Content-Type: multipart/mixed; boundary="-------------9907122044351" This is a multi-part message in MIME format. ---------------9907122044351 Content-Type: text/plain; name="99-267.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-267.comments" lchayes@math.ucla.edu rhs@math.ucla.edu http://www.math.ucla.edu/\~{ }rhs/ ---------------9907122044351 Content-Type: text/plain; name="99-267.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-267.keywords" Percolation, critical points, critical line, diluted potts models, differential inequalities, crossover exponent ---------------9907122044351 Content-Type: application/postscript; name="cs.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="cs.ps" %!PS-Adobe-2.0 %%Creator: dvips 5.58 Copyright 1986, 1994 Radical Eye Software %%Title: cs.dvi %%CreationDate: Mon Jul 12 18:05:42 1999 %%Pages: 18 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 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(of)g(maximal)f(de)n(gr)n(e)n(e)g Fk(D)i Ff(for)g(which)590 1664 y Fk(s)611 1673 y Fi(c)641 1664 y Fj(\025)c Fo(1)c Fj(\000)i Fo(2)790 1645 y Fh(b)p Fi(D)q(=)p Fg(2)p Fh(c)p Fg(+1)933 1664 y Fo(\(1)e Fj(\000)h Fk(b)1047 1673 y Fi(c)1064 1664 y Fo(\))1082 1645 y Fh(b)p Fi(D)q(=)p Fg(2)p Fh(c)1176 1664 y Fk(:)-48 1800 y Fo(Note)i(that)f(an)i (in\014nite)h(graph)d(of)h(maximal)h(degree)f(at)g(most)f(2)h(has)h Fk(b)1155 1809 y Fi(c)1184 1800 y Fo(=)g Fk(s)1253 1809 y Fi(c)1283 1800 y Fo(=)f(1,)h(so)f(that)g(the)g(assumption)-48 1874 y(that)g Fk(D)i Fj(\025)f Fo(3)g(is)g(harmless.)19 b(The)13 b(optimal)f(asp)q(ect)h(of)g(Theorem)f(1.1)g(is)h(the)g(iden)o (ti\014cation)h(of)e(the)h(exp)q(onen)o(t)-48 1947 y Fj(b)p Fk(D)q(=)p Fo(2)p Fj(c)p Fo(,)21 b(whic)o(h)f(is)h(the)f(same)f (in)i(parts)e(\(a\))g(and)h(\(b\).)34 b(The)20 b(v)m(alue)g(of)g(the)g (constan)o(t)e(whic)o(h)j(m)o(ultiplies)-48 2020 y(\(1)7 b Fj(\000)h Fk(b)62 2029 y Fi(c)79 2020 y Fo(\))97 2003 y Fh(b)p Fi(D)q(=)p Fg(2)p Fh(c)205 2020 y Fo(in)15 b(parts)d(\(a\))h (and)h(\(b\))f(can)h(b)q(e)g(sligh)o(tly)h(impro)o(v)o(ed,)d(using)j (our)e(tec)o(hniques)h(with)g(somewhat)-48 2094 y(more)e(w)o(ork.)18 b(But)12 b(since)i(these)f(tec)o(hniques)g(do)g(not)g(seem)g(to)f(b)q (e)i(su\016cien)o(t)f(to)f(iden)o(tify)i(the)e(optimal)h(v)m(alue)-48 2167 y(of)h(this)i(constan)o(t,)d(w)o(e)h(satis\014ed)h(ourselv)o(es)f (with)h(the)g(statemen)o(ts)e(ab)q(o)o(v)o(e.)-48 2264 y Fl(Question)23 b(1.)33 b Fo(Ho)o(w)19 b(w)o(ould)g(Theorem)f(1.1)g(b) q(e)i(mo)q(di\014ed)g(if)g(w)o(e)f(restricted)g(ourselv)o(es)f(to)h (the)g(class)h(of)-48 2338 y(transitiv)o(e)14 b(graphs)g(of)h(degree)f Fk(D)q Fo(?)-48 2480 y Fl(1.2)i(Crosso)o(v)o(er)h(exp)q(onen)o(t)e(for) h(diluted)g(P)o(otts)g(mo)q(dels.)k Fo(W)l(e)15 b(supp)q(ose)e(that)g (the)g(reader)g(is)h(familiar)-48 2554 y(with)20 b(the)h(basic)f (notions)g(and)h(terminology)e(on)h(the)h(statistical)e(mec)o(hanics)i (of)f(lattice)g(systems.)35 b(F)l(or)878 2678 y(3)p eop %%Page: 4 4 4 3 bop -48 199 a Fo(an)19 b(in)o(tro)q(duction)f(to)g(this)i(sub)s (ject,)e(consult,)i(e.g.,)e([7)o(].)31 b(The)19 b Fk(q)r Fo(-P)o(otts)e(mo)q(del)i(on)g(a)g(lo)q(cally)h(\014nite)f(\(not)-48 273 y(necessarily)13 b(in\014nite)g(or)f(connected\))g(graph)g Fk(G)g Fo(=)g(\()p Fk(V)s(;)c(E)s Fo(\))j(is)i(de\014ned)g(b)o(y)f (asso)q(ciating)g(to)g(eac)o(h)g(site)h Fk(x)f Fj(2)h Fk(V)22 b Fo(a)-48 346 y(random)10 b(v)m(ariable)j(\(spin\))e Fk(\033)r Fo(\()p Fk(x)p Fo(\))f(whic)o(h)i(tak)o(es)e(v)m(alues)j(1)p Fk(;)8 b Fo(2)p Fk(;)g(:)o(::)o(;)g(q)q Fo(.)16 b(T)l(o)11 b(eac)o(h)g(con\014guration)g Fk(\033)j Fj(2)e(f)p Fo(1)p Fk(;)c Fo(2)p Fk(;)g(::)o(:;)f(q)r Fj(g)1801 329 y Fi(V)-48 419 y Fo(an)15 b(energy)f(is)i(formally)e(asso)q(ciated)g(b)o(y)h(the)g (expression)607 525 y Fk(H)644 533 y Fi(J)667 525 y Fo(\()p Fk(\033)r Fo(\))c(=)i Fj(\000)p Fk(J)897 485 y Fe(X)862 576 y Fh(f)p Fi(x;y)q Fh(g2)p Fi(E)999 525 y Fk(\016)1019 533 y Fi(\033)q Fg(\()p Fi(x)p Fg(\))p Fi(;\033)q Fg(\()p Fi(y)q Fg(\))1160 525 y Fk(;)-48 655 y Fo(where)i Fk(J)k Fj(\025)14 b Fo(0)i(is)g(a)g(parameter)d(and)j Fk(\016)625 663 y Fi(a;b)684 655 y Fo(=)e(1)i(if)g Fk(a)e Fo(=)g Fk(b)p Fo(,)i Fk(\016)969 663 y Fi(a;b)1028 655 y Fo(=)e(0)h(if)i Fk(a)d Fj(6)p Fo(=)g Fk(b)p Fo(.)22 b(There)16 b(is)g(a)g(critical)g(v) m(alue)h Fk(J)1810 663 y Fi(c)-48 728 y Fo(\(p)q(ossibly)e(0)g(or)g Fj(1)p Fo(\))f(suc)o(h)h(that)f(if)i Fk(J)h(<)c(J)671 737 y Fi(c)703 728 y Fo(the)i(mo)q(del)g(has)g(a)g(unique)h(Gibbs)g (distribution,)f(but)g(if)g Fk(J)i(>)c(J)1810 737 y Fi(c)-48 802 y Fo(the)f(mo)q(del)h(has)f(more)f(than)h(one)g(Gibbs)i (distribution.)19 b(\(If)12 b Fk(G)g Fo(is)h(not)f(connected,)g(a)g (Gibbs)i(distribution)e(on)-48 875 y(it)k(is)h(obtained)f(b)o(y)f(asso) q(ciating)h(an)g(indep)q(enden)o(t)h(Gibbs)g(distribution)f(to)g(eac)o (h)g(connected)f(comp)q(onen)o(t,)-48 949 y(so)20 b(that)g(the)g(v)m (alue)i(of)e Fk(J)405 957 y Fi(c)442 949 y Fo(is)i(then)e(the)h (in\014m)o(um)f(of)g(those)g(of)g(its)h(connected)g(comp)q(onen)o (ts.\))35 b(F)l(or)20 b(an)-48 1022 y(in)o(tro)q(duction)c(to)h(the)f (P)o(otts)f(mo)q(del)i(and)g(the)g(asso)q(ciated)f(random-cluster)f(mo) q(dels,)i(see,)g(e.g.,)f([8)o(].)25 b(It)17 b(is)-48 1095 y(of)d(relev)m(ance)i(to)e(recall)i(that)e(the)h(case)f Fk(q)h Fo(=)d(2)j(corresp)q(onds)f(to)g(the)h(Ising)h(mo)q(del.)23 1191 y(Giv)o(en)k(an)f(in\014nite,)i(lo)q(cally)g(\014nite,)g (connected)e(graph)f Fk(G)i Fo(=)g(\()p Fk(V)s(;)8 b(E)s Fo(\))17 b(one)j(can)f(consider)g(the)h(b)q(ond-)-48 1265 y(diluted)d(and)e(the)g(site-diluted)i Fk(q)r Fo(-P)o(otts)c(mo)q (dels)j(on)f(this)g(graph.)20 b(In)c(eac)o(h)f(case)g(one)g(studies)h (the)f Fk(q)r Fo(-P)o(otts)-48 1338 y(mo)q(del)k(on)f(the)g(random)f (graph)g(resulting)h(from)f(p)q(erforming)h(b)q(ond)g(or)g(site)g(p)q (ercolation)g(on)g Fk(G)p Fo(.)29 b(\(Note)-48 1412 y(that)18 b(this)h(resulting)h(graph)e(is)h(in)h(general)f(not)f(connected,)i (but)f(as)g(observ)o(ed)f(in)i(the)f(last)g(paragraph,)-48 1485 y(this)f(is)h(not)f(an)g(imp)q(ortan)o(t)e(issue.\))30 b(The)18 b(study)g(of)f(statistical)h(mec)o(hanics)g(mo)q(dels)h(on)f (diluted)h(graphs)-48 1558 y(is)d(a)g(c)o(hapter)f(in)i(the)e(study)h (of)f(systems)g(in)i(random)d(en)o(vironmen)o(t,)h(a)h(v)o(ery)f(cen)o (tral)g(area)g(of)h(researc)o(h)e(in)-48 1632 y(ph)o(ysics)j(and)f(in)i (mathematical)d(ph)o(ysics.)25 b(F)l(or)17 b(some)e(of)i(the)f(bac)o (kground)f(on)i(the)f(sub)s(ject,)g(consult)h([1)o(],)-48 1705 y([13)o(])d(and)h(references)g(therein.)23 1801 y(One)21 b(can)f(sho)o(w,)g(using)h(Kolmogoro)o(v's)d(0-1)i(la)o(w)g (and)g(Theorem)g(7.33)f(of)h([7)o(],)h(or)f(Theorem)f(3.1)h(of)-48 1875 y([4)o(],)h(that)e(for)g(eac)o(h)h(graph)f(the)h(critical)h(v)m (alue)g Fk(J)841 1858 y Fi(B)837 1883 y(c)870 1875 y Fo(\()p Fk(b)p Fo(\))e(\()p Ff(r)n(esp.)35 b Fk(J)1119 1858 y Fi(S)1115 1883 y(c)1143 1875 y Fo(\()p Fk(s)p Fo(\)\))20 b(for)f(the)h Fk(q)r Fo(-P)o(otts)e(mo)q(del)i(on)g(the)-48 1948 y(corresp)q(onding)14 b(b)q(ond-diluted)j(\()p Ff(r)n(esp.)i Fo(site-diluted\))d(graph)e(is)i(w)o(ell)f(de\014ned)h(and)f(not)f (random.)23 2044 y(In)h([1)o(])g(the)g(follo)o(wing)g(w)o(as)e(pro)o(v) o(ed)h(\(see)h(displa)o(ys)g(\(9\))f(and)h(\(10\))f(in)h(that)f(pap)q (er\).)-48 2166 y Fl(Theorem)i(1.2)23 b Ff(F)m(or)c(e)n(ach)f (in\014nite,)i(lo)n(c)n(al)r(ly)d(\014nite,)j(c)n(onne)n(cte)n(d)e(gr)n (aph)h Fk(G)f Ff(which)h(has)h Fo(0)d Fk(<)h(b)1612 2175 y Fi(c)1647 2166 y Fk(<)f Fo(1)p Ff(,)j(and)-48 2240 y(e)n(ach)d Fk(q)r Ff(,)h(ther)n(e)f(exist)h(c)n(onstants,)h Fo(0)d Fk(<)f(A)667 2223 y Fi(B)667 2248 y Fg(1)711 2240 y Fj(\024)h Fk(A)796 2223 y Fi(B)796 2248 y Fg(2)840 2240 y Fk(<)g Fj(1)i Ff(such)h(that)f(for)f(the)h(b)n(ond-dilute)n(d)f Fk(q)r Ff(-Potts)h(mo)n(del)-48 2313 y(on)f Fk(G)p Ff(,)474 2387 y Fk(A)508 2367 y Fi(B)508 2395 y Fg(1)536 2387 y Fo(\()p Fk(b)10 b Fj(\000)g Fk(b)647 2395 y Fi(c)665 2387 y Fo(\))19 b Fj(\024)h Fo(exp\()p Fj(\000)p Fk(J)908 2367 y Fi(B)904 2395 y(c)937 2387 y Fo(\()p Fk(b)p Fo(\)\))f Fj(\024)h Fk(A)1118 2367 y Fi(B)1118 2395 y Fg(2)1147 2387 y Fo(\()p Fk(b)10 b Fj(\000)g Fk(b)1258 2395 y Fi(c)1275 2387 y Fo(\))p Fk(;)-48 2480 y Ff(for)19 b Fk(b)f(>)g(b)136 2489 y Fi(c)153 2480 y Ff(.)30 b(Given)19 b Fk(\016)g(>)f Fo(0)p Ff(,)i(one)f(c)n(an)g(take)g Fk(A)788 2463 y Fi(B)788 2488 y Fg(1)834 2480 y Fo(=)f(\(1)11 b Fj(\000)i Fk(\016)r Fo(\))p Fk(=)p Fo(\()p Fk(q)r(b)1109 2489 y Fi(c)1124 2480 y Fo(\))19 b Ff(and)g Fk(A)1285 2463 y Fi(B)1285 2488 y Fg(2)1332 2480 y Fo(=)f(1)p Fk(=b)1450 2489 y Fi(c)1466 2480 y Ff(,)i(pr)n(ovide)n(d)e(that)h(we)-48 2554 y(r)n(estrict)d(ourselves)g(to)g(values)g(of)g Fk(b)g Ff(close)g(enough)g(to)h Fk(b)916 2562 y Fi(c)933 2554 y Ff(.)878 2678 y Fo(4)p eop %%Page: 5 5 5 4 bop -48 199 a Fo(In)15 b(particular)g(the)g(crosso)o(v)o(er)d(exp)q (onen)o(t)j(\010)715 182 y Fi(B)743 199 y Fo(,)g(de\014ned)h(b)o(y)e (the)h(relation)226 318 y(exp\()p Fj(\000)p Fk(J)377 299 y Fi(B)373 326 y(c)405 318 y Fo(\()p Fk(b)p Fo(\)\))k Fj(\030)i Fo(\()p Fk(b)10 b Fj(\000)g Fk(b)664 327 y Fi(c)681 318 y Fo(\))699 299 y Fg(\010)723 287 y Fd(B)794 318 y Fo(in)16 b(the)f(logarithmic)f(sense)p Fk(;)52 b Fo(as)45 b Fk(b)13 b Fj(&)g Fk(b)1524 327 y Fi(c)1541 318 y Fk(;)-48 437 y Fo(tak)o(es)h(the)h(v)m(alue)g(\010)295 420 y Fi(B)336 437 y Fo(=)e(1.)23 536 y(The)d(metho)q(ds)g(in)i([1)o(]) f(are)f(based)g(on)h(the)f(relation)h(b)q(et)o(w)o(een)f(the)h(P)o (otts)e(mo)q(del)i(and)g(the)f(random-cluster)-48 609 y(mo)q(del)k(\(a)g(certain)g(dep)q(enden)o(t)g(b)q(ond)h(p)q (ercolation)f(mo)q(del\))g(and)g(on)g(comparisons)f(b)q(et)o(w)o(een)g (the)h(random-)-48 682 y(cluster)i(mo)q(del)h(and)f(indep)q(enden)o(t)i (b)q(ond)f(p)q(ercolation.)23 b(The)16 b(metho)q(d)g(do)q(es)h(not)e (extend)i(to)e(site)i(diluted)-48 756 y(P)o(otts)11 b(mo)q(dels,)i(b)q (ecause)h(the)e(random-cluster)g(mo)q(del)h(is)h(not)e(immediately)i (comparable)e(to)g(indep)q(enden)o(t)-48 829 y(p)q(ercolation.)29 b(Using)18 b(the)g(tec)o(hniques)h(from)e(the)h(curren)o(t)f(pap)q(er,) h(w)o(e)g(can)g(\014ll)h(this)g(gap)e(and)h(pro)o(v)o(e)f(the)-48 903 y(follo)o(wing.)-48 1038 y Fl(Theorem)f(1.3)23 b Ff(F)m(or)d(e)n(ach)f(in\014nite)h(c)n(onne)n(cte)n(d)f(gr)n(aph)h(of)g (b)n(ounde)n(d)f(de)n(gr)n(e)n(e)f Fk(G)i Ff(which)f(has)h Fk(b)1607 1047 y Fi(c)1644 1038 y Fk(<)f Fo(1)p Ff(,)i(and)-48 1112 y(e)n(ach)16 b(q,)g(ther)n(e)g(exist)h(c)n(onstants,)h Fo(0)13 b Fk(<)h(A)656 1095 y Fi(S)656 1120 y Fg(1)693 1112 y Fj(\024)g Fk(A)776 1095 y Fi(S)776 1120 y Fg(2)813 1112 y Fk(<)g Fj(1)p Ff(,)j(which)f(dep)n(end)g(only)h(on)g Fk(q)i Ff(and)e Fk(D)q Ff(,)g(such)g(that)g(for)-48 1185 y(the)f(site-dilute)n(d)f Fk(q)r Ff(-Potts)i(mo)n(dels)e(on)i Fk(G)p Ff(,)476 1304 y Fk(A)510 1285 y Fi(S)510 1312 y Fg(1)534 1304 y Fo(\()p Fk(s)10 b Fj(\000)g Fk(s)649 1312 y Fi(c)666 1304 y Fo(\))20 b Fj(\024)g Fo(exp\()p Fj(\000)p Fk(J)910 1285 y Fi(S)906 1312 y(c)934 1304 y Fo(\()p Fk(s)p Fo(\)\))f Fj(\024)h Fk(A)1117 1285 y Fi(S)1117 1312 y Fg(2)1141 1304 y Fo(\()p Fk(s)10 b Fj(\000)g Fk(s)1256 1312 y Fi(c)1273 1304 y Fo(\))p Fk(;)-48 1423 y Ff(for)16 b Fk(s)d(>)f(s)126 1431 y Fi(c)144 1423 y Ff(.)20 b(Given)c Fk(\016)e(>)f Fo(0)p Ff(,)i(one)h(c)n(an)g(take)g Fk(A)742 1406 y Fi(S)742 1431 y Fg(1)778 1423 y Fo(=)d(\(1)c Fj(\000)h Fk(\016)r Fo(\))p Fk(=)p Fo(\()p Fk(q)r(s)1045 1431 y Fi(c)1060 1423 y Fo(\))16 b Ff(and)g Fk(A)1215 1406 y Fi(S)1215 1431 y Fg(2)1252 1423 y Fo(=)d(\(1)c(+)g Fk(\016)r Fo(\))f(2)1465 1406 y Fi(D)1494 1423 y Fk(=)p Fo(\()p Fk(s)1556 1431 y Fi(c)1572 1423 y Fo(\(1)h Fj(\000)h Fk(s)1688 1431 y Fi(c)1705 1423 y Fo(\))1723 1406 y Fi(D)q Fh(\000)p Fg(2)1796 1423 y Fo(\))p Ff(,)-48 1497 y(wher)n(e)20 b Fk(D)i Ff(is)f(the)f(maximal)f(de)n(gr)n(e)n(e)g(of)h Fk(G)p Ff(,)h(pr)n(ovide)n(d)f(that)g(we)g(r)n(estrict)g(ourselves)h (to)f(values)g(of)h Fk(s)f Ff(close)-48 1570 y(enough)c(to)h Fk(s)181 1578 y Fi(c)198 1570 y Ff(.)-48 1706 y Fo(In)e(particular)g (the)g(crosso)o(v)o(er)d(exp)q(onen)o(t)j(\010)715 1688 y Fi(S)739 1706 y Fo(,)f(de\014ned)i(b)o(y)f(the)g(relation)225 1825 y(exp\()p Fj(\000)p Fk(J)376 1805 y Fi(S)372 1833 y(c)400 1825 y Fo(\()p Fk(s)p Fo(\)\))k Fj(\030)h Fo(\()p Fk(s)10 b Fj(\000)g Fk(s)664 1833 y Fi(c)681 1825 y Fo(\))699 1805 y Fg(\010)723 1793 y Fd(S)791 1825 y Fo(in)16 b(the)e(logarithmic) h(sense)p Fk(;)52 b Fo(as)45 b Fk(s)13 b Fj(&)f Fk(s)1524 1833 y Fi(c)1541 1825 y Fk(;)-48 1944 y Fo(tak)o(es)i(the)h(v)m(alue)g (\010)295 1927 y Fi(S)332 1944 y Fo(=)d(1.)23 2042 y(As)h(far)g(as)g(w) o(e)g(can)h(tell,)g(the)g(comparisons)e(b)q(et)o(w)o(een)i(indep)q (enden)o(t)g(b)q(ond)g(and)g(site)g(p)q(ercolation)g(a)o(v)m(ail-)-48 2115 y(able)h(in)g(the)f(literature)f(\(see)h([17)o(],)g([18)o(])f (\(Remark)g(6.2\))g(and)h([11)o(]\),)f(when)h(com)o(bined)h(with)f(the) g(argumen)o(ts)-48 2189 y(from)d([1)o(],)h(w)o(ould)g(only)h(allo)o(w)f (one)g(to)f(pro)o(v)o(e)g(a)h(w)o(eak)o(er)f(result,)h(whic)o(h)h(w)o (ould)f(only)g(imply)h(that)f(if)g(\010)1681 2172 y Fi(S)1717 2189 y Fo(exists)-48 2262 y(then)j(it)g(is)h(p)q(ositiv)o(e)f(and)g (\014nite.)20 b(Results)15 b(of)g(this)g(t)o(yp)q(e)g(are)f(con)o (tained)h(in)g([5])f(and)h([6)o(].)23 2360 y(Regarding)k(the)h(motiv)m (ation)f(b)q(ehind)j(the)e(study)f(of)h(systems)f(in)i(random)d(en)o (vironmen)o(t,)i(it)g(seems)-48 2434 y(natural)15 b(to)f(consider)h (site-diluted)i(P)o(otts)d(mo)q(dels)h(as)g(more)f(in)o(teresting)h (than)g(b)q(ond-diluted)i(ones,)d(since)-48 2507 y(site-dilution)19 b(corresp)q(onds)e(to)h(the)f(presence)h(of)g(impurities)h(in)f(a)g (crystal.)28 b(W)l(e)19 b(stress)d(also)i(that)f(while)878 2678 y(5)p eop %%Page: 6 6 6 5 bop -48 199 a Fo(Theorem)14 b(1.3)g(is)i(b)q(eing)g(stated)e(in)i (the)g(con)o(text)e(of)g(fairly)i(arbitrary)d(graphs,)h(as)h(far)f(as)h (w)o(e)g(kno)o(w,)f(it)i(is)f(a)-48 273 y(new)g(result)g(ev)o(en)g(in)g (the)g(sp)q(ecial)h(case)f(of)g(cubic)h(lattices.)-48 371 y Fl(1.3)23 b(Mixed)g(p)q(ercolation.)36 b Fo(In)20 b(the)f(mixed)h(p)q(ercolation)f(mo)q(del)h(eac)o(h)g(b)q(ond)f(is)h(k) o(ept)g(indep)q(enden)o(tly)-48 444 y(of)e(an)o(ything)g(else)i(with)f (probabilit)o(y)f Fk(b)h Fo(and)g(eac)o(h)g(site)g(is)g(k)o(ept)f (indep)q(enden)o(tly)i(of)f(an)o(ything)f(else)h(with)-48 518 y(probabilit)o(y)11 b Fk(s)p Fo(;)h(the)f(other)f(b)q(onds)h(and)f (sites)h(are)f(remo)o(v)o(ed)g(along)g(with)h(all)h(the)f(b)q(onds)f (inciden)o(t)i(to)f(remo)o(v)o(ed)-48 591 y(sites.)24 b(W)l(e)17 b(denote)f(b)o(y)g Fk(\031)393 574 y Fo(BS)451 591 y(\()p Fk(b;)8 b(s)p Fo(\))16 b(the)g(probabilit)o(y)g(that)g(the)g (remaining)g(graph)g(con)o(tains)f(some)h(in\014nite)-48 665 y(comp)q(onen)o(t)e(\(i.e.,)g(the)h(probabilit)o(y)g(that)f(p)q (ercolation)h(o)q(ccurs\).)j(It)d(is)h(natural)e(to)g(de\014ne,)i(for)e (0)e Fj(\024)g Fk(b)h Fj(\024)g Fo(1,)502 784 y Fk(s)523 792 y Fi(c)540 784 y Fo(\()p Fk(b)p Fo(\))19 b(=)h(sup)q Fj(f)p Fk(s)12 b Fj(2)h Fo([0)p Fk(;)8 b Fo(1])k(:)22 b Fk(\031)1005 764 y Fo(BS)1064 784 y(\()p Fk(b;)8 b(s)p Fo(\))j(=)i(0)p Fj(g)p Fk(:)-48 1063 y Fl(Theorem)j(1.4)23 b Ff(F)m(or)13 b(any)g(in\014nite)g(c)n(onne)n(cte)n(d)f(gr)n(aph)h(of) f(b)n(ounde)n(d)h(de)n(gr)n(e)n(e)e(which)h(has)i Fk(b)1467 1071 y Fi(c)1496 1063 y Fk(<)f Fo(1)p Ff(,)g(the)g(function)-48 1137 y Fk(s)-27 1145 y Fi(c)-10 1137 y Fo(\()p Fj(\001)p Fo(\))i Ff(has)i(the)f(fol)r(lowing)f(pr)n(op)n(erties:)-48 1235 y(a\))i(Ther)n(e)e(exist)h Fo(0)d Fk(<)f(C)359 1243 y Fg(1)391 1235 y Fj(\024)g Fk(C)470 1243 y Fg(2)502 1235 y Fk(<)g Fj(1)p Ff(,)17 b(which)e(dep)n(end)h(only)g(on)h(the)f (maximal)f(de)n(gr)n(e)n(e)f(of)i(the)g(gr)n(aph,)23 1333 y(such)g(that)g(for)h(any)f Fk(b)395 1316 y Fh(0)407 1333 y Fk(;)8 b(b)447 1316 y Fh(00)480 1333 y Fj(2)13 b Fo([)p Fk(b)555 1341 y Fi(c)571 1333 y Fk(;)8 b Fo(1])p Ff(,)471 1452 y Fk(C)503 1461 y Fg(1)522 1452 y Fj(j)p Fk(b)554 1433 y Fh(0)576 1452 y Fj(\000)i Fk(b)640 1433 y Fh(00)661 1452 y Fj(j)i(\024)h(j)p Fk(s)768 1461 y Fi(c)785 1452 y Fo(\()p Fk(b)822 1433 y Fh(0)833 1452 y Fo(\))d Fj(\000)g Fk(s)927 1461 y Fi(c)944 1452 y Fo(\()p Fk(b)981 1433 y Fh(00)1002 1452 y Fo(\))p Fj(j)i(\024)h Fk(C)1125 1461 y Fg(2)1143 1452 y Fj(j)p Fk(b)1175 1433 y Fh(0)1197 1452 y Fj(\000)d Fk(b)1261 1433 y Fh(00)1283 1452 y Fj(j)p Fk(:)-48 1596 y Ff(b\))16 b Fk(s)28 1604 y Fi(c)45 1596 y Fo(\()p Fj(\001)p Fo(\))f Ff(is)i(a)f(Lipschitz)h(c)n (ontinuous)g(function)g(on)f Fo([0)p Fk(;)8 b Fo(1])p Ff(.)-48 1694 y(c\))16 b Fk(s)28 1703 y Fi(c)45 1694 y Fo(\()p Fk(b)p Fo(\))c Fk(<)h Fo(1)32 b Ff(i\013)g Fk(b)13 b Fj(2)f Fo(\()p Fk(b)400 1703 y Fi(c)417 1694 y Fk(;)c Fo(1])p Ff(.)-48 1793 y(d\))17 b(If)f Fk(b)76 1801 y Fi(c)106 1793 y Fj(\024)c Fk(b)172 1776 y Fh(0)197 1793 y Fk(<)h(b)264 1776 y Fh(00)298 1793 y Fj(\024)f Fo(1)p Ff(,)k(then)g Fk(s)519 1801 y Fi(c)536 1793 y Fo(\()p Fk(b)573 1776 y Fh(00)594 1793 y Fo(\))c Fk(<)h(s)693 1801 y Fi(c)710 1793 y Fo(\()p Fk(b)747 1776 y Fh(0)759 1793 y Fo(\))p Ff(,)i(i.e.,)h Fk(s)920 1801 y Fi(c)937 1793 y Fo(\()p Fj(\001)p Fo(\))f Ff(is)h(strictly)g(de)n(cr)n(e)n (asing)f(on)h Fo([)p Fk(b)1512 1801 y Fi(c)1529 1793 y Fk(;)8 b Fo(1])p Ff(.)23 1928 y Fo(It)15 b(is)g(also)g(natural)f(to)g (de\014ne)197 2047 y Fk(b)216 2056 y Fi(c)233 2047 y Fo(\()p Fk(s)p Fo(\))e(=)g(inf)t Fj(f)p Fk(b)h Fj(2)f Fo([0)p Fk(;)c Fo(1])13 b(:)22 b Fk(\031)670 2028 y Fo(BS)728 2047 y(\()p Fk(b;)8 b(s)p Fo(\))k Fk(>)g Fo(0)p Fj(g)g Fo(=)h(inf)t Fj(f)p Fk(b)f Fj(2)h Fo([0)p Fk(;)8 b Fo(1])k(:)22 b Fk(\031)1310 2028 y Fo(BS)1369 2047 y(\()p Fk(b;)8 b(s)p Fo(\))j(=)i(1)p Fj(g)p Fk(:)-48 2167 y Fo(Theorem)h(1.4)g(can)i (b)q(e)f(rephrased)g(in)h(terms)e(of)h(the)g(function)g Fk(b)1073 2175 y Fi(c)1091 2167 y Fo(\()p Fj(\001)p Fo(\).)k(Note)14 b(also)h(that)g(from)f(this)h(theorem)-48 2240 y(w)o(e)i(obtain)g(that) f(the)h(function)g Fk(s)544 2248 y Fi(c)561 2240 y Fo(\()p Fj(\001)p Fo(\))f(restricted)h(to)f(the)h(domain)g([)p Fk(b)1162 2248 y Fi(c)1179 2240 y Fk(;)8 b Fo(1])16 b(is)h(the)g(in)o (v)o(erse)g(of)g(the)g(function)-48 2313 y Fk(b)-29 2322 y Fi(c)-12 2313 y Fo(\()p Fj(\001)p Fo(\))f(restricted)g(to)g(the)h (domain)g([)p Fk(s)590 2322 y Fi(c)607 2313 y Fk(;)8 b Fo(1].)24 b(Moreo)o(v)o(er)15 b(the)i(b)q(oundary)f(b)q(et)o(w)o(een) h(the)g(p)q(ercolativ)o(e)f(and)h(the)-48 2387 y(non-p)q(ercolativ)o(e) e(regions)f(in)i(the)f(plane)g(\()p Fk(b;)8 b(s)p Fo(\))14 b(is)i(precisely)f(the)g(curv)o(e)402 2506 y Fj(f)p Fo(\()p Fk(b;)8 b(s)504 2514 y Fi(c)519 2506 y Fo(\()p Fk(b)p Fo(\)\))k(:)g Fk(b)648 2514 y Fi(c)678 2506 y Fj(\024)h Fk(b)f Fj(\024)h Fo(1)p Fj(g)f Fo(=)g Fj(f)p Fo(\()p Fk(b)970 2514 y Fi(c)987 2506 y Fo(\()p Fk(s)p Fo(\))p Fk(;)c(s)p Fo(\))j(:)h Fk(s)1161 2514 y Fi(c)1190 2506 y Fj(\024)h Fk(s)g Fj(\024)f Fo(1)p Fj(g)p Fk(:)878 2678 y Fo(6)p eop %%Page: 7 7 7 6 bop 23 199 a Fo(An)11 b(imp)q(ortan)o(t)g(piece)i(of)e(information) g(con)o(tained)g(in)i(Theorem)e(1.4)f(refers)h(to)g(the)h(edges)g(of)f (this)h(curv)o(e.)-48 273 y(It)j(implies)h(that)e(for)g Fk(b)340 281 y Fi(c)370 273 y Fj(\024)f Fk(b)g Fj(\024)f Fo(1,)532 383 y Fk(C)564 391 y Fg(1)583 383 y Fo(\()p Fk(b)e Fj(\000)g Fk(b)694 391 y Fi(c)711 383 y Fo(\))20 b Fj(\024)g Fo(1)10 b Fj(\000)g Fk(s)903 391 y Fi(c)920 383 y Fo(\()p Fk(b)p Fo(\))20 b Fj(\024)g Fk(C)1082 391 y Fg(2)1101 383 y Fo(\()p Fk(b)10 b Fj(\000)g Fk(b)1212 391 y Fi(c)1229 383 y Fo(\))522 b(\(3\))-48 493 y(and)532 567 y Fk(C)564 575 y Fg(1)583 567 y Fo(\(1)9 b Fj(\000)h Fk(b)p Fo(\))20 b Fj(\024)g Fk(s)811 575 y Fi(c)828 567 y Fo(\()p Fk(b)p Fo(\))10 b Fj(\000)g Fk(s)959 575 y Fi(c)996 567 y Fj(\024)20 b Fk(C)1083 575 y Fg(2)1102 567 y Fo(\(1)10 b Fj(\000)g Fk(b)p Fo(\))p Fk(:)521 b Fo(\(4\))-48 663 y(In)15 b(particular)g(the)g(critical)h(exp)q(onen)o (ts)e(de\014ned)i(b)o(y)279 773 y(1)10 b Fj(\000)g Fk(s)378 781 y Fi(c)395 773 y Fo(\()p Fk(b)p Fo(\))20 b Fj(\030)g Fo(\()p Fk(b)562 781 y Fi(c)589 773 y Fj(\000)10 b Fk(b)p Fo(\))671 754 y Fg(\010)695 741 y Fc(0)753 773 y Fo(in)16 b(the)f(logarithmic)f(sense)p Fk(;)52 b Fo(as)45 b Fk(b)13 b Fj(&)g Fk(b)1483 781 y Fi(c)-48 883 y Fo(and)281 956 y Fk(s)302 965 y Fi(c)319 956 y Fo(\()p Fk(b)p Fo(\))c Fj(\000)h Fk(s)449 965 y Fi(c)487 956 y Fj(\030)20 b Fo(\(1)9 b Fj(\000)h Fk(b)p Fo(\))674 937 y Fg(\010)698 925 y Fc(00)765 956 y Fo(in)16 b(the)f(logarithmic)f(sense)p Fk(;)53 b Fo(as)44 b Fk(b)13 b Fj(\045)g Fo(1)-48 1052 y(tak)o(e)d(the)i(v)m(alue)g(\010)267 1035 y Fh(0)291 1052 y Fo(=)g(\010)371 1035 y Fh(00)405 1052 y Fo(=)g(1.)19 b(The)11 b(relations)g(in)h(\(4\))e(are)h(the)g(ones)g(that)g(will)i (allo)o(w)e(us)g(to)g(pro)o(v)o(e)f(Theorem)-48 1126 y(1.3.)-48 1223 y Fl(Question)18 b(2.)j Fo(Ho)o(w)14 b(smo)q(oth)g(is)i(the)f(function)g Fk(s)817 1231 y Fi(c)834 1223 y Fo(\()p Fj(\001)p Fo(\),)e(with)j(domain)f(restricted)f(to)g([)p Fk(b)1462 1231 y Fi(c)1479 1223 y Fk(;)8 b Fo(1]?)19 b(\(Presumably)-48 1296 y(it)c(is)h(analytic.\))j(Is)c(it)g(conca)o(v)o (e?)23 1393 y(In)g(Section)f(2)h(w)o(e)f(will)i(deriv)o(e)e(a)g (di\013eren)o(tial)i(inequalit)o(y)f(whic)o(h)g(will)h(b)q(e)f(used)g (in)g(Section)g(3)f(to)g(deriv)o(e)-48 1466 y(estimates)i(on)i(the)f (function)h Fk(s)500 1474 y Fi(c)517 1466 y Fo(\()p Fj(\001)p Fo(\),)e(and)i(so)f(pro)o(vide)g(the)g(pro)q(of)g(of)g(Theorem)f(1.4.) 26 b(In)18 b(Section)g(4,)f(these)-48 1540 y(estimates)c(on)h Fk(s)233 1548 y Fi(c)250 1540 y Fo(\()p Fj(\001)p Fo(\))f(will)j(b)q(e) f(used)f(to)g(pro)o(v)o(e)f(part)g(\(a\))g(of)h(Theorem)f(1.1;)g(also)h (the)g(example)g(whic)o(h)g(pro)o(v)o(es)-48 1613 y(part)g(\(b\))f(of)h (this)h(theorem)f(will)h(b)q(e)h(presen)o(ted)e(there.)19 b(Finally)d(in)f(Section)g(5)f(the)h(pro)q(of)e(of)h(Theorem)g(1.3,)-48 1686 y(based)h(on)g(\(4\))e(will)k(b)q(e)e(presen)o(ted.)-48 1971 y Fp(2)65 b(Di\013eren)n(tial)24 b(inequalit)n(y)-48 2113 y Fo(In)d(this)g(section)f(w)o(e)g(consider)h(mixed)g(p)q (ercolation)f(on)h(an)f(in\014nite)i(connected)f(graph)e Fk(G)h Fo(of)h(b)q(ounded)-48 2187 y(degree.)30 b(A)18 b(c)o(hain)h(is)g(a)f(\014nite)h(sequence)g(of)f(distinct)h(sites,)g Fk(x)1062 2195 y Fg(0)1080 2187 y Fk(;)8 b(:::;)g(x)1186 2195 y Fi(n)1206 2187 y Fo(,)19 b(suc)o(h)f(that)g(for)f Fk(i)i Fo(=)g(0)p Fk(;)8 b(:::;)f(n)i Fj(\000)k Fo(1,)-48 2260 y Fk(x)-22 2268 y Fi(i)9 2260 y Fo(and)k Fk(x)125 2268 y Fi(i)p Fg(+1)197 2260 y Fo(are)g(joined)g(b)o(y)g(an)f(edge.)26 b(These)17 b(edges)g(are)f(called)i(the)f(edges)g(of)f(the)h(c)o(hain.) 26 b(The)17 b(length)-48 2333 y(of)g(suc)o(h)g(a)g(c)o(hain)h(is)g Fk(n)p Fo(,)f(and)h(its)f(end-p)q(oin)o(ts)h(are)e Fk(x)860 2342 y Fg(0)896 2333 y Fo(and)i Fk(x)1013 2342 y Fi(n)1035 2333 y Fo(.)27 b(The)17 b(distance)g(b)q(et)o(w)o(een)g(t)o(w)o(o)f (sites)h(is)h(the)-48 2407 y(minimal)13 b(length)f(of)g(the)g(c)o (hains)h(whic)o(h)f(ha)o(v)o(e)g(these)g(sites)g(as)g(end-p)q(oin)o (ts.)19 b(An)12 b(arbitrary)f(site)h(of)g(the)g(graph)-48 2480 y(is)i(c)o(hosen)e(as)h(its)g(ro)q(ot,)f(and)h(denoted)g(b)o(y)g Fk(r)q Fo(.)20 b(The)13 b(ball)h(of)f(radius)g Fk(n)g Fo(cen)o(tered)g(at)f(the)h(ro)q(ot,)f(denoted)h Fk(V)d Fo(\()p Fk(n)p Fo(\),)-48 2554 y(is)15 b(the)g(set)g(of)f(sites)h (within)h(distance)f Fk(n)g Fo(of)g(the)g(ro)q(ot.)j(W)l(e)e(also)e (set)1147 2542 y(~)1140 2554 y Fk(V)c Fo(\()p Fk(n)p Fo(\))i(=)h Fk(V)d Fo(\()p Fk(n)p Fo(\))p Fj(nf)p Fk(r)q Fj(g)p Fo(.)878 2678 y(7)p eop %%Page: 8 8 8 7 bop 23 199 a Fo(Clusters)14 b(are)g(the)h(connected)h(comp)q(onen)o (ts)e(of)g(the)h(graph)f(obtained)h(from)f Fk(G)h Fo(after)f(sites)h (and)g(b)q(onds)-48 273 y(ha)o(v)o(e)h(b)q(een)h(deleted)h(in)f(the)g (p)q(ercolation)g(pro)q(cess.)24 b(W)l(e)18 b(denote)e(b)o(y)h Fk(\022)1189 256 y Fo(BS)1248 273 y(\()p Fk(b;)8 b(s)p Fo(\))16 b(the)g(probabilit)o(y)h(that)f(for)-48 346 y(mixed)f(p)q(ercolation)f(with)g(parameters)e(\()p Fk(b;)c(s)p Fo(\))13 b(on)h Fk(G)p Fo(,)g(the)g(ro)q(ot)f(b)q(elongs)i(to)e(an)h (in\014nite)i(cluster.)j(It)c(is)f(w)o(ell)-48 419 y(kno)o(wn)k(and)g (easy)h(to)e(see)i(that)f Fk(\031)559 402 y Fo(BS)617 419 y(\()p Fk(b;)8 b(s)p Fo(\))18 b(=)h(1)f(i\013)i Fk(\022)907 402 y Fo(BS)966 419 y(\()p Fk(b;)8 b(s)p Fo(\))18 b Fk(>)g Fo(0,)h(and)g(that)f(in)h(particular,)g(this)f(last)-48 493 y(statemen)o(t)13 b(is)i(indep)q(enden)o(t)i(of)d(the)h(c)o(hoice)g (of)g(the)g(ro)q(ot)e(of)i Fk(G)p Fo(.)23 591 y(As)10 b(usual,)i(b)q(onds)e(and)h(sites)g(whic)o(h)g(are)f(k)o(ept)g(in)i (the)e(p)q(ercolation)h(pro)q(cess)f(will)i(b)q(e)g(said)f(to)f(b)q(e)h (o)q(ccupied)-48 664 y(and)k(those)f(whic)o(h)i(are)e(remo)o(v)o(ed)f (will)k(b)q(e)e(said)g(to)g(b)q(e)g(v)m(acan)o(t.)23 762 y(A)i(c)o(hain)h(is)g(said)g(to)f(b)q(e)h(in)o(ternally)g(o)q (ccupied)h(if)f(all)g(its)g(sites,)g(except)f(p)q(ossibly)i(for)e(its)g (end-p)q(oin)o(ts,)-48 835 y(and)c(all)g(its)g(edges)g(are)f(o)q (ccupied.)20 b(Tw)o(o)11 b(sites)i(are)f(said)h(to)f(b)q(e)i(almost)d (connected)i(if)g(there)g(is)g(an)g(in)o(ternally)-48 908 y(o)q(ccupied)18 b(c)o(hain)f(whic)o(h)h(has)e(these)h(sites)g(as)f (end-p)q(oin)o(ts.)26 b(W)l(e)18 b(denote)e(b)o(y)h Fk(A)1325 917 y Fi(n)1365 908 y Fo(the)f(ev)o(en)o(t)h(that)f(the)g(ro)q(ot)-48 982 y(is)g(almost)f(connected)h(to)f(a)g(site)h(whic)o(h)g(is)g(at)f (distance)h Fk(n)11 b Fo(+)g(1)k(from)g(the)g(ro)q(ot.)21 b(Set)1451 970 y(~)1447 982 y Fk(\022)1469 965 y Fo(BS)1468 990 y Fi(n)1529 982 y Fo(\()p Fk(b;)8 b(s)p Fo(\))13 b(=)g Fk(P)6 b Fo(\()p Fk(A)1774 990 y Fi(n)1797 982 y Fo(\).)-48 1055 y(Denote,)16 b(as)g(usual,)h(b)o(y)g Fk(\022)399 1038 y Fo(BS)398 1063 y Fi(n)458 1055 y Fo(\()p Fk(b;)8 b(s)p Fo(\))16 b(the)g(probabilit)o(y)h(that)f Fk(A)1017 1063 y Fi(n)1056 1055 y Fo(happ)q(ens)h(and)g(also)f(the)h (ro)q(ot)e(is)i(o)q(ccupied.)-48 1129 y(Then)e(it)g(is)h(clear)f(that) 464 1202 y(lim)452 1227 y Fi(n)p Fh(!1)546 1202 y Fk(s)579 1190 y Fo(~)575 1202 y Fk(\022)597 1183 y Fo(BS)596 1210 y Fi(n)657 1202 y Fo(\()p Fk(b;)8 b(s)p Fo(\))j(=)25 b(lim)813 1227 y Fi(n)p Fh(!1)907 1202 y Fk(\022)929 1183 y Fo(BS)928 1210 y Fi(n)989 1202 y Fo(\()p Fk(b;)8 b(s)p Fo(\))j(=)h Fk(\022)1166 1183 y Fo(BS)1226 1202 y(\()p Fk(b;)c(s)p Fo(\))p Fk(:)433 b Fo(\(5\))-48 1457 y Fl(Prop)q(osition)19 b(2.1)k Ff(F)m(or)f(any)g(in\014nite)g(c)n(onne) n(cte)n(d)e(gr)n(aph)h(of)h(de)n(gr)n(e)n(e)d(b)n(ounde)n(d)i(ab)n(ove) g(by)g Fk(D)j(<)e Fj(1)p Ff(,)h(the)-48 1530 y(fol)r(lowing)15 b(di\013er)n(ential)g(ine)n(quality)g(holds)575 1634 y Fk(@)605 1622 y Fo(~)602 1634 y Fk(\022)624 1617 y Ff(BS)623 1642 y Fi(n)683 1634 y Fo(\()p Fk(b;)8 b(s)p Fo(\))p 575 1654 204 2 v 653 1694 a Fk(@)s(s)796 1665 y Fj(\024)854 1653 y Fo(~)844 1665 y Fk(C)s Fo(\()p Fk(b;)g(s)p Fo(\))988 1634 y Fk(@)1018 1622 y Fo(~)1015 1634 y Fk(\022)1037 1617 y Ff(BS)1036 1642 y Fi(n)1096 1634 y Fo(\()p Fk(b;)g(s)p Fo(\))p 988 1654 V 1067 1694 a Fk(@)s(b)1196 1665 y(;)558 b Ff(\(6\))-48 1780 y(wher)n(e)90 1769 y Fo(~)79 1780 y Fk(C)t Fo(\()p Fk(b;)8 b(s)p Fo(\))15 b Ff(c)n(an)h(b)n(e)f(chosen)i (as)f Fo(2)593 1763 y Fh(b)p Fi(D)q(=)p Fg(2)p Fh(c)687 1780 y Fk(=)p Fo(\()p Fk(s)8 b Fo(\(1)g Fj(\000)i Fk(b)p Fo(\))888 1763 y Fh(b)p Fi(D)q(=)p Fg(2)p Fh(c\000)p Fg(1)1025 1780 y Fo(\))16 b Ff(or)g(as)h Fo(2)1198 1763 y Fi(D)1227 1780 y Fk(=)p Fo(\()p Fk(s)8 b Fo(\(1)g Fj(\000)j Fk(s)p Fo(\))1431 1763 y Fi(D)q Fh(\000)p Fg(2)1503 1780 y Fo(\))p Ff(.)23 1912 y Fo(Under)j(the)h(same)g(assumptions)e(one)i (can)g(also)g(pro)o(v)o(e)f(a)g(di\013eren)o(tial)i(inequalit)o(y)g(of) f(the)f(form)575 2016 y Fk(@)s(\022)624 1999 y Fo(BS)623 2024 y Fi(n)683 2016 y Fo(\()p Fk(b;)8 b(s)p Fo(\))p 575 2036 V 653 2076 a Fk(@)s(s)796 2047 y Fj(\024)13 b Fk(C)s Fo(\()p Fk(b;)8 b(s)p Fo(\))988 2016 y Fk(@)s(\022)1037 1999 y Fo(BS)1036 2024 y Fi(n)1096 2016 y Fo(\()p Fk(b;)g(s)p Fo(\))p 988 2036 V 1067 2076 a Fk(@)s(b)1196 2047 y(:)560 b Fo(\(7\))-48 2162 y(But)16 b(the)f(corresp)q(onding)g(estimate)g(on)h Fk(C)s Fo(\()p Fk(b;)8 b(s)p Fo(\))15 b(turns)g(out)g(not)g(to)g(b)q(e) i(go)q(o)q(d)e(enough)h(to)f(pro)o(v)o(e)f(Theorem)-48 2236 y(1.1\(a\))f(\(it)h(w)o(ould)h(b)q(e)h(go)q(o)q(d)e(enough,)h (though,)e(to)i(pro)o(v)o(e)e(Theorems)h(1.4)g(and)h(1.3\).)23 2333 y(Note)f(that)g(t)o(w)o(o)f(alternativ)o(e)h(expressions)h(for)856 2322 y(~)846 2333 y Fk(C)s Fo(\()p Fk(b;)8 b(s)p Fo(\))14 b(app)q(ear)g(in)i(Prop)q(osition)e(2.1.)19 b(The)c(\014rst)f(one)h(is) -48 2407 y(imp)q(ortan)o(t)d(in)i(the)f(pro)q(of)f(of)g(Theorem)g (1.1\(a\))f(\(the)i(p)q(o)o(w)o(er)f(of)h(1)6 b Fj(\000)g Fk(b)14 b Fo(whic)o(h)f(app)q(ears)f(in)i(this)g(expression)f(is)-48 2480 y(of)h(ma)s(jor)f(relev)m(ance)j(then\).)j(The)c(second)g(one)g (is)g(imp)q(ortan)o(t)e(in)j(the)f(pro)q(of)f(of)g(Theorem)g(1.3)g(\(b) q(ecause)h(it)-48 2554 y(do)q(es)c(not)g(blo)o(w)g(up)g(as)g Fk(b)g Fo(approac)o(hes)f(1\).)17 b(Both)11 b(expressions)g(are)f (needed)i(to)e(pro)o(v)o(e)g(Theorem)g(1.4)g(\(b)q(ecause)878 2678 y(8)p eop %%Page: 9 9 9 8 bop -48 199 a Fo(they)15 b(allo)o(w)h(one)f(to)g(b)q(ound)462 188 y(~)451 199 y Fk(C)t Fo(\()p Fk(b;)8 b(s)p Fo(\))14 b(ab)q(o)o(v)o(e,)h(uniformly)g(on)h(a)f(domain)h(whic)o(h)g(excludes)g (a)f(neigh)o(b)q(orho)q(o)q(d)-48 273 y(of)f(the)h(p)q(oin)o(t)g(\()p Fk(b;)8 b(s)p Fo(\))k(=)g(\(1)p Fk(;)c Fo(1\))13 b(and)i(a)f(neigh)o(b) q(orho)q(o)q(d)h(of)g(the)g(line)h Fk(s)d Fo(=)f(0\).)23 371 y(Under)17 b(the)h(same)f(assumptions)g(of)h(Prop)q(osition)f(2.1,) g(one)h(can)g(also)f(pro)o(v)o(e)g(the)h(follo)o(wing)g(comple-)-48 444 y(men)o(tary)13 b(di\013eren)o(tial)j(inequalities:)616 551 y Fk(@)646 539 y Fo(~)643 551 y Fk(\022)665 534 y Fo(BS)664 560 y Fi(n)724 551 y Fo(\()p Fk(b;)8 b(s)p Fo(\))p 616 572 204 2 v 695 611 a Fk(@)s(b)838 583 y Fj(\024)890 551 y Fk(D)p 890 572 39 2 v 900 611 a(b)946 551 y(@)976 539 y Fo(~)973 551 y Fk(\022)995 534 y Fo(BS)994 560 y Fi(n)1054 551 y Fo(\()p Fk(b;)g(s)p Fo(\))p 946 572 204 2 v 1024 611 a Fk(@)s(s)1155 583 y(;)601 b Fo(\(8\))-48 702 y(and)616 754 y Fk(@)s(\022)665 737 y Fo(BS)664 762 y Fi(n)724 754 y Fo(\()p Fk(b;)8 b(s)p Fo(\))p 616 775 V 695 814 a Fk(@)s(b)838 785 y Fj(\024)890 754 y Fk(D)p 890 775 39 2 v 900 814 a(b)946 754 y(@)s(\022)995 737 y Fo(BS)994 762 y Fi(n)1054 754 y Fo(\()p Fk(b;)g(s)p Fo(\))p 946 775 204 2 v 1024 814 a Fk(@)s(s)1155 785 y(:)601 b Fo(\(9\))-48 886 y(All)15 b(the)e(applications)i(that)d(w)o (e)i(w)o(ould)f(ha)o(v)o(e)g(for)g(them)g(are)g(nev)o(ertheless)h(b)q (etter)f(accomplished)h(b)o(y)g(using)-48 959 y(\(1\).)23 1057 y(Di\013eren)o(tial)e(inequalities)i(in)f(the)f(spirit)g(of)g (\(6\),)f(\(7\),)g(\(8\))g(and)h(\(9\))f(ha)o(v)o(e)g(app)q(eared)h(in) g([19)o(],)g([2)o(],)g([3)o(],)g(and)-48 1131 y([10)o(].)18 b(They)c(ha)o(v)o(e)e(b)q(een)i(used)g(in)g(order)e(to)h(pro)o(v)o(e)f (strict)h(inequalities)i(b)q(et)o(w)o(een)e(critical)h(p)q(oin)o(ts,)f (equalities)-48 1204 y(b)q(et)o(w)o(een)18 b(critical)h(exp)q(onen)o (ts)f(obtained)h(b)o(y)f(approac)o(hing)f(a)h(critical)i(p)q(oin)o(t)e (from)f(di\013eren)o(t)i(directions,)-48 1278 y(and)e(Lipsc)o(hitz)h (con)o(tin)o(uit)o(y)e(of)g(critical)i(lines.)27 b(The)16 b(main)h(applications)h(in)f(our)f(pap)q(er)h(are)f(nev)o(ertheless)-48 1351 y(of)j(a)h(di\013eren)o(t)g(nature.)33 b(In)20 b(particular,)g(as) f(far)g(as)g(w)o(e)g(kno)o(w,)h(in)g(previous)g(uses)g(of)f(suc)o(h)g (inequalities)-48 1424 y(estimates)14 b(on)h(the)g(v)m(alues)h(of)f (the)g(constan)o(ts)e(whic)o(h)j(pla)o(y)o(ed)f(the)g(role)g(of)f(our) 1337 1413 y(~)1327 1424 y Fk(C)t Fo(\()p Fk(b;)8 b(s)p Fo(\))14 b(w)o(ere)g(not)h(a)f(crucial)-48 1498 y(issue.)-48 1596 y Fl(Pro)q(of)j(of)g(Prop)q(osition)h(2.1:)j Fo(W)l(e)15 b(will)h(use)e(the)g(notation)f Fk(E)1062 1604 y Fi(x)1098 1596 y Fo(to)h(denote)g(the)g(set)g(of)f(edges)h(inciden)o(t)i(to)-48 1669 y(the)h(v)o(ertex)f Fk(x)p Fo(,)h(and)h Fk(V)344 1678 y Fi(x)382 1669 y Fo(to)f(denote)f(the)i(set)e(of)h(v)o(ertices)g (whic)o(h)h(share)e(an)h(edge)g(with)g Fk(x)p Fo(.)27 b(Let)17 b(also)g Fk(E)s Fo(\()p Fk(n)p Fo(\))-48 1743 y(b)q(e)h(the)f(set)g(of)g(edges)g(whic)o(h)g(ha)o(v)o(e)g(at)f(least)h (one)g(end-p)q(oin)o(t)h(in)g(the)f(ball)h Fk(V)11 b Fo(\()p Fk(n)p Fo(\).)26 b(In)17 b(order)g(to)f(pro)o(v)o(e)g(the)-48 1816 y(prop)q(osition,)d(w)o(e)h(only)g(ha)o(v)o(e)f(to)g(consider)h (mixed)g(p)q(ercolation)g(on)g(the)g(\014nite)g(graph)f Fk(G)1480 1824 y Fi(n)1515 1816 y Fo(=)f(\()p Fk(V)e Fo(\()p Fk(n)p Fo(\))p Fk(;)e(E)s Fo(\()p Fk(n)p Fo(\)\).)-48 1890 y(A)14 b(realization)h(of)f(this)g(pro)q(cess)g(is)h(a)f(function) h Fk(!)f Fo(:)e Fk(V)f Fo(\()p Fk(n)p Fo(\))d Fj(\002)h Fk(E)s Fo(\()p Fk(n)p Fo(\))j Fj(!)g(f)p Fo(0)p Fk(;)c Fo(1)p Fj(g)p Fo(,)k(where)i(0's)f(are)h(asso)q(ciated)g(to)-48 1963 y(v)m(acan)o(t)g(b)q(onds)h(and)g(sites)g(and)g(1's)f(are)g(asso)q (ciated)h(to)f(o)q(ccupied)i(b)q(onds)f(and)g(sites.)23 2061 y(Giv)o(en)j(a)f(site)h Fk(x)e Fj(2)378 2050 y Fo(~)371 2061 y Fk(V)11 b Fo(\()p Fk(n)p Fo(\),)16 b(let)i Fk(\016)588 2070 y Fi(x)609 2061 y Fk(A)643 2070 y Fi(n)683 2061 y Fo(b)q(e)g(the)g(ev)o(en)o(t)f(that)f Fk(x)h Fo(is)h(piv)o(otal)g (for)e Fk(A)1400 2070 y Fi(n)1423 2061 y Fo(,)h(i.e.,)h(the)f(ev)o(en)o (t)g(that)-48 2135 y(if)g Fk(x)g Fo(is)g(o)q(ccupied)h(then)f Fk(A)413 2143 y Fi(n)453 2135 y Fo(happ)q(ens,)g(but)g(if)g Fk(x)g Fo(is)g(v)m(acan)o(t)f(then)h Fk(A)1147 2143 y Fi(n)1187 2135 y Fo(do)q(es)f(not)h(happ)q(en.)25 b(Giv)o(en)18 b(a)f(b)q(ond)-48 2208 y Fk(e)c Fj(2)f Fk(E)s Fo(\()p Fk(n)p Fo(\),)i(the)h(ev)o(en)o(t)f Fk(\016)371 2216 y Fi(e)389 2208 y Fk(A)423 2216 y Fi(n)445 2208 y Fo(,)h(that)f Fk(e)h Fo(is)h(piv)o(otal)e(for)g Fk(A)906 2216 y Fi(n)929 2208 y Fo(,)g(is)i(de\014ned)g(in)f(an)g(analogous)f(w)o(a)o(y)l(.)23 2306 y(Russo's)g(form)o(ula)f(\(see,)i(e.g.,)e([9)o(]\),)h(states)g (that)238 2416 y Fk(@)268 2404 y Fo(~)265 2416 y Fk(\022)287 2399 y Fo(BS)286 2424 y Fi(n)346 2416 y Fo(\()p Fk(b;)8 b(s)p Fo(\))p 238 2437 V 316 2476 a Fk(@)s(s)459 2447 y Fo(=)534 2407 y Fe(X)506 2503 y Fi(x)p Fh(2)553 2496 y Fg(~)547 2503 y Fi(V)f Fg(\()p Fi(n)p Fg(\))628 2447 y Fk(P)f Fo(\()p Fk(\016)701 2455 y Fi(x)722 2447 y Fk(A)756 2455 y Fi(n)779 2447 y Fo(\))982 2416 y Fk(@)1012 2404 y Fo(~)1009 2416 y Fk(\022)1031 2399 y Fo(BS)1030 2424 y Fi(n)1090 2416 y Fo(\()p Fk(b;)i(s)p Fo(\))p 982 2437 V 1061 2476 a Fk(@)s(b)1203 2447 y Fo(=)1276 2407 y Fe(X)1251 2498 y Fi(e)p Fh(2)p Fi(E)r Fg(\()p Fi(n)p Fg(\))1369 2447 y Fk(P)e Fo(\()p Fk(\016)1442 2455 y Fi(e)1460 2447 y Fk(A)1494 2455 y Fi(n)1516 2447 y Fo(\))p Fk(:)878 2678 y Fo(9)p eop %%Page: 10 10 10 9 bop 23 199 a Fo(The)14 b(pro)q(of)h(will)h(therefore)e(b)q(e)h (concluded)h(once)f(w)o(e)g(argue)f(that)g(for)g(eac)o(h)g Fk(x)f Fj(2)1430 188 y Fo(~)1423 199 y Fk(V)e Fo(\()p Fk(n)p Fo(\))464 333 y Fk(s)d(P)e Fo(\()p Fk(\016)566 342 y Fi(x)587 333 y Fk(A)621 342 y Fi(n)644 333 y Fo(\))19 b Fj(\024)736 274 y Fe(\022)809 302 y Fo(2)p 772 323 98 2 v 772 362 a(1)10 b Fj(\000)g Fk(b)874 274 y Fe(\023)905 282 y Fh(b)p Fi(D)q(=)p Fg(2)p Fh(c\000)p Fg(1)1059 293 y Fe(X)1049 382 y Fi(e)p Fh(2)p Fi(E)1111 386 y Fd(x)1137 333 y Fk(P)c Fo(\()p Fk(\016)1210 342 y Fi(e)1228 333 y Fk(A)1262 342 y Fi(n)1284 333 y Fo(\))p Fk(;)432 b Fo(\(10\))508 508 y Fk(s)8 b(P)e Fo(\()p Fk(\016)610 516 y Fi(x)631 508 y Fk(A)665 516 y Fi(n)688 508 y Fo(\))19 b Fj(\024)841 477 y Fo(2)864 460 y Fi(D)q Fh(\000)p Fg(1)p 785 498 207 2 v 785 537 a Fo(\(1)10 b Fj(\000)g Fk(s)p Fo(\))920 524 y Fi(D)q Fh(\000)p Fg(2)1015 468 y Fe(X)1005 557 y Fi(e)p Fh(2)p Fi(E)1067 561 y Fd(x)1093 508 y Fk(P)c Fo(\()p Fk(\016)1166 516 y Fi(e)1184 508 y Fk(A)1218 516 y Fi(n)1240 508 y Fo(\))p Fk(;)476 b Fo(\(11\))-48 638 y(and)15 b(use)g(the)g(fact)f(that)g(eac)o(h)h(b)q(ond)g(in)h Fk(E)s Fo(\()p Fk(n)p Fo(\))e(ob)o(viously)h(con)o(tains)f(at)g(most)g (2)h(end-p)q(oin)o(ts)g(in)1627 627 y(~)1620 638 y Fk(V)10 b Fo(\()p Fk(n)p Fo(\).)23 736 y(In)18 b(order)f(to)h(pro)o(v)o(e)f (\(10\),)g(\014rst)g(note)h(that)f Fk(s)8 b(P)e Fo(\()p Fk(\016)902 745 y Fi(x)923 736 y Fk(A)957 745 y Fi(n)979 736 y Fo(\))18 b(=)g Fk(P)6 b Fo(\()p Fk(A)1155 745 y Fi(x;n)1206 736 y Fo(\),)18 b(where)g Fk(A)1423 745 y Fi(x;n)1492 736 y Fo(is)h(the)f(ev)o(en)o(t)f(that)-48 810 y(the)g(site)h Fk(x)f Fo(is)h(o)q(ccupied)h(and)e(that)g Fk(\016)610 818 y Fi(x)631 810 y Fk(A)665 818 y Fi(n)705 810 y Fo(holds.)27 b(W)l(e)18 b(will)h(in)o(tro)q(duce)f(a)f (transformat)o(ion)e Fk(F)1605 793 y Fi(B)1599 818 y(x)1651 810 y Fo(:)h Fk(A)1714 818 y Fi(x;n)1782 810 y Fj(!)-48 883 y([)-18 891 y Fi(e)p Fh(2)p Fi(E)44 895 y Fd(x)65 883 y Fk(\016)85 891 y Fi(e)103 883 y Fk(A)137 891 y Fi(n)159 883 y Fo(,)f(with)g(the)g(follo)o(wing)g(prop)q(erties:)12 1019 y(i\))23 b Fk(P)6 b Fo(\()p Fk(F)153 1002 y Fi(B)147 1027 y(x)182 1019 y Fo(\()p Fk(!)r Fo(\)\))18 b Fj(\025)j Fo(\(1)9 b Fj(\000)h Fk(b)p Fo(\))472 1002 y Fh(b)p Fi(D)q(=)p Fg(2)p Fh(c\000)p Fg(1)616 1019 y Fk(P)c Fo(\()p Fk(!)r Fo(\).)0 1130 y(ii\))23 b(Eac)o(h)14 b(elemen)o(t)h(of)g Fj([)426 1138 y Fi(e)p Fh(2)p Fi(E)488 1142 y Fd(x)509 1130 y Fk(\016)529 1138 y Fi(e)547 1130 y Fk(A)581 1138 y Fi(n)618 1130 y Fo(is)h(the)f(image)f(of)h(at)f(most)g(2)1113 1112 y Fh(b)p Fi(D)q(=)p Fg(2)p Fh(c\000)p Fg(1)1264 1130 y Fo(elemen)o(ts)h(of)f Fk(A)1534 1138 y Fi(x;n)1586 1130 y Fo(.)-48 1265 y(The)k(existence)g(of)g(suc)o(h)g(a)g (transformat)o(ion)e(clearly)i(implies)i(\(10\).)27 b(T)l(o)18 b(pro)q(duce)g Fk(F)1461 1248 y Fi(B)1455 1273 y(x)1490 1265 y Fo(,)g(w)o(e)g(supp)q(ose)g(the)-48 1339 y(b)q(onds)d(in)h Fk(E)170 1347 y Fi(x)206 1339 y Fo(to)e(b)q(e)i(ordered)e(in)i(some)e (arbitrary)f(fashion,)i(and)f(w)o(e)h(pro)q(ceed)g(as)g(follo)o(ws.)23 1437 y(Giv)o(en)j Fk(!)g Fj(2)f Fk(A)283 1445 y Fi(x;n)334 1437 y Fo(,)g(let)h Fk(E)468 1420 y Fo(in)465 1445 y Fi(x;!)535 1437 y Fo(b)q(e)g(the)f(set)g(of)g(b)q(onds)h(in)g Fk(E)1029 1445 y Fi(x)1067 1437 y Fo(whic)o(h)g(in)g(the)f (con\014guration)f Fk(!)k Fo(ha)o(v)o(e)c(the)-48 1510 y(follo)o(wing)d(prop)q(ert)o(y)l(.)18 b(Their)c(end-p)q(oin)o(t)g (distinct)g(from)e Fk(x)i Fo(is)g(either)f(the)h(ro)q(ot,)e(or)h(else)h (it)g(is)g(a)f(site)h(whic)o(h)g(is)-48 1584 y(o)q(ccupied)f(and)f (connected)h(to)e(the)h(ro)q(ot)f(b)o(y)h(an)g(in)o(ternally)h(o)q (ccupied)g(c)o(hain)f(whic)o(h)h(do)q(es)f(not)g(pass)f(through)-48 1657 y Fk(x)p Fo(.)19 b(Let)12 b Fk(E)124 1640 y Fo(out)121 1665 y Fi(x;!)204 1657 y Fo(b)q(e)h(the)g(set)f(of)g(b)q(onds)h(in)g Fk(E)669 1665 y Fi(x)691 1657 y Fo(,)f(whic)o(h)h(in)h(the)e (con\014guration)f Fk(!)k Fo(ha)o(v)o(e)d(the)g(follo)o(wing)h(prop)q (ert)o(y)l(.)-48 1730 y(Their)k(end-p)q(oin)o(t)h(distinct)f(from)f Fk(x)h Fo(is)h(either)f(a)g(site)g(at)f(distance)i Fk(n)11 b Fo(+)h(1)17 b(from)e(the)i(ro)q(ot,)f(or)h(else)g(it)h(is)f(a)-48 1804 y(site)f(whic)o(h)f(is)h(o)q(ccupied)h(and)e(connected)h(to)e(a)i (site)f(at)g(distance)g Fk(n)c Fo(+)f(1)15 b(from)g(the)g(ro)q(ot)f(b)o (y)h(an)h(in)o(ternally)-48 1877 y(o)q(ccupied)i(c)o(hain)f(whic)o(h)g (do)q(es)g(not)f(pass)h(through)e Fk(x)p Fo(.)25 b(Since)18 b Fk(!)g Fj(2)d Fk(\016)1147 1886 y Fi(x)1168 1877 y Fk(A)1202 1886 y Fi(n)1225 1877 y Fo(,)i(w)o(e)f(ha)o(v)o(e)g Fk(E)1465 1860 y Fo(in)1462 1885 y Fi(x;!)1531 1877 y Fj(6)p Fo(=)g Fj(;)p Fo(,)h Fk(E)1671 1860 y Fo(out)1668 1885 y Fi(x;!)1754 1877 y Fj(6)p Fo(=)e Fj(;)-48 1951 y Fo(and)i Fk(E)78 1933 y Fo(in)75 1959 y Fi(x;!)140 1951 y Fj(\\)12 b Fk(E)218 1933 y Fo(out)215 1959 y Fi(x;!)302 1951 y Fo(=)17 b Fj(;)p Fo(.)27 b(If)18 b Fj(j)p Fk(E)514 1933 y Fo(in)511 1959 y Fi(x;!)563 1951 y Fj(j)e(\024)h(j)p Fk(E)693 1933 y Fo(out)690 1959 y Fi(x;!)760 1951 y Fj(j)p Fo(,)h(set)f Fk(E)910 1959 y Fi(x;!)980 1951 y Fo(=)f Fk(E)1067 1933 y Fo(in)1064 1959 y Fi(x;!)1118 1951 y Fo(;)i(otherwise)f(set)g Fk(E)1459 1959 y Fi(x;!)1529 1951 y Fo(=)f Fk(E)1616 1933 y Fo(out)1613 1959 y Fi(x;!)1684 1951 y Fo(.)27 b(Since)-48 2024 y Fj(j)p Fk(E)-2 2032 y Fi(x)19 2024 y Fj(j)17 b(\024)h Fk(D)q Fo(,)i(ob)o(viously)e Fj(j)p Fk(E)422 2032 y Fi(x;!)475 2024 y Fj(j)f(\024)h(b)p Fk(D)q(=)p Fo(2)p Fj(c)p Fo(.)30 b(De\014ne)18 b Fk(F)903 2007 y Fi(B)897 2032 y(x)932 2024 y Fo(\()p Fk(!)r Fo(\))f(as)h(the)g (con\014guration)f(obtained)h(from)f Fk(!)j Fo(b)o(y)-48 2097 y(making)c(v)m(acan)o(t)g(the)g(\014rst)g Fj(j)p Fk(E)478 2106 y Fi(x;!)530 2097 y Fj(j)11 b(\000)g Fo(1)16 b(b)q(onds)h(of)f Fk(E)859 2106 y Fi(x;!)912 2097 y Fo(.)24 b(In)17 b(the)f(con\014guration)g Fk(F)1397 2080 y Fi(B)1391 2106 y(x)1426 2097 y Fo(\()p Fk(!)r Fo(\))f(the)h(last)h(b)q(ond)f(in) -48 2171 y Fk(E)-15 2179 y Fi(x;!)51 2171 y Fo(is)e(piv)o(otal)f(for)f Fk(A)345 2179 y Fi(n)367 2171 y Fo(,)h(so)g(that)f Fk(F)578 2154 y Fi(B)572 2179 y(x)620 2171 y Fo(indeed)i(maps)f Fk(A)912 2179 y Fi(x;n)976 2171 y Fo(to)f Fj([)1059 2179 y Fi(e)p Fh(2)p Fi(E)1121 2183 y Fd(x)1142 2171 y Fk(\016)1162 2179 y Fi(e)1180 2171 y Fk(A)1214 2179 y Fi(n)1237 2171 y Fo(.)19 b(It)13 b(is)g(clear)h(that)e(also)g(\(i\))h(holds.)-48 2244 y(T)l(o)18 b(see)g(that)g(\(ii\))g(holds,)h(w)o(e)f(can)g(argue)g (as)f(follo)o(ws.)29 b(The)19 b(transformat)o(ion)d Fk(F)1373 2227 y Fi(B)1367 2252 y(x)1420 2244 y Fo(can)i(only)h(mo)q(dify)f(the) -48 2318 y(con\014guration)f(at)h(b)q(onds)g(inciden)o(t)h(to)f Fk(x)p Fo(.)29 b(Therefore,)17 b(b)o(y)i(lo)q(oking)f(at)g Fk(F)1268 2300 y Fi(B)1262 2326 y(x)1297 2318 y Fo(\()p Fk(!)r Fo(\))f(restricted)g(to)h(the)g(other)-48 2391 y(b)q(onds)f(and)f(sites)h(in)h Fk(E)s Fo(\()p Fk(n)p Fo(\))e(and)544 2379 y(~)537 2391 y Fk(V)11 b Fo(\()p Fk(n)p Fo(\))16 b(one)g(can)h(\014nd)g(what)f(the)h(set)f Fk(E)1214 2399 y Fi(x;!)1284 2391 y Fo(is.)25 b(If)17 b Fk(F)1434 2374 y Fi(B)1428 2399 y(x)1463 2391 y Fo(\()p Fk(!)1511 2374 y Fh(0)1522 2391 y Fo(\))e(=)h Fk(F)1641 2374 y Fi(B)1635 2399 y(x)1670 2391 y Fo(\()p Fk(!)1718 2374 y Fh(00)1738 2391 y Fo(\),)h Fk(!)1816 2374 y Fh(0)-48 2464 y Fo(and)e Fk(!)70 2447 y Fh(00)106 2464 y Fo(can)g(only)g (di\013er)h(at)e(the)h(b)q(onds)g(in)h Fk(E)756 2473 y Fi(x;!)807 2464 y Fc(0)h Fo(=)c Fk(E)914 2473 y Fi(x;!)965 2464 y Fc(00)t Fo(,)i(with)g(the)g(exception)g(of)g(the)g(last)f(suc)o (h)h(b)q(ond.)-48 2538 y(Since)h(the)f(cardinalit)o(y)g(of)f(this)i (set)e(where)h(they)f(can)h(di\013er)h(is)g(at)e(most)g Fj(b)p Fk(D)q(=)p Fo(2)p Fj(c)c(\000)g Fo(1,)k(this)h(pro)o(v)o(es)f (\(ii\).)867 2678 y(10)p eop %%Page: 11 11 11 10 bop 23 199 a Fo(In)15 b(order)g(to)f(pro)o(v)o(e)g(\(11\),)g(w)o (e)h(pro)q(ceed)g(in)h(a)f(similar)h(fashion.)k(W)l(e)c(will)h(in)o (tro)q(duce)e(a)g(transformation)-48 273 y Fk(F)-13 256 y Fi(S)-19 281 y(x)24 273 y Fo(:)d Fk(A)83 281 y Fi(x;n)147 273 y Fj(!)g([)234 281 y Fi(e)p Fh(2)p Fi(E)296 285 y Fd(x)318 273 y Fk(\016)338 281 y Fi(e)356 273 y Fk(A)390 281 y Fi(n)412 273 y Fo(,)j(with)g(the)f(follo)o(wing)h(prop)q(erties:) 12 408 y(i\))23 b Fk(P)6 b Fo(\()p Fk(F)153 391 y Fi(S)147 416 y(x)177 408 y Fo(\()p Fk(!)r Fo(\)\))19 b Fj(\025)h Fo(\(1)9 b Fj(\000)h Fk(s)p Fo(\))469 391 y Fi(D)q Fh(\000)p Fg(2)549 408 y Fk(P)c Fo(\()p Fk(!)r Fo(\).)0 519 y(ii\))23 b(Eac)o(h)14 b(elemen)o(t)h(of)g Fj([)426 527 y Fi(e)p Fh(2)p Fi(E)488 531 y Fd(x)509 519 y Fk(\016)529 527 y Fi(e)547 519 y Fk(A)581 527 y Fi(n)618 519 y Fo(is)h(the)f(image)f (of)h(at)f(most)g(2)1113 502 y Fi(D)q Fh(\000)p Fg(1)1200 519 y Fo(elemen)o(ts)h(of)f Fk(A)1470 527 y Fi(x;n)1522 519 y Fo(.)-48 655 y(The)e(existence)g(of)f(suc)o(h)h(a)f (transformation)e(clearly)j(implies)h(\(11\).)k(T)l(o)11 b(pro)q(duce)h Fk(F)1387 638 y Fi(S)1381 663 y(x)1411 655 y Fo(,)g(w)o(e)g(supp)q(ose)g(the)f(sites)-48 728 y(in)17 b Fk(V)32 736 y Fi(x)70 728 y Fo(to)f(b)q(e)h(ordered)f(in)h (some)f(arbitrary)e(fashion,)i(except)h(for)e(the)i(fact)e(that)h(if)h (the)f(ro)q(ot)f(is)i(con)o(tained)-48 801 y(in)i Fk(V)34 810 y Fi(x)55 801 y Fo(,)g(it)g(will)h(b)q(e)f(the)f(last)g(site)h(in)g (the)f(order.)29 b(Giv)o(en)19 b Fk(!)h Fj(2)f Fk(A)1095 810 y Fi(x;n)1146 801 y Fo(,)g(let)f Fk(V)1283 784 y Fo(in)1272 810 y Fi(x;!)1344 801 y Fo(b)q(e)h(the)f(set)g(of)g(sites)h (in)g Fk(V)1806 810 y Fi(x)-48 875 y Fo(whic)o(h)d(in)h(the)f (con\014guration)e Fk(!)k Fo(ha)o(v)o(e)d(the)h(follo)o(wing)g(prop)q (ert)o(y)l(.)21 b(The)16 b(site)g(is)g(either)g(the)g(ro)q(ot,)e(or)h (else)i(it)-48 948 y(is)h(o)q(ccupied,)h(the)f(b)q(ond)g(that)e(it)i (shares)f(with)h Fk(x)f Fo(is)i(o)q(ccupied)f(and)g(it)g(is)g (connected)g(to)f(the)g(ro)q(ot)g(b)o(y)g(an)-48 1022 y(in)o(ternally)h(o)q(ccupied)g(c)o(hain)g(whic)o(h)g(do)q(es)f(not)g (pass)g(through)f Fk(x)p Fo(.)27 b(Since)18 b Fk(!)g Fj(2)f Fk(\016)1361 1030 y Fi(x)1382 1022 y Fk(A)1416 1030 y Fi(n)1438 1022 y Fo(,)h(w)o(e)f(ha)o(v)o(e)g Fk(V)1681 1005 y Fo(in)1671 1030 y Fi(x;!)1740 1022 y Fj(6)p Fo(=)g Fj(;)p Fo(.)-48 1095 y(Let)46 1084 y(\026)39 1095 y Fk(V)75 1078 y Fo(out)65 1103 y Fi(x;!)164 1095 y Fo(b)q(e)22 b(the)f(set)f(of)h(sites)g(in)h Fk(V)642 1103 y Fi(x)685 1095 y Fo(whic)o(h)f(in)h(the)f(con\014guration)f Fk(!)j Fo(are)d(connected)h(to)g(a)g(site)g(at)-48 1168 y(distance)d Fk(n)12 b Fo(+)f(1)18 b(from)e(the)i(ro)q(ot)e(b)o(y)h(an)h(in)o 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b(s)p Fo(\))p 946 428 V 1025 468 a Fk(@)s(b)1744 439 y Ff(\(12\))-48 534 y(holds)16 b(for)g(arbitr)n(ary)g Fk(n)g Ff(and)h(arbitr)n(ary)e Fo(\()p Fk(b;)8 b(s)p Fo(\))k Fj(2)g Fo([)p Fk(b)830 543 y Fg(1)849 534 y Fk(;)c(b)889 543 y Fg(2)907 534 y Fo(])h Fj(\002)h Fo([)p Fk(s)1008 543 y Fg(1)1027 534 y Fk(;)e(s)1069 543 y Fg(2)1087 534 y Fo(])p Ff(.)10 661 y(i\))23 b(If)16 b Fo(\()p Fk(b;)8 b(s)p Fo(\))j Fj(2)i Fo(\([)p Fk(b)313 669 y Fg(1)331 661 y Fk(;)8 b(b)371 669 y Fg(2)389 661 y Fo(])h Fj(\002)h Fo([)p Fk(s)490 669 y Fg(1)508 661 y Fk(;)e(s)550 669 y Fg(2)568 661 y Fo(]\))h Fj(\\)h(P)679 669 y Fi(G)707 661 y Ff(,)16 b(then)g(for)g(al)r(l)f Fk(x)d Fj(\025)h Fo(0)j Ff(such)g(that)g Fo(\()p Fk(b)10 b Fo(+)f Fk(x;)f(s)h Fj(\000)h Fk(x=C)s Fo(\))i Fj(2)g Fo([)p Fk(b)1693 669 y Fg(1)1712 661 y Fk(;)c(b)1752 669 y Fg(2)1770 661 y Fo(])h Fj(\002)65 734 y Fo([)p Fk(s)99 743 y Fg(1)117 734 y Fk(;)f(s)159 743 y Fg(2)177 734 y Fo(])16 b Ff(we)g(have)g Fo(\()p Fk(b)10 b Fo(+)g Fk(x;)e(s)h Fj(\000)h Fk(x=C)s Fo(\))i Fj(2)h(P)779 743 y Fi(G)807 734 y Ff(.)-4 842 y(ii\))23 b(If)f Fo(\()p Fk(b;)8 b(s)p Fo(\))20 b Fj(2)j Fo(\([)p Fk(b)338 850 y Fg(1)356 842 y Fk(;)8 b(b)396 850 y Fg(2)414 842 y Fo(])13 b Fj(\002)h Fo([)p Fk(s)523 850 y Fg(1)541 842 y Fk(;)8 b(s)583 850 y Fg(2)601 842 y Fo(]\))13 b Fj(\\)h Fo(\()p Fj(P)738 850 y Fi(G)766 842 y Fo(\))784 825 y Fi(c)801 842 y Ff(,)22 b(then)g(for)f(al)r(l)g Fk(x)g Fj(\025)i Fo(0)e Ff(such)g(that)h Fo(\()p Fk(b)14 b Fj(\000)g Fk(x;)8 b(s)k Fo(+)i Fk(x=C)s Fo(\))22 b Fj(2)65 916 y Fo([)p Fk(b)97 924 y Fg(1)115 916 y Fk(;)8 b(b)155 924 y Fg(2)173 916 y Fo(])i Fj(\002)g Fo([)p Fk(s)275 924 y Fg(1)294 916 y Fk(;)e(s)336 924 y Fg(2)354 916 y Fo(])15 b Ff(we)h(have)g Fo(\()p Fk(b)10 b Fj(\000)g Fk(x;)e(s)h Fo(+)h Fk(x=C)s Fo(\))i Fj(2)h Fo(\()p Fj(P)973 924 y Fi(G)1001 916 y Fo(\))1019 898 y Fi(c)1035 916 y Ff(.)-48 1042 y Fl(Pro)q(of:)21 b Fo(F)l(rom)15 b(\(12\),)e(w)o(e)i (ha)o(v)o(e)f(under)h(the)g(conditions)g(in)h(\(i\):)569 1140 y(~)566 1152 y Fk(\022)588 1133 y Fo(BS)587 1160 y Fi(n)647 1152 y Fo(\()p Fk(b;)8 b(s)p Fo(\))k Fj(\024)807 1140 y Fo(~)803 1152 y Fk(\022)825 1133 y Fo(BS)824 1160 y Fi(n)885 1152 y Fo(\()p Fk(b)e Fo(+)g Fk(x;)e(s)h Fj(\000)h Fk(x=C)s Fo(\))p Fk(:)-48 1262 y Fo(Multiplying)19 b(b)o(y)e Fk(s)p Fo(,)g(letting)g Fk(n)f Fj(!)f(1)i Fo(and)g(using)g(\(5\),)f (giv)o(es)h Fk(\022)1059 1245 y Fo(BS)1118 1262 y(\()p Fk(b;)8 b(s)p Fo(\))15 b Fj(\024)h Fk(\022)1303 1245 y Fo(BS)1362 1262 y(\()p Fk(b)11 b Fo(+)h Fk(x;)c(s)i Fj(\000)h Fk(x=C)s Fo(\),)17 b(whic)o(h)-48 1335 y(pro)o(v)o(es)d(the)g (claim)i(in)g(\(i\).)23 1432 y(The)e(pro)q(of)h(of)f(\(ii\))h(is)h (analogous.)p 1797 1420 31 2 v 1797 1445 2 25 v 1825 1445 V 1797 1447 31 2 v 23 1529 a(In)f(com)o(bination)g(with)g(Prop)q (osition)g(2.1,)f(Prop)q(osition)g(3.1)g(pro)o(vides)h(estimates)f(on)i (the)f(b)q(eha)o(vior)g(of)-48 1602 y(the)d(function)g Fk(s)222 1611 y Fi(c)239 1602 y Fo(\()p Fj(\001)p Fo(\).)18 b(T)l(o)11 b(obtain)h(estimates)f(in)h(the)g(opp)q(osite)g(direction,)h (one)f(can)g(use)g(the)g(complemen)o(tary)-48 1676 y(inequalities)18 b(\(8\))d(or)g(\(9\).)22 b(F)l(or)16 b(our)g(purp)q(oses)g(one)g(can)g (do)g(b)q(etter,)f(though,)h(using)g(\(1\).)22 b(This)16 b(is)h(done)f(in)-48 1749 y(next)f(prop)q(osition,)f(in)i(whic)o(h)f(w) o(e)f(are)h(recalling)h(results)e(and)h(argumen)o(ts)e(from)h([15)o(].) -48 1973 y Fl(Prop)q(osition)19 b(3.2)k Ff(Supp)n(ose)17 b(that)f Fk(G)g Ff(is)h(an)f(in\014nite,)h(lo)n(c)n(al)r(ly)d (\014nite,)i(c)n(onne)n(cte)n(d)f(gr)n(aph.)10 2099 y(i\))23 b(If)16 b Fo(\()p Fk(b;)8 b(s)p Fo(\))k Fj(2)g(P)294 2107 y Fi(G)323 2099 y Ff(,)k(then)g(for)g(al)r(l)f Fo(0)e Fk(<)f(\013)h Fj(\024)f Fo(1)k Ff(such)h(that)f Fo(\()p Fk(b\013;)8 b(s=\013)p Fo(\))j Fj(2)i Fo([0)p Fk(;)8 b Fo(1])1320 2082 y Fg(2)1352 2099 y Ff(we)16 b(have)f Fo(\()p Fk(b\013;)8 b(s=\013)p Fo(\))j Fj(2)i(P)1785 2107 y Fi(G)1813 2099 y Ff(.)-4 2207 y(ii\))23 b(If)18 b Fo(\()p Fk(b;)8 b(s)p Fo(\))13 b Fj(2)i Fo(\()p Fj(P)318 2215 y Fi(G)346 2207 y Fo(\))364 2190 y Fi(c)380 2207 y Ff(,)j(then)f(for)h(al)r(l)e Fo(0)f Fk(<)g(\013)f Fj(\024)h Fo(1)i Ff(such)h(that)f Fo(\()p Fk(b=\013;)8 b(s\013)p Fo(\))13 b Fj(2)i Fo([0)p Fk(;)8 b Fo(1])1399 2190 y Fg(2)1433 2207 y Ff(we)17 b(have)g Fo(\()p Fk(b=\013;)8 b(s\013)p Fo(\))13 b Fj(2)65 2280 y Fo(\()p Fj(P)114 2289 y Fi(G)142 2280 y Fo(\))160 2263 y Fi(c)176 2280 y Ff(.)-48 2407 y Fl(Pro)q(of:)21 b Fo(T)l(o)14 b(pro)o(v)o(e)f(\(i\),) h(consider)g(the)h(random)e(graph)g Fk(G)955 2390 y Fh(0)981 2407 y Fo(obtaining)h(b)o(y)g(p)q(erforming)g(mixed)g(p)q(ercolation) -48 2480 y(on)i Fk(G)g Fo(with)g(parameters)e(\()p Fk(b;)8 b(s=\013)p Fo(\).)22 b(By)17 b(p)q(erforming)e(site)i(p)q(ercolation)f (on)g Fk(G)1317 2463 y Fh(0)1344 2480 y Fo(with)h(parameter)d Fk(\013)i Fo(\(inde-)-48 2554 y(p)q(enden)o(tly)d(of)g(the)f(pro)q (cess)h(that)e(randomly)h(originated)h Fk(G)983 2536 y Fh(0)994 2554 y Fo(\),)f(w)o(e)g(obtain)h(a)f(random)g(graph)g Fk(G)1601 2536 y Fh(00)1634 2554 y Fo(whic)o(h)h(has)867 2678 y(12)p eop %%Page: 13 13 13 12 bop -48 199 a Fo(the)15 b(la)o(w)h(of)f(a)g(random)f(graph)h (obtained)g(from)f(p)q(erforming)h(mixed)h(p)q(ercolation)f(with)h (parameters)d(\()p Fk(b;)8 b(s)p Fo(\))-48 273 y(on)14 b Fk(G)p Fo(.)19 b(Since)14 b(w)o(e)g(are)f(supp)q(osing)h(that)f(\()p Fk(b;)8 b(s)p Fo(\))j Fj(2)i(P)826 281 y Fi(G)868 273 y Fo(w)o(e)h(m)o(ust)f(ha)o(v)o(e)g(that)f Fk(G)1280 256 y Fh(00)1315 273 y Fo(con)o(tains)h(a.s.)19 b(some)13 b(in\014nite)-48 346 y(comp)q(onen)o(t.)22 b(This)16 b(means)f(that)g Fk(G)582 329 y Fh(0)610 346 y Fo(supp)q(orts)g(site)h (p)q(ercolation)g(at)f(parameter)f Fk(\013)p Fo(.)23 b(F)l(rom)16 b(\(1\))e(it)j(follo)o(ws)-48 419 y(that)d Fk(G)86 402 y Fh(0)113 419 y Fo(also)g(supp)q(orts)h(b)q(ond)g(p)q (ercolation)g(at)f(parameter)f Fk(\013)p Fo(.)20 b(In)c(other)e(w)o (ords,)g(the)h(random)e(graph)i Fk(G)1797 402 y Fh(000)-48 493 y Fo(obtained)e(b)o(y)g(p)q(erforming)f(b)q(ond)i(p)q(ercolation)f (on)g Fk(G)869 476 y Fh(0)893 493 y Fo(with)g(parameter)e Fk(\013)j Fo(\(indep)q(enden)o(tly)g(of)f(the)g(pro)q(cess)-48 566 y(that)k(randomly)g(originated)h Fk(G)511 549 y Fh(0)522 566 y Fo(\))g(con)o(tains)f(a.s.)28 b(some)18 b(in\014nite)h(comp)q (onen)o(t.)28 b(But)18 b Fk(G)1493 549 y Fh(000)1541 566 y Fo(has)g(the)g(la)o(w)g(of)-48 640 y(a)e(random)e(graph)h (obtained)h(from)f(p)q(erforming)g(mixed)h(p)q(ercolation)g(with)g (parameters)e(\()p Fk(b\013;)8 b(s=\013)p Fo(\))14 b(on)i Fk(G)p Fo(.)-48 713 y(This)f(pro)o(v)o(es)f(\(i\).)23 810 y(The)g(claim)i(in)g(\(ii\))f(follo)o(ws)g(from)e(that)h(in)i (\(i\).)p 1797 798 31 2 v 1797 823 2 25 v 1825 823 V 1797 825 31 2 v -48 908 a Fl(Pro)q(of)i(of)f(Theorem)f(1.4:)21 b Fo(This)15 b(theorem)f(is)i(immediate)f(from)e(Prop)q(ositions)h (2.1,)g(3.1,)f(and)i(3.2.)p 1797 895 V 1797 920 2 25 v 1825 920 V 1797 922 31 2 v -48 1194 a Fp(4)65 b(Pro)r(of)21 b(of)g(Theorem)f(1.1)-48 1336 y Fl(Pro)q(of)c(of)e(part)i(\(a\))f(of)g (Theorem)e(1.1:)20 b Fo(W)l(e)14 b(will)g(supp)q(ose,)f(with)g(no)f (loss,)h(that)f Fk(b)1427 1345 y Fi(c)1457 1336 y Fk(<)g Fo(1.)19 b(W)l(e)14 b(will)g(apply)-48 1410 y(Prop)q(osition)g(3.1)g (with)h Fk(b)390 1418 y Fg(1)421 1410 y Fo(=)e Fk(b)488 1418 y Fi(c)505 1410 y Fo(,)i Fk(b)552 1418 y Fg(2)583 1410 y Fo(=)e(\()p Fk(b)668 1418 y Fi(c)695 1410 y Fo(+)d(1\))p Fk(=)p Fo(2,)j Fk(s)874 1418 y Fg(1)905 1410 y Fo(=)g(1)p Fk(=)p Fo(2,)h Fk(s)1070 1418 y Fg(2)1101 1410 y Fo(=)f(1,)h(and)489 1535 y Fk(C)i Fo(=)695 1504 y(2)718 1487 y Fh(b)p Fi(D)q(=)p Fg(2)p Fh(c)p 590 1524 328 2 v 590 1564 a Fk(s)611 1573 y Fg(1)629 1564 y Fo(\(1)10 b Fj(\000)g Fk(b)744 1573 y Fg(2)763 1564 y Fo(\))781 1551 y Fh(b)p Fi(D)q(=)p Fg(2)p Fh(c\000)p Fg(1)934 1535 y Fo(=)1063 1504 y(2)1086 1487 y Fg(2)p Fh(b)p Fi(D)q(=)p Fg(2)p Fh(c)p 987 1524 286 2 v 987 1564 a Fo(\(1)f Fj(\000)h Fk(b)1101 1573 y Fi(c)1119 1564 y Fo(\))1137 1551 y Fh(b)p Fi(D)q(=)p Fg(2)p Fh(c\000)p Fg(1)1278 1535 y Fk(:)-48 1648 y Fo(Note)k(that)g (this)h(c)o(hoice)h(of)e Fk(C)19 b Fo(is)c(p)q(ossible)h(thanks)e(to)h (Prop)q(osition)f(2.1.)23 1745 y(Giv)o(en)e Fk(\017)h(>)g Fo(0,)e(set)g Fk(b)i Fo(=)g Fk(b)442 1754 y Fi(c)462 1745 y Fo(+)s Fk(\017)p Fo(,)g(and)e Fk(s)i Fo(=)g(1.)18 b(If)12 b Fk(\017)g Fo(is)g(small)f(enough,)h(then)g(\()p Fk(b;)c(s)p Fo(\))j Fj(2)i Fo(\([)p Fk(b)1459 1754 y Fg(1)1477 1745 y Fk(;)8 b(b)1517 1754 y Fg(2)1535 1745 y Fo(])s Fj(\002)s Fo([)p Fk(s)1623 1754 y Fg(1)1641 1745 y Fk(;)g(s)1683 1754 y Fg(2)1701 1745 y Fo(]\))s Fj(\\)s(P)1799 1754 y Fi(G)-48 1819 y Fo(and)j Fk(x)h Fo(=)h(\(1)s Fj(\000)s Fk(b)223 1827 y Fi(c)239 1819 y Fo(\))p Fk(=)p Fo(2)s Fj(\000)s Fk(\017)e(>)i Fo(0.)18 b(Since)12 b Fk(x)g Fj(\024)h Fo(1)e(and)g Fk(C)16 b Fj(\025)d Fo(2,)e(w)o(e)g(also)g(ha)o(v)o(e)g(\()p Fk(b)s Fo(+)s Fk(x;)d(s)s Fj(\000)s Fk(x=C)s Fo(\))h Fj(2)k Fo([)p Fk(b)1560 1827 y Fg(1)1579 1819 y Fk(;)8 b(b)1619 1827 y Fg(2)1637 1819 y Fo(])s Fj(\002)s Fo([)p Fk(s)1725 1827 y Fg(1)1742 1819 y Fk(;)g(s)1784 1827 y Fg(2)1802 1819 y Fo(].)-48 1892 y(Prop)q(osition)14 b(3.1\(i\))g(giv)o(es)h(no)o (w)f(\()p Fk(b)c Fo(+)g Fk(x;)e(s)h Fj(\000)h Fk(x=C)s Fo(\))i Fj(2)h(P)923 1900 y Fi(G)951 1892 y Fo(.)20 b(This)15 b(can)g(b)q(e)g(rewritten)f(as)490 1949 y Fe( )528 1990 y Fo(1)c(+)g Fk(b)625 1998 y Fi(c)p 528 2010 115 2 v 574 2050 a Fo(2)655 2021 y Fk(;)k Fo(1)c Fj(\000)765 1990 y Fo(\(1)f Fj(\000)h Fk(b)879 1998 y Fi(c)896 1990 y Fo(\))914 1973 y Fh(b)p Fi(D)q(=)p Fg(2)p Fh(c)p 765 2010 244 2 v 799 2050 a Fo(2)822 2037 y Fg(2)p Fh(b)p Fi(D)q(=)p Fg(2)p Fh(c)p Fg(+1)1023 2021 y Fo(+)1082 1990 y Fk(\017)p 1073 2010 36 2 v 1073 2050 a(C)1114 1949 y Fe(!)1167 2021 y Fj(2)20 b(P)1248 2029 y Fi(G)1276 2021 y Fk(:)-48 2147 y Fo(Since)c Fk(\017)d(>)g Fo(0)h(can)h(b)q(e)h (tak)o(en)e(arbitrarily)g(small,)h(w)o(e)f(obtain)572 2270 y Fk(s)593 2278 y Fi(c)618 2210 y Fe(\022)653 2239 y Fo(1)c(+)g Fk(b)750 2247 y Fi(c)p 653 2259 115 2 v 699 2299 a Fo(2)772 2210 y Fe(\023)816 2270 y Fj(\024)i Fo(1)e Fj(\000)946 2239 y Fo(\(1)f Fj(\000)h Fk(b)1060 2247 y Fi(c)1078 2239 y Fo(\))1096 2222 y Fh(b)p Fi(D)q(=)p Fg(2)p Fh(c)p 946 2259 244 2 v 980 2299 a Fo(2)1003 2286 y Fg(2)p Fh(b)p Fi(D)q(=)p Fg(2)p Fh(c)p Fg(+1)1194 2270 y Fk(:)-48 2383 y Fo(The)15 b(pro)q(of)f(is)h(therefore)f(complete,)g (since)i Fk(s)733 2391 y Fi(c)763 2383 y Fo(=)d Fk(s)832 2391 y Fi(c)849 2383 y Fo(\(1\))e Fj(\024)i Fk(s)988 2391 y Fi(c)1005 2383 y Fo(\(\(1)c(+)h Fk(b)1137 2391 y Fi(c)1154 2383 y Fo(\))p Fk(=)p Fo(2\).)p 1797 2370 31 2 v 1797 2395 2 25 v 1825 2395 V 1797 2397 31 2 v -48 2480 a Fl(Pro)q(of)24 b(of)f(part)g(\(b\))g(of)g(Theorem)f(1.1:)31 b Fo(In)20 b(order)f(to)g(construct)g(our)g(example,)i(w)o(e)e(need)i (to)e(\014rst)-48 2554 y(in)o(tro)q(duce)e(v)m(arious)g(t)o(yp)q(es)g (of)g(\014nite)g(graphs,)g(to)f(b)q(e)i(used)f(as)g(our)g(building)i (blo)q(c)o(ks.)26 b(Eac)o(h)16 b(one)i(of)e(these)867 2678 y(13)p eop %%Page: 14 14 14 13 bop -48 199 a Fo(\014nite)21 b(graphs)d(will)k(ha)o(v)o(e)d (exactly)h(2)f(v)o(ertices)h(whic)o(h)g(will)h(b)q(e)g(called)g(its)f (exterior)f(v)o(ertices,)h(the)g(other)-48 273 y(v)o(ertices)15 b(b)q(eing)g(called)h(in)o(terior)f(v)o(ertices.)23 371 y(A)e Fk(K)s Fo(-bridge)h(is)f(a)g(\014nite)h(graph)e(with)h Fk(K)d Fo(+)c(2)13 b(v)o(ertices,)f(2)h(of)g(them)f(b)q(eing)i(called)g (exterior)f(v)o(ertices)f(and)-48 444 y(the)j(others)g(b)q(eing)h (called)h(in)o(terior)e(v)o(ertices.)21 b(The)15 b(set)g(of)g(edges)h (is)g(de\014ned)g(b)o(y)f(declaring)h(that)f(eac)o(h)g(one)-48 518 y(of)h(the)h(2)g(exterior)f(v)o(ertices)g(is)h(connected)g(b)o(y)g (edges)f(to)h(eac)o(h)f(one)h(of)f(the)h(in)o(terior)f(v)o(ertices.)25 b(Note)16 b(that)-48 591 y(the)d(graph)e(has)i(2)p Fk(K)j Fo(edges,)c(that)g(the)g(t)o(w)o(o)f(exterior)h(v)o(ertices)g(pla)o(y)h (the)f(same)g(role)h(of)f(eac)o(h)h(other)e(and)i(ha)o(v)o(e)-48 665 y(degree)i Fk(K)s Fo(,)g(and)g(that)f(the)h Fk(K)j Fo(in)o(terior)d(v)o(ertices)f(pla)o(y)h(the)g(same)f(role)h(of)g(eac)o (h)g(other)f(and)g(ha)o(v)o(e)h(degree)f(2.)23 763 y(Supp)q(ose)h(that) f Fk(K)336 771 y Fg(1)368 763 y Fj(\024)e Fk(K)453 771 y Fg(2)472 763 y Fo(.)20 b(A)15 b(\()p Fk(K)610 771 y Fg(1)629 763 y Fk(;)8 b(K)688 771 y Fg(2)706 763 y Fo(\)-double-bridge) 16 b(is)f(a)g(graph)f(obtained)h(b)o(y)g(iden)o(tifying)h(one)f(of)-48 836 y(the)e(exterior)f(v)o(ertices)g(of)g(a)h Fk(K)477 845 y Fg(1)496 836 y Fo(-bridge)g(with)g(one)g(of)f(the)h(exterior)e(v) o(ertices)i(of)f(a)h Fk(K)1403 845 y Fg(2)1422 836 y Fo(-bridge.)19 b(The)13 b(in)o(terior)-48 910 y(v)o(ertices)k(of)g(the) h(t)o(w)o(o)e(original)i(graphs,)e(as)i(w)o(ell)g(as)f(the)g(v)o(ertex) g(resulting)h(from)e(the)i(iden)o(ti\014cation)g(just)-48 983 y(describ)q(ed)g(are)e(considered)h(to)f(b)q(e)h(in)o(terior)g(v)o (ertices)f(of)h(the)f(new)h(graph,)f(while)i(the)e(other)g(t)o(w)o(o)g (v)o(ertices)-48 1056 y(are)d(considered)i(to)e(b)q(e)i(exterior)e(v)o (ertices)g(of)h(the)g(new)g(graph.)k(Note)13 b(that)g(the)h(v)o(ertex)f (resulting)h(from)f(the)-48 1130 y(iden)o(ti\014cation)j(has)f(degree)g Fk(K)485 1138 y Fg(1)514 1130 y Fo(+)c Fk(K)598 1138 y Fg(2)617 1130 y Fo(,)k(the)g(other)f(in)o(terior)h(v)o(ertices)g(ha)o (v)o(e)g(degree)g(2,)g(one)g(of)g(the)g(exterior)-48 1203 y(v)o(ertices)e(has)h(degree)f Fk(K)371 1211 y Fg(1)404 1203 y Fo(\(it)h(will)h(b)q(e)f(called)h(the)f(\014rst)f(exterior)g(v)o (ertex)f(of)i(the)f(graph\))g(and)h(the)f(other)g(one)-48 1277 y(has)h(degree)g Fk(K)210 1285 y Fg(2)244 1277 y Fo(\(it)g(will)i(b)q(e)g(called)f(the)g(second)f(exterior)g(v)o(ertex)f (of)h(the)h(graph\).)j(\(If)c Fk(K)1498 1285 y Fg(1)1530 1277 y Fo(=)e Fk(K)1615 1285 y Fg(2)1634 1277 y Fo(,)j(then)f(the)-48 1350 y(t)o(w)o(o)h(exterior)h(v)o(ertices)h(pla)o(y)g(the)g(same)f (role)h(of)g(eac)o(h)f(other)h(and)g(the)f(c)o(hoices)i(of)e(the)h (\014rst)f(one)h(and)g(the)-48 1423 y(second)e(one)g(are)f(arbitrary)l (.\))23 1522 y(An)g Fk(l)q Fo(-c)o(hain)i(of)e(\()p Fk(K)352 1530 y Fg(1)371 1522 y Fk(;)8 b(K)430 1530 y Fg(2)448 1522 y Fo(\)-double-bridges)15 b(is)g(a)g(graph)e(obtained)i(as)f (follo)o(ws.)19 b(Start)14 b(with)g(a)h(sequence)-48 1595 y(of)e Fk(l)j Fo(disjoin)o(t)e(\()p Fk(K)247 1603 y Fg(1)265 1595 y Fk(;)8 b(K)324 1603 y Fg(2)342 1595 y Fo(\)-double-bridges.)20 b(No)o(w,)12 b(for)h Fk(i)g Fo(=)g(1)p Fk(;)8 b(:::;)g(l)e Fj(\000)i Fo(1)13 b(iden)o(tify)i(the)e (second)h(exterior)f(v)o(ertex)-48 1668 y(of)18 b(the)g Fk(i)p Fo(-th)g(graph)g(in)h(the)f(sequence)g(with)g(the)h(\014rst)e (exterior)g(v)o(ertex)g(of)h(the)g(\()p Fk(i)12 b Fo(+)h(1\)-th)k (graph)g(in)i(the)-48 1742 y(sequence.)g(The)14 b(in)o(terior)e(v)o (ertices)h(of)g(the)g Fk(l)i Fo(original)e(graphs,)f(as)h(w)o(ell)h(as) f(the)g(v)o(ertices)g(resulting)g(from)f(the)-48 1815 y(iden)o(ti\014cations)k(just)g(describ)q(ed)h(are)e(considered)i(to)e (b)q(e)i(in)o(terior)e(v)o(ertices)h(of)f(the)h(new)g(graph,)f(while)i (the)-48 1889 y(other)e(t)o(w)o(o)f(v)o(ertices)h(\(the)h(\014rst)f (exterior)f(v)o(ertex)h(of)g(the)h(\014rst)f(graph)g(and)h(the)f (second)h(exterior)f(v)o(ertex)g(of)-48 1962 y(the)e(last)h(graph\))e (are)h(considered)h(to)f(b)q(e)h(exterior)f(v)o(ertices)g(of)g(the)g (new)h(graph.)k(The)c(\014rst)f(exterior)f(v)o(ertex)-48 2035 y(of)17 b(the)h(\014rst)f(graph)g(is)h(called)h(the)f(\014rst)f (exterior)g(v)o(ertex)g(of)g(the)h(new)g(graph,)f(and)h(the)f(second)h (exterior)-48 2109 y(v)o(ertex)f(of)g(the)h(last)g(graph)f(is)h(called) h(the)f(second)g(exterior)f(v)o(ertex)g(of)g(the)h(new)g(graph.)27 b(Note)17 b(that)g(the)-48 2182 y(degrees)f(of)h(the)f(exterior)g(v)o (ertices)h(are)f Fk(K)695 2190 y Fg(1)731 2182 y Fo(and)h Fk(K)859 2190 y Fg(2)878 2182 y Fo(,)f(while)i(the)f(degrees)f(of)h (the)f(in)o(terior)h(v)o(ertices)f(are)g(2)-48 2256 y(and)f Fk(K)78 2264 y Fg(1)107 2256 y Fo(+)10 b Fk(K)190 2264 y Fg(2)209 2256 y Fo(.)23 2354 y(A)24 b(decorated)f Fk(l)q Fo(-c)o(hain)i(of)f(\()p Fk(K)571 2362 y Fg(1)589 2354 y Fk(;)8 b(K)648 2362 y Fg(2)667 2354 y Fo(\)-double-bridges)24 b(is)h(a)f(graph)f(obtained)h(from)f(an)i Fk(l)q Fo(-c)o(hain)g(of)-48 2427 y(\()p Fk(K)8 2436 y Fg(1)27 2427 y Fk(;)8 b(K)86 2436 y Fg(2)104 2427 y Fo(\)-double-bridges)21 b(b)o(y)f(adding)h(t)o (w)o(o)e(new)h(v)o(ertices:)31 b Fk(v)1070 2436 y Fh(\000)1118 2427 y Fo(and)21 b Fk(v)1234 2436 y Fg(+)1261 2427 y Fo(,)h(the)e(\014rst)g(of)g(them)h(is)g(joined)-48 2501 y(b)o(y)14 b(an)g(edge)h(to)f(the)g(\014rst)f(exterior)h(v)o(ertex)f (of)h(the)g Fk(l)q Fo(-c)o(hain)i(of)e(\()p Fk(K)1090 2509 y Fg(1)1108 2501 y Fk(;)8 b(K)1167 2509 y Fg(2)1185 2501 y Fo(\)-double-bridges,)15 b(and)f(the)g(second)867 2678 y(14)p eop %%Page: 15 15 15 14 bop -48 199 a Fo(one)14 b(is)h(joined)f(b)o(y)g(an)g(edge)g(to)g (the)g(second)g(exterior)f(v)o(ertex)g(of)h(the)g Fk(l)q Fo(-c)o(hain)h(of)f(\()p Fk(K)1400 208 y Fg(1)1418 199 y Fk(;)8 b(K)1477 208 y Fg(2)1495 199 y Fo(\)-double-bridges.)-48 273 y(The)16 b(v)o(ertices)h Fk(v)234 281 y Fh(\000)278 273 y Fo(and)g Fk(v)390 281 y Fg(+)434 273 y Fo(are)e(the)i(only)g (exterior)e(v)o(ertices)h(of)g(the)h(new)f(graph,)g(and)g(are)g(called) i(resp)q(ec-)-48 346 y(tiv)o(ely)h(its)g(\014rst)g(and)g(second)g (exterior)f(v)o(ertex.)30 b(Note)19 b(that)f(the)h(degree)f(of)h(the)g (exterior)f(v)o(ertices)g(is)i(1,)-48 419 y(while)c(the)f(degrees)f(of) h(the)g(in)o(terior)f(v)o(ertices)h(are)f Fk(K)872 428 y Fg(1)901 419 y Fo(+)c(1,)15 b Fk(K)1035 428 y Fg(2)1064 419 y Fo(+)10 b(1,)k(2)h(and)g Fk(K)1323 428 y Fg(1)1352 419 y Fo(+)10 b Fk(K)1435 428 y Fg(2)1454 419 y Fo(.)23 516 y(W)l(e)h(are)g(ready)f(no)o(w)g(to)g(in)o(tro)q(duce)h(an)g (in\014nite)h(graph)e(whic)o(h)h(will)i(serv)o(e)d(as)g(our)h(example.) 18 b(Its)11 b(maximal)-48 589 y(degree)i(will)i(b)q(e)e Fk(D)i Fo(and)f(for)e(it)h(to)g(satisfy)f(the)i(desired)f(relation)g(b) q(et)o(w)o(een)g Fk(s)1262 597 y Fi(c)1293 589 y Fo(and)g Fk(b)1398 597 y Fi(c)1415 589 y Fo(,)g(a)g(certain)g(parameter)-48 662 y Fk(l)19 b Fo(will)h(ha)o(v)o(e)d(to)g(b)q(e)h(tak)o(en)f(large)h (enough.)28 b(F)l(or)17 b(the)h(momen)o(t,)f(recall)h(that)f Fj(d)p Fk(x)p Fj(e)h Fo(is)g(the)g(smallest)g(in)o(teger)-48 736 y(whic)o(h)c(is)g(not)f(smaller)h(than)f Fk(x)p Fo(.)19 b(Note)13 b(that)g Fj(b)p Fk(D)q(=)p Fo(2)p Fj(c)8 b Fo(+)g Fj(d)p Fk(D)q(=)p Fo(2)p Fj(e)k Fo(=)g Fk(D)q Fo(.)20 b(Let)14 b Fk(T)1260 744 y Fg(2)1293 736 y Fo(denote)f(the)h (binary)g(tree,)f(i.e.,)-48 809 y(the)g(tree)g(in)h(whic)o(h)g(eac)o(h) f(v)o(ertex)f(has)h(degree)h(3.)k(Our)c(graph)e(will)j(b)q(e)f(denoted) f(b)o(y)g Fk(T)1422 817 y Fg(2)p Fi(;D)q(;l)1500 809 y Fo(,)g(and)g(is)h(obtained)-48 883 y(b)o(y)h(replacing)h(eac)o(h)g (edge)f(of)g Fk(T)493 891 y Fg(2)528 883 y Fo(with)g(a)h(decorated)e Fk(l)q Fo(-c)o(hain)j(of)e(\()p Fj(b)p Fk(D)q(=)p Fo(2)p Fj(c)p Fk(;)8 b Fj(d)p Fk(D)q(=)p Fo(2)p Fj(e)p Fo(\)-double-bridges,) 14 b(in)i(the)-48 956 y(sense)g(that)f(the)i(exterior)e(v)o(ertices)h (of)f(the)h(decorated)g Fk(l)q Fo(-c)o(hains)g(pla)o(y)h(the)f(role)g (of)g(the)g(v)o(ertices)f(of)h Fk(T)1730 964 y Fg(2)1749 956 y Fo(.)24 b(In)-48 1029 y(order)12 b(to)g(assure)g(that)g(the)h (graph)f Fk(T)579 1038 y Fg(2)p Fi(;D)q(;l)670 1029 y Fo(is)h(quasi-transitiv)o(e,)g(the)f(orien)o(tation)g(of)h(the)g (decorated)e Fk(l)q Fo(-c)o(hains)-48 1103 y(of)i(\()p Fj(b)p Fk(D)q(=)p Fo(2)p Fj(c)p Fk(;)8 b Fj(d)p Fk(D)q(=)p Fo(2)p Fj(e)p Fo(\)-double-bridges)k(is)h(relev)m(an)o(t,)g(and)g(a)g (prop)q(er)f(c)o(hoice)i(corresp)q(onds)e(to)g(iden)o(tifying)i(the)-48 1176 y(second)i(exterior)f(v)o(ertex)h(of)f(eac)o(h)h(suc)o(h)g (decorated)g Fk(l)q Fo(-c)o(hain)h(with)f(the)g(\014rst)f(exterior)h(v) o(ertices)f(of)h(2)g(other)-48 1250 y(suc)o(h)f(decorated)f Fk(l)q Fo(-c)o(hains.)20 b(Note)14 b(that)g(indeed)i(the)f(maximal)g (degree)f(of)h Fk(T)1275 1258 y Fg(2)p Fi(;D)q(;l)1367 1250 y Fo(is)h Fk(D)q Fo(.)23 1346 y(Before)f(w)o(e)g(pro)q(ceed,)h (recall)h(that)e(from)g(branc)o(hing)h(theory)l(,)f(one)h(kno)o(ws)e (that)h(the)h(critical)h(v)m(alue)g(for)-48 1419 y(b)q(ond)e(p)q (ercolation)g(on)g Fk(T)390 1427 y Fg(2)424 1419 y Fo(is)g(1)p Fk(=)p Fo(2.)k(This)c(allo)o(ws)g(us)g(to)f(write)h(for)f Fk(T)1132 1427 y Fg(2)p Fi(;D)q(;l)-48 1531 y Fk(b)-29 1512 y Fg(2)-29 1539 y Fi(c)-3 1484 y Fe(\020)22 1531 y Fo(1)c Fj(\000)g Fo(\(1)f Fj(\000)h Fk(b)214 1512 y Fg(2)214 1539 y Fi(c)233 1531 y Fo(\))251 1512 y Fh(b)p Fi(D)q(=)p Fg(2)p Fh(c)345 1484 y Fe(\021)370 1492 y Fi(l)390 1484 y Fe(\020)415 1531 y Fo(1)f Fj(\000)i Fo(\(1)e Fj(\000)h Fk(b)607 1512 y Fg(2)607 1539 y Fi(c)626 1531 y Fo(\))644 1512 y Fh(d)p Fi(D)q(=)p Fg(2)p Fh(e)738 1484 y Fe(\021)762 1492 y Fi(l)795 1531 y Fo(=)855 1500 y(1)p 855 1520 23 2 v 855 1560 a(2)903 1531 y(=)20 b Fk(s)979 1512 y Fg(2+2)p Fi(l)979 1539 y(c)1058 1484 y Fe(\020)1083 1531 y Fo(1)10 b Fj(\000)g Fo(\(1)f Fj(\000)h Fk(s)1277 1539 y Fi(c)1294 1531 y Fo(\))1312 1512 y Fh(b)p Fi(D)q(=)p Fg(2)p Fh(c)1406 1484 y Fe(\021)1431 1492 y Fi(l)1451 1484 y Fe(\020)1476 1531 y Fo(1)g Fj(\000)g Fo(\(1)f Fj(\000)h Fk(s)1670 1539 y Fi(c)1687 1531 y Fo(\))1705 1512 y Fh(d)p Fi(D)q(=)p Fg(2)p Fh(e)1799 1484 y Fe(\021)1824 1492 y Fi(l)1844 1531 y Fk(:)1747 1604 y Fo(\(13\))-48 1678 y(Note,)k(in)i(particular,)e(that)792 1751 y(lim)785 1778 y Fi(l)p Fh(!1)870 1751 y Fk(b)889 1759 y Fi(c)918 1751 y Fo(=)f(1)p Fk(;)745 b Fo(\(14\))-48 1845 y(since)16 b(otherwise)e(the)h(l.h.s.)k(of)c(\(13\))e(w)o(ould)i (b)q(ecome)g(smaller)g(than)f(1)p Fk(=)p Fo(2)g(for)g(some)g(large)h(v) m(alues)h(of)e Fk(l)q Fo(.)23 1941 y(Since)k Fj(b)p Fk(D)q(=)p Fo(2)p Fj(c)f(\024)g(d)p Fk(D)q(=)p Fo(2)p Fj(e)h Fo(and)g(0)e Fk(<)h(s)681 1949 y Fi(c)715 1941 y Fj(\024)h Fo(1,)f(w)o(e)g(can)h (extract)e(the)i(follo)o(wing)f(t)o(w)o(o)f(inequalities)j(from)-48 2015 y(\(13\).)543 2088 y Fk(b)562 2069 y Fg(2)562 2096 y Fi(c)588 2041 y Fe(\020)613 2088 y Fo(1)10 b Fj(\000)g Fo(\(1)f Fj(\000)h Fk(b)805 2069 y Fg(2)805 2096 y Fi(c)824 2088 y Fo(\))842 2069 y Fh(b)p Fi(D)q(=)p Fg(2)p Fh(c)936 2041 y Fe(\021)961 2049 y Fg(2)p Fi(l)1010 2088 y Fj(\024)1070 2057 y Fo(1)p 1070 2077 V 1070 2117 a(2)1118 2088 y Fj(\024)20 b Fk(s)1194 2069 y Fg(2)p Fi(l)1194 2096 y(c)1224 2088 y Fk(:)510 b Fo(\(15\))-48 2182 y(F)l(rom)15 b(the)f(\014rst)h (inequalit)o(y)g(in)h(\(15\))e(and)g(from)g(\(14\),)f(w)o(e)i(can)g (conclude)g(that)f(for)g(large)h Fk(l)q Fo(,)494 2316 y(1)9 b Fj(\000)h Fo(\(1)g Fj(\000)g Fk(b)686 2296 y Fg(2)686 2324 y Fi(c)705 2316 y Fo(\))723 2296 y Fh(b)p Fi(D)q(=)p Fg(2)p Fh(c)829 2316 y Fj(\024)877 2244 y Fe( )951 2284 y Fo(1)p 914 2305 97 2 v 914 2314 a Fj(p)p 952 2314 23 2 v 37 x Fo(2)p Fk(b)994 2360 y Fi(c)1016 2244 y Fe(!)1049 2252 y Fg(1)p Fi(=l)1108 2316 y Fj(\024)i Fo(\()p Fk(b)1192 2324 y Fi(c)1209 2316 y Fo(\))1227 2296 y Fg(1)p Fi(=l)1273 2316 y Fk(:)461 b Fo(\(16\))-48 2438 y(F)l(rom)15 b(\(15\))e(and)i(\(16\),)e(w)o(e)i(obtain)f(no)o(w)39 2544 y Fk(s)60 2553 y Fi(c)118 2544 y Fj(\025)42 b Fo(\()p Fk(b)232 2553 y Fi(c)249 2544 y Fo(\))267 2525 y Fg(1)p Fi(=l)320 2497 y Fe(\020)345 2544 y Fo(1)10 b Fj(\000)g Fo(\(1)f Fj(\000)h Fk(b)537 2525 y Fg(2)537 2552 y Fi(c)556 2544 y Fo(\))574 2525 y Fh(b)p Fi(D)q(=)p Fg(2)p Fh(c)668 2497 y Fe(\021)705 2544 y Fj(\025)753 2497 y Fe(\020)778 2544 y Fo(1)f Fj(\000)h Fo(\(1)g Fj(\000)g Fk(b)970 2525 y Fg(2)970 2552 y Fi(c)989 2544 y Fo(\))1007 2525 y Fh(b)p Fi(D)q(=)p Fg(2)p Fh(c)1100 2497 y Fe(\021)1125 2506 y Fg(2)867 2678 y Fo(15)p eop %%Page: 16 16 16 15 bop 118 199 a Fj(\025)42 b Fo(1)10 b Fj(\000)g Fo(2\(1)f Fj(\000)h Fk(b)410 180 y Fg(2)410 207 y Fi(c)429 199 y Fo(\))447 180 y Fh(b)p Fi(D)q(=)p Fg(2)p Fh(c)553 199 y Fo(=)j(1)c Fj(\000)h Fo(2\(1)f(+)h Fk(b)815 208 y Fi(c)833 199 y Fo(\))851 180 y Fh(b)p Fi(D)q(=)p Fg(2)p Fh(c)944 199 y Fo(\(1)g Fj(\000)g Fk(b)1059 208 y Fi(c)1076 199 y Fo(\))1094 180 y Fh(b)p Fi(D)q(=)p Fg(2)p Fh(c)1200 199 y Fj(\025)j Fo(1)c Fj(\000)i Fo(2)1349 180 y Fh(b)p Fi(D)q(=)p Fg(2)p Fh(c)p Fg(+1)1484 199 y Fo(\(1)f Fj(\000)g Fk(b)1599 208 y Fi(c)1616 199 y Fo(\))1634 180 y Fh(b)p Fi(D)q(=)p Fg(2)p Fh(c)1728 199 y Fk(:)p 1797 293 31 2 v 1797 317 2 25 v 1825 317 V 1797 319 31 2 v -48 401 a Fl(Remark:)24 b Fo(W)l(e)17 b(c)o(hose)f(the)h(binary)g(tree)f(in)h (the)g(construction)f(ab)q(o)o(v)o(e)f(for)h(con)o(v)o(enience.)25 b(This)17 b(pro)o(vided)-48 475 y(us)g(with)g(the)g(particularly)g (simple)h(expression)f(\(13\),)f(on)h(whic)o(h)g(the)g(rest)g(of)f(the) h(deriv)m(ation)g(w)o(as)g(built.)-48 548 y(But)d(there)g(is)h(great)e (\015exibilit)o(y)j(in)g(the)e(c)o(hoice)h(of)f(the)g(underlying)h (graph.)k(F)l(or)14 b(instance,)h(if)f(one)h(uses)f(the)-48 621 y(Euclidean)19 b(square)e(lattice)i Fk(Z)485 604 y Fg(2)521 621 y Fo(instead,)g(one)f(pro)q(duces)g(a)f(graph)h(whic)o (h)g(is)h(not)e(only)h(quasi-transitiv)o(e,)-48 695 y(but)c(is)g(also)g (a)g(p)q(erio)q(dic)h(graph)e(in)i(the)f(sense)g(of)f([16)o(].)19 b(The)14 b(\014rst)f(equalit)o(y)h(in)h(\(13\))d(is)j(then)f(still)h (true.)k(It)14 b(is)-48 768 y(also)f(not)f(hard)h(to)g(see)g(that)f (while)i(the)f(second)g(equalit)o(y)h(in)g(\(13\))d(is)j(no)f(longer)g (true,)f(it)i(can,)f(nev)o(ertheless,)-48 841 y(b)q(e)j(replaced)f (with)g(the)g(inequalit)o(y)411 947 y Fk(s)432 928 y Fg(2)p Fi(l)432 955 y(c)469 900 y Fe(\020)494 947 y Fo(1)10 b Fj(\000)g Fo(\(1)f Fj(\000)h Fk(s)688 956 y Fi(c)705 947 y Fo(\))723 928 y Fh(b)p Fi(D)q(=)p Fg(2)p Fh(c)817 900 y Fe(\021)842 909 y Fi(l)862 900 y Fe(\020)887 947 y Fo(1)g Fj(\000)g Fo(\(1)f Fj(\000)h Fk(s)1081 956 y Fi(c)1098 947 y Fo(\))1116 928 y Fh(d)p Fi(D)q(=)p Fg(2)p Fh(e)1210 900 y Fe(\021)1235 909 y Fi(l)1267 947 y Fj(\025)1328 916 y Fo(1)p 1328 937 23 2 v 1328 976 a(6)1355 947 y Fk(:)-48 1053 y Fo(This)18 b(is)g(su\016cien)o(t)f(to)g(follo)o(w)g (the)h(steps)f(in)h(the)g(pro)q(of)e(ab)q(o)o(v)o(e,)h(with)h(minor)f (mo)q(di\014cations,)g(and)h(deriv)o(e)-48 1126 y(also)d(in)g(this)h (case)e(the)h(inequalit)o(y)h(b)q(et)o(w)o(een)e Fk(b)760 1135 y Fi(c)793 1126 y Fo(and)h Fk(s)902 1135 y Fi(c)934 1126 y Fo(claimed)g(in)h(Theorem)e(1.1\(b\),)f(when)i Fk(l)h Fo(is)g(large.)-48 1409 y Fp(5)65 b(Pro)r(of)21 b(of)g(Theorem)f(1.3)-48 1550 y Fo(In)f(this)f(section,)h(the)f (notation)f(will)i(include)h(an)e(extra)f(subscript,)h(indicating)i (the)e(graph)f(to)g(whic)o(h)i(it)-48 1624 y(refers.)23 1720 y(In)13 b([1)o(])f(the)g(follo)o(wing)h(result)f(w)o(as)f(pro)o(v) o(ed)h(via)g(relations)h(b)q(et)o(w)o(een)f(P)o(otts)e(and)j (random-cluster)e(mo)q(dels)-48 1793 y(and)k(comparisons)e(of)h(those)g (with)h(indep)q(enden)o(t)g(b)q(ond)g(p)q(ercolation)g(mo)q(dels.)k(F)l (or)c(an)f(arbitrary)f(in\014nite,)-48 1867 y(lo)q(cally)j(\014nite)g (graph)e Fk(H)t Fo(,)468 1972 y Fk(b)487 1981 y Fi(c;H)564 1972 y Fj(\024)20 b Fo(1)10 b Fj(\000)g Fo(exp\()p Fj(\000)p Fk(J)844 1981 y Fi(c;H)900 1972 y Fo(\))19 b Fj(\024)1093 1941 y Fk(q)9 b(b)1141 1950 y Fi(c;H)p 998 1962 296 2 v 998 2001 a Fo(1)g(+)h(\()p Fk(q)i Fj(\000)e Fo(1\))e Fk(b)1238 2010 y Fi(c;H)1298 1972 y Fk(:)436 b Fo(\(17\))23 2106 y(Giv)o(en)11 b(an)g(in\014nite)i(connected)d(graph)h Fk(G)f Fo(of)h(b)q(ounded)g(degree,)g(let)h Fk(G)1211 2115 y Fi(s)1239 2106 y Fo(b)q(e)g(the)e(random)g(graph)g(obtained)-48 2180 y(from)k Fk(G)g Fo(b)o(y)h(p)q(erforming)f(site)h(p)q(ercolation)g (with)g(parameter)e Fk(s)i Fo(on)g(it.)20 b(Then,)15 b(a.s.,)511 2285 y Fk(J)540 2266 y Fi(S)536 2294 y(c;G)588 2285 y Fo(\()p Fk(s)p Fo(\))d(=)h Fk(J)734 2266 y Fi(S)730 2294 y(c;G)781 2298 y Fd(s)979 2285 y Fk(b)998 2294 y Fi(c;G)1051 2285 y Fo(\()p Fk(s)p Fo(\))f(=)h Fk(b)1187 2294 y Fi(c;G)1238 2298 y Fd(s)1256 2285 y Fk(:)478 b Fo(\(18\))23 2414 y(F)l(rom)14 b(\(17\),)f(with)i Fk(H)i Fo(=)12 b Fk(G)489 2422 y Fi(s)507 2414 y Fo(,)i(and)h(\(18\))f(w)o(e)g (obtain)389 2527 y Fk(b)408 2536 y Fi(c;G)461 2527 y Fo(\()p Fk(s)p Fo(\))19 b Fj(\024)i Fo(1)9 b Fj(\000)h Fo(exp)q(\()p Fj(\000)p Fk(J)822 2508 y Fi(S)818 2535 y(c;G)870 2527 y Fo(\()p Fk(s)p Fo(\)\))19 b Fj(\024)1119 2496 y Fk(q)10 b(b)1168 2504 y Fi(c;G)1221 2496 y Fo(\()p Fk(s)p Fo(\))p 1024 2517 349 2 v 1024 2556 a(1)g(+)g(\()p Fk(q)h Fj(\000)f Fo(1\))e Fk(b)1264 2565 y Fi(c;G)1316 2556 y Fo(\()p Fk(s)p Fo(\))1377 2527 y Fk(:)357 b Fo(\(19\))867 2678 y(16)p eop %%Page: 17 17 17 16 bop -48 199 a Fo(F)l(or)13 b(the)g(purp)q(ose)g(of)g(comparing)f (this)h(last)g(displa)o(y)h(with)f(\(4\),)f(w)o(e)g(rewrite)h(\(4\))f (in)i(terms)e(of)g Fk(b)1581 208 y Fi(c;G)1634 199 y Fo(\()p Fk(s)p Fo(\))h(\(recall)-48 273 y(that)h Fk(b)69 281 y Fi(c;G)122 273 y Fo(\()p Fk(s)p Fo(\))g(and)h(and)g Fk(s)390 281 y Fi(c;G)443 273 y Fo(\()p Fk(b)p Fo(\))f(are)h(in)o(v)o (erse)f(functions)h(for)f Fk(b)1020 281 y Fi(c)1050 273 y Fj(\024)f Fk(b)g Fj(\024)f Fo(1,)j Fk(s)1249 281 y Fi(c)1278 273 y Fj(\024)e Fk(s)g Fj(\024)f Fo(1\):)420 397 y(1)d Fj(\000)517 366 y Fo(1)p 503 386 51 2 v 503 426 a Fk(C)535 434 y Fg(1)558 397 y Fo(\()p Fk(s)h Fj(\000)g Fk(s)673 405 y Fi(c;G)726 397 y Fo(\))20 b Fj(\024)g Fk(b)838 405 y Fi(c;G)891 397 y Fo(\()p Fk(s)p Fo(\))f Fj(\024)i Fo(1)9 b Fj(\000)1120 366 y Fo(1)p 1105 386 V 1105 426 a Fk(C)1137 434 y Fg(2)1161 397 y Fo(\()p Fk(s)h Fj(\000)g Fk(s)1276 405 y Fi(c;G)1329 397 y Fo(\))p Fk(:)387 b Fo(\(20\))23 541 y(F)l(rom)14 b(\(19\))g(and)h(\(20\),)e(w)o (e)h(obtain)h(the)g(desired)g(inequalities)444 629 y(1)p 415 649 81 2 v 415 689 a Fk(q)10 b(C)477 697 y Fg(2)501 660 y Fo(\()p Fk(s)f Fj(\000)i Fk(s)616 668 y Fi(c;G)668 660 y Fo(\))20 b Fj(\024)g Fo(exp\()p Fj(\000)p Fk(J)912 641 y Fi(S)908 668 y(c;G)961 660 y Fo(\()p Fk(s)p Fo(\)\))f Fj(\024)1129 629 y Fo(1)p 1115 649 51 2 v 1115 689 a Fk(C)1147 697 y Fg(1)1171 660 y Fo(\()p Fk(s)9 b Fj(\000)i Fk(s)1286 668 y Fi(c;G)1338 660 y Fo(\))p Fk(:)-48 785 y Fo(The)18 b(v)m(alues)h(of)e(the)h(constan)o(ts)e Fk(A)558 768 y Fi(S)558 793 y Fg(1)599 785 y Fo(=)i(1)p Fk(=)p Fo(\()p Fk(q)r(C)770 793 y Fg(2)788 785 y Fo(\))f(and)h Fk(A)948 768 y Fi(S)948 793 y Fg(2)989 785 y Fo(=)g(1)p Fk(=C)1120 793 y Fg(1)1156 785 y Fo(whic)o(h)h(app)q(ear)e(in)i(the)f (statemen)o(t)e(of)-48 858 y(the)e(theorem)f(are)h(obtained)g(from)f(b) q(ounds)h(on)g Fk(C)820 866 y Fg(1)853 858 y Fo(and)g Fk(C)972 866 y Fg(2)1006 858 y Fo(in)h(\(4\))e(whic)o(h)h(deriv)o(e)g (from)f(Prop)q(ositions)h(2.1,)-48 931 y(3.1\(ii\))g(and)h(3.2\(i\).)p 1797 919 31 2 v 1797 944 2 25 v 1825 944 V 1797 946 31 2 v -48 1181 a Fp(References)20 1345 y Fo([1])21 b(Aizenman,)16 b(M.,)g(Cha)o(y)o(es,)e(J.,)h(Cha)o(y)o(es,)g(L.)g(and)h(Newman,)f(C.)g (M.)h(\(1987\))d(The)j(phase)g(b)q(oundary)90 1402 y(of)f(dilute)h(and) f(random)e(Ising)j(and)f(P)o(otts)e(ferromagnets,)f Ff(J.)j(Phys.)h(A)f Fl(20)p Fo(,)f(L313{L318.)20 1495 y([2])21 b(Aizenman,)j(M.)e(and)g (Grimmett,)h(G.)f(R.)h(\(1991\))c(Strict)j(monotonicit)o(y)f(in)i (critical)g(p)q(oin)o(ts)f(in)90 1552 y(p)q(ercolation)15 b(and)g(ferromagnetic)e(mo)q(dels,)i Ff(J.)g(Stat.)i(Phys.)e Fl(63)p Fo(,)f(817{835.)20 1646 y([3])21 b(Bezuidenhout,)e(C.)f(E.,)g (Grimmett,)f(G.)h(R.,)h(Kesten,)f(H.)g(\(1993\))e(Strict)h(inequalit)o (y)i(for)e(critical)90 1702 y(v)m(alues)f(of)e(P)o(otts)f(and)i (random-cluster)f(pro)q(cesses,)g Ff(Commun.)j(Math.)f(Phys.)p Fo(,)f Fl(158)p Fo(,)f(1{16.)20 1796 y([4])21 b(Georgii,)g(H.-O.)d (\(1981\))f(Sp)q(on)o(taneous)h(magnetization)h(of)f(randomly)h(dilute) h(ferromagnets,)d Ff(J.)90 1852 y(Stat.)g(Phys.)d Fl(25)p Fo(,)h(369{396.)20 1946 y([5])21 b(Georgii,)e(H.-O.)e(\(1984\))e(On)j (the)g(ferromagnetic)e(and)h(the)h(p)q(ercolativ)o(e)f(region)h(of)f (random)f(spin)90 2003 y(systems,)d Ff(A)n(dv.)i(in)i(Appl.)e(Pr)n(ob)n (ab.)f Fl(16)p Fo(,)g(732{765.)20 2097 y([6])21 b(Georgii,)d(H.-O.)f (\(1985\))e(Disordered)i(Ising)h(ferromagnets)c(and)k(p)q(ercolation,)f Ff(Particle)g(systems,)90 2153 y(r)n(andom)g(me)n(dia)f(and)h(lar)n(ge) e(deviations.)h(Contemp.)h(Math.,)g(41)e Fo(\(Durrett,)e(R.,)i (editor\),)f(135{147,)90 2209 y(Amer.)g(Math.)g(So)q(c.,)h(Pro)o (vidence,)f(R.I..)20 2303 y([7])21 b(Georgii,)15 b(H.-O.)g(\(1988\))d Ff(Gibbs)k(Me)n(asur)n(es)h(and)f(Phase)g(T)m(r)n(ansitions)p Fo(,)g(de)f(Gruyter,)f(New)h(Y)l(ork.)20 2397 y([8])21 b(Georgii,)14 b(H.-O.,)f(H\177)-23 b(aggstr\177)g(om,)11 b(O.)i(and)h(Maes,)f(C.)g(\(1999\))f Ff(The)j(r)n(andom)g(ge)n(ometry)f (of)h(e)n(quilibrium)90 2454 y(phases)p Fo(,)h(preprin)o(t.)20 2547 y([9])21 b(Grimmett,)14 b(G.)h(R.)g(\(1989\))e Ff(Per)n(c)n (olation)p Fo(,)g(Springer,)h(New)h(Y)l(ork.)867 2678 y(17)p eop %%Page: 18 18 18 17 bop -3 199 a Fo([10])21 b(Grimmett,)g(G.)f(R.)h(\(1995\))d (Comparison)h(and)h(disjoin)o(t-o)q(ccurrence)h(inequalities)h(for)d (random-)90 256 y(cluster)c(mo)q(dels,)g Ff(J.)g(Stat.)i(Phys.)p Fo(,)d Fl(78)p Fo(,)h(1311{1324)o(.)-3 350 y([11])21 b(Grimmett,)16 b(G.)g(R.)h(and)f(Stacey)l(,)g(A.)g(M.)g(\(1999\))f (Critical)h(probabilities)i(for)d(site)i(and)f(b)q(ond)h(p)q(er-)90 406 y(colation)e(mo)q(dels,)g Ff(A)o(nn.)h(Pr)n(ob)n(ab.)p Fo(,)d Fl(26)p Fo(,)h(1788{1812.)-3 500 y([12])21 b(H\177)-23 b(aggstr\177)g(om,)18 b(O.)h(\(1999\))f(Mark)o(o)o(v)g(random)h (\014elds)i(and)f(p)q(ercolation)f(on)h(general)g(graphs,)f Ff(A)n(dv.)90 556 y(Appl.)c(Pr)n(ob)n(ab.)p Fo(,)e(to)i(app)q(ear.)-3 650 y([13])21 b(H\177)-23 b(aggstr\177)g(om,)9 b(O.,)j(Sc)o(honmann,)g (R.)g(H.)g(and)g(Steif,)g(J.)g(\(1999\))e(The)i(Ising)g(mo)q(del)h(on)f (diluted)h(graphs)90 707 y(and)i(strong)e(amenabilit)o(y)l(,)i(preprin) o(t.)-3 800 y([14])21 b(Hammersley)l(,)e(J.)f(M.)h(\(1961\))e (Comparison)g(of)h(atom)g(and)g(b)q(ond)h(p)q(ercolation)g (probabilities,)h Ff(J.)90 857 y(Math.)d(Phys.)p Fo(,)d Fl(2)p Fo(,)h(728{733.)-3 951 y([15])21 b(Hammersley)l(,)13 b(J.)h(M.)g(\(1980\))e(A)i(generalization)g(of)g(McDiarmid's)f(theorem) g(for)g(mixed)i(Bernoulli)90 1007 y(p)q(ercolation,)g Ff(Math.)h(Pr)n(o)n(c.)f(Camb.)h(Phil.)f(So)n(c.)p Fo(,)f Fl(88)p Fo(,)h(167{170.)-3 1101 y([16])21 b(Kesten,)k(H.)f(\(1982\))d Ff(Per)n(c)n(olation)h(the)n(ory)i(for)g(mathematicians)p Fo(,)h(Birkh\177)-23 b(auser,)25 b(Boston-Basel-)90 1157 y(Stuttgard.)-3 1251 y([17])c(Liggett,)f(T.)g(M.,)h(Sc)o(honmann,)f(R.) g(H.)g(and)g(Stacey)l(,)g(A.)g(M.)g(\(1997\))e(Domination)h(b)o(y)h (pro)q(duct)90 1308 y(measures,)14 b Ff(A)o(nn.)i(Pr)n(ob)n(ab.)d Fl(25)p Fo(,)i(71{95.)-3 1401 y([18])21 b(Ly)o(ons,)i(R.)e(and)h(Sc)o (hramm,)g(O.)f(\(1999\))f(Indistinguishabilit)o(y)j(of)f(p)q (ercolation)f(clusters,)i Ff(A)o(nn.)90 1458 y(Pr)n(ob)n(ab.)p Fo(,)13 b(to)i(app)q(ear.)-3 1552 y([19])21 b(Menshik)o(o)o(v,)14 b(M.)f(\(1987\))e(Quan)o(titativ)o(e)i(estimates)f(and)i(rigorous)e (inequalities)j(for)e(critical)h(p)q(oin)o(ts)90 1608 y(of)h(a)f(graph)g(and)h(its)g(subgraphs,)f Ff(The)n(ory)i(of)g(Pr)n (ob)n(ability)e(and)i(its)h(Applic)n(ations)p Fo(,)d Fl(32)h Fo(544{547.)-48 1907 y Fb(Lincoln)h(Cha)m(yes)-48 1963 y Fo(Departmen)o(t)d(of)h(Mathematics,)g(Univ)o(ersit)o(y)g(of)h (California,)f(Los)h(Angeles.)-48 2020 y Fa(lchayes@math.)o(ucl)o(a.ed) o(u)-48 2191 y Fb(R)o(ober)m(to)i(H.)g(Schonmann)-48 2248 y Fo(Departmen)o(t)c(of)h(Mathematics,)g(Univ)o(ersit)o(y)g(of)h (California,)f(Los)h(Angeles.)-48 2304 y Fa(rhs@math.ucla)o(.ed)o(u)-48 2360 y(http://www.ma)o(th.)o(ucla)o(.ed)o(u/~r)o(hs/)867 2678 y Fo(18)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9907122044351--