Content-Type: multipart/mixed; boundary="-------------0212151103977" This is a multi-part message in MIME format. ---------------0212151103977 Content-Type: text/plain; name="02-519.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-519.keywords" Disordered systems, Gibbs measures, non-Gibbsian measures, joint measures, random field model ---------------0212151103977 Content-Type: application/postscript; name="fundamental111102.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="fundamental111102.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com) %%Title: fundamental111102.dvi %%Pages: 13 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips -o fundamental111102.ps fundamental111102.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.11.15:1402 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin /N{def}def /B{bind 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Fm(\000)22 b Fr(\010)1396 5344 y Fg(0)1396 5405 y(2)1441 5380 y Fr(.)p eop %%Page: 9 9 9 8 bop 950 50 a Fp(REGULARITY)39 b(PR)n(OPER)-7 b(TIES)36 b(OF)h(POTENTIALS)908 b(9)120 289 y Fr(The)37 b(con)m(trol)h(of)f(the)g (\014rst)g(t)m(w)m(o)g(terms)h(is)f(w)m(ell-kno)m(wn.)58 b(F)-8 b(rom)36 b([EFS93])g(Prop)s(osition)i(2.44)e(w)m(e)0 408 y(ha)m(v)m(e)e(for)f(the)h(\014rst)f(term)h(that)700 530 y Fe(\015)700 590 y(\015)700 650 y(\015)700 710 y(\015)700 770 y(\015)772 610 y(X)755 824 y Fi(A)p Fs(\032)p Fg(\003)949 705 y Fl(')1014 720 y Fg(1)p Fs(\000)p Fg(2)p Fi(;A)1244 530 y Fe(\015)1244 590 y(\015)1244 650 y(\015)1244 710 y(\015)1244 770 y(\015)1299 834 y Fi(B)1362 814 y Ff(0)1402 834 y Fg(\()1442 817 y(\026)1433 834 y(\012\))p Fi(=C)5 b(onst)1826 705 y Fr(=)28 b(inf)1974 765 y Fi(c)2069 705 y Fr(sup)2123 784 y Fi(\021)2219 530 y Fe(\014)2219 590 y(\014)2219 650 y(\014)2219 710 y(\014)2219 770 y(\014)2269 610 y(X)2252 824 y Fi(A)p Fs(\032)q Fg(\003)2446 705 y Fl(')2511 720 y Fg(1)p 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(prop)s(erties)f(for)2661 5230 y(\026)2647 5257 y Fl(\014)h Fr(=)35 b(0.)56 b(Here)37 b(w)m(e)h(kno)m(w)f(that)0 5376 y(exp)s(onen)m(tial)d(deca)m(y)e(of)g(sup)1070 5400 y Fi(\021)1135 5376 y Fm(j)p Fl(\026)1223 5333 y Fg(+)1223 5406 y Fi(\014)t(;h)1347 5376 y Fr([)p Fl(\021)t Fr(]\()p Fl(\033)1552 5391 y Fi(x)1601 5376 y Fl(\033)1658 5391 y Fi(y)1706 5376 y Fr(\))19 b Fm(\000)h Fl(\026)1921 5333 y Fg(+)1921 5406 y Fi(\014)t(;h)2045 5376 y Fr([)p Fl(\021)t Fr(]\()p Fl(\033)2250 5391 y Fi(x)2300 5376 y Fr(\))p Fl(\026)2399 5333 y Fg(+)2399 5406 y Fi(\014)t(;h)2522 5376 y Fr([)p Fl(\021)t Fr(]\()p Fl(\033)2727 5391 y Fi(y)2774 5376 y Fr(\))p Fm(j)32 b Fr(in)h(the)f(distance)i Fm(j)p Fl(x)19 b Fm(\000)h Fl(y)t Fm(j)p eop %%Page: 11 11 11 10 bop 950 50 a Fp(REGULARITY)39 b(PR)n(OPER)-7 b(TIES)36 b(OF)h(POTENTIALS)866 b(11)0 289 y Fr(implies)39 b(that)c(there)h (really)h(is)f(a)g(join)m(t)g(p)s(oten)m(tial)h(for)f Fl(K)2185 253 y Fg(+)2287 289 y Fr(in)g Fm(B)2473 253 y Fg(1)2519 289 y 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b(the)e(conditional)i(exp)s(ectations)f(of)f Fl(K)3128 970 y Fg(+)3232 1006 y Fr(and)g Fl(K)3523 970 y Fs(\000)3629 1006 y Fr(w)m(ere)0 1126 y(sho)m(wn)g(to)e(b)s(e)i(di\013eren)m(t)h(in) f(this)f(case,)i(ev)m(en)f(though)f(there)g(is)h(a)f(join)m(t)h(p)s (oten)m(tial)g(for)g Fl(K)3474 1090 y Fg(+)3577 1126 y Fr(and)g(a)0 1245 y(di\013eren)m(t)30 b(one)e(for)g Fl(K)804 1209 y Fs(\000)898 1245 y Fr(that)g(are)f(translation-in)m(v) -6 b(arian)m(t)32 b(and)c(deca)m(y)g(lik)m(e)h(a)f(stretc)m(hed)h(exp)s (onen)m(tial)0 1365 y(on)34 b(a)g(full)h(measure)g(set.)46 b(This)35 b(is)g(sho)m(wn)g(in)g([KLR02],)e(based)i(on)f(the)g(general) h(represen)m(tation)h(of)0 1484 y(the)28 b(join)m(t)i(p)s(oten)m(tial)f (giv)m(en)g(in)g([K01])f(and)g(the)h(renormalization)h(group)e (analysis)i(of)e(the)h(quenc)m(hed)0 1604 y(sp)s(eci\014cation)36 b(of)d(the)h(random)f(\014eld)i(Ising)g(mo)s(del)f(of)f([BK88].)120 1724 y(Let)45 b(us)h(come)f(to)g(the)g(case)h(of)f(dep)s(enden)m(t)j 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b(of)f([K01])f(ho)m(w)m(ev)m(er)i(w)m(as)f(to)f(sho)m(w)i (almost)f(sure)h(summabilit)m(y)h(of)e(the)g(p)s(oten)m(tial)h Fl(U)11 b Fr(\()p Fl(\021)t Fr(\))39 b(b)m(y)0 3529 y(abstract)f (argumen)m(ts)h(for)f(general)h(mo)s(dels,)h(ev)m(en)f(when)g(it)f (fails)i(to)d(b)s(e)h(summable)i(ev)m(erywhere.)0 3648 y(The)e(generalization)h(of)e Fh(this)g Fr(part)f(of)i([K01])e(to)h (dep)s(enden)m(t)i(disorder)g(v)-6 b(ariables)38 b(is)g(less)h(ob)m (vious,)0 3768 y(so)33 b(w)m(e)h(lea)m(v)m(e)g(a)f(discussion)k(of)c (the)h(non-Gibbsian)j(regime)d(to)f(a)g(future)h(publication.)120 3888 y(Let)28 b(us)h(no)m(w)g(illustrate)i(the)e(application)i(of)e (the)g(Theorems)g(2)f(and)h(3)g(to)f(the)g(presen)m(t)j(situation.)0 4068 y Fu(Corollary)26 b(from)g(Theorem)f(2.)100 b Fh(Assume)26 b(that)f(ther)-5 b(e)25 b(ar)-5 b(e)25 b(joint)g(p)-5 b(otentials)25 b Fr(\011)3125 4074 y Fg(\026)3114 4092 y Fi(\014)3159 4102 y Ff(1)3198 4092 y Fg(;)p Fi(\014)3266 4102 y Ff(1)3305 4092 y Fi(;h)3376 4102 y Ff(1)3419 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Fm(\000)23 b Fr(\011)1466 5065 y Fg(\026)1455 5083 y Fi(\014)1500 5093 y Ff(2)1539 5083 y Fg(;)p Fi(\014)1607 5093 y Ff(2)1646 5083 y Fi(;h)1717 5093 y Ff(2)1760 5059 y Fm(k)1810 5082 y Fs(B)1865 5062 y Ff(0)1903 5082 y Fg(\(\012)p Fs(\002)2062 5065 y Fg(\026)2053 5082 y(\012\))p Fi(=)p Fg(\()p Fs(J)13 b Fg(+)p Fi(C)5 b(onst)14 b Fg(\))851 5224 y Fm(\024)29 b Fl(d)p Fm(j)1051 5198 y Fr(\026)1037 5224 y Fl(\014)1093 5239 y Fg(1)1159 5224 y Fm(\000)1273 5198 y Fr(\026)1259 5224 y Fl(\014)1315 5239 y Fg(2)1360 5224 y Fm(j)22 b Fr(+)g Fl(d)p Fm(j)p Fl(\014)1645 5239 y Fg(1)1712 5224 y Fm(\000)h Fl(\014)1868 5239 y Fg(2)1913 5224 y Fm(j)f Fr(+)g Fm(j)p Fl(h)2147 5239 y Fg(1)2214 5224 y Fm(\000)h Fl(h)2371 5239 y Fg(2)2416 5224 y Fm(j)f Fr(+)g Fm(j)p Fl(h)2650 5239 y Fg(1)2695 5224 y Fm(j)g Fr(+)g Fm(j)p Fl(h)2929 5239 y Fg(2)2974 5224 y Fm(j)3621 5142 y Fr(\(5.2\))p eop %%Page: 12 12 12 11 bop 0 50 a Fp(12)1610 b(C.K)1856 30 y(\177)1845 50 y(ULSKE)0 289 y Fh(R)-5 b(emark.)50 b Fr(While)28 b(the)e(\014rst)g(b)s(ound)h(sho)m(ws)g(the)f(con)m(tin)m(uit)m(y)i (w.r.t.)40 b(the)26 b(parameters)g(of)g(the)g(de\014ning)0 408 y(p)s(oten)m(tial)48 b(and)f(p)s(oten)m(tial)h(of)e(a)g(priori)i (distribution,)53 b(the)46 b(second)i(b)s(ound)g(is)f(b)s(etter)g(for)f (small)0 528 y(magnetic)k(\014elds.)92 b(In)49 b(particular)i(for)e Fl(h)1623 543 y Fg(1)1721 528 y Fr(=)54 b Fl(h)1909 543 y Fg(2)2008 528 y Fr(=)g(0)48 b(the)h(disorder)i(v)-6 b(ariables)51 b(and)e(the)g(spin)0 648 y(v)-6 b(ariables)35 b(decouple,)g(the)f(join)m(t)g(p)s(oten)m(tial)h(is)f(just)g(the)g(sum) g Fl(')22 b Fr(+)h(\010,)33 b(and)g(w)m(e)h(ha)m(v)m(e)g(that)258 851 y Fm(k)p Fr(\011)396 857 y Fg(\026)385 875 y Fi(\014)430 885 y Ff(1)470 875 y Fg(;)p Fi(\014)538 885 y Ff(1)576 875 y Fi(;h)647 885 y Ff(1)685 875 y Fg(=0)813 851 y Fm(\000)23 b Fr(\011)1001 857 y Fg(\026)990 875 y Fi(\014)1035 885 y Ff(2)1074 875 y Fg(;)p Fi(\014)1142 885 y Ff(2)1181 875 y Fi(;h)1252 885 y Ff(2)1290 875 y Fg(=0)1396 851 y Fm(k)1446 874 y Fs(B)1501 854 y Ff(0)1539 874 y Fg(\(\012)p Fs(\002)1698 856 y Fg(\026)1689 874 y(\012\))p Fi(=)p Fg(\()p Fs(J)13 b Fg(+)p Fi(C)5 b(onst)14 b Fg(\))2274 851 y Fr(=)28 b Fl(d)p Fm(j)2473 824 y Fr(\026)2459 851 y Fl(\014)2515 866 y Fg(1)2582 851 y Fm(\000)2695 824 y Fr(\026)2681 851 y Fl(\014)2737 866 y Fg(2)2782 851 y Fm(j)22 b Fr(+)h Fl(d)p Fm(j)p Fl(\014)3068 866 y Fg(1)3134 851 y Fm(\000)g Fl(\014)3290 866 y Fg(2)3335 851 y Fm(j)3621 854 y Fr(\(5.3\))0 1063 y(\(see)34 b(b)s(elo)m(w\))g(and)g(so)f(the)g (b)s(ound)i(is)f(sharp.)0 1235 y Fh(Pr)-5 b(o)g(of)50 b(of)g(Cor)-5 b(ol)5 b(laries.)51 b Fr(T)-8 b(o)49 b(compute)h(the)g (norm)g(of)g(the)h(p)s(oten)m(tials)g(o)s(ccurring,)k(de\014ne)c(some)0 1354 y(corresp)s(onding)35 b(energy)f(densit)m(y)h(function)g(and)f (compute)g(its)g(sup-norm)g(mo)s(dulo)g(constan)m(ts.)120 1474 y(F)-8 b(or)21 b(the)i(de\014ning)h(p)s(oten)m(tial)g(of)e(the)h (quenc)m(hed)h(sp)s(eci\014cation)h(\010)2541 1489 y Fi(\014)t(;h)2665 1474 y Fr(\()p Fl(\033)t(\021)t Fr(\))c(w)m(e)h(put)h (e.g.)40 b Fl(f)3435 1438 y Fs(0)3424 1502 y Fi(\014)t(;h)3548 1474 y Fr(\()p Fl(\033)t(\021)t Fr(\))26 b(:=)0 1624 y Fm(\000)p Fl(\033)134 1639 y Fg(0)179 1624 y Fr([)p Fl(\014)285 1549 y Fe(P)390 1574 y Fi(d)390 1654 y(i)p Fg(=1)541 1624 y Fl(\033)598 1639 y Fi(e)636 1649 y Fj(i)678 1624 y Fm(\000)6 b Fl(h\021)867 1639 y Fg(0)914 1624 y Fr(])25 b(where)h(the)f(sum)h(is)g(o)m(v)m(er)g(all)g(unit)m(y)g(v)m (ectors)g Fl(e)2670 1639 y Fi(i)2729 1624 y Fr(spanning)h Fa(Z)3211 1588 y Fi(d)3280 1624 y Fr(and)f(p)s(oin)m(ting)0 1744 y(in)34 b(p)s(ositiv)m(e)h(directions.)46 b(Then)34 b(w)m(e)g(ha)m(v)m(e)g(that)452 1974 y Fm(k)p Fr(\010)574 1989 y Fi(\014)t(;h)698 1974 y Fm(k)748 1997 y Fs(B)803 1977 y Ff(0)841 1997 y Fg(\(\012)p Fs(\002)1000 1980 y Fg(\026)991 1997 y(\012\))p Fi(=)p Fg(\()p Fs(B)1206 1977 y Ff(1)1244 1997 y Fg(\()1284 1980 y(\026)1275 1997 y(\012\)+)p Fs(J)13 b Fg(+)p Fi(C)5 b(onst)14 b Fg(\))1849 1974 y Fr(=)28 b(inf)1997 2034 y Fi(c)2092 1974 y Fr(sup)2121 2053 y Fi(\033)s(\021)2242 1859 y Fe(\014)2242 1919 y(\014)2242 1979 y(\014)2275 1974 y Fl(f)2335 1933 y Fs(0)2324 1999 y Fi(\014)t(;h)2448 1974 y Fr(\()p Fl(\033)t(\021)t Fr(\))21 b Fm(\000)i Fl(c)2804 1859 y Fe(\014)2804 1919 y(\014)2804 1979 y(\014)2865 1974 y Fr(=)28 b Fl(d)p Fm(j)p Fl(\014)5 b Fm(j)22 b Fr(+)h Fm(j)p Fl(h)p Fm(j)3621 1993 y Fr(\(5.4\))0 2248 y(F)-8 b(or)22 b(the)g(\014rst)h(equalit)m(y)-8 b(,)24 b(see)f([EFS93])f(Prop)s(osition)h(A.11.)39 b(Similarly)25 b(w)m(e)d(ha)m(v)m(e)h Fm(k)p Fl(')3116 2254 y Fg(\026)3105 2272 y Fi(\014)3159 2248 y Fm(k)3209 2271 y Fs(B)3264 2251 y Ff(0)3302 2271 y Fg(\()3342 2254 y(\026)3333 2271 y(\012\))p Fi(=)p Fg(\()p Fs(J)12 b Fg(+)p Fi(C)5 b(onst)14 b Fg(\))3917 2248 y Fr(=)0 2382 y Fl(d)p Fm(j)94 2355 y Fr(\026)80 2382 y Fl(\014)5 b Fm(j)p Fr(.)44 b(Finally)35 b(w)m(e)e(ha)m(v)m(e)766 2618 y Fm(k)p Fr(\010)888 2633 y Fi(\014)t(;h)1011 2618 y Fm(k)1061 2641 y Fs(B)1116 2621 y Ff(0)1154 2641 y Fg(\(\012)p Fs(\002)1313 2623 y Fg(\026)1304 2641 y(\012\))p Fi(=)p Fg(\()p Fs(B)1519 2621 y Ff(1)1558 2641 y Fg(\(\012\)+)p Fs(J)12 b Fg(+)p Fi(C)5 b(onst)15 b Fg(\))2162 2618 y Fr(=)28 b(sup)2296 2697 y Fi(\033)s(\021)2417 2503 y Fe(\014)2417 2563 y(\014)2417 2623 y(\014)2451 2618 y Fm(\000)p Fl(h\033)2642 2633 y Fg(0)2687 2618 y Fl(\021)2736 2633 y Fg(0)2781 2503 y Fe(\014)2781 2563 y(\014)2781 2623 y(\014)2842 2618 y Fr(=)g Fm(j)p Fl(h)p Fm(j)3621 2636 y Fr(\(5.5\))0 2886 y(b)m(y)33 b(subtraction)i(of)e(the)h(purely)h Fl(\033)t Fr(-dep)s(enden)m(t)g(part.)p Fm(})0 3058 y Fu(Ac)m(kno)m(wledgmen)m (ts:)64 b Fr(The)40 b(author)g(thanks)g(A.v)-6 b(an)39 b(En)m(ter,)j(C.Maes,)f(A.Le)f(Ny)f(and)h(F.Redig)0 3177 y(for)33 b(v)-6 b(aluable)36 b(discussions)h(ab)s(out)c(generalized)j (Gibbs)f(measures.)0 3612 y Fu(REFERENCES:)50 3826 y Fr([BK88])49 b(J.Bricmon)m(t,)34 b(A.Kupiainen,)i Fh(\\Phase)g(tr)-5 b(ansition)35 b(in)h(the)g Fr(3)p Fl(d)f Fh(r)-5 b(andom)37 b(\014eld)f(Ising)g(mo)-5 b(d-)403 3945 y(el")p Fr(,)33 b(Comm)m(un.)45 b(Math.)f(Ph)m(ys.)h Fu(142)34 b Fr(539-572)e(\(1988\)) -12 4065 y([BKL98])49 b(J.Bricmon)m(t,)30 b(A.Kupiainen,)j(R.Lefev)m (ere,)e Fh(\\)i(R)-5 b(enormalization)31 b(gr)-5 b(oup)32 b(p)-5 b(atholo)g(gies)31 b(and)403 4184 y(the)36 b(de\014nition)e(of)h (Gibbs)g(states")p Fr(,)d(Comm)m(un.)45 b(Math.)f(Ph)m(ys.)g Fu(194)35 b Fr(359-388)d(\(1998\))10 4304 y([EFS93])49 b(A.C.D.)43 b(v)-6 b(an)45 b(En)m(ter,)j(R.)d(F)-8 b(ern\023)-50 b(andez)46 b(and)f(A.D.)f(Sok)-6 b(al,)48 b Fh(\\R)-5 b(e)g(gularity)46 b(pr)-5 b(op)g(erties)45 b(and)403 4423 y(p)-5 b(atholo)g(gies)32 b(of)g(p)-5 b(osition-sp)g(ac)g(e)30 b(r)-5 b(enormalization-gr)g(oup)32 b(tr)-5 b(ansformations:)43 b(Sc)-5 b(op)g(e)32 b(and)403 4543 y(limitations)j(of)g(Gibbsian)f(the) -5 b(ory")p Fr(,)33 b(J.)h(Stat.)43 b(Ph)m(ys.)h Fu(72)34 b Fr(879-1167)e(\(1993\))-83 4663 y([EFHR02])49 b(A.C.D.)26 b(v)-6 b(an)27 b(En)m(ter,)i(R.)e(F)-8 b(ern\023)-50 b(andez,)30 b(F.)d(den)h(Hollander,)h(F.)e(Redig,)j Fh(\\Possible)f (loss)h(and)403 4782 y(r)-5 b(e)g(c)g(overy)46 b(of)g(Gibbsianness)e (during)j(the)f(sto)-5 b(chastic)46 b(evolution)g(of)g(Gibbs)g(me)-5 b(asur)g(es")p Fr(,)403 4902 y(Comm.)44 b(Math.)g(Ph)m(ys.)g Fu(226)34 b Fr(101-130)f(\(2002\))-72 5021 y([EMSS00])50 b(A.C.D.)32 b(v)-6 b(an)34 b(En)m(ter,)f(C.)g(Maes,)h(R.H.)f(Sc)m (honmann)i(and)f(S.)g(Shlosman,)g Fh(\\The)i(Gri\016ths)403 5141 y(singularity)d(r)-5 b(andom)34 b(\014eld")p Fr(,)d(On)g (Dobrushin's)i(w)m(a)m(y:)42 b(from)31 b(probabilit)m(y)i(theory)d(to)h (sta-)403 5260 y(tistical)38 b(ph)m(ysics,)g(51{58,)d(Amer.)53 b(Math.)f(So)s(c.)g(T)-8 b(ransl.)53 b(Serie)37 b(2,)g Fu(198)p Fr(,)g(Amer.)52 b(Math.)403 5380 y(So)s(c.)44 b(Pro)m(vidence,)35 b(P)-8 b(.I.)32 b(\(2000\))p eop %%Page: 13 13 13 12 bop 950 50 a Fp(REGULARITY)39 b(PR)n(OPER)-7 b(TIES)36 b(OF)h(POTENTIALS)866 b(13)-38 289 y Fr([EKM00])49 b(A.C.D.)22 b(v)-6 b(an)23 b(En)m(ter,)i(C.)d(Maes)i(and)f(C.)f(K)s(\177)-53 b(ulsk)m(e,)27 b Fh(Comment)g(on)c Fr([KM94],)h(Ph)m(ys.Rev.Lett.)403 408 y Fu(84)p Fr(,)33 b(6134)g(\(2000\))26 528 y([Geo88])49 b(H.O.Georgii,)25 b Fh(\\Gibbs)g(me)-5 b(asur)g(es)26 b(and)f(Phase)g(tr)-5 b(ansitions")p Fr(,)23 b(de)g(Gruyter,)h(Berlin)g (\(1988\))65 648 y([Ku96])50 b(R.)41 b(K)s(\177)-53 b(uhn,)43 b Fh(\\Equilibrium)g(Ensemble)f(Appr)-5 b(o)g(ach)42 b(to)g(Disor)-5 b(der)g(e)g(d)41 b(Systems)i(I:)g(Gener)-5 b(al)403 767 y(The)g(ory,)35 b(Exact)h(R)-5 b(esults")p Fr(,)33 b(Z.)g(Ph)m(ys.)44 b Fu(100)p Fr(,)33 b(231-242)g(\(1996\))121 887 y([K99])49 b(C.K)s(\177)-53 b(ulsk)m(e,)34 b Fh(\(Non-\))f (Gibbsianness)g(and)h(Phase)g(T)-8 b(r)j(ansitions)33 b(in)i(R)-5 b(andom)34 b(L)-5 b(attic)g(e)34 b(Spin)403 1006 y(Mo)-5 b(dels)p Fr(,)33 b(Mark)m(o)m(v)g(Pro)s(c.)43 b(Rel.)j(Fields)35 b Fu(5)f Fr(357-383)e(\(1999\))121 1126 y([K01])49 b(C.K)s(\177)-53 b(ulsk)m(e,)32 b Fh(We)-5 b(akly)33 b(Gibbsian)f(R)-5 b(epr)g(esentations)31 b(for)i(joint)f(me) -5 b(asur)g(es)33 b(of)g(quenche)-5 b(d)33 b(lat-)403 1245 y(tic)-5 b(e)35 b(spin)g(mo)-5 b(dels)p Fr(,)33 b(Probab.)44 b(Theor.)g(Rel.)i(Fields)35 b Fu(119)p Fr(,)f(1-30)f (\(2001\).)-15 1365 y([KLR02])50 b(C.K)s(\177)-53 b(ulsk)m(e,)35 b(A.Le)f(Ny)-8 b(,)32 b(F.Redig,)j Fh(\\V)-8 b(ariational)36 b(principle)f(for)h(gener)-5 b(alize)g(d)35 b(Gibbs)g(me)-5 b(a-)403 1484 y(sur)g(es")p Fr(,)33 b(Eurandom)h(preprin)m(t)h (2002-035)d(\(2002\))29 1604 y([KM94])50 b(R.K)s(\177)-53 b(uhn)33 b(and)f(G.Mazzeo,)f Fh(Critic)-5 b(al)34 b(b)-5 b(ehavior)32 b(of)i(the)f(r)-5 b(andomly)35 b(spin)e(dilute)-5 b(d)34 b(2D)g(Ising)403 1724 y(mo)-5 b(del:)46 b(A)36 b(gr)-5 b(and)35 b(ensemble)g(appr)-5 b(o)g(ach")p Fr(,)33 b(Ph)m(ys.Rev.Lett.)44 b Fu(73)p Fr(,)33 b(2268-2271)f(\(1994\))29 1843 y([KM00])50 b(R.K)s(\177)-53 b(uhn)35 b(and)e(G.Mazzeo,)h Fh(R)-5 b(eply)36 b(to)f([EKM00])p Fr(,)d(Ph)m(ys.Rev.Lett.)44 b Fu(84)p Fr(,)33 b(6135)g(\(2000\))-196 1963 y([MMR)-8 b(TV00])49 b(C.Maes,)41 b(A.v)-6 b(an)38 b(Mo\013aert,)j(F.Redig,)g (F.T)-8 b(ak)m(ens)39 b(and)h(E.)f(V)-8 b(erbitsky)g(,)40 b Fh(\\Intermittency)403 2082 y(and)c(we)-5 b(ak)35 b(Gibbs)f(states")p Fr(,)f(Nonlinearit)m(y)i Fu(13)f Fr(no.5,)e(1681-1698)g(\(2000\))-97 2202 y([MRSV00])50 b(C.)39 b(Maes,)i(F.)e(Redig,)j(S.)e(Shlosman)h(and) f(A.)e(v)-6 b(an)40 b(Mo\013aert,)g Fh(Per)-5 b(c)g(olation,)42 b(p)-5 b(ath)41 b(lar)-5 b(ge)403 2321 y(deviations)34 b(and)i(we)-5 b(ak)35 b(Gibbsianity")p Fr(,)c(Comm.)44 b(Math.)g(Ph)m(ys.)h Fu(209)34 b Fr(517-45)e(\(2000\))57 2441 y([Mo64])49 b(T.)39 b(Morita,)i Fh(\\Statistic)-5 b(al)40 b(Me)-5 b(chanics)40 b(of)h(quenche)-5 b(d)41 b(solid)g(solutions)g(with)g(applic)-5 b(ation)403 2560 y(to)36 b(magnetic)-5 b(al)5 b(ly)36 b(dilute)g(al)5 b(loys")p Fr(,)34 b(J.)f(Math.)44 b(Ph)m(ys.)h Fu(5)p Fr(,)33 b(1402-1405)f(\(1964\))p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0212151103977--