Content-Type: multipart/mixed; boundary="-------------0109040817359" This is a multi-part message in MIME format. ---------------0109040817359 Content-Type: text/plain; name="01-317.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-317.keywords" Diffraction theory, random scatterers, random point sets, quasicrystals, large deviations, cluster expansions ---------------0109040817359 Content-Type: application/postscript; name="autocorrams.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="autocorrams.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com) %%Title: autocorrams.dvi %%Pages: 22 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips -o autocorrams.ps autocorrams.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2001.09.04:1445 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin /N{def}def /B{bind def}N 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b(Usually)35 b(it)e(is)h(assumed)h(to)e(b)s(e)g(a)h(real)g(Sc)m(h)m(w)m(artz)g (function.)46 b(The)34 b(corre-)0 5141 y(sp)s(onding)k(result)g(of)f (the)g(measuremen)m(t)h(is)f(then)h(giv)m(en)f(b)m(y)j(^)-53 b Fl(\015)2407 5105 y Fi(\021)2401 5165 y(r)2454 5141 y Ft(\()p Fl(')p Ft(\))33 b Fh(\021)2741 5060 y Fj(R)2828 5141 y Ft(^)-54 b Fl(\015)2882 5105 y Fi(\021)2876 5165 y(r)2930 5141 y Ft(\()p Fl(k)s Ft(\))p Fl(')p Ft(\()p Fl(k)s Ft(\))p Fl(dk)s Ft(.)52 b(Here)37 b(the)0 5260 y(F)-8 b(ourier-transform)32 b(of)e(a)g(temp)s(ered)h(distribution)i Fl(\015)i Ft(is)c(de\014ned)h(b)m(y)e(dualit)m(y)-8 b(,)35 b(^)-54 b Fl(\015)5 b Ft(\()p Fl(')p Ft(\))28 b(=)g Fl(\015)6 b Ft(\()16 b(^)-66 b Fl(')o Ft(\),)30 b(where)16 5380 y(^)-66 b Fl(')28 b Ft(denotes)h(the)f(F)-8 b(ourier-in)m(tegral)30 b(of)e(the)g(Sc)m(h)m(w)m(artz-function)j Fl(')d Ft(o)m(v)m(er)g Fg(R)2765 5344 y Fi(\027)2821 5380 y Ft(.)42 b(\(F)-8 b(or)27 b(a)g(quic)m(k)i(reminder)p eop %%Page: 5 5 5 4 bop 654 50 a Fr(SELF-A)-9 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y Fk(1)1461 2619 y Fh(k)1511 2634 y Fi(\027;a\033)1672 2619 y Ft(.)43 b(So)33 b(w)m(e)f(ha)m(v)m(e)h Fh(k)16 b Ft(^)-66 b Fl(')2370 2634 y Fi(\033)2423 2619 y Fh(k)2473 2634 y Fi(\027;a)2613 2619 y Fh(\030)28 b Ft(\()p Fl(a\033)t Ft(\))2910 2583 y Fu(\000)p Fi(\027)3037 2539 y Fj(R)3084 2654 y Fc(R)3137 2634 y Ff(\027)3200 2619 y Fh(j)16 b Ft(^)-66 b Fl(')3293 2634 y Fk(1)3338 2619 y Ft(\()p Fl(y)t Ft(\))p Fh(j)p Fl(dy)34 b Ft(with)0 2739 y Fl(\033)39 b Fh(#)d Ft(0,)j(when)g Fl(a)f Ft(is)h(\014xed)g (\(under)g(the)f(condition)j(that)c(the)i(higher)h(deriv)-6 b(ativ)m(es)40 b(are)e(in)m(tegrable\).)0 2859 y(This)25 b(immediately)h(con)m(trols)g(the)e(deterioration)i(of)e(our)g(large)h (deviation)g(estimate)g(when)g(w)m(e)g(mak)m(e)0 2978 y Fl(\033)30 b Ft(smaller)e(to)d(increase)k(the)d(precision)j(of)d (measuremen)m(t)i(of)f(the)f(scattering)h(image.)43 b(\(Without)27 b(loss)0 3098 y(w)m(e)37 b(could)g(ha)m(v)m(e)g(c)m(hosen)g(our)g (length-scale)i(in)e(suc)m(h)g(a)f(w)m(a)m(y)g(that)g Fl(a)c Ft(=)h(1)j(from)g(the)h(b)s(eginning,)i(so)0 3217 y(that)29 b(the)g(general)i(statemen)m(t)f(is)g(regained)g(b)m(y)g (rescaling)h(the)f(observ)-6 b(able)31 b(in)f Fl(k)s Ft(-space.)44 b(W)-8 b(e)30 b(b)s(eliev)m(e)0 3337 y(ho)m(w)m(ev)m(er)k (that)f(the)h(presen)m(t)g(form)g(of)f(the)h(theorem)g(is)g(more)f(in)m (tuitiv)m(e.\))0 3521 y Fm(R)-5 b(emark.)50 b Ft(The)43 b(norm)f(app)s(earing)h(is)h(\014nite)f(in)g(particular)h(for)e(the)h (commonly)g(used)g(Sc)m(h)m(w)m(artz-)0 3641 y(test-functions.)70 b(So)41 b(our)g(result)h(in)g(particular)g(implies)i(con)m(v)m (ergence-statemen)m(ts)g(in)e(the)f(sense)0 3761 y(of)36 b(temp)s(ered)h(distributions.)56 b(Supp)s(ose)38 b(w)m(e)e(are)g(giv)m (en)h(an)g(increasing)h(sequence)g(of)e(v)m(olumes)i(\000)3754 3776 y Fi(r)3798 3761 y Ft(.)0 3880 y(Then)25 b(w)m(e)f(immediately)i (obtain)f(the)f Fm(str)-5 b(ong)27 b(law)g(of)g(lar)-5 b(ge)27 b(numb)-5 b(ers)24 b Ft(as)g(a)g(consequence)i(of)e(Theorem)0 4000 y(1,)37 b(sa)m(ying)g(that)g(the)g(cen)m(tered)h(auto)s (correlation)g(measure)g(applied)h(to)d(a)h(test)f(function)j Fl(')e Ft(whose)0 4119 y(F)-8 b(ourier)28 b(transform)f(has)h(\014nite) g(norm)f(con)m(v)m(erges)h(to)f(zero,)h(for)f Fg(P)p Ft(-a.e.)44 b Fl(\021)t Ft(.)e(This)28 b(follo)m(ws)h(trivially)f(b)m (y)0 4239 y(summing)j(the)f(exp)s(onen)m(tial)i(b)s(ound)e(\(1.6\))e(o) m(v)m(er)i(the)g(v)m(olumes)h(using)g(the)f(Borel-Can)m(telli)j(Lemma.) 0 4423 y Fm(R)-5 b(emark.)50 b Ft(The)37 b(fact)g(that)f(the)h(dep)s (endence)i(on)e(the)g(observ)-6 b(able)38 b Fl(')p Ft(,)f(on)g(the)g(p) s(oin)m(t-set)h(\000)f(and)f(on)0 4543 y(the)43 b(distribution)k Fl(\026)c Ft(can)g(b)s(e)h(expressed)h(in)f(terms)f(of)g(the)h(handy)f (quan)m(tit)m(y)h Fl(K)7 b Fh(k)16 b Ft(^)-66 b Fl(')p Fh(k)3328 4558 y Fi(\027;a)3483 4543 y Ft(is)44 b Fm(not)f Ft(a)0 4663 y(priori)30 b(ob)m(vious.)44 b(The)29 b(o)s(ccurrence)i(of) e(the)g(norm)g(ho)m(w)m(ev)m(er)h(is)g(not)f(di\016cult)i(to)d (understand.)45 b(It)28 b(can)0 4782 y(b)s(e)35 b(motiv)-6 b(ated)36 b(b)m(y)f(noting)h(that)e(4\()p Fl(K)7 b Fh(k)16 b Ft(^)-66 b Fl(')o Fh(k)1621 4797 y Fi(\027;a)1733 4782 y Ft(\))1772 4746 y Fk(2)1817 4782 y Fl(=)p Fh(j)p Ft(\000)1957 4797 y Fi(r)2001 4782 y Fh(j)34 b Ft(is)i(an)f(upp)s(er)h(b)s(ound)g (for)f(the)h Fl(\026)p Ft(-exp)s(ectation)0 4902 y(of)h(the)h(square)f (of)g(the)h(mo)s(dulus)h(inside)g(the)f(probabilit)m(y)h(on)e(the)h (l.h.s.)56 b(of)37 b(\(1.6\).)54 b(\(This)38 b(is)g(seen)0 5021 y(using)25 b(the)e(indep)s(endence)k(of)d(the)f(scatterers)h(b)m (y)g(F)-8 b(ourier-transform,)26 b(and)e(substituting)i(the)d(norm-)0 5141 y(estimate)33 b(of)g(Prop)s(osition)h(3.\))43 b(Believing)35 b(in)e(the)g(corresp)s(onding)i(Gaussian)f(b)s(eha)m(vior,)f(the)g (small)7 5260 y(\026)-57 b Fl(")p Ft(-b)s(eha)m(vior)42 b(giv)m(en)f(in)h(\(1.7\))d(should)j(follo)m(w.)68 b(An)40 b(essen)m(tial)j(part)e(of)g(the)f(real)i(pro)s(of)e(consists)j(in)0 5380 y(estimating)29 b Fm(al)5 b(l)32 b(the)f(higher)f(moments)e Ft(con)m(tained)i(in)f(the)f(Laplace-transform)i(in)f(terms)g(of)f(p)s (o)m(w)m(ers)p eop %%Page: 7 7 7 6 bop 654 50 a Fr(SELF-A)-9 b(VERA)n(GING)39 b(OF)d(RANDOM)h(DIFFRA)n (CTION)i(MEASURES)611 b(7)0 289 y Ft(of)38 b Fl(K)7 b Fh(k)16 b Ft(^)-66 b Fl(')p Fh(k)375 304 y Fi(\027;a)487 289 y Ft(.)57 b(One)39 b(can)g(not)e(exp)s(ect)h(a)g(large)g(deviation) h(principle)j([whic)m(h)d(w)m(ould)g(in)g(particular)0 412 y(mean)30 b(the)g(existence)h(of)f(the)g(limit)h Fh(\000)17 b Ft(lim)1595 427 y Fi(r)1716 372 y Fk(1)p 1667 389 137 4 v 1667 446 a Fu(j)p Fk(\000)1740 456 y Ff(r)1780 446 y Fu(j)1832 412 y Ft(log)i Fl(\026)e Ft(\()o Fh(j)33 b Fl(:)17 b(:)g(:)31 b Fh(j)d(\025)g Fl(")p Ft(\)])h(without)h (an)m(y)g(assumptions)h(on)0 542 y(the)38 b(set)g(\000)g(other)g(than)g (minimal)i(distance.)60 b(In)38 b(fact,)g(without)h(further)g (assumptions)h(on)d(\000,)i(the)0 662 y(Laplace-transform)c(w)m(on't)e (con)m(v)m(erge.)0 1139 y Fw(Main)k(result)g(for)i(Mo)s(del)e(B)120 1336 y Ft(F)-8 b(or)33 b(the)h(Mo)s(del)h(B)f(of)g(thermal)h(dislo)s (cations)h(the)e(result)i(is)e(quite)h(analogous.)47 b(Here,)34 b(ho)m(w)m(ev)m(er)0 1455 y(the)42 b(Sob)s(olev-norm)h(of)e (the)h Fm(variation)f Ft(of)h(the)g(F)-8 b(ourier-transform)43 b(of)f(the)g(observ)-6 b(able)43 b(app)s(ears.)0 1575 y(Again,)32 b(there)g(will)i(b)s(e)e(a)g(sharp)s(er)g(v)m(ersion)h(of)f (this)h(result)g(in)g(Chapter)f(4)f(that)h(is)g(called)i Fm(`A)-5 b(ddition)0 1694 y(to)36 b(The)-5 b(or)g(em)35 b(2')p Ft(,)e(and)g(the)h Fm(Centr)-5 b(al)36 b(Limit)g(The)-5 b(or)g(em)p Ft(.)0 1889 y Fw(Theorem)40 b(2.)58 b Fm(Supp)-5 b(ose)37 b(again)h(that)g Ft(\000)1558 1904 y Fi(r)1602 1889 y Fh(\032)q Fg(R)1745 1853 y Fi(\027)1839 1889 y Fm(is)f(any)i(\014nite)e(set)h(and)f(denote)h(the)g(minimial)g(dis-)0 2009 y(tanc)-5 b(e)35 b(b)-5 b(etwe)g(en)35 b(its)g(p)-5 b(oints)35 b(by)h Fl(a)p Fm(.)120 2133 y(Supp)-5 b(ose)30 b(that)h(the)g(dislo)-5 b(c)g(ations)29 b Fl(!)j Ft(=)c(\()p Fl(!)1681 2148 y Fi(x)1731 2133 y Ft(\))1770 2148 y Fi(x)p Fu(2)p Fk(\000)1918 2158 y Ff(r)1994 2133 y Fm(have)i(indep)-5 b(endent,)31 b(not)g(ne)-5 b(c)g(essarily)30 b(identic)-5 b(al)0 2253 y(distribution)34 b Fl(\026)p Fm(,)i(such)f(that)h Fh(j)p Fl(!)1180 2268 y Fi(x)1230 2253 y Fh(j)27 b(\024)h Fl(\016)k(<)c(a=)p Ft(4)p Fm(,)35 b(for)g(al)5 b(l)36 b Fl(x)28 b Fh(2)g Ft(\000)2332 2268 y Fi(r)2376 2253 y Fm(,)36 b Fl(\026)p Fm(-a.s.)120 2378 y(Then)50 b(the)f(\014nite)h (volume)h(sc)-5 b(attering)49 b(image)k Ft(^)-54 b Fl(\015)2029 2342 y Fi(!)2023 2402 y(r)2086 2378 y Ft(\()p Fl(')p Ft(\))50 b Fm(ob)-5 b(eys)49 b(the)h(universal)g(lar)-5 b(ge)50 b(deviation)0 2497 y(estimate)252 2787 y Fl(\026)329 2646 y Fj(\022)402 2642 y(\014)402 2702 y(\014)402 2762 y(\014)402 2822 y(\014)439 2787 y Ft(^)-54 b Fl(\015)493 2746 y Fi(!)487 2811 y(r)550 2787 y Ft(\()p Fl(')p Ft(\))22 b Fh(\000)815 2651 y Fj(Z)931 2787 y Fl(\026)p Ft(\()p Fl(d!)t Ft(\))t(^)-54 b Fl(\015)1245 2746 y Fi(!)1239 2811 y(r)1300 2787 y Ft(\()p Fl(')p Ft(\))1443 2642 y Fj(\014)1443 2702 y(\014)1443 2762 y(\014)1443 2822 y(\014)1504 2787 y Fh(\025)28 b Fl(")1655 2646 y Fj(\023)1757 2787 y Fh(\024)g Ft(2)17 b(exp)2080 2676 y Fj(\020)2140 2787 y Fh(\000j)p Ft(\000)2307 2802 y Fi(r)2351 2787 y Fh(j)22 b(\002)2525 2762 y Ft(~)2501 2787 y Fl(J)2566 2676 y Fj(\020)2914 2719 y Fl(")p 2637 2764 601 4 v 2637 2855 a Ft(4)p Fl(\016)37 b Fh(k)p Fl(d)16 b Ft(^)-66 b Fl(')p Fh(k)2985 2870 y Fi(\027;a)p Fu(\000)p Fk(4)p Fi(\016)3250 2676 y Fj(\021\021)3621 2788 y Ft(\(1.8\))0 3120 y Fm(The)43 b(function)634 3095 y Ft(~)610 3120 y Fl(J)52 b Fm(has)42 b(the)h(same)f(form)h(as)f(the)g(function)g Fl(J)53 b Fm(fr)-5 b(om)42 b(The)-5 b(or)g(em)43 b(1)f(\(se)-5 b(e)42 b(\(1.7\)\),)g(but)0 3247 y(with)31 b(the)f(slightly)i(b)-5 b(etter)30 b(c)-5 b(onstant)1391 3222 y Ft(~)1368 3247 y Fl(D)31 b Ft(=)d(4)p Fl(:)p Ft(38)12 b Fh(\001)g Ft(10)1916 3211 y Fk(3)1958 3166 y Fj(\000)2004 3247 y Fh(\024)28 b Fl(D)2194 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b(the)h(sum)g(is)f(over)g(inde)-5 b(c)g(omp)g(osable)38 b(subsets)g Fh(C)6 b(\032)q(P)i Fm(.)54 b(`Inde)-5 b(c)g(omp)g(osable')38 b(me)-5 b(ans)38 b(that)h(ther)-5 b(e)0 3649 y(do)33 b(not)g(exist)f(nonempty)i Fh(C)1045 3664 y Fk(1)1123 3649 y Fm(and)f Fh(C)1366 3664 y Fk(2)1444 3649 y Fm(s.t.)44 b(the)33 b(p)-5 b(airs)32 b Fl(P)2094 3664 y Fk(1)2139 3649 y Fm(,)h Fl(P)2267 3664 y Fk(2)2345 3649 y Fm(ar)-5 b(e)33 b(always)g(c)-5 b(omp)g(atible)33 b(for)g Fl(P)3532 3664 y Fk(1)3604 3649 y Fh(2)28 b(C)3750 3664 y Fk(1)3795 3649 y Fm(,)0 3771 y Fl(P)64 3786 y Fk(2)143 3771 y Fh(2)35 b(C)296 3786 y Fk(2)341 3771 y Fm(.)56 b(The)39 b(weight)g Ft(\010)1019 3786 y Fu(C)1105 3771 y Ft(=)1217 3696 y Fj(P)1322 3720 y Fu(0)1322 3800 y Fi(I)5 b Fk(:)p Fi(I)g Fu(2)p Fc(N)1528 3780 y Fe(P)1611 3771 y Fl(c)1654 3786 y Fi(I)1716 3696 y Fj(Q)1810 3800 y Fi(P)11 b Fu(2P)2009 3771 y Fl(\032)2061 3726 y Fi(I)2096 3736 y Ff(P)2061 3800 y Fi(P)2197 3771 y Fm(is)39 b(the)g(sum)h(over)f(al)5 b(l)40 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Ft(0)p Fl(:)p Ft(633)3571 4434 y(\(2.16\))0 4740 y Fm(for)k(any)h(\014xe)-5 b(d)33 b Fl(P)14 b Fm(.)44 b(Her)-5 b(e)32 b(the)h(sum)g(is)f(over)g(al)5 b(l)34 b(clusters)f(inc)-5 b(omp)g(atible)32 b(with)g Fl(P)14 b Fm(,)33 b(i.e.)43 b(c)-5 b(ontaining)32 b(at)0 4859 y(le)-5 b(ast)35 b(one)h(p)-5 b(olymer)36 b(inc)-5 b(omp)g(atible)35 b(with)g Fl(P)49 b Fm(and)36 b(we)f(have)g(put)h Fh(jC)6 b(j)28 b Ft(=)2682 4784 y Fj(P)2787 4889 y Fi(P)11 b Fu(2C)2970 4859 y Fh(j)p Fl(P)j Fh(j)35 b Fm(wher)-5 b(e)35 b Fh(j)p Fl(P)14 b Fh(j)35 b Fm(is)g(the)0 4979 y(numb)-5 b(er)36 b(of)f(b)-5 b(onds)35 b(of)g(the)h(p)-5 b(olymer)36 b Fl(P)14 b Fm(.)120 5260 y Ft(The)31 b(pro)s(of)f(is)i (the)e(same)h(as)g(that)f(pro)m(vided)i(in)g([K01].)42 b(Only)32 b(the)e(result)i(w)m(as)f(form)m(ulated)h(for)f(a)0 5380 y(translation-in)m(v)-6 b(arian)m(t)38 b(setting,)e(and)f(applied) i(as)e(a)f(tec)m(hnical)j(to)s(ol)e(in)g(a)g(di\013eren)m(t)i (situation.)49 b(\(It)p eop %%Page: 12 12 12 11 bop 0 50 a Fr(12)1610 b(C.K)1856 30 y(\177)1845 50 y(ULSKE)0 289 y Ft(relies)33 b(on)d(the)h(the)g(general)h(Kotec)m (ky-Preiss)g(estimate)f([KP86].)42 b(A)30 b(simpler)j(pro)s(of)e(of)f (this)i(kind)f(of)0 408 y(result)k(is)f(giv)m(en)g(in)g([BoZa00].\))120 528 y(W)-8 b(e)33 b(note)h(that)e(in)j(our)e(case)h Fl(\025)f Ft(is)h(estimated)h(from)e(ab)s(o)m(v)m(e)h(b)m(y)1019 774 y Fl(\025)27 b Fh(\024)i Fl(\025)p Ft(\()p Fl(\013)p Ft(\))e(:=)h(sup)1571 857 y Fi(x)p Fu(2)p Fk(\000)1823 679 y Fj(X)1736 893 y Fi(y)s Fu(2)p Fk(\000:)p Fi(y)s Fu(6)p Fk(=)p Fi(x)2071 774 y Fl(g)2119 789 y Fk(1)2180 774 y Ft(\()o Fl(u)2275 789 y Fi(x;y)2392 774 y Ft(\))f Fh(\024)i Fl(g)2612 789 y Fk(1)2672 774 y Ft(\()p Fl(u)p Ft(\))3571 818 y(\(2.17\))0 1108 y(where)34 b(w)m(e)g(ha)m(v)m(e)f(put) 1467 1242 y Fl(u)27 b Ft(:=)h(sup)1685 1325 y Fi(x)p Fu(2)p Fk(\000)1938 1147 y Fj(X)1851 1361 y Fi(y)s Fu(2)p Fk(\000:)p Fi(y)s Fu(6)p Fk(=)p Fi(x)2185 1242 y Fl(u)2242 1257 y Fi(x;y)3571 1286 y Ft(\(2.18\))0 1536 y(The)40 b(second)g(inequalit)m(y)h(of)e(\(2.17\))f(follo)m(ws)j(from)e(the)h(p) s(ositivit)m(y)g(of)f(the)h(T)-8 b(a)m(ylor)38 b(co)s(e\016cien)m(ts)j (of)0 1655 y Fl(g)48 1670 y Fk(1)92 1655 y Ft(.)61 b(This)40 b(estimate)g(explains)h(the)e(o)s(ccurence)i(of)e(the)g(norm)g Fh(k)26 b(\001)f(k)2591 1670 y Fk(\000)2645 1655 y Ft(.)61 b(Suc)m(h)41 b(an)e(estimate)g(will)i(b)s(e)0 1775 y(used)34 b(o)m(v)m(er)g(and)g(o)m(v)m(er)f(b)s(elo)m(w.)120 1895 y(T)-8 b(o)41 b(compute)i(the)f(logarithm)h(of)f(the)g(Laplace)h (transform)g(up)f(to)g(quadratic)g(order)g(in)h Fl(\013)f Ft(w)m(e)0 2014 y(need)d(only)g(k)m(eep)g(clusters)h(with)f(at)e(most)h (t)m(w)m(o)g(b)s(onds.)60 b(W)-8 b(e)38 b(get)g(from)h(the)f(general)i (estimate)e(on)0 2134 y(cluster)d(sums)f(pro)m(vided)h(b)m(y)e (\(2.16\))f(the)i(b)s(ound)115 2280 y Fj(\014)115 2340 y(\014)115 2400 y(\014)115 2459 y(\014)115 2519 y(\014)115 2579 y(\014)115 2639 y(\014)148 2514 y Ft(log)18 b Fl(\026)370 2404 y Fj(\020)429 2514 y Fl(e)475 2473 y Fi(X)540 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2532 y Fu(g)3694 2280 y Fj(\014)3694 2340 y(\014)3694 2400 y(\014)3694 2459 y(\014)3694 2519 y(\014)3694 2579 y(\014)3694 2639 y(\014)126 2915 y Fh(\024)28 b Fl(a)284 2874 y Fu(\003)329 2915 y Fh(j)p Ft(\000)419 2930 y Fi(r)463 2915 y Fh(j)508 2775 y Fj(\022)593 2848 y Fl(g)641 2863 y Fk(1)685 2848 y Ft(\()p Fl(u)p Ft(\))p 593 2892 227 4 v 654 2983 a Fl(\025)712 2955 y Fu(\003)832 2775 y Fj(\023)905 2795 y Fk(3)1166 2915 y Ft(for)33 b Fl(u)28 b Fh(\024)g Ft(log)r(\(1)21 b(+)i Fl(\025)1905 2874 y Fu(\003)1951 2915 y Ft(\))3571 3094 y(\(2.19\))0 3214 y(The)36 b(hard)f(part)g(of)g(the)g(T)-8 b(a)m(ylor-expansion)37 b(is)f(no)m(w)f(done)h(b)m(y)f(the)g(general)h (estimate.)51 b(It)34 b(remains)0 3333 y(to)39 b(do)i(some)f(less)h (elegan)m(t)h(but)e(elemen)m(tary)h(w)m(ork:)57 b(W)-8 b(e)40 b(still)i(need)g(to)d(expand)i(the)f(three)h(sums)0 3453 y(app)s(earing)47 b(under)f(the)g(mo)s(dulus)h(on)e(the)h(l.h.s.) 81 b(up)46 b(to)f(quadratic)h(order)g(in)g Fl(\013)p Ft(,)i(estimate)e(the)0 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Fl(\032)3127 304 y Fi(P)3193 289 y Ft(\))k(+)h Fl(\032)3416 304 y Fi(P)3482 289 y Ft(.)69 b(Using)0 409 y Fl(\026)p Ft(\()p Fl(U)167 424 y Fi(x;y)284 409 y Ft(\))39 b(=)h(0)h(w)m(e)g(see) g(that)f(the)h(activit)m(y)g(of)g(the)g(single)h(b)s(ond)g(p)s(olymer)f Fl(P)53 b Ft(=)41 b Fh(f)p Fl(x;)17 b(y)t Fh(g)38 b Ft(is)k(in)f(fact)0 529 y(of)36 b(quadratic)h(order)g(in)g Fl(\013)p Ft(.)53 b([This)37 b(is)h(b)s(etter)e(than)g(the)h(application)i(of)d(the)h(b)s (ound)g(\(2.12\))e(whic)m(h)0 648 y(holds)g(for)e(all)i(p)s(olymers)f (w)m(ould)g(sho)m(w.])44 b(Indeed,)863 937 y Fh(j)p Fl(\032)943 952 y Fi(P)1009 937 y Ft(\()p Fl(\013)p Ft(\))p Fh(j)27 b Ft(=)1311 793 y Fj(\014)1311 853 y(\014)1311 913 y(\014)1311 972 y(\014)1356 870 y Fl(\026)p Ft(\()p Fl(U)1523 885 y Fi(x;y)1640 870 y Fl(g)1688 885 y Fk(1)1732 870 y Ft(\()p Fl(V)1829 885 y Fi(x)1901 870 y Ft(+)c Fl(V)2059 885 y Fi(y)2107 870 y Ft(\)\))p 1356 915 829 4 v 1554 1006 a Fl(\026)p Ft(\()p Fl(e)1699 975 y Fi(V)1746 985 y Ff(x)1791 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Fh(j)p Ft(\))33 b(to)g(get)635 2261 y Fj(\014)635 2321 y(\014)635 2380 y(\014)635 2440 y(\014)635 2500 y(\014)635 2560 y(\014)775 2370 y(X)669 2589 y Fi(P)11 b Fk(:)p Fu(j)p Fi(P)g Fu(j)p Fk(=1)p Fi(;)p Fk(2)1042 2384 y Fj(\000)1088 2465 y Ft(\010)1160 2483 y Fu(f)p Fi(P)g Fu(g)1330 2465 y Fh(\000)23 b Fl(\032)1482 2480 y Fi(P)1548 2384 y Fj(\001)1593 2261 y(\014)1593 2321 y(\014)1593 2380 y(\014)1593 2440 y(\014)1593 2500 y(\014)1593 2560 y(\014)1654 2465 y Fh(\024)1786 2370 y Fj(X)1771 2574 y Fe(f)p Ff(x;y)r Fe(g)1794 2631 y Ff(x)p Fe(6)p Fd(=)p Ff(y)1974 2465 y Fl(l)2022 2384 y Fj(\000)2068 2465 y Fl(u)2125 2480 y Fi(x;y)2241 2465 y Fl(g)2289 2480 y Fk(1)2350 2465 y Ft(\(2)p Fl(v)t Ft(\))16 b Fl(e)2592 2424 y Fk(2)p Fi(v)2701 2465 y Ft(+)23 b Fl(g)2849 2480 y Fk(2)2910 2465 y Ft(\()p Fl(u)3006 2480 y Fi(x;y)3122 2465 y Ft(\))3161 2384 y Fj(\001)741 2805 y Ft(+)861 2711 y Fj(X)840 2924 y Fi(y)s Fu(2)p Fk(\000)986 2934 y Ff(r)1196 2711 y Fj(X)1155 2910 y Ff(x;z)r Fe(2)p Fd(\000)1341 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Fl(g)2528 3595 y Fk(1)2572 3580 y Ft(\()p Fl(u)p Ft(\))2707 3538 y Fk(2)2751 3469 y Fj(\021)2811 3439 y(\023)3571 3580 y Ft(\(2.24\))0 3863 y(Next)35 b(w)m(e)h(need)g(the)g (error)f(terms)h(for)g(the)g(quadratic)g(appro)m(ximation)g(on)g(the)g (p)s(olymer)f(w)m(eigh)m(ts.)0 3983 y(Keeping)44 b(the)d(second)i (order)f(terms)g(and)g(using)h(similar)g(argumen)m(ts)g(as)e(b)s(efore) h(w)m(e)g(get)g(for)f(the)0 4102 y(single-b)s(ond)36 b(p)s(olymer)355 4242 y Fj(\014)355 4302 y(\014)355 4361 y(\014)355 4421 y(\014)388 4386 y Fl(\032)440 4401 y Fi(P)506 4386 y Ft(\()p Fl(\013)p Ft(\))22 b Fh(\000)h Fl(\026)17 b Ft(\()o Fl(U)953 4401 y Fi(x;y)1070 4386 y Ft(\()p Fl(V)1167 4401 y Fi(x)1239 4386 y Ft(+)23 b Fl(V)1397 4401 y Fi(y)1445 4386 y Ft(\)\))e Fh(\000)1656 4319 y Ft(1)p 1656 4363 V 1656 4455 a(2)1718 4386 y Fl(\026)1795 4306 y Fj(\000)1840 4386 y Fl(U)1919 4345 y Fk(2)1908 4411 y Fi(x;y)2025 4306 y Fj(\001)2070 4242 y(\014)2070 4302 y(\014)2070 4361 y(\014)2070 4421 y(\014)366 4652 y Fh(\024)28 b Fl(u)528 4667 y Fi(x;y)662 4572 y Fj(\002)703 4652 y Ft(2)p Fl(v)t(g)853 4667 y Fk(1)914 4652 y Ft(\(2)p Fl(v)t Ft(\))21 b(+)i Fl(g)1263 4667 y Fk(2)1324 4652 y Ft(\()o(2)p Fl(v)t Ft(\))16 b Fl(e)1565 4611 y Fk(2)p Fi(v)1653 4572 y Fj(\003)1716 4652 y Ft(+)1828 4585 y(1)p 1828 4629 V 1828 4721 a(2)1890 4652 y Fl(u)1947 4611 y Fk(2)1947 4677 y Fi(x;y)2064 4652 y Fl(g)2112 4667 y Fk(1)2172 4652 y Ft(\(2)p Fl(v)t Ft(\))g(\(1)22 b(+)g Fl(e)2624 4611 y Fk(2)p Fi(v)2712 4652 y Ft(\))g(+)g Fl(g)2920 4667 y Fk(3)2981 4652 y Ft(\()p Fl(u)3077 4667 y Fi(x;y)3194 4652 y Ft(\))3571 4506 y(\(2.25\))0 4915 y(F)-8 b(or)33 b(a)g(double-b)s(ond)j(p)s(olymer)d Fl(P)42 b Ft(=)28 b Fh(ff)p Fl(x;)17 b(y)t Fh(g)p Fl(;)g Fh(f)p Fl(y)t(;)g(z)t Fh(gg)29 b Ft(w)m(e)34 b(get)f(in)h(a)f(similar)j (fashion)273 5145 y Fh(j)p Fl(\032)353 5160 y Fi(P)419 5145 y Ft(\()p Fl(\013)p Ft(\))22 b Fh(\000)g Fl(\026)17 b Ft(\()p Fl(U)866 5160 y Fi(x;y)982 5145 y Fl(U)1050 5160 y Fi(y)s(;z)1163 5145 y Ft(\))p 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y Fk(3)p Fi(v)3124 5351 y Ft(\))3571 5265 y(\(2.26\))p eop %%Page: 14 14 14 13 bop 0 50 a Fr(14)1610 b(C.K)1856 30 y(\177)1845 50 y(ULSKE)0 289 y Ft(Summing)36 b(o)m(v)m(er)f(the)g(p)s(olymer)h(and) f(using)h(the)f(p)s(ositivit)m(y)h(of)f(the)h(T)-8 b(a)m(ylor)34 b(co)s(e\016cien)m(ts)j(of)e Fl(l)r(;)17 b(g)3646 304 y Fk(1)3689 289 y Fl(;)g(g)3782 304 y Fk(2)0 408 y Ft(w)m(e)34 b(obtain)221 484 y Fj(\014)221 544 y(\014)221 603 y(\014)221 663 y(\014)221 723 y(\014)221 783 y(\014)276 594 y(X)254 807 y Fi(x)p Fu(2)p Fk(\000)402 817 y Ff(r)459 688 y Ft(log)18 b Fl(\026)681 607 y Fj(\000)726 688 y Fl(e)772 647 y Fi(V)819 657 y Ff(x)869 607 y Fj(\001)937 688 y Ft(+)1143 594 y Fj(X)1037 812 y Fi(P)11 b Fk(:)p Fu(j)p Fi(P)g Fu(j)p Fk(=1)p Fi(;)p Fk(2)1410 688 y Fl(\032)1462 706 y Fu(f)p Fi(P)g Fu(g)1632 688 y Fh(\000)1744 621 y Ft(1)p 1744 665 50 4 v 1744 757 a(2)1805 688 y Fl(\026)1882 607 y Fj(\000)1928 688 y Fl(X)2011 703 y Fi(r)2054 688 y Ft(\()p Fl(\013)p Ft(\))2196 647 y Fk(2)2241 607 y Fj(\001)2286 484 y(\014)2286 544 y(\014)2286 603 y(\014)2286 663 y(\014)2286 723 y(\014)2286 783 y(\014)2347 688 y Fh(\024)28 b(j)p Ft(\000)2542 703 y Fi(r)2586 688 y Fh(j)2614 518 y Fj( )2693 688 y Fl(l)r Ft(\()p Fl(g)2812 703 y Fk(2)2856 688 y Ft(\()p Fl(v)t Ft(\)\))21 b(+)h Fl(g)3193 703 y Fk(3)3238 688 y Ft(\()p Fl(v)t Ft(\))226 1014 y(+)338 947 y(1)p 338 991 V 338 1082 a(2)400 1014 y Fl(u)474 933 y Fj(\002)515 1014 y Ft(2)p Fl(v)t(g)665 1029 y Fk(1)725 1014 y Ft(\(2)p Fl(v)t Ft(\))g(+)g Fl(g)1074 1029 y Fk(2)1135 1014 y Ft(\(2)p Fl(v)t Ft(\))16 b Fl(e)1377 973 y Fk(2)p Fi(v)1464 933 y Fj(\003)1528 1014 y Ft(+)1639 947 y(1)p 1639 991 V 1639 1082 a(4)1701 1014 y Fl(u)1758 973 y Fk(2)1803 1014 y Fl(g)1851 1029 y Fk(1)1912 1014 y Ft(\()o(2)p Fl(v)t Ft(\))h(\(1)k(+)i Fl(e)2364 973 y Fk(2)p Fi(v)2451 1014 y Ft(\))f(+)2624 947 y(1)p 2624 991 V 2624 1082 a(2)2685 1014 y Fl(g)2733 1029 y Fk(3)2794 1014 y Ft(\()p Fl(u)p Ft(\))226 1297 y(+)h(2)p Fl(ug)481 1312 y Fk(2)525 1297 y Ft(\()p Fl(u)p Ft(\))f(+)g Fl(g)829 1312 y Fk(2)873 1297 y Ft(\()p Fl(u)p Ft(\))1008 1256 y Fk(2)1075 1297 y Ft(+)g(2)p Fl(u)1298 1216 y Fj(\002)1339 1297 y Ft(2)p Fl(v)t(g)1489 1312 y Fk(1)1550 1297 y Ft(\(2)p Fl(v)t Ft(\))f(+)i Fl(g)1899 1312 y Fk(2)1960 1297 y Ft(\()o(2)p Fl(v)t Ft(\))16 b Fl(e)2201 1256 y Fk(2)p Fi(v)2289 1216 y Fj(\003)2352 1297 y Ft(+)2464 1229 y(1)p 2464 1274 V 2464 1365 a(2)2526 1297 y Fl(u)2583 1256 y Fk(2)2628 1297 y Fl(g)2676 1312 y Fk(1)2720 1297 y Ft(\(3)p Fl(v)t Ft(\)\(1)21 b(+)h Fl(e)3155 1256 y Fk(3)p Fi(v)3243 1297 y Ft(\))3282 1126 y Fj(!)3571 977 y Ft(\(2.27\))0 1571 y(Let)j(us)h(\014nally)g(treat)e(the)h(last)h(cluster)g(sum)g(under)g (the)f(mo)s(dulus)h(on)f(the)g(l.h.s.)42 b(of)25 b(\(2.19\))f(in)m(v)m (olving)0 1691 y(t)m(w)m(o)39 b(single-b)s(ond)j(p)s(olymers.)61 b(F)-8 b(or)39 b(a)f(pair)i(of)f(incompatible)j(p)s(olymers)d Fl(P)2908 1706 y Fk(1)2953 1691 y Ft(,)h Fl(P)3085 1706 y Fk(2)3169 1691 y Ft(one)f(alw)m(a)m(ys)g(has)0 1810 y(b)m(y)e(the)h(inclusion-exclusion)43 b(form)m(ula)c(that)e(\010)1829 1828 y Fu(f)p Fi(P)1921 1838 y Fd(1)1959 1828 y Fi(;P)2034 1838 y Fd(2)2072 1828 y Fu(g)2153 1810 y Ft(=)d(log)r(\(1)25 b(+)g Fl(\032)2661 1825 y Fi(P)2712 1835 y Fd(1)2780 1810 y Ft(+)g Fl(\032)2934 1825 y Fi(P)2985 1835 y Fd(2)3028 1810 y Ft(\))g Fh(\000)g Ft(log)r(\(1)f(+)h Fl(\032)3590 1825 y Fi(P)3641 1835 y Fd(1)3685 1810 y Ft(\))f Fh(\000)0 1931 y Ft(log)r(\(1)i(+)i Fl(\032)401 1946 y Fi(P)452 1956 y Fd(2)495 1931 y Ft(\).)66 b(The)42 b(easiest)g(w)m(a)m(y)e(to)g (treat)h(this)g(term)g(here)h(is)f(b)m(y)g(application)j(of)d(Prop)s (osition)0 2050 y(A.1)36 b(to)f(the)i(restricted)h(p)s(olymer)f(system) g(that)f(con)m(tains)h(only)g(single-b)s(ond)i(p)s(olymers.)55 b(W)-8 b(e)36 b(can)0 2170 y(use)e(the)g(impro)m(v)m(ed)i(second)e (order)h(b)s(ound)f(\(2.22\).)44 b(Denoting)34 b(b)m(y)g(\010)2639 2134 y Fu(0)2639 2197 y(C)2724 2170 y Ft(the)g(corresp)s(onding)i (cluster)0 2289 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5304 227 4 v 2097 5395 a Ft(\()p Fl(\025)2194 5366 y Fu(\003)2239 5395 y Ft(\))2278 5366 y Fk(2)2351 5246 y Fj(\000)2397 5326 y Fl(ug)2502 5341 y Fk(1)2563 5326 y Ft(\()o(2)p Fl(v)t Ft(\))16 b Fl(e)2804 5285 y Fk(2)p Fi(v)2914 5326 y Ft(+)23 b Fl(g)3062 5341 y Fk(2)3122 5326 y Ft(\()p Fl(u)p Ft(\))3257 5246 y Fj(\001)3303 5265 y Fk(2)3571 4956 y Ft(\(2.32\))p eop %%Page: 15 15 15 14 bop 654 50 a Fr(SELF-A)-9 b(VERA)n(GING)39 b(OF)d(RANDOM)h (DIFFRA)n(CTION)i(MEASURES)569 b(15)0 289 y Ft(No)m(w)22 b(w)m(e)h(use)g(that)f Fl(u)28 b Fh(\024)g Ft(2)p Fl(K)7 b Fh(k)p Fl(\013)p Fh(k)1214 304 y Fk(\000)1290 289 y Ft(and)23 b Fl(v)31 b Fh(\024)d Fl(K)7 b Fh(k)p Fl(\013)p Fh(k)1913 304 y Fk(\000)1967 289 y Ft(.)41 b(So,)24 b Fl(K)7 b Fh(k)p Fl(\013)p Fh(k)2449 304 y Fk(\000)2530 289 y Fh(\024)2647 250 y Fk(1)p 2647 266 40 4 v 2647 323 a(2)2716 289 y Ft(log)q(\(1+)p Fl(\025)3068 253 y Fu(\003)3115 289 y Ft(\))28 b(=:)f Fl(d)h Fh(\031)g Ft(0)p Fl(:)p Ft(05258)0 408 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Fj(\014)1529 3259 y(\014)1579 3284 y Fl(dt)k Fh(\024)h Ft(4)p Fl(\016)21 b Ft(sup)2064 3308 y Fi(w)r Fu(2)p Fi(B)2238 3319 y Fd(2)p Ff(\016)2310 3308 y Fk(\()p Fi(x)p Fu(\000)p Fi(y)s Fk(\))2545 3284 y Ft(sup)2695 3308 y Fu(j)p Fi(e)p Fu(j)p Fk(=1)2902 3169 y Fj(\014)2902 3229 y(\014)2902 3289 y(\014)2962 3245 y Fi(d)p 2947 3261 V 2947 3318 a(dt)3031 3169 y Fj(\014)3031 3229 y(\014)3031 3289 y(\014)3064 3354 y Fi(t)p Fk(=0)3200 3284 y Fl(g)t Ft(\()p Fl(w)j Ft(+)f Fl(te)p Ft(\))3607 3169 y Fj(\014)3607 3229 y(\014)3607 3289 y(\014)3640 3284 y Ft(.)120 3433 y(Using)40 b(the)f(estimate)i(in) f(terms)g(of)f(the)h(in)m(tegrals)h(o)m(v)m(er)f(balls)h(in)f(terms)g (of)g(deriv)-6 b(ativ)m(es)41 b(up)f(to)0 3552 y(order)33 b(of)g(the)g(dimension)i(pro)m(vided)g(in)e(the)g(pro)s(of)g(of)g(Prop) s(osition)h(3)e(w)m(e)h(get)g(for)g Fl(w)d Fh(2)e Fl(B)3436 3567 y Fk(2)p Fi(\016)3519 3552 y Ft(\()p Fl(x)21 b Fh(\000)g Fl(y)t Ft(\))0 3587 y Fj(\014)0 3647 y(\014)0 3707 y(\014)60 3663 y Fi(d)p 45 3679 V 45 3736 a(dt)129 3587 y Fj(\014)129 3647 y(\014)129 3707 y(\014)162 3772 y Fi(t)p Fk(=0)298 3702 y Fl(g)t Ft(\()p Fl(w)j Ft(+)e Fl(te)p Ft(\))704 3587 y Fj(\014)704 3647 y(\014)704 3707 y(\014)765 3702 y Fh(\024)936 3663 y Fk(1)p 883 3679 147 4 v 883 3736 a Fu(j)p Fi(B)967 3746 y Fd(1)1005 3736 y Fu(j)1058 3627 y Fj(P)1163 3652 y Fi(\027)1163 3732 y(k)r Fk(=0)1355 3663 y(1)p 1341 3679 67 4 v 1341 3736 a Fi(k)r Fk(!)1573 3663 y(1)p 1432 3679 323 4 v 1432 3737 a(\()r(~)-42 b Fi(a=)p Fk(2\))1618 3717 y Ff(\027)t Fe(\000)p Ff(k)1783 3622 y Fj(R)1830 3737 y Fi(B)1892 3750 y Fd(~)-36 b Ff(a=)p Fd(2)2004 3737 y Fk(\()p Fi(w)r Fk(\))2147 3702 y Fh(k)p Fl(d)2249 3666 y Fi(k)2325 3663 y(d)p 2310 3679 72 4 v 2310 3736 a(dt)2394 3587 y Fj(\014)2394 3647 y(\014)2394 3707 y(\014)2427 3772 y Fi(t)p Fk(=0)2563 3702 y Fl(g)t Ft(\()p Fl(u)21 b Ft(+)h Fl(te)p Ft(\))p Fh(k)p Fl(du)120 3856 y Ft(Here)43 b(w)m(e)g(ha)m(v)m(e)g(used)i(the)e(radius)j(~)-52 b Fl(a=)p Ft(2)42 b(b)s(ecause)j(this)f(implies)h(that)e Fl(B)2917 3874 y Fk(~)-42 b Fi(a=)p Fk(2)3043 3856 y Ft(\()p Fl(w)s Ft(\))p Fh(\032)p Fl(B)3348 3874 y Fi(a=)p Fk(2)3476 3856 y Ft(\()p Fl(x)29 b Fh(\000)g Fl(y)t Ft(\),)0 3976 y(indep)s(enden)m(tly)125 b(of)d Fl(w)s Ft(,)143 b(and)122 b(so)g(w)m(e)g(get)f(that)g(the)h(r.h.s)g(is)g(b)s(ounded)h (b)m(y)65 4056 y Fk(1)p 12 4073 147 4 v 12 4130 a Fu(j)p Fi(B)96 4140 y Fd(1)135 4130 y Fu(j)187 4021 y Fj(P)292 4045 y Fi(\027)292 4125 y(k)r Fk(=0)484 4056 y(1)p 471 4073 67 4 v 471 4130 a Fi(k)r Fk(!)702 4056 y(1)p 561 4073 323 4 v 561 4131 a(\()r(~)-42 b Fi(a=)p Fk(2\))747 4111 y Ff(\027)t Fe(\000)p Ff(k)912 4015 y Fj(R)959 4131 y Fi(B)1019 4144 y Ff(a=)p Fd(2)1133 4131 y Fk(\()p Fi(x)p Fu(\000)p Fi(y)s Fk(\))1368 4095 y Fh(k)p Fl(d)1470 4059 y Fi(k)r Fk(+1)1619 4095 y Fl(g)t Ft(\()p Fl(u)p Ft(\))p Fh(k)p Fl(du)p Ft(.)63 b(This)41 b(giv)m(es)g(the)f(desired)i(estimate) f(b)m(y)f(sum-)0 4241 y(ming)34 b(o)m(v)m(er)f Fl(y)k Ft(that)c(are)g(not)g(equal)h(to)f Fl(x)p Ft(,)g(and)h(extending)g(the) g(in)m(tegral)h(o)m(v)m(er)e(all)h(of)g Fg(R)3326 4205 y Fi(\027)3381 4241 y Fh(n)p Fl(B)3507 4259 y Fi(a=)p Fk(2)3635 4241 y Ft(\(0\).)0 5021 y Fw(Ac)m(kno)m(wledgmen)m(ts:)138 b Ft(I)43 b(am)h(grateful)g(to)g(M.Baak)m(e)f(for)g(in)m(triguing)k(me) d(with)g(random)g(dif-)0 5141 y(fraction)34 b(patterns)g(and)g(p)s(oin) m(ting)h(out)e(references)j([BaaHo)s(e00],)31 b([Hof95b].)43 b(This)35 b(w)m(ork)e(w)m(as)g(sup-)0 5260 y(p)s(orted)39 b(b)m(y)g(the)h(DF)m(G)f(Sc)m(h)m(w)m(erpunkt)j(`W)-8 b(ec)m(hselwirk)m(ende)42 b(sto)s(c)m(hastisc)m(he)g(Systeme)d(hoher)h (Kom-)0 5380 y(plexit\177)-50 b(at'.)p eop %%Page: 22 22 22 21 bop 0 50 a Fr(22)1610 b(C.K)1856 30 y(\177)1845 50 y(ULSKE)0 289 y Fw(REFERENCES:)-144 503 y Ft([BaaHo)s(e00])48 b(M.Baak)m(e,)40 b(M.H\177)-50 b(o\013e,)40 b Fm(\\Di\013r)-5 b(action)41 b(of)g(R)-5 b(andom)41 b(Tilings:)56 b(Some)42 b(R)m(igor)-5 b(ous)41 b(R)-5 b(esults")p Ft(,)403 623 y(J.)33 b(Stat.)44 b(Ph)m(ys.)g Fw(99)p Ft(,)33 b(Nos.)44 b(1/2,)32 b(219-261)h(\(2000\))-166 742 y([BaaMo)s(o98])48 b(M.Baak)m(e)29 b(and)h(R.V.Mo)s(o)s(dy)-8 b(,)28 b Fm(\\Di\013r)-5 b(active)31 b(p)-5 b(oint)32 b(sets)g(with)g(entr)-5 b(opy")p Ft(,)30 b(J.)f(Ph)m(ys.)43 b(A)29 b Fw(31)p Ft(,)403 862 y(9023-9039)j(\(1998\))-33 981 y([BoZa00])48 b(A.Bo)m(vier,)27 b(M.Zahradn)-11 b(\023)-39 b(\020k,)29 b Fm(\\A)h(simple)g(inductive)e(appr)-5 b(o)g(ach)29 b(to)h(the)f(pr)-5 b(oblem)31 b(of)e(c)-5 b(onver-)403 1101 y(genc)g(e)37 b(of)h(cluster)h(exp)-5 b(ansions)37 b(of)h(p)-5 b(olymer)39 b(mo)-5 b(dels")p Ft(,)37 b(J.)f(Stat.)52 b(Ph)m(ys.)g Fw(100)37 b Ft(Nos.)53 b(3/4,)403 1220 y(765-778)32 b(\(2000\))-44 1340 y([EnMi92])50 b(A.C.D.)56 b(v)-6 b(an)57 b(En)m(ter,)63 b(J.Miekisz,)i Fm(\\How)58 b(should)g(one)g (de\014ne)f(a)h(\(we)-5 b(ak\))56 b(crystal?")p Ft(,)403 1459 y(J.)33 b(Stat.)44 b(Ph)m(ys.)g Fw(66)p Ft(,)33 b(1147-1153)f(\(1992\))79 1579 y([He00])49 b Fm(\\Pr)-5 b(op)g(erties)54 b(of)i(Mo)-5 b(dels)55 b(for)h(ap)-5 b(erio)g(dic)54 b(solids")p Ft(,)59 b(D.J.L.Herrmann,)i(Ph.D.)54 b(thesis,)403 1698 y(Nijmegen)35 b(\(2000\))-7 1818 y([Hof95a])49 b(A.Hof,)39 b Fm(\\Di\013r)-5 b(action)40 b(by)h(ap)-5 b(erio)g(dic)39 b(structur)-5 b(es)41 b(at)f(high)g(temp)-5 b(er)g(atur)g(es")p Ft(,)40 b(J.)e(Ph)m(ys.)61 b(A)403 1938 y Fw(28)p Ft(,)33 b(57-62)g(\(1995\))-12 2057 y([Hof95b])49 b(A.Hof,)37 b Fm(\\On)i(di\013r)-5 b(action)38 b(by)h(ap)-5 b(erio)g(dic)37 b(structur)-5 b(es")p Ft(,)38 b(Comm)m(un.)55 b(Math.)f(Ph)m(ys.)h Fw(169)p Ft(,)403 2177 y(25-43)33 b(\(1995\))121 2296 y([K01])49 b(C.K)s(\177)-53 b(ulsk)m(e,)33 b Fm(\\On)h(the)g(Gibbsian)e(natur)-5 b(e)34 b(of)f(the)g(r)-5 b(andom)34 b(\014eld)g(Kac)f(mo)-5 b(del)34 b(under)g(blo)-5 b(ck-)403 2416 y(aver)g(aging")p Ft(,)32 b(J.)h(Stat.)43 b(Ph)m(ys.)i Fw(104)34 b Ft(Nos.)44 b(5/6,)32 b(991-1012)g(\(2001\))53 2535 y([KP86])49 b(R.Kotec)m(ky)-8 b(,)95 b(D.Preiss,)i Fm(\\Cluster)83 b(exp)-5 b(ansion)81 b(for)g(abstr)-5 b(act)81 b(p)-5 b(olymer)82 b(mo)-5 b(dels")p Ft(,)403 2655 y(Comm.Math.Ph)m(ys.)44 b Fw(103)p Ft(,)33 b(491-498)g(\(1986\)) 107 2774 y([M00])49 b(M.Sc)m(hlottmann,)41 b(Martin,)f Fm(\\Gener)-5 b(alize)g(d)39 b(mo)-5 b(del)41 b(sets)f(and)g(dynamic)-5 b(al)41 b(systems")p Ft(,)e(in:)403 2894 y(Directions)25 b(in)f(mathematical)h(quasicrystals,)h(143-159,)e(CRM)f(Monogr.Ser.,)i (13,)g(AMS,)403 3014 y(Pro)m(vidence,)35 b(\(2000\))p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0109040817359--