Content-Type: multipart/mixed; boundary="-------------0212181312363" This is a multi-part message in MIME format. ---------------0212181312363 Content-Type: text/plain; name="02-525.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-525.comments" 45 pages, 9 figures ---------------0212181312363 Content-Type: text/plain; name="02-525.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-525.keywords" step statistics, Airy kernel, determinantal process ---------------0212181312363 Content-Type: application/postscript; name="StepStat.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="StepStat.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: StepStat.dvi %%Pages: 45 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -ta4 -f StepStat.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.12.18:2009 %%BeginProcSet: texc.pro %! 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S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.& )S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!. &)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S! .&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S !.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&) S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.& )S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!. &)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!.&)S!s4I~> %%EOF %%EndDocument @endspecial 391 2566 a(Figure)32 b(1:)43 b Fs(Crystal)35 b(c)-5 b(orner)34 b(viewe)-5 b(d)34 b(fr)-5 b(om)34 b(the)h FB(\(111\))p Fs(-dir)-5 b(e)g(ction)34 b(for)g Fr(q)e FB(=)27 b(0)p Fr(:)p FB(98)p Fs(.)257 2965 y FB(b)s(ordering)43 b(the)h(crystal)g(facet)f(is)h(of)f(particular)f(in)m(terest.)77 b(T)-8 b(o)43 b(gain)g(some)g(insigh)m(t)257 3085 y(w)m(e)34 b(will)d(study)k(a)d(simpli\014ed)f(statistical)g(mec)m(hanics)j(mo)s (del)d(of)i(a)f(cubic)i(crystal.)45 b(Its)257 3206 y(equilibrium)38 b(shap)s(e)k(has)f(three)h(facets,)i(eac)m(h)d(consisting)g(of)f(a)h (part)g(of)f(one)h(of)g(the)257 3326 y(co)s(ordinate)31 b(planes.)44 b(The)33 b(facets)f(do)g(not)g(touc)m(h)h(eac)m(h)g(other) f(and)g(there)h(is)e(an)h(in)m(ter-)257 3446 y(p)s(olating)h(rounded)i (piece,)h(see)g(Figure)e(1.)50 b(F)-8 b(or)34 b(this)g(mo)s(del)f(the)i (step)h(statistics)e(will)257 3567 y(b)s(e)f(analyzed)g(in)f(great)g (detail.)404 3687 y(Let)j(us)h(\014rst)g(explain)e(our)h(mo)s(del)e (for)i(the)h(corner)f(of)g(a)g(crystal.)51 b(The)36 b(crystal)g(is)257 3808 y(assumed)31 b(to)f(b)s(e)h(simple)e(cubic)h(with)g(lattice)f Fq(Z)2033 3771 y Fp(3)2070 3808 y FB(.)43 b(W)-8 b(e)30 b(use)i(lattice)d(gas)h(language)f(and)257 3928 y(asso)s(ciate)41 b(to)f(eac)m(h)i(site)f Fr(x)h Fo(2)g Fq(Z)1493 3892 y Fp(3)1493 3953 y(+)1549 3928 y FB(,)h Fq(Z)1688 3943 y Fp(+)1786 3928 y FB(=)f Fo(f)p FB(0)p Fr(;)17 b FB(1)p Fr(;)g FB(2)p Fr(;)g(:)g(:)g(:)d Fo(g)p FB(,)43 b(the)e(o)s(ccupation)f (v)-5 b(ariable)257 4048 y Fr(n)315 4063 y Fw(x)400 4048 y FB(=)40 b(0)p Fr(;)17 b FB(1)39 b(with)h(1)f(standing)h(for)f(site)h Fr(x)g FB(o)s(ccupied)g(b)m(y)h(an)f(atom)f(and)h(0)g(for)f(site)h Fr(x)257 4169 y FB(empt)m(y)-8 b(.)44 b(Up)31 b(to)f(a)h(c)m(hemical)e (p)s(oten)m(tial)g(the)i(binding)f(energy)i(of)e(the)h(con\014guration) f Fr(n)257 4289 y FB(is)1104 4629 y Fr(H)8 b FB(\()p Fr(n)p FB(\))27 b(=)h Fr(J)1596 4535 y Fn(X)1537 4751 y Fm(j)p Fw(x)p Fm(\000)p Fw(y)r Fm(j)p Fp(=1)1799 4629 y FB(\()p Fr(n)1895 4644 y Fw(x)1961 4629 y Fo(\000)22 b Fr(n)2118 4644 y Fw(y)2160 4629 y FB(\))2198 4588 y Fp(2)2237 4629 y Fr(;)115 b(J)36 b(>)28 b FB(0)p Fr(:)646 b FB(\(1.1\))257 4954 y(W)-8 b(e)33 b(consider)g(v)m(ery)h(lo)m(w)e (temp)s(eratures,)g(meaning)f(that)i(all)d(allo)m(w)m(ed)i (con\014gurations)257 5075 y(ha)m(v)m(e)48 b(the)f(same)g(energy)-8 b(,)51 b(i.e.)46 b(the)h(same)f(n)m(um)m(b)s(er)h(of)f(brok)m(en)i(b)s (onds.)85 b(T)-8 b(o)47 b(de\014ne)257 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Fo(h)p Fr(a)1591 4019 y Fm(\003)1591 4085 y Fw(j)1620 4093 y Fh(n)1666 4060 y FB(\()p Fr(t)1739 4075 y Fw(n)1786 4060 y FB(\))p Fr(a)1875 4075 y Fw(j)1904 4083 y Fh(n)1950 4060 y FB(\()p Fr(t)2023 4075 y Fw(n)2070 4060 y FB(\))17 b Fo(\001)g(\001)g(\001)e Fr(a)2309 4019 y Fm(\003)2309 4085 y Fw(j)2338 4094 y Fk(1)2376 4060 y FB(\()p Fr(t)2449 4075 y Fp(1)2489 4060 y FB(\))p Fr(a)2578 4075 y Fw(j)2607 4084 y Fk(1)2645 4060 y FB(\()p Fr(t)2718 4075 y Fp(1)2758 4060 y FB(\))p Fo(i)2835 4075 y Fw(T)2890 4060 y Fr(:)329 b FB(\(2.46\))257 4367 y(Using)33 b(\(2.40\))e(the)i(left)f(hand)h(side)g(equals)665 4564 y Fn(X)607 4777 y Fw(k)644 4786 y Fk(1)678 4777 y Fw(;:::)10 b(;k)825 4785 y Fh(n)622 4845 y Fw(l)643 4854 y Fk(1)678 4845 y Fw(;:::)g(;l)809 4853 y Fh(n)926 4535 y Fw(n)884 4564 y Fn(Y)885 4774 y Fw(q)r Fp(=1)1028 4578 y Fn(\002)1069 4659 y Fr(e)1114 4618 y Fm(\000)p Fw(G)p Fp(\(0)p Fw(;t)1331 4626 y Fh(q)1366 4618 y Fp(\))1398 4578 y Fn(\003)1439 4698 y Fw(k)1476 4706 y Fh(q)1511 4698 y Fw(;j)1560 4706 y Fh(q)1597 4578 y Fn(\002)1639 4659 y Fr(e)1684 4618 y Fw(G)p Fp(\(0)p Fw(;t)1846 4626 y Fh(q)1881 4618 y Fp(\))1913 4578 y Fn(\003)1954 4698 y Fw(j)1983 4706 y Fh(q)2017 4698 y Fw(;l)2058 4706 y Fh(q)2096 4659 y Fo(h)p Fr(a)2186 4618 y Fm(\003)2186 4684 y Fw(k)2223 4692 y Fh(n)2270 4659 y Fr(a)2321 4674 y Fw(l)2342 4682 y Fh(n)2405 4659 y Fo(\001)17 b(\001)g(\001)e Fr(a)2589 4618 y Fm(\003)2589 4684 y Fw(k)2626 4693 y Fk(1)2665 4659 y Fr(a)2716 4674 y Fw(l)2737 4683 y Fk(1)2776 4659 y Fo(i)2815 4674 y Fw(T)2869 4659 y Fr(:)350 b FB(\(2.47\))257 5040 y(Let)33 b(us)g(set)903 5291 y Fr(R)977 5306 y Fw(T)1032 5291 y FB(\()p Fr(j;)17 b(t)p FB(;)g Fr(j)1279 5250 y Fm(0)1303 5291 y Fr(;)g(t)1382 5250 y Fm(0)1405 5291 y FB(\))27 b(=)1574 5151 y Fn(\032)1690 5229 y Fo(h)p Fr(a)1780 5193 y Fm(\003)1780 5253 y Fw(j)1820 5229 y FB(\()p Fr(t)p FB(\))p Fr(a)1982 5244 y Fw(j)2015 5225 y Fd(0)2041 5229 y FB(\()p Fr(t)2114 5193 y Fm(0)2137 5229 y FB(\))p Fo(i)2214 5244 y Fw(T)2379 5229 y FB(for)32 b Fr(t)c Fo(\025)g Fr(t)2731 5193 y Fm(0)2755 5229 y Fr(;)1690 5351 y Fo(\000h)p Fr(a)1857 5366 y Fw(j)1890 5347 y Fd(0)1916 5351 y FB(\()p Fr(t)1989 5314 y Fm(0)2013 5351 y FB(\))p Fr(a)2102 5314 y Fm(\003)2102 5375 y Fw(j)2142 5351 y FB(\()p Fr(t)p FB(\))p Fo(i)2292 5366 y Fw(T)2379 5351 y FB(for)k Fr(t)c(<)f(t)2729 5314 y Fm(0)2753 5351 y Fr(:)3246 5291 y FB(\(2.48\))p eop %%Page: 15 15 15 14 bop 257 266 a Ft(3)98 b(LIMIT)33 b(SHAPE)2339 b FB(15)257 573 y(Then)34 b(the)f(unequal)g(time)e(correlations)g(are)i (giv)m(en)g(b)m(y)825 848 y Fi(E)901 863 y Fw(T)972 678 y Fn( )1094 724 y Fw(n)1052 754 y Fn(Y)1051 966 y Fw(k)r Fp(=1)1197 848 y Fr(\021)t FB(\()p Fr(j)1327 863 y Fw(k)1369 848 y Fr(;)17 b(t)1448 863 y Fw(k)1491 848 y FB(\))1529 678 y Fn(!)1635 848 y FB(=)28 b(Det)16 b(\()p Fr(R)2023 863 y Fw(T)2079 848 y FB(\()p Fr(j)2157 863 y Fw(k)2199 848 y Fr(;)h(t)2278 863 y Fw(k)2321 848 y FB(;)g Fr(j)2405 863 y Fw(l)2431 848 y Fr(;)g(t)2510 863 y Fw(l)2536 848 y FB(\)\))2612 877 y Fp(1)p Fm(\024)p Fw(k)r(;l)q Fm(\024)p Fw(n)2901 848 y Fr(:)318 b FB(\(2.49\))404 1132 y(The)43 b(iden)m(tit)m(y)g(\(2.49\))e(has)i(b)s(een)h(deriv)m(ed)f(from)f(left) f(to)h(righ)m(t.)73 b(One)43 b(can)g(read)257 1252 y(it)k(also)g(from)f (righ)m(t)h(to)h(left.)88 b(Then)48 b Fr(R)1809 1267 y Fw(T)1912 1252 y FB(is)g(the)g(de\014ning)f(k)m(ernel,)53 b(resp.)48 b(Green's)257 1373 y(function,)e(whic)m(h)f(is)e(considered) h(to)g(b)s(e)g(giv)m(en)f(and)h(\(2.49\))f(de\014nes)j(the)e(momen)m (ts)257 1493 y(of)38 b(some)g(determinan)m(tal)f(space-time)h(random)f (\014eld)h(o)m(v)m(er)h Fq(Z)24 b Fo(\002)i Fq(Z)p FB(.)58 b(Of)38 b(course,)j Fr(R)3440 1508 y Fw(T)257 1614 y FB(cannot)47 b(b)s(e)f(c)m(hosen)i(arbitrarily)-8 b(,)47 b(since)g(the)g(righ)m(t)e(hand)i(side)f(of)g(\(2.49\))g(m)m(ust)g(b)s (e)257 1734 y(momen)m(ts)28 b(of)g(a)h(probabilit)m(y)d(measure.)42 b(F)-8 b(or)28 b(determinan)m(tal)f(random)g(\014elds)i(o)m(v)m(er)h (the)257 1854 y(space)c(co)s(ordinate)e(only)-8 b(,)26 b(compare)e(with)h(\(2.43\),)h(prop)s(er)e(conditions)g(on)h(the)g (de\014ning)257 1975 y(k)m(ernel)j(ha)m(v)m(e)g(b)s(een)g(studied)g(in) e(detail)f([17,)i(16].)42 b(The)28 b(space-time)e(v)-5 b(arian)m(t)26 b(is)h(less)g(w)m(ell)257 2095 y(understo)s(o)s(d,)33 b(see)h([7])f(for)f(a)g(discussion.)404 2215 y(The)27 b(determinan)m(tal)d(prop)s(ert)m(y)j(is)f(preserv)m(ed)i(under)f (limits.)39 b(Th)m(us)27 b(through)f(bulk)257 2336 y(and)34 b(edge)h(scaling)e(further)h(determinan)m(tal)f(space-time)g(random)g (\014elds)h(will)e(b)s(e)i(en-)257 2456 y(coun)m(tered)i(b)s(elo)m(w.) 48 b(One)35 b(of)f(them)g(is)f(o)m(v)m(er)j Fq(Z)20 b Fo(\002)k Fq(Z)32 b FB(with)i(equal-time)e(giv)m(en)i(through)257 2577 y(the)i(sine-k)m(ernel.)51 b(The)35 b(other)h(is)e(o)m(v)m(er)i Fq(Z)21 b Fo(\002)j Fq(R)46 b FB(with)35 b(equal-time)e(giv)m(en)i (through)g(the)257 2697 y(Airy)e(k)m(ernel.)257 3026 y Fu(3)161 b(Limit)55 b(shap)t(e)257 3245 y FB(On)47 b(the)h(macroscopic)e(scale,)51 b(in)46 b(the)h(limit)d Fr(T)66 b Fo(!)52 b(1)p FB(,)e(the)d(random)f(\014eld)h Fr(\021)t FB(\()p Fr(j;)17 b(t)p FB(\))257 3365 y(b)s(ecomes)33 b(deterministic)e(with)h(a)h(pro\014le)f(giv)m(en)h(b)m(y)357 3673 y Fr(\032)p FB(\()p Fr(\020)8 b(;)17 b(\034)11 b FB(\))28 b(=)762 3468 y Fn(8)762 3558 y(<)762 3737 y(:)892 3550 y FB(1)1406 b(for)32 b Fr(\020)j Fo(\024)28 b Fr(b)2720 3514 y Fm(\000)2720 3575 y(1)2795 3550 y FB(\()p Fr(\034)11 b FB(\))p Fr(;)906 3633 y Fp(1)p 902 3650 43 4 v 902 3707 a Fw(\031)972 3673 y FB(arccos)1249 3592 y Fn(\000)1295 3673 y FB(cosh\()p Fr(\034)g(=)p FB(2\))22 b Fo(\000)h Fr(e)1873 3636 y Fm(\000)p Fw(\020)5 b Fp(+)p Fm(j)o Fw(\034)j Fm(j)p Fw(=)p Fp(2)2172 3673 y Fr(=)p FB(2)2270 3592 y Fn(\001)2347 3673 y FB(for)32 b Fr(b)2537 3636 y Fm(\000)2537 3697 y(1)2613 3673 y FB(\()p Fr(\034)11 b FB(\))28 b Fr(<)f(\020)35 b(<)27 b(b)3095 3688 y Fm(1)3171 3673 y FB(\()p Fr(\034)11 b FB(\))p Fr(;)892 3793 y FB(0)1406 b(for)32 b Fr(\020)j Fo(\025)28 b Fr(b)2720 3808 y Fm(1)2795 3793 y FB(\()p Fr(\034)11 b FB(\))p Fr(;)3295 3922 y FB(\(3.1\))257 4109 y(with)693 4302 y Fr(b)734 4261 y Fm(\000)734 4327 y(1)809 4302 y FB(\()p Fr(\034)g FB(\))28 b(=)f Fo(\000)p FB(2)17 b(ln\(1)k(+)h Fr(e)1545 4261 y Fm(\000j)p Fw(\034)8 b Fm(j)p Fw(=)p Fp(2)1753 4302 y FB(\))p Fr(;)115 b(b)1974 4317 y Fm(1)2049 4302 y FB(\()p Fr(\034)11 b FB(\))28 b(=)f Fo(\000)p FB(2)17 b(ln\(1)k Fo(\000)i Fr(e)2787 4261 y Fm(\000j)p Fw(\034)8 b Fm(j)o Fw(=)p Fp(2)2995 4302 y FB(\))p Fr(:)235 b FB(\(3.2\))257 4496 y(More)39 b(precisely)-8 b(,)40 b(for)e(all)e(con)m(tin)m(uous)j (test)g(functions)g Fr(f)48 b FB(:)37 b Fq(R)2579 4460 y Fp(2)2662 4496 y Fo(!)g Fq(R)50 b FB(with)38 b(compact)257 4616 y(supp)s(ort)761 4853 y(lim)733 4915 y Fw(T)10 b Fm(!1)982 4785 y FB(1)p 951 4830 111 4 v 951 4921 a Fr(T)1022 4892 y Fp(2)1088 4758 y Fn(X)1123 4968 y Fw(j;t)1248 4853 y Fr(f)h FB(\()p Fr(j)6 b(=T)i(;)17 b(t=T)d FB(\))p Fr(\021)t FB(\()p Fr(j;)j(t)p FB(\))27 b(=)2120 4717 y Fn(Z)2236 4853 y FB(d)p Fr(\020)d FB(d)p Fr(\034)11 b(\032)p FB(\()p Fr(\020)d(;)17 b(\034)11 b FB(\))p Fr(f)g FB(\()p Fr(\020)d(;)17 b(\034)11 b FB(\))274 b(\(3.3\))257 5147 y(almost)32 b(surely)-8 b(.)45 b(\(3.3\))33 b(assumes)h(more)e (spatial)f(a)m(v)m(eraging)i(than)g(needed.)47 b(In)33 b(fact,)g(it)257 5268 y(su\016ces)42 b(to)e(c)m(ho)s(ose)g(a)g(test)g (function)f(whose)i(supp)s(ort)g(on)e(the)h(scale)g(of)f(the)i(lattice) 257 5388 y(div)m(erges)34 b(as)f Fr(T)41 b Fo(!)27 b(1)32 b FB(and)h(to)f(prop)s(erly)h(normalize.)p eop %%Page: 16 16 16 15 bop 257 266 a Ft(3)98 b(LIMIT)33 b(SHAPE)2339 b FB(16)404 573 y(As)32 b(a)g(consequence)j(of)d(\(3.3\))f(the)i(limit)28 b(\(1.9\))k(holds.)43 b Fr(h)2503 588 y Fp(ma)2633 573 y FB(can)32 b(b)s(e)h(read)f(o\013)f(from)257 693 y(\(3.1\))h(and)h(is) f(giv)m(en)h(in)f(parametric)f(form)g(through)506 954 y Fr(h)562 969 y Fp(ma)660 954 y FB(\()p Fr(u;)17 b(v)t FB(\))26 b(=)1017 813 y Fn(\032)1133 893 y FB(0)968 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Fo(\000)23 b Fr(\020)505 b FB(\(3.5\))257 1758 y(in)31 b(the)i(in)m(terv)-5 b(al)30 b([)p Fr(b)957 1722 y Fm(\000)957 1783 y(1)1032 1758 y FB(\()p Fr(\034)11 b FB(\))p Fr(;)17 b(b)1246 1773 y Fm(1)1322 1758 y FB(\()p Fr(\034)11 b FB(\)].)43 b(While)31 b(the)h(limit)c(\(3.3\))k(can)g(b)s (e)g(established)f(using)257 1879 y(transfer)39 b(matrices,)f(compare)f (with)h(Section)f(4,)i(the)f(existence)i(of)d(the)h(limit)d(shap)s(e) 257 1999 y(has)g(b)s(een)h(pro)m(v)m(ed)g(b)s(efore)e(b)m(y)i(Cerf)f (and)g(Ken)m(y)m(on)h([2)o(].)50 b(Instead)36 b(of)e(\(1.8\),)h(they)g (used)257 2120 y(the)h(\014xed)h(v)m(olume)e(constrain)m(t)h Fr(V)21 b FB(\()p Fr(h)p FB(\))33 b(=)g(2)p Fr(\020)1912 2135 y Fw(R)1969 2120 y FB(\(3\))p Fr(T)2165 2083 y Fp(3)2204 2120 y FB(,)k(resp.)f Fr(V)22 b FB(\()p Fr(h)p FB(\))33 b Fo(\024)g FB(2)p Fr(\020)2950 2135 y Fw(R)3007 2120 y FB(\(3\))p Fr(T)3203 2083 y Fp(3)3242 2120 y FB(,)k(with)257 2240 y Fr(\020)300 2255 y Fw(R)387 2240 y FB(the)30 b(Riemann)f(zeta)h 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5107 y Fo(\000)i Fr(j)6 b FB(\))22 b Fo(\000)h FB(\()p Fr(t)2431 5066 y Fm(0)2477 5107 y Fo(\000)f Fr(t)p FB(\))17 b(ln)o(\(1)22 b Fo(\000)h Fr(e)3001 5066 y Fm(\000j)o Fw(\034)3106 5075 y Fk(0)3141 5066 y Fm(j)p Fw(=)p Fp(2)3236 5107 y Fr(e)3281 5066 y Fw(ik)3348 5107 y FB(\))3386 5027 y Fn(\003)3295 5306 y FB(\(4.3\))p eop %%Page: 19 19 19 18 bop 257 266 a Ft(4)98 b(BULK)32 b(SCALING,)h(LOCAL)g(EQUILIBRIUM) 1122 b FB(19)257 573 y(for)32 b Fr(\034)448 588 y Fp(0)516 573 y Fr(<)27 b FB(0,)33 b(where)1120 837 y Fr(I)8 b FB(\()p Fr(t;)17 b(t)1323 796 y Fm(0)1347 837 y FB(\))27 b(=)1516 697 y Fn(\032)1632 776 y FB([)p Fo(\000)p Fr(\031)t(\032;)17 b(\031)t(\032)p FB(])p Fr(;)186 b FB(if)31 b Fr(t)d Fo(\025)g Fr(t)2530 740 y Fm(0)2554 776 y Fr(;)1632 897 y FB([)p Fr(\031)t(\032;)17 b FB(2)p Fr(\031)26 b Fo(\000)d Fr(\031)t(\032)p FB(])p Fr(;)33 b FB(if)e Fr(t)d(<)f(t)2528 861 y Fm(0)2552 897 y Fr(;)257 1107 y FB(and)34 b(sgn\()p Fr(t)23 b Fo(\000)g Fr(t)820 1071 y Fm(0)843 1107 y FB(\))29 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y(at)37 b Fr(k)g FB(=)e(0)h(the)h(starting)f(p)s(oin)m(t) g(of)g(a)h(branc)m(h)g(cut)g(of)f(the)i(term)e(in)g(the)h(exp)s(onen)m (tial,)257 5115 y(but)28 b(still)d(the)i(in)m(tegral)f(remains)g(unc)m (hanged)j(b)m(y)f(the)g(ab)s(o)m(v)m(e)g(deformation.)39 b(Since)28 b(the)257 5235 y(in)m(tegrands)j(are)f(2)p Fr(\031)t FB(-p)s(erio)s(dic)d(along)i(the)i(real)e(axis,)h(the)h (\014rst)f(and)g(the)h(last)e(in)m(tegrals)257 5356 y(cancels)35 b(exactly)-8 b(.)47 b Fr(\014)1014 5371 y Fp(1)1054 5356 y FB(\()p Fr(T)14 b FB(\))33 b(is)h(determined)f(suc)m(h)j(that)d(the)i (terms)e(in)g(the)i(exp)s(onen)m(tial)p eop %%Page: 27 27 27 26 bop 257 266 a Ft(5)98 b(EDGE)32 b(SCALING)2235 b FB(27)257 573 y(are)33 b(purely)g(imaginary)-8 b(.)40 b(W)-8 b(e)33 b(obtain)422 821 y Fr(\014)477 836 y Fw(i)506 821 y FB(\()p Fr(T)14 b FB(\))27 b(=)g Fo(\000)870 754 y FB(1)p 870 798 49 4 v 870 889 a(2)946 740 y Fn(\000)992 821 y Fr(s)1038 836 y Fw(i)1066 821 y Fr(T)1137 780 y Fp(2)p Fw(=)p Fp(3)1263 821 y 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b(same)g(strategy)g(has)g(b)s(een)g(used)257 5027 y(in)h([6].)80 b(In)45 b(the)g(case)g Fr(s)1171 5042 y Fp(2)1259 5027 y Fr(<)j(s)1429 5042 y Fp(1)1468 5027 y FB(,)g(for)c(large)f Fr(L)p FB(,)48 b(the)e(steep)s(est)g(descend)h(curv)m(e)f(do)s(es)257 5147 y(not)53 b(exist)h(an)m(ymore.)106 b(On)53 b(the)g(other)h(hand)f (the)h(term)f Fr(e)2598 5111 y Fm(\000)p Fw(L)p Fp(\()p Fw(s)2761 5120 y Fk(1)2796 5111 y Fm(\000)p Fw(s)2884 5120 y Fk(2)2918 5111 y Fp(\))3002 5147 y FB(serv)m(es)j(as)d(a)257 5268 y(con)m(v)m(ergence)36 b(factor)c(and)g(w)m(e)i(only)e(need)i(to)e (\014nd)h(b)s(ounds)g(for)f(the)h Fr(J)2875 5283 y Fw(i)2903 5268 y FB(\()p Fr(L)p FB(\).)p eop %%Page: 28 28 28 27 bop 257 266 a Ft(5)98 b(EDGE)32 b(SCALING)2235 b FB(28)257 573 y Fb(Con)m(v)m(ergence)38 b(for)g Fr(L)f Fb(in)g(a)h(b)s(ounded)h(set.)257 693 y FB(Let)33 b Fr(L)28 b Fo(2)g(B)s FB(.)44 b(The)34 b(in)m(tegral)d Fr(J)1369 708 y Fp(1)1408 693 y FB(\()p Fr(L)p FB(\))i(is)f(written)g(as)1100 930 y Fr(J)1154 945 y 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Fk(1)p Fh(=)p Fk(3)1188 824 y FB(d)p Fr(k)1296 680 y Fn(?)1296 740 y(?)1296 799 y(?)1296 859 y(?)1390 824 y Fo(\024)28 b Fr(e)1540 783 y Fm(\000)p Fw(\016)r(T)1701 689 y Fn(Z)1801 715 y Fw(\031)1756 914 y(")1916 757 y FB(2)p 1874 801 132 4 v 1874 821 a Fo(p)p 1957 821 49 4 v 83 x FB(3)2016 824 y Fr(e)2061 783 y Fp(\(Re)13 b Fw( )r Fp(\()p Fw(k)r Fp(\))p Fm(\000)p Fw(\016)r Fp(\))p Fw(T)2497 824 y Fr(e)2542 780 y Fm(O)2600 788 y FB(\()2638 780 y Fp(\()p Fw(L)p Fm(\000)p Fw(r)2800 789 y Fk(1)2834 780 y Fp(\))p Fw(T)2912 757 y Fk(1)p Fh(=)p Fk(3)3010 780 y Fp(+)p Fw(s)3098 789 y Fk(1)3132 780 y Fw(T)3183 757 y Fk(2)p Fh(=)p Fk(3)3279 788 y FB(\))3338 824 y(d)p Fr(x)q(:)3246 1017 y FB(\(5.31\))257 1235 y(F)-8 b(or)36 b(\()p Fr(L;)17 b(s)630 1250 y Fp(1)670 1235 y Fr(;)g(r)758 1250 y Fp(1)797 1235 y FB(\))34 b Fo(2)h(B)s FB(,)j(the)f(in)m(tegral)e (on)i(the)g(righ)m(t)f(side)g(is)g(uniformly)f(b)s(ounded)i(and)257 1356 y(therefore)1057 1463 y Fn(?)1057 1522 y(?)1057 1582 y(?)1057 1642 y(?)1123 1472 y(Z)1192 1680 y Fg(e)1179 1697 y Fw(C)1234 1667 y Fk(+)1229 1708 y Fh(")1305 1607 y Fr(e)1350 1566 y Fw( )1396 1577 y Fk(1)p Fh(;T)1495 1566 y Fp(\()p Fw(k)r Fp(\))p Fw(T)1643 1607 y Fr(e)1688 1566 y Fw(ik)r(LT)1850 1543 y Fk(1)p Fh(=)p Fk(3)1967 1607 y FB(d)p Fr(k)2075 1463 y Fn(?)2075 1522 y(?)2075 1582 y(?)2075 1642 y(?)2170 1607 y FB(=)27 b Fo(O)2372 1526 y Fn(\000)2417 1607 y Fr(e)2462 1566 y Fm(\000)p Fw(\016)r(T)2606 1526 y Fn(\001)2669 1607 y Fr(:)550 b FB(\(5.32\))257 1912 y(Similarly)29 b(for)j(the)h(in)m(tegral)e (along)1623 1887 y Fn(e)1604 1912 y Fr(C)1681 1876 y Fm(\000)1674 1936 y Fw(")1740 1912 y FB(.)p 3425 1912 4 66 v 3429 1849 59 4 v 3429 1912 V 3487 1912 4 66 v 257 2107 a Fb(Lemma)37 b(5.4.)50 b Fs(Uniformly)34 b(for)h FB(\()p Fr(L;)17 b(r)1690 2122 y Fp(1)1729 2107 y Fr(;)g(s)1819 2122 y Fp(1)1858 2107 y FB(\))28 b Fo(2)g(B)s Fs(,)35 b(one)g(has)813 2383 y Fr(J)867 2398 y Fp(1)906 2383 y FB(\()p Fr(L)p FB(\))28 b(=)f Fo(O)1278 2302 y Fn(\000)1324 2383 y Fr(e)1369 2342 y Fm(\000)p Fw(\016)r(T)1513 2302 y Fn(\001)1580 2383 y FB(+)22 b Fo(O)1777 2302 y Fn(\000)1823 2383 y Fr(T)1894 2342 y Fm(\000)p Fp(1)p Fw(=)p Fp(3)2058 2302 y Fn(\001)2126 2383 y FB(+)2234 2316 y(2)p Fr(\031)p 2234 2360 108 4 v 2260 2451 a(\024)2368 2383 y FB(Ai)2485 2243 y Fn(\022)2569 2316 y Fr(r)2613 2331 y Fp(1)2674 2316 y Fo(\000)h Fr(L)p 2569 2360 272 4 v 2676 2451 a(\024)2850 2243 y Fn(\023)3246 2383 y FB(\(5.33\))257 2668 y Fs(for)35 b(lar)-5 b(ge)34 b Fr(T)14 b Fs(,)35 b(with)g Fr(\024)27 b FB(=)1211 2630 y Fk(3)1186 2593 y Fo(p)p 1269 2593 151 4 v 75 x FB(2)p Fr(\013)1380 2683 y Fp(3)1419 2668 y Fs(.)257 2862 y Fa(Pr)n(oof:)50 b FB(By)32 b(Lemma)e(5.3)i(w)m(e)h (ha)m(v)m(e)g(to)f(ev)-5 b(aluate)31 b(the)h(con)m(tribution)f(of)g (the)i(in)m(tegral)257 2983 y(along)f Fr(C)588 2998 y Fw(")624 2983 y FB(.)43 b(F)-8 b(or)32 b Fr(k)k FB(close)c(to)h(0)f(w)m (e)i(ha)m(v)m(e)718 3241 y Fr( )781 3256 y Fp(1)p Fw(;T)891 3241 y FB(\()p Fr(k)s FB(\))p Fr(T)i FB(+)22 b Fr(ik)s(LT)1436 3200 y Fp(1)p Fw(=)p Fp(3)1629 3241 y FB(=)83 b Fo(\000)1875 3173 y FB(2)p 1875 3218 49 4 v 1875 3309 a(3)1935 3241 y Fr(i\013)2030 3256 y Fp(3)2069 3241 y Fr(k)2123 3200 y Fp(3)2163 3241 y Fr(T)36 b Fo(\000)22 b Fr(ik)s(T)2513 3200 y Fp(1)p Fw(=)p Fp(3)2623 3241 y FB(\()p Fr(r)2705 3256 y Fp(1)2767 3241 y Fo(\000)g Fr(L)p FB(\))276 b(\(5.34\))1788 3433 y(+)33 b Fo(O)1996 3353 y Fn(\000)2042 3433 y Fr(s)2088 3392 y Fp(2)2088 3458 y(1)2127 3433 y Fr(k)25 b FB(+)d Fr(s)2347 3448 y Fp(1)2387 3433 y Fr(k)2441 3392 y Fp(3)2480 3433 y Fr(T)2551 3392 y Fp(2)p Fw(=)p Fp(3)2683 3433 y FB(+)g Fr(k)2835 3392 y Fp(5)2875 3433 y Fr(T)2946 3353 y Fn(\001)3008 3433 y Fr(:)257 3653 y FB(Let)40 b Fr(C)516 3616 y Fp(+)509 3677 y Fw(")614 3653 y FB(b)s(e)f(the)h (part)f(of)f Fr(C)46 b FB(with)39 b Fr(x)g Fo(2)g FB([0)p Fr(;)17 b(")p FB(])39 b(and)g Fr(C)2312 3616 y Fm(\000)2305 3677 y Fw(")2410 3653 y FB(the)h(one)f(with)g Fr(x)g Fo(2)g FB([)p Fo(\000)p Fr(";)17 b FB(0].)257 3773 y(Then)514 3693 y Fn(R)561 3807 y Fw(C)611 3815 y Fh(")666 3773 y Fo(\001)g(\001)g(\001)29 b FB(=)920 3693 y Fn(R)967 3807 y Fw(C)1022 3778 y Fk(+)1017 3818 y Fh(")1094 3773 y Fo(\001)17 b(\001)g(\001)22 b FB(+)1333 3693 y Fn(R)1380 3807 y Fw(C)1435 3778 y Fd(\000)1430 3818 y Fh(")1508 3773 y Fo(\001)17 b(\001)g(\001)n Fr(:)35 b FB(W)-8 b(e)35 b(consider)g(explicitly)e(only)h(one)h(of)f(the)h(t)m(w)m(o)257 3893 y(in)m(tegrals,)d(the)h(second)h(b)s(eing)e(ev)-5 b(aluated)32 b(in)g(the)h(same)f(fashion,)547 4155 y Fr(T)618 4114 y Fp(1)p Fw(=)p Fp(3)745 4020 y Fn(Z)800 4245 y Fw(C)855 4215 y Fk(+)850 4256 y Fh(")926 4155 y Fr(e)971 4114 y Fw( )1017 4125 y Fk(1)p Fh(;T)1116 4114 y Fp(\()p Fw(k)r Fp(\))p Fw(T)10 b Fp(+)p Fw(ik)r(LT)1477 4091 y Fk(1)p Fh(=)p Fk(3)1594 4155 y FB(d)p Fr(k)720 4425 y FB(=)83 b Fr(T)950 4383 y Fp(1)p Fw(=)p Fp(3)1077 4289 y Fn(Z)1132 4515 y Fw(C)1187 4485 y Fk(+)1182 4525 y Fh(")1258 4425 y Fr(e)1303 4383 y Fm(\000)1368 4356 y Fk(2)p 1368 4368 31 3 v 1368 4409 a(3)1409 4383 y Fw(i\013)1478 4392 y Fk(3)1513 4383 y Fw(k)1552 4360 y Fk(3)1586 4383 y Fw(T)1641 4425 y Fr(e)1686 4383 y Fm(\000)p Fw(ik)r(T)1855 4360 y Fk(1)p Fh(=)p Fk(3)1951 4383 y Fp(\()p Fw(r)2010 4392 y Fk(1)2045 4383 y Fm(\000)p Fw(L)p Fp(\))2180 4425 y Fr(e)2225 4381 y Fm(O)2283 4389 y FB(\()2321 4381 y Fw(s)2354 4357 y Fk(2)2354 4401 y(1)2388 4381 y Fw(k)r Fp(+)p Fw(s)2515 4390 y Fk(1)2549 4381 y Fw(k)2588 4357 y Fk(3)2622 4381 y Fw(T)2673 4357 y Fk(2)p Fh(=)p Fk(3)2769 4381 y Fp(+)p Fw(k)2863 4357 y Fk(5)2897 4381 y Fw(T)2948 4389 y FB(\))3007 4425 y(d)p Fr(k)134 b FB(\(5.35\))720 4694 y(=)83 b Fr(T)950 4653 y Fp(1)p Fw(=)p Fp(3)1077 4558 y Fn(Z)1132 4784 y Fw(C)1187 4754 y Fk(+)1182 4794 y Fh(")1258 4694 y Fr(e)1303 4653 y Fm(\000)1368 4626 y Fk(2)p 1368 4638 V 1368 4679 a(3)1409 4653 y Fw(i\013)1478 4662 y Fk(3)1513 4653 y Fw(k)1552 4629 y Fk(3)1586 4653 y Fw(T)1641 4694 y Fr(e)1686 4653 y Fm(\000)p Fw(ik)r(T)1855 4629 y Fk(1)p Fh(=)p Fk(3)1951 4653 y Fp(\()p Fw(r)2010 4662 y Fk(1)2045 4653 y Fm(\000)p Fw(L)p Fp(\))2196 4694 y FB(d)q Fr(k)25 b FB(+)d Fr(E)2497 4709 y Fp(1)2536 4694 y FB(\()p Fr(L)p FB(\))p Fr(:)257 4978 y FB(The)34 b(error)e(term)g(is)g(the)h(in)m(tegral)e(along)h Fr(C)1888 4941 y Fp(+)1881 5002 y Fw(")1979 4978 y FB(with)g(in)m(tegrand)339 5220 y Fr(T)410 5179 y Fp(1)p Fw(=)p Fp(3)520 5220 y Fr(e)565 5179 y Fm(\000)630 5151 y Fk(2)p 630 5163 V 630 5205 a(3)671 5179 y Fw(i\013)740 5188 y Fk(3)775 5179 y Fw(k)814 5155 y Fk(3)848 5179 y Fw(T)903 5220 y Fr(e)948 5179 y Fm(\000)p Fw(ik)r(T)1117 5155 y Fk(1)p Fh(=)p Fk(3)1213 5179 y Fp(\()p Fw(r)1272 5188 y Fk(1)1307 5179 y Fm(\000)p Fw(L)p Fp(\))1441 5220 y FB(\()p Fr(e)1524 5176 y Fm(O)1582 5184 y FB(\()1620 5176 y Fw(s)1653 5152 y Fk(2)1653 5196 y(1)1688 5176 y Fw(k)r Fp(+)p Fw(s)1815 5185 y Fk(1)1848 5176 y Fw(k)1887 5152 y Fk(3)1921 5176 y Fw(T)1972 5152 y Fk(2)p Fh(=)p Fk(3)2069 5176 y Fp(+)p Fw(k)2163 5152 y Fk(5)2197 5176 y Fw(T)2248 5184 y FB(\))2312 5220 y Fo(\000)23 b FB(1\))747 b(\(5.36\))347 5388 y(=)33 b Fr(T)527 5347 y Fp(1)p Fw(=)p Fp(3)636 5388 y Fr(e)681 5347 y Fm(\000)746 5320 y Fk(2)p 746 5332 V 746 5373 a(3)787 5347 y Fw(i\013)856 5356 y Fk(3)891 5347 y Fw(k)930 5324 y Fk(3)964 5347 y Fw(T)1019 5388 y Fr(e)1064 5347 y Fm(\000)p Fw(ik)r(T)1233 5324 y Fk(1)p Fh(=)p Fk(3)1329 5347 y Fp(\()p Fw(r)1388 5356 y Fk(1)1423 5347 y Fm(\000)p Fw(L)p Fp(\))1558 5388 y Fr(e)1603 5344 y Fm(O)1661 5352 y FB(\()1699 5344 y Fw(s)1732 5321 y Fk(2)1732 5365 y(1)1766 5344 y Fw(k)r Fp(+)p Fw(s)1893 5353 y Fk(1)1927 5344 y Fw(k)1966 5321 y Fk(3)2000 5344 y Fw(T)2051 5321 y Fk(2)p Fh(=)p Fk(3)2147 5344 y Fp(+)p Fw(k)2241 5321 y Fk(5)2275 5344 y Fw(T)2326 5352 y FB(\))2368 5388 y Fo(O)2467 5307 y Fn(\000)2513 5388 y Fr(s)2559 5347 y Fp(2)2559 5413 y(1)2598 5388 y Fr(k)25 b FB(+)d Fr(s)2818 5403 y Fp(1)2858 5388 y Fr(k)2912 5347 y Fp(3)2951 5388 y Fr(T)3022 5347 y Fp(2)p Fw(=)p Fp(3)3154 5388 y FB(+)g Fr(k)3306 5347 y Fp(5)3346 5388 y Fr(T)3417 5307 y Fn(\001)3462 5388 y Fr(:)p eop %%Page: 31 31 31 30 bop 257 266 a Ft(5)98 b(EDGE)32 b(SCALING)2235 b FB(31)257 573 y(The)35 b(term)e(in)h(the)g(exp)s(onen)m(tial)f(is)g Fo(\000)1688 534 y Fp(2)p 1688 550 36 4 v 1688 607 a(3)1734 573 y Fr(i\013)1829 588 y Fp(3)1869 573 y Fr(k)1923 537 y Fp(3)1962 573 y Fr(T)14 b FB(\(1)23 b(+)g Fr(\037)2303 588 y Fp(1)2342 573 y FB(\))g Fo(\000)h Fr(ik)s(T)2662 537 y Fp(1)p Fw(=)p Fp(3)2772 573 y FB(\()p Fr(r)2854 588 y Fp(1)2916 573 y Fo(\000)g Fr(L)p FB(\)\(1)f(+)g Fr(\037)3391 588 y Fp(2)3430 573 y FB(\),)257 693 y(where)32 b Fr(\037)598 708 y Fp(1)669 693 y FB(and)f Fr(\037)918 708 y Fp(2)988 693 y FB(can)g(b)s(e)g(made)f(arbitrarily)e(small)h(b)m (y)j(taking)e Fr(")g FB(small)f(enough)i(\()p Fr(s)3456 708 y Fp(1)257 814 y FB(is)h(b)s(ounded\).)45 b(With)32 b(the)h(c)m(hange)g(of)f(v)-5 b(ariable)31 b Fr(z)h FB(=)c Fr(k)s(T)2355 777 y Fp(1)p Fw(=)p Fp(3)2497 814 y FB(w)m(e)34 b(obtain)341 1082 y Fr(E)413 1097 y Fp(1)453 1082 y FB(\()p Fr(L)p FB(\))28 b(=)802 1015 y(1)p 736 1059 181 4 v 736 1150 a Fr(T)807 1122 y Fp(1)p Fw(=)p Fp(3)944 946 y Fn(Z)999 1172 y Fw(T)1050 1152 y Fk(1)p Fh(=)p Fk(3)1147 1172 y Fw(C)1202 1142 y Fk(+)1197 1183 y Fh(")1084 1082 y Fr(e)1129 1041 y Fm(\000)p Fw(i)1218 1014 y Fk(2)p 1218 1026 31 3 v 1218 1067 a(3)1259 1041 y Fw(\013)1304 1050 y Fk(3)1339 1041 y Fp(\(1+)p Fw(\037)1500 1050 y Fk(1)1535 1041 y Fp(\))p Fw(z)1598 1017 y Fk(3)1632 1041 y Fm(\000)p Fw(i)p Fp(\(1+)p Fw(\037)1872 1050 y Fk(2)1907 1041 y Fp(\)\()p Fw(r)1993 1050 y Fk(1)2029 1041 y Fm(\000)p Fw(L)p Fp(\))p Fw(z)2199 1082 y Fo(O)2297 1001 y Fn(\000)2343 1082 y Fr(s)2389 1041 y Fp(2)2389 1107 y(1)2429 1082 y Fr(z)e FB(+)c Fr(s)2644 1097 y Fp(1)2684 1082 y Fr(z)2733 1041 y Fp(3)2795 1082 y FB(+)g Fr(z)2942 1041 y Fp(5)2982 1082 y Fr(T)3053 1041 y Fm(\000)p Fp(1)p Fw(=)p Fp(3)3218 1001 y Fn(\001)3280 1082 y FB(d)p Fr(z)5 b(:)3246 1275 y FB(\(5.37\))257 1488 y(Remark)23 b(that)g(at)g(the)g(b)s(oundary)h (of)f(the)g(in)m(tegration,)h(the)f(real)f(part)h(of)g(the)g(in)m (tegrand)257 1609 y(b)s(eha)m(v)m(es)31 b(as)d Fr(e)777 1572 y Fm(\000)842 1545 y Fk(2)p 842 1557 V 842 1598 a(3)882 1572 y Fw(\013)927 1581 y Fk(3)962 1572 y Fw(")995 1548 y Fk(3)1029 1572 y Fw(T)1085 1609 y FB(.)42 b(This)28 b(in)m(tegral)e(is)i(uniformly)e(b)s(ounded)j(in)e Fr(T)42 b FB(for)28 b(\()p Fr(L;)17 b(r)3195 1624 y Fp(1)3234 1609 y Fr(;)g(s)3324 1624 y Fp(1)3363 1609 y FB(\))28 b Fo(2)257 1729 y(B)s FB(.)42 b(The)27 b(same)f(holds)f(for)h(the)g(in) m(tegral)e(on)i Fr(C)1932 1693 y Fm(\000)1925 1754 y Fw(")1991 1729 y FB(.)41 b(Consequen)m(tly)28 b Fr(E)2729 1744 y Fp(1)2769 1729 y FB(\()p Fr(L)p FB(\))g(=)g Fo(O)3141 1648 y Fn(\000)3187 1729 y Fr(T)3258 1693 y Fm(\000)p Fp(1)p Fw(=)p Fp(3)3422 1648 y Fn(\001)3468 1729 y FB(.)404 1858 y(Next)56 b(w)m(e)g(extend)g(the)f(in)m(tegration)f(from)g Fr(C)2202 1873 y Fw(")2293 1858 y FB(to)h Fo(\000)p Fr(\031)t(T)2642 1822 y Fp(1)p Fw(=)p Fp(3)2752 1858 y FB(\(1)p Fr(;)17 b FB(cos\()p Fr(\031)t(=)p FB(6\)\))54 b(and)257 1979 y Fr(\031)t(T)387 1942 y Fp(1)p Fw(=)p Fp(3)497 1979 y FB(\(1)p Fr(;)17 b Fo(\000)g FB(cos\()p Fr(\031)t(=)p FB(6\)\),)42 b(obtaining)d(the)i(path)g Fr(D)2127 1994 y Fp(1)2166 1979 y FB(.)68 b(In)42 b(this)e(w)m(a)m(y)i(w)m(e)g(add)f (an)f(error)257 2099 y(of)g Fo(O)475 2018 y Fn(\000)521 2099 y Fr(e)566 2063 y Fm(\000)p Fw(\016)654 2039 y Fd(0)677 2063 y Fp(\()p Fw(")p Fp(\))p Fw(T)820 2018 y Fn(\001)906 2099 y FB(with)g Fr(\016)1183 2063 y Fm(0)1206 2099 y FB(\()p Fr(")p FB(\))h Fo(\030)h Fr(")1534 2063 y Fp(3)1573 2099 y FB(.)67 b(Similarly)37 b(w)m(e)k(can)g(complete)f(the)h(path)g (up)g(to)257 2233 y Fr(x)31 b FB(=)g Fo(\006)p Fr(N)10 b(\031)t(T)745 2196 y Fp(1)p Fw(=)p Fp(3)855 2233 y FB(,)35 b Fr(y)f FB(=)c Fo(\000)p Fr(N)10 b(\031)t(T)1400 2196 y Fp(1)p Fw(=)p Fp(3)1511 2233 y Fr(=)1560 2150 y Fo(p)p 1642 2150 49 4 v 1642 2233 a FB(3)34 b(b)m(y)i(straigh)m(t)d(lines.)48 b(The)35 b(in)m(tegral)e(is)g(equal)i(to)257 2353 y(the)e(in)m (tegration)d(from)h Fo(\000)p Fr(N)10 b(\031)t(T)1445 2317 y Fp(1)p Fw(=)p Fp(3)1587 2353 y FB(to)32 b Fr(N)10 b(\031)t(T)1924 2317 y Fp(1)p Fw(=)p Fp(3)2034 2353 y FB(,)32 b(since)g(the)g(function)g(is)f(2)p Fr(\031)t(T)3156 2317 y Fp(1)p Fw(=)p Fp(3)3297 2353 y FB(p)s(eri-)257 2473 y(o)s(dic)k(in)h(the)g(real)f(direction)g(and)i(the)f(error)g (added)h(b)m(y)g(completing)d(the)i(in)m(tegral)f(is)257 2594 y(exp)s(onen)m(tially)25 b(small)e(in)i Fr(T)14 b FB(,)26 b(for)f(all)f Fr(N)10 b FB(.)41 b(Therefore)27 b(w)m(e)g(ma)m(y)e(tak)m(e)h(the)g(limit)c Fr(N)38 b Fo(!)28 b(1)p FB(.)404 2714 y(Finally)i(w)m(e)j(obtain)f(\(5.33\),)g (since)958 2856 y Fn(Z)1058 2882 y Fm(1)1013 3081 y(\0001)1160 2991 y Fr(e)1205 2950 y Fm(\000)p Fw(i)1294 2923 y Fk(2)p 1294 2935 31 3 v 1294 2976 a(3)1334 2950 y Fw(\013)1379 2959 y Fk(3)1414 2950 y Fw(z)1450 2927 y Fk(3)1484 2950 y Fm(\000)p Fw(iz)s Fp(\()p Fw(r)1658 2959 y Fk(1)1693 2950 y Fm(\000)p Fw(L)p Fp(\))1844 2991 y FB(d)p Fr(z)g 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4261 151 4 v 75 x FB(2)p Fr(\013)853 4351 y Fp(3)892 4336 y Fs(.)257 4531 y Fa(Pr)n(oof:)50 b FB(Since)32 b(\()p Fr(r)986 4546 y Fp(1)1025 4531 y Fr(;)17 b(r)1113 4546 y Fp(2)1152 4531 y FB(\))28 b Fo(2)g(B)s FB(,)k(let)f(us)i(set)f Fr(L)1921 4546 y Fp(0)1993 4531 y FB(suc)m(h)h(that)f Fr(L)2489 4546 y Fp(0)2556 4531 y Fo(\024)d FB(2\()p Fo(j)o Fr(r)2820 4546 y Fp(1)2859 4531 y Fo(j)21 b FB(+)f Fo(j)p Fr(r)3076 4546 y Fp(2)3115 4531 y Fo(j)h FB(+)f(1\))32 b(for)257 4651 y(all)27 b Fr(r)433 4666 y Fp(1)473 4651 y Fr(;)17 b(r)561 4666 y Fp(2)600 4651 y FB(.)42 b Fr(K)752 4666 y Fw(T)836 4651 y FB(can)29 b(b)s(e)g(transformed)g(in)m(to)f(an)h(in)m(tegral)f (adding)g(an)h(error)g Fo(O)3141 4571 y Fn(\000)3187 4651 y Fr(T)3258 4615 y Fm(\000)p Fp(1)p Fw(=)p Fp(3)3422 4571 y Fn(\001)3468 4651 y FB(.)257 4772 y(Let)k(us)g(\014x)h(an)e Fr(")g FB(with)g(0)c Fr(<)f(")h Fo(\034)f FB(1.)43 b(Then)269 5054 y Fo(\000)p Fr(K)429 5069 y Fw(T)485 5054 y FB(\()p Fr(r)567 5069 y Fp(2)606 5054 y Fr(;)17 b(s)696 5069 y Fp(2)735 5054 y FB(;)g Fr(r)823 5069 y Fp(1)862 5054 y Fr(;)g(s)952 5069 y Fp(1)992 5054 y FB(\))83 b(=)1271 4918 y Fn(Z)1371 4944 y Fw(L)1419 4953 y Fk(0)1327 5144 y Fp(0)1474 5054 y Fr(J)1528 5069 y Fp(1)1568 5054 y FB(\()p Fr(L)p FB(\))p Fr(J)1764 5069 y Fp(2)1803 5054 y FB(\()p Fr(L)p FB(\))1955 4986 y Fr(e)2000 4950 y Fm(\000)p Fw(LX)p 1955 5031 216 4 v 1990 5122 a FB(4)p Fr(\031)2098 5093 y Fp(2)2198 5054 y FB(d)p Fr(L)22 b FB(+)2438 4918 y Fn(Z)2538 4944 y Fw("T)2622 4921 y Fk(2)p Fh(=)p Fk(3)2494 5144 y Fw(L)2542 5153 y Fk(0)2639 5054 y Fr(J)2693 5069 y Fp(1)2733 5054 y FB(\()p Fr(L)p FB(\))p Fr(J)2929 5069 y Fp(2)2968 5054 y FB(\()p Fr(L)p FB(\))3120 4986 y Fr(e)3165 4950 y Fm(\000)p Fw(LX)p 3120 5031 V 3155 5122 a FB(4)p Fr(\031)3263 5093 y Fp(2)3363 5054 y FB(d)p Fr(L)1271 5335 y FB(+)1397 5199 y Fn(Z)1497 5226 y Fm(1)1452 5425 y Fw("T)1536 5405 y Fk(2)p Fh(=)p Fk(3)1654 5335 y Fr(J)1708 5350 y Fp(1)1747 5335 y FB(\()p Fr(L)p FB(\))p Fr(J)1943 5350 y Fp(2)1982 5335 y FB(\()p Fr(L)p FB(\))2134 5268 y Fr(e)2179 5231 y Fm(\000)p Fw(LX)p 2134 5312 V 2169 5403 a FB(4)p Fr(\031)2277 5374 y Fp(2)2377 5335 y FB(d)p Fr(L)g FB(+)g Fo(O)2716 5254 y Fn(\000)2762 5335 y Fr(T)2833 5294 y Fm(\000)p Fp(1)p Fw(=)p Fp(3)2997 5254 y Fn(\001)3246 5335 y FB(\(5.40\))p eop %%Page: 32 32 32 31 bop 257 266 a Ft(5)98 b(EDGE)32 b(SCALING)2235 b FB(32)257 573 y(with)38 b Fr(X)44 b FB(=)733 534 y Fp(1)p 733 550 36 4 v 733 607 a(2)778 573 y FB(\()p Fr(s)862 588 y Fp(1)927 573 y Fo(\000)26 b Fr(s)1076 588 y Fp(2)1116 573 y FB(\)\(1)f(+)h Fo(O)19 b FB(\(1)p Fr(=T)14 b FB(\))o(\))37 b Fr(>)f FB(0.)60 b(Since)38 b Fo(j)o Fr(J)2374 588 y Fw(i)2402 573 y FB(\()p Fr(L)p FB(\))p Fo(j)f(\024)g Fr(T)2794 537 y Fp(1)p Fw(=)p Fp(3)2904 573 y FB(,)i Fr(i)e FB(=)f(1)p Fr(;)17 b FB(2,)39 b(the)257 712 y(third)e(term)f(is) h(b)s(ounded)h(b)m(y)g Fr(T)1460 676 y Fp(2)p Fw(=)p Fp(3)1570 712 y Fr(e)1615 676 y Fm(\000)p Fw("T)1754 653 y Fk(2)p Fh(=)p Fk(3)1850 676 y Fw(X)1917 712 y Fr(=X)43 b Fo(!)35 b FB(0)i(as)g Fr(T)50 b Fo(!)35 b(1)p FB(.)56 b(By)38 b(Lemma)e(5.4)257 833 y(the)d(\014rst)g(term)f(con)m(v)m (erges,)j(uniformly)c(for)h(\()p Fr(u)2011 848 y Fw(i)2038 833 y Fr(;)17 b(s)2128 848 y Fw(i)2156 833 y FB(\))28 b Fo(2)g(B)s FB(,)33 b(to)931 985 y Fn(Z)1030 1011 y Fw(L)1078 1020 y Fk(0)986 1210 y Fp(0)1134 1120 y FB(Ai)1250 980 y Fn(\022)1334 1053 y Fr(r)1378 1068 y Fp(1)1439 1053 y Fo(\000)23 b Fr(L)p 1334 1097 272 4 v 1441 1189 a(\024)1615 980 y Fn(\023)1705 1120 y FB(Ai)1822 980 y Fn(\022)1905 1053 y Fr(r)1949 1068 y Fp(2)2011 1053 y Fo(\000)f Fr(L)p 1905 1097 V 2013 1189 a(\024)2187 980 y Fn(\023)2277 1120 y Fr(e)2322 1079 y Fw(L)2380 1043 y Fh(s)2409 1058 y Fk(2)2444 1043 y Fd(\000)p Fh(s)2521 1058 y Fk(1)p 2380 1064 176 3 v 2452 1105 a(2)2603 1053 y FB(1)p 2579 1097 96 4 v 2579 1189 a Fr(\024)2635 1160 y Fp(2)2701 1120 y FB(d)q Fr(L)424 b FB(\(5.41\))257 1391 y(as)33 b Fr(T)42 b Fo(!)27 b(1)p FB(.)404 1511 y(W)-8 b(e)44 b(consider)f(the)h(second)h(term.)75 b(W)-8 b(e)44 b(ha)m(v)m(e)h(already)e(established)h(the)g(p)s(oin)m(t-)257 1632 y(wise)i(con)m(v)m(ergence)j(of)c Fr(J)1213 1647 y Fw(i)1241 1632 y FB(\()p Fr(L)p FB(\))h(to)1571 1593 y Fp(2)p Fw(\031)p 1571 1609 79 4 v 1590 1666 a(\024)1676 1632 y FB(Ai)1793 1551 y Fn(\000)1849 1591 y Fw(r)1881 1601 y Fh(i)1907 1591 y Fm(\000)p Fw(L)p 1849 1609 162 4 v 1909 1666 a(\024)2020 1551 y Fn(\001)2065 1632 y FB(.)83 b(If)46 b(w)m(e)h(obtain)d(that)i(for)f(large)f Fr(T)14 b FB(,)257 1761 y Fo(j)p Fr(J)339 1776 y Fw(i)367 1761 y FB(\()p Fr(L)p FB(\))p Fo(j)28 b(\024)g Fr(G)h FB(with)g(a)f(constan)m(t)i Fr(G)f FB(indep)s(enden)m(t)i(of)d Fr(r)2269 1776 y Fw(i)2297 1761 y Fr(;)17 b(s)2387 1776 y Fw(i)2444 1761 y FB(and)29 b Fr(L)g Fo(2)f FB([)p Fr(L)2912 1776 y Fp(0)2952 1761 y Fr(;)17 b("T)3113 1725 y Fp(2)p Fw(=)p Fp(3)3222 1761 y FB(],)30 b(then)257 1881 y(b)m(y)k(dominated)d (con)m(v)m(ergence)325 2179 y(lim)297 2241 y Fw(T)10 b Fm(!1)506 2043 y Fn(Z)605 2069 y Fw("T)689 2046 y Fk(2)p Fh(=)p Fk(3)561 2269 y Fw(L)609 2278 y Fk(0)665 2179 y Fr(J)719 2194 y Fp(1)758 2179 y FB(\()p Fr(L)p FB(\))p Fr(J)954 2194 y Fp(2)994 2179 y FB(\()p Fr(L)p FB(\))p Fr(e)1181 2138 y Fm(\000)p Fw(LX)1368 2179 y FB(d)p Fr(L)28 b FB(=)1620 2043 y Fn(Z)1719 2069 y Fm(1)1675 2269 y Fw(L)1723 2278 y Fk(0)1763 2179 y FB(Ai)1880 2038 y Fn(\022)1964 2111 y Fr(r)2008 2126 y Fp(1)2069 2111 y Fo(\000)23 b Fr(L)p 1964 2156 272 4 v 2071 2247 a(\024)2245 2038 y Fn(\023)2335 2179 y FB(Ai)2452 2038 y Fn(\022)2535 2111 y Fr(r)2579 2126 y Fp(2)2641 2111 y Fo(\000)f Fr(L)p 2535 2156 V 2643 2247 a(\024)2817 2038 y Fn(\023)2916 2111 y Fr(e)2971 2047 y Fk(1)p 2972 2059 31 3 v 2972 2101 a(2)3012 2075 y Fw(L)p Fp(\()p Fw(s)3120 2084 y Fk(2)3155 2075 y Fm(\000)p Fw(s)3243 2084 y Fk(1)3277 2075 y Fp(\))p 2916 2156 392 4 v 3065 2247 a Fr(\024)3121 2218 y Fp(2)3335 2179 y FB(d)p Fr(L)3246 2370 y FB(\(5.42\))257 2589 y(uniformly)f(for)h(\()p Fr(r)913 2604 y Fw(i)941 2589 y Fr(;)17 b(s)1031 2604 y Fw(i)1058 2589 y FB(\))28 b Fo(2)g(B)s FB(.)41 b(This)22 b(prop)s(ert)m(y)i(is)e(pro)m(v)m(en)i (in)d(the)i(follo)m(wing)d(lemma.)p 3425 2589 4 66 v 3429 2527 59 4 v 3429 2589 V 3487 2589 4 66 v 257 2784 a Fb(Lemma)37 b(5.6.)50 b Fs(F)-7 b(or)33 b Fr(L)28 b Fo(2)g FB([)p Fr(L)1340 2799 y Fp(0)1380 2784 y Fr(;)17 b("T)1541 2748 y Fp(2)p Fw(=)p Fp(3)1650 2784 y FB(])p Fs(,)34 b Fo(j)p Fr(J)1823 2799 y Fw(i)1851 2784 y FB(\()p Fr(L)p FB(\))p Fo(j)28 b(\024)g Fr(G)34 b Fs(with)g(the)g(c)-5 b(onstant)34 b Fr(G)h Fs(indep)-5 b(en-)257 2905 y(dent)35 b(of)g Fr(s)635 2920 y Fw(i)663 2905 y Fr(;)17 b(r)751 2920 y Fw(i)779 2905 y Fr(;)35 b Fs(and)f Fr(L)p Fs(,)h(pr)-5 b(ovide)g(d)34 b FB(0)27 b Fr(<)h(")f Fo(\034)g FB(1)35 b Fs(and)f Fr(T)49 b Fs(lar)-5 b(ge)34 b(enough.)257 3100 y Fa(Pr)n(oof:)50 b FB(The)26 b(exp)s(onen)m(tial)f(terms)h(in)f (\(5.20\))g(are)g(purely)h(imaginary)d(for)i Fr(k)k FB(real.)40 b(Let)257 3220 y(us)34 b(set)1192 3460 y Fr( )1259 3419 y Fw(I)1255 3484 y 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Fp(2)1441 4484 y FB(+)6 b Fr(s)1569 4499 y Fp(1)1608 4484 y Fr(T)1679 4443 y Fm(\000)p Fp(1)p Fw(=)p Fp(3)1844 4403 y Fn(\001)o(\001)1957 4484 y Fo(\000)23 b Fr(k)s FB(\()p Fr(r)2193 4499 y Fp(1)2254 4484 y Fo(\000)g Fr(L)p FB(\))p Fr(T)2529 4443 y Fm(\000)p Fp(2)p Fw(=)p Fp(3)2710 4403 y Fn(\000)2756 4484 y FB(1)f(+)g Fo(O)3024 4403 y Fn(\000)3069 4484 y Fr(s)3115 4443 y Fp(2)3115 4509 y(1)3155 4484 y Fr(T)3226 4443 y Fm(\000)p Fp(1)p Fw(=)p Fp(3)3390 4403 y Fn(\001\001)3498 4484 y Fr(:)3246 4645 y FB(\(5.45\))257 4886 y(Since)37 b(\()p Fr(r)598 4901 y Fp(1)662 4886 y Fo(\000)25 b Fr(L)p FB(\))p Fr(T)939 4850 y Fm(\000)p Fp(2)p Fw(=)p Fp(3)1137 4886 y Fo(\030)35 b(O)19 b FB(\()p Fr(")p FB(\))36 b(at)g(most,)h(w)m(e)g (set)2210 4861 y Fn(e)2202 4886 y Fr(L)d FB(=)g(\()p Fr(L)24 b Fo(\000)i Fr(r)2687 4901 y Fp(1)2726 4886 y FB(\))p Fr(T)2835 4850 y Fm(\000)p Fp(2)p Fw(=)p Fp(3)2999 4886 y FB(.)55 b Fr( )3148 4850 y Fw(I)3144 4911 y Fp(1)3188 4886 y FB(\()p Fr(k)s FB(\))36 b(has)257 5016 y(t)m(w)m(o)e(lo)s(cal)c (extrema)j(at)f Fo(\006)p Fr(k)s FB(\()1345 4991 y Fn(e)1337 5016 y Fr(L)q FB(\))g(with)1066 5271 y Fr(k)s FB(\()1166 5246 y Fn(e)1158 5271 y Fr(L)p FB(\))c(=)1393 5157 y Fn(p)p 1493 5157 67 4 v 1501 5246 a(e)1493 5271 y Fr(Lc)1601 5286 y Fp(0)1657 5160 y Fn(\020)1717 5271 y FB(1)22 b(+)g Fo(O)1984 5160 y Fn(\020)2052 5246 y(e)2044 5271 y Fr(L)g FB(+)g Fr(s)2276 5230 y Fp(2)2276 5295 y(1)2316 5271 y Fr(T)2387 5230 y Fm(\000)p Fp(1)p Fw(=)p Fp(3)2551 5160 y Fn(\021)q(\021)3246 5271 y FB(\(5.46\))p eop %%Page: 33 33 33 32 bop 257 266 a Ft(5)98 b(EDGE)32 b(SCALING)2235 b FB(33)257 573 y(and)38 b Fr(c)494 588 y Fp(0)569 573 y FB(=)e(\(2)p Fr(\013)830 588 y Fp(3)869 573 y FB(\))907 537 y Fm(\000)p Fp(1)p Fw(=)p Fp(2)1072 573 y FB(.)58 b(Moreo)m(v)m(er)39 b(for)e Fo(j)o Fr(k)s Fo(j)f(\025)g FB(2)p Fr(k)s FB(\()2151 548 y Fn(e)2143 573 y Fr(L)p FB(\),)j Fr( )2380 537 y Fw(I)2376 597 y Fp(1)2420 573 y FB(\()p Fr(k)s FB(\))e(is)g(strictly)g(decreasing.)257 723 y Fr(J)311 738 y Fp(1)351 723 y FB(\()p Fr(L)p FB(\))28 b(=)625 643 y Fn(R)691 669 y Fw(\031)672 757 y Fm(\000)p Fw(\031)791 723 y Fo(\001)17 b(\001)g(\001)26 b FB(=)1039 648 y Fn(P)1144 674 y Fp(4)1144 752 y Fw(i)p Fp(=1)1279 643 y Fn(R)1326 757 y Fw(I)1357 767 y Fh(i)1404 723 y Fo(\001)17 b(\001)g(\001)47 b FB(where)34 b Fr(I)1894 738 y Fp(1)1962 723 y FB(=)28 b([)p Fo(\000)p Fr(\031)t(;)17 b Fo(\000)p FB(2)p Fr(c)2441 738 y Fp(0)2480 614 y Fn(p)p 2580 614 67 4 v 2588 698 a(e)2580 723 y Fr(L)p FB(],)33 b Fr(I)2776 738 y Fp(2)2844 723 y FB(=)28 b([)p Fo(\000)p FB(2)p Fr(c)3143 738 y Fp(0)3183 614 y Fn(p)p 3282 614 V 3290 698 a(e)3282 723 y Fr(L)q(;)17 b FB(0],)257 893 y Fr(I)300 908 y Fp(3)386 893 y FB(=)45 b([0)p Fr(;)17 b FB(2)p Fr(c)718 908 y Fp(0)757 784 y Fn(p)p 856 784 V 864 868 a(e)856 893 y Fr(L)q FB(],)46 b(and)d Fr(I)1266 908 y Fp(4)1351 893 y FB(=)j([2)p Fr(c)1591 908 y Fp(0)1630 784 y Fn(p)p 1730 784 V 1738 868 a(e)1730 893 y Fr(L;)17 b(\031)t FB(].)75 b(The)44 b(in)m(tegrals)e(along)g Fr(I)2957 908 y Fp(1)3039 893 y FB(and)i Fr(I)3283 908 y Fp(4)3365 893 y FB(are)257 1013 y(ev)-5 b(aluated)37 b(similarly)c(and)k(so)g (are)g(the)g(in)m(tegrals)e(along)h Fr(I)2463 1028 y Fp(2)2539 1013 y FB(and)h Fr(I)2776 1028 y Fp(3)2815 1013 y FB(.)56 b(W)-8 b(e)37 b(presen)m(t)i(in)257 1158 y(detail)31 b(only)h(the)h(in)m(tegration)e(along)g Fr(I)1709 1173 y Fp(3)1781 1158 y FB(and)i Fr(I)2014 1173 y Fp(4)2054 1158 y FB(.)43 b(Let)33 b Fr(\015)f FB(=)2486 1049 y Fn(p)p 2586 1049 V 2594 1132 a(e)2586 1158 y Fr(L)p FB(.)43 b(Then)576 1289 y Fn(Z)631 1514 y Fw(I)662 1523 y Fk(4)717 1424 y Fo(\001)17 b(\001)g(\001)26 b FB(=)h Fr(T)1035 1383 y Fp(1)p Fw(=)p Fp(3)1162 1289 y Fn(Z)1261 1315 y Fw(\031)1217 1514 y Fp(2)p Fw(c)1283 1523 y Fk(0)1318 1514 y Fw(\015)1379 1424 y Fr(e)1424 1383 y Fw(i )1496 1360 y Fh(I)1494 1404 y Fk(1)1532 1383 y Fp(\()p Fw(k)r Fp(\))p Fw(T)1697 1424 y FB(d)p Fr(k)k FB(=)c Fr(T)2007 1383 y Fp(1)p Fw(=)p Fp(3)2133 1289 y Fn(Z)2233 1315 y Fw(u)p Fp(\()p Fw(\031)r Fp(\))2189 1514 y Fw(u)p Fp(\(2)p Fw(c)2323 1523 y Fk(0)2358 1514 y Fw(\015)t Fp(\))2446 1424 y Fr(f)11 b FB(\()p Fr(u)p FB(\))p Fr(e)2682 1383 y Fw(iuT)2818 1424 y FB(d)p Fr(u)318 b FB(\(5.47\))257 1717 y(where)34 b Fr(u)27 b FB(=)h Fr( )793 1680 y Fw(I)789 1741 y Fp(1)833 1717 y FB(\()p Fr(k)s FB(\))k(and)h Fr(f)11 b FB(\()p Fr(u)p FB(\))27 b(=)1516 1669 y Fp(d)p Fw(k)r Fp(\()p Fw(u)p Fp(\))p 1516 1694 174 4 v 1563 1751 a(d)p Fw(u)1700 1717 y FB(.)43 b(In)m(tegrating)32 b(b)m(y)i(parts)e(w)m(e)i (obtain)572 2007 y Fr(T)643 1966 y Fm(\000)p Fp(1)p Fw(=)p Fp(3)824 1872 y Fn(Z)879 2097 y Fw(I)910 2106 y Fk(4)965 2007 y Fo(\001)17 b(\001)g(\001)26 b FB(=)i Fr(f)11 b FB(\()p Fr(u)p FB(\))1414 1940 y Fr(e)1459 1904 y Fw(iuT)p 1414 1984 166 4 v 1443 2076 a Fr(iT)1588 1863 y Fn(?)1588 1923 y(?)1588 1982 y(?)1588 2042 y(?)1654 1889 y Fw(u)p Fp(\()p Fw(\031)r Fp(\))1654 2106 y Fw(u)p Fp(\(2)p Fw(c)1788 2115 y Fk(0)1823 2106 y Fw(\015)t Fp(\))1917 2007 y Fo(\000)2016 1872 y Fn(Z)2116 1898 y Fw(u)p Fp(\()p Fw(\031)r Fp(\))2072 2097 y Fw(u)p Fp(\(2)p Fw(c)2206 2106 y Fk(0)2241 2097 y Fw(\015)t Fp(\))2339 1940 y FB(d)p Fr(f)g FB(\()p Fr(u)p FB(\))p 2339 1984 245 4 v 2406 2076 a(d)p Fr(u)2603 1940 y(e)2648 1904 y Fw(iuT)p 2603 1984 166 4 v 2634 2076 a Fr(iT)2795 2007 y FB(d)p Fr(u:)314 b FB(\(5.48\))257 2308 y(F)-8 b(or)38 b Fr(k)j Fo(2)d Fr(I)677 2323 y Fp(4)754 2308 y FB(with)g Fo(j)p Fr(k)s Fo(j)f(\024)h Fr(")g FB(follo)m(ws)g (from)f(\(5.45\))g(that)2405 2269 y Fp(d)q Fw(u)p 2405 2285 81 4 v 2406 2342 a Fp(d)q Fw(k)2533 2308 y Fr(<)g FB(0)i(and)2939 2269 y Fp(d)2978 2243 y Fk(2)3013 2269 y Fw(u)p 2939 2285 115 4 v 2940 2342 a Fp(d)p Fw(k)3018 2323 y Fk(2)3101 2308 y Fo(\025)f FB(0.)61 b(F)-8 b(or)257 2428 y Fr(k)31 b(>)d(")p FB(,)554 2622 y(d)p Fr(u)p 554 2667 110 4 v 555 2758 a FB(d)p Fr(k)701 2689 y FB(=)813 2664 y Fn(e)805 2689 y Fr(L)22 b Fo(\000)h FB(ln)1091 2549 y Fn(\022)1174 2622 y FB(1)f(+)g Fr(e)1388 2586 y Fm(\000)p Fw(\034)1474 2595 y 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y(?)1266 3457 y(?)1266 3517 y(?)1360 3482 y Fo(\024)1541 3415 y FB(2)p 1475 3459 181 4 v 1475 3551 a Fr(T)1546 3522 y Fp(2)p Fw(=)p Fp(3)1683 3482 y FB(\()p Fo(j)o Fr(f)11 b FB(\()p Fr(u)p FB(\()p Fr(\031)t FB(\)\))p Fo(j)21 b FB(+)h Fo(j)p Fr(f)11 b FB(\()p Fr(u)p FB(\(2)p Fr(c)2531 3497 y Fp(0)2569 3482 y Fr(\015)5 b FB(\)\))p Fo(j)p FB(\))16 b Fr(:)436 b FB(\(5.50\))257 3745 y(Using)33 b(\(5.49\),)e(for)h Fr(T)47 b FB(su\016cien)m(tly)33 b(large,)325 3965 y Fo(j)p Fr(f)11 b FB(\()p Fr(u)p FB(\()p Fr(\031)t FB(\)\))p Fo(j)82 b FB(=)949 3881 y Fn(?)949 3940 y(?)1015 3965 y FB(2)17 b(ln)o(\(1)22 b Fo(\000)h Fr(e)1416 3924 y Fm(\000)p Fw(\034)1502 3933 y Fk(0)1537 3924 y Fw(=)p Fp(2)1611 3965 y FB(\))f Fo(\000)h FB(2)17 b(ln)o(\(1)22 b(+)g Fr(e)2170 3924 y Fm(\000)p Fw(\034)2256 3933 y Fk(0)2291 3924 y Fw(=)p Fp(2)2365 3965 y FB(\))g(+)g Fr(\015)2579 3924 y Fp(2)2641 3965 y FB(+)g Fo(O)2838 3885 y Fn(\000)2883 3965 y Fr(s)2929 3980 y Fp(1)2969 3965 y Fr(T)3040 3924 y Fm(\000)p Fp(1)p 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4670 y FB(+)g Fr(s)2160 4685 y Fp(1)2199 4670 y Fr(T)2270 4634 y Fm(\000)p Fp(1)p Fw(=)p Fp(3)2435 4589 y Fn(\001)p 1585 4721 896 4 v 1946 4812 a Fo(\000)p Fr(\015)2079 4783 y Fp(2)2490 4569 y Fn(?)2490 4629 y(?)2490 4689 y(?)2490 4749 y(?)2490 4808 y(?)2585 4744 y Fo(\024)28 b FB(2)p Fr(=\015)2844 4703 y Fp(2)3246 4744 y FB(\(5.52\))257 5031 y(for)k Fr(")h FB(small)d(and)j Fr(s)976 5046 y Fp(1)1043 5031 y Fo(2)28 b(B)s FB(.)44 b(Therefore)33 b(w)m(e)h(ha)m(v)m(e,)g(uniformly)c(in)i(\()p Fr(r)2753 5046 y Fp(1)2792 5031 y Fr(;)17 b(s)2882 5046 y Fp(1)2921 5031 y FB(\))28 b Fo(2)g(B)s FB(,)441 5145 y Fn(?)441 5204 y(?)441 5264 y(?)441 5324 y(?)508 5153 y(Z)563 5379 y Fw(I)594 5388 y Fk(4)649 5289 y Fo(\001)17 b(\001)g(\001)765 5145 y Fn(?)765 5204 y(?)765 5264 y(?)765 5324 y(?)859 5289 y FB(=)963 5145 y Fn(?)963 5204 y(?)963 5264 y(?)963 5324 y(?)1029 5153 y(Z)1085 5379 y Fw(I)1116 5388 y Fk(1)1171 5289 y Fo(\001)g(\001)g(\001)1287 5145 y Fn(?)1287 5204 y(?)1287 5264 y(?)1287 5324 y(?)1381 5289 y Fo(\024)1504 5222 y FB(2)p Fr(G)1630 5237 y Fp(1)p 1496 5266 181 4 v 1496 5357 a Fr(T)1567 5329 y Fp(2)p Fw(=)p Fp(3)1709 5289 y FB(+)1966 5222 y(2)p 1817 5266 348 4 v 1817 5357 a(\()p Fr(L)22 b Fo(\000)h Fr(r)2087 5372 y Fp(1)2126 5357 y FB(\))2202 5289 y Fo(\024)2325 5222 y FB(2)p Fr(G)2451 5237 y Fp(1)p 2317 5266 181 4 v 2317 5357 a Fr(T)2388 5329 y Fp(2)p Fw(=)p Fp(3)2530 5289 y FB(+)2807 5222 y(2)p 2638 5266 387 4 v 2638 5357 a(\()p Fr(L)2742 5372 y Fp(0)2804 5357 y Fo(\000)f Fr(r)2947 5372 y Fp(1)2987 5357 y FB(\))3035 5289 y Fr(:)184 b FB(\(5.53\))p eop %%Page: 34 34 34 33 bop 257 266 a Ft(5)98 b(EDGE)32 b(SCALING)2235 b FB(34)404 588 y(Next)33 b(w)m(e)h(estimate)1176 473 y Fn(?)1176 533 y(?)1176 593 y(?)1243 508 y(R)1290 622 y Fw(I)1321 631 y Fk(3)1376 588 y Fo(\001)17 b(\001)g(\001)1492 473 y Fn(?)1492 533 y(?)1492 593 y(?)1559 588 y FB(.)762 761 y Fn(Z)817 986 y Fw(I)848 995 y Fk(3)903 896 y Fo(\001)g(\001)g (\001)26 b FB(=)h Fr(T)1221 855 y Fp(1)p Fw(=)p Fp(3)1348 761 y Fn(Z)1447 787 y Fp(2)p Fw(c)1513 796 y Fk(0)1548 787 y Fw(\015)1403 986 y Fp(0)1609 896 y Fr(e)1654 855 y Fw(i )1726 832 y Fh(I)1724 876 y Fk(1)1762 855 y Fp(\()p Fw(k)r Fp(\))p Fw(T)1927 896 y FB(d)p Fr(k)k FB(=)c Fr(T)2237 855 y Fp(1)p Fw(=)p Fp(3)2364 761 y Fn(Z)2463 787 y Fw(c)2494 796 y Fk(2)2528 787 y Fw(\015)2419 986 y Fm(\000)p Fw(c)2505 995 y Fk(1)2539 986 y Fw(\015)2600 896 y Fr(e)2645 855 y Fw(i)2682 838 y Fg(e)2669 855 y Fw( )s Fp(\()p Fw(k)r Fp(\))p Fw(T)2883 896 y FB(d)p Fr(k)258 b FB(\(5.54\))257 1189 y(where)551 1163 y Fn(e)534 1189 y Fr( )t FB(\()p Fr(k)s FB(\))28 b(=)f Fr( )929 1153 y Fw(I)925 1214 y Fp(1)969 1189 y FB(\()p Fr(k)15 b Fo(\000)d Fr(k)s FB(\()1262 1164 y Fn(e)1254 1189 y Fr(L)q FB(\)\),)29 b Fr(c)1495 1204 y Fp(1)1562 1189 y FB(=)f Fr(c)1708 1204 y Fp(0)1747 1189 y FB(\(1)12 b(+)g Fo(O)19 b FB(\()p Fr(\015)5 b FB(\)\))28 b(and)g Fr(c)2457 1204 y Fp(2)2524 1189 y FB(=)f Fr(c)2669 1204 y Fp(0)2709 1189 y FB(\(1)12 b(+)g Fo(O)19 b FB(\()p Fr(\015)5 b FB(\)\).)42 b(Let)28 b(us)257 1310 y(de\014ne)35 b(the)e(paths)h Fr(\030)1018 1325 y Fp(0)1086 1310 y FB(=)28 b Fo(f)p Fr(k)j FB(=)e Fr(x;)17 b(x)29 b FB(:)g Fo(\000)p Fr(c)1785 1325 y Fp(1)1825 1310 y Fr(\015)k Fo(!)c Fr(c)2080 1325 y Fp(2)2119 1310 y Fr(\015)5 b Fo(g)p FB(,)33 b Fr(\030)2328 1325 y Fp(1)2396 1310 y FB(=)28 b Fo(f)p Fr(k)k FB(=)c Fo(\000)p Fr(c)2856 1325 y Fp(1)2896 1310 y Fr(\015)5 b(e)2997 1274 y Fm(\000)p Fw(i')3126 1310 y Fr(;)17 b(')29 b FB(:)f(0)h Fo(!)257 1430 y Fr(\031)t(=)p FB(4)p Fo(g)p FB(,)36 b Fr(\030)570 1445 y Fp(2)642 1430 y FB(=)d Fo(f)p Fr(k)k FB(=)c Fr(e)1043 1394 y Fm(\000)p Fw(i\031)r(=)p Fp(4)1239 1430 y Fr(x;)17 b(x)34 b FB(:)f Fo(\000)p Fr(c)1606 1445 y Fp(1)1646 1430 y Fr(\015)38 b Fo(!)33 b Fr(c)1910 1445 y Fp(2)1950 1430 y Fr(\015)5 b Fo(g)p FB(,)36 b Fr(\030)2162 1445 y Fp(3)2234 1430 y FB(=)d Fo(f)p Fr(k)j FB(=)d Fr(c)2631 1445 y Fp(2)2671 1430 y Fr(\015)5 b(e)2772 1394 y Fw(i')2846 1430 y Fr(;)17 b(')33 b FB(:)g Fr(\031)t(=)p FB(4)g Fo(!)g FB(0)p Fo(g)p FB(.)257 1550 y(Then)527 1470 y Fn(R)574 1585 y Fw(I)605 1594 y Fk(3)660 1550 y Fo(\001)17 b(\001)g(\001)51 b FB(=)958 1470 y Fn(R)1005 1585 y Fw(\030)1036 1594 y Fk(0)1092 1550 y Fo(\001)17 b(\001)g(\001)51 b FB(=)1390 1476 y Fn(P)1495 1502 y Fp(3)1495 1580 y Fw(i)p Fp(=1)1630 1470 y Fn(R)1677 1585 y Fw(\030)1708 1595 y Fh(i)1755 1550 y Fo(\001)17 b(\001)g(\001)e FB(.)88 b(The)48 b(in)m(tegrals)f (along)f Fr(\030)2945 1565 y Fp(1)3031 1550 y FB(and)i Fr(\030)3279 1565 y Fp(3)3365 1550 y FB(are)257 1671 y(estimated)32 b(in)g(the)h(same)g(w)m(a)m(y)-8 b(.)472 1941 y Fr(T)543 1900 y Fp(1)p Fw(=)p Fp(3)669 1806 y Fn(Z)724 2031 y Fw(\030)755 2040 y Fk(1)811 1941 y Fr(e)856 1900 y Fw(i)893 1883 y Fg(e)880 1900 y Fw( )s Fp(\()p Fw(k)r Fp(\))p Fw(T)1094 1941 y FB(d)p Fr(k)31 b FB(=)c Fr(T)1404 1900 y Fp(1)p Fw(=)p Fp(3)1530 1806 y Fn(Z)1630 1832 y Fw(\031)r(=)p Fp(4)1586 2031 y(0)1764 1941 y Fr(e)1809 1900 y Fw(i')1884 1941 y Fr(e)1929 1900 y Fw(i)1965 1883 y Fg(e)1953 1900 y Fw( )r Fp(\()p Fw(k)r Fp(\()p Fw(')p Fp(\)\))p Fw(T)2251 1941 y Fr(ic)2326 1956 y Fp(0)2365 1941 y Fr(\015)5 b FB(\(1)22 b(+)g Fo(O)e FB(\()p Fr(\015)5 b FB(\)\))17 b(d)p Fr(')214 b FB(\(5.55\))257 2205 y(and)33 b(therefore)590 2311 y Fn(?)590 2371 y(?)590 2431 y(?)590 2491 y(?)656 2456 y Fr(T)727 2415 y Fp(1)p Fw(=)p Fp(3)854 2320 y Fn(Z)909 2546 y Fw(\030)940 2555 y Fk(1)995 2456 y Fr(e)1040 2415 y Fw(i)1077 2397 y Fg(e)1064 2415 y Fw( )s Fp(\()p Fw(k)r Fp(\))p Fw(T)1278 2456 y FB(d)p Fr(k)1386 2311 y Fn(?)1386 2371 y(?)1386 2431 y(?)1386 2491 y(?)1480 2456 y Fo(\024)c FB(2)p Fr(T)1706 2415 y Fp(1)p Fw(=)p Fp(3)1815 2456 y Fr(\015)5 b(c)1913 2471 y Fp(0)1969 2320 y Fn(Z)2069 2347 y Fw(\031)r(=)p Fp(4)2024 2546 y(0)2203 2456 y Fr(e)2248 2415 y Fm(\000)p Fw(T)22 b Fp(Im)o(\()2490 2397 y Fg(e)2477 2415 y Fw( )s Fp(\()p Fw(k)r Fp(\()p Fw(')p Fp(\)\)\))2768 2456 y FB(d)q Fr(':)332 b FB(\(5.56\))257 2749 y(Since)525 2723 y Fn(e)508 2749 y Fr( )t FB(\()p Fr(k)s FB(\()p Fr(')p FB(\)\))27 b(=)993 2723 y Fn(e)976 2749 y Fr( )t FB(\(0\))14 b(+)1281 2710 y Fp(1)p 1281 2726 36 4 v 1281 2783 a(2)1343 2723 y Fn(e)1326 2749 y Fr( )1393 2713 y Fm(00)1435 2749 y FB(\(0\))p Fr(k)s FB(\()p Fr(')p FB(\))1754 2713 y Fp(2)1793 2749 y FB(\(1)g(+)g Fr(\016)2027 2764 y Fp(1)2066 2749 y FB(\()p Fr(')p FB(\)\))28 b(with)g Fr(\016)2533 2764 y Fp(1)2573 2749 y FB(\()p Fr(')p FB(\))f Fo(!)g FB(0)i(as)f Fr(")g Fo(!)f FB(0)h(and)257 2869 y Fr(k)s FB(\()p Fr(')p FB(\))451 2833 y Fp(2)518 2869 y FB(=)g Fr(c)664 2833 y Fp(2)664 2894 y(0)703 2869 y Fr(\015)759 2833 y Fp(2)799 2869 y FB(\(1)22 b(+)g Fo(O)d FB(\()p Fr(\015)5 b FB(\)\))p Fr(e)1319 2833 y Fm(\000)p Fp(2)p Fw(i')1484 2869 y FB(,)32 b(one)h(has)384 3117 y(Im)534 3091 y Fn(e)517 3117 y Fr( )t FB(\()p Fr(k)s FB(\()p Fr(')p FB(\)\))28 b(=)f Fo(\000)1072 3050 y FB(1)p 1072 3095 49 4 v 1072 3186 a(2)1148 3091 y Fn(e)1131 3117 y Fr( )1198 3076 y Fm(0)q(0)1241 3117 y FB(\(0\)\()p Fr(k)s FB(\()p Fr(')p FB(\)\))1636 3076 y Fp(2)1675 3117 y FB(\(1)22 b(+)g Fr(\016)1925 3132 y Fp(2)1964 3117 y FB(\()p Fr(')p FB(\)\))p Fr(c)2184 3076 y Fp(2)2184 3142 y(0)2223 3117 y Fr(\015)2279 3076 y Fp(2)2319 3117 y FB(\(1)g(+)g Fo(O)d FB(\()p Fr(\015)5 b FB(\)\))17 b(sin\(2)p Fr(')p FB(\))126 b(\(5.57\))257 3352 y(with)22 b Fr(\016)512 3367 y Fp(2)552 3352 y FB(\()p Fr(')p FB(\))27 b Fo(!)h FB(0)22 b(as)g Fr(")28 b Fo(!)f FB(0.)40 b(Moreo)m(v)m(er,)26 b(for)c Fr(")g FB(small)e(enough,)25 b(sin)o(\(2)p Fr(')p FB(\)\(1)q(+)q Fr(\016)3114 3367 y Fp(2)3154 3352 y FB(\()p Fr(')p FB(\)\)\(1)q(+)257 3473 y Fo(O)20 b FB(\()p Fr(\015)5 b FB(\)\))28 b Fo(\025)g Fr(')p FB(.)43 b(F)-8 b(rom)31 b(this)h(it)g(follo)m(ws)296 3604 y Fn(?)296 3664 y(?)296 3724 y(?)296 3783 y(?)362 3749 y Fr(T)433 3707 y Fp(1)p Fw(=)p Fp(3)559 3613 y Fn(Z)615 3838 y Fw(\030)646 3847 y Fk(1)701 3749 y Fr(e)746 3707 y Fw(i)783 3690 y Fg(e)770 3707 y Fw( )s Fp(\()p Fw(k)r Fp(\))p Fw(T)984 3749 y FB(d)p Fr(k)1092 3604 y Fn(?)1092 3664 y(?)1092 3724 y(?)1092 3783 y(?)1242 3749 y Fo(\024)83 b FB(2)p Fr(T)1522 3707 y Fp(1)p Fw(=)p Fp(3)1632 3749 y Fr(c)1674 3764 y Fp(0)1713 3749 y Fr(\015)1786 3613 y Fn(Z)1885 3639 y Fw(\031)r(=)p Fp(4)1841 3838 y(0)2020 3749 y Fr(e)2065 3707 y Fw(T)10 b(c)2147 3684 y Fk(2)2147 3728 y(0)2181 3707 y Fw(\015)2221 3684 y Fk(2)2268 3690 y Fg(e)2256 3707 y Fw( )2304 3684 y Fd(00)2345 3707 y Fp(\(0\))p Fw('=)p Fp(2)2572 3749 y FB(d)q Fr(')555 b FB(\(5.58\))1242 4038 y Fo(\024)83 b FB(2)p Fr(T)1522 3997 y Fp(1)p Fw(=)p Fp(3)1632 4038 y Fr(c)1674 4053 y Fp(0)1713 4038 y Fr(\015)1786 3902 y Fn(Z)1885 3929 y Fm(1)1841 4128 y Fp(0)1977 4038 y Fr(e)2022 3997 y Fw(T)10 b(c)2104 3973 y Fk(2)2104 4017 y(0)2138 3997 y Fw(\015)2178 3973 y Fk(2)2225 3979 y Fg(e)2213 3997 y Fw( )2261 3973 y Fd(0)q(0)2302 3997 y Fp(\(0\))p Fw('=)p Fp(2)2529 4038 y FB(d)q Fr(')27 b FB(=)2921 3971 y(4)p Fr(T)3041 3934 y Fp(1)p Fw(=)p Fp(3)3150 3971 y Fr(c)3192 3986 y Fp(0)3232 3971 y Fr(\015)p 2789 4015 632 4 v 2789 4142 a(T)14 b(c)2902 4107 y Fp(2)2902 4166 y(0)2941 4142 y Fr(\015)2997 4113 y Fp(2)3053 4027 y Fn(?)3053 4087 y(?)3053 4147 y(?)3136 4115 y(e)3119 4142 y Fr( )3186 4113 y Fm(00)3229 4142 y FB(\(0\))3354 4027 y Fn(?)3354 4087 y(?)3354 4147 y(?)3430 4038 y Fr(:)257 4396 y FB(W)-8 b(e)37 b(compute)845 4370 y Fn(e)828 4396 y Fr( )895 4360 y Fm(00)938 4396 y FB(\(0\))c(=)h Fo(\000)p FB(2)p Fr(\015)5 b(c)1430 4355 y Fm(\000)p Fp(1)1430 4421 y(0)1541 4316 y Fn(\000)1587 4396 y FB(1)22 b(+)g Fo(O)1854 4316 y Fn(\000)1900 4396 y Fr(\015)1956 4360 y Fp(2)2018 4396 y FB(+)g Fr(s)2162 4411 y Fp(1)2201 4396 y Fr(T)2272 4360 y Fm(\000)p Fp(1)p Fw(=)p Fp(3)2437 4316 y Fn(\001)o(\001)2528 4396 y FB(.)54 b(Therefore)38 b(for)d Fr(s)3253 4411 y Fp(1)3326 4396 y Fo(2)g(B)257 4517 y FB(and)e Fr(T)46 b FB(large)32 b(enough,)911 4628 y Fn(?)911 4688 y(?)911 4747 y(?)911 4807 y(?)978 4772 y Fr(T)1049 4731 y Fp(1)p Fw(=)p Fp(3)1175 4637 y Fn(Z)1230 4862 y Fw(\030)1261 4871 y Fk(1)1317 4772 y Fr(e)1362 4731 y Fw(i)1399 4714 y Fg(e)1386 4731 y Fw( )s Fp(\()p Fw(k)r Fp(\))p Fw(T)1600 4772 y FB(d)p Fr(k)1708 4628 y Fn(?)1708 4688 y(?)1708 4747 y(?)1708 4807 y(?)1802 4772 y Fo(\024)2066 4705 y FB(4)p 1917 4749 348 4 v 1917 4841 a(\()p Fr(L)23 b Fo(\000)f Fr(r)2187 4856 y Fp(1)2227 4841 y FB(\))2302 4772 y Fo(\024)2586 4705 y FB(4)p 2417 4749 387 4 v 2417 4841 a(\()p Fr(L)2521 4856 y Fp(0)2583 4841 y Fo(\000)h Fr(r)2727 4856 y Fp(1)2766 4841 y FB(\))2814 4772 y Fr(:)405 b FB(\(5.59\))404 5036 y(Next)33 b(w)m(e)h(need)f(to)g(ev)-5 b(aluate)32 b(the)h(in)m(tegral)e(along)g Fr(\030)2341 5051 y Fp(2)2380 5036 y FB(,)414 5285 y Fr(T)485 5244 y Fp(1)p Fw(=)p Fp(3)611 5150 y Fn(Z)667 5375 y Fw(\030)698 5384 y Fk(2)753 5285 y Fr(e)798 5244 y Fw(i)835 5227 y Fg(e)822 5244 y Fw( )s Fp(\()p Fw(k)r Fp(\))p Fw(T)1036 5285 y FB(d)p Fr(k)g FB(=)c Fr(T)1346 5244 y Fp(1)p Fw(=)p Fp(3)1456 5285 y Fr(e)1501 5244 y Fm(\000)p Fw(i\031)r(=)p Fp(4)1698 5285 y Fr(e)1743 5244 y Fw(iT)1831 5227 y Fg(e)1818 5244 y Fw( )s Fp(\(0\))1977 5150 y Fn(Z)2077 5176 y Fw(c)2108 5185 y Fk(2)2142 5176 y Fw(\015)2033 5375 y Fm(\000)p Fw(c)2119 5384 y Fk(1)2153 5375 y Fw(\015)2214 5285 y Fr(e)2259 5244 y Fw(T)2323 5227 y Fg(e)2310 5244 y Fw( )2358 5221 y Fd(0)q(0)2399 5244 y Fp(\(0\))p Fw(x)2528 5221 y Fk(2)2564 5244 y Fw(=)p Fp(2\(1+)p Fm(O)r Fp(\()q Fw(x)p Fp(\)\))2952 5285 y FB(d)q Fr(x:)157 b FB(\(5.60\))p eop %%Page: 35 35 35 34 bop 257 266 a Ft(5)98 b(EDGE)32 b(SCALING)2235 b FB(35)257 573 y(Then)34 b(for)e(su\016cien)m(tly)h(small)e Fr(\016)t FB(,)493 700 y Fn(?)493 760 y(?)493 820 y(?)493 879 y(?)560 845 y Fr(T)631 803 y Fp(1)p Fw(=)p Fp(3)757 709 y Fn(Z)813 935 y Fw(\030)844 944 y Fk(2)899 845 y Fr(e)944 803 y Fw(i)981 786 y Fg(e)968 803 y Fw( )s Fp(\()p Fw(k)r Fp(\))p Fw(T)1182 845 y FB(d)p Fr(k)1290 700 y Fn(?)1290 760 y(?)1290 820 y(?)1290 879 y(?)1384 845 y Fo(\024)d Fr(T)1560 803 y Fp(1)p Fw(=)p Fp(3)1687 709 y Fn(Z)1786 735 y Fw(c)1817 744 y Fk(2)1851 735 y Fw(\015)1742 935 y Fm(\000)p Fw(c)1828 944 y Fk(1)1862 935 y Fw(\015)1923 845 y Fr(e)1968 803 y Fw(T)2032 786 y Fg(e)2019 803 y Fw( )2067 780 y Fd(0)q(0)2108 803 y Fp(\(0\))p Fw(x)2237 780 y Fk(2)2273 803 y Fp(\(1+)p Fw(\016)r Fp(\))p Fw(=)p Fp(2)2543 845 y FB(d)p Fr(x)831 1119 y Fo(\024)84 b Fr(T)1063 1077 y Fp(1)p Fw(=)p Fp(3)1189 983 y Fn(Z)1289 1009 y Fm(1)1244 1209 y(\0001)1390 1119 y Fr(e)1435 1077 y Fw(T)1499 1060 y Fg(e)1486 1077 y Fw( )1534 1054 y Fd(0)q(0)1576 1077 y Fp(\(0\))p Fw(x)1705 1054 y Fk(2)1740 1077 y Fp(3)p Fw(=)p Fp(4)1867 1119 y FB(d)p Fr(x)28 b Fo(\024)g Fr(T)2180 1077 y Fp(1)p Fw(=)p Fp(3)2307 983 y Fn(Z)2406 1009 y Fm(1)2362 1209 y(\0001)2508 1119 y Fr(e)2553 1077 y Fm(\000)p Fw(x)2648 1054 y Fk(2)2683 1077 y Fw(T)10 b(\015)t(=c)2840 1086 y Fk(0)2895 1119 y FB(d)p Fr(x)242 b FB(\(5.61\))832 1399 y(=)1027 1257 y Fo(p)p 1110 1257 59 4 v 71 x Fr(\031)1169 1264 y Fo(p)p 1252 1264 82 4 v 64 x Fr(c)1294 1343 y Fp(0)p 1002 1376 358 4 v 1029 1433 a Fk(4)1005 1396 y Fo(p)p 1087 1396 272 4 v 1087 1472 a Fr(L)23 b Fo(\000)f Fr(r)1319 1487 y Fp(1)1397 1399 y FB(=)1555 1257 y Fo(p)p 1638 1257 59 4 v 71 x Fr(\031)1697 1264 y Fo(p)p 1780 1264 82 4 v 64 x Fr(c)1822 1343 y Fp(0)p 1510 1376 397 4 v 1538 1433 a Fk(4)1513 1396 y Fo(p)p 1596 1396 311 4 v 76 x Fr(L)1662 1487 y Fp(0)1724 1472 y Fo(\000)g Fr(r)1867 1487 y Fp(1)1917 1399 y Fr(:)257 1668 y FB(Th)m(us)34 b(w)m(e)g(ha)m(v)m(e,)g(uniformly)d(for)h Fr(s)1540 1683 y Fp(1)1607 1668 y Fo(2)c(B)36 b FB(and)c Fr(T)47 b FB(large)31 b(enough,)1129 1801 y Fn(?)1129 1861 y(?)1129 1921 y(?)1129 1980 y(?)1196 1810 y(Z)1251 2036 y Fw(I)1282 2045 y Fk(3)1337 1946 y Fo(\001)17 b(\001)g(\001)1453 1801 y Fn(?)1453 1861 y(?)1453 1921 y(?)1453 1980 y(?)1547 1946 y Fo(\024)1832 1878 y FB(8)p 1663 1923 387 4 v 1663 2014 a(\()p Fr(L)1767 2029 y Fp(0)1828 2014 y Fo(\000)23 b Fr(r)1972 2029 y Fp(1)2011 2014 y FB(\))2081 1946 y(+)2235 1804 y Fo(p)p 2318 1804 59 4 v 71 x Fr(\031)2377 1811 y Fo(p)p 2460 1811 82 4 v 64 x Fr(c)2502 1890 y Fp(0)p 2189 1923 397 4 v 2217 1980 a Fk(4)2192 1943 y Fo(p)p 2275 1943 311 4 v 76 x Fr(L)2341 2034 y Fp(0)2403 2019 y Fo(\000)g Fr(r)2547 2034 y Fp(1)2596 1946 y Fr(:)623 b FB(\(5.62\))257 2227 y(Therefore)34 b Fr(J)753 2242 y Fw(i)781 2227 y FB(\()p Fr(L)p FB(\))f(is)f(b)s(ounded)h(b)m(y)914 2507 y Fo(j)p Fr(J)996 2522 y Fw(i)1024 2507 y FB(\()p Fr(L)p FB(\))p Fo(j)28 b(\024)1344 2440 y FB(4)p Fr(G)1470 2455 y Fp(1)p 1337 2484 181 4 v 1337 2576 a Fr(T)1408 2547 y Fp(2)p Fw(=)p Fp(3)1549 2507 y FB(+)1796 2440 y(20)p 1657 2484 376 4 v 1657 2576 a(\()p Fr(L)1761 2591 y Fp(0)1823 2576 y Fo(\000)23 b Fr(r)1967 2591 y Fw(i)1995 2576 y FB(\))2065 2507 y(+)2392 2440 y(2)2441 2368 y Fo(p)p 2524 2368 59 4 v 72 x Fr(\031)p 2173 2484 629 4 v 2201 2552 a Fk(4)2176 2504 y Fn(p)p 2276 2504 526 4 v 86 x FB(2)p Fr(\013)2387 2605 y Fp(3)2426 2590 y FB(\()p Fr(L)2530 2605 y Fp(0)2592 2590 y Fo(\000)f Fr(r)2735 2605 y Fw(i)2763 2590 y FB(\))2811 2507 y Fr(:)408 b FB(\(5.63\))257 2809 y(Since)33 b Fr(L)578 2824 y Fp(0)640 2809 y Fo(\000)23 b Fr(r)784 2824 y Fw(i)840 2809 y Fo(\025)29 b FB(2)j(and)h Fr(X)j(>)28 b FB(0,)k(it)g(follo)m(ws)g(that)g Fr(J)2230 2824 y Fp(1)2269 2809 y FB(\()p Fr(L)p FB(\))p Fr(J)2465 2824 y Fp(2)2505 2809 y FB(\()p Fr(L)p FB(\))p Fr(e)2692 2773 y Fm(\000)p Fw(LX)2895 2809 y FB(is)h(b)s(ounded)g(b)m(y)257 2930 y(an)g(in)m(tegrable)e(function)h(on)h([)p Fr(L)1456 2945 y Fp(0)1496 2930 y Fr(;)17 b("T)1657 2894 y Fp(2)p Fw(=)p Fp(3)1766 2930 y FB(])33 b(for)f(0)27 b Fr(<)h(")f Fo(\034)g FB(1)32 b(and)h Fr(T)46 b FB(large)32 b(enough.)p 3425 2930 4 66 v 3429 2867 59 4 v 3429 2930 V 3487 2930 4 66 v 257 3125 a Fb(Con)m(v)m(ergence)38 b(of)g Fr(K)1125 3140 y Fw(T)1180 3125 y FB(\()p Fr(r)1262 3140 y Fp(2)1301 3125 y Fr(;)17 b(s)1391 3140 y Fp(2)1430 3125 y FB(;)g Fr(r)1518 3140 y Fp(1)1557 3125 y Fr(;)g(s)1647 3140 y Fp(1)1687 3125 y FB(\))37 b Fb(with)f Fr(s)2063 3140 y Fp(2)2130 3125 y Fo(\025)28 b Fr(s)2281 3140 y Fp(1)2321 3125 y Fb(.)257 3295 y(Lemma)37 b(5.7.)50 b Fs(Uniformly)34 b(for)h FB(\()p Fr(s)1582 3310 y Fw(i)1610 3295 y Fr(;)17 b(r)1698 3310 y Fw(i)1726 3295 y FB(\))27 b Fo(2)i(B)s Fs(,)35 b Fr(i)28 b FB(=)f(1)p Fr(;)17 b FB(2)p Fs(,)446 3580 y FB(lim)418 3642 y Fw(T)10 b Fm(!1)627 3580 y Fr(K)710 3595 y Fw(T)765 3580 y FB(\()p Fr(r)847 3595 y Fp(2)886 3580 y Fr(;)17 b(s)976 3595 y Fp(2)1015 3580 y FB(;)g Fr(r)1103 3595 y Fp(1)1142 3580 y Fr(;)g(s)1232 3595 y Fp(1)1271 3580 y FB(\))28 b(=)1441 3444 y Fn(Z)1540 3471 y Fp(0)1496 3670 y Fm(\0001)1642 3580 y Fr(e)1697 3512 y Fk(1)p 1697 3524 31 3 v 1697 3565 a(2)1738 3539 y Fw(L)p Fp(\()p Fw(s)1846 3548 y Fk(2)1880 3539 y Fm(\000)p Fw(s)1968 3548 y Fk(1)2003 3539 y Fp(\))2051 3580 y FB(Ai)2168 3440 y Fn(\022)2251 3513 y Fr(r)2295 3528 y Fp(1)2356 3513 y Fo(\000)23 b Fr(L)p 2251 3557 272 4 v 2359 3648 a(\024)2532 3440 y Fn(\023)2622 3580 y FB(Ai)2739 3440 y Fn(\022)2822 3513 y Fr(r)2866 3528 y Fp(2)2928 3513 y Fo(\000)g Fr(L)p 2822 3557 V 2930 3648 a(\024)3104 3440 y Fn(\023)3204 3513 y FB(d)p Fr(L)p 3204 3557 121 4 v 3216 3648 a(\024)3272 3620 y Fp(2)3246 3769 y FB(\(5.64\))257 3996 y Fs(with)35 b Fr(\024)28 b FB(=)684 3959 y Fk(3)659 3921 y Fo(p)p 742 3921 151 4 v 75 x FB(2)p Fr(\013)853 4011 y Fp(3)892 3996 y Fs(.)257 4192 y Fa(Pr)n(oof:)50 b FB(Let)36 b(us)h(set)h Fr(L)1180 4207 y Fp(0)1256 4192 y FB(suc)m(h)g(that)e Fr(L)1761 4207 y Fp(0)1835 4192 y Fo(\025)f FB(2\()p Fo(j)p Fr(r)2106 4207 y Fp(1)2145 4192 y Fo(j)25 b FB(+)f Fo(j)p Fr(r)2370 4207 y Fp(2)2409 4192 y Fo(j)h FB(+)g(1\))36 b(for)g(all)e(\()p Fr(r)3060 4207 y Fp(1)3099 4192 y Fr(;)17 b(r)3187 4207 y Fp(2)3227 4192 y FB(\))34 b Fo(2)h(B)s FB(.)257 4312 y(Then)i(the)f(sum)f(in)g Fr(K)1095 4327 y Fw(T)1185 4312 y FB(can)h(b)s(e)f(appro)m(ximated)g(b) m(y)h(an)g(in)m(tegral)d(at)i(the)h(exp)s(ense)i(of)257 4432 y(an)33 b(error)f Fo(O)730 4352 y Fn(\000)776 4432 y Fr(T)847 4396 y Fm(\000)p Fp(1)p Fw(=)p Fp(3)1011 4352 y Fn(\001)1057 4432 y FB(.)43 b(Let)33 b(us)g(\014x)h Fr(")p FB(,)e(0)27 b Fr(<)h(")f Fo(\034)g FB(1.)44 b(Then)303 4730 y Fr(K)386 4745 y Fw(T)441 4730 y FB(\()p Fr(r)523 4745 y Fp(2)562 4730 y Fr(;)17 b(s)652 4745 y Fp(2)691 4730 y FB(;)g Fr(r)779 4745 y Fp(1)818 4730 y Fr(;)g(s)908 4745 y Fp(1)947 4730 y FB(\))83 b(=)1227 4594 y Fn(Z)1327 4620 y Fp(0)1282 4820 y Fm(\000)p Fw(L)1385 4829 y Fk(0)1441 4730 y Fr(J)1495 4745 y Fp(1)1534 4730 y FB(\()p Fr(L)p FB(\))p Fr(J)1730 4745 y Fp(2)1770 4730 y FB(\()p Fr(L)p FB(\))1922 4662 y Fr(e)1967 4626 y Fw(LX)p 1922 4707 161 4 v 1929 4798 a FB(4)p Fr(\031)2037 4769 y Fp(2)2109 4730 y FB(d)p Fr(L)23 b FB(+)2350 4594 y Fn(Z)2449 4620 y Fm(\000)p Fw(L)2552 4629 y Fk(0)2405 4820 y Fm(\000)p Fw("T)2544 4800 y Fk(2)p Fh(=)p Fk(3)2661 4730 y Fr(J)2715 4745 y Fp(1)2754 4730 y FB(\()p Fr(L)p FB(\))p Fr(J)2950 4745 y Fp(2)2990 4730 y FB(\()p Fr(L)p FB(\))3142 4662 y Fr(e)3187 4626 y Fw(LX)p 3142 4707 V 3149 4798 a FB(4)p Fr(\031)3257 4769 y Fp(2)3329 4730 y FB(d)q Fr(L)1227 5032 y FB(+)1353 4897 y Fn(Z)1452 4923 y Fm(\000)p Fw("T)1591 4900 y Fk(2)p Fh(=)p Fk(3)1408 5122 y Fm(\0001)1708 5032 y Fr(J)1762 5047 y Fp(1)1802 5032 y FB(\()p Fr(L)p FB(\))p Fr(J)1998 5047 y Fp(2)2037 5032 y FB(\()p Fr(L)p FB(\))2189 4965 y Fr(e)2234 4929 y Fw(LX)p 2189 5009 V 2196 5101 a FB(4)p Fr(\031)2304 5072 y Fp(2)2377 5032 y FB(d)p Fr(L)f FB(+)g Fo(O)2716 4952 y Fn(\000)2762 5032 y Fr(T)2833 4991 y Fm(\000)p Fp(1)p Fw(=)p Fp(3)2997 4952 y Fn(\001)3060 5032 y Fr(;)159 b FB(\(5.65\))p eop %%Page: 36 36 36 35 bop 257 266 a Ft(5)98 b(EDGE)32 b(SCALING)2235 b FB(36)257 573 y(with)39 b Fr(X)45 b FB(=)736 534 y Fp(1)p 736 550 36 4 v 736 607 a(2)781 573 y FB(\()p Fr(s)865 588 y Fp(2)931 573 y Fo(\000)27 b Fr(s)1081 588 y Fp(1)1120 573 y FB(\)\(1)f(+)g Fo(O)20 b FB(\(1)p Fr(=T)14 b FB(\))o(\))38 b Fo(\025)g FB(0.)62 b(The)39 b(con)m(v)m(ergence)i(of)e(the)g(\014rst) g(term)257 713 y(has)c(already)f(b)s(een)h(pro)m(v)m(ed.)51 b(Let)35 b(us)g(set)1838 687 y Fn(e)1830 713 y Fr(L)c FB(=)g Fo(\000)p FB(\()p Fr(L)24 b Fo(\000)g Fr(r)2384 728 y Fp(1)2424 713 y FB(\))p Fr(T)2533 676 y Fm(\000)p Fp(2)p Fw(=)p Fp(3)2697 713 y FB(.)49 b(In)35 b(the)g(remainder)257 833 y(of)d(the)h(pro)s(of)f(w)m(e)i(set)1392 1058 y Fr( )t FB(\()p Fr(k)s FB(\))27 b(=)1730 991 y(1)p 1730 1035 49 4 v 1738 1126 a Fr(i)1788 1058 y( )1851 1073 y Fp(1)p Fw(;T)1962 1058 y FB(\()p Fr(k)s FB(\))22 b Fo(\000)g Fr(k)2276 1033 y Fn(e)2267 1058 y Fr(L)q(:)885 b FB(\(5.66\))404 1298 y(First)31 b(consider)1029 1273 y Fn(e)1021 1298 y Fr(L)d Fo(\025)g Fr(")p FB(.)1310 1523 y Fr(J)1364 1538 y Fp(1)1404 1523 y FB(\()p Fr(L)p FB(\))f(=)h Fr(T)1748 1482 y Fp(1)p Fw(=)p Fp(3)1874 1388 y Fn(Z)1974 1414 y Fw(\031)1930 1613 y Fm(\000)p Fw(\031)2048 1523 y Fr(e)2093 1482 y Fw( )r Fp(\()p Fw(k)r Fp(\))p Fw(T)2307 1523 y FB(d)p Fr(k)s(:)804 b FB(\(5.67\))257 1792 y(With)26 b(the)g(c)m(hange)h(of)e(v)-5 b(ariable)24 b Fr(u)j FB(=)h Fr( )t FB(\()p Fr(k)s FB(\),)f Fr(f)11 b FB(\()p Fr(u)p FB(\))27 b(=)2212 1744 y Fp(d)p Fw(k)r Fp(\()p Fw(u)p Fp(\))p 2212 1769 174 4 v 2259 1826 a(d)p Fw(u)2396 1792 y FB(,)g(and)f(in)m(tegration)e(b)m(y)j(parts,)257 1912 y(w)m(e)34 b(ha)m(v)m(e)956 2131 y Fo(j)p Fr(J)1038 2146 y Fp(1)1077 2131 y FB(\()p Fr(L)p FB(\))p Fo(j)28 b(\024)g Fr(T)1451 2089 y Fp(1)p Fw(=)p Fp(3)1570 2063 y FB(2)p Fr( )t FB(\()p Fr(\031)t FB(\))p 1570 2108 251 4 v 1660 2199 a Fr(T)2003 2131 y FB(max)1847 2197 y Fw(k)r Fm(2)p Fp([)p Fw( )r Fp(\()p Fm(\000)p Fw(\031)r Fp(\))p Fw(; )r Fp(\()p Fw(\031)r Fp(\)])2356 1986 y Fn(?)2356 2046 y(?)2356 2106 y(?)2356 2165 y(?)2432 2063 y FB(d)q Fr(f)11 b FB(\()p Fr(u)p FB(\))p 2432 2108 245 4 v 2500 2199 a(d)p Fr(u)2687 1986 y Fn(?)2687 2046 y(?)2687 2106 y(?)2687 2165 y(?)2770 2131 y Fr(:)449 b FB(\(5.68\))257 2414 y(T)-8 b(o)55 b(compute)840 2299 y Fn(?)840 2359 y(?)840 2419 y(?)916 2366 y Fp(d)p Fw(f)7 b Fp(\()p Fw(u)p Fp(\))p 916 2391 177 4 v 964 2448 a(d)p Fw(u)1102 2299 y Fn(?)1102 2359 y(?)1102 2419 y(?)1223 2414 y FB(w)m(e)55 b(use)1588 2366 y Fp(d)p Fw(f)7 b Fp(\()p Fw(u)p Fp(\))p 1588 2391 V 1636 2448 a(d)p Fw(u)1838 2414 y FB(=)64 b Fo(\000)p FB(\()2103 2375 y Fp(d)q Fw(u)p 2103 2391 81 4 v 2104 2448 a Fp(d)q Fw(k)2194 2414 y FB(\))2232 2378 y Fm(\000)p Fp(3)2336 2375 y(d)2375 2349 y Fk(2)2410 2375 y Fw(u)p 2336 2391 115 4 v 2337 2448 a Fp(d)p Fw(k)2415 2429 y Fk(2)2461 2414 y FB(.)2596 2329 y Fn(?)2596 2389 y(?)2672 2375 y Fp(d)p Fw(u)p 2672 2391 81 4 v 2673 2448 a Fp(d)p Fw(k)2762 2329 y Fn(?)2762 2389 y(?)2883 2414 y FB(is)53 b(\(5.49\))h(with)266 2559 y Fn(e)257 2584 y Fr(L)g FB(replaced)f(b)m (y)h Fo(\000)1023 2559 y Fn(e)1015 2584 y Fr(L)q FB(.)106 b(It)53 b(is)f(easy)j(then)f(to)e(see)j(that)e(uniformly)e(for)h Fr(s)3168 2599 y Fp(1)3270 2584 y Fo(2)64 b(B)s FB(,)257 2726 y(max)439 2742 y Fw(k)r Fm(2)p Fp([)p Fw( )r Fp(\()p Fm(\000)p Fw(\031)r Fp(\))p Fw(; )r Fp(\()p Fw(\031)r Fp(\)])952 2612 y Fn(?)952 2672 y(?)952 2731 y(?)1028 2679 y Fp(d)p Fw(f)7 b Fp(\()p Fw(u)p Fp(\))p 1028 2703 177 4 v 1076 2761 a(d)p Fw(u)1214 2612 y Fn(?)1214 2672 y(?)1214 2731 y(?)1327 2726 y Fo(\024)46 b Fr(G)1527 2741 y Fp(1)1575 2701 y Fn(e)1566 2726 y Fr(L)1632 2690 y Fm(\000)p Fp(1)1771 2726 y FB(for)c Fr(G)2007 2741 y Fp(1)2093 2726 y FB(=)k(2)p Fr(=)p FB(\(sinh)o(\()p Fr(\034)2604 2741 y Fp(0)2644 2726 y Fr(=)p FB(2\))p Fr(")p FB(\))2864 2690 y Fp(2)2949 2726 y Fr(<)g Fo(1)p FB(.)75 b(Then)257 2869 y(for)32 b(a)h(suitable)e(constan)m(t)j Fr(G)1322 2884 y Fp(2)1389 2869 y Fr(<)27 b Fo(1)p FB(,)1372 3055 y Fo(j)o Fr(J)1453 3070 y Fp(1)1493 3055 y FB(\()p Fr(L)p FB(\))p Fo(j)g(\024)i Fr(G)1873 3070 y Fp(2)1912 3055 y FB(\()p Fr(r)1994 3070 y Fp(1)2055 3055 y Fo(\000)23 b Fr(L)p FB(\))2259 3014 y Fm(\000)p Fp(1)2354 3055 y Fr(:)865 b FB(\(5.69\))257 3241 y(The)34 b(same)e(holds)h(for)f Fr(J)1161 3256 y Fp(2)1200 3241 y FB(,)h(therefore)g(the)g(third)f (term)g(in)f(\(5.65\))h(is)g(b)s(ounded)i(b)m(y)1248 3375 y Fn(Z)1348 3401 y Fm(\000)p Fw("T)1487 3377 y Fk(2)p Fh(=)p Fk(3)1304 3600 y Fm(\0001)1905 3443 y Fr(G)1982 3407 y Fp(2)1982 3467 y(2)p 1614 3487 700 4 v 1614 3578 a FB(\()p Fr(L)22 b FB(+)g Fo(j)p Fr(r)1910 3593 y Fp(1)1949 3578 y Fo(j)g FB(+)g Fo(j)p Fr(r)2169 3593 y Fp(2)2208 3578 y Fo(j)p FB(\))2274 3550 y Fp(2)2340 3510 y FB(d)p Fr(L;)759 b FB(\(5.70\))257 3750 y(whic)m(h)33 b(is)g(con)m(v)m(ergen)m (t)h(for)e Fr(T)47 b FB(\014nite)32 b(and)h(v)-5 b(anishes)33 b(for)f Fr(T)41 b Fo(!)27 b(1)p FB(.)404 3871 y(Finally)32 b(w)m(e)k(consider)g(0)31 b Fr(<)1463 3846 y Fn(e)1455 3871 y Fr(L)i Fo(\024)f Fr(")p FB(.)51 b(Let)35 b(us)h(set)g Fr(\014)h FB(=)2447 3786 y Fn(p)p 2547 3786 736 4 v 85 x FB(2\(cosh\()p Fr(\034)2898 3886 y Fp(0)2938 3871 y Fr(=)p FB(2\))21 b Fo(\000)i FB(1\).)51 b(W)-8 b(e)257 4024 y(in)m(tegrate)29 b(o)m(v)m(er)g Fr(C)35 b FB(=)1076 3913 y Fn(n)1142 4024 y Fr(k)c FB(=)c Fr(x)c FB(+)f Fr(iy)t FB(\()p Fr(x)p FB(\))p Fr(;)17 b(y)t FB(\()p Fr(x)p FB(\))26 b(=)i Fo(\000)2153 3939 y Fn(p)p 2253 3939 528 4 v 85 x Fr(y)t FB(\(0\))2430 3995 y Fp(2)2490 4024 y FB(+)22 b Fr(x)2643 3995 y Fp(2)2683 4024 y Fr(=)p FB(3)2781 3913 y Fn(o)2847 4024 y FB(,)30 b(with)e 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y FB(\))1727 1080 y Fr(T)1798 1039 y Fp(1)p Fw(=)p Fp(3)1924 945 y Fn(Z)1980 1170 y Fw(C)2030 1178 y Fh(")2084 1080 y FB(d)p Fr(k)s(e)2237 1034 y Fm(\000)2306 1007 y Fk(1)p 2303 1019 38 3 v 2303 1060 a Fh(\014)2350 952 y Fo(p)p 2433 952 48 4 v 2439 1017 a Fg(e)2433 1034 y Fw(L)p Fp(\()p Fw(k)r Fm(\000)p Fw(iy)r Fp(\(0\)\))2779 1011 y Fk(2)2815 1034 y Fw(T)2870 1080 y Fr(e)2915 1026 y Fm(\000)3020 999 y Fh(i)p 2980 1011 103 3 v 2980 1060 a Fk(3)p Fh(\014)3048 1046 y Fk(2)3093 1026 y Fp(\()p Fw(k)r Fm(\000)p Fw(iy)r Fp(\(0\)\))3391 1002 y Fk(3)3427 1026 y Fw(T)437 1382 y Fo(\002)p Fr(e)559 1308 y Fm(O)617 1215 y Fg(\022)676 1291 y(e)669 1308 y Fw(L)p Fp(\()p Fw(k)r Fm(\000)p Fw(iy)r Fp(\(0\)\))1015 1285 y Fk(2)1051 1308 y Fw(T)g Fp(+)1157 1226 y Fo(p)p 1240 1226 48 4 v 1246 1291 a Fg(e)1240 1308 y Fw(L)p Fp(\()p Fw(k)r Fm(\000)p Fw(iy)r Fp(\(0\)\))1586 1285 y Fk(2)1623 1308 y Fw(s)1656 1317 y Fk(1)1690 1308 y Fw(T)1741 1285 y Fk(2)p Fh(=)p Fk(3)1838 1308 y Fp(+)1899 1291 y Fg(e)1893 1308 y Fw(L)o Fp(\()p Fw(k)r Fm(\000)p Fw(iy)r Fp(\(0\)\))2238 1285 y Fk(3)2275 1308 y Fw(T)g Fp(+\()p Fw(k)r Fm(\000)p Fw(iy)r Fp(\(0\)\))2679 1285 y Fk(3)2715 1308 y Fw(s)2748 1317 y Fk(1)2782 1308 y Fw(T)2833 1285 y Fk(2)p Fh(=)p Fk(3)2930 1308 y Fp(+\()p Fw(k)r Fm(\000)p Fw(iy)r Fp(\(0\)\))3283 1285 y Fk(4)3319 1308 y Fw(T)3370 1215 y Fg(\023)3426 1382 y Fr(:)257 1592 y FB(Let)24 b Fr(\015)33 b FB(=)610 1515 y Fo(p)p 693 1515 272 4 v 77 x Fr(r)737 1607 y Fp(1)799 1592 y Fo(\000)22 b Fr(L)q FB(,)j(then)1230 1483 y Fn(p)p 1330 1483 67 4 v 1338 1567 a(e)1330 1592 y Fr(L)j FB(=)f Fr(\015)5 b(T)1654 1556 y Fm(\000)p Fp(1)p Fw(=)p Fp(3)1819 1592 y FB(.)40 b(Let)24 b Fr(k)2106 1556 y Fm(0)2157 1592 y FB(=)j Fr(k)7 b Fo(\000)s Fr(iy)t FB(\(0\),)25 b(then)f(the)g(in)m (tegration)257 1712 y(is)32 b(along)g Fr(C)693 1676 y Fm(0)686 1737 y Fw(")750 1712 y FB(=)27 b Fr(C)923 1727 y Fw(")982 1712 y FB(+)22 b Fr(iy)t FB(\(0\).)347 1957 y Fr(J)401 1972 y Fp(1)440 1957 y FB(\()p Fr(L)p FB(\))g Fo(\000)h 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y(k)1278 2721 y Fp(2)1317 2762 y Fr(T)1388 2721 y Fp(2)p Fw(=)p Fp(3)1498 2762 y FB(\(1)22 b(+)g Fr(\037)1766 2777 y Fp(1)1805 2762 y FB(\))g Fo(\000)2033 2694 y Fr(i)p 1975 2739 149 4 v 1975 2830 a FB(3)p Fr(\014)2085 2801 y Fp(2)2134 2762 y Fr(k)2188 2721 y Fp(3)2227 2762 y Fr(T)14 b FB(\(1)22 b(+)g Fr(\037)2566 2777 y Fp(2)2605 2762 y FB(\))p Fr(;)576 b FB(\(5.78\))257 3003 y(where)39 b(the)g Fr(\037)779 3018 y Fw(i)845 3003 y FB(can)f(b)s(e)g(made)f(as)h(small)d(as)j (desired)h(b)m(y)f(c)m(ho)s(osing)g Fr(")f FB(small)f(enough.)257 3124 y(After)d(the)g(c)m(hange)h(of)e(v)-5 b(ariable)30 b Fr(k)s(T)1604 3088 y Fp(1)p Fw(=)p Fp(3)1742 3124 y FB(=)d Fr(z)38 b FB(the)33 b(in)m(tegral)e(b)s(ecomes)1240 3228 y Fn(Z)1296 3453 y Fw(C)1351 3434 y Fd(0)1346 3470 y Fh(")1379 3453 y Fw(T)1430 3434 y Fk(1)p Fh(=)p Fk(3)1406 3363 y FB(d)p Fr(z)t(e)1554 3317 y Fm(\000)1620 3286 y Fh(\015)p 1620 3302 38 3 v 1620 3343 a(\014)1667 3317 y Fw(z)1703 3293 y Fk(2)1737 3317 y Fp(\(1+)p Fw(\037)1898 3326 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Fp(3)2983 5161 y Fr(s)3029 5120 y Fp(2)3029 5185 y(1)3068 5161 y Fr(T)3139 5120 y Fm(\000)p Fp(1)p Fw(=)p Fp(3)3304 5161 y FB(\))3342 5050 y Fn(\021)1843 5373 y Fo(\024)83 b(O)2102 5263 y Fn(\020)2162 5373 y Fr(T)2233 5332 y Fm(\000)p Fp(1)p Fw(=)p Fp(3)2397 5373 y Fr(e)2442 5332 y Fm(\000)2507 5301 y Fh(\014)p 2507 5317 V 2510 5358 a Fk(2)2555 5332 y Fp(\()p Fw(r)2614 5341 y Fk(1)2648 5332 y Fm(\000)p Fw(L)p Fp(\))2778 5309 y Fk(3)p Fh(=)p Fk(2)2880 5263 y Fn(\021)2956 5373 y Fr(:)263 b FB(\(5.82\))p eop %%Page: 39 39 39 38 bop 257 266 a Ft(6)98 b(THE)33 b(BORDER)f(STEP)-8 b(,)34 b(AIR)-8 b(Y)33 b(PR)m(OCESS)1248 b FB(39)257 573 y(After)32 b(this)f(step)h(w)m(e)g(can)g(also)e(remo)m(v)m(e)j(the) e(error)h(inside)e(the)i(in)m(tegral)e(\(5.77\).)42 b(As)32 b(in)257 693 y(the)38 b(case)h(of)e Fr(L)g Fo(2)f(B)s FB(,)j(the)f(remo)m(v)-5 b(al)37 b(of)g(this)g(error)g(leads)h(to)f(an) h(additional)c(error)k(of)257 814 y Fr(T)328 777 y Fm(\000)p Fp(1)p Fw(=)p Fp(3)526 814 y FB(with)32 b(the)h(prefactor)f Fr(e)1375 777 y Fm(\000)1440 750 y Fk(2)p 1440 762 31 3 v 1440 803 a(3)1481 777 y Fw(\014)s Fp(\()p Fw(r)1583 786 y Fk(1)1618 777 y Fm(\000)p Fw(L)p Fp(\))1748 753 y Fk(3)p Fh(=)p Fk(2)1849 814 y FB(.)43 b(Consequen)m(tly)35 b(w)m(e)f(ha)m(v)m(e)g(obtained)416 1068 y Fr(J)470 1083 y Fp(1)510 1068 y FB(\()p Fr(L)p FB(\))83 b(=)g Fr(e)939 1027 y Fm(\000)1004 999 y Fk(2)p 1004 1011 V 1004 1053 a(3)1044 1027 y Fw(\014)s Fp(\()p Fw(r)1146 1036 y Fk(1)1181 1027 y Fm(\000)p Fw(L)p Fp(\))1311 1003 y Fk(3)p Fh(=)p Fk(2)1429 932 y Fn(Z)1485 1158 y Fw(C)1540 1139 y Fd(0)1535 1174 y Fh(")1568 1158 y Fw(T)1619 1138 y Fk(1)p Fh(=)p Fk(3)1595 1068 y FB(d)p Fr(z)t(e)1743 1021 y Fm(\000)1809 991 y Fh(\015)p 1809 1006 38 3 v 1809 1047 a(\014)1856 1021 y Fw(z)1892 998 y Fk(2)1931 1068 y Fr(e)1976 1013 y Fm(\000)2081 986 y Fh(i)p 2041 998 103 3 v 2041 1047 a Fk(3)p Fh(\014)2109 1033 y Fk(2)2153 1013 y Fw(z)2189 990 y Fk(3)2250 1068 y FB(+)22 b Fo(O)2446 957 y Fn(\020)2506 1068 y 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b(ha)m(v)m(e)h(not)e(attempted)h(to)f(\014ll)f(this)h (gap.)257 5268 y(The)37 b(in)m(terested)f(reader)g(is)e(referred)i(to)f ([8],)h(where)g(tigh)m(tness)g(for)f(the)h(edge)f(scaling)257 5388 y(of)d(the)h(Aztec)h(diamond)d(is)h(pro)m(v)m(ed.)p eop %%Page: 42 42 42 41 bop 257 266 a Ft(A)98 b(FERMIONIC)34 b(CORRELA)-8 b(TIONS)1566 b FB(42)257 573 y Fu(A)162 b(F)-13 b(ermionic)53 b(correlations)257 821 y Fj(A.1)135 b(Tw)l(o-p)t(oin)l(t)45 b(function)257 1006 y FB(Let)446 980 y Fn(b)424 1006 y Fr(A)28 b FB(=)628 931 y Fn(P)733 1035 y Fw(k)r(;l)q Fm(2)p Fl(Z)926 1006 y Fr(A)999 1021 y Fw(k)r(;l)1083 1006 y Fr(a)1134 970 y Fm(\003)1134 1032 y Fw(k)1177 1006 y Fr(a)1228 1021 y Fw(l)1278 1006 y FB(b)s(e)d(the)g(second)g (quan)m(tization)f(of)g(the)h(one-particle)e(matrix)257 1141 y Fr(A)p FB(.)42 b(It)27 b(is)f(assumed)h(that)g Fr(e)1227 1104 y Fm(\000)p Fw(A)1366 1141 y FB(is)f(trace)h(class)g (and)g(Det)2275 1060 y Fn(\000)2321 1141 y FB(1)22 b(+)g Fr(e)2535 1104 y Fw(A)2592 1060 y Fn(\001)2665 1141 y Fo(6)p FB(=)28 b(0)e(\(see)i([15)o(],)h(Chap.)257 1261 y(XI)s(I)s(I\).)k(W)-8 b(e)33 b(use)h(the)f(iden)m(tities)805 1470 y Fr(e)850 1429 y Fm(\000)922 1412 y Fg(b)905 1429 y Fw(A)962 1470 y Fr(a)1013 1429 y Fm(\003)1013 1495 y Fw(i)1053 1470 y Fr(e)1114 1412 y Fg(b)1098 1429 y Fw(A)1182 1470 y FB(=)1286 1376 y Fn(X)1296 1587 y Fw(j)t Fm(2)p Fl(Z)1447 1470 y Fr(a)1498 1429 y Fm(\003)1498 1495 y Fw(j)1537 1470 y FB([)p Fr(e)1609 1429 y Fm(\000)p Fw(A)1721 1470 y FB(])1748 1485 y Fw(j;i)1825 1470 y Fr(;)114 b(e)2011 1429 y Fm(\000)2083 1412 y Fg(b)2066 1429 y Fw(A)2123 1470 y Fr(a)2174 1485 y Fw(i)2202 1470 y Fr(e)2264 1412 y Fg(b)2247 1429 y Fw(A)2332 1470 y FB(=)2436 1376 y Fn(X)2446 1587 y Fw(j)t Fm(2)p Fl(Z)2596 1470 y FB([)p Fr(e)2668 1429 y Fw(A)2726 1470 y FB(])2753 1485 y Fw(i;j)2833 1470 y Fr(a)2884 1485 y Fw(j)2921 1470 y Fr(:)322 b FB(\(A.1\))257 1758 y(Then)412 1968 y Fo(h)p Fr(a)502 1927 y Fm(\003)502 1993 y Fw(i)541 1968 y Fr(a)592 1983 y Fw(j)629 1968 y Fo(i)83 b FB(=)932 1901 y(1)p 920 1945 74 4 v 920 2036 a Fr(Z)1020 1968 y FB(T)-8 b(r)1137 1857 y Fn(\020)1197 1968 y Fr(e)1242 1927 y Fm(\000)1313 1910 y Fg(b)1297 1927 y Fw(A)1354 1968 y Fr(a)1405 1927 y Fm(\003)1405 1993 y Fw(i)1444 1968 y Fr(a)1495 1983 y Fw(j)1532 1857 y Fn(\021)1619 1968 y FB(=)1723 1873 y Fn(X)1728 2085 y Fw(n)p Fm(2)p Fl(Z)1906 1901 y FB(1)p 1893 1945 V 1893 2036 a Fr(Z)1994 1968 y FB(T)g(r)2111 1857 y Fn(\020)2170 1968 y Fr(a)2221 1927 y Fm(\003)2221 1993 y Fw(n)2268 1968 y FB([)p Fr(e)2340 1927 y Fm(\000)p Fw(A)2452 1968 y FB(])2479 1983 y Fw(n;i)2570 1968 y Fr(e)2615 1927 y Fm(\000)2687 1910 y Fg(b)2670 1927 y Fw(A)2727 1968 y Fr(a)2778 1983 y Fw(j)2815 1857 y Fn(\021)3270 1968 y FB(\(A.2\))751 2237 y(=)910 2142 y Fn(X)915 2354 y Fw(n)p Fm(2)p Fl(Z)1070 2237 y FB([)p Fr(e)1142 2196 y Fm(\000)p Fw(A)1254 2237 y FB(])1281 2252 y Fw(n;i)1372 2237 y FB(\()p Fo(\000h)p Fr(a)1577 2196 y Fm(\003)1577 2262 y Fw(n)1624 2237 y Fr(a)1675 2252 y Fw(j)1712 2237 y Fo(i)22 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2998 y FB(\))2585 2962 y Fm(\000)p Fp(1)2679 2998 y FB(])2706 3013 y Fw(i;m)2849 2998 y FB(w)m(e)34 b(obtain)1346 3188 y Fo(h)p Fr(a)1436 3147 y Fm(\003)1436 3213 y Fw(m)1502 3188 y Fr(a)1553 3203 y Fw(j)1590 3188 y Fo(i)28 b FB(=)f([\()p Fi(1)22 b FB(+)g Fr(e)2048 3147 y Fw(A)2106 3188 y FB(\))2144 3147 y Fm(\000)p Fp(1)2238 3188 y FB(])2265 3203 y Fw(j;m)2380 3188 y Fr(:)863 b FB(\(A.4\))257 3473 y Fj(A.2)135 b(Pro)t(of)45 b(of)g (\(2.43\)-\(2.45\))257 3658 y FB(W)-8 b(e)33 b(pro)m(v)m(e)h(recursiv)m (ely)g(that)998 3848 y Fo(h)p Fr(a)1088 3807 y Fm(\003)1088 3873 y Fw(i)1112 3882 y Fk(1)1151 3848 y Fr(a)1202 3863 y Fw(j)1231 3872 y Fk(1)1286 3848 y Fo(\001)17 b(\001)g(\001)e Fr(a)1470 3807 y Fm(\003)1470 3873 y Fw(i)1494 3881 y Fh(n)1541 3848 y Fr(a)1592 3863 y Fw(j)1621 3871 y Fh(n)1667 3848 y Fo(i)27 b FB(=)h(Det)16 b(\()p Fr(R)q FB(\()p Fr(i)2193 3863 y Fw(k)2236 3848 y Fr(;)h(j)2320 3863 y Fw(l)2346 3848 y FB(\)\))2422 3877 y Fp(1)p Fm(\024)p Fw(k)r(;l)q Fm(\024)p Fw(n)2711 3848 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y Fo(i)129 b(h)p Fr(a)2023 5026 y Fm(\003)2023 5087 y Fw(i)2047 5096 y Fk(2)2086 5062 y Fr(a)2137 5077 y Fw(j)2166 5086 y Fk(2)2205 5062 y Fo(i)c(\001)17 b(\001)g(\001)102 b(h)p Fr(a)2679 5026 y Fm(\003)2679 5087 y Fw(i)2703 5096 y Fk(2)2742 5062 y Fr(a)2793 5077 y Fw(j)2822 5085 y Fh(n)2868 5062 y Fo(i)1596 5158 y FB(.)1596 5191 y(.)1596 5224 y(.)2075 5158 y(.)2075 5191 y(.)2075 5224 y(.)2384 5166 y(.)2422 5191 y(.)2460 5216 y(.)2734 5158 y(.)2734 5191 y(.)2734 5224 y(.)1412 5345 y Fo(\000h)p Fr(a)1579 5360 y Fw(j)1608 5368 y Fh(n)1655 5345 y Fr(a)1706 5308 y Fm(\003)1706 5369 y Fw(i)1730 5378 y Fk(1)1769 5345 y Fo(i)83 b(\000h)p Fr(a)2058 5360 y Fw(j)2087 5368 y Fh(n)2133 5345 y Fr(a)2184 5308 y Fm(\003)2184 5369 y Fw(i)2208 5378 y Fk(2)2247 5345 y Fo(i)g(\001)17 b(\001)g(\001)98 b(h)p Fr(a)2675 5308 y Fm(\003)2675 5369 y Fw(i)2699 5377 y Fh(n)2745 5345 y Fr(a)2796 5360 y Fw(j)2825 5368 y Fh(n)2872 5345 y Fo(i)2952 4850 y Fn(?)2952 4910 y(?)2952 4970 y(?)2952 5029 y(?)2952 5089 y(?)2952 5149 y(?)2952 5209 y(?)2952 5269 y(?)2952 5328 y(?)3035 5144 y Fr(:)208 b FB(\(A.7\))p eop %%Page: 43 43 43 42 bop 257 266 a Ft(A)98 b(FERMIONIC)34 b(CORRELA)-8 b(TIONS)1566 b FB(43)257 573 y(W)-8 b(e)33 b(will)d(need)k(one)f(more)e (expression)j(for)e Fo(h\001)17 b(\001)g(\001)e(i)32 b FB(suc)m(h)i(that)e(in)g(the)g(\014rst)h Fr(k)j FB(pairs)c(the)257 693 y(annihilation)d(op)s(erator)j(precedes)j(the)e(creation)f(op)s (erator,)317 905 y Fo(h)p Fr(a)407 920 y Fw(j)436 929 y Fk(1)474 905 y Fr(a)525 869 y Fm(\003)525 929 y Fw(i)549 938 y Fk(1)605 905 y Fo(\001)17 b(\001)g(\001)d Fr(a)788 920 y Fw(j)817 932 y Fh(k)859 905 y Fr(a)910 869 y Fm(\003)910 929 y Fw(i)934 941 y Fh(k)977 905 y Fr(a)1028 869 y Fm(\003)1028 929 y Fw(i)1052 941 y Fh(k)q Fk(+1)1171 905 y Fr(a)1222 920 y Fw(j)1251 932 y Fh(k)q Fk(+1)1387 905 y Fr(:)j(:)g(:)f(a)1569 869 y Fm(\003)1569 929 y Fw(i)1593 937 y Fh(n)1640 905 y Fr(a)1691 920 y Fw(j)1720 928 y Fh(n)1766 905 y Fo(i)28 b FB(=)317 1484 y(=)f(\()p Fo(\000)p FB(1\))622 1448 y Fw(k)682 1041 y Fn(?)682 1100 y(?)682 1160 y(?)682 1220 y(?)682 1280 y(?)682 1339 y(?)682 1399 y(?)682 1459 y(?)682 1519 y(?)682 1579 y(?)682 1638 y(?)682 1698 y(?)682 1758 y(?)682 1818 y(?)830 1133 y Fo(\000h)p Fr(a)997 1148 y Fw(j)1026 1157 y Fk(1)1065 1133 y Fr(a)1116 1097 y Fm(\003)1116 1158 y Fw(i)1140 1167 y Fk(1)1179 1133 y Fo(i)123 b(\001)17 b(\001)g(\001)177 b(h)p Fr(a)1726 1097 y Fm(\003)1726 1158 y Fw(i)1750 1167 y Fk(1)1788 1133 y Fr(a)1839 1148 y Fw(j)1868 1160 y Fh(k)1910 1133 y Fo(i)205 b(h)p Fr(a)2244 1097 y Fm(\003)2244 1158 y Fw(i)2268 1167 y Fk(1)2306 1133 y Fr(a)2357 1148 y Fw(j)2386 1160 y Fh(k)q Fk(+1)2505 1133 y Fo(i)126 b(\001)17 b(\001)g(\001)138 b(h)p Fr(a)3016 1097 y Fm(\003)3016 1158 y Fw(i)3040 1167 y Fk(1)3079 1133 y Fr(a)3130 1148 y Fw(j)3159 1156 y Fh(n)3205 1133 y Fo(i)1010 1229 y FB(.)1010 1262 y(.)1010 1295 y(.)1355 1237 y(.)1394 1262 y(.)1432 1287 y(.)1779 1229 y(.)1779 1262 y(.)1779 1295 y(.)2335 1229 y(.)2335 1262 y(.)2335 1295 y(.)2685 1237 y(.)2723 1262 y(.)2761 1287 y(.)3071 1229 y(.)3071 1262 y(.)3071 1295 y(.)828 1416 y Fo(\000h)p Fr(a)995 1431 y Fw(j)1024 1443 y Fh(k)1066 1416 y Fr(a)1117 1379 y Fm(\003)1117 1440 y Fw(i)1141 1449 y Fk(1)1180 1416 y Fo(i)122 b(\001)17 b(\001)g(\001)136 b(\000h)p Fr(a)1762 1431 y Fw(j)1791 1443 y Fh(k)1833 1416 y Fr(a)1884 1379 y Fm(\003)1884 1440 y Fw(i)1908 1452 y Fh(k)1951 1416 y Fo(i)162 b(h)p Fr(a)2242 1379 y Fm(\003)2242 1440 y Fw(i)2266 1452 y Fh(k)2308 1416 y Fr(a)2359 1431 y Fw(j)2388 1443 y Fh(k)q Fk(+1)2507 1416 y Fo(i)124 b(\001)17 b(\001)g(\001)136 b(h)p Fr(a)3014 1379 y Fm(\003)3014 1440 y Fw(i)3038 1452 y Fh(k)3081 1416 y Fr(a)3132 1431 y Fw(j)3161 1439 y Fh(n)3207 1416 y Fo(i)790 1536 y(\000h)p Fr(a)957 1551 y Fw(j)986 1563 y Fh(k)q Fk(+1)1105 1536 y Fr(a)1156 1500 y Fm(\003)1156 1561 y Fw(i)1180 1570 y Fk(1)1219 1536 y Fo(i)83 b(\001)17 b(\001)g(\001)98 b(\000h)p Fr(a)1724 1551 y Fw(j)1753 1563 y Fh(k)q Fk(+1)1872 1536 y Fr(a)1923 1500 y Fm(\003)1923 1561 y Fw(i)1947 1573 y Fh(k)1989 1536 y Fo(i)85 b(h)p Fr(a)2203 1500 y Fm(\003)2203 1561 y Fw(i)2227 1573 y Fh(k)q Fk(+1)2347 1536 y Fr(a)2398 1551 y Fw(j)2427 1563 y Fh(k)q Fk(+1)2546 1536 y Fo(i)g(\001)17 b(\001)g(\001)98 b(h)p Fr(a)2976 1500 y Fm(\003)2976 1561 y Fw(i)3000 1573 y Fh(k)q Fk(+1)3119 1536 y Fr(a)3170 1551 y Fw(j)3199 1559 y Fh(n)3245 1536 y Fo(i)1010 1639 y FB(.)1010 1672 y(.)1010 1705 y(.)1355 1647 y(.)1394 1672 y(.)1432 1697 y(.)1779 1639 y(.)1779 1672 y(.)1779 1705 y(.)2335 1639 y(.)2335 1672 y(.)2335 1705 y(.)2685 1647 y(.)2723 1672 y(.)2761 1697 y(.)3071 1639 y(.)3071 1672 y(.)3071 1705 y(.)826 1826 y Fo(\000h)p Fr(a)993 1841 y Fw(j)1022 1849 y Fh(n)1069 1826 y Fr(a)1120 1790 y Fm(\003)1120 1850 y Fw(i)1144 1859 y Fk(1)1183 1826 y Fo(i)119 b(\001)17 b(\001)g(\001)134 b(\000h)p Fr(a)1760 1841 y Fw(j)1789 1849 y Fh(n)1835 1826 y Fr(a)1886 1790 y Fm(\003)1886 1850 y Fw(i)1910 1862 y Fh(k)1953 1826 y Fo(i)119 b(\000h)p Fr(a)2278 1841 y Fw(j)2307 1849 y Fh(n)2354 1826 y Fr(a)2405 1790 y Fm(\003)2405 1850 y Fw(i)2429 1862 y Fh(k)q Fk(+1)2548 1826 y Fo(i)83 b(\001)17 b(\001)g(\001)134 b(h)p Fr(a)3012 1790 y Fm(\003)3012 1850 y Fw(i)3036 1858 y Fh(n)3083 1826 y Fr(a)3134 1841 y Fw(j)3163 1849 y Fh(n)3209 1826 y Fo(i)3326 1041 y Fn(?)3326 1100 y(?)3326 1160 y(?)3326 1220 y(?)3326 1280 y(?)3326 1339 y(?)3326 1399 y(?)3326 1459 y(?)3326 1519 y(?)3326 1579 y(?)3326 1638 y(?)3326 1698 y(?)3326 1758 y(?)3326 1818 y(?)3409 1484 y Fr(:)3270 1970 y FB(\(A.8\))257 2189 y(Let)34 b(us)g(pro)m(v)m(e)g(this)f(form)m (ula.)43 b(F)-8 b(or)33 b Fr(k)e FB(=)e(0,)k(it)f(agrees)i(with)f (\(A.7\).)45 b(Supp)s(ose)34 b(it)e(to)h(b)s(e)257 2309 y(true)g(for)f(some)h Fr(k)s FB(.)43 b(Let)33 b(us)g(then)g(pro)m(v)m (e)h(that)f(the)g(form)m(ula)d(\(A.8\))j(holds)f(for)g Fr(k)25 b FB(+)d(1,)929 2529 y Fo(h)p Fr(a)1019 2544 y Fw(j)1048 2553 y Fk(1)1086 2529 y Fr(a)1137 2488 y Fm(\003)1137 2554 y Fw(i)1161 2563 y Fk(1)1216 2529 y Fo(\001)17 b(\001)g(\001)e Fr(a)1400 2544 y Fw(j)1429 2556 y Fh(k)q Fk(+1)1548 2529 y Fr(a)1599 2488 y Fm(\003)1599 2554 y Fw(i)1623 2566 y Fh(k)q Fk(+1)1743 2529 y Fr(a)1794 2488 y Fm(\003)1794 2554 y Fw(i)1818 2566 y Fh(k)q Fk(+2)1937 2529 y Fr(a)1988 2544 y Fw(j)2017 2556 y Fh(k)q Fk(+2)2153 2529 y Fr(:)i(:)g(:)f(a)2335 2488 y Fm(\003)2335 2554 y Fw(i)2359 2562 y Fh(n)2406 2529 y Fr(a)2457 2544 y Fw(j)2486 2552 y Fh(n)2532 2529 y Fo(i)28 b FB(=)929 2674 y(=)f Fo(\000h)p Fr(a)1199 2689 y Fw(j)1228 2698 y Fk(1)1267 2674 y Fr(a)1318 2633 y Fm(\003)1318 2699 y Fw(i)1342 2708 y Fk(1)1398 2674 y Fo(\001)17 b(\001)g(\001)d Fr(a)1581 2689 y Fw(j)1610 2701 y Fh(k)1652 2674 y Fr(a)1703 2633 y Fm(\003)1703 2699 y Fw(i)1727 2711 y Fh(k)1770 2674 y Fr(a)1821 2633 y Fm(\003)1821 2699 y Fw(i)1845 2711 y Fh(k)q Fk(+1)1964 2674 y Fr(a)2015 2689 y Fw(j)2044 2701 y Fh(k)q Fk(+1)2180 2674 y Fr(:)j(:)g(:)f(a)2362 2633 y Fm(\003)2362 2699 y Fw(i)2386 2707 y Fh(n)2433 2674 y Fr(a)2484 2689 y Fw(j)2513 2697 y Fh(n)2559 2674 y Fo(i)672 b FB(\(A.9\))1027 2820 y(+)22 b Fr(\016)1168 2835 y Fw(i)1192 2847 y Fh(k)q Fk(+1)1307 2835 y Fw(;j)1356 2847 y Fh(k)q Fk(+1)1474 2820 y Fo(h)p Fr(a)1564 2835 y Fw(j)1593 2844 y Fk(1)1632 2820 y Fr(a)1683 2779 y Fm(\003)1683 2844 y Fw(i)1707 2853 y Fk(1)1762 2820 y Fo(\001)17 b(\001)g(\001)e Fr(a)1946 2835 y Fw(j)1975 2847 y Fh(k)2017 2820 y Fr(a)2068 2779 y Fm(\003)2068 2844 y Fw(i)2092 2856 y Fh(k)2135 2820 y Fr(a)2186 2779 y Fm(\003)2186 2844 y Fw(i)2210 2856 y Fh(k)q Fk(+2)2329 2820 y Fr(a)2380 2835 y Fw(j)2409 2847 y Fh(k)q Fk(+2)2545 2820 y Fr(:)i(:)g(:)f(a)2727 2779 y Fm(\003)2727 2844 y Fw(i)2751 2852 y Fh(n)2798 2820 y Fr(a)2849 2835 y Fw(j)2878 2843 y Fh(n)2924 2820 y Fo(i)p Fr(:)257 3040 y FB(Using)28 b(the)g(expression)h(\(A.8\))e(and)h(considering)f(the)h (expansion)g(of)g(the)g(determinan)m(t)257 3160 y(in)40 b(the)h(\()p Fr(k)31 b FB(+)c(1\))865 3124 y Fp(th)977 3160 y FB(column)39 b(\(or)h(ro)m(w\),)j(it)d(is)g(easy)i(to)e(see)i (that)e(\(A.9\))h(corresp)s(onds,)257 3280 y(up)j(to)e(a)h(factor)f(of) g Fo(\000)p FB(1,)k(to)c(the)i(expression)g(\(A.8\))e(but)i(with)e(the) h(diagonal)e(term)257 3401 y Fr(a)308 3365 y Fm(\003)308 3425 y Fw(i)332 3437 y Fh(k)q Fk(+1)452 3401 y Fr(a)503 3416 y Fw(j)532 3428 y Fh(k)q Fk(+1)683 3401 y FB(replaced)33 b(b)m(y)h Fo(\000)p Fr(a)1332 3416 y Fw(j)1361 3428 y Fh(k)q Fk(+1)1480 3401 y Fr(a)1531 3365 y Fm(\003)1531 3425 y Fw(i)1555 3437 y Fh(k)q Fk(+1)1675 3401 y FB(.)43 b(Therefore)34 b(\(A.8\))e(holds)h(for)f Fr(k)25 b FB(+)d(1,)32 b(to)s(o.)404 3521 y(No)m(w)37 b(w)m(e)g(pro)m(v)m(e)h(\(A.7\))f(for)f Fr(n)25 b FB(+)f(1)37 b(b)m(y)g(using)f(\(A.7\))h(for)f Fr(n)g FB(and)h(\(A.8\))f(for)g Fr(n)h FB(and)257 3642 y Fr(k)31 b Fo(\024)d Fr(n)p FB(,)749 3901 y Fo(h)p Fr(a)839 3859 y Fm(\003)839 3925 y Fw(q)879 3901 y Fr(a)930 3916 y Fw(j)959 3925 y Fk(1)1014 3901 y Fo(\001)17 b(\001)g(\001)e Fr(a)1198 3859 y Fm(\003)1198 3925 y Fw(i)1222 3934 y Fh(n)p Fk(+1)1346 3901 y Fr(a)1397 3916 y Fw(j)1426 3925 y Fh(n)p Fk(+1)1549 3901 y Fo(i)27 b FB(=)1742 3833 y(1)p 1729 3878 74 4 v 1729 3969 a Fr(Z)1829 3901 y FB(T)-8 b(r)1946 3790 y Fn(\020)2006 3901 y Fr(e)2051 3859 y Fm(\000)2123 3843 y Fg(b)2106 3859 y Fw(A)2163 3901 y Fr(a)2214 3859 y Fm(\003)2214 3925 y Fw(q)2254 3901 y Fr(a)2305 3916 y Fw(j)2334 3925 y Fk(1)2389 3901 y Fo(\001)17 b(\001)g(\001)d Fr(a)2572 3859 y Fm(\003)2572 3925 y Fw(i)2596 3934 y Fh(n)p Fk(+1)2720 3901 y Fr(a)2771 3916 y Fw(j)2800 3925 y Fh(n)p Fk(+1)2924 3790 y Fn(\021)749 4134 y FB(=)858 4039 y Fn(X)853 4251 y Fw(m)p Fm(2)p Fl(Z)1046 4066 y FB(1)p 1033 4111 V 1033 4202 a Fr(Z)1117 4134 y FB([)p Fr(e)1189 4093 y Fm(\000)p Fw(A)1301 4134 y FB(])1328 4149 y Fw(m;q)1465 4134 y FB(T)-8 b(r)1582 4023 y Fn(\020)1641 4134 y Fr(e)1686 4093 y Fm(\000)1758 4076 y Fg(b)1741 4093 y Fw(A)1799 4134 y Fr(a)1850 4149 y Fw(j)1879 4158 y Fk(1)1934 4134 y Fo(\001)17 b(\001)g(\001)e Fr(a)2118 4093 y Fm(\003)2118 4158 y Fw(i)2142 4167 y Fh(n)p Fk(+1)2265 4134 y Fr(a)2316 4149 y Fw(j)2345 4158 y Fh(n)p Fk(+1)2469 4134 y Fr(a)2520 4093 y Fm(\003)2520 4158 y Fw(m)2587 4023 y Fn(\021)749 4403 y FB(=)28 b Fo(\000)952 4308 y Fn(X)947 4520 y Fw(m)p Fm(2)p Fl(Z)1117 4403 y FB([)p Fr(e)1189 4362 y Fm(\000)p Fw(A)1302 4403 y FB(])1329 4418 y Fw(m;q)1465 4403 y Fo(h)p Fr(a)1555 4362 y Fm(\003)1555 4427 y Fw(m)1622 4403 y Fr(a)1673 4418 y Fw(j)1702 4427 y Fk(1)1757 4403 y Fo(\001)17 b(\001)g(\001)e Fr(a)1941 4362 y Fm(\003)1941 4427 y Fw(i)1965 4436 y Fh(n)p Fk(+1)2089 4403 y Fr(a)2140 4418 y Fw(j)2169 4427 y Fh(n)p Fk(+1)2292 4403 y Fo(i)891 b FB(\(A.10\))847 4745 y(+)951 4620 y Fw(n)p Fp(+1)945 4650 y Fn(X)955 4859 y Fw(p)p Fp(=2)1106 4745 y FB([)p Fr(e)1178 4703 y Fm(\000)p Fw(A)1290 4745 y FB(])1317 4760 y Fw(j)1346 4768 y Fh(p)1382 4760 y Fw(;q)1456 4745 y Fo(h)p Fr(a)1546 4760 y Fw(j)1575 4769 y Fk(1)1613 4745 y Fr(a)1664 4703 y Fm(\003)1664 4769 y Fw(i)1688 4778 y Fk(2)1744 4745 y Fo(\001)17 b(\001)g(\001)e Fr(a)1928 4760 y Fw(j)1957 4769 y Fh(p)p Fd(\000)p Fk(1)2075 4745 y Fr(a)2126 4703 y Fm(\003)2126 4769 y Fw(i)2150 4777 y Fh(p)2191 4745 y Fr(a)2242 4703 y Fm(\003)2242 4769 y Fw(i)2266 4778 y Fh(p)p Fk(+1)2383 4745 y Fr(a)2434 4760 y Fw(j)2463 4769 y Fh(p)p Fk(+1)2597 4745 y Fr(:)i(:)g(:)f(a)2779 4703 y Fm(\003)2779 4769 y Fw(i)2803 4778 y Fh(n)p Fk(+1)2927 4745 y Fr(a)2978 4760 y Fw(j)3007 4769 y Fh(n)p Fk(+1)3130 4745 y Fo(i)847 5002 y FB(+)22 b([)p Fr(e)1017 4961 y Fm(\000)p Fw(A)1130 5002 y FB(])1157 5017 y Fw(j)1186 5026 y Fk(1)1220 5017 y Fw(;q)1294 5002 y Fo(h)p Fr(a)1384 4961 y Fm(\003)1384 5026 y Fw(i)1408 5035 y Fk(2)1447 5002 y Fr(a)1498 5017 y Fw(j)1527 5026 y Fk(2)1582 5002 y Fo(\001)17 b(\001)g(\001)e Fr(a)1766 4961 y Fm(\003)1766 5026 y Fw(i)1790 5035 y Fh(n)p Fk(+1)1914 5002 y Fr(a)1965 5017 y Fw(j)1994 5026 y Fh(n)p Fk(+1)2117 5002 y Fo(i)p Fr(:)257 5222 y FB(W)-8 b(e)35 b(tak)m(e)h(the)f(term)f(with)g(the)h (sum)f(o)m(v)m(er)i Fr(m)31 b Fo(2)g Fq(Z)h FB(together)j(with)f(the)h (\014rst)g(one)g(and)p eop %%Page: 44 44 44 43 bop 257 266 a Ft(REFERENCES)2472 b FB(44)257 573 y(m)m(ultiply)31 b(the)i(whole)f(expression)i(b)m(y)1697 498 y Fn(P)1802 602 y Fw(q)r Fm(2)p Fl(Z)1948 573 y FB([\()p Fi(1)23 b FB(+)f Fr(e)2237 537 y Fm(\000)p Fw(A)2349 573 y FB(\))2387 537 y Fm(\000)p Fp(1)2481 573 y FB(])2508 588 y Fw(q)r(;i)2586 597 y Fk(1)2657 573 y FB(to)32 b(obtain)807 793 y Fo(h)p Fr(a)897 752 y Fm(\003)897 817 y Fw(i)921 826 y Fk(1)960 793 y Fr(a)1011 808 y Fw(j)1040 817 y Fk(1)1095 793 y Fo(\001)17 b(\001)g(\001)e Fr(a)1279 752 y Fm(\003)1279 817 y Fw(i)1303 826 y Fh(n)p Fk(+1)1427 793 y Fr(a)1478 808 y Fw(j)1507 817 y Fh(n)p Fk(+1)1630 793 y Fo(i)27 b FB(=)h Fo(h)p Fr(a)1890 752 y Fm(\003)1890 817 y Fw(i)1914 826 y Fk(1)1953 793 y Fr(a)2004 808 y Fw(j)2033 817 y Fk(1)2071 793 y Fo(ih)p Fr(a)2200 752 y Fm(\003)2200 817 y Fw(i)2224 826 y Fk(2)2263 793 y Fr(a)2314 808 y Fw(j)2343 817 y Fk(2)2398 793 y Fo(\001)17 b(\001)g(\001)e Fr(a)2582 752 y Fm(\003)2582 817 y Fw(i)2606 826 y Fh(n)p Fk(+1)2729 793 y Fr(a)2780 808 y Fw(j)2809 817 y Fh(n)p Fk(+1)2933 793 y Fo(i)250 b FB(\(A.11\))807 1044 y(+)905 920 y Fw(n)p Fp(+1)899 949 y Fn(X)909 1159 y Fw(p)p Fp(=2)1060 1044 y Fo(h)p Fr(a)1150 1003 y Fm(\003)1150 1069 y Fw(i)1174 1078 y Fk(1)1213 1044 y Fr(a)1264 1059 y Fw(j)1293 1067 y Fh(p)1333 1044 y Fo(ih)p Fr(a)1462 1059 y Fw(j)1491 1068 y Fk(1)1529 1044 y Fr(a)1580 1003 y Fm(\003)1580 1069 y Fw(i)1604 1078 y Fk(2)1659 1044 y Fo(\001)17 b(\001)g(\001)e Fr(a)1843 1059 y Fw(j)1872 1068 y Fh(p)p Fd(\000)p Fk(1)1991 1044 y Fr(a)2042 1003 y Fm(\003)2042 1069 y Fw(i)2066 1077 y Fh(p)2106 1044 y Fr(a)2157 1003 y Fm(\003)2157 1069 y Fw(i)2181 1078 y Fh(p)p Fk(+1)2299 1044 y Fr(a)2350 1059 y Fw(j)2379 1068 y Fh(p)p Fk(+1)2512 1044 y Fr(:)i(:)g(:)f(a)2694 1003 y Fm(\003)2694 1069 y Fw(i)2718 1078 y Fh(n)p Fk(+1)2842 1044 y Fr(a)2893 1059 y Fw(j)2922 1068 y Fh(n)p Fk(+1)3046 1044 y Fo(i)p Fr(:)257 1364 y FB(Using)37 b(\(A.7\))g(and)g(\(A.8\))g (for)g Fr(n)g FB(terms)h(w)m(e)g(see)g(that)f(this)g(last)f(expression) j(is)e(noth-)257 1485 y(ing)43 b(else)i(than)f(the)h(expansion)f(with)g (resp)s(ect)h(to)f(the)h(\014rst)f(ro)m(w)h(of)e(\(A.7\))h(with)g Fr(n)257 1605 y FB(substituted)34 b(b)m(y)f Fr(n)23 b FB(+)f(1.)257 1938 y Fu(Ac)l(kno)l(wledgmen)l(ts)257 2157 y FB(W)-8 b(e)35 b(are)f(most)f(grateful)g(to)g(Mic)m(hael)h (Pr\177)-49 b(ahofer)33 b(for)h(his)f(constan)m(t)i(help)f(and)g (encour-)257 2277 y(agemen)m(t.)68 b(The)42 b(w)m(ork)g(of)e(P)-8 b(.L.)41 b(F)-8 b(errari)40 b(w)m(as)h(supp)s(orted)h(b)m(y)g(the)f (Swiss)g(fello)m(wship)257 2398 y(Sun)m(burst-F)-8 b(onds.)257 2730 y Fu(References)306 2949 y FB([1])49 b(H.W.J.)41 b(Bl\177)-49 b(ote)39 b(and)i(H.J.)g(Hilhorst,)g Fs(R)-5 b(oughening)41 b(tr)-5 b(ansitions)41 b(and)h(the)g(zer)-5 b(o-)458 3070 y(temp)g(er)g(atur)g(e)43 b(triangular)g(Ising)f (antiferr)-5 b(omagnet)p FB(,)43 b(J.)e(Ph)m(ys.)j(A:)d(Math.)h(Gen.) 458 3190 y Fb(15)32 b FB(\(1982\),)g(L631{L637.)306 3394 y([2])49 b(R.)c(Cerf)g(and)h(R.)e(Ken)m(y)m(on,)50 b Fs(The)c(low-temp)-5 b(er)g(atur)g(e)46 b(exp)-5 b(ansion)44 b(of)j(the)f(Wul\013)458 3514 y(crystal)32 b(in)f(the)h(3D)f(Ising)g (mo)-5 b(del)p FB(,)29 b(Comm.)f(Math.)i(Ph)m(ys)h Fb(222)e FB(\(2001\),)g(147{179.)306 3717 y([3])49 b(H.)39 b(Cohn,)i(N.)d (Elkies,)j(and)d(J.)h(Propp,)i Fs(L)-5 b(o)g(c)g(al)40 b(statistics)h(for)f(r)-5 b(andom)39 b(domino)458 3838 y(tilings)60 b(of)g(the)h(Azte)-5 b(c)61 b(diamond)p FB(,)66 b(Duk)m(e)c(Math.)f(J.)f Fb(85)h FB(\(1996\),)67 b(117{166,)458 3958 y(arXiv:math.CO/0008243.)306 4162 y([4])49 b(H.)36 b(Cohn,)i(M.)f(Larsen,)h(and)f(J.)f(Propp,)i Fs(The)g(shap)-5 b(e)37 b(of)h(a)g(typic)-5 b(al)39 b(b)-5 b(oxe)g(d)37 b(plane)458 4282 y(p)-5 b(artition)p FB(,)32 b(N.Y.)h(J.)g(Math.)g Fb(4)g FB(\(1998\),)e(137{165.)306 4485 y([5])49 b(T.)29 b(F)-8 b(unaki)27 b(and)h(H.)h(Sp)s(ohn,)g Fs(Motion)i(by)g(me)-5 b(an)30 b(curvatur)-5 b(e)31 b(fr)-5 b(om)31 b(the)f(Ginzbur)-5 b(g{)458 4606 y(Landau)34 b Fo(r)p Fr(')h Fs(interfac)-5 b(e)34 b(mo)-5 b(del)p FB(,)32 b(Comm.)g(Math.)g(Ph)m(ys.)j Fb(185)e FB(\(1997\),)e(1{36.)306 4809 y([6])49 b(J.)32 b(Gra)m(vner,)h(C.A.)g(T)-8 b(racy)g(,)33 b(and)f(H.)g(Widom,)f Fs(Limit)j(the)-5 b(or)g(ems)34 b(for)g(height)g(\015uc-)458 4929 y(tuations)44 b(in)g(a)g(class)f(of)h (discr)-5 b(ete)43 b(sp)-5 b(ac)g(e)44 b(and)f(time)h(gr)-5 b(owth)44 b(mo)-5 b(dels)p FB(,)44 b(J.)e(Stat.)458 5050 y(Ph)m(ys.)34 b Fb(102)f FB(\(2001\),)f(1085{1132.)306 5253 y([7])49 b(K.)30 b(Johansson,)h Fs(Discr)-5 b(ete)32 b(p)-5 b(olynucle)g(ar)32 b(gr)-5 b(owth)32 b(and)g(determinantal)f(pr) -5 b(o)g(c)g(esses)p FB(,)458 5374 y(\(2002\),)31 b(preprin)m(t,)i (arXiv:math.PR/0206208.)p eop %%Page: 45 45 45 44 bop 257 266 a Ft(REFERENCES)2472 b FB(45)306 573 y([8])49 b(K.)37 b(Johansson,)h Fs(Non-interse)-5 b(cting)38 b(p)-5 b(aths,)40 b(r)-5 b(andom)38 b(tilings)g(and)g(r)-5 b(andom)38 b(ma-)458 693 y(tric)-5 b(es)p FB(,)32 b(Probab.)h(Theory)h (Relat.)e(Fields)f Fb(123)i FB(\(2002\),)e(225{280.)306 897 y([9])49 b(P)-8 b(.W.)33 b(Kasteleyn,)g Fs(Dimer)h(statistics)h (and)f(phase)g(tr)-5 b(ansitions)p FB(,)33 b(J.)f(Math.)h(Ph)m(ys.)458 1017 y Fb(4)f FB(\(1963\),)g(287{293.)257 1220 y([10])49 b(R.)26 b(Ken)m(y)m(on,)i Fs(The)g(planar)g(dimer)g(mo)-5 b(del)28 b(with)g(b)-5 b(oundary:)42 b(a)28 b(survey)p FB(,)g(Directions)458 1341 y(in)k(mathematical)d(quasicrystals,)k(CRM)g (Monogr.)g(Ser.)g Fb(13)g FB(\(2000\),)e(307{328.)257 1544 y([11])49 b(R.)c(Ken)m(y)m(on,)k Fs(Dominos)c(and)g(Gaussian)h(fr) -5 b(e)g(e)46 b(\014eld)p FB(,)i(Ann.)d(Prob.)g Fb(29)g FB(\(2001\),)458 1665 y(1128{1137.)257 1868 y([12])k(A.)29 b(Ok)m(ounk)m(o)m(v)i(and)e(N.)g(Reshetikhin,)h Fs(Corr)-5 b(elation)31 b(function)g(of)h(Schur)f(pr)-5 b(o)g(c)g(ess)458 1988 y(with)48 b(applic)-5 b(ation)48 b(to)h(lo)-5 b(c)g(al)48 b(ge)-5 b(ometry)48 b(of)g(a)h(r)-5 b(andom)48 b FB(3)p Fs(-dimensional)e(Young)458 2109 y(diagr)-5 b(am)p FB(,)31 b(\(2001\),)h(preprin)m(t,)h(arXiv:math.CO/0107056.)257 2312 y([13])49 b(V.L.)38 b(P)m(okro)m(vsky)j(and)d(A.L.)g(T)-8 b(alap)s(o)m(v,)39 b Fs(Gr)-5 b(ound)41 b(state,)g(sp)-5 b(e)g(ctrum,)41 b(and)e(phase)458 2433 y(diagr)-5 b(am)36 b(of)h(two-dimensional)f(inc)-5 b(ommensur)g(ate)36 b(crystals)p FB(,)g(Ph)m(ys.)h(Rev.)f(Lett.)458 2553 y Fb(42)c FB(\(1978\),)g (65{68.)257 2756 y([14])49 b(M.)29 b(Pr\177)-49 b(ahofer)28 b(and)g(H.)g(Sp)s(ohn,)i Fs(Sc)-5 b(ale)30 b(invarianc)-5 b(e)29 b(of)i(the)g(PNG)g(dr)-5 b(oplet)31 b(and)f(the)458 2877 y(Airy)35 b(pr)-5 b(o)g(c)g(ess)p FB(,)32 b(J.)h(Stat.)f(Ph)m(ys.) j Fb(108)d FB(\(2002\),)g(1071{1106.)257 3080 y([15])49 b(M.)36 b(Reed)h(and)f(B.)g(Simon:)48 b Fs(Metho)-5 b(ds)38 b(of)g(Mo)-5 b(dern)37 b(Mathematic)-5 b(al)38 b(Physics)f(IV:)458 3200 y(A)n(nalysis)d(of)h(Op)-5 b(er)g(ators,)32 b FB(Academic)g (Press,)i(New)f(Y)-8 b(ork,)33 b(1978.)257 3404 y([16])49 b(T.)30 b(Shirai)d(and)j(Y.)f(T)-8 b(ak)j(ahashi,)30 b Fs(Random)h(p)-5 b(oint)31 b(\014elds)g(asso)-5 b(ciate)g(d)31 b(with)h(c)-5 b(ertain)458 3524 y(Fr)g(e)g(dholm)31 b(determinants)h (I:)g(Fermion,)g(Poisson)g(and)g(Boson)g(p)-5 b(oint)32 b(pr)-5 b(o)g(c)g(esses)p FB(,)458 3645 y(\(2002\),)31 b(preprin)m(t,)i(RIMS-1368.)257 3848 y([17])49 b(A.)30 b(Soshnik)m(o)m(v,)h Fs(Determinantal)h(r)-5 b(andom)31 b(p)-5 b(oint)32 b(\014elds)p FB(,)e(Russ.)g(Math.)h(Surv.)f Fb(55)458 3968 y FB(\(2000\),)h(923{976.)257 4172 y([18])49 b(C.A.)g(T)-8 b(racy)49 b(and)f(H.)g(Widom,)j Fs(L)-5 b(evel-sp)g(acing)48 b(distributions)h(and)f(the)i(Airy)458 4292 y(kernel)p FB(,)32 b(Comm.)f(Math.)i(Ph)m(ys.)i Fb(159)d FB(\(1994\),)g(151{174.)257 4496 y([19])49 b(F.Y.)54 b(W)-8 b(u,)59 b Fs(R)-5 b(emarks)54 b(on)g(the)h(mo)-5 b(di\014e)g(d)53 b(p)-5 b(otassium)54 b(dihydr)-5 b(o)g(gen)53 b(phosphate)458 4616 y(mo)-5 b(del)34 b(of)g(a)h(ferr)-5 b(o)g(ele)g(ctric)p FB(,)32 b(Ph)m(ys.)j(Rev.)e Fb(168)f FB(\(1968\),)g(539{543.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0212181312363--