%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: srf.dvi %%Pages: 55 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSCommandLine: dvips srf %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 1998.01.23:1206 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N 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ch-width 7 add 8 idiv string N /rc 0 N /gp 0 N /cp 0 N{rc 0 ne{rc 1 sub /rc X rw}{G}ifelse}imagemask restore}B /G{{id gp get /gp gp 1 add N dup 18 mod S 18 idiv pl S get exec}loop}B /adv{cp add /cp X}B /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp X adv}B /nd{/cp 0 N rw exit}B /lsh{rw cp 2 copy get dup 0 eq{pop 1}{ dup 255 eq{pop 254}{dup dup add 255 and S 1 and or}ifelse}ifelse put 1 adv}B /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv S 128 and or}ifelse}ifelse put 1 adv}B /clr{rw cp 2 index string putinterval adv}B /set{rw cp fillstr 0 4 index getinterval putinterval adv}B /fillstr 18 string 0 1 17{2 copy 255 put pop}for N /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc X nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall N /D{/cc X dup type /stringtype ne{] }if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 2 index 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put setpagedevice } { statusdict /a4tray known { statusdict begin a4tray end /a4 where { pop a4 } if } { /a4 where { pop a4 } { statusdict /setpage known { statusdict begin 595 842 0 setpage end } if } ifelse } ifelse } ifelse %%EndPaperSize %%EndSetup %%Page: 1 1 1 0 bop 700 100 a Fq(T)l(ypical)45 b(con\014gurations)g(for)h(one)f (dimensional)1195 310 y(random)g(\014eld)g(Kac)h(mo)t(del)2659 274 y Fp(\003)823 733 y Fo(Marzio)33 b(Cassandro)1641 697 y Fn(1)1776 733 y Fo(Enza)g(Orlandi)2392 697 y Fn(2)2470 733 y Fo(Pierre)g(Picco)3032 697 y Fn(3)120 1016 y Fm(Abstract.)41 b Fo(In)29 b(this)h(pap)s(er)g(w)m(e)g(study)g(the)f(t)m(ypical)g (pro\014les)h(of)g(a)f(random)g(\014eld)h(Kac)g(mo)s(del.)42 b(W)-8 b(e)0 1156 y(giv)m(e)33 b(upp)s(er)h(and)g(lo)m(w)m(er)f(b)s (ounds)i(of)e(the)h(space)g(scale)g(where)g(the)f(system)g(sta)m(ys)g (at)g(equilibrium.)0 1437 y Fq(1)45 b(In)l(tro)t(duction)120 1578 y Fo(In)j(this)h(pap)s(er)g(w)m(e)g(consider)g(a)f(one)h 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Fj(#)27 b Fo(0,)32 b(in)g(order)0 3533 y(to)h(get)h(non)g(trivial)e(limiting)f (distributions,)j(it)f(is)h(useful)h(to)e(in)m(tro)s(duce)i(the)f(so)g (called)f(blo)s(c)m(k-spins,)0 3672 y(that)h(are)h(space)h(a)m(v)m (erage)f(of)g(spins)h(o)m(v)m(er)e(regions)h(whose)h(size)f(div)m (erges)g(as)g Fl(\015)h Fj(#)30 b Fo(0)k(and)i(describ)s(e)g(the)0 3812 y(con\014gurations)25 b(of)g(the)g(system)f(in)h(term)e(of)i (these)h(magnetization)d(pro\014les.)42 b(In)25 b(the)g(one)g (dimensional)0 3951 y(case,)g(this)d(analysis)g([11])f(for)h(Ising)g (spin)h(and)g([7])e(for)i(more)e(general)h(spin,)j(allo)m(ws)d(to)f (get)h(a)g(satisfactory)p 0 4074 1200 4 v 7 4148 a Fp(\003)120 4184 y Fi(w)n(ork)36 b(supp)r(orted)i(b)n(y)e(the)g(CNR-GNFM,)i (INFM-sezione)g(di)f(Roma)g(and)g(the)g(Commission)h(of)f(the)g(Europ)r (ean)0 4324 y(Union)30 b(con)n(tract)d(No.)38 b(CHRX-CT93-0411)25 4407 y Fn(1)120 4443 y Fi(Dipartimen)n(to)51 b(di)i(Fisica,)59 b(Univ)n(ersit\023)-42 b(a)52 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b(~)-52 b Fl(c)p Fo(,)27 b Fl(p)h(>)g Fo(1,)f Fl(L)1749 3344 y Fn(1)1794 3329 y Fo(,)g(w)m(e)g(denote)g(b)m(y)f Fl(N)2514 3344 y Fk(\015)2593 3329 y Fo(=)i([)2740 3289 y Fn(~)-42 b Fk(c)p 2738 3306 36 4 v 2740 3363 a Fn(^)g Fk(c)2785 3329 y Fo(\(log)2984 3289 y Fn(1)p 2981 3306 47 4 v 2981 3363 a Fk(\015)3039 3329 y Fo(\))3078 3292 y Fk(p)3124 3329 y Fo(\(log)17 b(log)3469 3289 y Fn(1)p 3465 3306 V 3465 3363 a Fk(\015)3523 3329 y Fo(\)],)27 b(where)0 3468 y([)p Fl(x)p Fo(])32 b(is)i(the)f(in)m(teger)h(part)f(of)g Fl(x)p Fo(,)g(b)m(y)h Fl(`)1377 3483 y Fn(^)-42 b Fk(c)1415 3468 y Fo(\()p Fl(\015)6 b Fo(\))26 b(=)1857 3429 y Fn(^)-42 b Fk(c)p 1694 3445 358 4 v 1694 3508 a(\015)18 b Fn(log)d(log)2003 3481 y Fg(1)p 1999 3492 41 4 v 1999 3532 a Ff(\015)2097 3468 y Fo(and)34 b(b)m(y)735 3866 y Fj(R)819 3824 y Fk(\016)n(;\020)924 3866 y Fo(\()p Fl(L)1031 3881 y Fn(1)1075 3866 y Fl(;)18 b Fo(^)-51 b Fl(c)o(;)1221 3779 y Fo(~)f Fl(c)p Fo(\(log)1461 3740 y Fn(1)p 1458 3756 47 4 v 1458 3813 a Fk(\015)1516 3779 y Fo(\))1555 3743 y Fk(p)p 1219 3843 382 4 v 1381 3934 a Fl(\015)1612 3866 y Fo(\))28 b Fj(\021)1868 3730 y Fk(N)1931 3740 y Ff(\015)1867 3771 y Fd([)1784 3985 y Fk(k)r Fn(=)p Fp(\000)p Fk(N)2014 3995 y Ff(\015)2077 3866 y Fj(R)2161 3824 y Fk(\016)n(;\020)2265 3866 y Fo(\()p Fl(k)s(`)2403 3881 y Fn(^)-42 b Fk(c)2441 3866 y Fo(\()p Fl(\015)6 b Fo(\))p Fl(;)17 b(L)2690 3881 y Fn(1)2755 3866 y Fo(+)22 b Fl(k)s(`)2953 3881 y Fn(^)-42 b Fk(c)2991 3866 y Fo(\()p Fl(\015)6 b Fo(\)\))479 b(\(2)p Fl(:)p Fo(28\))0 4269 y(That)39 b(is)g(the)h(set)f(of)g(pro\014les)h(that)f (in)g(an)g(in)m(terv)-6 b(al)39 b(of)h(length)f(2)2517 4205 y Fn(~)-42 b Fk(c)o Fn(\(log)2713 4178 y Fg(1)p 2710 4189 41 4 v 2710 4229 a Ff(\015)2763 4205 y Fn(\))2794 4175 y Ff(p)p 2515 4247 321 4 v 2652 4304 a Fk(\015)2848 4269 y Fo(,)40 b(cen)m(tered)h(at)d(the)i(origin,)0 4409 y(ha)m(v)m(e)34 b(at)e(least)h(one)h(in)m(terv)-6 b(al)33 b(of)h(length)f Fl(L)1613 4424 y Fn(1)1691 4409 y Fo(that)g(it)f(is)h (rigid.)44 b(W)-8 b(e)33 b(ha)m(v)m(e)h(the)g(follo)m(wing)e(result.)0 4634 y Fm(Theorem)51 b(2.3)i Fe(Given)48 b Fo(~)-53 b Fl(c)48 b(>)g Fo(0)p Fe(,)h Fl(\014)k(>)48 b Fo(1)p Fe(,)h Fl(p)f(>)g Fo(1)p Fe(,)g Fl(\032)g(>)g Fo(0)p Fe(,)g Fl(c)2459 4649 y Fn(0)2552 4634 y Fl(>)g Fo(0)p Fe(,)h Fl(\014)5 b(\022)49 b Fe(smal)5 b(l)48 b(enough,)h(for)d(al)5 b(l)0 4774 y Fl(x)29 b(>)h Fo(0)p Fe(,)36 b Fl(\016)d(>)d(\016)542 4738 y Fp(\003)617 4774 y Fo(=)g Fl(c)767 4789 y Fn(0)828 4774 y Fo(log)17 b(log)1134 4735 y Fn(1)p 1131 4751 47 4 v 1131 4808 a Fk(\015)1225 4774 y Fe(ther)-5 b(e)36 b(exist)g(an)h(absolute)f(c)-5 b(onstant)36 b Fl(c)30 b(>)g Fo(0)36 b Fe(and)g(a)h(p)-5 b(ositive)34 b(c)-5 b(onstant)2 4913 y Fo(^)-52 b Fl(c)28 b Fo(=)j(^)-52 b Fl(c)p Fo(\()p Fl(\014)5 b(;)17 b(\022)s(;)g(x)p Fo(\))34 b Fe(such)i(that)g(if)f Fl(\015)e Fo(=)c(2)1359 4877 y Fp(\000)p Fk(n)1475 4913 y Fe(,)36 b Fl(I)-18 b(P)14 b Fe(-almost)36 b(sur)-5 b(ely,)36 b(for)f(al)5 b(l)37 b(but)g(a)e(\014nite)h(numb)-5 b(er)36 b(of)g(indic)-5 b(es)0 5053 y Fl(n)p Fe(,)35 b(if)860 5283 y Fl(L)928 5298 y Fn(1)1000 5283 y Fj(\025)1118 5196 y Fo(\(log)1317 5157 y Fn(1)p 1314 5173 V 1314 5231 a Fk(\015)1372 5196 y Fo(\)\(log)16 b(log)1755 5157 y Fn(1)p 1752 5173 V 1752 5231 a Fk(\015)1810 5196 y Fo(\))1849 5160 y Fn(2+)p Fk(\032)p 1118 5260 879 4 v 1528 5351 a Fl(\015)2024 5143 y Fd(\024)2345 5216 y Fl(c)p Fo(\()p Fl(x;)h(\032;)g(\015)6 b Fo(\))p 2089 5260 887 4 v 2089 5351 a(\()p Fl(\014)f(\022)s Fo(\))2278 5323 y Fn(2)2322 5351 y Fo(\()p Fl(m)2448 5366 y Fk(\014)t(;)p Fn(1)2588 5351 y Fo(+)22 b Fl(m)2774 5366 y Fk(\014)t(;)p Fn(2)2892 5351 y Fo(\))2931 5323 y Fn(2)2987 5143 y Fd(\025)3645 5283 y Fo(\(2)p Fl(:)p Fo(29\))0 5579 y Fg(23)p Ff(=j)s(anuar)r(y)r(=)p Fg(1998;)32 b(12:06)1145 b Fo(10)p eop %%Page: 11 11 11 10 bop 0 110 a Fe(wher)-5 b(e)35 b Fl(c)p Fo(\()p Fl(x;)17 b(\032;)g(\015)6 b Fo(\))25 b(=)1032 62 y Fn(2\(4+)p Fk(x)p Fn(\))1280 32 y Fg(2)p 801 88 750 4 v 801 203 a Fn(1+\(2+)1046 169 y Fg(3)p Ff(\032)p 1046 187 71 4 v 1064 227 a Fg(4)1128 203 y Fn(\))1171 143 y Fg(log)12 b(log)g(log)1489 124 y(1)p 1486 133 41 4 v 1486 169 a Ff(\015)p 1171 187 368 4 v 1221 242 a Fg(log)g(log)1439 223 y(1)p 1435 232 41 4 v 1435 269 a Ff(\015)1598 110 y Fe(then)1062 621 y Fl(\026)1122 636 y Fk(\014)t(;\022)r(;\015)1326 451 y Fd(")1384 621 y Fj(R)1468 580 y Fk(\016)n(;\020)1573 621 y Fo(\()p Fl(L)1680 636 y Fn(1)1724 621 y Fl(;)18 b Fo(^)-51 b Fl(c)o(;)1870 534 y Fo(~)f Fl(c)p Fo(\(log)2110 495 y Fn(1)p 2107 512 47 4 v 2107 569 a Fk(\015)2165 534 y Fo(\))2204 498 y Fk(p)p 1868 598 382 4 v 2030 690 a Fl(\015)2261 621 y Fo(\))2300 451 y Fd(#)2386 621 y Fj(\024)28 b Fl(e)2537 580 y Fp(\000)p Fk(\014)t(x\015)2739 550 y Fb(\000)p Fg(1)3645 621 y Fo(\(2)p Fl(:)p Fo(30\))0 989 y Fe(pr)-5 b(ovide)g(d)33 b(that)i(for)f(some)g(function)g Fl(g)1445 1004 y Fn(2)1489 989 y Fo(\(1)p Fl(=\020)7 b Fo(\))p Fe(,)33 b(with)h Fo(lim)2136 1004 y Fk(\020)5 b Fp(#)p Fn(0)2279 989 y Fl(g)2327 1004 y Fn(2)2371 989 y Fo(\(1)p Fl(=\020)i Fo(\))27 b(=)h Fj(1)p Fe(,)34 b(slow)5 b(ly)35 b(varying)f(at)h(in\014n-)0 1128 y(ity,)g Fo(lim)317 1143 y Fk(\020)5 b Fp(#)p Fn(0)459 1053 y Fj(p)p 543 1053 51 4 v 543 1128 a Fl(\020)h(g)641 1143 y Fn(2)686 1128 y Fo(\(1)p Fl(=\020)h Fo(\))26 b(=)j(0)35 b Fe(and)g Fl(\020)7 b(g)1424 1143 y Fn(2)1469 1128 y Fo(\(1)p Fl(=\020)g Fo(\))26 b Fl(<)j(\014)5 b(\022)s Fo(\()p Fl(m)2067 1143 y Fk(\014)t(;)p Fn(1)2206 1128 y Fo(+)23 b Fl(m)2393 1143 y Fk(\014)t(;)p Fn(2)2510 1128 y Fo(\))2549 1092 y Fn(2)2659 1089 y Fk(c)p 2606 1105 142 4 v 2606 1162 a(p)p Fn(+2)2760 1128 y Fe(.)120 1413 y Fo(T)-8 b(o)36 b(mak)m(e)f(the)i(previous)g(theorem)f(meaningful,)h(w)m(e)g(need)h(a)e (result)h(in)g(the)f(opp)s(osite)h(direction.)0 1553 y(That)31 b(is)g(to)g(pro)m(v)m(e)g(that)g(the)g(system)g(is)g(rigid)f (with)h(the)h Fe(same)f Fo(tolerance)g Fl(\020)38 b Fo(on)31 b(a)g(scale)g(smaller)f(that)0 1692 y Fl(L)68 1707 y Fn(1)112 1692 y Fo(.)44 b(As)33 b(w)m(e)h(will)d(see)j(later,)e(this)h (will)f(giv)m(e)h(a)f(constrain)m(t)h(from)g(b)s(elo)m(w)g(on)g Fl(\020)7 b Fo(.)43 b(F)-8 b(or)33 b(reasons)g(that)g(will)0 1832 y(b)s(ecome)e(clear)g(later)g(w)m(e)g(need)i(to)d(in)m(tro)s(duce) i(t)m(w)m(o)f(di\013eren)m(t)i(tolerance)e(parameters)g(that)g(w)m(e)g (call)g Fl(\020)3856 1847 y Fn(4)0 1971 y Fo(and)g Fl(\020)235 1986 y Fn(1)309 1971 y Fo(and)g(the)f(corresp)s(onding)i Fl(\016)1350 1986 y Fn(4)1425 1971 y Fo(and)e Fl(\016)1659 1986 y Fn(1)1704 1971 y Fo(.)43 b Fl(\020)1819 1986 y Fn(4)1894 1971 y Fo(will)29 b(pla)m(y)h(the)g(role)g(of)h Fl(\020)37 b Fo(in)30 b(the)g(previous)h(theorem.)120 2115 y(Giv)m(en)46 b Fl(`)463 2130 y Fn(1)556 2115 y Fj(2)j Fl(Z)-35 b(Z)7 b Fo(,)49 b Fl(`)898 2130 y Fn(2)991 2115 y Fj(2)g Fl(Z)-35 b(Z)53 b Fo(with)45 b Fl(`)1541 2130 y Fn(1)1634 2115 y Fl(<)50 b(`)1803 2130 y Fn(2)1847 2115 y Fo(,)f Fl(\016)1968 2130 y Fn(4)2061 2115 y Fl(>)h Fo(0,)e Fl(\020)2358 2130 y Fn(4)2451 2115 y Fl(>)i Fo(0,)e Fl(\016)2748 2130 y Fn(1)2842 2115 y Fl(>)h Fo(0,)g Fl(\020)3139 2130 y Fn(1)3232 2115 y Fl(>)g Fo(0,)g Fl(R)3561 2130 y Fn(1)3654 2115 y Fj(2)g Fl(I)-17 b(R)p Fo(,)0 2254 y Fl(x)52 b Fj(2)h Fo([)p Fl(`)298 2269 y Fn(1)374 2254 y Fo(+)33 b Fl(R)560 2269 y Fn(1)636 2254 y Fo(+)g(1)p Fl(;)17 b(`)883 2269 y Fn(2)958 2254 y Fj(\000)33 b Fo(3)p Fl(R)1194 2269 y Fn(1)1270 2254 y Fj(\000)f Fo(1])48 b(and)g Fl(\034)64 b Fj(2)53 b(f\000)p Fo(1)p Fl(;)17 b Fo(+1)p Fj(g)p Fo(,)51 b(w)m(e)d(de\014ne)i(a)e(fron)m(t)g(starting)g (at)g(the)0 2394 y(equilibrium)32 b Fl(\034)44 b Fo(at)33 b(the)g(p)s(oin)m(t)g Fl(x)h Fo(b)m(y)209 2675 y Fj(V)278 2634 y Fk(\016)314 2644 y Fg(1)353 2634 y Fk(;\020)413 2644 y Fg(1)452 2634 y Fk(;\016)512 2644 y Fg(4)551 2634 y Fk(;\020)611 2644 y Fg(4)671 2675 y Fo(\()p Fl(`)752 2690 y Fn(1)796 2675 y Fl(;)17 b(`)883 2690 y Fn(2)926 2675 y Fl(;)g(\034)6 b(;)17 b(x)p Fo(\))26 b(=)1293 2595 y Fd(\010)1351 2675 y Fl(m)1438 2634 y Fk(\016)1477 2604 y Fb(\003)1551 2675 y Fo(:)i Fj(8)p Fl(`)f Fj(2)h Fo([)p Fl(`)1895 2690 y Fn(1)1961 2675 y Fo(+)23 b Fl(R)2137 2690 y Fn(1)2181 2675 y Fl(;)17 b(x)p Fo(])p Fl(;)32 b(\021)2424 2634 y Fk(\016)2460 2644 y Fg(4)2499 2634 y Fk(;\020)2559 2644 y Fg(4)2603 2675 y Fo(\()p Fl(`)p Fo(\))27 b(=)h Fl(\034)38 b Fo(=)29 b Fl(\021)3096 2634 y Fk(\016)3132 2644 y Fg(1)3171 2634 y Fk(;\020)3231 2644 y Fg(1)3275 2675 y Fo(\()p Fl(x)p Fo(\))p Fl(;)408 2845 y Fj(8)p Fl(`)f Fj(2)g Fo([)p Fl(x)21 b Fo(+)i Fl(R)909 2860 y Fn(1)976 2845 y Fo(+)f(1)p Fl(;)17 b(`)1212 2860 y Fn(2)1278 2845 y Fj(\000)22 b Fl(R)1453 2860 y Fn(1)1498 2845 y Fo([)p Fl(;)32 b(\021)1639 2804 y Fk(\016)1675 2814 y Fg(4)1714 2804 y Fk(;\020)1774 2814 y Fg(4)1818 2845 y Fo(\()p Fl(`)p Fo(\))27 b(=)h Fj(\000)p Fl(\034)39 b Fo(=)28 b Fl(\021)2388 2804 y Fk(\016)2424 2814 y Fg(1)2463 2804 y Fk(;\020)2523 2814 y Fg(1)2567 2845 y Fo(\()p Fl(x)22 b Fo(+)g Fl(R)2860 2860 y Fn(1)2905 2845 y Fo(\))2944 2764 y Fd(\011)3645 2755 y Fo(\(2)p Fl(:)p Fo(31\))0 3132 y(and)42 b Fj(V)271 3096 y Fk(\016)307 3106 y Fg(1)347 3096 y Fk(;\020)407 3106 y Fg(1)446 3096 y Fk(;\016)506 3106 y Fg(4)544 3096 y Fk(;\020)604 3106 y Fg(4)665 3132 y Fo(\()o Fl(`)745 3147 y Fn(1)790 3132 y Fl(;)17 b(`)877 3147 y Fn(2)920 3132 y Fl(;)g(\034)11 b Fo(\))41 b(=)1220 3058 y Fd(S)1303 3162 y Fk(`)1336 3172 y Fg(1)1375 3162 y Fn(+)p Fk(R)1496 3172 y Fg(1)1535 3162 y Fn(+1)p Fp(\024)p Fk(x)p Fp(\024)p Fk(`)1838 3172 y Fg(2)1878 3162 y Fp(\000)p Fn(3)p Fk(R)2040 3172 y Fg(1)2079 3162 y Fp(\000)p Fn(1)2202 3132 y Fj(V)2271 3096 y Fk(\016)2307 3106 y Fg(1)2347 3096 y Fk(;\020)2407 3106 y Fg(1)2445 3096 y Fk(;\016)2505 3106 y Fg(4)2544 3096 y Fk(;\020)2604 3106 y Fg(4)2665 3132 y Fo(\()o Fl(`)2745 3147 y Fn(1)2790 3132 y Fl(;)17 b(`)2877 3147 y Fn(2)2920 3132 y Fl(;)g(\034)6 b(;)17 b(x)p Fo(\))n(.)70 b(W)-8 b(e)42 b(denote)h(the)0 3272 y(set)34 b(of)f(fron)m(ts)h(that)f(o)s(ccur)h(within)f([)p Fl(`)1417 3287 y Fn(1)1460 3272 y Fl(;)17 b(`)1547 3287 y Fn(2)1591 3272 y Fo(])33 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Fj(\021)g(R)1891 4038 y Fk(\016)1927 4048 y Fg(4)1966 4038 y Fk(;\020)2026 4048 y Fg(4)2087 4079 y Fo(\()o Fl(`)2167 4094 y Fn(1)2234 4079 y Fo(+)22 b(2)p Fl(R)2459 4094 y Fn(1)2504 4079 y Fl(;)17 b(`)2591 4094 y Fn(2)2656 4079 y Fj(\000)23 b Fo(2)p Fl(R)2882 4094 y Fn(1)2926 4079 y Fo(\))680 b(\(2)p Fl(:)p Fo(33\))0 4347 y(W)-8 b(e)34 b(ha)m(v)m(e)f(the)h(follo)m(wing)e(result.)0 4576 y Fm(Theorem)41 b(2.4)i Fe(Given)38 b Fl(\014)g(>)c Fo(1)p Fe(,)39 b Fl(\032)32 b(>)i Fo(0)k Fe(and)g Fd(b)-54 b Fl(c)38 b Fe(ther)-5 b(e)39 b(exists)e(an)i Fl(\017)2572 4591 y Fn(0)2656 4576 y Fe(such)f(that)h(if)f Fl(\014)5 b(\022)36 b Fj(\024)e Fl(\017)3487 4591 y Fn(0)3532 4576 y Fe(,)39 b(we)f(c)-5 b(an)0 4715 y(\014nd)36 b Fl(\015)251 4730 y Fn(0)325 4715 y Fl(>)30 b Fo(0)p Fe(,)36 b Fl(c)592 4730 y Fn(0)666 4715 y Fl(>)29 b Fo(0)37 b Fe(and)f(c)-5 b(onstants)36 b Fl(c)1535 4730 y Fk(i)1597 4715 y Fo(=)30 b Fl(c)1747 4730 y Fk(i)1780 4715 y Fo(\()p Fl(\014)5 b(;)17 b(\022)s Fo(\))35 b Fe(for)h 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Ff(=j)s(anuar)r(y)r(=)p Fg(1998;)32 b(12:06)1145 b Fo(11)p eop %%Page: 12 12 12 11 bop 0 111 a Fe(for)40 b Fl(R)240 126 y Fn(1)322 111 y Fo(=)e Fl(c)480 126 y Fn(2)525 111 y Fo(\()p Fl(\016)608 126 y Fn(1)653 111 y Fl(\020)704 75 y Fn(3)697 136 y(1)748 111 y Fo(\))787 75 y Fp(\000)p Fn(1)894 111 y Fe(,)k(for)e(any)i (interval)e Fl(I)45 b Fo(=)38 b([)p Fl(`)1972 126 y Fn(1)2016 111 y Fl(;)17 b(`)2103 126 y Fn(2)2146 111 y Fo(])41 b Fe(such)g(that)g Fo(4)p Fl(R)2782 126 y Fn(1)2864 111 y Fj(\024)c(j)p Fl(`)3048 126 y Fn(1)3118 111 y Fj(\000)27 b Fl(`)3264 126 y Fn(2)3308 111 y Fj(j)37 b(\024)3655 72 y Fd(b)-49 b Fk(c)p 3500 88 358 4 v 3500 151 a(\015)18 b Fn(log)c(log)3808 124 y Fg(1)p 3805 135 41 4 v 3805 175 a Ff(\015)3869 111 y Fe(,)0 328 y(ther)-5 b(e)35 b(exists)g Fo(\012)593 343 y Fn(1)665 328 y Fo(=)29 b(\012)843 343 y Fn(1)887 328 y Fo(\()p Fl(\014)5 b(;)17 b(\022)s(;)g(`)1169 343 y Fn(1)1212 328 y Fl(;)g(`)1299 343 y Fn(2)1342 328 y Fl(;)g(\015)6 b Fo(\))34 b Fe(such)i(that)f 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Ff(\020)3235 573 y Fg(3)3232 621 y(4)p 3056 629 218 4 v 3145 669 a Ff(\015)3645 689 y Fo(\(2)p Fl(:)p Fo(35\))0 1022 y Fe(in)43 b(p)-5 b(articular,)45 b(if)d Fl(\015)47 b Fo(=)c(2)999 986 y Fp(\000)p Fk(n)1115 1022 y Fe(,)i Fl(I)-18 b(P)14 b Fe(-almost)43 b(sur)-5 b(ely,)46 b(\(2.35\))c(o)-5 b(c)g(curs)43 b(for)g(al)5 b(l)45 b(but)e(a)h(\014nite)f(numb)-5 b(er)44 b(of)0 1161 y(indic)-5 b(es)34 b Fl(n)p Fe(.)120 1453 y Fo(Roughly)c(sp)s(eaking,)g(inside)h(an)f(in)m(terv)-6 b(al)30 b(of)h(length)h(^)-52 b Fl(c)p Fo(\()p Fl(\015)22 b Fo(log)17 b(log)2594 1414 y Fn(1)p 2591 1430 47 4 v 2591 1487 a Fk(\015)2649 1453 y Fo(\))2688 1417 y Fp(\000)p Fn(1)2825 1453 y Fo(cen)m(tered,)32 b(for)f(example,)e(at)0 1593 y(the)f(origin,)g(the)h(t)m(ypical)e(pro\014les)i(are)f(rigid)g (with)g(a)f(tolerance)i Fl(\020)2453 1608 y Fn(4)2525 1593 y Fo(around)g(one)f(of)g(the)h(t)m(w)m(o)f(equilibria)0 1732 y(or)33 b(mak)m(e)f(only)h(one)h(transition)f(b)s(et)m(w)m(een)h (the)g(t)m(w)m(o)f(equilibria.)44 b(Note)32 b(also)h(that)g(w)m(e)h(ha) m(v)m(e)g(allo)m(w)m(ed)f(a)0 1871 y(fuzzy)c(region)f(of)g(length)g(2)p Fl(R)1082 1886 y Fn(1)1154 1871 y Fo(around)h(the)f(extremes)g(of)g (the)h(in)m(terv)-6 b(al)27 b(w)m(e)i(ha)m(v)m(e)f(considered)i(and)e (also)0 2011 y(a)36 b(region)g Fl(R)464 2026 y Fn(1)545 2011 y Fo(around)h(the)g(fron)m(t.)54 b(As)36 b(it)g(will)f(b)s(e)i (pro)m(v)m(ed)g(in)g(the)f(c)m(hapter)i(5,)e(this)h Fl(R)3306 2026 y Fn(1)3386 2011 y Fo(corresp)s(onds)0 2150 y(to)32 b(the)h(longest)f(runs)h(of)g Fl(\021)1016 2114 y Fk(\016)1052 2124 y Fg(1)1091 2114 y Fk(;\020)1151 2124 y Fg(1)1223 2150 y Fo(=)28 b(0)k(whic)m(h)h(is)g(t)m(ypical)e(with)h(resp)s(ect)i (to)e(the)g(Gibbs)i(measure.)43 b(Note)0 2290 y(that)33 b(it)f(is)i(p)s(ossible)f(to)g(tak)m(e)g Fl(\020)1172 2305 y Fn(4)1249 2290 y Fo(in)h(the)f(Theorem)g(2.4)g(and)g Fl(\020)41 b Fo(in)33 b(the)g(Theorem)g(2.3)g(equal.)120 2440 y(The)49 b(Theorem)g(2.4)g(suggests)h(that)f(the)h(go)s(o)s(d)e 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Fo(\(log)18 b(1)p Fl(=\015)6 b Fo(\))1868 3241 y Fk(p)1960 3277 y Fo(for)48 b(some)f Fl(p)k(>)i Fo(1)47 b(since)i(it)e(is)g(the)h(scale)g(where)0 3416 y(w)m(e)d(kno)m(w)g(b)m(y)g(the)g(Theorem)g(2.3)f(that)g(suc)m(h)j (fron)m(ts)f(exists.)78 b(This)45 b(corresp)s(ond)i(also)d(to)g(giv)m (e)h(an)0 3556 y(upp)s(er)d(and)f(lo)m(w)m(er)g(b)s(ound)h(on)f(the)g (n)m(um)m(b)s(er)g(of)g(transitions)g(from)f(one)h(equilibrium)e(to)i (the)g(other)0 3695 y(one)50 b(in)g(suc)m(h)h(an)e(in)m(terv)-6 b(al.)93 b(T)-8 b(o)49 b(b)s(e)h(more)f(precise,)54 b(w)m(e)c(need)h (some)e(more)g(de\014nitions.)94 b(Giv)m(en)0 3835 y(an)50 b(in)m(terv)-6 b(al)49 b Fj(J)73 b Fo(=)55 b([)p Fj(\000)p Fl(j)950 3850 y Fn(1)995 3835 y Fl(;)17 b(j)1081 3850 y Fn(1)1125 3835 y Fo(],)53 b(cen)m(tered)f(at)d(the)h(origin,)i(and)e (p)s(ositiv)m(es)g(in)m(tegers)g Fl(k)i Fo(and)e Fl(L)p Fo(,)k(w)m(e)0 3974 y(de\014ne,)37 b(for)f Fl(`)515 3989 y Fn(1)559 3974 y Fl(;)17 b(`)646 3989 y Fn(2)721 3974 y Fj(2)32 b Fl(Z)-35 b(Z)7 b Fo(,)36 b Fl(\034)42 b Fj(2)32 b(f\000)p Fo(1)p Fl(;)17 b Fo(+1)p Fj(g)p Fo(,)35 b Fj(R)1721 3938 y Fk(\016)1757 3948 y Fg(4)1797 3938 y Fk(;\020)1857 3948 y Fg(4)1917 3974 y Fo(\()p Fl(`)1998 3989 y Fn(1)2042 3974 y Fl(;)17 b(`)2129 3989 y Fn(2)2172 3974 y Fl(;)g(R)2293 3989 y Fn(1)2337 3974 y Fl(;)g(\034)11 b Fo(\))30 b Fj(\021)i(R)2699 3938 y Fk(\016)2735 3948 y Fg(4)2775 3938 y Fk(;\020)2835 3948 y Fg(4)2895 3974 y Fo(\()p Fl(`)2976 3989 y Fn(1)3042 3974 y Fo(+)23 b(2)p Fl(R)3268 3989 y Fn(1)3312 3974 y Fl(;)17 b(`)3399 3989 y Fn(2)3465 3974 y Fj(\000)22 b Fo(2)p Fl(R)3690 3989 y Fn(1)3734 3974 y Fl(;)17 b(\034)11 b Fo(\))o(,)0 4114 y Fj(T)80 4078 y Fk(\016)116 4088 y Fg(4)155 4078 y Fk(;\020)215 4088 y Fg(4)258 4114 y Fo(\()p Fl(L;)17 b Fo(1)p Fl(;)g(\034)6 b(;)17 b Fj(J)e Fo(\))27 b Fj(\021)h(R)938 4078 y Fk(\016)974 4088 y Fg(4)1014 4078 y Fk(;\020)1074 4088 y Fg(4)1134 4114 y Fo(\()p Fj(\000)p Fl(j)1291 4129 y Fn(1)1336 4114 y Fl(;)17 b(j)1422 4129 y Fn(1)1466 4114 y Fl(;)g(R)1587 4129 y Fn(1)1631 4114 y Fl(;)g(\034)11 b Fo(\))32 b(and)380 4417 y Fj(T)460 4375 y Fk(\016)496 4385 y Fg(4)535 4375 y Fk(;\020)595 4385 y Fg(4)639 4417 y Fo(\()p Fl(L;)17 b(k)s(;)g(\034)6 b(;)17 b Fj(J)d Fo(\))28 b(=)495 4577 y Fk(j)528 4587 y Fg(1)475 4612 y Fd([)397 4826 y Fk(`)430 4836 y Fg(1)469 4826 y Fn(=)p Fp(\000)p Fk(j)625 4836 y Fg(1)819 4577 y Fk(j)852 4587 y Fg(1)799 4612 y Fd([)751 4826 y Fk(`)784 4836 y Fg(2)823 4826 y Fk(>`)918 4836 y Fg(1)693 4897 y Fk(`)726 4907 y Fg(2)765 4897 y Fp(\000)p Fk(`)860 4907 y Fg(1)899 4897 y Fk(>L)1045 4707 y Fl(:)17 b(:)g(:)1364 4577 y Fk(j)1397 4587 y Fg(1)1345 4612 y Fd([)1256 4813 y Ff(`)1286 4830 y(k)1329 4813 y(>`)1413 4830 y(k)q Fb(\000)p Fg(1)1190 4899 y Fk(`)1223 4910 y Ff(k)1267 4899 y Fp(\000)p Fk(`)1362 4910 y Ff(k)q Fb(\000)p Fg(1)1494 4899 y Fk(>L)1708 4582 y(k)1675 4612 y Fd(\\)1639 4826 y Fk(k)1681 4836 y Fg(1)1720 4826 y Fn(=1)1838 4707 y Fj(R)1922 4665 y Fk(\016)1958 4675 y Fg(4)1997 4665 y Fk(;\020)2057 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1768 y Fl(>)f(\016)3112 1732 y Fp(\003)3186 1768 y Fo(=)g Fl(c)3334 1783 y Fn(0)3379 1768 y Fl(\015)22 b Fo(log)17 b(log)g(1)p Fl(=\015)0 1907 y Fe(that)36 b(satisfy)f(\(2.34\),)g Fl(L)915 1922 y Fn(1)995 1907 y Fe(that)i(satis\014es)d(\(2.29\))h(and) h(for)g Fl(R)2305 1922 y Fn(1)2378 1907 y Fo(=)29 b Fl(c)2527 1922 y Fn(2)2572 1907 y Fo(\()p Fl(\016)2655 1922 y Fn(1)2699 1907 y Fl(\020)2750 1871 y Fn(3)2743 1932 y(1)2795 1907 y Fo(\))2834 1871 y Fp(\000)p Fn(1)2941 1907 y Fe(,)35 b(for)h(al)5 b(l)37 b(given)e(interval)0 2047 y Fj(J)57 b Fe(of)38 b(length)k Fo(~)-52 b Fl(c)p Fo(\(log)784 2008 y Fn(1)p 781 2024 47 4 v 781 2081 a Fk(\015)839 2047 y Fo(\))878 2011 y Fk(p)924 2047 y Fl(\015)982 2011 y Fp(\000)p Fn(1)1088 2047 y Fe(,)39 b(for)g(some)g(p)-5 b(ositive)37 b(c)-5 b(onstant)41 b Fo(~)-52 b Fl(c)p Fe(,)39 b(if)f Fl(\015)h Fo(=)34 b(2)2807 2011 y Fp(\000)p Fk(n)2924 2047 y Fe(,)39 b Fl(I)-18 b(P)14 b Fe(-almost)39 b(sur)-5 b(ely,)41 b(for)0 2186 y(al)5 b(l)37 b(but)f(a)f(\014nite)g (numb)-5 b(er)36 b(of)g(indic)-5 b(es)34 b Fl(n)p Fo(,)167 2540 y Fl(\026)227 2555 y Fk(\014)t(;\022)r(;\015)431 2369 y Fd( )509 2540 y Fj(T)589 2498 y Fk(\016)625 2508 y Fg(4)664 2498 y Fk(;\020)724 2508 y Fg(4)768 2540 y Fo(\()p Fl(`)851 2555 y Fn(^)-42 b Fk(c)889 2540 y Fo(\()p Fl(\015)6 b Fo(\))p Fl(;)31 b Fj(\024)1204 2472 y Fo(~)-53 b Fl(c)p 1201 2517 44 4 v 1203 2608 a Fo(^)h Fl(c)1256 2540 y Fo(\(log)1456 2472 y(1)p 1452 2517 58 4 v 1452 2608 a Fl(\015)1521 2540 y Fo(\))1560 2498 y Fk(p)1623 2540 y Fo(log)17 b(log)1929 2472 y(1)p 1925 2517 V 1925 2608 a Fl(\015)1994 2540 y(;)g Fj(J)h Fo(\))k Fj(n)f(T)2337 2498 y Fk(\016)2373 2508 y Fg(4)2412 2498 y Fk(;\020)2472 2508 y Fg(4)2516 2540 y Fo(\()p Fl(L)2623 2555 y Fn(1)2667 2540 y Fl(;)33 b Fj(\024)2920 2453 y Fo(~)-53 b Fl(c)q Fo(\(log)3160 2414 y Fn(1)p 3157 2430 47 4 v 3157 2487 a Fk(\015)3215 2453 y Fo(\))3254 2417 y Fk(p)p Fp(\000)p Fn(1)p 2845 2517 629 4 v 2847 2613 a Fo(^)h Fl(c)p Fo(\(log)17 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Fo(\))519 4090 y Fk(p)p Fp(\000)p Fn(1)666 4127 y Fo(\(log)17 b(log)g(1)p Fl(=\015)6 b Fo(\))1192 4090 y Fp(\000)p Fn(\(2+)p Fk(\032)p Fn(\))1462 4127 y Fo(.)0 5579 y Fg(23)p Ff(=j)s(anuar)r(y)r(=)p Fg(1998;)32 b(12:06)1145 b Fo(13)p eop %%Page: 14 14 14 13 bop 0 100 a Fq(3)45 b(The)g(analysis)h(of)f(the)g(blo)t(c)l(k)g (spin)g(represen)l(tation)120 363 y Fo(In)36 b(this)g(section)g(w)m(e)h (p)s(erform)e(the)h(blo)s(c)m(k)g(spin)h(transformation)e(on)h(the)g (scale)g Fl(\016)3248 327 y Fp(\003)3330 363 y Fo(men)m(tioned)g(in)0 503 y(the)e(previous)h(c)m(hapter)g(and)f(w)m(e)g(mak)m(e)f(a)h(rather) g(precise)h(analysis)f(of)g(the)g(sto)s(c)m(hastic)h(con)m(tribution)0 642 y(in)e(order)h(to)f(pro)m(v)m(e)g(our)h(theorems.)120 787 y(Giv)m(en)46 b(a)g(macroscopic)g(in)m(terv)-6 b(al)46 b Fl(I)57 b Fj(\021)49 b Fo([)p Fl(i)1752 751 y Fp(\000)1820 787 y Fl(;)17 b(i)1899 751 y Fn(+)1964 787 y Fo([)p Fj(\032)50 b Fl(I)-17 b(R)46 b Fo(with)g Fl(i)2542 751 y Fp(\006)2659 787 y Fj(2)k Fl(Z)-35 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Fj(\024)28 b Fl(x)g(<)0 1253 y(i)34 1217 y Fn(+)124 1253 y Fo(+)d(1)p Fj(g)36 b Fo(and)g Fl(@)617 1217 y Fp(\000)684 1253 y Fl(I)k Fj(\021)33 b(f)p Fl(x)f Fj(2)h Fl(I)-17 b(R;)17 b(i)1298 1217 y Fp(\000)1389 1253 y Fj(\000)24 b Fo(1)32 b Fj(\024)h Fl(x)f(<)h(i)1915 1217 y Fp(\000)1983 1253 y Fj(g)j Fo(the)g(t)m(w)m(o)g(macroscopic)g(in)m(terv)-6 b(als)36 b(of)g(length)h(1,)0 1392 y(that)c(are)g(on)h(the)f(righ)m(t)g (and)h(on)f(the)h(left)f(of)h Fl(I)8 b Fo(.)43 b(W)-8 b(e)34 b(call)e Fl(@)6 b(I)35 b Fo(=)28 b Fl(@)2504 1356 y Fn(+)2570 1392 y Fl(I)i Fj([)22 b Fl(@)2791 1356 y Fp(\000)2858 1392 y Fl(I)8 b Fo(.)120 1538 y(If)46 b Fl(F)310 1502 y Fk(\016)349 1472 y Fb(\003)441 1538 y Fo(is)f(a)h(\006)721 1502 y Fk(\016)760 1472 y Fb(\003)721 1565 y Fk(I)806 1538 y Fo(-measurable)g(b)s(ounded)h(function,)j(w)m(e) c(de\014ne)h(the)f(conditional)f(exp)s(ectation)0 1677 y(of)c Fl(F)199 1641 y Fk(\016)238 1611 y Fb(\003)284 1677 y Fo(,)i(giv)m(en)e(the)g Fl(\033)t Fo(-algebra)f(\006)1324 1641 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4102 y Fb(\003)1874 4088 y Fn(\()p Fk(I)5 b Fn(\))1993 3960 y Fl(J)2048 3975 y Fk(\016)2087 3955 y Fb(\003)2134 3960 y Fo(\()p Fl(x)22 b Fj(\000)g Fl(y)t Fo(\))c(~)-68 b Fl(m)2530 3919 y Fk(\016)2569 3889 y Fb(\003)2615 3960 y Fo(\()p Fl(x)p Fo(\))19 b(~)-69 b Fl(m)2837 3919 y Fk(\016)2876 3889 y Fb(\003)2923 3960 y Fo(\()p Fl(y)t Fo(\))641 b(\(3)p Fl(:)p Fo(3\))0 4371 y(and)559 4525 y Fl(E)6 b Fo(\()p Fl(m)765 4484 y Fk(\016)804 4454 y Fb(\003)765 4550 y Fk(I)850 4525 y Fl(;)17 b(m)982 4484 y Fk(\016)1021 4454 y Fb(\003)982 4553 y Fk(@)1029 4533 y Fb(\006)1088 4553 y Fk(I)1133 4525 y Fo(\))28 b Fj(\021)g(\000)p Fl(\016)1430 4484 y Fp(\003)1584 4430 y Fd(X)1493 4649 y Fk(x)p Fp(2C)1635 4663 y Ff(\016)1670 4648 y Fb(\003)1716 4649 y Fn(\()p Fk(I)5 b Fn(\))1978 4430 y Fd(X)1835 4649 y Fk(y)s Fp(2C)1975 4663 y Ff(\016)2010 4648 y Fb(\003)2056 4649 y Fn(\()p Fk(@)2134 4630 y Fb(\006)2193 4649 y Fk(I)g Fn(\))2281 4525 y Fl(J)2336 4540 y Fk(\016)2375 4520 y Fb(\003)2422 4525 y Fo(\()p Fl(x)22 b Fj(\000)h Fl(y)t Fo(\))18 b(~)-68 b Fl(m)2819 4484 y Fk(\016)2858 4454 y Fb(\003)2904 4525 y Fo(\()p Fl(x)p Fo(\))18 b(~)-68 b Fl(m)3126 4484 y Fk(\016)3165 4454 y Fb(\003)3211 4525 y Fo(\()p Fl(y)t Fo(\))353 b(\(3)p Fl(:)p Fo(4\))0 4891 y(On)34 b(the)g(set)f Fl(M)601 4854 y Fk(\016)640 4824 y Fb(\003)686 4891 y Fo(\()p Fl(m)812 4854 y Fk(\016)851 4824 y Fb(\003)898 4891 y Fo(\()p Fl(I)8 b Fo(\)\))26 b Fj(\021)i(f)p Fl(\033)j Fo(:)d Fl(m)1479 4854 y Fk(\016)1518 4824 y Fb(\003)1564 4891 y Fo(\()p Fl(x;)17 b(\033)t Fo(\))26 b(=)j Fl(m)2024 4854 y Fk(\016)2063 4824 y Fb(\003)2109 4891 y Fo(\()p Fl(x)p Fo(\))17 b Fj(8)2316 4909 y Fk(x)p Fp(2C)2458 4923 y Ff(\016)2493 4908 y Fb(\003)2539 4909 y Fn(\()p Fk(I)5 b Fn(\))2647 4891 y Fj(g)p Fo(,)32 b(w)m(e)i(ha)m(v)m (e)868 5232 y(sup)791 5326 y Fk(\033)837 5336 y Ff(I)877 5326 y Fp(2)p Fk(M)1014 5306 y Ff(\016)1049 5291 y Fb(\003)1095 5117 y Fd(\014)1095 5177 y(\014)1095 5237 y(\014)1128 5232 y Fl(H)8 b Fo(\()p Fl(\033)1315 5247 y Fk(I)1360 5232 y Fo(\))22 b(+)h Fl(\022)1646 5137 y Fd(X)1587 5356 y Fk(i)p Fp(2)p Fk(\015)1715 5336 y Fb(\000)p Fg(1)1808 5356 y Fk(I)1865 5232 y Fl(h)1922 5247 y Fk(i)1956 5232 y Fl(\033)2013 5247 y Fk(i)2068 5232 y Fj(\000)2183 5164 y Fo(1)p 2180 5209 58 4 v 2180 5300 a Fl(\015)2249 5232 y(E)6 b Fo(\()p Fl(m)2455 5191 y Fk(\016)2494 5161 y Fb(\003)2455 5256 y Fk(I)2540 5232 y Fo(\))2579 5117 y Fd(\014)2579 5177 y(\014)2579 5237 y(\014)2639 5232 y Fj(\024)28 b Fl(\016)2792 5191 y Fp(\003)2838 5232 y Fl(\015)2896 5191 y Fp(\000)p Fn(1)3002 5232 y Fj(j)p Fl(I)8 b Fj(j)585 b Fo(\(3)p Fl(:)p Fo(5\))0 5579 y Fg(23)p Ff(=j)s(anuar)r(y)r(=)p Fg(1998;)32 b(12:06)1145 b Fo(14)p eop %%Page: 15 15 15 14 bop 0 100 a Fo(here)34 b Fj(j)p Fl(I)8 b Fj(j)32 b Fo(is)h(the)h(length)f(of)h(the)g(macroscopic)e(in)m(terv)-6 b(al)33 b Fl(I)8 b Fo(.)44 b(Moreo)m(v)m(er)34 b(w)m(e)f(ha)m(v)m(e)h (also)859 425 y(sup)782 519 y Fk(\033)828 529 y Ff(I)868 519 y Fp(2)p Fk(M)1005 499 y Ff(\016)1040 484 y Fb(\003)1102 280 y Fd(\014)1102 340 y(\014)1102 400 y(\014)1102 460 y(\014)1135 425 y Fl(W)14 b Fo(\()p Fl(\033)1339 440 y Fk(I)1385 425 y Fj(j)p Fl(m)1500 384 y Fk(\016)1539 354 y Fb(\003)1585 425 y Fo(\()p Fl(@)1683 384 y Fp(\006)1750 425 y Fl(I)8 b Fo(\)\))21 b Fj(\000)2016 358 y Fo(1)p 2013 402 58 4 v 2013 493 a Fl(\015)2082 425 y(E)6 b Fo(\()p Fl(m)2288 384 y Fk(\016)2327 354 y Fb(\003)2288 450 y Fk(I)2373 425 y Fl(;)17 b(m)2505 384 y Fk(\016)2544 354 y Fb(\003)2505 453 y Fk(@)2552 433 y Fb(\006)2610 453 y Fk(I)2656 425 y Fo(\))2695 280 y Fd(\014)2695 340 y(\014)2695 400 y(\014)2695 460 y(\014)2755 425 y Fj(\024)29 b Fl(\016)2909 384 y Fp(\003)2954 425 y Fl(\015)3012 384 y Fp(\000)p Fn(1)3695 425 y Fo(\(3)p Fl(:)p Fo(6\))0 769 y(Therefore,)34 b(using)g(\(2.13\),)d(w)m(e)j(can)g(write)156 1117 y Fl(\026)216 1132 y Fk(\014)t(;\022)r(;\015)420 1006 y Fd(\020)480 1117 y Fl(F)558 1076 y Fk(\016)597 1046 y Fb(\003)671 1032 y Fd(\014)671 1092 y(\014)732 1117 y Fo(\006)804 1132 y Fk(@)9 b(I)901 1006 y Fd(\021)977 1117 y Fo(\()p Fl(m)1103 1076 y Fk(\016)1142 1046 y Fb(\003)1189 1117 y Fo(\()p Fl(@)i(I)d Fo(\)\))26 b(=)1704 1050 y Fl(e)1750 1013 y Fn(\()p Fp(\006)p Fk(\016)1882 983 y Fb(\003)1925 1013 y Fk(\015)1971 983 y Fb(\000)p Fg(1)2064 1013 y Fp(j)p Fk(I)5 b Fp(j)p Fn(\))p 1565 1094 763 4 v 1565 1185 a Fl(Z)1633 1200 y Fk(\014)t(;\022)r(;\015)t(;I)1885 1185 y Fo(\()p Fl(m)2011 1157 y Fk(\016)2050 1137 y Fb(\003)2096 1185 y Fo(\()p Fl(@)11 b(I)d Fo(\)\))394 1379 y Fd(X)173 1603 y Fk(m)244 1583 y Ff(\016)279 1568 y Fb(\003)325 1603 y Fn(\()p Fk(I)d Fn(\))p Fp(2M)575 1617 y Ff(\016)610 1602 y Fb(\003)656 1603 y Fn(\()p Fk(I)g Fn(\))775 1474 y Fl(F)853 1432 y Fk(\016)892 1402 y Fb(\003)939 1474 y Fo(\()p Fl(m)1065 1432 y Fk(\016)1104 1402 y Fb(\003)1150 1474 y Fo(\))p Fl(e)1235 1391 y Fp(\000)1310 1358 y Ff(\014)p 1310 1375 42 4 v 1311 1415 a(\015)1363 1288 y Fd(\020)1423 1391 y Fk(E)t Fn(\()p Fk(m)1587 1361 y Ff(\016)1622 1340 y Fb(\003)1587 1411 y Ff(I)1668 1391 y Fn(\)+)p Fk(E)t Fn(\()p Fk(m)1924 1361 y Ff(\016)1959 1340 y Fb(\003)1924 1411 y Ff(I)2005 1391 y Fk(;m)2100 1361 y Ff(\016)2135 1340 y Fb(\003)2100 1411 y Ff(@)j(I)2185 1391 y Fn(\)+)2290 1364 y Ff(\022)r(\016)2361 1344 y Fb(\003)p 2290 1375 112 4 v 2328 1415 a Fg(2)2426 1323 y Fd(P)2532 1428 y Ff(x)p Fb(2C)2657 1449 y Ff(\016)2692 1434 y Fb(\003)2738 1428 y Fg(\()p Ff(I)t Fg(\))2832 1391 y Fn(\()p Fk(m)2934 1361 y Ff(\016)2969 1340 y Fb(\003)2934 1411 y Fg(1)3014 1391 y Fn(\()p Fk(x)p Fn(\))p Fp(\000)p Fk(m)3254 1361 y Ff(\016)3289 1340 y Fb(\003)3254 1411 y Fg(2)3336 1391 y Fn(\()p Fk(x)p Fn(\)\))3474 1288 y Fd(\021)410 1733 y(X)372 1942 y Fk(\033)418 1966 y Ff(\015)458 1951 y Fb(\000)p Fg(1)552 1966 y Ff(I)609 1828 y Fo(1)-25 b(I)669 1851 y Fp(f)p Fk(m)781 1831 y Ff(\016)816 1816 y Fb(\003)862 1851 y Fn(\()p Fk(x;\033)s Fn(\)=)p Fk(m)1174 1831 y Ff(\016)1209 1816 y Fb(\003)1254 1851 y Fn(\()p Fk(x)p Fn(\))p Fk(;)p Fp(8)p Fk(x)p Fp(2C)1572 1865 y Ff(\016)1607 1850 y Fb(\003)1653 1851 y Fn(\()p Fk(I)5 b Fn(\))p Fp(g)1918 1733 y Fd(Y)1819 1951 y Fk(x)p Fp(2C)1961 1965 y Ff(\016)1996 1950 y Fb(\003)2042 1951 y Fn(\()p Fk(I)g Fn(\))2161 1828 y Fl(e)2207 1767 y Fn(2)p Fk(\014)t(\022)r(\025)p Fn(\()p Fk(x)p Fn(\))2505 1700 y Fd(P)2610 1805 y Ff(i)p Fb(2)p Ff(D)2741 1790 y(\025)2787 1805 y Fg(\()p Ff(x)p Fg(\))2899 1767 y Fk(\033)2945 1777 y Ff(i)3695 1493 y Fo(\(3)p Fl(:)p Fo(7\))0 2211 y(where,)31 b(this)g(equalit)m(y)e(has) i(to)e(b)s(e)i(in)m(terpreted)h(as)e(an)g(upp)s(er)i(b)s(ound)f(for)f Fj(\006)f Fo(=)f(1)i(and)h(a)f(lo)m(w)m(er)g(b)s(ound)0 2350 y(for)j Fj(\006)c Fo(=)f Fj(\000)p Fo(1)33 b(and)h(the)g(\014rst)f (sum)g(is)g(o)m(v)m(er)h Fl(m)1706 2314 y Fk(\016)1745 2284 y Fb(\003)1791 2350 y Fo(\()p Fl(x)p Fo(\))1926 2368 y Fk(x)p Fp(2C)2068 2382 y Ff(\016)2103 2367 y Fb(\003)2149 2368 y Fn(\()p Fk(I)5 b Fn(\))2285 2350 y Fj(2)28 b(M)2499 2365 y Fk(\016)2538 2345 y Fb(\003)2584 2350 y Fo(\()p Fl(I)8 b Fo(\).)120 2491 y(Note)27 b(that)g(the)h(random)f(terms)g(app) s(ear)h(only)f(in)g(the)h(last)f(pro)s(duct)2728 2416 y Fd(Q)2822 2521 y Fk(x)p Fp(2C)2964 2535 y Ff(\016)2999 2520 y Fb(\003)3045 2521 y Fn(\()p Fk(I)5 b Fn(\))3180 2491 y Fo(and)28 b(that)g(the)f(last)0 2630 y(sum)37 b(in)g(\(3.7\))f(factors)h(in)m(to)g(a)g(pro)s(duct)g(o)m(v)m(er)g(the) h(in)m(terv)-6 b(als)37 b(of)g(length)g Fl(\016)2844 2594 y Fp(\003)2890 2630 y Fl(\015)2948 2594 y Fp(\000)p Fn(1)3091 2630 y Fo(indexed)h(b)m(y)f Fj(C)3657 2645 y Fk(\016)3696 2625 y Fb(\003)3743 2630 y Fo(\()p Fl(I)8 b Fo(\).)0 2770 y(F)-8 b(or)45 b(all)f Fl(x)j Fj(2)h(C)611 2785 y Fk(\016)650 2765 y Fb(\003)697 2770 y Fo(\()p Fl(I)8 b Fo(\),)47 b(w)m(e)e(in)m(tro)s(duce)h(on)f Fj(f\000)p Fo(1)p Fl(;)17 b Fo(+1)p Fj(g)2064 2734 y Fk(\016)2103 2704 y Fb(\003)2144 2734 y Fk(\015)2190 2704 y Fb(\000)p Fg(1)2335 2770 y Fo(=)48 b Fj(S)2520 2789 y Fk(\016)2559 2769 y Fb(\003)2601 2789 y Fk(\015)2647 2769 y Fb(\000)p Fg(1)2790 2770 y Fo(the)d(measure)g(denoted)i(the)0 2909 y Fe(c)-5 b(anonic)g(al)32 b Fo(measure)i(in)f(ph)m(ysics)h(literature) 834 3262 y Fl(I)-17 b(E)940 3218 y Fk(\016)979 3188 y Fb(\003)940 3292 y Fk(x;m)1080 3272 y Ff(\016)1115 3257 y Fb(\003)1160 3292 y Fn(\()p Fk(x)p Fn(\))1272 3262 y Fo(\()p Fl(')p Fo(\))28 b(=)1560 3085 y Fd(P)1665 3189 y Fk(\033)s Fp(2S)1817 3213 y Ff(\016)1852 3198 y Fb(\003)1892 3213 y Ff(\015)1932 3198 y Fb(\000)p Fg(1)2052 3159 y Fl(')p Fo(\()p Fl(\033)t Fo(\)1)-25 b(I)2316 3183 y Fp(f)p Fk(m)2428 3163 y Ff(\016)2463 3148 y Fb(\003)2509 3183 y Fn(\()p Fk(x;\033)s Fn(\)=)p Fk(m)2821 3163 y Ff(\016)2856 3148 y Fb(\003)2901 3183 y Fn(\()p Fk(x)p Fn(\))p Fp(g)p 1560 3239 1495 4 v 1661 3255 a Fd(P)1767 3360 y Fk(\033)s Fp(2S)1919 3384 y Ff(\016)1954 3369 y Fb(\003)1994 3384 y Ff(\015)2034 3369 y Fb(\000)p Fg(1)2154 3330 y Fo(1)g(I)2215 3353 y Fp(f)p Fk(m)2327 3333 y Ff(\016)2362 3318 y Fb(\003)2407 3353 y Fn(\()p Fk(x;\033)s Fn(\)=)p Fk(m)2719 3333 y Ff(\016)2754 3318 y Fb(\003)2800 3353 y Fn(\()p Fk(x)p Fn(\))p Fp(g)3695 3262 y Fo(\(3)p Fl(:)p Fo(8\))0 3624 y(The)34 b(denominator)e(in)h(\(3.8\))f(is)1217 3828 y Fd(\022)1482 3884 y Fj(j)p Fl(B)1591 3848 y Fn(+)1657 3884 y Fj(j)1302 3999 y Fn(1+)p Fk(m)1474 3975 y Ff(\016)1509 3960 y Fb(\003)1474 4023 y Fg(1)1555 3999 y Fn(\()p Fk(x)p Fn(\))p 1302 4031 360 4 v 1462 4088 a(2)1674 4054 y Fj(j)p Fl(B)1783 4025 y Fn(+)1848 4054 y Fj(j)1876 3828 y Fd(\023\022)2215 3884 y Fj(j)p Fl(B)2324 3848 y Fp(\000)2390 3884 y Fj(j)2035 3999 y Fn(1+)p Fk(m)2207 3975 y Ff(\016)2242 3960 y Fb(\003)2207 4023 y Fg(2)2287 3999 y Fn(\()p Fk(x)p Fn(\))p 2035 4031 V 2195 4088 a(2)2407 4054 y Fj(j)p Fl(B)2516 4025 y Fp(\000)2582 4054 y Fj(j)2610 3828 y Fd(\023)3695 3968 y Fo(\(3)p Fl(:)p Fo(9\))0 4325 y(where)i Fj(j)p Fl(B)397 4289 y Fp(\006)464 4325 y Fj(j)27 b Fo(=)h Fj(j)p Fl(B)5 b Fj(j)27 b Fo(=)h Fl(\016)941 4289 y Fp(\003)987 4325 y Fo(\(2)p Fl(\015)6 b Fo(\))1173 4289 y Fp(\000)p Fn(1)1278 4325 y Fo(.)44 b(W)-8 b(e)34 b(set)323 4578 y Fd(b)299 4603 y Fj(F)9 b Fo(\()p Fl(m)506 4562 y Fk(\016)545 4532 y Fb(\003)506 4628 y Fk(I)592 4603 y Fl(;)17 b(m)724 4562 y Fk(\016)763 4532 y Fb(\003)724 4628 y Fk(@)9 b(I)821 4603 y Fo(\))27 b(=)q Fl(E)6 b Fo(\()p Fl(m)1171 4562 y Fk(\016)1210 4532 y Fb(\003)1171 4628 y Fk(I)1256 4603 y Fo(\))22 b(+)g Fl(E)6 b Fo(\()p Fl(m)1622 4562 y Fk(\016)1661 4532 y Fb(\003)1622 4628 y Fk(I)1707 4603 y Fl(;)17 b(m)1839 4562 y Fk(\016)1878 4532 y Fb(\003)1839 4628 y Fk(@)9 b(I)1936 4603 y Fo(\))987 4844 y(+)1099 4777 y Fl(\022)s(\016)1197 4741 y Fp(\003)p 1099 4821 144 4 v 1145 4913 a Fo(2)1361 4750 y Fd(X)1270 4968 y Fk(x)p Fp(2C)1412 4982 y Ff(\016)1447 4967 y Fb(\003)1493 4968 y Fn(\()p Fk(I)c Fn(\))1596 4844 y Fo(\()p Fl(m)1722 4803 y Fk(\016)1761 4773 y Fb(\003)1722 4869 y Fn(1)1808 4844 y Fo(\()p Fl(x)p Fo(\))22 b Fj(\000)g Fl(m)2151 4803 y Fk(\016)2190 4773 y Fb(\003)2151 4869 y Fn(2)2237 4844 y Fo(\()p Fl(x)p Fo(\)\))987 5229 y(+)h Fl(\016)1135 5188 y Fp(\003)1288 5134 y Fd(X)1197 5352 y Fk(x)p Fp(2C)1339 5366 y Ff(\016)1374 5351 y Fb(\003)1420 5352 y Fn(\()p Fk(I)5 b Fn(\))1601 5161 y Fl(\015)p 1552 5206 156 4 v 1552 5297 a(\014)g(\016)1661 5268 y Fp(\003)1736 5229 y Fo(log)1881 5088 y Fd(\022)2146 5144 y Fj(j)p Fl(B)2255 5108 y Fn(+)2320 5144 y Fj(j)1966 5259 y Fn(1+)p Fk(m)2138 5235 y Ff(\016)2173 5220 y Fb(\003)2138 5283 y Fg(1)2218 5259 y Fn(\()p Fk(x)p Fn(\))p 1966 5291 360 4 v 2126 5348 a(2)2338 5314 y Fj(j)p Fl(B)2447 5285 y Fn(+)2512 5314 y Fj(j)2540 5088 y Fd(\023\022)2878 5144 y Fj(j)p Fl(B)2987 5108 y Fp(\000)3054 5144 y Fj(j)2699 5259 y Fn(1+)p Fk(m)2871 5235 y Ff(\016)2906 5220 y Fb(\003)2871 5283 y Fg(2)2951 5259 y Fn(\()p Fk(x)p Fn(\))p 2699 5291 V 2859 5348 a(2)3070 5314 y Fj(j)p Fl(B)3179 5285 y Fp(\000)3246 5314 y Fj(j)3274 5088 y Fd(\023)3645 4968 y Fo(\(3)p Fl(:)p Fo(10\))0 5579 y Fg(23)p Ff(=j)s(anuar)r(y)r(=)p Fg(1998;)32 b(12:06)1145 b Fo(15)p eop %%Page: 16 16 16 15 bop 0 100 a Fo(W)-8 b(e)34 b(in)m(tro)s(duce)g(the)f(momen)m(t)f (generating)h(function)666 384 y Fl(L)734 343 y Fk(\016)773 313 y Fb(\003)734 416 y Fk(x;m)874 396 y Ff(\016)909 381 y Fb(\003)954 416 y Fn(\()p Fk(x)p Fn(\))1067 384 y Fo(\()p Fl(\025)p Fo(\()p Fl(x)p Fo(\))p Fl(\014)5 b(\022)s(;)17 b(D)1540 343 y Fk(\025)1591 384 y Fo(\()p Fl(x)p Fo(\)\))27 b Fj(\021)h Fl(I)-17 b(E)2003 340 y Fk(\016)2042 310 y Fb(\003)2003 414 y Fk(x;m)2143 394 y Ff(\016)2178 379 y Fb(\003)2223 414 y Fn(\()p Fk(x)p Fn(\))2336 384 y Fo(\()p Fl(e)2421 323 y Fn(2)p Fk(\014)t(\022)r(\025)p Fn(\()p Fk(x)p Fn(\))2718 256 y Fd(P)2823 361 y Ff(i)p Fb(2)p Ff(D)2954 346 y(\025)3000 361 y Fg(\()p Ff(x)p Fg(\))3112 323 y Fk(\033)3158 333 y Ff(i)3195 384 y Fo(\))411 b(\(3)p Fl(:)p Fo(11\))0 620 y(and)34 b(the)f(cum)m(ulan)m(t)h (generating)f(function)923 857 y Fj(G)982 880 y Fk(x;m)1122 860 y Ff(\016)1157 845 y Fb(\003)1202 880 y Fn(\()p Fk(x)p Fn(\))1331 857 y Fo(\()p Fl(\025)p Fo(\()p Fl(x)p Fo(\)\))27 b Fj(\021)h Fo(log)17 b Fl(L)1947 816 y Fk(\016)1986 786 y Fb(\003)1947 890 y Fk(x;m)2087 870 y Ff(\016)2122 855 y Fb(\003)2168 890 y Fn(\()p Fk(x)p Fn(\))2280 857 y Fo(\()p Fl(\025)p Fo(\()p Fl(x)p Fo(\))p Fl(\014)5 b(\022)s(;)17 b(D)2753 816 y Fk(\025)2804 857 y Fo(\()p Fl(x)p Fo(\)\))667 b(\(3)p Fl(:)p Fo(12\))120 1095 y(then)33 b(\(3.7\))f(b)s(ecomes)260 1371 y Fl(\026)320 1386 y Fk(\014)t(;\022)r(;\015)524 1260 y Fd(\020)583 1371 y Fl(F)661 1330 y Fk(\016)700 1300 y Fb(\003)774 1286 y Fd(\014)774 1346 y(\014)835 1371 y Fo(\006)907 1386 y Fk(@)9 b(I)1004 1260 y Fd(\021)1081 1371 y Fo(\()p Fl(m)1207 1330 y Fk(\016)1246 1300 y Fb(\003)1292 1371 y Fo(\()p Fl(@)i(I)d Fo(\)\))287 1672 y(=)544 1605 y Fl(e)590 1569 y Fn(\()p Fp(\006)p Fk(\016)722 1539 y Fb(\003)764 1569 y Fk(\015)810 1539 y Fb(\000)p Fg(1)903 1569 y Fp(j)p Fk(I)d Fp(j)p Fn(\))p 404 1649 763 4 v 404 1741 a Fl(Z)472 1756 y Fk(\014)t(;\022)r(;\015)t(;I)724 1741 y Fo(\()p Fl(m)850 1712 y Fk(\016)889 1692 y Fb(\003)936 1741 y Fo(\()p Fl(@)11 b(I)d Fo(\)\))1416 1578 y Fd(X)1196 1801 y Fk(m)1267 1782 y Ff(\016)1302 1767 y Fb(\003)1347 1801 y Fn(\()p Fk(I)d Fn(\))p Fp(2M)1597 1815 y Ff(\016)1632 1800 y Fb(\003)1678 1801 y Fn(\()p Fk(I)g Fn(\))1798 1672 y Fl(F)1876 1631 y Fk(\016)1915 1601 y Fb(\003)1961 1672 y Fo(\()p Fl(m)2087 1631 y Fk(\016)2126 1601 y Fb(\003)2173 1672 y Fo(\))p Fl(e)2258 1619 y Fp(\000)2335 1592 y Fg(1)p 2332 1603 41 4 v 2332 1643 a Ff(\015)2385 1546 y Fd(\010)2443 1619 y Fk(\014)2505 1614 y Fd(b)2492 1619 y Fp(F)i Fn(\()p Fk(m)2658 1589 y Ff(\016)2693 1569 y Fb(\003)2658 1639 y Ff(I)2738 1619 y Fk(;m)2833 1589 y Ff(\016)2868 1569 y Fb(\003)2833 1640 y Ff(@)h(I)2918 1619 y Fn(\)+)p Fk(\015)t Fp(G)t Fn(\()p Fk(m)3210 1589 y Ff(\016)3245 1569 y Fb(\003)3210 1639 y Ff(I)3291 1619 y Fn(\))3322 1546 y Fd(\011)3645 1568 y Fo(\(3)p Fl(:)p Fo(13\))0 2035 y(where)1224 2175 y Fj(G)e Fo(\()p Fl(m)1415 2133 y Fk(\016)1454 2103 y Fb(\003)1415 2199 y Fk(I)1500 2175 y Fo(\))28 b Fj(\021)1763 2080 y Fd(X)1672 2298 y Fk(x)p Fp(2C)1814 2312 y Ff(\016)1849 2297 y Fb(\003)1895 2298 y Fn(\()p Fk(I)5 b Fn(\))2014 2175 y Fj(G)2073 2198 y Fk(x;m)2213 2178 y Ff(\016)2248 2163 y Fb(\003)2294 2198 y Fn(\()p Fk(x)p Fn(\))2406 2175 y Fo(\()p Fl(\025)p Fo(\()p Fl(x)p Fo(\)\))968 b(\(3)p Fl(:)p Fo(14\))120 2526 y Fe(i.e)44 b Fo(up)h(to)g(the)g(error)h(terms) e Fl(e)1336 2490 y Fn(\()p Fp(\006)p Fk(c\016)1504 2459 y Fb(\003)1546 2490 y Fk(\015)1592 2459 y Fb(\000)p Fg(1)1685 2490 y Fp(j)p Fk(I)5 b Fp(j)p Fn(\))1809 2526 y Fo(,)48 b(w)m(e)d(ha)m(v)m(e)h(b)s(een)g(able)f(to)g(describ)s(e)h(our)g (system)e(in)0 2665 y(term)32 b(of)i(the)f(blo)s(c)m(k)g(spin)h(v)-6 b(ariables)33 b(giving)f(a)h(rather)h(explicit)e(form)h(to)g(the)g (deterministic)g(and)h(the)0 2805 y(sto)s(c)m(hastic)f(part.)120 2945 y(Note)j(that)h(the)g(sto)s(c)m(hastic)g(dep)s(endence)j(is)d(giv) m(en)f(only)h(b)m(y)g(the)g(\015uctuations)h(of)f(the)g(magnetic)0 3085 y(\014elds)d(on)g(eac)m(h)g(blo)s(c)m(k)f Fl(\025)p Fo(\()p Fl(x)p Fo(\))27 b(=)h(sgn)17 b(\()1406 3010 y Fd(P)1511 3114 y Fk(i)p Fp(2)p Fk(A)p Fn(\()p Fk(x)p Fn(\))1782 3085 y Fl(h)1839 3100 y Fk(i)1872 3085 y Fo(\))33 b(and)h(b)m(y)f Fj(j)p Fl(D)2389 3048 y Fk(\025)2442 3085 y Fo(\()p Fl(x)p Fo(\))p Fj(j)26 b Fo(=)2748 3036 y Fk(\025)p Fn(\()p Fk(x)p Fn(\))p 2748 3062 155 4 v 2806 3119 a(2)2932 3010 y Fd(P)3037 3114 y Fk(i)p Fp(2)p Fk(A)p Fn(\()p Fk(x)p Fn(\))3308 3085 y Fl(h)3365 3100 y Fk(i)3398 3085 y Fo(.)120 3225 y(As)33 b(w)m(e)h(will)e(see,)i(w)m(e) g(need)g(precise)h(estimates,)d(when)i Fl(\015)39 b Fo(small,)31 b(of)j Fj(G)2776 3248 y Fk(x;m)2916 3229 y Ff(\016)2951 3214 y Fb(\003)2996 3248 y Fn(\()p Fk(x)p Fn(\))3125 3225 y Fo(\()p Fl(\025)p Fo(\()p Fl(x)p Fo(\)\))o(,)g(see)g(\(3.12\);)0 3365 y(i.e.)42 b(of)31 b(the)f(cum)m(ulan)m(t)g(generating)g(function)h (of)f(an)g(Hyp)s(ergeometric.)41 b(T)-8 b(o)30 b(get)g(them)f(w)m(e)h (need)h(large)0 3504 y(deviation)e(estimates)f(in)i(the)f(strong)h (form)e(with)i(go)s(o)s(d)e(con)m(trol)i(of)f(the)h(p)s(olynomial)d (prefactors.)44 b(W)-8 b(e)0 3643 y(ha)m(v)m(e)35 b(to)e(consider)i (all)e(the)i(p)s(ossible)f(b)s(eha)m(viors)g(of)h(the)f(\015uctuations) h(of)f(an)g(Hyp)s(ergeometric.)45 b(It)34 b(is)0 3783 y(w)m(ell)c(kno)m(wn)g(that)g(there)g(are)g(three)h(p)s(ossible)f (regimes,)g(namely)f(a)h(gaussian)g(one,)h(a)e(binomial)g(and)h(a)0 3922 y(p)s(oissonian)i(one.)44 b(Classical)32 b(results)g(are)g (usually)g(giv)m(en)g(in)g(terms)f(of)i(con)m(v)m(ergence)g(in)f (distribution.)0 4062 y(Since)40 b(w)m(e)g(are)f(in)m(terested)i(in)e (the)h(cum)m(ulan)m(t)f(generating)g(function)i(and)e(in)h(con)m (trolling)e(the)i(error)0 4201 y(terms,)27 b(w)m(e)h(need)g(some)e (extra)h(w)m(ork.)41 b(Because)28 b(it)e(can)i(b)s(e)f(of)g(indep)s (enden)m(t)j(in)m(terest,)e(w)m(e)f(giv)m(e)g(a)g(short)0 4341 y(pro)s(of)36 b(of)f(the)h(estimates)e(w)m(e)i(need.)52 b(The)36 b(statemen)m(ts)f(of)h(them)e(are)i(giv)m(en)f(in)g(Prop)s (osition)g(3.4,)g(for)0 4480 y(the)f(gaussian)f(regime,)f(and)i(in)f 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Fl(\014)5 b(\022)s(;)17 b(D)1642 5280 y Fn(1)1686 5321 y Fo(\))28 b(=)g Fl(I)-17 b(E)1964 5278 y Fk(\016)2003 5248 y Fb(\003)1964 5356 y Fk(x;m)2104 5332 y Ff(\016)2139 5317 y Fb(\003)2104 5380 y Fg(2)2184 5356 y Fn(\()p Fk(x)p Fn(\))2297 5321 y Fo(\()p Fl(e)2382 5268 y Fn(2)p Fk(\014)t(\022)2524 5201 y Fd(P)2629 5306 y Ff(i)p Fb(2)p Ff(D)2760 5291 y Fg(1)2817 5268 y Fk(\033)2863 5278 y Ff(i)2900 5321 y Fo(\))706 b(\(3)p Fl(:)p Fo(15\))0 5579 y Fg(23)p Ff(=j)s(anuar)r(y)r (=)p Fg(1998;)32 b(12:06)1145 b Fo(16)p eop %%Page: 17 17 17 16 bop 0 100 a Fo(whic)m(h)34 b(dep)s(ends)h(only)e(on)g(the)h (second)g(co)s(ordinate)g(of)f Fl(m)2211 63 y Fk(\016)2250 33 y Fb(\003)2297 100 y Fo(\()p Fl(x)p Fo(\);)f(while)h(if)h Fl(\025)p Fo(\()p Fl(x)p Fo(\))27 b(=)h Fj(\000)p Fo(1,)872 334 y Fl(L)940 293 y Fk(\016)979 263 y Fb(\003)940 367 y Fk(x;m)1080 347 y Ff(\016)1115 332 y Fb(\003)1160 367 y Fn(\()p Fk(x)p Fn(\))1273 334 y Fo(\()p Fl(\014)5 b(\022)s(;)17 b(D)1553 293 y Fp(\000)p Fn(1)1659 334 y 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Fp(g)2939 1524 y Fd(i)p 1566 1715 1420 4 v 1858 1812 a Fl(I)-17 b(E)1964 1827 y Fk(\033)2010 1837 y Ff(B)2091 1731 y Fd(\002)2133 1812 y Fo(1)-25 b(I)2194 1829 y Fp(f)p Fk(m)2306 1839 y Ff(B)2364 1829 y Fn(\()p Fk(\033)s Fn(\)=)p Fk(m)p Fp(g)2653 1731 y Fd(\003)3645 1738 y Fo(\(3)p Fl(:)p Fo(18\))0 2014 y(Let)30 b Fl(I)-17 b(E)281 2029 y Fk(\027)360 2014 y Fo(b)s(e)29 b(the)h(grand)g(canonical)f(measure)h (with)f(c)m(hemical)g(p)s(oten)m(tial)f Fl(\027)6 b Fo(,)31 b(de\014ned)g(on)f Fj(f\000)p Fo(1)p Fl(;)17 b Fo(+1)p Fj(g)3785 1978 y Fp(j)p Fk(B)s Fp(j)1238 2352 y Fl(I)-17 b(E)1344 2367 y Fk(\027)1394 2352 y Fo(\()p Fl(')p Fo(\))27 b(=)1681 2249 y Fl(I)-17 b(E)1787 2264 y Fk(\033)1833 2274 y Ff(B)1914 2138 y Fd(h)1961 2249 y Fl(')p Fo(\()p Fl(\033)t Fo(\))p Fl(e)2211 2195 y Fk(\027)2269 2128 y Fd(P)2374 2233 y Ff(i)p Fb(2)p Ff(B)2520 2195 y Fk(\033)2566 2205 y Ff(i)2603 2138 y Fd(i)p 1681 2329 969 4 v 1783 2462 a Fl(I)g(E)1889 2477 y Fk(\033)1935 2487 y Ff(B)2016 2351 y Fd(h)2063 2462 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3146 1380 4 v 1289 3279 a Fl(I)-17 b(E)1395 3294 y Fk(\027)1435 3304 y Fg(2)1495 3168 y Fd(h)1542 3279 y Fl(e)1588 3225 y Fn(2)p Fk(b\022)r(\025)1764 3158 y Fd(P)1869 3263 y Ff(i)p Fb(2)p Ff(D)2019 3225 y Fk(\033)2065 3235 y Ff(i)2101 3168 y Fd(i)2812 3102 y Fo(1)p 2432 3146 810 4 v 2432 3243 a Fl(I)g(E)2538 3258 y Fk(\027)2578 3268 y Fg(1)2639 3162 y Fd(\002)2680 3243 y Fo(1)-25 b(I)2741 3261 y Fp(f)p Fk(m)2853 3271 y Ff(B)2912 3261 y Fn(\()p Fk(\033)s Fn(\)=)p Fk(m)p Fp(g)3200 3162 y Fd(\003)1017 3583 y Fl(e)1063 3542 y Fp(f)p Fk(m)p Fn(\()p Fk(\027)1246 3552 y Fg(1)1285 3542 y Fp(\000)p Fk(\027)1387 3552 y Fg(2)1426 3542 y Fn(\))p Fp(j)p Fk(B)s Fp(jg)1625 3515 y Fo(\(cosh)q(\()p Fl(\027)1941 3530 y Fn(2)1986 3515 y Fo(\)\))2063 3465 y Fp(j)p Fk(B)s Fp(n)p Fk(D)r Fp(j)2304 3515 y Fo(\()o(cosh)q(\()p Fl(\027)2619 3530 y Fn(2)2686 3515 y Fo(+)23 b(2)p Fl(\025\014)5 b(\022)s Fo(\)\))3082 3465 y Fp(j)p Fk(D)r Fp(j)p 1625 3560 1577 4 v 2137 3674 a Fo(\(cosh\()p Fl(\027)2452 3689 y Fn(1)2497 3674 y Fo(\)\))2574 3624 y Fp(j)p Fk(B)s Fp(j)3645 3347 y Fo(\(3)p Fl(:)p Fo(20\))0 3863 y(W)-8 b(e)34 b(c)m(ho)s(ose)g Fl(\027)537 3878 y Fn(1)609 3863 y Fj(\021)28 b Fl(\027)763 3878 y Fn(1)808 3863 y Fo(\()p Fl(m)p Fo(\))33 b(suc)m(h)i(that)e Fl(m)28 b Fo(=)g(tanh)17 b Fl(\027)1932 3878 y Fn(1)2010 3863 y Fo(and)34 b Fl(\027)2253 3878 y Fn(2)2326 3863 y Fj(\021)28 b Fl(\027)2480 3878 y Fn(2)2525 3863 y Fo(\()p Fl(m;)17 b(\013)p Fo(\))32 b(suc)m(h)j(that)1063 4143 y Fl(m)28 b Fo(=)g Fl(\013)17 b Fo(tanh\()p Fl(\027)1651 4158 y Fn(2)1718 4143 y Fo(+)23 b(2)p Fl(\025\014)5 b(\022)s Fo(\))22 b(+)g(\(1)g Fj(\000)h Fl(\013)p Fo(\))17 b(tanh)f Fl(\027)2792 4158 y Fn(2)3645 4143 y Fo(\(3)p Fl(:)p Fo(21\))0 4321 y(where)39 b Fl(\013)d Fo(=)519 4273 y Fp(j)p Fk(D)r Fp(j)p 519 4299 115 4 v 521 4356 a(j)p Fk(B)s Fp(j)645 4321 y Fo(,)j(0)d Fj(\024)h Fl(\013)f Fj(\024)g Fo(1.)59 b(Then)39 b(since)h Fj(f)p Fl(m)1914 4336 y Fk(B)1982 4321 y Fo(\()p Fl(\033)t Fo(\))27 b(=)h Fl(m)p Fj(g)37 b Fo(=)f Fj(f)p Fo(\()p Fj(j)p Fl(B)5 b Fj(j)p Fo(\))2805 4285 y Fp(\000)p Fn(1)p Fk(=)p Fn(2)3007 4247 y Fd(P)3113 4351 y Fk(i)p Fp(2)p Fk(B)3263 4321 y Fo(\()p Fl(\033)3359 4336 y Fk(i)3414 4321 y Fj(\000)23 b Fl(m)p Fo(\))28 b(=)g(0)p Fj(g)o Fo(,)0 4461 y(w)m(e)40 b(can)g(estimate)e(the)i(t)m(w)m(o)f(\014rst)h(ratios)e(in)i(\(3.20\))e (as)h(done)h(in)g(Y)-8 b(au)39 b([31],)g(using)h(a)f Fe(L)-5 b(o)g(c)g(al)40 b Fo(Cen)m(tral)0 4600 y(Limit)32 b(Theorem)h(\(LCL)-8 b(T\).)32 b(Therefore)j(denoting)681 4993 y(\011)758 5008 y Fk(\025\014)t(\022)r(;\013;m)1097 4993 y Fj(\021)1214 4859 y Fl(I)-17 b(E)1320 4874 y Fk(\027)1360 4884 y Fg(2)1420 4719 y Fd(\024)1473 4859 y Fl(e)1519 4806 y Fn(2)p Fk(\014)t(\022)r(\025)1709 4739 y Fd(P)1814 4843 y Ff(i)p Fb(2)p Ff(D)1963 4806 y Fk(\033)2009 4816 y Ff(i)2046 4859 y Fo(1)-25 b(I)2107 4892 y Fp(f)p Fn(\()p Fp(j)p Fk(B)s Fp(j)p Fn(\))2321 4872 y Fb(\000)p Fg(1)p Ff(=)p Fg(2)2498 4825 y Fd(P)2603 4929 y Ff(i)p Fb(2)p Ff(B)2735 4892 y Fn(\()p Fk(\033)2812 4902 y Ff(i)2844 4892 y Fp(\000)p Fk(m)p Fn(\)=0)p Fp(g)3155 4719 y Fd(\025)p 1214 4970 1994 4 v 1781 5102 a Fl(I)-17 b(E)1887 5117 y Fk(\027)1927 5127 y Fg(2)1987 4992 y Fd(h)2034 5102 y Fl(e)2080 5049 y Fn(2)p Fk(b\022)r(\025)2256 4982 y Fd(P)2361 5086 y Ff(i)p Fb(2)p Ff(D)2511 5049 y Fk(\033)2557 5059 y Ff(i)2593 4992 y Fd(i)3645 4993 y Fo(\(3)p Fl(:)p Fo(22\))p 0 5229 1200 4 v 25 5304 a Fn(1)120 5340 y Fi(W)-7 b(e)27 b(thank)i(Enzo)e(Olivieri)j(for)e(bringing)j(the)c(results)j(of) e([)p Fo(31)p Fi(])f(to)h(our)h(atten)n(tion.)0 5579 y Fg(23)p Ff(=j)s(anuar)r(y)r(=)p Fg(1998;)j(12:06)1145 b Fo(17)p eop %%Page: 18 18 18 17 bop 0 100 a Fo(and)551 309 y Fl(e)597 268 y Fk(\036)p Fn(\()p Fk(m;\025\014)t(\022)r(;\013)p Fn(\))1046 309 y Fj(\021)29 b Fl(e)1198 268 y Fp(f)p Fk(m)p Fn(\()p Fk(\027)1381 278 y Fg(1)1420 268 y Fp(\000)p Fk(\027)1522 278 y Fg(2)1561 268 y Fn(\))p Fp(j)p Fk(B)s Fp(jg)1760 242 y Fo(\(cosh)q(\()p Fl(\027)2076 257 y Fn(2)2121 242 y Fo(\)\))2198 192 y Fp(j)p Fk(B)s Fp(n)p Fk(D)r Fp(j)2439 242 y Fo(\()o(cosh)q(\()p Fl(\027)2754 257 y Fn(2)2821 242 y Fo(+)23 b(2)p Fl(\025\014)5 b(\022)s Fo(\)\))3217 192 y Fp(j)p Fk(D)r Fp(j)p 1760 286 1577 4 v 2272 401 a Fo(\(cosh\()p Fl(\027)2587 416 y Fn(1)2632 401 y Fo(\)\))2709 351 y Fp(j)p Fk(B)s Fp(j)3645 309 y Fo(\(3)p Fl(:)p Fo(23\))0 596 y(w)m(e)34 b(ha)m(v)m(e)1184 735 y Fl(L)1252 750 y Fk(m)1327 735 y Fo(\()p Fl(\025\014)5 b(\022)s(;)17 b(D)s Fo(\))27 b(=)1848 668 y(\011)1925 683 y Fk(\025\014)t(\022)r (;\013;m)p 1848 712 389 4 v 1903 803 a Fo(\011)1980 818 y Fn(0)p Fk(;)p Fn(0)p Fk(;m)2249 735 y Fl(e)2295 694 y Fk(\036)p Fn(\()p Fk(m;\025\014)t(\022)r(;\013)p Fn(\))3645 735 y Fo(\(3)p Fl(:)p Fo(24\))0 1002 y(The)36 b(next)f(step)h(is)f(to)f (estimate)g(eac)m(h)j(term)d(separately)-8 b(.)49 b(W)-8 b(e)36 b(start)e(with)h(\011)2969 1017 y Fk(\022)r(;\013;m)3185 1002 y Fo(.)49 b(W)-8 b(e)36 b(giv)m(e)f(\014rst)g(a)0 1141 y(lo)m(w)m(er)e(b)s(ound)i(for)e(the)h(v)-6 b(ariance)33 b(of)h Fl(m)1483 1156 y Fk(B)1552 1141 y Fo(\()p Fl(\033)t Fo(\))e(under)i Fl(I)-17 b(E)2111 1156 y Fk(\027)2151 1166 y Fg(2)2195 1141 y Fo(.)0 1282 y Fm(Lemma)38 b(3.1)h Fe(L)-5 b(et)36 b Fl(\027)812 1297 y Fn(2)892 1282 y Fe(b)-5 b(e)36 b(a)f(solution)h(of)f(\(3.21\),)f(and)i Fl(\033)2177 1297 y Fk(\025\014)t(\022)2353 1282 y Fe(given)f(by)936 1582 y Fl(\033)997 1540 y Fn(2)993 1606 y Fk(\025\014)t(\022)1162 1582 y Fo(=)28 b Fl(\013)1691 1514 y Fo(1)p 1343 1559 746 4 v 1343 1660 a(cosh)1532 1616 y Fn(2)1576 1660 y Fo(\()p Fl(\027)1664 1675 y Fn(2)1731 1660 y Fo(+)23 b(2)p Fl(\025\014)5 b(\022)s Fo(\))2123 1582 y(+)22 b(\(1)g Fj(\000)h Fl(\013)p Fo(\))2725 1514 y(1)p 2548 1559 405 4 v 2548 1660 a(cosh)2736 1616 y Fn(2)2781 1660 y Fo(\()p Fl(\027)2869 1675 y Fn(2)2913 1660 y Fo(\))3645 1582 y(\(3)p Fl(:)p Fo(25\))0 1899 y Fe(then)36 b(for)g(al)5 b(l)37 b Fl(m)f Fe(such)g(that)g Fj(j)p Fl(m)p Fj(j)29 b Fl(<)g Fo(1)p Fe(,)35 b(for)h(al)5 b(l)37 b Fl(\014)d(>)29 b Fo(1)p Fe(,)35 b(for)h(al)5 b(l)37 b Fl(\022)i Fe(such)d(that)g Fl(\014)5 b(\022)31 b(<)e(\017)p Fe(,)37 b(for)e(some)h Fl(\017)29 b(>)g Fo(0)0 2039 y Fe(smal)5 b(l)37 b(enough,)e(for)g(al)5 b(l)37 b Fl(\013)27 b Fj(2)i Fo([0)p Fl(;)17 b Fo(1])1331 2278 y Fl(\033)1392 2237 y Fn(2)1388 2303 y Fk(\025\014)t(\022)1557 2278 y Fl(>)28 b Fo(\(1)22 b Fj(\000)g Fl(m)1959 2237 y Fn(2)2005 2278 y Fo(\)\(1)f Fj(\000)i Fl(c)p Fo(\()p Fl(\014)5 b(\022)s Fo(\))2486 2237 y Fn(2)2530 2278 y Fo(\))1076 b(\(3)p Fl(:)p Fo(26\))0 2517 y Fe(for)35 b(some)h(p)-5 b(ositive)34 b(c)-5 b(onstant)35 b Fl(c)p Fe(.)0 2658 y Fa(Pro)s(of:)120 2798 y Fo(W)-8 b(e)24 b(ha)m(v)m(e)h Fl(\033)564 2762 y Fn(2)560 2826 y Fk(\025\014)t(\022) 729 2798 y Fo(=)j(1)5 b Fj(\000)g Fl(m)1058 2762 y Fn(2)1108 2798 y Fj(\000)g Fl(\013)p Fo(\(1)g Fj(\000)g Fl(\013)p Fo(\)\(tanh)o(\()p Fl(\027)1858 2813 y Fn(2)1907 2798 y Fo(+)g(2)p Fl(\025\014)g(\022)s Fo(\))g Fj(\000)g Fo(tanh\()p Fl(\027)2621 2813 y Fn(2)2666 2798 y Fo(\)\))2744 2762 y Fn(2)2788 2798 y Fo(.)41 b(No)m(w)24 b(calling)g Fl(\027)3431 2813 y Fn(2)3480 2798 y Fj(\000)5 b Fl(\027)3611 2813 y Fn(1)3684 2798 y Fj(\021)28 b Fo(\001,)0 2938 y(using)34 b Fl(m)28 b Fo(=)g(tanh)17 b Fl(\027)746 2953 y Fn(1)791 2938 y Fo(,)33 b(it)f(is)h(easy)h(to)e(see)i(that)259 3245 y(tanh\()p Fl(\027)546 3260 y Fn(2)613 3245 y Fo(+)23 b(2)p Fl(\025\014)5 b(\022)s Fo(\))22 b Fj(\000)g Fo(tanh)q(\()p Fl(\027)1380 3260 y Fn(2)1424 3245 y Fo(\))28 b(=)1671 3178 y(\(1)21 b Fj(\000)i Fl(m)1968 3142 y Fn(2)2013 3178 y Fo(\)\(tanh\(2)p Fl(\025\014)5 b(\022)25 b Fo(+)d(\001\))g Fj(\000)h Fo(tanh\(\001\)\))p 1608 3222 1767 4 v 1608 3313 a(\(1)e(+)i Fl(m)17 b Fo(tanh\(2)p Fl(\025\014)5 b(\022)25 b Fo(+)e(\001\)\)\(1)e(+)i Fl(m)17 b Fo(tanh\(\001\)\))3645 3245 y(\(3)p Fl(:)p Fo(27\))0 3553 y(On)34 b(the)g(other)f(hand,)h (since)g Fl(\027)1168 3568 y Fn(2)1240 3553 y Fo(=)29 b Fl(\027)1395 3568 y Fn(2)1440 3553 y Fo(\()p Fl(\022)s Fo(\))j(and)h Fl(\027)1842 3568 y Fn(2)1887 3553 y Fo(\(0\))27 b(=)i Fl(\027)2197 3568 y Fn(1)2241 3553 y Fo(,)k(and)h(see)g (\(3.21\),)1319 3786 y Fl(d\027)1420 3801 y Fn(2)p 1319 3831 146 4 v 1341 3922 a Fl(d\022)1504 3854 y Fo(=)1921 3786 y Fj(\000)p Fo(2)p Fl(\025\014)5 b(\022)s(\013)p 1621 3831 961 4 v 1621 3932 a(\033)1682 3898 y Fn(2)1678 3962 y Fk(\025\014)t(\022)1836 3932 y Fo(cosh)2024 3888 y Fn(2)2069 3932 y Fo(\()p Fl(\027)2157 3947 y Fn(2)2224 3932 y Fo(+)23 b(2)p Fl(\025\014)5 b(\022)s Fo(\))3645 3854 y(\(3)p Fl(:)p Fo(28\))0 4185 y(after)34 b(an)f(easy)g (computation,)f(w)m(e)i(get)130 4503 y Fl(\027)179 4518 y Fn(2)246 4503 y Fj(\000)23 b Fl(\027)395 4518 y Fn(1)467 4503 y Fo(=)573 4368 y Fd(Z)672 4392 y Fk(\022)628 4594 y Fn(0)746 4436 y Fl(d\027)847 4451 y Fn(2)p 746 4480 146 4 v 768 4571 a Fl(d\022)920 4503 y(d\022)1022 4462 y Fp(0)1077 4503 y Fo(=)28 b Fj(\000)p Fo(2)p Fl(\025\014)1446 4368 y Fd(Z)1546 4392 y Fk(\022)1501 4594 y Fn(0)2295 4436 y Fl(\013)17 b Fo(cosh)2565 4391 y Fn(2)2609 4436 y Fo(\()p Fl(\027)2697 4451 y Fn(2)2742 4436 y Fo(\()p Fl(\022)2831 4400 y Fp(0)2858 4436 y Fo(\)\))p 1619 4480 1993 4 v 1619 4582 a Fl(\013)g Fo(cosh)1889 4537 y Fn(2)1933 4582 y Fo(\()p Fl(\027)2021 4597 y Fn(2)2066 4582 y Fo(\()p Fl(\022)2155 4553 y Fp(0)2182 4582 y Fo(\)\))22 b(+)h(\(1)e Fj(\000)i Fl(\013)p Fo(\))17 b(cosh)2900 4537 y Fn(2)2945 4582 y Fo(\(2)p Fl(\025\014)5 b(\022)25 b Fo(+)d Fl(\027)3373 4597 y Fn(2)3418 4582 y Fo(\()p Fl(\022)3507 4553 y Fp(0)3534 4582 y Fo(\)\))3640 4503 y Fl(d\022)3742 4462 y Fp(0)3645 4721 y Fo(\(3)p Fl(:)p Fo(29\))0 4861 y(F)-8 b(rom)32 b(whic)m(h)i(it)e(is)i(easy)f(to)g(get)1621 5000 y Fj(j)p Fl(\027)1698 5015 y Fn(2)1764 5000 y Fj(\000)23 b Fl(\027)1913 5015 y Fn(1)1958 5000 y Fj(j)k(\024)h Fo(2)p Fl(\014)5 b(\022)1369 b Fo(\(3)p Fl(:)p Fo(30\))0 5201 y(Therefore,)55 b(w)m(e)49 b(ha)m(v)m(e)h(\(tanh\(2)p Fl(\025\014)5 b(\022)36 b Fo(+)d(\001\))g Fj(\000)h Fo(tanh\(\001\)\))54 b Fj(\024)h Fo([1)32 b Fj(\000)i Fo(tanh)2818 5156 y Fn(2)2863 5201 y Fo(\(2)p Fl(\014)5 b(\022)s Fo(\)])3130 5164 y Fp(\000)p Fn(1)3252 5201 y Fo(tanh)q(\(2)p Fl(\014)g(\022)s Fo(\))48 b(and)0 5340 y(\(1)22 b(+)g Fl(m)17 b Fo(tanh)q(\(2)p Fl(\025\014)5 b(\022)25 b Fo(+)d(\001\)\))27 b Fj(\025)i Fo(1)22 b Fj(\000)g Fo(tanh\(4)p Fl(\014)5 b(\022)s Fo(\).)44 b(Collecting)32 b(w)m(e)i(get)f(the)h(lemma.)p 3182 5340 70 70 v 0 5579 a Fg(23)p Ff(=j)s(anuar)r(y)r(=)p Fg(1998;)e(12:06)1145 b Fo(18)p eop %%Page: 19 19 19 18 bop 0 100 a Fm(Prop)s(osition)36 b(3.2)71 b(\(LCL)-10 b(T\))p Fe(Ther)-5 b(e)32 b(exists)g(an)h Fl(\017)28 b(>)h Fo(0)k Fe(such)g(that)g(if)g Fl(\014)5 b(\022)31 b Fj(\024)d Fl(\017)p Fe(,)34 b(for)f(al)5 b(l)34 b(given)e Fl(m)i Fe(such)0 239 y(that)i Fj(j)p Fl(m)p Fj(j)27 b Fl(<)h Fo(1)p Fe(,)35 b Fl(\013)28 b Fo(=)g Fj(j)p Fl(D)s Fj(j)p Fl(=)p Fj(j)p Fl(B)5 b Fj(j)34 b Fe(and)i Fl(\033)1405 254 y Fk(\025\014)t(\022)1581 239 y Fe(given)f(by)h(\(3.25\))1054 594 y Fo(\011)1131 609 y Fk(\025\014)t(\022)r(;\013;m)1470 594 y Fo(=)1834 526 y(1)p 1587 571 544 4 v 1587 591 a Fd(p)p 1686 591 247 4 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y(Therefore)35 b(after)e(some)g(computations,)f(w)m(e)i(get)934 4498 y Fl(L)1002 4513 y Fk(m)1077 4498 y Fo(\()p Fl(\025\014)5 b(\022)s(;)17 b(D)s Fo(\))27 b Fj(\024)h Fo(exp)17 b([2)p Fl(\025\014)5 b(\022)s(D)20 b Fo(\()o Fl(m)j Fo(+)38 b(^)-66 b Fl(')2416 4513 y Fk(p)2463 4498 y Fo(\()p Fl(\025\014)5 b(;)17 b(m;)g(\013)p Fo(\)\))o(])678 b(\(3)p Fl(:)p Fo(69\))0 4773 y(with)33 b Fj(j)16 b Fo(^)-66 b Fl(')320 4788 y Fk(p)366 4773 y Fo(\()p Fl(\025\014)5 b(;)17 b(m;)g(\013)p Fo(\))p Fj(j)27 b(\024)976 4724 y Fn(2)p Fk(g)1058 4694 y Fg(2)1096 4724 y Fn(\()p Fk(B)s Fn(\))p 976 4750 246 4 v 1067 4807 a Fk(B)1256 4773 y Fo(+)22 b(2)p Fl(\014)5 b(\022)1528 4724 y Fk(g)s Fn(\()p Fk(B)s Fn(\))p 1528 4750 168 4 v 1580 4807 a Fk(B)1707 4773 y Fo(.)120 4913 y(F)-8 b(or)32 b(the)i(lo)m(w)m(er)f(b)s(ound,)h(w)m(e)g(ha)m(v)m(e)332 5250 y Fl(L)400 5265 y Fk(k)449 5250 y Fo(\()p Fl(\025\014)5 b(\022)s(;)17 b(D)s Fo(\))27 b Fj(\025)h Fl(e)1004 5209 y Fn(2)p Fk(\025\014)t(\022)r(D)1269 5110 y Fd(\022)1354 5183 y Fl(B)f Fj(\000)c Fl(D)i Fj(\000)e Fl(k)p 1354 5227 465 4 v 1546 5318 a(B)1831 5110 y Fd(\023)1904 5130 y Fk(k)2008 5126 y(D)1970 5155 y Fd(X)1974 5370 y Fk(`)p Fn(=0)2152 5183 y Fo(1)p 2142 5227 70 4 v 2142 5318 a Fl(`)p Fo(!)2223 5250 y Fl(e)2269 5209 y Fp(\000)p Fn(4)p Fk(\025\014)t(\022)r(`)2580 5183 y Fo(\()p Fl(D)j Fj(\000)c Fl(`)p Fo(\))2907 5147 y Fk(`)2945 5183 y Fo(\()p Fl(k)j Fj(\000)d Fl(`)p Fo(\))3241 5147 y Fk(`)p 2558 5227 744 4 v 2558 5318 a Fo(\()p Fl(B)k Fj(\000)d Fl(D)i Fj(\000)e Fl(k)i Fo(+)e Fl(`)p Fo(\))3264 5290 y Fk(`)3645 5250 y Fo(\(3)p Fl(:)p Fo(70\))0 5579 y Fg(23)p Ff(=j)s(anuar)r(y)r(=)p Fg(1998;)32 b(12:06)1145 b Fo(24)p eop %%Page: 25 25 25 24 bop 0 100 a Fo(Keeping)34 b(the)g(\014rst)g(t)m(w)m(o)f(terms)f (in)i(the)f(previous)h(sum)f(giv)m(es)369 447 y Fl(L)437 462 y Fk(k)486 447 y Fo(\()p Fl(\025\014)5 b(\022)s(;)17 b(D)s Fo(\))27 b Fj(\025)h Fl(e)1041 406 y Fn(2)p Fk(\025\014)t(\022)r (D)1306 306 y Fd(\022)1391 379 y Fl(B)f 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Fk(B)s Fn(\))p 1068 1266 207 4 v 1140 1323 a Fk(B)1286 1289 y Fl(e)1332 1253 y Fn(8)p Fk(\014)t(\022)1466 1289 y Fo(.)p 1538 1289 70 70 v 0 5579 a Fg(23)p Ff(=j)s(anuar)r(y)r(=)p Fg(1998;)k(12:06)1145 b Fo(25)p eop %%Page: 26 26 26 25 bop 0 100 a Fq(4)45 b(Pro)t(of)g(of)g(the)g(Theorem)h(2.3)f(and)g (some)g(probabilit)l(y)i(estimates)120 358 y Fo(In)i(this)g(section)g (w)m(e)g(pro)m(v)m(e)h(the)f(Theorem)f(2.3.)91 b(T)-8 b(o)48 b(study)h(the)g(prop)s(erties)h(of)f(the)g(system,)0 498 y(uniformly)e(on)i(an)f(in)m(terv)-6 b(al)48 b Fl(V)71 b Fo(of)48 b(length)j(~)-52 b Fl(c)1783 449 y Fn(\(log)14 b(1)p Fk(=\015)t Fn(\))2088 419 y Ff(p)p 1783 475 348 4 v 1933 532 a Fk(\015)2142 498 y Fo(,)52 b Fl(p)h(>)g Fo(1,)e(w)m(e)e(start)f(considering)h(a)f(region)0 637 y Fl(V)58 652 y Fn(1)149 637 y Fj(\032)f Fl(V)67 b Fo(of)45 b(scale)g Fl(L)841 652 y Fn(1)932 637 y Fj(\031)1071 598 y Fn(1)p 1068 614 47 4 v 1068 672 a Fk(\015)1126 637 y Fo(\(log)17 b(1)p Fl(=\015)6 b Fo(\)\(log)16 b(log)h(1)p 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(the)g(sto)s(c)m(hastic)0 1475 y(part)e(is)g(smaller)f(than)i Fl(e)939 1431 y Fp(\000)1018 1404 y Ff(c)p 1013 1415 41 4 v 1013 1455 a(\015)1071 1475 y Fo(,)f Fl(c)28 b(>)g Fo(0.)120 1615 y(ii\))k Fe(for)j(al)5 b(l)34 b Fo(ab)s(o)m(v)m(e)f(men) m(tioned)h(in)m(terv)-6 b(als)33 b(the)h(con)m(tribution)f(of)h(the)f (second)i(factor)e(is)g(negligible)120 1756 y(iii\))f(this)h(can)g(b)s (e)h(done)g(uniformly)e(with)h(resp)s(ect)h(to)f(the)h(c)m(hoices)g(of) g Fl(V)2848 1771 y Fn(1)2926 1756 y Fo(in)f Fl(V)22 b Fo(.)120 1896 y(The)34 b(c)m(hoice)h(of)g(the)f(relativ)m(e)g(sizes)h (of)f(the)g(in)m(terv)-6 b(als)35 b(in)m(v)m(olv)m(ed)f(is)g(suggested) h(b)m(y)g(t)m(w)m(o)f(con\015icting)0 2036 y(conditions:)59 b(the)40 b(existence)i(of)e(a)g(large)g(enough)i(\015uctuation)f(of)f (the)h(magnetic)e(\014eld,)k(in)d(at)g(least)0 2175 y(one)30 b(small)e(in)m(terv)-6 b(al,)29 b(for)h(the)f(\014rst)h(factor)f(and)h (the)g(uniform)f(con)m(trol)g(of)h(the)f(second)i(factor)e(o)m(v)m(er) 59 b Fe(al)5 b(l)0 2315 y Fo(in)m(terv)-6 b(als)33 b(con)m(tained)h(in) g Fl(V)22 b Fo(.)120 2455 y(In)i(step)h(\(ii\))e(w)m(e)h(need)i(a)e (deviation)f(inequalit)m(y)g(for)i(a)f(Lipsc)m(hitz)h(function)g(of)g (symmetric)d(Bernoulli)0 2595 y(random)32 b(v)-6 b(ariables,)32 b(but)h(our)f(construction)h(of)g(the)f(sto)s(c)m(hastic)h(part,)f(in)g (section)h(3,)f(do)s(es)h(not)f(allo)m(w)0 2734 y(to)37 b(c)m(hec)m(k)i(the)e(con)m(v)m(exit)m(y)g(h)m(yp)s(othesis)i(assumed)e (in)h([21])e(or)i(in)f([30].)56 b(Therefore)39 b(w)m(e)e(giv)m(e)g(a)h (simple)0 2874 y(pro)s(of)33 b(of)h(suc)m(h)h(deviation)d(inequalit)m (y)g(without)h(an)m(y)g(con)m(v)m(exit)m(y)g(h)m(yp)s(othesis.)120 3014 y(W)-8 b(e)33 b(start)g(the)h(pro)s(of)f(of)g(the)h(Theorem)f (2.3.)43 b(Giv)m(en)36 b(~)-52 b Fl(c)27 b(>)i Fo(0,)j Fl(p)c(>)g Fo(1,)33 b(it)g(is)g(enough)h(to)f(pro)m(v)m(e)g(that)916 3227 y Fl(\026)976 3242 y Fk(\014)t(;\022)r(;\015)1180 3146 y Fd(\002)1221 3227 y Fj(R)1305 3186 y Fk(\016)n(;\020)1409 3227 y Fo(\([)p 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y Fn(2)3126 3579 y Fo(])52 b Fj(\032)g Fo([)p Fl(k)s(`)3462 3594 y Fn(^)-42 b Fk(c)3499 3579 y Fo(\()p Fl(\015)6 b Fo(\))p Fl(;)17 b(L)3748 3594 y Fn(1)3823 3579 y Fo(+)0 3718 y Fl(k)s(`)99 3733 y Fn(^)-42 b Fk(c)137 3718 y Fo(\()p Fl(\015)6 b Fo(\)])29 b(and)i(w)m(e)g(start)f(estimating)f Fl(\026)1446 3733 y Fk(\014)t(;\022)r(;\015)1650 3637 y Fd(\002)1691 3718 y Fj(R)1775 3682 y Fk(\016)n(;\020)1879 3718 y Fo(\()p Fl(`)1960 3733 y Fn(1)2004 3718 y Fl(;)17 b(`)2091 3733 y Fn(2)2135 3718 y Fl(;)g(\034)11 b Fo(\))2274 3637 y Fd(\003)2314 3718 y Fo(,)31 b(with)f Fl(`)2639 3733 y Fn(1)2714 3718 y Fo(and)h Fl(`)2947 3733 y Fn(2)3021 3718 y Fo(suc)m(h)h(that)e Fj(j)p Fl(`)3526 3733 y Fn(1)3587 3718 y Fj(\000)17 b Fl(`)3723 3733 y Fn(2)3767 3718 y Fj(j)28 b Fo(=)206 3818 y Fn(^)-42 b Fk(c)p 12 3835 420 4 v 12 3892 a(\015)18 b Fn(log)c(log)h(1)p Fk(=\015)471 3857 y Fo(=)28 b Fl(`)620 3872 y Fn(^)-42 b Fk(c)659 3857 y Fo(\()p Fl(\015)6 b Fo(\).)120 3998 y(The)45 b(\014rst)h(remark) 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(con\014gurations)g(in)g Fj(R)1798 4805 y Fk(\016)n(;\020)1902 4841 y Fo(\()p Fl(`)1983 4856 y Fn(1)2027 4841 y Fl(;)17 b(`)2114 4856 y Fn(2)2157 4841 y Fl(;)g Fj(\000)p Fl(\034)11 b Fo(\).)48 b(T)-8 b(aking)33 b(in)m(to)h(accoun)m(t)i(that)e(the)h(t)m (w)m(o)0 4981 y(normalization)c(factors)j(cancel,)f(w)m(e)h(ha)m(v)m(e) 529 5271 y Fl(\026)589 5286 y Fk(\014)t(;\022)r(;\015)793 5191 y Fd(\002)834 5271 y Fj(R)918 5230 y Fk(\016)n(;\020)1022 5271 y Fo(\()p Fl(`)1103 5286 y Fn(1)1147 5271 y Fl(;)17 b(`)1234 5286 y Fn(2)1278 5271 y Fl(;)g(\034)11 b Fo(\))1417 5191 y Fd(\003)1485 5271 y Fj(\024)28 b Fl(e)1636 5230 y Fn(4)p Fk(\014)t(\015)1771 5200 y Fb(\000)p Fg(1)1920 5198 y Fl(\026)1980 5213 y Fk(\014)t(;\022)r(;\015)2183 5117 y Fd(\000)2229 5198 y Fj(R)2313 5161 y Fk(\016)n(;\020)2417 5198 y Fo(\()p Fl(`)2498 5213 y Fn(1)2542 5198 y Fl(;)17 b(`)2629 5213 y Fn(2)2673 5198 y Fl(;)g(\034)11 b Fo(\))2838 5113 y Fd(\014)2838 5173 y(\014)2899 5198 y Fo(\006)2971 5213 y Fk(@)t(I)3053 5223 y Fg(12)3131 5117 y Fd(\001)3193 5198 y Fo(\(0\))p 1881 5248 1479 4 v 1881 5345 a Fl(\026)1941 5360 y Fk(\014)t(;\022)r(;\015)2145 5264 y Fd(\000)2190 5345 y Fj(R)2274 5316 y Fk(\016)n(;\020)2379 5345 y Fo(\()p Fl(`)2460 5360 y Fn(1)2504 5345 y Fl(;)17 b(`)2591 5360 y Fn(2)2634 5345 y Fl(;)g Fj(\000)p Fl(\034)11 b Fo(\))2877 5260 y Fd(\014)2877 5320 y(\014)2938 5345 y Fo(\006)3010 5360 y Fk(@)t(I)3092 5370 y Fg(12)3170 5264 y Fd(\001)3232 5345 y Fo(\(0\))3695 5271 y(\(4)p Fl(:)p Fo(3\))0 5579 y Fg(23)p Ff(=j)s(anuar)r(y)r(=)p Fg(1998;)32 b(12:06)1145 b Fo(26)p eop %%Page: 27 27 27 26 bop 0 100 a Fo(F)-8 b(or)35 b(simplicit)m(y)-8 b(,)33 b(let)i(us)h(denote)g Fj(R)p Fo(\()p Fl(\034)11 b Fo(\))31 b Fj(\021)g(R)1696 63 y Fk(\016)n(;\020)1800 100 y Fo(\()p Fl(`)1881 115 y Fn(1)1925 100 y Fl(;)17 b(`)2012 115 y Fn(2)2056 100 y Fl(;)g(\034)11 b Fo(\).)49 b(P)m(erforming)34 b(a)h(blo)s(c)m(k)g(spins)h(transforma-)0 239 y(tion)d(on)g(the)h(scale)f Fl(\016)801 203 y Fp(\003)847 239 y Fl(\015)905 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2635 y Fo(\))p 523 2679 386 4 v 523 2771 a Fl(Z)591 2786 y Fp(\000)p Fk(\034)704 2771 y Fo(\()p Fl(I)787 2786 y Fn(12)870 2771 y Fo(\))949 2702 y Fj(\021)1079 2541 y Fd(P)1184 2646 y Fk(m)1255 2626 y Ff(\016)1290 2611 y Fb(\003)1336 2646 y Fn(\()p Fk(I)1402 2656 y Fg(12)1475 2646 y Fn(\))p Fp(2M)1654 2660 y Ff(\016)1689 2645 y Fb(\003)1734 2646 y Fn(\()p Fk(I)1800 2656 y Fg(12)1874 2646 y Fn(\))1926 2616 y Fo(1)-25 b(I)1987 2634 y Fp(fR)p Fn(\()p Fk(\034)8 b Fn(\))p Fp(g)2249 2616 y Fl(e)2295 2563 y Fp(\000)2373 2536 y Fg(1)p 2370 2547 41 4 v 2370 2587 a Ff(\015)2423 2490 y Fd(\010)2481 2563 y Fk(\014)2542 2558 y Fd(b)2530 2563 y Fp(F)f Fn(\()p Fk(m)2696 2533 y Ff(\016)2731 2513 y Fb(\003)2696 2583 y Ff(I)2727 2598 y Fg(12)2804 2563 y Fk(;)p Fn(0\)+)p Fk(\015)t Fp(G)t Fn(\()p Fk(m)3160 2533 y Ff(\016)3195 2513 y Fb(\003)3160 2583 y Ff(I)3191 2598 y Fg(12)3269 2563 y Fn(\))3300 2490 y Fd(\011)p 1066 2679 2311 4 v 1066 2747 a(P)1171 2851 y Fk(m)1242 2832 y Ff(\016)1277 2817 y 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a Fk(Z)1894 3971 y Fb(\000)p Ff(\034)1993 3961 y Fn(\()p Fk(I)2059 3971 y Fg(12)2133 3961 y Fn(\))2176 3926 y Fo(\()p Fl(h)p Fo(\))g(is)g(a)g(symmetric)e(random)i(v)-6 b(ariable)30 b(and)0 4066 y(therefore)42 b(has)f(mean)f(zero.)68 b(The)41 b(next)g(step)g(is)g(to)f(extract)g(what)h(w)m(e)g(exp)s(ect)g (to)g(b)s(e)g(the)g(leading)0 4205 y(term)32 b(of)i(the)f(sto)s(c)m (hastic)h(part)f(coming)f(in)h(\(4.7\).)43 b(Recalling)33 b(\(3.14\),)f(w)m(e)h(in)m(tro)s(duce)989 4506 y(\001)p Fj(G)6 b Fo(\()p Fl(m)1263 4465 y Fk(\016)1302 4434 y Fb(\003)1263 4530 y Fk(\014)t(;I)1371 4540 y Fg(12)1449 4506 y Fl(;)17 b(\034)11 b Fo(\))27 b Fj(\021)h Fl(\034)1791 4395 y Fd(h)1838 4506 y Fj(G)6 b Fo(\()p Fl(m)2029 4465 y Fk(\016)2068 4434 y Fb(\003)2029 4530 y Fk(\014)t(;I)2137 4540 y Fg(12)2215 4506 y Fo(\))22 b Fj(\000)h(G)6 b Fo(\()p Fl(T)14 b(m)2639 4465 y Fk(\016)2678 4434 y Fb(\003)2639 4530 y Fk(\014)t(;I)2747 4540 y Fg(12)2825 4506 y Fo(\))2864 4395 y Fd(i)3695 4506 y Fo(\(4)p Fl(:)p Fo(9\))0 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2467 847 V 2514 887 a Fg(2)2612 904 y Fl(m)2699 863 y Fk(\016)2738 833 y Fb(\003)2699 929 y Fk(\014)t(;I)2807 939 y Fg(12)2885 904 y Fo(\))721 b(\(4)p Fl(:)p Fo(12\))0 1090 y(with)33 b Fl(T)299 1053 y Fn(0)371 1090 y Fo(=)c(1)-25 b(I)o(,)33 b(the)h(iden)m(tit)m(y)-8 b(.)120 1230 y(Recalling)37 b(that)g Fl(m)864 1245 y Fk(\014)t(;)p Fn(1)1019 1230 y Fo(and)h Fl(m)1304 1245 y Fk(\014)t(;)p Fn(2)1459 1230 y Fo(whic)m(h)h(are)e(de\014ned)j(b)s(efore)e(\(2.23\))f(are)g(b)s (ounded)i(a)m(w)m(a)m(y)f(from)0 1370 y(1.)44 b(F)-8 b(or)32 b Fl(\014)i(>)28 b Fo(1)33 b(and)h Fl(\014)5 b(\022)36 b Fo(small)c(enough,)h(w)m(e)h(can)g(use)g(the)f(Prop)s (osition)f(3.4)h(to)g(con)m(trol)g(\001)p Fj(G)6 b Fo(\()p Fl(m)3648 1333 y Fk(\016)3687 1303 y Fb(\003)3648 1398 y Fk(\014)t(;I)3756 1408 y Fg(12)3834 1370 y Fo(\).)0 1509 y(Recall)33 b(that)g(this)g(term)f(has)h(mean)g(zero.)44 b(Using)33 b(\(3.53\))e(and)j(the)f(de\014nition)h(of)f Fl(T)14 b(m)3287 1524 y Fk(\014)3374 1509 y Fo(giv)m(en)33 b(b)s(efore)0 1649 y(\(2.23\),)e(w)m(e)j(can)g(write)f(\001)p Fj(G)6 b Fo(\()p Fl(m)1174 1612 y Fk(\016)1213 1582 y Fb(\003)1174 1677 y Fk(\014)t(;I)1282 1687 y Fg(12)1360 1649 y Fl(;)17 b(\034)11 b Fo(\))26 b(=)i Fj(\000)p Fl(\034)1779 1574 y Fd(P)1884 1678 y Fk(x)p Fp(2C)2026 1692 y Ff(\016)2061 1677 y Fb(\003)2107 1678 y Fn(\()p Fk(I)2173 1688 y Fg(12)2247 1678 y Fn(\))2299 1649 y Fl(X)8 b Fo(\()p Fl(x)p Fo(\))32 b(with)86 1973 y Fl(X)8 b Fo(\()p Fl(x)p Fo(\))27 b Fj(\021)h(\000)p Fo(2)p Fl(\014)5 b(\022)s(\025)p Fo(\()p Fl(x)p Fo(\))p Fj(j)p Fl(D)s Fo(\()p Fl(x)p Fo(\))p Fj(j)1168 1862 y Fd(h)1213 1973 y Fl(m)1300 1932 y Fk(\016)1339 1902 y Fb(\003)1300 1997 y Fk(\014)t(;)p Fn(1)1440 1973 y Fo(+)23 b Fl(m)1627 1932 y Fk(\016)1666 1902 y Fb(\003)1627 1997 y Fk(\014)t(;)p Fn(2)1767 1973 y Fo(+)f(\004\()p Fl(x;)17 b(\014)5 b(\022)s(;)17 b(\013)p Fo(\))2332 1862 y Fd(i)2400 1973 y Fj(\000)23 b Fl(\025)p Fo(\()p Fl(x)p Fo(\))17 b(log)2866 1870 y(\011)2943 1898 y Fk(\014)t(\022)r(;\013)p 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y Fo(.)120 3143 y(Calling)32 b Fl(V)543 3107 y Fn(2)588 3143 y Fo(\()p Fl(N)11 b Fo(\))27 b(=)i Fl(V)970 3107 y Fn(2)1015 3143 y Fo(\()p Fl(N)11 b Fo(\()p Fl(w)s Fo(\)\))27 b Fj(\021)1468 3069 y Fd(P)1573 3173 y Fk(x)p Fp(2)p Fk(N)7 b Fn(\()p Fk(w)r Fn(\))1886 3143 y Fl(I)-17 b(E)5 b Fo([)p Fl(X)2111 3107 y Fn(2)2155 3143 y Fo(\()p Fl(x)p Fo(\)])32 b(for)i(1)28 b Fj(\024)h Fl(w)i Fj(\024)d(j)p Fl(I)2966 3158 y Fn(12)3050 3143 y Fj(j)p Fl(=)p Fo(\()p Fl(\016)3215 3107 y Fp(\003)3260 3143 y Fl(N)11 b Fo(\),)33 b(w)m(e)h(require)0 3283 y(that)f Fl(N)44 b Fo(satis\014es)34 b(also)717 3620 y Fl(\015)915 3526 y Fd(X)791 3744 y Fk(x)p Fp(2C)933 3758 y Ff(\016)968 3743 y Fb(\003)1013 3744 y Fn(\()p Fk(I)1079 3754 y Fg(12)1153 3744 y Fn(\))1200 3620 y Fl(X)8 b Fo(\()p Fl(x)p Fo(\))27 b(=)1724 3553 y Fl(N)p 1570 3597 399 4 v 1570 3689 a Fj(jC)1650 3704 y Fk(\016)1689 3684 y Fb(\003)1736 3689 y Fo(\()p Fl(I)1819 3704 y Fn(12)1902 3689 y Fo(\))p Fj(j)2009 3436 y Fb(jC)2071 3457 y Ff(\016)2106 3442 y Fb(\003)2152 3436 y Fg(\()p Ff(I)2210 3451 y Fg(12)2283 3436 y(\))p Fb(j)p 2009 3465 324 4 v 2142 3505 a Ff(N)2099 3526 y Fd(X)2091 3737 y Fk(w)r Fn(=1)2473 3553 y Fo(1)p 2373 3597 249 4 v 2373 3689 a Fl(V)c Fo(\()p Fl(N)11 b Fo(\))2724 3526 y Fd(X)2651 3744 y Fk(x)p Fp(2)p Fk(N)c Fn(\()p Fk(w)r Fn(\))2958 3620 y Fl(X)h Fo(\()p Fl(x)p Fo(\))461 b(\(4)p Fl(:)p Fo(15\))0 3947 y(Assuming)33 b(that)g(suc)m(h)h Fl(N)44 b Fo(can)34 b(b)s(e)g(found,)g(then)g(w)m(e) f(ha)m(v)m(e)595 4193 y Fd(n)661 4304 y Fl(\034)11 b(\015)914 4209 y Fd(X)790 4427 y Fk(x)p Fp(2C)932 4441 y Ff(\016)967 4426 y Fb(\003)1013 4427 y Fn(\()p Fk(I)1079 4437 y Fg(12)1152 4427 y Fn(\))1200 4304 y Fl(X)d Fo(\()p Fl(x)p Fo(\))26 b Fl(>)i(u)1614 4193 y Fd(o)1708 4304 y Fj(\033)1825 4120 y Fb(jC)1887 4141 y Ff(\016)1922 4126 y Fb(\003)1968 4120 y Fg(\()p Ff(I)2026 4135 y Fg(12)2099 4120 y(\))p Fb(j)p 1825 4149 324 4 v 1958 4188 a Ff(N)1932 4209 y Fd(\\)1907 4420 y Fk(w)r Fn(=1)2161 4193 y Fd(n)2336 4236 y Fl(\034)p 2239 4281 249 4 v 2239 4372 a(V)22 b Fo(\()p Fl(N)11 b Fo(\))2590 4209 y Fd(X)2516 4427 y Fk(x)p Fp(2)p Fk(N)c Fn(\()p Fk(w)r Fn(\))2824 4304 y Fl(X)h Fo(\()p Fl(x)p Fo(\))27 b Fl(>)h(u)3239 4193 y Fd(o)3645 4304 y Fo(\(4)p Fl(:)p Fo(16\))0 4630 y(Using)h(the)h(fact)f (that)g(the)h(ev)m(en)m(ts)g(in)g(the)f(righ)m(t)g(hand)i(side)e(are)h (indep)s(enden)m(t)h(w)m(e)f(apply)f(the)h(Cen)m(tral)0 4770 y(Limit)39 b(Theorem)h(to)f(estimate)g(their)i(individual)e (probabilities.)65 b(T)-8 b(o)40 b(c)m(hec)m(k)h(that)f(w)m(e)h(can)g (\014nd)g(an)0 4909 y(1)27 b Fj(\024)i Fl(N)38 b Fj(\024)418 4861 y Fp(j)p Fk(I)477 4871 y Fg(12)550 4861 y Fp(j)p 418 4886 156 4 v 456 4944 a Fk(\016)495 4924 y Fb(\003)619 4909 y Fo(suc)m(h)d(that)d(\(4.15\))g(is)h(true,)h(w)m(e)f(note)h(that) f(on)g(the)h(one)f(hand)211 5252 y Fl(I)-17 b(E)5 b Fo([)p Fl(X)436 5211 y Fn(2)479 5252 y Fo(\()p Fl(x)p Fo(\)])27 b Fj(\025)774 5142 y Fd(\020)834 5252 y Fl(\014)5 b(\022)s Fo(\()p Fl(m)1071 5211 y Fk(\016)1110 5181 y Fb(\003)1071 5277 y Fk(\014)t(;)p Fn(1)1210 5252 y Fo(+)23 b Fl(m)1397 5211 y Fk(\016)1436 5181 y Fb(\003)1397 5277 y Fk(\014)t(;)p Fn(2)1515 5252 y Fo(\))1554 5142 y Fd(\021)1613 5162 y Fn(2)1674 5252 y Fl(I)-17 b(E)5 b Fo([\()1847 5123 y Fk(\016)1886 5093 y Fb(\003)1927 5123 y Fk(\015)1973 5093 y Fb(\000)p Fg(1)1884 5158 y Fd(X)1892 5370 y Fk(i)p Fn(=1)2083 5252 y Fl(h)2140 5267 y Fk(i)2173 5252 y Fo(\))2212 5211 y Fn(2)2257 5252 y Fo(])27 b(=)2417 5142 y Fd(\020)2477 5252 y Fl(\014)5 b(\022)s Fo(\()p Fl(m)2714 5211 y Fk(\016)2753 5181 y Fb(\003)2714 5277 y Fk(\014)t(;)p Fn(1)2853 5252 y Fo(+)23 b Fl(m)3040 5211 y Fk(\016)3079 5181 y Fb(\003)3040 5277 y Fk(\014)t(;)p Fn(2)3157 5252 y Fo(\))3196 5142 y Fd(\021)3256 5162 y Fn(2)3329 5185 y Fl(\016)3377 5149 y Fp(\003)p 3329 5229 94 4 v 3347 5321 a Fl(\015)3645 5252 y Fo(\(4)p Fl(:)p Fo(17\))0 5579 y Fg(23)p Ff(=j)s(anuar)r(y)r(=)p Fg(1998;)32 b(12:06)1145 b Fo(28)p eop %%Page: 29 29 29 28 bop 0 100 a Fo(and)34 b(on)f(the)h(other)f(hand,)h(using)g(the)f (Prop)s(osition)f(3.4,)h(w)m(e)g(get)g(if)h Fl(g)t Fo(\()p Fj(j)p Fl(B)5 b Fj(j)p Fo(\))30 b(is)k(large)e(enough)646 393 y Fl(I)-17 b(E)5 b Fo([)p Fl(X)871 352 y Fn(2)914 393 y Fo(\()p Fl(x)p Fo(\)])27 b Fj(\024)1209 283 y Fd(\020)1269 393 y Fl(\014)5 b(\022)s Fo(\()p Fl(m)1506 352 y Fk(\016)1545 322 y Fb(\003)1506 418 y Fk(\014)t(;)p Fn(1)1646 393 y Fo(+)22 b Fl(m)1832 352 y Fk(\016)1871 322 y Fb(\003)1832 418 y Fk(\014)t(;)p Fn(2)1972 393 y Fo(+)g Fl(c\014)5 b(\022)s Fo(\))2264 283 y Fd(\021)2324 303 y Fn(2)2397 326 y Fl(\016)2445 290 y Fp(\003)p 2397 371 94 4 v 2416 462 a Fl(\015)2520 283 y Fd(\020)2579 393 y Fo(1)22 b(+)h(2)p Fl(c)2874 326 y(\015)p 2856 371 V 2856 462 a(\016)2904 433 y Fp(\003)2961 393 y Fo(\()p Fl(\014)5 b(\022)s Fo(\))3150 352 y Fn(2)3195 283 y Fd(\021)3645 393 y Fo(\(4)p Fl(:)p Fo(18\))0 683 y(for)33 b(some)g(p)s(ositiv)m(e)g(constan)m(t)h Fl(c)p Fo(.)44 b(It)33 b(is)g(easy)g(to)g(c)m(hec)m(k)h(that)f Fl(N)44 b Fo(m)m(ust)33 b(satisfy)60 872 y Fd(\020)119 983 y Fl(\014)5 b(\022)s Fo(\()p Fl(m)356 942 y Fk(\016)395 912 y Fb(\003)356 1007 y Fk(\014)t(;)p Fn(1)496 983 y Fo(+)22 b Fl(m)682 942 y Fk(\016)721 912 y Fb(\003)682 1007 y Fk(\014)t(;)p Fn(2)800 983 y Fo(\))839 872 y Fd(\021)898 893 y Fn(2)971 915 y Fl(\015)6 b Fj(j)p Fl(I)1101 930 y Fn(12)1184 915 y Fj(j)1212 879 y Fn(2)p 971 960 286 4 v 1067 1051 a Fl(\016)1115 1022 y Fp(\003)1296 983 y Fj(\024)29 b Fl(N)38 b Fj(\024)1625 872 y Fd(\020)1685 983 y Fl(\014)5 b(\022)s Fo(\()p Fl(m)1922 942 y Fk(\016)1961 912 y Fb(\003)1922 1007 y Fk(\014)t(;)p Fn(1)2062 983 y Fo(+)22 b Fl(m)2248 942 y Fk(\016)2287 912 y Fb(\003)2248 1007 y Fk(\014)t(;)p Fn(2)2366 983 y Fo(\))2405 872 y Fd(\021)2464 893 y Fn(2)2537 915 y Fl(\015)6 b Fj(j)p Fl(I)2667 930 y Fn(12)2750 915 y Fj(j)2778 879 y Fn(2)p 2537 960 V 2633 1051 a Fl(\016)2681 1022 y Fp(\003)2851 872 y Fd(\020)2911 983 y Fo(1)22 b(+)g(2)p Fl(c)3205 915 y(\015)p 3187 960 94 4 v 3187 1051 a(\016)3235 1022 y Fp(\003)3293 983 y Fo(\()p Fl(\014)5 b(\022)s Fo(\))3482 942 y Fn(2)3526 872 y Fd(\021)3645 983 y Fo(\(4)p Fl(:)p Fo(19\))0 1318 y(Therefore,)37 b Fl(N)k Fj(\024)31 b(j)p Fl(I)783 1333 y Fn(12)867 1318 y Fj(j)p Fl(=\016)993 1282 y Fp(\003)1073 1318 y Fo(pro)m(vided)36 b Fj(j)p Fl(I)1557 1333 y Fn(12)1641 1318 y Fj(j)30 b(\024)1823 1279 y Fn(1)p 1819 1295 47 4 v 1819 1353 a Fk(\015)1894 1208 y Fd(\020)1954 1318 y Fl(\014)5 b(\022)s Fo(\()p Fl(m)2191 1282 y Fk(\016)2230 1252 y Fb(\003)2191 1347 y Fk(\014)t(;)p Fn(1)2330 1318 y Fo(+)23 b Fl(m)2517 1282 y Fk(\016)2556 1252 y Fb(\003)2517 1347 y Fk(\014)t(;)p Fn(2)2634 1318 y Fo(\))2673 1208 y Fd(\021)2733 1228 y Fp(\000)p Fn(2)2856 1238 y Fd(\000)2902 1318 y Fo(1)f(+)g(2)p Fl(c)3195 1274 y Fk(\015)p 3178 1295 81 4 v 3178 1353 a(\016)3217 1333 y Fb(\003)3271 1318 y Fo(\()p Fl(\014)5 b(\022)s Fo(\))3460 1282 y Fn(2)3504 1238 y Fd(\001)3550 1258 y Fp(\000)p Fn(1)3657 1318 y Fo(.)49 b(Ob-)0 1458 y(viously)32 b Fl(N)39 b Fj(\025)28 b Fo(1)33 b(pro)m(vided)1217 1782 y Fj(j)p Fl(I)1289 1797 y Fn(12)1373 1782 y Fj(j)27 b(\025)1533 1641 y Fd(\022)1619 1714 y Fl(\016)1667 1678 y Fp(\003)p 1619 1759 94 4 v 1637 1850 a Fl(\015)1724 1641 y Fd(\023)1798 1662 y Fn(1)p Fk(=)p Fn(2)2286 1714 y Fo(1)p 1952 1759 720 4 v 1952 1853 a Fl(\014)5 b(\022)s Fo(\()p Fl(m)2189 1819 y Fk(\016)2228 1799 y Fb(\003)2189 1883 y Fk(\014)t(;)p Fn(1)2328 1853 y Fo(+)23 b Fl(m)2515 1819 y Fk(\016)2554 1799 y Fb(\003)2515 1883 y Fk(\014)t(;)p Fn(2)2632 1853 y Fo(\))3645 1782 y(\(4)p Fl(:)p Fo(20\))120 2105 y(T)-8 b(o)32 b(con)m(tin)m(ue,)i(using)g(\(4.17\))e(and)i (\(4.19\))d(w)m(e)j(ha)m(v)m(e)1173 2397 y Fl(V)1253 2356 y Fn(2)1298 2397 y Fo(\()p Fl(N)11 b Fo(\))27 b Fj(\025)1599 2287 y Fd(\020)1659 2397 y Fl(\014)5 b(\022)s Fo(\()p Fl(m)1896 2356 y Fk(\016)1935 2326 y Fb(\003)1896 2422 y Fk(\014)t(;)p Fn(1)2035 2397 y Fo(+)23 b Fl(m)2222 2356 y Fk(\016)2261 2326 y Fb(\003)2222 2422 y Fk(\014)t(;)p Fn(2)2339 2397 y Fo(\))2378 2287 y Fd(\021)2438 2307 y Fn(4)2499 2397 y Fj(j)p Fl(I)2571 2412 y Fn(12)2655 2397 y Fj(j)2683 2356 y Fn(2)3645 2397 y Fo(\(4)p Fl(:)p Fo(21\))0 2669 y(Since)36 b(lim)400 2684 y Fk(\015)t Fp(#)p Fn(0)548 2669 y Fj(j)p Fl(I)620 2684 y Fn(12)704 2669 y Fj(j)30 b Fo(=)h Fj(1)p Fo(,)k(it)f(is)h(clear)g(that)f(w)m(e)h (are)g(in)g(the)g(domain)f(of)h(application)g(of)g(the)g(Cen)m(tral)0 2808 y(Limit)d(Theorem.)43 b(Therefore,)35 b(w)m(e)e(ha)m(v)m(e,)h(for) f(all)g(~)-51 b Fl(\017)29 b(>)f Fo(0)33 b(and)g Fl(u)28 b(>)g Fo(0)118 3164 y Fl(I)-17 b(P)239 2964 y Fd(2)239 3143 y(4)414 3097 y Fl(\034)p 317 3141 249 4 v 317 3233 a(V)23 b Fo(\()p Fl(N)11 b Fo(\))668 3070 y Fd(X)594 3288 y Fk(x)p Fp(2)p Fk(N)c Fn(\()p Fk(w)r Fn(\))902 3164 y Fl(X)h Fo(\()p Fl(x)p Fo(\))27 b Fl(>)h(u)1317 2964 y Fd(3)1317 3143 y(5)1411 3164 y Fj(\025)g Fl(I)-17 b(P)1637 2964 y Fd(2)1637 3143 y(4)1704 3164 y Fl(u)p Fo(\(1)21 b(+)j(~)-51 b Fl(\017)q Fo(\))27 b Fj(\025)2292 3097 y Fl(\034)p 2195 3141 V 2195 3233 a(V)22 b Fo(\()p Fl(N)11 b Fo(\))2546 3070 y Fd(X)2472 3288 y Fk(x)p Fp(2)p Fk(N)c Fn(\()p Fk(w)r Fn(\))2780 3164 y Fl(X)h Fo(\()p Fl(x)p Fo(\))27 b Fl(>)h(u)3195 2964 y Fd(3)3195 3143 y(5)3289 3164 y Fj(\025)g Fl(e)3440 3123 y Fp(\000)3515 3087 y Ff(u)3557 3067 y Fg(2)3596 3087 y(\(1+)r(~)-36 b Ff(\017)o Fg(\))p 3515 3108 251 4 v 3623 3147 a(2)3645 3433 y Fo(\(4)p Fl(:)p Fo(22\))0 3573 y(and,)47 b(using)e(the)f(lo)m(w) m(er)h(b)s(ound)g(for)f Fl(N)11 b Fo(,)47 b(see)e(\(4.19\),)g(and)g (for)f Fl(V)23 b Fo(\()p Fl(N)11 b Fo(\),)46 b(see)f(\(4.21\),)g (together)f(with)0 3712 y(\(4.16\),)31 b(w)m(e)j(get)373 4074 y Fl(I)-17 b(P)494 3873 y Fd(2)494 4053 y(4)560 4074 y Fl(\034)11 b(\015)813 3979 y Fd(X)689 4197 y Fk(x)p Fp(2C)831 4211 y Ff(\016)866 4196 y Fb(\003)912 4197 y Fn(\()p Fk(I)978 4207 y Fg(12)1051 4197 y Fn(\))1099 4074 y Fl(X)d Fo(\()p Fl(x)p Fo(\))26 b Fj(\025)i Fl(u)1513 3873 y Fd(3)1513 4053 y(5)1607 4074 y Fj(\025)h Fo(exp)1881 3903 y Fd( )1960 4074 y Fj(\000)2420 4006 y Fl(u)2477 3970 y Fn(2)2522 4006 y Fo(\(1)22 b(+)h(~)-51 b Fl(\017)q Fo(\))p 2049 4051 1133 4 v 2049 4145 a(2\()p Fl(\014)5 b(\022)s Fo(\()p Fl(m)2375 4111 y Fk(\016)2414 4091 y Fb(\003)2375 4176 y Fk(\014)t(;)p Fn(1)2515 4145 y Fo(+)22 b Fl(m)2701 4111 y Fk(\016)2740 4091 y Fb(\003)2701 4176 y Fk(\014)t(;)p Fn(2)2819 4145 y Fo(\)\))2897 4117 y Fn(2)2941 4145 y Fl(\015)6 b Fj(j)p Fl(I)3071 4160 y Fn(12)3154 4145 y Fj(j)3194 3903 y Fd(!)3645 4074 y Fo(\(4)p Fl(:)p Fo(23\))120 4436 y(No)m(w)32 b(to)g(end)h(the)g(pro)s(of)g(of)g (the)g(Theorem)f(2.3,)f(\014rst)i(w)m(e)g(use)h(\(4.23\),)d(for)h Fl(M)43 b Fo(consecutiv)m(e)34 b(blo)s(c)m(ks)0 4576 y(of)29 b(length)g Fl(`)453 4591 y Fn(^)-42 b Fk(c)491 4576 y Fo(\()p Fl(\015)6 b Fo(\),)28 b(that)h(w)m(e)g(denote)h(b)m(y)f Fl(L)p Fo(\(1\))p Fl(;)17 b(:)g(:)g(:)c(L)p Fo(\()p Fl(M)e Fo(\).)41 b(Using)29 b(indep)s(endence)j(o)m(v)m(er)d(disjoin)m(t)g (blo)s(c)m(ks)0 4715 y(and)34 b(1)22 b Fj(\000)g Fl(x)28 b Fj(\024)g Fl(e)601 4679 y Fp(\000)p Fk(x)714 4715 y Fo(,)33 b(considering)h(the)f(t)m(w)m(o)h(cases)g Fl(\034)k Fo(=)28 b(1)33 b(and)h Fl(\034)39 b Fo(=)28 b Fj(\000)p Fo(1)33 b(separately)-8 b(,)33 b(w)m(e)g(get)55 5071 y Fl(I)-17 b(P)176 4871 y Fd(2)176 5050 y(4)385 5071 y Fo(inf)243 5140 y Fk(\034)8 b Fp(2f\000)p Fn(1)p Fk(;)p Fn(+1)p Fp(g)731 5071 y Fo(sup)666 5155 y Fn(1)p Fp(\024)p Fk(`)p Fp(\024)p Fk(M)963 5071 y Fl(\034)j(\015)1238 4976 y Fd(X)1092 5195 y Fk(x)p Fp(2C)1234 5209 y Ff(\016)1269 5194 y Fb(\003)1315 5195 y Fn(\()p Fk(L)p Fn(\()p Fk(`)p Fn(\)\))1544 5071 y Fl(X)d Fo(\()p Fl(x)p Fo(\))27 b Fj(\025)h Fl(u)1959 4871 y Fd(3)1959 5050 y(5)2053 5071 y Fj(\025)g Fo(1)22 b Fj(\000)h Fo(2)17 b(exp)2565 4901 y Fd(")2623 5071 y Fj(\000)p Fl(M)11 b(e)2854 4974 y Fp(\000)3224 4938 y Ff(u)3266 4918 y Fg(2)3305 4938 y(\(1+)r(~)-36 b Ff(\017)p Fg(\))p 2929 4958 842 4 v 2929 5017 a(2\()p Ff(\014)s(\022)r Fg(\()p Ff(m)3156 5000 y(\016)3191 4985 y Fb(\003)3156 5051 y Ff(\014)s(;)p Fg(1)3259 5017 y(+)p Ff(m)3371 5000 y(\016)3406 4985 y Fb(\003)3371 5051 y Ff(\014)s(;)p Fg(2)3475 5017 y(\)\))3529 5002 y(2)3568 5017 y Ff(\015)s(`)3641 5035 y Fg(^)f Ff(c)3675 5017 y Fg(\()p Ff(\015)s Fg(\))3787 4901 y Fd(#)3645 5340 y Fo(\(4)p Fl(:)p Fo(24\))0 5579 y Fg(23)p Ff(=j)s(anuar)r(y)r(=)p Fg(1998;)32 b(12:06)1145 b Fo(29)p eop %%Page: 30 30 30 29 bop 120 100 a Fo(Moreo)m(v)m(er)39 b(it)e(follo)m(ws)i(from)e (the)i(next)g(prop)s(osition,)g(see)g(\(4.27\),)f(that)g(for)h(all)e Fl(\017)g(>)g Fo(0,)i(pro)m(vided)0 239 y Fl(g)48 254 y Fn(2)92 239 y Fo(\(1)p Fl(=\020)7 b Fo(\))32 b(is)i(a)f(div)m (erging,)f(slo)m(wly)h(v)-6 b(arying)32 b(function)i(at)f(in\014nit)m (y)-8 b(,)33 b(lim)2663 254 y Fk(\020)5 b Fp(#)p Fn(0)2806 164 y Fj(p)p 2889 164 51 4 v 75 x Fl(\020)i(g)2988 254 y Fn(2)3032 239 y Fo(\(1)p Fl(=\020)g Fo(\))27 b(=)h(0,)33 b(then)212 574 y Fl(I)-17 b(P)334 434 y Fd(\024)452 574 y Fo(sup)386 659 y Fn(1)p Fp(\024)p Fk(`)p Fp(\024)p Fk(M)667 460 y Fd(\014)667 519 y(\014)667 579 y(\014)700 574 y Fo(log)858 507 y Fl(Z)926 522 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y Fj(k)2403 1166 y Fn(1)2469 1151 y Fo(+)2946 1084 y Fl(c)p 2581 1128 774 4 v 2581 1233 a(g)t Fo(\()p Fj(j)p Fl(B)5 b Fj(j)p Fo(\))2848 1148 y Fd(p)p 2945 1148 410 4 v 2945 1233 a Fo(log)17 b Fl(g)t Fo(\()p Fj(j)p Fl(B)5 b Fj(j)p Fo(\))3645 1054 y(\(4)p Fl(:)p Fo(58\))0 1514 y Fe(on)36 b(the)f(set)463 1403 y Fd(n)530 1514 y Fj(j)p Fl(m)645 1478 y Fk(\016)684 1448 y Fb(\003)730 1514 y Fo(\()p Fl(x)p Fo(\()p Fl(i)p Fo(\)\))p Fj(j)27 b(\024)h Fo(1)22 b Fj(\000)1321 1466 y Fk(g)s Fn(\()p Fp(j)p Fk(B)s Fp(j)p Fn(\))p 1321 1491 215 4 v 1372 1548 a Fp(j)p Fk(B)s Fp(j)1547 1403 y Fd(o)1614 1514 y Fe(.)120 1654 y(While,)93 1958 y Fj(j)p Fo(\001)204 1973 y Fn(0)248 1958 y Fj(G)313 1917 y Fk(h)365 1958 y Fo([)p Fl(m)480 1917 y Fk(\016)519 1887 y Fb(\003)566 1958 y Fo(\()p Fl(x)p Fo(\()p Fl(i)p Fo(\)\)])f Fj(\000)h Fo(\001)1044 1973 y Fn(0)1089 1958 y Fj(G)1155 1899 y Fn(~)1154 1917 y Fk(h)1201 1927 y Ff(i)1237 1958 y Fo([)p Fl(m)1352 1917 y Fk(\016)1391 1887 y Fb(\003)1438 1958 y Fo(\()p Fl(x)p 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b(pro)m(v)m(e)i(\(4.58\),)0 2623 y(remem)m(b)s(ering)32 b(\(3.53\),)g(w)m(e)h(ha)m(v)m(e)h(to)f(study)g(three)h(terms.)44 b(The)33 b(\014rst)h(one)g(is)f(the)g(simplest:)161 2899 y(\001)244 2857 y Fn(1)244 2923 y(0)289 2899 y Fj(G)6 b Fo([)p Fl(m)469 2857 y Fk(\016)508 2827 y Fb(\003)555 2899 y Fo(\()p Fl(x)p Fo(\()p Fl(i)p Fo(\)\)])26 b Fj(\021)j Fo(2)p Fl(\014)5 b(\022)1139 2788 y Fd(\020)1199 2899 y Fl(\025)1257 2857 y Fk(h)1308 2899 y Fj(j)p Fl(D)1421 2857 y Fk(h)1473 2899 y Fj(j)22 b(\000)g Fl(\025)1681 2839 y Fn(~)1680 2857 y Fk(h)1727 2867 y Ff(i)1764 2899 y Fj(j)p Fl(D)1878 2839 y Fn(~)1877 2857 y Fk(h)1924 2867 y Ff(i)1960 2899 y Fj(j)1988 2788 y Fd(\021)17 b(h)2111 2899 y Fl(m)2198 2857 y Fk(\016)2237 2827 y Fb(\003)2198 2926 y Fk(\023)p Fn(\()p Fk(x)p Fn(\()p Fk(i)p Fn(\)\))2430 2899 y Fo(\()p Fl(x)p Fo(\()p Fl(i)p Fo(\)\))22 b Fj(\000)g Fl(m)2885 2857 y Fk(\016)2924 2827 y Fb(\003)2885 2926 y Fk(\014)t(;\023)p Fn(\()p Fk(x)p Fn(\()p Fk(i)p Fn(\)\))3191 2899 y Fo(\()p Fl(x)p 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4193 y Fo(\()p Fl(x)p Fo(\))p Fl(\014)5 b(\022)s(;)17 b(\013)3460 4134 y Fn(~)3458 4152 y Fk(h)3505 4162 y Ff(i)3540 4193 y Fo(\))758 4422 y Fj(\000)858 4312 y Fd(\020)917 4422 y Fl(\036)p Fo(\()p Fl(m)1102 4381 y Fk(\016)1141 4351 y Fb(\003)1102 4449 y Fk(\014)t(;\023)p Fn(\()p Fk(x)p Fn(\()p Fk(i)p Fn(\)\))1408 4422 y Fo(\()p Fl(x)p Fo(\()p Fl(i)p Fo(\)\))p Fl(;)g(\025)1758 4381 y Fk(h)1808 4422 y Fo(\()p Fl(x)p Fo(\))p Fl(\014)5 b(\022)s(;)17 b(\013)2163 4381 y Fk(h)2213 4422 y Fo(\))22 b Fj(\000)h Fl(\036)p Fo(\()p Fl(m)2559 4381 y Fk(\016)2598 4351 y Fb(\003)2559 4449 y Fk(\014)t(;\023)p Fn(\()p Fk(x)p Fn(\()p Fk(i)p Fn(\)\))2864 4422 y Fo(\()p Fl(x)p Fo(\()p Fl(i)p Fo(\)\))p Fl(;)17 b(\025)3215 4363 y Fn(~)3214 4381 y Fk(h)3261 4391 y Ff(i)3296 4422 y Fo(\()p Fl(x)p Fo(\))p Fl(\014)5 b(\022)s(;)17 b(\013)3653 4363 y Fn(~)3651 4381 y Fk(h)3698 4391 y Ff(i)3733 4422 y Fo(\))3772 4312 y Fd(\021)758 4651 y Fj(\000)23 b Fo(\001)941 4610 y Fn(1)941 4676 y(0)985 4651 y Fj(G)6 b Fo([)p 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Ff(=j)s(anuar)r(y)r(=)p Fg(1998;)32 b(12:06)1145 b Fo(36)p eop %%Page: 37 37 37 36 bop 120 100 a Fo(It)34 b(remains)g(to)g(consider)h(the)g(last)f (term)g(in)h(\(3.53\).)46 b(W)-8 b(e)35 b(use)g(that)g(\011)2806 115 y Fn(0)p Fk(;)p Fn(0)p Fk(;m)3043 100 y Fo(do)s(es)g(not)g(dep)s (end)h(on)0 239 y Fl(\013)d Fo(and)h(de\014ne)33 502 y(\001)116 461 y Fn(3)116 527 y(0)161 502 y Fj(G)6 b Fo(\()p Fl(m)352 461 y Fk(\016)391 431 y Fb(\003)438 502 y Fo(\()p Fl(x)p Fo(\()p Fl(i)p Fo(\)\)\))26 b Fj(\021)117 b Fo(log)17 b(\011)1166 530 y Fk(\025)1213 510 y Ff(h)1260 530 y Fn(\()p Fk(x)p Fn(\()p Fk(i)p Fn(\)\))p Fk(\014)t(\022)r(;\013) 1622 510 y Ff(h)1669 530 y Fk(;m)1764 506 y Ff(\016)1799 491 y Fb(\003)1764 559 y Ff(\023)p Fg(\()p Ff(x)p Fg(\()p Ff(i)p Fg(\)\))1969 530 y Fn(\()p Fk(x)p Fn(\()p Fk(i)p Fn(\)\))2194 502 y Fj(\000)23 b Fo(log)17 b(\011)2516 538 y Fk(\025)2565 502 y Fg(~)2563 515 y Ff(h)2604 531 y(i)2642 538 y Fn(\()p Fk(x)p Fn(\()p Fk(i)p Fn(\)\))p Fk(\014)t(\022)r(;\013)3006 502 y 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Fp(\000)p Fn(4)p Fp(\000)p Fn(2)p Fk(cR)p Fn(\()p Fk(\016)2202 4568 y Fb(\003)2244 4598 y Fn(+)2335 4565 y Ff(\015)p 2317 4583 76 4 v 2317 4624 a(\016)2352 4609 y Fb(\003)2419 4598 y Fn(log)2547 4571 y Ff(\016)2582 4551 y Fb(\003)p 2547 4582 V 2565 4622 a Ff(\015)2635 4598 y Fn(\))p Fp(\000)p Fn(2)p Fk(R\022)2868 4534 y Fj(p)p 2952 4534 100 4 v 2981 4565 a Ff(\015)p 2964 4583 76 4 v 2964 4624 a(\016)2999 4609 y Fb(\003)3052 4598 y Fp(\000)3114 4531 y Fj(p)p 3197 4531 107 4 v 67 x Fk(R\015)t Fn(4)p Fk(\022)r(z)3421 4608 y Fg(1)3460 4525 y Fd(\003)3695 4651 y Fo(\(5)p Fl(:)p Fo(1\))0 4945 y Fe(and)35 b Fl(I)-18 b(P)14 b Fo([\012)397 4960 y Fn(1)442 4945 y Fo(])27 b Fj(\025)h Fo(1)22 b Fj(\000)h Fl(e)820 4909 y Fp(\000)894 4858 y Ff(z)930 4838 y Fg(2)928 4886 y(1)p 894 4894 76 4 v 898 4933 a(64)986 4945 y Fe(.)0 5086 y Fa(Pro)s(of:)52 b Fo(By)32 b(the)i(v)m(ery)f(same)f(argumen)m(t)h(that)g(leads)h(to)f (\(4.2\))f(w)m(e)h(ha)m(v)m(e)696 5315 y Fl(\026)756 5330 y Fk(\014)t(;\022)r(;\015)960 5204 y Fd(\020)1020 5315 y Fj(O)1102 5267 y Fk(\016)1138 5277 y Fg(1)1177 5267 y Fk(;\020)1237 5277 y Fg(1)1099 5342 y Fn(0)1281 5315 y Fo(\(\001)1403 5330 y Fk(R)1468 5315 y Fo(\))1507 5204 y Fd(\021)1594 5315 y Fj(\024)28 b Fl(e)1745 5271 y Fn(4)1797 5238 y Ff(\014)p 1797 5255 42 4 v 1798 5295 a(\015)1856 5315 y Fl(\026)1916 5330 y Fk(\014)t(;\022)r(;\015)2120 5204 y Fd(\020)2179 5315 y Fj(O)2261 5267 y Fk(\016)2297 5277 y Fg(1)2336 5267 y Fk(;\020)2396 5277 y Fg(1)2258 5342 y Fn(0)2440 5315 y Fo(\(\001)2562 5330 y Fk(R)2627 5315 y Fo(\))2694 5230 y Fd(\014)2694 5290 y(\014)2755 5315 y Fo(\006)2827 5330 y Fk(@)t Fn(\001)2939 5340 y Ff(R)3000 5204 y Fd(\021)3076 5315 y Fo(\(0\))491 b(\(5)p Fl(:)p Fo(2\))0 5579 y Fg(23)p Ff(=j)s(anuar)r(y)r(=)p Fg(1998;)32 b(12:06)1145 b Fo(40)p eop %%Page: 41 41 41 40 bop 0 100 a Fo(P)m(erforming)33 b(a)g(blo)s(c)m(k)g(spin)g (transformation)g(on)g(the)h(scale)f Fl(\016)2349 63 y Fp(\003)2395 100 y Fo(,)g(recalling)g(\(3.13\),)e(w)m(e)j(ha)m(v)m(e) 267 391 y Fl(\026)327 406 y Fk(\014)t(;\022)r(;\015)531 280 y Fd(\020)590 391 y Fj(O)672 343 y Fk(\016)708 353 y Fg(1)747 343 y Fk(;\020)807 353 y Fg(1)669 417 y Fn(0)851 391 y Fo(\(\001)973 406 y Fk(R)1038 391 y Fo(\))1105 306 y Fd(\014)1105 366 y(\014)1165 391 y Fo(\006)1237 406 y Fk(@)t Fn(\001)1349 416 y Ff(R)1411 280 y Fd(\021)1487 391 y Fo(\(0\))294 692 y(=)423 625 y Fl(e)469 588 y Fn(\()p Fp(\006)p Fk(\014)t(\016)650 558 y Fb(\003)692 588 y Fk(\015)738 558 y Fb(\000)p Fg(1)831 588 y Fk(R)p Fn(\))p 411 669 529 4 v 411 760 a Fl(Z)479 775 y Fk(\014)t(;\022)r(;\015)t(;)p Fn(\001)751 785 y Ff(R)812 760 y Fo(\(0\))1270 597 y Fd(X)968 821 y Fk(m)1039 801 y Ff(\016)1074 786 y Fb(\003)1120 821 y Fn(\(\001)1216 831 y Ff(R)1272 821 y Fn(\))p Fp(2M)1451 835 y Ff(\016)1486 820 y Fb(\003)1532 821 y Fn(\(\001)1628 831 y Ff(R)1685 821 y Fn(\))1732 692 y Fo(1)-25 b(I)1793 731 y Fp(fO)1899 692 y Ff(\016)1932 707 y Fg(1)1971 692 y Ff(;\020)2027 707 y Fg(1)1897 755 y(0)2071 731 y Fn(\(\001)2167 741 y Ff(R)2224 731 y Fn(\))p Fp(g)2301 692 y Fl(e)2347 639 y Fp(\000)2425 612 y Fg(1)p 2421 623 41 4 v 2421 663 a Ff(\015)2474 566 y Fd(\010)2532 639 y Fk(\014)2594 634 y Fd(b)2581 639 y Fp(F)7 b Fn(\()p Fk(m)2747 609 y Ff(\016)2782 589 y Fb(\003)2747 659 y Fg(\001)2802 676 y Ff(R)2863 639 y Fk(;)p Fn(0\)+)p Fk(\015)t Fp(G)t Fn(\()p Fk(m)3219 609 y Ff(\016)3254 589 y Fb(\003)3219 659 y Fg(\001)3274 676 y Ff(R)3334 639 y Fn(\))3365 566 y Fd(\011)3695 587 y Fo(\(5)p Fl(:)p Fo(3\))120 1076 y(T)-8 b(o)37 b(estimate)g(the)h(sto)s(c)m(hastic)g(part,)h(w)m(e)g (mak)m(e)e(a)g(rough)i(upp)s(er)g(b)s(ound,)h(see)e(\(3.15\))f(and)h (\(3.16\))0 1131 y Fd(\014)0 1190 y(\014)33 1215 y Fj(G)92 1239 y Fk(x;m)232 1219 y Ff(\016)267 1204 y Fb(\003)318 1215 y Fo(\()p Fl(x)p Fo(\)\()p Fl(\025)p Fo(\()p Fl(x)p Fo(\)\))724 1131 y Fd(\014)724 1190 y(\014)792 1215 y Fj(\024)f Fo(2)p Fl(\014)5 b(\022)s Fj(j)p Fl(D)s Fo(\()p Fl(x)p Fo(\))p Fj(j)37 b Fo(whic)m(h)j(corresp)s(onds)g(to)e(the)h (situation)f(where)h(all)f(the)h(spins)g(in)0 1355 y Fl(D)85 1319 y Fk(\025)137 1355 y Fo(\()p Fl(x)p Fo(\))33 b(are)g(equal)g(to)g Fj(\000)p Fl(\025)p Fo(\()p Fl(x)p Fo(\).)44 b(This)34 b(giv)m(es)f(us)h(a)f(factor)1079 1606 y(\004\(2)p Fl(\014)5 b(\022)s(;)17 b Fo(\001)1473 1621 y Fk(R)1538 1606 y Fo(\))27 b Fj(\021)h Fo(exp)q Fj(f)2043 1511 y Fd(X)1912 1730 y Fk(x)p Fp(2C)2054 1744 y Ff(\016)2089 1729 y Fb(\003)2134 1730 y Fn(\(\001)2230 1740 y Ff(R)2287 1730 y Fn(\))2335 1606 y Fo(2)p Fl(\014)5 b(\022)s Fj(j)p Fl(D)s Fo(\()p Fl(x)p Fo(\))p Fj(jg)873 b Fo(\(5)p Fl(:)p Fo(4\))0 1983 y(that)26 b(w)m(e)g(extract)g(from)f (the)i(n)m(umerator)e(in)h(the)h(left)f(hand)h(side)g(of)f(\(5.3\).)40 b(T)-8 b(o)26 b(estimate)f(the)h(com)m(bina-)0 2123 y(torial)i(factor)h (that)f(app)s(ears)i(in)1255 2098 y Fd(b)1231 2123 y Fj(F)10 b Fo(,)29 b(see)h(\(3.10\),)e(w)m(e)h(use)h(the)g(Stirling)e (form)m(ula)g(in)h(the)h(form)e(giv)m(en)h(b)m(y)0 2262 y(Robbins)h([26])f(whic)m(h)h(is,)f Fj(8)p Fl(N)39 b Fj(\025)28 b Fo(1,)i Fl(N)11 b Fo(!)27 b(=)1613 2180 y Fj(p)p 1696 2180 111 4 v 82 x Fo(2)p Fl(\031)t(N)1898 2226 y Fk(N)7 b Fn(+)2041 2199 y Fg(1)p 2041 2210 34 4 v 2041 2250 a(2)2092 2262 y Fl(e)2138 2226 y Fp(\000)p Fk(N)2276 2262 y Fl(e)2322 2226 y Fk(\017)2355 2236 y Ff(N)2454 2262 y Fo(with)29 b(1)p Fl(=)p Fo(12)p Fl(N)38 b Fj(\024)28 b Fl(\017)3140 2277 y Fk(N)3244 2262 y Fj(\024)g Fo(1)p Fl(=)p Fo(\(12)p Fl(N)c Fo(+)14 b(1\).)0 2402 y(Let)33 b(us)h(denote)688 2665 y Fd(e)664 2690 y Fj(F)9 b Fo(\()p Fl(m)871 2649 y Fk(\016)910 2619 y Fb(\003)871 2714 y Fn(\001)936 2724 y Ff(R)998 2690 y Fo(\))28 b(=)1154 2622 y Fl(\016)1202 2586 y Fp(\003)p 1154 2667 94 4 v 1176 2758 a Fo(2)1508 2595 y Fd(X)1277 2817 y Fn(\()p Fk(x;y)s Fn(\))p Fp(2C)1551 2793 y Fg(2)1548 2847 y(1)p Ff(=\016)1654 2832 y Fb(\003)1699 2817 y Fn(\(\001)1795 2827 y Ff(R)1852 2817 y Fn(\))1900 2690 y Fl(J)1955 2705 y Fk(\016)1994 2685 y Fb(\003)2040 2690 y Fo(\()p Fl(x)22 b Fj(\000)h Fl(y)t Fo(\))2367 2579 y Fd(h)2431 2690 y Fo(~)-68 b Fl(m)2500 2649 y Fk(\016)2539 2619 y Fb(\003)2585 2690 y Fo(\()p Fl(x)p Fo(\))22 b Fj(\000)41 b Fo(~)-68 b Fl(m)2929 2649 y Fk(\016)2968 2619 y Fb(\003)3014 2690 y Fo(\()p Fl(y)t Fo(\))3145 2579 y Fd(i)3191 2600 y Fn(2)1164 3044 y Fo(+)23 b Fl(\016)1312 3003 y Fp(\003)1506 2949 y Fd(X)1375 3168 y Fk(x)p Fp(2C)1517 3182 y Ff(\016)1552 3167 y Fb(\003)1597 3168 y Fn(\(\001)1693 3178 y Ff(R)1750 3168 y Fn(\))1798 3044 y Fl(f)1847 3059 y Fk(\014)t(;\022)1964 3044 y Fo(\()p Fl(m)2090 3003 y Fk(\016)2129 2973 y Fb(\003)2176 3044 y Fo(\()p Fl(x)p Fo(\)\))3695 2899 y(\(5)p Fl(:)p Fo(5\))0 3421 y(where)38 b Fl(f)341 3436 y Fk(\014)t(;\022)496 3421 y Fo(is)f(the)g(canonical)h(free)g(energy)f(of)h(the)f(RF)m(CW)i (mo)s(del,)d(see)i(\(2.17\).)55 b(It)37 b(is)g(easy)g(to)g(see)0 3561 y(that)h(restricting)g(the)h(con\014gurations)f(to)g(those)h(that) f(are)g(constan)m(tly)g(equal)g(to)g Fl(m)3281 3525 y Fk(\016)3320 3494 y Fb(\003)3281 3589 y Fk(\014)3367 3561 y Fo(,)h(where)g Fl(m)3814 3525 y Fk(\016)3853 3494 y Fb(\003)3814 3589 y Fk(\014)0 3752 y Fo(is)33 b(the)g(nearest)h(p)s (oin)m(t)e(to)h Fl(m)1083 3767 y Fk(\014)1170 3752 y Fo(b)s(elonging)f(to)h(the)g(set)2068 3671 y Fd(\002)2109 3752 y Fj(\000)p Fo(1)p Fl(;)17 b Fj(\000)p Fo(1)22 b(+)2542 3708 y Fn(4)p Fk(\015)p 2542 3729 86 4 v 2545 3786 a(\016)2584 3766 y Fb(\003)2640 3752 y Fl(;)17 b Fj(\000)p Fo(1)22 b(+)2945 3708 y Fn(8)p Fk(\015)p 2945 3729 V 2948 3786 a(\016)2987 3766 y Fb(\003)3043 3752 y Fl(;)17 b(:)g(:)g(:)d(;)j Fo(1)k Fj(\000)3448 3708 y Fn(4)p Fk(\015)p 3448 3729 V 3451 3786 a(\016)3490 3766 y Fb(\003)3546 3752 y Fl(;)c Fo(1)3641 3671 y Fd(\003)3681 3687 y Fn(2)3726 3752 y Fo(,)33 b(w)m(e)0 3891 y(get)g(a)g(lo)m(w)m(er)g(b)s(ound)h(for)f(the)g (normalization)e(factor)i Fl(Z)2123 3906 y Fk(\014)t(;\022)r(;\015)t(;) p Fn(\001)2395 3916 y Ff(R)2456 3891 y Fo(\(0\))o(.)44 b(On)34 b(the)f(other)g(hand)h(using)g(the)0 4031 y(fact)f(that)1321 4157 y Fd(X)981 4381 y Fn(\()p Fk(m)1083 4361 y Ff(\016)1118 4346 y Fb(\003)1164 4381 y Fn(\()p Fp(\006)p Fn(1)p Fk(;x)p Fn(\)\))1428 4394 y Ff(x)p Fb(2C)1553 4415 y Ff(\016)1588 4400 y Fb(\003)1634 4394 y Fg(\(\001)1716 4411 y Ff(R)1772 4394 y Fg(\))1821 4251 y Fo(1)27 b Fj(\024)2004 4111 y Fd(\022)2095 4184 y Fl(\016)2143 4148 y Fp(\003)p 2089 4229 107 4 v 2089 4320 a Fo(2)p Fl(\015)2208 4111 y Fd(\023)2293 4105 y Fg(2)p Ff(R)p 2293 4116 86 4 v 2298 4158 a(\016)2333 4143 y Fb(\003)2423 4251 y Fo(=)h Fl(e)2586 4180 y Fg(2)p Ff(R)p 2586 4191 V 2591 4233 a(\016)2626 4218 y Fb(\003)2697 4207 y Fn(log)2826 4180 y Ff(\016)2861 4160 y Fb(\003)p 2826 4191 76 4 v 2827 4231 a Fg(2)p Ff(\015)3695 4251 y Fo(\(5)p Fl(:)p Fo(6\))0 4605 y(to)38 b(con)m(trol)f(the)i(n)m(um)m (b)s(er)f(of)h(terms)e(that)h(o)s(ccurs)h(in)f(the)g(sum)g(in)g (\(5.3\),)f(after)i(the)f(cancellation)g(of)0 4745 y(some)33 b(constan)m(ts)h(w)m(e)f(get)519 5030 y Fl(\026)579 5045 y Fk(\014)t(;\022)r(;\015)783 4920 y Fd(\020)842 5030 y Fj(O)924 4982 y Fk(\016)960 4992 y Fg(1)999 4982 y Fk(;\020)1059 4992 y Fg(1)921 5057 y Fn(0)1103 5030 y Fo(\(\001)1225 5045 y Fk(R)1290 5030 y Fo(\))1357 4946 y Fd(\014)1357 5005 y(\014)1417 5030 y Fo(\006)1489 5045 y Fk(@)t Fn(\001)1601 5055 y Ff(R)1663 4920 y Fd(\021)1739 5030 y Fo(\(0\))27 b Fj(\024)q Fl(e)2030 4953 y Ff(\014)p 2030 4970 42 4 v 2031 5010 a(\015)2084 4986 y Fn(\()p Fk(R\016)2214 4956 y Fb(\003)2256 4986 y Fn(+4+2)p Fk(R)2547 4953 y Ff(\015)p 2530 4970 76 4 v 2530 5012 a(\016)2565 4997 y Fb(\003)2631 4986 y Fn(log)2760 4959 y Ff(\016)2795 4939 y Fb(\003)p 2760 4970 V 2777 5010 a Ff(\015)2848 4986 y Fn(\))2884 5030 y Fo(\004\(4)p Fl(\014)5 b(\022)s(;)17 b Fo(\001)3278 5045 y Fk(R)3343 5030 y Fo(\))1972 5345 y Fl(e)2018 5242 y Fp(\000)2092 5209 y Ff(\014)p 2092 5227 42 4 v 2093 5266 a(\015)2160 5242 y Fn(inf)2257 5285 y Ff(m)2318 5268 y(\016)2353 5253 y Fb(\003)2318 5317 y Fg(\001)2373 5334 y Ff(R)2434 5285 y Fb(2O)2537 5253 y Ff(\016)2570 5268 y Fg(1)2608 5253 y Ff(;\020)2664 5268 y Fg(1)2535 5316 y(0)2713 5169 y Fd(\010)2771 5242 y Fp(F)7 b Fn(\()p Fk(m)2937 5212 y Ff(\016)2972 5192 y Fb(\003)2937 5262 y Fg(\001)2992 5279 y Ff(R)3053 5242 y Fn(\))3084 5169 y Fd(\011)3695 5173 y Fo(\(5)p Fl(:)p Fo(7\))0 5579 y Fg(23)p Ff(=j)s(anuar)r(y)r(=)p Fg(1998;)32 b(12:06)1145 b Fo(41)p eop %%Page: 42 42 42 41 bop 0 100 a Fo(where)34 b Fj(F)10 b Fo(\()p Fl(m)496 63 y Fk(\016)535 33 y Fb(\003)496 127 y Fn(\001)561 137 y Ff(R)623 100 y Fo(\))27 b Fj(\021)818 74 y Fd(e)794 100 y Fj(F)10 b Fo(\()p Fl(m)1002 63 y Fk(\016)1041 33 y Fb(\003)1002 127 y Fn(\001)1067 137 y Ff(R)1129 100 y Fo(\))22 b Fj(\000)1313 74 y Fd(e)1289 100 y Fj(F)10 b Fo(\()p Fl(m)1497 63 y Fk(\016)1536 33 y Fb(\003)1497 128 y Fk(\014)t(;)p Fn(\001)1635 138 y Ff(R)1696 100 y Fo(\).)120 240 y(T)-8 b(o)34 b(giv)m(e)h(a)g(lo)m(w)m(er)h(b)s(ound)g (on)f(the)h(previous)f(in\014m)m(um,)g(w)m(e)g(use)i(the)e(fact)h(that) f(if)g Fl(x)3328 255 y Fk(i)3396 240 y Fo(are)h(p)s(ositiv)m(e)0 380 y(n)m(um)m(b)s(ers,)e(b)s(ounded)h(from)e(ab)s(o)m(v)m(e)g(b)m(y)h (a)f(constan)m(t)h Fl(c)g Fo(then)g(if)f(the)h(arithmetic)e(mean)h(of)h Fl(N)44 b Fo(terms)33 b Fl(x)3867 395 y Fk(i)0 519 y Fo(is)j(b)s(ounded)i(from)d(b)s(elo)m(w)i(b)m(y)f(some)f Fl(\020)1474 534 y Fn(1)1551 519 y Fj(\024)e Fl(c)k Fo(then)f(there)h (are)f(at)g(least)2804 475 y Fk(N)7 b(\020)2910 485 y Fg(1)p 2770 496 213 4 v 2770 553 a Fn(2)p Fk(c)p Fp(\000)p Fk(\020)2944 563 y Fg(1)3031 519 y Fo(terms)35 b Fl(x)3368 534 y Fk(i)3438 519 y Fo(among)g(the)0 700 y Fl(N)11 b Fo(,)33 b(suc)m(h)i(that)d Fl(x)649 715 y Fk(i)710 700 y Fl(>)827 655 y Fk(\020)863 665 y Fg(1)p 827 677 76 4 v 845 734 a Fn(2)914 700 y Fo(.)44 b(Using)33 b(\(2.25\))f(w)m(e)i (get)f(after)g(some)g(easy)g(computations,)749 1004 y(inf)571 1094 y Fk(m)642 1070 y Ff(\016)677 1055 y Fb(\003)642 1120 y Fg(\001)697 1137 y Ff(R)758 1094 y Fp(2O)877 1055 y Ff(\016)910 1070 y Fg(1)948 1055 y Ff(;\020)1004 1070 y Fg(1)875 1118 y(0)1065 893 y Fd(n)1131 1004 y Fj(F)10 b Fo(\()p Fl(m)1339 963 y Fk(\016)1378 933 y Fb(\003)1339 1028 y Fn(\001)1404 1038 y Ff(R)1466 1004 y Fo(\))1505 893 y Fd(o)1598 1004 y Fj(\025)29 b Fl(R)q(c)p Fo(\()p Fl(\014)5 b(;)17 b(\022)s Fo(\))p Fl(\020)2109 963 y Fn(3)2102 1028 y(1)2152 1004 y Fl(\016)2403 936 y Fo(1)p 2212 981 432 4 v 2212 1072 a(4\(4)k Fj(\000)i Fl(\020)2516 1087 y Fn(1)2560 1072 y Fo(\))2599 1043 y Fn(2)2683 1004 y Fj(\025)28 b Fl(R)q(c)2908 1019 y Fn(1)2952 1004 y Fo(\()p Fl(\014)5 b(;)17 b(\022)s Fo(\))p Fl(\020)3237 963 y Fn(3)3230 1028 y(1)3281 1004 y Fl(\016)370 b Fo(\(5)p Fl(:)p Fo(8\))120 1360 y(It)36 b(remains)g(to)g(estimate)f(\004\(4)p Fl(\014)5 b(\022)s(;)17 b Fo(\001)1527 1375 y Fk(R)1592 1360 y Fo(\).)54 b(Let)37 b(us)g(denote)h Fl(X)8 b Fo(\(\001)2564 1375 y Fk(R)2628 1360 y Fo(\))33 b Fj(\021)h Fo(4)p Fl(\015)2934 1286 y Fd(P)3040 1390 y Fk(x)p Fp(2C)3182 1404 y Ff(\016)3217 1389 y Fb(\003)3263 1390 y Fn(\(\001)3359 1400 y Ff(R)3415 1390 y Fn(\))3468 1360 y Fj(j)p Fl(D)s Fo(\()p Fl(x)p Fo(\))p Fj(j)p Fo(.)53 b(It)0 1541 y(is)39 b(easy)g(to)g(see)h(that)f Fl(I)-17 b(E)5 b Fo(\()p Fl(X)j Fo(\(\001)1206 1556 y Fk(R)1270 1541 y Fo(\))38 b Fj(\024)g Fl(cR)1582 1455 y Fd(p)p 1681 1455 201 4 v 1681 1541 a Fl(\015)6 b(=\016)1837 1512 y Fp(\003)1882 1541 y Fo(.)62 b(Using)39 b(the)h(Lemma)d(4.2,)j (setting)f Fl(t)f Fo(=)g(2)3565 1466 y Fj(p)p 3648 1466 225 4 v 75 x Fl(R)q(\015)6 b(z)3829 1556 y Fn(1)3872 1541 y Fo(,)0 1680 y(where)34 b Fl(z)334 1695 y Fn(1)412 1680 y Fo(is)g(a)f(p)s(ositiv)m(e)f(real)h(n)m(um)m(b)s(er,)h(and)f (regrouping,)h(w)m(e)f(get)g(\(5.1\).)p 2902 1680 70 70 v 120 1931 a(With)23 b(the)h(Prop)s(osition)e(5.1)h(w)m(e)h(can)g (con)m(trol)f(the)h(Gibbs-Probabilit)m(y)f(to)g(ha)m(v)m(e)h(a)f(run)h (of)f Fl(\021)3538 1895 y Fk(\016)3574 1905 y Fg(1)3614 1895 y Fk(;\020)3674 1905 y Fg(1)3745 1931 y Fo(=)28 b(0)0 2070 y(an)m(ywhere)33 b(on)g(in)m(terv)-6 b(als)32 b(that)g(are)h(rather)f(long.)43 b(Ho)m(w)m(ev)m(er)33 b(their)f(lengths)h(dep)s(end)h(on)f(the)f(parame-)0 2210 y(ters)i Fl(\016)239 2225 y Fn(1)284 2210 y Fl(;)17 b(\020)373 2225 y Fn(1)416 2210 y Fl(;)g(\016)509 2173 y Fp(\003)554 2210 y Fo(.)0 2350 y Fm(Corollary)53 b(5.2)f Fe(Given)46 b Fl(\014)54 b(>)49 b Fo(1)d Fe(and)h Fl(\014)5 b(\022)50 b Fe(that)d(satis\014es)e(\(2.21\),)j(then)f(ther)-5 b(e)47 b(exists)f(a)g(c)-5 b(onstant)2 2490 y Fo(~)-52 b Fl(c)28 b Fo(=)i(~)-52 b Fl(c)p Fo(\()p Fl(\014)5 b(;)17 b(\022)s Fo(\))35 b Fe(such)g(that,)g(if)g Fl(\016)1093 2453 y Fp(\003)1156 2490 y Fo(log)1316 2450 y Fn(1)p 1313 2467 47 4 v 1313 2524 a Fk(\015)1399 2490 y Fj(#)27 b Fo(0)36 b Fe(when)f Fl(\015)e Fj(#)27 b Fo(0)p Fe(,)35 b(for)h(al)5 b(l)36 b Fl(\016)2440 2505 y Fn(1)2513 2490 y Fl(>)28 b(\016)2666 2453 y Fp(\003)2740 2490 y Fl(>)g Fo(0)p Fe(,)35 b Fl(\020)3005 2505 y Fn(1)3077 2490 y Fl(>)28 b Fo(0)p Fe(,)35 b(that)g(satisfy)1361 2829 y Fl(\016)1405 2844 y Fn(1)1450 2829 y Fl(\020)1501 2788 y Fn(3)1494 2854 y(1)1573 2829 y Fj(\025)30 b Fo(~)-52 b Fl(c)p Fo(\()p Fl(\014)5 b(;)17 b(\022)s Fo(\))1972 2689 y Fd(\022)2044 2684 y(r)p 2144 2684 118 4 v 2174 2762 a Fl(\015)p 2156 2806 94 4 v 2156 2898 a(\016)2204 2869 y Fp(\003)2284 2829 y Fj(_)22 b Fl(\016)2420 2788 y Fp(\003)2466 2689 y Fd(\023)3695 2829 y Fo(\(5)p Fl(:)p Fo(9\))0 3131 y Fe(for)37 b(al)5 b(l)39 b Fl(x)32 b(>)g Fo(0)p Fe(,)38 b(for)f(al)5 b(l)39 b(intervals)e Fo(\001)1420 3146 y Fk(R)1524 3131 y Fe(of)g(macr)-5 b(osc)g(opic)37 b(length)h Fl(R)g Fe(that)g(ar)-5 b(e)37 b(include)-5 b(d)38 b(in)f(a)h(macr)-5 b(o-)0 3271 y(sc)g(opic)34 b(interval)i Fl(I)43 b Fe(c)-5 b(ontaining)34 b(the)i(origin,)e(with)h Fj(j)p Fl(I)8 b Fj(j)27 b(\024)h Fl(\015)2236 3235 y Fp(\000)p Fn(2)2378 3271 y Fe(and)35 b(satisfy)1477 3576 y Fl(R)28 b Fj(\025)g Fl(R)1762 3591 y Fn(1)1834 3576 y Fj(\021)1972 3508 y Fo(4)p Fl(\014)5 b Fo(\(1)23 b(+)f Fl(x)p Fo(\))p 1951 3553 461 4 v 1951 3644 a Fl(c)p Fo(\()p Fl(\014)5 b(;)17 b(\022)s Fo(\))p Fl(\016)2272 3659 y Fn(1)2316 3644 y Fl(\020)2367 3610 y Fn(3)2360 3671 y(1)3645 3576 y Fo(\(5)p Fl(:)p Fo(10\))0 3888 y Fe(if)35 b Fl(\015)e Fo(=)28 b(2)337 3852 y Fp(\000)p Fk(n)453 3888 y Fe(,)35 b(with)h Fl(I)-18 b(P)14 b Fe(-pr)-5 b(ob)g(ability)34 b(one,)h(for)g(al)5 b(l)37 b(but)f(a)f(\014nite)h(numb)-5 b(er)36 b(of)f(indic)-5 b(es)34 b Fl(n)p Fe(,)829 4126 y Fl(\026)889 4141 y Fk(\014)t(;\022)r(;\015)1093 4015 y Fd(\020)1153 4126 y Fj(9)1208 4144 y Fk(R)p Fn(:)p Fk(R)1351 4154 y Fg(1)1390 4144 y Fp(\024j)p Fk(R)p Fp(j\024j)p Fk(I)5 b Fp(j)1743 4126 y Fj(9)1798 4141 y Fn(\001)1863 4151 y Ff(R)1920 4141 y Fp(\032)p Fk(I)2083 4126 y Fo(:)28 b Fj(O)2221 4078 y Fk(\016)2257 4088 y Fg(1)2296 4078 y Fk(;\020)2356 4088 y Fg(1)2218 4152 y Fn(0)2400 4126 y Fo(\(\001)2522 4141 y Fk(R)2587 4126 y Fo(\))2626 4015 y Fd(\021)2713 4126 y Fj(\024)g Fl(e)2864 4081 y Fp(\000)2938 4048 y Fg(4)p Ff(\014)s(x)p 2938 4066 116 4 v 2976 4105 a(\015)3645 4126 y Fo(\(5)p Fl(:)p Fo(11\))0 4504 y Fa(Pro)s(of:)120 4645 y Fo(Let)39 b(us)i(\014rst)e(remark)g(that)g(for)h(a)f(giv)m(en)g Fl(R)q Fo(,)i(the)e(n)m(um)m(b)s(er)i(of)e(in)m(terv)-6 b(als)40 b(\001)3035 4660 y Fk(R)3140 4645 y Fo(that)f(are)h(included)0 4784 y(in)e Fl(I)8 b Fo(,)37 b(is)h(b)s(ounded)h(from)e(ab)s(o)m(v)m(e) g(b)m(y)h Fj(j)p Fl(I)8 b Fj(j)1531 4748 y Fn(2)1575 4784 y Fo(,)38 b(therefore)h(if)e(w)m(e)h(tak)m(e)f Fl(z)2576 4799 y Fn(1)2656 4784 y Fo(=)2769 4676 y Fd(q)p 2868 4676 622 4 v 2868 4784 a Fo(64\(5)22 b(+)g Fl(\017)p Fo(\))17 b(log)3434 4745 y Fn(1)p 3431 4761 47 4 v 3431 4818 a Fk(\015)3527 4784 y Fo(for)38 b(some)0 4924 y(p)s(ositiv)m(e)33 b Fl(\017)p Fo(,)g(w)m(e)h(get)f(easily)63 5255 y Fl(I)-17 b(P)185 5085 y Fd(")419 5255 y Fo(sup)243 5344 y Fk(R)p Fn(:)p Fk(R)386 5354 y Fg(1)424 5344 y Fp(\024j)p Fk(R)p Fp(j\024j)p Fk(I)5 b Fp(j)799 5255 y Fo(sup)761 5339 y Fn(\001)826 5349 y Ff(R)883 5339 y Fp(\032)p Fk(I)1070 5188 y Fo(1)p 1015 5232 160 4 v 1015 5252 a Fj(p)p 1098 5252 77 4 v 85 x Fl(R)1203 5255 y Fo(\()o Fl(X)j Fo(\(\001)1454 5270 y Fk(R)1519 5255 y Fo(\))22 b Fj(\000)g Fl(I)-17 b(E)22 b Fo([)p Fl(X)8 b Fo(\(\001)2043 5270 y Fk(R)2107 5255 y Fo(\)]\))27 b Fj(\025)2345 5108 y Fd(r)p 2445 5108 751 4 v 147 x Fo(64\(5)21 b(+)i Fl(\017)p Fo(\))p Fl(\015)f Fo(log\()3091 5188 y(1)p 3087 5232 58 4 v 3087 5324 a Fl(\015)3156 5255 y Fo(\))3195 5085 y Fd(#)3281 5255 y Fj(\024)28 b Fl(\015)3444 5214 y Fn(1+)p Fk(\017)3645 5255 y Fo(\(5)p Fl(:)p Fo(12\))0 5579 y Fg(23)p Ff(=j)s(anuar)r(y)r(=)p Fg(1998;)k(12:06)1145 b Fo(42)p eop %%Page: 43 43 43 42 bop 0 100 a Fo(and)43 b(the)f Fl(I)-17 b(P)14 b Fo(-probabilistic)42 b(statemen)m(t)g(follo)m(ws)g(from)g(the)g (\014rst)h(Borel-Can)m(telli)e(Lemma.)70 b(Let)42 b(us)0 239 y(consider)35 b(the)f(term)f(in)m(to)h(the)g(brac)m(k)m(et)h(in)e (the)i(exp)s(onen)m(t)f(in)g(the)h(righ)m(t)e(hand)i(side)g(of)f (\(5.1\).)44 b(Notice)0 379 y(\014rst)34 b(that,)f(since)707 339 y Fk(\016)746 309 y Fb(\003)p 707 356 81 4 v 724 413 a Fk(\015)827 379 y Fj(")c(1)k Fo(when)h Fl(\015)g Fj(#)28 b Fo(0,)1574 298 y Fd(p)p 1674 298 105 4 v 1703 334 a Fk(\015)p 1686 356 81 4 v 1686 413 a(\016)1725 393 y Fb(\003)1807 379 y Fj(\025)1942 334 y Fk(\015)p 1925 356 V 1925 413 a(\016)1964 393 y Fb(\003)2034 379 y Fo(log)2191 339 y Fk(\016)2230 309 y Fb(\003)p 2191 356 V 2208 413 a Fk(\015)2283 379 y Fo(,)34 b(if)f Fl(\015)39 b Fo(small)32 b(enough)j(w)m(e)f(can)g(ignore)g(the)0 518 y(corresp)s(onding)h(term)e(in)g(\(5.1\))g(and)h(k)m(eep)g(just)h (the)f(square)g(ro)s(ot.)44 b(T)-8 b(o)33 b(get)h(a)f(negativ)m(e)h (term)e(in)i(this)0 658 y(exp)s(onen)m(t,)g(w)m(e)f(imp)s(ose,)f(since) i Fl(\014)5 b(\022)37 b Fo(is)c(small,)757 1002 y Fl(c)p Fo(\()p Fl(\014)5 b(;)17 b(\022)s Fo(\))p Fl(\020)1085 961 y Fn(3)1078 1027 y(1)1129 1002 y Fl(\016)1173 1017 y Fn(1)1240 1002 y Fj(\000)22 b Fo(4\()p Fl(\016)1476 961 y Fp(\003)1544 1002 y Fj(_)1633 856 y Fd(r)p 1732 856 118 4 v 1763 935 a Fl(\015)p 1744 979 94 4 v 1744 1071 a(\016)1792 1042 y Fp(\003)1850 1002 y Fo(\))g Fj(\000)g Fo(256\(5)g(+)g Fl(\017)p Fo(\))p Fl(\014)5 b(\022)2560 855 y Fd(r)p 2660 855 301 4 v 147 x Fl(\015)22 b Fo(log)2895 935 y(1)p 2891 979 58 4 v 2891 1071 a Fl(\015)2988 1002 y Fj(\025)28 b Fo(0)502 b(\(5)p Fl(:)p Fo(13\))0 1350 y(Using)36 b Fl(\016)331 1314 y Fp(\003)394 1350 y Fo(log)554 1311 y Fn(1)p 551 1328 47 4 v 551 1385 a Fk(\015)642 1350 y Fj(#)d Fo(0)j(when)i Fl(\015)g Fj(#)33 b Fo(0,)k(this)f(b)s (ecomes)h Fl(c)p Fo(\()p Fl(\014)5 b(;)17 b(\022)s Fo(\))p Fl(\020)2288 1314 y Fn(3)2281 1375 y(1)2331 1350 y Fl(\016)2375 1365 y Fn(1)2445 1350 y Fj(\000)24 b Fo(4)p Fl(c)p Fo(\()p Fl(\016)2726 1314 y Fp(\003)2796 1350 y Fj(_)2887 1270 y Fd(p)p 2987 1270 105 4 v 3016 1306 a Fk(\015)p 2999 1328 81 4 v 2999 1385 a(\016)3038 1365 y Fb(\003)3091 1350 y Fo(\))33 b Fj(\025)g Fo(0)k(b)m(y)f(enlarging)0 1490 y(the)f(constan)m(t)g Fl(c)g Fo(if)g(necessary)-8 b(.)50 b(T)-8 b(o)34 b(cancel)h(the)g(constan)m(t)h(term)e(4)p Fl(\014)5 b Fo(,)35 b(in)g(\(5.1\))e(and)i(get)g(the)g(factor)g Fl(x)0 1629 y Fo(in)e(\(5.11\))f(w)m(e)i(just)g(imp)s(ose)e(\(5.10\).)p 1410 1629 70 70 v 120 1908 a(The)44 b(second)i(family)c(of)i(ev)m(en)m (ts)h(w)m(e)f(consider)i(are)e(roughly)f(sp)s(eaking)h(those)h(one)f (ha)m(ving)g(t)m(w)m(o)0 2048 y(blo)s(c)m(ks,)30 b(far)f(apart)h(but)f (not)h(to)s(o)e(m)m(uc)m(h,)j(at)e(the)g(same)g(equilibrium)f(and)i (somewhere)g(b)s(et)m(w)m(een)h(them)0 2187 y(there)j(is)f(a)g(blo)s(c) m(k)g(of)g(macroscopic)g(length)h(at)e(least)h(1,)g(close)h(to)f(the)g (other)g(equilibrium.)120 2331 y(Let)38 b(\001)386 2346 y Fk(L)481 2331 y Fo(=)f([)p Fl(`)665 2346 y Fn(1)709 2331 y Fl(;)17 b(`)796 2346 y Fn(2)839 2331 y Fo(])38 b(with)g Fl(`)1179 2346 y Fk(i)1247 2331 y Fj(2)f Fl(Z)-35 b(Z)45 b Fo(for)38 b Fl(i)e Fo(=)h(1)p Fl(;)17 b Fo(2)37 b(b)s(e)h(a)g(macroscopic)g(in)m(terv)-6 b(al)38 b(of)g(length)h Fl(L)p Fo(,)f(and)0 2471 y Fl(\016)44 2486 y Fn(2)117 2471 y Fl(>)28 b Fo(0)f Fl(;)44 b(\020)415 2486 y Fn(2)487 2471 y Fl(>)28 b Fo(0)33 b(b)s(e)h(t)m(w)m(o)f(real)g(p)s(ositiv)m(e)g (n)m(um)m(b)s(ers,)h(let)e(us)i(de\014ne)h(for)f Fl(\021)d Fo(=)d(+1)34 b(or)f Fl(\021)f Fo(=)c Fj(\000)p Fo(1)116 2741 y Fj(W)222 2700 y Fk(\016)258 2710 y Fg(2)298 2700 y Fk(;\020)358 2710 y Fg(2)402 2741 y Fo(\(\001)524 2756 y Fk(L)583 2741 y Fl(;)17 b(\021)t Fo(\))27 b Fj(\021)852 2630 y Fd(n)918 2741 y Fl(\021)971 2700 y Fk(\016)1007 2710 y Fg(2)1047 2700 y Fk(;\020)1107 2710 y Fg(2)1150 2741 y Fo(\()p Fl(`)1231 2756 y Fn(1)1275 2741 y Fo(\))h(=)g Fl(\021)1500 2700 y Fk(\016)1536 2710 y Fg(2)1575 2700 y Fk(;\020)1635 2710 y Fg(2)1679 2741 y Fo(\()p Fl(`)1760 2756 y Fn(2)1804 2741 y Fo(\))f(=)h Fl(\021)t(;)44 b Fj(9)2162 2715 y Fo(~)2155 2741 y Fl(`;)g(`)2311 2756 y Fn(1)2383 2741 y Fl(<)2495 2715 y Fo(~)2488 2741 y Fl(`)27 b(<)i(`)2705 2756 y Fn(2)2848 2741 y Fl(\021)2901 2700 y Fk(\016)2937 2710 y Fg(2)2977 2700 y Fk(;\020)3037 2710 y Fg(2)3080 2741 y Fo(\()3126 2715 y(~)3119 2741 y Fl(`)p Fo(\))e(=)h Fj(\000)p Fl(\021)3462 2630 y Fd(o)3645 2741 y Fo(\(5)p Fl(:)p Fo(14\))0 3061 y(and)34 b Fj(W)300 3025 y Fk(\016)336 3035 y Fg(2)375 3025 y Fk(;\020)435 3035 y Fg(2)479 3061 y Fo(\(\001)601 3076 y Fk(L)661 3061 y Fo(\))27 b Fj(\021)h(W)938 3025 y Fk(\016)974 3035 y Fg(2)1014 3025 y Fk(;\020)1074 3035 y Fg(2)1118 3061 y Fo(\(\001)1240 3076 y Fk(L)1299 3061 y Fl(;)17 b Fo(+\))22 b Fj([)g(W)1676 3025 y Fk(\016)1712 3035 y Fg(2)1752 3025 y Fk(;\020)1812 3035 y Fg(2)1856 3061 y Fo(\(\001)1978 3076 y Fk(L)2037 3061 y Fl(;)17 b Fj(\000)p Fo(\).)44 b(Our)34 b(second)g(result)g(is)0 3205 y Fm(Prop)s(osition)50 b(5.3)g Fe(Given)45 b Fl(\014)50 b(>)45 b Fo(1)g Fe(and)f Fl(\014)5 b(\022)49 b Fe(that)c(satis\014es)e(\(2.21\),)j Fl(\016)2793 3220 y Fn(2)2882 3205 y Fl(>)f(\016)3052 3169 y Fp(\003)3143 3205 y Fl(>)g Fo(0)p Fe(,)h Fl(\020)3436 3220 y Fn(2)3525 3205 y Fl(>)f Fo(0)g Fe(and)0 3345 y Fl(z)46 3360 y Fn(2)119 3345 y Fl(>)28 b Fo(0)p Fe(,)35 b(then)g(ther)-5 b(e)36 b(exists)e Fo(\012)1154 3360 y Fn(2)1227 3345 y Fo(=)28 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2422 3641 a Ff(\015)2493 3617 y Fn(\))p Fp(\000)p Fn(2)p Fk(L\014)t(\022)2770 3553 y Fj(p)p 2852 3553 100 4 v 2882 3584 a Ff(\015)p 2864 3602 76 4 v 2864 3644 a(\016)2899 3629 y Fb(\003)2952 3617 y Fp(\000)3014 3550 y Fj(p)p 3097 3550 101 4 v 67 x Fk(L\015)t Fn(4)p Fk(\014)t(\022)r(z)3365 3627 y Fg(2)3404 3544 y Fd(\003)3645 3670 y Fo(\(5)p Fl(:)p Fo(15\))0 4016 y Fe(for)35 b(a)h(strictly)f(p)-5 b(ositive)34 b(c)-5 b(onstant)36 b Fo(\001)p Fj(F)h Fo(=)28 b(\001)p Fj(F)10 b Fo(\()p Fl(\014)5 b(;)17 b(\022)s Fo(\))34 b Fe(and)i Fl(I)-18 b(P)14 b Fo([\012)2465 4031 y Fn(1)2509 4016 y Fo(])27 b Fj(\025)h Fo(1)22 b Fj(\000)h Fl(e)2887 3980 y Fp(\000)2962 3928 y Ff(z)2998 3908 y Fg(2)2996 3956 y(2)p 2962 3964 76 4 v 2966 4004 a(64)3089 4016 y Fe(.)0 4160 y Fa(Pro)s(of:)60 b Fo(The)38 b(pro)s(of)f(is)g(similar)e(to)h (the)i(pro)s(of)f(of)g(the)g(Prop)s(osition)f(\(5.1\),)g(w)m(e)i(p)s (oin)m(t)f(out)f(only)h(the)0 4299 y(main)32 b(di\013erences.)46 b(Let)33 b(us)h(call)f(\001)1345 4257 y 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Fl(;)17 b(\021)t Fo(\))2544 5255 y Fd(\014)2544 5315 y(\014)2605 5340 y Fo(\006)2677 5355 y Fk(@)t Fn(\001)2789 5365 y Ff(L)2847 5259 y Fd(\001)2909 5340 y Fo(\()p Fl(m)3035 5355 y Fk(\014)t(;\021)s(;@)t Fn(\001)3288 5365 y Ff(L)3345 5340 y Fo(\))261 b(\(5)p Fl(:)p Fo(17\))0 5579 y Fg(23)p Ff(=j)s(anuar)r(y)r(=)p Fg(1998;)32 b(12:06)1145 b Fo(43)p eop %%Page: 44 44 44 43 bop 0 100 a Fo(Then)34 b(making)c(a)j(blo)s(c)m(k)f(spin)h (transformation)f(on)h(the)g(scale)g Fl(\016)2439 63 y Fp(\003)2517 100 y Fo(inside)g(the)g(v)m(olume)f(\001)3396 57 y Fp(\000)3396 129 y Fk(L)3463 100 y Fo(,)h(denoting)328 422 y Fj(F)10 b Fo(\()p Fl(m)536 381 y Fk(\016)575 351 y Fb(\003)536 463 y Fn(\001)601 430 y Fb(\000)601 489 y Ff(L)665 422 y Fl(;)17 b(m)797 381 y Fk(\016)836 351 y Fb(\003)797 447 y Fk(@)t Fn(\001)909 457 y Ff(L)966 422 y Fo(\))28 b(=)g Fj(F)10 b Fo(\()p Fl(m)1346 381 y Fk(\016)1385 351 y Fb(\003)1346 463 y Fn(\001)1411 430 y Fb(\000)1411 489 y Ff(L)1475 422 y Fo(\))22 b(+)1648 355 y 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Fl(\015)1002 3999 y Fp(\000)p Fn(2)1143 4035 y Fe(and)36 b(satisfy)1237 4359 y Fl(L)27 b Fj(\024)h Fl(L)1505 4374 y Fn(2)1578 4359 y Fj(\021)1695 4292 y Fo(\001)p Fj(F)10 b Fo(\(1)21 b Fj(\000)i Fl(x)p Fo(\))e Fj(\000)k Fo(~)-52 b Fl(c)p Fo(\()p Fl(\014)5 b(;)17 b(\022)s Fo(\))p Fl(\020)2608 4307 y Fn(2)p 1695 4336 957 4 v 1792 4437 a Fl(c)p Fo(\()p Fl(\014)5 b(;)17 b(\022)s Fo(\)\()p Fl(\016)2156 4408 y Fp(\003)2223 4437 y Fj(_)2311 4356 y Fd(p)p 2411 4356 105 4 v 2440 4392 a Fk(\015)p 2423 4414 81 4 v 2423 4471 a(\016)2462 4451 y Fb(\003)2515 4437 y Fo(\))3645 4359 y(\(5)p Fl(:)p Fo(22\))0 4708 y Fe(if)35 b Fl(\015)e Fo(=)28 b(2)337 4672 y Fp(\000)p Fk(n)453 4708 y Fe(,)35 b(with)h Fl(I)-18 b(P)14 b Fe(-pr)-5 b(ob)g(ability)34 b(one,)h(for)g(al)5 b(l)37 b(but)f(a)f(\014nite)h(numb)-5 b(er)36 b(of)f(indic)-5 b(es)34 b Fl(n)p Fo(,)831 4964 y Fl(\026)891 4979 y Fk(\014)t(;\022)r(;\015)1095 4884 y Fd(\000)1140 4964 y Fj(9)1195 4982 y Fk(L)p Fn(:2)p Fp(\024j)p Fk(L)p Fp(j\024)p Fk(L)1595 4992 y Fg(2)1665 4964 y Fj(9)1720 4979 y Fn(\001)1785 4989 y Ff(L)1839 4979 y Fp(\032)p Fk(I)2002 4964 y Fo(:)27 b Fj(W)2163 4923 y Fk(\016)2199 4933 y Fg(2)2239 4923 y Fk(;\020)2299 4933 y Fg(2)2343 4964 y Fo(\(\001)2465 4979 y Fk(L)2524 4964 y Fo(\))2563 4884 y Fd(\001)2636 4964 y Fj(\024)h Fl(e)2787 4920 y Fp(\000)2862 4887 y Ff(\014)s(x)p Fg(\001)p Fb(F)p 2862 4905 191 4 v 2937 4944 a Ff(\015)3645 4964 y Fo(\(5)p Fl(:)p Fo(23\))120 5340 y(The)33 b(third)h(family)d(of)j(ev) m(en)m(ts)g(describ)s(es)h(\015uctuations)f(around)f(one)h (equilibrium.)0 5579 y Fg(23)p Ff(=j)s(anuar)r(y)r(=)p Fg(1998;)e(12:06)1145 b Fo(44)p eop %%Page: 45 45 45 44 bop 120 100 a Fo(Let)38 b(\001)386 115 y Fk(L)481 100 y Fo(=)f([)p Fl(`)665 115 y Fn(1)709 100 y Fl(;)17 b(`)796 115 y Fn(2)839 100 y Fo(])38 b(with)g Fl(`)1179 115 y Fk(i)1247 100 y Fj(2)f Fl(Z)-35 b(Z)45 b Fo(for)38 b Fl(i)e Fo(=)h(1)p Fl(;)17 b Fo(2)37 b(b)s(e)h(a)g(macroscopic)g(in)m (terv)-6 b(al)38 b(of)g(length)h Fl(L)p Fo(,)f(and)0 239 y Fl(\016)44 254 y Fn(4)117 239 y Fl(>)28 b(\016)266 254 y Fn(1)338 239 y Fl(>)h Fo(0)e Fl(;)44 b(\020)637 254 y Fn(4)709 239 y Fl(>)28 b(\020)858 254 y Fn(1)930 239 y Fl(>)g Fo(0)h(b)s(e)h(four)g(real)f(p)s(ositiv)m(e)f(n)m(um)m(b)s (ers,)j(let)d(us)i(de\014ne)h(for)e Fl(\021)j Fo(=)c(+1)i(or)f Fl(\021)j Fo(=)c Fj(\000)p Fo(1)248 501 y Fj(R)332 453 y Fk(\016)368 463 y Fg(1)407 453 y Fk(;\020)467 463 y Fg(1)506 453 y Fk(;\016)566 463 y Fg(4)605 453 y Fk(;\020)665 463 y Fg(4)332 527 y Fn(0)p Fk(;\021)709 501 y Fo(\(\001)831 516 y Fk(L)890 501 y Fl(;)946 476 y Fo(~)935 501 y Fl(L)o Fo(\))f Fj(\021)248 589 y Fd(\010)306 670 y Fl(\021)359 629 y Fk(\016)395 639 y Fg(1)434 629 y Fk(;\020)494 639 y Fg(1)538 670 y Fo(\()p Fl(`)619 685 y Fn(1)663 670 y Fo(\))g(=)h Fl(\021)887 629 y Fk(\016)923 639 y Fg(1)962 629 y Fk(;\020)1022 639 y Fg(1)1066 670 y Fo(\()p Fl(`)1147 685 y Fn(2)1191 670 y Fo(\))g(=)g Fl(\021)t(;)44 b Fj(8)p Fl(`)27 b Fj(2)h Fo(\()p Fl(`)1787 685 y Fn(1)1831 670 y Fl(;)17 b(`)1918 685 y Fn(2)1962 670 y Fo(\))p Fl(;)43 b(\021)2125 629 y Fk(\016)2161 639 y Fg(1)2201 629 y Fk(;\020)2261 639 y Fg(1)2332 670 y Fo(=)28 b(0)p Fl(;)17 b Fj(9)2594 644 y Fo(~)2587 670 y Fl(`)2629 685 y Fn(1)2673 670 y Fl(;)2724 644 y Fo(~)2718 670 y Fl(`)2760 685 y Fn(2)2803 670 y Fl(;)2882 644 y Fo(~)2875 670 y Fl(`)2917 685 y Fn(2)2984 670 y Fj(\000)3090 644 y Fo(~)3083 670 y Fl(`)3125 685 y Fn(1)3197 670 y Fo(=)3314 645 y(~)3302 670 y Fl(L)347 840 y(`)389 855 y Fn(1)461 840 y Fl(<)573 813 y Fo(~)566 840 y Fl(`)608 855 y Fn(1)680 840 y Fl(<)792 813 y Fo(~)785 840 y Fl(`)827 855 y Fn(2)899 840 y Fj(\024)28 b Fl(`)1046 855 y Fn(2)1146 840 y Fl(\021)1199 798 y Fk(\016)1235 808 y Fg(4)1274 798 y Fk(;\020)1334 808 y Fg(4)1378 840 y Fo(\()1424 813 y(~)1417 840 y Fl(`)o Fo(\))g(=)g(0)f Fj(8)1769 813 y Fo(~)1762 840 y Fl(`)h Fo(:)1922 813 y(~)1915 840 y Fl(`)1957 855 y Fn(1)2029 840 y Fj(\024)2141 813 y Fo(~)2134 840 y Fl(`)f Fj(\024)2315 813 y Fo(~)2308 840 y Fl(`)2350 855 y Fn(2)2394 759 y Fd(\011)3645 664 y Fo(\(5)p Fl(:)p Fo(24\))0 1117 y(and)34 b Fj(R)278 1069 y Fk(\016)314 1079 y Fg(1)353 1069 y Fk(;\020)413 1079 y Fg(1)452 1069 y Fk(;\016)512 1079 y Fg(4)551 1069 y Fk(;\020)611 1079 y Fg(4)278 1143 y Fn(0)655 1117 y Fo(\(\001)777 1132 y Fk(L)836 1117 y Fl(;)892 1092 y Fo(~)881 1117 y Fl(L)o Fo(\))27 b Fj(\021)i(R)1204 1069 y Fk(\016)1240 1079 y Fg(1)1279 1069 y Fk(;\020)1339 1079 y Fg(1)1378 1069 y Fk(;\016)1438 1079 y Fg(4)1477 1069 y Fk(;\020)1537 1079 y Fg(4)1204 1143 y Fn(0)p Fk(;)p Fn(+)1580 1117 y Fo(\(\001)1702 1132 y Fk(L)1762 1117 y Fl(;)1818 1092 y Fo(~)1807 1117 y Fl(L)o Fo(\))22 b Fj([)g(R)2107 1069 y Fk(\016)2143 1079 y Fg(1)2183 1069 y Fk(;\020)2243 1079 y Fg(1)2282 1069 y Fk(;\016)2342 1079 y Fg(4)2380 1069 y Fk(;\020)2440 1079 y Fg(4)2107 1143 y Fn(0)p Fk(;)p Fp(\000)2484 1117 y Fo(\(\001)2606 1132 y Fk(L)2666 1117 y Fl(;)2722 1092 y Fo(~)2711 1117 y Fl(L)o Fo(\).)0 1399 y Fm(Prop)s(osition)44 b(5.5)h Fe(Given)39 b Fl(\014)j(>)37 b Fo(1)j Fe(and)g Fl(\014)5 b(\022)43 b Fe(that)e(satis\014es)e(\(2.21\),)g Fl(\016)2730 1414 y Fn(4)2811 1399 y Fl(>)e(\016)2969 1414 y Fn(1)3050 1399 y Fl(>)g(\016)3212 1363 y Fp(\003)3294 1399 y Fl(;)52 b(\020)3418 1414 y Fn(4)3499 1399 y Fl(>)36 b(\020)3656 1414 y Fn(1)3737 1399 y Fl(>)g Fo(0)0 1538 y Fe(and)f Fl(z)239 1553 y Fn(3)312 1538 y Fl(>)28 b Fo(0)36 b Fe(then)f(ther)-5 b(e)36 b(exists)e Fo(\012)1317 1553 y Fn(3)1390 1538 y Fo(=)28 b(\012)1567 1553 y Fn(3)1611 1538 y Fo(\()p Fl(\015)6 b(;)17 b(\016)1801 1502 y Fp(\003)1845 1538 y Fl(;)g(\016)1934 1553 y Fn(1)1979 1538 y Fl(;)g(\016)2068 1553 y Fn(4)2112 1538 y Fl(;)g(\020)2201 1553 y Fn(1)2245 1538 y Fl(;)g(\020)2334 1553 y Fn(4)2377 1538 y Fl(;)g(z)2468 1553 y Fn(3)2512 1538 y Fl(;)g Fo(\001)2640 1553 y Fk(L)2699 1538 y Fl(;)2755 1513 y Fo(~)2744 1538 y Fl(L)o Fo(\))36 b Fe(such)f(that)h(on)g Fo(\012)3531 1553 y Fn(3)225 1824 y Fl(\026)285 1839 y Fk(\014)t(;\022)r(;\015)488 1713 y Fd(\020)548 1824 y Fj(R)632 1776 y Fk(\016)668 1786 y Fg(1)708 1776 y Fk(;\020)768 1786 y Fg(1)806 1776 y Fk(;\016)866 1786 y Fg(4)905 1776 y Fk(;\020)965 1786 y Fg(4)632 1850 y Fn(0)p Fk(;\021)1009 1824 y Fo(\(\001)1131 1839 y Fk(L)1190 1824 y Fl(;)1246 1799 y Fo(~)1235 1824 y Fl(L)o Fo(\))1341 1713 y Fd(\021)252 2089 y Fj(\024)28 b Fl(e)403 2036 y Fp(\000)p Fk(\015)511 2005 y Fb(\000)p Fg(1)605 1962 y Fd(\002)647 2036 y Fk(c)p Fn(\()p Fk(\014)t(;\022)r Fn(\)\()p Fk(\020)930 2005 y Fg(3)925 2055 y(4)968 2036 y Fk(\016)1004 2046 y Fg(4)1052 2018 y Fn(~)1043 2036 y Fk(L)o Fn(+)p Fk(\020)1199 2005 y Fg(3)1194 2055 y(1)1238 2036 y Fk(\016)1274 2046 y Fg(1)1313 2036 y Fn(\()p Fk(L)p Fp(\000)1471 2018 y Fn(~)1461 2036 y Fk(L)p Fn(\)\))p Fp(\000)p Fn(4)p Fk(\014)t(\020)1765 2046 y Fg(1)1804 2036 y Fp(\000)p Fn(2)p Fk(L)p Fn(\()p Fk(\016)2031 2005 y Fb(\003)2072 2036 y Fn(+)2163 2002 y Ff(\015)p 2146 2020 76 4 v 2146 2062 a(\016)2181 2047 y Fb(\003)2247 2036 y Fn(log)2376 2009 y Ff(\016)2411 1989 y Fb(\003)p 2376 2020 V 2393 2060 a Ff(\015)2463 2036 y Fn(\))p Fp(\000)p Fn(2)p Fk(L\014)t(\022)2740 1971 y Fj(p)p 2823 1971 100 4 v 2853 2002 a Ff(\015)p 2835 2020 76 4 v 2835 2062 a(\016)2870 2047 y Fb(\003)2923 2036 y Fp(\000)2985 1968 y Fj(p)p 3068 1968 101 4 v 68 x Fk(L\015)t Fn(4)p Fk(\014)t(\022)r(z) 3336 2046 y Fg(3)3374 1962 y Fd(\003)3645 1941 y Fo(\(5)p Fl(:)p Fo(25\))0 2409 y Fe(for)35 b(some)h(p)-5 b(ositive)34 b(c)-5 b(onstants)35 b Fl(c)p Fo(\()p Fl(\014)5 b(;)17 b(\022)s Fo(\))34 b Fe(and)i Fl(c)g Fe(and)f Fl(P)30 b Fo([\012)2182 2424 y Fn(3)2227 2409 y Fo(])d Fj(\025)h Fo(1)22 b Fj(\000)h Fl(e)2605 2373 y Fp(\000)2679 2322 y Ff(z)2715 2302 y Fg(2)2713 2350 y(3)p 2679 2358 76 4 v 2683 2397 a(64)2807 2409 y Fe(.)0 2550 y Fa(Pro)s(of:)52 b Fo(The)34 b(pro)s(of)f(is)g(similar)f(to)h(the)g(pro)s(ofs)h(of)f (the)h(Prop)s(ositions)e(5.1)h(and)h(5.3.)p 3261 2550 70 70 v 120 2804 a(An)f(immediate)e(consequence)k(of)f(this)f(result)h (is)f(the)0 2944 y Fm(Corollary)45 b(5.6)h Fe(Given)40 b Fl(\014)j(>)37 b Fo(1)k Fe(and)g Fl(\014)5 b(\022)44 b Fe(that)d(satis\014es)f(\(2.21\),ther)-5 b(e)38 b(exists)j(two)g(c)-5 b(onstants)42 b Fo(~)-52 b Fl(c)3752 2959 y Fk(i)3823 2944 y Fo(=)2 3084 y(~)g Fl(c)43 3099 y Fk(i)76 3084 y Fo(\()p Fl(\014)5 b(;)17 b(\022)s Fo(\))36 b Fe(for)g Fl(i)30 b Fo(=)g(1)p Fl(;)17 b Fo(2)36 b Fe(such)g(that)h(if)f Fl(\016)1436 3048 y Fp(\003)1498 3084 y Fo(log)1659 3044 y Fn(1)p 1655 3061 47 4 v 1655 3118 a Fk(\015)1743 3084 y Fj(#)30 b Fo(0)36 b Fe(when)h Fl(\015)d Fj(#)c Fo(0)p Fe(,)36 b(for)g(al)5 b(l)38 b Fl(\016)2796 3099 y Fn(4)2870 3084 y Fl(>)30 b(\016)3021 3099 y Fn(1)3096 3084 y Fl(>)g(\016)3251 3048 y Fp(\003)3327 3084 y Fl(;)45 b(\020)3444 3099 y Fn(4)3518 3084 y Fl(>)30 b(\020)3669 3099 y Fn(1)3743 3084 y Fl(>)g Fo(0)0 3223 y Fe(that)36 b(satisfy)1193 3436 y Fl(\016)1237 3451 y Fn(4)1282 3436 y Fl(\020)1333 3395 y Fn(3)1326 3461 y(4)1405 3436 y Fj(\025)28 b Fl(\016)1554 3451 y Fn(1)1599 3436 y Fl(\020)1650 3395 y Fn(3)1643 3461 y(1)1723 3436 y Fj(\025)i Fo(~)-52 b Fl(c)1871 3451 y Fn(1)1932 3296 y Fd(\022)2006 3290 y(r)p 2105 3290 118 4 v 2136 3369 a Fl(\015)p 2117 3413 94 4 v 2117 3505 a(\016)2165 3476 y Fp(\003)2245 3436 y Fj(_)23 b Fl(\015)e Fo(log)2568 3369 y(1)p 2565 3413 58 4 v 2565 3505 a Fl(\015)2634 3296 y Fd(\023)3645 3436 y Fo(\(5)p Fl(:)p Fo(26\))0 3716 y Fe(and)45 b Fl(\016)247 3731 y Fn(4)292 3716 y Fl(\020)343 3679 y Fn(3)336 3740 y(4)432 3716 y Fj(\025)i Fo(~)-52 b Fl(c)597 3731 y Fn(1)642 3716 y Fl(\020)686 3731 y Fn(1)730 3716 y Fe(,)47 b(for)e(al)5 b(l)46 b Fo(1)e Fl(>)h(x)g(>)g Fo(0)p Fe(,)h(for)f(al)5 b(l)46 b(intervals)e Fo(\001)2515 3731 y Fk(L)2620 3716 y Fe(of)g(macr)-5 b(osc)g(opic)44 b(length)h Fl(L)g Fe(that)0 3855 y(ar)-5 b(e)44 b(include)-5 b(d)44 b(in)f(an)h(interval)g Fl(I)51 b Fe(that)44 b(c)-5 b(ontains)43 b(the)h(origin,)h(with)f Fj(j)p Fl(I)8 b Fj(j)41 b(\024)j Fl(\015)3001 3819 y Fp(\000)p Fn(2)3151 3855 y Fe(if)f Fl(\015)48 b Fo(=)43 b(2)3526 3819 y Fp(\000)p Fk(n)3643 3855 y Fe(,)i(with)0 3995 y Fl(I)-18 b(P)14 b Fe(-pr)-5 b(ob)g(ability)35 b(one,)f(for)i(al)5 b(l)36 b(but)g(a)g(\014nite)f(numb)-5 b(er)36 b(of)f(indic)-5 b(es)34 b Fl(n)p Fe(,)h(for)g(al)5 b(l)2895 3969 y Fo(~)2883 3995 y Fl(L)27 b Fj(\025)i Fo(1,)579 4308 y Fl(\026)639 4323 y Fk(\014)t(;\022)r(;\015)843 4197 y Fd(\020)903 4308 y Fj(9)958 4326 y Fk(L)p Fn(:2)p Fp(\024)p Fk(L)p Fp(\024j)p Fk(I)5 b Fp(j)1347 4308 y Fj(9)1402 4323 y Fn(\001)1467 4333 y Ff(L)1520 4323 y Fp(\032)p Fk(I)1656 4308 y Fo(:)55 b Fj(R)1823 4260 y Fk(\016)1859 4270 y Fg(1)1899 4260 y Fk(;\020)1959 4270 y Fg(1)1997 4260 y Fk(;\016)2057 4270 y Fg(4)2096 4260 y Fk(;\020)2156 4270 y Fg(4)1823 4334 y Fn(0)2200 4308 y Fo(\(\001)2322 4323 y Fk(L)2381 4308 y Fl(;)2437 4283 y Fo(~)2426 4308 y Fl(L)p Fo(\))2533 4197 y Fd(\021)2619 4308 y Fj(\024)29 b Fl(e)2771 4264 y Fp(\000)2848 4212 y Fg(~)-37 b Ff(c)2877 4227 y Fg(2)2916 4212 y(\()p Ff(\014)s(;\022)r Fg(\))3079 4199 y(~)3071 4212 y Ff(L)o(x\020)3193 4192 y Fg(3)3190 4240 y(4)3232 4212 y Ff(\016)3265 4227 y Fg(4)p 2845 4248 459 4 v 3054 4288 a Ff(\015)3645 4308 y Fo(\(5)p Fl(:)p Fo(27\))120 4555 y(Therefore)34 b(if)g(w)m(e)f (denote)1064 4800 y Fj(O)1146 4752 y Fk(\016)1182 4762 y Fg(1)1221 4752 y Fk(;\020)1281 4762 y Fg(1)1143 4827 y Fn(0)1325 4800 y Fo(\()p Fl(I)8 b Fo(\))26 b Fj(\021)1770 4706 y Fd([)1586 4924 y Fk(R)p Fn(:)d Fk(R)1752 4934 y Fg(1)1791 4924 y Fp(\024)p Fk(R)p Fp(\024j)p Fk(I)5 b Fp(j)2165 4706 y Fd([)2108 4919 y Fn(\001)2173 4929 y Ff(R)2230 4919 y Fp(\032)p Fk(I)2350 4800 y Fj(O)2432 4752 y Fk(\016)2468 4762 y Fg(1)2507 4752 y Fk(;\020)2567 4762 y Fg(1)2429 4827 y Fn(0)2611 4800 y Fo(\(\001)2733 4815 y Fk(R)2798 4800 y Fo(\))808 b(\(5)p Fl(:)p Fo(28\))1076 5238 y Fj(W)1182 5197 y Fk(\016)1218 5207 y Fg(2)1258 5197 y Fk(;\020)1318 5207 y Fg(2)1362 5238 y Fo(\()p Fl(I)8 b Fo(\))27 b Fj(\021)1775 5143 y Fd([)1624 5357 y Fk(L)p Fn(:)22 b(2)p Fp(\024)p Fk(L)p Fp(\024)p Fk(L)1998 5367 y Fg(2)2136 5143 y Fd([)2080 5357 y Fn(\001)2145 5367 y Ff(L)2198 5357 y Fp(\032)p Fk(I)2318 5238 y Fj(W)2424 5197 y Fk(\016)2460 5207 y Fg(2)2500 5197 y Fk(;\020)2560 5207 y Fg(2)2603 5238 y Fo(\(\001)2725 5253 y Fk(L)2785 5238 y Fo(\))821 b(\(5)p Fl(:)p Fo(29\))0 5579 y Fg(23)p Ff(=j)s(anuar)r(y)r(=)p Fg(1998;)32 b(12:06)1145 b Fo(45)p eop %%Page: 46 46 46 45 bop 0 100 a Fo(and)639 241 y Fj(R)723 193 y Fk(\016)759 203 y Fg(1)798 193 y Fk(;\020)858 203 y Fg(1)897 193 y Fk(;\016)957 203 y Fg(4)996 193 y Fk(;\020)1056 203 y Fg(4)723 268 y Fn(0)1100 241 y Fo(\()p Fl(I)8 b Fo(\))26 b Fj(\021)1510 147 y Fd([)1362 365 y Fk(L)p Fn(:)c(2)p Fp(\024)p Fk(L)p Fp(\024j)p Fk(I)5 b Fp(j)1868 147 y Fd([)1813 360 y Fn(\001)1878 370 y Ff(L)1931 360 y Fp(\032)p Fk(I)2209 147 y Fd([)2088 360 y Fn(~)2078 377 y Fk(L)p Fn(:)22 b(1)p Fp(\024)2290 360 y Fn(~)2280 377 y Fk(L)p Fp(\024)p Fk(L)2468 241 y Fj(R)2552 193 y Fk(\016)2588 203 y Fg(1)2628 193 y Fk(;\020)2688 203 y Fg(1)2726 193 y Fk(;\016)2786 203 y Fg(4)2825 193 y Fk(;\020)2885 203 y Fg(4)2552 268 y Fn(0)2929 241 y Fo(\(\001)3051 256 y Fk(L)3110 241 y Fl(;)3166 216 y Fo(~)3155 241 y Fl(L)o Fo(\))384 b(\(5)p Fl(:)p Fo(30\))120 583 y(then,)50 b(for)e(an)f (appropriate)g(c)m(hoice)h(of)g(v)-6 b(arious)47 b(parameters,)j Fl(\016)2659 598 y Fk(i)2692 583 y Fl(;)17 b(\020)2781 598 y Fk(i)2860 583 y Fo(for)47 b Fl(i)k Fo(:)102 b(1)50 b Fj(\024)i Fl(i)f Fj(\024)g Fo(4,)f(as)0 722 y(a)e(consequence)j(of)e (the)f(Corollaries)g(5.2,)j(5.4)d(and)h(5.6,)i(all)d(the)h(previous)g (sets)g(ha)m(v)m(e)g(a)f(Gibbs-)0 862 y(Probabilit)m(y)g(that)h(go)s (es)g(to)g(zero,)54 b Fl(I)-17 b(P)13 b Fo(-almost)48 b(surely)-8 b(.)92 b(It)49 b(is)g(con)m(v)m(enien)m(t)i(to)e(mak)m(e)f (the)i(c)m(hoices)0 1001 y Fl(\016)44 1016 y Fn(1)125 1001 y Fo(=)36 b Fl(\016)282 1016 y Fn(2)327 1001 y Fo(,)j Fl(\020)438 1016 y Fn(2)518 1001 y Fo(=)d Fl(\020)675 1016 y Fn(1)719 1001 y Fo(,)j Fl(\020)830 1016 y Fn(4)910 1001 y Fl(>)e(\020)1068 1016 y Fn(1)1150 1001 y Fo(and)i Fl(\016)1393 1016 y Fn(4)1473 1001 y Fj(\025)e Fl(\016)1631 1016 y Fn(1)1675 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Fk(;\020)1450 1393 y Fg(1)1493 1419 y Fo(\()p Fl(`)p Fo(\))g(=)h(0)f(b)s(et)m(w)m(een)h(adjacen)m(t)h(blo)s(c)m(ks)e(of)g (the)g(same)f(equilibrium)0 1559 y(can)34 b(b)s(e)f(only)g Fl(\021)590 1523 y Fk(\016)626 1533 y Fg(4)665 1523 y Fk(;\020)725 1533 y Fg(4)796 1559 y Fo(=)c Fl(\021)t Fo(,)k(since)h(\(5.27\).)120 1700 y(The)i(next)g(step)h(is)e(to)h(pro)m (v)m(e)g(that)g(the)g(length)g(of)h(the)f(previous)g(run)h(of)f Fl(\021)3011 1664 y Fk(\016)3047 1674 y Fg(4)3086 1664 y Fk(;\020)3146 1674 y Fg(4)3222 1700 y Fo(=)d Fl(\021)40 b Fo(whic)m(h)d(is)e(at)0 1840 y(least)h Fl(L)304 1855 y Fn(2)385 1840 y Fo(is)g(in)g(fact)h(b)s(ounded)g(from)f(b)s(elo)m(w)g (b)m(y)g(a)g(m)m(uc)m(h)h(larger)f(quan)m(tit)m(y)-8 b(.)52 b(W)-8 b(e)36 b(de\014ne,)j(see)e(\(5.14\),)0 1979 y(for)c Fl(\021)f Fj(2)c(f)p Fo(+1)p Fl(;)17 b Fj(\000)p Fo(1)p Fj(g)p Fo(,)33 b Fl(`)829 1994 y Fn(1)901 1979 y Fl(<)1013 1953 y Fo(~)1006 1979 y Fl(`)1048 1994 y Fn(1)1120 1979 y Fl(<)1232 1953 y Fo(~)1225 1979 y Fl(`)1267 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Fk(;\020)931 2397 y Fg(4)975 2428 y Fo(\()p Fl(`)1056 2443 y Fn(2)1100 2428 y Fo(\))e(=)h Fl(\021)t(;)44 b(\021)1449 2387 y Fk(\016)1485 2397 y Fg(4)1524 2387 y Fk(;\020)1584 2397 y Fg(4)1628 2428 y Fo(\()1674 2401 y(~)1667 2428 y Fl(`)1709 2443 y Fn(1)1775 2428 y Fj(\000)23 b Fo(1\))k(=)h Fl(\021)2149 2387 y Fk(\016)2185 2397 y Fg(4)2224 2387 y Fk(;\020)2284 2397 y Fg(4)2328 2428 y Fo(\()2374 2401 y(~)2367 2428 y Fl(`)2409 2443 y Fn(1)2453 2428 y Fo(\))f(=)i Fj(\000)p Fl(\021)t(;)56 2597 y(\021)109 2556 y Fk(\016)145 2566 y Fg(4)185 2556 y Fk(;\020)245 2566 y Fg(4)288 2597 y Fo(\()p Fl(`)p Fo(\))e(=)h Fj(\000)p Fl(\021)k Fj(8)p Fl(`)c Fo(:)885 2571 y(~)878 2597 y Fl(`)920 2612 y Fn(1)987 2597 y Fo(+)22 b(1)28 b Fj(\024)g Fl(`)f Fj(\024)1450 2571 y Fo(~)1443 2597 y Fl(`)1485 2612 y Fn(2)1552 2597 y Fj(\000)22 b Fo(1)p Fl(;)44 b(\021)1826 2556 y Fk(\016)1862 2566 y Fg(4)1901 2556 y Fk(;\020)1961 2566 y Fg(4)2005 2597 y Fo(\()2051 2571 y(~)2044 2597 y Fl(`)2086 2612 y Fn(2)2130 2597 y 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Fo(1)p Fe(,)36 b Fo(0)28 b Fl(<)h(x)f(<)h Fo(1)p Fe(,)35 b Fl(p)28 b(>)h Fo(1)p Fe(,)38 b Fo(^)-52 b Fl(c)28 b(>)h Fo(0)p Fe(,)35 b Fl(\032)28 b(>)h Fo(0)p Fe(,)35 b(if)g Fl(\022)c Fj(\024)3287 3127 y Fk(x)3332 3097 y Fg(2)3371 3127 y Fn(\001)p Fp(F)p 3122 3144 545 4 v 3122 3223 a Fn(48)3202 3156 y Fj(p)p 3284 3156 383 4 v 3286 3223 a Fn(^)-42 b Fk(c)p Fn(\()p Fk(p)p Fn(+1+2)p Fk(\032)p Fn(\))3714 3167 y Fe(then)0 3393 y(ther)-5 b(e)35 b(exist)f Fl(\015)531 3408 y Fn(0)603 3393 y Fl(>)28 b Fo(0)35 b Fe(and)g Fl(c)1029 3408 y Fn(0)1102 3393 y Fl(>)28 b Fo(0)35 b Fe(such)g(that)g(for)g Fl(\015)e Fj(\024)28 b Fl(\015)2122 3408 y Fn(0)2166 3393 y Fe(,)35 b(if)f Fl(\020)2372 3408 y Fn(4)2417 3393 y Fl(g)2465 3408 y Fn(2)2509 3393 y Fo(\(1)p Fl(=\020)2692 3408 y Fn(4)2735 3393 y Fo(\))28 b Fj(\024)2919 3353 y Fk(x)2964 3323 y Fg(2)3003 3353 y Fn(\001)p Fp(F)p 2919 3370 214 4 v 2986 3427 a Fn(96)3145 3393 y Fo(\(1)20 b Fj(^)3562 3348 y Fk(\014)p 3354 3370 466 4 v 3354 3382 a Fj(p)p 3437 3382 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b Fl(c)p Fe(,)46 b(on)e(a)g(set)g Fo(\012)3735 3756 y Fn(4)3823 3741 y Fo(=)0 3880 y(\012)72 3895 y Fn(4)117 3880 y Fo(\()p Fj(J)17 b Fl(;)g(\014)5 b(;)17 b(\022)s(;)g(\015)6 b Fo(\))33 b Fe(that)i(satis\014es)965 4201 y Fl(I)-18 b(P)14 b Fo([\012)1169 4216 y Fn(4)1213 4201 y Fo(])28 b Fj(\025)g Fo(1)22 b Fj(\000)1557 4133 y Fo(2)r(~)-52 b Fl(c)p 1557 4178 93 4 v 1585 4269 a Fo(^)f Fl(c)1662 4201 y Fo(\(log)1862 4133 y(1)p 1858 4178 58 4 v 1858 4269 a Fl(\015)1927 4201 y Fo(\))1966 4159 y Fk(p)p Fn(+)p Fk(\032)2115 4201 y Fl(e)2161 4156 y Fp(\000)p Fn(\(log)15 b(log)2503 4129 y Fg(1)p 2500 4140 41 4 v 2500 4180 a Ff(\015)2552 4156 y Fn(\)\()p Fk(p)p Fn(+2)p Fk(\032)p Fn(+1\))3645 4201 y Fo(\(5)p Fl(:)p Fo(32\))0 4543 y Fe(we)36 b(have,)e(uniformly)j(on)e(al)5 b(l)37 b(intervals)e Fo([)1588 4516 y(~)1581 4543 y Fl(`)1623 4558 y Fn(1)1667 4543 y Fl(;)1718 4516 y Fo(~)1712 4543 y Fl(`)1754 4558 y Fn(2)1797 4543 y Fo(])28 b Fj(\032)g Fl(I)43 b Fe(and)36 b(uniformly)g(on)g Fl(I)f Fj(\032)28 b(J)18 b Fe(,)790 4873 y Fl(\026)850 4888 y Fk(\014)t(;\022)r(;\015)1054 4763 y Fd(\020)1150 4848 y Fo(~)1113 4873 y Fj(W)1219 4832 y Fk(\016)1255 4842 y Fg(4)1295 4832 y Fk(;\020)1355 4842 y Fg(4)1211 4898 y Fk(\021)1398 4873 y Fo(\()p Fl(`)1479 4888 y Fn(1)1523 4873 y Fl(;)1575 4847 y Fo(~)1568 4873 y Fl(`)1610 4888 y Fn(1)1654 4873 y Fl(;)1705 4847 y Fo(~)1699 4873 y Fl(`)1741 4888 y Fn(2)1784 4873 y Fl(;)f(`)1871 4888 y Fn(2)1915 4873 y Fo(\))1954 4763 y Fd(\021)2041 4873 y Fj(\024)28 b Fo(exp)2315 4733 y Fd(\024)2367 4873 y Fj(\000)2456 4806 y Fl(\014)p 2456 4851 62 4 v 2458 4942 a(\015)2530 4873 y(x)p Fo(\(1)22 b Fj(\000)h Fl(x)p Fo(\)\001)p Fj(F)3059 4733 y Fd(\025)3645 4873 y Fo(\(5)p Fl(:)p Fo(33\))0 5199 y Fe(for)35 b Fl(\021)d Fo(=)c Fj(\006)p Fo(1)0 5340 y Fa(Pro)s(of:)0 5579 y Fg(23)p Ff(=j)s(anuar)r(y)r(=)p Fg(1998;)k(12:06)1145 b Fo(46)p eop %%Page: 47 47 47 46 bop 120 100 a Fo(The)33 b(\014rst)g(step)h(is)e(to)h(restrict)g (ourself)g(to)g(a)f(\014nite)h(v)m(olume)f(Gibbs)i(measure.)44 b(Since)33 b Fl(\021)3452 63 y Fk(\016)3488 73 y Fg(4)3527 63 y Fk(;\020)3587 73 y Fg(4)3631 100 y Fo(\()p Fl(`)3712 115 y Fn(1)3756 100 y Fo(\))28 b(=)0 239 y Fl(\021)53 203 y Fk(\016)89 213 y Fg(4)128 203 y Fk(;\020)188 213 y Fg(4)232 239 y Fo(\()p Fl(`)313 254 y Fn(2)357 239 y Fo(\))f(=)h Fl(\021)t Fo(,)33 b(w)m(e)h(get)178 549 y Fl(\026)238 564 y Fk(\014)t(;\022)r(;\015)442 439 y Fd(\020)538 524 y Fo(~)501 549 y Fj(W)607 508 y Fk(\016)643 518 y Fg(4)683 508 y Fk(;\020)743 518 y Fg(4)599 574 y Fk(\021)787 549 y Fo(\()p Fl(`)868 564 y Fn(1)912 549 y Fl(;)963 523 y Fo(~)957 549 y Fl(`)999 564 y Fn(1)1042 549 y Fl(;)1093 523 y Fo(~)1087 549 y Fl(`)1129 564 y Fn(2)1173 549 y Fl(;)17 b(`)1260 564 y Fn(2)1303 549 y Fo(\))1342 439 y Fd(\021)1429 549 y Fj(\024)28 b Fl(e)1580 505 y Fn(4)p Fk(\014)1681 466 y Ff(\020)1714 481 y Fg(4)p 1681 489 73 4 v 1697 529 a Ff(\015)1770 549 y Fl(\026)1830 564 y Fk(\014)t(;\022)r(;\015)2034 439 y Fd(\020)2130 524 y Fo(~)2093 549 y Fj(W)2199 508 y Fk(\016)2235 518 y Fg(4)2275 508 y Fk(;\020)2335 518 y Fg(4)2191 574 y Fk(\021)2379 549 y Fo(\()p Fl(`)2460 564 y Fn(1)2504 549 y Fl(;)2555 523 y Fo(~)2549 549 y Fl(`)2591 564 y Fn(1)2634 549 y Fl(;)2685 523 y Fo(~)2679 549 y Fl(`)2721 564 y Fn(2)2765 549 y Fl(;)17 b(`)2852 564 y Fn(2)2895 549 y Fo(\))2962 464 y Fd(\014)2962 524 y(\014)3022 549 y Fo(\006)3094 564 y Fk(@)t Fn(\001)3206 574 y Ff(L)3264 439 y Fd(\021)3340 549 y Fo(\(0\))177 b(\(5)p Fl(:)p Fo(34\))0 874 y(Using)38 b(the)g(fact)h(that)e Fl(\021)936 838 y Fk(\016)972 848 y Fg(4)1011 838 y Fk(;\020)1071 848 y Fg(4)1115 874 y Fo(\()1161 848 y(~)1154 874 y Fl(`)1196 889 y Fn(1)1240 874 y Fo(\))f(=)g Fl(\021)1481 838 y Fk(\016)1517 848 y Fg(4)1556 838 y Fk(;\020)1616 848 y Fg(4)1660 874 y Fo(\()1706 848 y(~)1699 874 y Fl(`)1741 889 y Fn(1)1810 874 y Fj(\000)26 b Fo(1\))38 b(and)g Fl(\021)2291 838 y Fk(\016)2327 848 y Fg(4)2366 838 y Fk(;\020)2426 848 y Fg(4)2470 874 y Fo(\()2516 848 y(~)2509 874 y Fl(`)2551 889 y Fn(2)2620 874 y Fo(+)26 b(1\))36 b(=)g Fl(\021)3014 838 y Fk(\016)3050 848 y Fg(4)3089 838 y Fk(;\020)3149 848 y Fg(4)3193 874 y Fo(\()3239 848 y(~)3232 874 y Fl(`)3274 889 y Fn(2)3318 874 y Fo(\))h(w)m(e)i(can) f(also)0 1014 y(decouple)46 b(the)f(in)m(terv)-6 b(al)45 b([)1011 988 y(~)1004 1014 y Fl(`)1046 1029 y Fn(1)1120 1014 y Fj(\000)30 b Fo(1)p Fl(;)1328 988 y Fo(~)1322 1014 y Fl(`)1364 1029 y Fn(2)1438 1014 y Fo(+)g(1])44 b(from)g(the)i(in)m(terv)-6 b(al)44 b([)p Fl(`)2539 1029 y Fn(1)2583 1014 y Fl(;)17 b(`)2670 1029 y Fn(2)2714 1014 y Fo(].)78 b(This)45 b(will)f(pro)s(duce)i(three)0 1153 y(adjacen)m(t)34 b(in)m(terv)-6 b(als.)44 b(W)-8 b(e)34 b(asso)s(ciate,)e(the)h(in)m(teraction)g(b)s(et)m(w)m(een)i(the) e(\014rst)g(and)h(the)f(second)h(in)m(terv)-6 b(al)0 1293 y(to)30 b(the)i(\014rst)f(one,)g(and)h(the)f(in)m(teraction)g(b)s (et)m(w)m(een)h(the)f(second)h(and)f(the)h(third)f(in)m(terv)-6 b(al)30 b(to)h(the)g(third)0 1432 y(one.)43 b(This)31 b(will)d(giv)m(e,)i(up)h(to)e(a)h(factor)g Fl(e)1526 1388 y Fn(8)p Fk(\014)1627 1350 y Ff(\020)1660 1365 y Fg(4)p 1627 1373 V 1643 1412 a Ff(\015)1716 1432 y Fo(,)g(a)g(pro)s (duct)h(of)f(three)h(terms)e(eac)m(h)i(one)f(b)s(eing)g(lo)s(calized)0 1572 y(on)38 b(one)g(of)g(the)f(three)i(in)m(terv)-6 b(als.)57 b(W)-8 b(e)38 b(mak)m(e)e(a)i(rough)g(estimate)e(for)i(the)g (terms)f(corresp)s(onding)h(to)0 1711 y(the)44 b(\014rst)g(and)g(the)g (third)f(in)m(terv)-6 b(al.)75 b(Applying)43 b(an)h(argumen)m(t)f (similar)f(to)h(the)h(one)g(giv)m(en)f(in)h(the)0 1851 y(Corollary)31 b(5.4,)i(w)m(e)g(get)g(that,)g(with)f(a)h Fl(I)-17 b(P)14 b Fo(-probabilit)m(y)32 b(1,)h(uniformly)f(with)h(resp) s(ect)h(to)f(all)f(in)m(terv)-6 b(als)0 1990 y([)35 1964 y(~)28 1990 y Fl(`)70 2005 y Fn(1)114 1990 y Fl(;)165 1964 y Fo(~)159 1990 y Fl(`)201 2005 y Fn(2)244 1990 y Fo(])33 b(included)i(in)e(an)g(in)m(terv)-6 b(al)33 b Fj(J)52 b Fo(con)m(taining)33 b(the)g(origin,)g(with)g Fj(jJ)17 b(j)28 b(\024)2939 1951 y Fn(1)p 2917 1967 86 4 v 2917 2025 a Fk(\015)2963 2005 y Fg(2)3014 1990 y Fo(,)289 2323 y Fl(\026)349 2338 y Fk(\014)t(;\022)r(;\015)553 2213 y Fd(\020)649 2298 y Fo(~)612 2323 y Fj(W)718 2282 y Fk(\016)754 2292 y Fg(4)794 2282 y Fk(;\020)854 2292 y Fg(4)710 2348 y Fk(\021)898 2323 y Fo(\()p Fl(`)979 2338 y Fn(1)1023 2323 y Fl(;)1074 2297 y Fo(~)1068 2323 y Fl(`)1110 2338 y Fn(1)1153 2323 y Fl(;)1204 2297 y Fo(~)1198 2323 y Fl(`)1240 2338 y Fn(2)1284 2323 y Fl(;)17 b(`)1371 2338 y Fn(2)1414 2323 y Fo(\))1480 2239 y Fd(\014)1480 2298 y(\014)1541 2323 y Fo(\006)1613 2338 y Fk(@)t Fn(\001)1725 2348 y Ff(L)1783 2213 y Fd(\021)1843 2323 y Fo(\(0\))27 b Fj(\024)h Fl(e)2149 2279 y Fn(12)p Fk(\014)2289 2241 y Ff(\020)2322 2256 y Fg(4)p 2289 2263 73 4 v 2305 2303 a Ff(\015)2378 2323 y Fl(e)2424 2279 y Fp(\000)2499 2246 y Ff(\014)s(x)p Fg(\001)p Fb(F)p 2499 2263 191 4 v 2574 2303 a Ff(\015)1865 2572 y Fj(\002)22 b Fl(\026)2024 2603 y Fk(\014)t(;\022)r(;\015)t(;)2236 2585 y Fn(~)2231 2603 y Fk(`)2264 2613 y Fg(1)2303 2603 y Fk(;)2332 2585 y Fn(~)2327 2603 y Fk(`)2360 2613 y Fg(2)2420 2462 y Fd(\020)2516 2547 y Fo(~)2480 2572 y Fj(W)2586 2531 y Fk(\016)2622 2541 y Fg(4)2661 2531 y Fk(;\020)2721 2541 y Fg(4)2578 2597 y Fk(\021)2765 2572 y Fo(\()2811 2546 y(~)2804 2572 y Fl(`)2846 2587 y Fn(1)2890 2572 y Fl(;)2941 2546 y Fo(~)2935 2572 y Fl(`)2977 2587 y Fn(2)3020 2572 y Fo(\))3059 2462 y Fd(\021)3135 2572 y Fo(\(0)p Fl(;)17 b Fo(0\))3645 2446 y(\(5)p Fl(:)p Fo(35\))0 2898 y(where)50 b Fl(\026)364 2929 y Fk(\014)t(;\022)r(;\015)t(;)576 2910 y Fn(~)571 2929 y Fk(`)604 2939 y Fg(1)642 2929 y Fk(;)671 2910 y Fn(~)666 2929 y Fk(`)699 2939 y Fg(2)743 2898 y Fo(\()p Fl(:)p Fo(\)\(0)p Fl(;)17 b Fo(0\))47 b(is)i(the)g(Gibbs)h(measure)f(in)g(the)h(macroscopic)e(v)m(olume)g([) 3420 2871 y(~)3413 2898 y Fl(`)3455 2913 y Fn(1)3499 2898 y Fl(;)3550 2871 y Fo(~)3544 2898 y Fl(`)3586 2913 y Fn(2)3630 2898 y Fo(])g(with)0 3037 y(empt)m(y)32 b(b)s(oundary)i (conditions,)f(see)h(\(2.4\),)e(with)38 b(~)-55 b Fl(\033)31 b Fo(=)d(0.)120 3180 y(It)38 b(remains)h(to)f(estimate)g(the)h(last)g (term.)60 b(Recalling)39 b(\(4.3\),)g(\(4.4\))f(and)h(\(4.5\))f(it)h (is)g(easy)g(to)f(see)0 3320 y(that)33 b(if)g(w)m(e)h(denote)787 3295 y(~)775 3320 y Fl(I)819 3335 y Fn(12)931 3320 y Fj(\021)28 b Fo([)1071 3294 y(~)1064 3320 y Fl(`)1106 3335 y Fn(1)1150 3320 y Fl(;)1201 3294 y Fo(~)1195 3320 y Fl(`)1237 3335 y Fn(2)1280 3320 y Fo(])33 b(w)m(e)h(ha)m(v)m(e)568 3672 y Fl(\026)628 3703 y Fk(\014)t(;\022)r(;\015)t(;)840 3684 y Fn(~)835 3703 y Fk(`)868 3713 y Fg(1)906 3703 y Fk(;)935 3684 y Fn(~)930 3703 y Fk(`)963 3713 y Fg(2)1023 3561 y Fd(\020)1120 3647 y Fo(~)1083 3672 y Fj(W)1189 3631 y Fk(\016)1225 3641 y Fg(4)1265 3631 y Fk(;\020)1325 3641 y Fg(4)1181 3696 y Fk(\021)1368 3672 y Fo(\()1414 3645 y(~)1407 3672 y Fl(`)1449 3687 y Fn(1)1493 3672 y Fl(;)1544 3645 y Fo(~)1538 3672 y Fl(`)1580 3687 y Fn(2)1624 3672 y Fo(\))1663 3561 y Fd(\021)1739 3672 y Fo(\(0)p Fl(;)17 b Fo(0\))26 b Fj(\024)i Fl(e)2139 3631 y Fk(\014)t(\015)2234 3600 y Fb(\000)p Fg(1)2327 3631 y Fn([)p Fk(\016)2389 3600 y Fb(\003)2430 3631 y Fn(\()2466 3612 y(~)2461 3631 y Fk(`)2494 3641 y Fg(2)2533 3631 y Fp(\000)2600 3612 y Fn(~)2595 3631 y Fk(`)2628 3641 y Fg(1)2668 3631 y Fn(\)])2738 3604 y Fl(Z)2806 3619 y Fp(\000)p Fk(\021)s(;\016)2972 3629 y Fg(4)3011 3619 y Fk(;\020)3071 3629 y Fg(4)3115 3604 y Fo(\()3166 3579 y(~)3154 3604 y Fl(I)3198 3619 y Fn(12)3282 3604 y Fo(\))p 2738 3649 583 4 v 2770 3752 a Fl(Z)2838 3767 y Fk(\021)s(;\016)2942 3777 y Fg(4)2980 3767 y Fk(;\020)3040 3777 y Fg(4)3084 3752 y Fo(\()3135 3727 y(~)3123 3752 y Fl(I)3167 3767 y Fn(12)3251 3752 y Fo(\))3645 3672 y(\(5)p Fl(:)p Fo(36\))0 4042 y(where)42 b(the)f(last)f(term)g(is)g (similar)f(to)h(the)h(one)g(de\014ned)i(in)e(\(4.5\),)g(with)f Fj(R)p Fo(\()p Fl(\034)11 b Fo(\))40 b(=)h Fj(R)3300 4005 y Fk(\016)3336 4015 y Fg(4)3375 4005 y Fk(;\020)3435 4015 y Fg(4)3479 4042 y Fo(\()3525 4015 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y Fj(\024)28 b Fo(exp)2136 5115 y Fd(\022)2209 5255 y Fj(\000)2730 5188 y Fl(\017)2770 5152 y Fn(2)p 2299 5232 948 4 v 2299 5336 a Fo(212)p Fl(\015)6 b Fj(j)2547 5311 y Fo(~)2535 5336 y Fl(I)2579 5351 y Fn(12)2661 5336 y Fj(j)p Fl(\014)f(\022)s(\020)2844 5351 y Fn(4)2888 5336 y Fl(g)2936 5351 y Fn(2)2981 5336 y Fo(\(1)p Fl(=\020)3164 5351 y Fn(4)3207 5336 y Fo(\))3258 5115 y Fd(\023)3645 5255 y Fo(\(5)p Fl(:)p Fo(38\))0 5579 y Fg(23)p Ff(=j)s(anuar)r(y)r(=)p Fg(1998;)32 b(12:06)1145 b Fo(47)p eop %%Page: 48 48 48 47 bop 120 100 a Fo(T)-8 b(o)40 b(get)i(a)f(result)h(whic)m(h)g(is)f (true)h Fe(uniformly)h Fo(with)e(resp)s(ect)h(to)f(all)g(subin)m(terv) -6 b(als)3337 74 y(~)3325 100 y Fl(I)3369 115 y Fn(12)3495 100 y Fo(of)42 b Fl(I)8 b Fo(,)42 b(and)0 239 y(for)c(an)m(y)f Fl(I)45 b Fo(in)37 b(a)g(giv)m(en)g(in)m(terv)-6 b(al)38 b Fj(J)55 b Fo(of)37 b(length)j(~)-52 b Fl(c)p Fo(\()p Fl(\015)6 b Fo(\))2002 203 y Fp(\000)p Fn(1)2108 239 y Fo(\(log)17 b(1)p Fl(=\015)6 b Fo(\))2489 203 y Fk(p)2571 239 y Fo(con)m(taining)37 b(the)h(origin,)f(w)m(e)h(need)0 379 y(a)45 b(mo)s(di\014cation)g(of)h(the)g(Otta)m(viani)f(inequalit)m (y)f([29])h(that)g(tak)m(es)h(in)m(to)f(accoun)m(t)i(that)e(w)m(e)h(do) g(not)0 518 y(ha)m(v)m(e)29 b(sum)g(of)g(random)g(v)-6 b(ariables)29 b Fe(i.e.)41 b Fo(an)29 b(additiv)m(e)g(pro)s(cess)h(but) g(merely)d(an)j(appro)m(ximate)d(additiv)m(e)0 658 y(pro)s(cess.)120 839 y(T)-8 b(o)32 b(simplify)g(notations,)h(giv)m(en)g(an)g(in)m(terv) -6 b(al)1879 814 y(~)1867 839 y Fl(I)35 b Fj(\032)29 b Fl(I)8 b Fo(,)32 b(let)h(us)h(call)f Fl(Y)21 b Fo(\()2749 814 y(~)2737 839 y Fl(I)8 b Fo(\))27 b Fj(\021)h Fo(log)3118 782 y Fk(Z)3172 793 y Fb(\000)p Ff(\021)r(;)p Fg(0)p Ff(;\016)3378 808 y Fg(4)3416 793 y Ff(;\020)3472 808 y Fg(4)3516 782 y Fn(\()3556 765 y(~)3547 782 y Fk(I)6 b Fn(\))p 3118 816 502 4 v 3145 884 a Fk(Z)3199 895 y Ff(\021)r(;)p Fg(0)p Ff(;\016)3351 910 y Fg(4)3389 895 y Ff(;\020)3445 910 y Fg(4)3489 884 y Fn(\()3529 867 y(~)3520 884 y Fk(I)g Fn(\))3631 839 y Fo(.)0 1020 y Fm(Lemma)38 b(5.8)h Fe(F)-8 b(or)36 b(any)g(given)e(interval)i Fl(I)550 1408 y(I)-17 b(P)671 1238 y Fd(")753 1408 y Fo(max)738 1472 y Fn(~)729 1489 y Fk(I)764 1499 y Fg(1)p Ff(;)p Fg(2)860 1489 y Fp(\032)p Fk(I)979 1294 y Fd(\014)979 1353 y(\014)979 1413 y(\014)1013 1408 y Fl(Y)22 b Fo(\()1144 1383 y(~)1132 1408 y Fl(I)1176 1423 y Fn(12)1260 1408 y Fo(\))1299 1294 y Fd(\014)1299 1353 y(\014)1299 1413 y(\014)1359 1408 y Fj(\025)28 b Fl(\014)1538 1341 y Fo(4)p Fl(\017)22 b Fo(+)h(12)p Fl(\020)1894 1356 y Fn(4)p 1538 1385 400 4 v 1709 1476 a Fl(\015)1950 1238 y Fd(#)2036 1408 y Fj(\024)2373 1305 y Fl(I)-17 b(P)2495 1194 y Fd(h)2542 1305 y Fj(j)o Fl(Y)22 b Fo(\()p Fl(I)8 b Fo(\))p Fj(j)27 b(\025)h Fl(\014)3019 1265 y Fk(\017)p 3012 1282 47 4 v 3012 1339 a(\015)3071 1194 y Fd(i)p 2153 1385 1186 4 v 2153 1512 a Fo(inf)2283 1525 y Fn(~)2274 1542 y Fk(I)2309 1552 y Fg(12)2382 1542 y Fp(\032)p Fk(I)2507 1512 y Fl(I)-17 b(P)2628 1401 y Fd(h)2675 1397 y(\014)2675 1457 y(\014)2675 1517 y(\014)2708 1512 y Fl(Y)22 b Fo(\()2839 1487 y(~)2827 1512 y Fl(I)2871 1527 y Fn(12)2955 1512 y Fo(\))2994 1397 y Fd(\014)2994 1457 y(\014)2994 1517 y(\014)3054 1512 y Fj(\024)29 b Fl(\014)3240 1473 y Fk(\017)p 3233 1489 47 4 v 3233 1546 a(\015)3291 1401 y Fd(i)3645 1408 y Fo(\(5)p Fl(:)p Fo(39\))0 1821 y Fa(Pro)s(of:)49 b Fo(Recall)29 b(that)g([)p Fl(`)936 1836 y Fn(1)980 1821 y Fl(;)17 b(`)1067 1836 y Fn(2)1110 1821 y Fo(])27 b Fj(\021)i Fl(I)36 b Fo(and)29 b(in)m(terv)-6 b(als)1947 1796 y(~)1935 1821 y Fl(I)1979 1836 y Fn(12)2091 1821 y Fo(=)28 b([)2231 1795 y(~)2224 1821 y Fl(`)2266 1836 y Fn(1)2310 1821 y Fl(;)2361 1795 y Fo(~)2355 1821 y Fl(`)2397 1836 y Fn(2)2440 1821 y Fo(].)43 b(Using)28 b(the)h(fact)g(that)g(for)g(all)3679 1796 y(~)3667 1821 y Fl(I)3711 1836 y Fn(12)3823 1821 y Fj(\032)0 1961 y Fl(I)8 b Fo(,)29 b Fj(j)p Fl(Y)21 b Fo(\()267 1935 y(~)255 1961 y Fl(I)299 1976 y Fn(12)383 1961 y Fo(\))p Fj(j)27 b(\024)i(j)p Fl(Y)21 b Fo(\([)p Fl(`)799 1976 y Fn(1)843 1961 y Fl(;)894 1934 y Fo(~)888 1961 y Fl(`)930 1976 y Fn(1)973 1961 y Fo(]\))p Fj(j)13 b Fo(+)g Fj(j)p Fl(Y)21 b Fo(\([)p Fl(`)1387 1976 y Fn(1)1431 1961 y Fl(;)1482 1934 y Fo(~)1476 1961 y Fl(`)1518 1976 y Fn(2)1562 1961 y Fo(]\))p Fj(j)13 b Fo(+)g Fl(\014)1832 1916 y Fn(4)p Fk(\020)1908 1926 y Fg(4)p 1832 1938 115 4 v 1867 1995 a Fk(\015)1959 1961 y Fo(,)29 b(w)m(e)g(get)g Fj(j)p Fl(Y)22 b Fo(\()2479 1935 y(~)2467 1961 y Fl(I)2511 1976 y Fn(12)2594 1961 y Fo(\))p Fj(j)28 b(\024)g Fo(2)17 b(max)3046 1992 y Fk(`)3079 2002 y Fg(1)3118 1992 y Fp(\024)3185 1973 y Fn(~)3180 1992 y Fk(`)p Fp(\024)p Fk(`)3308 2002 y Fg(2)3369 1961 y Fj(j)p Fl(Y)22 b Fo(\([)p Fl(`)3586 1976 y Fn(1)3630 1961 y Fl(;)3681 1934 y Fo(~)3675 1961 y Fl(`)o Fo(]\))p Fj(j)13 b Fo(+)0 2141 y Fl(\014)74 2097 y Fn(4)p Fk(\020)150 2107 y Fg(4)p 74 2118 V 108 2176 a Fk(\015)200 2141 y Fo(.)44 b(Therefore)245 2514 y Fl(I)-17 b(P)366 2344 y Fd(")448 2514 y Fo(max)433 2578 y Fn(~)424 2595 y Fk(I)459 2605 y Fg(1)p Ff(;)p Fg(2)555 2595 y Fp(\032)p Fk(I)674 2400 y Fd(\014)674 2459 y(\014)674 2519 y(\014)708 2514 y Fl(Y)22 b Fo(\()839 2489 y(~)827 2514 y Fl(I)871 2529 y Fn(12)955 2514 y Fo(\))994 2400 y Fd(\014)994 2459 y(\014)994 2519 y(\014)1054 2514 y Fj(\025)28 b Fl(\014)1233 2447 y Fo(4)p Fl(\017)22 b Fo(+)h(12)p Fl(\020)1589 2462 y Fn(4)p 1233 2491 400 4 v 1404 2582 a Fl(\015)1645 2344 y Fd(#)1731 2514 y Fj(\024)28 b Fl(I)-17 b(P)1957 2374 y Fd(\024)2068 2514 y Fo(max)2009 2596 y Fk(`)2042 2606 y Fg(1)2082 2596 y Fp(\024)2149 2578 y Fn(~)2144 2596 y Fk(`)p Fp(\024)p Fk(`)2272 2606 y Fg(2)2328 2400 y Fd(\014)2328 2459 y(\014)2328 2519 y(\014)2362 2514 y Fl(Y)22 b Fo(\([)p Fl(`)2551 2529 y Fn(1)2594 2514 y Fl(;)2645 2488 y Fo(~)2639 2514 y Fl(`)o Fo(]\))2747 2400 y Fd(\014)2747 2459 y(\014)2747 2519 y(\014)2807 2514 y Fj(\025)28 b Fl(\014)2986 2447 y Fo(2)p Fl(\017)22 b Fo(+)h(4)p Fl(\020)3292 2462 y Fn(4)p 2986 2491 351 4 v 3133 2582 a Fl(\015)3348 2374 y Fd(\025)3645 2514 y Fo(\(5)p Fl(:)p Fo(40\))0 2904 y(Let)52 b Fl(\034)69 b Fo(=)58 b(inf)583 2793 y Fd(n)649 2904 y Fl(t)28 b Fj(\025)g Fl(`)860 2919 y Fn(1)932 2904 y Fo(;)44 b Fj(j)p Fl(Y)22 b Fo(\([)p Fl(`)1221 2919 y Fn(1)1264 2904 y Fl(;)17 b(t)p Fo(]\))p Fj(j)26 b(\025)j Fl(\014)1645 2860 y Fn(2)p Fk(\017)p Fn(+4)p Fk(\020)1855 2870 y Fg(4)p 1645 2881 249 4 v 1747 2938 a Fk(\015)1906 2793 y Fd(o)1972 2904 y Fo(,)56 b(inf)8 b(\()p Fj(;)p Fo(\))58 b(=)g Fj(1)p Fo(.)99 b(Since,)56 b(for)c(all)f Fl(k)61 b Fj(2)d Fo([)p Fl(`)3670 2919 y Fn(1)3714 2904 y Fl(;)17 b(`)3801 2919 y Fn(2)3845 2904 y Fo(],)0 3101 y Fj(j)p Fl(Y)22 b Fo(\()p Fl(I)8 b Fo(\))p Fj(j)26 b(\025)i(j)p Fl(Y)22 b Fo(\([)p Fl(`)614 3116 y Fn(1)658 3101 y Fl(;)17 b(k)s Fo(]\))p Fj(j)j(\000)j(j)p Fl(Y)e Fo(\([)p Fl(k)k Fo(+)e(1)p Fl(;)17 b(`)1461 3116 y Fn(2)1504 3101 y Fo(]\))p Fj(j)k(\000)i Fl(\014)1793 3057 y Fn(4)p Fk(\020)1869 3067 y Fg(4)p 1793 3079 115 4 v 1828 3136 a Fk(\015)1920 3101 y Fo(,)33 b(w)m(e)h(ha)m(v)m(e)812 3330 y Fd(\010)870 3410 y Fl(\034)39 b Fo(=)28 b Fl(k)1113 3330 y Fd(\011)1193 3410 y Fj(\\)1282 3330 y Fd(\010)1340 3410 y Fj(j)p Fl(Y)22 b Fo(\([)p Fl(k)i Fo(+)f(1)p Fl(;)17 b(`)1828 3425 y Fn(2)1871 3410 y Fo(]\))p Fj(j)27 b(\024)h Fl(\014)2180 3343 y(\017)p 2172 3387 58 4 v 2172 3479 a(\015)2241 3330 y Fd(\011)2326 3410 y Fj(\032)2432 3330 y Fd(\010)2490 3410 y Fj(j)p Fl(Y)21 b Fo(\()p Fl(I)8 b Fo(\))p Fj(j)27 b(\025)h Fl(\014)2969 3343 y(\017)p 2961 3387 V 2961 3479 a(\015)3030 3330 y Fd(\011)3645 3410 y Fo(\(5)p Fl(:)p Fo(41\))0 3718 y(Therefore,)i(making)d(a)h(partition)g(o)m(v)m (er)g(the)h(p)s(ossible)g(v)-6 b(alues)29 b(of)f Fl(\034)40 b Fo(and)29 b(using)f(indep)s(endence,)k(w)m(e)d(get)330 4072 y Fl(I)-17 b(P)451 3932 y Fd(\024)504 4072 y Fj(j)o Fl(Y)22 b Fo(\()p Fl(I)8 b Fo(\))p Fj(j)27 b(\025)h Fl(\014)983 4005 y(\017)p 975 4049 V 975 4140 a(\015)1044 3932 y Fd(\025)1124 4072 y Fj(\025)124 b Fo(inf)1229 4137 y Fk(`)1262 4147 y Fg(1)1301 4137 y Fp(\024)p Fk(k)r Fp(\024)p Fk(`)1502 4147 y Fg(2)1559 4072 y Fl(I)-17 b(P)1680 3932 y Fd(\024)1732 4072 y Fj(j)p Fl(Y)22 b Fo(\([)p Fl(k)j Fo(+)d(1)p Fl(;)17 b(`)2220 4087 y Fn(2)2263 4072 y Fo(]\))p Fj(j)27 b(\024)h Fl(\014)2572 4005 y(\017)p 2564 4049 V 2564 4140 a(\015)2633 3932 y Fd(\025)2755 3946 y Fk(`)2788 3956 y Fg(2)2719 3977 y Fd(X)2702 4192 y Fk(k)r Fn(=)p Fk(`)2840 4202 y Fg(1)2896 4072 y Fl(I)-17 b(P)30 b Fo([)p Fl(\034)39 b Fo(=)28 b Fl(k)s Fo(])329 b(\(5)p Fl(:)p Fo(42\))0 4437 y(Using)33 b(the)h(de\014nition)f(of)h Fl(\034)11 b Fo(,)33 b(w)m(e)g(get)g(\(5.39\).)p 1763 4437 70 70 v 120 4694 a(W)-8 b(e)38 b(assume)g(without)g(loss)g(of)h (generalit)m(y)-8 b(,)38 b(that)g Fj(J)56 b Fo(is)38 b(cen)m(tered)i(at)d(the)i(origin)e(and)i(that)f Fj(j)p Fl(I)8 b Fj(j)35 b Fo(=)2 4833 y(^)-52 b Fl(c)214 4794 y Fn(1)p 55 4810 358 4 v 55 4873 a Fk(\015)18 b Fn(log)c(log)363 4846 y Fg(1)p 360 4857 41 4 v 360 4897 a Ff(\015)425 4833 y Fo(,)34 b(for)h(a)f(giv)m(en)i(^)-52 b Fl(c)p Fo(.)48 b(W)-8 b(e)34 b(mak)m(e)g(a)g(blo)s(c)m(k)g(decomp)s(osition)f (of)i(the)g(in)m(terv)-6 b(al)34 b Fj(J)52 b Fo(in)m(to)34 b(blo)s(c)m(ks)h(of)0 5061 y(length)j(^)-52 b Fl(c)p Fo(\(2)p Fl(\015)22 b Fo(log)17 b(log)818 5022 y Fn(1)p 815 5038 47 4 v 815 5095 a Fk(\015)873 5061 y Fo(\))912 5025 y Fp(\000)p Fn(1)1019 5061 y Fo(,)36 b(that)g(is)g Fj(J)50 b Fo(=)33 b Fj([)1699 5076 y Fp(\000)p Fk(j)1794 5086 y Fg(1)1833 5076 y Fp(\024)p Fk(j)t Fp(\024)p Fk(j)2027 5086 y Fg(1)2081 5036 y Fd(b)2072 5061 y Fl(I)2116 5076 y Fk(j)2194 5061 y Fo(with)i(2)p Fl(j)2514 5076 y Fn(1)2583 5061 y Fo(+)24 b(1)32 b(=)2876 4950 y Fd(h)2935 5022 y Fn(2)r(~)-42 b Fk(c)p 2935 5038 76 4 v 2957 5095 a Fn(^)g Fk(c)3022 5061 y Fo(\(log)18 b(1)p Fl(=\015)6 b Fo(\))3404 5025 y Fk(p)3465 5061 y Fo(log)17 b(log)3770 5022 y Fn(1)p 3767 5038 47 4 v 3767 5095 a Fk(\015)3825 4950 y Fd(i)3872 5061 y Fo(.)0 5201 y(Note)30 b(that)f(an)m(y)h(in)m (terv)-6 b(al)30 b Fl(I)37 b Fo(w)m(e)31 b(consider)g(is)f(included)h (in)f(the)g(union)h(of)f(three)h(consecutiv)m(e)g(in)m(terv)-6 b(als)9 5315 y Fd(b)0 5340 y Fl(I)44 5358 y Fn([)p Fk(j;j)t Fn(+2])316 5340 y Fj(\021)430 5315 y Fd(b)421 5340 y Fl(I)465 5355 y Fk(j)528 5340 y Fj([)626 5315 y Fd(b)617 5340 y Fl(I)661 5355 y Fk(j)t Fn(+1)825 5340 y Fj([)922 5315 y Fd(b)913 5340 y Fl(I)957 5355 y Fk(j)t Fn(+2)1133 5340 y Fo(for)33 b(some)f Fj(\000)p Fl(j)1652 5355 y Fn(1)1725 5340 y Fj(\024)c Fl(j)34 b Fj(\024)28 b Fl(j)2051 5355 y Fn(1)2117 5340 y Fj(\000)23 b Fo(2.)43 b(Therefore)35 b(w)m(e)e(get,)g(denoting)g(max)3724 5304 y Fp(\003)3724 5367 y Fk(I)5 b Fp(\032J)0 5579 y Fg(23)p Ff(=j)s(anuar)r(y)r(=)p Fg(1998;)32 b(12:06)1145 b Fo(48)p eop %%Page: 49 49 49 48 bop 0 100 a Fo(the)40 b(maxim)m(um)d(o)m(v)m(er)j(the)g(in)m (terv)-6 b(als)40 b Fl(I)47 b Fo(suc)m(h)42 b(that)d Fj(j)p Fl(I)8 b Fj(j)38 b Fo(=)j(^)-52 b Fl(c)p Fo(\()p Fl(\015)21 b Fo(log)d(log)2722 60 y Fn(1)p 2719 77 47 4 v 2719 134 a Fk(\015)2777 100 y Fo(\))2816 63 y Fp(\000)p Fn(1)2962 100 y Fo(that)40 b(are)g(in)g Fj(J)18 b Fo(,)41 b(for)f(all)0 239 y Fl(\017)28 b(>)g Fo(0,)33 b(setting)h(~)-51 b Fl(\017)28 b Fo(=)g(4)p Fl(\017)23 b Fo(+)f(12)p Fl(\020)1140 254 y Fn(4)1184 239 y Fo(,)33 b(w)m(e)h(ha)m(v)m(e)159 545 y Fl(I)-17 b(P)280 405 y Fd(\024)332 545 y Fo(max)340 609 y Fk(I)5 b Fp(\032J)518 504 y(\003)593 545 y Fo(max)589 609 y Fn(~)580 626 y Fk(I)615 636 y Fg(12)688 626 y Fp(\032)p Fk(I)808 545 y Fj(j)p Fl(Y)22 b Fo(\()967 520 y(~)955 545 y Fl(I)999 560 y Fn(12)1083 545 y Fo(\))p Fj(j)27 b(\025)h Fl(\014)1365 478 y Fo(~)-51 b Fl(\017)p 1355 522 58 4 v 1355 613 a(\015)1424 405 y Fd(\025)1505 545 y Fj(\024)1622 458 y Fo(2)r(~)f Fl(c)p Fo(\(log)17 b(1)p Fl(=\015)6 b Fo(\))2096 422 y Fk(p)2157 458 y Fo(log)17 b(log)2463 419 y Fn(1)p 2459 435 47 4 v 2459 493 a Fk(\015)p 1622 522 896 4 v 2050 613 a Fo(^)-52 b Fl(c)2529 545 y(I)-17 b(P)2651 375 y Fd(")2787 545 y Fo(max)2718 621 y Fn(~)2709 638 y Fk(I)2744 648 y Fg(12)2817 638 y Fp(\032)2880 633 y Fd(b)2879 638 y Fk(I)2914 651 y Fg([0)p Ff(;)p Fg(2])3067 545 y Fj(j)p Fl(Y)21 b Fo(\()3225 520 y(~)3213 545 y Fl(I)3257 560 y Fn(12)3341 545 y Fo(\))p Fj(j)27 b(\025)i Fl(\014)3623 478 y Fo(~)-51 b Fl(\017)p 3614 522 58 4 v 3614 613 a(\015)3683 375 y Fd(#)3645 789 y Fo(\(5)p Fl(:)p Fo(43\))120 929 y(Using)33 b(\(5.38\))e(and)j(\(5.39\)) e(w)m(e)i(ha)m(v)m(e)209 1241 y Fl(I)-17 b(P)330 1070 y Fd(")388 1241 y Fo(max)395 1305 y Fk(I)5 b Fp(\032J)573 1199 y(\003)649 1241 y Fo(max)645 1304 y Fn(~)636 1322 y Fk(I)671 1332 y Fg(12)744 1322 y Fp(\032)p Fk(I)863 1066 y Fd(\014)863 1126 y(\014)863 1186 y(\014)863 1246 y(\014)863 1305 y(\014)897 1241 y Fo(log)1054 1173 y Fl(Z)1122 1188 y Fp(\000)p Fk(\021)s(;)p Fn(0)p Fk(;\016)1352 1198 y Fg(4)1390 1188 y Fk(;\020)1450 1198 y Fg(4)1494 1173 y Fo(\()1545 1148 y(~)1533 1173 y Fl(I)1577 1188 y Fn(12)1660 1173 y Fo(\))p 1054 1218 646 4 v 1085 1321 a Fl(Z)1153 1336 y Fk(\021)s(;)p Fn(0)p Fk(;\016)1321 1346 y Fg(4)1359 1336 y Fk(;\020)1419 1346 y Fg(4)1462 1321 y Fo(\()1513 1296 y(~)1501 1321 y Fl(I)1545 1336 y Fn(12)1629 1321 y Fo(\))1711 1066 y Fd(\014)1711 1126 y(\014)1711 1186 y(\014)1711 1246 y(\014)1711 1305 y(\014)1772 1241 y Fj(\025)28 b Fl(\014)1960 1173 y Fo(~)-51 b Fl(\017)p 1951 1218 58 4 v 1951 1309 a(\015)2020 1070 y Fd(#)2106 1241 y Fj(\024)2223 1154 y Fo(2)r(~)f Fl(c)p Fo(\(log)2515 1115 y Fn(1)p 2512 1131 47 4 v 2512 1188 a Fk(\015)2570 1154 y Fo(\))2609 1118 y Fk(p)p Fn(+)p Fk(\032)p 2223 1218 535 4 v 2471 1309 a Fo(^)f Fl(c)2867 1173 y(e)2913 1129 y Fp(\000)p Fk(u)14 b Fn(log)h(log)3285 1102 y Fg(1)p 3281 1113 41 4 v 3281 1153 a Ff(\015)p 2781 1218 644 4 v 2781 1333 a Fo(1)22 b Fj(\000)h Fl(e)2999 1289 y Fp(\000)p Fk(u)14 b Fn(log)h(log)3371 1262 y Fg(1)p 3367 1273 41 4 v 3367 1313 a Ff(\015)3645 1241 y Fo(\(5)p Fl(:)p Fo(44\))0 1584 y(where)32 b Fl(u)c Fj(\021)716 1539 y Fn(~)-41 b Fk(\017)748 1509 y Fg(2)787 1539 y Fk(\014)836 1509 y Fg(2)p 488 1561 615 4 v 488 1618 a Fn(212)r(^)f Fk(c)n(\014)t(\022)r(\020)767 1628 y Fg(4)806 1618 y Fk(g)845 1628 y Fg(2)884 1618 y Fn(\(1)p Fk(=\020)1032 1628 y Fg(4)1071 1618 y Fn(\))1145 1584 y Fo(and)31 b Fl(\032)d(>)g Fo(0)j(is)g(small)e(as)i(w)m(e)g(w)m(an)m(t.)44 b(W)-8 b(e)31 b(assume)g(for)h(the)f(momen)m(t)e(that)0 1723 y(the)j(v)-6 b(arious)32 b(parameters)g(are)g(c)m(hosen)h(suc)m(h) h(that)d Fl(u)d Fj(\025)g Fl(p)20 b Fo(+)g(1)g(+)g(2)p Fl(\032)p Fo(.)43 b(Using)31 b(the)i(\014rst)f(Borel-Can)m(telli)0 1863 y(lemma,)40 b(recalling)h(that)f Fl(\015)46 b Fo(=)40 b(2)1253 1827 y Fp(\000)p Fk(n)1370 1863 y Fo(,)i(w)m(e)f(get)g(that)f (with)h(a)g Fl(I)-17 b(P)13 b Fo(-probabilit)m(y)40 b(1,)j(for)e(all)f (but)h(a)g(\014nite)0 2002 y(n)m(um)m(b)s(er)34 b(of)f(indices)h Fl(n)1121 2201 y Fo(max)1129 2266 y Fk(I)5 b Fp(\032J)1307 2160 y(\003)1382 2201 y Fo(max)1378 2265 y Fn(~)1369 2282 y Fk(I)1404 2292 y Fg(12)1477 2282 y Fp(\032)p Fk(I)1609 2134 y Fl(Z)1677 2149 y Fp(\000)p Fk(\021)s(;)p Fn(0)p Fk(;\016)1907 2159 y Fg(4)1945 2149 y Fk(;\020)2005 2159 y Fg(4)2049 2134 y Fo(\()2100 2109 y(~)2088 2134 y Fl(I)2132 2149 y Fn(12)2216 2134 y Fo(\))p 1609 2178 646 4 v 1640 2282 a Fl(Z)1708 2297 y Fk(\021)s(;)p Fn(0)p Fk(;\016)1876 2307 y Fg(4)1914 2297 y Fk(;\020)1974 2307 y Fg(4)2017 2282 y Fo(\()2068 2257 y(~)2056 2282 y Fl(I)2100 2297 y Fn(12)2184 2282 y Fo(\))2294 2201 y Fj(\024)28 b Fl(e)2445 2157 y Fk(\014)2506 2119 y Fg(4)p Ff(\017)p Fg(+12)p Ff(\020)2723 2134 y Fg(4)p 2506 2141 256 4 v 2614 2181 a Ff(\015)3645 2201 y Fo(\(5)p Fl(:)p Fo(45\))120 2475 y(It)k(remains)h(to)g(estimate)f(the)i(\014rst)f(term)f(in)i(the)f (righ)m(t)g(hand)h(side)g(of)g(\(5.37\).)42 b(W)-8 b(e)34 b(ha)m(v)m(e)120 2615 y(\001)p Fj(G)6 b Fo(\()p Fl(m)394 2579 y Fk(\016)433 2549 y Fb(\003)394 2660 y Fk(\014)t(;)476 2642 y Fn(~)467 2660 y Fk(I)502 2670 y Fg(12)579 2615 y Fo(\))28 b(=)g Fj(\000)p Fl(\021)898 2541 y Fd(P)1003 2646 y Fk(x)p Fp(2C)1145 2660 y Ff(\016)1180 2645 y Fb(\003)1226 2646 y Fn(\()1266 2628 y(~)1257 2646 y Fk(I)1292 2656 y Fg(12)1365 2646 y Fn(\))1418 2615 y Fl(X)8 b Fo(\()p Fl(x)p Fo(\))32 b(where)86 2956 y Fl(X)8 b Fo(\()p Fl(x)p Fo(\))27 b Fj(\021)h(\000)p Fo(2)p Fl(\014)5 b(\022)s(\025)p Fo(\()p Fl(x)p Fo(\))p Fj(j)p Fl(D)s Fo(\()p Fl(x)p Fo(\))p Fj(j)1168 2846 y Fd(h)1213 2956 y Fl(m)1300 2915 y Fk(\016)1339 2885 y Fb(\003)1300 2981 y Fk(\014)t(;)p Fn(1)1440 2956 y Fo(+)23 b Fl(m)1627 2915 y Fk(\016)1666 2885 y Fb(\003)1627 2981 y Fk(\014)t(;)p Fn(2)1767 2956 y Fo(+)f(\004\()p Fl(x;)17 b(\014)5 b(\022)s(;)17 b(\013)p Fo(\))2332 2846 y Fd(i)2400 2956 y Fj(\000)23 b Fl(\025)p Fo(\()p Fl(x)p Fo(\))17 b(log)2866 2854 y(\011)2943 2881 y Fk(\014)t(\022)r(;\013)p Fn(\()p Fk(x)p Fn(\))p Fk(;m)3310 2857 y Ff(\016)3345 2842 y Fb(\003)3310 2908 y Ff(\014)s(;)p Fg(2)3418 2854 y Fo(\011)3495 2881 y Fn(0)p Fk(;)p Fn(0)p Fk(;m)3694 2857 y Ff(\016)3729 2842 y Fb(\003)3694 2908 y Ff(\014)s(;)p Fg(1)p 2866 2933 936 4 v 2866 3024 a Fo(\011)2943 3052 y Fk(\014)t(\022)r(;\013)p Fn(\()p Fk(x)p Fn(\))p Fk(;m)3310 3028 y Ff(\016)3345 3013 y Fb(\003)3310 3079 y Ff(\014)s(;)p Fg(1)3418 3024 y Fo(\011)3495 3052 y Fn(0)p Fk(;)p Fn(0)p Fk(;m)3694 3028 y Ff(\016)3729 3013 y Fb(\003)3694 3079 y Ff(\014)s(;)p Fg(2)3645 3203 y Fo(\(5)p Fl(:)p Fo(46\))0 3342 y(with)22 b(\004\()p Fl(x;)17 b(\014)5 b(\022)s(;)17 b(\013)p Fo(\))27 b Fj(\021)h Fo([)13 b Fd(b)-68 b Fl(')p Fo(\()p Fl(m)1033 3306 y Fk(\016)1072 3276 y Fb(\003)1033 3371 y Fk(\014)t(;)p Fn(1)1150 3342 y Fl(;)17 b(\025)p Fo(\()p Fl(x)p Fo(\))p Fl(\014)5 b(\022)s(;)17 b(\013)p Fo(\))q Fj(\000)d Fd(b)-68 b Fl(')o Fo(\()p Fl(m)1916 3306 y Fk(\016)1955 3276 y Fb(\003)1916 3371 y Fk(\014)t(;)p Fn(2)2033 3342 y Fl(;)17 b(\025)p Fo(\()p Fl(x)p Fo(\))p Fl(\014)5 b(\022)s(;)17 b(\013)p Fo(\)].)38 b(Therefore)24 b(w)m(e)f(need)h(to)e(estimate)0 3482 y(from)33 b(ab)s(o)m(v)m(e)g(the) g(probabilit)m(y)g(of)1062 3694 y Fj(A)28 b(\021)1275 3583 y Fd(n)1341 3694 y Fo(max)1349 3758 y Fk(I)5 b Fp(\032J)1527 3652 y(\003)1602 3694 y Fo(max)1598 3757 y Fn(~)1589 3775 y Fk(I)1624 3785 y 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b(E)5 b Fo(\()p Fl(X)j Fo(\))26 b(=)j(0,)j(w)m(e)i(get)971 580 y Fl(I)-17 b(E)1094 469 y Fd(h)1141 580 y Fl(e)1187 539 y Fk(t\015)t(X)5 b Fn(\()p Fk(x)p Fn(\))1447 469 y Fd(i)1522 580 y Fj(\024)28 b Fo(1)22 b(+)g(\()p Fl(t\015)6 b Fo(\))1970 539 y Fn(2)2026 512 y Fl(I)-17 b(E)5 b Fo([)p Fl(X)2251 476 y Fn(2)2294 512 y Fo(\()p Fl(x)p Fo(\)])p 2026 557 431 4 v 2216 648 a(2)2468 580 y Fl(e)2514 539 y Fk(t\015)t Fp(k)p Fk(X)g Fn(\()p Fk(x)p Fn(\))p Fp(k)2849 549 y Fb(1)3645 580 y Fo(\(5)p Fl(:)p Fo(49\))0 845 y(Using)37 b(the)h(Prop)s(osition)f (3.4,)g(if)h Fl(\015)k Fo(is)c(small)e(enough,)j(and)f(ho)m(w)g(small)e (dep)s(ends)j(on)f Fl(\014)5 b(\022)40 b Fo(to)d(absorb)0 984 y(the)30 b(last)g(term)f(in)h(\(3.54\),)f(w)m(e)h(ha)m(v)m(e)h(for) f(some)f(p)s(ositiv)m(e)h(constan)m(t)g Fl(c)p Fo(,)g Fj(k)p Fl(X)8 b Fo(\()p Fl(x)p Fo(\))p Fj(k)2971 999 y Fp(1)3082 984 y Fj(\024)28 b Fo(4)p Fl(\014)5 b(\022)3360 945 y Fk(\016)3399 915 y Fb(\003)p 3360 962 81 4 v 3377 1019 a Fk(\015)3469 984 y Fo(\(1)22 b(+)g Fl(c\014)5 b(\022)s Fo(\).)0 1165 y(On)32 b(the)f(other)h(hand)f(it)g(is)g(easy)g (to)f(c)m(hec)m(k)j(that,)e(calling)f Fl(I)-17 b(E)5 b Fo([)p Fj(j)p Fl(D)s Fo(\()p Fl(x)p Fo(\))p Fj(j)2602 1129 y Fn(2)2645 1165 y Fo(])28 b(=)g Fl(V)2886 1129 y Fn(2)2931 1165 y Fo(\()p Fl(x)p Fo(\))f(=)3210 1125 y Fk(\016)3249 1095 y Fb(\003)p 3210 1142 V 3227 1199 a Fk(\015)3302 1165 y Fo(,)k(w)m(e)h(ha)m(v)m(e)f(also)0 1345 y(for)g(some)g(p)s(ositiv)m(e)f(constan)m(t)i Fl(c)p Fo(,)f(if)h Fl(\015)k Fo(is)31 b(small)e(enough)j Fl(I)-17 b(E)5 b Fo([)p Fl(X)2366 1309 y Fn(2)2410 1345 y Fo(\()p Fl(x)p Fo(\)])27 b Fj(\024)h Fo(16\()p Fl(\014)5 b(\022)s Fo(\))2994 1309 y Fn(2)3038 1345 y Fo(\(1)18 b(+)g Fl(c\014)5 b(\022)s Fo(\))3433 1309 y Fn(2)3490 1306 y Fk(\016)3529 1276 y Fb(\003)p 3490 1322 V 3507 1379 a Fk(\015)3582 1345 y Fo(.)43 b(Using)0 1546 y(1)22 b(+)h Fl(x)k Fj(\024)h Fl(e)407 1509 y Fk(x)491 1546 y Fj(8)p Fl(x)g Fj(2)g Fl(I)-17 b(R)33 b Fo(and)h Fj(jC)1135 1561 y Fk(\016)1174 1541 y Fb(\003)1221 1546 y Fo(\()1272 1520 y(~)1260 1546 y Fl(I)1304 1561 y Fn(12)1388 1546 y Fo(\))p Fj(j)27 b Fo(=)1599 1497 y Fp(j)1632 1480 y Fn(~)1623 1497 y Fk(I)1658 1507 y Fg(12)1731 1497 y Fp(j)p 1599 1523 156 4 v 1636 1580 a Fk(\016)1675 1560 y Fb(\003)1766 1546 y Fo(,)33 b(w)m(e)h(get)f(easily)671 1765 y Fd(Y)538 1996 y Fk(x)p Fp(2C)680 2010 y Ff(\016)715 1995 y Fb(\003)761 1996 y Fn(\()801 1978 y(~)792 1996 y Fk(I)827 2006 y Fg(12)901 1996 y Fn(\))948 1860 y Fl(I)-17 b(E)1071 1749 y Fd(h)1118 1860 y Fl(e)1164 1819 y Fk(tX)5 b Fn(\()p Fk(x)p Fn(\))1378 1749 y Fd(i)1452 1860 y Fj(\024)29 b Fo(exp)1726 1749 y Fd(h)1773 1860 y Fl(\015)6 b Fo(8\()p Fl(t\014)f(\022)s Fo(\))2106 1819 y Fn(2)2150 1860 y Fo(\(1)21 b(+)i Fl(c\014)5 b(\022)s Fo(\))p Fj(j)2593 1835 y Fo(~)2581 1860 y Fl(I)2625 1875 y Fn(12)2709 1860 y Fj(j)p Fl(e)2783 1819 y Fk(t\016)2852 1789 y Fb(\003)2893 1819 y Fn(4)p Fk(\014)t(\022)r Fn(\(1+)p Fk(c\014)t(\022)r Fn(\))3315 1749 y Fd(i)3645 1860 y Fo(\(5)p Fl(:)p 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Fk(\016)2386 2427 y Fb(\003)p 2294 2474 187 4 v 2294 2531 a Fk(g)2333 2541 y Fg(3)2372 2531 y Fn(\()p Fk(\015)t Fn(\))2492 2496 y Fo(,)k(w)m(e)f(c)m(ho)s(ose) h Fl(t)c Fo(=)3343 2432 y Fk(s)13 b Fn(log)i(log)3643 2405 y Fg(1)p 3639 2416 41 4 v 3639 2456 a Ff(\015)p 3221 2474 593 4 v 3221 2531 a Fn(32)r(^)-42 b Fk(c)o Fn(\()p Fk(\014)t(\022)r Fn(\))3487 2511 y Fg(2)3526 2531 y Fn(\(1+)p Fk(c\014)t(\022)r Fn(\))3826 2496 y Fl(s)p Fo(.)0 2684 y(Assuming)33 b(that)g Fl(g)722 2699 y Fn(3)766 2684 y Fo(\()p Fl(\015)6 b Fo(\))32 b(is)h(suc)m(h)i(that)d Fl(\015)6 b Fo(\(log)17 b(log)1876 2644 y Fn(1)p 1873 2661 47 4 v 1873 2718 a Fk(\015)1931 2684 y Fo(\))1970 2647 y Fn(2)2042 2684 y Fj(\024)28 b Fo(\()p Fl(g)2234 2699 y Fn(3)2278 2684 y Fo(\()p Fl(\015)6 b Fo(\)\))2453 2647 y Fn(2)2496 2684 y Fo(,)33 b(in)g(b)s(oth)g(the)h(cases,)g(w)m(e)g (get)330 3101 y Fl(I)-17 b(P)452 2901 y Fd(2)452 3080 y(4)518 2987 y(\014)518 3046 y(\014)518 3106 y(\014)551 3101 y Fl(\015)750 3007 y Fd(X)625 3237 y Fk(x)p Fp(2C)767 3251 y Ff(\016)802 3236 y Fb(\003)848 3237 y Fn(\()888 3220 y(~)879 3237 y Fk(I)914 3247 y Fg(12)987 3237 y Fn(\))1035 3101 y Fl(X)8 b Fo(\()p Fl(x)p Fo(\))1261 2987 y Fd(\014)1261 3046 y(\014)1261 3106 y(\014)1321 3101 y Fj(\025)28 b Fl(\014)5 b(s)1534 2901 y Fd(3)1534 3080 y(5)1628 3101 y Fj(\024)28 b Fo(2)17 b(exp)1968 2931 y Fd(")2027 3101 y Fj(\000)2116 3014 y Fl(s)2163 2978 y Fn(2)2224 3014 y Fo(log)g(log)2530 2975 y Fn(1)p 2527 2992 V 2527 3049 a Fk(\015)2585 3014 y Fo(\(1)k Fj(\000)i Fo(2)p Fl(s)2892 3029 y Fn(0)2936 3014 y Fl(cg)3027 3029 y Fn(3)3071 3014 y Fo(\()p Fl(\015)6 b Fo(\)\))p 2116 3078 1129 4 v 2322 3170 a(32)r(^)-52 b Fl(c)o Fo(\()p Fl(\022)s Fo(\))2592 3141 y Fn(2)2636 3170 y Fo(\(1)22 b(+)g Fl(c\014)5 b(\022)s Fo(\))3257 2931 y Fd(#)3645 3101 y Fo(\(5)p Fl(:)p Fo(51\))0 3511 y(for)39 b Fl(s)d Fj(\024)h Fl(s)402 3526 y Fn(0)484 3511 y Fo(and)i(for)g(some)f (constan)m(t)h Fl(c)1584 3475 y Fn(1)1629 3511 y Fo(.)59 b(T)-8 b(o)38 b(get)h(uniformit)m(y)e(with)h(resp)s(ect)h(to)f(all)g (subin)m(terv)-6 b(als)0 3651 y(that)33 b(are)g(in)g Fl(I)8 b Fo(,)33 b(w)m(e)h(write)e(simply)939 4019 y(max)935 4082 y Fn(~)926 4100 y Fk(I)961 4110 y Fg(12)1034 4100 y Fp(\032)p Fk(I)1137 3904 y Fd(\014)1137 3964 y(\014)1137 4024 y(\014)1170 4019 y Fl(\015)1369 3924 y Fd(X)1244 4154 y Fk(x)p Fp(2C)1386 4168 y Ff(\016)1421 4153 y Fb(\003)1467 4154 y Fn(\()1507 4137 y(~)1498 4154 y Fk(I)1533 4164 y Fg(12)1606 4154 y Fn(\))1654 4019 y Fl(X)8 b Fo(\()p Fl(x)p Fo(\))1880 3904 y Fd(\014)1880 3964 y(\014)1880 4024 y(\014)1940 4019 y Fj(\024)28 b Fo(2)75 b(max)2112 4084 y Fk(`)2145 4094 y Fg(1)2184 4084 y Fp(\024)p Fk(`)p Fp(\024)p Fk(`)2374 4094 y Fg(2)2414 3904 y Fd(\014)2414 3964 y(\014)2414 4024 y(\014)2447 4019 y Fl(\015)2593 3894 y Fk(`)2538 3924 y Fd(X)2521 4138 y Fk(x)p Fn(=)p Fk(`)2660 4148 y Fg(1)2716 4019 y Fl(X)8 b Fo(\()p Fl(x)p Fo(\))2942 3904 y Fd(\014)2942 3964 y(\014)2942 4024 y(\014)3645 4019 y Fo(\(5)p Fl(:)p Fo(52\))0 4417 y(Therefore,)34 b(using)g(the)g(Otta)m(viani)e(inequalit)m(y)255 4820 y Fl(I)-17 b(P)376 4649 y Fd(")492 4820 y Fo(max)434 4885 y Fk(`)467 4895 y Fg(1)506 4885 y Fp(\024)p Fk(`)p Fp(\024)p Fk(`)696 4895 y Fg(2)753 4645 y Fd(\014)753 4705 y(\014)753 4765 y(\014)753 4825 y(\014)753 4884 y(\014)786 4820 y Fl(\015)933 4695 y Fk(`)877 4725 y Fd(X)860 4940 y Fk(x)p Fn(=)p Fk(`)999 4950 y Fg(1)1055 4820 y Fl(X)8 b Fo(\()p Fl(x)p Fo(\))1281 4645 y Fd(\014)1281 4705 y(\014)1281 4765 y(\014)1281 4825 y(\014)1281 4884 y(\014)1341 4820 y Fj(\025)28 b Fo(2)p Fl(\014)5 b(s)1604 4649 y Fd(#)1690 4820 y Fj(\024)2029 4716 y Fl(I)-17 b(P)2151 4606 y Fd(h)2198 4602 y(\014)2198 4661 y(\014)2198 4721 y(\014)2231 4716 y Fl(\015)2305 4641 y Fd(P)2410 4666 y Fk(`)2443 4676 y Fg(2)2410 4746 y Fk(x)p Fn(=)p Fk(`)2549 4756 y Fg(1)2610 4716 y Fl(X)8 b Fo(\()p Fl(x)p Fo(\))2836 4602 y Fd(\014)2836 4661 y(\014)2836 4721 y(\014)2896 4716 y Fj(\025)28 b Fl(\014)5 b(s)3109 4606 y Fd(i)p 1807 4797 1572 4 v 1807 4923 a Fo(inf)1928 4938 y Fk(`)1961 4948 y Fg(1)2000 4938 y Fp(\024)p Fk(`)p Fp(\024)p Fk(`)2190 4948 y Fg(2)2252 4923 y Fl(I)-17 b(P)2373 4813 y Fd(h)2420 4809 y(\014)2420 4869 y(\014)2420 4928 y(\014)2453 4923 y Fl(\015)2527 4849 y Fd(P)2632 4873 y Fk(`)2632 4953 y(x)p Fn(=)p Fk(`)2771 4963 y Fg(1)2832 4923 y Fl(X)8 b Fo(\()p Fl(x)p Fo(\))3058 4809 y Fd(\014)3058 4869 y(\014)3058 4928 y(\014)3118 4923 y Fj(\024)28 b Fl(\014)5 b(s)3331 4813 y Fd(i)3645 4820 y Fo(\(5)p Fl(:)p Fo(53\))p 0 5090 1200 4 v 25 5164 a Fn(1)120 5201 y Fi(Remark)32 b(that)g(giv)n(en)g Fl(s)902 5216 y Fn(0)983 5201 y Fl(>)k Fo(0)c Fi(it)h(is)h(alw)n(a)n(ys)f(p)r(ossible)h(to)f(\014nd)g Fl(\015)2288 5216 y Fn(0)2368 5201 y Fl(>)j Fo(0)c Fi(suc)n(h)h(that)f (for)h Fl(\015)41 b Fj(\024)36 b Fl(\015)3338 5216 y Fn(0)3382 5201 y Fi(,)d(the)f(quan)n(tit)n(y)0 5340 y Fo(\(1)22 b Fj(\000)g Fo(2)p Fl(s)307 5355 y Fn(0)352 5340 y Fl(cg)443 5355 y Fn(3)487 5340 y Fo(\()p Fl(\015)6 b Fo(\)\))26 b Fi(is)k(strictly)e(p)r(ositiv)n(e.)0 5579 y Fg(23)p Ff(=j)s(anuar)r(y)r(=)p Fg(1998;)k(12:06)1145 b Fo(50)p eop %%Page: 51 51 51 50 bop 0 101 a Fo(w)m(e)34 b(get,)e(setting)40 b(~)-57 b Fl(u)28 b Fo(=)890 52 y Fn(\(1)p Fp(\000)p Fn(2)p Fk(s)1101 62 y Fg(0)1139 52 y Fk(cg)1214 62 y Fg(3)1253 52 y Fn(\()p Fk(\015)t Fn(\)\))p 869 78 544 4 v 869 135 a(32)r(^)-42 b Fk(c)o Fn(\()p Fk(\022)r Fn(\))1086 115 y Fg(2)1126 135 y Fn(\(1+)p Fk(c\014)t(\022)r Fn(\))1458 101 y Fo(b)m(y)34 b(an)f(argumen)m(t)g(similar)e(to)i(the)h(one)f(that)g(giv)m(es)g (\(5.43\),)339 516 y Fl(I)-17 b(P)460 316 y Fd(2)460 495 y(4)526 516 y Fo(max)534 580 y Fk(I)5 b Fp(\032J)712 475 y(\003)787 516 y Fo(max)783 580 y Fn(~)774 597 y Fk(I)809 607 y Fg(12)882 597 y Fp(\032)p Fk(I)1002 312 y Fd(\014)1002 372 y(\014)1002 432 y(\014)1002 491 y(\014)1002 551 y(\014)1002 611 y(\014)1035 516 y Fl(\015)1200 422 y Fd(X)1109 640 y Fk(x)p Fp(2C)1251 654 y Ff(\016)1286 639 y Fb(\003)1332 640 y Fn(\()p Fk(I)g Fn(\))1451 516 y Fl(X)j Fo(\()p Fl(x)p Fo(\))1677 312 y Fd(\014)1677 372 y(\014)1677 432 y(\014)1677 491 y(\014)1677 551 y(\014)1677 611 y(\014)1737 516 y Fj(\025)28 b Fo(2)p Fl(\014)5 b(s)2000 316 y Fd(3)2000 495 y(5)2094 516 y Fj(\024)2212 429 y Fo(4)r(~)-52 b Fl(c)o Fo(\(log)2504 390 y Fn(1)p 2500 407 47 4 v 2500 464 a Fk(\015)2559 429 y Fo(\))2598 393 y Fk(p)p Fn(+)p Fk(\032)p 2212 493 535 4 v 2459 585 a Fo(^)g Fl(c)2758 516 y(e)2804 472 y Fp(\000)p Fk(s)2904 442 y Fg(2)2949 472 y Fn(~)-46 b Fk(u)14 b Fn(log)g(log)3252 445 y Fg(1)p 3249 456 41 4 v 3249 496 a Ff(\015)3645 516 y Fo(\(5)p Fl(:)p Fo(54\))120 906 y(W)-8 b(e)33 b(then)h(collect)f (\(5.34\),)e(\(5.36\),)h(\(5.37\),)f(\(5.45\))h(obtaining)138 1235 y Fl(\026)198 1250 y Fk(\014)t(;\022)r(;\015)402 1124 y Fd(\020)462 1235 y Fj(W)568 1194 y Fk(\016)604 1204 y Fg(1)643 1194 y Fk(;\020)703 1204 y Fg(1)742 1194 y Fk(;\016)802 1204 y Fg(4)841 1194 y Fk(;\020)901 1204 y Fg(4)560 1260 y Fk(\021)945 1235 y Fo(\()p Fl(`)1026 1250 y Fn(1)1070 1235 y Fl(;)1121 1209 y Fo(~)1115 1235 y Fl(`)1157 1250 y Fn(1)1200 1235 y Fl(;)1251 1209 y Fo(~)1245 1235 y Fl(`)1287 1250 y Fn(2)1331 1235 y Fl(;)17 b(`)1418 1250 y Fn(2)1461 1235 y Fo(\))1500 1124 y Fd(\021)1587 1235 y Fj(\024)28 b Fo(exp)1861 1094 y Fd(\024)1914 1235 y Fj(\000)2003 1168 y Fl(\014)p 2003 1212 62 4 v 2005 1303 a(\015)2093 1235 y Fo(\()p Fl(x)p Fo(\001)p Fj(F)j(\000)23 b Fo(24)p Fl(\020)2619 1250 y Fn(4)2685 1235 y Fj(\000)g Fo(4)p Fl(\017)f Fj(\000)h Fo(4)p Fl(s)e Fj(\000)i Fl(\016)3263 1194 y Fp(\003)3309 1235 y Fj(j)p Fl(I)8 b Fj(j)p Fo(\))3455 1094 y Fd(\025)3645 1235 y Fo(\(5)p Fl(:)p Fo(55\))120 1592 y(W)-8 b(e)39 b(mak)m(e)e(the)i(follo)m(wing)f(c)m(hoices)i Fl(s)c Fj(\024)i Fl(s)1751 1607 y Fn(0)1832 1592 y Fo(=)1958 1552 y Fk(x)2003 1522 y Fg(2)2042 1552 y Fn(\001)p Fp(F)p 1958 1569 214 4 v 2025 1626 a Fn(16)2183 1592 y Fo(,)i Fl(\017)d Fo(=)2474 1552 y Fn(1)p 2454 1569 80 4 v 2454 1626 a(16)2546 1592 y Fl(x)2603 1555 y Fn(2)2647 1592 y Fo(\001)p Fj(F)10 b Fo(,)39 b Fl(c)2922 1607 y Fn(0)3004 1592 y Fo(=)3130 1552 y Fk(x)3175 1522 y Fg(2)3214 1552 y Fn(\001)p Fp(F)p 3130 1569 214 4 v 3199 1626 a Fn(4)r(^)-42 b Fk(c)3356 1592 y Fo(,)39 b Fl(\020)3467 1607 y Fn(4)3548 1592 y Fj(\024)3674 1552 y Fk(x)3719 1522 y Fg(2)3758 1552 y Fn(\001)p Fp(F)p 3674 1569 V 3742 1626 a Fn(96)0 1731 y Fo(this)49 b(will)f(giv)m(e)g(us)h(\(5.33\).)89 b(W)-8 b(e)49 b(tak)m(e)g Fl(\015)1621 1746 y Fn(0)1714 1731 y Fo(suc)m(h)h(that)e(\(1)33 b Fj(\000)f Fo(2)p Fl(s)2513 1746 y Fn(0)2558 1731 y Fl(cg)2649 1746 y Fn(3)2693 1731 y Fo(\()p Fl(\015)2784 1746 y Fn(0)2828 1731 y Fo(\)\))53 b(=)3102 1692 y Fn(1)p 3102 1708 40 4 v 3102 1765 a(2)3154 1731 y Fo(.)90 b(T)-8 b(o)48 b(b)s(e)i(able)e(to)0 1871 y(satisfy)g Fl(s)373 1834 y Fn(2)423 1871 y Fo(~)-56 b Fl(u)52 b Fj(\025)h Fl(p)32 b Fo(+)g(2)p Fl(\032)f Fo(+)i(1)47 b Fe(and)h Fl(s)k Fj(\024)g Fl(s)1668 1886 y Fn(0)1713 1871 y Fo(,)f(w)m(e)d(imp)s(ose)f Fl(\022)55 b Fj(\024)2712 1831 y Fk(x)2757 1801 y Fg(2)2796 1831 y Fn(\001)p Fp(F)p 2546 1848 545 4 v 2546 1927 a Fn(48)2626 1860 y Fj(p)p 2709 1860 383 4 v 2711 1927 a Fn(^)-42 b Fk(c)o Fn(\()p Fk(p)p Fn(+2)p Fk(\032)p Fn(+1\))3150 1871 y Fo(and)49 b(w)m(e)f(can)g(tak)m(e)0 2088 y Fl(s)27 b Fo(=)i(16)p Fl(\022)330 2003 y Fd(p)p 428 2003 566 4 v 85 x Fo(^)-52 b Fl(c)p Fo(\()p Fl(p)22 b Fo(+)h(2)p Fl(\032)f Fo(+)g(1\).)40 b(Recalling)23 b(that)f(w)m(e)h(need)h(also)f Fl(u)k Fj(\025)i Fl(p)q Fo(+)q(2)p Fl(\032)q Fo(+)q(1,)c(w)m(e)e(imp)s (ose)f(that)h Fl(\020)3552 2103 y Fn(4)3619 2088 y Fo(is)f(suc)m(h)0 2228 y(that)33 b Fl(\020)260 2243 y Fn(4)304 2228 y Fl(g)352 2243 y Fn(2)396 2228 y Fo(\(1)p Fl(=\020)579 2243 y Fn(4)623 2228 y Fo(\))27 b Fj(\024)948 2183 y Fk(x)993 2153 y Fg(2)1032 2183 y Fk(\014)t Fn(\001)p Fp(F)p 807 2205 545 4 v 807 2285 a Fn(72)887 2217 y Fj(p)p 969 2217 383 4 v 971 2285 a Fn(^)-42 b Fk(c)p Fn(\()p Fk(p)p Fn(+2)p Fk(\032)p Fn(+1\))1363 2228 y Fo(,)33 b(that)g(is)g(with)g(the)g (condition)h(ab)s(o)m(v)m(e)f(w)m(e)h(assume)f Fl(\020)3393 2243 y Fn(4)3437 2228 y Fl(g)3485 2243 y Fn(2)3529 2228 y Fo(\(1)p Fl(=\020)3712 2243 y Fn(4)3756 2228 y Fo(\))28 b Fj(\024)12 2457 y Fk(x)57 2427 y Fg(2)96 2457 y Fn(\001)p Fp(F)p 12 2474 214 4 v 79 2531 a Fn(96)254 2356 y Fd(\024)307 2497 y Fo(1)22 b Fj(^)687 2452 y Fk(\014)p 479 2474 466 4 v 479 2486 a Fj(p)p 562 2486 383 4 v 564 2553 a Fn(^)-42 b Fk(c)p Fn(\()p Fk(p)p Fn(+2)p Fk(\032)p Fn(+1\))956 2356 y Fd(\025)1042 2497 y Fo(and)34 b(w)m(e)f(get)g(\(5.32\).)43 b(This)33 b(ends)i(the)e(pro)s(of)h(of)f(the)h(Prop)s(osition)e(5.7.)p 3783 2497 70 70 v 0 2821 a Fm(Pro)s(of)39 b(of)g(Theorem)e(2.4:)120 2962 y Fo(W)-8 b(e)36 b(pro)m(v)m(e)g(that)g(the)h(complemen)m(tary)d (of)i(the)h(set)f Fj(R)2192 2926 y Fk(\016)2228 2936 y Fg(4)2268 2926 y Fk(;\020)2328 2936 y Fg(4)2388 2962 y Fo(\()p Fl(`)2469 2977 y Fn(1)2513 2962 y Fl(;)17 b(`)2600 2977 y Fn(2)2643 2962 y Fl(;)g(R)2764 2977 y Fn(1)2808 2962 y Fo(\))2863 2887 y Fd(S)2963 2962 y Fj(V)3032 2926 y Fk(\016)3068 2936 y Fg(1)3107 2926 y 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3759 y Fn(1)3823 3744 y Fo(+)0 3884 y(2)p Fl(R)126 3899 y Fn(1)170 3884 y Fl(;)17 b(`)257 3899 y Fn(2)330 3884 y Fj(\000)29 b Fo(2)p Fl(R)562 3899 y Fn(1)607 3884 y Fo(]\),)45 b(While)f Fj(A)1128 3899 y Fn(1)1216 3884 y Fo(and)g Fj(A)1500 3899 y Fn(2)1588 3884 y Fo(refer)h(to)e(the)h(b)s(eha)m(viour)g(of)g (the)g(pro\014les)g(in)g(the)g(bulk)g(of)0 4023 y(the)h(in)m(terv)-6 b(al,)47 b Fj(A)665 4038 y Fn(3)753 4023 y Fo(and)e Fj(A)1038 4038 y Fn(4)1127 4023 y Fo(consider)h(the)f(b)s(eha)m(viour)f(of)h(the) g(pro\014les)g(in)g(a)f(region)h(close)g(to)f(the)0 4163 y(b)s(oundaries.)65 b(Namely)38 b(for)j(a)f(giv)m(en)g Fl(\021)j Fj(2)c(f\000)p Fo(1)p Fl(;)17 b Fo(+1)p Fj(g)p Fo(,)41 b(w)m(e)f(can)h(b)s(e)f(in)2728 4082 y Fd(\000)2774 4163 y Fj(R)2858 4127 y Fk(\016)2894 4137 y Fg(4)2933 4127 y Fk(;\020)2993 4137 y Fg(4)3054 4163 y Fo(\()p Fl(`)3135 4178 y Fn(1)3179 4163 y Fl(;)17 b(`)3266 4178 y Fn(2)3309 4163 y Fl(;)g(R)3430 4178 y Fn(1)3474 4163 y Fl(;)g(\021)t Fo(\))3610 4082 y Fd(\001)3656 4103 y Fk(c)3736 4163 y Fo(just)0 4302 y(b)s(ecause)35 b(w)m(e)e(ha)m(v)m(e)h Fl(\021)798 4266 y Fk(\016)834 4276 y Fg(4)873 4266 y Fk(;\020)933 4276 y Fg(4)977 4302 y Fo(\()p Fl(`)1058 4317 y Fn(1)1124 4302 y Fo(+)23 b(2)p Fl(R)1350 4317 y Fn(1)1394 4302 y Fo(\))k Fj(6)p Fo(=)h Fl(\021)38 b Fo(or)33 b Fl(\021)1827 4266 y Fk(\016)1863 4276 y Fg(4)1902 4266 y Fk(;\020)1962 4276 y Fg(4)2006 4302 y Fo(\()p Fl(`)2087 4317 y Fn(2)2153 4302 y Fj(\000)22 b Fo(2)p Fl(R)2378 4317 y Fn(1)2422 4302 y Fo(\))28 b Fj(6)p Fo(=)g Fl(\021)t Fo(.)44 b(Let)33 b(us)h(de\014ne)1297 4601 y Fj(A)1377 4553 y Fk(\021)1377 4628 y Fn(3)1425 4601 y Fo(\()p Fl(`)p Fo(\))27 b Fj(\021)1677 4491 y Fd(n)1744 4601 y Fl(m)1831 4560 y Fk(\016)1870 4530 y Fb(\003)1944 4601 y Fo(:)g Fl(\021)2052 4560 y Fk(\016)2088 4570 y Fg(4)2128 4560 y Fk(;\020)2188 4570 y Fg(4)2231 4601 y Fo(\()p Fl(`)p Fo(\))g Fj(6)p Fo(=)h Fl(\021)2536 4491 y Fd(o)3645 4601 y Fo(\(5)p Fl(:)p Fo(57\))0 4861 y(and)891 5000 y Fj(A)971 5015 y Fn(3)1043 5000 y Fo(=)1362 4906 y Fd([)1149 5124 y Fk(\021)s(;\021)1261 5104 y Fb(0)1286 5124 y Fp(2f\000)p Fn(1)p Fk(;)p Fn(+1)p Fp(g)1649 5104 y Fg(2)1704 5000 y Fj(A)1784 4952 y Fk(\021)1784 5027 y Fn(3)1832 5000 y Fo(\()p Fl(`)1913 5015 y Fn(1)1979 5000 y Fo(+)22 b(2)p Fl(R)2204 5015 y Fn(1)2248 5000 y Fo(\))g Fj([)h(A)2478 4952 y Fk(\021)2522 4922 y Fb(0)2478 5027 y Fn(3)2553 5000 y Fo(\()p Fl(`)2634 5015 y Fn(2)2700 5000 y Fj(\000)g Fo(2)p Fl(R)2926 5015 y Fn(1)2970 5000 y Fo(\))636 b(\(5)p Fl(:)p Fo(58\))120 5340 y(Supp)s(ose)34 b(that)f(a)g(pro\014le)h(is)f(in)g Fj(A)1410 5292 y Fk(\021)1410 5367 y Fn(3)1458 5340 y Fo(\()p Fl(`)1539 5355 y Fn(1)1605 5340 y Fo(+)23 b(2)p Fl(R)1831 5355 y Fn(1)1875 5340 y Fo(\))33 b(then)h(w)m(e)g(can)f(ha)m(v)m(e)h(four)g(alternativ)m(es.) 0 5579 y Fg(23)p Ff(=j)s(anuar)r(y)r(=)p Fg(1998;)e(12:06)1145 b Fo(51)p eop %%Page: 52 52 52 51 bop 120 100 a Fo(The)25 b(blo)s(c)m(k)g Fl(`)613 115 y Fn(1)664 100 y Fo(+)6 b(2)p Fl(R)873 115 y Fn(1)943 100 y Fo(has)26 b Fl(\021)1166 63 y Fk(\016)1202 73 y Fg(4)1241 63 y Fk(;\020)1301 73 y Fg(4)1344 100 y Fo(\()p Fl(`)1425 115 y Fn(1)1476 100 y Fo(+)6 b(2)p Fl(R)1685 115 y Fn(1)1729 100 y Fo(\))28 b(=)g(0)d(or)g Fl(\021)2143 63 y Fk(\016)2179 73 y Fg(4)2218 63 y Fk(;\020)2278 73 y Fg(4)2322 100 y Fo(\()p Fl(`)2403 115 y Fn(1)2453 100 y Fo(+)6 b(2)p Fl(R)2662 115 y Fn(1)2707 100 y Fo(\))28 b(=)g Fj(\000)p Fl(\021)h Fo(and)d(it)f(is)g(sandwic)m(hed)0 239 y(at)37 b(a)f(distance)i(smaller)e(than)h(2)p Fl(R)1314 254 y Fn(1)1395 239 y Fo(b)m(y)g(t)m(w)m(o)g(blo)s(c)m(ks)g(with)g(the) g(same)g Fl(\021)t Fo('s)g(or)g(with)f(di\013eren)m(t)i Fl(\021)t Fo('s.)56 b(In)0 379 y(this)33 b(last)g(case)h(the)f (pro\014les)h(are)g(fron)m(ts.)120 519 y(It)e(is)i(easy)f(to)g(see)h (that)233 780 y Fj(A)313 795 y Fn(3)358 780 y Fj(\\)441 669 y Fd(\020)500 780 y Fj(O)582 732 y Fk(\016)618 742 y Fg(1)657 732 y Fk(;\020)717 742 y Fg(1)579 806 y Fn(0)761 780 y Fo(\([)p Fl(`)870 795 y Fn(1)936 780 y Fo(+)22 b Fl(R)1111 795 y Fn(1)1156 780 y Fl(;)17 b(`)1243 795 y Fn(2)1308 780 y Fj(\000)23 b Fl(R)1484 795 y Fn(1)1528 780 y Fo(]\))1595 669 y Fd(\021)1654 690 y Fk(c)1717 780 y Fj(\\)g(A)k(\032)128 b(R)2202 732 y Fk(\016)2238 742 y Fg(1)2277 732 y Fk(;\020)2337 742 y Fg(1)2376 732 y Fk(;\016)2436 742 y Fg(4)2475 732 y Fk(;\020)2535 742 y Fg(4)2202 806 y Fn(0)2578 780 y Fo(\([)p Fl(`)2687 795 y Fn(1)2753 780 y Fo(+)23 b Fl(R)2929 795 y Fn(1)2973 780 y Fl(;)17 b(`)3060 795 y Fn(2)3126 780 y Fj(\000)22 b Fl(R)3301 795 y Fn(1)3346 780 y Fo(]\))441 924 y Fd([)568 1019 y Fj(W)674 977 y Fk(\016)710 987 y Fg(4)750 977 y Fk(;\020)810 987 y Fg(4)853 1019 y Fo(\([)p Fl(`)962 1034 y Fn(1)1028 1019 y Fo(+)h Fl(R)1204 1034 y Fn(1)1248 1019 y Fl(;)17 b(`)1335 1034 y Fn(2)1401 1019 y Fj(\000)22 b Fl(R)1576 1034 y Fn(1)1621 1019 y Fo(]\))1705 924 y Fd([)1831 1019 y Fj(V)1900 977 y Fk(\016)1936 987 y Fg(1)1975 977 y Fk(;\020)2035 987 y Fg(1)2074 977 y Fk(;\016)2134 987 y Fg(4)2173 977 y Fk(;\020)2233 987 y Fg(4)2277 1019 y Fo(\([)p Fl(`)2386 1034 y Fn(1)2451 1019 y Fo(+)h Fl(R)2627 1034 y Fn(1)2671 1019 y Fl(;)17 b(`)2758 1034 y Fn(2)2824 1019 y Fj(\000)23 b Fl(R)3000 1034 y Fn(1)3044 1019 y Fo(]\))3645 891 y(\(5)p Fl(:)p Fo(59\))120 1265 y(It)35 b(remains)g(to)g(consider)i(what)e(is)g(left)h(in)g Fj(A)p Fo(.)50 b(The)36 b(presence)i(of)e Fj(A)2731 1280 y Fn(4)2811 1265 y Fo(comes)f(from)g(the)h(fact)f(that)0 1404 y(in)h(the)g (de\014nition)h(of)f Fj(A)935 1419 y Fn(1)979 1404 y Fo(,)g(there)h(are)f(four)g(parameters)f Fl(\016)2242 1419 y Fn(4)2287 1404 y Fl(;)17 b(\020)2376 1419 y Fn(4)2420 1404 y Fl(;)g(\016)2509 1419 y Fn(1)2553 1404 y Fl(;)g(\020)2642 1419 y Fn(1)2722 1404 y Fo(and)36 b(since)h Fl(\016)3209 1419 y Fn(4)3254 1404 y Fl(\020)3305 1368 y Fn(3)3298 1429 y(4)3381 1404 y Fj(\025)e Fo(~)-52 b Fl(c)3534 1419 y Fn(1)3579 1404 y Fo(\()p Fl(\014)5 b(;)17 b(\022)s Fo(\))p Fl(\020)3857 1419 y Fn(1)0 1544 y Fo(w)m(e)34 b(can)f(ha)m(v)m(e)h(blo)s(c)m(ks)f(suc)m(h)i(that)e Fl(\021)1355 1507 y Fk(\016)1391 1517 y Fg(1)1430 1507 y Fk(;\020)1490 1517 y Fg(1)1561 1544 y Fo(=)c(0)k(but)g Fl(\021)1985 1507 y Fk(\016)2021 1517 y Fg(4)2060 1507 y Fk(;\020)2120 1517 y Fg(4)2192 1544 y Fo(=)28 b(1.)44 b(Let)33 b(us)h(de\014ne)1008 1764 y Fj(A)1088 1717 y Fk(\021)1088 1791 y Fn(4)1136 1764 y Fo(\()p Fl(`)p Fo(\))27 b Fj(\021)1388 1654 y Fd(n)1455 1764 y Fl(m)1542 1723 y Fk(\016)1581 1693 y Fb(\003)1655 1764 y Fo(:)h Fl(\021)1764 1723 y Fk(\016)1800 1733 y Fg(4)1839 1723 y Fk(;\020)1899 1733 y Fg(4)1942 1764 y Fo(\()p Fl(`)p Fo(\))f(=)h Fl(\021)t(;)17 b(\021)2345 1723 y Fk(\016)2381 1733 y Fg(1)2420 1723 y Fk(;\020)2480 1733 y Fg(1)2524 1764 y Fo(\()p Fl(`)p Fo(\))26 b(=)i(0)2825 1654 y Fd(o)3645 1764 y Fo(\(5)p Fl(:)p Fo(60\))0 1985 y(and)891 2125 y Fj(A)971 2140 y Fn(4)1043 2125 y Fo(=)1362 2030 y Fd([)1149 2248 y Fk(\021)s(;\021)1261 2228 y Fb(0)1286 2248 y Fp(2f\000)p Fn(1)p Fk(;)p Fn(+1)p Fp(g)1649 2228 y Fg(2)1704 2125 y Fj(A)1784 2077 y Fk(\021)1784 2151 y Fn(4)1832 2125 y Fo(\()p Fl(`)1913 2140 y Fn(1)1979 2125 y Fo(+)22 b(2)p Fl(R)2204 2140 y Fn(1)2248 2125 y Fo(\))g Fj([)h(A)2478 2077 y Fk(\021)2522 2047 y Fb(0)2478 2151 y Fn(4)2553 2125 y Fo(\()p Fl(`)2634 2140 y Fn(2)2700 2125 y Fj(\000)g Fo(2)p Fl(R)2926 2140 y Fn(1)2970 2125 y Fo(\))636 b(\(5)p Fl(:)p Fo(61\))120 2438 y(Arguing)32 b(as)i(b)s(efore)g(w)m(e)g(get)541 2739 y Fj(A)621 2754 y Fn(4)688 2739 y Fj(\\)776 2628 y Fd(\020)836 2739 y Fj(O)918 2691 y Fk(\016)954 2701 y Fg(1)993 2691 y Fk(;\020)1053 2701 y Fg(1)915 2765 y Fn(0)1097 2739 y Fo(\([)p Fl(`)1206 2754 y Fn(1)1271 2739 y Fo(+)23 b Fl(R)1447 2754 y Fn(1)1491 2739 y Fl(;)17 b(`)1578 2754 y Fn(2)1644 2739 y Fj(\000)23 b Fl(R)1820 2754 y Fn(1)1864 2739 y Fo(]\))1931 2628 y Fd(\021)1990 2649 y Fk(c)2053 2739 y Fj(\\)f(A)27 b(\032)641 2978 y(R)725 2930 y Fk(\016)761 2940 y Fg(1)800 2930 y Fk(;\020)860 2940 y Fg(1)899 2930 y Fk(;\016)959 2940 y Fg(4)998 2930 y Fk(;\020)1058 2940 y Fg(4)725 3004 y Fn(0)1102 2978 y Fo(\([)p Fl(`)1211 2993 y Fn(1)1276 2978 y Fo(+)c Fl(R)1452 2993 y Fn(1)1496 2978 y Fl(;)17 b(`)1583 2993 y Fn(2)1649 2978 y Fj(\000)23 b Fl(R)1825 2993 y Fn(1)1869 2978 y Fo(]\))1953 2883 y Fd([)2079 2978 y Fj(V)2148 2937 y Fk(\016)2184 2947 y Fg(1)2224 2937 y Fk(;\020)2284 2947 y Fg(1)2322 2937 y Fk(;\016)2382 2947 y Fg(4)2421 2937 y Fk(;\020)2481 2947 y Fg(4)2525 2978 y Fo(\([)p Fl(`)2634 2993 y Fn(1)2700 2978 y Fo(+)f Fl(R)2875 2993 y Fn(1)2920 2978 y Fl(;)17 b(`)3007 2993 y Fn(2)3072 2978 y Fj(\000)23 b Fl(R)3248 2993 y Fn(1)3292 2978 y Fo(]\))3645 2851 y(\(5)p Fl(:)p Fo(62\))0 3194 y(It)33 b(is)g(no)m(w)h(clear)f(that)g(w) m(e)g(ha)m(v)m(e)621 3435 y Fj(A)22 b(\\)812 3355 y Fd(\000)857 3435 y Fj(V)926 3394 y Fk(\016)962 3404 y Fg(1)1002 3394 y Fk(;\020)1062 3404 y Fg(1)1100 3394 y Fk(;\016)1160 3404 y Fg(4)1199 3394 y Fk(;\020)1259 3404 y Fg(4)1303 3435 y Fo(\()p Fl(`)1384 3450 y Fn(1)1428 3435 y Fl(;)17 b(`)1515 3450 y Fn(2)1558 3435 y Fo(\))1597 3355 y Fd(\001)1643 3371 y Fk(c)1705 3435 y Fj(\\)1794 3325 y Fd(\020)1854 3435 y Fj(O)1936 3388 y Fk(\016)1972 3398 y Fg(1)2011 3388 y Fk(;\020)2071 3398 y Fg(1)1933 3462 y Fn(0)2114 3435 y Fo(\([)p Fl(`)2223 3450 y Fn(1)2289 3435 y Fo(+)23 b Fl(R)2465 3450 y Fn(1)2509 3435 y Fl(;)17 b(`)2596 3450 y Fn(2)2662 3435 y Fj(\000)22 b Fl(R)2837 3450 y Fn(1)2882 3435 y Fo(]\))2949 3325 y Fd(\021)3008 3346 y Fk(c)3076 3435 y Fj(\032)721 3675 y(R)805 3627 y Fk(\016)841 3637 y Fg(1)880 3627 y Fk(;\020)940 3637 y Fg(1)979 3627 y Fk(;\016)1039 3637 y Fg(4)1078 3627 y Fk(;\020)1138 3637 y Fg(4)805 3701 y Fn(0)1182 3675 y Fo(\([)p Fl(`)1291 3690 y Fn(1)1357 3675 y Fo(+)g Fl(R)1532 3690 y Fn(1)1577 3675 y Fl(;)17 b(`)1664 3690 y Fn(2)1729 3675 y Fj(\000)23 b Fl(R)1905 3690 y Fn(1)1949 3675 y Fo(]\))2033 3580 y Fd([)2159 3675 y Fj(W)2265 3633 y Fk(\016)2301 3643 y Fg(4)2341 3633 y Fk(;\020)2401 3643 y Fg(4)2445 3675 y Fo(\([)p Fl(`)2554 3690 y Fn(1)2620 3675 y Fo(+)f Fl(R)2795 3690 y Fn(1)2840 3675 y Fl(;)17 b(`)2927 3690 y Fn(2)2992 3675 y Fj(\000)23 b Fl(R)3168 3690 y Fn(1)3212 3675 y Fo(]\))3645 3547 y(\(5)p Fl(:)p Fo(63\))0 3925 y(and)39 b(\(2.35\))e(follo)m(ws)h(immediatly)d(from)j(the)g(Corollaries)g(5.2,) g(5.4,)h(5.6)e(and)i(the)g(Prop)s(osition)e(5.7.)p 0 4065 70 70 v 0 4449 a Fm(Pro)s(of)i(of)g(Theorem)e(2.5:)120 4590 y Fo(T)-8 b(aking)33 b(in)m(to)h(accoun)m(t)i(\(2.35\),)d(w)m(e)i (m)m(ust)f(c)m(hec)m(k)i(that)e(for)h Fl(`)2422 4605 y Fn(1)2496 4590 y Fj(\024)30 b Fl(`)2645 4605 y Fn(2)2719 4590 y Fj(\024)h Fl(`)2869 4605 y Fn(3)2947 4590 y Fo(that)j(b)s (elongs)h(to)f Fj(J)18 b Fo(,)35 b(an)0 4729 y(ev)m(en)m(t)f(of)f(the)h (form)1057 4869 y Fj(V)1126 4828 y Fk(\016)1162 4838 y Fg(1)1202 4828 y Fk(;\020)1262 4838 y Fg(1)1301 4828 y Fk(;\016)1361 4838 y Fg(4)1399 4828 y Fk(;\020)1459 4838 y Fg(4)1503 4869 y Fo(\()p Fl(`)1584 4884 y Fn(1)1628 4869 y Fl(;)17 b(`)1715 4884 y Fn(2)1759 4869 y Fl(;)g(\021)t Fo(\))k Fj(\\)h(V)2074 4828 y Fk(\016)2110 4838 y Fg(1)2150 4828 y Fk(;\020)2210 4838 y Fg(1)2248 4828 y Fk(;\016)2308 4838 y Fg(4)2347 4828 y Fk(;\020)2407 4838 y Fg(4)2451 4869 y Fo(\()p Fl(`)2532 4884 y Fn(2)2576 4869 y Fl(;)17 b(`)2663 4884 y Fn(3)2706 4869 y Fl(;)g(\021)t Fo(\))802 b(\(5)p Fl(:)p Fo(64\))0 5061 y(with)37 b Fl(`)273 5076 y Fn(2)343 5061 y Fj(\000)25 b Fl(`)487 5076 y Fn(1)567 5061 y Fj(\024)36 b Fl(`)724 5076 y Fn(^)-42 b Fk(c)762 5061 y Fo(\()p Fl(\015)6 b Fo(\))36 b(and)i Fl(`)1174 5076 y Fn(3)1244 5061 y Fj(\000)25 b Fl(`)1388 5076 y Fn(2)1468 5061 y Fj(\024)35 b Fl(`)1624 5076 y Fn(^)-42 b Fk(c)1662 5061 y Fo(\()p Fl(\015)6 b Fo(\))37 b(has)h(small)e (Gibbs-probabilit)m(y)i(and)g(moreo)m(v)m(er)f(that)0 5201 y(this)k(is)f(true)h(with)f(a)h(v)m(ery)f(high)h Fl(I)-17 b(P)13 b Fo(-probabilit)m(y)-8 b(,)41 b(uniformly)f(for)h Fl(`)2646 5216 y Fn(1)2730 5201 y Fj(\024)f Fl(`)2889 5216 y Fn(2)2973 5201 y Fj(\024)h Fl(`)3133 5216 y Fn(3)3217 5201 y Fo(in)g Fj(J)18 b Fo(.)66 b(But)40 b(it)g(is)0 5340 y(immediate)31 b(to)i(see)h(that)f(those)g(ev)m(en)m(ts)i(are)e (con)m(trolled)g(b)m(y)h(the)f(Prop)s(osition)g(5.7.)0 5579 y Fg(23)p Ff(=j)s(anuar)r(y)r(=)p Fg(1998;)f(12:06)1145 b Fo(52)p eop %%Page: 53 53 53 52 bop 120 115 a Fo(>F)-8 b(rom)38 b(the)h(Theorem)g(2.3,)h (denoting)g(b)m(y)f Fl(c)1838 130 y Fn(2)1921 115 y Fo(=)2037 4 y Fd(h)2307 66 y Fk(c)p Fn(\()p Fk(x;\032;\015)t Fn(\))p 2096 92 702 4 v 2096 149 a(\()p Fk(\014)t(\022)r Fn(\))2247 129 y Fg(2)2286 149 y Fn(\()p Fk(m)2388 160 y Ff(\014)s(;)p Fg(1)2491 149 y Fn(+)p Fk(m)2623 160 y Ff(\014)s(;)p Fg(2)2727 149 y Fn(\))2758 129 y Fg(2)2809 4 y Fd(i)2856 115 y Fo(,)h(see)g(\(2.29\),)f(w)m(e)h(end)h(the)0 254 y(pro)s(of)33 b(of)h(the)f(Theorem)g(2.5.)p 1166 254 70 70 v 120 585 a Fe(A)-5 b(cknow)5 b(le)-5 b(dgements.)60 b Fo(W)-8 b(e)39 b(w)m(ould)g(lik)m(e)f(to)g(thank)h(Enzo)g(Olivieri)f (and)h(Errico)f(Presutti)h(for)g(en-)0 724 y(ligh)m(tening)33 b(discussions.)0 5579 y Fg(23)p Ff(=j)s(anuar)r(y)r(=)p Fg(1998;)f(12:06)1145 b Fo(53)p eop %%Page: 54 54 54 53 bop 1681 100 a Fm(References)-35 318 y Fo([1])49 b(A.)30 b(de)i(Acosta,)e Fe(A)k(new)g(pr)-5 b(o)g(of)32 b(of)h(the)h(Hartman-Wintner)e(law)j(of)e(the)g(iter)-5 b(ate)g(d)33 b(lo)-5 b(garithm.)43 b Fo(Ann.)120 458 y(Prob.)g Fm(11)p Fo(,)34 b(270{276)e(\(1983\).)-35 605 y([2])49 b(A.)d(Aharon)m(y)-8 b(,)49 b Fe(T)-8 b(ricritic)j(al)47 b(p)-5 b(oints)47 b(in)h(systems)g(with)f(r)-5 b(andom)48 b(\014elds.)167 b Fo(Ph)m(ys.)84 b(Rev.)h(B)46 b Fm(18)p Fo(,)120 745 y(3318{3327)31 b(\(1978\).)-35 893 y([3])49 b(M.)39 b(Aizenman)h(and)g(J.)f(W)-8 b(ehr,)42 b Fe(R)-5 b(ounding)41 b(of)g(\014rst)h(order)f(phase)g(tr)-5 b(ansitions)40 b(in)i(systems)f(with)120 1032 y(quenche)-5 b(d)35 b(disor)-5 b(der.)42 b Fo(Com.)h(Math.)h(Ph)m(ys.,)33 b Fm(130)p Fo(,)h(489{528)e(\(1990\).)-35 1180 y([4])49 b(J.M.)36 b(G.)h(Amaro)f(de)i(Matos)f(and)g(J.)g(F.)g(P)m(erez,)i Fe(Fluctuations)g(in)g(the)g(Curie-Weiss)e(version)i(of)120 1319 y(the)c(r)-5 b(andom)36 b(\014eld)g(Ising)f(mo)-5 b(del.)44 b Fo(J.)33 b(Stat.)44 b(Ph)m(ys.)g Fm(62)p Fo(,)34 b(587{608)d(\(1991\).)-35 1467 y([5])49 b(J)41 b(M)g(G)g(Amaro)e(de)i(Matos,)i(A)d(E)h(P)m(atric)m(k,)h(and)f(V)g(A)f (Zagrebno)m(v,)j Fe(R)-5 b(andom)42 b(in\014nite)g(volume)120 1606 y(Gibbs)32 b(states)h(for)f(the)i(Curie-Weiss)d(r)-5 b(andom)33 b(\014eld)h(Ising)f(mo)-5 b(del.)p Fo(,)31 b(J.)f(Stat.)43 b(Ph)m(ys.,)31 b Fm(66)p Fo(,)g(139{164)120 1746 y(\(1992\).)-35 1894 y([6])49 b(A.)32 b(Beretti,)g Fe(Some)k(pr)-5 b(op)g(erties)34 b(of)i(r)-5 b(andom)35 b(Ising)h(mo)-5 b(dels.)44 b Fo(J.)33 b(Stat.)43 b(Ph)m(ys,)33 b Fm(38)p Fo(,)h(483)f(\(1985\).)-35 2041 y([7])49 b(T.)28 b(Bo)s(dineau,)h Fe(Interfac)-5 b(e)31 b(in)g(a)g(one-dimensional)f (Ising)h(spin)g(system.)42 b Fo(Sto)s(c)m(h.)h(Pro)s(c.)f(Appl.)h Fm(61)p Fo(,)120 2181 y(1{23)32 b(\(1996\).)-35 2329 y([8])49 b(P)-8 b(.)26 b(Bleher,)i(J.)f(Ruiz,)i(and)e(V.)g(Zagrebno)m (v,)h Fe(One-dimensional)h(r)-5 b(andom-\014eld)30 b(Ising)g(mo)-5 b(del:)43 b(Gibbs)120 2468 y(states)35 b(and)g(structur)-5 b(e)36 b(of)f(gr)-5 b(ound)36 b(states.)43 b Fo(J.)33 b(Stat.)44 b(Ph)m(ys.)g Fm(84)34 b Fo(1077{1093)d(\(1996\).)-35 2616 y([9])49 b(A.)27 b(Bo)m(vier,)i(V.)f(Ga)m(yrard)g(and)h(P)-8 b(.)28 b(Picco,)h Fe(Distribution)h(of)h(pr)-5 b(o\014les)31 b(for)g(the)g(Kac-Hop\014eld)g(mo)-5 b(del.)120 2755 y Fo(Comm.)41 b(Math.)j(Ph)m(ys.)h Fm(186)34 b Fo(323{379)e(\(1997\).) -85 2903 y([10])49 b(J.)34 b(Bricmon)m(t)g(and)h(A.)g(Kupiainen,)h Fe(Phase)g(tr)-5 b(ansition)36 b(in)h(the)g(thr)-5 b(e)g(e-dimensional) 35 b(r)-5 b(andom)38 b(\014eld)120 3042 y(Ising)d(mo)-5 b(del.)44 b Fo(Com.)f(Math.)h(Ph)m(ys.,)p Fm(116)p Fo(,)33 b(539{572)f(\(1988\).)-85 3190 y([11])49 b(M.)34 b(Cassandro,)h(E.)f (Orlandi,)h(and)g(E.)f(Presutti,)g Fe(Interfac)-5 b(es)36 b(and)h(typic)-5 b(al)37 b(Gibbs)f(c)-5 b(on\014gur)g(ations)120 3330 y(for)35 b(one-dimensional)f(Kac)h(p)-5 b(otentials.)43 b Fo(Prob.)h(Theor.)h(Rel.)f(Fields)33 b Fm(96)p Fo(,)h(57-96)f (\(1993\).)-85 3477 y([12])49 b(J.)26 b(T.)h(Chalk)m(er,)h Fe(On)i(the)f(lower)i(critic)-5 b(al)29 b(dimensionality)g(of)g(the)h (Ising)f(mo)-5 b(del)30 b(in)g(a)g(r)-5 b(andom)30 b(\014eld.)p Fo(,)120 3617 y(J.)j(Ph)m(ys.)44 b(C:)33 b(Sol.)43 b(State)34 b(Ph)m(ys.,)e Fm(16)p Fo(,)i(6615{6622)d(\(1983\).)-85 3765 y([13])49 b(Y.S.)33 b(Cho)m(w)h(and)g(H.)f(T)-8 b(eic)m(her,)34 b Fe(Pr)-5 b(ob)g(ability)35 b(The)-5 b(ory:)47 b(Indep)-5 b(endenc)g(e,)35 b(Inter)-5 b(change)g(ability,)35 b(Mar-)120 3904 y(tingales.)43 b Fo(Springer-V)-8 b(erlag,)33 b(Berlin)g(and)g(New)g(Y)-8 b(ork,)32 b(\(1978\).)-85 4052 y([14])49 b(P)-8 b(.)28 b(Da)h(Pra)f(and)i(F.)f(den)h(Hollander,)f Fe(McKe)-5 b(an-Vlasov)31 b(limit)h(for)f(inter)-5 b(acting)31 b(r)-5 b(andom)32 b(pr)-5 b(o)g(c)g(esses)120 4191 y(in)35 b(r)-5 b(andom)36 b(me)-5 b(dia.)43 b Fo(J.)33 b(Stat.)44 b(Ph)m(ys.)g Fm(84)p Fo(,)33 b(735{772)f(\(1996\).)-85 4339 y([15])49 b(D.S.)33 b(Fisher,)h(J.)f(F)-8 b(r\177)-50 b(ohlic)m(h,)34 b(and)g(T.)f(Sp)s(encer,)i Fe(The)h(Ising)g(mo)-5 b(del)36 b(in)g(a)g(r)-5 b(andom)36 b(magnetic)g(\014eld.)p Fo(,)120 4478 y(J.)d(Stat.)43 b(Ph)m(ys.,)33 b Fm(34)p Fo(,)h(863{870)d(\(1984\).)-85 4626 y([16])49 b(J.)36 b(Im)m(brie,)h Fe(The)i(gr)-5 b(ound)40 b(states)e(of)h(the)f(thr)-5 b(e)g(e-dimensional)38 b(r)-5 b(andom)39 b(\014eld)h(Ising)e(mo)-5 b(del.)56 b Fo(Com.)120 4766 y(Math.)44 b(Ph)m(ys.,)33 b Fm(98)p Fo(,)g(145{176)f(\(1985\).)-85 4913 y([17])49 b(Y.)34 b(Imry)f(and)i(S.K.)g(Ma,)g Fe(R)-5 b(andom-\014eld)37 b(instability)g(of)f(the)h(or)-5 b(der)g(e)g(d)36 b(state)h(of)g(c)-5 b(ontinuous)37 b(sym-)120 5053 y(metry.)44 b Fo(Ph)m(ys.)g(Rev.)h (Lett.)f Fm(35)p Fo(,)33 b(1399{1401)e(\(1975\).)-85 5201 y([18])49 b(M.)27 b(Kac,)h(G.)f(Uhlen)m(b)s(ec)m(k,)i(and)f(P)-8 b(.C.)26 b(Hemmer,)g Fe(On)31 b(the)f(van)g(der)f(Waals)h(the)-5 b(ory)30 b(of)g(vap)-5 b(our-liquid)120 5340 y(e)g(quilibrium.)76 b(I.)46 b(Discussion)f(of)h(a)g(one-dimensional)e(mo)-5 b(del)p Fo(,)48 b(J.)c(Math.)79 b(Ph)m(ys.)f Fm(4)p Fo(,)47 b(216{228)0 5579 y Fg(23)p Ff(=j)s(anuar)r(y)r(=)p Fg(1998;)32 b(12:06)1145 b Fo(54)p eop %%Page: 55 55 55 54 bop 120 100 a Fo(\(1963\);)33 b Fe(II.)k(Discussion)e(of)h(the)h (distribution)e(functions)p Fo(,)e(J.)h(Math.)48 b(Ph)m(ys.)f Fm(4)p Fo(,)35 b(229{247)e(\(1963\);)120 239 y Fe(III.)j(Discussion)e (of)h(the)h(critic)-5 b(al)35 b(r)-5 b(e)g(gion)p Fo(,)32 b(J.)h(Math.)44 b(Ph)m(ys.)g Fm(5)p Fo(,)33 b(60{74)g(\(1964\).)-85 380 y([19])49 b(C.)31 b(K)s(\177)-53 b(ulsk)m(e,)32 b Fe(Metastates)i(in)f(disor)-5 b(der)g(e)g(d)33 b(me)-5 b(an)34 b(\014eld)g(mo)-5 b(dels:)45 b(r)-5 b(andom)34 b(\014eld)g(and)g(Hop\014eld)g(mo)-5 b(d-)120 519 y(els.)43 b Fo(T)-8 b(o)33 b(app)s(ear)h(J.)f(Stat.)43 b(Ph)m(ys.)i(\(1998\).)-85 660 y([20])k(J.)34 b(Leb)s(o)m(witz)i(and)f(O.)h(P)m(enrose,)g Fe(R)m(igor)-5 b(ous)37 b(tr)-5 b(e)g(atment)37 b(of)g(the)g(V)-8 b(an)38 b(der)f(Waals)g(Maxwel)5 b(l)39 b(the)-5 b(ory)120 799 y(of)35 b(the)g(liquid-vap)-5 b(our)35 b(tr)-5 b(ansition)p Fo(,)32 b(J.)h(Math.)44 b(Ph)m(ys.)g Fm(7)p Fo(,)34 b(98{113)e (\(1966\).)-85 939 y([21])49 b(M.)31 b(Ledoux)h Fe(On)j(T)-8 b(alagr)j(and's)34 b(deviation)e(ine)-5 b(qualities)33 b(for)h(pr)-5 b(o)g(duct)34 b(me)-5 b(asur)g(es)31 b Fo(ESAIM:Probab.)120 1079 y(and)i(Statist.)44 b Fm(1)33 b Fo(63{87)f(\(1996\).)-85 1219 y([22])49 b(M.)40 b(Ledoux)h(and)g(M.)g (T)-8 b(alagrand,)41 b Fe(Pr)-5 b(ob)g(ability)41 b(in)h(Banach)g(Sp)-5 b(ac)g(es)p Fo(,)41 b(Springer,)i(Berlin-Heidel-)120 1359 y(b)s(erg-New)33 b(Y)-8 b(ork,)32 b(\(1991\).)-85 1499 y([23])49 b(P)-8 b(.)32 b(Mathieu)i(and)h(P)-8 b(.)32 b(Picco,)i Fe(Metastability)h(and)h(c)-5 b(onver)g(genc)g(e)35 b(to)h(e)-5 b(quilibrium)36 b(for)f(the)h(R)-5 b(andom)120 1639 y(\014eld)35 b(Curie-Weiss)f(mo)-5 b(del)p Fo(,)34 b(preprin)m(t)g(\(1997\).)-85 1779 y([24])49 b(O.)30 b(P)m(enrose)i(and)f(J.L.)g(Leb)s(o)m(witz,)g Fe(T)-8 b(owar)j(ds)33 b(a)h(rigor)-5 b(ous)32 b(mole)-5 b(cular)35 b(the)-5 b(ory)33 b(of)g(metastability.)43 b Fo(in)120 1919 y(Fluctuation)33 b(Phenomena)g(\()g(e.W.)h(Mon)m(troll)f(and)g (J.L.)h(Leb)s(o)m(witz)f(ed\))h(North-Holland)e(Ph)m(ysics)120 2058 y(Publishing)h(\(1987\).)-85 2199 y([25])49 b(A.)32 b(R)m(\023)-47 b(en)m(yi.)45 b Fe(Pr)-5 b(ob)g(ability)34 b(The)-5 b(ory)34 b Fo(North-Holland.)43 b(Amsterdam-London,)32 b(\(1970\).)-85 2339 y([26])49 b(H.)32 b(Robbins,)i Fe(A)i(r)-5 b(emark)36 b(on)f(Stirling's)g(formula)f Fo(Amer.)43 b(Math.)h(Mon)m(tlh)m(y)33 b Fm(62)p Fo(,)h(26{29)e(\(1955\))-85 2480 y([27])49 b(Y.)37 b(Sinai)h Fe(The)i(limiting)f(b)-5 b(ehavior)39 b(of)g(a)h(one-dimensional)f(r)-5 b(andom)40 b(walk)h(in)e(r)-5 b(andom)40 b(envir)-5 b(on-)120 2619 y(ment)44 b Fo(Theory)h(of)f(Prob.)78 b(and)45 b(its)f(applications)g Fm(27)h Fo(256-268)e(\(1982\);)49 b(F.)44 b(Salomon)f Fe(R)-5 b(andom)120 2759 y(walks)35 b(in)h(r)-5 b(andom)35 b(envir)-5 b(onment)33 b Fo(Ann.)44 b(Prob.)g Fm(3)34 b Fo(1{31)e(\(1975\).)-85 2899 y([28])49 b(S.R.)32 b(Salinas)g(and)h (W.F.)f(W)-8 b(reszinski,)33 b Fe(On)j(the)e(me)-5 b(an)35 b(\014eld)g(Ising)g(mo)-5 b(del)35 b(in)g(a)g(r)-5 b(andom)35 b(external)120 3039 y(\014eld.)44 b Fo(J.)33 b(Stat.)43 b(Ph)m(ys.)h Fm(41)p Fo(,)34 b(299{313)e(\(1985\).)-85 3179 y([29])49 b(W.)33 b(J.)g(Stout)g Fe(A)m(lmost)k(sur)-5 b(e)36 b(c)-5 b(onver)g(genc)g(e.)42 b Fo(Academic)32 b(Press,)i(New)f(Y)-8 b(ork)32 b(\(1974\).)-85 3320 y([30])49 b(M.)29 b(T)-8 b(alagrand,)29 b Fe(Conc)-5 b(entr)g(ation)31 b(of)h(me)-5 b(asur)g(e)32 b(and)f(isop)-5 b(erimetric)31 b(ine)-5 b(qualities)30 b(in)i(pr)-5 b(o)g(duct)32 b(sp)-5 b(ac)g(e)120 3459 y Fo(Publ.)44 b(Math.)g(I.H.E.S.)32 b Fm(81)p Fo(,)h(73{205)f(\(1995\).)-85 3600 y([31])49 b(H-T.)23 b(Y)-8 b(au.)40 b Fe(L)-5 b(o)g(garithmic)27 b(Sob)-5 b(olev)27 b(Ine)-5 b(quality)28 b(for)e(L)-5 b(attic)g(e)27 b(Gases)g(with)f(mixing)h(c)-5 b(onditions)23 b Fo(Com.)120 3739 y(Math.)44 b(Ph)m(ys.)g Fm(181)34 b Fo(367{408\(1996\).)-85 3879 y([32])49 b(V.)38 b(V.)h(Y)-8 b(urinskii,)39 b Fe(Exp)-5 b(onential)41 b(ine)-5 b(qualities)40 b(for)g(sums)i(of)e(r)-5 b(andom)42 b(ve)-5 b(ctors)p Fo(,)39 b(J.)g(Multiv)-6 b(ariate.)120 4019 y(Anal.)43 b Fm(6)p Fo(:)h(473{499)32 b(\(1976\))0 5579 y Fg(23)p Ff(=j)s(anuar)r(y)r(=)p Fg(1998;)g(12:06)1145 b Fo(55)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF