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Fm(3)50 513 y Fp(W)-8 b(e)40 b(call)d(this)h(latter)h(class)f(the)h(SK-mo)s (dels,)g(since)f(it)g(comprises)g(in)f(particular)g(the)i(standard)e (SK-)50 658 y(mo)s(del)d(and)g(the)h Fi(p)p Fp(-spin)e(SK)h(mo)s(dels)g (\(for)h(recen)m(t)h(progress)e(on)h(these)h(mo)s(dels,)f(see)g ([T1,T2,GT,Gu].)50 804 y(In)m(terestingly)-8 b(,)33 b(w)m(e)g(sill)d (see)j(that)g(in)e(our)i(class)f(of)h(mo)s(dels,)f(up)s(on)f(suitable)g (c)m(hoices)i(of)g(the)f(co)m(v)-5 b(ariance)50 949 y(function)33 b Fi(A)i Fo(any)g Fp(structure)f(that)h(emerges)h(as)e(the)h(solution)e (of)i(a)g(mean)g(\014eld)e(spin)f(glass)j(in)e(P)m(arisi's)50 1095 y(theory)g(can)f(b)s(e)g(obtained.)47 b(In)31 b(particular,)h(for) g(some)h(c)m(hoice)h(of)e Fi(A)h Fp(the)g(ensuing)d(limiting)g(ob)5 b(ject)33 b(will)50 1240 y(ha)m(v)m(e)j(the)e(same)h(prop)s(erties)e (as)i(the)f(standard)g(SK)g(mo)s(del.)52 b(Ho)m(w)m(ev)m(er,)38 b(in)33 b(con)m(trast)j(to)f(the)g(standard)50 1385 y(class)25 b(of)g(spin)e(glasses)j(de\014ned)e(via)h(the)g(Hamming)g(distance,)h (where)f(w)m(e)g(ha)m(v)m(e)i(no)e(clue)g(on)g(ho)m(w)g(P)m(arisi's)50 1531 y(solution)33 b(arises)g(ev)m(en)i(on)f(a)h(v)m(ery)f(qualitativ)m (e)g(lev)m(el)g(\(with)f(the)i(exception)f(of)g(a)h(rare)f(glimpse)f (at)i(the)50 1676 y(emergence)f(of)e(non-trivial)e(structure)i(in)g (the)g Fi(p)p Fp(-spin)f(mo)s(dels)g([T1,T2])i(or)g(the)f(Hop\014eld)g (mo)s(dels)f(\(see)50 1822 y(e.g.)49 b([Bo]\)\),)36 b(in)31 b(our)i(mo)s(dels)e(all)h(this)g(can)h(b)s(e)f(p)s(erfectly)g(understo) s(o)s(d)f(b)s(oth)h(on)h(a)g(mathematical)g(and)50 1967 y(in)m(tuitiv)m(e)22 b(lev)m(el.)39 b(F)-8 b(rom)24 b(that)g(p)s(ersp)s (ectiv)m(e,)h(w)m(e)f(feel)g(that)g(our)f(results)g(constitute)h(a)g (ma)5 b(jor)24 b(con)m(tribution)50 2113 y(to)m(w)m(ards)31 b(a)g(partial)f(de-m)m(ysti\014cation)h(of)g(the)f(P)m(arisi)g (solution.)40 b(Moreo)m(v)m(er,)34 b(as)d(had)f(b)s(een)g(p)s(oin)m (ted)f(out)50 2258 y(already)g(b)m(y)h(Derrida)g(and)f(Gardner)g ([DG1],)j(one)f(ma)m(y)f(argue)h(that)f(the)g(particular)f(c)m(hoice)i (of)f(the)g(dis-)50 2404 y(tance)c(concept)h(used)d(ma)m(y)i(not)g(ha)m (v)m(e)h(dramatic)e(consequences)h(on)f(the)h(prop)s(erties)d(of)j(the) g(system,)g(and)50 2549 y(that)e(the)g(real)g(k)m(ey)g(parameter)h(of)f (the)g(mo)s(del)e(migh)m(t)i(just)f(b)s(e)g(the)i(en)m(trop)m(y)f(as)g (a)g(function)f(of)h(the)g(co)m(v)-5 b(ari-)50 2695 y(ance.)40 b(With)26 b(suc)m(h)h(a)g(univ)m(ersalit)m(y)e(h)m(yp)s(othesis,)h(our) g(results)g(suggest)h(that)h(indeed)d(the)i(P)m(arisi)e(solution)50 2840 y(is)k(v)m(ery)i(natural)e(and)h(that)h(indeed)e(the)i(structures) f(w)m(e)g(exhibit)f(here)h(ough)m(t)h(to)g(b)s(e)f(canonical)g(for)g (an)m(y)50 2985 y(\(Gaussian\))j(mean)g(\014eld)f(spin)f(glass.)50 b(The)32 b(natural)h(relation)f(b)s(et)m(w)m(een)i(mo)s(dels)e(of)h (the)h(SK)e(class)h(and)50 3131 y(the)c(Derrida)f(class)h(w)m(ould)f(b) s(e)g(that)i(if)e(a)h(mo)s(del)f(in)g(the)h(SK)f(class)h(has)g(a)g(co)m (v)-5 b(ariance)30 b(function)e Fi(B)5 b Fp(,)29 b(then)50 3276 y(the)24 b(analogue)g(mo)s(del)f(in)g(the)h(Derrida)g(class)f(m)m (ust)h(ha)m(v)m(e)i(co)m(v)-5 b(ariance)25 b(function)e Fi(A)p Fp(\()p Fi(x)p Fp(\))j(=)f Fi(B)5 b Fp(\()p Fi(I)3293 3243 y Ff(\000)p Fs(1)3390 3276 y Fp(\(ln)14 b(2)p Fi(x)p Fp(\)\),)50 3422 y(where)36 b Fi(I)44 b Fp(is)36 b(Cram)m(\023)-43 b(er's)37 b(en)m(trop)m(y)g(function.)59 b(In)36 b(particular,)h(the)g (classical)f Fi(p)p Fp(-spin)f(SK)h(mo)s(dels)f(corre-)50 3567 y(sp)s(ond)h(to)k(Derrida)d(mo)s(dels)g(with)g Fi(A)p Fp(\()p Fi(x)p Fp(\))j(=)e(\()p Fi(I)1733 3534 y Ff(\000)p Fs(1)1831 3567 y Fp(\(ln)14 b(2)p Fi(x)p Fp(\)\))2124 3534 y Fh(p)2167 3567 y Fp(.)65 b(Note)40 b(for)e(instance)g(that)h (this)e(function)50 3713 y(is)d(con)m(v)m(ex)k(if)c Fi(p)g Fp(=)g(2,)j(while)d(for)h Fi(p)f Fg(\025)g Fp(3,)j(it)e(is)g(conca)m(v) 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b(that)f(do)s(es)f(not)h(hold)f (in)f(the)i(corresp)s(onding)e(appro)m(ximation.)39 b(Th)m(us,)29 b(a)i(fundamen)m(tal)d(di\013erence)50 4876 y(b)s(et)m(w)m(een)36 b(the)f(t)m(w)m(o)i(mo)s(del)d(classes)h(remains)f(and)g(a)i(profound)d (understanding)g(of)i(the)h(SK)e(mo)s(dels)g(is)50 5022 y(still)28 b(amiss.)50 5240 y Fq(Ac)m(kno)m(wledgemen)m(ts:)57 b Fp(W)-8 b(e)36 b(are)f(indebted)f(to)h(Erwin)e(Bolthausen)h(for)h (helpful)d(discussions.)52 b(A.B.)50 5385 y(thanks)24 b(F)-8 b(rancis)24 b(Comets)h(and)f(the)h(Univ)m(ersit)m(\023)-43 b(e)25 b(P)m(aris)e(7)i(for)g(kind)d(hospitalit)m(y)i(during)e(the)j(p) s(erio)s(d)d(when)p eop %%Page: 4 4 4 3 bop 50 295 a Fm(4)3309 b Fj(Se)l(ction)27 b Fm(2)50 513 y Fp(this)i(w)m(ork)i(w)m(as)f(completed.)50 712 y Ft(2.)55 b(Limiting)40 b(form)m(ula)g(and)h(heuristic)g (considerations.)159 930 y Fp(As)22 b(had)g(already)f(b)s(een)h(noted)g (b)m(y)g(Derrida)g(and)f(Gardner)h([DG1,DG2],)27 b(thermo)s(dynamic)20 b(quan)m(tities)50 1076 y(in)30 b(the)i(GREMs)g(p)s(ossess)f(limits)e (as)j(the)g(n)m(um)m(b)s(er)e(of)i(lev)m(els)f(tends)g(to)i(in\014nit)m (y)-8 b(.)42 b(In)31 b(fact,)i(the)f(form)m(ulae)50 1221 y(for)h(the)g(free)g(energy)h(and)f(the)g(o)m(v)m(erlap)h(distribution) 29 b(can)34 b(b)s(e)e(written)g(in)g(a)i(closed)f(form)g(in)e(terms)j (of)50 1367 y(the)c(function)f Fi(A)i Fp(that)g(mak)m(es)g(this)e(fact) j(manifest.)159 1585 y(Let)j(us)g(denote)g(the)g(con)m(v)m(ex)i(h)m (ull)32 b(of)j(the)h(function)d Fi(A)p Fp(\()p Fi(x)p Fp(\))j(b)m(y)2403 1562 y(\026)2379 1585 y Fi(A)p Fp(\()p Fi(x)p Fp(\).)55 b(W)-8 b(e)36 b(will)c(also)j(need)g(the)g(left-)50 1730 y(deriv)-5 b(ativ)m(e)32 b(of)h(this)f(function,)i(\026)-46 b Fi(a)p Fp(\()p Fi(x)p Fp(\))30 b Fg(\021)g Fp(lim)1567 1744 y Fh(\017)p Ff(#)p Fs(0)1690 1730 y Fi(\017)1727 1697 y Ff(\000)p Fs(1)1825 1730 y Fp(\()1884 1707 y(\026)1860 1730 y Fi(A)p Fp(\()p Fi(x)p Fp(\))23 b Fg(\000)2190 1707 y Fp(\026)2166 1730 y Fi(A)p Fp(\()p Fi(x)f Fg(\000)g Fi(\017)p Fp(\)\))33 b(whic)m(h)f(exists)h(for)f(all)g(v)-5 b(alues)33 b(of)50 1876 y Fi(x)25 b Fg(2)g Fp(\(0)p Fi(;)15 b Fp(1].)159 2094 y(It)33 b(is)g(a)h(v)m(ery)f(simple)f(matter)i(to)g (realize)f(that)h(the)g(form)m(ula)f(of)g(Cap)s(o)s(caccia)h(et)g(al.) 50 b([CaCaPi])33 b(\(see)50 2239 y(Theorem)d(1.2)h(in)e([BK2]\))j(can)f (b)s(e)f(written)f(as)899 2527 y Fi(F)957 2541 y Fh(\014)1032 2527 y Fp(=)1128 2445 y Fg(p)p 1203 2445 197 4 v 1203 2527 a Fp(2)15 b(ln)g(2)p Fi(\014)1472 2403 y Fc(Z)1563 2426 y Fh(x)p Fs(\()p Fh(\014)s Fs(\))1522 2610 y(0)1725 2445 y Fc(p)p 1816 2445 171 4 v 1817 2527 a Fp(\026)-46 b Fi(a)p Fp(\()p Fi(x)p Fp(\))p Fi(dx)21 b Fp(+)2208 2466 y Fi(\014)2264 2433 y Fs(2)p 2208 2506 98 4 v 2234 2589 a Fp(2)2316 2527 y(\(1)g Fg(\000)2532 2504 y Fp(\026)2508 2527 y Fi(A)p Fp(\()p Fi(x)p Fp(\()p Fi(\014)5 b Fp(\)\)\))664 b(\(2)p Fi(:)p Fp(1\))50 2789 y(where)1293 2934 y Fi(x)p Fp(\()p Fi(\014)5 b Fp(\))26 b Fg(\021)f Fp(sup)1745 2806 y Fc(\022)1812 2934 y Fi(x)p Fg(j)q Fp(\026)-46 b Fi(a)p Fp(\()p Fi(x)p Fp(\))26 b Fi(>)2192 2873 y Fp(2)15 b(ln)f(2)p 2192 2913 197 4 v 2242 2997 a Fi(\014)2298 2970 y Fs(2)2400 2806 y Fc(\023)3523 2934 y Fp(\(2)p Fi(:)p Fp(2\))50 3156 y(The)30 b(mean)g(distance)g(distribution)d (function)1213 3408 y Fi(f)1258 3422 y Fh(\014)1306 3408 y Fp(\()p Fi(x)p Fp(\))f Fg(\021)49 b Fp(lim)1550 3467 y Fh(N)7 b Ff("1)1740 3408 y Fi(\026)1795 3369 y Ff(\012)p Fs(2)1795 3435 y Fh(\014)s(;N)1945 3408 y Fp(\()p Fi(d)2027 3422 y Fh(N)2096 3408 y Fp(\()p Fi(\033)n(;)15 b(\033)2276 3370 y Ff(0)2303 3408 y Fp(\))25 b Fi(<)g(x)p Fp(\))977 b(\(2)p Fi(:)p Fp(3\))50 3673 y(as)30 b(giv)m(en)h(in)e(Derrida)g(and)h (Gardner)g([DG2])i(or)e(Prop)s(osition)f(1.7)i(of)38 b([BK2])31 b(can)g(b)s(e)f(re-written)f(as)1022 3952 y Fe(E)11 b Fi(f)1122 3966 y Fh(\014)1177 3952 y Fp(\()p Fi(x)p Fp(\))26 b(=)1420 3823 y Fc(\032)1519 3888 y Fi(\014)1575 3854 y Ff(\000)p Fs(1)1673 3810 y Fg(p)p 1749 3810 V 77 x Fp(2)15 b(ln)f(2)1946 3888 y Fi(=)1991 3810 y Fc(p)p 2082 3810 171 4 v 2083 3887 a Fp(\026)-46 b Fi(a)p Fp(\()p Fi(x)p Fp(\))2253 3888 y Fi(;)91 b Fp(if)14 b Fi(x)25 b Fg(\024)g Fi(x)2662 3902 y Fh(\014)1519 4024 y Fp(1)p Fi(;)780 b Fp(if)14 b Fi(x)25 b(>)g(x)2662 4038 y Fh(\014)3523 3952 y Fp(\(2)p Fi(:)p Fp(4\))50 4225 y(These)h(app)s(ealing)e(form)m (ulas)h(of)i(course)f(suggest)h(that)g(they)f(migh)m(t)g(b)s(e)f(true)h (also)g(when)f Fi(A)i Fp(is)e(not)h(a)h(step)50 4371 y(function)i(with)g(\014nitely)g(man)m(y)h(steps.)41 b(W)-8 b(e)31 b(will)d(sho)m(w)i(that)h(this)e(is)h(indeed)f(the)h (case.)159 4589 y(Note)h(that)f(one)f(cannot)h(write)f(suc)m(h)g (closed)g(expressions)f(for)h(the)h(sub-leading)d Fo(c)-5 b(orr)g(e)g(ctions)63 b Fp(to)31 b(the)50 4735 y(free)e(energy)-8 b(.)41 b(In)29 b(fact,)h(it)f(is)f(to)i(b)s(e)f(exp)s(ected)g(that)h (these)g(will)c(in)i(general)h(not)h(only)e(dep)s(end)f(on)3484 4712 y(\026)3460 4735 y Fi(A)j Fp(\(see)50 4880 y(e.g.)57 b(the)36 b(analysis)e(of)i(the)g(maxim)m(um)f(of)g(branc)m(hing)g(Bro)m (wnian)f(motion)i(b)m(y)f(Bramson)h([BR,BDG]\).)50 5025 y(In)d(particular,)g(w)m(e)h(cannot)g(construct)g(immediately)e (sensible)g(limits)f(of)j(the)f(P)m(oisson)h(cascades)h Fg(P)3608 4992 y Fs(\()p Fh(k)r Fs(\))50 5171 y Fp(\(see)f([BK1]\).)50 b(This)32 b(migh)m(t)h(discourage)g(us)f(from)h(inquiring)c(in)m(to)34 b(the)f(\014ner)f(prop)s(erties)g(of)h(the)g(Gibbs)50 5316 y(measures)e(that)g(w)m(ere)h(analysed)e(in)g([BK2].)44 b(Ho)m(w)m(ev)m(er,)34 b(the)d(situation)f(is)g(actually)h(b)s(etter)g (than)g(migh)m(t)50 5462 y(b)s(e)f(feared.)p eop %%Page: 5 5 5 4 bop 50 295 a Fj(GREM,)28 b(p)l(art)f(3)3127 b Fm(5)159 517 y Fp(Let)32 b(us)e(lo)s(ok)h(at)h(the)g(in)m(terpretation)f(of)g (the)h(pro)s(cesses)f Fg(W)2241 470 y Fs(\()p Fh(m)p Fs(\))2234 544 y Fh(\014)2398 517 y Fp(that)h(w)m(e)g(considered)e(in)g ([BK2].)45 b(W)-8 b(e)50 662 y(ha)m(v)m(e)31 b(though)m(t)g(of)g(them)f (as)h(p)s(oin)m(t)e(pro)s(cesses)h(on)h(\(0)p Fi(;)15 b Fp(1])1995 629 y Fh(m)2066 662 y Fp(.)40 b(Ho)m(w)m(ev)m(er,)33 b(the)d(v)m(ectors)50 808 y(\()p Fi(\026)140 822 y Fh(\014)s(;N)275 808 y Fp(\()p Fi(B)379 822 y Fs(1)420 808 y Fp(\()p Fi(\033)s Fp(\)\))p Fi(;)15 b(:)g(:)g(:)j(;)d(\026)838 822 y Fh(\014)s(;N)973 808 y Fp(\()p Fi(B)1077 822 y Fh(m)1147 808 y Fp(\()p Fi(\033)s Fp(\)\)\))32 b(are)f(in)f(fact)i(more)f(reasonably)f(though)m (t)h(of)g(as)g(the)g(v)-5 b(alues)31 b(of)g(the)50 953 y(mass)f(distribution)1290 1104 y Fi(m)1370 1118 y Fh(\033)1419 1104 y Fp(\()p Fi(x)p Fp(\))c Fg(\021)f Fi(\026)1718 1118 y Fh(\014)s(;N)1868 1104 y Fp(\()p Fi(d)1950 1118 y Fh(N)2019 1104 y Fp(\()p Fi(\033)n(;)15 b(\033)2199 1067 y Ff(0)2226 1104 y Fp(\))25 b Fi(<)g(x)p Fp(\))1054 b(\(2)p Fi(:)p Fp(5\))50 1319 y(at)39 b(the)g(v)-5 b(alues)38 b Fi(q)654 1333 y Fh(i)723 1319 y Fp(for)g(whic)m(h)g(the)h(limiting)c (distribution)g(will)h(ha)m(v)m(e)k(jumps.)64 b(Seen)39 b(in)e(this)h(ligh)m(t,)i(it)50 1464 y(w)m(ould)d(b)s(e)h(most)h (appropriate)e(to)j(in)m(tro)s(duce)d(the)i(follo)m(wing)e(ob)5 b(jects,)41 b(that)e(w)m(e)g(ma)m(y)g(call)f Fo(empiric)-5 b(al)50 1610 y(distanc)g(e)34 b(distribution)f(function)1338 1870 y Fg(K)1407 1884 y Fh(\014)s(;N)1567 1870 y Fg(\021)1696 1784 y Fc(X)1663 1979 y Fh(\033)r Ff(2S)1801 1988 y Fk(N)1875 1870 y Fi(\026)1930 1884 y Fh(\014)s(;N)2065 1870 y Fp(\()p Fi(\033)s Fp(\))p Fi(\016)2230 1887 y Fh(m)2295 1896 y Fk(\033)2339 1887 y Fs(\()p Ff(\001)p Fs(\))3523 1870 y Fp(\(2)p Fi(:)p Fp(6\))50 2189 y(as)f(a)g(random)e(measure)i(on)f (the)h(set)g(of)g(probabilit)m(y)d(measures)i(on)h([0)p Fi(;)15 b Fp(1].)46 b(This)30 b(ob)5 b(ject)32 b(is)f(an)g(elemen)m(t) 50 2335 y(of)h(the)g Fo(c)-5 b(omp)g(act)34 b Fp(space)f Fg(M)1017 2349 y Fs(1)1057 2335 y Fp(\()p Fg(M)1201 2349 y Fs(1)1243 2335 y Fp(\()p Fg(M)1387 2349 y Fs(1)1428 2335 y Fp(\([0)p Fi(;)15 b Fp(1]\)\)\))35 b(where)d(all)e(sets)j(of)f (measures)f(are)i(endo)m(w)m(ed)f(with)f(the)50 2480 y(top)s(ology)23 b(of)g(w)m(eak)h(con)m(v)m(ergence.)40 b(Note)25 b(that)e(the)g(\014rst)f(momen)m(t)i(of)f Fg(K)2507 2494 y Fh(\014)s(;N)2665 2480 y Fp(is)f(the)h(distance)f(distribution) 50 2625 y(function)29 b Fi(f)451 2639 y Fh(\014)s(;N)585 2625 y Fp(.)159 2844 y(In)g(the)h(case)h(of)f(the)g(GREM)g(with)e (\014nitely)g(man)m(y)i(lev)m(els,)g(the)g(results)f(in)f([BK2])j (imply)c(readily)i(the)50 2989 y(con)m(v)m(ergence)f(of)e Fg(K)716 3003 y Fh(\014)s(;N)877 2989 y Fp(to)h(a)f(measure)g(that)g (is)f(concen)m(trated)j(on)e(atomic)g(distributions)c(with)j(jumps)f (at)50 3135 y(the)j(v)-5 b(alues)27 b Fi(q)512 3149 y Fh(i)569 3135 y Fp(those)h(v)-5 b(alues)27 b(are)h(con)m(trolled)f(b)m (y)g(the)h(p)s(oin)m(t)e(pro)s(cess)h Fg(W)2562 3087 y Fs(\()p Fh(m)p Fs(\))2555 3162 y Fh(\014)2688 3135 y Fp(,)h(and)f(whence)g(giv)m(en)h(\014nally)50 3280 y(in)h(terms)h(of)g(the)h(pro)s(cesses)f Fg(P)1132 3247 y Fs(\()p Fh(m)p Fs(\))1258 3280 y Fp(.)41 b(This)29 b(approac)m(h)h(clearly)g(fails)e(in)h(the)i(con)m(tin)m(uous)f(case.) 41 b(Ho)m(w)m(ev)m(er,)50 3425 y(w)m(e)29 b(ha)m(v)m(e)h(already)f (seen)g(in)f(the)h(standard)f(GREM,)h(that)h(the)f Fo(Ghirlanda-Guerr) -5 b(a)33 b Fp(iden)m(tities)27 b([GG,A)m(C])50 3571 y(pro)m(vide)f(an)g(alternativ)m(e)h(approac)m(h)g(to)h(the)f (construction)f(of)h(the)g(in\014nite)d(v)m(olume)j(limit.)37 b(Here)27 b(w)m(e)g(will)50 3716 y(exploit)32 b(this)g(fully)-8 b(.)46 b(Let)33 b(us)g(brie\015y)e(outline)g(the)i(strategy)h(that)g(w) m(e)f(will)e(follo)m(w)h(in)f(the)i(remainder)f(of)50 3862 y(the)e(pap)s(er.)17 4080 y(\(i\))46 b(First)32 b(w)m(e)i(will)c(pro)m(v)m(e)k(that)f(the)h(free)f(energy)g(and)f(the)i (mean)f(distance)f(distribution)e(function)h(will)159 4225 y(con)m(v)m(erge)42 b(to)f(the)f(limits)e(giv)m(en)i(b)m(y)g (\(2.1\))i(and)d(\(2.2\).)72 b(This)38 b(will)g(rely)h(on)h(the)g (computations)g(in)159 4371 y(the)c(mo)s(del)e(with)g(\014nitely)g(man) m(y)h(hierarc)m(hies,)h(and)f(standard)g(comparison)f(theorems)i(that)g (allo)m(w)159 4516 y(to)j(con)m(trol)f(con)m(v)m(ergence)j(of)d(the)h (free)f(energy)g(of)h(a)f(general)g(mo)s(del)f(through)h(those)g(of)h (standard)159 4662 y(GREM's)31 b(with)e(\014nitely)f(man)m(y)j(hierarc) m(hies.)-8 4880 y(\(ii\))45 b(Next)40 b(w)m(e)f(observ)m(e)h(that)f (the)g(random)f(measures)h Fg(K)2045 4894 y Fh(\014)s(;N)2219 4880 y Fp(are)g(completely)g(determined)f(b)m(y)g(the)h(so)159 5025 y(called)29 b('m)m(ulti-o)m(v)m(erlap)g(distribution)d (functions'.)39 b(W)-8 b(e)31 b(will)c(then)j(sho)m(w)f(that)i(the)e (latter)h(satisfy)g(the)159 5171 y(Ghirlanda-Guerra)c(iden)m(tities,)i (and)g(that)h(these)g(allo)m(w)f(to)i(c)m(haracterize)g(the)e(p)s (ossible)f(limit)f(p)s(oin)m(ts)159 5316 y(up)35 b(to)i(function)d Fe(E)11 b Fi(f)874 5330 y Fh(\014)928 5316 y Fp(.)58 b(By)36 b(compactness,)j(this)34 b(implies)g(con)m(v)m(ergence)k(of)e Fg(K)2901 5330 y Fh(\014)s(;;N)3094 5316 y Fp(to)h(a)f Fg(K)3361 5330 y Fh(\014)3446 5316 y Fp(that)h(is)159 5462 y(completely)30 b(determined)f(b)m(y)h Fe(E)11 b Fi(f)1317 5476 y Fh(\014)1371 5462 y Fp(.)p eop %%Page: 6 6 6 5 bop 50 295 a Fm(6)3309 b Fj(Se)l(ction)27 b Fm(3)-33 513 y Fp(\(iii\))44 b(Finally)-8 b(,)44 b(w)m(e)f(will)c(in)m(tro)s (duce)i(certain)i(marginals)d(of)j(the)f(measures)g Fg(K)2743 527 y Fh(\014)2835 513 y Fp(and)f(sho)m(w)i(that,)j(again)159 658 y(due)36 b(to)h(the)g(Ghirlanda-Guerra)e(iden)m(tities,)i(these)g (are)g(iden)m(tical)f(to)h(the)g(corresp)s(onding)e(ob)5 b(jects)159 804 y(in)26 b(certain)i(GREM's)g(with)f(\014nitely)e(man)m (y)j(hierarc)m(hies,)f(and)h(are)g(th)m(us)f(explicitly)e (constructable)j(in)159 949 y(terms)i(of)h(P)m(oisson)f(cascades.)159 1167 y(As)g(a)h(result,)f(w)m(e)g(obtain)g(a)h(complete)g(and)f(quite)f (explicit)g(description)f(of)j(the)f(asymptotics)h(of)g(the)50 1313 y(Gibbs)e(measures)h(in)f(this)g(class)h(of)h(mo)s(dels.)50 1513 y Ft(3.)55 b(Ground)41 b(state)g(energy)g(and)g(free)f(energy)-10 b(.)159 1732 y Fp(The)29 b(basis)f(of)i(all)e(our)h(results)f(is)h(con) m(trol)h(of)g(the)f(con)m(v)m(ergence)j(of)e(the)f(free)h(energy)-8 b(.)41 b(As)30 b(a)g(w)m(arm-up)50 1877 y(let)40 b(us)f(\014rst)g (consider)f(the)i(ground)f(state)i(energy)f(densit)m(y)-8 b(.)69 b(Recall)39 b(that)i(the)f(leading)e(term)i(in)f(the)50 2022 y(function)29 b Fi(U)468 2036 y Fh(J)n(;N)597 2022 y Fp(\()p Fi(x)p Fp(\))i(tak)m(es)h(the)e(form)1441 2228 y Fg(p)p 1516 2228 280 4 v 1516 2310 a Fp(2)p Fi(N)c Fp(ln)14 b(2)1811 2186 y Fc(Z)1902 2209 y Fs(1)1862 2393 y(0)1958 2228 y Fc(p)p 2049 2228 171 4 v 2050 2310 a Fp(\026)-46 b Fi(a)p Fp(\()p Fi(x)p Fp(\))q Fi(dx)1204 b Fp(\(3)p Fi(:)p Fp(1\))50 2575 y(whic)m(h)29 b(pro)m(vides)g(a)i (closed)f(form)g(for)g(the)h(ground)e(state)j(energy)f(for)f(our)g(mo)s (dels.)159 2793 y(In)g(fact,)h(this)e(form)m(ula)h(holds)f(in)g(the)h (general)h(case.)50 3012 y Fb(Theorem)44 b(3.1:)115 b Fo(L)-5 b(et)37 b Fi(X)1072 3026 y Fh(\033)1159 3012 y Fo(b)-5 b(e)37 b(a)h(c)-5 b(enter)g(e)g(d)39 b(Gaussian)g(pr)-5 b(o)g(c)g(ess)39 b(on)f Fg(S)2621 3026 y Fh(N)2728 3012 y Fo(with)g(c)-5 b(ovarianc)g(e)39 b(given)e(by)50 3157 y(\(1.2\).)43 b(Then)1018 3332 y Fp(lim)994 3391 y Fh(N)7 b Ff("1)1183 3332 y Fi(N)1266 3295 y Ff(\000)p Fs(1)p Fh(=)p Fs(2)1437 3332 y Fe(E)46 b Fp(max)1507 3391 y Fh(\033)r Ff(2S)1645 3400 y Fk(N)1725 3332 y Fi(X)1800 3346 y Fh(\033)1874 3332 y Fp(=)1970 3250 y Fg(p)p 2046 3250 197 4 v 82 x Fp(2)15 b(ln)f(2)2258 3208 y Fc(Z)2349 3231 y Fs(1)2308 3415 y(0)2405 3250 y Fc(p)p 2496 3250 171 4 v 2497 3332 a Fp(\026)-46 b Fi(a)p Fp(\()p Fi(x)p Fp(\))p Fi(dx)758 b Fp(\(3)p Fi(:)p Fp(2\))50 3760 y Fa(Pro)s(of:)67 b Fp(Let)40 b Fi(A)639 3774 y Fh(n)727 3760 y Fp(denote)g(a)f(sequence)g(of)g(piecewise)g(constan)m(t)h (distributions)35 b(functions)i(suc)m(h)i(that)50 3906 y Fi(A)118 3920 y Fh(n)167 3906 y Fp(\()p Fi(x)p Fp(\))26 b Fg(\024)f Fi(A)p Fp(\()p Fi(x)p Fp(\))h(for)f(all)f Fi(x)h Fg(2)g Fp([0)p Fi(;)15 b Fp(1])27 b(that)e(con)m(v)m(erge)i(to)f Fi(A)f Fp(in)f(the)h(sup-norm.)37 b(Denote)27 b(the)e(corresp)s(onding) 50 4051 y(pro)s(cesses)30 b(b)m(y)g Fi(X)651 4004 y Fs(\()p Fh(n)p Fs(\))644 4062 y Fh(\033)758 4051 y Fp(.)41 b(Then,)29 b(b)m(y)i(Slepian's)d(lemma)i([LLR],)1375 4307 y Fe(E)46 b Fp(max)1445 4365 y Fh(\033)r Ff(2S)1583 4374 y Fk(N)1663 4307 y Fi(X)1745 4269 y Fs(\()p Fh(n)p Fs(\))1738 4329 y Fh(\033)1877 4307 y Fg(\025)25 b Fe(E)45 b Fp(max)2042 4365 y Fh(\033)s Ff(2S)2181 4374 y Fk(N)2260 4307 y Fi(X)2335 4321 y Fh(\033)3523 4307 y Fp(\(3)p Fi(:)p Fp(3\))50 4566 y(On)29 b(the)i(other)g(hand,)e(for)h(all)f Fi(n)p Fp(,)1018 4845 y(lim)994 4905 y Fh(N)7 b Ff("1)1183 4845 y Fi(N)1266 4808 y Ff(\000)p Fs(1)p Fh(=)p Fs(2)1437 4845 y Fe(E)46 b Fp(max)1507 4904 y Fh(\033)r Ff(2S)1645 4913 y Fk(N)1725 4845 y Fi(X)1800 4859 y Fh(\033)1874 4845 y Fp(=)1970 4763 y Fg(p)p 2046 4763 197 4 v 82 x Fp(2)15 b(ln)f(2)2258 4721 y Fc(Z)2349 4744 y Fs(1)2308 4928 y(0)2405 4763 y Fc(p)p 2496 4763 171 4 v 2497 4845 a Fp(\026)-46 b Fi(a)p Fp(\()p Fi(x)p Fp(\))p Fi(dx)758 b Fp(\(3)p Fi(:)p Fp(4\))50 5115 y(Moreo)m(v)m(er,)33 b(if)c Fi(A)626 5129 y Fh(n)706 5115 y Fp(con)m(v)m(erges)j(to)f Fi(A)p Fp(,)g(then)g(\026)-46 b Fi(a)1603 5129 y Fh(x)1649 5115 y Fp(\()p Fi(x)p Fp(\))31 b(con)m(v)m(erges)h(to)h(\026)-47 b Fi(a)p Fp(\()p Fi(x)p Fp(\).)42 b(Th)m(us)942 5400 y(lim)15 b(sup)994 5477 y Fh(N)7 b Ff("1)1235 5400 y Fi(N)1318 5362 y Ff(\000)p Fs(1)p Fh(=)p Fs(2)1489 5400 y Fe(E)46 b Fp(max)1559 5458 y Fh(\033)r Ff(2S)1697 5467 y Fk(N)1777 5400 y Fi(X)1852 5414 y Fh(\033)1926 5400 y Fg(\024)2022 5318 y(p)p 2098 5318 197 4 v 82 x Fp(2)15 b(ln)f(2)2310 5276 y Fc(Z)2401 5298 y Fs(1)2361 5482 y(0)2457 5317 y Fc(p)p 2548 5317 171 4 v 2549 5400 a Fp(\026)-46 b Fi(a)p Fp(\()p Fi(x)p Fp(\))q Fi(dx)705 b Fp(\(3)p Fi(:)p Fp(5\))p eop %%Page: 7 7 7 6 bop 50 295 a Fj(GREM,)28 b(p)l(art)f(3)3127 b Fm(7)50 513 y Fp(It)29 b(remains)e(to)j(pro)m(v)m(e)g(a)f(corresp)s(onding)e (lo)m(w)m(er)i(b)s(ound.)38 b(T)-8 b(o)29 b(do)g(so,)g(w)m(e)h(need)e (to)i(construct)f(a)g(sequence)50 658 y(of)23 b(piecewise)g(constan)m (t)h(distribution)c(functions)i Fi(B)1836 672 y Fh(n)1909 658 y Fp(suc)m(h)h(that)h Fi(B)2366 672 y Fh(n)2415 658 y Fp(\()p Fi(x)p Fp(\))i Fg(\025)f Fi(A)p Fp(\()p Fi(x)p Fp(\))f(for)f(all)f Fi(x)k Fg(2)e Fp([0)p Fi(;)15 b Fp(1].)41 b(This)50 804 y(in)m(tro)s(duces)26 b(only)h(one)h(small)e(problem,)h (namely)g(that)h(in)f(general)g(suc)m(h)h(an)f(upp)s(er)f(appro)m (ximation)g(will)50 949 y(ha)m(v)m(e)f(to)g(ha)m(v)m(e)g(a)f(p)s (ositiv)m(e)f(atom)i(at)f(zero,)j(and)c(the)h(last)g(\\atom",)j Fi(b)2371 963 y Fh(n)2421 949 y Fp(,)e(ma)m(y)g(ha)m(v)m(e)g(to)f(b)s (e)g(zero.)39 b(The)23 b(latter)50 1095 y(fact)35 b(has)f(no)h (consequence,)h(while)d(the)h(former)h(requires)e(us)g(to)j(sligh)m (tly)c(mo)s(dify)h(the)i(represen)m(tation)50 1240 y(form)m(ula)29 b(Eq.)41 b(\(1.4\))32 b(in)d([BK1])j(in)m(to)963 1469 y Fc(e)939 1492 y Fi(X)1014 1506 y Fh(\033)1089 1492 y Fp(=)1185 1408 y Fc(p)p 1276 1408 80 4 v 84 x Fi(b)1315 1506 y Fs(0)1355 1492 y Fi(X)1430 1506 y Fs(0)1492 1492 y Fp(+)1583 1408 y Fc(p)p 1673 1408 V 1673 1492 a Fi(b)1712 1506 y Fs(1)1753 1492 y Fi(X)1828 1506 y Fh(\033)1870 1515 y Fl(1)1931 1492 y Fp(+)20 b Fg(\001)15 b(\001)g(\001)21 b Fp(+)2239 1411 y Fc(p)p 2330 1411 182 4 v 81 x Fi(b)2369 1506 y Fh(n)p Ff(\000)p Fs(1)2511 1492 y Fi(X)2586 1506 y Fh(\033)2628 1515 y Fl(1)2664 1506 y Fh(:::)o(\033)2771 1515 y Fk(n)3523 1492 y Fp(\(3)p Fi(:)p Fp(6\))50 1744 y(where)28 b(no)m(w)499 1676 y Fc(P)595 1698 y Fh(n)p Ff(\000)p Fs(1)595 1771 y Fh(k)r Fs(=1)752 1744 y Fi(b)791 1758 y Fh(k)861 1744 y Fp(=)d(1)17 b Fg(\000)g Fi(b)1146 1758 y Fs(0)1187 1744 y Fp(,)29 b(and)f Fi(X)1491 1758 y Fs(0)1561 1744 y Fp(is)f(a)i(new)f(standard)g(Gaussian,)h(indep)s (enden)m(t)d(of)j(all)e(others.)50 1889 y(Th)m(us)f(the)h(mean)g(of)g (the)g(maxim)m(um)e(of)1466 1866 y Fc(e)1443 1889 y Fi(X)1518 1903 y Fh(\033)1594 1889 y Fp(is)h(equal)g(to)i(the)f(mean)g(of)g(the)g (maxim)m(um)e(of)3215 1866 y Fc(e)3191 1889 y Fi(X)3266 1903 y Fh(\033)3329 1889 y Fg(\000)3413 1819 y(p)p 3489 1819 80 4 v 70 x Fi(b)3528 1903 y Fs(0)3568 1889 y Fi(X)3643 1903 y Fs(0)3685 1889 y Fp(.)50 2035 y(This)g(v)-5 b(ariable)26 b(has)g(the)h(same)h(form)e(as)h(b)s(efore,)h(except)f(that)h(the)f (total)h(v)-5 b(ariance)26 b(is)g(diminished)d(b)m(y)k Fi(b)3644 2049 y Fs(0)3685 2035 y Fp(.)50 2180 y(Then)i(the)i(form)m (ula)e(for)h(its)g(mean)h(tak)m(es)h(the)e(same)h(form)f(as)g(b)s (efore.)41 b(Th)m(us)998 2438 y(lim)974 2497 y Fh(N)7 b Ff("1)1163 2438 y Fi(N)1246 2400 y Ff(\000)p Fs(1)p Fh(=)p Fs(2)1417 2438 y Fe(E)45 b Fp(max)1487 2497 y Fh(\033)r Ff(2S)1625 2506 y Fk(N)1728 2415 y Fc(e)1704 2438 y Fi(X)1779 2452 y Fh(\033)1854 2438 y Fp(=)1950 2356 y Fg(p)p 2026 2356 197 4 v 82 x Fp(2)15 b(ln)f(2)2238 2314 y Fc(Z)2329 2337 y Fs(1)2288 2521 y(0)2385 2325 y Fc(q)p 2476 2325 212 4 v 2473 2414 a Fp(\026)2476 2438 y Fi(b)2515 2452 y Fh(n)2564 2438 y Fp(\()p Fi(x)p Fp(\))q Fi(dx)737 b Fp(\(3)p Fi(:)p Fp(7\))50 2699 y(and)30 b(Slepian's)e (lemma)i(no)m(w)g(sho)m(ws)g(that)955 2975 y(lim)15 b(inf)994 3034 y Fh(N)7 b Ff("1)1222 2975 y Fi(N)1305 2937 y Ff(\000)p Fs(1)p Fh(=)p Fs(2)1476 2975 y Fe(E)46 b Fp(max)1546 3034 y Fh(\033)r Ff(2S)1684 3043 y Fk(N)1764 2975 y Fi(X)1839 2989 y Fh(\033)1913 2975 y Fg(\025)2009 2893 y(p)p 2085 2893 197 4 v 82 x Fp(2)15 b(ln)f(2)2297 2851 y Fc(Z)2388 2874 y Fs(1)2347 3058 y(0)2444 2893 y Fc(p)p 2535 2893 171 4 v 2536 2975 a Fp(\026)-46 b Fi(a)p Fp(\()p Fi(x)p Fp(\))p Fi(dx)719 b Fp(\(3)p Fi(:)p Fp(8\))50 3236 y(This)28 b(pro)m(v)m(es)j(the)g(theorem.)41 b Fg(})159 3455 y Fp(In)c(the)i(same)g(w)m(a)m(y)-8 b(,)41 b(w)m(e)e(get)g(a)g(corresp)s (onding)d(result)h(for)h(the)h(free)f(energy)h(in)e(the)h(general)g (case.)50 3600 y(Essen)m(tially)h(all)g(w)m(e)i(need)f(is)g(to)h (replace)g(Slepian's)d(lemma)i(b)m(y)h(the)f(more)h(general)g(result)e (giv)m(en)h(as)50 3746 y(Theorem)30 b(3.11)i(in)d([L)-8 b(T],)31 b(Chapter)f(3,)h(page)g(74)50 3964 y Fb(Lemma)k(3.2:)89 b Fo(L)-5 b(et)31 b Fi(X)37 b Fo(and)32 b Fi(Y)50 b Fo(b)-5 b(e)31 b Fi(n)p Fo(-dimensional)g(Gaussian)h(ve)-5 b(ctors.)42 b(L)-5 b(et)31 b Fi(D)2934 3978 y Fs(1)3005 3964 y Fo(and)h Fi(D)3255 3978 y Fs(2)3326 3964 y Fo(b)-5 b(e)31 b(subsets)50 4109 y(of)i Fg(f)p Fp(1)p Fi(;)15 b(:)g(:)g(:)i(;)e(n)p Fg(g)21 b(\002)f(f)p Fp(1)p Fi(;)15 b(:)g(:)g(:)i(;)e(n)p Fg(g)p Fo(.)42 b(Assume)33 b(that)1106 4325 y Fe(E)11 b Fi(X)1236 4339 y Fh(i)1272 4325 y Fi(X)1347 4339 y Fh(j)1411 4325 y Fg(\024)25 b Fe(E)11 b Fi(Y)1615 4339 y Fh(i)1651 4325 y Fi(Y)1704 4339 y Fh(j)1742 4325 y Fi(;)108 b Fo(if)93 b Fp(\()p Fi(i;)15 b(j)5 b Fp(\))27 b Fg(2)e Fi(D)2395 4339 y Fs(1)1106 4498 y Fe(E)11 b Fi(X)1236 4512 y Fh(i)1272 4498 y Fi(X)1347 4512 y Fh(j)1411 4498 y Fg(\025)25 b Fe(E)11 b Fi(Y)1615 4512 y Fh(i)1651 4498 y Fi(Y)1704 4512 y Fh(j)1742 4498 y Fi(;)108 b Fo(if)93 b Fp(\()p Fi(i;)15 b(j)5 b Fp(\))27 b Fg(2)e Fi(D)2395 4512 y Fs(2)1106 4670 y Fe(E)11 b Fi(X)1236 4684 y Fh(i)1272 4670 y Fi(X)1347 4684 y Fh(j)1411 4670 y Fp(=)25 b Fe(E)11 b Fi(Y)1615 4684 y Fh(i)1651 4670 y Fi(Y)1704 4684 y Fh(j)1742 4670 y Fi(;)108 b Fo(if)93 b Fp(\()p Fi(i;)15 b(j)5 b Fp(\))27 b Fg(62)e Fi(D)2395 4684 y Fs(1)2456 4670 y Fg([)20 b Fi(D)2612 4684 y Fs(2)3523 4498 y Fp(\(3)p Fi(:)p Fp(9\))50 4881 y Fo(L)-5 b(et)33 b Fi(f)42 b Fo(b)-5 b(e)32 b(a)h(function)g(on)g Fe(R)1032 4848 y Fh(n)1120 4881 y Fo(such)f(that)i(its)f(se)-5 b(c)g(ond)34 b(derivatives)f (satisfy)1334 5088 y Fi(@)1387 5055 y Fs(2)p 1242 5129 280 4 v 1242 5212 a Fi(@)5 b(x)1347 5226 y Fh(i)1377 5212 y Fi(@)g(x)1482 5226 y Fh(j)1532 5150 y Fi(f)10 b Fp(\()p Fi(x)p Fp(\))25 b Fg(\025)g Fp(0)93 b Fo(if)g Fp(\()p Fi(i;)15 b(j)5 b Fp(\))27 b Fg(2)e Fi(D)2488 5164 y Fs(1)1334 5348 y Fi(@)1387 5315 y Fs(2)p 1242 5389 V 1242 5472 a Fi(@)5 b(x)1347 5486 y Fh(i)1377 5472 y Fi(@)g(x)1482 5486 y Fh(j)1532 5410 y Fi(f)10 b Fp(\()p Fi(x)p Fp(\))25 b Fg(\024)g Fp(0)93 b Fo(if)g Fp(\()p Fi(i;)15 b(j)5 b Fp(\))27 b Fg(2)e Fi(D)2488 5424 y Fs(2)3477 5279 y Fp(\(3)p Fi(:)p Fp(10\))p eop %%Page: 8 8 8 7 bop 50 295 a Fm(8)3309 b Fj(Se)l(ction)27 b Fm(3)50 513 y Fo(Then)1539 662 y Fe(E)11 b Fi(F)i Fp(\()p Fi(X)8 b Fp(\))31 b Fg(\024)25 b Fe(E)11 b Fi(F)i Fp(\()p Fi(Y)26 b Fp(\))1257 b(\(3)p Fi(:)p Fp(11\))50 1093 y Fb(Theorem)37 b(3.3:)91 b Fo(Under)33 b(the)g(assumptions)i(of)e(The)-5 b(or)g(em)34 b(3.1,)f(let)1293 1365 y Fi(x)p Fp(\()p Fi(\014)5 b Fp(\))26 b Fg(\021)f Fp(sup)1745 1237 y Fc(\022)1812 1365 y Fi(x)p Fg(j)q Fp(\026)-46 b Fi(a)p Fp(\()p Fi(x)p Fp(\))26 b Fi(>)2192 1303 y Fp(2)15 b(ln)f(2)p 2192 1344 197 4 v 2242 1427 a Fi(\014)2298 1401 y Fs(2)2400 1237 y Fc(\023)50 1637 y Fo(Then)610 1799 y Fp(lim)586 1858 y Fh(N)7 b Ff("1)775 1799 y Fi(N)858 1761 y Ff(\000)p Fs(1)956 1799 y Fe(E)25 b Fp(ln)20 b Fi(Z)1184 1813 y Fh(\014)s(;N)1345 1799 y Fp(=)1441 1717 y Fg(p)p 1516 1717 V 1516 1799 a Fp(2)15 b(ln)g(2)p Fi(\014)1785 1675 y Fc(Z)1876 1697 y Fh(x)p Fs(\()p Fh(\014)s Fs(\))1835 1881 y(0)2038 1716 y Fc(p)p 2129 1716 171 4 v 2130 1799 a Fp(\026)-46 b Fi(a)p Fp(\()p Fi(x)p Fp(\))q Fi(dx)20 b Fp(+)2521 1737 y Fi(\014)2577 1704 y Fs(2)p 2521 1778 98 4 v 2547 1861 a Fp(2)2629 1799 y(\(1)h Fg(\000)2845 1776 y Fp(\026)2821 1799 y Fi(A)p Fp(\()p Fi(x)p Fp(\()p Fi(\014)5 b Fp(\)\)\))305 b(\(3)p Fi(:)p Fp(12\))50 2021 y Fo(Note)32 b(that)i(this)f(formula)i(implies)e(that)h(the)f(critic)-5 b(al)33 b(temp)-5 b(er)g(atur)g(e)35 b Fi(\014)2468 2035 y Fh(c)2538 2021 y Fo(is)e(given)f(by)1503 2328 y Fi(\014)1554 2342 y Fh(c)1617 2328 y Fp(=)1713 2168 y Fc(s)p 1803 2168 454 4 v 1932 2266 a Fp(2)15 b(ln)f(2)p 1814 2307 432 4 v 1814 2390 a(lim)1941 2404 y Fh(x)p Ff(#)p Fs(0)2076 2390 y Fp(\026)-46 b Fi(a)p Fp(\()p Fi(x)p Fp(\))50 2804 y Fa(Pro)s(of:)75 b Fp(W)-8 b(e)43 b(ma)m(y)g(easily)f(c)m(hec)m(k)i (that)f(the)f(function)g(ln)2140 2736 y Fc(P)2236 2832 y Fh(\033)2300 2804 y Fi(e)2342 2771 y Fh(\014)2386 2717 y Ff(p)p 2447 2717 65 4 v 2447 2771 a Fh(N)6 b(X)2571 2780 y Fk(\033)2661 2804 y Fp(satis\014es)42 b(the)h(h)m(yp)s(othesis)e (of)50 2950 y(Lemma)34 b(3.2)i(with)d Fi(D)815 2964 y Fs(1)887 2950 y Fp(=)f Fg(f)p Fi(\033)n(;)15 b(\033)1180 2917 y Ff(0)1238 2950 y Fg(2)31 b(S)1392 2917 y Fs(2)1385 2975 y Fh(N)1486 2950 y Fp(:)h Fi(\033)j Fg(6)p Fp(=)d Fi(\033)1788 2917 y Ff(0)1813 2950 y Fg(g)j Fp(and)f Fi(D)2149 2964 y Fs(2)2221 2950 y Fp(=)e Fg(;)p Fp(.)53 b(Th)m(us)33 b(taking)h Fi(A)3033 2917 y Ff(\006)3033 2972 y Fh(n)3094 2950 y Fp(\(1\))f(=)f Fi(A)p Fp(\(1\))j(and)50 3095 y Fi(A)118 3062 y Ff(\006)118 3118 y Fh(n)179 3095 y Fp(\()p Fi(x)p Fp(\))26 b(smaller)e(resp)s(ectiv)m(ely)h(larger)g (than)g Fi(A)h Fp(elsewhere,)g(w)m(e)g(can)g(indeed)d(construct)j(upp)s (er)e(and)g(lo)m(w)m(er)50 3241 y(b)s(ounds)k(for)i Fi(F)563 3255 y Fh(\014)643 3241 y Fp(that)h(con)m(v)m(erge)h(to)f(the)g(same)g (limit)d(when)h Fi(n)c Fg(")h(1)p Fp(.)40 b Fg(})159 3459 y Fp(As)27 b(w)m(e)g(ha)m(v)m(e)h(seen)e(in)g(the)g(discrete)h (case,)h(a)g(crucial)d(observ)-5 b(ation)26 b(is)g(that)h(the)g (distance)f(distribution)50 3604 y(function)31 b(can)i(b)s(e)f (expressed)h(as)g(a)g(deriv)-5 b(ativ)m(e)32 b(of)h(the)g(free)g (energy)g(as)g(a)g(function)e(of)i(the)g(co)m(v)-5 b(ariance.)50 3750 y(In)35 b(the)h(con)m(tin)m(uous)f(case,)k(this)c(will)e(b)s(e)i (a)h(little)f(more)h(subtle)e(and)h(will)f(require)g(to)j(in)m(tro)s (duce)d(some)50 3895 y(more)c(structure.)159 4114 y(In)g(the)i (discrete)f(case)h(w)m(e)f(needed)g(to)h(b)s(e)f(able)f(to)i (di\013eren)m(tiate)f(with)f(resp)s(ect)h(to)h(the)f(v)-5 b(alue)31 b(of)g(one)50 4259 y(atom,)j(and)d(then)h(to)h(use)f(an)g(in) m(tegration)h(b)m(y)f(parts)g(form)m(ula.)45 b(Both)33 b(is)e(not)i(immediately)d(ob)m(vious)i(in)50 4404 y(the)37 b(general)h(case.)62 b(In)37 b(fact,)j(to)e(do)f(so)h(it)e(will)f(b)s (e)i(necessary)g(to)h(add)f(some)h(temp)s(oral)e(structure)h(to)50 4550 y(our)c(Gaussian)f(pro)s(cess)h Fi(X)1003 4564 y Fh(\033)1086 4550 y Fp(and)f(to)i(de\014ne)f(the)g(Gaussian)g(pro)s (cess)g Fi(X)2593 4564 y Fh(\033)2642 4550 y Fp(\()p Fi(t)p Fp(\),)i Fi(\033)e Fg(2)d(S)3036 4564 y Fh(N)3105 4550 y Fp(,)k Fi(t)c Fg(2)g Fp([0)p Fi(;)15 b Fp(1])35 b(with)50 4695 y(co)m(v)-5 b(ariance)1033 4845 y(co)m(v)16 b(\()p Fi(X)1289 4859 y Fh(\033)1339 4845 y Fp(\()p Fi(t)p Fp(\))p Fi(;)f(X)1557 4859 y Fh(\033)1601 4840 y Fd(0)1632 4845 y Fp(\()p Fi(s)p Fp(\)\))26 b(=)f Fi(A)p Fp(\()p Fi(t)20 b Fg(^)g Fi(s)g Fg(^)g Fp([)p Fi(d)2355 4859 y Fh(N)2424 4845 y Fp(\()p Fi(\033)n(;)15 b(\033)2604 4807 y Ff(0)2631 4845 y Fp(\)]\))751 b(\(3)p Fi(:)p Fp(13\))50 5058 y(Note)36 b(that)f Fi(X)550 5072 y Fh(\033)600 5058 y Fp(\()p Fi(t)p Fp(\))g(is)f(a)h(martingale)f(in)g(the)h(v)-5 b(ariable)33 b Fi(t)i Fp(and)f(that)i(its)e(incremen)m(ts)g(with)g (resp)s(ect)g(to)i Fi(t)50 5203 y Fp(are)31 b(indep)s(enden)m(t.)38 b(It)31 b(is)e(also)h(useful)f(to)i(realize)f(that)h(w)m(e)g(ma)m(y)g (represen)m(t)f Fi(X)2774 5217 y Fh(\033)2824 5203 y Fp(\()p Fi(t)p Fp(\))g(as)1533 5462 y Fi(X)1608 5476 y Fh(\033)1657 5462 y Fp(\()p Fi(t)p Fp(\))c(=)f Fi(Y)1935 5476 y Fh(\033)1984 5462 y Fp(\()p Fi(A)p Fp(\()p Fi(t)p Fp(\)\))1252 b(\(3)p Fi(:)p Fp(14\))p eop %%Page: 9 9 9 8 bop 50 295 a Fj(GREM,)28 b(p)l(art)f(3)3127 b Fm(9)50 513 y Fp(where)30 b Fi(Y)366 527 y Fh(\033)414 513 y Fp(\()p Fi(t)p Fp(\))h(is)f(a)g(con)m(tin)m(uous)g(Gaussian)g (martingale)g(with)f(co)m(v)-5 b(ariance)1055 770 y(co)m(v)17 b(\()p Fi(Y)1290 784 y Fh(\033)1339 770 y Fp(\()p Fi(t)p Fp(\))p Fi(;)e(Y)1535 784 y Fh(\033)1579 766 y Fd(0)1609 770 y Fp(\()p Fi(s)p Fp(\)\))26 b(=)f Fi(t)20 b Fg(^)g Fi(s)g Fg(^)g Fi(A)p Fp(\([)p Fi(d)2332 784 y Fh(N)2402 770 y Fp(\()p Fi(\033)n(;)15 b(\033)2582 733 y Ff(0)2608 770 y Fp(\)]\))774 b(\(3)p Fi(:)p Fp(15\))50 1027 y(Observ)m(e)30 b(that)h(there)g(is)e(the)i(follo)m(wing)d(in)m(tegration)j(b)m(y)f (parts)g(form)m(ula:)50 1246 y Fb(Lemma)38 b(3.4:)51 b Fo(F)-7 b(or)33 b(any)h Fi(t)25 b Fg(2)g Fp(\(0)p Fi(;)15 b Fp(1])34 b Fo(and)g Fi(\017)25 b(>)g Fp(0)p Fo(,)322 1544 y Fe(E)10 b(E)437 1558 y Fh(\033)503 1483 y Fp(\()p Fi(X)613 1497 y Fh(\033)663 1483 y Fp(\()p Fi(t)p Fp(\))20 b Fg(\000)g Fi(X)952 1497 y Fh(\033)1001 1483 y Fp(\()p Fi(t)h Fg(\000)f Fi(\017)p Fp(\)\))p Fi(e)1330 1450 y Fh(\014)1374 1396 y Ff(p)p 1435 1396 65 4 v 1435 1450 a Fh(N)6 b(X)1559 1459 y Fk(\033)p 503 1523 1104 4 v 816 1621 a Fe(E)871 1635 y Fh(\033)915 1617 y Fd(0)950 1621 y Fi(e)992 1595 y Fh(\014)1036 1541 y Ff(p)p 1097 1541 65 4 v 1097 1595 a Fh(N)g(X)1221 1612 y Fk(\033)1260 1599 y Fd(0)1643 1544 y Fp(=)25 b Fi(\014)1795 1463 y Fg(p)p 1871 1463 83 4 v 81 x Fi(N)1969 1421 y Fc(Z)2060 1443 y Fh(t)2019 1627 y(t)p Ff(\000)p Fh(\017)2154 1544 y Fi(dA)p Fp(\()p Fi(s)p Fp(\))p Fe(E)11 b Fi(\026)2492 1505 y Ff(\012)p Fs(2)2492 1572 y Fh(\014)s(;N)2633 1544 y Fp([)p Fi(d)2705 1558 y Fh(N)2774 1544 y Fp(\()p Fi(\033)n(;)k(\033) 2954 1507 y Ff(0)2981 1544 y Fp(\))26 b Fi(<)f(s)p Fp(])271 b(\(3)p Fi(:)p Fp(16\))50 2020 y Fa(Pro)s(of:)47 b Fp(Let)31 b(us)f(in)m(tro)s(duce)f(the)i(in\014nitesimal)26 b(incremen)m(ts)k(of) h(the)f(pro)s(cess)g Fi(Y)2823 2034 y Fh(\033)2872 2020 y Fp(\()p Fi(t)p Fp(\).)41 b(Clearly)1008 2277 y Fe(E)11 b Fi(d)5 b(Y)1168 2291 y Fh(\033)1223 2277 y Fp(\()p Fi(t)p Fp(\))p Fi(d)g(Y)1431 2291 y Fh(\033)1475 2273 y Fd(0)1505 2277 y Fp(\()p Fi(s)p Fp(\))26 b(=)f Fi(dtds\016)s Fp(\()p Fi(s)c Fg(\000)f Fi(t)p Fp(\)1)-23 b(I)2267 2294 y Fh(A)p Fs(\()p Fh(d)2388 2303 y Fk(N)2446 2294 y Fs(\()p Fh(\033)n(;\033)2580 2276 y Fd(0)2605 2294 y Fs(\)\))p Ff(\025)p Fh(t)3477 2277 y Fp(\(3)p Fi(:)p Fp(17\))50 2535 y(The)30 b(pro)s(of)f(mak)m(es)j(use)e(of)g(the)h(Gaussian)e(in)m (tegration)i(b)m(y)f(parts)g(form)m(ula)650 2805 y Fe(E)11 b Fi(d)5 b(Y)810 2819 y Fh(\033)865 2805 y Fp(\()p Fi(t)p Fp(\))p Fi(f)1038 2677 y Fc(\022)1105 2682 y(Z)1211 2805 y Fi(d)g(Y)1316 2819 y Fh(\033)1360 2801 y Fd(0)1389 2805 y Fp(\()p Fi(s)p Fp(\))1502 2677 y Fc(\023)1595 2805 y Fp(=)25 b Fe(E)10 b Fi(f)1800 2768 y Ff(0)1846 2677 y Fc(\022)1913 2682 y(Z)2019 2805 y Fi(d)5 b(Y)2124 2819 y Fh(\033)2168 2801 y Fd(0)2198 2805 y Fp(\()p Fi(s)p Fp(\))2311 2677 y Fc(\023)2393 2682 y(Z)2499 2805 y Fe(E)11 b Fi(d)5 b(Y)2659 2819 y Fh(\033)2714 2805 y Fp(\()p Fi(t)p Fp(\))p Fi(d)g(Y)2922 2819 y Fh(\033)2966 2801 y Fd(0)2996 2805 y Fp(\()p Fi(s)p Fp(\))1595 3005 y(=)25 b Fe(E)10 b Fi(f)1800 2968 y Ff(0)1846 3005 y Fp(\()q Fi(Y)1935 3019 y Fh(\033)1979 3001 y Fd(0)2008 3005 y Fp(\))15 b(1)-23 b(I)2114 3022 y Fh(A)p Fs(\()p Fh(d)2235 3031 y Fk(N)2293 3022 y Fs(\()p Fh(\033)n(;\033)2427 3004 y Fd(0)2453 3022 y Fs(\)\))p Ff(\025)p Fh(t)2599 3005 y Fi(dt)3477 2878 y Fp(\(3)p Fi(:)p Fp(18\))50 3227 y(where)30 b Fi(f)39 b Fp(is)30 b(an)m(y)g(di\013eren)m(tiable)f (function.)40 b(Using)29 b(\(3.18\))k(in)c(\(3.17\),)k(w)m(e)e(get)340 3532 y Fe(E)11 b(E)455 3546 y Fh(\033)521 3470 y Fi(d)5 b(Y)626 3484 y Fh(\033)675 3470 y Fp(\()p Fi(t)p Fp(\))p Fi(e)820 3437 y Fh(\014)864 3383 y Ff(p)p 925 3383 65 4 v 925 3437 a Fh(N)i(Y)1033 3446 y Fk(\033)p 521 3511 560 4 v 571 3609 a Fe(E)625 3623 y Fh(\033)670 3604 y Fd(0)705 3609 y Fi(e)747 3582 y Fh(\014)791 3528 y Ff(p)p 851 3528 65 4 v 54 x Fh(N)g(Y)959 3599 y Fk(\033)998 3586 y Fd(0)1116 3532 y Fp(=)25 b Fi(dt)p Fe(E)11 b(E)1408 3546 y Fh(\033)1463 3532 y Fe(E)1517 3546 y Fh(\033)1562 3527 y Fd(0)1608 3390 y Fc(\000)1649 3464 y Fp(1)21 b Fg(\000)f Fp(1)-23 b(I)1862 3480 y Fh(A)p Fs(\()p Fh(d)1983 3489 y Fk(N)2041 3480 y Fs(\()p Fh(\033)n(;\033)2175 3462 y Fd(0)2200 3480 y Fs(\)\))p Ff(\025)p Fh(t)2346 3390 y Fc(\001)2403 3464 y Fi(e)2445 3431 y Fh(\014)2489 3377 y Ff(p)p 2549 3377 V 54 x Fh(N)7 b Fs(\()p Fh(X)2702 3440 y Fk(\033)2745 3431 y Fs(+)p Fh(X)2861 3448 y Fk(\033)2900 3435 y Fd(0)2928 3431 y Fs(\))p 1608 3511 1354 4 v 1971 3544 a Fc(\020)2026 3645 y Fe(E)2080 3659 y Fh(\033)2125 3640 y Fd(0)2160 3645 y Fi(e)2202 3618 y Fh(\014)2246 3564 y Ff(p)p 2306 3564 65 4 v 54 x Fh(N)g(X)2431 3635 y Fk(\033)2470 3622 y Fd(0)2503 3544 y Fc(\021)2557 3563 y Fs(2)2972 3532 y Fi(\014)3028 3450 y Fg(p)p 3104 3450 83 4 v 82 x Fi(N)300 b Fp(\(3)p Fi(:)p Fp(19\))50 3888 y(Th)m(us)702 4055 y Fe(E)10 b(E)817 4069 y Fh(\033)883 3994 y Fi(dX)1005 4008 y Fh(\033)1054 3994 y Fp(\()p Fi(t)p Fp(\))p Fi(e)1199 3961 y Fh(\014)1243 3906 y Ff(p)p 1304 3906 65 4 v 1304 3961 a Fh(N)d(X)1429 3970 y Fk(\033)p 883 4034 594 4 v 941 4132 a Fe(E)996 4146 y Fh(\033)1040 4128 y Fd(0)1075 4132 y Fi(e)1117 4106 y Fh(\014)1161 4051 y Ff(p)p 1222 4051 65 4 v 1222 4106 a Fh(N)f(X)1346 4122 y Fk(\033)1385 4109 y Fd(0)1512 4055 y Fp(=)25 b Fi(dA)p Fp(\()p Fi(t)p Fp(\))p Fe(E)12 b Fi(\026)1937 4016 y Ff(\012)p Fs(2)1937 4083 y Fh(\014)s(;N)2093 3981 y Fc(\000)2135 4055 y Fp(1)-23 b(I)2190 4071 y Fh(A)p Fs(\()p Fh(d)2311 4080 y Fk(N)2370 4071 y Fs(\()p Fh(\033)n(;\033)2504 4053 y Fd(0)2529 4071 y Fs(\)\))p Fh()g Fp(0)30 b(\014xed)g(the)h (random)e(pro)s(cess)1228 5056 y Fi(X)1310 5019 y Fh(u)1303 5079 y(\033)1383 5056 y Fg(\021)c Fi(X)1554 5070 y Fh(\033)1624 5056 y Fp(+)19 b Fi(u)p Fp([)p Fi(X)1866 5070 y Fh(\033)1916 5056 y Fp(\()p Fi(t)p Fp(\))i Fg(\000)f Fi(X)2206 5070 y Fh(\033)2255 5056 y Fp(\()p Fi(t)g Fg(\000)g Fi(\017)p Fp(\)])946 b(\(3)p Fi(:)p Fp(21\))50 5313 y(Clearly)1279 5462 y(co)m(v)16 b(\()p Fi(X)1542 5424 y Fh(u)1535 5484 y(\033)1590 5462 y Fi(;)f(X)1712 5424 y Fh(u)1705 5484 y(\033)1749 5466 y Fd(0)1780 5462 y Fp(\))25 b(=)g Fi(A)2004 5424 y Fh(u)2052 5462 y Fp(\()p Fi(d)2134 5476 y Fh(N)2204 5462 y Fp(\()p Fi(\033)n(;)15 b(\033)2384 5424 y Ff(0)2410 5462 y Fp(\)\))997 b(\(3)p Fi(:)p Fp(22\))p eop %%Page: 10 10 10 9 bop 50 295 a Fm(10)3270 b Fj(Se)l(ction)27 b Fm(3)50 513 y Fp(where)594 735 y Fi(A)662 698 y Fh(u)710 735 y Fp(\()p Fi(x)p Fp(\))f(=)954 522 y Fc(8)954 604 y(>)954 631 y(<)954 795 y(>)954 822 y(:)1065 599 y Fi(A)p Fp(\()p Fi(x)p Fp(\))p Fi(;)1268 b Fp(if)29 b Fi(x)c(<)g(t)20 b Fg(\000)g Fi(\017)1065 735 y(A)p Fp(\()p Fi(x)p Fp(\))h(+)f(\(2)p Fi(u)h Fp(+)f Fi(u)1663 702 y Fs(2)1704 735 y Fp(\)\()p Fi(A)p Fp(\()p Fi(x)p Fp(\))i Fg(\000)e Fi(A)p Fp(\()p Fi(t)g Fg(\000)g Fi(\017)p Fp(\)\))p Fi(;)92 b Fp(if)29 b Fi(t)20 b Fg(\000)g Fi(\017)25 b Fg(\024)g Fi(x)g(<)g(t)1065 872 y(A)p Fp(\()p Fi(x)p Fp(\))c(+)f(\(2)p Fi(u)h Fp(+)f Fi(u)1663 839 y Fs(2)1704 872 y Fp(\)\()p Fi(A)p Fp(\()p Fi(t)p Fp(\))h Fg(\000)f Fi(A)p Fp(\()p Fi(t)h Fg(\000)f Fi(\017)p Fp(\)\))p Fi(;)111 b Fp(if)29 b Fi(x)c Fg(\025)g Fi(t)3477 735 y Fp(\(3)p Fi(:)p Fp(23\))50 1027 y(Let)41 b(us)g(denote)g(the)g(partition)f(function)g(corresp)s(onding)f(to)i (the)h(pro)s(cess)e(with)g(co)m(v)-5 b(ariance)42 b Fi(A)3525 994 y Fh(u)3614 1027 y Fp(b)m(y)50 1173 y Fi(Z)119 1140 y Fh(u)112 1199 y(\014)s(;N)247 1173 y Fp(,)30 b(etc.)42 b(Clearly)29 b(w)m(e)i(ha)m(v)m(e)h(that)702 1457 y Fi(\014)758 1376 y Fg(p)p 834 1376 83 4 v 81 x Fi(N)10 b Fe(E)h(E)1032 1471 y Fh(\033)1098 1395 y Fp(\()p Fi(X)1208 1409 y Fh(\033)1258 1395 y Fp(\()p Fi(t)p Fp(\))21 b Fg(\000)f Fi(X)1548 1409 y Fh(\033)1597 1395 y Fp(\()p Fi(t)g Fg(\000)g Fi(\017)p Fp(\)\))p Fi(e)1925 1362 y Fh(\014)1969 1308 y Ff(p)p 2030 1308 65 4 v 2030 1362 a Fh(N)7 b(X)2155 1371 y Fk(\033)p 1098 1436 1104 4 v 1412 1534 a Fe(E)1466 1548 y Fh(\033)1511 1529 y Fd(0)1546 1534 y Fi(e)1588 1507 y Fh(\014)1632 1453 y Ff(p)p 1692 1453 65 4 v 54 x Fh(N)g(X)1817 1524 y Fk(\033)1856 1511 y Fd(0)2238 1457 y Fp(=)2371 1395 y Fi(d)p 2345 1436 100 4 v 2345 1519 a(du)2471 1383 y Fc(\000)2512 1457 y Fe(E)26 b Fp(ln)20 b Fi(Z)2748 1419 y Fh(u)2741 1479 y(\014)s(;N)2876 1383 y Fc(\001)2918 1496 y Fh(u)p Fs(=0)3477 1457 y Fp(\(3)p Fi(:)p Fp(24\))50 1707 y(This)28 b(yields)h(the)i(generalization)f(of)g(Prop)s(osition)f (4.1)i(of)38 b([BK2])31 b(in)e(the)i(discrete)f(case:)50 1925 y Fb(Lemma)39 b(3.5:)53 b Fo(With)34 b(the)f(notation)j(intr)-5 b(o)g(duc)g(e)g(d)36 b(ab)-5 b(ove)34 b(we)f(have)h(for)g(any)g Fi(t)27 b Fg(2)f Fp(\(0)p Fi(;)15 b Fp(1])35 b Fo(and)f(any)g Fi(\017)27 b(>)f Fp(0)50 2070 y Fo(that)700 2270 y Fi(\014)756 2232 y Ff(\000)p Fs(2)854 2270 y Fi(N)937 2232 y Ff(\000)p Fs(1)1071 2208 y Fi(d)p 1045 2249 V 1045 2332 a(du)1170 2196 y Fc(\000)1212 2270 y Fe(E)g Fp(ln)20 b Fi(Z)1448 2232 y Fh(u)1441 2292 y(\014)s(;N)1576 2196 y Fc(\001)1618 2309 y Fh(u)p Fs(=0)1782 2270 y Fp(=)1942 2128 y Fh(t)1890 2146 y Fc(Z)1858 2402 y Fh(t)p Ff(\000)p Fh(\017)2008 2270 y Fi(dA)p Fp(\()p Fi(s)p Fp(\))p Fe(E)11 b Fi(\026)2346 2231 y Ff(\012)p Fs(2)2346 2297 y Fh(\014)t(;N)2487 2270 y Fp([)p Fi(d)2559 2284 y Fh(N)2629 2270 y Fp(\()p Fi(\033)n(;)k(\033) 2809 2232 y Ff(0)2835 2270 y Fp(\))26 b Fi(<)f(s)p Fp(])417 b(\(3)p Fi(:)p Fp(25\))159 2698 y(This)28 b(allo)m(ws)i(us)g(to)h (obtain)f(an)g(explicit)f(form)m(ula)g(for)h(the)h(distance)f (distribution)d(function.)50 2916 y Fb(Theorem)37 b(3.6:)91 b Fo(Under)33 b(the)g(assumptions)i(of)e(The)-5 b(or)g(em)34 b(3.1,)662 3165 y Fp(lim)638 3224 y Fh(N)7 b Ff("1)827 3165 y Fe(E)j Fi(\026)936 3126 y Ff(\012)p Fs(2)936 3192 y Fh(\014)t(;N)1077 3165 y Fp(\()p Fi(d)1159 3179 y Fh(N)1229 3165 y Fp(\()p Fi(\033)n(;)15 b(\033)1409 3127 y Ff(0)1436 3165 y Fp(\))25 b Fi(<)g(x)p Fp(\))h(=)1801 3037 y Fc(\032)1899 3101 y Fi(\014)1955 3068 y Ff(\000)p Fs(1)2053 3023 y Fg(p)p 2129 3023 197 4 v 78 x Fp(2)15 b(ln)f(2)q Fi(=)2371 3023 y Fc(p)p 2462 3023 171 4 v 2463 3101 a Fp(\026)-46 b Fi(a)p Fp(\()p Fi(x)p Fp(\))q Fi(;)93 b Fp(if)14 b Fi(x)25 b Fg(\024)g Fi(x)3044 3115 y Fh(\014)1899 3237 y Fp(1)p Fi(;)782 b Fo(if)15 b Fi(x)25 b(>)g(x)3047 3251 y Fh(\014)3477 3165 y Fp(\(3)p Fi(:)p Fp(26\))84 3425 y Fa(Pro)s(of:)57 b Fp(Observ)m(e)35 b(that)g Fi(F)1049 3392 y Fh(u)1036 3451 y(\014)s(;N)1203 3425 y Fg(\021)d Fi(N)1389 3392 y Ff(\000)p Fs(1)1487 3425 y Fe(E)26 b Fp(ln)20 b Fi(Z)1723 3392 y Fh(u)1716 3451 y(\014)s(;N)1885 3425 y Fp(is)34 b(a)h(con)m(v)m(ex)h(function)d(of)i Fi(u)p Fp(.)53 b(A)34 b(trivial)f(extension)50 3570 y(of)c(Theorem)f (3.1)i(sho)m(ws)e(that)h Fi(F)1204 3537 y Fh(u)1191 3596 y(\014)s(;N)1355 3570 y Fp(con)m(v)m(erges)i(to)e(the)g(function)e Fi(F)2450 3537 y Fh(u)2437 3596 y(\014)2526 3570 y Fp(giv)m(en)i(b)m(y) f(the)h(expression)e(\(3.12\))50 3716 y(when)36 b Fi(A)i Fp(is)f(replaced)g(b)m(y)g Fi(A)1066 3683 y Fh(u)1114 3716 y Fp(.)62 b(By)38 b(con)m(v)m(exit)m(y)-8 b(,)42 b(this)36 b(implies)f(that)j(lim)2613 3730 y Fh(N)7 b Ff("1)2839 3680 y Fh(d)p 2817 3695 81 4 v 2817 3747 a(du)2909 3716 y Fi(F)2967 3730 y Fh(\014)s(;N)3140 3716 y Fp(=)3280 3680 y Fh(d)p 3259 3695 V 3259 3747 a(du)3351 3716 y Fi(F)3409 3730 y Fh(\014)3496 3716 y Fp(at)38 b(all)50 3861 y(p)s(oin)m(ts)31 b Fi(u)i Fp(where)f Fi(F)746 3828 y Fh(u)733 3887 y(\014)827 3861 y Fp(is)f(di\013eren)m(tiable.)46 b(Th)m(us)32 b(w)m(e)h(only)f(ha)m(v)m(e)i(to)g(compute)f(this)e(deriv) -5 b(ativ)m(e.)48 b(W)-8 b(e)33 b(can)50 4007 y(write)675 4144 y Fi(F)746 4107 y Fh(u)733 4167 y(\014)814 4144 y Fg(\000)20 b Fi(F)963 4158 y Fh(\014)1037 4144 y Fp(=)1133 4062 y Fg(p)p 1209 4062 197 4 v 82 x Fp(2)15 b(ln)f(2)q Fi(\014)1477 4021 y Fc(Z)1568 4043 y Fh(x)1609 4053 y Fk(\014)1528 4227 y Fs(0)1672 4144 y Fi(dx)1786 4043 y Fc(\020)1841 4062 y(p)p 1932 4062 219 4 v 1933 4144 a Fp(\026)-46 b Fi(a)1980 4118 y Fh(u)2027 4144 y Fp(\()p Fi(x)p Fp(\))21 b Fg(\000)2261 4062 y Fc(p)p 2352 4062 171 4 v 2353 4144 a Fp(\026)-46 b Fi(a)p Fp(\()p Fi(x)p Fp(\))2523 4043 y Fc(\021)1032 4412 y Fp(+)1123 4330 y Fg(p)p 1199 4330 197 4 v 82 x Fp(2)15 b(ln)f(2)q Fi(\014)1467 4288 y Fc(Z)1558 4311 y Fh(x)1599 4283 y Fk(u)1599 4329 y(\014)1518 4495 y Fh(x)1559 4505 y Fk(\014)1662 4412 y Fi(dx)1761 4330 y Fc(p)p 1852 4330 219 4 v 1853 4412 a Fp(\026)-46 b Fi(a)1900 4386 y Fh(u)1948 4412 y Fp(\()p Fi(x)p Fp(\))21 b(+)2193 4350 y Fi(\014)2249 4317 y Fs(2)p 2193 4391 98 4 v 2218 4474 a Fp(2)2316 4338 y Fc(\000)2381 4389 y Fp(\026)2357 4412 y Fi(A)2425 4374 y Fh(u)2473 4412 y Fp(\()p Fi(x)2560 4426 y Fh(\014)2609 4412 y Fp(\))g Fg(\000)2780 4389 y Fp(\026)2756 4412 y Fi(A)2824 4374 y Fh(u)2871 4412 y Fp(\()p Fi(x)2958 4374 y Fh(u)2958 4434 y(\014)3008 4412 y Fp(\))3043 4338 y Fc(\001)1032 4685 y Fp(+)1134 4624 y Fi(\014)1190 4591 y Fs(2)p 1134 4664 V 1160 4747 a Fp(2)1257 4611 y Fc(\000)1299 4685 y Fp(\()1358 4662 y(\026)1334 4685 y Fi(A)1402 4648 y Fh(u)1450 4685 y Fp(\(1\))g Fg(\000)1701 4662 y Fp(\026)1677 4685 y Fi(A)1745 4648 y Fh(u)1793 4685 y Fp(\()p Fi(x)1880 4699 y Fh(\014)1929 4685 y Fp(\)\))g Fg(\000)f Fp(\()2170 4662 y(\026)2146 4685 y Fi(A)q Fp(\(1\))h Fg(\000)2466 4662 y Fp(\026)2442 4685 y Fi(A)p Fp(\()p Fi(x)2597 4699 y Fh(\014)2646 4685 y Fp(\)\))2716 4611 y Fc(\001)2774 4685 y Fi(:)3477 4404 y Fp(\(3)p Fi(:)p Fp(27\))159 4949 y(If)30 b Fi(x)302 4963 y Fh(\014)376 4949 y Fi(<)25 b(t)p Fp(,)30 b(then)g Fi(x)819 4963 y Fh(\014)894 4949 y Fi(<)25 b(t)20 b Fg(\000)g Fi(\017)30 b Fp(for)g Fi(\017)25 b(>)g Fp(0)31 b(small)e(enough)h(and)g(\(3.27\))i(simpli\014es)27 b(to)646 5202 y Fi(F)717 5164 y Fh(u)704 5224 y(\014)784 5202 y Fg(\000)20 b Fi(F)933 5216 y Fh(\014)1008 5202 y Fp(=)1115 5140 y Fi(\014)1171 5107 y Fs(2)p 1115 5181 V 1141 5264 a Fp(2)1238 5128 y Fc(\000)1280 5202 y Fp(\()1339 5179 y(\026)1315 5202 y Fi(A)1383 5164 y Fh(u)1431 5202 y Fp(\(1\))h Fg(\000)1682 5179 y Fp(\026)1658 5202 y Fi(A)1726 5164 y Fh(u)1774 5202 y Fp(\()p Fi(x)1861 5216 y Fh(\014)1910 5202 y Fp(\)\))g Fg(\000)f Fp(\()2151 5179 y(\026)2127 5202 y Fi(A)p Fp(\(1\))i Fg(\000)2447 5179 y Fp(\026)2423 5202 y Fi(A)p Fp(\()p Fi(x)2578 5216 y Fh(\014)2627 5202 y Fp(\)\))2697 5128 y Fc(\001)1008 5436 y Fp(=)1115 5374 y Fi(\014)1171 5341 y Fs(2)p 1115 5415 V 1141 5498 a Fp(2)1238 5362 y Fc(\000)1304 5413 y Fp(\026)1280 5436 y Fi(A)1348 5398 y Fh(u)1395 5436 y Fp(\(1\))g Fg(\000)1647 5413 y Fp(\026)1623 5436 y Fi(A)p Fp(\(1\))1806 5362 y Fc(\001)1874 5436 y Fp(=)j(\(2)p Fi(u)c Fp(+)f Fi(u)2266 5398 y Fs(2)2307 5436 y Fp(\))2353 5374 y Fi(\014)2409 5341 y Fs(2)p 2353 5415 V 2379 5498 a Fp(2)2461 5436 y([)p Fi(A)p Fp(\()p Fi(t)p Fp(\))h Fg(\000)f Fi(A)p Fp(\()p Fi(t)h Fg(\000)e Fi(\017)p Fp(\)])3477 5305 y(\(3)p Fi(:)p Fp(28\))p eop %%Page: 11 11 11 10 bop 50 295 a Fj(GREM,)28 b(p)l(art)f(3)3089 b Fm(11)50 513 y Fp(Hence)31 b(in)e(this)g(case)1242 597 y Fi(d)p 1216 637 100 4 v 1216 721 a(du)1342 585 y Fc(\000)1383 658 y Fi(F)1454 621 y Fh(u)1441 681 y(\014)1502 585 y Fc(\001)1544 698 y Fh(u)p Fs(=0)1708 658 y Fp(=)c Fi(\014)1860 621 y Fs(2)1901 658 y Fp([)p Fi(A)p Fp(\()p Fi(t)p Fp(\))c Fg(\000)f Fi(A)p Fp(\()p Fi(t)h Fg(\000)f Fi(\017)p Fp(\)])923 b(\(3)p Fi(:)p Fp(29\))50 846 y(Inserting)29 b(this)g(in)m(to)i (\(3.25\))h(and)e(letting)g Fi(\017)g Fp(tend)g(to)h(zero,)h(w)m(e)e (obtain)g(that)h(for)f Fi(t)25 b(>)g(x)3082 860 y Fh(\014)3131 846 y Fp(,)50 991 y(lim)176 1005 y Fh(N)7 b Ff("1)370 991 y Fe(E)k Fi(\026)480 952 y Ff(\012)p Fs(2)480 1018 y Fh(N)s(;\014)616 991 y Fp([)p Fi(d)688 1005 y Fh(N)758 991 y Fp(\()p Fi(\033)n(;)k(\033)938 958 y Ff(0)965 991 y Fp(\))25 b Fi(<)g(t)p Fp(])g(=)g(1.)159 1209 y(If)i Fi(x)299 1223 y Fh(\014)374 1209 y Fi(>)e(t)p Fp(,)j(w)m(e)h(m)m(ust)e (distinguish)e(t)m(w)m(o)k(cases:)40 b(\(a\))30 b Fi(A)e 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2314 y Fs(1)338 2300 y Fp(\()p Fi(u)p Fp(\))g(and)e(to)i Fi(A)876 2267 y Fh(u)923 2300 y Fp(\()p Fi(x)p Fp(\))g(at)g Fi(x)g Fp(=)g Fi(z)1454 2314 y Fs(2)1495 2300 y Fp(\()p Fi(u)p Fp(\).)70 b(In)39 b(other)i(w)m(ords)e Fi(a)p Fp(\()p Fi(z)2478 2314 y Fs(1)2520 2300 y Fp(\()p Fi(u)p Fp(\)\))j(=)f Fi(a)p Fp(\()p Fi(z)2956 2314 y Fs(2)2997 2300 y Fp(\()p Fi(u)p Fp(\)\)\(1)29 b(+)d(2)p Fi(u)h Fp(+)g Fi(u)3634 2267 y Fs(2)3674 2300 y Fp(\))50 2446 y(and)32 b Fi(A)p Fp(\()p Fi(z)374 2460 y Fs(2)416 2446 y Fp(\()p Fi(u)p Fp(\)\))23 b(+)f(\(2)p Fi(u)h Fp(+)e Fi(u)988 2413 y Fs(2)1029 2446 y Fp(\)\()p Fi(A)p Fp(\()p Fi(z)1244 2460 y Fs(2)1286 2446 y Fp(\()p Fi(u)p Fp(\)\))i Fg(\000)f Fi(A)p Fp(\()p Fi(t)h Fg(\000)e Fi(\017)p Fp(\)\))31 b(=)e Fi(A)p Fp(\()p Fi(z)2193 2460 y Fs(1)2235 2446 y Fp(\()p Fi(u)p Fp(\)\))23 b(+)e(\()p Fi(z)2584 2460 y Fs(2)2626 2446 y Fp(\()p Fi(u)p Fp(\))i Fg(\000)e Fi(z)2905 2460 y Fs(1)2946 2446 y Fp(\()p Fi(u)p Fp(\)\))p Fi(a)p Fp(\()p Fi(z)3228 2460 y Fs(1)3271 2446 y Fp(\()p Fi(u)p Fp(\)\).)50 b(Then)74 2568 y(\026)50 2591 y Fi(A)118 2558 y Fh(u)166 2591 y Fp(\()p Fi(x)p Fp(\))30 b(coincides)f(with)f (this)h(straigh)m(t)g(line)g(for)g Fi(x)c Fg(2)g Fp([)p Fi(z)1960 2605 y Fs(1)2001 2591 y Fp(\()p Fi(u)p Fp(\))p Fi(;)15 b(z)2205 2605 y Fs(2)2248 2591 y Fp(\()p Fi(u)p Fp(\)],)2475 2568 y(\026)2451 2591 y Fi(A)2519 2558 y Fh(u)2566 2591 y Fp(\()p Fi(x)p Fp(\))26 b(=)2834 2568 y(\026)2810 2591 y Fi(A)p Fp(\()p Fi(x)p Fp(\))31 b(for)e Fi(x)c Fg(2)g Fp([0)p Fi(;)15 b(z)3484 2605 y Fs(1)3526 2591 y Fp(\()p Fi(u)p Fp(\)\),)74 2713 y(\026)50 2736 y Fi(A)118 2703 y Fh(u)166 2736 y Fp(\()p Fi(x)p Fp(\))32 b(=)f Fi(A)490 2703 y Fh(u)538 2736 y Fp(\()p Fi(x)p Fp(\))j(for)g Fi(x)e Fg(2)f Fp(\()p Fi(z)1090 2750 y Fs(2)1131 2736 y Fp(\()p Fi(u)p Fp(\))p Fi(;)15 b(t)p Fp(],)1437 2713 y(\026)1413 2736 y Fi(A)1481 2703 y Fh(u)1529 2736 y Fp(\()p Fi(x)p Fp(\))32 b(=)1809 2713 y(\026)1785 2736 y Fi(A)p Fp(\()p Fi(x)p Fp(\))24 b(+)e(\(2)p Fi(u)i Fp(+)e Fi(u)2393 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3813 y Fc(\020)1339 3832 y(p)p 1430 3832 700 4 v 82 x Fi(a)p Fp(\()p Fi(x)p Fp(\)\(1)21 b(+)f(2)p Fi(u)h Fp(+)f Fi(u)2053 3888 y Fs(2)2094 3914 y Fp(\))g Fg(\000)2240 3832 y Fc(p)p 2331 3832 171 4 v 82 x Fi(a)p Fp(\()p Fi(x)p Fp(\))2502 3813 y Fc(\021)3477 3512 y Fp(\(3)p Fi(:)p Fp(30\))50 4142 y(Note)29 b(that)f(the)g(straigh)m(t)g (line)d(tangen)m(t)30 b(to)e Fi(A)p Fp(\()p Fi(x)p Fp(\))g(at)h(the)e (p)s(oin)m(t)g Fi(x)e Fp(=)g Fi(z)t Fp(\()p Fi(u)p Fp(\))h Fi(<)f(t)15 b Fg(\000)g Fi(\017)27 b Fp(suc)m(h)g(that)h Fi(a)p Fp(\()p Fi(z)t Fp(\()p Fi(u)p Fp(\)\))f(=)50 4287 y Fi(a)p Fp(\()p Fi(t)22 b Fg(\000)g Fi(\017)p Fp(\)\(1)i(+)e(2)p Fi(u)g Fp(+)g Fi(u)814 4254 y Fs(2)855 4287 y Fp(\))34 b(do)s(es)f(not)h(cross)f Fi(A)1591 4254 y Fh(u)1639 4287 y Fp(\()p Fi(x)p Fp(\))h(for)f Fi(x)e(>)f(z)t Fp(\()p Fi(u)p Fp(\).)51 b(Then)32 b Fi(z)t Fp(\()p Fi(u)p Fp(\))f Fi(<)g(z)2948 4301 y Fs(1)2989 4287 y Fp(\()p Fi(u)p Fp(\))g Fi(<)f(t)22 b Fg(\000)g Fi(\017)p Fp(.)50 b(Since)50 4433 y Fi(z)t 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v 82 x Fi(a)p Fp(\()p Fi(t)g Fg(\000)g Fi(\017)p Fp(\))26 b(=)f Fi(O)s Fp(\()p Fi(u)p Fp(\))3477 5374 y(\(3)p Fi(:)p Fp(32\))p eop %%Page: 12 12 12 11 bop 50 295 a Fm(12)3270 b Fj(Se)l(ction)27 b Fm(3)50 513 y Fp(Since)21 b Fi(z)321 527 y Fs(2)362 513 y Fp(\()p Fi(u)p Fp(\))t Fg(\000)t Fi(z)605 527 y Fs(1)647 513 y Fp(\()p Fi(u)p Fp(\))26 b(=)f Fi(O)s Fp(\()p Fi(u)p Fp(\),)g(b)s(oth)d(of)g(these)h(terms)g(are)g(of)f(the)h(order)f Fi(O)s Fp(\()p Fi(u)2685 480 y Fs(2)2726 513 y Fp(\))g(and)g(do)s(es)g (not)h(con)m(tribute)50 658 y(to)31 b(the)f(deriv)-5 b(ativ)m(e)30 b(of)h Fi(F)909 625 y Fh(u)896 684 y(\014)957 658 y Fp(.)40 b(The)30 b(in)m(tegral)g(of)h(the)f(\014rst)g(term)g(can) h(b)s(e)f(written)f(as)558 955 y(\()593 866 y Fc(p)p 684 866 458 4 v 89 x Fp(1)20 b(+)g(2)p Fi(u)h Fp(+)f Fi(u)1101 929 y Fs(2)1162 955 y Fg(\000)g Fp(1\))1348 831 y Fc(Z)1440 854 y Fh(t)1399 1038 y(t)p Ff(\000)p Fh(\017)1534 873 y Fc(p)p 1625 873 171 4 v 82 x Fi(a)p Fp(\()p Fi(x)p Fp(\))p Fi(dx)26 b Fp(=)f Fi(u)p Fp(\(1)c(+)f Fi(o)p Fp(\(1\)\))2469 831 y Fc(Z)2561 854 y Fh(t)2521 1038 y(t)p Ff(\000)p Fh(\017)2774 893 y Fp(1)p 2666 934 262 4 v 2666 952 a Fc(p)p 2757 952 171 4 v 2758 1030 a Fp(\026)-46 b Fi(a)p Fp(\()p Fi(x)p Fp(\))2939 955 y Fi(dA)p Fp(\()p Fi(x)p Fp(\))p Fi(:)276 b Fp(\(3)p Fi(:)p Fp(33\))50 1244 y(Therefore,)30 b(b)m(y)g(\(3.25\),)j(w)m(e)e (get)h(that)388 1443 y Fg(p)p 464 1443 197 4 v 82 x Fp(2)15 b(ln)g(2)p Fi(\014)717 1488 y Ff(\000)p Fs(1)830 1402 y Fc(Z)921 1424 y Fh(t)881 1608 y(t)p Ff(\000)p Fh(\017)1015 1525 y Fi(dA)p Fp(\()p Fi(x)p Fp(\))1372 1464 y(1)p 1263 1504 262 4 v 1263 1523 a Fc(p)p 1355 1523 171 4 v 1356 1600 a Fp(\026)-46 b Fi(a)p Fp(\()p Fi(x)p Fp(\))1562 1525 y(=)1658 1402 y Fc(Z)1749 1424 y Fh(t)1708 1608 y(t)p Ff(\000)p Fh(\017)1843 1525 y Fi(dA)p Fp(\()p Fi(x)p Fp(\))40 b(lim)2095 1585 y Fh(N)7 b Ff("1)2285 1525 y Fe(E)k Fi(\026)2395 1486 y Ff(\012)p Fs(2)2395 1553 y Fh(N)s(;\014)2532 1525 y Fp([)p Fi(d)2604 1539 y Fh(N)2673 1525 y Fp(\()p Fi(\033)n(;)k(\033)2853 1488 y Ff(0)2880 1525 y Fp(\))25 b Fi(<)g(x)p Fp(])p Fi(:)339 b Fp(\(3)p Fi(:)p Fp(34\))50 1814 y(Since)29 b(this)g(is)h(true)g(for)g(an)m(y)h Fi(\017)25 b(>)g Fp(0,)31 b(\(3.26\))i(follo)m(ws.)50 2033 y(\(b\))i(W)-8 b(e)36 b(no)m(w)f(consider)f(the)h(case)h(when)1543 2010 y(\026)1519 2033 y Fi(A)f Fp(is)f(linear)g(to)h(the)g(left)g(of)g Fi(t)p Fp(.)54 b(The)34 b(deformation)h(of)3472 2010 y(\026)3448 2033 y Fi(A)g Fp(no)m(w)50 2178 y(extends)e(further)e(do)m (wn)i(to)g(the)g(b)s(eginning)e(of)i(the)g(linear)e(piece)i(of)2507 2155 y(\026)2483 2178 y Fi(A)p Fp(.)49 b(Assume)32 b(that)3188 2155 y(\026)3164 2178 y Fi(A)h Fp(is)f(linear)f(on)50 2323 y(the)f(in)m(terv)-5 b(al)29 b(\()p Fi(y)s(;)15 b(z)t Fp(])27 b Fg(\033)d Fp(\()p Fi(t)c Fg(\000)f Fi(\017;)c(t)p Fp(].)41 b(Then,)29 b(for)h(small)f(enough)g Fi(u)p Fp(,)i(the)f(slop)s (e)f(of)2801 2300 y(\026)2777 2323 y Fi(A)2845 2290 y Fh(u)2923 2323 y Fp(will)e(di\013er)i(from)g(that)50 2469 y(of)177 2446 y(\026)153 2469 y Fi(A)i Fp(only)e(in)g(\()p Fi(y)s(;)15 b(z)t Fp(].)42 b(Moreo)m(v)m(er,)1266 2446 y(\026)1242 2469 y Fi(A)1310 2436 y Fh(u)1358 2469 y Fp(\()p Fi(y)s Fp(\))26 b(=)1622 2446 y(\026)1598 2469 y Fi(A)p Fp(\()p Fi(y)s Fp(\))31 b(and)2015 2446 y(\026)1991 2469 y Fi(A)2059 2436 y Fh(u)2107 2469 y Fp(\()p Fi(z)t Fp(\))26 b(=)2369 2446 y(\026)2345 2469 y Fi(A)p Fp(\()p Fi(z)t Fp(\))21 b(+)f(\(2)p Fi(u)h Fp(+)e Fi(u)2936 2436 y Fs(2)2977 2469 y Fp(\)\()p Fi(A)p Fp(\()p Fi(t)p Fp(\))i Fg(\000)f Fi(A)p Fp(\()p Fi(t)g Fg(\000)g Fi(\017)p Fp(\)\).)50 2614 y(Let)758 2760 y Fi(z)804 2722 y Ff(\003)871 2760 y Fg(\021)75 b Fp(sup)967 2840 y Fh(x)p Ff(2)p Fs(\()p Fh(y)r(;z)s Fs(])1204 2760 y Fg(f)1274 2737 y Fp(\026)1249 2760 y Fi(A)1317 2722 y Fh(u)1365 2760 y Fp(\()p Fi(x)p Fp(\))21 b Fg(\000)1623 2737 y Fp(\026)1599 2760 y Fi(A)p Fp(\()p Fi(x)p Fp(\))26 b Fi(<)f Fp(\(2)p Fi(u)c Fp(+)f Fi(u)2207 2722 y Fs(2)2248 2760 y Fp(\)\()p Fi(A)p Fp(\()p Fi(t)p Fp(\))i Fg(\000)d Fi(A)p Fp(\()p Fi(t)i Fg(\000)f Fi(\017)p Fp(\)\))p Fg(g)476 b Fp(\(3)p Fi(:)p Fp(35\))50 3006 y(Ob)m(viously)-8 b(,)29 b Fi(x)547 3020 y Fh(\014)621 3006 y Fg(62)c Fp(\()p Fi(y)s(;)15 b(z)t Fp(\).)42 b(If)30 b Fi(t)25 b Fg(\024)g Fi(x)1275 3020 y Fh(\014)1324 3006 y Fp(,)30 b(w)m(e)h(get)h(that)122 3300 y Fi(F)193 3263 y Fh(u)180 3323 y(\014)261 3300 y Fg(\000)20 b Fi(F)410 3314 y Fh(\014)484 3300 y Fp(=)580 3218 y Fg(p)p 656 3218 197 4 v 82 x Fp(2)15 b(ln)g(2)p Fi(\014)924 3177 y Fc(Z)1015 3199 y Fh(z)1053 3172 y Fd(\003)975 3383 y Fh(y)1110 3300 y Fi(dx)1224 3199 y Fc(\020)1279 3218 y(p)p 1370 3218 1685 4 v 1371 3300 a Fp(\026)-46 b Fi(a)p Fp(\()p Fi(z)t Fp(\))21 b(+)f(\(2)p Fi(u)h Fp(+)f Fi(u)1942 3274 y Fs(2)1983 3300 y Fp(\)\()p Fi(A)p Fp(\()p Fi(t)p Fp(\))i Fg(\000)d Fi(A)p Fp(\()p Fi(t)i Fg(\000)f Fi(\017)p Fp(\)\))p Fi(=)p Fp(\()p Fi(z)2817 3274 y Ff(\003)2881 3300 y Fg(\000)g Fi(y)s Fp(\))g Fg(\000)3166 3218 y Fc(p)p 3257 3218 166 4 v 3258 3300 a Fp(\026)-46 b Fi(a)p Fp(\()p Fi(z)t Fp(\))3422 3199 y Fc(\021)484 3582 y Fp(=)580 3500 y Fg(p)p 656 3500 197 4 v 82 x Fp(2)15 b(ln)g(2)p Fi(\014)909 3500 y Fc(p)p 1000 3500 166 4 v 1001 3582 a Fp(\026)-46 b Fi(a)p Fp(\()p Fi(z)t Fp(\))1180 3458 y Fc(Z)1271 3481 y Fh(z)1309 3453 y Fd(\003)1231 3665 y Fh(y)1366 3582 y Fi(dx)1480 3481 y Fc(\020)1535 3500 y(p)p 1626 3500 1801 4 v 82 x Fp(1)21 b(+)f(\(2)p Fi(u)h Fp(+)e Fi(u)2078 3556 y Fs(2)2119 3582 y Fp(\)\()p Fi(A)p Fp(\()p Fi(t)p Fp(\))j Fg(\000)e Fi(A)p Fp(\()p Fi(t)g Fg(\000)g Fi(\017)p Fp(\)\))p Fi(=)p Fp(\()q(\026)-46 b Fi(a)r Fp(\()p Fi(z)t Fp(\)\()p Fi(z)3154 3556 y Ff(\003)3217 3582 y Fg(\000)20 b Fi(y)s Fp(\)\))h Fg(\000)f Fp(1)3583 3481 y Fc(\021)484 3803 y Fp(=)580 3721 y Fg(p)p 656 3721 197 4 v 82 x Fp(2)15 b(ln)g(2)p Fi(\014)5 b Fp(\()q(\026)-46 b Fi(a)q Fp(\()p Fi(z)t Fp(\)\))1144 3765 y Ff(\000)p Fs(1)p Fh(=)p Fs(2)1316 3803 y Fi(u)p Fp(\()p Fi(A)p Fp(\()p Fi(t)p Fp(\))21 b Fg(\000)f Fi(A)p Fp(\()p Fi(t)h Fg(\000)f Fi(\017)p Fp(\)\))h(+)f Fi(O)s Fp(\()p Fi(u)2312 3765 y Fs(2)2353 3803 y Fp(\))3477 3912 y(\(3)p Fi(:)p Fp(36\))50 4057 y(Again)30 b(\(3.26\))j(follo)m(ws)c(no)m(w)h(from)g (\(3.25\).)43 b Fg(})50 4275 y Fq(Remark:)g Fp(It)26 b(is)f(clear)g(from)h(the)g(ab)s(o)m(v)m(e)g(consideration)f(that)h(w)m (e)h(ma)m(y)f(rep)s(eat)g(the)g(same)g(computation)50 4421 y(with)32 b Fi(u)e(<)g Fp(0)k(to)g(compute)g(the)f(left-deriv)-5 b(ativ)m(e)33 b(of)h Fi(F)1924 4388 y Fh(u)1911 4446 y(\014)2004 4421 y Fp(at)h(zero.)50 b(The)33 b(results)f(coincide,)h (except)i(when)50 4566 y Fi(t)28 b Fp(=)h Fi(x)263 4580 y Fh(\014)312 4566 y Fp(.)46 b(Similarly)-8 b(,)30 b(one)j(sho)m(ws)f (that)h(the)f(second)h(deriv)-5 b(ativ)m(e)32 b(of)g Fi(F)2470 4533 y Fh(u)2457 4592 y(\014)2550 4566 y Fp(is)f(\014nite)h (in)f(a)i(neigh)m(b)s(orho)s(o)s(d)d(of)50 4712 y(zero)h(whenev)m(er)f Fi(t)25 b Fg(6)p Fp(=)g Fi(x)847 4726 y Fh(\014)896 4712 y Fp(.)p eop %%Page: 13 13 13 12 bop 50 295 a Fj(GREM,)28 b(p)l(art)f(3)3089 b Fm(13)50 513 y Ft(4.)55 b(Ruelle's)41 b(pro)s(cesses)i(and)e(Ghirlanda-Guerra)f (iden)m(tities.)159 731 y Fp(W)-8 b(e)31 b(need)g(to)g(b)s(egin)e(with) g(some)i(more)f(notation.)41 b(Let)31 b(us)f(denote)g(b)m(y)h(\006)2710 745 y Fh(\033)2758 731 y Fp(\()p Fi(t)p Fp(\))g(the)g(sets)1238 976 y(\006)1304 990 y Fh(\033)1353 976 y Fp(\()p Fi(t)p Fp(\))25 b Fg(\021)g(f)q Fi(\033)1678 938 y Ff(0)1729 976 y Fg(2)g(S)1870 990 y Fh(N)1939 976 y Fg(j)p Fi(d)2011 990 y Fh(N)2080 976 y Fp(\()p Fi(\033)n(;)15 b(\033)2260 938 y Ff(0)2287 976 y Fp(\))25 b Fi(>)g(t)p Fg(g)1002 b Fp(\(4)p Fi(:)p Fp(1\))50 1220 y(Note)35 b(that)f(b)m(y)f (ultrametricit)m(y)-8 b(,)34 b(w)m(e)g(ha)m(v)m(e)g(that)g(for)f(all)g Fi(\033)2081 1187 y Ff(0)2137 1220 y Fg(2)d Fp(\006)2294 1234 y Fh(\033)2342 1220 y Fp(\()p Fi(t)p Fp(\),)35 b(\006)2571 1234 y Fh(\033)2615 1216 y Fd(0)2644 1220 y Fp(\()p Fi(t)p Fp(\))c(=)f(\006)2945 1234 y Fh(\033)2994 1220 y Fp(\()p Fi(t)p Fp(\).)50 b(The)33 b(set)h(of)g(all)50 1366 y(distinct)29 b(sets)h(\006)622 1380 y Fh(\033)671 1366 y Fp(\()p Fi(t)p Fp(\),)h Fi(\033)e Fg(2)24 b(S)1051 1380 y Fh(N)1151 1366 y Fp(is)29 b(isomorphic)f(to)k Fg(f\000)p Fp(1)p Fi(;)15 b Fp(1)p Fg(g)2105 1333 y Fs([)p Fh(tN)7 b Fs(])2269 1366 y Fg(\021)25 b(S)2427 1333 y Fh(t)2420 1391 y(N)2489 1366 y Fp(.)159 1584 y(W)-8 b(e)23 b(will)c(no)m(w)j(reconsider)e(the)i (p)s(oin)m(t)f(pro)s(cess)1768 1561 y Fc(c)1758 1584 y Fg(W)1848 1598 y Fh(\014)s(;N)2004 1584 y Fp(in)m(tro)s(duced)f(in)h ([BK2])i(in)d(a)i(somewhat)g(di\013eren)m(t)50 1729 y(w)m(a)m(y)-8 b(.)49 b(Note)34 b(that)f(the)g(atoms)g(of)g(the)g(pro)s(cess)1714 1706 y Fc(c)1703 1729 y Fg(W)1793 1743 y Fh(\014)s(;N)1928 1729 y Fp(,)1987 1656 y Fc(\000)2028 1729 y Fi(\026)2083 1743 y Fh(\014)s(;N)2218 1729 y Fp(\(\006)2319 1743 y Fh(\033)2368 1729 y Fp(\()p Fi(q)2444 1743 y Fs(1)2485 1729 y Fp(\)\))p Fi(;)15 b(:)g(:)g(:)i(;)e(\026)2812 1743 y Fh(\014)s(;N)2947 1729 y Fp(\(\006)3048 1743 y Fh(\033)3097 1729 y Fp(\()p Fi(q)3173 1746 y Fh(l)p Fs(\()p Fh(\014)s Fs(\))3302 1729 y Fp(\)\))3372 1656 y Fc(\001)3447 1729 y Fp(can)33 b(b)s(e)50 1875 y(seen)20 b(as)h(the)f(incremen)m(ts)g (of)h(the)f(mass)g(distribution)d(around)i Fi(\033)24 b Fp(measured)19 b(in)g(the)i(hierarc)m(hical)e(distance,)50 2020 y(i.e.)40 b(if)30 b(w)m(e)h(de\014ne)e(the)i(distribution)26 b(functions)1230 2265 y Fi(m)1310 2279 y Fh(\033)1358 2265 y Fp(\()p Fi(x)p Fp(\))g Fg(\021)f Fi(\026)1657 2279 y Fh(\014)s(;N)1792 2265 y Fp(\()p Fi(\033)k Fp(:)c Fi(d)2005 2279 y Fh(N)2075 2265 y Fp(\()p Fi(\033)2165 2227 y Ff(0)2191 2265 y Fi(;)15 b(\033)s Fp(\))26 b Fi(>)f(x)p Fp(\))993 b(\(4)p Fi(:)p Fp(2\))50 2509 y(w)m(e)21 b(can)g(think)f(of)h (the)g(pro)s(cess)f(as)1223 2487 y Fc(c)1212 2509 y Fg(W)1302 2523 y Fh(\014)s(;N)1458 2509 y Fp(as)h(a)g(p)s(oin)m(t)f(pro)s(cess)h (on)g(the)g(space)g(of)g(probabilit)m(y)e(distributions)50 2655 y(on)30 b([0)p Fi(;)15 b Fp(1],)1496 2777 y Fc(c)1485 2800 y Fg(W)1575 2814 y Fh(\014)s(;N)1736 2800 y Fp(=)1864 2714 y Fc(X)1831 2909 y Fh(\033)r Ff(2S)1969 2918 y Fk(N)2043 2800 y Fi(\016)2083 2817 y Fh(m)2148 2826 y Fk(\033)2191 2817 y Fs(\()p Ff(\001)p Fs(\))3523 2800 y Fp(\(4)p Fi(:)p Fp(3\))50 3059 y(Of)32 b(course,)i(in)e(the)h(case)h(of)g(\014nitely)d (man)m(y)i(hierarc)m(hies,)g(all)e(measures)i(are)h(supp)s(orted)d(on)i (the)g(set)h(of)50 3204 y(atomic)k(distributions)33 b(with)j(atoms)i (at)g(\014xed)e(p)s(oin)m(ts.)60 b(As)37 b(a)h(matter)g(of)f(fact,)j (this)c(de\014nition)f(turns)50 3350 y(out)f(not)g(to)g(b)s(e)g(to)s(o) g(con)m(v)m(enien)m(t.)52 b(Already)33 b(in)g(the)h(case)h(of)f(the)g (discrete)f(GREM,)i(w)m(e)f(ha)m(v)m(e)h(follo)m(w)m(ed)50 3495 y(Ruelle's)f(prescription)e(only)j(b)m(y)f(con)m(v)m(en)m(tion.)56 b(In)35 b(the)g(con)m(tin)m(uous)g(case,)i(it)e(will)d(b)s(ecome)k (essen)m(tially)50 3641 y(imp)s(erativ)m(e)29 b(to)i(replace)932 3618 y Fc(c)922 3641 y Fg(W)1012 3655 y Fh(\014)s(;N)1177 3641 y Fp(b)m(y)f(the)g(pro)s(cess)1338 3885 y Fg(K)1407 3899 y Fh(\014)s(;N)1567 3885 y Fg(\021)1696 3799 y Fc(X)1663 3994 y Fh(\033)r Ff(2S)1801 4003 y Fk(N)1875 3885 y Fi(\026)1930 3899 y Fh(\014)s(;N)2065 3885 y Fp(\()p Fi(\033)s Fp(\))p Fi(\016)2230 3902 y Fh(m)2295 3911 y Fk(\033)2339 3902 y Fs(\()p Ff(\001)p Fs(\))3523 3885 y Fp(\(4)p Fi(:)p Fp(4\))50 4183 y(The)42 b(adv)-5 b(an)m(tage)45 b(is)d(that)i(this)e (ob)5 b(ject)44 b(is)e(a)h(random)g Fo(pr)-5 b(ob)g(ability)45 b Fp(measure)e(on)g(the)g(space)h(of)f(mass)50 4329 y(distributions.)e (Let)33 b(us)e(men)m(tion)g(that)i(in)d(spirit)g(this)g(measure)i(is)f (rather)h(close)g(to)h(the)f(\\metastates")50 4474 y(in)m(tro)s(duced) 39 b(b)m(y)h(Aizenman)g(and)g(W)-8 b(ehr)41 b([A)-10 b(W])41 b(and)f(promoted)h(in)e(the)i(con)m(text)h(of)f(spin)d(glasses) j(b)m(y)50 4620 y(Newman)29 b(and)f(Stein)g([N,NS1,NS2].)41 b(It)30 b(has)e(a)i(v)m(ery)f(app)s(ealing)e(ph)m(ysical)h(in)m (terpretation:)39 b(it)28 b(tells,)h(for)50 4765 y(a)c(\014xed)f (realization)g(of)h(the)g(disorder,)g(with)e(whic)m(h)h(probabilit)m(y) e(an)j(observ)m(er)g(that)g(is)f(itself)g(distributed)50 4911 y(with)29 b(the)h(Gibbs)f(distribution,)e(will)h(see)j(a)g(giv)m (en)f(distribution)d(of)j(mass)g(around)g(himself.)159 5129 y(Let)k(us)g(sa)m(y)h(a)f(w)m(ord)g(more)g(on)g(the)g(in)m (terpretation)g(of)g(these)g(pro)s(cesses.)52 b(Note)35 b(that)g(the)f(distance)50 5274 y(distribution)27 b(function)1076 5420 y Fi(\026)1131 5381 y Ff(\012)p Fs(2)1131 5447 y Fh(\014)s(;N)1266 5420 y Fp(\()p Fi(d)1348 5434 y Fh(N)1417 5420 y Fp(\()p Fi(\033)n(;)15 b(\033)1597 5382 y Ff(0)1624 5420 y Fp(\))26 b Fi(>)f(q)s Fp(\))g(=)1981 5296 y Fc(Z)2087 5420 y Fg(K)2156 5434 y Fh(\014)s(;N)2291 5420 y Fp(\()p Fi(dm)p Fp(\))p Fi(m)p Fp(\()p Fi(q)s Fp(\))841 b(\(4)p Fi(:)p Fp(5\))p eop %%Page: 14 14 14 13 bop 50 295 a Fm(14)3270 b Fj(Se)l(ction)27 b Fm(4)50 513 y Fp(is)f(the)i(\014rst)e(momen)m(t)i(of)g Fg(K)994 527 y Fh(\014)s(;N)1129 513 y Fp(.)40 b(Th)m(us)26 b Fg(K)1490 527 y Fh(\014)s(;N)1652 513 y Fp(will)f(b)s(e)i (asymptotically)f(concen)m(trated)j(on)f(distributions)50 658 y(for)k(whic)m(h)e Fi(m)p Fp(\()p Fi(q)s Fp(\))f(=)f(0)k(if)f Fi(q)g(>)d(x)p Fp(\()p Fi(\014)5 b Fp(\).)46 b(In)31 b(other)i(w)m(ords,)f(the)g(smallest)f(blo)s(c)m(ks)h(\006)2852 672 y Fh(\033)2932 658 y Fp(around)f(an)m(y)i(p)s(oin)m(t)e Fi(\033)50 804 y Fp(that)f(ha)m(v)m(e)h(p)s(ositiv)m(e)e(mass)h(are)g (of)h(size)e(2)1485 771 y Fs(\(1)p Ff(\000)p Fh(x)p Fs(\()p Fh(\014)s Fs(\)\))p Fh(N)1847 804 y Fp(.)41 b(Since)28 b(the)j(mass)e(distribution)e(around)h(an)m(y)j(p)s(oin)m(t)50 949 y(within)d(suc)m(h)i(a)h(\\massiv)m(e")g(blo)s(c)m(k)f(is)g(iden)m (tical,)f(suc)m(h)h(a)h(blo)s(c)m(k)f(con)m(tributes)g(with)g(mass)g Fi(\026)3242 963 y Fh(\014)s(;N)3377 949 y Fp(\()p Fi(\033)s Fp(\))h(with)50 1095 y(a)g(Dirac)f(measure)g(on)h(the)f(empirical)e (mass)j(distribution)26 b(around)k(itself.)159 1313 y(While)g(in)g(the) h(discrete)g(case,)i(the)e(con)m(v)m(ergence)i(of)f(these)f(pro)s (cesses)g(could)f(b)s(e)h(pro)m(v)m(en)g(directly)-8 b(,)31 b(it)50 1458 y(is)k(a)h(priori)e(not)i(clear)g(ho)m(w)g(this)e (could)h(b)s(e)h(ac)m(hiev)m(ed)g(in)f(the)h(general)g(case.)58 b(But)36 b(recall)f(that)i(in)e(the)50 1604 y(discrete)h(case,)k (instead)c(of)h(constructing)f(the)h(limiting)d(pro)s(cesses)j (directly)-8 b(,)38 b(the)f(Ghirlanda-Guerra)50 1749 y(iden)m(tities)32 b(pro)m(vide)g(an)i(alternativ)m(e)g(w)m(a)m(y)-8 b(.)51 b(W)-8 b(e)35 b(will)c(see)j(that)g(this)e(path)h(is)g(still)e (op)s(en)i(in)f(the)i(general)50 1895 y(case.)50 2040 y Fq(P)m(oin)m(t)h(pro)s(cesses)i(on)e(the)f(space)i(of)f(probabilit)m (y)g(distributions.)159 2258 y Fp(W)-8 b(e)24 b(will)c(think)i(of)h (the)g(quan)m(tities)f Fg(K)1431 2272 y Fh(\014)s(;N)1590 2258 y Fp(as)h(probabilit)m(y)d(distributions)g(on)i(the)i(space)f(of)g (probabilit)m(y)50 2404 y(distributions)c(on)j([0)p Fi(;)15 b Fp(1],)26 b(that)d(is)f(to)h(sa)m(y)g(as)g(elemen)m(ts)g(of)f(the)h (space)g Fg(M)2504 2418 y Fs(1)2545 2404 y Fp(\()p Fg(M)2689 2418 y Fs(1)2730 2404 y Fp(\()p Fg(M)2874 2418 y Fs(1)2916 2404 y Fp(\([0)p Fi(;)15 b Fp(1]\)\)\).)41 b(Equipping)50 2549 y(the)32 b(spaces)g Fg(M)597 2563 y Fs(1)638 2549 y Fp(\([0)p Fi(;)15 b Fp(1]\))34 b(and)e Fg(M)1210 2563 y Fs(1)1251 2549 y Fp(\()p Fg(M)1395 2563 y Fs(1)1436 2549 y Fp(\([0)p Fi(;)15 b Fp(1]\)\))35 b(with)30 b(the)i(top)s (ologies)g(of)g(w)m(eak)h(con)m(v)m(ergence,)i(there)d(is)50 2695 y(no)25 b(obstacle)g(to)h(de\014ne)e(w)m(eak)i(con)m(v)m(ergence)i (of)d(our)g(ob)5 b(jects)25 b(in)f(m)m(uc)m(h)h(the)g(same)h(w)m(a)m(y) g(as)f(in)f(the)h(discrete)50 2840 y(case)36 b(\(when)e Fg(M)633 2854 y Fs(1)674 2840 y Fp(\([0)p Fi(;)15 b Fp(1]\))37 b(w)m(as)f(replaced)e(b)m(y)h(a)g(\014nite)f(dimensional)e(simplex)h (in)h Fe(R)2963 2807 y Fh(n)3019 2840 y Fp(\).)54 b(Just)35 b(note)g(that)50 2985 y(the)e(con)m(tin)m(uous)g(functions)f(of)h(a)h (measure)f Fi(m)d Fg(2)g(M)1914 2999 y Fs(1)1955 2985 y Fp(\([0)p Fi(;)15 b Fp(1]\))35 b(can)f(b)s(e)e(appro)m(ximated)h (arbitrarily)e(w)m(ell)50 3131 y(b)m(y)40 b(monomials)f(in)g(\014nite)g (collections)h(of)g(in)m(tegrals)g(with)f(resp)s(ect)h(to)h Fi(m)f Fp(of)g(indicator)f(functions)g(of)50 3276 y(\(disjoin)m(t\))c (in)m(terv)-5 b(als)36 b(\001)897 3290 y Fs(1)937 3276 y Fi(;)15 b(:)g(:)g(:)i(;)e Fp(\001)1215 3290 y Fh(l)1278 3276 y Fg(\032)35 b Fp([0)p Fi(;)15 b Fp(1],)40 b(and)c(that)h(in)e (turn)g(con)m(tin)m(uous)h(functions)f(of)i(a)g(measure)50 3422 y Fg(W)32 b(2)25 b(M)367 3436 y Fs(1)408 3422 y Fp(\()p Fg(M)552 3436 y Fs(1)593 3422 y Fp(\([0)p Fi(;)15 b Fp(1]\)\))31 b(can)d(b)s(e)e(appro)m(ximated)i(b)m(y)f(p)s (olynomials)e(in)h(a)i(collection)f(of)g(in)m(tegrals)g(of)h(suc)m(h)50 3567 y(functions.)37 b(Th)m(us,)23 b(if)f(w)m(e)h(sho)m(w)g(that)g(for) g(an)m(y)g(collection)g(\001)2102 3581 y Fh(ij)2191 3567 y Fg(\032)i Fp([0)p Fi(;)15 b Fp(1])24 b(and)f(in)m(tegers)g Fi(q)3031 3581 y Fh(i)3061 3567 y Fi(;)15 b(r)3142 3581 y Fh(ij)3206 3567 y Fp(,)25 b Fi(i)g Fp(=)g(1)p Fi(;)15 b(:)g(:)g(:)i(;)e(l)r Fp(,)50 3713 y Fi(j)31 b Fp(=)25 b(1)p Fi(;)15 b(:)g(:)g(:)i(;)e(k)508 3727 y Fh(i)539 3713 y Fp(,)856 3920 y Fe(E)911 3765 y Fc( )989 3792 y(\022)1056 3796 y(Z)1162 3920 y Fg(K)1231 3934 y Fh(\014)s(;N)1366 3920 y Fp(\()p Fi(dm)p Fp(\))p Fi(m)p Fp(\(\001)1754 3934 y Fs(11)1832 3920 y Fp(\))1867 3883 y Fh(r)1901 3892 y Fl(11)1987 3920 y Fi(:)g(:)g(:)h(m)p Fp(\(\001)2299 3934 y Fs(1)p Fh(j)2365 3943 y Fl(1)2406 3920 y Fp(\))2441 3883 y Fh(r)2475 3892 y Fl(1)p Fk(j)2533 3906 y Fl(1)2577 3792 y Fc(\023)2644 3811 y Fh(q)2677 3820 y Fl(1)2732 3920 y Fi(:)f(:)g(:)1053 4229 y(:)g(:)g(:)1174 4101 y Fc(\022)1241 4106 y(Z)1347 4229 y Fg(K)1416 4243 y Fh(\014)s(;N)1552 4229 y Fp(\()p Fi(dm)p Fp(\))p Fi(m)p Fp(\(\001)1940 4243 y Fh(l)p Fs(1)2005 4229 y Fp(\))2040 4192 y Fh(r)2074 4202 y Fk(l)p Fl(1)2151 4229 y Fi(:)g(:)g(:)h(m)p Fp(\(\001)2463 4243 y Fh(lj)2516 4253 y Fk(l)2547 4229 y Fp(\))2582 4192 y Fh(r)2616 4202 y Fk(lj)2665 4217 y(l)2700 4101 y Fc(\023)2767 4120 y Fh(q)2800 4130 y Fk(l)2831 4074 y Fc(!)3523 4075 y Fp(\(4)p Fi(:)p Fp(6\))50 4500 y(con)m(v)m(erges,)32 b(then)e(the)h(p)s(oin)m(t)e(pro)s(cess)h Fg(K)1469 4514 y Fh(N)s(;\014)1631 4500 y Fp(con)m(v)m(erges)i(w)m(eakly)f(to)g(a)g (limit)d Fg(K)2812 4514 y Fh(\014)2861 4500 y Fp(.)159 4719 y(A)i(somewhat)h(simpler)d(w)m(a)m(y)j(of)g(lo)s(oking)e(at)i (this)f(pro)s(cess)f(is)h(to)h(simply)d(consider)1312 4995 y Fi(K)1389 5009 y Fh(\014)s(;N)1524 4995 y Fp(\()p Fi(t)p Fp(\))e Fg(\021)1749 4908 y Fc(X)1792 5099 y Fh(\033)1895 4995 y Fi(\026)1950 5009 y Fh(\014)s(;N)2085 4995 y Fp(\()p Fi(\033)s Fp(\))p Fi(\016)2250 5011 y Fh(m)2315 5020 y Fk(\033)2359 5011 y Fs(\()p Fh(t)p Fs(\))3523 4995 y Fp(\(4)p Fi(:)p Fp(7\))50 5316 y(and)35 b(to)i(think)e(of)h(it)g(as)g (a)h(measure)f(v)-5 b(alued)35 b(pro)s(cess)g(on)h([0)p Fi(;)15 b Fp(1].)60 b(This)35 b(pro)s(cess)g(is)g(usually)f(called)i (the)50 5462 y(coalescen)m(t)c(and)d(has)g(b)s(een)g(studied)g(in)f (the)i(spin)e(glass)i(con)m(text)h(b)m(y)f(Bolthausen)f(and)h(Sznitman) e([BS].)p eop %%Page: 15 15 15 14 bop 50 295 a Fj(GREM,)28 b(p)l(art)f(3)3089 b Fm(15)50 513 y Fp(An)32 b(alternativ)m(e)h(construction)g(based)f(on)h(sub)s (ordinators)d(w)m(as)j(later)g(giv)m(en)g(b)m(y)g(Bertoin)f(and)h(Le)g (Gall)50 658 y([BL].)e(It)g(clearly)e(con)m(tains)i(somewhat)g(less)e (information)g(than)h Fg(K)2389 672 y Fh(\014)2469 658 y Fp(itself.)50 876 y Fq(Multi-o)m(v)m(erlap)35 b(distributions.)159 1095 y Fp(Our)29 b(task)i(is)e(no)m(w)i(to)g(pro)m(v)m(e)g(the)g (Ghirlanda-Guerra)d(iden)m(tities)h(in)g(the)i(general)f(case.)50 1313 y Fb(Theorem)37 b(4.1:)50 b Fo(F)-7 b(or)34 b(any)f Fi(n)25 b Fg(2)f Fe(N)45 b Fo(and)34 b(any)f Fi(x)25 b Fg(2)g Fp([0)p Fi(;)15 b Fp(1])p Fg(n)p Fi(x)2162 1327 y Fh(\014)2245 1313 y Fo(for)33 b(which)g(the)g(derivative)g(of)f Fi(A)p Fp(\()p Fi(x)p Fp(\))i Fo(do)-5 b(es)50 1458 y(not)33 b(vanish)158 1726 y Fp(lim)134 1785 y Fh(N)7 b Ff("1)308 1566 y Fc(\014)308 1621 y(\014)308 1676 y(\014)308 1730 y(\014)308 1785 y(\014)339 1726 y Fe(E)j Fi(\026)448 1687 y Ff(\012)p Fh(n)p Fs(+1)448 1753 y Fh(\014)s(;N)667 1652 y Fc(\000)709 1726 y Fi(h)p Fp(\()p Fi(\033)851 1688 y Fs(1)893 1726 y Fi(;)15 b(:)g(:)g(:)h(;)f(\033)1149 1688 y Fh(n)1199 1726 y Fp(\)1)-23 b(I)1290 1744 y Fh(d)1328 1753 y Fk(N)1386 1744 y Fs(\()p Fh(\033)1458 1726 y Fk(k)1499 1744 y Fh(;\033)1565 1726 y Fk(n)p Fl(+1)1687 1744 y Fs(\))p Ff(\025)p Fh(x)1818 1652 y Fc(\001)328 2062 y Fg(\000)435 2000 y Fp(1)p 430 2041 55 4 v 430 2124 a Fi(n)496 2062 y Fe(E)10 b Fi(\026)605 2023 y Ff(\012)p Fh(n)p Fs(+1)605 2089 y Fh(\014)t(;N)825 1879 y Fc(0)825 2043 y(@)904 2062 y Fi(h)p Fp(\()p Fi(\033)1046 2024 y Fs(1)1088 2062 y Fi(;)15 b(:)g(:)g(:)h(;)f(\033)1344 2024 y Fh(n)1395 2062 y Fp(\))1445 1879 y Fc(0)1445 2043 y(@)1568 1948 y Fh(n)1525 1976 y Fc(X)1531 2171 y Fh(l)p Ff(6)p Fs(=)p Fh(k)1671 2062 y Fp(1)-23 b(I)1726 2080 y Fh(d)1764 2089 y Fk(N)1822 2080 y Fs(\()p Fh(\033)1894 2062 y Fk(k)1935 2080 y Fh(;\033)2001 2062 y Fk(l)2027 2080 y Fs(\))p Ff(\025)p Fh(x)2179 2062 y Fp(+)19 b Fe(E)11 b Fi(\026)2379 2023 y Ff(\012)p Fs(2)2379 2089 y Fh(\014)s(;N)2520 2062 y Fp(\(1)-23 b(I)2610 2078 y Fh(d)2648 2087 y Fk(N)2706 2078 y Fs(\()p Fh(\033)2778 2060 y Fl(1)2815 2078 y Fh(;\033)2881 2060 y Fl(2)2916 2078 y Fs(\))p Ff(\025)p Fh(x)3047 2062 y Fp(\))3082 1879 y Fc(1)3082 2043 y(A)3162 1879 y(1)3162 2043 y(A)3242 1903 y(\014)3242 1957 y(\014)3242 2012 y(\014)3242 2066 y(\014)3242 2121 y(\014)3297 2062 y Fp(=)25 b(0)3523 1907 y(\(4)p Fi(:)p Fp(8\))50 2529 y Fa(Pro)s(of:)47 b Fp(One)29 b(of)g(the)h(pillars)c(of)k(the)f (Ghirlanda-Guerra)e(iden)m(tities)h(is)h(concen)m(tration)h(of)g (measure)f(for)50 2675 y(the)h(free)h(energy)-8 b(,)31 b(that)g(holds)e(also)h(here:)50 2893 y Fb(Lemma)40 b(4.2:)56 b Fo(Assume)34 b(that)i Fi(A)e Fo(is)h(non-critic)-5 b(al,)36 b(i.e.)46 b(whenever)35 b Fi(x)f Fo(is)h(not)g(in)f(the)h(set) f(of)h(extr)-5 b(emal)50 3038 y(p)g(oints)34 b(of)448 3015 y Fp(\026)424 3038 y Fi(A)p Fo(,)e Fi(A)p Fp(\()p Fi(x)p Fp(\))27 b Fi(<)888 3015 y Fp(\026)864 3038 y Fi(A)q Fp(\()p Fi(x)p Fp(\))p Fo(.)159 3257 y(F)-7 b(or)34 b(any)f Fi(\014)38 b Fo(and)33 b(for)g(any)h Fi(\017)25 b Fg(\025)g Fp(0)940 3518 y Fe(P)15 b Fp([)q Fg(j)p Fp(ln)f Fi(Z)1214 3532 y Fh(\014)s(;N)1369 3518 y Fg(\000)20 b Fe(E)26 b Fp(ln)20 b Fi(Z)1689 3532 y Fh(\014)s(;N)1824 3518 y Fg(j)25 b Fi(>)g(\017)p Fp(])h Fg(\024)f Fp(2)15 b(exp)2368 3390 y Fc(\022)2435 3518 y Fg(\000)2591 3457 y Fi(\017)2628 3424 y Fs(2)p 2517 3497 226 4 v 2517 3580 a Fp(2)p Fi(\014)2618 3554 y Fs(2)2659 3580 y Fi(N)2753 3390 y Fc(\023)3523 3518 y Fp(\(4)p Fi(:)p Fp(9\))50 3981 y Fa(Pro)s(of:)68 b Fp(This)38 b(follo)m(ws,)j(e.g.,)i(from)c(the) h(standard)e(Gaussian)h(concen)m(tration)h(of)g(measure)f(theorem)50 4126 y(\(see)k([L)-8 b(T],)42 b(page)h(21\))g(and)f(the)g(represen)m (tation)g(of)g(ln)14 b Fi(Z)2099 4140 y Fh(\014)s(;N)2276 4126 y Fp(as)42 b(a)h(Lipshitz)c(function)i(of)h(2)3376 4093 y Fh(N)7 b Fs(+1)3566 4126 y Fg(\000)27 b Fp(2)50 4272 y(indep)s(enden)m(t)h(standard)i(Gaussian)f(random)h(v)-5 b(ariables)29 b(with)g(Lipshitz)f(constan)m(t)k Fi(\014)3026 4195 y Fg(p)p 3102 4195 83 4 v 77 x Fi(N)10 b Fp(.)40 b Fg(})159 4490 y Fp(As)30 b(a)h(\014rst)f(step)g(w)m(e)h(need)f(the)h (generalization)f(of)g(Lemma)h(3.4)g(as)g(in)e(Lemma)h(4.2)i(of)37 b([BK2].)50 4708 y Fb(Lemma)h(4.3:)52 b Fo(F)-7 b(or)34 b(any)g Fi(t)26 b Fg(2)f Fp(\(0)p Fi(;)15 b Fp(1])35 b Fo(and)f Fi(\017)26 b(>)g Fp(0)p Fo(,)33 b(and)h(let)f Fi(h)27 b Fp(:)f Fg(S)2341 4675 y Fh(n)2334 4733 y(N)2429 4708 y Fg(!)g Fe(R)41 b Fo(b)-5 b(e)33 b(any)h(b)-5 b(ounde)g(d)34 b(function)f(of)50 4854 y Fi(n)f Fo(spin-c)-5 b(on\014gur)g(ations:)265 5038 y Fp(1)p 208 5079 159 4 v 208 5097 a Fg(p)p 284 5097 83 4 v 77 x Fi(N)378 5100 y Fe(E)11 b Fi(\026)488 5061 y Ff(\012)p Fh(n)488 5127 y(\014)s(;N)643 5026 y Fc(\000)685 5100 y Fp(\()p Fi(X)795 5118 y Fh(\033)839 5100 y Fk(k)h Fp(\()p Fi(t)p Fp(\))20 b Fg(\000)g Fi(X)1174 5118 y Fh(\033)1218 5100 y Fk(k)11 b Fp(\()p Fi(t)21 b Fg(\000)e Fi(\017)p Fp(\)\))p Fi(h)p Fp(\()p Fi(\033)1691 5062 y Fs(1)1734 5100 y Fi(;)c(:)g(:)g(:)h(;)f(\033)1990 5062 y Fh(n)2041 5100 y Fp(\))2076 5026 y Fc(\001)223 5380 y Fp(=)25 b Fi(\014)390 5256 y Fc(Z)481 5278 y Fh(t)440 5463 y(t)p Ff(\000)p Fh(\017)575 5380 y Fi(dA)p Fp(\()p Fi(s)p Fp(\))p Fe(E)11 b Fi(\026)913 5341 y Ff(\012)p Fh(n)p Fs(+1)913 5407 y Fh(\014)s(;N)1132 5224 y Fc( )1204 5380 y Fi(h)p Fp(\()p Fi(\033)1346 5342 y Fs(1)1388 5380 y Fi(;)k(:)g(:)g(:)i(;)e(\033)1645 5342 y Fh(n)1695 5380 y Fp(\))1745 5224 y Fc( )1861 5266 y Fh(n)1817 5293 y Fc(X)1825 5489 y Fh(l)p Fs(=1)1964 5380 y Fp(1)-23 b(I)2019 5398 y Fh(d)2057 5407 y Fk(N)2115 5398 y Fs(\()p Fh(\033)2187 5380 y Fk(k)2228 5398 y Fh(;\033)2294 5380 y Fk(l)2320 5398 y Fs(\))p Ff(\025)p Fh(s)2464 5380 y Fg(\000)20 b Fi(n)p Fp(1)-23 b(I)2665 5398 y Fh(d)2703 5407 y Fk(N)2761 5398 y Fs(\()p Fh(\033)2833 5380 y Fk(k)2874 5398 y Fh(;\033)2940 5380 y Fk(n)p Fl(+1)3062 5398 y Fs(\))p Ff(\025)p Fh(s)3186 5224 y Fc(!!)3477 5262 y Fp(\(4)p Fi(:)p Fp(10\))p eop %%Page: 16 16 16 15 bop 50 295 a Fm(16)3270 b Fj(Se)l(ction)27 b Fm(4)50 731 y Fa(Pro)s(of:)47 b Fp(Exactly)31 b(analogous)g(to)g(the)f(pro)s (of)g(of)g(Lemma)h(3.4.)42 b Fg(})159 949 y Fp(The)30 b(more)g(imp)s(ortan)m(t)g(step)g(is)g(the)g(pro)s(of)g(of)g(the)h (analogue)g(of)f(Lemma)h(4.3)g(of)38 b([BK2].)50 1167 y Fb(Lemma)g(4.4:)51 b Fo(L)-5 b(et)33 b Fi(h)g Fo(b)-5 b(e)32 b(as)h(in)g(the)g(pr)-5 b(evious)33 b(lemma.)43 b(Exc)-5 b(ept)33 b(p)-5 b(ossibly)34 b(when)g Fi(t)25 b Fp(=)g Fi(x)3166 1181 y Fh(\014)3215 1167 y Fo(,)708 1436 y Fp(lim)684 1495 y Fh(N)7 b Ff("1)941 1374 y Fp(1)p 884 1415 159 4 v 884 1433 a Fg(p)p 960 1433 83 4 v 77 x Fi(N)1054 1277 y Fc(\014)1054 1331 y(\014)1054 1386 y(\014)1054 1440 y(\014)1054 1495 y(\014)1084 1436 y Fe(E)k Fi(\026)1194 1397 y Ff(\012)p Fh(n)1194 1463 y(\014)s(;N)1350 1362 y Fc(\000)1392 1436 y Fp(\()p Fi(X)1502 1454 y Fh(\033)1546 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Fi(x)1754 5330 y Fh(\014)1838 5316 y Fp(=)f Fi(t)p Fp(.)58 b(Th)m(us,)37 b(the)g(\014rst)e(term)h(in)f (\(4.13\))k(will)33 b(v)-5 b(anish)50 5462 y(when)35 b(divided)f(b)m(y)i Fi(N)10 b Fp(.)58 b(T)-8 b(o)36 b(see)h(that)g(the) f(co)s(e\016cien)m(t)h(of)g Fi(N)46 b Fp(of)36 b(the)g(second)h(term)f (giv)m(es)h(a)f(v)-5 b(anishing)p eop %%Page: 17 17 17 16 bop 50 295 a Fj(GREM,)28 b(p)l(art)f(3)3089 b Fm(17)50 513 y Fp(con)m(tribution,)32 b(w)m(e)h(use)f(the)h(general)g(fact)g (that)h(if)d(the)i(v)-5 b(ariance)33 b(of)g(family)e(of)i(a)g(con)m(v)m (ex)h(\(or)f(conca)m(v)m(e\))50 658 y(functions)22 b(tends)h(to)i (zero,)h(then)d(the)h(same)g(is)f(true)h(for)f(its)g(deriv)-5 b(ativ)m(e,)25 b(pro)m(vided)d(the)i(second)g(deriv)-5 b(ativ)m(e)50 804 y(of)30 b(the)h(exp)s(ectation)g(is)e(b)s(ounded)f (\(see)k(e.g.)42 b(Lemma)30 b(8.9)h(in)f([BG],)h(or)g(Prop)s(osition)d (4.3)k(in)d([T3]\).)159 1022 y(But)j(b)m(y)g(Lemma)g(4.2)h(the)f(v)-5 b(ariance)32 b(of)g Fi(F)1622 1036 y Fh(\014)s(;N)1790 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Fp(\))g(=)g(1)p Fi(=\013)h(>)f Fp(0)p Fo(.)67 b(Then)41 b(the)50 1022 y(r)-5 b(andom)35 b(me)-5 b(asur)g(e)34 b Fi(K)810 1036 y Fh(\014)859 1022 y Fp(\()p Fi(t)p Fp(\))f Fo(is)g(a)g (Dirichlet-Poisson)g(pr)-5 b(o)g(c)g(ess)35 b(\(se)-5 b(e)33 b(e.g.)41 b Fp([Ru,T1])p Fo(\))33 b(with)g(p)-5 b(ar)g(ameter)36 b Fi(\013)d Fo(.)159 1240 y Fp(In)e(fact)i(m)m(uc)m(h) f(more)g(is)f(true.)46 b(W)-8 b(e)33 b(can)f(consider)f(the)h(pro)s (cesses)g(on)g(arbitrary)f(\014nite)g(dimensional)50 1385 y(marginals,)e(i.e.)909 1542 y Fi(K)986 1556 y Fh(\014)s(;N)1121 1542 y Fp(\()p Fi(t)1189 1556 y Fs(1)1230 1542 y Fi(;)15 b(:)g(:)g(:)i(;)e(t)1465 1556 y Fh(m)1534 1542 y Fp(\))26 b Fg(\021)1723 1455 y Fc(X)1691 1650 y Fh(\033)r Ff(2S)1829 1659 y Fk(N)1902 1542 y Fi(\026)1957 1556 y Fh(\014)s(;N)2092 1542 y Fp(\()p Fi(\033)s Fp(\))p Fi(\016)2257 1558 y Fh(m)2322 1567 y Fk(\033)2366 1558 y Fs(\()p Fh(t)2421 1567 y Fl(1)2457 1558 y Fs(\))p Fh(;:::)o(;m)2659 1567 y Fk(\033)2702 1558 y Fs(\()p Fh(t)2757 1567 y Fk(m)2818 1558 y Fs(\))3477 1542 y Fp(\(4)p Fi(:)p Fp(18\))50 1815 y(for)39 b(0)h Fi(<)g(t)427 1829 y Fs(1)508 1815 y Fi(<)g Fg(\001)15 b(\001)g(\001)41 b Fi(<)f(t)909 1829 y Fh(m)1018 1815 y Fi(<)g Fp(1.)67 b(The)39 b(p)s(oin)m(t)f(is)h(that)h(this)e(pro) s(cess)h(is)f(en)m(tirely)g(determined)g(b)m(y)i(the)50 1960 y(expressions)27 b(\(4.6\))j(with)c(the)j(\001)1172 1974 y Fh(ij)1264 1960 y Fp(all)e(of)h(the)g(form)g(\()p Fi(t)1924 1974 y Fh(i)1955 1960 y Fi(;)15 b Fp(1])29 b(for)f Fi(t)2264 1974 y Fh(i)2322 1960 y Fp(in)f(the)h(\014xed)g(set)g (of)h(v)-5 b(alues)27 b Fi(t)3340 1974 y Fs(1)3381 1960 y Fi(;)15 b(:)g(:)g(:)h(;)f(t)3615 1974 y Fh(m)3685 1960 y Fp(.)50 2106 y(This)23 b(in)g(turn)h(implies)d(that)26 b(the)e(pro)s(cess)h(is)e(determined)h(b)m(y)g(the)h(m)m(ulti-replica)d (distribution)f(functions)50 2251 y Fe(Q)115 2204 y Fs(\()p Fh(n)p Fs(\))115 2279 y Fh(\014)s(;N)277 2251 y Fp(restricted)f(to)i (the)f(discrete)g(set)h(of)f(ev)m(en)m(ts)h Fg(f)p Fi(d)1831 2265 y Fh(N)1901 2251 y Fp(\()p Fi(\033)1991 2218 y Fh(i)2022 2251 y Fi(;)15 b(\033)2117 2218 y Fh(j)2156 2251 y Fp(\))26 b Fi(>)e(t)2345 2265 y Fh(k)2390 2251 y Fg(g)p Fp(.)38 b(Since)20 b(these)i(n)m(um)m(b)s(ers)d(are)j(totally)50 2397 y(determined)33 b(through)g(the)i(Ghirlanda-Guerra)d(iden)m (tities,)i(they)h(are)f(iden)m(tically)f(to)i(those)f(obtained)50 2542 y(in)d(a)i(GREM)g(with)e Fi(m)h Fp(lev)m(els,)h(i.e.)47 b(a)33 b(function)e Fi(A)i Fp(ha)m(ving)f(steps)g(at)h(the)g(v)-5 b(alues)32 b Fi(t)2947 2556 y Fh(i)2977 2542 y Fp(,)h(those)g(t)m(w)m (o-replica)50 2688 y(distribution)f(function)j(tak)m(es)i(the)f(same)h (v)-5 b(alues)35 b(as)h(that)h(of)f(the)g(mo)s(del)f(with)f(con)m(tin)m (uous)i Fi(A)g Fp(at)h(the)50 2833 y(p)s(oin)m(ts)29 b Fi(t)356 2847 y Fh(i)416 2833 y Fp(and)h(is)g(constan)m(t)h(b)s(et)m (w)m(een)g(those)g(v)-5 b(alues.)40 b(In)30 b(fact)50 3051 y Fb(Theorem)39 b(4.7:)56 b Fo(L)-5 b(et)35 b Fp(0)29 b Fi(<)f(t)1137 3065 y Fs(1)1206 3051 y Fi(<)g Fg(\001)15 b(\001)g(\001)29 b Fi(<)f(t)1571 3065 y Fh(k)1644 3051 y Fg(\024)g Fi(q)1784 3065 y Fh(max)1934 3051 y Fp(\()p Fi(\014)5 b Fp(\))35 b Fo(b)-5 b(e)34 b(p)-5 b(oints)36 b(of)f(incr)-5 b(e)g(ase)35 b(of)g Fe(E)10 b Fi(f)3145 3065 y Fh(\014)3200 3051 y Fo(.)47 b(Consider)35 b(a)50 3197 y(GREM)23 b(with)i Fi(k)i Fo(levels)d(and)h(p)-5 b(ar)g(ameters)27 b Fi(\013)1531 3211 y Fh(i)1561 3197 y Fi(;)15 b(a)1649 3211 y Fh(i)1704 3197 y Fo(and)25 b(temp)-5 b(er)g(atur)g(e)2373 3173 y Fp(~)2360 3197 y Fi(\014)29 b Fo(that)c(satisfy)g Fp(ln)14 b Fi(\013)3041 3211 y Fh(i)3071 3197 y Fi(=)h Fp(ln)g(2)26 b(=)e Fi(t)3422 3211 y Fh(i)3454 3197 y Fg(\000)q Fi(t)3559 3211 y Fh(i)p Ff(\000)p Fs(1)3682 3197 y Fo(,)63 3318 y Fp(~)50 3342 y Fi(\014)106 3309 y Ff(\000)p Fs(1)204 3265 y Fc(p)p 295 3265 364 4 v 77 x Fp(2)15 b(ln)f Fi(\013)504 3356 y Fh(i)534 3342 y Fi(=a)627 3356 y Fh(i)684 3342 y Fp(=)24 b Fe(E)11 b Fi(f)879 3356 y Fh(\014)934 3342 y Fp(\()p Fi(t)1002 3356 y Fh(i)1032 3342 y Fp(\))p Fo(.)42 b(Then)1335 3608 y Fp(lim)1311 3667 y Fh(N)7 b Ff("1)1500 3608 y Fg(K)1569 3622 y Fh(\014)s(;N)1705 3608 y Fp(\()p Fi(t)1773 3622 y Fs(1)1814 3608 y Fi(;)15 b(:)g(:)g(:)h(;)f(t)2048 3622 y Fh(k)2093 3608 y Fp(\))26 b(=)f Fg(W)2347 3560 y Fs(\()p Fh(k)r Fs(\))2350 3634 y(~)2340 3650 y Fh(\014)3477 3608 y Fp(\(4)p Fi(:)p Fp(19\))159 4091 y(Th)m(us,)39 b(if)e(the)h Fi(t)712 4105 y Fh(i)779 4091 y Fp(are)h(c)m(hosen)f(in)e (suc)m(h)i(a)g(w)m(a)m(y)h(that)f(for)g(all)f(of)h(them)f Fe(E)11 b Fi(f)2773 4105 y Fh(\014)2828 4091 y Fp(\()p Fi(t)2896 4105 y Fh(i)2926 4091 y Fp(\))38 b Fi(>)f Fp(0,)k(then)c(w)m (e)h(can)50 4236 y(construct)30 b(an)h(explicit)d(represen)m(tation)i (of)h(the)f(limiting)e(marginal)g(pro)s(cess)i Fg(K)2853 4250 y Fh(\014)2903 4236 y Fp(\()p Fi(t)2971 4250 y Fs(1)3011 4236 y Fi(;)15 b(:)g(:)g(:)i(;)e(t)3246 4250 y Fh(m)3315 4236 y Fp(\))31 b(in)e(terms)50 4382 y(of)24 b(a)g(P)m(oisson-cascade)h (pro)s(cess)f(via)f(the)h(corresp)s(onding)e(form)m(ulae)i(in)e(the)i (asso)s(ciated)g Fi(m)p Fp(-lev)m(el)g(GREM.)50 4527 y(In)29 b(this)f(sense)i(w)m(e)g(obtain)f(an)h(explicit)e(description)f (of)j(the)g(limiting)c(mass)k(distribution)c(function)i Fg(K)3635 4541 y Fh(\014)3685 4527 y Fp(.)50 4735 y Ft(References.)-68 4953 y Fp([A)m(C])46 b(M.)e(Aizenmann)e(and)h(P)-8 b(.)44 b(Con)m(tucci,)f(On)g(the)h(stabilit)m(y)e(of)h(the)h(quenc)m(hed)f (state)i(in)d(mean)h(\014eld)159 5098 y(spin-glass)29 b(mo)s(dels,)g(J.)h(Statist.)h(Ph)m(ys.)f Fq(92)p Fp(,)h(765-783)i (\(1998\).)-89 5316 y([A)-10 b(W])47 b(M.)42 b(Aizenman,)g(J.)f(W)-8 b(ehr,)43 b(Rounding)c(e\013ects)44 b(of)e(quenc)m(hed)f(randomness)g (on)g(\014rst-order)g(phase)159 5462 y(transitions,)29 b(Comm.)h(Math.)h(Ph)m(ys.)g Fq(130)p Fp(,)g(489-528)i(\(1990\).)p eop %%Page: 20 20 20 19 bop 50 295 a Fm(20)3225 b Fj(R)l(efer)l(enc)l(es)-58 513 y Fp([BL])46 b(J.)38 b(Bertoin)g(and)f(J.F.)i(Le)f(Gall,)f(The)h (Bolthausen-Snitzman)f(coalescen)m(t)j(and)d(the)h(genealogy)i(of)159 658 y(con)m(tin)m(uous)30 b(state)i(branc)m(hing)d(pro)s(cesses,)h (Probab.)g(Theor.)g(T)-8 b(el.)31 b(Fields)d Fq(117)p Fp(,)k(249-266)h(\(2000\).)-142 876 y([BDG])47 b(E.)28 b(Bolthausen,)f(J.-D.)i(Deusc)m(hel,)f(and)g(G.)g(Giacomin,)f(En)m (tropic)h(repulsion)d(and)i(the)h(maxim)m(um)f(of)159 1022 y(the)j(t)m(w)m(o-dimensional)g(harmonic)f(crystal,)h(Ann.)g (Probab.)g Fq(29)p Fp(,)h(1670-1692)j(\(2001\).)-52 1240 y([BS])46 b(E.)23 b(Bolthausen)f(and)g(A.-S.)h(Snitzman,)f(On)g (Ruelle's)f(probabilit)m(y)f(cascades)k(and)e(an)h(abstract)g(ca)m(vit) m(y)159 1385 y(metho)s(d,)30 b(Comm)m(un.)g(Math.)h(Ph)m(ys.)f Fq(107)p Fp(,)h(247-276)j(\(1998\).)-118 1604 y([BK1])47 b(A.)21 b(Bo)m(vier)g(and)f(I.)g(Kurk)m(o)m(v)-5 b(a,)21 b(Derrida's)f(Generalized)g(random)g(Energy)g(mo)s(dels)f(1.)i(P)m (oisson)f(cascades)159 1749 y(and)30 b(the)g(extremal)h(pro)s(cess,)f (preprin)m(t)e(\(2002\).)-118 1967 y([BK2])47 b(A.)21 b(Bo)m(vier)h(and)e(I.)i(Kurk)m(o)m(v)-5 b(a,)21 b(Derrida's)g (Generalized)f(Random)h(Energy)g(mo)s(dels)f(2.)h(Gibbs)f(measures)159 2113 y(and)30 b(probabilit)m(y)e(cascades,)j(preprin)m(t)e(\(2002\).) -47 2331 y([Bo])47 b(A.)21 b(Bo)m(vier,)g(Statistical)e(mec)m(hanics)h (of)h(disordered)d(systems,)j(MaPh)m(ySto)g(Lecture)g(Notes)g(10,)g (Aarh)m(us,)159 2476 y(2001.)-73 2695 y([BG])47 b(A.)30 b(Bo)m(vier,)g(V.)g(Ga)m(yrard,)g(The)f(Hop\014eld)f(mo)s(del)g(as)i(a) f(generalized)g(random)g(mean)h(\014eld)e(mo)s(del,)g(in)159 2840 y(Mathematics)37 b(of)f(spin)d(glasses)j(and)f(neural)f(net)m(w)m (orks,)j(A.)f(Bo)m(vier,)h(P)-8 b(.)36 b(Picco.)g(Eds.,)f(Progress)h (in)159 2985 y(Probablit)m(y)-8 b(,)30 b(Birkh\177)-45 b(auser,)29 b(Boston,)i(\(1997\).)-68 3204 y([BR])46 b(M.D.)25 b(Bramson,)f(Maximal)f(displacemen)m(t)g(of)h(branc)m(hing)f (Bro)m(wnian)g(motion,)h(Comm)m(un.)f(Pure)g(and)159 3349 y(Appl.)29 b(Math.)i Fq(31)p Fp(,)g(531-581)j(\(1978\).)-135 3567 y([CCR])45 b(D.)29 b(Cap)s(o)s(caccia,)h(M.)f(Cassandro,)f(and)g (P)-8 b(.)30 b(Picco,)f(On)f(the)h(existence)g(of)g(Thermo)s(dynamics)d (for)j(the)159 3713 y(Generalised)g(Random)h(Energy)g(Mo)s(del.)61 b(J.)30 b(Stat.)h(Ph)m(ysics)f Fq(46)p Fp(,)h(493{505)i(\(1987\).)-7 3931 y([D])47 b(B.)21 b(Derrida,)f(A)h(generalization)f(of)g(the)h (random)f(energy)h(mo)s(del)e(that)i(includes)d(correlations)i(b)s(et)m (w)m(een)159 4076 y(the)30 b(energies,)h(J.)f(Ph)m(ysique.)g(Lett.)h Fq(46)p Fp(,)g(401{407)j(\(1985\).)-123 4295 y([DG1])47 b(B.)30 b(Derrida)g(and)f(E.)h(Gardner,)f(Solution)f(of)j(the)f (generalized)f(random)g(energy)i(mo)s(del,)e(J.)g(Ph)m(ys.)h(C)159 4440 y Fq(19)p Fp(,)h(2253{2274)j(\(1986\).)-123 4658 y([DG2])47 b(B.)e(Derrida)f(and)g(E.)g(Gardner,)g(Magnetic)i(prop)s (erties)d(and)h(function)f Fi(q)s Fp(\()p Fi(x)p Fp(\))i(of)g(the)g (generalised)159 4804 y(random)29 b(energy)i(mo)s(del,)f(J.)g(Ph)m(ys.) g(C)g Fq(19)p Fp(,)h(5783-5798)j(\(1986\))-123 5022 y([DG3])47 b(E.)34 b(Gardner)f(and)h(B.)g(Derrida,)g(The)f(probabilit)m(y)f (distribution)e(of)k(the)g(partition)f(function)f(of)j(the)159 5167 y(random)29 b(energy)i(mo)s(del,)f(J.)g(Ph)m(ys.)g(A)p Fq(22)p Fp(,)h(1975-1981)k(\(1989\).)-80 5385 y([GG])47 b(S.)38 b(Ghirlanda,)e(F.)j(Guerra.)76 b(General)39 b(prop)s(erties)d (of)j(the)f(o)m(v)m(erlap)h(probabilit)m(y)d(distributions)e(in)p eop %%Page: 21 21 21 20 bop 50 295 a Fj(GREM,)28 b(p)l(art)f(3)3089 b Fm(21)159 513 y Fp(disordered)28 b(spin)h(systems.)h(T)-8 b(o)m(w)m(ards)31 b(P)m(arisi)e(ultrametricit)m(y,)h(J.)g(Ph)m(ys.)g(A)h Fq(31)p Fp(,)g(9144-9155)j(\(1998\).)-59 731 y([Gu])46 b(F.)33 b(Guerra,)f(Brok)m(en)h(replica)f(symmetry)f(b)s(ounds)g(in)g (the)h(mean)h(\014eld)e(spin)f(glass)i(mo)s(del,)g(preprin)m(t)159 876 y(2002.)-74 1095 y([GT])46 b(F.)29 b(Guerra)f(and)g(F.)h(T)-8 b(oninelli,)26 b(The)i(thermo)s(dynamics)f(limit)f(in)h(mean)i(\014eld) e(spin)f(glass)j(mo)s(dels,)e(to)159 1240 y(app)s(ear)j(in)f(Comm)m (un.)g(Math.)j(Ph)m(ys.)e(\(2002\).)-118 1458 y([LLR])46 b(M.)26 b(R.)f(Leadb)s(etter,)h(G.)g(Lindgren,)d(H.)j(Ro)s(otz)m(\023) -43 b(en,)53 b(Extremes)25 b(and)g(Related)h(Prop)s(erties)e(of)h (Random)159 1604 y(Sequences)30 b(and)g(Pro)s(cesses,)h(Springer,)d (Berlin-Heidelb)s(erg-New)g(Y)-8 b(ork,)31 b(1983.)-52 1822 y([L)-8 b(T])46 b(M.)22 b(Ledoux)g(and)f(L.)h(T)-8 b(alagrand,)21 b(Probabilit)m(y)f(in)g(Banac)m(h)j(space.)g(Spinger,)d (Berlin-Heidelb)s(erg-New)159 1967 y(Y)-8 b(ork,)31 b(1991.)-5 2185 y([N])46 b(Ch.M.)f(Newman,)h(T)-8 b(opics)45 b(in)g(disordered)e (systems,)j(Lectures)g(in)e(Mathematics)j(ETH)e(Z)s(\177)-48 b(uric)m(h.)159 2331 y(Birkh\177)j(auser)29 b(V)-8 b(erlag,)31 b(Basel,)g(1997.)-101 2549 y([NS1])46 b(Ch.M.)34 b(Newman)g(and)g(D.L.) h(Stein,)f(Thermo)s(dynamic)e(c)m(haos)j(and)f(the)h(structure)f(of)g (short)g(range)159 2695 y(spin)24 b(glasses,)h(in)g(\\Mathematical)i (asp)s(ects)e(of)h(spin)e(glasses)h(and)g(neural)f(net)m(w)m(orks",)j (A.)f(Bo)m(vier)g(and)159 2840 y(P)-8 b(.)31 b(Picco)f(\(Eds.\),)h (Progress)g(in)e(Probabilit)m(y)-8 b(,)29 b(Birkh\177)-45 b(auser,)29 b(Boston)i(\(1997\).)-101 3058 y([NS2])46 b(C.M.)38 b(Newman,)g(D.L.)g(Stein,)f(Metastate)k(approac)m(h)d(to)g (thermo)s(dynamic)e(c)m(haos.)j(Ph)m(ys.)f(Rev.)g(E)159 3204 y Fq(55)p Fp(,)31 b(5194-5211)j(\(1997\).)-49 3422 y([Ro])46 b(R.T.)30 b(Ro)s(c)m(k)-5 b(afellar,)31 b(Con)m(v)m(ex)g (Analysis,)e(Princeton)h(Univ)m(ersit)m(y)f(Press,)h(Princeton,)g(N.J.) h(1970.)-54 3640 y([Ru])45 b(D.)26 b(Ruelle,)f(A)h(mathematical)g (reform)m(ulation)e(of)i(Derrida's)f(REM)h(and)f(GREM.)52 b(Comm)m(un.)25 b(Math.)159 3785 y(Ph)m(ys.)60 b Fq(108)p Fp(,)32 b(225-239)h(\(1987\).)-58 4004 y([SK])45 b(D.)28 b(Sherrington)e(and)h(S.)g(Kirkpatric)m(k,)f(\\Solv)-5 b(able)27 b(mo)s(del)g(of)g(a)h(spin)e(glass",)56 b(Ph)m(ys.)27 b(Rev.)h(Lett.)h Fq(35)p Fp(,)159 4149 y(1792-1796)34 b(\(1972\).)-48 4367 y([T1])46 b(M.)30 b(T)-8 b(alagrand,)30 b(Rigorous)f(lo)m(w)g(temp)s(erature)h(results)e(for)i(mean)f(\014eld)g (p-spin)e(in)m(teraction)j(mo)s(dels.)159 4513 y(Probab.)g(Theor.)g (Rel.)g(Fields.)f Fq(117)p Fp(,)j(303-360)h(\(2000\).)-48 4731 y([T2])46 b(M.)c(T)-8 b(alagrand,"Self)41 b(organization)h(in)f (the)h(lo)m(w-temp)s(erature)g(region)f(of)h(a)h(spin)d(glass)h(mo)s (del",)159 4876 y(preprin)m(t)28 b(2002.)-48 5095 y([T3])46 b(M.)21 b(T)-8 b(alagrand,)21 b(Rigorous)g(results)f(for)h(the)g (Hop\014eld)f(mo)s(del)f(with)h(man)m(y)h(patterns,)h(Probab.)e(Theory) 159 5240 y(Related)31 b(Fields)d Fq(110)p Fp(,)k(177-276)h(\(1998\).)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF