Content-Type: multipart/mixed; boundary="-------------0107251348948" This is a multi-part message in MIME format. ---------------0107251348948 Content-Type: text/plain; name="01-287.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-287.comments" lrenato@ime.usp.br rhs@math.ucla.edu vladas@impa.br www.math.ucla.edu/~rhs ---------------0107251348948 Content-Type: text/plain; name="01-287.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-287.keywords" Stochastic Ising model, Glauber dynamics, voter model, fixation, relaxation, stretched exponential, multiscale analysis, bootstrap percolation ---------------0107251348948 Content-Type: application/postscript; name="fss.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="fss.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: fss.dvi %%Pages: 25 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips fss -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2001.07.12:1027 %%BeginProcSet: texc.pro %! 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1557 y Fl(<)25 b(L)1076 1401 y Fi(!)1204 1557 y Fp(=)56 b FA(P)1418 1401 y Fi( )1513 1443 y Fy(4)p Fm(L)1490 1470 y Fi(X)1497 1665 y Fm(i)p Fy(=1)1649 1534 y Fp(\026)1638 1557 y Fl(I)1678 1571 y Fm(i)1734 1557 y Fl(<)25 b(L)1892 1401 y Fi(!)1204 1910 y Fp(=)56 b FA(P)1418 1754 y Fi( )1490 1910 y Fp(exp)1645 1754 y Fi( )1718 1910 y Fo(\000)p Fl(\022)1872 1796 y Fy(4)p Fm(L)1849 1824 y Fi(X)1856 2018 y Fm(i)p Fy(=1)2007 1887 y Fp(\026)1996 1910 y Fl(I)2036 1924 y Fm(i)2067 1754 y Fi(!)2165 1910 y Fl(>)25 b Fp(exp)o(\()p Fo(\000)p Fl(\022)s(L)p Fp(\))2651 1754 y Fi(!)2778 1910 y Fo(\024)2916 1815 y FA(E)3000 1714 y Fi(\020)3055 1815 y Fp(exp)3209 1714 y Fi(\020)3264 1815 y Fo(\000)p Fl(\022)3396 1747 y Fi(P)3492 1769 y Fy(4)p Fm(L)3492 1843 y(i)p Fy(=1)3641 1792 y Fp(\026)3630 1815 y Fl(I)3670 1829 y Fm(i)3701 1714 y Fi(\021)o(\021)p 2916 1889 894 4 v 3168 1973 a Fp(exp\()p Fo(\000)p Fl(\022)s(L)p Fp(\))3821 1910 y Fl(:)3717 2020 y Fp(\(3.4)-7 b(\))0 2256 y(Let)182 2233 y Fi(e)158 2256 y 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Fl(e)1849 5125 y Fm(\022)1905 5034 y Fi(\022)1983 5101 y Fl(e)2026 5068 y Fh(\000)p Fm(\022)2144 5101 y Fp(+)20 b(2)p 1983 5142 V 2109 5225 a(3)2292 5034 y Fi(\023)2359 5046 y Fy(4)2400 5007 y Fi(!)2472 5026 y Fm(L)2583 5163 y Fp(=)56 b Fl(e)2753 5125 y Fh(\000)p Fm(\015)t Fy(\()p Fm(\022)r Fy(\))p Fm(L)3000 5163 y Fl(;)685 b Fp(\(3)p Fl(:)p Fp(7\))p eop %%Page: 8 8 8 7 bop 0 46 a Fw(8)1150 b(F)n(ONTES,)33 b(SCHONMANN,)i(SIDORA)-9 b(VICIUS)0 265 y Fp(for)30 b Fl(L)25 b Fo(\025)h Fl(L)386 279 y Fy(0)427 265 y Fp(.)41 b(Elemen)m(tary)28 b(calculus)g(sho)m(ws)i (that)1063 513 y(lim)1053 572 y Fm(\022)r Fh(&)p Fy(0)1230 513 y Fl(\015)5 b Fp(\()p Fl(\022)s Fp(\))54 b(=)i(0)182 b(and)207 b(lim)2154 572 y Fm(\022)r Fh(&)p Fy(0)2342 451 y Fl(d\015)p 2342 492 100 4 v 2345 576 a(d\022)2452 513 y Fp(\()p Fl(\022)s Fp(\))55 b(=)2763 451 y(1)p 2763 492 46 4 v 2763 576 a(3)2819 513 y Fl(:)0 768 y Fp(Therefore)28 b Fl(\015)5 b Fp(\()p Fl(\022)s Fp(\))24 b Fl(>)i Fp(0,)k(for)g(small)f Fl(\022)f(>)d Fp(0)30 b(and)g(Claim)f(\(a\))g(follo)m(ws)g(from)g (\(3.7\),)h(in)f(the)h(case)f Fl(d)c Fp(=)g(2.)0 981 y Ff(Pr)-5 b(o)g(of)32 b(of)h(Claim)f(\(a\).)43 b(Step)33 b(2:)41 b(c)-5 b(ase)33 b Fl(d)25 b(>)h Fp(2)p Ff(.)40 b Fp(F)-8 b(or)30 b Fl(i)c Fp(=)f(\()p Fl(i)1998 995 y Fy(3)2039 981 y Fl(;)15 b(i)2110 995 y Fy(4)2152 981 y Fl(;)g(:::;)g(i)2338 995 y Fm(d)2383 981 y Fp(\))25 b Fo(2)g(f)p Fp(1)p Fl(;)15 b(:::;)g(L)p Fo(g)2885 948 y Fm(d)p Fh(\000)p Fy(2)3023 981 y Fp(,)31 b(set)253 1186 y Fl(R)322 1200 y Fm(i)378 1186 y Fp(=)25 b Fo(f)p Fp(\()p Fl(x)608 1200 y Fy(1)649 1186 y Fl(;)15 b(x)741 1200 y Fy(2)782 1186 y Fl(;)g(i)853 1200 y Fy(3)895 1186 y Fl(;)g(:::;)g(i)1081 1200 y Fm(d)1127 1186 y Fp(\))25 b(:)g Fl(x)1290 1200 y Fy(1)1332 1186 y Fl(;)15 b(x)1424 1200 y Fy(2)1491 1186 y Fo(2)25 b(f)p Fp(1)p Fl(;)15 b(:::;)g(L)p Fo(gg)p Fl(;)199 b(S)2258 1200 y Fm(i)2314 1186 y Fp(=)25 b Fo(f)p Fp(\()p Fl(x)2544 1200 y Fy(1)2585 1186 y Fl(;)15 b(x)2677 1200 y Fy(2)2718 1186 y Fl(;)g(i)2789 1200 y Fy(3)2831 1186 y Fl(;)g(:::;)g(i)3017 1200 y Fm(d)3063 1186 y Fp(\))25 b(:)g Fl(x)3226 1200 y Fy(1)3268 1186 y Fl(;)15 b(x)3360 1200 y Fy(2)3427 1186 y Fo(2)25 b Fn(Z)p Fo(g)p Fl(:)0 1392 y Fp(Then)f(for)g(eac)m(h)g Fl(i)i Fo(2)g(f)p Fp(1)p Fl(;)15 b(:::;)g(L)p Fo(g)1066 1359 y Fm(d)p Fh(\000)p Fy(2)1203 1392 y Fp(,)27 b Fl(R)1324 1406 y Fm(i)1380 1392 y Fo(\032)e Fl(S)1532 1406 y Fm(i)1562 1392 y Fp(,)i Fl(S)1670 1406 y Fm(i)1725 1392 y Fp(is)e(isomorphic)d (to)i Fn(Z)2440 1359 y Fy(2)2477 1392 y Fp(,)i(and)e Fl(R)2769 1406 y Fm(i)2825 1392 y Fp(is)h(isomorphic)d(to)i Fo(f)p Fp(1)p Fl(;)15 b(:::;)g(L)p Fo(g)3830 1359 y Fy(2)3872 1392 y Fp(.)0 1501 y(Note)30 b(also)f(that)g Fo(f)p Fl(R)717 1515 y Fm(i)773 1501 y Fp(:)c Fl(i)h Fo(2)f(f)p Fp(1)p Fl(;)15 b(:::;)g(L)p Fo(g)1321 1468 y Fm(d)p Fh(\000)p Fy(2)1459 1501 y Fo(g)30 b Fp(is)g(a)g(partition)d(of)j Fl(R)q Fp(.)110 1611 y(Let)k Fl(i)p Fp(\(1\))g Fl(<)g(i)p Fp(\(2\))f Fl(<)h(:::)h(<)f(i)p Fp(\()p Fl(N)10 b Fp(\),)37 b Fl(N)45 b Fp(=)34 b Fl(L)1601 1578 y Fm(d)p Fh(\000)p Fy(2)1737 1611 y Fp(,)j(b)s(e)e(the)f(lexicographic)d(order)j(in)h Fo(f)p Fp(1)p Fl(;)15 b(:::;)g(L)p Fo(g)3358 1578 y Fm(d)p Fh(\000)p Fy(2)3496 1611 y Fp(.)56 b(Observ)m(e)0 1721 y(that)32 b(for)g(eac)m(h)g Fl(x)e Fo(2)g Fn(Z)789 1687 y Fm(d)827 1721 y Fp(,)k(in)f(the)f(lexicographic)c(order,)33 b(half)e(of)i(the)f(neigh)m(b)s(ors)e(of)j Fl(x)g Fp(are)g(larger)d (than)i Fl(x)i Fp(and)0 1830 y(half)f(of)h(the)f(neigh)m(b)s(ors)f(of)i Fl(x)g Fp(are)f(smaller)g(than)g Fl(x)p Fp(.)53 b(It)33 b(follo)m(ws)g(that)g(for)h(eac)m(h)f Fl(j)k Fo(2)31 b(f)p Fp(1)p Fl(;)15 b(:::;)g(L)p Fo(g)3369 1797 y Fm(d)p Fh(\000)p Fy(2)3541 1830 y Fp(and)34 b(eac)m(h)0 1940 y Fl(x)k Fo(2)f Fl(R)257 1954 y Fm(j)296 1940 y Fp(,)j(among)d(the)f(2) p Fl(d)h Fp(neigh)m(b)s(ors)e(of)j Fl(x)p Fp(,)i(4)d(are)g(in)g Fl(S)2019 1954 y Fm(j)2057 1940 y Fp(,)j Fl(d)25 b Fo(\000)g Fp(2)38 b(are)f(in)g(\()p Fo([)2746 1954 y Fm(k)r(j)342 2053 y Fl(R)411 2067 y Fm(k)457 2053 y Fp(\))22 b Fo([)h Fl(R)669 2019 y Fm(C)730 2053 y Fp(.)52 b(Therefore,)33 b(for)g(eac)m(h)g Fl(i)f Fo(2)f(f)p Fp(1)p Fl(;)15 b(:::;)g(L)p Fo(g)2113 2019 y Fm(d)p Fh(\000)p Fy(2)2251 2053 y Fp(,)35 b(once)e(all)f(spins)h(in)g(the)g(sets)g Fl(R)3408 2067 y Fm(j)3447 2053 y Fp(,)i Fl(j)h(<)c(i)p Fp(,)k(are)0 2162 y(+1,)k(they)d(will)f(sta)m(y)h(in)g(this)g(state)f (forev)m(er.)62 b(Moreo)m(v)m(er,)39 b(the)e(sites)f(in)h Fl(R)2661 2176 y Fm(i)2730 2162 y Fp(will)f(ha)m(v)m(e)h(then)g(at)g (least)f Fl(d)25 b Fo(\000)g Fp(2)0 2272 y(neigh)m(b)s(ors)34 b(whic)m(h)h(are)g(not)h(in)f Fl(S)1180 2286 y Fm(i)1247 2272 y Fp(and)h(whic)m(h)f(are)g(frozen)g(in)g(the)h(state)e(+1.)59 b(Therefore,)36 b(the)f(spins)g(in)h Fl(R)3867 2286 y Fm(i)0 2381 y Fp(will)d(b)s(eha)m(v)m(e)i(then)f(in)g(a)h(w)m(a)m(y)h (that)e(fa)m(v)m(ors)i(spins)e(+1)h(as)g(compared)e(to)i(the)f(2)h (dimensional)d(v)m(ersion)i(of)h(the)0 2491 y(dynamics)c(restricted)e (to)i Fl(S)984 2505 y Fm(i)1047 2491 y Fp(and)g(started)f(from)i(the)f (con\014guration)e(whic)m(h)i(has)h(spins)f Fo(\000)p Fp(1)h(in)f Fl(R)3456 2505 y Fm(i)3519 2491 y Fp(and)h(spins)0 2601 y(+1)26 b(in)g Fl(S)302 2615 y Fm(i)332 2601 y Fo(n)p Fl(R)447 2615 y Fm(i)504 2601 y Fp(\(meaning)e(that)h(one)g(can)h (construct)e(a)i(coupling)d(of)j(these)f(ev)m(olutions)f(in)i(whic)m(h) f(the)g(dynamics)0 2710 y(whic)m(h)k(is)h(said)f(to)h(fa)m(v)m(or)h (spins)e(+1)h(has)g(a)g(spin)f(+1)h(wherev)m(er)f(the)h(other)e(one)i (has)g(one\).)110 2820 y(Motiv)-5 b(ated)38 b(b)m(y)k(the)e(observ)-5 b(ations)38 b(ab)s(o)m(v)m(e,)44 b(mo)s(dify)c(no)m(w)h(the)f (dynamics,)j(b)m(y)f(only)e(allo)m(wing)f(spins)h Fo(\000)p Fp(1)0 2929 y(in)e(eac)m(h)h(set)f Fl(R)550 2943 y Fm(i)620 2929 y Fp(to)h(\015ip)f(after)g(all)g(the)g(spins)g(in)g(the)h(sets)f Fl(R)2210 2943 y Fm(j)2249 2929 y Fp(,)k Fl(j)j(<)40 b(i)p Fp(,)j(are)38 b(in)g(state)g(+1.)67 b(Let)39 b Fl(T)3553 2943 y Fm(k)3638 2929 y Fp(b)s(e)f(the)0 3039 y(time)c(in)g(this)f(mo)s(di\014ed)f(dynamics)i(when)g(all)f(spins)h (in)g Fl(R)2080 3055 y Fm(i)p Fy(\()p Fm(k)r Fy(\))2208 3039 y Fp(,)j Fl(k)e Fp(=)e(1)p Fl(;)15 b(:::;)g(N)10 b Fp(,)38 b(are)c(\014rst)g(in)g(state)g(+1.)54 b(Then)0 3148 y Fl(T)53 3162 y Fy(1)94 3148 y Fl(;)15 b(T)187 3162 y Fy(2)245 3148 y Fo(\000)h Fl(T)385 3162 y Fy(1)427 3148 y Fl(;)f(:::;)g(T)635 3162 y Fm(N)723 3148 y Fo(\000)h Fl(T)863 3162 y Fm(N)7 b Fh(\000)p Fy(1)1054 3148 y Fp(are)28 b(m)m(utually)e(indep)s(enden)m(t,)f(and)j(eac)m(h)f(one)g(is)h(sto)s (c)m(hastically)c(smaller)i(than)h(a)0 3260 y(random)i(v)-5 b(ariable)29 b Fl(T)745 3227 y Fy(\(2\))873 3260 y Fp(whic)m(h)h (corresp)s(onds)e(to)i(the)f(time)h(to)g(ero)s(de)e(an)i Fl(L)21 b Fo(\002)g Fl(L)31 b Fp(square)e(in)h(the)g(2)g(dimension)0 3370 y(v)m(ersion)f(of)h(the)g(dynamics.)39 b(F)-8 b(rom)30 b(Step)f(1,)h(w)m(e)h(kno)m(w)f(that)f(there)g(exists)g Fl(C)q(;)15 b(\015)32 b Fo(2)25 b Fp(\(0)p Fl(;)15 b Fo(1)p Fp(\))30 b(suc)m(h)g(that)1360 3630 y FA(P)1447 3501 y Fi(\022)1514 3630 y Fl(T)1580 3592 y Fy(\(2\))1704 3630 y Fl(>)1827 3568 y(C)p 1811 3609 105 4 v 1811 3693 a Fp(2)p Fl(\013)1926 3630 y(L)1988 3592 y Fy(2)2029 3501 y Fi(\023)2153 3630 y Fo(\024)55 b Fl(e)2322 3592 y Fh(\000)p Fy(2)p Fm(\015)t(L)2512 3630 y Fl(;)1173 b Fp(\(3)p Fl(:)p Fp(8\))0 3885 y(for)30 b(large)e Fl(L)p Fp(.)110 3995 y(A)m(ttractiv)m(eness)c(allo)m(ws)i(the)g(follo)m(wing)e (comparison)h(b)s(et)m(w)m(een)g(our)h(original)e(dynamics)i(and)g(the) g(mo)s(di\014ed)0 4104 y(one:)1136 4214 y FA(P)p Fp(\()p Fl(T)38 b(>)26 b Fp(\()p Fl(C)t(=\013)p Fp(\))p Fl(L)1739 4176 y Fm(d)1781 4214 y Fp(\))55 b Fo(\024)h FA(P)p Fp(\()p Fl(T)2160 4228 y Fm(N)2254 4214 y Fl(>)26 b Fp(\()p Fl(C)t(=\013)p Fp(\))p Fl(L)2658 4176 y Fm(d)2700 4214 y Fp(\))p Fl(:)949 b Fp(\(3)p Fl(:)p Fp(9\))110 4381 y(No)m(w)38 b(w)m(e)g(mo)s(dify)e (further)g(the)h(mo)s(di\014ed)e(dynamics,)j(b)m(y)g(observing)e(it)h (at)g(times)f(\()p Fl(C)t(=)p Fp(\(2)p Fl(\013)p Fp(\)\))p Fl(L)3524 4348 y Fy(2)3564 4381 y Fl(m)p Fp(,)k Fl(m)e Fp(=)0 4491 y(1)p Fl(;)15 b Fp(2)p Fl(;)g(:::)p Fp(,)29 b(and)c(resetting)e(to)i Fo(\000)p Fp(1)h(at)f(those)f(times)h(all)g (the)f(spins)h(in)g(eac)m(h)g(of)h(the)f Fl(R)2802 4507 y Fm(i)p Fy(\()p Fm(k)r Fy(\))2930 4491 y Fp(,)i Fl(k)h Fp(=)e(1)p Fl(;)15 b(:::;)g(N)10 b Fp(,)29 b(whic)m(h)c(still)0 4601 y(has)34 b(some)f(spin)g Fo(\000)p Fp(1)h(at)g(this)f(time.)51 b(\(This)33 b(will)f(a\013ect)h(at)g(most)h(one)f(of)h(the)f Fl(R)2826 4618 y Fm(i)p Fy(\()p Fm(k)r Fy(\))2988 4601 y Fp(at)h(eac)m(h)f(of)h(these)f(times,)0 4711 y(since)c(the)h(others)g (will)f(either)g(b)s(e)h(full)f(of)i(spins)f(+1)g(or)h(full)e(of)i (spins)f Fo(\000)p Fp(1.\))42 b(Using)31 b(attractiv)m(eness)c(again,)j (w)m(e)0 4821 y(ha)m(v)m(e)k(the)f(follo)m(wing)e(comparison,)i(where)g (\()p Fl(S)1637 4835 y Fm(m)1706 4821 y Fp(\))1742 4835 y Fm(m)p Fy(=0)p Fm(;)p Fy(1)p Fm(;:::)2082 4821 y Fp(is)g(a)h(random)f (w)m(alk)h(whic)m(h)e(starts)h(from)h(0)f(and)g(at)0 4937 y(eac)m(h)27 b(unit)g(of)g(time)g(mo)m(v)m(es)i(1)e(unit)g(up,)h (with)f(probabilit)m(y)e FA(P)o Fp(\()p Fl(T)2234 4904 y Fy(\(2\))2357 4937 y Fo(\024)h Fp(\()p Fl(C)t(=)p Fp(\(2)p Fl(\013)p Fp(\)\))p Fl(L)2879 4904 y Fy(2)2918 4937 y Fp(\),)j(and)e(otherwise)e(remains)0 5047 y(still:)600 5188 y FA(P)p Fp(\()p Fl(T)761 5202 y Fm(N)856 5188 y Fl(>)g Fp(\()p Fl(C)t(=\013)p Fp(\))p Fl(L)1259 5151 y Fm(d)1302 5188 y Fp(\))55 b Fo(\024)h FA(P)1607 5059 y Fi(\022)1674 5188 y Fl(S)1780 5202 y Fe(\()p Fj(C)r(=\013)p Fe(\))p Fj(L)1997 5189 y(d)p 1741 5220 332 4 v 1741 5266 a Fe(\()p Fj(C)r(=)p Fe(\(2)p Fj(\013)p Fe(\)\))p Fj(L)2039 5253 y Fe(2)2113 5188 y Fl(<)26 b(N)2293 5059 y Fi(\023)2416 5188 y Fp(=)56 b FA(P)p Fp(\()p Fl(S)2707 5207 y Fy(2)p Fm(L)2793 5189 y Fj(d)p Fd(\000)p Fe(2)2942 5188 y Fl(<)25 b(L)3100 5151 y Fm(d)p Fh(\000)p Fy(2)3236 5188 y Fp(\))p Fl(:)367 b Fp(\(3)p Fl(:)p Fp(10\))p eop %%Page: 9 9 9 8 bop 661 46 a Fw(STRETCHED)34 b(EXPONENTIAL)g(FIXA)-6 b(TION)34 b(A)-6 b(T)33 b(ZER)n(O)h(TEMPERA)-6 b(TURE)622 b(9)0 265 y Fp(Using)30 b(\(3.8\),)f(w)m(e)i(ha)m(v)m(e)57 578 y FA(P)p Fp(\()p Fl(S)221 596 y Fy(2)p Fm(L)307 578 y Fj(d)p Fd(\000)p Fe(2)k Fl(<)25 b(L)614 540 y Fm(d)p Fh(\000)p Fy(2)750 578 y Fp(\))56 b Fo(\024)968 476 y Fi(\020)1038 543 y Fy(2)p Fm(L)1124 516 y Fj(d)p Fd(\000)p Fe(2)1056 626 y Fm(L)1106 598 y Fj(d)p Fd(\000)p Fe(2)1259 476 y Fi(\021)1329 449 y(\022)1396 578 y FA(P)1483 449 y Fi(\022)1550 578 y Fl(T)1616 540 y Fy(\(2\))1740 578 y Fl(>)1863 516 y(C)p 1847 557 105 4 v 1847 640 a Fp(2)p Fl(\013)1962 578 y(L)2024 540 y Fy(2)2065 449 y Fi(\023)q(\023)2200 468 y Fm(L)2250 440 y Fj(d)p Fd(\000)p Fe(2)2429 578 y Fo(\024)g Fp(2)2602 540 y Fy(2)p Fm(L)2688 512 y Fj(d)p Fd(\000)p Fe(2)2827 504 y Fi(\000)2869 578 y Fl(e)2912 540 y Fh(\000)p Fy(2)p Fm(\015)t(L)3102 504 y Fi(\001)3144 519 y Fm(L)3194 491 y Fj(d)p Fd(\000)p Fe(2)3373 578 y Fo(\024)g Fl(e)3543 540 y Fh(\000)p 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col0 sh gr /Palatino-Italic ff 450.00 scf sf 6375 5565 m gs 1 -1 sc (k) col0 sh gr /Palatino-Italic ff 555.00 scf sf 6660 5265 m gs 1 -1 sc (/) col0 sh gr /Palatino-Italic ff 555.00 scf sf 6825 5310 m gs 1 -1 sc (3) col0 sh gr /Palatino-Italic ff 555.00 scf sf 8520 7440 m gs 1 -1 sc (L) col0 sh gr /Palatino-Italic ff 450.00 scf sf 8835 7695 m gs 1 -1 sc (k) col0 sh gr $F2psEnd rs %%EndDocument @endspecial 524 5000 a Fz(Figure)34 b(1)p Fp(.)47 b(Solid)28 b(lines)h(separate)g(blo)s(c)m(ks)g Fl(B)2144 4967 y Fm(i)2139 5026 y(k)2184 5000 y Fp(,)j Fl(i)26 b Fo(2)g Fn(Z)2450 4967 y Fm(d)2488 5000 y Fp(;)32 b(dashed)d(lines)g(indicate) 524 5125 y(the)f(blo)s(c)m(k)938 5101 y Fi(e)919 5125 y Fl(B)988 5139 y Fm(k)1033 5125 y Fp(,)h(asso)s(ciated)c(with)j(the)f (blo)s(c)m(k)g Fl(B)2190 5139 y Fm(k)2236 5125 y Fp(,)i(whic)m(h)e(is)h (shaded.)39 b(The)28 b(outer)524 5240 y(corridors)g(b)s(et)m(w)m(een)h (the)g(b)s(oundaries)e(of)2004 5217 y Fi(e)1985 5240 y Fl(B)2054 5254 y Fm(k)2129 5240 y Fp(and)j Fl(B)2376 5254 y Fm(k)2452 5240 y Fp(are)f(of)h(width)f Fl(L)3029 5254 y Fm(k)3074 5240 y Fl(=)p Fp(3.)p eop %%Page: 12 12 12 11 bop 0 46 a Fw(12)1111 b(F)n(ONTES,)33 b(SCHONMANN,)i(SIDORA)-9 b(VICIUS)110 265 y Fp(Theorem)19 b(1.1)i(will)f(b)s(e)g(obtained)e (through)i(a)h(comparison)d(of)j(the)f(original)e(system)j(\(1.1-2\))f (with)g(a)h(mo)s(di\014ed)0 374 y(dynamics,)k(further)f(referred)f(to)i (as)g(blo)s(c)m(k-dynamics,)g(coupled)e(to)h(the)h(original)d(dynamics) i(in)h(that)f(it)g(will)g(b)s(e)0 484 y(constructed)j(on)i(the)h(same)f (probabilit)m(y)e(space,)i(using)g(the)g(same)h(graphical)d(marks,)j (and)g(de\014ned)e(b)m(y)j(Rules)0 594 y(1-3)f(b)s(elo)m(w.)39 b(The)30 b(blo)s(c)m(k-dynamics)e(will)h(b)s(e)g(so)h(constructed)d(as) j(to)g(ha)m(v)m(e)g(the)f(follo)m(wing)f(prop)s(erties.)0 741 y(\(A\))g(The)h(blo)s(c)m(k-dynamics)d(fa)m(v)m(ors)j Fo(\000)p Fp(1)g(spins,)f(in)g(the)g(sense)g(that)f(at)h(an)m(y)h(site) e(and)h(time)g(where)f(the)h(original)0 850 y(dynamics)h(has)h(a)g Fo(\000)p Fp(1)g(spin,)g(also)f(the)g(blo)s(c)m(k-dynamics)f(has)i(a)g Fo(\000)p Fp(1)g(spin.)0 997 y(\(B\))e(In)h(the)g(blo)s(c)m(k-dynamics) d(at)j(the)f(time)g Fl(T)1605 1011 y Fm(k)1680 997 y Fp(all)g(h)m(yp)s(ercub)s(es)e(of)j(the)g Fl(k)2593 964 y Fy(th)2695 997 y Fp(scale)f(will)f(b)s(e)h(`mono)s(c)m(hromatic',)0 1107 y(i.e.,)h(they)e(will)g(b)s(e)g(en)m(tirely)f(\014lled)g(in)i (either)e(b)m(y)j(+1)f(or)g Fo(\000)p Fp(1)g(spins.)39 b(In)28 b(the)g(former)f(case)g(w)m(e)i(will)d(sa)m(y)j(that)e(the)0 1216 y(blo)s(c)m(k)i(is)h(in)f(state)g(+1,)i(and)e(in)h(the)f(latter)f (case)h(that)g(it)g(is)h(in)g(state)e Fo(\000)p Fp(1.)0 1363 y(\(C\))35 b(F)-8 b(or)35 b(eac)m(h)f Fl(k)i Fo(\025)e Fp(1,)j(the)e(random)f(\014eld)g Fl(\021)1600 1377 y Fm(k)1681 1363 y Fp(that)g(asso)s(ciates)e(to)j(eac)m(h)f Fl(i)h Fo(2)f Fn(Z)2862 1330 y Fm(d)2936 1363 y Fp(a)h(random)f(v)-5 b(ariable)33 b Fl(\021)3750 1377 y Fm(k)3795 1363 y Fp(\()p Fl(i)p Fp(\))0 1473 y(whic)m(h)f(tak)m(es)h(the)g(v)-5 b(alue)32 b(+1)h(\(resp.)49 b Fo(\000)p Fp(1\))32 b(if)h(at)g(time)f Fl(T)1976 1487 y Fm(k)2055 1473 y Fp(the)g(blo)s(c)m(k)g Fl(B)2533 1440 y Fm(i)2528 1499 y(k)2607 1473 y Fp(is)g(in)h(state)f (+1)h(\(resp.)49 b Fo(\000)p Fp(1\))32 b(is)h(a)h(1-)0 1585 y(dep)s(enden)m(t)24 b(random)h(\014eld.)38 b(This)26 b(means)f(that)g(for)h(eac)m(h)g Fl(n)p Fp(,)h(if)f Fl(i)2231 1599 y Fy(1)2273 1585 y Fl(;)15 b(:::;)g(i)2459 1599 y Fm(n)2537 1585 y Fo(2)25 b Fn(Z)2688 1552 y Fm(d)2753 1585 y Fp(are)h(suc)m(h)g(that)f(dist)n(\()p Fl(x)3533 1599 y Fm(i)3563 1585 y Fl(;)15 b(x)3655 1599 y Fm(j)3695 1585 y Fp(\))25 b Fl(>)g Fp(1)0 1695 y(for)30 b Fl(i)c Fo(6)p Fp(=)f Fl(j)5 b Fp(,)31 b(then)e(the)g(random)g(v)-5 b(ariables)28 b Fl(\021)1516 1709 y Fm(k)1561 1695 y Fp(\()p Fl(i)1628 1709 y Fy(1)1669 1695 y Fp(\))p Fl(;)15 b(:::;)g(\021)1905 1709 y Fm(k)1952 1695 y Fp(\()p Fl(i)2019 1709 y Fm(n)2069 1695 y Fp(\))30 b(are)f(indep)s(enden)m(t.)110 1842 y(T)-8 b(o)32 b(assure)f(that)f(\(C\))h(ab)s(o)m(v)m(e)g(is)h (satis\014ed,)e(in)i(the)e(blo)s(c)m(k-dynamics,)h(the)g(ev)m(olution)e (of)j(the)e(spins)h(in)h(eac)m(h)0 1952 y(b)s(o)m(x)k Fl(B)255 1918 y Fm(i)250 1977 y(k)295 1952 y Fp(,)j Fl(i)d Fo(2)f Fn(Z)587 1918 y Fm(d)625 1952 y Fp(,)k(during)34 b(the)i(in)m(terv)-5 b(al)34 b(of)i(time)f([)p Fl(T)1886 1966 y Fm(k)r Fh(\000)p Fy(1)2025 1952 y Fl(;)15 b(T)2118 1966 y Fm(k)2164 1952 y Fp(],)39 b(will)34 b(dep)s(end)g(only)i(on)g (the)f(con\014guration)d(at)0 2076 y(time)d Fl(T)261 2090 y Fm(k)r Fh(\000)p Fy(1)430 2076 y Fp(and)h(the)f(graphical)e (marks)j(inside)f(the)g(corresp)s(onding)d(b)s(o)m(x)2631 2053 y Fi(e)2612 2076 y Fl(B)2686 2042 y Fm(i)2681 2101 y(k)2726 2076 y Fp(.)110 2185 y(T)-8 b(o)32 b(de\014ne)f(the)h(blo)s(c) m(k-dynamics)e(so)i(that)f(\(A\))h(ab)s(o)m(v)m(e)g(is)g(also)f (satis\014ed,)g(w)m(e)i(\014rst)e(in)m(tro)s(duce)f(a)i(notion)e(of)0 2310 y Ff(`in\015uenc)-5 b(e)34 b(time')p Fp(,)d(asso)s(ciated)c(with)j (the)f(b)s(o)m(x)1671 2287 y Fi(e)1652 2310 y Fl(B)1726 2277 y Fm(i)1721 2336 y(k)1766 2310 y Fp(.)42 b(Let)30 b(\()p Fl(\033)2088 2262 y Fm(\030)r(;T)2188 2272 y Fj(k)q Fd(\000)p Fe(1)2099 2340 y Fc(e)2085 2356 y Fm(B)2143 2334 y Fj(i)2140 2378 y(k)2180 2356 y Fm(;\020)t Fy(;)p Fm(s)2313 2310 y Fp(\))2349 2324 y Fm(s)p Fh(\025)p Fm(T)2483 2334 y Fj(k)q Fd(\000)p Fe(1)2640 2310 y Fp(b)s(e)f(the)h(ev)m(olution) e(in)i(the)g(b)s(o)m(x)3777 2287 y Fi(e)3758 2310 y Fl(B)3832 2277 y Fm(i)3827 2336 y(k)3872 2310 y Fp(,)0 2450 y(with)h(b)s(oundary) f(condition)e Fl(\020)39 b Fp(outside)29 b(this)i(b)s(o)m(x,)i(started) c(at)j(time)e Fl(T)2490 2464 y Fm(k)r Fh(\000)p Fy(1)2661 2450 y Fp(from)i(the)f(con\014guration)d Fl(\030)36 b Fp(inside)0 2560 y(the)29 b(b)s(o)m(x)h(and)g(obtained)d(using)i(the)h (graphical)d(marks.)40 b(Set,)30 b(no)m(w,)123 2815 y Fl(\034)173 2777 y Fm(i)163 2838 y(k)248 2815 y Fp(=)41 b(inf)486 2714 y Fi(n)547 2815 y Fl(s)25 b Fo(\025)h Fl(T)765 2829 y Fm(k)r Fh(\000)p Fy(1)929 2815 y Fp(:)g Fl(\033)1035 2766 y Fm(\030)r(;T)1135 2776 y Fj(k)q Fd(\000)p Fe(1)1046 2844 y Fc(e)1032 2861 y Fm(B)1090 2839 y Fj(i)1087 2883 y(k)1127 2861 y Fm(;)p Fy(+;)p Fm(s)1264 2815 y Fp(\()p Fl(x)p Fp(\))50 b Fo(6)p Fp(=)h Fl(\033)1615 2766 y Fm(\030)r(;T)1715 2776 y Fj(k)q Fd(\000)p Fe(1)1627 2844 y Fc(e)1612 2861 y Fm(B)1670 2839 y Fj(i)1667 2883 y(k)1707 2861 y Fm(;)p Fh(\000)p Fy(;)p Fm(s)1846 2815 y Fp(\()p Fl(x)p Fp(\))24 b(for)30 b(some)25 b Fl(x)h Fo(2)f Fl(B)2596 2777 y Fm(i)2591 2838 y(k)2661 2815 y Fp(and)30 b(some)g Fl(\030)f Fo(2)d(f\000)p Fp(1)p Fl(;)15 b Fp(+1)p Fo(g)3604 2761 y Fc(e)3590 2777 y Fm(B)3648 2750 y Fj(i)3645 2796 y(k)3688 2714 y Fi(o)3749 2815 y Fl(:)3710 2960 y Fp(\(4)p Fl(:)p Fp(5\))0 3070 y(Note)22 b(that)g(b)m(y)i(attractiv)m(eness,)d(ev)m(olutions)f(with)i(other)f(b) s(oundary)g(conditions)f(will)h(b)s(e)h(b)s(ounded)f(from)h(ab)s(o)m(v) m(e)0 3179 y(and)32 b(b)s(elo)m(w)g(b)m(y)i(the)e(t)m(w)m(o)h(ev)m (olutions)e(whic)m(h)h(app)s(ear)f(in)i(the)f(de\014nition)d(of)k Fl(\034)2746 3146 y Fm(i)2736 3205 y(k)2781 3179 y Fp(.)50 b(Therefore)31 b(w)m(e)i(can)f(think)g(of)0 3290 y Fl(\034)50 3257 y Fm(i)40 3316 y(k)115 3290 y Fp(as)d(the)g(\014rst)g(\(random\))e (time)h(after)h Fl(T)1452 3304 y Fm(k)r Fh(\000)p Fy(1)1590 3290 y Fp(,)i(when)e(spins)f(in)h Fl(B)2292 3257 y Fm(i)2287 3316 y(k)2362 3290 y Fp(can)g(su\013er)g(an)m(y)h(in\015uence)d(from)i (the)g(spins)0 3414 y(outside)f(of)438 3391 y Fi(e)419 3414 y Fl(B)493 3381 y Fm(i)488 3440 y(k)564 3414 y Fp(at)h(time)g Fl(T)936 3428 y Fm(k)r Fh(\000)p Fy(1)1075 3414 y Fp(.)110 3526 y(In)i(order)f(to)h(de\014ne)f(the)h(ev)m(olution)e(in)i(the)f (blo)s(c)m(k-dynamics)g(of)h(the)f(spins)h(inside)f(eac)m(h)g(b)s(o)m (x)i Fl(B)3524 3493 y Fm(i)3519 3552 y(k)3564 3526 y Fp(,)g Fl(i)c Fo(2)g Fn(Z)3834 3493 y Fm(d)3872 3526 y Fp(,)0 3636 y(from)i Fl(T)269 3650 y Fm(k)r Fh(\000)p Fy(1)438 3636 y Fp(to)g Fl(T)603 3650 y Fm(k)648 3636 y Fp(,)h(w)m(e)f(use)g(no)m(w)g(the)g(follo)m(wing)d(rules.)0 3793 y Ff(R)n(ule)j(1.)40 b Fp(During)27 b(the)g(in)m(terv)-5 b(al)26 b(of)i(time)f([)p Fl(T)1500 3807 y Fm(k)r Fh(\000)p Fy(1)1639 3793 y Fl(;)15 b(T)1732 3807 y Fm(k)1778 3793 y Fp(\))28 b(w)m(e)g(observ)m(e)g(the)f(ev)m(olution)f(inside)g(the)h (b)s(o)m(x)3451 3770 y Fi(e)3432 3793 y Fl(B)3506 3760 y Fm(i)3501 3819 y(k)3575 3793 y Fp(with)g(+1)0 3903 y(b)s(oundary)33 b(conditions.)53 b(W)-8 b(e)35 b(assign)f(to)g(the)h (spins)f(in)g(the)h(b)s(o)m(x)g Fl(B)2382 3870 y Fm(i)2377 3929 y(k)2457 3903 y Fp(up)g(to)g(time)67 b(min)p Fo(f)p Fl(\034)3206 3870 y Fm(i)3196 3929 y(k)3240 3903 y Fl(;)49 b(T)3367 3917 y Fm(k)3413 3903 y Fo(g)35 b Fp(the)f(v)-5 b(alues)0 4013 y(that)29 b(w)m(e)h(see)g(in)g(that)e(ev)m(olution.)0 4160 y Ff(R)n(ule)35 b(2.)50 b Fp(If)33 b(it)g(is)g(the)f(case)h(that)f Fl(\034)1231 4127 y Fm(i)1221 4186 y(k)1311 4160 y Fl(<)46 b(T)1481 4174 y Fm(k)1527 4160 y Fp(,)34 b(then)f(at)f(the)h(time)f Fl(\034)2334 4127 y Fm(i)2324 4186 y(k)2402 4160 y Fp(all)g(spins)h(in) f Fl(B)2948 4127 y Fm(i)2943 4186 y(k)3022 4160 y Fp(will)g(b)s(e)g (declared)f(to)i(b)s(e)0 4270 y Fo(\000)p Fp(1,)e(and)e(p)s(ersist)f (at)i(this)f(state)g(without)e(c)m(hange)i(up)h(to)g(time)f Fl(T)2303 4284 y Fm(k)2348 4270 y Fp(.)0 4416 y Ff(R)n(ule)35 b(3.)51 b Fp(If,)35 b(follo)m(wing)c(the)i(t)m(w)m(o)h(rules)e(ab)s(o)m (v)m(e,)j(there)d(is)h(an)m(y)h(spin)f(in)g(state)f Fo(\000)p Fp(1)h(in)g Fl(B)3074 4383 y Fm(i)3069 4442 y(k)3148 4416 y Fp(at)g(times)g(whic)m(h)g(are)0 4527 y(arbitrarily)20 b(close)h(to)i Fl(T)802 4541 y Fm(k)847 4527 y Fp(,)i(then)d(at)g(time) g Fl(T)1456 4541 y Fm(k)1525 4527 y Fp(all)g(the)g(spins)g(in)g Fl(B)2190 4494 y Fm(i)2185 4553 y(k)2254 4527 y Fp(are)g(declared)f(to) h(b)s(e)g(in)h(state)e Fo(\000)p Fp(1.)39 b(Otherwise,)0 4637 y(at)30 b(time)f Fl(T)373 4651 y Fm(k)448 4637 y Fp(all)g(the)g(spins)h(in)f Fl(B)1142 4604 y Fm(i)1137 4663 y(k)1213 4637 y Fp(are)g(declared)f(to)i(b)s(e)f(in)g(state)g(+1.) 0 4784 y(It)24 b(is)g(immediate)e(from)i Ff(R)n(ules)j(1-3)d Fp(that)f(the)g(blo)s(c)m(k-dynamics)f(satis\014es)h(prop)s(erties)e (\(A\),)j(\(B\))g(and)f(\(C\))h(ab)s(o)m(v)m(e.)0 4894 y(\(Note)k(that)f(for)i(this)e(w)m(e)i(need)f(the)g(initial)d (distribution)g(to)j(b)s(e)g(1-dep)s(enden)m(t,)e(whic)m(h)i(is)h(the)f (case,)g(since)f(it)h(is)0 5004 y(a)i(pro)s(duct)e(measure.)39 b(The)30 b(prop)s(erties)d(can)i(then)g(b)s(e)g(v)m(eri\014ed)g(b)m(y)i (induction)c(in)j Fl(k)s Fp(.\))110 5113 y(Let)j Fi(e)-55 b Fl(q)314 5127 y Fm(k)358 5113 y Fp(,)31 b Fl(k)d Fo(\025)d Fp(0,)30 b(denote)e(the)h(probabilit)m(y)d(that)j(at)g(the)g(time)f Fl(T)2330 5127 y Fm(k)2406 5113 y Fp(the)h(blo)s(c)m(k)f Fl(B)2872 5127 y Fm(k)2947 5113 y Fp(is)i(in)f(the)g(state)f Fo(\000)p Fp(1.)40 b(Note)0 5223 y(that)33 b Fi(e)-55 b Fl(q)239 5237 y Fy(0)305 5223 y Fp(=)25 b Fl(q)442 5237 y Fy(0)483 5223 y Fp(.)41 b(The)30 b(follo)m(wing)d(is)j(our)g (main)f(tec)m(hnical)e(estimate)h(in)i(order)f(to)g(pro)m(v)m(e)h (Theorem)f(1.1.)p eop %%Page: 13 13 13 12 bop 661 46 a Fw(STRETCHED)34 b(EXPONENTIAL)g(FIXA)-6 b(TION)34 b(A)-6 b(T)33 b(ZER)n(O)h(TEMPERA)-6 b(TURE)583 b(13)0 265 y FA(Lemma)34 b(4.1.)52 b Ff(If)33 b Fl(q)j Ff(is)d(smal)5 b(l)33 b(enough,)f(then)37 b Fi(e)-55 b Fl(q)1684 279 y Fm(m)1794 265 y Fo(\024)25 b Fl(q)1931 279 y Fm(m)2033 265 y Ff(for)32 b(al)5 b(l)33 b Fl(m)25 b Fo(\025)h Fp(0)p Ff(.)110 433 y Fp(The)j(follo)m(wing)f(estimates)g (will)g(b)s(e)i(needed)e(in)h(the)h(pro)s(of)e(of)i(Lemma)g(4.1)h(and)e (Theorem)g(1.1.)41 b(Let)855 655 y(^)-53 b Fl(q)43 b Fp(=)e(sup)15 b Fo(f)p Fl(x)25 b(>)g Fp(0)g(:)h(if)k Fl(q)e Fo(2)d Fp(\(0)p Fl(;)15 b(x)p Fp(\),)31 b(then)e Fl(q)2292 669 y Fm(k)2362 655 y Fo(\024)c Fl(q)s Fp(,)31 b Fl(k)d Fp(=)d(0)p Fl(;)15 b Fp(1)p Fl(;)g(:::)r Fo(g)p Fl(:)0 878 y Fp(Note)29 b(that)g(if)g Fl(q)f(>)e Fp(0)j(is)g(small)g (enough,)f(then)h Fl(q)1660 892 y Fy(1)1726 878 y Fo(\024)c Fl(q)1863 892 y Fy(0)1930 878 y Fp(=)g Fl(q)33 b Fp(and)c(then,)g(b)m (y)h(induction,)d Fl(q)3108 892 y Fm(k)3183 878 y Fp(is)i(decreasing)e (in)i Fl(k)s Fp(.)0 987 y(Therefore)38 b(^)-53 b Fl(q)33 b(>)d Fp(0.)50 b(Sev)m(eral)31 b(times)h(w)m(e)i(will)d(need)h(to)g (tak)m(e)h Fl(q)g Fo(2)d Fp(\(0)p Fl(;)22 b Fp(^)-53 b Fl(q)s Fp(\),)34 b(to)f(assure)f(that)g(certain)e(inequalities)0 1097 y(hold)g(regardless)g(of)h(the)g(v)-5 b(alue)30 b(of)i Fl(k)s Fp(.)45 b(In)32 b(other)e(w)m(ords,)i(w)m(e)g(will)e (need)h(this)g(assumption)e(in)i(order)g(to)g(assure)0 1206 y(uniformit)m(y)d(o)m(v)m(er)j(scales)e(in)g(the)g(c)m(hoice)g(of) h(constan)m(ts)e(in)i(the)f(m)m(ultiscale)e(sc)m(heme.)110 1317 y(W)-8 b(e)30 b(claim)e(that)h(for)h(arbitrary)e Fl(\016)g(>)e Fp(0,)k(if)g Fl(q)e Fo(2)e Fp(\(0)p Fl(;)21 b Fp(^)-52 b Fl(q)r Fp(\))30 b(is)g(small)f(enough,)g(then)1417 1544 y Fl(q)1458 1558 y Fm(k)r Fh(\000)p Fy(1)1596 1544 y Fl(q)1637 1558 y Fm(k)r Fh(\000)p Fy(2)1791 1544 y Fo(\001)15 b(\001)g(\001)h Fl(q)1953 1558 y Fy(1)1994 1544 y Fl(q)2035 1558 y Fy(0)2132 1544 y Fo(\025)55 b Fp(\()p Fl(q)2335 1558 y Fm(k)2380 1544 y Fp(\))2416 1506 y Fm(\016)2455 1544 y Fl(;)1230 b Fp(\(4)p Fl(:)p Fp(6\))0 1782 y(for)33 b Fl(k)f Fo(\025)e Fp(1.)49 b(Indeed,)32 b(for)h Fl(k)f Fp(=)e(1)j(w)m(e)g(ha)m(v)m(e,)i(when)d Fl(q)k Fp(is)d(small,)g Fl(q)2271 1796 y Fy(0)2341 1782 y Fo(\025)d Fp(exp\()p Fo(\000)p Fl(\016)s Fn({)t Fl(=)p Fp(\()p Fl(q)2921 1796 y Fy(0)2960 1782 y Fp(\))3048 1718 y Fe(1)p 3006 1728 115 4 v 3006 1765 a Fj(d)p Fd(\000)p Fe(1)3136 1782 y Fp(\))g(=)g(\()p Fl(q)3380 1796 y Fy(1)3420 1782 y Fp(\))3456 1749 y Fm(\016)3496 1782 y Fp(.)49 b(Supp)s(ose)0 1892 y(no)m(w)31 b(that)e(\(4.6\))g(is)h(true)g(for)g Fl(k)23 b Fo(\000)d Fp(1)31 b(in)f(place)e(of)j Fl(k)s Fp(,)f(i.e.,)h Fl(q)2037 1906 y Fm(k)r Fh(\000)p Fy(2)2175 1892 y Fl(q)2216 1906 y Fm(k)r Fh(\000)p Fy(3)2370 1892 y Fo(\001)15 b(\001)g(\001)i Fl(q)2533 1906 y Fy(1)2573 1892 y Fl(q)2614 1906 y Fy(0)2681 1892 y Fo(\025)26 b Fp(\()p Fl(q)2855 1906 y Fm(k)r Fh(\000)p Fy(1)2993 1892 y Fp(\))3029 1859 y Fm(\016)3068 1892 y Fp(.)42 b(Then,)30 b(for)h Fl(q)e Fo(2)c Fp(\(0)p Fl(;)d Fp(^)-53 b Fl(q)s Fp(\))0 2017 y(small,)30 b Fl(q)306 2031 y Fm(k)r Fh(\000)p Fy(1)444 2017 y Fl(q)485 2031 y Fm(k)r Fh(\000)p Fy(2)639 2017 y Fo(\001)15 b(\001)g(\001)h Fl(q)801 2031 y Fy(0)868 2017 y Fo(\025)26 b Fp(\()p Fl(q)1042 2031 y Fm(k)r Fh(\000)p Fy(1)1180 2017 y Fp(\))1216 1984 y Fm(\016)r Fy(+1)1374 2017 y Fo(\025)g Fp(exp\()p Fo(\000)p Fl(\016)s Fn({)t Fl(=)p Fp(\()p Fl(q)1950 2031 y Fm(k)r Fh(\000)p Fy(1)2086 2017 y Fp(\))2175 1953 y Fe(1)p 2133 1963 V 2133 2000 a Fj(d)p Fd(\000)p Fe(1)2263 2017 y Fp(\))f(=)i(\()p Fl(q)2499 2031 y Fm(k)2543 2017 y Fp(\))2579 1984 y Fm(\016)2618 2017 y Fp(,)32 b(whic)m(h)e(is)g(\(4.6\).)41 b(This)30 b(pro)m(v)m(es,)h(b)m(y)0 2127 y(induction,)c(that)i(\(4.6\))g(holds)g (for)h(all)f Fl(k)f Fo(\025)d Fp(1.)110 2238 y(F)-8 b(rom)30 b(the)f(de\014nitions)e(of)j Fl(L)1115 2252 y Fm(k)1190 2238 y Fp(and)g Fl(l)1395 2252 y Fm(k)1440 2238 y Fp(,)h(it)e(is)h (clear)e(that)h(for)h(small)f Fl(q)f Fo(2)e Fp(\(0)p Fl(;)21 b Fp(^)-52 b Fl(q)s Fp(\),)1363 2552 y Fl(L)1425 2566 y Fm(k)1525 2552 y Fo(\025)56 b Fl(l)1679 2566 y Fm(k)1780 2552 y Fo(\025)1918 2491 y Fp(1)p 1918 2531 46 4 v 1918 2615 a(2)1990 2424 y Fi(\022)2135 2491 y Fp(1)p 2068 2531 180 4 v 2068 2615 a Fl(q)2109 2629 y Fm(k)r Fh(\000)p Fy(1)2258 2424 y Fi(\023)2336 2414 y Fj(d)p Fe(+2)p Fj(")p 2336 2428 143 4 v 2351 2464 a(d)p Fd(\000)p Fe(1)2510 2552 y Fl(;)1175 b Fp(\(4)p Fl(:)p Fp(7\))0 2830 y(for)24 b Fl(k)k Fo(\025)d Fp(1.)39 b(F)-8 b(or)24 b(a)h(b)s(ound)e(in)g(the)h(opp)s(osite)e(direction,)g(w)m(e)j (use)f(\(4.6\))g(to)g(obtain)e(the)i(follo)m(wing.)36 b(F)-8 b(or)24 b(arbitrary)0 2939 y Fl(\016)k(>)e Fp(0,)k(if)g Fl(q)e Fo(2)e Fp(\(0)p Fl(;)21 b Fp(^)-52 b Fl(q)r Fp(\))30 b(is)g(small)f(enough,)572 3309 y Fl(L)634 3323 y Fm(k)734 3309 y Fo(\024)861 3180 y Fi(\022)1246 3247 y Fp(1)p 939 3288 659 4 v 939 3371 a Fl(q)980 3385 y Fm(k)r Fh(\000)p Fy(1)1119 3371 y Fl(q)1160 3385 y Fm(k)r Fh(\000)p Fy(2)1313 3371 y Fo(\001)15 b(\001)g(\001)i Fl(q)1476 3385 y Fy(1)1517 3371 y Fl(q)1558 3385 y Fy(0)1609 3180 y Fi(\023)1687 3171 y Fj(d)p Fe(+2)p Fj(")p 1687 3185 143 4 v 1702 3221 a(d)p Fd(\000)p Fe(1)1901 3309 y Fo(\024)2028 3125 y Fi(0)2028 3290 y(@)2108 3180 y(\022)2206 3247 y Fp(1)p 2186 3288 86 4 v 2186 3371 a Fl(q)2227 3385 y Fm(k)2283 3180 y Fi(\023)2361 3167 y Fe(\()p Fj(d)p Fd(\000)p Fe(1\))p Fj(\016)p 2361 3184 196 4 v 2387 3221 a(d)p Fe(+2)p Fj(")2572 3125 y Fi(1)2572 3290 y(A)2663 3116 y Fj(d)p Fe(+2)p Fj(")p 2663 3130 143 4 v 2677 3166 a(d)p Fd(\000)p Fe(1)2877 3309 y Fp(=)3003 3180 y Fi(\022)3102 3247 y Fp(1)p 3082 3288 86 4 v 3082 3371 a Fl(q)3123 3385 y Fm(k)3178 3180 y Fi(\023)3245 3199 y Fm(\016)3301 3309 y Fl(;)384 b Fp(\(4)p Fl(:)p Fp(8\))0 3636 y(for)30 b Fl(k)e Fo(\025)d Fp(1.)0 3860 y Ff(Pr)-5 b(o)g(of)30 b(of)g(L)-5 b(emma)31 b(4.1.)46 b Fp(W)-8 b(e)28 b(use)f(induction)e(on)j Fl(m)p Fp(.)40 b(The)27 b(statemen)m(t)f(is)i(ob)m(viously)f(true)f(for)i Fl(m)d Fp(=)h(0.)40 b(Assume)0 3970 y(no)m(w)30 b(that)f(it)g(is)h (true)f(for)h Fl(m)c Fp(=)f Fl(k)s Fp(,)30 b(and)g(w)m(e)g(will)f(sho)m (w)h(that)j Fi(e)-55 b Fl(q)2162 3984 y Fm(k)r Fy(+1)2340 3970 y Fo(\024)25 b Fl(q)2477 3984 y Fm(k)r Fy(+1)2614 3970 y Fp(.)110 4091 y(F)-8 b(ollo)m(wing)29 b(Rule)i(1,)h(w)m(e)g (observ)m(e)f(the)g(ev)m(olution)f(inside)f(the)i(b)s(o)m(x)2475 4068 y Fi(e)2456 4091 y Fl(B)2525 4105 y Fm(k)r Fy(+1)2695 4091 y Fp(with)f(+1)i(b)s(oundary)e(conditions,)0 4200 y(during)i(the)g(in)m(terv)-5 b(al)32 b(of)h(time)f([)p Fl(T)1183 4214 y Fm(k)1229 4200 y Fl(;)15 b(T)1322 4214 y Fm(k)r Fy(+1)1460 4200 y Fp(\).)51 b(Let)33 b Fl(F)1798 4214 y Fm(k)r Fy(+1)1969 4200 y Fp(b)s(e)f(the)h(ev)m(en)m(t)g(that)f (in)h(this)f(ev)m(olution)g Fo(\000)p Fp(1)h(spins)g(are)0 4310 y(presen)m(t)c(in)g(the)h(b)s(o)m(x)g Fl(B)826 4324 y Fm(k)r Fy(+1)994 4310 y Fp(at)f(times)g(whic)m(h)h(are)f(arbitrarily) e(close)i(to)g Fl(T)2586 4324 y Fm(k)r Fy(+1)2724 4310 y Fp(.)41 b(W)-8 b(e)30 b(will)f(sho)m(w)h(that)1554 4554 y FA(P)1626 4568 y Fm(p)1668 4554 y Fp(\()p Fl(F)1763 4568 y Fm(k)r Fy(+1)1900 4554 y Fp(\))55 b Fo(\024)2129 4492 y Fl(q)2170 4506 y Fm(k)r Fy(+1)p 2129 4533 179 4 v 2195 4617 a Fp(2)2318 4554 y Fl(:)1367 b Fp(\(4)p Fl(:)p Fp(9\))0 4802 y(W)-8 b(e)30 b(will)f(also)g(sho)m(w)h(that)1392 4916 y FA(P)1464 4930 y Fm(p)1506 4916 y Fp(\()p Fl(\034)1582 4930 y Fm(k)r Fy(+1)1760 4916 y Fl(<)40 b(T)1924 4930 y Fm(k)r Fy(+1)2062 4916 y Fp(\))55 b Fo(\024)2291 4854 y Fl(q)2332 4868 y Fm(k)r Fy(+1)p 2291 4895 V 2357 4979 a Fp(2)2480 4916 y Fl(;)1159 b Fp(\(4)p Fl(:)p Fp(10\))0 5129 y(where)31 b Fl(\034)306 5143 y Fm(k)r Fy(+1)473 5129 y Fp(=)e Fl(\034)623 5096 y Fy(0)613 5155 y Fm(k)r Fy(+1)751 5129 y Fp(.)48 b(Com)m(bining)30 b(\(4.9\))i(with)f(\(4.10\)) g(yields)g(the)h(desired)e(inequalit)m(y)35 b Fi(e)-56 b Fl(q)3185 5143 y Fm(k)r Fy(+1)3367 5129 y Fo(\024)29 b Fl(q)3508 5143 y Fm(k)r Fy(+1)3646 5129 y Fp(,)k(since)4 5240 y Fi(e)-55 b Fl(q)41 5254 y Fm(k)r Fy(+1)204 5240 y Fo(\024)25 b FA(P)372 5254 y Fm(p)414 5240 y Fp(\()p Fl(F)509 5254 y Fm(k)r Fy(+1)646 5240 y Fp(\))19 b(+)i FA(P)865 5254 y Fm(p)907 5240 y Fp(\()p Fl(\034)983 5254 y Fm(k)r Fy(+1)1160 5240 y Fl(<)41 b(T)1325 5254 y Fm(k)r Fy(+1)1463 5240 y Fp(\).)p eop %%Page: 14 14 14 13 bop 0 46 a Fw(14)1111 b(F)n(ONTES,)33 b(SCHONMANN,)i(SIDORA)-9 b(VICIUS)110 265 y Fp(The)30 b(pro)s(of)g(of)h(\(4.9\))g(will)e(b)s(e)i (divided)e(in)m(to)i(t)m(w)m(o)g(steps.)43 b(In)32 b(the)e(\014rst)h (step,)g(w)m(e)h(will)d(analyze)h(the)g(random)0 386 y(con\014guration)21 b(inside)821 363 y Fi(e)802 386 y Fl(B)871 400 y Fm(k)r Fy(+1)1034 386 y Fp(at)k(the)f(time)g Fl(T)1549 400 y Fm(k)1594 386 y Fp(,)j(and)d(use)h(metho)s(ds)e(from)h (the)h(study)f(of)h(b)s(o)s(otstrap)c(p)s(ercolation)0 495 y(to)28 b(sho)m(w)h(that)f(all)f(the)h Fo(\000)p Fp(1)h(spins)e(in)i(this)e(con\014guration)e(are)j(lik)m(ely)g(to)g(b)s (e)g(con)m(tained)e(in)i(a)h(collection)24 b(of)29 b(w)m(ell)0 605 y(separated)19 b(rectangles)f(whic)m(h)i(are)g(not)h(to)s(o)e (large.)36 b(In)21 b(the)g(second)e(step,)k(w)m(e)e(will)f(analyze)f (the)h(ev)m(olution)f(inside)0 721 y(the)28 b(b)s(o)m(x)348 698 y Fi(e)329 721 y Fl(B)398 735 y Fm(k)r Fy(+1)564 721 y Fp(with)f(+1)i(b)s(oundary)d(conditions,)g(during)h(the)g(in)m (terv)-5 b(al)26 b(of)j(time)e([)p Fl(T)2944 735 y Fm(k)2990 721 y Fl(;)15 b(T)3083 735 y Fm(k)r Fy(+1)3221 721 y Fp(\),)29 b(conditioned)24 b(on)0 831 y(the)29 b(con\014guration)e(at)i (time)g Fl(T)1083 845 y Fm(k)1159 831 y Fp(ha)m(ving)g(this)g(prop)s (ert)m(y)-8 b(.)110 940 y(The)29 b(third)g(step)g(in)h(the)f(pro)s(of)g (of)h(Lemma)g(4.1)g(will)e(b)s(e)i(the)f(pro)s(of)g(of)h(\(4.10\).)0 1162 y Ff(Step)j(1.)41 b(Contr)-5 b(ol)33 b(of)g(b)-5 b(o)g(otstr)g(apping)32 b(at)h(time)f Fl(T)1678 1176 y Fm(k)1723 1162 y Ff(.)41 b Fp(W)-8 b(e)30 b(will)e(co)m(v)m(er)i(all) f(the)g(sites)f(in)2970 1139 y Fi(e)2951 1162 y Fl(B)3020 1176 y Fm(k)r Fy(+1)3188 1162 y Fp(whic)m(h)h(ha)m(v)m(e)h(a)g(spin)0 1272 y Fo(\000)p Fp(1)f(at)g(time)f Fl(T)517 1286 y Fm(k)591 1272 y Fp(with)g(a)h(collection)c Fl(R)1347 1286 y Fy(1)1388 1272 y Fp(,)30 b Fl(R)1512 1286 y Fy(2)1554 1272 y Fp(,)g(...,)h Fl(R)1809 1286 y Fm(N)1908 1272 y Fp(of)e(w)m(ell)f(separated)e (rectangles.)37 b(\(Recall)27 b(that)h(a)h(family)0 1381 y(of)h(disjoin)m(t)e(subsets)h(of)i Fn(Z)915 1348 y Fm(d)984 1381 y Fp(is)e(w)m(ell)h(separated)d(if)j(there)f(is)h(no)g(v)m(ertex)g (in)f Fn(Z)2657 1348 y Fm(d)2726 1381 y Fp(at)h(distance)d(less)j(than) f(or)h(equal)0 1491 y(to)37 b(1)g(from)g(t)m(w)m(o)h(sets)e(in)h(the)f (family)-8 b(.\))61 b(A)m(t)38 b(time)e Fl(T)1839 1505 y Fm(k)1922 1491 y Fp(all)g(blo)s(c)m(ks)g Fl(B)2414 1458 y Fm(i)2409 1517 y(k)2492 1491 y Fp(of)h(scale)f Fl(k)k Fp(are)d(mono)s(c)m(hromatic,)f(i.e.,)0 1601 y(en)m(tirely)e(o)s (ccupied)f(b)m(y)k(spins)f Fo(\000)p Fp(1)g(or)g(+1,)j(and)c(no)m(w)i (they)f(will)e(pla)m(y)i(the)g(role)f(of)h(\\renormalized)c(sites")j (of)19 1693 y Fi(e)0 1716 y Fl(B)69 1730 y Fm(k)r Fy(+1)207 1716 y Fp(.)41 b(F)-8 b(or)29 b(the)g(sak)m(e)h(of)g(notation,)d(w)m(e) k(iden)m(tify)d(these)g(\\renormalized)e(sites")i(with)h(elemen)m(ts)f (of)3530 1693 y(\026)3508 1716 y Fl(B)3577 1730 y Fm(k)r Fy(+1)3745 1716 y Fp(\(see)0 1826 y(\(4.4\)\).)43 b(F)-8 b(or)31 b Fl(i)c Fo(2)626 1803 y Fp(\026)604 1826 y Fl(B)673 1840 y Fm(k)r Fy(+1)811 1826 y Fp(,)32 b(let)e Fl(\021)1046 1840 y Fm(k)1091 1826 y Fp(\()p Fl(i)p Fp(\))i(b)s(e)e(+1)h(\(resp.)43 b Fo(\000)p Fp(1\))31 b(if)g(the)f(blo)s(c)m(k)g Fl(B)2510 1793 y Fm(i)2505 1852 y(k)2582 1826 y Fp(is)h(in)g(state)e(+1)j (\(resp.)43 b Fo(\000)p Fp(1\))30 b(at)h(time)0 1936 y Fl(T)53 1950 y Fm(k)98 1936 y Fp(.)64 b(W)-8 b(e)38 b(obtain)e(\014rst)h(a)h(the)f(collection)c(of)38 b(rectangles)2068 1913 y(\026)2049 1936 y Fl(R)2118 1950 y Fy(1)2159 1936 y Fp(,)2244 1913 y(\026)2224 1936 y Fl(R)2293 1950 y Fy(2)2335 1936 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(sidelengths)d(of)j(the)f(rectangle.)p eop %%Page: 15 15 15 14 bop 661 46 a Fw(STRETCHED)34 b(EXPONENTIAL)g(FIXA)-6 b(TION)34 b(A)-6 b(T)33 b(ZER)n(O)h(TEMPERA)-6 b(TURE)583 b(15)110 265 y Fp(W)-8 b(e)29 b(claim)e(that)h(if)h Fl(R)d Fo(\032)1001 242 y Fp(\026)980 265 y Fl(B)1049 279 y Fm(k)1123 265 y Fp(is)j(a)g(rectangle)d(of)j(size)f Fl(n)2009 279 y Fy(1)2068 265 y Fo(\002)19 b Fl(n)2213 279 y Fy(2)2272 265 y Fo(\002)f(\001)d(\001)g(\001)20 b(\002)e Fl(n)2630 279 y Fm(d)2673 265 y Fp(,)30 b(with)e Fl(n)2990 279 y Fy(1)3056 265 y Fo(\024)d Fl(n)3207 279 y Fy(2)3273 265 y Fo(\024)h Fl(:::)g Fo(\024)g Fl(n)3623 279 y Fm(d)3665 265 y Fp(,)k(then)0 374 y(for)g(the)f(b)s(o)s(otstrap)e(p)s(ercolation) e(pro)s(cess)k(that)g(w)m(e)h(are)g(considering,)813 570 y FA(P)884 584 y Fm(p)927 570 y Fp(\()p Fl(R)h Fp(is)e(in)m (ternally)f(spanned\))53 b Fo(\024)j Fp(\(2)p Fl(n)2246 584 y Fy(1)2286 570 y Fl(n)2341 584 y Fy(2)2397 570 y Fo(\001)15 b(\001)g(\001)i Fl(n)2574 584 y Fm(d)p Fh(\000)p 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b(can)g(b)s(e)h(obtained)d(on)j(this)f(probabilit)m(y)f(b)m(y)j (starting)c(the)j(ev)m(olution)e(inside)3045 4279 y Fi(e)3026 4302 y Fl(B)3095 4316 y Fm(k)r Fy(+1)3265 4302 y Fp(at)i(time)f Fl(T)3642 4316 y Fm(k)3720 4302 y Fp(with)0 4412 y Fo(\000)p Fp(1)40 b(spins)f(at)g(all)f(sites)h(of)g(the)g(rectangles)e Fl(R)1653 4426 y Fy(1)1694 4412 y Fl(;)15 b(:::;)g(R)1918 4426 y Fm(N)2030 4412 y Fp(describ)s(ed)37 b(in)i(Step)f(1.)70 b(Clearly)-8 b(,)41 b(no)e(spin)g Fo(\000)p Fp(1)h(can)0 4521 y(app)s(ear)24 b(in)h(this)g(ev)m(olution)f(outside)g(of)i(the)f (rectangles)e Fl(R)2022 4535 y Fy(1)2063 4521 y Fl(;)15 b(:::;)g(R)2287 4535 y Fm(N)2359 4521 y Fp(.)40 b(Also,)27 b(once)e(a)h(rectangle)c(is)k(\\destro)m(y)m(ed",)0 4631 y(meaning)33 b(that)g(it)h(con)m(tains)e(no)j Fo(\000)p Fp(1)f(spins,)h(no)g Fo(\000)p Fp(1)f(spins)g(will)f(ev)m(er)h(b)s(e)g (created)f(inside)f(of)j(it)e(again,)i(in)f(the)0 4741 y(ev)m(olution)28 b(that)h(w)m(e)h(are)g(considering)c(here.)110 4850 y(If)21 b Fl(E)259 4864 y Fm(k)r Fy(+1)419 4850 y Fp(o)s(ccurs,)g(then)g(eac)m(h)f(rectangle)f Fl(R)1567 4864 y Fm(n)1617 4850 y Fp(,)k Fl(n)j Fp(=)f(1)p Fl(;)15 b(:::;)g(N)10 b Fp(,)26 b(is)21 b(con)m(tained)e(in)i(a)g(cub)s(e)f(of) h(sidelength)d(b)s(ounded)0 4960 y(ab)s(o)m(v)m(e)30 b(b)m(y)1295 5007 y Fi($)1494 5101 y Fl(b)p 1365 5142 298 4 v 1365 5246 a Fp(\()p Fl(q)1442 5260 y Fm(k)1486 5246 y Fp(\))1574 5182 y Fe(1)p 1532 5192 115 4 v 1532 5229 a Fj(d)p Fd(\000)p Fe(1)1673 5007 y Fi(\045)1747 5163 y Fl(L)1809 5177 y Fm(k)1910 5163 y Fo(\024)2036 5034 y Fi(\022)2135 5101 y Fp(1)p 2115 5142 86 4 v 2115 5225 a Fl(q)2156 5239 y Fm(k)2211 5034 y Fi(\023)2289 5021 y Fe(1+)p Fj("=)p Fe(\(2)p Fj(d)p Fe(\))p 2289 5039 257 4 v 2361 5075 a Fj(d)p Fd(\000)p Fe(1)2577 5163 y Fl(;)p eop %%Page: 16 16 16 15 bop 0 46 a Fw(16)1111 b(F)n(ONTES,)33 b(SCHONMANN,)i(SIDORA)-9 b(VICIUS)0 265 y Fp(for)42 b(small)g Fl(q)49 b Fo(2)d Fp(\(0)p Fl(;)21 b Fp(^)-52 b Fl(q)r Fp(\),)46 b(where)c(w)m(e)h(used)e (\(4.8\).)77 b(By)43 b(attractiv)m(eness,)g(the)e(time)h(needed)f(to)h (ero)s(de)e Fl(R)3743 279 y Fm(n)3836 265 y Fp(is)0 374 y(therefore)29 b(sto)s(c)m(hastically)e(b)s(ounded)i(ab)s(o)m(v)m(e)i (b)m(y)i(the)d(time)h(needed)e(to)i(ero)s(de)f(a)i(cub)s(e)e(with)g (sidelength)e(equal)0 484 y(to)34 b(this)g(upp)s(er)f(b)s(ound.)52 b(F)-8 b(rom)34 b(Theorem)g(1.3\(a\))f(w)m(e)i(conclude)d(then)h(that,) i(for)f(small)f Fl(q)j Fo(2)d Fp(\(0)p Fl(;)21 b Fp(^)-52 b Fl(q)s Fp(\),)35 b(for)g(eac)m(h)0 597 y Fl(n)29 b Fp(=)g(1)p Fl(;)15 b(:::;)g(N)10 b Fp(,)36 b(the)c(probabilit)m(y)d (that)i(at)h(time)g Fl(T)1736 611 y Fm(k)r Fy(+1)1902 597 y Fp(=)e Fl(T)2056 611 y Fm(k)2123 597 y Fp(+)22 b Fl(t)2249 611 y Fm(k)r Fy(+1)2415 597 y Fp(=)30 b Fl(T)2569 611 y Fm(k)2636 597 y Fp(+)21 b(\(1)p Fl(=q)2897 611 y Fm(k)2941 597 y Fp(\))2977 564 y Fy(\()p Fm(d)p Fy(+)p Fm(")p Fy(\))p Fm(=)p Fy(\()p Fm(d)p Fh(\000)p Fy(1\))3426 597 y Fp(there)30 b(is)j(an)m(y)0 707 y(spin)c Fo(\000)p Fp(1)h(inside)f 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179 4 v 2519 1719 a Fl(q)2560 1733 y Fm(k)r Fy(+1)2708 1656 y Fl(;)0 1931 y Fp(b)m(y)31 b(\(4.8\).)40 b(F)-8 b(rom)29 b(the)h(b)s(ounds)e(in)i(the)f(last)g(t)m(w)m(o)i(displa)m (ys,)e(w)m(e)i(obtain)736 2260 y FA(P)807 2274 y Fm(p)850 2260 y Fp(\()p Fl(F)945 2274 y Fm(k)r Fy(+1)1081 2260 y Fo(j)p Fl(E)1173 2274 y Fm(k)r Fy(+1)1311 2260 y Fp(\))56 b Fo(\024)1607 2198 y Fp(1)p 1541 2239 V 1541 2323 a Fl(q)1582 2337 y Fm(k)r Fy(+1)1775 2260 y Fp(exp)1930 2077 y Fi(0)1930 2241 y(@)2010 2260 y Fo(\000)p Fl(\015)2149 2131 y Fi(\022)2247 2198 y Fp(1)p 2227 2239 86 4 v 2227 2323 a Fl(q)2268 2337 y Fm(k)2324 2131 y Fi(\023)2402 2118 y Fe(1+)p Fj("=)p Fe(\(2)p Fj(d)p Fe(\))p 2402 2136 257 4 v 2473 2172 a Fj(d)p Fd(\000)p Fe(1)2674 2077 y Fi(1)2674 2241 y(A)2810 2260 y Fo(\024)2947 2198 y Fl(q)2988 2212 y Fm(k)r Fy(+1)p 2947 2239 179 4 v 3014 2323 a Fp(4)3137 2260 y Fl(;)0 2589 y Fp(for)30 b(small)f Fl(q)f Fo(2)d Fp(\(0)p Fl(;)d Fp(^)-53 b Fl(q)s Fp(\).)41 b(This)29 b(completes)f(the)h(pro)s(of)g(of)h(\(4.17\),)f(and)h(hence)e(of)i (\(4.9\).)0 2809 y Ff(Step)j(3.)42 b(Contr)-5 b(ol)33 b(of)f(the)h(outer)f(in\015uenc)-5 b(e.)42 b Fp(In)30 b(this)f(step)g(w)m(e)i(will)d(pro)m(v)m(e)i(\(4.10\).)110 2930 y(W)-8 b(e)44 b(b)s(egin)d(with)i(the)g(notion)f(of)i(a)g Ff(discr)-5 b(ep)g(ancy)45 b(pr)-5 b(o)g(c)g(ess)p Fp(.)83 b(Consider)42 b(t)m(w)m(o)i(ev)m(olutions)d Fl(\033)3405 2886 y Fm(\030)r(;T)3505 2896 y Fj(k)3417 2959 y Fc(e)3402 2975 y Fm(B)3457 2985 y Fj(k)q Fe(+1)3576 2975 y Fm(;)p Fy(+;)p Fm(t)3751 2930 y Fp(and)0 3091 y Fl(\033)55 3047 y Fm(\030)r(;T)155 3057 y Fj(k)67 3120 y Fc(e)52 3136 y Fm(B)107 3146 y Fj(k)q Fe(+1)225 3136 y Fm(;)p Fh(\000)p Fy(;)p Fm(t)357 3091 y Fp(,)36 b Fl(t)d Fo(\025)g Fl(T)641 3105 y Fm(k)686 3091 y Fp(,)j(starting)c(at)i(time)g Fl(T)1474 3105 y Fm(k)1554 3091 y Fp(from)g(the)g(same)g (con\014guration)c Fl(\030)39 b Fp(in)2936 3068 y Fi(e)2916 3091 y Fl(B)2985 3105 y Fm(k)r Fy(+1)3123 3091 y Fp(,)d(with)e(\(+\))g 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4071 y(at)e(time)f Fl(s)p Fo(\000)i Fp(there)e(m)m(ust)h(b)s(e)g(at)g (least)f(one)g(neigh)m(b)s(oring)e(v)m(ertex)j(of)g Fl(z)42 b Fp(o)s(ccupied)33 b(b)m(y)k(a)f(discrepancy)-8 b(.)58 b(This)0 4181 y(follo)m(ws)38 b(straigh)m(tforw)m(ardly)d(from)j(the)g (fact)g(that)g(the)g(t)m(w)m(o)h(ev)m(olutions)d(that)i(w)m(e)h(are)f (considering)d(use)k(the)0 4297 y(same)30 b(graphical)d(marks.)41 b(F)-8 b(or)29 b Fl(x)d Fo(2)1279 4273 y Fi(e)1260 4297 y Fl(B)1329 4311 y Fm(k)r Fy(+1)1467 4297 y Fp(,)31 b(w)m(e)f(de\014ne) 667 4542 y Fl(T)13 b Fp(\()p Fl(x)p Fp(\))55 b(=)h(inf)1180 4441 y Fi(n)1241 4542 y Fl(s)25 b Fo(\025)h Fl(T)1459 4556 y Fm(k)1529 4542 y Fp(:)51 b Fl(\033)1660 4498 y Fm(\030)r(;T)1760 4508 y Fj(k)1672 4571 y Fc(e)1657 4588 y Fm(B)1712 4598 y Fj(k)q Fe(+1)1831 4588 y Fm(;)p Fy(+;)p Fm(t)1961 4542 y Fp(\()p Fl(x)p Fp(\))25 b Fo(6)p Fp(=)g Fl(\033)2261 4498 y Fm(\030)r(;T)2361 4508 y Fj(k)2273 4571 y Fc(e)2258 4588 y Fm(B)2313 4598 y Fj(k)q Fe(+1)2432 4588 y Fm(;)p Fh(\000)p 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Fm(n)739 5021 y Fo(g)e Fp(suc)m(h)g(that)1023 5240 y Fl(T)1076 5254 y Fm(k)1162 5240 y Fl(<)41 b(T)13 b Fp(\()p Fl(x)1428 5254 y Fy(1)1469 5240 y Fp(\))40 b Fl(<)g(T)13 b Fp(\()p Fl(x)1810 5254 y Fy(2)1851 5240 y Fp(\))40 b Fl(<)25 b Fo(\001)15 b(\001)g(\001)27 b Fl(<)41 b(T)13 b Fp(\()p Fl(x)2421 5254 y Fm(n)2470 5240 y Fp(\))41 b Fl(<)f(T)2711 5254 y Fm(k)r Fy(+1)2849 5240 y Fl(:)p eop %%Page: 17 17 17 16 bop 661 46 a Fw(STRETCHED)34 b(EXPONENTIAL)g(FIXA)-6 b(TION)34 b(A)-6 b(T)33 b(ZER)n(O)h(TEMPERA)-6 b(TURE)583 b(17)0 265 y Fp(In)30 b(this)f(case)h(w)m(e)g(sa)m(y)h(that)e(the)g(c)m (hronological)d(path)j(starts)g(at)g Fl(x)2315 279 y Fy(1)2387 265 y Fp(and)h(ends)f(at)h Fl(x)2937 279 y Fm(n)2987 265 y Fp(.)110 374 y(In)g(what)f(follo)m(ws,)g(for)h(an)m(y)h (v)m(olume)e(\003)798 585 y Fl(@)5 b Fp(\003)27 b(=)e Fo(f)p Fl(x)g Fo(2)h Fp(\003)f(:)57 b(there)28 b(is)i(a)g(nearest)f (neigh)m(b)s(or)e(of)j Fl(x)h Fp(not)e(in)h(\003)p Fo(g)p Fp(;)810 710 y(\026)798 734 y Fl(@)6 b Fp(\003)26 b(=)f Fo(f)p Fl(x)36 b(=)-57 b Fo(2)26 b Fp(\003)f(:)57 b(there)28 b(is)i(a)g(nearest)f(neigh)m(b)s(or)e(of)j Fl(x)h Fp(in)e(\003)q Fo(g)p Fl(:)0 956 y Fp(Pro)s(ceeding)24 b(with)j(the)g(argumen)m(t,)g (w)m(e)h(claim)f(that)f(if)h(some)h(v)m(ertex)f Fl(x)f Fo(2)f Fl(@)5 b(B)2703 923 y Fm(i)2698 982 y(k)r Fy(+1)2865 956 y Fp(at)27 b(time)f Fl(t)g(>)f(T)3387 970 y Fm(k)3461 956 y Fp(is)i(o)s(ccupied)0 1067 y(b)m(y)32 b(a)g(discrepancy)-8 b(,)30 b(then)g(there)g(exists)h(a)g(c)m(hronological)d(path)i(in)h (the)g(ep)s(o)s(c)m(h)e(\()p Fl(T)2881 1081 y Fm(k)2926 1067 y Fl(;)15 b(t)p Fp(\))32 b(whic)m(h)f(starts)f(at)h(some)0 1188 y(v)m(ertex)h(of)397 1164 y(\026)386 1188 y Fl(@)458 1165 y Fi(e)439 1188 y Fl(B)513 1155 y Fm(i)508 1214 y(k)r Fy(+1)679 1188 y Fp(and)g(ends)g(at)g Fl(x)e Fo(2)f Fl(@)5 b(B)1483 1155 y Fm(i)1478 1214 y(k)r Fy(+1)1616 1188 y Fp(.)48 b(Note)33 b(that)e(the)h(ab)s(o)m(v)m(e)g(claim)f(is)h (a)h(deterministic)28 b(statemen)m(t)0 1299 y(and)f(do)s(es)g(not)g 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(b)s(oring)d(v)m(ertex)j(and)g(at)g(time)f Fl(T)2146 2819 y Fm(k)2191 2805 y Fp(,)j(the)d(discrepancies)d(are)k(lo)s(cated)d (only)j(outside)19 2898 y Fi(e)0 2921 y Fl(B)74 2888 y Fm(i)69 2947 y(k)r Fy(+1)207 2921 y Fp(.)41 b(So)29 b(our)h(path)f(m)m(ust)h(b)s(e)g(traced)e(bac)m(k)i(to)g(some)g Fl(x)2011 2935 y Fm(r)2077 2921 y Fo(2)2174 2897 y Fp(\026)2163 2921 y Fl(@)2236 2898 y Fi(e)2216 2921 y Fl(B)2290 2888 y Fm(i)2285 2947 y(k)r Fy(+1)2423 2921 y Fp(.)110 3031 y(T)-8 b(o)28 b(get)f(a)h(forw)m(ards)g(in)f(time)g(path,)h(w)m(e)h(in) m(v)m(ert)e(the)h(order)f(and)g(set)h Fl(x)2536 3045 y Fm(j)2600 3031 y Fp(=)j(~)-52 b Fl(x)2748 3045 y Fm(r)r Fy(+1)p Fh(\000)p Fm(j)2972 3031 y Fl(;)41 b Fp(1)25 b Fo(\024)h Fl(j)k Fo(\024)25 b Fl(r)s Fp(,)k(and)f(obtain)0 3155 y(a)i(c)m(hronological)c(path)j(starting)f(at)h Fl(x)1344 3169 y Fy(1)1411 3155 y Fo(2)1509 3131 y Fp(\026)1497 3155 y Fl(@)1570 3132 y Fi(e)1551 3155 y Fl(B)1625 3122 y Fm(i)1620 3181 y(k)r Fy(+1)1788 3155 y Fp(and)g(ending)g(at)g Fl(x)2423 3169 y Fm(r)2489 3155 y Fo(2)c Fl(@)5 b(B)2702 3122 y Fm(i)2697 3181 y(k)r Fy(+1)2836 3155 y Fp(.)110 3265 y(F)-8 b(rom)23 b(the)g(claim)f(it)h(follo)m(ws)g(that,)h(if)f Fl(\034)1445 3279 y Fm(k)r Fy(+1)1623 3265 y Fo(\024)41 b Fl(T)1788 3279 y Fm(k)r Fy(+1)1925 3265 y Fp(,)26 b(then)d(there)f (exists)h(a)g(c)m(hronological)d(path)i(connecting)12 3360 y(\026)0 3384 y Fl(@)73 3361 y Fi(e)54 3384 y Fl(B)123 3398 y Fm(k)r Fy(+1)291 3384 y Fp(to)29 b Fl(@)5 b(B)524 3398 y Fm(k)r Fy(+1)663 3384 y Fp(,)30 b(th)m(us)g(co)m(v)m(ering)f(a)h (distance)e(whic)m(h,)i(for)f(small)g Fl(q)g Fo(2)c Fp(\(0)p Fl(;)d Fp(^)-53 b Fl(q)s Fp(\),)30 b(is)g(at)g(least)1269 3687 y Fl(r)1310 3701 y Fm(k)1411 3687 y Fp(=)1538 3558 y Fi(\026)1602 3625 y Fp(1)p 1602 3666 46 4 v 1602 3750 a(4)1659 3687 y Fl(L)1721 3701 y Fm(k)r Fy(+1)1858 3558 y Fi(\027)1967 3687 y Fo(\025)2105 3625 y Fp(1)p 2105 3666 V 2105 3750 a(9)2177 3558 y Fi(\022)2275 3625 y Fp(1)p 2255 3666 86 4 v 2255 3750 a 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b(estimate)h(for)i(P)m(oisson)f(random)g(v)-5 b(ariables)30 b(\(see,)i(e.g.,)h(\(A.2\))f(in)g([KS],)h(p.467\))e FA(P)p Fo(f)p Fl(Z)h Fo(\025)25 b Fl(r)e Fo(\000)d Fp(1)p Fo(g)44 b(\024)0 5240 y Fl(e)43 5207 y Fh(\000)p Fy(\(log)q(\()p Fm(r)r(=t)353 5217 y Fj(k)q Fe(+1)470 5207 y Fy(\))p Fh(\000)p Fy(1\))p Fm(r)661 5240 y Fl(;)39 b Fp(together)d(with)i(the)g (upp)s(er)f(b)s(ound)g(\(2)p Fl(d)p Fp(\))2203 5207 y Fm(r)2280 5240 y Fp(on)i(the)e(n)m(um)m(b)s(er)h(of)h(self-a)m(v)m (oiding)d(paths)i(of)p eop %%Page: 18 18 18 17 bop 0 46 a Fw(18)1111 b(F)n(ONTES,)33 b(SCHONMANN,)i(SIDORA)-9 b(VICIUS)0 265 y Fp(length)24 b Fl(r)29 b Fp(starting)23 b(from)j(a)g(giv)m(en)f(v)m(ertex,)i(w)m(e)f(obtain)e(the)i(follo)m (wing)d(estimate.)37 b(If)26 b Fl(q)j Fo(2)c Fp(\(0)p Fl(;)c Fp(^)-52 b Fl(q)s Fp(\))25 b(is)h(c)m(hosen)f(small)0 374 y(enough,)k(then)g(for)h(appropriate)c(constan)m(ts)j Fl(C)1657 341 y Fh(0)1713 374 y Fp(and)g Fl(C)1962 341 y Fh(00)2009 374 y Fp(,)h(for)g(an)m(y)h Fl(k)d 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b(TURE)583 b(19)0 265 y Fp(for)29 b(the)f(corresp)s(onding)c(v)-5 b(alues)28 b(of)h Fl(q)1295 279 y Fm(k)1340 265 y Fp(,)h Fl(t)1428 279 y Fm(k)1502 265 y Fp(and)e Fl(T)1731 279 y Fm(k)1806 265 y Fp(de\014ned)f(b)m(y)j(\(4.2\))e(and)g(\(4.3\))g (with)g Fl(q)3103 279 y Fy(0)3169 265 y Fp(=)d Fl(q)s Fp(.)41 b(Summarizing)0 374 y(the)26 b(conclusion)d(in)j(\(4.22\),)g(w) m(e)h(kno)m(w)g(that)e(there)g(exists)30 b Fi(e)-55 b Fl(q)28 b Fo(2)d Fp(\(0)p Fl(;)d Fp(^)-53 b Fl(q)s Fp(\))26 b(and)g Fl(C)32 b(>)26 b Fp(0,)h(suc)m(h)g(that)e(for)h(all)f Fl(q)k Fo(2)c Fp(\(0)p Fl(;)19 b Fi(e)-55 b Fl(q)s Fp(],)0 484 y(if)30 b(for)f(some)h Fl(k)e Fo(\025)d Fp(1,)31 b Fl(t)25 b Fp(=)h Fl(T)934 498 y Fm(k)979 484 y Fp(\()p Fl(q)s Fp(\),)k(then)1418 715 y FA(P)1490 729 y Fm(p)1532 715 y Fp(\()p Fl(\033)1620 729 y Fm(t)1652 715 y Fp(\(0\))24 b(=)i Fo(\000)p Fp(1\))55 b Fo(\024)g Fl(e)2268 677 y Fh(\000)p Fm(C)t(t)2409 650 y Fj(a)2454 715 y Fl(:)1185 b Fp(\(4)p Fl(:)p Fp(23\))0 954 y(W)-8 b(e)35 b(will)f(write)k Fi(e)-56 b Fl(q)619 968 y Fm(k)698 954 y Fp(=)33 b Fl(q)843 968 y Fm(k)888 954 y Fp(\()t Fi(e)-55 b Fl(q)s Fp(\),)1064 937 y Fi(e)1065 954 y Fl(t)1098 968 y Fm(k)1177 954 y Fp(=)33 b Fl(t)1314 968 y Fm(k)1359 954 y Fp(\()t Fi(e)-55 b Fl(q)s Fp(\))35 b(and)1708 930 y Fi(e)1692 954 y Fl(T)1745 968 y Fm(k)1824 954 y Fp(=)f Fl(T)1982 968 y Fm(k)2027 954 y Fp(\()t Fi(e)-55 b Fl(q)s Fp(\).)55 b(Since)38 b Fi(e)-56 b Fl(q)37 b Fo(2)d Fp(\(0)p Fl(;)21 b Fp(^)-52 b Fl(q)r Fp(\),)41 b Fi(e)-55 b Fl(q)2943 968 y Fm(k)3023 954 y Fp(decreases)33 b(with)h Fl(k)s Fp(,)j(and)0 1068 y(therefore)382 1051 y Fi(e)384 1068 y Fl(t)417 1082 y Fm(k)492 1068 y Fp(increases)28 b(with)h Fl(k)s Fp(.)110 1179 y(Note)j(that,)g(for)g(eac)m(h)g(\014xed)g Fl(k)g Fo(\025)d Fp(1,)34 b(if)e(w)m(e)h(imagine)d(the)i(parameter)e Fl(q)36 b Fp(decreasing)29 b(con)m(tin)m(uously)h(from)37 b Fi(e)-55 b Fl(q)0 1295 y Fp(to)36 b Fi(e)-55 b Fl(q)155 1309 y Fy(1)196 1295 y Fp(,)34 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Fm(k)r Fy(+1)2696 1416 y Fo(\000)2784 1400 y Fi(e)2785 1416 y Fl(t)2818 1430 y Fy(1)2859 1416 y Fp(.)41 b(Th)m(us,)30 b(b)m(y)g(con)m(tin)m(uit)m(y)-8 b(,)29 b(an)m(y)0 1537 y Fl(t)i(>)g Fp(0)i(whic)m(h)g(is)g(not)g(in)g Fo([)942 1551 y Fm(k)r Fh(\025)p Fy(1)1080 1537 y Fp([)1121 1514 y Fi(e)1105 1537 y Fl(T)1158 1551 y Fm(k)1226 1537 y Fo(\000)1319 1521 y Fi(e)1320 1537 y Fl(t)1353 1551 y Fy(1)1394 1537 y Fl(;)1465 1514 y Fi(e)1450 1537 y Fl(T)1503 1551 y Fm(k)1548 1537 y Fp(\),)h(can)f(b)s(e)g(written)e(as)i Fl(t)e Fp(=)g Fl(T)2599 1554 y Fm(k)r Fy(\()p Fm(t)p Fy(\))2729 1537 y Fp(\()p Fl(q)s Fp(\()p Fl(t)p Fp(\)\),)i(for)h(some)f Fl(k)s Fp(\()p Fl(t)p Fp(\))c Fo(\025)i Fp(1)j(and)0 1647 y(some)c Fl(q)e Fp(=)e Fl(q)s Fp(\()p Fl(t)p Fp(\))e Fo(2)i Fp(\()t Fi(e)-55 b Fl(q)732 1661 y Fy(1)772 1647 y Fl(;)20 b Fi(e)-56 b Fl(q)t Fp(].)41 b(Set)30 b Fl(p)p Fp(\()p Fl(t)p Fp(\))24 b(=)i(1)20 b Fo(\000)g Fl(q)s Fp(\()p Fl(t)p Fp(\).)40 b(Then)30 b(for)g(an)m(y)g Fl(q)f Fo(2)c Fp(\(0)p Fl(;)19 b Fi(e)-55 b Fl(q)2619 1661 y Fy(1)2659 1647 y Fp(\))31 b(w)m(e)f(ha)m(v)m(e)h Fl(p)25 b Fp(=)h(1)20 b Fo(\000)g Fl(q)29 b Fo(\025)c Fp(1)20 b Fo(\000)25 b Fi(e)-56 b Fl(q)3760 1661 y Fy(1)3827 1647 y Fo(\025)0 1757 y Fp(1)20 b Fo(\000)g Fl(q)s Fp(\()p Fl(t)p Fp(\))25 b(=)g Fl(p)p Fp(\()p Fl(t)p Fp(\).)40 b(Therefore,)28 b(b)m(y)j(attractiv)m(eness)c(and)i(\(4.23\),)h(w)m(e)g (ha)m(v)m(e)972 1988 y FA(P)1044 2002 y Fm(p)1086 1988 y Fp(\()p Fl(\033)1174 2002 y Fm(t)1206 1988 y Fp(\(0\))24 b(=)i Fo(\000)p Fp(1\))55 b Fo(\024)h FA(P)1851 2005 y Fm(p)p Fy(\()p Fm(t)p Fy(\))1978 1988 y Fp(\()p Fl(\033)2066 2002 y Fm(t)2098 1988 y Fp(\(0\))24 b(=)i Fo(\000)p Fp(1\))55 b Fo(\024)g Fl(e)2714 1950 y Fh(\000)p Fm(C)t(t)2855 1923 y Fj(a)2900 1988 y Fl(:)0 2233 y Fp(This)32 b(establishes)e(the)i (v)-5 b(alidit)m(y)31 b(of)i(\(4.19\))e(for)i Fl(q)f(<)j Fi(e)-56 b Fl(q)1888 2247 y Fy(1)1962 2233 y Fp(and)33 b Fl(t)c(>)h Fp(0)j(whic)m(h)f(is)h(not)f(in)g Fo([)3079 2247 y Fm(k)r Fh(\025)p 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b(the)0 2591 y(Mark)m(o)m(v)31 b(prop)s(ert)m(y)-8 b(,)29 b(w)m(e)h(obtain)e(then)481 2852 y FA(P)552 2866 y Fm(p)595 2852 y Fp(\()p Fl(\033)683 2866 y Fm(t)714 2852 y Fp(\(0\))c(=)i Fo(\000)p Fp(1\))55 b Fo(\024)1360 2791 y Fp(1)p 1299 2831 168 4 v 1299 2927 a Fl(e)1342 2900 y Fh(\000)1398 2888 y Fc(e)1399 2900 y Fm(t)1427 2909 y Fe(1)1492 2852 y FA(P)1564 2866 y Fm(p)1606 2852 y Fp(\()p Fl(\033)1706 2867 y Fc(e)1694 2883 y Fm(T)1737 2893 y Fj(k)1782 2852 y Fp(\(0\))24 b(=)h Fo(\000)p Fp(1\))55 b Fo(\024)h Fl(C)2427 2815 y Fh(0)2452 2852 y Fl(e)2495 2815 y Fh(\000)p Fm(C)2608 2787 y Fd(0)q(0)2654 2815 y Fy(\()2694 2799 y Fc(e)2683 2815 y Fm(T)2726 2825 y Fj(k)2765 2815 y Fy(\))2794 2787 y Fj(a)2894 2852 y Fo(\024)g Fl(C)3093 2815 y Fh(0)3118 2852 y Fl(e)3161 2815 y Fh(\000)p Fm(C)3274 2787 y Fd(0)q(0)3320 2815 y Fm(t)3348 2787 y Fj(a)3392 2852 y Fl(;)247 b Fp(\(4)p Fl(:)p Fp(24\))0 3141 y(where)35 b(the)h(term)f Fl(e)701 3108 y Fh(\000)757 3096 y Fc(e)758 3108 y Fm(t)786 3117 y 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4704 y Fm(t)3045 4621 y Fl(e)3088 4583 y Fh(\000)p Fm(C)3197 4592 y Fe(2)3232 4583 y Fm(s)3266 4556 y Fj(a)3327 4621 y Fl(ds:)0 4923 y Fp(Observ)m(e)30 b(no)m(w)g(that)f(for)h(arbitrary)e Fl(\016)g(>)d Fp(0,)1397 5069 y Fi(Z)1488 5092 y Fh(1)1447 5277 y Fm(t)1580 5194 y Fl(e)1623 5156 y Fh(\000)p Fm(C)1732 5165 y Fe(2)1768 5156 y Fm(s)1802 5129 y Fj(a)1862 5194 y Fl(ds)55 b Fo(\024)h Fl(e)2178 5156 y Fh(\000)p Fm(C)2287 5165 y Fe(2)2323 5156 y Fm(t)2351 5129 y Fj(a)p Fd(\000)p Fj(\016)2476 5194 y Fl(;)p eop %%Page: 20 20 20 19 bop 0 46 a Fw(20)1111 b(F)n(ONTES,)33 b(SCHONMANN,)i(SIDORA)-9 b(VICIUS)0 266 y Fp(for)31 b(large)e Fl(t)p Fp(.)44 b(T)-8 b(o)31 b(see)f(this,)h(note)f(that)g Fl(e)1401 233 y Fh(\000)p Fm(C)1510 242 y Fe(2)1545 233 y Fm(s)1579 205 y Fj(a)1682 266 y Fo(\024)58 b Fl(C)1876 280 y Fy(2)1933 266 y Fp(\()p Fl(a)21 b Fo(\000)f Fl(\016)s Fp(\))p Fl(s)2252 233 y Fm(a)p Fh(\000)p Fm(\016)r Fh(\000)p Fy(1)2497 266 y Fl(e)2540 233 y Fh(\000)p Fm(C)2649 242 y Fe(2)2685 233 y 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b(A.)g(J.,)g(\(1994\))d(The)i (theory)f(of)h(phase-ordering)d(kinetics.)38 b Ff(A)-5 b(dvanc)g(es)33 b(in)g(Physics)e FA(93)p Fp(,)f(357{459.)-240 1196 y FA([CDN])53 b Fp(Camia,)24 b(F.,)h(De)f(San)m(tis,)g(E.,)i (Newman,)f(C.M.,)h(\(2001\))c(Clusters)f(and)i(recurrence)e(in)i(the)g (t)m(w)m(o-dimensional)110 1306 y(zero-temp)s(erature)i(sto)s(c)m (hastic)i(Ising)j(mo)s(del.)38 b Ff(Pr)-5 b(eprint)p Fp(.)-194 1470 y FA([CSS])54 b Fp(Cha)m(y)m(es,)46 b(L.,)g(Sc)m (honmann,)e(R.)f(H.,)k(Swindle,)d(G.,)i(\(1995\))40 b(Lifsc)m(hitz')h (la)m(w)h(for)g(the)f(v)m(olume)h(of)g(a)g(t)m(w)m(o-)110 1580 y(dimensional)26 b(droplet)i(at)i(zero)f(temp)s(erature.)37 b Ff(J.)c(of)f(Statist.)43 b(Phys.)e FA(79)p Fp(,)30 b(821{831.)-236 1744 y FA([Che1])53 b Fp(Chen,)29 b(H.-N.)i(\(1992\))c (On)j(the)e(stabilit)m(y)e(of)j(a)h(p)s(opulation)25 b(gro)m(wth)j(mo)s(del)f(with)h(sexual)g(repro)s(duction)d(on)110 1854 y Fn(Z)175 1821 y Fy(2)211 1854 y Fp(.)41 b Ff(A)n(nn.)i(Pr)-5 b(ob)g(ab.)40 b FA(20)p 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b(New-Y)-8 b(ork,)32 b(Berlin.)-224 4100 y FA([NNS])54 b Fp(Nanda,)31 b(S.,)g(Newman,)h(C.)f(M.,)h(Stein,)e (D.,)i(\(2000\))c(Dynamics)i(of)h(Ising)f(spin)g(systems)h(at)f(zero)g (temp)s(era-)110 4210 y(ture.)45 b(In)33 b Ff(On)h(Dobrushin)-9 b('s)35 b(way)f(\(F)-7 b(r)i(om)34 b(Pr)-5 b(ob)g(ability)35 b(The)-5 b(ory)34 b(to)h(Statistic)-5 b(al)36 b(Me)-5 b(chanics\))33 b Fp(R.)g(Minlos,)f(S.)110 4320 y(Shlosman)c(and)i(Y.)h (Suho)m(v,)f(eds.,)h(Am.)41 b(Math.)f(So)s(c.)f(T)-8 b(ransl.)40 b(\(2\))29 b FA(198)p Fp(,)h(183{194.)-142 4484 y FA([NS])54 b Fp(Newman,)29 b(C.)g(M.,)h(Stein,)e(D.,)h(\(2000\)) e(Zero-temp)s(erature)e(dynamics)i(of)i(Ising)e(spin)h(systems)h(follo) m(wing)d(a)110 4594 y(deep)j(quenc)m(h:)40 b(results)28 b(and)i(op)s(en)e(problems.)39 b Ff(Physic)-5 b(a)33 b(A)e FA(279)p Fp(,)e(159-168.)-162 4758 y FA([Sc)m(h])54 b Fp(Sc)m(honmann,)44 b(R.)f(H.,)48 b(\(1992\))40 b(On)j(the)e(b)s(eha) m(vior)g(of)h(some)h(cellular)c(automata)h(related)g(to)i(b)s(o)s (otstrap)110 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