Content-Type: multipart/mixed; boundary="-------------9811101839278" This is a multi-part message in MIME format. ---------------9811101839278 Content-Type: text/plain; name="98-703.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="98-703.comments" rhs@math.ucla.edu http://www.math.ucla.edu/~rhs/ ---------------9811101839278 Content-Type: text/plain; name="98-703.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="98-703.keywords" Percolation, transitive graphs, Cayley graphs, number of infinite clusters, critical behavior ---------------9811101839278 Content-Type: application/postscript; name="ipo.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="ipo.ps" %!PS-Adobe-2.0 %%Creator: dvips 5.58 Copyright 1986, 1994 Radical Eye Software %%Title: ipo.dvi %%CreationDate: Mon Nov 9 17:23:56 1998 %%Pages: 17 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: dvips ipo.dvi 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y(254)h(of)h([Dur2]\))f(that)g(for)h(some)g(random)f(v)m(ariable)j Fp(X)h Fs(with)e(mean)f Fo(E)r Fs(\()p Fp(X)t Fs(\))g(=)e(1)465 2609 y Fp(Z)496 2616 y Fn(j)p 436 2630 107 2 v 436 2671 a Fs(\()p Fp(\026)481 2678 y Fn(n)506 2671 y Fs(\))524 2658 y Fn(j)561 2640 y Ft(!)g Fp(X)94 b Fs(a.s.)c(as)15 b Fp(j)g Ft(!)e(1)p Fp(:)p eop %%Page: 10 10 10 9 bop 0 111 a Fm(10)1081 b Fk(R)n(ob)n(erto)15 b(H.)g(Schonmann)0 220 y Fs(In)h(particular,)f(there)h(is)f Fp(\017)e(>)g Fs(0)i(suc)o(h)h(that)125 369 y Fo(P)164 305 y Ff(\022)204 357 y Fs(\026)194 369 y Fp(Z)225 376 y Fn(l)252 369 y Ft(\025)305 338 y Fs(\()p Fp(ba)p Fs(\))385 322 y Fn(nl)p 305 358 116 2 v 351 400 a Fs(2)426 305 y Ff(\023)472 369 y Ft(\025)d Fo(P)558 305 y Ff(\022)589 369 y Fp(Z)620 376 y Fn(l)646 369 y Ft(\025)700 338 y Fs(\()p Fp(ba)p Fs(\))780 322 y Fn(nl)p 700 358 V 746 400 a Fs(2)820 305 y Ff(\023)867 369 y Ft(\025)g Fo(P)953 305 y Ff(\022)983 369 y Fp(Z)1014 376 y Fn(l)1041 369 y Ft(\025)1094 338 y Fs(\()p Fp(\026)1139 345 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Ff(\023)1424 1962 y Fp(:)-1437 b Fs(\(2.8)l(\))0 2108 y(W)l(e)17 b(will)h(no)o(w)e (lo)q(ok)h(separately)g(at)f(eac)o(h)g(one)h(of)f(the)h(factors)e(in)j (the)f(righ)o(t)f(hand)h(side)0 2167 y(of)e(\(2.8\).)k(First,)c(noting) h(that)e(the)i(graphs)f Fo(T)802 2174 y Fn(b)818 2167 y Fs(\()p Fp(x)p Fs(\))9 b Ft(\002)i Fs([)p Fp(l)q(s;)d Fs(\(2)p Fp(i)g Fs(+)i(1\))p Fp(l)q(s)p Fs(],)k Fp(x)f Ft(2)h Fp(S)1343 2181 y Fe(T)1361 2167 y Fc(+)1361 2192 y Fg(b)1392 2167 y Fs(\()p Fp(nl)q Fs(\))g(are)0 2236 y(disjoin)o(t)i(and)f(isomorphic)h(to)f Fo(T)569 2217 y Fj(+)569 2250 y Fn(b)607 2236 y Ft(\002)10 b Fs([0)p Fp(;)e Fs(2)p Fp(il)q(s)p Fs(],)k(w)o(e)j(can)g(use)h(indep)q(endence)i (to)d(write)45 2388 y Fo(P)84 2324 y Ff(\022)114 2388 y Fs(\(0)p Fp(;)8 b Fs(0\))228 2355 y Fe(T)246 2340 y Fc(+)246 2366 y Fg(b)275 2355 y Fi(\002)p Fj([0)p Fn(;)p Fj(\(2)p Fn(i)p Fj(+1\))p Fn(ls)p Fj(])316 2388 y Ft( )-8 b(!)102 b Fp(S)528 2402 y Fe(T)546 2387 y Fc(+)546 2413 y Fg(b)577 2388 y Fs(\()p Fp(nl)q 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y(variant)14 b(me)n(asur)n(e)h(of)f(the)h(c)n(ontact)f(pr)n(o)n(c) n(ess)p Fs(,)e(The)i(Annals)g(of)f(Probabilit)o(y)i Fq(25)188 2465 y Fs(\(1997\),)e(1846{1871.)0 2524 y([SaS2])66 b(M.)15 b(Salzano)i(and)g(R.)f(H.)g(Sc)o(honmann,)h Fh(A)g(new)g(pr)n(o)n(of)g (that)h(for)g(the)f(c)n(ontact)188 2583 y(pr)n(o)n(c)n(ess)e(on)h(homo) n(gene)n(ous)g(tr)n(e)n(es)f(lo)n(c)n(al)g(survival)i(implies)e(c)n (omplete)h(c)n(onver-)188 2643 y(genc)n(e)p Fs(,)d(The)j(Annals)g(of)f (Probabilit)o(y)h Fq(26)f Fs(\(1998\),)e(1251{1258.)p eop %%Page: 17 17 17 16 bop 15 111 a Fk(Per)n(c)n(olation)16 b(in)f Ft(1)10 b Fs(+)h(1)j Fk(dimensions)i(at)g(the)g(uniqueness)e(thr)n(eshold)367 b Fm(17)0 220 y Fs([SaS3])66 b(M.)18 b(Salzano)i(and)g(R.)f(H.)g(Sc)o (honmann,)i Fh(The)f(se)n(c)n(ond)e(lowest)i(extr)n(emal)g(in-)188 279 y(variant)c(me)n(asur)n(e)g(of)h(the)f(c)n(ontact)g(pr)n(o)n(c)n (ess)f(II)p Fs(,)f(preprin)o(t)i(\(1997\).)0 338 y([Sc)o(h1])70 b(R.)20 b(H.)g(Sc)o(honmann,)i Fh(The)f(triangle)g(c)n(ondition)f(for)i (c)n(ontact)f(pr)n(o)n(c)n(esses)e(on)188 397 y(homo)n(gene)n(ous)h(tr) n(e)n(es)p Fs(,)f(Journal)h(of)e(Statistical)j(Ph)o(ysics)f Fq(90)h Fs(\(1998\),)d(1429{)188 456 y(1440.)0 515 y([Sc)o(h2])70 b(R.)15 b(H.)f(Sc)o(honmann,)h Fh(Stability)h(of)g(in\014nite)e (clusters)i(in)f(sup)n(er)n(critic)n(al)g(p)n(er)n(c)n(o-)188 574 y(lation)p Fs(,)f(Probabilit)o(y)i(Theory)f(and)h(Related)g(Fields) g(\(to)f(app)q(ear\).)0 633 y([ST])104 b(R.)17 b(H.)g(Sc)o(honmann)h (and)f(N.)g(I.)g(T)l(anak)m(a,)g Fh(L)n(ack)h(of)g(monotonicity)g(in)g (ferr)n(o-)188 693 y(magnetic)f(Ising)e(mo)n(del)j(phase)f(diagr)n(ams) p Fs(,)f(The)h(Annals)g(of)f(Applied)i(Prob-)188 752 y(abilit)o(y)e Fq(8)f Fs(\(1998\),)e(234{245.)0 811 y([SeS])92 b(C.)10 b(M.)g(Series)i(and)f(Y)l(a.)18 b(G.)10 b(Sinai,)j Fh(Ising)d(mo)n(dels)i(on)g(the)g(L)n(ob)n(achevsky)f(Plane)p Fs(,)188 870 y(Comm)o(unications)k(in)h(Mathematical)f(Ph)o(ysics)h Fq(128)g Fs(\(1990\),)d(63{76.)0 929 y([Sta])96 b(A.)13 b(M.)g(Stacey,)h Fh(The)h(existenc)n(e)e(of)j(an)f(interme)n(diate)f (phase)i(for)f(the)g(c)n(ontact)188 988 y(pr)n(o)n(c)n(ess)g(on)h(tr)n (e)n(es)p Fs(,)d(The)j(Annals)g(of)f(Probabilit)o(y)h Fq(24)f Fs(\(1996\),)e(1711{1726.)0 1047 y([W)l(u1])71 b(C.)24 b(C.)h(W)l(u,)i Fh(Critic)n(al)d(b)n(ehavior)i(of)g(p)n(er)n(c) n(olation)e(and)h(Markov)h(\014elds)e(on)188 1106 y(br)n(anching)13 b(planes)p Fs(,)e(Journal)i(of)f(Applied)j(Probabilit)o(y)f Fq(30)f Fs(\(1993\),)e(538{547.)0 1165 y([W)l(u2])71 b(C.)20 b(C.)g(W)l(u,)h Fh(The)g(c)n(ontact)g(pr)n(o)n(c)n(ess)f(on)h (a)h(tr)n(e)n(e)e({)i(b)n(ehavior)f(ne)n(ar)g(the)h(\014rst)188 1224 y(tr)n(ansition)p Fs(,)g(Sto)q(c)o(hastic)g(Pro)q(cesses)g(and)g (their)g(Applications)j Fq(57)f Fs(\(1995\),)188 1283 y(99{112.)0 1343 y([W)l(u3])71 b(C.)18 b(C.)f(W)l(u,)i Fh(Ising)f(mo)n(dels)h(on)g(hyp)n(erb)n(olic)f(gr)n(aphs)p Fs(,)h(Journal)g(of)f(Statistical)188 1402 y(Ph)o(ysics)d Fq(85)h Fs(\(1996\),)d(251{259.)0 1461 y([Zha])86 b(Y.)17 b(Zhang,)g Fh(The)h(c)n(omplete)g(c)n(onver)n(genc)n(e)e(the)n(or)n(em) i(of)h(the)f(c)n(ontact)g(pr)n(o)n(c)n(ess)188 1520 y(on)e(tr)n(e)n(es) p Fs(,)e(The)h(Annals)h(of)f(Probabilit)o(y)h Fq(24)f Fs(\(1996\),)e(1408{1443.)13 1629 y Fa(Mathematics)d(Departmen)o(t,)j (Univ)o(ersit)o(y)f(of)h(California)f(at)h(Los)g(Angeles,)e(Los)h (Angeles,)f(CA)0 1688 y(90095)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9811101839278--