This is a postscript file of the paper. Just cut after 'body' and print. 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Fy(\()p Fo(t)p Fy(\))p 1033 1505 85 2 v 1063 1551 a(2)1123 1517 y(\))-66 1621 y(and)16 b(th)o(us)h(the)f(main)f(theorem)f(of)j ([1])f(is)g(reco)o(v)o(ered)e(b)o(y)i(using)h(purely)e(probabilistic)g (argumen)o(ts.)124 b Fa(2)7 1729 y Fy(The)20 b(other)g(mathematical)d(result) i(kno)o(wn)h(ab)q(out)i(the)d(Sherrington-Kirkpatric)o(k)g(mo)q(del)g (concerns)-66 1789 y(the)d(w)o(eak)g(self-a)o(v)o(eraging)g(prop)q(ert)o(y)h (of)f(the)h(sp)q(eci\014c)f(free)f(energy)l(.)22 b Fs(Self-aver)n(aging)d Fy(is)d(a)h(v)o(ery)e(imp)q(ortan)o(t)-66 1849 y(prop)q(ert)o(y)f(of)g (thermo)q(dynamic)d(functions;)k(when)f(it)g(is)f(true,)h(it)g(states)g(that) h(the)f(corresp)q(onding)h(function)-66 1909 y(is)g(a)h(trivial)e(random)h(v) m(ariable)g(in)g(the)h(sense)f(that)h(it)f(do)q(es)h(not)g(\015uctuate)g (from)e(sample)g(to)i(sample.)k(The)-66 1969 y(\014rst)13 b(result,)g(v)m (alid)g(on)h(the)f(whole)g(region)g(of)h(temp)q(erature,)e(w)o(as)i (established)f(b)o(y)f(P)o(astur)i(and)g(Shc)o(herbina)-66 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Fo(F)1418 2547 y Fp(N)1463 2540 y Fn(\000)11 b Ff(E)c Fo(F)1578 2547 y Fp(N)1627 2540 y Fy(=)1680 2506 y Fd(P)1724 2550 y Fp(K)1766 2540 y Fo( )1798 2547 y Fp(K)1832 2540 y Fy(,)17 b(and)-66 2600 y Fn(j)p Fo( )-20 2607 y Fp(K)13 2600 y Fn(j)d(\024)120 2580 y FC(1)p 99 2588 61 2 v 99 2593 a Fq(p)p 128 2593 32 2 v 28 x Fp(N)165 2600 y Fy(\()p Fn(j)p Fo(J)225 2607 y Fp(K)258 2600 y Fn(j)d Fy(+)g Ff(E)d Fn(j)p Fo(J)407 2607 y Fp(K)441 2600 y Fn(j)p Fy(\))16 b(the)g(result)f(is)h(obtained)h(b)o(y)f(optimising)e (the)i(Mark)o(o)o(v)g(inequalit)o(y)l(.)155 b Fa(2)913 2823 y Fy(16)p eop %%Page: 17 17 bop 7 24 a Fy(In)15 b(parallel)g(to)h(these)f(ma)s(jor)f(results)i (concerning)f(the)g(mo)q(del)f(there)h(exist)g(also)h(some)e(partial)i (results)-66 84 y(concerning)g(v)m(arious)h(quan)o(tities.)22 b(Let)16 b(us)h(men)o(tion)e(the)h(result)h(on)g(the)f Fs(supr)n(emum)g Fy(of)h(the)f(Hamiltonian)-66 144 y(o)o(v)o(er)f(con\014gurations)j(obtained) e(in)g([128)q(])g(that)g(states:)223 279 y(0)e Fo(<)g(C)348 286 y FC(1)381 279 y Fn(\024)g Fy(lim)8 b(inf)486 309 y Fp(N)582 245 y Fy(sup)656 257 y Fp(\033)688 245 y Fo(H)728 252 y Fp(N)762 245 y Fy(\()p Fo(\033)r Fy(\))p 582 268 247 2 v 684 313 a Fo(N)848 279 y Fn(\024)13 b Fy(lim)8 b(sup)959 318 y Fp(N)1063 245 y Fy(sup)1137 257 y Fp(\033)1168 245 y Fo(H)1208 252 y Fp(N)1242 245 y Fy(\()p Fo(\033)r Fy(\))p 1063 268 V 1164 313 a Fo(N)1329 279 y Fn(\024)13 b Fo(C)1416 286 y FC(2)1450 279 y Fo(<)1501 227 y Fd(q)p 1543 227 96 2 v 52 x Fy(log)c(2)q Fo(:)-66 412 y Fy(This)15 b(result)g(is)g(w)o(eak)o(er)f(than)i(the)f(corresp)q(onding)i (result)d(con)o(tained)h(in)g(equations)h(2.20)g(and)g(2.23)g(of)f([1].)-66 472 y(Its)c(in)o(terest)g(stems)f(in)i(the)g(metho)q(d)f(with)g(whic)o(h)g (it)h(w)o(as)g(obtained:)19 b(a)13 b(comparison)e(of)h(Gaussian)h(pro)q (cesses)-66 532 y(b)o(y)g(Slepian's)g(lemm)o(a)e([88])i(and)i(a)f(Gilb)q (ert-V)l(arshamo)o(v)e(b)q(ound)i([92)q(])f(from)f(information)h(theory)g (are)h(used.)-66 592 y(In)e(m)o(y)f(opinion)h(this)h(is)f(an)h(in)o (teresting)f(direction)f(to)i(searc)o(h)g(for)f(obtaining)i(new)e(results)h (in)f(the)g(spin)h(glass)-66 652 y(mo)q(dels.)20 b(In)15 b(particular,)g(the) g(b)q(eha)o(viour)h(of)f(Gaussian)i(pro)q(cesses)f(indexed)f(b)o(y)g(the)g (spin)h(con\014gurations,)-66 712 y(in)g(terms)e(of)j(geometrical)d(prop)q (erties)i(of)h(the)f(con\014guration)h(space,)f(m)o(ust)f(b)q(e)h(understo)q (o)q(d.)7 820 y(Other)h(results)g(concern)g(b)q(ounds)i(for)f(the)f Fs(limes)i(in\014mum)f Fy(and)g Fs(limes)h(sup)n(erior)e Fy(of)g(the)g(sp)q (eci\014c)g(free)-66 880 y(energy)g(v)m(alid)h(in)f(the)h(lo)o(w)g(temp)q (erature)e(region)i(\()p Fo(\014)h(>)e Fy(1\).)27 b(These)18 b(results)g(are)g(scattered)f(through)i(four)-66 940 y(di\013eren)o(t)c (sources)i([1,)e(83)q(,)h(31)q(,)f(99)q(])h(that)h(are)f(presen)o(ted)f(here) h(as)h(a)g(single)e(theorem:)-66 1087 y FD(Theorem)35 b(4.6)24 b Fs(F)l(or)17 b(al)r(l)i Fo(\014)d(>)d Fy(1)p Fs(,)18 b(we)g(have,)g(in)g (distribution,)g(the)g(fol)r(lowing)i(ine)n(qualities)496 1195 y Fn(\000)p Fo(g)r Fy(\()p Fo(\014)s Fy(\))13 b Fn(\024)h Fy(lim)8 b(inf)746 1224 y Fp(N)838 1195 y Fo(f)862 1202 y Fp(N)910 1195 y Fn(\024)13 b Fy(lim)8 b(sup)1021 1234 y Fp(N)1120 1195 y Fo(f)1144 1202 y Fp(N)1191 1195 y Fn(\024)14 b(\000)p Fo(h)p Fy(\()p Fo(\014)s Fy(\))-66 1327 y Fs(wher)n(e)j Fo(g)r Fy(\()p Fo(\014)s Fy(\))c(=)h Fo(g)253 1334 y FC(1)273 1327 y Fy(\()p Fo(\014)s Fy(\))d Fn(^)g Fo(g)420 1334 y FC(2)440 1327 y Fy(\()p Fo(\014)s Fy(\))17 b Fs(and)691 1459 y Fo(g)714 1466 y FC(1)734 1459 y Fy(\()p Fo(\014)s Fy(\))c(=)868 1407 y Fd(q)p 909 1407 V 909 1459 a Fy(log)d(2)h(+)1070 1426 y(log)e(2)p 1070 1448 V 1103 1494 a Fo(\014)1171 1459 y(;)605 1623 y(g)628 1630 y FC(2)648 1623 y Fy(\()p Fo(\014)s Fy(\))k(=)h(1)d(+)871 1589 y(log)f(2)p 871 1611 V 904 1657 a Fo(\014)983 1623 y Fn(\000)1038 1589 y Fy(log)f Fo(\014)p 1038 1611 102 2 v 1061 1657 a Fy(2)p Fo(\014)1156 1623 y Fn(\000)1226 1589 y Fy(3)p 1210 1611 55 2 v 1210 1657 a(4)p Fo(\014)-66 1732 y Fs(and)494 1820 y Fo(h)p Fy(\()p Fo(\014)s Fy(\))k(=)h Fn(\000)700 1786 y Fy(log)9 b(2)p 700 1808 96 2 v 732 1854 a Fo(\014)811 1820 y Fn(\000)869 1786 y Fy(1)p 866 1808 31 2 v 866 1854 a Fo(\014)910 1761 y Fd(Z)951 1774 y FC(1)933 1856 y(0)979 1820 y Fy(log)g(cosh)q(\()p Fo(\014)1193 1739 y Fd(s)p 1234 1739 63 2 v 1239 1786 a Fy(2)p Fo(u)p 1239 1808 53 2 v 1250 1854 a(\031)1296 1820 y Fy(\))p Fo(du:)-66 1989 y Fy(In)k(spite)h(of)g(the)g(e\013orts)h(to)f(pro)o(v)o(e)f(the)h (existence)f(of)h(the)g(sp)q(eci\014c)f(free)g(energy)h(for)g(this)g(mo)q (del)f(up)h(to)h(no)o(w,)-66 2049 y(suc)o(h)i(a)h(pro)q(of)g(is)g(still)e (missing)g(no)o(w)o(ada)o(ys.)25 b(But)17 b(if)g(this)h(limit)c(exists,)j(it) g(m)o(ust)f(necessarily)g(b)q(e)i(a)g(trivial)-66 2110 y(random)f(v)m (ariable)f(in)h(the)g(sense)h(that)f(it)g(coincides)f(with)i(its)f(exp)q (ectation)g(as)g(it)g(is)g(sho)o(wn)h(in)f([13)q(].)24 b(The)-66 2170 y(b)q(ounds)17 b(obtained)f(in)g(this)g(theorem)e(are)i(used)g(to)g (delimit)d(the)j(gra)o(y)g(region)g(of)h(\014gure)f(3;)g(as)g(a)h(matter)d (of)-66 2230 y(fact,)h(there)g(is)g(an)h(in\014nite)f(family)f(of)i(a)o(v)m (ailable)e(b)q(ounds)j(for)f(the)g(v)o(ery)e(lo)o(w)i(temp)q(erature)e (regime)f(and)j Fo(g)1921 2237 y FC(1)-66 2290 y Fy(m)o(ust)f(b)q(e)h (optimised)f(o)o(v)o(er)g(all)h(these)g(b)q(ounds.)23 b(Not)16 b(to)h(burden)f(the)g(presen)o(tation,)g(I)g(giv)o(e)f(here)h(only)g(the)-66 2350 y(easier)f(linear)h(b)q(ound)h(for)g Fo(g)453 2357 y FC(1)473 2350 y Fy(.)-66 2458 y Fs(Pr)n(o)n(of)f(of)i(the)h(the)n(or)n(em:)j Fy(The)c(simplest)d(b)q(ound)j Fo(g)876 2465 y FC(1)913 2458 y Fy(from)e(those)h(in)g([83])g(is)g(obtained)g(as)h(follo)o(ws:)23 b(denote)-66 2518 y(b)o(y)15 b Fo(A)38 2525 y Fp(N)72 2518 y Fy(\()p Fo(\033)r Fy(\))g(the)h(con\014guration)i(dep)q(enden)o(t)d(ev)o (en)o(t)386 2655 y Fo(A)423 2662 y Fp(N)456 2655 y Fy(\()p Fo(\033)r Fy(\))f(=)f Fn(f)p Fy(exp\()732 2621 y Fo(\014)760 2628 y FC(1)p 713 2643 86 2 v 713 2652 a Fn(p)p 754 2652 45 2 v 754 2694 a Fo(N)812 2614 y Fd(X)814 2705 y Fp(i)f Fy(2)928 47 y Fn(p)p 970 47 96 2 v 37 x Fy(log)9 b(2,)18 b(hence)f Fo(B)1274 66 y Fp(c)1271 96 y(N)1323 84 y Fy(has)i(full)e(measure.)25 b(On)18 b(the)g(set)-66 144 y Fo(B)-26 126 y Fp(c)-29 156 y(N)21 144 y Fy(w)o(e)d(ha)o(v)o(e,)338 229 y Fo(Z)371 236 y Fp(N)405 229 y Fy(\()p Fo(\014)s Fy(\))e(=)539 187 y Fd(X)558 275 y Fp(\033)607 229 y Fy(exp\()733 195 y Fo(\014)p 706 217 86 2 v 706 226 a Fn(p)p 747 226 45 2 v 747 268 a Fo(N)805 187 y Fd(X)807 278 y Fp(i)p FC(2)690 438 y Fn(p)p 732 438 70 2 v 33 x FC(log)5 b(2)801 427 y Fy(\()825 393 y Fo(\014)853 400 y FC(1)p 825 415 48 2 v 836 461 a Fy(2)888 427 y(+)942 393 y(log)10 b(2)p 942 415 96 2 v 975 461 a Fo(\014)1043 427 y Fy(\))k(=)1127 374 y Fd(q)p 1169 374 V 53 x Fy(log)9 b(2)j(+)1330 393 y(log)d(2)p 1330 415 V 1362 461 a Fo(\014)1430 427 y(:)7 605 y Fy(The)22 b(b)q(ound)i(for)f Fo(g)374 612 y FC(2)416 605 y Fy(is)g(obtained)g(in)f([31],)h(some)f(ideas)g(are)h(also)g(in)f([99,)g (61)q(].)40 b(The)22 b(sp)q(eci\014c)g(free)-66 665 y(energy)17 b(of)h(the)g(mo)q(del)e(is)i(not)g(c)o(hanged)g(b)o(y)f(adding)i(a)f (\014nite,)f(con\014guration)i(indep)q(enden)o(t)e(term)f(in)i(the)-66 725 y(Hamiltonian.)34 b(Th)o(us,)23 b(w)o(e)d(use)i(the)f(Hamiltonian)e Fo(H)972 732 y Fp(N)1006 725 y Fy(,)j(de\014ned)f(on)h(\006)1323 732 y Fp(N)1379 725 y Fy(=)h Fn(f\000)p Fy(1)p Fo(;)8 b Fy(1)p Fn(g)1599 707 y Fp(N)1633 725 y Fy(,)22 b(giv)o(en)e(b)o(y)h(the)-66 785 y(form)o(ula)469 855 y Fo(H)509 862 y Fp(N)543 855 y Fy(\()p Fo(\033)r Fy(\))14 b(=)f Fn(\000)751 822 y Fy(1)p 720 844 86 2 v 720 853 a Fn(p)p 762 853 45 2 v 42 x Fo(N)819 814 y Fd(X)821 905 y Fp(i)h(j)787 1373 y FC(1)p 757 1381 79 2 v 757 1386 a Fq(p)p 786 1386 18 2 v 27 x FC(2)p Fp(N)840 1393 y Fo(J)867 1400 y Fp(ii)939 1393 y Fy(if)41 b Fo(i)13 b Fy(=)h Fo(j:)-66 1505 y Fy(This)22 b(matrix)e(has)i (almost)f(surely)g(simple)f(eigen)o(v)m(alues)g 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y(it)j(should)i(b)q(e)f(uniformly)e (distributed)h(on)i(the)f(set)g Fo(O)q Fy(\()p Fo(N)5 b Fy(\))19 b(of)f(orthogonal)i(matrices.)25 b(In)18 b(particular,)f(for)-66 2032 y(ev)o(ery)d(p)q(ositiv)o(e)i(measurable)f(function)h Fo(F)7 b Fy(,)15 b(w)o(e)h(ha)o(v)o(e)520 2153 y Ff(E)7 b Fy([)p Fo(F)g Fy(\()p FD(M)p Fy(\))p Fn(j)p Fy(\003])13 b(=)823 2094 y Fd(Z)873 2153 y Fo(F)7 b Fy(\()p Fo(\036)p Fy(\003)p Fo(\036)1023 2132 y Fp(T)1050 2153 y Fy(\))p Fo(\024)p Fy(\()p Fo(d\036)p Fy(\))49 b Ff(P)-7 b Fy(-a.s.)-66 2276 y(where)23 b Fo(\024)h Fy(is)f(the)h(uniform)e(probabilit)o(y)h(on)h Fo(O)q Fy(\()p Fo(N)5 b Fy(\).)45 b(Change)25 b(no)o(w)f(the)f(con\014guration)i(space)f (\006)1843 2283 y Fp(N)1904 2276 y Fy(=)-66 2337 y Fn(f\000)p Fy(1)p Fo(;)8 b Fy(1)p Fn(g)93 2319 y Fp(N)149 2337 y Fy(in)o(to)23 b(the)f(sphere)h Fo(S)534 2344 y Fp(N)592 2337 y Fy(=)i Fn(f)p Fo(s)g Fn(2)g Ff(R)822 2315 y Fp(N)880 2337 y Fy(:)g Fn(j)p Fo(s)p Fn(j)970 2319 y FC(2)1014 2337 y Fy(=)1077 2303 y Fd(P)1120 2317 y Fp(N)1120 2349 y(i)p FC(=1)1188 2337 y Fo(s)1211 2319 y FC(2)1211 2349 y Fp(i)1256 2337 y Fy(=)f Fo(N)5 b Fn(g)23 b Fy(and)h(de\014ne)e(a)h(new)g(mo)q(del)-66 2397 y(\(termed)15 b(spherical)h(mo)q(del\))g(with)h(the)f(same)g(functional)h(form)f(for)h(the) g(Hamiltonian)e(but)j(de\014ned)e(o)o(v)o(er)-66 2457 y(the)g (con\014guration)h(space)f Fo(S)473 2464 y Fp(N)507 2457 y Fy(.)21 b(Its)16 b(partition)g(function)h(is)f(giv)o(en)f(b)o(y)597 2578 y Fo(Z)634 2554 y FC(sph)630 2590 y Fp(N)703 2578 y Fy(=)755 2519 y Fd(Z)804 2578 y Fy(exp\()p Fn(\000)p Fo(\014)s(H)1008 2585 y Fp(N)1041 2578 y Fy(\()p Fo(s)p Fy(\)\))p Fo(\027)1145 2585 y Fp(N)1179 2578 y Fy(\()p Fo(ds)p Fy(\))p Fo(;)-66 2699 y Fy(where)h Fo(\027)99 2706 y Fp(N)149 2699 y Fy(is)g(the)g(uniform)e (probabilit)o(y)i(on)h(the)f(sphere)g Fo(S)1048 2706 y Fp(N)1081 2699 y Fy(.)913 2823 y(18)p eop %%Page: 19 19 bop 7 24 a Fy(The)20 b(main)e(p)q(oin)o(t)i(is)g(the)g(domination)f(of)h(the) 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b(J)15 b(R)g(L)g(Almeida,)e(D)i(J)h(Thouless,)g (Stabilit)o(y)e(of)g(the)i(Sherrington-Kirkpatric)o(k)g(solution)e(of)h(a)g (spin)g(glass)g(mo)q(del,)f Fu(J.)23 434 y(Phys.)h(A:)f(Math.)h(Gen.)p FB(,)f Fb(11)p FB(,)f(983{990)f(\(1978\).)-46 509 y([4])24 b(J)19 b(M)g(G)g(Amaro)e(de)j(Matos,)g(A)f(E)g(P)o(atric)o(k,)h(V)f(A)g (Zagrebno)o(v,)h(Random)d(in\014nite)i(v)o(olume)e(Gibbs)i(states)h(for)f (the)23 559 y(Curie-W)m(eiss)14 b(random)e(\014eld)i(Ising)f(mo)q(del,)f Fu(J.)j(Stat.)f(Phys.)p FB(,)g Fb(66)p FB(,)f(139{164)f(\(1992\).)-46 634 y([5])24 b(D)10 b(J)h(Amit,)e(H)i(Gutfreund,)g(H)g(Somp)q(olinsky)m(,)d (Spin-glass)i(mo)q(dels)f(of)h(neural)h(net)o(w)o(orks,)g Fu(Phys.)h(R)n(ev.) p FB(,)f Fb(A32)p FB(,)g(1007{1018)23 683 y(\(1985\).)-46 758 y([6])24 b(L)9 b(A)h(Bassalygo,)g(R)f(L)h(Dobrushin,)g(Uniqueness)h(of)e(a)g (Gibbs)h(\014eld)f(with)h(random)e(p)q(oten)o(tial:)15 b(an)10 b(elemen)o(tary)f(approac)o(h,)23 808 y Fu(Th.)14 b(Pr)n(ob.)h(Appl.)p FB(,)e Fb(31)p FB(,)g(572{589)f(\(1988\).)-46 883 y([7])24 b(C)16 b(E)h(Bezuidenhout,)i(G)d(R)h(Grimm)o(ett,)e(H)i(Kesten,)h Fu(Strict)f(ine)n(qualities)g(for)g(critic)n(al)g(values)g(of)h(Potts)f(mo)n (dels)h(and)23 932 y(r)n(andom)d(cluster)f(pr)n(o)n(c)n(esses)p FB(,)f(preprin)o(t)i(Bristol)e(Univ)o(ersit)o(y)h(\(1992\).)-46 1007 y([8])24 b(K)17 b(Binder,)i(A)e(P)h(Y)m(oung,)f(Spin)g(glasses:)25 b(exp)q(erimen)o(tal)17 b(facts,)h(theoretical)g(concepts,)h(and)e(op)q(en)h (questions,)g Fu(R)n(ev.)23 1057 y(Mo)n(d.)d(Phys.)p FB(,)e Fb(58)p FB(,)g(801{976)g(\(1988\).)-46 1132 y([9])24 b(K)10 b(Binder,)h(Y)f(Imry)m(,)f(E)h(Pytte,)h(In)o(terface)g(roughening)f(and)g (random)e(\014eld)i(instabilities,)g Fu(Phys.)h(R)n(ev.)p FB(,)g Fb(B24)p FB(,)f(6736{6739)23 1181 y(\(1981\).)-66 1256 y([10])23 b(P)13 b(M)h(Bleher,)g(The)g(Bethe)h(lattice)e(spin)g(glass)h(at)f(zero)h (temp)q(erature,)g Fu(A)o(nn.)h(Inst.)f(H.)g(Poinc)n(ar)o(\023)-20 b(e)p FB(,)13 b Fb(54)p FB(,)g(89{113)f(\(1991\).)-66 1331 y([11])23 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y(Math.)i(Phys.)p FB(,)e Fb(116)p FB(,)g(539{572)f (\(1988\).)-66 1854 y([16])23 b(J)18 b(Bricmon)o(t,)g(A)g(Kupiainen,)h (Random)d(w)o(alk)h(in)h(asymmetric)e(random)h(en)o(vironmen)o(ts,)h Fu(Commun.)h(Math.)g(Phys.)p FB(,)23 1904 y Fb(142)p FB(,)13 b(345{420)f(\(1991\).)-66 1978 y([17])23 b(E)11 b(Bu\013et,)i(A)e(P)o(atric)o (k,)h(J)f(V)h(Pul)o(\023)-20 b(e,)11 b(Directed)i(p)q(olymers)d(on)h(trees:) 19 b(a)11 b(martingale)e(approac)o(h,)i Fu(J.)h(Phys.)h(A:)f(Math.)h(Gen.)p FB(,)23 2028 y Fb(26)p FB(,)g(1823{1834)e(\(1993\).)-66 2103 y([18])23 b(M)12 b(Campanino,)f(E)i(Olivieri,)e(One)j(dimensional)c(random)i (Ising)g(system)h(with)f(in)o(teraction)h(deca)o(y)g Fj(r)1643 2088 y Fc(\000)p Fv(\(1+)p Fi(\017)p Fv(\))1753 2103 y FB(;)g(a)f(con)o(v)o (er-)23 2153 y(gen)o(t)i(cluster)h(expansion,)e Fu(Commun.)i(Math.)g(Phys.)p FB(,)f Fb(111)p FB(,)f(555{577)f(\(1987\).)-66 2228 y([19])23 b(M)13 b(Campanino,)e(E)j(Olivieri,)e(A)i(C)f(D)g(v)n(an)g(En)o(ter,)i(One)f 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y(glasses)f(in)f(a)h(magnetic)f(\014eld,)g Fu(J.)i(Phys.)g(F)m (r)n(anc)n(e)p FB(,)e Fb(51)p FB(,)g(1877{1895)f(\(1990\).)913 2823 y Fy(24)p eop %%Page: 25 25 bop -66 24 a FB([23])23 b(J)16 b(T)g(Chalk)o(er,)g(On)g(the)h(lo)o(w)o(er)e (critical)h(dimensionalit)o(y)d(of)i(the)i(Ising)f(mo)q(del)e(in)i(a)f (random)g(\014eld,)h Fu(J.)g(Phys.)h(C:)g(Sol.)23 74 y(State)e(Phys.)p FB(,)e Fb(16)p FB(,)g(6615{6622)f(\(1983\).)-66 147 y([24])23 b(B)11 b(Chauvin,)g(A)g(Rouault,)f Fu(Boltmann-Gibbs)j(wights)e(in)i(the)f (br)n(anching)h(r)n(andom)g(walk)p FB(,)d(IMA)i(w)o(orkshop)f(on)g(branc)o (hing)23 196 y(pro)q(cesses,)16 b(June)e(1994,)f(to)h(app)q(ear)g(in)f(the)i (pro)q(ceedings.)-66 269 y([25])23 b(D)13 b(Cho)o(wdh)o(ury)m(,)g Fu(Spin)i(glasses)g(and)h(other)e(frustr)n(ate)n(d)g(systems)p FB(,)f(W)m(orld)g(Scien)o(ti\014c,)h(Singap)q(ore)g(\(1986\).)-66 342 y([26])23 b(J)15 b(Cha)o(y)o(es,)g(L)g(Cha)o(y)o(es,)g(J)g(F)m(r\177)-21 b(ohlic)o(h,)14 b(The)i(lo)o(w)e(temp)q(erature)i(b)q(eha)o(viour)f(of)f (disordered)i(magnets,)e Fu(Commun.)i(Math.)23 392 y(Phys.)p FB(,)d Fb(100)p FB(,)g(399{437)f(\(1985\).)-66 465 y([27])23 b(J)18 b(Cha)o(y)o(es,)h(L)e(Cha)o(y)o(es,)i(J)f(P)h(Sethna,)g(D)e(J)i (Thouless,)g(A)f(mean)f(\014eld)h(spin)g(glass)f(with)h(short-range)h(in)o (teractions,)23 515 y Fu(Commun.)c(Math.)g(Phys.)p FB(,)e Fb(106)p FB(,)g(41{89)g(\(1986\).)-66 588 y([28])23 b(P)17 b(Collet,)g(J)h(P)g(Ec)o (kmann,)f(Spin)g(glass)g(mo)q(del)f(with)h(random)f(couplings,)h Fu(Commun.)h(Math.)h(Phys.)p FB(,)f Fb(93)p FB(,)f(379{406)23 638 y(\(1984\).)-66 711 y([29])23 b(P)14 b(Collet,)g(F)g(Koukiou,)g(Large)h (deviations)f(for)g(the)h(m)o(ultiplicativ)n(e)d(c)o(haos,)j Fu(Commun.)g(Math.)h(Phys.)p FB(,)e Fb(147)p FB(,)g(329{342)23 761 y(\(1992\).)-66 834 y([30])23 b(F)16 b(Comets,)g(J)g(Nev)o(eu,)i Fu(The)f(Sherrington-Kirkp)n(atrick)f(mo)n(del)h(of)g(spin)g(glasses)h(and)f (sto)n(chastic)h(c)n(alculus:)23 b(the)17 b(high)23 883 y(temp)n(er)n(atur)n (e)d(c)n(ase)p FB(,)f(preprin)o(t)i(CMAP)655 873 y(\023)651 883 y(Ecole)f(P)o(olytec)o(hnique)g(\(1993\).)-66 956 y([31])23 b(F)9 b(Comets,)h Fu(A)h(spheric)n(al)f(b)n(ound)i(for)f(the)g (Sherrington-Kirkp)n(atrick)f(mo)n(del)p FB(,)g(preprin)o(t)g(Univ)o(ersit)o (\023)-20 b(e)11 b(de)f(P)o(aris)g(VI)q(I)g(\(1994\).)-66 1029 y([32])23 b(P)10 b(Con)o(tucci,)h Fu(On)h(the)g(strict)e(ine)n(quality)i(b)n (etwe)n(en)g(quenche)n(d)h(and)f(anne)n(ale)n(d)h(Ising)f(spin)g(glass)p FB(,)e(preprin)o(ts)h(SISSA)g(\(1993\))23 1079 y(to)i(app)q(ear)i(in)e Fu(L)n(ett.)h(Math.)h(Phys.)-66 1152 y FB([33])23 b(B)18 b(Derrida,)g(Random) e(energy)j(mo)q(del:)24 b(limit)16 b(of)h(a)h(family)d(of)i(disordered)i(mo)q (dels,)e Fu(Phys.)i(R)n(ev.)g(L)n(ett.)p FB(,)f Fb(45)p FB(,)g(79{82)23 1202 y(\(1980\).)-66 1275 y([34])23 b(B)15 b(Derrida,)f(H)h(Sp)q(ohn,)g(P)o (olymers)f(on)h(disordered)h(trees,)g(spin)f(glasses,)g(and)g(tra)o(v)o (elling)e(w)o(a)o(v)o(es,)i Fu(J.)g(Stat.)g(Phys.)p FB(,)g Fb(51)p FB(,)23 1325 y(817{840)d(\(1988\).)-66 1398 y([35])23 b(B)12 b(Derrida,)g(M)g(R)g(Ev)n(ans,)g(E)g(R)g(Sp)q(eer,)h(Mean-\014eld)g (theory)f(of)g(directed)h(p)q(olyumers)f(with)g(random)e(complex)h(w)o(eigh)o (ts,)23 1448 y(preprin)o(t)j(SPhT)g(\(1992\))f(to)h(app)q(ear)h(in)e Fu(Commun.)i(Math.)g(Phys.)-66 1521 y FB([36])23 b(R)12 b(Dobrushin,)g(The)i (description)f(of)f(a)h(random)e(\014eld)i(b)o(y)g(means)f(of)g(conditional)f (probabilities)h(and)h(conditions)f(of)g(its)23 1570 y(regularit)o(y)m(,)g Fu(The)n(or.)i(Pr)n(ob.)h(Appl.)p FB(,)e Fb(13)p FB(,)g(197{224)f(\(1968\).) -66 1643 y([37])23 b(C)13 b(de)i(Dominicis,)c(M)j(Hilhorst,)f(Random)e(free)k (energies)g(in)f(spin)f(glasses,)h Fu(J.)h(Phys.)g(L)n(ett.)p FB(,)e Fb(46)p FB(,)g(L909{L914)f(\(1985\).)-66 1716 y([38])23 b(H)d(v)o(on)f(Dreifus,)i(A)f(Klein,)g(J)g(F)m(ernando)g(P)o(erez,)j Fu(T)m(aming)d(Gri\016ths')f(singularities:)28 b(in\014nite)21 b(di\013er)n(entiability)f(of)23 1766 y(quenche)n(d)c(c)n(orr)n(elation)e (functions)p FB(,)g(preprin)o(t)g(UCLA)g(\(1994\))f(to)h(app)q(ear)g(in)g Fu(Commun.)h(Math.)g(Phys.)-66 1839 y FB([39])23 b(S)14 b(F)f(Edw)o(ards,)h (P)g(W)g(Anderson,)h(Theory)f(of)f(spin)h(glasses,)g Fu(J.)g(Phys.)h(F:)g (Metal)g(Phys.)p FB(,)e Fb(5)p FB(,)h(965{974)e(\(1975\).)-66 1912 y([40])23 b(A)11 b(C)g(D)g(v)n(an)f(En)o(ter,)i(One-dimensional)e(spin)h (glasses;)h(uniqueness)g(and)f(cluster)h(prop)q(erties,)h Fu(J.)f(Phys.)g(A:) g(Math.)h(Gen.)p FB(,)23 1962 y Fb(21)p FB(,)g(1781{1786)e(\(1988\).)-66 2035 y([41])23 b(A)15 b(C)g(D)f(v)n(an)g(En)o(ter,)i(J)f(L)g(v)n(an)g (Hemmen,)e(Statistical)h(mec)o(hanical)f(formalism)e(for)k(spin)g(glasses,)g Fu(Phys.)h(R)n(ev.)g(A)p FB(,)e Fb(29)p FB(,)23 2085 y(355{365)e(\(1984\).) -66 2158 y([42])23 b(A)14 b(C)f(D)h(v)n(an)f(En)o(ter,)h(J)g(L)g(v)n(an)f (Hemmen,)f(Thermo)q(dynamic)f(limit)g(for)j(long-range)f(spin)h(glasses,)g Fu(J.)g(Stat.)h(Phys.)p FB(,)e Fb(32)p FB(,)23 2208 y(141{152)f(\(1985\).)-66 2281 y([43])23 b(A)d(C)h(D)f(v)n(an)g(En)o(ter,)j(J)e(F)m(r\177)-21 b(ohlic)o(h,)21 b(Absence)i(of)d(symmetry)f(breaking)h(for)h Fj(N)5 b FB(-v)o(ector)21 b(spin)g(glass)f(mo)q(dels)g(in)g(t)o(w)o(o)23 2330 y(dimensions,)12 b Fu(Commun.)j(Math.)g(Phys.)p FB(,)e Fb(98)p FB(,)h(425{432)e(\(1985\).)-66 2403 y([44])23 b(A)15 b(C)h(D)f(v)n(an)g(En)o(ter,)h(R)f(Gri\016ths,)g(The)h(order)g(parameter)f (in)g(a)g(spin)h(glass,)f Fu(Commun.)h(Math.)h(Phys.)p FB(,)e Fb(90)p FB(,)g(319{327)23 2453 y(\(1983\).)-66 2526 y([45])23 b(A)11 b(C)h(D)f(v)n(an)g(En)o(ter,)h(J)g(Mi\030)-18 b(ekisz,)11 b(Breaking)h(of)f(p)q(erio)q(dicit)o(y)g(at)g(p)q(ositiv)o(e)h(temp)q (eratures,)g Fu(Commun.)h(Math.)g(Phys.)p FB(,)e Fb(134)p FB(,)23 2576 y(647{651)h(\(1990\).)-66 2649 y([46])23 b(D)15 b(Espriu,)h(M)f(Gross,)h (P)g(E)g(L)f(Rak)o(o)o(w,)f(J)i(F)f(Wheater,)i(Con)o(tin)o(uum)c(limit)g(on)i (a)g(t)o(w)o(o-dimensional)e(random)h(lattice,)23 2699 y Fu(Nucl.)g(Phys.)p FB(,)g Fb(B265)h([FS)h(15])p FB(,)d(92{112)g(\(1986\).)913 2823 y Fy(25)p eop %%Page: 26 26 bop -66 24 a FB([47])23 b(D)11 b(S)h(Fisher,)g(J)h(F)m(r\177)-21 b(ohlic)o(h,)10 b(T)i(Sp)q(encer,)i(The)e(Ising)g(mo)q(del)e(in)h(a)h(random) e(magnetic)h(\014eld,)g Fu(J.)i(Stat.)g(Phys.)p FB(,)f Fb(34)p FB(,)f(863{870)23 74 y(\(1984\).)-66 148 y([48])23 b(D)18 b(S)h(Fisher,)h(D)e (A)h(Huse,)i(Pure)e(phases)h(in)f(spin)f(glasses,)i Fu(J.)f(Phys.)h(A:)f (Math.)g(Gen.)p FB(,)h Fb(20)p FB(,)f(L997{L1003)e(\(1987\);)23 198 y(Absence)e(of)f(man)o(y)e(states)j(in)e(magnetic)g(spin)h(glasses,)g Fu(J.)g(Phys.)h(A:)g(Math.)g(Gen.)p FB(,)e Fb(20)p FB(,)g(L1005{L1010)f (\(1987\).)-66 273 y([49])23 b(J)d(F)m(r\177)-21 b(ohlic)o(h,)20 b Fu(Mathematic)n(al)g(asp)n(e)n(cts)h(of)f(the)h(physics)g(of)f(disor)n(der) n(e)n(d)g(systems)p FB(,)h(in)e Fu(Critic)n(al)g(phenomena,)k(r)n(andom)23 323 y(systems,)14 b(and)i(gauge)f(the)n(ories)p FB(,)e(K)h(Osterw)o(alder,)h (R)f(Stora)f(eds.)i(725{894,)c(North-Holland,)i(Amsterdam)f(\(1986\).)-66 397 y([50])23 b(J)13 b(F)m(r\177)-21 b(ohlic)o(h,)13 b(J)g(Im)o(brie,)f (Impro)o(v)o(ed)g(p)q(erturbation)i(expansion)g(for)f(disordered)i(systems:)j (b)q(eating)13 b(Gri\016ths)g(singulari-)23 447 y(ties,)g Fu(Commun.)i(Math.) h(Phys.)p FB(,)d Fb(96)p FB(,)g(145{180)f(\(1984\).)-66 522 y([51])23 b(J)14 b(F)m(r\177)-21 b(ohlic)o(h,)13 b(B)h(Zegarlinski,)f(The)i (disordered)g(phase)g(of)e(long-range)g(Ising)h(spin)g(glasses,)g Fu(Eur)n(ophys.)h(L)n(ett.)p FB(,)e Fb(2)p FB(,)h(53{60)23 572 y(\(1986\).)-66 646 y([52])23 b(J)14 b(F)m(r\177)-21 b(ohlic)o(h,)13 b(B)i(Zegarlinski,)f(The)g(high)g(temp)q(erature)h(phase)g(of)f(long)f(range) i(spin)f(glasses,)h Fu(Commun.)g(Math.)h(Phys.)p FB(,)23 696 y Fb(110)p FB(,)d(121{155)f(\(1987\).)-66 771 y([53])23 b(J)12 b(F)m(r\177)-21 b(ohlic)o(h,)11 b(B)i(Zegarlinski,)e(Some)h(commen)o(ts)e(on) i(the)h(Sherrington-Kirkpatric)o(k)f(mo)q(del)f(of)g(spin)i(glasses,)f Fu(Commun.)23 821 y(Math.)j(Phys.)p FB(,)e Fb(112)p FB(,)g(553{566)f (\(1987\).)-66 895 y([54])23 b(J)16 b(F)m(r\177)-21 b(ohlic)o(h,)16 b(B)g(Zegarlinski,)g(Spin)g(glasses)h(and)f(other)h(lattice)f(systems)g(with) g(long)f(range)i(in)o(teractions,)f Fu(Commun.)23 945 y(Math.)f(Phys.)p FB(,)e Fb(120)p FB(,)g(665{688)f(\(1989\).)-66 1020 y([55])23 b(A)15 b(Galv)o(es,)f(S)h(Mart)-5 b(\023)-16 b(\020nez,)16 b(P)f(Picco,)h(Fluctuations)f(in)g(Derrida's)g(random)e(energy)j(and)f (generalised)h(random)e(energy)23 1070 y(mo)q(dels,)e Fu(J.)i(Stat.)h(Phys.)p FB(,)f Fb(54)p FB(,)f(515{529)f(\(1989\).)-66 1145 y([56])23 b(E)14 b(Gardner,)f(A)h(spin)g(glass)g(on)g(a)f(hierarc)o(hical)h(lattice,)f Fu(J.)i(Phys.)g(F)m(r)n(anc)n(e)p FB(,)e Fb(45)p FB(,)g(1755{1763)f (\(1984\).)-66 1219 y([57])23 b(C)15 b(P)h(M)f(Geerse,)i(A)f(Hof,)f(Lattice)h (gas)f(mo)q(dels)g(on)g(self-similar)e(ap)q(erio)q(dic)j(tilings,)e Fu(R)n(ev.)j(Math.)f(Phys.)p FB(,)g Fb(3)p FB(,)f(163{221)23 1269 y(\(1991\).)-66 1344 y([58])23 b(A)15 b(Georges,)g(M)g(M)o(\023)-20 b(ezard,)16 b(J)f(Y)m(edidia,)f(Lo)o(w)g(temp)q(erature)i(phase)g(of)e(the)i (Ising)f(spin)g(glass)f(on)h(a)g(h)o(yp)q(ercubic)h(lattice,)23 1394 y Fu(Phys.)f(R)n(ev.)g(L)n(ett.)p FB(,)d Fb(64)p FB(,)h(2937{2940)f (\(1990\).)-66 1468 y([59])23 b(A)14 b(Georges,)g(D)g(Hansel,)g(P)g(Le)g (Doussal,)g(J)g(M)g(Maillard,)e(J)j(Bouc)o(haud,)f(Rigorous)f(b)q(ounds)h (for)g(2D)g(disordered)h(Ising)23 1518 y(mo)q(dels,)d Fu(J.)i(Phys.)h(F)m(r)n (anc)n(e)p FB(,)f Fb(47)p FB(,)f(947{953)f(\(1986\).)-66 1593 y([60])23 b(H)18 b(O)g(Georgii,)g(Sp)q(on)o(taneous)g(magnetisation)e(of)h (randomly)f(dilute)i(ferromagnets,)g Fu(J.)h(Stat.)f(Phys.)p FB(,)h Fb(25)p FB(,)f(369{396)23 1643 y(\(1981\).)-66 1717 y([61])23 b(V)14 b(L)f(Girk)o(o,)g Fu(The)n(ory)h(of)h(r)n(andom)g (determinants)p FB(,)f(Klu)o(w)o(er)g(Academic)f(Publ.,)g(Do)q(drec)o(h)o(t)h (\(1990\).)-66 1792 y([62])23 b(E)13 b(Goles,)g(S)g(Mart)-5 b(\023)-16 b(\020nez,)14 b(The)g(one-site)g(distribution)f(of)g(Gibbs)g (states)h(on)f(the)h(Bethe)h(lattice)f(are)g(probabilit)o(y)d(v)o(ectors)23 1842 y(of)i(p)q(erio)q(d)h Fh(\024)e FB(2)i(for)f(a)h(nonlinear)f (transformation,)e Fu(J.)k(Stat.)g(Phys.)p FB(,)e Fb(52)p FB(,)g(267{285)f (\(1988\).)-66 1917 y([63])23 b(S)c(Goulart-Rosa,)g(The)i(thermo)q(dynamic)c (limit)g(of)i(quenc)o(hed)i(free)g(energy)f(of)f(magnetic)g(systems)h(with)f (random)23 1966 y(in)o(teractions,)13 b Fu(J.)i(Phys.)g(A:)f(Math.)h(Gen.)p FB(,)f Fb(15)p FB(,)f(L51{L54)g(\(1982\).)-66 2041 y([64])23 b(R)17 b(B)i(Gri\016ths,)f(Nonalytic)f(b)q(eha)o(viour)h(ab)q(o)o(v)o(e)g (the)h(critical)e(p)q(oin)o(t)h(in)g(a)f(random)g(Ising)h(ferromagnet,)f Fu(Phys.)i(R)n(ev.)23 2091 y(L)n(ett.)p FB(,)12 b Fb(23)p FB(,)h(17{19)g (\(1969\).)-66 2166 y([65])23 b(G)15 b(R)g(Grimmett,)e(The)j(sto)q(c)o (hastic)h(random)d(cluster)j(pro)q(cess)h(and)e(the)g(uniqueness)h(for)e (random)g(cluster)i(measures,)23 2215 y(preprin)o(t)d(Cam)o(bridge)e (\(1994\).)-66 2290 y([66])23 b(F)13 b(Guerra,)h Fu(Fluctuations)h(and)g (thermo)n(dynamic)g(variables)g(in)f(me)n(an)h(\014eld)g(spin)g(glass)f FB(mo)q(dels,)e(preprin)o(t)i(Roma)d(-)j(La)23 2340 y(Sapienza)g(\(1992\).) -66 2415 y([67])23 b(F)15 b(Guerra,)g Fu(The)h(c)n(avity)f(metho)n(d)i(in)e (the)h(me)n(an)h(\014eld)f(spin)g(glass)g(mo)n(del.)f(F)m(unctional)h(r)n (epr)n(esentations)g(of)g(thermo)n(dy-)23 2465 y(namic)f(variables)p FB(,)e(preprin)o(t)h(Roma)e(-)h(La)h(Sapienza)g(\(1994\).)-66 2539 y([68])23 b(F)11 b(Guerra,)g Fu(F)m(unctional)i(or)n(der)f(p)n(ar)n (ameters)f(for)h(the)h(quenche)n(d)g(fr)n(e)n(e)f(ener)n(gy)g(in)h(me)n(an)f (\014eld)h(spin)g(glass)f(mo)n(del)p FB(,)f(preprin)o(t)23 2589 y(Roma)g(-)j(La)f(Sapienza)h(\(1992\).)-66 2664 y([69])23 b(J)14 b(L)g(v)n(an)f(Hemmen,)e(Classical)j(spin)f(glasses)i(mo)q(del,)d Fu(Phys.)j(R)n(ev.)g(L)n(ett.)p FB(,)e Fb(49)p FB(,)g(409{412)f(\(1982\).)913 2823 y Fy(26)p eop %%Page: 27 27 bop -66 24 a FB([70])23 b(J)10 b(L)h(v)n(an)f(Hemmen,)f(I)h(Morgenstern)i (eds.,)f(Heidelb)q(erg)h(collo)q(quium)c(on)i(spin)g(glasses,)i Fu(L)n(e)n(ct.)f(Notes)h(Phys.)p FB(,)f Fb(192)f FB(\(1983\).)-66 98 y([71])23 b(J)16 b(J)g(Hop\014eld,)g(Neural)g(net)o(w)o(orks)h(and)f(ph)o (ysical)f(systems)i(with)e(emergen)o(t)h(collectiv)o(e)g(computational)e (capabilities,)23 148 y Fu(Pr)n(o)n(c.)g(Natl.)g(A)n(c)n(ad.)h(Sci.)f(USA)p FB(,)g Fb(79)p FB(,)f(2554{2558)f(\(1982\).)-66 223 y([72])23 b(J)17 b(Im)o(brie,)e(The)j(ground)e(states)i(of)e(the)i(three-dimensional)d (random)h(\014eld)g(Ising)h(mo)q(del,)e Fu(Commun.)j(Math.)g(Phys.)p FB(,)23 273 y Fb(98)p FB(,)13 b(145{176)f(\(1985\).)-66 348 y([73])23 b(Y)14 b(Imry)m(,)d(Random)h(external)i(\014elds,)g Fu(J.)g(Stat.)h(Phys.)p FB(,)f Fb(34)p FB(,)f(849{862)f(\(1984\).)-66 422 y([74])23 b(J)17 b(P)f(Kahane,)i(Multiplications)d(al)o(\023)-20 b(eatoires)16 b(et)h(dimension)e(de)i(Hausdor\013,)h Fu(A)o(nn.)f(Inst.)h(H.) f(Poinc)n(ar)o(\023)-20 b(e)p FB(,)17 b Fb(23)p FB(,)f(289{296)23 472 y(\(1987\).)-66 547 y([75])23 b(K)14 b(M)g(Khanin,)g(Absence)i(of)d (phase)i(transition)f(in)f(one-dimensional)f(long-range)i(spin)g(systems)g (with)g(random)e(in)o(ter-)23 597 y(actions,)h Fu(The)n(or.)h(Math.)h(Phys.)p FB(,)f Fb(43)p FB(,)f(445{449)f(\(1980\).)-66 671 y([76])23 b(K)14 b(M)g(Khanin,)f(Y)m(a)g(Sinai,)f(Existence)k(of)d(free)i(energy)g(for) e(mo)q(dels)g(with)h(long-range)f(random)f(Hamiltonian,)e Fu(J.)15 b(Stat.)23 721 y(Phys.)p FB(,)e Fb(20)p FB(,)g(573{584)f(\(1979\).)-66 796 y([77])23 b(S)14 b(Kirkpatric)o(k,)f(D)g(Sherrington,)h (In\014nite-ranged)g(mo)q(dels)f(of)g(spin)h(glasses)h Fu(Phys.)g(R)n(ev.)g (B)p FB(,)e Fb(17)p FB(,)h(4384{4403)d(\(1978\).)-66 871 y([78])23 b(H)14 b(Ko)q(c)o(h,)g(A)f(free)i(energy)g(b)q(ound)f(for)f(the)i(Hop\014eld) f(mo)q(del,)e Fu(J.)i(Phys.)h(A:)g(Math.)g(Gen.)p FB(,)e Fb(26)p FB(,)g(L353{L355)f(\(1993\).)-66 945 y([79])23 b(R)11 b(Kotec)o(k)q(\023)-21 b(y,)11 b(E)g(Olivieri,)g(Droplet)g(dynamics)f(for)h(asymmetric)f(Ising)h(mo) q(del,)f Fu(J.)i(Stat.)h(Phys.)p FB(,)e Fb(70)p FB(,)g(1121{1148)f(\(1993\).) -66 1020 y([80])23 b(A)14 b(M)g(Khorunzy)m(,)h(B)f(A)h(Khoruzhenk)o(o,)f(L)h (A)f(P)o(astur,)h(M)f(Shc)o(herbina,)g Fu(L)n(ar)n(ge)h Fj(n)h Fu(limit)e(in)h(statistic)n(al)f(me)n(chanics)i(and)23 1070 y(sp)n(e)n(ctr)n(al)g(the)n(ory)i(of)g(disor)n(der)n(e)n(d)f(systems)p FB(,)g(in)f Fu(Phase)j(tr)n(ansitions)e(and)h(critic)n(al)f(phenomena)h FB(\(Dom)o(b)d(and)i(Leb)q(o)o(witz)23 1120 y(eds.\),)d Fb(15)p FB(,)f(74{241,)f(Academic)h(Press,)i(London)e(\(1992\).)-66 1194 y([81])23 b(F)12 b(Koukiou,)g(Rigorous)g(b)q(ounds)h(for)g(the)g(free)h (energy)f(of)f(short)i(range)f(Ising)f(spin)h(glass)f(mo)q(del,)f Fu(Eur)n(ophys.)j(L)n(ett.)p FB(,)e Fb(17)p FB(,)23 1244 y(669{671)g (\(1992\).)-66 1319 y([82])23 b(F)11 b(Koukiou,)g(A)g(random)f(co)o(v)o (ering)i(in)o(terpretation)f(for)g(the)h(phase)h(transition)e(of)f(the)i (random)e(energy)j(mo)q(del,)d Fu(J.)i(Stat.)23 1369 y(Phys.)p FB(,)h Fb(60)p FB(,)g(669{674)f(\(1990\).)-66 1443 y([83])23 b(F)14 b(Koukiou,)e(priv)n(ate)i(comm)o(unicatio)o(n)d(\(1994\).)-66 1518 y([84])23 b(F)18 b(Koukiou,)f(D)h(P)o(etritis,)h(M)f(Zahradn)-5 b(\023)-16 b(\020k,)18 b(Extension)g(of)g(the)g(Pirogo)o(v-Sinai)e(theory)i (to)g(a)g(class)g(of)f(quasip)q(erio)q(dic)23 1568 y(in)o(teractions,)c Fu(Commun.)i(Math.)g(Phys.)p FB(,)f Fb(118)p FB(,)f(365{383)f(\(1988\).)-66 1643 y([85])23 b(F)15 b(Koukiou,)g(P)h(Picco,)g(P)o(oisson)g(p)q(oin)o(t)f (pro)q(cesses,)j(cascades,)f(and)f(random)e(co)o(v)o(erings)i(of)f Ff(R)1553 1619 y Fi(n)1575 1643 y FB(,)h Fu(J.)g(Stat.)g(Phys.)p FB(,)g Fb(62)p FB(,)23 1692 y(481{489)c(\(1991\).)-66 1767 y([86])23 b(W)14 b(Krauth,)h(O)g(Pluc)o(hery)m(,)g(A)g(rapid)f(dynamical)e (Mon)o(te)j(Carlo)f(algorithm)f(for)h(glassy)g(systems,)h Fu(J.)g(Phys.)h(A:) f(Math.)23 1817 y(Gen.)p FB(,)e Fb(27)p FB(,)g(L715{L720)f(\(1994\).)-66 1892 y([87])23 b(O)15 b(E)h(Lanford,)f(D)h(Ruelle,)f(Observ)n(able)h(at)g (in\014nit)o(y)f(and)g(states)i(with)e(short)h(range)g(correlations)g(in)f (statistical)h(me-)23 1942 y(c)o(hanics,)e Fu(Commun.)h(Math.)g(Phys.)p FB(,)e Fb(13)p FB(,)g(194{215)f(\(1969\).)-66 2016 y([88])23 b(M)14 b(Ledoux,)f(M)h(T)m(alagrand,)e Fu(Pr)n(ob)n(ability)i(in)h(Banach)h (sp)n(ac)n(es)p FB(,)d(Springer-V)m(erlag,)h(Berlin)g(\(1991\).)-66 2091 y([89])23 b(F)d(Ledrappier,)h(Pressure)i(and)c(v)n(ariational)f (principle)i(for)f(random)g(Ising)g(mo)q(del,)h Fu(Commun.)g(Math.)h(Phys.)p FB(,)g Fb(56)p FB(,)23 2141 y(297{302)12 b(\(1977\).)-66 2215 y([90])23 b(L)13 b(Lo)o(v\023)-21 b(asz,)14 b Fu(Combinatorial)g(pr)n(oblems) g(and)i(exer)n(cises)p FB(,)d(North-Holland,)f(Amsterdam)h(\(1993\).)-66 2290 y([91])23 b(T)13 b(C)g(Lub)q(ensky)m(,)g(Critical)g(prop)q(erties)h(of)f (random)f(spin)h(mo)q(dels)f(from)f Fj(\017)i FB(expansions,)g Fu(Phys.)i(R)n(ev.)f(B)p FB(,)f Fb(11)p FB(,)g(3573{3580)23 2340 y(\(1975\).)-66 2415 y([92])23 b(R)10 b(J)h(McEliece,)i Fu(The)f(the)n(ory)g(of)g(information)g(and)h(c)n(o)n(ding)p FB(,)e(Encyclop)q(edia)h(of)e(Mathematics,)h(Addison-W)m(esley)m(,)f(Read-)23 2465 y(ing)j(\(1977\).)-66 2539 y([93])23 b(E)17 b(Marinari,)f(G)g(P)o (arisi,)h(F)g(Ritort,)f Fu(R)n(eplic)n(a)i(\014eld)f(the)n(ory)h(for)f (deterministic)f(mo)n(dels:)24 b(binary)18 b(se)n(quenc)n(es)g(with)f(low)23 2589 y(auto)n(c)n(orr)n(elation)p FB(,)11 b(preprin)o(t)i(Roma)d(-)i(La)f (Sapienza)i(\(1994\),)e(a)o(v)n(ailable)f(from)g Fz(hep-th@babbage.sissa)o (.it)f FB(under)k(the)23 2639 y(reference)j(9405148.)913 2823 y Fy(27)p eop %%Page: 28 28 bop -66 24 a FB([94])23 b(E)12 b(Marinari,)f(G)h(P)o(arisi,)g(F)g(Ritort,)f Fu(R)n(eplic)n(a)j(\014eld)f(the)n(ory)g(for)g(deterministic)g(mo)n(dels:)18 b(non)c(r)n(andom)f(spin)h(glasses)f(with)23 74 y(glassy)g(b)n(ehaviour)p FB(,)g(preprin)o(t)h(Roma)c(-)j(La)f(Sapienza)h(\(1994\),)f(a)o(v)n(ailable)e (from)h Fz(cond-mat@babbage.s)o(issa.)o(it)f FB(under)23 123 y(the)k(reference)i(9406074.)-66 198 y([95])23 b(E)14 b(Marinari,)f(G)g(P)o (arisi,)g(F)h(Ritort,)f Fu(On)i(the)g(3d)h(Ising)f(spin)g(glass)p FB(,)e(preprin)o(t)i(Roma)c(-)j(La)g(Sapienza)g(\(1993\),)f(a)o(v)n(ailable) 23 248 y(from)f Fz(cond-mat@babbage)o(.siss)o(a.it)e FB(under)15 b(the)g(reference)h(9310041.)-66 323 y([96])23 b(F)13 b(Martinelli,)f (Dynamical)f(analysis)h(of)h(the)h(lo)o(w)f(temp)q(erature)h(cluster)h (algorithms,)10 b Fu(J.)k(Stat.)h(Phys.)p FB(,)e Fb(66)p FB(,)g(1245{1276)23 372 y(\(1992\).)-66 447 y([97])23 b(F)14 b(Martinelli,)e(E)i(Olivieri,)e(E)j (Scopp)q(ola,)e(On)h(the)g(Sw)o(endsen-W)m(ang)g(dynamics)f(I:)g(Exp)q(onen)o (tial)g(con)o(v)o(ergence)j(to)e(the)23 497 y(equilibrium,)c Fu(J.)15 b(Stat.)g(Phys.)p FB(,)e Fb(62)p FB(,)g(117{133)f(\(1991\).)-66 572 y([98])23 b(F)14 b(Martinelli,)g(E)h(Olivieri,)e(E)i(Scopp)q(ola,)f(On)h (the)g(Sw)o(endsen-W)m(ang)g(dynamics)e(I)q(I:)i(Critical)e(droplets)j(and)e (homoge-)23 621 y(neous)g(n)o(ucleation)g(at)f(lo)o(w)g(temp)q(erature,)h Fu(J.)h(Stat.)g(Phys.)p FB(,)e Fb(62)p FB(,)g(135{159)f(\(1991\).)-66 696 y([99])23 b(J)14 b(M)o(\023)-20 b(emin,)12 b(priv)n(ate)h(comm)o (unication)e(\(1994\).)-66 771 y([100])22 b(M)14 b(M)o(\023)-20 b(ezard,)15 b(G)e(P)o(arisi,)g(M)h(Virasoro,)f Fu(Spin)j(glass)e(the)n(ory)h (and)h(b)n(eyond)p FB(,)e(W)m(orld)f(Scien)o(ti\014c,)h(Singap)q(ore)f (\(1987\).)-66 846 y([101])22 b(M)14 b(M)o(\023)-20 b(ezard,)13 b(G)g(P)o(arisi,)f(M)i(Virasoro,)e(The)i(replica)f(solution)f(without)h (replicas,)g Fu(Eur)n(ophys.)i(L)n(ett.)p FB(,)d Fb(1)p FB(,)h(77{82)f (\(1986\).)-66 920 y([102])22 b(M)12 b(M)o(\023)-20 b(ezard,)13 b(G)f(P)o(arisi,)f(N)h(Sourlas,)f(G)h(T)m(oulouse,)f(M)h(Virasoro,)g(Nature)g (of)g(the)g(spin-glass)g(phase,)g Fu(Phys.)i(R)n(ev.)f(L)n(ett.)p FB(,)23 970 y Fb(52)p FB(,)g(1156{1159)e(\(1984\).)-66 1045 y([103])22 b(J)15 b(Mi\030)-18 b(ekisz,)13 b(A)h(microscopic)f(mo)q(del)g (with)g(quasicrystalline)h(prop)q(erties,)g Fu(J.)h(Stat.)g(Phys.)p FB(,)e Fb(58)p FB(,)g(1137{1146)f(\(1990\).)-66 1120 y([104])22 b(C)12 b(M)f(Newman,)f(Memory)g(capacit)o(y)h(in)g(neural)g(net)o(w)o(ork)h (mo)q(del:)j(rigorous)c(lo)o(w)o(er)g(b)q(ounds,)h Fu(Neur)n(al)f(Net.)p FB(,)g Fb(1)p FB(,)g(223{238)23 1169 y(\(1988\).)-66 1244 y([105])22 b(C)d(M)f(Newman,)g(D)f(L)i(Stein,)g(Multiple)e(states)j(and)e(thermo)q (dynamic)e(limits)g(in)h(short)i(ranged)g(Ising)f(spin)g(glass)23 1294 y(mo)q(dels,)12 b Fu(Phys.)j(R)n(ev.)g(B)p FB(,)f Fb(46)p FB(,)f(973{)f(\(1992\).)-66 1369 y([106])22 b(A)14 b(Ogielski,)e(I)h (Morgenstern,)h(Critical)e(b)q(eha)o(viour)h(of)g(the)h(three)g(dimensional)d (Ising)i(spin)g(glass)g(mo)q(del,)e Fu(Phys.)j(R)n(ev.)23 1418 y(L)n(ett.)p FB(,)e Fb(54)p FB(,)h(928{931)g(\(1985\).)-66 1493 y([107])22 b(E)14 b(Olivieri,)f(P)h(Picco,)f(On)i(the)f(existence)i(of)d (thermo)q(dynamics)f(for)h(the)i(random)d(energy)j(mo)q(del,)c Fu(Commun.)k(Math.)23 1543 y(Phys.)p FB(,)e Fb(96)p FB(,)g(125{144)f (\(1984\).)-66 1618 y([108])22 b(G)14 b(P)o(arisi,)f(In\014nite)h(n)o(um)o(b) q(er)f(of)g(order)i(parameters)f(for)f(spin)h(glasses,)g Fu(Phys.)h(R)n(ev.)g (L)n(ett.)p FB(,)e Fb(43)p FB(,)g(1754{1756)f(\(1979\).)-66 1692 y([109])22 b(G)13 b(P)o(arisi,)g(A)h(sequence)h(of)e(appro)o(ximated)f (solutions)h(to)h(the)g(Sherrington-Kirkpatric)o(k)f(mo)q(del)g(for)g(spin)g (glasses,)h Fu(J.)23 1742 y(Phys.)h(A:)f(Math.)h(Gen.)p FB(,)f Fb(13)p FB(,)f(L115{L121)f(\(1980\).)-66 1817 y([110])22 b(G)14 b(P)o(arisi,)f(The)h(order)h(parameter)f(for)f(spin)h(glasses:)19 b(a)14 b(function)f(on)h(the)h(in)o(terv)n(al)e([0)p Fj(;)7 b FB(1],)k Fu(J.)k(Phys.)g(A:)g(Math.)g(Gen.)p FB(,)23 1867 y Fb(13)p FB(,)e(1101{1112)e(\(1980\).)-66 1942 y([111])22 b(G)17 b(P)o(arisi,)f(Magnetic)h(prop)q(erties)i(of)d(spin)h(glasses)g(in)g (a)f(new)h(mean)f(\014eld)h(theory)m(,)g Fu(J.)g(Phys.)h(A:)f(Math.)h(Gen.)p FB(,)f Fb(13)p FB(,)23 1991 y(1887{1895)11 b(\(1980\).)-66 2066 y([112])22 b(L)11 b(P)o(astur,)h Fu(Statistic)n(al)g(physics)h(and)g(sp) n(e)n(ctr)n(al)f(the)n(ory)g(o)n(d)g(disor)n(der)n(e)n(d)g(systems:)18 b(some)12 b(r)n(e)n(c)n(ent)g(developments)p FB(,)g(in)f(Pro)q(c.)23 2116 y(Xth)j(In)o(ternational)f(conference)j(of)d(mathematical)e(ph)o(ysics)j (\(Leipzig)g(1991\),)e(Sringer-V)m(erlag,)h(Berlin)i(\(1992\).)-66 2191 y([113])22 b(L)14 b(P)o(astur,)g(A)g(Figotin,)e(Theory)j(of)e (disordered)i(spin)f(systems,)f Fu(The)n(or.)i(Math.)g(Phys.)p FB(,)e Fb(35)p FB(,)g(403{414)g(\(1978\).)-66 2265 y([114])22 b(L)14 b(P)o(astur,)g(A)g(Figotin.)e Fu(Sp)n(e)n(ctr)n(a)j(of)g(r)n(andom)g (and)h(almost)e(p)n(erio)n(dic)h(op)n(er)n(ators)p FB(,)e(Springer-V)m (erlag,)g(Berlin)h(\(1992\).)-66 2340 y([115])22 b(L)13 b(P)o(astur,)g(M)g (Shc)o(herbina,)g(Absence)i(of)d(self-a)o(v)o(eraging)g(of)g(the)h(order)h (parameter)f(in)f(the)h(Sherrington-Kirkpatric)o(k)23 2390 y(mo)q(del,)f Fu(J.)i(Stat.)h(Phys.)p FB(,)e Fb(62)p FB(,)g(1{19)g(\(1991\).) -66 2465 y([116])22 b(L)14 b(P)o(astur,)f(M)g(Shc)o(herbina,)h(B)f(Tirozzi,)g (The)h(replica)f(symmetric)f(solution)g(without)h(replica)g(tric)o(k)h(for)f (the)g(Hop\014eld)23 2514 y(mo)q(del,)f Fu(J.)i(Stat.)h(Phys.)p FB(,)e Fb(74)p FB(,)g(1161{1183)f(\(1994\).)-66 2589 y([117])22 b(D)14 b(P)o(etritis,)g(The)g(zero-lev)o(el)h(set)g(of)e(the)h(Bro)o(wnian)g (bridge)g(as)g(a)g(mo)q(del)e(of)h(disordered)j(ph)o(ysical)d(systems,)h(in)f (Pro)q(c.)23 2639 y(Prob.)g(Theory)i(and)e(Math.)h(Statistics)g(V)m(ol.)e(2)i (\(Vilnius)f(1989\),)g(Mosklas,)g(Vilnius)g(\(1990\).)913 2823 y Fy(28)p eop %%Page: 29 29 bop -66 24 a FB([118])22 b(D)13 b(P)o(etritis,)g Fu(Simulations)g(num)o(\023) -20 b(eriques)14 b(Monte)h(Carlo)p FB(,)c(preprin)o(t)j(Univ)o(ersit)o(\023) -20 b(e)13 b(de)g(Rennes)g(I)g(\(1993\),)f(to)g(b)q(e)h(published)23 74 y(b)o(y)g(Masson,)h(P)o(aris)g(\(1995\).)-66 148 y([119])22 b(P)14 b(Picco,)g Fu(V)m(err)n(es)g(de)h(spin)p FB(,)f(Th)o(\022)-20 b(ese)15 b(d')653 138 y(\023)649 148 y(Etat,)e(Univ)o(ersit)o(\023)-20 b(e)15 b(de)f(Marseille)g(I)q(I)g(\(1988\).)-66 223 y([120])22 b(P)14 b(Picco,)g(Rigorous)f(results)i(on)f(spin)g(glasses:)k(a)c(review,)g Fu(R)n(ev.)h(Math.)g(Appl.)p FB(,)e Fb(12)p FB(,)g(157{169)f(\(1991\).)-66 298 y([121])22 b(P)16 b(Picco,)h(Upp)q(er)f(b)q(ound)g(on)g(the)g(deca)o(y)h (of)e(correlations)h(in)f(the)i(plane)e(rotator)h(mo)q(del)f(with)g (long-range)g(random)23 348 y(in)o(teraction,)e Fu(J.)h(Stat.)h(Phys.)p FB(,)f Fb(36)p FB(,)f(489{516)f(\(1984\).)-66 422 y([122])22 b(L)14 b(E)g(P)o(on)o(tin,)f(J)h(A)g(Ba)o(^)-20 b(eta)14 b(Segundo,)f(J)h(F)m (ernando)g(P)o(erez,)h(Dilute)e(an)o(tiferromagnets:)k(Imry-Ma)12 b(argumen)o(t,)g(hierar-)23 472 y(c)o(hical)h(mo)q(del,)f(and)i(equiv)n (alence)g(to)g(random)e(\014eld)i(Ising)f(mo)q(dels,)g Fu(J.)h(Stat.)h(Phys.) p FB(,)e Fb(75)p FB(,)g(51{65)g(\(1994\).)-66 547 y([123])22 b(P)14 b(R)o(\023)-20 b(ev)o(esz,)15 b Fu(R)n(andom)h(walks)e(in)h(r)n(andom) g(and)h(non)g(r)n(andom)f(envir)n(onments)p FB(,)e(W)m(orld)g(Scien)o (ti\014c,)h(Singap)q(ore)g(\(199?\).)-66 621 y([124])22 b(M)15 b(A)g(Ruderman,)f(C)h(Kittel,)f(Indirect)i(exc)o(hange)g(coupling)e(of)g(n)o (uclear)i(magnetic)d(momen)o(ts)g(b)o(y)i(conduction)g(elec-)23 671 y(trons,)f Fu(Phys.)h(R)n(ev.)p FB(,)e Fb(96)p FB(,)g(99{102)g(\(1954\).) -66 746 y([125])22 b(D)c(Ruelle,)g(A)g(mathematical)d(reform)o(ulation)h(of)h (Derrida's)h(REM)g(and)g(GREM,)f Fu(Commun.)i(Math.)g(Phys.)p FB(,)f Fb(108)p FB(,)23 796 y(225{239)12 b(\(1987\).)-66 871 y([126])22 b(M)11 b(Shc)o(herbina,)h(B)g(Tirozzi,)f(The)g(free)h(energy)g(of) e(a)h(class)h(of)e(Hop\014eld)h(mo)q(dels,)f Fu(J.)i(Stat.)g(Phys.)p FB(,)g Fb(72)p FB(,)e(113{125)g(\(1993\).)-66 945 y([127])22 b(D)14 b(Sherrington,)g(S)g(Kirkpatric)o(k,)f(Solv)n(able)f(mo)q(del)h(of)g (a)h(spin)g(glass,)f Fu(Phys.)i(R)n(ev.)g(L)n(ett.)p FB(,)d Fb(35)p FB(,)i(1792{1796)d(\(1975\).)-66 1020 y([128])22 b(D)14 b(Sz\023)-21 b(asz,)14 b(comm)o(unication)d(\(1992\).)-66 1095 y([129])22 b(S)14 b(R)g(S)g(V)m(aradhan,)e(priv)n(ate)i(comm)o(unicatio)o(n)d (\(1992\).)-66 1169 y([130])22 b(P)13 b(V)m(uillermot,)c(Thermo)q(dynamics)i (of)g(quenc)o(hed)j(random)d(spin)h(systems)g(and)h(applications)e(to)h(the)h (proble)f(of)g(phase)23 1219 y(transitions)h(in)h(magnetic)e(spin)i(glass,)f Fu(J.)i(Phys.)g(A:)f(Math.)h(Gen.)p FB(,)f Fb(10)p FB(,)f(1319{1333)f (\(1977\).)-66 1294 y([131])22 b(F)12 b(V)m(ermet,)f Fu(Asymptotic)h(study)i (of)e(a)h(neur)n(al)g(network:)k(the)c(Hop\014eld)g(deterministic)f(se)n (quential)h(dynamics)p FB(,)f(preprin)o(t)23 1344 y(Univ)o(ersit)o(\023)-20 b(e)14 b(de)g(Rennes)h(I)f(\(1994\).)-66 1418 y([132])22 b(F)14 b(V)m(ermet,)247 1408 y Fu(\023)241 1418 y(Etude)h(asymptotique)g(d'un)g(r)o (\023)-20 b(ese)n(au)14 b(neur)n(onal)p FB(,)g(Th)o(\022)-20 b(ese)15 b(de)f(do)q(ctorat,)f(Univ)o(ersit)o(\023)-20 b(e)15 b(de)f(Rennnes)g(I)g(\(1994\).)-66 1493 y([133])22 b(E)c(C)f(W)m(a)o(ymire,)f (S)h(C)h(William)o(s,)d Fu(Corr)n(elate)n(d)i(spin)h(glasses)g(on)h(tr)n(e)n (es)p FB(,)e(IMA)g(w)o(orkshop)h(on)f(branc)o(hing)g(pro)q(cesses,)23 1543 y(June)d(1994,)f(to)g(app)q(ear)i(in)e(the)i(pro)q(ceedings.)-66 1618 y([134])22 b(P)14 b(Whittle,)f(Random)f(\014elds)i(on)g(random)e (graphs,)i Fu(A)n(dv.)h(Appl.)f(Pr)n(ob.)p FB(,)f Fb(24)p FB(,)g(455{473)f (\(1992\).)-66 1692 y([135])22 b(H)14 b(Wilf,)e Fu(Gener)n(atingfunctionolo)n (gy)p FB(,)i(Academic)g(Press,)h(Boston)f(\(1994\).)-66 1767 y([136])22 b(B)12 b(Zegarlinski,)f(In)o(teractions)g(and)g(pressure)j (functionals)c(for)h(disordered)h(lattice)f(systems,)h Fu(Commun.)g(Math.)h (Phys.)p FB(,)23 1817 y Fb(139)p FB(,)g(305{339)f(\(1991\).)-66 1892 y([137])22 b(B)17 b(Zegarlinski,)f(Spin)g(glass)g(with)g(long)f(range)i (in)o(teractions)g(at)f(high)g(temp)q(erature,)h Fu(J.)f(Stat.)h(Phys.)p FB(,)g Fb(17)p FB(,)f(911-930)23 1942 y(\(1987\).)913 2823 y Fy(29)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF