Content-Type: multipart/mixed; boundary="-------------9811191753332" This is a multi-part message in MIME format. ---------------9811191753332 Content-Type: text/plain; name="98-724.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="98-724.comments" olleh@math.chalmers.se peres@stat.berkeley.edu rhs@math.ucla.edu URL http://www.math.chalmers.se/~olleh URL http://www.stat.Berkeley.edu/~peres/ URL http://www.math.ucla.edu/~rhs/ ---------------9811191753332 Content-Type: text/plain; name="98-724.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="98-724.keywords" Percolation, transitive graphs, uniqueness of infinite cluster, ends of infinite clusters, indistinguishability of infinite clusters, critical points. ---------------9811191753332 Content-Type: application/postscript; name="hps.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="hps.ps" %!PS-Adobe-2.0 %%Creator: dvips 5.58 Copyright 1986, 1994 Radical Eye Software %%Title: hps.dvi %%CreationDate: Thu Nov 19 15:47:08 1998 %%Pages: 31 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: dvips hps.dvi -o hps.ps %DVIPSParameters: dpi=300, comments removed %DVIPSSource: TeX output 1998.11.19:1546 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N 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Fo(Clearly)l(,)g(a)j(transitiv)o(e)e(graph)i(is)f (quasi-transitiv)o(e.)144 1227 y(Benjamini)8 b(and)j(Sc)o(hramm)d([9])j (conjectured)f(that)h(when)g Fm(G)g Fo(is)g(quasi-transitiv)o(e,)71 1305 y(a.s.)k(uniqueness)g(of)g(the)g(in\014nite)g(cluster)g(holds)g (for)h(all)f Fm(p)f(>)g(p)1242 1312 y Fu(u)1265 1305 y Fo(.)21 b(This)15 b(w)o(as)h(pro)o(v)o(ed)71 1383 y(for)22 b(Ca)o(yley)g(graphs)h(\(and,)h(more)e(generally)l(,)g(for)h (quasi-transitiv)o(e)e(unimo)q(dular)71 1461 y(graphs;)16 b(see)f(De\014nition)f(6.1\))i(b)o(y)f(H\177)-24 b(aggstr\177)g(om)15 b(and)g(P)o(eres)g([18],)g(and)g(in)g(full)g(gener-)71 1540 y(alit)o(y)g(b)o(y)g(Sc)o(honmann)h([29].)71 1664 y Fk(Theorem)g(1.2)i(\([18],)g([29)o(]\))24 b Fj(Consider)17 b(b)n(ond)f(p)n(er)n(c)n(olation)h(on)f(a)h(c)n(onne)n(cte)n(d,)h(in-) 71 1742 y(\014nite,)23 b(lo)n(c)n(al)r(ly)e(\014nite,)i(quasi-tr)n (ansitive)f(gr)n(aph)e Fm(G)p Fj(.)33 b(Then)21 b Fk(P)1246 1724 y Fu(G)1246 1754 y(p)1276 1742 y Fj(-a.s.,)h(the)f(numb)n(er)71 1820 y Fm(N)h Fj(of)c(in\014nite)h(clusters)f(satis\014es)559 2016 y Fm(N)i Fo(=)669 1891 y Fi(8)669 1928 y(>)669 1941 y(>)669 1953 y(>)669 1966 y(<)669 2041 y(>)669 2053 y(>)669 2065 y(>)669 2078 y(:)727 1941 y Fo(0)67 b Fj(if)42 b Fm(p)14 b Fl(2)g Fo([0)p Fm(;)8 b(p)1059 1948 y Fu(c)1077 1941 y Fo(\))727 2019 y Fl(1)41 b Fj(if)h Fm(p)14 b Fl(2)g Fo(\()p Fm(p)1018 2026 y Fu(c)1036 2019 y Fm(;)8 b(p)1082 2026 y Fu(u)1105 2019 y Fo(\))727 2097 y(1)67 b Fj(if)42 b Fm(p)14 b Fl(2)g Fo(\()p Fm(p)1018 2104 y Fu(u)1041 2097 y Fm(;)8 b Fo(1])16 b Fm(:)71 2215 y Fo(The)21 b(parameter)f (space)h([0)p Fm(;)8 b Fo(1])22 b(is)f(th)o(us)g(split)g(in)o(to)g (three)g(qualitativ)o(ely)d(di\013eren)o(t)71 2293 y(in)o(terv)m(als,)f (separated)i(b)o(y)f(the)g(t)o(w)o(o)g(critical)f(v)m(alues)i Fm(p)1091 2300 y Fu(c)1127 2293 y Fo(and)g Fm(p)1248 2300 y Fu(u)1271 2293 y Fo(.)28 b(Some)17 b(of)i(the)f(in-)71 2371 y(terv)m(als)e(ma)o(y)f(b)q(e)i(degenerate)f(or)h(empt)o(y)d (\(e.g.,)i(for)g Fk(Z)1084 2353 y Fu(d)1121 2371 y Fo(w)o(e)h(ha)o(v)o (e)e Fm(p)1330 2378 y Fu(c)1363 2371 y Fo(=)f Fm(p)1439 2378 y Fu(u)1462 2371 y Fo(,)i(and)h(for)71 2449 y(trees)e(w)o(e)h(ha)o (v)o(e)f Fm(p)395 2456 y Fu(u)432 2449 y Fo(=)f(1\).)22 b(Grimme)o(tt)13 b(and)k(Newman)e([16])h(presen)o(ted)f(the)h(\014rst)h (ex-)71 2528 y(ample)12 b(of)j(a)g(transitiv)o(e)e(graph)i(where)f(all) g(three)g(regimes)e(are)i(nondegenerate:)21 b(the)71 2606 y(pro)q(duct)16 b(of)g(a)g(regular)f(tree)g(and)h Fk(Z)p Fo(.)22 b(Other)15 b(examples)f(w)o(ere)g(giv)o(en)h(b)o(y)g (Benjamini)71 2684 y(and)h(Sc)o(hramm)e([9])i(and)g(Lalley)g([22].)p eop %%Page: 4 4 4 3 bop 257 120 a Fo(4)820 b Fn(H\177)-24 b(aggstr\177)g(om,)16 b(P)o(eres)f(and)i(Sc)o(honmann)331 274 y Fo(There)f(is)h(a)g(natural)h (w)o(a)o(y)f(to)g(couple)g(the)f(p)q(ercolation)h(pro)q(cesses)h(for)f (all)g Fm(p)g Fo(si-)257 352 y(m)o(ultaneously)l(.)h(Equip)12 b(the)f(edges)h(of)g Fm(G)h Fo(with)f(i.i.d.)d(random)j(v)m(ariables)f Fl(f)p Fm(U)5 b Fo(\()p Fm(e)p Fo(\))p Fl(g)1749 359 y Fu(e)p Fh(2)p Fu(E)1819 352 y Fo(,)257 430 y(uniform)12 b(in)h([0)p Fm(;)8 b Fo(1],)13 b(and)h(write)e(\011)867 412 y Fu(G)910 430 y Fo(for)h(the)g(resulting)g(pro)q(duct)h(measure)e (on)h([0)p Fm(;)8 b Fo(1])1789 412 y Fu(E)1819 430 y Fo(.)257 509 y(F)l(or)21 b(eac)o(h)e Fm(p)p Fo(,)j(the)e(edge)g(set)g Fl(f)p Fm(e)g Fl(2)h Fm(E)j Fo(:)k Fm(U)5 b Fo(\()p Fm(e)p Fo(\))21 b Fl(\024)f Fm(p)p Fl(g)h Fo(has)g(the)f(same)f(distribution) 257 587 y(as)f(the)e(set)h(of)g(op)q(en)h(edges)e(under)h Fk(P)961 569 y Fu(G)961 599 y(p)991 587 y Fo(.)23 b(This)17 b(yields)f(a)h Fj(c)n(o)n(alesc)n(ent)i(pr)n(o)n(c)n(ess)c Fo(whic)o(h)257 665 y(has)23 b(turned)f(out)g(to)h(b)q(e)f(a)g (fruitful)f(ob)s(ject)h(of)g(study)g(in)g(Erd})-24 b(os{R)o(\023)h(en)o (yi)22 b(random)257 743 y(graph)e(theory)f(\(see)g(e.g.)f([19,)h(2]\))g (and)g(whic)o(h)f(has)i(recen)o(tly)d(attracted)i(more)f(at-)257 822 y(ten)o(tion)g(also)g(in)g(p)q(ercolation)g(theory)g(\(e.g.)f ([10]\):)24 b(When)18 b Fm(p)g Fo(=)e(0)j(ev)o(ery)d(v)o(ertex)h(is)257 900 y(its)g(o)o(wn)g(connected)f(comp)q(onen)o(t.)21 b(As)c(the)f(parameter)f Fm(p)i Fo(is)g(increased,)e(more)h(and)257 978 y(more)e(edges)h(b)q(ecome)f(op)q(en,)h(causing)h(connected)e(comp) q(onen)o(ts)h(to)g(coalesce,)f(un)o(til)257 1056 y(\014nally)l(,)i (when)g Fm(p)e Fo(=)g(1)j(all)e(edges)i(are)f(op)q(en.)331 1138 y(By)e(Theorem)g(1.2)h(and)h(F)l(ubini's)e(Theorem,)f(w)o(e)i(ha)o (v)o(e)f(\011)1404 1120 y Fu(G)1433 1138 y Fo(-a.s.)h(that)h(the)f(n)o (um-)257 1216 y(b)q(er)j(of)f(in\014nite)g(clusters)g(is)g Fl(1)g Fo(for)h(\(Leb)q(esgue-\)a.e.)f Fm(p)f Fl(2)g Fo(\()p Fm(p)1409 1223 y Fu(c)1427 1216 y Fm(;)8 b(p)1473 1223 y Fu(u)1496 1216 y Fo(\),)17 b(and)h(1)g(for)f(a.e.)257 1294 y Fm(p)k Fl(2)f Fo(\()p Fm(p)398 1301 y Fu(u)421 1294 y Fm(;)8 b Fo(1].)31 b(Ho)o(w)o(ev)o(er,)19 b(it)g(is)g(not)i(ob)o (vious)e(that)i(the)e(quan)o(ti\014er)g(\\a.e.")32 b(can)20 b(b)q(e)257 1373 y(strengthened)e(to)f(\\ev)o(ery")g(in)g(these)g (statemen)o(ts.)22 b(Alexander)16 b([3])h(demonstrated)257 1451 y(that)e(this)g(strengthening)f(holds)h(for)g Fm(G)f Fo(=)g Fk(Z)1096 1433 y Fu(d)1131 1451 y Fo(and)h(other)g(Euclidean)e (lattices,)h(and)257 1529 y(H\177)-24 b(aggstr\177)g(om)14 b(and)g(P)o(eres)f([18)q(])g(handled)h(the)f(case)h(where)f Fm(G)h Fo(is)g(quasi-transitiv)o(e)f(and)257 1607 y(unimo)q(dular.)30 b(Here)18 b(w)o(e)h(pro)o(v)o(e)f(the)h(sim)o(ultaneous)f(v)o(ersion)h (of)h(Theorem)d(1.2)j(for)257 1686 y(all)c(quasi-transitiv)o(e)f (graphs.)257 1824 y Fk(Theorem)i(1.3)24 b Fj(L)n(et)17 b Fm(G)h Fj(b)n(e)g(an)f(in\014nite,)i(lo)n(c)n(al)r(ly)g(\014nite,)f (c)n(onne)n(cte)n(d,)h(quasi-tr)n(ansi-)257 1902 y(tive)g(gr)n(aph,)e (and)h(let)h Fm(p)687 1909 y Fu(c)722 1902 y Fj(and)f Fm(p)841 1909 y Fu(u)882 1902 y Fj(b)n(e)g(as)g(in)g(The)n(or)n(em)e (1.2.)23 b(Consider)18 b(the)g(c)n(oupling)257 1981 y Fo(\011)295 1963 y Fu(G)350 1981 y Fj(of)24 b(the)i(p)n(er)n(c)n (olation)e(pr)n(o)n(c)n(esses)g(on)h Fm(G)g Fj(for)f(al)r(l)i Fm(p)i Fl(2)g Fo([0)p Fm(;)8 b Fo(1])25 b Fj(simultane)n(ously,)257 2059 y(and)19 b(let)g Fm(N)5 b Fo(\()p Fm(p)p Fo(\))19 b Fj(b)n(e)f(the)h(numb)n(er)f(of)g(in\014nite)i(clusters)f(determine)n (d)g(by)f(the)h(e)n(dge)g(set)257 2137 y Fl(f)p Fm(e)14 b Fl(2)g Fm(E)j Fo(:)k Fm(U)5 b Fo(\()p Fm(e)p Fo(\))14 b Fl(\024)g Fm(p)p Fl(g)p Fj(.)22 b(With)17 b Fo(\011)866 2119 y Fu(G)896 2137 y Fj(-pr)n(ob)n(ability)g Fo(1)p Fj(,)h(we)g(then)h(have)669 2343 y Fm(N)5 b Fo(\()p Fm(p)p Fo(\))14 b(=)841 2218 y Fi(8)841 2256 y(>)841 2268 y(>)841 2280 y(>)841 2293 y(<)841 2368 y(>)841 2380 y(>)841 2393 y(>)841 2405 y(:)898 2268 y Fo(0)68 b Fj(for)17 b(al)r(l)42 b Fm(p)15 b Fl(2)f Fo([0)p Fm(;)8 b(p)1331 2275 y Fu(c)1348 2268 y Fo(\))898 2346 y Fl(1)42 b Fj(for)17 b(al)r(l)42 b Fm(p)15 b Fl(2)f Fo(\()p Fm(p)1290 2353 y Fu(c)1308 2346 y Fm(;)8 b(p)1354 2353 y Fu(u)1376 2346 y Fo(\))898 2424 y(1)68 b Fj(for)17 b(al)r(l)42 b Fm(p)15 b Fl(2)f Fo(\()p Fm(p)1290 2431 y Fu(u)1313 2424 y Fm(;)8 b Fo(1])g Fm(:)257 2549 y Fo(This)21 b(is)g(an)h(imm)o(ediate)c(consequence)i(of) h(the)g(follo)o(wing)f(result)h(in)f(conjunction)257 2627 y(with)c(Theorem)f(1.2.)p eop %%Page: 5 5 5 4 bop 71 120 a Fn(P)o(ercolation)15 b(on)i(transitiv)o(e)e(graphs)869 b Fo(5)71 274 y Fk(Theorem)16 b(1.4)24 b Fj(L)n(et)17 b Fm(G)h Fj(b)n(e)g(an)g(in\014nite,)h(lo)n(c)n(al)r(ly)f(\014nite,)h (c)n(onne)n(cte)n(d,)f(quasi-tr)n(ansi-)71 352 y(tive)h(gr)n(aph.)25 b(With)18 b Fo(\011)482 334 y Fu(G)512 352 y Fj(-pr)n(ob)n(ability)g Fo(1)p Fj(,)h(for)f(al)r(l)i Fm(p)997 359 y Fr(1)1033 352 y Fm(<)c(p)1111 359 y Fr(2)1150 352 y Fj(in)j Fo(\()p Fm(p)1254 359 y Fu(c)1271 352 y Fm(;)8 b Fo(1])p Fj(,)19 b(every)g(in\014nite)71 430 y Fm(p)95 437 y Fr(2)115 430 y Fj(-cluster)g(c)n(ontains)f(an)g(in\014nite)h Fm(p)745 437 y Fr(1)765 430 y Fj(-cluster.)71 552 y Fo(This)e(sharp)q(ens)h(a)f (result)g(of)g(Sc)o(honmann)g([29],)f(whic)o(h)h(giv)o(es)f(the)h(same) f(assertion)71 631 y(except)g(that)i(the)g(order)f(of)h(the)g(quan)o (ti\014ers)f(\\with)h(\011)1104 612 y Fu(G)1133 631 y Fo(-probabilit)o(y)f(1")i(and)f(\\for)71 709 y(all)d Fm(p)162 716 y Fr(1)196 709 y Fm(<)f(p)272 716 y Fr(2)292 709 y Fo(")j(is)f(in)o(terc)o(hanged.)144 787 y(Theorems)f(1.3)h(and)h (1.4)f(imply)e(that)i(as)h(the)f(parameter)f Fm(p)h Fo(increases,)g (in\014nite)71 866 y(clusters)j(are)h(\\b)q(orn")i(only)e(at,)h(or)f (immedi)o(ately)d(after,)j(lev)o(el)e Fm(p)1310 873 y Fu(c)1328 866 y Fo(.)32 b(F)l(or)21 b(larger)f Fm(p)p Fo(,)71 944 y(in\014nite)d(clusters)h(gro)o(w)h(and)g(merge,)f(but)g (no)h(new)g(ones)g(are)f(formed)g(from)f(\014nite)71 1022 y(clusters.)i(Our)12 b(next)g(result)g(sho)o(ws)h(that)g (in\014nite)e(clusters)h(\\merge)f(relen)o(tlessly")f(in)71 1100 y(the)17 b(in)o(termediate)e(regime)h(\()p Fm(p)643 1107 y Fu(c)661 1100 y Fm(;)8 b(p)707 1107 y Fu(u)730 1100 y Fo(\).)26 b(W)l(e)18 b(can)g(only)g(pro)o(v)o(e)g(this)f(result) h(for)g(quasi-)71 1179 y(transitiv)o(e)h(graphs)j(under)e(the)h (additional)f(assumption)g(of)h(unimo)q(dularit)o(y)e(\(see)71 1257 y(De\014nition)c(6.1\),)i(but)f(w)o(e)g(b)q(eliev)o(e)e(it)i (holds)g(for)h(all)f(quasi-transitiv)o(e)f(graphs.)71 1379 y Fk(Theorem)h(1.5)24 b Fj(L)n(et)17 b Fm(G)h Fj(b)n(e)g(an)g (in\014nite,)h(lo)n(c)n(al)r(ly)f(\014nite,)h(c)n(onne)n(cte)n(d,)f (quasi-tr)n(ansi-)71 1457 y(tive)e(unimo)n(dular)g(gr)n(aph,)f(and)h (let)h Fm(p)748 1464 y Fu(c)782 1457 y Fj(and)f Fm(p)899 1464 y Fu(u)937 1457 y Fj(b)n(e)g(as)g(in)g(The)n(or)n(em)f(1.2.)21 b(Then,)c(with)71 1536 y Fo(\011)109 1518 y Fu(G)138 1536 y Fj(-pr)n(ob)n(ability)f Fo(1)p Fj(,)g(for)g(any)g Fm(p)637 1543 y Fr(1)671 1536 y Fm(<)e(p)747 1543 y Fr(2)783 1536 y Fj(in)i Fo(\()p Fm(p)884 1543 y Fu(c)902 1536 y Fm(;)8 b(p)948 1543 y Fu(u)971 1536 y Fo(\))p Fj(,)16 b(any)g(in\014nite)i(cluster)f(at)f(level)i Fm(p)1625 1543 y Fr(2)71 1614 y Fj(c)n(ontains)g(in\014nitely)h(many)e (in\014nite)i(clusters)g(at)e(level)j Fm(p)1139 1621 y Fr(1)1159 1614 y Fj(.)144 1736 y Fo(The)13 b(next)f(result)h(also)h (concerns)f(the)f(in)o(termediate)e(regime)h(\()p Fm(p)1327 1743 y Fu(c)1345 1736 y Fm(;)d(p)1391 1743 y Fu(u)1414 1736 y Fo(\).)20 b(Sa)o(y)13 b(that)71 1814 y(t)o(w)o(o)j(in\014nite)f (self-a)o(v)o(oiding)g(paths)i Fm(\030)761 1821 y Fr(1)797 1814 y Fo(and)g Fm(\030)913 1821 y Fr(2)950 1814 y Fo(in)e(the)h(same)g (in\014nite)f(cluster)g Fl(C)20 b Fo(are)71 1892 y Fj(e)n(quivalent)i Fo(if)f(for)g(an)o(y)f(\014nite)g(set)h Fl(f)p Fm(e)783 1899 y Fr(1)802 1892 y Fm(;)8 b(:)g(:)g(:)f(;)h(e)934 1899 y Fu(n)957 1892 y Fl(g)21 b Fo(of)g(edges)g(in)f Fl(C)s Fo(,)h(b)q(oth)h(paths)g(are)71 1971 y(ev)o(en)o(tually)16 b(in)i(the)g(same)f(connected)h(comp)q(onen)o(t)f(of)i Fl(C)d(n)c(f)p Fm(e)1241 1978 y Fr(1)1260 1971 y Fm(;)c(:)g(:)g(:)g(;)g (e)1393 1978 y Fu(n)1416 1971 y Fl(g)p Fo(.)27 b(Equiv)m(a-)71 2049 y(lence)13 b(classes)j(of)f(self-a)o(v)o(oiding)g(paths)h(in)e Fl(C)19 b Fo(are)c(called)f Fj(ends)i Fo(of)g Fl(C)s Fo(.)21 b(The)15 b(follo)o(wing)71 2127 y(theorem)f(is)i(pro)o(v)o(ed)g (in)g(Section)f(4.)71 2249 y Fk(Theorem)h(1.6)24 b Fj(L)n(et)17 b Fm(G)h Fj(b)n(e)g(an)g(in\014nite,)h(lo)n(c)n(al)r(ly)f(\014nite,)h (c)n(onne)n(cte)n(d,)f(quasi-tr)n(ansi-)71 2328 y(tive)i(gr)n(aph.)27 b(Then,)20 b Fo(\011)506 2309 y Fu(G)536 2328 y Fj(-a.s.,)g(for)e(al)r (l)j Fm(p)c Fl(2)h Fo(\()p Fm(p)949 2335 y Fu(c)967 2328 y Fm(;)8 b(p)1013 2335 y Fu(u)1036 2328 y Fo(\))19 b Fj(every)h(in\014nite)h Fm(p)p Fj(-cluster)g(has)71 2406 y(pr)n(e)n(cisely)c Fo(2)291 2388 y Fh(@)313 2393 y Fg(0)349 2406 y Fj(many)h(ends.)71 2528 y Fo(The)e(pro)q(of)h(extends)f(to)h (the)f(case)h Fm(p)d Fo(=)g Fm(p)838 2535 y Fu(u)878 2528 y Fo(if)h(there)h(are)h(m)o(ultiple)c(in\014nite)i(clusters)71 2606 y(at)k(that)g(lev)o(el;)e(th)o(us,)i(this)f(theorem)f(con\014rms)h (Conjecture)g(5)h(of)g(Benjamini)d(and)71 2684 y(Sc)o(hramm)c([9])j (\(except)f(for)h(the)g(case)g Fm(p)f Fo(=)g Fm(p)895 2691 y Fu(c)913 2684 y Fo(,)h(if)g(in\014nitely)e(man)o(y)h(in\014nite) g(clusters)p eop %%Page: 6 6 6 5 bop 257 120 a Fo(6)820 b Fn(H\177)-24 b(aggstr\177)g(om,)16 b(P)o(eres)f(and)i(Sc)o(honmann)257 274 y Fo(can)g(exist)f(there\).)21 b(The)16 b(\014xed-)p Fm(p)h Fo(unimo)q(dular)e(case)i(w)o(as)g (\014rst)f(pro)o(v)o(ed)g(in)g(an)h(early)257 352 y(v)o(ersion)f(of)g ([18)q(].)331 430 y(When)j(there)g(are)g(in\014nitely)f(man)o(y)f (in\014nite)i(clusters,)g(can)g(they)g(b)q(e)g(qualita-)257 509 y(tiv)o(ely)14 b(di\013eren)o(t?)21 b(T)l(o)c(mak)o(e)e(this)h (question)g(precise,)e(w)o(e)i(need)g(some)f(de\014nitions.)257 627 y Fk(De\014nition)j(1.7)24 b Fj(L)n(et)13 b Fm(G)i Fo(=)e(\()p Fm(V)s(;)8 b(E)s Fo(\))14 b Fj(b)n(e)g(a)g(quasi-tr)n (ansitive)h(gr)n(aph.)20 b(By)14 b(a)g Fo(subgraph)257 705 y Fj(of)20 b Fm(G)p Fj(,)g(we)h(me)n(an)e(a)h(c)n(ol)r(le)n(ction)h (of)f(e)n(dges.)29 b(A)20 b(set)g Fm(Q)f Fj(of)h(sub)n(gr)n(aphs)e(of)i Fm(G)g Fj(is)f(c)n(al)r(le)n(d)257 783 y(a)h Fk(prop)r(ert)n(y)f Fj(if)g(for)g(every)h Fm(p)e Fl(2)h Fo(\(0)p Fm(;)8 b Fo(1\))20 b Fj(and)g(every)g(vertex)h Fm(x)p Fj(,)e(the)i(event)g(that) f(the)257 862 y(op)n(en)e(cluster)g(of)g Fm(x)f Fj(at)g(level)j Fm(p)e Fj(b)n(elongs)h(to)e Fm(Q)g Fj(is)h Fk(P)1214 869 y Fu(p)1234 862 y Fj(-me)n(asur)n(able.)330 980 y Fl(\017)24 b Fm(Q)d Fj(is)g(an)h Fk(in)n(v)m(arian)n(t)f Fj(pr)n(op)n(erty)e(if)j(for)e(every)i Fm(\015)i Fl(2)d Fo(Aut\()p Fm(G)p Fo(\))g Fj(and)h Fm(E)1684 987 y Fr(0)1725 980 y Fl(2)f Fm(Q)p Fj(,)379 1058 y(ne)n(c)n(essarily)d Fm(\015)s Fo(\()p Fm(E)706 1065 y Fr(0)726 1058 y Fo(\))13 b Fl(2)h Fm(Q)p Fj(.)330 1177 y Fl(\017)24 b Fm(Q)14 b Fj(is)h Fk(monotone)e Fj(if)h(whenever)i Fm(E)1026 1184 y Fr(1)1060 1177 y Fl(2)e Fm(Q)g Fj(and)h Fm(E)1288 1184 y Fr(1)1322 1177 y Fl(\032)e Fm(E)1410 1184 y Fr(2)1430 1177 y Fj(,)i(then)g(also)g Fm(E)1698 1184 y Fr(2)1732 1177 y Fl(2)f Fm(Q)p Fj(.)330 1296 y Fl(\017)24 b Fm(Q)18 b Fj(is)h Fk(robust)f Fj(if)h(for)f(for)g(every)h(in\014nite)h(c)n (onne)n(cte)n(d)f(sub)n(gr)n(aph)f Fl(C)k Fj(of)c Fm(G)h Fj(and)379 1375 y(every)i(e)n(dge)g Fm(e)d Fl(2)i(C)s Fj(,)h(we)g(have)f(the)h(e)n(quivalenc)n(e:)39 b Fl(C)22 b(2)d Fm(Q)28 b Fj(i\013)49 b(ther)n(e)20 b(is)g(an)379 1453 y(in\014nite)f(c)n(onne)n(cte)n(d)g(c)n(omp)n(onent)f(of)f Fl(C)d(n)d(f)p Fm(e)p Fl(g)17 b Fj(that)h(satis\014es)g Fm(Q)p Fj(.)257 1571 y Fo(Supp)q(ose)g(that)f Fm(Q)f Fo(is)g(a)g(robust)i(prop)q(ert)o(y)e(and)h Fl(C)i Fo(is)e(an)f (in\014nite)g(cluster)f(satisfying)257 1649 y Fm(Q)p Fo(.)20 b(If)13 b(an)h(edge)f(adjacen)o(t)h(to)f Fl(C)k Fo(is)c(op)q(ened,)h(then)f(the)h(resulting)e(cluster)h(will)f(satisfy) 257 1728 y Fm(Q)p Fo(,)17 b(and)h(if)e(an)i(edge)f(in)g Fl(C)j Fo(is)d(closed,)g(then)g(at)h(least)f(one)g(of)h(the)f (resulting)f(in\014nite)257 1806 y(clusters)g(will)f(satisfy)h Fm(Q)p Fo(.)331 1884 y(T)l(ransience)h(\(for)i(simple)d(random)i(w)o (alk\))g(is)g(a)h(robust,)g(monotone,)f(in)o(v)m(arian)o(t)257 1962 y(prop)q(ert)o(y)24 b(of)f(subgraphs)i(that)f(has)g(b)q(een)f (studied)g(extensiv)o(ely)l(.)40 b(An)23 b(in)o(v)m(arian)o(t)257 2041 y(prop)q(ert)o(y)17 b(of)f(in)o(terest,)f(that)h(is)g(robust)h (but)g(not)g(monotone,)e(is)562 2167 y Fl(fC)25 b Fo(:)d Fl(9)16 b Fo(in\014nitely)e(man)o(y)h(encoun)o(ter)h(p)q(oin)o(ts)g(in) g Fl(C)s(g)257 2293 y Fo(\(see)21 b([11],)g([8],)g(or)g(the)f(end)h(of) g(the)f(presen)o(t)h(pap)q(er)g(for)g(the)f(de\014nition)h(and)g(sig-) 257 2371 y(ni\014cance)e(of)g(encoun)o(ter)g(p)q(oin)o(ts\).)30 b(A)18 b(monotone)h(in)o(v)m(arian)o(t)g(prop)q(ert)o(y)g(for)g(whic)o (h)257 2449 y(robustness)f(is)e(not)g(kno)o(wn)h(is)f Fl(fC)25 b Fo(:)d Fm(p)972 2456 y Fu(c)990 2449 y Fo(\()p Fl(C)14 b(\002)d Fk(Z)p Fo(\))j Fm(<)f(p)1241 2456 y Fr(0)1262 2449 y Fl(g)p Fo(,)i(where)h Fm(p)1481 2456 y Fr(0)1515 2449 y Fm(<)e Fo(1)j(is)f(\014xed.)331 2528 y(F)l(ollo)o(wing)f(a)h(question)g(of)h(O.)e(Sc)o(hramm)e(\(p)q (ersonal)k(comm)o(unic)o(ation\),)c(H\177)-24 b(agg-)257 2606 y(str\177)g(om)18 b(and)h(P)o(eres)e([18)q(])g(sho)o(w)o(ed)i (that)f(for)h Fm(G)f Fo(quasi-transitiv)o(e)g(and)h(unimo)q(dular,)257 2684 y(if)c Fm(Q)f Fo(is)h(an)o(y)g(monotone)g(in)o(v)m(arian)o(t)f (prop)q(ert)o(y)l(,)h(then)g Fk(P)1276 2691 y Fu(p)1296 2684 y Fo(-a.s,)g(in\014nite)f(clusters)h(with)p eop %%Page: 7 7 7 6 bop 71 120 a Fn(P)o(ercolation)15 b(on)i(transitiv)o(e)e(graphs)869 b Fo(7)71 274 y(and)22 b(without)g Fm(Q)g Fo(cannot)h(co)q(exist,)f (except)f(p)q(ossibly)h(at)g(one)h(v)m(alue)e(of)i Fm(p)p Fo(.)38 b(This)71 352 y(result)12 b(w)o(as)h(substan)o(tially)f(impro)o (v)o(ed)e(b)o(y)i(Ly)o(ons)h(and)h(Sc)o(hramm)9 b([24)q(],)j(who)i(sho) o(w)o(ed)71 430 y(there)k(is)g(no)h(exceptional)e Fm(p)p Fo(,)j(and)f(the)f(monotonicit)o(y)f(assumption)h(on)h Fm(Q)g Fo(can)f(b)q(e)71 509 y(dropp)q(ed.)44 b(Th)o(us)24 b(on)g(quasi-transitiv)o(e)f(unimo)q(dular)g(graphs,)j([24])e(sho)o(ws) g(that)71 587 y(in\014nite)f(clusters)g(are)h Fj(indistinguishable)j (by)d(invariant)h(pr)n(op)n(erties)p Fo(.)44 b(As)23 b(noted)71 665 y(there,)15 b(this)h(strong)h(result)f(fails)g(without)g (unimo)q(dularit)o(y;)e(see)i(Section)g(6.)144 743 y(Nev)o(ertheless,) 24 b(on)i(an)o(y)f(quasi-transitiv)o(e)f(graph,)k(in\014nite)c (clusters)h(cannot)71 822 y(b)q(e)e(distinguished)g(b)o(y)f Fj(r)n(obust)h Fo(in)o(v)m(arian)o(t)f(prop)q(erties.)42 b(The)23 b(follo)o(wing)g(theorem)71 900 y(is)16 b(pro)o(v)o(ed)f(in)h (Section)g(5.)71 1013 y Fk(Theorem)g(1.8)24 b Fj(L)n(et)17 b Fm(G)h Fj(b)n(e)g(an)g(in\014nite,)h(lo)n(c)n(al)r(ly)f(\014nite,)h (c)n(onne)n(cte)n(d,)f(quasi-tr)n(ansi-)71 1092 y(tive)24 b(gr)n(aph,)h(and)f(let)h Fm(p)i Fl(2)f Fo(\()p Fm(p)655 1099 y Fu(c)672 1092 y Fm(;)8 b(p)718 1099 y Fu(u)741 1092 y Fo(])p Fj(.)42 b(If)23 b Fm(Q)h Fj(is)g(a)g(r)n(obust)f (invariant)i(pr)n(op)n(erty)d(of)71 1170 y(sub)n(gr)n(aphs)16 b(of)h Fm(G)h Fj(such)g(that)71 1248 y Fk(P)109 1255 y Fu(p)129 1248 y Fo(\()p Fl(9)35 b Fo(an)16 b(in\014nite)g(cluster)f 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2414 y Fo(\))p Fj(.)71 2528 y Fo(In)22 b(particular,)i(if)f Fm(G)475 2535 y Fr(1)495 2528 y Fm(;)8 b(:)g(:)g(:)f(;)h(G)642 2535 y Fu(d)686 2528 y Fo(are)23 b(in\014nite)f(connected)h(graphs)h (with)f(b)q(ounded)71 2606 y(degree,)15 b(then)560 2684 y Fm(p)584 2691 y Fu(u)607 2684 y Fo(\()p Fm(G)664 2691 y Fr(1)695 2684 y Fl(\002)c(\001)d(\001)g(\001)j(\002)g Fm(G)902 2691 y Fu(d)923 2684 y Fo(\))j Fl(\024)f Fm(p)1032 2691 y Fu(c)1050 2684 y Fo(\()p Fk(Z)1103 2664 y Fu(d)1124 2684 y Fo(\))p Fm(:)p eop %%Page: 8 8 8 7 bop 257 120 a Fo(8)820 b Fn(H\177)-24 b(aggstr\177)g(om,)16 b(P)o(eres)f(and)i(Sc)o(honmann)331 274 y Fo(The)i(rest)g(of)h(the)g (pap)q(er)g(is)f(organized)h(as)g(follo)o(ws.)31 b(In)19 b(the)h(next)f(section)g(w)o(e)257 352 y(state)f(an)h(extension)e(of)h (Theorem)e(1.4.)26 b(W)l(e)18 b(pro)o(v)o(e)f(this)h(extension)f(in)g (Section)g(3,)257 430 y(b)o(y)j(com)o(bining)e(the)i(approac)o(h)h(of)f (Sc)o(honmann)f([29)q(])g(with)h(in)o(v)m(asion)g(p)q(ercolation)257 509 y(ideas.)41 b(Theorem)21 b(1.6)i(on)g(ends)g(is)f(established)g(b)o (y)g(similar)f(means)h(in)g(Section)257 587 y(4.)40 b(W)l(e)21 b(pro)o(v)o(e)h(Theorem)f(1.8)h(\(indistinguishabilit)o(y)e(b)o(y)i (robust)h(prop)q(erties\))f(in)257 665 y(Section)f(5.)34 b(In)21 b(Section)f(6,)i(w)o(e)e(de\014ne)g(unimo)q(dularit)o(y)f(and)j (recall)d(a)i(tec)o(hnique)257 743 y(kno)o(wn)k(as)g(the)f(mass-transp) q(ort)i(metho)q(d,)f(whic)o(h)f(w)o(e)f(then)i(use)f(in)g(Section)g(7) 257 822 y(to)e(pro)o(v)o(e)e(Theorem)f(1.5.)36 b(In)20 b(Section)g(8)i(w)o(e)e(pro)o(v)o(e)g(Theorem)g(1.9,)i(building)e(on) 257 900 y(classical)e(results)g(for)g(p)q(ercolation)g(in)g Fk(Z)1028 882 y Fu(d)1048 900 y Fo(,)h(and)f(a)h(result)e(in)h([29].)27 b(Lo)o(w)o(er)18 b(b)q(ounds)257 978 y(on)f Fm(p)349 985 y Fu(u)388 978 y Fo(are)g(also)f(discussed)h(there.)331 1058 y(Section)e(9)i(con)o(tains)f(examples,)e(remarks,)g(and)j(unsolv) o(ed)f(problems.)257 1263 y FB(2)81 b(Uniform)37 b(p)r(ercolation)i 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Fj(-cluster)f(interse)n(cts)f Fm(B)s Fo(\()p Fm(x;)8 b(R)p Fo(\)\))14 b(=)f(1)8 b Fm(:)92 b Fo(\(1\))331 2225 y(It)17 b(is)i(easy)f(to)h(see)f(that)h(an)o(y)f(quasi-transitiv)o(e)f (graph)j(exhibits)d(uniform)g(p)q(er-)257 2303 y(colation)24 b(at)g(all)f(lev)o(els)f Fm(p)27 b(>)g(p)869 2310 y Fu(c)887 2303 y Fo(.)43 b(In)24 b(fact,)h(this)f(holds)g(in)f(the)h(larger)f (class)h(of)257 2381 y Fj(semi-tr)n(ansitive)18 b Fo(graphs.)257 2528 y Fk(De\014nition)g(2.2)24 b Fj(A)19 b(gr)n(aph)e Fm(G)f Fo(=)g(\()p Fm(V)s(;)8 b(E)s Fo(\))18 b Fj(is)h(c)n(al)r(le)n(d) g Fk(semi-transitiv)m(e)d Fj(if)i(ther)n(e)g(is)257 2606 y(a)d(\014nite)h(set)f Fm(V)521 2613 y Fu(F)564 2606 y Fl(\032)f Fm(V)26 b Fj(such)15 b(that)f(for)g(any)h(vertex)h Fm(x)d Fl(2)h Fm(V)e Fj(,)j(ther)n(e)f(is)h(a)f(vertex)i Fm(y)g Fl(2)e Fm(V)1803 2613 y Fu(F)257 2684 y Fj(and)k(an)g(inje)n (ctive)h(gr)n(aph)d(homomorphism)g(of)i Fm(G)f Fj(that)h(maps)f Fm(y)i Fj(to)f Fm(x)p Fj(.)p eop %%Page: 9 9 9 8 bop 71 120 a Fn(P)o(ercolation)15 b(on)i(transitiv)o(e)e(graphs)869 b Fo(9)144 274 y(The)25 b(simplest)e(examples)g(of)i(semi-transitiv)o (e)d(graph)k(that)g(are)f(not)g(quasi-)71 352 y(transitiv)o(e)15 b(are)h(the)g(nearest-neigh)o(b)q(or)h(graph)g(on)g(the)f(p)q(ositiv)o (e)g(in)o(tegers)f Fk(Z)1507 359 y Fr(+)1537 352 y Fo(,)h(and)71 430 y Fm(d)p Fo(-ary)h(trees)f(where)h(the)g(ro)q(ot)g(has)h(degree)e Fm(d)h Fo(and)h(all)e(other)h(v)o(ertices)e(ha)o(v)o(e)h(degree)71 509 y Fm(d)c Fo(+)g(1.)26 b(More)18 b(generally)l(,)e(the)i(\\sup)q (er-p)q(erio)q(dic")h(trees)e(that)h(discussed)g(in)f(Ly)o(ons)71 587 y(and)f(P)o(eres)f([23)q(])g(are)h(semi-transitiv)o(e;)d(an)j (example)e(is)i(the)f(subtree)h(of)g(the)g(binary)71 665 y(tree)f(consisting)i(of)g(all)f(v)o(ertices)f(suc)o(h)h(that)h (the)f(path)h(from)e(the)i(ro)q(ot)g(to)g Fm(v)h Fo(has)f(at)71 743 y(least)h(as)g(man)o(y)f(left)h(turns)g(as)h(righ)o(t)f(turns.)27 b(These)19 b(trees)e(are)i(closely)e(related)g(to)71 822 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b(for)i(eac)o(h)e(\014xed)h Fm(x)e Fl(2)g Fm(V)27 b Fo(that)273 2020 y(lim)258 2049 y Fu(R)p Fh(!1)364 2020 y Fk(P)402 2027 y Fu(p)422 2020 y Fo(\(some)15 b(in\014nite)h Fm(p)753 2027 y Fr(1)773 2020 y Fo(-cluster)f(in)o (tersects)g Fm(B)s Fo(\()p Fm(x;)8 b(R)p Fo(\)\))14 b(=)g(1)8 b Fm(:)71 2136 y Fo(The)17 b(in\014m)o(um)d(in)j(\(1\))g(is)g(attained) h(for)f(some)g Fm(y)h Fo(in)f(the)g(\014nite)g(set)g Fm(V)1360 2143 y Fu(F)1407 2136 y Fo(sp)q(eci\014ed)g(in)71 2215 y(De\014nition)h(2.2,)i(and)g(it)e(follo)o(ws)h(that)h(\(1\))f (holds)h(for)f(an)o(y)g Fm(p)g(>)g(p)1325 2222 y Fu(c)1343 2215 y Fo(\()p Fm(G)p Fo(\).)30 b(In)o(v)o(oking)71 2293 y(Theorem)15 b(2.3)h(completes)e(the)i(pro)q(of.)p 1616 2280 31 2 v 1616 2305 2 25 v 1644 2305 V 1616 2307 31 2 v 144 2371 a(Theorem)g(1.3)j(ma)o(y)d(fail)h(in)h(the)g (semi-transitiv)o(e)d(setting,)i(b)q(ecause)i(there)e(ex-)71 2449 y(ist)i(semi-transitiv)o(e)d(graphs)21 b(where)e(with)g(p)q 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Fo(the)f(n)o(um)o(b)q(er)f(of)h(in\014nite)f(clusters)h (is)g(0)p Fm(;)8 b Fo(1)17 b(or)g Fl(1)f Fo(a.s.)257 644 y FB(3)81 b(In)n(v)l(asion)26 b(hits)g(in\014nite)g(p)r(ercolation) g(clusters)257 778 y Fo(A)20 b(k)o(ey)f(idea)g(in)h(pro)o(ving)g (Theorem)e(2.3)j(is)e(to)i(use)e Fj(invasion)j(p)n(er)n(c)n(olation)p Fo(,)e(whic)o(h)257 856 y(is)e(a)h(sequen)o(tial)e(construction)i (based)f(on)h(the)f(same)g(uniform)f(random)g(v)m(ariables)257 934 y Fl(f)p Fm(U)5 b Fo(\()p Fm(e)p Fo(\))p Fl(g)406 941 y Fu(e)p Fh(2)p Fu(E)491 934 y Fo(as)16 b(the)f(canonical)f (coupling)h(of)h(the)f(ordinary)g(p)q(ercolation)g(pro)q(cesses.)257 1012 y(Here)k(w)o(e)g(giv)o(e)f(only)i(a)g(brief)e(description)h(of)h (in)o(v)m(asion)f(p)q(ercolation;)i(w)o(e)e(refer)g(to)257 1091 y(Cha)o(y)o(es,)c(Cha)o(y)o(es)g(and)h(Newman)e([12])h(for)h(a)g (general)f(in)o(tro)q(duction)g(to)h(the)f(mo)q(del,)257 1169 y(and)21 b(to)f([25,)f(28)q(])h(for)g(some)e(in)o(teresting)h 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Fo(is)f(the)h(edge)f(whic)o(h)g(minimi)o(zes) e Fm(U)5 b Fo(\()p Fm(e)p Fo(\))18 b(among)h(all)f(edges)h Fm(e)f Fo(that)i(are)e(not)i(in)e Fm(I)1811 1624 y Fu(x)1807 1654 y(i)257 1720 y Fo(but)f(are)g(adjacen)o(t)f(to)h(some)e(edge)i(in) f Fm(I)1001 1702 y Fu(x)997 1732 y(i)1022 1720 y Fo(.)22 b(The)17 b(in)o(v)m(asion)f(cluster)g(of)h Fm(x)f Fo(is)g(the)h(edge) 257 1798 y(set)915 1886 y Fm(I)941 1865 y Fu(x)937 1898 y Fh(1)988 1886 y Fo(=)1051 1832 y Fh(1)1045 1844 y Fi([)1040 1935 y Fu(i)p Fr(=1)1105 1886 y Fm(I)1131 1865 y Fu(x)1127 1898 y(i)1161 1886 y Fm(:)257 2060 y Fk(Prop)r(osition)h(3.1)24 b Fj(L)n(et)h Fm(G)k Fo(=)g(\()p Fm(V)s(;)8 b(E)s Fo(\))25 b Fj(b)n(e)g(an)h(in\014nite,)j(c)n(onne)n(cte)n(d)d(gr)n(aph)f(with) 257 2139 y(b)n(ounde)n(d)d(de)n(gr)n(e)n(es.)33 b(If)22 b Fm(G)f Fj(exhibits)i(uniform)e(p)n(er)n(c)n(olation)g(at)g(level)j Fm(p)1578 2146 y Fh(\003)1598 2139 y Fj(,)e(then)g Fo(\011)1785 2121 y Fu(G)1815 2139 y Fj(-)257 2217 y(a.s.)g(for)14 b(any)h Fm(p)f(>)g(p)634 2224 y Fh(\003)669 2217 y Fj(and)i(any)f Fm(x)e Fl(2)h Fm(V)e Fj(,)j(the)g(invasion)h(cluster)g Fm(I)1462 2199 y Fu(x)1458 2229 y Fh(1)1510 2217 y Fj(interse)n(cts)g (some)257 2295 y(in\014nite)j Fm(p)p Fj(-cluster.)257 2449 y Fo(This)e(prop)q(osition)i(w)o(as)e(pro)o(v)o(ed)f(b)o(y)h(Cha)o (y)o(es,)f(Cha)o(y)o(es)h(and)h(Newman)d([12)q(])h(for)h Fk(Z)1798 2431 y Fu(d)1819 2449 y Fo(,)257 2528 y(b)o(y)k(Alexander)f ([4])h(for)h(other)g(Euclidean)e(graphs,)k(and)e(b)o(y)f(O.)f(Sc)o (hramm)f(\(p)q(er-)257 2606 y(sonal)k(comm)o(unic)o(ation\))c(for)j (transitiv)o(e)f(unimo)q(dular)g(graphs.)39 b(Before)20 b(pro)o(ving)257 2684 y(Prop)q(osition)e(3.1,)e(w)o(e)g(explain)f(ho)o (w)i(it)e(implies)f(Theorem)h(2.3.)p eop %%Page: 11 11 11 10 bop 71 120 a Fn(P)o(ercolation)15 b(on)i(transitiv)o(e)e(graphs) 845 b Fo(11)71 274 y Fk(Pro)r(of)19 b(of)h(Theorem)d(2.3:)23 b Fo(F)l(or)17 b Fm(p)f Fl(2)f Fo([0)p Fm(;)8 b Fo(1])17 b(and)h(a)f(v)o(ertex)f Fm(x)p Fo(,)g(let)g Fl(C)s Fo(\()p 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Fu(x)1075 678 y Fh(1)1112 665 y Fo(,)g(and)h(hence)e Fl(C)s Fo(\()p Fm(x;)8 b(p)1488 672 y Fr(2)1508 665 y Fo(\))14 b(m)o(ust)71 743 y(in)o(tersect)f(some)h(in\014nite)g Fm(p)p Fo(-cluster)h Fl(C)s Fo(\()p Fm(y)r(;)8 b(p)p Fo(\).)21 b(Ob)o(viously)l(,)14 b Fl(C)s Fo(\()p Fm(x;)8 b(p)1282 750 y Fr(2)1302 743 y Fo(\))15 b(then)g(in)o(tersects)71 822 y(some)i(in\014nite)h Fm(p)387 829 y Fr(1)408 822 y Fo(-cluster)g(for)h(an)o(y)g Fm(p)780 829 y Fr(1)818 822 y Fl(2)g Fo([)p Fm(p;)8 b(p)954 829 y Fr(2)974 822 y Fo(\).)29 b(Prop)q(osition)20 b(3.1)f(ensures)g (that)71 900 y(\011)109 882 y Fu(G)138 900 y Fo(\()p Fl(\\)190 907 y Fu(x;p)240 900 y Fo(\012)275 907 y Fu(x;p)324 900 y Fo(\))k(=)f(1,)h(where)e(the)g(in)o(tersection)f(ranges)i(o)o(v)o (er)e(all)h Fm(x)h Fl(2)h Fm(V)32 b Fo(and)22 b(all)71 978 y(rational)16 b Fm(p)e(>)g(p)366 985 y Fh(\003)386 978 y Fo(,)i(and)h(this)f(pro)o(v)o(es)g(the)g(theorem.)p 1616 966 31 2 v 1616 991 2 25 v 1644 991 V 1616 993 31 2 v 144 1058 a(The)23 b(pro)q(of)i(of)f(Prop)q(osition)g(3.1)g(is)g (based)g(on)g(an)g(adaptation)h(to)f(in)o(v)m(asion)71 1136 y(p)q(ercolation)15 b(of)h(the)g(pro)q(of)h(of)f(the)f(main)g (result)g(in)g([29].)21 b(The)16 b(follo)o(wing)f(lemma)e(is)71 1214 y(needed.)71 1342 y Fk(Lemm)o(a)j(3.2)24 b Fj(L)n(et)18 b Fm(G)d Fo(=)f(\()p Fm(V)s(;)8 b(E)s Fo(\))18 b Fj(b)n(e)g(an)h (in\014nite,)g(c)n(onne)n(cte)n(d)g(gr)n(aph)e(with)h(b)n(ounde)n(d)71 1421 y(de)n(gr)n(e)n(es,)g(and)h(let)g Fm(R)e(>)f Fo(0)j Fj(b)n(e)g(an)g(inte)n(ger.)26 b(With)19 b Fo(\011)1046 1403 y Fu(G)1075 1421 y Fj(-pr)n(ob)n(ability)f Fo(1)p Fj(,)i(the)f(invasion)71 1499 y(cluster)f Fm(I)254 1481 y Fu(x)250 1511 y Fh(1)304 1499 y Fj(c)n(ontains)g(a)g(b)n(al)r(l)g(of) g(r)n(adius)e Fm(R)p Fj(.)71 1626 y Fk(Pro)r(of:)k Fo(Denote)c(b)o(y)g Fm(D)h Fo(the)f(maxim)o(um)11 b(degree)k(of)i(v)o(ertices)c(in)j Fm(G)p Fo(.)22 b(The)15 b(standard)71 1704 y(inequalit)o(y)652 1786 y Fm(p)676 1793 y Fu(c)694 1786 y Fo(\()p Fm(G)p Fo(\))f Fl(\025)893 1753 y Fo(1)p 842 1775 128 2 v 842 1820 a Fm(D)f Fl(\000)e Fo(1)988 1786 y Fm(>)j Fo(0)519 b(\(2\))71 1896 y(\(see,)20 b(e.g.,)g([15]\),)g(will)f(b)q(e)h(used)h (at)f(the)g(end)g(of)h(the)f(pro)q(of.)34 b(Let)20 b(\()p Fm(v)1393 1903 y Fr(1)1412 1896 y Fm(;)8 b(v)1458 1903 y Fr(2)1477 1896 y Fm(;)g(:)g(:)g(:)o Fo(\))20 b(b)q(e)71 1975 y(an)e(arbitrary)g(en)o(umeration)f(of)h(the)g(v)o(ertex)f(set)h Fm(V)11 b Fo(.)27 b(F)l(or)19 b Fm(n)e Fo(=)g(1)p Fm(;)8 b Fo(2)p Fm(;)g(:)g(:)g(:)p Fo(,)18 b(set)g Fm(L)1567 1982 y Fu(n)1608 1975 y Fo(=)71 2053 y Fm(n)p Fo(\(2)p Fm(R)12 b Fo(+)f(1\),)16 b(and)h(de\014ne)83 2185 y Fm(\034)104 2192 y Fu(n)141 2185 y Fo(:=)d(min)n Fl(f)p Fm(k)i Fo(:)22 b Fm(I)416 2164 y Fu(x)412 2197 y(k)453 2185 y Fo(comes)15 b(within)h(distance)g Fm(R)h Fo(from)e(some)g Fm(y)g Fl(2)f Fm(V)23 b Fl(n)11 b Fm(B)s Fo(\()p Fm(x;)d(L)1545 2192 y Fu(n)1567 2185 y Fo(\))p Fl(g)g Fm(:)71 2317 y Fo(Since)18 b(the)i(in)o(v)m(asion)f(cluster)g Fm(I)666 2299 y Fu(x)662 2330 y 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b(P)o(eres)f(and)i(Sc)o(honmann)257 274 y Fo(Since)j(there)g(are)h (a.s.)f(no)h(in\014nite)f Fm(p)p Fo(-clusters)h(for)g Fm(p)h(<)f(p)1367 281 y Fu(c)1385 274 y Fo(,)g(on)g Fm(A)1529 281 y Fu(n)1573 274 y Fo(the)f(in)o(v)m(asion)257 352 y(cluster)e Fm(I)443 334 y Fu(x)439 365 y Fh(1)495 352 y Fo(m)o(ust)f(con)o(tain)h(the)h(ball)f Fm(B)s Fo(\()p Fm(y)1059 359 y Fu(n)1081 352 y Fm(;)8 b(R)p Fo(\).)29 b(Th)o(us)19 b(it)f(su\016ces)g(to)h(pro)o(v)o(e)f(that)257 430 y(\011)295 412 y Fu(G)325 382 y Fi(\020)362 430 y Fl(\\)395 412 y Fh(1)395 443 y Fu(i)p Fr(=1)466 430 y Fm(A)503 412 y Fu(c)503 443 y(i)520 382 y Fi(\021)562 430 y Fo(=)e(0.)27 b(Note)17 b(that)h Fm(I)933 412 y Fu(x)929 443 y(\034)945 447 y Ff(n)986 430 y Fo(and)g Fm(B)s Fo(\()p Fm(y)1165 437 y Fu(n)1188 430 y Fm(;)8 b(R)p Fo(\))18 b(\\touc)o(h")h(but)f(do)g(not)g(in)o(ter-)257 509 y(sect,)12 b(and)h(that)f(the)g(in)o(v)m(asion)g(pro)q(cess)h(up)f (to)g(time)e Fm(\034)1235 516 y Fu(n)1270 509 y Fo(giv)o(es)i(no)g 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Fu(G)517 1576 y Fo(\()p Fm(I)562 1555 y Fu(x)558 1588 y Fh(1)611 1576 y Fo(in)o(tersects)i(some)g(in\014nite) h Fm(p)p Fo(-cluster)g Fl(j)8 b Fm(\030)1358 1555 y Fu(x)1356 1588 y(p)1374 1592 y Fe(\003)1408 1576 y Fo(=)14 b Fl(1)p Fo(\))f(=)h(1)8 b Fm(:)130 b Fo(\(4\))257 1704 y(Letting)17 b Fm(p)d Fl(#)g Fm(p)530 1711 y Fh(\003)566 1704 y Fo(through)k(a)e (coun)o(table)g(sequence)f(then)h(pro)o(v)o(es)g(the)g(prop)q(osition.) 331 1782 y(By)f(the)g(uniform)g(p)q(ercolation)g(assumption,)h(w)o(e)f (can,)g(for)h(an)o(y)g Fm(")e(>)g Fo(0,)h(pic)o(k)g(an)257 1860 y Fm(R)i Fo(so)g(large)f(that)418 1988 y(inf)413 2017 y Fu(y)q Fh(2)p Fu(V)491 1988 y Fo(\011)529 1967 y Fu(G)559 1988 y Fo(\(some)f(in\014nite)g Fm(p)889 1995 y Fh(\003)910 1988 y Fo(-cluster)g(in)o(tersects)g Fm(B)s Fo(\()p Fm(y)r(;)8 b(R)p Fo(\)\))14 b Fl(\025)f Fo(1)f Fl(\000)e Fm(")e(:)93 b Fo(\(5\))257 2116 y(Let)20 b Fm(\034)25 b Fo(denote)19 b(the)g(smallest)f Fm(k)j Fo(for)f(whic)o(h)e Fm(I)1124 2098 y Fu(x)1120 2128 y(k)1165 2116 y Fo(con)o(tains)h(a)h (ball)f(of)g(radius)h Fm(R)p Fo(;)h(b)o(y)257 2194 y(Lemma)14 b(3.2,)j Fm(\034)i(<)14 b Fl(1)i Fo(a.s.)331 2272 y(F)l(or)g(an)h(edge) f(set)g Fm(E)709 2279 y Fr(0)743 2272 y Fl(\032)d Fm(E)s Fo(,)j(set)483 2400 y Fm(V)11 b Fo(\()p Fm(E)577 2407 y Fr(0)597 2400 y Fo(\))j(=)f Fl(f)p Fm(y)j Fl(2)e Fm(V)25 b Fo(:)c Fm(y)d Fo(is)e(an)h(endp)q(oin)o(t)f(of)h(some)e Fm(e)e Fl(2)i Fm(E)1541 2407 y Fr(0)1560 2400 y Fl(g)8 b Fm(:)257 2528 y Fo(If)22 b Fm(E)348 2535 y Fr(0)389 2528 y Fo(is)f(\014nite)h(and)g(con)o(tains)f(some)g(ball)g(of)h (radius)g Fm(R)p Fo(,)h(then)f(b)o(y)f(\(5\))h(w)o(e)f(ha)o(v)o(e)257 2606 y(with)f(probabilit)o(y)e(at)i(least)f(1)14 b Fl(\000)f Fm(")20 b Fo(that)f(some)g(v)o(ertex)f(in)h Fm(V)11 b Fo(\()p Fm(E)1476 2613 y Fr(0)1496 2606 y Fo(\))1515 2588 y Fu(c)1552 2606 y Fo(at)20 b(distance)f(1)257 2684 y(from)d Fm(V)11 b Fo(\()p Fm(E)467 2691 y Fr(0)487 2684 y Fo(\))16 b(has)h(an)g(op)q(en)g(path)g(to)g(in\014nit)o(y)e(at)i(lev) o(el)d Fm(p)1332 2691 y Fh(\003)1368 2684 y Fo(via)i(v)o(ertices)f(in)h Fm(V)11 b Fo(\()p Fm(E)1776 2691 y Fr(0)1796 2684 y Fo(\))1815 2666 y Fu(c)p eop %%Page: 13 13 13 12 bop 71 120 a Fn(P)o(ercolation)15 b(on)i(transitiv)o(e)e(graphs) 845 b Fo(13)71 274 y(only)l(.)29 b(Since)18 b(the)g(in)o(v)m(asion)h (cluster)g(up)g(to)g(time)e Fm(\034)25 b Fo(giv)o(es)18 b(no)h(information)f(ab)q(out)71 352 y(the)j(set)h(of)g(edges)g(not)h (adjacen)o(t)f(to)g Fm(I)824 334 y Fu(x)820 365 y(\034)845 352 y Fo(,)h(w)o(e)f(ma)o(y)e(apply)i(the)g(ab)q(o)o(v)o(e)g(reasoning) 71 430 y(with)14 b Fm(E)216 437 y Fr(0)250 430 y Fo(=)f Fm(I)327 412 y Fu(x)323 443 y(\034)363 430 y Fo(to)i(deduce)f(that)h (the)f(conditional)g(probabilit)o(y)g(that)g(there)g(is)h(some)71 509 y(in\014nite)i Fm(p)262 516 y Fh(\003)282 509 y Fo(-cluster)h (within)f(distance)h(1)h(from)e Fm(I)989 491 y Fu(x)985 521 y(\034)1028 509 y Fo(is)h(at)h(least)f(1)13 b Fl(\000)f Fm(")p Fo(.)26 b(This)19 b(sho)o(ws)71 587 y(that)d(\011)214 569 y Fu(G)244 587 y Fo(\()p Fm(\030)286 569 y Fu(x)284 599 y(p)302 603 y Fe(\003)336 587 y Fo(=)d(0\))i Fl(\024)e Fm(")p Fo(,)j(and)h(since)e Fm(")h Fo(w)o(as)h(arbitrary)f(w)o(e)g(ha)o (v)o(e)675 720 y(\011)713 699 y Fu(G)742 720 y Fo(\()p Fm(\030)784 699 y Fu(x)782 732 y(p)800 736 y Fe(\003)834 720 y Fo(=)e(0\))g(=)g(0)8 b Fm(:)542 b Fo(\(6\))71 852 y(The)15 b(next)h(step)g(is)f(to)i(rule)e(out)h(the)g(p)q(ossibilit)o (y)f(of)h(ha)o(ving)g Fm(\030)1225 834 y Fu(x)1223 865 y(p)1241 869 y Fe(\003)1275 852 y Fo(=)d Fm(n)j Fo(for)g(an)o(y)g (\014nite)71 931 y Fm(n)p Fo(.)23 b(Note)16 b(that)h(on)h(the)e(ev)o (en)o(t)g(\()p Fm(\030)686 912 y Fu(x)684 943 y(p)702 947 y Fe(\003)736 931 y Fo(=)f Fm(n)p Fo(\))i(w)o(e)g(can)g(mo)o(v)o(e) d(in)o(to)j(the)f(ev)o(en)o(t)g(\()p Fm(\030)1499 912 y Fu(x)1497 943 y(p)1515 947 y Fe(\003)1549 931 y Fo(=)f(0\))71 1009 y(b)o(y)j(c)o(hanging)h(the)g(status)g(of)g(\014nitely)f(man)o(y)f (edges.)29 b(It)18 b(is)h(easy)g(to)g(see)f(that)i(this)71 1087 y(implies)14 b(that)k(if)e(\011)427 1069 y Fu(G)457 1087 y Fo(\()p Fm(\030)499 1069 y Fu(x)497 1099 y(p)515 1103 y Fe(\003)550 1087 y Fo(=)f Fm(n)p Fo(\))h Fm(>)f Fo(0,)i(then)g(\011)925 1069 y Fu(G)955 1087 y Fo(\()p Fm(\030)997 1069 y Fu(x)995 1099 y(p)1013 1103 y Fe(\003)1048 1087 y Fo(=)f(0\))f Fm(>)h Fo(0)h(holds)h(as)f(w)o(ell.)23 b(But)71 1165 y(this)16 b(w)o(ould)g(con)o(tradict)g(\(6\),)g(so)h(w)o (e)e(ha)o(v)o(e)683 1298 y(\011)721 1277 y Fu(G)751 1298 y Fo(\()p Fm(\030)793 1277 y Fu(x)791 1310 y(p)809 1314 y Fe(\003)843 1298 y Fo(=)f Fm(n)p Fo(\))g(=)f(0)551 b(\(7\))71 1431 y(for)16 b(an)o(y)g Fm(n)e(<)g Fl(1)p Fo(,)i(and)g(\(3\))h(is)f(established.)144 1510 y(T)l(o)f(pro)o(v)o(e)g (\(4\),)g(consider)g(the)g(follo)o(wing)g(\\coloring)g(follo)o(w)o(ed)f (b)o(y)h(in)o(v)m(asion)g(p)q(er-)71 1589 y(colation")i(pro)q(cedure.) 25 b(First)17 b(mark)f(ev)o(ery)g(edge)h(blue)f(whic)o(h)h(is)g(in)g (some)g(in\014nite)71 1667 y Fm(p)95 1674 y Fh(\003)115 1667 y Fo(-cluster.)k(Then)c(mark)e(ev)o(ery)g(edge)h(red)g(whic)o(h)g (is)g(not)h(blue)f(but)h(is)f(adjacen)o(t)g(to)71 1745 y(some)c(blue)h(edge.)20 b(Giv)o(en)13 b(the)h(coloring)f(information,) g(start)h(to)g(build)f(the)h(in)o(v)m(asion)71 1823 y(cluster)20 b(at)h Fm(x)g Fo(in)g(the)g(usual)g(w)o(a)o(y)l(.)35 b(F)l(or)21 b(the)g(ev)o(en)o(t)f(\()p Fm(\030)1114 1805 y Fu(x)1112 1836 y(p)1130 1840 y Fe(\003)1172 1823 y Fo(=)i Fl(1)p Fo(\))f(to)g(happ)q(en,)i(the)71 1902 y(in)o(v)m(asion)17 b(cluster)f(has)i(to)f(meet)e(\(b)q(ecome)h(adjacen)o(t)g(to\))i (in\014nitely)d(man)o(y)h(colored)71 1980 y(edges.)k(If)14 b(the)h(in)o(v)m(asion)f(cluster)g(ev)o(er)f(meets)g(an)o(y)h(of)h(the) g(blue)f(edges,)g(then)g(w)o(e)h(are)71 2058 y(done)20 b(\(i.e.)e(the)i(in)o(v)m(asion)g(cluster)f(in)o(tersects)f(some)h (in\014nite)g Fm(p)p Fo(-cluster\),)h(b)q(ecause)71 2136 y(the)11 b(in)o(v)m(asion)g(cluster)g(m)o(ust)f(then)h(ev)o(en)o (tually)f(con)o(tain)h(the)g(encoun)o(tered)g(blue)g(edge)71 2215 y(unless)16 b(it)h(p)q(enetrates)g(some)f(in\014nite)g Fm(p)816 2197 y Fh(\003)836 2215 y Fo(-cluster)g(elsewhere.)22 b(Otherwise)16 b(\(still)g(on)71 2293 y(the)h(ev)o(en)o(t)e(\()p Fm(\030)327 2275 y Fu(x)325 2305 y(p)343 2309 y Fe(\003)379 2293 y Fo(=)h Fl(1)p Fo(\)\))h(the)g(in)o(v)m(asion)g(cluster)f(has)i (to)g(meet)d(in\014nitely)h(man)o(y)g(red)71 2371 y(edges.)k(Supp)q (ose)c(no)o(w)f(that)g(a)g(giv)o(en)f(red)g(edge)h Fm(e)f Fo(is)h(met)e(for)i(the)f(\014rst)h(time.)j(Then)71 2449 y(the)12 b(conditional)g(distribution)g(of)g Fm(U)5 b Fo(\()p Fm(e)p Fo(\))13 b(\(giv)o(en)e(the)h(coloring)h(information)e (and)i(the)71 2528 y(in)o(v)m(asion)i(cluster)f(so)i(far\))f(is)g (uniform)f(on)i(\()p Fm(p)904 2535 y Fh(\003)924 2528 y Fm(;)8 b Fo(1].)21 b(Th)o(us)15 b(the)g(ev)o(en)o(t)f(that)i Fm(U)5 b Fo(\()p Fm(e)p Fo(\))13 b Fm(<)h(p)71 2606 y Fo(\(whic)o(h)22 b(ob)o(viously)h(implies)d(that)k Fm(I)770 2588 y Fu(x)766 2618 y Fh(1)826 2606 y Fo(in)o(tersects)e(some)g (in\014nite)h Fm(p)p Fo(-cluster\))g(has)71 2684 y(conditional)17 b(probabilit)o(y)579 2663 y Fu(p)p Fh(\000)p Fu(p)642 2667 y Fe(\003)p 579 2673 81 2 v 579 2701 a Fr(1)p Fh(\000)p Fu(p)642 2705 y Fe(\003)681 2684 y Fm(>)f Fo(0.)26 b(Since)16 b(this)i(conditional)f(probabilit)o(y)g(is)g(the)p eop %%Page: 14 14 14 13 bop 257 120 a Fo(14)796 b Fn(H\177)-24 b(aggstr\177)g(om,)16 b(P)o(eres)f(and)i(Sc)o(honmann)257 274 y Fo(same)g(ev)o(ery)f(time)f (a)j(red)g(edge)f(is)g(encoun)o(tered)g(b)o(y)g(the)g(in)o(v)m(asion)h (cluster)f(for)h(the)257 352 y(\014rst)f(time,)d(w)o(e)h(ha)o(v)o(e)h (\(4\),)g(and)h(the)f(pro)q(of)h(is)f(complete.)p 1802 340 31 2 v 1802 365 2 25 v 1831 365 V 1802 367 31 2 v 257 545 a FB(4)81 b(Uncoun)n(tably)26 b(man)n(y)f(ends)257 672 y Fk(Pro)r(of)c(of)g(Theorem)e(1.6:)25 b Fo(W)l(e)18 b(shall)g(pro)o(v)o(e)f(that)i(for)f(an)o(y)g Fm(p)1470 679 y Fr(1)1507 672 y Fm(<)g(p)1587 679 y Fr(2)1625 672 y Fo(in)g(\()p Fm(p)1727 679 y Fu(c)1745 672 y Fm(;)8 b(p)1791 679 y Fu(u)1813 672 y 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y(Theorem)e(1.4\))i(enough)g(to)f(sho)o(w,)h(for)f(an)o(y)h Fm(x)1118 1232 y Fr(0)1151 1225 y Fl(2)d Fm(V)d Fo(,)16 b(that)864 1345 y(\011)902 1324 y Fu(G)931 1345 y Fo(\()p Fm(H)994 1324 y Fu(x)1014 1329 y Fg(0)990 1357 y Fu(p)1008 1362 y Fg(0)1025 1357 y Fu(;p)1053 1362 y Fg(1)1070 1357 y Fu(;p)1098 1362 y Fg(2)1117 1345 y Fo(\))e(=)g(0)544 b(\(9\))257 1464 y(where)15 b Fm(H)441 1446 y Fu(x)461 1451 y Fg(0)437 1477 y Fu(p)455 1482 y Fg(0)473 1477 y Fu(;p)501 1482 y Fg(1)518 1477 y Fu(;p)546 1482 y Fg(2)580 1464 y Fo(is)g(the)g(ev)o(en)o(t)f(that)i Fm(x)972 1471 y Fr(0)1007 1464 y Fo(is)f(in)g(an)h(in\014nite)e Fm(p)1366 1471 y Fr(0)1386 1464 y Fo(-cluster)h(and)h(for)f(some)257 1542 y Fm(p)27 b Fl(2)g Fo([)p Fm(p)406 1549 y Fr(1)426 1542 y Fm(;)8 b(p)472 1549 y Fr(2)492 1542 y Fo(])23 b(it)g(is)h(in)f(an)h(in\014nite)f Fm(p)p Fo(-cluster)h(with)f(less)h (than)g(2)1521 1524 y Fh(@)1543 1529 y Fg(0)1586 1542 y Fo(ends.)44 b(Also)257 1621 y(de\014ne)419 1608 y(~)406 1621 y Fm(H)450 1603 y Fu(x)470 1608 y Fg(0)446 1633 y Fu(p)464 1638 y Fg(0)481 1633 y Fu(;p)509 1638 y Fg(1)526 1633 y Fu(;p)554 1638 y Fg(2)597 1621 y Fo(as)24 b(the)g(ev)o(en)o(t)e (that)j Fm(x)1034 1628 y Fr(0)1077 1621 y Fo(is)f(in)f(an)h(in\014nite) f Fm(p)1470 1628 y Fr(0)1490 1621 y Fo(-cluster)h(and)g(for)257 1699 y(some)g Fm(p)j Fl(2)h Fo([)p Fm(p)538 1706 y Fr(1)558 1699 y Fm(;)8 b(p)604 1706 y Fr(2)624 1699 y Fo(])24 b(it)f(is)h(in)g(an)h(in\014nite)e Fm(p)p Fo(-cluster)h(with)g(just)h (one)f(end.)45 b(If)24 b(a)257 1777 y(giv)o(en)16 b(realization)g Fm(\021)g Fl(2)f Fo([0)p Fm(;)8 b Fo(1])810 1759 y Fu(E)856 1777 y Fo(of)17 b(the)f(v)m(ariables)g Fl(f)p Fm(U)5 b Fo(\()p Fm(e)p Fo(\))p Fl(g)1347 1784 y Fu(e)p Fh(2)p Fu(E)1433 1777 y Fo(is)17 b(in)f Fm(H)1584 1759 y Fu(x)1604 1764 y Fg(0)1580 1790 y Fu(p)1598 1795 y Fg(0)1616 1790 y Fu(;p)1644 1795 y Fg(1)1660 1790 y Fu(;p)1688 1795 y Fg(2)1707 1777 y Fo(,)h(then)257 1855 y(\(arguing)h(as)g(in)f (Benjamini)d(and)k(Sc)o(hramm)c([9],)i(p.)h(76\))g(one)h(can)f(c)o (hange)g(\014nitely)257 1934 y(man)o(y)d(of)i(the)f(v)m(ariables)g(to)g (obtain)h(a)g(realization)e Fm(\021)1240 1916 y Fh(0)1267 1934 y Fo(whic)o(h)g(is)h(in)1522 1921 y(~)1509 1934 y Fm(H)1553 1916 y Fu(x)1573 1921 y Fg(0)1549 1946 y Fu(p)1567 1951 y Fg(0)1585 1946 y Fu(;p)1613 1951 y Fg(1)1630 1946 y Fu(;p)1658 1951 y Fg(2)1677 1934 y Fo(.)21 b(Th)o(us,)257 2012 y(\(9\))c(follo)o(ws)f(easily)f(once)i(w)o(e)e(sho)o(w)853 2132 y(\011)891 2111 y Fu(G)920 2132 y Fo(\()952 2119 y(~)939 2132 y Fm(H)983 2111 y Fu(x)1003 2116 y Fg(0)979 2144 y Fu(p)997 2149 y Fg(0)1015 2144 y Fu(;p)1043 2149 y Fg(1)1059 2144 y Fu(;p)1087 2149 y Fg(2)1106 2132 y Fo(\))f(=)g(0)8 b Fm(:)509 b Fo(\(10\))331 2251 y(Let)23 b Fm(L)458 2258 y Fu(x;R;k)569 2251 y Fo(b)q(e)g(the)g(ev)o(en)o(t)f (that)h(there)g(are)g(at)h(least)f Fm(k)i Fo(in\014nite)d Fm(p)1635 2258 y Fr(2)1656 2251 y Fo(-clusters)257 2330 y(whic)o(h)16 b(con)o(tain)h(in\014nite)f 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Fo(b)q(e)g(the)f(set)h(of)g(edges)g (in)g Fm(E)d Fl(n)d Fm(I)1226 569 y Fu(x)1246 574 y Fg(0)1222 599 y Fu(\034)1281 587 y Fo(that)16 b(are)g(adjacen)o(t)71 665 y(to)g Fm(I)156 647 y Fu(x)176 652 y Fg(0)152 678 y Fu(\034)195 665 y Fo(.)21 b(Using)16 b(\(11\))h(and)f(arguing)h(as)g (in)f(the)g(pro)q(of)h(of)f(Prop)q(osition)i(3.1,)e(w)o(e)f(ha)o(v)o(e) 71 743 y(that)300 822 y(\011)338 801 y Fu(G)367 822 y Fo(\()p Fm(A)423 800 y Fu(x)443 805 y Fg(0)423 834 y Fu(k)470 822 y Fl(j)8 b Fo(the)16 b(in)o(v)m(asion)g(pro)q(cess)h(up)g (to)f(time)f Fm(\034)6 b Fo(\))13 b Fl(\025)h Fo(1)d Fl(\000)g Fm(")d(;)71 925 y Fo(where)16 b Fm(A)249 903 y Fu(x)269 908 y Fg(0)249 937 y Fu(k)306 925 y Fo(is)h(the)g(ev)o(en)o (t)e(that)j(the)f(p)q(ercolation)g(pro)q(cess)h(restricted)e(to)i(the)f (edge)71 1003 y(set)f Fm(E)e Fl(n)d Fo(\()p Fm(I)278 985 y Fu(x)298 990 y Fg(0)274 1015 y Fu(\034)327 1003 y Fl([)h Fm(@)s(I)427 985 y Fu(x)447 990 y Fg(0)423 1015 y Fu(\034)465 1003 y Fo(\))k(has)h(at)g(least)f Fm(k)i Fo(in\014nite)d Fm(p)993 1010 y Fr(1)1014 1003 y Fo(-clusters)h(that)95 1118 y(\(i\))g(are)g(con)o(tained)g(in)g(separate)g Fm(p)738 1125 y Fr(2)759 1118 y Fo(-clusters,)f(and)95 1236 y(\(ii\))g(con)o (tain)h(some)g(v)o(ertex)e(at)j(distance)f(1)g(from)g Fm(I)1050 1218 y Fu(x)1070 1223 y Fg(0)1046 1249 y Fu(\034)1088 1236 y Fo(.)71 1351 y(If)f Fm(x)147 1358 y Fr(0)183 1351 y Fo(is)h(in)g(an)h(in\014nite)e Fm(p)546 1358 y Fr(0)566 1351 y Fo(-cluster,)g(then)559 1474 y Fm(U)5 b Fo(\()p Fm(e)p Fo(\))13 b Fl(\024)h Fm(p)748 1481 y Fr(0)784 1474 y Fo(for)j(ev)o(ery)e Fm(e)e Fl(2)h Fm(I)1097 1454 y Fu(x)1117 1459 y Fg(0)1093 1486 y Fu(\034)1144 1474 y Fm(:)401 b Fo(\(12\))71 1597 y(Giv)o(en)14 b(\(12\))h(and)h(the)f(in) o(v)m(asion)g(pro)q(cess)h(up)f(to)h(time)d Fm(\034)6 b Fo(,)14 b(eac)o(h)h Fm(e)e Fl(2)h Fm(@)s(I)1384 1579 y Fu(x)1404 1584 y Fg(0)1380 1609 y Fu(\034)1438 1597 y Fo(is)g(op)q(en)i(at)71 1675 y(lev)o(el)h Fm(p)209 1682 y Fr(1)248 1675 y Fo(indep)q(enden)o(tly)g(with)i(conditional)g (probabilit)o(y)f(at)i(least)1372 1654 y Fu(p)1390 1659 y Fg(1)1407 1654 y Fh(\000)p Fu(p)1452 1659 y Fg(0)p 1372 1663 98 2 v 1380 1692 a Fr(1)p Fh(\000)p Fu(p)1443 1697 y Fg(0)1474 1675 y Fo(.)30 b(If)19 b Fm(A)1607 1653 y Fu(x)1627 1658 y Fg(0)1607 1688 y Fu(k)71 1753 y Fo(happ)q(ens,)h(w)o (e)e(ma)o(y)g(pic)o(k)f Fm(e)590 1760 y Fr(1)610 1753 y Fm(;)8 b(:)g(:)g(:)f(e)720 1760 y Fu(k)760 1753 y Fl(2)18 b Fm(@)s(I)866 1735 y Fu(x)886 1740 y Fg(0)862 1766 y Fu(\034)923 1753 y Fo(adjacen)o(t)h(to)h Fm(k)h Fo(di\013eren)o(t)d Fm(p)1449 1760 y Fr(1)1469 1753 y Fo(-clusters)71 1832 y(with)h(the)g(prop)q(erties)g(\(i\),)g(\(ii\))f(ab)q(o)o(v)o(e.)30 b(These)19 b(prop)q(erties)g(guaran)o(tee)h(that)f(if)g Fm(x)1626 1839 y Fr(0)71 1910 y Fo(is)c(in)h(an)g(in\014nite)f Fm(p)432 1917 y Fr(0)452 1910 y Fo(-cluster)g(and)h(at)g(least)g(t)o(w) o(o)g(of)g(the)f(edges)h Fm(e)1275 1917 y Fr(1)1294 1910 y Fm(;)8 b(:)g(:)g(:)g(;)g(e)1427 1917 y Fu(k)1463 1910 y Fo(are)16 b(op)q(en)71 1988 y(at)g(lev)o(el)f Fm(p)266 1995 y Fr(1)286 1988 y Fo(,)h(then)g Fm(x)455 1995 y Fr(0)491 1988 y Fo(is)h(in)f(an)h(in\014nite)e Fm(p)p Fo(-cluster)i(with)f(at)h(least)f(t)o(w)o(o)h(ends)f(for)h(all)71 2066 y Fm(p)d Fl(2)g Fo([)p Fm(p)194 2073 y Fr(1)214 2066 y Fm(;)8 b(p)260 2073 y Fr(2)280 2066 y Fo(].)21 b(Hence)219 2212 y(\011)257 2191 y Fu(G)286 2212 y Fo(\()318 2199 y(~)305 2212 y Fm(H)349 2191 y Fu(x)369 2196 y Fg(0)345 2224 y Fu(p)363 2229 y Fg(0)381 2224 y Fu(;p)409 2229 y Fg(1)425 2224 y Fu(;p)453 2229 y Fg(2)472 2212 y Fo(\))14 b Fl(\024)g Fm(")d Fo(+)641 2139 y Fi( )678 2178 y Fo(1)h Fl(\000)f Fm(p)788 2185 y Fr(1)p 678 2200 130 2 v 678 2246 a Fo(1)h Fl(\000)f Fm(p)788 2253 y Fr(0)813 2139 y Fi(!)846 2150 y Fu(k)878 2212 y Fo(+)g Fm(k)963 2139 y Fi( )1000 2178 y Fm(p)1024 2185 y Fr(1)1056 2178 y Fl(\000)f Fm(p)1129 2185 y Fr(0)p 1000 2200 150 2 v 1010 2246 a Fo(1)i Fl(\000)f Fm(p)1120 2253 y Fr(0)1155 2139 y Fi(!)d( )1234 2178 y Fo(1)j Fl(\000)g Fm(p)1343 2185 y Fr(1)p 1234 2200 130 2 v 1234 2246 a Fo(1)g Fl(\000)g Fm(p)1343 2253 y Fr(0)1368 2139 y Fi(!)1401 2150 y Fu(k)q Fh(\000)p Fr(1)1484 2212 y Fm(:)71 2348 y Fo(Sending)17 b Fm(")f Fl(!)f Fo(0)j(and)g Fm(k)g Fl(!)e(1)h Fo(pro)o(v)o(es)g (\(10\),)h(and)g(th)o(us)f(also)h(\(9\))g(and)g(\(8\),)g(so)g(the)71 2426 y(pro)q(of)f(is)f(complete.)p 1616 2414 31 2 v 1616 2438 2 25 v 1644 2438 V 1616 2440 31 2 v 71 2528 a(W)l(e)g(end)h(this)g (section)g(b)o(y)g(noting)g(the)g(follo)o(wing)g(v)o(ery)f(simple)f (corollary)i(to)g(The-)71 2606 y(orem)h(1.4.)32 b(It)19 b(is)g(the)h(natural)g(analogue)g(for)g(the)g(uniqueness)f(regime)e(\() p Fm(p)1498 2613 y Fu(u)1521 2606 y Fm(;)8 b Fo(1\))20 b(of)71 2684 y(Theorem)15 b(1.6.)p eop %%Page: 16 16 16 15 bop 257 120 a Fo(16)796 b Fn(H\177)-24 b(aggstr\177)g(om,)16 b(P)o(eres)f(and)i(Sc)o(honmann)257 274 y Fk(Corollary)h(4.1)24 b Fj(L)n(et)17 b Fm(G)h Fj(b)n(e)g(an)g(in\014nite,)h(lo)n(c)n(al)r(ly) f(\014nite,)h(c)n(onne)n(cte)n(d,)f(semi-tr)n(ansi-)257 352 y(tive)f(gr)n(aph.)k(Then,)c Fo(\011)680 334 y Fu(G)709 352 y Fj(-a.s.,)g(for)e(al)r(l)i Fm(p)d Fl(2)g Fo(\()p Fm(p)1105 359 y Fu(u)1128 352 y Fm(;)8 b Fo(1\))16 b Fj(the)h(\(unique\))g(in\014nite)g Fm(p)p Fj(-cluster)257 430 y(has)h(a)f(single)i(end.)257 557 y Fk(Pro)r(of:)i Fo(Supp)q(ose)16 b(for)f(con)o(tradiction)g(that)g(with)g(p)q(ositiv)o (e)f(\011)1420 539 y Fu(G)1450 557 y Fo(-probabilit)o(y)l(,)g(there)257 635 y(exists)21 b(some)f Fm(p)i Fl(2)g Fo(\()p Fm(p)668 642 y Fu(u)691 635 y Fm(;)8 b Fo(1\))22 b(for)f(whic)o(h)f(the)h (in\014nite)f(cluster)g(has)i(more)e(than)h(one)257 714 y(end.)46 b(An)o(y)23 b(realization)h Fm(\021)29 b Fl(2)f Fo([0)p Fm(;)8 b Fo(1])964 695 y Fu(E)1018 714 y Fo(of)24 b(the)h Fl(f)p Fm(U)5 b Fo(\()p Fm(e)p Fo(\))p Fl(g)1323 721 y Fu(e)p Fh(2)p Fu(E)1416 714 y Fo(v)m(ariables)24 b(for)h(whic)o(h)257 792 y(this)c(happ)q(ens)g(at)g(lev)o(el)d Fm(p)j Fo(can)f(b)q(e)h(mo)q(di\014ed)e(in)o(to)h(a)h(con\014guration)g Fm(\021)1616 774 y Fh(0)1648 792 y Fo(in)f(whic)o(h)257 870 y(uniqueness)h(of)g(the)f(in\014nite)g(cluster)g(fails)h(at)g(lev)o (el)d Fm(p)p Fo(,)k(b)o(y)f(c)o(hanging)g(the)f(status)257 948 y(of)g(just)g(\014nitely)e(man)o(y)h(edges.)31 b(It)19 b(follo)o(ws)h(that)g(with)f(p)q(ositiv)o(e)g(\011)1544 930 y Fu(G)1574 948 y Fo(-probabilit)o(y)l(,)257 1027 y(there)f(is)g(some)e Fm(p)i Fl(2)f Fo(\()p Fm(p)693 1034 y Fu(u)716 1027 y Fm(;)8 b Fo(1\))18 b(for)g(whic)o(h)f (uniqueness)h(of)g(the)g(in\014nite)f(cluster)g(fails,)257 1105 y(con)o(tradicting)f(Theorem)f(1.4.)p 1802 1092 31 2 v 1802 1117 2 25 v 1831 1117 V 1802 1119 31 2 v 257 1307 a FB(5)81 b(Indistinguishabilit)n(y)23 b(b)n(y)j(robust)h (prop)r(erties)257 1437 y Fk(Pro)r(of)17 b(of)g(Theorem)e(1.8:)29 b Fo(Fix)13 b Fm(p)934 1444 y Fr(0)969 1437 y Fl(2)h Fo(\()p Fm(p)1059 1444 y Fu(c)1076 1437 y Fm(;)8 b(p)p Fo(\).)21 b(Since,)14 b(b)o(y)g(Theorem)g(1.4,)h(\011)1716 1419 y Fu(G)1745 1437 y Fo(-a.s.)257 1515 y(an)o(y)g(in\014nite)e Fm(p)p Fo(-cluster)h(con)o(tains)h(an)f(in\014nite)g Fm(p)1150 1522 y Fr(0)1170 1515 y Fo(-cluster,)g(it)f(su\016ces)h(to)h (sho)o(w)g(that)257 1594 y(for)i(all)f Fm(x)d Fl(2)h Fm(V)e Fo(,)574 1726 y(\011)612 1705 y Fu(G)642 1726 y Fo([)p Fl(C)s Fo(\()p Fm(x;)c(p)778 1733 y Fr(0)797 1726 y Fo(\))24 b(is)17 b(in\014nite)e(and)i Fl(C)s Fo(\()p Fm(x;)8 b(p)p Fo(\))19 b Fm(=)-29 b Fl(2)14 b Fm(Q)p Fo(])f(=)g(0)8 b Fm(:)231 b Fo(\(13\))331 1859 y(De\014ne)13 b(the)h(random)f(v)m(ariable)g Fm(\030)940 1841 y Fu(x)977 1859 y Fo(as)h(the)g(n)o(um)o(b)q(er)e(of)i(edges)g(that)g(are)g (adjacen)o(t)257 1938 y(to,)23 b(or)f(con)o(tained)g(in,)g Fm(I)732 1920 y Fu(x)728 1950 y Fh(1)787 1938 y Fo(and)g(are)g(also)g (adjacen)o(t)g(to,)h(or)f(con)o(tained)f(in,)h(some)257 2016 y(in\014nite)c Fm(p)p Fo(-cluster)g(whic)o(h)g(satis\014es)g Fm(Q)p Fo(.)27 b(W)l(e)18 b(will)g(establish)g(\(13\))h(b)o(y)e(pro)o (ving)i(the)257 2094 y(follo)o(wing)d(t)o(w)o(o)g(statemen)o(ts:)861 2226 y(\011)899 2206 y Fu(G)929 2226 y Fo([)p Fm(\030)966 2206 y Fu(x)1002 2226 y Fm(<)d Fl(1)p Fo(])h(=)f(0)8 b Fm(;)518 b Fo(\(14\))257 2358 y(and)474 2441 y(\011)512 2420 y Fu(G)542 2441 y Fo([)p Fl(C)s Fo(\()p Fm(x;)8 b(p)678 2448 y Fr(0)697 2441 y Fo(\))25 b(is)16 b(in\014nite,)f Fm(\030)992 2420 y Fu(x)1028 2441 y Fo(=)f Fl(1)24 b Fo(and)17 b Fl(C)s Fo(\()p Fm(x;)8 b(p)p Fo(\))20 b Fm(=)-30 b Fl(2)14 b Fm(Q)p Fo(])f(=)h(0)8 b Fm(:)131 b Fo(\(15\))331 2552 y(W)l(e)14 b(\014rst)h(pro)o(v)o(e)f(\(14\).)22 b(By)14 b(the)h(0-1)g(la)o(w)g(for)g(automorphism-in)o(v)m(arian)o(t)e (ev)o(en)o(ts,)569 2684 y(\011)607 2664 y Fu(G)637 2684 y Fo([)p Fl(9)29 b Fo(an)17 b(in\014nite)e Fm(p)p Fo(-cluster)h (satisfying)h Fm(Q)p Fo(])c(=)g(1)8 b Fm(:)p eop %%Page: 17 17 17 16 bop 71 120 a Fn(P)o(ercolation)15 b(on)i(transitiv)o(e)e(graphs) 845 b Fo(17)71 274 y(Therefore,)15 b(for)h(an)o(y)h Fm(")c(>)h Fo(0,)i(w)o(e)g(can)g(pic)o(k)f(an)i Fm(R)g Fo(so)g(that)123 379 y(inf)118 409 y Fu(y)q Fh(2)p Fu(V)196 379 y Fo(\011)234 358 y Fu(G)264 331 y Fi(\020)289 379 y Fo(no)f(in\014nite)g Fm(p)p Fo(-cluster)g(with)g(prop)q(ert)o(y)g Fm(Q)g Fo(in)o(tersects)f Fm(B)s Fo(\()p Fm(y)r(;)8 b(R)p Fo(\))1464 331 y Fi(\021)1502 379 y Fm(<)14 b(")8 b(:)1559 457 y Fo(\(16\))71 535 y(Let)k Fm(\034)18 b Fo(b)q(e)13 b(the)f(smallest)f Fm(m)h Fo(for)h(whic)o(h)f Fm(I)808 517 y Fu(x)804 548 y(m)849 535 y Fo(con)o(tains)g(a)h(ball)f (of)h(radius)g Fm(R)p Fo(;)h(b)o(y)e(Lemma)71 614 y(3.2,)k Fm(\034)j(<)14 b Fl(1)i Fo(a.s.)144 692 y(F)l(or)j(an)h(edge)f(set)h Fm(E)535 699 y Fr(0)574 692 y Fl(\032)e Fm(E)s Fo(,)i(let)f Fm(@)s(E)843 699 y Fr(0)881 692 y Fo(b)q(e)h(the)f(set)g(of)h(edges)g (outside)f Fm(E)1517 699 y Fr(0)1556 692 y Fo(that)71 770 y(are)e(adjacen)o(t)g(to)h Fm(E)446 777 y Fr(0)465 770 y Fo(,)g(and)f(denote)h(b)o(y)e Fl(S)t Fo(\()p Fm(E)908 777 y Fr(0)928 770 y Fo(\))h(the)g(set)g(of)h(edges)f(in)g Fm(@)s(E)1437 777 y Fr(0)1474 770 y Fo(that)h(are)71 848 y(adjacen)o(t)j(to)h(an)g(in\014nite)e(connected)h(comp)q(onen)o(t) f(of)i Fl(f)p Fm(e)28 b(=)-30 b Fl(2)23 b Fm(@)s(E)1318 855 y Fr(0)1369 848 y Fo(:)30 b Fm(U)5 b Fo(\()p Fm(e)p Fo(\))23 b Fl(\024)f Fm(p)p Fl(g)71 927 y Fo(whic)o(h)15 b(has)i(prop)q(ert)o(y)f Fm(Q)p Fo(.)144 1005 y(If)g(a)h(\014nite)f (edge)h(set)f Fm(E)583 1012 y Fr(0)620 1005 y Fo(in)o(tersects)f(an)j (in\014nite)d Fm(p)p Fo(-cluster)i(that)g(has)h(prop)q(ert)o(y)71 1083 y Fm(Q)p Fo(,)13 b(then)h(robustness)g(of)g Fm(Q)g Fo(implies)d(that)j Fl(S)t Fo(\()p Fm(E)943 1090 y Fr(0)963 1083 y Fo(\))g Fl(6)p Fo(=)f Fl(;)p Fo(.)21 b(Therefore,)13 b(an)o(y)h(\014nite)f(edge)71 1161 y(set)19 b Fm(E)186 1168 y Fr(0)225 1161 y Fo(that)g(con)o(tains)h(a)f(ball)g(of)h(radius)f Fm(R)p Fo(,)h(satis\014es)g(\011)1169 1143 y Fu(G)1198 1113 y Fi(\020)1223 1161 y Fl(S)t Fo(\()p Fm(E)1312 1168 y Fr(0)1332 1161 y Fo(\))f(=)g Fl(;)1452 1113 y Fi(\021)1495 1161 y Fm(<)g(")g Fo(b)o(y)71 1240 y(\(16\).)144 1318 y(Since)e(the)g(in)o(v)m(asion)h(cluster)g Fm(I)733 1300 y Fu(x)729 1330 y(\034)772 1318 y Fo(giv)o(es)f(no)h(information)f(ab)q (out)j(the)d(lab)q(els)h(on)71 1396 y(edges)e(not)h(in)f Fm(I)371 1378 y Fu(x)367 1409 y(\034)403 1396 y Fl([)c Fm(@)s(I)503 1378 y Fu(x)499 1409 y(\034)524 1396 y Fo(,)k(w)o(e)g(ma)o (y)e(apply)j(the)f(ab)q(o)o(v)o(e)g(reasoning)h(with)g Fm(E)1453 1403 y Fr(0)1486 1396 y Fo(=)d Fm(I)1564 1378 y Fu(x)1560 1409 y(\034)1602 1396 y Fo(to)71 1474 y(deduce)h(that)i (\011)377 1456 y Fu(G)406 1426 y Fi(\020)431 1474 y Fl(S)t Fo(\()p Fm(I)510 1456 y Fu(x)506 1487 y(\034)531 1474 y Fo(\))d(=)g Fl(;)641 1426 y Fi(\021)679 1474 y Fm(<)g(")p Fo(.)21 b(In)16 b(particular,)f(\011)1126 1456 y Fu(G)1156 1474 y Fo(\()p Fm(\030)1198 1456 y Fu(x)1234 1474 y Fo(=)f(0\))g Fm(<)g(")p Fo(,)i(and)g(since)71 1553 y Fm(")e Fo(w)o(as)h(arbitrary)l (,)g(\011)454 1535 y Fu(G)484 1553 y Fo(\()p Fm(\030)526 1535 y Fu(x)562 1553 y Fo(=)f(0\))g(=)g(0)8 b Fm(:)15 b Fo(On)g(the)f(ev)o(en)o(t)f(\()p Fm(\030)1115 1535 y Fu(x)1152 1553 y Fo(=)g Fm(n)p Fo(\),)i(w)o(e)f(can)h(mo)o(v)o(e)e (in)o(to)71 1631 y(the)k(ev)o(en)o(t)g(\()p Fm(\030)329 1613 y Fu(x)368 1631 y Fo(=)g(0\))h(b)o(y)f(c)o(hanging)h(the)g(lab)q (els)g Fm(U)5 b Fo(\()p Fm(e)p Fo(\))18 b(on)g(\014nitely)f(man)o(y)f (edges)i(to)71 1709 y(b)q(e)d(greater)g(than)g Fm(p)p Fo(;)h(it)e(follo)o(ws)h(that)h(\011)815 1691 y Fu(G)844 1709 y Fo(\()p Fm(\030)886 1691 y Fu(x)923 1709 y Fo(=)e Fm(n)p Fo(\))f(=)h(0)i(for)f(an)o(y)g Fm(n)f(<)g Fl(1)p Fo(,)g(and)i(\(14\))71 1787 y(is)g(established.)144 1866 y(T)l(o)k(pro)o(v)o(e)e(\(15\),)i(observ)o(e)f(that)h(if)f Fl(C)s Fo(\()p Fm(x;)8 b(p)929 1873 y Fr(0)948 1866 y Fo(\))20 b(is)f(in\014nite,)f(then)i Fm(I)1362 1848 y Fu(x)1358 1878 y Fh(1)1413 1866 y Fl(\022)f(C)s Fo(\()p Fm(x;)8 b(p)1593 1873 y Fr(0)1613 1866 y Fo(\).)71 1944 y(Therefore,)14 b(if)h(also)g Fm(\030)467 1926 y Fu(x)504 1944 y Fo(=)e Fl(1)i Fo(and)h Fl(C)s Fo(\()p Fm(x;)8 b(p)p Fo(\))20 b Fm(=)-30 b Fl(2)14 b Fm(Q)p Fo(,)g(then)h(the)g(follo) o(wing)g(ev)o(en)o(t,)e(whic)o(h)71 2022 y(w)o(e)g(call)g Fm(F)7 b Fo(,)13 b(m)o(ust)g(happ)q(en:)21 b Fl(C)s Fo(\()p Fm(x;)8 b(p)p Fo(\))19 b Fm(=)-29 b Fl(2)14 b Fm(Q)f Fo(but)h(there)f(are)h(in\014nitely)f(man)o(y)f(edges)i(in)71 2100 y Fm(@)s Fl(C)s Fo(\()p Fm(x;)8 b(p)222 2107 y Fr(0)241 2100 y Fo(\))14 b(whic)o(h)f(are)g(adjacen)o(t)h(to)g Fm(p)p Fo(-clusters)g Fl(C)j(2)d Fm(Q)f Fo(suc)o(h)h(that)g Fl(C)9 b(\\)d(C)s Fo(\()p Fm(x;)i(p)1503 2107 y Fr(0)1523 2100 y Fo(\))13 b(=)h Fl(;)p Fo(.)71 2179 y(T)l(o)i(establish)g (\(15\),)g(it)f(su\016ces)h(to)g(sho)o(w)g(that)h(\011)996 2161 y Fu(G)1025 2179 y Fo(\()p Fm(F)7 b Fo(\))13 b(=)h(0.)21 b(T)l(o)c(pro)o(v)o(e)e(this,)g(write)71 2257 y Fl(C)97 2264 y Fu(`)113 2257 y Fo(\()p Fm(x;)8 b(p)206 2264 y Fr(0)226 2257 y Fo(\))18 b(for)f(the)h(collection)e(of)i(edges)g(that)g (can)f(b)q(e)h(reac)o(hed)f(from)g Fm(x)g Fo(b)o(y)g(a)h(path)71 2335 y(whic)o(h)c(is)h(op)q(en)g(at)h(lev)o(el)c Fm(p)565 2342 y Fr(0)601 2335 y Fo(and)j(is)g(con)o(tained)f(in)h(the)g(ball)f Fm(B)s Fo(\()p Fm(x;)8 b(`)p Fo(\).)21 b(Consider)15 b(the)71 2413 y(ev)o(en)o(t)d Fm(F)229 2420 y Fu(`;k)287 2413 y Fo(that)i Fl(C)s Fo(\()p Fm(x;)8 b(p)p Fo(\))20 b Fm(=)-30 b Fl(2)14 b Fm(Q)f Fo(and)h Fm(@)s Fl(C)791 2420 y Fu(`)808 2413 y Fo(\()p Fm(x;)8 b(p)901 2420 y Fr(0)920 2413 y Fo(\))14 b(con)o(tains)f(at)h(least)f Fm(k)j Fo(edges)e(that)g(are)71 2492 y(adjacen)o(t)k(to)h(in\014nite)f Fm(p)p Fo(-clusters)h Fl(C)i(2)d Fm(Q)g Fo(suc)o(h)g(that)h Fl(C)d(\\)d(C)s Fo(\()p Fm(x;)8 b(p)1303 2499 y Fr(0)1323 2492 y Fo(\))18 b(=)g Fl(;)p Fo(.)28 b(Clearly)l(,)71 2570 y(for)16 b(ev)o(ery)f Fm(k)r Fo(,)349 2675 y Fm(F)20 b Fl(\022)14 b([)487 2654 y Fh(1)487 2687 y Fu(`)p Fr(=1)548 2675 y Fm(F)580 2682 y Fu(`;k)634 2675 y Fm(;)32 b Fo(whence)24 b(\011)897 2654 y Fu(G)927 2675 y Fo(\()p Fm(F)7 b Fo(\))13 b Fl(\024)22 b Fo(lim)1070 2705 y Fu(`)p Fh(!1)1163 2675 y Fo(\011)1201 2654 y Fu(G)1230 2675 y Fo(\()p Fm(F)1281 2682 y Fu(`;k)1326 2675 y Fo(\))8 b Fm(:)p eop %%Page: 18 18 18 17 bop 257 120 a Fo(18)796 b Fn(H\177)-24 b(aggstr\177)g(om,)16 b(P)o(eres)f(and)i(Sc)o(honmann)257 274 y Fo(The)g(pro)q(of)g(will)e (therefore)h(b)q(e)g(complete)e(once)i(w)o(e)g(establish)g(that)h(for)f (an)o(y)g Fm(`;)8 b(k)r Fo(,)796 424 y(\011)834 404 y Fu(G)864 424 y Fo(\()p Fm(F)915 431 y Fu(`;k)960 424 y Fo(\))14 b Fl(\024)1045 351 y Fi( )1093 391 y Fo(1)d Fl(\000)g Fm(p)p 1083 413 130 2 v 1083 459 a Fo(1)g Fl(\000)g Fm(p)1192 466 y Fr(0)1217 351 y Fi(!)1250 363 y Fu(k)1280 424 y Fm(:)452 b Fo(\(17\))257 565 y(Clearly)l(,)567 645 y Fm(F)599 652 y Fu(`;k)658 645 y Fl(\022)710 597 y Fi(n)738 645 y Fl(C)s Fo(\()p Fm(x;)8 b(p)p Fo(\))20 b Fm(=)-30 b Fl(2)14 b Fm(Q)24 b Fo(and)h Fl(jS)t Fo(\()p Fl(C)1199 652 y Fu(`)1216 645 y Fo(\()p Fm(x;)8 b(p)1309 652 y Fr(0)1328 645 y Fo(\)\))p Fl(j)14 b(\025)g Fm(k)1474 597 y Fi(o)1510 645 y Fm(:)257 753 y Fo(\(In)22 b(fact)f(these)g(ev)o (en)o(ts)g(coincide,)g(but)h(w)o(e)f(do)h(not)g(need)f(this.\))37 b(Therefore,)22 b(b)o(y)257 831 y(robustness)c(of)e Fm(Q)p Fo(,)344 962 y Fm(F)376 969 y Fu(`;k)435 962 y Fl(\022)488 914 y Fi(n)515 962 y Fl(jS)t Fo(\()p Fl(C)608 969 y Fu(`)625 962 y Fo(\()p Fm(x;)8 b(p)718 969 y Fr(0)738 962 y Fo(\)\))p Fl(j)13 b(\025)h Fm(k)32 b Fo(and)f Fl(8)p Fm(e)12 b Fl(2)i(S)t Fo(\()p Fl(C)1212 969 y Fu(`)1228 962 y Fo(\()p Fm(x;)8 b(p)1321 969 y Fr(0)1341 962 y Fo(\)\))49 b Fm(U)5 b Fo(\()p Fm(e)p Fo(\))14 b Fl(\025)f Fm(p)1617 914 y Fi(o)1645 962 y Fm(:)87 b Fo(\(18\))257 1092 y(Denote)23 b(b)o(y)f Fl(F)541 1099 y Fu(`)580 1092 y Fo(the)g Fm(\033)r Fo(-\014eld)g(generated)g(b)o(y)g Fl(C)1156 1099 y Fu(`)1173 1092 y Fo(\()p Fm(x;)8 b(p)1266 1099 y Fr(0)1286 1092 y Fo(\))22 b(and)i(the)e(lab)q(els)g Fl(f)p Fm(U)5 b Fo(\()p Fm(e)p Fo(\))33 b(:)257 1170 y Fm(e)28 b(=)-30 b Fl(2)23 b Fm(@)s Fl(C)413 1177 y Fu(`)429 1170 y Fo(\()p Fm(x;)8 b(p)522 1177 y Fr(0)542 1170 y Fo(\))p 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b(up)i(naturally)g(are)f(unimo)q(dular.)21 b(In)15 b(particular,)g(the)g(Ca)o(yley)g(graph)i(of)e(an)o(y)257 2371 y(\014nitely)i(generated)h(group)g(is)g(transitiv)o(e)f(and)h (unimo)q(dular.)25 b(A)18 b(transitiv)o(e)e(graph)268 2433 y Fi(b)257 2449 y Fm(T)286 2456 y Fu(d)326 2449 y Fo(whic)o(h)j(is)h Fj(not)25 b Fo(unimo)q(dular)19 b(can)h(b)q(e)g(constructed)f(b)o(y)h(considering)f(a)h(regular)257 2528 y(tree)g Fm(T)388 2535 y Fu(d)429 2528 y Fo(with)g(degree)g Fm(d)h Fl(\025)g Fo(3,)h(\014xing)e(an)h(end)g Fm(\030)i Fo(of)e Fm(T)1306 2535 y Fu(d)1326 2528 y Fo(,)g(and)g(for)g(eac)o(h)f (v)o(ertex)f Fm(x)257 2606 y Fo(adding)i(an)f(edge)g(b)q(et)o(w)o(een)f Fm(x)g Fo(and)i(its)e Fm(\030)r Fo(-grandparen)o(t;)k(see)d([31])f(or)i ([7)o(].)32 b(In)20 b(this)257 2684 y(example,)d Fm(p)489 2691 y Fu(u)529 2684 y Fo(=)h(1,)h(and)g(for)f Fm(p)g Fl(2)g Fo(\()p Fm(p)951 2691 y Fu(c)969 2684 y Fm(;)8 b Fo(1\))19 b(ev)o(ery)d(in\014nite)i(cluster)g Fl(C)j Fo(has)e(a)g(unique)p eop %%Page: 19 19 19 18 bop 71 120 a Fn(P)o(ercolation)15 b(on)i(transitiv)o(e)e(graphs) 845 b Fo(19)71 274 y(v)o(ertex)16 b Fm(v)r Fo(\()p Fl(C)s Fm(;)8 b(\030)r Fo(\))18 b(that)h(is)f(\\closest")g(to)h Fm(\030)r Fo(.)27 b(Th)o(us,)19 b(as)f(noted)h(in)e([24)q(],)g(the)h (in)o(v)m(arian)o(t)71 352 y(prop)q(ert)o(y)h Fl(fC)31 b Fo(:)d Fm(v)r Fo(\()p Fl(C)s Fm(;)8 b(\030)r Fo(\))16 b(has)h(degree)f(1)h(in)f Fl(C)s(g)k Fo(distinguishes)f(some)g (in\014nite)g(clus-)71 430 y(ters)c(in)231 414 y Fi(b)221 430 y Fm(T)250 437 y Fu(d)285 430 y Fo(from)f(others.)21 b(By)15 b(utilizing)f(more)g(of)i(the)f(lo)q(cal)g(structure)g(of)h(a)f (cluster)71 509 y Fl(C)21 b Fo(as)e(seen)f(from)g Fm(v)r Fo(\()p Fl(C)s Fm(;)8 b(\030)r Fo(\),)18 b Fj(any)23 b Fo(t)o(w)o(o)18 b(in\014nite)f(clusters)h(in)1185 493 y Fi(b)1174 509 y Fm(T)1203 516 y Fu(d)1242 509 y Fo(ma)o(y)e(b)q(e)j (in)o(v)m(arian)o(tly)71 587 y(distinguished.)144 674 y(The)13 b(signi\014cance)f(of)i(unimo)q(dularit)o(y)d(for)i(us)g(is)g 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Fm(m)p Fo(\()p Fl(\001)p Fm(;)8 b Fl(\001)p Fm(;)g Fl(\001)p Fo(\))16 b Fj(as)h(ab)n(ove,)h(let)471 1866 y Fm(M)5 b Fo(\()p Fm(x;)j(y)r Fo(\))13 b(=)702 1807 y Fi(Z)725 1902 y Fh(f)p Fr(0)p Fu(;)p Fr(1)p Fh(g)807 1892 y Ff(E)842 1866 y Fm(m)p Fo(\()p Fm(x;)8 b(y)r(;)g(!)r Fo(\))g Fm(d)p Fk(P)1124 1845 y Fu(G)1124 1878 y(p)1153 1866 y Fo(\()p Fm(!)r Fo(\))g Fm(:)71 2026 y Fj(Then)23 b(the)g(exp)n(e)n(cte)n(d)g (total)g(mass)f(tr)n(ansp)n(orte)n(d)f(out)i(of)f(any)h(vertex)h Fm(x)e Fj(e)n(quals)h(the)71 2105 y(exp)n(e)n(cte)n(d)18 b(total)g(mass)f(tr)n(ansp)n(orte)n(d)f(into)h Fm(x)p Fj(,)h(i.e.,)463 2260 y Fl(8)p Fm(x)12 b Fl(2)i Fm(V)682 2219 y Fi(X)677 2311 y Fu(y)q Fh(2)p Fu(V)756 2260 y Fm(M)5 b Fo(\()p Fm(x;)j(y)r Fo(\))13 b(=)992 2219 y Fi(X)987 2311 y Fu(y)q Fh(2)p Fu(V)1066 2260 y Fm(M)5 b Fo(\()p Fm(y)r(;)j(x)p Fo(\))g Fm(:)305 b Fo(\(19\))71 2449 y(W)l(e)21 b(remark)f(that)i(\(19\))h(fails)e(in)h(the)f(non)o (unimo)q(dular)g(case;)j(see)e([7].)37 b(The)22 b(k)o(ey)71 2528 y(elemen)o(t)f(in)i(a)i(successful)e(application)h(of)h(the)f (mass-transp)q(ort)h(metho)q(d)e(is)h(to)71 2606 y(mak)o(e)13 b(a)i(suitable)g(c)o(hoice)f(of)i(the)e(transp)q(ort)j(function)e Fm(m)p Fo(\()p Fl(\001)p Fm(;)8 b Fl(\001)p Fm(;)g Fl(\001)p Fo(\);)14 b(examples)f(can)i(b)q(e)71 2684 y(found)h(e.g.)g(in)g([17,)g (7,)g(8,)g(18)q(],)f(and)i(also)g(in)f(Section)g(7)g(b)q(elo)o(w.)p eop %%Page: 20 20 20 19 bop 257 120 a Fo(20)796 b Fn(H\177)-24 b(aggstr\177)g(om,)16 b(P)o(eres)f(and)i(Sc)o(honmann)257 274 y FB(7)81 b(Relen)n(tless)25 b(merging)257 452 y Fo(The)c(main)f(step)h(in)g(pro)o(ving)g(Theorem)f (1.5)h(is)g(sho)o(wing)h(that)f(for)g Fm(p)i Fl(2)f Fo(\()p Fm(p)1713 459 y Fu(c)1731 452 y Fm(;)8 b(p)1777 459 y Fu(u)1800 452 y Fo(\),)257 530 y(an)o(y)20 b(in\014nite)f Fm(p)p Fo(-cluster)h(will)f(a.s.)h(come)e(within)i(distance)g(1)g(from) f(other)h(in\014nite)257 608 y Fm(p)p Fo(-clusters)d(in)e(in\014nitely) g(man)o(y)g(places,)g(as)i(stated)f(in)g(the)g(follo)o(wing)g(prop)q (osition.)257 686 y(The)d(result)f(assumes)g(quasi-transitivit)o(y)f (and)i(unimo)q(dularit)o(y;)e(w)o(e)h(conjecture)g(the)257 765 y(latter)k(condition)g(to)h(b)q(e)f(remo)o(v)m(able.)257 1033 y Fk(Prop)r(osition)i(7.1)24 b Fj(Consider)19 b(b)n(ond)h(p)n(er)n (c)n(olation)f(on)h(an)f(in\014nite,)j(lo)n(c)n(al)r(ly)e(\014nite,)257 1111 y(c)n(onne)n(cte)n(d,)k(quasi-tr)n(ansitive)e(unimo)n(dular)f(gr)n (aph)g Fm(G)h Fj(with)f(r)n(etention)i(p)n(ar)n(ameter)257 1189 y Fm(p)15 b Fl(2)g Fo(\()p Fm(p)387 1196 y Fu(c)405 1189 y Fm(;)8 b(p)451 1196 y Fu(u)473 1189 y Fo(\))p Fj(.)23 b(Then,)c Fk(P)711 1171 y Fu(G)711 1201 y(p)741 1189 y Fj(-a.s.,)e(for)h(any)f(in\014nite)j(cluster)e Fl(C)j Fj(ther)n(e)d(exist)h(in\014nitely)257 1267 y(many)c(close)n(d)g (e)n(dges)h Fm(e)e Fj(with)h(one)h(endp)n(oint)f(in)h Fl(C)i Fj(and)d(the)g(other)g(endp)n(oint)g(in)g(some)257 1346 y(other)j(in\014nite)h(cluster)f(\(which)g(may)f(dep)n(end)h(on)g Fm(e)p Fj(\).)257 1589 y Fk(Pro)r(of:)24 b Fo(Assume)16 b(for)i(con)o(tradiction)e(that)i(with)g(p)q(ositiv)o(e)e Fk(P)1427 1571 y Fu(G)1427 1601 y(p)1457 1589 y Fo(-probabilit)o(y)h (there)257 1667 y(is)h(some)f(in\014nite)g(cluster)g Fl(C)k Fo(whic)o(h)c(comes)f(within)i(distance)f(exactly)g(1)h(from)f (the)257 1745 y(set)24 b(of)g(other)g(in\014nite)f(clusters)g(in)g(at)h (most)f(\014nitely)g(man)o(y)f(lo)q(cations.)44 b(As)24 b(in)257 1823 y(the)19 b(argumen)o(t)e(for)i(\(7\))g(in)g(the)f(pro)q (of)i(of)f(Prop)q(osition)h(3.1,)f(it)f(follo)o(ws)g(that)i(with)257 1902 y(p)q(ositiv)o(e)g(probabilit)o(y)f(there)h(is)g(some)f (in\014nite)g(cluster)h Fl(C)j Fo(in)d(whic)o(h)f(exactly)g(one)257 1980 y(v)o(ertex)e Fm(x)h Fo(is)f(at)i(distance)e(1)i(from)e(the)h(set) f(of)i(other)f(in\014nite)f(clusters.)26 b(Call)18 b(suc)o(h)257 2058 y(an)g(in\014nite)f(cluster)f Fl(C)21 b Fo(a)c Fj(kingdom)p Fo(,)h(call)f Fm(x)g Fo(its)g Fj(king)p Fo(,)i(and)f(consider)f(the)g (follo)o(wing)257 2136 y(mass)g(transp)q(ort.)25 b(If)17 b(a)h(v)o(ertex)d Fm(y)k Fo(is)e(in)g(a)g(kingdom,)f(and)i(its)f(king)g (is)g(in)g(the)f(same)257 2215 y(orbit)i(as)h Fm(y)h Fo(\(recall)d(De\014nition)h(1.1\),)g(then)g Fm(y)i Fo(sends)e(unit)g (mass)g(to)h(its)e(king.)27 b(If)18 b Fm(y)257 2293 y Fo(is)h(in)f(a)h(kingdom)e(but)i(not)g(in)f(the)g(same)f(orbit)i(as)g (the)f(king,)g(then)h Fm(y)h Fo(sends)f(unit)257 2371 y(mass)c(whic)o(h)g(it)g(distributes)g(equally)f(among)i(the)f(v)o (ertices)f(in)h(its)g(orbit)h(whic)o(h)e(are)257 2449 y(closest)i(in)f Fm(G)h Fo(to)g(the)g(king.)21 b(Otherwise,)14 b(no)i(mass)g(is)f(sen)o(t)g(from)g Fm(y)r Fo(.)21 b(The)15 b(exp)q(ected)257 2528 y(mass)i(sen)o(t)g(from)f(eac)o(h)h(v)o(ertex)e (is)i(then)g(at)h(most)e(1,)h(whereas)h(the)f(exp)q(ected)f(mass)257 2606 y(receiv)o(ed)f(has)k(to)e(b)q(e)h Fl(1)f Fo(for)h(some)e(v)o (ertices.)22 b(By)17 b(the)g(Mass-T)l(ransp)q(ort)j(Principle)257 2684 y(\(Theorem)15 b(6.2\))i(w)o(e)f(ha)o(v)o(e)f(the)h(desired)g(con) o(tradiction.)p 1802 2672 31 2 v 1802 2697 2 25 v 1831 2697 V 1802 2699 31 2 v eop %%Page: 21 21 21 20 bop 71 120 a Fn(P)o(ercolation)15 b(on)i(transitiv)o(e)e(graphs) 845 b Fo(21)71 274 y Fk(Pro)r(of)16 b(of)i(Theorem)c(1.5:)21 b Fo(W)l(e)14 b(\014rst)h(pro)o(v)o(e)f(the)h(assertion)g(of)g(the)g (theorem)e(with)71 352 y(the)j(quan)o(ti\014ers)f(in)o(terc)o(hanged,)g (i.e.)g(that)124 477 y(for)i(all)f Fm(p)291 484 y Fr(1)325 477 y Fm(<)d(p)400 484 y Fr(2)437 477 y Fo(in)j(\()p Fm(p)537 484 y Fu(c)555 477 y Fm(;)8 b(p)601 484 y Fu(u)623 477 y Fo(\),)16 b(w)o(e)g(ha)o(v)o(e)f(\011)894 459 y Fu(G)924 477 y Fo(-a.s.)h(that)h(an)o(y)f(in\014nite)f Fm(p)1414 484 y Fr(2)1434 477 y Fo(-cluster)124 555 y(con)o(tains)i (in\014nitely)d(man)o(y)h(in\014nite)g Fm(p)842 562 y Fr(1)862 555 y Fo(-clusters.)1559 620 y(\(20\))71 698 y(W)l(e)f(kno)o(w)h(from)f(Theorem)g(1.2)h(that)g(an)o(y)g(in\014nite)f Fm(p)1058 705 y Fr(2)1078 698 y Fo(-cluster)h(con)o(tains)g(some)f (in\014-)71 777 y(nite)f Fm(p)190 784 y Fr(1)210 777 y Fo(-cluster.)20 b(Hence,)13 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b(graphs)g(and)g(estimates)e(on)i Fd(p)1426 2485 y Fm(u)71 2606 y Fo(Let)d(us)h(collect)d(the)j(results)f (from)f(the)h(literature)f(that)i(are)f(needed)g(to)g(pro)o(v)o(e)71 2684 y(Theorem)17 b(1.9.)28 b(The)18 b(\014rst)h(one)g(concerns)f(p)q (ercolation)h(on)g(the)f(orthan)o(t)h Fk(Z)1503 2666 y Fu(d)1503 2696 y Fr(+)1533 2684 y Fo(.)27 b(F)l(or)p eop %%Page: 22 22 22 21 bop 257 120 a Fo(22)796 b Fn(H\177)-24 b(aggstr\177)g(om,)16 b(P)o(eres)f(and)i(Sc)o(honmann)257 274 y Fm(d)h Fo(=)g(2,)h(it)f (follo)o(ws)h(from)e(the)h(w)o(ork)h(of)g(Kesten)f([20],)g(while)g(for) h(general)f Fm(d)h Fo(it)f(w)o(as)257 352 y(\014rst)f(obtained)f(b)o(y) g(Barsky)l(,)g(Grimme)o(tt)d(and)k(Newman)e([6].)257 476 y Fk(Theorem)i(8.1)h(\([20],)f([6]\))24 b Fj(F)l(or)17 b(b)n(ond)g(p)n(er)n(c)n(olation)g(on)h Fk(Z)1389 458 y Fu(d)1389 488 y Fr(+)1419 476 y Fj(,)f Fm(d)d Fl(\025)g Fo(2)p Fj(,)k(we)g(have)282 599 y(\(a\))34 b Fm(p)405 606 y Fu(c)422 599 y Fo(\()p Fk(Z)475 581 y Fu(d)475 612 y Fr(+)505 599 y Fo(\))14 b(=)g Fm(p)614 606 y Fu(c)632 599 y Fo(\()p Fk(Z)685 581 y Fu(d)705 599 y Fo(\))p Fj(,)j(and)282 722 y(\(b\))g(for)g Fm(p)e(>)e(p)553 729 y Fu(c)571 722 y Fo(\()p Fk(Z)624 704 y Fu(d)624 735 y Fr(+)654 722 y Fo(\))p Fj(,)k(ther)n(e)h(is)f Fm(P)917 691 y Fc(Z)942 679 y Ff(d)942 702 y Fg(+)910 726 y Fu(p)970 722 y Fj(-a.s.)g(a)g (unique)i(in\014nite)h(cluster.)257 845 y Fo(The)g(follo)o(wing)g (result)f(is)g(due)h(to)g(Sc)o(honmann)f([29)q(].)31 b(There)19 b(it)h(w)o(as)g(form)o(ulated)257 924 y(in)e(the)g(setting)g (of)g(quasi-transitiv)o(e)f(graphs,)i(but)f(that)h(pro)q(of)g(go)q(es)g (through)g(un-)257 1002 y(c)o(hanged)e(in)f(the)g(generalit)o(y)f (stated)h(here.)257 1126 y Fk(Theorem)h(8.2)h(\([29]\))23 b Fj(L)n(et)f Fm(G)h Fj(b)n(e)f(any)g(b)n(ounde)n(d)h(de)n(gr)n(e)n(e)e (gr)n(aph,)i(and)f(pick)h Fm(p)g Fl(2)257 1204 y Fo([0)p Fm(;)8 b Fo(1])p Fj(.)22 b(If)641 1284 y Fo(lim)627 1314 y Fu(R)p Fh(!1)753 1284 y Fo(inf)732 1314 y Fu(x;y)q Fh(2)p Fu(V)841 1284 y Fk(P)879 1264 y Fu(G)879 1297 y(p)909 1284 y Fo(\()p Fm(B)s Fo(\()p Fm(x;)8 b(R)p Fo(\))13 b Fl($)h Fm(B)s Fo(\()p Fm(y)r(;)8 b(R)p Fo(\)\))13 b(=)h(1)8 b Fm(;)283 b Fo(\(21\))257 1399 y Fj(then)19 b(for)e(al)r(l)h Fm(p)537 1381 y Fh(0)563 1399 y Fm(>)c(p)p Fj(,)k(ther)n(e)f(is)h Fk(P)884 1381 y Fu(G)884 1411 y(p)902 1402 y Fe(0)915 1399 y Fj(-a.s.)f(exactly)i(one)g(in\014nite)g(cluster.)257 1522 y Fk(Remark.)g Fo(The)e(conclusion)f(of)g(this)g(theorem)f(can)h (no)o(w)h(b)q(e)f(strengthened)h(to)257 1600 y Fj(\\)p Fo(\011)320 1582 y Fu(G)350 1600 y Fj(-a.s.)g(ther)n(e)h(is)f(exactly)i (one)f(in\014nite)h(cluster)g(at)e(e)n(ach)h(level)25 b Fm(p)1520 1582 y Fh(0)1546 1600 y Fm(>)13 b(p)p Fj(.")257 1678 y Fo(Indeed,)g(\(21\))h(implies)d(uniform)h(p)q(ercolation)h(at)h (lev)o(el)d Fm(p)p Fo(,)j(so)g(Theorem)e(2.3)h(applies.)331 1784 y(In)19 b(the)g(follo)o(wing)g(pro)q(of,)h(w)o(e)f(shall)g(w)o (ork)h(with)f(more)f(than)i(one)f(graph,)i(and)257 1862 y(therefore)f(write)f Fm(B)631 1869 y Fu(G)661 1862 y Fo(\()p Fm(x;)8 b(R)p Fo(\))20 b(for)g Fm(B)s Fo(\()p Fm(x;)8 b(R)p Fo(\))20 b(to)g(indicate)f(in)h(whic)o(h)f(graph)i(the)f (ball)257 1940 y(sits.)257 2045 y Fk(Pro)r(of)f(of)g(Theorem)e(1.9:)k Fo(W)l(e)16 b(\014rst)h(pro)o(v)o(e)f(uniqueness)g(of)g(the)h (in\014nite)e(cluster)257 2124 y(on)20 b Fm(G)g Fo(for)f(\014xed)g Fm(p)g(>)g(p)708 2131 y Fu(c)725 2124 y Fo(\()p Fk(Z)778 2106 y Fu(d)799 2124 y Fo(\).)30 b(By)18 b(Theorem)g(8.2,)h(it)g(is)g (su\016cien)o(t)f(to)h(sho)o(w)h(that)257 2202 y(\(21\))d(holds)g(for)f (all)g Fm(p)f(>)e(p)744 2209 y Fu(c)762 2202 y Fo(\()p Fk(Z)815 2184 y Fu(d)836 2202 y Fo(\).)21 b(Fix)15 b(suc)o(h)h(a)h Fm(p)p Fo(.)22 b(Theorem)14 b(8.1)j(implies)d(that)717 2332 y(lim)703 2362 y Fu(R)p Fh(!1)808 2332 y Fk(P)846 2301 y Fc(Z)871 2289 y Ff(d)871 2311 y Fg(+)846 2336 y Fu(p)899 2332 y Fo(\()p Fm(B)955 2344 y Fc(Z)980 2332 y Ff(d)980 2355 y Fg(+)1007 2332 y Fo(\(0)p Fm(;)8 b(R)p Fo(\))15 b Fl($)e(1)p Fo(\))h(=)g(1)8 b Fm(;)359 b Fo(\(22\))257 2462 y(where)17 b(\()p Fm(B)455 2474 y Fc(Z)480 2462 y Ff(d)480 2485 y Fg(+)507 2462 y Fo(\(0)p Fm(;)8 b(R)p Fo(\))16 b Fl($)e(1)p Fo(\))j(is)g(the)f(ev)o(en)o(t)f(that)j(there)e (is)h(an)g(in\014nite)f(op)q(en)h(cluster)257 2540 y(in)o(tersecting)e Fm(B)556 2552 y Fc(Z)581 2540 y Ff(d)581 2563 y Fg(+)608 2540 y Fo(\(0)p Fm(;)8 b(R)p Fo(\).)22 b(Pic)o(k)15 b Fm(")f(>)g Fo(0,)i(and)h Fm(R)g Fo(large)f(enough)h(so)g(that)713 2683 y Fk(P)751 2651 y Fc(Z)776 2640 y Ff(d)776 2662 y Fg(+)751 2687 y Fu(p)804 2683 y Fo(\()p Fm(B)860 2694 y Fc(Z)885 2683 y Ff(d)885 2706 y Fg(+)912 2683 y Fo(\(0)p Fm(;)8 b(R)p Fo(\))15 b Fl($)e(1)p Fo(\))h Fl(\025)g Fo(1)d Fl(\000)g Fm(")d(:)p eop %%Page: 23 23 23 22 bop 71 120 a Fn(P)o(ercolation)15 b(on)i(transitiv)o(e)e(graphs) 845 b Fo(23)71 274 y(No)o(w)19 b(let)f Fm(x)h Fo(=)f(\()p Fm(x)408 281 y Fr(1)428 274 y Fm(;)8 b(:)g(:)g(:)f(;)h(x)565 281 y Fu(d)585 274 y Fo(\))19 b(and)h Fm(y)g Fo(=)f(\()p Fm(y)865 281 y Fr(1)885 274 y Fm(;)8 b(:)g(:)g(:)f(;)h(y)1018 281 y Fu(d)1038 274 y Fo(\))19 b(b)q(e)g(arbitrary)h(v)o(ertices)d(of)j Fm(G)p Fo(.)71 352 y(F)l(or)f Fm(i)f Fo(=)h(1)p Fm(;)8 b(:)g(:)g(:)f(;)h(d)p Fo(,)20 b(let)e Fm(T)554 334 y Fu(i)547 365 y(x)588 352 y Fo(b)q(e)h(some)f(in\014nite)g(self-a)o(v)o (oiding)g(path)i(in)e Fm(G)1446 359 y Fu(i)1480 352 y Fo(starting)71 430 y(in)f Fm(x)157 437 y Fu(i)171 430 y Fo(.)27 b(Also)18 b(let)f Fm(T)432 412 y Fu(i)425 443 y(y)464 430 y Fo(b)q(e)h(some)f(in\014nite)g(self-a)o(v)o(oiding)g (path)i(in)f Fm(G)1317 437 y Fu(i)1349 430 y Fo(from)f Fm(y)1490 437 y Fu(i)1522 430 y Fo(whic)o(h)71 509 y(ev)o(en)o(tually) 10 b(coincides)i(with)h Fm(T)647 491 y Fu(i)640 521 y(x)661 509 y Fo(.)20 b(Suc)o(h)13 b(a)g(path)h(is)e(easily)g(seen)h(to)g (exist:)19 b(just)13 b(tak)o(e)f(a)71 587 y(path)k(from)f Fm(y)323 594 y Fu(i)352 587 y Fo(to)h Fm(x)439 594 y Fu(i)453 587 y Fo(,)f(concatenate)h(it)f(with)h Fm(T)945 569 y Fu(i)938 599 y(x)960 587 y Fo(,)f(and)h(erase)g(an)o(y)g (circuits.)j(Finally)l(,)71 665 y(let)d Fm(z)165 672 y Fu(i)197 665 y Fo(b)q(e)i(the)f(\014rst)h(v)o(ertex)e(on)i Fm(T)706 647 y Fu(i)699 678 y(x)738 665 y Fo(with)g(the)f(prop)q(ert)o (y)g(that)h Fm(T)1279 647 y Fu(i)1272 678 y(x)1312 665 y Fo(and)g Fm(T)1444 647 y Fu(i)1437 678 y(y)1475 665 y Fo(coincide)71 743 y(from)c Fm(z)208 750 y Fu(i)237 743 y Fo(to)i(in\014nit)o(y)l(,)d(and)j(de\014ne)f Fm(T)742 725 y Fu(i)735 756 y(z)771 743 y Fo(to)h(b)q(e)f(the)g(self-a)o(v)o (oiding)f(path)i(starting)g(at)g Fm(z)1632 750 y Fu(i)71 822 y Fo(that)i(is)g(con)o(tained)f(in)h Fm(T)546 804 y Fu(i)539 834 y(x)560 822 y Fo(.)26 b(De\014ne)18 b(the)g(pro)q(duct)g (graph)h Fm(G)1201 804 y Fh(\003)1201 834 y Fu(x)1240 822 y Fo(=)e Fm(T)1331 804 y Fr(1)1324 834 y Fu(x)1362 822 y Fl(\002)12 b(\001)c(\001)g(\001)h Fm(T)1516 804 y Fu(d)1509 834 y(x)1535 822 y Fo(,)18 b(and)71 900 y(de\014ne)13 b Fm(G)247 882 y Fh(\003)247 912 y Fu(y)282 900 y Fo(and)i Fm(G)413 882 y Fh(\003)413 912 y Fu(z)447 900 y Fo(analogously)l(.)22 b(Note)13 b(that)h Fm(G)982 882 y Fh(\003)982 912 y Fu(x)1005 900 y Fo(,)g Fm(G)1071 882 y Fh(\003)1071 912 y Fu(y)1106 900 y Fo(and)g Fm(G)1236 882 y Fh(\003)1236 912 y Fu(z)1271 900 y Fo(are)g(all)f(isomorphic)71 978 y(to)j Fk(Z)164 960 y Fu(d)164 991 y Fr(+)194 978 y Fo(,)g(and)h(furthermore)e(that)i (they)f(are)g(all)g(subgraphs)i(of)f Fm(G)g Fo(and)g(that)g Fm(G)1535 960 y Fh(\003)1535 991 y Fu(z)1572 978 y Fo(is)f(a)71 1056 y(subgraph)h(b)q(oth)g(of)g Fm(G)491 1038 y Fh(\003)491 1069 y Fu(x)530 1056 y Fo(and)f(of)h Fm(G)718 1038 y Fh(\003)718 1069 y Fu(y)739 1056 y Fo(.)144 1136 y(Let)e Fm(D)270 1143 y Fu(R;x)345 1136 y Fo(b)q(e)g(the)h(ev)o(en)o(t)e(that)i (some)e(v)o(ertex)g(in)h Fm(B)1088 1143 y Fu(G)1115 1134 y Fe(\003)1115 1151 y Ff(x)1137 1136 y Fo(\()p Fm(x;)8 b(R)p Fo(\))16 b(has)g(an)g(op)q(en)g(path)71 1214 y(to)i(in\014nit)o (y)e(in)i Fm(G)398 1196 y Fh(\003)398 1226 y Fu(x)420 1214 y Fo(.)26 b(De\014ne)18 b Fm(D)653 1221 y Fu(R;y)729 1214 y Fo(analogously)l(,)h(and)f(set)g Fm(D)1219 1221 y Fu(R;x;y)1323 1214 y Fo(=)f Fm(D)1418 1221 y Fu(R;x)1489 1214 y Fl(\\)12 b Fm(D)1574 1221 y Fu(R;y)1632 1214 y Fo(.)71 1292 y(Using)k(\(22\),)g(w)o(e)g(get)631 1373 y Fk(P)669 1353 y Fu(G)669 1385 y(p)699 1373 y Fo(\()p Fm(D)758 1380 y Fu(R;x;y)845 1373 y Fo(\))e Fl(\025)g Fo(1)d Fl(\000)g Fo(2)p Fm(")d(:)71 1482 y Fo(By)22 b(Theorem)f(8.1,)k (w)o(e)d(ha)o(v)o(e)g Fk(P)702 1464 y Fu(G)702 1494 y(p)732 1482 y Fo(-a.s.)h(that)g Fm(G)992 1464 y Fh(\003)992 1494 y Fu(x)1037 1482 y Fo(and)h Fm(G)1177 1464 y Fh(\003)1177 1494 y Fu(y)1221 1482 y Fo(eac)o(h)e(ha)o(v)o(e)g(a)h(unique)71 1560 y(in\014nite)e(cluster,)h(and)g(that)g(b)q(oth)h(these)f (in\014nite)f(clusters)g(con)o(tain)h(the)f(unique)71 1638 y(in\014nite)f(cluster)h(of)h Fm(G)503 1620 y Fh(\003)503 1650 y Fu(z)523 1638 y Fo(.)37 b(Hence,)21 b(w)o(e)h(ha)o(v)o(e)e Fk(P)972 1620 y Fu(G)972 1650 y(p)1002 1638 y Fo(-a.s.)i(on)g(the)f(ev) o(en)o(t)f Fm(D)1447 1645 y Fu(R;x;y)1556 1638 y Fo(that)71 1716 y(there)15 b(is)h(an)h(op)q(en)g(path)g(in)f Fm(G)g Fo(connecting)g Fm(B)936 1723 y Fu(G)966 1716 y Fo(\()p Fm(x;)8 b(R)p Fo(\))16 b(and)h Fm(B)1239 1723 y Fu(G)1269 1716 y Fo(\()p Fm(y)r(;)8 b(R)p Fo(\),)15 b(so)i(that)465 1847 y Fk(P)503 1827 y Fu(G)503 1860 y(p)533 1847 y Fo(\()p Fm(B)589 1854 y Fu(G)619 1847 y Fo(\()p Fm(x;)8 b(R)p Fo(\))14 b Fl($)f Fm(B)858 1854 y Fu(G)888 1847 y Fo(\()p Fm(y)r(;)8 b(R)p Fo(\)\))14 b Fl(\025)f Fo(1)f Fl(\000)f Fo(2)p Fm(")d(:)71 1978 y Fo(Note)18 b(that)i(this)f(is)g(a)h(uniform)e (b)q(ound)i(for)f(all)g(v)o(ertices)e Fm(x)i Fo(and)h Fm(y)h Fo(in)e Fm(G)p Fo(.)30 b(Since)19 b Fm(")71 2056 y Fo(w)o(as)d(arbitrary)h(w)o(e)e(ha)o(v)o(e)h(\(21\),)g(so)h(the)f (pro)q(of)i(for)e(\014xed)g Fm(p)h Fo(is)f(complete.)144 2135 y(The)f(asserted)h(sim)o(ultaneous)f(uniqueness)g(is)g(implied)e (b)o(y)i(the)h(remark)e(follo)o(w-)71 2214 y(ing)i(Theorem)f(8.2.)p 1616 2201 31 2 v 1616 2226 2 25 v 1644 2226 V 1616 2228 31 2 v 144 2293 a(Next,)23 b(w)o(e)g(discuss)g(lo)o(w)o(er)f(b)q(ounds) j(for)e Fm(p)950 2300 y Fu(u)973 2293 y Fo(.)42 b(Benjamini)21 b(and)j(Sc)o(hramm)c([9,)71 2371 y(Theorem)h(4])h(pro)o(v)o(ed)g(that)h (an)o(y)f(quasi-transitiv)o(e)g(graph)h Fm(G)g Fo(satis\014es)g Fm(p)1483 2378 y Fu(u)1506 2371 y Fo(\()p Fm(G)p Fo(\))i Fl(\025)71 2449 y Fo(\()p Fm(D)q(\032)156 2456 y Fu(G)186 2449 y Fo(\))205 2431 y Fh(\000)p Fr(1)252 2449 y Fo(,)f(where)f Fm(D)i Fo(is)d(the)h(maximal)d(degree)i(in)h Fm(G)p Fo(,)h(and)g Fm(\032)1280 2456 y Fu(G)1332 2449 y Fo(is)f(the)g(sp)q(ectral)71 2528 y(radius)18 b(for)h(simple)d(random)j(w)o(alk)f(on)h Fm(G)p Fo(.)28 b(Their)18 b(pro)q(of)i(w)o(as)f(based)g(on)g(coupling) 71 2606 y(the)f(p)q(ercolation)h(pro)q(cess)g(with)f(a)h(branc)o(hing)g (random)f(w)o(alk.)28 b(In)18 b(fact,)h(a)g(simple)71 2684 y(coun)o(ting)d(argumen)o(t)f(yields)g(a)i(sligh)o(tly)e(b)q (etter)h(b)q(ound.)p eop %%Page: 24 24 24 23 bop 257 120 a Fo(24)796 b Fn(H\177)-24 b(aggstr\177)g(om,)16 b(P)o(eres)f(and)i(Sc)o(honmann)331 274 y Fo(Giv)o(en)h(a)h(lo)q(cally) f(\014nite)g(connected)h(graph)h Fm(G)p Fo(,)f(let)g Fm(A)1356 256 y Fu(n)1356 286 y(x;y)1425 274 y Fo(denote)f(the)h(n)o (um)o(b)q(er)257 352 y(of)e(paths)g(of)g(length)f Fm(n)g Fo(whic)o(h)f(connect)h Fm(x)g Fo(to)h Fm(y)r Fo(.)k(It)16 b(is)g(easy)g(to)g(see)g(that)810 477 y Fm(\025)838 484 y Fu(G)882 477 y Fo(:=)d(lim)8 b(sup)976 512 y Fu(n)p Fh(!1)1096 477 y Fo(\()p Fm(A)1152 456 y Fu(n)1152 489 y(x;y)1202 477 y Fo(\))1221 456 y Fr(1)p Fu(=n)257 602 y Fo(do)q(es)17 b(not)g(dep)q(end)f(on)h Fm(x)f Fo(and)h Fm(y)r Fo(.)257 719 y Fk(Prop)r(osition)h(8.3)24 b Fj(Supp)n(ose)e (that)g Fm(G)h Fj(is)e(an)i(in\014nite,)h(lo)n(c)n(al)r(ly)f(\014nite,) h(c)n(onne)n(cte)n(d)257 797 y(gr)n(aph.)e(Then)c(for)f Fm(p)d(<)g(\025)733 776 y Fh(\000)p Fr(1)733 809 y Fu(G)780 797 y Fj(,)k(for)f(any)g Fm(x;)8 b(y)15 b Fl(2)f Fm(V)d Fj(,)779 936 y Fk(P)817 916 y Fu(G)817 949 y(p)847 936 y Fo([)p Fm(x)i Fl($)g Fm(y)r Fo(])g Fl(\024)1076 903 y Fo(\()p Fm(p\025)1147 910 y Fu(G)1177 903 y Fo(\))1196 885 y Fu(d)p Fr(\()p Fu(x;y)q Fr(\))p 1076 925 217 2 v 1100 971 a Fo(1)f Fl(\000)f Fm(p\025)1238 978 y Fu(G)1297 936 y Fm(:)257 1073 y Fj(If)18 b Fm(G)f Fj(is)h(also)f(quasi-tr)n (ansitive,)i(then)f Fm(p)1004 1080 y Fu(u)1027 1073 y Fo(\()p Fm(G)p Fo(\))c Fl(\025)g Fm(\025)1198 1052 y Fh(\000)p Fr(1)1198 1085 y Fu(G)1246 1073 y Fj(.)257 1190 y Fo(This)e(b)q(ound)h(on)g Fm(p)598 1197 y Fu(u)632 1190 y Fo(coincides)e(with)h(the)f(b)q(ound)i(in)f([9])f(for)h (quasi-transitiv)o(e)f(graphs)257 1268 y(with)16 b(constan)o(t)h (degree,)e(and)i(impro)o(v)o(es)d(up)q(on)j(it)f(if)g(the)g(degree)g (is)g(nonconstan)o(t.)257 1346 y Fk(Pro)r(of:)22 b Fo(Clearly)l(,)865 1425 y Fm(A)902 1404 y Fu(m)p Fr(+)p Fu(n)902 1437 y(x;x)998 1425 y Fl(\025)13 b Fm(A)1087 1404 y Fu(m)1087 1437 y(x;y)1137 1425 y Fm(A)1174 1404 y Fu(n)1174 1437 y(y)q(;x)1746 1425 y Fo(\(23\))257 1528 y(for)k(an)o(y)f(t)o(w)o(o)g(sites)g Fm(x;)8 b(y)r Fo(,)15 b(and)i(an)o(y)f(non-negativ)o(e)g(in)o(tegers)f Fm(m;)8 b(n)p Fo(.)21 b(Since)16 b Fm(A)1678 1510 y Fr(2)p Fu(k)1678 1541 y(x;x)1743 1528 y Fm(>)d Fo(0,)257 1607 y(it)j(follo)o(ws)g(\(see,)g(e.g.,)f(Section)g(8)i(of)g([21]\))f(that) 694 1731 y(lim)683 1761 y Fu(k)q Fh(!1)773 1731 y Fo(\()p Fm(A)829 1711 y Fr(2)p Fu(k)829 1744 y(x;x)880 1731 y Fo(\))913 1697 y Fg(1)p 904 1703 33 2 v 904 1724 a(2)p Ff(k)957 1731 y Fo(=)1011 1718 y(~)1009 1731 y Fm(\025)1037 1738 y Fu(G)1081 1731 y Fo(=)e(sup)1137 1771 y Fu(k)q Fh(\025)p Fr(1)1214 1731 y Fo(\()p Fm(A)1270 1711 y Fr(2)p Fu(k)1270 1744 y(x;x)1321 1731 y Fo(\))1354 1697 y Fg(1)p 1345 1703 V 1345 1724 a(2)p Ff(k)1393 1731 y Fm(:)257 1869 y Fo(Using)i(symmetry)d(and)k(\(23\),)g(w)o(e)f(obtain)576 1993 y Fl(8)p Fm(x;)8 b(y)13 b Fl(2)h Fm(V)60 b Fl(8)p Fm(k)15 b Fl(\025)f Fo(1)49 b Fm(A)1059 1973 y Fu(k)1059 2006 y(x;y)1122 1993 y Fl(\024)1175 1945 y Fi(\020)1200 1993 y Fm(A)1237 1973 y Fr(2)p Fu(k)1237 2006 y(x;x)1288 1945 y Fi(\021)1313 1957 y Fr(1)p Fu(=)p Fr(2)1382 1993 y Fl(\024)1436 1980 y Fo(~)1434 1993 y Fm(\025)1462 1973 y Fu(k)1462 2006 y(G)1501 1993 y Fm(:)231 b Fo(\(24\))257 2118 y(Note)19 b(that)g(\(24\))g(implies)d(that)j Fm(\025)895 2125 y Fu(G)944 2118 y Fo(=)1002 2105 y(~)1000 2118 y Fm(\025)1028 2125 y Fu(G)1058 2118 y Fo(.)28 b(Denote)19 b(b)o(y)f([)p Fm(x)g Fl($)f Fm(y)r Fo(])h(the)h(ev)o(en)o(t)e(that)257 2196 y(there)d(is)f(an)h(op)q(en)h(path)f(connecting)g(the)f(sites)h Fm(x)f Fo(and)i Fm(y)r Fo(,)e(and)i(let)e([)p Fm(x)g Fl($)g(1)p Fo(])g(denote)257 2275 y(the)k(ev)o(en)o(t)f(that)i Fm(x)f Fo(b)q(elongs)h(to)g(an)f(in\014nite)g Fm(p)p Fo(-cluster.)24 b(Let)17 b Fl(N)1452 2257 y Fr(\()p Fu(k)q Fr(\))1445 2287 y Fu(x;y)1519 2275 y Fo(b)q(e)g(the)g(n)o(um)o(b)q(er) 257 2353 y(of)g(self-a)o(v)o(oiding)e(paths)i(of)g(length)f Fm(k)i Fo(whic)o(h)e(connect)g Fm(x)g Fo(to)g Fm(y)r Fo(.)21 b(Then)556 2486 y Fk(P)594 2493 y Fu(p)614 2486 y Fo([)p Fm(x)13 b Fl($)h Fm(y)r Fo(])41 b Fl(\024)947 2432 y Fh(1)934 2444 y Fi(X)894 2538 y Fu(k)q Fr(=)p Fu(d)p Fr(\()p Fu(x;y)q Fr(\))1043 2486 y Fl(N)1091 2465 y Fr(\()p Fu(k)q Fr(\))1084 2498 y Fu(x;y)1140 2486 y Fm(p)1164 2465 y Fu(k)1199 2486 y Fl(\024)1305 2432 y Fh(1)1292 2444 y Fi(X)1252 2538 y Fu(k)q Fr(=)p Fu(d)p Fr(\()p Fu(x;y)q Fr(\))1401 2486 y Fm(p)1425 2465 y Fu(k)1447 2486 y Fm(A)1484 2465 y Fu(k)1484 2498 y(x;y)814 2644 y Fl(\024)947 2590 y Fh(1)934 2603 y Fi(X)894 2697 y Fu(k)q Fr(=)p Fu(d)p Fr(\()p Fu(x;y)q Fr(\))1034 2644 y Fo(\()p Fm(p\025)1105 2651 y Fu(G)1136 2644 y Fo(\))1155 2623 y Fu(k)1190 2644 y Fo(=)1247 2610 y(\()p Fm(p\025)1318 2617 y Fu(G)1348 2610 y Fo(\))1367 2592 y Fu(d)p Fr(\()p Fu(x;y)q Fr(\))p 1247 2632 217 2 v 1271 2678 a Fo(1)12 b Fl(\000)e Fm(p\025)1408 2685 y Fu(G)1477 2644 y Fm(;)255 b Fo(\(25\))p eop %%Page: 25 25 25 24 bop 71 120 a Fn(P)o(ercolation)15 b(on)i(transitiv)o(e)e(graphs) 845 b Fo(25)71 274 y(pro)o(vided)15 b Fm(p)f(<)g(\025)389 253 y Fh(\000)p Fr(1)389 286 y Fu(G)437 274 y Fo(.)144 352 y(Supp)q(ose)j(no)o(w)g(that)f Fm(G)h Fo(is)f(quasi-transitiv)o(e.) k(In)c(this)g(case)579 478 y Fm(\022)q Fo(\()p Fm(p)p Fo(\))f(=)k(inf)731 508 y Fu(z)q Fh(2)p Fu(V)809 478 y Fk(P)847 485 y Fu(p)868 478 y Fo([)p Fm(z)c Fl($)e(1)p Fo(])h Fm(>)f Fo(0)71 604 y(for)j(an)o(y)g Fm(p)f(>)e(p)351 611 y Fu(c)369 604 y Fo(.)21 b(If)16 b Fm(p)e(>)g(p)567 611 y Fu(u)590 604 y Fo(,)i(then)g(for)h(all)e Fm(x;)8 b(y)15 b Fl(2)f Fm(V)e Fo(,)95 730 y Fk(P)133 737 y Fu(p)153 730 y Fo([)p Fm(x)h Fl($)h Fm(y)r Fo(])f Fl(\025)h Fk(P)416 737 y Fu(p)436 730 y Fo([)p Fm(x)f Fl($)g(1)p Fm(;)8 b(y)16 b Fl($)d(1)p Fo(])g Fl(\025)h Fk(P)897 737 y Fu(p)917 730 y Fo([)p Fm(x)f Fl($)h(1)p Fo(])p Fk(P)1138 737 y Fu(p)1157 730 y Fo([)p Fm(y)h Fl($)f(1)p Fo(])f(=)h Fm(\022)q Fo(\()p Fm(p)p Fo(\))1489 710 y Fr(2)1523 730 y Fm(>)g Fo(0)8 b Fm(;)1559 809 y Fo(\(26\))71 887 y(where)17 b(the)g(second)g(inequalit)o(y)f(is)h(an)h(instance)f(of)g(the)g (Harris)h(inequalit)o(y)l(.)k(Com-)71 965 y(paring)16 b(\(25\))h(to)g(\(26\))g(giv)o(es)e Fm(p)631 972 y Fu(u)668 965 y Fl(\025)f Fm(\025)749 944 y Fh(\000)p Fr(1)749 977 y Fu(G)796 965 y Fo(.)p 1616 953 31 2 v 1616 977 2 25 v 1644 977 V 1616 979 31 2 v 71 1043 a Fk(Remark.)19 b Fo(O.)d(Sc)o(hramm)e(\(p)q(ersonal)j(comm)o(unic)o(ation\))d(has)j (obtained)g(a)g(sharp)q(er)71 1122 y(lo)o(w)o(er)e(b)q(ound)i(for)g Fm(p)447 1129 y Fu(u)470 1122 y Fo(.)k(He)16 b(sho)o(w)o(ed)g(that)g Fm(p)877 1129 y Fu(u)914 1122 y Fl(\025)e Fm(\015)995 1101 y Fh(\000)p Fr(1)992 1134 y Fu(G)1042 1122 y Fo(,)i(where)597 1252 y Fm(\015)622 1259 y Fu(G)666 1252 y Fo(:=)d(sup)732 1291 y Fu(x)p Fh(2)p Fu(V)813 1252 y Fo(lim)8 b(sup)843 1292 y Fu(k)q Fh(!1)971 1204 y Fi(\020)996 1252 y Fl(N)1044 1231 y Fu(k)1037 1264 y(x)1065 1204 y Fi(\021)1096 1202 y Fg(1)p 1095 1208 17 2 v 1095 1228 a Ff(k)71 1391 y Fo(and)16 b Fl(N)213 1372 y Fu(k)206 1403 y(x)251 1391 y Fo(is)g(the)g(n)o(um)o(b)q(er)f(of)h(self-a)o(v)o(oiding)g(cycles)e (that)j(start)g(and)g(end)f(at)h Fm(x)p Fo(.)71 1493 y(The)g(\014nal)h(topic)f(of)h(this)g(section)f(is)h(the)f(relation)g (b)q(et)o(w)o(een)g(the)h(n)o(um)o(b)q(er)d(of)j(ends)71 1571 y(of)d(a)g(quasi-transitiv)o(e)f(graph)i(and)g(the)e(critical)g (parameters)g Fm(p)1256 1578 y Fu(c)1288 1571 y Fo(and)i Fm(p)1406 1578 y Fu(u)1429 1571 y Fo(.)21 b(It)14 b(is)h(w)o(ell)71 1650 y(kno)o(wn)i(that)h(a)f(quasi-transitiv)o(e)g(graph)h Fm(G)f Fo(can)h(only)f(ha)o(v)o(e)f(1,)i(2)f(or)h(uncoun)o(tably)71 1728 y(man)o(y)12 b(ends)j(\(see,)e(e.g.,)h(Section)f(6)i(in)f([26]\).) 20 b(In)14 b(case)g(the)g(n)o(um)o(b)q(er)e(of)j(ends)f(is)g(more)71 1806 y(than)g(1,)h(one)f(can)g(use)h(the)e(con)o(v)o(erse)g(of)i (Theorem)e(8.2)h(to)h(sho)o(w)f(that)h Fm(p)1403 1813 y Fu(u)1440 1806 y Fo(=)f(1.)20 b(This)71 1884 y(con)o(v)o(erse)15 b(states)h(that)h(for)g(quasi-transitiv)o(e)e(graphs,)318 2010 y Fl(8)p Fm(p)e(>)g(p)459 2017 y Fu(u)565 2010 y Fo(lim)551 2040 y Fu(R)p Fh(!1)677 2010 y Fo(inf)656 2040 y Fu(x;y)q Fh(2)p Fu(V)765 2010 y Fk(P)803 1990 y Fu(G)803 2023 y(p)833 1962 y Fi(\020)857 2010 y Fm(B)s Fo(\()p Fm(x;)8 b(R)p Fo(\))14 b Fl($)f Fm(B)s Fo(\()p Fm(y)r(;)8 b(R)p Fo(\))1262 1962 y Fi(\021)1301 2010 y Fo(=)13 b(1)8 b Fm(:)161 b Fo(\(27\))71 2136 y(\(This)18 b(is)h(a)g(consequence)e(of)i(the)g(Harris)f(inequalit)o(y)l(,)f(see)h (Theorem)f(3.1)i(of)g([29)q(]\).)71 2215 y(The)c(remo)o(v)m(al)e(of)j (an)f(appropriate)h(\014nite)e(set)h(of)h(sites)f(and)g(edges)g(from)g (a)g(graph)h Fm(G)71 2293 y Fo(whic)o(h)i(has)h(more)e(than)i(1)h(end)e (breaks)h(the)f(graph)i(in)o(to)e(more)f(than)j(one)f(in\014nite)71 2371 y(connected)13 b(comp)q(onen)o(t.)19 b(Therefore)14 b(the)f(limit)f(in)h(\(27\))i(cannot)f(b)q(e)g(1)h(when)f Fm(p)g(<)g Fo(1,)71 2449 y(and)h(w)o(e)g(m)o(ust)e(ha)o(v)o(e)h Fm(p)489 2456 y Fu(u)526 2449 y Fo(=)g(1.)21 b(If)15 b(the)f(n)o(um)o(b)q(er)f(of)j(ends)f(of)g Fm(G)g Fo(is)g(uncoun)o (table,)g(then)71 2528 y Fm(p)95 2535 y Fu(c)112 2528 y Fo(\()p Fm(G)p Fo(\))i Fm(<)e Fo(1,)j(since)f(then)g Fm(G)h Fo(has)g(a)g(p)q(ositiv)o(e)e(Cheeger)i(constan)o(t)f(\(see)g (Prop)q(osition)71 2606 y(6.2)f(in)g([26)q(])g(and)h(Theorem)e(2)h(in)g ([9]\).)21 b(If)16 b Fm(G)h Fo(has)g(2)g(ends,)f(then)g Fm(G)h Fo(is)f(just)h(a)g(\\\014nite)71 2684 y(extension")e(of)h Fk(Z)p Fo(,)g(so)h Fm(p)513 2691 y Fu(c)531 2684 y Fo(\()p Fm(G)p Fo(\))d(=)g(1.)21 b(More)16 b(precisely)l(,)d(b)o(y)j(the)f(pro) q(of)i(of)f(Prop)q(osition)p eop %%Page: 26 26 26 25 bop 257 120 a Fo(26)796 b Fn(H\177)-24 b(aggstr\177)g(om,)16 b(P)o(eres)f(and)i(Sc)o(honmann)257 274 y Fo(6.1)g(in)g([26],)f(there)g (is)h(a)g(doubly)f(in\014nite)g(sequence)g(\()p Fm(:::;)8 b(A)1366 281 y Fh(\000)p Fr(2)1411 274 y Fm(;)g(A)1470 281 y Fh(\000)p Fr(1)1517 274 y Fm(;)g(A)1576 281 y Fr(0)1595 274 y Fm(;)g(A)1654 281 y Fr(1)1673 274 y Fm(;)g(A)1732 281 y Fr(2)1751 274 y Fm(;)g(:::)p Fo(\))257 352 y(of)21 b(pairwise)f(disjoin)o(t)f(and)i(isomorphic)d(\014nite)i(subgraphs)i (of)e Fm(G)p Fo(,)h(suc)o(h)f(that)g(an)o(y)257 430 y(in\014nite)13 b(self-a)o(v)o(oiding)h(path)g(in)g Fm(G)h Fo(m)o(ust)d(in)o(tersect)h (either)g(all)h(the)f(graphs)j Fm(A)1706 437 y Fu(j)1738 430 y Fo(with)257 509 y(large)h(enough)g Fm(j)s Fo(,)e(or)i(all)f(the)g (graphs)h Fm(A)1003 516 y Fh(\000)p Fu(j)1065 509 y Fo(with)f(large)g (enough)h Fm(j)s Fo(.)331 592 y(The)j(case)g(of)g(a)g(single)g(end)g (is)f(more)g(delicate.)31 b(Babson)21 b(and)f(Benjamini)e([5])257 670 y(pro)o(v)o(ed)12 b(that)h Fm(p)538 677 y Fu(u)561 670 y Fo(\()p Fm(G)p Fo(\))h Fm(<)g Fo(1)e(if)g Fm(G)h Fo(is)f(the)g(Ca)o(yley)f(graph)j(of)e(a)h(\014nitely)e(presen)o(ted)g (group)257 748 y(with)17 b(one)h(end.)24 b(The)17 b(question)g(stated)g (in)g([9],)f(whether)h Fm(p)1369 755 y Fu(u)1392 748 y Fo(\()p Fm(G)p Fo(\))f Fm(<)f Fo(1)j(whenev)o(er)e Fm(G)257 827 y Fo(is)g(a)h(quasi-transitiv)o(e)e(graph)j(with)e(one)g (end,)g(is)g(still)f(unsolv)o(ed.)257 1050 y FB(9)81 b(Examples)24 b(and)j(questions)257 1188 y Fo(A)16 b(v)m(arian)o(t)g (of)h(the)f(follo)o(wing)g(example)e(w)o(as)j(sho)o(wn)g(to)f(us)h(b)o (y)f(O.)f(Sc)o(hramm.)257 1271 y Fk(Example:)27 b(A)21 b(semi-transitiv)m(e)d(graph)j(where)g(exactly)e Fo(2)i Fk(in\014nite)f(clus-)257 1349 y(ters)e(can)g(co)r(exist.)h Fo(Let)c Fm(T)22 b Fo(b)q(e)16 b(a)g(binary)f(tree)g(with)g(ro)q(ot)h Fm(\032)p Fo(,)g(i.e.)e Fm(T)21 b Fo(is)16 b(the)f(tree)g(in)257 1427 y(whic)o(h)f Fm(\032)h Fo(is)f(inciden)o(t)f(to)i(exactly)f(t)o(w) o(o)g(edges,)h(and)g(all)f(other)h(v)o(ertices)d(are)j(inciden)o(t)257 1506 y(to)j(exactly)e(three)h(edges.)25 b(Let)18 b Fm(H)k Fo(b)q(e)17 b(the)g(pro)q(duct)i(graph)f Fm(T)g Fl(\002)12 b Fk(Z)18 b Fo(with)f(an)h(addi-)257 1584 y(tional)h(distinguished)g(v) o(ertex)e Fm(v)873 1566 y Fh(\003)911 1584 y Fo(joined)h(b)o(y)h(a)g (single)f(edge)h(to)g(the)g(v)o(ertex)e(\()p Fm(\032;)8 b Fo(0\))257 1662 y(of)18 b Fm(T)h Fl(\002)12 b Fk(Z)p Fo(.)26 b(Theorem)16 b(1.9)i(implies)d(that)k Fm(p)1073 1669 y Fu(u)1096 1662 y Fo(\()p Fm(H)t Fo(\))d(=)g Fm(p)1272 1669 y Fu(u)1295 1662 y Fo(\()p Fm(T)j Fl(\002)12 b Fk(Z)p Fo(\))k Fm(<)h Fo(1.)26 b(Finally)l(,)16 b(let)257 1740 y Fm(G)h Fo(consist)g(of)g(t)o(w)o(o)f(copies)g Fm(H)805 1747 y Fr(1)825 1740 y Fm(;)8 b(H)887 1747 y Fr(2)923 1740 y Fo(of)17 b Fm(H)t Fo(,)f(glued)h(together)f(at)h(their)f (distinguished)257 1819 y(v)o(ertices)d(\(so)j(these)e(v)o(ertices)f Fm(v)834 1800 y Fh(\003)832 1831 y Fr(1)868 1819 y Fo(and)i Fm(v)987 1800 y Fh(\003)985 1831 y Fr(2)1021 1819 y Fo(are)g(iden)o (ti\014ed,)e(and)i(the)g(resulting)f(v)o(ertex)257 1897 y(of)20 b Fm(G)g Fo(is)f(denoted)g Fm(w)649 1879 y Fh(\003)669 1897 y Fo(\).)31 b(It)19 b(is)g(easy)g(to)h(see)f(that)h Fm(G)f Fo(is)h(semi-transitiv)o(e,)c(with)k Fm(V)1803 1904 y Fu(F)257 1975 y Fo(consisting)14 b(of)h(the)e(distinguished)h(v) o(ertex)e Fm(w)1090 1957 y Fh(\003)1124 1975 y Fo(only)l(.)20 b(F)l(or)14 b(all)f Fm(p)h(>)g(p)1509 1982 y Fu(u)1532 1975 y Fo(\()p Fm(H)t Fo(\),)g(it)f(follo)o(ws)257 2053 y(that)j(b)q(ond)g(p)q(ercolation)f(on)h Fm(G)g Fo(can)f(ha)o(v)o(e)f (one)i(or)f(t)o(w)o(o)g(in\014nite)f(clusters)h(with)g(p)q(os-)257 2132 y(itiv)o(e)h(probabilit)o(y:)22 b(If)17 b(at)h(least)f(one)h(of)f (the)h(t)o(w)o(o)f(edges)g(inciden)o(t)f(to)i Fm(w)1604 2113 y Fh(\003)1641 2132 y Fo(is)f(closed,)257 2210 y(then)i(a.s.)g (there)f(are)h(exactly)f(t)o(w)o(o)h(in\014nite)f(clusters,)g(one)i (con)o(tained)e(in)h Fm(H)1715 2217 y Fr(1)1754 2210 y Fo(and)257 2288 y(the)d(other)f(in)h Fm(H)564 2295 y Fr(2)584 2288 y Fo(.)21 b(On)15 b(the)h(other)f(hand,)h(clearly)e (the)i(in\014nite)e(clusters)h(of)h Fm(H)1718 2295 y Fr(1)1754 2288 y Fo(and)257 2366 y Fm(H)297 2373 y Fr(2)334 2366 y Fo(can)g(connect)g(to)h(eac)o(h)e(other)i(with)f(p)q(ositiv)o(e) f(probabilit)o(y)l(.)p 1802 2354 31 2 v 1802 2379 2 25 v 1831 2379 V 1802 2381 31 2 v 331 2449 a(Next,)25 b(w)o(e)f(discuss)h (brie\015y)f(y)o(et)g(another)h(phase)h(transition.)47 b(Let)25 b Fm(G)g Fo(b)q(e)g(a)257 2528 y(non)o(unimo)q(dular)14 b(quasi-transitiv)o(e)g(graph,)h(and)g(denote)g(b)o(y)f Fm(\026)h Fo(the)f(left)g(Haar)g(mea-)257 2606 y(sure)f(on)g(Aut\()p Fm(G)p Fo(\).)20 b(Recall)12 b(the)g(de\014nition)g(of)h(stabilizer)e Fm(S)s Fo(\()p Fm(x)p Fo(\))h(from)g(Section)g(6.)20 b(Sa)o(y)257 2684 y(that)13 b(a)g(cluster)f Fl(C)j Fo(with)e(v)o(ertex) e(set)h Fm(V)f Fo(\()p Fl(C)s Fo(\))i(is)f Fk(hea)n(vy)g Fo(if)1274 2651 y Fi(P)1318 2695 y Fu(x)p Fh(2)p Fu(V)7 b Fr(\()p Fh(C)r Fr(\))1448 2684 y Fm(\026)p Fo([)p Fm(S)s Fo(\()p Fm(x)p Fo(\)])13 b(=)h Fl(1)p Fo(;)f(oth-)p eop %%Page: 27 27 27 26 bop 71 120 a Fn(P)o(ercolation)15 b(on)i(transitiv)o(e)e(graphs) 845 b Fo(27)71 274 y(erwise,)17 b(w)o(e)g(sa)o(y)h(that)h Fl(C)i Fo(is)d Fk(ligh)n(t)p Fo(.)26 b(Theorem)16 b(1.8)j(implies)c (that)j(hea)o(vy)g(and)g(ligh)o(t)71 352 y(in\014nite)12 b(clusters)h(cannot)h(co)q(exist)f(at)h(an)o(y)g(\014xed)f(lev)o(el)e Fm(p)p Fo(,)j(so)h(it)e(is)g(natural)h(to)g(de\014ne)260 485 y Fm(p)284 492 y Fu(h)321 485 y Fo(:=)f(inf)453 437 y Fi(n)481 485 y Fm(p)h Fo(:)22 b Fk(P)593 492 y Fu(p)613 485 y Fo([)16 b(there)f(is)i(a)f(hea)o(vy)g(in\014nite)f(cluster)o(])f Fm(>)f Fo(0)1406 437 y Fi(o)1443 485 y Fm(:)102 b Fo(\(28\))71 617 y(The)16 b(mass)g(transp)q(ort)i(metho)q(d)e(can)h(b)q(e)f(used)h (e\013ectiv)o(ely)d(in)i(hea)o(vy)g(clusters.)21 b(F)l(or)71 696 y(instance,)f(Theorem)f(1.5)h(can)h(b)q(e)f(easily)f(extended)h(to) g(non)o(unimo)q(dular)g(graphs,)71 774 y(pro)o(vided)d(that)i(the)f (parameters)g Fm(p)746 781 y Fr(1)766 774 y Fm(;)8 b(p)812 781 y Fr(2)850 774 y Fo(considered)18 b(there)g(are)h(greater)f(than)h Fm(p)1609 781 y Fu(h)1632 774 y Fo(.)71 852 y(F)l(or)i(the)f(non)o (unimo)q(dular)h(example)801 836 y Fi(b)790 852 y Fm(T)819 859 y Fu(d)860 852 y Fo(men)o(tioned)e(in)i(Section)f(6)h(\(a)h(tree)e (with)71 930 y(additional)14 b(edges)g(leading)g(to)h Fm(\030)r Fo(-grandparen)o(ts\))h(it)e(is)g(easy)g(to)h(see)f(that)g Fm(p)1463 937 y Fu(c)1495 930 y Fm(<)g(p)1571 937 y Fu(h)1608 930 y Fo(=)71 1009 y Fm(p)95 1016 y Fu(u)131 1009 y Fo(=)g(1.)21 b(On)14 b(the)f(other)h(hand,)g(let)f Fm(T)757 1016 y Fu(k)792 1009 y Fo(denote)g(the)h Fm(k)r Fo(-regular)g(tree.)19 b(F)l(or)14 b(an)o(y)g(graph)71 1087 y Fm(G)109 1094 y Fr(0)145 1087 y Fo(with)i(b)q(ounded)h(degrees,)f(if)g Fm(k)i Fo(is)e(large)g(enough,)h(then)f Fm(G)e Fo(=)g Fm(G)1331 1094 y Fr(0)1362 1087 y Fl(\002)d Fm(T)1441 1094 y Fu(k)1478 1087 y Fo(satis\014es)691 1220 y Fm(p)715 1227 y Fu(h)738 1220 y Fo(\()p Fm(G)p Fo(\))j Fm(<)g(p)904 1227 y Fu(u)927 1220 y Fo(\()p Fm(G)p Fo(\))8 b Fm(:)534 b Fo(\(29\))71 1352 y(The)14 b(pro)q(of)i(is)f(similar)d(to)j(an)h (argumen)o(t)d(in)i([9,)f(Sect.)g(4].)20 b(Let)15 b Fm(D)1280 1359 y Fr(0)1315 1352 y Fo(b)q(e)g(the)g(maximal)71 1430 y(degree)20 b(in)g Fm(G)326 1437 y Fr(0)346 1430 y Fo(,)i(and)f(let)f Fm(\032)581 1437 y Fu(G)631 1430 y Fo(b)q(e)h(the)g(sp)q(ectral)f (radius)h(for)g(simple)e(random)h(w)o(alk)71 1509 y(on)g Fm(G)p Fo(.)34 b(It)20 b(is)g(easy)g(to)h(see)f(that)g(if)g Fm(p)h(>)g(p)882 1516 y Fu(c)900 1509 y Fo(\()p Fm(T)948 1516 y Fu(k)968 1509 y Fo(\),)g(then)f(an)o(y)g(in\014nite)g Fm(p)p Fo(-cluster)g(in)71 1587 y Fm(G)14 b Fo(=)g Fm(G)213 1594 y Fr(0)244 1587 y Fl(\002)d Fm(T)323 1594 y Fu(k)360 1587 y Fo(is)16 b(hea)o(vy)l(.)k(Therefore,)95 1720 y Fm(\025)123 1727 y Fu(G)158 1720 y Fl(\001)5 b Fm(p)201 1727 y Fu(h)224 1720 y Fo(\()p Fm(G)p Fo(\))14 b Fl(\024)g Fo(\()p Fm(D)426 1727 y Fr(0)451 1720 y Fo(+)5 b Fm(k)r Fo(\))g Fl(\001)g Fm(\032)589 1727 y Fu(G)622 1720 y Fl(\001)g Fm(p)665 1727 y Fu(h)688 1720 y Fo(\()p Fm(G)p Fo(\))14 b Fl(\024)g Fo(\()p Fm(D)890 1727 y Fr(0)915 1720 y Fo(+)5 b Fm(k)r Fo(\))g Fl(\001)g Fm(\032)1053 1727 y Fu(G)1086 1720 y Fl(\001)g Fm(p)1129 1727 y Fu(c)1147 1720 y Fo(\()p Fm(T)1195 1727 y Fu(k)1216 1720 y Fo(\))13 b(=)1305 1686 y Fm(D)1345 1693 y Fr(0)1376 1686 y Fo(+)e Fm(k)p 1305 1708 148 2 v 1323 1754 a(k)i Fl(\000)d Fo(1)1457 1720 y Fm(\032)1482 1727 y Fu(G)1520 1720 y Fm(:)25 b Fo(\(30\))71 1852 y(Since)15 b Fm(\032)223 1859 y Fu(G)267 1852 y Fo(=)f Fm(\032)344 1859 y Fu(G)371 1864 y Fg(0)389 1859 y Fh(\002)p Fu(T)437 1865 y Ff(k)472 1852 y Fl(!)g Fo(0)i(as)h Fm(k)f Fl(!)e(1)p Fo(,)i(it)g(follo)o(ws)g(from)f([9,)h (Theorem)f(4],)h(or)h(from)71 1930 y(Prop)q(osition)i(8.3)f(ab)q(o)o(v) o(e,)g(that)g(for)g(large)g(enough)g Fm(k)i Fo(\(when)e(the)g(righ)o(t) f(hand)i(side)71 2009 y(of)e(\(30\))h(is)f(less)f(than)i(1\))f(w)o(e)g (ha)o(v)o(e)f(\(29\).)25 b(W)l(e)16 b(exp)q(ect)h(that)g(there)g(exist) f(transitiv)o(e)71 2087 y(graphs)h(where)f Fm(p)393 2094 y Fu(c)425 2087 y Fm(<)d(p)500 2094 y Fu(h)537 2087 y Fm(<)h(p)613 2094 y Fu(u)650 2087 y Fm(<)f Fo(1,)k(but)f(w)o(e)g(do)h (not)f(ha)o(v)o(e)g(an)h(explicit)d(example.)71 2195 y(W)l(e)i(end)g(the)g(pap)q(er)h(with)f(some)f Fk(questions:)130 2322 y Fo(1.)25 b(Is)i Fm(p)281 2329 y Fu(c)319 2322 y Fm(<)19 b(p)400 2329 y Fu(h)451 2322 y Fo(for)h(ev)o(ery)e(non)o (unimo)q(dular)i(quasi-transitiv)o(e)e(graph?)33 b(\()p Fm(p)1570 2329 y Fu(h)1613 2322 y Fo(is)193 2400 y(de\014ned)16 b(in)f(\(28\).\))193 2479 y(What)h(geometric)f(prop)q(erties)h(of)g Fm(G)h Fo(guaran)o(tee)g(that)25 b Fm(p)1253 2486 y Fu(h)1289 2479 y Fm(<)14 b(p)1365 2486 y Fu(u)1388 2479 y Fo(?)130 2606 y(2.)25 b(Can)20 b(one)g(drop)g(the)g(unimo)q(dularit)o(y)d (assumption)j(made)e(in)i(Theorem)e(1.5)193 2684 y(and)e(Prop)q (osition)i(7.1?)p eop %%Page: 28 28 28 27 bop 257 120 a Fo(28)796 b Fn(H\177)-24 b(aggstr\177)g(om,)16 b(P)o(eres)f(and)i(Sc)o(honmann)317 274 y Fo(3.)24 b(A)e(graph)i Fm(G)f Fo(is)f(said)h(to)g(exhibit)e Fk(cluster)k(repulsion)d Fo(at)h(lev)o(el)d Fm(p)j Fo(if,)g(for)379 352 y(i.i.d.)17 b(p)q(ercolation)i(with)g(reten)o(tion)f(parameter)f Fm(p)j Fo(on)f Fm(G)p Fo(,)h(an)o(y)f(t)o(w)o(o)f(in\014nite)379 430 y(clusters)i(can)h(come)d(within)i(unit)h(distance)f(from)f(eac)o (h)h(other)g(in)g(at)h(most)379 509 y(\014nitely)10 b(man)o(y)f (places.)19 b(Do)q(es)12 b(a)f(quasi-transitiv)o(e)f(graph)i (necessarily)e(exhibit)379 587 y(cluster)23 b(repulsion)g(for)h(an)o(y) g Fm(p)i Fl(2)h Fo([0)p Fm(;)8 b Fo(1]?)43 b(It)23 b(is)h(not)g(hard)g (to)g(sho)o(w)g(that)379 665 y(cluster)19 b(repulsion)h(at)g(lev)o(el)e Fm(p)i Fo(follo)o(ws)g(if)f(the)h(pair)g(connectivit)o(y)e(function)379 743 y Fm(p)d Fl(7!)e Fk(P)519 725 y Fu(G)519 756 y(p)549 743 y Fo([)p Fm(u)g Fl($)h Fm(v)r Fo(])e(is)g(con)o(tin)o(uous)h(at)g Fm(p)h Fo(for)f(all)f Fm(u;)c(v)r Fo(.)19 b(Hence)12 b(cluster)g(repulsion)379 822 y(can)17 b(fail)g(at)g(most)f(for)i(coun) o(tably)e(man)o(y)g(v)m(alues)h(of)g Fm(p)p Fo(.)24 b(If)17 b(cluster)f(repulsion)379 900 y(holds,)f(then)g(it)f(follo)o(ws)g(that) h(for)g(ev)o(ery)e Fm(R)p Fo(,)i(an)o(y)g(t)o(w)o(o)f(in\014nite)g (clusters)g(come)379 978 y(within)22 b(distance)g Fm(R)h Fo(from)e(eac)o(h)h(other)g(at)h(most)f(\014nitely)f(often)h(a.s.)40 b(W)l(e)379 1056 y(ha)o(v)o(e)19 b(an)h(example)e(of)i(a)g (\(non-semi-transitiv)o(e\))e(graph)i(for)g(whic)o(h)f(cluster)379 1135 y(repulsion)d(fails)g(for)h(certain)e Fm(p)p Fo(.)317 1279 y(4.)24 b(Let)18 b Fm(G)f Fo(b)q(e)h(a)f(quasi-transitiv)o(e)f (graph.)25 b(A)17 b(site)g Fm(x)g Fo(is)g(an)g Fk(encoun)n(ter)j(p)r (oin)n(t)379 1358 y Fo(of)i(the)g(cluster)e Fl(C)25 b Fo(if)c(remo)o(ving)f(the)h(edges)h(inciden)o(t)e(to)i Fm(x)f Fo(from)g Fl(C)k Fo(yields)379 1436 y(at)f(least)e(three)h (in\014nite)e(connected)i(comp)q(onen)o(ts.)40 b(F)l(or)23 b Fm(p)j Fl(2)f Fo(\()p Fm(p)1656 1443 y Fu(c)1674 1436 y Fm(;)8 b(p)1720 1443 y Fu(u)1743 1436 y Fo(\),)24 b(is)379 1514 y Fk(P)417 1521 y Fu(p)437 1514 y Fo([)p Fl(9)16 b Fo(an)h(in\014nite)e(cluster)g(with)h(in\014nitely)f(man)o(y)g (encoun)o(ter)g(p)q(oin)o(ts)q(])22 b Fm(>)h Fo(0?)379 1592 y(By)g(Theorem)g(1.8,)i(this)f(w)o(ould)g(imply)d(that)k Fj(every)f Fo(in\014nite)f(cluster)g(has)379 1671 y(in\014nitely)13 b(man)o(y)f(encoun)o(ter)i(p)q(oin)o(ts)g Fk(P)1116 1678 y Fu(p)1136 1671 y Fo(-a.s.)21 b(This)14 b(question)g(has)h(a)f(p)q (ositiv)o(e)379 1749 y(answ)o(er)g(in)g(the)f(unimo)q(dular)h(case)f (\(see)h([8])f(or)h([24)q(]\),)f(whic)o(h)g(readily)g(extends)379 1827 y(to)g(hea)o(vy)f(clusters)h(in)f(the)g(non)o(unimo)q(dular)h (case.)20 b(A)12 b(general)g(answ)o(er)h(w)o(ould)379 1905 y(b)q(e)k(a)f(signi\014can)o(t)g(step)h(to)o(w)o(ard)f (determining)e(whether)758 2048 y Fk(P)796 2055 y Fu(p)814 2059 y Ff(c)832 2048 y Fo([)p Fl(9)i Fo(in\014nite)f(op)q(en)i (clusters)f(])d(=)h(0)293 b(\(31\))379 2192 y(for)13 b(quasi-transitiv)o(e)e(graphs)j(with)e(a)h(nonamenable)e(automorphism) g(group.)379 2270 y(\(Under)j(the)f(additional)h(assumption)g(of)g (unimo)q(dularit)o(y)l(,)e(\(31\))j(is)f(pro)o(v)o(ed)f(in)379 2348 y([8].\))257 2528 y Fk(Ac)n(kno)n(wledgmen)n(ts.)21 b Fo(W)l(e)15 b(thank)h(Itai)f(Benjamini,)e(Russ)j(Ly)o(ons)g(and)g (Je\013)g(Steif)257 2606 y(for)h(stim)o(ulating)e(discussions.)22 b(W)l(e)17 b(are)f(esp)q(ecially)f(indebted)h(to)h(Oded)f(Sc)o(hramm) 257 2684 y(for)h(man)o(y)e(helpful)g(and)i(insigh)o(tful)e(commen)o (ts.)p eop %%Page: 29 29 29 28 bop 71 120 a Fn(P)o(ercolation)15 b(on)i(transitiv)o(e)e(graphs) 845 b Fo(29)71 274 y FB(References)144 385 y Fo([1])24 b(Aizenman,)11 b(M.,)h(Kesten)h(H.)f(and)i(Newman,)d(C.)i(M.)g (\(1987\))h(Uniqueness)e(of)220 445 y(the)e(in\014nite)h(cluster)f(and) i(con)o(tin)o(uit)o(y)d(of)i(connectivit)o(y)e(functions)i(for)h (short-)220 505 y(and)k(long-range)i(p)q(ercolation,)e Fj(Commun.)h(Math.)g(Phys.)f Fk(111)p Fo(,)g(505{532.)144 610 y([2])24 b(Aldous,)14 b(D.)f(\(1997\))j(Bro)o(wnian)e(excursions,)g (critical)e(random)i(graphs)h(and)220 670 y(the)h(m)o(ultipli)o(cativ)n (e)d(coalescen)o(t,)i Fj(A)o(nn.)j(Pr)n(ob)n(ab.)e Fk(25)p Fo(,)f(812{854.)144 776 y([3])24 b(Alexander,)c(K.)g(\(1995\))j(Sim)o (ultaneous)c(uniqueness)h(of)h(in\014nite)f(clusters)220 836 y(in)e(stationary)i(random)f(lab)q(eled)f(graphs,)j Fj(Commun.)f(Math.)f(Phys.)g Fk(168)p Fo(,)220 896 y(39{55.)144 1001 y([4])24 b(Alexander,)17 b(K.)h(\(1995\))i(P)o(ercolation)e(and)h (minim)o(al)d(spanning)j(forests)g(in)220 1061 y(in\014nite)c(graphs.)i Fj(A)o(nn.)h(Pr)n(ob)n(ab.)d Fk(23)p Fo(,)h(87{104.)144 1166 y([5])24 b(Babson,)f(E.)e(and)i(Benjamini,)d(I.)h(\(1998\))i(Cut)f (sets)g(and)g(normed)f(coho-)220 1226 y(mology)13 b(with)i(application) f(to)h(p)q(ercolation,)g Fj(Pr)n(o)n(c.)f(A)o(mer.)i(Math.)g(So)n(c.)p Fo(,)e(to)220 1287 y(app)q(ear.)144 1392 y([6])24 b(Barsky)l(,)11 b(D.)g(J.,)g(Grimmett,)e(G.)i(R.)g(and)h(Newman,)e(C.)h(M.)f(\(1991\))j (Dynamic)220 1452 y(renormalization)k(and)i(con)o(tin)o(uit)o(y)e(of)i (the)f(p)q(ercolation)h(transition)g(in)f(or-)220 1512 y(than)o(ts,)e Fj(Sp)n(atial)h(Sto)n(chastic)h(Pr)n(o)n(c)n(esses)p Fo(,)d(pp)h(37{55,)i(Birkh\177)-24 b(auser,)14 b(Boston.)144 1617 y([7])24 b(Benjamini,)11 b(I.,)i(Ly)o(ons,)h(R.,)g(P)o(eres,)f(Y.) g(and)h(Sc)o(hramm,)d(O.)i(\(1997\))i(Group-)220 1677 y(in)o(v)m(arian)o(t)g(p)q(ercolation)i(on)f(graphs,)h Fj(Ge)n(om.)g(F)l(unc.)h(A)o(nalysis)p Fo(,)e(to)h(app)q(ear.)144 1782 y([8])24 b(Benjamini,)10 b(I.,)i(Ly)o(ons,)i(R.,)f(P)o(eres,)f(Y.) g(and)i(Sc)o(hramm,)c(O.)i(\(1998\))j(Critical)220 1843 y(p)q(ercolation)d(on)h(an)o(y)f(nonamenable)f(group)i(has)g(no)g (in\014nite)e(clusters,)h Fj(A)o(nn.)220 1903 y(Pr)n(ob)n(ab.)p Fo(,)j(to)h(app)q(ear.)144 2008 y([9])24 b(Benjamini,)j(I.)f(and)i(Sc)o (hramm,)f(O.)g(\(1996\))h(P)o(ercolation)f(b)q(ey)o(ond)h Fk(Z)1612 1990 y Fu(d)1632 2008 y Fo(:)220 2068 y(man)o(y)21 b(questions)j(and)g(a)f(few)g(answ)o(ers,)i Fj(Ele)n(ctr.)g(Commun.)f (Pr)n(ob)n(ab.)e Fk(1)p Fo(,)220 2128 y(71{82.)119 2233 y([10])j(Borgs,)12 b(C.,)f(Cha)o(y)o(es,)h(J.,)f(Kesten,)g(H.)g(and)g (Sp)q(encer,)h(J.)f(\(1998\))h(The)f(birth)g(of)220 2293 y(the)j(in\014nite)f(cluster:)19 b(\014nite-size)13 b(scaling)h(in)g(p) q(ercolation,)h Fj(in)h(pr)n(ep)n(ar)n(ation.)119 2399 y Fo([11])25 b(Burton,)15 b(R.)f(and)i(Keane,)e(M.)h(\(1989\))h(Densit) o(y)e(and)i(uniqueness)e(in)h(p)q(erco-)220 2459 y(lation,)g Fj(Commun.)j(Math.)f(Phys.)f Fk(121)p Fo(,)f(501{505.)119 2564 y([12])25 b(Cha)o(y)o(es,)12 b(J.)g(T.,)g(Cha)o(y)o(es,)h(L.)f (and)h(Newman,)e(C.)i(M.)e(\(1985\))j(The)e(sto)q(c)o(hastic)220 2624 y(geometry)h(of)j(in)o(v)m(asion)f(p)q(ercolation,)g Fj(Commun.)h(Math.)g(Phys.)f Fk(101)p Fo(,)g(383{)220 2684 y(407.)p eop %%Page: 30 30 30 29 bop 257 120 a Fo(30)796 b Fn(H\177)-24 b(aggstr\177)g(om,)16 b(P)o(eres)f(and)i(Sc)o(honmann)306 274 y Fo([13])24 b(F)l(alconer,)16 b(K.)f(\(1997\))j Fj(T)l(e)n(chniques)h(in)f(F)l(r)n (actal)f(Ge)n(ometry)p Fo(,)e(Wiley)l(.)306 381 y([14])24 b(Gandol\014,)14 b(A.,)d(Grimmett,)f(G.)i(R.,)g(and)g(Russo,)i(L.)e (\(1988\))i(On)e(the)g(unique-)406 441 y(ness)j(of)g(the)f(in\014nite)f (op)q(en)i(cluster)f(in)g(the)g(p)q(ercolation)g(mo)q(del,)f Fj(Commun.)406 501 y(Math.)k(Phys.)f Fk(114)p Fo(,)g(549{552.)306 608 y([15])24 b(Grimmett,)13 b(G.)j(R.)g(\(1989\))h Fj(Per)n(c)n (olation)p Fo(,)f(Springer,)g(New)g(Y)l(ork.)306 714 y([16])24 b(Grimmett,)13 b(G.)j(R.)g(and)h(Newman,)e(C.)h(M.)f (\(1990\))j(P)o(ercolation)e(in)g Fl(1)11 b Fo(+)g(1)406 775 y(dimensions,)k Fj(Disor)n(der)h(in)h(Physic)n(al)h(Systems)e Fo(\(G.)g(R.)g(Grimmett)d(and)k(D.)406 835 y(J.)f(A.)g(W)l(elsh,)f (editors\),)h(219{240,)i(Clarendon)f(Press,)f(Oxford.)306 941 y([17])24 b(H\177)-24 b(aggstr\177)g(om,)16 b(O.)g(\(1997\))h (In\014nite)f(clusters)f(in)h(dep)q(enden)o(t)g(automorphism)406 1002 y(in)o(v)m(arian)o(t)g(p)q(ercolation)g(on)h(trees,)e Fj(A)o(nn.)k(Pr)n(ob)n(ab.)c Fk(25)p Fo(,)h(1423{1436.)306 1108 y([18])24 b(H\177)-24 b(aggstr\177)g(om,)14 b(O.)g(and)h(P)o (eres,)e(Y.)g(\(1998\))j(Monotonicit)o(y)d(of)i(uniqueness)e(for)406 1169 y(p)q(ercolation)i(on)h(Ca)o(yley)e(graphs:)21 b(all)15 b(in\014nite)f(clusters)g(are)h(b)q(orn)h(sim)o(ulta-)406 1229 y(neously)l(,)g Fj(Pr)n(ob)n(ab.)h(Th.)g(R)n(el.)g(Fields)p Fo(,)f(to)h(app)q(ear.)306 1335 y([19])24 b(Janson,)14 b(S.,)e(Kn)o(uth,)h(D.)f(E.,)g(Luczak,)h(T.)e(and)i(Pittel,)f(B.)f (\(1993\))j(The)e(birth)406 1396 y(of)17 b(the)f(gian)o(t)g(comp)q (onen)o(t,)f Fj(R)n(and.)i(Struct.)h(A)o(lg.)f Fk(4)p Fo(,)f(233{358.)306 1502 y([20])24 b(Kesten,)18 b(H.)f(\(1980\))j(The)e (critical)e(probabilit)o(y)h(for)i(b)q(ond)g(p)q(ercolation)f(on)406 1563 y(the)e(square)h(lattice)e(equals)946 1543 y Fr(1)p 946 1551 18 2 v 946 1580 a(2)968 1563 y Fo(,)h Fj(Commun.)i(Math.)f (Phys.)e Fk(74)p Fo(,)h(41{59.)306 1669 y([21])24 b(Kingman,)18 b(J.)g(F.)g(C.)h(\(1963\))h(The)e(ergo)q(dic)h(deca)o(y)f(of)g(Mark)o (o)o(v)g(transition)406 1729 y(probabilities,)d Fj(Pr)n(o)n(c.)i(L)n (ondon)g(Math.)g(So)n(c.)f Fk(13)p Fo(,)g(337{358.)306 1836 y([22])24 b(Lalley)l(,)i(S.)e(\(1996\))i(P)o(ercolation)d(on)i(F)l (uc)o(hsian)f(groups,)j Fj(A)o(nn.)f(Inst.)f(H.)406 1896 y(Poinc)n(ar)o(\023)-24 b(e,)18 b(Pr)n(ob)n(ab.)f(Stat.)f Fk(34)p Fo(,)g(151{178.)306 2003 y([23])24 b(Ly)o(ons,)17 b(R.)f(and)h(P)o(eres,)f(Y.)f(\(1998\))j Fj(Pr)n(ob)n(ability)f(on)h(T) l(r)n(e)n(es)g(and)f(Networks.)406 2063 y Fo(Bo)q(ok)g(in)f (preparation,)g(draft)h(a)o(v)m(ailable)e(at)i(URL)406 2123 y Fb(http://php.)o(ind)o(ian)o(a.)o(edu)o(/~r)o(dl)o(yon)o(s)306 2230 y Fo([24])24 b(Ly)o(ons,)18 b(R.)e(and)i(Sc)o(hramm,)13 b(O.)k(\(1998\))h(Indistinguishabilit)o(y)d(of)j(P)o(ercola-)406 2290 y(tion)f(Clusters,)e Fj(in)j(pr)n(ep)n(ar)n(ation)p Fo(.)306 2397 y([25])24 b(Mac)o(h)o(ta,)d(J.,)f(Choi,)h(Y.)e(S.,)h(Luc) o(k)o(e,)g(A.,)f(Sc)o(h)o(w)o(eizer,)g(T.)g(and)i(Cha)o(y)o(es,)f(L.) 406 2457 y(\(1995\))26 b(In)o(v)m(aded)d(cluster)g(algorithms)g(for)h (equilibrium)c(critical)i(p)q(oin)o(ts,)406 2517 y Fj(Phys.)17 b(R)n(ev.)h(L)n(ett.)e Fk(75)p Fo(,)f(2792{279)q(5.)306 2624 y([26])24 b(Mohar,)e(B.)e(\(1991\))j(Some)c(relations)i(b)q(et)o (w)o(een)f(analytic)g(and)h(geometric)406 2684 y(prop)q(erties)c(of)f (in\014nite)f(graphs,)i Fj(Discr.)g(Math.)f Fk(95)p Fo(,)g(193{219.)p eop %%Page: 31 31 31 30 bop 71 120 a Fn(P)o(ercolation)15 b(on)i(transitiv)o(e)e(graphs) 845 b Fo(31)119 274 y([27])25 b(Newman,)f(C.)g(M.)g(and)h(Sc)o(h)o (ulman,)f(L.)g(S.)g(\(1981\))i(In\014nite)d(clusters)h(in)220 334 y(p)q(ercolation)16 b(mo)q(dels,)f Fj(J.)i(Statist.)h(Phys.)e Fk(26)p Fo(,)f(613{628.)119 436 y([28])25 b(Newman,)12 b(C.)h(M.)g(and)h(Stein,)f(D.)g(L.)g(\(1996\))i(Ground-state)g (structure)e(in)g(a)220 496 y(highly)g(disordered)h(spin-glass)i(mo)q (del,)d Fj(J.)i(Statist.)h(Phys.)e Fk(82)p Fo(,)h(1113{1132.)119 598 y([29])25 b(Sc)o(honmann,)10 b(R.)h(H.)f(\(1998\))i(Stabilit)o(y)e (of)h(in\014nite)f(clusters)g(in)h(sup)q(ercritical)220 658 y(p)q(ercolation,)16 b Fj(Pr)n(ob)n(ab.)g(Th.)h(R)n(el.)h(Fields)p Fo(,)e(to)g(app)q(ear.)119 760 y([30])25 b(Sc)o(honmann,)18 b(R.)g(H.)g(\(1998\))i(P)o(ercolation)e(in)g Fl(1)13 b Fo(+)f(1)20 b(dimensions)d(at)i(the)220 820 y(uniqueness)c (threshold,)h Fj(this)i(volume)p Fo(.)119 922 y([31])25 b(T)l(ro\014mo)o(v,)14 b(V.)i(I.)f(\(1985\))j(Automorphism)c(groups)j (of)g(graphs)g(as)g(top)q(olog-)220 982 y(ical)e(groups,)i Fj(Math.)g(Notes)g Fk(38)p Fo(,)f(717{720.)71 1186 y Fa(Olle)h(H)253 1182 y(\177)251 1186 y(aggstr)430 1182 y(\177)428 1186 y(om)71 1246 y Fo(Mathematical)d(Statistics,)h (Chalmers)g(Univ)o(ersit)o(y)e(of)k(T)l(ec)o(hnology)l(,)e(Gothen)o (burg.)71 1306 y Fb(olleh@mat)o(h.c)o(ha)o(lme)o(rs.)o(se)71 1391 y Fa(Yuv)l(al)h(Peres)71 1452 y Fo(Departmen)o(t)i(of)j (Statistics,)f(Univ)o(ersit)o(y)e(of)j(California,)g(Berk)o(eley)c(and) k(Depart-)71 1512 y(men)o(t)14 b(of)i(Mathematics,)f(Hebrew)g(Univ)o (ersit)o(y)l(,)e(Jerusalem.)71 1572 y Fb(peres@sta)o(t.b)o(er)o(kel)o (ey.)o(ed)o(u)71 1657 y Fa(R)o(ober)m(to)k(H.)h(Schonmann)71 1717 y Fo(Departmen)o(t)c(of)j(Mathematics,)d(Univ)o(ersit)o(y)f(of)k (California,)f(Los)h(Angeles.)71 1777 y Fb(rhs@math.)o(ucl)o(a.)o(edu)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9811191753332--