Content-Type: multipart/mixed; boundary="-------------0305300704531" This is a multi-part message in MIME format. ---------------0305300704531 Content-Type: text/plain; name="03-247.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-247.keywords" Gaussian processes, generalized random energy model, continuous state branching process, subordinators, coalescent processes, genealogy, Ghirlanda-Guerra identities. ---------------0305300704531 Content-Type: application/postscript; name="bk4.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="bk4.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com) %%Title: bk4.dvi %%Pages: 27 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips bk4 -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.05.28:1245 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin /N{def}def /B{bind def}N 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h(little)f(more)h(than)g(com-)50 1313 y(bining)25 b(our)i(results)f (from)h([BK3])h(with)e(those)i(of)g(Bertoin)f(and)g(Le)h(Gall)f ([BeLG],)i(but)d(w)m(e)i(feel)f(that)i(the)50 1458 y(emerging)h (complete)h(picture)e(is)h(w)m(ell)f(w)m(orth)h(to)h(b)s(e)f(put)g(in)f (evidence.)159 1676 y(Let)43 b(us)f(insist)f(that)i(the)g(main)e(purp)s (ose)g(of)i(this)e(and)h(our)h(preceding)e(pap)s(ers)h(is)f(to)j(sho)m (w)e(that)50 1822 y(meaningful)33 b Fv(in\014nite)38 b(volume)g(limits)f Fw(exist)e(in)g(highly)f(disordered)g(mean)i (\014eld)e(mo)s(dels,)i(con)m(trary)h(to)50 1967 y(what)24 b(is)f(sometimes)g(claimed.)38 b(Quite)23 b(on)h(the)g(con)m(trary)-8 b(,)26 b(there)e(exist)g(as)g(w)m(e)h(will)c(see)j Fv(universal)h Fw(limiting)50 2113 y(random)33 b(ob)5 b(jects)34 b(that)g(serv)m(e)h (as)f(go)s(o)s(d)f(appro)m(ximations)g(of)g(the)h(\\large)g(but)f (\014nite")g(systems,)i(in)e(the)50 2258 y(b)s(est)e(spirit)f(of)j (statistical)e(mec)m(hanics.)46 b(The)32 b(fact)h(that)f(these)h(ob)5 b(jects)33 b(turn)e(out)h(to)h(b)s(e)e(random,)h(and)50 2404 y(that)g(con)m(v)m(ergence)h(tends)e(to)h(b)s(e)f(in)f(the)h (sense)h(of)f(probabilit)m(y)e(distributions)e(is)k(certainly)f (unfamiliar)50 2549 y(to)g(the)f(traditionally)e(trained)h (mathematical)h(statistical)g(ph)m(ysicist,)f(while)f(this)h(will)e (hardly)h(come)j(as)50 2695 y(a)25 b(surprise)d(to)j(probabilists)d(or) i(statisticians.)38 b(Let)25 b(us)f(men)m(tion)g(that)h(the)g(imp)s (ortance)f(of)h(ditributional)50 2840 y(limits)31 b(of)j(random)f (measures)g(in)g(the)h(con)m(text)h(of)f(spin)e(glasses)i(w)m(as)g (strongly)f(adv)m(o)s(cated)i(in)d(a)i(series)50 2985 y(of)d(pap)s(ers)e(b)m(y)i(Newman)g(and)f(Stein,)g(see)i(e.g.)43 b([NS1,NS2,NS3].)g(When)31 b(sa)m(ying)g(that)g(limits)e(are)i(go)s(o)s (d)50 3131 y(appro)m(ximations,)22 b(care)g(has)g(to)g(b)s(e)f(tak)m (en)h(of)g(the)g(top)s(ology)g(used)e(when)h(constructing)g(limits.)35 b(There)21 b(are)50 3276 y(indeed)h(man)m(y)i(pitfalls)d(p)s(ossible,)i (and)g(great)i(care)f(m)m(ust)g(b)s(e)f(tak)m(en)i(in)d(order)h(to)h (get)h(meaningful)d(results.)50 3422 y(Rather)k(unsurprisingly)-8 b(,)22 b(the)j(ingenious)f(analysis)g(of)i(this)f(problem)f(in)m(tro)s (duced)g(in)g(the)i(con)m(text)i(of)e(the)50 3567 y(replica)j(metho)s (d)h([MPV])h(is)e(largely)h(equiv)-5 b(alen)m(t)30 b(to)h(formalism)e (used)g(in)g([BK3].)159 3785 y(Let)f(us)f(recall)g(the)g(cen)m(tral)h (problem)e(one)i(is)e(faced)i(with)f(when)f(analysing)g(mean)i(\014eld) e(spin)f(glasses.)50 3931 y(What)31 b(w)m(e)f(w)m(an)m(t)h(to)g(do)e (is)h(to)g(describ)s(e)f(the)h(geometric)h(structure)e(of)i(a)f(random) f(probabilit)m(y)f(measure)50 4076 y(on)39 b(a)h(set)f Fo(S)475 4090 y Fn(N)544 4076 y Fw(.)68 b(One)39 b(exp)s(ects)g(that)h (this)e(measure)h(will)e(concen)m(trate)42 b(\(at)e(lo)m(w)f(temp)s (eratures\))g(on)g(a)50 4222 y(relativ)m(ely)i(v)m(ery)h(small)f (subset)g(with)g(rather)h(complicated)f(structure.)75 b(Since)41 b(due)g(to)i(randomness)50 4367 y(and)36 b(symmetries)f (there)h(are)h(no)f(external)h(references,)h(w)m(e)f(need)f(a)g(w)m(a)m (y)i(to)f(describ)s(e)d(the)j(structural)50 4513 y(geometric)29 b(prop)s(erties)e(of)h(suc)m(h)g(measures)g(in)f(an)h(in)m(trinsic,)e (reference-free)j(w)m(a)m(y)-8 b(.)41 b(On)28 b(the)g(other)h(hand,)50 4658 y(w)m(e)i(need)f(to)h(allo)m(w)f(su\016cien)m(t)g(compactness)h (for)f(limits)e(to)j(exist.)159 4876 y(T)-8 b(o)26 b(resolv)m(e)h(this) e(problem,)h(w)m(e)g(in)m(tro)s(duced)f(in)g([BK3])i(w)m(as)f(what)g(w) m(e)h(called)e(the)h Fv(empiric)-5 b(al)31 b(distanc)-5 b(e)50 5022 y(distribution)34 b(function)p Fw(,)c(i.e.)41 b(the)30 b(random)g(measure)1338 5371 y Fo(K)1407 5385 y Fn(\014)s(;N)1567 5371 y Fo(\021)1696 5285 y Fk(X)1663 5480 y Fn(\033)r Fm(2S)1801 5489 y Fr(N)1875 5371 y Fp(\026)1930 5385 y Fn(\014)s(;N)2065 5371 y Fw(\()p Fp(\033)s Fw(\))p Fp(\016)2230 5388 y Fn(m)2295 5397 y Fr(\033)2339 5388 y Fz(\()p Fm(\001)p Fz(\))3523 5371 y Fw(\(1)p Fp(:)p Fw(4\))p eop %%Page: 5 5 5 4 bop 50 295 a Fq(GREM,)28 b(p)l(art)f(4)3127 b Ft(5)50 513 y Fw(where)1231 658 y Fp(m)1311 672 y Fn(\033)1360 658 y Fw(\()p Fp(t)p Fw(\))25 b Fo(\021)g Fp(\026)1639 672 y Fn(\014)s(;N)1789 658 y Fw(\()q Fp(\033)1880 621 y Fm(0)1931 658 y Fw(:)g Fp(d)2028 672 y Fn(N)2097 658 y Fw(\()p Fp(\033)n(;)15 b(\033)2277 621 y Fm(0)2304 658 y Fw(\))26 b Fp(>)f(t)p Fw(\))994 b(\(1)p Fp(:)p Fw(5\))50 863 y(This)25 b(ob)5 b(ject)27 b(describ)s(es)f(the)h (probabilit)m(y)d(of)j(a)g(mass)g(distribution)c(around)i(a)j(randomly) d(\(according)i(to)50 1008 y(the)j(Gibbs)f(measure\))i(dra)m(wn)e(p)s (oin)m(t)h(on)g Fo(S)1554 1022 y Fn(N)1623 1008 y Fw(.)159 1226 y(A)g(k)m(ey)i(ob)5 b(ject)31 b(is)e(the)i(mean)f(\014rst)g(momen) m(t)h(of)f(this)g(random)f(measure,)1254 1352 y Fk(Z)1360 1475 y Fo(K)1429 1489 y Fn(\014)s(;N)1564 1475 y Fw(\()p Fp(dm)p Fw(\))p Fp(m)p Fw(\()p Fp(t)p Fw(\))d Fo(\021)f Fw(1)c Fo(\000)f Fp(f)2268 1489 y Fn(\014)s(;N)2402 1475 y Fw(\()p Fp(t)p Fw(\))1018 b(\(1)p Fp(:)p Fw(6\))50 1730 y(whic)m(h)37 b(is)g(nothing)g(but)g(probabilit)m(y)e(that)k(t)m (w)m(o)g(con\014gurations,)h Fp(\033)n(;)15 b(\033)2555 1697 y Fm(0)2581 1730 y Fw(,)40 b(dra)m(wn)d(indep)s(enden)m(tly)e (form)50 1875 y(the)30 b(Gibbs)f(sample)h(satisfy)g Fp(d)1105 1889 y Fn(N)1174 1875 y Fw(\()p Fp(\033)n(;)15 b(\033)1354 1842 y Fm(0)1380 1875 y Fw(\))26 b Fp(>)f(t)p Fw(.)40 b(The)30 b(function)1284 2119 y Fp(f)1329 2133 y Fn(\014)s(;N)1463 2119 y Fw(\()p Fp(t)p Fw(\))c Fo(\021)f Fp(\026)1743 2080 y Fm(\012)p Fz(2)1743 2146 y Fn(\014)s(;N)1893 2119 y Fw(\()p Fp(d)1975 2133 y Fn(N)2045 2119 y Fw(\()p Fp(\033)n(;)15 b(\033)2225 2082 y Fm(0)2251 2119 y Fw(\))26 b Fo(\024)f Fp(t)p Fw(\))1047 b(\(1)p Fp(:)p Fw(7\))50 2363 y(is)29 b(no)m(w)i(the)f(analog)h(of)g(P)m(arisi's)e(order)h(parameter)1839 2330 y Fz(4)1880 2363 y Fw(.)41 b(In)29 b([BK3])j(w)m(e)e(pro)m(v)m(ed) h(that)985 2607 y Fp(f)1030 2621 y Fn(\014)s(;N)1164 2607 y Fw(\()p Fp(t)p Fw(\))26 b Fo(!)f Fl(E)11 b Fp(f)1509 2621 y Fn(\014)1588 2607 y Fw(=)25 b(min)1851 2506 y Fk(\020)1905 2607 y Fp(\014)1961 2570 y Fm(\000)p Fz(1)2059 2525 y Fo(p)p 2135 2525 197 4 v 82 x Fw(2)15 b(ln)f(2)q Fp(=)2377 2525 y Fk(p)p 2468 2525 152 4 v 2469 2607 a Fw(\026)-46 b Fp(a)p Fw(\()p Fp(t)p Fw(\))q Fp(;)30 b Fw(1)2720 2506 y Fk(\021)3523 2607 y Fw(\(1)p Fp(:)p Fw(8\))50 2851 y(where)38 b(\026)-46 b Fp(a)38 b Fw(is)f(the)h(righ)m (t-deriv)-5 b(ativ)m(e)37 b(of)h(the)g(con)m(v)m(ex)h(h)m(ull)d(of)i (the)g(function)e Fp(A)p Fw(.)63 b(Con)m(v)m(ergence)39 b(in)e(\(1.8\))50 2997 y(holds)29 b(b)s(oth)g(in)g(mean)i(and)f(almost) g(surely)-8 b(.)40 b(W)-8 b(e)31 b(also)g(sho)m(w)m(ed)f(that)1635 3241 y Fo(K)1704 3255 y Fn(\014)s(;N)1881 3189 y Fm(D)1865 3241 y Fo(!)25 b(K)2050 3255 y Fn(\014)2099 3241 y Fp(:)1399 b Fw(\(1)p Fp(:)p Fw(9\))50 3485 y(The)20 b(limit)e(is)i(uniquely)e (determined)h(b)m(y)i(Ghirlanda-Guerra)d(relations,)k(whic)m(h)e(giv)m (e)h(recursiv)m(e)f(form)m(ulas)50 3630 y(to)31 b(compute)g(all)e (momen)m(ts)i(of)f Fo(K)1215 3644 y Fn(\014)1295 3630 y Fw(starting)g(from)g(the)g(function)g Fp(f)2407 3644 y Fn(\014)2455 3630 y Fw(.)159 3848 y(In)i(fact,)i(while)c(the)j (random)e(measures)h Fo(K)1676 3862 y Fn(\014)s(;N)1844 3848 y Fw(ma)m(y)h(lo)s(ok)f(somewhat)h(unfamiliar,)d(their)h(momen)m (ts)50 3994 y(are)24 b(closely)e(link)m(ed)g(and)h(ev)m(en)h(equiv)-5 b(alen)m(t)23 b(to)h(the)g(more)f(con)m(v)m(en)m(tional)h Fp(n)p Fw(-replica)e(distance)h(distribution)50 4139 y Fl(Q)115 4092 y Fz(\()p Fn(n)p Fz(\))115 4167 y Fn(\014)s(;N)256 4139 y Fw(.)40 b(These)30 b(are)h(measures)f(on)h(the)f(space)h([0)p Fp(;)15 b Fw(1])1832 4106 y Fn(n)p Fz(\()p Fn(n)p Fm(\000)p Fz(1\))p Fn(=)p Fz(2)1064 4391 y Fl(Q)1129 4344 y Fz(\()p Fn(n)p Fz(\))1129 4419 y Fn(\014)s(;N)1270 4391 y Fw(\()p Fo(A)p Fw(\))26 b Fo(\021)f Fl(E)10 b Fp(\026)1644 4352 y Fm(\012)p Fn(n)1644 4419 y(\014)s(;N)1800 4290 y Fk(\020)1854 4391 y Fw(\()q Fp(d)p Fw(\()p Fp(i;)15 b(j)5 b Fw(\)\))2157 4418 y Fz(1)p Fm(\024)p Fn(i;j)t Fm(\024)p Fn(N)2483 4391 y Fo(2)24 b(A)2641 4290 y Fk(\021)3452 4391 y Fw(\(1)p Fp(:)p Fw(10\))s Fp(:)50 4635 y Fw(Note)32 b(that)g(these)g(measures)f (do)g(of)g(course)h(giv)m(e)g(full)d(measure)i(to)h(sets)f(that)h(resp) s(ect)f(the)h(ultrametric)50 4781 y(triangle)27 b(relations.)40 b(In)27 b([BK3])j(w)m(e)e(pro)m(v)m(ed)h(their)f(con)m(v)m(ergence)i (to)f(a)g(limiting)d(distribution)e Fl(Q)3376 4733 y Fz(\()p Fn(n)p Fz(\))3376 4808 y Fn(\014)3488 4781 y Fw(.)40 b(The)50 4926 y(Ghirlanda-Guerra)31 b(iden)m(tities)i (\(together)i(with)d(the)i(fact)g(that)h(1)22 b Fo(\000)g Fp(d)2545 4940 y Fn(N)2648 4926 y Fw(is)33 b(an)g(ultrametric)g (distance\))50 5072 y(allo)m(w)25 b(to)h(compute)g Fl(Q)815 5033 y Fn(n)p Fz(+1)815 5099 y Fn(\014)988 5072 y Fw(in)e(terms)i(of)g Fl(Q)1501 5039 y Fn(n)1501 5097 y(\014)1582 5072 y Fw(recursiv)m(ely)-8 b(,)26 b(while)e Fl(Q)2347 5039 y Fz(2)2347 5097 y Fn(\014)2427 5072 y Fw(has)i(distribution)c(function)i Fl(E)10 b Fp(f)3526 5086 y Fn(\014)3581 5072 y Fw(\()p Fp(t)p Fw(\).)p 50 5192 546 4 v 159 5254 a Fu(4)197 5279 y Ft(In)36 b(the)h(con)n(text)f (of)g(the)h(SK)f(mo)r(dels,)k(this)d(function)g(is)g(usually)g (de\014ned)g(with)g Fj(d)2752 5291 y Fi(N)2851 5279 y Ft(replaced)h(b)n(y)e(the)h(\\o)n(v)n(erlap)50 5370 y(parameter)23 b Fj(R)475 5382 y Fi(N)538 5370 y Ft(\()p Fj(\033)n(;)14 b(\033)692 5345 y Fh(0)716 5370 y Ft(\))22 b Fg(\021)f Fj(N)918 5345 y Fh(\000)p Fu(1)1022 5307 y Fk(P)1118 5402 y Fi(i)1159 5370 y Fj(\033)1203 5381 y Fi(i)1232 5370 y Fj(\033)1279 5345 y Fh(0)1276 5395 y Fi(i)1305 5370 y Ft(.)34 b(In)23 b([BK2])h(w)n(e)f(ha)n(v)n(e)g(sho)n(wn)f(that)h (in)h(the)f(GREM,)g(the)g(c)n(hoice)i(of)d(the)i(distance)50 5462 y(used)i(in)g(the)g(de\014nition)h(of)e Fj(f)918 5475 y Fi(\014)s(;N)1069 5462 y Ft(do)r(es)h(not)f(a\013ect)h(the)h (result)g(in)f(the)g(limit)h Fj(N)i Fg(")21 b(1)p Ft(.)p eop %%Page: 6 6 6 5 bop 50 295 a Ft(6)3309 b Fq(Se)l(ction)27 b Ft(1)50 517 y Fw(On)c(the)h(other)h(hand,)f(the)g(full)e(set)j(of)f (distributions)c Fl(Q)1950 470 y Fz(\()p Fn(n)p Fz(\))1950 544 y Fn(\014)2086 517 y Fw(determines)j(the)i(limiting)c(random)i (measures)50 662 y Fo(K)119 676 y Fn(\014)198 662 y Fw(through)30 b(its)g(momen)m(ts.)159 880 y(It)41 b(no)m(w)g(remains)e(to)j(in)m (terpret)e(these)i(limiting)37 b(ob)5 b(jects)42 b(in)e(the)h(con)m (text)h(of)f(Nev)m(eu's)h(branc)m(hing)50 1026 y(pro)s(cess.)57 b(It)37 b(will)c(turn)i(out)i(that)f(b)s(oth)g Fo(K)1569 1040 y Fn(\014)1654 1026 y Fw(and)g Fl(Q)1901 979 y Fz(\()q Fn(n)p Fz(\))1901 1053 y Fn(\014)2050 1026 y Fw(ha)m(v)m(e)h(natural)e (in)m(terpretations.)58 b(The)35 b(former)50 1171 y(will)23 b(b)s(e)j(in)m(terpreted)f(in)g(the)i(language)f(of)h(the)f(con)m(tin)m (uous)g(state)h(branc)m(hing)e(pro)s(cess,)i(while)d(the)i(latter)50 1317 y(are)31 b(naturally)d(in)m(terpreted)i(in)f(the)i(corresp)s (onding)d(coalescen)m(t)k(pro)s(cess)e(on)g(in)m(teger)h(partitions.) 159 1535 y(The)h(remainder)f(of)i(the)g(pap)s(er)f(is)f(organised)h(as) h(follo)m(ws.)47 b(In)32 b(Section)h(2)g(w)m(e)g(describ)s(e)e(a)i (canonical)50 1680 y(construction)j(of)g(the)g(genealogy)i(asso)s (ciated)f(to)g(a)f(\015o)m(w)g(of)h(probabilit)m(y)c(measures)j Fp(\026)3134 1647 y Fn(t)3202 1680 y Fw(on)g([0)p Fp(;)15 b Fw(1])38 b(in)d(a)50 1826 y(general)e(setting.)50 b(In)32 b(this)h(pro)s(cess)g(w)m(e)g(in)m(tro)s(duce)g(the)g(empirical)e (distribution)f Fo(K)2994 1840 y Fn(t)3059 1826 y Fw(on)k(the)f (functions)50 1971 y Fp(m)130 1985 y Fn(x)175 1971 y Fw(\()p Fp(t;)15 b Fo(\001)p Fw(\))31 b(whic)m(h)d(describ)s(e)g(the)h (dep)s(endence)f(of)i(the)f(family)f(size)h(of)h(the)f(individual)c Fp(x)k Fw(as)h(a)g(function)e(of)50 2117 y(the)d(degree)i(of)e (relatedness.)39 b(W)-8 b(e)27 b(also)e(de\014ne)g(the)g(genealogical)i (distance)e(b)s(et)m(w)m(een)h(an)m(y)g(t)m(w)m(o)h(p)s(oin)m(ts)d(on) 50 2262 y([0)p Fp(;)15 b Fw(1])31 b(as)e(the)h(last)f(time)g(they)g (had)g(an)g(ancestor)h(in)e(common.)41 b(W)-8 b(e)30 b(are)g(mostly)f(in)m(terested)g(in)f(the)i(case)50 2408 y(when)e(the)h(measures)g(in)f(the)h(\015o)m(w)g(are)g(random.)40 b(Consequen)m(tly)-8 b(,)29 b(w)m(e)g(de\014ne)g(the)g(distance)g(on)f (in)m(tegers)50 2553 y Fp(i;)15 b(j)43 b Fw(as)38 b(the)g(distance)f(b) s(et)m(w)m(een)h(indep)s(enden)m(t)d(uniformly)f(distributed)h(random)h (v)-5 b(ariables)36 b Fp(U)3405 2567 y Fn(i)3436 2553 y Fp(;)15 b(U)3538 2567 y Fn(j)3614 2553 y Fw(on)50 2699 y([0)p Fp(;)g Fw(1])28 b(and)e(consider)f(the)i(partitions)e(of)i(in)m (tegers)g(in)e(blo)s(c)m(ks)h(whose)g(distance)g(is)g(most)h Fp(s)p Fw(.)39 b(The)26 b(family)f(of)50 2844 y(these)h(partitions)d (as)j(a)f(pro)s(cess)g(of)h Fp(s)e Fw(form)h(a)h(coalescen)m(t)h(pro)s (cess.)39 b(It)25 b(turns)f(out)h(that)h Fo(K)3139 2858 y Fn(t)3197 2844 y Fw(is)e(completely)50 2989 y(determined)k(b)m(y)i (this)e(coalescen)m(t,)k(as)e(w)m(e)h(giv)m(e)f(explicit)e(expressions) g(for)i(all)e(its)h(momen)m(ts)i(in)d(terms)i(of)50 3135 y(probabilities)d(of)j(random)g(partitions.)159 3353 y(In)h(Section)h(3)h(w)m(e)g(sho)m(w)f(ho)m(w)g(this)f(construction)h (w)m(orks)g(for)h(a)f(\015o)m(w)g(of)h(the)f(Gibbs)f(measures)h Fp(\026)3575 3367 y Fn(\014)s(;N)50 3499 y Fw(on)37 b Fo(S)238 3513 y Fn(N)345 3499 y Fw(\(iden)m(ti\014ed)f(with)h([0)p Fp(;)15 b Fw(1])39 b(via)f(the)g(canonical)f(map)h(\(2.1\)\))h(of)f (the)g(CREM)g(with)e(an)i(arbitrary)50 3644 y(function)g Fp(A)p Fw(.)67 b(Namely)-8 b(,)42 b Fo(K)1006 3658 y Fn(\014)s(;N)1181 3644 y Fw(of)d(\(1.4\))i(is)d(precisely)g Fo(K)2069 3658 y Fz(1)2149 3644 y Fw(de\014ned)g(in)g(Section)h(1)h (for)f(this)f(\015o)m(w.)68 b(W)-8 b(e)50 3789 y(also)30 b(explain)f(ho)m(w)h(this)g(construction)g(can)g(b)s(e)g(reform)m (ulated)g(for)g(the)h(Gaussian)e(pro)s(cess)h(on)h(the)f(unit)50 3935 y(in)m(terv)-5 b(al.)159 4153 y(In)31 b(Section)h(4)g(w)m(e)h (form)m(ulate)f(our)f(main)g(theorem.)46 b(It)32 b(iden)m(ti\014es)f (the)h(limit)d(as)k Fp(N)38 b Fo(")28 b(1)k Fw(of)g Fp(K)3467 4167 y Fn(\014)s(;N)3634 4153 y Fw(in)50 4299 y(terms)20 b(of)g(the)h(\015o)m(w)f(of)h(measures)f(corresp)s(onding)e(to)j(Nev)m (eu's)g(branc)m(hing)e(pro)s(cess:)35 b Fp(X)7 b Fw(\()p Fp(t)p Fw(\()p Fp(y)s Fw(\))p Fp(;)15 b(x)p Fw(\))p Fp(=X)7 b Fw(\()p Fp(t)p Fw(\()p Fp(y)s Fw(\))p Fp(;)15 b Fw(1\),)50 4444 y(where)23 b Fp(X)7 b Fw(\()p Fp(t;)15 b(x)p Fw(\))24 b(is)f(the)g(size)h(of)f(the)h(p)s(opulation)d(of)i(this)g(pro)s(cess)g (at)h(time)f Fp(t)p Fw(,)i(pro)m(vided)d(that)i(at)g(the)f(initial)50 4589 y(momen)m(t)j(it)g(w)m(as)g Fp(x)p Fw(.)40 b(Here)26 b Fp(t)p Fw(\()p Fp(y)s Fw(\))h(is)e(an)h(appropriate)f(time)g(c)m (hange)j(de\014ned)c(only)i(b)m(y)g Fl(E)10 b Fp(f)3077 4603 y Fn(\014)3132 4589 y Fw(\()p Fp(y)s Fw(\))26 b(of)g(\(1.9\).)41 b(In)50 4735 y(fact,)32 b(conceptually)f(w)m(e)g(do)h(sligh)m(tly)d (more:)42 b(w)m(e)32 b(sho)m(w)f(that)h(there)f(is)f(a)i(\015o)m(w)f (of)g(probabilit)m(y)e(measures)57 4880 y(~)-52 b Fp(\026)105 4847 y Fn(t)105 4906 y(\014)s(;N)266 4880 y Fw(constructed)27 b(via)f(an)h(em)m(b)s(edding)e(of)i(Gibbs)e(measures)h Fp(\026)2278 4847 y Fn(t)2278 4906 y(\014)s(;N)2440 4880 y Fw(that)h(con)m(v)m(erges)i(to)e(a)g(limiting)d(\015o)m(w)50 5026 y(of)i(measures)g(\002)606 4993 y Fn(t)606 5052 y(f)641 5062 y Fr(\014)715 5026 y Fw(constructed)g(from)g(a)g(time)g(c) m(hanged)h(Nev)m(eu)g(branc)m(hing)d(pro)s(cess,)j(in)e(the)h(sense)g (that)50 5171 y(the)k(genealogies)i(of)e(the)h(\015o)m(w)f(con)m(v)m (erges.)159 5389 y(In)44 b(Sections)g(5)h(and)e(6)i(w)m(e)g(pro)m(v)m (e)g(this)f(theorem.)83 b(Since)44 b Fo(K)2376 5403 y Fn(t)2452 5389 y Fw(is)g(determined)f(b)m(y)h(its)g(momen)m(ts,)p eop %%Page: 7 7 7 6 bop 50 295 a Fq(GREM,)28 b(p)l(art)f(4)3127 b Ft(7)50 513 y Fw(or)40 b(equiv)-5 b(alen)m(tly)d(,)41 b(b)m(y)f(the)g (genealogical)g(distances)f(of)h(in)m(tegers)g(for)g(the)g(corresp)s (onding)e(coalescen)m(t,)50 658 y(as)g(w)m(as)g(established)e(at)i(the) g(end)f(of)h(Section)f(2,)j(its)d(su\016ces)h(to)g(sho)m(w)g(that)g (the)g Fp(n)p Fw(-replica)e(distance)50 804 y(distribution)e(functions) j(\(1.10\))k(of)d(our)g(spin)f(glass)h(mo)s(del)f(con)m(v)m(erge)j(to)g (the)e(genealogical)h(distance)50 949 y(distribution)20 b(function)i(of)i(the)g(Bolthausen-Sznitman)f(coalescen)m(t)i(\(whic)m (h)e(corresp)s(onds)g(to)h Fo(K)3382 963 y Fz(1)3423 949 y Fw(\))g(under)50 1095 y(an)g(appropriate)f(time)g(c)m(hange.)40 b(One)23 b(w)m(a)m(y)i(\(short)f(but)f(indirect\))g(to)h(pro)m(v)m(e)h (this)e(is)g(indicated)f(in)h(Section)50 1240 y(5)36 b(and)e(relies)g(on)h(the)h(connection)f(b)s(et)m(w)m(een)h(Nev)m(eu's) h(branc)m(hing)d(pro)s(cess)g(and)h(Ruelle's)f(probabilit)m(y)50 1385 y(cascades)d(established)e(in)g([BeLG].)159 1604 y(The)46 b(second)h(w)m(a)m(y)h(\(more)g(direct\))e(is)g(giv)m(en)h(in) f(Section)g(6:)75 b(it)46 b(consists)g(in)g(sho)m(wing)g(that)i(the)50 1749 y(Bolthausen-Sznitman)34 b(coalescen)m(t)j(satis\014es)d (Ghirlanda-Guerra)g(iden)m(tities.)53 b(F)-8 b(or)36 b(that)g(purp)s(ose)d(w)m(e)50 1895 y(use)d(the)g(Chinese)f(restauran)m (t)i(pro)s(cess)f(of)h(J.)f(Pitman)f([P].)159 2113 y(W)-8 b(e)34 b(hop)s(e)f(that)h(the)f(results)f(presen)m(ted)h(in)f(this)g (class)h(of)g(mo)s(dels)f(elucidate)h(in)f(a)h(mathematically)50 2258 y(comprehensible)39 b(con)m(text)k(the)e(fundamen)m(tal)f(and)g (univ)m(ersal)f(r^)-45 b(ole)41 b(pla)m(y)m(ed)g(b)m(y)g(Nev)m(eu's)h (con)m(tin)m(uous)50 2404 y(state)33 b(branc)m(hing)e(pro)s(cess)g(as)h (a)h(univ)m(ersal)d(random)h(mec)m(hanism)g(go)m(v)m(erning)i(the)f (extremal)g(pro)s(cesses)50 2549 y(for)39 b(a)h(wide)f(class)g(of)h (sto)s(c)m(hastic)g(pro)s(cesses.)68 b(If)39 b(one)h(accepts)h(the)f (common)g(b)s(eliev)m(e)f(of)g(theoretical)50 2695 y(ph)m(ysicists,)32 b(its)h(r^)-45 b(ole)33 b(go)s(es)g(w)m(ell)g(b)s(ey)m(ond)f(the)h (class)g(of)h(mo)s(dels)d(w)m(e)j(discuss)d(here.)49 b(Ev)m(en)34 b(on)f(a)g(sligh)m(tly)50 2840 y(less)39 b(sp)s(eculativ)m(e)g(lev)m(el,)j(Nev)m(eu's)f(pro)s(cess)e(will)e 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b(on)h(con)m(tin)m(uous)f(state)i(branc)m(hing,)e (sub)s(ordinators,)g(and)g(coalescen)m(ts.)58 b(A.B.)36 b(also)g(ac)m(kno)m(wl-)50 4222 y(edges)k(inspiring)c(discussions)h (with)i(An)m(ton)h(W)-8 b(ak)m(olbinger.)70 b(I.K.)40 b(thanks)f(Ph.)h(Marc)m(hal)g(for)g(helpful)50 4367 y(discussions.)47 b(P)m(art)34 b(of)g(this)e(w)m(ork)i(w)m(as)g(done)f(while)f(A.B.)i(w)m (as)g(residen)m(t)f(at)h(the)g(Erwin-Sc)m(hro)s(edinger)50 4513 y(Institute)f(for)g(Mathematical)i(Ph)m(ysics)e(\(ESI\))g(in)g (Vienna)g(in)f(the)i(programme)f(\\Mathematical)j(p)s(op-)50 4658 y(ulation)j(genetics)j(and)e(statistical)g(ph)m(ysics".)71 b(He)42 b(thanks)e(the)h(organisers,)i(Ellen)c(Baak)m(e,)46 b(Mic)m(hael)50 4804 y(Baak)m(e,)35 b(and)d(Reinhard)f(B)s(\177)-48 b(urger)33 b(for)f(the)h(kind)e(in)m(vitation)g(and)h(the)h(ESI)f(for)g (\014nancial)f(supp)s(ort.)46 b(I.K.)50 4949 y(thanks)39 b(the)g(W)-8 b(eierstrass)40 b(Institute)f(for)g(Applied)d(Analysis)i 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Fm(\000)p Fn(i)1985 1266 y Fw(\(1)21 b(+)f Fp(\033)2229 1280 y Fn(i)2259 1266 y Fw(\))p Fp(=)p Fw(2)p Fp(:)1114 b Fw(\(2)p Fp(:)p Fw(1\))50 1500 y(It)41 b(w)m(as)h(sho)m(wn)f(that)h(the)f(phase) h(transition)d(in)i(the)g(REM)h(manifests)e(itself)g(b)m(y)i(the)f (fact)i(that)f(the)50 1645 y(resulting)27 b(image)h(measure)h(con)m(v)m (erges)i(to)e(Leb)s(esgue)f(measure)h(in)e(the)i(high)f(temp)s(erature) g(phase)g(\()p Fp(\014)j Fo(\024)50 1713 y(p)p 126 1713 197 4 v 78 x Fw(2)15 b(ln)f(2)q(\))25 b(and)g(to)m(w)m(ards)h(a)f (dense)g(pure)g(p)s(oin)m(t)f(measure)h(in)f(the)i(lo)m(w-temp)s (erature)f(phase)g(\()p Fp(\014)31 b(>)3376 1713 y Fo(p)p 3452 1713 V 78 x Fw(2)15 b(ln)g(2\).)50 1936 y(While)30 b(in)h(the)g(REM)h(this)f(app)s(eared)f(to)j(giv)m(e)f(a)g(rather)f (nice)h(picture,)f(at)h(\014rst)f(glance)h(it)f(seems)h(to)g(b)s(e)50 2082 y(di\016cult)42 b(to)i(enco)s(de)g(the)g(far)g(more)g(complex)f (structure)h(of)g(the)g(Gibbs)e(measures)h(of)h(the)g(GREM)50 2227 y(and)37 b(CREM)g(in)g(suc)m(h)g(a)h(simple)e(em)m(b)s(edding.)61 b(Our)36 b(purp)s(ose)g(w)m(ould)h(b)s(e)g(to)h(iden)m(tify)e(a)i (measure)g(on)50 2373 y([0)p Fp(;)15 b Fw(1])42 b(that)g(represen)m(ts) f(the)g(limiting)d(Gibbs)h(measure.)72 b(It)41 b(will)e(b)s(e)h (instructiv)m(e)g(to)h(explain)e(ho)m(w)i(a)50 2518 y(naiv)m(e)g (approac)m(h)g(to)h(do)f(so)g(fails.)72 b(On)40 b(the)h(h)m(yp)s(ercub) s(e)f(w)m(e)h(are)h(in)m(terested)f(in)f(the)h(masses)g(of)g(sets)50 2663 y Fp(b)89 2677 y Fn(\033)138 2663 y Fw(\()p Fp(t)p Fw(\))31 b Fo(\021)f(f)p Fp(\033)473 2630 y Fm(0)530 2663 y Fw(:)g Fp(d)632 2677 y Fn(N)702 2663 y Fw(\()p Fp(\033)n(;)15 b(\033)882 2630 y Fm(0)908 2663 y Fw(\))31 b Fp(>)f(t)p Fo(g)p Fw(.)50 b(If)33 b(w)m(e)h(map)g(suc)m(h)f(sets)h (on)f(the)h(unit)e(in)m(terv)-5 b(al)32 b(via)i Fp(r)3063 2677 y Fn(N)3131 2663 y Fw(,)h(w)m(e)f(obtain)f(an)50 2809 y(in)m(terv)-5 b(al)36 b(of)h(length)f(2)824 2776 y Fm(\000)p Fz([)p Fn(tN)7 b Fz(])1019 2809 y Fw(.)60 b(In)36 b(fact)i(there)f(is)f(no)g(di\016cult)m(y)g(to)h(express)f (e.g.)62 b Fo(K)2971 2823 y Fn(\014)s(;N)3143 2809 y Fw(for)36 b Fp(N)47 b Fw(\014xed)36 b(in)50 2954 y(terms)d(of)h(quan)m (tities)f(de\014ned)f(with)h(resp)s(ect)g(to)h(the)g(image)g(measure)g (on)f(the)h(h)m(yp)s(ercub)s(e.)48 b(Ho)m(w)m(ev)m(er,)50 3100 y(the)40 b(construction)g(in)m(v)m(olv)m(es)h(masses)f(of)h(in)m (terv)-5 b(als)39 b(of)i(exp)s(onen)m(tially)e(small)g(size)h(\(in)f Fp(N)10 b Fw(\).)72 b(So)40 b(what)50 3245 y(should)31 b(one)j(do)g(in)e(the)i(limit)e(when)g Fp(N)44 b Fw(is)33 b(in\014nite?)48 b(W)-8 b(e)35 b(cannot)f(analyse)f(the)h(structure)f (b)m(y)h(lo)s(oking)50 3391 y(at)d(in)m(terv)-5 b(als)29 b(of)i(the)f(size)h(2)1003 3358 y Fm(\000)p Fn(t)p Fm(1)1164 3391 y Fw(.)159 3609 y(What)25 b(is)e(needed)h(is)f(clearly)g(a)i (construction)f(that)g(do)s(es)g(not)h(refer)f(explicitly)d(to)k (masses)f(of)h(in)m(terv)-5 b(als)50 3754 y(of)37 b(exp)s(onen)m (tially)d(small)h(size)i(while)d(still)h(rev)m(ealing)h(the)h(\014ne)f (structure)g(of)h(the)f(measure)h(at)g(suc)m(h)f(a)50 3900 y(scale.)k(In)27 b(this)f(section)i(w)m(e)g(sho)m(w)g(that)g(a)g (canonical)f(construction)h(exists)f(when)g(w)m(e)h(consider)e(a)i (family)50 4045 y(of)i(probabilit)m(y)e(measures)i(on)h([0)p Fp(;)15 b Fw(1].)159 4263 y(Let)40 b Fo(f)p Fp(\026)431 4230 y Fn(t)463 4263 y Fo(g)508 4281 y Fn(t)p Fm(2)p Ff(R)634 4262 y Fs(+)727 4263 y Fw(b)s(e)f(a)h(family)e(of)h (probabilit)m(y)f(measures)h(on)h([0)p Fp(;)15 b Fw(1].)70 b(Denote)41 b(b)m(y)f(\002)3157 4277 y Fn(t)3228 4263 y Fw(their)f(\(righ)m(t-)50 4423 y(con)m(tin)m(uous\))f(probabilit)m(y) e(distribution)e(functions)j(\002)1995 4437 y Fn(t)2026 4423 y Fw(\()p Fp(x)p Fw(\))i(=)2315 4332 y Fn(x)2296 4350 y Fk(R)2299 4507 y Fz(0)2372 4423 y Fp(\026)2427 4390 y Fn(t)2459 4423 y Fw(\()p Fp(dx)p Fw(\).)64 b(Let)39 b(us)e(note)i(that)g(w)m(e)f(will)50 4597 y(mostly)29 b(b)s(e)f(in)m(terested)i(in)d(cases)k(when)d(the)h(measures)g Fp(\026)2051 4564 y Fn(t)2112 4597 y Fw(are)h(getting)g(more)f(and)g (more)g(irregular)f(as)h Fp(t)50 4743 y Fw(increases.)40 b(W)-8 b(e)32 b(will)c(consider)h(\002)1219 4757 y Fn(t)1281 4743 y Fw(as)i(a)g(map)f([0)p Fp(;)15 b Fw(1])27 b Fo(!)e Fw([0)p Fp(;)15 b Fw(1],)33 b(so)e(that)g Fo(f)p Fw(\002)2657 4757 y Fn(t)2689 4743 y Fo(g)2734 4760 y Fn(t)p Fm(2)p Ff(R)2860 4742 y Fs(+)2943 4743 y Fw(represen)m(ts)f(a)h(\015o)m(w)g (of)50 4888 y(maps)f(on)g(the)h(unit)d(in)m(terv)-5 b(al.)40 b(De\014ne)31 b(for)f Fp(t)1580 4855 y Fm(0)1631 4888 y Fp(<)25 b(t)1400 5079 y(S)1461 5042 y Fz(\()p Fn(t)1516 5015 y Fe(0)1542 5042 y Fn(;t)p Fz(\))1624 5079 y Fw(\()p Fp(x)p Fw(\))g Fo(\021)g Fw(\002)1938 5093 y Fn(t)1985 5006 y Fk(\000)2027 5079 y Fw(\002)2098 5041 y Fm(\000)p Fz(1)2098 5106 y Fn(t)2125 5088 y Fe(0)2195 5079 y Fw(\()p Fp(x)p Fw(\))2317 5006 y Fk(\001)3523 5079 y Fw(\(2)p Fp(:)p Fw(2\))50 5271 y(where)30 b(the)g(in)m(v)m(erse)h(of)f(a)h(righ) m(t-con)m(tin)m(uous,)f(non-decreasing)g(function)f(\002)h(is)f (de\014ned)h(as)1317 5462 y(\002)1388 5424 y Fm(\000)p Fz(1)1486 5462 y Fw(\()p Fp(x)p Fw(\))c(=)f(inf)5 b Fo(f)p Fp(y)29 b Fo(j)c Fw(\002\()p Fp(y)s Fw(\))h Fo(\025)f Fp(x)p Fo(g)p Fp(:)1081 b Fw(\(2)p Fp(:)p Fw(3\))p eop %%Page: 9 9 9 8 bop 50 295 a Fq(GREM,)28 b(p)l(art)f(4)3127 b Ft(9)50 513 y Fw(W)-8 b(e)30 b(will)25 b(need)k(some)g(elemen)m(tary)g(prop)s (erties)e(of)h(the)h(in)m(v)m(erse)g(function.)38 b(W)-8 b(e)30 b(sa)m(y)f(that)h(\002)e(increases)g(at)50 658 y Fp(x)p Fw(,)i(if)g(for)g(an)m(y)h Fp(\017)25 b(>)g Fw(0,)31 b(\002\()p Fp(x)p Fw(\))25 b Fp(>)g Fw(\002\()p Fp(x)20 b Fo(\000)g Fp(\017)p Fw(\).)50 876 y Fd(Lemma)45 b(2.1:)118 b Fv(L)-5 b(et)38 b Fw(\002)g Fv(b)-5 b(e)38 b(a)g(non-de)-5 b(cr)g(e)g(asing)40 b(function.)59 b(Then)38 b(for)h(any)f(p)-5 b(oint)40 b Fp(x)e Fv(at)g(which)h Fw(\002)f Fv(is)50 1022 y(incr)-5 b(e)g(asing,)1577 1167 y Fw(\002)1648 1130 y Fm(\000)p Fz(1)1745 1167 y Fw(\(\002\()p Fp(x)p Fw(\)\))27 b(=)e Fp(x)1340 b Fw(\(2)p Fp(:)p Fw(4\))50 1585 y Fc(Pro)s(of:)47 b Fw(W)-8 b(e)32 b(ha)m(v)m(e)1189 1730 y(\002)1260 1693 y Fm(\000)p Fz(1)1357 1730 y Fw(\(\002\()p Fp(x)p Fw(\)\))26 b(=)f(inf)6 b Fo(f)p Fp(y)28 b Fw(:)e(\002\()p Fp(y)s Fw(\))f Fo(\025)g Fw(\002\()p Fp(x)p Fw(\))p Fo(g)953 b Fw(\(2)p Fp(:)p Fw(5\))50 1930 y(Since)34 b(\002)h(is)f (non-decreasing,)i(for)f(an)m(y)g Fp(y)h Fo(\025)d Fp(x)p Fw(,)k(\002\()p Fp(y)s Fw(\))c Fo(\025)g Fw(\002\()p Fp(x)p Fw(\))j(and)e(th)m(us)h(\002)2773 1897 y Fm(\000)p Fz(1)2870 1930 y Fw(\(\002\()p Fp(x)p Fw(\)\))f Fo(\024)f Fp(x)p Fw(.)55 b(Assume)50 2075 y(that)32 b(\002)319 2042 y Fm(\000)p Fz(1)416 2075 y Fw(\(\002\()p Fp(x)p Fw(\)\))c(=)f Fp(y)j(<)d(x)p Fw(.)43 b(This)30 b(implies)f(that)j(for)f (some)h Fp(y)d(<)e(x)p Fw(,)32 b(\002\()p Fp(y)s Fw(\))27 b(=)g(\002\()p Fp(x)p Fw(\),)32 b(con)m(tradicting)g(the)50 2221 y(assumption)d(that)i Fp(x)f Fw(is)f(a)i(p)s(oin)m(t)e(of)i (increase.)40 b Fo(})50 2439 y Fd(Lemma)29 b(2.2:)47 b Fv(F)-7 b(or)28 b(a)e(given)g(family)i(of)e(me)-5 b(asur)g(es)28 b Fp(\026)1948 2406 y Fn(t)1980 2439 y Fv(,)f(let)g Fo(I)2207 2453 y Fn(t)2265 2439 y Fv(denote)g(of)g(the)g(set)f(of)h(p)-5 b(oints)28 b(of)f(incr)-5 b(e)g(ase)50 2585 y(of)33 b(the)g(function)f Fw(\002)736 2599 y Fn(t)768 2585 y Fv(.)42 b(Assume)32 b(that)i(for)f Fp(t)1541 2552 y Fm(00)1613 2585 y Fp(<)25 b(t)1742 2552 y Fm(0)1792 2585 y Fp(<)g(t)p Fv(,)33 b Fo(I)2032 2599 y Fn(t)2059 2580 y Fe(00)2133 2585 y Fo(\032)25 b(I)2279 2599 y Fn(t)2306 2580 y Fe(0)2335 2585 y Fv(.)42 b(Then)1286 2818 y Fp(S)1347 2780 y Fz(\()p Fn(t)1402 2753 y Fe(0)q(0)1448 2780 y Fn(;t)p Fz(\))1530 2818 y Fw(\()p Fp(x)p Fw(\))26 b(=)e Fp(S)1834 2780 y Fz(\()p Fn(t)1889 2753 y Fe(0)1915 2780 y Fn(;t)p Fz(\))2017 2818 y Fo(\016)d Fp(S)2144 2780 y Fz(\()p Fn(t)2199 2753 y Fe(0)q(0)2244 2780 y Fn(;t)2293 2753 y Fe(0)2318 2780 y Fz(\))2351 2818 y Fw(\()p Fp(x)p Fw(\))1050 b(\(2)p Fp(:)p Fw(6\))50 3051 y Fv(for)33 b(any)g Fp(x)25 b Fo(2)g Fw([0)p Fp(;)15 b Fw(1])p Fv(.)50 3269 y Fc(Pro)s(of:)47 b Fw(By)31 b(de\014nition,)1018 3502 y Fp(S)1079 3464 y Fz(\()p Fn(t)1134 3437 y Fe(0)1160 3464 y Fn(;t)p Fz(\))1262 3502 y Fo(\016)21 b Fp(S)1389 3464 y Fz(\()p Fn(t)1444 3437 y Fe(0)q(0)1489 3464 y Fn(;t)1538 3437 y Fe(0)1563 3464 y Fz(\))1596 3502 y Fw(\()p Fp(x)p Fw(\))26 b(=)f(\002)1911 3516 y Fn(t)1942 3502 y Fw(\(\002)2048 3463 y Fm(\000)p Fz(1)2048 3529 y Fn(t)2075 3510 y Fe(0)2146 3502 y Fw(\(\002)2252 3516 y Fn(t)2279 3497 y Fe(0)2309 3502 y Fw(\(\002)2415 3463 y Fm(\000)p Fz(1)2415 3529 y Fn(t)2442 3510 y Fe(00)2512 3502 y Fw(\()p Fp(x)p Fw(\)\)\)\))784 b(\(2)p Fp(:)p Fw(7\))50 3735 y(Note)33 b(\014rst)f(that)h(since)f Fp(y)f Fw(=)d(\002)1131 3696 y Fm(\000)p Fz(1)1131 3762 y Fn(t)1158 3743 y Fe(00)1228 3735 y Fw(\()p Fp(x)p Fw(\))33 b(is)f(the)g(smallest) g(v)-5 b(alue)31 b(for)h(whic)m(h)g(\002\()p Fp(y)s Fw(\))c Fo(\025)g Fp(x)p Fw(,)33 b(for)f(an)m(y)h Fp(y)3416 3702 y Fm(0)3470 3735 y Fp(<)28 b(y)35 b Fw(it)50 3880 y(m)m(ust)30 b(b)s(e)g(true)g(that)h(\002)859 3894 y Fn(t)886 3876 y Fe(00)935 3880 y Fw(\()p Fp(y)1018 3847 y Fm(0)1044 3880 y Fw(\))26 b Fp(<)f Fw(\002)1272 3894 y Fn(t)1299 3876 y Fe(00)1348 3880 y Fw(\()p Fp(y)s Fw(\).)41 b(Th)m(us)29 b(\002)1833 3842 y Fm(\000)p Fz(1)1833 3907 y Fn(t)1860 3889 y Fe(0)q(0)1931 3880 y Fw(\()p Fp(x)p Fw(\))d Fo(2)e(I)2214 3894 y Fn(t)2241 3876 y Fe(00)2291 3880 y Fw(.)159 4099 y(But)34 b(if)f Fp(y)k Fw(is)c(a)h(p)s(oin)m(t)f(of)i(increase)f(of)g (\002)1561 4113 y Fn(t)1588 4094 y Fe(00)1637 4099 y Fw(,)h(b)m(y)f(assumption)f Fp(y)j Fw(is)e(also)g(a)g(p)s(oin)m(t)f(of) h(increase)g(of)g(\002)3628 4113 y Fn(t)3655 4094 y Fe(0)3685 4099 y Fw(,)50 4244 y(hence)1115 4390 y Fp(S)1176 4352 y Fz(\()p Fn(t)1231 4325 y Fe(0)1256 4352 y Fn(;t)p Fz(\))1359 4390 y Fo(\016)20 b Fp(S)1485 4352 y Fz(\()p Fn(t)1540 4325 y Fe(0)q(0)1586 4352 y Fn(;t)1635 4325 y Fe(0)1659 4352 y Fz(\))1692 4390 y Fw(\()p Fp(x)p Fw(\))26 b(=)f(\002)2007 4404 y Fn(t)2039 4390 y Fw(\()p Fp(y)s Fw(\))h(=)f Fp(S)2340 4352 y Fz(\()p Fn(t)2395 4325 y Fe(0)q(0)2440 4352 y Fn(;t)p Fz(\))2522 4390 y Fw(\()p Fp(x)p Fw(\))879 b(\(2)p Fp(:)p Fw(8\))50 4589 y(whic)m(h)29 b(pro)m(v)m(es)i(the)g(lemma.)40 b Fo(})50 4807 y Fx(Remark:)55 b Fw(W)-8 b(e)36 b(see)f(that)g(the)f (construction)g(of)h(the)f Fp(S)2024 4774 y Fz(\()p Fn(t)2079 4747 y Fe(0)2105 4774 y Fn(;t)p Fz(\))2221 4807 y Fw(is)f(b)s(est)h (suited)f(in)h(situations)f(where)h(the)50 4953 y(distribution)41 b(functions)i(\002)1037 4967 y Fn(t)1114 4953 y Fw(are)i(ev)m(erywhere) g(increasing.)83 b(W)-8 b(e)46 b(will)c(ho)m(w)m(ev)m(er)k(encoun)m (ter)f(a)h(more)50 5098 y(delicate)30 b(situation)f(where)h(\002)1096 5065 y Fn(t)1158 5098 y Fw(are)h(step)g(functions)d(with)i Fo(I)2152 5112 y Fn(t)2179 5094 y Fe(0)2233 5098 y Fo(\032)25 b(I)2379 5112 y Fn(t)2441 5098 y Fw(for)30 b(all)f(0)d Fo(\024)f Fp(t)2906 5065 y Fm(0)2956 5098 y Fo(\024)g Fp(t)g(<)g Fo(1)p Fw(.)50 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1702 l 2706 1721 l 2705 1739 l 2703 1758 l 2700 1778 l 2696 1798 l 2692 1819 l 2686 1839 l 2680 1860 l 2673 1881 l 2665 1901 l 2657 1922 l 2648 1942 l 2639 1961 l 2630 1981 l 2620 2000 l 2610 2019 l 2600 2038 l 2590 2055 l 2581 2072 l 2570 2090 l 2560 2109 l 2549 2128 l 2538 2147 l 2526 2167 l 2514 2188 l 2502 2209 l 2490 2230 l 2478 2252 l 2467 2273 l 2455 2295 l 2444 2316 l 2433 2337 l 2423 2358 l 2414 2378 l 2405 2398 l 2396 2418 l 2389 2437 l 2382 2456 l 2375 2475 l 2368 2496 l 2362 2517 l 2357 2538 l 2352 2559 l 2347 2581 l 2343 2604 l 2339 2627 l 2336 2650 l 2333 2673 l 2330 2697 l 2328 2720 l 2326 2742 l 2324 2764 l 2322 2786 l 2320 2807 l 2319 2827 l 2317 2846 l 2316 2865 l 2314 2883 l 2313 2900 l 2310 2919 l 2308 2938 l 2306 2957 l 2303 2975 l 2299 2995 l 2295 3014 l 2291 3033 l 2286 3052 l 2281 3072 l 2275 3090 l 2268 3109 l 2262 3127 l 2254 3145 l 2247 3162 l 2239 3178 l 2231 3194 l 2222 3210 l 2213 3225 l 2203 3240 l 2192 3256 l 2181 3272 l 2168 3288 l 2155 3304 l 2141 3321 l 2126 3339 l 2111 3356 l 2095 3373 l 2078 3391 l 2062 3408 l 2045 3424 l 2029 3440 l 2013 3456 l 1996 3471 l 1981 3485 l 1965 3499 l 1950 3513 l 1935 3526 l 1919 3539 l 1903 3552 l 1886 3566 l 1868 3581 l 1849 3597 l 1828 3613 l 1805 3631 l 1781 3650 l 1756 3669 l 1731 3688 l 1708 3706 l 1687 3722 l 1671 3734 l 1660 3743 l 1653 3748 l 1650 3750 l gs col2 s gr % Polyline n 3105 885 m 3103 888 l 3098 894 l 3090 906 l 3078 921 l 3063 941 l 3046 964 l 3029 988 l 3011 1011 l 2994 1034 l 2979 1056 l 2965 1075 l 2952 1093 l 2941 1108 l 2931 1123 l 2921 1136 l 2913 1148 l 2905 1160 l 2895 1174 l 2887 1189 l 2878 1203 l 2870 1217 l 2862 1230 l 2856 1244 l 2849 1258 l 2844 1272 l 2840 1285 l 2836 1298 l 2833 1311 l 2831 1323 l 2830 1335 l 2830 1348 l 2830 1360 l 2831 1373 l 2833 1387 l 2836 1401 l 2840 1417 l 2844 1433 l 2849 1449 l 2856 1466 l 2862 1482 l 2870 1499 l 2878 1515 l 2887 1530 l 2895 1545 l 2905 1560 l 2914 1573 l 2923 1586 l 2934 1600 l 2945 1614 l 2956 1628 l 2968 1643 l 2981 1659 l 2994 1674 l 3007 1690 l 3021 1706 l 3034 1722 l 3046 1737 l 3059 1753 l 3070 1768 l 3082 1783 l 3093 1798 l 3103 1813 l 3114 1828 l 3124 1845 l 3135 1862 l 3145 1879 l 3155 1898 l 3165 1916 l 3175 1935 l 3184 1954 l 3193 1972 l 3201 1991 l 3208 2008 l 3214 2025 l 3220 2042 l 3225 2057 l 3230 2073 l 3235 2090 l 3239 2107 l 3242 2124 l 3245 2142 l 3248 2160 l 3250 2178 l 3252 2196 l 3253 2214 l 3254 2231 l 3254 2248 l 3255 2264 l 3255 2280 l 3255 2295 l 3255 2310 l 3255 2323 l 3255 2336 l 3255 2350 l 3254 2364 l 3254 2379 l 3253 2395 l 3252 2411 l 3250 2428 l 3248 2445 l 3245 2463 l 3242 2480 l 3238 2499 l 3234 2517 l 3229 2535 l 3224 2553 l 3218 2573 l 3212 2588 l 3206 2605 l 3200 2622 l 3192 2639 l 3185 2658 l 3176 2677 l 3167 2697 l 3158 2717 l 3148 2738 l 3137 2758 l 3127 2779 l 3116 2799 l 3105 2819 l 3094 2838 l 3083 2856 l 3072 2874 l 3061 2890 l 3051 2906 l 3040 2921 l 3030 2935 l 3017 2952 l 3004 2968 l 2990 2983 l 2976 2998 l 2962 3012 l 2947 3026 l 2931 3039 l 2916 3051 l 2900 3063 l 2884 3074 l 2868 3084 l 2853 3093 l 2837 3101 l 2822 3109 l 2807 3116 l 2793 3123 l 2777 3129 l 2762 3135 l 2745 3141 l 2728 3147 l 2711 3154 l 2692 3160 l 2674 3166 l 2655 3173 l 2636 3179 l 2618 3185 l 2599 3191 l 2582 3198 l 2565 3204 l 2548 3210 l 2533 3216 l 2518 3223 l 2500 3230 l 2483 3238 l 2466 3247 l 2449 3256 l 2432 3266 l 2415 3276 l 2398 3287 l 2381 3298 l 2366 3308 l 2352 3319 l 2338 3330 l 2326 3340 l 2315 3350 l 2305 3360 l 2294 3372 l 2284 3383 l 2274 3396 l 2265 3409 l 2257 3423 l 2249 3437 l 2243 3451 l 2238 3466 l 2235 3480 l 2232 3494 l 2230 3508 l 2230 3523 l 2230 3537 l 2232 3552 l 2234 3568 l 2237 3585 l 2240 3602 l 2244 3619 l 2248 3635 l 2253 3651 l 2257 3665 l 2261 3677 l 2264 3688 l 2268 3698 l 2272 3712 l 2276 3722 l 2278 3729 l 2279 3733 l 2280 3735 l gs col2 s gr % Polyline 7.500 slw n 3075 900 m 3075 901 l 3072 905 l 3066 914 l 3056 929 l 3042 949 l 3027 971 l 3012 994 l 2997 1015 l 2984 1034 l 2972 1051 l 2962 1066 l 2953 1079 l 2945 1090 l 2938 1100 l 2929 1111 l 2921 1122 l 2913 1132 l 2905 1142 l 2897 1152 l 2889 1163 l 2882 1173 l 2875 1183 l 2868 1193 l 2862 1203 l 2856 1214 l 2850 1225 l 2845 1235 l 2840 1246 l 2835 1258 l 2831 1271 l 2826 1285 l 2823 1300 l 2819 1316 l 2817 1332 l 2816 1348 l 2815 1364 l 2816 1380 l 2818 1395 l 2821 1410 l 2825 1425 l 2830 1438 l 2836 1451 l 2843 1465 l 2851 1479 l 2860 1494 l 2871 1509 l 2882 1525 l 2894 1541 l 2907 1558 l 2920 1574 l 2934 1591 l 2947 1608 l 2961 1625 l 2974 1641 l 2987 1658 l 3000 1675 l 3010 1689 l 3021 1703 l 3031 1718 l 3042 1734 l 3053 1751 l 3064 1768 l 3075 1787 l 3086 1806 l 3097 1826 l 3107 1846 l 3118 1867 l 3128 1888 l 3137 1910 l 3146 1932 l 3155 1953 l 3162 1975 l 3170 1997 l 3176 2019 l 3182 2040 l 3188 2063 l 3192 2083 l 3196 2104 l 3200 2126 l 3203 2149 l 3206 2172 l 3208 2197 l 3210 2222 l 3212 2248 l 3213 2274 l 3213 2301 l 3213 2328 l 3212 2356 l 3211 2383 l 3209 2411 l 3206 2438 l 3203 2464 l 3200 2490 l 3196 2516 l 3191 2541 l 3186 2565 l 3181 2589 l 3175 2613 l 3168 2636 l 3161 2659 l 3153 2683 l 3145 2707 l 3135 2731 l 3125 2755 l 3114 2779 l 3103 2804 l 3090 2828 l 3077 2852 l 3063 2875 l 3048 2898 l 3033 2921 l 3018 2942 l 3002 2963 l 2986 2982 l 2970 3000 l 2954 3017 l 2938 3033 l 2921 3048 l 2905 3062 l 2888 3075 l 2868 3088 l 2848 3101 l 2828 3112 l 2806 3124 l 2784 3134 l 2761 3144 l 2738 3154 l 2713 3162 l 2689 3171 l 2664 3179 l 2640 3186 l 2616 3193 l 2592 3200 l 2569 3206 l 2547 3212 l 2525 3217 l 2505 3222 l 2486 3227 l 2468 3232 l 2450 3238 l 2429 3244 l 2409 3251 l 2389 3258 l 2370 3266 l 2351 3275 l 2333 3283 l 2316 3292 l 2300 3302 l 2284 3311 l 2270 3321 l 2258 3330 l 2246 3340 l 2236 3349 l 2227 3358 l 2219 3366 l 2213 3375 l 2205 3387 l 2198 3398 l 2192 3411 l 2187 3423 l 2183 3437 l 2180 3450 l 2178 3463 l 2177 3477 l 2176 3489 l 2175 3502 l 2175 3513 l 2175 3525 l 2175 3539 l 2175 3553 l 2175 3568 l 2176 3583 l 2177 3598 l 2178 3613 l 2180 3627 l 2182 3640 l 2184 3652 l 2188 3663 l 2191 3673 l 2196 3682 l 2202 3692 l 2210 3703 l 2219 3715 l 2230 3727 l 2239 3738 l 2246 3746 l 2249 3749 l 2250 3750 l gs col0 s gr % Polyline n 2250 900 m 2252 903 l 2255 909 l 2262 919 l 2271 935 l 2283 955 l 2297 979 l 2313 1005 l 2330 1032 l 2346 1059 l 2362 1085 l 2377 1110 l 2390 1132 l 2403 1153 l 2415 1172 l 2425 1190 l 2435 1206 l 2445 1222 l 2454 1236 l 2463 1250 l 2473 1267 l 2484 1284 l 2494 1300 l 2505 1317 l 2515 1334 l 2526 1350 l 2536 1366 l 2547 1382 l 2556 1398 l 2566 1413 l 2575 1427 l 2584 1440 l 2591 1453 l 2599 1465 l 2606 1477 l 2613 1488 l 2621 1502 l 2629 1516 l 2637 1530 l 2645 1544 l 2652 1559 l 2659 1573 l 2665 1587 l 2671 1601 l 2676 1614 l 2681 1627 l 2684 1638 l 2688 1650 l 2690 1662 l 2692 1673 l 2694 1686 l 2695 1698 l 2696 1712 l 2697 1725 l 2696 1738 l 2695 1752 l 2694 1764 l 2692 1777 l 2690 1788 l 2688 1800 l 2684 1812 l 2681 1823 l 2677 1836 l 2672 1849 l 2666 1863 l 2661 1877 l 2655 1891 l 2649 1906 l 2643 1920 l 2637 1934 l 2631 1948 l 2625 1963 l 2620 1975 l 2615 1988 l 2610 2002 l 2604 2016 l 2598 2032 l 2592 2048 l 2586 2064 l 2579 2081 l 2572 2097 l 2566 2114 l 2559 2130 l 2552 2145 l 2545 2160 l 2538 2175 l 2531 2188 l 2524 2201 l 2517 2215 l 2509 2228 l 2501 2243 l 2492 2257 l 2484 2272 l 2475 2288 l 2466 2303 l 2458 2318 l 2449 2332 l 2441 2347 l 2433 2360 l 2426 2374 l 2419 2387 l 2413 2400 l 2405 2415 l 2398 2430 l 2391 2446 l 2385 2462 l 2378 2479 l 2372 2496 l 2366 2514 l 2360 2532 l 2355 2550 l 2351 2568 l 2347 2586 l 2343 2603 l 2340 2620 l 2338 2638 l 2335 2653 l 2334 2668 l 2332 2685 l 2330 2702 l 2329 2719 l 2328 2738 l 2326 2756 l 2325 2775 l 2324 2794 l 2322 2812 l 2321 2831 l 2320 2848 l 2318 2865 l 2316 2882 l 2315 2897 l 2313 2913 l 2310 2927 l 2308 2942 l 2305 2958 l 2301 2973 l 2297 2989 l 2293 3005 l 2288 3021 l 2282 3038 l 2276 3054 l 2270 3070 l 2263 3086 l 2256 3102 l 2249 3117 l 2241 3133 l 2233 3148 l 2225 3163 l 2217 3176 l 2209 3190 l 2200 3204 l 2190 3219 l 2180 3234 l 2169 3250 l 2157 3267 l 2145 3283 l 2132 3300 l 2119 3317 l 2106 3333 l 2093 3350 l 2079 3366 l 2066 3381 l 2052 3396 l 2039 3410 l 2026 3424 l 2013 3438 l 1999 3451 l 1985 3464 l 1971 3477 l 1956 3491 l 1941 3504 l 1926 3518 l 1910 3531 l 1894 3544 l 1878 3557 l 1863 3570 l 1848 3582 l 1833 3594 l 1820 3605 l 1807 3615 l 1794 3625 l 1783 3634 l 1772 3642 l 1763 3650 l 1749 3661 l 1736 3671 l 1725 3681 l 1713 3691 l 1701 3702 l 1689 3713 l 1677 3724 l 1667 3734 l 1658 3742 l 1653 3747 l 1650 3750 l gs col0 s gr /Times-Roman ff 240.00 scf sf 2250 750 m gs 1 -1 sc (x) col0 sh gr /Times-Roman ff 240.00 scf sf 2700 750 m gs 1 -1 sc (y) col0 sh gr /Times-Roman ff 240.00 scf sf 3150 750 m gs 1 -1 sc (z) col0 sh gr % Polyline n 1200 3750 m 4800 3750 l gs col0 s gr /Symbol ff 240.00 scf sf 5100 1350 m gs 1 -1 sc (g) col0 sh gr /Times-Roman ff 240.00 scf sf 6600 4620 m gs 1 -1 sc (Here the genealogy is ultrametric) col0 sh gr /Symbol ff 240.00 scf sf 5250 1275 m gs 1 -1 sc (e) col0 sh gr /Symbol ff 240.00 scf sf 5100 1800 m gs 1 -1 sc (g) col0 sh gr /Symbol ff 240.00 scf sf 5250 1725 m gs 1 -1 sc (e) col0 sh gr /Symbol ff 240.00 scf sf 5025 3375 m gs 1 -1 sc (g) col0 sh gr /Symbol ff 240.00 scf sf 5175 3300 m gs 1 -1 sc (e) col0 sh gr /Times-Roman ff 240.00 scf sf 5400 1350 m gs 1 -1 sc (\(y,z\)) col0 sh gr /Times-Roman ff 240.00 scf sf 5400 1800 m gs 1 -1 sc (\(x,y\)) col0 sh gr /Times-Roman ff 240.00 scf sf 5325 3450 m gs 1 -1 sc (\(z,x\)) col0 sh gr /Times-Roman ff 240.00 scf sf 1425 4275 m gs 1 -1 sc (Ancestrol lines of three points and ) col0 sh gr /Times-Roman ff 240.00 scf sf 1425 4545 m gs 1 -1 sc (their neighbohoods. Note that the) col0 sh gr /Symbol ff 240.00 scf sf 1425 4875 m gs 1 -1 sc (e) col0 sh gr /Times-Roman ff 240.00 scf sf 1575 4875 m gs 1 -1 sc (genealogies are not ultrametric) col0 sh gr /Times-Roman ff 240.00 scf sf 7275 750 m gs 1 -1 sc (x) col0 sh gr /Times-Roman ff 240.00 scf sf 8175 750 m gs 1 -1 sc (y) col0 sh gr /Times-Roman ff 240.00 scf sf 8625 750 m gs 1 -1 sc (z) col0 sh gr /Times-Roman ff 240.00 scf sf 825 900 m gs 1 -1 sc (t) col0 sh gr /Times-Roman ff 240.00 scf sf 6225 975 m gs 1 -1 sc (t) col0 sh gr /Times-Roman ff 240.00 scf sf 975 3750 m gs 1 -1 sc (0) col0 sh gr /Times-Roman ff 240.00 scf sf 6300 3750 m gs 1 -1 sc (0) col0 sh gr /Symbol ff 240.00 scf sf 10425 1575 m gs 1 -1 sc (g\() col0 sh gr /Symbol ff 240.00 scf sf 10425 1950 m gs 1 -1 sc (g\() col0 sh gr /Times-Roman ff 240.00 scf sf 10650 1575 m gs 1 -1 sc (y,z\)) col0 sh gr /Times-Roman ff 240.00 scf sf 10650 1950 m gs 1 -1 sc (x,y\)=) col0 sh gr /Symbol ff 240.00 scf sf 11250 1950 m gs 1 -1 sc (g) col0 sh gr /Times-Roman ff 240.00 scf sf 11400 1950 m gs 1 -1 sc (\(x,,z\)) col0 sh gr /Times-Roman ff 240.00 scf sf 6600 4350 m gs 1 -1 sc (Ancestral lines in a flow with jumps.) col0 sh gr $F2psEnd rs %%EndDocument @endspecial 50 3297 a Fc(Pro)s(of:)47 b Fw(Let)31 b(us)f(note)h (\014rst)e(o\013)i(all)e(that)1137 3541 y Fp(y)f Fo(2)d Fw(lim)1307 3600 y Fn(\017)p Fm(#)p Fz(0)1437 3541 y Fp(m)1517 3503 y Fn(\017)1517 3563 y(x)1563 3541 y Fw(\()p Fp(t;)15 b(t)1704 3503 y Fm(00)1751 3541 y Fw(\))91 b Fo(8)p Fp(t)1961 3503 y Fm(0)o(0)2032 3541 y Fp(<)25 b(t)2161 3503 y Fm(0)2212 3541 y Fw(=)g Fp(\015)2355 3555 y Fn(t)2387 3541 y Fw(\()p Fp(x;)15 b(y)s Fw(\))p Fp(:)855 b Fw(\(2)p Fp(:)p Fw(15\))50 3792 y(In)29 b(fact)j(it)e (su\016ces)g(to)h(see)g(that)761 4035 y(lim)773 4094 y Fn(\017)p Fm(#)p Fz(0)903 4035 y Fp(S)964 3997 y Fz(\()p Fn(t)1019 3970 y Fe(0)1044 3997 y Fn(;t)p Fz(\))1126 4035 y Fw(\()p Fp(V)1214 4049 y Fn(\017)1249 4035 y Fw(\()p Fp(S)1345 3997 y Fz(\()p Fn(t)1400 3970 y Fe(0)1426 3997 y Fn(;t)p Fz(\))1508 4035 y Fw(\))1543 3997 y Fm(\000)p Fz(1)1641 4035 y Fw(\()p Fp(x)p Fw(\)\))26 b Fo(\032)f Fw(lim)1932 4094 y Fn(\017)p Fm(#)p Fz(0)2062 4035 y Fp(S)2123 3997 y Fz(\()p Fn(t)2178 3970 y Fe(0)q(0)2223 3997 y Fn(;t)p Fz(\))2305 4035 y Fw(\()p Fp(V)2393 4049 y Fn(\017)2429 4035 y Fw(\()p Fp(S)2525 3997 y Fz(\()p Fn(t)2580 3970 y Fe(0)q(0)2625 3997 y Fn(;t)p Fz(\))2707 4035 y Fw(\))2742 3997 y Fm(\000)p Fz(1)2840 4035 y Fw(\()p Fp(x)p Fw(\)\))480 b(\(2)p Fp(:)p Fw(16\))50 4291 y(whic)m(h)29 b(is)g(b)m(y)i(compatibilit)m(y)d(\(2.6\))k(equiv)-5 b(alen)m(t)30 b(to)878 4534 y(lim)890 4593 y Fn(\017)p Fm(#)p Fz(0)1020 4534 y Fp(V)1073 4548 y Fn(\017)1108 4534 y Fw(\()p Fp(S)1204 4496 y Fz(\()p Fn(t)1259 4469 y Fe(0)1284 4496 y Fn(;t)p Fz(\))1366 4534 y Fw(\))1401 4496 y Fm(\000)p Fz(1)1499 4534 y Fw(\()p Fp(x)p Fw(\)\))d Fo(\032)e Fw(lim)1790 4593 y Fn(\017)p Fm(#)p Fz(0)1920 4534 y Fp(S)1981 4496 y Fz(\()p Fn(t)2036 4469 y Fe(0)q(0)2082 4496 y Fn(;t)2131 4469 y Fe(0)2155 4496 y Fz(\))2188 4534 y Fw(\()p Fp(V)2276 4548 y Fn(\017)2311 4534 y Fw(\()p Fp(S)2407 4496 y Fz(\()p Fn(t)2462 4469 y Fe(0)r(0)2508 4496 y Fn(;t)p Fz(\))2590 4534 y Fw(\))2625 4496 y Fm(\000)p Fz(1)2723 4534 y Fw(\()p Fp(x)p Fw(\)\))597 b(\(2)p Fp(:)p Fw(17\))50 4807 y(This)47 b(last)i(equalit)m(y)g(again)g(holds)e(b)m(y) i(\(2.6\).)99 b(It)49 b(is)f(indeed)g(trivial)f(if)h Fp(S)2789 4774 y Fz(\()p Fn(t)2844 4747 y Fe(0)q(0)2890 4774 y Fn(;t)2939 4747 y Fe(0)2963 4774 y Fz(\))3045 4807 y Fw(is)g(con)m(tin)m(uous)h(at)50 4953 y(\()p Fp(S)146 4920 y Fz(\()p Fn(t)201 4892 y Fe(0)q(0)247 4920 y Fn(;t)p Fz(\))329 4953 y Fw(\))364 4920 y Fm(\000)p Fz(1)462 4953 y Fw(\()p Fp(x)p Fw(\).)159 5171 y(It)42 b(follo)m(ws)e(from)i (\(2.15\))h(that)g(for)e(an)m(y)h Fp(x;)15 b(y)s(;)g(z)49 b Fo(2)44 b Fw([0)p Fp(;)15 b Fw(1])43 b(if)e Fp(\015)2361 5185 y Fn(t)2393 5171 y Fw(\()p Fp(x;)15 b(y)s Fw(\))45 b Fo(6)p Fw(=)e Fp(\015)2809 5185 y Fn(t)2842 5171 y Fw(\()p Fp(x;)15 b(z)t Fw(\))42 b(then)g 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10935 2325 m 10965 2325 l gs col4 s gr % Polyline n 11085 2325 m 11115 2325 l gs col4 s gr % Polyline n 11160 2325 m 11190 2325 l gs col4 s gr % Polyline n 11235 2325 m 11265 2325 l gs col4 s gr % Polyline n 11385 2325 m 11415 2325 l gs col4 s gr % Polyline n 11460 2325 m 11490 2325 l gs col4 s gr % Polyline n 11685 2325 m 11715 2325 l gs col4 s gr % Polyline n 11910 2325 m 11940 2325 l gs col4 s gr % Polyline n 11760 2325 m 11790 2325 l gs col4 s gr % Polyline n 11985 2325 m 12015 2325 l gs col4 s gr % Polyline n 12210 2325 m 12240 2325 l gs col4 s gr % Polyline n 12360 2325 m 12390 2325 l gs col4 s gr % Polyline n 12585 2325 m 12615 2325 l gs col4 s gr % Polyline n 12660 2325 m 12690 2325 l gs col4 s gr % Polyline n 12885 2325 m 12915 2325 l gs col4 s gr % Polyline n 12810 2325 m 12840 2325 l gs col4 s gr % Polyline n 13110 2325 m 13140 2325 l gs col4 s gr % Polyline n 13185 2325 m 13215 2325 l gs col4 s gr % Polyline n 13410 2325 m 13440 2325 l gs col4 s gr % Polyline n 13260 2325 m 13290 2325 l gs col4 s gr % Polyline n 13485 2325 m 13515 2325 l gs col4 s gr % Polyline n 13710 2325 m 13740 2325 l gs col4 s gr % Polyline n 13860 2325 m 13890 2325 l gs col4 s gr % Polyline n 14010 2325 m 14040 2325 l gs col4 s gr % Polyline n 14160 2325 m 14190 2325 l gs col4 s gr % Polyline n 14460 2325 m 14490 2325 l gs col4 s gr % Polyline n 14685 2325 m 14715 2325 l gs col4 s gr % Polyline n 14310 2325 m 14340 2325 l gs col4 s gr % Polyline n 8835 3900 m 8865 3900 l gs col4 s gr % Polyline n 8910 3900 m 8940 3900 l gs col4 s gr % Polyline n 8985 3900 m 9015 3900 l gs col4 s gr % Polyline n 9135 3900 m 9165 3900 l gs col4 s gr % Polyline n 9285 3900 m 9315 3900 l gs col4 s gr % Polyline n 9360 3900 m 9390 3900 l gs col4 s gr % Polyline n 9450 3900 m 9525 3900 l gs col4 s gr % Polyline n 9735 3900 m 9765 3900 l gs col4 s gr % Polyline n 9885 3900 m 9915 3900 l gs col4 s gr % Polyline n 10035 3900 m 10065 3900 l gs col4 s gr % Polyline n 10035 3900 m 10065 3900 l gs col4 s gr % Polyline n 10275 3900 m 10350 3900 l gs col4 s gr % Polyline n 10410 3900 m 10440 3900 l gs col4 s gr % Polyline n 10560 3900 m 10590 3900 l gs col4 s gr % Polyline n 10710 3900 m 10740 3900 l gs col4 s gr % Polyline n 10860 3900 m 10890 3900 l gs col4 s gr % Polyline n 11010 3900 m 11040 3900 l gs col4 s gr % Polyline n 11085 3900 m 11115 3900 l gs col4 s gr % Polyline n 11160 3900 m 11190 3900 l gs col4 s gr % Polyline n 11310 3900 m 11340 3900 l gs col4 s gr % Polyline n 11385 3900 m 11415 3900 l gs col4 s gr % Polyline n 11610 3900 m 11640 3900 l gs col4 s gr % Polyline n 11835 3900 m 11865 3900 l gs col4 s gr % Polyline n 1785 600 m 1815 600 l gs col4 s gr % Polyline n 11685 3900 m 11715 3900 l gs col4 s gr /Times-Roman ff 240.00 scf sf 2250 5025 m gs 1 -1 sc (Genealogy of a flow of measures) col0 sh gr % Polyline n 11910 3900 m 11940 3900 l gs col4 s gr % Polyline n 12135 3900 m 12165 3900 l gs col4 s gr % Polyline n 12285 3900 m 12315 3900 l gs col4 s gr % Polyline n 12510 3900 m 12540 3900 l gs col4 s gr % Polyline n 12585 3900 m 12615 3900 l gs col4 s gr % Polyline n 12810 3900 m 12840 3900 l gs col4 s gr % Polyline n 12735 3900 m 12765 3900 l gs col4 s gr % Polyline n 13035 3900 m 13065 3900 l gs col4 s gr % Polyline n 13110 3900 m 13140 3900 l gs col4 s gr % Polyline n 13335 3900 m 13365 3900 l gs col4 s gr % Polyline n 13185 3900 m 13215 3900 l gs col4 s gr % Polyline n 13410 3900 m 13440 3900 l gs col4 s gr % Polyline n 13635 3900 m 13665 3900 l gs col4 s gr % Polyline n 13785 3900 m 13815 3900 l gs col4 s gr % Polyline n 13935 3900 m 13965 3900 l gs col4 s gr % Polyline n 14085 3900 m 14115 3900 l gs col4 s gr % Polyline n 14385 3900 m 14415 3900 l gs col4 s gr % Polyline n 14610 3900 m 14640 3900 l gs col4 s gr % Polyline n 14235 3900 m 14265 3900 l gs col4 s gr % Polyline n 8835 4575 m 8865 4575 l gs col4 s gr % Polyline n 8910 4575 m 8940 4575 l gs col4 s gr % Polyline n 8985 4575 m 9015 4575 l gs col4 s gr % Polyline n 9135 4575 m 9165 4575 l gs col4 s gr % Polyline n 9285 4575 m 9315 4575 l gs col4 s gr % Polyline n 9360 4575 m 9390 4575 l gs col4 s gr % Polyline n 9450 4575 m 9525 4575 l gs col4 s gr % Polyline n 9735 4575 m 9765 4575 l gs col4 s gr % Polyline n 9885 4575 m 9915 4575 l gs col4 s gr % Polyline n 10035 4575 m 10065 4575 l gs col4 s gr % Polyline n 10035 4575 m 10065 4575 l gs col4 s gr % Polyline n 10275 4575 m 10350 4575 l gs col4 s gr % Polyline n 10410 4575 m 10440 4575 l gs col4 s gr % Polyline n 10560 4575 m 10590 4575 l gs col4 s gr % Polyline n 10710 4575 m 10740 4575 l gs col4 s gr % Polyline n 10860 4575 m 10890 4575 l gs col4 s gr % Polyline n 11010 4575 m 11040 4575 l gs col4 s gr % Polyline n 11085 4575 m 11115 4575 l gs col4 s gr % Polyline n 11160 4575 m 11190 4575 l gs col4 s gr % Polyline n 11310 4575 m 11340 4575 l gs col4 s gr % Polyline n 11385 4575 m 11415 4575 l gs col4 s gr % Polyline n 11610 4575 m 11640 4575 l gs col4 s gr % Polyline n 11835 4575 m 11865 4575 l gs col4 s gr % Polyline n 11685 4575 m 11715 4575 l gs col4 s gr % Polyline n 11910 4575 m 11940 4575 l gs col4 s gr % Polyline n 12135 4575 m 12165 4575 l gs col4 s gr % Polyline n 12285 4575 m 12315 4575 l gs col4 s gr % Polyline n 12510 4575 m 12540 4575 l gs col4 s gr % Polyline n 12585 4575 m 12615 4575 l gs col4 s gr % Polyline n 12810 4575 m 12840 4575 l gs col4 s gr % Polyline n 12735 4575 m 12765 4575 l gs col4 s gr % Polyline n 13035 4575 m 13065 4575 l gs col4 s gr % Polyline n 13110 4575 m 13140 4575 l gs col4 s gr % Polyline n 13335 4575 m 13365 4575 l gs col4 s gr % Polyline n 13185 4575 m 13215 4575 l gs col4 s gr % Polyline n 13410 4575 m 13440 4575 l gs col4 s gr % Polyline n 13635 4575 m 13665 4575 l gs col4 s gr % Polyline n 13785 4575 m 13815 4575 l gs col4 s gr % Polyline n 13935 4575 m 13965 4575 l gs col4 s gr % Polyline n 14085 4575 m 14115 4575 l gs col4 s gr % Polyline n 14385 4575 m 14415 4575 l gs col4 s gr % Polyline n 14610 4575 m 14640 4575 l gs col4 s gr % Polyline n 14235 4575 m 14265 4575 l gs col4 s gr % Polyline 15.000 slw n 8775 600 m 8775 750 l gs col14 s gr % Polyline n 9375 600 m 9375 750 l gs col14 s gr % Polyline n 9675 600 m 9675 750 l gs col14 s gr % Polyline n 10200 600 m 10200 750 l gs col14 s gr % Polyline n 10575 600 m 10575 750 l gs col14 s gr % Polyline n 10725 600 m 10725 750 l gs col14 s gr % Polyline n 10875 600 m 10875 750 l gs col14 s gr % Polyline n 11325 600 m 11325 750 l gs col14 s gr % Polyline n 11700 600 m 11700 750 l gs col14 s gr % Polyline n 12525 600 m 12525 750 l gs col14 s gr % Polyline n 13650 600 m 13650 750 l gs col14 s gr % Polyline n 14250 600 m 14250 750 l gs col14 s gr % Polyline n 8775 1650 m 8775 1800 l gs col14 s gr % Polyline n 9375 1650 m 9375 1800 l gs col14 s gr % Polyline n 9675 1650 m 9675 1800 l gs col14 s gr % Polyline n 10200 1650 m 10200 1725 l gs col14 s gr % Polyline n 10200 1725 m 10200 1800 l gs col14 s gr % Polyline n 10575 1650 m 10575 1800 l gs col14 s gr % Polyline n 11325 1650 m 11325 1800 l gs col14 s gr % Polyline n 11700 1650 m 11700 1800 l gs col14 s gr % Polyline n 13650 1650 m 13650 1800 l gs col14 s gr % Polyline n 14250 1650 m 14250 1800 l gs col14 s gr % Polyline n 8775 2250 m 8775 2400 l gs col14 s gr % Polyline n 9375 2250 m 9375 2400 l gs col14 s gr % Polyline n 9675 2250 m 9675 2400 l gs col14 s gr % Polyline n 10575 2250 m 10575 2400 l gs col14 s gr % Polyline n 11325 2250 m 11325 2325 l gs col14 s gr % Polyline n 11325 2325 m 11325 2400 l gs col14 s gr % Polyline n 11700 2250 m 11700 2400 l gs col14 s gr % Polyline n 13650 2250 m 13650 2400 l gs col14 s gr % Polyline n 14250 2250 m 14250 2400 l gs col14 s gr % Polyline n 10575 3825 m 10575 3900 l gs col14 s gr % Polyline n 10575 3900 m 10575 3975 l gs col14 s gr % Polyline n 11700 3825 m 11700 3975 l gs col14 s gr % Polyline n 13650 3825 m 13650 3900 l gs col14 s gr % Polyline n 13650 3825 m 13650 3975 l gs col14 s gr % Polyline n 14250 3825 m 14250 3900 l gs col14 s gr % Polyline n 14250 3900 m 14250 3975 l gs col14 s gr % Polyline 7.500 slw n 8775 1725 m 14775 1725 l gs col0 s gr % Polyline n 8775 3900 m 14775 3900 l gs col0 s gr % Polyline n 8775 2325 m 14775 2325 l gs col0 s gr % Polyline n 8850 4575 m 14850 4575 l gs col0 s gr % Polyline n 8775 675 m 14550 675 l gs col0 s gr % Polyline n 8775 225 m 8775 4575 l gs col0 s gr % Polyline n 8775 4575 m 8925 4575 l gs col0 s gr % Polyline n 8775 2325 m 14775 2325 l gs col0 s gr % Polyline 15.000 slw n 9375 675 m 9375 2325 l gs col24 s gr % Polyline n 9675 675 m 9675 2325 l gs col24 s gr % Polyline n 10200 675 m 10200 1725 l gs col24 s gr % Polyline n 10575 675 m 10575 3900 l gs col24 s gr % Polyline n 11325 675 m 11325 2325 l 11400 2325 l gs col24 s gr % Polyline n 11700 675 m 11700 3900 l gs col24 s gr % Polyline n 13650 675 m 13650 3900 l gs col24 s gr % Polyline n 14250 675 m 14250 3900 l gs col24 s gr % Polyline n 12525 3825 m 12525 3975 l gs col14 s gr % Polyline n 12525 675 m 12525 3900 l gs col18 s gr % Polyline n 10725 1650 m 10725 1800 l gs col14 s gr % Polyline n 10725 675 m 10725 1650 l gs col18 s gr % Polyline n 14475 675 m 14475 1725 l gs col18 s gr % Polyline n 12825 675 m 12825 1725 l gs col18 s gr % Polyline 7.500 slw n 14700 675 m 14700 4575 l gs col0 s gr /Times-Roman ff 240.00 scf sf 8250 4650 m gs 1 -1 sc (0) col0 sh gr /Times-Roman ff 240.00 scf sf 8325 3975 m gs 1 -1 sc (t) col0 sh gr /Times-Roman ff 240.00 scf sf 8325 2325 m gs 1 -1 sc (t') col0 sh gr /Times-Roman ff 240.00 scf sf 8400 675 m gs 1 -1 sc (1) col0 sh gr /Times-Roman ff 240.00 scf sf 8325 1725 m gs 1 -1 sc (t) col0 sh gr /Times-Roman ff 240.00 scf sf 8400 3900 m gs 1 -1 sc ('') col0 sh gr /Times-Roman ff 240.00 scf sf 9450 5100 m gs 1 -1 sc (Associated partions of random i.i.d. variables \(dots\).) col0 sh gr % Polyline 0.000 slw n 3225 1050 m 3225 1200 l % Polyline n 3225 1050 m 3225 1200 l % Polyline 7.500 slw n 1650 4500 m 1725 4500 l gs col0 s gr % Polyline n 1650 4500 m 1650 225 l gs col0 s gr % Polyline n 1650 600 m 7650 600 l gs col0 s gr % Polyline n 1650 1650 m 7650 1650 l gs col0 s gr % Polyline n 1650 3825 m 7650 3825 l gs col0 s gr % Polyline n 1650 2250 m 7650 2250 l gs col0 s gr % Polyline n 1725 4500 m 7725 4500 l gs col0 s gr % Polyline n 7650 600 m 7650 4500 l gs col0 s gr % Polyline n 2250 600 m 1800 1650 l gs col1 s gr % Polyline n 1650 600 m 1800 1650 l gs col1 s gr % Polyline n 2175 1650 m 1800 2250 l gs col1 s gr % Polyline n 1650 1650 m 1800 2250 l gs col1 s gr % Polyline n 2550 600 m 2325 1650 l gs col1 s gr % Polyline n 2250 600 m 2325 1650 l gs col1 s gr % Polyline n 2475 1650 m 2250 2250 l gs col1 s gr % Polyline n 2175 1650 m 2250 2250 l gs col1 s gr % Polyline n 3450 2250 m 2250 3825 l gs col1 s gr % Polyline n 1650 2250 m 2250 3825 l gs col1 s gr % Polyline n 3075 600 m 2775 1650 l 2550 600 l gs col1 s gr % Polyline n 3450 600 m 3150 1650 l 3075 600 l gs col1 s gr % Polyline n 2475 1650 m 2775 2250 l 3300 1650 l gs col1 s gr % Polyline n 3600 600 m 3525 1650 l 3450 600 l gs col1 s gr % Polyline n 4200 600 m 3825 1650 l 3600 600 l gs col1 s gr % Polyline n 3300 1650 m 3600 2250 l 4125 1650 l gs col1 s gr % Polyline n 4500 600 m 4275 1650 l 4200 600 l gs col1 s gr % Polyline n 4425 1650 m 4200 2250 l 4125 1650 l gs col1 s gr % Polyline n 3450 2250 m 3750 3825 l 4350 2250 l gs col1 s gr % Polyline n 4500 600 m 4875 1650 l 5400 600 l gs col1 s gr % Polyline n 4425 1650 m 4650 2250 l gs col1 s gr % Polyline n 4950 1650 m 4650 2250 l gs col1 s gr % Polyline n 4350 2325 m 4575 3825 l 4875 2250 l gs col1 s gr % Polyline n 5400 600 m 5475 1650 l gs col1 s gr % Polyline n 5475 1650 m 5700 600 l gs col1 s gr % Polyline n 6525 600 m 6000 1650 l 5700 600 l gs col1 s gr % Polyline n 5250 1650 m 5475 2250 l gs col1 s gr % Polyline n 7050 600 m 6825 1650 l 6525 600 l gs col1 s gr % Polyline n 6900 1650 m 6975 2250 l 7125 1650 l gs col1 s gr % Polyline n 7125 1650 m 7275 2250 l 7575 1650 l gs col1 s gr % Polyline n 4875 2250 m 5175 3825 l 5700 2250 l gs col1 s gr % Polyline n 5700 2250 m 6075 3825 l gs col1 s gr % Polyline n 5475 2250 m 5625 1650 l gs col1 s gr % Polyline n 5625 1650 m 5925 2250 l 6300 1650 l gs col1 s gr % Polyline n 6300 1650 m 6450 2250 l 6900 1650 l gs col1 s gr % Polyline n 6300 2250 m 6075 3825 l gs col1 s gr % Polyline n 6750 2250 m 6675 3825 l gs col1 s gr % Polyline n 6300 2325 m 6675 3825 l gs col1 s gr % Polyline n 7050 600 m 6975 1650 l gs col1 s gr % Polyline n 6975 1650 m 7425 600 l gs col1 s gr % Polyline n 7425 675 m 7350 1575 l 7350 1650 l gs col1 s gr % Polyline n 7350 1650 m 7650 600 l gs col1 s gr % Polyline n 6750 2250 m 7200 3825 l gs col1 s gr % Polyline n 7650 2250 m 7200 3825 l gs col1 s gr % Polyline n 1650 3825 m 1650 4500 l gs col1 s gr % Polyline n 1650 4500 m 7650 3825 l gs col1 s gr % Polyline n 4950 1650 m 5100 2250 l gs col1 s gr % Polyline n 5100 2250 m 5250 1650 l gs col1 s gr /Times-Roman ff 240.00 scf sf 1125 4575 m gs 1 -1 sc (0) col0 sh gr /Times-Roman ff 240.00 scf sf 1200 3900 m gs 1 -1 sc (t) col0 sh gr /Times-Roman ff 240.00 scf sf 1200 2250 m gs 1 -1 sc (t') col0 sh gr /Times-Roman ff 240.00 scf sf 1275 600 m gs 1 -1 sc (1) col0 sh gr /Times-Roman ff 240.00 scf sf 1200 1650 m gs 1 -1 sc (t) col0 sh gr /Times-Roman ff 240.00 scf sf 1275 3825 m gs 1 -1 sc ('') col0 sh gr $F2psEnd rs %%EndDocument @endspecial 159 2000 a Fw(The)23 b(k)m(ey)i(observ)-5 b(ation)24 b(for)g(our)f(purp)s(oses)f(is)i(the)g(p)s(ossibilit)m(y)c (to)25 b(express)f Fp(m)2791 2014 y Fn(x)2836 2000 y Fw(\()p Fp(t;)15 b(t)2977 1967 y Fm(0)3003 2000 y Fw(\))24 b(and)g(its)f(momen)m(ts)50 2145 y(in)29 b(terms)h(of)h(this)e (coalescen)m(t)j([Be1].)43 b(Namely)-8 b(,)31 b(it)f(is)f(plain)f(that) 1066 2392 y(lim)1052 2451 y Fn(n)p Fm("1)1222 2392 y Fp(n)1277 2354 y Fm(\000)p Fz(1)1433 2278 y Fn(n)1389 2306 y Fk(X)1392 2499 y Fn(j)t Fz(=1)1536 2392 y Fw(1)-23 b(I)1591 2408 y Fn(j)t Fm(2)p Fn(B)1729 2418 y Fr(k)1769 2408 y Fz(\()p Fn(t)1824 2390 y Fe(0)1849 2408 y Fz(\))1908 2392 y Fw(=)24 b Fp(m)2083 2406 y Fn(U)2133 2415 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b(us)f(note)i(\014rst)e(that)h(in)f(these)h (expressions)f(con)m(tributions)f(from)h(terms)h(where)f(t)m(w)m(o)i (indices)d(are)50 658 y(equal)30 b(can)g(b)s(e)g(neglected.)41 b(Second,)31 b(since)917 870 y Fp(B)986 887 y Fn(k)r Fz(\()p Fn(i;t)1130 896 y Fr(p)1167 887 y Fz(\))1200 870 y Fw(\()p Fp(t)1268 884 y Fn(p)1310 870 y Fw(\))26 b Fo(\032)f Fp(B)1536 887 y Fn(k)r Fz(\()p Fn(i;t)1680 896 y Fr(p)p Fe(\000)p Fs(1)1798 887 y Fz(\))1831 870 y Fw(\()p Fp(t)1899 884 y Fn(p)p Fm(\000)p Fz(1)2034 870 y Fw(\))h Fo(\032)e(\001)15 b(\001)g(\001)27 b(\032)e Fp(B)2487 887 y Fn(k)r Fz(\()p Fn(i;t)2631 896 y Fs(1)2666 887 y Fz(\))2698 870 y Fw(\()p Fp(t)2766 884 y Fz(1)2807 870 y Fw(\))635 b(\(2)p Fp(:)p Fw(25\))50 1083 y(the)30 b(summand)f(in)g(\(2.24\))k(is)c(the)i(same)f(as)802 1295 y(1)-23 b(I)857 1315 y Fn(j)891 1293 y Fs(1)887 1337 y(1)926 1315 y Fn(;:::)o(;j)1069 1293 y Fs(1)1065 1339 y Fr(k)1099 1353 y Fs(11)1170 1315 y Fn(;:::)n(;j)1312 1284 y Fr(p)1308 1337 y Fs(1)1350 1315 y Fn(;:::)n(;j)1492 1284 y Fr(p)1488 1339 y(k)1522 1353 y Fs(1)p Fr(p)1596 1315 y Fm(2)p Fn(B)1700 1327 y Fr(k)q Fs(\()p Fr(i;t)1831 1341 y Fs(1)1866 1327 y(\))1895 1315 y Fz(\()p Fn(t)1950 1324 y Fs(1)1986 1315 y Fz(\))2034 1295 y Fp(:)15 b(:)g(:)i Fw(1)-23 b(I)2211 1314 y Fn(j)2245 1283 y Fr(p)2241 1336 y Fs(1)2283 1314 y Fn(;:::)n(;j)2425 1283 y Fr(p)2421 1338 y(k)2455 1352 y Fs(1)p Fr(p)2529 1314 y Fm(2)p Fn(B)2633 1326 y Fr(k)q Fs(\()p Fr(i;t)2764 1335 y(p)2802 1326 y Fs(\))2831 1314 y Fz(\()p Fn(t)2886 1323 y Fr(p)2925 1314 y Fz(\))3477 1295 y Fw(\(2)p Fp(:)p Fw(26\))50 1507 y(Then)188 1679 y Fp(M)10 b Fw(\()p Fp(p;)15 b(t)p 407 1694 33 4 v 1 w(;)g(k)p 481 1694 51 4 v 481 1711 V 3 w Fw(\))26 b(=)39 b(lim)688 1738 y Fn(n)p Fm("1)858 1679 y Fp(n)913 1641 y Fm(\000)p Fn(`)p Fm(\000)p Fn(k)1096 1650 y Fs(11)1161 1641 y Fm(\000\001\001\001)o(\000)p Fn(k)1379 1651 y Fr(`p)1448 1679 y Fl(E)1572 1592 y Fk(X)1518 1786 y Fn(i)1544 1795 y Fs(1)1585 1786 y Fn(;:::)o(;i)1720 1796 y Fr(`)2121 1592 y Fk(X)1766 1804 y Fn(j)1800 1773 y Fs(1)p Fr(;)p Fs(1)1796 1826 y(1)1887 1804 y Fn(;:::)n(;j)2029 1773 y Fs(1)p Fr(;)p Fs(1)2025 1828 y Fr(k)2059 1842 y Fs(11)2130 1804 y Fn(;:::)o(;j)2273 1773 y Fr(p;)p Fs(1)2269 1826 y(1)2362 1804 y Fn(;:::)n(;j)2504 1773 y Fr(p;)p Fs(1)2500 1828 y Fr(k)2534 1842 y Fs(1)p Fr(p)2623 1679 y Fo(\001)15 b(\001)g(\001)3092 1592 y Fk(X)2744 1806 y Fn(j)2778 1775 y Fs(1)p Fr(;`)2774 1828 y Fs(1)2862 1806 y Fn(;:::)n(;j)3004 1775 y Fs(1)p Fr(;`)3000 1830 y(k)3034 1846 y(`)p Fs(1)3101 1806 y Fn(;:::)o(;j)3244 1775 y Fr(p;`)3240 1828 y Fs(1)3329 1806 y Fn(;:::)n(;j)3471 1775 y Fr(p;`)3467 1830 y(k)3501 1846 y(`p)188 1978 y Fw(1)-23 b(I)244 2007 y Fn(j)278 1976 y Fs(1)p Fr(;)p Fs(1)274 2028 y(1)365 2007 y Fn(;:::)n(;j)507 1976 y Fs(1)p Fr(;)p Fs(1)503 2031 y Fr(k)537 2045 y Fs(11)608 2007 y Fn(;:::)o(;j)751 1976 y Fr(p;)p Fs(1)747 2028 y(1)840 2007 y Fn(;:::)n(;j)982 1976 y Fr(p;)p Fs(1)978 2031 y Fr(k)1012 2045 y Fs(1)p Fr(p)1086 2007 y Fm(2)p Fn(B)1190 2019 y Fr(k)q Fs(\()p Fr(i)1274 2033 y Fs(1)1309 2019 y Fr(;t)1356 2033 y Fs(1)1391 2019 y(\))1421 2007 y Fz(\()p Fn(t)1476 2016 y Fs(1)1512 2007 y Fz(\))1560 1978 y Fp(:)15 b(:)g(:)h(:)f(:)g(:)h Fw(1)-23 b(I)1858 2009 y Fn(j)1892 1978 y Fs(1)p Fr(;`)1888 2031 y Fs(1)1975 2009 y Fn(;:::)o(;j)2118 1978 y Fs(1)p Fr(;`)2114 2033 y(k)2148 2049 y(`)p Fs(1)2215 2009 y Fn(;:::)n(;j)2357 1978 y Fr(p;`)2353 2031 y Fs(1)2443 2009 y Fn(;:::)n(;j)2585 1978 y Fr(p;`)2581 2033 y(k)2615 2049 y(`p)2685 2009 y Fm(2)p Fn(B)2789 2021 y Fr(k)q Fs(\()p Fr(i)2873 2037 y(`)2905 2021 y(;t)2952 2035 y Fs(1)2987 2021 y(\))3016 2009 y Fz(\()p Fn(t)3071 2018 y Fs(1)3107 2009 y Fz(\))203 2171 y Fp(:)15 b(:)g(:)i Fw(1)-23 b(I)380 2200 y Fn(j)414 2169 y Fr(p;)p Fs(1)410 2222 y(1)504 2200 y Fn(;:::)n(;j)646 2169 y Fr(p;)p Fs(1)642 2224 y Fr(k)676 2238 y Fs(1)p Fr(p)750 2200 y Fm(2)p Fn(B)854 2212 y Fr(k)q Fs(\()p Fr(i)938 2226 y Fs(1)973 2212 y Fr(;t)1020 2221 y(p)1058 2212 y Fs(\))1087 2200 y Fz(\()p Fn(t)1142 2209 y Fr(p)1181 2200 y Fz(\))1229 2171 y Fp(:)15 b(:)g(:)i(:)e(:)g(:)h Fw(1)-23 b(I)1527 2202 y Fn(j)1561 2172 y Fr(p;`)1557 2224 y Fs(1)1647 2202 y Fn(;:::)o(;j)1790 2172 y Fr(p;`)1786 2227 y(k)1820 2243 y(`p)1890 2202 y Fm(2)p Fn(B)1994 2214 y Fr(k)q Fs(\()p Fr(i)2078 2231 y(`)2109 2214 y(;t)2156 2223 y(p)2194 2214 y Fs(\))2223 2202 y Fz(\()p Fn(t)2278 2211 y Fr(p)2317 2202 y Fz(\))213 2365 y Fw(=)25 b Fl(P)15 b Fw([)q Fp(J)455 2379 y Fz(11)558 2365 y Fo(2)25 b Fp(B)713 2379 y Fz(1)754 2365 y Fw(\()p Fp(t)822 2379 y Fz(1)863 2365 y Fw(\))p Fp(;)15 b(:)g(:)g(:)i(;)e(J)1150 2379 y Fz(1)p Fn(p)1254 2365 y Fo(2)25 b Fp(B)1409 2379 y Fz(1)1450 2365 y Fw(\()p Fp(t)1518 2379 y Fn(p)1560 2365 y Fw(\))p Fp(;)15 b(:)g(:)g(:)i(J)1807 2379 y Fn(`)p Fz(1)1904 2365 y Fo(2)24 b Fp(B)2058 2379 y Fn(`)2093 2365 y Fw(\()p Fp(t)2161 2379 y Fz(1)2202 2365 y Fw(\))p Fp(;)15 b(:)g(:)g(:)i(;)e(J)2489 2379 y Fn(`p)2588 2365 y Fo(2)25 b Fp(B)2743 2379 y Fn(`)2778 2365 y Fw(\()p Fp(t)2846 2379 y Fn(p)2888 2365 y 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Fq(GREM,)28 b(p)l(art)f(4)3089 b Ft(15)159 513 y Fw(Recall)30 b(that)h(our)f(basic)g(ob)5 b(jects)31 b(are)17 731 y(\(i\))46 b Fp(\033)c Fo(2)c(S)407 745 y Fn(N)515 731 y Fo(\021)h(f\000)p Fw(1)p Fp(;)15 b Fw(1)p Fo(g)916 698 y Fn(N)1025 731 y Fw(equipp)s(ed)36 b(with)i Fp(d)1685 745 y Fn(N)1754 731 y Fw(\()p Fp(\033)n(;)15 b(\034)10 b Fw(\))40 b Fo(\021)f Fp(N)2197 698 y Fm(\000)p Fz(1)2309 731 y Fw(\()q(min)13 b(\()q Fp(i)25 b Fw(:)h Fp(\033)2706 745 y Fn(i)2761 731 y Fo(6)p Fw(=)f Fp(\034)2897 745 y Fn(i)2927 731 y Fw(\))c Fo(\000)f Fw(1\))q(.)65 b(Due)39 b(to)g(this)159 876 y(notion,)d(w)m(e)f(can)h(think)d(of)i Fo(S)1190 890 y Fn(N)1294 876 y Fw(as)h(b)s(eing)d(the)j(lea)m(v)m(es)g (of)f(a)g(binary)f(tree)h Fo(T)2802 890 y Fn(N)2871 876 y Fw(,)h(obtained)f(b)m(y)g(setting)159 1022 y Fp(\033)s Fw(\()p Fp(t)p Fw(\))47 b Fo(\021)f Fw(\()p Fp(\033)568 1036 y Fn(i)594 1045 y Fs(1)634 1022 y Fp(;)15 b(\033)726 1036 y Fn(i)752 1045 y Fs(2)792 1022 y Fp(;)g(:)g(:)g(:)i(;)e(\033)1046 1036 y Fn(i)1072 1048 y Fs([)p Fr(tN)5 b Fs(])1197 1022 y Fw(\).)79 b(It)43 b(will)e(b)s(e)h(con)m(v)m(enien)m(t)i(to)g(b)s(e)f (able)f(to)i(represen)m(t)f Fp(\033)s Fw(,)k(for)c(giv)m(en)159 1167 y(0)26 b Fp(<)f(t)359 1181 y Fz(1)424 1167 y Fp(<)g(t)553 1181 y Fz(2)619 1167 y Fp(<)g Fo(\001)15 b(\001)g(\001)27 b Fp(<)d(t)975 1181 y Fn(n)1050 1167 y Fp(<)h Fw(1)31 b(in)e(the)h(form)1286 1431 y Fp(\033)e Fw(=)d(\()p Fp(\033)s Fw(\()p Fp(t)1620 1445 y Fz(1)1662 1431 y Fw(\))p Fp(;)15 b(\033)s Fw(\()p Fp(t)1860 1445 y Fz(2)1902 1431 y Fp(;)g(t)1975 1445 y Fz(1)2016 1431 y Fw(\))p Fp(;)g(:)g(:)g(:)i(;)e(\033)s Fw(\()p Fp(t)2376 1445 y Fn(n)2426 1431 y Fp(;)g Fw(1\)\))942 b(\(3)p Fp(:)p Fw(1\))159 1695 y(where)1322 1849 y Fp(\033)s Fw(\()p Fp(t)1445 1863 y Fn(i)1476 1849 y Fp(;)15 b(t)1549 1863 y Fn(i)p Fz(+1)1672 1849 y Fw(\))25 b Fo(2)g(f\000)p Fw(1)p Fp(;)15 b Fw(1)p Fo(g)2109 1812 y Fz([)p Fn(N)7 b Fz(\()p Fn(t)2250 1821 y Fr(i)p Fs(+1)2358 1812 y Fm(\000)p Fn(t)2442 1821 y Fr(i)2471 1812 y Fz(\)])p Fn(:)3523 1849 y Fw(\(3)p Fp(:)p Fw(2\))-8 2140 y(\(ii\))45 b(F)-8 b(or)32 b(non-decreasing)e(function)g Fp(A)c Fw(:)g([0)p Fp(;)15 b Fw(1])28 b Fo(!)e Fw([0)p Fp(;)15 b Fw(1],)34 b(w)m(e)d(ha)m(v)m(e)h(a)f(Gaussian)f(pro)s(cess)h Fp(X)3210 2154 y Fn(\033)3290 2140 y Fw(with)f(mean)159 2286 y(zero)h(and)f(co)m (v)-5 b(ariance)1475 2440 y Fl(E)11 b Fp(X)1605 2454 y Fn(\033)1660 2440 y Fp(X)1735 2454 y Fn(\034)1806 2440 y Fw(=)25 b Fp(A)p Fw(\()p Fp(d)2052 2454 y Fn(N)2122 2440 y Fw(\()p Fp(\033)n(;)15 b(\034)10 b Fw(\)\))p Fp(:)1131 b Fw(\(3)p Fp(:)p Fw(3\))159 2658 y(W)-8 b(e)31 b(extend)g(this)e(to)i (a)g(Gaussian)f(pro)s(cess)g(indexed)e(b)m(y)j(the)f(tree)h Fo(T)2533 2672 y Fn(N)2632 2658 y Fw(b)m(y)f(setting)1656 2922 y Fp(X)1731 2938 y Fn(\033)r Fz(\()p Fn(t)p Fz(\))1889 2922 y Fw(=)25 b Fp(X)2060 2936 y Fn(\033)2110 2922 y Fw(\()p Fp(t)p Fw(\))1310 b(\(3)p Fp(:)p Fw(4\))159 3186 y(where)38 b Fp(X)505 3200 y Fn(\033)555 3186 y Fw(\()p Fp(t)p 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b(through)g(the)g(map)g Fp(r)3215 5085 y Fn(N)3318 5071 y Fw(de\014ned)f(in)159 5217 y(\(2.1\))g(via)1372 5371 y(~)-52 b Fp(\026)1420 5334 y Fn(t)1420 5394 y(\014)s(;N)1580 5371 y Fo(\021)1708 5285 y Fk(X)1676 5480 y Fn(\033)r Fm(2S)1814 5489 y Fr(N)1888 5371 y Fp(\016)1928 5388 y Fn(r)1962 5400 y Fs([)p Fr(tN)5 b Fs(])2082 5388 y Fz(\()p Fn(\033)r Fz(\))2188 5371 y Fp(\026)2243 5334 y Fn(t)2243 5394 y(\014)s(;N)2378 5371 y Fw(\()p Fp(\033)s Fw(\))1020 b(\(3)p Fp(:)p Fw(8\))p eop %%Page: 16 16 16 15 bop 50 295 a Ft(16)3270 b Fq(Se)l(ction)27 b Ft(3)159 513 y Fw(W)-8 b(e)31 b(will)d(apply)h(the)i(construction)f(of)g (Section)g(2)h(to)g(this)f(family)e(of)j(measures.)40 b(Let)31 b(us)f(denote)h(b)m(y)1466 765 y Fk(e)1456 788 y Fw(\002)1527 750 y Fn(t)1527 810 y(\014)s(;N)1662 788 y Fw(\()p Fp(x)p Fw(\))25 b(=)1905 664 y Fk(Z)1996 687 y Fn(x)1956 871 y Fz(0)2062 788 y Fk(e)-56 b Fp(\026)2112 750 y Fn(t)2112 810 y(\014)s(;N)2247 788 y Fw(\()p Fp(dy)s Fw(\))1111 b(\(3)p Fp(:)p Fw(9\))159 1088 y(Note)46 b(that)f(b)m(y)g (construction)f(the)h(functions)1867 1065 y Fk(e)1857 1088 y Fw(\002)1928 1055 y Fn(t)1928 1114 y(\014)s(;N)2108 1088 y Fw(increase)f(on)h(the)g(set)g Fp(r)2975 1105 y Fz([)p Fn(tN)7 b Fz(])3113 1088 y Fw(\()p Fo(S)3203 1102 y Fn(N)3272 1088 y Fw(\))45 b(and)f(that)159 1234 y Fp(r)200 1250 y Fz([)p Fn(t)248 1232 y Fe(0)273 1250 y Fn(N)7 b Fz(])362 1234 y Fw(\()p Fo(S)452 1248 y Fn(N)521 1234 y Fw(\))35 b Fo(\032)e Fp(r)736 1250 y Fz([)p Fn(tN)7 b Fz(])874 1234 y Fw(\()p Fo(S)964 1248 y Fn(N)1033 1234 y Fw(\),)37 b(for)e(all)g(0)f Fp(<)g(t)1623 1201 y Fm(0)1682 1234 y Fp(<)f(t)h(<)f Fo(1)p Fw(,)k(th)m(us)e(the)h(assumptions)e(of)h (Lemma)h(2.2)g(are)159 1379 y(satis\014ed)29 b(for)i(all)e(v)-5 b(alues)30 b(of)g Fp(t;)15 b(t)1254 1346 y Fm(0)1280 1379 y Fp(;)g(t)1353 1346 y Fm(00)1399 1379 y Fw(.)41 b(It)30 b(is)g(also)g(useful)f(to)i(realise)e(that)1087 1621 y Fk(e)1077 1644 y Fw(\002)1148 1607 y Fn(t)1148 1667 y(\014)s(;N)1282 1644 y Fw(\()p Fp(r)1358 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Fw(\()p Fp(\033)s Fw(\)\))32 b(and)d(will)f(b)s(e)i(b)m(y)g(an)g(amoun) m(t)h Fp(\026)2180 625 y Fn(t)2180 684 y(\014)s(;N)2315 658 y Fw(\()p Fp(\033)s Fw(\()p Fp(t)2473 625 y Fm(0)2499 658 y Fw(\)\))g(and)f(so)1292 933 y Fp(m)1382 946 y Fz(~)1372 962 y(\002)1428 940 y Fr(t)1454 927 y Fe(0)1428 986 y Fr(\014)r(;N)1545 962 y Fz(\()p Fn(r)1607 971 y Fr(N)1666 962 y Fz(\()p Fn(\033)r Fz(\)\))1800 933 y Fw(\()p Fp(t;)15 b(t)1941 895 y Fm(0)1967 933 y Fw(\))26 b(=)f Fp(m)2204 947 y Fn(\033)2252 933 y Fw(\()p Fp(t)2320 895 y Fm(0)2346 933 y Fp(;)15 b Fw(1\))1011 b(\(3)p Fp(:)p Fw(15\))50 1207 y(where)36 b Fp(m)399 1221 y Fn(\033)484 1207 y Fw(is)f(de\014ned)g(in)g(\(1.5\).)61 b(This)35 b(observ)-5 b(ation)36 b(allo)m(ws)f(us)h(to)h(express)f(the)h(measure)f Fo(K)3417 1221 y Fn(\014)s(;N)3589 1207 y Fw(de-)50 1353 y(\014ned)d(in)g(\(1.4\))k(describing)32 b(the)j(geometry)g(of)g(the)g (Gibbs)e(measures)h(on)g(the)h(h)m(yp)s(ercub)s(e)e(in)g(terms)i(of)50 1498 y(the)h(genealogical)g(distance)f(distribution)d(functions)i Fo(K)2049 1465 y Fn(\017)2048 1521 y(t)2120 1498 y Fw(de\014ned)g(in)g (Section)h(2,)j(with)c(the)i(family)e(of)50 1644 y(measures)c Fp(\032)487 1611 y Fn(t)549 1644 y Fw(giv)m(en)h(b)m(y)f(the)g (measures)35 b Fk(e)-55 b Fp(\026)1515 1611 y Fn(t)1515 1669 y(\014)s(;N)1649 1644 y Fw(.)41 b(Namely:)50 1862 y Fd(Prop)s(osition)i(3.2:)p Fv(L)-5 b(et)36 b Fo(K)1082 1829 y Fn(\017)1081 1884 y(t)1154 1862 y Fv(b)-5 b(e)36 b(as)h(de\014ne)-5 b(d)37 b(in)g(\(2.11\))g(for)g(the)g(family)g(of)g (me)-5 b(asur)g(es)45 b Fw(~)-52 b Fp(\026)3235 1829 y Fn(t)3235 1888 y(\014)s(;N)3369 1862 y Fv(,)37 b(and)h(let)50 2007 y Fo(K)119 2021 y Fn(\014)s(;N)287 2007 y Fv(b)-5 b(e)32 b(de\014ne)-5 b(d)34 b(in)e(\(1.4\).)43 b(Then)1591 2173 y Fw(lim)1603 2232 y Fn(\017)p Fm(#)p Fz(0)1733 2173 y Fo(K)1803 2135 y Fn(\017)1802 2195 y Fz(1)1868 2173 y Fw(=)25 b Fo(K)2033 2187 y Fn(\014)s(;N)3477 2173 y Fw(\(3)p Fp(:)p Fw(16\))50 2409 y Fv(In)32 b(p)-5 b(articular,)1159 2598 y Fp(f)1204 2612 y Fn(\014)s(;N)1339 2598 y Fw(\()p Fp(t)1407 2561 y Fm(0)1433 2598 y Fw(\))25 b(=)g(1)c Fo(\000)f Fw(lim)1757 2658 y Fn(\017)p Fm(#)p Fz(0)1887 2598 y Fl(E)1957 2475 y Fk(Z)2054 2497 y Fz(1)2014 2681 y(0)2110 2598 y Fp(dx)p Fo(j)p Fp(m)2314 2561 y Fn(\017)2314 2621 y(x)2360 2598 y Fw(\(1)p Fp(;)15 b(t)2513 2561 y Fm(0)2540 2598 y Fw(\))p Fo(j)877 b Fw(\(3)p Fp(:)p Fw(17\))50 2869 y Fv(is)32 b(expr)-5 b(esse)g(d)35 b(in)d(terms)i(of)f(the)g(size) f(biase)-5 b(d)34 b(aver)-5 b(age)33 b(gap)g(size)g(in)f(the)h (function)2920 2846 y Fw(~)2904 2869 y Fp(S)2965 2822 y Fz(\()p Fn(t;)p Fz(1\))2960 2896 y Fn(\014)s(;N)3112 2869 y Fv(.)159 3087 y Fw(Prop)s(osition)22 b(3.2)j(is)e(obtained)g(in) g(a)h(purely)e(algebraic)i(manner)f(and)g(simply)e(sho)m(ws)j(that)h (the)f(general)50 3232 y(formalism)34 b(in)m(tro)s(duced)g(in)g (Section)i(2)g(allo)m(ws,)h(in)d(the)i(case)h(of)f(the)g(\014nite)f(v)m (olume)h(Gibbs)e(measures,)50 3378 y(to)c(express)f(the)g(measures)g Fo(K)1089 3392 y Fn(\014)s(;N)1254 3378 y Fw(in)f(terms)h(of)g(the)h (genealogy)h(of)e(the)h(em)m(b)s(edded)e(family)f(of)j(measures)57 3523 y(~)-52 b Fp(\026)105 3490 y Fn(t)105 3549 y(\014)s(;N)240 3523 y Fw(.)79 b(This)42 b(op)s(ens)g(the)i(w)m(a)m(y)g(to)g(express)f (the)g(thermo)s(dynamic)f(limit)f(of)i Fo(K)2903 3537 y Fn(\014)s(;N)3082 3523 y Fw(in)f(terms)h(of)h(the)50 3669 y(genealogy)31 b(of)g(a)g(suitable)e(family)f(of)j(measures)f(on)g ([0)p Fp(;)15 b Fw(1].)50 3955 y Fx(An)35 b(alternativ)m(e)f(setting.) 159 4173 y Fw(Before)26 b(turning)e(to)i(this)e(question,)i(let)f(us)g (note)g(that)h(the)g(ab)s(o)m(v)m(e)g(setting)g(of)f(Gaussian)g(pro)s (cesses)g(on)50 4319 y Fo(f\000)p Fw(1)p Fp(;)15 b Fw(1)p Fo(g)341 4286 y Fn(N)437 4319 y Fw(can)26 b(b)s(e)f(completely)h (reform)m(ulated)f(in)g(terms)g(of)h(Gaussian)f(pro)s(cesses)h(on)f (the)h(unit)e(in)m(terv)-5 b(al,)50 4464 y(resp)s(ectiv)m(ely)24 b(unit)g(square.)38 b(T)-8 b(o)26 b(see)f(this,)g(let)1656 4440 y(~)1640 4464 y Fp(d)1687 4478 y Fz(1)1753 4464 y Fw(denote)h(the)f(standard)f(dy)m(adic)h(v)-5 b(aluation)24 b(on)h([0)p Fp(;)15 b Fw(1],)28 b(i.e.)50 4609 y(if)34 b Fp(x)e Fw(=)326 4541 y Fk(P)422 4637 y Fn(i)p Fm(2f)p Fz(0)p Fn(;)p Fz(1)p Fm(g)685 4609 y Fp(x)737 4623 y Fn(i)767 4609 y Fw(2)812 4576 y Fm(\000)p Fn(i)899 4609 y Fw(,)37 b(then)1188 4585 y(~)1172 4609 y Fp(d)1219 4623 y Fz(1)1260 4609 y Fw(\()p Fp(x;)15 b(x)1439 4576 y Fm(0)1465 4609 y Fw(\))34 b(=)e(inf)21 b(\()p Fp(i)26 b Fw(:)f Fp(x)1957 4623 y Fn(i)2013 4609 y Fo(6)p Fw(=)g Fp(x)2161 4576 y Fm(0)2161 4633 y Fn(i)2191 4609 y Fw(\))e Fo(\000)g Fw(1.)55 b(Set)2645 4585 y(~)2629 4609 y Fp(d)2676 4623 y Fn(N)2745 4609 y Fw(\()p Fp(x)p Fw(\))34 b Fo(\021)e Fp(N)3087 4576 y Fm(\000)p Fz(1)3201 4585 y Fw(~)3185 4609 y Fp(d)3232 4623 y Fn(N)3301 4609 y Fw(\()p Fp(x)p Fw(\).)55 b(Then)50 4755 y(let)30 b Fp(X)256 4769 y Fn(N)325 4755 y Fw(\()p Fp(t;)15 b(x)p Fw(\))32 b(b)s(e)d(the)i(cen)m(tered)g 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5316 y Fw(\))26 b Fo(\025)f Fp(t)p Fw(,)50 5462 y(i.e.)40 b Fp(X)281 5476 y Fn(N)351 5462 y Fw(\()p Fp(t;)15 b(x)p Fw(\))31 b(is)e(piecewise)h(constan)m(t)h(as)g(a)g (function)e(of)h Fp(x)h Fw(on)f(blo)s(c)m(ks)g(of)g(size)g(2)2881 5429 y Fm(\000)p Fz([)p Fn(N)7 b(t)p Fz(])3076 5462 y Fw(.)p eop %%Page: 18 18 18 17 bop 50 295 a Ft(18)3270 b Fq(Se)l(ction)27 b Ft(4)159 513 y Fw(While)g(it)h(is)g(plain)e(that)j(the)g(pro)s(cess)f Fp(X)1566 527 y Fn(N)1664 513 y Fw(do)s(es)g(not)h(con)m(v)m(erge)i(to) e(a)g(sensible)d(limit,)h(it)h(is)g(a)h(sensible)50 658 y(question)k(to)i(ask)g(for)f(the)h(extremal)f(prop)s(erties)f(of)h (the)h(pro)s(cess)f Fp(X)2474 672 y Fn(N)2543 658 y Fw(\()p Fp(t;)15 b(x)p Fw(\))35 b(as)g Fp(N)42 b Fo(")32 b(1)p Fw(.)52 b(T)-8 b(o)35 b(this)e(end)50 804 y(one)d(ma)m(y)h(in)m(tro)s (duce)f(the)g(measures)1290 1101 y(^)-52 b Fp(\026)1338 1063 y Fn(t)1338 1123 y(\014)s(;N)1473 1101 y Fw(\()p Fp(dx)p Fw(\))26 b Fo(\021)1830 1039 y Fp(e)1872 1006 y Fn(\014)1916 952 y Fm(p)p 1977 952 65 4 v 1977 1006 a Fn(N)6 b(X)2101 1015 y Fr(N)2159 1006 y Fz(\()p Fn(t;x)p Fz(\))2311 1039 y Fp(dx)p 1775 1080 691 4 v 1775 1109 a Fk(R)1836 1132 y Fz(1)1818 1215 y(0)1892 1182 y Fp(dy)s(e)2029 1156 y Fn(\014)2073 1102 y Fm(p)p 2133 1102 65 4 v 54 x Fn(N)h(X)2258 1165 y Fr(N)2316 1156 y Fz(\()p Fn(t;y)r Fz(\))3477 1101 y Fw(\(3)p Fp(:)p Fw(19\))50 1402 y(While)26 b(these)h(measures)g(are)g(absolutely)f(con)m(tin)m(uous)g(with)g(resp) s(ect)h(to)h(Leb)s(esgue)e(measure,)i(they)f(ma)m(y)50 1548 y(con)m(v)m(erge)32 b(to)f(singular)e(measures)h(as)g Fp(N)41 b Fw(tends)30 b(to)h(in\014nit)m(y)-8 b(.)159 1766 y(W)g(e)32 b(can)f(naturally)e(apply)g(the)i(construction)f(of)h (Section)g(2)g(to)g(the)g(family)e(of)i(measures)38 b(^)-52 b Fp(\026)3398 1733 y Fn(t)3398 1792 y(\014)s(;N)3563 1766 y Fw(and)50 1911 y(de\014ne)27 b(in)g(particular)f(the)j(corresp)s (onding)d(measures)1971 1889 y Fk(b)1956 1911 y Fo(K)2026 1878 y Fn(\017)2025 1937 y(\014)s(;N)s(;t)2205 1911 y Fw(,)j(for)f Fp(\017)d(>)g Fw(0.)40 b(Of)28 b(course,)h(for)e Fp(N)38 b Fw(\014xed,)29 b(the)50 2057 y(limits)c(as)j Fp(\017)g Fw(tends)f(to)i(zero)g(will)c(b)s(e)i(trivial)f(since)h(the)h (corresp)s(onding)e(functions)g Fp(S)2977 2024 y Fn(t)3004 1997 y Fe(0)3029 2024 y Fn(;t)3111 2057 y Fw(are)i(con)m(tin)m(uous.)50 2202 y(But)41 b(k)m(eeping)h Fp(\017)h(>)g Fw(0)f(\014xed,)i(letting)d Fp(N)51 b Fw(tend)41 b(to)h(in\014nit)m(y)-8 b(,)43 b(and)d(taking)i (then)f Fp(\017)g Fw(to)h(zero,)j(w)m(e)d(get)g(a)50 2348 y(non-trivial)24 b(answ)m(er)i(that)h(coincides)e(with)g(the)i (one)f(w)m(e)h(will)d(obtain)i(from)f(the)i(construction)f(describ)s (ed)50 2493 y(in)j(the)h(\014rst)g(part)g(of)h(this)e(section.)50 2874 y FA(4.)55 b(Con)m(tin)m(uous)41 b(state)g(branc)m(hing,)g (bridges,)h(and)f(all)g(that.)159 3092 y Fw(W)-8 b(e)38 b(will)c(no)m(w)i(giv)m(e)h(a)h(brief)d(summary)g(of)i(the)g (constructions)f(in)f([BeLG])j(related)f(to)g(con)m(tin)m(uous)50 3237 y(state)42 b(branc)m(hing)d(pro)s(cesses.)72 b(The)41 b(basic)f(ob)5 b(ject)41 b(here)g(is)f(a)h(con)m(tin)m(uous)g(state)h (branc)m(hing)d(pro)s(cess)50 3383 y(c)m(haracterized)34 b(b)m(y)f(its)f(Laplace)h(functional)e Fp(u)1682 3397 y Fn(t)1714 3383 y Fw(\()p Fp(\025)p Fw(\).)49 b(The)33 b(pro)s(cess)f(started)i(in)d Fp(a)f Fo(\025)f Fw(0)k(will)d(b)s(e)j (denoted)50 3528 y Fp(X)7 b Fw(\()p Fo(\001)p Fp(;)15 b(a)p Fw(\).)65 b(This)36 b(can)i(b)s(e)f(extended)h(to)h(a)f(gen)m (uine)f(t)m(w)m(o)j(parameter)e(pro)s(cess)f(using)g(the)h(fundamen)m (tal)50 3674 y(branc)m(hing)21 b(prop)s(ert)m(y)h(that)i(state)g(that)g (if)d Fp(X)1580 3641 y Fm(0)1606 3674 y Fw(\()p Fo(\001)p Fp(;)15 b(b)p Fw(\))25 b(and)d Fp(X)7 b Fw(\()p Fo(\001)p Fp(;)15 b(a)p Fw(\))25 b(are)e(indep)s(enden)m(t)d(copies,)25 b(then)d Fp(X)7 b Fw(\()p Fo(\001)p Fp(;)15 b(a)5 b Fw(+)50 3819 y Fp(b)p Fw(\))35 b(has)g(the)g(same)h(la)m(w)f(as)g Fp(X)1087 3786 y Fm(0)1113 3819 y Fw(\()p Fo(\001)p Fp(;)15 b(b)p Fw(\))25 b(+)e Fp(X)7 b Fw(\()p Fo(\001)p Fp(;)15 b(a)p Fw(\).)56 b(Th)m(us)34 b(they)h(construct)h(their)e(pro)s(cess)h (b)m(y)f(demanding)50 3965 y(that)26 b(for)g(an)m(y)g Fp(a;)15 b(b)25 b Fo(\025)g Fw(0,)j Fp(X)7 b Fw(\()p Fo(\001)p Fp(;)15 b(a)c Fw(+)g Fp(b)p Fw(\))g Fo(\000)g Fp(X)c Fw(\()p Fo(\001)p Fp(;)15 b(a)p Fw(\))28 b(is)d(indep)s(enden)m (t)e(of)j(the)g(pro)s(cesses)g Fp(X)7 b Fw(\()p Fo(\001)p Fp(;)15 b(c)p Fw(\),)29 b(for)c(all)g Fp(c)h Fo(\024)f Fp(a)p Fw(,)50 4110 y(and)34 b(its)f(la)m(w)i(is)e(the)i(same)f(as)h (that)g(of)g Fp(X)7 b Fw(\()p Fo(\001)p Fp(;)15 b(b)p Fw(\).)54 b(The)34 b(righ)m(t)g(con)m(tin)m(uous)g(v)m(ersion)g(of)h Fp(X)7 b Fw(\()p Fp(t;)15 b Fo(\001)p Fw(\))36 b(is)d(then)h(a)50 4256 y(sub)s(ordinator.)j(The)26 b(Mark)m(o)m(v)j(prop)s(ert)m(y)d(of)i (the)f(branc)m(hing)e(pro)s(cess)i(leads)f(to)i(the)f(follo)m(wing)f (extended)50 4401 y(construction,)k(due)g(to)h(Bertoin)f(and)g(Le)g (Gall)g([BeLG]:)50 4619 y Fd(Prop)s(osition)38 b(4.1:)p Fv(Ther)-5 b(e)33 b(exists)g(a)g(pr)-5 b(o)g(c)g(ess)1759 4596 y Fw(\026)1744 4619 y Fp(S)1805 4586 y Fz(\()p Fn(s;t)p Fz(\))1950 4619 y Fw(\()p Fp(a)p Fw(\))p Fp(;)15 b Fw(0)27 b Fo(\024)d Fp(s)h Fo(\024)g Fp(t;)15 b(a)26 b Fo(\025)f Fw(0)p Fv(,)33 b(such)f(that)10 4837 y(\(i\))47 b(F)-7 b(or)34 b(any)f Fw(0)26 b Fo(\024)f Fp(s)f Fo(\024)h Fp(t)p Fv(,)942 4814 y Fw(\026)927 4837 y Fp(S)988 4804 y Fz(\()p Fn(s;t)p Fz(\))1165 4837 y Fv(is)32 b(a)h(sub)-5 b(or)g(dinator)36 b(with)e(L)-5 b(aplac)g(e)34 b(exp)-5 b(onent)34 b Fp(u)2819 4851 y Fn(t)p Fm(\000)p Fn(s)2942 4837 y Fw(\()p Fp(\025)p Fw(\))p Fv(.)-18 5056 y(\(ii\))47 b(F)-7 b(or)24 b(any)g(inte)-5 b(ger)24 b Fp(p)h Fo(\025)g Fw(2)e Fv(and)i Fw(0)g Fo(\024)g Fp(t)1378 5070 y Fz(1)1444 5056 y Fo(\024)g Fp(t)1573 5070 y Fz(2)1639 5056 y Fo(\024)g(\001)15 b(\001)g(\001)26 b(\024)f Fp(t)1995 5070 y Fn(p)2037 5056 y Fv(,)g(the)f(sub)-5 b(or)g(dinators)2798 5033 y Fw(\026)2782 5056 y Fp(S)2843 5023 y Fz(\()p Fn(t)2898 5032 y Fs(1)2935 5023 y Fn(;t)2984 5032 y Fs(2)3019 5023 y Fz(\))3052 5056 y Fp(;)3108 5033 y Fw(\026)3092 5056 y Fp(S)3153 5023 y Fz(\()p Fn(t)3208 5032 y Fs(2)3245 5023 y Fn(;t)3294 5032 y Fs(3)3330 5023 y Fz(\))3362 5056 y Fp(;)15 b(:)g(:)g(:)i(;)3580 5033 y Fw(\026)3564 5056 y Fp(S)3625 5023 y Fz(\()p Fn(t)3680 5032 y Fr(p)p Fe(\000)p Fs(1)3800 5023 y Fn(;t)3849 5032 y Fr(p)3887 5023 y Fz(\))159 5201 y Fv(ar)-5 b(e)33 b(indep)-5 b(endent)35 b(and)708 5439 y Fw(\026)693 5462 y Fp(S)754 5424 y Fz(\()p Fn(t)809 5433 y Fs(1)845 5424 y Fn(;t)894 5433 y Fr(p)932 5424 y Fz(\))965 5462 y Fw(\()p Fp(a)p Fw(\))26 b(=)1220 5439 y(\026)1205 5462 y Fp(S)1266 5424 y Fz(\()p Fn(t)1321 5433 y Fs(1)1357 5424 y Fn(;t)1406 5433 y Fs(2)1442 5424 y Fz(\))1495 5462 y Fo(\016)1576 5439 y Fw(\026)1561 5462 y Fp(S)1622 5424 y Fz(\()p Fn(t)1677 5433 y Fs(2)1713 5424 y Fn(;t)1762 5433 y Fs(3)1798 5424 y Fz(\))1851 5462 y Fo(\016)20 b(\001)15 b(\001)g(\001)22 b(\016)2124 5439 y Fw(\026)2108 5462 y Fp(S)2169 5424 y Fz(\()p Fn(t)2224 5433 y Fr(p)p Fe(\000)p Fs(1)2344 5424 y Fn(;t)2393 5433 y Fr(p)2431 5424 y Fz(\))2464 5462 y Fw(\()p Fp(a)p Fw(\))p Fp(;)15 b Fo(8)2673 5476 y Fn(a)p Fm(\025)p Fz(0)2810 5462 y Fp(;)109 b Fv(a.s.)440 b Fw(\(4)p Fp(:)p Fw(1\))p eop %%Page: 19 19 19 18 bop 50 295 a Fq(GREM,)28 b(p)l(art)f(4)3089 b Ft(19)-46 513 y Fv(\(iii\))47 b(The)33 b(pr)-5 b(o)g(c)g(esses)747 490 y Fw(\026)732 513 y Fp(S)793 480 y Fz(\(0)p Fn(;t)p Fz(\))939 513 y Fw(\()p Fp(a)p Fw(\))34 b Fv(and)f Fp(X)7 b Fw(\()p Fp(t;)15 b(a)p Fw(\))34 b Fv(have)f(the)g(same)g(\014nite)g (dimensional)i(mar)-5 b(ginals.)159 731 y Fw(It)25 b(is)f(plain)f(that) i(the)917 708 y(\026)902 731 y Fp(S)963 698 y Fz(\()p Fn(t)1018 670 y Fe(0)1043 698 y Fn(;t)p Fz(\))1150 731 y Fw(constructed)g(in)f([BeLG])i(are)f(closely)g(related)g(the)g(the)g Fp(S)3137 698 y Fz(\()p Fn(t)3192 670 y Fe(0)3217 698 y Fn(;t)p Fz(\))3324 731 y Fw(of)g(Section)50 876 y(2.)40 b(In)26 b(fact,)j(w)m(e)f(can)g(\014rst)e(asso)s(ciate)i(to)g(the)g (increasing)e(pro)s(cess)h Fp(X)7 b Fw(\()p Fp(t;)15 b(a)p Fw(\))28 b(the)g(probabilit)m(y)c(distribution)50 1022 y(function)1556 1167 y(\002)1627 1130 y Fn(t)1659 1167 y Fw(\()p Fp(x)p Fw(\))i Fo(\021)1914 1106 y Fp(X)7 b Fw(\()p Fp(t;)15 b(x)p Fw(\))p 1914 1146 279 4 v 1917 1230 a Fp(X)7 b Fw(\()p Fp(t;)15 b Fw(1\))3523 1167 y(\(4)p Fp(:)p Fw(2\))50 1385 y(for)30 b Fp(x)25 b Fo(2)g Fw([0)p Fp(;)15 b Fw(1].)42 b(Then)30 b(the)g(normalized)f(v)m(ersions)h(of)h (the)f(pro)s(cess)2395 1362 y(\026)2380 1385 y Fp(S)2441 1352 y Fz(\()p Fn(s;t)p Fz(\))2586 1385 y Fw(\()p Fp(a)p Fw(\))1203 1601 y Fp(S)1264 1563 y Fz(\()p Fn(s;t)p Fz(\))1409 1601 y Fw(\()p Fp(x)p Fw(\))c Fo(\021)1777 1539 y Fw(1)p 1664 1580 272 4 v 1664 1663 a Fp(X)7 b Fw(\()p Fp(t;)15 b Fw(1\))1962 1578 y(\026)1946 1601 y Fp(S)2007 1563 y Fz(\()p Fn(s;t)p Fz(\))2152 1601 y Fw(\()p Fp(X)7 b Fw(\()p Fp(s;)15 b Fw(1\))p Fp(x)p Fw(\))969 b(\(4)p Fp(:)p Fw(3\))50 1821 y(can)30 b(b)s(e)g(represen)m(ted)h(as)1356 1967 y Fp(S)1417 1929 y Fz(\()p Fn(s;t)p Fz(\))1562 1967 y Fw(\()p Fp(a)p Fw(\))26 b(=)f(\002)1873 1929 y Fn(t)1920 1893 y Fk(\000)1961 1967 y Fw(\(\002)2067 1929 y Fn(s)2106 1967 y Fw(\))2141 1929 y Fm(\000)p Fz(1)2239 1967 y Fw(\()p Fp(x)p Fw(\))2361 1893 y Fk(\001)3523 1967 y Fw(\(4)p Fp(:)p Fw(4\))50 2155 y(as)k(in)e(Section)i(2.)40 b(Note)30 b(that)g(the)e(h)m(yp)s(othesis)g(of)h(Lemma)f(2.2)i(are)f(alw)m(a)m (ys)g(satis\014ed)f(in)g(this)f(case)j(with)50 2301 y(probabilit)m(y)k (one,)40 b(since)c(the)h(sub)s(ordinators)1711 2278 y(\026)1696 2301 y Fp(S)1757 2268 y Fz(\()p Fn(s;t)p Fz(\))1938 2301 y Fw(increase)g(is)f(dense.)60 b(The)37 b(asso)s(ciated)g(genealogy)50 2446 y(of)42 b(this)g(family)e(of)j(probabilit)m(y)d(measures)i(is)f (then)h(directly)f(equiv)-5 b(alen)m(t)42 b(to)h(the)f(genealogy)i(of)f (the)50 2592 y(underlying)27 b(branc)m(hing)i(pro)s(cess.)159 2810 y(W)-8 b(e)31 b(denote)g(b)m(y)752 2787 y(^)735 2810 y Fo(K)804 2824 y Fn(t)866 2810 y Fw(empirical)d(genealogical)j (pro)s(cess)f(asso)s(ciated)h(to)g(this)f(measure.)159 3028 y(No)m(w)46 b(let)f(us)f(consider)g(the)h(ab)s(o)m(v)m(e)h (construction)f(for)g(the)g(sp)s(ecial)f(case)i(of)f(Nev)m(eu's)h (branc)m(hing)50 3173 y(pro)s(cess.)40 b(This)29 b(is)g(the)i(pro)s (cess)f(with)f(Laplace)h(functional)f(\011\()p Fp(u)p Fw(\))d(=)f Fp(u)15 b Fw(ln)f Fp(u)p Fw(.)159 3392 y(Bertoin)35 b(and)h(Le)f(Gall)g([BeLG])i(sho)m(w)m(ed)f(that)h(the)f(coalescen)m(t) h(pro)s(cess)e(on)h(the)g(in)m(tegers)g(induced)50 3537 y(b)m(y)30 b(Nev)m(eu's)i(pro)s(cess)e(\(as)h(explained)d(in)h(Section) i(2\))1937 3514 y(^)1919 3537 y Fo(K)1988 3551 y Fn(t)2051 3537 y Fw(coincides)e(with)g(the)i(coalescen)m(t)h(pro)s(cess)e(con-)50 3683 y(structed)23 b(b)m(y)g(Bolthausen)g(and)g(Sznitman)e([BoS].)k (They)d(also)i(pro)m(v)m(ed)f(the)h(follo)m(wing)d(remark)-5 b(able)23 b(result)50 3828 y(connecting)31 b(the)g(collection)f(of)h (sub)s(ordinators)d(to)k(Ruelle's)d(mo)s(del.)41 b(Let)31 b(us)f(state)i(this)e(result)g(for)g(our)50 3973 y(con)m(v)m(enience.) 41 b(T)-8 b(ak)m(e)31 b(the)e(parameters)h(0)c Fp(<)e(x)1644 3987 y Fz(1)1710 3973 y Fp(<)h Fo(\001)15 b(\001)g(\001)27 b Fp(<)e(x)2086 3987 y Fn(p)2153 3973 y Fp(<)g Fw(1)k(and)g(0)d Fp(<)f(t)2699 3987 y Fz(1)2765 3973 y Fp(<)g Fo(\001)15 b(\001)g(\001)26 b Fp(<)f(t)3121 3987 y Fn(p)3192 3973 y Fw(link)m(ed)i(b)m(y)j(the)50 4119 y(iden)m(tities)1493 4264 y Fp(t)1526 4278 y Fn(k)1596 4264 y Fw(=)25 b(ln)14 b Fp(x)1835 4278 y Fn(k)r Fz(+1)1992 4264 y Fo(\000)20 b Fw(ln)14 b Fp(x)2226 4278 y Fz(1)3523 4264 y Fw(\(4)p Fp(:)p Fw(5\))50 4453 y(for)29 b Fp(k)f Fw(=)d(0)p Fp(;)15 b(:)g(:)g(:)i(;)e(p)j Fo(\000)f Fw(1,)29 b(and)g Fp(t)1066 4467 y Fn(p)1133 4453 y Fw(=)c Fo(\000)15 b Fw(ln)f Fp(x)1458 4467 y Fz(1)1498 4453 y Fw(.)41 b(Then)28 b(the)h(la)m(w)f(of)i(the)f (family)e(of)i(jumps)e(of)j(the)f(normalised)50 4599 y(sub)s(ordinators)36 b Fp(S)694 4566 y Fz(\()p Fn(t)749 4576 y Fr(k)790 4566 y Fn(;t)839 4575 y Fr(p)877 4566 y Fz(\))910 4599 y Fw(,)42 b(for)d Fp(k)k Fw(=)c(0)p Fp(;)15 b(:)g(:)g(:)i(;)e(p)26 b Fo(\000)g Fw(1,)42 b(is)c(the)i(same)f (as)h(the)f(la)m(w)g(of)g(Ruelle's)f(probabilit)m(y)50 4744 y(cascades)31 b(with)f(parameters)g Fp(x)1144 4758 y Fn(i)1174 4744 y Fw(,)h Fp(i)26 b Fw(=)f(1)p Fp(;)15 b(:)g(:)g(:)i(;)e(p)p Fw(.)159 4962 y(No)m(w)26 b(consider)f(a)h(GREM)g (with)e(\014nitely)g(man)m(y)h(hierarc)m(hies)g(and)g(parameters)h(suc) m(h)f(that)h(the)g(p)s(oin)m(ts)50 5108 y Fp(y)95 5122 y Fz(0)167 5108 y Fw(=)32 b(0)j(and)f Fp(y)576 5122 y Fn(i)606 5108 y Fw(,)i Fp(i)c Fw(=)g(1)p Fp(;)15 b(:)g(:)g(:)i(;)e(p)35 b Fw(are)g(the)g(extremal)f(p)s(oin)m(ts)g(of)g(the)h(con)m(v)m(ex)h(h) m(ull)d(of)i Fp(A)p Fw(.)53 b(Let)35 b(us)f(remind)50 5253 y(that)d(lim)373 5267 y Fn(N)7 b Fm(!1)602 5253 y Fl(E)k Fp(f)702 5267 y Fn(\014)s(;N)842 5253 y Fw(\()p Fp(y)s Fw(\))26 b(=)f Fl(E)11 b Fp(f)1182 5267 y Fn(\014)1236 5253 y Fw(\()p Fp(y)s Fw(\))31 b(can)g(b)s(e)e(computed)i(b)m(y)f (\(1.8\))i(for)e(an)m(y)h Fp(y)d Fo(2)d Fw([0)p Fp(;)15 b Fw(1].)42 b(No)m(w)31 b(set)1628 5462 y Fl(E)11 b Fp(f)1728 5476 y Fn(\014)1782 5462 y Fw(\()p Fp(y)1862 5476 y Fn(i)1893 5462 y Fw(\))25 b(=)g Fp(x)2101 5476 y Fn(i)3523 5462 y Fw(\(4)p Fp(:)p Fw(6\))p eop %%Page: 20 20 20 19 bop 50 295 a Ft(20)3270 b Fq(Se)l(ction)27 b Ft(5)50 513 y Fw(where)48 b(all)g(of)h(the)f Fp(x)824 527 y Fn(i)910 513 y Fp(<)56 b Fw(1.)96 b(In)48 b(Theorem)h(1.5)g(of)56 b([BK2])50 b(w)m(e)f(pro)m(v)m(ed)g(that)h(the)e(p)s(oin)m(t)g(pro)s (cess)50 590 y Fk(P)146 685 y Fn(\033)210 658 y Fp(\016)250 675 y Fn(\026)294 685 y Fr(\014)r(;N)412 675 y Fz(\()p Fn(\033)484 656 y Fe(0)509 675 y Fz(:)p Fn(d)568 684 y Fr(N)626 675 y Fz(\()p Fn(\033)n(;\033)760 656 y Fe(0)786 675 y Fz(\))p Fn(>y)908 684 y Fs(1)943 675 y Fz(\))p Fn(;:::)11 b(;\026)1136 685 y Fr(\014)r(;N)1254 675 y Fz(\()p Fn(\033)1326 656 y Fe(0)1351 675 y Fz(:)p Fn(d)1410 684 y Fr(N)1468 675 y Fz(\()p Fn(\033)n(;\033)1602 656 y Fe(0)1628 675 y Fz(\))p Fn(>y)1750 684 y Fr(p)1788 675 y Fz(\))1859 658 y Fw(in)38 b([0)p Fp(;)15 b Fw(1])2154 625 y Fn(p)2236 658 y Fw(con)m(v)m(erge)41 b(to)f(Ruelle's)d (probabilit)m(y)g(cas-)50 804 y(cades)31 b(with)e(parameters)i Fp(x)1023 818 y Fn(i)1053 804 y Fw(,)f Fp(i)c Fw(=)f(1)p Fp(;)15 b(:)g(:)g(:)32 b(;)15 b(p)p Fw(.)159 1022 y(Com)m(bining)28 b(these)j(t)m(w)m(o)h(results)d(yields)50 1240 y Fd(Prop)s(osition)35 b(4.2:)p Fv(L)-5 b(et)32 b Fp(\026)1055 1254 y Fn(\014)s(;N)1220 1240 y Fv(b)-5 b(e)31 b(the)g(Gibbs)g(me)-5 b(asur)g(e)32 b(asso)-5 b(ciate)g(d)34 b(to)d(a)g(GREM)g(with)g(\014nitely)h(many)50 1385 y(hier)-5 b(ar)g(chies)40 b(satisfying)f(\(4.6\))h(at)f(the)g (extr)-5 b(emal)40 b(p)-5 b(oints)40 b Fp(y)2119 1399 y Fn(i)2148 1385 y Fv(,)g Fp(i)c Fw(=)g(1)p Fp(;)15 b(:)g(:)g(:)i(;)e (k)42 b Fv(of)c(the)h(c)-5 b(onvex)39 b(hul)5 b(l)39 b(of)f(the)50 1531 y(function)h Fp(A)p Fv(.)61 b(Then)40 b(the)f(family)h(of)f(functions)1783 1508 y Fk(e)1770 1531 y Fp(S)1831 1482 y Fz(\()p Fn(y)1896 1492 y Fr(k)1936 1482 y Fn(;y)1995 1491 y Fr(p)2033 1482 y Fz(\))1826 1558 y Fn(\014)s(;N)2065 1531 y Fv(,)i Fp(k)f Fw(=)d(0)p Fp(;)15 b Fw(1)p Fp(;)g(:)g(:)g(:)j(;)d(p)25 b Fo(\000)f Fw(1)40 b Fv(de\014ne)-5 b(d)40 b(ac)-5 b(c)g(or)g(ding)41 b(to)50 1676 y(\(3.11\))c(with)f(r)-5 b(esp)g(e)g(ct)36 b(to)g Fp(\026)992 1690 y Fn(\014)s(;N)1162 1676 y Fv(c)-5 b(onver)g(ges)36 b(in)f(law,)i(and)f(the)g(limit)f(has)i(the)e(same)h (distribution)h(as)e(the)50 1822 y(family)42 b(of)f(normalize)-5 b(d)43 b(sub)-5 b(or)g(dinators)44 b(\(4.3\))e(or)g(\(4.4\))g Fp(S)2146 1789 y Fz(\()p Fn(t)2201 1799 y Fr(k)2241 1789 y Fn(;t)2290 1798 y Fr(p)2328 1789 y Fz(\))2361 1822 y Fv(,)h Fp(k)g Fw(=)d(0)p Fp(;)15 b Fw(1)p Fp(;)g(:)g(:)g(:)j(;)d(p)27 b Fo(\000)e Fw(1)42 b Fv(in)e(the)i(sense)50 1967 y(that)c(the)f(joint) g(distribution)g(of)g(their)g(jumps)g(has)h(the)f(same)g(law,)i(pr)-5 b(ovide)g(d)39 b Fp(t)2858 1981 y Fn(k)2939 1967 y Fv(is)d(chosen)i(ac) -5 b(c)g(or)g(ding)50 2113 y(to)33 b(\(4.6\),)h(\(4.5\).)159 2331 y Fw(W)-8 b(e)30 b(can)g(actually)f(pro)m(v)m(e)h(a)f(more)g (general)h(result.)39 b(F)-8 b(rom)30 b(the)f(preceding)f(theorem)i(w)m (e)f(exp)s(ect)h(that)50 2476 y(Nev)m(eu's)h(pro)s(cess)e(will)f(pro)m (vide)h(the)h(univ)m(ersal)e(limit)g(for)i(all)f(of)h(our)f(mo)s(dels;) g(the)i(dep)s(endence)e(on)h(the)50 2622 y(particular)c(mo)s(del)g (\(i.e.)40 b(the)27 b(function)g Fp(A)p Fw(\))g(and)g(on)h(the)f(temp)s (erature)h(m)m(ust)f(come)h(from)f(a)h(rescaling)f(of)50 2767 y(time.)44 b(Set)32 b Fp(x)p Fw(\()p Fp(y)s Fw(\))c Fo(\021)f Fl(E)11 b Fp(f)850 2781 y Fn(\014)904 2767 y Fw(\()p Fp(y)s Fw(\),)33 b(where)e Fl(E)11 b Fp(f)1444 2781 y Fn(\014)1498 2767 y Fw(\()p Fp(y)s Fw(\))33 b(is)d(de\014ned)h (b)m(y)g(the)h(function)f Fp(A)g Fw(through)g(\(1.8\).)47 b(Set)32 b(also)50 2913 y Fp(T)38 b Fw(=)25 b Fo(\000)15 b Fw(ln)f Fp(x)p Fw(\(0\),)31 b(and)f(for)g Fp(y)e(>)d Fw(0)1512 3058 y Fp(t)p Fw(\()p Fp(y)s Fw(\))g(=)g Fp(T)33 b Fw(+)20 b(ln)14 b Fp(x)p Fw(\()p Fp(y)s Fw(\))p Fp(:)1276 b Fw(\(4)p Fp(:)p Fw(7\))50 3249 y(De\014ne)30 b(the)h(family)e(of)h (distribution)d(functions)1472 3472 y(\002)1543 3429 y Fn(y)1543 3500 y(f)1578 3510 y Fr(\014)1625 3472 y Fw(\()p Fp(x)p Fw(\))f Fo(\021)1880 3411 y Fp(X)7 b Fw(\()p Fp(t)p Fw(\()p Fp(y)s Fw(\))p Fp(;)15 b(x)p Fw(\))p 1880 3451 397 4 v 1883 3535 a Fp(X)7 b Fw(\()p Fp(t)p Fw(\()p Fp(y)s Fw(\))p Fp(;)15 b Fw(1\))3523 3472 y(\(4)p Fp(:)p Fw(8\))50 3728 y(where)23 b Fp(X)31 b Fw(is)23 b(Nev)m(eu's)i(pro)s (cess.)38 b(Construct)23 b(the)h(genealogical)h(pro)s(cess)2572 3705 y(^)2554 3728 y Fo(K)2624 3678 y Fn(f)2659 3688 y Fr(\014)2623 3738 y Fn(y)2733 3728 y Fw(=)g(lim)2955 3742 y Fn(\017)p Fm(#)p Fz(0)3096 3705 y Fw(^)3078 3728 y Fo(K)3148 3678 y Fn(f)3183 3688 y Fr(\014)3227 3678 y Fn(;\017)3147 3738 y(y)3307 3728 y Fw(asso)s(ciated)50 3873 y(to)31 b(this)e(family)g(of)i(measures.)40 b(Then)50 4091 y Fd(Theorem)k(4.3:)67 b Fv(Consider)39 b(a)f(Derrida)h(mo)-5 b(del)40 b(with)f(gener)-5 b(al)39 b(function)g Fp(A)f Fv(such)g(that)h Fp(A)g Fv(do)-5 b(es)39 b(not)50 4237 y(touch)33 b(its)g(c)-5 b(onvex)33 b(hul)5 b(l)920 4214 y Fw(\026)896 4237 y Fp(A)33 b Fv(in)f(the)h(interior)h(of)e(any)i (interval)f(wher)-5 b(e)2489 4214 y Fw(\026)2465 4237 y Fp(A)33 b Fv(is)f(line)-5 b(ar.)43 b(Then)1630 4449 y Fo(K)1699 4463 y Fn(\014)s(;N)1876 4398 y Fm(D)1860 4449 y Fo(!)1994 4426 y Fw(^)1976 4449 y Fo(K)2046 4399 y Fn(f)2081 4409 y Fr(\014)2045 4473 y Fz(1)3523 4449 y Fw(\(4)p Fp(:)p Fw(9\))159 4880 y(Theorem)34 b(4.3)i(is)d(the)i(main) e(result)h(of)g(this)g(pap)s(er.)52 b(The)34 b(basis)f(of)i(the)f(pro)s (of)g(are)h(the)g(Ghirlanda-)50 5025 y(Guerra)41 b(iden)m(tities.)72 b(In)40 b(fact)i(w)m(e)g(will)d(giv)m(e)i(t)m(w)m(o)i(pro)s(ofs.)72 b(In)41 b(Section)g(5)h(w)m(e)f(will)e(use)i(a)g(somewhat)50 5171 y(indirect)34 b(argumen)m(t)i(that)g(use)f(Prop)s(osition)e(4.2)k (to)f(sho)m(w)g(in)e(a)i(somewhat)g(con)m(torted)h(w)m(a)m(y)f(that)h (the)50 5316 y(same)25 b(Ghirlanda-Guerra)e(iden)m(tities)g(hold)h(for) g(the)h(limit)e(of)i Fo(K)2252 5330 y Fn(\014)s(;N)2412 5316 y Fw(and)f(for)h Fo(K)2787 5267 y Fn(f)2822 5277 y Fr(\014)2786 5341 y Fz(1)2870 5316 y Fw(.)39 b(In)24 b(Section)h(6)g(w)m(e)g(giv)m(e)50 5462 y(a)31 b(direct)e(pro)s(of)h (of)h(the)f(Ghirlanda-Guerra)f(iden)m(tities)f(for)j(the)f (Bolthausen-Snitzman)f(coalescen)m(t.)p eop %%Page: 21 21 21 20 bop 50 295 a Fq(GREM,)28 b(p)l(art)f(4)3089 b Ft(21)50 513 y FA(5.)55 b(Coalescence)40 b(and)h(Ghirlanda-Guerra)f(iden)m (tities.)159 731 y Fw(In)i(this)f(section)h(w)m(e)h(will)d(pro)m(v)m(e) j(Theorem)f(4.3.)78 b(As)42 b(it)g(w)m(as)h(remark)m(ed)f(in)f(Section) i(2)f Fo(K)3442 745 y Fn(t)3517 731 y Fw(asso-)50 876 y(ciated)c(with)e(a)j(\015o)m(w)e(of)h(measures)g(is)f(completely)g (determined)g(b)m(y)g(its)h(momen)m(ts)g(\(2.23\))i(whic)m(h)c(can)50 1022 y(b)s(e)c(expressed)f(via)h(genealogical)h(distance)g (distributions)28 b(of)33 b(the)f(corresp)s(onding)f(coalescen)m(t.)48 b(So,)33 b(w)m(e)50 1167 y(will)f(pro)m(v)m(e)k(that)g(the)g(momen)m (ts)g(of)f Fo(K)1409 1181 y Fn(\014)s(;N)1544 1167 y Fw(,)i(whic)m(h)d(are)i(the)f Fp(n)p Fw(-replica)f(distance)h (distributions)c(in)j(our)50 1313 y(spin)i(glass)j(mo)s(del)e(\(1.10\)) k(con)m(v)m(erge)f(to)g(the)e(genealogical)i(distance)e(distributions)c (on)39 b(the)f(in)m(tegers)50 1458 y(constructed)f(from)g(the)g(family) f(of)h(measures)g(\002)1795 1414 y Fn(y)1795 1486 y(f)1830 1496 y Fr(\014)1915 1458 y Fw(based)g(on)g(the)g(time-c)m(hanged)h(Nev) m(eu)g(branc)m(hing)50 1604 y(pro)s(cess.)159 1822 y(It)26 b(will)d(b)s(e)i(done)g(b)m(y)h(sho)m(wing)f(that)h(the)g (Ghirlanda-Guerra)e(relations,)i(whic)m(h)e(w)m(e)j(ha)m(v)m(e)f (established)50 1967 y(in)31 b([BK3])j(to)g(hold)e(for)g(the)h(family)f (of)h(limiting)d(measures)i(and)h(whic)m(h)e(determine)h(them)h (completely)-8 b(,)50 2113 y(are)31 b(satis\014ed)e(for)h(the)h(prop)s (osed)e(limit.)159 2331 y(In)c(addition,)h(it)g(giv)m(es)g(us)g(the)g 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Fz(2)3072 3107 y Fk(\001)672 3376 y Fw(=)f Fl(P)837 3302 y Fk(\000)880 3376 y Fp(\032)927 3390 y Fz(1)968 3376 y Fw(\(1)p Fp(;)15 b Fw(2\))27 b Fo(\024)e Fp(t)p Fw(\()p Fp(y)1404 3390 y Fz(1)1444 3376 y Fw(\))p Fp(;)15 b(:)g(:)g(:)i(;)e(\032)1728 3390 y Fz(1)1770 3376 y Fw(\()p Fp(n)20 b Fo(\000)g Fw(1)p Fp(;)15 b(n)p Fw(\))26 b Fo(\024)f Fp(t)p Fw(\()p Fp(y)2381 3392 y Fn(n)p Fz(\()p Fn(n)p Fm(\000)p Fz(1\))p Fn(=)p Fz(2)2698 3376 y Fw(\))2733 3302 y Fk(\001)3523 3276 y Fw(\(5)p Fp(:)p Fw(1\))50 3716 y Fv(wher)-5 b(e)42 b Fp(t)p Fw(\()p Fp(y)s Fw(\))f Fv(is)g(de\014ne)-5 b(d)41 b(in)g(The)-5 b(or)g(em)43 b(4.3.)67 b(The)41 b(distanc)-5 b(e)42 b Fp(\032)2248 3730 y Fz(1)2329 3716 y Fv(is)f(the)g(distanc)-5 b(e)42 b(on)g(inte)-5 b(gers)41 b(for)g(the)50 3862 y (Bolthausen-Sznitman)31 b(c)-5 b(o)g(alesc)g(ent,)31 b(induc)-5 b(e)g(d)29 b(thr)-5 b(ough)30 b(\(2.19\))g(by)f(the)f(gene) -5 b(alo)g(gic)g(al)31 b(distanc)-5 b(e)29 b Fp(\015)3419 3876 y Fz(1)3489 3862 y Fv(of)f(the)50 4007 y(\015ow)34 b(of)f(me)-5 b(asur)g(es)34 b Fp(X)7 b Fw(\()p Fp(t;)15 b(x)p Fw(\))p Fp(=X)7 b Fw(\()p Fp(t;)15 b Fw(1\))36 b Fv(of)d(Neveu's)e(br)-5 b(anching)34 b(pr)-5 b(o)g(c)g(ess.)50 4225 y Fc(Pro)s(of:)80 b Fw(The)44 b(pro)s(of)g(of)g(this)g(theorem,)k (and)c(in)g(fact)h(the)g(en)m(tire)f(iden)m(ti\014cations)f(of)i(the)g (limiting)50 4371 y(pro)s(cesses)32 b(with)f(ob)5 b(jects)33 b(constructed)g(from)f(Nev)m(eu's)h(branc)m(hing)e(pro)s(cess)h(relies) f(on)h(the)h(Ghirlanda-)50 4516 y(Guerra)25 b(iden)m(tities)g([GG])i (that)f(w)m(ere)g(deriv)m(ed)f(for)g(the)h(mo)s(dels)f(considered)f (here)i(in)e([BK3].)40 b(W)-8 b(e)27 b(restate)50 4662 y(this)40 b(result)h(in)f(a)i(sligh)m(tly)d(mo)s(di\014ed)h(form.)73 b(Let)42 b(us)f(remind)f(that)i(the)f(family)f(of)i(measures)f Fl(Q)3569 4629 y Fn(n)3569 4688 y(\014)s(;N)50 4807 y Fw(is)35 b(determined)g(on)i(the)f(space)h([0)p Fp(;)15 b Fw(1])1352 4774 y Fn(n)p Fz(\()p Fn(n)p Fm(\000)p Fz(1\))p Fn(=)p Fz(2)1709 4807 y Fw(as)36 b Fl(E)11 b Fp(\026)1936 4768 y Fm(\012)p Fn(n)1936 4835 y Ff(N)p Fn(;\014)2058 4807 y Fw(\()p Fp(d)2140 4821 y Fn(N)p 2093 4835 117 4 v 2245 4807 a Fo(2)35 b(\001)p Fw(\))i(where)f Fp(d)2754 4821 y Fn(N)p 2707 4835 V 2860 4807 a Fw(denotes)h(the)f(v)m(ector)j (of)50 4953 y(replica)30 b(distances)i Fp(d)783 4967 y Fn(N)852 4953 y Fw(\()p Fp(\033)942 4920 y Fn(k)987 4953 y Fp(;)15 b(\033)1082 4920 y Fn(l)1111 4953 y Fw(\),)32 b(1)c Fo(\024)f Fp(k)k(<)c(l)i Fo(\024)f Fp(n)p Fw(.)44 b(Denote)33 b(b)m(y)f Fo(B)2330 4967 y Fn(k)2406 4953 y Fw(the)f(v)m(ector)j(of)e(the)g(\014rst)e Fp(k)s Fw(\()p Fp(k)25 b Fo(\000)c Fw(1\))p Fp(=)p Fw(2)50 5098 y(co)s(ordinates.)50 5316 y Fd(Theorem)36 b(5.2:)50 b Fw([BK3])p Fv(The)34 b(family)e(of)h(me)-5 b(asur)g(es)33 b Fl(Q)2015 5269 y Fz(\()p Fn(n)p Fz(\))2015 5344 y Fn(\014)s(;N)2188 5316 y Fv(c)-5 b(onver)g(ge)32 b(to)h(limiting)f(me)-5 b(asur)g(es)33 b Fl(Q)3453 5269 y Fz(\()p Fn(n)p Fz(\))3453 5344 y Fn(\014)3597 5316 y Fv(for)50 5462 y(al)5 b(l)39 b(\014nite)f Fp(n)p Fv(,)h(as)g Fp(N)45 b Fo(")36 b(1)p Fv(.)58 b(Mor)-5 b(e)g(over,)41 b(these)e(me)-5 b(asur)g(es)39 b(ar)-5 b(e)39 b(uniquely)f(determine)-5 b(d)40 b(by)e(the)h(distanc)-5 b(e)p eop %%Page: 22 22 22 21 bop 50 295 a Ft(22)3270 b Fq(Se)l(ction)27 b Ft(6)50 513 y Fv(distribution)34 b(functions)e Fp(f)976 527 y Fn(\014)1025 513 y Fv(.)42 b(They)33 b(satisfy,)g(for)g(any)g Fp(y)c Fo(2)24 b Fw([0)p Fp(;)15 b Fw(1])p Fv(,)35 b Fp(n)24 b Fo(2)h Fl(N)45 b Fv(and)34 b Fp(k)28 b Fo(\024)d Fp(n)p Fv(,)559 834 y Fl(Q)624 787 y Fz(\()q Fn(n)p Fz(+1\))624 862 y Fn(\014)844 834 y Fw(\()p Fp(d)p Fw(\()p Fp(k)s(;)15 b(n)21 b Fw(+)f(1\))26 b Fo(\024)f Fp(y)s Fo(jB)1553 848 y Fn(n)1602 834 y Fw(\))h(=)1774 773 y(1)p 1770 813 55 4 v 1770 897 a Fp(n)1835 834 y(f)1880 848 y Fn(\014)1929 834 y Fw(\()p Fp(y)s Fw(\))20 b(+)2174 773 y(1)p 2169 813 V 2169 897 a Fp(n)2293 721 y Fn(n)2250 748 y Fk(X)2256 944 y Fn(l)p Fm(6)p Fz(=)p Fn(k)2396 834 y Fl(Q)2461 787 y Fz(\()p Fn(n)p Fz(\))2461 862 y Fn(\014)2589 834 y Fw(\()p Fp(d)p Fw(\()p Fp(k)s(;)15 b(l)r Fw(\))27 b Fo(\024)e Fp(y)s Fo(jB)3116 848 y Fn(n)3165 834 y Fw(\))323 b(\(5)p Fp(:)p Fw(2\))159 1334 y(Let)37 b(us)g(recall)f(that)i(due)e (to)i(the)f(ultrametric)f(prop)s(ert)m(y)g(of)h Fp(d)2381 1348 y Fn(N)2451 1334 y Fw(,)h(these)g(iden)m(tities)d(determine)i(the) 50 1480 y(measures)c Fl(Q)508 1432 y Fz(\()p Fn(n)p Fz(\))508 1507 y Fn(\014)654 1480 y Fw(uniquely)-8 b(.)48 b(Th)m(us,)34 b(w)m(e)g(m)m(ust)f(sho)m(w)h(that)g(the)g(righ)m(t-hand)e(side)h(of)g (\(5.1\))j(satis\014es,)e(for)50 1625 y Fp(t)25 b(<)g Fw(1,)606 1799 y Fl(P)15 b Fw(\()q Fp(\032)759 1813 y Fz(1)800 1799 y Fw(\()p Fp(k)s(;)g(n)20 b Fw(+)g(1\))26 b Fo(\024)f Fp(t)g Fo(j)h(B)1462 1813 y Fn(n)1511 1799 y Fw(\))g(=)1683 1738 y(1)p 1679 1778 V 1679 1862 a Fp(n)1744 1799 y(e)1786 1762 y Fn(t)p Fm(\000)p Fz(1)1932 1799 y Fw(+)2038 1738 y(1)p 2034 1778 V 2034 1862 a Fp(n)2182 1713 y Fk(X)2114 1909 y Fn(l)p Fm(\024)p Fn(n;l)p Fm(6)p Fz(=)p Fn(k)2395 1799 y Fl(P)15 b Fw(\()q Fp(\032)2548 1813 y Fz(1)2589 1799 y Fw(\()p Fp(k)s(;)g(l)r Fw(\))27 b Fo(\024)e Fp(t)g Fo(j)g(B)3069 1813 y Fn(n)3118 1799 y Fw(\))370 b(\(5)p Fp(:)p Fw(3\))159 2171 y(There)30 b(are)h Fv(two)g Fw(w)m(a)m(ys)g(to)g(v)m(erify)f(that)h(\(5.3\))h (holds)d(for)h(the)h(Bolthausen-Sznitman)e(coalescen)m(t.)159 2389 y(The)f(\014rst)f(one)h(is)f(to)i(observ)m(e)g(that)g(relation)e (\(5.3\))j(in)m(v)m(olv)m(es)e(only)f(the)i(marginals)d(of)j(the)f (coalescen)m(t)50 2534 y(at)k(a)g(\014nite)f(set)h(of)g(times)f Fp(t)998 2548 y Fn(i)1029 2534 y Fw(.)44 b(By)32 b(Theorem)g(5)g(of)g (Bertoin-Le)g(Gall)f([BeLG],)i(these)f(can)g(b)s(e)f(expressed)50 2680 y(in)d(terms)i(of)g(Ruelle's)e(cascades)j(mo)s(dulo)d(the)i (appropriate)f(time)g(c)m(hange.)42 b(Th)m(us,)29 b(b)m(y)g(Theorem)h (1.5)h(of)50 2825 y([BK2])j(these)g(probabilities)c(can)k(b)s(e)f (expressed)g(as)g(limits)e(of)j(a)g(suitably)d(constructed)j(GREM)g (\(with)50 2971 y(\014nitely)e(man)m(y)i(hierarc)m(hies\))f(for)h(whic) 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4294 y Fn(n)3263 4280 y Fw(denote)g(thet)50 4425 y(restriction)k(of)i(\005)f(to)h(the)g(\014rst)e Fp(n)h Fw(p)s(ositiv)m(e)g(in)m(tegers.)40 b(Then,)28 b(conditionally)e(giv)m(en)i(\005)3024 4439 y Fn(n)3099 4425 y Fw(=)d Fo(f)p Fp(A)3308 4439 y Fz(1)3349 4425 y Fp(;)15 b(:)g(:)g(:)i(;)e(A)3619 4439 y Fn(k)3664 4425 y Fo(g)50 4571 y Fw(for)28 b(an)m(y)g(particular)e(partition)h(of)h Fo(f)p Fw(1)p Fp(;)15 b Fw(2)p Fp(;)g(:)g(:)g(:)k(;)c(n)p Fo(g)28 b Fw(in)m(to)g Fp(k)j Fw(subsets)d(\(tables\))g Fp(A)2730 4585 y Fn(i)2789 4571 y Fw(of)g(sizes)g Fp(n)3151 4585 y Fn(i)3181 4571 y Fw(,)g Fp(i)e Fw(=)f(1)p Fp(;)15 b(:)g(:)g(:)i(;)e(k)s Fw(,)50 4716 y(the)41 b(partition)f(\005)675 4730 y Fn(n)p Fz(+1)857 4716 y Fw(is)g(an)h(extension)g(of)g(\005)1691 4730 y Fn(n)1782 4716 y Fw(suc)m(h)f(that)i(the)f(n)m(um)m(b)s(er)f Fp(n)27 b Fw(+)g(1)41 b(\(new)g(customer\))h(is)50 4862 y(attac)m(hed)h(to)e(the)g(class)g(\(table\))g Fp(A)1320 4876 y Fn(i)1392 4862 y Fw(with)e(probabilit)m(y)g(\()p Fp(n)2175 4876 y Fn(i)2232 4862 y Fo(\000)27 b Fp(x)p Fw(\))p Fp(=n)p Fw(,)44 b(and)c(forms)h(a)g(new)f(class)h(\(sits)50 5007 y(at)e(a)g(new)f(table\))h(with)e(probabilit)m(y)g Fp(k)s(x=n)p Fw(.)65 b(Let)39 b(us)f(denote)h(b)m(y)g Fp(p)p Fw(\()p Fp(n)2568 5021 y Fz(1)2608 5007 y Fp(;)15 b(:)g(:)g(:)32 b(;)15 b(n)2880 5021 y Fn(k)2925 5007 y Fw(\))39 b(the)f(probabilit)m(y)f(of)50 5153 y(partitions)27 b(\005)h(with)g(\005)833 5167 y Fn(n)911 5153 y Fw(a)g(particular)f (partition)g(of)i Fp(k)j Fw(classes)c(of)h(sizes)f Fp(n)2610 5167 y Fz(1)2650 5153 y Fp(;)15 b(:)g(:)g(:)i(;)e(n)2907 5167 y Fn(k)2980 5153 y Fw(resp)s(ectiv)m(ely)-8 b(.)40 b(Then)1050 5436 y Fp(p)p Fw(\()p Fp(n)1186 5450 y Fz(1)1246 5436 y Fo([)20 b Fw(1)p Fp(;)15 b(n)1467 5450 y Fz(2)1508 5436 y Fp(;)g(:)g(:)g(:)i(;)e(n)1765 5450 y Fn(k)1810 5436 y Fw(\))25 b(=)1977 5374 y(1)c Fo(\000)f Fp(x)p 1977 5415 209 4 v 2054 5498 a(n)2197 5436 y(p)p Fw(\()p Fp(n)2333 5450 y Fz(1)2373 5436 y Fp(;)15 b(:)g(:)g(:)i(;)e(n)2630 5450 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Fw(\()p Fp(k)s(;)15 b(n)20 b Fw(+)g(1\))26 b Fp(>)f(t)g Fo(j)h(B)1390 813 y Fn(n)1439 725 y Fk(\001)1506 799 y Fw(=)1613 737 y(1)21 b Fo(\000)e Fp(e)1811 704 y Fn(t)p Fm(\000)p Fn(t)1922 713 y Fr(p)p 1613 778 354 4 v 1762 861 a Fp(n)1997 799 y Fw(+)2104 737 y(1)p 2099 778 55 4 v 2099 861 a Fp(n)2247 712 y Fk(X)2180 908 y Fn(l)p Fm(\024)p Fn(n;l)p Fm(6)p Fz(=)p Fn(k)2461 799 y Fl(P)2531 725 y Fk(\000)2573 799 y Fp(\032)2620 813 y Fn(t)2647 822 y Fr(p)2691 799 y Fw(\()p Fp(k)s(;)c(l)r Fw(\))26 b Fp(>)f(t)g Fo(j)h(B)3171 813 y Fn(n)3220 725 y Fk(\001)3523 799 y Fw(\(6)p Fp(:)p Fw(6\))50 1124 y(whic)m(h)j(is)g (equiv)-5 b(alen)m(t)30 b(to)h(\(5.3\).)159 1342 y(Recen)m(tly)i(Ph.)e (Marc)m(hal)i(found)e(another)h(b)s(eautiful)e(w)m(a)m(y)j(to)g(iden)m (tify)e(the)h(limiting)d(frequencies)j(of)50 1488 y(the)g(Chinese)f (restauran)m(t)i(pro)s(cess)f(with)g(Ruelle's)f(REM)h(\(or,)i(equiv)-5 b(alen)m(tly)d(,)32 b(the)h(range)g(of)f(the)h(stable)50 1633 y(sub)s(ordinator\))21 b(and)h(also)h(the)g(iterated)h(Chinese)d (restauran)m(t)j(pro)s(cess)e(with)g(the)h(Bolthausen-Sznitman)50 1778 y(coalescen)m(t,)32 b(see)f([M].)1615 2116 y Fb(References)-34 2335 y Ft([A)n(C])39 b(M.)24 b(Aizenmann)g(and)f(P)-6 b(.)24 b(Con)n(tucci,)h(On)f(the)g(stabilit)n(y)i(of)d(the)h(quenc)n (hed)g(state)h(in)f(mean)g(\014eld)g(spin-glass)h(mo)r(dels,)h(J.)159 2425 y(Statist.)h(Ph)n(ys.)f Fa(92)p Ft(,)f(765-783)e(\(1998\).)-50 2589 y([Be1])39 b(J.)26 b(Bertoin,)g(Homogeneous)f(fragmen)n(tation)f (pro)r(cesses,)k(Probab.)d(Theory)h(Related)g(Fields)i Fa(121)c Ft(,)j(301{318)c(\(2001\).)-50 2753 y([Be2])39 b(J.)26 b(Bertoin,)g(Self-similar)h(fragmen)n(tations,)e(Ann.)h(Inst.)h (H.)f(P)n(oincar)n(\023)-36 b(e)24 b(\(Probab.)h(Statist.\))i Fa(38)p Ft(,)f(319{340)d(\(2002\).)-121 2916 y([BeLG])40 b(J.)28 b(Bertoin)h(and)e(J.F.)j(Le)e(Gall,)i(The)e (Bolthausen-Snitzman)g(coalescen)n(t)i(and)d(the)i(genealogy)f(of)f (con)n(tin)n(uous)h(state)159 3007 y(branc)n(hing)d(pro)r(cesses,)k (Probab.)c(Theor.)h(T)-6 b(el.)26 b(Fields)i Fa(117)p Ft(,)d(249-266)e(\(2000\).)-86 3171 y([BePi])40 b(J.)29 b(Bertoin)f(and)g(J.)h(Pitman,)g(Tw)n(o)e(coalescen)n(ts)k(deriv)n(ed)e (from)e(the)i(ranges)g(of)f(stable)h(sub)r(ordinators,)h(Electron.)g (J.)159 3262 y(Probab.)25 b Fa(5)p Ft(,)h(no.)f(7)h(\(2000\).)-102 3425 y([BeY)-6 b(o])39 b(J.)32 b(Bertoin)g(and)f(M.)i(Y)-6 b(or,)33 b(On)f(sub)r(ordinators,)i(self-similar)f(Mark)n(o)n(v)e(pro)r (cesses)j(and)d(some)h(factorizations)g(of)f(the)159 3516 y(exp)r(onen)n(tial)26 b(v)l(ariable,)g(Electron.)h(Comm.)e (Probab.)g Fa(6)p Ft(,)g(95{106)f(\(2001\))-59 3680 y([BoS])39 b(E.)e(Bolthausen)g(and)g(A.-S.)h(Snitzman,)i(On)d(Ruelle's)j (probabilit)n(y)d(cascades)i(and)e(an)f(abstract)i(ca)n(vit)n(y)f (metho)r(d,)159 3771 y(Comm)n(un.)24 b(Math.)i(Ph)n(ys.)f Fa(107)p Ft(,)g(247-276)f(\(1998\).)-76 3935 y([BK1])39 b(A.)e(Bo)n(vier)f(and)f(I.)i(Kurk)n(o)n(v)l(a,)h(Derrida's)g (Generalized)f(random)e(Energy)h(mo)r(dels)h(1.)f(P)n(oisson)g (cascades)i(and)d(the)159 4025 y(extremal)26 b(pro)r(cess,)h(preprin)n (t)g(727,)e(U.)h(P)n(aris)g(6)f(\(2002\).)-76 4189 y([BK2])39 b(A.)18 b(Bo)n(vier)g(and)g(I.)g(Kurk)n(o)n(v)l(a,)h(Derrida's)g (Generalized)h(Random)c(Energy)i(mo)r(dels)g(2.)h(Gibbs)f(measures)h (and)f(probabilit)n(y)159 4280 y(cascades,)27 b(preprin)n(t)g(728,)e (U.)h(P)n(aris)g(6)f(\(2002\).)-76 4444 y([BK3])39 b(A.)g(Bo)n(vier)f (and)f(I.)i(Kurk)n(o)n(v)l(a,)h(Derrida's)g(Generalized)g(Random)d (Energy)g(mo)r(dels)i(3.)f(Mo)r(dels)h(with)g(con)n(tin)n(uous)159 4535 y(hierarc)n(hies,)28 b(preprin)n(t)f(729,)e(U.)h(P)n(aris)g(6)f (\(2002\).)-86 4698 y([BKL])40 b(A.)20 b(Bo)n(vier,)h(I.)f(Kurk)n(o)n (v)l(a,)g(and)f(M.)h(L\177)-39 b(ow)n(e,)21 b(Fluctuations)g(of)e(the)h (free)g(energy)f(in)h(the)g(REM)f(and)g(the)h Fj(p)p Ft(-spin)g(SK-mo)r(dels,)159 4789 y(Ann.)26 b(Probab.)f Fa(30)p Ft(,)g(605-651)f(\(2002\).)23 4953 y([B])39 b(A.)26 b(Bo)n(vier,)g(Statistical)h(mec)n(hanics)g(of)e(disordered)i(systems,) g(MaPh)n(ySto)d(Lecture)j(Notes)f(10,)g(Aarh)n(us,)g(2001.)-87 5116 y([CCP])39 b(D.)34 b(Cap)r(o)r(caccia,)j(M.)e(Cassandro,)h(and)e (P)-6 b(.)33 b(Picco,)k(On)d(the)h(existence)h(of)d(thermo)r(dynamics)h (for)g(the)g(Generalised)159 5207 y(Random)24 b(Energy)h(Mo)r(del.)52 b(J.)26 b(Stat.)g(Ph)n(ysics)h Fa(46)p Ft(,)e(493{505)e(\(1987\).)18 5371 y([D])40 b(B.)25 b(Derrida,)h(A)e(generalization)i(of)e(the)h (random)f(energy)h(mo)r(del)f(that)h(includes)h(correlations)g(b)r(et)n (w)n(een)g(the)f(energies,)159 5462 y(J.)h(Ph)n(ysique.)g(Lett.)h Fa(46)p Ft(,)e(401{407)f(\(1985\).)p eop %%Page: 26 26 26 25 bop 50 295 a Ft(26)3225 b Fq(R)l(efer)l(enc)l(es)-81 513 y Ft([DG1])39 b(B.)27 b(Derrida)g(and)g(E.)g(Gardner,)g(Solution)g (of)f(the)h(generalized)i(random)c(energy)i(mo)r(del,)h(J.)f(Ph)n(ys.)g (C)f Fa(19)p Ft(,)h(2253{2274)159 604 y(\(1986\).)-81 767 y([DG2])39 b(B.)19 b(Derrida)g(and)g(E.)g(Gardner,)h(Magnetic)f (prop)r(erties)h(and)f(function)g Fj(q)s Ft(\()p Fj(x)p Ft(\))e(of)h(the)h(generalised)i(random)c(energy)i(mo)r(del,)159 858 y(J.)26 b(Ph)n(ys.)g(C)f Fa(19)p Ft(,)h(5783-5798)c(\(1986\))-44 1022 y([GG])39 b(S.)31 b(Ghirlanda,)i(F.)f(Guerra.)63 b(General)31 b(prop)r(erties)i(of)d(the)i(o)n(v)n(erlap)f(probabilit)n (y)g(distributions)i(in)f(disordered)g(spin)159 1113 y(systems.)27 b(T)-6 b(o)n(w)n(ards)25 b(P)n(arisi)h(ultrametricit)n (y,)h(J.)f(Ph)n(ys.)g(A)g Fa(31)p Ft(,)f(9144-9155)d(\(1998\).)-26 1276 y([Gu])39 b(F.)27 b(Guerra,)f(Brok)n(en)f(replica)i(symmetry)e(b)r (ounds)h(in)g(the)g(mean)f(\014eld)h(spin)h(glass)f(mo)r(del,)g (preprin)n(t)h(2002.)-39 1440 y([GT])39 b(F.)28 b(Guerra)f(and)f(F.)i (T)-6 b(oninelli,)30 b(The)d(thermo)r(dynamics)f(limit)i(in)g(mean)e (\014eld)i(spin)g(glass)g(mo)r(dels,)g(Comm)n(un.)e(Math.)159 1531 y(Ph)n(ys.)p Fa(230)p Ft(,)f(71-79)f(\(2002\).)-107 1695 y([GMP])39 b(A.)29 b(Galv)n(ez,)h(S.)g(Martinez,)g(and)f(P)-6 b(.)28 b(Picco,)j(Fluctuations)f(in)f(Derrida's)i(random)c(energy)i (and)g(generalized)h(random)159 1785 y(enery)c(mo)r(dels,)53 b(J.)25 b(Stat.)h(Ph)n(ys.)g Fa(54)p Ft(,)f(515-529)f(\(1989\).)-5 1949 y([Le])40 b(M.)31 b(Ledoux.)h(On)f(the)h(distribution)h(of)d(o)n (v)n(erlaps)i(in)f(the)h(Sherrington-Kirkpatric)n(k)g(spin)g(glass)g (mo)r(del.)f(J.)h(Statist.)159 2040 y(Ph)n(ys.)26 b(100)e(\(2000\),)h (871-892.)7 2204 y([M])39 b(Ph.)26 b(Marc)n(hal)f(Regenerativ)n(e)i (sets,)g(random)e(partitions)h(and)f(the)h(Bolthausen-Sznitman)g (coalescen)n(t.)h(Preprin)n(t.)-104 2367 y([MPV])39 b(M.)26 b(M)n(\023)-36 b(ezard,)25 b(G.)h(P)n(arisi,)h(M.A.)f(Virasoro,)g (Spin-glass)h(theory)e(and)h(b)r(ey)n(ond.)51 b(W)-6 b(orld)26 b(Scien)n(ti\014c,)i(Singap)r(ore)d(\(1988\).)-15 2531 y([Ne])40 b(J.)30 b(Nev)n(eu,)i(A)e(con)n(tin)n(uous)h(state)g (branc)n(hing)f(pro)r(cess)h(in)g(relation)g(with)g(the)f(GREM)f(mo)r (del)h(of)g(spin)h(glass)g(theory)-6 b(,)159 2627 y(Rapp)r(ort)25 b(in)n(terne)i(no)e(267,)1004 2609 y(\023)997 2627 y(Ecole)h(P)n (olytec)n(hnique)g(\(1992\).)-62 2791 y([NS1])39 b(Ch.M.)29 b(Newman)f(and)h(D.L.)h(Stein,)h(Thermo)r(dynamic)c(c)n(haos)i(and)g (the)g(structure)h(of)e(short)h(range)g(spin)h(glasses,)h(in)159 2882 y(\\Mathematical)d(asp)r(ects)i(of)f(spin)g(glasses)i(and)d (neural)i(net)n(w)n(orks",)f(A.)g(Bo)n(vier)g(and)g(P)-6 b(.)28 b(Picco)h(\(Eds.\),)i(Progress)e(in)159 2972 y(Probabilit)n(y)-6 b(,)26 b(Birkh\177)-39 b(auser,)26 b(Boston)f(\(1997\).)-62 3136 y([NS2])39 b(C.M.)34 b(Newman,)h(D.L.)f(Stein,)j(Metastate)c (approac)n(h)g(to)g(thermo)r(dynamic)f(c)n(haos.)i(Ph)n(ys.)f(Rev.)h(E) f Fa(55)p Ft(,)i(5194-5211)159 3227 y(\(1997\).)-62 3391 y([NS3])k(Ch.M.)21 b(Newman)g(and)f(D.L.)i(Stein,)i(Ultrademo)r(cracy)c (of)h(ground)f(states)i(in)g(the)f(in\014nite)h(ranged)f(spin)h(glass,) h(preprin)n(t)159 3482 y(cond-mat/0301202,)f(2003.)-22 3645 y([Ru])39 b(D.)31 b(Ruelle,)i(A)d(mathematical)f(reform)n(ulation) h(of)f(Derrida's)j(REM)d(and)h(GREM.)60 b(Comm)n(un.)28 b(Math.)i(Ph)n(ys.)61 b Fa(108)p Ft(,)159 3736 y(225-239)23 b(\(1987\).)25 3905 y([P])39 b(J.)31 b(Pitman,)h(Com)n(binatorial)e (sto)r(c)n(hastic)j(pro)r(cesses,)i(Lecture)d(Notes)g(of)e(the)i(2002) 2726 3887 y(\023)2719 3905 y(Ecole)g(des)g(Probabilit)n(\023)-36 b(es)31 b(de)g(St.)159 3996 y(Flour;)c(to)e(app)r(ear)h(as)f(Springer)i (Lecture)g(Notes)f(in)g(Mathematics.)-35 4160 y([Pi1])39 b(J.)33 b(Pitman,)i(Exc)n(hangeable)e(and)g(partially)g(exc)n (hangeable)h(random)e(partitions,)71 b(Probab.)33 b(Theor.)h(Relat.)g (Fields)159 4250 y Fa(102)p Ft(,)25 b(145{158)e(\(1995\).)-86 4414 y([PPY])39 b(M.)25 b(P)n(erman,)g(J.)h(Pitman,)e(M.)i(Y)-6 b(or,)25 b(Size-biased)i(sampling)e(of)g(P)n(oisson)g(p)r(oin)n(t)g (pro)r(cesses)j(and)d(excursions,)53 b(Probab.)159 4505 y(Theor.)26 b(Relat.)h(Fields)g Fa(92)p Ft(,)f(21{39)e(\(1992\).)-26 4669 y([SK])40 b(D.)23 b(Sherrington)g(and)f(S.)i(Kirkpatric)n(k,)g (Solv)l(able)f(mo)r(del)g(of)f(a)g(spin)i(glass,)48 b(Ph)n(ys.)23 b(Rev.)g(Lett.)h Fa(35)p Ft(,)f(1792-1796)d(\(1972\).)-17 4832 y([T1])39 b(M.)23 b(T)-6 b(alagrand,)22 b(Rigorous)g(lo)n(w)h (temp)r(erature)g(results)i(for)d(mean)g(\014eld)h(p-spin)g(in)n (teraction)h(mo)r(dels,)48 b(Probab.)22 b(Theor.)159 4923 y(Rel.)27 b(Fields.)h Fa(117)p Ft(,)d(303-360)e(\(2000\).)-17 5087 y([T2])39 b(M.)26 b(T)-6 b(alagrand,)24 b(Self)j(organization)e (in)h(the)g(lo)n(w-temp)r(erature)g(region)g(of)f(a)g(spin)i(glass)f (mo)r(del,)g(preprin)n(t)h(2002.)-17 5256 y([T3])39 b(M.)34 b(T)-6 b(alagrand,)34 b(Mean)g(\014eld)g(mo)r(dels)g(for)f(spin)i (glasses:)52 b(a)33 b(\014rst)i(course,)i(Lecture)e(Notes)f(of)f(the)h (2000)3394 5237 y(\023)3387 5256 y(Ecole)g(des)159 5347 y(Probabilit)n(\023)-36 b(es)26 b(de)g(St.)g(Flour;)h(to)e(app)r(ear)h (as)g(Springer)g(Lecture)h(Notes)f(in)g(Mathematics.)p eop %%Page: 27 27 27 26 bop 50 295 a Fq(GREM,)28 b(p)l(art)f(4)3089 b Ft(27)-17 513 y([T4])39 b(M.)24 b(T)-6 b(alagrand,)23 b(Spin)h(Glasses:)35 b(a)24 b(Challenge)g(to)g(Mathematicians,)g(Ergebnisse)i(der)e (Mathematik)f(und)g(ihrer)i(Gren-)159 604 y(zgebiete)i(V)-6 b(ol.)26 b(46,)f(Springer,)i(Berlin-Heidelb)r(erg-New)h(Y)-6 b(ork,)25 b(2003.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0305300704531--