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b(form)f(is)h(really)g(quite)h(general.)19 b(It)12 b(is)g(a)f(simple)i(matter)e(to)0 973 y(write)i(the)g(random)g (\014eld)h(Ising)g(mo)q(del)g(or)f(an)f(Ising)i(mo)q(del)g(with)g (disordered)g(nearest)f(neigh)o(b)q(or)g(couplings)0 1046 y(in)j(the)f(ab)q(o)o(v)o(e)g(form.)91 1155 y(Next,)g(w)o(e)g (supp)q(ose)h(from)e(the)h(b)q(eginning)j(that)c(w)o(e)h(ha)o(v)o(e)g (the)g(existence)i(of)d(a)h(w)o(eak)g(limit)798 1286 y(lim)783 1320 y Fo(\003)p Fp(")p Fi(Z)-20 b(Z)858 1309 y Fh(d)884 1286 y Fj(\026)911 1261 y Fi(\033)933 1247 y Ff(b.c.)932 1270 y Fh(@)r Ff(\003)911 1299 y Fo(\003)988 1286 y Fl([)p Fj(\021)r Fl(])11 b(=)i Fj(\026)p Fl([)p Fj(\021)r Fl(])1857 1292 y(\(2.2\))0 1429 y(for)h Fj(I)-8 b(P)6 b Fl(-a.e.)19 b Fj(\021)14 b Fg(\021)f Fj(\021)327 1440 y Fi(Z)-20 b(Z)358 1429 y Fh(d)19 b Fl(with)14 b(a)g(nonrandom)g (b)q(oundary)h(condition)g Fj(\033)1202 1413 y Ff(b.c.)1254 1429 y Fl(.)k(In)c(ferromagnetic)f(examples)h(lik)o(e)0 1502 y(the)e(random)f(\014eld)j(Ising)e(mo)q(del)h(this)g(can)f(b)q(e)g (concluded)i(b)o(y)e(monotonicit)o(y)g(argumen)o(ts.)18 b(Note)13 b(that)f(there)0 1575 y(is)i(ho)o(w)o(ev)o(er)e(no)h(general) g(argumen)o(t)f(that)h(w)o(ould)g(giv)o(e)g(the)g(existence)i(of)d (this)i(limit)g(-)f(indeed)i(it)e(is)h(exp)q(ected)0 1647 y(to)h(fail)h(e.g.)j(for)c(lo)o(w)g(temp)q(erature)g(spinglasses.) 857 1631 y Fo(2)91 1756 y Fl(Assuming)k(\(2.2\))e(it)h(follo)o(ws)g (that)g Fj(\026)744 1763 y Fp(1)783 1756 y Fl([)p Fj(\021)819 1767 y Fi(Z)-20 b(Z)849 1756 y Fh(d)5 b Fl(])18 b(is)g(an)h (in\014nite-v)o(olume)h(Gibbs)f(measure)g(for)e Fj(P)6 b Fl(-a.e.)30 b Fj(\021)0 1829 y Fl(that)16 b(dep)q(ends)j(measurably)e (on)g Fj(\021)r Fl(.)25 b(W)l(e)17 b(lo)q(ok)g(at)f(spins)i(and)f (disorder)h(v)m(ariables)g(at)f(the)g(same)f(time)i(and)0 1902 y(de\014ne)h Fm(join)o(t)i(spin)f(v)m(ariables)e Fj(\030)609 1909 y Fi(x)649 1902 y Fl(=)f(\()p Fj(\033)745 1909 y Fi(x)767 1902 y Fj(;)8 b(\021)811 1909 y Fi(x)833 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b(It)18 b(pro)o(vides)h(a)f (description)0 524 y(of)h(the)g(join)o(t)h(measure)f(as)g(an)g (\\annealed)h(system".)32 b(This)20 b(notion)g(should)g(not)f(b)q(e)h (confused)g(with)g(the)0 597 y(\\trivial")c(annealed)g(system)f(app)q (earing)h(in)g(the)f(next)g(de\014nition.)91 706 y(W)l(e)h(call)h(a)e (p)q(oten)o(tial)h Fj(U)520 690 y Fi(ann)603 706 y Fl(on)g(the)f(join)o (t)h(v)m(ariables)h(a)e Fm(p)q(oten)o(tial)21 b(for)c(the)i(annealed)g (system)14 b Fl(if)0 779 y(it)h(is)h(\014nite)g(range)f(and)h(pro)q (duces)g(the)f(annealed)h(lo)q(cal)h(sp)q(eci\014cation,)f(i.e.)141 911 y Fj(e)162 885 y Fp(\000)197 854 y Fe(P)245 902 y Fh(A)p Fd(\\)p Ff(\003)p Fd(6)p Ff(=)p Fd(;)359 885 y Fi(U)387 871 y Fh(ann)384 894 y(A)447 885 y Fo(\()p Fi(\033)482 890 y Ff(\003)505 885 y Fi(\033)527 871 y Ff(b.c.)526 901 y Fh(Z)-21 b(Z)550 894 y(d)569 901 y Fd(n)p Ff(\003)609 885 y Fi(;\021)639 890 y Ff(\003)661 885 y Fi(\021)681 871 y Ff(b.c.)680 901 y Fh(Z)g(Z)703 894 y(d)722 901 y Fd(n)p Ff(\003)762 885 y 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1588 y(Finally)l(,)j(a)e(p)q(oten)o(tial)h Fj(U)k Fl(is)c(called)h Fm(summable)g(for)g Fj(\030)g Fl(if,)f(for)f(an)o(y)g(\003)p Fg(\032)p Fj(Z)-19 b(Z)1399 1568 y Fi(d)1421 1588 y Fl(,)14 b(w)o(e)i(ha)o(v)o(e)e(that)h(the)h(limit)0 1660 y(lim)63 1671 y Fo(\001)p Fp(")p Fi(Z)-20 b(Z)143 1660 y Fh(d)172 1626 y Fe(P)220 1674 y Fi(A)p Fo(:)p Fi(A)p Fp(\\)p Fo(\003)p Fp(6)p Fo(=)p Fp(;)p Fi(;A)p Fp(\032)q Fo(\001)487 1660 y Fj(U)518 1667 y Fi(A)548 1660 y Fl(\()p Fj(\030)r Fl(\))12 b(=:)678 1626 y Fe(P)726 1674 y Fi(A)p Fo(:)p Fi(A)p Fp(\\)p Fo(\003)p Fp(6)p Fo(=)p Fp(;)897 1660 y Fj(U)928 1667 y Fi(A)957 1660 y Fl(\()p Fj(\030)r Fl(\))i(exists)g(and)g(is)g (indep)q(enden)o(t)i(of)d(the)h(sequence)h(of)0 1733 y(\001's.)k(This)14 b(is)g(needed)g(for)f(the)h(sums)f(in)h(\(2.4\))e (to)h(mak)o(e)g(sense.)19 b Fj(U)g Fl(is)14 b(called)h Fm(absolutely)h(summable)f(for)0 1806 y Fj(\030)i Fl(if,)e(for)g(an)o (y)g(\003)p Fg(\032)p Fj(Z)-19 b(Z)360 1786 y Fi(d)396 1806 y Fl(w)o(e)15 b(ha)o(v)o(e)g(that)f(sup)735 1817 y Fo(\001)p Fp(\032)p Fi(Z)-20 b(Z)824 1806 y Fh(d)853 1772 y Fe(P)901 1819 y Fi(A)p Fo(:)p Fi(A)p Fp(\\)p Fo(\003)p Fp(6)p Fo(=)p Fp(;)p Fi(;A)p Fp(\032)q Fo(\001)1168 1806 y Fg(j)p Fj(U)1212 1813 y Fi(A)1241 1806 y Fl(\()p Fj(\030)r Fl(\))p Fg(j)12 b Fj(<)h Fg(1)p Fl(.)91 1915 y(No)o(w,)g(the)g(most)g (natural)h(approac)o(h)f(to)g(\014nd)h(a)f(p)q(oten)o(tial)h(for)f(the) h(join)o(t)f(measure)g(is)h(to)f(write)h(do)o(wn)f(a)0 1988 y(formal)h(v)m(acuum)h(p)q(oten)o(tial)g(on)g(the)f(join)o(t)h (space)f(and)h(ask)f(what)g(w)o(e)g(can)h(sa)o(y)e(ab)q(out)i(its)f (con)o(v)o(ergence)h(\(see)0 2060 y(Theorem)e(2.2\).)18 b(W)l(e)13 b(remind)h(the)f(reader)g(that)g(a)g(p)q(oten)o(tial)h Fj(U)k Fl(is)13 b(called)i(v)m(acuum)f(p)q(oten)o(tial)g(with)f(v)m (acuum)5 2121 y(^)0 2133 y Fj(\030)r Fl(,)j(if)h Fj(U)125 2140 y Fi(A)155 2133 y Fl(\()p Fj(\030)193 2141 y Fi(A)p Fp(n)p Fi(x)266 2121 y Fl(^)261 2133 y Fj(\030)281 2140 y Fi(x)304 2133 y 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(tial)f(in)0 1173 y(detail)c(w)o(e)f(need)h(some)f(more)g(notation.)k (F)l(or)c(a)g(subset)g Fj(V)10 b Fg(\032)p Fj(Z)-19 b(Z)1103 1153 y Fi(d)1124 1173 y Fl(,)15 b(w)o(e)g(call)h(the)g(expression)260 1336 y(\001)p Fj(H)336 1343 y Fi(V)367 1336 y Fl(\()p Fj(\033)p 411 1324 32 2 v 15 x Fi(V)442 1336 y Fj(;)8 b(\021)488 1317 y Fo(1)486 1347 y Fi(V)516 1336 y Fj(;)g(\021)562 1317 y Fo(2)560 1347 y Fi(V)591 1336 y Fj(;)g(\021)635 1343 y Fi(@)r(V)686 1336 y Fl(\))13 b(:=)827 1293 y Fe(X)777 1392 y Fi(A)p Fo(:)p Fi(A)p Fp(\\)p Fi(V)d Fp(6)p Fo(=)p Fp(;)943 1285 y Fe(\020)970 1336 y Fl(\010)1003 1343 y Fi(A)1040 1299 y Fe(\000)1061 1336 y Fj(\033)p 1087 1324 V 15 x Fi(V)1118 1336 y Fj(;)e(\021)1164 1317 y Fo(1)1162 1347 y Fi(V)1192 1336 y Fj(\021)1215 1343 y Fi(@)r(V)1268 1299 y Fe(\001)1299 1336 y Fg(\000)i Fl(\010)1377 1343 y Fi(A)1414 1299 y Fe(\000)1435 1336 y Fj(\033)p 1461 1324 V 15 x Fi(V)1493 1336 y Fj(;)e(\021)1539 1317 y Fo(2)1537 1347 y Fi(V)1567 1336 y Fj(\021)1590 1343 y Fi(@)r(V)1643 1299 y Fe(\001)1663 1285 y(\021)1857 1355 y Fl(\(2.6\))0 1463 y(the)16 b Fj(V)9 b Fm(-v)m(ariation)21 b(of)d(the)g(Hamiltonian)i(w.r.t.)k(the)18 b(disorder)g(v)m(ariables)p Fl(.)k(T)l(o)15 b(denote)h(the)g(corre-)0 1536 y(sp)q(onding)j (function)g(on)e(the)h(spin-v)m(ariables)i(obtained)f(b)o(y)e(\014xing) i(the)f(disorder)g(v)m(ariables)h(w)o(e)f(will)h(drop)0 1609 y(the)d(spin-v)m(ariable)i Fj(\033)f Fl(on)f(the)f(l.h.s.)21 b(of)16 b(\(2.6\).)j(In)d(particular,)g(for)f Fj(V)23 b Fl(=)14 b Fg(f)p Fj(x)p Fg(g)p Fl(,)h(the)g(expression)i(\(2.6\))d (is)i(the)0 1682 y(observ)m(able)g(conjugate)f(to)g(the)g(indep)q (enden)o(t)j(disorder)d(v)m(ariable)i Fj(\021)1187 1689 y Fi(x)1209 1682 y Fl(.)j(W)l(e)15 b(put)476 1795 y Fj(Q)512 1802 y Fi(x)535 1795 y Fl(\()p Fj(\021)578 1777 y Fo(1)576 1807 y Fi(x)598 1795 y Fj(;)8 b(\021)644 1777 y Fo(2)642 1807 y Fi(x)663 1795 y Fj(;)g(\021)707 1806 y Fi(Z)-20 b(Z)737 1795 y Fh(d)756 1806 y Fp(n)p Fi(x)798 1795 y Fl(\))12 b(:=)h Fj(\026)p 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Ff(fe,abs,in)o(v)817 57 y Fi(\026)953 45 y Fl(\()p Fj(\021)r Fl(\))f Fb(is)g(tr)n(anslation) h(invariant,)f(to)n(o.)91 155 y Fm(Remark:)k Fl(Again,)11 b(the)f(p)q(oten)o(tial)h(is)g(ev)o(en)g(in)o(tegrable.)19 b(Moreo)o(v)o(er,)9 b(for)h(an)o(y)f(nonnegativ)o(e)i(translation)0 227 y(in)o(v)m(arian)o(t)16 b(function)g Fj(w)q Fl(\()p Fj(A)p Fl(\))f(giving)i(w)o(eigh)o(t)e(to)g(a)g(subset)h Fj(A)p Fg(\032)p Fj(Z)-19 b(Z)1117 207 y Fi(d)1154 227 y Fl(w)o(e)15 b(ha)o(v)o(e)g(the)h(follo)o(wing)g(estimate)f(on)h(its)0 300 y(deca)o(y)340 345 y Fe(X)309 443 y Fi(A)p Fo(:)p Fi(A)p Fp(3)p Fi(x)418 448 y Ff(0)444 388 y Fj(w)q Fl(\()p Fj(A)p Fl(\))556 327 y Fe(Z)608 388 y Fj(I)-8 b(P)6 b Fl(\()p Fj(d)s Fl(~)-26 b Fj(\021)q Fl(\))747 336 y Fe(\014)747 363 y(\014)747 391 y(\014)762 388 y Fj(U)798 368 y Ff(fe,abs,in)o(v)793 402 y Fi(\026)p Fo(;)p Fi(A)929 388 y Fl(\()s(~)g Fj(\021)q Fl(\))989 336 y Fe(\014)989 363 y(\014)989 391 y(\014)1016 388 y Fg(\024)13 b Fj(C)1097 395 y Fo(1)1127 388 y Fl(+)e Fj(C)1206 395 y Fo(2)1254 332 y Fp(1)1239 345 y Fe(X)1233 442 y Fi(m)p Fo(=2)1319 388 y Fj(m)1359 370 y Fo(2)p Fi(d)p Fp(\000)p Fo(1)1454 388 y Fl(\026)-32 b Fj(w)p Fl(\()p Fj(m)p Fl(\))q(\026)-24 b Fj(c)o Fl(\()p Fj(m)p Fl(\))1834 394 y(\(2.11\))0 536 y(where)30 b(\026)-33 b Fj(w)q Fl(\()p Fj(m)p Fl(\))20 b(:=)h Fj(w)376 485 y Fe(\020)404 536 y Fg(f)p Fj(z)14 b Fg(2)f Fj(Z)-19 b(Z)551 516 y Fi(d)572 536 y Fl(;)8 b Fj(z)14 b Fg(\025)f Fl(0)p Fj(;)8 b Fg(j)p Fj(z)r Fg(j)j(\024)i Fj(m)p Fg(g)891 485 y Fe(\021)938 536 y Fl(where)20 b Fg(\025)h Fl(denotes)f(the)g (lexicographic)i(order.)35 b(The)0 608 y(constan)o(ts)14 b Fj(C)234 615 y Fo(1)254 608 y Fl(,)h Fj(C)315 615 y Fo(2)350 608 y Fl(are)g(related)g(to)g(a-priori)h(b)q(ounds)g(on)f (\001)p Fj(H)1092 615 y Fi(x)1115 608 y Fl(.)91 717 y(Under)i(the)f (stronger)g(condition)i(that)d(w)o(e)i(ha)o(v)o(e)f(b)q(ounds)h(of)f (the)g(same)h(form)e(on)i(the)f(sup)1696 718 y Fh(x;y)q Ff(:)p Fd(j)p Fh(x)p Fd(\000)p Fh(y)q Fd(j)p Ff(=)p Fh(m)1714 746 y(\021)1731 751 y(x)1752 746 y(;\021)1779 751 y(y)1799 746 y Fd(2H)1847 753 y Ff(0)0 790 y Fl(sup)71 801 y Fo(~)-21 b Fi(\021)98 790 y Fg(j)p Fj(c)131 797 y Fi(x;y)184 790 y Fl(\()p Fj(\021)225 797 y Fi(x)247 790 y Fj(;)8 b(\021)291 797 y Fi(y)311 790 y Fj(;)j Fl(~)-26 b Fj(\021)q Fl(\))p Fg(j)15 b Fl(the)h(absolute)h(con)o(v)o(ergence)f(is)h(not)e(only)i (a.s.)k(but)c(ev)o(erywhere,)f(and)g(\(2.11\))e(holds)0 863 y(for)h(all)h(realizations)g(without)f(the)g Fj(I)-8 b(P)6 b Fl(-in)o(tegral)17 b(\(with)e(non-random)g(constan)o(ts\).)0 1135 y Fq(I)r(I)r(I.)k(The)i(in\014nite)f(v)n(olume)h(conditional)g (exp)r(ectations)91 1244 y Fl(W)l(e)d(start)e(with)i(a)f(suitable)i (represen)o(tation)f(of)f(the)h(in\014nite)i(v)o(olume)e(conditional)h (exp)q(ectations)f(of)0 1316 y(the)d(join)o(t)g(measure.)91 1425 y(W)l(e)k(write)f Fj(\030)j Fl(=)d(\()p Fj(\033)o(;)8 b(\021)r Fl(\))17 b(here)i(and)f(b)q(elo)o(w,)i(so)e(that,)h(for)e(an)o (y)i(set)f Fj(A)p Fg(\032)q Fj(Z)-19 b(Z)1375 1406 y Fi(d)1415 1425 y Fl(w)o(e)18 b(ha)o(v)o(e)h Fj(\030)1614 1432 y Fi(A)1661 1425 y Fl(=)g(\()p Fj(\033)1759 1432 y Fi(A)1788 1425 y Fj(;)8 b(\021)1832 1432 y Fi(A)1861 1425 y Fl(\).)0 1498 y(Recall)18 b(that)d Fj(r)i Fl(is)f(the)g(range)g (of)g(the)g(de\014ning)h(p)q(oten)o(tial)g(\010.)22 b(W)l(e)16 b(write)p 1286 1462 35 2 v 16 w Fj(A)e Fl(=)h Fg(f)p Fj(y)g Fg(2)f Fj(Z)-19 b(Z)1534 1478 y Fi(d)1555 1498 y Fj(;)8 b(d)p Fl(\()p Fj(y)r(;)g(A)p Fl(\))k Fg(\024)i Fj(r)q Fg(g)i Fl(for)0 1571 y(the)f Fj(r)q Fl(-neigh)o(b)q(orho)q(o)q (d)i(of)d(a)h(set)g Fj(A)p Fl(,)g(and)h(put)f Fj(@)s(A)d Fl(=)p 916 1534 V 13 w Fj(A)p Fg(n)p Fj(A)p Fl(.)0 1680 y Fc(Prop)r(osition)i(3.1:)22 b Fb(Assume)12 b(ther)n(e)h(is)f(a)h(set) g(of)g(r)n(e)n(alizations)e Fg(H)1165 1664 y Fo(0)1186 1680 y Fg(\032)q(H)i Fb(of)g Fj(I)-8 b(P)6 b Fb(-me)n(asur)n(e)14 b(one)e(such)h(that)h(the)0 1753 y(quenche)n(d)i(in\014nite)e(volume)i (Gibbs)f(me)n(asur)n(e)h Fj(\026)p Fl([)p Fj(\021)r Fl(])e Fb(is)i(a)g(we)n(ak)f(limit)h(\(2.2\))f(of)h(the)g(quenche)n(d)g (\014nite)f(volume)0 1825 y(me)n(asur)n(es)g(\(2.1\))h(for)g(al)r(l)f Fj(\021)f Fg(2)f(H)562 1809 y Fo(0)583 1825 y Fb(.)21 b(Then,)15 b(a)h(version)g(of)g(the)g(in\014nite)e(volume)i(c)n (onditional)f(exp)n(e)n(ctation)h(of)0 1898 y(the)h(c)n(orr)n(esp)n (onding)d(joint)i(me)n(asur)n(e)g Fj(K)s Fl(\()p Fj(d\033)o(;)8 b(d\021)r Fl(\))i(=)j Fj(I)-8 b(P)6 b Fl(\()p Fj(d\021)r Fl(\))p Fj(\026)p Fl([)p Fj(\021)r Fl(]\()p Fj(d\033)q Fl(\))13 b Fb(is)j(given)g(by)g(the)g(formula)482 2032 y Fj(K)531 1996 y Fe(\002)550 2032 y Fj(\030)570 2039 y Fo(\003)597 1994 y Fe(\014)597 2022 y(\014)613 2032 y Fj(\030)633 2043 y Fi(Z)-21 b(Z)663 2032 y Fh(d)682 2043 y Fp(n)p Fo(\003)728 1996 y Fe(\003)760 2032 y Fl(=)1023 2002 y Fj(\026)1050 1980 y Ff(ann,)q Fi(\030)1126 1985 y Fh(@)r Ff(\003)1050 2015 y Fo(\003)1170 2002 y Fl(\()p Fj(\030)1208 2009 y Fo(\003)1235 2002 y Fl(\))p 813 2022 650 2 v 813 2035 a Fe(R)851 2072 y Fj(\026)878 2050 y Ff(ann,)q Fi(\030)954 2055 y Fh(@)r Ff(\003)878 2085 y Fo(\003)997 2072 y Fl(\()p Fj(d)s Fl(~)-26 b Fj(\021)1062 2079 y Fo(\003)1088 2072 y Fl(\))p Fj(Q)1142 2079 y Fo(\003)1169 2072 y Fl(\()p Fj(\021)1210 2079 y Fo(\003)1237 2072 y Fj(;)10 b Fl(~)-25 b Fj(\021)1281 2079 y Fo(\003)1307 2072 y Fj(;)8 b(\021)1351 2082 y Fi(Z)-21 b(Z)1380 2072 y Fh(d)1399 2082 y Fp(n)p Fo(\003)1445 2072 y Fl(\))1857 2035 y(\(3.1\))0 2183 y Fb(Her)n(e)16 b Fj(\026)136 2161 y Ff(ann,)q Fi(\030)212 2166 y Fh(@)r Ff(\003)136 2196 y Fo(\003)256 2183 y Fl(\()p Fj(\030)294 2190 y Fo(\003)320 2183 y Fl(\))h Fb(is)f(the)h Fm(annealed)i(lo)q(cal)i(sp)q (eci\014cation)e Fb(given)d(by)h(\(2.7\),)f(which)h(c)n(an)f(b)n(e)h (written)f(in)0 2255 y(terms)g(of)h(the)f(sp)n(e)n(cial)f(anne)n(ale)n (d)g(p)n(otential)h Fj(U)808 2239 y Fi(ann)803 2268 y(A)875 2255 y Fl(\()p Fj(\033)o(;)8 b(\021)r Fl(\))i(=)j(\010)1073 2262 y Fi(A)1103 2255 y Fl(\()p Fj(\033)o(;)8 b(\021)r Fl(\))g Fg(\000)i Fl(1)1286 2264 y Fi(A)p Fo(=)p Fp(f)p Fi(x)p Fp(g)1409 2255 y Fl(log)e Fj(\027)s Fl(\()p Fj(\021)1541 2262 y Fi(x)1564 2255 y Fl(\))p Fb(.)91 2364 y(F)m(urther)16 b(we)h(have)f(put)468 2484 y Fj(Q)504 2491 y Fo(\003)531 2484 y Fl(\()p Fj(\021)574 2465 y Fo(1)572 2495 y(\003)599 2484 y Fj(;)8 b(\021)645 2465 y Fo(2)643 2495 y(\003)669 2484 y Fj(;)g(\021)713 2494 y Fi(Z)-21 b(Z)742 2484 y Fh(d)761 2494 y Fp(n)p Fo(\003)807 2484 y Fl(\))12 b(:=)h Fj(\026)p Fl([)p Fj(\021)963 2465 y Fo(2)961 2495 y(\003)988 2484 y Fj(\021)1011 2494 y Fi(Z)-21 b(Z)1041 2484 y Fh(d)1060 2494 y Fp(n)p Fo(\003)1106 2484 y Fl(]\()p Fj(e)1158 2465 y Fp(\000)p Fo(\001)p Fi(H)1246 2470 y Ff(\003)1268 2465 y Fo(\()p Fi(\021)1302 2451 y Ff(1)1301 2474 y(\003)1324 2465 y Fi(;\021)1355 2451 y Ff(2)1354 2474 y(\003)1377 2465 y Fi(;\021)1407 2470 y Fh(@)r Ff(\003)1448 2465 y Fo(\))1464 2484 y Fl(\))1857 2481 y(\(3.2\))0 2597 y Fb(A)n(c)n(c)n(or)n(ding)17 b(to)j(our)g(assumption)f(on)g(the)g(me)n (asur)n(ability)g(on)g Fj(\026)p Fl([)p Fj(\021)r Fl(])p Fb(,)g Fj(Q)1237 2604 y Fo(\003)1283 2597 y Fb(dep)n(ends)f(me)n(asur)n (ably)h(on)g Fj(\021)1782 2608 y Fi(Z)-21 b(Z)1812 2597 y Fh(d)1831 2608 y Fp(n)p Fo(\003)1877 2597 y Fb(.)0 2670 y(We)17 b(note)f(the)g(fol)r(lowing)g(pr)n(op)n(erties)952 2779 y Fl(11)p eop %%Page: 12 12 12 11 bop 17 48 a Fb(\(i\))22 b Fj(Q)127 55 y Fo(\003)154 48 y Fl(\()p Fj(\021)197 31 y Fo(1)195 60 y(\003)221 48 y Fj(;)8 b(\021)267 31 y Fo(2)265 60 y(\003)291 48 y Fj(;)g(\021)335 59 y Fi(Z)-21 b(Z)365 48 y Fh(d)384 59 y Fp(n)p Fo(\003)430 48 y Fl(\))12 b(=)508 11 y Fe(\002)527 48 y Fj(Q)563 55 y Fo(\003)590 48 y Fl(\()p Fj(\021)633 31 y Fo(2)631 60 y(\003)657 48 y Fj(;)c(\021)703 31 y Fo(1)701 60 y(\003)727 48 y Fj(;)g(\021)771 59 y Fi(Z)-21 b(Z)801 48 y Fh(d)820 59 y Fp(n)p Fo(\003)866 48 y Fl(\))884 11 y Fe(\003)902 21 y Fp(\000)p Fo(1)3 157 y Fb(\(ii\))22 b(F)m(or)16 b(any)g Fl(\001)p Fg(\033)p Fl(\003)g Fb(we)g(have)h Fj(Q)592 164 y Fo(\001)624 157 y Fl(\()p Fj(\021)667 141 y Fo(1)665 170 y(\003)691 157 y Fj(\021)714 165 y Fo(\001)p Fp(n)p Fo(\003)790 157 y Fj(;)8 b(\021)836 141 y Fo(2)834 170 y(\003)859 157 y Fj(\021)882 165 y Fo(\001)p Fp(n)p Fo(\003)958 157 y Fj(;)g(\021)1002 168 y Fi(Z)-21 b(Z)1031 157 y Fh(d)1050 168 y Fp(n)p Fo(\001)1101 157 y Fl(\))12 b(=)h Fj(Q)1215 164 y Fo(\003)1242 157 y Fl(\()p Fj(\021)1285 141 y Fo(1)1283 170 y(\003)1310 157 y Fj(;)8 b(\021)1356 141 y Fo(2)1354 170 y(\003)1380 157 y Fj(;)g(\021)1424 168 y Fi(Z)-21 b(Z)1453 157 y Fh(d)1472 168 y Fp(n)p Fo(\003)1518 157 y Fl(\))-11 276 y Fb(\(iii\))22 b(F)m(or)16 b(any)g Fj(\021)288 260 y Fo(3)286 289 y(\003)329 276 y Fb(we)g(have)506 245 y Fi(Q)535 250 y Ff(\003)558 245 y Fo(\()p Fi(\021)592 231 y Ff(1)591 254 y(\003)614 245 y Fi(;\021)645 231 y Ff(3)644 254 y(\003)666 245 y Fi(;\021)696 256 y Fh(Z)-22 b(Z)719 250 y(d)738 256 y Fd(n)p Ff(\003)777 245 y Fo(\))p 506 266 286 2 v 506 293 a Fi(Q)535 298 y Ff(\003)558 293 y Fo(\()p Fi(\021)592 282 y Ff(2)591 305 y(\003)614 293 y Fi(;\021)645 282 y Ff(3)644 305 y(\003)666 293 y Fi(;\021)696 305 y Fh(Z)g(Z)719 298 y(d)738 305 y Fd(n)p Ff(\003)777 293 y Fo(\))810 276 y Fl(=)13 b Fj(Q)894 283 y Fo(\003)921 276 y Fl(\()p Fj(\021)964 260 y Fo(1)962 289 y(\003)988 276 y Fj(;)8 b(\021)1034 260 y Fo(2)1032 289 y(\003)1058 276 y Fj(;)g(\021)1102 287 y Fi(Z)-21 b(Z)1131 276 y Fh(d)1151 287 y Fp(n)p Fo(\003)1196 276 y Fl(\))0 385 y Fb(whenever)16 b Fj(\021)e Fg(2)f(H)316 369 y Fo(0)337 385 y Fb(.)91 494 y Fm(Remark:)32 b Fl(Note)19 b(that,)g(b)o(y)h(our)f(assumption)h(on)f(the)h(a.s.)32 b(con)o(v)o(ergence)20 b(of)f(the)g(in\014nite)j(v)o(olume)0 567 y(Gibbs)16 b(measures,)f Fj(Q)376 574 y Fo(\003)418 567 y Fl(can)g(b)q(e)h(written)f(in)h(the)g(form)60 743 y Fj(Q)96 750 y Fo(\003)123 743 y Fl(\()p Fj(\021)166 724 y Fo(1)164 754 y(\003)191 743 y Fj(;)8 b(\021)237 724 y Fo(2)235 754 y(\003)260 743 y Fj(;)g(\021)304 753 y Fi(Z)-20 b(Z)335 743 y Fh(d)354 753 y Fp(n)p Fo(\003)399 743 y Fl(\))13 b(=)43 b(lim)478 777 y Fo(\003)503 782 y Fh(N)532 777 y Fp(")p Fi(Z)-20 b(Z)582 766 y Fh(d)608 743 y Fj(\026)635 710 y Fi(\033)657 696 y Ff(b.c.)656 719 y Fh(@)r Ff(\003)696 727 y Fh(N)635 756 y Fo(\003)660 761 y Fh(N)730 743 y Fl([)p Fj(\021)768 724 y Fo(2)787 743 y Fj(\021)p 810 731 57 2 v 15 x Fo(\003)835 763 y Fh(N)866 743 y Fl(])887 692 y Fe(\020)913 743 y Fj(e)934 724 y Fp(\000)p Fo(\001)p Fi(H)1022 729 y Ff(\003)1045 724 y Fo(\()p Fi(\021)1079 710 y Ff(1)1078 733 y(\003)1101 724 y Fi(;\021)1132 710 y Ff(2)1131 733 y(\003)1154 724 y Fi(;\021)1184 729 y Fh(@)r Ff(\003)1225 724 y Fo(\))1241 692 y Fe(\021)1281 743 y Fl(=)43 b(lim)1329 777 y Fo(\003)1354 782 y Fh(N)1383 777 y Fp(")p Fi(Z)-20 b(Z)1433 766 y Fh(d)1465 702 y Fj(Z)1499 669 y Fi(\033)1521 656 y Ff(b.c.)1520 679 y Fh(@)r Ff(\003)1560 686 y Fh(N)1496 715 y Fo(\003)1521 720 y Fh(N)1594 702 y Fl([)p Fj(\021)1632 685 y Fo(1)1630 714 y(\003)1656 702 y Fj(\021)p 1679 690 55 2 v 15 x Fo(\003)1704 722 y Fh(N)1733 717 y Fp(n)p Fo(\003)1778 702 y Fl(])p 1465 732 326 2 v 1465 800 a Fj(Z)1499 765 y Fi(\033)1521 754 y Ff(b.c.)1520 777 y Fh(@)r Ff(\003)1560 784 y Fh(N)1496 813 y Fo(\003)1521 818 y Fh(N)1594 800 y Fl([)p Fj(\021)1632 784 y Fo(2)1630 813 y(\003)1656 800 y Fj(\021)p 1679 788 55 2 v 15 x Fo(\003)1704 820 y Fh(N)1733 815 y Fp(n)p Fo(\003)1778 800 y Fl(])1857 743 y(\(3.3\))0 919 y(with)16 b(the)f(quenc)o(hed)h(partition)g (function)564 1048 y Fj(Z)598 1023 y Fi(\033)620 1010 y Ff(b.c.)619 1033 y Fh(@)r Ff(\003)595 1062 y Fo(\003)674 1048 y Fl([)p Fj(\021)p 710 1036 28 2 v 16 x Fo(\003)736 1048 y Fl(])d(=)810 1005 y Fe(X)820 1101 y Fi(\033)841 1106 y Ff(\003)883 1048 y Fj(e)904 1022 y Fp(\000)939 991 y Fe(P)987 1039 y Fh(A)p Ff(:)p Fh(A)p Fd(\\)p Ff(\003)p Fd(6)p Ff(=)p Fd(;)1134 1022 y Fo(\010)1160 1027 y Fh(A)1185 1022 y Fo(\()p Fi(\033)1220 1027 y Ff(\003)1244 1022 y Fi(\033)1266 1008 y Ff(b.c.)1265 1031 y Fh(@)r Ff(\003)1317 1022 y Fi(;\021)p 1347 1015 23 2 v 14 x Ff(\003)1370 1022 y Fo(\))1857 1062 y Fl(\(3.4\))0 1211 y(whenev)o(er)19 b Fj(\021)g Fg(2)f(H)331 1194 y Fo(0)352 1211 y Fl(.)29 b(Morally)l(,)19 b Fj(Q)612 1218 y Fo(\003)658 1211 y Fl(is)g(th)o(us)f(a)g(fraction)g(b)q(et)o(w)o(een)h(in\014nite)h(v)o (olume)e(partition)h(functions)0 1283 y(whose)c(disorder)h(v)m (ariables)g(di\013er)g(in)g(the)f(v)o(olume)h(\003.)91 1392 y Fm(Remark:)30 b Fl(W)l(e)19 b(note)f(that)g(form)o(ulas)g(for)g (the)h Fb(\014nite)f(volume)i(c)n(onditional)f(exp)n(e)n(ctations)f Fl(ha)o(v)o(e)g(ap-)0 1465 y(p)q(eared)h(in)f([K6])f([see)h(Lemma)g (2.1,)f(\(2.4\))g(therein].)28 b(They)18 b(seem)g(to)f(lo)q(ok)h(more)g (complicated)h(than)f(the)0 1538 y(in\014nite)f(v)o(olume)e(expression) g(\(3.1\).)j(In)e(that)e(pap)q(er)h(w)o(e)f(w)o(an)o(ted)g(to)g(b)q(e)i (able)f(to)f(deal)i(also)e(with)i(the)e(more)0 1611 y(general)f(case)f (in)h(whic)o(h)h(w)o(e)e(do)g(not)g(assume)g Fj(I)-8 b(P)6 b Fl(-a.s.)19 b(con)o(v)o(ergence)13 b(of)f(the)g(\014nite)i(v)o (olume)e(Gibbs)h(measures,)0 1683 y(but)i(only)h(con)o(v)o(ergence)f (of)g(the)h(\014nite)g(v)o(olume)f(join)o(t)g(measures.)20 b(Then)c(\(3.1\))d(is)j(not)f(a)o(v)m(ailable.)0 1792 y Fc(Pro)r(of:)23 b Fl(Prop)q(erties)16 b(\(i\),\(ii\),\(iii\))g(are)f (clear)h(from)e(\(3.3\).)0 1907 y(T)l(o)c(get)g(\(3.1\))e(w)o(e)i(will) i(sho)o(w)d(at)h(\014rst)g(that,)g(for)f(the)h(measure)h Fj(K)1085 1874 y Fi(\033)1107 1861 y Ff(b.c.)1106 1884 y Fh(@)r Ff(\003)1146 1891 y Fh(N)1082 1920 y Fo(\003)1107 1925 y Fh(N)1179 1907 y Fl(\()p Fj(\033)1223 1914 y Fo(\003)1248 1919 y Fh(N)1279 1907 y Fj(;)d(\021)p 1323 1895 57 2 v 15 x Fo(\003)1348 1927 y Fh(N)1378 1907 y Fl(\))k(:=)h Fj(I)-8 b(P)6 b Fl(\()p Fj(\021)p 1557 1895 V 15 x Fo(\003)1582 1927 y Fh(N)1613 1907 y Fl(\))p Fj(\026)1658 1874 y Fi(\033)1680 1861 y Ff(b.c.)1679 1884 y Fh(@)r Ff(\003)1719 1891 y Fh(N)1658 1920 y Fo(\003)1683 1925 y Fh(N)1752 1907 y Fl([)p Fj(\021)p 1788 1895 V 15 x Fo(\003)1813 1927 y Fh(N)1844 1907 y Fl(]\()p Fj(\033)1901 1914 y Fo(\003)1926 1919 y Fh(N)1956 1907 y Fl(\))0 1980 y(on)15 b(\012)96 1987 y Fo(\003)133 1980 y Fg(\002)c(H)p 217 1968 28 2 v 15 x Fo(\003)259 1980 y Fl(w)o(e)k(ha)o(v)o(e,)g(for)f(\014nite)i (\003)p Fj(;)8 b Fl(\001)p Fj(;)g Fl(\003)773 1987 y Fi(N)820 1980 y Fl(with)16 b(\003)p Fg(\032)p Fl(\001)f(and)p 1132 1943 38 2 v 15 w(\001)p Fg(\032)q Fl(\003)1238 1987 y Fi(N)1272 1980 y Fl(,)f(the)i(form)o(ula)25 2144 y Fj(K)67 2111 y Fi(\033)89 2097 y Ff(b.c.)88 2121 y Fh(@)r Ff(\003)128 2128 y Fh(N)64 2157 y Fo(\003)89 2162 y Fh(N)168 2107 y Fe(\002)187 2144 y Fj(\030)207 2151 y Fo(\003)234 2105 y Fe(\014)234 2133 y(\014)249 2144 y Fj(\030)269 2152 y Fo(\001)p Fp(n)p Fo(\003)345 2107 y Fe(\003)376 2144 y Fl(=)424 2082 y Fe(Z)477 2144 y Fj(K)519 2111 y Fi(\033)541 2097 y Ff(b.c.)540 2121 y Fh(@)r Ff(\003)580 2128 y Fh(N)516 2157 y Fo(\003)541 2162 y Fh(N)621 2093 y Fe(h)642 2144 y Fj(d)r Fl(\026)-25 b Fj(\033)692 2152 y Fo(\003)717 2157 y Fh(N)746 2152 y Fp(n)p Fo(\001)797 2144 y Fj(d)s Fl(\026)f Fj(\021)p 844 2132 55 2 v 15 x Fo(\003)869 2164 y Fh(N)897 2159 y Fp(n)p Fo(\001)948 2105 y Fe(\014)948 2133 y(\014)963 2144 y Fj(\030)983 2152 y Fo(\001)p Fp(n)p Fo(\003)1058 2093 y Fe(i)1319 2113 y Fj(\026)1346 2085 y Ff(ann,)q Fi(\030)1422 2090 y Fh(@)r Ff(\003)p Fd(\\)p Ff(\001)1513 2077 y Fo(\026)1510 2085 y Fi(\030)1527 2091 y Fh(@)r Ff(\003)p Fd(n)p Ff(\001)1346 2126 y Fo(\003)1612 2113 y Fl(\()p Fj(\030)1650 2120 y Fo(\003)1676 2113 y Fl(\))p 1093 2133 828 2 v 1093 2195 a Fe(R)1131 2232 y Fj(\026)1158 2204 y Ff(ann,)q Fi(\030)1234 2209 y Fh(@)r Ff(\003)p Fd(\\)p Ff(\001)1325 2196 y Fo(\026)1321 2204 y Fi(\030)1338 2210 y Fh(@)r Ff(\003)p Fd(n)p Ff(\001)1158 2245 y Fo(\003)1424 2232 y Fl(\()p Fj(d)s Fl(~)g Fj(\021)1489 2239 y Fo(\003)1515 2232 y Fl(\))1538 2196 y Fi(Z)1565 2163 y Fh(\033)1585 2155 y Ff(b.c.)1584 2178 y Fh(@)r Ff(\003)1624 2185 y Fh(N)1563 2208 y Ff(\003)1584 2216 y Fh(N)1656 2196 y Fo([)p Fi(\021)1685 2201 y Ff(\003)1708 2196 y Fi(\021)1727 2202 y Ff(\001)p Fd(n)p Ff(\003)1795 2196 y Fo(\026)-21 b Fi(\021)p 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eop %%Page: 13 13 13 12 bop 0 45 a Fl(and)15 b(note)g(that)g(the)g(term)g(under)h(the)f (in)o(tegral)h(on)f(the)g(r.h.s.)k(equals)478 172 y Fj(K)520 140 y Fi(\033)542 126 y Ff(b.c.)541 149 y Fh(@)r Ff(\003)581 157 y Fh(N)517 186 y Fo(\003)542 191 y Fh(N)622 122 y Fe(h)643 172 y Fj(\030)663 179 y Fo(\003)690 172 y Fj(\030)710 181 y Fo(\001)p Fp(n)p Fo(\003)788 172 y Fl(\026)-25 b Fj(\033)812 181 y Fo(\003)837 186 y Fh(N)866 181 y Fp(n)p Fo(\001)920 172 y Fl(\026)f Fj(\021)p 940 160 55 2 v 16 x Fo(\003)965 193 y Fh(N)993 188 y Fp(n)p Fo(\001)1044 122 y Fe(i)p 430 209 685 2 v 430 243 a(P)481 283 y Fo(~)478 291 y Fi(\030)495 296 y Ff(\003)527 277 y Fj(K)569 241 y Fi(\033)591 231 y Ff(b.c.)590 254 y Fh(@)r Ff(\003)630 261 y Fh(N)566 290 y Fo(\003)591 295 y Fh(N)671 226 y Fe(h)697 265 y Fl(~)692 277 y Fj(\030)712 284 y Fo(\003)739 277 y Fj(\030)759 285 y Fo(\001)p Fp(n)p Fo(\003)837 277 y Fl(\026)h Fj(\033)861 285 y Fo(\003)886 290 y Fh(N)915 285 y Fp(n)p Fo(\001)969 277 y Fl(\026)e 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1683 y Fi(Z)-20 b(Z)1645 1673 y Fh(d)1664 1683 y Fp(n)p Fo(\001)1715 1673 y Fl(\))1857 1626 y(\(3.9\))0 1781 y(T)l(o)17 b(see)h(this,)g(write)f(do)o(wn)g (\(3.5\))f(explicitly)k(in)e(terms)f(of)f(the)i(quenc)o(hed)g(lo)q(cal) h(sp)q(eci\014cations)g(and)f(\(3.9\))0 1854 y(in)e(terms)f(of)g(the)h (in\014nite)h(v)o(olume)f(Gibbs)h(measure.)k(Note)15 b(that)g(the)g(dep)q(endence)j(on)e(those)f(measures)g(is)0 1927 y(completely)k(lo)q(cal-)g(therefore)e(\(3.9\))f(follo)o(ws)h(b)o (y)h(the)g(assumption)f(of)g Fj(I)-8 b(P)6 b Fl(-a.s.)27 b(lo)q(cal)19 b(con)o(v)o(ergence)f(of)f(the)0 1999 y(\014nite)j(v)o (olume)g(Gibbs)g(measures.)33 b(But)19 b(from)g(\(3.9\))f(w)o(e)h(can)g (conclude)j(no)o(w,)d(that)g(what)g(is)h(under)g(the)0 2072 y(in)o(tegral)15 b(on)g(the)g(r.h.s.)20 b(m)o(ust)14 b(b)q(e)i(the)f(in\014nite)i(v)o(olume)e(conditional)i(exp)q(ectation.) j(More)15 b(precisely)l(,)i(\(3.1\))0 2145 y(follo)o(ws)e(from)g(the)g (follo)o(wing)h(general)g(measure-theoretic)91 2254 y Fm(F)l(act:)23 b Fb(Assume)17 b(that)i Fj(\030)512 2265 y Fi(Z)-21 b(Z)542 2254 y Fh(d)22 b Fb(is)17 b(a)h(r)n(andom)g(\014eld) f(with)h(distribution)g Fj(K)s Fb(,)g Fj(\030)1377 2261 y Fi(x)1417 2254 y Fb(taking)g(values)f(in)g(a)h(\014nite)0 2327 y(set,)e(and)185 2315 y Fl(~)173 2327 y Fj(K)223 2290 y Fe(\002)241 2327 y Fj(\030)261 2334 y Fo(\003)288 2288 y Fe(\014)288 2315 y(\014)304 2327 y Fj(\030)324 2337 y Fi(Z)-21 b(Z)354 2327 y Fh(d)373 2337 y Fp(n)p Fo(\003)419 2290 y Fe(\003)454 2327 y Fb(is)16 b(a)h(Bor)n(el)e(pr)n (ob)n(ability)h(kernel)g(that)h(satis\014es)438 2463 y Fj(K)488 2426 y Fe(\002)506 2463 y Fj(\030)526 2470 y Fo(\003)553 2424 y Fe(\014)553 2452 y(\014)569 2463 y Fj(\030)589 2471 y Fo(\001)p Fp(n)p Fo(\003)664 2426 y Fe(\003)696 2463 y Fl(=)744 2401 y Fe(Z)797 2463 y Fj(K)846 2426 y Fe(\002)865 2463 y Fj(d)894 2451 y Fl(\026)889 2463 y Fj(\030)909 2474 y Fi(Z)-21 b(Z)939 2463 y Fh(d)958 2474 y Fp(n)p Fo(\001)1009 2424 y Fe(\014)1009 2452 y(\014)1024 2463 y Fj(\030)1044 2471 y Fo(\001)p Fp(n)p Fo(\003)1120 2426 y Fe(\003)1159 2452 y Fl(~)1146 2463 y Fj(K)1196 2426 y Fe(\002)1215 2463 y Fj(\030)1235 2470 y Fo(\003)1262 2424 y Fe(\014)1262 2452 y(\014)1277 2463 y Fj(\030)1297 2471 y Fo(\001)p Fp(n)p Fo(\003)1377 2451 y Fl(\026)1372 2463 y Fj(\030)1392 2474 y Fi(Z)g(Z)1423 2463 y Fh(d)1442 2474 y Fp(n)p Fo(\001)1493 2426 y Fe(\003)1834 2463 y Fl(\(3.10\))0 2597 y Fb(for)20 b(al)r(l)f(\014nite)g Fl(\001)p Fg(\033)p Fl(\003)p Fb(,)h(wher)n(e)f Fj(K)585 2560 y Fe(\002)604 2597 y Fj(d)633 2585 y Fl(\026)628 2597 y Fj(\030)648 2607 y Fi(Z)-21 b(Z)678 2597 y Fh(d)697 2607 y Fp(n)p Fo(\001)748 2558 y Fe(\014)748 2585 y(\014)763 2597 y Fj(\030)783 2605 y Fo(\001)p Fp(n)p Fo(\003)858 2560 y Fe(\003)897 2597 y Fb(is)19 b(a)h(version)e(of)i(the)g(c)n (onditional)f(exp)n(e)n(ctation.)30 b(Then)12 2658 y Fl(~)0 2670 y Fj(K)49 2633 y Fe(\002)68 2670 y Fj(\030)88 2677 y Fo(\003)115 2631 y Fe(\014)115 2658 y(\014)131 2670 y Fj(\030)151 2680 y Fi(Z)-21 b(Z)181 2670 y Fh(d)200 2680 y Fp(n)p Fo(\003)246 2633 y Fe(\003)281 2670 y Fb(is)16 b(a)h(version)e(of)i(the)f(in\014nite)f(volume)h(c)n(onditional)g(exp)n (e)n(ctation)g Fj(K)1486 2633 y Fe(\002)1505 2670 y Fj(\030)1525 2677 y Fo(\003)1552 2631 y Fe(\014)1552 2658 y(\014)1567 2670 y Fj(\030)1587 2680 y Fi(Z)-21 b(Z)1618 2670 y Fh(d)1637 2680 y Fp(n)p Fo(\003)1683 2633 y Fe(\003)1702 2670 y Fb(.)952 2779 y Fl(13)p eop %%Page: 14 14 14 13 bop 91 45 a Fl(W)l(e)15 b(include)j(a)d(pro)q(of)f(for)h(the)g (con)o(v)o(enience)i(of)e(the)g(reader:)103 143 y(~)91 155 y Fj(K)140 118 y Fe(\002)159 155 y Fj(\030)179 162 y Fo(\003)206 116 y Fe(\014)206 143 y(\014)221 155 y Fj(\030)241 165 y Fi(Z)-20 b(Z)273 155 y Fh(d)292 165 y Fp(n)p Fo(\003)337 118 y Fe(\003)376 155 y Fl(is)19 b(assumed)h(to)e(b)q(e)i Fj(\033)772 118 y Fe(\000)793 155 y Fj(\030)813 165 y Fi(Z)-20 b(Z)844 155 y Fh(d)863 165 y Fp(n)p Fo(\003)909 118 y Fe(\001)930 155 y Fl(-measurable.)33 b(So,)20 b(to)e(v)o(erify)i(the)f(de\014nition)i(of)e(the)0 227 y(conditional)e(exp)q(ectation)f(w)o(e)f(ha)o(v)o(e)g(to)g(sho)o(w) f(that,)h(for)f(all)j(ev)o(en)o(ts)e Fj(C)g Fg(2)e Fj(\033)1329 190 y Fe(\000)1350 227 y Fj(\030)1370 238 y Fi(Z)-20 b(Z)1401 227 y Fh(d)1420 238 y Fp(n)p Fo(\003)1466 190 y Fe(\001)1502 227 y Fl(and)16 b Fj(A)d Fg(2)g Fj(\033)c Fl(\()p Fj(\030)1754 238 y Fi(Z)-20 b(Z)1785 227 y Fh(d)t Fl(\))15 b(w)o(e)0 300 y(ha)o(v)o(e)g(that)318 318 y Fe(Z)343 421 y Fi(C)382 315 y Fe(\022)415 318 y(Z)440 421 y Fi(A)490 368 y Fl(~)477 379 y Fj(K)527 329 y Fe(h)548 379 y Fj(d\030)592 386 y Fo(\003)619 341 y Fe(\014)619 368 y(\014)634 379 y Fj(\030)656 361 y Fp(0)654 394 y Fi(Z)-20 b(Z)685 384 y Fh(d)704 394 y Fp(n)p Fo(\003)750 329 y Fe(i)782 379 y Fg(\012)10 b Fj(\016)847 387 y Fi(\030)865 376 y Fd(0)864 406 y Fh(Z)-21 b(Z)887 399 y(d)906 406 y Fd(n)p Ff(\003)948 379 y Fl(\()p Fj(d\030)1010 390 y Fi(Z)h(Z)1041 379 y Fh(d)1060 390 y Fp(n)p Fo(\003)1106 379 y Fl(\))1124 315 y Fe(\023)1164 379 y Fj(K)s Fl(\()p Fj(d\030)1270 361 y Fp(0)1268 394 y Fi(Z)g(Z)1299 384 y Fh(d)1318 394 y Fp(n)p Fo(\003)1364 379 y Fl(\))12 b(=)h Fj(K)s Fl(\()p Fj(A)d Fg(\\)g Fj(C)s Fl(\))194 b(\(3.11\))0 504 y(W)l(riting)23 b Fj(A)g Fl(in)h(the)f(form)e Fj(A)26 b Fl(=)612 470 y Fe(P)660 518 y Fi(\030)677 523 y Ff(\003)710 504 y Fl(\()o Fg(f)p Fj(\030)770 511 y Fo(\003)797 504 y Fg(g)10 b(\002)g Fj(A)909 511 y Fi(\030)926 516 y Ff(\003)951 504 y Fl(\))23 b(where)g Fj(A)1165 511 y Fi(\030)1182 516 y Ff(\003)1232 504 y Fg(2)i Fj(\033)1322 467 y Fe(\000)1343 504 y Fj(\030)1363 515 y Fi(Z)-20 b(Z)1394 504 y Fh(d)1413 515 y Fp(n)p Fo(\003)1459 467 y Fe(\001)1503 504 y Fl(w)o(e)22 b(see)h(that)f(this)h(is)0 577 y(equiv)m(alen)o(t)d(to)277 543 y Fe(P)325 591 y Fi(\030)342 596 y Ff(\003)374 540 y Fe(R)396 593 y Fi(C)446 565 y Fl(~)434 577 y Fj(K)483 527 y Fe(h)505 577 y Fj(\030)525 584 y Fo(\003)552 538 y Fe(\014)552 566 y(\014)567 577 y Fj(\030)589 560 y Fp(0)587 594 y Fi(Z)-20 b(Z)618 584 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y Fo(\001)1798 2119 y Fj(;)10 b Fl(~)-25 b Fj(\021)1842 2129 y Fi(Z)1869 2120 y Fh(d)1887 2129 y Fp(n)p Fo(\001)1937 2119 y Fl(\))1955 2069 y Fe(\021)1857 2197 y Fl(\(4.6\))0 2270 y(for)15 b(an)o(y)f(\014xed)i Fj(\021)290 2253 y Fp(0)302 2270 y Fl(.)k(The)c(last)f(t)o(w)o(o)f(terms)g(cancel,)i(due)g(to)f(Prop)q (osition)h(3.1)e(\(i\))h(and)g(the)h(\014rst)e(term)h(equals)0 2343 y Fj(E)37 2326 y Fi(\013)34 2355 y Fo(\003)62 2343 y Fl(\()p Fj(\021)103 2350 y Fo(\003)129 2343 y Fl(\),)h(due)h(to)e (\(ii\),)i(as)f(desired.)23 b(Note)16 b(that)f(this)i(w)o(orks)e(also)h (in)h(the)f(case)g Fj(\013)f Fl(=)f Fj(\016)1508 2350 y Fo(^)-21 b Fi(\021)1544 2343 y Fl(since)17 b(w)o(e)f(assumed)0 2415 y(w)o(eak)f(con)o(v)o(ergence)g(for)g(the)g(p)q(oin)o(t)k(^)-26 b Fj(\021)q Fl(!)91 2525 y(F)l(rom)17 b(this)h(follo)o(ws)f(easily)i (from)e(the)g(usual)h(pla)o(y)g(with)g(signed)h(sums)e(that,)g(in)h (fact,)g(the)f(p)q(oten)o(tial)0 2597 y Fj(U)36 2581 y Ff(fe)p Fi(;\013)117 2597 y Fl(is)j Fj(\013)p 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Fp(\032)q Fo(\001)p Fp(n)p Fo(\003)1407 137 y Fj(U)1443 115 y Fi(\013;f)t(e)1438 150 y(A)1518 137 y Fl(\()p Fj(\021)r Fl(\))385 262 y(=)13 b Fj(E)470 244 y Fi(\013)467 274 y Fo(\001)498 262 y Fl(\()p Fj(\021)r Fl(\))c Fg(\000)h Fj(E)650 244 y Fi(\013)647 275 y Fo(\001)p Fp(n)p Fo(\003)723 262 y Fl(\()p Fj(\021)r Fl(\))385 371 y(=)433 309 y Fe(Z)486 371 y Fj(\013)p Fl(\()p Fj(d)s Fl(~)-26 b Fj(\021)p Fl(\))8 b(log)724 335 y Fj(Q)760 342 y Fo(\001)792 335 y Fl(\()p Fj(\021)833 342 y Fo(\001)864 335 y Fj(;)j Fl(~)-26 b Fj(\021)908 342 y Fo(\001)939 335 y Fj(;)11 b Fl(~)-26 b Fj(\021)983 346 y Fi(Z)-20 b(Z)1013 335 y Fh(d)1032 346 y Fp(n)p Fo(\001)1083 335 y Fl(\))p 677 361 471 2 v 677 402 a Fj(Q)713 409 y Fo(\001)745 402 y Fl(\()s(~)-26 b Fj(\021)786 409 y Fo(\003)813 402 y Fj(\021)836 411 y Fo(\001)p Fp(n)p Fo(\003)911 402 y Fj(;)11 b Fl(~)-26 b Fj(\021)955 409 y Fo(\001)986 402 y Fj(;)10 b Fl(~)-25 b Fj(\021)1030 413 y Fi(Z)-20 b(Z)1060 402 y Fh(d)1079 413 y Fp(n)p Fo(\001)1130 402 y Fl(\))385 501 y(=)433 439 y Fe(Z)486 501 y Fj(\013)p Fl(\()p Fj(d)s Fl(~)-26 b Fj(\021)p Fl(\))8 b(log)g Fj(Q)708 508 y Fo(\003)735 501 y Fl(\()p Fj(\021)776 508 y Fo(\003)802 501 y Fj(;)j Fl(~)-26 b Fj(\021)846 508 y Fo(\003)872 501 y Fj(;)8 b(\021)916 509 y Fo(\001)p Fp(n)p Fo(\003)994 501 y Fl(~)-26 b Fj(\021)1014 512 y Fi(Z)-20 b(Z)1045 501 y Fh(d)1064 512 y Fp(n)p Fo(\001)1114 501 y Fl(\))1857 326 y(\(4.7\))0 609 y(The)19 b(second)h(equalit)o(y)f (is)h(\(4.4\))d(and)i(for)g(the)g(next)g(t)o(w)o(o)e(equalities)k(w)o (e)e(ha)o(v)o(e)f(used)i(prop)q(erties)f(\(ii\))h(and)0 682 y(\(iii\))e(for)f Fj(Q)p Fl(.)25 b(The)17 b(imp)q(ortan)o(t)g(p)q (oin)o(t)g(that)g(exploits)h(the)f(nature)g(of)g Fj(\013)g Fl(b)q(eing)h(a)f(pro)q(duct)g(measure)g(is)h(the)0 754 y(con)o(v)o(ergence)d(statemen)o(t)154 859 y(lim)136 893 y Fo(\001)p Fp(")p Fi(Z)-20 b(Z)216 883 y Fh(d)242 798 y Fe(Z)295 859 y Fj(\013)p Fl(\()p Fj(d)s Fl(~)-26 b Fj(\021)q Fl(\))8 b(log)g Fj(Q)518 866 y Fo(\003)545 859 y Fl(\()p Fj(\021)588 841 y Fo(1)586 871 y(\003)612 859 y Fj(;)g(\021)658 841 y Fo(2)656 871 y(\003)682 859 y Fj(;)g(\021)726 868 y Fo(\001)p Fp(n)p Fo(\003)804 859 y Fl(~)-26 b Fj(\021)824 870 y Fi(Z)-20 b(Z)854 859 y Fh(d)874 870 y Fp(n)p Fo(\001)924 859 y Fl(\))13 b(=)g(log)8 b Fj(Q)1105 866 y Fo(\003)1132 859 y Fl(\()p Fj(\021)1175 841 y Fo(1)1173 871 y(\003)1199 859 y Fj(;)g(\021)1245 841 y Fo(2)1243 871 y(\003)1269 859 y Fj(;)g(\021)1313 870 y Fi(Z)-20 b(Z)1343 859 y Fh(d)1362 870 y Fp(n)p Fo(\003)1408 859 y Fl(\))45 b(for)g Fj(\013)p Fl(-a.e.)20 b Fj(\021)1857 862 y Fl(\(4.8\))0 981 y(This)15 b(follo)o(ws)g(b)o(y)f (the)h(martingale)f(con)o(v)o(ergence)h(theorem,)f(since,)h(for)f(an)o (y)h(\014xed)g(\014nite)g(\003)p Fg(\032)p Fj(Z)-19 b(Z)1673 961 y Fi(d)1709 981 y Fl(and)14 b(\014xed)0 1053 y Fj(\021)25 1037 y Fo(1)23 1066 y(\003)50 1053 y Fj(;)8 b(\021)96 1037 y Fo(2)94 1066 y(\003)135 1053 y Fl(the)15 b(expression)h(under)g (the)g(limit)g(on)g(the)f(l.h.s)g(indexed)i(b)o(y)e(\014nite)i(subsets) e(\001)p Fg(\032)p Fj(Z)-19 b(Z)1610 1033 y Fi(d)1646 1053 y Fl(s.t.)20 b(\001)p Fg(\033)p Fl(\003,)14 b(is)0 1126 y(a)h(martingale)g(w.r.t)f(the)h(distribution)i(giv)o(en)f(b)o(y)f Fj(\013)p Fl(.)0 1235 y Fm(Theorem)i(2.1:)j Fl(W)l(e)c(put)g Fj(\013)d Fl(=)g Fj(I)-8 b(P)6 b Fl(.)22 b(Then)16 b(w)o(e)f(see)h (from)e(\(4.7\))g(and)i(\(4.8\))e(that)h(the)g(p)q(oten)o(tial)h(con)o (v)o(erges)0 1308 y(with)g(\001)d Fg(")g Fj(Z)-19 b(Z)237 1288 y Fi(d)274 1308 y Fl(for)15 b Fj(I)-8 b(P)6 b Fl(-a.e.)22 b Fj(\021)r Fl(.)e(Since)d Fj(I)-8 b(P)23 b Fl(is)16 b(the)g(marginal)g(of)f Fj(K)j Fl(on)e(the)g(disorder-space,)g(this)g (is)h(exactly)0 1381 y(what)e(w)o(e)g(w)o(an)o(t.)0 1490 y Fm(Theorem)i(2.2:)j Fl(W)l(e)c(put)g Fj(\013)d Fl(=)h Fj(\016)601 1497 y Fo(^)-21 b Fi(\021)635 1490 y Fl(where)19 b(^)-26 b Fj(\021)17 b Fl(is)f(the)f(assumed)h(direction)h(of)e(con)o (tin)o(uit)o(y)l(.)21 b(In)16 b(this)g(case)g(the)0 1562 y(r.h.s.)j(of)13 b(\(4.7\))g(is)h(just)g Fj(Q)446 1569 y Fo(\003)473 1562 y Fl(\()p Fj(\021)514 1569 y Fo(\003)540 1562 y Fj(;)d Fl(^)-26 b Fj(\021)584 1569 y Fo(\003)610 1562 y Fj(;)8 b(\021)654 1571 y Fo(\001)p Fp(n)p Fo(\003)732 1562 y Fl(^)-26 b Fj(\021)752 1573 y Fi(Z)-20 b(Z)783 1562 y Fh(d)802 1573 y Fp(n)p Fo(\001)852 1562 y Fl(\).)20 b(Using)14 b(prop)q(ert)o(y)g(\(iii\))h(for)f Fj(Q)1406 1569 y Fo(\003)1447 1562 y Fl(w)o(e)f(ma)o(y)h(rewrite)g(this)g(as)0 1635 y(a)g(telescoping)i(sum)365 1601 y Fe(P)413 1649 y Fi(x)p Fp(2)p Fo(\003)493 1635 y Fj(Q)529 1642 y Fo(\003)556 1635 y Fl(\()p Fj(\021)597 1642 y Fo(\003)622 1647 y Fd(\024)p Fh(x)668 1635 y Fj(;)8 b(\021)712 1642 y Fo(\003)737 1647 y Fh(r)q Fp(g)1676 935 y Fl(\))196 1055 y Fg(\000)241 993 y Fe(Z)294 1055 y Fj(I)-8 b(P)6 b Fl(\()p Fj(d)s Fl(~)-26 b Fj(\021)r Fl(\))8 b(log)f Fj(Q)535 1062 y Fi(x)558 1055 y Fl(\()p Fj(\021)599 1062 y Fi(x)621 1055 y Fj(;)k Fl(~)-26 b Fj(\021)665 1062 y Fi(x)687 1055 y Fj(;)10 b Fl(~)-25 b Fj(\021)731 1063 y Fp(f)p Fi(y)q Fo(:1)p Fp(\024)p Fo(#\()p Fi(y)q Fo(\))p Fi(<)p Fo(#\()p Fi(x)p Fo(\))p Fp(g)1031 1055 y Fj(\021)1054 1063 y Fp(f)p Fi(y)q Fo(:#\()p Fi(y)q Fo(\))p Fi(>)p Fo(#\()p Fi(x)p Fo(\))p Fp(g)1308 1055 y Fl(\))1326 1002 y Fe(\014)1326 1030 y(\014)1326 1057 y(\014)1353 1055 y Fl(=:)13 b Fj(\017)1432 1062 y Fi(x)1455 1055 y Fl(\()p Fj(r)q Fl(\))f Fg(#)g Fl(0)1834 995 y(\(4.12\))0 1178 y(with)h Fj(r)g Fg(")g(1)p Fl(,)g(for)f(an)o(y)g(\014xed)h Fj(x)p Fl(.)19 b(This)13 b(is)g(clear,)h(since)f(the)g(\014rst)f(line)j (of)d(the)g(expression)i(under)f(the)g(mo)q(dulus)0 1251 y(is)j(a)f(martingale)g(w.r.t.)j(to)d(the)g(parameter)g(r,)f(for)h(an)o (y)g(\014xed)g Fj(x)h Fl(and)f(\014xed)h Fj(\021)1362 1258 y Fi(x)1384 1251 y Fl(.)91 1360 y(T)l(ak)o(e)f(some)h(subsequence) h Fj(r)q Fl(\()p Fj(n)p Fl(\))f(of)f(the)h(in)o(tegers,)g(to)f(b)q(e)i (de\014ned)g(b)q(elo)o(w.)22 b(F)l(or)15 b Fj(x)f Fg(\025)g Fl(1,)i Fj(m)d Fg(\025)h Fl(1)i(de\014ne)0 1433 y Fj(A)34 1440 y Fi(x;m)113 1433 y Fl(:=)c Fg(f)p Fj(z)j Fg(2)e Fj(Z)-19 b(Z)321 1413 y Fi(d)342 1433 y Fj(;)8 b Fl(#\()p Fj(x)p Fl(\))j Fg(\024)i Fl(#\()p Fj(z)r Fl(\))f Fg(\024)h Fj(r)q Fl(\()p Fj(x)c Fl(+)h Fj(m)p Fl(\))p Fg(g)p Fl(,)k(put)h(also)g Fj(A)1115 1440 y Fi(x;m)p Fo(=0)1240 1433 y Fl(=)e Fg(;)p Fl(.)19 b(Starting)c(from)f(general)i Fj(\013)p Fl(,)e(let)0 1506 y(us)h(de\014ne)i(the)e(resummed)g(p)q(oten)o(tial)h(b)o(y)g(the)f (form)o(ula)g(corresp)q(onding)h(to)e(\(4.10\),)f(i.e.)289 1618 y Fj(U)325 1596 y Fi(\013;)p Ff(fe,)d(abs)320 1631 y Fi(A)347 1636 y Fh(x;m)450 1618 y Fl(\()p Fj(\021)r Fl(\))i(:=)721 1575 y Fe(X)653 1666 y Fh(A)p Ff(:)p Fh(x)p Fd(2)p Fh(A)p Fd(\032)o Fh(A)797 1671 y(x;m)589 1697 y(A)p Fd(\\)p Ff(\()p Fh(A)670 1702 y(x;m)728 1697 y Fd(n)p Fh(A)769 1704 y(x;m)p Fd(\000)p Ff(1)867 1697 y(\))p Fd(6)p Ff(=)p Fd(;)933 1618 y Fj(U)969 1596 y Fi(\013;)p Ff(fe)964 1631 y Fi(A)1029 1618 y Fl(\()p Fj(\021)r Fl(\))p Fj(;)51 b(U)1190 1596 y Fi(\013;)p Ff(fe,)10 b(abs)1185 1631 y Fi(C)1316 1618 y Fl(\()p Fj(\021)r Fl(\))h(=)i(0)i(otherwise)1834 1653 y(\(4.13\))0 1809 y(for)g(all)h Fj(x)c Fg(2)h Fj(Z)-19 b(Z)260 1789 y Fi(d)296 1809 y Fl(and)16 b Fj(m)c Fg(\025)h Fl(1.)20 b(Then)c(w)o(e)e(ha)o(v)o(e)h(for)g Fj(m)d Fg(\025)h Fl(2)8 1907 y Fj(U)44 1885 y Fi(\013;)p Ff(fe,)d(abs)39 1920 y Fi(A)66 1925 y Fh(x;m)169 1907 y Fl(\()p Fj(\021)r Fl(\))h(=)i Fj(E)326 1888 y Fi(\013)323 1918 y(A)350 1923 y Fh(x;m)411 1907 y Fl(\()p Fj(\021)r Fl(\))c Fg(\000)h Fj(E)563 1888 y Fi(\013)560 1918 y(A)587 1923 y Fh(x;m)p Fd(\000)p Ff(1)688 1907 y Fl(\()p Fj(\021)r Fl(\))e Fg(\000)j Fj(E)840 1888 y Fi(\013)837 1919 y(A)864 1924 y Fh(x;m)922 1919 y Fp(n)p Fi(x)964 1907 y Fl(\()p Fj(\021)r Fl(\))e(+)h Fj(E)1116 1888 y Fi(\013)1113 1919 y(A)1140 1924 y Fh(x;m)p Fd(\000)p Ff(1)1239 1919 y Fp(n)p Fi(x)1280 1907 y Fl(\()p Fj(\021)r Fl(\))20 2018 y(=)68 1957 y Fe(Z)121 2018 y Fj(\013)p Fl(\()p Fj(d)s Fl(~)-26 b Fj(\021)215 2029 y Fi(Z)-20 b(Z)246 2018 y Fh(d)t Fl(\))8 b(log)364 1981 y Fj(Q)400 1988 y Fi(A)427 1993 y Fh(x;m)488 1981 y Fl(\()p Fj(\021)529 1988 y Fi(A)556 1993 y Fh(x;m)617 1981 y Fj(;)i Fl(~)-25 b Fj(\021)661 1988 y Fi(A)688 1993 y Fh(x;m)748 1981 y Fj(;)10 b Fl(~)-25 b Fj(\021)792 1992 y Fi(Z)-20 b(Z)822 1981 y Fh(d)841 1992 y Fp(n)p Fi(A)887 1997 y Fh(x;m)947 1981 y Fl(\))p Fj(Q)1001 1988 y Fi(A)1028 1993 y Fh(x;m)1089 1981 y Fl(\()p Fj(\021)1130 1989 y Fi(A)1157 1994 y Fh(x;m)p Fd(\000)p Ff(1)1256 1989 y Fp(n)p Fi(x)1300 1981 y Fl(~)-26 b Fj(\021)1320 1989 y Fi(A)1347 1994 y Fh(x;m)1406 1989 y Fp(n)p Fo(\()p Fi(A)1466 1994 y Fh(x;m)p Fd(\000)p Ff(1)1564 1989 y Fp(n)p Fi(x)p Fo(\))1620 1981 y Fj(;)11 b Fl(~)-26 b Fj(\021)1664 1988 y Fi(A)1691 1993 y Fh(x;m)1751 1981 y Fj(;)11 b Fl(~)-26 b Fj(\021)1795 1992 y Fi(Z)-20 b(Z)1825 1981 y Fh(d)1844 1992 y Fp(n)p Fi(A)1890 1997 y Fh(x;m)1951 1981 y Fl(\))p 364 2008 1605 2 v 375 2050 a Fj(Q)411 2057 y Fi(A)438 2062 y Fh(x;m)499 2050 y Fl(\()p Fj(\021)540 2057 y Fi(A)567 2062 y Fh(x;m)p Fd(\000)p Ff(1)671 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Fj(\026)643 2058 y Fp(1)682 2051 y Fl([)p Fj(J)720 2058 y Fo(\003)745 2049 y Fd(\003)777 2040 y Fl(^)766 2051 y Fj(J)791 2062 y Fo(\()p Fi(Z)g(Z)836 2051 y Fh(d)856 2062 y Fo(\))870 2053 y Fd(\003)889 2062 y Fp(n)p Fo(\003)933 2053 y Fd(\003)954 2051 y Fl(]\()r(~)-25 b Fj(\033)1011 2058 y Fi(x)1035 2051 y Fl(~)g Fj(\033)1059 2058 y Fi(y)1081 2051 y Fl(\))9 b Fg(\000)i Fj(\026)1181 2058 y Fp(1)1220 2051 y Fl([)p Fj(J)1258 2062 y Fo(\()p Fi(Z)-20 b(Z)1303 2051 y Fh(d)1322 2062 y Fo(\))1336 2053 y Fd(\003)1358 2051 y Fl(]\()r(~)-25 b Fj(\033)1415 2058 y Fi(x)1439 2051 y Fl(~)g Fj(\033)1463 2058 y Fi(y)1484 2051 y Fl(\))1502 1999 y Fe(\014)1502 2026 y(\014)1502 2054 y(\014)1530 2051 y Fl(=)13 b(0)1857 2062 y(\(5.7\))0 2172 y(for)i(nearest)g(neigh)o(b)q(ors.)20 b(Finally)l(,)d(the)e(quan)o(tit)o(y)g(giving)h(the)f(deca)o(y)g(of)g (the)g(p)q(oten)o(tial)h(is)117 2262 y(\026)-24 b Fj(c)p Fl(\()p Fj(m)p Fl(\))129 2375 y Fg(\024)13 b Fj(C)s(onst)81 b Fl(sup)320 2409 y Fh(x;y)q Ff(:)p Fd(j)p 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