Content-Type: multipart/mixed; boundary="-------------0308260542604" This is a multi-part message in MIME format. ---------------0308260542604 Content-Type: text/plain; name="03-385.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-385.keywords" Wulff construction, spectral gap, interfaces, Hausdorff stability, wetting ---------------0308260542604 Content-Type: application/postscript; name="BI.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="BI.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: BI_gap_July.dvi %%Pages: 38 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips BI_gap_July -o BI.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.08.26:1116 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S 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b(on)g(the)g(equilibrium)d(measure.)69 b(The)41 b(analysis)f(of)h([Ma])g(has)g(b)s(een)g(based)h(on)f(v)m(ery) 118 1180 y(sp)s(eci\014c)36 b(facts)g(ab)s(out)f(the)h(Hausdor\013)g (stabilit)m(y)e(of)h(the)h(2D)e(nearest)j(neigh)m(b)s(or)e(Ising)g(mo)s (del)118 1296 y(in)m(terfaces,)j(on)e(the)h(closely)f(related)g(exp)s (onen)m(tial)f(relaxation)g(prop)s(erties)h(of)g(\014nite)g(v)m(olume) 118 1413 y(pure)f(state)g(and)f(on)h(exact)g(surface)g(order)f(large)g (deviation)f(asymptotics)h(for)g(the)g(magneti-)118 1529 y(zation)k(inside)h(the)g(phase)i(co)s(existence)f(region.)62 b(In)40 b(higher)f(dimensions,)h(w)m(e)g(are)f(going)f(to)118 1645 y(use)31 b(the)f(large)e(deviation)h(estimates)g(of)g(the)h Fx(L)1866 1609 y Fw(1)1912 1645 y Fz(-theory)f(and)h(the)g(Hausdor\013) g(stabilit)m(y)e(of)h(the)118 1761 y(random)j(in)m(terface)g(on)h(the)g (macroscopic)f(scale)g(in)g(order)h(to)f(extend)i(the)f(results)g(of)f ([Ma)q(].)218 1877 y(A)37 b(more)g(complete)f(discussion)i(on)f(the)h (in)m(terpla)m(y)f(b)s(et)m(w)m(een)i(the)f(metastabilit)m(y)d(and)i (the)118 1994 y(w)m(etting)c(is)f(p)s(ostp)s(oned)h(to)f(Subsection)h (3.2.)218 2153 y(Apart)d(from)f(b)s(eing)g(dep)s(enden)m(t)j(on)e(the)h (v)-5 b(alidit)m(y)28 b(of)i(Pisztora)g(coarse)h(graining)c(\(c.f.)43 b(Sub-)118 2269 y(section)37 b(2.2\))g(our)g(pro)s(of)f(of)h(the)g(in)m (terface)g(stabilit)m(y)f(relies)g(on)h(strict)g(con)m(v)m(exit)m(y)i (of)e(surface)118 2385 y(tension.)43 b(The)33 b(analysis)e(of)h(the)g (sp)s(ectral)g(gap)f(asymptotics)h(for)f(the)h(Glaub)s(er)f(dynamics)h (re-)118 2501 y(quires)f(an)f(additional)d(assumption)i(on)h(exp)s (onen)m(tial)g(mixing)e(prop)s(erties)i(of)f(the)i(underlying)118 2618 y(FK)h(measures.)44 b(Both)32 b(assumptions)f(are)h(describ)s(ed)h (and)f(discussed)i(in)d(Subsection)i(2.5)e(and)118 2734 y(are)i(exp)s(ected)h(to)e(hold)g(for)g(a)g(wide)h(range)g(of)f (sub-critical)e(temp)s(eratures.)218 2850 y(While)40 b(completing)g(this)i(pap)s(er,)i(w)m(e)f(learn)m(t)e(ab)s(out)g(the)i (recen)m(t)g(w)m(ork)g(b)m(y)f(N.)g(Sugimine)118 2966 y([Su1,)29 b(Su2])f(on)g(upp)s(er)h(and)f(lo)m(w)m(er)g(b)s(ounds)h (for)e(sp)s(ectral)h(gap)g(for)g(the)g(three)h(dimensional)d(lo)m(w)118 3083 y(temp)s(erature)33 b(Ising)f(mo)s(del)f(with)h(mixed)g(+)p Ft(=)p Fs(;)g Fz(b)s(oundary)h(conditions.)1131 3286 y(2.)49 b Fy(Not)-7 b(a)g(tions)39 b(and)f(Assumptions)118 3461 y Fz(2.1.)49 b FE(The)j(microscopic)h(mo)s(del.)d Fz(W)-8 b(e)46 b(consider)g(the)f(nearest)i(neigh)m(b)s(or)e (ferromagnetic)118 3577 y(Ising)33 b(mo)s(del)f(in)h(dimension)f Ft(d)d Fr(>)g Fz(2.)46 b(F)-8 b(or)33 b(an)m(y)h(domain)e(\001)d Fs(\032)h Fx(Z)2526 3541 y Fq(d)2597 3577 y Fz(and)j(b)s(oundary)h (conditions)118 3693 y Ft(\021)f Fz(outside)c(\001,)h(the)f(Gibbs)f (measure)i(on)f Fs(f\006)p Fz(1)p Fs(g)1848 3657 y Fw(\001)1939 3693 y Fz(at)g(in)m(v)m(erse)h(temp)s(erature)f Ft(\014)35 b Fz(will)26 b(b)s(e)j(denoted)118 3809 y(b)m(y)34 b Ft(\026)313 3762 y Fq(\021)313 3837 y(\014)s(;)p Fw(\001)438 3809 y Fz(.)43 b(Th)m(us,)35 b(giv)m(en)d Ft(\033)g Fs(2)c(f\006)p Fz(1)p Fs(g)1444 3773 y Fw(\001)1507 3809 y Fz(,)1158 4033 y Ft(\026)1217 3986 y Fq(\021)1217 4061 y(\014)s(;)p Fw(\001)1342 4033 y Fz(\()p Ft(\033)t Fz(\))45 b(=)1724 3965 y(1)p 1652 4010 195 4 v 1652 4101 a FE(Z)1720 4054 y Fq(\021)1720 4129 y(\014)s(;)p Fw(\001)1872 4033 y Fz(exp)18 b Fs(f)e(\000)p Ft(\014)6 b FE(H)2330 3986 y Fq(\021)2330 4060 y Fw(\001)2393 4033 y Fz(\()p Ft(\033)t Fz(\))p Fs(g)16 b Ft(;)118 4269 y Fz(where)34 b(the)f(nearest)g(neigh)m (b)s(our)g(Hamiltonian)c FE(H)2009 4222 y Fq(\021)2009 4296 y Fw(\001)2104 4269 y Fz(is)j(giv)m(en)g(b)m(y)1068 4479 y FE(H)1156 4431 y Fq(\021)1156 4506 y Fw(\001)1219 4479 y Fz(\()p Ft(\033)t Fz(\))44 b(=)g Fs(\000)1605 4411 y Fz(1)p 1605 4456 49 4 v 1605 4547 a(2)1718 4384 y Fp(X)1681 4595 y Fq(i)p Fo(\030)p Fq(j)t Fo(2)p Fw(\001)1915 4479 y Ft(\033)1970 4494 y Fq(i)1998 4479 y Ft(\033)2053 4494 y Fq(j)2129 4479 y Fs(\000)2300 4384 y Fp(X)2307 4595 y Fq(i)p Fo(2)p Fw(\001)2245 4667 y Fq(j)t Fo(2)p Fn(Z)2375 4643 y Fm(d)2405 4667 y Fo(n)p Fw(\001)2516 4479 y Ft(\021)2564 4494 y Fq(j)2601 4479 y Ft(\033)2656 4494 y Fq(i)2684 4479 y Ft(:)118 4813 y Fz(There)31 b(exists)f(a)f (critical)e(v)-5 b(alue)28 b Ft(\014)1366 4828 y Fq(c)1430 4813 y Fz(suc)m(h)j(for)e(an)m(y)h Ft(\014)j(>)28 b(\014)2221 4828 y Fq(c)2285 4813 y Fz(a)h(phase)h(transition,)f(c)m(haracterized) 118 4930 y(b)m(y)c(symmetry)f(breaking,)i(o)s(ccurs.)41 b(Throughout)25 b(the)f(pap)s(er,)i(w)m(e)f(alw)m(a)m(ys)g(consider)f (an)g(in)m(v)m(erse)118 5046 y(temp)s(erature)29 b Ft(\014)34 b Fz(for)28 b(whic)m(h)h(the)f(system)i(is)e(in)g(a)g(phase)h(co)s (existence)h(regime)e(and,)h(w)m(e)h(denote)118 5162 y(b)m(y)k Ft(m)339 5126 y Fo(\003)378 5162 y Fz(\()p Ft(\014)6 b Fz(\))32 b(the)h(sp)s(on)m(taneous)h(magnetization)c(in)i (the)h(+)f(phase.)218 5364 y(It)26 b(will)e(b)s(e)j(con)m(v)m(enien)m 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627 y Fz(.)53 b(W)-8 b(e)36 b(shall)f(suppress)j(the)e(domain)e(sub-index)j(and)f(write)118 744 y(\004)d(whenev)m(er)i(\001)28 b(=)f Fx(Z)925 707 y Fq(d)963 744 y Fz(.)218 860 y(Giv)m(en)36 b Ft(\030)i Fs(2)c Fz(\004)i(and)g(a)g(b)s(ond)g Ft(b)e Fs(2)g(E)1551 824 y Fq(d)1591 860 y Fz(,)j(w)m(e)g(sa)m(y)g(that)f Ft(b)h Fz(is)e(op)s(en)i(if)e Ft(\030)5 b Fz(\()p Ft(b)p Fz(\))33 b(=)h(1.)53 b(Tw)m(o)37 b(sites)g(of)118 976 y Fx(Z)187 940 y Fq(d)257 976 y Fz(are)c(said)f(to)g(b)s(e)g(connected) i(if)e(one)g(can)h(b)s(e)g(reac)m(hed)g(from)f(another)g(via)g(a)g(c)m (hain)g(of)g(op)s(en)118 1092 y(b)s(onds.)69 b(Th)m(us,)45 b(eac)m(h)d Ft(\030)k Fs(2)d Fz(\004)e(splits)f Fx(Z)1616 1056 y Fq(d)1695 1092 y Fz(in)m(to)g(the)h(disjoin)m(t)f(union)g(of)h (maximal)d(connected)118 1209 y(comp)s(onen)m(ts,)d(whic)m(h)g(are)f (called)e(the)j(op)s(en)f(clusters)h(of)e(\004.)48 b(Giv)m(en)34 b(a)f(\014nite)h(subset)i Ft(B)f Fs(\032)c Fx(Z)3624 1172 y Fq(d)118 1325 y Fz(w)m(e)k(use)g Ft(c)475 1340 y Fq(B)535 1325 y Fz(\()p Ft(\030)5 b Fz(\))34 b(to)f(denote)i(the)f(n) m(um)m(b)s(er)g(of)g(di\013eren)m(t)g(op)s(en)g(\014nite)f(clusters)i (of)e Ft(\030)39 b Fz(whic)m(h)34 b(ha)m(v)m(e)118 1441 y(a)e(non-empt)m(y)h(in)m(tersection)g(with)f Ft(B)5 b Fz(.)218 1557 y(Belo)m(w)31 b(w)m(e)i(giv)m(e)f(a)f(general)g (de\014nition)g(of)g(FK)g(measures)i(whic)m(h)f(are)g(related)f(to)h (the)g(\014nite)118 1673 y(v)m(olume)39 b(spin)h(Gibbs)f(states)i(on)f (\001)g Fs(\032)h Fx(Z)1701 1637 y Fq(d)1739 1673 y Fz(.)65 b(W)-8 b(e)40 b(use)h(a)f(pro)m(visional)e(notation)g(in)m(tended)j(to) 118 1790 y(illustrate)35 b(the)j(connection)g(b)s(et)m(w)m(een)h(the)f (Gibbs)f(states)h(and)f(the)h(FK)f(measures.)59 b(A)37 b(more)118 1906 y(precise)g(notation)e(will)e(b)s(e)k(in)m(tro)s(duced) f(later)f(for)h(particular)e(cases)j(whic)m(h)g(sho)m(w)g(up)g(in)e (the)118 2022 y(main)c(b)s(o)s(dy)i(of)f(the)h(pap)s(er.)218 2138 y(The)44 b(set)g(of)f(b)s(onds)g(connecting)h(\001)f(to)g(\001) 1843 2102 y Fq(c)1921 2138 y Fz(will)e(b)s(e)i(denoted)i(b)m(y)f Fs(E)2838 2153 y Fq(@)t Fw(\001)2941 2138 y Fz(.)76 b(The)44 b(b)s(oundary)118 2255 y(conditions)29 b(are)g(sp)s(eci\014ed)i(b)m(y)f (a)g(frozen)g(p)s(ercolation)e(con\014guration)g Ft(\031)k Fs(2)c Fz(\004)16 b Fs(n)g Fz(\004)3114 2270 y Fw(\001)3207 2255 y Fz(and)29 b(b)m(y)i(the)118 2371 y(collection)i FE(p)e Fs(2)h Fz([0)p Ft(;)17 b Fz(1)23 b Fs(\000)h Fz(e)1081 2335 y Fo(\000)p Fw(2)p Fq(\014)1219 2371 y Fz(])1246 2335 y Fo(E)1283 2347 y Fm(@)s Fl(\001)1377 2371 y Fz(,)36 b(whic)m(h)f(describ)s(es)h(the)f("activit)m(y")f(of)g(the)h(b)s(ounds) g(on)g(the)118 2487 y(b)s(oundary)e(of)f(\001.)218 2603 y(W)-8 b(e)37 b(write)g Ft(\030)29 b Fs(_)d Ft(\031)40 b Fz(for)d(the)g(join)m(t)f(con\014guration)g(in)g(\004)h(and)g (de\014ne)h(the)f(\014nite)g(v)m(olume)f(FK)118 2720 y(measure)d(on)f(\001)h(with)g(the)g(b)s(oundary)f(conditions)g Ft(\031)37 b Fz(and)32 b FE(p)h Fz(as:)683 2981 y(\010)753 2933 y Fq(\031)r(;)p Fk(p)753 3008 y Fw(\001)882 2981 y Fz(\()p Ft(\030)5 b Fz(\))1074 2925 y Fw(\001)1065 2981 y Fz(=)1253 2913 y(1)p 1211 2958 132 4 v 1211 3049 a FE(Z)1279 3015 y Fq(\031)1279 3076 y Fw(\001)1370 2810 y Fp( )1448 2886 y(Y)1497 3098 y Fq(b)1592 2900 y Fp(\000)1638 2981 y Fz(1)22 b Fs(\000)g Ft(p)p Fz(\()p Ft(b)p Fz(\))1974 2900 y Fp(\001)2020 2922 y Fw(1)p Fo(\000)p Fq(\030)2141 2934 y Fm(b)2177 2900 y Fp(\000)2223 2981 y Ft(p)p Fz(\()p Ft(b)p Fz(\))2389 2900 y Fp(\001)2435 2922 y Fq(\030)2466 2934 y Fm(b)2501 2810 y Fp(!)2614 2981 y Fz(2)2663 2939 y Fq(c)2694 2950 y Fl(\001)2747 2939 y Fw(\()p Fq(\030)s Fo(_)p Fq(\031)r Fw(\))2929 2981 y Fz(\()p Ft(\030)5 b Fz(\))17 b Ft(;)364 b Fz(\(2.1\))118 3250 y(where,)34 b(for)e(a)g(b)s(ond)h Ft(b)28 b Fz(=)f(\()p Ft(i)17 b(;)33 b(j)6 b Fz(\))33 b(with)f(\()p Ft(i)17 b(;)33 b(j)6 b Fz(\))27 b Fs(2)i Fz(\001,)j(w)m(e)i(set)f Ft(p)p Fz(\()p Ft(b)p Fz(\))28 b(=)g Ft(p)p Fz(\()p Ft(\014)6 b(;)17 b(b)p Fz(\))27 b(=)g(1)22 b Fs(\000)h Fz(exp)q(\()p Fs(\000)p Fz(2)p Ft(\014)6 b Fz(\);)118 3366 y(otherwise)35 b(if)e Ft(b)i Fz(is)f(in)f Fs(E)989 3381 y Fq(@)t Fw(\001)1127 3366 y Fz(the)i(corresp)s(onding)f(p)s(ercolation)f(probabilit)m(y)f Ft(p)p Fz(\()p Ft(b)p Fz(\))f(=)g FE(p)p Fz(\()p Ft(b)p Fz(\).)49 b(The)118 3482 y(t)m(w)m(o)35 b(extreme)f(con\014gurations)g (suc)m(h)h(that)f FE(p)p Fz(\()p Ft(b)p Fz(\))d Fs(\021)f Fz(1)23 b Fs(\000)h Fz(exp)q(\()p Fs(\000)p Fz(2)p Ft(\014)6 b Fz(\))34 b(and)g Ft(\031)g Fs(\021)c Fz(1)k(and,)h(resp)s(ec-)118 3599 y(tiv)m(ely)-8 b(,)38 b FE(p)p Fz(\()p Ft(b)p Fz(\))e Fs(\021)g Fz(0)h(or)f Ft(\031)k Fs(\021)35 b Fz(0)i(for)g(all)e Ft(b)i Fz(in)g Fs(E)1785 3614 y Fq(@)t Fw(\001)1925 3599 y Fz(lead)g(to)f(the)i(FK)e(measures)i(with)f(wired)g(\010)3598 3563 y Fw(w)3598 3624 y(\001)118 3715 y Fz(or)44 b(free)g(\010)517 3679 y Fw(f)517 3740 y(\001)625 3715 y Fz(b)s(oundary)g(conditions)f (resp)s(ectiv)m(ely)-8 b(.)79 b(The)45 b(in)m(termediate)d(v)-5 b(alues)44 b(of)g FE(p)g Fz(ha)m(v)m(e)118 3831 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b(Next)45 b(sample)d(a)h(b)s(ond)h (con\014guration)e Ft(\030)50 b Fs(2)d Fz(\004)2996 4494 y Fw(\001)3102 4479 y Fz(from)42 b(the)i(FK)118 4595 y(measure)33 b(\010)568 4548 y Fq(w)r(;)p Fk(p)568 4623 y Fq(\014)s(;)p Fw(\001)694 4595 y Fz(,)f(and)g(pain)m(t)g(with)g(1)g (all)e(the)j(clusters)g(of)e Ft(\030)37 b Fz(connected)d(to)e(the)g (regions)g(of)g(the)118 4713 y(b)s(oundary)39 b(where)h Ft(\021)899 4728 y Fq(i)966 4713 y Ft(>)e Fz(0,)i(whereas)g(all)d(the)i (remaining)d(clusters)k(of)e Ft(\030)43 b Fz(are)c(to)f(b)s(e)h(pain)m (ted)118 4829 y(in)m(to)e Fs(\006)p Fz(1)g(with)g(probabilit)m(y)e(1)p Ft(=)p Fz(2)i(eac)m(h.)58 b(The)38 b(corresp)s(onding)f(join)m(t)g(b)s (ond-spin)f(probabilit)m(y)118 4946 y(measure)d(is)f(denoted)i(b)m(y)f Fx(P)1159 4898 y Fq(\021)1159 4973 y(\014)s(;)p Fw(\001)1287 4946 y Fz(.)1189 5147 y Fx(P)1248 5100 y Fq(\021)1248 5175 y(\014)s(;)p Fw(\001)1375 5147 y Fz(\()p Ft(\033)n(;)17 b(\030)5 b Fz(\))28 b(=)f Ft(P)1804 5100 y Fq(\021)r(;\030)1790 5174 y Fw(\001)1899 5147 y Fz(\()p Ft(\033)t Fz(\))g(\010)2131 5100 y Fq(\031)r(;)p Fk(p)2131 5175 y Fq(\014)s(;)p Fw(\001)2257 5147 y Fz(\()p Ft(\030)5 b Fz(\))17 b Ft(;)118 5348 y Fz(where)31 b Ft(P)474 5301 y Fq(\021)r(;\030)460 5375 y Fw(\001)597 5348 y Fz(denotes)g(the)f(pain)m(ting)d(measure.)43 b(The)30 b(Gibbs)f(state)h Ft(\026)2679 5301 y Fq(\021)2679 5376 y(\014)s(;)p Fw(\001)2833 5348 y Fz(is)f(then)h(reco)m(v)m(ered)h (as)118 5473 y(the)i Ft(\033)t Fz(-marginal)c(of)j Fx(P)954 5426 y Fq(\021)954 5501 y(\014)s(;)p Fw(\001)1082 5473 y Fz(.)p eop %%Page: 5 5 5 4 bop 3623 48 a Fu(5)218 366 y Fz(The)45 b(represen)m(tation)h(for)e (more)g(general)g(b)s(oundary)h(conditions)f(whic)m(h)h(corresp)s(ond)h (to)118 483 y(sign)32 b(c)m(hanging)g Ft(\021)37 b Fz(will)30 b(b)s(e)j(discussed)h(later.)118 720 y(2.2.)49 b FE(Scales)37 b(and)g(coarse)f(graining.)50 b Fz(All)30 b(scales)i(are)g(binary)-8 b(.)42 b(The)33 b(running)e(microscopic)118 836 y(scale)40 b(will)d(b)s(e)j Ft(N)51 b Fz(=)40 b(2)984 800 y Fq(n)1070 836 y Fz(and)g(the)g(asso)s(ciated)g(renormalization)c(scale)k Ft(K)47 b Fz(=)40 b(2)3156 800 y Fq(k)3198 836 y Fz(.)66 b(W)-8 b(e)40 b(shall)118 953 y(w)m(ork)g(either)f(with)g(\014xed)i (\014nite)e(scales)g Ft(K)47 b Fz(or)39 b(else)g(w)m(e)h(shall)e (explicitly)g(relate)g Ft(K)47 b Fz(to)38 b Ft(N)50 b Fz(as)118 1069 y Ft(K)35 b Fz(=)27 b Ft(N)427 1033 y Fq(a)470 1069 y Fz(,)k(where)i(the)e(\014xed)h(p)s(ositiv)m(e)f(n)m(um) m(b)s(er)g Ft(a)d Fz(=)g Ft(a)p Fz(\()p Ft(n)p Fz(\))j(\(the)h(dep)s (endence)h(on)e Ft(n)h Fz(is)e(only)h(in)118 1185 y(order)i(to)f(b)s(e) h(compatible)d(with)j(the)g(binary)f(notation\))f(satis\014es)1397 1374 y(0)d Ft(<)f(a)1628 1389 y Fw(1)1696 1374 y Fr(6)h Ft(a)g Fr(6)g Ft(a)2036 1389 y Fw(2)2103 1374 y Ft(<)g Fz(1)p Ft(=d:)1078 b Fz(\(2.2\))118 1562 y(All)37 b(our)i(computations) e(go)i(through)f(if)g(instead)g(of)h Ft(K)45 b Fz(=)38 b Ft(N)2474 1526 y Fq(a)2555 1562 y Fz(w)m(e)i(c)m(ho)s(ose)f(the)g (mesoscopic)118 1678 y(scale)33 b Ft(K)i Fz(=)27 b Ft(C)d Fz(log)16 b Ft(N)43 b Fz(for)32 b Ft(C)40 b Fz(large)31 b(enough.)218 1795 y(W)-8 b(e)34 b(in)m(tro)s(duce)f(no)m(w)h(the)g (mesoscopic)g(partitions)e(of)h Fx(D)2319 1810 y Fq(N)2421 1795 y Fz(=)d Fs(f)p Fz(1)p Ft(;)17 b(:)g(:)g(:)e(;)i(N)10 b Fs(g)2982 1758 y Fq(d)3023 1795 y Fz(.)46 b(A)m(t)34 b(eac)m(h)g(\014xed)118 1911 y(mesoscopic)28 b(scale)f Ft(K)35 b Fz(=)28 b(2)1112 1875 y Fq(k)1182 1911 y Fz(w)m(e)g(split)e (the)i(microscopic)e(domain)g Fx(D)2624 1926 y Fq(N)2725 1911 y Fz(in)m(to)h(the)h(disjoin)m(t)e(union)118 2050 y(of)k(shifts)h(of)g(the)g(mesoscopic)g(b)s(o)m(x)g Fx(B)1504 2065 y Fq(K)1614 1994 y Fw(\001)1606 2050 y Fz(=)c Fs(f\000)1846 2010 y Fw(1)p 1846 2027 36 4 v 1846 2084 a(2)1892 2050 y Ft(K)e Fz(+)19 b(1)p Ft(;)e(:)g(:)g(:)f(;)2372 2010 y Fw(1)p 2372 2027 V 2372 2084 a(2)2417 2050 y Ft(K)7 b Fs(g)2557 2014 y Fq(d)2598 2050 y Fz(.)43 b(These)32 b(shifted)f(b)s(o)m(xes)h(are)118 2204 y(cen)m(tered)24 b(at)d(the)h(lattice)f(p)s(oin)m(ts)g(from)f(the)j(rescaled)f(set)g Fx(D)2272 2219 y Fq(N)s(;K)2462 2148 y Fw(\001)2453 2204 y Fz(=)28 b Ft(K)2663 2123 y Fp(\000)2709 2204 y Fx(D)2775 2219 y Fq(N)q(=K)2964 2204 y Fs(\000)22 b Fz(\(1)p Ft(=)p Fz(2)p Ft(;)17 b(:)g(:)g(:)f(;)h Fz(1)p Ft(=)p Fz(2\))3652 2123 y Fp(\001)3696 2204 y Fz(:)1455 2424 y Fx(D)1521 2439 y Fq(N)1654 2424 y Fz(=)1858 2329 y Fp(_)1790 2541 y Fq(i)p Fo(2)p Fn(D)1903 2552 y Fm(N)s(;K)2054 2424 y Fx(B)2114 2439 y Fq(K)2189 2424 y Fz(\()p Ft(i)p Fz(\))p Ft(;)1136 b Fz(\(2.3\))118 2763 y(where)34 b Fx(B)460 2778 y Fq(K)535 2763 y Fz(\()p Ft(i)p Fz(\))680 2708 y Fw(\001)672 2763 y Fz(=)27 b Ft(i)22 b Fz(+)g Fx(B)989 2778 y Fq(K)1091 2763 y Fz(=)27 b(\()p Ft(i)c Fz(+)f Fx(B)1446 2778 y Fq(K)1521 2763 y Fz(\))g Fs(\\)g Fx(Z)1738 2727 y Fq(d)1776 2763 y Fz(.)218 2952 y(As)k(explained)f(in)g(the)i(in) m(tro)s(duction)d(a)i(k)m(ey)h(to)s(ol)d(to)h(understand)i(the)g(in)m (terface)e(b)s(eha)m(vior)h(is)118 3068 y(a)d(renormalization)c(pro)s (cedure.)42 b(In)23 b(this)g(pap)s(er)g(w)m(e)h(will)d(use)j(a)e (coarse)i(graining)d(implemen)m(ted)118 3184 y(b)m(y)47 b(Pizstora)f([Pi)o(])g(b)m(y)h(means)f(of)g(the)h(FK)e(represen)m (tation.)85 b(W)-8 b(e)46 b(recall)f(b)s(elo)m(w)h(the)g(main)118 3301 y(features)33 b(of)g(this)f(coarse)h(graining.)218 3417 y(First)h(of)h(all)e(w)m(e)j(shall)e(set)i(up)f(the)h(notion)e(of) g(go)s(o)s(d)g(b)s(o)m(x)i(on)f(the)h Ft(K)7 b Fz(-scale)35 b(whic)m(h)g(c)m(harac-)118 3533 y(terizes)e(a)f(lo)s(cal)f (equilibrium)e(in)j(a)g(pure)i(phase)118 3685 y FE(De\014nition)45 b(2.1.)g Fj(L)-5 b(et)41 b(us)f(say)g(that)h(a)f(K-b)-5 b(ox)40 b Fx(B)1990 3700 y Fq(K)2064 3685 y Fz(\()p Ft(i)p Fz(\))e Fs(\032)h Fx(D)2393 3700 y Fq(N)2466 3685 y Fj(,)i(c)-5 b(enter)g(e)g(d)40 b(in)g Ft(i)p Fj(,)i(is)e(go)-5 b(o)g(d)40 b(with)118 3802 y(r)-5 b(esp)g(e)g(ct)35 b(to)g(the)g(p)-5 b(er)g(c)g(olation)34 b(c)-5 b(on\014gur)g(ation)34 b Ft(\030)e Fs(2)c Fz(\004)35 b Fj(if)118 3918 y(\(1\))d(Ther)-5 b(e)31 b(exists)h(a)g(cr)-5 b(ossing)31 b(cluster)h Ft(C)1664 3882 y Fo(\003)1731 3918 y Fz(=)c Ft(C)1912 3882 y Fo(\003)1951 3918 y Fz(\()p Fx(B)2049 3933 y Fw(2)q Fq(K)2159 3918 y Fz(\()p Ft(i)p Fz(\)\))k Fj(c)-5 b(onne)g(cte)g(d)31 b(to)i(al)5 b(l)31 b(the)h(fac)-5 b(es)32 b(of)f(the)118 4034 y(inner)j(vertex)h(b)-5 b(oundary)35 b(of)f(the)h(2K-b)-5 b(ox)34 b Fx(B)1752 4049 y Fw(2)p Fq(K)1861 4034 y Fz(\()p Ft(i)p Fz(\))p Fj(.)118 4150 y(\(2\))c(A)n(ny)h(FK-c)-5 b(onne)g(cte)g(d)29 b(cluster)h(of)h(vertic)-5 b(es)30 b(of)g Fx(B)2030 4165 y Fw(2)q Fq(K)2140 4150 y Fz(\()p Ft(i)p Fz(\))h Fj(which)e(has)h(a)h(diameter)e(lar)-5 b(ger)31 b(than)118 4267 y Ft(K)r(=)p Fz(10)j Fj(is)h(ne)-5 b(c)g(essarily)34 b(c)-5 b(onne)g(cte)g(d)33 b(to)i Ft(C)1610 4230 y Fo(\003)1650 4267 y Fj(.)218 4419 y Fz(F)-8 b(undamen)m(tal)35 b(tec)m(hniques)j(dev)m(elop)s(ed)f(b)m(y)h(Pizstora)e(in)f([Pi])h (imply)f(that)h(there)h(exists)g Fs(B)118 4535 y Fz(a)32 b(subset)i(of)e(])p Ft(\014)691 4550 y Fq(c)726 4535 y Ft(;)17 b Fs(1)p Fz([)32 b(for)g(whic)m(h)h(the)f(follo)m(wing)e (holds:)43 b(for)32 b(an)m(y)h Ft(\014)g Fs(2)28 b(B)s Fz(,)33 b(there)g(is)f(a)g(constan)m(t)118 4651 y Ft(c)c(>)f Fz(0,)i(suc)m(h)h(that)e(for)g(all)e(scales)i Ft(K)35 b Fr(>)28 b Ft(K)1668 4666 y Fw(0)1736 4651 y Fz(large)f(enough)i (\(and,)g(in)e(particular,)h(for)f(our)i(basic)118 4768 y(scale)k Ft(K)i Fz(=)27 b Ft(N)661 4731 y Fq(a)703 4768 y Fz(\),)427 4956 y(inf)465 5016 y Fq(\031)562 4956 y Fz(\010)632 4909 y Fq(\031)r(;)p Fk(p)632 4988 y Fn(B)671 4999 y Fl(2)p Fm(K)766 4988 y Fw(\()p Fq(i)p Fw(\))866 4956 y Fz(\()p Ft(\030)37 b Fz(is)32 b(a)g(go)s(o)s(d)f (con\014guration)h(in)g Fx(B)2166 4971 y Fq(K)2241 4956 y Fz(\()p Ft(i)p Fz(\)\))44 b Fr(>)28 b Fz(1)22 b Fs(\000)h Fz(exp)q(\()p Fs(\000)p Ft(cK)7 b Fz(\))17 b Ft(;)275 b Fz(\(2.4\))118 5176 y(uniformly)21 b(in)i(the)h(b)s(oundary)g (conditions)e Ft(\031)t Fz(,)k(b)s(oundary)e(b)s(ond)f(activities)f FE(p)i Fz(and)f(in)g Ft(i)28 b Fs(2)g Fx(D)3481 5191 y Fq(N)s(;K)3634 5176 y Fz(.)218 5364 y(The)37 b(imp)s(ortan)m(t)d(p)s (oin)m(t)i(is)g(that)g(the)h(set)g Fs(B)i Fz(is)d(de\014ned)i(in)e(a)g (non)g(p)s(erturbativ)m(e)h(w)m(a)m(y)g(\(see)118 5480 y([Pi]\).)43 b(In)33 b(particular,)e(it)h(is)g(conjectured)i(to)e (coincide)g(with)g(])p Ft(\014)2489 5495 y Fq(c)2524 5480 y Ft(;)17 b Fs(1)p Fz([.)p eop %%Page: 6 6 6 5 bop 118 51 a Fu(6)1186 b(T.)26 b(BODINEA)n(U)e(AND)g(D.)i(IOFFE)118 366 y Fz(2.3.)49 b FE(Equilibrium)40 b(setting.)49 b Fz(In)35 b(equilibrium,)c(our)j(result)h(on)f(the)h(lo)s(calization)30 b(concerns)118 483 y(primarily)24 b(in)m(terfaces)j(imp)s(osed)f(b)m(y) i(mixed)f(b)s(oundary)g(conditions.)41 b(W)-8 b(e)27 b(also)f(in)m(v)m(estigate)h(the)118 599 y(consequences)i(of)c(the)h (lo)s(calization)c(on)j(the)h(structure)h(of)e(the)h(mesoscopic)f (droplet)g(when)i(t)m(w)m(o)118 715 y(phases)34 b(co)s(exist.)44 b(W)-8 b(e)33 b(de\014ne)h(b)s(elo)m(w)e(the)h(t)m(w)m(o)g(corresp)s (onding)g(framew)m(orks.)118 892 y(2.3.1.)49 b Fj(Pur)-5 b(e)36 b(b)-5 b(oundary)36 b(c)-5 b(onditions.)48 b Fz(The)35 b(Gibbs)f(measure)g(on)g(the)h(set)g Fx(D)2956 907 y Fq(N)3059 892 y Fz(=)c Fs(f)p Fz(1)p Ft(;)17 b(:)g(:)g(:)e(;)i(N)10 b Fs(g)3621 856 y Fq(d)118 1009 y Fz(with)30 b(+)f(b)s(oundary)i (conditions)e(will)e(b)s(e)j(denoted)h(b)m(y)g Ft(\026)2213 967 y Fw(+)2213 1036 y Fq(N)2310 1009 y Fz(and)f(the)g(corresp)s (onding)g(FK)f(mea-)118 1125 y(sure)34 b(b)m(y)f(\010)530 1089 y Fw(w)530 1150 y Fq(N)598 1125 y Fz(.)218 1241 y(An)28 b(imp)s(ortan)m(t)e(quan)m(tit)m(y)i(to)f(study)i(phase)g(co)s (existence)g(is)f(the)g(surface)g(tension)g(whic)m(h)g(w)m(e)118 1357 y(no)m(w)33 b(in)m(tro)s(duce.)44 b(Let)30 b Ft(~)-46 b(n)28 b Fs(2)g Fx(S)1209 1321 y Fq(d)p Fo(\000)p Fw(1)1366 1357 y Fz(b)s(e)33 b(a)g(unit)f(normal)e(and)j(assume)g(for)f(the)h (de\014niteness)i(that)118 1474 y(\()m Ft(~)-46 b(n;)13 b(~)-45 b(e)303 1489 y Fq(d)344 1474 y Fz(\))27 b Ft(>)h Fz(1)p Ft(=)611 1389 y Fs(p)p 693 1389 51 4 v 693 1474 a Ft(d)p Fz(.)41 b(Let)25 b Ft(Z)1053 1432 y Fw(+)1046 1501 y Fq(N)1138 1474 y Fz(and)g Ft(Z)1394 1432 y Fo(\006)1387 1501 y Fq(N)1454 1474 y Fz(\()m Ft(~)-46 b(n)q Fz(\))24 b(b)s(e)i(the)f(partition)e(functions)i(on)g Fs(f\000)p Ft(N)5 b(;)17 b(:)g(:)g(:)f(;)h(N)10 b Fs(g)3406 1437 y Fq(d)3472 1474 y Fz(with)118 1590 y(resp)s(ectiv)m(ely)37 b(\\+")e(and)g(mixed)g(b)s(oundary)h(conditions,)g(the)f(latter)g(b)s (eing)g(de\014ned)i(b)m(y)f Ft(\033)3524 1605 y Fq(i)3586 1590 y Fz(=)118 1706 y(sign\(\()m Ft(~)-46 b(n;)17 b(i)p Fz(\)\),)35 b(with)f(sign\(0\))d(=)g(1.)50 b(The)35 b(bulk)g(surface)h (tension)e(in)g(the)i(direction)d(orthogonal)118 1822 y(to)c Ft(~)-46 b(n)33 b Fz(is)1143 1981 y Ft(\034)11 b Fz(\()m Ft(~)-46 b(n)q Fz(\))1400 1925 y Fw(\001)1391 1981 y Fz(=)94 b(lim)1527 2043 y Fq(N)7 b Fo(!1)1748 1981 y Fs(\000)1875 1913 y Fz(\()m Ft(~)-46 b(n)q(;)13 b(~)-45 b(e)2061 1928 y Fq(d)2101 1913 y Fz(\))p 1835 1958 344 4 v 1835 2049 a(\(2)p Ft(N)10 b Fz(\))2048 2020 y Fq(d)p Fo(\000)p Fw(1)2206 1981 y Fz(log)2358 1913 y Ft(Z)2432 1872 y Fo(\006)2425 1940 y Fq(N)2492 1913 y Fz(\()m Ft(~)-46 b(n)p Fz(\))p 2358 1958 269 4 v 2425 2051 a Ft(Z)2499 2010 y Fw(+)2492 2078 y Fq(N)3461 1981 y Fz(\(2.5\))218 2177 y(The)33 b(equilibrium)d(crystal)i(shap)s(e)i(of) e(v)m(olume)g Ft(a)c(>)f Fz(0)32 b(is)h(the)g(W)-8 b(ul\013)31 b(shap)s(e)j Fs(K)3118 2192 y Fq(a)3192 2177 y Fz(de\014ned)g(b)m(y)531 2415 y Fs(K)607 2430 y Fq(a)676 2415 y Fz(=)780 2275 y Fp(\022)904 2348 y Ft(a)p 863 2392 133 4 v 863 2484 a Fs(jK)q(j)1006 2275 y Fp(\023)1079 2297 y Fw(1)p Fq(=d)1207 2415 y Fs(K)q Ft(;)114 b Fz(where)99 b Fs(K)29 b Fz(=)1981 2321 y Fp(\\)2013 2534 y Fq(~)-33 b(n)2108 2335 y Fp(\010)2166 2415 y Ft(x)28 b Fs(2)g Fx(R)2409 2374 y Fq(d)2456 2415 y Ft(;)82 b Fz(\()p Ft(x)22 b Fs(\001)d Ft(~)-46 b(n)p Fz(\))28 b Fr(6)g Ft(\034)11 b Fz(\()m Ft(~)-46 b(n)q Fz(\))3147 2335 y Fp(\011)3221 2415 y Ft(:)213 b Fz(\(2.6\))218 2900 y(F)-8 b(or)34 b(our)h(purp)s(ose,)h(it)e(will)f(b)s(e)i(more)f (con)m(v)m(enien)m(t)j(to)e(recall)e(the)j(phase)g(co)s(existence)g (The-)118 3016 y(orem)c(directly)g(in)g(the)i(FK)e(coarse)h(grained)f (setting.)44 b(F)-8 b(or)32 b(an)m(y)i(con\014guration)d Ft(\030)38 b Fz(in)32 b(\004)3381 3031 y Fn(D)3423 3042 y Fm(N)3490 3016 y Fz(,)h(w)m(e)118 3132 y(partition)e(the)i(set)g Fx(D)911 3147 y Fq(N)s(;K)1096 3132 y Fz(in)m(to)f(3)h(sets)g(\(not)g (necessarily)g(connected\))1158 3315 y Fx(D)1224 3330 y Fq(N)s(;K)1405 3315 y Fz(=)28 b Ft(C)1586 3274 y Fw(+)1579 3342 y Fq(N)1646 3315 y Fz(\()p Ft(\030)5 b Fz(\))1787 3221 y Fp(_)1913 3315 y Ft(C)1990 3274 y Fw(0)1983 3340 y Fq(N)2050 3315 y Fz(\()p Ft(\030)g Fz(\))2191 3221 y Fp(_)2317 3315 y Ft(C)2394 3274 y Fo(\000)2387 3342 y Fq(N)2454 3315 y Fz(\()p Ft(\030)g Fz(\))17 b Ft(;)839 b Fz(\(2.7\))118 3502 y(where)30 b Ft(C)473 3461 y Fw(+)466 3529 y Fq(N)533 3502 y Fz(\()p Ft(\030)5 b Fz(\))28 b(denotes)i(the)f (set)g(of)f(go)s(o)s(d)g(b)s(o)m(xes)i 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4237 y(\014)1818 4262 y Ft(C)1895 4221 y Fo(\000)1888 4289 y Fq(N)1955 4262 y Fz(\()p Ft(\030)g Fz(\))2079 4178 y Fp(\014)2079 4237 y(\014)2139 4262 y Fr(>)29 b Ft(a)2307 4195 y(N)2395 4159 y Fq(d)p 2306 4239 131 4 v 2306 4331 a Ft(K)2396 4302 y Fq(d)2446 4122 y Fp(\033)2554 4262 y Ft(:)118 4456 y Fz(This)35 b(is)f(the)h(set)h(of)e(con\014gurations)g (for)g(whic)m(h)h(there)h(is)e(a)g(densit)m(y)i(at)e(least)g Ft(a)e(>)f Fz(0)j(of)g(go)s(o)s(d)118 4572 y(blo)s(c)m(ks)f(detac)m (hed)h(from)e(the)h(b)s(oundary)f(\(i.e.)43 b(of)33 b(the)g Fs(\000)g Fz(phase)g(in)f(the)h(spin)f(language\).)218 4689 y(The)26 b Fx(L)472 4653 y Fw(1)517 4689 y Fz(-approac)m(h)g (\(see)h([CePi,)f(BIV1]\))g(implies)d(that)j(for)f(an)m(y)i Ft(\014)33 b Fs(2)28 b(B)h Fz(there)e(is)e(a)h(sequence)118 4805 y(\()p Ft(\016)199 4820 y Fq(N)266 4805 y Fz(\))33 b(v)-5 b(anishing)31 b(to)i(0)f(suc)m(h)i(that)293 5085 y(lim)258 5147 y Fq(N)7 b Fo(!1)479 5085 y Fz(\010)549 5043 y Fw(w)549 5109 y Fq(N)633 4884 y Fp(0)633 5064 y(@)789 4990 y([)720 5202 y Fq(i)p Fo(2)p Fn(D)834 5213 y Fm(N)r(;K)984 4944 y Fp(\032)1059 5085 y Ft(\030)e Fz(;)1247 5000 y Fp(\014)1247 5060 y(\014)1297 5004 y(\000)1343 5085 y Fx(Z)1412 5043 y Fq(d)1472 5085 y Fs(\\)22 b Fz(\()p Ft(i)h Fz(+)f Fs(K)1828 5104 y Fq(a)p Fw(\()p Fq(N)q(=K)5 b Fw(\))2076 5085 y Fm(d)k Fz(\))2155 5004 y Fp(\001)2217 5085 y Fz(\001)p Ft(C)2375 5043 y Fo(\000)2368 5112 y Fq(N)2436 5000 y Fp(\014)2436 5060 y(\014)2497 5085 y Ft(<)27 b(\016)2643 5100 y Fq(N)2721 5017 y Ft(N)2809 4981 y Fq(d)p 2720 5062 V 2720 5153 a Ft(K)2810 5124 y Fq(d)2861 4944 y Fp(\033)2952 4970 y(\014)2952 5030 y(\014)2952 5089 y(\014)3018 5085 y Fx(V)3084 5100 y Fq(N)s(;a)3210 4884 y Fp(1)3210 5064 y(A)3325 5085 y Fz(=)h(1)17 b Ft(;)118 5364 y Fz(where)35 b(\001)g(denotes)g(the)f (symmetric)f(di\013erence)i(of)f(the)g(sets)h(in)e Fx(D)2609 5379 y Fq(N)s(;K)2796 5364 y Fz(and)h Fs(j)23 b(\001)g(j)34 b Fz(the)g(cardinal)118 5480 y(of)e(a)h(set.)p eop %%Page: 7 7 7 6 bop 3623 48 a Fu(7)118 366 y Fz(2.3.2.)49 b Fj(Mixe)-5 b(d)26 b(b)-5 b(oundary)26 b(c)-5 b(onditions.)47 b Fz(Let)24 b Fx(L)1819 330 y Fq(d)1889 366 y Fz(b)s(e)f(the)h(lattice)e (half-space)g Fs(f)p Ft(i)28 b Fs(2)g Fx(Z)3177 330 y Fq(d)3259 366 y Fz(:)45 b Ft(i)3364 381 y Fq(d)3432 366 y Ft(>)28 b Fz(0)p Fs(g)p Fz(.)118 483 y(The)34 b(exteriour)g(b)s (oundary)g(of)f Fx(D)1352 498 y Fq(N)1455 483 y Fz(=)c Fs(f)p Fz(1)p Ft(;)17 b(:)g(:)g(:)e(N)10 b Fs(g)1971 447 y Fq(d)2045 483 y Fz(in)33 b Fx(L)2220 447 y Fq(d)2300 483 y Fz(will)e(b)s(e)j(denoted)g(b)m(y)g Ft(@)3180 447 y Fw(ext)3281 483 y Fx(D)3347 498 y Fq(N)3421 483 y Fz(.)45 b(The)118 599 y(b)s(ottom)29 b(face)h(of)g Fx(D)830 614 y Fq(N)934 599 y Fz(is)f(denoted)i(b)m(y)h Ft(@)1585 563 y Fw(in)n(t)1580 625 y(b)1674 599 y Fx(D)1740 614 y Fq(N)1841 599 y Fz(=)27 b Fs(f)p Fz(1)p Ft(;)17 b(:)g(:)g(:)f(;)h(N) 10 b Fs(g)2400 563 y Fq(d)p Fo(\000)p Fw(1)2548 599 y Fs(\002)18 b(f)p Fz(1)p Fs(g)p Fz(.)42 b(W)-8 b(e)30 b(consider)h(mixed)118 715 y(b)s(oundary)f(conditions)e(equal)i(to)f(1) g(in)g Ft(@)1633 679 y Fw(ext)1734 715 y Fx(D)1800 730 y Fq(N)1902 715 y Fz(and)h(to)f Fs(\000)p Fz(1)g(outside)h Fx(L)2754 679 y Fq(d)2801 715 y Fz(.)42 b(The)30 b(corresp)s(onding)118 832 y(Ising)i(measure)h(will)e(b)s(e)h(denoted)i Ft(\026)1479 790 y Fo(\006)1479 859 y Fq(N)1578 832 y Fz(accordingly)-8 b(.)218 1064 y(The)48 b(FK)f(represen)m(tation)h(of)f(the)h(mixed)e(b)s (oundary)i(conditions)f(requires)h(some)f(care.)118 1180 y(Consider)29 b(the)f(graph)956 1100 y Fp(\000)1002 1180 y Fx(L)1062 1144 y Fq(d)1108 1180 y Ft(;)17 b Fs(L)1221 1144 y Fq(d)1261 1100 y Fp(\001)1307 1180 y Fz(,)29 b(where)g(the)g (edge)f(set)h Fs(L)2238 1144 y Fq(d)2306 1180 y Fz(consists)g(of)f(all) e(\(unorien)m(ted\))i(pairs)118 1306 y(of)40 b(nearest)h(neigh)m(b)s (our)f(v)m(ertices)i(\()p Ft(i;)17 b(j)6 b Fz(\))41 b Fs(\032)g Fx(L)1824 1270 y Fq(d)1911 1306 y Fz(.)67 b(Let)40 b Fs(L)2256 1270 y Fq(d)2256 1331 y(N)2363 1306 y Fz(denote)i(the)e 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Ft(h)390 3629 y Fq(i)456 3614 y Fz(applied)f(on)i(the)f Ft(i)1148 3578 y Fw(th)1219 3614 y Fz(-face)h(of)f(the)g(cub)s(e)i Fx(D)2044 3629 y Fq(N)2155 3614 y Fz(is)e(denoted)h(b)m(y)h Ft(\026)2831 3578 y Fk(h)2831 3640 y Fq(N)2898 3614 y Fz(.)58 b(In)37 b(this)g(w)m(a)m(y)i(free)118 3731 y(b)s(oundary)f (conditions)e(corresp)s(ond)i(to)f FE(h)f Fz(=)g(\(0)p Ft(;)17 b(:)g(:)g(:)e(;)i Fz(0\),)38 b(whereas)h(pure)f(+)f(b)s (oundary)h(con-)118 3847 y(ditions)d(corresp)s(ond)i(to)f FE(h)f Fz(=)f(\(1)p Ft(;)17 b(:)g(:)g(:)e(;)i Fz(1\).)55 b(As)37 b(w)m(e)g(shall)e(explain)g(b)s(elo)m(w)i(our)f(results)h(on)f (the)118 3963 y(relaxation)30 b(sp)s(eed)i(are)f(non-trivial)d(only)j (when)h(the)g(b)s(oundary)f(magnetic)f(\014elds)h Ft(h)3271 3978 y Fw(1)3311 3963 y Ft(;)17 b(:)g(:)g(:)f(;)h(h)3586 3978 y Fw(2)p Fq(d)118 4079 y Fz(are)37 b(in)g(the)h(partial)d(w)m (etting)i(regime,)h(whic)m(h)g(is)f(the)g(case)i(for)d(example)h(for)g (free)h(b)s(oundary)118 4196 y(conditions,)32 b(but)h(not)f(for)g(the)h (pure)g(+)g(ones.)218 4428 y(In)43 b(a)h(metastable)e(regime,)j(the)f (rescaled)g(ev)m(olution)e(of)h(the)h(system)g(can)g(b)s(e)g(describ)s (ed)118 4544 y(b)m(y)37 b(an)e(energy)i(landscap)s(e)e(whic)m(h)h(is)f (related)h(to)f(equilibrium)d(thermo)s(dynamic)i(quan)m(tities.)118 4661 y(Therefore)46 b(w)m(e)g(\014rst)f(pro)s(ceed)g(in)f(recalling)e (the)j(basic)g(framew)m(ork)g(of)f(equilibrium)d(phase)118 4777 y(co)s(existence)47 b(\(w)m(e)f(refer)f(to)g([BIV1])h(for)e(a)h (detailed)f(review\).)82 b(A)46 b(heuristic)e(discussion)i(of)118 4893 y(the)33 b(in)m(terpla)m(y)f(b)s(et)m(w)m(een)j(the)e(equilibrium) c(prop)s(erties)k(and)f(the)h(dynamics)g(is)f(p)s(ostp)s(oned)h(to)118 5009 y(subsection)42 b(3.2.)69 b(The)43 b(basic)e(thermo)s(dynamic)f (quan)m(tities)h(in)f(this)h(con)m(text)i(are)e(the)g(bulk)118 5125 y(surface)34 b(tension)e(\(2.5\))g(and)h(the)g(b)s(oundary)g(free) g(energy)-8 b(.)218 5242 y(The)40 b(in\015uence)g(of)f(a)g(magnetic)g (\014eld)g Ft(h)g Fs(2)h Fx(R)50 b Fz(applied)39 b(along)f(the)h(b)s (oundary)h(leads)g(to)f(a)118 5363 y(sp)s(eci\014c)i(surface)f(energy) -8 b(.)66 b(Consider)40 b(the)h(partition)c(functions)j Ft(Z)2668 5315 y Fo(\000)p Fq(;h)2661 5390 y(N)2827 5363 y Fz(and)g Ft(Z)3098 5315 y Fw(+)p Fq(;h)3091 5390 y(N)3257 5363 y Fz(with)f(\\)p Fs(\000)p Fz(")118 5480 y(and)j(\\+")g(b)s (oundary)h(conditions)e(on)h Fx(L)1663 5444 y Fq(d)1738 5480 y Fs(n)29 b Fx(D)1883 5495 y Fq(N)1998 5480 y Fz(and)43 b Ft(h)f Fz(outside)g Fx(L)2702 5444 y Fq(d)2749 5480 y Fz(.)72 b(The)43 b(b)s(oundary)g(free)p eop %%Page: 8 8 8 7 bop 118 51 a Fu(8)1186 b(T.)26 b(BODINEA)n(U)e(AND)g(D.)i(IOFFE)118 366 y Fz(energy)42 b(\001)519 381 y Fq(h)606 366 y Fz(is)e(de\014ned)j (as)e(the)h(di\013erence)g(b)s(et)m(w)m(een)h(the)f(in)m(terfacial)d (free)i(energies)h(of)f(the)118 483 y(co)s(existing)32 b(phases:)1299 728 y(\001)1380 743 y Fq(h)1494 672 y Fw(\001)1485 728 y Fz(=)95 b(lim)1621 790 y Fq(N)7 b Fo(!1)1937 661 y Fz(1)p 1852 705 220 4 v 1852 796 a Ft(N)1940 768 y Fq(d)p Fo(\000)p Fw(1)2098 728 y Fz(log)2250 661 y Ft(Z)2324 614 y Fw(+)p Fq(;h)2317 688 y(N)p 2250 705 194 4 v 2250 811 a Ft(Z)2324 763 y Fo(\000)p Fq(;h)2317 838 y(N)2454 728 y Ft(:)980 b Fz(\(2.9\))118 998 y(W)-8 b(e)31 b(refer)f(the)h(reader)f(to)g([FP1])g(and)g([FP2])g(for)g(a)g (detailed)f(study)i(of)f(the)g(b)s(oundary)h(surface)118 1114 y(tension)i(as)f(w)m(ell)g(as)h(related)f(phenomena.)218 1230 y(On)d(the)h(macroscopic)e(lev)m(el)h(the)h(equilibrium)c(phase)k (co)s(existence)g(is)f(go)m(v)m(erned)i(b)m(y)f(a)f(v)-5 b(ari-)118 1347 y(ational)35 b(principle)g(in)m(v)m(olving)g(the)j (bulk)e(surface)i(tension)f(and)g(the)g(w)m(all)e(free)j(energies)f (\001)3563 1362 y Fq(h)3604 1372 y Fm(i)3634 1347 y Fz(.)118 1463 y(As)42 b(it)e(has)h(b)s(een)h(realized)e(in)g([ABCP)r(])h(the)g (appropriate)f(macroscopic)h(setting)f(is)h(that)g(of)118 1579 y(the)g(functions)f(of)g(b)s(ounded)h(v)-5 b(ariations)39 b(and)h(w)m(e)h(shall)e(rep)s(eatedly)i(refer)g(to)f([EG])g(for)g(the) 118 1695 y(necessary)35 b(bac)m(kground.)218 1821 y(The)42 b(microscopic)e(system)i(is)f(em)m(b)s(edded)i(in)d(the)i(con)m(tin)m (uous)g(domain)3031 1795 y Fp(b)3023 1821 y Fx(D)67 b Fz(=)43 b([0)p Ft(;)17 b Fz(1])3452 1785 y Fq(d)3492 1821 y Fz(.)69 b(A)118 1952 y(macroscopic)26 b(distribution)f(of)i (phases)h(in)1712 1926 y Fp(b)1704 1952 y Fx(D)52 b Fz(is)26 b(represen)m(ted)j(b)m(y)f(a)f(signed)g(indicator)e(function)118 2068 y Ft(u)31 b Fz(taking)g(v)-5 b(alues)32 b Fs(f\006)p Fz(1)p Fs(g)f Fz(according)g(to)h(the)g(lo)s(cal)d(equilibrium.)40 b(Let)32 b Fs(O)j Fz(b)s(e)d(an)f(op)s(en)h(smo)s(oth)118 2199 y(neigh)m(b)s(orho)s(o)s(d)47 b(of)881 2173 y Fp(b)873 2199 y Fx(D)25 b Fz(.)90 b(F)-8 b(or)47 b Fj(non)i(ne)-5 b(gative)47 b Fz(b)s(oundary)i(\014elds,)j(it)47 b(is)h(enough)g(to)g (consider)118 2315 y(functions)40 b Ft(u)g Fz(taking)f(v)-5 b(alues)40 b(in)f(the)i(set)g(of)e(b)s(ounded)i(v)-5 b(ariation)38 b(functions)i(BV)18 b(\()p Fs(O)s Ft(;)f Fs(f\006)p Fz(1)p Fs(g)p Fz(\))118 2446 y(and)33 b(equal)f(to)g(1)h (outside)1111 2420 y Fp(b)1102 2446 y Fx(D)25 b Fz(.)218 2571 y(Let)30 b Fs(P)459 2586 y Fq(i)518 2571 y Fz(b)s(e)h(the)g Ft(i)848 2535 y Fw(th)919 2571 y Fz(-face)f(of)g(the)h(cub)s(e)1658 2545 y Fp(b)1650 2571 y Fx(D)25 b Fz(.)43 b(The)31 b(b)s(oundary)g(of) 2544 2545 y Fp(b)2536 2571 y Fx(D)55 b Fz(is)30 b(denoted)h(b)m(y)h Fs(P)k Fz(=)27 b Fs([)3508 2586 y Fq(i)3537 2571 y Fs(P)3606 2586 y Fq(i)3634 2571 y Fz(.)118 2687 y(The)34 b(in)m(terfacial)c (energy)j(asso)s(ciated)g(to)f Ft(u)g Fz(is)g(de\014ned)i(b)m(y)590 2993 y Fs(W)688 3008 y Fk(h)737 2993 y Fz(\()p Ft(u)p Fz(\))27 b(=)1000 2858 y Fp(Z)1055 3083 y Fq(@)1096 3064 y Fh(\003)1132 3083 y Fq(u)p Fo(nP)1284 2993 y Ft(\034)11 b Fz(\()m Ft(~)-46 b(n)1433 3008 y Fq(x)1478 2993 y Fz(\))p Ft(d)p Fs(H)1652 2952 y Fw(\()p Fq(d)p Fo(\000)p Fw(1\))1651 3018 y Fq(x)1859 2993 y Fz(+)1993 2869 y Fw(2)p Fq(d)1957 2899 y Fp(X)1972 3109 y Fq(i)p Fw(=1)2118 2993 y Fz(\001)2199 3008 y Fq(h)2240 3018 y Fm(i)2314 2858 y Fp(Z)2370 3083 y Fq(@)2411 3064 y Fh(\003)2447 3083 y Fq(u)p Fo(\\P)2584 3093 y Fm(i)2659 2993 y Ft(d)p Fs(H)2795 2952 y Fw(\()p Fq(d)p Fo(\000)p Fw(1\))2794 3018 y Fq(x)2997 2993 y Ft(;)388 b Fz(\(2.10\))118 3299 y(where)43 b Ft(@)465 3263 y Fo(\003)505 3299 y Ft(u)e Fz(is)g(the)g(reduced)i(b)s(oundary)f ([EG])f(of)g Fs(f)p Ft(u)h Fz(=)g Fs(\000)p Fz(1)p Fs(g)p Fz(.)70 b(In)41 b(the)h(particular)e(case)i(of)118 3416 y(b)s(oundary)f(magnetic)e(\014eld)i(acting)e(only)h(on)h(one)f(of)g (the)h(faces)h(of)e Fx(D)2787 3431 y Fq(N)2900 3416 y Fz(the)h(probabilit)m(y)e(of)118 3532 y(observing)30 b(spin)g(con\014gurations)g(whic)m(h)h(are)f(close)g(\(in)f(the)i Fx(L)2426 3496 y Fw(1)2502 3532 y Fz(sense\))g(to)f(some)g(macroscopic) 118 3648 y(con\014guration)35 b Ft(u)h Fz(w)m(as)h(pro)m(v)m(en)g(to)f (deca)m(y)h(exp)s(onen)m(tially)e(fast)h(with)g(the)g(order)g Ft(N)3251 3612 y Fq(d)p Fo(\000)p Fw(1)3382 3648 y Fs(W)3480 3663 y Fk(h)3530 3648 y Fz(\()p Ft(u)p Fz(\))118 3764 y(\(see)e([BIV2]\).)218 3997 y(Finally)-8 b(,)30 b(the)j(optimal)c(in)m (terfacial)i(energy)i(under)h(a)e(v)m(olume)g(constrain)m(t)g(is)g (de\014ned)i(as)770 4267 y Fs(F)852 4225 y Fk(h)901 4267 y Fz(\()p Ft(m)p Fz(\))27 b(=)64 b(inf)1193 4329 y Fq(u)p Fo(2)p Fw(BV)1400 4126 y Fp(\032)1475 4267 y Fs(W)1573 4282 y Fk(h)1623 4267 y Fz(\()p Ft(u)p Fz(\))1852 4182 y Fp(\014)1852 4242 y(\014)1999 4131 y(Z)2059 4339 y Fg(b)2054 4357 y Fn(D)2123 4267 y Ft(u)p Fz(\()p Ft(x)p Fz(\))17 b Ft(dx)27 b Fz(=)2594 4199 y Ft(m)p 2574 4244 125 4 v 2574 4335 a(m)2659 4306 y Fo(\003)2709 4126 y Fp(\033)2817 4267 y Ft(:)568 b Fz(\(2.11\))118 4611 y(2.5.)49 b FE(The)38 b(assumptions.)51 b Fz(There)34 b(are)g(t)m(w)m(o)g(main)d (assumptions.)45 b(The)34 b(\014rst)g(one)g(is)e(of)h(geo-)118 4727 y(metric)k(nature)h(and)f(will)f(pla)m(y)h(a)g(crucial)g(role)f (in)h(the)h(lo)s(calization)c(of)j(the)h(in)m(terface.)58 b(The)118 4843 y(second)36 b(assumption)f(is)f(a)h(mixing)d(prop)s(ert) m(y)k(for)e(the)i(FK)e(measure)h(and)g(will)e(b)s(e)i(only)f(used)118 4960 y(in)e(the)h(estimation)d(of)i(the)h(sp)s(ectral)g(gap.)118 5248 y(2.5.1.)49 b Fj(Strict)40 b(c)-5 b(onvexity)40 b(of)g(the)g(surfac)-5 b(e)40 b(tension.)48 b Fz(Recall)36 b(that)i(a)g Ft(d)p Fz(-dimensional)d(simplex)118 5364 y(is)44 b(the)h(con)m(v)m(ex)h(en)m(v)m(elop)f Ft(S)54 b Fz(=)47 b Ft(S)6 b Fz(\()p Ft(u)1497 5328 y Fw(1)1536 5364 y Ft(;)17 b(:)g(:)g(:)f(;)h(u)1811 5328 y Fq(d)p Fw(+1)1941 5364 y Fz(\))44 b(of)f(\()p Ft(d)30 b Fz(+)g(1\))44 b(p)s(oin)m(ts)g Ft(u)2862 5328 y Fw(1)2901 5364 y Ft(;)17 b(:)g(:)g(:)f(u)3132 5328 y Fq(d)p Fw(+1)3309 5364 y Fs(2)48 b Fx(R)3489 5328 y Fq(d)3580 5364 y Fz(in)118 5480 y(general)37 b(p)s(osition.)55 b(The)38 b(latter)f(means)g(that)g Ft(S)43 b Fz(has)37 b(a)g(non-empt)m(y)h(in)m(teriour.)56 b(Giv)m(en)37 b(suc)m(h)p eop %%Page: 9 9 9 8 bop 3623 48 a Fu(9)118 366 y Fz(a)37 b Ft(d)p Fz(-dimensional)d (simplex)i Ft(S)43 b Fz(let)36 b Ft(F)1500 381 y Fw(1)1540 366 y Ft(;)17 b(:)g(:)g(:)f(;)h(F)1822 381 y Fq(d)p Fw(+1)1990 366 y Fz(b)s(e)37 b(its)g(faces)g(and)e Ft(~)-46 b(n)2761 381 y Fw(1)2800 366 y Ft(;)17 b(:)g(:)g(:)f(;)e(~)-46 b(n)3077 381 y Fq(d)p Fw(+1)3245 366 y Fz(the)38 b(corre-)118 483 y(sp)s(onding)32 b(outer)h(normals.)42 b(By)33 b(the)g(Gauss-Green) g(theorem)f([EG],)1575 629 y Fq(d)p Fw(+1)1567 659 y Fp(X)1574 871 y Fq(k)r Fw(=1)1711 754 y Fs(j)p Ft(F)1802 769 y Fq(k)1844 754 y Fs(j)m Ft(~)-46 b(n)1930 769 y Fq(k)2017 754 y Fz(=)44 b(0)p Ft(:)1199 b Fz(\(2.12\))118 1029 y(Giv)m(en)24 b(an)g(axis)g(direction)18 b Ft(~)-44 b(e)1152 1044 y Fq(d)1216 1029 y Fz(let)23 b(us)i(sa)m(y)g(that)f(a)f (simplex)g Ft(S)34 b Fz(=)27 b Ft(S)6 b Fz(\()p Ft(u)2604 993 y Fw(1)2643 1029 y Ft(;)17 b(:)g(:)g(:)f(;)h(u)2918 993 y Fq(d)p Fw(+1)3048 1029 y Fz(\))24 b(is)19 b Ft(~)-45 b(e)3244 1044 y Fq(d)3285 1029 y Fz(-orien)m(ted)118 1145 y(if:)218 1290 y(\(i\))64 b Ft(u)442 1254 y Fw(1)481 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Fq(d)3285 1667 y Fz(-orien)m(ted)118 1783 y(simplex)32 b Ft(S)6 b Fz(,)32 b(\(2.12\))g(yields)g(a)g (represen)m(tation)i(of)28 b Ft(~)-45 b(e)2030 1798 y Fq(d)2103 1783 y Fz(as)33 b(a)f(non-trivial)e(linear)h(com)m(bination) 1469 2054 y Fs(j)p Ft(F)1560 2069 y Fw(1)1599 2054 y Fs(j)l Ft(~)-45 b(e)1672 2069 y Fq(d)1757 2054 y Fz(=)1886 1930 y Fq(d)p Fw(+1)1877 1960 y Fp(X)1885 2172 y Fq(k)r Fw(=2)2021 2054 y Fs(j)p Ft(F)2112 2069 y Fq(k)2155 2054 y Fs(j)m Ft(~)f(n)2241 2069 y Fq(k)2283 2054 y Ft(:)118 2392 y Fz(W)-8 b(e)44 b(sa)m(y)g(that)f(the)g(surface)h(tension)g Ft(\034)1605 2407 y Fq(\014)1695 2392 y Fz(is)f(strictly)f(con)m(v)m (ex)k(at)39 b Ft(~)-45 b(e)2654 2407 y Fq(d)2738 2392 y Fz(if)42 b(the)h(follo)m(wing)e(strict)118 2508 y(inequalit)m(y)1353 2733 y Fs(j)p Ft(F)1444 2748 y Fw(1)1484 2733 y Fs(j)p Ft(\034)11 b Fz(\()l Ft(~)-45 b(e)1648 2748 y Fq(d)1688 2733 y Fz(\))45 b Ft(<)1900 2609 y Fq(d)p Fw(+1)1891 2639 y Fp(X)1898 2851 y Fq(k)r Fw(=2)2035 2733 y Fs(j)p Ft(F)2126 2748 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Fw(1)2541 3337 y Ft(;)17 b(:)g(:)g(:)f(;)d(~)-45 b(v)2807 3352 y Fq(d)2847 3337 y Fz(,)32 b(that)g(is)f(there)i(exist)118 3459 y(p)s(ositiv)m(e)f(n)m (um)m(b)s(ers)h Ft(\025)929 3474 y Fw(1)969 3459 y Ft(;)17 b(:)g(:)g(:)f(;)h(\025)1245 3474 y Fq(d)1285 3459 y Fz(,)32 b(suc)m(h)i(that)29 b Ft(~)-45 b(e)1821 3474 y Fq(d)1906 3459 y Fz(=)2026 3384 y Fp(P)2131 3411 y Fq(d)2131 3488 y(k)r Fw(=1)2280 3459 y Ft(\025)2337 3474 y Fq(k)2376 3459 y Ft(~)g(v)2427 3474 y Fq(k)2470 3459 y Fz(,)32 b(then,)1447 3734 y Ft(\034)11 b Fz(\()l Ft(~)-45 b(e)1583 3749 y Fq(d)1624 3734 y Fz(\))44 b Ft(<)1880 3610 y Fq(d)1826 3639 y Fp(X)1834 3852 y Fq(k)r Fw(=1)1987 3734 y Ft(\025)2044 3749 y Fq(k)2086 3734 y Ft(\034)11 b Fz(\()l Ft(~)-45 b(v)2224 3749 y Fq(k)2268 3734 y Fz(\))p Ft(:)1079 b Fz(\(2.14\))118 4030 y FE(Assumption)38 b(\(SC\):)33 b Ft(\034)44 b Fz(is)32 b(strictly)f(con)m(v)m(ex)k(at)28 b Ft(~)-45 b(e)2035 4045 y Fq(d)2076 4030 y Fz(.)218 4261 y(This)42 b(assumption)f(is)h(of)f(course)i(true)g(at)e(lo)m(w)h (enough)g(temp)s(eratures)h(when)g(the)f(W)-8 b(ul\013)118 4377 y(shap)s(e)29 b(exhibits)g(a)f(\015at)g(facet)h(in)f(the)d Ft(~)-45 b(e)1548 4392 y Fq(d)1617 4377 y Fz(direction.)41 b(Presumably)29 b(it)e(is)h(true)h(for)f(ev)m(ery)j Ft(\014)i(>)28 b(\014)3627 4392 y Fq(c)118 4493 y Fz(since,)h(at)d(least)g(on)h(the)g (heuristic)f(lev)m(el,)i(kinks)g(on)e(the)h(b)s(oundary)h(of)e(the)h(W) -8 b(ul\013)26 b(shap)s(e)h(w)m(ould)118 4610 y(corresp)s(ond)34 b(to)g(pathological)c(large)i(in)m(terface)i(\015uctuations.)46 b(W)-8 b(e)34 b(could)f(ha)m(v)m(e)i(a)m(v)m(oided)f(this)118 4726 y(assumption)23 b(b)m(y)i(simply)d(considering)h(a)h(direction)f (whic)m(h)h(is)f(orthogonal)f(to)h(smo)s(oth)g(p)s(ortions)118 4842 y(of)g(the)i(W)-8 b(ul\013)22 b(shap)s(e.)42 b(This)24 b(w)m(ould)f(not)h(mend)f(the)i(situation)d(and)i(w)m(e)g(prefer)h (this)e(assumption)118 4958 y(for)32 b(the)h(notational)d(simplicit)m 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y Fp(Z)1292 5487 y Fq(@)1333 5468 y Fh(\003)1329 5504 y Fm(g)1369 5487 y Fq(u)p Fo(n)p Fq(@)1482 5495 y Fl(s)1515 5470 y Fg(b)1510 5487 y Fn(D)1579 5397 y Ft(\034)11 b Fz(\()m Ft(~)-46 b(n)1728 5412 y Fq(x)1773 5397 y Fz(\))p Ft(d)p Fs(H)1947 5356 y Fw(\()p Fq(d)p Fo(\000)p Fw(1\))1946 5422 y Fq(x)2189 5397 y Fz(+)57 b(\001)2403 5412 y Fq(")2456 5262 y Fp(Z)2512 5487 y Fq(@)2553 5468 y Fh(\003)2549 5504 y Fm(g)2589 5487 y Fq(u)p Fo(\\)p Fq(@)2714 5495 y Fl(s)2747 5470 y Fg(b)2742 5487 y Fn(D)2811 5397 y Ft(d)p Fs(H)2947 5356 y Fw(\()p Fq(d)p Fo(\000)p Fw(1\))2946 5422 y Fq(x)3132 5397 y Ft(:)p eop %%Page: 11 11 11 10 bop 3585 48 a Fu(11)118 366 y Fj(Then)38 b Ft(u)33 b Fz(=)h(1)-25 b(I\()p Fs(\001)p Fz(\))38 b Fj(is)g(the)h(stable)f (minimum)f(of)h Fs(W)1985 381 y Fq(")2022 366 y Fz(\()p Fs(\001j)p Ft(g)t Fz(\))f Fj(in)h(the)g(fol)5 b(lowing)37 b(sense:)51 b(F)-7 b(or)38 b(every)118 483 y Ft(\027)c(>)28 b Fz(0)34 b Fj(ther)-5 b(e)35 b(exists)g Ft(c)939 498 y Fw(2)1006 483 y Fz(=)27 b 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b(the)i(lines)f(of)g(the)h(pro)s(of)f(of)g(Theorem)h(5.3)f (\(c\))h(of)f([Gri)n(].)218 4659 y(W)-8 b(e)41 b(conjecture)h(that)f (the)h(relaxation)d(prop)s(ert)m(y)j(should)f(b)s(e)g(v)-5 b(alid)39 b(on)i(])p Ft(\014)3070 4674 y Fq(c)3105 4659 y Ft(;)17 b Fs(1)p Fz([.)68 b(In)42 b(fact)118 4775 y FE(\(RP\))c Fz(can)i(b)s(e)g(related)f(to)h(the)g(notion)f(of)g(strong) h(mixing)d(whic)m(h)j(w)m(as)h(in)m(tro)s(duced)f(in)f(the)118 4891 y(con)m(text)f(of)f(Ising)g(mo)s(del)e(b)m(y)j(Dobrushin)e(and)i (Shlosman)e([DS3)o(])h(\(see)h(also)e(Martinelli)e(and)118 5008 y(Olivieri)21 b([MO])j(for)f(the)h(regular)e(strong)i(mixing)d (prop)s(ert)m(y\).)41 b(The)25 b(coun)m(terpart)f(of)f(this)g(notion) 118 5124 y(for)36 b(the)h(FK)g(mo)s(del)e(can)i(b)s(e)g(stated)g(as)g (follo)m(ws:)50 b(there)38 b(exists)f Ft(c)2590 5139 y Fw(1)2630 5124 y Fz(\()p Ft(\014)6 b Fz(\))p Ft(;)17 b(c)2853 5139 y Fw(2)2891 5124 y Fz(\()p Ft(\014)6 b Fz(\))34 b Ft(>)h Fz(0)h(suc)m(h)j(that)118 5240 y(for)32 b(an)m(y)h(cub)s(e)h(\001)f(of)f Fx(Z)976 5204 y Fq(d)1014 5240 y Fz(,)g(an)m(y)h(pair)f(of)g(b)s(oundary)h(conditions)f Ft(\031)t(;)17 b(\031)2637 5204 y Fo(0)553 5449 y Fs(8)p Ft(b)28 b Fs(2)g Fz(\004)836 5464 y Fw(\001)899 5449 y Ft(;)1138 5334 y Fp(\014)1138 5394 y(\014)1138 5454 y(\014)1171 5449 y Fz(\010)1241 5408 y Fq(\031)1241 5473 y Fw(\001)1305 5449 y Fz(\()p Ft(\030)1386 5464 y Fq(b)1419 5449 y Fz(\))23 b Fs(\000)f Fz(\010)1649 5408 y Fq(\031)1692 5384 y Fh(0)1649 5473 y Fw(\001)1719 5449 y Fz(\()p Ft(\030)1800 5464 y Fq(b)1834 5449 y Fz(\))1872 5334 y Fp(\014)1872 5394 y(\014)1872 5454 y(\014)1933 5449 y Fr(6)28 b Ft(c)2080 5464 y Fw(2)2136 5449 y Fz(exp)2302 5368 y Fp(\000)2369 5449 y Fs(\000)23 b Ft(c)2511 5464 y Fw(1)2567 5449 y Fz(dist\()p Ft(b;)17 b(\031)26 b Fs(^)c Ft(\031)3076 5408 y Fo(0)3100 5449 y Fz(\))3138 5368 y Fp(\001)3200 5449 y Ft(;)p eop %%Page: 12 12 12 11 bop 118 51 a Fu(12)1148 b(T.)26 b(BODINEA)n(U)e(AND)g(D.)i(IOFFE) 118 366 y Fz(where)32 b Ft(\031)22 b Fs(^)d Ft(\031)619 330 y Fo(0)673 366 y Fz(refers)31 b(to)f(the)i(region)d(where)j Ft(\031)j Fz(and)30 b Ft(\031)2124 330 y Fo(0)2178 366 y Fz(di\013er.)43 b(This)30 b(prop)s(ert)m(y)i(implies)c FE(\(RP\))118 483 y Fz(and)36 b(w)m(e)h(conjecture)g(that)f(it)e(holds) i(for)f(the)h(parameters)g Ft(\014)42 b Fz(for)35 b(whic)m(h)h(the)g (FK)g(measure)g(is)118 599 y(unique)d(in)f(the)h(thermo)s(dynamic)e (limit.)218 762 y(Finally)-8 b(,)30 b(w)m(e)k(stress)h(the)e(fact)g (that)g FE(\(RP\))e Fz(do)s(es)i(not)g(apply)g(directly)f(to)h(the)g (Ising)g(mo)s(del,)118 879 y(nev)m(ertheless)k(com)m(bined)e(with)f (the)h(lo)s(calization)c(of)k(the)g(in)m(terface,)g(it)f(will)f(ha)m(v) m(e)j(useful)f(im-)118 995 y(plications)j(on)i(the)g(relaxation)e(of)h (the)i(spin)e(system.)66 b(This)40 b(will)e(b)s(e)i(discussed)h(in)f (Section)118 1111 y(5.)1525 1320 y(3.)49 b Fy(The)38 b(resul)-7 b(ts)218 1495 y Fz(Throughout)28 b(the)h(pap)s(er,)g(the)f (dimension)f Ft(d)h 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y(croscopic)j(in)m(terfaces)f(are)g(not)g(the)h(appropriate)e (ob)5 b(jects)36 b(to)f(describ)s(e)h(stabilit)m(y)d(prop)s(erties)118 2491 y(of)45 b(phase)h(b)s(oundaries.)80 b(In)46 b(an)m(y)f(case,)50 b(ho)m(w)m(ev)m(er,)g(the)c(phases)g(are)f(c)m(haracterized)h(b)m(y)g (the)118 2607 y(order)37 b(parameter)f(\(sp)s(on)m(taneous)i (magnetization\))c Fs(\006)p Ft(m)2281 2571 y Fo(\003)2321 2607 y Fz(\()p Ft(\014)6 b Fz(\))36 b(in)g(the)g(sense)j(that)d(lo)s (cal)e(spin)118 2723 y(a)m(v)m(erages,)d(or)d(lo)s(cal)e(magnetization) g(pro\014les,)k(inside)d(what)i(is)f(exp)s(ected)j(to)d(b)s(e)g(")14 b(+)g(")28 b(or)g(")14 b Fs(\000)g Fz(")118 2840 y(phases)37 b(should)f(con)m(v)m(erge,)i(as)e(the)h(a)m(v)m(eraging)e(scale)h(gro)m (ws,)h(to)f Ft(m)2657 2803 y Fo(\003)2696 2840 y Fz(\()p Ft(\014)6 b Fz(\))36 b(or)f Fs(\000)p Ft(m)3153 2803 y Fo(\003)3193 2840 y Fz(\()p Ft(\014)6 b Fz(\))36 b(resp)s(ec-)118 2956 y(tiv)m(ely)-8 b(.)54 b(Our)36 b(main)e(stabilit)m(y)h(result)h(b) 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3398 y Fp(\014)1791 3457 y(\014)1791 3517 y(\014)1791 3577 y(\014)1791 3637 y(\014)1791 3696 y(\014)1875 3534 y Fz(1)p 1834 3579 131 4 v 1834 3670 a Ft(K)1924 3641 y Fq(d)2050 3507 y Fp(X)1992 3723 y Fq(j)t Fo(2)p Fn(B)2109 3734 y Fm(K)2174 3723 y Fw(\()p Fq(i)p Fw(\))2270 3602 y Ft(\033)2325 3617 y Fq(j)2384 3602 y Fs(\000)22 b Ft(m)2568 3561 y Fo(\003)2608 3602 y Fz(\()p Ft(\014)6 b Fz(\))2745 3398 y Fp(\014)2745 3457 y(\014)2745 3517 y(\014)2745 3577 y(\014)2745 3637 y(\014)2745 3696 y(\014)2839 3602 y Fr(6)44 b Ft(\032)17 b(;)1297 3998 y Fs(\000)p Fz(1)g Ft(;)211 b Fz(if)1791 3794 y Fp(\014)1791 3853 y(\014)1791 3913 y(\014)1791 3973 y(\014)1791 4033 y(\014)1791 4092 y(\014)1875 3930 y Fz(1)p 1834 3975 V 1834 4066 a Ft(K)1924 4037 y Fq(d)2050 3903 y Fp(X)1992 4119 y Fq(j)t Fo(2)p Fn(B)2109 4130 y Fm(K)2174 4119 y Fw(\()p Fq(i)p Fw(\))2270 3998 y Ft(\033)2325 4013 y Fq(j)2384 3998 y Fz(+)22 b Ft(m)2567 3957 y Fo(\003)2606 3998 y Fz(\()p Ft(\014)6 b Fz(\))2743 3794 y Fp(\014)2743 3853 y(\014)2743 3913 y(\014)2743 3973 y(\014)2743 4033 y(\014)2743 4092 y(\014)2837 3998 y Fr(6)45 b Ft(\032)17 b(;)1374 4259 y Fz(0)g Ft(;)211 b Fz(otherwise)17 b Ft(:)3412 3873 y Fz(\(3.18\))118 4427 y(Th)m(us,)45 b(~)-55 b Ft(u)453 4379 y Fq(\032)453 4454 y(N)s(;K)636 4427 y Fz(uses)37 b(the)g(resolution)e Ft(\032)h Fz(to)g(lab)s(el)e(the)j(pro)m(ximit)m (y)e(of)g(the)i(lo)s(cal)d(magnetization)118 4557 y(pro\014le)27 b(to)f(the)i(order)f(parameter)f Fs(\006)p Ft(m)1558 4521 y Fo(\003)1598 4557 y Fz(\()p Ft(\014)6 b Fz(\))27 b(on)g(the)g(renormalization)c(scale)k Ft(K)35 b Fz(=)28 b(2)3246 4521 y Fq(k)3288 4557 y Fz(.)42 b(F)-8 b(or)26 b(the)118 4673 y(spin)i(mo)s(del)e(on)i Fx(D)802 4688 y Fq(N)903 4673 y Fz(with)g(mixed)f(b)s(oundary)h(conditions)f(describ) s(ed)i(in)e(Subsection)i(2.3.2)e(w)m(e)118 4812 y(shall)d(\(b)m(y)i (abuse)h(of)e(notation\))f(extend)33 b(~)-55 b Ft(u)1661 4765 y Fq(\032)1661 4839 y(N)s(;K)1833 4812 y Fz(to)25 b(the)h(whole)f(of)g Fx(Z)2548 4776 y Fq(d)2548 4837 y(K)2650 4756 y Fw(\001)2642 4812 y Fz(=)i Ft(K)2852 4732 y Fp(\000)2898 4812 y Fx(Z)2967 4776 y Fq(d)3027 4812 y Fs(\000)22 b Fz(\(1)p Ft(=)p Fz(2)p Ft(;)17 b(:)g(:)g(:)e(;)i Fz(1\))3616 4732 y Fp(\001)118 4930 y Fz(as)33 b(follo)m(ws:)834 5116 y(~)-55 b Ft(u)884 5069 y Fq(\032)884 5143 y(N)s(;K)1031 5116 y Fz(\()p Ft(i)p Fz(\))44 b(=)1304 4946 y Fp(\()1479 5034 y Fz(1)195 b(if)38 b Ft(i)28 b Fs(2)g Fx(Z)2043 4993 y Fq(d)2043 5058 y(K)2131 5034 y Fs(n)22 b Fx(D)2269 5049 y Fq(N)s(;K)2454 5034 y Fz(and)33 b Ft(i)2677 5049 y Fq(d)2745 5034 y Ft(>)28 b Fz(0)17 b Ft(;)1401 5197 y Fs(\000)q Fz(1)195 b(if)38 b Ft(i)28 b Fs(2)g Fx(Z)2043 5156 y Fq(d)2043 5222 y(K)2131 5197 y Fs(n)22 b Fx(D)2269 5212 y Fq(N)s(;K)2454 5197 y Fz(and)33 b Ft(i)2677 5212 y Fq(d)2745 5197 y Ft(<)28 b Fz(0)17 b Ft(:)118 5350 y Fz(It)45 b(happ)s(ens)h(to)f(b)s(e)g(more)f(con)m(v)m(enien)m(t)i(to) f(w)m(ork)h(with)e(the)i(adjusted)f(phase)h(lab)s(els)e Ft(u)3488 5302 y Fq(\032)3488 5377 y(N)s(;K)3634 5350 y Fz(:)118 5474 y Ft(u)174 5426 y Fq(\032)174 5501 y(N)s(;K)321 5474 y Fz(\()p Ft(i)p Fz(\))h(=)g(1)d(\(resp)s(ectiv)m(ely)i Fs(\000)p Fz(1\))f(if)k(~)-55 b Ft(u)1629 5426 y Fq(\032)1629 5501 y(N)s(;K)1776 5474 y Fz(\()p Ft(i)p Fz(\))45 b(=)g(1)d(\(resp)s (ectiv)m(ely)i(to)f Fs(\000)p Fz(1\))f(b)s(oth)h(at)f Ft(i)h Fz(and)p eop %%Page: 13 13 13 12 bop 3585 48 a Fu(13)118 366 y Fz(at)37 b(all)e Fs(\003)p Fz(-neigh)m(b)s(ours)i(of)f Ft(i)i Fz(in)e Fx(Z)1336 330 y Fq(d)1336 392 y(K)1402 366 y Fz(.)57 b(Otherwise,)39 b Ft(u)2036 319 y Fq(\032)2036 393 y(N)s(;K)2183 366 y Fz(\()p Ft(i)p Fz(\))e(is)g(set)h(to)f(b)s(e)g(equal)g(to)g (zero.)57 b(The)118 484 y(adv)-5 b(an)m(tage)28 b(of)f(suc)m(h)i (adjustmen)m(t)f(is)f(that)g(an)m(y)h(nearest)h(neigh)m(b)s(our)e(path) g(of)h(v)m(ertices)g(of)f Fx(D)3508 499 y Fq(N)s(;K)118 600 y Fz(whic)m(h)d(connects)i(regions)d(with)g(di\013eren)m(t)h(phase) h(lab)s(els)d(is)i(forced)g(to)f(con)m(tain)h(a)f(site)h(with)f(zero) 118 716 y(phase)30 b(lab)s(el.)41 b(Accordingly)28 b(let)h(us)h (de\014ne)g(the)g(collection)d(of)i(phase)h(b)s(oundaries)f(induced)g (b)m(y)118 833 y Ft(u)174 785 y Fq(\032)174 860 y(N)s(;K)321 833 y Fz(\()p Ft(i)p Fz(\))j(as)1151 1001 y Ft(@)p 1151 1017 57 4 v 1208 954 a Fq(\032)1208 1028 y(N)s(;K)1408 945 y Fw(\001)1400 1001 y Fz(=)44 b Fs(f)p Ft(i)28 b Fs(2)g Fx(D)1791 1016 y Fq(N)s(;K)1988 1001 y Fz(:)44 b Ft(u)2115 954 y Fq(\032)2115 1028 y(N)s(;K)2262 1001 y Fz(\()p Ft(i)p Fz(\))28 b(=)f(0)p Fs(g)p Ft(:)118 1154 y Fz(The)g(set)g Ft(@)p 458 1170 V 514 1107 a Fq(\032)514 1181 y(N)s(;K)687 1154 y Fz(is)f(in)f(general)g(disconnected)j(and)e (for)f(\014xed)i(\014nite)f(v)-5 b(alues)26 b(of)f(the)h(renormaliza-) 118 1277 y(tion)g(scale)h Ft(K)33 b Fz(con)m(tains)27 b(\(for)f(en)m(tropic)h(reasons\))h(man)m(y)e(small)f(comp)s(onen)m(ts) i(ev)m(en)h(in)e(the)h(case)118 1394 y(of)39 b(pure)i(b)s(oundary)f (conditions.)64 b(In)40 b(the)g(case)h(of)e(mixed)g(b)s(oundary)h (conditions,)h(ho)m(w)m(ev)m(er)118 1510 y Ft(@)p 118 1526 V 175 1463 a Fq(\032)175 1537 y(N)s(;K)356 1510 y Fz(con)m(tains)34 b(a)g(unique)g(un)m(b)s(ounded)i(connected)f(comp)s (onen)m(t)f(whic)m(h)h(w)m(e)g(shall)d(denote)j(as)118 1634 y Ft(@)174 1587 y Fq(\032)169 1661 y(N)s(;K)317 1634 y Fz(.)51 b(By)35 b(the)h(construction)f Ft(@)1338 1587 y Fq(\032)1333 1661 y(N)s(;K)1516 1634 y Fz(con)m(tains)g(an)g (in\014nite)f(\015at)h(double)g(la)m(y)m(er)g(outside)g Fx(D)3508 1649 y Fq(N)s(;K)118 1758 y Fz(and,)k(in)d(fact,)i(all)e(the) h(non-trivial)e(geometry)i(of)f Ft(@)2083 1711 y Fq(\032)2078 1785 y(N)s(;K)2263 1758 y Fz(is)h(con\014ned)h(to)f Fx(D)2945 1773 y Fq(N)s(;K)3098 1758 y Fz(.)58 b(Here)38 b(is)f(our)118 1882 y(Hausdor\013)c(stabilit)m(y)e(result)h(in)g(terms)h(of)f(phase)h (lab)s(els)f Ft(u)2313 1835 y Fq(\032)2313 1909 y(N)s(;K)2459 1882 y Fz(:)118 2030 y FE(Theorem)46 b(3.1.)g Fj(Assume)41 b(that)h Ft(\014)k Fs(2)40 b(B)45 b Fj(and)c(that)h(the)f(Assumption)h FE(\(SC\))f Fj(holds,)h(that)g(is)118 2146 y(the)36 b(surfac)-5 b(e)36 b(tension)g Ft(\034)1000 2161 y Fq(\014)1083 2146 y Fj(is)g(strictly)h(c)-5 b(onvex)35 b(at)d Ft(~)-45 b(e)1996 2161 y Fq(d)2037 2146 y Fj(.)49 b(Then)35 b(for)h(any)g Ft(\027)h(>)30 b Fz(0)36 b Fj(and)f Ft(\032)c(>)f Fz(0)35 b Fj(ther)-5 b(e)118 2262 y(exists)37 b(a)h(\014nite)f(sc)-5 b(ale)37 b Ft(K)1049 2277 y Fw(0)1121 2262 y Fz(=)32 b Ft(K)1312 2277 y Fw(0)1352 2262 y Fz(\()p Ft(\027)q(;)17 b(\032)p Fz(\))37 b Fj(and)g(a)h(p)-5 b(ositive)36 b(c)-5 b(onstant)38 b Ft(c)32 b Fz(=)h Ft(c)p Fz(\()p Ft(\027)q(;)17 b(\032)p Fz(\))37 b Fj(such)h(that)g(for)118 2378 y(every)d Ft(K)g Fr(>)28 b Ft(K)682 2393 y Fw(0)756 2378 y Fj(and)34 b(for)h(al)5 b(l)35 b Ft(N)45 b Fj(su\016ciently)35 b(lar)-5 b(ge,)945 2547 y Ft(\026)1004 2506 y Fo(\006)1004 2574 y Fq(N)1088 2466 y Fp(\000)1150 2547 y Ft(@)1206 2500 y Fq(\032)1201 2574 y(N)s(;K)1371 2547 y Fs(\\)22 b(f)p Ft(i)45 b Fz(:)f Ft(i)1691 2562 y Fq(d)1760 2547 y Ft(>)27 b(\027)6 b(N)k Fs(g)29 b(6)p Fz(=)e Fs(;)2237 2466 y Fp(\001)2344 2547 y Fr(6)44 b Fz(e)2508 2506 y Fo(\000)p Fq(c)p Fw(\()p Fq(\027;\032)p Fw(\))p Fq(N)2807 2547 y Ft(:)578 b Fz(\(3.19\))218 2714 y(The)37 b(ab)s(o)m(v)m(e)g(statemen) m(t)f(asserts)i(that)e(on)g(large)e(enough,)k(though)e(still)e (\014nite,)j(renormal-)118 2830 y(ization)26 b(scales)j Ft(K)35 b Fz(the)29 b(in)m(terface)f(is)g(macroscopically)d(stable)j (in)g(the)g(sense)i(that)e(the)h(order)f(of)118 2946 y(its)d(\015uctuations)h(is)f(smaller)e(than)j(the)g(linear)e(size)i (of)f(the)h(system)g Ft(N)10 b Fz(.)42 b(Since)25 b(the)h (\015uctuations)118 3063 y(in)40 b(question)h(are)g(exp)s(ected)h(to)f (b)s(e)f(of)h(the)g(log)16 b Ft(N)10 b Fz(-size)41 b(for)f(mo)s (derately)g(lo)m(w)g(temp)s(eratures)118 3179 y(and,)f(at)e(least)g (for)g(axis)h(orien)m(ted)f(in)m(terfaces,)j(are)d(kno)m(wn)i(to)f(b)s (e)f(b)s(ounded)i(for)e(su\016cien)m(tly)118 3295 y(lo)m(w)43 b(ones)i([DS1],)h(the)e(result)g(is)f(far)h(from)e(b)s(eing)h(optimal)e (and,)47 b(in)c(a)g(w)m(a)m(y)-8 b(,)48 b(it)43 b(illustrates)118 3411 y(limitations)28 b(of)k(the)i Fx(L)946 3426 y Fw(1)991 3411 y Fz(-approac)m(h.)218 3527 y(The)h(pro)s(of)e(of)g(Theorem)i(3.1) e(is)h(based)h(on)f(the)g(follo)m(wing)e(result)i(on)g(the)g(stabilit)m (y)e(of)i(the)118 3644 y(FK)g(in)m(terfaces.)50 b(In)35 b(the)g(sequel)h(w)m(e)f(emplo)m(y)f(the)h(notation)e(in)m(tro)s(duced) i(in)f(Subsections)i(2.2)118 3760 y(and)d(2.3.)118 3901 y FE(Theorem)g(3.2.)38 b Fj(Assume)31 b(that)g Ft(\014)i Fs(2)28 b(B)34 b Fj(and)c(that)h(the)g(surfac)-5 b(e)30 b(tension)g Ft(\034)2881 3916 y Fq(\014)2959 3901 y Fj(is)g(strictly)h (c)-5 b(onvex)118 4017 y(at)31 b Ft(~)-45 b(e)280 4032 y Fq(d)321 4017 y Fj(.)45 b(L)-5 b(et)35 b Ft(K)g Fz(=)27 b Ft(N)873 3981 y Fq(a)915 4017 y Fj(,)35 b(wher)-5 b(e)34 b Ft(a)h Fj(is)g(chosen)f(ac)-5 b(c)g(or)g(ding)33 b(to)42 b Fz(\(2.2\))o Fj(.)j(Then)34 b(for)h(any)f Ft(\027)g(>)28 b Fz(0)p Fj(,)787 4183 y Fz(lim)753 4245 y Fq(N)7 b Fo(!1)974 4183 y Fz(\010)1044 4142 y Fo(\006)1044 4210 y Fq(N)1128 4183 y Fz(\()p Fj(Ther)-5 b(e)34 b(is)h(a)f(b)-5 b(ad)35 b(blo)-5 b(ck)34 b(in)g Fs(f)p Ft(i)45 b Fz(:)f Ft(i)2400 4198 y Fq(d)2469 4183 y Ft(>)27 b(\027)6 b(N)k Fs(g)p Fz(\))29 b(=)e(0)17 b Ft(:)385 b Fz(\(3.20\))118 4388 y Fj(Mor)-5 b(e)35 b(pr)-5 b(e)g(cisely,)34 b(ther)-5 b(e)35 b(exists)g Ft(c)1345 4403 y Fw(1)1412 4388 y Fz(=)27 b Ft(c)1557 4403 y Fw(1)1597 4388 y Fz(\()p Ft(\027)6 b Fz(\))28 b Ft(>)f Fz(0)p Fj(,)35 b(such)f(that)i(uniformly)e(in)h Ft(N)45 b Fj(lar)-5 b(ge)35 b(enough,)1088 4562 y Fz(\010)1158 4520 y Fo(\006)1158 4589 y Fq(N)1243 4562 y Fz(\()p Fi(g)28 b Fs($)f(f)p Ft(i)44 b Fz(:)h Ft(i)1718 4577 y Fq(d)1786 4562 y Fr(>)28 b Ft(\027)6 b(N)k Fs(g)p Fz(\))62 b Fr(6)h Fz(e)2366 4520 y Fo(\000)p Fq(c)2452 4529 y Fl(1)2486 4520 y Fw(\()p Fq(\027)t Fw(\))p Fq(N)2664 4562 y Ft(:)721 b Fz(\(3.21\))218 4727 y(Notice)41 b(that)g(the)i(second)f(part)g(of)f (the)h(statemen)m(t)g(implies)e(that)h(in)g(the)h(FK)f(represen-)118 4844 y(tation)34 b(the)i(in)m(terface)f(is)g(lo)s(calized)e(ev)m(en)k (on)e(a)g(microscopic)f(lev)m(el)h(without)g(an)m(y)h(additional)118 4960 y(renormalization)29 b(pro)s(cedures.)218 5126 y(Let)j(us)i(no)m (w)f(pro)m(v)m(e)h(Theorem)f(3.1)f(as)h(a)f(consequence)j(of)e (inequalit)m(y)f(\(3.21\).)118 5292 y Fj(Pr)-5 b(o)g(of)35 b(of)f(The)-5 b(or)g(em)34 b(3.1.)43 b Fz(Let)33 b(us)g(decomp)s(ose)g (the)g(p)s(ercolation)d(ev)m(en)m(t)1064 5480 y Fs(T)1118 5495 y Fq(N)s(;\027)1293 5424 y Fw(\001)1285 5480 y Fz(=)44 b Fs(f)p Ft(\030)32 b Fs(2)c Fi(J)1679 5495 y Fq(N)1791 5480 y Fz(:)44 b Fi(g)28 b Fs(6$)f(f)p Ft(i)45 b Fz(:)f Ft(i)2299 5495 y Fq(d)2367 5480 y Fr(>)28 b Ft(\027)6 b(N)k Fs(gg)p eop %%Page: 14 14 14 13 bop 118 51 a Fu(14)1148 b(T.)26 b(BODINEA)n(U)e(AND)g(D.)i(IOFFE) 118 366 y Fz(according)31 b(to)g(the)h(realization)c(of)j(the)h (maximal)c(connected)33 b(comp)s(onen)m(t)e Fs(C)6 b Fz(\()p Fi(g)p Fz(\))32 b(of)f(the)g(ghost)118 483 y(site)h Fi(g)p Fz(.)44 b(By)33 b(the)g(v)m(ery)h(de\014nition,)1159 664 y Fs(C)6 b Fz(\()p Fi(g)p Fz(\))28 b Fs(\022)g Fx(D)1542 623 y Fq(\027)1542 689 y(N)1652 609 y Fw(\001)1643 664 y Fz(=)g Fx(D)1813 679 y Fq(N)1908 664 y Fs(\\)23 b(f)p Ft(i)44 b Fz(:)g Ft(i)2228 679 y Fq(d)2297 664 y Ft(<)27 b(\027)6 b(N)k Fs(g)p Ft(;)118 824 y Fz(on)41 b Fs(T)316 839 y Fq(N)s(;\027)438 824 y Fz(.)69 b(Consequen)m(tly)-8 b(,)45 b(for)c(ev)m(ery)h(p)s(ercolation)e(ev)m(en)m(t)i Ft(A)f Fz(whic)m(h)h(dep)s(ends)h(only)d(on)h(the)118 940 y(b)s(onds)h(connected)h(to)e(the)h(upp)s(er)g(sub-b)s(o)m(x)g Fx(D)1910 955 y Fq(N)2011 940 y Fs(n)28 b Fx(D)2155 904 y Fq(\027)2155 965 y(N)2228 940 y Fz(,)44 b(the)e(follo)m(wing)d (decoupling)h(b)s(ound)118 1056 y(holds:)1137 1174 y(\010)1207 1133 y Fo(\006)1207 1201 y Fq(N)1291 1094 y Fp(\000)1354 1174 y Ft(A)1444 1090 y Fp(\014)1444 1149 y(\014)1493 1174 y Fs(T)1547 1189 y Fq(N)s(;\027)1686 1094 y Fp(\001)1793 1174 y Fr(6)77 b Fz(max)2016 1234 y Fq(\031)2145 1174 y Fz(\010)2215 1133 y Fq(\031)2215 1199 y Fn(D)2257 1210 y Fm(N)2321 1199 y Fo(n)p Fn(D)2398 1176 y Fm(\027)2398 1222 y(N)2466 1174 y Fz(\()p Ft(A)p Fz(\))p Ft(:)770 b Fz(\(3.22\))218 1344 y(W)-8 b(e)26 b(stress)h(the)g(fact)e(that)h(in) f(this)h(Theorem)g(the)g(coarse)g(graining)e(scale)i Ft(K)33 b Fz(is)25 b(indep)s(enden)m(t)118 1460 y(of)35 b Ft(N)10 b Fz(,)36 b(unlik)m(e)e(in)g(\(3.20\).)50 b(As)36 b(in)e([BIV1,)h(BIV2])g(de\014ne)i(the)e(join)m(t)f(spin-b)s(ond)g(lab) s(el)g Ft(v)3411 1413 y Fq(\032)3407 1487 y(N)s(;K)3586 1460 y Fz(=)118 1590 y Ft(v)169 1543 y Fq(\032)165 1617 y(N)s(;K)312 1590 y Fz([)p Ft(\033)n(;)17 b(\030)5 b Fz(])28 b Fs(2)g(f)p Fz(0)p Ft(;)17 b Fz(1)p Fs(g)875 1554 y Fn(D)916 1565 y Fm(N)r(;K)1087 1590 y Fz(via:)779 1837 y Ft(v)830 1790 y Fq(\032)826 1864 y(N)s(;K)973 1837 y Fz(\()p Ft(i)p Fz(\))60 b(=)1278 1697 y Fp(\032)1370 1757 y Fz(1)p Ft(;)49 b Fz(if)38 b Fs(j)p Ft(u)1675 1710 y Fq(\032)1675 1784 y(N)s(;K)1821 1757 y Fs(j)28 b Fz(=)f(1)33 b(and)f Fx(B)2312 1772 y Fw(2)p Fq(K)2421 1757 y Fz(\()p Ft(i)p Fz(\))h(is)f Ft(\030)37 b Fz(go)s(o)s(d)16 b Ft(;)1370 1904 y Fz(0)p Ft(;)49 b Fz(otherwise)17 b Ft(:)3412 1837 y Fz(\(3.23\))118 2063 y(Clearly)-8 b(,)990 2181 y Ft(@)p 990 2197 57 4 v 1047 2134 a Fq(\032)1047 2208 y(N)s(;K)1216 2181 y Fs(\\)23 b Fx(D)1371 2196 y Fq(N)s(;K)1568 2181 y Fs(\022)45 b(f)p Ft(i)27 b Fs(2)i Fx(D)1961 2196 y Fq(N)s(;K)2158 2181 y Fz(:)44 b Ft(v)2280 2134 y Fq(\032)2276 2208 y(N)s(;K)2424 2181 y Fz(\()p Ft(i)p Fz(\))27 b(=)h(0)p Fs(g)p Ft(:)118 2328 y Fz(On)44 b(the)g(other)g(hand,)j(it)c(follo)m (ws)g(from)g(\(3.22\))g(that)g(the)i(distribution)d(of)h(the)h(\014eld) g Ft(v)3518 2281 y Fq(\032)3514 2355 y(N)s(;K)118 2469 y Fz(on)39 b Fs(f)p Fz(0)p Ft(;)17 b Fz(1)p Fs(g)502 2426 y Fn(D)543 2437 y Fm(N)r(;K)677 2426 y Fo(n)p Fn(D)754 2403 y Fm(\027)754 2449 y(N)r(;K)931 2469 y Fz(under)40 b Fx(P)1273 2428 y Fo(\006)1273 2497 y Fq(\014)s(;)p Fn(D)1380 2508 y Fm(N)1447 2469 y Fz(\()17 b Fs(\001)g(jT)1629 2484 y Fq(N)s(;\027)1750 2469 y Fz(\))39 b(sto)s(c)m(hastically)f (dominates)g(the)i(Bernoulli)d(site)118 2600 y(p)s(ercolation)e(pro)s (cess)k Fx(P)1033 2563 y Fw(Bern)1033 2624 y Fq(p)1187 2600 y Fz(,)f(where)g(the)f(probabilit)m(y)e Ft(p)h Fz(=)f Ft(p)p Fz(\()p Ft(\014)6 b(;)17 b(\032;)g(K)7 b Fz(\))36 b(of)h(a)f(particular)g(site)118 2720 y(to)h(b)s(e)h(o)s(ccupied)g (satis\014es)g(lim)1289 2735 y Fq(K)5 b Fo(!1)1515 2720 y Ft(p)p Fz(\()p Ft(\014)h(;)17 b(\032;)g(K)7 b Fz(\))35 b(=)h(1,)j(see)g(Section)e(3.2)g(of)g([BIV2])h(for)f(more)118 2836 y(details)43 b(and)h(references.)80 b(As)44 b(a)g(result,)i (\(3.19\))e(follo)m(ws)e(from)h(the)h(exp)s(onen)m(tial)g(deca)m(y)h (of)118 2953 y(connectivities)38 b(for)g(the)h(sub-critical)d(site)i(p) s(ercolation)f(once)i(\(1)26 b Fs(\000)g Ft(p)p Fz(\))39 b(is)f(su\016cien)m(tly)h(small)118 3069 y(\(or,)32 b(equiv)-5 b(alen)m(tly)d(,)32 b(once)h Ft(K)40 b Fz(is)32 b(su\016cien)m(tly)i (large\).)1517 b Fr(\003)218 3228 y Fz(W)-8 b(e)40 b(turn)f(no)m(w)h (to)f(the)h(case)g(of)f(the)h(W)-8 b(ul\013)39 b(shap)s(e)h(for)f(whic) m(h)h(the)g(phase)g(co)s(existence)h(is)118 3344 y(imp)s(osed)32 b(in)g(a)g(more)g(indirect)f(w)m(a)m(y)j(via)e(a)g(v)m(olume)g (constrain)m(t.)44 b(A)33 b(straigh)m(tforw)m(ard)f(mo)s(di\014-)118 3460 y(cation)h(of)g(the)h(tec)m(hniques)h(whic)m(h)f(w)m(e)h(shall)d (emplo)m(y)h(for)g(the)h(pro)s(of)f(of)g(Theorem)h(3.2)f(yields)118 3577 y(\(see)h(\(2.7\))o(\):)118 3714 y FE(Theorem)43 b(3.3.)h Fj(Assume)39 b(that)g Ft(\014)h Fs(2)35 b(B)s Fj(.)57 b(L)-5 b(et)39 b Ft(K)j Fz(=)35 b Ft(N)2234 3678 y Fq(a)2276 3714 y Fj(,)40 b(wher)-5 b(e)38 b Ft(a)h Fj(is)f(chosen)g(ac)-5 b(c)g(or)g(ding)37 b(to)118 3830 y Fz(\(2.2\))p Fj(.)44 b(Then)34 b(for)h(any)g Ft(\027)f(>)28 b Fz(0)p Fj(,)737 4081 y Fz(lim)703 4143 y Fq(N)7 b Fo(!1)923 4081 y Fz(\010)993 4040 y Fw(w)993 4105 y Fq(N)1078 3910 y Fp( )1190 3986 y([)1157 4198 y Fq(i)p Fo(2)p Fn(D)1270 4209 y Fm(N)1350 4000 y Fp(\010)1408 4081 y Fz(\()p Ft(i)22 b Fz(+)g Fs(K)1675 4100 y Fq(a)p Fw(\(1)p Fo(\000)p Fq(\027)t Fw(\)\()p Fq(N)q(=K)5 b Fw(\))2106 4081 y Fm(d)11 b Fz(\))27 b Fs(\032)i(C)2378 4040 y Fo(\000)2372 4108 y Fq(N)2439 4000 y Fp(\011)2514 3966 y(\014)2514 4026 y(\014)2514 4086 y(\014)2582 4081 y Fx(V)2648 4096 y Fq(N)s(;a)2774 3910 y Fp(!)2881 4081 y Fz(=)e(1)17 b Ft(:)218 4339 y Fz(Notice)45 b(that)g(w)m(e)h(made)f(no)g(additional)d(assumptions)j (on)g(the)h(strict)f(con)m(v)m(exit)m(y)i(of)e Ft(\034)3587 4354 y Fq(\014)3634 4339 y Fz(.)118 4456 y(Indeed,)g(directions)c Ft(v)k Fz(at)c(whic)m(h)h Ft(\034)1477 4471 y Fq(\014)1566 4456 y Fz(is)f(not)g(strictly)g(con)m(v)m(ex)i(corresp)s(ond)g(to)e (non-smo)s(oth)118 4572 y(p)s(ortions)32 b(of)g(the)h(b)s(oundary)g Ft(@)5 b Fs(K)35 b Fz(whic)m(h)e(ha)m(v)m(e)h(zero)f(surface)g (measure.)218 4688 y(Theorem)27 b(3.3)g(implies)e(that)i(there)h(is)f (no)h(p)s(ercolation)d(of)i(the)h(+)f(phase)h(inside)f(the)h(in)m (terior)118 4804 y(of)j(the)h Fs(\000)g Fz(droplet.)43 b(In)32 b(this)f(w)m(a)m(y)i(usual)e(conclusion)g(of)g(the)h Fx(L)2448 4819 y Fw(1)2493 4804 y Fz(-theory)g(is)f(clearly)g (upgraded.)118 4921 y(On)22 b(the)h(other)f(hand)h(w)m(e)g(are)g(not)f (able)f(to)h(establish)g(a)g(complete)f(statemen)m(t)i(on)f(the)h (Hausdor\013)118 5037 y(lo)s(calization,)29 b(i.e.)43 b(that)32 b(there)h(alw)m(a)m(ys)h(exists)f Ft(i)g Fz(in)f Fx(D)2118 5052 y Fq(N)2224 5037 y Fz(suc)m(h)i(that)897 5196 y(\()p Ft(i)22 b Fz(+)g Fs(K)1164 5215 y Fq(a)p Fw(\(1)p Fo(\000)p Fq(\027)t Fw(\)\()p Fq(N)q(=K)5 b Fw(\))1595 5196 y Fm(d)10 b Fz(\))28 b Fs(\032)g(C)1866 5155 y Fo(\000)1860 5223 y Fq(N)1955 5196 y Fs(\032)h Fz(\()p Ft(i)22 b Fz(+)g Fs(K)2328 5215 y Fq(a)p Fw(\(1+)p Fq(\027)t Fw(\)\()p Fq(N)q(=K)5 b Fw(\))2759 5196 y Fm(d)10 b Fz(\))17 b Ft(:)118 5364 y Fz(This)32 b(w)m(ould)g(imply)d(that)j (the)g(in)m(terface)g(b)s(et)m(w)m(een)i Fs(C)2103 5323 y Fo(\000)2097 5391 y Fq(N)2196 5364 y Fz(and)e Fs(C)2443 5323 y Fw(+)2437 5391 y Fq(N)2536 5364 y Fz(is)f(alw)m(a)m(ys)i(lo)s (calized)c(close)j(to)118 5480 y(the)f(b)s(oundary)f(of)g(the)g(W)-8 b(ul\013)29 b(shap)s(e.)44 b(This)30 b(limitation)25 b(is)30 b(due)g(to)g(our)g(metho)s(d)f(of)h(pro)s(of:)41 b(w)m(e)p eop %%Page: 15 15 15 14 bop 3585 48 a Fu(15)118 366 y Fz(are)27 b(able)g(to)g(pro)m(v)m (e)i(that)e(large)f(protub)s(erances)j(of)e(the)h(in)m(terface)f(are)g (not)h(statistically)d(fa)m(v)m(or-)118 483 y(able)35 b(and)h(therefore)g(can)g(b)s(e)g(c)m(hopp)s(ed.)54 b(Ho)m(w)m(ev)m (er,)39 b(the)d(v)m(olume)f(constrain)m(t)h Fx(V)3144 498 y Fq(N)s(;a)3306 483 y Fz(prev)m(en)m(ts)118 599 y(us)i(to)f(con)m(trol)f(the)i(p)s(ercolation)d(of)i(the)h Fs(\000)g Fz(phase)g(inside)e(the)i(+)f(phase)h(b)s(ecause)h(erasing)e (a)118 715 y(\014lamen)m(t)32 b(of)g Fs(\000)h Fz(blo)s(c)m(ks)g(migh)m (t)e(b)s(e)i(in)e(con\015ict)i(with)f(the)h(v)m(olume)f(constrain)m(t.) 118 933 y(3.2.)49 b FE(Sp)s(ectral)35 b(gap.)50 b Fz(W)-8 b(e)31 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1691 y Fp(\000)2275 1772 y Fs(j)p Ft(f)11 b Fz(\()p Ft(\033)2459 1731 y Fq(x)2502 1772 y Fz(\))22 b Fs(\000)h Ft(f)11 b Fz(\()p Ft(\033)t Fz(\))p Fs(j)2884 1731 y Fw(2)2922 1691 y Fp(\001)2985 1772 y Ft(;)118 2063 y Fz(where)30 b Ft(\033)455 2027 y Fq(x)528 2063 y Fz(is)f(the)h(spin)f (con\014guration)f(deduced)j(from)d Ft(\033)j Fs(2)d(f\006)p Fz(1)p Fs(g)2587 2027 y Fn(D)2629 2038 y Fm(N)2726 2063 y Fz(b)m(y)i(\015ipping)e(the)i(spin)f(at)118 2179 y(site)j Ft(x)p Fz(.)44 b(The)33 b(reader)f(is)g(referred)h(to)e(the)i(lecture)f (notes)h(b)m(y)g(Martinelli)c([Ma])j(and)g(Guionnet,)118 2295 y(Zegarlinski)23 b([GZ)o(])i(for)f(a)h(precise)h(de\014nition)e (and)h(related)f(results)i(on)e(the)i(Glaub)s(er)d(dynamics.)118 2412 y(In)37 b(the)f(phase)h(transition)e(regime,)h(the)g(t)m(w)m(o)h (phases)h(segregate)f(and)f(the)h(relaxation)d(of)i(the)118 2528 y(system)e(is)e(related)g(to)g(the)h(slo)m(w)g(motion)d(of)i(the)h (in)m(terfaces.)218 2644 y(A)48 b(con)m(v)m(enien)m(t)i(parameter)e(to) 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b(:)g(:)g(:)f(;)h(h)1595 4367 y Fw(2)p Fq(d)1671 4352 y Fz(\))35 b(with)f(non)h(negativ)m(e)g (comp)s(onen)m(ts,)h(w)m(e)g(in)m(tro)s(duce)118 4469 y(the)d(functional)1218 4720 y Fs(G)1283 4679 y Fk(h)1333 4720 y Fz(\()p Ft(m)p Fz(\))27 b(=)h Fs(F)1707 4679 y Fk(h)1756 4720 y Fz(\()p Ft(m)p Fz(\))22 b Fs(\000)2075 4596 y Fw(2)p Fq(d)2039 4625 y Fp(X)2054 4835 y Fq(i)p Fw(=1)2199 4720 y Fz(\001)2280 4735 y Fq(h)2321 4745 y Fm(i)2368 4720 y Ft(;)1017 b Fz(\(3.26\))118 4999 y(where)34 b Fs(F)482 4962 y Fk(h)563 4999 y Fz(w)m(as)g(in)m(tro)s(duced)f(in)e (\(2.11\).)43 b(W)-8 b(e)33 b(get)118 5146 y FE(Theorem)38 b(3.5.)k Fj(F)-7 b(or)34 b(any)h Ft(\014)e Fs(2)28 b(B)s Fj(,)35 b(the)g(fol)5 b(lowing)34 b(asymptotic)g(hold)727 5368 y Fz(lim)17 b(sup)774 5449 y Fq(N)7 b Fo(!1)1154 5301 y Fz(1)p 1069 5345 V 1069 5436 a Ft(N)1157 5408 y Fq(d)p Fo(\000)p Fw(1)1342 5368 y Fz(log)17 b(SG\()p Ft(N)5 b(;)17 b FE(h)p Fz(\))27 b Fr(6)51 b Fs(\000)159 b Fz(sup)2135 5454 y Fq(m)p Fo(2)p Fw(])p Fo(\000)p Fq(m)2381 5435 y Fh(\003)2418 5454 y Fq(;m)2500 5435 y Fh(\003)2536 5454 y Fw([)2572 5368 y Fs(G)2637 5327 y Fq(h)2682 5368 y Fz(\()p Ft(m)p Fz(\))17 b Ft(:)525 b Fz(\(3.27\))p eop %%Page: 16 16 16 15 bop 118 51 a Fu(16)1148 b(T.)26 b(BODINEA)n(U)e(AND)g(D.)i(IOFFE) 118 366 y FE(Remark)39 b(3.1.)j Fj(In)35 b Fz([Ma])g Fj(only)h(the)f(fr)-5 b(e)g(e)35 b(b)-5 b(oundary)35 b(c)-5 b(onditions)35 b(have)f(b)-5 b(e)g(en)35 b(c)-5 b(onsider)g(e)g(d,)34 b(but,)118 483 y(in)39 b(view)g(of)g(the)g(r)-5 b(esults)39 b(of)g Fz([PV)q(])p Fj(,)h(the)g(pr)-5 b(o)g(of)38 b(p)-5 b(ertains)39 b(to)g(the)h(c)-5 b(ase)38 b(of)h(b)-5 b(oundary)39 b(magnetic)118 599 y(\014elds)e(in)g(the)g(p)-5 b(artial)5 b(ly)38 b(wetting)f(r)-5 b(e)g(gime.)51 b(In)37 b(dimension)f(2,)i(Pisztor)-5 b(a's)37 b(c)-5 b(o)g(arse)36 b(gr)-5 b(aining)37 b(is)118 715 y(not)44 b(valid,)i(but)f(an)e (alternative)h(c)-5 b(o)g(arse)43 b(gr)-5 b(aining)44 b(for)f(which)h(the)g Fx(L)2764 679 y Fw(1)2810 715 y Fj(-appr)-5 b(o)g(ach)43 b(holds)g(has)118 831 y(b)-5 b(e)g(en)37 b(devise)-5 b(d)36 b(in)h Fz([BoMa])p Fj(.)52 b(Notic)-5 b(e)37 b(that)h(in)f(2)g(dimensions)e(the)i(r)-5 b(elaxation)37 b(pr)-5 b(op)g(erty)37 b FE(\(RP\))118 948 y Fj(is)j(known)g(to)h(b)-5 b(e)40 b(valid)g(up)g(to)h(the)g (critic)-5 b(al)40 b(temp)-5 b(er)g(atur)g(e.)62 b(Ther)-5 b(efor)g(e)39 b(the)i(statement)f(holds)118 1064 y(also)34 b(in)h(dimension)e(2)i(for)g(any)f Ft(\014)g(>)27 b(\014)1579 1079 y Fq(c)1614 1064 y Fj(.)218 1213 y Fz(The)32 b(functional)e Fs(G)939 1177 y Fk(h)1020 1213 y Fz(should)h(b)s(e)g(in)m(terpreted)h (as)g(an)f(energy)i(landscap)s(e)e(parametrized)f(b)m(y)118 1329 y(the)37 b(a)m(v)m(eraged)i(magnetization.)54 b(The)38 b(time)d(for)i(a)f(con\014guration)g(starting)g(in)h(the)g Fs(\000)g Fz(phase)118 1445 y(to)g(relax)h(to)f(the)h(+)g(phase)h(pro)m (vides)f(an)g(estimation)d(of)j(the)g(sp)s(ectral)f(gap.)59 b(This)38 b(explains)118 1562 y(wh)m(y)e(the)e(suprem)m(um)h(is)e(tak)m (en)i(o)m(v)m(er)h(the)e(v)-5 b(alues)34 b(of)g Ft(m)g Fz(in)g(])23 b Fs(\000)h Ft(m)2594 1525 y Fo(\003)2634 1562 y Ft(;)17 b(m)2763 1525 y Fo(\003)2802 1562 y Fz([.)48 b(The)36 b(suprem)m(um)e(of)118 1678 y Fs(G)183 1642 y Fk(h)270 1678 y Fz(is)k(related)f(to)g(the)h(energy)h(of)e(the)h(b)s (ottlenec)m(k)h(and)f(Theorem)g(3.5)f(asserts)i(that)f(if)e(it)h(is)118 1794 y(p)s(ositiv)m(e)i(then)g(the)h(system)g(has)f(a)g(metastable)f(b) s(eha)m(vior)h(and)g(ev)m(olv)m(es)i(extremely)e(slo)m(wly)-8 b(.)118 1910 y(In)37 b(this)f(case)h(the)f(system)i(has)e(time)f(to)h (equilibrate)f(and)h(equilibrium)d(parameters)j(should)118 2027 y(b)s(e)d(relev)-5 b(an)m(t)32 b(as)g(w)m(ell)f(to)h(describ)s(e)h (the)g(in\015uence)g(of)f(the)g(b)s(oundary)h(\014eld)f(on)g(the)g (dynamics.)118 2143 y(W)-8 b(e)34 b(exp)s(ect)h(that)f(inequalit)m(y)g (\(3.27\))e(is,)i(in)f(fact,)h(an)f(equalit)m(y)-8 b(.)47 b(F)-8 b(or)33 b(appropriate)f(c)m(hoices)j(of)118 2259 y FE(h)p Fz(,)30 b(an)e(explicit)f(upp)s(er)i(b)s(ound)f(can)h(b)s(e)g 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b(b)s(eha)m(vior)118 3022 y(of)42 b(the)h(dynamics)f(is)g(v)m(ery)i(sensitiv)m(e)f(to)f(the) g(b)s(oundary)h(conditions.)72 b(In)43 b(dimension)e(t)m(w)m(o,)118 3138 y(Martinelli)32 b(sho)m(w)m(ed)k(that)f(for)f(free)h(b)s(oundary)h (conditions)d(or,)i(equiv)-5 b(alen)m(tly)d(,)35 b(in)f(the)h(case)h (of)118 3254 y(zero)41 b(b)s(oundary)g(magnetic)e(\014elds)i FE(h)g Fs(\021)h Fz(0,)g(log)16 b(SG\()p Ft(N)5 b(;)17 b FE(h)p Fz(\))40 b(scales)h(lik)m(e)f Fs(\000)p Ft(N)10 b(\034)h Fz(\()l Ft(~)-45 b(e)3148 3269 y Fq(d)3190 3254 y Fz(\).)67 b(Instead,)118 3371 y(when)39 b(at)e(least)h(one)g(side)g (of)f(the)h(square)h(has)f(all)e(+)h(b)s(oundary)h(conditions)f(and)h (the)g(other)118 3487 y(sides)i(free)g(b)s(oundary)h(conditions)d (\(i.e.)65 b FE(h)40 b Fz(=)g(\(0)p Ft(;)17 b(:)g(:)g(:)e(;)i Fz(0)p Ft(;)g Fz(1\)\),)40 b(then)h(for)e(an)m(y)h Ft(")g(>)g Fz(0)f(and)h Ft(N)118 3603 y Fz(large)32 b(enough)956 3787 y(log)17 b(SG\()p Ft(N)5 b(;)17 b(h)p Fz(\))27 b Fr(>)51 b Fs(\000)22 b Ft(N)1831 3746 y Fw(1)p Fq(=)p Fw(2+)p Fq(")2029 3787 y Fz(\()p Fs(\035)27 b(\000)p Ft(N)10 b(\034)h Fz(\()l Ft(~)-45 b(e)2495 3802 y Fq(d)2538 3787 y Fz(\)\))17 b Ft(:)118 3969 y Fz(In)26 b(this)g(case)h(the)f(sp)s (ectral)g(gap)f(is)h(conjectured)h(to)f(deca)m(y)h(p)s(olynomially)22 b(fast.)41 b(An)26 b(app)s(ealing)118 4086 y(in)m(terpretation)35 b(of)h(this)g(result)g(w)m(ould)g(b)s(e)h(to)f(relate)f(the)i(dynamics) f(of)g(the)g(Ising)g(mo)s(del)f(to)118 4202 y(an)k(e\013ectiv)m(e)h(mo) s(del)d(ev)m(olving)i(in)f(a)h(one)g(w)m(ell)f(p)s(oten)m(tial)g(\(+)h (b)s(c\))g(or)f(a)h(t)m(w)m(o)h(w)m(ell)e(p)s(oten)m(tial)118 4318 y(\(free)c(b)s(c\):)45 b(the)34 b(p)s(ositiv)m(e)f(magnetic)g (\014eld)g FE(h)h Fz(enforces)g(a)f(unique)h(ground)g(state)g(whereas)h (the)118 4434 y(transitions)c(b)s(et)m(w)m(een)j(t)m(w)m(o)e(symmetric) f(w)m(ells)h(on)f(the)i(energy)g(landscap)s(e)f(in)f(the)h(case)h(of)e (the)118 4550 y(free)g(b)s(oundary)g(conditions)f(ha)m(v)m(e)i(to)f 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(arbitrary)118 5480 y(small)26 b(densit)m(y)k(of)e(spins.)42 b(Roughly)28 b(sp)s(eaking,)h(they)h(sho)m(w)m(ed)g(that)e(in)g(t)m(w)m (o)h(dimensions)f(if)f(the)p eop %%Page: 17 17 17 16 bop 3585 48 a Fu(17)118 366 y Fz(b)s(oundary)37 b(conditions)f(ha)m(v)m(e)i(an)f(a)m(v)m(erage)h(magnetization)c(less)j (than)g(1,)g(there)h(exists)g(some)118 483 y(in)m(v)m(erse)25 b(temp)s(erature)e Ft(\014)1033 498 y Fw(0)1096 483 y Fz(large)g(enough)g(ab)s(o)m(v)m(e)i(whic)m(h)f(the)g(dynamics)f (exhibits)g(a)g(metastable)118 599 y(phase.)43 b(Our)29 b(result)g(w)m(as)g(originally)c(motiv)-5 b(ated)27 b(b)m(y)j([A)-8 b(Y];)30 b(the)f(magnetic)f(\014eld)g Ft(h)g(<)g Fz(1)g(can)h(b)s(e)118 715 y(in)m(terpreted)39 b(as)f(an)g(e\013ectiv)m(e)h(b)s(oundary)g (condition)d(after)i(a)m(v)m(eraging)g(the)g(spins.)61 b(F)-8 b(or)37 b(sim-)118 831 y(plicit)m(y)-8 b(,)29 b(let)g(us)h(fo)s(cus)g(on)g(the)g(case)h FE(h)d Fz(=)f(\(0)p Ft(;)17 b(:)g(:)g(:)f(;)h Fz(0)p Ft(;)g(h)2090 846 y Fq(d)2130 831 y Ft(;)g Fz(0)p Ft(;)g(:)g(:)g(:)e(;)i Fz(0\).)42 b(Extrap)s(olating)28 b(the)i(results)118 948 y(of)g([A)-8 b(Y])30 b(to)f(this)h(setting,)g(one)h(can)f(state)g (that)g(in)f(t)m(w)m(o)i(dimensions)e(and)h(for)f(an)m(y)i Ft(h)3257 963 y Fq(d)3328 948 y Fz(in)e([0)p Ft(;)17 b Fz(1[,)118 1064 y(there)43 b(exists)g Ft(\014)k Fz(large)41 b(enough)i(suc)m(h)g(that)f(the)h(sp)s(ectral)f(gap)f(deca)m(ys)j(exp)s (onen)m(tially)d(fast.)118 1180 y(Theorems)34 b(3.4)e(and)h(3.5)f(will) e(enable)j(us)g(to)g(in)m(terpret)f(these)i(results)g(in)d(a)i(more)f (qualitativ)m(e)118 1296 y(w)m(a)m(y)-8 b(.)45 b(In)33 b(order)f(to)h(do)f(so,)h(w)m(e)h(\014rst)f(recall)e(some)h(statemen)m (ts)i(on)e(the)h(w)m(etting)g(transition.)218 1413 y(It)i(w)m(as)i (deriv)m(ed)f(in)f([FP2])g(that)h(\001)1542 1428 y Fq(h)1583 1440 y Fm(d)1656 1413 y Fs(2)d Fz([)p Fs(\000)p Ft(\034)11 b Fz(\()l Ft(~)-45 b(e)1995 1428 y Fq(d)2037 1413 y Fz(\))p Ft(;)17 b(\034)11 b Fz(\()l Ft(~)-45 b(e)2255 1428 y Fq(d)2295 1413 y Fz(\)].)53 b(In)36 b(fact)f(there)h(exists)h(a)e (critical)118 1529 y(v)-5 b(alue)40 b Ft(h)434 1544 y Fq(c)511 1529 y Fr(6)i Fz(1)e(suc)m(h)j(that)d(if)g Ft(h)1321 1544 y Fq(d)1403 1529 y Ft(<)h(h)1576 1544 y Fq(c)1652 1529 y Fz(then)g(\001)1963 1544 y Fq(h)2004 1556 y Fm(d)2086 1529 y Ft(<)h(\034)11 b Fz(\()l Ft(~)-45 b(e)2340 1544 y Fq(d)2381 1529 y Fz(\).)68 b(The)41 b(critical)e(v)-5 b(alue)40 b Ft(h)3372 1544 y Fq(c)3447 1529 y Fz(c)m(har-)118 1645 y(acterizes)j(the)f(in\015uence)h(of)e(the)i(b)s(oundary)f (\014eld)g(on)g(the)g(thermo)s(dynamic)f(prop)s(erties)h(of)118 1761 y(the)c(Gibbs)g(measure.)60 b(More)38 b(precisely)-8 b(,)39 b(one)f(should)g(also)f(distinguish)g(the)h(partial)e(drying)118 1877 y(regime)d(\(0)d Fr(6)h Ft(h)715 1892 y Fq(d)786 1877 y Ft(<)f(h)948 1892 y Fq(c)983 1877 y Fz(\))k(from)f(the)h (partial)e(w)m(etting)i(regime)f(\(0)d Fr(>)h Ft(h)2720 1892 y Fq(d)2791 1877 y Ft(>)f Fs(\000)p Ft(h)3030 1892 y Fq(c)3066 1877 y Fz(\).)48 b(W)-8 b(e)34 b(refer)h(to)118 1994 y(P\014ster,)f(V)-8 b(elenik)32 b([PV)q(])g(or)h(to)f([BIV1])h (for)f(further)h(discussions)g(on)f(the)h(equilibrium)d(issues.)218 2110 y(The)g(previous)f(Theorem)g(sho)m(ws)h(that)f Ft(h)1747 2125 y Fq(c)1811 2110 y Fz(is)f(also)g(related)g(to)h(the)g(metastable) f(b)s(eha)m(vior)h(of)118 2226 y(the)f(system)g(and)f(th)m(us)h(it)e (also)g(pla)m(ys)h(the)h(role)e(of)h(a)f(critical)f(v)-5 b(alue)27 b(in)f(the)h(dynamical)f(setting.)118 2342 y(Nev)m(ertheless,)43 b(for)38 b(an)m(y)h Ft(h)1116 2357 y Fq(d)1195 2342 y Ft(>)f Fz(0,)i(the)f(Gibbs)f(measure)h(is)f(unique)h (in)f(the)h(thermo)s(dynamic)118 2459 y(limit.)e(This)22 b(con\014rms)g(the)g(fact)g(that)g(the)g(dynamical)f(prop)s(erties)h (cannot)g(b)s(e)g(deduced)i(simply)118 2575 y(from)31 b(the)h(bulk)g(prop)s(erties,)g(but)h(that)f(the)g(metastabilit)m(y)e (is)h(related)h(to)g(surface)g(prop)s(erties)118 2691 y(\(the)j(picture)g(of)f(the)h(e\013ectiv)m(e)g(magnetization)d(ev)m (olving)i(in)g(a)g(one)h(w)m(ell)f(p)s(oten)m(tial)f(w)m(as)i(to)s(o) 118 2807 y(simplistic\).)616 3177 y(4.)49 b Fy(Ha)n(usdorff)38 b(localiza)-7 b(tion:)51 b(Pr)n(oof)38 b(of)h(Theorem)e(3.2)118 3351 y Fz(4.1.)49 b FE(FK)33 b(phase)h(lab)s(els.)50 b Fz(W)-8 b(e)30 b(use)f(the)g(notation)f(in)m(tro)s(duced)h(in)f (Subsections)i(2.2)e(and)h(2.3.)118 3467 y(De\014ne)k(the)g(follo)m (wing)d(dep)s(enden)m(t)35 b(p)s(ercolation)30 b(pro)s(cess)k(on)f Fx(D)2520 3482 y Fq(N)s(;K)2673 3467 y Fz(:)118 3844 y Ft(X)207 3803 y Fw(FK)199 3869 y Fq(N)s(;K)346 3844 y Fz(\()p Ft(i)p Fz(\))50 b(=)630 3610 y Fp(8)630 3700 y(>)630 3730 y(<)630 3909 y(>)630 3939 y(:)813 3686 y Fz(1)p Ft(;)81 b Fz(if)38 b Fx(B)1127 3701 y Fw(2)p Fq(K)1237 3686 y Fz(\()p Ft(i)p Fz(\))32 b(is)g(FK)g(go)s(o)s(d)g(and)g Ft(C)2153 3645 y Fo(\003)2193 3686 y Fz(\()p Fx(B)2291 3701 y Fw(2)p Fq(K)2401 3686 y Fz(\()p Ft(i)p Fz(\)\))g(is)g(connected) j(to)d Ft(@)3309 3645 y Fw(ext)3410 3686 y Fx(D)3476 3701 y Fq(N)735 3844 y Fs(\000)q Fz(1)p Ft(;)81 b Fz(if)38 b Fx(B)1127 3859 y Fw(2)p Fq(K)1237 3844 y Fz(\()p Ft(i)p Fz(\))32 b(is)g(FK)g(go)s(o)s(d)g(and)g Ft(C)2153 3803 y Fo(\003)2193 3844 y Fz(\()p Fx(B)2291 3859 y Fw(2)p Fq(K)2401 3844 y Fz(\()p Ft(i)p Fz(\)\))g(is)g(not)h(connected)h(to)e Ft(@)3482 3803 y Fw(ext)3583 3844 y Fx(D)3649 3859 y Fq(N)813 3995 y Fz(0)p Ft(;)81 b Fz(if)38 b Fx(B)1127 4010 y Fw(2)p Fq(K)1237 3995 y Fz(\()p Ft(i)p Fz(\))32 b(is)g(FK)g(bad)3412 4118 y(\(4.28\))118 4234 y(W)-8 b(e)33 b(recall)e(the)i(c)m(hoice)g Ft(K)i Fz(=)28 b Ft(N)1314 4198 y Fq(a)1388 4234 y Fz(for)k(some)h Ft(a)28 b Fs(2)p Fz(]0)p Ft(;)17 b Fz(1)p Ft(=d)p Fz([.)218 4466 y(Exactly)33 b(as)g(in)e([BIV1)q(])h(the)h(stabilit)m(y)e(assumption)h (implies:)118 4633 y FE(Lemma)39 b(4.1.)i Fj(F)-7 b(or)34 b(every)g Ft(\013)29 b(>)e Fz(0)35 b Fj(ther)-5 b(e)34 b(exists)g(a)h(p)-5 b(ositive)33 b(c)-5 b(onstant)35 b Ft(c)2821 4648 y Fw(4)2888 4633 y Fz(=)27 b Ft(c)3033 4648 y Fw(4)3073 4633 y Fz(\()p Ft(\013)q Fz(\))p Fj(,)34 b(such)g(that)444 4971 y Fz(\010)514 4930 y Fo(\006)514 4998 y Fq(N)599 4771 y Fp(0)599 4950 y(@)737 4876 y(X)686 5088 y Fq(i)p Fo(2)p Fn(D)799 5099 y Fm(N)r(;K)949 4971 y Ft(X)1038 4930 y Fw(FK)1030 4996 y Fq(N)s(;K)1177 4971 y Fz(\()p Ft(i)p Fz(\))28 b Fr(6)g Fz(\(1)22 b Fs(\000)h Ft(\013)q Fz(\))1746 4831 y Fp(\022)1829 4904 y Ft(N)p 1828 4948 91 4 v 1828 5039 a(K)1928 4831 y Fp(\023)2002 4853 y Fq(d)2042 4771 y Fp(1)2042 4950 y(A)2209 4971 y Fr(6)44 b Fz(exp)2496 4831 y Fp(\022)2569 4971 y Fs(\000)p Ft(c)2688 4986 y Fw(4)2739 4904 y Ft(N)2827 4867 y Fq(d)p Fo(\000)p Fw(1)p 2738 4948 221 4 v 2738 5039 a Ft(K)2828 5011 y Fq(d)p Fo(\000)p Fw(1)2969 4831 y Fp(\023)3059 4971 y Ft(;)326 b Fz(\(4.29\))118 5304 y Fj(for)35 b(al)5 b(l)34 b Ft(N)46 b Fj(lar)-5 b(ge)34 b(enough.)218 5479 y Fz(Recall)d(that)h(the)h(total)e(n)m(um)m(b)s(er)i(of)f(mesoscopic)h (b)s(o)m(xes)h(in)d Fx(D)2538 5494 y Fq(N)2644 5479 y Fz(is)2753 5440 y Fq(N)2816 5416 y Fm(d)p 2752 5456 101 4 v 2752 5514 a Fq(K)2817 5495 y Fm(d)2863 5479 y Fz(.)p eop %%Page: 18 18 18 17 bop 118 51 a Fu(18)1148 b(T.)26 b(BODINEA)n(U)e(AND)g(D.)i(IOFFE) 118 366 y Fz(4.2.)49 b FE(Logic)i(of)g(the)g(pro)s(of.)f Fz(Bac)m(k)45 b(to)f(Theorem)h(3.2)f(w)m(e)i(shall)d(use)i(Lemma)e(4.1) h(in)g(the)118 483 y(regime)31 b(0)d Ft(<)f(\013)i Fs(\034)e Ft(\027)34 b Fs(\034)28 b Fz(1.)43 b(W)-8 b(e)33 b(argue)f(that)h(on)f (the)h(ev)m(en)m(t)849 586 y Fp(8)849 676 y(<)849 855 y(:)938 790 y Ft(\030)f Fs(2)c Fz(\004)1172 805 y Fq(N)s(;)p Fo(\006)1370 790 y Fz(:)1509 696 y Fp(X)1457 907 y Fq(i)p Fo(2)p Fn(D)1570 918 y Fm(N)r(;K)1721 790 y Fz(1)-25 b(I)1780 813 y Fo(f)p Fq(X)1878 790 y Fl(FK)1873 836 y Fm(N)r(;K)2001 813 y Fw(\()p Fq(i)p Fw(\))p Fo(6)p Fw(=1)p Fo(g)2209 790 y Fz(\()p Ft(\030)5 b Fz(\))27 b Fr(6)i Ft(\013)2545 650 y Fp(\022)2629 723 y Ft(N)p 2628 767 91 4 v 2628 859 a(K)2728 650 y Fp(\023)2802 672 y Fq(d)2842 586 y Fp(9)2842 676 y(=)2842 855 y(;)3412 790 y Fz(\(4.30\))118 1098 y Ft(\027)6 b(N)k Fz(-long)27 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b(pro)s(of)g(of)g(Theo-)118 4857 y(rem)d(3.2)g(relies)g(on) h(the)g(follo)m(wing)d(uniform)g(upp)s(er)j(b)s(ound:)118 5011 y FE(Theorem)38 b(4.1.)k Fj(Ther)-5 b(e)34 b(exists)h Ft(c)1398 5026 y Fw(5)1465 5011 y Fz(=)27 b Ft(c)1610 5026 y Fw(5)1650 5011 y Fz(\()p Ft(\014)6 b Fz(\))27 b Ft(>)g Fz(0)35 b Fj(such)g(that)1343 5234 y Fz(\010)1413 5193 y Fo(\006)1413 5261 y Fq(N)1498 5124 y Fp(\020)1557 5234 y Fs(F)10 b Fz([)1680 5208 y Ft(~)1666 5234 y(f)1714 5249 y Fq(N)s(;K)1861 5234 y Fz(])1888 5124 y Fp(\021)2027 5234 y Fr(6)63 b Fz(e)2210 5193 y Fo(\000)p Fq(c)2296 5202 y Fl(5)2330 5193 y Fq(\027)t(N)3412 5234 y Fz(\(4.32\))118 5480 y Fj(uniformly)35 b(in)f(strictly)i(p)-5 b(ositive)34 b(\014nger)g(lab)-5 b(els)1909 5454 y Ft(~)1895 5480 y(f)1943 5495 y Fq(N)s(;K)2125 5480 y Fj(and)35 b(in)f Ft(N)46 b Fj(su\016ciently)34 b(lar)-5 b(ge.)p eop %%Page: 19 19 19 18 bop 3585 48 a Fu(19)218 366 y Fz(Since)22 b(the)g(total)f(n)m(um) m(b)s(er)h(of)g(di\013eren)m(t)g(\014nger)h(lab)s(els)d(\(recall)h(the) h(c)m(hoice)h(of)e(scales)i Ft(K)35 b Fz(=)27 b Ft(N)3581 330 y Fq(a)3623 366 y Fz(\))118 483 y(is)32 b(b)s(ounded)i(ab)s(o)m(v)m (e)f(as)902 560 y Fp( )981 590 y(\022)1064 663 y Fz(2)p Ft(N)p 1064 707 138 4 v 1088 799 a(K)1211 590 y Fp(\023)1285 612 y Fw(\()p Fq(d)p Fo(\000)p Fw(1\))1470 560 y Fp(!)1549 582 y Fq(R)1684 730 y Fr(6)60 b Fz(exp)1987 650 y Fp(\010)2045 730 y Ft(c)2087 745 y Fw(5)2127 730 y Fz(\()p Ft(d)p Fz(\))p Ft(\027)6 b(N)2396 689 y Fw(1)p Fo(\000)p Fq(a)2545 730 y Fz(log)16 b Ft(N)2775 650 y Fp(\011)2850 730 y Ft(;)535 b Fz(\(4.33\))118 978 y(Theorem)33 b(3.2)f(instan)m(tly)g (follo)m(ws.)118 1163 y(4.5.)49 b FE(Splitting)39 b(of)h Fs(L)980 1178 y Fq(N)s(;)p Fo(\006)1157 1163 y FE(with)f(resp)s(ect)g (to)g Ft(l)r FE(-th)h(mesoscopic)g(la)m(y)m(er.)50 b Fz(Giv)m(en)34 b(a)g(meso-)118 1279 y(scopic)f(la)m(y)m(er)g Ft(l)d Fz(=)d(1)p Ft(;)17 b Fz(2)p Ft(;)g(:)g(:)g(:)e(;)i(R)33 b Fz(de\014ne)h(the)f(follo)m(wing)d(mesoscopic)j(sets:)1090 1484 y Fx(H)1161 1437 y Fo(\000)p Fq(;l)1161 1511 y(N)s(;K)1367 1428 y Fw(\001)1359 1484 y Fz(=)44 b Fs(f)p Ft(i)28 b Fs(2)g Fx(D)1750 1499 y Fq(N)s(;K)1947 1484 y Fz(:)44 b Ft(i)2051 1499 y Fq(d)2120 1484 y Fr(6)28 b Fz(\()p Ft(l)c Fs(\000)2426 1416 y Fz(1)p 2426 1461 49 4 v 2426 1552 a(2)2485 1484 y(\))p Ft(K)7 b Fs(g)p Ft(;)1090 1721 y Fx(H)1161 1680 y Fq(l)1161 1746 y(N)s(;K)1367 1665 y Fw(\001)1359 1721 y Fz(=)44 b Fs(f)p Ft(i)28 b Fs(2)g Fx(D)1750 1736 y Fq(N)s(;K)1947 1721 y Fz(:)44 b Ft(i)2051 1736 y Fq(d)2120 1721 y Fz(=)27 b(\()p Ft(l)e Fs(\000)2424 1654 y Fz(1)p 2424 1698 V 2424 1790 a(2)2483 1721 y(\))p Ft(K)7 b Fs(g)p Ft(;)1090 1958 y Fx(H)1161 1911 y Fw(+)p Fq(;l)1161 1985 y(N)s(;K)1367 1903 y Fw(\001)1359 1958 y Fz(=)44 b Fs(f)p Ft(i)28 b Fs(2)g Fx(D)1750 1973 y Fq(N)s(;K)1947 1958 y Fz(:)44 b Ft(i)2051 1973 y Fq(d)2120 1958 y Ft(>)27 b Fz(\()p Ft(l)e Fs(\000)2424 1891 y Fz(1)p 2424 1936 V 2424 2027 a(2)2483 1958 y(\))p Ft(K)7 b Fs(g)p Ft(:)118 2174 y Fz(Their)27 b(microscopic)f(coun)m(terparts)j(are)e (denoted)h(b)m(y)g Fs(H)2196 2127 y Fo(\000)p Fq(;l)2195 2201 y(N)s(;K)2342 2174 y Ft(;)17 b Fs(H)2471 2138 y Fq(l)2470 2199 y(N)s(;K)2644 2174 y Fz(and,)29 b(resp)s(ectiv)m(ely)-8 b(,)29 b Fs(H)3488 2127 y Fw(+)p Fq(;l)3487 2201 y(N)s(;K)3634 2174 y Fz(,)118 2292 y(where)347 2454 y Fs(H)432 2406 y Fo(\000)p Fq(;l)431 2481 y(N)s(;K)606 2454 y Fz(=)780 2359 y Fp([)710 2592 y Fq(i)p Fo(2)p Fn(H)827 2560 y Fh(\000)p Fm(;l)827 2615 y(N)r(;K)977 2454 y Fx(B)1037 2469 y Fw(2)q Fq(K)1147 2454 y Fz(\()p Ft(i)p Fz(\))p Ft(;)82 b Fs(H)1450 2413 y Fq(l)1449 2478 y(N)s(;K)1624 2454 y Fz(=)1797 2359 y Fp([)1727 2583 y Fq(i)p Fo(2)p Fn(H)1844 2560 y Fm(l)1844 2606 y(N)s(;K)1995 2454 y Fx(B)2055 2469 y Fw(2)p Fq(K)2165 2454 y Fz(\()p Ft(i)p Fz(\))32 b(and)h Fs(H)2581 2406 y Fw(+)p Fq(;l)2580 2481 y(N)s(;K)2755 2454 y Fz(=)2929 2359 y Fp([)2859 2592 y Fq(i)p Fo(2)p Fn(H)2975 2560 y Fl(+)p Fm(;l)2975 2615 y(N)s(;K)3126 2454 y Fx(B)3186 2469 y Fw(2)q Fq(K)3296 2454 y Fz(\()p Ft(i)p Fz(\))p Ft(:)118 2771 y Fz(Accordingly)-8 b(,)32 b(w)m(e)i(split)d(the)i(set)g(of)f(all)f(edges)i Fs(L)1936 2735 y Fq(d)1936 2797 y(N)s(;)p Fo(\006)2107 2771 y Fz(in)m(to)e(the)i(disjoin)m(t)f(union)1419 2968 y Fs(L)1488 2927 y Fq(d)1488 2993 y(N)s(;)p Fo(\006)1670 2968 y Fz(=)44 b Fs(E)1852 2921 y Fo(\000)p Fq(;l)1843 2995 y(N)s(;K)2006 2873 y Fp(_)2134 2968 y Fs(E)2196 2921 y Fw(+)p Fq(;l)2187 2995 y(N)s(;K)2333 2968 y Ft(;)118 3146 y Fz(where)756 3274 y Fs(E)818 3227 y Fo(\000)p Fq(;l)809 3301 y(N)s(;K)1009 3218 y Fw(\001)1000 3274 y Fz(=)1120 3164 y Fp(n)1187 3274 y Fz(\()p Ft(i)17 b(;)33 b(j)6 b Fz(\))44 b(:)g(either)33 b Ft(i)g Fz(or)f Ft(j)38 b Fz(b)s(elong)32 b(to)65 b Fs(H)2623 3227 y Fo(\000)p Fq(;l)2622 3301 y(N)s(;K)2791 3274 y Fs([)23 b Fi(g)2930 3164 y Fp(o)2996 3274 y Ft(;)118 3466 y Fz(and)33 b Fs(E)370 3419 y Fw(+)p Fq(;l)361 3493 y(N)s(;K)535 3466 y Fz(=)28 b Fs(L)708 3430 y Fq(d)708 3491 y(N)s(;)p Fo(\006)867 3466 y Fs(n)22 b(E)1001 3419 y Fo(\000)p Fq(;l)992 3493 y(N)s(;K)1139 3466 y Fz(.)218 3583 y(The)33 b(induced)g(notation)f(for) g(the)h(splitting)d(of)i(the)h(p)s(ercolation)e(con\014gurations)h Ft(\030)g Fs(2)d Fz(\004)3524 3598 y Fq(N)s(;)p Fo(\006)118 3702 y Fz(is)j Ft(\030)g Fz(=)c Ft(\030)443 3666 y Fq(l)438 3726 y Fo(\000)519 3702 y Fs(_)22 b Ft(\030)655 3666 y Fq(l)650 3726 y Fw(+)741 3702 y Fz(with)723 3899 y Ft(\030)771 3858 y Fq(l)766 3923 y Fo(\000)852 3899 y Fs(2)29 b Fz(\004)1012 3851 y Fo(\000)p Fq(;l)1012 3926 y(N)s(;K)1195 3843 y Fw(\001)1186 3899 y Fz(=)f Fs(f)p Fz(0)p Ft(;)17 b Fz(1)p Fs(g)1532 3855 y Fo(E)1575 3823 y Fh(\000)p Fm(;l)1569 3878 y(N)r(;K)1798 3899 y Fz(and)97 b Ft(\030)2100 3858 y Fq(l)2095 3923 y Fw(+)2182 3899 y Fs(2)28 b Fz(\004)2341 3851 y Fw(+)p Fq(;l)2341 3926 y(N)s(;K)2524 3843 y Fw(\001)2516 3899 y Fz(=)f Fs(f)p Fz(0)p Ft(;)17 b Fz(1)p Fs(g)2861 3855 y Fo(E)2904 3823 y Fl(+)p Fm(;l)2898 3878 y(N)r(;K)3029 3899 y Ft(:)118 4086 y Fz(Giv)m(en)23 b(a)f(p)s(ercolation)f(con\014guration)g Ft(\030)1587 4050 y Fq(l)1582 4111 y Fo(\000)1669 4086 y Fs(2)28 b Fz(\004)1828 4039 y Fo(\000)p Fq(;l)1828 4113 y(N)s(;K)1975 4086 y Fz(,)d(w)m(e)e(use)h Fs(C)2377 4039 y Fo(\000)p Fq(;l)2371 4113 y(N)s(;K)2518 4086 y Fz(\()p Fi(g)p Fz(\))f(to)f(denote)i(the)f(connected)118 4228 y(cluster)33 b(of)f(the)h(ghost)g(site)f Fi(g)h Fj(inside)e Fs(H)1601 4180 y Fo(\000)p Fq(;l)1600 4255 y(N)s(;K)1748 4228 y Fz(.)118 4442 y(4.6.)49 b FE(The)34 b(ev)m(en)m(t)f Fs(A)918 4457 y Fq(l)944 4442 y Fz(\()996 4415 y Ft(~)982 4442 y(f)1030 4457 y Fq(N)s(;K)1177 4442 y Fz(\))p FE(.)49 b Fz(Let)1480 4415 y Ft(~)1466 4442 y(f)1514 4457 y Fq(N)s(;K)1691 4442 y Fz(b)s(e)29 b(a)g(collection)f (of)h(strictly)f(p)s(ositiv)m(e)h(\014nger)h(lab)s(els.)118 4575 y(F)-8 b(or)32 b(eac)m(h)h(mesoscopic)g(la)m(y)m(er)g Ft(l)r Fz(,)g(w)m(e)g(de\014ne)h(the)f(asso)s(ciated)g(ev)m(en)m(t)h Fs(A)2734 4590 y Fq(l)2759 4575 y Fz(\()2811 4549 y Ft(~)2797 4575 y(f)2845 4590 y Fq(N)s(;K)2993 4575 y Fz(\))27 b Fs(\032)h Fz(\004)3228 4528 y Fo(\000)p Fq(;l)3228 4602 y(N)s(;K)3376 4575 y Fz(:)281 4889 y Fs(A)361 4904 y Fq(l)386 4889 y Fz(\()438 4862 y Ft(~)424 4889 y(f)472 4904 y Fq(N)s(;K)619 4889 y Fz(\))60 b(=)853 4685 y Fp(8)853 4774 y(<)853 4953 y(:)975 4889 y Ft(\030)1023 4848 y Fq(l)1018 4913 y Fo(\000)1104 4889 y Fs(2)28 b Fz(\004)1263 4842 y Fo(\000)p Fq(;l)1263 4916 y(N)s(;K)1443 4685 y Fp(\014)1443 4744 y(\014)1443 4804 y(\014)1443 4864 y(\014)1443 4924 y(\014)1443 4983 y(\014)1558 4762 y Fz(\()p Ft(i)p Fz(\))65 b Fi(g)28 b Fs(6$)f Ft(@)1993 4721 y Fw(ext)2095 4762 y Fx(D)2161 4777 y Fq(N)2266 4762 y FE(inside)34 b Fs(H)2666 4715 y Fo(\000)p Fq(;l)2665 4789 y(N)s(;K)1558 5019 y Fz(\()p Ft(ii)p Fz(\))f(#)p Fs(f)p Ft(@)1920 4978 y Fw(in)n(t)1915 5043 y Fq(t)2010 5019 y Fs(H)2095 4972 y Fo(\000)p Fq(;l)2094 5046 y(N)s(;K)2263 5019 y Fs(\\)23 b(C)2410 4972 y Fo(\000)p Fq(;l)2404 5046 y(N)s(;K)2551 5019 y Fz(\()p Fi(g)p Fz(\))p Fs(g)28 b Fr(6)g Ft(f)2919 4978 y Fq(l)2908 5043 y(N)s(;K)3055 5019 y Fz(\(2)p Ft(K)7 b Fz(\))3270 4978 y Fq(d)p Fo(\000)p Fw(1)3410 4685 y Fp(9)3410 4774 y(=)3410 4953 y(;)118 5182 y Fz(where)978 5307 y Ft(@)1034 5265 y Fw(in)n(t)1029 5331 y Fq(t)1124 5307 y Fs(H)1209 5259 y Fo(\000)p Fq(;l)1208 5334 y(N)s(;K)1391 5251 y Fw(\001)1383 5307 y Fz(=)27 b Fs(f)p Ft(u)g Fs(2)h(H)1798 5259 y Fo(\000)p Fq(;l)1797 5334 y(N)s(;K)1989 5307 y Fz(:)44 b Ft(u)2116 5322 y Fq(d)2184 5307 y Fz(=)28 b(\()p Ft(l)c Fz(+)e(1)p Ft(=)p Fz(2\))p Ft(K)7 b Fs(g)118 5474 y Fz(is)32 b(the)h(top)g(la)m(y)m(er)f(of)g(the)h(b)s(o)m(x)g Fs(H)1344 5426 y Fo(\000)p Fq(;l)1343 5501 y(N)s(;K)1491 5474 y Fz(.)p eop %%Page: 20 20 20 19 bop 118 51 a Fu(20)1148 b(T.)26 b(BODINEA)n(U)e(AND)g(D.)i(IOFFE) 218 381 y Fz(Notice)37 b(that)g(if)f Ft(\030)k Fz(=)c Ft(\030)1082 345 y Fq(l)1077 405 y Fo(\000)1161 381 y Fs(_)26 b Ft(\030)1301 345 y Fq(l)1296 405 y Fw(+)1390 381 y Fs(2)36 b(F)10 b Fz(\()1626 354 y Ft(~)1612 381 y(f)1660 396 y Fq(N)s(;K)1807 381 y Fz(\))37 b(then,)j(necessarily)-8 b(,)39 b Ft(\030)2697 345 y Fq(l)2692 405 y Fo(\000)2786 381 y Fs(2)d(A)2968 396 y Fq(l)2994 381 y Fz(\()3046 354 y Ft(~)3032 381 y(f)3080 396 y Fq(N)s(;K)3227 381 y Fz(\).)58 b(Indeed,)118 514 y(the)31 b(connected)g(cluster)g Fs(C)1107 467 y Fo(\000)p Fq(;l)1101 541 y(N)s(;K)1248 514 y Fz(\()p Fi(g)p Fz(\))f(is)g(capable)g(of)f(hitting)g(the)h(top)g (la)m(y)m(er)g Ft(@)2901 478 y Fw(in)n(t)2896 539 y Fq(t)2991 514 y Fs(H)3076 467 y Fo(\000)p Fq(;l)3075 541 y(N)s(;K)3252 514 y Fz(only)g(from)118 631 y(within)i(the)h(set)1673 657 y Fp([)1495 881 y Fq(i)p Fo(2)p Fn(F)1602 892 y Fm(N)r(;K)1735 881 y Fo(\\)p Fn(H)1827 858 y Fm(l)1827 904 y(N)s(;K)1978 752 y Fx(B)2038 767 y Fw(2)q Fq(K)2148 752 y Fz(\()p Ft(i)p Fz(\))p Ft(:)118 1081 y Fz(4.7.)49 b FE(Surgery)41 b(in)g Ft(l)r FE(-th)h(mesoscopic)g(la)m(y)m(er.)49 b Fz(Giv)m(en)36 b(a)f(p)s(ercolation)f(con\014guration)h Ft(\030)3508 1044 y Fq(l)3503 1105 y Fo(\000)3595 1081 y Fs(2)118 1214 y Fz(\004)183 1167 y Fo(\000)p Fq(;l)183 1241 y(N)s(;K)363 1214 y Fz(de\014ne)f(a)e(new)h(con\014guration)1527 1188 y(\026)1517 1214 y Ft(\030)1565 1178 y Fq(l)1560 1238 y Fo(\000)1651 1214 y Fz(as)410 1575 y(\026)400 1601 y Ft(\030)448 1560 y Fq(l)443 1626 y Fo(\000)502 1601 y Fz(\()p Ft(b)p Fz(\))44 b(=)784 1307 y Fp(8)784 1397 y(>)784 1427 y(>)784 1456 y(>)784 1486 y(<)784 1666 y(>)784 1696 y(>)784 1725 y(>)784 1755 y(:)889 1405 y Fz(0)p Ft(;)422 b Fz(if)32 b Ft(b)c Fz(=)f(\()p Ft(u;)17 b(v)t Fz(\))32 b(with)g Ft(u)27 b Fs(2)h Ft(@)2363 1364 y Fw(in)n(t)2358 1430 y Fq(t)2453 1405 y Fs(H)2538 1358 y Fo(\000)p Fq(;l)2537 1432 y(N)s(;K)2706 1405 y Fs(\\)23 b(C)2853 1358 y Fo(\000)p Fq(;l)2847 1432 y(N)s(;K)2994 1405 y Fz(\()p Fi(g)p Fz(\))1387 1552 y(and)33 b Ft(v)1624 1567 y Fq(d)1692 1552 y Fz(=)28 b Ft(u)1852 1567 y Fq(d)1914 1552 y Fz(+)22 b(1)p Ft(;)914 1801 y(\030)962 1760 y Fq(l)957 1826 y Fo(\000)1015 1801 y Fz(\()p Ft(b)p Fz(\))p Ft(;)228 b Fz(otherwise)q Ft(:)3412 1601 y Fz(\(4.34\))118 1981 y(In)39 b(other)g(w)m(ords,)i(the)e(surgery)g Ft(\030)1393 1945 y Fq(l)1388 2005 y Fo(\000)1485 1981 y Fs(7!)1633 1954 y Fz(\026)1622 1981 y Ft(\030)1670 1945 y Fq(l)1665 2005 y Fo(\000)1762 1981 y Fz(cuts)h(o\013)e(all)f(the)h(v)m(ertical)g (b)s(onds)h(whic)m(h)g(emerge)118 2114 y(from)d(those)h(p)s(oin)m(ts)g (of)f(the)i(top)e(la)m(y)m(er)h Ft(@)1671 2078 y Fw(in)n(t)1666 2139 y Fq(t)1761 2114 y Fs(H)1846 2067 y Fo(\000)p Fq(;l)1845 2141 y(N)s(;K)2029 2114 y Fz(whic)m(h)g(are)g Ft(\030)2527 2078 y Fq(l)2522 2139 y Fo(\000)2581 2114 y Fz(-connected)h(\(inside)e Fs(H)3477 2067 y Fo(\000)p Fq(;l)3476 2141 y(N)s(;K)3623 2114 y Fz(\))118 2231 y(to)c(the)h(ghost)g(site)f Fi(g)p Fz(.)218 2356 y(F)-8 b(or)41 b(ev)m(ery)i Ft(l)j Fz(=)d(1)p Ft(;)17 b Fz(2)p Ft(;)g(:)g(:)g(:)e(;)i(R)43 b Fz(and)f(for)f(eac)m(h) 1938 2330 y Ft(~)1925 2356 y(f)1973 2371 y Fq(N)s(;K)2161 2356 y Fz(the)i(map)e Ft(\030)2613 2320 y Fq(l)2608 2381 y Fo(\000)2710 2356 y Fs(7!)2863 2330 y Fz(\026)2853 2356 y Ft(\030)2901 2320 y Fq(l)2896 2381 y Fo(\000)2996 2356 y Fz(lea)m(v)m(es)i Fs(A)3365 2371 y Fq(l)3390 2356 y Fz(\()3442 2330 y Ft(~)3428 2356 y(f)3476 2371 y Fq(N)s(;K)3623 2356 y Fz(\))118 2497 y(in)m(v)-5 b(arian)m(t)31 b(and)i(is)f(at)g (most)g(2)1222 2454 y Fq(f)1263 2431 y Fm(l)1256 2477 y(N)r(;K)1384 2454 y Fw(\(2)p Fq(K)5 b Fw(\))1538 2431 y Fm(d)p Fh(\000)p Fl(1)1689 2497 y Fz(to)33 b(1)f(on)g(the)h(latter)f (set.)218 2613 y(F)-8 b(urthermore,)47 b(giv)m(en)1105 2587 y Ft(~)1091 2613 y(f)1139 2628 y Fq(N)s(;K)1287 2613 y Fz(,)g(a)e(mesoscopic)f(la)m(y)m(er)h Ft(l)i Fz(and)d(a)h(p)s (ercolation)d(con\014guration)118 2747 y Ft(\030)166 2711 y Fq(l)161 2772 y Fo(\000)247 2747 y Fs(2)29 b(A)422 2762 y Fq(l)447 2747 y Fz(\()499 2721 y Ft(~)485 2747 y(f)533 2762 y Fq(N)s(;K)680 2747 y Fz(\),)c(the)f(concatenation)e Ft(\030)32 b Fz(=)1731 2721 y(\026)1721 2747 y Ft(\030)1769 2711 y Fq(l)1764 2772 y Fo(\000)1825 2747 y Fs(_)s Ft(\021)27 b Fz(b)s(elongs)c(to)f Fi(J)2474 2762 y Fq(N)2569 2747 y Fz(=)28 b Fs(f)p Ft(\030)k Fs(2)c Fz(\004)2957 2762 y Fq(N)s(;)p Fo(\006)3139 2747 y Fz(:)44 b Fi(g)28 b Fs(6$)f Ft(@)5 b Fx(D)3538 2762 y Fq(N)3612 2747 y Fs(g)118 2884 y Fz(for)26 b(ev)m(ery)i Ft(\021)k Fs(2)c Fz(\004)751 2837 y Fw(+)p Fq(;l)751 2911 y(N)s(;K)898 2884 y Fz(.)41 b(In)27 b(particular,)f(giv)m(en)h(an)m(y)g(ev)m(en)m(t)i Fs(B)i(\032)d Fz(\004)2499 2837 y Fw(+)p Fq(;l)2499 2911 y(N)s(;K)2672 2884 y Fz(and)f(an)m(y)g Ft(\030)3082 2848 y Fq(l)3077 2909 y Fo(\000)3164 2884 y Fs(2)h(A)3338 2899 y Fq(l)3363 2884 y Fz(\()3415 2858 y Ft(~)3401 2884 y(f)3449 2899 y Fq(N)s(;K)3596 2884 y Fz(\),)1203 3064 y(\010)1273 3023 y Fo(\006)1273 3091 y Fq(N)1357 2984 y Fp(\000)1419 3064 y Fs(B)1505 2980 y Fp(\014)1505 3040 y(\014)1564 3038 y Fz(\026)1554 3064 y Ft(\030)1602 3023 y Fq(l)1597 3089 y Fo(\000)1656 2984 y Fp(\001)1763 3064 y Fr(6)61 b Fz(max)1970 3124 y Fq(\031)2099 3064 y Fz(\010)2169 3023 y Fq(\031)2169 3106 y Fo(E)2212 3074 y Fl(+)p Fm(;l)2206 3129 y(N)r(;K)2355 3064 y Fz(\()17 b Fs(B)j Fz(\))d Ft(;)835 b Fz(\(4.35\))118 3286 y(where)44 b(\010)480 3250 y Fq(\031)480 3333 y Fo(E)523 3300 y Fl(+)p Fm(;l)517 3356 y(N)r(;K)693 3286 y Fz(is)f(the)h(FK-measure)e(on)h(\004)1754 3239 y Fw(+)p Fq(;l)1754 3313 y(N)s(;K)1944 3286 y Fz(with)g(wired)g(b)s (oundary)h(conditions)e(on)h(the)118 3467 y(lateral)31 b(sides)j(and)g(on)f(the)h(top)f(of)g Fs(H)1524 3420 y Fw(+)p Fq(;l)1523 3494 y(N)s(;K)1703 3467 y Fz(and,)h(resp)s(ectiv)m (ely)-8 b(,)34 b(with)f Ft(\031)38 b Fz(b)s(oundary)33 b(conditions)118 3609 y(on)25 b(the)h(b)s(ottom)e(side)i(of)f Fs(H)1122 3562 y Fw(+)p Fq(;l)1121 3636 y(N)s(;K)1268 3609 y Fz(.)41 b(Th)m(us,)28 b(the)e(surgery)h(op)s(eration)d(\(4.34\)) g(e\013ectiv)m(ely)i(decouples)118 3750 y(the)31 b(p)s(ercolation)e (con\014guration)h Ft(\030)1426 3714 y Fq(l)1421 3775 y Fw(+)1510 3750 y Fz(on)g(the)h(edges)h Fs(E)2130 3703 y Fw(+)p Fq(;l)2121 3777 y(N)s(;K)2298 3750 y Fz(of)f(the)g(upp)s(er)g (b)s(o)m(x)g Fs(H)3121 3703 y Fw(+)p Fq(;l)3120 3777 y(N)s(;K)3298 3750 y Fz(from)e(the)118 3868 y(global)h(constrain)m(t)j Fi(J)919 3883 y Fq(N)986 3868 y Fz(.)118 4051 y(4.8.)49 b FE(Upp)s(er)g(b)s(ound)h(in)f(one)h(mesoscopic)f(la)m(y)m(er.)h Fz(Giv)m(en)42 b(a)g(mesoscopic)h(la)m(y)m(er)g Fx(H)3508 4014 y Fq(m)3508 4076 y(N)s(;K)118 4167 y Fz(and)35 b(a)g(p)s (ercolation)e(con\014guration)h Ft(\030)i Fs(2)c Fz(\004)1738 4182 y Fq(N)s(;)p Fo(\006)1876 4167 y Fz(,)j(let)f(us)i(de\014ne)g(the) f(site)g(p)s(ercolation)e(pro)s(cess)118 4283 y Ft(Y)196 4247 y Fq(m)175 4308 y(N)s(;K)354 4283 y Fz(on)g Fx(H)561 4247 y Fq(m)561 4308 y(N)s(;K)747 4283 y Fz(as:)447 4614 y Ft(Y)525 4573 y Fq(m)504 4639 y(N)s(;K)651 4614 y Fz(\()p Ft(i)p Fz(\))44 b(=)924 4380 y Fp(8)924 4470 y(>)924 4500 y(<)924 4679 y(>)924 4709 y(:)1107 4456 y Fz(1)17 b Ft(;)194 b Fz(if)32 b Fx(B)1544 4471 y Fw(2)p Fq(K)1654 4456 y Fz(\()p Ft(i)p Fz(\))g(is)h(go)s(o)s(d)e(and)i Ft(C)2399 4415 y Fo(\003)2466 4456 y Fs($)27 b Ft(@)2649 4415 y Fw(ext)2750 4456 y Fx(D)2816 4471 y Fq(N)2922 4456 y FE(in)33 b Fs(H)3133 4415 y Fq(m)3132 4481 y(N)s(;K)3296 4456 y Ft(;)1029 4615 y Fs(\000)p Fz(1)17 b Ft(;)195 b Fz(if)32 b Fx(B)1544 4630 y Fw(2)p Fq(K)1654 4615 y Fz(\()p Ft(i)p Fz(\))g(is)h(go)s(o)s(d)e(and)i Ft(C)2399 4574 y Fo(\003)2466 4615 y Fs(6$)27 b Ft(@)2649 4574 y Fw(ext)2750 4615 y Fx(D)2816 4630 y Fq(N)2922 4615 y FE(in)33 b Fs(H)3133 4574 y Fq(m)3132 4639 y(N)s(;K)3296 4615 y Ft(;)1074 4766 y Fz(0)17 b Ft(;)227 b Fz(if)32 b Fx(B)1544 4781 y Fw(2)p Fq(K)1654 4766 y Fz(\()p Ft(i)p Fz(\))g(is)h(bad)16 b Ft(;)118 4933 y Fz(where)34 b Ft(C)477 4897 y Fo(\003)549 4933 y Fz(denotes)g(the)f(crossing)f(cluster)h(of)f Fx(B)1926 4948 y Fw(2)p Fq(K)2035 4933 y Fz(\()p Ft(i)p Fz(\))h(\(see)h(Subsection)f(2.2\).)218 5049 y(Clearly)-8 b(,)47 b(the)e(p)s(ercolation)d(pro)s(cess)k(\()p Ft(Y)1758 5013 y Fq(m)1736 5074 y(N)s(;K)1883 5049 y Fz(\))e(dominates)g(the)h (restriction)f(of)g(\()p Ft(X)3353 5013 y Fw(FK)3345 5074 y Fq(N)s(;K)3492 5049 y Fz(\))h(to)118 5165 y Fx(H)190 5129 y Fq(m)190 5190 y(N)s(;K)1091 5226 y Fp(X)1037 5438 y Fq(i)p Fo(2)p Fn(H)1154 5415 y Fm(m)1154 5461 y(N)r(;K)1305 5320 y Fz(1)-25 b(I)1364 5336 y Fo(f)p Fq(Y)1456 5313 y Fm(m)1440 5359 y(N)r(;K)1568 5336 y Fw(\()p Fq(i)p Fw(\))p Fo(6)p Fw(=1)p Fo(g)1821 5320 y Fr(>)2012 5226 y Fp(X)1959 5438 y Fq(i)p Fo(2)p Fn(H)2076 5415 y Fm(m)2076 5461 y(N)r(;K)2226 5320 y Fz(1)g(I)2285 5343 y Fo(f)p Fq(X)2383 5320 y Fl(FK)2378 5366 y Fm(N)r(;K)2507 5343 y Fw(\()p Fq(i)p Fw(\))p Fo(6)p Fw(=1)p Fo(g)2715 5320 y Ft(:)670 b Fz(\(4.36\))p eop %%Page: 21 21 21 20 bop 3585 48 a Fu(21)118 366 y FE(Lemma)39 b(4.2.)j Fj(Ther)-5 b(e)34 b(exists)h(a)f(p)-5 b(ositive)35 b(c)-5 b(onstant)34 b Ft(c)2144 381 y Fw(6)2211 366 y Ft(>)28 b Fz(0)p Fj(,)34 b(such)h(that)169 648 y Fz(max)238 708 y Fq(\031)367 648 y Fz(\010)437 607 y Fq(\031)437 673 y Fo(E)480 650 y Fm(m)474 696 y(N)r(;K)624 448 y Fp(0)624 627 y(@)764 554 y(X)711 765 y Fq(i)p Fo(2)p Fn(H)827 743 y Fm(m)827 789 y(N)s(;K)978 648 y Fz(1)-25 b(I)1037 664 y Fo(f)p Fq(Y)1129 641 y Fm(m)1113 687 y(N)r(;K)1241 664 y Fw(\()p Fq(i)p Fw(\))p Fo(6)p Fw(=1)p Fo(g)1494 648 y Fr(>)45 b Ft(f)1675 448 y Fp(1)1675 627 y(A)1822 648 y Fr(6)g Fz(exp)2110 538 y Fp(\020)2169 648 y Fs(\000)p Ft(c)2288 663 y Fw(6)2345 648 y Fz(min)2524 538 y Fp(n)2590 648 y Ft(f)11 b(K)r(;)17 b(f)2846 577 y Fm(d)p Fh(\000)p Fl(2)p 2846 590 111 3 v 2846 631 a Fm(d)p Fh(\000)p Fl(1)2971 648 y Ft(K)3061 607 y Fq(d)p Fo(\000)p Fw(1)3192 538 y Fp(o\021)3334 648 y Ft(;)51 b Fz(\(4.37\))118 936 y Fj(uniformly)35 b(in)f(p)-5 b(ositive)34 b(inte)-5 b(gers)35 b Ft(f)45 b Fj(and)35 b Ft(N)45 b Fj(su\016ciently)35 b(lar)-5 b(ge.)218 1077 y Fz(A)27 b(section)g(of)f(a)h(\014nger)g(on)g (the)g(la)m(y)m(er)g Ft(m)g Fz(leads)g(to)f(an)h(exp)s(onen)m(tial)f (deca)m(y)j(of)d(order)h Ft(f)11 b(K)34 b Fz(if)25 b(it)118 1211 y(con)m(tains)31 b(at)g(least)854 1167 y Fq(f)p 854 1188 42 4 v 857 1245 a Fw(2)936 1211 y Fz(blo)s(c)m(ks)g(with)g (lab)s(el)e(0)i(or)g(of)f(order)i Ft(f)2314 1139 y Fm(d)p Fh(\000)p Fl(2)p 2314 1152 111 3 v 2314 1193 a Fm(d)p Fh(\000)p Fl(1)2439 1211 y Ft(K)2529 1175 y Fq(d)p Fo(\000)p Fw(1)2691 1211 y Fz(if)e(it)g(con)m(tains)h(at)g(least)3610 1167 y Fq(f)p 3610 1188 42 4 v 3613 1245 a Fw(2)118 1328 y Fz(blo)s(c)m(ks)23 b(with)e(lab)s(el)g Fs(\000)p Fz(1.)40 b(The)23 b(claim)c(of)j(the)h(lemma)c(follo)m(ws)i(from)g(\(2.4\))h (via)f(a)h(straigh)m(tforw)m(ard)118 1444 y(mo)s(di\014cation)30 b(of)i(the)h(argumen)m(t)f(presen)m(ted)j(in)d(the)h(App)s(endix)g(A)g (in)f([BIV1].)218 1561 y(W)-8 b(e)32 b(stress)i(the)e(fact)g(that)g (the)h(scaling)d Ft(K)35 b Fz(=)28 b Ft(N)2019 1524 y Fq(a)2093 1561 y Fz(will)i(b)s(e)i(used)h(only)f(in)f(the)h(deriv)-5 b(ation)31 b(of)118 1677 y(\(4.37\))o(.)118 1884 y(4.9.)49 b FE(Upp)s(er)i(b)s(ound)g(on)g Fs(F)10 b Fz(\()1341 1858 y Ft(~)1327 1884 y(f)1375 1899 y Fq(N)s(;K)1522 1884 y Fz(\))p FE(.)49 b Fz(W)-8 b(e)44 b(shall)e(consider)j(only)e(ev) m(en)j(mesoscopic)e(la)m(y)m(ers)118 2017 y Ft(m)52 b Fz(=)g(2)p Ft(;)17 b Fz(4)p Ft(;)g(:)g(:)g(:)e(;)i(R)q Fz(.)86 b(F)-8 b(or)46 b(ev)m(ery)i(suc)m(h)g Ft(m)f Fz(and)g(ev)m(ery)i(p)s(ositiv)m(e)d(collection)3069 1991 y Ft(~)3056 2017 y(f)3104 2032 y Fq(N)s(;K)3298 2017 y Fz(of)g(\014nger)118 2134 y(lab)s(els,)d(de\014ne)g(the)f(ev)m (en)m(t)h Fs(B)1230 2149 y Fq(m)1297 2134 y Fz(,)h(whic)m(h)e(dep)s (ends)i(only)d(on)g(the)h(p)s(ercolation)e(con\014guration)118 2250 y(inside)32 b(the)h(slab)f Fs(H)849 2214 y Fq(m)848 2275 y(N)s(;K)995 2250 y Fz(:)763 2451 y Fs(B)828 2466 y Fq(m)939 2451 y Fz(=)44 b Fs(B)1124 2466 y Fq(m)1191 2451 y Fz(\()1243 2424 y Ft(~)1229 2451 y(f)1277 2466 y Fq(N)s(;K)1424 2451 y Fz(\))h(=)f Fs(f)p Ft(\030)k Fz(:)1893 2356 y Fp(X)1840 2568 y Fq(i)p Fo(2)p Fn(H)1956 2545 y Fm(m)1956 2591 y(N)s(;K)2107 2451 y Fz(1)-25 b(I)2166 2466 y Fo(f)p Fq(Y)2258 2443 y Fm(m)2242 2489 y(N)r(;K)2370 2466 y Fw(\()p Fq(i)p Fw(\))p Fo(6)p Fw(=1)p Fo(g)2623 2451 y Fr(>)45 b Ft(f)2804 2410 y Fq(m)2793 2475 y(N)s(;K)2940 2451 y Fs(g)p Ft(:)118 2738 y Fz(Giv)m(en)33 b(an)f(ev)m(en)i (mesoscopic)f(la)m(y)m(er)g Ft(l)d Fz(=)d(2)p Ft(;)17 b Fz(4)p Ft(;)g(:)g(:)g(:)e(;)i(R)q Fz(,)1032 2996 y Fs(F)10 b Fz(\()1166 2969 y Ft(~)1152 2996 y(f)1200 3011 y Fq(N)s(;K)1347 2996 y Fz(\))44 b Fs(\022)1639 2871 y Fq(R)1610 2901 y Fp(\\)1551 3113 y Fq(m)p Fw(=)p Fq(l)q Fw(+2)1797 2996 y Fs(B)1862 3011 y Fq(m)1929 2996 y Fz(\()1981 2969 y Ft(~)1967 2996 y(f)2015 3011 y Fq(N)s(;K)2162 2996 y Fz(\))2217 2901 y Fp(\\)2344 2996 y Fs(A)2424 3011 y Fq(l)2449 2996 y Fz(\()2501 2969 y Ft(~)2487 2996 y(f)2535 3011 y Fq(N)s(;K)2682 2996 y Fz(\))p Ft(:)218 3257 y Fz(W)-8 b(e)37 b(will)d(compute)j(the)g(balance)f(b)s(et)m(w)m (een)j(the)e(energetic)g(cost)g(of)g(c)m(hopping)f(a)h(\014nger)g(at) 118 3374 y(the)32 b(section)g Ft(l)i Fz(and)d(the)h(gain)f(of)g (erasing)g(this)g(\014nger)h(ab)s(o)m(v)m(e)g(the)g(lev)m(el)g Ft(l)r Fz(.)43 b(Consequen)m(tly)-8 b(,)34 b(the)118 3490 y(estimates)e(\(4.35\))g(and)h(\(4.37\))f(imply:)156 3753 y(\010)226 3712 y Fo(\006)226 3780 y Fq(N)310 3642 y Fp(\020)387 3753 y Fs(F)10 b Fz(\()521 3726 y Ft(~)507 3753 y(f)555 3768 y Fq(N)s(;K)701 3753 y Fz(\))756 3642 y Fp(\021)860 3753 y Fr(6)44 b Fz(\010)1051 3712 y Fo(\006)1051 3780 y Fq(N)1136 3582 y Fp( )1319 3628 y Fq(R)1290 3658 y Fp(\\)1231 3870 y Fq(m)p Fw(=)p Fq(l)q Fw(+2)1477 3753 y Fs(B)1542 3768 y Fq(m)1609 3753 y Fz(\()1661 3726 y Ft(~)1647 3753 y(f)1695 3768 y Fq(N)s(;K)1842 3753 y Fz(\))1897 3658 y Fp(\\)2024 3753 y Fs(A)2104 3768 y Fq(l)2130 3753 y Fz(\()2182 3726 y Ft(~)2168 3753 y(f)2216 3768 y Fq(N)s(;K)2363 3753 y Fz(\))2418 3582 y Fp(!)860 4104 y Fz(=)1013 4010 y Fp(X)980 4233 y Fq(\030)1014 4210 y Fm(l)1011 4254 y Fh(\000)1063 4233 y Fo(2A)1166 4245 y Fm(l)1207 4104 y Fz(\010)1277 4063 y Fo(\006)1277 4131 y Fq(N)1361 3934 y Fp( )1528 3980 y Fq(R)1500 4010 y Fp(\\)1440 4222 y Fq(m)p Fw(=)p Fq(l)q Fw(+2)1686 4104 y Fs(B)1751 4119 y Fq(m)1818 4104 y Fz(\()1870 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Ft(K)3304 5183 y Fq(d)p Fo(\000)p Fw(1)3434 5113 y Fp(o)3501 5054 y(!)3596 5224 y Ft(:)3412 5441 y Fz(\(4.38\))p eop %%Page: 22 22 22 21 bop 118 51 a Fu(22)1148 b(T.)26 b(BODINEA)n(U)e(AND)g(D.)i(IOFFE) 118 366 y Fz(4.10.)48 b FE(Pro)s(of)25 b(of)g(Theorem)h(4.1.)49 b Fz(W)-8 b(e)22 b(claim)e(that)h(there)i(exists)g(a)e(p)s(ositiv)m(e)g (constan)m(t)i Ft(c)3415 381 y Fw(8)3482 366 y Ft(>)28 b Fz(0,)118 483 y(suc)m(h)34 b(that)332 723 y(min)228 790 y Fw(2)20 b Ff(6)g Fq(l)g Ff(6)g Fq(R=)p Fw(2)614 553 y Fp(\()695 723 y Ft(c)737 738 y Fw(7)776 723 y Ft(f)835 682 y Fq(l)824 748 y(N)s(;K)971 723 y Ft(K)1061 682 y Fq(d)p Fo(\000)p Fw(1)1231 723 y Fs(\000)39 b Ft(c)1389 738 y Fw(6)1533 599 y Fq(R)1488 629 y Fp(X)1445 841 y Fq(m)p Fw(=)p Fq(l)q Fw(+2)1691 723 y Fz(min)1870 613 y Fp(n)1937 723 y Ft(f)1996 682 y Fq(m)1985 748 y(N)s(;K)2132 723 y Ft(K)r(;)17 b Fz(\()p Ft(f)2358 682 y Fq(m)2347 748 y(N)s(;K)2493 723 y Fz(\))2541 652 y Fm(d)p Fh(\000)p Fl(2)p 2541 665 111 3 v 2541 706 a Fm(d)p 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b(or)39 b(some)g Ft(c)f(>)h Fz(0,)i(consider)e(the)h(sequence)j(\026) -50 b Ft(a)3215 4753 y Fw(0)3255 4738 y Ft(;)17 b(:)g(:)g(:)e(;)j Fz(\026)-50 b Ft(a)3524 4753 y Fq(n)p Fo(\000)p Fw(1)118 4855 y Fz(giv)m(en)33 b(b)m(y)1175 4984 y(\026)-51 b Ft(a)1224 4999 y Fq(i)1280 4984 y Fz(=)28 b(\(1)22 b(+)g Ft(ic)p Fz(\))1704 4942 y Fw(1)p Fq(=)p Fw(\(1)p Fo(\000)p Fq(\015)t Fw(\))2008 4984 y Fz(=)44 b(\(1)22 b(+)g Ft(ic)p Fz(\))2448 4942 y Fq(d)p Fo(\000)p Fw(1)2579 4984 y Ft(:)118 5130 y Fz(By)33 b(con)m(v)m(exit)m(y:)497 5304 y(\026)-50 b Ft(a)547 5319 y Fq(i)p Fw(+1)688 5304 y Fs(\000)24 b Fz(\026)-51 b Ft(a)838 5319 y Fq(i)911 5304 y Fr(6)45 b Fz(\()p Ft(d)21 b Fs(\000)i Fz(1\))p Ft(c)17 b Fz(\()o(1)22 b(+)g(\()p Ft(i)h Fz(+)f(1\))p Ft(c)p Fz(\))1953 5255 y Fq(d)p Fo(\000)p Fw(2)911 5475 y Fr(6)28 b Fz(\()p Ft(d)22 b Fs(\000)h Fz(1\))p Ft(c)p Fz(\(1)e(+)h Ft(c)p Fz(\))1642 5433 y Fq(d)p Fo(\000)p Fw(2)1789 5475 y Fz(\(1)g(+)g Ft(ic)p Fz(\))2109 5426 y Fq(d)p Fo(\000)p Fw(2)2284 5475 y Fz(=)44 b(\()p Ft(d)22 b Fs(\000)h Fz(1\))p Ft(c)p Fz(\(1)e(+)h Ft(c)p Fz(\))3030 5433 y Fq(d)p Fo(\000)p Fw(2)3162 5475 y Fz(\026)-50 b Ft(a)3212 5427 y Fq(\015)3212 5500 y(i)3256 5475 y Ft(:)p eop %%Page: 24 24 24 23 bop 118 51 a Fu(24)1148 b(T.)26 b(BODINEA)n(U)e(AND)g(D.)i(IOFFE) 118 366 y Fz(Let)33 b(us)g(c)m(ho)s(ose)h Ft(c)27 b Fz(=)h Ft(c)p Fz(\()p Ft(\037;)17 b(d)p Fz(\))32 b(according)g(to)g(\()p Ft(d)22 b Fs(\000)g Fz(1\))p Ft(c)p Fz(\(1)g(+)g Ft(c)p Fz(\))2391 330 y Fq(d)p Fo(\000)p Fw(2)2549 366 y Fz(=)28 b Ft(\037)p Fz(.)43 b(Then,)1503 633 y(\026)-50 b Ft(a)1553 648 y Fq(i)1626 633 y Fr(6)45 b Ft(\037)1840 509 y Fq(i)p Fo(\000)p Fw(1)1825 539 y Fp(X)1836 749 y Fq(j)t Fw(=0)1987 633 y Fz(\026)-50 b Ft(a)2037 586 y Fq(\015)2037 659 y(j)2104 633 y Fz(+)22 b(1)p Ft(;)118 917 y Fz(for)35 b(all)e Ft(j)38 b Fz(=)32 b(1)p Ft(;)17 b(:)g(:)g(:)e(;)i(n)24 b Fs(\000)g Fz(1.)51 b(Comparing)34 b(with)g(\(4.46\))h(w)m(e)h (readily)e(infer)g(that)h(\(1)24 b(+)g Ft(ic)p Fz(\))3423 881 y Fq(d)p Fo(\000)p Fw(1)3586 917 y Fz(=)119 1033 y(\026)-50 b Ft(a)169 1048 y Fq(i)225 1033 y Fr(6)28 b Ft(a)381 1048 y Fq(n)p Fo(\000)p Fq(i)540 1033 y Fz(and,)33 b(consequen)m(tly)-8 b(,)34 b(\(4.47\))e(follo)m(ws)f(with,)i(for)f (example,)1273 1284 y Ft(c)1315 1299 y Fw(9)1399 1284 y Fz(=)1551 1216 y(1)p 1529 1261 93 4 v 1529 1352 a Ft(dc)1648 1284 y Fz(min)1663 1343 y Fq(n>)p Fw(1)1837 1216 y Fz(\(1)22 b(+)g Ft(c)p Fz(\()p Ft(n)h Fs(\000)f Fz(1\)\))2429 1168 y Fq(d)p 1837 1261 632 4 v 2104 1352 a Ft(n)2162 1323 y Fq(d)2479 1284 y Ft(:)1332 1556 y Fz(5.)48 b Fy(Exponential)38 b(mixing)218 1731 y Fz(The)43 b(analysis)f(of)g(the)g(sp)s(ectral)h (gap)f(will)e(rely)i(on)g(t)m(w)m(o)h(prop)s(erties)g(of)f(the)g (equilibrium)118 1847 y(measure:)g(the)30 b(lo)s(calization)25 b(of)k(the)h(in)m(terface)g(and)f(the)h(relaxation)e(of)g(the)i(system) h(to)e(a)g(pure)118 1963 y(phase)35 b(a)m(w)m(a)m(y)g(from)e(the)i(in)m (terface.)47 b(These)36 b(estimates)e(will)d(b)s(e)k(used)g(in)e 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Fw(ext)1922 3486 y Fq(t)2029 3462 y Fx(D)2095 3425 y Fq(\016)2095 3487 y(N)2168 3462 y Fz(,)e(where)f Fj(t)g Fz(stands)g(for)f Fj(top)g Fz(and)g Fj(s)g Fz(stands)118 3578 y(for)32 b Fj(sides)p Fz(.)43 b(The)33 b(b)s(ottom)e(face)i(of)f Fx(D)1461 3542 y Fq(\016)1461 3603 y(N)1567 3578 y Fz(is)g(denoted)i(b) m(y)f Ft(@)2225 3542 y Fw(in)n(t)2220 3604 y Fq(b)2315 3578 y Fx(D)2381 3542 y Fq(\016)2381 3603 y(N)2454 3578 y Fz(.)218 3694 y(W)-8 b(e)23 b(are)h(going)d(to)i(established)h(the)f (Hausdor\013)h(stabilit)m(y)d(of)i(the)h(in)m(terface)f(when)h(a)f (negativ)m(e)118 3810 y(\(resp)s(ectiv)m(ely)29 b(p)s(ositiv)m(e\))f (magnetic)f(\014eld)h(is)g(applied)f(on)i Ft(@)2317 3774 y Fw(in)n(t)2312 3836 y Fq(b)2406 3810 y Fx(D)2472 3774 y Fq(\016)2472 3836 y(N)2573 3810 y Fz(\(resp)h Ft(@)2870 3774 y Fw(ext)2865 3835 y Fq(t)2971 3810 y Fx(D)3037 3774 y Fq(\016)3037 3836 y(N)3110 3810 y Fz(\))e(and)h(a)f(small)118 3927 y(p)s(ositiv)m(e)k(\014eld)h Ft(")28 b(>)g Fz(0)k(is)h(applied)e 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y Fs(\000)55 b Ft(")2718 4180 y Fp(X)2713 4396 y Fw(\()p Fq(i)12 b(;)g(j)t Fw(\))2535 4479 y Fq(i)p Fo(2)p Fn(D)2648 4456 y Fm(\016)2648 4502 y(N)2712 4479 y Fq(;j)t Fo(2)p Fq(@)2853 4456 y Fl(ext)2849 4496 y Fm(s)2940 4479 y Fn(D)2982 4456 y Fm(\016)2982 4502 y(N)3062 4274 y Ft(\033)3117 4289 y Fq(i)810 4654 y Fz(+)1016 4559 y Fp(X)940 4783 y Fq(i)p Fo(2)p Fq(@)1052 4760 y Fl(in)n(t)1048 4806 y Fm(b)1130 4783 y Fn(D)1172 4760 y Fm(\016)1172 4806 y(N)1253 4654 y Ft(\033)1308 4669 y Fq(i)1336 4654 y Ft(;)3412 4504 y Fz(\(5.51\))118 4954 y(where)28 b(the)g(\014rst)f(three)h(sums)g(are) f(o)m(v)m(er)h(\(subsets)h(of)e(\))g(nearest)h(neigh)m(b)s(our)e(b)s (onds)i(\()p Ft(i)17 b(;)33 b(j)6 b Fz(\).)41 b(F)-8 b(ol-)118 5075 y(lo)m(wing)26 b(notation)f(in)m(tro)s(duced)j(in)e (Subsection)i(2.3,)g(w)m(e)g(denote)g(b)m(y)g Ft(\026)2693 5028 y Fw(+)p Fq(;";)p Fo(\000)2693 5102 y Fq(N)2905 5075 y Fz(the)f(corresp)s(onding)118 5192 y(Gibbs)32 b(measure)h(and)g(b)m(y)g(\010)1176 5150 y Fo(\006)1176 5219 y Fq(N)s(;")1325 5192 y Fz(the)g(FK)f(measure.)218 5364 y(The)g(coun)m(terpart)f(of)g(Theorem)g(3.2)g(in)f(this)g(con)m (text)i(relies)f(also)f(on)g(the)i(strict)e(con)m(v)m(exit)m(y)118 5480 y(assumption)i(of)g(the)h(surface)h(tension)e FE(\(SC\))p Fz(.)p eop %%Page: 25 25 25 24 bop 3585 48 a Fu(25)118 366 y FE(Theorem)35 b(5.1.)40 b Fj(Assume)32 b FE(\(SC\))f Fj(and)h(let)h Ft(\016)e(>)d Fz(0)p Fj(,)k Ft(")c(>)f Fz(0)32 b Fj(b)-5 b(e)32 b(\014xe)-5 b(d.)43 b(F)-7 b(or)32 b(any)g Ft(\014)h Fs(2)28 b(B)36 b Fj(and)31 b(any)118 483 y Ft(\027)j(>)28 b Fz(0)34 b Fj(ther)-5 b(e)35 b(exists)g Ft(c)939 498 y Fw(1)1006 483 y Fz(=)27 b Ft(c)1151 498 y Fw(1)1191 483 y Fz(\()p Ft(\027)6 b Fz(\))28 b Ft(>)f Fz(0)p Fj(,)35 b(such)g(that)g(uniformly) f(in)h Ft(N)45 b Fj(lar)-5 b(ge)35 b(enough,)1050 675 y Fz(\010)1120 634 y Fo(\006)1120 702 y Fq(N)s(;")1253 675 y Fz(\()o Fi(g)28 b Fs($)g(f)p Ft(i)44 b Fz(:)g Ft(i)1727 690 y Fq(d)1796 675 y Ft(>)27 b(\027)6 b(\016)t(N)k Fs(g)p Fz(\))62 b Fr(6)h Fz(e)2421 634 y Fo(\000)p Fq(c)2507 643 y Fl(1)2541 634 y Fw(\()p Fq(\027)t Fw(\))p Fq(N)2702 675 y Ft(:)683 b Fz(\(5.52\))218 860 y(The)33 b(pro)s(of)f(is)g (similar)d(to)k(the)g(one)f(of)h(Theorem)f(3.2.)118 1089 y(5.2.)49 b FE(Screening.)g Fz(Com)m(bining)27 b(the)i(lo)s(calization) c(of)j(the)h(in)m(terface)g(\(Theorem)g(5.1\))f(and)h(the)118 1205 y(mixing)i(assumption)g FE(\(RP\))p Fz(,)g(w)m(e)j(obtain)e(a)g (screening)h(prop)s(ert)m(y)g(for)f(the)h(Gibbs)f(measure.)118 1356 y FE(Prop)s(osition)42 b(5.1.)i Fj(Fix)38 b Ft(\014)j Fs(2)35 b(B)1385 1371 y Fw(1)1425 1356 y Fj(.)56 b(Then)37 b(ther)-5 b(e)39 b(is)f Ft(c)2165 1371 y Fq(\014)2247 1356 y Ft(>)d Fz(0)j Fj(such)h(that)g(for)f(any)h(function)f Ft(g)118 1476 y Fj(with)26 b(supp)-5 b(ort)26 b(include)-5 b(d)25 b(in)g Fs(S)1199 1491 y Fq(N)s(;\016)1344 1476 y Fz(=)1447 1396 y Fp(\010)1506 1476 y Ft(i)j Fs(2)g Fx(D)1727 1440 y Fq(\016)1727 1502 y(N)1800 1476 y Fz(;)116 b Ft(i)1976 1491 y Fq(d)2045 1476 y Fz(=)2158 1437 y Fw(1)p 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Fz(\()p Ft(j)33 b Fs($)28 b Ft(@)2945 5161 y Fw(ext)3057 5176 y Fz(~)3046 5202 y Fx(D)3112 5161 y Fq(\016)3112 5226 y(N)3185 5202 y Fz(\))2332 5475 y(+)p Ft(N)2496 5433 y Fq(d)p Fo(\000)p Fw(1)2660 5475 y Fz(exp)q(\()p Fs(\000)p Ft(c)2966 5490 y Fw(2)3006 5475 y Ft(\016)t(N)10 b Fz(\))17 b Ft(:)p eop %%Page: 27 27 27 26 bop 3585 48 a Fu(27)118 366 y Fz(By)33 b(the)g(FK)m(G)f(prop)s (ert)m(y)h(of)f(the)h(random)f(cluster)h(measures,)1119 542 y(\010)1189 501 y Fw(w)1189 567 y Fq(N)s(;")1305 542 y Fz(\()p Ft(j)h Fs($)27 b Ft(@)5 b Fx(D)1667 501 y Fq(\016)1667 567 y(N)1740 542 y Fz(\))28 b Fr(6)g Fz(\010)1981 501 y Fw(w)1987 562 y(~)1981 579 y Fn(D)2023 556 y Fm(\016)2023 602 y(N)2087 579 y Fq(;")2144 542 y Fz(\()p Ft(j)33 b Fs($)27 b Ft(@)2451 517 y Fz(~)2438 542 y Fx(D)2505 501 y Fq(\016)2505 567 y(N)2579 542 y Fz(\))17 b Ft(:)218 741 y Fz(A)m(t)39 b(this)f(stage,)i(it)e(will)e(b)s(e)j(enough)g(to)f (apply)h(the)g(strong)f(mixing)f(inequalit)m(y)i(\(5.54\))f(to)118 858 y(conclude.)218 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Fz(\()p Fs(A)p Fz(\))22 b Fs(\000)g Fz(\010)2365 5423 y Fw(w)q Fq(=)p Fw(f)2365 5501 y Fq(N)s(;")p Fw(=0)2572 5474 y Fz(\()p Fs(A)p Fz(\))f(+)h Ft(o)2894 5393 y Fp(\000)2956 5474 y Fz(exp)q(\()p Fs(\000)p Ft(c\016)t(N)10 b Fz(\))3435 5393 y Fp(\001)3498 5474 y Ft(:)p eop %%Page: 28 28 28 27 bop 118 51 a Fu(28)1148 b(T.)26 b(BODINEA)n(U)e(AND)g(D.)i(IOFFE) 118 366 y Fz(The)40 b(previous)f(FK)g(measures)h(are)e(ordered)i(\(in)e (the)i(FK)m(G)e(sense\).)64 b(W)-8 b(e)39 b(can)h(consider)f(the)118 483 y(join)m(t)28 b(measure)i Ft(\027)767 498 y Fq(N)834 483 y Fz(\()p Ft(\030)5 b(;)17 b(\030)1012 447 y Fo(0)1034 483 y Fz(\))29 b(suc)m(h)h(that)f(the)g(\014rst)h(marginal)c(is)i(\010) 2454 447 y Fw(w)2454 508 y Fq(N)s(;")p Fw(=0)2660 483 y Fz(,)i(the)g(second)g(marginal)c(is)118 630 y(\010)188 579 y Fw(w)q Fq(=)p Fw(f)188 657 y Fq(N)s(;")p Fw(=0)427 630 y Fz(and)32 b(the)h(measure)g(is)f(supp)s(orted)i(b)m(y)f(the)g (con\014gurations)g Ft(\030)f Fr(>)c Ft(\030)2882 594 y Fo(0)2905 630 y 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Fq(b)2006 1136 y Fs(\000)23 b Ft(\030)2154 1095 y Fo(0)2149 1161 y Fq(b)2183 1056 y Fp(\001)2256 1136 y Fz(=)2480 1042 y Fp(X)2359 1257 y Fq(b)p Fo(2)p Fn(B)2475 1271 y Fm(\016)r(N)q(=)p Fl(10)2656 1257 y Fw(\()p Fq(j)t Fw(\))2760 1136 y Fz(\010)2830 1095 y Fw(w)2830 1161 y Fq(N)s(;")p Fw(=0)3036 1136 y Fz(\()p Ft(\030)3117 1151 y Fq(b)3151 1136 y Fz(\))f Fs(\000)h Fz(\010)3381 1085 y Fw(w)q Fq(=)p Fw(f)3381 1163 y Fq(N)s(;")p Fw(=0)3587 1136 y Fz(\()p Ft(\030)3673 1095 y Fo(0)3668 1161 y Fq(b)3702 1136 y Fz(\))17 b Ft(:)118 1425 y Fz(F)-8 b(or)31 b(an)m(y)i(b)s(ound)f Ft(b)p Fz(,)g(the)g(probabilit)m(y)e(on)i(the)g(LHS)g(can)g(b)s(e)g (estimated)g(thanks)h(to)e(the)h(relax-)118 1542 y(ation)f(prop)s(ert)m (y)j(\(see)f(De\014nition)e(2.2\))913 1729 y(\010)983 1688 y Fw(w)983 1754 y Fq(N)s(;")p Fw(=0)1189 1729 y Fz(\()p Fs(A)p Fz(\))22 b Fs(\000)h Fz(\010)1537 1678 y Fw(w)q Fq(=)p Fw(f)1537 1756 y Fq(N)s(;")p Fw(=0)1743 1729 y Fz(\()p Fs(A)p Fz(\))k Fr(6)h Ft(N)2119 1688 y Fq(d)2177 1729 y Fz(exp)q(\()p Fs(\000)p Ft(c\016)t(N)10 b Fz(\))17 b Ft(;)118 1905 y Fz(for)32 b(some)h Ft(c)27 b(>)h Fz(0.)43 b(This)33 b(completes)f(the)h(deriv)-5 b(ation)31 b(of)39 b(\(5.54\))o(.)1087 b Fr(\003)1229 2139 y Fz(6.)49 b Fy(Spectral)36 b(gap)h(estima)-7 b(tes)118 2314 y Fz(6.1.)49 b FE(Lo)m(w)m(er)28 b(b)s(ound.)51 b Fz(W)-8 b(e)25 b(turn)g(no)m(w)h(to)f(the)g(deriv)-5 b(ation)24 b(of)g(the)i(lo)m(w)m(er)f(b)s(ound)h(\(3.25\))e(on)h(the) 118 2430 y(sp)s(ectral)30 b(gap.)43 b(The)31 b(pro)s(of)f(follo)m(ws)f (closely)h(the)g(strategy)i(dev)m(elop)s(ed)f(b)m(y)g(Martinelli)c([Ma) q(])j(in)118 2546 y(the)j(t)m(w)m(o)g(dimensional)d(case.)45 b(W)-8 b(e)33 b(will)d(brie\015y)i(recall)g(the)g(main)f(steps)j(of)e (the)h(pro)s(of)f(as)g(they)118 2662 y(are)d(exp)s(osed)h(in)d(the)i (Chapter)h(6)e(of)g([Ma])h(and)f(fo)s(cus)h(only)f(on)g(the)h(c)m (hanges.)44 b(This)28 b(comprises)118 2779 y(a)37 b(more)f(careful)g (analysis)h(of)f(the)i(b)s(oundary)f(e\013ects)h(to)f(tak)m(e)h(in)m 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Ft(\016)t(N)10 b Fz(])28 b Fr(6)g Ft(x)2056 3814 y Fq(d)2125 3799 y Ft(<)g Fz(\()2284 3731 y Ft(i)p 2277 3776 V 2277 3867 a Fz(2)2357 3799 y(+)22 b(1\)[)p Ft(\016)t(N)10 b Fz(])2731 3658 y Fp(\033)2840 3799 y Ft(:)118 4030 y Fz(The)33 b(total)d(n)m(um)m(b)s(er)i(of)f(sets)i Fs(f)p Ft(R)1328 4045 y Fq(i)1356 4030 y Fs(g)1406 4045 y Fq(i)1466 4030 y Fz(is)e(indep)s(enden)m(t)i(of)e Ft(N)42 b Fz(and)32 b(denoted)h(b)m(y)f Ft(L)d Fz(=)e Ft(L)p Fz(\()p Ft(\016)t Fz(\).)43 b(The)118 4146 y(sets)34 b Ft(R)383 4161 y Fq(i)444 4146 y Fz(are)f(simply)f(shifts)h(of)f(the)h(set)h Fx(D)1677 4110 y Fq(\016)1677 4171 y(N)1783 4146 y Fz(in)m(tro)s(duced) f(in)f(\(5.50\))o(.)45 b(The)33 b(blo)s(c)m(k)g(dynamics)g(is)118 4262 y(de\014ned)h(in)e(terms)g(of)h(the)g(generator)680 4458 y Fs(8)p Ft(f)39 b Fs(2)28 b Fx(L)977 4417 y Fw(2)1022 4458 y Fz(\()p Ft(\026)1119 4417 y Fk(h)1119 4482 y Fq(N)1186 4458 y Fz(\))p Ft(;)212 b Fs(L)1532 4417 y Fk(h)1532 4482 y Fq(N)s(;\016)1648 4458 y Ft(f)11 b Fz(\()p 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Ft(\016)t(N)1994 5082 y Fq(d)p Fo(\000)p Fw(1)2125 5123 y Fz(\))27 b(SG\()p Fs(L)2428 5082 y Fk(h)2428 5148 y Fq(N)s(;\016)2544 5123 y Fz(\))17 b Ft(;)786 b Fz(\(6.59\))118 5305 y(where)34 b(SG\()p Fs(L)638 5269 y Fk(h)638 5331 y Fq(N)s(;\016)754 5305 y Fz(\))e(denotes)i(the)f(sp)s(ectral)g(gap)f(of)g(the)h(blo)s(c)m(k)f (dynamics.)118 5480 y FE(Step)38 b(2.)43 b Fz(Thanks)34 b(to)f(\(6.59\))o(,)f(it)g(is)g(enough)h(to)f(deriv)m(e)p eop %%Page: 29 29 29 28 bop 3585 48 a Fu(29)118 366 y FE(Lemma)39 b(6.1.)j Fj(L)-5 b(et)35 b FE(h)28 b Fz(=)g(\(0)p Ft(;)17 b(:)g(:)g(:)e(;)i Fz(0)p Ft(;)g(h)1547 381 y Fq(d)1587 366 y Ft(;)g Fz(0)p Ft(;)g(:)g(:)g(:)e(;)i Fz(0\))p Fj(.)45 b(Then,)33 b(for)i(any)g Ft(\016)d(>)27 b Fz(0)p Fj(,)870 591 y Fz(lim)17 b(inf)903 653 y Fq(N)7 b Fo(!1)1252 523 y Fz(1)p 1167 568 220 4 v 1167 659 a Ft(N)1255 630 y Fq(d)p Fo(\000)p Fw(1)1413 591 y Fz(log)16 b(SG\()p Fs(L)1793 550 y Fk(h)1793 615 y Fq(N)s(;\016)1909 591 y Fz(\))28 b Fr(>)50 b Fs(\000)23 b Ft(\034)11 b Fz(\()l Ft(~)-45 b(e)2338 606 y Fq(d)2379 591 y Fz(\))22 b(+)g(\001)2618 606 y Fq(h)2659 618 y Fm(d)2716 591 y Ft(:)218 813 y Fz(The)39 b(pro)s(of)e(b)s(oils)g(do)m (wn)i(to)f(c)m(hec)m(k)j(that)d(the)g(semi-group)f(asso)s(ciated)h(to)g Fs(L)3137 776 y Fk(h)3137 838 y Fq(N)s(;\016)3292 813 y Fz(is)g(a)g(con-)118 929 y(traction)32 b(for)g(some)g(time)f Ft(T)14 b Fz(,)32 b(i.e.)43 b(that)33 b(there)g(is)f Ft(r)2003 944 y Fq(N)2098 929 y Ft(>)27 b Fz(0)33 b(suc)m(h)h(that)e (for)g(all)f Ft(N)43 b Fz(large)31 b(enough)1107 1104 y(sup)1161 1184 y Fq(\021)1270 1020 y Fp(\014)1270 1080 y(\014)1303 1104 y Fx(E)1364 1024 y Fp(\000)1416 1104 y Ft(f)11 b Fz(\()p Ft(\033)1572 1057 y Fq(\021)1568 1131 y(T)1623 1104 y Fz(\))1661 1024 y Fp(\001)1706 1020 y(\014)1706 1080 y(\014)1767 1104 y Fr(6)28 b Fz(\(1)22 b Fs(\000)h Ft(r)2125 1119 y Fq(N)2192 1104 y Fz(\))p Fs(k)p Ft(f)11 b Fs(k)2389 1119 y Fo(1)2480 1104 y Ft(;)905 b Fz(\(6.60\))118 1356 y(for)32 b(an)m(y)h Ft(f)44 b Fz(suc)m(h)34 b(that)e Ft(\026)1033 1320 y Fk(h)1033 1382 y Fq(N)1100 1356 y Fz(\()p Ft(f)11 b Fz(\))27 b(=)h(0.)43 b(In)33 b(our)f(con)m(text)i Ft(r)2171 1371 y Fq(N)2271 1356 y Fz(will)c(b)s(e)j(suc)m(h)h(that)1101 1581 y(lim)1067 1643 y Fq(N)7 b Fo(!1)1383 1514 y Fz(1)p 1297 1558 V 1297 1649 a Ft(N)1385 1620 y Fq(d)p Fo(\000)p Fw(1)1543 1581 y Fz(log)17 b Ft(r)1730 1596 y Fq(N)1825 1581 y Fz(=)27 b Fs(\000)p Ft(\034)11 b Fz(\()l Ft(~)-45 b(e)2141 1596 y Fq(d)2183 1581 y Fz(\))22 b(+)g(\001)2422 1596 y Fq(h)2463 1608 y Fm(d)2520 1581 y Ft(:)865 b Fz(\(6.61\))218 1795 y(Iterating)33 b(\(6.60\))o(,)g(w)m(e)g(get)g(for)f(an)m(y)h Ft(f)484 2032 y Fs(8)p Ft(t)c Fr(>)f Fz(0)p Ft(;)995 1948 y Fp(\015)995 2007 y(\015)1051 2032 y Fx(E)1111 1952 y Fp(\000)1163 2032 y Ft(f)11 b Fz(\()p Ft(\033)1319 1985 y Fq(\021)1315 2055 y(t)1360 2032 y Fz(\))1398 1952 y Fp(\001)1466 2032 y Fs(\000)23 b Ft(\026)1625 1991 y Fk(h)1625 2057 y Fq(N)1692 2032 y Fz(\()p Ft(f)11 b Fz(\))1827 1948 y Fp(\015)1827 2007 y(\015)1882 2071 y Fo(1)2001 2032 y Fr(6)28 b Fs(k)p Ft(f)11 b Fs(k)2265 2047 y Fo(1)2355 2032 y Fz(exp)2521 1892 y Fp(\022)2594 2032 y Fs(\000)p Ft(r)2715 2047 y Fq(N)2800 1892 y Fp(\024)2880 1965 y Ft(t)p 2862 2009 71 4 v 2862 2101 a(T)2943 1892 y Fp(\025\023)3102 2032 y Ft(:)283 b Fz(\(6.62\))118 2269 y(This)33 b Fx(L)401 2233 y Fo(1)515 2269 y Fz(con)m(traction)f (and)g(\(6.61\))g(imply)f(Lemma)g(6.1.)218 2502 y(W)-8 b(e)40 b(turn)h(no)m(w)g(to)f(the)h(deriv)-5 b(ation)38 b(of)47 b(\(6.60\))o(.)67 b(F)-8 b(or)40 b(tec)m(hnical)f(reasons,)44 b(it)39 b(will)f(b)s(e)j(con-)118 2618 y(v)m(enien)m(t)f(to)e(replace)g (the)h(free)g(b)s(oundary)g(conditions)f(b)m(y)h(a)f(small)f(coupling)g Ft(")g(>)h Fz(0)g(and)h(to)118 2734 y(consider)f(the)g(ev)m(olution)e (of)h(the)h(Glaub)s(er)e(dynamics)h(asso)s(ciated)h(to)f(the)g (generator)h(whic)m(h)118 2855 y(tak)m(es)h(in)m(to)d(accoun)m(t)i(the) g(new)g(b)s(oundary)f(conditions.)56 b(Let)38 b(us)g(denote)g(b)m(y)g Ft(\026)3107 2808 y Fk(h)p Fq(;")3107 2882 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5327 y(\014)1330 5387 y(\014)1407 5411 y Fr(6)29 b Fz(\(1)21 b Fs(\000)i Ft(p)p Fz(\))p Fs(k)p Ft(f)11 b Fs(k)1967 5426 y Fo(1)2063 5411 y Fz(+)22 b(sup)2216 5491 y Fq(\021)2325 5327 y Fp(\014)2325 5387 y(\014)2358 5411 y Fx(E)2418 5331 y Fp(\000)2470 5411 y Ft(f)2518 5426 y Fw(0)2557 5411 y Fz(\()p Ft(\033)2654 5364 y Fq(\021)2650 5436 y(t)p Fw(=1)2771 5411 y Fz(\))2809 5331 y Fp(\001)2871 5327 y(\014)2871 5387 y(\014)2937 5411 y Ft(:)p eop %%Page: 30 30 30 29 bop 118 51 a Fu(30)1148 b(T.)26 b(BODINEA)n(U)e(AND)g(D.)i(IOFFE) 118 366 y Fz(Th)m(us)38 b(\(6.60\))d(will)f(follo)m(w)g(if)h(one)h(can) h(deriv)m(e)f(that)g(for)g(an)m(y)g Ft( )k Fz(whic)m(h)d(do)s(es)f(not) g(dep)s(end)h(on)118 493 y(the)c(spins)g(in)f Ft(R)719 508 y Fw(0)791 493 y Fz(and)h(has)g(zero)g(mean)f(under)h Ft(\026)1956 446 y Fk(h)p Fq(;")1956 520 y(N)2057 493 y Fz(,)1025 663 y(sup)1079 742 y Fq(\021)1188 579 y Fp(\014)1188 638 y(\014)1222 663 y Fx(E)1282 583 y Fp(\000)1334 663 y Ft( )t Fz(\()p Ft(\033)1498 616 y 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3585 48 a Fu(31)218 366 y Fz(As)33 b Ft( )k Fz(do)s(es)c(not)g(dep)s(end)h(on)f(the)g(spins)h(in)e Ft(R)1933 381 y Fw(0)1972 366 y Fz(,)i(the)f(supp)s(ort)g(of)g Ft(g)2721 381 y Fw(1)2793 366 y Fz(is)f(included)h(in)f Fs(S)3456 381 y Fq(N)s(;\016)3595 366 y Fs([)118 487 y Fz(\()p Ft(R)230 502 y Fw(0)294 487 y Fs([)25 b Ft(R)459 502 y Fw(1)499 487 y Fz(\))537 451 y Fq(c)572 487 y Fz(,)37 b(where)g Fs(S)981 502 y Fq(N)s(;\016)1132 487 y Fz(=)1242 407 y Fp(\010)1300 487 y Ft(i)d Fs(2)g Ft(R)1541 502 y Fw(0)1581 487 y Fz(;)114 b Ft(i)1755 502 y Fq(d)1830 487 y Fz(=)1949 448 y Fw(1)p 1949 464 36 4 v 1949 522 a(2)1995 487 y Ft(\016)t(N)2130 407 y Fp(\011)2188 487 y Fz(.)54 b(W)-8 b(e)37 b(consider)g(the)f(ev)m(en)m(t)i Fs(A)3337 502 y Fq(\021)3415 487 y Fz(whic)m(h)118 604 y(decouples)c(the)f(spins)g(in)e Fs(S)1144 619 y Fq(N)s(;\016)1294 604 y Fz(from)g(the)i(b)s(oundary)g(conditions)f Ft(\021)k Fz(outside)d Ft(R)3093 619 y Fw(0)1018 814 y Fs(A)1098 829 y Fq(\021)1167 814 y Fz(=)1270 733 y 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Fs([)g Ft(R)1697 1514 y Fw(1)1737 1499 y Fz(\))1775 1463 y Fq(c)1809 1499 y Fz(,)33 b(w)m(e)h(can)e(apply)h(Prop)s(osition)d(5.1)j(to)f(get)841 1585 y Fp(\014)841 1645 y(\014)874 1670 y Ft(\026)933 1622 y Fw(+)p Fq(;";\021)933 1697 y(R)986 1706 y Fl(0)1102 1670 y Fz(\()p Ft(g)1187 1685 y Fw(1)1226 1670 y Fz(\))22 b Fs(\000)h Ft(\026)1445 1622 y Fw(+)p Fq(;";\021)1445 1697 y(R)1498 1706 y Fl(0)1532 1697 y Fo([)p Fq(R)1632 1706 y Fl(1)1671 1670 y Fz(\()p Ft(g)1756 1685 y Fw(1)1795 1670 y Fz(\))1833 1585 y Fp(\014)1833 1645 y(\014)1894 1670 y Fr(6)45 b Fz(exp)q(\()p Fs(\000)p Ft(c\016)t(N)10 b Fz(\))p Fs(k)p Ft(f)h Fs(k)2654 1685 y Fo(1)2745 1670 y Ft(:)640 b Fz(\(6.70\))118 1849 y(Since)33 b Ft(\026)432 1801 y Fw(+)p Fq(;";\021)432 1876 y(R)485 1885 y Fl(0)519 1876 y Fo([)p Fq(R)619 1885 y Fl(1)658 1849 y Fz(\()p Ft(g)743 1864 y Fw(1)782 1849 y Fz(\))28 b(=)f Ft(\026)1010 1801 y Fw(+)p Fq(;";\021)1010 1876 y(R)1063 1885 y Fl(0)1098 1876 y Fo([)p Fq(R)1198 1885 y Fl(1)1237 1849 y Fz(\()p Ft(g)1322 1864 y Fw(2)1361 1849 y Fz(\),)32 b(the)h(previous)g(argumen) m(t)g(can)g(b)s(e)f(iterated)449 2120 y Ft(\026)508 2073 y Fw(+)p Fq(;";\021)508 2147 y(R)561 2156 y Fl(0)677 2120 y Fz(\()p Ft(g)762 2135 y Fw(1)801 2120 y Fz(\))27 b(=)973 1996 y Fq(L)p Fo(\000)p Fw(1)970 2026 y Fp(X)985 2236 y Fq(i)p Fw(=0)1130 2010 y Fp(\020)1190 2120 y Ft(\026)1249 2073 y Fw(+)p Fq(;";\021)1249 2147 y(R)1302 2156 y Fl(0)1336 2147 y Fo([\001\001\001)o([)p Fq(R)1542 2157 y Fm(i)1573 2120 y Fz(\()p Ft(g)1658 2135 y Fq(i)p Fw(+1)1776 2120 y Fz(\))22 b Fs(\000)g Ft(\026)1994 2073 y Fw(+)p Fq(;";\021)1994 2147 y(R)2047 2156 y Fl(0)2082 2147 y Fo([\001\001\001)o([)p Fq(R)2288 2157 y Fm(i)p Fl(+1)2395 2120 y Fz(\()p Ft(g)2480 2135 y Fq(i)p Fw(+1)2598 2120 y Fz(\))2636 2010 y Fp(\021)2718 2120 y Fz(+)g Ft(\026)2875 2073 y Fw(+)p Fq(;")2875 2147 y(N)2986 2120 y Fz(\()p Ft(f)11 b Fz(\))17 b Ft(;)118 2370 y Fz(where)30 b(w)m(e)g(used)g(that)f Ft(f)38 b Fz(=)28 b Ft(g)1200 2385 y Fq(L)p Fw(+1)1370 2370 y Fz(and)h Fx(D)1622 2385 y Fq(N)1723 2370 y Fz(=)f Ft(R)1901 2385 y Fw(0)1955 2370 y Fs([)15 b(\001)i(\001)g(\001)c([)i Ft(R)2322 2385 y Fq(L)2374 2370 y Fz(.)42 b(Using)29 b(the)g(fact)g(that)f(for)g Ft(i)g Fr(>)h Fz(1,)118 2487 y(the)f(restriction)e(of)g Ft(g)894 2502 y Fq(i)p Fw(+1)1039 2487 y Fz(to)h Ft(R)1227 2502 y Fw(0)1277 2487 y Fs([)11 b(\001)17 b(\001)g(\001)9 b([)i Ft(R)1632 2502 y Fq(i)p Fw(+1)1778 2487 y Fz(is)26 b(measurable)h(wrt)g Ft(R)2622 2502 y Fw(0)2672 2487 y Fs([)11 b(\001)17 b(\001)g(\001)10 b([)h Ft(R)3028 2502 y Fq(i)p Fo(\000)p Fw(1)3173 2487 y Fz(an)27 b(estimate)118 2603 y(similar)i(to)k(\(6.70\))e(holds.)218 2719 y(F)-8 b(or)31 b Ft(")d(>)f Fz(0,)33 b(an)f(argumen)m(t)h(similar) c(to)j(the)h(one)g(used)h(in)d(Prop)s(osition)g(5.1)h(implies)962 2803 y Fp(\014)962 2862 y(\014)995 2887 y Ft(\026)1054 2840 y Fw(+)p Fq(;")1054 2914 y(N)1165 2887 y Fz(\()p Ft(f)11 b Fz(\))22 b Fs(\000)h Ft(\026)1481 2840 y Fq(h;")1481 2914 y(N)1577 2887 y Fz(\()p Ft(f)11 b Fz(\))1712 2803 y Fp(\014)1712 2862 y(\014)1773 2887 y Fr(6)45 b Fz(exp)q(\()p Fs(\000)p Ft(c\016)t(N)10 b Fz(\))p Fs(k)p Ft(f)h Fs(k)2533 2902 y Fo(1)2624 2887 y Ft(:)218 3071 y Fz(By)40 b(construction)g Ft(\026)1003 3024 y Fk(h)p Fq(;")1003 3098 y(N)1104 3071 y Fz(\()p Ft(f)11 b Fz(\))39 b(=)h(0.)65 b(Summarizing)37 b(the)j(previous)g(estimates,)h(there)g(exists)118 3187 y Ft(c)28 b(>)f Fz(0)33 b(suc)m(h)h(that)893 3351 y(sup)948 3430 y Fq(\021)1057 3266 y Fp(\014)1057 3326 y(\014)1090 3351 y Ft(\026)1149 3304 y Fw(+)p Fq(;";\021)1149 3378 y(R)1202 3387 y Fl(0)1317 3351 y Fz(\()p Ft(g)1402 3366 y Fw(1)1441 3351 y Fz(\))1479 3266 y Fp(\014)1479 3326 y(\014)1557 3351 y Fr(6)28 b Ft(LN)1816 3310 y Fq(d)p Fo(\000)p Fw(1)1964 3351 y Fz(exp)q(\()p Fs(\000)p Ft(c\016)t(N)10 b Fz(\))p Fs(k)p Ft(f)h Fs(k)2602 3366 y Fo(1)2693 3351 y Ft(:)218 3583 y Fz(Th)m(us)34 b(\(6.67\))e(holds.)118 3771 y(6.2.)49 b FE(Upp)s(er)34 b(Bound)h(on)g(the)f(sp)s(ectral)g (gap.)50 b Fz(W)-8 b(e)30 b(turn)g(no)m(w)h(to)e(the)h(deriv)-5 b(ation)29 b(of)g(The-)118 3887 y(orem)j(3.5.)43 b(F)-8 b(or)32 b(an)m(y)h Ft(m)28 b Fs(2)p Fz(])23 b Fs(\000)f Ft(m)1329 3851 y Fo(\003)1369 3887 y Fz(\()p Ft(\014)6 b Fz(\))p Ft(;)17 b(m)1635 3851 y Fo(\003)1674 3887 y Fz(\()p Ft(\014)6 b Fz(\)[,)32 b(w)m(e)i(set)1104 4054 y Ft(A)1177 4069 y Fq(m)1271 4054 y Fz(=)1375 3973 y Fp(\010)1433 4054 y Ft(\033)d Fs(2)d(f\000)p Fz(1)p Ft(;)17 b Fz(1)p Fs(g)1932 4013 y Fn(D)1974 4024 y Fm(N)2069 4054 y Fs(j)98 b Fx(M)2283 4069 y Fq(N)2384 4054 y Fr(6)28 b Ft(m)2574 3973 y Fp(\011)2649 4054 y Ft(;)118 4227 y Fz(where)43 b Fx(M)497 4242 y Fq(N)612 4227 y Fz(denotes)g(the)g(a)m (v)m(eraged)g(magnetization)c Fx(M)2296 4242 y Fq(N)2413 4227 y Fz(=)k(1)p Ft(=)-5 b(N)2713 4191 y Fq(d)2787 4152 y Fp(P)2892 4256 y Fq(i)p Fo(2)p Fn(D)3005 4267 y Fm(N)3089 4227 y Ft(\033)3144 4242 y Fq(i)3173 4227 y Fz(.)71 b(Applying)118 4350 y(form)m(ula)45 b(\(3.24\))g(to)h(the)h(test)g(function)f Ft(f)11 b Fz(\()p Ft(\033)t Fz(\))51 b(=)g(1)2121 4366 y Fo(f)p Fq(\033)r Fo(2)p Fq(A)2298 4374 y Fm(m)2357 4366 y Fo(g)2397 4350 y Fz(,)e(w)m(e)f(get)e(the)h(follo)m(wing)d(upp)s (er)118 4466 y(b)s(ound)33 b(on)f(the)h(sp)s(ectral)g(gap)919 4700 y(SG\()p Ft(N)5 b(;)17 b FE(h)p Fz(\))28 b Fr(6)g Fz(\(2)p Ft(N)10 b Fz(\))1661 4659 y Fq(d)1969 4626 y Ft(\026)2028 4590 y Fk(h)2028 4651 y Fq(N)2095 4545 y Fp(\000)2141 4626 y Ft(@)5 b(A)2270 4641 y Fq(m)2337 4545 y Fp(\001)p 1711 4677 930 4 v 1711 4769 a Ft(\026)1770 4735 y Fk(h)1770 4796 y Fq(N)1837 4769 y Fz(\()p Ft(A)1948 4784 y Fq(m)2015 4769 y Fz(\)\(1)22 b Fs(\000)g Ft(\026)2320 4735 y Fk(h)2320 4796 y Fq(N)2387 4769 y Fz(\()p Ft(A)2498 4784 y Fq(m)2565 4769 y Fz(\)\))2667 4700 y Ft(;)718 b Fz(\(6.71\))118 4921 y(where)34 b Ft(@)5 b(A)529 4936 y Fq(m)629 4921 y Fz(is)32 b(the)h(b)s(oundary)g(of)f(the)h(set)h Ft(A)1839 4936 y Fq(m)118 5084 y Ft(@)5 b(A)247 5099 y Fq(m)342 5084 y Fz(=)446 5004 y Fp(\010)504 5084 y Ft(\033)32 b Fs(2)c(f\000)p Fz(1)p Ft(;)17 b Fz(1)p Fs(g)1004 5043 y Fn(D)1045 5054 y Fm(N)1141 5084 y Fs(j)32 b(9)p Ft(x)c Fs(2)g Fx(D)1499 5099 y Fq(N)1573 5084 y Ft(;)114 b(\033)32 b Fs(2)c Ft(A)1968 5099 y Fq(m)2034 5084 y Ft(;)50 b(\033)2170 5043 y Fq(x)2241 5084 y Fs(62)28 b Ft(A)2408 5099 y Fq(m)2475 5084 y Ft(;)114 b Fz(or)97 b Ft(\033)32 b Fs(62)c Ft(A)3054 5099 y Fq(m)3121 5084 y Ft(;)49 b(\033)3256 5043 y Fq(x)3328 5084 y Fs(2)28 b Ft(A)3495 5099 y Fq(m)3562 5004 y Fp(\011)3636 5084 y Ft(:)118 5248 y Fz(Optimizing)c(this)j(inequalit)m(y)f(o)m(v)m(er)i Ft(m)f Fz(will)e(enables)i(us)h(to)e(b)s(ound)h(the)h(sp)s(ectral)f (gap)f(in)g(terms)118 5364 y(of)32 b(equilibrium)e(quan)m(tities.)p eop %%Page: 34 34 34 33 bop 118 51 a Fu(34)1148 b(T.)26 b(BODINEA)n(U)e(AND)g(D.)i(IOFFE) 218 366 y Fz(F)-8 b(or)35 b(an)m(y)i Ft(\014)k Fz(in)36 b Fs(B)s Fz(,)h(the)g(measure)f Ft(\026)1544 330 y Fk(h)1544 392 y Fq(N)1647 366 y Fz(con)m(v)m(erges)j(to)c(the)i(pure)g(phase)g Ft(\026)2942 330 y Fw(+)3037 366 y Fz(in)e(the)h(thermo-)118 483 y(dynamic)e(limit)d(as)k(so)s(on)g(as)g(one)g(of)f(the)h(co)s (ordinates)f(of)g Ft(h)h Fz(is)f(p)s(ositiv)m(e.)50 b(As)35 b(a)f(consequence)118 599 y Ft(\026)177 563 y Fk(h)177 624 y Fq(N)244 599 y Fz(\()p Ft(A)355 614 y Fq(m)422 599 y Fz(\))e(v)-5 b(anishes)33 b(as)g Ft(N)43 b Fz(go)s(es)33 b(to)f(in\014nit)m(y)g(for)g Ft(m)h Fz(in)f([)p Fs(\000)p Ft(m)2356 563 y Fo(\003)2396 599 y Fz(\()p Ft(\014)6 b Fz(\))p Ft(;)17 b(m)2662 563 y Fo(\003)2701 599 y Fz(\()p Ft(\014)6 b Fz(\)[.)218 715 y(The)35 b(set)h Ft(A)648 730 y Fq(m)749 715 y Fz(con)m(tains)e(the)h(con\014gurations)f(in)g (the)h Fs(\000)g Fz(phase)h(whic)m(h)f(can)g(b)s(e)f(asso)s(ciated,)118 831 y(on)44 b(the)g(macroscopic)f(lev)m(el,)j(to)e(the)g(function)f Ft(u)g Fz(uniformly)f(equal)h(to)h Fs(\000)p Fz(1.)77 b(In)44 b(this)f(case)118 948 y(there)31 b(is)f(no)g(in)m(terface)g(in) g(the)h(bulk)f(and)g(the)h(in)m(terfacial)d(energy)j(is)f(concen)m (trated)i(along)d(the)118 1064 y(b)s(oundary)-8 b(.)44 b(A)33 b(straigh)m(tforw)m(ard)f(adaptation)f(of)h(prop)s(osition)f (4.1)h(of)g([BIV2])h(implies)580 1350 y(lim)17 b(inf)613 1412 y Fq(N)7 b Fo(!1)979 1283 y Fz(1)p 894 1327 220 4 v 894 1419 a Ft(N)982 1390 y Fq(d)p Fo(\000)p Fw(1)1167 1350 y Fz(log)17 b Ft(\026)1369 1309 y Fk(h)1369 1375 y Fq(N)1436 1350 y Fz(\()p Ft(A)1547 1365 y Fq(m)1613 1350 y Fz(\))28 b Fr(>)50 b Fs(\000)23 b(F)1988 1309 y Fk(h)2037 1350 y Fz(\()p Fs(\000)p Ft(m)2237 1309 y Fo(\003)2277 1350 y Fz(\()p Ft(\014)6 b Fz(\)\))27 b(=)g Fs(\000)2713 1226 y Fw(2)p Fq(d)2676 1256 y Fp(X)2691 1466 y Fq(i)p Fw(=1)2837 1350 y Fz(\001)2918 1365 y Fq(h)2959 1375 y Fm(i)3006 1350 y Ft(:)379 b Fz(\(6.72\))218 1631 y(Prop)s(osition)31 b(4.2)h(of)g([BIV2])h(implies)d(the)j(follo)m(wing) d(upp)s(er)j(b)s(ound)375 1869 y(lim)17 b(sup)455 1951 y Fq(\016)r Fo(!)p Fw(0)691 1869 y Fz(lim)g(sup)738 1950 y Fq(N)7 b Fo(!1)1118 1801 y Fz(1)p 1033 1846 V 1033 1937 a Ft(N)1121 1908 y Fq(d)p Fo(\000)p Fw(1)1306 1869 y Fz(log)17 b Ft(\026)1508 1827 y Fk(h)1508 1893 y Fq(N)1575 1788 y Fp(\000)1621 1869 y Fs(f)p Fx(M)1758 1884 y Fq(N)1859 1869 y Fs(2)28 b Fz([)p Ft(m)23 b Fs(\000)g Ft(\016)n(;)17 b(m)22 b Fz(+)g Ft(\016)t Fz(])p Fs(g)2602 1788 y Fp(\001)2676 1869 y Fr(6)50 b Fs(\000)22 b(F)2984 1827 y Fk(h)3033 1869 y Fz(\()p Ft(m)p Fz(\))17 b Ft(:)118 2111 y Fz(The)37 b(previous)f(inequalit)m(y)f(rests)i(up)s(on)f(the)g(lo)m(w)m(er)g (semi-con)m(tin)m(uit)m(y)f(of)g(the)h(functional)f Fs(W)3612 2126 y Fk(h)118 2227 y Fz(whic)m(h,)e(for)f(the)h(sak)m(e)h(of)e (completeness,)i(is)e(pro)m(v)m(en)i(in)e(the)h(App)s(endix.)218 2343 y(F)-8 b(or)31 b(an)m(y)j Ft(\016)d(>)d Fz(0,)k Ft(@)5 b(A)992 2358 y Fq(m)1092 2343 y Fz(is)33 b(included)f(in)g(the)h (set)g Fs(f)p Fx(M)2153 2358 y Fq(N)2254 2343 y Fs(2)28 b Fz([)p Ft(m)22 b Fs(\000)h Ft(\016)n(;)17 b(m)23 b Fz(+)f Ft(\016)t Fz(])p Fs(g)p Fz(,)32 b(th)m(us)325 2581 y Fs(8)p Ft(m)d Fs(2)p Fz(])22 b Fs(\000)h Ft(m)794 2540 y Fo(\003)834 2581 y Fz(\()p Ft(\014)6 b Fz(\))p Ft(;)17 b(m)1100 2540 y Fo(\003)1139 2581 y Fz(\()p Ft(\014)6 b Fz(\)[)p Ft(;)211 b Fz(lim)17 b(sup)1589 2662 y Fq(N)7 b Fo(!1)1968 2513 y Fz(1)p 1883 2558 V 1883 2649 a Ft(N)1971 2620 y Fq(d)p Fo(\000)p Fw(1)2157 2581 y Fz(log)16 b Ft(\026)2358 2540 y Fk(h)2358 2605 y Fq(N)2425 2581 y Fz(\()p Ft(@)5 b(A)2592 2596 y Fq(m)2660 2581 y Fz(\))27 b Fr(6)51 b Fs(\000)22 b(F)3034 2540 y Fk(h)3083 2581 y Fz(\()p Ft(m)p Fz(\))17 b Ft(:)124 b Fz(\(6.73\))218 2828 y(Com)m(bining)31 b(estimates)h(\(6.72\))g(and)g(\(6.73\),)g(w)m (e)i(conclude)f(Theorem)g(3.5.)118 2980 y FE(Remark)49 b(6.1.)d Fj(T)-7 b(o)43 b(our)h(b)-5 b(est)43 b(know)5 b(le)-5 b(dge,)44 b(in)f(dimension)f Ft(d)i Fr(>)g Fz(3)p Fj(,)h(even)e(the)h(solutions)f(of)118 3096 y(the)38 b(isop)-5 b(erimetric)36 b(pr)-5 b(oblem)44 b Fz(\(2.11\))37 b Fj(in)g(a)g(cub)-5 b(e)37 b(with)h("fr)-5 b(e)g(e")37 b(b)-5 b(oundary)37 b(c)-5 b(onditions)36 b(ar)-5 b(e)37 b(not)118 3213 y(known)44 b(for)h(gener)-5 b(al)44 b(volume)g(c)-5 b(onstr)g(aints.)74 b(In)44 b(the)h(p)-5 b(articular)45 b(c)-5 b(ase)44 b(of)h(isotr)-5 b(opic)44 b(surfac)-5 b(e)118 3329 y(tension)38 b(and)g(for)h(a)g(volume)f(c)-5 b(onstr)g(aint)38 b(which)g(is)h(the)g(half)f(of)g(the)h(volume)f(of)h (the)g(cub)-5 b(e,)39 b(it)118 3445 y(was)33 b(only)h(r)-5 b(e)g(c)g(ently)34 b(pr)-5 b(oven)33 b(by)h(Barthe)g(and)f(Maur)-5 b(ey)35 b Fz([BaMa])f Fj(that)g(the)g(solution)f(is)h(the)g(half)118 3561 y(cub)-5 b(e.)218 3678 y(This)35 b(explains)g(why,)h(in)g(The)-5 b(or)g(em)35 b(3.5,)h(the)g(de)-5 b(c)g(ay)36 b(of)f(the)h(sp)-5 b(e)g(ctr)g(al)36 b(gap)g(c)-5 b(annot)35 b(b)-5 b(e)36 b(char-)118 3794 y(acterize)-5 b(d)34 b(mor)-5 b(e)35 b(pr)-5 b(e)g(cisely.)1596 4083 y Fz(7.)49 b Fy(Appendix)118 4257 y Fz(7.1.)g FE(Lo)m(w)m(er)27 b(semi-con)m(tin)m(uit)m(y.)50 b Fz(By)25 b(considering)e(appropriate)g(b)s(oundary)h(conditions,)h(w) m(e)118 4373 y(are)34 b(going)e(to)h(reduce)i Fs(W)1079 4337 y Fk(h)1162 4373 y Fz(to)e(a)h(functional)d(whic)m(h)j(do)s(es)g (not)g(explicitly)d(tak)m(e)k(in)m(to)e(accoun)m(t)118 4489 y(the)g(b)s(oundary)g(\014eld.)43 b(W)-8 b(e)33 b(set)657 4869 y Ft(g)708 4827 y Fk(h)757 4869 y Fz(\()p Ft(x)p Fz(\))61 b(=)1085 4605 y Fp(8)1085 4694 y(>)1085 4724 y(>)1085 4754 y(<)1085 4933 y(>)1085 4963 y(>)1085 4993 y(:)1254 4664 y Fs(j)p Fz(\001)1363 4679 y Fq(h)1404 4689 y Fm(i)1434 4664 y Fs(j)p 1232 4709 252 4 v 1232 4800 a Fz(2)p Ft(\034)11 b Fz(\()l Ft(~)-45 b(e)1417 4815 y Fq(i)1446 4800 y Fz(\))1494 4732 y Ft(;)195 b Fz(if)38 b Ft(x)1867 4747 y Fq(i)1918 4732 y Fs(\000)23 b Fz(1)p Ft(=)p Fz(2)k Fr(>)44 b Fz(max)2388 4792 y Fq(j)2495 4732 y Fs(fj)p Ft(x)2628 4747 y Fq(j)2687 4732 y Fs(\000)22 b Fz(1)p Ft(=)p Fz(2)p Fs(jg)17 b Ft(;)1239 4937 y Fs(j)p Fz(\001)1348 4952 y Fq(h)1389 4962 y Fl(2)p Fm(i)1449 4937 y Fs(j)p 1232 4982 V 1232 5073 a Fz(2)p Ft(\034)11 b Fz(\()l Ft(~)-45 b(e)1417 5088 y Fq(i)1446 5073 y Fz(\))1494 5005 y Ft(;)195 b Fz(if)38 b Ft(x)1867 5020 y Fq(i)1918 5005 y Fs(\000)23 b Fz(1)p Ft(=)p Fz(2)k Fr(6)44 b Fz(min)2379 5065 y Fq(j)2476 5005 y Fs(f\000j)p Ft(x)2686 5020 y Fq(j)2745 5005 y Fs(\000)23 b Fz(1)p Ft(=)p Fz(2)p Fs(jg)17 b Ft(:)218 5248 y Fz(T)-8 b(o)26 b(an)m(y)i(function)e Ft(u)g Fz(of)h(b)s(ounded)g(v)-5 b(ariation,)26 b(w)m(e)h(asso)s(ciate) g Ft(D)s(u)f Fz(the)h(v)m(ector)h(measure)f(of)f(its)118 5364 y(\014rst)34 b(partial)d(deriv)-5 b(ativ)m(es)34 b(and)f Fs(j)p Ft(D)s(u)p Fs(j)f Fz(the)i(p)s(ositiv)m(e)f(measure)h (obtained)f(b)m(y)h(taking)f(the)h(total)118 5480 y(v)-5 b(ariation)28 b(of)i Ft(D)s(u)p Fz(.)42 b(Finally)-8 b(,)28 b(w)m(e)j(denote)g(b)m(y)d Ft(~)-46 b(n)30 b Fz(the)h(v)m(ector) g(function)f(obtained)g(as)g(the)h(Radon)p eop %%Page: 35 35 35 34 bop 3585 48 a Fu(35)118 381 y Fz(Nyk)m(o)s(dim)32 b(deriv)-5 b(ativ)m(e)32 b(of)1126 342 y Fq(dD)r(u)p 1106 358 177 4 v 1106 416 a(d)p Fo(j)p Fq(D)r(u)p Fo(j)1292 381 y Fz(.)44 b(F)-8 b(or)32 b(an)m(y)h(function)f Ft(u)g Fz(in)g(BV)q(\(in)m(t)2612 355 y Fp(b)2604 381 y Fx(D)25 b Ft(;)17 b Fs(f\006)p Fz(1)p Fs(g)p Fz(\))o(,)33 b(one)g(sets)1123 617 y Fi(W)1227 576 y Fk(h)1277 617 y Fz(\()p Ft(u)p Fz(\))27 b(=)1549 550 y(1)p 1549 594 49 4 v 1549 686 a(2)1624 482 y Fp(Z)1680 707 y Fo(O)1758 617 y Ft(\034)1800 632 y Fq(\014)1848 617 y Fz(\()m Ft(~)-46 b(n)1944 632 y Fq(x)1988 617 y Fz(\))17 b Ft(d)p Fs(j)p Ft(D)s(u)j Fs(_)j Ft(g)2422 576 y Fk(h)2470 617 y Fs(j)p Fz(\()p Ft(x)p Fz(\))p Ft(:)756 b Fz(\(7.74\))118 865 y(This)33 b(functional)e(is)h(lo)m(w)m(er)h(semi-con)m(tin)m(uous)f(w.r.t.)44 b Fx(L)2194 829 y Fw(1)2240 865 y Fz(-con)m(v)m(ergence.)h(Let)33 b Ft(s)3070 880 y Fq(i)3126 865 y Fz(=)3268 807 y Fo(j)p Fw(\001)3347 819 y Fm(h)3383 834 y(i)3413 807 y Fo(j)p 3240 842 223 4 v 3240 899 a Fw(2)p Fq(\034)3306 911 y Fm(\014)3348 899 y Fw(\()n Fq(~)-33 b(e)3408 909 y Fm(i)3434 899 y Fw(\))3504 865 y Fz(and)118 1027 y(recall)41 b(that)h Fs(P)53 b Fz(=)44 b Fs([)917 991 y Fw(2)p Fq(d)917 1051 y(i)p Fw(=1)1036 1027 y Fs(P)1105 1042 y Fq(i)1176 1027 y Fz(denotes)f(the)g(faces)g(of)f(the)h(cub)s(e)2509 1001 y Fp(b)2501 1027 y Fx(D)25 b Fz(.)72 b(This)43 b(functional)e(can) h(b)s(e)118 1143 y(rewritten)33 b(as)g(follo)m(ws)338 1328 y Fi(W)442 1287 y Fk(h)491 1328 y Fz(\()p Ft(u)p Fz(\))82 b(=)864 1193 y Fp(Z)920 1418 y Fq(@)961 1395 y Fh(\003)957 1456 y Fm(g)989 1442 y Fe(h)1036 1418 y Fq(u)p Fo(nP)1187 1328 y Ft(\034)1229 1343 y Fq(\014)1277 1328 y Fz(\()m Ft(~)-46 b(n)1373 1343 y Fq(x)1417 1328 y Fz(\))p Ft(d)p Fs(H)1591 1287 y Fw(\()p Fq(d)p Fo(\000)p Fw(1\))1590 1353 y Fq(x)1059 1608 y Fz(+)1152 1513 y Fp(X)1212 1723 y Fq(i)1296 1608 y Fz(\()p Fs(jP)1431 1623 y Fq(i)1459 1608 y Fs(j)22 b(\000)g Ft(S)1668 1623 y Fq(i)1697 1608 y Fz(\)\(1)f Fs(\000)i Ft(s)1989 1623 y Fq(i)2017 1608 y Fz(\))p Ft(\034)2097 1623 y Fq(\014)2145 1608 y Fz(\()l Ft(~)-45 b(e)2228 1623 y Fq(i)2256 1608 y Fz(\))22 b(+)g Ft(S)2474 1623 y Fq(i)2502 1608 y Fz(\(1)g(+)g Ft(s)2755 1623 y Fq(i)2783 1608 y Fz(\))p Ft(\034)2863 1623 y Fq(\014)2911 1608 y Fz(\()l Ft(~)-45 b(e)2994 1623 y Fq(i)3022 1608 y Fz(\))22 b(+)g Ft(C)7 b Fz(\()p Fs(O)s Fz(\))p Ft(;)118 1865 y Fz(where)34 b Ft(S)460 1880 y Fq(i)521 1865 y Fz(stands)f(for)f(the)i(Hausdor\013)e(measure)h (of)g Ft(@)2135 1829 y Fo(\003)2175 1865 y Fs(ff)p Ft(u)27 b Fz(=)h Fs(\000)p Fz(1)p Fs(g)22 b(_)g Ft(g)2799 1829 y Fk(h)2848 1865 y Fs(g)g(\\)h(P)3078 1880 y Fq(i)3139 1865 y Fz(and)32 b Ft(C)7 b Fz(\()p Fs(O)s Fz(\))33 b(is)118 1996 y(the)g(v)-5 b(ariation)30 b(of)i Ft(g)k Fz(in)c Fs(O)25 b(n)1187 1970 y Fp(b)1179 1996 y Fx(D)g Fz(.)44 b(W)-8 b(e)33 b(reco)m(v)m(er)h Fs(W)1929 1960 y Fk(h)1979 1996 y Fz(\()p Ft(u)p Fz(\))e(up)h(to)f(a)g(constan)m(t)777 2175 y Fs(W)883 2134 y Fk(h)933 2175 y Fz(\()p Ft(u)p Fz(\))27 b(=)h Fi(W)1300 2134 y Fk(h)1349 2175 y Fz(\()p Ft(u)p Fz(\))22 b Fs(\000)1602 2080 y Fp(X)1662 2290 y Fq(i)1763 2175 y Fs(jP)1860 2190 y Fq(i)1888 2175 y Fs(j)17 b Fz(\()o Ft(\034)2012 2190 y Fq(\014)2060 2175 y Fz(\()l Ft(~)-45 b(e)2143 2190 y Fq(i)2171 2175 y Fz(\))22 b Fs(\000)h Fz(\001)2412 2190 y Fq(h)2453 2200 y Fm(i)2483 2175 y Ft(=)p Fz(2\))f Fs(\000)g Ft(C)7 b Fz(\()p Fs(O)s Fz(\))p Ft(:)410 b Fz(\(7.75\))118 2432 y(This)33 b(implies)d(that)i (the)h(functional)e Fs(W)1615 2396 y Fk(h)1665 2432 y Fz(\()p Ft(u)p Fz(\))h(is)g(lo)m(w)m(er)h(semi-con)m(tin)m(uous.)118 2607 y(7.2.)49 b FE(Pro)s(of)37 b(of)g(Prop)s(osition)h(2.1.)49 b Fz(.)218 2724 y(W)-8 b(e)33 b(split)e(the)i(pro)s(of)f(in)m(to)g(sev) m(eral)h(steps:)118 2956 y FE(Step)58 b(1.)46 b Fz(If)34 b Ft(\034)669 2971 y Fq(\014)750 2956 y Fz(is)f(strictly)g(con)m(v)m (ex)j(at)29 b Ft(~)-45 b(e)1670 2971 y Fq(d)1711 2956 y Fz(,)34 b(then)g(also)f Ft(\034)2245 2920 y Fq(\017)2234 2982 y(\014)2315 2956 y Fz(is)g(strictly)g(con)m(v)m(ex)j(at)29 b Ft(~)-45 b(e)3235 2971 y Fq(d)3276 2956 y Fz(.)47 b(Indeed,)118 3078 y(de\014ne)41 b FE(x)g Fz(=)g(\(0)p Ft(;)17 b(:)g(:)g(:)e(;)i Fz(0)p Ft(;)g(\034)1064 3093 y Fq(\014)1111 3078 y Fz(\()l Ft(~)-45 b(e)1194 3093 y Fq(d)1234 3078 y Fz(\)\).)66 b(Th)m(us,)44 b FE(x)c Fz(b)s(elongs)g(to)g Ft(@)5 b Fs(K)q Fz(,)43 b(it)c(is)h(just)g(a)g(p)s(oin)m(t)g(where)h(the)114 3194 y Ft(~)-45 b(e)163 3209 y Fq(d)204 3194 y Fz(-orthogonal)35 b(h)m(yp)s(erplane)j(touc)m(hes)h Ft(@)5 b Fs(K)q Fz(.)58 b(Of)37 b(course,)j Ft(\034)2336 3158 y Fq(\017)2325 3220 y(\014)2373 3194 y Fz(\()l Ft(~)-45 b(e)2456 3209 y Fq(d)2496 3194 y Fz(\))36 b(=)f(\()p FE(x)p Ft(;)13 b(~)-45 b(e)2867 3209 y Fq(d)2908 3194 y Fz(\))37 b(for)g(ev)m(ery)i Ft(\017)d(>)g Fz(0.)118 3316 y(The)i(inequalit)m(y)f(\(2.14\))g(can)g (b)s(e)g(equiv)-5 b(alen)m(tly)37 b(reform)m(ulated)f(as)h(follo)m(ws:) 51 b(at)37 b(least)g(for)f(one)118 3432 y(of)c(the)h(v)m(ectors)d Ft(~)-45 b(v)775 3447 y Fq(k)818 3432 y Fz(,)33 b Ft(\034)920 3447 y Fq(\014)967 3432 y Fz(\()l Ft(~)-45 b(v)1052 3447 y Fq(k)1095 3432 y Fz(\))28 b Ft(>)f Fz(\()p FE(x)p Ft(;)13 b(~)-45 b(v)1452 3447 y Fq(k)1495 3432 y Fz(\),)32 b(or,)h(in)f(other)g (w)m(ords,)1014 3597 y Fs(f)p Ft(x)27 b Fs(2)i Fx(R)1307 3556 y Fq(d)1397 3597 y Fz(:)45 b(\()p Ft(x)22 b Fs(\000)h FE(x)p Ft(;)13 b(~)-45 b(v)1834 3612 y Fq(k)1876 3597 y Fz(\))28 b(=)g(0)p Fs(g)21 b(\\)i Fz(in)m(t)o(\()p Fs(K)q Fz(\))45 b Fs(6)p Fz(=)f Fs(;)p Ft(:)118 3754 y Fz(Since)j(the)g(W)-8 b(ul\013)45 b(shap)s(e)j Fs(K)f Fz(is)f(con)m(v)m(ex)j(and)e(has)g(a)f(non-empt)m(y)h(in)m(teriour,)h (the)f(latter)f(is)118 3871 y(equiv)-5 b(alen)m(t)32 b(to)1011 3987 y Fs(f)p Ft(x)c Fs(2)g Fx(R)1304 3946 y Fq(d)1394 3987 y Fz(:)45 b(\()p Ft(x)22 b Fs(\000)h FE(x)p Ft(;)13 b(~)-45 b(v)1831 4002 y Fq(k)1874 3987 y Fz(\))27 b(=)h(0)p Fs(g)22 b(\\)g Fz(in)m(t\()p Fs(K)2484 3946 y Fq(\017)2517 3987 y Fz(\))44 b Fs(6)p Fz(=)g Fs(;)118 4124 y Fz(for)32 b(ev)m(ery)j Ft(\017)28 b(>)f Fz(0.)43 b(Hence)34 b Ft(\034)1157 4088 y Fq(\017)1146 4150 y(\014)1194 4124 y Fz(\()l Ft(~)-45 b(v)1279 4139 y Fq(k)1322 4124 y Fz(\))27 b Ft(>)h Fz(\()p FE(x)p Ft(;)13 b(~)-45 b(v)1679 4139 y Fq(k)1721 4124 y Fz(\))33 b(as)g(w)m(ell.)118 4267 y FE(Step)71 b(2.)81 b Fz(Belo)m(w)45 b(w)m(e)h(use)g(a)e (simpli\014ed)f(notation)h Ft(\034)60 b Fs(\021)50 b Ft(\034)2444 4231 y Fq(\017)2433 4293 y(\014)2480 4267 y Fz(.)81 b(Let)45 b Ft(E)55 b Fs(\032)49 b(f)p Ft(x)66 b Fz(:)g Ft(x)3347 4282 y Fq(d)3436 4267 y Fr(>)50 b Fz(0)p Fs(g)118 4389 y Fz(b)s(e)35 b(a)f(b)s(ounded)h(set)g(of)f (\014nite)g(p)s(erimeter)f(and)i(p)s(ositiv)m(e)e(v)m(olume)h(\()p Ft(d)p Fz(-dimensional)d(Hausdor\013)118 4505 y(measure\).)59 b(Let)37 b Ft(@)824 4469 y Fo(\003)865 4505 y Ft(E)43 b Fz(b)s(e)38 b(the)g(reduced)h(b)s(oundary)e([EG])h(of)f Ft(E)6 b Fz(.)58 b(Let)38 b(us)g(split)e(it)g(as)i Ft(@)3431 4469 y Fo(\003)3471 4505 y Ft(E)43 b Fz(=)118 4621 y Ft(A)22 b Fs([)h Fz(\006,)33 b(where)1014 4738 y(\006)28 b(=)g Ft(@)1272 4697 y Fo(\003)1312 4738 y Ft(E)h Fs(n)21 b(f)p Ft(x)28 b Fs(2)g Fx(R)1777 4697 y Fq(d)1868 4738 y Fz(:)45 b Ft(x)1995 4753 y Fq(d)2063 4738 y Fz(=)28 b(0)p Fs(g)f Fz(=)g Ft(@)2452 4697 y Fo(\003)2493 4738 y Ft(E)h Fs(n)22 b Ft(A:)118 4875 y Fz(W)-8 b(e)33 b(claim)d(that)1192 4884 y Fp(Z)1247 5110 y Fw(\006)1319 5020 y Ft(\034)11 b Fz(\()m Ft(~)-46 b(n)1468 5035 y Fq(x)1512 5020 y Fz(\))p Ft(d)p Fs(H)1686 4979 y Fq(d)p Fo(\000)p Fw(1)1685 5044 y Fq(x)1861 5020 y Ft(>)44 b Fs(H)2066 4979 y Fq(d)p Fo(\000)p Fw(1)2197 5020 y Fz(\()p Ft(A)p Fz(\))p Ft(\034)11 b Fz(\()l Ft(~)-45 b(e)2482 5035 y Fq(d)2523 5020 y Fz(\))p Ft(:)824 b Fz(\(7.76\))118 5214 y(By)33 b(the)g(Gauss-Green)g(form)m (ula)e([EG],)1321 5424 y Fs(H)1406 5382 y Fq(d)p Fo(\000)p Fw(1)1537 5424 y Fz(\()p Ft(A)p Fz(\))l Ft(~)-45 b(e)1731 5439 y Fq(d)1816 5424 y Fz(=)1936 5288 y Fp(Z)1991 5514 y Fw(\006)2060 5424 y Ft(~)f(n)2121 5439 y Fq(x)2165 5424 y Ft(d)p Fs(H)2301 5382 y Fq(d)p Fo(\000)p Fw(1)2300 5448 y Fq(x)2431 5424 y Ft(:)954 b Fz(\(7.77\))p eop %%Page: 36 36 36 35 bop 118 51 a Fu(36)1148 b(T.)26 b(BODINEA)n(U)e(AND)g(D.)i(IOFFE) 118 366 y Fz(In)35 b(view)g(of)41 b(\(2.14\))34 b(it)g(is)g(enough)h (to)g(sho)m(w)h(that)e(one)h(can)g(\014nd)g(a)g(decomp)s(osition)e(of)h (\006)h(in)m(to)118 483 y(a)d(disjoin)m(t)g(union)g(\006)c(=)f(\006) 1089 498 y Fw(1)1151 483 y Fs([)c(\001)17 b(\001)g(\001)j([)j Fz(\006)1537 498 y Fq(d)1578 483 y Fz(,)32 b(suc)m(h)i(that)f(the)g(v)m (ectors)1178 688 y Ft(~)-45 b(v)1229 703 y Fq(k)1324 633 y Fw(\001)1316 688 y Fz(=)1436 553 y Fp(Z)1491 778 y Fw(\006)1542 790 y Fm(k)1598 688 y Ft(~)f(n)1659 703 y Fq(x)1703 688 y Ft(d)p Fs(H)1839 647 y Fq(d)p Fo(\000)p Fw(1)1838 713 y Fq(x)2067 688 y Ft(k)31 b Fz(=)d(1)p Ft(;)17 b(:)g(:)g(:)e(;)i(d;)814 b Fz(\(7.78\))118 910 y(are)37 b(in)g(the)h(general)f(p)s(osition.)56 b(A)m(t)37 b(this)g(stage)h(the)f(p)s(ositivit)m(y)f(of)h(the)h(v)m(olume)f(of)g Ft(E)43 b Fz(en)m(ters)118 1026 y(the)36 b(picture.)51 b(By)36 b(the)g(con)m(tin)m(uit)m(y)f(one)g(can)h(pic)m(k)g(p)s(ositiv) m(e)e(n)m(um)m(b)s(ers)i(0)c(=)g Ft(a)3026 1041 y Fw(0)3098 1026 y Ft(<)g(a)3257 1041 y Fw(1)3329 1026 y Ft(<)g Fs(\001)17 b(\001)g(\001)31 b Ft(<)118 1143 y(a)169 1158 y Fq(d)p Fo(\000)p Fw(1)328 1143 y Ft(<)c(a)482 1158 y Fq(d)551 1143 y Fz(=)g Fs(1)32 b Fz(suc)m(h)i(that)218 1306 y(\(i\))64 b(min)548 1321 y Fq(k)21 b Ff(6)f Fq(d)738 1306 y Fs(H)823 1270 y Fq(d)880 1306 y Fz(\()p Ft(E)28 b Fs(\\)23 b(f)p Ft(x)44 b Fz(:)h Ft(a)1379 1321 y Fq(k)r Fo(\000)p Fw(1)1539 1306 y Ft(<)28 b(x)1698 1321 y Fq(d)1767 1306 y Ft(<)f(a)1921 1321 y Fq(k)1964 1306 y Fs(g)p Fz(\))g Ft(>)h Fz(0.)218 1444 y(\(ii\))i(min)543 1459 y Fq(k)21 b Ff(6)f Fq(d)p Fo(\000)p Fw(1)823 1444 y Fs(H)908 1407 y Fq(d)p Fo(\000)p Fw(1)1055 1444 y Fz(\()p Ft(E)28 b Fs(\\)23 b(f)p Ft(x)44 b Fz(:)h Ft(x)1558 1459 y Fq(d)1626 1444 y Fz(=)28 b Ft(a)1781 1459 y Fq(k)1824 1444 y Fs(g)p Fz(\))f Ft(>)h Fz(0.)218 1607 y(Of)42 b(course,)k Ft(E)35 b Fs(\\)30 b(f)p Ft(x)61 b Fz(:)h Ft(a)1213 1622 y Fq(k)r Fo(\000)p Fw(1)1391 1607 y Ft(<)45 b(x)1567 1622 y Fq(d)1652 1607 y Ft(<)g(a)1824 1622 y Fq(k)1867 1607 y Fs(g)d Fz(is)g(just)h(the)h (part)e(of)g Ft(E)49 b Fz(whic)m(h)43 b(is)f(c)m(hopp)s(ed)118 1752 y(out)i(b)m(y)c Ft(~)-45 b(e)494 1767 y Fq(d)535 1752 y Fz(-orthogonal)41 b(h)m(yp)s(erplanes)k(through)f(the)g(p)s(oin) m(ts)f FE(x)2539 1767 y Fq(k)r Fo(\000)p Fw(1)2727 1696 y(\001)2719 1752 y Fz(=)j(\(0)p Ft(;)17 b(:)g(:)g(:)e(;)i Fz(0)p Ft(;)g(a)3290 1767 y Fq(k)r Fo(\000)p Fw(1)3423 1752 y Fz(\))43 b(and)118 1896 y FE(x)177 1911 y Fq(k)281 1840 y Fw(\001)272 1896 y Fz(=)52 b(\(0)p Ft(;)17 b(:)g(:)g(:)f(;)h Fz(0)p Ft(;)g(a)850 1911 y Fq(k)892 1896 y Fz(\))47 b(resp)s(ectiv)m (ely)-8 b(.)88 b(Since)47 b Ft(E)53 b Fz(is)46 b(a)h(set)h(of)e (\014nite)h(p)s(erimeter)f(w)m(e)i(ma)m(y)f(in)118 2012 y(addition)21 b(assume)j(that)f(small)e(p)s(erturbations)i(of)f(these)j (h)m(yp)s(erplanes)f(retain)f(b)s(oth)f(prop)s(erties)118 2128 y(ab)s(o)m(v)m(e.)44 b(Sp)s(eci\014cally)-8 b(,)31 b(there)i(exist)f(p)s(ositiv)m(e)f(n)m(um)m(b)s(ers)i Ft(\016)2235 2143 y Fw(1)2274 2128 y Ft(;)17 b(:)g(:)g(:)f(;)h(\016) 2536 2143 y Fq(d)p Fo(\000)p Fw(1)2694 2128 y Ft(>)28 b Fz(0,)k(suc)m(h)h(that)f(the)g(sets)118 2245 y(\()p Ft(k)f Fz(=)c(2)p Ft(;)17 b(:)g(:)g(:)f(;)h(d)k Fs(\000)i Fz(1\))444 2432 y Ft(S)504 2447 y Fq(k)599 2376 y Fw(\001)591 2432 y Fz(=)44 b Ft(E)28 b Fs(\\)23 b(f)p Ft(x)44 b Fz(:)h(\()p Ft(x)22 b Fs(\000)h FE(x)1395 2447 y Fq(k)1438 2432 y Ft(;)13 b(~)-45 b(e)1527 2447 y Fq(d)1589 2432 y Fz(+)22 b Ft(\016)1730 2447 y Fq(k)1769 2432 y Ft(~)-45 b(e)1818 2447 y Fq(k)1861 2432 y Fz(\))27 b Ft(<)h Fz(0)f Ft(<)h Fz(\()p Ft(x)23 b Fs(\000)f FE(x)2484 2447 y Fq(k)r Fo(\000)p Fw(1)2617 2432 y Ft(;)13 b(~)-45 b(e)2706 2447 y Fq(d)2769 2432 y Fz(+)22 b Ft(\016)2910 2447 y Fq(k)r Fo(\000)p Fw(1)3039 2432 y Ft(~)-45 b(e)3088 2447 y Fq(k)r Fo(\000)p Fw(1)3221 2432 y Fz(\))p Fs(g)p Ft(;)118 2618 y(S)178 2633 y Fw(1)255 2563 y(\001)246 2618 y Fz(=)29 b Fs(f)p Ft(x)45 b Fz(:)h(\()p Ft(x)23 b Fs(\000)g FE(x)849 2633 y Fw(1)889 2618 y Ft(;)13 b(~)-45 b(e)978 2633 y Fq(d)1040 2618 y Fz(+)23 b Ft(\016)1182 2633 y Fq(k)1221 2618 y Ft(~)-45 b(e)1270 2633 y Fw(1)1309 2618 y Fz(\))29 b Ft(<)g Fz(0)f Ft(<)h Fz(\()p Ft(x;)13 b(~)-45 b(e)1845 2633 y Fq(d)1886 2618 y Fz(\))p Fs(g)32 b Fz(and)i Ft(S)2257 2633 y Fq(d)2326 2618 y Fz(=)29 b Ft(E)f Fs(n)23 b([)2670 2577 y Fq(d)p Fo(\000)p Fw(1)2670 2646 y Fq(k)r Fw(=1)2803 2618 y Ft(S)2863 2633 y Fq(k)2906 2618 y Fz(,)33 b(are)g(disjoin)m(t)f (and,)118 2735 y(furthermore,)38 b(eac)m(h)f(and)g(ev)m(ery)m(one)j(of) c(the)h(corresp)s(onding)g(p)s(ortions)f(of)g(their)g(b)s(oundaries,) 118 2851 y(whic)m(h)d(w)m(e)h(denote)f(as)g(\006)1045 2866 y Fq(k)1116 2851 y Fz(=)27 b(\006)c Fs(\\)f Ft(S)1460 2866 y Fq(k)1503 2851 y Fz(;)49 b Ft(k)31 b Fz(=)d(1)p Ft(;)17 b(:)g(:)g(:)e(;)i(d)32 b Fz(and)669 3010 y Ft(A)742 3025 y Fq(k)812 3010 y Fz(=)c Ft(E)g Fs(\\)22 b(f)p Ft(x)45 b Fz(:)f(\()p Ft(x)23 b Fs(\000)f FE(x)1599 3025 y Fq(k)1642 3010 y Ft(;)13 b(~)-45 b(e)1731 3025 y Fq(d)1794 3010 y Fz(+)22 b Ft(\016)1935 3025 y Fq(k)1974 3010 y Ft(~)-45 b(e)2023 3025 y Fq(k)2065 3010 y Fz(\))28 b(=)g(0)p Fs(g)p Fz(;)49 b Ft(k)31 b Fz(=)c(1)p Ft(;)17 b(:)g(:)g(:)f(;)h(d)k Fs(\000)i Fz(1)p Ft(;)118 3168 y Fz(has)33 b(a)f(p)s(ositiv)m(e)g(\()p Ft(d)22 b Fs(\000)h Fz(1\)-dimensional)29 b(Hausdor\013)k(measure.)218 3285 y(Subsequen)m(t)f(application)27 b(of)i(the)h(Gauss-Green)g(form)m (ula)e(on)i(eac)m(h)g(of)f(the)h(sets)h Ft(S)3303 3300 y Fw(1)3342 3285 y Ft(;)17 b(:)g(:)g(:)f(;)h(S)3621 3300 y Fq(d)118 3401 y Fz(leads)41 b(no)m(w)g(to)f(the)i(follo)m(wing)c(c)m (hain)i(of)g(equalities)g(for)g(the)h(v)m(ectors)e Ft(~)-45 b(v)2850 3416 y Fw(1)2889 3401 y Ft(;)17 b(:)g(:)g(:)f(;)d(~)-45 b(v)3155 3416 y Fq(d)3236 3401 y Fz(de\014ned)42 b(in)118 3517 y(\(7.78\))o(:)392 3680 y(0)28 b Fs(6)p Fz(=)g Ft(~)-45 b(v)624 3695 y Fq(d)758 3680 y Fz(=)27 b(\()l Ft(~)-45 b(e)944 3695 y Fq(d)1007 3680 y Fz(+)22 b Ft(\016)1148 3695 y Fq(d)p Fo(\000)p Fw(1)1275 3680 y Ft(~)-45 b(e)1324 3695 y Fq(d)p Fo(\000)p Fw(1)1455 3680 y Fz(\))16 b Fs(H)1594 3639 y Fq(d)p Fo(\000)p Fw(1)1725 3680 y Fz(\()p Ft(A)1836 3695 y Fq(d)p Fo(\000)p Fw(1)1966 3680 y Fz(\))392 3844 y(0)28 b Fs(6)p Fz(=)g Ft(~)-45 b(v)624 3859 y Fq(d)p Fo(\000)p Fw(1)783 3844 y Fz(=)27 b Fs(\000)17 b Fz(\()l Ft(~)-45 b(e)1063 3859 y Fq(d)1126 3844 y Fz(+)22 b Ft(\016)1267 3859 y Fq(d)p Fo(\000)p Fw(1)1394 3844 y Ft(~)-45 b(e)1443 3859 y Fq(d)p Fo(\000)p Fw(1)1574 3844 y Fz(\))16 b Fs(H)1713 3803 y Fq(d)p Fo(\000)p Fw(1)1844 3844 y Fz(\()p Ft(A)1955 3859 y Fq(d)p Fo(\000)p Fw(1)2086 3844 y Fz(\))22 b(+)g(\()l Ft(~)-45 b(e)2327 3859 y Fq(d)2390 3844 y Fz(+)22 b Ft(\016)2531 3859 y Fq(d)p Fo(\000)p Fw(2)2657 3844 y Ft(~)-45 b(e)2706 3859 y Fq(d)p Fo(\000)p Fw(2)2837 3844 y Fz(\))17 b Fs(H)2977 3803 y Fq(d)p Fo(\000)p Fw(1)3108 3844 y Fz(\()p Ft(A)3219 3859 y Fq(d)p Fo(\000)p Fw(2)3349 3844 y Fz(\))594 3989 y Fs(\001)g(\001)g(\001)392 4153 y Fz(0)28 b Fs(6)p Fz(=)g Ft(~)-45 b(v)624 4168 y Fw(1)757 4153 y Fz(=)27 b Fs(\000)17 b Fz(\()l Ft(~)-45 b(e)1037 4168 y Fq(d)1100 4153 y Fz(+)22 b Ft(\016)1241 4168 y Fw(1)1277 4153 y Ft(~)-45 b(e)1326 4168 y Fw(1)1365 4153 y Fz(\))17 b Fs(H)1505 4111 y Fq(d)p Fo(\000)p Fw(1)1635 4153 y Fz(\()p Ft(A)1746 4168 y Fw(1)1786 4153 y Fz(\))22 b(+)c Ft(~)-45 b(e)1989 4168 y Fq(d)2030 4153 y Fs(H)2115 4111 y Fq(d)p Fo(\000)p Fw(1)2245 4153 y Fz(\()p Ft(A)p Fz(\))p Ft(:)118 4337 y Fz(Recall)40 b(that)i(b)m(y)i(\(7.77\))1079 4262 y Fp(P)1184 4289 y Fq(d)1184 4366 y Fw(1)1237 4337 y Ft(~)-45 b(v)1288 4352 y Fq(k)1375 4337 y Fz(=)39 b Ft(~)-45 b(e)1539 4352 y Fq(d)1580 4337 y Fs(H)1665 4301 y Fq(d)p Fo(\000)p Fw(1)1795 4337 y Fz(\()p Ft(A)p Fz(\).)72 b(Consequen)m(tly)-8 b(,)42 b Ft(~)-45 b(v)2725 4352 y Fw(1)2765 4337 y Ft(;)17 b(:)g(:)g(:)f(;)d(~)-45 b(v)3031 4352 y Fq(d)3113 4337 y Fz(span)42 b Fx(R)3416 4301 y Fq(d)3504 4337 y Fz(and)118 4453 y(\(7.76\))32 b(follo)m(ws.)118 4622 y FE(Step)44 b(3.)c Fz(Finally)19 b(w)m(e)k(turn)f(to)f(the)h(pro)s(of)f(of)g(Prop)s (osition)f(2.1)i(prop)s(er.)40 b(Let)22 b Ft(u)27 b Fs(2)h Fz(BV)3285 4512 y Fp(\020)3345 4622 y Fz(in)m(t)3486 4596 y Fp(b)3478 4622 y Fx(D)d Ft(;)17 b Fs(f\006)p Fz(1)p Fs(g)3820 4512 y Fp(\021)3879 4622 y Fz(.)118 4770 y(Set)43 b Ft(E)50 b Fz(=)44 b Fs(f)p Ft(x)61 b Fz(:)g Ft(u)p Fz(\()p Ft(x)p Fz(\))45 b(=)f Fs(\000)p Fz(1)p Fs(g)p Fz(.)72 b(As)43 b(in)f FE(Step)49 b(2)p Fz(,)c(split)c Ft(@)2376 4734 y Fo(\003)2416 4770 y Ft(E)50 b Fz(=)44 b(\006)30 b Fs([)f Ft(A)p Fz(.)73 b(The)43 b(functional)130 4878 y Fp(c)118 4903 y Fs(W)216 4918 y Fq(\014)s(;\017)328 4903 y Fz(\()p Ft(u)p Fs(j)p Ft(g)t Fz(\))31 b(can)i(b)s(e)g(then)g (written)g(\(in)e(the)i(notation)e Ft(\034)40 b Fs(\021)28 b Ft(\034)2395 4867 y Fq(\017)2384 4929 y(\014)2431 4903 y Fz(\))33 b(as)751 5103 y Fp(c)739 5128 y Fs(W)837 5143 y Fq(\014)s(;\017)949 5048 y Fp(\000)995 5128 y Ft(u)1068 5044 y Fp(\014)1068 5103 y(\014)1117 5128 y Ft(g)1168 5048 y Fp(\001)1257 5128 y Fz(=)1377 4993 y Fp(Z)1432 5218 y Fw(\006)1504 5128 y Ft(\034)11 b Fz(\()m Ft(~)-46 b(n)1653 5143 y Fq(x)1698 5128 y Fz(\))p Ft(d)p Fs(H)1872 5087 y Fq(d)p Fo(\000)p Fw(1)1871 5153 y Fq(x)2041 5128 y Fz(+)2155 5048 y Fp(\000)2201 5128 y Fz(1)22 b Fs(\000)h(H)2457 5087 y Fq(d)p Fo(\000)p Fw(1)2587 5128 y Fz(\()p Ft(A)p Fz(\))2736 5048 y Fp(\001)2799 5128 y Ft(\034)11 b Fz(\()l Ft(~)-45 b(e)2935 5143 y Fq(d)2976 5128 y Fz(\))p Ft(:)118 5344 y Fz(By)35 b(\(7.76\))o(,)1270 5453 y Fp(c)1258 5479 y Fs(W)1356 5494 y Fq(\014)s(;\017)1469 5398 y Fp(\000)1514 5479 y Ft(u)1587 5394 y Fp(\014)1587 5454 y(\014)1636 5479 y Ft(g)1687 5398 y Fp(\001)1776 5479 y Ft(>)1908 5453 y Fp(c)1897 5479 y Fs(W)1995 5494 y Fq(\014)s(;\017)2107 5398 y Fp(\000)2153 5479 y Fz(1)-25 b(I\()p Fs(\001)p Fz(\))2333 5394 y Fp(\014)2333 5454 y(\014)2382 5479 y Ft(g)2433 5398 y Fp(\001)2494 5479 y Ft(;)p eop %%Page: 37 37 37 36 bop 3585 48 a Fu(37)118 366 y Fz(as)33 b(so)s(on)f(as)h Fs(H)668 330 y Fq(d)709 366 y Fz(\()p Ft(E)6 b Fz(\))27 b Ft(>)h Fz(0.)43 b(But)33 b(this)f(is)g(precisely)h(the)g(claim)d(of)i (the)h(Prop)s(osition.)3584 483 y Fr(\003)1599 740 y Fy(References)118 898 y FD([ABCP])41 b(G.)26 b(Alb)r(erti,)e(G.)i (Bellettini,)c(M.)j(Cassandro,)f(E.)h(Presutti,)f Fv(Surfac)l(e)k (tension)f(in)h(Ising)f(system)h(with)289 997 y(Kac)i(p)l(otentials)7 b FD(,)28 b(J.)g(Stat.)g(Ph)n(ys.)e Fd(82)p FD(,)h(743{796)e(\(1996\).) 118 1097 y([A])63 b(K.)34 b(Alexander,)h Fv(The)i(Sp)l(e)l(ctr)l(al)f (Gap)i(of)f(the)f(2-D)g(Sto)l(chastic)h(Ising)g(Mo)l(del)g(with)g(Ne)l (arly)g(Single-Spin)289 1196 y(Boundary)31 b(Conditions)p FD(,)e(J.)e(Stat.)h(Ph)n(ys.)f Fd(104)p FD(,)g(no.)g(1-2,)g(59{87)e (\(2001\).)118 1296 y([ABL])42 b(Aizenman,)22 b(Bricmon)n(t,)f(Leb)r(o) n(witz,)h Fv(Per)l(c)l(olation)27 b(of)f(the)g(minority)g(spins)f(in)h (high-dimensional)i(Ising)289 1396 y(mo)l(dels)7 b FD(,)28 b(J.)g(Stat.)g(Ph)n(ys.)e Fd(49)p FD(,)h(859{865)e(\(1987\).)118 1495 y([A)-7 b(Y])42 b(K.)e(Alexander,)i(N.)f(Y)-7 b(oshida,)42 b Fv(The)g(sp)l(e)l(ctr)l(al)g(gap)h(of)f(the)f(2D)h(Ising)g(mo)l(del)g (with)g(mixe)l(d)g(b)l(oundary)289 1595 y(c)l(onditions)7 b FD(,)28 b(J.)g(Stat.)g(Ph)n(ys.)e Fd(104)p FD(,)h(no.)h(1-2,)e (89{109)f(\(2001\).)118 1695 y([BaMa])41 b(F.)29 b(Barthe,)f(B.)g (Maurey)-7 b(,)28 b Fv(Some)j(r)l(emarks)g(on)f(isop)l(erimetry)j(of)e (Gaussian)g(typ)l(e)p FD(,)f(Ann.)f(Inst.)g(Henri)289 1794 y(P)n(oincare,)24 b(Prob.)j(Stat.)h(36,)e(No.4,)h(419-434)e (\(2000\).)118 1894 y([BoMa])41 b(T.)28 b(Bo)r(dineau,)g(F.)h (Martinelli,)24 b Fv(Some)31 b(new)g(r)l(esults)e(on)i(the)g(kinetic)g (Ising)g(mo)l(del)h(in)e(a)h(pur)l(e)g(phase)p FD(,)289 1993 y(J.)c(Statist.)g(Ph)n(ys.)g Fd(109)p FD(,)g(no.)g(1-2,)g(207{235) d(\(2002\).)118 2093 y([BBBP])41 b(O.)31 b(Benois,)f(T.)h(Bo)r(dineau,) g(P)-7 b(.)31 b(Butta,)h(E.)f(Presutti,)g Fv(On)h(the)i(validity)h(of)f (van)f(der)h(Waals)g(the)l(ory)289 2193 y(of)c(surfac)l(e)h(tension)6 b FD(,)28 b(Mark.)e(Pro)r(c.)h(and)g(Rel.)g(Fields)f Fd(3)p FD(,)h(175{198)e(\(1997\).)118 2292 y([BCK])41 b(M.)47 b(Biskup,)i(L.)e(Cha)n(y)n(es,)i(R.)e(Kotec)n(ky)-7 b(,)50 b Fv(Critic)l(al)e(r)l(e)l(gion)f(for)h(dr)l(oplet)g(formation)g (in)f(the)h(two-)289 2392 y(dimensional)31 b(Ising)f(mo)l(del)p FD(,)f(preprin)n(t,)d(\(2002\).)118 2492 y([BIV1])41 b(T.)e(Bo)r(dineau,)h(D.)f(Io\013e,)i(Y.)e(V)-7 b(elenik,)39 b Fv(R)n(igor)l(ous)h(pr)l(ob)l(abilistic)i(analysis)f(of)g(e)l (quilibrium)g(crystal)289 2591 y(shap)l(es)7 b FD(,)28 b(J.)g(Math.)g(Ph)n(ys.)e Fd(41)p FD(,)h(No.3,)h(1033{1098)23 b(\(2000\).)118 2691 y([BIV2])41 b(T.)22 b(Bo)r(dineau,)f(D.)h (Io\013e,)g(Y.)g(V)-7 b(elenik,)21 b Fv(Winterb)l(ottom)j(c)l (onstruction)f(for)i(\014nite)f(r)l(ange)g(ferr)l(omagnetic)289 2790 y(mo)l(dels:)39 b(A)n(n)29 b Fc(L)787 2802 y FA(1)824 2790 y Fv(-appr)l(o)l(ach)6 b FD(,)30 b(J.)e(Stat.)f(Ph)n(ys.)g Fd(105)p FD(,)g(No.)h(1-2,)e(93{131)f(\(2001\).)118 2890 y([CePi])40 b(R.)d(Cerf,)h(A.)f(Pisztora,)e Fv(On)i(the)h(Wul\013)g (crystal)g(in)g(the)g(Ising)g(mo)l(del)9 b FD(,)39 b(Ann.)e(Probab.)e Fd(28)p FD(,)j(no.)e(3,)289 2990 y(947{1017)24 b(\(2000\).)118 3089 y([DH])42 b(R.L.)25 b(Dobrushin,)f(O.)g(Hryniv,)g Fv(Fluctuations)i(of)i(the)e(phase)j(b)l(oundary)e(in)g(the)g FD(2)p Fv(D)f(Ising)h(ferr)l(omagnet)p FD(,)289 3189 y(Comm.)e(Math.)j(Ph)n(ys.)f Fd(189)p FD(,)g(no.)g(2,)g(395{445,)e (\(1997\).)118 3289 y([DKS])42 b(R.L.)35 b(Dobrushin,)g(R.)g(Kotec)n(k) q(\023)-43 b(y,)36 b(S.)f(Shlosman,)f Fv(Wul\013)i(c)l(onstruction:)51 b(a)37 b(glob)l(al)h(shap)l(e)g(fr)l(om)e(lo)l(c)l(al)289 3388 y(inter)l(action)6 b FD(,)28 b(AMS)g(translations)d(series,)g(v)n (ol)h(104,)g(Pro)n(vidence)f(R.I.)j(\(1992\).)118 3488 y([DS1])42 b(R.L)37 b(Dobrushin,)g(S.B.)g(Shlosman,)f Fv(The)j(pr)l(oblem)g(of)g(tr)l(anslation)f(invarianc)l(e)i(of)f(Gibbs) g(states)e(at)289 3587 y(low)30 b(temp)l(er)l(atur)l(es)p FD(,)d(Mathematical)d(ph)n(ysics)i(reviews,)f Fd(5)p FD(,)i(53{195,)e(Harw)n(o)r(o)r(d)h(Academic)f(Publ.,)i(Ch)n(ur,)289 3687 y(1985.)118 3787 y([DS2])42 b(R.L)d(Dobrushin,)i(S.B.)e(Shlosman,) g Fv(Thermo)l(dynamic)k(ine)l(qualities)e(for)g(surfac)l(e)f(tension)h (and)f(the)289 3886 y(ge)l(ometry)e(of)g(the)g(Wul\013)f(c)l (onstruction)p FD(,)h(Ideas)d(and)h(metho)r(ds)f(in)h(quan)n(tum)f(and) h(statistical)c(ph)n(ysics)289 3986 y(\(Oslo,)26 b(1988\),)g(461{483,)e (Cam)n(bridge)g(Univ.)j(Press,)f(Cam)n(bridge,)f(\(1992\).)118 4086 y([DS3])42 b(R.L)33 b(Dobrushin,)h(S.B.)f(Shlosman,)f Fv(Completely)37 b(analytic)l(al)f(Gibbs)g(\014elds)p FD(,)f(Statistical)30 b(ph)n(ysics)i(and)289 4185 y(dynamical)g (systems)j(\(K\177)-42 b(oszeg,)36 b(1984\),)h(371{403,)e(Progr.)e(Ph)n (ys.,)38 b(10,)f(Birkh\177)-42 b(auser)33 b(Boston,)k(Boston,)289 4285 y(MA,)28 b(\(1985\).)118 4384 y([El])44 b(R.)c(Ellis,)g Fv(Entr)l(opy,)45 b(lar)l(ge)d(deviations,)k(and)c(statistic)l(al)g(me) l(chanics)7 b FD(,)44 b(Springer-V)-7 b(erlag,)39 b(New)h(Y)-7 b(ork,)289 4484 y(\(1985\).)118 4584 y([EG])41 b(L.)22 b(Ev)-5 b(ans,)22 b(R.)g(Gariep)n(y)-7 b(,)21 b Fv(Me)l(asur)l(e)j(The) l(ory)i(and)e(Fine)h(Pr)l(op)l(erties)g(of)g(F)-6 b(unctions)7 b FD(,)22 b(CR)n(C)g(Press,)f(London)289 4683 y(\(1992\).)118 4783 y([ES])41 b(R.)28 b(Edw)n(ards,)e(A.)i(Sok)-5 b(al)26 b(:)37 b(Generalization)23 b(of)28 b(the)g(F)-7 b(ortuin-Kasteleyn-Sw)n (endsen-W)g(ang)23 b(represen)n(ta-)289 4883 y(tion)j(and)i(Mon)n(te)f (Carlo)e(algorithm,)f Fv(Phys.)31 b(R)l(ev.)f(D)p FD(,)d Fd(38)p FD(,)h(No.)f(6,)g(2009-2012,)d(\(1988\).)118 4982 y([FP1])41 b(J.)26 b(F)-7 b(r\177)-42 b(ohlic)n(h,)24 b(C.-E.)h(P\014ster,)h Fv(Semi{in\014nite)j(Ising)g(mo)l(del)g(I.)g (Thermo)l(dynamic)i(functions)d(and)h(phase)289 5082 y(diagr)l(am)i(in)f(absenc)l(e)g(of)h(magnetic)f(\014eld)9 b FD(,)28 b(Comm.)e(Math.)h(Ph)n(ys.)g Fd(109)p FD(,)g(493{523)d (\(1987\).)118 5181 y([FP2])41 b(J.)22 b(F)-7 b(r\177)-42 b(ohlic)n(h,)20 b(C.-E.)i(P\014ster,)g Fv(Semi{in\014nite)j(Ising)g(mo) l(del)h(II.)f(The)h(wetting)e(and)h(layering)i(tr)l(ansitions)7 b FD(,)289 5281 y(Comm.)25 b(Math.)j(Ph)n(ys.)f Fd(112)p FD(,)g(51{74)e(\(1987\).)118 5381 y([Gri])40 b(G.)33 b(Grimmett,)e Fv(The)j(sto)l(chastic)h(r)l(andom)g(cluster)e(pr)l(o)l (c)l(ess)h(and)h(the)e(uniqueness)h(of)g(r)l(andom)h(cluster)289 5480 y(me)l(asur)l(es)7 b FD(,)27 b(Ann.)h(Prob.)f Fd(23)p FD(,)g(1461{1510)c(\(1995\).)p eop %%Page: 38 38 38 37 bop 118 51 a Fu(38)1148 b(T.)26 b(BODINEA)n(U)e(AND)g(D.)i(IOFFE) 118 366 y FD([GZ])42 b(A.)24 b(Guionnet,)g(B.)f(Zegarlinski)c Fv(L)l(e)l(ctur)l(es)25 b(on)h(L)l(o)l(garithmic)i(Sob)l(olev)f(ine)l (qualities)p FD(,)f(S)n(\023)-39 b(eminaire)19 b(de)24 b(Prob-)289 466 y(abilit)n(\023)-39 b(es,)23 b(XXXVI,)29 b(1{134,)c(Lecture)j(Notes)f(in)g(Math.,)h(1801,)d(Springer,)h(Berlin,) f(\(2003\).)118 566 y([ISc])42 b(D.)29 b(Io\013e,)h(R.)f(Sc)n(honmann,) f Fv(Dobrushin-Kote)l(ck)o(\023)-41 b(y-Shlosman)32 b(the)l(ory)f(up)g (to)g(the)h(critic)l(al)g(temp)l(er)l(atur)l(e)6 b FD(,)289 665 y(Comm.)25 b(Math.)j(Ph)n(ys.)f Fd(199)p FD(,)g(117{167)d (\(1998\).)118 765 y([Ma])41 b(F.)29 b(Martinelli)c Fv(L)l(e)l(ctur)l (es)30 b(on)g(Glaub)l(er)i(dynamics)g(for)f(discr)l(ete)h(spin)f(mo)l (dels)p FD(,)f(93{191,)c(Lecture)j(Notes)289 865 y(in)e(Math.,)g(1717,) f(Springer,)g(Berlin,)f(\(1999\).)118 964 y([MO])41 b(F.)f (Martinelli,E.)35 b(Olivieri,)i Fv(Appr)l(o)l(ach)42 b(to)e(e)l(quilibrium)h(of)h(Glaub)l(er)f(dynamics)g(in)g(the)f(one)h (phase)289 1064 y(r)l(e)l(gion.)30 b(I.)h(The)f(attr)l(active)g(c)l (ase)p FD(,)f(Comm.)c(Math.)j(Ph)n(ys.)f Fd(161)p FD(,)g(no.)g(3,)g (447{486)d(\(1994\).)118 1163 y([MMR])42 b(A.)30 b(Messager,)e(S.)i (Miracle-Sol,)c(J.)j(Ruiz,)h Fv(Convexity)i(pr)l(op)l(erty)h(of)f(the)g (surfac)l(e)g(tension)f(and)i(e)l(qui-)289 1263 y(librium)d(crystals)7 b FD(,)28 b(J.)g(Stat.)g(Ph)n(ys.)e Fd(67)p FD(,)i(No.)f(3/4,)f (449{470)f(\(1992\).)118 1363 y([Pf)6 b(])42 b(C.-E.)22 b(P\014ster,)g Fv(L)l(ar)l(ge)k(deviations)g(and)g(phase)g(sep)l(ar)l (ation)g(in)f(the)g(two-dimensional)i(Ising)e(mo)l(del)p FD(,)f(Helv.)289 1462 y(Ph)n(ys.)i(Acta)i Fd(64)p FD(,)f(no.)h(7,)f (953{1054,)d(\(1991\).)118 1562 y([PV])42 b(C.-E.)24 b(P\014ster,)g(Y)50 b(V)-7 b(elenik,)24 b Fv(L)l(ar)l(ge)k(deviations)h (and)e(c)l(ontinuum)f(limit)i(in)g(the)f(2D)g(Ising)h(mo)l(del)9 b FD(,)26 b(Prob.)289 1662 y(Th.)i(Rel.)e(Fields)g Fd(109)p FD(,)h(435{506)d(\(1997\).)118 1761 y([Pi])44 b(A.)29 b(Pisztora,)e Fv(Surfac)l(e)32 b(or)l(der)g(lar)l(ge)g(deviations)h(of) f(Ising,)g(Potts)f(and)h(p)l(er)l(c)l(olation)h(mo)l(dels)7 b FD(,)30 b(Prob.)e(Th.)289 1861 y(Rel.)f(Fields)e Fd(104)p FD(,)i(427{466)e(\(1996\).)118 1961 y([Ro])41 b(A.Ros,)30 b Fv(The)i(Isop)l(erimetric)g(and)g(Wil)t(lmor)l(e)g(Pr)l(oblems)p FD(,)f(Lecture)d(at)h(the)h(In)n(ternational)c(Congress)h(on)289 2060 y(Di\013eren)n(tial)d(Geometry)-7 b(,)26 b(in)h(memory)e(of)j (Alfred)f(Gra)n(y)f(\(2000\).)118 2160 y([Su1])42 b(N.)28 b(Sugimine,)e Fv(Extension)31 b(of)g(Thomas's)i(r)l(esult)c(and)i(upp)l (er)g(b)l(ound)g(on)g(the)f(sp)l(e)l(ctr)l(al)h(gap)h(of)f(d\()25 b Fb(>)f FD(3)p Fv(\)-)289 2259 y(dimensional)31 b(sto)l(chastic)g (Ising)f(mo)l(del)p FD(,)f(preprin)n(t)d(\(2002\).)118 2359 y([Su2])42 b(N.)26 b(Sugimine,)e Fv(A)k(lower)i(b)l(ound)f(on)f (the)h(sp)l(e)l(ctr)l(al)g(gap)g(of)h(the)f(3)g(dimensional)h(sto)l (chastic)g(Ising)e(mo)l(del)p FD(,)289 2459 y(preprin)n(t)e(\(2002\).) 218 2645 y FC(Universit)621 2638 y(\023)621 2645 y(e)36 b(P)-7 b(aris)36 b(7,)h(D)1137 2638 y(\023)1137 2645 y(ep)-6 b(ar)g(tement)39 b(de)d(Ma)-6 b(th)2021 2638 y(\023)2021 2645 y(ema)g(tiques,)39 b(Case)e(7012,)f(2)h(pla)n(ce)g (Jussieu,)118 2745 y(F-75251)31 b(P)-7 b(aris,)31 b(France)218 2845 y Fv(E-mail)f(addr)l(ess)7 b FD(:)38 b Fa(Thierry.Bodineau@)o(gau) o(ss)o(.m)o(ath)o(.j)o(uss)o(ie)o(u.)o(fr)218 3014 y FC(F)-10 b(a)n(cul)k(ty)32 b(of)f(Industrial)g(Engineering,)g (Technion,)g(Haif)-7 b(a)31 b(32000,)g(Israel)218 3114 y Fv(E-mail)f(addr)l(ess)7 b FD(:)38 b Fa(ieioffe@ie.techni)o(on.)o(ac) o(.i)o(l)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF 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