Content-Type: multipart/mixed; boundary="-------------0110190534440" This is a multi-part message in MIME format. ---------------0110190534440 Content-Type: text/plain; name="01-388.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-388.keywords" random walk in random environment, diffusion coefficient ---------------0110190534440 Content-Type: application/postscript; name="capio.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="capio.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: capio_submitted.dvi %%Pages: 17 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips capio_submitted -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2001.10.19:1114 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul 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3472 y(Subsection)34 b(4.5,)j(our)d(pro)s(of)g(has)g(an)h(additional)e(adv) -5 b(an)m(tage)36 b(of)f(giving)f(a)h(natural)f(in)m(terpretation)118 3580 y(of)i(the)h(translation)e(co)m(v)-5 b(arian)m(t)37 b(states)h(for)e(a)g(harmonic)f(in)m(terface)i(in)e(a)h(random)g(en)m (vironmen)m(t)g(in)118 3688 y(terms)30 b(of)h(the)f(F)-8 b(unaki-Sp)s(ohn)29 b(states)i([5)q(].)218 3796 y(W)-8 b(e)24 b(w)m(ould)d(lik)m(e)h(to)h(men)m(tion)g(that)g(appro)m (ximation)e(results)h(of)h(the)f(t)m(yp)s(e)h(\(1.4\))i(or)d(\(1.5\))j (go)e(b)s(ey)m(ond)118 3904 y(the)30 b(general)g(sp)s(ectral)f (analysis)f(of)i(the)g(Kipnis-V)-8 b(aradhan)27 b(approac)m(h)j(as)g (dev)m(elop)s(ed)f(in)f([10)r(])h(or)h([3)q(].)118 4012 y(This)d(issue)h(is)g(brie\015y)f(addressed)h(in)g(Subsection)g(4.4.)41 b(Similar)26 b(appro)m(ximation)i(results)g(ha)m(v)m(e)i(b)s(een)118 4120 y(recen)m(tly)g(deriv)m(ed)e(in)g([12)r(])h(in)f(a)i(m)m(uc)m(h)f (more)h(delicate)f(case)i(of)e(the)h(self-di\013usion)c(co)s(e\016cien) m(t)31 b(of)e(the)118 4228 y(tagged)j(particle)e(in)f(the)h(exclusion)f (pro)s(cess.)218 4357 y(Our)f(main)h(motiv)-5 b(ation)30 b(for)g(this)f(w)m(ork)h(came)h(from)e(the)h(theory)h(of)f(massless)f (gradien)m(t)h(\014elds)e(on)118 4465 y Fp(Z)183 4432 y Fo(d)219 4465 y Fu(.)41 b(These)30 b(are)h(sp)s(eci\014ed)d(b)m(y)j (the)f(formal)g(Hamiltonian)958 4703 y Fq(H)q Fu(\()p Fs(\036)p Fu(\))c(=)1282 4617 y Fi(X)1327 4808 y Fo(x)1476 4589 y(d)1428 4617 y Fi(X)1437 4812 y Fo(i)p Fk(=1)1575 4703 y Fs(U)f Fu(\()p Fq(r)1773 4717 y Fo(i)1801 4703 y Fs(\036)p Fu(\()p Fs(x)p Fu(\)\))h(=)2134 4617 y Fi(X)2180 4808 y Fo(x)2328 4589 y(d)2281 4617 y Fi(X)2289 4812 y Fo(i)p Fk(=1)2427 4703 y Fs(U)f Fu(\()q Fs(\021)2595 4717 y Fo(i)2623 4703 y Fu(\()p Fs(x)p Fu(\)\))16 b Fs(;)118 4941 y Fu(where)41 b Fq(r)468 4955 y Fo(i)538 4941 y Fu(is)f(the)i(discrete)f(lattice)h(gradien)m(t;)48 b Fs(\021)1886 4955 y Fo(i)1914 4941 y Fu(\()p Fs(x)p Fu(\))d(=)e Fq(r)2271 4955 y Fo(i)2299 4941 y Fs(\036)p Fu(\()p Fs(x)p Fu(\))i(=)f Fs(\036)p Fu(\()p Fs(x)28 b Fu(+)f Fs(e)2944 4955 y Fo(i)2973 4941 y Fu(\))h Fq(\000)f Fs(\036)p Fu(\()p Fs(x)p Fu(\).)75 b(In)41 b(the)118 5049 y(uniformly)32 b(elliptic)g(case,)37 b(1)p Fs(=c)d Fr(6)f Fs(U)1401 5016 y Fn(00)1443 5049 y Fu(\()p Fq(\001)p Fu(\))h Fr(6)e Fs(c)p Fu(,)k(the)f(in\014nite)e(v)m(olume)h(gradien)m(t)h(states)h (exist)f(in)e(an)m(y)118 5157 y(dimension)f Fs(d)g Fr(>)f Fu(1,)36 b(and,)f(as)f(it)g(has)g(b)s(een)g(established)e(in)h(the)h (pap)s(er)g(b)m(y)g(F)-8 b(unaki)33 b(and)h(Sp)s(ohn)e([5)q(],)118 5264 y(an)m(y)j(translation)f(in)m(v)-5 b(arian)m(t)35 b(in\014nite)d(v)m(olume)j(gradien)m(t)g(state)h(is)e(decomp)s(osable)h (in)m(to)g(the)g(con)m(v)m(ex)118 5372 y(com)m(bination)42 b(of)g(the)h(extremal)g(states)g(\(the)g(so)g(called)f(F)-8 b(unaki-Sp)s(ohn)40 b(states\),)47 b(whic)m(h)41 b(can)i(b)s(e)118 5480 y(constructed)25 b(as)g(limits)d(of)i(\014nite)g(v)m(olume)g(p)s (erio)s(dized)e(measures)i(with)f(appropriately)g(tilted)g(slop)s(es.)p eop %%Page: 3 3 3 2 bop 3626 43 a Fk(3)118 366 y Fu(The)27 b(v)m(ery)h(same)f(appro)m (ximation)f(b)m(y)i(the)f(p)s(erio)s(dized)e(states)j(leads)f([5])h(to) g(a)g(meaningful)d(de\014nition)118 474 y(of)30 b(the)f(\(slop)s(e-dep) s(enden)m(t\))f(surface)i(tension.)39 b(The)29 b(latter)h(is)e(kno)m (wn)h(to)h(b)s(e)f(strictly)f(con)m(v)m(ex)j([4)q(],[7)q(].)218 582 y(On)d(the)h(mesoscopic)g(lev)m(el,)g(the)h(in)m(tegral)e(of)h(the) g(surface)g(tension)f(happ)s(ens)g(to)h(b)s(e)g(precisely)e(the)118 690 y(large)32 b(deviation)e(rate)i(function)f(for)g(the)h (appropriately)d(scaled)j(heigh)m(t)f(\014eld)f([4)q(],)i(and)f(it)g (has)g(b)s(een)118 798 y(conjectured)f(in)e([7)q(])i(that)g(the)f (Hessian)g(of)h(the)g(surface)f(tension)g(go)m(v)m(erns)i(the)f (equilibrium)24 b(\015uctua-)118 906 y(tions)h(in)e(the)j(corresp)s (onding)d(F)-8 b(unaki-Sp)s(ohn)23 b(state.)40 b(In)24 b(view)h(of)g(the)g(random)f(w)m(alk)h(represen)m(tation)118 1014 y(dev)m(elop)s(ed)h(in)e([4)q(],)j(this)e(conjecture)i(w)m(ould)e (follo)m(w)g(from)h(an)g(appro)m(ximation)f(result)g(along)h(the)g (lines)118 1122 y(of)h(Theorem)f(1.1,)j(but)d(for)h(a)g(di\013eren)m(t) f(t)m(yp)s(e)h(of)g(en)m(vironmen)m(t.)39 b(W)-8 b(e)28 b(refer)e(to)i([7])f(for)g(further)e(details.)118 1230 y(T)-8 b(o)31 b(the)g(b)s(est)e(of)i(our)f(kno)m(wledge)g(this)g(is)f (still)f(an)i(op)s(en)g(problem.)218 1369 y(The)e(pap)s(er)g(is)g (organized)g(as)h(follo)m(ws:)39 b(In)28 b(Section)h(2)g(w)m(e)g (brie\015y)e(recall)h(the)h(v)-5 b(ariational)28 b(form)m(ula)118 1477 y(for)34 b(the)h(in\014nite)d(v)m(olume)i(e\013ectiv)m(e)i (di\013usion)31 b(matrix)j(and)f(express)h(it)g(in)f(terms)h(of)h(the)f (corrector)118 1585 y(\014eld.)39 b(The)27 b(relev)-5 b(an)m(t)28 b(prop)s(erties)e(of)i(the)g(corrector)h(\014eld)d(are)i (listed)e(in)h(Prop)s(osition)e(2.1)k(and,)f(other)118 1693 y(w)m(a)m(y)33 b(around,)e(w)m(e)h(sho)m(w)f(that)i(for)e(ev)m (ery)h Fs(v)f Fq(2)c Fp(R)1803 1660 y Fo(d)1881 1693 y Fu(the)k(corrector)i(\014eld)d Fs( )2681 1707 y Fo(v)2754 1693 y Fu(is)h(uniquely)e(determined)118 1801 y(b)m(y)38 b(these)g(prop)s(erties.)62 b(Subsequen)m(tly)-8 b(,)38 b Fs( )1617 1815 y Fo(v)1696 1801 y Fu(is)f(reco)m(v)m(ered)j(as)e(the) g Fs(L)2550 1768 y Fk(2)2589 1801 y Fu(\()p Fs(\026)p Fu(\))g(limit)e(of)i(the)g(p)s(erio)s(dized)118 1916 y(corrector)28 b(\014elds)746 1892 y(~)727 1916 y Fs( )786 1930 y Fo(N)s(;v)932 1916 y Fu(in)d(Prop)s(osition)g(3.1)i(of)g (Section)f(3.)39 b(This)25 b(already)h(leads)f(to)i(the)g(con)m(v)m (ergence)118 2024 y(of)34 b(the)g(a)m(v)m(eraged)i(\014nite)d(v)m (olume)g(e\013ectiv)m(e)j(di\013usivities)30 b Fp(E)2210 2038 y Fo(\026)2263 2024 y Fs(D)2338 2038 y Fo(N)2405 2024 y Fu(\()p Fs(\030)t Fu(\),)36 b(Corollary)c(3.1.)52 b(In)33 b(Section)h(4)118 2132 y(w)m(e)28 b(sho)m(w)g(that)g Fs(D)741 2146 y Fo(N)809 2132 y Fu(\()p Fs(\030)t Fu(\))g(concen)m (trates)i(around)d(its)g(exp)s(ected)h(v)-5 b(alue)27 b Fp(E)2565 2146 y Fo(\026)2617 2132 y Fs(D)2692 2146 y Fo(N)2760 2132 y Fu(\()p Fs(\030)t Fu(\).)40 b(The)27 b(concen)m(tration)118 2240 y(estimates)34 b(of)f(Subsection)f(4.3)i (pro)m(v)m(e)g(the)f(main)f(claim)g(in)g(Theorem)h(1.1.)50 b(The)33 b(k)m(ey)g(ingredien)m(t)f(of)118 2348 y(the)e(pro)s(of)e(is)g (an)h(a-priori)f Fs(L)1110 2315 y Fo(p)1179 2348 y Fu(estimate)i(of)f (Subsection)f(4.1,)j(whic)m(h)d(is)g(used)h(to)h(deriv)m(e)e (appropriate)118 2456 y(b)s(ounds)20 b(on)i(the)h(Hamming)e(distance)h (in)f(Subsection)g(4.2.)39 b(By)22 b(Mey)m(ers)h(argumen)m(t)g(suc)m(h) f Fs(L)3266 2423 y Fo(p)3328 2456 y Fu(estimate)118 2564 y(follo)m(ws)38 b(from)h(discrete)g(v)m(ersion)g(of)g(Calderon-Zygm)m (und)e(inequalit)m(y)-8 b(.)66 b(Since)38 b(w)m(e)h(could)g(not)g (\014nd)118 2672 y(direct)30 b(references)h(to)g(the)f(latter,)h(the)g (pro)s(of)e(is)h(sk)m(etc)m(hed)i(in)d(the)h(App)s(endix.)1043 2925 y(2.)51 b Ft(Diffusivity)34 b(in)h(the)e(infinite)i(medium)118 3087 y Fu(2.1.)54 b Fz(The)35 b(v)-6 b(ariational)35 b(form)m(ula.)45 b Fu(W)-8 b(e)32 b(will)c(denote)j Fs(\030)2130 3101 y Fo(i)2158 3087 y Fu(\()p Fs(x)p Fu(\),)h Fs(i)26 b Fu(=)g(1)p Fs(;)15 b(:)g(:)g(:)i(;)e(d)p Fu(,)32 b Fs(x)26 b Fq(2)f Fp(Z)3071 3054 y Fo(d)3107 3087 y Fu(,)31 b(the)g(rate)h Fs(\030)t Fu(\()p Fs(b)p Fu(\))118 3195 y(at)d(the)f(b)s(ond)e Fs(b)f Fu(=)g(\()p Fs(x;)15 b(x)g Fu(+)g Fs(e)1090 3209 y Fo(i)1119 3195 y Fu(\),)29 b(with)d Fs(e)1454 3209 y Fo(i)1510 3195 y Fu(the)i(unit)e(v)m(ector)k(in)c(the) i Fs(i)p Fu(-th)g(direction.)39 b(When)27 b(w)m(e)i(write)d Fs(\030)3633 3209 y Fo(i)118 3305 y Fu(only)i(w)m(e)h(mean)g Fs(\030)729 3319 y Fo(i)757 3305 y Fu(\(0\).)41 b Fs(\034)978 3319 y Fo(y)1020 3305 y Fs(\030)32 b Fu(denotes)e(the)f(shifted)e (con\014guration)h(\()p Fs(\034)2493 3319 y Fo(y)2535 3305 y Fs(\030)t Fu(\))2614 3319 y Fo(i)2642 3305 y Fu(\()p Fs(x)p Fu(\))e(=)f Fs(\030)2926 3319 y Fo(i)2954 3305 y Fu(\()p Fs(x)17 b Fq(\000)g Fs(y)s Fu(\),)30 b Fs(x;)15 b(y)28 b Fq(2)d Fp(Z)3600 3272 y Fo(d)3636 3305 y Fu(,)118 3413 y Fs(i)h Fu(=)f(1)p Fs(;)15 b(:)g(:)g(:)i(;)e(d)p Fu(.)41 b(W)-8 b(e)32 b(also)e(use)g(the)h(notation,)g(for)f(an)m(y)g Fs(f)35 b Fu(:)25 b(\012)g Fq(!)g Fp(R)s Fu(,)897 3583 y Fs(D)972 3597 y Fo(i)1001 3583 y Fs(f)10 b Fu(\()p Fs(\030)t Fu(\))25 b(=)g Fs(f)10 b Fu(\()p Fs(\034)1421 3597 y Fn(\000)p Fo(e)1509 3607 y Fv(i)1538 3583 y Fs(\030)t Fu(\))21 b Fq(\000)f Fs(f)10 b Fu(\()p Fs(\030)t Fu(\))15 b Fs(;)91 b(D)2107 3545 y Fn(\003)2104 3605 y Fo(i)2146 3583 y Fs(f)10 b Fu(\()p Fs(\030)t Fu(\))25 b(=)g Fq(\000)p Fs(D)2582 3597 y Fo(i)2610 3583 y Fs(f)10 b Fu(\()p Fs(\034)2740 3597 y Fo(e)2773 3607 y Fv(i)2803 3583 y Fs(\030)t Fu(\))118 3753 y(In)28 b(the)g(ab)s(o)m(v)m(e)h(notation)g(the)f(asso)s(ciated)h (pro)s(cess)e(of)i(the)f(en)m(vironmen)m(t)g Fs(\030)2717 3767 y Fo(t)2772 3753 y Fu(=)d Fs(\034)2908 3771 y Fo(X)5 b Fk(\()p Fo(t)p Fk(\))3055 3753 y Fs(\030)32 b Fu(seen)c(from)g(the) 118 3864 y(particle)i(\(see)h([10)q(],)g([3)q(]\))g(is)e(a)i(Mark)m(o)m (v)h(pro)s(cess)e(with)f(generator)1031 4126 y Fq(L)p Fs(f)10 b Fu(\()p Fs(\030)t Fu(\))25 b(=)f Fq(\000)1517 4013 y Fo(d)1469 4040 y Fi(X)1478 4235 y Fo(i)p Fk(=1)1616 4126 y Fs(D)1694 4089 y Fn(\003)1691 4149 y Fo(i)1748 4126 y Fu(\()q Fs(\030)1824 4140 y Fo(i)1852 4126 y Fs(D)1927 4140 y Fo(i)1955 4126 y Fs(f)10 b Fu(\))15 b(\()p Fs(\030)t Fu(\))g Fs(;)92 b(f)34 b Fu(:)26 b(\012)f Fq(!)g Fp(R)18 b Fs(:)118 4378 y Fu(By)28 b(translation)f(in)m(v)-5 b(ariance)27 b Fs(\026)h Fu(is)f(a)h(rev)m(ersible)f(measure)h(for)g (this)e(pro)s(cess)i(and)f Fq(L)h Fu(is)f(self)g(adjoin)m(t)h(in)118 4486 y Fs(L)180 4453 y Fk(2)219 4486 y Fu(\()p Fs(\026)p Fu(\).)218 4625 y(It)45 b(is)f(w)m(ell)g(kno)m(wn)g([3)q(])h(that)g (the)g(annealed)g(e\013ectiv)m(e)h(di\013usion)d(matrix)h Fs(D)j Fu(in)d(\(1.1\))j(can)e(b)s(e)118 4733 y(reco)m(v)m(ered)32 b(from)e(the)h(v)-5 b(ariational)29 b(form)m(ula:)1072 4972 y(\()p Fs(D)s(v)s(;)15 b(v)s Fu(\))27 b(=)e(2)94 b(inf)1537 5038 y Fo(f)7 b Fn(2)p Fo(L)1673 5019 y Fg(2)1708 5038 y Fk(\()p Fo(\026)p Fk(\))1868 4859 y Fo(d)1821 4886 y Fi(X)1829 5081 y Fo(i)p Fk(=1)1967 4972 y Fp(E)2022 4986 y Fo(\026)2090 4899 y Fi(\000)2131 4972 y Fs(\030)2171 4986 y Fo(i)2199 4972 y Fu(\()p Fs(v)2278 4986 y Fo(i)2327 4972 y Fu(+)20 b Fs(D)2493 4986 y Fo(i)2521 4972 y Fs(f)10 b Fu(\))2611 4935 y Fk(2)2651 4899 y Fi(\001)3475 4972 y Fu(\(2.1\))118 5264 y(2.2.)54 b Fz(Corrector)37 b(\014eld.)45 b Fu(In)31 b(general,)i(the)f(ab)s(o)m(v)m(e)h(v)-5 b(ariational)31 b(problem)f(cannot)i(b)s(e)g(solv)m(ed)f(for)h Fs(f)10 b Fu(.)118 5372 y(Ho)m(w)m(ev)m(er)33 b(the)e(next)f(prop)s(osition)f (sho)m(ws)h(that)h(if)f(w)m(e)h(only)f(lo)s(ok)g(at)h(\\gradien)m(ts")h (of)e Fs(f)40 b Fu(the)31 b(problem)118 5480 y(has)f(a)h(unique)d (solution.)p eop %%Page: 4 4 4 3 bop 118 45 a Fk(4)1045 b(PIETR)n(O)31 b(CAPUTO)f(AND)g(DMITR)-6 b(Y)30 b(IOFFE)118 366 y Fz(Prop)s(osition)36 b(2.1.)46 b Fj(F)-7 b(or)34 b(every)f Fs(v)28 b Fq(2)d Fp(R)1520 333 y Fo(d)1599 366 y Fj(we)32 b(have)799 630 y Fu(\()p Fs(D)s(v)s(;)15 b(v)s Fu(\))27 b(=)e(2)1312 517 y Fo(d)1265 544 y Fi(X)1273 740 y Fo(i)p Fk(=1)1411 630 y Fs(v)1455 644 y Fo(i)1483 630 y Fp(E)1538 644 y Fo(\026)1590 630 y Fs(\030)1630 644 y Fo(i)1674 557 y Fi(\000)1715 630 y Fs(v)1759 644 y Fo(i)1808 630 y Fu(+)20 b Fs( )1961 593 y Fo(i)1958 653 y(v)1999 557 y Fi(\001)2066 630 y Fu(=)25 b(2)2270 517 y Fo(d)2222 544 y Fi(X)2231 740 y Fo(i)p Fk(=1)2369 630 y Fp(E)2423 644 y Fo(\026)2476 630 y Fs(\030)2516 644 y Fo(i)2559 557 y Fi(\000)2601 630 y Fs(v)2645 644 y Fo(i)2693 630 y Fu(+)20 b Fs( )2846 593 y Fo(i)2843 653 y(v)2884 557 y Fi(\001)2926 575 y Fk(2)3475 630 y Fu(\(2.2\))118 903 y Fj(wher)-5 b(e)34 b Fs( )437 870 y Fo(i)434 925 y(v)475 903 y Fj(,)e Fs(i)26 b Fu(=)f(1)p Fs(;)15 b(:)g(:)g(:)i(;)e(d)33 b Fj(ar)-5 b(e)34 b(the)f(unique)e(elements)i(of)g Fs(L)2156 870 y Fk(2)2196 903 y Fu(\()p Fs(\026)p Fu(\))g Fj(such)f(that)243 1045 y Fq(\017)42 b Fp(E)385 1059 y Fo(\026)437 1045 y Fs( )499 1012 y Fo(i)496 1068 y(v)563 1045 y Fu(=)24 b(0)16 b Fj(,)243 1168 y Fq(\017)330 1100 y Fi(P)426 1126 y Fo(d)426 1195 y(i)p Fk(=1)559 1168 y Fs(D)637 1135 y Fn(\003)634 1194 y Fo(i)692 1095 y Fi(\002)730 1168 y Fs(\030)770 1182 y Fo(i)813 1095 y Fi(\000)854 1168 y Fs(v)898 1182 y Fo(i)947 1168 y Fu(+)k Fs( )1100 1135 y Fo(i)1097 1191 y(v)1138 1168 y Fu(\()p Fs(\030)t Fu(\))1252 1095 y Fi(\001\003)1357 1168 y Fu(=)25 b(0)16 b Fj(,)110 b Fs(\026)p Fj(-)32 b(a.s.)16 b(,)243 1288 y Fq(\017)42 b Fs(D)405 1303 y Fo(k)448 1288 y Fs( )510 1255 y Fo(i)507 1310 y(v)573 1288 y Fu(=)25 b Fs(D)744 1302 y Fo(i)773 1288 y Fs( )835 1255 y Fo(k)832 1310 y(v)893 1288 y Fj(,)110 b Fs(\026)p Fj(-)32 b(a.s.)16 b(,)103 b Fs(i;)15 b(k)29 b Fu(=)c(1)p Fs(;)15 b(:)g(:)g(:)i(;)e(d)h Fj(.)118 1469 y(Pr)-5 b(o)g(of.)43 b Fu(F)-8 b(or)31 b(an)m(y)g(v)m(ector)h Fs(v)c Fq(2)d Fp(R)1225 1436 y Fo(d)1302 1469 y Fu(w)m(e)31 b(de\014ne)e(the)i(lo)s(cal)e(drift)g(or)i (curren)m(t)f(along)g Fs(v)k Fu(as)918 1733 y Fs(')977 1747 y Fo(v)1019 1733 y Fu(\()p Fs(\030)t Fu(\))26 b(=)1302 1620 y Fo(d)1254 1647 y Fi(X)1263 1842 y Fo(i)p Fk(=1)1401 1733 y Fs(v)1445 1747 y Fo(i)1488 1733 y Fu(\()q Fs(\030)1564 1747 y Fo(i)1592 1733 y Fu(\(0\))21 b Fq(\000)f Fs(\030)1859 1747 y Fo(i)1887 1733 y Fu(\()p Fq(\000)p Fs(e)2035 1747 y Fo(i)2063 1733 y Fu(\)\))26 b(=)f Fq(\000)2389 1620 y Fo(d)2341 1647 y Fi(X)2350 1842 y Fo(i)p Fk(=1)2488 1733 y Fs(v)2532 1747 y Fo(i)2560 1733 y Fs(D)2638 1696 y Fn(\003)2635 1756 y Fo(i)2677 1733 y Fs(\030)2717 1747 y Fo(i)2745 1733 y Fu(\(0\))615 b(\(2.3\))118 1992 y(Let)31 b(us)f(no)m(w)g(de\014ne)g(the)g(functions)1121 2169 y Fs(\037)1178 2132 y Fo(\025)1178 2192 y(v)1248 2169 y Fu(=)25 b(\()p Fq(\000L)20 b Fu(+)g Fs(\025)p Fu(\))1712 2132 y Fn(\000)p Fk(1)1807 2169 y Fs(')1866 2183 y Fo(v)1922 2169 y Fs(;)91 b(\025)26 b(>)f Fu(0)15 b Fs(;)41 b(v)28 b Fq(2)d Fp(R)2557 2132 y Fo(d)2619 2169 y Fs(;)831 b Fu(\(2.4\))118 2346 y(or,)31 b(in)e(terms)h(of)h(the)f(pro)s(cess:)1219 2567 y Fs(\037)1276 2530 y Fo(\025)1276 2590 y(v)1321 2567 y Fu(\()p Fs(\030)t Fu(\))c(=)f Fp(E)1612 2523 y Fo(\030)1612 2593 y Fk(0)1672 2444 y Fi(Z)1763 2470 y Fn(1)1723 2650 y Fk(0)1853 2567 y Fu(e)1893 2530 y Fn(\000)p Fo(\025t)2019 2567 y Fs(')2078 2581 y Fo(v)2119 2567 y Fu(\()p Fs(\034)2194 2586 y Fo(X)5 b Fk(\()p Fo(t)p Fk(\))2342 2567 y Fs(\030)t Fu(\))15 b(d)p Fs(t)g(:)118 2795 y Fu(F)-8 b(ollo)m(wing)27 b(Kipnis)d(and)i(V)-8 b(aradhan)27 b([10)q(],)i(Theorem)d(1.3)j(-)e(see)h(also)f([16)q(])g(-) g(one)h(can)f(use)g(the)g(sp)s(ectral)118 2903 y(resolution)i(of)i(the) f(non-negativ)m(e)i(op)s(erator)e Fq(\000L)g Fu(on)g Fs(L)2052 2870 y Fk(2)2091 2903 y Fu(\()p Fs(\026)p Fu(\))h(to)g(pro)m (v)m(e)1404 3087 y Fs(\025)p Fp(E)1512 3101 y Fo(\026)1564 3087 y Fu(\()p Fs(\037)1656 3050 y Fo(\025)1656 3110 y(v)1702 3087 y Fu(\))1737 3050 y Fk(2)1802 3087 y Fq(!)25 b Fu(0)15 b Fs(;)92 b(\025)25 b Fq(!)g Fu(0)15 b Fs(:)1101 b Fu(\(2.5\))118 3278 y(Moreo)m(v)m(er)33 b(one)d(can)h(pro)m(v)m(e)g (that)g(there)g(exist)f(functions)f Fs( )2195 3245 y Fo(i)2192 3301 y(v)2258 3278 y Fq(2)c Fs(L)2406 3245 y Fk(2)2445 3278 y Fu(\()p Fs(\026)p Fu(\),)31 b Fs(i)26 b Fu(=)f(1)p Fs(;)15 b(:)g(:)g(:)i(;)e(d)31 b Fu(suc)m(h)f(that)1206 3433 y Fo(d)1159 3461 y Fi(X)1167 3656 y Fo(i)p Fk(=1)1305 3547 y Fp(E)1360 3561 y Fo(\026)1412 3547 y Fs(\030)1452 3561 y Fo(i)1480 3547 y Fu(\()p Fs(D)1590 3561 y Fo(i)1619 3547 y Fs(\037)1676 3510 y Fo(\025)1676 3570 y(v)1741 3547 y Fq(\000)20 b Fs( )1894 3510 y Fo(i)1891 3570 y(v)1932 3547 y Fu(\))1967 3510 y Fk(2)2033 3547 y Fq(!)25 b Fu(0)15 b Fs(;)91 b(\025)26 b Fq(!)f Fu(0)15 b Fs(;)870 b Fu(\(2.6\))118 3822 y(whic)m(h)27 b(in)f(turn)h(b)m(y)g(the)h(ellipticit)m(y)e(of)i Fs(\030)j Fu(implies)25 b(strong)j(con)m(v)m(ergence)i(of)e Fs(D)2788 3836 y Fo(i)2817 3822 y Fs(\037)2874 3789 y Fo(\025)2874 3844 y(v)2947 3822 y Fu(to)g Fs( )3117 3789 y Fo(i)3114 3844 y(v)3183 3822 y Fu(in)f Fs(L)3349 3789 y Fk(2)3388 3822 y Fu(\()p Fs(\026)p Fu(\).)40 b(In)118 3930 y(particular,)29 b(w)m(e)i(ha)m(v)m(e)386 4194 y Fp(E)440 4208 y Fo(\026)508 4120 y Fi(\000)550 4194 y Fs(')609 4208 y Fo(v)650 4194 y Fu(\()p Fq(\000L)p Fu(\))854 4156 y Fn(\000)p Fk(1)948 4194 y Fs(')1007 4208 y Fo(v)1049 4120 y Fi(\001)1116 4194 y Fu(=)k(lim)1212 4255 y Fo(\025)p Fn(!)p Fk(0)1374 4194 y Fp(E)1429 4208 y Fo(\026)1481 4194 y Fs(')1540 4208 y Fo(v)1581 4194 y Fs(\037)1638 4156 y Fo(\025)1638 4216 y(v)1709 4194 y Fu(=)25 b Fq(\000)g Fu(lim)1891 4255 y Fo(\025)p Fn(!)p Fk(0)2100 4080 y Fo(d)2053 4107 y Fi(X)2061 4303 y Fo(i)p Fk(=1)2199 4194 y Fs(v)2243 4208 y Fo(i)2272 4194 y Fp(E)2326 4208 y Fo(\026)2379 4194 y Fs(\030)2419 4208 y Fo(i)2447 4194 y Fs(D)2522 4208 y Fo(i)2550 4194 y Fs(\037)2607 4156 y Fo(\025)2607 4216 y(v)2678 4194 y Fu(=)g Fq(\000)2907 4080 y Fo(d)2859 4107 y Fi(X)2868 4303 y Fo(i)p Fk(=1)3006 4194 y Fs(v)3050 4208 y Fo(i)3078 4194 y Fp(E)3133 4208 y Fo(\026)3185 4194 y Fs(\030)3225 4208 y Fo(i)3253 4194 y Fs( )3315 4156 y Fo(i)3312 4216 y(v)3369 4194 y Fs(:)118 4457 y Fu(Since)k(\(2.1\))k(is)c(equiv)-5 b(alen)m(t)30 b(to)959 4726 y(\()p Fs(D)s(v)s(;)15 b(v)s Fu(\))27 b(=)e(2)1472 4613 y Fo(d)1424 4640 y Fi(X)1433 4836 y Fo(i)p Fk(=1)1571 4726 y Fp(E)1625 4740 y Fo(\026)1678 4726 y Fu(\()p Fs(\030)1753 4740 y Fo(i)1781 4726 y Fu(\))p Fs(v)1863 4689 y Fk(2)1860 4749 y Fo(i)1924 4726 y Fq(\000)20 b Fu(2)p Fp(E)2115 4740 y Fo(\026)2182 4653 y Fi(\000)2224 4726 y Fs(')2283 4740 y Fo(v)2324 4726 y Fu(\()p Fq(\000L)p Fu(\))2528 4689 y Fn(\000)p Fk(1)2623 4726 y Fs(')2682 4740 y Fo(v)2723 4653 y Fi(\001)2780 4726 y Fs(;)670 b Fu(\(2.7\))118 4990 y(w)m(e)31 b(obtain)f(the)g(\014rst)g(iden)m(tit)m(y)g(in)f (\(2.2\).)218 5098 y(T)-8 b(o)31 b(pro)m(v)m(e)g(the)f(second)h(one)f (w)m(e)h(observ)m(e)g(that)1301 5249 y Fo(d)1253 5276 y Fi(X)1262 5471 y Fo(i)p Fk(=1)1400 5362 y Fp(E)1454 5376 y Fo(\026)1507 5362 y Fs(\030)1547 5376 y Fo(i)1575 5362 y Fu(\()p Fs( )1672 5325 y Fo(i)1669 5385 y(v)1710 5362 y Fu(\))1745 5325 y Fk(2)1810 5362 y Fu(=)25 b Fq(\000)2040 5249 y Fo(d)1992 5276 y Fi(X)2001 5471 y Fo(i)p Fk(=1)2139 5362 y Fs(v)2183 5376 y Fo(i)2211 5362 y Fp(E)2265 5376 y Fo(\026)2318 5362 y Fs(\030)2358 5376 y Fo(i)2386 5362 y Fs( )2448 5325 y Fo(i)2445 5385 y(v)2501 5362 y Fs(:)949 b Fu(\(2.8\))p eop %%Page: 5 5 5 4 bop 3626 43 a Fk(5)118 366 y Fu(This)29 b(follo)m(ws)g(from)h (\(2.5\))i(and)e(the)h(con)m(v)m(ergence)h(of)f Fs(D)2071 380 y Fo(i)2099 366 y Fs(\037)2156 333 y Fo(\025)2156 389 y(v)2232 366 y Fu(to)g Fs( )2405 333 y Fo(i)2402 389 y(v)2473 366 y Fu(\(2.6\))i(since)702 499 y Fo(d)654 526 y Fi(X)663 721 y Fo(i)p Fk(=1)801 612 y Fp(E)855 626 y Fo(\026)908 612 y Fs(\030)948 626 y Fo(i)976 612 y Fu(\()p Fs( )1073 575 y Fo(i)1070 635 y(v)1111 612 y Fu(\))1146 575 y Fk(2)1211 612 y Fu(=)j(lim)1307 674 y Fo(\025)p Fn(!)p Fk(0)1517 499 y Fo(d)1469 526 y Fi(X)1478 721 y Fo(i)p Fk(=1)1616 612 y Fp(E)1670 626 y Fo(\026)1723 612 y Fs(\030)1763 626 y Fo(i)1791 612 y Fu(\()p Fs(D)1901 626 y Fo(i)1930 612 y Fs(\037)1987 575 y Fo(\025)1987 635 y(v)2032 612 y Fu(\))2067 575 y Fk(2)2132 612 y Fu(=)f(lim)2228 674 y Fo(\025)p Fn(!)p Fk(0)2390 612 y Fp(E)2445 626 y Fo(\026)2512 512 y Fi(\020)2567 612 y Fs(\037)2624 575 y Fo(\025)2624 635 y(v)2669 612 y Fu(\()p Fq(\000L)p Fu(\))p Fs(\037)2930 575 y Fo(\025)2930 635 y(v)2975 512 y Fi(\021)1211 935 y Fu(=)h(lim)1307 996 y Fo(\025)p Fn(!)p Fk(0)1469 834 y Fi(n)1530 935 y Fp(E)1585 949 y Fo(\026)1637 935 y Fs(\037)1694 897 y Fo(\025)1694 957 y(v)1739 935 y Fs(')1798 949 y Fo(v)1860 935 y Fq(\000)20 b Fs(\025)p Fp(E)2058 949 y Fo(\026)2111 935 y Fu(\()p Fs(\037)2203 897 y Fo(\025)2203 957 y(v)2249 935 y Fu(\))2284 897 y Fk(2)2323 834 y Fi(o)2409 935 y Fu(=)25 b Fq(\000)2639 821 y Fo(d)2591 848 y Fi(X)2600 1044 y Fo(i)p Fk(=1)2738 935 y Fs(v)2782 949 y Fo(i)2810 935 y Fp(E)2864 949 y Fo(\026)2917 935 y Fs(\030)2957 949 y Fo(i)2985 935 y Fs( )3047 897 y Fo(i)3044 957 y(v)3100 935 y Fs(:)218 1170 y Fu(Next)32 b(w)m(e)f(sho)m(w)g(that)g(the)g(functions)f Fs( )1607 1184 y Fo(v)1679 1170 y Fu(satisfy)g(the)h(three)h(prop)s (erties)d(in)g(the)j(statemen)m(t.)44 b(The)118 1278 y(\014rst)d(prop)s(ert)m(y)g(is)g(ob)m(vious)g(since)g Fs( )1439 1245 y Fo(i)1436 1301 y(v)1519 1278 y Fu(is)f(the)i Fs(L)1851 1245 y Fk(2)1890 1278 y Fu(\()p Fs(\026)p Fu(\)-limit)f(of)h (a)g(gradien)m(t)f(and)g Fs(\026)h Fu(is)e(translation)118 1386 y(in)m(v)-5 b(arian)m(t.)40 b(T)-8 b(o)31 b(pro)m(v)m(e)g(the)g (second)f(iden)m(tit)m(y)g(recall)g(that,)h(in)e(view)g(of)i(\(2.5\)) 1041 1547 y(lim)1031 1608 y Fo(\025)p Fn(!)p Fk(0)1193 1547 y Fp(E)1247 1561 y Fo(\026)1300 1547 y Fu(\()p Fs(f)10 b Fu(\()p Fq(\000L)p Fu(\))p Fs(\037)1651 1509 y Fo(\025)1651 1569 y(v)1696 1547 y Fu(\))26 b(=)e Fp(E)1907 1561 y Fo(\026)1960 1547 y Fu(\()p Fs(f)10 b(')2109 1561 y Fo(v)2150 1547 y Fu(\))15 b Fs(;)91 b(f)35 b Fq(2)25 b Fs(L)2544 1509 y Fk(2)2583 1547 y Fu(\()p Fs(\026)p Fu(\))15 b Fs(:)118 1747 y Fu(Therefore)30 b(for)g(an)m(y)h Fs(f)k Fq(2)24 b Fs(L)1068 1714 y Fk(2)1108 1747 y Fu(\()p Fs(\026)p Fu(\))31 b(w)m(e)f(ha)m(v)m(e)625 1880 y Fo(d)578 1907 y Fi(X)586 2102 y Fo(i)p Fk(=1)724 1993 y Fp(E)779 2007 y Fo(\026)831 1993 y Fs(f)10 b(D)964 1956 y Fn(\003)961 2016 y Fo(i)1018 1920 y Fi(\002)1056 1993 y Fs(\030)1096 2007 y Fo(i)1139 1920 y Fi(\000)1181 1993 y Fs(v)1225 2007 y Fo(i)1273 1993 y Fu(+)20 b Fs( )1426 1956 y Fo(i)1423 2016 y(v)1464 1993 y Fu(\()p Fs(\030)t Fu(\))1578 1920 y Fi(\001)q(\003)1684 1993 y Fu(=)25 b Fq(\000)p Fp(E)1905 2007 y Fo(\026)1958 1993 y Fu(\()p Fs(f)10 b(')2107 2007 y Fo(v)2148 1993 y Fu(\))20 b(+)30 b(lim)2294 2055 y Fo(\025)p Fn(!)p Fk(0)2456 1993 y Fp(E)2511 2007 y Fo(\026)2563 1993 y Fu(\()p Fs(f)10 b Fu(\()p Fq(\000L)p Fu(\))p Fs(\037)2914 1956 y Fo(\025)2914 2016 y(v)2959 1993 y Fu(\))26 b(=)f(0)15 b Fs(:)118 2245 y Fu(The)30 b(third)f(iden)m(tit)m(y)g(follo)m(ws)h(in) f(a)i(similar)c(fashion)j(from)f Fs(D)2248 2260 y Fo(k)2291 2245 y Fs(D)2366 2259 y Fo(i)2395 2245 y Fs(\037)2452 2212 y Fo(\025)2452 2267 y(v)2522 2245 y Fu(=)c Fs(D)2693 2259 y Fo(i)2722 2245 y Fs(D)2797 2260 y Fo(k)2840 2245 y Fs(\037)2897 2212 y Fo(\025)2897 2267 y(v)2942 2245 y Fu(,)30 b Fs(\025)c(>)f Fu(0.)218 2353 y(Finally)-8 b(,)26 b(to)i(pro)m(v)m(e)g(uniqueness)d(w)m(e)j(adapt)g(the)f(argumen) m(t)h(of)f(Theorem)g(2)h(in)e([18)q(])i(and)e(of)i(Prop)s(o-)118 2463 y(sition)e(3)h(in)e([2)q(]:)39 b(In)26 b(the)h(language)h(of)f ([2])g(the)g(\014eld)f Fs( )1964 2477 y Fo(v)2005 2463 y Fu(\()p Fs(\030)t Fu(\))g(=)f(\()p Fs( )2338 2430 y Fk(1)2335 2486 y Fo(v)2378 2463 y Fu(\()p Fs(\030)t Fu(\))p Fs(;)15 b(:)g(:)g(:)i(;)e( )2756 2430 y Fo(d)2753 2486 y(v)2798 2463 y Fu(\()p Fs(\030)t Fu(\)\))28 b(is,)e(b)m(y)h(the)g (third)e(of)118 2571 y(the)34 b(prop)s(erties)e(in)g(\(2.2\))r(,)j(an)e Fs(L)1256 2538 y Fk(2)1296 2571 y Fu(\()p Fs(\026)p Fu(\)-co)s(cycle.) 52 b(A)33 b(straigh)m(tforw)m(ard)h(mo)s(di\014cation)e(of)h(the)h(pro) s(of)f(of)118 2681 y(Prop)s(osition)22 b(3)j(in)d([2)q(])i(rev)m(eals)h (that)f(an)m(y)g Fs(L)1600 2648 y Fk(2)1640 2681 y Fu(\()p Fs(\026)p Fu(\)-co)s(cycle)h(\()p Fs(u)2189 2648 y Fk(1)2229 2681 y Fu(\()p Fs(\030)t Fu(\))p Fs(;)15 b(:)g(:)g(:)i(;)e(u)2597 2648 y Fo(d)2638 2681 y Fu(\()p Fs(\030)t Fu(\)\))25 b(is,)g(in)e(fact,)j(an)e Fs(L)3405 2648 y Fk(2)3444 2681 y Fu(-limit)118 2789 y(of)31 b(gradien)m(ts:)40 b(There)30 b(exists)g(a)h(sequence)g Fs(g)1659 2803 y Fo(\015)1729 2789 y Fq(2)24 b Fs(L)1876 2756 y Fk(2)1916 2789 y Fu(\()p Fs(\026)p Fu(\),)31 b Fs(\015)f(>)25 b Fu(0,)31 b(with)1003 2954 y Fp(E)1058 2968 y Fo(\026)1110 2954 y Fu(\()p Fs(D)1220 2969 y Fo(k)1264 2954 y Fs(g)1307 2968 y Fo(\015)1372 2954 y Fq(\000)19 b Fs(u)1514 2916 y Fo(k)1557 2954 y Fu(\))1592 2916 y Fk(2)1657 2954 y Fq(!)25 b Fu(0)15 b Fs(;)67 b(\015)30 b Fq(!)25 b Fu(0)15 b Fs(;)67 b(k)28 b Fu(=)d(1)p Fs(;)15 b(:)g(:)g(:)i(;)e(d)g(:)700 b Fu(\(2.9\))118 3124 y(Uniqueness)37 b(of)i Fs( )773 3138 y Fo(v)852 3124 y Fu(is)f(an)g(easy)h(consequence.)66 b(Indeed,)39 b(\014x)f Fs(v)k Fu(and)c(let)g Fs(u)2796 3091 y Fo(k)2878 3124 y Fu(=)g Fs( )3049 3091 y Fo(k)3046 3148 y Fk(1)p Fo(;v)3168 3124 y Fq(\000)25 b Fs( )3326 3091 y Fo(k)3323 3148 y Fk(2)p Fo(;v)3419 3124 y Fu(,)41 b(with)118 3232 y Fs( )177 3246 y Fk(1)p Fo(;v)273 3232 y Fs(;)15 b( )372 3246 y Fk(2)p Fo(;v)499 3232 y Fu(b)s(oth)29 b(satisfying)h(the)g(required)f(conditions.)39 b(W)-8 b(e)31 b(then)f(ha)m(v)m(e)1270 3368 y Fo(d)1223 3396 y Fi(X)1224 3593 y Fo(k)r Fk(=1)1369 3482 y Fp(E)1424 3496 y Fo(\026)1477 3482 y Fs(\030)1517 3497 y Fo(k)1559 3482 y Fs(u)1611 3445 y Fo(k)1654 3482 y Fs(D)1729 3497 y Fo(k)1772 3482 y Fs(f)k Fu(=)25 b(0)15 b Fs(;)92 b(f)35 b Fq(2)24 b Fs(L)2351 3445 y Fk(2)2391 3482 y Fu(\()p Fs(\026)p Fu(\))15 b Fs(:)873 b Fu(\(2.10\))118 3725 y(Cho)s(osing)29 b Fs(f)35 b Fu(=)25 b Fs(g)734 3739 y Fo(\015)809 3725 y Fu(in)k(\(2.10\),)k(with)c Fs(g)1453 3739 y Fo(\015)1528 3725 y Fu(as)h(in)f(\(2.9\),)k(w)m(e)e(obtain)1104 3861 y Fo(d)1056 3888 y Fi(X)1058 4086 y Fo(k)r Fk(=1)1203 3975 y Fp(E)1257 3989 y Fo(\026)1310 3975 y Fs(\030)1350 3990 y Fo(k)1392 3975 y Fu(\()p Fs(u)1479 3937 y Fo(k)1522 3975 y Fu(\))1557 3937 y Fk(2)1622 3975 y Fu(=)k(lim)1718 4033 y Fo(\015)t Fn(!)p Fk(0)1927 3861 y Fo(d)1880 3888 y Fi(X)1881 4086 y Fo(k)r Fk(=1)2026 3975 y Fp(E)2081 3989 y Fo(\026)2133 3975 y Fs(\030)2173 3990 y Fo(k)2216 3975 y Fs(u)2268 3937 y Fo(k)2311 3975 y Fs(D)2386 3990 y Fo(k)2428 3975 y Fs(g)2471 3989 y Fo(\015)2541 3975 y Fu(=)25 b(0)15 b Fs(;)118 4229 y Fu(whic)m(h,)29 b(b)m(y)i(the)f (ellipticit)m(y)e(of)j Fs(\030)t Fu(,)f(implies)e Fs(u)1654 4196 y Fo(k)1722 4229 y Fu(=)d(0)31 b Fs(\026)p Fu(-a.e.,)h Fs(k)c Fu(=)d(1)p Fs(;)15 b(:)g(:)g(:)i(;)e(d)p Fu(.)p 3595 4229 4 62 v 3599 4171 55 4 v 3599 4229 V 3653 4229 4 62 v 118 4429 a Fj(R)-5 b(emark)45 b(1)p Fu(.)77 b(Using)42 b(the)g(linearit)m(y)f(in)g Fs(v)46 b Fu(implicit)39 b(in)i(the)i(de\014nition)d(\(2.4\))k(w)m(e)f(ma)m(y)g(de\014ne)e(the) 118 4537 y(functions)29 b Fs( )572 4504 y Fo(ij)658 4537 y Fu(=)c Fs( )816 4504 y Fo(i)813 4559 y(e)846 4569 y Fv(j)883 4537 y Fu(,)30 b Fs(i;)15 b(j)32 b Fu(=)25 b(1)p Fs(;)15 b(:)g(:)g(:)i(;)e(d)p Fu(,)31 b(so)g(that)1415 4802 y Fs( )1477 4765 y Fo(i)1474 4825 y(v)1540 4802 y Fu(=)25 b(\(\011)p Fs(v)s Fu(\))1824 4816 y Fo(i)1884 4751 y Fk(\001)1878 4802 y Fu(=)2022 4688 y Fo(d)1974 4716 y Fi(X)1979 4911 y Fo(j)t Fk(=1)2121 4802 y Fs( )2183 4765 y Fo(ij)2244 4802 y Fs(v)2288 4816 y Fo(j)2339 4802 y Fs(:)1065 b Fu(\(2.11\))118 5051 y(The)30 b(di\013usivit)m(y)e Fs(D)33 b Fu(is)c(then)h(giv)m(en)h(b)m(y)f(the)h(matrix)1518 5205 y Fs(D)d Fu(=)d(2)p Fp(E)1817 5219 y Fo(\026)1870 5205 y Fu(\003)15 b(\()q(1)20 b(+)g(\011\))1183 b(\(2.12\))118 5372 y(with)33 b(\003)i(denoting)f(the)h(diagonal)e(matrix)h(\003)1696 5339 y Fo(ij)1789 5372 y Fu(=)e Fs(\030)1932 5386 y Fo(i)1960 5372 y Fs(\016)2000 5386 y Fo(ij)2061 5372 y Fu(.)53 b(The)34 b(functions)f Fs( )2788 5339 y Fo(ij)2884 5372 y Fu(are)h(often)h(called)f(the)118 5480 y(corrector)e(\014elds)d(and)g (\011)h(the)h(stream)g(matrix.)p eop %%Page: 6 6 6 5 bop 118 45 a Fk(6)1045 b(PIETR)n(O)31 b(CAPUTO)f(AND)g(DMITR)-6 b(Y)30 b(IOFFE)118 366 y Fj(R)-5 b(emark)36 b(2)p Fu(.)49 b(The)33 b(di\013usion)d(constan)m(t)k Fs(D)i Fu(has)c(an)h(explicit)e (expression)h(if)g Fs(d)e Fu(=)f(1.)48 b(Indeed,)33 b(for)g(an)m(y)118 474 y Fs(v)c Fq(2)c Fp(R)r Fu(,)37 b Fs(\030)t Fu(\()p Fs(v)24 b Fu(+)c Fs( )695 488 y Fo(v)736 474 y Fu(\()p Fs(\030)t Fu(\)\))31 b(m)m(ust)f(b)s(e)g(constan)m(t)i Fs(\026)p Fu(-a.s.)41 b(and)30 b(normalizing)e(w)m(e)j(ha)m(v)m(e)869 650 y Fs( )928 664 y Fo(v)969 650 y Fu(\()p Fs(\030)t Fu(\))26 b(=)f Fs(v)1267 549 y Fi(h)1310 650 y Fs(\030)1354 613 y Fn(\000)p Fk(1)1464 576 y Fi(\000)1505 650 y Fp(E)1560 664 y Fo(\026)1613 650 y Fs(\030)1657 613 y Fn(\000)p Fk(1)1751 576 y Fi(\001)1793 596 y Fn(\000)p Fk(1)1907 650 y Fq(\000)20 b Fu(1)2043 549 y Fi(i)2117 650 y Fs(;)91 b(D)28 b Fu(=)2432 576 y Fi(\000)2474 650 y Fp(E)2528 664 y Fo(\026)2581 650 y Fs(\030)2625 613 y Fn(\000)p Fk(1)2719 576 y Fi(\001)2761 596 y Fn(\000)p Fk(1)2885 650 y Fs(:)519 b Fu(\(2.13\))118 821 y(By)37 b(ellipticit)m(y)d(w)m(e)j (see)g(that)h(the)e(corrector)i(\014eld)d Fs( )1989 835 y Fo(v)2067 821 y Fu(is)h(uniformly)d(b)s(ounded)i(in)g(this)g(case.)61 b(The)118 929 y(situation)35 b(is)g(di\013eren)m(t)g(if)g Fs(d)g Fr(>)g Fu(2.)57 b(There)36 b(is)f(no)h(longer)f(an)h(expression) f(for)h Fs( )2924 943 y Fo(v)3001 929 y Fu(and)f(all)g(w)m(e)h(kno)m(w) 118 1037 y(apriori)e(is)g(that)i Fs( )i Fq(2)33 b Fs(L)968 1004 y Fk(2)1007 1037 y Fu(\()p Fs(\026)p Fu(\))j(.)56 b(F)-8 b(or)36 b Fs(d)e Fu(=)g(2)i(this)e(has)h(b)s(een)g(upgraded)g (in)f([1])i(to)g Fs( )i Fq(2)33 b Fs(L)3269 1004 y Fk(2+)p Fo(\017)3392 1037 y Fu(\()p Fs(\026)p Fu(\))j(for)118 1144 y(some)31 b Fs(\017)26 b(>)g Fu(0,)31 b(whic)m(h)f(\(in)g(t)m(w)m (o)i(dimensions\))c(leads)i(to)i(an)e(almost)h(sure)f(homogenization)h (result.)41 b(In)118 1252 y(the)32 b(sequel,)f(c.f.)43 b(the)32 b(remark)f(follo)m(wing)e(Theorem)i(4.1,)i(w)m(e)f(shall)d (establish)h(that)i Fs( )e Fq(2)c Fs(L)3306 1219 y Fk(2+)p Fo(\017)3429 1252 y Fu(\()p Fs(\026)p Fu(\))32 b(in)118 1360 y(an)m(y)f(dimension)d Fs(d)d Fr(>)g Fu(2.)1027 1550 y(3.)51 b Ft(Diffusivity)34 b(in)h(the)e(periodic)h(medium)218 1712 y Fu(In)i(the)i(case)g(of)g(the)f(p)s(erio)s(dized)e(en)m (vironmen)m(t)i(\(1.2\))r(,)i(the)e(pro)s(cess)g(of)h(the)f(en)m (vironmen)m(t)g(seen)118 1827 y(from)h(the)h(particle)e(is)h(equiv)-5 b(alen)m(t)38 b(to)h(the)f(pro)5 b(jected)39 b(random)f(w)m(alk)2673 1804 y(_)2637 1827 y Fs(X)7 b Fu(\()p Fs(t;)15 b(\031)2879 1841 y Fo(N)2947 1827 y Fs(\030)t Fu(\))39 b(on)f Fs(T)3252 1841 y Fo(N)3319 1827 y Fu(,)j(i.e.)64 b(an)118 1935 y(irreducible)26 b(Mark)m(o)m(v)k(c)m(hain)f(with)e(\014nite)h(state)i (space)g(and)e(unique)f(in)m(v)-5 b(arian)m(t)28 b(measure)h Fs(\032)3283 1949 y Fo(N)3379 1935 y Fu(de\014ned)118 2043 y(b)m(y)1116 2229 y Fp(E)1171 2243 y Fo(\032)1207 2254 y Fv(N)1275 2229 y Fs(f)34 b Fu(=)1523 2168 y(1)p 1460 2208 171 4 v 1460 2291 a Fq(j)p Fs(T)1538 2305 y Fo(N)1606 2291 y Fq(j)1683 2143 y Fi(X)1656 2340 y Fo(x)p Fn(2)p Fo(T)1784 2351 y Fv(N)1857 2229 y Fs(f)10 b Fu(\()p Fs(x)p Fu(\))15 b Fs(;)91 b(f)35 b Fu(:)25 b Fs(T)2348 2243 y Fo(N)2441 2229 y Fq(!)g Fp(R)18 b Fs(:)772 b Fu(\(3.14\))118 2471 y(Then,)30 b(in)m(tro)s(ducing)e(the)i(p)s(erio)s(dic)e(function) 1448 2620 y Fs(')1507 2583 y Fo(N)1507 2643 y(v)1575 2620 y Fu(\()p Fs(x)p Fu(\))e(=)f Fs(')1878 2634 y Fo(v)1919 2620 y Fu(\()p Fs(\034)1994 2634 y Fn(\000)p Fo(x)2093 2620 y Fs(\031)2145 2634 y Fo(N)2212 2620 y Fs(\030)t Fu(\))15 b Fs(;)118 2768 y Fu(where)30 b Fs(')440 2782 y Fo(v)512 2768 y Fu(is)f(de\014ned)g(b)m(y)h(\(2.3\),)j(w)m(e)e(arriv) m(e)f(as)g(in)f(\(2.7\))k(at)e(the)f(expression)653 3008 y(\()p Fs(D)763 3022 y Fo(N)831 3008 y Fu(\()p Fs(\030)t Fu(\))p Fs(v)s(;)15 b(v)s Fu(\))28 b(=)d(2)1346 2895 y Fo(d)1298 2922 y Fi(X)1307 3117 y Fo(i)p Fk(=1)1445 3008 y Fp(E)1499 3022 y Fo(\032)1535 3033 y Fv(N)1603 3008 y Fu(\(\()p Fs(\031)1725 3022 y Fo(N)1793 3008 y Fs(\030)t Fu(\))1872 3022 y Fo(i)1901 3008 y Fu(\))p Fs(v)1983 2971 y Fk(2)1980 3031 y Fo(i)2043 3008 y Fq(\000)20 b Fu(2)p Fp(E)2234 3022 y Fo(\032)2270 3033 y Fv(N)2353 2935 y Fi(\000)2395 3008 y Fs(')2454 2971 y Fo(N)2454 3031 y(v)2521 3008 y Fu(\()p Fq(\000L)2690 3022 y Fo(N)2757 3008 y Fu(\))2792 2971 y Fn(\000)p Fk(1)2887 3008 y Fs(')2946 2971 y Fo(N)2946 3031 y(v)3014 2935 y Fi(\001)3086 3008 y Fs(;)318 b Fu(\(3.15\))118 3254 y(where)27 b(the)h(generator)h Fq(L)997 3268 y Fo(N)1091 3254 y Fu(describ)s(es)d(the)i(jumps)e(on)h Fs(T)2074 3268 y Fo(N)2142 3254 y Fu(,)h(i.e.)40 b(for)27 b(an)m(y)h(p)s(erio)s(dic)d(function)h Fs(f)37 b Fu(on)27 b Fp(Z)3625 3221 y Fo(d)405 3402 y Fq(L)468 3416 y Fo(N)535 3402 y Fs(f)10 b Fu(\()p Fs(x)p Fu(\))25 b(=)g(\()p Fs(\031)920 3416 y Fo(N)987 3402 y Fs(\030)t Fu(\))1066 3416 y Fo(i)1095 3402 y Fu(\()p Fs(x)p Fu(\))15 b([)q Fs(f)10 b Fu(\()p Fs(x)20 b Fu(+)f Fs(e)1552 3416 y Fo(i)1581 3402 y Fu(\))i Fq(\000)e Fs(f)10 b Fu(\()p Fs(x)p Fu(\)])21 b(+)f(\()p Fs(\031)2128 3416 y Fo(N)2195 3402 y Fs(\030)t Fu(\))2274 3416 y Fo(i)2303 3402 y Fu(\()p Fs(x)g Fq(\000)g Fs(e)2543 3416 y Fo(i)2572 3402 y Fu(\))15 b([)p Fs(f)10 b Fu(\()p Fs(x)20 b Fq(\000)g Fs(e)2942 3416 y Fo(i)2971 3402 y Fu(\))g Fq(\000)g Fs(f)10 b Fu(\()p Fs(x)p Fu(\)])31 b Fs(:)118 3550 y Fu(Alternativ)m(ely)-8 b(,)31 b(iden)m(tifying)d(p)s (erio)s(dic)f(functions)i(with)g(functions)g(on)h Fs(T)2657 3564 y Fo(N)2755 3550 y Fu(w)m(e)h(ma)m(y)g(write)785 3785 y Fq(L)848 3799 y Fo(N)915 3785 y Fs(f)10 b Fu(\()p Fs(x)p Fu(\))25 b(=)g Fq(\000)1347 3671 y Fo(d)1299 3699 y Fi(X)1308 3894 y Fo(i)p Fk(=1)1461 3762 y Fu(~)1446 3785 y Fq(r)1522 3748 y Fn(\003)1522 3808 y Fo(i)1576 3684 y Fi(\020)1630 3785 y Fu(\()p Fs(\031)1717 3799 y Fo(N)1785 3785 y Fs(\030)t Fu(\))1864 3799 y Fo(i)1892 3785 y Fu(\()p Fs(x)p Fu(\))2030 3762 y(~)2014 3785 y Fq(r)2090 3799 y Fo(i)2119 3785 y Fs(f)10 b Fu(\()p Fs(x)p Fu(\))2296 3684 y Fi(\021)2380 3785 y Fs(;)91 b(f)35 b Fu(:)26 b Fs(T)2680 3799 y Fo(N)2772 3785 y Fq(!)f Fp(R)18 b Fs(;)118 4035 y Fu(where)390 4012 y(~)375 4035 y Fq(r)24 b Fu(are)g(the)h(discrete)f(gradien)m(ts)g(on)g(the)g(torus)g Fs(T)2035 4049 y Fo(N)2103 4035 y Fu(.)38 b(W)-8 b(e)26 b(then)e(ha)m(v)m(e)h(the)g(v)-5 b(ariational)23 b(principle)822 4270 y(\()p Fs(v)s(;)15 b(D)1019 4284 y Fo(N)1087 4270 y Fu(\()p Fs(\030)t Fu(\))p Fs(v)s Fu(\))27 b(=)e(2)99 b(inf)1466 4332 y Fo(f)7 b Fk(:)p Fo(T)1568 4343 y Fv(N)1626 4332 y Fn(!)p Fm(R)1807 4157 y Fo(d)1760 4184 y Fi(X)1768 4379 y Fo(i)p Fk(=1)1906 4270 y Fp(E)1961 4284 y Fo(\032)1997 4295 y Fv(N)2080 4169 y Fi(\020)2134 4270 y Fu(\()p Fs(\031)2221 4284 y Fo(N)2289 4270 y Fs(\030)t Fu(\))2368 4284 y Fo(i)2396 4270 y Fu(\()p Fs(v)2475 4284 y Fo(i)2524 4270 y Fu(+)2630 4247 y(~)2615 4270 y Fq(r)2691 4284 y Fo(i)2719 4270 y Fs(f)j Fu(\))2809 4233 y Fk(2)2848 4169 y Fi(\021)2933 4270 y Fs(:)471 b Fu(\(3.16\))118 4500 y(In)35 b(this)g (\014nite-dimensional)d(setting)k(the)f(ab)s(o)m(v)m(e)i(problem)d(ma)m (y)j(b)s(e)e(solv)m(ed)g(directly)f(without)h(the)118 4608 y(appro)m(ximation)29 b(pro)s(cedure)h(outlined)e(in)h(the)i(pro)s (of)e(of)i(Prop)s(osition)d(2.1.)42 b(Namely)-8 b(,)31 b(de\014ne)1174 4758 y Fs(\037)1231 4772 y Fo(N)s(;v)1351 4758 y Fu(\()p Fs(x)p Fu(\))26 b(=)f(\()p Fq(\000L)1764 4772 y Fo(N)1831 4758 y Fu(\))1866 4720 y Fn(\000)p Fk(1)1960 4758 y Fs(')2019 4720 y Fo(N)2019 4780 y(v)2087 4758 y Fu(\()p Fs(x)p Fu(\))15 b Fs(;)92 b(x)25 b Fq(2)g Fp(Z)2569 4720 y Fo(d)3429 4758 y Fu(\(3.17\))118 4906 y(and)36 b(observ)m(e)h(that)g Fs(\037)891 4920 y Fo(N)s(;v)1047 4906 y Fu(is)e(a)i(b)s(ounded)d(p)s(erio)s(dic)g(function)h(for)h(eac)m (h)h Fs(N)10 b Fu(.)59 b(Boundedness)35 b(follo)m(ws)118 5019 y(from)26 b(the)h(exp)s(onen)m(tial)e(mixing)g(prop)s(erties)f(of) j(the)f(pro)s(cess)g(generated)i(b)m(y)e Fq(L)2846 5033 y Fo(N)2913 5019 y Fu(,)h(since)f Fp(E)3239 5033 y Fo(\032)3275 5044 y Fv(N)3343 5019 y Fs(')3402 4986 y Fo(N)3402 5042 y(v)3495 5019 y Fu(=)f(0.)118 5127 y(As)30 b(in)g(the)g(previous)f (section)h(w)m(e)h(then)f(ha)m(v)m(e)269 5362 y(\()p Fs(v)s(;)15 b(D)466 5376 y Fo(N)535 5362 y Fu(\()p Fs(\030)t Fu(\))p Fs(v)s Fu(\))26 b(=)f(2)961 5249 y Fo(d)914 5276 y Fi(X)923 5471 y Fo(i)p Fk(=1)1060 5362 y Fs(v)1104 5376 y Fo(i)1133 5362 y Fp(E)1187 5376 y Fo(\032)1223 5387 y Fv(N)1291 5362 y Fu(\()p Fs(\031)1378 5376 y Fo(N)1446 5362 y Fs(\030)t Fu(\))1525 5376 y Fo(i)1568 5289 y Fi(\000)1610 5362 y Fs(v)1654 5376 y Fo(i)1702 5362 y Fu(+)20 b Fs( )1855 5325 y Fo(i)1852 5385 y(N)s(;v)1972 5289 y Fi(\001)2039 5362 y Fu(=)25 b(2)2243 5249 y Fo(d)2196 5276 y Fi(X)2204 5471 y Fo(i)p Fk(=1)2342 5362 y Fp(E)2397 5376 y Fo(\032)2433 5387 y Fv(N)2501 5362 y Fu(\()p Fs(\031)2588 5376 y Fo(N)2655 5362 y Fs(\030)t Fu(\))2734 5376 y Fo(i)2778 5289 y Fi(\000)2819 5362 y Fs(v)2863 5376 y Fo(i)2912 5362 y Fu(+)20 b Fs( )3065 5325 y Fo(i)3062 5385 y(N)s(;v)3182 5289 y Fi(\001)3223 5307 y Fk(2)3429 5362 y Fu(\(3.18\))p eop %%Page: 7 7 7 6 bop 3626 43 a Fk(7)118 366 y Fu(where)30 b Fs( )443 333 y Fo(i)440 393 y(N)s(;v)560 366 y Fu(,)g Fs(i)c Fu(=)f(1)p Fs(;)15 b(:)g(:)g(:)i(;)e(d)31 b Fu(are)g(de\014ned)e(b)m(y)971 543 y Fs( )1033 505 y Fo(i)1030 565 y(N)s(;v)1150 543 y Fu(\()p Fs(x)p Fu(\))c(=)g Fs(\037)1450 557 y Fo(N)s(;v)1570 543 y Fu(\()p Fs(x)20 b Fu(+)g Fs(e)1810 557 y Fo(i)1839 543 y Fu(\))g Fq(\000)g Fs(\037)2042 557 y Fo(N)s(;v)2162 543 y Fu(\()p Fs(x)p Fu(\))26 b(=)2421 520 y(~)2406 543 y Fq(r)2482 557 y Fo(i)2510 543 y Fs(\037)2567 557 y Fo(N)s(;v)2686 543 y Fu(\()p Fs(x)p Fu(\))621 b(\(3.19\))118 709 y(Clearly)24 b Fs( )491 676 y Fo(i)488 736 y(N)s(;v)633 709 y Fu(are)i(p)s(erio)s(dic)c(functions.)38 b(The)25 b(dep)s(endence)f(on)h Fs(\030)k Fu(b)s(ecomes)d(explicit)e(in)g(the)h (notation)1086 884 y Fs( )1148 846 y Fo(i)1145 906 y(N)s(;v)1265 884 y Fu(\()p Fs(x;)15 b(\030)t Fu(\))26 b(=)f Fs( )1655 846 y Fo(i)1652 906 y(N)s(;v)1772 884 y Fu(\()p Fs(x)p Fu(\))h(=)f Fs( )2078 846 y Fo(i)2075 906 y(N)s(;v)2195 884 y Fu(\(0)p Fs(;)15 b(\034)2355 898 y Fn(\000)p Fo(x)2455 884 y Fs(\031)2507 898 y Fo(N)2574 884 y Fs(\030)t Fu(\))g Fs(:)118 1072 y Fu(3.1.)54 b Fz(Con)m(v)m(ergence.)46 b Fu(F)-8 b(rom)31 b(no)m(w)f(on)h Fs( )1582 1039 y Fo(i)1579 1099 y(N)s(;v)1729 1072 y Fu(will)d(b)s(e)h(regarded)i(as)f(a)h (function)e(of)i Fs(\030)t Fu(,)f(b)m(y)g(setting)1202 1225 y(~)1183 1249 y Fs( )1245 1212 y Fo(i)1242 1272 y(N)s(;v)1362 1249 y Fu(\()p Fs(\030)t Fu(\))c(=)f Fs( )1660 1212 y Fo(i)1657 1272 y(N)s(;v)1777 1249 y Fu(\(0)p Fs(;)15 b(\030)t Fu(\))g Fs(;)93 b(i)25 b Fu(=)g(1)p Fs(;)15 b(:)g(:)g(:)i(;)e(d)g(:)834 b Fu(\(3.20\))118 1437 y Fz(Prop)s(osition)36 b(3.1.)46 b Fj(F)-7 b(or)40 b(e)-5 b(ach)38 b Fs(v)h Fq(2)34 b Fp(R)1514 1404 y Fo(d)1560 1437 y Fj(,)1646 1413 y Fu(~)1628 1437 y Fs( )1687 1451 y Fo(N)s(;v)1841 1437 y Fq(!)h Fs( )2026 1451 y Fo(v)2105 1437 y Fj(in)j Fs(L)2284 1404 y Fk(2)2324 1437 y Fu(\()p Fs(\026)p Fu(\))p Fj(,)h(as)g Fs(N)45 b Fq(!)35 b(1)p Fj(,)k(wher)-5 b(e)39 b Fs( )3361 1451 y Fo(v)3440 1437 y Fj(is)f(the)118 1545 y(\014eld)33 b(de\014ne)-5 b(d)34 b(in)e(Pr)-5 b(op)g(osition)36 b(2.1)118 1710 y(Pr)-5 b(o)g(of.)43 b Fu(Notice)31 b(that,)h(as)e(in)f(\(2.8\))976 1841 y Fo(d)929 1869 y Fi(X)938 2064 y Fo(i)p Fk(=1)1075 1955 y Fp(E)1130 1969 y Fo(\032)1166 1980 y Fv(N)1234 1955 y Fu(\()p Fs(\031)1321 1969 y Fo(N)1388 1955 y Fs(\030)t Fu(\))1467 1969 y Fo(i)1496 1955 y Fu(\()p Fs( )1593 1917 y Fo(i)1590 1977 y(N)s(;v)1710 1955 y Fu(\))1745 1917 y Fk(2)1810 1955 y Fu(=)c Fq(\000)2040 1841 y Fo(d)1992 1869 y Fi(X)2001 2064 y Fo(i)p Fk(=1)2139 1955 y Fs(v)2183 1969 y Fo(i)2211 1955 y Fp(E)2265 1969 y Fo(\032)2301 1980 y Fv(N)2369 1955 y Fu(\()p Fs(\031)2456 1969 y Fo(N)2524 1955 y Fs(\030)t Fu(\))2603 1969 y Fo(i)2631 1955 y Fs( )2693 1917 y Fo(i)2690 1977 y(N)s(;v)2825 1955 y Fs(:)579 b Fu(\(3.21\))118 2189 y(Then)29 b(b)m(y)i(ellipticit)m(y)d(and)i(using)e (Sc)m(h)m(w)m(arz')k(inequalit)m(y)d(w)m(e)i(obtain)1393 2342 y(max)1465 2401 y Fo(i)1577 2342 y Fp(E)1631 2356 y Fo(\032)1668 2367 y Fv(N)1735 2342 y Fu(\()p Fs( )1832 2304 y Fo(i)1829 2364 y(N)s(;v)1950 2342 y Fu(\))1985 2304 y Fk(2)2050 2342 y Fr(6)25 b Fs(c)2185 2304 y Fk(2)2224 2342 y Fq(j)p Fs(v)s Fq(j)2321 2304 y Fk(2)2362 2342 y Fs(:)1042 b Fu(\(3.22\))118 2526 y(No)m(w)27 b(w)m(e)g(observ)m(e)g (that)h(since)d(the)i(b)s(ond)e(v)-5 b(ariables)25 b Fs(\030)31 b Fu(are)26 b Fj(exchange)-5 b(able)28 b Fu(under)d Fs(\026)p Fu(,)i(the)g(distribution)118 2634 y(of)k Fs(\031)274 2648 y Fo(N)341 2634 y Fs(\030)j Fu(coincides)29 b(with)g(that)i(of)g Fs(\034)1347 2648 y Fo(x)1390 2634 y Fs(\031)1442 2648 y Fo(N)1510 2634 y Fs(\030)j Fu(for)c(an)m(y)h Fs(x)25 b Fq(2)g Fp(Z)2123 2601 y Fo(d)2159 2634 y Fu(.)40 b(In)30 b(particular)f(w)m(e)i(see)g(that)600 2825 y Fp(E)654 2839 y Fo(\026)707 2825 y Fu(\()761 2801 y(~)742 2825 y Fs( )804 2787 y Fo(i)801 2847 y(N)s(;v)921 2825 y Fu(\))956 2787 y Fk(2)1021 2825 y Fu(=)1190 2763 y(1)p 1127 2804 171 4 v 1127 2887 a Fq(j)p Fs(T)1205 2901 y Fo(N)1273 2887 y Fq(j)1350 2738 y Fi(X)1323 2935 y Fo(x)p Fn(2)p Fo(T)1451 2946 y Fv(N)1524 2825 y Fp(E)1579 2839 y Fo(\026)1631 2825 y Fu(\()p Fs( )1728 2787 y Fo(i)1725 2847 y(N)s(;v)1845 2825 y Fu(\(0)p Fs(;)15 b(\034)2005 2839 y Fo(x)2050 2825 y Fs(\031)2102 2839 y Fo(N)2169 2825 y Fs(\030)t Fu(\)\))2283 2787 y Fk(2)2349 2825 y Fu(=)25 b Fp(E)2499 2839 y Fo(\026)2552 2825 y Fp(E)2606 2839 y Fo(\032)2643 2850 y Fv(N)2710 2825 y Fu(\()p Fs( )2807 2787 y Fo(i)2804 2847 y(N)s(;v)2925 2825 y Fu(\()p Fq(\001)p Fs(;)15 b(\030)t Fu(\)\))3139 2787 y Fk(2)3429 2825 y Fu(\(3.23\))118 3093 y(and)42 b(\(3.22\))k(pro)m(vides)41 b(a)j(uniform)c(b)s(ound)h (on)i(the)g Fs(L)2059 3060 y Fk(2)2099 3093 y Fu(\()p Fs(\026)p Fu(\)-norm)f(of)2639 3069 y(~)2621 3093 y Fs( )2683 3060 y Fo(i)2680 3120 y(N)s(;v)2799 3093 y Fu(,)47 b(whic)m(h)41 b(implies)f(w)m(eak)118 3204 y(con)m(v)m(ergence)29 b(along)d (subsequences.)38 b(Next)27 b(w)m(e)f(sho)m(w)g(that)h(an)m(y)f(w)m (eak)h(limit)d(p)s(oin)m(t)h(m)m(ust)h(satisfy)f(the)118 3315 y(conditions)32 b(of)h(Prop)s(osition)e(2.1,)j(th)m(us)f (establishing)e(the)i(w)m(eak)h(con)m(v)m(ergence)h(of)3049 3291 y(~)3030 3315 y Fs( )3089 3329 y Fo(N)s(;v)3242 3315 y Fu(to)e Fs( )3414 3329 y Fo(v)3456 3315 y Fu(.)48 b(Let)137 3411 y(~)118 3435 y Fs( )180 3402 y Fo(i)177 3457 y Fn(1)p Fo(;v)339 3435 y Fu(b)s(e)30 b(a)h(w)m(eak)g(limit)e(of) 1106 3411 y(~)1088 3435 y Fs( )1150 3402 y Fo(i)1147 3462 y(N)s(;v)1267 3435 y Fu(.)41 b Fp(E)1388 3449 y Fo(\026)1459 3411 y Fu(~)1440 3435 y Fs( )1502 3402 y Fo(i)1499 3457 y Fn(1)p Fo(;v)1656 3435 y Fu(=)25 b(0)31 b(is)f(a)h(consequence)h(of)e Fp(E)2668 3449 y Fo(\026)2739 3411 y Fu(~)2721 3435 y Fs( )2783 3402 y Fo(i)2780 3462 y(N)s(;v)2925 3435 y Fu(=)25 b(0,)32 b(whic)m(h)d(in)g(turn)118 3546 y(follo)m(ws)35 b(from)g(exc)m(hangeabilit)m(y)-8 b(.)58 b(T)-8 b(o)36 b(c)m(hec)m(k)i(the)e(second)f(condition)g(w)m(e)h (sho)m(w)g(that)g(for)g(an)m(y)g(lo)s(cal)118 3654 y(function)29 b Fs(f)35 b Fq(2)25 b Fs(L)702 3621 y Fk(2)741 3654 y Fu(\(\012\))31 b(w)m(e)f(ha)m(v)m(e)1238 3785 y Fo(d)1190 3812 y Fi(X)1199 4008 y Fo(i)p Fk(=1)1336 3899 y Fp(E)1391 3913 y Fo(\026)1459 3798 y Fi(h)1502 3899 y Fs(D)1577 3913 y Fo(i)1605 3899 y Fs(f)10 b(\030)1700 3913 y Fo(i)1743 3798 y Fi(\020)1797 3899 y Fs(v)1841 3913 y Fo(i)1889 3899 y Fu(+)1999 3875 y(~)1980 3899 y Fs( )2042 3861 y Fo(i)2039 3921 y Fn(1)p Fo(;v)2171 3899 y Fu(\()p Fs(\030)t Fu(\))2285 3798 y Fi(\021)q(i)2408 3899 y Fu(=)25 b(0)15 b Fs(:)840 b Fu(\(3.24\))118 4147 y(F)-8 b(rom)31 b(the)g(de\014nition) d(of)i Fs( )1083 4114 y Fo(i)1080 4174 y(N)s(;v)1200 4147 y Fu(,)h(writing)d(out)j(\(3.17\))h(explicitly)c(w)m(e)j(ha)m(v)m (e)959 4295 y Fo(d)912 4322 y Fi(X)920 4517 y Fo(i)p Fk(=1)1073 4385 y Fu(~)1058 4408 y Fq(r)1134 4371 y Fn(\003)1134 4431 y Fo(i)1188 4335 y Fi(\000)1230 4408 y Fu(\()p Fs(\031)1317 4422 y Fo(N)1384 4408 y Fs(\030)t Fu(\))1463 4422 y Fo(i)1492 4408 y Fu(\()p Fs(x)p Fu(\))p Fs( )1676 4371 y Fo(i)1673 4431 y(N)s(;v)1794 4408 y Fu(\()p Fs(x;)15 b(\030)t Fu(\))2000 4335 y Fi(\001)2067 4408 y Fu(=)25 b Fs(')2222 4371 y Fo(N)2222 4431 y(v)2290 4408 y Fu(\()p Fs(x)p Fu(\))15 b Fs(;)92 b(x)25 b Fq(2)g Fs(T)2760 4422 y Fo(N)2843 4408 y Fs(:)118 4642 y Fu(A)m(t)31 b Fs(x)26 b Fu(=)f(0)30 b(this)g(yields)734 4768 y Fo(d)686 4796 y Fi(X)695 4991 y Fo(i)p Fk(=1)833 4781 y Fi(h)876 4882 y Fu(\()p Fs(\034)951 4896 y Fo(e)984 4906 y Fv(i)1014 4882 y Fs(\030)t Fu(\))1093 4896 y Fo(i)1137 4781 y Fi(\020)1191 4882 y Fs(v)1235 4896 y Fo(i)1284 4882 y Fu(+)1393 4858 y(~)1374 4882 y Fs( )1436 4844 y Fo(i)1433 4904 y(N)s(;v)1553 4882 y Fu(\()p Fs(\034)1628 4896 y Fo(e)1661 4906 y Fv(i)1692 4882 y Fs(\031)1744 4896 y Fo(N)1811 4882 y Fs(\030)t Fu(\))1890 4781 y Fi(\021)1965 4882 y Fq(\000)19 b Fs(\030)2095 4896 y Fo(i)2139 4781 y Fi(\020)2193 4882 y Fs(v)2237 4896 y Fo(i)2285 4882 y Fu(+)2395 4858 y(~)2376 4882 y Fs( )2438 4844 y Fo(i)2435 4904 y(N)s(;v)2555 4882 y Fu(\()p Fs(\031)2642 4896 y Fo(N)2709 4882 y Fs(\030)t Fu(\))2788 4781 y Fi(\021)q(i)2911 4882 y Fu(=)25 b(0)15 b Fs(:)337 b Fu(\(3.25\))118 5129 y(If)35 b Fs(f)42 b Fq(2)33 b Fs(L)457 5096 y Fk(2)496 5129 y Fu(\(\012\))i(is)f(lo)s(cal,) i(then)f Fs(f)10 b Fu(\()p Fs(\030)t Fu(\))33 b(=)g Fs(f)10 b Fu(\()p Fs(\031)1669 5143 y Fo(N)1735 5129 y Fs(\030)t Fu(\))36 b(and)e Fs(f)10 b Fu(\()p Fs(\034)2161 5143 y Fn(\006)p Fo(e)2249 5153 y Fv(j)2285 5129 y Fs(\030)t Fu(\))33 b(=)g Fs(f)10 b Fu(\()p Fs(\034)2631 5143 y Fn(\006)p Fo(e)2719 5153 y Fv(j)2755 5129 y Fs(\031)2807 5143 y Fo(N)2874 5129 y Fs(\030)t Fu(\))35 b(when)f Fs(N)45 b Fu(is)34 b(large,)118 5237 y(and)c(b)m(y)g(the)h(exc)m(hangeabilit)m (y)f(w)m(e)h(ha)m(v)m(e)525 5307 y Fi(Z)575 5514 y Fk(\012)645 5431 y Fs(\026)p Fu(\()15 b(d)p Fs(\030)t Fu(\))p Fs(f)10 b Fu(\()p Fs(\031)1022 5445 y Fo(N)1089 5431 y Fs(\030)t Fu(\)\()p Fs(\034)1243 5445 y Fo(e)1276 5455 y Fv(i)1307 5431 y Fs(\030)t Fu(\))1386 5445 y Fo(i)1433 5407 y Fu(~)1414 5431 y Fs( )1476 5394 y Fo(i)1473 5454 y(N)s(;v)1593 5431 y Fu(\()p Fs(\034)1668 5445 y Fo(e)1701 5455 y Fv(i)1732 5431 y Fs(\031)1784 5445 y Fo(N)1851 5431 y Fs(\030)t Fu(\))25 b(=)2051 5307 y Fi(Z)2102 5514 y Fk(\012)2172 5431 y Fs(\026)p Fu(\()15 b(d)p Fs(\030)t Fu(\))p Fs(f)10 b Fu(\()p Fs(\034)2537 5445 y Fn(\000)p Fo(e)2625 5455 y Fv(i)2655 5431 y Fs(\031)2707 5445 y Fo(N)2774 5431 y Fs(\030)t Fu(\))p Fs(\030)2893 5445 y Fo(i)2939 5407 y Fu(~)2921 5431 y Fs( )2983 5394 y Fo(i)2980 5454 y(N)s(;v)3100 5431 y Fu(\()p Fs(\030)t Fu(\))15 b Fs(:)p eop %%Page: 8 8 8 7 bop 118 45 a Fk(8)1045 b(PIETR)n(O)31 b(CAPUTO)f(AND)g(DMITR)-6 b(Y)30 b(IOFFE)118 366 y Fu(Multiplying)d(\(3.25\))33 b(with)c Fs(f)39 b Fu(and)30 b(in)m(tegrating)g(w)m(e)h(see)g(that)g (when)f Fs(N)40 b Fu(is)29 b(large)1301 519 y Fo(d)1253 546 y Fi(X)1262 742 y Fo(i)p Fk(=1)1400 633 y Fp(E)1454 647 y Fo(\026)1522 532 y Fi(h)1565 633 y Fs(D)1640 647 y Fo(i)1668 633 y Fs(f)10 b(\030)1763 647 y Fo(i)1806 532 y Fi(\020)1860 633 y Fs(v)1904 647 y Fo(i)1952 633 y Fu(+)2062 609 y(~)2043 633 y Fs( )2105 595 y Fo(i)2102 655 y(N)s(;v)2222 532 y Fi(\021i)2345 633 y Fu(=)25 b(0)15 b Fs(;)118 893 y Fu(and)26 b(the)h(claim)f(\(3.24\))j(follo)m(ws)d (from)h(w)m(eak)g(con)m(v)m(ergence.)42 b(The)27 b(last)f(condition)g (in)f(Prop)s(osition)g(2.1)118 1001 y(is)30 b(pro)m(v)m(en)g(b)m(y)g (similar)e(reasoning,)i(using)f(also)1014 1154 y(~)999 1177 y Fq(r)1075 1191 y Fo(i)1103 1177 y Fs( )1165 1140 y Fo(k)1162 1200 y(N)s(;v)1307 1177 y Fu(=)1418 1154 y(~)1403 1177 y Fq(r)1479 1191 y Fo(i)1522 1154 y Fu(~)1507 1177 y Fq(r)1583 1192 y Fo(k)1625 1177 y Fs(\037)1682 1191 y Fo(N)s(;v)1827 1177 y Fu(=)1938 1154 y(~)1923 1177 y Fq(r)1999 1192 y Fo(k)2056 1154 y Fu(~)2041 1177 y Fq(r)2117 1191 y Fo(i)2145 1177 y Fs(\037)2202 1191 y Fo(N)s(;v)2347 1177 y Fu(=)2458 1154 y(~)2443 1177 y Fq(r)2519 1192 y Fo(k)2561 1177 y Fs( )2623 1140 y Fo(i)2620 1200 y(N)s(;v)2755 1177 y Fs(:)218 1355 y Fu(By)38 b(the)g(orthogonalit)m(y)g(relations)e(\(2.8\))k(and)d(\(3.21\))j(w)m (e,)g(using)c(again)i(exc)m(hangeabilit)m(y)g(as)g(in)118 1463 y(\(3.23\))r(,)31 b(infer:)1096 1691 y(lim)1057 1752 y Fo(N)7 b Fn(!1)1324 1578 y Fo(d)1277 1605 y Fi(X)1285 1800 y Fo(i)p Fk(=1)1423 1691 y Fp(E)1478 1705 y Fo(\026)1530 1691 y Fs(\030)1570 1705 y Fo(i)1613 1590 y Fi(\020)1686 1667 y Fu(~)1668 1691 y Fs( )1730 1654 y Fo(i)1727 1714 y(N)s(;v)1847 1590 y Fi(\021)1901 1613 y Fk(2)1996 1691 y Fu(=)2170 1578 y Fo(d)2122 1605 y Fi(X)2131 1800 y Fo(i)p Fk(=1)2269 1691 y Fp(E)2323 1705 y Fo(\026)2376 1691 y Fs(\030)2416 1705 y Fo(i)2459 1618 y Fi(\000)2501 1691 y Fs( )2563 1654 y Fo(i)2560 1714 y(v)2601 1618 y Fi(\001)2642 1636 y Fk(2)2697 1691 y Fs(:)118 1950 y Fu(In)31 b(view)f(of)h(the)h(uniform)d(ellipticit)m(y)g(condition,)h Fs(\030)1917 1964 y Fo(i)1971 1950 y Fr(>)d Fu(1)p Fs(=c)h(>)e Fu(0,)32 b(this)e(implies)e(that)3129 1926 y(~)3110 1950 y Fs( )3172 1917 y Fo(i)3169 1977 y(N)s(;v)3320 1950 y Fu(con)m(v)m(erge)118 2074 y(strongly)i(in)f Fs(L)638 2041 y Fk(2)677 2074 y Fu(\()p Fs(\026)p Fu(\).)p 3595 2074 4 62 v 3599 2016 55 4 v 3599 2074 V 3653 2074 4 62 v 218 2215 a(An)36 b(immediate)f(corollary)h(of)g(the)h(ab)s(o)m(v)m (e)g(results)e(is)h(the)g(mean)g(con)m(v)m(ergence)j(of)e(the)f (di\013usion)118 2323 y(constan)m(ts)c Fq(f)p Fs(D)641 2337 y Fo(N)709 2323 y Fu(\()p Fs(\030)t Fu(\))p Fq(g)p Fu(:)118 2503 y Fz(Corollary)37 b(3.1.)46 b Fj(F)-7 b(or)33 b(every)g Fs(v)28 b Fq(2)d Fp(R)1421 2470 y Fo(d)1467 2503 y Fj(,)1271 2677 y Fu(lim)1232 2738 y Fo(N)7 b Fn(!1)1452 2677 y Fp(E)1506 2691 y Fo(\026)1574 2677 y Fu(\()p Fs(v)s(;)15 b(D)1771 2691 y Fo(N)1840 2677 y Fu(\()p Fs(\030)t Fu(\))p Fs(v)s Fu(\))59 b(=)e(\()q Fs(v)s(;)15 b(D)s(v)s Fu(\))h Fs(:)118 2893 y Fj(Pr)-5 b(o)g(of.)43 b Fu(F)-8 b(ollo)m(ws)28 b(from)g(the)g(represen)m(tation)h(form)m(ulas)e(\(2.2\))j(and)d (\(3.18\))s(,)h(exc)m(hangeabilit)m(y)h(and)e(the)118 3009 y Fs(L)180 2976 y Fk(2)219 3009 y Fu(\()p Fs(\026)p Fu(\))k(con)m(v)m(ergence)i(of)1000 2985 y(~)982 3009 y Fs( )1044 2976 y Fo(i)1041 3036 y(N)s(;v)1191 3009 y Fu(to)e Fs( )1364 2976 y Fo(i)1361 3032 y(v)1402 3009 y Fu(.)p 3595 3009 V 3599 2951 55 4 v 3599 3009 V 3653 3009 4 62 v 1066 3276 a(4.)51 b Ft(Almost)33 b(sure)g(conver)n(gence)f (of)i Fs(D)2646 3290 y Fo(N)218 3438 y Fu(In)k(this)f(section)i(w)m(e)g (sho)m(w)g(that)g(the)g(p)s(erio)s(dized)d(\014nite)i(v)m(olume)g (e\013ectiv)m(e)i(di\013usion)c(matrices)118 3546 y Fs(D)193 3560 y Fo(N)261 3546 y Fu(\()p Fs(\030)t Fu(\))28 b(con)m(v)m(erge)i (to)e(the)g(in\014nite)e(v)m(olume)h(e\013ectiv)m(e)j(di\013usivit)m(y) 25 b Fs(D)30 b Fu(almost)e(surely)f(in)f(the)i(en)m(viron-)118 3661 y(men)m(t,)i(that)g(is)e(\(1.5\))i(holds.)39 b Fs(L)1208 3628 y Fo(p)1277 3661 y Fu(b)s(ounds)26 b(on)j(the)g(gradien)m(ts)2277 3638 y(~)2262 3661 y Fq(r)p Fs(\037)2395 3675 y Fo(N)s(;v)2543 3661 y Fu(of)g(the)h(solutions)d(to)j(\(3.17\))h(set)118 3768 y(up)h(stage)j(for)d(an)h(application)f(of)h(T)-8 b(alagrand's)33 b(concen)m(tration)h(estimates,)g(whic)m(h)e(imply)f (that)j(for)118 3876 y(ev)m(ery)d Fs(v)e Fq(2)c Fp(R)576 3843 y Fo(d)623 3876 y Fu(:)860 4050 y Fs(\026)20 b Fq(\000)g Fu(a)p Fs(:)p Fu(s)145 b(lim)1238 4111 y Fo(N)7 b Fn(!1)1458 4050 y Fq(j)p Fu(\()p Fs(D)1593 4064 y Fo(N)1661 4050 y Fu(\()p Fs(\030)t Fu(\))p Fs(v)s(;)15 b(v)s Fu(\))22 b Fq(\000)e Fp(E)2111 4064 y Fo(\026)2179 4050 y Fu(\()q Fs(D)2290 4064 y Fo(N)2357 4050 y Fu(\()p Fs(\030)t Fu(\))p Fs(v)s(;)15 b(v)s Fu(\))r Fq(j)56 b Fu(=)f(0)p Fs(:)556 b Fu(\(4.1\))118 4285 y(In)39 b(the)g(sequel)g(w)m(e)h(shall)d(use)i Fs(\037)1273 4241 y Fo(\030)1273 4314 y(N)s(;v)1432 4285 y Fu(instead)f(of)i Fs(\037)1924 4299 y Fo(N)s(;v)2043 4285 y Fu(,)i(just)d(in)f(order)h(to)h(stress)f(the)g(dep)s(endence)118 4413 y(of)g(solutions)e(of)46 b(\(3.17\))41 b(on)e(a)g(particular)e (realization)h Fs(\030)43 b Fq(2)c Fu([1)p Fs(=c;)15 b(c)p Fu(])2549 4380 y Fo(T)2590 4391 y Fv(N)2694 4413 y Fu(=)39 b(\012)2870 4427 y Fo(N)2975 4413 y Fu(of)g(the)g(p)s(erio)s (dized)118 4521 y(en)m(vironmen)m(t.)118 4740 y(4.1.)54 b Fs(L)374 4707 y Fo(p)448 4740 y Fz(estimates.)44 b Fu(The)30 b(equation)g(\(3.17\))j(is)c(uniformly)f(elliptic)g(in)h Fs(N)35 b Fq(2)25 b Fp(N)43 b Fu(and)30 b Fs(\030)f Fq(2)c Fu(\012)3368 4754 y Fo(N)3435 4740 y Fu(.)118 4918 y Fz(Theorem)34 b(4.1.)46 b Fj(Ther)-5 b(e)38 b(exists)f(a)h(p)-5 b(ower)39 b Fs(p)33 b Fu(=)g Fs(p)p Fu(\()p Fs(c)p Fu(\))h Fs(>)f Fu(2)k Fj(and)h(a)g(c)-5 b(onstant)39 b Fs(\013)33 b Fu(=)g Fs(\013)p Fu(\()p Fs(c)p Fu(\))i Fs(<)e Fq(1)p Fj(,)38 b(such)118 5028 y(that)c(uniformly)g(in)e Fs(v)d Fq(2)c Fp(R)1048 4995 y Fo(d)1095 5028 y Fj(,)32 b Fs(N)j Fq(2)25 b Fp(N)45 b Fj(and)34 b Fs(\030)29 b Fq(2)c Fu(\012)1845 5042 y Fo(N)1912 5028 y Fj(,)838 5339 y Fq(k)899 5316 y Fu(~)883 5339 y Fq(r)q Fs(\037)1017 5295 y Fo(\030)1017 5368 y(N)s(;v)1136 5339 y Fq(k)1181 5353 y Fo(N)s(;p)1358 5339 y Fu(=)1486 5157 y Fi(0)1486 5320 y(@)1639 5278 y Fu(1)p 1576 5318 171 4 v 1576 5402 a Fq(j)p Fs(T)1654 5416 y Fo(N)1721 5402 y Fq(j)1799 5253 y Fi(X)1772 5450 y Fo(x)p Fn(2)p Fo(T)1900 5461 y Fv(N)1973 5235 y Fi(\014)1973 5289 y(\014)1973 5344 y(\014)2018 5316 y Fu(~)2003 5339 y Fq(r)p Fs(\037)2136 5295 y Fo(\030)2136 5368 y(N)s(;v)2255 5339 y Fu(\()p Fs(x)p Fu(\))2377 5235 y Fi(\014)2377 5289 y(\014)2377 5344 y(\014)2408 5261 y Fo(p)2448 5157 y Fi(1)2448 5320 y(A)2538 5152 y Fg(1)p 2537 5164 33 3 v 2537 5205 a Fv(p)2656 5339 y Fr(6)33 b Fs(\013)p Fq(j)p Fs(v)s Fq(j)p Fs(:)535 b Fu(\(4.2\))p eop %%Page: 9 9 9 8 bop 3626 43 a Fk(9)118 366 y Fj(R)-5 b(emark.)42 b Fu(By)31 b(the)f(exc)m(hangeabilit)m(y)g(of)h(the)g Fs(\030)t Fu(-en)m(vironmen)m(t,)f(\(4.2\))i(implies)1464 619 y(sup)1501 697 y Fo(N)1616 619 y Fp(E)1671 633 y Fo(\026)1738 514 y Fi(\014)1738 569 y(\014)1738 624 y(\014)1787 595 y Fu(~)1769 619 y Fs( )1828 633 y Fo(N)s(;v)1948 514 y Fi(\014)1948 569 y(\014)1948 624 y(\014)1978 541 y Fo(p)2073 619 y Fs(<)55 b Fq(1)p Fs(;)118 914 y Fu(where)411 890 y(~)392 914 y Fs( )451 928 y Fo(N)s(;v)613 914 y Fu(has)41 b(b)s(een)g(de\014ned)f(in)h(\(3.20\))r(.)74 b(Since)41 b(the)h(in\014nite)d(v)m(olume)i(corrector)i(\014eld)d (could)118 1034 y(b)s(e)d(obtained)g(as)g(an)h(almost)f(sure)g(limit)e (of)1735 1010 y(~)1716 1034 y Fs( )1775 1048 y Fo(N)1831 1060 y Fv(k)1870 1048 y Fo(;v)1968 1034 y Fu(along)i(a)h(subsequence)e Fq(f)p Fs(N)2938 1049 y Fo(k)2982 1034 y Fq(g)p Fu(,)j(it)e(follo)m(ws) g(that)118 1148 y Fs( )177 1162 y Fo(v)243 1148 y Fq(2)25 b Fs(L)391 1115 y Fo(p)431 1148 y Fu(\()p Fs(\026)p Fu(\))31 b(for)f(ev)m(ery)h Fs(v)e Fq(2)24 b Fp(R)1184 1115 y Fo(d)1231 1148 y Fu(.)218 1280 y(Because)h(of)f(the)h(uniform)c (ellipticit)m(y)h(of)i Fq(L)1699 1294 y Fo(N)1790 1280 y Fu(the)g(inequalit)m(y)g(\(4.2\))i(follo)m(ws)d(b)m(y)h(Mey)m(ers)h (argumen)m(t)118 1388 y([15)q(])31 b(\(see)h(also)f(Step)g(1)g(in)f (Subsection)f(3.3)j(in)e([4)q(])h(for)g(the)g(discrete)f(case\))j(once) e(the)g(corresp)s(onding)118 1496 y Fs(L)180 1463 y Fo(q)248 1496 y Fu(estimate)g(holds)e(for)h(the)h(P)m(oisson)f(equation)1305 1692 y Fo(d)1257 1719 y Fi(X)1266 1914 y Fo(i)p Fk(=1)1419 1782 y Fu(~)1404 1805 y Fq(r)1480 1768 y Fn(\003)1480 1828 y Fo(i)1534 1782 y Fu(~)1519 1805 y Fq(r)1595 1819 y Fo(i)1623 1805 y Fs(u)p Fu(\()p Fs(x)p Fu(\))56 b(=)f Fq(\000)2113 1692 y Fo(d)2065 1719 y Fi(X)2074 1914 y Fo(i)p Fk(=1)2227 1782 y Fu(~)2212 1805 y Fq(r)2288 1768 y Fn(\003)2288 1828 y Fo(i)2327 1805 y Fs(f)2372 1819 y Fo(i)2400 1805 y Fu(\()p Fs(x)p Fu(\))953 b(\(4.3\))118 2115 y(for)33 b(some)g Fs(q)f(>)d Fu(2.)48 b(Namely)-8 b(,)34 b(w)m(e)f(ha)m(v)m(e)h(to)f(sho)m(w)g(that)g(there)g(exists)f Fs(\014)j Fu(=)29 b Fs(\014)5 b Fu(\()p Fs(q)s Fu(\))30 b Fs(<)e Fq(1)p Fu(,)34 b(suc)m(h)e(that)i(for)118 2235 y(an)m(y)d Fs(N)k Fq(2)25 b Fp(N)43 b Fu(and)30 b(ev)m(ery)h(v)m(ector) h(\014eld)1479 2211 y Fs(~)1467 2235 y(f)i Fu(=)25 b(\()p Fs(f)1722 2249 y Fk(1)1762 2235 y Fs(;)15 b(:)g(:)g(:)h(;)f(f)2008 2250 y Fo(d)2049 2235 y Fu(\))30 b(on)g Fs(T)2293 2249 y Fo(N)2391 2235 y Fu(solutions)f Fs(u)h Fu(of)38 b(\(4.3\))32 b(satisfy:)1027 2500 y Fq(k)1088 2477 y Fu(~)1072 2500 y Fq(r)q Fs(u)p Fq(k)1246 2456 y Fo(q)1246 2529 y(N)s(;q)1419 2500 y Fu(=)1587 2439 y(1)p 1524 2479 171 4 v 1524 2563 a Fq(j)p Fs(T)1602 2577 y Fo(N)1670 2563 y Fq(j)1748 2414 y Fi(X)1720 2611 y Fo(x)p Fn(2)p Fo(T)1848 2622 y Fv(N)1921 2500 y Fq(j)1962 2477 y Fu(~)1946 2500 y Fq(r)p Fs(u)p Fq(j)2099 2463 y Fo(q)2193 2500 y Fr(6)e Fs(\014)5 b Fu(\()p Fs(q)s Fu(\))p Fq(k)2522 2476 y Fs(~)2509 2500 y(f)11 b Fq(k)2610 2456 y Fo(q)2610 2529 y(N)s(;q)2727 2500 y Fs(:)723 b Fu(\(4.4\))118 2822 y(\(4.4\))41 b(is)c(a)i(discrete) g(v)m(ersion)f(of)h(Calderon-Zygm)m(und)d(inequalit)m(y)-8 b(.)64 b(W)-8 b(e)40 b(could)e(not)h(\014nd)e(a)i(direct)118 2930 y(reference)29 b(whic)m(h)f(w)m(ould)g(co)m(v)m(er)j(the)e(case)h (w)m(e)f(consider)f(here.)40 b(F)-8 b(or)30 b(the)f(con)m(v)m(enience)h (of)f(the)g(reader,)118 3038 y(a)i(brief)e(sk)m(etc)m(h)i(of)g(the)g (pro)s(of)e(is)h(giv)m(en)g(in)f(the)i(app)s(endix.)118 3354 y(4.2.)54 b Fz(Bounds)42 b(on)f(the)g(Hamming)e(distance.)46 b Fu(F)-8 b(or)37 b Fs(\030)t(;)15 b(\030)2265 3321 y Fn(0)2323 3354 y Fq(2)34 b Fu(\012)2484 3368 y Fo(N)2586 3354 y Fu(let)i(us)f(de\014ne)g(the)h(Hamming)118 3462 y(distance)1132 3716 y(d)1183 3730 y Fk(Ham)o Fo(;N)1432 3642 y Fi(\010)1485 3716 y Fs(\030)t(;)15 b(\030)1613 3678 y Fn(0)1636 3642 y Fi(\011)1745 3716 y Fu(=)1898 3629 y Fi(X)1871 3826 y Fo(x)p Fn(2)p Fo(T)1999 3837 y Fv(N)2120 3602 y Fo(d)2072 3629 y Fi(X)2081 3825 y Fo(i)p Fk(=1)2218 3716 y Fs(\016)2258 3735 y Fo(\030)2289 3745 y Fv(i)2316 3735 y Fk(\()p Fo(x)p Fk(\))p Fn(6)p Fk(=)p Fo(\030)2499 3712 y Fl(0)2496 3757 y Fv(i)2523 3735 y Fk(\()p Fo(x)p Fk(\))2622 3716 y Fs(:)118 4019 y Fu(Set)34 b Fs(q)g Fu(=)d Fs(p=)p Fu(\()p Fs(p)23 b Fq(\000)f Fu(2\))32 b Fs(<)f Fq(1)p Fu(,)j(where)g Fs(p)d Fu(=)f Fs(p)p Fu(\()p Fs(c)p Fu(\))i Fs(>)f Fu(2)j(is)f(as)h(in)f(the)h (statemen)m(t)i(on)d(Theorem)h(4.1.)52 b(W)-8 b(e)118 4127 y(claim)30 b(that)h(uniformly)c(in)i Fs(v)34 b Fu(and)29 b(in)g Fs(\030)t(;)15 b(\030)1566 4094 y Fn(0)1615 4127 y Fq(2)25 b Fu(\012)1767 4141 y Fo(N)1834 4127 y Fu(,)625 4345 y Fi(\014)625 4400 y(\014)655 4423 y Fu(\()q Fs(D)766 4437 y Fo(N)833 4423 y Fu(\()p Fs(\030)t Fu(\))p Fs(v)s(;)15 b(v)s Fu(\))23 b Fq(\000)1229 4349 y Fi(\000)1271 4423 y Fs(D)1346 4437 y Fo(N)1414 4423 y Fu(\()p Fs(\030)1493 4385 y Fn(0)1516 4423 y Fu(\))p Fs(v)s(;)15 b(v)1685 4349 y Fi(\001)1728 4345 y(\014)1728 4400 y(\014)1829 4423 y Fr(6)30 b Fs(c)1969 4437 y Fk(1)2009 4423 y Fq(j)p Fs(v)s Fq(j)2106 4385 y Fk(2)2162 4294 y Fi(\024)2282 4361 y Fu(1)p 2219 4402 V 2219 4485 a Fq(j)p Fs(T)2297 4499 y Fo(N)2365 4485 y Fq(j)2400 4423 y Fu(d)2451 4437 y Fk(Ham)o Fo(;N)2700 4349 y Fi(\010)2753 4423 y Fs(\030)t(;)15 b(\030)2881 4385 y Fn(0)2904 4349 y Fi(\011)2957 4294 y(\025)3005 4317 y Fk(1)p Fo(=q)3129 4423 y Fs(:)321 b Fu(\(4.5\))118 4700 y(Indeed,)30 b(b)m(y)h(\(3.16\))i(and)d(\(3.18\)) r(,)759 4953 y(1)p 759 4994 46 4 v 759 5077 a(2)829 5015 y(\()q Fs(D)940 5029 y Fo(N)1007 5015 y Fu(\()p Fs(\030)t Fu(\))p Fs(v)s(;)15 b(v)s Fu(\))58 b(=)118 b(min)1474 5076 y Fo(f)7 b Fk(:)p Fo(T)1576 5087 y Fv(N)1634 5076 y Fn(7!)p Fm(R)1840 4953 y Fu(1)p 1777 4994 171 4 v 1777 5077 a Fq(j)p Fs(T)1855 5091 y Fo(N)1923 5077 y Fq(j)2000 4928 y Fi(X)1973 5125 y Fo(x)p Fn(2)p Fo(T)2101 5136 y Fv(N)2221 4901 y Fo(d)2174 4928 y Fi(X)2182 5124 y Fo(i)p Fk(=1)2320 5015 y Fs(\030)2360 5029 y Fo(i)2388 5015 y Fu(\()p Fs(x)p Fu(\))2525 4914 y Fi(\020)2580 5015 y Fs(v)2624 5029 y Fo(i)2673 5015 y Fu(+)2779 4992 y(~)2764 5015 y Fq(r)2840 5029 y Fo(i)2868 5015 y Fs(f)2923 4914 y Fi(\021)2976 4936 y Fk(2)1408 5350 y Fu(=)1607 5288 y(1)p 1545 5329 V 1545 5412 a Fq(j)p Fs(T)1623 5426 y Fo(N)1690 5412 y Fq(j)1768 5263 y Fi(X)1740 5460 y Fo(x)p Fn(2)p Fo(T)1868 5471 y Fv(N)1989 5236 y Fo(d)1941 5263 y Fi(X)1950 5459 y Fo(i)p Fk(=1)2088 5350 y Fs(\030)2128 5364 y Fo(i)2156 5350 y Fu(\()p Fs(x)p Fu(\))2293 5249 y Fi(\020)2348 5350 y Fs(v)2392 5364 y Fo(i)2440 5350 y Fu(+)2546 5327 y(~)2531 5350 y Fq(r)2607 5364 y Fo(i)2635 5350 y Fs(\037)2692 5305 y Fo(\030)2692 5378 y(N)s(;v)2812 5249 y Fi(\021)2866 5271 y Fk(2)2920 5350 y Fs(;)3475 5186 y Fu(\(4.6\))p eop %%Page: 10 10 10 9 bop 118 45 a Fk(10)1010 b(PIETR)n(O)31 b(CAPUTO)f(AND)g(DMITR)-6 b(Y)30 b(IOFFE)118 366 y Fu(for)37 b(ev)m(ery)h Fs(v)h Fq(2)d Fp(R)751 333 y Fo(d)835 366 y Fu(and)g Fs(\030)k Fq(2)c Fu(\012)1261 380 y Fo(N)1328 366 y Fu(.)61 b(Consequen)m(tly)-8 b(,)38 b(in)e(view)g(of)i(the)f(a-priori)e(estimate)j(\(4.2\))r(,)h(w)m (e)118 474 y(infer:)151 592 y(1)p 151 633 46 4 v 151 716 a(2)206 577 y Fi(\014)206 631 y(\014)237 654 y Fu(\()p Fs(D)347 668 y Fo(N)415 654 y Fu(\()p Fs(\030)t Fu(\))p Fs(v)s(;)15 b(v)s Fu(\))22 b Fq(\000)e Fu(\()p Fs(D)921 668 y Fo(N)989 654 y Fu(\()p Fs(\030)1068 616 y Fn(0)1091 654 y Fu(\))p Fs(v)s(;)15 b(v)s Fu(\))1295 577 y Fi(\014)1295 631 y(\014)725 920 y Fr(6)899 859 y Fu(1)p 836 900 171 4 v 836 983 a Fq(j)p Fs(T)914 997 y Fo(N)982 983 y Fq(j)1059 834 y Fi(X)1032 1031 y Fo(x)p Fn(2)p Fo(T)1160 1042 y Fv(N)1280 807 y Fo(d)1233 834 y Fi(X)1241 1029 y Fo(i)p Fk(=1)1379 843 y Fi(\014)1379 898 y(\014)1409 920 y Fs(\030)1449 934 y Fo(i)1477 920 y Fu(\()p Fs(x)p Fu(\))21 b Fq(\000)f Fs(\030)1755 883 y Fn(0)1751 943 y Fo(i)1779 920 y Fu(\()p Fs(x)p Fu(\))1901 843 y Fi(\014)1901 898 y(\014)1947 792 y(\024)1995 820 y(\020)2049 920 y Fs(v)2093 934 y Fo(i)2142 920 y Fu(+)2248 897 y(~)2233 920 y Fq(r)2309 934 y Fo(i)2337 920 y Fs(\037)2394 876 y Fo(\030)2394 949 y(N)s(;v)2513 920 y Fu(\()p Fs(x)p Fu(\))2635 820 y Fi(\021)2690 842 y Fk(2)2750 920 y Fu(+)2841 820 y Fi(\020)2895 920 y Fs(v)2939 934 y Fo(i)2988 920 y Fu(+)3094 897 y(~)3078 920 y Fq(r)3154 934 y Fo(i)3182 920 y Fs(\037)3239 876 y Fo(\030)3273 853 y Fl(0)3239 949 y Fo(N)s(;v)3359 920 y Fu(\()p Fs(x)p Fu(\))3481 820 y Fi(\021)3536 842 y Fk(2)3575 792 y Fi(\025)846 1296 y Fr(6)30 b Fs(c)986 1310 y Fk(2)1026 1296 y Fq(j)p Fs(v)s Fq(j)1123 1259 y Fk(2)1178 1114 y Fi(2)1178 1277 y(4)1312 1235 y Fu(1)p 1249 1275 V 1249 1359 a Fq(j)p Fs(T)1327 1373 y Fo(N)1395 1359 y Fq(j)1472 1210 y Fi(X)1445 1407 y Fo(x)p Fn(2)p Fo(T)1573 1418 y Fv(N)1693 1183 y Fo(d)1646 1210 y Fi(X)1654 1405 y Fo(i)p Fk(=1)1792 1219 y Fi(\014)1792 1274 y(\014)1822 1296 y Fs(\030)1862 1310 y Fo(i)1890 1296 y Fu(\()p Fs(x)p Fu(\))21 b Fq(\000)f Fs(\030)2168 1259 y Fn(0)2164 1319 y Fo(i)2192 1296 y Fu(\()p Fs(x)p Fu(\))2314 1219 y Fi(\014)2314 1274 y(\014)2345 1245 y Fo(q)2383 1114 y Fi(3)2383 1277 y(5)2444 1136 y Fk(1)p Fo(=q)2567 1296 y Fs(:)118 1668 y Fu(Ob)m(viously)-8 b(,)1024 1790 y Fi(X)997 1986 y Fo(x)p Fn(2)p Fo(T)1125 1997 y Fv(N)1246 1762 y Fo(d)1198 1790 y Fi(X)1207 1985 y Fo(i)p Fk(=1)1345 1799 y Fi(\014)1345 1853 y(\014)1375 1876 y Fs(\030)1415 1890 y Fo(i)1443 1876 y Fu(\()p Fs(x)p Fu(\))21 b Fq(\000)f Fs(\030)1721 1838 y Fn(0)1717 1898 y Fo(i)1745 1876 y Fu(\()p Fs(x)p Fu(\))1867 1799 y Fi(\014)1867 1853 y(\014)1897 1825 y Fo(q)2006 1876 y Fr(6)30 b Fs(c)2146 1838 y Fo(q)2185 1876 y Fu(d)2236 1890 y Fk(Ham)n Fo(;N)2484 1802 y Fi(\010)2537 1876 y Fs(\030)t(;)15 b(\030)2665 1838 y Fn(0)2689 1802 y Fi(\011)2757 1876 y Fs(;)118 2117 y Fu(uniformly)28 b(in)h Fs(\030)t(;)15 b(\030)767 2084 y Fn(0)815 2117 y Fq(2)25 b Fu(\012)967 2131 y Fo(N)1034 2117 y Fu(,)31 b(and)e(the)i(b)s(ound)d(\(4.5\))33 b(follo)m(ws.)118 2306 y(4.3.)54 b Fz(Concen)m(tration)35 b(of)g Fs(D)1191 2320 y Fo(N)1258 2306 y Fu(\()p Fs(\030)t Fu(\))p Fz(.)46 b Fu(By)32 b(\(4.6\))r(;)1363 2468 y(0)56 b Fr(6)45 b Fu(\()p Fs(D)1690 2482 y Fo(N)1758 2468 y Fu(\()p Fs(\030)t Fu(\))p Fs(v)s(;)15 b(v)s Fu(\))73 b Fr(6)30 b Fs(c)p Fq(j)p Fs(v)s Fq(j)2351 2431 y Fk(2)2391 2468 y Fs(;)118 2630 y Fu(for)g(ev)m(ery)i Fs(v)d Fq(2)c Fp(R)716 2597 y Fo(d)793 2630 y Fu(and)30 b(for)g(eac)m(h)i(realization)e(of)h(the)f (en)m(vironmen)m(t)g Fs(\030)g Fq(2)25 b Fu(\012)2764 2644 y Fo(N)2831 2630 y Fu(.)41 b(Th)m(us,)30 b(giv)m(en)g Fs(\017)3427 2644 y Fo(N)3520 2630 y Fs(>)25 b Fu(0)118 2737 y(one)41 b(can)g(\014nd)f(a)h(v)-5 b(alue)1018 2714 y(\026)996 2737 y Fs(D)1071 2751 y Fo(N)1139 2737 y Fu(\()p Fs(v)s Fu(\))44 b Fq(2)e Fu([0)p Fs(;)15 b(c)p Fq(j)p Fs(v)s Fq(j)1649 2704 y Fk(2)1691 2737 y Fu(],)44 b(suc)m(h)c(that)i Fs(\026)15 b Fu(\()p Fs(A)2381 2751 y Fo(N)2449 2737 y Fu(\()p Fs(v)s Fu(\)\))59 b Fr(>)42 b Fu(2)p Fs(\017)2855 2751 y Fo(N)2923 2737 y Fs(=)p Fu(\()p Fs(c)p Fq(j)p Fs(v)s Fq(j)3139 2704 y Fk(2)3180 2737 y Fu(\),)i(where)d(w)m(e)118 2845 y(de\014ne)30 b(the)g(set)h Fs(A)747 2859 y Fo(N)815 2845 y Fu(\()p Fs(v)s Fu(\))26 b Fq(\032)f Fu(\012)1120 2859 y Fo(N)1217 2845 y Fu(as:)826 3005 y Fs(A)894 3019 y Fo(N)962 3005 y Fu(\()p Fs(v)s Fu(\))56 b(=)1262 2931 y Fi(\010)1315 3005 y Fs(\030)29 b Fq(2)c Fu(\012)1536 3019 y Fo(N)1658 3005 y Fu(:)1739 2928 y Fi(\014)1739 2982 y(\014)1769 3005 y Fu(\()q Fs(D)1880 3019 y Fo(N)1947 3005 y Fu(\()p Fs(\030)t Fu(\))p Fs(v)s(;)15 b(v)s Fu(\))23 b Fq(\000)2365 2982 y Fu(\026)2343 3005 y Fs(D)2418 3019 y Fo(N)2486 3005 y Fu(\()p Fs(v)s Fu(\))2603 2928 y Fi(\014)2603 2982 y(\014)2660 3005 y Fs(<)h(\017)2792 3019 y Fo(N)2860 2931 y Fi(\011)2928 3005 y Fs(:)118 3166 y Fu(Ev)m(ery)31 b Fs(\030)424 3133 y Fn(0)472 3166 y Fq(2)25 b Fu(\012)624 3180 y Fo(N)721 3166 y Fu(suc)m(h)30 b(that)1276 3210 y Fi(\014)1276 3264 y(\014)1306 3214 y(\000)1348 3287 y Fs(D)1423 3301 y Fo(N)1491 3287 y Fu(\()p Fs(\030)1570 3250 y Fn(0)1593 3287 y Fu(\))p Fs(v)s(;)15 b(v)1762 3214 y Fi(\001)1826 3287 y Fq(\000)1938 3264 y Fu(\026)1916 3287 y Fs(D)1991 3301 y Fo(N)2059 3287 y Fu(\()p Fs(v)s Fu(\))2176 3210 y Fi(\014)2176 3264 y(\014)2233 3287 y Fs(>)24 b Fu(2)p Fs(\017)2410 3301 y Fo(N)2478 3287 y Fs(;)118 3427 y Fu(certainly)30 b(satis\014es)1084 3549 y(min)1018 3614 y Fo(\030)s Fn(2)p Fo(A)1152 3625 y Fv(N)1210 3614 y Fk(\()p Fo(v)r Fk(\))1316 3471 y Fi(\014)1316 3526 y(\014)1347 3549 y Fu(\()p Fs(D)1457 3563 y Fo(N)1525 3549 y Fu(\()p Fs(\030)t Fu(\))p Fs(v)s(;)15 b(v)s Fu(\))22 b Fq(\000)1921 3475 y Fi(\000)1962 3549 y Fs(D)2037 3563 y Fo(N)2105 3549 y Fu(\()p Fs(\030)2184 3511 y Fn(0)2208 3549 y Fu(\))p Fs(v)s(;)15 b(v)2377 3475 y Fi(\001)2420 3471 y(\014)2420 3526 y(\014)2505 3549 y Fs(>)56 b(\017)2669 3563 y Fo(N)2736 3549 y Fs(:)118 3750 y Fu(By)32 b(\(4.5\))g(w)m(e)f (arriv)m(e)f(to)h(the)g(follo)m(wing)e(b)s(ound)f(on)i(suc)m(h)g Fs(\030)2178 3717 y Fn(0)2232 3750 y Fu(in)f(terms)h(of)h(the)f (Hamming)g(distance:)596 3966 y(d)647 3980 y Fk(Ham)o Fo(;N)896 3892 y Fi(\010)949 3966 y Fs(\030)993 3928 y Fn(0)1016 3966 y Fs(;)15 b(A)1124 3980 y Fo(N)1192 3966 y Fu(\()p Fs(v)s Fu(\))1309 3892 y Fi(\011)1418 3966 y Fu(=)121 b(min)1545 4031 y Fo(\030)s Fn(2)p Fo(A)1679 4042 y Fv(N)1736 4031 y Fk(\()p Fo(v)r Fk(\))1843 3966 y Fu(d)1894 3980 y Fk(Ham)n Fo(;N)2142 3892 y Fi(\010)2195 3966 y Fs(\030)2239 3928 y Fn(0)2262 3966 y Fs(;)15 b(\030)2346 3892 y Fi(\011)2455 3966 y Fs(>)55 b(c)2620 3980 y Fk(4)2675 3838 y Fi(\022)2769 3904 y Fs(\017)2806 3918 y Fo(N)p 2752 3945 138 4 v 2752 4028 a Fq(j)p Fs(v)s Fq(j)2849 4002 y Fk(2)2900 3838 y Fi(\023)2966 3860 y Fo(q)3035 3966 y Fs(N)3118 3928 y Fo(d)3158 3966 y Fs(:)118 4184 y Fu(W)-8 b(e)32 b(are)e(no)m(w)h(in)e(p)s(osition)f(to)j(use)g(the)f (concen)m(tration)i(estimate)f(\(5.2\))h(in)d([19)q(]:)605 4401 y Fs(\026)675 4272 y Fi(\022)742 4401 y Fs(\030)786 4363 y Fn(0)864 4401 y Fu(:)56 b(d)996 4415 y Fk(Ham)o Fo(;N)1245 4327 y Fi(\010)1298 4401 y Fs(\030)1342 4363 y Fn(0)1365 4401 y Fs(;)15 b(A)1473 4415 y Fo(N)1541 4401 y Fu(\()p Fs(v)s Fu(\))1658 4327 y Fi(\011)1737 4401 y Fs(>)25 b(c)1872 4415 y Fk(4)1927 4272 y Fi(\022)2020 4339 y Fs(\017)2057 4353 y Fo(N)p 2004 4380 V 2004 4463 a Fq(j)p Fs(v)s Fq(j)2101 4437 y Fk(2)2151 4272 y Fi(\023)2218 4295 y Fo(q)2286 4401 y Fs(N)2369 4363 y Fo(d)2410 4272 y Fi(\023)685 4681 y Fr(6)963 4620 y Fu(1)p 796 4660 379 4 v 796 4744 a Fs(\026)p Fu(\()p Fs(A)954 4758 y Fo(N)1021 4744 y Fu(\()p Fs(v)s Fu(\)\))1185 4681 y(exp)1339 4526 y Fi(\()1412 4681 y Fq(\000)p Fs(c)1522 4695 y Fk(5)1602 4617 y Fs(\017)1639 4573 y Fk(2)p Fo(q)1639 4646 y(N)p 1571 4660 172 4 v 1571 4744 a Fq(j)p Fs(v)s Fq(j)1668 4717 y Fk(4)p Fo(q)1768 4681 y Fs(N)1851 4644 y Fo(d)1891 4526 y Fi(\))2035 4681 y Fr(6)2146 4620 y Fs(c)p Fq(j)p Fs(v)s Fq(j)2282 4587 y Fk(2)p 2146 4660 177 4 v 2160 4744 a Fu(2)p Fs(\017)2242 4758 y Fo(N)2333 4681 y Fu(exp)2487 4580 y Fi(n)2547 4681 y Fq(\000)p Fs(c)2657 4695 y Fk(6)2697 4681 y Fu(\()p Fq(j)p Fs(v)s Fq(j)p Fu(\))p Fs(\017)2901 4637 y Fk(2)p Fo(q)2901 4710 y(N)2991 4681 y Fs(N)3074 4644 y Fo(d)3114 4580 y Fi(o)3475 4553 y Fu(\(4.7\))118 4946 y(It)36 b(remains)f(to)h(c)m(ho)s(ose)h Fs(\017)1014 4960 y Fo(N)1115 4946 y Fu(=)1220 4862 y Fq(p)p 1296 4862 417 4 v 84 x Fs(N)1379 4919 y Fn(\000)p Fk(\()p Fo(d)p Fn(\000)p Fk(\001\))p Fo(=q)1748 4946 y Fu(for)e(some)h(\001)e Fs(>)g Fu(0:)52 b(Since)35 b(the)h(sequence)g(of)g(random)118 5054 y(v)-5 b(ariables)29 b(\()p Fs(D)605 5068 y Fo(N)673 5054 y Fu(\()p Fs(\030)t Fu(\))p Fs(v)s(;)15 b(v)s Fu(\))32 b(is)e(uniformly)d(b)s(ounded,)i(w)m(e)h(readily)f(infer)g(from)h (\(4.7\))i(that)1215 5136 y Fi(\014)1215 5190 y(\014)1246 5213 y Fp(E)1300 5227 y Fo(\026)1368 5213 y Fu(\()p Fs(D)1478 5227 y Fo(N)1546 5213 y Fu(\()p Fs(\030)t Fu(\))p Fs(v)s(;)15 b(v)s Fu(\))22 b Fq(\000)1963 5190 y Fu(\026)1942 5213 y Fs(D)2017 5227 y Fo(N)2084 5213 y Fu(\()p Fs(v)s Fu(\))2201 5136 y Fi(\014)2201 5190 y(\014)2288 5213 y Fs(<)56 b Fu(3)p Fs(\017)2497 5227 y Fo(N)118 5372 y Fu(for)43 b(all)e(large)i(enough)g(v)-5 b(alues)42 b(of)h Fs(N)10 b Fu(,)46 b(and)c(the)h(assertion)g(\(1.3\))h(of)f(Theorem)g(1.1)h (follo)m(ws)e(with)118 5480 y Fs(\027)31 b Fu(=)25 b(min)o Fq(f)p Fu(\001)p Fs(;)15 b Fu(\()p Fs(d)21 b Fq(\000)f Fu(\001\))p Fs(=)p Fu(2)p Fs(q)s Fq(g)p Fu(.)p eop %%Page: 11 11 11 10 bop 3591 43 a Fk(11)118 366 y Fu(4.4.)54 b Fz(Sp)s(ectral)33 b(measures)f(of)h(the)g(lo)s(cal)g(drift)g Fs(')2007 380 y Fo(v)2048 366 y Fz(.)46 b Fu(W)-8 b(e)30 b(w)m(ould)d(lik)m(e)h (to)i(stipulate)d(the)i(implica-)118 474 y(tion)21 b(of)29 b(\(1.5\))24 b(for)d(the)h(sp)s(ectral)f(prop)s(erties)f(of)i(the)g(lo) s(cal)f(drifts)f Fs(')2377 488 y Fo(v)2440 474 y Fu(as)i(compared)g(to) g(the)g(information)118 582 y(whic)m(h)29 b(could)h(b)s(e)f(extracted)j (from)e(the)h(general)f(Kipnis-V)-8 b(aradhan)28 b(approac)m(h:)218 690 y(In)h(the)i(notation)g(of)f(Section)g(3)h(let)f(us)g(in)m(tro)s (duce)f(the)i(empirical)d(measures)1253 891 y Fs(\026)1308 905 y Fo(N)1400 891 y Fu(=)d Fs(\026)1551 847 y Fo(\030)1551 919 y(N)1674 891 y Fu(=)1873 829 y(1)p 1810 870 171 4 v 1810 953 a Fq(j)p Fs(T)1888 967 y Fo(N)1956 953 y Fq(j)2033 804 y Fi(X)2006 1001 y Fo(x)p Fn(2)p Fo(T)2134 1012 y Fv(N)2207 891 y Fs(\016)2247 906 y Fo(\034)2278 915 y Fl(\000)p Fv(x)2365 906 y Fo(\031)2406 917 y Fv(N)2464 906 y Fo(\030)2501 891 y Fs(:)118 1148 y Fu(F)-8 b(or)44 b(ev)m(ery)g Fs(\030)51 b Fq(2)46 b Fu(\012)d(\014xed)g(or,)k(equiv)-5 b(alen)m(tly)d(,)45 b(for)f(eac)m(h)g Fs(\031)2200 1162 y Fo(N)2267 1148 y Fs(\030)51 b Fq(2)46 b Fu(\012)2531 1162 y Fo(N)2641 1148 y Fu(\014xed)d(there)g(is)g(an)g(ob)m(vious)118 1256 y(corresp)s(ondence)31 b(b)s(et)m(w)m(een)g(the)g(spaces)h Fs(L)1604 1223 y Fk(2)1643 1256 y Fu(\()p Fs(T)1731 1270 y Fo(N)1799 1256 y Fs(;)15 b(\032)1886 1270 y Fo(N)1954 1256 y Fu(\))31 b(and)f Fs(L)2259 1223 y Fk(2)2299 1256 y Fu(\(\012)2400 1270 y Fo(N)2467 1256 y Fs(;)15 b(\026)2562 1270 y Fo(N)2629 1256 y Fu(\).)43 b(By)33 b(\(2.7\))f(and)f(\(3.15\))i (the)118 1363 y(limiting)27 b(relation)j(\(1.5\))i(could)e(b)s(e)f (written)h(as)650 1529 y Fs(\026)20 b Fq(\000)g Fs(a:s:)236 b Fu(lim)1154 1590 y Fo(N)7 b Fn(!1)1373 1529 y Fp(E)1428 1543 y Fo(\026)1470 1554 y Fv(N)1553 1456 y Fi(\000)1595 1529 y Fs(')1654 1543 y Fo(v)1695 1529 y Fu(\()p Fq(\000L)1864 1543 y Fo(N)1931 1529 y Fu(\))1966 1492 y Fn(\000)p Fk(1)2061 1529 y Fs(')2120 1543 y Fo(v)2161 1456 y Fi(\001)2258 1529 y Fu(=)56 b Fp(E)2439 1543 y Fo(\026)2507 1456 y Fi(\000)2549 1529 y Fs(')2608 1543 y Fo(v)2649 1529 y Fu(\()p Fq(\000L)p Fu(\))2853 1492 y Fn(\000)p Fk(1)2947 1529 y Fs(')3006 1543 y Fo(v)3048 1456 y Fi(\001)3105 1529 y Fs(:)345 b Fu(\(4.8\))118 1752 y(Fix)29 b(no)m(w)g Fs(v)g Fq(2)c Fp(R)686 1719 y Fo(d)733 1752 y Fu(.)40 b(Let)30 b Fs(\027)35 b Fu(and,)29 b(resp)s(ectiv)m(ely)-8 b(,)30 b Fs(\027)1799 1766 y Fo(N)1891 1752 y Fu(=)25 b Fs(\027)2038 1708 y Fo(\030)2032 1781 y(N)2128 1752 y Fu(b)s(e)k(the)g(sp)s(ectral)g(measures)g(of)h Fs(')3298 1766 y Fo(v)3368 1752 y Fu(relativ)m(e)118 1865 y(to)40 b(the)f(op)s(erator)g(\()p Fq(\000L)p Fu(\))g(on)f Fs(L)1217 1832 y Fk(2)1257 1865 y Fu(\(\012)p Fs(;)15 b(\026)p Fu(\))39 b(and,)i(resp)s(ectiv)m(ely)-8 b(,)41 b(relativ)m(e)e(to)g (the)g(op)s(erator)g(\()p Fq(\000L)3424 1879 y Fo(N)3491 1865 y Fu(\))g(on)118 1986 y Fs(L)180 1953 y Fk(2)219 1986 y Fu(\(\012)320 2000 y Fo(N)388 1986 y Fs(;)15 b(\026)483 1942 y Fo(\030)483 2015 y(N)550 1986 y Fu(\).)69 b(Both)41 b Fq(\000L)e Fu(and)g(all)f(of)i Fq(\000L)1656 2000 y Fo(N)1763 1986 y Fu(are)g(self-adjoin)m(t)f(and)g(b)s(ounded)f(on)h (the)h(resp)s(ectiv)m(e)118 2094 y(spaces.)50 b(Let)34 b Fs(K)40 b Fu(b)s(e)33 b(a)h(common)f(upp)s(er)f(b)s(ound)f(on)i(the)h (sp)s(ectral)e(radiuses.)49 b(In)32 b(terms)i(of)f(sp)s(ectral)118 2202 y(measures)d(the)h(limit)d(in)h(\(4.8\))j(reads)e(as)984 2442 y Fs(\026)20 b Fq(\000)g Fs(a:s:)236 b Fu(lim)1488 2503 y Fo(N)7 b Fn(!1)1707 2318 y Fi(Z)1798 2345 y Fo(K)1758 2524 y Fk(0)1892 2378 y Fs(\027)1943 2334 y Fo(\030)1937 2407 y(N)2004 2378 y Fu(\(d)p Fs(r)s Fu(\))p 1892 2421 277 4 v 2008 2504 a Fs(r)2234 2442 y Fu(=)2360 2318 y Fi(Z)2451 2345 y Fo(K)2411 2524 y Fk(0)2545 2381 y Fs(\027)f Fu(\(d)p Fs(r)s Fu(\))p 2545 2421 216 4 v 2631 2504 a Fs(r)2770 2442 y(:)680 b Fu(\(4.9\))118 2706 y(Kipnis-V)-8 b(aradhan)20 b(approac)m(h)j(is)e(based)h(on)h(the)f(fact)i(that)2153 2633 y Fi(R)2213 2659 y Fo(K)2196 2738 y Fk(0)2307 2661 y Fo(\027)t Fk(\(d)p Fo(r)r Fk(\))p 2307 2685 167 4 v 2373 2737 a Fo(r)2506 2706 y Fu(\(or,)h(resp)s(ectiv)m(ely)-8 b(,)3179 2633 y Fi(R)3240 2659 y Fo(K)3222 2738 y Fk(0)3333 2654 y Fo(\027)3372 2622 y Fv(\030)3368 2678 y(N)3425 2654 y Fk(\(d)p Fo(r)r Fk(\))p 3333 2685 221 4 v 3426 2737 a Fo(r)3586 2706 y Fu(in)118 2814 y(the)27 b(case)g(of)g(the)f(p)s (erio)s(dized)e(en)m(vironmen)m(t\))i(is)f(b)s(ounded)g(ab)s(o)m(v)m (e.)40 b(W)-8 b(e)28 b(claim)d(that)i(our)f(con)m(v)m(ergence)118 2950 y(result)40 b(\(4.9\))j(is)e(equiv)-5 b(alen)m(t)40 b(to)i(the)g Fs(\026)p Fu(-a.s.)g(uniform)d(in)m(tegrabilit)m(y)h(of)h (the)g(family)3149 2849 y Fi(n)3210 2950 y Fs(\027)3261 2906 y Fo(\030)3255 2978 y(N)3322 2950 y Fu(\(d)p Fs(r)s Fu(\))p Fs(=r)3576 2849 y Fi(o)3636 2950 y Fu(.)118 3080 y(Suc)m(h)34 b(equiv)-5 b(alence)34 b(w)m(ould)g(follo)m(w)f(if,)i(for) g(example,)g(w)m(e)h(are)f(able)f(to)h(sho)m(w)g(that)g Fs(\026)p Fu(-almost)g(surely)118 3211 y(the)c(sequence)651 3110 y Fi(n)712 3211 y Fs(\027)763 3167 y Fo(\030)757 3239 y(N)824 3110 y Fi(o)915 3211 y Fu(w)m(eakly)15 b Fq(\003)31 b Fu(con)m(v)m(erges)i(to)e Fs(\027)6 b Fu(.)40 b(The)30 b(latter)h(is)e(a)i(consequence)g(of)118 3404 y Fz(Lemma)i(4.1.)1109 3610 y Fu(lim)1070 3670 y Fo(N)7 b Fn(!1)1289 3486 y Fi(Z)1380 3512 y Fo(K)1340 3692 y Fk(0)1464 3610 y Fu(e)1504 3572 y Fn(\000)p Fo(nr)1640 3610 y Fs(\027)1691 3566 y Fo(\030)1685 3638 y(N)1752 3610 y Fu(\(d)p Fs(r)s Fu(\))58 b(=)2103 3486 y Fi(Z)2194 3512 y Fo(K)2154 3692 y Fk(0)2278 3610 y Fu(e)2318 3572 y Fn(\000)p Fo(nr)2454 3610 y Fs(\027)6 b Fu(\(d)p Fs(r)s Fu(\))p Fs(;)734 b Fu(\(4.10\))118 3818 y Fs(\026)p Fj(-a.s)32 b(for)i(al)5 b(l)33 b Fs(n)25 b Fq(2)f Fp(N)7 b Fj(.)118 3990 y(Pr)-5 b(o)g(of.)43 b Fu(The)33 b(claim)g(of)h(the)g(lemma)f (essen)m(tially)g(follo)m(ws)g(from)g(the)h(strong)g(la)m(w)g(of)g (large)g(n)m(um)m(b)s(ers:)118 4098 y(Since)29 b(the)i(lo)s(cal)f (drift)e Fs(')990 4112 y Fo(v)1062 4098 y Fu(is)i(b)s(ounded,)257 4197 y Fi(Z)348 4223 y Fo(K)308 4403 y Fk(0)417 4321 y Fu(e)457 4283 y Fn(\000)p Fo(nr)593 4321 y Fs(\027)644 4277 y Fo(\030)638 4349 y(N)705 4321 y Fu(\(d)p Fs(r)s Fu(\))2574 b(\(4.11\))442 4570 y(=)775 4509 y(1)p 548 4549 500 4 v 548 4633 a([2\()p Fs(N)31 b Fq(\000)20 b Fs(M)10 b Fu(\)])1006 4607 y Fo(d)1416 4484 y Fi(X)1072 4689 y Fo(x)p Fn(2f\000)p Fo(N)d Fk(+)p Fo(M)s(;:::)o(;N)g Fn(\000)p Fo(M)g Fn(\000)p Fk(1)p Fn(g)1855 4670 y Fv(d)1906 4570 y Fs(')1965 4584 y Fo(v)2006 4570 y Fu(\()p Fs(\034)2081 4584 y Fn(\000)p Fo(x)2180 4570 y Fs(\030)t Fu(\))p Fp(E)2314 4533 y Fo(\030)2314 4593 y(x)2364 4570 y Fs(')2423 4533 y Fo(N)2423 4593 y(v)2490 4570 y Fu(\()2562 4547 y(_)2525 4570 y Fs(X)h Fu(\()p Fs(n;)15 b(\031)2790 4584 y Fo(N)2858 4570 y Fs(\030)t Fu(\)\))51 b(+)f(O)3230 4442 y Fi(\022)3307 4509 y Fs(M)p 3307 4549 99 4 v 3315 4633 a(N)3415 4442 y Fi(\023)3497 4570 y Fs(;)118 4861 y Fu(where)430 4838 y(_)394 4861 y Fs(X)8 b Fu(\()p Fs(n;)15 b(\031)659 4875 y Fo(N)726 4861 y Fs(\030)t Fu(\))44 b(is)e(the)i(wrapping)e(round)g (the)i(torus)f Fs(T)2281 4875 y Fo(N)2392 4861 y Fu(of)g(the)h(random)f (w)m(alk)g Fs(X)7 b Fu(\()p Fs(n;)15 b(\031)3514 4875 y Fo(N)3582 4861 y Fs(\030)t Fu(\))118 4983 y(mo)m(ving)39 b(in)e(the)i(p)s(erio)s(dic)d(en)m(vironmen)m(t)j Fs(\031)1665 4997 y Fo(N)1732 4983 y Fs(\030)j Fu(and)c Fp(P)2054 4938 y Fo(\030)2054 4994 y(x)2138 4983 y Fu(denotes)h(the)g(la)m(w)f (of)h(suc)m(h)g(random)f(w)m(alk)118 5092 y(with)31 b(the)i(starting)f (p)s(oin)m(t)1103 5069 y(_)1067 5092 y Fs(X)7 b Fu(\(0)p Fs(;)15 b(\031)1321 5106 y Fo(N)1389 5092 y Fs(\030)t Fu(\))29 b(=)g Fs(x)p Fu(.)47 b(Since)31 b(the)i(jump)e(rates)i(of)g Fs(X)7 b Fu(\()p Fs(t;)15 b(\030)t Fu(\))34 b(are,)f(uniformly)d(in)118 5200 y Fs(\030)f Fq(2)c Fu(\012,)30 b(b)s(ounded)f(ab)s(o)m(v)m(e)i(b)m (y)f Fs(c)p Fu(,)778 5413 y(max)770 5473 y Fo(x)p Fn(2)p Fo(T)898 5484 y Fv(N)971 5413 y Fu(sup)973 5492 y Fo(\030)s Fn(2)p Fk(\012)1123 5413 y Fp(P)1178 5375 y Fo(\030)1178 5435 y(x)1237 5284 y Fi(\022)1366 5413 y Fu(max)1304 5472 y Fk(0)20 b Ff(6)g Fo(t)g Ff(6)f Fo(n)1611 5413 y Fq(j)q Fs(X)7 b Fu(\()p Fs(t;)15 b(\030)t Fu(\))21 b Fq(\000)f Fs(x)p Fq(j)25 b Fs(>)g(M)2314 5284 y Fi(\023)2452 5413 y Fr(6)30 b Fs(c)2592 5427 y Fk(5)2632 5413 y Fu(e)2672 5375 y Fn(\000)p Fo(c)2758 5384 y Fg(6)2792 5375 y Fo(M)2867 5352 y Fg(2)2902 5375 y Fo(=n)2984 5413 y Fs(:)p eop %%Page: 12 12 12 11 bop 118 45 a Fk(12)1010 b(PIETR)n(O)31 b(CAPUTO)f(AND)g(DMITR)-6 b(Y)30 b(IOFFE)118 366 y Fu(Consequen)m(tly)-8 b(,)802 542 y Fp(E)856 505 y Fo(\030)856 565 y(x)906 542 y Fs(')965 505 y Fo(N)965 565 y(v)1033 542 y Fu(\()1105 519 y(_)1068 542 y Fs(X)8 b Fu(\()p Fs(n;)15 b(\031)1333 556 y Fo(N)1400 542 y Fs(\030)t Fu(\)\))56 b(=)g Fp(E)1751 505 y Fo(\030)1751 565 y(x)1801 542 y Fs(')1860 556 y Fo(v)1902 542 y Fu(\()p Fs(X)7 b Fu(\()p Fs(n;)15 b(\030)t Fu(\)\))22 b(+)d(O)2461 441 y Fi(\020)2516 542 y Fu(e)2556 505 y Fn(\000)p Fo(c)2642 514 y Fg(6)2676 505 y Fo(M)2751 481 y Fg(2)2786 505 y Fo(=n)2868 441 y Fi(\021)2937 542 y Fs(;)467 b Fu(\(4.12\))118 734 y(uniformly)23 b(in)h Fs(x)h Fq(2)g(f\000)p Fs(N)c Fu(+)11 b Fs(M)5 b(;)15 b(:)g(:)g(:)h(;)f(N)21 b Fq(\000)11 b Fs(M)20 b Fq(\000)11 b Fu(1)p Fq(g)1834 701 y Fo(d)1874 734 y Fu(.)39 b(Substituting)24 b(\(4.12\))k(in)m(to)e(\(4.11\))i(and)d (c)m(ho)s(osing)118 849 y Fs(M)35 b Fu(=)25 b Fs(M)10 b Fu(\()p Fs(N)g Fu(\))26 b(=)710 772 y Fq(p)p 786 772 83 4 v 77 x Fs(N)10 b Fu(,)31 b(w)m(e)g(arriv)m(e)f(to)h(the)f(claim)g (of)g(the)h(lemma.)p 3595 849 4 62 v 3599 791 55 4 v 3599 849 V 3653 849 4 62 v 118 1026 a(4.5.)54 b Fz(Some)43 b(remarks)g(on)h(a)f(massless)g(Gaussian)h(\014eld)g(with)f(b)s(ond)h (disorder.)i Fu(An)38 b(har-)118 1134 y(monic)f(in)m(terface)h(in)d (the)j(quenc)m(hed)f(random)f(en)m(vironmen)m(t)h Fs(\030)k Fq(2)36 b Fu(\012)h(is)f(describ)s(ed)g(b)m(y)h(the)g(formal)118 1242 y(Hamiltonian)1230 1465 y Fq(H)q Fu(\()p Fs(\036)p Fu(\))26 b(=)1564 1403 y(1)p 1564 1444 46 4 v 1564 1527 a(2)1634 1378 y Fi(X)1680 1569 y Fo(x)1828 1351 y(d)1781 1378 y Fi(X)1789 1574 y Fo(i)p Fk(=1)1927 1465 y Fs(\030)1967 1479 y Fo(i)1995 1465 y Fu(\()p Fs(x)p Fu(\)\()p Fq(r)2228 1479 y Fo(i)2257 1465 y Fs(\036)p Fu(\()p Fs(x)p Fu(\)\))2468 1427 y Fk(2)2524 1465 y Fs(:)880 b Fu(\(4.13\))118 1699 y(W)-8 b(e)29 b(shall)e(see)h(b)s(elo)m(w)g(ho)m(w)g(the)g(corrector)h (\014elds)e(in)m(tro)s(duced)f(in)h(previous)f(sections)j(can)f(b)s(e)f (used)h(to)118 1807 y(c)m(haracterize)k(an)e(in)m(teresting)g(class)g (of)h(Gibbs)e(measures)h(for)g(the)g(in)m(teraction)h(\(4.13\).)218 1938 y(In)36 b(order)h(to)h(lo)s(calize)e(the)i(in)m(terface)f(in)f Fs(d)h Fu(=)f(1)p Fs(;)15 b Fu(2)39 b(w)m(e)e(ma)m(y)h(consider)e(the)i (\014eld)d(pinned)g(at)j(the)118 2046 y(origin,)28 b(i.e.)i(w)m(e)g (imp)s(ose)e Fs(\036)p Fu(\(0\))f(=)e(0.)41 b(Since)28 b(the)i(in)m(teraction)f(\(4.13\))j(is)c(quadratic)h(a)h(full)d (description)118 2156 y(of)38 b(the)g(set)g Fq(G)601 2123 y Fo(\030)677 2156 y Fu(of)g(in\014nite-v)m(olume)e(Gibbs)g (measures)h(for)h(a)g(giv)m(en)g(t)m(ypical)f(realization)g Fs(\030)42 b Fq(2)37 b Fu(\012)g(is)118 2264 y(a)m(v)-5 b(ailable)28 b(\([6)q(],)h(c)m(hap.)g(13\).)42 b(Namely)-8 b(,)29 b(it)f(is)g(w)m(ell)f(kno)m(wn)h(that)h(the)g(set)g(of)g (extremal)f(elemen)m(ts)h(of)g Fq(G)3598 2231 y Fo(\030)3636 2264 y Fu(,)118 2386 y(denoted)h(ext)q Fq(G)644 2353 y Fo(\030)682 2386 y Fu(,)g(coincides)f(with)f(the)i(set)g(of)g (Gaussian)f(\014elds)f(on)i Fp(R)2534 2353 y Fm(Z)2590 2330 y Fv(d)2655 2386 y Fu(with)e(a)i Fs(\030)t Fq(\000)p Fu(harmonic)e(mean)118 2494 y(v)m(ector)k Fs(\037)25 b Fu(:)h Fp(Z)589 2461 y Fo(d)650 2494 y Fq(!)f Fp(R)39 b Fu(satisfying)960 2574 y Fi(X)1014 2770 y Fo(i)1106 2661 y Fq(r)1182 2623 y Fn(\003)1182 2683 y Fo(i)1222 2661 y Fu(\()p Fs(\030)1297 2675 y Fo(i)1325 2661 y Fu(\()p Fs(x)p Fu(\))p Fq(r)1523 2675 y Fo(i)1551 2661 y Fs(\037)p Fu(\()p Fs(x)p Fu(\)\))27 b(=)e(0)15 b Fs(;)41 b(x)25 b Fq(2)g Fp(Z)2242 2623 y Fo(d)2293 2661 y Fu(;)106 b Fs(\037)p Fu(\(0\))27 b(=)e(0)15 b Fs(;)625 b Fu(\(4.14\))118 2912 y(and)20 b(co)m(v)-5 b(ariance)21 b Fs(G)785 2879 y Fo(\030)824 2912 y Fu(\()p Fs(x;)15 b(y)s Fu(\))21 b(giv)m(en)f(b)m(y)g(the)h(Green)f(function)f(of)h(the)h(random)e(w)m (alk)h(in)f(the)i Fs(\030)t Fu(-en)m(vironmen)m(t)118 3022 y(killed)28 b(up)s(on)h(hitting)g(the)h(origin.)40 b(ext)p Fq(G)1523 2989 y Fo(\030)1592 3022 y Fu(is)29 b(th)m(us)h(c)m(haracterized)i(b)m(y)e(solutions)f(to)i(\(4.14\).)218 3152 y(T)-8 b(o)28 b(study)f(the)i(tilted)e(states)i(asso)s(ciated)g (to)f(\(4.13\))j(it)c(is)g(con)m(v)m(enien)m(t)j(to)e(w)m(ork)h (directly)d(with)h(the)118 3289 y(gradien)m(t)32 b(\014eld)f Fs(\021)720 3303 y Fo(i)782 3238 y Fk(\001)777 3289 y Fu(=)c Fq(r)951 3303 y Fo(i)979 3289 y Fs(\036)p Fu(.)47 b(This)30 b(is)h(no)h(loss)g(of)g(information)f(in)f(view)i(of)g(the)g (condition)f Fs(\036)p Fu(\(0\))f(=)e(0.)118 3397 y(In)i(particular,)f (with)g(the)h(corresp)s(ondence)g Fs(\033)1727 3411 y Fo(i)1781 3397 y Fu(=)25 b Fq(r)1953 3411 y Fo(i)1980 3397 y Fs(\037)p Fu(,)31 b(\(4.14\))i(is)c(equiv)-5 b(alen)m(t)30 b(to)330 3483 y Fi(X)384 3678 y Fo(i)476 3569 y Fq(r)552 3531 y Fn(\003)552 3591 y Fo(i)592 3569 y Fu(\()p Fs(\030)667 3583 y Fo(i)695 3569 y Fu(\()p Fs(x)p Fu(\))p Fs(\033)869 3583 y Fo(i)898 3569 y Fu(\()p Fs(x)p Fu(\)\))c(=)f(0)15 b Fs(;)107 b Fq(r)1445 3584 y Fo(k)1487 3569 y Fs(\033)1539 3583 y Fo(i)1567 3569 y Fu(\()p Fs(x)p Fu(\))26 b(=)f Fq(r)1887 3583 y Fo(i)1915 3569 y Fs(\033)1967 3584 y Fo(k)2010 3569 y Fu(\()p Fs(x)p Fu(\))15 b Fs(;)107 b(i;)15 b(k)29 b Fu(=)c(1)p Fs(;)15 b(:)g(:)g(:)i(;)e(d)g(;)42 b(x)25 b Fq(2)g Fp(Z)3126 3531 y Fo(d)3177 3569 y Fs(;)227 b Fu(\(4.15\))118 3820 y(and)27 b(ext)q Fq(G)475 3787 y Fo(\030)541 3820 y Fu(is)f(c)m(haracterized)j(b)m(y)f(solutions)e(to) i(\(4.15\),)j(so)c(that)i Fs(\027)i Fq(2)24 b Fu(ext)q Fq(G)2722 3787 y Fo(\030)2788 3820 y Fu(is)j(a)h(Gaussian)e(measure)118 3942 y(on)k(\()p Fp(R)339 3909 y Fo(d)386 3942 y Fu(\))421 3909 y Fm(Z)471 3885 y Fv(d)536 3942 y Fu(with)f(mean)i Fs(\033)i Fu(satisfying)c(\(4.15\))k(and)d(co)m(v)-5 b(ariance)462 4124 y Fs(C)534 4080 y Fo(\030)527 4151 y(ij)588 4124 y Fu(\()p Fs(x;)15 b(y)s Fu(\))830 4073 y Fk(\001)824 4124 y Fu(=)25 b(co)m(v)1052 4138 y Fo(\027)1096 4124 y Fu(\()p Fs(\021)1176 4138 y Fo(i)1204 4124 y Fu(\()p Fs(x)p Fu(\))p Fs(;)15 b(\021)1411 4138 y Fo(j)1449 4124 y Fu(\()p Fs(y)s Fu(\)\))824 4283 y(=)25 b Fs(G)991 4246 y Fo(\030)1029 4283 y Fu(\()p Fs(x)c Fu(+)f Fs(e)1270 4297 y Fo(i)1298 4283 y Fs(;)15 b(y)23 b Fu(+)d Fs(e)1539 4297 y Fo(j)1576 4283 y Fu(\))h(+)f Fs(G)1794 4246 y Fo(\030)1832 4283 y Fu(\()p Fs(x;)15 b(y)s Fu(\))21 b Fq(\000)f Fs(G)2225 4246 y Fo(\030)2263 4283 y Fu(\()p Fs(x)h Fu(+)f Fs(e)2504 4297 y Fo(i)2532 4283 y Fs(;)15 b(y)s Fu(\))21 b Fq(\000)f Fs(G)2838 4246 y Fo(\030)2876 4283 y Fu(\()p Fs(x;)15 b(y)24 b Fu(+)c Fs(e)3205 4297 y Fo(j)3242 4283 y Fu(\))15 b Fs(:)118 4459 y Fj(T)-7 b(r)i(anslation)39 b(c)-5 b(ovarianc)g(e)38 b(and)f(tilte)-5 b(d)37 b(states.)52 b Fu(A)35 b(random)e(Gibbs)g(measure)h(is)f(a)h (measurable)g(map)118 4567 y Fs(\027)f Fu(:)42 b(\012)26 b Fq(!)h(fG)577 4534 y Fo(\030)615 4567 y Fs(;)42 b(\030)31 b Fq(2)26 b Fu(\012)p Fq(g)32 b Fu(suc)m(h)f(that)h Fs(\027)1438 4534 y Fo(\030)1502 4567 y Fq(2)26 b(G)1648 4534 y Fo(\030)1717 4567 y Fu(for)31 b(ev)m(ery)h Fs(\030)f Fq(2)26 b Fu(\012.)43 b(The)31 b(map)g Fs(\027)2832 4534 y Fn(\001)2887 4567 y Fu(is)f(called)g Fj(tr)-5 b(anslation)118 4677 y(c)g(ovariant)44 b Fu(when)c Fs(\027)825 4644 y Fo(\030)890 4677 y Fq(\016)28 b Fs(\022)1006 4691 y Fo(x)1094 4677 y Fu(=)43 b Fs(\027)1259 4644 y Fo(\034)1290 4652 y Fv(x)1329 4644 y Fo(\030)1409 4677 y Fu(for)e(ev)m(ery)h Fs(\030)48 b Fq(2)c Fu(\012,)g Fs(x)g Fq(2)f Fp(Z)2403 4644 y Fo(d)2440 4677 y Fu(,)h(where)d Fs(\022)2826 4691 y Fo(x)2911 4677 y Fu(denotes)h(the)g(action)118 4785 y(of)36 b(translation)f(group)g(and)g Fs(\034)1178 4799 y Fo(x)1258 4785 y Fu(is)g(the)h(en)m(vironmen)m(t)f(shift.)56 b(W)-8 b(e)37 b(de\014ne)e(the)h Fj(tilte)-5 b(d)38 b(states)f Fu(for)f(our)118 4893 y(in)m(terface)22 b(in)e(random)g(en)m(vironmen)m (t)h(as)g(the)g(set)h(of)f(translation)f(co)m(v)-5 b(arian)m(t)23 b(random)d(Gibbs)g(measures)118 5001 y Fs(\027)169 4968 y Fn(\001)235 5001 y Fu(suc)m(h)43 b(that)h Fs(\027)714 4968 y Fo(\030)798 5001 y Fq(2)h Fu(ext)q Fq(G)1087 4968 y Fo(\030)1125 5001 y Fu(.)79 b(It)43 b(is)f(not)h(di\016cult)e(no)m(w) i(to)h(use)f(Prop)s(osition)e(2.1)j(to)g(giv)m(e)f(a)h(full)118 5117 y(c)m(haracterization)36 b(of)e(tilted)g(states.)54 b(Indeed,)34 b(since)g Fs(C)2071 5073 y Fo(\030)2064 5144 y(ij)2159 5117 y Fu(is)f(translation)h(co)m(v)-5 b(arian)m(t,)37 b(w)m(e)e(only)e(ha)m(v)m(e)118 5245 y(to)g(c)m(haracterize)h(the)f(translation)e(co)m(v)-5 b(arian)m(t)34 b(maps)e Fs(\030)g Fq(!)d Fs(\033)2238 5212 y Fo(\030)2308 5245 y Fu(with)i Fs(\033)2572 5212 y Fo(\030)2642 5245 y Fu(ob)s(eying)h(\(4.15\).)48 b(F)-8 b(or)34 b(ev)m(ery)118 5355 y Fs(v)29 b Fq(2)c Fp(R)336 5322 y Fo(d)413 5355 y Fu(w)m(e)31 b(kno)m(w)f(that)1465 5480 y Fs(\033)1520 5443 y Fo(\030)1517 5503 y(v)1558 5480 y Fu(\()p Fs(x)p Fu(\))1712 5429 y Fk(\001)1706 5480 y Fu(=)25 b Fs(v)f Fu(+)c Fs( )2020 5494 y Fo(v)2061 5480 y Fu(\()p Fs(\034)2136 5494 y Fn(\000)p Fo(x)2235 5480 y Fs(\030)t Fu(\))p eop %%Page: 13 13 13 12 bop 3591 43 a Fk(13)118 366 y Fu(has)38 b(the)h(righ)m(t)f(prop)s (erties.)64 b(On)38 b(the)g(other)h(hand)f(if)f(w)m(e)i(additionally)d (require)h(that)i Fs(\033)3293 333 y Fo(\030)3331 366 y Fu(\(0\))h(is)e(in)118 488 y Fs(L)180 455 y Fk(2)219 488 y Fu(\(\012)p Fs(;)15 b(\026)p Fu(\))42 b(then)e(the)h(uniqueness)d (statemen)m(t)43 b(in)c(Prop)s(osition)g(2.1)i(allo)m(ws)f(to)i (conclude)e(that)h Fs(\033)3598 444 y Fo(\030)3595 500 y(v)3636 488 y Fu(,)118 598 y Fs(v)f Fq(2)d Fp(R)360 565 y Fo(d)444 598 y Fu(are)g(the)h(only)f(maps)g(with)f(these)i(prop)s (erties.)60 b(In)37 b(other)g(w)m(ords,)i(for)e(ev)m(ery)i Fs(v)h Fq(2)d Fp(R)3451 565 y Fo(d)3535 598 y Fu(the)118 714 y(Gaussian)28 b(measure)g Fs(\027)910 670 y Fo(\030)904 726 y(v)976 714 y Fu(with)f(co)m(v)-5 b(ariance)30 b Fs(C)1691 681 y Fo(\030)1756 714 y Fu(and)e(mean)h Fs(\033)2227 670 y Fo(\030)2224 726 y(v)2293 714 y Fu(is)e(the)i(unique)d(tilted)i (state)i(suc)m(h)e(that)118 831 y Fp(E)173 845 y Fo(\026)225 831 y Fu([)p Fs(\027)301 786 y Fo(\030)295 842 y(v)339 831 y Fu(\()p Fs(\021)s Fu(\(0\)\)])35 b(=)d Fs(v)37 b Fu(and)d Fp(E)1052 845 y Fo(\026)1105 831 y Fu([)p Fs(\027)1181 786 y Fo(\030)1175 842 y(v)1218 831 y Fu(\()p Fq(j)p Fs(\021)s Fu(\(0\))p Fq(j)1466 798 y Fk(2)1508 831 y Fu(\)])f Fs(<)f Fq(1)p Fu(.)53 b(The)34 b(ab)s(o)m(v)m(e)i (measures)e(are)h(the)g(analogue)g(in)e(our)118 939 y(setting)e(of)f(F) -8 b(unaki-Sp)s(ohn)28 b(states)k(for)e(translation)f(in)m(v)-5 b(arian)m(t)30 b(massless)f(gradien)m(t)i(\014elds)e(\([5)q(]\).)118 1068 y Fj(Tilte)-5 b(d)35 b(states)g(on)f(the)g(torus)h(and)g(surfac)-5 b(e)34 b(tension.)45 b Fu(T)-8 b(o)32 b(carry)g(this)e(analogy)j(a)f (little)e(further)g(here)118 1176 y(w)m(e)37 b(mimic)d(the)i (construction)g(of)g([5)q(])g(to)h(describ)s(e)e(tilted)g(states)i(as)f (the)h(in\014nite)c(v)m(olume)j(limit)e(of)118 1284 y(tilted)c(measure) g(on)g(the)h(torus)f Fs(T)1284 1298 y Fo(N)1351 1284 y Fu(.)218 1392 y(De\014ne)k(\000)560 1406 y Fo(N)662 1392 y Fu(as)h(the)f(set)h(of)42 b(~)-52 b Fs(\021)36 b Fq(2)31 b Fu(\()p Fp(R)1460 1359 y Fo(d)1507 1392 y Fu(\))1542 1359 y Fo(T)1583 1370 y Fv(N)1679 1392 y Fu(suc)m(h)k(that) 41 b(~)-51 b Fs(\021)35 b Fu(=)2288 1369 y(~)2273 1392 y Fq(r)p Fs(\036)f Fu(for)h(some)g Fs(\036)d Fq(2)g Fp(R)3051 1359 y Fo(T)3092 1370 y Fv(N)3160 1392 y Fu(,)3236 1369 y(~)3221 1392 y Fq(r)i Fu(b)s(eing)f(as)118 1499 y(usual)28 b(the)i(discrete)f(gradien)m(t)g(on)h(the)f(torus.)40 b(Consider)28 b(the)i(probabilit)m(y)c(measure)k(on)f(\000)3282 1513 y Fo(N)3379 1499 y Fu(de\014ned)118 1607 y(b)m(y)444 1796 y(~)-50 b Fs(\027)490 1752 y Fo(\030)484 1825 y(N)s(;v)603 1796 y Fu(\()15 b(d)7 b(~)-52 b Fs(\021)t Fu(\))25 b(=)988 1734 y(1)p 919 1775 182 4 v 919 1876 a Fs(Z)988 1832 y Fo(\030)981 1905 y(N)s(;v)1126 1796 y Fu(exp)1280 1722 y Fi(\000)1342 1796 y Fq(\000)1443 1734 y Fu(1)p 1443 1775 46 4 v 1443 1858 a(2)1541 1710 y Fi(X)1514 1906 y Fo(x)p Fn(2)p Fo(T)1642 1917 y Fv(N)1714 1710 y Fi(X)1768 1905 y Fo(i)1846 1796 y Fu(\()p Fs(\031)1933 1810 y Fo(N)2000 1796 y Fs(\030)t Fu(\))2079 1810 y Fo(i)2108 1796 y Fu(\()p Fs(x)p Fu(\)\()7 b(~)-52 b Fs(\021)2310 1810 y Fo(i)2339 1796 y Fu(\()p Fs(x)p Fu(\))21 b(+)f Fs(v)2617 1810 y Fo(i)2645 1796 y Fu(\))2680 1758 y Fk(2)2720 1722 y Fi(\001)2776 1796 y Fs(m)2856 1810 y Fo(N)2924 1796 y Fu(\()15 b(d)6 b(~)-51 b Fs(\021)s Fu(\))321 b(\(4.16\))118 2061 y(where)27 b Fs(m)458 2075 y Fo(N)525 2061 y Fu(\()15 b(d)7 b(~)-52 b Fs(\021)s Fu(\))28 b(stands)f(for)g(the)g(image)h(of)g(Leb)s(esgue)f (measure)g(under)f(the)h(map)g Fs(\036)f Fq(!)3227 2038 y Fu(~)3212 2061 y Fq(r)p Fs(\036)p Fu(.)39 b(Notice)118 2183 y(that)k(b)m(y)e(symmetry)h Fp(E)959 2218 y Fk(~)-39 b Fo(\027)994 2186 y Fv(\030)990 2242 y(N)r(;)p Fg(0)1104 2183 y Fu(\()7 b(~)-52 b Fs(\021)t Fu(\()p Fs(x)p Fu(\)\))45 b(=)f(0,)h Fs(x)g Fq(2)f Fs(T)1875 2197 y Fo(N)1942 2183 y Fu(.)75 b(Th)m(us)41 b(when)g Fs(v)47 b Fu(=)d(0,)51 b(~)-50 b Fs(\027)2906 2138 y Fo(\030)2900 2211 y(N)s(;)p Fk(0)3059 2183 y Fu(is)41 b(the)h(cen)m(tered)118 2350 y(Gaussian)20 b(measure)h(on)g(\000)1018 2364 y Fo(N)1106 2350 y Fu(with)f(co)m(v)-5 b(ariance)22 b Fs(C)1806 2306 y Fo(\030)1799 2379 y(N)s(;ij)1960 2350 y Fu(giv)m(en)f(b)m(y)g(the)g (gradien)m(ts)g(of)g(the)g(Green)h(function)118 2490 y Fs(G)189 2446 y Fo(\030)189 2519 y(N)257 2490 y Fu(\()p Fs(x;)15 b(y)s Fu(\))33 b(of)f(the)g(p)s(erio)s(dized)e(random)h(w)m (alk)h(on)g Fs(T)1935 2504 y Fo(N)2034 2490 y Fu(with)f(killing)e(up)s (on)i(hitting)g(the)h(origin)e(\([4)r(]\).)118 2598 y(No)m(w,)g(the)f (linear)e(tilt)g Fs(v)32 b Fu(only)c(a\013ects)i(the)e(mean)h(in)e (\(4.16\))k(and)d(it)g(remains)f(to)j(compute)f(the)f(mean)118 2706 y Fp(E)177 2742 y Fk(~)-39 b Fo(\027)212 2709 y Fv(\030)208 2765 y(N)r(;v)324 2706 y Fu(\()7 b(~)-52 b Fs(\021)t Fu(\()p Fs(x)p Fu(\)\).)41 b(W)-8 b(e)32 b(ma)m(y)f(pro)s(ceed)f(as)h(follo)m(ws.)1167 2966 y Fp(E)1226 3001 y Fk(~)-39 b Fo(\027)1261 2969 y Fv(\030)1257 3025 y(N)r(;v)1373 2966 y Fu(\()7 b(~)-52 b Fs(\021)t Fu(\(0\)\))27 b(=)1730 2842 y Fi(Z)1820 2868 y Fk(1)1780 3048 y(0)1917 2904 y Fu(d)p 1885 2945 99 4 v 1900 3028 a(d)p Fs(t)1993 2966 y Fp(E)2052 3001 y Fk(~)-39 b Fo(\027)2087 2969 y Fv(\030)2083 3025 y(N)r(;tv)2223 2966 y Fu(\()7 b(~)-52 b Fs(\021)t Fu(\(0\)\))32 b(d)o Fs(t)15 b(:)118 3161 y Fu(But)243 3275 y(d)p 211 3315 V 226 3399 a(d)p Fs(t)320 3336 y Fp(E)378 3372 y Fk(~)-39 b Fo(\027)413 3340 y Fv(\030)409 3395 y(N)r(;tv)549 3336 y Fu(\()7 b(~)-52 b Fs(\021)629 3350 y Fo(i)658 3336 y Fu(\(0\)\))27 b(=)e Fq(\000)p Fu(co)m(v)1138 3372 y Fk(~)-39 b Fo(\027)1173 3340 y Fv(\030)1169 3395 y(N)r(;tv)1303 3263 y Fi(\000)1351 3336 y Fu(~)-52 b Fs(\021)1389 3350 y Fo(i)1418 3336 y Fu(\(0\))p Fs(;)1601 3250 y Fi(X)1573 3447 y Fo(x)p Fn(2)p Fo(T)1701 3458 y Fv(N)1775 3250 y Fi(X)1825 3445 y Fo(j)1906 3336 y Fu(\()p Fs(\031)1993 3350 y Fo(N)2061 3336 y Fs(\030)t Fu(\))2140 3350 y Fo(j)2177 3336 y Fu(\()p Fs(x)p Fu(\)\()7 b(~)-52 b Fs(\021)2379 3350 y Fo(j)2416 3336 y Fu(\()p Fs(x)p Fu(\))21 b(+)f Fs(v)2694 3350 y Fo(j)2731 3336 y Fu(\))p Fs(v)2810 3350 y Fo(j)2847 3263 y Fi(\001)835 3598 y Fu(=)25 b Fq(\000)1044 3512 y Fi(X)1017 3709 y Fo(x)p Fn(2)p Fo(T)1145 3720 y Fv(N)1217 3512 y Fi(X)1267 3707 y Fo(j)1349 3598 y Fu(\()p Fs(\031)1436 3612 y Fo(N)1503 3598 y Fs(\030)t Fu(\))1582 3612 y Fo(j)1619 3598 y Fu(\()p Fs(x)p Fu(\))p Fs(v)1785 3612 y Fo(j)1822 3598 y Fs(C)1887 3612 y Fo(N)s(;ij)2026 3598 y Fu(\(0)p Fs(;)15 b(x)p Fu(\))27 b(=)e Fs(\037)2413 3612 y Fo(N)s(;v)2533 3598 y Fu(\()p Fs(e)2610 3612 y Fo(i)2638 3598 y Fu(\))c Fq(\000)f Fs(\037)2842 3612 y Fo(N)s(;v)2961 3598 y Fu(\(0\))27 b(=)e Fs( )3261 3561 y Fo(i)3258 3621 y(N)s(;v)3378 3598 y Fu(\(0)p Fs(;)15 b(\030)t Fu(\))118 3848 y(where)30 b(in)f(the)i(last)f(line)f(w)m(e)h(used)g(the)h(iden)m(tities)d (\(3.17\))33 b(and)d(\(3.19\))17 b(.)218 3991 y(No)m(w)44 b(for)f(ev)m(ery)h Fs(v)50 b Fq(2)d Fp(R)1104 3958 y Fo(d)1194 3991 y Fu(let)c Fs(\027)1389 3947 y Fo(\030)1383 4019 y(N)s(;v)1546 3991 y Fu(b)s(e)g(the)g(probabilit)m(y)e(measure)j (induced)d(b)m(y)48 b(~)-50 b Fs(\027)3239 3947 y Fo(\030)3233 4019 y(N)s(;v)3396 3991 y Fu(on)43 b(the)118 4147 y(\014eld)35 b Fs(\021)410 4095 y Fk(\001)404 4147 y Fu(=)41 b(~)-51 b Fs(\021)27 b Fu(+)d Fs(v)s Fu(.)58 b(The)35 b(ab)s(o)m(v)m(e)j (computation)e(sho)m(ws)g(that)g Fs(\027)2319 4102 y Fo(\030)2313 4175 y(N)s(;v)2469 4147 y Fu(is)f(the)h(Gaussian)g (measure)g(with)118 4287 y(co)m(v)-5 b(ariance)32 b Fs(C)630 4243 y Fo(\030)623 4315 y(N)s(;ij)792 4287 y Fu(and)e(mean)h(v)m(ector) 1313 4447 y Fp(E)1368 4483 y Fo(\027)1407 4451 y Fv(\030)1403 4506 y(N)r(;v)1519 4447 y Fu(\()p Fs(\021)s Fu(\()p Fs(x)p Fu(\)\))c(=)e Fs(v)e Fu(+)d Fs( )2099 4461 y Fo(N)s(;v)2219 4447 y Fu(\()p Fs(x;)15 b(\030)t Fu(\))g Fs(:)964 b Fu(\(4.17\))118 4657 y(F)-8 b(rom)40 b(the)f(con)m(v)m(ergence)i(of)f Fs(C)1227 4613 y Fo(\030)1220 4686 y(N)s(;ij)1398 4657 y Fu(w)m(e)g(then)e(infer,)i(b)m(y)f(Prop)s(osition)e(3.1,)43 b(that)d(at)f(least)h(along)f(a)118 4797 y(subsequence)30 b Fs(N)715 4764 y Fn(0)763 4797 y Fq(!)25 b(1)p Fu(,)31 b Fs(\027)1077 4753 y Fo(\030)1071 4827 y(N)1134 4808 y Fl(0)1156 4827 y Fo(;v)1247 4797 y Fs(\026)p Fq(\000)p Fu(almost)f(surely)f(con)m(v)m(erges)j(w)m(eakly)f(to)g(the)f Fs(v)s Fq(\000)p Fu(tilted)g(state)i Fs(\027)3542 4753 y Fo(\030)3536 4809 y(v)3579 4797 y Fu(.)218 4933 y(A)40 b(last)f(observ)-5 b(ation)40 b(is)f(a)h(simple)e(iden)m(tit)m(y)i (connecting)g(the)g(di\013usion)d(co)s(e\016cien)m(t)k(to)f(surface)118 5041 y(tension.)f(This)24 b(t)m(yp)s(e)j(of)f(relation)f(w)m(as)i (discussed)d(in)h([5)q(,)h(7)q(])g(in)f(the)h(con)m(text)j(of)d (translation)f(in)m(v)-5 b(arian)m(t)118 5151 y(massless)30 b(\014elds.)39 b(Recalling)29 b(\(4.16\))k(w)m(e)e(de\014ne)e(the)i (surface)f(tension)g(on)g Fs(T)2757 5165 y Fo(N)2855 5151 y Fu(under)e(tilt)i Fs(v)f Fq(2)24 b Fp(R)3482 5118 y Fo(d)3559 5151 y Fu(as)1420 5392 y Fs(\033)1475 5347 y Fo(\030)1472 5420 y(N)1539 5392 y Fu(\()p Fs(v)s Fu(\))j(=)1851 5330 y(1)p 1789 5371 171 4 v 1789 5454 a Fq(j)p Fs(T)1867 5468 y Fo(N)1934 5454 y Fq(j)1985 5392 y Fu(log)2128 5315 y Fs(Z)2197 5270 y Fo(\030)2190 5343 y(N)s(;)p Fk(0)p 2127 5371 182 4 v 2127 5472 a Fs(Z)2196 5428 y Fo(\030)2189 5501 y(N)s(;v)2334 5392 y Fs(:)p eop %%Page: 14 14 14 13 bop 118 45 a Fk(14)1010 b(PIETR)n(O)31 b(CAPUTO)f(AND)g(DMITR)-6 b(Y)30 b(IOFFE)118 366 y Fu(By)36 b(p)s(erforming)d(the)i(Gaussian)f (in)m(tegrals)h(and)g(using)e(iden)m(tities)h(\(3.17\))k(and)c (\(3.15\))k(one)d(obtains)118 474 y(the)c(sough)m(t)g(relation)1426 681 y Fs(\033)1481 637 y Fo(\030)1478 710 y(N)1545 681 y Fu(\()p Fs(v)s Fu(\))c(=)1795 620 y(1)p 1795 660 46 4 v 1795 743 a(4)1850 681 y(\()p Fs(D)1960 695 y Fo(N)2028 681 y Fu(\()p Fs(\030)t Fu(\))p Fs(v)s(;)15 b(v)s Fu(\))g Fs(:)1078 b Fu(\(4.18\))118 903 y(An)28 b(immediate)g(corollary)g(of)g (Theorem)g(1.1)i(is)d(then)h(the)h(a.s.)g(con)m(v)m(ergence)i(of)e Fs(\033)2978 858 y Fo(\030)2975 931 y(N)3042 903 y Fu(\()p Fs(v)s Fu(\))g(to)3307 867 y Fk(1)p 3307 882 36 4 v 3307 934 a(4)3353 903 y Fu(\()p Fs(D)s(v)s(;)15 b(v)s Fu(\).)118 1011 y(As)32 b(w)m(e)h(ha)m(v)m(e)h(already)e(men)m(tioned,)g(the)h (corresp)s(onding)d(problem)h(raised)g(in)g([7)q(])h(in)f(the)i(con)m (text)h(of)118 1119 y(an-harmonic)c(mo)s(dels)f(remains)g(op)s(en.)1618 1367 y(5.)51 b Ft(Appendix)218 1529 y Fu(The)26 b(pro)s(of)h(of)g (Theorem)g(4.1)h(is)e(a)i(straigh)m(tforw)m(ard)f(adjustmen)m(t)g(to)h (the)f(discrete)g(P)m(oisson)g(equa-)118 1637 y(tion)d(\(4.3\))j(on)e Fs(T)685 1651 y Fo(N)777 1637 y Fu(of)g(the)g(argumen)m(ts)g(emplo)m(y) m(ed)g(in)f(Chapter)g(9)h(of)g([8)q(]:)38 b(Giv)m(en)25 b Fs(N)35 b Fq(2)25 b Fp(N)37 b Fu(and)25 b(a)g(v)m(ector)118 1745 y(\014eld)g Fs(~)-41 b(g)34 b Fu(on)c Fs(T)571 1759 y Fo(N)669 1745 y Fu(de\014ne)g(the)g(distribution)d(function)i Fs(\021)1985 1761 y Fo(N)s(;)m(~)-32 b(g)2134 1745 y Fu(of)27 b Fs(~)-41 b(g)34 b Fu(as:)1062 1963 y Fs(\021)1107 1980 y Fo(N)s(;)m(~)-32 b(g)1226 1963 y Fu(\()p Fs(t)p Fu(\))56 b(=)1584 1902 y(1)p 1521 1942 171 4 v 1521 2026 a Fq(j)p Fs(T)1599 2040 y Fo(N)1667 2026 y Fq(j)1702 1963 y Fu(#)15 b Fq(f)p Fs(x)25 b Fq(2)g Fs(T)2054 1977 y Fo(N)2177 1963 y Fu(:)56 b Fq(j)l Fs(~)-41 b(g)t Fu(\()p Fs(x)p Fu(\))p Fq(j)26 b Fs(>)f(t)p Fq(g)16 b Fs(:)118 2206 y Fu(By)33 b(the)g(Marcinkiewicz)e(in)m(terp)s(olation)g(theorem)i (applied)d(for)i(the)h(solution)e(map)h Fs(T)42 b Fu(:)3250 2182 y Fs(~)3238 2206 y(f)c Fq(7!)3456 2183 y Fu(~)3441 2206 y Fq(r)o Fs(u)33 b Fu(it)118 2314 y(w)m(ould)c(b)s(e)h(enough)g (to)h(c)m(hec)m(k)h(that)f(there)g(exist)f(constan)m(ts)h Fs(c)2262 2328 y Fk(1)2332 2314 y Fu(and)f Fs(c)2548 2328 y Fk(2)2618 2314 y Fu(suc)m(h)g(that)h(for)g(ev)m(ery)g Fs(t)25 b Fr(>)g Fu(0)1445 2552 y Fs(\021)1490 2582 y Fo(N)s(;)1580 2565 y Fk(~)1569 2582 y Fn(r)o Fo(u)1672 2552 y Fu(\()p Fs(t)p Fu(\))56 b Fr(6)30 b Fs(c)1971 2566 y Fk(1)2021 2490 y Fq(k)2079 2466 y Fs(~)2066 2490 y(f)10 b Fq(k)2166 2504 y Fo(N)s(;)p Fk(1)p 2021 2531 264 4 v 2136 2614 a Fs(t)2295 2552 y(;)1155 b Fu(\(5.1\))118 2747 y(and,)1445 2997 y Fs(\021)1490 3027 y Fo(N)s(;)1580 3011 y Fk(~)1569 3027 y Fn(r)o Fo(u)1672 2997 y Fu(\()p Fs(t)p Fu(\))56 b Fr(6)30 b Fs(c)1971 3011 y Fk(2)2021 2922 y Fq(k)2079 2898 y Fs(~)2066 2922 y(f)10 b Fq(k)2166 2889 y Fk(2)2166 2949 y Fo(N)s(;)p Fk(2)p 2021 2976 V 2117 3060 a Fs(t)2150 3033 y Fk(2)2295 2997 y Fs(;)1155 b Fu(\(5.2\))118 3218 y(Since)40 b Fq(k)427 3195 y Fu(~)411 3218 y Fq(r)p Fs(u)p Fq(k)584 3232 y Fo(N)s(;)p Fk(2)746 3218 y Fr(6)j Fq(k)918 3194 y Fs(~)905 3218 y(f)10 b Fq(k)1005 3232 y Fo(N)s(;)p Fk(2)1124 3218 y Fu(,)44 b(the)d(second)g(estimate)h(trivially)c(follo)m(ws)j(from)f(Mark)m(o)m (v)j(inequalit)m(y)118 3326 y(\(with)30 b Fs(c)400 3340 y Fk(2)465 3326 y Fu(=)25 b(1\).)41 b(It)30 b(remains)g(to)h(c)m(hec)m (k)h(\(5.1\))218 3433 y(T)-8 b(o)34 b(facilitate)f(the)h(exp)s(osition) e(w)m(e)i(shall)d(consider)i(only)f(the)i(dy)m(adic)f(case)h Fs(N)41 b Fu(=)30 b(2)3150 3400 y Fo(n)3197 3433 y Fu(.)50 b(Also,)34 b(it)f(is)118 3548 y(enough)d(to)h(deriv)m(e)f(the)h(b)s (ound)d(only)i(for)g(the)g(v)m(ector)i(\014elds)2261 3524 y Fs(~)2248 3548 y(f)40 b Fu(of)30 b(the)h(form)2820 3524 y Fs(~)2808 3548 y(f)9 b Fu(\()p Fs(x)p Fu(\))26 b(=)f Fs(f)10 b Fu(\()p Fs(x)p Fu(\))l Fs(~)-41 b(e)3325 3562 y Fk(1)3365 3548 y Fu(.)218 3656 y(F)-8 b(or)31 b(ev)m(ery)g Fs(k)d Fu(=)d(0)p Fs(;)15 b Fu(1)p Fs(;)g(:)g(:)g(:)j(;)d (n)21 b Fq(\000)e Fu(1)31 b(de\014ne)f(the)g(decomp)s(osition)1507 3838 y Fs(T)1560 3852 y Fo(N)1683 3838 y Fu(=)1925 3752 y Fi(_)1810 3949 y Fo(x)p Fn(2)p Fo(K)5 b(T)2003 3963 y Fv(N)q(=K)2156 3838 y Fs(B)2230 3801 y Fo(k)2225 3861 y(x)118 4133 y Fu(of)29 b(the)h(torus)e Fs(T)660 4147 y Fo(N)757 4133 y Fu(in)m(to)h(the)g(lattice)h(b)s(o)m(xes)f Fs(B)1695 4100 y Fo(k)1690 4155 y(x)1762 4133 y Fu(=)c Fs(x)18 b Fu(+)f Fq(f)p Fu(0)p Fs(;)e(:)g(:)g(:)i(;)e Fu(2)2353 4100 y Fo(k)2415 4133 y Fq(\000)i Fu(1)p Fq(g)2593 4100 y Fo(d)2663 4133 y Fu(of)29 b(the)h(linear)d(size)i Fs(K)j Fu(=)25 b(2)3593 4100 y Fo(k)3636 4133 y Fu(.)118 4243 y(The)39 b(b)s(o)m(xes)h(on)g(di\013eren)m(t)f Fs(k)s Fu(-scales)h(are)g(naturally)e(ordered)h(b)m(y)h(the)g(inclusion:)56 b(w)m(e)40 b(sa)m(y)g Fs(B)3439 4210 y Fo(l)3434 4266 y(y)3515 4243 y Fu(is)f(a)118 4366 y(predecessor)33 b(of)g Fs(B)782 4333 y Fo(k)777 4388 y(x)856 4366 y Fu(if)f Fs(l)f(>)e(k)36 b Fu(and)c Fs(B)1436 4333 y Fo(k)1431 4388 y(x)1507 4366 y Fq(\032)d Fs(B)1681 4333 y Fo(l)1676 4388 y(y)1717 4366 y Fu(.)48 b(By)33 b(de\014nition)d Fs(T)2394 4380 y Fo(N)2494 4366 y Fu(is)i(a)h(predecessor)f(of)h(an)m (y)g(of)g(the)118 4489 y(b)s(o)m(xes)f Fs(B)444 4456 y Fo(k)439 4511 y(x)519 4489 y Fu(on)g(an)m(y)g(of)g(the)h Fs(k)s Fu(-scales.)46 b(Let)33 b(us)e(\014x)h(a)g(n)m(um)m(b)s(er)f Fs(t)d(>)g Fq(k)p Fs(f)10 b Fq(k)2587 4503 y Fo(N)s(;)p Fk(1)2705 4489 y Fu(.)46 b(W)-8 b(e)33 b(sa)m(y)g(that)g(a)f(b)s(o)m(x) g Fs(B)3619 4456 y Fo(k)3614 4511 y(x)118 4597 y Fu(is)e Fz(correct)h Fu(if)1164 4686 y(1)p 1104 4727 167 4 v 1104 4810 a Fq(j)p Fs(B)1203 4784 y Fo(k)1198 4833 y(x)1245 4810 y Fq(j)1319 4661 y Fi(X)1295 4867 y Fo(y)r Fn(2)p Fo(B)1435 4848 y Fv(k)1432 4883 y(x)1489 4748 y Fq(j)p Fs(f)10 b Fu(\()p Fs(y)s Fu(\))p Fq(j)56 b Fu(=)1944 4686 y(1)p 1905 4727 125 4 v 1905 4810 a Fs(K)1989 4784 y Fo(d)2078 4661 y Fi(X)2054 4867 y Fo(y)r Fn(2)p Fo(B)2194 4848 y Fv(k)2191 4883 y(x)2248 4748 y Fq(j)p Fs(f)10 b Fu(\()p Fs(y)s Fu(\))p Fq(j)56 b Fr(6)30 b Fs(t:)118 5021 y Fu(By)39 b(the)g(c)m(hoice)g(of)f Fs(t)h Fu(the)f(common)h (ancestor)h Fs(T)1852 5035 y Fo(N)1957 5021 y Fu(is)e(alw)m(a)m(ys)h (correct.)66 b(An)38 b(incorrect)g(b)s(o)m(x)h Fs(B)3520 4988 y Fo(k)3515 5044 y(x)3600 5021 y Fu(is)118 5129 y(called)30 b Fz(mark)m(ed)f Fu(if)h(it)f(is)h(incorrect,)g(but)g(all)f (his)g(predecessors)h(are)h(correct.)42 b(By)31 b(construction,)1315 5336 y Fs(t)55 b(<)1600 5274 y Fu(1)p 1540 5315 167 4 v 1540 5398 a Fq(j)p Fs(B)1639 5372 y Fo(k)1634 5421 y(x)1681 5398 y Fq(j)1755 5250 y Fi(X)1731 5455 y Fo(y)r Fn(2)p Fo(B)1871 5436 y Fv(k)1868 5471 y(x)1925 5336 y Fq(j)p Fs(f)10 b Fu(\()p Fs(y)s Fu(\))p Fq(j)56 b Fr(6)30 b Fu(2)2350 5298 y Fo(d)2391 5336 y Fs(t;)1026 b Fu(\(5.3\))p eop %%Page: 15 15 15 14 bop 3591 43 a Fk(15)118 366 y Fu(for)28 b(ev)m(ery)i(mark)m(ed)f (b)s(o)m(x)f Fs(B)1062 333 y Fo(l)1057 389 y(x)1101 366 y Fu(.)40 b(Since)27 b(all)h(the)h(mark)m(ed)f(b)s(o)m(xes)h(are)g (disjoin)m(t,)f(the)g(\014rst)g(of)h(the)g(inequal-)118 474 y(ities)h(in)f(\(5.3\))j(implies)27 b(that)k(the)g(total)g(mark)m (ed)g(v)m(olume)1238 629 y(1)p 1175 669 171 4 v 1175 753 a Fq(j)p Fs(T)1253 767 y Fo(N)1321 753 y Fq(j)1604 604 y Fi(X)1371 809 y Fk(\()p Fo(k)r(;x)p Fk(\):)p Fo(B)1600 790 y Fv(k)1597 826 y(x)1669 809 y Fk(is)23 b(mark)n(ed)1982 690 y Fq(j)p Fs(B)2081 653 y Fo(k)2076 713 y(x)2124 690 y Fq(j)55 b Fr(6)2315 628 y Fq(k)p Fs(f)10 b Fq(k)2460 642 y Fo(N)s(;)p Fk(1)p 2315 669 264 4 v 2431 753 a Fs(t)2589 690 y(:)861 b Fu(\(5.4\))118 960 y(Let)31 b(us)f(ren)m(um)m(b)s(er)f (all)g(the)i(mark)m(ed)f(b)s(o)m(xes)h(as)f Fs(B)1842 974 y Fk(1)1882 960 y Fs(;)15 b(:)g(:)g(:)h(;)f(B)2152 975 y Fo(l)2209 960 y Fu(and)30 b(decomp)s(ose)g Fs(f)40 b Fu(as)642 1209 y Fs(f)10 b Fu(\()p Fs(x)p Fu(\))25 b(=)g Fs(f)10 b Fu(\()p Fs(x)p Fu(\)1)-25 b(I)1170 1228 y Fo(T)1211 1239 y Fv(N)1269 1228 y Fn(n[)p Fo(B)1404 1238 y Fv(i)1435 1209 y Fu(\()p Fs(x)p Fu(\))21 b(+)1723 1096 y Fo(l)1669 1123 y Fi(X)1677 1319 y Fo(i)p Fk(=1)1815 1209 y Fs(f)10 b Fu(\()p Fs(x)p Fu(\)1)-25 b(I)2045 1223 y Fo(B)2098 1233 y Fv(i)2129 1209 y Fu(\()p Fs(x)p Fu(\))2313 1158 y Fk(\001)2307 1209 y Fu(=)56 b Fs(f)2489 1172 y Fk(0)2527 1209 y Fu(\()p Fs(x)p Fu(\))21 b(+)2816 1096 y Fo(l)2761 1123 y Fi(X)2770 1319 y Fo(i)p Fk(=1)2908 1209 y Fs(f)2963 1172 y Fo(i)2990 1209 y Fu(\()p Fs(x)p Fu(\))p Fs(:)118 1478 y Fu(Accordingly)-8 b(,)30 b(w)m(e)h(decomp)s (ose)g(solutions)d Fs(u)j Fu(of)37 b(\(4.3\))32 b(as)f Fs(u)25 b Fu(=)g Fs(u)2359 1445 y Fk(0)2419 1478 y Fu(+)2510 1410 y Fi(P)2606 1436 y Fo(l)2606 1505 y Fk(1)2660 1478 y Fs(u)2712 1445 y Fo(j)2749 1478 y Fu(,)31 b(where)1338 1623 y Fo(d)1291 1650 y Fi(X)1299 1846 y Fo(i)p Fk(=1)1452 1714 y Fu(~)1437 1736 y Fq(r)1513 1699 y Fn(\003)1513 1759 y Fo(i)1567 1714 y Fu(~)1552 1736 y Fq(r)1628 1750 y Fo(i)1656 1736 y Fs(u)1708 1699 y Fo(j)1745 1736 y Fu(\()p Fs(x)p Fu(\))56 b(=)f Fq(\000)2135 1714 y Fu(~)2120 1736 y Fq(r)2196 1699 y Fn(\003)2196 1759 y Fk(1)2235 1736 y Fs(f)2290 1699 y Fo(j)2326 1736 y Fu(\()p Fs(x)p Fu(\))p Fs(;)1002 b Fu(\(5.5\))118 1980 y(for)30 b Fs(j)h Fu(=)25 b(0)p Fs(;)15 b Fu(1)p Fs(;)g(:)g(:)g(:)j(;)d(l)r Fu(.)41 b(Eviden)m(tly)-8 b(,)1127 2143 y Fs(\021)1172 2173 y Fo(N)s(;)1263 2156 y Fk(~)1251 2173 y Fn(r)p Fo(u)1355 2143 y Fu(\(2)p Fs(t)p Fu(\))56 b Fr(6)30 b Fs(\021)1705 2173 y Fo(N)s(;)1796 2156 y Fk(~)1784 2173 y Fn(r)p Fo(u)1884 2154 y Fg(0)1923 2143 y Fu(\()p Fs(t)p Fu(\))20 b(+)g Fs(\021)2182 2175 y Fo(N)s(;)2273 2158 y Fk(~)2261 2175 y Fn(r)2332 2125 y Fe(P)2406 2146 y Fv(l)2406 2196 y Fg(1)2452 2175 y Fo(u)2493 2156 y Fv(i)2524 2143 y Fu(\()p Fs(t)p Fu(\))p Fs(:)118 2334 y Fu(Since)29 b Fq(j)p Fs(f)435 2301 y Fk(0)474 2334 y Fq(j)i Fu(is)e(b)s(ounded)g(ab)s(o)m(v)m(e)i(b)m (y)f Fs(t)p Fu(,)h(it)f(follo)m(ws)f(from)h(\(5.2\))i(that)1215 2571 y Fs(\021)1260 2601 y Fo(N)s(;)1351 2584 y Fk(~)1339 2601 y Fn(r)p Fo(u)1439 2582 y Fg(0)1477 2571 y Fu(\()p Fs(t)p Fu(\))56 b Fr(6)1747 2495 y Fq(k)p Fs(f)1847 2462 y Fk(0)1886 2495 y Fq(k)1931 2462 y Fk(2)1931 2522 y Fo(N)s(;)p Fk(2)p 1747 2550 303 4 v 1863 2633 a Fs(t)1896 2607 y Fk(2)2116 2571 y Fr(6)2227 2508 y Fq(k)p Fs(f)2327 2475 y Fk(0)2366 2508 y Fq(k)2411 2522 y Fo(N)s(;)p Fk(1)p 2227 2550 V 2362 2633 a Fs(t)2539 2571 y(:)911 b Fu(\(5.6\))118 2771 y(The)42 b(bulk)f(of)i(the)g(w)m(ork)g(is,)i(th)m(us,)h(to)d (deriv)m(e)g(the)g Fs(L)2062 2738 y Fk(1)2143 2771 y Fu(estimate)h(on)f(the)g(distribution)c(function)118 2886 y Fs(\021)163 2919 y Fo(N)s(;)254 2902 y Fk(~)242 2919 y Fn(r)313 2869 y Fe(P)387 2890 y Fv(l)387 2940 y Fg(1)433 2919 y Fo(u)474 2900 y Fv(i)546 2886 y Fu(corresp)s(onding)g (to)i(the)g(irregular)e(part)2017 2818 y Fi(P)2113 2844 y Fo(l)2113 2913 y Fk(1)2168 2886 y Fs(f)2223 2853 y Fo(l)2289 2886 y Fu(of)i(the)g(v)m(ector)h(\014eld)3073 2862 y Fs(~)3061 2886 y(f)52 b Fu(=)43 b Fs(f)6 b(~)-41 b(e)3369 2900 y Fk(1)3408 2886 y Fu(.)72 b(The)118 3027 y(equation)38 b(\(5.5\))h(feels)f(the)g(righ)m(t)f(hand)g(side)g Fs(f)1800 2994 y Fo(i)1865 3027 y Fu(only)g(inside)f(the)i(b)s(o)m(x)f Fs(B)2752 3041 y Fo(i)2780 3027 y Fu(.)64 b(F)-8 b(or)38 b Fs(B)3108 3041 y Fo(i)3174 3027 y Fu(=)f Fs(B)3356 2994 y Fo(k)3351 3049 y(x)3436 3027 y Fu(let)h(us)118 3137 y(de\014ne)30 b(the)g(enlargemen)m(t)1072 3114 y(\026)1050 3137 y Fs(B)1119 3151 y Fo(i)1173 3137 y Fu(=)1290 3114 y(\026)1269 3137 y Fs(B)1343 3104 y Fo(k)1338 3160 y(x)1415 3137 y Fu(via)1218 3295 y(\026)1197 3318 y Fs(B)1266 3332 y Fo(i)1319 3318 y Fu(=)1436 3295 y(\026)1415 3318 y Fs(B)1489 3280 y Fo(k)1484 3340 y(x)1556 3318 y Fu(=)1989 3231 y Fi([)1652 3433 y Fo(y)r Fn(2)p Fo(K)5 b(T)1842 3447 y Fv(N)q(=K)1981 3433 y Fk(:)p Fn(k)p Fo(y)r Fn(\000)p Fo(x)p Fn(k)2203 3441 y Fl(1)2288 3433 y Ff(6)19 b Fo(K)2442 3318 y Fs(B)2516 3280 y Fo(k)2511 3340 y(y)2558 3318 y Fs(:)118 3610 y Fu(In)33 b(other)g(w)m(ords,)786 3587 y(\026)765 3610 y Fs(B)839 3577 y Fo(k)834 3633 y(x)914 3610 y Fu(is)f(the)h(union)f(of)h Fs(B)1603 3577 y Fo(k)1598 3633 y(x)1678 3610 y Fu(with)f(all)g(the)i Fq(\003)p Fu(-con)m(tiguous)g(b)s(o)m(xes)f(on)g(the)g Fs(k)s Fu(-th)h(scale.)118 3718 y(By)e(\(5.4\))r(,)1476 3911 y(1)p 1414 3951 171 4 v 1414 4035 a Fq(j)p Fs(T)1492 4049 y Fo(N)1559 4035 y Fq(j)1664 3859 y Fo(l)1610 3886 y Fi(X)1618 4081 y Fo(i)p Fk(=1)1756 3972 y Fq(j)1803 3949 y Fu(\026)1781 3972 y Fs(B)1850 3986 y Fo(i)1878 3972 y Fq(j)56 b Fr(6)30 b Fu(3)2105 3935 y Fo(d)2156 3911 y Fq(k)p Fs(f)10 b Fq(k)2301 3925 y Fk(1)p 2156 3951 185 4 v 2232 4035 a Fs(t)2351 3972 y(:)1099 b Fu(\(5.7\))118 4228 y(In)30 b(order)g(to)h(estimate)g Fs(\021)990 4258 y Fo(N)s(;)1081 4241 y Fk(~)1069 4258 y Fn(r)p Fo(u)1169 4239 y Fv(i)1229 4228 y Fu(outside)1564 4205 y(\026)1543 4228 y Fs(B)1612 4242 y Fo(i)1670 4228 y Fu(write:)598 4416 y(~)583 4439 y Fq(r)659 4453 y Fo(j)695 4439 y Fs(u)747 4402 y Fo(i)775 4439 y Fu(\()p Fs(x)p Fu(\))26 b(=)1036 4353 y Fi(X)1019 4550 y Fo(y)r Fn(2)p Fo(B)1156 4560 y Fv(i)1214 4416 y Fu(~)1198 4439 y Fq(r)1274 4453 y Fk(1)1329 4416 y Fu(~)1314 4439 y Fq(r)1390 4453 y Fo(j)1426 4439 y Fs(G)1497 4453 y Fo(N)1565 4439 y Fu(\()p Fs(x)20 b Fq(\000)g Fs(y)s Fu(\))p Fs(f)1901 4402 y Fo(i)1929 4439 y Fu(\()p Fs(y)s Fu(\))2109 4388 y Fk(\001)2103 4439 y Fu(=)2245 4353 y Fi(X)2229 4550 y Fo(y)r Fn(2)p Fo(B)2366 4560 y Fv(i)2423 4416 y Fu(~)2408 4439 y Fq(r)2484 4402 y Fk(2)2484 4462 y(1)p Fo(;j)2575 4439 y Fs(G)2646 4453 y Fo(N)2714 4439 y Fu(\()p Fs(x)g Fq(\000)g Fs(y)s Fu(\))p Fs(f)3050 4402 y Fo(i)3078 4439 y Fu(\()p Fs(y)s Fu(\))279 b(\(5.8\))118 4697 y(where)1338 4851 y(~)1323 4874 y Fq(r)1399 4836 y Fk(2)1399 4896 y(1)p Fo(;j)1490 4874 y Fs(G)1561 4888 y Fo(N)1629 4874 y Fu(\()p Fs(z)t Fu(\))26 b(=)1908 4760 y Fn(1)1878 4787 y Fi(X)1867 4981 y Fo(m)p Fk(=0)2050 4851 y Fu(~)2035 4874 y Fq(r)2111 4836 y Fk(2)2111 4896 y(1)p Fo(;j)2202 4874 y Fs(p)2248 4888 y Fo(m)2314 4874 y Fu(\()p Fs(z)t Fu(\))p Fs(;)118 5097 y Fu(and)k Fs(p)341 5111 y Fo(m)437 5097 y Fu(is)g(the)g Fs(m)p Fu(-step)h(transition)e (function)g(of)i(the)f(simple)e(random)i(w)m(alk)g(on)g Fs(T)3027 5111 y Fo(N)3095 5097 y Fu(.)218 5205 y(There)g(is)f(no)h (loss)g(to)h(assume)f(that)h Fs(f)1548 5172 y Fo(i)1606 5205 y Fu(has)f(zero)h(a)m(v)m(erage:)1588 5295 y Fi(X)1571 5492 y Fo(y)r Fn(2)p Fo(B)1708 5502 y Fv(i)1750 5381 y Fs(f)1805 5343 y Fo(i)1833 5381 y Fu(\()p Fs(x)p Fu(\))56 b(=)f(0)p Fs(:)p eop %%Page: 16 16 16 15 bop 118 45 a Fk(16)1010 b(PIETR)n(O)31 b(CAPUTO)f(AND)g(DMITR)-6 b(Y)30 b(IOFFE)118 366 y Fu(Th)m(us,)g(for)g Fs(B)582 380 y Fo(i)635 366 y Fu(=)25 b Fs(B)805 333 y Fo(k)800 389 y(y)835 398 y Fg(0)874 366 y Fu(,)30 b(w)m(e)h(can)g(rewrite)e (\(5.8\))k(as)731 542 y(~)716 565 y Fq(r)792 579 y Fo(j)828 565 y Fs(u)880 527 y Fo(i)908 565 y Fu(\()p Fs(x)p Fu(\))56 b(=)1229 478 y Fi(X)1213 675 y Fo(y)r Fn(2)p Fo(B)1350 685 y Fv(i)1392 464 y Fi(\020)1461 542 y Fu(~)1446 565 y Fq(r)1522 527 y Fk(2)1522 587 y(1)p Fo(;j)1613 565 y Fs(G)1684 579 y Fo(N)1752 565 y Fu(\()p Fs(x)20 b Fq(\000)g Fs(y)s Fu(\))h Fq(\000)2160 542 y Fu(~)2145 565 y Fq(r)2221 527 y Fk(2)2221 587 y(1)p Fo(;j)2312 565 y Fs(G)2383 579 y Fo(N)2450 565 y Fu(\()p Fs(x)g Fq(\000)f Fs(y)2694 579 y Fk(0)2733 565 y Fu(\))2768 464 y Fi(\021)2838 565 y Fs(f)2893 527 y Fo(i)2920 565 y Fu(\()p Fs(y)s Fu(\))p Fs(:)412 b Fu(\(5.9\))118 824 y(By)31 b(Theorem)f(1.5.5)i(in)d([13)q(]) i(\()p Fs(d)26 b Fr(>)f Fu(3\))31 b(or)f(b)m(y)g(Theorem)h(1.6.5)h(in)d ([13)q(])h(\()p Fs(d)c Fu(=)f(2\),)777 924 y Fi(\014)777 979 y(\014)777 1033 y(\014)823 1006 y Fu(~)808 1029 y Fq(r)884 991 y Fk(2)884 1051 y(1)p Fo(;j)975 1029 y Fs(G)1046 1043 y Fo(N)1114 1029 y Fu(\()p Fs(x)20 b Fq(\000)g Fs(y)s Fu(\))g Fq(\000)1521 1006 y Fu(~)1506 1029 y Fq(r)1582 991 y Fk(2)1582 1051 y(1)p Fo(;j)1673 1029 y Fs(G)1744 1043 y Fo(N)1812 1029 y Fu(\()p Fs(x)h Fq(\000)f Fs(y)2056 1043 y Fk(0)2095 1029 y Fu(\))2130 924 y Fi(\014)2130 979 y(\014)2130 1033 y(\014)2231 1029 y Fr(6)30 b Fs(c)2371 1043 y Fk(3)2411 1029 y Fu(\()p Fs(d)p Fu(\))2711 967 y Fs(K)p 2538 1008 429 4 v 2538 1091 a Fq(j)p Fs(x)21 b Fq(\000)f Fs(y)2772 1105 y Fk(0)2811 1091 y Fq(j)2836 1065 y Fo(d)p Fk(+1)2977 1029 y Fs(:)118 1235 y Fu(It)31 b(follo)m(ws)e(that)1362 1274 y Fi(X)1277 1481 y Fo(x)p Fn(2)p Fo(T)1405 1492 y Fv(N)1463 1481 y Fn(n)1514 1464 y Fk(\026)1498 1481 y Fo(B)1551 1491 y Fv(i)1593 1256 y Fi(\014)1593 1311 y(\014)1593 1365 y(\014)1638 1338 y Fu(~)1623 1361 y Fq(r)p Fs(u)1751 1323 y Fo(i)1779 1256 y Fi(\014)1779 1311 y(\014)1779 1365 y(\014)1880 1361 y Fr(6)h Fs(c)2020 1375 y Fk(4)2091 1274 y Fi(X)2075 1471 y Fo(y)r Fn(2)p Fo(B)2212 1481 y Fv(i)2254 1361 y Fq(j)p Fs(f)10 b Fu(\()p Fs(y)s Fu(\))p Fq(j)p Fs(;)118 1609 y Fu(for)30 b(all)g Fs(i)25 b Fu(=)g(1)p Fs(;)15 b(:)g(:)g(:)i(;)e(l)r Fu(.)41 b(Since)29 b(the)i(mark)m(ed)g(b)s(o)m (xes)f Fs(B)1915 1623 y Fk(1)1954 1609 y Fs(;)15 b(:)g(:)g(:)i(;)e(B) 2225 1624 y Fo(l)2282 1609 y Fu(are)30 b(disjoin)m(t,)1309 1768 y Fi(X)1200 1974 y Fo(x)p Fn(2)p Fo(T)1328 1985 y Fv(N)1386 1974 y Fn(n[)1485 1957 y Fk(\026)1468 1974 y Fo(B)1521 1984 y Fv(i)1563 1695 y Fi(\014)1563 1749 y(\014)1563 1804 y(\014)1563 1858 y(\014)1563 1913 y(\014)1609 1831 y Fu(~)1593 1854 y Fq(r)1739 1740 y Fo(l)1684 1768 y Fi(X)1693 1963 y Fo(i)p Fk(=1)1831 1854 y Fs(u)1883 1816 y Fo(i)1911 1695 y Fi(\014)1911 1749 y(\014)1911 1804 y(\014)1911 1858 y(\014)1911 1913 y(\014)2012 1854 y Fr(6)2154 1768 y Fi(X)2128 1964 y Fo(y)r Fn(2)p Fo(T)2253 1975 y Fv(N)2326 1854 y Fq(j)p Fs(f)10 b Fu(\()p Fs(x)p Fu(\))p Fq(j)p Fs(:)118 2121 y Fu(As)30 b(a)h(result,)871 2267 y(1)p 808 2307 171 4 v 808 2391 a Fq(j)p Fs(T)886 2405 y Fo(N)954 2391 y Fq(j)989 2328 y Fu(#)1080 2173 y Fi(\()1153 2328 y Fs(x)25 b Fq(2)g Fs(T)1369 2342 y Fo(N)1457 2328 y Fq(n)c([)1605 2305 y Fu(\026)1584 2328 y Fs(B)1653 2342 y Fo(i)1736 2328 y Fu(:)1817 2169 y Fi(\014)1817 2224 y(\014)1817 2278 y(\014)1817 2333 y(\014)1817 2387 y(\014)1862 2305 y Fu(~)1847 2328 y Fq(r)1993 2215 y Fo(l)1938 2242 y Fi(X)1946 2437 y Fo(i)p Fk(=1)2084 2328 y Fs(u)2136 2291 y Fo(i)2165 2169 y Fi(\014)2165 2224 y(\014)2165 2278 y(\014)2165 2333 y(\014)2165 2387 y(\014)2220 2328 y Fs(>)k(t)2349 2173 y Fi(\))2493 2328 y Fr(6)30 b Fs(c)2633 2342 y Fk(4)2683 2266 y Fq(k)p Fs(f)10 b Fq(k)2828 2280 y Fo(N)s(;)p Fk(1)p 2683 2307 264 4 v 2798 2391 a Fs(t)2956 2328 y(;)118 2574 y Fu(whic)m(h,)27 b(b)m(y)h(\(5.7\))r(,)f(leads)g(to)g(the)g(desired)f(estimated)h(on)g (the)g(distribution)c(function)j(of)h(the)3355 2551 y(~)3340 2574 y Fq(r)3431 2506 y Fi(P)3527 2532 y Fo(l)3527 2601 y Fk(1)3581 2574 y Fs(u)3633 2541 y Fo(i)118 2682 y Fu(part:)1373 2851 y Fs(\021)1418 2883 y Fo(N)s(;)1508 2867 y Fk(~)1497 2883 y Fn(r)1567 2834 y Fe(P)1642 2854 y Fv(l)1642 2904 y Fg(1)1688 2883 y Fo(u)1729 2865 y Fv(i)1759 2851 y Fu(\()p Fs(t)p Fu(\))56 b Fr(6)30 b Fs(c)2058 2865 y Fk(5)2108 2788 y Fq(k)p Fs(f)10 b Fq(k)2253 2802 y Fo(N)s(;)p Fk(1)p 2108 2830 V 2224 2913 a Fs(t)2382 2851 y(:)118 3016 y Fu(The)30 b(pro)s(of)g(of)g(Theorem)g(4.1)i(is)d(concluded.)1625 3235 y Ft(References)157 3380 y Fy([1])39 b(D.)21 b(Boivin,)j Fh(We)l(ak)g(c)l(onver)l(genc)l(e)i(for)e(r)l(eversible)h(r)l(andom)g (walks)f(in)f(a)h(r)l(andom)h(envir)l(onment)p Fy(,)f(Ann.)c(Probab.)j Fd(21)p Fy(,)276 3471 y(1427{1440,)29 b(1993)157 3563 y([2])39 b(D.Bo)n(vin,)h(Y.)c(Derriennic,)k Fh(The)e(er)l(go)l(dic)h (the)l(or)l(em)g(for)f(additive)g(c)l(o)l(cycles)i(of)d Fw(Z)2759 3531 y Fv(d)2827 3563 y Fh(or)h Fw(R)2987 3531 y Fv(d)3063 3563 y Fy(,)i(Ergo)r(dic)e(Theory)276 3654 y(Dynam.)24 b(Systems)h Fd(11)p Fy(,)h(19{39,)i(1991)157 3745 y([3])39 b(A.)e(De)h(Masi,)k(P)-6 b(.)38 b(F)-6 b(errari,)42 b(S.)37 b(Goldstein,)42 b(W.D.)c(Wic)n(k,)j Fh(A)n(n)e(invarianc)l(e)g(principle)g(for)f(r)l(eversible)i(Markov)276 3836 y(pr)l(o)l(c)l(esses.)c(Applic)l(ations)e(to)g(r)l(andom)g (motions)f(in)g(r)l(andom)h(envir)l(onments)p Fy(,)i(J.)d(Statist.)f (Ph)n(ys.)g Fd(55)p Fy(,)i(787{855,)276 3928 y(1989)157 4019 y([4])39 b(J-D.)26 b(Deusc)n(hel,)g(G.)g(Giacomin,)h(D.)e (Io\013e,)h Fh(L)l(ar)l(ge)j(deviations)g(and)f(c)l(onc)l(entr)l(ation) j(pr)l(op)l(erties)e(for)f Fc(r)p Fb(\036)f Fh(interfac)l(e)276 4110 y(mo)l(dels)p Fy(,)f(Probab.)h(Theory)f(Related)f(Fields)i Fd(117)p Fy(,)f(49{111,)j(2000)157 4202 y([5])39 b(T.)29 b(F)-6 b(unaki)27 b(and)g(H.)h(Sp)r(ohn,)g Fh(Motion)i(by)g(me)l(an)g (curvatur)l(e)i(fr)l(om)d(the)i(Ginzbur)l(g-L)l(andau)h Fc(r)p Fb(\036)d Fh(interfac)l(e)i(mo)l(del)p Fy(,)276 4293 y(Comm.)25 b(Math.)h(Ph)n(ys.)g Fd(185)p Fy(,)g(1{36,)i(1997)157 4384 y([6])39 b(H.O.)26 b(Georgii,)h Fh(Gibbs)i(me)l(asur)l(es)g(and)f (phase)h(tr)l(ansitions)p Fy(,)e(de)f(Gruyter)f(Studies)g(in)h(Math.)g (9,)g(1988.)157 4476 y([7])39 b(G.)28 b(Giacomin,)g(S.)f(Olla,)h(H.)f (Sp)r(ohn,)g Fh(Equilibrium)h(Fluctuation)i(for)f(a)g(Ginzbur)l(g-L)l (andau)i(Interfac)l(e)f(Mo)l(del)p Fy(,)e(to)276 4567 y(app)r(ear)e(in)g(Ann.)e(Probab.)157 4658 y([8])39 b(D.Gilbarg,)27 b(N.S.T)-6 b(rudinger,)26 b Fh(El)t(liptic)h(Partial)h(Di\013er)l (ential)f(Equations)i(of)e(Se)l(c)l(ond)i(Or)l(der)p Fy(,)e(Springer)f(1983)157 4750 y([9])39 b(V.V.)27 b(Jik)n(o)n(v,)h (S.M.)f(Kozlo)n(v,)h(O.A.)f(Oleinik,)h Fh(Homo)l(genization)i(of)f (Di\013er)l(ential)g(Op)l(er)l(ators)i(and)e(Inte)l(gr)l(al)h(F)-6 b(unc-)276 4841 y(tionals)p Fy(,)26 b(Springer)g(1994)118 4932 y([10])40 b(C.)28 b(Kipnis,)h(S.R.S.)f(V)-6 b(aradhan,)27 b Fh(Centr)l(al)k(limit)d(the)l(or)l(em)j(for)f(additive)g(functionals) g(of)f(r)l(eversible)i(Markov)f(pr)l(o-)276 5024 y(c)l(esses)f(and)f (applic)l(ations)h(to)f(simple)f(exclusions)p Fy(,)g(Comm.)e(Math.)h (Ph)n(ys.)g Fd(104)p Fy(,)h(1{19,)g(1986)118 5115 y([11])40 b(S.M.)c(Kozlo)n(v,)j Fh(The)d(metho)l(d)i(of)e(aver)l(aging)i(and)f (walks)g(in)f(inhomo)l(gene)l(ous)i(envir)l(onments)p Fy(,)i(Russian)c(Math.)276 5206 y(Surv)n(eys)24 b Fd(40:2)p Fy(,)i(73-145,)i(1985)118 5298 y([12])40 b(C.)20 b(Landim,)g(S.)f (Olla,)i(S.R.S.)e(V)-6 b(aradhan,)21 b Fh(Finite-dimensional)h(appr)l (oximation)g(of)g(the)g(self-di\013usion)g(c)l(o)l(e\016cient)276 5389 y(for)27 b(the)i(exclusion)f(pr)l(o)l(c)l(ess)p Fy(,)h(to)c(app)r(ear)h(in)g(Ann.)f(Probab.)118 5480 y([13])40 b(G.F.La)n(wler,)25 b Fh(Interse)l(ction)h(of)e(R)l(andom)g (Walks)p Fy(,)f(Probabilit)n(y)g(and)f(its)g(Applications.)h(Birkhuser) f(Boston,)h(1991.)p eop %%Page: 17 17 17 16 bop 3591 43 a Fk(17)118 366 y Fy([14])40 b(R.)33 b(K)r(\177)-41 b(unnemann,)34 b Fh(The)g(di\013usion)h(limit)e(for)h(r) l(eversible)i(jump)e(pr)l(o)l(c)l(esses)k(on)c Fb(Z)2711 335 y Fv(d)2782 366 y Fh(with)h(er)l(go)l(dic)g(r)l(andom)h(b)l(ond)276 458 y(c)l(onductivities)p Fy(,)28 b(Comm.)d(Math.)h(Ph)n(ys.)f Fd(90)p Fy(,)i(27{68,)h(1983)118 549 y([15])40 b(N.C.)21 b(Mey)n(ers,)h Fh(A)n(n)h Fw(L)898 557 y Fv(p)962 549 y Fh(estimate)h(for)e(the)i(gr)l(adient)g(of)e(solutions)i(of)e(se)l(c) l(ond)j(or)l(der)f(el)t(liptic)e(diver)l(genc)l(e)i(e)l(quations)p Fy(,)276 640 y(Ann.)h(Sc.)h(Norm.)e(Sup.)h(Pisa)i Fd(17)p Fy(,)g(189-206,)h(1963)118 732 y([16])40 b(S.)27 b(Olla,)h Fh(L)l(e)l(ctur)l(es)j(on)e(Homo)l(genization)g(of)g(Di\013usion)f(Pr)l (o)l(c)l(esses)j(in)d(R)l(andom)h(Fields)p Fy(,)e(publ.)g(Ecole)h(P)n (olytec)n(h-)276 823 y(nique,)d(1994)118 914 y([17])40 b(H.)25 b(Owhadi,)i(priv)l(ate)e(comm)n(unication.)118 1006 y([18])40 b(G.)23 b(P)n(apanicolaou,)i(S.R.S.)d(V)-6 b(aradhan,)23 b Fh(Di\013usions)i(with)f(r)l(andom)i(c)l(o)l (e\016cients)p Fy(,)f(in:)33 b(Statistics)23 b(and)f(probabilit)n(y:) 276 1097 y(essa)n(ys)k(in)g(honor)g(of)g(C.)h(R.)e(Rao,)h(pp.)g (547{552,)i(North-Holland,)e(Amsterdam,)e(1982)118 1188 y([19])40 b(M.)26 b(T)-6 b(alagrand,)28 b Fh(A)f(new)h(lo)l(ok)g(at)g (indep)l(endenc)l(e)p Fy(,)g(Ann.)d(Probab.)h(24,)h(1{34,)g(1996)218 1360 y Fx(Dip.)36 b(Ma)-5 b(tema)g(tica)27 b(Universit)-5 b(a)27 b(di)g(R)n(oma)h(Tre,)g(L.go)g(S.)g(Murialdo)g(1,)g(00146)i(R)n (oma,)e(It)-5 b(al)g(y)218 1451 y Fh(E-mail)27 b(addr)l(ess)6 b Fy(:)36 b Fa(caputo@mat.uniroma3.it)218 1606 y Fx(F)-9 b(a)n(cul)k(ty)29 b(of)g(Industrial)f(Engineering,)e(Technion,)h(Haif) -7 b(a)28 b(3200,)h(Israel)218 1698 y Fh(E-mail)e(addr)l(ess)6 b Fy(:)36 b Fa(ieioffe@ie.technion.ac.il)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0110190534440--