Content-Type: multipart/mixed; boundary="-------------9905101904796" This is a multi-part message in MIME format. ---------------9905101904796 Content-Type: text/plain; name="99-160.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-160.comments" olleh@math.chalmers.se rhs@math.ucla.edu steif@math.chalmers.se URL http://www.math.chalmers.se/~olleh/ URL http://www.math.ucla.edu/~rhs/ URL http://www.math.chalmers.se/~steif/ ---------------9905101904796 Content-Type: text/plain; name="99-160.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-160.keywords" diluted Ising model, percolation, anchored Cheeger constant, strong amenability, graphs, phase diagram ---------------9905101904796 Content-Type: application/postscript; name="hss.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="hss.ps" %!PS-Adobe-2.0 %%Creator: dvips 5.58 Copyright 1986, 1994 Radical Eye Software %%Title: hss.dvi 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Fq(1])j(:)20 b Fk(P)854 1855 y Fl(G)854 1885 y(p)883 1873 y Fq(\()p Fm(9)15 b Fq(an)g(in\014nite)i(cluster)q(\))12 b(=)h(1)p Fm(g)p Fn(:)23 2016 y Fq(In)k(this)g(pap)q(er)h(w)o(e)e(will)j(also)e (consider)g(more)g(general)g Fj(stationary)g Fq(\(or)f Fj(invariant)p Fq(\))g(p)q(ercolation)i(pro-)-48 2090 y(cesses)g(on)g(a)g(transitiv)o(e)g(graph.)27 b(Suc)o(h)19 b(a)f(pro)q(cess)g(can)g(b)q(e)g(seen)h(as)e(simply)i(a)f(probabilit)o (y)h(measure)f(on)-48 2163 y Fm(f)p Fq(0)p Fn(;)8 b Fq(1)p Fm(g)65 2147 y Fl(V)111 2163 y Fq(whic)o(h)18 b(is)f(in)o(v)m(arian)o (t)h(under)g(Aut\()p Fn(G)p Fq(\).)25 b(This)17 b(binary)h(random)f (\014eld)h(is)g(then)f(in)o(terpreted)h(in)g(the)-48 2236 y(same)f(w)o(a)o(y)g(as)h(in)g(the)g(case)g(of)f(Bernoulli)j(p)q (ercolation,)f(with)f(clusters)g(ha)o(ving)g(the)g(same)g(de\014nition) h(as)-48 2310 y(ab)q(o)o(v)o(e.)36 b(One)22 b(sa)o(ys)e(that)g(the)h Fj(\014nite)g(ener)n(gy)f(c)n(ondition)h Fq(holds)g(if)h(on)e(an)o(y)h 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b(prop)q(erties)g(of)f(the)h(in\014nite)h(cluster\(s\))e(that)g(arise)h (from)f(p)q(ercolation.)28 b(F)l(or)17 b(example,)h(in)h([47)o(])e(and) -48 703 y([6)o(],)f(it)h(is)f(studied)i(whether)e(clusters)h(are)f (transien)o(t)g(for)g(simple)h(random)f(w)o(alk.)23 b(F)l(or)16 b(another)f(example,)-48 777 y(in)f([30)o(])f(and)g([51])f(it)i(is)g (sho)o(wn)f(that)f(for)h(Bernoulli)i(p)q(ercolation)g(on)e(a)g (transitiv)o(e)g(graph)g Fn(G)g Fq(with)h(reten)o(tion)-48 850 y(parameter)j Fn(p)f(>)h(p)286 857 y Fl(c)304 850 y Fq(\()p Fn(G)p Fq(\))f(ev)o(ery)i(in\014nite)h(cluster)f Fn(C)j Fq(has)c Fn(p)980 857 y Fl(c)997 850 y Fq(\()p Fn(C)s Fq(\))f(=)h Fn(p)1160 857 y Fl(c)1177 850 y Fq(\()p Fn(G)p Fq(\))p Fn(=p)p Fq(.)26 b(Other)18 b(results)g(concerning)-48 923 y(prop)q(erties)d(of)e(in\014nite)j(clusters)f(can)f(b)q(e)h (found,)f(e.g.,)g(in)h([27)o(],)e([4],)g([31)o(])h(and)g([47)o(])g(\(n) o(um)o(b)q(er)g(of)g Fj(top)n(olo)n(gic)n(al)-48 997 y(ends)k 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y(of)h(result)g(in)h(Section)h(4.)23 1535 y(It)d(is)h(natural)f (to)g(ask)g(whether)h(the)f(amenabilit)o(y)i(or)e(nonamenabilit)o(y)i (of)e(a)g(graph)g(is)h(\\inherited")h(b)o(y)-48 1609 y(the)g(in\014nite)i(clusters)e(on)g(this)g(graph,)g(at)f(least)h(in)h (the)f(Bernoulli)i(case.)k(The)16 b(answ)o(er)g(is)g(trivially)i(\\y)o (es")-48 1682 y(when)e Fn(p)c Fq(=)h(1,)h(but)i(otherwise,)f(and)g(if)g (the)h(graph)e(is)i(of)f(b)q(ounded)h(degree,)f(it)h(is)f(negativ)o(e,) g(since)h(it)g(is)f(not)-48 1756 y(hard)e(to)f(see)h(that)g(all)g(the)g (in\014nite)i(clusters)f(then)f(a.s.)f(con)o(tain)h(arbitrarily)g (large)g(c)o(hains)h(of)e(v)o(ertices)i(with)-48 1829 y(degree)i(2)e(\(see,)h(e.g.,)f([51)o(,)h(pro)q(of)g(of)g(Lemma)g (1]\),)f(and)h(therefore)g(are)g(alw)o(a)o(ys)f(amenable.)23 1927 y(In)i([6,)f(Question)i(6.5],)e(it)h(is)h(ask)o(ed)f(whether)g (for)f(Bernoulli)j(p)q(ercolation)f(on)f(a)g(nonamenable)h(graph)-48 2001 y(all)d(the)g(in\014nite)h(clusters)f(are)f(alw)o(a)o(ys)g(w)o 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(to)f(the)g(random-)-48 2359 y(cluster)19 b(measure)f(with)h (parameters)e Fn(r)i Fq(and)f Fn(q)i Fq(=)e(2)g(on)g(the)g(subgraph)h (of)e Fn(G)h Fq(obtained)h(b)o(y)f(deleting)i(all)-48 2433 y(edges)15 b(with)h(v)m(alue)g Fm(\000)p Fq(1.)867 2581 y(19)p eop %%Page: 20 20 20 19 bop 23 311 a Fq(Also)20 b(let)210 299 y(~)200 311 y Fn( )232 289 y Fl(r)o(;)p Fh(2)p Fl(;p)230 325 y(G)324 311 y Fq(b)q(e)g(the)h(probabilit)o(y)g(measure)f(on)g Fm(f)p Fq(0)p Fn(;)8 b Fq(1)p Fm(g)1074 295 y Fl(E)1122 311 y Fq(corresp)q(onding)21 b(to)e(\014rst)h(pic)o(king)h Fn(Y)32 b Fm(2)-48 385 y(f\000)p Fq(1)p Fn(;)8 b Fq(0)p Fn(;)g Fq(1)p Fm(g)144 368 y Fl(E)184 385 y Fq(according)14 b(to)f Fn( )473 363 y Fl(r)o(;)p Fh(2)p Fl(;p)471 398 y(G)544 385 y Fq(,)g(and)h(then)f(obtaining)965 373 y(~)958 385 y Fn(Y)23 b Fm(2)13 b(f)p Fq(0)p Fn(;)8 b Fq(1)p Fm(g)1163 368 y Fl(E)1204 385 y Fq(b)o(y)13 b(letting)1414 373 y(~)1407 385 y Fn(Y)e Fq(\()p Fn(e)p Fq(\))h(=)h(max)p Fm(f)p Fq(0)p Fn(;)8 b(Y)h Fq(\()p Fn(e)p 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y(with)d(the)f(same)g(parameter)f(v)m(alues)i(on)g(the)f(graph)g (obtained)h(from)e Fn(H)19 b Fq(b)o(y)c(remo)o(ving)g Fn(e)1503 2264 y Fw(0)1515 2281 y Fq(.)20 b(Hence,)351 2384 y Fn( )383 2368 y Fl(r)400 2356 y Fp(0)410 2368 y Fl(;)p Fh(2)p Fl(;p)468 2384 y Fq(\()p Fn(Y)10 b Fq(\()p Fn(E)i Fm(n)d(f)p Fn(e)p Fm(g)p Fq(\))j(=)h Fn(\021)c Fm(j)f Fn(Y)i Fq(\()p Fn(e)p Fq(\))i(=)h(0)p Fn(;)1019 2362 y Fq(~)1019 2373 y(~)1013 2384 y Fn(Y)c Fq(\()p Fn(E)j Fm(n)e(f)p Fn(e)p Fm(g)p Fq(\))i(=)1291 2373 y(~)1293 2384 y(~)-26 b Fn(\021)q Fq(\))p 333 2404 1017 2 v 333 2462 a Fn( )365 2449 y Fl(r)382 2439 y Fp(0)393 2449 y Fl(;)p Fh(2)p Fl(;p)450 2462 y Fq(\()p Fn(Y)10 b Fq(\()p Fn(E)i Fm(n)e(f)p Fn(e)p Fm(g)p Fq(\))i(=)h Fn(\021)8 b Fm(j)805 2440 y Fq(~)805 2450 y(~)798 2462 y Fn(Y)i Fq(\()p Fn(e)p Fq(\))i(=)h Fm(\000)p Fq(1)p Fn(;)1036 2440 y Fq(~)1036 2450 y(~)1030 2462 y Fn(Y)d Fq(\()p Fn(E)i Fm(n)e(f)p Fn(e)p Fm(g)p Fq(\))i(=)1308 2451 y(~)1311 2462 y(~)-26 b Fn(\021)q Fq(\))1367 2415 y(=)13 b(1)p Fn(:)867 2581 y Fq(21)p eop %%Page: 22 22 22 21 bop -48 311 a Fq(Note)15 b(also)g(that)376 385 y Fn( )408 366 y Fl(r)425 354 y Fp(0)435 366 y Fl(;)p Fh(2)p Fl(;p)492 385 y Fq(\()p Fn(Y)c Fq(\()p Fn(e)p Fq(\))h(=)h(0)8 b Fm(j)721 362 y Fq(~)721 373 y(~)716 385 y Fn(Y)h Fq(\()p Fn(e)p Fq(\))j(=)h(0)p Fn(;)918 362 y Fq(~)918 373 y(~)912 385 y Fn(Y)c Fq(\()p Fn(E)j Fm(n)e(f)p Fn(e)p Fm(g)p Fq(\))i(=)1190 374 y(~)1192 385 y(~)-26 b Fn(\021)q Fq(\))12 b Fm(\025)h Fq(1)d Fm(\000)h Fn(r)1392 366 y Fw(0)-48 481 y Fq(using)16 b(\(9\).)j(Com)o(bining)d (these)f(observ)m(ations,)g(w)o(e)g(get)-43 587 y Fn( )-11 570 y Fl(r)6 559 y Fp(0)16 570 y Fl(;)p Fh(2)p Fl(;p)73 587 y Fq(\()p Fn(Y)c Fq(\()p Fn(e)p Fq(\))h(=)h(0)p Fn(;)8 b(Y)h Fq(\()p Fn(E)j Fm(n)e(f)p Fn(e)p Fm(g)p Fq(\))i(=)h Fn(\021)c Fm(j)625 565 y Fq(~)625 575 y(~)619 587 y Fn(Y)g Fq(\()p Fn(e)p Fq(\))j(=)h(0)p Fn(;)821 565 y Fq(~)821 575 y(~)815 587 y Fn(Y)d Fq(\()p Fn(E)i Fm(n)e(f)p Fn(e)p Fm(g)p Fq(\))h(=)1093 576 y(~)1096 587 y(~)-27 b Fn(\021)r Fq(\))p -43 607 1178 2 v 38 665 a Fn( )70 652 y Fl(r)87 642 y Fp(0)97 652 y Fl(;)p Fh(2)p Fl(;p)154 665 y Fq(\()p Fn(Y)10 b Fq(\()p Fn(E)i Fm(n)e(f)p Fn(e)p Fm(g)p Fq(\))i(=)h Fn(\021)c Fm(j)509 642 y Fq(~)509 653 y(~)503 665 y Fn(Y)g Fq(\()p Fn(e)p Fq(\))j(=)h Fm(\000)p Fq(1)p Fn(;)741 642 y Fq(~)741 653 y(~)734 665 y Fn(Y)d Fq(\()p Fn(E)i Fm(n)e(f)p Fn(e)p Fm(g)p Fq(\))i(=)1013 654 y(~)1015 665 y(~)-26 b Fn(\021)q Fq(\))-6 776 y(=)75 745 y Fn( )107 729 y Fl(r)124 717 y Fp(0)135 729 y Fl(;)p Fh(2)p Fl(;p)192 745 y Fq(\()p Fn(Y)10 b Fq(\()p Fn(E)i Fm(n)e(f)p Fn(e)p Fm(g)p Fq(\))i(=)h Fn(\021)8 b Fm(j)g Fn(Y)i Fq(\()p Fn(e)p Fq(\))i(=)h(0)p Fn(;)743 723 y Fq(~)743 734 y(~)737 745 y Fn(Y)d Fq(\()p Fn(E)i Fm(n)d(f)p Fn(e)p Fm(g)p Fq(\))j(=)1015 735 y(~)1018 745 y(~)-27 b Fn(\021)r Fq(\))p Fn( )1089 729 y Fl(r)1106 717 y Fp(0)1116 729 y Fl(;)p Fh(2)p Fl(;p)1173 745 y Fq(\()p Fn(Y)10 b Fq(\()p Fn(e)p Fq(\))i(=)h(0)8 b Fm(j)1401 723 y Fq(~)1401 734 y(~)1396 745 y Fn(Y)h 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Fp(0)16 1128 y Fl(;)p Fh(2)p Fl(;p)73 1142 y Fq(\()p Fn(Y)e Fq(\()p Fn(e)p Fq(\))h(=)h Fm(\000)p Fq(1)8 b Fm(j)g Fn(Y)h Fq(\()p Fn(E)j Fm(n)e(f)p Fn(e)p Fm(g)p Fq(\))i(=)h Fn(\021)r Fq(\))676 1110 y(=)754 1080 y Fn( )786 1063 y Fl(r)803 1051 y Fp(0)813 1063 y Fl(;)p Fh(2)p Fl(;p)870 1080 y Fq(\()p Fn(Y)e Fq(\()p Fn(e)p Fq(\))h(=)h(0)p Fn(;)8 b(Y)h Fq(\()p Fn(E)j Fm(n)e(f)p Fn(e)p Fm(g)p Fq(\))i(=)h Fn(\021)r Fq(\)\))p 736 1100 704 2 v 736 1142 a Fn( )768 1128 y Fl(r)785 1119 y Fp(0)796 1128 y Fl(;)p Fh(2)p Fl(;p)853 1142 y Fq(\()p Fn(Y)d Fq(\()p Fn(e)p Fq(\))i(=)h Fm(\000)p Fq(1)p Fn(;)8 b(Y)i Fq(\()p Fn(E)i Fm(n)e(f)p Fn(e)p Fm(g)p Fq(\))h(=)i Fn(\021)r Fq(\)\))-6 1253 y(=)75 1222 y Fn( )107 1206 y Fl(r)124 1194 y Fp(0)135 1206 y Fl(;)p Fh(2)p Fl(;p)192 1222 y Fq(\()217 1200 y(~)217 1211 y(~)210 1222 y Fn(Y)d Fq(\()p Fn(e)p Fq(\))i(=)h(0)p Fn(;)413 1200 y Fq(~)413 1211 y(~)407 1222 y Fn(Y)c Fq(\()p Fn(E)j Fm(n)e(f)p Fn(e)p Fm(g)p Fq(\))i(=)685 1211 y(~)687 1222 y(~)-26 b Fn(\021)q 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1289 y(~)1289 1300 y Fn(Y)i Fq(\()p Fn(e)p Fq(\))i(=)h Fm(\000)p Fq(1)p Fn(;)1527 1278 y Fq(~)1527 1289 y(~)1521 1300 y Fn(Y)d Fq(\()p Fn(E)i Fm(n)e(f)p Fn(e)p Fm(g)p Fq(\))i(=)1799 1289 y(~)1802 1300 y(~)-26 b Fn(\021)q Fq(\))-6 1394 y Fm(\025)75 1363 y Fn(p)p Fq(\(1)10 b Fm(\000)g Fn(r)216 1346 y Fw(0)228 1363 y Fq(\))p 75 1383 170 2 v 110 1425 a(1)g Fm(\000)g Fn(p)250 1394 y(:)-48 1510 y Fq(Com)o(bining)16 b(this)g(with)f (\(12\),)f(w)o(e)h(get)219 1621 y Fn( )251 1602 y Fl(r)268 1590 y Fp(0)278 1602 y Fl(;)p Fh(2)p Fl(;p)335 1621 y Fq(\()p Fn(Y)10 b Fq(\()p Fn(e)p Fq(\))i(=)h(1)8 b Fm(j)g Fn(Y)h Fq(\()p Fn(E)j Fm(n)e(f)p Fn(e)p Fm(g)p Fq(\))i(=)h Fn(\021)r Fq(\))f Fm(\025)1133 1590 y Fq(1)p 943 1610 404 2 v 943 1657 a(1)d(+)1026 1639 y Fh(1)p Fw(\000)p Fl(r)1088 1629 y Fp(0)p 1026 1646 74 2 v 1048 1672 a Fl(r)1065 1663 y Fp(0)1104 1657 y Fq(\(1)g(+)1233 1636 y Fh(1)p Fw(\000)p Fl(p)p 1205 1646 119 2 v 1205 1673 a(p)p Fh(\(1)p Fw(\000)p Fl(r)1299 1663 y Fp(0)1310 1673 y Fh(\))1328 1657 y Fq(\))1364 1621 y(=)k Fn(pr)1457 1602 y Fw(0)1480 1621 y Fm(\025)g Fn(r)o(;)-48 1757 y Fq(where)g(w)o(e)g(used)h(\(8\).)k(So)13 b(\(10\))e(is)j(established.) 21 b(A)13 b(similar)h(calculation)g(sho)o(ws)f(that)f(\(11\))g(holds)i (whenev)o(er)-48 1830 y(\(8\))g(holds,)i(so)f(that)f(the)h(prop)q (osition)h(follo)o(ws.)p 1797 1818 31 2 v 1797 1843 2 25 v 1825 1843 V 1797 1845 31 2 v 23 1927 a(No)o(w)h(let)i Fn(H)i Fq(=)d(\()p Fn(V)s(;)8 b(E)s Fq(\))15 b(b)q(e)k(an)f(in\014nite) i(lo)q(cally)g(\014nite)f(connected)h(graph)e(as)f(in)j(Theorem)e(6.1.) 28 b(Let)-48 2001 y Fn(F)-19 2008 y Fh(1)1 2001 y Fn(;)8 b(F)51 2008 y Fh(2)70 2001 y Fn(;)g(:)g(:)g(:)14 b Fq(b)q(e)j(a)f (sequence)i(of)e(\014nite)h(subsets)g(of)f Fn(V)26 b Fq(suc)o(h)17 b(that)f Fn(F)1120 2008 y Fh(1)1154 2001 y Fm(\032)g Fn(F)1234 2008 y Fh(2)1268 2001 y Fm(\032)f(\001)8 b(\001)g(\001)15 b Fq(and)1478 1969 y Fi(S)1512 1982 y Fw(1)1512 2012 y Fl(i)p Fh(=1)1579 2001 y Fn(F)1608 2008 y Fl(i)1637 2001 y Fq(=)g Fn(V)10 b Fq(.)24 b(F)l(or)-48 2074 y(eac)o(h)17 b Fn(i)p Fq(,)g(let)h(\003)202 2081 y Fl(i)233 2074 y Fq(b)q(e)g(the)f(\(\014nite\))g(graph)g(whose)g(v)o (ertex)g(set)g Fn(V)1037 2081 y Fh(\003)1061 2086 y Fg(i)1093 2074 y Fq(is)h Fn(F)1170 2081 y Fl(i)1196 2074 y Fm([)12 b(f)p Fq(\001)p Fm(g)p Fq(,)k(where)i(\001)f(is)g(an)h(auxiliary)-48 2148 y(v)o(ertex,)c(and)i(whose)f(edge)g(set)g Fn(E)533 2155 y Fh(\003)557 2160 y Fg(i)588 2148 y Fq(consists)g(of)-25 2275 y(\(i\))g(the)g(edges)h(in)g Fn(E)h Fq(that)d(connect)i(t)o(w)o(o) e(v)o(ertices)h(in)i Fn(F)940 2282 y Fl(i)954 2275 y Fq(,)e(and)-25 2383 y(\(ii\))e(for)f(eac)o(h)h(edge)g(in)h Fn(E)h Fq(that)d(connects)h(some)g Fn(x)f Fm(2)h Fn(F)916 2390 y Fl(i)943 2383 y Fq(to)g(some)f Fn(y)j Fm(2)e Fn(V)i Fm(n)5 b Fn(F)1287 2390 y Fl(i)1301 2383 y Fq(,)13 b(w)o(e)f(include)k (an)c(edge)i Fm(h)p Fn(x;)8 b Fq(\001)p Fm(i)66 2457 y Fq(in)16 b Fn(E)153 2464 y Fh(\003)177 2469 y Fg(i)192 2457 y Fq(.)867 2581 y(22)p eop %%Page: 23 23 23 22 bop -48 311 a Fq(Fix)18 b Fn(x)e Fm(2)h Fn(F)154 318 y Fh(1)174 311 y Fq(.)27 b(It)18 b(is)g(w)o(ell)g(kno)o(wn)g(\(see) f(e.g.)g([28)o(]\))g(and)h(easy)f(to)g(sho)o(w)g(using)h(Prop)q (osition)g(6.3,)f(that)g(the)-48 385 y(Ising)f(mo)q(del)g(on)e Fn(H)19 b Fq(with)c(coupling)i(constan)o(t)d Fn(J)19 b Fq(and)c(zero)g(external)g(\014eld,)h(has)f(phase)g(transition)g(if)h (and)-48 458 y(only)g(if)692 532 y(lim)682 560 y Fl(i)p Fw(!1)772 532 y Fn(\036)799 509 y Fl(r)o(;)p Fh(2)799 545 y(\003)823 550 y Fg(i)844 532 y Fq(\()p Fn(x)c Fm($)h Fq(\001\))f Fn(>)h Fq(0)649 b(\(13\))-48 632 y(where)15 b Fn(r)e Fq(=)g(1)c Fm(\000)h Fn(e)263 615 y Fw(\000)p Fh(2)p Fl(J)333 632 y Fq(.)19 b(By)c(similar)h(argumen)o(ts,)e(it)h(is) g(easy)g(to)f(see)h(that)f(the)h(Ising)h(mo)q(del)f(with)g(coupling)-48 705 y(constan)o(t)h Fn(J)166 689 y Fw(0)196 705 y Fq(and)h(zero)h (external)f(\014eld)i(on)e(the)h(union)g(of)f(the)g(clusters)h(of)f Fn(H)k Fq(obtained)d(with)g(Bernoulli)-48 779 y(p)q(ercolation)e(with)g (parameter)e Fn(p)p Fq(,)h(has)g(phase)g(transition)h(if)g(and)f(only)h (if)669 898 y(lim)660 927 y Fl(i)p Fw(!1)759 886 y Fq(~)750 898 y Fn( )782 876 y Fl(r)799 864 y Fp(0)809 876 y Fl(;)p Fh(2)p Fl(;p)780 911 y Fh(\003)804 916 y Fg(i)866 898 y Fq(\()p Fn(x)c Fm($)i Fq(\001\))e Fn(>)h Fq(0)626 b(\(14\))-48 1023 y(where)16 b Fn(r)106 1006 y Fw(0)132 1023 y Fq(=)e(1)d Fm(\000)g Fn(e)282 1006 y Fw(\000)p Fh(2)p Fl(J)349 994 y Fp(0)363 1023 y Fq(.)22 b(\(That)16 b(the)g(limits)h(in)g(\(13\))e (and)h(\(14\))f(exist)i(follo)o(ws)f(from)f(standard)h(sto)q(c)o (hastic)-48 1096 y(monotonicit)o(y)f(argumen)o(ts)g(based)g(on)g (Holley's)h(Lemma.\))23 1194 y(Supp)q(ose)g(no)o(w)f(that)606 1281 y Fn(J)635 1262 y Fw(0)660 1281 y Fm(\025)e(\000)748 1250 y Fq(1)p 748 1270 23 2 v 748 1312 a(2)784 1281 y(log)850 1209 y Fi( )888 1250 y Fn(e)909 1234 y Fw(\000)p Fh(2)p Fl(J)988 1250 y Fq(+)e Fn(p)f Fm(\000)g Fq(1)p 888 1270 248 2 v 1000 1312 a Fn(p)1140 1209 y Fi(!)-48 1395 y Fq(whic)o(h)16 b(is)g(equiv)m(alen)o(t)h(to)d Fn(p)f Fm(\025)493 1377 y Fl(r)p 488 1384 29 2 v 488 1411 a(r)505 1401 y Fp(0)521 1395 y Fq(.)20 b(By)15 b(Prop)q(osition)h(6.4,)d(w)o(e) i(then)h(ha)o(v)o(e)f(that)589 1514 y Fn(\036)616 1492 y Fl(r)o(;)p Fh(2)616 1528 y(\003)640 1533 y Fg(i)661 1514 y Fq(\()p Fn(x)d Fm($)h Fq(\001\))f Fm(\024)900 1502 y Fq(~)891 1514 y Fn( )923 1492 y Fl(r)940 1480 y Fp(0)950 1492 y Fl(;)p Fh(2)p Fl(;p)921 1528 y Fh(\003)945 1533 y Fg(i)1007 1514 y Fq(\()p Fn(x)h Fm($)g Fq(\001\))p Fn(:)-48 1633 y Fq(By)k(letting)g Fn(i)d Fm(!)i(1)h Fq(and)g(using)g (\(13\))e(and)i(\(14\),)e(w)o(e)h(arriv)o(e)h(at)f(the)g(conclusion)j (that)c(if)i(the)g(\(0)p Fn(;)8 b(J)t Fq(\)-Ising)-48 1707 y(mo)q(del)22 b(on)e Fn(H)k Fq(has)d(phase)g(transition,)h(then)f (so)f(do)q(es)i(the)e(\(0)p Fn(;)8 b(J)1122 1690 y Fw(0)1133 1707 y Fq(\)-Ising)22 b(mo)q(del)f(on)g(the)g(union)h(of)e(the)-48 1780 y(clusters)i(of)f Fn(H)k Fq(obtained)d(from)f(Bernoulli)j(\()p Fn(p)p Fq(\))d(b)q(ond)h(p)q(ercolation.)40 b(This)22 b(immediately)h(implies)h(the)-48 1853 y(follo)o(wing)16 b(result.)-48 1989 y Fk(Lemm)o(a)e(6.6)23 b Fj(L)n(et)12 b Fn(H)k Fq(=)d(\()p Fn(V)s(;)8 b(E)s Fq(\))h Fj(b)n(e)j(an)h (in\014nite,)f(lo)n(c)n(al)r(ly)f(\014nite)h(c)n(onne)n(cte)n(d)f(gr)n (aph)i(satisfying)f Fn(J)1613 1996 y Fl(c)1630 1989 y Fq(\()p Fn(H)t Fq(\))g Fn(<)h Fm(1)p Fj(.)-48 2062 y(F)m(or)j(any)g Fn(p)c(>)h Fq(1)d Fm(\000)h Fn(e)307 2046 y Fw(\000)p Fh(2)p Fl(J)371 2050 y Fg(c)387 2046 y Fh(\()p Fl(H)s Fh(\))448 2062 y Fj(,)17 b(we)f(have)444 2195 y Fn(J)469 2202 y Fl(c)487 2195 y Fq(\()p Fn(H)q(;)8 b(p;)g(bond)p Fq(\))i Fm(\024)j(\000)818 2164 y Fq(1)p 818 2184 23 2 v 818 2226 a(2)853 2195 y(log)919 2123 y Fi( )957 2164 y Fn(e)978 2148 y Fw(\000)p Fh(2)p Fl(J)1042 2152 y Fg(c)1059 2148 y Fh(\()p Fl(H)s Fh(\))1130 2164 y Fq(+)d Fn(p)g Fm(\000)h Fq(1)p 957 2184 320 2 v 1106 2226 a Fn(p)1282 2123 y Fi(!)1322 2195 y Fn(:)411 b Fq(\(15\))-48 2327 y Fj(In)15 b(p)n(articular,)622 2400 y Fq(lim)618 2430 y Fl(p)p Fw(\045)p Fh(1)696 2400 y Fn(J)721 2407 y Fl(c)739 2400 y Fq(\()p Fn(H)q(;)8 b(p;)g(bond)p Fq(\))h(=)k Fn(J)1054 2407 y Fl(c)1072 2400 y Fq(\()p Fn(H)t Fq(\))p Fn(:)867 2581 y Fq(23)p eop %%Page: 24 24 24 23 bop -48 311 a Fk(Pro)q(of)17 b(of)h(Theorem)c(6.1:)20 b Fq(Con)o(tin)o(uit)o(y)15 b(of)f Fn(J)782 318 y Fl(c)800 311 y Fq(\()p Fn(H)q(;)8 b(p;)g(bond)p Fq(\))k(at)i Fn(p)f Fq(=)g(1)i(w)o(as)f(established)i(in)g(Lemma)f(6.6;)-48 385 y(what)e(remains)i(is)f(to)g(pro)o(v)o(e)f(con)o(tin)o(uit)o(y)h (for)g Fn(p)e Fm(2)h Fq(\()p Fn(p)854 392 y Fl(c)871 385 y Fn(;)8 b Fq(1\).)18 b(Fix)c(suc)o(h)h(a)e Fn(p)p Fq(.)20 b(Let)14 b Fn(")f(>)g Fq(0)g(b)q(e)i(arbitrary)e(and)h(let)-48 458 y Fn(J)j Fq(=)c Fn(J)67 465 y Fl(c)85 458 y Fq(\()p Fn(H)q(;)8 b(p;)g(bond)p Fq(\))f(+)j Fn(")p Fq(.)20 b(No)o(w)15 b(\014x)g(an)g(arbitrary)g Fn(x)e Fm(2)g Fn(V)24 b Fq(and)16 b(de\014ne)g(the)f(ev)o(en)o(t)664 577 y Fn(E)698 584 y Fl(x)732 577 y Fq(:=)d Fm(f)p Fn(J)840 584 y Fl(c)858 577 y Fq(\()p Fn(C)s Fq(\()p Fn(x)p Fq(\)\))f Fm(\024)i Fn(J)t Fm(g)p Fn(:)-48 696 y Fq(Then,)h(b)o(y)f(\(3\),)f(w)o(e)h(ha)o (v)o(e)g Fk(P)431 680 y Fl(H)431 709 y(p;bond)532 696 y Fq(\()p Fn(E)584 703 y Fl(x)605 696 y Fq(\))f Fn(>)h Fq(0.)19 b(Note)13 b(that)g(for)f Fn(p)1033 680 y Fw(0)1057 696 y Fn(<)h(p)p Fq(,)g(Bernoulli)j(\()p Fn(p)1390 680 y Fw(0)1401 696 y Fq(\))d(b)q(ond)h(p)q(ercolation)g(on)-48 770 y Fn(H)i Fq(can)d(b)q(e)g(obtained)g(b)o(y)g(\014rst)f(p)q (erforming)h(Bernoulli)i(\()p Fn(p)p Fq(\))d(p)q(ercolation)h(on)g Fn(H)t Fq(,)f(and)h(then)g(doing)g(Bernoulli)-48 843 y(\()-25 823 y Fl(p)-7 811 y Fp(0)p -25 832 29 2 v -20 859 a Fl(p)9 843 y Fq(\))g(p)q(ercolation)h(on)g(the)f(clusters)h(of)f (the)h(\014rst)f(p)q(ercolation.)20 b(If)14 b Fn(\032)e(>)h Fq(1)7 b Fm(\000)g Fn(e)1268 827 y Fw(\000)p Fh(2)p Fl(J)1337 843 y Fq(,)14 b(then)g Fn(\032)e(>)h Fq(1)7 b Fm(\000)g Fn(e)1643 827 y Fw(\000)p Fh(2)p Fl(J)1707 831 y Fg(c)1723 827 y Fh(\()p Fl(C)r Fh(\()p Fl(x)p Fh(\)\))-48 916 y Fq(for)15 b Fn(!)f Fm(2)f Fn(E)141 923 y Fl(x)178 916 y Fq(and)i(hence)h(b)o(y)f(\(15\),)f(w)o(e)h(ha)o(v)o(e)g(that)f(for)h Fn(!)f Fm(2)f Fn(E)1023 923 y Fl(x)1045 916 y Fq(,)130 1051 y Fn(J)155 1058 y Fl(c)172 1051 y Fq(\()p Fn(C)s Fq(\()p Fn(x)p Fq(\))p Fn(;)8 b(\032;)g(bond)p Fq(\))h Fm(\024)k(\000)562 1020 y Fq(1)p 562 1041 23 2 v 562 1082 a(2)597 1051 y(log)664 979 y Fi( )702 1020 y Fn(e)723 1004 y Fw(\000)p Fh(2)p Fl(J)787 1008 y Fg(c)803 1004 y Fh(\()p Fl(C)r Fh(\()p Fl(x)p Fh(\)\))918 1020 y Fq(+)d Fn(\032)g Fm(\000)g Fq(1)p 702 1041 364 2 v 871 1082 a Fn(\032)1070 979 y Fi(!)1115 1051 y Fm(\024)j(\000)1203 1020 y Fq(1)p 1203 1041 23 2 v 1203 1082 a(2)1239 1051 y(log)1305 979 y Fi( )1343 1020 y Fn(e)1364 1004 y Fw(\000)p Fh(2)p Fl(J)1444 1020 y Fq(+)d Fn(\032)g Fm(\000)g Fq(1)p 1343 1041 249 2 v 1455 1082 a Fn(\032)1596 979 y Fi(!)1636 1051 y Fn(:)-48 1186 y Fq(Hence)16 b(b)o(y)f(\(3\),)f(for)h Fn(!)f Fm(2)f Fn(E)425 1193 y Fl(x)447 1186 y Fq(,)353 1321 y Fk(P)389 1297 y Fl(C)r Fh(\()p Fl(x)p Fh(\))389 1335 y Fl(\032;bond)498 1249 y Fi( )531 1321 y Fn(J)556 1328 y Fl(c)574 1321 y Fq(\()602 1309 y(~)592 1321 y Fn(C)r Fq(\()p Fn(x)p Fq(\)\))f Fm(\024)h(\000)807 1290 y Fq(1)p 807 1310 23 2 v 807 1352 a(2)843 1321 y(log)909 1249 y Fi( )947 1290 y Fn(e)968 1274 y Fw(\000)p Fh(2)p Fl(J)1047 1290 y Fq(+)e Fn(\032)f Fm(\000)g Fq(1)p 947 1310 249 2 v 1059 1352 a Fn(\032)1200 1249 y Fi(!)1243 1321 y Fq(+)g Fn(")1309 1249 y Fi(!)1355 1321 y Fn(>)j Fq(0)-48 1463 y(where)94 1452 y(~)83 1463 y Fn(C)s Fq(\()p Fn(x)p Fq(\))i(is)g(the)h(cluster)f(of)g Fn(x)g Fq(when)h(p)q (erforming)g Fn(\032)e Fq(p)q(ercolation)j(on)e Fn(C)s Fq(\()p Fn(x)p Fq(\))f(and)h(so)338 1598 y Fk(P)374 1579 y Fl(H)374 1609 y(p\032;bond)501 1526 y Fi( )534 1598 y Fn(J)559 1605 y Fl(c)576 1598 y Fq(\()p Fn(C)s Fq(\()p Fn(x)p Fq(\)\))c Fm(\024)i(\000)809 1567 y Fq(1)p 809 1588 23 2 v 809 1629 a(2)845 1598 y(log)911 1526 y Fi( )949 1567 y Fn(e)970 1551 y Fw(\000)p Fh(2)p Fl(J)1050 1567 y Fq(+)e Fn(\032)e Fm(\000)i Fq(1)p 949 1588 249 2 v 1062 1629 a Fn(\032)1202 1526 y Fi(!)1245 1598 y Fq(+)g Fn(")1312 1526 y Fi(!)1357 1598 y Fn(>)i Fq(0)p Fn(:)-48 1730 y Fq(This)j(giv)o(es)f(that)434 1817 y Fn(J)459 1824 y Fl(c)476 1817 y Fq(\()p Fn(H)q(;)8 b(p\032;)g(bond)p Fq(\))h Fm(\024)k(\000)830 1786 y Fq(1)p 830 1807 23 2 v 830 1848 a(2)866 1817 y(log)932 1745 y Fi( )970 1786 y Fn(e)991 1770 y Fw(\000)p Fh(2)p Fl(J)1071 1786 y Fq(+)d Fn(\032)g Fm(\000)g Fq(1)p 970 1807 249 2 v 1082 1848 a Fn(\032)1223 1745 y Fi(!)1266 1817 y Fq(+)h Fn(";)-48 1931 y Fq(whic)o(h)16 b(establishes)h(the)e(left)g(con)o(tin)o(uit)o(y) l(,)h(as)f Fn(")d(>)h Fq(0)i(is)h(arbitrary)l(.)23 2029 y(As)f(for)f(the)i(righ)o(t)f(con)o(tin)o(uit)o(y)l(,)g(assume)g(for)g (con)o(tradiction)g(that)g(for)f(some)h Fn(p)e(>)g(p)1463 2036 y Fl(c)1480 2029 y Fq(,)i(w)o(e)g(ha)o(v)o(e)f(that)417 2148 y Fn(\016)g Fq(:=)f Fn(J)537 2155 y Fl(c)554 2148 y Fq(\()p Fn(H)q(;)8 b(p;)g(bond)p Fq(\))g Fm(\000)13 b Fq(lim)840 2178 y Fl(")p Fw(&)p Fh(0)917 2148 y Fn(J)942 2155 y Fl(c)959 2148 y Fq(\()p Fn(H)q(;)8 b(p)h Fq(+)i Fn(";)d(bond)p Fq(\))j Fn(>)i Fq(0)p Fn(:)-48 2273 y Fq(F)l(or)i(an)o(y)f Fn(")f(>)g Fq(0,)i(w)o(e)g(then)g(ha)o(v)o(e)g(b)o (y)g(\(3\))f(that)408 2396 y Fk(P)444 2377 y Fl(H)444 2407 y(p)p Fh(+)p Fl(";bond)596 2337 y Fi(\022)627 2396 y Fn(J)652 2403 y Fl(c)669 2396 y Fq(\()p Fn(C)s Fq(\()p Fn(x)p Fq(\)\))d Fm(\024)i Fn(J)887 2403 y Fl(c)905 2396 y Fq(\()p Fn(H)q(;)8 b(p;)g(bond)p Fq(\))f Fm(\000)1195 2365 y Fq(3)p Fn(\016)p 1195 2386 45 2 v 1206 2427 a Fq(4)1245 2337 y Fi(\023)1288 2396 y Fn(>)13 b Fq(0)p Fn(:)867 2581 y Fq(24)p eop %%Page: 25 25 25 24 bop -48 311 a Fq(Cho)q(ose)15 b Fn(")e(>)g Fq(0)i(suc)o(h)g(that) 630 354 y Fn(p)p 592 374 100 2 v 592 416 a(p)10 b Fq(+)g Fn(")709 385 y(>)j Fq(1)d Fm(\000)g Fn(e)856 366 y Fw(\000)p Fh(2[)p Fl(J)930 370 y Fg(c)947 366 y Fh(\()p Fl(H)q(;p;bond)p Fh(\))p Fw(\000)1146 352 y Fc(3)p Fg(\016)p 1145 358 31 2 v 1153 379 a Fc(4)1180 366 y Fh(])-48 485 y Fq(and)241 577 y Fm(\000)281 547 y Fq(1)p 281 567 23 2 v 281 609 a(2)316 577 y(log)383 493 y Fi(0)383 568 y(@)424 538 y Fn(e)445 520 y Fw(\000)p Fh(2[)p Fl(J)519 524 y Fg(c)535 520 y Fh(\()p Fl(H)q(;p;bond)p Fh(\))p Fw(\000)734 507 y Fc(3)p Fg(\016)p 734 513 31 2 v 741 533 a Fc(4)769 520 y Fh(])791 538 y Fq(+)863 518 y Fl(p)p 841 528 62 2 v 841 554 a(p)p Fh(+)p Fl(")918 538 y Fm(\000)g Fq(1)p 424 567 563 2 v 696 588 a Fl(p)p 674 598 62 2 v 674 624 a(p)p Fh(+)p Fl(")991 493 y Fi(1)991 568 y(A)1038 577 y Fq(+)g Fn(")j(<)g(J)1190 584 y Fl(c)1207 577 y Fq(\()p Fn(H)q(;)8 b(p;)g(bond)p Fq(\))g Fm(\000)1498 547 y Fn(\016)p 1498 567 23 2 v 1498 609 a Fq(2)1526 577 y Fn(:)-48 706 y Fq(Let)18 b Fn(E)j Fq(b)q(e)e(the)f(ev)o(en)o(t)g(that)f(under)i(\()p Fn(p)12 b Fq(+)g Fn(")p Fq(\))18 b(p)q(ercolation,)i Fn(J)1028 713 y Fl(c)1045 706 y Fq(\()p Fn(C)s Fq(\()p Fn(x)p Fq(\)\))c Fm(\024)i Fn(J)1273 713 y Fl(c)1291 706 y Fq(\()p Fn(H)q(;)8 b(p;)g(bond)p Fq(\))h Fm(\000)j Fq(3)p Fn(\016)r(=)p Fq(4.)28 b(On)19 b Fn(E)-48 779 y Fq(\(whic)o(h)d(has)f(p)q(ositiv)o(e)h(probabilit)o(y\),)g(w)o(e)f (ha)o(v)o(e)f(that)699 868 y Fn(p)p 661 888 100 2 v 661 930 a(p)c Fq(+)g Fn(")778 898 y(>)j Fq(1)d Fm(\000)g Fn(e)925 880 y Fw(\000)p Fh(2)p Fl(J)989 884 y Fg(c)1006 880 y Fh(\()p Fl(C)r Fh(\()p Fl(x)p Fh(\)\))1110 898 y Fn(:)-48 1023 y Fq(Hence,)16 b(on)f(the)g(ev)o(en)o(t)g Fn(E)s Fq(,)f(\(15\))g(\(together)g(with)i(\(3\)\))e(implies)j(that)e (when)h(w)o(e)f(p)q(erform)g(Bernoulli)i(\()1765 1003 y Fl(p)p 1743 1012 62 2 v 1743 1038 a(p)p Fh(+)p Fl(")1809 1023 y Fq(\))-48 1096 y(p)q(ercolation)f(on)f Fn(C)s Fq(\()p Fn(x)p Fq(\),)f(the)h(\(new,)g(smaller\))h(cluster)g(of)e Fn(x)i Fq(has)f(a)g Fn(J)1137 1103 y Fl(c)1169 1096 y Fq(v)m(alue)i(smaller)f(than)533 1243 y Fm(\000)573 1213 y Fq(1)p 573 1233 23 2 v 573 1275 a(2)609 1243 y(log)675 1159 y Fi(0)675 1234 y(@)717 1204 y Fn(e)738 1188 y Fw(\000)p Fh(2)p Fl(J)802 1192 y Fg(c)818 1188 y Fh(\()p Fl(C)r Fh(\()p Fl(x)p Fh(\)\))932 1204 y Fq(+)1005 1184 y Fl(p)p 983 1194 62 2 v 983 1220 a(p)p Fh(+)p Fl(")1059 1204 y Fm(\000)11 b Fq(1)p 717 1233 412 2 v 913 1254 a Fl(p)p 891 1264 62 2 v 891 1290 a(p)p Fh(+)p Fl(")1133 1159 y Fi(1)1133 1234 y(A)1179 1243 y Fq(+)g Fn(")-48 1388 y Fq(with)16 b(p)q(ositiv)o(e)g(probabilit)o(y)l(.)21 b(On)16 b Fn(E)s Fq(,)e(the)h(latter)g(is)h(at)e(most)458 1535 y Fm(\000)498 1504 y Fq(1)p 498 1524 23 2 v 498 1566 a(2)533 1535 y(log)600 1451 y Fi(0)600 1525 y(@)641 1496 y Fn(e)662 1478 y Fw(\000)p Fh(2[)p Fl(J)736 1482 y Fg(c)752 1478 y Fh(\()p Fl(H)q(;p;bond)p Fh(\))p Fw(\000)951 1464 y Fc(3)p Fg(\016)p 951 1470 31 2 v 958 1491 a Fc(4)986 1478 y Fh(])1008 1496 y Fq(+)1080 1475 y Fl(p)p 1058 1485 62 2 v 1058 1511 a(p)p Fh(+)p Fl(")1135 1496 y Fm(\000)c Fq(1)p 641 1524 563 2 v 913 1546 a Fl(p)p 891 1556 62 2 v 891 1582 a(p)p Fh(+)p Fl(")1208 1451 y Fi(1)1208 1525 y(A)1255 1535 y Fq(+)g Fn(")-48 1687 y Fq(whic)o(h)20 b(in)h(turn)f(is)g(less)g(than)f Fn(J)523 1694 y Fl(c)541 1687 y Fq(\()p Fn(H)q(;)8 b(p;)g(bond)p Fq(\))i Fm(\000)k Fn(\016)r(=)p Fq(2.)32 b(Hence,)21 b(under)g Fk(P)1269 1671 y Fl(H)1269 1701 y(p;bond)1370 1687 y Fq(,)f Fn(J)1428 1694 y Fl(c)1446 1687 y Fq(\()p Fn(C)s Fq(\()p Fn(x)p Fq(\)\))e(is)i(less)g(than)-48 1761 y Fn(J)-23 1768 y Fl(c)-5 1761 y Fq(\()p Fn(H)q(;)8 b(p;)g(bond)p Fq(\))f Fm(\000)j Fn(\016)r(=)p Fq(2)15 b(with)g(p)q(ositiv)o(e)h(probabilit)o (y)l(,)h(con)o(tradicting)e(the)h(de\014nition)h(of)d Fn(J)1509 1768 y Fl(c)1527 1761 y Fq(\()p Fn(H)q(;)8 b(p;)g(bond)p Fq(\).)p 1797 1748 31 2 v 1797 1773 2 25 v 1825 1773 V 1797 1775 31 2 v 23 1859 a(W)l(e)14 b(\014nally)h(w)o(an) o(t)f(to)f(bring)i(Lemma)f(6.6)f(and)i(Theorem)f(6.1)f(o)o(v)o(er)g(to) h(a)g(site)g(p)q(ercolation)i(setting.)j(T)l(o)-48 1932 y(this)12 b(end,)h(w)o(e)e(w)o(an)o(t)g(to)g(b)q(e)i(able)f(to)f(sto)q (c)o(hastically)i(compare)e(site)i(p)q(ercolation)f(on)g Fn(H)j Fq(to)c(b)q(ond)i(p)q(ercolation)-48 2006 y(on)k Fn(H)t Fq(.)26 b(It)18 b(is)f(not)g(immediately)j(clear)d(what)g(suc)o (h)h(a)f(sto)q(c)o(hastic)g(domination)h(should)h(mean,)e(since)i(the) -48 2079 y(measures)g Fk(P)187 2063 y Fl(H)187 2090 y(p)239 2079 y Fq(and)g Fk(P)367 2063 y Fl(H)367 2092 y(p;bond)487 2079 y Fq(are)g(de\014ned)h(on)f(di\013eren)o(t)g(spaces)g(\()p Fm(f)p Fq(0)p Fn(;)8 b Fq(1)p Fm(g)1251 2063 y Fl(V)1298 2079 y Fq(and)20 b Fm(f)p Fq(0)p Fn(;)8 b Fq(1)p Fm(g)1504 2063 y Fl(E)1532 2079 y Fq(,)19 b(resp)q(ectiv)o(ely\).)-48 2153 y(Ho)o(w)o(ev)o(er,)i(a)g(site)g(p)q(ercolation)h(pro)q(cess)g(is) f(naturally)h(iden)o(ti\014ed)h(with)e(a)g(\(usually)h(dep)q(enden)o (t\))g(b)q(ond)-48 2226 y(p)q(ercolation)c(pro)q(cess)e(in)i(whic)o(h)f (a)f(b)q(ond)h Fn(e)e Fq(=)g Fm(h)p Fn(x;)8 b(y)r Fm(i)15 b Fq(is)i(considered)h(to)e(b)q(e)h(op)q(en)g(if)g(and)g(only)g(if)g (the)f(t)o(w)o(o)-48 2299 y(v)o(ertices)g Fn(x)f Fq(and)h Fn(y)h Fq(are)f(op)q(en.)21 b(W)l(e)16 b(denote)f(b)o(y)786 2288 y(~)780 2299 y Fk(P)816 2283 y Fl(H)816 2313 y(p)834 2303 y Fp(0)865 2299 y Fq(the)g(probabilit)o(y)i(measure)e(for)g(a)g(b) q(ond)i(mo)q(del)f(whic)o(h)-48 2373 y(arises)h(from)f(an)g(i.i.d.)h (site)g(mo)q(del)h(with)f(parameter)e Fn(p)924 2356 y Fw(0)953 2373 y Fq(in)i(this)g(w)o(a)o(y)l(.)23 b(Note)17 b(that)e(the)i(comp)q(onen)o(ts)g(in)g(a)-48 2446 y(site)h(mo)q(del)g (are)e(alw)o(a)o(ys)h(induced)i(subgraphs)e(while)i(for)d(b)q(ond)i(mo) q(dels,)g(this)g(is)g(not)e(true.)26 b(Ho)o(w)o(ev)o(er,)16 b(a)867 2581 y(25)p eop %%Page: 26 26 26 25 bop -48 311 a Fq(b)q(ond)19 b(mo)q(del)g(constructed)f(from)g(a)g (site)g(mo)q(del)h(as)f(ab)q(o)o(v)o(e)g(has)g(all)h(of)f(its)g(comp)q (onen)o(ts)g(b)q(eing)i(induced)-48 385 y(subgraphs)15 b(and)g(moreo)o(v)o(er,)e(the)i(in\014nite)h(comp)q(onen)o(ts)f(in)h (this)f(b)q(ond)g(mo)q(del)h(are)f(precisely)h(the)f(same)f(as)-48 458 y(the)k(in\014nite)h(comp)q(onen)o(ts)f(in)h(the)f(original)h(site) f(mo)q(del.)28 b(With)18 b(this)h(iden)o(ti\014cation)g(in)g(mind,)g(w) o(e)e(ha)o(v)o(e)-48 532 y(the)e(follo)o(wing)h(sto)q(c)o(hastic)f (domination.)-48 657 y Fk(Lemm)o(a)f(6.7)23 b Fj(L)n(et)18 b Fn(H)j Fq(=)e(\()p Fn(V)s(;)8 b(E)s Fq(\))15 b Fj(b)n(e)k(a)g(b)n (ounde)n(d)g(de)n(gr)n(e)n(e)f(gr)n(aph)i(in)f(which)g(e)n(ach)g (vertex)g(has)g(at)h(most)f Fn(D)-48 731 y Fj(neighb)n(ors.)h(If)c Fn(p)g Fj(and)g Fn(p)369 714 y Fw(0)397 731 y Fj(satisfy)693 804 y Fn(p)716 785 y Fw(0)740 804 y Fq(=)d(1)d Fm(\000)h Fq(\(1)e Fm(\000)963 774 y(p)p 1001 774 23 2 v 30 x Fn(p)o Fq(\))1041 785 y Fl(D)1073 804 y Fn(;)660 b Fq(\(16\))-48 900 y Fj(then)746 973 y Fk(P)782 954 y Fl(H)782 984 y(p;bond)899 936 y Fw(D)896 973 y Fm(\026)950 961 y Fq(~)944 973 y Fk(P)980 954 y Fl(H)980 984 y(p)998 975 y Fp(0)1020 973 y Fn(:)713 b Fq(\(17\))-48 1099 y Fk(Pro)q(of:)20 b Fq(See)15 b([47,)f(Remark)i(6.2],)d(whic)o(h)j(w)o(as)e(motiv)m(ated)i(b)o(y)f (the)g(results)h(in)g([41)o(].)p 1797 1086 31 2 v 1797 1111 2 25 v 1825 1111 V 1797 1113 31 2 v -48 1212 a Fk(Lemm)o(a)e(6.8) 23 b Fj(L)n(et)15 b Fn(H)h Fq(=)d(\()p Fn(V)s(;)8 b(E)s Fq(\))13 b Fj(b)n(e)j(an)g(in\014nite,)f(lo)n(c)n(al)r(ly)g(\014nite)g (gr)n(aph)i(with)g(de)n(gr)n(e)n(es)e(b)n(ounde)n(d)h(by)g Fn(D)h Fj(and)-48 1286 y Fn(J)-23 1293 y Fl(c)-5 1286 y Fq(\()p Fn(H)t Fq(\))11 b Fn(<)i Fm(1)p Fj(.)21 b(F)m(or)16 b(any)g Fn(p)407 1269 y Fw(0)435 1286 y Fj(satisfying)566 1348 y Fi(\020)591 1395 y Fq(1)9 b Fm(\000)i Fq(\(1)e Fm(\000)i Fn(p)788 1376 y Fw(0)799 1395 y Fq(\))827 1363 y Fc(1)p 822 1369 26 2 v 822 1389 a Fg(D)855 1348 y Fi(\021)879 1359 y Fh(2)912 1395 y Fn(>)i Fq(1)d Fm(\000)g Fn(e)1059 1376 y Fw(\000)p Fh(2)p Fl(J)1123 1380 y Fg(c)1140 1376 y Fh(\()p Fl(H)s Fh(\))1201 1395 y Fn(;)-48 1504 y Fj(we)16 b(have)336 1608 y Fn(J)361 1615 y Fl(c)379 1608 y Fq(\()p Fn(H)q(;)8 b(p)480 1590 y Fw(0)490 1608 y Fq(\))k Fm(\024)h(\000)608 1578 y Fq(1)p 608 1598 23 2 v 608 1640 a(2)644 1608 y(log)710 1512 y Fi(0)710 1585 y(B)710 1611 y(@)752 1565 y Fn(e)773 1549 y Fw(\000)p Fh(2)p Fl(J)837 1553 y Fg(c)853 1549 y Fh(\()p Fl(H)s Fh(\))924 1565 y Fq(+)970 1518 y Fi(\020)994 1565 y Fq(1)d Fm(\000)h Fq(\(1)e Fm(\000)i Fn(p)1192 1549 y Fw(0)1203 1565 y Fq(\))1231 1534 y Fc(1)p 1226 1540 26 2 v 1226 1560 a Fg(D)1259 1518 y Fi(\021)1283 1530 y Fh(2)1313 1565 y Fm(\000)g Fq(1)p 752 1598 630 2 v 900 1613 a Fi(\020)925 1660 y Fq(1)e Fm(\000)i Fq(\(1)e Fm(\000)i Fn(p)1122 1647 y Fw(0)1133 1660 y Fq(\))1161 1628 y Fc(1)p 1156 1634 26 2 v 1156 1655 a Fg(D)1189 1613 y Fi(\021)1213 1624 y Fh(2)1386 1512 y Fi(1)1386 1585 y(C)1386 1611 y(A)1430 1608 y Fn(:)303 b Fq(\(18\))-48 1744 y Fj(In)15 b(p)n(articular,)672 1817 y Fq(lim)663 1847 y Fl(p)681 1838 y Fp(0)692 1847 y Fw(\045)p Fh(1)752 1817 y Fn(J)777 1824 y Fl(c)795 1817 y Fq(\()p Fn(H)q(;)8 b(p)896 1798 y Fw(0)906 1817 y Fq(\))k(=)h Fn(J)1009 1824 y Fl(c)1027 1817 y Fq(\()p Fn(H)t Fq(\))p Fn(:)-48 1964 y Fk(Pro)q(of:)23 b Fq(Cho)q(ose)16 b Fn(p)h Fq(so)f(that)g (\(16\))g(holds.)25 b(A)17 b(simple)h(computation)f(sho)o(ws)f(that)g Fn(p)f Fq(=)1494 1917 y Fi(\020)1518 1964 y Fq(1)10 b Fm(\000)h Fq(\(1)e Fm(\000)i Fn(p)1716 1948 y Fw(0)1727 1964 y Fq(\))1755 1933 y Fc(1)p 1750 1939 V 1750 1959 a Fg(D)1782 1917 y Fi(\021)1807 1929 y Fh(2)-48 2038 y Fq(and)k(so)g(Lemma)g(6.6)g(giv)o(es)454 2160 y Fn(J)479 2167 y Fl(c)497 2160 y Fq(\()p Fn(H)q(;)8 b(p;)g(bond)p Fq(\))i Fm(\024)j(\000)828 2129 y Fq(1)p 828 2150 23 2 v 828 2191 a(2)863 2160 y(log)930 2088 y Fi( )967 2129 y Fn(e)988 2113 y Fw(\000)p Fh(2)p Fl(J)1052 2117 y Fg(c)1069 2113 y Fh(\()p Fl(H)s Fh(\))1140 2129 y Fq(+)e Fn(p)f Fm(\000)g Fq(1)p 967 2150 320 2 v 1116 2191 a Fn(p)1292 2088 y Fi(!)-48 2283 y Fq(whic)o(h)16 b(equals)452 2396 y Fm(\000)492 2365 y Fq(1)p 492 2385 23 2 v 492 2427 a(2)528 2396 y(log)594 2299 y Fi(0)594 2372 y(B)594 2399 y(@)635 2353 y Fn(e)656 2336 y Fw(\000)p Fh(2)p Fl(J)720 2340 y Fg(c)737 2336 y Fh(\()p Fl(H)s Fh(\))808 2353 y Fq(+)854 2306 y Fi(\020)878 2353 y Fq(1)10 b Fm(\000)h Fq(\(1)e Fm(\000)i Fn(p)1076 2336 y Fw(0)1087 2353 y Fq(\))1115 2321 y Fc(1)p 1110 2327 26 2 v 1110 2348 a Fg(D)1142 2306 y Fi(\021)1167 2317 y Fh(2)1197 2353 y Fm(\000)g Fq(1)p 635 2385 630 2 v 784 2400 a Fi(\020)808 2447 y Fq(1)f Fm(\000)h Fq(\(1)e Fm(\000)i Fn(p)1006 2434 y Fw(0)1017 2447 y Fq(\))1045 2416 y Fc(1)p 1040 2422 26 2 v 1040 2442 a Fg(D)1073 2400 y Fi(\021)1097 2412 y Fh(2)1270 2299 y Fi(1)1270 2372 y(C)1270 2399 y(A)1314 2396 y Fn(:)867 2581 y Fq(26)p eop %%Page: 27 27 27 26 bop -48 311 a Fq(By)11 b(Lemma)f(6.7)g(\(and)g(the)h(commen)o(ts) f(ab)q(o)o(v)o(e)g(it\),)h(w)o(e)f(ha)o(v)o(e)g(that)g Fn(J)1106 318 y Fl(c)1124 311 y Fq(\()p Fn(H)q(;)e(p)1225 295 y Fw(0)1235 311 y Fq(\))k Fm(\024)h Fn(J)1338 318 y Fl(c)1356 311 y Fq(\()p Fn(H)q(;)8 b(p;)g(bond)p Fq(\),)g(completing) -48 385 y(the)15 b(pro)q(of.)p 1797 372 31 2 v 1797 397 2 25 v 1825 397 V 1797 399 31 2 v -48 482 a Fk(Pro)q(of)21 b(of)f(Theorem)f(4.4:)25 b Fq(F)l(ollo)o(ws)18 b(b)o(y)g(applying)h (Lemma)f(6.8)f(in)i(the)f(same)g(w)o(a)o(y)f(that)g(Lemma)h(6.6)-48 556 y(w)o(as)c(used)i(to)f(pro)o(v)o(e)f(Theorem)h(6.1.)p 1797 543 V 1797 568 2 25 v 1825 568 V 1797 570 31 2 v -48 744 a Fr(7)67 b(App)r(endix:)56 b(Remarks)33 b(on)h(the)h (de\014nition)i(of)d(amenabilit)n(y)i(and)53 841 y(strong)22 b(amenabilit)n(y)-6 b(,)23 b(and)g(a)f(mean)e(ergo)r(dic)j(theorem)-48 984 y Fq(It)13 b(is)g(clear)g(that)g(according)g(to)f(De\014nition)i (2.2)e(an)h(in\014nite,)h(lo)q(cally)h(\014nite,)f(connected)f(graph)g Fn(G)f Fq(=)h(\()p Fn(V)s(;)8 b(E)s Fq(\))-48 1057 y(is)17 b(amenable)h(i\013)f(there)g(exists)g(a)f(F\034lner)h(sequence)h(for)e (it,)h(i.e.,)g(a)g(sequence)h(\()p Fn(W)1388 1064 y Fl(n)1411 1057 y Fq(\))1429 1064 y Fl(n)p Fw(\025)p Fh(1)1514 1057 y Fq(of)e(\014nite)i(subsets)-48 1131 y(of)d Fn(V)25 b Fq(suc)o(h)15 b(that)742 1204 y(lim)728 1229 y Fl(n)p Fw(!1)832 1173 y Fm(j)p Fn(@)s(W)915 1180 y Fl(n)938 1173 y Fm(j)p 832 1194 119 2 v 845 1235 a(j)p Fn(W)901 1242 y Fl(n)925 1235 y Fm(j)968 1204 y Fq(=)e(0)p Fn(:)694 b Fq(\(19\))-48 1309 y(Note)17 b(that)g(clearly)i(w)o(e)e(m)o(ust)g(ha) o(v)o(e)g(lim)667 1316 y Fl(n)p Fw(!1)769 1309 y Fm(j)p Fn(W)825 1316 y Fl(n)848 1309 y Fm(j)f Fq(=)h Fm(1)p Fq(,)h(for)f(\(19\))f(to)h(hold.)27 b(The)18 b(prop)q(ositions)g(b)q (elo)o(w)-48 1383 y(address)g(the)f(issue)i(of)e(whether)h(additional)h (conditions)f(can)g(b)q(e)g(imp)q(osed)h(on)f(the)f(sequence)i(\()p Fn(W)1705 1390 y Fl(n)1728 1383 y Fq(\))1746 1390 y Fl(n)p Fw(\025)p Fh(1)1814 1383 y Fq(.)-48 1456 y(As)c(b)q(efore,)g(w)o(e)g (write)g Fn(W)394 1463 y Fl(n)431 1456 y Fm(\045)e Fn(V)24 b Fq(if)16 b Fn(W)625 1463 y Fh(1)657 1456 y Fm(\032)d Fn(W)748 1463 y Fh(2)781 1456 y Fm(\032)g Fn(:::)h Fq(and)h Fm([)1000 1439 y Fw(1)1000 1468 y Fl(n)p Fh(=1)1069 1456 y Fn(W)1112 1463 y Fl(n)1148 1456 y Fq(=)e Fn(V)c Fq(.)-48 1574 y Fk(Prop)q(osition)19 b(7.1)j Fj(Supp)n(ose)16 b(that)h Fn(G)12 b Fq(=)h(\()p Fn(V)s(;)8 b(E)s Fq(\))13 b Fj(is)j(an)g(in\014nite,)f(lo)n(c)n(al)r(ly)g(\014nite,)g(c)n(onne)n (cte)n(d)g(gr)n(aph.)21 b(Then)-48 1647 y Fn(G)15 b Fj(is)f(amenable)h (i\013)g(ther)n(e)g(exists)f(a)h(se)n(quenc)n(e)f Fq(\()p Fn(W)824 1654 y Fl(n)847 1647 y Fq(\))865 1654 y Fl(n)p Fw(\025)p Fh(1)948 1647 y Fj(of)h(\014nite)f(subsets)h(of)g Fn(V)24 b Fj(such)16 b(that)f Fn(W)1609 1654 y Fl(n)1645 1647 y Fm(\045)e Fn(V)25 b Fj(and)-48 1721 y Fq(\(19\))15 b Fj(holds.)-48 1839 y Fk(Pro)q(of:)20 b Fq(W)l(e)15 b(kno)o(w)g(that)f(there)h(exists)h(a)f(sequence)h(of)f(\014nite)h (subsets)f(of)g Fn(V)10 b Fq(,)15 b(\()p Fn(W)1386 1822 y Fw(0)1380 1850 y Fl(n)1403 1839 y Fq(\))1421 1846 y Fl(n)p Fw(\025)p Fh(1)1489 1839 y Fq(,)g(suc)o(h)g(that)477 1961 y(lim)463 1986 y Fl(n)p Fw(!1)567 1930 y Fm(j)p Fn(@)s(W)656 1914 y Fw(0)650 1942 y Fl(n)673 1930 y Fm(j)p 567 1951 V 581 1992 a(j)p Fn(W)643 1979 y Fw(0)637 2004 y Fl(n)660 1992 y Fm(j)703 1961 y Fq(=)e(0)76 b(and)98 b(lim)1006 1986 y Fl(n)p Fw(!1)1106 1961 y Fm(j)p Fn(W)1168 1942 y Fw(0)1162 1972 y Fl(n)1185 1961 y Fm(j)12 b Fq(=)h Fm(1)p Fn(:)430 b Fq(\(20\))-48 2082 y(Fix)19 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Fq(\))f(will)h(b)q(e)g(a)f(F\034lner)h (sequence.)p 1797 2444 31 2 v 1797 2469 2 25 v 1825 2469 V 1797 2471 31 2 v 867 2581 a(27)p eop %%Page: 28 28 28 27 bop -48 311 a Fk(Prop)q(osition)19 b(7.2)j Fj(Supp)n(ose)c(that)h Fn(G)e Fq(=)g(\()p Fn(V)s(;)8 b(E)s Fq(\))15 b Fj(is)j(an)g (in\014nite,)f(c)n(onne)n(cte)n(d)g(gr)n(aph)i(of)g(b)n(ounde)n(d)f(de) n(gr)n(e)n(e.)-48 385 y(Then)f Fn(G)h Fj(is)f(amenable)h(i\013)f(ther)n (e)h(exists)f(a)h(se)n(quenc)n(e)f Fq(\()p Fn(W)967 392 y Fl(n)990 385 y Fq(\))1008 392 y Fl(n)p Fw(\025)p Fh(1)1094 385 y Fj(of)h(c)n(onne)n(cte)n(d)e(\014nite)h(subsets)g(of)h Fn(V)28 b Fj(such)-48 458 y(that)17 b(\(19\))f(holds.)-48 578 y Fk(Pro)q(of:)29 b Fq(W)l(e)20 b(kno)o(w)g(that)f(there)h(exists)h (a)e(sequence)j(\()p Fn(W)988 562 y Fw(0)982 589 y Fl(n)1005 578 y Fq(\))1023 585 y Fl(n)p Fw(\025)p Fh(1)1111 578 y Fq(suc)o(h)e(that)g(\(20\))e(holds.)35 b(W)l(rite)21 b Fn(W)1754 562 y Fw(0)1748 589 y Fl(n)1792 578 y Fq(=)-48 651 y Fm([)-18 658 y Fl(i)p Fw(2)p 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Fm(j)p Fn(F)768 1349 y Fl(n)791 1342 y Fm(j)829 1270 y Fi(X)816 1362 y Fl(v)q Fw(2)p Fl(F)880 1366 y Fg(n)910 1311 y Fn(\021)933 1318 y Fl(v)965 1311 y Fq(=)d Fn(E)s Fq([)p Fn(\021)1086 1318 y Fh(0)1104 1311 y Fm(jI)s Fq(])-48 1459 y Fj(in)j Fn(L)39 1442 y Fh(2)58 1459 y Fq(\()p Fn(S)107 1442 y Fl(V)137 1459 y Fn(;)8 b Fm(S)189 1442 y Fl(V)219 1459 y Fn(;)g(\026)p Fq(\))p Fj(,)15 b(wher)n(e)h Fm(I)k Fj(is)c(the)g Fn(\033)r Fj({algebr)n(a)g(of)g(invariant)g(events)f(in)h Fm(S)1291 1442 y Fl(V)1322 1459 y Fj(.)-48 1581 y Fk(Pro)q(of:)j Fq(Let)13 b(\000)g(b)q(e)h(the)f(set)f(of)h(automorphisms)f(of)h Fn(G)p Fq(,)g(with)g(the)g(top)q(ology)f(of)h(p)q(oin)o(t)o(wise)h(con) o(v)o(ergence.)19 b(It)-48 1655 y(is)c(w)o(ell)g(kno)o(wn)f(\(see,)g (e.g.,)f([57)o(]\))h(that)f(\000)i(is)g(a)f(lo)q(cally)i(compact)e (group)g(with)g(a)g(subbasis)h(for)f(its)g(top)q(ology)-48 1728 y(giv)o(en)i(b)o(y)f(the)g(coun)o(table)h(collection)h(of)e(op)q (en-compact)g(sets)g(\000)1080 1735 y Fl(u;v)1144 1728 y Fq(=)e Fm(f)p Fn(\015)h Fm(2)f Fn(G)8 b Fq(:)14 b Fn(\015)s Fq(\()p Fn(u)p Fq(\))d(=)i Fn(v)r Fm(g)p Fq(,)i Fn(u;)8 b(v)13 b Fm(2)g Fn(V)d Fq(.)23 1826 y(In)19 b([55)o(])g(\(see)g(also)g ([4)o(]\),)g(it)g(is)h(sho)o(wn)e(that)h Fn(G)f Fq(b)q(eing)j(amenable) e(and)h(transitiv)o(e)f(implies)i(that)d(\000)h(is)-48 1900 y(unimo)q(dular,)g(meaning)e(that)f(a)h(left)g(Haar)f(measure,)h Fn(m)p Fq(\()p Fm(\001)p Fq(\),)f(is)h(also)g(a)g(righ)o(t)f(Haar)h (measure.)24 b(Cho)q(ose)17 b(a)-48 1973 y(represen)o(tativ)o(e,)e Fn(g)279 1980 y Fl(v)299 1973 y Fq(,)g(from)f(eac)o(h)h(\000)564 1980 y Fh(0)p Fl(;v)613 1973 y Fq(.)20 b(Since)c(\000)792 1980 y Fl(u;v)856 1973 y Fq(=)d Fn(g)926 1980 y Fl(v)946 1973 y Fq(\000)974 1980 y Fh(0)p Fl(;)p Fh(0)1022 1973 y Fn(g)1046 1957 y Fw(\000)p Fh(1)1044 1984 y Fl(u)1092 1973 y Fq(,)i(it)g(follo)o(ws)h(that)519 2091 y Fn(m)p Fq(\(\000)605 2098 y Fl(u;v)656 2091 y Fq(\))f(do)q(es)g(not)g(dep)q (end)i(on)e Fn(u;)8 b(v)13 b Fm(2)g Fn(V)s(:)485 b Fq(\(21\))-48 2210 y(Set)15 b(\000)58 2217 y Fl(v)92 2210 y Fq(=)e(\000)168 2217 y Fh(0)p Fl(;v)231 2210 y Fq(and)487 2272 y(~)477 2283 y Fn(F)506 2290 y Fl(n)542 2283 y Fq(:=)g Fm(f)p Fn(\015)h Fm(2)f Fq(\000)g(:)f Fn(\015)800 2264 y Fw(\000)p Fh(1)847 2283 y Fq(\(0\))f Fm(2)i Fn(F)989 2290 y Fl(n)1013 2283 y Fm(g)f Fq(=)h Fm([)1126 2290 y Fl(v)q Fw(2)p Fl(F)1190 2294 y Fg(n)1214 2283 y Fq(\000)1242 2264 y Fw(\000)p Fh(1)1242 2294 y Fl(v)1290 2283 y Fn(:)-48 2383 y Fq(Then)k(eac)o(h)186 2372 y(~)176 2383 y Fn(F)205 2390 y Fl(n)245 2383 y Fq(is)g(a)g (compact)f(set)g(and)688 2372 y(~)677 2383 y Fn(F)706 2390 y Fl(n)745 2383 y Fm(\045)g Fq(\000.)24 b(Moreo)o(v)o(er,)15 b(using)j(\(21\))d(and)i(the)g(fact)f(that)g(the)h Fn(F)1773 2390 y Fl(n)1796 2383 y Fq('s)-48 2457 y(are)d(a)f(F\034lner)h (sequence)h(for)f Fn(G)p Fq(,)f(one)h(can)g(sho)o(w)f(that)h Fm(f)928 2445 y Fq(~)919 2457 y Fn(F)948 2464 y Fl(n)971 2457 y Fm(g)994 2464 y Fl(n)p Fw(\025)p Fh(1)1076 2457 y Fq(is)h(a)e(F\034lner)h(sequence)h(for)f(\000)g(in)h(the)e(sense)867 2581 y(29)p eop %%Page: 30 30 30 29 bop -48 311 a Fq(that)670 385 y(lim)655 410 y Fl(n)p Fw(!1)760 354 y Fn(m)p Fq(\()p Fn(K)869 342 y Fq(~)860 354 y Fn(F)889 361 y Fl(n)912 354 y Fm(4)963 342 y Fq(~)953 354 y Fn(F)982 361 y Fl(n)1005 354 y Fq(\))p 760 374 263 2 v 827 421 a Fn(m)p Fq(\()895 410 y(~)885 421 y Fn(F)914 428 y Fl(n)938 421 y Fq(\))1040 385 y(=)13 b(0)p Fn(;)-48 494 y Fq(for)i(an)o(y)g(compact)g(set)h Fn(K)i Fq(in)e(\000.)22 b(A)15 b(pro)q(of)g(of)h(this)g(claim)g(is)g(con)o (tained)g(in)h(the)e(pro)q(of)h(of)f(the)g(\\if)t(")g(part)g(of)-48 567 y([4)o(,)g(Theorem)g(3.9])f(\(in)i(this)f(pro)q(of)g(the)h(authors) e(assume)h(that)g(the)g(set)g Fn(K)j Fq(con)o(tains)d(the)h(iden)o(tit) o(y;)f(to)g(see)-48 641 y(that)f(no)i(generalit)o(y)f(is)h(lost)f(see)h ([23)o(,)e(Lemma)i(4.2]\).)23 739 y(W)l(e)f(can)g(no)o(w)g(apply)h([23) o(,)f(Corollary)g(3.4])f(to)g(get)506 855 y(lim)492 880 y Fl(n)p Fw(!1)649 825 y Fq(1)p 596 845 128 2 v 596 892 a Fn(m)p Fq(\()664 880 y(~)654 892 y Fn(F)683 899 y Fl(n)707 892 y Fq(\))737 798 y Fi(Z)768 884 y Fh(~)760 892 y Fl(F)782 896 y Fg(n)813 855 y Fn(\021)836 865 y Fl(\015)856 856 y Fp(\000)p Fc(1)897 865 y Fh(\(0\))944 855 y Fn(dm)p Fq(\()p Fn(\015)s Fq(\))d(=)i Fn(E)s Fq([)p Fn(\021)1203 862 y Fh(0)1221 855 y Fm(jI)s Fq(])p Fn(;)-48 988 y Fq(in)j Fn(L)36 972 y Fh(2)56 988 y Fq(.)21 b(\([23)o(,)15 b(Corollary)g(3.4])f (do)q(es)i(not)f(iden)o(tify)i(the)e(limit)i(in)g(this)f(fashion,)f (but)h(b)o(y)f(applying)i(it)f(to)f(the)-48 1062 y(random)i(v)m (ariables)j Fn(\021)336 1069 y Fh(0)362 1062 y Fk(1)388 1069 y Fl(A)417 1062 y Fq(,)e(with)g Fn(A)g Fm(2)f(I)s Fq(,)h(and)g(taking)g(exp)q(ectations,)h(one)f(can)g(see)g(that)f(the)h (limit)h(\014ts)-48 1135 y(the)c(de\014nition)i(of)e Fn(E)s Fq([)p Fn(\021)357 1142 y Fh(0)375 1135 y Fm(jI)s Fq(].\))23 1233 y(By)e(decomp)q(osing)369 1222 y(~)359 1233 y Fn(F)388 1240 y Fl(n)425 1233 y Fq(in)o(to)g(a)g(union)h(of)f 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(on)f(R)n(andom)h(Walks)91 1757 y(and)h(Discr)n(ete)g(Potential)g(The)n (ory)p Fq(,)e(Cam)o(bridge)i(Univ.)f(Press,)g(Cam)o(bridge,)g(to)f(app) q(ear.)20 1851 y([7])22 b(Benjamini,)f(I.)d(and)h(Sc)o(hramm,)g(O.)g (\(1996\))e(P)o(ercolation)h(b)q(ey)o(ond)i Fk(Z)1331 1835 y Fl(d)1351 1851 y Fq(,)f(man)o(y)f(questions)h(and)g(a)91 1908 y(few)c(answ)o(ers,)f Fj(Ele)n(ctr.)i(Commun.)g(Pr)n(ob)n(ab.)f Fk(1)p Fq(,)f(71{82.)20 2002 y([8])22 b(Benjamini,)43 b(I.)37 b(and)g(Sc)o(hramm,)42 b(O.,)g(Recen)o(t)c(progress)e(on)h(p)q (ercolation)h(b)q(ey)o(ond)f Fk(Z)1794 1985 y Fl(d)1814 2002 y Fq(.)91 2058 y Fb(http://www.wisdom.weizman)o(n.ac.il/)o (~schram)o(m/paper)o(s/pyond-)o(rep/ind)o(ex.html)20 2152 y Fq([9])22 b(Benjamini,)16 b(I.)g(and)f(Sc)o(hramm,)f(O.,)h(P)o (ercolation)h(in)g(the)f(h)o(yp)q(erb)q(olic)i(plane,)f(in)g (preparation.)-3 2246 y([10])22 b(Berg,)g(J.)e(v)m(an)h(den)h(and)f (Steif,)h(J.)f(\(1999\))e(On)i(the)g(existence)h(and)f(non-existence)h (of)f(\014nitary)91 2302 y(co)q(dings)16 b(for)f(a)f(class)i(of)f (random)g(\014elds,)h Fj(A)o(nn.)f(Pr)n(ob)n(ab.)p Fq(,)f(to)g(app)q (ear.)-3 2396 y([11])22 b(Bleher,)15 b(P)l(.)e(M.,)g(Ruiz,)i(J.)f(and)g (Zagrebno)o(v,)f(V.)g(A.)g(\(1995\))f(On)j(the)e(purit)o(y)h(of)g(the)g (limiting)h(Gibbs)91 2452 y(state)f(for)h(the)g(Ising)h(mo)q(del)g(on)f (the)h(Bethe)f(lattice,)h Fj(J.)g(Stat.)g(Phys.)f Fk(79)p Fq(,)g(473{482.)867 2581 y(31)p eop %%Page: 32 32 32 31 bop -3 311 a Fq([12])22 b(Burton,)h(R.)e(and)h(Keane,)i(M.)d (\(1989\))f(Densit)o(y)h(and)h(uniqueness)i(in)e(p)q(ercolation,)i Fj(Commun.)91 368 y(Math.)16 b(Phys.)f Fk(121)p Fq(,)g(501{505.)-3 461 y([13])22 b(Burton,)16 b(R.)g(and)h(Keane,)g(M.)e(\(1991\))g(T)l (op)q(ological)i(and)g(metric)f(prop)q(erties)h(of)f(in\014nite)i (clusters)91 517 y(in)e(stationary)e(t)o(w)o(o-dimensional)i(site)g(p)q (ercolation,)g Fj(Isr)n(ael)f(J.)h(Math.)f Fk(76)p Fq(,)g(299{316.)-3 610 y([14])22 b(Cha)o(y)o(es,)17 b(J.,)g(Cha)o(y)o(es,)g(L.)g(and)g(F)l (r\177)-23 b(ohlic)o(h,)19 b(J.)e(\(1985\))e(The)j(lo)o(w-temp)q (erature)f(b)q(eha)o(vior)h(of)f(disor-)91 666 y(dered)f(magnets,)e Fj(Commun.)i(Math.)h(Phys.)e Fk(100)p Fq(,)g(399{437.)-3 759 y([15])22 b(Chen,)15 b(D.)g(and)g(P)o(eres,)g(Y.,)f(Anc)o(hored)i (expansion,)g(p)q(ercolation)g(and)f(sp)q(eed,)h(preprin)o(t.)-3 852 y([16])22 b(Ellis,)27 b(R.)d(S.)g(\(1985\))e Fj(Entr)n(opy,)27 b(L)n(ar)n(ge)c(Deviations,)j(and)f(Statistic)n(al)e(Me)n(chanics)p Fq(,)h(Springer{)91 908 y(V)l(erlag,)15 b(New)g(Y)l(ork.)-3 1001 y([17])22 b(Ev)m(ans,)17 b(W.,)g(Ken)o(y)o(on,)g(C.,)g(P)o(eres,)g (Y.)f(and)i(Sc)o(h)o(ulman,)g(L.)f(J.,)g(Broadcasting)g(on)g(trees)g (and)g(the)91 1057 y(Ising)f(mo)q(del,)g(preprin)o(t.)-3 1150 y([18])22 b(F)l(alk,)g(H.)e(and)g(Gehring,)i(G.)e(A.)g(\(1975\))f (Correlation)h(function)i(and)e(transition)h(temp)q(erature)91 1207 y(b)q(ounds)16 b(for)f(b)q(ond-disordered)i(Ising)f(systems,)e Fj(J.)i(phys.)g(C)f Fk(8)p Fq(,)g(L298{L301.)-3 1299 y([19])22 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(theorems)g(and)g(the)g(construction)g(of)g(summing)g(sequences)h(in)91 1953 y(amenable)h(lo)q(cally)h(compact)e(groups,)f Fj(Commun.)i(Pur)n (e)h(Appl.)f(Math.)f Fk(26)p Fq(,)g(29{46.)-3 2045 y([24])22 b(Gri\016ths,)13 b(R.)h(B.)f(and)h(Leb)q(o)o(witz,)h(J.)e(L.)h (\(1968\))e(Random)i(spin)h(systems:)j(some)c(rigorous)f(results,)91 2102 y Fj(J.)j(Math.)g(phys.)g Fk(9)p Fq(,)f(1284{1292.)-3 2195 y([25])22 b(Grimmett,)13 b(G.)g(R.)h(and)g(Newman,)g(C.)f(M.)g (\(1990\))f(P)o(ercolation)i(in)g Fm(1)7 b Fq(+)g(1)15 b(dimensions,)h Fj(Disor)n(der)91 2251 y(in)d(Physic)n(al)g(Systems)e Fq(\(Grimmett,)h(G.)f(R.)i(and)f(W)l(elsh,)i(D.)e(J.)g(A.,)g (editors\),)h(219{240,)d(Clarendon)91 2307 y(Press,)k(Oxford.)-3 2400 y([26])22 b(Grimmett,)15 b(G.)g(R.)h(and)g(Stacey)l(,)g(A.)g(M.)f (\(1999\))f(Critical)j(probabilities)h(for)d(site)i(and)f(b)q(ond)g(p)q (er-)91 2457 y(colation)g(mo)q(dels,)f Fj(A)o(nn.)g(Pr)n(ob)n(ab.)p Fq(,)g Fk(26)p Fq(,)g(1788{1812.)867 2581 y(32)p eop %%Page: 33 33 33 32 bop -3 311 a Fq([27])22 b(H\177)-23 b(aggstr\177)g(om,)11 b(O.)i(\(1997\))e(In\014nite)j(clusters)g(in)g(dep)q(enden)o(t)g (automorphism)e(in)o(v)m(arian)o(t)i(p)q(ercolation)91 368 y(on)h(trees,)g Fj(A)o(nn.)g(Pr)n(ob)n(ab.)f Fk(25)p Fq(,)h(1423{1436.)-3 462 y([28])22 b(H\177)-23 b(aggstr\177)g(om,)12 b(O.)h(\(1998\))f(Random-cluster)j(represen)o(tations)f(in)g(the)g (study)g(of)f(phase)h(transitions,)91 518 y Fj(Markov)i(Pr)n(o)n(c.)g (R)n(el.)g(Fields)e Fk(4)p Fq(,)h(275{321.)-3 612 y([29])22 b(H\177)-23 b(aggstr\177)g(om,)18 b(O.)h(\(1999\))e(Mark)o(o)o(v)h (random)h(\014elds)h(and)f(p)q(ercolation)h(on)f(general)h(graphs,)f Fj(A)n(dv.)91 668 y(Appl.)d(Pr)n(ob)n(ab.)p Fq(,)e(to)h(app)q(ear.)-3 762 y([30])22 b(H\177)-23 b(aggstr\177)g(om,)12 b(O.)h(and)h(P)o(eres,) f(Y.)h(\(1999\))d(Monotonicit)o(y)j(of)f(uniqueness)i(for)e(p)q (ercolation)i(on)f(Ca)o(y-)91 819 y(ley)20 b(graphs:)27 b(all)19 b(in\014nite)i(clusters)f(are)e(b)q(orn)h(sim)o(ultaneously)l (,)j Fj(Pr)n(ob)n(ab.)d(Th.)g(R)n(el.)g(Fields)p Fq(,)g Fk(113)91 875 y Fq(273{285.)-3 969 y([31])j(H\177)-23 b(aggstr\177)g(om,)11 b(O.,)i(P)o(eres,)g(Y.)g(and)g(Sc)o(honmann,)g (R.)g(H.)g(\(1999\))e(P)o(ercolation)i(on)g(transitiv)o(e)g(graphs)91 1025 y(as)h(a)g(coalescen)o(t)i(pro)q(cess:)j(Relen)o(tless)e(Merging)d (F)l(ollo)o(w)o(ed)h(b)o(y)g(Sim)o(ultaneous)h(Uniqueness,)g Fj(Per-)91 1082 y(plexing)22 b(Pr)n(ob)n(ability)f(Pr)n(oblems:)32 b(Pap)n(ers)23 b(in)e(Honor)i(of)f(Harry)h(Kesten)p Fq(,)f(\(Bramson,)h (M.)e(and)91 1138 y(Durrett,)14 b(R.,)h(editors\),)f(Birkhauser,)i(to)e (app)q(ear.)-3 1232 y([32])22 b(Holley)l(,)14 b(R.)e(\(1974\))f (Remarks)h(on)g(the)h(FK)o(G)f(inequalities,)j Fj(Commun.)f(Math.)g (Phys.)d Fk(36)p Fq(,)i(227{231.)-3 1326 y([33])22 b(Io\013e,)c(D.)f (\(1996\))f(On)j(the)f(extremalit)o(y)g(of)g(the)g(disordered)h(state)e (for)g(the)h(Ising)h(mo)q(del)g(on)f(the)91 1382 y(Bethe)e(lattice,)f Fj(L)n(ett.)h(Math.)g(Phys.)f Fk(37)p Fq(,)g(137{143.)-3 1476 y([34])22 b(Io\013e,)j(D.)d(\(1996\))g(Extremalit)o(y)h(of)g(the)g (disordered)h(state)f(for)f(the)i(Ising)g(mo)q(del)g(on)f(general)91 1533 y(trees,)17 b Fj(T)m(r)n(e)n(es)f(\(V)m(ersail)r(les,)g(1995\))i Fq(\(Chauvin,)f(B.,)g(Cohen,)h(S.)f(and)g(Roualt,)h(A.,)e(editors\),)h (3{14,)91 1589 y(Birkh\177)-23 b(auser,)15 b(Basel.)-3 1683 y([35])22 b(Jonasson,)12 b(J.)h(\(1999\))e(The)i(random)f(cluster) h(mo)q(del)h(on)e(a)g(general)h(graph)g(and)g(a)f(phase)h(transition)91 1739 y(c)o(haracterization)i(of)g(nonamenabilit)o(y)l(,)i Fj(Sto)n(ch.)e(Pr)n(o)n(c.)h(Appl.)p Fq(,)f Fk(79)p Fq(,)g(335{354.)-3 1833 y([36])22 b(Jonasson,)17 b(J.)g(and)g(Steif,)h(J.)f(\(1999\))e (Amenabilit)o(y)k(and)f(phase)f(transition)g(in)h(the)f(Ising)h(mo)q (del,)91 1890 y Fj(J.)e(The)n(or.)g(Pr)n(ob)n(ab.)p Fq(,)e(to)h(app)q (ear.)-3 1983 y([37])22 b(Kaimano)o(vic)o(h,)c(V.)g(and)f(V)l(ershik,)i (A.)e(\(1983\))f(Random)i(w)o(alks)f(on)h(discrete)g(groups:)24 b(b)q(oundary)91 2040 y(and)15 b(en)o(trop)o(y)l(,)g Fj(A)o(nn.)g(Pr)n(ob)n(ab.)f Fk(11)p Fq(,)h(457{490.)-3 2134 y([38])22 b(Kesten,)e(H.)f(\(1959\))f(Symmetric)h(random)g(w)o (alks)g(on)g(groups,)h Fj(T)m(r)n(ans.)e(A)o(mer.)h(Math.)i(So)n(c.)d Fk(92)p Fq(,)91 2190 y(336{354.)-3 2284 y([39])k(Kesten,)15 b(H.)g(\(1959\))f(F)l(ull)i(Banac)o(h)f(mean)h(on)f(coun)o(table)h (groups,)e Fj(Math.)i(Sc)n(and.)f Fk(7)p Fq(,)g(146{156.)-3 2378 y([40])22 b(Lalley)l(,)16 b(S.)f(\(1998\))f(P)o(ercolation)h(on)g (F)l(uc)o(hsian)h(groups,)e Fj(A)o(nn.)h(Inst.)g(H.)h(Poinc)n(ar)o (\023)-22 b(e,)15 b(Pr)n(ob)n(ab.)h(Stat.)91 2434 y Fk(34)p Fq(,)f(151{178.)867 2581 y(33)p eop %%Page: 34 34 34 33 bop -3 311 a Fq([41])22 b(Liggett,)e(T.)f(M.,)h(Sc)o(honmann,)g (R.)g(H.)f(and)h(Stacey)l(,)h(A.)e(M.)g(\(1997\))e(Domination)j(b)o(y)g (pro)q(duct)91 368 y(measures,)15 b Fj(A)o(nn.)g(Pr)n(ob)n(ab.)f Fk(25)p Fq(,)h(71{95.)-3 461 y([42])22 b(Ly)o(ons,)11 b(R.)f(\(1989\))e(The)j(Ising)g(mo)q(del)g(and)f(p)q(ercolation)i(on)e (trees)g(and)g(tree-lik)o(e)i(graphs,)e Fj(Commun.)91 517 y(Math.)16 b(Phys.)f Fk(125)p Fq(,)g(337{353.)-3 610 y([43])22 b(Ly)o(ons,)15 b(R.)g(\(1990\))e(Random)j(w)o(alk)f(and)g (p)q(ercolation)h(on)g(trees,)e Fj(A)o(nn.)h(Pr)n(ob)n(ab.)g Fk(18)p Fq(,)g(931{958.)-3 702 y([44])22 b(Ly)o(ons,)e(R.)g(\(1992\))e (Random)i(w)o(alk,)g(capacit)o(y)g(and)g(p)q(ercolation)h(on)f(trees,)g Fj(A)o(nn.)f(Pr)n(ob)n(ab.)g Fk(20)p Fq(,)91 759 y(2043{2088.)-3 852 y([45])j(Ly)o(ons,)15 b(R.)h(\(1995\))e(Random)i(w)o(alks)f(and)h (the)g(gro)o(wth)e(of)h(groups,)g Fj(C.)h(R.)h(A)n(c)n(ad.)f(Sci.)g (Paris)g(Ser.)91 908 y(I)g(Math.)f Fk(320)p Fq(,)g(1361{1366.)-3 1001 y([46])22 b(Ly)o(ons,)15 b(R.,)g(P)o(eman)o(tle,)g(R.)g(and)h(P)o (eres,)f(Y.)g(\(1996\))f(Random)h(w)o(alks)g(on)h(the)f(lampligh)o(ter) i(group,)91 1057 y Fj(A)o(nn.)e(Pr)n(ob)n(ab.)g Fk(24)p Fq(,)g(1993{2006.)-3 1150 y([47])22 b(Ly)o(ons,)g(R.)f(and)g(Sc)o (hramm,)h(O.)f(\(1999\))e(Indistinguishabil)q(i)q(t)o(y)k(of)e(p)q (ercolation)h(clusters,)g Fj(A)o(nn.)91 1207 y(Pr)n(ob)n(ab.)p Fq(,)14 b(to)h(app)q(ear.)-3 1299 y([48])22 b(Newman,)17 b(C.)g(M.)g(and)h(Sc)o(h)o(ulman,)g(L.)g(S.)f(\(1981\))f(In\014nite)j (clusters)f(in)g(p)q(ercolation)h(mo)q(dels,)f Fj(J.)91 1356 y(Stat.)e(Phys.)f Fk(26)p Fq(,)g(613{628.)-3 1448 y([49])22 b(Newman,)16 b(C.)g(M.)f(and)i(W)l(u,)f(C.)g(C.)g(\(1990\))e (Mark)o(o)o(v)h(\014elds)j(on)e(branc)o(hing)h(planes,)g Fj(Pr)n(ob)n(ab.)g(Th.)91 1505 y(R)n(el.)e(Fields)g Fk(85)p Fq(,)g(539{552.)-3 1598 y([50])22 b(Propp,)14 b(J.)g(G.)f(and)h (Wilson,)h(D.)e(B.)h(\(1996\))e(Exact)i(sampling)h(with)f(coupled)i (Mark)o(o)o(v)c(c)o(hains)j(and)91 1654 y(applications)i(to)d (statistical)i(mec)o(hanics,)g Fj(R)n(andom)g(Structur)n(es)g(A)o (lgorithms)p Fq(,)f Fk(9)p Fq(,)g(223{252.)-3 1747 y([51])22 b(Sc)o(honmann,)12 b(R.)e(H.)h(\(1999\))d(Stabilit)o(y)k(of)e (in\014nite)j(clusters)e(in)g(sup)q(ercritical)i(p)q(ercolation,)f Fj(Pr)n(ob)n(ab.)91 1803 y(Th.)k(R)n(el.)f(Fields)g Fk(113)p Fq(,)g(287{300.)-3 1896 y([52])22 b(Sc)o(honmann,)15 b(R.)f(H.)h(\(1999\))d(P)o(ercolation)j(in)h Fm(1)9 b Fq(+)g(1)14 b(dimensions)j(at)d(the)g(uniqueness)j(threshold,)91 1953 y Fj(Perplexing)f(Pr)n(ob)n(ability)g(Pr)n(oblems:)21 b(Pap)n(ers)c(in)f(Honor)h(of)g(Harry)g(Kesten)p Fq(,)e(\(Bramson,)g (M.)g(and)91 2009 y(Durrett,)f(R.,)h(editors\),)f(Birkhauser,)i(to)e (app)q(ear.)-3 2102 y([53])22 b(Sc)o(honmann,)14 b(R.)f(H.)g(and)h(T)l (anak)m(a,)f(N.)h(I.)f(\(1998\))f(Lac)o(k)h(of)g(monotonicit)o(y)h(in)g (ferromagnetic)f(Ising)91 2158 y(mo)q(del)j(phase)g(diagrams,)e Fj(A)o(nn.)h(Appl.)h(Pr)n(ob)n(ab.)f Fk(8)p Fq(,)g(234{245.)-3 2251 y([54])22 b(Series,)14 b(C.)f(M.)g(and)g(Sinai,)i(Y.)e(G.)g (\(1990\))f(Ising)i(mo)q(dels)g(on)g(the)f(Lobac)o(hevsky)h(Plane,)g Fj(Commun.)91 2307 y(Math.)i(Phys.)f Fk(128)p Fq(,)g(63{76)-3 2400 y([55])22 b(Soardi,)13 b(P)l(.)f(M.)g(and)h(W)l(o)q(ess,)f(W.)g (\(1990\))f(Amenabilit)o(y)l(,)j(unimo)q(dularit)o(y)l(,)h(and)e(the)f (sp)q(ectral)h(radius)91 2457 y(of)i(random)f(w)o(alks)h(on)h (in\014nite)h(graphs,)d Fj(Math)j(Z.)e Fk(205)p Fq(,)g(471{486.)867 2581 y(34)p eop %%Page: 35 35 35 34 bop -3 311 a Fq([56])22 b(Vir\023)-23 b(ag,)15 b(B.,)f(On)i(the)f(anc)o(hored)h(expansion)g(prop)q(ert)o(y)l(,)e (preprin)o(t.)-3 405 y([57])22 b(W)l(o)q(ess,)15 b(W.)f(\(1991\))g(T)l (op)q(ological)i(groups)f(and)g(in\014nite)i(graphs,)e Fj(Discr)n(ete)g(Math.)h Fk(95)p Fq(,)f(373{384.)-3 499 y([58])22 b(W)l(u,)12 b(C.)g(C.)f(\(1993\))f(Critical)j(b)q(eha)o(vior) g(of)e(p)q(ercolation)i(and)f(Mark)o(o)o(v)f(\014elds)i(on)f(branc)o (hing)g(planes,)91 555 y Fj(J.)k(Appl.)g(Pr)n(ob)n(ab.)f Fk(30)p Fq(,)g(538{547.)-3 649 y([59])22 b(W)l(u,)15 b(C.)f(C.)h(\(1996\))e(Ising)k(mo)q(dels)f(on)f(h)o(yp)q(erb)q(olic)i (graphs,)e Fj(J.)h(Stat.)g(Phys.)f Fk(85)p Fq(,)g(251{259.)-3 743 y([60])22 b(Zhang,)c(Y.)g(\(1996\))e(Con)o(tin)o(uit)o(y)i(of)g(p)q (ercolation)h(probabilit)o(y)g(in)g Fm(1)12 b Fq(+)h(1)k(dimensions,)k Fj(J.)d(Appl.)91 800 y(Pr)n(ob)n(ab.)d 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