Content-Type: multipart/mixed; boundary="-------------0009200614863" This is a multi-part message in MIME format. ---------------0009200614863 Content-Type: text/plain; name="00-372.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-372.keywords" Kawasaki dynamics, logarithmic Sobolev constant, equivalence of ensembles, concentration inequalities. ---------------0009200614863 Content-Type: application/postscript; name="totale.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="totale.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: totale.dvi %%Pages: 52 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o totale.ps totale.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.09.20:1302 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S 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Fp(\(2)p Fo(:)p Fp(10\))0 1955 y(A)23 b(second)g(relev)-5 b(an)m(t)23 b(quan)m(tit)m(y)g(is)f(the)h (logarithmic)e(Sob)s(olev)h(constan)m(t)i Fo(c)2503 1922 y Fl(\034)2503 1980 y(V)t(;N)2663 1955 y Fp(de\014ned)e(as)h(the)g (smallest)f(constan)m(t)0 2082 y Fo(c)31 b Fp(suc)m(h)f(that)1424 2209 y(En)m(t)1569 2170 y Fl(\034)1569 2232 y(V)t(;N)1707 2209 y Fp(\()p Fo(f)1797 2172 y Fs(2)1837 2209 y Fp(\))c Fk(\024)2008 2148 y Fo(c)p 2004 2189 46 4 v 2004 2272 a Fp(2)2076 2209 y Fk(E)2132 2172 y Fl(\034)2124 2232 y(V)t(;N)2261 2209 y Fp(\()p Fo(f)5 b(;)15 b(f)10 b Fp(\))1192 b(\(2)p Fo(:)p Fp(11\))0 2442 y(for)44 b(all)f(non)h(negativ)m(e)i (functions)d Fo(f)53 b Fp(with)43 b Fo(\027)1633 2409 y Fl(\034)1627 2467 y(V)t(;N)1765 2442 y Fp(\()p Fo(f)1855 2409 y Fs(2)1895 2442 y Fp(\))49 b(=)f(1,)h(where)44 b(En)m(t)2639 2403 y Fl(\034)2639 2465 y(V)t(;N)2777 2442 y Fp(\()p Fo(f)2867 2409 y Fs(2)2907 2442 y Fp(\))49 b(=)f Fo(\027)3161 2409 y Fl(\034)3155 2467 y(V)t(;N)3293 2442 y Fp(\()p 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y([Ce]\),)30 b(that,)f(v)m(ery)f(roughly)e(sp)s(eaking,)i (allo)m(ws)f(us)g(to)i(b)s(ound)c(the)j(m)m(ulticanonical)e(en)m(trop)m (y)j(in)d(terms)i(of)227 3511 y(the)33 b(m)m(ulticanonical)e(a)m(v)m (erage)k(of)d(the)h(sum)e(of)i(lo)s(cal)f(en)m(tropies)g(and)g(of)g (the)h(m)m(ulticanonical)e(v)-5 b(ariance.)227 3638 y(F)d(or)26 b(the)f(exp)s(ert)g(reader)g(w)m(e)g(just)f(men)m(tion)h(that)g(the)h (imp)s(ortance)e(of)h(suc)m(h)f(an)h(inequalit)m(y)e(resides)h(in)g (the)227 3766 y(fact)30 b(that)g(one)f(is)f(spared)g(from)g(the)h(cum)m (b)s(ersome)g(computation)g(of)g(quan)m(tities)f(lik)m(e)3245 3692 y Ff(\002)3283 3766 y Fk(r)3359 3780 y Fl(xy)3459 3766 y Fo(\027)3510 3733 y Fl(\034)3504 3791 y Fs(\003)3558 3766 y Fp(\()p Fo(f)3648 3733 y Fs(2)3688 3766 y Fp(\))3734 3708 y Fi(1)p 3735 3718 31 4 v 3735 3755 a(2)3796 3692 y Ff(\003)3834 3711 y Fs(2)3875 3766 y Fp(.)136 3955 y Fk(\017)46 b Fp(In)27 b(paragraphs)g Fk(x)q Fp(3)p Fo(:)p Fp(4)p Fo(;)k Fk(x)q Fp(3)p Fo(:)p Fp(5)p Fo(;)g 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5340 y(for)e(suc)m(h)g(a) h(measure)g(via)f(Hardy)g(inequalit)m(y)e(\(see)k(the)e(app)s(endix\).) 0 5558 y Fi(20)p Fh(=settembr)r(e=)p Fi(2000)f Fs([10])1204 b Fp(3.1)p eop %%Page: 11 11 11 10 bop 0 91 a Ft(The)34 b(general)f(setting.)0 311 y Fp(Throughout)c(all)h(this)f(section)h(our)g(setting)h(and)e (notation)i(will)d(b)s(e)i(as)g(follo)m(ws.)0 471 y(Fix)i Fo(\016)205 485 y Fs(0)274 471 y Fk(2)c Fp(\(0)p Fo(;)15 b Fp(1\))34 b(and)e(t)m(w)m(o)h(in)m(tegers)g Fo(j)1325 485 y Fs(max)1464 471 y Fo(;)15 b(l)35 b Fp(suc)m(h)c(that)i Fo(\016)2011 485 y Fs(0)2052 471 y Fo(j)2089 485 y Fs(max)2257 471 y Fo(<)28 b Fp(1.)47 b(Let)32 b Fo(L)2699 485 y Fs(1)2740 471 y Fo(;)15 b(:)g(:)g(:)i(;)e(L)3004 485 y Fl(j)3034 494 y Fi(max)3188 471 y Fp(b)s(e)32 b(large)g(m)m(ultiples)0 598 y(of)j(the)g(basic)f(length)g(scale)h Fo(l)r Fp(,)h(let)f Fo(L)e Fp(=)1429 530 y Ff(P)1525 626 y Fl(j)1578 598 y Fo(L)1640 612 y Fl(j)1713 598 y Fp(and)h(assume)h(that)g Fo(L)2476 612 y Fl(j)2547 598 y Fk(\025)d 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Fc(small)c(enough)h(indep)s(enden)m(t)f(of)i Fk(f)p Fo(\032)1295 2779 y Fl(j)1333 2765 y Fk(g)1378 2721 y Fl(j)1408 2730 y Fi(max)1378 2790 y Fl(i)p Fs(=1)1531 2765 y Fc(,)i(there)e(exists)f Fo(L)2159 2779 y Fs(0)2239 2765 y Fp(=)g Fo(L)2410 2779 y Fs(0)2451 2765 y Fp(\()p Fo(C)q(;)15 b(m;)g Fk(k)p Fp(\010)p Fk(k)p Fo(;)g(l)r(;)g(M)5 b(;)15 b(\016)3150 2779 y Fs(0)3194 2765 y Fo(;)g(\017)p Fp(\))39 b Fc(suc)m(h)g(that,)i(for)0 2892 y(an)m(y)h Fo(L)i Fk(\025)f Fo(L)465 2906 y Fs(0)547 2892 y Fc(the)f(follo)m(wing) e(holds.)73 b(Fix)41 b(t)m(w)m(o)i Fo(l)r Fc({regular)e(sets)h Fp(\001)2433 2906 y Fs(1)2474 2892 y Fo(;)15 b Fp(\001)2590 2906 y Fs(2)2672 2892 y Fc(in)40 b Fp(\003)i Fc(suc)m(h)f(that)h Fk(j)p Fp(\001)3419 2906 y Fl(i)3449 2892 y Fk(j)j(\024)e(j)p Fp(\003)p Fk(j)3746 2859 y Fs(1)p Fg(\000)p Fl(\017)3875 2892 y Fc(,)0 3019 y Fo(i)26 b Fp(=)f(1)p Fo(;)15 b Fp(2)p Fc(,)31 b(and)f(t)m(w)m(o)i(functions)d Fo(f)5 b(;)15 b(g)33 b Fc(with)c Fk(k)p Fo(f)10 b Fk(k)1598 3033 y Fg(1)1701 3019 y Fp(=)25 b Fk(k)p Fo(g)s Fk(k)1933 3033 y Fg(1)2037 3019 y Fp(=)g(1)p Fc(,)31 b Fp(\001)2310 2972 y Fs(\()p Fl(l)p Fs(\))2310 3047 y Fl(f)2419 3019 y Fp(=)25 b(\001)2591 3033 y Fs(1)2632 3019 y Fc(,)30 b Fp(\001)2763 2972 y Fs(\()p Fl(l)p Fs(\))2763 3030 y Fl(g)2873 3019 y Fp(=)25 b(\001)3045 3033 y Fs(2)3086 3019 y Fc(.)66 3175 y(\(1\))46 b(Assume)30 b Fo(j)601 3189 y Fs(max)766 3175 y Fp(=)24 b(1)p Fc(.)42 b(Then)1260 3455 y Fk(j)15 b Fo(\027)6 b Fp(\()p Fo(f)f(;)15 b(g)s Fp(\))g Fk(j)27 b(\024)e Fo(C)7 b Fp(\()p Fo(f)e(;)15 b(g)s Fp(\))p Fo(\032)2060 3354 y Ff(h)2148 3394 y Fp(1)p 2114 3434 114 4 v 2114 3518 a Fk(j)p Fp(\003)p Fk(j)2259 3455 y Fp(+)20 b Fo(e)2392 3418 y Fg(\000)p Fl(md)p Fs(\(\001)2640 3427 y Fi(1)2675 3418 y Fl(;)p Fs(\001)2757 3427 y Fi(2)2792 3418 y Fs(\))2824 3354 y Ff(i)227 3752 y Fc(where)30 b Fo(C)7 b Fp(\()p Fo(f)e(;)15 b(g)s Fp(\))26 b(=)f Fo(k)18 b Fk(j)p Fp(\001)1056 3766 y Fs(1)1097 3752 y Fk(j)1122 3719 y Fs(2)1178 3752 y Fk(j)p Fp(\001)1279 3766 y Fs(2)1320 3752 y Fk(j)1345 3719 y Fs(2)1386 3752 y Fc(.)41 b(Moreo)m(v)m(er)1610 4042 y Fk(j)15 b Fo(\027)6 b Fp(\()p Fo(f)k Fp(\))20 b Fk(\000)g Fo(\026)p Fp(\()p Fo(f)10 b Fp(\))15 b Fk(j)25 b(\024)g Fo(k)2339 3981 y Fk(j)p Fp(\001)2440 3995 y Fs(1)2481 3981 y Fk(j)p 2339 4021 168 4 v 2366 4105 a(j)p Fp(\003)p Fk(j)66 4361 y Fc(\(2\))46 b(Assume)30 b Fo(j)601 4375 y Fs(max)766 4361 y Fk(\025)24 b Fp(2)p Fc(.)42 b(Then)306 4735 y Fk(j)15 b Fo(\027)6 b Fp(\()p Fo(f)f(;)15 b(g)s Fp(\))g Fk(j)27 b(\024)e Fo(C)7 b Fp(\()p Fo(f)e(;)15 b(g)s Fp(\))p Fo(A)p Fp(\(\001)1223 4749 y Fs(1)1264 4735 y Fo(;)g Fp(\001)1380 4749 y Fs(2)1421 4735 y Fp(\))1471 4522 y Ff(8)1471 4604 y(>)1471 4631 y(<)1471 4795 y(>)1471 4822 y(:)1567 4590 y Fo(\027)6 b Fp(\()p Fk(j)p Fo(f)k Fk(j)p Fp(\))21 b Fk(^)f Fo(\027)6 b Fp(\()p Fk(j)p Fo(g)s Fk(j)p Fp(\))107 b Fc(if)30 b Fp(\001)2379 4604 y Fs(1)2449 4590 y Fc(and)g Fp(\001)2702 4604 y Fs(2)2773 4590 y Fc(are)h(b)s(oth)e(go)s(o)s(d)i(or)f(bad)1567 4726 y Fo(\027)6 b Fp(\()p Fk(j)p Fo(f)k Fk(j)p Fp(\))426 b Fc(if)30 b Fp(\001)2379 4740 y Fs(1)2449 4726 y Fc(is)g(bad)1567 4881 y Fo(\027)6 b Fp(\()p Fk(j)p Fo(g)s Fk(j)p Fp(\))435 b Fc(if)30 b Fp(\001)2379 4895 y Fs(2)2449 4881 y Fc(is)g(bad)594 5208 y(where)121 b Fo(A)p Fp(\(\001)1127 5222 y Fs(1)1168 5208 y Fo(;)15 b Fp(\001)1284 5222 y Fs(2)1325 5208 y Fp(\))26 b(=)1471 4995 y Ff(8)1471 5076 y(>)1471 5104 y(<)1471 5267 y(>)1471 5295 y(:)1607 5048 y Fs(1)p 1578 5063 93 4 v 1578 5115 a Fg(j)p Fs(\003)p Fg(j)1702 5084 y Fp(+)20 b Fo(e)1835 5051 y Fg(\000)p Fl(md)p Fs(\(\001)2083 5060 y Fi(1)2118 5051 y Fl(;)p Fs(\001)2200 5060 y Fi(2)2235 5051 y Fs(\))3009 5084 y Fc(if)29 b Fp(\001)3168 5098 y Fs(1)3239 5084 y Fc(or)h Fp(\001)3426 5098 y Fs(2)3497 5084 y Fc(is)f(go)s(o)s(d)1607 5273 y Fs(1)p 1578 5288 V 1578 5340 a Fg(j)p Fs(\003)p Fg(j)1697 5208 y Ff(\020)1763 5265 y Fg(j)1796 5249 y Fs(\026)1785 5265 y(\001)1845 5274 y Fi(2)1879 5265 y Fg(j)p 1763 5288 139 4 v 1763 5340 a(j)p Fs(\001)1845 5349 y Fi(2)1879 5340 y Fg(j)1912 5208 y Ff(\021)1966 5227 y Fs(2)2022 5208 y Ff(\020)2087 5265 y Fg(j)2121 5249 y Fs(\026)2109 5265 y(\001)2169 5274 y Fi(1)2204 5265 y Fg(j)p 2087 5288 V 2087 5340 a(j)p Fs(\001)2169 5349 y Fi(1)2204 5340 y Fg(j)2237 5208 y Ff(\021)2291 5227 y Fs(2)2352 5309 y Fp(+)20 b Fo(e)2485 5276 y Fg(\000)p Fl(md)p Fs(\(\001)2733 5285 y Fi(1)2768 5276 y Fl(;)p Fs(\001)2850 5285 y Fi(2)2885 5276 y Fs(\))3009 5309 y Fc(otherwise)0 5558 y Fi(20)p Fh(=settembr)r(e=)p Fi(2000)29 b Fs([12])1204 b Fp(3.3)p eop %%Page: 13 13 13 12 bop 66 91 a Fc(\(3\))46 b(Assume)30 b Fo(j)601 105 y Fs(max)766 91 y Fk(\025)24 b Fp(2)31 b Fc(and)f Fp(\001)1190 105 y Fs(1)1256 91 y Fk(\032)25 b Fp(\003)1415 105 y Fl(j)1453 91 y Fc(,)31 b Fo(j)g Fk(\024)25 b Fo(j)1710 105 y Fs(max)1849 91 y Fc(.)40 b(Then)436 483 y Fp(sup)401 560 y Fl(\034)442 541 y Fe(0)466 560 y Fg(2)p Fs(\012)567 569 y Fh(\034)623 483 y Fk(j)p Fo(\027)699 446 y Fl(\034)744 483 y Fp(\()p Fo(f)10 b Fp(\))21 b Fk(\000)e Fo(\027)1031 446 y Fl(\034)1072 418 y Fe(0)1101 483 y Fp(\()p Fo(f)10 b Fp(\))p Fk(j)26 b(\024)f Fo(k)1438 270 y Ff(8)1438 351 y(>)1438 379 y(<)1438 542 y(>)1438 570 y(:)1534 362 y Fk(j)p Fp(\001)1635 376 y Fs(1)1676 362 y Fk(j)1716 261 y Ff(h)1798 326 y Fs(1)p 1770 341 93 4 v 1770 394 a Fg(j)p Fs(\003)p Fg(j)1894 362 y Fp(+)2106 326 y Fs(1)p 1996 341 257 4 v 1996 395 a Fl(L)2046 376 y Fh(k)q Fe(\000)p Fh(n)p Fi(+1)2263 261 y Ff(i)3172 362 y Fc(if)k Fp(\001)3331 376 y Fs(1)3402 362 y Fc(is)g(go)s(o)s(d)1545 560 y Fg(j)p Fs(\001)1627 569 y Fi(1)1662 560 y Fg(j)p 1545 583 139 4 v 1568 636 a(j)p Fs(\003)p Fg(j)1694 531 y Ff(\000)1747 560 y Fg(j)1780 544 y Fs(\026)1769 560 y(\001)1829 569 y Fi(1)1863 560 y Fg(j)p 1747 583 V 1747 636 a(j)p Fs(\001)1829 645 y Fi(1)1863 636 y Fg(j)1896 531 y Ff(\001)1938 550 y Fs(2)1999 604 y Fp(+)20 b(max)2259 618 y Fl(j)t Fs(=)p Fl(n;n)p Fg(\006)p Fs(1)2588 503 y Ff(h)2655 558 y Fg(j)2688 542 y Fs(\026)2677 558 y(\001)2737 567 y Fi(1)2772 558 y Fg(\\)p Fs(\003)2871 567 y Fh(j)2904 558 y Fg(j)2937 510 y Fi(1)p 2937 518 31 4 v 2937 551 a(2)p 2642 583 356 4 v 2642 636 a Fl(L)2692 618 y Fh(j)2719 627 y Fi(max)2836 618 y Fe(\000)p Fh(j)s Fi(+1)3038 503 y Ff(i)3172 604 y Fc(if)29 b Fp(\001)3331 618 y Fs(1)3402 604 y Fc(is)g(bad)227 873 y(where)70 b Fp(\012)596 887 y Fl(\034)711 873 y Fc(is)f(the)h(set)h(of)f(con\014gurations)f Fo(\034)2039 840 y Fg(0)2135 873 y Fc(that)h(coincide)g(with)e Fo(\034)80 b Fc(in)69 b(the)h(half)f(space)227 1000 y Fk(f)p Fo(x)26 b Fk(2)f Fm(Z)501 967 y Fl(d)594 1000 y Fp(:)56 b Fo(x)727 1014 y Fl(d)795 1000 y Fo(<)25 b(L)p Fk(g)p Fc(.)0 1288 y Fj(R)-5 b(emark.)41 b Fp(Actually)27 b(the)i(\014rst)e(part)h(of)h (the)f(prop)s(osition)e(holds)h(in)g(a)h(m)m(uc)m(h)g(more)h(general)f (geometric)h(con)m(text)0 1416 y(\(see)i Fk(x)q Fp(7)p Fo(:)p Fp(3)g(of)g([CM1]\).)0 1651 y(F)-8 b(or)31 b(future)d(purp)s (oses)g(the)i(next)g(result)f(is)g(stated)h(in)f(a)h(sligh)m(tly)e (more)i(general)f(form.)40 b(The)30 b(pro)s(of)f(is)f(giv)m(en)i(in)0 1778 y([CM2].)0 1960 y Fq(Prop)s(osition)44 b(3.3.)53 b Fc(Assume)36 b(condition)g Fo(U)10 b(S)5 b(M)10 b(T)j Fp(\()p Fo(C)q(;)i(m;)g(l)r Fp(\))38 b Fc(and)e(let)h Fo(f)46 b Fc(b)s(e)36 b(suc)m(h)h(that)g Fk(j)p Fp(\003)3242 1974 y Fl(j)3306 1960 y Fk(n)25 b Fp(\001)3452 1974 y Fl(f)3499 1960 y Fk(j)36 b(\025)g Fo(\016)3707 1974 y Fs(0)3748 1960 y Fk(j)p Fp(\003)3836 1974 y Fl(j)3875 1960 y Fk(j)0 2087 y Fc(for)31 b(an)m(y)h Fo(j)h Fp(=)27 b(1)p Fo(;)15 b(:)g(:)g(:)i(j)725 2101 y Fs(max)864 2087 y Fc(.)45 b(Then,)31 b(for)g(an)m(y)h Fo(l)h Fc(large)f(enough,)g (there)g(exists)f(a)h(constan)m(t)h Fo(A)e Fc(dep)s(ending)e(only)i(on) 0 2214 y Fo(C)q(;)15 b(m;)g Fk(k)p Fp(\010)p Fk(k)p Fo(;)g(l)r(;)g(j) 528 2228 y Fs(max)669 2214 y Fo(;)g(\016)749 2228 y Fs(0)821 2214 y Fc(suc)m(h)30 b(that)1619 2341 y Fo(\027)6 b Fp(\()p Fk(j)p Fo(f)k Fk(j)p Fp(\))26 b Fk(\024)f Fo(A)15 b(\026)p Fp(\()p Fk(j)p Fo(f)10 b Fk(j)p Fp(\))0 2530 y Fc(In)30 b(particular)1437 2657 y Fp(V)-8 b(ar)1579 2671 y Fl(\027)1624 2657 y Fp(\()p Fo(f)5 b(;)15 b(f)10 b Fp(\))25 b Fk(\024)g Fo(A)30 b Fp(V)-8 b(ar)2200 2671 y Fl(\026)2249 2657 y Fp(\()p Fo(f)5 b(;)15 b(f)10 b Fp(\))0 2847 y(W)-8 b(e)31 b(conclude)d(this)h(paragraph)g(with)f(a)i(\014nal)e(result)g (that)i(pla)m(ys)f(a)h(crucial)e(role)i(in)e(our)h(approac)m(h)g(\(see) i Fk(x)p Fp(3)p Fo(:)p Fp(3\).)0 2974 y(F)-8 b(or)23 b(simplicit)m(y)c(w)m(e)k(consider)e(only)g(the)h(t)m(w)m(o)h (dimensional)c(case)k(and)f(at)h(the)f(end)f(w)m(e)i(explain)d(ho)m(w)i (to)h(generalize)0 3101 y(it)30 b(to)h(higher)e(dimensions.)0 3256 y(Assume)k(that)h(the)g(n)m(um)m(b)s(er)f(of)g(la)m(y)m(ers)i Fo(j)1438 3270 y Fs(max)1610 3256 y Fp(is)e(greater)i(than)e(4,)i (\014x)f(3)d Fk(\024)f Fo(j)2684 3270 y Fs(0)2756 3256 y Fk(\024)h Fo(j)2895 3270 y Fs(max)3067 3256 y Fp(and)i(let)h Fo(A)d Fp(=)f Fk([)3643 3212 y Fl(j)3673 3221 y Fi(0)3643 3281 y Fl(j)t Fs(=1)3773 3256 y Fp(\003)3836 3270 y Fl(j)3875 3256 y Fp(,)0 3383 y Fo(B)35 b Fp(=)30 b Fk([)266 3340 y Fl(j)296 3349 y Fi(max)266 3409 y Fl(j)296 3418 y Fi(0)332 3409 y Fg(\000)p Fs(1)429 3383 y Fp(\003)492 3397 y Fl(j)564 3383 y Fp(and)j Fo(S)j Fp(=)30 b(\003)1000 3397 y Fl(j)1030 3406 y Fi(0)1066 3397 y Fg(\000)p Fs(1)1185 3383 y Fk([)22 b Fp(\003)1331 3397 y Fl(j)1361 3406 y Fi(0)1402 3383 y Fp(.)50 b(Let)34 b(also)f Fo(\027)1873 3397 y Fl(X)1942 3383 y Fp(\()p Fk(\001)p Fp(\))f(:=)f Fo(\027)6 b Fp(\()p Fk(\001)15 b(j)g(F)2427 3397 y Fl(X)2492 3379 y Fh(c)2531 3383 y Fp(\))34 b(for)f Fo(X)38 b Fp(=)30 b Fo(A;)15 b(B)5 b(;)15 b(S)5 b Fp(.)51 b(Notice)35 b(that)f Fo(\027)6 b Fp({)0 3511 y(almost)30 b(surely)f Fo(N)632 3525 y Fl(A)692 3511 y Fp(,)h Fo(N)820 3525 y Fl(B)913 3511 y Fp(and)g Fo(N)1163 3525 y Fl(S)1246 3511 y Fp(are)h(constan)m(t.)0 3692 y Fq(Lemma)e(3.4.)53 b Fc(Assume)26 b(condition)g Fo(U)10 b(S)5 b(M)10 b(T)j Fp(\()p Fo(C)q(;)i(m;)g(l)r 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3827 y Fs(\003)747 3836 y Fh(j)811 3813 y Fp(=)25 b Fo(N)980 3827 y Fl(j)1018 3813 y Fp(.)41 b(Let)31 b Fo(u)25 b Fp(=)g Fk(b)p Fo(\032)1507 3827 y Fl(j)1546 3813 y Fk(j)p Fo(V)20 b Fk(jc)p Fp(,)32 b(where)d Fo(\032)2075 3827 y Fl(j)2139 3813 y Fp(=)2264 3767 y Fl(N)2322 3776 y Fh(j)p 2246 3792 127 4 v 2246 3844 a Fg(j)p Fs(\003)2318 3853 y Fh(j)2351 3844 y Fg(j)2384 3813 y Fp(,)h(and)g(let,)h(for)f Fo(n)25 b Fk(2)f Fp(\()q Fo(n)3168 3827 y Fs(min)3292 3813 y Fo(;)15 b(n)3387 3827 y Fs(max)3526 3813 y Fp(\),)460 4204 y Fo(A)p Fp(\()p Fo(n)p Fp(\))25 b(=)774 3990 y Ff(8)774 4072 y(>)774 4099 y(<)774 4263 y(>)774 4290 y(:)902 4028 y Fs(1)p 881 4043 78 4 v 881 4095 a Fl(c)913 4104 y Fh(n)980 4020 y Fl(\015)t Fs(\()p Fl(n)p Fg(\000)p Fs(1\))p 980 4043 237 4 v 1027 4095 a Fl(\015)t Fs(\()p Fl(n)p Fs(\))1243 3996 y Ff(P)1358 4076 y Fl(x)p Fg(2)p Fl(V)1350 4135 y(z)s Fg(2)p Fl(W)1545 4064 y Fo(\027)1611 3990 y Ff(\000)1653 4064 y Fp(\()p Fk(r)1764 4078 y Fl(z)s(x)1847 4064 y Fo(f)1902 4031 y Fs(2)1942 4064 y Fp(\)1)-25 b(I)2030 4078 y Fl(E)2083 4087 y Fh(z)r(x)2162 4064 y Fo(e)2204 4031 y Fg(\000r)2322 4040 y Fh(xz)2395 4031 y Fl(H)2455 4040 y Fi(\003)2520 4064 y Fk(j)16 b Fo(n)k Fk(\000)f Fp(1)2771 3990 y Ff(\001)2929 4064 y Fp(if)29 b Fo(n)c Fk(\024)g Fo(u)992 4278 y Fs(1)p 881 4293 258 4 v 881 4345 a Fl(c)913 4323 y Fe(0)913 4369 y Fh(N)5 b Fe(\000)p Fh(n)p Fi(+1)1207 4269 y Fl(\015)t Fs(\()p Fl(n)p Fs(\))p 1161 4293 237 4 v 1161 4345 a Fl(\015)t Fs(\()p Fl(n)p Fg(\000)p Fs(1\))1424 4245 y Ff(P)1539 4326 y Fl(x)p Fg(2)p Fl(V)1531 4385 y(z)s Fg(2)p Fl(W)1726 4314 y Fo(\027)1792 4240 y Ff(\000)1833 4314 y Fp(\()p Fk(r)1944 4328 y Fl(xz)2028 4314 y Fo(f)2083 4281 y Fs(2)2123 4314 y Fp(\)1)-25 b(I)2211 4328 y Fl(E)2264 4337 y Fh(xz)2342 4314 y Fo(e)2384 4281 y Fg(\000r)2502 4290 y Fh(xz)2576 4281 y Fl(H)2636 4290 y Fi(\003)2701 4314 y Fk(j)15 b Fo(n)2796 4240 y Ff(\001)2929 4314 y Fp(otherwise,)454 4794 y Fo(B)5 b Fp(\()p Fo(n)p Fp(\))25 b(=)774 4580 y Ff(8)774 4662 y(>)774 4690 y(<)774 4853 y(>)774 4880 y(:)902 4618 y Fs(1)p 881 4633 78 4 v 881 4686 a Fl(c)913 4695 y Fh(n)980 4610 y Fl(\015)t Fs(\()p Fl(n)p Fg(\000)p Fs(1\))p 980 4633 237 4 v 1027 4686 a Fl(\015)t Fs(\()p Fl(n)p Fs(\))1243 4586 y Ff(P)1358 4666 y Fl(x)p Fg(2)p Fl(V)1350 4726 y(z)s Fg(2)p Fl(W)1545 4654 y Fo(\027)1611 4581 y Ff(\000\000)1694 4654 y Fo(e)1736 4621 y Fg(\000r)1854 4630 y Fh(xz)1928 4621 y Fl(H)1988 4630 y Fi(\003)2058 4654 y Fk(\000)20 b Fp(1)2194 4581 y Ff(\001)2251 4654 y Fp(1)-25 b(I)2304 4668 y Fl(E)2357 4677 y Fh(z)r(x)2436 4654 y Fo(;)15 b(f)2531 4621 y Fs(2)2586 4654 y Fk(j)h Fo(n)j Fk(\000)h Fp(1)2837 4581 y Ff(\001)2995 4654 y Fp(if)29 b Fo(n)c Fk(\024)g Fo(u)992 4868 y Fs(1)p 881 4883 258 4 v 881 4935 a Fl(c)913 4914 y Fe(0)913 4959 y Fh(N)5 b Fe(\000)p Fh(n)p Fi(+1)1207 4860 y Fl(\015)t Fs(\()p Fl(n)p Fs(\))p 1161 4883 237 4 v 1161 4935 a Fl(\015)t Fs(\()p Fl(n)p Fg(\000)p Fs(1\))1424 4836 y Ff(P)1539 4916 y Fl(x)p Fg(2)p Fl(V)1531 4975 y(z)s Fg(2)p Fl(W)1726 4904 y Fo(\027)1792 4830 y Ff(\000)o(\000)1875 4904 y Fo(e)1917 4871 y Fg(\000r)2035 4880 y Fh(xz)2108 4871 y Fl(H)2168 4880 y Fi(\003)2239 4904 y Fk(\000)20 b Fp(1)2375 4830 y Ff(\001)2432 4904 y Fp(1)-25 b(I)2485 4918 y Fl(E)2538 4927 y Fh(xz)2616 4904 y Fo(;)15 b(f)2711 4871 y Fs(2)2767 4904 y Fk(j)g Fo(n)2862 4830 y Ff(\001)2995 4904 y Fp(otherwise.)3713 4499 y(\(3)p Fo(:)p Fp(9\))0 5213 y(With)42 b(this)f(de\014nition,)i(and)e(using)g(\()1353 5148 y Fk(p)p 1429 5148 52 4 v 65 x Fo(x)28 b Fk(\000)1608 5156 y(p)p 1684 5156 48 4 v 57 x Fo(y)s Fp(\))1767 5180 y Fs(2)1853 5213 y Fp(=)1968 5112 y Ff(\020)2093 5173 y Fl(x)p Fg(\000)p Fl(y)p 2033 5192 256 4 v 2033 5202 a Fg(p)p 2093 5202 42 4 v 46 x Fl(x)p Fs(+)2190 5208 y Fg(p)p 2250 5208 40 4 v 40 x Fl(y)2300 5112 y Ff(\021)2354 5131 y Fs(2)2440 5213 y Fk(\024)2566 5169 y Fs(\()p Fl(x)p Fg(\000)p Fl(y)r Fs(\))2759 5142 y Fi(2)p 2566 5192 230 4 v 2617 5244 a Fl(x)p Fg(_)p Fl(y)2807 5213 y Fp(,)45 b(w)m(e)e(ha)m(v)m(e)g(immediately)e(the)0 5340 y(follo)m(wing)29 b(corollary)0 5558 y Fi(20)p Fh(=settembr)r(e=)p Fi(2000)g Fs([16])1204 b Fp(3.7)p eop %%Page: 17 17 17 16 bop 0 91 a Fq(Corollary)35 b(3.7.)53 b Fc(In)30 b(the)g(same)h(setting)f(of)h(prop)s(osition)d(3.6)306 257 y Ff(X)349 447 y Fl(n)453 343 y Fo(\015)5 b Fp(\()p Fo(n)p Fp(\))20 b Fk(^)g Fo(\015)5 b Fp(\()p Fo(n)20 b Fk(\000)g Fp(1\))1079 242 y Ff(\020)1135 261 y(p)p 1225 261 327 4 v 1225 343 a Fo(\027)6 b Fp(\()p Fo(f)1366 317 y Fs(2)1422 343 y Fk(j)15 b Fo(n)p Fp(\))20 b Fk(\000)1663 261 y Ff(p)p 1754 261 484 4 v 82 x Fo(\027)6 b Fp(\()p Fo(f)1895 317 y Fs(2)1950 343 y Fk(j)16 b Fo(n)j Fk(\000)h Fp(1\))2252 242 y Ff(\021)2307 261 y Fs(2)2373 343 y Fk(\024)25 b Fp(2)2544 257 y Ff(X)2588 447 y Fl(n)2691 343 y Fo(\015)2738 357 y Fl(f)2785 343 y Fp(\()p Fo(n)p Fp(\))2925 269 y Ff(\000)2967 343 y Fo(A)p Fp(\()p Fo(n)p Fp(\))3160 305 y Fs(2)3222 343 y Fp(+)20 b Fo(B)5 b Fp(\()p Fo(n)p Fp(\))3512 305 y Fs(2)3552 269 y Ff(\001)0 660 y Fc(where)30 b Fo(\015)310 674 y Fl(f)357 660 y Fp(\()p Fo(n)p Fp(\))c(:=)781 616 y Fl(\015)t Fs(\()p Fl(n)p Fs(\))p Fg(^)p Fl(\015)t Fs(\()p Fl(n)p Fg(\000)p Fs(1\))p 640 639 712 4 v 640 691 a Fl(\027)t Fs(\()p Fl(f)750 673 y Fi(2)802 691 y Fg(j)15 b Fl(n)p Fs(\))e Fg(_)f Fl(\027)t Fs(\()p Fl(f)1096 673 y Fi(2)1148 691 y Fg(j)j Fl(n)p Fg(\000)p Fs(1\))0 1057 y Ft(3.5)35 b(Bound)e(on)642 989 y Ff(P)738 1084 y Fl(n)803 1057 y Fo(\015)850 1071 y Fl(f)897 1057 y Fp(\()p Fo(n)p Fp(\))p Fo(A)p Fp(\()p Fo(n)p Fp(\))1215 1024 y Fs(2)1257 1057 y Ft(.)0 1256 y Fp(Here)e(w)m(e)g(sho)m(w)f(ho)m(w)g(to)h(b)s(ound)e(from)h(ab)s(o)m (v)m(e)h(the)g(term)2001 1188 y Ff(P)2097 1283 y Fl(n)2162 1256 y Fo(\015)2209 1270 y Fl(f)2256 1256 y Fp(\()p Fo(n)p Fp(\))p Fo(A)p Fp(\()p Fo(n)p Fp(\))2574 1223 y Fs(2)2646 1256 y Fp(of)f(corollary)g(3.7.)0 1431 y Fq(Prop)s(osition)36 b(3.8.)1266 1534 y Ff(X)1309 1725 y Fl(n)1412 1620 y Fo(\015)1459 1634 y Fl(f)1506 1620 y Fp(\()p Fo(n)p Fp(\))15 b Fo(A)p Fp(\()p Fo(n)p Fp(\))1839 1583 y Fs(2)1906 1620 y Fk(\024)25 b Fo(k)2146 1559 y(L)2208 1526 y Fs(2)p 2078 1599 238 4 v 2078 1683 a Fo(\032)2125 1697 y Fl(j)2164 1683 y Fk(j)p Fp(\003)2252 1697 y Fl(j)2290 1683 y Fk(j)2327 1620 y(E)2375 1634 y Fl(\027)2420 1620 y Fp(\()p Fo(f)5 b(;)15 b(f)10 b Fp(\))0 1896 y Fj(Pr)-5 b(o)g(of.)30 b Fp(Let)f Fo(u)c Fp(=)g Fk(b)p Fo(\032)689 1910 y Fl(j)728 1896 y Fk(j)p Fo(V)20 b Fk(jc)29 b Fp(and)f(assume,)h(without)f(loss)g (of)g(generalit)m(y)-8 b(,)30 b Fo(\032)2525 1910 y Fl(j)2589 1896 y Fk(\024)2695 1861 y Fs(1)p 2695 1876 37 4 v 2695 1928 a(2)2771 1896 y Fp(and)e Fo(n)d Fk(\024)g Fo(u)p Fp(.)40 b(Then)27 b(w)m(e)i(observ)m(e)0 2023 y(that,)i(b)s(ecause)g (of)f(the)h(conserv)-5 b(ation)30 b(la)m(w,)1053 2220 y(1)p 1031 2261 89 4 v 1031 2344 a Fo(c)1070 2358 y Fl(n)1172 2220 y Fo(\015)5 b Fp(\()p Fo(n)21 b Fk(\000)e Fp(1\))p 1172 2261 334 4 v 1251 2344 a Fo(\015)5 b Fp(\()p Fo(n)p Fp(\))1560 2196 y Ff(X)1551 2391 y Fl(x)p Fg(2)p Fl(V)1543 2450 y(z)s Fg(2)p Fl(W)1734 2282 y Fo(\027)1785 2208 y Ff(\000)1826 2282 y Fo(e)1868 2244 y Fg(\000r)1986 2253 y Fh(xz)2060 2244 y Fl(H)2120 2253 y Fi(\003)2170 2282 y Fp(1)-25 b(I)2223 2296 y Fl(E)2276 2305 y Fh(z)r(x)2369 2282 y Fk(j)16 b Fo(n)j Fk(\000)h Fp(1)2620 2208 y Ff(\001)2688 2282 y Fp(=)25 b(1)g Fo(;)0 2638 y Fp(so)31 b(that)936 2804 y Fo(e)978 2767 y Fg(\000)p Fs(2)p Fg(k)p Fs(\010)p Fg(k)1385 2743 y Fp(1)p 1243 2783 330 4 v 1243 2866 a Fo(c)1282 2835 y Fg(0)1282 2893 y Fl(N)1340 2902 y Fh(j)1374 2893 y Fg(\000)p Fl(n)p Fs(+1)1609 2804 y Fk(\024)1737 2743 y Fp(1)p 1716 2783 89 4 v 1716 2866 a Fo(c)1755 2880 y Fl(n)1857 2743 y Fo(\015)5 b Fp(\()p Fo(n)20 b Fk(\000)g Fp(1\))p 1857 2783 334 4 v 1935 2866 a Fo(\015)5 b Fp(\()p Fo(n)p Fp(\))2227 2804 y Fk(\024)25 b Fo(e)2365 2767 y Fs(2)p Fg(k)p Fs(\010)p Fg(k)2715 2743 y Fp(1)p 2573 2783 330 4 v 2573 2866 a Fo(c)2612 2835 y Fg(0)2612 2893 y Fl(N)2670 2902 y Fh(j)2704 2893 y Fg(\000)p Fl(n)p Fs(+1)2939 2804 y Fo(:)0 3070 y Fp(No)m(w,)k(using)d(t)m(wice)j(the)f (Cauc)m(h)m(y-Sc)m(h)m(w)m(arz)h(inequalit)m(y)d(together)j(with)e(the) h(iden)m(tit)m(y)f Fo(a)3041 3037 y Fs(2)3097 3070 y Fk(\000)15 b Fo(b)3222 3037 y Fs(2)3287 3070 y Fp(=)25 b(\()p Fo(a)15 b Fk(\000)g Fo(b)p Fp(\)\()p Fo(a)g Fp(+)g Fo(b)p Fp(\))0 3197 y(on)30 b Fk(r)202 3211 y Fl(xz)285 3197 y Fo(f)340 3164 y Fs(2)411 3197 y Fp(giv)m(es)265 3455 y Fo(A)p Fp(\()p Fo(n)p Fp(\))458 3418 y Fs(2)525 3455 y Fp(=)621 3355 y Ff(h)696 3394 y Fp(1)p 674 3435 89 4 v 674 3518 a Fo(c)713 3532 y Fl(n)785 3394 y Fo(\015)5 b Fp(\()p Fo(n)20 b Fk(\000)g Fp(1\))p 785 3435 334 4 v 863 3518 a Fo(\015)5 b Fp(\()p Fo(n)p Fp(\))1279 3369 y Ff(X)1145 3568 y Fs(\()p Fl(x;z)s Fs(\))p Fg(2)p Fl(V)16 b Fg(\002)p Fl(W)1560 3455 y Fo(\027)1611 3382 y Ff(\000)1667 3455 y Fo(e)1709 3418 y Fg(\000r)1827 3427 y Fh(xz)1901 3418 y Fl(H)1970 3455 y Fk(r)2046 3469 y Fl(xz)2128 3455 y Fo(f)2183 3418 y Fs(2)2239 3455 y Fk(j)f Fo(n)20 b Fk(\000)g Fp(1)p Fo(;)15 b(E)2597 3469 y Fl(z)s(x)2696 3382 y Ff(\001)2738 3455 y Fo(\027)2789 3382 y Ff(\000)2845 3455 y Fo(E)2912 3469 y Fl(z)s(x)3011 3455 y Fk(j)g Fo(n)20 b Fk(\000)g Fp(1)3277 3382 y Ff(\001)3319 3355 y(i)3362 3373 y Fs(2)525 3777 y Fk(\024)121 b Fo(k)935 3716 y Fp(1)p 793 3756 330 4 v 793 3839 a Fo(c)832 3808 y Fg(0)832 3866 y Fl(N)890 3875 y Fh(j)924 3866 y Fg(\000)p Fl(n)p Fs(+1)1282 3691 y Ff(X)1148 3890 y Fs(\()p Fl(x;z)s Fs(\))p Fg(2)p Fl(V)16 b Fg(\002)p Fl(W)1563 3777 y Fo(\027)1614 3703 y Ff(\000)1670 3777 y Fp(\()p Fk(r)1781 3791 y Fl(xz)1865 3777 y Fo(f)10 b Fp(\))1955 3739 y Fs(2)2010 3777 y Fk(j)16 b Fo(n)j Fk(\000)h Fp(1)p Fo(;)15 b(E)2368 3791 y Fl(z)s(x)2467 3703 y Ff(\001)2509 3777 y Fo(\027)2560 3703 y Ff(\000)2617 3777 y Fo(E)2684 3791 y Fl(z)s(x)2782 3777 y Fk(j)g Fo(n)20 b Fk(\000)g Fp(1)3048 3703 y Ff(\001)568 4108 y Fk(\002)742 4047 y Fp(1)p 720 4087 89 4 v 720 4171 a Fo(c)759 4185 y Fl(n)831 4047 y Fo(\015)5 b Fp(\()p Fo(n)20 b Fk(\000)g Fp(1\))p 831 4087 334 4 v 909 4171 a Fo(\015)5 b Fp(\()p Fo(n)p Fp(\))1325 4022 y Ff(X)1191 4221 y Fs(\()p Fl(x;z)s Fs(\))p Fg(2)p Fl(V)15 b Fg(\002)p Fl(W)1605 4108 y Fo(\027)1656 4035 y Ff(\000)1713 4108 y Fo(f)1768 4071 y Fs(2)1828 4108 y Fp(+)20 b(\()p Fo(T)2007 4122 y Fl(xz)2091 4108 y Fo(f)10 b Fp(\))2181 4071 y Fs(2)2236 4108 y Fk(j)16 b Fo(n)j Fk(\000)h Fp(1)p Fo(;)15 b(E)2594 4122 y Fl(z)s(x)2693 4035 y Ff(\001)2735 4108 y Fo(\027)2786 4035 y Ff(\000)2843 4108 y Fo(E)2910 4122 y Fl(z)s(x)3008 4108 y Fk(j)g Fo(n)20 b Fk(\000)g Fp(1)3274 4035 y Ff(\001)3668 3809 y Fp(\(3)p Fo(:)p Fp(10\))0 4426 y(W)-8 b(e)32 b(recall)d(that)i Fo(\027)21 b Fp(\()p Fk(\001)16 b(j)f Fo(n)20 b Fk(\000)g Fp(1\))q({almost)31 b(surely)-8 b(,)1647 4358 y Ff(P)1762 4438 y Fl(x)p Fg(2)p Fl(V)1754 4497 y(z)s Fg(2)p Fl(W)1949 4426 y Fp(1)-25 b(I)2002 4440 y Fl(E)2055 4449 y Fh(z)r(x)2159 4426 y Fp(=)25 b Fo(c)2294 4393 y Fg(0)2294 4451 y Fl(N)7 b Fg(\000)p Fl(n)p Fs(+1)2557 4426 y Fp(.)41 b(So,)30 b(one)h(can)f(see)h(that)958 4681 y(1)p 936 4722 89 4 v 936 4805 a Fo(c)975 4819 y Fl(n)1047 4681 y Fo(\015)5 b Fp(\()p Fo(n)20 b Fk(\000)g Fp(1\))p 1047 4722 334 4 v 1125 4805 a Fo(\015)5 b Fp(\()p Fo(n)p Fp(\))1541 4657 y Ff(X)1407 4856 y Fs(\()p Fl(x;z)s Fs(\))p Fg(2)p Fl(V)15 b Fg(\002)p Fl(W)1821 4743 y Fo(\027)1872 4669 y Ff(\000)1929 4743 y Fo(f)1984 4705 y Fs(2)2039 4743 y Fk(j)g Fo(n)20 b Fk(\000)g Fp(1)p Fo(;)15 b(E)2397 4757 y Fl(z)s(x)2496 4669 y Ff(\001)2538 4743 y Fo(\027)2589 4669 y Ff(\000)2645 4743 y Fo(E)2712 4757 y Fl(z)s(x)2811 4743 y Fk(j)g Fo(n)20 b Fk(\000)g Fp(1)3077 4669 y Ff(\001)1417 5064 y Fk(\024)25 b Fo(k)1716 5003 y Fp(1)p 1574 5044 330 4 v 1574 5127 a Fo(c)1613 5096 y Fg(0)1613 5154 y Fl(N)1671 5163 y Fh(j)1705 5154 y Fg(\000)p Fl(n)p Fs(+1)2063 4978 y Ff(X)1929 5177 y Fs(\()p Fl(x;z)s Fs(\))p Fg(2)p Fl(V)16 b Fg(\002)p Fl(W)2344 5064 y Fo(\027)2395 4991 y Ff(\000)2451 5064 y Fo(f)2506 5027 y Fs(2)2546 5064 y Fp(1)-25 b(I)2599 5078 y Fl(E)2652 5087 y Fh(z)r(x)2746 5064 y Fk(j)15 b Fo(n)20 b Fk(\000)g Fp(1)3012 4991 y Ff(\001)1417 5345 y Fp(=)25 b Fo(k)s(\027)1614 5271 y Ff(\000)1670 5345 y Fo(f)1725 5307 y Fs(2)1781 5345 y Fk(j)15 b Fo(n)20 b Fk(\000)g Fp(1)2047 5271 y Ff(\001)2114 5345 y Fo(:)3668 5017 y Fp(\(3)p Fo(:)p Fp(11\))0 5558 y Fi(20)p Fh(=settembr)r(e=)p Fi(2000)29 b Fs([17])1204 b Fp(3.8)p eop %%Page: 18 18 18 17 bop 0 91 a Fp(On)20 b(the)h(other)f(hand,)i(using)d(the)i(c)m (hange)h(of)f(v)-5 b(ariable)19 b Fo(\033)29 b Fk(7!)c Fo(\033)2104 58 y Fl(xz)2208 91 y Fp(and)20 b(the)g(equalit)m(y)h Fo(\027)6 b Fp(\()p Fo(f)k(g)s Fp(\))25 b(=)g Fo(\027)6 b Fp(\()p Fo(f)f(;)15 b(g)s Fp(\))q(+)q Fo(\027)6 b Fp(\()p Fo(f)k Fp(\))p Fo(\027)c Fp(\()p Fo(g)s Fp(\),)0 218 y(w)m(e)31 b(get)782 340 y(1)p 760 381 89 4 v 760 464 a Fo(c)799 478 y Fl(n)871 340 y Fo(\015)5 b Fp(\()p Fo(n)20 b Fk(\000)g Fp(1\))p 871 381 334 4 v 949 464 a Fo(\015)5 b Fp(\()p Fo(n)p Fp(\))1365 316 y Ff(X)1231 515 y Fs(\()p Fl(x;z)s Fs(\))p Fg(2)p Fl(V)16 b Fg(\002)p Fl(W)1646 402 y Fo(\027)1697 328 y Ff(\000)1753 402 y Fp(\()p Fo(T)1841 416 y Fl(xz)1925 402 y Fo(f)10 b Fp(\))2015 364 y Fs(2)2070 402 y Fk(j)16 b Fo(n)k Fk(\000)f Fp(1)p Fo(;)c(E)2428 416 y Fl(z)s(x)2528 328 y Ff(\001)2569 402 y Fo(\027)2620 328 y Ff(\000)2677 402 y Fo(E)2744 416 y Fl(z)s(x)2842 402 y Fk(j)h Fo(n)j Fk(\000)h Fp(1)3108 328 y Ff(\001)775 733 y Fp(=)903 672 y(1)p 882 712 89 4 v 882 795 a Fo(c)921 809 y Fl(n)992 672 y Fo(\015)5 b Fp(\()p Fo(n)20 b Fk(\000)g Fp(1\))p 992 712 334 4 v 1070 795 a Fo(\015)5 b Fp(\()p Fo(n)p Fp(\))1486 647 y Ff(X)1352 846 y Fs(\()p Fl(x;z)s Fs(\))p Fg(2)p Fl(V)16 b Fg(\002)p Fl(W)1767 733 y Fo(\027)1818 659 y Ff(\000)1874 733 y Fp(\()p Fo(T)1962 747 y Fl(xz)2046 733 y Fo(f)10 b Fp(\))2136 696 y Fs(2)2176 733 y Fp(1)-25 b(I)2229 747 y Fl(E)2282 756 y Fh(z)r(x)2376 733 y Fk(j)15 b Fo(n)20 b Fk(\000)g Fp(1)2642 659 y Ff(\001)775 1055 y Fk(\024)903 993 y Fp(1)p 882 1034 89 4 v 882 1117 a Fo(c)921 1131 y Fl(n)1131 968 y Ff(X)996 1168 y Fs(\()p Fl(x;z)s Fs(\))p Fg(2)p Fl(V)c Fg(\002)p Fl(W)1411 1055 y Fo(\027)1462 981 y Ff(\000)1519 1055 y Fo(f)1574 1017 y Fs(2)1614 1055 y Fp(1)-25 b(I)1667 1069 y Fl(E)1720 1078 y Fh(xz)1798 1055 y Fo(e)1840 1017 y Fg(\000r)1958 1026 y Fh(z)r(x)2032 1017 y Fl(H)2116 1055 y Fk(j)15 b Fo(n)2226 981 y Ff(\001)775 1335 y Fk(\024)25 b Fo(k)s(\027)972 1261 y Ff(\000)1028 1335 y Fo(f)1083 1297 y Fs(2)1139 1335 y Fk(j)15 b Fo(n)1249 1261 y Ff(\001)3668 842 y Fp(\(3)p Fo(:)p Fp(12\))0 1534 y(where)30 b(for)g(the)h(last)f (inequalit)m(y)-8 b(,)29 b(w)m(e)i(exploited)e(the)i(fact)g(that)g Fo(\027)21 b Fp(\()p Fk(\001)16 b(j)f Fo(n)p Fp(\){almost)31 b(surely)3105 1466 y Ff(P)3220 1546 y Fl(x)p Fg(2)p Fl(V)3212 1605 y(z)s Fg(2)p Fl(W)3407 1534 y Fp(1)-25 b(I)3460 1548 y Fl(E)3513 1557 y Fh(xz)3617 1534 y Fp(=)25 b Fo(c)3752 1548 y Fl(n)3801 1534 y Fp(.)0 1715 y(No)m(w,)31 b(since)f Fo(c)494 1682 y Fg(0)494 1737 y Fl(n)574 1715 y Fp(is)g(increasing)f (in)g Fo(n)p Fp(,)h(w)m(e)h(obtain)e(from)h(\(3.10\),)j(\(3.11\))g(and) d(\(3.12\))i(that)338 2002 y Fo(A)p Fp(\()p Fo(n)p Fp(\))531 1964 y Fs(2)597 2002 y Fk(\024)25 b Fo(k)895 1940 y Fp(1)p 754 1981 328 4 v 754 2064 a Fo(c)793 2033 y Fg(0)793 2091 y Fl(N)851 2100 y Fh(j)885 2091 y Fg(\000)p Fl(u)p Fs(+1)1092 2002 y Fk(f)p Fo(\027)6 b Fp(\()p Fo(f)1278 1964 y Fs(2)1334 2002 y Fk(j)15 b Fo(n)p Fp(\))21 b Fk(_)e Fo(\027)6 b Fp(\()p Fo(f)1706 1964 y Fs(2)1762 2002 y Fk(j)15 b Fo(n)20 b Fk(\000)g Fp(1\))p Fk(g)2243 1915 y Ff(X)2108 2115 y Fs(\()p Fl(x;z)s Fs(\))p Fg(2)p Fl(V)d Fg(\002)p Fl(W)2524 2002 y Fo(\027)2575 1928 y Ff(\000)2631 2002 y Fp(\()p Fk(r)2742 2016 y Fl(xz)2826 2002 y Fo(f)10 b Fp(\))2916 1964 y Fs(2)2971 2002 y Fk(j)16 b Fo(n)j Fk(\000)h Fp(1)3237 1928 y Ff(\001)3305 2002 y Fo(;)338 b Fp(\(3)p Fo(:)p Fp(13\))0 2358 y(and)30 b(similarily)c(for)k Fo(n)25 b(>)g(u)p Fp(.)41 b(Notice)31 b(that)g Fo(c)1528 2325 y Fg(0)1528 2383 y Fl(N)1586 2392 y Fh(j)1620 2383 y Fg(\000)p Fl(u)p Fs(+1)1841 2358 y Fk(\025)25 b Fo(c)1976 2372 y Fl(u)2050 2358 y Fp(=)g Fo(k)s(\032)2243 2372 y Fl(j)2281 2358 y Fk(j)p Fp(\003)2369 2372 y Fl(j)2408 2358 y Fk(j)2433 2325 y Fs(2)2474 2358 y Fp(.)40 b(Th)m(us,)30 b(it)g(follo)m(ws)f(from)h(\(3.13\))j(that)947 2578 y Ff(X)990 2769 y Fl(n)1094 2665 y Fo(\015)1141 2679 y Fl(f)1188 2665 y Fp(\()p Fo(n)p Fp(\))15 b Fo(A)p Fp(\()p Fo(n)p Fp(\))1521 2627 y Fs(2)1588 2665 y Fk(\024)25 b Fo(k)1869 2603 y Fp(1)p 1745 2644 294 4 v 1745 2727 a Fo(\032)1792 2741 y Fl(j)1845 2727 y Fk(j)p Fp(\003)1933 2741 y Fl(j)1972 2727 y Fk(j)1997 2701 y Fs(2)2198 2578 y Ff(X)2064 2778 y Fs(\()p Fl(x;z)s Fs(\))p Fg(2)p Fl(V)16 b Fg(\002)p Fl(W)2479 2665 y Fo(\027)2530 2591 y Ff(\000)2586 2665 y Fp(\()p Fk(r)2697 2679 y Fl(xz)2781 2665 y Fo(f)10 b Fp(\))2871 2627 y Fs(2)2911 2591 y Ff(\001)1588 3002 y Fk(\024)25 b Fo(k)1812 2940 y(L)1874 2907 y Fs(2)p 1745 2981 238 4 v 1745 3064 a Fo(\032)1792 3078 y Fl(j)1830 3064 y Fk(j)p Fp(\003)1918 3078 y Fl(j)1957 3064 y Fk(j)1993 3002 y(E)2041 3016 y Fl(\027)2086 3002 y Fp(\()p Fo(f)5 b(;)15 b(f)10 b Fp(\))3668 2840 y(\(3)p Fo(:)p Fp(14\))0 3297 y(b)s(ecause)1302 3338 y Ff(X)1168 3537 y Fs(\()p Fl(x;z)s Fs(\))p Fg(2)p Fl(V)16 b Fg(\002)p Fl(W)1583 3424 y Fo(\027)1634 3351 y Ff(\000)1690 3424 y Fp(\()p Fk(r)1801 3438 y Fl(xz)1884 3424 y Fo(f)10 b Fp(\))1974 3387 y Fs(2)2015 3351 y Ff(\001)2082 3424 y Fk(\024)25 b Fo(k)s(L)2290 3387 y Fl(d)p Fs(+2)2424 3424 y Fk(E)2472 3438 y Fl(\027)2517 3424 y Fp(\()p Fo(f)5 b(;)15 b(f)10 b Fp(\))936 b(\(3)p Fo(:)p Fp(15\))0 3730 y(In)30 b(order)g(to)h(pro)m (v)m(e)g(\(3.15\))h(w)m(e)f(need)f(the)h(follo)m(wing)e(de\014nition.)0 3912 y Fq(De\014nition)42 b(3.9.)53 b Fp(Giv)m(en)37 b(a)g(\014nite)e(connected)i(subset)f(\003)h(of)g Fm(Z)2290 3879 y Fl(d)2364 3912 y Fp(a)g Fj(p)-5 b(ath)40 b(choic)-5 b(e)39 b(in)k Fp(\003)37 b(is)f(a)h(collection)f Fo(\025)f Fp(=)0 4039 y Fk(f)p Fo(\025)98 4053 y Fl(xz)207 4039 y Fp(:)25 b(\()p Fo(x;)15 b(z)t Fp(\))27 b Fk(2)e Fp(\003)20 b Fk(\002)g Fp(\003)p Fk(g)31 b Fp(suc)m(h)f(that)h Fo(\025)1346 4053 y Fl(xz)1459 4039 y Fp(is)f(a)h(self{a)m(v)m(oiding)e(path)i(from) e Fo(x)i Fp(to)g Fo(z)j Fp(inside)28 b(\003.)0 4249 y(Giv)m(en)i(a)h (path)f(c)m(hoice)h Fo(\025)g Fp(in)e(\003)1074 4263 y Fl(j)1142 4249 y Fp(and)h Fo(V)5 b(;)15 b(W)44 b Fp(as)30 b(ab)s(o)m(v)m(e,)i(w)m(e)f(let)1112 4496 y Fk(G)1166 4510 y Fl(V)1229 4496 y Fp(\()p Fo(\025)p Fp(\))26 b(=)f(max)1475 4555 y Fl(e)p Fg(2E)1597 4564 y Fi(\003)1659 4496 y Fp(#)p Fk(f)p Fp(\()p Fo(x;)15 b(z)t Fp(\))26 b Fk(2)f Fo(V)41 b Fk(\002)20 b Fo(W)37 b Fp(:)26 b Fo(\025)2512 4510 y Fl(xz)2620 4496 y Fb(A)f Fo(e)p Fk(g)1096 4714 y(D)1166 4728 y Fl(V)1229 4714 y Fp(\()p Fo(\025)p Fp(\))h(=)140 b(max)1474 4777 y Fs(\()p Fl(x;z)s Fs(\))p Fg(2)p Fl(V)16 b Fg(\002)p Fl(W)1889 4714 y Fk(j)p Fo(\025)1967 4728 y Fl(xz)2050 4714 y Fk(j)0 5014 y Fp(Let)31 b(no)m(w)f Fo(\025)h Fp(b)s(e)e(an)m(y)i(path)f(c)m(hoice.)42 b(Thanks)29 b(to)i(Lemma)g(4.3)g(in)e([Y])i(w)m(e)g(get)g(that)1185 5249 y Fo(\027)1236 5175 y Ff(\000)1292 5249 y Fp(\()p Fk(r)1403 5263 y Fl(xz)1487 5249 y Fo(f)10 b Fp(\))1577 5211 y Fs(2)1632 5175 y Ff(\001)1699 5249 y Fk(\024)25 b Fo(k)18 b Fk(j)p Fo(\025)1938 5263 y Fl(xz)2022 5249 y Fk(j)2116 5162 y Ff(X)2077 5358 y Fl(e)p Fa(@)p Fl(\025)2211 5367 y Fh(xz)2300 5249 y Fo(\027)2351 5175 y Ff(\000)2393 5249 y Fp(\()p Fk(r)2504 5263 y Fl(e)2543 5249 y Fo(f)10 b Fp(\))2633 5211 y Fs(2)2673 5175 y Ff(\001)0 5558 y Fi(20)p Fh(=settembr)r(e=)p Fi(2000)29 b Fs([18])1204 b Fp(3.9)p eop %%Page: 19 19 19 18 bop 0 91 a Fp(whic)m(h,)29 b(together)j(with)d(the)i (de\014nition)d(of)i Fk(G)1569 105 y Fl(V)1633 91 y Fp(\()p Fo(\025)p Fp(\))h(and)e Fk(D)2033 105 y Fl(V)2097 91 y Fp(\()p Fo(\025)p Fp(\),)i(implies)1144 233 y Ff(X)1010 432 y Fs(\()p Fl(x;z)s Fs(\))p Fg(2)p Fl(V)15 b Fg(\002)p Fl(W)1424 319 y Fo(\027)1475 246 y Ff(\000)1532 319 y Fp(\()p Fk(r)1643 333 y Fl(xz)1726 319 y Fo(f)10 b Fp(\))1816 282 y Fs(2)1856 246 y Ff(\001)1923 319 y Fk(\024)25 b Fo(k)19 b Fk(G)2139 333 y Fl(V)2202 319 y Fp(\()p Fo(\025)p Fp(\))p Fk(D)2395 333 y Fl(V)2459 319 y Fp(\()p Fo(\025)p Fp(\))p Fk(E)2630 333 y Fl(\027)2676 319 y Fp(\()p Fo(f)5 b(;)15 b(f)10 b Fp(\))0 663 y(A)24 b(sensible)e(path)h(c)m(hoice)i(is)e (the)h(follo)m(wing.)37 b(Giv)m(en)24 b Fo(x)h Fk(2)g Fo(V)44 b Fp(and)23 b Fo(z)30 b Fk(2)25 b Fo(W)13 b Fp(,)25 b(start)f(increasing)e(\(or)j(decreasing\))f(the)0 790 y(\014rst)c(co)s(ordinate)h(of)h Fo(x)f Fp(un)m(til)e(it)i(is)f(equal)h (to)g(the)h(\014rst)e(co)s(ordinate)h(of)g Fo(z)t Fp(.)38 b(Rep)s(eat)22 b(for)f(the)g(remaining)e(co)s(ordinates)0 918 y(un)m(til)25 b(y)m(ou)i(get)h(to)f Fo(z)t Fp(.)40 b(With)26 b(this)g(particular)f(path)h(c)m(hoice)i(it)e(is)f(easy)j(to) f(see)g(that)h Fk(G)2910 932 y Fl(V)2973 918 y Fp(\()p Fo(\025)p Fp(\))p Fk(D)3166 932 y Fl(V)3230 918 y Fp(\()p Fo(\025)p Fp(\))e Fk(\024)f Fo(k)s(L)3587 885 y Fl(d)p Fs(+2)3748 918 y Fp(\(see)0 1045 y(section)31 b(5.2)g(in)e([CM2]\))j (and)d(the)i(pro)s(of)f(of)g(\(3.15\))j(follo)m(ws)c(at)i(once.)92 b Fb(\003)0 1407 y Ft(3.6)35 b(Bound)e(on)642 1339 y Ff(P)738 1434 y Fl(n)803 1407 y Fo(\015)850 1421 y Fl(f)897 1407 y Fp(\()p Fo(n)p Fp(\))p Fo(B)5 b Fp(\()p Fo(n)p Fp(\))1221 1374 y Fs(2)1262 1407 y Ft(.)0 1615 y Fp(Here)31 b(w)m(e)g(sho)m(w)f(ho)m(w)g(to)h(b)s(ound)e(from)h(ab)s(o)m(v)m(e)h (the)g(term)2001 1547 y Ff(P)2097 1642 y Fl(n)2162 1615 y Fo(\015)2209 1629 y Fl(f)2256 1615 y Fp(\()p Fo(n)p Fp(\))p Fo(B)5 b Fp(\()p Fo(n)p Fp(\))2580 1582 y Fs(2)2651 1615 y Fp(of)31 b(corollary)f(3.7.)0 1796 y Fq(Prop)s(osition)i(3.10.) 53 b Fc(Assume)26 b(prop)s(ert)m(y)g Fo(U)10 b(S)5 b(M)10 b(T)j Fp(\()p Fo(C)q(;)i(m;)g(l)r Fp(\))p Fc(.)41 b(Then)26 b(for)g(an)m(y)h Fo(\017)e(>)g Fp(0)i Fc(and)f Fo(\032)3204 1810 y Fs(0)3270 1796 y Fk(\024)3377 1760 y Fs(1)p 3377 1775 37 4 v 3377 1827 a(2)3450 1796 y Fc(there)h(exists)0 1923 y Fo(C)65 1937 y Fl(\017)130 1923 y Fc(suc)m(h)j(that,)h(for)g(an) m(y)f Fo(f)40 b Fc(with)29 b Fo(\027)6 b Fp(\()p Fo(f)1301 1890 y Fs(2)1341 1923 y Fp(\))26 b(=)f(1)p Fc(,)86 2131 y Fp(\(i\))45 b(if)30 b Fo(\032)358 2145 y Fl(j)421 2131 y Fk(\024)25 b Fo(\032)564 2145 y Fs(0)1201 2228 y Ff(X)1244 2418 y Fl(n)1347 2314 y Fo(\015)1394 2328 y Fl(f)1441 2314 y Fp(\()p Fo(n)p Fp(\))15 b Fo(B)5 b Fp(\()p Fo(n)p Fp(\))1780 2276 y Fs(2)1846 2314 y Fk(\024)2064 2252 y Fp(1)p 1968 2293 238 4 v 1968 2376 a Fo(\032)2015 2390 y Fl(j)2054 2376 y Fk(j)p Fp(\003)2142 2390 y Fl(j)2180 2376 y Fk(j)2232 2240 y Ff(\002)2270 2314 y Fo(C)2335 2328 y Fl(\017)2390 2314 y Fp(+)20 b Fo(\017)15 b Fp(En)m(t)2678 2328 y Fl(\027)2723 2314 y Fp(\()p Fo(f)2813 2276 y Fs(2)2853 2314 y Fp(\))2888 2240 y Ff(\003)61 2602 y Fp(\(ii\))44 b(if)30 b Fo(\032)358 2616 y Fs(0)424 2602 y Fk(\024)25 b Fo(\032)567 2616 y Fl(j)630 2602 y Fk(\024)737 2566 y Fs(1)p 737 2581 37 4 v 737 2633 a(2)881 2805 y Ff(X)924 2995 y Fl(n)1027 2891 y Fo(\015)1074 2905 y Fl(f)1122 2891 y Fp(\()p Fo(n)p Fp(\))15 b Fo(B)5 b Fp(\()p Fo(n)p Fp(\))1461 2853 y Fs(2)1527 2891 y Fk(\024)1746 2829 y Fp(1)p 1649 2870 240 4 v 1649 2953 a Fo(\032)1696 2922 y Fs(2)1696 2979 y Fl(j)1737 2953 y Fk(j)p Fp(\003)1825 2967 y Fl(j)1863 2953 y Fk(j)1915 2817 y Ff(\002)1953 2891 y Fo(C)2018 2905 y Fl(\017)2073 2891 y Fp(+)20 b Fo(C)2229 2905 y Fl(\017)2278 2891 y Fo(L)2340 2853 y Fs(2)2381 2891 y Fk(E)2429 2905 y Fl(\027)2474 2891 y Fp(\()p Fo(f)5 b(;)15 b(f)10 b Fp(\))20 b(+)g Fo(\017)15 b Fp(En)m(t)2997 2905 y Fl(\027)3043 2891 y Fp(\()p Fo(f)3133 2853 y Fs(2)3173 2891 y Fp(\))3208 2817 y Ff(\003)0 3206 y Fj(Pr)-5 b(o)g(of.)0 3333 y Fp(\()p Fo(i)p Fp(\))164 b(Let)35 b Fo(u)c Fp(=)h Fk(b)p Fo(\032)705 3347 y Fl(j)743 3333 y Fk(j)p Fo(V)21 b Fk(jc)34 b Fp(and)g(assume,)h(without)e(loss)h (of)g(generalit)m(y)-8 b(,)36 b Fo(n)31 b Fk(\024)g Fo(u)p Fp(.)52 b(Then,)34 b(b)m(y)g(pro)s(ceeding)f(as)h(in)0 3461 y(the)d(pro)s(of)e(of)i(prop)s(osition)d(3.8,)788 3739 y Fo(B)5 b Fp(\()p Fo(n)p Fp(\))987 3702 y Fs(2)1053 3739 y Fk(\024)25 b Fo(k)18 b(\027)1265 3638 y Ff(\020)1319 3739 y Fo(f)1374 3702 y Fs(2)1414 3739 y Fo(;)1606 3678 y 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b(is)g(then)g(easy)h(to)g (c)m(hec)m(k)h(that)f Fo(\015)k Fp(can)c(b)s(e)f(written)f(in)g(the)i (Gibbsian)d(form,)i Fo(\015)5 b Fp(\()p Fo(n)p Fp(\))26 b(=)f Fo(e)3070 1258 y Fg(\000)p Fl(H)5 b Fs(\()p Fl(n)p Fs(\))3297 1291 y Fo(')p Fp(\()p Fo(n)p Fp(\),)32 b(where)321 1575 y Fo(H)7 b Fp(\()p Fo(n)p Fp(\))26 b(:=)676 1489 y Ff(X)684 1684 y Fl(i)p Fg(6)p Fs(=)p Fl(j)822 1575 y Fp(\()q Fo(\025)911 1589 y Fl(i)941 1575 y Fp(\()p Fo(n)p Fp(\))21 b Fk(\000)f Fo(\025)1231 1589 y Fl(i)1261 1575 y Fp(\))15 b Fo(N)1384 1589 y Fl(i)1435 1575 y Fp(+)20 b Fo(\025)1579 1589 y Fl(V)1657 1575 y Fo(n)g Fp(+)g Fo(\025)1876 1589 y Fl(W)1958 1575 y Fp(\()p Fo(N)2066 1589 y Fl(j)2125 1575 y Fk(\000)g Fo(n)p Fp(\))g Fk(\000)g Fo(\025)2470 1589 y Fl(j)2509 1575 y Fo(N)2582 1589 y Fl(j)2640 1575 y Fk(\000)g Fp(log)2864 1419 y Ff( )2947 1513 y Fo(Z)3016 1463 y Fl(\034)s(;\025)p 3075 1478 V -1 w Fs(\()p Fl(n)p Fs(\))3009 1539 y(\003)p 2947 1554 277 4 v 2997 1659 a Fo(Z)3066 1609 y Fl(\034)s(;\025)p 3125 1624 44 4 v 3059 1685 a Fs(\003)3234 1419 y Ff(!)3321 1575 y Fo(;)345 1929 y(')p Fp(\()p Fo(n)p Fp(\))26 b(:=)687 1867 y Fo(\026)742 1817 y Fl(\034)s(;\025)p 801 1832 V -1 w Fs(\()p Fl(n)p Fs(\))742 1893 y(\003)965 1867 y Fp(\()p Fq(N)1082 1881 y Fs(\003)1162 1867 y Fp(=)e Fq(N)15 b Fp(;)31 b Fo(N)1483 1881 y Fl(V)1571 1867 y Fp(=)25 b Fo(n)p Fp(\))p 687 1908 1071 4 v 929 2012 a Fo(\026)984 1963 y Fl(\034)s(;\025)p 1043 1978 44 4 v 984 2039 a Fs(\003)1105 2012 y Fp(\()q Fq(N)1223 2026 y Fs(\003)1302 2012 y Fp(=)g Fq(N)p Fp(\))1768 1929 y Fo(:)3668 1750 y Fp(\(3)p Fo(:)p Fp(21\))0 2209 y(Finally)-8 b(,)35 b(giv)m(en)g Fo(\017)e Fk(2)f Fp(\(0)p Fo(;)15 b Fp(1\),)38 b(w)m(e)e(consider)e(for)g(tec)m(hnical)h(reasons)g(the)h (\\)p Fo(\017)p Fp({regularization")f(\()p Fo(\017)g Fp(here)g(m)m(ust)g(b)s(e)0 2336 y(thougth)30 b(to)i(b)s(e)d(close)i (to)g(one\))g(of)g Fo(\015)k Fp(de\014ned)29 b(b)m(y)1314 2647 y Fo(\015)1361 2661 y Fl(\017)1396 2647 y Fp(\()p Fo(n)p Fp(\))d(:=)1668 2461 y Ff(8)1668 2543 y(<)1668 2706 y(:)1784 2509 y Fs(1)p 1775 2524 54 4 v 1775 2576 a Fl(Z)1840 2544 y Fo(e)1882 2511 y Fg(\000)p Fl(H)5 b Fs(\()p Fl(n)p Fs(\))2200 2544 y Fp(if)30 b Fo(n)25 b Fk(2)f Fo(I)2489 2558 y Fl(\017)1764 2763 y Fo(\015)5 b Fp(\()p Fo(n)p Fp(\))259 b(otherwise)3668 2647 y(\(3)p Fo(:)p Fp(22\))0 2953 y(where)1551 3142 y Fo(Z)32 b Fp(:)25 b(=)1802 3002 y Ff(P)1898 3098 y Fl(n)p Fg(2)p Fl(I)2024 3107 y Fh(\017)2076 3070 y Fo(e)2118 3037 y Fg(\000)p Fl(H)5 b Fs(\()p Fl(n)p Fs(\))p 1802 3121 545 4 v 1848 3136 a Ff(P)1944 3232 y Fl(n)p Fg(2)p Fl(I)2070 3241 y Fh(\017)2123 3204 y Fo(\015)g Fp(\()p Fo(n)p Fp(\))1477 3382 y Fo(n)1532 3344 y Fl(\017)1532 3404 y Fs(max)1695 3382 y Fp(=)30 b(\026)-50 b Fo(n)20 b Fp(+)g Fo(\017)p Fp(\()p Fo(n)2084 3396 y Fs(max)2243 3382 y Fk(\000)k Fp(\026)-50 b Fo(n)p Fp(\))1490 3536 y Fo(n)1545 3499 y Fl(\017)1545 3559 y Fs(min)1695 3536 y Fp(=)30 b(\026)-50 b Fo(n)20 b Fp(+)g Fo(\017)p Fp(\()p Fo(n)2084 3550 y Fs(min)2229 3536 y Fk(\000)k Fp(\026)-49 b Fo(n)o Fp(\))1545 3691 y Fo(I)1585 3705 y Fl(\017)1645 3691 y Fp(:)25 b(=)g([)p Fo(n)1871 3653 y Fl(\017)1871 3713 y Fs(min)1996 3691 y Fo(;)15 b(n)2091 3653 y Fl(\017)2091 3713 y Fs(max)2230 3691 y Fp(])20 b Fk(\\)g Fo(I)0 3871 y Fp(W)-8 b(e)25 b(no)m(w)e(recall,)h(without)f(an)m(y)g(pro)s(of,)i(some)f(results)e (on)h Fo(\015)2017 3885 y Fl(\017)2075 3871 y Fp(established)f(in)g (section)i(3.4)g(and)f(4)h(of)f([CM2])h(under)0 3998 y(the)36 b(mixing)f(h)m(yp)s(othesis)g Fo(U)10 b(S)5 b(M)10 b(T)j Fp(\()p Fo(C)q(;)i(m;)g(l)r Fp(\))37 b(that)g(will)d(turn) h(out)h(to)h(b)s(e)f(useful.)56 b(Then,)37 b(w)m(e)g(complete)f(these)0 4125 y(prop)s(erties)e(b)m(y)i(some)h(others)f(in)e(order)i(to)h(pro)m (v)m(e)g(that)f Fo(\015)2050 4139 y Fl(\017)2121 4125 y Fp(satis\014es)f(a)i Fj(c)-5 b(onvex)38 b(c)-5 b(ondition)p Fp(.)59 b(Once)36 b(con)m(v)m(exit)m(y)0 4252 y(is)k(established,)j(w)m (e)e(can)h(app)s(eal)e(to)i(prop)s(osition)d(A.5)j(in)e(the)h(app)s 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Fp(\026)g Fo(n)2590 5068 y Fp(for)30 b(all)f Fo(n)c Fk(\025)30 b Fp(\026)-50 b Fo(n)828 5252 y Fp(1)p 825 5293 51 4 v 825 5376 a Fo(k)917 5252 y Fp(\026)h Fo(n)19 b Fk(\000)h Fo(n)p 913 5293 221 4 v 1000 5376 a Fp(\026)-50 b Fo(n)1169 5314 y Fk(\024)25 b Fo(H)7 b Fp(\()p Fo(n)20 b Fk(\000)g Fp(1\))h Fk(\000)f Fo(H)7 b Fp(\()p Fo(n)p Fp(\))25 b Fk(\024)g Fo(k)2151 5252 y Fp(\026)-50 b Fo(n)20 b Fk(\000)g Fo(n)p 2146 5293 V 2234 5376 a Fp(\026)-50 b Fo(n)2590 5314 y Fp(for)30 b(all)f Fo(n)c Fk(\024)30 b Fp(\026)-50 b Fo(n)0 5558 y Fi(20)p Fh(=settembr)r(e=)p Fi(2000)29 b Fs([21])1181 b Fp(3.12)p eop %%Page: 22 22 22 21 bop 0 91 a Fp(The)36 b(next)g(t)m(w)m(o)h(results)e(sho)m(w)h (that)h(the)f(relativ)m(e)h(densit)m(y)e(b)s(et)m(w)m(een)i Fo(\015)k Fp(and)35 b(its)h(regularization)f(is)g(b)s(ounded)0 218 y(uniformly)27 b(in)i(the)i(size)f(of)h Fo(I)37 b Fp(and)30 b(that)h(the)f(tails)g(of)g Fo(\015)5 b Fp(\()p Fo(n)p Fp(\))31 b(are)g(at)g(least)g(exp)s(onen)m(tial.)39 b(More)32 b(precisely)0 390 y Fq(Lemma)h(3.12.)1342 494 y Fp(1)p 1340 535 51 4 v 1340 618 a Fo(k)1426 556 y Fk(\024)i Fp(inf)1522 614 y Fl(n)p Fg(2)p Fl(I)1694 494 y Fo(\015)5 b Fp(\()p Fo(n)p Fp(\))p 1679 535 208 4 v 1679 618 a Fo(\015)1726 632 y Fl(\017)1761 618 y Fp(\()p Fo(n)p Fp(\))1923 556 y Fk(\024)25 b Fp(sup)2022 632 y Fl(n)p Fg(2)p Fl(I)2196 494 y Fo(\015)5 b Fp(\()p Fo(n)p Fp(\))p 2182 535 V 2182 618 a Fo(\015)2229 632 y Fl(\017)2263 618 y Fp(\()p Fo(n)p Fp(\))2425 556 y Fk(\024)25 b Fo(k)0 883 y Fq(Lemma)33 b(3.13.)53 b Fc(There)30 b(exists)g(a)h(p)s(ositiv)m (e)e(constan)m(t)j Fo(\017)1968 897 y Fs(0)2033 883 y Fp(=)25 b Fo(\017)2166 897 y Fs(0)2207 883 y Fp(\()p Fo(\016)n(;)15 b Fk(k)p Fp(\010)p Fk(k)2476 897 y Fg(1)2555 883 y Fp(\))31 b Fc(suc)m(h)f(that)h Fk(8)p Fo(\017)25 b Fk(2)f Fp(\()p Fo(\017)3293 897 y Fs(0)3334 883 y Fo(;)15 b Fp(1\))1246 1073 y Fo(\015)5 b Fp(\()p Fo(n)20 b Fp(+)g(1\))p 1246 1114 334 4 v 1325 1197 a Fo(\015)5 b Fp(\()p Fo(n)p Fp(\))1616 1135 y Fk(\024)1723 1073 y Fp(1)p 1723 1114 46 4 v 1723 1197 a(2)1871 1135 y(if)97 b Fo(n)24 b Fk(2)h Fp([)p Fo(n)2267 1097 y Fl(\017)2267 1157 y Fs(max)2406 1135 y Fo(;)15 b(n)2501 1149 y Fs(max)2639 1135 y Fp(])1246 1351 y Fo(\015)5 b Fp(\()p Fo(n)20 b Fk(\000)g Fp(1\))p 1246 1392 334 4 v 1325 1475 a Fo(\015)5 b Fp(\()p Fo(n)p Fp(\))1616 1413 y Fk(\024)1723 1351 y Fp(1)p 1723 1392 46 4 v 1723 1475 a(2)1871 1413 y(if)97 b Fo(n)24 b Fk(2)h Fp([)p Fo(n)2267 1427 y Fs(min)2392 1413 y Fo(;)15 b(n)2487 1375 y Fl(\017)2487 1435 y Fs(min)2612 1413 y Fp(])0 1661 y Fq(Corollary)35 b(3.14.)1290 1783 y Fo(\015)1337 1797 y Fl(\017)1372 1783 y Fp(\()p Fo(n)20 b Fp(+)g(1\))p 1290 1824 364 4 v 1368 1907 a Fo(\015)1415 1921 y Fl(\017)1450 1907 y Fp(\()p Fo(n)p Fp(\))1690 1845 y Fk(\024)25 b Fo(k)s(e)1878 1807 y Fg(\000)p Fl(k)1999 1779 y Fh(n)p Fe(\000)5 b Fi(\026)-36 b Fh(n)p 1999 1794 130 4 v 2048 1829 a Fi(\026)g Fh(n)2325 1845 y Fk(8)15 b Fo(n)25 b Fk(\025)k Fp(\026)-49 b Fo(n)1046 b Fp(\(3)p Fo(:)p Fp(23\))0 2101 y Fc(and)30 b(similarily)c(for)724 2057 y Fl(\015)762 2066 y Fh(\017)795 2057 y Fs(\()p Fl(n)p Fg(\000)p Fs(1\))p 724 2080 266 4 v 771 2133 a Fl(\015)809 2142 y Fh(\017)841 2133 y Fs(\()p Fl(n)p Fs(\))1000 2101 y Fc(,)31 b Fo(n)25 b Fk(\024)k Fp(\026)-49 b Fo(n)o Fc(.)0 2318 y Fp(The)31 b(next)g(result)f(sho)m(ws)g(that)i(the)f(normalization)f(factor)i Fo(Z)37 b Fp(app)s(earing)30 b(in)g(the)h(de\014nition)e(of)i Fo(\015)3443 2332 y Fl(\017)3509 2318 y Fp(is)f(close)h(to)5 2444 y(\026)-50 b Fo(n)66 2387 y Fi(1)p 65 2397 31 4 v 65 2433 a(2)112 2444 y Fp(.)0 2616 y Fq(Lemma)33 b(3.15.)1616 2681 y Fp(1)p 1614 2722 51 4 v 1614 2805 a Fo(k)1675 2667 y Fk(p)p 1751 2667 55 4 v 1756 2743 a Fp(\026)-50 b Fo(n)25 b Fk(\024)g Fo(Z)31 b Fk(\024)25 b Fo(k)2166 2667 y Fk(p)p 2242 2667 V 2247 2743 a Fp(\026)-50 b Fo(n)0 2966 y Fj(Pr)-5 b(o)g(of.)32 b Fp(W)-8 b(e)31 b(m)m(ust)g(consider)e(t) m(w)m(o)j(terms,)1456 2898 y Ff(P)1552 2994 y Fl(n)p Fg(2)p Fl(I)1678 3003 y Fh(\017)1730 2966 y Fo(e)1772 2933 y Fg(\000)p Fl(H)5 b Fs(\()p Fl(n)p Fs(\))2030 2966 y Fp(and)2207 2898 y Ff(P)2303 2994 y Fl(n)p Fg(2)p Fl(I)2429 3003 y Fh(\017)2481 2966 y Fo(\015)g Fp(\()p Fo(n)p Fp(\).)0 3093 y(By)31 b(using)e(lemma)g(3.11)j(one)f(can)g(easily)e(get)j(for)e (all)f Fo(n)c Fk(2)g Fo(I)2072 3107 y Fl(\017)1216 3269 y Fp(1)p 1214 3309 51 4 v 1214 3392 a Fo(k)1275 3330 y(e)1317 3293 y Fg(\000)1388 3268 y Fh(k)p 1385 3278 41 4 v 1390 3314 a Fi(\026)-36 b Fh(n)1436 3293 y Fs(\()t(\026)c Fl(n)p Fg(\000)p Fl(n)p Fs(\))1639 3265 y Fi(2)1705 3330 y Fk(\024)25 b Fo(e)1843 3293 y Fg(\000)p Fl(H)5 b Fs(\()p Fl(n)p Fs(\))2096 3330 y Fk(\024)25 b Fo(k)18 b(e)2299 3293 y Fg(\000)2389 3268 y Fi(1)p 2367 3278 76 4 v 2367 3314 a Fh(k)5 b Fi(\026)-35 b Fh(n)2453 3293 y Fs(\()t(\026)-40 b Fl(n)q Fg(\000)p Fl(n)p Fs(\))2657 3265 y Fi(2)3668 3330 y Fp(\(3)p Fo(:)p Fp(24\))0 3520 y(Th)m(us,)30 b(using)e(a)j (comparison)f(with)f(in)m(tegrals)h(and)g(a)g(c)m(hange)i(of)e(v)-5 b(ariable,)30 b(it)g(easily)f(follo)m(ws)h(that)1429 3695 y(1)p 1426 3736 51 4 v 1426 3819 a Fo(k)1488 3681 y Fk(p)p 1563 3681 55 4 v 1568 3757 a Fp(\026)-50 b Fo(n)25 b Fk(\024)1753 3670 y Ff(X)1739 3865 y Fl(n)p Fg(2)p Fl(I)1865 3874 y Fh(\017)1913 3757 y Fo(e)1955 3719 y Fg(\000)p Fl(H)5 b Fs(\()p Fl(n)p Fs(\))2208 3757 y Fk(\024)25 b Fo(k)2354 3681 y Fk(p)p 2430 3681 V 2435 3757 a Fp(\026)-50 b Fo(n)1183 b Fp(\(3)p Fo(:)p Fp(25\))0 4046 y(On)30 b(the)g(other)h(hand,)e(it)h(follo)m(ws)g(from)g(lemma)f(3.12)j(that) 1132 4219 y Fl(n)1177 4191 y Fh(\017)1177 4237 y Fi(min)1142 4256 y Ff(X)1081 4447 y Fl(n)p Fs(=)p Fl(n)1227 4456 y Fi(min)1349 4343 y Fo(\015)5 b Fp(\()p Fo(n)p Fp(\))26 b(=)f Fo(\015)5 b Fp(\()p Fo(n)1790 4305 y Fl(\017)1790 4365 y Fs(min)1915 4343 y Fp(\))2016 4219 y Fl(n)2061 4191 y Fh(\017)2061 4237 y Fi(min)2026 4256 y Ff(X)1965 4447 y Fl(n)p Fs(=)p Fl(n)2111 4456 y Fi(min)2306 4281 y Fo(\015)g Fp(\()p Fo(n)p Fp(\))p 2244 4322 303 4 v 2244 4405 a Fo(\015)g Fp(\()p Fo(n)2386 4374 y Fl(\017)2386 4431 y Fs(min)2511 4405 y Fp(\))1552 4709 y(=)25 b Fo(\015)5 b Fp(\()p Fo(n)1790 4671 y Fl(\017)1790 4731 y Fs(min)1915 4709 y Fp(\))2016 4585 y Fl(n)2061 4558 y Fh(\017)2061 4603 y Fi(min)2026 4622 y Ff(X)1965 4813 y Fl(n)p Fs(=)p Fl(n)2111 4822 y Fi(min)2270 4585 y Fl(n)2315 4558 y Fh(\017)2315 4603 y Fi(min)2288 4622 y Ff(Y)2233 4816 y Fl(j)t Fs(=)p Fl(n)p Fs(+1)2486 4647 y Fo(\015)g Fp(\()p Fo(j)26 b Fk(\000)20 b Fp(1\))p 2486 4688 323 4 v 2564 4771 a Fo(\015)5 b Fp(\()p Fo(j)g Fp(\))1552 5081 y Fk(\024)25 b Fo(\015)5 b Fp(\()p Fo(n)1790 5043 y Fl(\017)1790 5103 y Fs(min)1915 5081 y Fp(\))2016 4957 y Fl(n)2061 4930 y Fh(\017)2061 4975 y Fi(min)2026 4995 y Ff(X)1965 5185 y Fl(n)p Fs(=)p Fl(n)2111 5194 y Fi(min)2233 4953 y Ff(\022)2311 5019 y Fp(1)p 2311 5060 46 4 v 2311 5143 a(2)2367 4953 y Ff(\023)2434 4972 y Fl(n)2479 4944 y Fh(\017)2479 4990 y Fi(min)2586 4972 y Fg(\000)p Fl(n)1552 5345 y Fk(\024)25 b Fp(2)15 b Fo(\015)5 b Fp(\()p Fo(n)1850 5307 y Fl(\017)1850 5367 y Fs(min)1975 5345 y Fp(\))0 5558 y Fi(20)p Fh(=settembr)r(e=)p Fi(2000)29 b Fs([22])1181 b Fp(3.13)p eop %%Page: 23 23 23 22 bop 0 91 a Fp(A)30 b(similar)e(statemen)m(t)k(for)959 23 y Ff(P)1055 45 y Fl(n)1100 54 y Fi(max)1055 118 y Fl(n)p Fs(=)p Fl(n)1201 100 y Fh(\017)1201 136 y Fi(max)1338 91 y Fo(\015)5 b Fp(\()p Fo(n)p Fp(\))30 b(yields)1219 230 y Ff(X)1205 425 y Fl(n)p Fg(2)p Fl(I)1331 434 y Fh(\017)1379 317 y Fo(\015)5 b Fp(\()p Fo(n)p Fp(\))25 b Fk(\025)g Fp(1)c Fk(\000)f Fp(2)15 b(\()q Fo(\015)5 b Fp(\()p Fo(n)2072 279 y Fl(\017)2072 339 y Fs(min)2197 317 y Fp(\))21 b(+)e Fo(\015)5 b Fp(\()p Fo(n)2485 279 y Fl(\017)2485 339 y Fs(max)2624 317 y Fp(\)\))0 596 y(W)-8 b(e)28 b(are)f(left)f(with)f (the)i(problem)e(of)h(b)s(ounding)e(from)i(ab)s(o)m(v)m(e)i Fo(\015)5 b Fp(\()p Fo(n)2266 563 y Fl(\017)2266 621 y Fs(min)2391 596 y Fp(\))26 b(and)g Fo(\015)5 b Fp(\()p Fo(n)2767 563 y Fl(\017)2767 619 y Fs(max)2906 596 y Fp(\).)40 b(Because)28 b(of)e(lemma)g(3.13)0 723 y(it's)35 b(enough)f(to)i(con)m(trol)f Fo(\015)947 737 y Fl(\017)982 723 y Fp(\()p Fo(n)1072 690 y Fl(\017)1072 747 y Fs(min)1197 723 y Fp(\))g(and)f Fo(\015)1495 737 y Fl(\017)1530 723 y Fp(\()p Fo(n)1620 690 y Fl(\017)1620 745 y Fs(max)1759 723 y Fp(\).)54 b(By)36 b(using)d(lemma)h(3.11)j(together)f(with)e (corollary)g(3.14,)0 849 y(w)m(e)d(get)1251 1063 y Fo(\015)1298 1077 y Fl(\017)1333 1063 y Fp(\()p Fo(n)1423 1025 y Fl(\017)1423 1085 y Fs(min)1548 1063 y Fp(\))25 b(=)g Fo(\015)1751 1077 y Fl(\017)1786 1063 y Fp(\()5 b(\026)-50 b Fo(n)p Fp(\))2075 949 y Fs(\026)-40 b Fl(n)2035 976 y Ff(Y)1926 1170 y Fl(j)t Fs(=)p Fl(n)2061 1148 y Fh(\017)2061 1194 y Fi(min)2168 1170 y Fs(+1)2286 1001 y Fo(\015)2333 1015 y Fl(\017)2368 1001 y Fp(\()p Fo(j)26 b Fk(\000)20 b Fp(1\))p 2286 1042 352 4 v 2364 1125 a Fo(\015)2411 1139 y Fl(\017)2446 1125 y Fp(\()p Fo(j)5 b Fp(\))1608 1361 y Fk(\024)25 b Fo(k)19 b(e)1812 1323 y Fg(\000)1898 1299 y Fh(k)p 1880 1309 71 4 v 1880 1345 a Fi(2)5 b(\026)-36 b Fh(n)1962 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e(set)i(and,)h(for)e(a)h(giv)m(en)f(b)s(oundary)e(con\014guration)i Fo(\034)0 3358 y Fp(and)d(\(p)s(ossible)e(v)m(ector\))k(c)m(hemical)e (p)s(oten)m(tial)g Fo(\025)p 1642 3373 54 4 v Fp(,)i(let)e Fo(\026)e Fp(:=)h Fo(\026)2166 3309 y Fl(\034)s(;\025)p 2225 3324 44 4 v 2166 3385 a Fs(\003)2272 3358 y Fp(.)55 b(Let)36 b Fk(f)p Fo(V)2618 3372 y Fl(\013)2670 3358 y Fk(g)2715 3372 y Fl(\013)p Fg(2)p Fl(I)2889 3358 y Fp(b)s(e)f(a)g(collection)g(of)h(subsets)0 3486 y(of)44 b(\003)f(suc)m(h)h(that)g(dist)o(\()p Fo(V)887 3500 y Fl(\013)939 3486 y Fo(;)15 b(V)1032 3500 y Fl(\014)1081 3486 y Fp(\))48 b Fk(\025)f Fo(r)32 b Fp(+)c(1)44 b(for)g Fo(\013)j Fk(6)p Fp(=)g Fo(\014)5 b Fp(,)48 b Fo(r)e Fp(b)s(eing)c(the)i(range)f(of)h(the)g(in)m(teraction,)j(and)c(let)0 3613 y Fo(V)i Fp(=)25 b Fk([)255 3627 y Fl(\013)307 3613 y Fo(V)360 3627 y Fl(\013)412 3613 y Fp(.)40 b(Let)31 b(also)f Fo(f)k Fp(:)26 b(\012)1018 3627 y Fs(\003)1097 3613 y Fk(7!)f Fm(R)39 b Fp(b)s(e)29 b(suc)m(h)h(that)g Fo(\026)p 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Fl(t)587 198 y Fi(0)635 189 y Fs(Osc\()p Fl(f)d Fs(\))856 222 y Fp(.)42 b(Suc)m(h)30 b(a)i(c)m(hoice)f(mak)m(es)h(sense)f(when)f(Osc\()p Fo(f)10 b Fp(\))31 b(is)f(indep)s(enden)m(t)f(of)i Fk(j)p Fp(\003)p Fk(j)p Fp(.)43 b(Ho)m(w)m(ev)m(er,)33 b(in)0 349 y(most)g(of)g(the)g(sub)s(equen)m(t)f(application,)g(Osc\()p Fo(f)10 b Fp(\))30 b(=)f Fo(O)s Fp(\()p Fk(j)p Fp(\003)p Fk(j)p Fp(\))34 b(and)e(a)i(more)f(re\014ned)f(c)m(hoice)h(of)g(the)h (sets)f Fo(V)3674 363 y Fl(\013)3759 349 y Fp(will)0 477 y(b)s(e)d(necessary)-8 b(.)0 731 y Fj(Pr)j(o)g(of.)28 b Fp(In)e(order)g(to)h(pro)m(v)m(e)g(part)g(\()p Fo(i)p Fp(\))g(w)m(e)g(app)s(eal)f(to)h(the)g(follo)m(wing)d(lemma)j(kno)m(wn) f(as)g(the)h(Herbst's)f(argumen)m(t)0 859 y(\(see)31 b(for)f(istance)h(section)g(6)f(in[An]\).)0 1113 y Fq(Lemma)f(4.2.)53 b Fc(Let)27 b Fp(\(\012)p Fo(;)15 b Fk(F)9 b Fo(;)15 b(\026)p Fp(\))29 b Fc(b)s(e)d(a)h(\014nite)f(probabilit)m(y)f(space)j (and)e Fo(f)36 b Fc(a)28 b(function)d(on)i Fp(\012)p Fc(.)40 b(Assume)26 b(that)i(there)0 1240 y(exists)i Fo(K)i(>)25 b Fp(0)31 b Fc(suc)m(h)f(that)h(for)f(all)f Fo(t)c Fk(2)g Fp([0)p Fo(;)15 b(t)1486 1254 y Fs(0)1528 1240 y Fp(])p Fc(,)1499 1487 y Fp(En)m(t)1644 1501 y Fl(\026)1693 1487 y Fp(\()p Fo(e)1770 1449 y Fl(tf)1845 1487 y Fp(\))26 b Fk(\024)f Fo(K)7 b(t)2119 1449 y Fs(2)2159 1487 y Fo(\026)p Fp(\()p Fo(e)2291 1449 y Fl(tf)2366 1487 y Fp(\))0 1734 y Fc(Then,)30 b(for)g(all)f Fo(t)c Fk(2)g Fp([0)p Fo(;)15 b(t)815 1748 y Fs(0)857 1734 y Fp(])p Fc(,)1567 1871 y Fo(\026)p Fp(\()p Fo(e)1699 1834 y Fl(tf)1774 1871 y Fp(\))26 b Fk(\024)f Fo(e)1973 1834 y Fl(t\026)p Fs(\()p Fl(f)7 b Fs(\)+)p Fl(K)t(t)2290 1806 y Fi(2)0 2077 y Fj(Pr)-5 b(o)g(of)39 b(of)g(the)f(L)-5 b(emma.)95 b Fp(Let)37 b Fo(H)7 b Fp(\()p Fo(t)p Fp(\))35 b(=)g Fo(\026)p Fp(\()p Fo(e)1557 2044 y Fl(tf)1632 2077 y Fp(\))i(b)s(e)e(the)i(Laplace)f(transform)g(of)g Fo(f)10 b Fp(.)58 b(A)36 b(simple)e(computation)0 2204 y(giv)m(es)1048 2342 y Fo(tH)1164 2304 y Fg(0)1189 2342 y Fp(\()p Fo(t)p Fp(\))21 b Fk(\000)f Fo(H)7 b Fp(\()p Fo(t)p Fp(\))15 b(ln)f Fo(H)7 b Fp(\()p Fo(t)p Fp(\))26 b(=)f(En)m(t\()p Fo(e)2226 2304 y Fl(tf)2301 2342 y Fp(\))h Fk(\024)f Fo(K)7 b(t)2575 2304 y Fs(2)2615 2342 y Fo(H)g Fp(\()p Fo(t)p Fp(\))26 b Fo(:)0 2543 y Fp(Dividing)i(b)m(y)i Fo(t)529 2510 y Fs(2)570 2543 y Fo(H)7 b Fp(\()p Fo(t)p Fp(\))31 b(and)e(writing)g Fo(K)7 b Fp(\()p Fo(t)p Fp(\))25 b(=)1596 2499 y Fs(ln)12 b Fl(H)5 b Fs(\()p Fl(t)p Fs(\))p 1596 2522 222 4 v 1693 2574 a Fl(t)1829 2543 y Fp(,)30 b(w)m(e)h(get)1716 2789 y Fo(K)1800 2752 y Fg(0)1825 2789 y Fp(\()p Fo(t)p Fp(\))26 b Fk(\024)f Fo(K)32 b(:)0 3036 y Fp(Notice)e(no)m(w)g(that)f Fo(K)7 b Fp(\()p Fo(t)p Fp(\))26 b Fk(!)f Fo(\026)p Fp(\()p Fo(f)10 b Fp(\))29 b(as)g Fo(t)d Fk(!)f Fp(0.)40 b(Consequen)m(tly)-8 b(,)30 b(a)f(simple)f(in)m(tegration)h(ac)m(hiev)m(es)h(the)g(pro)s(of.)90 b Fb(\003)0 3290 y Fp(Bac)m(k)30 b(to)e(the)h(pro)s(of)e(of)h(the)g (prop)s(osition,)f(if)g(w)m(e)h(apply)f(the)h(de\014nition)e(of)i(the)g 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Fo(c)299 3870 y Fl(x)345 3856 y Fp(\()p Fo(\033)s Fp(\))g(=)e Fo(\026)648 3823 y Fl(\033)648 3885 y Fg(f)p Fl(x)p Fg(g)768 3856 y Fp(\()p Fo(\033)858 3823 y Fl(x)904 3856 y Fp(\))j(are)g(the)g(\015ip)e(rates)j (for)e(the)h(Heat)i(Bath)e(dynamics.)39 b(Th)m(us,)27 b(thanks)h(to)g(lemma)g(4.2,)0 3984 y(part)i(\()p Fo(i)p Fp(\))i(follo)m(ws)d(at)i(once.)0 4146 y(In)37 b(order)g(to)i(pro)m(v)m (e)g(part)e(\()p Fo(ii)p Fp(\))i(w)m(e)f(pro)s(ceed)g(as)g(follo)m(ws.) 62 b(Denote)39 b(b)m(y)f Fo(\026)2581 4113 y Fl(\034)2581 4171 y(V)2682 4146 y Fp(the)g(Gibbs)e(measure)i(on)f Fo(V)58 b Fp(with)0 4274 y(particle)30 b(con\014guration)g Fo(\034)41 b Fp(in)30 b(\003)20 b Fk(n)h Fo(V)g Fp(.)42 b(By)31 b(construction)f Fo(\026)2077 4241 y Fl(\034)2077 4299 y(V)2171 4274 y Fp(is)g(a)h(pro)s(duct)f(measure)h(o)m(v)m(er)h (the)f(sets)g Fk(f)p Fo(V)3664 4288 y Fl(\013)3717 4274 y Fk(g)3762 4288 y Fl(\013)p Fg(2)p Fl(I)0 4401 y Fp(that,)g(for)f (simplicit)m(y)e(and)i(omitting)f(the)i(sup)s(erscript)d Fo(\034)10 b Fp(,)30 b(w)m(e)h(write)f(as)2530 4333 y Ff(Q)2616 4428 y Fl(\013)2683 4401 y Fo(\026)2738 4415 y Fl(\013)2790 4401 y Fp(.)40 b(Then)30 b(w)m(e)h(ha)m(v)m(e)1432 4688 y Fo(\026)p Fp(\()p Fo(e)1564 4651 y Fl(tf)1639 4688 y Fp(\))26 b(=)f Fo(\026)1851 4587 y Ff(\020)1905 4615 y(\000)1946 4602 y(Y)1981 4793 y Fl(\013)2078 4688 y Fo(\026)2133 4702 y Fl(\013)2184 4615 y Ff(\001)2226 4688 y Fp(\()p Fo(e)2303 4651 y Fl(tf)2378 4688 y Fp(\))2413 4587 y Ff(\021)0 5035 y Fp(Let)31 b(us)f(examine)g(a)g(generic)h(term)f Fo(\026)1292 5049 y Fl(\013)1344 5035 y Fp(\()p Fo(e)1421 5002 y Fl(tf)1496 5035 y Fp(\).)41 b(A)30 b(simple)f(T)-8 b(a)m(ylor)30 b(expansion)g(up)f(to)i(the)g(second)f(order)g(giv)m(es) 1027 5317 y Fo(\026)1082 5331 y Fl(\013)1134 5317 y Fp(\()p Fo(e)1211 5280 y Fl(tf)1286 5317 y Fp(\))c Fk(\024)f Fp(exp)1597 5216 y Ff(\020)1651 5317 y Fo(t\026)1739 5331 y Fl(\013)1790 5317 y Fp(\()p Fo(f)10 b Fp(\))21 b(+)f Fo(t)2060 5280 y Fs(2)2115 5317 y Fp(V)-8 b(ar)2257 5331 y Fl(\013)2309 5317 y Fp(\()p Fo(f)10 b Fp(\))p Fo(e)2476 5280 y Fl(t)2503 5289 y Fi(0)2551 5280 y Fs(Osc)2668 5289 y Fh(\013)2714 5280 y Fs(\()p Fl(f)d Fs(\))2818 5216 y Ff(\021)3713 5317 y Fp(\(4)p Fo(:)p Fp(2\))0 5558 y Fi(20)p Fh(=settembr)r(e=)p Fi(2000)29 b Fs([25])1204 b Fp(4.2)p eop %%Page: 26 26 26 25 bop 0 91 a Fp(In)38 b(order)g(to)h(iterate)h(\(o)m(v)m(er)g(the)f (index)e Fo(\013)p Fp(\))i(the)g(ab)s(o)m(v)m(e)h(b)s(ound,)f(w)m(e)g (simply)d(observ)m(e)j(that)g(V)-8 b(ar)3439 105 y Fl(\014)3488 91 y Fp(\()p Fo(\026)3578 105 y Fl(\013)3630 91 y Fp(\()p Fo(f)10 b Fp(\)\))39 b Fk(\024)0 218 y Fp(V)-8 b(ar)142 232 y Fl(\014)191 218 y Fp(\()p Fo(f)10 b Fp(\))30 b(and)g(Osc)670 232 y Fl(\014)719 218 y Fp(\()p Fo(\026)809 232 y Fl(\013)861 218 y Fp(\()p Fo(f)10 b Fp(\)\))25 b Fk(\024)g Fp(Osc)1289 232 y Fl(\014)1338 218 y Fp(\()p Fo(f)10 b Fp(\).)41 b(Th)m(us)897 448 y Fo(\026)952 348 y Ff(\020)1006 375 y(\000)1048 362 y(Y)1082 553 y Fl(\013)1179 448 y Fo(\026)1234 462 y Fl(\013)1286 375 y Ff(\001)1327 448 y Fp(\()p Fo(e)1404 411 y Fl(tf)1479 448 y Fp(\))1514 348 y Ff(\021)1594 448 y Fk(\024)25 b Fo(e)1732 405 y Fl(t)1759 377 y Fi(2)1820 343 y Ff(P)1916 439 y Fh(\013)1975 405 y Fs(V)-6 b(ar)2087 414 y Fh(\013)2133 405 y Fs(\()p Fl(f)7 b Fs(\))p Fl(e)2265 377 y Fh(t)2291 391 y Fi(0)2340 377 y(Osc)2439 386 y Fh(\013)2485 377 y Fi(\()p Fh(f)e Fi(\))2580 448 y Fo(\026)2635 375 y Ff(\000)2676 448 y Fo(e)2730 411 y Fl(t\026)2801 384 y Fh(\034)2801 429 y(V)2857 411 y Fs(\()p Fl(f)i Fs(\))2961 375 y Ff(\001)3713 448 y Fp(\(4)p Fo(:)p Fp(3\))0 742 y(It)30 b(su\016ces)h(no)m(w)f(to)h(apply)e(part)h(\()p Fo(i)p Fp(\))i(to)f(the)f(new)g(term)h Fo(\026)p Fp(\()p Fo(e)2104 709 y Fl(t\026)2175 682 y Fh(\034)2175 727 y(V)2231 709 y Fs(\()p Fl(f)7 b Fs(\))2335 742 y Fp(\).)91 b Fb(\003)0 959 y Fj(R)-5 b(emark.)42 b Fp(It)30 b(is)f(imp)s(ortan)m (t)h(to)h(understand)d(the)j(di\013erence)e(b)s(et)m(w)m(een)i(the)g(t) m(w)m(o)g(results)e(giv)m(en)h(in)f(prop)s(osition)0 1085 y(4.1.)55 b(Assume)34 b(that)h(condition)e Fo(U)10 b(M)g(S)5 b(T)13 b Fp(\()p Fo(C)q(;)i(m;)g(l)r Fp(\))36 b(holds)d(so)i(that)h(the)e(logarithmic)g(Sob)s(olev)f(constan)m(t)j Fo(c)3755 1099 y Fl(\026)3839 1085 y Fp(is)0 1212 y(b)s(ounded)f(from)h (ab)s(o)m(v)m(e)i(uniformly)c(in)i(\003)h(\(see)h(e.g.)61 b([M])38 b(and)e(references)h(therein\))f(and)h(let)g Fo(f)45 b Fp(=)3529 1144 y Ff(P)3625 1239 y Fl(x)p Fg(2)p Fs(\003)3784 1212 y Fo(f)3829 1226 y Fl(x)3875 1212 y Fp(,)0 1339 y(where)d Fo(f)320 1353 y Fl(x)408 1339 y Fp(is)g(a)h(mean)g(zero)g(lo)s(cal)f(function)g(with)f(small)h(supp)s (ort)f(around)g Fo(x)p Fp(.)78 b(F)-8 b(or)44 b(small)d(v)-5 b(alues)42 b(of)h(the)0 1466 y(parameter)31 b Fo(t)f Fp(one)h(exp)s(ects,)g(on)f(the)g(basis)g(of)g(a)h(second)f(order)g(T) -8 b(a)m(ylor)31 b(expansion,)e(a)i(b)s(ound)d(of)j(the)g(form)1648 1655 y Fo(\026)1718 1582 y Ff(\000)1760 1655 y Fo(e)1802 1618 y Fl(tf)1877 1582 y Ff(\001)1944 1655 y Fk(\024)25 b Fo(e)2082 1618 y Fl(t)2109 1590 y Fi(2)2145 1618 y Fl(K)2206 1628 y Fh(f)0 1845 y Fp(with)32 b Fo(K)287 1859 y Fl(f)365 1845 y Fk(\031)f Fo(\026)p Fp(\()p Fo(f)5 b(;)15 b(f)10 b Fp(\))30 b Fk(\031)868 1777 y Ff(P)964 1873 y Fl(x)1025 1845 y Fo(\026)p Fp(\()p Fo(f)1160 1859 y Fl(x)1205 1845 y Fo(;)15 b(f)1290 1859 y Fl(x)1336 1845 y Fp(\))34 b(b)s(ecause)g(of)f(the)h(mixing)e(assumption.)49 b(P)m(art)34 b(\()p Fo(i)p Fp(\))h(of)e(the)h(prop)s(osition)0 1972 y(pro)m(v)m(es)k(the)g(ab)s(o)m(v)m(e)h(Gaussian)e(b)s(ound)e(for) i Fj(al)5 b(l)39 b Fo(t)p Fp(,)g(but)e(with)f(a)i(new)f(constan)m(t)i Fo(K)2897 1986 y Fl(f)2982 1972 y Fk(\031)e Fo(c)3129 1986 y Fl(\026)3193 1904 y Ff(P)3289 1999 y Fl(x)3350 1972 y Fk(kr)3471 1986 y Fl(x)3517 1972 y Fo(f)3562 1986 y Fl(x)3607 1972 y Fk(k)3652 1939 y Fs(2)3652 1994 y Fg(1)3729 1972 y Fp(.)63 b(In)0 2099 y(\\normal")24 b(situations)f (that)h(is)f(quite)g(satisfactory)i(b)s(ecause)f Fo(K)2187 2113 y Fl(f)2258 2099 y Fp(b)s(ecomes)h(of)f(the)g(same)g(order)g(of)g (the)g(heuristic)0 2226 y(guess.)40 b(The)29 b(problem)f(arises)h (when,)g(anomalous,)g(v)m(ery)h(large)f(v)-5 b(alues)29 b(\(ma)m(yb)s(e)h(dep)s(ending)d(on)i(the)h(size)f(of)h(\003\))0 2352 y(of)d(the)h(c)m(hemical)f(p)s(oten)m(tial)f Fo(\025)p 1002 2367 54 4 v 27 w Fp(are)i(considered.)38 b(In)26 b(that)i(case)g(the)g(Gibbs)d(measure)i Fo(\026)g Fp(b)s(ecomes)g(v)m (ery)h(close)f(to)0 2479 y(the)k(pro)s(duct)f(of)i(highly)d(unfair)g (Bernoulli)g(measures)i(and)f(its)h(logarithmic)f(Sob)s(olev)g(constan) m(t)i Fo(c)3462 2493 y Fl(\026)3542 2479 y Fp(gets)h(v)m(ery)0 2606 y(large)26 b(\(see)g(e.g.)40 b([DiSa]\))26 b(so)g(that)g(the)g (Herbst's)f(b)s(ound)f(deteriorates.)39 b(P)m(art)27 b(\()p Fo(ii)p Fp(\))f(of)g(the)g(prop)s(osition)d(partially)0 2733 y(tak)m(es)33 b(care)f(of)f(these)h(extreme)g(situation,)e(at)i (least)g(for)f(not)g(to)s(o)h(large)g(v)-5 b(alues)30 b(of)i Fo(t)p Fp(,)f(as)h(w)m(e)f(no)m(w)g(explain)f(\(see)0 2859 y(also)f(prop)s(osition)e(4.3)j(b)s(elo)m(w)e(for)h(more)g(tec)m (hnical)g(details\).)40 b(Assume)29 b(that)g(eac)m(h)h(set)g Fo(V)3090 2873 y Fl(\013)3171 2859 y Fp(is)e(a)h(cub)s(e)g(of)g(side)f Fo(l)3859 2873 y Fs(0)0 2986 y Fp(m)m(ultiple)j(of)i(the)g(basic)f (length)h(scale)g Fo(l)i Fp(and)d(supp)s(ose)g(that)i Fo(f)39 b Fp(=)2325 2918 y Ff(P)2421 3013 y Fl(\013)2488 2986 y Fo(f)2533 3000 y Fl(\013)2617 2986 y Fp(where)33 b Fo(f)2928 3000 y Fl(\013)3008 2986 y Fp(=)3246 2900 y Ff(X)3230 3082 y Fh(x)p Fe(2)p Fh(V)3347 3091 y(\013)3120 3149 y(d)p Fi(\()p Fh(x;V)3285 3134 y(c)3274 3168 y(\013)3319 3149 y Fi(\))p Fe(\025)3404 3132 y Fi(1)p 3404 3140 31 4 v 3404 3173 a(2)3446 3149 y Fh(l)3468 3163 y Fi(0)3529 2986 y Fo(f)3574 3000 y Fl(x)3619 2986 y Fp(.)49 b(Then)0 3287 y(the)31 b(constan)m(t)g Fo(K)607 3254 y Fg(0)600 3313 y Fl(f)678 3287 y Fp(app)s(earing)e(in)g(part)h(\()p Fo(ii)p Fp(\))i(of)e(the)h(prop)s(osition)d(b)s(ecomes)682 3520 y Fo(K)766 3483 y Fg(0)759 3543 y Fl(f)831 3520 y Fp(=)d Fo(e)969 3483 y Fs(2)p Fg(k)p Fs(\010)p Fg(k)1137 3520 y Fo(c)1176 3534 y Fl(\026)1255 3434 y Ff(X)1297 3625 y Fl(\013)1425 3434 y Ff(X)1401 3633 y Fl(x)8 b(=)-45 b Fg(2)q Fl(V)1535 3642 y Fh(\013)1596 3520 y Fk(kr)1717 3534 y Fl(x)1763 3520 y Fo(\026)1818 3534 y Fl(V)1861 3543 y Fh(\013)1911 3520 y Fp(\()p Fo(f)1991 3534 y Fl(\013)2043 3520 y Fp(\))p Fk(k)2123 3483 y Fs(2)2123 3543 y Fg(1)2221 3520 y Fp(+)2312 3434 y Ff(X)2354 3625 y Fl(\013)2458 3520 y Fp(V)-8 b(ar)2600 3534 y Fl(\013)2652 3520 y Fp(\()p Fo(f)2732 3534 y Fl(\013)2783 3520 y Fp(\))15 b Fo(e)2875 3483 y Fl(t)2902 3492 y Fi(0)2952 3483 y Fs(Osc)3068 3492 y Fh(\013)3115 3483 y Fs(\()p Fl(f)7 b Fs(\))0 3831 y Fp(The)34 b(second)h(term)g(in)f(the)h(ab)s(o)m(v)m(e)h(expression)e (is,)h(apart)g(from)g(the)g(constan)m(t)h Fo(e)2851 3798 y Fl(t)2878 3807 y Fi(0)2926 3798 y Fs(Osc)3043 3807 y Fh(\013)3089 3798 y Fs(\()p Fl(f)7 b Fs(\))3193 3831 y Fp(,)37 b(exactly)e(the)g(con-)0 3958 y(tribution)30 b(giv)m(en)j(b)m(y)g(a)g(naiv)m(e)f(T)-8 b(a)m(ylor)33 b(expansion.)47 b(The)32 b(\014rst)g(term)h(is)e(instead)h(similar)e (to)k(the)f(constan)m(t)h Fo(K)3853 3972 y Fl(f)0 4085 y Fp(discussed)29 b(ab)s(o)m(v)m(e)i(but)f(with)f(the)i(follo)m(wing)e (imp)s(ortan)m(t)g(di\013erence.)41 b(The)30 b(logarithmic)f(Sob)s (olev)g(constan)m(t)j Fo(c)3851 4099 y Fl(\026)0 4212 y Fp(is)27 b(no)m(w)i(m)m(ultiplied)c(a)j(term,)1021 4143 y Ff(P)1117 4239 y Fl(x)1178 4212 y Fk(kr)1299 4226 y Fl(x)1345 4212 y Fo(\026)1400 4226 y Fl(V)1443 4235 y Fh(\013)1494 4212 y Fp(\()p Fo(f)1574 4226 y Fl(\013)1625 4212 y Fp(\))p Fk(k)1705 4179 y Fs(2)1705 4234 y Fg(1)1783 4212 y Fp(,)h(whic)m(h,)f(apart)h(from)e(trivial)g(constan)m(ts,)j(is)e (di\013eren)m(t)f(from)0 4338 y(zero)f(only)f(if)g Fo(x)35 b(=)-55 b Fk(2)25 b Fo(V)677 4352 y Fl(\013)754 4338 y Fp(and)g(in)g(that)h(case)h(is)d(prop)s(ortional)g(to)i Fk(j)p Fo(V)21 b Fk(j)15 b Fo(e)2300 4305 y Fg(\000)p Fl(ml)2445 4314 y Fi(0)2510 4338 y Fp(b)s(ecause)26 b(of)g(the)g (mixing)d(assumptions.)0 4465 y(Moreo)m(v)m(er,)30 b(for)d(extreme)i(v) -5 b(alues)26 b(of)i(the)f(c)m(hemical)h(p)s(oten)m(tial,)f(the)h (\\mass")g Fo(m)f Fp(can)h(b)s(e)f(tak)m(en)h(prop)s(ortional)d(to)0 4592 y(the)31 b(c)m(hemical)f(p)s(oten)m(tial)g(itself)f(so)h(that)h (the)g(\\e\013ectiv)m(e")i(logarithmic)c(Sob)s(olev)g(constan)m(t)j Fo(e)3266 4559 y Fg(\000)p Fl(ml)3411 4568 y Fi(0)3450 4592 y Fo(c)3489 4606 y Fl(\026)3569 4592 y Fp(b)s(ecomes)0 4719 y(v)m(ery)f(small)e(and)g(the)i(Gaussian)e(b)s(ound)g(b)s(ecomes)h (more)h(precise.)0 5063 y Ft(4.1)k(An)f(applica)-6 b(tion)33 b(a)-6 b(t)34 b(lo)n(w)f(density.)0 5213 y Fp(Here)e(w)m(e)h(discuss)d (an)h(application)f(of)i(our)g(b)s(ound)e(to)i(a)g(concrete)i(case)e (that)h(will)c(b)s(e)i(imp)s(ortan)m(t)g(in)g(the)h(next)0 5340 y(section.)0 5558 y Fi(20)p Fh(=settembr)r(e=)p Fi(2000)e Fs([26])1204 b Fp(4.3)p eop %%Page: 27 27 27 26 bop 0 91 a Fp(The)30 b(setting)g(is)g(the)g(standard)g(m)m (ulticanonical)e(setting)j(describ)s(ed)d(at)j(the)g(b)s(eginning)d(of) i(section)h(3.)0 218 y(Giv)m(en)h(no)m(w)h(a)g(lo)s(cal)e(function)g Fo(g)36 b Fp(with)c(supp)s(ort)e(\003)1799 232 y Fl(g)1874 218 y Fp(con)m(taining)i(the)h(origin)e(and)g(of)i(diameter)f(smaller)f (than)0 345 y(2)p Fo(r)s Fp(,)i Fo(r)j Fp(b)s(eing)31 b(the)i(range)g(of)g(in)m(teraction,)g(and)f(an)h(in)m(teger)g Fo(j)i Fk(\024)28 b Fo(j)2290 359 y Fs(max)2430 345 y Fp(,)33 b(w)m(e)g(de\014ne)f Fo(G)e Fp(=)3089 277 y Ff(P)3185 372 y Fl(x)p Fg(2)p Fs(\003)3325 381 y Fh(j)3379 345 y Fp([)15 b Fo(g)3462 359 y Fl(x)3529 345 y Fk(\000)20 b Fo(\026)p Fp(\()p Fo(g)3753 359 y Fl(x)3799 345 y Fp(\))15 b(])q(,)0 516 y(where)31 b Fo(g)307 530 y Fl(x)384 516 y Fp(is)f(the)h(translate)h(of)f Fo(g)k Fp(b)m(y)c Fo(x)p Fp(.)43 b(Let)31 b(also)g Fo(\032)1836 530 y Fl(j)1901 516 y Fp(:=)2052 470 y Fl(N)2110 479 y Fh(j)p 2035 495 127 4 v 2035 547 a Fg(j)p Fs(\003)2107 556 y Fh(j)2140 547 y Fg(j)2204 516 y Fp(and)f(assume)h(without)f(loss)h(of)g (generalit)m(y)g(that)0 643 y Fo(\032)47 657 y Fl(j)111 643 y Fo(<)25 b Fp(1)p Fo(=)p Fp(2.)0 867 y Fq(Prop)s(osition)g(4.3.)53 b Fc(Assume)20 b Fo(U)10 b(S)5 b(M)10 b(T)j Fp(\()p Fo(C)q(;)i(m;)g(l)r Fp(\))22 b Fc(and)e(that)i(there)e(exists)h(a)g(constan)m(t)h Fo(k)28 b Fp(=)d Fo(k)s Fp(\()p Fk(k)p Fp(\010)p Fk(k)3394 881 y Fg(1)3472 867 y Fo(;)15 b(C)q(;)g(m;)g(l)r Fp(\))27 b Fo(>)0 994 y Fp(0)k Fc(suc)m(h)f(that)777 1109 y Fo(\026)p Fp(\()p Fk(j)p Fo(g)935 1123 y Fl(x)982 1109 y Fk(j)p Fp(\))c Fk(\024)f Fo(k)18 b(\032)1276 1072 y Fs(2)1276 1132 y Fl(j)1332 1109 y Fk(k)p Fo(g)s Fk(k)1468 1123 y Fg(1)1728 1109 y Fk(8)25 b Fo(x)g Fk(2)g Fp(\003)2030 1123 y Fl(j)2099 1109 y Fc(suc)m(h)30 b(that)h Fp(dist)o(\()p Fo(x;)15 b Fp(\003)2838 1072 y Fl(c)2838 1132 y(j)2877 1109 y Fp(\))26 b Fk(\025)f Fp(2)p Fo(r)777 1264 y(\026)p Fp(\()p Fk(j)p Fo(g)935 1278 y Fl(x)982 1264 y Fk(j)p Fp(\))h Fk(\024)f Fo(k)18 b(\032)1276 1278 y Fl(j)1330 1264 y Fk(k)p Fo(g)s Fk(k)1466 1278 y Fg(1)1726 1264 y Fk(8)25 b Fo(x)g Fk(2)g Fp(\003)2028 1278 y Fl(j)2096 1264 y Fc(suc)m(h)30 b(that)h Fp(dist)o(\()p Fo(x;)15 b Fp(\003)2835 1226 y Fl(c)2835 1286 y(j)2875 1264 y Fp(\))25 b Fk(\024)g Fp(2)p Fo(r)0 1454 y Fc(Then,)34 b(for)g(an)m(y)g Fo(t)618 1468 y Fs(0)690 1454 y Fo(>)c Fp(0)35 b Fc(there)f(exist)f(p)s(ositiv)m(e)g(constan)m(ts)i Fo(A)d Fp(=)e Fo(A)p Fp(\()p Fk(k)p Fp(\010)p Fk(k)2531 1468 y Fg(1)2609 1454 y Fo(;)15 b(C)q(;)g(m;)g(l)r(;)g(t)2977 1468 y Fs(0)3020 1454 y Fo(;)g(r)m(;)g Fk(k)p Fo(g)s Fk(k)3274 1468 y Fg(1)3354 1454 y Fo(;)g(\016)3434 1468 y Fs(0)3475 1454 y Fp(\))34 b Fc(and)g Fo(B)h Fp(=)0 1581 y Fo(B)5 b Fp(\()p Fk(k)p Fp(\010)p Fk(k)265 1595 y Fg(1)342 1581 y Fo(;)15 b(C)q(;)g(m;)g(l)r(;)g(r)m(;)g(\016)795 1595 y Fs(0)839 1581 y Fp(\))31 b Fc(suc)m(h)f(that)1097 1821 y Fo(\026)p Fp(\()p Fo(e)1229 1784 y Fl(t)13 b(G)1330 1821 y Fp(\))26 b Fk(\024)f Fo(e)1529 1782 y Fl(t)1556 1755 y Fi(2)1592 1782 y Fl(AL)1697 1755 y Fh(d)1735 1782 y Fl(\032)1773 1791 y Fh(j)1807 1789 y Fp([)1832 1782 y Fl(\032)1870 1791 y Fh(j)1904 1782 y Fs(+)1970 1758 y Fi(ln)12 b Fh(L)p 1970 1768 107 4 v 2002 1804 a(L)2088 1789 y Fp(])2300 1821 y Fk(8)25 b Fo(t)g Fk(2)g Fp([0)p Fo(;)15 b(t)2663 1835 y Fs(0)2704 1821 y Fp(])1101 2018 y Fo(\027)6 b Fp(\()p Fo(e)1229 1980 y Fl(t)13 b(G)1330 2018 y Fp(\))26 b Fk(\024)f Fo(B)5 b(e)1603 1978 y Fl(t)1630 1951 y Fi(2)1666 1978 y Fl(AL)1771 1951 y Fh(d)1809 1978 y Fl(\032)1847 1987 y Fh(j)1880 1985 y Fp([)1905 1978 y Fl(\032)1943 1987 y Fh(j)1977 1978 y Fs(+)2044 1954 y Fi(ln)11 b Fh(L)p 2044 1964 V 2076 2000 a(L)2162 1985 y Fp(])2373 2018 y Fk(8)25 b Fo(t)g Fk(2)g Fp([0)p Fo(;)15 b(t)2736 2032 y Fs(0)2778 2018 y Fp(])0 2244 y Fj(R)-5 b(emark.)39 b Fp(Although)22 b(the)h(prop)s(osition)d(is)i(stated)h (for)f(an)m(y)h(v)-5 b(alue)22 b(of)h(the)g(densit)m(y)f Fo(\032)2852 2258 y Fl(j)2915 2244 y Fk(\024)3022 2208 y Fs(1)p 3022 2223 37 4 v 3022 2275 a(2)3069 2244 y Fp(,)j(the)e(most)g (in)m(teresting)0 2371 y(application)29 b(is)g(when)g Fo(\032)845 2385 y Fl(j)914 2371 y Fp(is)h(v)m(ery)g(small)f(\(p)s (ossibly)f(dep)s(ending)g(on)i Fo(L)p Fp(\).)0 2595 y Fj(Pr)-5 b(o)g(of.)50 b Fp(In)e(what)g(follo)m(ws)g Fo(k)k Fp(will)46 b(alw)m(a)m(ys)j(denote)g(a)g(generic)f(n)m(umerical)f (constan)m(t)j(dep)s(ending)c(only)i(on)0 2722 y Fk(k)p Fp(\010)p Fk(k)p Fo(;)15 b(C)q(;)g(m;)g(l)r(;)g(t)524 2736 y Fs(0)567 2722 y Fo(;)g(r)m(;)g(d;)g(\016)812 2736 y Fs(0)855 2722 y Fo(;)g Fk(k)p Fo(g)s Fk(k)1031 2736 y Fg(1)1140 2722 y Fp(and)30 b(whose)g(v)-5 b(alue)30 b(ma)m(y)h(v)-5 b(ary)30 b(in)g(di\013eren)m(t)f(estimates.)0 2849 y(Let)e Fk(f)p Fo(C)269 2863 y Fl(\013)321 2849 y Fk(g)366 2863 y Fl(\013)p Fg(2)p Fl(I)530 2849 y Fp(b)s(e)f(a)g(c)m (hessb)s(oard{lik)m(e)g(partition)f(of)h(\003)1897 2863 y Fl(j)1961 2849 y Fp(in)m(to)h(cub)s(es)e(of)h(side)f Fo(l)2692 2863 y Fs(0)2758 2849 y Fp(=)g(6)p Fo(l)r Fp(,)j Fo(l)g Fp(b)s(eing)d(the)h(basic)g(length)0 2976 y(scale.)39 b(A)24 b(momen)m(t)h(of)g(though)m(t)f(su\016ces)g(to)h(con)m(vince)g 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Fh(d)2386 1799 y Fl(Al)2464 1753 y Fe(\000)2524 1735 y Fi(1)p 2524 1743 31 4 v 2524 1777 a(2)2464 1821 y(0)2757 1836 y Fk(8)15 b Fo(t)634 2030 y Fp(\()p Fo(ii)p Fp(\))770 b Fo(\026)1591 1957 y Ff(\000)1633 2030 y Fo(e)1675 1993 y Fl(t)13 b Fs(\()p Fl(G)1800 2002 y Fh(\027)1840 1993 y Fg(\000)p Fl(G)1954 2002 y Fh(\026)1997 1993 y Fs(\))2030 1957 y Ff(\001)2096 2030 y Fk(\024)25 b Fo(e)2234 1993 y Fl(t)2261 1965 y Fi(2)2298 1993 y Fl(L)2348 1965 y Fh(d)2386 1993 y Fl(Al)2464 1963 y Fe(\000)p Fh(d)2464 2015 y Fi(0)2738 2030 y Fk(8)15 b Fo(t)24 b Fk(2)h Fp([0)p Fo(;)15 b(t)3090 2044 y Fs(0)3132 2030 y Fp(])600 2233 y(\()p Fo(iii)p Fp(\))770 b Fo(\026)1588 2159 y Ff(\000)1630 2233 y Fo(e)1672 2195 y Fl(t)13 b Fs(\()p Fl(G)1797 2155 y Fi(0)p 1797 2170 36 4 v 1797 2207 a Fh(\026)1840 2195 y Fg(\000)p Fl(G)1954 2204 y Fh(\026)1997 2195 y Fs(\))2030 2159 y Ff(\001)2096 2233 y Fk(\024)25 b Fo(e)2234 2195 y Fl(t)2261 2168 y Fi(2)2298 2195 y Fl(L)2348 2168 y Fh(d)2386 2195 y Fl(Al)2464 2166 y Fe(\000)p Fi(1)2464 2217 y(0)2735 2233 y Fk(8)15 b Fo(t)605 2435 y Fp(\()p Fo(iv)s Fp(\))994 b Fo(\026)1802 2361 y Ff(\000)1843 2435 y Fo(e)1885 2398 y Fl(t)13 b(G)1982 2357 y Fi(0)p 1982 2372 V 1982 2410 a Fh(\026)2030 2361 y Ff(\001)2096 2435 y Fk(\024)25 b Fo(e)2234 2398 y Fl(t)2261 2370 y Fi(2)2298 2398 y Fl(L)2348 2370 y Fh(d)2386 2398 y Fl(Al)2464 2368 y Fe(\000)p Fh(d)2464 2419 y Fi(0)2738 2435 y Fk(8)15 b Fo(t)24 b Fk(2)h Fp([0)p Fo(;)15 b(t)3090 2449 y Fs(0)3147 2435 y Fo(l)3176 2396 y Fg(\000)p Fl(d)3174 2459 y Fs(0)3275 2435 y Fp(])0 2620 y Fc(The)34 b(same)i(b)s(ounds)c (hold,)j(but)f(with)g(an)g(extra)i(factor)g Fo(B)h Fp(=)32 b Fo(B)5 b Fp(\()p Fk(k)p Fp(\010)p Fk(k)2469 2634 y Fg(1)2547 2620 y Fo(;)15 b(C)q(;)g(m;)g(l)r(;)g(r)m(;)g(\016)s Fp(\))38 b Fc(in)c(fron)m(t)h(of)g(the)g(exp)s(o-)0 2746 y(nen)m(tial,)30 b(if)f Fo(\026)h Fc(is)g(replaced)f(b)m(y)i(the)f(m)m (ulticanonical)f(measure)h Fo(\027)6 b Fc(.)0 2915 y Fj(Pr)-5 b(o)g(of.)50 b Fp(In)e(what)g(follo)m(ws)g 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y(one.)41 b(As)29 b(w)m(e)h(will)c(see,)31 b(conditioning)c(on)i (the)h(exterior)f(of)h(\003)2111 3974 y Fl(j)2178 3960 y Fp(is)f(a)g(simple)f(w)m(a)m(y)i(to)g(tak)m(e)h(care)f(of)g(suc)m(h)f (a)g(nast)m(y)0 4087 y(p)s(ossibilit)m(y)-8 b(.)0 4213 y(Since)1435 4254 y Ff(X)1414 4449 y Fl(x)p Fg(2)p Fs(\003)1554 4458 y Fh(j)1603 4340 y Fk(kr)1724 4354 y Fl(x)1770 4340 y Fo(G)1841 4303 y Fs(ext)1945 4340 y Fk(k)1990 4303 y Fs(2)1990 4363 y Fg(1)2093 4340 y Fk(\024)25 b Fo(k)s(L)2301 4303 y Fl(d)2343 4340 y Fo(l)2372 4287 y Fg(\000)2440 4262 y Fi(1)p 2440 4272 31 4 v 2440 4309 a(2)2370 4364 y Fs(0)0 4605 y Fp(w)m(e)31 b(get)1198 4789 y Fo(\026)1253 4716 y Ff(\000)1294 4789 y Fo(\026)1349 4803 y Fl(j)1387 4789 y Fp(\()p Fo(e)1464 4752 y Fl(t)14 b(G)1562 4724 y Fi(ext)1655 4789 y Fp(\))1690 4716 y Ff(\001)1758 4789 y Fk(\024)25 b Fo(\026)p Fp(\()p Fo(e)1986 4752 y Fl(t\026)2057 4761 y Fh(j)2092 4752 y Fs(\()p Fl(G)2177 4724 y Fi(ext)2266 4752 y Fs(\))2299 4789 y Fp(\))p Fo(e)2376 4752 y Fl(t)2403 4724 y Fi(2)2440 4752 y Fl(k)r(L)2531 4724 y Fh(d)2568 4752 y Fl(l)2591 4706 y Fe(\000)2651 4688 y Fi(1)p 2651 4696 V 2651 4730 a(2)2591 4774 y(0)3713 4789 y Fp(\(4)p Fo(:)p Fp(9\))0 4952 y(The)33 b(term)g Fo(\026)p Fp(\()p Fo(e)542 4919 y Fl(t\026)613 4928 y Fh(j)648 4919 y Fs(\()p Fl(G)733 4892 y Fi(ext)822 4919 y Fs(\))855 4952 y Fp(\))h(can)f(also)g (b)s(e)g(estimated)g(b)m(y)h(the)f(Herbst)g(argumen)m(t)h(if)e(w)m(e)h (recall)g(the)g(observ)-5 b(ation)0 5079 y(that,)31 b(under)e(the)i (mixing)d(assumption,)h Fo(c)1482 5093 y Fl(\026)1557 5079 y Fk(\024)24 b Fo(k)19 b Fp(ln)14 b Fo(L)30 b Fp(\(see)h(the)g (pro)s(of)e(of)i(prop)s(osition)d(4.3\).)42 b(W)-8 b(e)32 b(get)1227 5254 y Fo(c)1266 5268 y Fl(\026)1350 5167 y Ff(X)1346 5362 y Fl(x)p Fg(2)p Fs(\003)1501 5254 y Fk(kr)1622 5268 y Fl(x)1668 5254 y Fo(\026)1723 5216 y Fl(\034)1723 5276 y(j)1768 5254 y Fp(\()p Fo(G)1874 5216 y Fs(ext)1978 5254 y Fp(\))p Fk(k)2058 5216 y Fs(2)2058 5276 y Fg(1)2161 5254 y Fk(\024)25 b Fo(k)s(L)2369 5216 y Fl(d)p Fg(\000)p Fs(1)2520 5254 y Fp(ln)14 b Fo(L)0 5558 y Fi(20)p Fh(=settembr)r(e=)p Fi(2000)29 b Fs([30])1204 b Fp(4.7)p eop %%Page: 31 31 31 30 bop 0 91 a Fp(so)31 b(that)1416 218 y Fo(\026)1471 144 y Ff(\000)1512 218 y Fo(e)1554 180 y Fl(t\026)1625 189 y Fh(j)1660 180 y Fs(\()p Fl(G)1745 153 y Fi(ext)1835 180 y Fs(\))1868 144 y Ff(\001)1935 218 y Fk(\024)25 b Fo(e)2073 180 y Fl(t)2100 153 y Fi(2)2136 180 y Fl(k)r(L)2227 153 y Fh(d)p Fe(\000)p Fi(1)2357 180 y Fs(ln)12 b Fl(L)3668 218 y Fp(\(4)p Fo(:)p Fp(10\))0 396 y(Clearly)29 b(\(4.9\))j(and)e (\(4.10\))i(complete)f(the)g(pro)s(of.)0 572 y(\()p Fo(ii)p Fp(\).)41 b(First)26 b(w)m(e)i(observ)m(e)f(that,)i(b)s(ecause)e(of)g (the)g(equiv)-5 b(alence)27 b(of)g(ensem)m(bles)f(\(see)i(3.2)g(p)s (oin)m(t)e(\(1\)\),)k(sup)3567 594 y Fl(\013)3634 572 y Fk(k)3679 499 y Ff(\000)3721 572 y Fo(g)3767 539 y Fl(\016)3764 595 y(\013)3829 572 y Fk(\000)0 747 y Fo(\030)44 714 y Fl(\016)40 770 y(\013)92 674 y Ff(\001)133 747 y Fk(k)178 761 y Fg(1)291 747 y Fk(\024)36 b Fo(k)s Fp(.)59 b(Next,)39 b(giv)m(en)e Fo(\013)p Fp(,)i(let)d Fo(V)1343 761 y Fl(\013)1431 747 y Fp(b)s(e)g(an)h(\()1740 711 y Fs(1)p 1740 726 37 4 v 1740 779 a(2)1787 747 y Fo(l)1827 670 y Fi(1)p 1827 680 31 4 v 1827 716 a(2)1814 772 y Fs(0)1898 747 y Fk(\000)24 b Fp(2)p Fo(r)s Fp(\){neigh)m(b)s(orho)s(o)s (d)35 b(of)h(the)h(cub)s(e)f Fo(C)3298 761 y Fl(\013)3350 747 y Fp(.)59 b(F)-8 b(or)37 b Fo(l)3630 761 y Fs(0)3708 747 y Fp(large)0 874 y(enough)30 b(the)h(sets)f Fk(f)p Fo(V)745 888 y Fl(\013)798 874 y Fk(g)h Fp(satisfy)e(the)i(condition)e (of)h(prop)s(osition)e(4.1.)42 b(Moreo)m(v)m(er)956 1155 y Fk(kr)1077 1169 y Fl(x)1123 1155 y Fo(\026)1178 1169 y Fl(V)1221 1178 y Fh(\013)1272 1081 y Ff(\000)1313 1155 y Fo(g)1359 1117 y Fl(\016)1356 1177 y(\013)1429 1155 y Fk(\000)19 b Fo(\030)1563 1117 y Fl(\016)1559 1177 y(\013)1611 1081 y Ff(\001)1653 1155 y Fk(k)1698 1169 y Fg(1)1800 1155 y Fk(\024)1896 1026 y Ff(\032)1980 1141 y Fo(k)f(e)2087 1107 y Fg(\000)2155 1083 y Fi(1)p 2155 1093 V 2155 1129 a(2)2197 1107 y Fl(ml)2296 1044 y Fi(1)p 2296 1052 V 2296 1085 a(2)2285 1129 y(0)2438 1141 y Fp(if)29 b Fo(x)c Fk(2)g Fo(@)2737 1107 y Fs(+)2732 1163 y Fl(r)2798 1141 y Fo(V)2851 1155 y Fl(\013)1980 1249 y Fp(0)413 b(otherwise)0 1430 y(for)30 b Fo(l)166 1444 y Fs(0)237 1430 y Fp(large)h(enough.)0 1557 y(Th)m(us,)d(part)h(\()p Fo(ii)p Fp(\))g(of)g(prop)s(osition)d(4.1)k(together)g(the)f(usual)e (conditioning)f Fo(\026)p Fp(\()p Fk(\001)p Fp(\))g(=)f Fo(\026)2927 1483 y Ff(\000)2968 1557 y Fo(\026)3023 1571 y Fl(j)3062 1557 y Fp(\()p Fk(\001)p Fp(\))3157 1483 y Ff(\001)3228 1557 y Fp(giv)m(e)k(\(see)g(also)g(the)0 1684 y(argumen)m(t)i(leading)e(to)i(form)m(ula)f(\(4.10\)\))1146 1922 y Fo(\026)1201 1849 y Ff(\000)1242 1922 y Fo(e)1284 1885 y Fl(t)p Fs(\()p Fl(G)1396 1894 y Fh(\027)1437 1885 y Fg(\000)p Fl(G)1551 1894 y Fh(\026)1594 1885 y Fs(\))1627 1849 y Ff(\001)1694 1922 y Fk(\024)25 b Fo(e)1832 1885 y Fl(t)1859 1857 y Fi(2)1895 1885 y Fl(L)1945 1857 y Fh(d)1983 1885 y Fl(k)13 b(l)2058 1855 y Fe(\000)p Fh(d)2058 1907 y Fi(0)2151 1922 y Fo(\026)2206 1849 y Ff(\000)2247 1922 y Fo(e)2289 1885 y Fl(t\026)2360 1894 y Fh(j)2395 1885 y Fs(\()p Fl(G)2480 1894 y Fh(\027)2520 1885 y Fg(\000)p Fl(G)2634 1894 y Fh(\026)2677 1885 y Fs(\))2710 1849 y Ff(\001)1694 2116 y Fk(\024)25 b Fo(e)1832 2079 y Fl(t)1859 2052 y Fi(2)1895 2079 y Fl(L)1945 2052 y Fh(d)1983 2079 y Fl(k)13 b Fs(\()p Fl(l)2086 2049 y Fe(\000)p Fh(d)2086 2101 y Fi(0)2175 2079 y Fs(+)p Fl(L)2281 2052 y Fe(\000)p Fi(1)2377 2079 y Fs(ln)g Fl(L)p Fs(\))1694 2311 y Fk(\024)25 b Fo(e)1832 2273 y Fl(t)1859 2246 y Fi(2)1895 2273 y Fl(L)1945 2246 y Fh(d)1983 2273 y Fl(Al)2061 2244 y Fe(\000)p Fh(d)2061 2295 y Fi(0)2335 2311 y Fk(8)15 b Fo(t)24 b Fk(2)h Fp([0)p Fo(;)15 b(t)2687 2325 y Fs(0)2729 2311 y Fp(])3668 2106 y(\(4)p Fo(:)p Fp(11\))0 2523 y(for)30 b Fo(L)g Fp(large)h(enough.)0 2699 y(\()p Fo(iii)p Fp(\).)42 b(Notice)32 b(that)965 2890 y Fk(kr)1086 2904 y Fl(x)1132 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Fp(0)h(b)s(e)f(an)g(arbitrary)-8 b(,)39 b(large)f(constan)m(t)h(that) f(will)d(b)s(e)i(\014xed)0 3216 y(further)29 b(on)h(and)g(let)g Fo(t)773 3183 y Fs(2)773 3238 y Fg(\003)840 3216 y Fp(=)975 3180 y Fs(1)p 947 3195 92 4 v 947 3247 a Fl(N)1005 3256 y Fh(j)1065 3142 y Ff(\000)1106 3216 y Fp(1)21 b Fk(_)f Fo(M)25 b Fp(En)m(t)1511 3230 y Fl(\027)1557 3216 y Fp(\()p Fo(f)1647 3183 y Fs(2)1687 3216 y Fp(\))1722 3142 y Ff(\001)1764 3216 y Fp(.)41 b(W)-8 b(e)31 b(distinguish)c(b)s(et)m(w)m(een)k(t)m(w)m (o)h(cases.)0 3386 y(In)h(the)h(\014rst)f(one,)j(w)m(e)e(supp)s(ose)f Fo(t)1181 3400 y Fg(\003)1253 3386 y Fk(\025)e Fo(M)10 b Fp(.)52 b(According)33 b(to)i(the)f(general)g(discussion)d(w)m(e)k (can)f(safely)f(apply)g(an)0 3514 y Fo(L)62 3528 y Fg(1)169 3514 y Fp(b)s(ound)28 b(to)k(get)1039 3713 y Fo(\027)1105 3639 y Ff(\000)1147 3713 y Fo(f)1202 3675 y Fs(2)1242 3713 y Fo(;)15 b(G)1353 3639 y Ff(\001)1395 3657 y Fs(2)1461 3713 y Fk(\024)25 b Fp(2)p Fk(k)p Fo(G)p Fk(k)1763 3675 y Fs(2)1763 3735 y Fg(1)1867 3713 y Fk(\024)g Fo(k)s(N)2096 3675 y Fs(2)2086 3735 y Fl(j)2162 3713 y Fk(\024)2293 3651 y Fo(k)p 2269 3692 99 4 v 2269 3775 a(M)2378 3713 y(N)2451 3727 y Fl(j)2505 3713 y Fp(En)m(t)2650 3727 y Fl(\027)2695 3713 y Fp(\()p Fo(f)2785 3675 y Fs(2)2826 3713 y Fp(\))852 b(\(5)p Fo(:)p Fp(5\))0 3964 y(where)30 b(w)m(e)h(ha)m(v)m(e)g(used)f(the)g(h)m(yp)s(otheses)h(on)f Fk(k)p Fo(G)p Fk(k)1721 3978 y Fg(1)1830 3964 y Fp(and)f(the)i (de\014nition)d(of)i Fo(t)2703 3978 y Fg(\003)2745 3964 y Fp(.)0 4134 y(In)g(the)g(second)h(case,)g(w)m(e)g(supp)s(ose)e Fo(t)1290 4148 y Fg(\003)1357 4134 y Fk(\024)c Fo(M)40 b Fp(and)30 b(w)m(e)h(\014rst)f(write)1309 4391 y Fo(\027)6 b Fp(\()p Fo(f)1450 4354 y Fs(2)1490 4391 y Fo(;)15 b(G)p Fp(\))1636 4354 y Fs(2)1703 4391 y Fk(\024)25 b Fp(2)p Fo(\027)1910 4317 y Ff(\000)1952 4391 y Fo(f)2007 4354 y Fs(2)2047 4391 y Fo(G)2118 4317 y Ff(\001)2160 4335 y Fs(2)2221 4391 y Fp(+)20 b(2)p Fo(\027)6 b Fp(\()p Fo(G)p Fp(\))2549 4354 y Fs(2)3713 4391 y Fp(\(5)p Fo(:)p Fp(6\))0 4649 y(Using)35 b(the)h(equiv)-5 b(alence)35 b(of)h(ensem)m(bles,)g(\(see)h(p)s(oin)m(t)e(1)h(of)g(prop)s(osition)d (3.3\))k(and)e(the)h(fact)h(that)f Fo(\026)p Fp(\()p Fo(G)p Fp(\))f(=)f(0,)0 4777 y Fk(j)p Fo(\027)6 b Fp(\()p Fo(G)p Fp(\))p Fk(j)27 b(\024)e Fo(k)33 b Fp(for)d(some)h(constan)m(t)h Fo(k)s Fp(.)41 b(Th)m(us,)29 b(w)m(e)i(can)g(fo)s(cus)e(our)h(atten)m (tion)i(on)e(the)h(\014rst)e(term)i Fo(\027)3390 4703 y Ff(\000)3432 4777 y Fo(f)3487 4744 y Fs(2)3527 4777 y Fo(G)3598 4703 y Ff(\001)3640 4722 y Fs(2)3681 4777 y Fp(.)0 4905 y(By)38 b(applying)d(prop)s(osition)h(4.3)j(with)d Fo(t)1415 4919 y Fs(0)1493 4905 y Fp(=)h Fo(M)10 b Fp(,)40 b(w)m(e)e(get)h(that)f(there)g(exist)f(constan)m(ts)i Fo(A;)15 b(B)43 b Fp(dep)s(ending)35 b(on)0 5032 y Fk(k)p Fp(\010)p Fk(k)156 5046 y Fg(1)234 5032 y Fo(;)15 b(C)q(;)g(m;)g(l)r(;) g(t)602 5046 y Fs(0)644 5032 y Fo(;)g(r)m(;)g Fk(k)p Fo(g)s Fk(k)898 5046 y Fg(1)978 5032 y Fo(;)g(\016)1058 5046 y Fs(0)1130 5032 y Fp(suc)m(h)30 b(that)985 5340 y Fo(\027)6 b Fp(\()p Fo(e)1113 5302 y Fl(t)1140 5311 y Fe(\003)1179 5302 y Fl(G)1240 5340 y Fp(\))26 b Fk(\024)f Fo(B)5 b(e)1513 5298 y Fl(t)1540 5271 y Fi(2)1540 5317 y Fe(\003)1578 5298 y Fl(AL)1683 5271 y Fh(d)1721 5298 y Fl(\032)1759 5307 y Fh(j)1793 5225 y Ff(\002)1831 5298 y Fl(\032)1869 5307 y Fh(j)1902 5298 y Fs(+)1969 5274 y Fi(ln)11 b Fh(L)p 1969 5284 107 4 v 2001 5320 a(L)2087 5225 y Ff(\003)2155 5340 y Fp(=)24 b Fo(B)5 b(e)2366 5298 y Fl(t)2393 5271 y Fi(2)2393 5317 y Fe(\003)2432 5298 y Fl(AN)2545 5307 y Fh(j)2578 5225 y Ff(\002)2616 5298 y Fl(\032)2654 5307 y Fh(j)2688 5298 y Fs(+)2754 5274 y Fi(ln)12 b Fh(L)p 2754 5284 V 2786 5320 a(L)2872 5225 y Ff(\003)0 5558 y Fi(20)p Fh(=settembr)r(e=)p Fi(2000)29 b Fs([34])1204 b Fp(5.3)p eop %%Page: 35 35 35 34 bop 0 91 a Fp(Th)m(us,)30 b(b)m(y)g(applying)e(lemma)i(5.1)h (with)e Fo(t)d Fp(=)f Fo(t)1587 105 y Fg(\003)1628 91 y Fp(,)31 b(w)m(e)g(get)775 396 y Fo(\027)841 322 y Ff(\000)882 396 y Fo(f)937 358 y Fs(2)977 396 y Fo(G)1048 322 y Ff(\001)1091 340 y Fs(2)1157 396 y Fk(\024)1253 268 y Ff(\022)1345 334 y Fp(1)p 1330 375 75 4 v 1330 458 a Fo(t)1363 472 y Fg(\003)1431 396 y Fp(ln)14 b Fo(B)25 b Fp(+)20 b Fo(t)1740 410 y Fg(\003)1781 396 y Fo(AN)1922 410 y Fl(j)1961 322 y Ff(\002)1999 396 y Fo(\032)2046 410 y Fl(j)2104 396 y Fp(+)2206 334 y(ln)14 b Fo(L)p 2206 375 153 4 v 2251 458 a(L)2370 322 y Ff(\003)2428 396 y Fp(+)2544 334 y(1)p 2529 375 75 4 v 2529 458 a Fo(t)2562 472 y Fg(\003)2630 396 y Fp(En)m(t)2775 410 y Fl(\027)2821 396 y Fp(\()p Fo(f)2911 358 y Fs(2)2951 396 y Fp(\))2986 268 y Ff(\023)3053 286 y Fs(2)1157 678 y Fk(\024)25 b Fo(k)s(N)1376 692 y Fl(j)1429 549 y Ff(\022)1496 678 y Fp(1)c(+)1653 577 y Ff(\020)1707 678 y Fo(M)10 b(A)1873 640 y Fs(2)1914 604 y Ff(\002)1952 678 y Fo(\032)1999 692 y Fl(j)2057 678 y Fp(+)2159 616 y(ln)k Fo(L)p 2159 657 153 4 v 2205 740 a(L)2323 604 y Ff(\003)2361 623 y Fs(2)2422 678 y Fp(+)2550 616 y(1)p 2524 657 99 4 v 2524 740 a Fo(M)2633 577 y Ff(\021)2702 678 y Fp(En)m(t)2847 692 y Fl(\027)2893 678 y Fp(\()p Fo(f)2983 640 y Fs(2)3023 678 y Fp(\))3058 549 y Ff(\023)3713 527 y Fp(\(5)p Fo(:)p Fp(7\))0 969 y(No)m(w,)37 b(from)d(\(5.6\),)39 b(\(5.5\))d(and)f(\(5.7\),)j(one)d (can)g(c)m(ho)s(ose)h(\014rst)e Fo(M)45 b Fp(large)35 b(enough)g(and)f(then)h Fo(\032)3302 983 y Fl(j)3375 969 y Fp(small)e(enough)0 1096 y(and)d Fo(L)g Fp(large)g(enough)h(in)e (order)g(to)j(conclude)d(the)i(pro)s(of.)90 b Fb(\003)0 1458 y Ft(5.2)35 b(Normal)e(density.)0 1612 y Fp(Here)i(w)m(e)h(treat)g (instead)e(the)h(case)h(of)e(\\normal")h(densit)m(y)f Fo(\032)2137 1626 y Fl(j)2176 1612 y Fp(,)i(namely)e(w)m(e)h(assume)g (that)g Fo(\032)3262 1626 y Fl(j)3333 1612 y Fk(\025)d Fo(\032)3483 1626 y Fs(0)3559 1612 y Fp(for)j(some)0 1740 y(constan)m(t)d Fo(\032)414 1754 y Fs(0)485 1740 y Fp(indep)s(enden)m(t)c(of)i Fo(L)p Fp(.)0 1973 y Fq(Prop)s(osition)36 b(5.3.)53 b Fc(Assume)29 b Fo(U)10 b(S)5 b(M)10 b(T)j Fp(\()p Fo(C)q(;)i(m;)g(l)r Fp(\))p Fc(.)42 b(Then,)29 b(for)h(an)m(y)g Fo(\017)25 b(>)g Fp(0)31 b Fc(and)e(an)m(y)h Fp(0)c Fo(<)f(\032)3172 1987 y Fs(0)3238 1973 y Fo(<)3345 1937 y Fs(1)p 3345 1952 37 4 v 3345 2004 a(2)3392 1973 y Fc(,)30 b(there)g(exists)0 2100 y Fo(C)65 2114 y Fl(\017)130 2100 y Fc(suc)m(h)g(that)h(for)f(all)g Fo(\032)845 2114 y Fs(0)911 2100 y Fk(\024)25 b Fo(\032)1054 2114 y Fl(j)1117 2100 y Fk(\024)1224 2064 y Fs(1)p 1224 2079 V 1224 2131 a(2)1271 2100 y Fc(,)1013 2328 y Fo(\027)6 b Fp(\()p Fo(f)1154 2290 y Fs(2)1194 2328 y Fo(;)15 b(G)p Fp(\))1340 2290 y Fs(2)1407 2328 y Fk(\024)25 b(j)p Fp(\003)1591 2342 y Fl(j)1630 2328 y Fk(j)1670 2254 y Ff(\002)1708 2328 y Fo(C)1773 2342 y Fl(\017)1828 2328 y Fp(+)20 b Fo(\017)15 b Fp(En)m(t)2116 2342 y Fl(\027)2162 2328 y Fp(\()p Fo(f)2252 2290 y Fs(2)2292 2328 y Fp(\))21 b(+)f Fo(L)2501 2290 y Fs(2)2541 2328 y Fk(E)2589 2342 y Fl(\027)2634 2328 y Fp(\()p Fo(f)5 b(;)15 b(f)10 b Fp(\))2849 2254 y Ff(\003)0 2556 y Fj(Pr)-5 b(o)g(of.)34 b Fp(Fix)f Fo(f)43 b Fp(with)31 b Fo(\027)6 b Fp(\()p Fo(f)876 2523 y Fs(2)917 2556 y Fp(\))30 b(=)f(1,)35 b(together)g(with)c Fo(\017)f(>)g Fp(0,)k Fo(\032)2077 2570 y Fs(0)2148 2556 y Fo(>)c Fp(0)j(and)g Fo(M)43 b Fp(large)33 b(enough)g(\(ho)m(w)h(large)f(will)e(b)s(e)0 2684 y(sp)s(eci\014ed)e(later)h(on\).)0 2838 y(As)k(in)e(prop)s (osition)g(5.2)j(w)m(e)f(de\014ne)f Fo(t)1315 2805 y Fs(2)1315 2860 y Fg(\003)1388 2838 y Fp(=)1546 2802 y Fs(1)p 1501 2817 127 4 v 1501 2869 a Fg(j)p Fs(\003)1573 2878 y Fh(j)1606 2869 y Fg(j)1654 2764 y Ff(\000)1696 2838 y Fp(1)20 b Fk(_)g Fo(M)25 b Fp(En)m(t)2100 2852 y Fl(\027)2146 2838 y Fp(\()p Fo(f)2236 2805 y Fs(2)2276 2838 y Fp(\))2311 2764 y Ff(\001)2387 2838 y Fp(where)33 b(the)h(n)m(um)m(b)s(er)f(of)h(particles)f Fo(N)3696 2852 y Fl(j)3768 2838 y Fp(has)0 2965 y(b)s(een)d(replaced)f(b)m(y)i (the)f(v)m(olume)g Fk(j)p Fp(\003)1257 2979 y Fl(j)1296 2965 y Fk(j)h Fp(b)s(ecause)f(w)m(e)h(are)g(assuming)e(that)i Fo(\032)2613 2979 y Fl(j)2676 2965 y Fk(\025)25 b Fo(\032)2819 2979 y Fs(0)2860 2965 y Fp(.)0 3146 y(If)30 b Fo(t)124 3160 y Fg(\003)191 3146 y Fk(\025)25 b Fo(M)10 b Fp(,)30 b(then)g(w)m(e)h(can)g(pro)s(ceed)f(as)h(in)e(\(5.5\))j(to)f(get)1369 3430 y Fo(\027)1435 3357 y Ff(\000)1477 3430 y Fo(f)1532 3393 y Fs(2)1572 3430 y Fo(;)15 b(G)1683 3357 y Ff(\001)1725 3374 y Fs(2)1791 3430 y Fk(\024)1922 3369 y Fo(k)p 1898 3409 99 4 v 1898 3493 a(M)2007 3430 y Fk(j)p Fp(\003)2095 3444 y Fl(j)2134 3430 y Fk(j)g Fp(En)m(t)2320 3444 y Fl(\027)2365 3430 y Fp(\()p Fo(f)2455 3393 y Fs(2)2495 3430 y Fp(\))1183 b(\(5)p Fo(:)p Fp(8\))0 3658 y(and)30 b(the)g(pro)s(of)g(is)f(\014nished)f(pro)m(vided)h(that)i Fo(M)41 b Fp(w)m(as)30 b(tak)m(en)i(large)e(enough.)0 3839 y(Let)h(us)f(no)m(w)g(examine)g(the)h(m)m(uc)m(h)f(more)h (complicate)f(case)h(of)g Fo(t)2235 3853 y Fg(\003)2302 3839 y Fk(\024)25 b Fo(M)10 b Fp(.)0 3993 y(W)-8 b(e)32 b(de\014ne)d Fk(f)p Fo(C)529 4007 y Fl(\013)581 3993 y Fk(g)626 4007 y Fl(\013)p Fg(2)p Fl(I)795 3993 y Fp(to)i(b)s(e)f(a)h (collection)f(of)g(cub)s(es)g(of)g(side)g Fo(l)2177 4007 y Fs(0)2243 3993 y Fk(\035)25 b Fo(r)33 b Fp(m)m(ultiple)28 b(of)i Fo(l)j Fp(inside)28 b(\003)3273 4007 y Fl(j)3311 3993 y Fp(,)j(suc)m(h)f(that)1224 4263 y Fo(i)p Fp(\))259 b(dist)o(\()p Fo(C)1796 4277 y Fl(\013)1848 4263 y Fo(;)15 b(C)1953 4277 y Fl(\014)2003 4263 y Fp(\))25 b Fk(\025)g Fo(l)2199 4185 y Fi(1)p 2199 4195 31 4 v 2199 4231 a(2)2186 4287 y Fs(0)2427 4263 y Fk(8)g Fo(\013)h Fk(6)p Fp(=)f Fo(\014)1192 4465 y(ii)q Fp(\))259 b(dist)o(\()p Fo(C)1796 4479 y Fl(\013)1848 4465 y Fo(;)15 b(@)5 b Fp(\003)2004 4479 y Fl(j)2043 4465 y Fp(\))26 b Fk(\025)f Fo(l)2240 4388 y Fi(1)p 2240 4398 V 2240 4434 a(2)2227 4489 y Fs(0)2468 4465 y Fk(8)g Fo(\013)1161 4668 y(iii)q Fp(\))259 b Fk(j)p Fp(\003)1637 4682 y Fl(j)1696 4668 y Fk(n)21 b([)1823 4682 y Fl(\013)1874 4668 y Fo(C)1939 4682 y Fl(\013)1991 4668 y Fk(j)k(\024)g Fo(l)2166 4614 y Fg(\000)2234 4590 y Fi(1)p 2234 4600 V 2234 4636 a(2)2164 4692 y Fs(0)2280 4668 y Fk(j)p Fp(\003)2368 4682 y Fl(j)2407 4668 y Fk(j)0 4931 y Fp(Clearly)k(suc)m(h)h(a)h(construction)f(is)g(alw)m(a)m(ys)h(p) s(ossible.)39 b(De\014ne)30 b(also)h Fo(C)2415 4898 y Fs(in)n(t)2408 4954 y Fl(\013)2532 4931 y Fp(=)25 b Fk(f)p Fo(x)h Fk(2)f Fo(C)2902 4945 y Fl(\013)2994 4931 y Fp(:)51 b Fo(d)p Fp(\()p Fo(x;)15 b(C)3316 4898 y Fl(c)3309 4954 y(\013)3362 4931 y Fp(\))26 b Fk(\025)f Fo(l)3548 4884 y Fs(1)p Fl(=)p Fs(4)3546 4955 y(0)3678 4931 y Fk(g)31 b Fp(and)0 5058 y Fo(C)72 5025 y Fs(in)n(t)188 5058 y Fp(=)25 b Fk([)345 5072 y Fl(\013)397 5058 y Fo(C)469 5025 y Fs(in)n(t)462 5081 y Fl(\013)560 5058 y Fp(.)0 5213 y(Next)f(w)m(e)f(observ)m(e)h(that,)h(without)d(loss)g(of)h (generalit)m(y)-8 b(,)26 b(w)m(e)d(can)g(replace)g Fo(G)g Fp(b)m(y)g Fo(G)5 b Fk(\000)g Fo(\016)20 b(N)2982 5227 y Fs(\003)3032 5236 y Fh(j)3070 5213 y Fp(,)25 b Fo(\016)i Fp(b)s(eing)21 b(an)i(arbitrary)0 5340 y(constan)m(t)k(indep)s(enden)m (t)c(of)i Fo(x)p Fp(,)h(b)s(ecause)g Fo(N)1478 5354 y Fs(\003)1528 5363 y Fh(j)1591 5340 y Fp(=)f Fo(N)1760 5354 y Fl(j)1824 5340 y Fp(almost)g(surely)-8 b(.)38 b(Our)24 b(c)m(hoice)i(of)f Fo(\016)k Fp(will)23 b(b)s(e)h(made)i (later)f(\(see)0 5558 y Fi(20)p Fh(=settembr)r(e=)p Fi(2000)k Fs([35])1204 b Fp(5.4)p eop %%Page: 36 36 36 35 bop 0 91 a Fp(\(5.16\)\))33 b(but)d(w)m(e)i(an)m(ticipate)f(that) g(it)f(will)f(b)s(e)h(almost)g(indep)s(enden)m(t)f(of)i Fo(l)2547 105 y Fs(0)2619 91 y Fp(and)f(that)h(under)e Fo(U)10 b(S)5 b(M)10 b(T)j Fp(\()p Fo(C)q(;)i(m;)g(l)r Fp(\),)0 218 y(there)33 b(exists)f(a)g(constan)m(t)i Fo(\016)s Fp(\()p Fo(C)q(;)15 b(m;)g(l)r(;)g Fk(k)p Fo(g)s Fk(k)1443 232 y Fg(1)1524 218 y Fp(\))33 b(suc)m(h)f(that)h Fo(\016)f Fk(\024)c Fo(\016)s Fp(\()p Fo(C)q(;)15 b(m;)g(l)r(;)g Fk(k)p Fo(g)s Fk(k)2678 232 y Fg(1)2760 218 y Fp(\))32 b(uniformly)e(in)h Fo(L)p Fp(.)46 b(Finally)30 b(w)m(e)0 345 y(set)h Fo(g)188 312 y Fl(\016)185 367 y(x)231 345 y Fp(\()p Fo(\033)s Fp(\))26 b(:=)f Fo(g)546 359 y Fl(x)593 345 y Fp(\()p Fo(\033)s Fp(\))c Fk(\000)f Fo(\016)s(\033)s Fp(\()p Fo(x)p Fp(\))32 b(and)1368 577 y Fo(G)1439 540 y Fs(ext)1568 577 y Fp(=)1772 491 y Ff(X)1664 691 y Fl(x)p Fg(2)p Fs(\003)1804 700 y Fh(j)1838 691 y Fg(n)p Fl(C)1931 673 y Fi(in)n(t)2011 504 y Ff(\002)2065 577 y Fo(g)2111 540 y Fl(\016)2108 600 y(x)2174 577 y Fk(\000)20 b Fo(\026)p Fp(\()p Fo(g)2401 540 y Fl(\016)2398 600 y(x)2444 577 y Fp(\))2494 504 y Ff(\003)1379 878 y Fo(G)1450 840 y Fs(in)n(t)1568 878 y Fp(=)1711 791 y Ff(X)1664 992 y Fl(x)p Fg(2)p Fl(C)1810 974 y Fi(in)n(t)1891 804 y Ff(\002)1944 878 y Fo(g)1990 840 y Fl(\016)1987 900 y(x)2053 878 y Fk(\000)g Fo(\026)p Fp(\()p Fo(g)2280 840 y Fl(\016)2277 900 y(x)2323 878 y Fp(\))2373 804 y Ff(\003)0 1183 y Fp(Then)29 b(w)m(e)i(write)812 1317 y Fo(\027)878 1243 y Ff(\000)920 1317 y Fo(f)975 1279 y Fs(2)1015 1317 y Fo(;)15 b(G)1126 1279 y Fs(ext)1230 1243 y Ff(\001)1272 1261 y Fs(2)1338 1317 y Fk(\024)25 b Fp(2)15 b Fo(\027)1560 1243 y Ff(\000)1602 1317 y Fo(f)1657 1279 y Fs(2)1697 1317 y Fo(G)1768 1279 y 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Fo(l)1714 1968 y Fs(0)1786 1954 y Fp(but)g(p)s(ossibly)d(dep)s(enden)m(t)i(on)h Fo(M)42 b Fp(and)30 b(the)i(last)f(inequalit)m(y)0 2080 y(holds)e(if)g Fo(M)41 b Fp(and)29 b Fo(l)653 2094 y Fs(0)725 2080 y Fp(are)h(c)m(hosen)h(appropriately)-8 b(.)0 2207 y(Ab)s(o)m(v)m(e)30 b(w)m(e)f(ha)m(v)m(e)h(applied)d(lemma)h (5.1)i(with)d Fo(t)e Fp(=)g Fo(t)1764 2221 y Fg(\003)1835 2207 y Fp(together)30 b(with)e(prop)s(osition)e(4.4)k(p)s(oin)m(t)e(\() p Fo(i)p Fp(\))h(to)h(b)s(ound)d(the)0 2334 y(\014rst)33 b(term)g(and)g(the)h(equiv)-5 b(alence)33 b(of)h(ensem)m(bles)f(\(see)h (prop)s(osition)d(3.2)k(p)s(oin)m(t)d(1\))j(together)g(with)d Fo(iii)p Fp(\))j(ab)s(o)m(v)m(e)0 2461 y(to)c(b)s(ound)d(the)j(second)f (one.)0 2637 y(W)-8 b(e)32 b(no)m(w)e(turn)f(to)i(the)g(relev)-5 b(an)m(t)31 b(term)f Fo(\027)6 b Fp(\()p Fo(f)1518 2604 y Fs(2)1558 2637 y Fo(;)15 b(G)1669 2604 y Fs(in)n(t)1762 2637 y Fp(\))1797 2604 y Fs(2)1838 2637 y Fp(.)0 2788 y(Let)31 b Fk(F)228 2802 y Fs(0)299 2788 y Fp(b)s(e)e(the)h 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3165 y Fk(j)g(F)1986 3179 y Fs(0)2027 3091 y Ff(\001)g(\001)2126 3107 y Fs(2)2187 3165 y Fp(+)20 b(2)15 b Fo(\027)2404 3091 y Ff(\000)2446 3165 y Fo(f)2501 3127 y Fs(2)2541 3165 y Fo(;)g(\027)2647 3091 y Ff(\000)2689 3165 y Fo(G)2760 3127 y Fs(in)n(t)2867 3165 y Fk(j)h(F)2973 3179 y Fs(0)3014 3091 y Ff(\001)e(\001)3112 3107 y Fs(2)3193 3165 y Fo(:)495 b Fp(\(5)p Fo(:)p Fp(9\))0 3370 y(F)-8 b(or)37 b(simplicit)m(y)c(let)i Fo(\027)770 3384 y Fs(0)811 3370 y Fp(\()p Fk(\001)p Fp(\))g(:=)g Fo(\027)6 b Fp(\()p Fk(\001)15 b(j)h(F)1304 3384 y Fs(0)1345 3370 y Fp(\),)37 b Fo(f)1497 3337 y Fs(2)1487 3392 y(0)1572 3370 y Fp(:=)d Fo(f)1757 3337 y Fs(2)1797 3370 y Fo(=\027)1887 3384 y Fs(0)1928 3370 y Fp(\()p Fo(f)2018 3337 y Fs(2)2059 3370 y Fp(\))i(and)f Fo(s)2355 3337 y Fs(2)2355 3392 y Fg(\003)2431 3370 y Fp(:=)2597 3334 y Fl(M)p 2572 3349 127 4 v 2572 3401 a Fg(j)p Fs(\003)2644 3410 y Fh(j)2677 3401 y Fg(j)2725 3370 y Fp(En)m(t)2870 3384 y Fl(\027)2906 3393 y Fi(0)2947 3370 y Fp(\()p 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4007 y Fp(is)f(the)g(pro)s(duct)f(of)i(standard)f (canonical)g(measures)g(on)g(eac)m(h)h(cub)s(e)f Fo(C)3226 4021 y Fl(\013)3320 4007 y Fp(with)f(a)i(certain)0 4134 y(n)m(um)m(b)s(er)29 b(of)i(particles)e(and)h(b)s(oundary)e (conditions.)39 b(Th)m(us)30 b(a)g(simple)f(T)-8 b(a)m(ylor)30 b(expansion)g(giv)m(es)1311 4339 y Fo(\027)1356 4353 y Fs(0)1412 4238 y Ff(\020)1467 4339 y Fo(e)1509 4301 y Fl(s)1543 4310 y Fe(\003)1581 4301 y Fs(\()p Fl(G)1666 4274 y Fi(in)n(t)1746 4301 y Fg(\000)p Fl(\027)1839 4310 y Fi(0)1875 4301 y Fs(\()p Fl(G)1960 4274 y Fi(in)n(t)2040 4301 y Fs(\)\))2101 4238 y Ff(\021)2181 4339 y Fk(\024)25 b Fo(e)2319 4301 y Fl(K)t(s)2418 4274 y Fi(2)2418 4320 y Fe(\003)2457 4301 y Fg(j)p Fs(\003)2529 4310 y Fh(j)2562 4301 y Fg(j)0 4544 y Fp(where)30 b Fo(K)37 b Fp(no)m(w)30 b(ma)m(y)h(dep)s(end)e(on)h Fo(l)1232 4558 y Fs(0)1273 4544 y Fp(.)41 b(Th)m(us)1220 4796 y Fo(\027)1265 4810 y Fs(0)1305 4796 y Fp(\()p Fo(f)1395 4758 y Fs(2)1385 4818 y(0)1436 4796 y Fo(;)15 b(G)1547 4758 y Fs(in)n(t)1639 4796 y Fp(\))26 b Fk(\024)f Fo(K)7 b Fk(j)p Fp(\003)1968 4810 y Fl(j)2006 4796 y Fk(j)p Fo(s)2074 4810 y Fg(\003)2136 4796 y Fp(+)2257 4734 y(1)p 2238 4775 V 2238 4858 a Fo(s)2281 4872 y Fg(\003)2348 4796 y Fp(En)m(t)2494 4810 y Fl(\027)2530 4819 y Fi(0)2585 4796 y Fo(f)2640 4758 y Fs(2)2630 4818 y(0)0 5062 y Fp(and)30 b(similarily)c(for)k Fo(G)784 5029 y Fs(in)n(t)907 5062 y Fp(replaced)f(b)m(y)i Fk(\000)p Fo(G)1534 5029 y Fs(in)n(t)1626 5062 y Fp(.)40 b(The)30 b(de\014nition)e(of)j Fo(s)2429 5076 y Fg(\003)2500 5062 y Fp(yields)1073 5314 y Fo(\027)1118 5328 y Fs(0)1159 5314 y Fp(\()p Fo(f)1249 5276 y Fs(2)1239 5336 y(0)1289 5314 y Fo(;)15 b(G)1400 5276 y Fs(in)n(t)1493 5314 y Fp(\))1528 5276 y Fs(2)1594 5314 y Fk(\024)25 b Fp(2)p Fo(K)1819 5276 y Fs(2)1860 5240 y Ff(\000)1902 5314 y Fo(M)30 b Fp(+)2148 5252 y(1)p 2122 5293 99 4 v 2122 5376 a Fo(M)2231 5240 y Ff(\001)2272 5314 y Fk(j)p Fp(\003)2360 5328 y Fl(j)2399 5314 y Fk(j)15 b Fp(En)m(t)2585 5328 y Fl(\027)2621 5337 y Fi(0)2661 5314 y Fp(\()p Fo(f)2751 5276 y Fs(2)2741 5336 y(0)2792 5314 y Fp(\))0 5558 y Fi(20)p Fh(=settembr)r(e=)p Fi(2000)29 b Fs([36])1204 b Fp(5.5)p eop %%Page: 37 37 37 36 bop 0 91 a Fp(Finally)-8 b(,)48 b(since)d Fo(\027)629 105 y Fs(0)716 91 y Fp(is)f(a)i(pro)s(duct)f(measure)g(it)h(certainly)e (satis\014es)i(the)g(logarithmic)e(Sob)s(olev)h(inequalit)m(y)0 218 y(En)m(t)145 232 y Fl(\027)181 241 y Fi(0)222 218 y Fp(\()p Fo(f)312 185 y Fs(2)302 241 y(0)352 218 y Fp(\))26 b Fk(\024)f Fo(C)7 b Fp(\()p Fo(l)643 232 y Fs(0)683 218 y Fp(\))p Fk(E)766 232 y Fl(\027)802 241 y Fi(0)843 218 y Fp(\()p Fo(f)923 232 y Fs(0)964 218 y Fo(;)15 b(f)1049 232 y Fs(0)1090 218 y Fp(\),)30 b(and)g(w)m(e)h(can)g(conclude)e(that) 1281 443 y Fo(\027)1347 370 y Ff(\000)1389 443 y Fo(\027)1440 370 y Ff(\000)1481 443 y Fo(f)1536 406 y Fs(2)1576 443 y Fo(;)15 b(G)1687 406 y Fs(in)n(t)1795 443 y Fk(j)g(F)1900 457 y Fs(0)1941 370 y Ff(\001)g(\001)2040 386 y Fs(2)2106 443 y Fk(\024)25 b Fo(K)2286 406 y Fg(0)2311 443 y Fk(E)2359 457 y Fl(\027)2404 443 y Fp(\()p Fo(f)5 b(;)15 b(f)10 b Fp(\))1049 b(\(5)p Fo(:)p Fp(10\))0 669 y(where)30 b Fo(K)347 636 y Fg(0)397 669 y Fp(=)25 b Fo(K)577 636 y Fg(0)603 669 y Fp(\()p Fk(k)p Fp(\010)p Fk(k)794 683 y Fg(1)871 669 y Fo(;)15 b Fk(k)p Fo(g)s Fk(k)1047 683 y Fg(1)1126 669 y Fo(;)g(r)m(;)g(l)1271 683 y Fs(0)1314 669 y Fo(;)g(M)10 b Fp(\).)0 849 y(The)28 b(second)h(term)g(in)f(the)h (r.h.s.)39 b(of)29 b(\(5.9\))i(needs)e(some)g(more)g(reductions.)39 b(W)-8 b(e)30 b(recall)e(\014rst)g(some)h(de\014nitions)0 976 y(in)m(tro)s(duced)g(in)g(section)h(4.)0 1103 y(Giv)m(en)42 b Fo(\021)49 b Fk(2)c Fp(\012)541 1117 y Fs(\003)591 1126 y Fh(j)671 1103 y Fp(and)d Fo(n)j Fk(2)g Fp([0)p Fo(;)15 b(:)g(:)g(:)j(;)d(l)1368 1070 y Fl(d)1366 1126 y Fs(0)1410 1103 y Fp(],)46 b(write)c Fo(\026)1806 1070 y Fl(\021)r(;n)1806 1126 y(\013)1917 1103 y Fp(\()p Fk(\001)p Fp(\))k(:=)f Fo(\026)2254 1056 y Fl(\021)r(;\025)p Fs(\()p Fl(\021)r(;n)p Fs(\))2254 1130 y Fl(C)2306 1139 y Fh(\013)2526 1103 y Fp(\()p Fk(\001)p Fp(\))e(to)h(denote)f(the)f(grand)g(canonical) 0 1230 y(Gibbs)25 b(measure)h(on)h Fo(C)798 1244 y Fl(\013)876 1230 y Fp(with)e(b)s(oundary)g(condition)g Fo(\021)30 b Fp(and)c(constan)m(t)i(c)m(hemical)e(p)s(oten)m(tial)g Fo(\025)p Fp(\()p Fo(\021)s(;)15 b(n)p Fp(\))28 b(suc)m(h)e(that)0 1358 y Fo(\026)55 1310 y Fl(\021)r(;\025)p Fs(\()p Fl(\021)r(;n)p Fs(\))55 1384 y Fl(C)107 1393 y Fh(\013)327 1358 y Fp(\()p Fo(N)435 1372 y Fl(C)487 1381 y Fh(\013)538 1358 y Fp(\))g(=)f Fo(n)p Fp(.)38 b(W)-8 b(e)27 b(will)22 b(use)j(the)h(standard)e (notation)i Fo(\027)2219 1314 y Fl(\021)2213 1384 y(C)2265 1393 y Fh(\013)2311 1384 y Fl(;n)2408 1358 y Fp(for)f(the)g(corresp)s (onding)f(canonical)h(Gibbs)0 1485 y(measure.)0 1612 y(With)30 b(this)f(notation)i(w)m(e)g(de\014ne)e(\(see)j(\(4.7\)\),)698 1865 y Fo(\030)742 1827 y Fl(\016)738 1887 y(\013)789 1865 y Fp(\()p Fo(\021)s(;)15 b(n)p Fp(\))27 b(=)1172 1778 y Ff(X)1125 1979 y Fl(x)p Fg(2)p Fl(C)1271 1961 y Fi(in)n(t)1267 1997 y Fh(\013)1367 1865 y Fo(\026)1422 1827 y Fl(\021)r(;n)1422 1887 y(\013)1532 1865 y Fp(\()p Fo(g)1610 1879 y Fl(x)1677 1865 y Fk(\000)20 b Fo(\016)s(\033)s Fp(\()p Fo(x)p Fp(\)\))93 b(;)106 b Fo(\030)2291 1827 y Fl(\016)2287 1887 y(\013)2339 1865 y Fp(\()p Fo(\021)s Fp(\))26 b(:=)f Fo(\030)2648 1827 y Fl(\016)2644 1887 y(\013)2696 1865 y Fp(\()p Fo(\021)s(;)15 b(N)2892 1879 y Fl(C)2944 1888 y Fh(\013)2996 1865 y Fp(\()p Fo(\021)s Fp(\)\))694 2165 y Fo(g)740 2128 y Fl(\016)737 2188 y(\013)789 2165 y Fp(\()p Fo(\021)s(;)g(n)p Fp(\))27 b(=)1172 2079 y Ff(X)1125 2280 y Fl(x)p Fg(2)p Fl(C)1271 2261 y Fi(in)n(t)1267 2298 y Fh(\013)1367 2165 y Fo(\027)1418 2122 y Fl(\021)1412 2192 y(C)1464 2201 y Fh(\013)1510 2192 y Fl(;n)1581 2165 y Fp(\()p Fo(g)1659 2179 y Fl(x)1725 2165 y Fk(\000)20 b Fo(\016)s(\033)s Fp(\()p Fo(x)p Fp(\)\))93 b(;)107 b Fo(g)2342 2128 y Fl(\016)2339 2188 y(\013)2391 2165 y Fp(\()p Fo(\021)s Fp(\))26 b(:=)f Fo(g)2702 2128 y Fl(\016)2699 2188 y(\013)2751 2165 y Fp(\()p Fo(\021)s(;)15 b(N)2947 2179 y Fl(C)2999 2188 y Fh(\013)3051 2165 y Fp(\()p Fo(\021)s Fp(\)\))0 2504 y(and)1076 2636 y Fo(G)1147 2650 y Fl(\027)1193 2636 y Fp(\()p Fo(\021)s Fp(\))26 b(:=)g Fo(\027)1524 2563 y Ff(\000)1566 2636 y Fo(G)1637 2599 y Fs(in)n(t)1744 2636 y Fk(j)16 b(F)1850 2650 y Fs(0)1891 2563 y Ff(\001)1958 2636 y Fp(=)2054 2550 y Ff(X)2096 2741 y Fl(\013)2200 2636 y Fp([)f Fo(g)2286 2599 y Fl(\016)2283 2659 y(\013)2336 2636 y Fp(\()p Fo(\021)s Fp(\))21 b Fk(\000)f Fo(\026)p Fp(\()p Fo(g)2702 2599 y Fl(\016)2699 2659 y(\013)2751 2636 y Fp(\))15 b(])1073 2909 y Fo(G)1144 2923 y Fl(\026)1193 2909 y Fp(\()p Fo(\021)s Fp(\))26 b(:=)1459 2823 y Ff(X)1501 3014 y Fl(\013)1605 2909 y Fp([)15 b Fo(\030)1689 2872 y Fl(\016)1685 2932 y(\013)1737 2909 y Fp(\()p Fo(\021)s Fp(\))21 b Fk(\000)f Fo(\026)p Fp(\()p Fo(\030)2101 2872 y Fl(\016)2097 2932 y(\013)2149 2909 y Fp(\))15 b(])0 3181 y(Then)29 b(w)m(e)i(write)349 3413 y Fo(\027)6 b Fp(\()p Fo(f)490 3375 y Fs(2)530 3413 y Fo(;)15 b(\027)6 b Fp(\()p Fo(G)727 3375 y Fs(in)n(t)835 3413 y Fk(j)16 b(F)941 3427 y Fs(0)982 3413 y Fp(\)\))1052 3375 y Fs(2)1118 3413 y Fp(=)25 b Fo(\027)1280 3339 y Ff(\000)1322 3413 y Fo(f)1377 3375 y Fs(2)1417 3413 y Fo(;)15 b(G)1528 3427 y Fl(\027)1574 3339 y Ff(\001)1616 3357 y Fs(2)1118 3605 y Fk(\024)25 b Fp(4)p Fo(\027)1326 3532 y Ff(\000)1367 3605 y Fo(f)1422 3568 y Fs(2)1462 3605 y Fp(\()p Fo(G)1568 3619 y Fl(\027)1635 3605 y Fk(\000)20 b Fo(G)1797 3619 y Fl(\026)1846 3605 y Fp(\))1881 3532 y Ff(\001)1923 3549 y Fs(2)1984 3605 y Fp(+)g(4)15 b(\()q Fo(\027)6 b Fp(\()p Fo(G)2328 3619 y Fl(\027)2394 3605 y Fk(\000)20 b Fo(G)2556 3619 y Fl(\026)2605 3605 y Fp(\)\))2675 3559 y Fs(2)2736 3605 y Fp(+)g(2)p Fo(\027)2939 3532 y Ff(\000)2980 3605 y Fo(f)3035 3568 y Fs(2)3075 3605 y Fo(;)15 b(G)3186 3619 y Fl(\026)3236 3532 y Ff(\001)3278 3549 y Fs(2)3668 3499 y Fp(\(5)p Fo(:)p Fp(11\))0 3835 y(Let)31 b(us)f(examine)g(the)g(three)h(terms)f(separately)-8 b(.)0 3962 y(Using)28 b(lemma)g(5.1)i(with)d Fo(t)e Fp(=)g Fo(t)1082 3976 y Fg(\003)1152 3962 y Fp(com)m(bined)j(with)f(\()p Fo(ii)p Fp(\))j(of)f(prop)s(osition)d(4.4,)k(w)m(e)f(can)g(b)s(ound)e (the)i(\014rst)e(term)i(b)m(y)823 4255 y(4)p Fo(\027)934 4181 y Ff(\000)976 4255 y Fo(f)1031 4217 y Fs(2)1071 4255 y Fp(\()p Fo(G)1177 4269 y Fl(\027)1243 4255 y Fk(\000)20 b Fo(G)1405 4269 y Fl(\026)1454 4255 y Fp(\))1489 4181 y Ff(\001)1531 4199 y Fs(2)1597 4255 y Fk(\024)25 b Fo(k)s Fk(j)p Fp(\003)1831 4269 y Fl(j)1870 4255 y Fk(j)1910 4127 y Ff(\024)1958 4255 y Fp(1)c(+)2115 4181 y Ff(\000)2167 4193 y Fo(M)10 b(A)2333 4160 y Fs(2)p 2167 4234 208 4 v 2217 4320 a Fo(l)2246 4289 y Fs(2)p Fl(d)2244 4344 y Fs(0)2406 4255 y Fp(+)2534 4193 y(1)p 2507 4234 99 4 v 2507 4317 a Fo(M)2616 4181 y Ff(\001)2673 4255 y Fp(En)m(t)2818 4269 y Fl(\027)2864 4255 y Fp(\()p Fo(f)2954 4217 y Fs(2)2994 4255 y Fp(\))3029 4127 y Ff(\025)3668 4255 y Fp(\(5)p Fo(:)p Fp(12\))0 4547 y(where)30 b Fo(A)25 b Fp(=)g Fo(A)p Fp(\()p Fo(C)q(;)15 b(m;)g(l)r(;)g(M)5 b(;)15 b Fk(k)p Fo(g)s Fk(k)1119 4561 y Fg(1)1200 4547 y Fo(;)g(\032)1287 4561 y Fs(0)1328 4547 y Fp(\))31 b(is)e(the)i (constan)m(t)h(app)s(earing)d(in)g(prop)s(osition)f(4.4.)0 4674 y(Because)h(of)f(the)f(equiv)-5 b(alence)27 b(of)h(ensem)m(bles)f (\(see)h(p)s(oin)m(t)f(\(1\))i(of)e(prop)s(osition)f(3.2\))j(and)e(the) g(fact)i(that)f Fo(\026)p Fp(\()p Fo(G)3769 4688 y Fl(\027)3829 4674 y Fk(\000)0 4802 y Fo(G)71 4816 y Fl(\026)120 4802 y Fp(\))e(=)f(0)1490 4929 y(4)15 b(\()q Fo(\027)6 b Fp(\()p Fo(G)1743 4943 y Fl(\027)1809 4929 y Fk(\000)20 b Fo(G)1971 4943 y Fl(\026)2020 4929 y Fp(\)\))2091 4883 y Fs(2)2157 4929 y Fk(\024)25 b Fo(k)s(l)2332 4891 y Fs(2)p Fl(d)2330 4951 y Fs(0)3668 4929 y Fp(\(5)p Fo(:)p Fp(13\))0 5115 y(In)30 b(conclusion,)f(b)m(y)h(a)h(suitable)e(c)m(hoice)i(of)f Fo(M)41 b Fp(and)30 b Fo(l)1826 5129 y Fs(0)1897 5115 y Fp(w)m(e)h(get)731 5340 y(4)p Fo(\027)843 5266 y Ff(\000)884 5340 y Fo(f)939 5302 y Fs(2)979 5340 y Fp(\()p Fo(G)1085 5354 y Fl(\027)1152 5340 y Fk(\000)20 b Fo(G)1314 5354 y Fl(\026)1363 5340 y Fp(\))1398 5266 y Ff(\001)1440 5284 y Fs(2)1501 5340 y Fp(+)g(4)15 b(\()q Fo(\027)6 b Fp(\()p Fo(G)1845 5354 y Fl(\027)1911 5340 y Fk(\000)20 b Fo(G)2073 5354 y Fl(\026)2122 5340 y Fp(\)\))2193 5294 y Fs(2)2259 5340 y Fk(\024)25 b(j)p Fp(\003)2443 5354 y Fl(j)2481 5340 y Fk(j)2521 5266 y Ff(\000)2563 5340 y Fo(C)2628 5354 y Fl(\017)2683 5340 y Fp(+)20 b Fo(\017)15 b Fp(En)m(t)2971 5354 y Fl(\027)3032 5340 y Fo(f)3087 5302 y Fs(2)3127 5266 y Ff(\001)3668 5340 y Fp(\(5)p Fo(:)p Fp(14\))0 5558 y Fi(20)p Fh(=settembr)r(e=)p Fi(2000)29 b Fs([37])1204 b Fp(5.6)p eop %%Page: 38 38 38 37 bop 0 91 a Fp(for)30 b(a)h(suitable)e(constan)m(t)i Fo(C)985 105 y Fl(\017)1020 91 y Fp(.)0 245 y(So,)36 b(it)d(remains)h(to)h(b)s(ound)d(the)j(third)d(term)j(in)e(the)h (r.h.s.)52 b(of)35 b(\(5.11\))i Fo(\027)2569 171 y Ff(\000)2610 245 y Fo(f)2665 212 y Fs(2)2705 245 y Fo(;)15 b(G)2816 259 y Fl(\026)2866 171 y Ff(\001)2908 190 y Fs(2)2948 245 y Fp(.)53 b(Here)35 b(w)m(e)g(are)g(\(unfortu-)0 372 y(nately\))c(forced)f(to)h(distinguish)c(b)s(et)m(w)m(een)k(t)m(w)m (o)h(sub)s(cases.)0 552 y Fj(a\))f Fo(t)147 566 y Fg(\003)214 552 y Fk(\024)321 516 y Fl(M)p 321 531 77 4 v 328 588 a(l)351 566 y Fh(d)351 610 y Fi(0)408 552 y Fp(.)0 706 y(W)-8 b(e)32 b(can)e(app)s(eal)g(to)h(lemma)f(5.1)h(with)e Fo(t)c Fp(=)g Fo(t)1558 720 y Fg(\003)1630 706 y Fp(com)m(bined)30 b(with)f(\()p Fo(iii)p Fp(\))p Fo(;)j Fp(\()p Fo(iv)s Fp(\))g(of)e(prop)s(osition)e(4.4)k(to)f(get)832 960 y Fo(\027)898 886 y Ff(\000)940 960 y Fo(f)995 922 y Fs(2)1035 960 y Fo(;)15 b(G)1146 974 y Fl(\026)1196 886 y Ff(\001)1237 904 y Fs(2)1303 960 y Fk(\024)25 b Fp(2)p Fo(\027)1511 886 y Ff(\000)1552 960 y Fo(f)1607 922 y Fs(2)1662 960 y Fo(G)1733 974 y Fl(\026)1783 886 y Ff(\001)1824 904 y Fs(2)1885 960 y Fp(+)20 b(2)p Fk(j)p Fo(\027)6 b Fp(\()p Fo(G)2203 974 y Fl(\026)2253 960 y Fp(\))21 b Fk(\000)f Fo(\026)p Fp(\()p Fo(G)2561 974 y Fl(\026)2610 960 y Fp(\))p Fk(j)2670 922 y Fs(2)1303 1191 y Fk(\024)25 b Fo(k)s Fk(j)p Fp(\003)1537 1205 y Fl(j)1576 1191 y Fk(j)1616 1063 y Ff(\024)1664 1191 y Fp(1)c(+)1821 1117 y Ff(\000)1874 1129 y Fo(M)10 b(A)2040 1096 y Fs(2)p 1874 1170 208 4 v 1943 1253 a Fo(l)1970 1267 y Fs(0)2112 1191 y Fp(+)2240 1129 y(1)p 2214 1170 99 4 v 2214 1253 a Fo(M)2323 1117 y Ff(\001)2379 1191 y Fp(En)m(t)2525 1205 y Fl(\027)2570 1191 y Fp(\()p Fo(f)2660 1153 y Fs(2)2700 1191 y Fp(\))2735 1063 y Ff(\025)2804 1191 y Fp(+)20 b Fo(k)e(l)2989 1153 y Fs(2)p Fl(d)2987 1213 y Fs(0)1303 1419 y Fk(\024)25 b(j)p Fp(\003)1487 1433 y Fl(j)1526 1419 y Fk(j)1566 1346 y Ff(\000)1608 1419 y Fo(C)1673 1433 y Fl(\017)1728 1419 y Fp(+)20 b Fo(\017)15 b Fp(En)m(t)2016 1433 y Fl(\027)2077 1419 y Fo(f)2132 1382 y Fs(2)2172 1346 y Ff(\001)3668 1180 y Fp(\(5)p Fo(:)p Fp(15\))0 1648 y(for)30 b(a)h(suitable)e(c)m(hoice)i(of)g Fo(M)10 b Fp(,)30 b Fo(l)1108 1662 y Fs(0)1179 1648 y Fp(and)g(for)g(all)g Fo(L)g Fp(large)g(enough.)0 1828 y Fj(b\))120 1793 y Fl(M)p 120 1808 77 4 v 127 1865 a(l)150 1843 y Fh(d)150 1886 y Fi(0)233 1828 y Fk(\024)24 b Fo(t)361 1842 y Fg(\003)428 1828 y Fk(\024)h Fo(M)10 b Fp(.)0 2027 y(In)20 b(this)f(case)j(w)m(e)f (can)g(assume,)h(without)e(loss)g(of)h(generalit)m(y)-8 b(,)23 b(that)e Fk(j)p Fp(\003)2356 2041 y Fl(j)2395 2027 y Fk(j)f Fp(is)g(so)h(large)f(that)i Fo(t)3057 1994 y Fs(2)3057 2049 y Fg(\003)3123 2027 y Fp(=)3276 1991 y Fs(1)p 3230 2006 127 4 v 3230 2058 a Fg(j)p Fs(\003)3302 2067 y Fh(j)3336 2058 y Fg(j)3368 2027 y Fo(M)j Fp(En)m(t)3627 2041 y Fl(\027)3672 2027 y Fp(\()p Fo(f)3762 1994 y Fs(2)3802 2027 y Fp(\).)0 2154 y(W)-8 b(e)35 b(\014rst)e(need)h(to)h(simplify)c (a)j(bit)f(the)h(random)g(v)-5 b(ariable)32 b Fo(G)2167 2168 y Fl(\026)2251 2154 y Fp(b)m(y)h(reducing)g(it)h(to)g(a)h(sum)e (of)h(v)-5 b(ariables)33 b(eac)m(h)0 2281 y(of)f(whic)m(h)e(dep)s(ends) g(only)h(on)g(the)h(n)m(um)m(b)s(er)f(of)h(particles)e(in)h Fo(C)2188 2248 y Fs(in)n(t)2181 2303 y Fl(\013)2311 2281 y Fp(and)g(not)h(also)f(on)h(the)g(b)s(oundary)d(condition)0 2408 y(outside)h Fo(C)379 2422 y Fl(\013)430 2408 y Fp(.)0 2562 y(Let)j Fo(\030)209 2513 y Fl(\016)n(;)12 b Fs(0)p 274 2528 37 4 v 205 2573 a Fl(\013)315 2562 y Fp(\()p Fo(n)p Fp(\))29 b(:=)g Fo(\030)638 2529 y Fl(\016)634 2584 y(\013)685 2562 y Fp(\(0)p Fo(;)15 b(n)p Fp(\))34 b(and)e(let)g Fo(G)1312 2513 y Fs(0)p 1312 2528 41 4 v 1312 2573 a Fl(\026)1361 2562 y Fp(\()p Fo(\021)s Fp(\))e(=)1608 2494 y Ff(P)1704 2589 y Fl(\013)1771 2562 y Fp([)15 b Fo(\030)1855 2513 y Fl(\016)n(;)e Fs(0)p 1921 2528 37 4 v 1851 2573 a Fl(\013)1962 2562 y Fp(\()p Fo(N)2070 2576 y Fl(C)2122 2585 y Fh(\013)2173 2562 y Fp(\()p Fo(\021)s Fp(\)\))23 b Fk(\000)e Fo(\026)p Fp(\()p Fo(\030)2575 2513 y Fl(\016)n(;)13 b Fs(0)p 2641 2528 V 2571 2573 a Fl(\013)2682 2562 y Fp(\))i(].)47 b(Notice)34 b(that,)g(b)m(y)e (translation)0 2689 y(in)m(v)-5 b(ariance)30 b(inside)e(\003)751 2703 y Fl(j)789 2689 y Fp(,)j Fo(\030)889 2640 y Fl(\016)n(;)12 b Fs(0)p 954 2655 V 885 2700 a Fl(\013)995 2689 y Fp(\()p Fo(n)p Fp(\))30 b(is)g(indep)s(enden)m(t)e(of)i Fo(\013)p Fp(.)0 2816 y(Using)g(once)h(more)f(prop)s(osition)e(3.2)k(w)m(e)e(ha)m (v)m(e)i Fk(j)p Fo(\027)6 b Fp(\()p Fo(G)1840 2830 y Fl(\026)1910 2816 y Fk(\000)20 b Fo(G)2072 2767 y Fs(0)p 2072 2782 41 4 v 2072 2827 a Fl(\026)2121 2816 y Fp(\))p Fk(j)26 b(\024)f Fo(k)s(l)2382 2783 y Fl(d)2380 2839 y Fs(0)2424 2816 y Fp(.)41 b(Therefore)779 3108 y Fo(\027)845 3034 y Ff(\000)887 3108 y Fo(f)942 3071 y Fs(2)982 3108 y Fo(;)15 b(G)1093 3122 y Fl(\026)1142 3034 y Ff(\001)1184 3052 y Fs(2)1250 3108 y Fk(\024)25 b Fo(k)1426 2980 y Ff(\024)1474 3108 y Fo(\027)1540 3007 y Ff(\020)1595 3108 y Fo(f)1650 3071 y Fs(2)1690 3108 y Fp(\()p Fo(G)1796 3122 y Fl(\026)1865 3108 y Fk(\000)20 b Fo(G)2027 3059 y Fs(0)p 2027 3074 V 2027 3119 a Fl(\026)2077 3108 y Fp(\))2112 3007 y Ff(\021)2166 3026 y Fs(2)2227 3108 y Fp(+)g Fo(k)s(l)2397 3071 y Fs(2)p Fl(d)2395 3131 y Fs(0)2496 3108 y Fp(+)g Fo(\027)2653 3007 y Ff(\020)2707 3108 y Fo(f)2762 3071 y Fs(2)2802 3108 y Fo(;)15 b(G)2913 3059 y Fs(0)p 2913 3074 V 2913 3119 a Fl(\026)2963 3007 y Ff(\021)3017 3026 y Fs(2)3073 2980 y Ff(\025)0 3397 y Fp(No)m(w,)31 b(using)e(Lemma)i(5.1)g(with)e Fo(t)c Fp(=)g Fo(t)1335 3411 y Fg(\003)1407 3397 y Fp(together)32 b(with)d(Prop)s(osition)g(4.4)i(p)s(oin)m(t)e(\()p Fo(iii)p Fp(\),)k(w)m(e)d(get)902 3689 y Fo(\027)968 3588 y Ff(\020)1022 3689 y Fo(f)1077 3651 y Fs(2)1117 3689 y Fp(\()p Fo(G)1223 3703 y Fl(\026)1293 3689 y Fk(\000)20 b Fo(G)1455 3639 y Fs(0)p 1455 3654 V 1455 3699 a Fl(\026)1504 3689 y Fp(\))1539 3588 y Ff(\021)1593 3607 y Fs(2)1660 3689 y Fk(\024)k Fo(k)s(N)1878 3703 y Fl(j)1932 3560 y Ff(\022)1999 3689 y Fp(1)d(+)2167 3627 y Fo(M)10 b(A)2333 3594 y Fs(2)p 2167 3668 208 4 v 2235 3751 a Fo(l)2264 3720 y Fs(2)2262 3775 y(0)2405 3689 y Fp(+)2533 3627 y(1)p 2507 3668 99 4 v 2507 3751 a Fo(M)2631 3689 y Fp(En)m(t)2776 3703 y Fl(\027)2836 3689 y Fo(f)2891 3651 y Fs(2)2931 3560 y Ff(\023)0 3972 y Fp(In)30 b(conclusion)936 4099 y Fo(\027)1002 4026 y Ff(\000)1043 4099 y Fo(f)1098 4062 y Fs(2)1138 4099 y Fo(;)15 b(G)1249 4113 y Fl(\026)1299 4026 y Ff(\001)1341 4043 y Fs(2)1407 4099 y Fk(\024)25 b(j)p Fp(\003)1591 4113 y Fl(j)1629 4099 y Fk(j)1669 4026 y Ff(\000)1711 4099 y Fo(C)1776 4113 y Fl(\017)1831 4099 y Fp(+)20 b Fo(\017)15 b Fp(En)m(t)2120 4113 y Fl(\027)2180 4099 y Fo(f)2235 4062 y Fs(2)2275 4026 y Ff(\001)2337 4099 y Fp(+)20 b Fo(k)e(\027)2559 3999 y Ff(\020)2613 4099 y Fo(f)2668 4062 y Fs(2)2708 4099 y Fo(;)d(G)2819 4050 y Fs(0)p 2819 4065 41 4 v 2819 4110 a Fl(\026)2869 3999 y Ff(\021)2923 4017 y Fs(2)0 4285 y Fp(for)30 b(suitably)f(c)m(hosen)i Fo(M)40 b Fp(and)30 b Fo(l)1109 4299 y Fs(0)1150 4285 y Fp(.)0 4515 y(W)-8 b(e)27 b(are)g(left)f(with)f(the)h(estimate)h(of)g (the)f(term)h Fo(\027)1701 4414 y Ff(\020)1756 4515 y Fo(f)1811 4482 y Fs(2)1851 4515 y Fo(;)15 b(G)1962 4466 y Fs(0)p 1962 4481 V 1962 4526 a Fl(\026)2011 4414 y Ff(\021)2066 4433 y Fs(2)2106 4515 y Fp(.)40 b(It)26 b(is)f(at)i(this)e(p)s(oin)m(t)h(that)h(the)f(substraction)f(with)0 4642 y(the)31 b(free)f(parameter)h Fo(\016)j Fp(made)c(at)i(the)e(b)s (eginning)e(b)s(ecomes)i(imp)s(ortan)m(t.)0 4796 y(Let)35 b Fo(\016)j Fp(b)s(e)c(suc)m(h)h(that)g(for)f(one)h(\(and)f(therefore)h (all\))f(cub)s(e)g Fo(C)2150 4810 y Fl(\013)2202 4796 y Fp(,)2291 4761 y Fl(d)p 2273 4776 73 4 v 2273 4828 a(ds)2356 4796 y Fo(\030)2400 4747 y Fl(\016)n(;)13 b Fs(0)p 2466 4762 37 4 v 2396 4807 a Fl(\013)2507 4796 y Fp(\()p Fo(s)p Fp(\))p Fk(j)2645 4810 y Fl(s)p Fs(=)t(\026)-40 b Fl(n)2817 4796 y Fp(=)32 b(0)j(where)k(\026)-50 b Fo(n)32 b Fp(=)g Fo(\032)3504 4810 y Fl(j)3542 4796 y Fk(j)p Fo(C)3632 4810 y Fl(\013)3684 4796 y Fk(j)p Fp(.)54 b(By)0 4923 y(de\014nition,)28 b Fo(\016)34 b Fp(is)c(indep)s(enden)m(t)e(of)j Fo(\013)f Fp(and)g(it)g(is)f(giv)m(en)i(b)m(y)f(the)h(follo)m(wing)d (expression)1337 5253 y Fo(\016)h Fp(=)1513 5172 y Fo(\026)1568 5123 y Fs(0)p 1568 5138 V Fl(;\025)p Fs(\(0)p 1697 5138 V 1 w Fl(;)t Fs(\026)-40 b Fl(n)o Fs(\))1568 5199 y Fl(C)1620 5208 y Fh(\013)1833 5172 y Fp(\()1868 5104 y Ff(P)1965 5200 y Fl(x)p Fg(2)p Fl(C)2111 5181 y Fi(in)n(t)2107 5218 y Fh(\013)2211 5172 y Fo(g)2254 5186 y Fl(x)2300 5172 y Fo(;)15 b(N)2413 5186 y Fl(C)2465 5195 y Fh(\013)2516 5172 y Fp(\))p 1513 5232 1039 4 v 1622 5340 a Fo(\026)1677 5291 y Fs(0)p 1677 5306 37 4 v Fl(;\025)p Fs(\(0)p 1806 5306 V Fl(;)t Fs(\026)-40 b Fl(n)p Fs(\))1677 5367 y Fl(C)1729 5376 y Fh(\013)1942 5340 y Fp(\()p Fo(N)2050 5358 y Fl(C)2106 5340 y Fi(in)n(t)2102 5376 y Fh(\013)2191 5340 y Fo(;)15 b(N)2304 5354 y Fl(C)2356 5363 y Fh(\013)2408 5340 y Fp(\))3668 5253 y(\(5)p Fo(:)p Fp(16\))0 5558 y Fi(20)p Fh(=settembr)r(e=)p Fi(2000)29 b Fs([38])1204 b Fp(5.7)p eop %%Page: 39 39 39 38 bop 0 91 a Fp(so)31 b(that,)g(thanks)f(to)h(the)f(mixing)f(h)m (yp)s(othesis,)g Fk(j)p Fo(\016)s Fk(j)e(\024)e Fo(k)33 b Fp(uniformly)28 b(in)h Fo(L)p Fp(.)0 219 y(Notice)i(that)g(with)e (this)h(c)m(hoice,)738 535 y Fk(j)15 b Fo(\030)822 485 y Fl(\016)n(;)d Fs(0)p 887 500 37 4 v 818 545 a Fl(\013)928 535 y Fp(\()p Fo(n)p Fp(\))21 b Fk(\000)f Fo(\030)1209 485 y Fl(\016)n(;)12 b Fs(0)p 1274 500 V 1205 545 a Fl(\013)1315 535 y Fp(\()5 b(\026)-50 b Fo(n)p Fp(\))15 b Fk(j)26 b Fp(=)f Fk(j)1657 411 y Ff(Z)1748 433 y Fl(n)1712 617 y Fs(\026)-40 b Fl(n)1813 535 y Fo(ds)1918 411 y Ff(Z)2009 433 y Fl(s)1973 617 y Fs(\026)f Fl(n)2063 535 y Fo(dt)2185 473 y(d)2232 440 y Fs(2)p 2169 514 121 4 v 2169 597 a Fo(dt)2249 571 y Fs(2)2301 535 y Fo(\030)2345 497 y Fl(\016)2341 557 y(\013)2392 535 y Fp(\(0)p Fo(;)15 b(t)p Fp(\))g Fk(j)27 b(\024)e Fo(k)2820 473 y Fp(\()p Fo(n)20 b Fk(\000)k Fp(\026)-49 b Fo(n)o Fp(\))3110 440 y Fs(2)p 2820 514 332 4 v 2963 597 a Fp(\026)f Fo(n)3668 535 y Fp(\(5)p Fo(:)p Fp(17\))0 848 y(b)s(ecause)30 b(of)h(lemma)f(3.1)h(\(p)s(oin)m (t)f(\()p Fo(ii)p Fp(\))i(of)e(\(2\)\).)42 b(In)30 b(turn)f(\(5.17\))k (implies,)28 b(in)h(particular,)g(that)1015 1138 y Fo(\027)1081 1037 y Ff(\020)1135 1138 y Fk(j)p Fo(\030)1204 1088 y Fl(\016)n(;)13 b Fs(0)p 1270 1103 37 4 v 1200 1149 a Fl(\013)1331 1138 y Fk(\000)20 b Fo(\030)1466 1088 y Fl(\016)n(;)12 b Fs(0)p 1531 1103 V 1462 1149 a Fl(\013)1572 1138 y Fp(\()5 b(\026)-50 b Fo(n)p Fp(\))p Fk(j)1722 1037 y Ff(\021)1802 1138 y Fk(\024)25 b Fo(k)s(\026)2018 1037 y Ff(\020)2072 1138 y Fk(j)p Fo(\030)2141 1088 y Fl(\016)n(;)13 b Fs(0)p 2207 1103 V 2137 1149 a Fl(\013)2268 1138 y Fk(\000)20 b Fo(\030)2403 1088 y Fl(\016)n(;)12 b Fs(0)p 2468 1103 V 2399 1149 a Fl(\013)2509 1138 y Fp(\()5 b(\026)-50 b Fo(n)p Fp(\))p Fk(j)2659 1037 y Ff(\021)2739 1138 y Fk(\024)25 b Fo(k)786 b Fp(\(5)p Fo(:)p Fp(18\))0 1392 y(again)30 b(b)s(ecause)h(of)f(the)h(mixing)d (condition.)0 1592 y Fq(Lemma)34 b(5.4.)53 b Fc(Assume)32 b Fo(U)10 b(S)5 b(M)10 b(T)j Fp(\()p Fo(C)q(;)i(m;)g(l)r Fp(\))p Fc(.)46 b(Then,)32 b(for)f(an)m(y)h Fo(\017)c(>)f Fp(0)p Fc(,)33 b(there)f(exists)g(a)g(constan)m(t)h Fo(C)3460 1606 y Fl(\017)3527 1592 y Fc(suc)m(h)e(that)0 1720 y(for)f(all)f (function)g Fo(f)10 b Fc(,)905 1915 y Fo(\027)970 1814 y Ff(\020)1025 1915 y Fo(f)1080 1877 y Fs(2)1120 1915 y Fo(;)15 b(G)1231 1865 y Fs(0)p 1231 1880 41 4 v 1231 1925 a Fl(\026)1280 1814 y Ff(\021)1335 1832 y Fs(2)1401 1915 y Fk(\024)25 b(j)p Fp(\003)1585 1929 y Fl(j)1623 1915 y Fk(j)1663 1841 y Ff(\002)1702 1915 y Fo(C)1767 1929 y Fl(\017)1822 1915 y Fp(+)20 b Fo(C)1978 1929 y Fl(\017)2028 1915 y Fo(L)2090 1877 y Fs(2)2130 1915 y Fk(E)2178 1929 y Fl(\027)2223 1915 y Fp(\()p Fo(f)5 b(;)15 b(f)10 b Fp(\))20 b(+)g Fo(\017)15 b Fp(En)m(t)2746 1929 y Fl(\027)2792 1915 y Fp(\()p Fo(f)2882 1877 y Fs(2)2922 1915 y Fp(\))2957 1841 y Ff(\003)0 2166 y Fj(Pr)-5 b(o)g(of.)32 b Fp(First)d(write)h(\()p Fo(n)814 2180 y Fl(\013)891 2166 y Fk(\021)25 b Fo(N)1060 2180 y Fl(C)1112 2189 y Fh(\013)1163 2166 y Fp(\()p Fo(\021)s Fp(\)\))329 2474 y Fo(\027)395 2373 y Ff(\020)449 2474 y Fo(f)504 2436 y Fs(2)545 2474 y Fo(;)15 b(G)656 2424 y Fs(0)p 656 2439 V 656 2485 a Fl(\026)705 2373 y Ff(\021)759 2392 y Fs(2)825 2474 y Fk(\024)25 b Fp(2)p Fo(\027)1017 2373 y Ff(\020)1072 2474 y Fo(f)1127 2436 y Fs(2)1183 2388 y Ff(X)1182 2583 y Fl(\013)p Fg(2)p Fl(I)1331 2400 y Ff(\002)1384 2474 y Fo(\030)1428 2424 y Fl(\016)n(;)12 b Fs(0)p 1493 2439 37 4 v 1424 2485 a Fl(\013)1534 2474 y Fp(\()p Fo(n)1624 2488 y Fl(\013)1676 2474 y Fp(\))20 b Fk(\000)g Fo(\030)1866 2424 y Fl(\016)n(;)12 b Fs(0)p 1931 2439 V 1862 2485 a Fl(\013)1972 2474 y Fp(\()5 b(\026)-50 b Fo(n)p Fp(\))2112 2400 y Ff(\003)2151 2373 y(\021)2205 2392 y Fs(2)2266 2474 y Fp(+)20 b(2)p Fo(\027)2453 2373 y Ff(\020)2508 2388 y(X)2507 2583 y Fl(\013)p Fg(2)p Fl(I)2656 2400 y Ff(\002)2709 2474 y Fo(\030)2753 2424 y Fl(\016)n(;)12 b Fs(0)p 2818 2439 V 2749 2485 a Fl(\013)2859 2474 y Fp(\()p Fo(n)2949 2488 y Fl(\013)3001 2474 y Fp(\))20 b Fk(\000)g Fo(\030)3191 2424 y Fl(\016)n(;)13 b Fs(0)p 3257 2439 V 3187 2485 a Fl(\013)3297 2474 y Fp(\()5 b(\026)-50 b Fo(n)p Fp(\))3437 2400 y Ff(\003)3476 2373 y(\021)3530 2392 y Fs(2)825 2783 y Fk(\024)25 b Fp(2)p Fo(\027)1017 2682 y Ff(\020)1072 2783 y Fo(f)1127 2746 y Fs(2)1183 2697 y Ff(X)1182 2892 y Fl(\013)p Fg(2)p Fl(I)1331 2710 y Ff(\002)1384 2783 y Fo(\030)1428 2734 y Fl(\016)n(;)12 b Fs(0)p 1493 2749 V 1424 2794 a Fl(\013)1534 2783 y Fp(\()p Fo(n)1624 2797 y Fl(\013)1676 2783 y Fp(\))20 b Fk(\000)g Fo(\030)1866 2734 y Fl(\016)n(;)12 b Fs(0)p 1931 2749 V 1862 2794 a Fl(\013)1972 2783 y Fp(\()5 b(\026)-50 b Fo(n)p Fp(\))2112 2710 y Ff(\003)2151 2682 y(\021)2205 2701 y Fs(2)2266 2783 y Fp(+)20 b Fo(k)s Fk(j)p Fo(I)7 b Fk(j)2504 2746 y Fs(2)825 3099 y Fk(\024)25 b Fp(2)p Fo(\027)1017 2998 y Ff(\020)1072 3099 y Fo(f)1127 3062 y Fs(2)1183 3013 y Ff(X)1182 3208 y Fl(\013)p Fg(2)p Fl(I)1331 3025 y Ff(\002)1384 3099 y Fo(\030)1428 3050 y Fl(\016)n(;)12 b Fs(0)p 1493 3065 V 1424 3110 a Fl(\013)1534 3099 y Fp(\()p Fo(n)1624 3113 y Fl(\013)1676 3099 y Fp(\))20 b Fk(\000)g Fo(\030)1866 3050 y Fl(\016)n(;)12 b Fs(0)p 1931 3065 V 1862 3110 a Fl(\013)1972 3099 y Fp(\()5 b(\026)-50 b Fo(n)p Fp(\))2112 3025 y Ff(\003)2151 2998 y(\021)2205 3017 y Fs(2)2266 3099 y Fp(+)20 b Fo(k)2433 3038 y Fk(j)p Fp(\003)2521 3052 y Fl(j)2560 3038 y Fk(j)p 2433 3078 152 4 v 2460 3161 a Fo(M)2611 3099 y Fp(En)m(t)2756 3113 y Fl(\027)2801 3099 y Fp(\()p Fo(f)2891 3062 y Fs(2)2932 3099 y Fp(\))0 3452 y(where)37 b(w)m(e)h(ha)m(v)m(e)g(used)f(once)h (more)f(the)h(equiv)-5 b(alence)37 b(of)g(ensem)m(bles,)i(prop)s (osition)c(3.2)k(p)s(oin)m(t)d(\(1\),)k(together)0 3580 y(with)29 b(\(5.17\))k(and)c(the)i(fact)g(that)g Fo(t)1212 3594 y Fg(\003)1279 3580 y Fk(\025)1386 3544 y Fl(M)p 1386 3559 77 4 v 1393 3616 a(l)1416 3594 y Fh(d)1416 3638 y Fi(0)1473 3580 y Fp(.)0 3837 y(W)-8 b(e)31 b(can)e(th)m(us)h(fo) s(cus)e(our)h(atten)m(tion)i(on)e(the)h(relev)-5 b(an)m(t)30 b(term)f Fo(\027)2210 3736 y Ff(\020)2264 3837 y Fo(f)2319 3804 y Fs(2)2375 3769 y Ff(P)2471 3865 y Fl(\013)p Fg(2)p Fl(I)2624 3764 y Ff(\002)2677 3837 y Fo(\030)2721 3788 y Fl(\016)n(;)12 b Fs(0)p 2786 3803 37 4 v 2717 3848 a Fl(\013)2827 3837 y Fp(\()p Fo(n)2917 3851 y Fl(\013)2969 3837 y Fp(\))20 b Fk(\000)g Fo(\030)3159 3788 y Fl(\016)n(;)13 b Fs(0)p 3225 3803 V 3155 3848 a Fl(\013)3265 3837 y Fp(\()5 b(\026)-50 b Fo(n)p Fp(\))3405 3764 y Ff(\003)3444 3736 y(\021)3498 3755 y Fs(2)3539 3837 y Fp(.)40 b(Thanks)0 4001 y(to)31 b(\(5.17\))i(w)m(e)e(can)f(write)785 4322 y Fo(\027)836 4221 y Ff(\020)890 4322 y Fo(f)945 4284 y Fs(2)1002 4235 y Ff(X)1000 4430 y Fl(\013)p Fg(2)p Fl(I)1149 4248 y Ff(\002)1202 4322 y Fo(\030)1246 4272 y Fl(\016)n(;)12 b Fs(0)p 1311 4287 V 1242 4332 a Fl(\013)1352 4322 y Fp(\()p Fo(n)1442 4336 y Fl(\013)1494 4322 y Fp(\))21 b Fk(\000)e Fo(\030)1684 4272 y Fl(\016)n(;)13 b Fs(0)p 1750 4287 V 1680 4332 a Fl(\013)1791 4322 y Fp(\()5 b(\026)-50 b Fo(n)p Fp(\))1931 4248 y Ff(\003)1969 4221 y(\021)2023 4240 y Fs(2)2089 4322 y Fk(\024)25 b Fo(k)19 b(\027)2302 4221 y Ff(\020)2356 4322 y Fo(f)2411 4284 y Fs(2)2467 4235 y Ff(X)2466 4430 y Fl(\013)p Fg(2)p Fl(I)2625 4260 y Fp(\()p Fo(n)2715 4274 y Fl(\013)2787 4260 y Fk(\000)25 b Fp(\026)-50 b Fo(n)p Fp(\))2968 4227 y Fs(2)p 2625 4301 384 4 v 2794 4384 a Fp(\026)h Fo(n)3020 4221 y Ff(\021)3074 4240 y Fs(2)3668 4322 y Fp(\(5)p Fo(:)p Fp(19\))0 4675 y(W)-8 b(e)35 b(no)m(w)f(analyze)g(the)g(r.h.s.)50 b(of)34 b(\(5.19\).)53 b(Let)35 b Fk(h\001i)f Fp(denote)h(the)f(a)m(v)m(erage)i (\(normalized)d(sum\))g(o)m(v)m(er)i(the)f(cub)s(es)0 4803 y Fk(f)p Fo(C)110 4817 y Fl(\014)159 4803 y Fk(g)204 4817 y Fl(\014)s Fg(2)p Fl(I)340 4803 y Fp(.)41 b(Thanks)29 b(to)j(the)e(conserv)-5 b(ation)31 b(la)m(w)f(w)m(e)h(exp)s(ect)g Fk(h)p Fo(n)2200 4817 y Fl(\014)2249 4803 y Fk(i)g Fp(to)g(b)s(e)f (quite)g(close)h(to)k(\026)-49 b Fo(n)o Fp(.)41 b(In)30 b(fact,)i(it)e(is)f(easy)0 4931 y(to)i(c)m(hec)m(k)h(that)f Fk(h)p Fo(n)643 4945 y Fl(\014)692 4931 y Fk(i)21 b(\000)j Fp(\026)-50 b Fo(n)25 b Fp(=)1047 4895 y Fs(1)p 1025 4910 81 4 v 1025 4962 a Fg(j)p Fl(I)5 b Fg(j)1132 4863 y Ff(P)1228 4958 y Fl(x)p Fg(2)p Fs(\003)1368 4967 y Fh(j)1401 4958 y Fg(n)p Fl(C)1500 4857 y Ff(\002)1538 4931 y Fo(\032)1585 4945 y Fl(j)1643 4931 y Fk(\000)20 b Fo(\021)s Fp(\()p Fo(x)p Fp(\))1904 4857 y Ff(\003)1943 4931 y Fp(.)41 b(Notice)31 b(that)1054 5150 y Ff(X)1100 5341 y Fl(y)1200 5236 y Fk(kr)1321 5250 y Fl(y)1365 5163 y Ff(\002)1471 5150 y(X)1403 5349 y Fl(x)p Fg(2)p Fs(\003)1543 5358 y Fh(j)1577 5349 y Fg(n)p Fl(C)1671 5236 y Fp(\()p Fo(\032)1753 5250 y Fl(j)1811 5236 y Fk(\000)20 b Fo(\021)s Fp(\()p Fo(x)p Fp(\)\))2107 5163 y Ff(\003)2147 5182 y Fs(2)2187 5236 y Fk(k)2232 5199 y Fs(2)2232 5259 y Fg(1)2335 5236 y Fk(\024)25 b Fo(k)18 b Fk(j)p Fp(\003)2584 5250 y Fl(j)2643 5236 y Fk(n)j Fo(C)7 b Fk(j)2806 5199 y Fs(3)0 5558 y Fi(20)p Fh(=settembr)r(e=)p Fi(2000)29 b Fs([39])1204 b Fp(5.8)p eop %%Page: 40 40 40 39 bop 0 91 a Fp(Moreo)m(v)m(er,)33 b(using)c(the)h(mixing)f (condition)g(together)j(with)d(the)h(P)m(oincar)m(\023)-43 b(e)32 b(inequalit)m(y)d(for)h Fo(\026)p Fp(,)g(w)m(e)h(also)f(ha)m(v)m (e)1282 387 y Fk(j)p Fo(\026)1362 313 y Ff(\000)1472 301 y(X)1404 500 y Fl(x)p Fg(2)p Fs(\003)1544 509 y Fh(j)1578 500 y Fg(n)p Fl(C)1671 313 y Ff(\002)1709 387 y Fo(\032)1756 401 y Fl(j)1815 387 y Fk(\000)20 b Fo(\021)s Fp(\()p Fo(x)p Fp(\))2076 313 y Ff(\003)q(\001)2156 387 y Fk(j)26 b(\024)f Fo(k)2364 326 y Fp(ln)14 b Fo(L)p 2364 366 153 4 v 2409 449 a(L)2528 387 y Fk(j)p Fp(\003)2616 401 y Fl(j)2654 387 y Fk(j)1197 686 y Fp(V)-8 b(ar)1339 700 y Fl(\026)1387 612 y Ff(\000)1497 600 y(X)1429 799 y Fl(x)p Fg(2)p Fs(\003)1569 808 y Fh(j)1603 799 y Fg(n)p Fl(C)1697 612 y Ff(\002)1735 686 y Fo(\032)1782 700 y Fl(j)1840 686 y Fk(\000)20 b Fo(\021)s Fp(\()p Fo(x)p Fp(\))2101 612 y Ff(\003)q(\001)2207 686 y Fk(\024)25 b Fo(k)s Fk(j)p Fp(\003)2441 700 y Fl(j)2500 686 y Fk(n)c Fo(C)7 b Fk(j)2663 649 y Fs(3)0 1043 y Fp(Th)m(us)586 1251 y Fo(\027)637 1150 y Ff(\020)691 1251 y Fo(f)746 1213 y Fs(2)801 1165 y Ff(X)843 1355 y Fl(\013)959 1110 y Ff(\000)1000 1184 y Fk(h)p Fo(n)1090 1198 y Fl(\014)1139 1184 y Fk(i)21 b(\000)j Fp(\026)-49 b Fo(n)1340 1110 y Ff(\001)1382 1129 y Fs(2)p 959 1230 465 4 v 1168 1313 a Fp(\026)f Fo(n)1434 1150 y Ff(\021)1488 1169 y Fs(2)1554 1251 y Fk(\024)25 b Fo(k)1785 1189 y Fp(1)p 1711 1230 193 4 v 1711 1313 a Fk(j)p Fp(\003)1799 1327 y Fl(j)1838 1313 y Fk(j)1863 1287 y Fs(2)1930 1251 y Fo(\027)1981 1150 y Ff(\020)2035 1251 y Fo(f)2090 1213 y Fs(2)2130 1177 y Ff(\002)2236 1165 y(X)2168 1364 y Fl(x)p Fg(2)p Fs(\003)2308 1373 y Fh(j)2342 1364 y Fg(n)p Fl(C)2435 1177 y Ff(\000)2477 1251 y Fo(\032)2524 1265 y Fl(j)2583 1251 y Fk(\000)20 b Fo(\021)s Fp(\()p Fo(x)p Fp(\))2844 1177 y Ff(\001\003)2924 1196 y Fs(2)2965 1150 y Ff(\021)3019 1169 y Fs(2)1554 1579 y Fk(\024)25 b Fo(k)1700 1478 y Ff(\020)1765 1518 y Fp(ln)14 b Fo(L)p 1765 1558 153 4 v 1811 1641 a(L)1929 1478 y Ff(\021)1983 1497 y Fs(4)2024 1579 y Fk(j)p Fp(\003)2112 1593 y Fl(j)2151 1579 y Fk(j)2176 1542 y Fs(2)2237 1579 y Fp(+)20 b Fo(k)2463 1518 y Fp(1)p 2389 1558 193 4 v 2389 1641 a Fk(j)p Fp(\003)2477 1655 y Fl(j)2516 1641 y Fk(j)2541 1615 y Fs(2)2593 1579 y Fk(j)p Fp(\003)2681 1593 y Fl(j)2739 1579 y Fk(n)h Fo(C)7 b Fk(j)2902 1542 y Fs(3)2958 1579 y Fp(En)m(t)3103 1593 y Fl(\027)3148 1579 y Fp(\()p Fo(f)3238 1542 y Fs(2)3279 1579 y Fp(\))1554 1810 y Fk(\024)25 b Fo(\017)15 b Fk(j)p Fp(\003)1790 1824 y Fl(j)1829 1810 y Fk(j)g Fp(En)m(t)2014 1824 y Fl(\027)2059 1810 y Fp(\()p Fo(f)2149 1772 y Fs(2)2190 1810 y Fp(\))3668 1487 y(\(5)p Fo(:)p Fp(20\))0 2013 y(pro)m(vided)29 b(that)i Fo(l)598 2027 y Fs(0)669 2013 y Fp(and)f Fo(L)g Fp(are)h(large)f(enough.)0 2141 y(Ab)s(o)m(v)m(e)40 b(w)m(e)f(ha)m(v)m(e)h(used)e(lemma)h(5.1)h(together)g(with)d(part)i (i\))f(of)i(prop)s(osition)c(4.1)k(applied)d(to)i(the)g(function)20 2245 y(^)0 2268 y Fo(F)53 b Fp(=)39 b Fo(F)g Fk(\000)26 b Fo(\026)p Fp(\()p Fo(F)13 b Fp(\),)42 b Fo(F)53 b Fp(=)899 2195 y Ff(\002)937 2200 y(P)1033 2295 y Fl(x)p Fg(2)p Fs(\003)1173 2304 y Fh(j)1207 2295 y Fg(n)p Fl(C)1305 2195 y Ff(\000)1347 2268 y Fo(\032)1394 2282 y Fl(j)1458 2268 y Fk(\000)26 b Fo(\021)s Fp(\()p Fo(x)p Fp(\))1725 2195 y Ff(\001)q(\003)1806 2213 y Fs(2)1846 2268 y Fp(,)42 b(the)d(b)s(ound)e Fo(\027)6 b Fp(\()p Fo(e)2495 2235 y Fl(t)2539 2219 y Fs(^)2522 2235 y Fl(F)2583 2268 y Fp(\))40 b Fk(\024)g Fo(A\026)p Fp(\()p Fo(e)2969 2235 y Fl(t)3013 2219 y Fs(^)2996 2235 y Fl(F)3057 2268 y Fp(\))g(and)e(the)h(h)m(yp)s(othesis)0 2396 y Fo(t)33 2410 y Fg(\003)100 2396 y Fk(\025)207 2360 y Fl(M)p 207 2375 77 4 v 214 2432 a(l)237 2410 y Fh(d)237 2454 y Fi(0)294 2396 y Fp(.)0 2557 y(Therefore)33 b(w)m(e)g(can)g(safely)g(add)g(and)f (subtract)h Fk(h)p Fo(n)1787 2571 y Fl(\014)1836 2557 y Fk(i)g Fp(inside)e(the)i(r.h.s.)48 b(of)33 b(\(5.19\))i(and)e (con\014ne)f(ourselv)m(es)h(to)0 2743 y(the)e(estimate)g(of)f Fo(\027)675 2642 y Ff(\020)729 2743 y Fo(f)784 2710 y Fs(2)839 2674 y Ff(P)935 2770 y Fl(\013)p Fg(2)p Fl(I)1099 2696 y Fs(\()p Fl(n)1172 2705 y Fh(\013)1219 2696 y Fg(\000h)p Fl(n)1350 2706 y Fh(\014)1393 2696 y Fg(i)p Fs(\))1450 2668 y Fi(2)p 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Fk(\024)25 b Fo(k)s Fp(,)k(the)0 3607 y(second)i(term)g(in)e(the)i(r.h.s.)42 b(of)31 b(\(5.21\))i(is)d(b)s(ounded)e(b)m(y)j Fo(k)s Fk(j)p Fo(I)7 b Fk(j)2119 3574 y Fs(2)2187 3607 y Fk(\024)25 b Fo(k)2344 3561 y Fg(j)p Fs(\003)2416 3570 y Fh(j)2450 3561 y Fg(j)p 2345 3586 127 4 v 2370 3638 a Fl(M)2498 3607 y Fp(En)m(t)2643 3621 y Fl(\027)2688 3607 y Fp(\()p Fo(f)2778 3574 y Fs(2)2818 3607 y Fp(\))31 b(b)s(ecause)g(of)g(the)g(h) m(yp)s(othesis)0 3734 y Fo(t)33 3748 y Fg(\003)100 3734 y Fk(\025)207 3698 y Fl(M)p 207 3713 77 4 v 214 3770 a(l)237 3749 y Fh(d)237 3792 y Fi(0)294 3734 y Fp(.)0 3926 y(It)h(no)m(w)h(remains)d(to)j(b)s(ound)e(the)h(\014rst)f(term)i (of)f(\(5.21\).)48 b(A)m(t)33 b(this)f(stage)h(w)m(e)g(cannot)g(app)s (eal)e(to)i(the)f(same)h(old)0 4053 y(argumen)m(t)e(based)f(on)g(the)h (en)m(trop)m(y)g(inequalit)m(y)d(and)i(w)m(e)h(m)m(ust)f(pro)s(ceed)g (di\013eren)m(tly)-8 b(.)0 4215 y(F)g(ollo)m(wing)78 b([Y)-8 b(au])79 b(w)m(e)g(in)m(tro)s(duce)e Fk(F)1440 4229 y 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Fl(\013;\014)2234 4593 y Ff(\001)2275 4566 y(\021)2330 4666 y Fk(j)20 b Fp(+)g Fk(j)p Fo(\027)2542 4566 y Ff(\020)2596 4666 y Fo(f)2651 4629 y Fs(2)2691 4666 y Fo(;)15 b(\027)2782 4593 y Ff(\000)2835 4605 y Fp(\()p Fo(n)2925 4619 y Fl(\013)2997 4605 y Fk(\000)20 b Fo(n)3143 4619 y Fl(\014)3191 4605 y Fp(\))3226 4572 y Fs(2)p 2835 4646 V 3029 4729 a Fp(\026)-50 b Fo(n)3294 4666 y Fk(j)15 b(F)3399 4680 y Fl(\013;\014)3517 4593 y Ff(\001)3559 4566 y(\021)3613 4666 y Fk(j)1086 4928 y(\024)25 b(j)p Fo(\027)1258 4827 y Ff(\020)1313 4928 y Fo(\027)1364 4854 y Ff(\000)1405 4928 y Fo(f)1460 4890 y Fs(2)1500 4928 y Fo(;)1551 4866 y Fp(\()p Fo(n)1641 4880 y Fl(\013)1713 4866 y Fk(\000)20 b Fo(n)1859 4880 y Fl(\014)1908 4866 y Fp(\))1943 4833 y Fs(2)p 1551 4907 V 1745 4990 a Fp(\026)-50 b Fo(n)2010 4928 y Fk(j)15 b(F)2115 4942 y Fl(\013;\014)2234 4854 y Ff(\001)2275 4827 y(\021)2330 4928 y Fk(j)20 b Fp(+)g Fo(k)0 5212 y Fp(where)29 b(w)m(e)g(ha)m(v)m(e)i(used)d(once)i(more)f(the)g(equiv) -5 b(alence)29 b(of)g(ensem)m(bles)g(\(Prop)s(osition)e(3.2)k(p)s(oin)m (t)d(\(1\),)i(the)g(mixing)0 5340 y(assumption)d(and)h(the)h(fact)h (that)g Fo(n)1238 5354 y Fl(\013)1318 5340 y Fp(\()p Fo(n)1408 5354 y Fl(\014)1457 5340 y Fp(\))f(b)s(eing)e(the)i(n)m(um)m (b)s(er)f(of)h(particles)e(in)h Fo(C)2883 5354 y Fl(\013)2963 5340 y Fp(\()p Fo(C)3063 5354 y Fl(\014)3113 5340 y Fp(\))h(is)f(w)m (eakly)h(sensitiv)m(e)0 5558 y Fi(20)p Fh(=settembr)r(e=)p Fi(2000)g Fs([40])1204 b Fp(5.9)p eop %%Page: 41 41 41 40 bop 0 91 a Fp(to)31 b(the)g(b)s(oundary)d(conditions)h(outside)h Fo(C)1491 105 y Fl(\013)1573 91 y Fp(\()p Fo(C)1673 105 y Fl(\014)1722 91 y Fp(\),)h(to)g(b)s(ound)d(the)j(second)f(term.)0 218 y(De\014ne)h(no)m(w)1126 344 y Fk(E)1174 358 y Fl(\013;\014)1292 344 y Fp(\()p Fo(f)5 b(;)15 b(f)10 b Fp(\))25 b(=)1759 258 y Ff(X)1628 453 y Fl(x;y)r Fg(2)p Fl(C)1831 462 y Fh(\013)1877 453 y Fg([)p Fl(C)1978 463 y Fh(\014)2036 344 y Fo(\027)2087 243 y Ff(\020)2141 271 y(\000)2183 344 y Fk(r)2259 358 y Fl(xy)2344 344 y Fo(f)2399 271 y Ff(\001)2440 290 y Fs(2)2496 344 y Fk(j)15 b(F)2601 358 y Fl(\013;\014)2719 243 y Ff(\021)0 626 y Fp(Ob)m(viously)28 b(there)j(exists)f(a)h(\\sp)s(ectral)f(gap")h(constan)m(t)h Fo(C)7 b Fp(\()p Fo(l)2092 640 y Fs(0)2132 626 y Fp(\))31 b(suc)m(h)f(that)1180 811 y Fo(\027)1231 737 y Ff(\000)1272 811 y Fo(f)5 b(;)15 b(f)24 b Fk(j)16 b(F)1537 825 y Fl(\013;\014)1655 737 y Ff(\001)1722 811 y Fk(\024)25 b Fo(C)7 b Fp(\()p Fo(l)1952 825 y Fs(0)1992 811 y Fp(\))p Fk(E)2075 825 y Fl(\013;\014)2194 811 y Fp(\()p Fo(f)e(;)15 b(f)10 b Fp(\))181 b Fk(8)25 b Fo(f)0 996 y Fp(Th)m(us)391 1226 y Fk(j)p Fo(\027)467 1153 y Ff(\000)508 1226 y Fo(f)563 1189 y Fs(2)603 1226 y Fo(;)655 1165 y Fp(\()p Fo(n)745 1179 y Fl(\013)817 1165 y Fk(\000)20 b Fo(n)963 1179 y Fl(\014)1011 1165 y Fp(\))1046 1132 y Fs(2)p 655 1205 433 4 v 848 1289 a Fp(\026)-49 b Fo(n)1113 1226 y Fk(j)16 b(F)1219 1240 y Fl(\013;\014)1337 1153 y Ff(\001)1379 1226 y Fk(j)25 b(\024)g Fp(2)p Fk(k)1626 1165 y Fp(\()p Fo(n)1716 1179 y Fl(\013)1789 1165 y Fk(\000)20 b Fo(n)1935 1179 y Fl(\014)1983 1165 y Fp(\))2018 1132 y Fs(2)p 1627 1205 V 1820 1289 a Fp(\026)-49 b Fo(n)2070 1226 y Fk(k)2115 1240 y Fg(1)2193 1125 y Ff(\020)2247 1226 y Fo(\027)2298 1153 y Ff(\000)2340 1226 y Fo(f)5 b(;)15 b(f)24 b Fk(j)15 b(F)2604 1240 y Fl(\013;\014)2722 1153 y Ff(\001)2764 1125 y(\021)2829 1120 y Fi(1)p 2829 1130 31 4 v 2829 1166 a(2)2876 1125 y Ff(\020)2930 1226 y Fo(\027)2981 1153 y Ff(\000)3022 1226 y Fo(f)3077 1189 y Fs(2)3132 1226 y Fk(j)h(F)3238 1240 y Fl(\013;\014)3356 1153 y Ff(\001)3398 1125 y(\021)3463 1120 y Fi(1)p 3463 1130 V 3463 1166 a(2)1429 1488 y Fk(\024)25 b Fo(l)1554 1450 y Fl(d)1552 1510 y Fs(0)1596 1488 y Fo(C)7 b Fp(\()p Fo(l)1730 1502 y Fs(0)1771 1488 y Fp(\))1817 1426 y Fi(1)p 1817 1436 V 1817 1472 a(2)1864 1387 y Ff(\020)1918 1488 y Fk(E)1966 1502 y Fl(\013;\014)2084 1488 y Fp(\()p Fo(f)e(;)15 b(f)10 b Fp(\))2299 1387 y Ff(\021)2364 1381 y Fi(1)p 2364 1391 V 2364 1428 a(2)2410 1387 y Ff(\020)2464 1488 y Fo(\027)2515 1414 y Ff(\000)2557 1488 y Fo(f)2612 1450 y Fs(2)2667 1488 y Fk(j)15 b(F)2772 1502 y Fl(\013;\014)2891 1414 y Ff(\001)2932 1387 y(\021)2998 1381 y Fi(1)p 2998 1391 V 2998 1428 a(2)0 1705 y Fp(Sc)m(h)m(w)m(arz)31 b(inequalit)m(y)e(yields)741 1952 y Fo(\027)792 1852 y Ff(\020)846 1952 y Fk(j)p Fp(\()p Fo(\027)957 1879 y Ff(\000)999 1952 y Fo(f)1054 1915 y Fs(2)1094 1952 y Fo(;)1145 1891 y Fp(\()p Fo(n)1235 1905 y Fl(\013)1307 1891 y Fk(\000)20 b Fo(n)1453 1905 y Fl(\014)1501 1891 y Fp(\))1536 1858 y Fs(2)p 1145 1932 433 4 v 1339 2015 a Fp(\026)-50 b Fo(n)1604 1952 y Fk(j)15 b(F)1709 1966 y Fl(\013;\014)1827 1879 y Ff(\001)1869 1952 y Fk(j)1894 1852 y Ff(\021)1949 1870 y Fs(2)2015 1952 y Fk(\024)25 b Fo(C)2183 1915 y Fg(0)2207 1952 y Fp(\()p Fo(l)2269 1966 y Fs(0)2311 1952 y Fp(\))2492 1866 y Ff(X)2361 2061 y Fl(x;y)r Fg(2)p Fl(C)2564 2070 y Fh(\013)2610 2061 y Fg([)p Fl(C)2711 2071 y Fh(\014)2770 1952 y Fo(\027)2821 1879 y Ff(\000)2862 1952 y Fk(r)2938 1966 y Fl(xy)3023 1952 y Fo(f)3078 1879 y Ff(\001)3119 1898 y Fs(2)3668 1952 y Fp(\(5)p Fo(:)p Fp(22\))0 2251 y(Finally)-8 b(,)29 b(from)h(\(5.22\),)j(w)m(e)e(get)g(that)g(the)g(\014rst)e(term)i(in)e (the)h(r.h.s.)40 b(of)31 b(\(5.21\))i(is)c(b)s(ounded)f(from)i(ab)s(o)m (v)m(e)i(b)m(y)526 2443 y(2)p 480 2483 139 4 v 480 2567 a Fk(j)p Fo(I)7 b Fk(j)577 2540 y Fs(2)629 2403 y Ff(h)687 2418 y(X)696 2613 y Fl(\013;\014)834 2504 y Fo(\027)885 2431 y Ff(\000)926 2504 y Fo(f)981 2467 y Fs(2)1021 2504 y Fo(;)1073 2443 y Fp(\()p Fo(n)1163 2457 y Fl(\013)1235 2443 y Fk(\000)19 b Fo(n)1380 2457 y Fl(\014)1429 2443 y Fp(\))1464 2410 y Fs(2)p 1073 2483 433 4 v 1266 2567 a Fp(\026)-49 b Fo(n)1516 2431 y Ff(\001)1558 2403 y(i)1601 2422 y Fs(2)1667 2504 y Fk(\024)25 b Fo(C)1835 2467 y Fg(00)1881 2504 y Fp(\()p Fo(l)1943 2518 y Fs(0)1984 2504 y Fp(\))2086 2418 y Ff(X)2034 2613 y Fl(x;y)r Fg(2)p Fs(\003)2235 2622 y Fh(j)2284 2504 y Fo(\027)c Fp(\()p Fk(r)2461 2518 y Fl(xy)2546 2504 y 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b(Applica)-6 b(tions.)0 4188 y Fp(Here)28 b(w)m(e)g(discuss)e(an)h(application)f(of)h(our)g(results)f(whic)m(h)h (is)f(directly)g(relev)-5 b(an)m(t)28 b(for)f(the)h(pro)s(of)e(of)i (the)g(di\013usiv)m(e)0 4315 y(scaling)h(of)i(the)g(logarithmic)e(Sob)s (olev)g(constan)m(t)j(for)e(the)g(Ka)m(w)m(asaki)h(dynamics.)0 4485 y(Fix)g Fo(i;)15 b(j)34 b Fk(2)27 b(f)p Fp(1)p Fo(;)15 b(:)g(:)g(:)i(;)e(j)723 4499 y Fs(max)863 4485 y Fk(g)32 b Fp(with)e Fo(i)e Fk(6)p Fp(=)f Fo(j)37 b Fp(and)31 b(let)h Fo(\032)27 b Fp(=)1909 4439 y Fl(N)1967 4448 y Fh(j)2000 4439 y Fs(+)p Fl(N)2114 4448 y Fh(j)p 1873 4464 310 4 v 1873 4517 a Fg(j)p Fs(\003)1945 4526 y Fh(j)1979 4517 y Fg(j)p Fs(+)p Fg(j)p Fs(\003)2129 4526 y Fh(j)2161 4517 y Fg(j)2194 4485 y Fp(.)44 b(Clearly)2592 4448 y Fl(\016)2625 4457 y Fi(0)p 2592 4464 69 4 v 2608 4517 a Fs(2)2671 4485 y Fp(\()p Fo(\032)2753 4499 y Fl(i)2805 4485 y Fp(+)20 b Fo(\032)2943 4499 y Fl(j)2982 4485 y Fp(\))43 b Fk(\024)f Fo(\032)g Fk(\024)27 b Fo(\032)3407 4499 y Fl(i)3459 4485 y Fp(+)21 b Fo(\032)3598 4499 y Fl(j)3667 4485 y Fp(where)0 4612 y Fo(\032)47 4626 y Fl(i)109 4612 y Fp(and)31 b Fo(\032)334 4626 y Fl(j)405 4612 y Fp(are)h(the)g(densities)e(in)h(\003)1258 4626 y Fl(i)1320 4612 y Fp(and)g(\003)1561 4626 y Fl(j)1600 4612 y Fp(.)45 b(Without)32 b(loss)f(of)h(generalit)m(y)g(w)m(e)h (assume)e(that)i Fo(\032)3435 4626 y Fl(i)3493 4612 y Fk(\024)27 b Fo(\032)32 b Fp(whic)m(h)0 4739 y(implies)c Fo(\032)356 4753 y Fl(j)419 4739 y Fk(\025)d Fo(\032)562 4753 y Fl(i)592 4739 y Fp(.)41 b(Let)31 b(also)1042 4955 y Fo(g)1085 4969 y Fl(x)1131 4955 y Fp(\()p Fo(\033)s Fp(\))26 b(=)1378 4881 y Ff(\002)1416 4955 y Fo(e)1458 4917 y Fg(\000r)1576 4926 y Fh(x)1616 4917 y Fl(H)1681 4890 y Fh(\034)1721 4917 y Fs(\()p Fl(\033)r Fs(\))1847 4955 y Fk(\000)20 b Fp(1)1983 4881 y Ff(\003)2022 4955 y Fo(\033)s Fp(\()p Fo(x)p Fp(\))182 b Fo(;)198 b(G)25 b Fp(:=)2840 4868 y Ff(X)2822 5063 y Fl(x)p Fg(2)p Fs(\003)2962 5072 y Fh(i)3005 4955 y Fo(g)3048 4969 y Fl(x)793 5241 y Fo(h)845 5255 y Fl(x)891 5241 y Fp(\()p Fo(\033)s Fp(\))h(=)1138 5167 y Ff(\002)1176 5241 y Fo(e)1218 5203 y Fg(\000r)1336 5212 y Fh(x)1376 5203 y Fl(H)1441 5176 y Fh(\034)1482 5203 y Fs(\()p Fl(\033)r Fs(\))1608 5241 y Fk(\000)19 b Fp(1)1743 5167 y Ff(\003)q(\000)1823 5241 y Fp(1)i Fk(\000)f Fo(\033)s Fp(\()p Fo(x)p Fp(\))2157 5167 y Ff(\001)2381 5241 y Fo(;)198 b Fk(H)26 b Fp(:=)2847 5154 y Ff(X)2828 5349 y Fl(z)s Fg(2)p Fs(\003)2965 5358 y Fh(j)3013 5241 y Fo(h)3065 5255 y Fl(z)0 5558 y Fi(20)p Fh(=settembr)r(e=)p Fi(2000)j Fs([41])1181 b Fp(5.10)p eop %%Page: 42 42 42 41 bop 0 91 a Fp(Notice)37 b(that)f Fo(g)536 105 y Fl(x)618 91 y Fp(satis\014es)f(the)h(h)m(yp)s(otheses)f(of)h(prop)s (osition)d(4.3)k(simply)c(b)s(ecause)j Fo(g)3013 105 y Fl(x)3093 91 y Fp(=)e(0)i(if)f(there)h(are)g(less)0 218 y(than)d(t)m(w)m(o)i(particles)d(\(spins)g(equal)h(to)h(1\))g (inside)d(its)i(supp)s(ort.)48 b(Similarly)30 b Fo(h)2709 232 y Fl(z)2782 218 y 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Fp(ev)m(ery)j(thing)f(is)g(the)h(same)g(except) h(that)f(w)m(e)g(apply)0 3132 y(prop)s(osition)28 b(5.2)j(to)h Fo(\032)p Fk(H)q Fp(.)40 b(F)-8 b(rom)31 b(no)m(w)f(on)g(w)m(e)h(will)d (write)i Fo(G)g Fp(for)h Fo(G)20 b Fk(\000)g Fo(\026)p Fp(\()p Fo(G)p Fp(\))31 b(and)f(similarily)c(for)31 b Fk(H)q Fp(.)0 3288 y(Next)h(w)m(e)f(write)f Fk(H)d Fp(=)f Fk(H)869 3255 y Fs(in)954 3288 y Fp(+)21 b Fk(H)1124 3255 y Fs(ext)1258 3288 y Fp(where)30 b Fk(H)1599 3255 y Fs(in)1695 3288 y Fp(is)g(the)h(sum)f(o)m(v)m(er)i(those)f Fo(z)t Fp('s)h(in)d(\003)2878 3302 y Fl(j)2947 3288 y Fp(suc)m(h)i(that)h(\003)3414 3302 y Fl(h)3457 3311 y Fh(z)3524 3288 y Fk(\032)26 b Fp(\003)3684 3302 y Fl(j)3754 3288 y Fp(and)0 3415 y Fk(H)78 3382 y Fs(ext)211 3415 y Fp(the)31 b(rest.)41 b(Similarily)26 b(for)k Fo(G)p Fp(.)41 b(Th)m(us)29 b(w)m(e)i(ha)m(v)m(e)h(to)f(b)s(ound)d(four)i (terms)444 3651 y Fo(\027)495 3578 y Ff(\000)536 3651 y Fo(f)591 3614 y Fs(2)631 3651 y Fo(;)15 b(G)742 3614 y Fs(in)n(t)835 3651 y 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Fg(\000)p Fs(2)1469 5199 y Ff(\001)1536 5272 y Fo(>)25 b Fp(0)30 b(is)g(a)h(trivial)d(consequence)j(of)g(the)f (recursiv)m(e)g(b)s(ound.)89 b Fb(\003)0 5558 y Fi(20)p Fh(=settembr)r(e=)p Fi(2000)29 b Fs([47])1193 b Fp(A.1)p eop %%Page: 48 48 48 47 bop 0 91 a Fq(App)s(endix.)47 b(Logarithmic)34 b(Sob)s(olev)i(inequalit)m(y)f(for)g(one)g(dimensional)g(discrete)g (random)f(w)m(alks)0 250 y Fp(Let)21 b(\012)k(=)g Fk(f)p Fo(n)440 264 y Fs(min)565 250 y Fo(;)15 b(n)660 264 y Fs(min)785 250 y Fp(+1)p Fo(;)g(:)g(:)g(:)i(;)e(n)1158 264 y Fs(max)1296 250 y Fk(g)21 b Fp(b)s(e)f(a)h(\014nite)e(subset)h (of)g(the)h(in)m(tegers)f(and)g(let)g Fo(\015)26 b Fp(b)s(e)19 b(a)i(p)s(ositiv)m(e)e(probabilit)m(y)0 378 y(measure)34 b(on)g(\012.)51 b(W)-8 b(e)35 b(presen)m(t)f(a)g(result,)g(based)g(on)g (Hardy)f(t)m(yp)s(e)i(inequalities,)d(to)j(b)s(ound)d(in)g(a)j(precise) e(w)m(a)m(y)0 505 y(the)e(logarithmic)f(Sob)s(olev)f(constan)m(t)k(of)e Fo(\015)36 b Fp(with)29 b(resp)s(ect)i(to)g(the)h(Diric)m(hlet)d(form)i (of)g(discrete)f(random)g(w)m(alks)0 633 y(rev)m(ersible)38 b(w.r.t.)66 b Fo(\015)5 b Fp(.)67 b(After)39 b(that)h(w)m(e)g(discuss)d (a)i(concrete)i(example)d(related)h(to)h(the)f(distribution)d(of)j(the) 0 760 y(n)m(um)m(b)s(er)29 b(of)i(particles)e(under)g(a)i(m)m (ulticanonical)d(measure)i(\(see)i Fk(x)p Fp(7\).)0 948 y Fq(Prop)s(osition)k(A.1.)53 b Fc(F)-8 b(or)31 b(all)e(functions)g Fo(f)40 b Fc(on)30 b Fp(\012)g Fc(w)m(e)h(ha)m(v)m(e)720 1268 y Fp(En)m(t)865 1282 y Fl(\015)912 1268 y Fp(\()p Fo(f)1002 1231 y Fs(2)1042 1268 y Fp(\))26 b Fk(\024)f Fp(20)p Fo(B)1485 1153 y Fl(n)1530 1162 y Fi(max)1500 1182 y Ff(X)1394 1374 y Fl(n)p Fs(=)p Fl(n)1540 1383 y Fi(min)1646 1374 y Fs(+1)1738 1268 y Fp(\()p Fo(\015)5 b Fp(\()p Fo(n)p Fp(\))21 b Fk(^)f Fo(\015)5 b Fp(\()p Fo(n)20 b Fk(\000)g Fp(1\)\))2435 1194 y Ff(\002)2474 1268 y Fo(f)10 b Fp(\()p Fo(n)p Fp(\))20 b Fk(\000)g Fo(f)10 b Fp(\()p Fo(n)19 b Fk(\000)h Fp(1\))3100 1194 y Ff(\003)3139 1213 y Fs(2)0 1612 y Fc(where)30 b Fo(B)g Fp(:=)25 b(inf)593 1626 y Fl(i)654 1612 y Fp([)p Fo(B)748 1626 y Fs(+)809 1612 y Fp(\()p Fo(i)p Fp(\))c Fk(_)f Fo(B)1081 1626 y Fg(\000)1142 1612 y Fp(\()p Fo(i)p Fp(\)])q Fc(,)31 b(with)479 1944 y Fo(B)548 1958 y Fs(+)609 1944 y Fp(\()p Fo(i)p Fp(\))26 b(:=)67 b(sup)857 2021 y Fl(n)p Fg(\025)p Fl(i)p Fs(+1)1092 1789 y Ff( )1164 1830 y Fl(n)1209 1839 y Fi(max)1180 1858 y Ff(X)1175 2054 y Fl(k)r Fs(=)p Fl(n)1342 1944 y Fo(\015)5 b Fp(\()p Fo(k)s Fp(\))1514 1789 y Ff(!)1602 1944 y Fp(ln)1693 1816 y Ff(\022)1974 1883 y Fp(1)p 1771 1923 452 4 v 1771 1939 a Ff(P)1867 1962 y Fl(n)1912 1971 y Fi(max)1867 2035 y Fl(k)r Fs(=)p Fl(n)2049 2007 y Fo(\015)g Fp(\()p Fo(k)s Fp(\))2233 1816 y Ff(\023)2315 1789 y( )2472 1831 y Fl(n)2428 1858 y Ff(X)2387 2054 y Fl(k)r Fs(=)p Fl(i)p Fs(+1)2906 1883 y Fp(1)p 2627 1923 604 4 v 2627 2007 a Fo(\015)g Fp(\()p Fo(k)s Fp(\))21 b Fk(^)f Fo(\015)5 b Fp(\()p Fo(k)24 b Fk(\000)c Fp(1\))3241 1789 y Ff(!)478 2299 y Fo(B)547 2313 y Fg(\000)609 2299 y Fp(\()p Fo(i)p Fp(\))26 b(:=)67 b(sup)857 2375 y Fl(n)p Fg(\024)p Fl(i)p Fg(\000)p Fs(1)1093 2143 y Ff( )1266 2185 y Fl(n)1223 2212 y Ff(X)1165 2408 y Fl(k)r Fs(=)p Fl(n)1307 2417 y Fi(min)1428 2299 y Fo(\015)5 b Fp(\()p Fo(k)s Fp(\))1600 2143 y Ff(!)1688 2299 y Fp(ln)1779 2143 y Ff( )2107 2237 y Fp(1)p 1862 2278 537 4 v 1862 2294 a Ff(P)1958 2316 y Fl(n)1958 2389 y(k)r Fs(=)p Fl(n)2100 2398 y Fi(min)2225 2362 y Fo(\015)g Fp(\()p Fo(k)s Fp(\))2409 2143 y Ff(!)15 b( )2579 2185 y Fl(i)p Fg(\000)p Fs(1)2573 2212 y Ff(X)2568 2408 y Fl(k)r Fs(=)p Fl(n)3014 2237 y Fp(1)p 2735 2278 604 4 v 2735 2361 a Fo(\015)5 b Fp(\()p Fo(k)s Fp(\))21 b Fk(^)f Fo(\015)5 b Fp(\()p Fo(k)24 b Fp(+)c(1\))3350 2143 y Ff(!)0 2648 y Fj(Pr)-5 b(o)g(of.)35 b Fp(The)f(pro)s(of)f(is)g(a)h(straigh)m(t)g(forw)m(ard)g(application)e (of)i(Hardy)g(t)m(yp)s(e)g(inequalities)d(\(see)k([Mi])f(section)g (3\).)0 2776 y(Indeed,)43 b(it's)e(enough)g(to)g(consider)f(the)i 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4869 y(\015)5 b Fp(\()p Fo(n)20 b Fk(\000)g Fp(1\))p 1149 4909 334 4 v 1227 4993 a Fo(\015)5 b Fp(\()p Fo(n)p Fp(\))1519 4930 y Fk(\024)25 b Fo(c)15 b(e)1711 4893 y Fg(\000)1779 4860 y Fi(\()5 b(\026)-36 b Fh(n)p Fe(\000)p Fh(n)p Fi(\))p 1779 4879 179 4 v 1834 4915 a Fh(c)5 b Fi(\026)-36 b Fh(n)2246 4930 y Fp(for)30 b(all)g Fo(n)24 b Fk(\024)30 b Fp(\026)-50 b Fo(n)917 5230 y Fp(1)p 855 5271 170 4 v 855 5361 a Fo(c)894 5289 y Fk(p)p 970 5289 55 4 v 975 5361 a Fp(\026)g Fo(n)1036 5292 y(e)1078 5254 y Fg(\000)1146 5222 y Fh(c)p Fi(\()5 b(\026)-36 b Fh(n)o Fe(\000)p Fh(n)p Fi(\))1354 5203 y(2)p 1146 5240 244 4 v 1252 5276 a(\026)h Fh(n)1430 5292 y Fk(\024)25 b Fo(\015)5 b Fp(\()p Fo(n)p Fp(\))26 b Fk(\024)1881 5230 y Fo(c)p 1836 5271 131 4 v 1836 5289 a Fk(p)p 1911 5289 55 4 v 1916 5361 a Fp(\026)-50 b Fo(n)1977 5292 y(e)2019 5254 y Fg(\000)2087 5222 y Fi(\()5 b(\026)-36 b Fh(n)p Fe(\000)p Fh(n)p Fi(\))2266 5203 y(2)p 2087 5240 215 4 v 2159 5276 a Fh(c)5 b Fi(\026)-36 b Fh(n)2559 5292 y Fp(for)30 b(all)f Fo(n)c Fk(2)g Fp(\012)634 b(\()p Fo(A:)p Fp(2\))0 5558 y Fi(20)p Fh(=settembr)r(e=)p Fi(2000)29 b Fs([48])1193 b Fp(A.2)p eop %%Page: 49 49 49 48 bop 0 91 a Fj(R)-5 b(emark)p Fp(.)60 b(Since)35 b Fo(\015)42 b Fp(is)35 b(assumed)h(to)h(b)s(e)f(a)h(p)s(ositiv)m(e)e (probabilit)m(y)f(measure,)k(w)m(e)f(can)f(alw)m(a)m(ys)h(write)f(it)g (in)f(the)0 218 y(Gibbsian)h(form)h Fo(\015)5 b Fp(\()p Fo(n)p Fp(\))39 b(=)f Fo(Z)1009 185 y Fg(\000)p Fs(1)1121 218 y Fp(exp\()p Fk(\000)p Fo(H)7 b Fp(\()p Fo(n)p Fp(\)\).)64 b(Then,)40 b(if)d Fo(H)45 b Fp(is)37 b(strictly)g(con)m(v)m(ex)j(with)c (\\second)j(deriv)-5 b(ativ)m(e")0 346 y(b)s(ounded)28 b(from)i(b)s(elo)m(w)g(and)f(from)h(ab)s(o)m(v)m(e)i(b)m(y)e Fo(c)1658 313 y Fg(\000)p Fs(1)1786 346 y Fp(and)g Fo(c)h Fp(resp)s(ectiv)m(ely)-8 b(,)30 b(and)g(the)h(minim)m(um)c(in)34 b(\026)-50 b Fo(n)p Fp(,)30 b Fj(i.e.)926 569 y Fp(1)p 894 609 110 4 v 894 693 a Fo(c)20 b Fp(\026)-50 b Fo(n)1039 630 y Fk(\024)25 b Fo(H)7 b Fp(\()p Fo(n)20 b Fp(+)g(2\))h Fk(\000)f Fp(2)p Fo(H)7 b Fp(\()p Fo(n)21 b Fp(+)e(1\))i(+)f Fo(H)7 b Fp(\()p Fo(n)p Fp(\))26 b Fk(\024)2480 569 y Fo(c)p 2473 609 55 4 v 2478 693 a Fp(\026)-50 b Fo(n)2720 630 y Fk(8)15 b Fo(n)24 b Fk(2)h Fp(\012)1135 834 y Fo(H)7 b Fp(\()e(\026)-50 b Fo(n)20 b Fp(+)g(1\))26 b(=)f Fo(H)7 b Fp(\()e(\026)-50 b Fo(n)p Fp(\))3690 711 y(\()p Fo(A:)p Fp(3\))0 1081 y(then,)27 b(if)f(\()p Fo(c;)21 b Fp(\026)-51 b Fo(n)q(;)15 b(n)574 1095 y Fs(max)712 1081 y Fo(;)g(n)807 1095 y Fs(min)932 1081 y Fp(\))27 b(are)h(in)d(the)i(righ)m(t)f(prop)s (ortion)f(describ)s(ed)g(in)g Fo(C)7 b(O)s(N)j(V)20 b Fp(\()p Fo(c;)g Fp(\026)-50 b Fo(n)p Fp(\),)28 b Fo(\015)k Fp(satis\014es)26 b(condition)0 1208 y Fo(C)7 b(O)s(N)j(V)20 b Fp(\()p Fo(c;)g Fp(\026)-50 b Fo(n)p Fp(\).)0 1448 y(Our)37 b(aim)h(no)m(w)g(is)f(to)i(pro)m(v)m(e)g(that)g(under)d Fo(C)7 b(O)s(N)j(V)20 b Fp(\()p Fo(c;)h Fp(\026)-51 b Fo(n)q Fp(\))38 b(the)g(measure)h Fo(\015)k Fp(satis\014es)37 b(a)i(logarithmic)e(Sob)s(olev)0 1575 y(inequalit)m(y)30 b(with)h(optimal)g(constan)m(t)i(prop)s(ortional)d(\(dep)s(ending)f(on) j Fo(c)p Fp(\))h(to)k(\026)-50 b Fo(n)32 b Fp(\(see)g(Prop)s(osition)e (A.5)j(b)s(elo)m(w\).)0 1702 y(F)-8 b(or)24 b(this)f(purp)s(ose,)g(let) h(us)f(\014rst)f(discuss)g(some)i(simple)e(lemma)h(that)h(deal)f(with)f (the)i(di\013eren)m(t)f(terms)g(app)s(earing)0 1830 y(in)j(the)i (constan)m(ts)h Fo(B)726 1844 y Fs(+)786 1830 y Fp(\()p Fo(i)p Fp(\))g(and)e Fo(B)1159 1844 y Fg(\000)1220 1830 y Fp(\()p Fo(i)p Fp(\))i(of)f(Prop)s(osition)d(A.1.)41 b(Since)26 b(ob)m(viously)h Fo(B)i Fk(\024)c Fo(B)3038 1844 y Fs(+)3099 1830 y Fp(\()5 b(\026)-50 b Fo(n)p Fp(\))15 b Fk(_)g Fo(B)3384 1844 y Fg(\000)3444 1830 y Fp(\()5 b(\026)-50 b Fo(n)p Fp(\)\))29 b(w)m(e)f(can)0 1957 y(restrict)i(us)g (to)h(the)f(case)i Fo(i)25 b Fp(=)30 b(\026)-50 b Fo(n)p Fp(.)0 2197 y Fq(Lemma)33 b(A.2.)52 b Fc(Let)31 b Fo(\015)36 b Fc(satis\014es)30 b Fo(C)7 b(O)s(N)j(V)19 b Fp(\()p Fo(c;)i Fp(\026)-51 b Fo(n)q Fp(\))p Fc(.)41 b(Then)1066 2403 y Fl(n)1111 2412 y Fi(max)1082 2431 y Ff(X)1077 2627 y Fl(k)r Fs(=)p Fl(n)1244 2518 y Fo(\015)5 b Fp(\()p Fo(k)s Fp(\))26 b Fk(\024)f Fo(C)1708 2456 y Fp(\026)-50 b Fo(n)p 1621 2497 221 4 v 1621 2580 a(n)19 b Fk(\000)25 b Fp(\026)-50 b Fo(n)1852 2518 y(\015)5 b Fp(\()p Fo(n)p Fp(\))126 b(for)31 b(all)89 b Fo(n)25 b Fk(\025)30 b Fp(\026)-50 b Fo(n)20 b Fp(+)g(1)p Fo(;)1083 2736 y Fl(n)1039 2763 y Ff(X)981 2959 y Fl(k)r Fs(=)p Fl(n)1123 2968 y Fi(min)1244 2850 y Fo(\015)5 b Fp(\()p Fo(k)s Fp(\))26 b Fk(\024)f Fo(C)1708 2788 y Fp(\026)-50 b Fo(n)p 1621 2829 V 1626 2912 a Fp(\026)g Fo(n)19 b Fk(\000)h Fo(n)1852 2850 y(\015)5 b Fp(\()p Fo(n)p Fp(\))126 b(for)31 b(all)89 b Fo(n)25 b Fk(\024)30 b Fp(\026)-50 b Fo(n)20 b Fk(\000)g Fp(1)26 b Fo(;)0 3187 y Fc(for)k(a)h(suitable)e(n)m(umerical)g(constan) m(t)i Fo(C)37 b Fc(dep)s(ending)28 b(only)i(on)g Fo(c)g Fc(and)g(not)h(on)k Fp(\026)-50 b Fo(n)p Fc(.)0 3372 y Fj(Pr)-5 b(o)g(of.)38 b Fp(Fix)f Fo(n)f Fk(\025)k Fp(\026)-49 b Fo(n)24 b Fp(+)g(1.)62 b(By)37 b(using)e(the)j(de\014nition)c(of)k Fo(C)7 b(O)s(N)j(V)19 b Fp(\()p Fo(c;)i Fp(\026)-51 b Fo(n)q Fp(\))37 b(together)i(with)c(a)j(simple)d(telescopic)0 3499 y(argumen)m(t)c(w)m(e)g(get)1356 3607 y Fl(n)1401 3616 y Fi(max)1372 3635 y Ff(X)1367 3831 y Fl(k)r Fs(=)p Fl(n)1547 3660 y Fo(\015)5 b Fp(\()p Fo(k)s Fp(\))p 1545 3701 178 4 v 1545 3784 a Fo(\015)g Fp(\()p Fo(n)p Fp(\))1759 3721 y(=)1855 3607 y Fl(n)1900 3616 y Fi(max)1870 3635 y Ff(X)1865 3831 y Fl(k)r Fs(=)p Fl(n)2033 3608 y(k)r Fg(\000)p Fs(1)2042 3635 y Ff(Y)2032 3829 y Fl(j)t Fs(=)p Fl(n)2208 3660 y Fo(\015)g Fp(\()p Fo(j)26 b Fp(+)20 b(1\))p 2208 3701 323 4 v 2286 3784 a Fo(\015)5 b Fp(\()p Fo(j)g Fp(\))1759 4081 y Fk(\024)25 b Fo(c)1909 3966 y Fl(n)1954 3975 y Fi(max)1925 3994 y Ff(X)1920 4190 y Fl(k)r Fs(=)p Fl(n)2088 3967 y(k)r Fg(\000)p Fs(1)2096 3994 y Ff(Y)2087 4188 y Fl(j)t Fs(=)p Fl(n)2237 4081 y Fo(e)2279 4043 y Fg(\000)2347 4011 y Fi(\()p Fh(j)s Fe(\000)5 b Fi(\026)-36 b Fh(n)o Fi(\))p 2347 4029 168 4 v 2396 4065 a Fh(c)5 b Fi(\026)-36 b Fh(n)1759 4377 y Fk(\024)25 b Fo(c)1914 4291 y Ff(X)1909 4487 y Fl(k)r Fg(\025)p Fl(n)2066 4377 y Fo(e)2108 4340 y Fg(\000)2176 4307 y Fi(\()p Fh(k)q Fe(\000)p Fh(n)p Fi(\)\()p Fh(n)p Fe(\000)5 b Fi(\026)-36 b Fh(n)p Fi(\))p 2176 4326 353 4 v 2302 4362 a(2)p Fh(c)5 b Fi(\026)-36 b Fh(n)1759 4691 y Fp(=)25 b Fo(c)2112 4630 y Fp(1)p 1905 4671 461 4 v 1905 4780 a(1)20 b Fk(\000)g Fo(e)2103 4748 y Fg(\000)2171 4716 y Fi(\()p Fh(n)p Fe(\000)5 b Fi(\026)-36 b Fh(n)p Fi(\))p 2171 4734 179 4 v 2210 4770 a(2)p Fh(c)5 b Fi(\026)-36 b Fh(n)0 5012 y Fp(But,)33 b(\()p Fo(n)22 b Fk(\000)j Fp(\026)-49 b Fo(n)o Fp(\))p Fo(=)5 b Fp(\026)-50 b Fo(n)29 b Fk(\024)f Fp(\()p Fo(n)819 5026 y Fs(max)979 5012 y Fk(\000)e Fp(\026)-50 b Fo(n)p Fp(\))p Fo(=)5 b Fp(\026)-50 b Fo(n)29 b Fk(\024)f Fo(c)k Fp(and)g(therefore)2023 4911 y Ff(\020)2077 5012 y Fp(1)21 b Fk(\000)f Fo(e)2276 4979 y Fg(\000)2344 4946 y Fi(\()p Fh(n)p Fe(\000)5 b Fi(\026)-36 b Fh(n)p Fi(\))p 2344 4965 V 2383 5001 a(2)p Fh(c)5 b Fi(\026)-36 b Fh(n)2538 4911 y Ff(\021)2592 4930 y Fg(\000)p Fs(1)2718 5012 y Fk(\024)28 b Fo(C)2955 4976 y Fs(\026)-41 b Fl(n)p 2899 4991 147 4 v 2899 5043 a(n)p Fg(\000)t Fs(\026)h Fl(n)3089 5012 y Fp(for)32 b(some)h(constan)m(t)h Fo(C)0 5139 y Fp(dep)s(ending)f(only)i(on)h Fo(c)p Fp(.)58 b(A)36 b(similar)d(computation)j(giv)m(es)g(the)g (corresp)s(onding)e(result)g(for)i Fo(n)e Fk(\024)39 b Fp(\026)-50 b Fo(n)24 b Fk(\000)f Fp(1.)58 b(This)0 5267 y(ac)m(hiev)m(es)32 b(the)e(pro)s(of.)91 b Fb(\003)0 5558 y Fi(20)p Fh(=settembr)r(e=)p Fi(2000)29 b Fs([49])1193 b Fp(A.3)p eop %%Page: 50 50 50 49 bop 0 91 a Fq(Lemma)33 b(A.3.)52 b Fc(Let)31 b Fo(\015)36 b Fc(satis\014es)30 b Fo(C)7 b(O)s(N)j(V)19 b Fp(\()p Fo(c;)i Fp(\026)-51 b Fo(n)q Fp(\))p Fc(.)41 b(Then)1086 242 y Fl(n)1043 270 y Ff(X)992 465 y Fl(k)r Fs(=)t(\026)-40 b Fl(n)o Fs(+1)1315 295 y Fp(1)p 1251 335 174 4 v 1251 418 a Fo(\015)5 b Fp(\()p Fo(k)s Fp(\))1460 356 y Fk(\024)25 b Fo(C)1726 295 y Fp(\026)-50 b Fo(n)p 1639 335 221 4 v 1639 418 a(n)19 b Fk(\000)25 b Fp(\026)-50 b Fo(n)1947 295 y Fp(1)p 1881 335 178 4 v 1881 418 a Fo(\015)5 b Fp(\()p Fo(n)p Fp(\))2195 356 y(for)30 b(all)90 b Fo(n)25 b Fk(\025)k Fp(\026)-49 b Fo(n)20 b Fp(+)g(1)1090 596 y Fs(\026)-41 b Fl(n)p Fg(\000)p Fs(1)1089 623 y Ff(X)1084 819 y Fl(k)r Fs(=)p Fl(n)1315 648 y Fp(1)p 1251 689 174 4 v 1251 772 a Fo(\015)5 b Fp(\()p Fo(k)s Fp(\))1460 709 y Fk(\024)25 b Fo(C)1726 648 y Fp(\026)-50 b Fo(n)p 1639 689 221 4 v 1644 772 a Fp(\026)g Fo(n)19 b Fk(\000)h Fo(n)1947 648 y Fp(1)p 1881 689 178 4 v 1881 772 a Fo(\015)5 b Fp(\()p Fo(n)p Fp(\))2195 709 y(forall)90 b Fo(n)25 b Fk(\024)k Fp(\026)-49 b Fo(n)19 b Fk(\000)h Fp(1)0 983 y Fc(for)30 b(a)h(suitable)e(n)m(umerical)g(constan)m(t)i Fo(C)37 b Fc(dep)s(ending)28 b(only)i(on)g Fo(c)p Fc(.)0 1154 y Fj(Pr)-5 b(o)g(of.)32 b Fp(The)e(pro)s(of)f(is)h(practically)f (the)h(same)h(as)g(that)g(of)f(lemma)g(A.2)h(and)f(it)g(is)g(omitted.) 91 b Fb(\003)0 1367 y Fq(Lemma)33 b(A.4.)52 b Fc(Let)31 b Fo(\015)36 b Fc(satis\014es)30 b Fo(C)7 b(O)s(N)j(V)19 b Fp(\()p Fo(c;)i Fp(\026)-51 b Fo(n)q Fp(\))p Fc(.)41 b(Then)738 1617 y Fp(ln)829 1489 y Ff(\022)1110 1556 y Fp(1)p 907 1596 452 4 v 907 1612 a Ff(P)1003 1635 y Fl(n)1048 1644 y Fi(max)1003 1708 y Fl(k)r Fs(=)p Fl(n)1185 1681 y Fo(\015)5 b Fp(\()p Fo(k)s Fp(\))1369 1489 y Ff(\023)1461 1617 y Fk(\024)25 b Fo(C)1644 1489 y Ff(\022)1711 1617 y Fp(1)20 b(+)1878 1556 y(\()5 b(\026)-50 b Fo(n)20 b Fk(\000)g Fo(n)p Fp(\))2169 1523 y Fs(2)p 1878 1596 332 4 v 2021 1680 a Fp(\026)-49 b Fo(n)2221 1489 y Ff(\023)2429 1617 y Fp(for)30 b(all)120 b Fo(n)25 b Fk(\025)30 b Fp(\026)-50 b Fo(n)20 b Fp(+)3127 1541 y Fk(p)p 3203 1541 55 4 v 3208 1617 a Fp(\026)-50 b Fo(n)643 1931 y Fp(ln)734 1775 y Ff( )1062 1869 y Fp(1)p 817 1910 537 4 v 817 1925 a Ff(P)913 1948 y Fl(n)913 2021 y(k)r Fs(=)p Fl(n)1055 2030 y Fi(min)1180 1994 y Fo(\015)5 b Fp(\()p Fo(k)s Fp(\))1364 1775 y Ff(!)1461 1931 y Fk(\024)25 b Fo(C)1644 1802 y Ff(\022)1711 1931 y Fp(1)20 b(+)1878 1869 y(\()5 b(\026)-50 b Fo(n)20 b Fk(\000)g Fo(n)p Fp(\))2169 1836 y Fs(2)p 1878 1910 332 4 v 2021 1993 a Fp(\026)-49 b Fo(n)2221 1802 y Ff(\023)2429 1931 y Fp(for)30 b(all)120 b Fo(n)25 b Fk(\024)30 b Fp(\026)-50 b Fo(n)20 b Fk(\000)3127 1854 y(p)p 3203 1854 55 4 v 3208 1931 a Fp(\026)-50 b Fo(n)0 2200 y Fc(where)30 b Fo(C)37 b Fc(is)29 b(a)i(suitable)e(n)m (umerical)g(constan)m(t)i(dep)s(ending)d(only)i(on)g Fo(c)p Fc(.)0 2370 y Fj(Pr)-5 b(o)g(of.)44 b Fp(If)i(\026)-49 b Fo(n)42 b Fp(is)f(smaller)g(than)h(some)h(large)f(constan)m(t)i(dep)s (ending)c(on)i Fo(c)h Fp(there)g(is)e(nothing)g(to)j(b)s(e)d(pro)m(v)m (ed.)0 2497 y(Therefore)30 b(w)m(e)h(can)g(assume)f(that)35 b(\026)-49 b Fo(n)30 b Fp(is)f(large)h(enough)h(and)e(w)m(e)i(consider) f(t)m(w)m(o)h(cases.)0 2646 y(In)f(the)g(\014rst)g(one,)g(w)m(e)h(supp) s(ose)j(\026)-50 b Fo(n)20 b Fp(+)1290 2574 y Fk(p)p 1366 2574 V 1371 2646 a Fp(\026)-50 b Fo(n)25 b Fk(\024)g Fo(n)g Fk(\024)g Fo(n)1773 2660 y Fs(max)1931 2646 y Fk(\000)2022 2574 y(p)p 2097 2574 V 2102 2646 a Fp(\026)-50 b Fo(n)p Fp(.)40 b(By)31 b(using)e(the)h(de\014nition)e(of)j Fo(C)7 b(O)s(N)j(V)19 b Fp(\()p Fo(c;)i Fp(\026)-51 b Fo(n)q Fp(\))30 b(w)m(e)0 2772 y(get)1257 2863 y Fl(n)1302 2872 y Fi(max)1272 2891 y Ff(X)1267 3087 y Fl(k)r Fs(=)p Fl(n)1434 2977 y Fo(\015)5 b Fp(\()p Fo(k)s Fp(\))27 b Fk(\025)1740 2916 y Fp(1)p 1740 2957 46 4 v 1743 3040 a Fo(c)1811 2863 y Fl(n)1856 2872 y Fi(max)1827 2891 y Ff(X)1822 3087 y Fl(k)r Fs(=)p Fl(n)2042 2916 y Fp(1)p 2000 2957 131 4 v 2000 2975 a Fk(p)p 2076 2975 55 4 v 2081 3046 a Fp(\026)-50 b Fo(n)2141 2977 y(e)2183 2940 y Fg(\000)2257 2916 y Fh(c)p 2251 2926 41 4 v 2256 2962 a Fi(\026)-36 b Fh(n)2302 2940 y Fs(\()t(\026)c Fl(n)q Fg(\000)p Fl(k)r Fs(\))2502 2912 y Fi(2)1633 3341 y Fk(\025)1740 3279 y Fp(1)p 1740 3320 46 4 v 1743 3403 a Fo(c)1811 3226 y Fl(n)p Fs(+)1912 3175 y Fg(p)p 1972 3175 45 4 v 1976 3226 a Fs(\026)g Fl(n)1848 3255 y Ff(X)1843 3450 y Fl(k)r Fs(=)p Fl(n)2085 3279 y Fp(1)p 2043 3320 131 4 v 2043 3338 a Fk(p)p 2118 3338 55 4 v 2123 3410 a Fp(\026)-50 b Fo(n)2184 3341 y(e)2226 3303 y Fg(\000)2299 3279 y Fh(c)p 2294 3289 41 4 v 2299 3325 a Fi(\026)-36 b Fh(n)2345 3303 y Fs(\()t(\026)c Fl(n)p Fg(\000)p Fl(k)r Fs(\))2544 3276 y Fi(2)1633 3643 y Fk(\025)1740 3582 y Fp(1)p 1740 3622 46 4 v 1743 3705 a Fo(c)1796 3567 y Fk(p)p 1872 3567 55 4 v 1877 3643 a Fp(\026)-50 b Fo(n)1980 3582 y Fp(1)p 1937 3622 131 4 v 1937 3640 a Fk(p)p 2013 3640 55 4 v 2018 3712 a Fp(\026)g Fo(n)2078 3643 y(e)2120 3606 y Fg(\000)2194 3581 y Fh(c)p 2189 3591 41 4 v 2194 3627 a Fi(\026)-36 b Fh(n)2239 3606 y Fs(\()p Fl(n)p Fs(+)2368 3555 y Fg(p)p 2428 3555 45 4 v 2432 3606 a Fs(\026)c Fl(n)p Fg(\000)t Fs(\026)g Fl(n)p Fs(\))2603 3578 y Fi(2)1633 3911 y Fk(\025)1740 3850 y Fp(1)p 1740 3890 46 4 v 1743 3974 a Fo(c)1796 3911 y(e)1838 3874 y Fg(\000)1906 3849 y Fi(4)p Fh(c)p 1906 3859 61 4 v 1921 3896 a Fi(\026)k Fh(n)1977 3874 y Fs(\()p Fl(n)p Fg(\000)t Fs(\026)c Fl(n)q Fs(\))2181 3846 y Fi(2)2246 3911 y Fo(;)3690 3423 y Fp(\()p Fo(A:)p Fp(4\))0 4129 y(where,)24 b(in)e(the)i(last)f(inequalit)m(y)-8 b(,)24 b(w)m(e)f(ha)m(v)m(e)i(used)d(the)i(fact)g(that)g(\()p Fo(n)6 b Fp(+)2344 4057 y Fk(p)p 2419 4057 55 4 v 2424 4129 a Fp(\026)-50 b Fo(n)6 b Fk(\000)11 b Fp(\026)-50 b Fo(n)o Fp(\))2646 4096 y Fs(2)2712 4129 y Fk(\024)25 b Fp(4\()p Fo(n)6 b Fk(\000)11 b Fp(\026)-50 b Fo(n)p Fp(\))3116 4096 y Fs(2)3180 4129 y Fp(for)23 b(all)f Fo(n)j Fk(\025)30 b Fp(\026)-50 b Fo(n)6 b Fp(+)3745 4057 y Fk(p)p 3820 4057 V 3825 4129 a Fp(\026)-50 b Fo(n)p Fp(.)0 4278 y(In)30 b(the)g(second)h(case,)g(w)m(e)g(supp)s(ose)e Fo(n)c Fk(\025)g Fo(n)1488 4292 y Fs(max)1646 4278 y Fk(\000)1737 4206 y(p)p 1813 4206 V 1818 4278 a Fp(\026)-50 b Fo(n)p Fp(.)40 b(An)30 b(easy)h(computation)g(giv)m(es)584 4425 y Fl(n)629 4434 y Fi(max)599 4453 y Ff(X)595 4649 y Fl(k)r Fs(=)p Fl(n)762 4539 y Fo(\015)5 b Fp(\()p Fo(k)s Fp(\))26 b Fk(\025)1067 4478 y Fp(1)p 1067 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b(ac)m(hiev)m(es)i(the)g(pro)s(of.)90 b Fb(\003)0 5558 y Fi(20)p Fh(=settembr)r(e=)p Fi(2000)29 b Fs([51])1193 b Fp(A.5)p eop %%Page: 52 52 52 51 bop 0 91 a Fq(References)13 303 y Fp([An])45 b(C.)30 b(An)m(\023)-43 b(e)32 b(et)f(Al:)f Fj(A)n(utour)i(de)h(l'in)n(\023)-44 b(egalit)n(\023)g(e)33 b(de)g(Sob)-5 b(olev)33 b(lo)-5 b(garithmique)p Fp(.)33 b(Preprin)m(t)c(1999.)-64 431 y([BCC])45 b(L.)24 b(Bertini,)f(N.)i(Cancrini,)d(F.)i(Cesi:)g Fj(The)j(sp)-5 b(e)g(ctr)g(al)29 b(gap)f(for)f(a)g(Glaub)-5 b(er{typ)g(e)30 b(dynamics)e(in)f(a)g(c)-5 b(ontinuous)227 558 y(gas.)31 b Fp(Preprin)m(t)e(n.)h(00-249)j(on)d(h)m (ttp://rene.ma.utexas.edu/mp)p 2515 558 28 4 v 65 w(arc)-69 686 y([BCO])45 b(L.)37 b(Bertini,)e(E.)h(N.)h(M.)g(Cirillo)c(and)j(E.)g (Olivieri:)e Fj(R)-5 b(enormalization{Gr)g(oup)43 b(tr)-5 b(ansformations)42 b(under)227 814 y(str)-5 b(ong)26 b(mixing)e(c)-5 b(onditions:)26 b(Gibbsianness)f(and)g(c)-5 b(onver)g(genc)g(e)25 b(of)f(r)-5 b(enormalize)g(d)27 b(inter)-5 b(actions)p 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y(r)-5 b(andom)35 b(\014elds)p Fp(,)d(Preprin)m(t)d(\(2000\),)k(\(h)m (ttp://isis.roma1.infn.it/)c(cesi/publ.h)m(tml\).)-63 2091 y([CM1])46 b(N.)39 b(Cancrini)c(and)j(F.)g(Martinelli:)e Fj(Comp)-5 b(arison)43 b(of)d(\014nite)f(volume)i(c)-5 b(anonic)g(al)41 b(and)g(gr)-5 b(and)41 b(c)-5 b(anonic)g(al)227 2218 y(Gibbs)33 b(me)-5 b(asur)g(es)34 b(under)f(a)g(mixing)g(c)-5 b(ondition.)32 b Fp(Mark)m(o)m(v)g(Pro)s(c.)e(Rel.)g(Fields)f Fq(6)p Fp(,)i(1-49)h(\(2000\).)-63 2346 y([CM2])46 b(N.)40 b(Cancrini)e(and)h(F.)h(Martinelli:)e Fj(On)j(the)g(sp)-5 b(e)g(ctr)g(al)44 b(gap)e(of)f(Kawasaki)i(dynamics)g(under)e(a)h (mixing)227 2474 y(c)-5 b(ondition)35 b(r)-5 b(evisite)g(d.)31 b Fp(J.)f(Math.)h(Ph)m(ys.)g Fq(41)p Fp(,)g(1391-1423)j(\(2000\).)-63 2602 y([CM3])46 b(N.)29 b(Cancrini)d(and)i(F.)h(Martinelli:)d Fj(Di\013usive)j(sc)-5 b(aling)32 b(of)f(the)g(sp)-5 b(e)g(ctr)g(al)33 b(gap)f(for)f(the)g(dilute)h(I)e(sing)h(lattic)-5 b(e)227 2729 y(gas.)p Fp(,)31 b(Roma,)g(preprin)m(t)e(\(2000\).)-59 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b(Y)-8 b(au:)36 b Fj(L)-5 b(o)g(garithmic)40 b(Sob)-5 b(olev)38 b(Ine)-5 b(quality)38 b(for)g(gener)-5 b(alize)g(d)39 b(simple)f(exclusion)g(pr)-5 b(o)g(c)g(esses)38 b Fp(Proba-)227 4773 y(bilit)m(y)29 b(Theory)h(and)f(Related)i(Fields,) e Fq(109)p Fp(,)i(No.4,)h(507-538,)i(\(1997\).)0 5558 y Fi(20)p Fh(=settembr)r(e=)p Fi(2000)29 b Fs([52])1193 b Fp(A.6)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0009200614863--