Ft(\()p Fo(\014)s(;k)r Ft(\))2423 232 y Fn(p)p 2482 232 50 4 v 2482 286 a Fo(L)12 b Ft(\(log)i Fo(L)p Ft(\))2757 259 y Fh(2)0 556 y Fc(for)30 b(an)m(y)h Fp(L)f Fc(large)h(enough.)0 738 y Fk(Pr)-5 b(o)g(of.)40 b Fq(De\014ne)e(a)h (new)f(scale)g(for)g(large)h(con)m(tours)g(as)f Fp(s)p Fq(\()p Fp(L)p Fq(\))h(=)f Fp(k)18 b Fq(log)f Fp(L)p Fq(.)64 b(F)-8 b(or)39 b(an)m(y)g(con)m(tour)g Fp(\015)44 b Fm(2)38 b Fp(B)3569 695 y Fn(\003)p Fo(;lb)3564 765 y(L)3725 738 y Fq(suc)m(h)0 866 y(that)g Fp(\015)k Fm(\032)37 b Fp(T)454 880 y Fo(L)508 866 y Fq(,)i(w)m(e)f(set)g(\003)f(=)g Fp(A)1138 880 y Fo(\015)1210 866 y Fm([)24 b Fp(B)1364 880 y Fo(\015)1448 866 y Fq(as)38 b(explained)d(in)h(Section)h(6.1)i (and)e(de\014ne)f(the)i(sets)g Fp(A)3442 833 y Fn(\016)3442 888 y Fo(\015)3488 866 y Fq(,)i Fp(B)3627 833 y Fn(\016)3622 888 y Fo(\015)3706 866 y Fq(as)d(in)0 993 y(\(6.7\).)42 b(W)-8 b(e)28 b(can)g(write)f(the)g(ev)m(en)m(t)i Fp(E)1256 1007 y Fo(\015)1330 993 y Fq(\(remem)m(b)s(er)f(\(6.8\)\))h(as)f(the)f (disjoin)m(t)f(union)g Fp(E)2937 1007 y Fo(\015)3009 993 y Fq(=)f Fp(E)3177 960 y Ft(+)3172 1015 y Fo(\015)3252 993 y Fm([)14 b Fp(E)3399 960 y Fn(\000)3394 1015 y Fo(\015)3461 993 y Fq(,)28 b(where)f(the)0 1120 y(sup)s(erscript)h(+)j(means)g(that) h Fp(\033)s Fq(\()p Fp(x)p Fq(\))27 b(=)f(+1)31 b(for)g(all)f Fp(x)d Fm(2)f Fq(\001)2005 1087 y Fo(s)2005 1142 y(\015)2072 1120 y Fm(\\)20 b Fp(B)2222 1134 y Fo(\015)2300 1120 y Fq(and)30 b Fp(\033)s Fq(\()p Fp(x)p Fq(\))d(=)g Fm(\000)p Fq(1)k(for)g(all)e Fp(x)e Fm(2)f Fq(\001)3434 1087 y Fo(s)3434 1142 y(\015)3501 1120 y Fm(\\)20 b Fp(A)3650 1134 y Fo(\015)3697 1120 y Fq(,)32 b(and)0 1247 y(vicev)m(ersa)f(for)f Fp(E)598 1214 y Fn(\000)593 1270 y Fo(\015)660 1247 y Fq(.)41 b(Th)m(us)29 b(w)m(e)i(ha)m(v)m(e)115 1544 y Fp(\026)170 1500 y Fo(\014)s(;)p Fn(;)170 1571 y Fo(Q)227 1580 y Fi(L)279 1544 y Fq(\()15 b Fp(N)402 1558 y Fo(Q)459 1567 y Fi(L)538 1544 y Fq(=)25 b Fm(N)722 1506 y Fo(m)709 1566 y(L)806 1544 y Fm(j)15 b Fp(E)918 1506 y Fo(k)913 1566 y(L)983 1544 y Fq(\))26 b Fm(\025)109 b Fq(inf)1150 1621 y Fo(\015)t Fn(2)p Fo(B)1299 1591 y Fj(\003)p Fi(;lb)1296 1645 y(L)1185 1700 y Fo(\015)t Fn(\032)p Fo(T)1327 1709 y Fi(L)1435 1544 y Fp(\026)1490 1500 y Fo(\014)s(;)p Fn(;)1490 1571 y Fo(Q)1547 1580 y Fi(L)1600 1544 y Fq(\()15 b Fp(N)1723 1558 y Fo(Q)1780 1567 y Fi(L)1858 1544 y Fq(=)25 b Fm(N)2042 1506 y Fo(m)2029 1566 y(L)2126 1544 y Fm(j)15 b Fp(E)2233 1558 y Fo(\015)2296 1544 y Fq(\))25 b(=)2463 1483 y(1)p 2463 1523 46 4 v 2463 1606 a(2)2620 1544 y(inf)2546 1621 y Fo(\015)t Fn(2)p Fo(B)2695 1591 y Fj(\003)p Fi(;lb)2692 1645 y(L)2580 1700 y Fo(\015)t Fn(\032)p Fo(T)2722 1709 y Fi(L)2830 1544 y Fp(\026)2885 1500 y Fo(\014)s(;)p Fn(;)2885 1571 y Fo(Q)2942 1580 y Fi(L)2995 1544 y Fq(\()15 b Fp(N)3118 1558 y Fo(Q)3175 1567 y Fi(L)3253 1544 y Fq(=)25 b Fm(N)3437 1506 y Fo(m)3424 1566 y(L)3521 1544 y Fm(j)15 b Fp(E)3633 1506 y Ft(+)3628 1566 y Fo(\015)3710 1544 y Fq(\))g Fp(:)0 1940 y Fq(By)31 b(prop)s(ert)m(y)e(\()p Fp(E)616 1954 y Ft(1)658 1940 y Fq(\))h(in)f(the)i(de\014nition)d(of)j Fp(E)1566 1907 y Fo(k)1561 1965 y(L)1645 1940 y Fq(w)m(e)g(kno)m(w)g(that)g Fm(j)p Fp(\015)5 b Fm(j)25 b(\024)g Fp(k)s(L)15 b Fq(log)i Fp(L)p Fq(.)41 b(Therefore)1599 2173 y Fm(j)p Fq(\001)1700 2135 y Fo(s)1700 2195 y(\015)1747 2173 y Fm(j)25 b(\024)g Fq(3)p Fp(k)s(L)15 b Fq(log)j Fp(L)d(:)1367 b Fq(\(7)p Fp(:)p Fq(15\))0 2406 y(F)-8 b(urthermore,)27 b(since)e Fp(\015)31 b Fm(\032)25 b Fp(T)994 2420 y Fo(L)1074 2406 y Fq(and)g(since)h Fp(A)1533 2373 y Fn(\016)1533 2429 y Fo(\015)1575 2407 y Fi(w)1571 2452 y(L)1656 2406 y Fq(=)f Fp(L)1814 2373 y Ft(2)1855 2406 y Fq(\(1)11 b Fm(\000)g Fp(m=m)2233 2373 y Fn(\003)2276 2406 y Fq(\))p Fp(=)p Fq(2)27 b(and)f Fp(B)2675 2373 y Fn(\016)2670 2429 y Fo(\015)2712 2407 y Fi(w)2708 2452 y(L)2793 2406 y Fq(=)f Fp(L)2951 2373 y Ft(2)2992 2406 y Fq(\(1)11 b(+)g Fp(m=m)3370 2373 y Fn(\003)3413 2406 y Fq(\))p Fp(=)p Fq(2,)28 b(w)m(e)f(ha)m(v)m(e)0 2533 y(that,)k(for)f(constan)m (t)i Fp(C)793 2547 y Ft(1)834 2533 y Fq(,)1208 2588 y Fg(\014)1208 2643 y(\014)1254 2665 y Fm(j)p Fp(A)1347 2628 y Fn(\016)1347 2688 y Fo(\015)1394 2665 y Fm(j)20 b(\000)g Fp(L)1592 2628 y Ft(2)1633 2665 y Fq(\(1)h Fm(\000)f Fp(m=m)2030 2628 y Fn(\003)2072 2665 y Fq(\))p Fp(=)p Fq(2)2212 2588 y Fg(\014)2212 2643 y(\014)2269 2665 y Fm(\024)25 b Fp(C)2430 2679 y Ft(1)2471 2665 y Fp(Lw)2598 2679 y Fo(L)1208 2743 y Fg(\014)1208 2797 y(\014)1253 2820 y Fm(j)p Fp(B)1352 2782 y Fn(\016)1347 2842 y Fo(\015)1394 2820 y Fm(j)20 b(\000)g Fp(L)1592 2782 y Ft(2)1633 2820 y Fq(\(1)h(+)f Fp(m=m)2030 2782 y Fn(\003)2072 2820 y Fq(\))p Fp(=)p Fq(2)2212 2743 y Fg(\014)2212 2797 y(\014)2269 2820 y Fm(\024)25 b Fp(C)2430 2834 y Ft(1)2471 2820 y Fp(Lw)2598 2834 y Fo(L)2667 2820 y Fp(:)3668 2744 y Fq(\(7)p Fp(:)p Fq(16\))0 3021 y(Let,)30 b(for)e(an)m(y)h(\014nite)f(v)m(olume)h Fp(V)20 b Fq(,)29 b Fm(N)1252 2988 y Fo(m)1239 3046 y(V)1346 3021 y Fq(=)c Fm(b)p Fq(\(1)18 b(+)f Fp(m)p Fq(\))p Fm(j)p Fp(V)k Fm(j)p Fp(=)p Fq(2)p Fm(c)p Fq(,)30 b(so)f(that)h Fm(N)2486 2988 y Fo(m)2473 3046 y(L)2580 3021 y Fq(=)25 b Fm(N)2764 2988 y Fo(m)2751 3046 y(Q)2808 3055 y Fi(L)2860 3021 y Fq(.)41 b(By)29 b(\(7.16\))i(and)d(\(7.15\))j(one)0 3149 y(easily)f(sees)g(that,)i(if)d(w)m(e)i(de\014ne)e Fp(#)p Fq(\()p Fp(\015)5 b Fq(\))31 b(b)m(y)1398 3381 y Fm(N)1486 3344 y Fo(m)1473 3404 y(L)1580 3381 y Fq(=)25 b Fm(N)1764 3342 y Fn(\000)p Fo(m)1886 3315 y Fj(\003)1751 3408 y Fo(A)1806 3390 y Fj(\016)1806 3426 y Fi(\015)1949 3381 y Fq(+)19 b Fm(N)2127 3342 y Ft(+)p Fo(m)2248 3315 y Fj(\003)2114 3408 y Fo(B)2172 3390 y Fj(\016)2169 3426 y Fi(\015)2310 3381 y Fq(+)h Fp(#)2455 3344 y Fo(a)2455 3404 y(\015)0 3614 y Fq(then)30 b(w)m(e)h(ha)m(v)m(e)h Fm(j)p Fp(#)630 3581 y Fo(a)630 3637 y(\015)676 3614 y Fm(j)26 b(\024)f Fq(3)p Fp(C)933 3628 y Ft(1)974 3614 y Fp(Lw)1101 3628 y Fo(L)1155 3614 y Fq(.)41 b(Moreo)m(v)m(er,)33 b(w)m(e)d(can)h(write)1463 3847 y Fp(N)1536 3861 y Fo(Q)1593 3870 y Fi(L)1671 3847 y Fq(=)25 b Fp(N)1840 3861 y Fo(A)1895 3842 y Fj(\016)1895 3879 y Fi(\015)1961 3847 y Fq(+)20 b Fp(N)2125 3861 y Fo(B)2183 3842 y Fj(\016)2180 3879 y Fi(\015)2246 3847 y Fq(+)g Fp(#)2391 3809 y Fo(b)2391 3869 y(\015)0 4080 y Fq(where)37 b Fp(#)324 4047 y Fo(b)324 4102 y(\015)407 4080 y Fq(=)f Fp(N)587 4094 y Ft(\001)647 4075 y Fi(s)647 4111 y(\015)730 4080 y Fq(is)g(a)i(n)m(um)m(b)s(er)d (whic)m(h)h(do)s(es)h(not)h(dep)s(end)d(on)i(the)g(sp)s(eci\014c)f (con\014guration)h Fp(\033)j Fm(2)c Fp(E)3656 4047 y Ft(+)3651 4102 y Fo(\015)3754 4080 y Fq(and)0 4207 y(satis\014es)30 b Fp(#)389 4174 y Fo(b)389 4229 y(\015)461 4207 y Fm(\024)24 b(j)p Fq(\001)657 4174 y Fo(s)657 4229 y(\015)704 4207 y Fm(j)i(\024)f Fp(C)916 4221 y Ft(1)956 4207 y Fp(Lw)1083 4221 y Fo(L)1138 4207 y Fq(.)40 b(Therefore,)31 b(b)m(y)f(an)g(ob)m (vious)g(v)-5 b(ariation)30 b(on)g(Prop)s(osition)e(6.1,)k(w)m(e)f(get) 124 4475 y Fp(\026)179 4431 y Fo(\014)s(;)p Fn(;)179 4502 y Fo(Q)236 4511 y Fi(L)288 4475 y Fq(\()15 b Fp(N)411 4489 y Fo(Q)468 4498 y Fi(L)547 4475 y Fq(=)25 b Fm(N)731 4437 y Fo(m)718 4497 y(L)815 4475 y Fm(j)15 b Fp(E)927 4437 y Ft(+)922 4497 y Fo(\015)1003 4475 y Fq(\))26 b(=)f Fp(\026)1215 4431 y Fo(\014)s(;)p Fn(;)1215 4502 y Fo(Q)1272 4511 y Fi(L)1324 4475 y Fq(\()15 b Fp(N)1447 4489 y Fo(A)1502 4470 y Fj(\016)1502 4507 y Fi(\015)1569 4475 y Fq(+)20 b Fp(N)1733 4489 y Fo(B)1791 4470 y Fj(\016)1788 4507 y Fi(\015)1859 4475 y Fq(=)25 b Fm(N)2043 4436 y Fn(\000)p Fo(m)2165 4409 y Fj(\003)2030 4502 y Fo(A)2085 4483 y Fj(\016)2085 4520 y Fi(\015)2227 4475 y Fq(+)20 b Fm(N)2406 4436 y Ft(+)p Fo(m)2527 4409 y Fj(\003)2393 4502 y Fo(B)2451 4483 y Fj(\016)2448 4520 y Fi(\015)2589 4475 y Fq(+)g Fp(#)2734 4437 y Fo(a)2734 4497 y(\015)2801 4475 y Fm(\000)g Fp(#)2946 4437 y Fo(b)2946 4497 y(\015)3007 4475 y Fm(j)c Fp(E)3120 4437 y Ft(+)3115 4497 y Fo(\015)3195 4475 y Fq(\))26 b Fm(\025)1064 4694 y(\025)f Fp(\026)1215 4651 y Fo(\014)s(;)p Fn(;)1318 4623 y Fj(\000)p Fi(;Q)1437 4639 y(L)1215 4721 y Fo(A)1270 4703 y Fj(\016)1270 4739 y Fi(\015)1311 4721 y Fo(;s)1494 4694 y Fq(\()15 b Fp(N)1617 4708 y Fo(A)1672 4690 y Fj(\016)1672 4726 y Fi(\015)1744 4694 y Fq(=)25 b Fm(N)1928 4656 y Fn(\000)p Fo(m)2050 4628 y Fj(\003)1915 4721 y Fo(A)1970 4703 y Fj(\016)1970 4739 y Fi(\015)2112 4694 y Fq(+)20 b Fp(#)2257 4657 y Fo(a)2257 4717 y(\015)2318 4694 y Fq(\))15 b Fp(\026)2423 4651 y Fo(\014)s(;)p Fn(;)2526 4623 y Fh(+)p Fi(;Q)2643 4639 y(L)2423 4721 y Fo(B)2481 4703 y Fj(\016)2478 4739 y Fi(\015)2520 4721 y Fo(;s)2701 4694 y Fq(\()g Fp(N)2824 4708 y Fo(B)2882 4690 y Fj(\016)2879 4726 y Fi(\015)2951 4694 y Fq(=)25 b Fm(N)3135 4657 y Fo(m)3200 4630 y Fj(\003)3122 4717 y Fo(B)3180 4699 y Fj(\016)3177 4735 y Fi(\015)3262 4694 y Fm(\000)20 b Fp(#)3407 4657 y Fo(b)3407 4717 y(\015)3468 4694 y Fq(\))15 b Fp(:)3668 4589 y Fq(\(7)p Fp(:)p Fq(17\))0 4958 y(W)-8 b(e)38 b(no)m(w)g(ha)m(v)m(e)g(to)g(\014nd)d(appropriate)i (lo)m(w)m(er)g(b)s(ounds)e(on)i(b)s(oth)g(factors)h(of)f(the)g(RHS)g (of)g(\(7.17\).)64 b(Since)36 b(the)0 5086 y(treatmen)m(t)27 b(is)c(iden)m(tical)h(\(mo)s(dulo)g(spin{\015ip\),)f(w)m(e)j(consider)d (only)i(the)g(second)g(factor.)40 b(Let)25 b Fp(B)3275 5053 y Fn(\016\016)3270 5108 y Fo(\015)3379 5086 y Fq(=)g Fp(B)3549 5053 y Fn(\016)3544 5108 y Fo(\015)3590 5086 y Fm(n)p Fp(@)3683 5100 y Ft(2)p Fo(s)3759 5086 y Fp(B)3833 5053 y Fn(\016)3828 5108 y Fo(\015)3875 5086 y Fq(.)0 5213 y(The)36 b(b.c.)60 b Fm(;)439 5180 y Ft(+)p Fo(;Q)574 5189 y Fi(L)663 5213 y Fq(is)36 b(of)h(t)m(yp)s(e)g(\()p Fp(s;)15 b Fq(+\),)39 b(so)e(w)m(e)g(pro)s(ceed)f(as)h(in)f(the)h(pro)s (of)f(of)h(Lemma)f(6.2)i(and)e(sho)m(w)h(that)g(to)0 5340 y(\(almost\))32 b(ev)m(ery)g Fp(\033)i Fq(w)m(e)e(can)g(asso)s (ciate)g(a)g Fm(\045)p Fq({path)f Fm(C)5 b Fq(\()p Fp(\033)s Fq(\))32 b(whic)m(h)e(is)h(the)g(most)h(external)f Fm(\045)p Fq({path)g(of)h(+1)f(spins)0 5558 y Fh(15)p Fi(=apr)r(ile=)p Fh(1999)f Ft([45])1317 b Fq(7.7)p eop %%Page: 46 46 46 45 bop 0 91 a Fq(surrounding)20 b Fp(B)573 58 y Fn(\016\016)568 113 y Fo(\015)651 91 y Fq(.)39 b(It)23 b(also)g(clear)g(that)h(this)e (path,)i(b)s(eing)e(the)h(\\most)h(external",)i(can)d(b)s(e)f(written)h (as)g(a)h(disjoin)m(t)0 218 y(union)1535 345 y Fm(C)5 b Fq(\()p Fp(\033)s Fq(\))26 b(=)f Fm(C)1883 359 y Ft(1)1924 345 y Fq(\()p Fp(\033)s Fq(\))c Fm([)f(C)2199 359 y Ft(2)2239 345 y Fq(\()p Fp(\033)s Fq(\))1304 b(\(7)p Fp(:)p Fq(18\))0 536 y(where)30 b Fp(C)328 550 y Ft(1)369 536 y Fq(\()p Fp(\033)s Fq(\))c Fm(\032)f Fq(\001)692 503 y Fo(s)692 559 y(\015)768 536 y Fq(and)30 b Fm(C)993 550 y Ft(2)1034 536 y Fq(\()p Fp(\033)s Fq(\))c Fm(\032)f Fp(@)1329 550 y Ft(2)p Fo(s)1404 536 y Fp(Q)1476 550 y Fo(L)1531 536 y Fq(.)40 b(Therefore)217 773 y Fm(jC)5 b Fq(\()p Fp(\033)s Fq(\))p Fm(j)27 b Fq(=)e Fm(jC)641 787 y Ft(1)682 773 y Fq(\()p Fp(\033)s Fq(\))p Fm(j)c Fq(+)f Fm(jC)1017 787 y Ft(2)1058 773 y Fq(\()p Fp(\033)s Fq(\))p Fm(j)26 b(\024)f(j)p Fq(\001)1431 735 y Fo(s)1431 795 y(g)1474 773 y Fm(j)20 b Fq(+)g Fm(j)p Fp(@)1683 787 y Ft(2)p Fo(s)1758 773 y Fp(Q)1830 787 y Fo(L)1885 773 y Fm(j)25 b(\024)g Fq(3)p Fp(k)s(L)15 b Fq(log)i Fp(L)k Fq(+)e(8)p Fp(k)s(L)c Fq(log)j Fp(L)25 b Fq(=)g(11)p Fp(k)s(L)15 b Fq(log)i Fp(L)e(:)218 b Fq(\(7)p Fp(:)p Fq(19\))0 1009 y(W)-8 b(e)33 b(can)f(also)g(de\014ne)f(the)h(sets)h Fp(X)1194 976 y Fo(i)1225 1009 y Fq(\()p Fp(\033)s Fq(\))f(whic)m(h)f (is)g(the)h(set)g(of)h(all)d(sites)i Fp(x)g Fq(surrounded)d(b)m(y)j Fm(C)5 b Fq(\()p Fp(\033)s Fq(\))33 b(and)e Fp(X)3636 976 y Fo(e)3676 1009 y Fq(\()p Fp(\033)s Fq(\))d(=)0 1136 y Fp(B)74 1103 y Fn(\016)69 1159 y Fo(\015)116 1136 y Fm(n)p Fp(X)243 1103 y Fo(i)274 1136 y Fq(\()p Fp(\033)s Fq(\).)41 b(Then,)30 b(again)g(w)m(e)h(ha)m(v)m(e)h Fp(X)1396 1103 y Fo(e)1435 1136 y Fq(\()p Fp(\033)s Fq(\))26 b Fm(\032)f Fq(\001)1758 1103 y Fo(s)1758 1159 y(\015)1825 1136 y Fm([)20 b Fp(@)1954 1150 y Ft(2)p Fo(s)2029 1136 y Fp(Q)2101 1150 y Fo(L)2156 1136 y Fq(,)30 b(so)396 1373 y Fm(j)p Fp(X)503 1335 y Fo(e)543 1373 y Fq(\()p Fp(\033)s Fq(\))p Fm(j)c(\024)f Fq(11)p Fp(k)s(L)15 b Fq(log)j Fp(L)d(;)197 b Fm(j)p Fp(@)5 b(X)1625 1335 y Fo(e)1665 1373 y Fq(\()p Fp(\033)s Fq(\))p Fm(j)26 b(\024)f Fq(11)p Fp(k)s(L)15 b Fq(log)j Fp(L)d(;)197 b Fm(j)p Fp(@)5 b(X)2747 1335 y Fo(i)2779 1373 y Fq(\()p Fp(\033)s Fq(\))p Fm(j)26 b(\024)f Fq(11)p Fp(k)s(L)15 b Fq(log)j Fp(L)d(:)0 1609 y Fq(The)33 b(idea)g(is)f(then)h(to)h(condition)e(on)h (what)g(happ)s(ens)e(\\outside")j Fm(C)5 b Fq(\()p Fp(\033)s Fq(\),)35 b(and)e(to)h(use)f(the)g(DLR)h(prop)s(ert)m(y)-8 b(.)49 b(In)0 1736 y(this)29 b(w)m(a)m(y)j(w)m(e)e(get)232 2002 y Fp(\026)287 1958 y Fo(\014)s(;)p Fn(;)390 1931 y Fh(+)p Fi(;Q)507 1946 y(L)287 2028 y Fo(B)345 2010 y Fj(\016)342 2047 y Fi(\015)384 2028 y Fo(;s)564 2002 y Fq(\()15 b Fp(N)687 2016 y Fo(B)745 1997 y Fj(\016)742 2033 y Fi(\015)814 2002 y Fq(=)25 b Fm(N)998 1964 y Fo(m)1063 1937 y Fj(\003)985 2024 y Fo(B)1043 2006 y Fj(\016)1040 2042 y Fi(\015)1125 2002 y Fm(\000)20 b Fp(#)1270 1964 y Fo(b)1270 2024 y(\015)1332 2002 y Fq(\))25 b Fm(\025)g Fq(inf)1494 2060 y Fo(X)1559 2042 y Fi(e)1627 2002 y Fq(inf)1614 2056 y Fo(\033)1656 2067 y Fi(X)1710 2054 y(e)1765 2002 y Fp(\026)1820 1958 y Fo(\014)s(;)p Fn(;)1923 1931 y Fh(+)p Fi(;Q)2040 1946 y(L)1820 2028 y Fo(B)1878 2010 y Fj(\016)1875 2047 y Fi(\015)1917 2028 y Fo(;s)2097 2002 y Fq(\()15 b Fp(N)2220 2016 y Fo(B)2278 1997 y Fj(\016)2275 2033 y Fi(\015)2347 2002 y Fq(=)25 b Fm(N)2531 1964 y Fo(m)2596 1937 y Fj(\003)2518 2024 y Fo(B)2576 2006 y Fj(\016)2573 2042 y Fi(\015)2658 2002 y Fm(\000)20 b Fp(#)2803 1964 y Fo(b)2803 2024 y(\015)2865 2002 y Fm(j)15 b Fp(X)2987 1964 y Fo(e)3027 2002 y Fq(\()p Fp(\033)s Fq(\))26 b(=)f Fp(X)3356 1964 y Fo(e)3395 2002 y Fp(;)15 b(\033)3487 2016 y Fo(X)3552 1997 y Fi(e)3607 2002 y Fq(\))26 b(=)161 2242 y(=)15 b(inf)253 2301 y Fo(X)318 2283 y Fi(e)386 2242 y Fq(inf)373 2297 y Fo(\033)415 2308 y Fi(X)469 2295 y(e)524 2242 y Fp(\026)579 2198 y Fo(\014)s(;)p Fn(;)682 2171 y Fh(+)p Fi(;Q)799 2186 y(L)579 2269 y Fo(B)637 2251 y Fj(\016)634 2287 y Fi(\015)676 2269 y Fo(;s)856 2242 y Fq(\()p Fp(N)964 2259 y Fo(X)1029 2241 y Fi(i)1088 2242 y Fq(=)25 b Fm(N)1272 2205 y Fo(m)1337 2177 y Fj(\003)1259 2265 y Fo(B)1317 2246 y Fj(\016)1314 2283 y Fi(\015)1399 2242 y Fm(\000)20 b Fp(#)1544 2205 y Fo(b)1544 2265 y(\015)1611 2242 y Fm(\000)f Fp(N)1774 2256 y Fo(X)1839 2238 y Fi(e)1894 2242 y Fm(j)c Fp(X)2016 2205 y Fo(e)2056 2242 y Fq(\()p Fp(\033)s Fq(\))26 b(=)f Fp(X)2385 2205 y Fo(e)2425 2242 y Fp(;)15 b(\033)2517 2256 y Fo(X)2582 2238 y Fi(e)2621 2242 y Fq(\))26 b Fm(\025)161 2468 y(\025)15 b Fq(inf)253 2527 y Fo(X)318 2509 y Fi(e)386 2468 y Fq(inf)373 2523 y Fo(\033)415 2534 y Fi(X)469 2521 y(e)524 2468 y Fp(\026)579 2425 y Fo(\014)s(;)p Ft(+)579 2500 y Fo(X)644 2482 y Fi(i)672 2500 y Fo(;s)733 2468 y Fq(\()p Fp(N)841 2486 y Fo(X)906 2467 y Fi(i)964 2468 y Fq(=)25 b Fm(N)1148 2431 y Fo(m)1213 2403 y Fj(\003)1135 2491 y Fo(B)1193 2473 y Fj(\016)1190 2509 y Fi(\015)1276 2468 y Fm(\000)20 b Fp(#)1421 2431 y Fo(b)1421 2491 y(\015)1487 2468 y Fm(\000)g Fp(N)1651 2482 y Fo(X)1716 2464 y Fi(e)1756 2468 y Fq(\()p Fp(\033)s Fq(\)\))26 b Fm(\025)161 2695 y(\025)240 b Fq(inf)278 2760 y Fo(X)5 b Fn(\032)p Fo(Q)457 2769 y Fi(L)505 2760 y Ft(:)p Fo(X)g Fn(\033)p Fo(B)706 2732 y Fj(\016\016)703 2778 y Fi(\015)258 2845 y Fn(j)p Fo(@)t(X)g Fn(j\024)p Ft(11)p Fo(k)r(L)10 b Ft(log)k Fo(L)982 2695 y Fq(inf)838 2757 y Fn(j)p Fo(n)p Fn(j\024)p Ft(2)p Fo(C)1072 2766 y Fh(1)1107 2757 y Fo(Lw)1210 2767 y Fi(l)1251 2695 y Fp(\026)1306 2651 y Fo(\014)s(;)p Ft(+)1306 2721 y Fo(X)q(;s)1432 2695 y Fq(\()p Fp(N)1540 2709 y Fo(X)1635 2695 y Fq(=)25 b Fm(N)1819 2657 y Fo(m)1884 2630 y Fj(\003)1806 2717 y Fo(B)1864 2699 y Fj(\016)1861 2735 y Fi(\015)1946 2695 y Fq(+)20 b Fp(n)p Fq(\))25 b Fm(\025)161 3022 y(\025)263 b Fq(inf)258 3094 y Fo(X)5 b Fn(\032)p Fo(Q)437 3103 y Fi(L)485 3094 y Ft(:)p Fn(j)p Fo(X)g Fn(j\025)p Fo(C)724 3103 y Fh(2)758 3094 y Fo(L)808 3066 y Fh(2)281 3168 y Fn(j)p Fo(@)t(X)g Fn(j\024)p Ft(11)p Fo(k)r(L)11 b Ft(log)i Fo(L)1123 3022 y Fq(inf)884 3089 y Fn(j)p Fo(n)p Fn(j\024)p Ft(2)p Fo(C)1118 3098 y Fh(1)1153 3089 y Fo(L)1203 3071 y Fh(5)p Fi(=)p Fh(4)1315 3089 y Ft(log)h Fo(L)1487 3022 y Fp(\026)1542 2978 y Fo(\014)s(;)p Ft(+)1542 3048 y Fo(X)q(;s)1668 3022 y Fq(\()p Fp(N)1776 3036 y Fo(X)1871 3022 y Fq(=)25 b Fm(N)2055 2984 y Fo(m)2120 2957 y Fj(\003)2042 3044 y Fo(B)2100 3026 y Fj(\016)2097 3062 y Fi(\015)2182 3022 y Fq(+)20 b Fp(n)p Fq(\))15 b Fp(:)0 3407 y Fq(A)m(t)47 b(this)d(p)s(oin)m(t)h(w)m(e)h(can)g(use)f (inequalities)e(\(1.1.1\))48 b(and)d(Lemma)h(2.3.3)i(in)c([IS],)i(and)f (the)h(prop)s(osition)d(is)0 3534 y(pro)m(v)m(en.)92 b Fd(\003)0 5558 y Fh(15)p Fi(=apr)r(ile=)p Fh(1999)30 b Ft([46])1317 b Fq(7.8)p eop %%Page: 47 47 47 46 bop 0 91 a Fr(References)63 244 y Fq([A])46 b(D.)30 b(B.)g(Abraham:)f Fk(Surfac)-5 b(e)31 b(structur)-5 b(es)33 b(and)f(phase)h(tr)-5 b(ansitions{Exact)34 b(r)-5 b(esults)p Fq(,)31 b(in)d(Phase)h(T)-8 b(ransitions)227 371 y(and)29 b(Critical)e(Phenomena,)i(V)-8 b(ol.)30 b(10,)g(C.)f(Dom)m(b)h(and)e (J.)h(L.)g(Leb)s(o)m(witz)g(eds,)g(Academic)h(Press,)f(London)227 498 y(1987)-107 626 y([ABSZ])45 b(J.)25 b(E.)g(Avron,)g(H.)h(v)-5 b(an)24 b(Beijeren,)h(L.)g(S.)g(Sc)m(h)m(ulman)f(and)g(R.)h(K.)g(P)-8 b(.)26 b(Zia:)e Fk(R)-5 b(oughening)29 b(tr)-5 b(ansition,)29 b(surfac)-5 b(e)227 753 y(tension)34 b(and)g(e)-5 b(quilibrium)33 b(dr)-5 b(oplet)35 b(shap)-5 b(es)35 b(in)e(a)g(two{dimensiona)j(Ising) d(system)p Fq(.)f(J.)e(Ph)m(ys.)h(A:)g(Math.)227 880 y(Gen.)g Fr(15)p Fq(,)g(L81)g(\(1982\))-51 1008 y([CCS])45 b(J.)37 b(Cha)m(y)m(es,)g(L.)g(Cha)m(y)m(es)g(and)f(R.)g(H.)h(Sc)m (honmann:)f Fk(Exp)-5 b(onential)40 b(de)-5 b(c)g(ay)39 b(of)g(c)-5 b(onne)g(ctivity)39 b(in)f(the)g(two{)227 1135 y(dimensional)d(Ising)d(mo)-5 b(del.)32 b Fq(J.)e(Stat.)i(Ph)m (ys.)e Fr(49)p Fq(,)h(433)h(\(1987\))-140 1263 y([CGMS])46 b(F.)26 b(Cesi,)f(G.)g(Guadagni,)g(F.)h(Martinelli)d(and)h(R.)i(H.)f (Sc)m(honmann:)g Fk(On)i(the)i(two-dimensional)h(sto)-5 b(chastic)227 1390 y(Ising)28 b(mo)-5 b(del)30 b(in)e(the)g(phase)i(c) -5 b(o)g(existenc)g(e)29 b(r)-5 b(e)g(gion)29 b(ne)-5 b(ar)29 b(the)f(critic)-5 b(al)29 b(p)-5 b(oint)p Fq(.)27 b(J.)f(Stat.)g(Ph)m(ys.)f Fr(85)p Fq(,)h(Nos.)g(1/2,)227 1517 y(55{102)33 b(\(1996\))-101 1645 y([CMM])46 b(F.)30 b(Cesi,)f(C.)h(Maes)g(and)g(F.)g(Martinelli:)d Fk(R)-5 b(elaxation)34 b(of)f(disor)-5 b(der)g(e)g(d)35 b(magnets)e(in)e(the)i (Gri\016ths)g(r)-5 b(e)g(gime)p Fq(.)227 1772 y(Comm)m(un.)30 b(Math.)h(Ph)m(ys.)f Fr(188)p Fq(,)i(135{173)h(\(1997\))-59 1900 y([DKS])45 b(R.)39 b(Dobrushin,)d(R.)j(Kotec)m(k)q(\023)-47 b(y)41 b(and)d(S.)g(Shlosman:)f Fk(Wul\013)j(c)-5 b(onstruction.)42 b(A)d(glob)-5 b(al)42 b(shap)-5 b(e)42 b(fr)-5 b(om)41 b(lo)-5 b(c)g(al)227 2027 y(inter)g(action)p Fq(.)32 b(T)-8 b(ranslation)30 b(of)g(Mathematical)h(Monographs,)g(V)-8 b(ol.)31 b(104,)h(AMS,)e(1992)1 2154 y([FK])46 b(C.)31 b(M.)g(F)-8 b(ortuin)30 b(and)g(P)-8 b(.)31 b(W.)g(Kasteleyn,)f(P)-8 b(.W:)32 b Fk(On)g(the)i(r)-5 b(andom)35 b(cluster)e(mo)-5 b(del.)34 b(I.)f(Intr)-5 b(o)g(duction)35 b(and)227 2281 y(r)-5 b(elation)35 b(to)e(other)h(mo)-5 b(dels)p Fq(.)32 b(Ph)m(ysica)e Fr(57)h Fq(536-564)i(\(1972\))-68 2409 y([FK)m(G])47 b(C.)39 b(M.)f(F)-8 b(ortuin,)39 b(P)-8 b(.)38 b(W.)h(Kasteleyn)g(and)f(J.)g(Ginibre:)e Fk(Corr)-5 b(elation)43 b(ine)-5 b(qualities)40 b(on)h(some)g(p)-5 b(artial)5 b(ly)227 2536 y(or)-5 b(der)g(e)g(d)36 b(sets.)30 b Fq(Comm)m(un.)g(Math.)h(Ph)m(ys.)f Fr(22)p Fq(,)h(89)h(\(1971\))98 2664 y([I])46 b(D.)28 b(Io\013e:)h Fk(Exact)h(lar)-5 b(ge)31 b(deviation)g(b)-5 b(ounds)31 b(up)f(to)g Fp(T)2014 2678 y Fo(c)2081 2664 y Fk(for)h(the)f(Ising)g(mo)-5 b(del)32 b(in)e(two)g(dimension)p Fq(.)f(Probab.)227 2791 y(Theory)h(Related)h(Fields,)e Fr(102)p Fq(,)i(313{330)j(\(1995\)) 48 2919 y([IS])45 b(D.)32 b(Io\013e,)f(R.)g(H.)h(Sc)m(honmann:)e Fk(Dobrushin{Kote)-5 b(cky{Shlosman)37 b(the)-5 b(or)g(em)35 b(up)e(to)h(the)f(critic)-5 b(al)34 b(temp)-5 b(er-)227 3046 y(atur)g(e)p Fq(.)32 b(Preprin)m(t)d(\(1997\),)k(to)e(app)s(ear)f (in)f(Comm)m(un.)g(Math.)j(Ph)m(ys.)24 3174 y([LS])45 b(G.)36 b(F.)g(La)m(wler)f(and)g(A.)g(D.)h(Sok)-5 b(al:)35 b(Bounds)g(on)g(the)h Fp(L)2179 3141 y Ft(2)2255 3174 y Fq(sp)s(ectrum)e(for)h(Mark)m(o)m(v)i(c)m(hains)e(and)f(Mark)m(o)m(v) 227 3301 y(pro)s(cesses:)d(A)g(generalization)g(of)g(Cheeger's)g (inequalit)m(y)-8 b(.)30 b(T)-8 b(rans.)31 b(Amer.)g(Math.)h(So)s(c.)f Fr(309)p Fq(,)h(No)f(2,)h(557)227 3428 y(\(1988\))14 3556 y([L)-8 b(Y])46 b(S.)35 b(L.)g(Lu)g(and)f(H.)i(T.Y)-8 b(au:)35 b Fk(Sp)-5 b(e)g(ctr)g(al)40 b(Gap)e(and)g(L)-5 b(o)g(garithmic)39 b(Sob)-5 b(olev)38 b(Ine)-5 b(quality)38 b(for)f(Kawasaki)i(and)227 3683 y(Glaub)-5 b(er)34 b(Dynamics)p Fq(.)d(Comm)m(un.)f(Math.)h(Ph)m(ys.)f Fr(156)p Fq(,)i(399-433)h (\(1993\))3 3811 y([M1])46 b(F.)32 b(Martinelli:)d Fk(L)-5 b(e)g(ctur)g(es)34 b(on)g(Glaub)-5 b(er)34 b(dynamics)h(for)f(discr)-5 b(ete)34 b(spin)g(mo)-5 b(dels)p Fq(.)33 b(Pro)s(ceedings)d(of)i(the)f ('97)227 3938 y(Sain)m(t)26 b(Flour)g(summer)g(sc)m(ho)s(ol)g(in)g (probabilit)m(y)e(theory)-8 b(,)27 b(to)h(app)s(ear)e(in)f(Lecture)i (Notes)h(in)e(Mathematics.)3 4066 y([M2])46 b(F.)31 b(Martinelli:)d(Ka) m(w)m(asaki)j(dynamics)e(at)j(high)d(temp)s(erature)h(revisited,)f(in)g (preparation)-73 4193 y([MOS])45 b(F.)25 b(Martinelli,)c(E.)j(Olivieri) d(and)i(R.H.Sc)m(honmann:)h Fk(F)-7 b(or)27 b(Gibbs)g(state)g(of)g(2D)g (lattic)-5 b(e)27 b(spin)h(systems)f(we)-5 b(ak)227 4320 y(mixing)33 b(implies)g(str)-5 b(ong)34 b(mixing.)d Fq(Comm)m(un.)e (Math.)j(Ph)m(ys.)e Fr(165)p Fq(,)h(33)g(\(1994\))69 4448 y([P])46 b(C.)d(E.)g(P\014ster:)g Fk(L)-5 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3376 4951 l 3376 5176 l 2971 5176 l 2926 5221 l 2926 5401 l 2701 5401 l clp gs col7 1.00 shd ef gr gs col-1 s gr % Polyline n 3376 4276 m 3601 4276 l 3601 4726 l 3376 4726 l clp gs col7 1.00 shd ef gr gs col-1 s gr % Polyline n 1351 1351 m 1351 1576 l 1171 1576 l 1126 1621 l 1126 1801 l 1351 1801 l 1351 2026 l 901 2026 l 901 1801 l 676 1801 l 676 1576 l 1081 1576 l 1126 1531 l 1126 1351 l clp gs col7 1.00 shd ef gr gs col-1 s gr % Polyline n 676 2251 m 901 2251 l 901 2476 l 676 2476 l clp gs col7 1.00 shd ef gr gs col-1 s gr 0.000 slw % Polyline n 3601 5176 m 3826 5176 l 3826 5401 l 3601 5401 l clp gs col7 0.50 shd ef gr 7.500 slw % Polyline n 3826 5176 m 4276 5176 l 4276 5626 l 3826 5626 l clp gs col7 1.00 shd ef gr gs col-1 s gr 0.000 slw % Polyline n 901 2926 m 1126 2926 l 1126 3376 l 901 3376 l clp gs col7 0.50 shd ef gr 7.500 slw % Polyline n 451 2926 m 901 2926 l 901 3376 l 451 3376 l clp gs col7 1.00 shd ef gr gs col-1 s gr 45.000 slw % Polyline n 676 676 m 2026 676 l 2026 901 l 2476 901 l 2476 1351 l 3151 1351 l 3151 2026 l 3601 2026 l 3601 1801 l 4051 1801 l 4051 2926 l 4501 2926 l 4501 3826 l 3826 3826 l 3826 4051 l 4051 4051 l 4051 4276 l 4276 4276 l 4276 4501 l 4726 4501 l 4726 5176 l 4276 5176 l 4276 5626 l 3151 5626 l 3151 5401 l 2701 5401 l 2701 5176 l 1126 5176 l 1126 4726 l 451 4726 l 451 2926 l 676 2926 l 676 901 l clp gs col-1 s gr 7.500 slw % Polyline n 2250 450 m 2025 450 l 450 450 l 450 2700 l 225 2700 l 225 4950 l 900 4950 l 900 5400 l 2475 5400 l 2475 5625 l 2925 5625 l 2925 5850 l 4500 5850 l 4500 5625 l 4500 5400 l 4950 5400 l 4950 4275 l 4500 4275 l 4500 4050 l 4725 4050 l 4725 2700 l 4275 2700 l 4275 2475 l gs col-1 s gr % Polyline n 2250 450 m 2250 675 l 2700 675 l 2700 1125 l 3375 1125 l 3375 1575 l 4275 1575 l 4275 2475 l gs col-1 s gr $F2psEnd restore %%EndDocument @endspecial 530 2600 a Fq(Fig.)31 b(3.)61 b(An)30 b(example)g(of)h(ho) m(w)f(to)h(construct)g(the)f Fm(\045)p Fq({path)h(of)f(Prop)s(osition)f (6.2)0 5558 y Fh(15)p Fi(=apr)r(ile=)p Fh(1999)h Ft([51])1294 b Fq(7.13)p eop %%Page: 52 52 52 51 bop 0 91 a Fr(Figure)35 b(captions)-49 329 y Fq(Fig.)30 b(1)46 b(The)30 b(set)h Fp(U)-49 457 y Fq(Fig.)f(2)46 b(A)31 b(collection)f(of)g(edges)h(\(left\))g(is)e(split)g(in)m(to)h(a) h(collection)f(of)h(con)m(tours)g(\(righ)m(t\))-49 585 y(Fig.)f(3)46 b(An)30 b(example)g(of)h(ho)m(w)f(to)h(construct)g(the)g Fm(\045)p Fq({path)f(of)h(Prop)s(osition)d(6.2)0 5558 y Fh(15)p Fi(=apr)r(ile=)p Fh(1999)i Ft([52])1294 b Fq(7.14)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ENDBODY