Content-Type: multipart/mixed; boundary="-------------9907300717774" This is a multi-part message in MIME format. ---------------9907300717774 Content-Type: text/plain; name="99-285.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-285.comments" dunlop@ptm.u-cergy.fr, topolski@math.uni.wroc.pl ---------------9907300717774 Content-Type: text/plain; name="99-285.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-285.keywords" Ising model, Correlation inequalities, Surface tension, Disorder ---------------9907300717774 Content-Type: application/postscript; name="fk.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="fk.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.85 Copyright 1999 Radical Eye Software %%Title: fk.dvi %%Pages: 16 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o fk.ps fk.dvi %DVIPSParameters: dpi=600, 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Fm(\022)31 b(\034)1526 4398 y Fv(+)p Fk(\000)1663 4439 y Fl(\000)22 b Fo(sin)17 b Fm(\022)2008 4371 y(@)p 1984 4415 106 4 v 1984 4507 a(@)5 b(\022)2127 4439 y(\034)2180 4398 y Fv(+)p Fk(\000)2322 4439 y Fo(=)28 b Fm(\034)2479 4398 y Fk(\000)p Fj(W)2637 4439 y Fl(\000)22 b Fm(\034)2789 4398 y Fv(+)p Fj(W)2953 4439 y Fm(;)240 4664 y Fo(where)31 b Fm(\034)572 4628 y Fv(+)p Fk(\000)715 4664 y Fo(is)e(the)h(in)m(terfacial)d(tension,)j (whic)m(h)g(is)f(a)g(function)g(of)g(the)g(in)m(terface)h(orien)m (tation,)e Fm(\034)3854 4628 y Fk(\000)p Fj(W)240 4784 y Fo(is)34 b(the)h(w)m(all)f(tension)h(of)f(the)h Fl(\000)p Fo(phase)h(on)f(the)g(w)m(all-substrate)g Fm(W)14 b Fo(,)35 b(and)f Fm(\034)3039 4748 y Fv(+)p Fj(W)3210 4784 y Fo(is)g(the)i(w)m (all)d(tension)240 4905 y(of)c(the)i(+phase)f(on)g(the)g(same)g(w)m (all-substrate.)42 b(The)31 b(prop)s(erties)f(of)f(the)h(substrate)h (en)m(ter)g(the)f(w)m(all)240 5025 y(free)j(energies)g(or)f(w)m(all)f (tensions)j Fm(\034)1547 4989 y Fv(+)p Fj(W)1682 5025 y Fo(,)f Fm(\034)1795 4989 y Fk(\000)p Fj(W)1963 5025 y Fo(and)g(the)g(di\013eren)m(tial)e(w)m(all)g(tension)1641 5212 y(\001)p Fm(\034)11 b Fo(\()p Fm(W)j Fo(\))28 b(=)g Fm(\034)2142 5171 y Fk(\000)p Fj(W)2300 5212 y Fl(\000)23 b Fm(\034)2453 5171 y Fv(+)p Fj(W)360 5400 y Fo(Let)32 b(us)h(denote)f(\001)p Fm(\034)11 b Fo(\()p Fn(\022a\023)18 b Fo(\))32 b(and)g(\001)p Fm(\034)11 b Fo(\()p Fn(\022b\023)18 b Fo(\))32 b(the)g(di\013eren)m(tial)e(w)m(all)h(tensions)h(asso)s (ciated)g(to)f(pure)240 5520 y Fn(\022a\023)17 b Fo(and)33 b(pure)g Fn(\022b\023)18 b Fo(resp)s(ectiv)m(ely)-8 b(.)44 b(Cassie's)34 b(la)m(w)e([1])g(giv)m(es)h(the)g(prediction)f(that)1343 5707 y(\001)p Fm(\034)11 b Fo(\()p Fm(W)j Fo(\))28 b(=)f Fm(c)1832 5722 y Fj(a)1874 5707 y Fo(\001)p Fm(\034)11 b Fo(\()p Fn(\022a\023)18 b Fo(\))k(+)g Fm(c)2417 5722 y Fj(b)2451 5707 y Fo(\001)p Fm(\034)11 b Fo(\()p Fn(\022b\023)19 b Fo(\))27 b Fm(;)978 b Fo(\(1\))2091 5956 y(1)p eop %%Page: 2 2 2 1 bop 240 107 a Fo(where)37 b Fm(c)567 122 y Fj(a)644 107 y Fo(and)f Fm(c)879 122 y Fj(b)946 107 y Fo(=)d(1)24 b Fl(\000)h Fm(c)1272 122 y Fj(a)1349 107 y Fo(are)36 b(the)g(relativ)m(e)f(concen)m(trations)h(of)f Fn(\022a\023)53 b Fo(and)36 b Fn(\022b\023)18 b Fo(,)36 b(whatev)m(er)i(the)240 227 y(geometrical)e(arrangemen)m(t)i(of)g Fn(\022a\023)18 b Fo(with)38 b Fn(\022b\023)17 b Fo(.)61 b(This)39 b(equation)f(simply) f(states)j(that)e(the)h(di\013er-)240 348 y(en)m(tial)34 b(w)m(all)h(tension)h(for)f(a)g(comp)s(ound)h(w)m(all)e(is)h(the)i(w)m (eigh)m(ted)f(a)m(v)m(erage)h(of)e(the)i(pure)f(comp)s(onen)m(t)240 468 y(quan)m(tities.)44 b(It)33 b(w)m(as)h(prop)s(osed)g(b)m(y)f (Cassie)h(on)f(the)g(basis)g(of)f(measuremen)m(ts)i(of)e(con)m(tact)i (angles)e(of)240 588 y(w)m(ater)h(droplets)g(on)f(fabrics.)360 709 y(Equation)j(\(1\))g(has)h(b)s(een)g(v)m(eri\014ed)g(to)g(a)f(go)s (o)s(d)f(appro)m(ximation)f(b)m(y)j(Mon)m(te-Carlo)f(sim)m(ulations)240 829 y(in)c(t)m(w)m(o)i(dimensions,)e(for)h(ordered)g(or)g(disordered)g (substrates)i([2].)43 b(It)32 b(can)h(also)e(b)s(e)h(tested)h(against) 240 949 y(a)e(lo)m(w-temp)s(erature)f(expansion)i(for)f(the)h(Ising)f (mo)s(del:)41 b(for)31 b(a)g(p)s(erio)s(dic)e(substrate)k(in)d(t)m(w)m (o)i(dimen-)240 1070 y(sions,)26 b(one)f(\014nds)h(agreemen)m(t)f(up)g (to)g(order)g(exp)q(\()p Fl(\000)p Fo(6)p Fm(\014)6 b(J)j Fo(\),)26 b(but)f(disagreemen)m(t)g(at)f(order)h(exp)q(\()p Fl(\000)p Fo(8)p Fm(\014)6 b(J)j Fo(\),)240 1190 y(whic)m(h)36 b(ma)m(y)g(of)f(course)i(b)s(e)f(a)g(v)m(ery)h(small)d(correction)h(to) h(Cassie's)h(la)m(w;)g(this)e(computation,)g(with)240 1311 y(no)d(claim)f(to)h(mathematical)d(rigour,)j(is)g(giv)m(en)g(at)h (the)g(end)g(of)f(the)h(presen)m(t)h(pap)s(er.)360 1431 y(Our)f(main)d(result)j(is)f(the)h(follo)m(wing)d(conca)m(vit)m(y)j (prop)s(ert)m(y)-8 b(,)1342 1647 y(\001)p Fm(\034)11 b Fo(\()p Fm(W)j Fo(\))28 b Fl(\025)g Fm(c)1833 1662 y Fj(a)1875 1647 y Fo(\001)p Fm(\034)11 b Fo(\()p Fn(\022a\023)18 b Fo(\))k(+)g Fm(c)2418 1662 y Fj(b)2452 1647 y Fo(\001)p Fm(\034)11 b Fo(\()p Fn(\022b\023)18 b Fo(\))28 b Fm(;)977 b Fo(\(2\))240 1863 y(whic)m(h)49 b(w)m(e)g(pro)m(v)m(e)h(b)m(y)f(the)g (metho)s(d)f(of)g(correlation)e(inequalities,)51 b(for)d(p)s(erio)s (dic)f(or)h(stationary)240 1983 y(ergo)s(dic)27 b(random)h(substrates,) j(in)c(an)m(y)i(dimensions,)g(under)g(the)f(condition)f(that)h Fm(a)g Fl(\025)g Fo(0)g(and)h Fm(b)f Fl(\025)g Fo(0.)360 2104 y(Since)j(\001)p Fm(\034)43 b Fo(is)30 b(an)h(o)s(dd)f(function)h (of)f(the)h(b)s(oundary)h(\014eld,)f(inequalit)m(y)f(\(2\))g Fi(rev)m(ersed)j Fo(holds)e(when)240 2224 y Fm(a)46 b Fl(\024)h Fo(0)c(and)g Fm(b)k Fl(\024)f Fo(0.)76 b(When)44 b Fm(a)j Fl(\024)f Fo(0)g Fl(\024)h Fm(b)c Fo(and)h Fm(a)30 b Fo(+)f Fm(b)47 b Fl(\025)f Fo(0,)g(w)m(e)e(exp)s(ect)h(conca)m(vit)m (y)f(of)f(\001)p Fm(\034)55 b Fo(for)240 2344 y Fm(c)282 2359 y Fj(a)354 2344 y Fl(2)30 b Fo([0)p Fm(;)17 b Fo(1)p Fm(=)p Fo(2],)33 b(but)h(w)m(e)h(pro)m(v)m(e)g(\(2\))f(only)f(for)h(p)s (erio)s(dic)e(or)i(Bernoulli)d(distributed)j(substrates)h(with)240 2465 y Fm(c)282 2480 y Fj(a)351 2465 y Fo(=)28 b(1)p Fm(=n)p Fo(,)k Fm(n)c Fl(2)g Fh(N)t Fo(.)360 2585 y(Of)40 b(course)i(\(2\))e(and)g(\(1\))g(are)h(compatible.)65 b(Cassie's)42 b(la)m(w)e(\(1\))g(should)g(b)s(ecome)g(exact)i(in)e(the) 240 2706 y(limit)g(whereb)m(y)47 b(the)d(w)m(all)f(is)g(made)h(from)f (pure)h(patc)m(hes)i(of)d Fn(\022a\023)18 b Fo(and)44 b Fn(\022b\023)17 b Fo(whose)46 b(size)e(is)f(m)m(uc)m(h)240 2826 y(larger)31 b(than)i(the)g(bulk)f(Ising)h(correlation)d(length.) 240 3158 y Fp(2)161 b(Mo)t(del)54 b(and)g(basic)g(result)240 3377 y Fo(W)-8 b(e)39 b(consider)g(the)g Fm(d)p Fo(-dimensional)c (Ising)j(mo)s(del)f(de\014ned)j(on)e(the)h(semi-in\014nite)e(lattice)g Fh(L)45 b Fl(\032)38 b Fh(Z)3928 3341 y Fj(d)3963 3377 y Fm(;)240 3498 y Fh(L)51 b Fo(=)43 b Fh(Z)519 3461 y Fj(d)p Fk(\000)p Fv(1)672 3498 y Fl(\002)29 b Fh(Z)839 3461 y Fv(+)892 3498 y Fm(:)42 b Fo(The)h(p)s(oin)m(ts)e(of)g Fh(L)49 b Fo(are)42 b(denoted)h(b)m(y)f Fm(i)i Fo(=)f(\()p Fm(i)2651 3513 y Fv(1)2690 3498 y Fm(;)17 b(:::;)g(i)2892 3513 y Fj(d)2933 3498 y Fo(\),)44 b(where)f Fm(i)3366 3513 y Fj(k)3452 3498 y Fl(2)h Fh(Z)35 b Fo(for)41 b Fm(k)46 b Fo(=)240 3618 y(1)p Fm(;)17 b(:::;)g(d)24 b Fl(\000)i Fo(1)36 b(and)h Fm(i)948 3633 y Fj(d)1023 3618 y Fl(2)e Fh(Z)1186 3582 y Fv(+)1239 3618 y Fm(:)h Fo(F)-8 b(or)36 b(eac)m(h)i Fm(i)d Fl(2)g Fh(L)q Fo(,)44 b Fm(\033)2049 3633 y Fj(i)2112 3618 y Fo(=)35 b Fl(\006)p Fo(1)i(denotes)g(an)g (Ising)f(spin.)56 b(The)38 b(system)f(of)240 3738 y(spins)c Fm(\033)544 3702 y Fv(\003)625 3738 y Fo(=)27 b Fl(f)p Fm(\033)833 3753 y Fj(i)862 3738 y Fl(g)912 3753 y Fj(i)p Fk(2)p Fv(\003)1036 3738 y Fo(,)32 b(con\014ned)i(in)e(a)g(b)s(o)m(x)h (\003)1027 3954 y(\003)27 b(=)h Fl(f)p Fm(i)g Fl(2)g Fh(L)51 b Fo(:)28 b Fl(j)p Fm(i)1646 3969 y Fj(k)1689 3954 y Fl(j)f(\024)h Fm(L;)34 b(k)d Fo(=)c(1)p Fm(;)17 b(:::;)g(d)22 b Fl(\000)g Fo(1)p Fm(;)33 b Fo(0)28 b Fl(\024)g Fm(i)2875 3969 y Fj(d)2943 3954 y Fl(\024)g Fm(M)10 b Fl(g)240 4170 y Fo(with)32 b(b)s(oundary)h(condition)e Fm(\033)1388 4134 y Fv(\003)1437 4111 y Fg(c)1506 4170 y Fo(in)m(teracts)i(via)f(the)h(Hamiltonian)735 4425 y Fm(H)816 4440 y Fv(\003)869 4425 y Fo(\()p Fm(J)n(;)17 b(h;)g(\033)1162 4384 y Fv(\003)1211 4360 y Fg(c)1247 4425 y Fo(\)\()p Fm(\033)1382 4384 y Fv(\003)1435 4425 y Fo(\))27 b(=)h Fl(\000)1691 4358 y Fo(1)p 1691 4402 49 4 v 1691 4493 a(2)1793 4342 y Ff(X)1767 4526 y Fj(i;j)t Fk(2)p Fv(\003)1939 4425 y Fm(J)1993 4440 y Fj(ij)2053 4425 y Fm(\033)2108 4440 y Fj(i)2137 4425 y Fm(\033)2192 4440 y Fj(j)2267 4425 y Fl(\000)2440 4342 y Ff(X)2350 4526 y Fj(i)p Fk(2)p Fv(\003)p Fj(;j)t Fk(2)p Fv(\003)2619 4507 y Fg(c)2650 4425 y Fm(J)2704 4440 y Fj(ij)2765 4425 y Fm(\033)2820 4440 y Fj(i)2848 4425 y Fm(\033)2903 4440 y Fj(j)2979 4425 y Fl(\000)3124 4342 y Ff(X)3062 4526 y Fj(i)p Fk(2)p Fv(\003)p Fk(\\)p Fj(W)3272 4425 y Fm(h)3328 4440 y Fj(i)3357 4425 y Fm(\033)3412 4440 y Fj(i)3468 4425 y Fm(;)370 b Fo(\(3\))240 4733 y(where)37 b Fm(W)48 b Fo(=)34 b Fl(f)p Fm(i)g Fl(2)g Fh(Z)1053 4697 y Fj(d)1122 4733 y Fo(:)50 b Fm(i)1232 4748 y Fj(d)1307 4733 y Fo(=)34 b(0)p Fl(g)h Fo(and)i(\003)1813 4697 y Fj(c)1881 4733 y Fo(=)d Fh(L)e Fl(n)24 b Fo(\003.)54 b(The)37 b(corresp)s(onding)g (partition)d(function)h(is)240 4853 y(de\014ned)f(as)1423 4974 y Fl(Z)1495 4989 y Fv(\003)1548 4974 y Fo(\()p Fm(J)n(;)17 b(h;)g(\033)1841 4933 y Fv(\003)1890 4909 y Fg(c)1926 4974 y Fo(\))28 b(=)2095 4891 y Ff(X)2111 5082 y Fj(\033)2153 5063 y Fe(\003)2232 4974 y Fo(exp)2381 4878 y Ff(\020)2431 4974 y Fl(\000)p Fm(\014)6 b(H)2650 4989 y Fv(\003)2703 4878 y Ff(\021)2780 4974 y Fm(:)240 5226 y Fo(This)37 b(mo)s(del)f(con)m(tains)h(three)h(sets)g(of)f(parameters:)52 b Fm(J)45 b Fo(=)35 b Fl(f)p Fm(J)2579 5241 y Fj(ij)2639 5226 y Fl(g)2689 5241 y Fj(i;j)t Fk(2)p Fv(\003)2902 5226 y Fo(-)i(the)g(set)h(of)e(coupling)g(con-)240 5346 y(stan)m(ts)41 b(b)s(et)m(w)m(een)i(spins)e(in)e Fh(L)r Fm(;)46 b Fo(the)41 b(b)s(oundary)g(condition)e Fm(\033)2533 5310 y Fv(\003)2582 5286 y Fg(c)2658 5346 y Fo(and)i Fm(h)g Fo(=)g Fl(f)p Fm(h)3176 5361 y Fj(i)3204 5346 y Fl(g)3254 5361 y Fj(i)p Fk(2)p Fj(W)3445 5346 y Fo(-)f(the)h(surface) 240 5466 y(\014eld)32 b(acting)g(on)g(spins)h(in)f Fm(W)m(:)360 5587 y Fo(Our)68 b(starting)g(p)s(oin)m(t)f(is)h(an)h(inequalit)m(y)e (for)h(pro)s(ducts)h(of)f(four)g(partition)f(functions)240 5707 y Fl(Z)312 5722 y Fv(\003)365 5707 y Fo(\()p Fm(J)n(;)17 b(h)555 5671 y Fv(\()p Fj(\013)p Fv(\))660 5707 y Fm(;)g Fo(\()p Fm(\033)801 5671 y Fv(\()p Fj(\013)p Fv(\))905 5707 y Fo(\))943 5671 y Fv(\003)992 5648 y Fg(c)1028 5707 y Fo(\))p Fm(;)38 b(\013)g Fo(=)f(1)p Fm(;)17 b Fo(2)p Fm(;)g Fo(3)p Fm(;)g Fo(4)p Fm(;)37 b Fo(asso)s(ciated)h(with)g (Hamiltonians)d Fm(H)3120 5722 y Fv(\003)3173 5707 y Fo(\()p Fm(J)n(;)17 b(h)3363 5671 y Fv(\()p Fj(\013)p Fv(\))3467 5707 y Fm(;)g Fo(\()p Fm(\033)3608 5671 y Fv(\()p Fj(\013)p Fv(\))3712 5707 y Fo(\))3750 5671 y Fv(\003)3799 5648 y Fg(c)3835 5707 y Fo(\))38 b(of)2091 5956 y(2)p eop %%Page: 3 3 3 2 bop 240 107 a Fo(ferromagnetic)31 b(t)m(yp)s(e,)360 348 y(\(0\))h(for)g(all)f Fm(i;)17 b(j)33 b Fl(2)28 b Fh(L)1844 588 y Fm(J)1898 603 y Fj(ij)1986 588 y Fo(=)g Fm(J)2144 603 y Fj(j)t(i)2232 588 y Fl(\025)g Fo(0)240 763 y(with)k(surface)i(\014elds)e Fl(f)p Fm(h)1146 727 y Fv(\()p Fj(\013)p Fv(\))1251 763 y Fl(g)g Fo(and)h(b)s(oundary)g (conditions)e(\()p Fm(\033)2525 727 y Fv(\()p Fj(\013)p Fv(\))2630 763 y Fo(\))2668 727 y Fv(\003)2717 703 y Fg(c)2785 763 y Fo(satisfying)360 1027 y(\(H\))i(for)f(all)e Fm(i)e Fl(2)g Fm(W)1515 1060 y Ff(8)1515 1135 y(<)1515 1284 y(:)1630 1162 y Fm(h)1686 1111 y Fv(\(1\))1686 1185 y Fj(i)1803 1162 y Fo(+)22 b Fm(h)1957 1111 y Fv(\(2\))1957 1185 y Fj(i)2079 1162 y Fl(\025)28 b(j)p Fm(h)2268 1111 y Fv(\(3\))2268 1185 y Fj(i)2384 1162 y Fo(+)22 b Fm(h)2538 1111 y Fv(\(4\))2538 1185 y Fj(i)2633 1162 y Fl(j)1630 1322 y Fm(h)1686 1271 y Fv(\(4\))1686 1345 y Fj(i)1803 1322 y Fl(\000)g Fm(h)1958 1271 y Fv(\(3\))1958 1345 y Fj(i)2080 1322 y Fl(\025)29 b(j)p Fm(h)2270 1271 y Fv(\(1\))2270 1345 y Fj(i)2386 1322 y Fl(\000)23 b Fm(h)2542 1271 y Fv(\(2\))2542 1345 y Fj(i)2636 1322 y Fl(j)360 1496 y Fo(\(B\))32 b(for)g(all)f Fm(i)d Fl(2)g Fo(\003)1045 1460 y Fj(c)1509 1529 y Ff(8)1509 1604 y(<)1509 1754 y(:)1624 1631 y Fm(\033)1683 1580 y Fv(\(1\))1679 1654 y Fj(i)1800 1631 y Fo(+)22 b Fm(\033)1957 1580 y Fv(\(2\))1953 1654 y Fj(i)2079 1631 y Fl(\025)28 b(j)p Fm(\033)2271 1580 y Fv(\(3\))2267 1654 y Fj(i)2387 1631 y Fo(+)22 b Fm(\033)2544 1580 y Fv(\(4\))2540 1654 y Fj(i)2639 1631 y Fl(j)1624 1791 y Fm(\033)1683 1740 y Fv(\(4\))1679 1814 y Fj(i)1800 1791 y Fl(\000)g Fm(\033)1958 1740 y Fv(\(3\))1954 1814 y Fj(i)2080 1791 y Fl(\025)29 b(j)p Fm(\033)2273 1740 y Fv(\(1\))2269 1814 y Fj(i)2389 1791 y Fl(\000)22 b Fm(\033)2547 1740 y Fv(\(2\))2543 1814 y Fj(i)2642 1791 y Fl(j)240 1990 y Fo(Examples)33 b(of)f(surface)h (\014elds)g(satisfying)f(\(H\))g(are)883 2224 y Fm(h)939 2173 y Fv(\(1\))939 2247 y Fj(i)1061 2224 y Fl(\025)c Fo(0)17 b Fm(;)212 b(h)1527 2173 y Fv(\(2\))1527 2247 y Fj(i)1649 2224 y Fl(\025)28 b Fo(0)17 b Fm(;)211 b(h)2114 2173 y Fv(\(3\))2114 2247 y Fj(i)2236 2224 y Fo(=)28 b(0)17 b Fm(;)211 b(h)2700 2173 y Fv(\(4\))2700 2247 y Fj(i)2822 2224 y Fo(=)28 b Fm(h)2982 2173 y Fv(\(1\))2982 2247 y Fj(i)3098 2224 y Fo(+)22 b Fm(h)3252 2173 y Fv(\(2\))3252 2247 y Fj(i)240 2444 y Fo(or)466 2664 y Fm(h)522 2613 y Fv(\(1\))522 2687 y Fj(i)617 2664 y Fm(;)49 b(h)749 2613 y Fv(\(2\))749 2687 y Fj(i)941 2664 y Fo(suc)m(h)34 b(that)97 b Fm(h)1493 2613 y Fv(\(1\))1493 2687 y Fj(i)1610 2664 y Fo(+)22 b Fm(h)1764 2613 y Fv(\(2\))1764 2687 y Fj(i)1886 2664 y Fl(\025)28 b Fo(0)p Fm(;)114 b(h)2237 2613 y Fv(\(3\))2237 2687 y Fj(i)2359 2664 y Fo(=)28 b Fm(h)2519 2613 y Fv(\(1\))2519 2687 y Fj(i)2636 2664 y Fl(^)22 b Fm(h)2780 2613 y Fv(\(2\))2780 2687 y Fj(i)2875 2664 y Fm(;)114 b(h)3072 2613 y Fv(\(4\))3072 2687 y Fj(i)3194 2664 y Fo(=)27 b Fm(h)3353 2613 y Fv(\(1\))3353 2687 y Fj(i)3470 2664 y Fl(_)c Fm(h)3615 2613 y Fv(\(2\))3615 2687 y Fj(i)3737 2664 y Fm(;)240 2884 y Fo(where)34 b Fl(^)f Fo(is)f(min)f(and)i Fl(_)g Fo(is)f(max)o(.)44 b(A)32 b(sligh)m(t)g(generalization)e(of)i(the)h(previous)g(t)m(w)m(o)g (examples)g(is)1298 3104 y Fm(h)1354 3053 y Fv(\(1\))1354 3127 y Fj(i)1449 3104 y Fm(;)49 b(h)1581 3053 y Fv(\(2\))1581 3127 y Fj(i)1773 3104 y Fo(suc)m(h)34 b(that)130 b Fm(h)2358 3053 y Fv(\(1\))2358 3127 y Fj(i)2475 3104 y Fo(+)22 b Fm(h)2629 3053 y Fv(\(2\))2629 3127 y Fj(i)2751 3104 y Fl(\025)28 b Fo(0)p Fm(;)1059 3403 y(h)1115 3352 y Fv(\(3\))1115 3426 y Fj(i)1238 3403 y Fo(=)1351 3335 y Fm(h)1407 3284 y Fv(\(1\))1407 3358 y Fj(i)1524 3335 y Fo(+)22 b Fm(h)1678 3284 y Fv(\(2\))1678 3358 y Fj(i)p 1351 3379 422 4 v 1537 3471 a Fo(2)1821 3403 y Fl(\000)h Fm(\017)1960 3418 y Fj(i)2005 3403 y Fm(;)212 b(h)2300 3352 y Fv(\(4\))2300 3426 y Fj(i)2422 3403 y Fo(=)2535 3335 y Fm(h)2591 3284 y Fv(\(1\))2591 3358 y Fj(i)2708 3335 y Fo(+)22 b Fm(h)2862 3284 y Fv(\(2\))2862 3358 y Fj(i)p 2535 3379 V 2721 3471 a Fo(2)3005 3403 y(+)g Fm(\017)3142 3418 y Fj(i)240 3602 y Fo(with)1766 3779 y Fm(\017)1805 3794 y Fj(i)1861 3779 y Fl(\025)1966 3679 y Ff(\014)1966 3729 y(\014)1966 3779 y(\014)2004 3711 y Fm(h)2060 3661 y Fv(\(1\))2060 3734 y Fj(i)2176 3711 y Fl(\000)h Fm(h)2332 3661 y Fv(\(2\))2332 3734 y Fj(i)p 2004 3756 423 4 v 2191 3847 a Fo(2)2437 3679 y Ff(\014)2437 3729 y(\014)2437 3779 y(\014)240 3984 y Fo(An)33 b(example)f(of)g(b)s (oundary)h(conditions)f(satisfying)f(\(B\))i(is)1379 4218 y Fm(\033)1438 4167 y Fv(\(1\))1434 4241 y Fj(i)1560 4218 y Fo(=)28 b Fm(\033)1723 4167 y Fv(\(2\))1719 4241 y Fj(i)1845 4218 y Fo(=)g(1)17 b Fm(;)113 b(\033)2214 4167 y Fv(\(3\))2210 4241 y Fj(i)2336 4218 y Fo(=)28 b Fm(\033)2499 4167 y Fv(\(4\))2495 4241 y Fj(i)2621 4218 y Fo(=)f Fl(\000)p Fo(1)240 4488 y Fd(Lemma)37 b(1)p Fo(.)44 b Fc(If)34 b(assumptions)g(\(0\),)g(\(H\))h(and)f(\(B\))h(ar)-5 b(e)34 b(ful\014l)5 b(le)-5 b(d)34 b(then)1055 4667 y Fv(4)1019 4692 y Ff(Y)1005 4873 y Fj(\013)p Fv(=1)1157 4775 y Fl(Z)1229 4790 y Fv(\003)1282 4775 y Fo(\()p Fm(J)n(;)17 b(h)1472 4734 y Fv(\()p Fj(\013)p Fv(\))1576 4775 y Fm(;)g Fo(\()p Fm(\033)1717 4734 y Fv(\()p Fj(\013)p Fv(\))1821 4775 y Fo(\))1859 4734 y Fv(\003)1908 4711 y Fg(c)1945 4775 y Fo(\))27 b Fl(\025)2165 4667 y Fv(4)2130 4692 y Ff(Y)2115 4873 y Fj(\013)p Fv(=1)2267 4775 y Fl(Z)2339 4790 y Fv(\003)2393 4775 y Fo(\()p Fm(J)n(;)17 b Fl(\000)p Fm(h)2660 4734 y Fv(\()p Fj(\013)p Fv(\))2765 4775 y Fm(;)g Fo(\()p Fm(\033)2906 4734 y Fv(\()p Fj(\013)p Fv(\))3010 4775 y Fo(\))3048 4734 y Fv(\003)3097 4711 y Fg(c)3133 4775 y Fo(\))27 b Fm(:)240 5118 y Fd(Pro)s(of)p Fo(.)41 b(The)28 b(pro)s(of)d(is)h(similar)d(to)k(the)f(pro)s(of)g(of)g (the)h(GHS)f(and)h(Leb)s(o)m(witz)f(inequalities)f([3,)h(4,)h(5,)f(6].) 240 5239 y(W)-8 b(e)33 b(sho)m(w)h(that)1676 5298 y Fv(4)1641 5323 y Ff(Y)1626 5504 y Fj(\013)p Fv(=1)1778 5406 y Fl(Z)1850 5421 y Fv(\003)1903 5406 y Fo(\()p Fm(J)n(;)17 b(h)2093 5365 y Fv(\()p Fj(\013)p Fv(\))2198 5406 y Fm(;)g Fo(\()p Fm(\033)2339 5365 y Fv(\()p Fj(\013)p Fv(\))2443 5406 y Fo(\))2481 5365 y Fv(\003)2530 5342 y Fg(c)2566 5406 y Fo(\))2091 5956 y(3)p eop %%Page: 4 4 4 3 bop 240 108 a Fo(has)35 b(a)f(p)s(ositiv)m(e)g(expansion)i(in)e Fm(J)1488 123 y Fj(ij)1583 108 y Fo(with)g Fm(i;)17 b(j)37 b Fl(2)32 b Fo(\003,)j Fm(h)2245 57 y Fv(\(1\))2245 131 y Fj(i)2363 108 y Fo(+)23 b Fm(h)2518 57 y Fv(\(2\))2518 131 y Fj(i)2636 108 y Fl(\006)h Fo(\()p Fm(h)2831 57 y Fv(\(3\))2831 131 y Fj(i)2949 108 y Fo(+)g Fm(h)3105 57 y Fv(\(4\))3105 131 y Fj(i)3199 108 y Fo(\))35 b(and)f Fm(h)3519 57 y Fv(\(4\))3519 131 y Fj(i)3637 108 y Fl(\000)24 b Fm(h)3794 57 y Fv(\(3\))3794 131 y Fj(i)3913 108 y Fl(\006)240 245 y Fo(\()p Fm(h)334 194 y Fv(\(1\))334 268 y Fj(i)455 245 y Fl(\000)j Fm(h)615 194 y Fv(\(2\))615 268 y Fj(i)710 245 y Fo(\))39 b(with)f Fm(i)i Fl(2)f Fm(W)14 b Fo(,)40 b Fm(\033)1425 194 y Fv(\(1\))1421 268 y Fj(i)1546 245 y Fo(+)27 b Fm(\033)1708 194 y Fv(\(2\))1704 268 y Fj(i)1829 245 y Fl(\006)g Fo(\()p Fm(\033)2030 194 y Fv(\(3\))2026 268 y Fj(i)2151 245 y Fo(+)f Fm(\033)2312 194 y Fv(\(4\))2308 268 y Fj(i)2407 245 y Fo(\))39 b(and)g Fm(\033)2739 194 y Fv(\(4\))2735 268 y Fj(i)2860 245 y Fl(\000)27 b Fm(\033)3023 194 y Fv(\(3\))3019 268 y Fj(i)3144 245 y Fl(\006)g Fo(\()p Fm(\033)3345 194 y Fv(\(1\))3341 268 y Fj(i)3466 245 y Fl(\000)g Fm(\033)3629 194 y Fv(\(2\))3625 268 y Fj(i)3723 245 y Fo(\))39 b(with)240 365 y Fm(i)28 b Fl(2)g Fo(\003)463 329 y Fj(c)497 365 y Fo(.)44 b(It)33 b(is)f(therefore)h(larger)e(or)h(equal)h(than)1610 544 y Fv(4)1575 569 y Ff(Y)1560 750 y Fj(\013)p Fv(=1)1712 652 y Fl(Z)1784 667 y Fv(\003)1837 652 y Fo(\()p Fm(J)n(;)17 b Fl(\000)p Fm(h)2104 610 y Fv(\()p Fj(\013)p Fv(\))2209 652 y Fm(;)g Fo(\()p Fm(\033)2350 610 y Fv(\()p Fj(\013)p Fv(\))2454 652 y Fo(\))2492 610 y Fv(\003)2541 587 y Fg(c)2577 652 y Fo(\))28 b Fm(;)240 952 y Fo(whic)m(h)i(pro)m(v)m(es)i (the)e(Lemma.)42 b(F)-8 b(or)29 b(this)g(w)m(e)i(consider)2243 885 y Ff(Q)2322 912 y Fv(4)2322 977 y Fj(\013)p Fv(=1)2478 952 y Fl(Z)2550 967 y Fv(\003)2603 952 y Fo(\()p Fm(J)n(;)17 b(h)2793 916 y Fv(\()p Fj(\013)p Fv(\))2898 952 y Fm(;)g Fo(\()p Fm(\033)3039 916 y Fv(\()p Fj(\013)p Fv(\))3143 952 y Fo(\))3181 916 y Fv(\003)3230 892 y Fg(c)3266 952 y Fo(\))30 b(as)g(the)g(partition)240 1086 y(function)i(asso)s(ciated)h (with)f(a)g(collection)f(of)h(spins)h(\()p Fm(\033)2277 1035 y Fv(\(1\))2273 1109 y Fj(i)2371 1086 y Fm(;)17 b(\033)2474 1035 y Fv(\(2\))2470 1109 y Fj(i)2568 1086 y Fm(;)g(\033)2671 1035 y Fv(\(3\))2667 1109 y Fj(i)2765 1086 y Fm(;)g(\033)2868 1035 y Fv(\(4\))2864 1109 y Fj(i)2962 1086 y Fo(\))p Fm(;)33 b Fo(with)f(the)h(Hamiltonian)890 1378 y Fm(H)8 b Fo(\()p Fm(\033)1076 1337 y Fv(\(1\))1170 1378 y Fm(;)17 b(\033)1273 1337 y Fv(\(2\))1367 1378 y Fm(;)g(\033)1470 1337 y Fv(\(3\))1564 1378 y Fm(;)g(\033)1667 1337 y Fv(\(4\))1761 1378 y Fo(\))28 b(=)1980 1270 y Fv(4)1938 1295 y Ff(X)1930 1476 y Fj(\013)p Fv(=1)2082 1378 y Fm(H)2163 1393 y Fv(\003)2216 1378 y Fo(\()p Fm(J)n(;)17 b(h)2406 1337 y Fv(\()p Fj(\013)p Fv(\))2511 1378 y Fm(;)g Fo(\()p Fm(\033)2652 1337 y Fv(\()p Fj(\013)p Fv(\))2756 1378 y Fo(\))2794 1337 y Fv(\003)2843 1313 y Fg(c)2879 1378 y Fo(\)\(\()p Fm(\033)3052 1337 y Fv(\()p Fj(\013)p Fv(\))3156 1378 y Fo(\))3194 1337 y Fv(\003)3247 1378 y Fo(\))28 b Fm(:)240 1765 y Fo(W)-8 b(e)33 b(then)g(de\014ne)h(the)f (transformed)f(v)-5 b(ariables)240 1933 y Fm(\021)288 1948 y Fj(i)344 1933 y Fo(=)457 1894 y Fv(1)p 457 1910 36 4 v 457 1967 a(2)503 1933 y Fo(\()p Fm(\033)600 1882 y Fv(\(1\))596 1956 y Fj(i)716 1933 y Fo(+)22 b Fm(\033)873 1882 y Fv(\(2\))869 1956 y Fj(i)990 1933 y Fl(\000)g Fm(\033)1148 1882 y Fv(\(3\))1144 1956 y Fj(i)1265 1933 y Fl(\000)g Fm(\033)1423 1882 y Fv(\(4\))1419 1956 y Fj(i)1518 1933 y Fo(\))27 b Fm(;)261 b(\014)1926 1948 y Fj(i)1981 1933 y Fo(=)2095 1894 y Fv(1)p 2095 1910 V 2095 1967 a(2)2140 1933 y Fo(\()p Fm(\033)2237 1882 y Fv(\(1\))2233 1956 y Fj(i)2354 1933 y Fo(+)22 b Fm(\033)2511 1882 y Fv(\(2\))2507 1956 y Fj(i)2627 1933 y Fo(+)g Fm(\033)2784 1882 y Fv(\(3\))2780 1956 y Fj(i)2901 1933 y Fo(+)g Fm(\033)3058 1882 y Fv(\(4\))3054 1956 y Fj(i)3152 1933 y Fo(\))28 b Fm(;)240 2147 y(\015)291 2162 y Fj(i)346 2147 y Fo(=)460 2108 y Fv(1)p 460 2124 V 460 2182 a(2)505 2147 y Fo(\()p Fm(\033)602 2096 y Fv(\(1\))598 2170 y Fj(i)719 2147 y Fl(\000)22 b Fm(\033)877 2096 y Fv(\(2\))873 2170 y Fj(i)994 2147 y Fo(+)g Fm(\033)1151 2096 y Fv(\(4\))1147 2170 y Fj(i)1267 2147 y Fl(\000)h Fm(\033)1426 2096 y Fv(\(3\))1422 2170 y Fj(i)1520 2147 y Fo(\))28 b Fm(;)260 b(\016)1916 2162 y Fj(i)1972 2147 y Fo(=)2086 2108 y Fv(1)p 2086 2124 V 2086 2182 a(2)2131 2147 y Fo(\()p Fl(\000)p Fm(\033)2305 2096 y Fv(\(1\))2301 2170 y Fj(i)2422 2147 y Fo(+)22 b Fm(\033)2579 2096 y Fv(\(2\))2575 2170 y Fj(i)2695 2147 y Fo(+)g Fm(\033)2852 2096 y Fv(\(4\))2848 2170 y Fj(i)2969 2147 y Fl(\000)g Fm(\033)3127 2096 y Fv(\(3\))3123 2170 y Fj(i)3222 2147 y Fo(\))27 b Fm(:)240 2315 y Fo(This)33 b(transformation)d(of)i(v)-5 b(ariables)32 b(is)g(orthogonal)e(so)830 2675 y Fl(\000)p Fm(H)8 b Fo(\()p Fm(\033)1093 2634 y Fv(\(1\))1188 2675 y Fm(;)17 b(\033)1291 2634 y Fv(\(2\))1385 2675 y Fm(;)g(\033)1488 2634 y Fv(\(3\))1582 2675 y Fm(;)g(\033)1685 2634 y Fv(\(4\))1779 2675 y Fo(\))27 b(=)1990 2607 y(1)p 1990 2651 49 4 v 1990 2743 a(2)2119 2592 y Ff(X)2093 2775 y Fj(i;j)t Fk(2)p Fv(\003)2282 2675 y Fm(J)2336 2690 y Fj(ij)2396 2675 y Fo(\()p Fm(\021)2482 2690 y Fj(i)2511 2675 y Fm(\021)2559 2690 y Fj(j)2618 2675 y Fo(+)22 b Fm(\014)2771 2690 y Fj(i)2799 2675 y Fm(\014)2854 2690 y Fj(j)2912 2675 y Fo(+)h Fm(\015)3062 2690 y Fj(i)3089 2675 y Fm(\015)3140 2690 y Fj(j)3199 2675 y Fo(+)f Fm(\016)3340 2690 y Fj(i)3368 2675 y Fm(\016)3411 2690 y Fj(j)3448 2675 y Fo(\))1873 2921 y(+)2072 2838 y Ff(X)1982 3022 y Fj(i)p Fk(2)p Fv(\003)p Fj(;j)t Fk(2)p Fv(\003)2251 3003 y Fg(c)2282 2921 y Fm(J)2336 2936 y Fj(ij)2396 2921 y Fo(\()p Fm(\021)2482 2936 y Fj(i)2511 2921 y Fm(\021)2559 2936 y Fj(j)2618 2921 y Fo(+)g Fm(\014)2771 2936 y Fj(i)2799 2921 y Fm(\014)2854 2936 y Fj(j)2912 2921 y Fo(+)h Fm(\015)3062 2936 y Fj(i)3089 2921 y Fm(\015)3140 2936 y Fj(j)3199 2921 y Fo(+)f Fm(\016)3340 2936 y Fj(i)3368 2921 y Fm(\016)3411 2936 y Fj(j)3448 2921 y Fo(\))578 3208 y(+)697 3141 y(1)p 697 3185 V 697 3276 a(2)861 3125 y Ff(X)800 3309 y Fj(i)p Fk(2)p Fv(\003)p Fk(\\)p Fj(W)1043 3112 y Ff(n)1126 3208 y Fo(\()p Fm(h)1220 3157 y Fv(\(1\))1220 3231 y Fj(i)1337 3208 y Fo(+)g Fm(h)1491 3157 y Fv(\(2\))1491 3231 y Fj(i)1607 3208 y Fl(\000)h Fm(h)1763 3157 y Fv(\(3\))1763 3231 y Fj(i)1880 3208 y Fl(\000)f Fm(h)2035 3157 y Fv(\(4\))2035 3231 y Fj(i)2130 3208 y Fo(\))p Fm(\021)2216 3223 y Fj(i)2266 3208 y Fo(+)g(\()p Fm(h)2458 3157 y Fv(\(1\))2458 3231 y Fj(i)2575 3208 y Fo(+)g Fm(h)2729 3157 y Fv(\(2\))2729 3231 y Fj(i)2845 3208 y Fo(+)g Fm(h)2999 3157 y Fv(\(3\))2999 3231 y Fj(i)3116 3208 y Fo(+)g Fm(h)3270 3157 y Fv(\(4\))3270 3231 y Fj(i)3364 3208 y Fo(\))p Fm(\014)3457 3223 y Fj(i)898 3455 y Fo(+\()p Fm(h)1068 3404 y Fv(\(1\))1068 3478 y Fj(i)1184 3455 y Fl(\000)h Fm(h)1340 3404 y Fv(\(2\))1340 3478 y Fj(i)1456 3455 y Fo(+)f Fm(h)1610 3404 y Fv(\(4\))1610 3478 y Fj(i)1727 3455 y Fl(\000)h Fm(h)1883 3404 y Fv(\(3\))1883 3478 y Fj(i)1977 3455 y Fo(\))p Fm(\015)2066 3470 y Fj(i)2116 3455 y Fo(+)f(\()p Fl(\000)p Fm(h)2385 3404 y Fv(\(1\))2385 3478 y Fj(i)2502 3455 y Fo(+)g Fm(h)2656 3404 y Fv(\(2\))2656 3478 y Fj(i)2773 3455 y Fo(+)g Fm(h)2927 3404 y Fv(\(4\))2927 3478 y Fj(i)3043 3455 y Fl(\000)h Fm(h)3199 3404 y Fv(\(3\))3199 3478 y Fj(i)3293 3455 y Fo(\))p Fm(\016)3374 3470 y Fj(i)3430 3359 y Ff(o)3865 3455 y Fo(\(4\))240 3675 y(This)36 b(is)g(a)g(p)s (olynomial)c(with)k(non-negativ)m(e)g(co)s(e\016cien)m(ts.)55 b(By)37 b(expanding)f(the)g(exp)s(onen)m(tial)g(and)240 3796 y(factoring)28 b(the)i(resulting)f(sums)h(o)m(v)m(er)h(the)f(v)m (ertices)h Fm(i)d Fl(2)g Fo(\003,)i(the)g(pro)s(of)f(is)h(reduced)h(to) e(sho)m(wing)h(that)240 3916 y(for)i(eac)m(h)i(v)m(ertex)g Fm(i)f Fo(and)f(all)f(nonnegativ)m(e)i(in)m(tegers)g Fm(k)s(;)17 b(l)r(;)g(m)32 b Fo(and)h Fm(n)p Fo(,)1730 4053 y Ff(X)1636 4237 y Fj(\021)1671 4247 y Fg(i)1698 4237 y Fj(;\014)1758 4247 y Fg(i)1784 4237 y Fj(;\015)1840 4247 y Fg(i)1866 4237 y Fj(;\016)1917 4247 y Fg(i)1960 4136 y Fm(\021)2012 4095 y Fj(k)2008 4161 y(i)2054 4136 y Fm(\014)2115 4095 y Fj(l)2109 4161 y(i)2141 4136 y Fm(\015)2197 4095 y Fj(m)2192 4161 y(i)2263 4136 y Fm(\016)2310 4095 y Fj(n)2306 4161 y(i)2385 4136 y Fl(\025)28 b Fo(0)g Fm(:)1271 b Fo(\(5\))240 4441 y(By)32 b(the)g(spin)f(\015ip)g (symmetries)h(corresp)s(onding)f(to)g(p)s(erm)m(utations)g(of)g(the)h (original)c(\()p Fm(\013)q Fo(\))j(index)h(set,)240 4562 y(the)40 b(sum)f(\(5\))g(can)g(b)s(e)h(nonzero)f(only)g(if)f Fm(k)30 b Fo(+)d Fm(m)p Fo(,)41 b Fm(k)30 b Fo(+)c Fm(n)p Fo(,)41 b(and)f Fm(m)27 b Fo(+)f Fm(n)40 b Fo(are)f(ev)m(en.)65 b(By)40 b(symmetry)240 4682 y(under)35 b(global)e(spin)h(\015ip,)h(the) g(sum)f(\(5\))g(can)h(b)s(e)g(nonzero)g(only)f(if)g Fm(k)26 b Fo(+)e Fm(l)h Fo(+)f Fm(m)g Fo(+)f Fm(n)35 b Fo(is)f(ev)m(en.)51 b(When)240 4802 y Fm(k)s(;)17 b(l)r(;)g(m)33 b Fo(and)g Fm(n)g Fo(are)g(all)d(ev)m(en)35 b(the)e(sum)g(\(5\))f(is)h(clearly)e (p)s(ositiv)m(e.)44 b(When)34 b Fm(k)s(;)17 b(l)r(;)g(m)33 b Fo(and)g Fm(n)g Fo(are)f(all)f(o)s(dd)240 4923 y(w)m(e)j(ha)m(v)m(e) 1300 5043 y Fm(\021)1352 5002 y Fj(k)1348 5068 y(i)1394 5043 y Fm(\014)1455 5002 y Fj(l)1449 5068 y(i)1481 5043 y Fm(\015)1537 5002 y Fj(m)1532 5068 y(i)1604 5043 y Fm(\016)1651 5002 y Fj(n)1647 5068 y(i)1725 5043 y Fo(=)28 b Fm(\021)1881 5002 y Fj(k)r Fk(\000)p Fv(1)1877 5068 y Fj(i)2013 5043 y Fm(\014)2074 5002 y Fj(l)q Fk(\000)p Fv(1)2068 5068 y Fj(i)2190 5043 y Fm(\015)2246 5002 y Fj(m)p Fk(\000)p Fv(1)2241 5068 y Fj(i)2403 5043 y Fm(\016)2450 5002 y Fj(n)p Fk(\000)p Fv(1)2446 5068 y Fj(i)2620 5043 y Fm(\021)2668 5058 y Fj(i)2696 5043 y Fm(\014)2751 5058 y Fj(i)2780 5043 y Fm(\015)2831 5058 y Fj(i)2858 5043 y Fm(\016)2901 5058 y Fj(i)240 5218 y Fo(where)j Fm(\021)571 5176 y Fj(k)r Fk(\000)p Fv(1)567 5240 y Fj(i)704 5218 y Fm(\014)765 5176 y Fj(l)q Fk(\000)p Fv(1)759 5240 y Fj(i)880 5218 y Fm(\015)936 5176 y Fj(m)p Fk(\000)p Fv(1)931 5240 y Fj(i)1093 5218 y Fm(\016)1140 5176 y Fj(n)p Fk(\000)p Fv(1)1136 5240 y Fj(i)1307 5218 y Fo(is)f(p)s(ositiv)m(e)f(b)s(ecause)i Fm(k)20 b Fl(\000)d Fo(1)p Fm(;)g(l)h Fl(\000)g Fo(1)p Fm(;)f(m)f Fl(\000)h Fo(1)30 b(and)g Fm(n)17 b Fl(\000)g Fo(1)30 b(are)f(all)f(ev)m(en)k(and)240 5365 y Fm(\021)288 5380 y Fj(i)316 5365 y Fm(\014)371 5380 y Fj(i)400 5365 y Fm(\015)451 5380 y Fj(i)478 5365 y Fm(\016)521 5380 y Fj(i)577 5365 y Fo(=)691 5326 y Fv(1)p 691 5342 36 4 v 691 5399 a(4)736 5269 y Ff(\020)786 5365 y Fm(\033)845 5314 y Fv(\(1\))841 5388 y Fj(i)939 5365 y Fm(\033)998 5314 y Fv(\(2\))994 5388 y Fj(i)1115 5365 y Fl(\000)22 b Fm(\033)1273 5314 y Fv(\(3\))1269 5388 y Fj(i)1368 5365 y Fm(\033)1427 5314 y Fv(\(4\))1423 5388 y Fj(i)1521 5269 y Ff(\021)1570 5292 y Fv(2)1610 5365 y Fm(:)32 b Fo(Hence)i(the)f(sum)g(\(5\))f(is)g(non-negativ)m(e.)2091 5956 y(4)p eop %%Page: 5 5 5 4 bop 240 107 a Fp(3)161 b(Di\013eren)l(tial)55 b(w)l(all)g(tension) 240 326 y Fo(The)23 b(di\013eren)m(tial)d(w)m(all)g(tension)i(has)g(b)s (een)g(studied)h(mathematically)18 b(b)m(y)k(F)-8 b(r\177)-49 b(ohlic)m(h)21 b(and)g(P\014ster)i([7,)f(8])240 446 y(for)k(uniform)g (substrates.)43 b(W)-8 b(e)28 b(shall)d(recall)h(some)h(de\014nitions)f (and)h(results,)i(and)e(pro)m(v)m(e)h(some)f(com-)240 567 y(plemen)m(tary)f(results)h(for)f(random)g(substrates.)43 b(W)-8 b(e)27 b(consider)g(t)m(w)m(o)g(Hamiltonians)d Fm(H)3429 582 y Fv(\003)3481 567 y Fo(\()p Fm(J)n(;)17 b(h;)g Fo(\(+\))3867 530 y Fv(\003)3916 507 y Fg(c)3952 567 y Fo(\))240 687 y(and)35 b Fm(H)513 702 y Fv(\003)566 687 y Fo(\()p Fm(J)n(;)17 b(h;)g Fo(\()p Fl(\000)p Fo(\))953 651 y Fv(\003)1002 627 y Fg(c)1038 687 y Fo(\))34 b(whic)m(h)h(are)g (di\013eren)m(t)f(only)g(b)m(y)i(the)f(c)m(hoice)g(of)f(b)s(oundary)h (conditions.)48 b(The)240 807 y(surface)38 b(\014eld)e Fm(h)f Fo(=)f Fl(f)p Fm(h)1095 822 y Fj(i)1124 807 y Fl(g)1174 822 y Fj(i)p Fk(2)p Fj(W)1362 807 y Fo(describ)s(es)k(the)f (prop)s(erties)f(of)h(the)g(w)m(all.)54 b(If)37 b Fm(h)3115 822 y Fj(i)3180 807 y Fo(is)f(p)s(ositiv)m(e)g(the)h(w)m(all)240 928 y(adsorbs)30 b(preferen)m(tially)e(the)i(+phase)f(and)h(if)e Fm(h)2017 943 y Fj(i)2074 928 y Fo(is)h(negativ)m(e)g(the)h(w)m(all)e (adsorbs)i(preferen)m(tially)e(the)240 1048 y Fl(\000)p Fo(phase.)360 1169 y(The)34 b(di\013eren)m(tial)c(w)m(all)h(tension)i (is)f(\014rst)h(de\014ned)h(in)e(\014nite)g(v)m(olume)g(b)m(y)i(the)f (form)m(ula)825 1438 y Fm(\014)6 b Fo(\001)p Fm(\034)1009 1453 y Fv(\003)1062 1438 y Fo(\()p Fd(h)p Fo(\))28 b(=)f Fm(\014)6 b(\034)1445 1397 y Fk(\000)p Fj(W)1434 1463 y Fv(\003)1581 1438 y Fo(\()p Fd(h)p Fo(\))22 b Fl(\000)h Fm(\014)6 b(\034)1955 1397 y Fv(+)p Fj(W)1944 1463 y Fv(\003)2090 1438 y Fo(\()p Fd(h)p Fo(\))28 b(=)g Fl(\000)2592 1371 y Fo(1)p 2447 1415 340 4 v 2447 1506 a Fl(j)p Fm(W)36 b Fl(\\)22 b Fo(\003)p Fl(j)2813 1438 y Fo(ln)2921 1371 y Fl(Z)2993 1386 y Fv(\003)3046 1371 y Fo(\()p Fm(J)n(;)17 b(h;)g Fl(\000)p Fo(\))p 2921 1415 475 4 v 2922 1506 a Fl(Z)2994 1521 y Fv(\003)3047 1506 y Fo(\()p Fm(J)n(;)g(h;)g Fo(+\))3865 1438 y(\(6\))240 1721 y(where)34 b Fl(j)p Fm(W)h Fl(\\)23 b Fo(\003)p Fl(j)k Fo(=)g(\(2)p Fm(L)c Fo(+)f(1\))1353 1685 y Fj(d)p Fk(\000)p Fv(1)1516 1721 y Fo(for)32 b(b)s(o)m(xes)h(\003)g(as)f(de\014ned)i(in)e(section)h(2.) 43 b(Then)1476 1941 y Fm(\014)6 b Fo(\001)p Fm(\034)11 b Fo(\()p Fd(h)p Fo(\))28 b(=)54 b(lim)1940 2001 y Fj(L)p Fk(!1)2186 1941 y Fo(lim)2146 2001 y Fj(M)7 b Fk(!1)2379 1941 y Fm(\014)f Fo(\001)p Fm(\034)2563 1956 y Fv(\003)2616 1941 y Fo(\()p Fd(h)p Fo(\))1111 b(\(7\))240 2192 y(whenev)m(er)49 b(the)d(limit)d(exists.)85 b(Existence)48 b(of)d(this)h(limit)d(has)j (b)s(een)h(pro)m(v)m(en)h(in)d([8])h(for)g(nearest)240 2312 y(neigh)m(b)s(our)31 b(in)m(teractions,)f Fm(J)1302 2327 y Fj(ij)1390 2312 y Fo(=)e(0)j(if)e Fl(j)p Fm(i)19 b Fl(\000)g Fm(j)6 b Fl(j)27 b Fm(>)h Fo(1,)j(and)g(uniform)e Fd(h)p Fo(.)43 b(The)32 b(metho)s(d)e(also)g(applies)g(to)240 2432 y(p)s(erio)s(dic)35 b Fd(h)p Fo(.)55 b(F)-8 b(or)35 b(random)h Fd(h)p Fo(,)h(w)m(e)g(can)g(pro)m(v)m(e)h(existence)f(and)g (self-a)m(v)m(eraging)e(of)h(the)h(di\013eren)m(tial)240 2553 y(w)m(all)31 b(tension)i(only)f(under)h(the)g(h)m(yp)s(othesis)h (that)e(the)h(random)f(b)s(oundary)h(\014eld)f(is)g(p)s(ositiv)m(e:)240 2773 y Fd(Prop)s(osition)42 b(1.)60 b Fc(L)-5 b(et)41 b Fm(J)1227 2788 y Fj(ij)1327 2773 y Fc(b)-5 b(e)40 b(a)g(tr)-5 b(anslation)40 b(invariant)f(ferr)-5 b(omagnetic)39 b(\014nite)g(r)-5 b(ange)40 b(c)-5 b(oupling:)240 2893 y Fm(J)294 2908 y Fj(ij)400 2893 y Fo(=)46 b Fm(J)9 b Fo(\()p Fm(i)29 b Fl(\000)h Fm(j)6 b Fo(\))46 b Fl(\025)g Fo(0)e Fc(and,)j(for)d(some)g Fm(r)s Fc(,)i Fm(J)9 b Fo(\()p Fm(k)s Fo(\))46 b(=)g(0)e Fc(whenever)f Fl(j)p Fm(k)s Fl(j)i Fm(>)h(r)s Fc(.)73 b(L)-5 b(et)45 b Fl(f)p Fm(h)3516 2908 y Fj(i)3545 2893 y Fl(g)3595 2912 y Fj(i)p Fk(2)p Fb(Z)3716 2893 y Fg(d)p Fa(\000)p Fe(1)r Fc(,)i(b)-5 b(e)240 3013 y(a)44 b(stationary)f(er)-5 b(go)g(dic)43 b(non-ne)-5 b(gative)43 b(r)-5 b(andom)42 b(\014eld,)k(such)d(that)h Fh(E)9 b Fm(h)2881 3028 y Fv(1)2970 3013 y Fm(<)44 b Fl(1)p Fm(:)g Fc(Then)f Fo(\001)p Fm(\034)55 b Fc(is)44 b(self-)240 3134 y(aver)-5 b(aging,)34 b(namely)h(it)h(is)f(uniquely)h(de\014ne)-5 b(d)35 b(by)g(\(7\))g(and)g (is)h(almost)f(sur)-5 b(ely)35 b(e)-5 b(qual)36 b(to)f(its)h(aver)-5 b(age,)240 3254 y Fo(\001)p Fm(\034)11 b Fo(\()p Fd(h)p Fo(\))29 b(=)e Fh(E)8 b Fo(\001)q Fm(\034)52 b Fc(almost)34 b(sur)-5 b(ely.)240 3400 y Fd(Pro)s(of.)43 b Fo(De\014nition)31 b(\(6\))h(can)h(b)s(e)g(written)f(as)h([8,)g(2)o(])885 3682 y Fm(\014)6 b Fo(\001)p Fm(\034)1069 3697 y Fv(\003)1123 3682 y Fo(\()p Fd(h)p Fo(\))27 b(=)h Fl(\000)1700 3614 y Fo(1)p 1479 3659 491 4 v 1479 3750 a(\(2)p Fm(L)23 b Fo(+)f(1\))1840 3721 y Fj(d)p Fk(\000)p Fv(1)1997 3682 y Fo(ln)2095 3536 y Ff(*)2161 3682 y Fo(exp)2327 3536 y Ff( )2393 3682 y Fl(\000)p Fo(2)p Fm(\014)2658 3599 y Ff(X)2596 3783 y Fj(i)p Fk(2)p Fj(W)10 b Fk(\\)p Fv(\003)2856 3682 y Fm(h)2912 3697 y Fj(i)2941 3682 y Fm(\033)2996 3697 y Fj(i)3024 3536 y Ff(!+)3157 3799 y Fv(\003)p Fj(;h;)p Fv(+)3865 3682 y Fo(\(8\))240 3989 y(F)-8 b(or)34 b(simplicit)m(y)e(of) i(notation,)f(w)m(e)j(no)m(w)f(tak)m(e)g Fm(d)c Fo(=)g(2)j(and)h(range) f Fm(r)g Fo(=)d(1.)49 b(The)35 b(pro)s(of)f(w)m(orks)i(in)e(the)240 4110 y(same)e(w)m(a)m(y)i(in)e(higher)g(dimension)f(and)i(for)f(an)m(y) h(\014nite)g(range.)43 b(Let)33 b(for)f Fm(k)s(;)17 b(n)28 b Fl(2)g Fh(N)1376 4330 y Fo(\003)1444 4345 y Fj(k)r(;n)1576 4330 y Fo(=)g Fl(f)p Fm(i)g Fl(2)g Fh(L)51 b Fo(:)28 b Fm(k)j Fl(\024)d Fm(i)2259 4345 y Fv(1)2326 4330 y Fl(\024)h Fm(n;)k Fo(0)28 b Fl(\024)g Fm(i)2765 4345 y Fv(2)2804 4330 y Fl(g)240 4550 y Fo(F)-8 b(or)39 b(arbitrary)h (\014xed)h Fm(k)i Fo(and)d Fm(n)p Fo(,)j(setting)d(to)f(+1)h(all)e(the) j(spins)f(in)g(\003)2887 4565 y Fv(0)p Fj(;k)3011 4550 y Fl(\\)28 b Fo(\003)3173 4565 y Fj(k)r(;n)3318 4550 y Fo(is)40 b(an)g(increasing)240 4670 y(function;)27 b(the)d(observ)-5 b(able)24 b(in)g(the)g(brac)m(k)m(ets)j(in)c(\(8\))h (is)g(a)f(decreasing)i(function)f(of)f(the)i(spins)g(b)s(ecause)240 4790 y Fm(h)296 4805 y Fj(i)352 4790 y Fl(\025)j Fo(0)17 b(;)32 b(therefore)h(b)m(y)h(the)f(FK)m(G)f(inequalit)m(y)-8 b(,)656 4921 y Ff(*)723 5067 y Fo(exp)872 4946 y Ff(\022)933 5067 y Fl(\000)p Fo(2)p Fm(\014)1208 4984 y Ff(X)1136 5168 y Fj(W)10 b Fk(\\)p Fv(\003)1309 5177 y Fe(0)p Fg(;n)1417 5067 y Fm(h)1473 5082 y Fj(i)1501 5067 y Fm(\033)1556 5082 y Fj(i)1585 4946 y Ff(\023)1646 4921 y(+)1713 5199 y Fv(\003)1762 5208 y Fe(0)p Fg(;n)1854 5199 y Fj(;h;)p Fv(+)2020 5067 y Fl(\025)2125 4921 y Ff(*)2192 5067 y Fo(exp)2341 4946 y Ff(\022)2402 5067 y Fl(\000)p Fo(2)p Fm(\014)2678 4984 y Ff(X)2606 5168 y Fj(W)g Fk(\\)p Fv(\003)2779 5177 y Fe(0)p Fg(;n)2887 5067 y Fm(h)2943 5082 y Fj(i)2971 5067 y Fm(\033)3026 5082 y Fj(i)3055 4946 y Ff(\023)3116 4921 y(+)3183 5199 y Fv(\003)3232 5208 y Fe(0)p Fg(;n)3323 5199 y Fj(;h;)p Fv(+)p Fj(;\033)3521 5165 y Fe(\003)3562 5182 y(0)p Fg(;k)3649 5165 y Fa(\\)p Fe(\003)3732 5182 y Fg(k)q(;n)3832 5199 y Fv(=+1)240 5472 y Fo(=)28 b(exp)17 b(\()p Fl(\000)p Fo(2)p Fm(\014)6 b(h)790 5487 y Fj(k)833 5472 y Fo(\))22 b Fl(\001)943 5326 y Ff(*)1010 5472 y Fo(exp)1159 5350 y Ff(\022)1220 5472 y Fl(\000)p Fo(2)p Fm(\014)1532 5389 y Ff(X)1423 5572 y Fj(W)10 b Fk(\\)p Fv(\003)1596 5584 y Fe(0)p Fg(;k)q Fa(\000)p Fe(1)1778 5472 y Fm(h)1834 5487 y Fj(i)1862 5472 y Fm(\033)1917 5487 y Fj(i)1946 5350 y Ff(\023)2007 5326 y(+)2074 5606 y Fv(\003)2123 5618 y Fe(0)p Fg(;k)q Fa(\000)p Fe(1)2289 5606 y Fj(;h;)p Fv(+)2450 5472 y Fl(\001)2500 5326 y Ff(*)2567 5472 y Fo(exp)2716 5350 y Ff(\022)2777 5472 y Fl(\000)p Fo(2)p Fm(\014)3093 5389 y Ff(X)2980 5572 y Fj(W)g Fk(\\)p Fv(\003)3153 5584 y Fg(k)q Fe(+1)p Fg(;n)3341 5472 y Fm(h)3397 5487 y Fj(i)3426 5472 y Fm(\033)3481 5487 y Fj(i)3509 5350 y Ff(\023)3571 5326 y(+)3637 5606 y Fv(\003)3686 5618 y Fg(k)q Fe(+1)p Fg(;n)3859 5606 y Fj(;h;)p Fv(+)2091 5956 y Fo(5)p eop %%Page: 6 6 6 5 bop 240 107 a Fo(whic)m(h)33 b(giv)m(es)834 321 y(\()p Fm(n)22 b Fo(+)g(1\)\001)p Fm(\034)1260 336 y Fv(\003)1309 345 y Fe(0)p Fg(;n)1405 321 y Fo(\()p Fd(h)p Fo(\))28 b Fl(\024)g Fo(2)p Fm(h)1781 336 y Fj(k)1846 321 y Fo(+)22 b Fm(k)s Fo(\001)p Fm(\034)2121 336 y Fv(\003)2170 348 y Fe(0)p Fg(;k)q Fa(\000)p Fe(1)2341 321 y Fo(\()p Fd(h)p Fo(\))g(+)g(\()p Fm(n)g Fl(\000)h Fm(k)s Fo(\)\001)p Fm(\034)3032 336 y Fv(\003)3081 348 y Fg(k)q Fe(+1)p Fg(;n)3258 321 y Fo(\()p Fd(h)p Fo(\))240 535 y(or)309 749 y Fm(h)365 764 y Fk(\000)p Fv(1)482 749 y Fo(+)f(\()p Fm(n)g Fo(+)g(1\)\001)p Fm(\034)1006 764 y Fv(\003)1055 773 y Fe(0)p Fg(;n)1174 749 y Fo(+)g Fm(h)1328 764 y Fj(n)p Fv(+1)1493 749 y Fl(\024)28 b Fm(h)1654 764 y Fk(\000)p Fv(1)1771 749 y Fo(+)22 b Fm(k)s 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y Fv(0)p Fj(;n)3539 3126 y Fo(imply)e(that)240 3259 y(\001)p Fm(\034)11 b Fo(\()p Fd(h)p Fo(\))33 b(is)g(self-a)m(v)m(eraging,)e (whic)m(h)i(concludes)h(the)f(pro)s(of)e(of)i(Prop)s(osition)d(1.)240 3475 y Fd(Prop)s(osition)i(2.)42 b Fc(The)32 b(di\013er)-5 b(ential)31 b(wal)5 b(l)31 b(tension)h Fo(\001)p Fm(\034)2320 3490 y Fv(\003)2373 3475 y Fo(\()p Fd(h)p Fo(\))g Fc(is)g(an)g(incr)-5 b(e)g(asing)30 b(function)i(of)g Fd(h)p Fc(,)g(i.e.)240 3596 y Fm(h)296 3611 y Fj(i)352 3596 y Fl(\025)c Fm(h)513 3559 y Fk(0)513 3620 y Fj(i)573 3596 y Fl(8)p Fm(i)60 b Fl(\))27 b Fo(\001)p Fm(\034)971 3611 y Fv(\003)1025 3596 y Fo(\()p Fd(h)p Fo(\))h Fl(\025)g Fo(\001)p Fm(\034)1419 3611 y Fv(\003)1473 3596 y Fo(\()p Fd(h)1573 3559 y Fk(0)1596 3596 y Fo(\))p Fc(.)44 b(It)31 b(is)h(an)f(o)-5 b(dd)31 b(function)g(of)h Fd(h)p Fc(,)g Fo(\001)p Fm(\034)2962 3611 y Fv(\003)3016 3596 y Fo(\()p Fl(\000)p Fd(h)p Fo(\))c(=)g Fl(\000)p Fo(\001)p Fm(\034)3563 3611 y Fv(\003)3617 3596 y Fo(\()p Fd(h)p Fo(\))p 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5956 y(6)p eop %%Page: 7 7 7 6 bop 240 107 a Fo(where)32 b Fl(h\001i)626 122 y Fv(\003)p Fj(;h;)p Fk(\000)843 107 y Fo(is)f(the)g(exp)s(ectation)f(with)h(resp)s (ect)h(to)e(the)h(Gibbs)f(measure)h(generated)h(b)m(y)f(Hamil-)240 227 y(tonian)45 b(\(1\))h(with)g Fl(\000)g Fo(b)s(oundary)h(condition.) 83 b(The)47 b(FK)m(G)e(inequalit)m(y)g(implies)f(that)i(\(11\))f(is)h (an)240 348 y(increasing)c(function)g(of)g Fd(h)p Fo(,)j(so)e(that)g (the)g(di\013eren)m(tial)d(w)m(all)i(tension)g(\(7\))g(is)h(also)e(an)i (increasing)240 468 y(function)32 b(of)g Fd(h)p Fo(.)360 588 y(The)g(second)h(part)e(of)g(the)h(prop)s(osition)e(follo)m(ws)g (easily)h(b)m(y)h(considering)f(the)h(case)g Fm(h)3530 603 y Fj(i)3586 588 y Fo(=)c Fm(a)3741 603 y Fj(i)3800 588 y Fo(with)240 709 y(probabilit)m(y)f Fm(p)784 724 y Fj(i)842 709 y Fo(and)i Fm(h)1084 724 y Fj(i)1140 709 y Fo(=)e Fm(b)1284 724 y Fj(i)1341 709 y Fm(>)g(a)1495 724 y Fj(i)1553 709 y Fo(with)i(probabilit)m(y)e(1)15 b Fl(\000)g Fm(p)2472 724 y Fj(i)2501 709 y Fo(,)30 b(and)f(v)-5 b(arying)28 b(one)i Fm(p)3312 724 y Fj(i)3369 709 y Fo(at)f(a)g(time.) 41 b(The)240 829 y(last)32 b(part)g(follo)m(ws)f(from)h(the)h (de\014nition)e(of)h(the)h(FK)m(G)f(ordering)g(of)g(measures)i([11)o (].)240 1159 y Fp(4)161 b(Conca)l(vit)l(y)53 b(inequalities)240 1399 y Fd(Lemma)37 b(2.)44 b Fc(L)-5 b(et)35 b Fm(J)44 b Fc(and)34 b Fl(f)p Fm(h)1329 1363 y Fv(\()p Fj(\013)p Fv(\))1433 1399 y Fl(g)1483 1363 y Fv(4)1483 1424 y Fj(\013)p Fv(=1)1658 1399 y Fc(ob)-5 b(ey)35 b(\(0\))f(and)g(\(H\).)h(Then)f(for) h(any)g Fo(\003)1095 1596 y(\001)p Fm(\034)1218 1611 y Fv(\003)1272 1596 y Fo(\()p Fm(h)1366 1555 y Fv(\(1\))1460 1596 y Fo(\))22 b(+)g(\001)p Fm(\034)1741 1611 y Fv(\003)1795 1596 y Fo(\()p Fm(h)1889 1555 y Fv(\(2\))1983 1596 y Fo(\))28 b Fl(\025)g Fo(\001)p Fm(\034)2277 1611 y Fv(\003)2331 1596 y Fo(\()p Fm(h)2425 1555 y Fv(\(3\))2519 1596 y Fo(\))22 b(+)g(\001)p Fm(\034)2800 1611 y Fv(\003)2854 1596 y Fo(\()p Fm(h)2948 1555 y Fv(\(4\))3042 1596 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Fv(\003)1619 2253 y Fo(\()p Fm(J)n(;)g(h)1809 2212 y Fv(\(2\))1903 2253 y Fm(;)g Fo(+\))p Fl(Z)2133 2268 y Fv(\003)2186 2253 y Fo(\()p Fm(J)n(;)g(h)2376 2212 y Fv(\(3\))2471 2253 y Fm(;)g Fl(\000)p Fo(\))p Fl(Z)2702 2268 y Fv(\003)2755 2253 y Fo(\()p Fm(J)n(;)g(h)2945 2212 y Fv(\(4\))3039 2253 y Fm(;)g Fl(\000)p Fo(\))28 b Fl(\025)797 2450 y(Z)869 2465 y Fv(\003)922 2450 y Fo(\()p Fm(J)n(;)17 b Fl(\000)p Fm(h)1189 2409 y Fv(\(1\))1284 2450 y Fm(;)g Fo(+\))p Fl(Z)1514 2465 y Fv(\003)1567 2450 y Fo(\()p Fm(J)n(;)g Fl(\000)p Fm(h)1834 2409 y Fv(\(2\))1928 2450 y Fm(;)g Fo(+\))p Fl(Z)2158 2465 y Fv(\003)2211 2450 y Fo(\()p Fm(J)n(;)g Fl(\000)p Fm(h)2478 2409 y Fv(\(3\))2573 2450 y Fm(;)g Fl(\000)p Fo(\))p Fl(Z)2804 2465 y Fv(\003)2858 2450 y Fo(\()p Fm(J)n(;)g Fl(\000)p Fm(h)3125 2409 y Fv(\(4\))3220 2450 y Fm(;)g Fl(\000)p Fo(\))27 b Fm(:)240 2615 y Fo(By)33 b(the)g(spin)g(\015ip)f(symmetry)-8 b(,)836 2812 y Fl(Z)908 2827 y Fv(\003)961 2812 y Fo(\()p Fm(J)n(;)17 b Fl(\000)p Fm(h)1228 2771 y Fv(\(1\))1323 2812 y Fm(;)g Fo(+\))p Fl(Z)1553 2827 y Fv(\003)1606 2812 y Fo(\()p Fm(J)n(;)g Fl(\000)p Fm(h)1873 2771 y Fv(\(2\))1968 2812 y Fm(;)g Fo(+\))27 b(=)g Fl(Z)2328 2827 y Fv(\003)2382 2812 y Fo(\()p Fm(J)n(;)17 b(h)2572 2771 y Fv(\(1\))2666 2812 y Fm(;)g Fl(\000)p Fo(\))p Fl(Z)2897 2827 y Fv(\003)2951 2812 y Fo(\()p Fm(J)n(;)g(h)3141 2771 y Fv(\(2\))3235 2812 y Fm(;)g Fl(\000)p Fo(\))240 3009 y(and)836 3129 y Fl(Z)908 3144 y Fv(\003)961 3129 y Fo(\()p Fm(J)n(;)g Fl(\000)p Fm(h)1228 3088 y Fv(\(3\))1323 3129 y Fm(;)g Fl(\000)p Fo(\))p Fl(Z)1554 3144 y Fv(\003)1607 3129 y Fo(\()p Fm(J)n(;)g Fl(\000)p Fm(h)1874 3088 y Fv(\(4\))1969 3129 y Fm(;)g Fl(\000)p Fo(\))28 b(=)g Fl(Z)2332 3144 y Fv(\003)2385 3129 y Fo(\()p Fm(J)n(;)17 b(h)2575 3088 y Fv(\(3\))2669 3129 y Fm(;)g Fo(+\))p Fl(Z)2899 3144 y Fv(\003)2952 3129 y Fo(\()p Fm(J)n(;)g(h)3142 3088 y Fv(\(4\))3237 3129 y Fm(;)g Fo(+\))240 3294 y(so)882 3388 y Fl(Z)954 3403 y Fv(\003)1007 3388 y Fo(\()p Fm(J)n(;)g(h)1197 3352 y Fv(\(1\))1292 3388 y Fm(;)g Fo(+\))p 881 3432 569 4 v 881 3524 a Fl(Z)953 3539 y Fv(\003)1006 3524 y Fo(\()p Fm(J)n(;)g(h)1196 3495 y Fv(\(1\))1291 3524 y Fm(;)g Fl(\000)p Fo(\))1471 3388 y Fl(Z)1543 3403 y Fv(\003)1596 3388 y Fo(\()p Fm(J)n(;)g(h)1786 3352 y Fv(\(2\))1880 3388 y Fm(;)g Fo(+\))p 1470 3432 V 1470 3524 a Fl(Z)1542 3539 y Fv(\003)1595 3524 y Fo(\()p Fm(J)n(;)g(h)1785 3495 y Fv(\(2\))1879 3524 y Fm(;)g Fl(\000)p Fo(\))2076 3455 y Fl(\025)2192 3388 y(Z)2264 3403 y Fv(\003)2317 3388 y Fo(\()p Fm(J)n(;)g(h)2507 3352 y Fv(\(3\))2602 3388 y Fm(;)g Fo(+\))p 2191 3432 V 2191 3524 a Fl(Z)2263 3539 y Fv(\003)2317 3524 y Fo(\()p Fm(J)n(;)g(h)2507 3495 y Fv(\(3\))2601 3524 y Fm(;)g Fl(\000)p Fo(\))2781 3388 y Fl(Z)2853 3403 y Fv(\003)2906 3388 y Fo(\()p Fm(J)n(;)g(h)3096 3352 y Fv(\(4\))3191 3388 y Fm(;)g Fo(+\))p 2780 3432 V 2780 3524 a Fl(Z)2852 3539 y Fv(\003)2905 3524 y Fo(\()p Fm(J)n(;)g(h)3095 3495 y Fv(\(4\))3190 3524 y Fm(;)g Fl(\000)p Fo(\))3817 3455 y(\(12\))240 3683 y(T)-8 b(aking)32 b(logarithms)e(and)j(m)m(ultiplying)c(b)m(y)k(\(2)p Fm(L)23 b Fo(+)f(1\))2254 3647 y Fk(\000)p Fj(d)p Fv(+1)2471 3683 y Fo(yields)33 b(Lemma)e(2.)240 3890 y Fd(Prop)s(osition)i(3.)49 b([7,)36 b(8])30 b Fc(The)i(di\013er)-5 b(ential)32 b(wal)5 b(l)33 b(tension)f(of)h(a)f(homo)-5 b(gene)g(ous)32 b(substr)-5 b(ate,)33 b Fo(\001)p Fm(\034)11 b Fo(\()p Fm(h)p Fo(\))p Fc(,)240 4011 y(is)35 b(a)f(c)-5 b(onc)g(ave)34 b(function)h(of)f Fm(h)28 b Fl(2)g Fo([0)p Fm(;)17 b Fl(1)p Fo(\))p Fc(.)240 4166 y Fd(Pro)s(of.)65 b Fo(Let)41 b Fm(\017)f(>)g Fo(0,)i(then)f Fm(h)1390 4115 y Fv(\(1\))1390 4189 y Fj(i)1524 4166 y Fo(=)f Fm(h)1696 4115 y Fv(\(2\))1696 4189 y Fj(i)1831 4166 y Fo(=)g Fm(h)p Fo(,)i Fm(h)2128 4115 y Fv(\(3\))2128 4189 y Fj(i)2263 4166 y Fo(=)e Fm(h)27 b Fl(\000)h Fm(\017)p Fo(,)42 b Fm(h)2731 4115 y Fv(\(4\))2731 4189 y Fj(i)2866 4166 y Fo(=)e Fm(h)28 b Fo(+)f Fm(\017)40 b Fl(8)p Fm(i)g Fo(ob)s(ey)h(\(H\),)f(and)240 4287 y(Lemma)31 b(2)h(giv)m(es)1390 4407 y(2\001)p Fm(\034)11 b Fo(\()p Fm(h)p Fo(\))28 b Fl(\025)g Fo(\001)p Fm(\034)11 b Fo(\()p Fm(h)23 b Fl(\000)g Fm(\017)p Fo(\))g(+)f(\001)p Fm(\034)11 b Fo(\()p Fm(h)23 b Fo(+)f Fm(\017)p Fo(\))p Fm(;)240 4572 y Fo(whic)m(h)33 b(implies)d(conca)m(vit)m(y)-8 b(.)360 4735 y(Let)31 b(us)g(no)m(w)h(consider)f(the)g(simplest)f(comp)s(ound)g(w)m(all,)g (namely)g(a)h(c)m(hec)m(k)m(erb)s(oard)i(w)m(all)d(de\014ned)240 4856 y(as)360 5134 y Fm(h)416 5083 y Fv(\(1\))416 5157 y Fj(i)538 5134 y Fo(=)642 4988 y Ff(\()750 5073 y Fm(a)83 b Fo(if)32 b Fl(j)p Fm(i)p Fl(j)g Fo(ev)m(en)750 5193 y Fm(b)93 b Fo(if)32 b Fl(j)p Fm(i)p Fl(j)g Fo(o)s(dd)1531 5134 y Fm(h)1587 5083 y Fv(\(2\))1587 5157 y Fj(i)1709 5134 y Fo(=)1813 4988 y Ff(\()1921 5073 y Fm(b)93 b Fo(if)32 b Fl(j)p Fm(i)p Fl(j)g Fo(ev)m(en)1921 5193 y Fm(a)83 b Fo(if)32 b Fl(j)p Fm(i)p Fl(j)g Fo(o)s(dd)240 5390 y(where)37 b Fl(j)p Fm(i)p Fl(j)c Fo(=)756 5323 y Ff(P)844 5348 y Fj(d)p Fk(\000)p Fv(1)844 5415 y Fj(k)r Fv(=1)993 5390 y Fm(i)1026 5405 y Fj(k)1069 5390 y Fo(.)54 b(Let)36 b(us)g(assume)h Fm(a)c Fl(\024)h Fm(b)i Fo(and)g Fm(a)25 b Fo(+)f Fm(b)34 b Fl(\025)g Fo(0.)53 b(Let)36 b Fm(h)2987 5339 y Fv(\(3\))2987 5413 y Fj(i)3115 5390 y Fo(=)d Fm(a)j Fo(and)g Fm(h)3560 5339 y Fv(\(4\))3560 5413 y Fj(i)3688 5390 y Fo(=)d Fm(b)j Fl(8)p Fm(i:)240 5510 y Fo(F)-8 b(or)32 b(suc)m(h)i Fm(h)691 5474 y Fv(\(1\))785 5510 y Fm(;)f(h)901 5474 y Fv(\(2\))995 5510 y Fo(,)g Fm(h)1111 5474 y Fv(\(3\))1238 5510 y Fo(and)g Fm(h)1484 5474 y Fv(\(4\))1611 5510 y Fo(assumption)f(\(H\))g(is)g(ful\014lled,)f(and)i (Lemma)e(2)h(giv)m(es)1173 5707 y(\001)p Fm(\034)11 b Fo(\()p Fm(h)1401 5666 y Fv(\(1\))1496 5707 y Fo(\))22 b(+)g(\001)p Fm(\034)11 b Fo(\()p Fm(h)1882 5666 y Fv(\(2\))1977 5707 y Fo(\))28 b Fl(\025)g Fo(\001)p Fm(\034)11 b Fo(\()p Fn(\022a\023)18 b Fo(\))k(+)g(\001)p Fm(\034)11 b Fo(\()p Fn(\022b\023)18 b Fo(\))p Fm(;)2091 5956 y Fo(7)p eop %%Page: 8 8 8 7 bop 240 107 a Fo(or,)32 b(taking)g(in)m(to)g(acoun)m(t)h(that)f (\001)p Fm(\034)11 b Fo(\()p Fm(h)1641 71 y Fv(\(1\))1737 107 y Fo(\))27 b(=)h(\001)p Fm(\034)11 b Fo(\()p Fm(h)2134 71 y Fv(\(2\))2229 107 y Fo(\))1345 367 y(\001)p Fm(\034)g Fo(\()p Fm(h)1573 326 y Fv(\(1\))1668 367 y Fo(\))27 b Fl(\025)1849 300 y Fo(1)p 1849 344 49 4 v 1849 435 a(2)1907 367 y(\001)p Fm(\034)11 b Fo(\()p Fn(\022a\023)18 b Fo(\))k(+)2418 300 y(1)p 2418 344 V 2418 435 a(2)2477 367 y(\001)p Fm(\034)11 b Fo(\()p Fn(\022b\023)18 b Fo(\))p Fm(:)360 613 y Fo(F)-8 b(or)28 b(general)f(p)s(erio)s(dic)g(patterns,)j (the)f(picture)g(is)f(clear)f(when)j Fm(a;)17 b(b)28 b Fl(\025)g Fo(0,)h(but)g(not)f(so)h(clear)f(when)240 733 y Fm(a)g(<)f Fo(0)h Fm(<)f(b)33 b Fo(:)240 955 y Fd(Theorem)k(1)49 b Fc(L)-5 b(et)29 b Fm(a)f(<)f(b)p Fc(,)j Fm(a)8 b Fo(+)g Fm(b)29 b Fl(\025)f Fo(0)p Fc(,)h(and)f(let)h Fd(h)f Fo(=)f Fm(h)2254 970 y Fj(i)2278 979 y Fe(1)2313 970 y Fj(;:::i)2417 982 y Fg(d)p Fa(\000)p Fe(1)2563 955 y Fc(b)-5 b(e)28 b(a)g(p)-5 b(erio)g(dic)28 b(function)g(of)g Fm(i)3631 970 y Fv(1)3671 955 y Fm(;)17 b(:::i)3829 970 y Fj(d)p Fk(\000)p Fv(1)3960 955 y Fc(,)240 1076 y(taking)35 b(values)h Fm(a)g Fc(or)g Fm(b)p Fc(.)48 b(L)-5 b(et)37 b Fm(f)46 b Fc(b)-5 b(e)36 b(the)g(r)-5 b(elative)36 b(fr)-5 b(e)g(quency)35 b(of)h(o)-5 b(c)g(curr)g(enc)g(e)35 b(of)h Fm(a)p Fc(.)48 b(Assume)36 b(at)g(le)-5 b(ast)240 1196 y(one)34 b(of)h(the)g(fol)5 b(lowing)33 b(c)-5 b(onditions:)240 1316 y(\(p1\):)44 b Fm(a)28 b Fl(\025)g Fo(0)240 1437 y Fc(\(p2\):)44 b Fm(f)38 b Fo(=)28 b(1)p Fm(=)p Fo(2)34 b Fc(and)g Fo(\001)p Fm(\034)11 b Fo(\()p Fd(h)p Fo(\))29 b(=)f(\001)p Fm(\034)11 b Fo(\()p Fd(a)23 b Fo(+)f Fd(b)g Fl(\000)h Fd(h)p Fo(\))p Fc(,)35 b(wher)-5 b(e)34 b Fo(\()p Fd(a)23 b Fo(+)f Fd(b)g Fl(\000)h Fd(h)p Fo(\))2927 1452 y Fj(i)2983 1437 y Fo(=)k Fm(a)c Fo(+)f Fm(b)g Fl(\000)h Fm(h)3477 1452 y Fj(i)240 1557 y Fc(\(p3\):)69 b(ther)-5 b(e)48 b(exists)f(a)h(\014nite)f(se)-5 b(quenc)g(e)47 b(of)g(shifts)g(on)g(the)h(lattic)-5 b(e,)51 b Fm(T)2940 1572 y Fv(1)2979 1557 y Fc(,)g Fm(T)3117 1572 y Fv(2)3156 1557 y Fc(,...)83 b Fm(T)3416 1572 y Fj(n)3463 1557 y Fc(,)51 b(such)c(that)240 1677 y Fd(h)22 b Fl(^)h Fm(T)470 1692 y Fv(1)510 1677 y Fd(h)f Fl(^)h Fm(:)17 b(:)g(:)k Fl(^)i Fm(T)965 1692 y Fj(n)1012 1677 y Fd(h)28 b Fo(=)f Fd(a)p Fc(,)36 b(and)e Fl(8)p Fm(k)d Fo(=)d(1)p Fm(;)17 b(:)g(:)g(:)e(n)p Fc(,)35 b Fo(\()p Fd(h)22 b Fl(^)h Fm(T)2369 1692 y Fv(1)2409 1677 y Fd(h)f Fl(^)h Fm(:)17 b(:)g(:)k Fl(^)i Fm(T)2864 1692 y Fj(k)r Fk(\000)p Fv(1)2997 1677 y Fd(h)p Fo(\))f Fl(_)h Fm(T)3265 1692 y Fj(k)3308 1677 y Fd(h)k Fo(=)h Fd(b)240 1918 y Fc(Then)34 b(the)h(di\013er)-5 b(ential)34 b(wal)5 b(l)34 b(tension)g Fo(\001)p Fm(\034)11 b Fo(\()p Fd(h)p Fo(\))36 b Fc(ob)-5 b(eys)1289 2133 y Fo(\001)p Fm(\034)11 b Fo(\()p Fd(h)p Fo(\))29 b Fl(\025)f Fm(f)11 b Fo(\001)p Fm(\034)g Fo(\()p Fn(\022a\023)18 b Fo(\))k(+)g(\(1)g Fl(\000)g Fm(f)11 b Fo(\)\001)p Fm(\034)g Fo(\()p Fn(\022b\023)18 b Fo(\))240 2403 y Fd(Examples.)42 b Fo(An)32 b(example)f(for)g(\()p Fm(p)p Fo(2\))g(or)g(\()p Fm(p)p 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Fo(1\))p Fd(.)41 b Fo(Let)24 b Fm(h)1254 3151 y Fv(\(1\))1376 3187 y Fo(=)k Fd(h)d Fo(and)f(let)g Fm(h)1937 3151 y Fv(\(2\))2056 3187 y Fo(b)s(e)h(its)f(image)f(under)j(a)e(unit)g(shift,)i(e.g.)41 b Fm(h)3607 3136 y Fv(\(2\))3607 3210 y Fj(i)3631 3219 y Fe(1)3665 3210 y Fj(;:::i)3769 3222 y Fg(d)p Fa(\000)p Fe(1)3914 3187 y Fo(=)240 3338 y Fm(h)296 3287 y Fv(\(1\))296 3360 y Fj(i)320 3369 y Fe(1)355 3360 y Fk(\000)p Fv(1)p Fj(;:::i)549 3372 y Fg(d)p Fa(\000)p Fe(1)666 3338 y Fo(.)h(Let)29 b Fm(h)962 3287 y Fv(\(3\))962 3360 y Fj(i)1084 3338 y Fo(=)e Fm(h)1243 3287 y Fv(\(1\))1243 3360 y Fj(i)1351 3338 y Fl(^)13 b Fm(h)1486 3287 y Fv(\(2\))1486 3360 y Fj(i)1581 3338 y Fo(,)29 b Fm(h)1693 3287 y Fv(\(4\))1693 3360 y Fj(i)1816 3338 y Fo(=)e Fm(h)1975 3287 y Fv(\(1\))1975 3360 y Fj(i)2083 3338 y Fl(_)13 b Fm(h)2218 3287 y Fv(\(2\))2218 3360 y Fj(i)2313 3338 y Fo(.)42 b(Then,)31 b(from)c(Lemma)f(2)i(with)g (\001)p Fm(\034)11 b Fo(\()p Fm(h)3753 3301 y Fv(\(2\))3849 3338 y Fo(\))27 b(=)240 3473 y(\001)p Fm(\034)11 b Fo(\()p Fm(h)468 3437 y Fv(\(1\))563 3473 y Fo(\),)1365 3638 y(\001)p Fm(\034)g Fo(\()p Fm(h)1593 3597 y Fv(\(1\))1688 3638 y Fo(\))27 b Fl(\025)1868 3571 y Fo(1)p 1868 3615 V 1868 3707 a(2)1927 3638 y(\001)p Fm(\034)11 b Fo(\()p Fm(h)2155 3597 y Fv(\(3\))2250 3638 y Fo(\))22 b(+)2418 3571 y(1)p 2418 3615 V 2418 3707 a(2)2477 3638 y(\001)p Fm(\034)11 b Fo(\()p Fm(h)2705 3597 y Fv(\(4\))2800 3638 y Fo(\))p Fm(:)240 3836 y Fo(If)32 b(w)m(e)h(iterate)f(this)f(pro)s (cedure)j Fm(n)e Fo(times,)f(alw)m(a)m(ys)i(with)f(the)g(same)g(shift,) g(w)m(e)h(get)f(a)g(con)m(v)m(ex)i(com)m(bi-)240 3956 y(nation)29 b(of)g(2)698 3920 y Fj(n)775 3956 y Fo(terms.)43 b(The)31 b(surface)g(\014eld)e(of)h(one)g(of)g(these)h(terms)f(is)f (lab)s(elled)f(b)m(y)j(a)f(w)m(alk)g(of)f(length)240 4076 y Fm(n)p Fo(,)37 b(with)e(step)h(lab)s(els)e(taking)h(v)-5 b(alue)35 b(\(3\))g(or)g(\(4\).)52 b(A)m(t)36 b(eac)m(h)h(step,)g(the)f (fraction)e(of)h Fm(a)p Fo('s)i(increases)f(if)240 4197 y(the)j(lab)s(el)e(is)h(\(3\),)i(or)e(decreases)j(if)d(the)h(lab)s(el)d (is)j(\(4\),)g(b)m(y)h(a)e(non-v)-5 b(anishing)37 b(amoun)m(t,)j (unless)g(the)240 4317 y(preceding)34 b(surface)h(\014eld)f(w)m(as)h (in)m(v)-5 b(arian)m(t)32 b(under)j(the)g(shift,)f(in)f(whic)m(h)h (case)h(it)e(remains)g(in)m(v)-5 b(arian)m(t.)240 4437 y(This)37 b(random)e(w)m(alk)h(is)g(th)m(us)i(absorb)s(ed)f(b)m(y)g (the)g(t)m(w)m(o)g(b)s(oundaries,)g(after)f(an)g(a)m(v)m(erage)i(n)m (um)m(b)s(er)e(of)240 4558 y(iterations)d(of)h(the)h(order)g(of)f(at)h (most)f(the)h(square)g(of)g(the)g(corresp)s(onding)f(p)s(erio)s(d.)49 b(Applying)34 b(the)240 4678 y(pro)s(cedure)d(in)e(eac)m(h)i(direction) e(and)h(taking)g(suitable)f(limits,)e(noting)i(that)h(at)g(eac)m(h)h (step)g(w)m(e)g(had)f(a)240 4799 y(con)m(v)m(ex)35 b(com)m(bination,)30 b(w)m(e)k(get)1306 5013 y(\001)p Fm(\034)11 b Fo(\()p Fd(h)p Fo(\))28 b Fl(\025)h Fm(c)p Fo(\001)p Fm(\034)11 b Fo(\()p Fn(\022a\023)18 b Fo(\))k(+)g(\(1)g Fl(\000)g Fm(c)p 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5666 y Fn(\022b\023)2610 5707 y Fo(=)k Fm(c)2091 5956 y Fo(8)p eop %%Page: 9 9 9 8 bop 240 107 a Fo(b)s(ecause)34 b Fm(f)660 71 y Fn(\022a\023)862 107 y Fo(=)27 b(1)33 b(and)f Fm(f)1295 71 y Fn(\022b\023)1497 107 y Fo(=)c(0.)240 277 y Fd(Pro)s(of)35 b(with)g Fo(\()p Fm(p)p Fo(2\))p Fd(.)42 b Fo(Essen)m(tially)31 b(the)g(same)g(as)g(the) g(c)m(hec)m(k)m(erb)s(oard)i(example)e(giv)m(en)g(b)s(efore)g(Theo-)240 397 y(rem)h(1.)240 568 y Fd(Pro)s(of)37 b(with)f Fo(\()p Fm(p)p Fo(3\))p Fd(.)44 b Fo(Lemma)31 b(2)h(giv)m(es)1199 788 y(\001)p Fm(\034)11 b Fo(\()p Fd(h)p Fo(\))24 b(+)e(\001)p Fm(\034)11 b Fo(\()p Fm(T)1822 803 y Fv(1)1862 788 y Fd(h)p Fo(\))28 b Fl(\025)g Fo(\001)p Fm(\034)11 b Fo(\()p Fd(h)23 b Fl(^)g Fm(T)2498 803 y Fv(1)2537 788 y Fd(h)p Fo(\))g(+)f(\001)p Fm(\034)11 b Fo(\()p Fd(b)p Fo(\))930 1008 y(\001)p Fm(\034)g Fo(\()p Fd(h)23 b Fl(^)g Fm(T)1333 1023 y Fv(1)1372 1008 y Fd(h)p Fo(\))g(+)f(\001)p Fm(\034)11 b Fo(\()p Fm(T)1822 1023 y Fv(2)1862 1008 y Fd(h)p Fo(\))28 b 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y Fm(n)p 2384 1772 V 2384 1863 a(n)f Fo(+)g(1)2621 1795 y(\001)p Fm(\034)11 b Fo(\()p Fd(b)p Fo(\))240 2054 y(whic)m(h)31 b(is)f(the)h(required)g (result.)43 b(Indeed)32 b Fm(f)38 b Fo(=)28 b(1)p Fm(=)p Fo(\()p Fm(n)18 b Fo(+)g(1\))30 b(b)m(y)h(the)g(same)g(argumen)m(t)f (as)h(in)f(the)h(pro)s(of)240 2174 y(with)h(\(p1\).)360 2394 y(Let)k(us)h(no)m(w)f(turn)g(to)g(the)g(random)f(case.)55 b(Let)36 b Fl(f)p Fm(h)2287 2409 y Fj(i)2315 2394 y Fl(g)2365 2413 y Fj(i)p Fk(2)p Fb(Z)2486 2394 y Fg(d)p Fa(\000)p Fe(1)j Fo(b)s(e)d(a)g(stationary)f(ergo)s(dic)g(random)240 2514 y(\014eld,)40 b(let)e Fl(f)p Fm(Y)740 2529 y Fj(n)786 2514 y Fl(g)836 2478 y Fk(1)836 2539 y Fv(1)949 2514 y Fo(b)s(e)h(a)g(sequence)i(of)d(0{1)g(v)-5 b(alued)38 b(i.i.d.)60 b(random)38 b(v)-5 b(ariables)37 b(and)i(let)f Fl(f)p Fm(\013)3680 2529 y Fj(n)3727 2514 y Fl(g)3777 2478 y Fk(1)3777 2539 y Fv(1)3890 2514 y Fo(b)s(e)240 2635 y(a)43 b(sequence)j(of)c(i.i.d.)74 b(random)42 b(v)-5 b(ariables)42 b(taking)g(v)-5 b(alue)42 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3512 y Fj(k)r Fv(+)p Fj(\013)1969 3520 y Fg(n)2030 3390 y Ff(\021)2101 3487 y Fo(+)f(\(1)g Fl(\000)h Fm(Y)2465 3502 y Fj(n)2511 3487 y Fo(\))2549 3390 y Ff(\020)2599 3487 y Fm(X)2688 3436 y Fv(\()p Fj(n)p Fk(\000)p Fv(1\))2680 3512 y Fj(k)2902 3487 y Fl(_)f Fm(X)3079 3436 y Fv(\()p Fj(n)p Fk(\000)p Fv(1\))3071 3512 y Fj(k)r Fv(+)p Fj(\013)3210 3520 y Fg(n)3271 3390 y Ff(\021)3321 3487 y Fm(:)240 3739 y Fo(Let)33 b Fl(f)p Fd(X)550 3702 y Fv(\()p Fj(n)p Fv(\))651 3739 y Fl(g)701 3702 y Fk(1)701 3763 y Fj(n)p Fv(=1)871 3739 y Fo(b)s(e)g(the)h (sequence)h(of)e(random)f(elemen)m(ts)h(of)g(the)g(space)h Fh(R)3041 3702 y Fb(Z)3097 3679 y Fg(d)p Fa(\000)p Fe(1)3243 3739 y Fo(de\014ned)g(as)f Fd(X)3784 3702 y Fv(\()p Fj(n)p Fv(\))3914 3739 y Fo(=)240 3873 y Fl(f)p Fm(X)379 3822 y Fv(\()p Fj(n)p Fv(\))371 3898 y Fj(k)480 3873 y Fl(g)530 3892 y Fj(k)r Fk(2)p Fb(Z)666 3873 y Fg(d)p Fa(\000)p Fe(1)s Fm(:)e Fo(The)i(de\014nition)d(is)h(suc)m(h)i(that)e Fd(X)2080 3836 y Fv(\()p Fj(n)p Fv(\))2213 3873 y Fo(is)g(for)f(eac)m (h)j Fm(n)e Fo(a)g(stationary)-8 b(,)31 b(ergo)s(dic)g(random)240 4007 y(\014eld,)h(suc)m(h)i(that)f Fm(P)14 b Fl(f)p Fm(X)1126 3956 y Fv(\()p Fj(n)p Fv(\))1118 4032 y Fj(k)1254 4007 y Fo(=)28 b Fm(a)p Fl(g)g Fo(=)f Fm(p;)33 b(P)14 b Fl(f)p Fm(X)1915 3956 y Fv(\()p Fj(n)p Fv(\))1907 4032 y Fj(k)2043 4007 y Fo(=)28 b Fm(b)p Fl(g)g Fo(=)f(1)22 b Fl(\000)h Fm(p)p Fo(.)240 4127 y(Let)44 b Fd(X)f Fo(denote)h(the)g(random)e (elemen)m(t)h(of)g(the)h(space)h Fh(R)2424 4091 y Fb(Z)2480 4067 y Fg(d)p Fa(\000)p Fe(1)2636 4127 y Fo(suc)m(h)g(that)e Fm(P)14 b Fl(f)p Fd(X)45 b Fo(=)h Fd(a)p Fl(g)h Fo(=)f Fm(p)d Fo(and)240 4247 y Fm(P)14 b Fl(f)p Fd(X)27 b Fo(=)g Fd(b)p Fl(g)h Fo(=)f(1)22 b Fl(\000)h Fm(p)32 b Fo(where)i Fd(a)f Fo(and)g Fd(b)g Fo(are)f(the)h(constan)m(t)h(\014elds)f(of)f(v) -5 b(alue)32 b Fm(a)g Fo(and)h Fm(b)g Fo(resp)s(ectiv)m(ely)-8 b(.)240 4467 y Fd(Lemma)31 b(3.)42 b Fi(The)28 b(sequence)j Fl(f)p Fd(X)1490 4431 y Fv(\()p Fj(n)p Fv(\))1591 4467 y Fl(g)1641 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y(Let)h(us)g(\014rst)g(giv)m(e)g(the)g(pro)s(of)e(in)h(dimension)f Fm(d)d Fo(=)f(2.)43 b(F)-8 b(or)32 b Fm(x)2590 5323 y Fv(1)2630 5308 y Fm(;)17 b(x)2729 5323 y Fv(2)2769 5308 y Fm(;)g(:)g(:)g(:)e(;)i(x)3042 5323 y Fj(l)3096 5308 y Fl(2)28 b(f)p Fm(a;)17 b(b)p Fl(g)p Fo(,)33 b(let)875 5541 y Fm(p)924 5556 y Fj(n)971 5541 y Fo(\()p Fm(x)1064 5556 y Fv(1)1104 5541 y Fm(x)1159 5556 y Fv(2)1215 5541 y Fm(:)17 b(:)g(:)f(x)1401 5556 y Fj(l)1428 5541 y Fo(\))27 b(=)h Fm(P)14 b Fl(f)p Fm(X)1813 5490 y Fv(\()p Fj(n)p Fv(\))1805 5566 y Fj(k)r Fv(+1)1965 5541 y Fo(=)27 b Fm(x)2123 5556 y Fv(1)2163 5541 y Fm(;)17 b(X)2296 5490 y Fv(\()p Fj(n)p Fv(\))2288 5566 y Fj(k)r Fv(+2)2448 5541 y Fo(=)28 b Fm(x)2607 5556 y Fv(2)2647 5541 y Fm(;)17 b(:)g(:)g(:)e(;)i(X)2954 5490 y Fv(\()p Fj(n)p Fv(\))2946 5566 y Fj(k)r Fv(+)p Fj(l)3093 5541 y Fo(=)28 b Fm(x)3252 5556 y Fj(l)3278 5541 y Fl(g)p Fm(:)2091 5956 y Fo(9)p eop %%Page: 10 10 10 9 bop 240 107 a Fo(As)33 b(a)f(simple)f(consequence)36 b(of)c(the)h(de\014nition)f(of)g Fl(f)p Fd(X)2279 71 y Fv(\()p Fj(n)p Fv(\))2380 107 y Fl(g)g Fo(w)m(e)i(ha)m(v)m(e)1446 327 y Fm(p)1495 342 y Fj(n)p Fv(+1)1632 327 y Fo(\()p Fm(ab)p Fo(\))28 b(=)1942 288 y Fv(1)p 1942 304 36 4 v 1942 361 a(2)1987 327 y Fm(p)2036 342 y Fj(n)2083 327 y Fo(\()p Fm(aab)p Fo(\))23 b(+)2433 288 y Fv(1)p 2433 304 V 2433 361 a(2)2478 327 y Fm(p)2527 342 y Fj(n)2574 327 y Fo(\()p Fm(abb)p Fo(\))1034 b(\(13\))240 547 y(whic)m(h)33 b(giv)m(es)749 767 y Fm(p)798 782 y Fj(n)p Fv(+1)935 767 y Fo(\()p Fm(ab)p Fo(\))28 b Fl(\024)1246 728 y Fv(1)p 1246 744 V 1246 801 a(2)1291 767 y Fo([)p Fm(p)1367 782 y Fj(n)1414 767 y Fo(\()p Fm(aab)p Fo(\))23 b(+)f Fm(p)1803 782 y Fj(n)1850 767 y Fo(\()p Fm(abb)p Fo(\)])h(+)2217 728 y Fv(1)p 2217 744 V 2217 801 a(2)2263 767 y Fo([)p Fm(p)2339 782 y Fj(n)2386 767 y Fo(\()p Fm(aba)p Fo(\))g(+)f Fm(p)2775 782 y Fj(n)2821 767 y Fo(\()p Fm(bab)p Fo(\)])29 b(=)f Fm(p)3239 782 y Fj(n)3286 767 y Fo(\()p Fm(ab)p Fo(\))p Fm(:)240 987 y Fo(The)37 b(sequence)h Fl(f)p Fm(p)950 1002 y Fj(n)997 987 y Fo(\()p Fm(ab)p Fo(\))p Fl(g)1215 951 y Fk(1)1215 1011 y Fv(1)1325 987 y Fo(is)e(decreasing)g (and)g(b)s(ounded)g(so)g(the)g(limit)c(lim)3162 1002 y Fj(n)p Fk(!1)3366 987 y Fm(p)3415 1002 y Fj(n)3462 987 y Fo(\()p Fm(ab)p Fo(\))i(=)e Fm(p)p Fo(\()p Fm(ab)p Fo(\))240 1107 y(exists.)44 b(No)m(w)33 b(adding)f(side)h(b)m(y)g(side) g(the)g(equalities)1546 1327 y Fm(p)1595 1342 y Fj(n)1642 1327 y Fo(\()p Fm(ab)p Fo(\))28 b(=)g Fm(p)1991 1342 y Fj(n)2038 1327 y Fo(\()p Fm(aab)p Fo(\))23 b(+)f Fm(p)2427 1342 y Fj(n)2474 1327 y Fo(\()p Fm(bab)p Fo(\))1546 1547 y Fm(p)1595 1562 y Fj(n)1642 1547 y Fo(\()p Fm(ab)p Fo(\))28 b(=)g Fm(p)1991 1562 y Fj(n)2038 1547 y Fo(\()p Fm(abb)p Fo(\))23 b(+)f Fm(p)2417 1562 y Fj(n)2464 1547 y Fo(\()p Fm(aba)p Fo(\))240 1722 y(and)33 b(using)f(\(13\))g(w)m(e)h(get)1254 1942 y Fm(p)1303 1957 y Fj(n)1350 1942 y Fo(\()p Fm(ab)p Fo(\))28 b(=)1660 1902 y Fv(1)p 1660 1918 V 1660 1976 a(2)1705 1942 y Fo([)p Fm(p)1781 1957 y Fj(n)1828 1942 y Fo(\()p Fm(bab)p Fo(\))23 b(+)f Fm(p)2207 1957 y Fj(n)2254 1942 y Fo(\()p Fm(aba)p Fo(\)])h(+)f Fm(p)2670 1957 y Fj(n)p Fv(+1)2807 1942 y Fo(\()p Fm(ab)p Fo(\))240 2162 y(whic)m(h)33 b(as)g(a)f(consequence)k(giv)m(es)d(us)1796 2382 y(lim)1772 2431 y Fj(n)p Fk(!1)1972 2382 y Fm(p)2021 2397 y Fj(n)2068 2382 y Fo(\()p Fm(bab)p Fo(\))c(=)e(0)1778 2622 y(lim)1753 2672 y Fj(n)p Fk(!1)1954 2622 y Fm(p)2003 2637 y Fj(n)2050 2622 y Fo(\()p Fm(aba)p Fo(\))h(=)g(0)p Fm(:)240 2829 y Fo(Denote)33 b Fm(p)624 2793 y Fv(\()p Fj(k)r Fv(\))624 2853 y Fj(n)749 2829 y Fo(=)27 b Fm(p)901 2844 y Fj(n)948 2829 y Fo(\()p Fm(b)17 b(a)g(:)g(:)g(:)f(a)1044 2864 y Ff(|)p 1081 2864 51 10 v 51 w({z)p 1206 2864 V 51 w(})1150 2948 y Fj(k)1311 2829 y Fm(b)p Fo(\))33 b(and)g(observ)m(e) h(that)1796 3165 y Fm(p)1845 3115 y Fv(\()p Fj(k)r Fv(\))1845 3187 y Fj(n)p Fv(+1)2010 3165 y Fl(\025)2125 3126 y Fv(1)p 2125 3142 36 4 v 2125 3200 a(2)2170 3165 y Fm(p)2219 3124 y Fv(\()p Fj(k)r Fv(+1\))2219 3190 y Fj(n)2407 3165 y Fm(:)240 3385 y Fo(No)m(w)f(b)m(y)h(induction)d(w)m(e)j(conclude)f (that)f(for)g(all)f Fm(k)f Fl(2)f Fh(N)1723 3605 y Fo(lim)1699 3655 y Fj(n)p Fk(!1)1900 3605 y Fm(p)1949 3564 y Fv(\()p Fj(k)r Fv(\))1949 3630 y Fj(n)2074 3605 y Fo(=)e Fm(p)2226 3564 y Fv(\()p Fj(k)r Fv(\))2352 3605 y Fo(=)g(0)p Fm(;)240 3841 y Fo(whic)m(h)33 b(giv)m(es)g(us)g(that)1417 4013 y(lim)1393 4063 y Fj(n)p Fk(!1)1594 4013 y Fm(p)1643 4028 y Fj(n)1690 4013 y Fo(\()p Fm(ab)p Fo(\))28 b(=)f Fm(p)p Fo(\()p Fm(ab)p Fo(\))i(=)2367 3905 y Fk(1)2342 3930 y Ff(X)2338 4115 y Fj(k)r Fv(=1)2483 4013 y Fm(p)2532 3972 y Fv(\()p Fj(k)r Fv(\))2658 4013 y Fo(=)e(0)p Fm(:)240 4254 y Fo(W)-8 b(e)40 b(ha)m(v)m(e)g(used)g(stationarit)m(y)-8 b(,)40 b(giving)d(us)j(that)f Fm(k)j Fo(=)c Fl(1)h Fo(has)h(zero)f(w)m (eigh)m(t.)64 b(T)-8 b(o)39 b(observ)m(e)i(this)e(let)240 4375 y(us)e(notice)g(that)f(if)f Fd(X)i Fo(is)f(limit)d(in)j (distribution)e(of)j(a)f(con)m(v)m(ergen)m(t)j(subsequence)h(of)c(the)h (sequence)240 4495 y Fl(f)p Fd(X)375 4459 y Fv(\()p Fj(n)p Fv(\))476 4495 y Fl(g)526 4459 y Fk(1)526 4520 y Fj(n)p Fv(=1)696 4495 y Fo(then)c Fd(X)p Fo(,)f(as)h(a)f(limit)d(of)j (stationary)g(sequences,)k(is)c(stationary)-8 b(.)43 b(F)-8 b(rom)31 b(this,)h(denoting)1039 4715 y Fm(A)1112 4730 y Fj(k)1183 4715 y Fo(=)27 b Fl(f)p Fm(!)k Fo(:)61 b Fm(X)1597 4730 y Fj(i)1625 4715 y Fo(\()p Fm(!)t Fo(\))27 b(=)h Fm(b)98 b Fl(8)33 b Fm(i)28 b(<)f(k)101 b Fo(and)d Fm(X)2776 4730 y Fj(k)2818 4715 y Fo(\()p Fm(!)t Fo(\))27 b(=)h Fm(a)p Fl(g)240 4935 y Fo(w)m(e)34 b(ha)m(v)m(e)1699 5056 y Fm(P)14 b Fl(f)p Fm(A)1899 5071 y Fj(k)1941 5056 y Fl(g)27 b Fo(=)h Fm(P)14 b Fl(f)p Fm(A)2322 5071 y Fj(k)r Fv(+1)2454 5056 y Fl(g)p Fm(:)240 5230 y Fo(Let)725 5350 y Fm(A)28 b Fo(=)f Fl(f)p Fm(!)k Fo(:)61 b Fl(9)33 b Fm(k)101 b Fo(suc)m(h)34 b(that)f Fm(X)1912 5365 y Fj(k)1954 5350 y Fo(\()p Fm(!)t Fo(\))27 b(=)h Fm(b)98 b Fo(and)f Fl(8)33 b Fm(i)28 b(<)g(k)100 b(X)3104 5365 y Fj(i)3133 5350 y Fo(\()p Fm(!)t Fo(\))26 b(=)i Fm(a)p Fl(g)240 5525 y Fo(then)33 b Fm(A)28 b Fo(=)g Fl([)733 5540 y Fj(k)r Fk(2)p Fb(Z)867 5525 y Fm(A)940 5540 y Fj(k)1015 5525 y Fo(and)33 b Fm(p)1254 5488 y Fv(\()p Fk(1)p Fv(\))1411 5525 y Fo(=)28 b Fm(P)14 b Fl(f)p Fm(A)p Fl(g)p Fm(:)2066 5956 y Fo(10)p eop %%Page: 11 11 11 10 bop 360 107 a Fo(Because)34 b Fm(A)806 122 y Fj(k)871 107 y Fl(\\)23 b Fm(A)1033 122 y Fj(l)1086 107 y Fo(=)28 b Fl(;)k Fo(for)g Fm(k)f Fl(6)p Fo(=)d Fm(l)r(;)k Fo(w)m(e)i(ha)m(v)m (e)1447 313 y Fm(p)1496 272 y Fv(\()p Fk(1)p Fv(\))1653 313 y Fo(=)28 b Fm(P)14 b Fl(f)p Fm(A)p Fl(g)27 b Fo(=)2142 230 y Ff(X)2137 415 y Fj(k)r Fk(2)p Fb(Z)2284 313 y Fm(P)14 b Fl(f)p Fm(A)2484 328 y Fj(k)2526 313 y Fl(g)27 b Fo(=)h(0)p Fm(:)360 590 y Fo(In)33 b(a)f(similar)d(w)m(a)m(y)34 b(w)m(e)g(get)1629 711 y(lim)1605 761 y Fj(n)p Fk(!1)1805 711 y Fm(p)1854 726 y Fj(n)1901 711 y Fo(\()p Fm(ba)p Fo(\))28 b(=)g Fm(p)p Fo(\()p Fm(ba)p Fo(\))g(=)g(0)p Fm(:)360 894 y Fo(This)50 b(means)f(that)g(if)g Fd(X)g Fo(is)g(limit)d(in)i(distribution)g(of)h(a)g(con)m(v)m(ergen)m(t)j (subsequence)h(of)c(the)240 1015 y(sequence)i Fl(f)p Fd(X)795 979 y Fv(\()p Fj(n)p Fv(\))896 1015 y Fl(g)946 979 y Fk(1)946 1039 y Fj(n)p Fv(=1)1131 1015 y Fo(then)e Fd(X)e Fo(is)h(distributed)f(on)h(constan)m(t)h(sequences)i Fd(a)k Fo(=)e Fl(f)p Fm(a)p Fl(g)3536 1034 y Fj(k)r Fk(2)p Fb(Z)3672 1015 y Fg(d)p Fa(\000)p Fe(1)e Fo(and)240 1135 y Fd(b)43 b Fo(=)f Fl(f)p Fm(b)p Fl(g)604 1154 y Fj(k)r Fk(2)p Fb(Z)740 1135 y Fg(d)p Fa(\000)p Fe(1)s Fm(:)f Fo(F)-8 b(urthermore)40 b(w)m(e)j(ha)m(v)m(e)f(that)f Fm(P)14 b Fl(f)p Fd(X)41 b Fo(=)i Fd(a)p Fl(g)f Fo(=)h Fm(p)p Fo(,)g Fm(P)14 b Fl(f)p Fd(X)41 b Fo(=)i Fd(b)p Fl(g)f Fo(=)g(1)28 b Fl(\000)h Fm(p:)41 b Fo(This)240 1256 y(concludes)34 b(the)f(pro)s(of)e(of)h(the)h(lemma)e(for)h Fm(d)27 b Fo(=)h(2.)360 1376 y(Let)33 b(us)g(no)m(w)g(consider)g(d=3,)g (and)f(de\014ne)805 1582 y Fm(p)854 1597 y Fj(n)901 1582 y Fo(\()p Fm(x)994 1597 y Fv(1)1034 1582 y Fm(x)1089 1597 y Fv(2)1145 1582 y Fm(:)17 b(:)g(:)f(x)1331 1597 y Fj(l)1358 1582 y Fo(\))28 b(=)f Fm(P)14 b Fl(f)p Fm(X)1743 1531 y Fv(\()p Fj(n)p Fv(\))1735 1607 y Fj(k)r Fv(+)p Fj(e)1862 1616 y Fe(1)1927 1582 y Fo(=)27 b Fm(x)2085 1597 y Fv(1)2125 1582 y Fm(;)17 b(X)2258 1531 y Fv(\()p Fj(n)p Fv(\))2250 1607 y Fj(k)r Fv(+2)p Fj(e)2412 1616 y Fe(1)2478 1582 y Fo(=)27 b Fm(x)2636 1597 y Fv(2)2676 1582 y Fm(;)17 b(:)g(:)g(:)f(;)h(X)2984 1531 y Fv(\()p Fj(n)p Fv(\))2976 1607 y Fj(k)r Fv(+)p Fj(l)q(e)3125 1616 y Fe(1)3190 1582 y Fo(=)28 b Fm(x)3349 1597 y Fj(l)3375 1582 y Fl(g)853 1966 y Fm(p)902 1981 y Fj(n)966 1720 y Ff(0)966 1866 y(B)966 1916 y(B)966 1966 y(B)966 2019 y(@)1069 1785 y Fm(x)1124 1800 y Fv(1)1069 1906 y Fm(x)1124 1921 y Fv(2)1059 2026 y Fm(:)17 b(:)g(:)1055 2146 y(x)1110 2161 y Fj(m)1194 1720 y Ff(1)1194 1866 y(C)1194 1916 y(C)1194 1966 y(C)1194 2019 y(A)1294 1966 y Fo(=)28 b Fm(P)14 b Fl(f)p Fm(X)1614 1915 y Fv(\()p Fj(n)p Fv(\))1606 1991 y Fj(k)r Fv(+)p Fj(e)1733 2000 y Fe(2)1797 1966 y Fo(=)28 b Fm(x)1956 1981 y Fv(1)1996 1966 y Fm(;)17 b(X)2129 1915 y Fv(\()p Fj(n)p Fv(\))2121 1991 y Fj(k)r Fv(+2)p Fj(e)2283 2000 y Fe(2)2348 1966 y Fo(=)28 b Fm(x)2507 1981 y Fv(2)2547 1966 y Fm(;)17 b(:)g(:)g(:)f(;)h(X)2855 1915 y Fv(\()p Fj(n)p Fv(\))2847 1991 y Fj(k)r Fv(+)p Fj(me)3036 2000 y Fe(2)3101 1966 y Fo(=)28 b Fm(x)3260 1981 y Fj(m)3327 1966 y Fl(g)804 2427 y Fm(p)853 2442 y Fj(n)916 2231 y Ff(0)916 2377 y(B)916 2430 y(@)1033 2307 y Fm(x)1088 2322 y Fv(11)1179 2307 y Fm(:)17 b(:)g(:)g(x)1366 2322 y Fj(l)q Fv(1)1173 2427 y Fm(:)g(:)g(:)1006 2547 y(x)1061 2562 y Fv(1)p Fj(m)1179 2547 y Fm(:)g(:)g(:)g(x)1366 2562 y Fj(l)q(m)1471 2231 y Ff(1)1471 2377 y(C)1471 2430 y(A)1571 2427 y Fo(=)28 b Fm(P)14 b Fl(f)p Fm(X)1891 2376 y Fv(\()p Fj(n)p Fv(\))1883 2452 y Fj(k)r Fv(+)p Fj(e)2010 2461 y Fe(1)2043 2452 y Fv(+)p Fj(e)2131 2461 y Fe(2)2197 2427 y Fo(=)27 b Fm(x)2355 2442 y Fv(11)2431 2427 y Fm(;)17 b(:)g(:)g(:)e(;)i(X)2738 2376 y Fv(\()p Fj(n)p Fv(\))2730 2452 y Fj(k)r Fv(+)p Fj(l)q(e)2879 2461 y Fe(1)2913 2452 y Fv(+)p Fj(me)3063 2461 y Fe(2)3129 2427 y Fo(=)28 b Fm(x)3288 2442 y Fj(l)q(m)3376 2427 y Fl(g)240 2702 y Fo(Then)946 2842 y Fm(p)995 2857 y Fj(n)p Fv(+1)1132 2842 y Fo(\()p Fm(ab)p Fo(\))g(=)1442 2774 y(1)p 1442 2819 49 4 v 1442 2910 a(2)1500 2842 y Fm(p)1549 2857 y Fj(n)p Fv(+1)1687 2842 y Fo(\()p Fm(ab)p Fl(j)p Fm(\013)1907 2857 y Fj(n)p Fv(+1)2072 2842 y Fo(=)f Fm(e)2220 2857 y Fv(1)2260 2842 y Fo(\))22 b(+)2428 2774 y(1)p 2428 2819 V 2428 2910 a(2)2487 2842 y Fm(p)2536 2857 y Fj(n)p Fv(+1)2673 2842 y Fo(\()p Fm(ab)p Fl(j)p Fm(\013)2893 2857 y Fj(n)p Fv(+1)3058 2842 y Fo(=)28 b Fm(e)3207 2857 y Fv(2)3246 2842 y Fo(\))1211 3098 y Fm(p)1260 3113 y Fj(n)p Fv(+1)1397 3098 y Fo(\()p Fm(ab)p Fl(j)p Fm(\013)1617 3113 y Fj(n)p Fv(+1)1782 3098 y Fo(=)g Fm(e)1931 3113 y Fv(1)1971 3098 y Fo(\))f(=)2150 3031 y(1)p 2150 3075 V 2150 3167 a(2)2208 3098 y Fm(p)2257 3113 y Fj(n)2304 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a(2)3533 3358 y Fm(p)3582 3373 y Fj(n)3645 3237 y Ff(\022)3723 3298 y Fm(a)g(b)3728 3418 y(b)g(b)3849 3237 y Ff(\023)925 3634 y Fo(=)28 b Fm(p)1078 3649 y Fj(n)1141 3513 y Ff(\022)1219 3574 y Fm(a)17 b(b)1219 3694 y(a)g(b)1345 3513 y Ff(\023)1428 3634 y Fo(+)22 b Fm(p)1575 3649 y Fj(n)1638 3513 y Ff(\022)1716 3574 y Fm(a)17 b(a)1721 3694 y(a)g(b)1852 3513 y Ff(\023)1935 3634 y Fo(+)22 b Fm(p)2082 3649 y Fj(n)2146 3513 y Ff(\022)2228 3574 y Fm(b)17 b(b)2223 3694 y(a)g(b)2349 3513 y Ff(\023)2438 3634 y Fo(=)28 b Fm(p)2591 3649 y Fj(n)2637 3634 y Fo(\()p Fm(ab)p Fo(\))23 b Fl(\000)g Fm(p)2977 3649 y Fj(n)3040 3513 y Ff(\022)3118 3574 y Fm(b)17 b(a)3118 3694 y(a)g(b)3244 3513 y Ff(\023)240 3842 y Fo(giving)922 4004 y Fm(p)971 4019 y Fj(n)p Fv(+1)1108 4004 y Fo(\()p Fm(ab)p Fo(\))28 b(=)1418 3937 y(1)p 1418 3981 V 1418 4073 a(4)1477 4004 y Fm(p)1526 4019 y Fj(n)1572 4004 y Fo(\()p Fm(aab)p Fo(\))23 b(+)1922 3937 y(1)p 1922 3981 V 1922 4073 a(4)1981 4004 y Fm(p)2030 4019 y Fj(n)2077 4004 y Fo(\()p Fm(abb)p Fo(\))g(+)2417 3937 y(1)p 2417 3981 V 2417 4073 a(2)2476 4004 y Fm(p)2525 4019 y Fj(n)2572 4004 y Fo(\()p Fm(ab)p Fo(\))f Fl(\000)2872 3937 y Fo(1)p 2872 3981 V 2872 4073 a(2)2931 4004 y Fm(p)2980 4019 y Fj(n)3043 3883 y Ff(\022)3121 3944 y Fm(a)17 b(b)3121 4065 y(b)g(a)3247 3883 y Ff(\023)1146 4281 y Fo(=)27 b Fm(p)1298 4296 y Fj(n)1345 4281 y Fo(\()p Fm(ab)p Fo(\))c Fl(\000)1646 4213 y Fo(1)p 1646 4257 V 1646 4349 a(4)1704 4281 y Fm(p)1753 4296 y Fj(n)1800 4281 y Fo(\()p Fm(aba)p Fo(\))g Fl(\000)2152 4213 y Fo(1)p 2152 4257 V 2152 4349 a(4)2210 4281 y Fm(p)2259 4296 y Fj(n)2306 4281 y Fo(\()p Fm(bab)p Fo(\))h Fl(\000)2648 4213 y Fo(1)p 2648 4257 V 2648 4349 a(2)2707 4281 y Fm(p)2756 4296 y Fj(n)2819 4159 y Ff(\022)2897 4220 y Fm(a)17 b(b)2897 4341 y(b)g(a)3023 4159 y Ff(\023)240 4497 y Fo(whic)m(h)33 b(implies)d(that)j(lim)1197 4512 y Fj(n)p Fk(!1)1402 4497 y Fm(p)1451 4512 y Fj(n)1498 4497 y Fo(\()p Fm(ab)p Fo(\))28 b(=)f Fm(p)p Fo(\()p Fm(ab)p Fo(\))34 b(exists)f(and)g(that) 1118 4751 y(lim)1093 4801 y Fj(n)p Fk(!1)1294 4751 y Fm(p)1343 4766 y Fj(n)1390 4751 y Fo(\()p Fm(aba)p Fo(\))28 b(=)52 b(lim)1741 4801 y Fj(n)p Fk(!1)1942 4751 y Fm(p)1991 4766 y Fj(n)2037 4751 y Fo(\()p Fm(bab)p Fo(\))29 b(=)52 b(lim)2379 4801 y Fj(n)p Fk(!1)2579 4751 y Fm(p)2628 4766 y Fj(n)2692 4630 y Ff(\022)2770 4691 y Fm(a)17 b(b)2770 4811 y(b)g(a)2895 4630 y Ff(\023)2984 4751 y Fo(=)28 b(0)240 4999 y(Next)33 b(w)m(e)h(observ)m(e)g(that)f(using)f(the)h (same)f(notation)g(as)g(in)g(the)h Fm(d)28 b Fo(=)f(2)32 b(case)i(w)m(e)g(ha)m(v)m(e)1789 5244 y Fm(p)1838 5193 y Fv(\()p Fj(k)r Fv(\))1838 5266 y Fj(n)p Fv(+1)2003 5244 y Fl(\025)2118 5177 y Fo(1)p 2118 5221 V 2118 5313 a(4)2177 5244 y Fm(p)2226 5203 y Fv(\()p Fj(k)r Fv(+1\))2226 5269 y Fj(n)2414 5244 y Fm(;)240 5476 y Fo(and)f(b)m(y)g(induction)f(w) m(e)h(conclude)g(that)g(for)f(all)e Fm(k)h Fl(2)d Fh(N)1723 5682 y Fo(lim)1699 5732 y Fj(n)p Fk(!1)1900 5682 y Fm(p)1949 5641 y Fv(\()p Fj(k)r Fv(\))1949 5707 y Fj(n)2074 5682 y Fo(=)f Fm(p)2226 5641 y Fv(\()p Fj(k)r Fv(\))2352 5682 y Fo(=)g(0)p Fm(;)2066 5956 y Fo(11)p eop %%Page: 12 12 12 11 bop 240 107 a Fo(whic)m(h)33 b(giv)m(es)g(us,)g(in)f(a)g(similar) e(w)m(a)m(y)k(as)e(for)g(d=2,)h(that)1629 322 y(lim)1605 371 y Fj(n)p Fk(!1)1805 322 y Fm(p)1854 337 y Fj(n)1901 322 y Fo(\()p Fm(ab)p Fo(\))28 b(=)g Fm(p)p Fo(\()p Fm(ab)p Fo(\))g(=)g(0)p Fm(;)240 551 y Fo(and)1629 672 y(lim)1605 722 y Fj(n)p Fk(!1)1805 672 y Fm(p)1854 687 y Fj(n)1901 672 y Fo(\()p Fm(ba)p Fo(\))g(=)g Fm(p)p Fo(\()p Fm(ba)p Fo(\))g(=)g(0)p Fm(:)240 858 y Fo(By)33 b(symmetry)1450 1021 y(lim)1426 1070 y Fj(n)p Fk(!1)1627 1021 y Fm(p)1676 1036 y Fj(n)1739 899 y Ff(\022)1817 960 y Fm(a)1822 1081 y(b)1885 899 y Ff(\023)1974 1021 y Fo(=)51 b(lim)2077 1070 y Fj(n)p Fk(!1)2278 1021 y Fm(p)2327 1036 y Fj(n)2390 899 y Ff(\022)2473 960 y Fm(b)2468 1081 y(a)2536 899 y Ff(\023)2624 1021 y Fo(=)28 b(0)p Fm(:)240 1235 y Fo(Therefore)34 b(the)f(limit)c(is)j(distributed)g(on)g(pure)i Fm(a)p Fo('s)f(or)f(pure)h Fm(b)p Fo('s.)360 1452 y(By)g(Lemma)e(2)i(and)f (Lemma)f(3)i(w)m(e)g(get)g(the)g(follo)m(wing.)240 1651 y Fd(Theorem)k(2)49 b Fc(L)-5 b(et)35 b Fo(0)29 b Fl(\024)f Fm(a)h(<)f(b;)36 b Fc(and)e(let)i Fd(h)28 b Fo(=)h Fl(f)p Fm(h)2087 1666 y Fj(k)2129 1651 y Fl(g)2179 1670 y Fj(k)r Fk(2)p Fb(Z)2315 1651 y Fg(d)p Fa(\000)p Fe(1)38 b Fc(b)-5 b(e)35 b(a)g(stationary)g(er)-5 b(go)g(dic)34 b(r)-5 b(andom)35 b(\014eld)240 1771 y(taking)f(values)h Fm(a)g Fc(and)f Fm(b)p Fc(,)h(with)g(pr)-5 b(ob)g(ability)35 b Fm(p)f Fc(and)h Fo(\(1)22 b Fl(\000)g Fm(p)p Fo(\))p Fm(;)35 b Fc(r)-5 b(esp)g(e)g(ctively.)44 b(Then)1285 1986 y Fo(\001)p Fm(\034)11 b Fo(\()p Fd(h)p Fo(\))29 b Fl(\025)f Fm(p)p Fo(\001)p Fm(\034)11 b Fo(\()p Fn(\022a\023)18 b Fo(\))k(+)g(\(1)g Fl(\000)h Fm(p)p Fo(\)\001)p Fm(\034)11 b Fo(\()p Fn(\022b\023)18 b Fo(\))p Fm(:)240 2297 y Fd(Pro)s(of.)54 b Fo(F)-8 b(or)35 b(simplicit)m(y)e(of)j(notation,)f(w)m(e)j(consider)e (the)g Fm(d)e Fo(=)f(2)j(case.)55 b(The)37 b(pro)s(of)e(w)m(orks)i(in)f (the)240 2417 y(same)c(w)m(a)m(y)i(in)e(higher)g(dimension.)42 b(F)-8 b(rom)31 b(Lemma)g(2)i(for)f(\014xed)i Fd(h)p Fo(\()p Fm(!)t Fo(\))e(w)m(e)h(ha)m(v)m(e)585 2632 y(\001)p Fm(\034)708 2647 y Fv(\003)762 2632 y Fo(\()p Fd(h)p Fo(\()p Fm(!)t Fo(\)\))22 b(+)g(\001)p Fm(\034)1284 2647 y Fv(\003)1337 2632 y Fo(\()p Fm(T)14 b Fd(h)p Fo(\()p Fm(!)t Fo(\)\))27 b Fl(\025)h Fo(\001)p Fm(\034)1942 2647 y Fv(\003)1996 2632 y Fo(\()p Fd(h)p Fo(\()p Fm(!)t Fo(\))22 b Fl(_)g Fm(T)14 b Fd(h)p Fo(\()p Fm(!)t Fo(\)\))21 b(+)h(\001)p Fm(\034)2901 2647 y Fv(\003)2955 2632 y Fo(\()p Fd(h)p Fo(\()p Fm(!)t Fo(\))g Fl(^)g Fm(T)14 b Fd(h)p Fo(\()p Fm(!)t Fo(\)\))p Fm(:)240 2847 y Fo(Observ)m(e)34 b(that)f(the)g(last)f(inequalit)m(y)f(ma)m(y)i(b)s(e)f(written)h(as) 1117 3033 y(1)p 1117 3077 49 4 v 1117 3169 a(2)1175 3100 y([\001)p Fm(\034)1325 3115 y Fv(\003)1379 3100 y Fo(\()p Fd(h)p Fo(\()p Fm(!)t Fo(\)\))22 b(+)g(\001)p Fm(\034)1901 3115 y Fv(\003)1954 3100 y Fo(\()p Fm(T)14 b Fd(h)p Fo(\()p Fm(!)t Fo(\)\)])27 b Fl(\025)h Fh(E)2513 3116 y Fj(h)p Fv(\()p Fj(!)r Fv(\))2665 3100 y Fo(\001)p Fm(\034)2788 3115 y Fv(\003)2841 3100 y Fo(\()p Fd(X)2964 3059 y Fv(\(1\))3058 3100 y Fo(\))p Fm(;)240 3357 y Fo(where)37 b Fh(E)574 3373 y Fj(h)p Fv(\()p Fj(!)s Fv(\))762 3357 y Fo(denotes)f(that)g(exp)s (ectation)g(is)f(tak)m(en)h(for)f Fd(X)2467 3321 y Fv(\(1\))2597 3357 y Fo(with)g(initial)d(condition)i Fd(h)p Fo(\()p Fm(!)t Fo(\))p Fm(:)h Fo(Iter-)240 3478 y(ating)c(the)i(last)f (inequalit)m(y)g Fm(n)g Fo(times)g(w)m(e)i(get)1242 3682 y(1)p 1219 3726 96 4 v 1219 3817 a(2)1268 3789 y Fj(n)1379 3641 y(n)1341 3666 y Ff(X)1344 3848 y Fj(i)p Fv(=0)1477 3603 y Ff( )1543 3682 y Fm(n)1556 3817 y(i)1601 3603 y Ff(!)1667 3749 y Fo(\001)p Fm(\034)1790 3764 y Fv(\003)1844 3749 y Fo(\()p Fm(T)1953 3708 y Fj(i)1981 3749 y Fd(h)p Fo(\()p Fm(!)t Fo(\)\))27 b Fl(\025)h Fh(E)2403 3765 y Fj(h)p Fv(\()p Fj(!)s Fv(\))2555 3749 y Fo(\001)p Fm(\034)2678 3764 y Fv(\003)2732 3749 y Fo(\()p Fd(X)2855 3708 y Fv(\()p Fj(n)p Fv(\))2956 3749 y Fo(\))p Fm(:)796 b Fo(\(14\))240 4034 y(T)-8 b(aking)32 b(exp)s(ectation)h(of)f(b)s(oth)g(sides)h(of)f (inequalit)m(y)g(\(14\))g(w)m(e)i(get)e(that)h(for)f(all)e Fm(n)e Fl(\025)g Fo(1)1603 4249 y Fh(E)8 b Fo(\001)p Fm(\034)1776 4264 y Fv(\003)1835 4249 y Fo(\()p Fd(h)p Fo(\))28 b Fl(\025)g Fh(E)8 b Fo(\001)p Fm(\034)2279 4264 y Fv(\003)2338 4249 y Fo(\()p Fd(X)2461 4208 y Fv(\()p Fj(n)p Fv(\))2562 4249 y Fo(\))p Fm(:)1190 b Fo(\(15\))240 4483 y(F)-8 b(rom)28 b(Lemma)h(3)h(w)m(e)h(kno)m(w)g(that)f Fd(X)1605 4447 y Fv(\()p Fj(n)p Fv(\))1786 4430 y Fk(D)1734 4483 y Fl(\000)-16 b(!)27 b Fd(X)p Fm(;)66 b Fo(as)34 b Fm(n)28 b Fl(!)f(1)p Fm(:)j Fo(Hence)h(from)e([12)o(,)i(Theorem)f (5.1])g(w)m(e)240 4603 y(conclude)j(that)1342 4724 y(\001)p Fm(\034)1465 4739 y Fv(\003)1519 4724 y Fo(\()p Fd(X)1642 4683 y Fv(\()p Fj(n)p Fv(\))1743 4724 y Fo(\))1861 4671 y Fk(D)1809 4724 y Fl(\000)-16 b(!)27 b Fo(\001)p Fm(\034)2120 4739 y Fv(\003)2174 4724 y Fo(\()p Fd(X)p Fo(\))p Fm(;)66 b Fo(as)33 b Fm(n)28 b Fl(!)f(1)p Fm(:)240 4896 y Fo(No)m(w)i(b)s (ecause)g(from)e(Prop)s(osition)g(2)h(w)m(e)h(ha)m(v)m(e)h(that)e(\001) p Fm(\034)2328 4911 y Fv(\003)2382 4896 y Fo(\()p Fd(X)2505 4860 y Fv(\()p Fj(n)p Fv(\))2606 4896 y Fo(\))g Fl(\024)g Fo(\001)p Fm(\034)2900 4911 y Fv(\003)2954 4896 y Fo(\()p Fn(\022b\023)17 b Fo(\))28 b(w)m(e)h(ma)m(y)f(apply)g([12,)240 5016 y(Theorem)33 b(5.4])f(to)g(get)795 5231 y(lim)771 5281 y Fj(n)p Fk(!1)972 5231 y Fh(E)8 b Fo(\001)p Fm(\034)1145 5246 y Fv(\003)1204 5231 y Fo(\()p Fd(X)1327 5190 y Fv(\()p Fj(n)p Fv(\))1428 5231 y Fo(\))28 b(=)f Fh(E)9 b Fo(\001)p Fm(\034)1770 5246 y Fv(\003)1829 5231 y Fo(\()p Fd(X)p Fo(\))28 b(=)f Fm(p)p Fo(\001)p Fm(\034)2293 5246 y Fv(\003)2347 5231 y Fo(\()p Fn(\022a\023)17 b Fo(\))22 b(+)g(\(1)g Fl(\000)h Fm(p)p Fo(\)\001)p Fm(\034)3132 5246 y Fv(\003)3185 5231 y Fo(\()p Fn(\022b\023)18 b Fo(\))p Fm(:)358 b Fo(\(16\))240 5466 y(The)34 b(last)f(equalit)m(y)f(follo)m(ws)g(from)g(the)i(form)e (of)g(the)i(distribution)d(of)i Fd(X)p Fm(:)g Fo(F)-8 b(rom)32 b(\(15\))g(and)h(\(16\))g(w)m(e)240 5587 y(get)1195 5707 y Fh(E)8 b Fo(\001)p Fm(\034)1368 5722 y Fv(\003)1427 5707 y Fo(\()p Fd(h)p Fo(\))28 b Fl(\025)g Fm(p)p Fo(\001)p Fm(\034)1870 5722 y Fv(\003)1923 5707 y Fo(\()p Fn(\022a\023)18 b Fo(\))k(+)g(\(1)g Fl(\000)g Fm(p)p Fo(\)\001)p Fm(\034)2708 5722 y Fv(\003)2762 5707 y Fo(\()p Fn(\022b\023)17 b Fo(\))p Fm(:)782 b Fo(\(17\))2066 5956 y(12)p eop %%Page: 13 13 13 12 bop 240 107 a Fo(No)m(w)37 b(in)e(order)i(to)f(complete)f(the)i (pro)s(of)e(w)m(e)j(use)f(Prop)s(osition)d(1)i(and)h(tak)m(e)g(the)f (thermo)s(dynamic)240 227 y(limit)29 b(in)j(of)g(b)s(oth)g(sides)h(of)f (\(17\))g(to)h(obtain)1299 440 y(\001)p Fm(\034)11 b Fo(\()p Fd(h)p Fo(\))28 b Fl(\025)h Fm(p)p Fo(\001)p Fm(\034)11 b Fo(\()p Fn(\022a\023)18 b Fo(\))k(+)g(\(1)g Fl(\000)g Fm(p)p Fo(\)\001)p Fm(\034)11 b Fo(\()p Fn(\022b\023)18 b Fo(\))240 653 y(whic)m(h)34 b(ends)g(the)g(pro)s(of)e(of)h(Theorem)g (2)g(for)g Fm(d)c Fo(=)f(2.)45 b(In)34 b(the)g(case)g Fm(d)28 b(>)h Fo(2)k(the)h(pro)s(of)e(of)h(Theorem)g(2)240 774 y(is)f(similar:)40 b(Let)33 b Fm(T)928 789 y Fj(k)1003 774 y Fo(denotes)h(the)f(shift)f(in)g(the)h Fm(e)2071 789 y Fj(k)2146 774 y Fo(direction)f(de\014ned)i(for)e Fd(x)c Fo(=)f Fl(f)p Fm(x)3333 789 y Fj(i)3362 774 y Fl(g)3412 793 y Fj(i)p Fk(2)p Fb(Z)3533 774 y Fg(d)p Fa(\000)p Fe(1)35 b Fo(as)1647 987 y Fm(T)1704 1002 y Fj(k)1747 987 y Fo(\()p Fd(x)p Fo(\))27 b(=)h Fl(f)p Fm(x)2118 1002 y Fj(i)p Fv(+)p Fj(e)2230 1014 y Fg(k)2272 987 y Fl(g)2322 1006 y Fj(i)p Fk(2)p Fb(Z)2443 987 y Fg(d)p Fa(\000)p Fe(1)s Fm(:)240 1200 y Fo(F)-8 b(rom)31 b(Lemma)g(2)i(for)f(\014xed)h Fd(h)p Fo(\()p Fm(!)t Fo(\))f(w)m(e)i(ha) m(v)m(e)335 1380 y Fj(d)p Fk(\000)p Fv(1)338 1405 y Ff(X)334 1590 y Fj(k)r Fv(=1)479 1488 y Fo([\001)p Fm(\034)629 1503 y Fv(\003)683 1488 y Fo(\()p Fd(h)p Fo(\()p Fm(!)t Fo(\)\))22 b(+)g(\001)p Fm(\034)1205 1503 y Fv(\003)1258 1488 y Fo(\()p Fm(T)1353 1503 y Fj(k)1396 1488 y Fd(h)p Fo(\()p Fm(!)t Fo(\)\)])27 b Fl(\025)1798 1380 y Fj(d)p Fk(\000)p Fv(1)1801 1405 y Ff(X)1796 1590 y Fj(k)r Fv(=1)1942 1488 y Fo([\001)p Fm(\034)2092 1503 y Fv(\003)2146 1488 y Fo(\()p Fd(h)p Fo(\()p Fm(!)t Fo(\))21 b Fl(_)i Fm(T)2554 1503 y Fj(k)2597 1488 y Fd(h)p Fo(\()p Fm(!)t Fo(\)\))e(+)h(\001)p Fm(\034)3080 1503 y Fv(\003)3134 1488 y Fo(\()p Fd(h)p Fo(\()p Fm(!)t Fo(\))f Fl(^)i Fm(T)3542 1503 y Fj(k)3585 1488 y Fd(h)p Fo(\()p Fm(!)t Fo(\)\)])16 b Fm(:)240 1768 y Fo(The)34 b(rest)f(of)f(the)h(pro)s(of)f(is)g(the)h(same)f(as)h(for)f Fm(d)27 b Fo(=)h(2.)240 1989 y Fd(Theorem)37 b(3)49 b Fc(L)-5 b(et)35 b Fm(a)29 b(<)f(b)p Fc(,)36 b Fm(a)23 b Fo(+)f Fm(b)29 b Fl(\025)g Fo(0)p Fc(,)35 b(and)f(let)i Fm(h)2124 2004 y Fj(i)2152 1989 y Fc(,)f Fm(i)29 b Fl(2)g Fh(Z)2435 1953 y Fj(d)p Fk(\000)p Fv(1)2560 1989 y Fc(,)35 b(b)-5 b(e)35 b(i.i.d.)45 b(variables)34 b(taking)h(value)g Fm(a)240 2109 y Fc(with)g(pr)-5 b(ob)g(ability)34 b Fm(p)h Fc(and)f(value)h Fm(b)g Fc(with)g(pr)-5 b(ob)g(ability)34 b Fo(1)22 b Fl(\000)h Fm(p)p Fc(.)45 b(Then)240 2230 y(\(i\))34 b(if)h Fm(a)28 b Fl(\025)g Fo(0)p Fc(,)35 b(then)f(the)h(di\013er)-5 b(ential)34 b(wal)5 b(l)34 b(tension)h(is)f(a)h(c)-5 b(onc)g(ave)34 b(function)g(of)h Fm(p)27 b Fl(2)h Fo([0)p Fm(;)17 b Fo(1])p Fc(,)240 2350 y(\(ii\))34 b(if)h Fm(a)28 b(<)f Fo(0)p Fc(,)35 b(then)1031 2607 y Fo(\001)p Fm(\034)1154 2622 y Fv(\003)1208 2607 y Fo(\()1261 2540 y(1)p 1256 2584 59 4 v 1256 2676 a Fm(n)1324 2607 y Fo(\))28 b Fl(\025)1509 2540 y Fo(1)p 1505 2584 V 1505 2676 a Fm(n)1573 2607 y Fo(\001)p Fm(\034)1696 2622 y Fv(\003)1750 2607 y Fo(\()p Fn(\022a\023)17 b Fo(\))22 b(+)2126 2540 y Fm(n)g Fl(\000)h Fo(1)p 2126 2584 229 4 v 2211 2676 a Fm(n)2365 2607 y Fo(\001)p Fm(\034)2488 2622 y Fv(\003)2542 2607 y Fo(\()p Fn(\022b\023)17 b Fo(\))28 b Fm(;)116 b(n)28 b Fl(2)g Fh(N)240 2895 y Fd(Pro)s(of)37 b(\(i\).)k Fo(Let)33 b Fd(h)978 2859 y Fv(\(1\))1105 2895 y Fo(and)f Fd(h)1356 2859 y Fv(\(1)1418 2836 y Fa(0)1441 2859 y Fv(\))1505 2895 y Fo(b)s(e)h(dra)m(wn)g(indep)s(enden)m(tly)g (from)e(the)h(same)g(Bernoulli)e(distribu-)240 3016 y(tion,)36 b(at)g(concen)m(tration)h Fm(p)f Fo(of)g Fm(a)p Fo('s.)55 b(F)-8 b(or)35 b(an)m(y)i Fm(\017)g Fo(suc)m(h)h(that)e(0)e Fl(\024)g Fm(\017)g Fl(\024)h Fm(p)25 b Fl(\000)g Fm(p)3092 2980 y Fv(2)3131 3016 y Fo(,)37 b(let)f Fm(\017)3379 3031 y Fj(i)3408 3016 y Fo(,)h Fm(i)d Fl(2)h Fh(Z)3701 2980 y Fj(d)p Fk(\000)p Fv(1)3825 3016 y Fo(,)j(b)s(e)240 3136 y(i.i.d.)45 b(v)-5 b(ariables)32 b(taking)h(v)-5 b(alue)33 b(1)g(with)h(probabilit)m(y)d Fm(\017)j Fo(and)g(v)-5 b(alue)33 b(0)g(with)g(probabilit)m(y)f(1)23 b Fl(\000)g Fm(\017)p Fo(.)47 b(Let)240 3257 y Fd(h)302 3220 y Fv(\(2\))397 3257 y Fo(,)32 b Fd(h)518 3220 y Fv(\(3\))645 3257 y Fo(and)h Fd(h)897 3220 y Fv(\(4\))1024 3257 y Fo(b)s(e)g(de\014ned)h(b) m(y)1562 3479 y Fm(h)1618 3428 y Fv(\(2\))1618 3502 y Fj(i)1740 3479 y Fo(=)27 b(\(1)22 b Fl(\000)h 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b(\(H\))i(requires)g Fm(h)1287 190 y Fv(\(1\))1287 264 y Fj(i)1407 241 y Fo(+)26 b Fm(h)1565 190 y Fv(\(2\))1565 264 y Fj(i)1696 241 y Fl(\025)37 b Fo(0)g Fl(8)p Fm(i)p Fo(.)60 b(Supp)s(ose)38 b Fd(h)2520 205 y Fv(\(1\))2653 241 y Fo(follo)m(ws)e(a)h(Bernoulli)f(distribution)240 361 y(with)d(concen)m(tration)h Fm(p)1123 376 y Fv(1)1196 361 y Fo(of)f Fm(a)g Fo(and)h(1)22 b Fl(\000)i Fm(p)1804 376 y Fv(1)1876 361 y Fo(of)34 b Fm(b)p Fo(,)g(and)f(similarly)d Fd(h)2743 325 y Fv(\(2\))2871 361 y Fo(with)j(concen)m(tration)h Fm(p)3754 376 y Fv(2)3827 361 y Fo(of)f Fm(a)240 482 y Fo(and)g(1)22 b Fl(\000)g Fm(p)649 497 y Fv(2)721 482 y Fo(of)32 b Fm(b)p Fo(.)44 b(Let)33 b Fm(p)1168 497 y Fv(1)1229 482 y Fo(+)22 b Fm(p)1376 497 y Fv(2)1444 482 y Fl(\024)28 b Fo(1.)43 b(Then)34 b(w)m(e)f(can)g(pro)m(v)m(e)1101 698 y(\001)p Fm(\034)1224 713 y Fv(\003)1278 698 y Fo(\()p Fm(p)1365 713 y Fv(1)1404 698 y Fo(\))22 b(+)g(\001)p Fm(\034)1685 713 y Fv(\003)1739 698 y Fo(\()p Fm(p)1826 713 y Fv(2)1865 698 y Fo(\))28 b Fl(\025)g Fo(\001)p Fm(\034)2159 713 y Fv(\003)2213 698 y Fo(\()p Fm(p)2300 713 y Fv(1)2361 698 y Fo(+)22 b Fm(p)2508 713 y Fv(2)2548 698 y Fo(\))g(+)g(\001)p Fm(\034)2829 713 y Fv(\003)2883 698 y Fo(\()p Fn(\022b\023)17 b Fo(\))688 b(\(18\))240 914 y(Indeed)33 b(let)d Fm(\021)743 929 y Fj(i)772 914 y Fo(,)h Fm(i)d Fl(2)g Fh(Z)1047 878 y Fj(d)p Fk(\000)p Fv(1)1171 914 y Fo(,)k(b)s(e)f(i.i.d.)42 b(v)-5 b(ariables)30 b(taking)g(v)-5 b(alue)31 b(0)g(with)g(probabilit)m(y)e Fm(q)j Fo(=)27 b Fm(p)3572 929 y Fv(2)3612 914 y Fm(=)p Fo(\(1)19 b Fl(\000)h Fm(p)3913 929 y Fv(1)3952 914 y Fo(\))240 1035 y(and)33 b(v)-5 b(alue)32 b(1)g(with)g(probabilit)m(y)f (1)22 b Fl(\000)g Fm(q)t Fo(.)44 b(Then)33 b(w)m(e)h(ma)m(y)e(write,)h (in)e(la)m(w,)1434 1251 y Fm(h)1490 1200 y Fv(\(2\))1490 1274 y Fj(i)1612 1251 y Fo(=)c Fm(a)c Fo(+)f Fm(b)g Fl(\000)h Fm(h)2106 1200 y Fv(\(1\))2106 1274 y Fj(i)2223 1251 y Fo(+)f Fm(\021)2369 1266 y Fj(i)2397 1251 y Fo(\()p Fm(h)2491 1200 y Fv(\(1\))2491 1274 y Fj(i)2608 1251 y Fl(\000)g Fm(a)p Fo(\))240 1481 y(Moreo)m(v)m(er)34 b Fm(h)724 1430 y Fv(\(1\))724 1504 y Fj(i)841 1481 y Fo(+)22 b Fm(h)995 1430 y Fv(\(2\))995 1504 y Fj(i)1117 1481 y Fl(\025)28 b Fo(0.)43 b(Let)1768 1613 y Fm(h)1824 1562 y Fv(\(3\))1824 1636 y Fj(i)1946 1613 y Fo(=)28 b Fm(h)2106 1562 y Fv(\(1\))2106 1636 y Fj(i)2223 1613 y Fl(^)22 b Fm(h)2367 1562 y Fv(\(2\))2367 1636 y Fj(i)1768 1797 y Fm(h)1824 1746 y Fv(\(4\))1824 1820 y Fj(i)1946 1797 y Fo(=)28 b Fm(h)2106 1746 y Fv(\(1\))2106 1820 y Fj(i)2223 1797 y Fl(_)22 b Fm(h)2367 1746 y Fv(\(2\))2367 1820 y Fj(i)240 1970 y Fo(Then)34 b(Lemma)d(2)h(applies)g(and)h(giv)m (es)g(\(18\),)f(whic)m(h)h(can)g(then)g(b)s(e)g(applied)e(as)i(follo)m (ws:)1539 2231 y(2\001)p Fm(\034)11 b Fo(\()1775 2163 y(1)p 1770 2207 59 4 v 1770 2299 a Fm(n)1838 2231 y Fo(\))28 b Fl(\025)g Fo(\001)p Fm(\034)11 b Fo(\()2196 2163 y(2)p 2191 2207 V 2191 2299 a Fm(n)2260 2231 y Fo(\))22 b(+)g(\001)p Fm(\034)11 b Fo(\()p Fd(b)p Fo(\))1359 2535 y(\001)p Fm(\034)g Fo(\()1546 2467 y(2)p 1541 2512 V 1541 2603 a Fm(n)1610 2535 y Fo(\))22 b(+)g(\001)p Fm(\034)11 b Fo(\()1955 2467 y(1)p 1950 2512 V 1950 2603 a Fm(n)2019 2535 y Fo(\))27 b Fl(\025)h Fo(\001)p Fm(\034)11 b Fo(\()2376 2467 y(3)p 2371 2512 V 2371 2603 a Fm(n)2440 2535 y Fo(\))22 b(+)g(\001)p Fm(\034)11 b Fo(\()p Fd(b)p Fo(\))2058 2708 y Fm(:)17 b(:)g(:)1285 2900 y Fo(\001)p Fm(\034)11 b Fo(\()1467 2832 y Fm(n)23 b Fl(\000)g Fo(1)p 1467 2877 229 4 v 1553 2968 a Fm(n)1707 2900 y Fo(\))f(+)g(\001)p Fm(\034)11 b Fo(\()2052 2832 y(1)p 2047 2877 59 4 v 2047 2968 a Fm(n)2116 2900 y Fo(\))27 b Fl(\025)h Fo(\001)p Fm(\034)11 b Fo(\()p Fd(a)p Fo(\))24 b(+)e(\001)p Fm(\034)11 b Fo(\()p Fd(b)p Fo(\))240 3097 y(Summing)31 b(up)h(giv)m(es)1412 3256 y(\001)p Fm(\034)11 b Fo(\()1599 3189 y(1)p 1594 3233 V 1594 3325 a Fm(n)1663 3256 y Fo(\))27 b Fl(\025)1848 3189 y Fo(1)p 1843 3233 V 1843 3325 a Fm(n)1911 3256 y Fo(\001)p Fm(\034)11 b Fo(\()p Fd(a)p Fo(\))24 b(+)2307 3189 y Fm(n)e Fl(\000)h Fo(1)p 2307 3233 229 4 v 2392 3325 a Fm(n)2545 3256 y Fo(\001)p Fm(\034)11 b Fo(\()p Fd(b)p Fo(\))240 3454 y(whic)m(h)33 b(concludes)h(the)f(pro)s(of)e(of)h (Theorem)h(3.)360 3575 y(W)-8 b(e)38 b(exp)s(ect)h(that)f(conca)m(vit)m (y)g(should)g(hold)f(for)g Fm(p)f Fl(2)h Fo(\(0)p Fm(;)17 b Fo(1)p Fm(=)p Fo(2\),)37 b(and)h(ev)m(en)h(in)e(a)h(larger)e(in)m (terv)-5 b(al)240 3695 y(dep)s(ending)33 b(up)s(on)f Fm(a)h Fo(and)g Fm(b)p Fo(,)g(e.g.)44 b(a)32 b(condition)f(lik)m(e)h Fm(pa)22 b Fo(+)g(\(1)g Fl(\000)h Fm(p)p Fo(\))p Fm(b)28 b Fl(\025)g Fo(0.)240 4027 y Fp(5)161 b(Strict)54 b(conca)l(vit)l(y:)70 b(lo)l(w)54 b(temp)t(erature)e(expansion)240 4246 y Fo(A)m(t)33 b(lo)m(w)g(enough)g(temp)s(erature,)g(di\013eren)m(tial)e(w)m(all)g (tensions)j(can)f(b)s(e)g(estimated)f(b)m(y)i(a)f(lo)m(w-temp)s(e-)240 4367 y(rature)k(expansion.)55 b(Remainders)36 b(can)h(in)e(principle)g (b)s(e)i(con)m(trolled)e(rigorously)-8 b(,)36 b(as)h(w)m(as)g(done)g (to)240 4487 y(b)s(ound)e(the)h(p)s(oin)m(t)e(of)h(w)m(etting)g (transition)e(in)i([13)o(])g(or)g(to)g(study)h(rough)f(substrates)i(in) d([14].)51 b(Here)240 4608 y(w)m(e)34 b(do)e(only)g(the)h(formal)d (expansion,)k(up)e(to)h(order)f(exp)q(\()p Fl(\000)p Fo(8)p Fm(\014)6 b(J)j Fo(\).)360 4728 y(Let)33 b(us)g(consider)g (again)e(the)i(c)m(hec)m(k)m(erb)s(oard)i(w)m(all,)c(no)m(w)j(with)e Fm(a)c Fo(=)f(0)32 b(and)h Fm(b)28 b Fo(=)g Fm(h)g(>)f Fo(0)33 b(:)1028 5001 y Fm(h)1084 4950 y Fv(\(1\))1084 5024 y Fj(i)1206 5001 y Fo(=)1310 4855 y Ff(\()1418 4940 y Fm(h)83 b Fo(if)32 b Fl(j)p Fm(i)p Fl(j)g Fo(o)s(dd)1418 5060 y(0)90 b(if)32 b Fl(j)p Fm(i)p Fl(j)g Fo(ev)m(en)2221 5001 y Fm(h)2277 4950 y Fv(\(2\))2277 5024 y Fj(i)2399 5001 y Fo(=)2503 4855 y Ff(\()2611 4940 y Fo(0)90 b(if)32 b Fl(j)p Fm(i)p Fl(j)g Fo(o)s(dd)2611 5060 y Fm(h)83 b Fo(if)32 b Fl(j)p Fm(i)p Fl(j)g Fo(ev)m(en)240 5291 y(Then)j Fm(h)552 5255 y Fv(\(3\))677 5291 y Fo(=)30 b Fm(h)839 5255 y Fv(\(1\))956 5291 y Fl(^)24 b Fm(h)1102 5255 y Fv(\(2\))1226 5291 y Fo(=)30 b(0)k(and)g Fm(h)1662 5255 y Fv(\(4\))1786 5291 y Fo(=)c Fm(h)1948 5255 y Fv(\(1\))2066 5291 y Fl(_)23 b Fm(h)2211 5255 y Fv(\(2\))2336 5291 y Fo(=)30 b Fm(h)p Fo(.)48 b(Let)34 b(\001)p Fm(\034)11 b Fo(\()p Fm(h)p Fo(\))35 b(denote)f(the)h(di\013eren)m(tial)240 5411 y(w)m(all)c(tension)i(for)f(the)h(pure)g Fm(h)g Fo(w)m(all.)42 b(Of)32 b(course)i(\001)p Fm(\034)11 b Fo(\(0\))28 b(=)f(0,)33 b(so)f(that)h(Theorem)g(1)f(giv)m(es)1454 5672 y(\001)p Fm(\034)11 b Fo(\()p Fm(h)1682 5631 y Fv(\(1\))1777 5672 y Fo(\))28 b(=)f(\001)p Fm(\034)11 b Fo(\()p Fm(h)2174 5631 y Fv(\(2\))2270 5672 y Fo(\))27 b Fl(\025)2450 5605 y Fo(1)p 2450 5649 49 4 v 2450 5740 a(2)2509 5672 y(\001)p Fm(\034)11 b Fo(\()p Fm(h)p Fo(\))2066 5956 y(14)p eop %%Page: 15 15 15 14 bop 360 107 a Fo(In)33 b(order)g(to)f(sho)m(w)h(strict)g(conca)m (vit)m(y)-8 b(,)33 b(w)m(e)h(no)m(w)f(expand)h(the)f(t)m(w)m(o)g (di\013eren)m(tial)e(w)m(all)g(tensions)i(in)240 227 y(p)s(o)m(w)m(ers)i(of)f(exp)q(\()p Fl(\000)p Fm(\014)6 b(J)j Fo(\).)47 b(Let)34 b Fm(\014)6 b(J)39 b Fl(\035)29 b Fo(1)34 b(and)g Fm(J)39 b Fl(\035)29 b Fm(h)h(>)g Fo(0.)47 b(F)-8 b(or)33 b(simplicit)m(y)e(of)i(notation)g(w)m(e)i(do)f(the)240 348 y(computation)d(in)h(t)m(w)m(o)h(dimensions.)43 b(First)453 568 y Fm(\014)6 b Fo(\001)p Fm(\034)11 b Fo(\()p Fm(h)p Fo(\))84 b(=)g(2)p Fm(\014)6 b(h)21 b Fo(+)h(exp)q(\()p Fl(\000)p Fo(6)p Fm(\014)6 b(J)31 b Fo(+)22 b(2)p Fm(\014)6 b(h)p Fo(\))22 b(+)g(exp)q(\()p Fl(\000)p Fo(6)p Fm(\014)6 b(J)31 b Fl(\000)23 b Fo(2)p Fm(\014)6 b(h)p Fo(\))863 725 y Fl(\000)84 b Fo(exp)q(\()p Fl(\000)p Fo(8)p Fm(\014)6 b(J)31 b Fo(+)22 b(4)p Fm(\014)6 b(h)p Fo(\))21 b(+)h(exp)q(\()p Fl(\000)p Fo(8)p Fm(\014)6 b(J)31 b Fl(\000)23 b Fo(4)p Fm(\014)6 b(h)p Fo(\))22 b(+)g Fl(O)2869 629 y Ff(\020)2918 725 y Fo(exp)q(\()p Fl(\000)p Fo(10)p Fm(\014)6 b(J)31 b Fo(+)22 b(4)p Fm(\014)6 b(h)p Fo(\))3728 629 y Ff(\021)3817 725 y Fo(\(19\))240 945 y(Then)257 1185 y Fm(\014)g Fo(\001)p Fm(\034)11 b Fo(\()p Fm(h)546 1144 y Fv(\(1\))641 1185 y Fo(\))28 b(=)f Fm(\014)6 b(h)22 b Fl(\000)1059 1117 y Fo(1)p 1059 1162 49 4 v 1059 1253 a(2)1134 1185 y(exp)q(\()p Fl(\000)p Fo(6)p Fm(\014)6 b(J)31 b Fo(+)22 b(2)p Fm(\014)6 b(h)p Fo(\))22 b(+)2025 1117 y(1)p 2025 1162 V 2025 1253 a(2)2100 1185 y(exp)q(\()p Fl(\000)p Fo(6)p Fm(\014)6 b(J)31 b Fl(\000)23 b Fo(2)p Fm(\014)6 b(h)p Fo(\))21 b(+)h Fl(O)3064 1089 y Ff(\020)3114 1185 y Fo(exp)q(\()p Fl(\000)p Fo(10)p Fm(\014)6 b(J)31 b Fo(+)22 b(4)p Fm(\014)6 b(h)p Fo(\))3924 1089 y Ff(\021)240 1430 y Fo(The)37 b(p)s(oin)m(t)f(is)g(that)g(an)h(excitation)e(made)h(of)g(t)m(w)m(o)h (neigh)m(b)s(oring)e(Ising)h(spins)h(along)e(an)i(alternate)240 1551 y(+)p Fl(\000)42 b Fo(w)m(all)f(has)h(the)h(same)e(w)m(eigh)m(t)i (in)e(the)h(+phase)h(and)f(the)g Fl(\000)p Fo(phase,)k(and)c(therefore) h(do)s(es)f(not)240 1671 y(con)m(tribute)33 b(to)f(the)h(di\013eren)m (tial)e(w)m(all)g(tension.)43 b(Therefore)617 1911 y Fm(\014)6 b Fo(\001)p Fm(\034)11 b Fo(\()p Fm(h)906 1869 y Fv(\(1\))1001 1911 y Fo(\))83 b(=)1291 1843 y(1)p 1291 1887 V 1291 1979 a(2)1350 1911 y Fm(\014)6 b Fo(\001)p Fm(\034)11 b Fo(\()p Fm(h)p Fo(\))22 b(+)1807 1843 y(1)p 1807 1887 V 1807 1979 a(2)1883 1911 y(exp)q(\()p Fl(\000)p Fo(8)p Fm(\014)6 b(J)31 b Fo(+)22 b(4)p Fm(\014)6 b(h)p Fo(\))21 b Fl(\000)2775 1843 y Fo(1)p 2775 1887 V 2775 1979 a(2)2850 1911 y(exp)q(\()p Fl(\000)p Fo(8)p Fm(\014)6 b(J)31 b Fl(\000)23 b Fo(4)p Fm(\014)6 b(h)p Fo(\))1122 2112 y(+)83 b Fl(O)1363 2015 y Ff(\020)1413 2112 y Fo(exp)q(\()p Fl(\000)p Fo(10)p Fm(\014)6 b(J)31 b Fo(+)22 b(4)p Fm(\014)6 b(h)p Fo(\))2223 2015 y Ff(\021)1122 2326 y Fm(>)1291 2259 y Fo(1)p 1291 2303 V 1291 2395 a(2)1350 2326 y Fm(\014)g Fo(\001)p Fm(\034)11 b Fo(\()p Fm(h)p Fo(\))2140 b(\(20\))240 2572 y(whic)m(h)33 b(is)f(strict)g(conca)m(vit)m(y)-8 b(.)240 2792 y Fi(Ac)m(kno)m(wledgemen)m(ts:)53 b Fo(It)36 b(is)g(a)g(pleasure)g(to)g(thank)h(J.)f(De)g(Coninc)m(k)h(for)f(stim)m (ulating)d(discussions)240 2912 y(ab)s(out)28 b(surface)i(tensions)f (in)e(presence)k(of)d(disorder.)42 b(The)30 b(presen)m(t)g(w)m(ork)g(w) m(as)f(partially)d(supp)s(orted)240 3033 y(b)m(y)35 b(a)f(con)m(tract)g (of)g(the)g(Cop)s(ernicus)h(programme)d(of)h(the)i(Europ)s(ean)f(Comm)m (unit)m(y)g(\(CIP)-8 b(A-CT92-)240 3153 y(4016\).)240 3486 y Fp(References)289 3705 y Fo([1])49 b(A.)36 b(B.)h(D.)g(Cassie,)h (Con)m(tact)g(Angles,)f Fi(Discuss.)h(F)-8 b(arada)m(y)36 b(So)s(c.)h Fd(3)p Fo(,)h(11{)e(16)g(\(1948\);)i(Con)m(tact)441 3825 y(Angles)k(and)g(the)g(Absorption)g(of)f(Liquids,)j(pp)f(166{171)d (in)h(Surface)i(Phenomena)g(Chem.)441 3946 y(Biol.,)30 b(Danielli,)f(P)m(ankh)m(urst,)35 b(Riddford)c(eds.,)j(P)m(ergamon)e (\(1958\).)289 4149 y([2])49 b(D.)28 b(Urban,)j(K.)e(T)-8 b(op)s(olski,)29 b(J.)g(De)g(Coninc)m(k,)i(W)-8 b(all)27 b(T)-8 b(ension)30 b(and)g(Heterogeneous)g(Substrates,)441 4269 y Fi(Ph)m(ys.)k(Rev.)f(Lett.)g Fd(76)p Fo(,)g(4388{4391)d (\(1996\).)289 4473 y([3])49 b(J.)c(L.)h(Leb)s(o)m(witz,)i(G.H.S.)e (and)g(other)f(inequalities,)i Fi(Comm)m(un.)e(Math.)h(Ph)m(ys.)h Fd(35)p Fo(,)i(87{92)441 4593 y(\(1974\).)289 4797 y([4])g(F.)32 b(Dunlop,)f(Correlation)g(Inequalities)h(for)g(Multicomp)s(onen)m(t)f (Rotators,)h Fi(Comm)m(un.)h(Math.)441 4917 y(Ph)m(ys.)h Fd(49)p Fo(,)f(247{256)e(\(1976\).)289 5120 y([5])49 b(G.S.)35 b(Sylv)m(ester,)j(Inequalities)c(for)h(Con)m(tin)m(uous-Spin) h(Ising)f(F)-8 b(erromagnets,)35 b Fi(J.)h(Stat.)f(Ph)m(ys.)441 5241 y Fd(15)p Fo(,)d(327{341)f(\(1976\).)289 5444 y([6])49 b(A.)32 b(Messager)i(and)e(S.)h(Miracle-Sole,)d(Correlation)h (functions)h(and)h(b)s(oundary)f(conditions)g(in)441 5565 y(the)h(Ising)f(ferromagnet,)g Fi(J.)g(Stat.)h(Ph)m(ys.)h Fd(17)p Fo(,)f(245{)e(262)h(\(1977\).)2066 5956 y(15)p eop %%Page: 16 16 16 15 bop 289 107 a Fo([7])49 b(J.)41 b(F)-8 b(r\177)-49 b(ohlic)m(h)39 b(and)j(C.)f(E.)h(P\014ster,)i(The)e(W)-8 b(etting)41 b(and)g(La)m(y)m(ering)g(T)-8 b(ransitions)41 b(in)f(the)i(Half-)441 227 y(In\014nite)32 b(Ising)g(Mo)s(del,)h Fi(Europh)m(ys.)h(Lett.)f Fd(3)p Fo(,)g(845{852)d(\(1987\).)289 431 y([8])49 b(J.)25 b(F)-8 b(r\177)-49 b(ohlic)m(h)25 b(and)h(C.)g(E.)g(P\014ster,)i(Semi-In\014nite)d(Ising)g(Mo)s(del)g(I.) h(Thermo)s(dynamic)f(F)-8 b(unctions)441 551 y(and)27 b(Phase)i(Diagram)24 b(in)j(Absence)j(of)d(Magnetic)g(Field,)g Fi(Comm)m(un.)g(Math.)h(Ph)m(ys.)h Fd(109)p Fo(,)g(493{)441 671 y(523)j(\(1987\);)h(I)s(I.)g(The)i(W)-8 b(etting)32 b(and)i(La)m(y)m(ering)f(T)-8 b(ransitions,)33 b Fi(Comm)m(un.)g(Math.) h(Ph)m(ys.)h Fd(112)p Fo(,)441 792 y(51{74)c(\(1987\).)289 995 y([9])49 b(L.)32 b(Breiman,)f Fc(Pr)-5 b(ob)g(ability.)33 b Fo(Addison-W)-8 b(esley)g(,)33 b(Reading)e(\(1968\).)240 1199 y([10])49 b(T.)33 b(M.)g(Liggett)e Fc(Inter)-5 b(acting)34 b(Particle)h(Systems.)d Fo(Springer,)g(New)i(Y)-8 b(ork)32 b(\(1985\).)240 1402 y([11])49 b(R.)39 b(Holley)-8 b(,)40 b(Remarks)g(on)g(the)g(FK)m(G)f(inequalities,)g Fi(Comm)m(un.)g(Math.)h (Ph)m(ys.)i Fd(36)p Fo(,)g(227{231)441 1522 y(\(1974\).)240 1726 y([12])49 b(P)-8 b(.)32 b(Billingsley)-8 b(,)30 b Fc(Conver)-5 b(genc)g(e)33 b(of)h(Pr)-5 b(ob)g(ability)35 b(Me)-5 b(asur)g(es.)33 b Fo(John)f(Wiley)-8 b(,)32 b(New)i(Y)-8 b(ork)32 b(\(1968\).)240 1929 y([13])49 b(C.)35 b(E.)h(P\014ster,)h(O.) d(P)m(enrose,)k(Analyticit)m(y)c(Prop)s(erties)h(of)g(the)g(Surface)h (F)-8 b(ree)35 b(Energy)h(of)f(the)441 2050 y(Ising)d(Mo)s(del,)g Fi(Comm)m(un.)g(Math.)h(Ph)m(ys.)h Fd(115)p Fo(,)f(691{699)e(\(1988\).) 240 2253 y([14])49 b(C.)38 b(Borgs,)i(J.)f(De)f(Coninc)m(k,)j(R.)d (Kotec)m(k)q(\023)-50 b(y,)42 b(An)c(equilibrium)e(lattice)h(mo)s(del)g (of)h(w)m(etting)g(on)441 2373 y(rough)32 b(substrates,)i Fd(mp)p 1369 2373 34 4 v 40 w(arc)f Fo(98{698.)2066 5956 y(16)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9907300717774--