Content-Type: multipart/mixed; boundary="-------------0107251329426" This is a multi-part message in MIME format. ---------------0107251329426 Content-Type: text/plain; name="01-286.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-286.comments" rhs@math.ucla.edu www.math.ucla.edu/~rhs ---------------0107251329426 Content-Type: text/plain; name="01-286.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-286.keywords" Percolation, transitive graphs, non-amenable graphs, phase transitions, critical behavior, critical exponents, mean-field behavior, planar graphs, number of ends ---------------0107251329426 Content-Type: application/postscript; name="mfp.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="mfp.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: mfp.dvi %%Pages: 12 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSWebPage: 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Fj(z)s Ff(2)p Fj(V)457 4197 y Fk(P)516 4212 y Fj(p;q)626 4197 y Fs(\()p Fo(x)722 4212 y Fi(1)794 4197 y Fh($)28 b Fo(z)t(;)17 b(x)1074 4212 y Fi(1)1146 4197 y Fh(6$)27 b Fo(Q;)17 b(x)1454 4212 y Fi(2)1526 4197 y Fh($)27 b Fo(Q;)17 b(x)1834 4212 y Fi(3)1905 4197 y Fh($)28 b Fo(Q;)17 b(x)2214 4212 y Fi(2)2285 4197 y Fh(6$)27 b Fo(x)2469 4212 y Fi(3)2514 4197 y Fs(\))63 b Fh(\025)h Fo(c)17 b(\037)p Fs(\()p Fo(p;)g(q)t Fs(\))g(\()p Fo(\022)s Fs(\()p Fo(p;)g(q)t Fs(\)\))3466 4156 y Fi(2)3507 4197 y Fo(:)0 4506 y Fl(Then)36 b Fo(\016)31 b Fs(=)e(2)35 b Fl(and)h Fo(\014)d Fs(=)28 b(1)p Fl(.)120 4680 y Fs(The)36 b(role)f(of)h(unimo)s(dularit)m(y)e(in)i(the)f(deriv)-6 b(ation)36 b(of)g(the)g(t)m(w)m(o)f(lemmas)f(ab)s(o)m(v)m(e)i(is)f (explained)h(in)0 4800 y(Section)e(3.2)e(of)i([Sc)m(h].)120 4920 y(In)c(the)g(remainder)g(of)g(this)g(section,)g(w)m(e)h(will)e (reduce)i(the)f(lemmas)e(ab)s(o)m(v)m(e)i(to)g(further)h(su\016cien)m (t)0 5039 y(conditions)44 b(for)g(statemen)m(ts)g(of)g(mean-\014eld)h (criticalit)m(y)-8 b(.)74 b(The)44 b(reader)h(can)g(either)f(study)g (these)0 5159 y(lemmas)d(in)j(the)f(order)h(in)f(whic)m(h)h(they)f (will)g(b)s(e)g(presen)m(ted,)48 b(or)43 b(alternativ)m(ely)-8 b(,)43 b(study)h(\014rst)g(the)0 5278 y(lemmas)38 b(lab)s(eled)j(with)f (\\A",)f(whic)m(h)j(refer)f(to)f(the)h(exp)s(onen)m(ts)g Fo(\015)46 b Fs(and)40 b(\001,)i(and)f(later)f(study)h(the)0 5398 y(lemmas)35 b(lab)s(eled)i(with)g(\\B",)f(whic)m(h)i(refer)g(to)f (the)g(exp)s(onen)m(ts)h Fo(\016)j Fs(and)d Fo(\014)5 b Fs(,)38 b(and)f(whic)m(h)h(ha)m(v)m(e)g(more)0 5517 y(in)m(v)m(olv)m(ed)33 b(pro)s(ofs.)p eop %%Page: 5 5 5 4 bop 1325 50 a Fq(MEAN-FIELD)37 b(CRITICALITY)1286 b(5)0 289 y Fu(Lemma)46 b(2.2.A.)56 b Fl(Supp)-5 b(ose)42 b(that)f Fo(G)e Fs(=)f(\()p Fo(V)6 b(;)17 b(E)6 b Fs(\))40 b Fl(is)g(an)i(in\014nite,)f(lo)-5 b(c)g(al)5 b(ly)43 b(\014nite,)f(c)-5 b(onne)g(cte)g(d)40 b(tr)-5 b(an-)0 408 y(sitive)49 b(unimo)-5 b(dular)52 b(gr)-5 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Fs(\()p Fo(p)p Fs(\))203 b(\()p Fo(i)28 b Fs(=)g(1)p Fo(;)17 b Fs(2\))p Fo(:)0 1312 y Fl(Then)36 b Fo(\015)c Fs(=)d(1)35 b Fl(and)h Fs(\001)27 b(=)h(2)p Fl(.)0 1490 y(Pr)-5 b(o)g(of.)49 b Fs(The)33 b(ev)m(en)m(t)f Fh(f)p Fo(V)886 1505 y Fi(1)958 1490 y Fh($)27 b Fo(V)1143 1505 y Fi(2)1188 1490 y Fh(g)32 b Fs(dep)s(ends)h(only)f(on)f(the)h (state)g(of)g(o)s(ccupancy)g(of)g(the)g(edges)h(whic)m(h)0 1640 y(ha)m(v)m(e)d(at)f(least)h(one)g(endp)s(oin)m(t)g(in)g(\()p Fo(V)1374 1655 y Fi(1)1433 1640 y Fh([)15 b Fo(V)1572 1655 y Fi(2)1617 1640 y Fs(\))1656 1604 y Fj(c)1696 1640 y Fs(,)30 b(while)g(for)g Fo(i)e Fs(=)g(1)p Fo(;)17 b Fs(2,)29 b(the)h(ev)m(en)m(ts)g Fh(f)p Fo(x)3102 1655 y Fj(i)3215 1583 y(V)3262 1593 y Fd(i)3163 1640 y Fh( )-17 b(!)27 b Fo(z)t Fh(g)j Fs(dep)s(end)0 1760 y(only)g(on)h(the)g(state)f (of)h(o)s(ccupancy)g(of)g(the)g(edges)g(whic)m(h)h(ha)m(v)m(e)f(b)s (oth)f(endp)s(oin)m(ts)i(in)f Fo(V)3266 1775 y Fj(i)3299 1760 y Fs(.)43 b(Therefore,)0 1879 y(b)m(y)33 b(indep)s(endence,)242 2042 y Fg(X)166 2256 y Fj(z)204 2266 y Fe(1)243 2256 y Fj(;z)305 2266 y Fe(2)343 2256 y Ff(2)p Fj(V)478 2137 y Fk(P)537 2152 y Fj(p)585 2137 y Fs(\()p Fo(x)681 2152 y Fi(1)753 2137 y Fh($)27 b Fo(z)926 2152 y Fi(1)971 2137 y Fo(;)17 b(x)1073 2152 y Fi(2)1145 2137 y Fh($)27 b Fo(z)1318 2152 y Fi(2)1363 2137 y Fo(;)17 b(x)1465 2152 y Fi(1)1537 2137 y Fh(6$)27 b Fo(x)1721 2152 y Fi(2)1766 2137 y Fs(\))60 b Fh(\025)2079 2042 y Fg(X)2004 2256 y Fj(z)2042 2266 y Fe(1)2081 2256 y Fj(;z)2143 2266 y Fe(2)2181 2256 y Ff(2)p Fj(V)2316 2137 y Fk(P)2375 2152 y Fj(p)2423 2137 y Fs(\()p Fo(x)2519 2152 y Fi(1)2639 2079 y Fj(V)2686 2089 y Fe(1)2591 2137 y Fh( )-17 b(!)27 b Fo(z)2847 2152 y Fi(1)2892 2137 y Fo(;)17 b(x)2994 2152 y Fi(2)3114 2079 y Fj(V)3161 2089 y Fe(2)3066 2137 y Fh( )-17 b(!)27 b Fo(z)3322 2152 y Fi(2)3367 2137 y Fo(;)17 b(V)3470 2152 y Fi(1)3542 2137 y Fh(6$)27 b Fo(V)3727 2152 y Fi(2)3772 2137 y Fs(\))39 2435 y(=)242 2340 y Fg(X)166 2554 y Fj(z)204 2564 y Fe(1)243 2554 y Fj(;z)305 2564 y Fe(2)343 2554 y Ff(2)p Fj(V)478 2435 y Fk(P)537 2450 y Fj(p)585 2435 y Fs(\()p Fo(x)681 2450 y Fi(1)801 2377 y Fj(V)848 2387 y Fe(1)753 2435 y Fh( )-17 b(!)27 b Fo(z)1009 2450 y Fi(1)1054 2435 y Fs(\))17 b Fk(P)1169 2450 y Fj(p)1217 2435 y Fs(\()p Fo(x)1313 2450 y Fi(2)1433 2377 y Fj(V)1480 2387 y Fe(2)1385 2435 y Fh( )-17 b(!)27 b Fo(z)1641 2450 y Fi(2)1686 2435 y Fs(\))17 b Fk(P)1801 2450 y Fj(p)1848 2435 y Fs(\()p Fo(V)1945 2450 y Fi(1)2017 2435 y Fh(6$)28 b Fo(V)2203 2450 y Fi(2)2248 2435 y Fs(\))39 2791 y(=)166 2621 y Fg( )281 2697 y(X)245 2910 y Fj(z)283 2920 y Fe(1)322 2910 y Ff(2)p Fj(V)423 2920 y Fe(1)478 2791 y Fk(P)537 2806 y Fj(p)585 2791 y Fs(\()p Fo(x)681 2806 y Fi(1)801 2734 y Fj(V)848 2744 y Fe(1)753 2791 y Fh( )-17 b(!)27 b Fo(z)1009 2806 y Fi(1)1054 2791 y Fs(\))1093 2621 y Fg(!)33 b( )1320 2697 y(X)1284 2910 y Fj(z)1322 2920 y Fe(2)1361 2910 y Ff(2)p Fj(V)1462 2920 y Fe(2)1517 2791 y Fk(P)1576 2806 y Fj(p)1624 2791 y Fs(\()p Fo(x)1720 2806 y Fi(2)1840 2734 y Fj(V)1887 2744 y Fe(2)1792 2791 y Fh( )-17 b(!)27 b Fo(z)2048 2806 y Fi(2)2093 2791 y Fs(\))2132 2621 y Fg(!)2244 2791 y Fk(P)2303 2806 y Fj(p)2351 2791 y Fs(\()p Fo(V)2448 2806 y Fi(1)2520 2791 y Fh(6$)h Fo(V)2706 2806 y Fi(2)2751 2791 y Fs(\))60 b Fh(\025)i Fo(c)3032 2750 y Fi(3)3093 2791 y Fs(\()p Fo(\037)p Fs(\()p Fo(p)p Fs(\)\))3361 2750 y Fi(2)3405 2791 y Fo(:)0 3128 y Fs(And)38 b(the)g(claim)f(follo)m (ws)g(from)g(Lemma)f(2.1.A.)h(\(The)h(h)m(yp)s(othesis)g(in)g(that)f (lemma)f(that)h Fo(p)3577 3143 y Fj(c)3653 3128 y Fo(<)f Fs(1)0 3248 y(m)m(ust)d(hold,)g(since)h(otherwise)g Fk(P)1243 3263 y Fj(p)1291 3248 y Fs(\()p Fo(V)1388 3263 y Fi(1)1460 3248 y Fh($)27 b Fo(V)1645 3263 y Fi(2)1690 3248 y Fs(\))g Fh(!)h Fs(1,)k(as)i Fo(p)27 b Fh(\045)h Fo(p)2372 3263 y Fj(c)2412 3248 y Fs(.\))p 3756 3238 60 50 v 0 3425 a Fu(Lemma)46 b(2.2.B.)58 b Fl(Supp)-5 b(ose)41 b(that)h Fo(G)d Fs(=)h(\()p Fo(V)6 b(;)17 b(E)6 b Fs(\))39 b Fl(is)i(an)h (in\014nite,)g(lo)-5 b(c)g(al)5 b(ly)43 b(\014nite,)g(c)-5 b(onne)g(cte)g(d)40 b(tr)-5 b(an-)0 3545 y(sitive)49 b(unimo)-5 b(dular)52 b(gr)-5 b(aph.)91 b(Supp)-5 b(ose)51 b(also)f(that)h(ther)-5 b(e)51 b(ar)-5 b(e)50 b Fo(\017;)17 b(c)55 b(>)h Fs(0)p Fl(,)e(disjoint)49 b(sets)i(of)f(sites)0 3665 y Fo(V)58 3680 y Fi(1)103 3665 y Fo(;)17 b(V)206 3680 y Fi(2)250 3665 y Fo(;)g(V)353 3680 y Fi(3)431 3665 y Fh(\032)35 b Fo(V)62 b Fl(and)39 b(sites)g Fo(x)1148 3680 y Fi(1)1227 3665 y Fh(2)c Fo(V)1386 3680 y Fi(1)1431 3665 y Fl(,)40 b Fo(x)1559 3680 y Fi(2)1638 3665 y Fh(2)35 b Fo(V)1797 3680 y Fi(2)1841 3665 y Fl(,)40 b Fo(x)1969 3680 y Fi(3)2048 3665 y Fh(2)35 b Fo(V)2207 3680 y Fi(3)2292 3665 y Fl(such)k(that)g(for)g(every)h Fo(p)34 b Fh(2)h Fs(\()p Fo(p)3434 3680 y Fj(c)3499 3665 y Fh(\000)26 b Fo(\017;)17 b(p)3737 3680 y Fj(c)3777 3665 y Fs(\))0 3784 y Fl(and)35 b Fo(q)d Fh(2)c Fs(\(0)p Fo(;)17 b(\017)p Fs(\))p Fl(,)1141 4023 y Fk(P)1200 4038 y Fj(p;q)1310 4023 y Fs(\()p Fo(V)1407 4038 y Fj(i)1468 4023 y Fh($)27 b Fo(V)1653 4038 y Fj(j)1695 4023 y Fs(\))63 b Fh(\024)h Fs(1)22 b Fh(\000)h Fo(c)203 b Fs(\()p Fo(i)28 b Fh(6)p Fs(=)g Fo(j)6 b Fs(\))p Fo(;)1032 4149 y Fg(X)1014 4363 y Fj(z)s Ff(2)p Fj(V)1156 4373 y Fe(1)1211 4244 y Fk(P)1270 4259 y Fj(p;q)1381 4244 y Fs(\()p Fo(x)1477 4259 y Fi(1)1597 4186 y Fj(V)1644 4196 y Fe(1)1549 4244 y Fh( )-17 b(!)27 b Fo(z)t(;)17 b(x)1911 4259 y Fi(1)1983 4244 y Fh(6$)28 b Fo(Q)p Fs(\))63 b Fh(\025)g Fo(c)17 b(\037)p Fs(\()p Fo(p;)g(q)t Fs(\))p Fo(;)1004 4575 y Fk(P)1063 4590 y Fj(p;q)1174 4575 y Fs(\()p Fo(x)1270 4590 y Fj(i)1383 4517 y(V)1430 4527 y Fd(i)1331 4575 y Fh( )-17 b(!)27 b Fo(Q)p Fs(\))63 b Fh(\025)g Fo(c)17 b(\022)s Fs(\()p Fo(p;)g(q)t Fs(\))202 b(\()p Fo(i)28 b Fs(=)g(2)p Fo(;)17 b Fs(3\))p Fo(:)0 4761 y Fl(Then)36 b Fo(\016)31 b Fs(=)e(2)35 b Fl(and)h Fo(\014)d Fs(=)28 b(1)p Fl(.)0 4939 y(Pr)-5 b(o)g(of.)49 b Fs(Set)1024 5083 y Fo(A)1099 5042 y Fj(z)1099 5108 y Fi(1)1189 5083 y Fs(=)c Fh(f)p Fo(x)1418 5098 y Fi(1)1538 5026 y Fj(V)1585 5036 y Fe(1)1490 5083 y Fh( )-17 b(!)27 b Fo(z)t Fh(g)p Fo(;)2070 5058 y Fs(~)2044 5083 y Fo(A)2119 5098 y Fi(1)2208 5083 y Fs(=)44 b Fh(f)p Fo(x)2436 5098 y Fi(1)2509 5083 y Fh(6$)27 b Fo(Q)p Fh(g)p Fo(;)1342 5275 y Fs(~)1316 5300 y Fo(A)1391 5259 y Fj(z)1391 5325 y Fi(1)1481 5300 y Fs(=)45 b Fh(f)p Fo(x)1710 5315 y Fi(1)1830 5243 y Fj(V)1877 5253 y Fe(1)1782 5300 y Fh( )-17 b(!)27 b Fo(z)t(;)17 b(x)2144 5315 y Fi(1)2216 5300 y Fh(6$)28 b Fo(Q)p Fh(g)p Fo(;)969 5517 y(A)1044 5532 y Fi(2)1133 5517 y Fs(=)44 b Fh(f)p Fo(x)1361 5532 y Fi(2)1482 5460 y Fj(V)1529 5470 y Fe(2)1434 5517 y Fh( )-17 b(!)27 b Fo(Q)p Fh(g)p Fo(;)215 b(A)2091 5532 y Fi(3)2180 5517 y Fs(=)44 b Fh(f)p Fo(x)2408 5532 y Fi(3)2529 5460 y Fj(V)2576 5470 y Fe(3)2481 5517 y Fh( )-17 b(!)27 b Fo(Q)p Fh(g)p Fo(;)p eop %%Page: 6 6 6 5 bop 0 50 a Fq(6)1289 b(R)n(OBER)-7 b(TO)36 b(H.)h(SCHONMANN)1136 289 y Fo(B)49 b Fs(=)44 b Fh(f)p Fo(V)1490 304 y Fi(1)1563 289 y Fh(6$)27 b Fo(V)1748 304 y Fi(2)1793 289 y Fo(;)17 b(V)1896 304 y Fi(2)1968 289 y Fh(6$)27 b Fo(V)2153 304 y Fi(3)2198 289 y Fo(;)17 b(V)2301 304 y Fi(3)2373 289 y Fh(6$)27 b Fo(V)2558 304 y Fi(1)2603 289 y Fh(g)p Fo(:)0 474 y Fs(F)-8 b(or)22 b(eac)m(h)h(set)g Fo(E)606 438 y Ff(0)661 474 y Fh(\032)28 b Fo(E)6 b Fs(,)23 b(w)m(e)g(will)e(denote) i(b)m(y)g Fh(F)1720 489 y Fj(E)1782 469 y Fc(0)1836 474 y Fs(the)g Fo(\033)t Fs(-\014eld)f(generated)h(b)m(y)g(the)f(state)h (of)f(o)s(ccupancy)0 594 y(of)35 b(the)f(edges)i(in)e Fo(E)752 557 y Ff(0)779 594 y Fs(.)47 b(Let)35 b Fo(E)1108 609 y Fi(1)1182 594 y Fs(=)30 b Fh(ff)p Fo(u;)17 b(v)t Fh(g)28 b(2)i Fo(E)35 b Fs(:)29 b Fo(u;)17 b(v)33 b Fh(2)c Fo(V)2219 609 y Fi(1)2264 594 y Fh(g)p Fs(,)34 b(and)h(let)f(\()p Fo(E)2835 557 y Fj(k)2829 618 y Fi(1)2883 594 y Fs(\))2922 609 y Fj(k)r Ff(\025)p Fi(1)3107 594 y Fs(b)s(e)h(an)f(increasing)0 716 y(sequence)h(of)e(subsets)i(of)f Fo(E)1059 731 y Fi(1)1136 716 y Fs(whic)m(h)g(con)m(v)m(erges)h(to)d(this)i(set)f (\(i.e.,)f Fh([)2632 731 y Fj(k)2681 716 y Fo(E)2761 680 y Fj(k)2755 741 y Fi(1)2837 716 y Fs(=)d Fo(E)3017 731 y Fi(1)3061 716 y Fs(\).)120 852 y(F)-8 b(or)21 b(an)m(y)h Fo(k)s Fs(,)i(and)f(an)m(y)f(con\014guration)h Fo(!)1586 867 y Fi(1)1658 852 y Fh(2)28 b(f)p Fs(0)p Fo(;)17 b Fs(1)p Fh(g)1997 816 y Fj(E)2059 785 y Fd(k)2055 835 y Fe(1)2107 852 y Fs(,)24 b(the)e(set)h(of)f(con\014gurations)h(in)f Fh(f)p Fs(0)p Fo(;)17 b Fs(1)p Fh(g)3548 816 y Fj(E)t Ff(n)p Fj(E)3713 785 y Fd(k)3709 835 y Fe(1)3760 852 y Fh(\002)0 978 y(f)p Fs(0)p Fo(;)g Fs(1)p Fh(g)245 942 y Fj(V)345 978 y Fs(whic)m(h)33 b(in)f(com)m(bination)f(with)h Fo(!)1598 993 y Fi(1)1675 978 y Fs(pro)s(duce)h(a)f(con\014guration)g (in)2879 953 y(~)2852 978 y Fo(A)2927 993 y Fi(1)3004 978 y Fs(is)g(a)g(decreasing)h(set.)0 1098 y(Similarly)e(for)i Fo(B)5 b Fs(.)44 b(Therefore,)34 b(b)m(y)f(the)h(Harris')e(inequalit)m (y)-8 b(,)337 1329 y Fk(P)396 1344 y Fj(p;q)507 1329 y Fs(\()572 1304 y(~)546 1329 y Fo(A)621 1344 y Fi(1)665 1329 y Fo(B)5 b Fh(jF)846 1353 y Fj(E)908 1329 y Fd(k)904 1375 y Fe(1)956 1329 y Fs(\))60 b Fh(\025)i Fo(P)1258 1344 y Fj(p;q)1366 1329 y Fs(\()1431 1304 y(~)1405 1329 y Fo(A)1480 1344 y Fi(1)1525 1329 y Fh(jF)1625 1353 y Fj(E)1687 1329 y Fd(k)1683 1375 y Fe(1)1735 1329 y Fs(\))17 b Fk(P)1850 1344 y Fj(p;q)1960 1329 y Fs(\()p Fo(B)5 b Fh(jF)2180 1353 y Fj(E)2242 1329 y Fd(k)2238 1375 y Fe(1)2289 1329 y Fs(\))61 b(=)g Fo(P)2591 1344 y Fj(p;q)2700 1329 y Fs(\()2765 1304 y(~)2739 1329 y Fo(A)2814 1344 y Fi(1)2858 1329 y Fh(jF)2958 1353 y Fj(E)3020 1329 y Fd(k)3016 1375 y Fe(1)3068 1329 y Fs(\))17 b Fk(P)3183 1344 y Fj(p;q)3293 1329 y Fs(\()p Fo(B)5 b Fs(\))p Fo(;)0 1564 y Fs(where)37 b(in)e(the)i(last)e(step)h(w)m(e)h(used)g(the)f (fact)g(that)f Fo(B)40 b Fs(dep)s(ends)e(only)d(on)h(the)g(state)g(of)g (o)s(ccupancy)0 1684 y(of)c(the)g(edges)h(whic)m(h)f(ha)m(v)m(e)g(at)f (least)h(one)g(endp)s(oin)m(t)h(in)e(\()p Fo(V)2216 1699 y Fi(1)2280 1684 y Fh([)19 b Fo(V)2423 1699 y Fi(2)2487 1684 y Fh([)g Fo(V)2630 1699 y Fi(3)2675 1684 y Fs(\))2714 1647 y Fj(c)2782 1684 y Fh(\032)28 b Fs(\()p Fo(V)2984 1699 y Fi(1)3029 1684 y Fs(\))3068 1647 y Fj(c)3140 1684 y Fs(and)k(therefore)h(is)0 1803 y(indep)s(enden)m(t)j(of)d Fh(F)750 1828 y Fj(E)812 1803 y Fd(k)808 1849 y Fe(1)860 1803 y Fs(.)44 b(Letting)34 b Fo(k)c Fh(!)d(1)p Fs(,)33 b(and)h(using)f(\(5.9\))f(on)i(p.264)e(of)i([Dur],)f(yields)1012 2061 y Fk(P)1071 2076 y Fj(p;q)1182 2061 y Fs(\()1247 2036 y(~)1221 2061 y Fo(A)1296 2076 y Fi(1)1340 2061 y Fo(B)5 b Fh(jF)1521 2076 y Fj(E)1579 2086 y Fe(1)1622 2061 y Fs(\))61 b Fh(\025)g Fo(P)1924 2076 y Fj(p;q)2033 2061 y Fs(\()2098 2036 y(~)2072 2061 y Fo(A)2147 2076 y Fi(1)2191 2061 y Fh(jF)2291 2076 y Fj(E)2349 2086 y Fe(1)2393 2061 y Fs(\))17 b Fk(P)2508 2076 y Fj(p;q)2618 2061 y Fs(\()p Fo(B)5 b Fs(\))p Fo(:)0 2291 y Fs(In)m(tegration)33 b(o)m(v)m(er)g Fo(A)803 2254 y Fj(z)803 2315 y Fi(1)877 2291 y Fh(2)28 b(F)1043 2306 y Fj(E)1101 2316 y Fe(1)1145 2291 y Fs(,)33 b(yields)g(no)m(w)316 2527 y Fk(P)375 2542 y Fj(p;q)485 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b(x)1646 3016 y Fi(3)1717 3001 y Fh($)27 b Fo(Q;)17 b(x)2025 3016 y Fi(2)2097 3001 y Fh(6$)27 b Fo(x)2281 3016 y Fi(3)2326 3001 y Fs(\))60 b Fh(\025)2582 2906 y Fg(X)2564 3120 y Fj(z)s Ff(2)p Fj(V)2706 3130 y Fe(1)2761 3001 y Fk(P)2820 3016 y Fj(p;q)2930 3001 y Fs(\()2996 2976 y(~)2969 3001 y Fo(A)3044 2960 y Fj(z)3044 3026 y Fi(1)3090 3001 y Fo(A)3165 3016 y Fi(2)3209 3001 y Fo(A)3284 3016 y Fi(3)3329 3001 y Fo(B)5 b Fs(\))296 3284 y(=)420 3190 y Fg(X)401 3403 y Fj(z)s Ff(2)p Fj(V)543 3413 y Fe(1)599 3284 y Fo(P)663 3299 y Fj(p;q)771 3284 y Fs(\()836 3259 y(~)810 3284 y Fo(A)885 3243 y Fj(z)885 3309 y Fi(1)931 3284 y Fo(B)g Fs(\))17 b Fo(P)1132 3299 y Fj(p;q)1239 3284 y Fs(\()p Fo(A)1353 3299 y Fi(2)1397 3284 y Fs(\))g Fo(P)1517 3299 y Fj(p;q)1625 3284 y Fs(\()p Fo(A)1739 3299 y Fi(3)1784 3284 y Fs(\))60 b Fh(\025)2040 3190 y Fg(X)2022 3403 y Fj(z)s Ff(2)p Fj(V)2164 3413 y Fe(1)2219 3284 y Fo(P)2283 3299 y Fj(p;q)2391 3284 y Fs(\()2456 3259 y(~)2430 3284 y Fo(A)2505 3243 y Fj(z)2505 3309 y Fi(1)2551 3284 y Fs(\))17 b Fo(P)2671 3299 y Fj(p;q)2779 3284 y Fs(\()p Fo(A)2893 3299 y Fi(2)2937 3284 y Fs(\))g Fo(P)3057 3299 y Fj(p;q)3165 3284 y Fs(\()p Fo(A)3279 3299 y Fi(3)3323 3284 y Fs(\))g Fk(P)3438 3299 y Fj(p;q)3548 3284 y Fs(\()p Fo(B)5 b Fs(\))296 3549 y Fh(\025)62 b Fo(c)478 3508 y Fi(6)539 3549 y Fo(\037)p Fs(\()p Fo(p;)17 b(q)t Fs(\))g(\()p Fo(\022)s Fs(\()p Fo(p;)g(q)t Fs(\)\))1188 3508 y Fi(2)1230 3549 y Fo(:)0 3801 y Fs(In)34 b(the)g(second)i(step)e (ab)s(o)m(v)m(e)g(w)m(e)h(used)g(the)f(fact)g(that)2102 3776 y(~)2076 3801 y Fo(A)2151 3765 y Fj(z)2151 3826 y Fi(1)2196 3801 y Fo(B)39 b Fs(dep)s(ends)d(only)d(on)h(the)g(state)g (of)g(o)s(ccu-)0 3920 y(pancy)29 b(of)f(the)h(edges)h(whic)m(h)f(ha)m (v)m(e)g(at)f(least)g(one)h(endp)s(oin)m(t)g(in)f(\()p Fo(V)2469 3935 y Fi(2)2527 3920 y Fh([)13 b Fo(V)2664 3935 y Fi(3)2708 3920 y Fs(\))2747 3884 y Fj(c)2816 3920 y Fs(and)29 b(on)f(the)h(state)g(\(green)0 4040 y(or)40 b(not\))g(of)g(the)h(v)m(ertices)g(in)f(\()p Fo(V)1239 4055 y Fi(2)1310 4040 y Fh([)27 b Fo(V)1461 4055 y Fi(3)1506 4040 y Fs(\))1545 4004 y Fj(c)1586 4040 y Fs(,)41 b(while,)h(for)e Fo(i)g Fs(=)g(2)p Fo(;)17 b Fs(3,)40 b Fo(A)2590 4055 y Fj(i)2663 4040 y Fs(dep)s(ends)j(only)c(on)i(the)f(state)0 4160 y(of)e(o)s(ccupancy)h(of)g(the)f(edges)h(whic)m(h)g(ha)m(v)m(e)g (b)s(oth)f(endp)s(oin)m(ts)h(in)f Fo(V)2579 4175 y Fj(i)2650 4160 y Fs(and)h(on)f(the)h(state)f(\(green)g(or)0 4279 y(not\))30 b(of)h(the)h(v)m(ertices)f(in)g Fo(V)1024 4294 y Fj(i)1057 4279 y Fs(.)43 b(In)31 b(the)g(last)g(step)g(ab)s(o)m (v)m(e)g(w)m(e)g(used)h(the)f(Harris')f(inequalit)m(y)g(to)g(obtain)0 4399 y Fk(P)59 4414 y Fj(p;q)170 4399 y Fs(\()p Fo(B)5 b Fs(\))26 b Fh(\025)i Fo(c)503 4363 y Fi(3)548 4399 y Fs(.)43 b(The)30 b(claim)e(follo)m(ws)i(no)m(w)g(from)e(Lemma)g (2.1.B.)g(\(The)i(h)m(yp)s(othesis)g(in)g(that)f(lemma)0 4518 y(that)k Fo(p)266 4533 y Fj(c)334 4518 y Fo(<)28 b Fs(1)33 b(m)m(ust)g(hold,)g(since)h(otherwise)g Fk(P)1765 4533 y Fj(p;q)1876 4518 y Fs(\()p Fo(V)1973 4533 y Fj(i)2033 4518 y Fh($)28 b Fo(V)2219 4533 y Fj(j)2261 4518 y Fs(\))f Fh(!)g Fs(1,)33 b(as)g Fo(p)28 b Fh(\045)g Fo(p)2943 4533 y Fj(c)2983 4518 y Fs(.\))p 3756 4508 60 50 v 0 4695 a Fu(Lemma)39 b(2.3.A.)57 b Fl(Supp)-5 b(ose)36 b(that)h Fo(G)30 b Fs(=)f(\()p Fo(V)6 b(;)17 b(E)6 b Fs(\))34 b Fl(is)i(an)h(in\014nite,)e(lo)-5 b(c)g(al)5 b(ly)38 b(\014nite,)e(c)-5 b(onne)g(cte)g(d)35 b(tr)-5 b(ansi-)0 4815 y(tive)36 b(unimo)-5 b(dular)39 b(gr)-5 b(aph.)49 b(Supp)-5 b(ose)37 b(also)g(that)g(ther)-5 b(e)37 b(ar)-5 b(e)37 b(disjoint)e(sets)i(of)f(sites)h Fo(V)3211 4830 y Fi(1)3255 4815 y Fo(;)17 b(V)3358 4830 y Fi(2)3433 4815 y Fh(\032)31 b Fo(V)59 b Fl(and)0 4935 y(sites)35 b Fo(x)284 4950 y Fi(1)356 4935 y Fh(2)28 b Fo(V)508 4950 y Fi(1)553 4935 y Fl(,)35 b Fo(x)676 4950 y Fi(2)748 4935 y Fh(2)28 b Fo(V)900 4950 y Fi(2)981 4935 y Fl(such)35 b(that)1476 5164 y Fk(P)1535 5179 y Fj(p)1576 5189 y Fd(c)1620 5164 y Fs(\()p Fo(V)1717 5179 y Fi(1)1790 5164 y Fh($)27 b Fo(V)1975 5179 y Fi(2)2020 5164 y Fs(\))63 b Fo(<)g Fs(1)p Fo(;)1131 5323 y Fg(X)1064 5537 y Fj(v)s Ff(2)p Fj(@)1204 5547 y Fe(in)1265 5537 y Fj(V)1312 5547 y Fd(i)1360 5417 y Fk(P)1419 5432 y Fj(p)1460 5442 y Fd(c)1504 5417 y Fs(\()p Fo(x)1600 5432 y Fj(i)1661 5417 y Fh($)27 b Fo(v)t Fs(\))63 b Fo(<)h Fs(1)203 b(\()p Fo(i)28 b Fs(=)g(1)p Fo(;)17 b Fs(2\))p Fo(:)p eop %%Page: 7 7 7 6 bop 1325 50 a Fq(MEAN-FIELD)37 b(CRITICALITY)1286 b(7)0 289 y Fl(Then)36 b Fo(\015)c Fs(=)d(1)35 b Fl(and)h Fs(\001)27 b(=)h(2)p Fl(.)0 487 y(Pr)-5 b(o)g(of.)49 b Fs(W)-8 b(e)45 b(will)f(v)m(erify)h(the)g(h)m(yp)s(othesis)g(of)h (Lemma)d(2.2.A)g(with)i Fo(\017)i Fs(=)h Fo(p)2938 502 y Fj(c)3023 487 y Fs(and)d Fo(c)j Fs(=)f(min)o Fh(f)p Fs(1)30 b Fh(\000)0 607 y Fk(P)59 622 y Fj(p)100 632 y Fd(c)144 607 y Fs(\()p Fo(V)241 622 y Fi(1)317 607 y Fh($)g Fo(V)505 622 y Fi(2)550 607 y Fs(\))p Fo(;)j Fs(1)23 b Fh(\000)824 532 y Fg(P)929 636 y Fj(v)s Ff(2)p Fj(@)1069 646 y Fe(in)1130 636 y Fj(V)1177 646 y Fe(1)1238 607 y Fk(P)1297 622 y Fj(p)1338 632 y Fd(c)1382 607 y Fs(\()p Fo(x)1478 622 y Fi(1)1553 607 y Fh($)31 b Fo(v)t Fs(\))p Fo(;)h Fs(1)24 b Fh(\000)2009 532 y Fg(P)2115 636 y Fj(v)s Ff(2)p Fj(@)2255 646 y Fe(in)2316 636 y Fj(V)2363 646 y Fe(2)2423 607 y Fk(P)2482 622 y Fj(p)2523 632 y Fd(c)2567 607 y Fs(\()p Fo(x)2663 622 y Fi(2)2739 607 y Fh($)30 b Fo(v)t Fs(\))p Fh(g)p Fs(.)50 b(By)34 b(monotonicit)m(y)0 726 y(in)g Fo(p)p Fs(,)g(only)f(the)h(second)h (displa)m(y)f(in)g(the)g(h)m(yp)s(othesis)g(of)g(Lemma)f(2.2.A)f (requires)i(an)m(y)g(non-trivial)0 871 y(argumen)m(tation.)62 b(T)-8 b(o)39 b(v)m(erify)g(it,)h(w)m(e)g(note)f(that)g(if)h Fh(f)p Fo(x)2103 886 y Fj(i)2174 871 y Fh($)e Fo(z)t Fh(g)h Fs(o)s(ccurs,)j(then)e(either)g Fh(f)p Fo(x)3424 886 y Fj(i)3547 813 y(V)3594 823 y Fd(i)3495 871 y Fh( )-17 b(!)37 b Fo(z)t Fh(g)0 990 y Fs(o)s(ccurs,)j(or)d(else)i(there)f(is)g (some)f(v)m(ertex)h Fo(v)h Fh(2)d Fo(@)1829 1005 y Fi(in)1901 990 y Fo(V)1959 1005 y Fj(i)2030 990 y Fs(for)i(whic)m(h)h(the)f(ev)m (en)m(t)g Fh(f)p Fo(x)3027 1005 y Fj(i)3096 990 y Fh($)d Fo(v)t Fh(g)17 b Fb(\003)g Fh(f)p Fo(v)38 b Fh($)d Fo(z)t Fh(g)0 1110 y Fs(o)s(ccurs.)60 b(F)-8 b(rom)37 b(the)h(v)-6 b(an)38 b(den)i(Berg)e(-)g(Kesten)i(-)e(Fiebig)g(-)g(Reimer)g (inequalit)m(y)-8 b(,)38 b(w)m(e)h(obtain)f(then,)0 1229 y(for)33 b Fo(p)28 b(<)g(p)385 1244 y Fj(c)426 1229 y Fs(,)575 1512 y Fk(P)634 1527 y Fj(p)682 1512 y Fs(\()p Fo(x)778 1527 y Fj(i)839 1512 y Fh($)f Fo(z)t Fs(\))61 b Fh(\024)h Fk(P)1314 1527 y Fj(p)1362 1512 y Fs(\()p Fo(x)1458 1527 y Fj(i)1570 1454 y(V)1617 1464 y Fd(i)1518 1512 y Fh( )-17 b(!)27 b Fo(z)t Fs(\))40 b(+)2024 1417 y Fg(X)1956 1632 y Fj(v)s Ff(2)p Fj(@)2096 1642 y Fe(in)2157 1632 y Fj(V)2204 1642 y Fd(i)2252 1512 y Fk(P)2311 1527 y Fj(p)2359 1512 y Fs(\()p Fo(x)2455 1527 y Fj(i)2516 1512 y Fh($)27 b Fo(v)t Fs(\))p Fk(P)2793 1527 y Fj(p)2841 1512 y Fs(\()p Fo(v)k Fh($)c Fo(z)t Fs(\))1116 1807 y Fh(\024)62 b Fk(P)1314 1822 y Fj(p)1362 1807 y Fs(\()p Fo(x)1458 1822 y Fj(i)1570 1749 y(V)1617 1759 y Fd(i)1518 1807 y Fh( )-17 b(!)27 b Fo(z)t Fs(\))40 b(+)2024 1712 y Fg(X)1956 1927 y Fj(v)s Ff(2)p Fj(@)2096 1937 y Fe(in)2157 1927 y Fj(V)2204 1937 y Fd(i)2252 1807 y Fk(P)2311 1822 y Fj(p)2352 1832 y Fd(c)2397 1807 y Fs(\()p Fo(x)2493 1822 y Fj(i)2553 1807 y Fh($)28 b Fo(v)t Fs(\))p Fk(P)2831 1822 y Fj(p)2878 1807 y Fs(\()p Fo(v)j Fh($)d Fo(z)t Fs(\))p Fo(:)0 2163 y Fs(Summing)k(o)m(v)m(er)h Fo(z)g Fh(2)28 b Fo(V)22 b Fs(,)743 2446 y Fo(\037)p Fs(\()p Fo(p)p Fs(\))61 b Fh(\024)1147 2351 y Fg(X)1132 2565 y Fj(z)s Ff(2)p Fj(V)1274 2575 y Fd(i)1322 2446 y Fk(P)1381 2461 y Fj(p)1429 2446 y Fs(\()p Fo(x)1525 2461 y Fj(i)1638 2388 y(V)1685 2398 y Fd(i)1586 2446 y Fh( )-17 b(!)27 b Fo(z)t Fs(\))39 b(+)2092 2351 y Fg(X)2024 2566 y Fj(v)s Ff(2)p Fj(@)2164 2576 y Fe(in)2225 2566 y Fj(V)2272 2576 y Fd(i)2320 2446 y Fk(P)2379 2461 y Fj(p)2420 2471 y Fd(c)2464 2446 y Fs(\()p Fo(x)2560 2461 y Fj(i)2621 2446 y Fh($)27 b Fo(v)t Fs(\))17 b Fo(\037)p Fs(\()p Fo(p)p Fs(\))p Fo(:)0 2803 y Fs(Therefore,)1308 2872 y Fg(X)1293 3085 y Fj(z)s Ff(2)p Fj(V)1435 3095 y Fd(i)1483 2966 y Fk(P)1542 2981 y Fj(p)1590 2966 y Fs(\()p Fo(x)1686 2981 y Fj(i)1798 2909 y(V)1845 2919 y Fd(i)1746 2966 y Fh( )-17 b(!)28 b Fo(z)t Fs(\))61 b Fh(\025)g Fo(c)17 b(\037)p Fs(\()p Fo(p)p Fs(\))p Fo(:)p 3756 3253 60 50 v 0 3461 a Fu(Lemma)40 b(2.3.B.)58 b Fl(Supp)-5 b(ose)36 b(that)h Fo(G)30 b Fs(=)h(\()p Fo(V)6 b(;)17 b(E)6 b Fs(\))34 b Fl(is)i(an)h(in\014nite,)f(lo)-5 b(c)g(al)5 b(ly)38 b(\014nite,)e(c)-5 b(onne)g(cte)g(d)36 b(tr)-5 b(ansi-)0 3581 y(tive)39 b(unimo)-5 b(dular)41 b(gr)-5 b(aph.)58 b(Supp)-5 b(ose)39 b(also)h(that)g(ther)-5 b(e)40 b(ar)-5 b(e)39 b(disjoint)f(sets)h(of)h(sites)f Fo(V)3249 3596 y Fi(1)3293 3581 y Fo(;)17 b(V)3396 3596 y Fi(2)3441 3581 y Fo(;)g(V)3544 3596 y Fi(3)3623 3581 y Fh(\032)36 b Fo(V)0 3700 y Fl(and)f(sites)g Fo(x)477 3715 y Fi(1)549 3700 y Fh(2)28 b Fo(V)701 3715 y Fi(1)746 3700 y Fl(,)35 b Fo(x)869 3715 y Fi(2)942 3700 y Fh(2)28 b Fo(V)1094 3715 y Fi(2)1139 3700 y Fl(,)35 b Fo(x)1262 3715 y Fi(3)1334 3700 y Fh(2)28 b Fo(V)1486 3715 y Fi(3)1567 3700 y Fl(such)35 b(that)1236 3958 y Fk(P)1295 3973 y Fj(p)1336 3983 y Fd(c)1380 3958 y Fs(\()p Fo(V)1477 3973 y Fj(i)1538 3958 y Fh($)27 b Fo(V)1723 3973 y Fj(j)1765 3958 y Fs(\))63 b Fo(<)h Fs(1)203 b(\()p Fo(i)28 b Fh(6)p Fs(=)g Fo(j)6 b Fs(\))p Fo(;)1084 4164 y Fg(X)1017 4378 y Fj(v)s Ff(2)p Fj(@)1157 4388 y Fe(in)1218 4378 y Fj(V)1265 4388 y Fd(i)1313 4258 y Fk(P)1372 4273 y Fj(p)1413 4283 y Fd(c)1457 4258 y Fs(\()p Fo(x)1553 4273 y Fj(i)1614 4258 y Fh($)27 b Fo(v)t Fs(\))63 b Fo(<)h Fs(1)203 b(\()p Fo(i)28 b Fs(=)g(1)p Fo(;)17 b Fs(2)p Fo(;)g Fs(3\))p Fo(:)0 4566 y Fl(Then)36 b Fo(\016)31 b Fs(=)e(2)35 b Fl(and)h Fo(\014)d Fs(=)28 b(1)p Fl(.)0 4764 y(Pr)-5 b(o)g(of.)49 b Fs(W)-8 b(e)46 b(will)e(v)m(erify)g(the)i(h)m(yp)s (othesis)g(of)f(Lemma)e(2.2.B)h(with)h Fo(\017)i Fs(=)h Fo(p)2936 4779 y Fj(c)3022 4764 y Fs(and)e Fo(c)h Fs(=)h(min)o Fh(f)p Fs(1)30 b Fh(\000)0 4883 y Fk(P)59 4898 y Fj(p)100 4908 y Fd(c)144 4883 y Fs(\()p Fo(V)241 4898 y Fi(1)334 4883 y Fh($)47 b Fo(V)539 4898 y Fi(2)584 4883 y Fs(\))p Fo(;)32 b Fs(1)e Fh(\000)h Fk(P)930 4898 y Fj(p)971 4908 y Fd(c)1015 4883 y Fs(\()p Fo(V)1112 4898 y Fi(2)1205 4883 y Fh($)47 b Fo(V)1410 4898 y Fi(3)1455 4883 y Fs(\))p Fo(;)33 b Fs(1)d Fh(\000)g Fk(P)1801 4898 y Fj(p)1842 4908 y Fd(c)1887 4883 y Fs(\()p Fo(V)1984 4898 y Fi(3)2076 4883 y Fh($)47 b Fo(V)2281 4898 y Fi(1)2326 4883 y Fs(\))p Fo(;)33 b Fs(1)d Fh(\000)2613 4809 y Fg(P)2719 4913 y Fj(v)s Ff(2)p Fj(@)2859 4923 y Fe(in)2920 4913 y Fj(V)2967 4923 y Fe(1)3027 4883 y Fk(P)3086 4898 y Fj(p)3127 4908 y Fd(c)3171 4883 y Fs(\()p Fo(x)3267 4898 y Fi(1)3360 4883 y Fh($)47 b Fo(v)t Fs(\))p Fo(;)33 b Fs(1)d Fh(\000)0 4933 y Fg(P)105 5038 y Fj(v)s Ff(2)p Fj(@)245 5048 y Fe(in)306 5038 y Fj(V)353 5048 y Fe(2)414 5008 y Fk(P)473 5023 y Fj(p)514 5033 y Fd(c)558 5008 y Fs(\()p Fo(x)654 5023 y Fi(2)750 5008 y Fh($)50 b Fo(v)t Fs(\))p Fo(;)33 b Fs(1)e Fh(\000)1242 4933 y Fg(P)1348 5038 y Fj(v)s Ff(2)p Fj(@)1488 5048 y Fe(in)1549 5038 y Fj(V)1596 5048 y Fe(3)1656 5008 y Fk(P)1715 5023 y Fj(p)1756 5033 y Fd(c)1800 5008 y Fs(\()p Fo(x)1896 5023 y Fi(3)1992 5008 y Fh($)51 b Fo(v)t Fs(\))p Fh(g)p Fs(.)86 b(By)46 b(monotonicit)m(y)g (in)h Fo(p)p Fs(,)k(only)c(the)0 5127 y(second)j(and)f(third)g(displa)m (ys)g(in)g(the)g(h)m(yp)s(othesis)h(of)f(Lemma)e(2.2.B)g(require)i(an)m (y)g(non-trivial)0 5247 y(argumen)m(tation.)120 5398 y(T)-8 b(o)33 b(v)m(erify)g(the)h(third)g(displa)m(y)-8 b(,)34 b(w)m(e)g(note)g(that)f(if)h Fh(f)p Fo(x)2109 5413 y Fj(i)2170 5398 y Fh($)29 b Fo(Q)p Fh(g)k Fs(o)s(ccurs,)h(then)h (either)f Fh(f)p Fo(x)3415 5413 y Fj(i)3528 5340 y(V)3575 5350 y Fd(i)3476 5398 y Fh( )-17 b(!)28 b Fo(Q)p Fh(g)0 5517 y Fs(o)s(ccurs,)38 b(or)e(else)i(there)f(is)f(some)g(v)m(ertex)h Fo(v)g Fh(2)d Fo(@)1816 5532 y Fi(in)1887 5517 y Fo(V)1945 5532 y Fj(i)2015 5517 y Fs(for)j(whic)m(h)g(the)g(ev)m(en)m(t)h Fh(f)p Fo(x)3008 5532 y Fj(i)3074 5517 y Fh($)33 b Fo(v)t Fh(g)17 b Fb(\003)g Fh(f)p Fo(v)36 b Fh($)d Fo(Q)p Fh(g)p eop %%Page: 8 8 8 7 bop 0 50 a Fq(8)1289 b(R)n(OBER)-7 b(TO)36 b(H.)h(SCHONMANN)0 289 y Fs(o)s(ccurs.)60 b(F)-8 b(rom)37 b(the)h(v)-6 b(an)38 b(den)i(Berg)e(-)g(Kesten)i(-)e(Fiebig)g(-)g(Reimer)g(inequalit)m(y)-8 b(,)38 b(w)m(e)h(obtain)f(then,)0 408 y(for)33 b Fo(p)28 b(<)g(p)385 423 y Fj(c)459 408 y Fs(and)34 b Fo(q)e Fh(2)c Fs(\(0)p Fo(;)17 b Fs(1],)231 664 y Fo(\022)s Fs(\()p Fo(p;)g(q)t Fs(\))26 b(=)i Fk(P)692 679 y Fj(p;q)803 664 y Fs(\()p Fo(x)899 679 y Fj(i)959 664 y Fh($)g Fo(Q)p Fs(\))60 b Fh(\024)h Fk(P)1462 679 y Fj(p;q)1573 664 y Fs(\()p Fo(x)1669 679 y Fj(i)1781 606 y(V)1828 616 y Fd(i)1729 664 y Fh( )-17 b(!)28 b Fo(Q)p Fs(\))38 b(+)2263 569 y Fg(X)2196 784 y Fj(v)s Ff(2)p Fj(@)2336 794 y Fe(in)2397 784 y Fj(V)2444 794 y Fd(i)2492 664 y Fk(P)2551 679 y Fj(p;q)2661 664 y Fs(\()p Fo(x)2757 679 y Fj(i)2818 664 y Fh($)27 b Fo(v)t Fs(\))17 b Fk(P)3112 679 y Fj(p;q)3222 664 y Fs(\()p Fo(v)31 b Fh($)d Fo(Q)p Fs(\))1265 959 y Fh(\024)61 b Fk(P)1462 974 y Fj(p;q)1573 959 y Fs(\()p Fo(x)1669 974 y Fj(i)1781 901 y(V)1828 911 y Fd(i)1729 959 y Fh( )-17 b(!)28 b Fo(Q)p Fs(\))38 b(+)2263 864 y Fg(X)2196 1079 y Fj(v)s Ff(2)p Fj(@)2336 1089 y Fe(in)2397 1079 y Fj(V)2444 1089 y Fd(i)2492 959 y Fk(P)2551 974 y Fj(p)2592 984 y Fd(c)2636 959 y Fs(\()p Fo(x)2732 974 y Fj(i)2793 959 y Fh($)27 b Fo(v)t Fs(\))17 b Fo(\022)s Fs(\()p Fo(p;)g(q)t Fs(\))p Fo(:)0 1283 y Fs(Therefore,)1009 1428 y Fk(P)1068 1443 y Fj(p;q)1179 1428 y Fs(\()p Fo(x)1275 1443 y Fj(i)1387 1370 y(V)1434 1380 y Fd(i)1335 1428 y Fh( )-17 b(!)28 b Fo(Q)p Fs(\))60 b Fh(\025)h Fo(c)17 b(\022)s Fs(\()p Fo(p;)g(q)t Fs(\))198 b(\()p Fo(i)27 b Fs(=)i(2)p Fo(;)17 b Fs(3\))p Fo(:)120 1611 y Fs(T)-8 b(o)40 b(v)m(erify)i(the)f(second)i(displa)m(y)e(in)h(the)f(h)m(yp)s (othesis)h(of)g(Lemma)e(2.2.B)g(note)h(that)g(if)h Fh(f)p Fo(x)3631 1626 y Fi(1)3716 1611 y Fh($)0 1761 y Fo(z)t(;)17 b(x)152 1776 y Fi(1)227 1761 y Fh(6$)30 b Fo(Q)p Fh(g)k Fs(o)s(ccurs,)i(then)g(either)f Fh(f)p Fo(x)1478 1776 y Fi(1)1601 1703 y Fj(V)1648 1713 y Fe(1)1553 1761 y Fh( )-17 b(!)30 b Fo(z)t(;)17 b(x)1918 1776 y Fi(1)1993 1761 y Fh(6$)30 b Fo(Q)p Fh(g)k Fs(o)s(ccurs,)i(or)e(else)i(there)f(is) g(some)f(v)m(ertex)0 1880 y Fo(v)e Fh(2)c Fo(@)227 1895 y Fi(in)299 1880 y Fo(V)357 1895 y Fj(i)423 1880 y Fs(for)34 b(whic)m(h)g(the)g(ev)m(en)m(t)g Fh(f)p Fo(x)1403 1895 y Fi(1)1475 1880 y Fh($)28 b Fo(v)t Fh(g)17 b Fb(\003)g Fh(f)p Fo(v)31 b Fh($)c Fo(z)t(;)17 b(v)32 b Fh(6$)c Fo(Q)p Fh(g)33 b Fs(o)s(ccurs)h(\(this)f(is)g(sligh)m(tly)g(subtle;)0 2000 y(recall)g(that)f(our)h(sample)g(space)h(is)f Fh(f)p Fs(0)p Fo(;)17 b Fs(1)p Fh(g)1602 1964 y Fj(E)1689 2000 y Fh(\002)22 b(f)p Fs(0)p Fo(;)17 b Fs(1)p Fh(g)2033 1964 y Fj(V)2134 2000 y Fs(and,)33 b(using)g(the)g(notation)g(in)g ([Gri],)e(p.)45 b(38,)0 2120 y(tak)m(e)32 b(for)h(the)f(set)h Fo(K)39 b Fs(in)33 b(the)g(de\014nition)f(of)h Fb(\003)g Fs(a)f(set)h(of)g(edges)g(whic)m(h)g(pro)s(duce)h(a)e(path)h(from)e Fo(x)3650 2135 y Fi(1)3727 2120 y Fs(to)0 2239 y Fo(v)37 b Fs({)32 b(do)h(not)f(include)i(an)m(y)f(v)m(ertex)f(in)h Fo(K)7 b Fs(\).)43 b(F)-8 b(rom)31 b(the)i(v)-6 b(an)33 b(den)h(Berg)e(-)h(Kesten)h(-)f(Fiebig)g(-)f(Reimer)0 2359 y(inequalit)m(y)-8 b(,)32 b(w)m(e)h(obtain)g(then,)h(for)f Fo(p)28 b(<)g(p)1578 2374 y Fj(c)1652 2359 y Fs(and)34 b Fo(q)e Fh(2)c Fs(\(0)p Fo(;)17 b Fs(1],)291 2584 y Fk(P)350 2599 y Fj(p;q)461 2584 y Fs(\()p Fo(x)557 2599 y Fi(1)629 2584 y Fh($)27 b Fo(z)t(;)17 b(x)908 2599 y Fi(1)981 2584 y Fh(6$)27 b Fo(Q)p Fs(\))518 2779 y Fh(\024)62 b Fk(P)716 2794 y Fj(p;q)826 2779 y Fs(\()p Fo(x)922 2794 y Fi(1)1043 2721 y Fj(V)1090 2731 y Fe(1)994 2779 y Fh( )-17 b(!)28 b Fo(z)t(;)17 b(x)1357 2794 y Fi(1)1429 2779 y Fh(6$)27 b Fo(Q)p Fs(\))38 b(+)1884 2684 y Fg(X)1812 2898 y Fj(v)s Ff(2)p Fj(@)1952 2908 y Fe(in)2013 2898 y Fj(V)2060 2908 y Fe(1)2116 2779 y Fk(P)2175 2794 y Fj(p;q)2285 2779 y Fs(\()p Fo(x)2381 2794 y Fi(1)2453 2779 y Fh($)28 b Fo(v)t Fs(\))17 b Fk(P)2748 2794 y Fj(p;q)2857 2779 y Fs(\()p Fo(v)32 b Fh($)27 b Fo(z)t(;)17 b(v)32 b Fh(6$)27 b Fo(Q)p Fs(\))518 3073 y Fh(\024)62 b Fk(P)716 3088 y Fj(p;q)826 3073 y Fs(\()p Fo(x)922 3088 y Fi(1)1043 3016 y Fj(V)1090 3026 y Fe(1)994 3073 y Fh( )-17 b(!)28 b Fo(z)t(;)17 b(x)1357 3088 y Fi(1)1429 3073 y Fh(6$)27 b Fo(Q)p Fs(\))38 b(+)1884 2979 y Fg(X)1812 3193 y Fj(v)s Ff(2)p Fj(@)1952 3203 y Fe(in)2013 3193 y Fj(V)2060 3203 y Fe(1)2116 3073 y Fk(P)2175 3088 y Fj(p)2216 3098 y Fd(c)2260 3073 y Fs(\()p Fo(x)2356 3088 y Fi(1)2428 3073 y Fh($)27 b Fo(v)t Fs(\))17 b Fk(P)2722 3088 y Fj(p;q)2832 3073 y Fs(\()p Fo(v)31 b Fh($)d Fo(z)t(;)17 b(v)31 b Fh(6$)c Fo(Q)p Fs(\))p Fo(:)0 3397 y Fs(Summing)32 b(o)m(v)m(er)h Fo(z)g Fh(2)28 b Fo(V)22 b Fs(,)411 3647 y Fo(\037)p Fs(\()p Fo(p;)17 b(q)t Fs(\))60 b Fh(\024)911 3552 y Fg(X)892 3766 y Fj(z)s Ff(2)p Fj(V)1034 3776 y Fe(1)1089 3647 y Fk(P)1148 3662 y Fj(p;q)1259 3647 y Fs(\()p Fo(x)1355 3662 y Fi(1)1475 3589 y Fj(V)1522 3599 y Fe(1)1427 3647 y Fh( )-17 b(!)27 b Fo(z)t(;)17 b(x)1789 3662 y Fi(1)1862 3647 y Fh(6$)27 b Fo(Q)p Fs(\))38 b(+)2316 3552 y Fg(X)2245 3767 y Fj(v)s Ff(2)p Fj(@)2385 3777 y Fe(in)2446 3767 y Fj(V)2493 3777 y Fe(1)2548 3647 y Fk(P)2607 3662 y Fj(p)2648 3672 y Fd(c)2693 3647 y Fs(\()p Fo(x)2789 3662 y Fi(1)2861 3647 y Fh($)27 b Fo(v)t Fs(\))17 b Fo(\037)p Fs(\()p Fo(p;)g(q)t Fs(\))p Fo(:)0 3971 y Fs(Therefore,)1035 4021 y Fg(X)1016 4235 y Fj(z)s Ff(2)p Fj(V)1158 4245 y Fe(1)1214 4116 y Fk(P)1273 4131 y Fj(p;q)1383 4116 y Fs(\()p Fo(x)1479 4131 y Fi(1)1600 4058 y Fj(V)1647 4068 y Fe(1)1551 4116 y Fh( )-17 b(!)28 b Fo(z)t(;)17 b(x)1914 4131 y Fi(1)1986 4116 y Fh(6$)27 b Fo(Q)p Fs(\))61 b Fh(\025)g Fo(c)17 b(\037)p Fs(\()p Fo(p;)g(q)t Fs(\))p Fo(:)p 3756 4378 60 50 v 1149 4621 a Ft(3.)55 b(The)36 b(case)g(of)h(planar)g(graphs)120 4800 y Fs(In)44 b(this)f(section)h(w)m(e)g(supp)s(ose)i(that)d Fo(G)i Fs(=)h(\()p Fo(V)6 b(;)17 b(E)6 b Fs(\))41 b(is)j(an)g (in\014nite,)i(lo)s(cally)c(\014nite,)k(connected)0 4920 y(transitiv)m(e)32 b(non-amenable)i(planar)f(single-ended)i(graph.)44 b(Prop)s(osition)32 b(2.1)h(of)g([BS2])f(states)h(that)0 5039 y Fo(G)27 b Fs(is)g(unimo)s(dular)f(and)i(that)f(it)f(can)h(b)s(e) h(em)m(b)s(edded)g(in)f(the)g(h)m(yp)s(erb)s(olic)h(plane)f Fk(H)3076 5003 y Fi(2)3153 5039 y Fs(in)g(the)h(follo)m(wing)0 5159 y(w)m(a)m(y)-8 b(.)73 b(Eac)m(h)44 b(v)m(ertex)f(of)g Fo(G)h Fs(is)f(mapp)s(ed)g(in)m(to)g(a)g(p)s(oin)m(t)g(of)h Fk(H)2346 5122 y Fi(2)2440 5159 y Fs(and)g(eac)m(h)g(edge)g(of)f Fo(G)h Fs(is)f(mapp)s(ed)0 5278 y(in)m(to)35 b(a)f(geo)s(desic)i(line)f (segmen)m(t)f(with)h(endp)s(oin)m(ts)h(at)e(the)i(p)s(oin)m(ts)f(of)g Fk(H)2726 5242 y Fi(2)2812 5278 y Fs(whic)m(h)g(are)g(images)f(of)h (its)0 5398 y(endp)s(oin)m(ts;)29 b(moreo)m(v)m(er)c(the)h(group)f(of)h (automorphisms)e(of)i Fo(G)g Fs(is)f(mapp)s(ed)h(in)f(this)h(w)m(a)m(y) f(in)m(to)g(a)h(group)0 5517 y(of)39 b(isometries)e(of)i Fk(H)780 5481 y Fi(2)831 5517 y Fs(.)59 b(It)38 b(is)g(clear)h(that,)g (b)m(y)f(adjusting)h(the)g(length)g(scale,)g(suc)m(h)h(an)f(em)m(b)s (edding)p eop %%Page: 9 9 9 8 bop 1325 50 a Fq(MEAN-FIELD)37 b(CRITICALITY)1286 b(9)0 289 y Fs(can)30 b(b)s(e)g(c)m(hosen)g(so)g(that)f(eac)m(h)h(face) g(in)g(the)f(em)m(b)s(edding)h(has)g(diameter)f(less)h(than)f(1.)43 b(In)29 b(particular)0 408 y(an)m(y)j(p)s(oin)m(t)g(of)h Fk(H)630 372 y Fi(2)713 408 y Fs(is)f(then)h(within)f(distance)h(1)f (of)g(a)g(p)s(oin)m(t)g(whic)m(h)h(represen)m(ts)i(a)d(v)m(ertex)g(of)g Fo(G)p Fs(,)g(and)0 528 y(all)d(the)h(geo)s(desic)h(line)e(segmen)m(ts) h(whic)m(h)h(represen)m(t)h(edges)f(of)f Fo(G)g Fs(ha)m(v)m(e)g(length) g(at)g(most)e(1.)43 b(W)-8 b(e)30 b(will)0 648 y(refer)k(to)f(suc)m(h)i (an)e(em)m(b)s(edding)h(as)f(a)g(\\nice)h(em)m(b)s(edding".)120 771 y(One)44 b(con)m(v)m(enien)m(t)i(w)m(a)m(y)d(to)h(describ)s(e)h (the)f(dual)g Fo(G)2077 734 y Ff(y)2164 771 y Fs(=)i(\()p Fo(V)2406 734 y Ff(y)2447 771 y Fo(;)17 b(E)2572 734 y Ff(y)2613 771 y Fs(\))43 b(of)h Fo(G)g Fs(is)g(to)f(represen)m(t)j (eac)m(h)0 890 y(elemen)m(t)29 b(of)h Fo(V)554 854 y Ff(y)625 890 y Fs(b)m(y)f(a)g(face)h(\(tile\))f(in)g(the)h(em)m(b)s (edding)f(of)h Fo(G)p Fs(,)g(describ)s(ed)h(ab)s(o)m(v)m(e,)f(and)f (represen)m(t)j(ele-)0 1010 y(men)m(ts)c(of)g Fo(E)474 974 y Ff(y)542 1010 y Fs(b)m(y)g(pairs)g(of)g(faces)h(whose)g(top)s (ological)d(b)s(oundaries)j(in)m(tersect)f(on)g(a)g(non-degenerate)0 1129 y(geo)s(desic)43 b(line)g(segmen)m(t)g(\(whic)m(h)g(represen)m(ts) i(an)d(edge)i(of)f Fo(G)p Fs(\).)72 b(This)43 b(establishes)h(a)e (one-to-one)0 1249 y(corresp)s(ondence)e(b)s(et)m(w)m(een)e Fo(E)43 b Fs(and)37 b Fo(E)1482 1213 y Ff(y)1523 1249 y Fs(,)h(and)g(the)f(image)f(of)h Fo(e)e Fh(2)f Fo(E)43 b Fs(under)38 b(this)f(corresp)s(ondence)0 1368 y(will)32 b(b)s(e)i(denoted)g(b)m(y)g Fo(e)885 1332 y Ff(y)926 1368 y Fs(.)44 b(Since)35 b Fo(G)e Fs(is)g(transitiv)m(e,)g Fo(G)2020 1332 y Ff(y)2095 1368 y Fs(is)g(quasi-transitiv)m(e.)120 1491 y(An)m(y)26 b(b)s(ond)i(p)s(ercolation)e(pro)s(cess)i(on)f Fo(G)g Fs(is)g(coupled)h(to)e(a)h(b)s(ond)g(p)s(ercolation)f(pro)s (cess)i(on)f Fo(G)3613 1455 y Ff(y)3655 1491 y Fs(,)h(b)m(y)0 1611 y(declaring)34 b(eac)m(h)h(edge)g Fo(e)926 1575 y Ff(y)1002 1611 y Fs(v)-6 b(acan)m(t)34 b(\()p Fl(r)-5 b(esp.)46 b Fs(o)s(ccupied\))35 b(if)f Fo(e)h Fs(is)f(o)s(ccupied)h(\() p Fl(r)-5 b(esp.)46 b Fs(v)-6 b(acan)m(t\).)46 b(Indep)s(en-)0 1731 y(den)m(t)33 b(p)s(ercolation)f(at)g(densit)m(y)h Fo(p)f Fs(on)h Fo(G)f Fs(is)h(coupled)g(in)f(this)h(fashion)g(to)f (indep)s(enden)m(t)i(p)s(ercolation)0 1850 y(at)f(densit)m(y)g(1)22 b Fh(\000)h Fo(p)33 b Fs(on)h Fo(G)937 1814 y Ff(y)979 1850 y Fs(.)120 1973 y(The)28 b(follo)m(wing)g(lemma)e(is)i(a)g(basic)h (building)f(blo)s(c)m(k)g(in)g(our)g(argumen)m(tation)g(in)g(this)g (section.)43 b(In)0 2093 y(the)32 b(statemen)m(t)g(of)g(this)h(lemma,)d (w)m(e)i(iden)m(tify)h(a)e(path)i(in)f(the)g(dual)h(graph)f(with)g(the) h(union)f(of)h(the)0 2212 y(tiles)g(that)g(corresp)s(ond)h(to)f(the)h (endp)s(oin)m(ts)g(of)f(the)h(dual)f(edges)i(in)e(this)g(path)h(in)f (the)g(em)m(b)s(edding.)0 2402 y Fu(Lemma)57 b(3.1.)g Fl(Supp)-5 b(ose)51 b(that)f Fo(G)h Fl(is)f(an)g(in\014nite,)j(lo)-5 b(c)g(al)5 b(ly)52 b(\014nite,)h(c)-5 b(onne)g(cte)g(d)49 b(tr)-5 b(ansitive)49 b(non-)0 2522 y(amenable)c(planar)g(single-ende) -5 b(d)43 b(gr)-5 b(aph,)47 b(nic)-5 b(ely)44 b(emb)-5 b(e)g(dde)g(d)44 b(in)h Fk(H)2592 2486 y Fi(2)2643 2522 y Fl(.)73 b(If)45 b Fo(p)f(<)h(p)3127 2537 y Fj(u)3179 2522 y Fl(,)i(then)d(ther)-5 b(e)45 b(is)0 2641 y Fo(C)71 2656 y Fi(0)155 2641 y Fo(>)39 b Fs(0)j Fl(such)f(that)h(the)g(fol)5 b(lowing)41 b(happ)-5 b(ens.)63 b(L)-5 b(et)42 b Fh(L)g Fl(b)-5 b(e)41 b(an)h(arbitr)-5 b(ary)41 b(ge)-5 b(o)g(desic)40 b(line)i(in)f Fk(H)3617 2605 y Fi(2)3667 2641 y Fl(,)i Fo(s)3788 2605 y Ff(0)0 2761 y Fl(and)36 b Fo(s)241 2725 y Ff(00)327 2761 y Fl(b)-5 b(e)36 b(two)g(p)-5 b(oints)35 b(on)h Fh(L)p Fl(,)g(sep)-5 b(ar)g(ate)g(d)34 b(by)j(distanc)-5 b(e)34 b Fo(l)c(>)f Fs(2)p Fl(,)36 b(and)g Fh(L)2700 2725 y Ff(0)2763 2761 y Fl(and)g Fh(L)3026 2725 y Ff(00)3113 2761 y Fl(b)-5 b(e)35 b(ge)-5 b(o)g(desic)34 b(lines)0 2880 y(p)-5 b(erp)g(endicular)35 b(to)h Fh(L)f Fl(thr)-5 b(ough)36 b Fo(s)1245 2844 y Ff(0)1308 2880 y Fl(and)g Fo(s)1549 2844 y Ff(0)o(0)1599 2880 y Fl(,)f(r)-5 b(esp)g(e)g(ctively.) 45 b(Then)499 3130 y Fk(P)558 3145 y Fj(p)606 3130 y Fs(\()p Fl(ther)-5 b(e)35 b(is)g(an)h(o)-5 b(c)g(cupie)g(d)35 b(dual)h(p)-5 b(ath)35 b(sep)-5 b(ar)g(ating)35 b Fh(L)2512 3089 y Ff(0)2575 3130 y Fl(fr)-5 b(om)35 b Fh(L)2879 3089 y Ff(00)2930 3130 y Fs(\))63 b Fo(>)h(C)3244 3145 y Fi(0)3289 3130 y Fo(:)0 3450 y Fl(Pr)-5 b(o)g(of.)49 b Fs(This)29 b(w)m(as)h(pro)m(v)m(ed)f(in)g(a)g(somewhat)f(more)g (restricted)i(setting)f(and)g(for)h(site)e(p)s(ercolation)h(in)0 3569 y([Lal2],)i(Lemma)h(2.15.)43 b(The)33 b(more)f(general)h(case)g (considered)i(here)e(can)h(b)s(e)f(handled)h(in)e(the)i(same)0 3689 y(w)m(a)m(y)-8 b(,)35 b(b)m(y)g(using)g(results)h(in)f([BS2].)48 b(First,)35 b(from)f(Theorem)h(3.7)f(of)h([BS2],)f(w)m(e)i(learn)f (that)f(there)i(is)0 3809 y(p)s(ercolation)h(in)g(the)h(dual)g(pro)s (cess)g(when)h Fo(p)34 b(<)i(p)1907 3824 y Fj(u)1959 3809 y Fs(.)56 b(F)-8 b(rom)36 b(the)i(generalization)f(of)h(Corollary) e(4.4)0 3928 y(of)i([BS2])g(to)g(quasi-transitiv)m(e)f(tilings)h(of)g Fk(H)1716 3892 y Fi(2)1766 3928 y Fs(,)h(w)m(e)g(learn)f(then)h(that)f (p)s(ercolation)g(also)g(o)s(ccurs)h(in)0 4048 y(this)32 b(dual)f(pro)s(cess)i(on)e(h)m(yp)s(erb)s(olic)h(half-spaces.)45 b(This)32 b(enables)g(us)h(to)e(use)h(the)g(argumen)m(ts)f(in)h(the)0 4167 y(pro)s(ofs)i(of)f(Lemma)f(2.14)g(and)i(2.15)e(in)h([Lal2])g(to)f (conclude)j(the)e(pro)s(of.)p 3756 4157 60 50 v 120 4357 a(Giv)m(en)g(a)h(nice)g(em)m(b)s(edding)g(of)g Fo(G)f Fs(in)h Fk(H)1614 4321 y Fi(2)1699 4357 y Fs(and)g(a)f(set)h Fh(S)i(\032)28 b Fk(H)2405 4321 y Fi(2)2456 4357 y Fs(,)33 b(w)m(e)h(will)e(use)j(the)f(notation)3618 4332 y(\026)3595 4357 y Fh(S)42 b Fs(for)0 4477 y(the)d(set)g(of)f(v)m(ertices)h(of)g Fo(G)f Fs(whic)m(h)i(are)e(endp)s(oin)m(ts)i(of)e(edges)i(represen)m (ted)h(in)d(the)h(em)m(b)s(edding)g(b)m(y)0 4596 y(geo)s(desic)34 b(line)f(segmen)m(ts)g(whic)m(h)i(in)m(tersect)f Fh(S)7 b Fs(.)0 4786 y Fu(Lemma)57 b(3.2.)g Fl(Supp)-5 b(ose)51 b(that)f Fo(G)h Fl(is)f(an)g(in\014nite,)j(lo)-5 b(c)g(al)5 b(ly)52 b(\014nite,)h(c)-5 b(onne)g(cte)g(d)49 b(tr)-5 b(ansitive)49 b(non-)0 4906 y(amenable)42 b(planar)f(single-ende)-5 b(d)40 b(gr)-5 b(aph,)42 b(nic)-5 b(ely)41 b(emb)-5 b(e)g(dde)g(d)41 b(in)g Fk(H)2567 4869 y Fi(2)2618 4906 y Fl(.)63 b(If)41 b Fo(p)d(<)h(p)3076 4921 y Fj(u)3128 4906 y Fl(,)j(then)f(ther)-5 b(e)41 b(ar)-5 b(e)0 5025 y Fo(C)71 5040 y Fi(1)116 5025 y Fo(;)17 b(C)232 5040 y Fi(2)306 5025 y Fh(2)31 b Fs(\(0)p Fo(;)17 b Fh(1)p Fs(\))p Fl(,)35 b(such)i(that)g(the)g(fol)5 b(lowing)37 b(happ)-5 b(ens.)48 b(L)-5 b(et)38 b Fh(L)e Fl(b)-5 b(e)37 b(an)g(arbitr)-5 b(ary)36 b(ge)-5 b(o)g(desic)36 b(line)g(in)0 5145 y Fk(H)71 5108 y Fi(2)122 5145 y Fl(,)41 b Fo(s)241 5108 y Ff(0)309 5145 y Fl(and)g Fo(s)555 5108 y Ff(0)o(0)646 5145 y Fl(b)-5 b(e)40 b(two)g(p)-5 b(oints)40 b(on)g Fh(L)p Fl(,)h(sep)-5 b(ar)g(ate)g(d)39 b(by)i(distanc)-5 b(e)39 b Fo(L)p Fl(,)i(and)f Fh(L)2916 5108 y Ff(0)2984 5145 y Fl(and)h Fh(L)3252 5108 y Ff(0)o(0)3343 5145 y Fl(b)-5 b(e)40 b(ge)-5 b(o)g(desic)0 5264 y(lines)35 b(p)-5 b(erp)g(endicular)35 b(to)h Fh(L)f Fl(thr)-5 b(ough)36 b Fo(s)1479 5228 y Ff(0)1542 5264 y Fl(and)g Fo(s)1783 5228 y Ff(00)1834 5264 y Fl(,)f(r)-5 b(esp)g(e)g(ctively.)44 b(Then)1324 5517 y Fk(P)1383 5532 y Fj(p)1431 5517 y Fs(\()1493 5492 y(\026)1470 5517 y Fh(L)1539 5476 y Ff(0)1594 5517 y Fh($)1745 5492 y Fs(\026)1721 5517 y Fh(L)1790 5476 y Ff(00)1841 5517 y Fs(\))63 b Fh(\024)h Fo(C)2155 5532 y Fi(1)2200 5517 y Fo(e)2246 5476 y Ff(\000)p Fj(C)2365 5486 y Fe(2)2405 5476 y Fj(L)2464 5517 y Fo(:)p eop %%Page: 10 10 10 9 bop 0 50 a Fq(10)1247 b(R)n(OBER)-7 b(TO)36 b(H.)h(SCHONMANN)0 289 y Fl(Pr)-5 b(o)g(of.)49 b Fs(T)-8 b(ak)m(e)34 b Fo(l)d(>)e Fs(2)34 b(and)h(consider)g(the)g(set)f(of)g(geo)s(desic)h(lines)f(whic) m(h)h(separate)g Fh(L)3203 253 y Ff(0)3265 289 y Fs(from)e Fh(L)3570 253 y Ff(00)3621 289 y Fs(,)h(are)0 408 y(p)s(erp)s (endicular)45 b(to)e Fh(L)g Fs(and)g(cross)i(it)d(at)h(p)s(oin)m(ts)h (whic)m(h)g(are)f(at)g(distance)h Fo(j)6 b(l)r Fs(,)45 b Fo(j)51 b Fs(=)45 b(1)p Fo(;)17 b Fs(2)p Fo(;)g(:::;)g Fh(b)p Fo(L=l)r Fh(c)0 528 y Fs(from)30 b Fo(s)280 492 y Ff(0)307 528 y Fs(.)43 b(Since)32 b(an)m(y)e(path)h(from)1307 503 y(\026)1284 528 y Fh(L)1353 492 y Ff(0)1411 528 y Fs(to)1553 503 y(\026)1530 528 y Fh(L)1599 492 y Ff(00)1680 528 y Fs(has)g(to)f(cross)h(all)f(these)i(lines,)e(the)h(claim)e(follo) m(ws)i(from)0 648 y(Lemma)h(3.1.)p 3756 638 60 50 v 0 824 a Fu(Lemma)57 b(3.3.)g Fl(Supp)-5 b(ose)51 b(that)f Fo(G)h Fl(is)f(an)g(in\014nite,)j(lo)-5 b(c)g(al)5 b(ly)52 b(\014nite,)h(c)-5 b(onne)g(cte)g(d)49 b(tr)-5 b(ansitive)49 b(non-)0 944 y(amenable)42 b(planar)f(single-ende)-5 b(d)40 b(gr)-5 b(aph,)42 b(nic)-5 b(ely)41 b(emb)-5 b(e)g(dde)g(d)41 b(in)g Fk(H)2567 908 y Fi(2)2618 944 y Fl(.)63 b(If)41 b Fo(p)d(<)h(p)3076 959 y Fj(u)3128 944 y Fl(,)j(then)f(ther)-5 b(e)41 b(ar)-5 b(e)0 1063 y Fo(C)71 1078 y Fi(3)116 1063 y Fo(;)17 b(C)232 1078 y Fi(4)306 1063 y Fh(2)31 b Fs(\(0)p Fo(;)17 b Fh(1)p Fs(\))p Fl(,)35 b(such)i(that)g(the)g(fol)5 b(lowing)37 b(happ)-5 b(ens.)48 b(L)-5 b(et)38 b Fh(L)e Fl(b)-5 b(e)37 b(an)g(arbitr)-5 b(ary)36 b(ge)-5 b(o)g(desic)36 b(line)g(in)0 1183 y Fk(H)71 1147 y Fi(2)122 1183 y Fl(,)44 b Fo(s)e Fl(and)h Fo(s)534 1147 y Ff(0)604 1183 y Fl(b)-5 b(e)42 b(two)h(p)-5 b(oints)42 b(on)g Fh(L)p Fl(,)i(sep)-5 b(ar)g(ate)g(d)41 b(by)i(distanc)-5 b(e)41 b Fo(L)p Fl(,)j(and)e Fh(L)2897 1147 y Ff(0)2968 1183 y Fl(b)-5 b(e)42 b(the)g(ge)-5 b(o)g(desic)41 b(line)0 1302 y(p)-5 b(erp)g(endicular)35 b(to)g Fh(L)g Fl(thr)-5 b(ough)35 b Fo(s)1243 1266 y Ff(0)1271 1302 y Fl(.)45 b(L)-5 b(et)36 b Fo(x)f Fl(b)-5 b(e)35 b(a)g(vertex)g(of)g Fo(G)h Fl(which)e(in)h(the)g(emb)-5 b(e)g(dding)35 b(is)f(mapp)-5 b(e)g(d)0 1422 y(into)35 b(a)h(p)-5 b(oint)35 b(of)g Fk(H)733 1386 y Fi(2)820 1422 y Fl(at)g(distanc)-5 b(e)35 b(at)h(most)f Fs(1)h Fl(fr)-5 b(om)35 b Fo(s)p Fl(.)45 b(Then)1297 1568 y Fg(X)1280 1792 y Fj(y)s Ff(2)1395 1775 y Fi(\026)1377 1792 y Ff(L)1432 1772 y Fc(0)1475 1663 y Fk(P)1534 1678 y Fj(p)1582 1663 y Fs(\()p Fo(x)28 b Fh($)f Fo(y)t Fs(\))62 b Fh(\024)i Fo(C)2199 1678 y Fi(3)2244 1663 y Fo(e)2290 1622 y Ff(\000)p Fj(C)2409 1632 y Fe(4)2449 1622 y Fj(L)2508 1663 y Fo(:)0 2065 y Fl(Pr)-5 b(o)g(of.)49 b Fs(Let)28 b Fh(L)555 2029 y Ff(0)555 2089 y Fi(+)649 2065 y Fs(and)h Fh(L)907 2029 y Ff(0)907 2089 y(\000)1001 2065 y Fs(b)s(e)g(the)f(t)m (w)m(o)g(half-lines)g(in)m(to)g(whic)m(h)h Fo(s)2423 2029 y Ff(0)2478 2065 y Fs(partitions)e Fh(L)2996 2029 y Ff(0)3024 2065 y Fs(.)42 b(T)-8 b(ak)m(e)28 b(some)f Fo(l)i(>)g Fs(2.)0 2184 y(Set)42 b Fo(s)227 2199 y Fi(0)314 2184 y Fs(=)h Fo(s)481 2148 y Ff(0)509 2184 y Fs(,)h(and)f(for)f Fo(k)j Fh(2)e(f)p Fs(1)p Fo(;)17 b Fs(2)p Fo(;)g(:::)p Fh(g)39 b Fs(let)j Fo(s)1764 2199 y Fj(k)1855 2184 y Fs(\()p Fl(r)-5 b(esp.)70 b Fo(s)2217 2199 y Ff(\000)p Fj(k)2328 2184 y Fs(\))41 b(b)s(e)i(the)f(p)s(oin)m(t)g(on)h Fh(L)3219 2148 y Ff(0)3219 2209 y Fi(+)3327 2184 y Fs(\()p Fl(r)-5 b(esp.)69 b Fh(L)3710 2148 y Ff(0)3710 2209 y(\000)3777 2184 y Fs(\))0 2304 y(at)43 b(distance)i Fo(k)s(l)h Fs(from)d Fo(s)955 2268 y Ff(0)982 2304 y Fs(.)76 b(F)-8 b(or)44 b Fo(k)k Fh(2)e(f)p Fs(1)p Fo(;)17 b Fs(2)p Fo(;)g(:::)p Fh(g)41 b Fs(let)j Fh(I)2112 2319 y Fj(k)2205 2304 y Fs(\()p Fl(r)-5 b(esp.)75 b Fh(I)2579 2319 y Ff(\000)p Fj(k)2734 2304 y Fs(b)s(e)44 b(the)h(geo)s(desic)f(segmen)m(t)0 2423 y(\(con)m(tained)35 b(in)g Fh(L)678 2387 y Ff(0)706 2423 y Fs(\))f(with)h(endp)s(oin)m(ts)h Fo(s)1511 2438 y Fj(k)r Ff(\000)p Fi(1)1696 2423 y Fs(and)f Fo(s)1938 2438 y Fj(k)2022 2423 y Fs(\()p Fl(r)-5 b(esp.)48 b Fo(s)2362 2438 y Ff(\000)p Fj(k)r Fi(+1)2608 2423 y Fs(and)35 b Fo(s)2850 2438 y Ff(\000)p Fj(k)2961 2423 y Fs(\).)49 b(F)-8 b(or)34 b Fo(j)i Fh(2)31 b Fk(Z)p Fs(,)h(let)i Fh(L)3774 2438 y Fj(j)0 2543 y Fs(b)s(e)g(the)f(geo)s(desic)h(line)f(p) s(erp)s(endicular)i(to)d Fh(L)1708 2507 y Ff(0)1769 2543 y Fs(through)i Fo(s)2193 2558 y Fj(j)2235 2543 y Fs(.)44 b(Then)993 2688 y Fg(X)976 2911 y Fj(y)s Ff(2)1091 2894 y Fi(\026)1073 2911 y Ff(L)1128 2891 y Fc(0)1171 2782 y Fk(P)1230 2797 y Fj(p)1278 2782 y Fs(\()p Fo(x)27 b Fh($)h Fo(y)t Fs(\))60 b Fh(\024)1897 2688 y Fg(X)1819 2906 y Fj(j)t Ff(2)p Fa(Z)-9 b Ff(nf)p Fi(0)p Ff(g)2186 2688 y Fg(X)2168 2911 y Fj(y)s Ff(2)2272 2894 y Fi(\026)2265 2911 y Ff(I)2310 2921 y Fd(j)2364 2782 y Fk(P)2423 2797 y Fj(p)2471 2782 y Fs(\()p Fo(x)27 b Fh($)g Fo(y)t Fs(\))p Fo(:)0 3084 y Fs(Let)35 b Fo(D)i Fs(b)s(e)e(the)g(degree)g(of)g Fo(G)p Fs(.)48 b(It)34 b(is)h(easy)f(to)g(see)h(that)f(for)h(some)f (small)f Fo(\017)d(>)g Fs(0)k(an)m(y)h(ball)f(of)h(radius)0 3204 y Fo(\017)e Fs(in)g Fk(H)261 3168 y Fi(2)344 3204 y Fs(can)g(in)m(tersect)h(at)f(most)e Fo(D)36 b Fs(edges)e(of)f(the)h (em)m(b)s(edding)f(of)g Fo(G)g Fs(in)g Fk(H)2877 3168 y Fi(2)2927 3204 y Fs(.)44 b(Therefore)34 b(it)f(is)f(also)0 3323 y(easy)k(to)f(see)i(that)e(an)m(y)h(geo)s(desic)g(line)g(segmen)m (t)f(of)h(length)g Fo(d)g Fs(can)g(in)m(tersect)h(at)e(most)f Fo(dD)s(=\017)i Fs(suc)m(h)0 3443 y(edges.)45 b(Therefore,)34 b(from)f(the)g(previous)h(displa)m(y)f(w)m(e)h(obtain,)e(for)i (arbitrary)e Fo(J)10 b Fs(,)574 3621 y Fg(X)556 3845 y Fj(y)s Ff(2)671 3828 y Fi(\026)653 3845 y Ff(L)708 3825 y Fc(0)752 3716 y Fk(P)811 3731 y Fj(p)859 3716 y Fs(\()p Fo(x)27 b Fh($)g Fo(y)t Fs(\))60 b Fh(\024)1412 3649 y Fo(l)r(D)p 1412 3693 117 4 v 1450 3784 a(\017)1614 3621 y Fg(X)1557 3840 y Fj(j)t Fi(:)p Ff(j)p Fj(j)t Ff(j)p Fj(>J)1831 3716 y Fk(P)1890 3731 y Fj(p)1938 3716 y Fs(\()p Fo(x)27 b Fh($)2197 3691 y Fs(\026)2188 3716 y Fh(I)2242 3731 y Fj(j)2284 3716 y Fs(\))56 b(+)2523 3649 y(2)p Fo(l)r(J)10 b(D)p 2523 3693 232 4 v 2619 3784 a(\017)2767 3716 y Fk(P)2826 3731 y Fj(p)2874 3716 y Fs(\()p Fo(x)27 b Fh($)3148 3691 y Fs(\026)3124 3716 y Fh(L)p Fs(\))p Fo(:)0 4068 y Fs(When)f Fo(j)34 b(>)28 b Fs(1)d(\()p Fl(r)-5 b(esp.)40 b Fo(j)33 b(<)28 b Fs(1\))d(an)m(y)g(path)g(from)f Fo(x)h Fs(to)1950 4042 y(\026)1941 4068 y Fh(I)1995 4083 y Fj(j)2063 4068 y Fs(has)g(to)f(cross)i(the)f(lines)g Fh(L)3033 4083 y Fj(i)3066 4068 y Fs(,)i Fo(i)h Fs(=)g(1)p Fo(;)17 b Fs(2)p Fo(;)g(:::;)g(j)11 b Fh(\000)6 b Fs(1,)0 4187 y(\()p Fl(r)-5 b(esp.)43 b Fo(i)28 b Fs(=)g Fh(\000)p Fs(1)p Fo(;)17 b Fh(\000)p Fs(2)p Fo(;)g(:::;)g Fh(\000)p Fo(j)26 b Fs(+)d(1\).)43 b(Hence,)34 b(Lemma)d(3.1)i(implies)1365 4418 y Fo(P)1429 4433 y Fj(p)1475 4418 y Fs(\()p Fo(x)28 b Fh($)1735 4393 y Fs(\026)1726 4418 y Fh(I)1780 4433 y Fj(j)1822 4418 y Fs(\))61 b Fh(\024)g Fo(C)2131 4433 y Fi(5)2176 4418 y Fo(e)2222 4377 y Ff(\000)p Fj(C)2341 4387 y Fe(6)2381 4377 y Fj(j)2423 4418 y Fo(;)0 4645 y Fs(for)33 b(some)g Fo(C)473 4660 y Fi(5)518 4645 y Fo(;)17 b(C)634 4660 y Fi(6)706 4645 y Fh(2)28 b Fs(\(0)p Fo(;)17 b Fh(1)p Fs(\).)42 b(Therefore,)34 b(using)g(Lemma)e(3.2)g(and) i(taking)e Fo(J)38 b Fs(=)28 b Fh(b)p Fo(L)p Fh(c)p Fs(,)33 b(w)m(e)h(obtain)611 4797 y Fg(X)594 5021 y Fj(y)s Ff(2)709 5003 y Fi(\026)691 5021 y Ff(L)746 5001 y Fc(0)789 4892 y Fk(P)848 4907 y Fj(p)896 4892 y Fs(\()p Fo(x)27 b Fh($)h Fo(y)t Fs(\))60 b Fh(\024)h Fo(C)1508 4907 y Fi(7)1553 4892 y Fo(e)1599 4850 y Ff(\000)p Fj(C)1718 4860 y Fe(6)1758 4850 y Ff(b)p Fj(L)p Ff(c)1946 4892 y Fs(+)56 b Fo(C)2150 4907 y Fi(8)2194 4892 y Fh(b)p Fo(L)p Fh(c)p Fo(e)2396 4850 y Ff(\000)p Fj(C)2515 4860 y Fe(2)2555 4850 y Fj(L)2676 4892 y Fh(\024)61 b Fo(C)2885 4907 y Fi(3)2930 4892 y Fo(e)2976 4850 y Ff(\000)p Fj(C)3095 4860 y Fe(4)3135 4850 y Fj(L)3195 4892 y Fo(:)p 3756 5211 60 50 v 0 5398 a Fl(Pr)-5 b(o)g(of)30 b(of)g(The)-5 b(or)g(em)32 b(1.1\(i\).)49 b Fs(W)-8 b(e)28 b(will)f(c)m(hec)m(k)j(that)e(the)h(h)m(yp)s(othesis)g (of)f(Lemma)f(2.3.A)g(and)i(Lemma)0 5517 y(2.3.B)j(are)h(satis\014ed)h (\(note)f(that)g(the)h(former)e(are)i(con)m(tained)g(in)f(the)h (latter\).)p eop %%Page: 11 11 11 10 bop 1325 50 a Fq(MEAN-FIELD)37 b(CRITICALITY)1243 b(11)120 289 y Fs(Supp)s(ose)37 b(that)f Fo(G)g Fs(is)g(nicely)g(em)m (b)s(edded)h(in)f Fk(H)1901 253 y Fi(2)1951 289 y Fs(.)53 b(Let)36 b Fh(L)g Fs(b)s(e)g(a)g(geo)s(desic)h(line)f(and)g Fo(s)3370 304 y Fi(1)3415 289 y Fo(;)17 b(:::;)g(s)3636 304 y Fi(7)3714 289 y Fs(b)s(e)0 408 y(distinct)44 b(p)s(oin)m(ts)g(on) g Fh(L)p Fs(,)h(suc)m(h)g(that)f(for)g Fo(i)h Fs(=)h(1)p Fo(;)17 b(:::;)g Fs(6,)43 b(the)h(distance)g(b)s(et)m(w)m(een)h Fo(s)3166 423 y Fj(i)3243 408 y Fs(and)f Fo(s)3494 423 y Fj(i)p Fi(+1)3672 408 y Fs(has)0 528 y(the)c(same)f(common)e(v)-6 b(alue)40 b Fo(L)p Fs(.)63 b(F)-8 b(or)39 b(eac)m(h)h Fo(i)p Fs(,)h(let)e Fh(L)1998 543 y Fj(i)2070 528 y Fs(b)s(e)h(the)g (geo)s(desic)g(line)g(p)s(erp)s(endicular)g(to)f Fh(L)0 648 y Fs(through)c Fo(r)423 663 y Fj(i)456 648 y Fs(.)47 b(The)35 b(remo)m(v)-6 b(al)33 b(of)i Fh(L)1293 663 y Fi(2)1360 648 y Fh([)24 b(L)1519 663 y Fi(3)1586 648 y Fh([)f(L)1744 663 y Fi(5)1811 648 y Fh([)h(L)1970 663 y Fi(6)2049 648 y Fs(breaks)34 b Fk(H)2435 611 y Fi(2)2520 648 y Fs(in)m(to)g(5)g(connected)i(comp)s(onen)m(ts.)0 767 y(F)-8 b(or)33 b Fo(i)28 b Fs(=)g(1)p Fo(;)17 b Fs(4)p Fo(;)g Fs(7,)31 b(let)i Fh(V)850 782 y Fj(i)916 767 y Fs(b)s(e)h(the)f(connected)i(comp)s(onen)m(t)e(whic)m(h)h(con)m(tains)g Fo(s)2920 782 y Fj(i)2953 767 y Fs(.)120 887 y(Set)42 b Fo(V)358 902 y Fi(1)445 887 y Fs(=)577 861 y(\026)565 887 y Fh(V)626 902 y Fi(1)670 887 y Fs(,)i Fo(V)800 902 y Fi(2)887 887 y Fs(=)1019 861 y(\026)1007 887 y Fh(V)1068 902 y Fi(4)1112 887 y Fs(,)g Fo(V)1242 902 y Fi(3)1329 887 y Fs(=)1461 861 y(\026)1449 887 y Fh(V)1510 902 y Fi(7)1555 887 y Fs(.)70 b(Let)42 b Fo(x)1897 902 y Fi(1)1942 887 y Fs(,)h Fo(x)2070 902 y Fi(2)2157 887 y Fs(and)f Fo(x)2416 902 y Fi(3)2503 887 y Fs(b)s(e)g(v)m(ertices)h(of)f Fo(G)g Fs(whic)m(h)h(in)f(the)0 1006 y(em)m(b)s(edding)47 b(are)f(mapp)s(ed)g(in)m(to)h(p)s(oin)m(ts)f(of)h Fk(H)1814 970 y Fi(2)1911 1006 y Fs(at)f(distance)h(at)f(most)f(1)h(from)g Fo(s)3231 1021 y Fi(1)3275 1006 y Fs(,)j Fo(s)3399 1021 y Fi(4)3490 1006 y Fs(and)e Fo(s)3744 1021 y Fi(7)3788 1006 y Fs(,)0 1126 y(resp)s(ectiv)m(ely)-8 b(.)49 b(With)35 b(these)h(c)m(hoices,)g(the)f(h)m(yp)s(othesis)h(of)f(Lemma)e(2.3.B)h (are)h(satis\014ed,)g(pro)m(vided)0 1245 y(that)c Fo(L)h Fs(is)g(large)f(enough,)i(as)f(can)g(b)s(e)g(seen)h(from)e(Lemma)g (3.2,)g(Lemma)f(3.3)h(and)h(Theorem)g(1.1)f(of)0 1365 y([BS2],)h(whic)m(h)i(states)f(that)g Fo(p)1126 1380 y Fj(c)1195 1365 y Fo(<)28 b(p)1350 1380 y Fj(u)1402 1365 y Fs(.)p 3756 1355 60 50 v 668 1579 a Ft(4.)55 b(The)36 b(case)g(of)i(graphs)e(with)h(infinitel)-7 b(y)37 b(many)h(ends)120 1758 y Fs(W)-8 b(e)31 b(will)e(need)j(some)e(notation)g(and)h (terminology)e(related)i(to)f(the)h(binary)f(homogeneous)h(tree,)0 1878 y Fk(T)63 1893 y Fi(2)111 1878 y Fs(,)j(i.e.,)f(the)h(tree)g(in)g (whic)m(h)g(ev)m(ery)g(v)m(ertex)g(has)g(degree)h(3.)45 b(The)35 b(set)f(of)g(v)m(ertices)g(of)g(this)g(tree)g(will)0 1997 y(b)s(e)h(denoted)g(b)m(y)g Fo(V)22 b Fs(\()p Fk(T)836 2012 y Fi(2)884 1997 y Fs(\).)47 b(Giv)m(en)35 b Fo(i;)17 b(j;)g(k)32 b Fh(2)e Fo(V)22 b Fs(\()p Fk(T)1815 2012 y Fi(2)1863 1997 y Fs(\))34 b(w)m(e)h(will)e(sa)m(y)i(that)f Fo(k)j Fs(is)d(b)s(et)m(w)m(een)i Fo(i)f Fs(and)g Fo(j)40 b Fs(if)35 b(the)0 2117 y(shortest)f(path)f(from)g Fo(i)g Fs(to)g Fo(j)39 b Fs(passes)34 b(through)g Fo(k)s Fs(.)120 2236 y(The)29 b(follo)m(wing)g(prop)s(osition)g(will)g(b)s(e)g(used)i (in)e(this)h(section;)g(it)f(can)h(b)s(e)g(easily)e(pro)m(v)m(ed)j (with)e(the)0 2356 y(argumen)m(ts)39 b(in)h(the)g(pro)s(of)f(of)h(Prop) s(ositions)f(6.1)g(in)g([Moh2].)63 b(\(Compare)38 b(with)h(Prop)s (osition)g(2.1)0 2476 y(in)34 b([Sc)m(h].\))45 b(Belo)m(w)33 b Fo(B)5 b Fs(\()p Fo(u;)17 b(n)p Fs(\))31 b(will)i(denote)h(the)g (ball)f(of)h(radius)g Fo(n)f Fs(cen)m(tered)j(at)d Fo(u)28 b Fh(2)g Fo(V)56 b Fs(in)34 b(the)g(graph)0 2595 y Fo(G)28 b Fs(=)g(\()p Fo(V)6 b(;)17 b(E)6 b Fs(\).)0 2762 y Fu(Prop)s(osition) 37 b(4.1.)58 b Fl(Supp)-5 b(ose)34 b(that)g Fo(G)28 b Fs(=)g(\()p Fo(V)6 b(;)17 b(E)6 b Fs(\))33 b Fl(is)g(an)i(in\014nite,)e (lo)-5 b(c)g(al)5 b(ly)36 b(\014nite,)e(c)-5 b(onne)g(cte)g(d)33 b(tr)-5 b(an-)0 2881 y(sitive)30 b(gr)-5 b(aph.)43 b(If)31 b Fo(G)g Fl(has)g(in\014nitely)g(many)h(ends,)f(then)f(ther)-5 b(e)31 b(is)f(a)h(p)-5 b(ositive)30 b(inte)-5 b(ger)30 b Fo(n)h Fl(and)f(vertic)-5 b(es)0 3001 y Fo(u)57 3016 y Fj(k)138 3001 y Fh(2)32 b Fo(V)22 b Fl(,)38 b Fo(k)c Fh(2)e Fo(V)23 b Fs(\()p Fk(T)752 3016 y Fi(2)800 3001 y Fs(\))37 b Fl(such)h(that)g(the)f(b)-5 b(al)5 b(ls)39 b Fo(B)5 b Fs(\()p Fo(u)1882 3016 y Fj(k)1930 3001 y Fo(;)17 b(n)p Fs(\))p Fl(,)37 b Fo(k)d Fh(2)e Fo(Z)45 b Fl(ar)38 b(disjoint)e(and)i(have)f(the)h(fol)5 b(low-)0 3121 y(ing)39 b(pr)-5 b(op)g(erty.)59 b(F)-8 b(or)40 b(e)-5 b(ach)39 b Fo(i;)17 b(j)41 b Fh(2)36 b Fo(V)22 b Fs(\()p 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Fs(,)g Fo(x)3771 4020 y Fi(2)0 4124 y Fs(and)k Fo(x)251 4139 y Fi(3)295 4124 y Fs(.)44 b(Call)33 b(them,)f(resp)s(ectiv)m(ely) -8 b(,)33 b Fo(V)1481 4139 y Fi(1)1525 4124 y Fs(,)g Fo(V)1644 4139 y Fi(2)1722 4124 y Fs(and)h Fo(V)1974 4139 y Fi(3)2019 4124 y Fs(.)120 4244 y(Since)44 b Fo(G)f Fs(has)h(in\014nitely)f(man)m(y)f(ends,)47 b(it)c(is)g(non-amenable)g (and)h(hence,)j(b)m(y)d(Theorem)f(2)g(of)0 4363 y([BS1])32 b(\(adapted)i(to)f(b)s(ond)h(p)s(ercolation\),)e(it)g(has)i Fo(p)1970 4378 y Fj(c)2038 4363 y Fo(<)29 b Fs(1.)120 4483 y(T)-8 b(o)36 b(v)m(erify)h(the)g(h)m(yp)s(othesis)h(of)f(Lemma)e (2.3.B,)g(let)i Fo(K)44 b Fs(b)s(e)37 b(the)g(n)m(um)m(b)s(er)h(of)f (edges)h(of)f Fo(G)g Fs(whic)m(h)0 4602 y(ha)m(v)m(e)d(at)e(least)h (one)h(endp)s(oin)m(t)g(in)f Fo(B)5 b Fs(\()p Fo(u)1475 4617 y Fi(0)1520 4602 y Fo(;)17 b(n)p Fs(\),)31 b(and)j(note)f(that,)g (for)g(1)28 b Fh(\024)g Fo(i)g(<)g(j)34 b Fh(\024)28 b Fs(3,)1092 4789 y Fk(P)1151 4804 y Fj(p)1192 4814 y Fd(c)1236 4789 y Fs(\()p Fo(V)1333 4804 y Fj(i)1394 4789 y Fh($)f Fo(V)1579 4804 y Fj(j)1621 4789 y Fs(\))61 b Fh(\024)g Fs(1)22 b Fh(\000)h Fs(\(1)f Fh(\000)g Fo(p)2291 4804 y Fj(c)2332 4789 y Fs(\))2371 4748 y Fj(K)2508 4789 y Fo(<)62 b Fs(1)p Fo(;)0 4972 y Fs(and,)33 b(for)h Fo(i)28 b Fs(=)g(1)p Fo(;)17 b Fs(2)p Fo(;)g Fs(3,)409 5074 y Fg(X)341 5289 y Fj(v)s Ff(2)p Fj(@)481 5299 y Fe(in)542 5289 y Fj(V)589 5299 y Fd(i)637 5169 y Fk(P)696 5184 y Fj(p)737 5194 y Fd(c)781 5169 y Fs(\()p Fo(x)877 5184 y Fj(i)938 5169 y Fh($)27 b Fo(v)t Fs(\))61 b Fh(\024)g(j)p Fo(@)1436 5184 y Fi(in)1507 5169 y Fo(V)1565 5184 y Fj(i)1598 5169 y Fh(j)33 b Fk(P)1718 5184 y Fj(p)1759 5194 y Fd(c)1803 5169 y Fs(\()p Fo(x)1899 5184 y Fj(i)1960 5169 y Fh($)27 b Fo(@)2140 5184 y Fi(in)2212 5169 y Fo(V)2270 5184 y Fj(i)2303 5169 y Fs(\))61 b Fh(\024)g Fo(K)23 b Fs(\(1)f Fh(\000)h Fs(\(1)e Fh(\000)i Fo(p)3120 5184 y Fj(c)3161 5169 y Fs(\))3200 5128 y Fj(K)3276 5169 y Fs(\))3315 5128 y Fj(l)p Ff(\000)p Fi(1)3448 5169 y Fo(:)0 5451 y Fs(The)34 b(last)f(expression)h(can)f(b)s(e)h(made)f(arbitrarily)e (small)h(b)m(y)h(taking)f Fo(l)j Fs(su\016cien)m(tly)f(large.)p 3756 5441 V eop %%Page: 12 12 12 11 bop 0 50 a Fq(12)1247 b(R)n(OBER)-7 b(TO)36 b(H.)h(SCHONMANN)1627 289 y Ft(references)0 493 y Fs([AB])209 b(M.)35 b(Aizenman)g(and)g(D.)g (Barsky,)f Fl(Sharpness)i(of)h(the)g(phase)f(tr)-5 b(ansition)36 b(in)h(p)-5 b(er)g(c)g(olation)411 612 y(mo)g(dels)p Fs(,)34 b(Comm)m(unications)d(in)i(Mathematical)f(Ph)m(ysics)i Fu(108)g Fs(\(1987\),)d(489{526.)0 732 y([AN])205 b(M.)30 b(Aizenman)g(and)g(C.)g(M.)g(Newman,)f Fl(T)-8 b(r)j(e)g(e)33 b(gr)-5 b(aph)32 b(ine)-5 b(qualities)32 b(and)g(critic)-5 b(al)32 b(b)-5 b(ehavior)411 851 y(in)36 b(p)-5 b(er)g(c)g(olation)34 b(mo)-5 b(dels)p Fs(,)34 b(Journal)f(of)g(Statistical)f(Ph)m(ysics)i Fu(16)g Fs(\(1983\),)e(811{828.)0 971 y([BA])209 b(D.)30 b(J.)g(Barsky)e(and)i(M.)g(Aizenman,)g Fl(Per)-5 b(c)g(olation)31 b(critic)-5 b(al)32 b(exp)-5 b(onents)32 b(under)g(the)h(trian-)411 1091 y(gle)e(c)-5 b(ondition)p Fs(,)27 b(Comm)m(unications)g(in)h (Mathematical)e(Ph)m(ysics)j Fu(19)h Fs(\(1991\),)d(1520{1536.)0 1210 y([BLPS1])49 b(I.)28 b(Benjamini,)f(R.)g(Ly)m(ons,)i(Y.)e(P)m (eres)i(and)f(O.)f(Sc)m(hramm,)h Fl(Gr)-5 b(oup-invariant)29 b(p)-5 b(er)g(c)g(olation)411 1330 y(on)36 b(gr)-5 b(aphs)p Fs(,)32 b(Geometric)h(and)g(F)-8 b(unctional)33 b(Analysis)g Fu(9)g Fs(\(1999\),)f(29{66.)0 1449 y([BLPS2])49 b(I.)32 b(Benjamini,)g(R.)g(Ly)m(ons,)h(Y.)e(P)m(eres)i(and)g(O.)f(Sc)m(hramm,) g Fl(Critic)-5 b(al)34 b(p)-5 b(er)g(c)g(olation)34 b(on)h(any)411 1569 y(non-amenable)50 b(gr)-5 b(oup)49 b(has)g(no)h(in\014nite)e (clusters)p Fs(,)k(The)d(Annals)f(of)h(Probabilit)m(y)h Fu(27)411 1688 y Fs(\(1999\),)32 b(1347-1356.)0 1808 y([BS1])179 b(I.)38 b(Benjamini)e(and)j(O.)e(Sc)m(hramm,)h Fl(Per)-5 b(c)g(olation)39 b(b)-5 b(eyond)39 b Fk(Z)2739 1772 y Fj(d)2783 1808 y Fl(,)h(many)h(questions)e(and)g(a)411 1927 y(few)d(answers)p Fs(,)c(Electronic)h(Comm)m(unications)f(in)h (Probabilit)m(y)f Fu(1)h Fs(\(1996\),)f(71{82.)0 2047 y([BS2])179 b(I.)35 b(Benjamini)g(and)h(O.)f(Sc)m(hramm,)g Fl(Per)-5 b(c)g(olation)37 b(in)g(the)g(hyp)-5 b(erb)g(olic)37 b(plane)p Fs(,)f(Journal)g(of)411 2166 y(the)e(American)e(Mathematical) g(So)s(ciet)m(y)h Fu(14)h Fs(\(2000\),)d(487{507.)0 2286 y([Dur])185 b(R.)29 b(Durrett,)g Fl(Pr)-5 b(ob)g(ability:)42 b(The)-5 b(ory)32 b(and)f(Examples)p Fs(,)e(Second)h(edition,)f (Duxbury)g(Press,)411 2406 y(1996.)0 2525 y([Gri])210 b(G.)32 b(R.)g(Grimmett,)e Fl(Per)-5 b(c)g(olation)p Fs(,)30 b(Springer-V)-8 b(erlag,)32 b(New)g(Y)-8 b(ork-Berlin)31 b(\(2nd)h(edition\),)411 2645 y(1999.)0 2764 y([Lal1])165 b(S.)23 b(P)-8 b(.)21 b(Lalley)-8 b(,)23 b Fl(Per)-5 b(c)g(olation)24 b(on)i(F)-8 b(uchsian)25 b(gr)-5 b(oups)p Fs(,)24 b(Annales)e(de)h(L'Institut)f(Henri)g(P)m(oincar)m(\023)-47 b(e)411 2884 y(\(Probabilit)m(y)32 b(and)i(Statistics\))e Fu(34)i Fs(\(1998\),)e(151{177.)0 3003 y([Lal2])165 b(S.)23 b(P)-8 b(.)22 b(Lalley)-8 b(,)24 b Fl(Per)-5 b(c)g(olation)25 b(clusters)h(in)g(hyp)-5 b(erb)g(olic)25 b(tesselations)p Fs(,)f(Geometric)e(and)h(F)-8 b(unc-)411 3123 y(tional)33 b(Analysis)f(\(to)h(app)s(ear\).)0 3242 y([Ly)m(o])193 b(R.)35 b(Ly)m(ons,)h Fl(Phase)g(tr)-5 b(ansition)36 b(on)h(non-amenable)g(gr)-5 b(aphs)p Fs(,)34 b(Journal)h(of)g (Mathematical)411 3362 y(Ph)m(ysics)f Fu(41)g Fs(\(2000\),)e(1099)g({)h (1126.)0 3482 y([Moh])159 b(B.)31 b(Mohar,)g Fl(Some)j(r)-5 b(elations)34 b(b)-5 b(etwe)g(en)33 b(analytic)h(and)f(ge)-5 b(ometric)33 b(pr)-5 b(op)g(erties)33 b(of)g(in\014nite)411 3601 y(gr)-5 b(aphs)p Fs(,)33 b(Discrete)g(Mathematics)g Fu(95)h Fs(\(1991\),)d(193{219.)0 3721 y([New])164 b(C.)35 b(M.)g(Newman,)f Fl(A)m(nother)j(critic)-5 b(al)36 b(exp)-5 b(onent)37 b(ine)-5 b(quality)37 b(for)g(p)-5 b(er)g(c)g(olation:)47 b Fo(\014)36 b Fh(\025)31 b Fs(2)p Fo(=\016)t Fs(,)411 3840 y(Journal)j(of)f(Statistical)f(Ph)m(ysics)i Fu(47)g Fs(\(1987\),)d(695{699.)0 3960 y([Ngu])175 b(B.)51 b(Nguy)m(en,)56 b Fl(Gap)c(exp)-5 b(onent)52 b(for)g(p)-5 b(er)g(c)g(olation)52 b(pr)-5 b(o)g(c)g(esses)50 b(with)j(triangle)f(c)-5 b(ondition)p Fs(,)411 4079 y(Journal)34 b(of)f(Statistical)f(Ph)m(ysics)i Fu(49)g Fs(\(1987\),)d(235{243.)0 4199 y([Sc)m(h])204 b(R.)42 b(H.)e(Sc)m(honmann,)k Fl(Multiplicity)g(of)e(phase)g(tr)-5 b(ansitions)42 b(and)h(me)-5 b(an-\014eld)43 b(critic)-5 b(ality)411 4318 y(on)37 b(highly)f(non-amenable)g(gr)-5 b(aphs)p Fs(,)33 b(Comm)m(unications)f(in)i(Mathematical)e(Ph)m(ysics)j (\(to)411 4438 y(app)s(ear\).)0 4558 y([W)-8 b(u])206 b(C.)51 b(C.)f(W)-8 b(u,)55 b Fl(Critic)-5 b(al)51 b(b)-5 b(ehavior)51 b(of)g(p)-5 b(er)g(c)g(olation)51 b(and)g(Markov)h (\014elds)g(on)f(br)-5 b(anching)411 4677 y(planes)p Fs(,)33 b(Journal)h(of)f(Applied)h(Probabilit)m(y)e Fu(30)h Fs(\(1993\),)f(538{547.)120 4846 y(Mathematics)g(Departmen)m(t)120 4965 y(Univ)m(ersit)m(y)g(of)i(California)e(at)h(Los)g(Angeles)120 5085 y(Los)g(Angeles,)g(CA)g(90095)120 5204 y(U.S.A.)120 5373 y(rhs@math.ucla.edu)120 5492 y(www.math.ucla.edu/)8 b(~)f(rhs)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0107251329426--