Content-Type: multipart/mixed; boundary="-------------0111261105364" This is a multi-part message in MIME format. ---------------0111261105364 Content-Type: text/plain; name="01-434.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-434.keywords" phase segregation, self-avoiding polygons, fluctuation of phase boundaries, Wulff construction, local limit theorems ---------------0111261105364 Content-Type: application/postscript; name="sap.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="sap.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: sap.dvi %%Pages: 50 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o sap.ps sap.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2001.11.26:1508 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro %! 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(y)n(ed)f(b)n(y)h(an)f(accurate)g(analysis)f(of)i(the)456 923 y(statistical)j(prop)r(erties)h(of)g(random)f(in)n(terfaces.)48 b(In)31 b(fact,)h(suc)n(h)f(an)g(analysis)f(is)i(piv)n(otal)e(for)456 1023 y(the)i(DKS)g(theory)g(in)g(the)h(sense)e(that)i(the)f(sharpness)f (of)h(all)g(other)f(estimates)h(\(e.g.,)h(lo)r(cal)456 1123 y(limit)26 b(description)f(of)h(the)h(magnetisation)d(in)i(large)f (\014nite)h(v)n(olumes\))f(directly)h(hinges)f(on)h(it.)555 1222 y(Random)40 b(phase)f(b)r(oundary)g(is,)k(arguably)-7 b(,)41 b(the)f(cen)n(tral)f(ob)5 b(ject)39 b(in)h(the)g(probabilistic) 456 1322 y(theory)c(of)h(phase)f(segregation.)63 b(When)38 b(apply)-7 b(,)39 b(the)e(results)g(and)g(metho)r(ds)g(of)g(the)g(DKS) 456 1421 y(approac)n(h)f(certainly)i(giv)n(e)f(m)n(uc)n(h)h(more)g (than)g(just)h(a)f(re\014nemen)n(t)g(of)g(limit)i(theorems)d(as)456 1521 y(pro)n(vided)27 b(b)n(y)h(the)h FE(L)1107 1533 y FD(1)1179 1521 y FG(theory)-7 b(.)38 b(Apart)28 b(from)g(giving)g(a)g (scaled)f(description)h(of)g(the)h(in)n(terface)456 1621 y FA(p)l(er)k(se)39 b FG(\(see,)32 b(e.g.)49 b([22)o(,)32 b(23)o(]\))g(and)f(apart)g(from)h(pro)n(viding)e(explicit)i(estimates)f (in)h(terms)g(of)456 1720 y(sizes)23 b(of)h(\014nite)h(systems)f(under) g(consideration,)f(they)h(suggest)f(a)h(probabilistic)f(approac)n(h)g (to)456 1820 y(a)h(v)-5 b(ariet)n(y)24 b(of)i(in)n(teresting)e(and)h (delicate)g(issues)f(suc)n(h)h(as)f(metastabilit)n(y)h(and)g(\014ne)g (prop)r(erties)456 1920 y(of)h(sto)r(c)n(hastic)f(dynamics)g(in)i(the)f (phase)f(co-existence)g(regime)g([38,)h(46)o(],)g(w)n(etting)g (transition)456 2019 y(in)k(the)h(vicinit)n(y)g(of)f(the)h(w)n(etting)f (p)r(oin)n(t)h([32)o(])f(and)h(analytic)f(prop)r(erties)f(of)i(the)f (free)h(energy)456 2119 y(and)c(the)h(surface)f(tension)g([45)o(,)h(13) o(].)555 2218 y(A)n(t)19 b(least)g(for)f(\014nite)i(range)d(mo)r(dels)i (in)g(t)n(w)n(o)f(dimensions,)i(random)e(in)n(terfaces)g(are)g(exp)r (ected)456 2318 y(to)25 b(exhibit)i(the)f(in)n(trinsic)g(Bro)n(wnian)e (statistical)h(b)r(eha)n(viour)g(based)h(up)r(on)g(a)f(rapid)h(deca)n (y)f(of)456 2418 y(correlations.)33 b(In)24 b(particular,)g(the)g(lo)r (cal)f(p)r(osition)h(of)g(the)g(in)n(terface)g(is)g(b)r(eliev)n(ed)f (to)h(ha)n(v)n(e)f(the)456 2517 y(Gaussian)31 b(scaling,)h(whereas)e (the)j(global)d(description)i(of)g(the)g(\015uctuating)g(random)f (phase)456 2617 y(b)r(oundaries)36 b(is)h(supp)r(osed)g(to)g(rely)f(up) r(on)h(the)h(appropriate)d(v)n(ersions)h(of)h(the)g(conditional)456 2717 y(in)n(v)-5 b(ariance)26 b(principle)h(\(e.g.,)g(under)g(the)h (Dobrushin)f(t)n(yp)r(e)g(b)r(oundary)g(conditions)f(or)h(under)456 2816 y(the)h(\014xed)f(magnetisation)g(constrain)n(t\).)555 2916 y(So)d(far)f(rigorous)e(results)i(in)h(this)g(direction)f(ha)n(v)n (e)f(b)r(een)i(obtained)g(mainly)f(in)h(the)g(con)n(text)456 3015 y(of)j(the)i(nearest)d(neigh)n(b)r(our)h(v)n(ery)g(lo)n(w)g(temp)r (erature)g(Ising)h(mo)r(del)f([27,)g(28,)g(21)o(,)h(23)o(,)g(30)o(,)g (31)o(].)456 3115 y(T)-7 b(ec)n(hnically)25 b(all)h(these)g(w)n(orks)f (are)g(based)h(up)r(on)g(the)h(lo)n(w)e(temp)r(erature)h(cluster)g (expansions)456 3215 y(and)j(emplo)n(y)f(non-probabilistic)g (computations,)h(suc)n(h)g(as)f(v)-5 b(arious)28 b(estimates)h(on)g (generat-)456 3314 y(ing)34 b(functions)g(of)g(distributions)g(with)h (small)f(negativ)n(e)f(w)n(eigh)n(ts.)56 b(F)-7 b(urthermore,)35 b(the)f(lo)n(w)456 3414 y(temp)r(erature)e(assumption)h(w)n(as)f (frequen)n(tly)h(used)g(to)f(discoun)n(t)h(the)h(en)n(trop)n(y)d(b)r (ey)n(ond)i(the)456 3514 y(in)n(trinsic)27 b(probabilistic)g(structure) g(of)g(the)h(mo)r(del.)555 3613 y(The)c(main)f(ob)5 b(jectiv)n(e)22 b(of)i(our)e(w)n(ork)g(here)h(is)g(to)g(dev)n(elop)g(a)g(\\mo)r(del")f (purely)h(probabilistic)456 3713 y(analysis)40 b(of)h(microscopic)f (random)h(in)n(terfaces)f(in)i(t)n(w)n(o)f(dimensions.)78 b(W)-7 b(e)42 b(consider)e(the)456 3812 y(simplest)19 b(non-trivial)f(ensem)n(ble)h(of)g(microscopic)f(phase)h(b)r (oundaries,)h(namely)f(the)h(ensem)n(ble)456 3912 y(of)32 b(self-a)n(v)n(oiding)f FE(Z)1098 3882 y FD(2)1129 3912 y FG(-lattice)i(p)r(olygons)e Fz(\015)38 b FG(with)33 b(the)g(asso)r(ciated)f(w)n(eigh)n(ts)g(e)2929 3882 y Fy(\000)p FC(\014)s Fy(j)p FC(\015)t Fy(j)3103 3912 y FG(,)j Fz(\014)h(>)c(\014)3388 3924 y FC(c)3421 3912 y FG(,)456 4012 y(where)f Fz(\014)747 4024 y FC(c)814 4012 y FG(is)h(the)h(critical)e(p)r(oin)n(t)i(for)f(the)g(self-a)n(v)n (oiding)f(planar)g(w)n(alks)g(\(see)h(\(1.3\))g(b)r(elo)n(w\).)456 4111 y(This)38 b(collection)g(of)h(p)r(olygons)e(corresp)r(onds)g(to)h (the)h(ensem)n(ble)f(of)h(phase)f(b)r(oundaries)g(of)456 4211 y(the)33 b(Ising)g(mo)r(del)g(in)h(the)f(corner)f FE(Z)1635 4181 y FD(2)1635 4232 y(+)1718 4211 y FG(with)h(free)g(b)r (oundary)g(conditions)f([19)o(].)54 b(W)-7 b(e)34 b(obtain)456 4311 y(the)e(expansion)f(up)h(to)g(zero)e(order)h(terms)g(of)h(the)h (canonical)d(\(with)j(\014xed)f(area)e(inside)i(the)456 4410 y(p)r(olygons\))e(partition)h(function.)49 b(As)32 b(a)f(b)n(ypro)r(duct)g(of)g(these)g(considerations)f(\(and)i(a)f(v)n (ery)456 4510 y(w)n(eak)19 b(v)n(ersion)g(of)h(our)g(main)g(result\))h (one)f(can)g(easily)g(deduce)g(the)h(asymptotic)f(concen)n(tration)456 4610 y(prop)r(ert)n(y)k(of)h(the)h(appropriately)e(rescaled)g(p)r (olygons)g(on)h(the)h(quarter)e(of)i(the)g(related)e(W)-7 b(ul\013)456 4709 y(shap)r(e.)40 b(Of)29 b(course,)f(enclosed)h(area)e (of)i(self-a)n(v)n(oiding)e(p)r(olygons)h(is)g(an)h(in)n(teresting)f (topic)h(in)456 4809 y(its)e(o)n(wn)g(righ)n(t,)g(see)g(e.g.,)h([26)o (])g(or)e([14)o(])i(for)f(a)g(more)g(ph)n(ysical-st)n(yle)f(treatmen)n (t.)555 4908 y(The)k(mo)r(del)g(is)f(describ)r(ed)g(in)h(Sect.)g(1.1)f (and)g(the)h(result)g(is)f(stated)g(in)h(Sect.)g(1.2.)42 b(As)30 b(w)n(e)456 5008 y(men)n(tion)g(in)h(Remark)f(1.1.1)f(b)r(elo)n (w)h(the)h(tec)n(hniques)g(w)n(e)f(dev)n(elop)g(lead)g(to)g(the)h (sharp)f(lo)r(cal)456 5108 y(limit)21 b(description)f(of)h (\015uctuations)g(of)g(random)f(in)n(terfaces)g(\(p)r(olygons\))g (around)g(the)h(limiting)456 5207 y(quarter)26 b(W)-7 b(ul\013)29 b(shap)r(e.)p eop %%Page: 3 3 3 2 bop 1488 226 a FD(SELF-A)-7 b(V)n(OIDING)29 b(POL)-5 b(YGONS)999 b(3)456 425 y FG(1.1.)46 b FO(The)30 b(mo)s(del.)39 b FG(Consider)24 b(the)j(family)e Fx(Z)1944 446 y FD(+)2026 425 y FG(of)g(all)h(self-a)n(v)n(oiding)d FE(Z)2767 395 y FD(2)2798 425 y FG(-lattice)i(paths)h Fz(!)g FG(=)456 525 y(\()p Fz(!)s FG(\(0\))p Fz(;)14 b(:::;)g(!)s FG(\()p Fz(n)p FG(\)\))28 b(whic)n(h)f(cross)f(the)i(p)r(ositiv)n(e)f(quadran)n (t)1170 670 y Fw(Quad)1356 685 y FD(+)1435 670 y FG(=)1522 603 y Fv(\010)1571 670 y FG(\()p Fz(x;)14 b(y)s FG(\))23 b Fx(2)h FE(R)1919 636 y FD(2)1962 600 y Fv(\014)1962 649 y(\014)2017 670 y Fz(x)g(>)e FG(0)55 b Fu(and)e Fz(y)26 b(>)d FG(0)2646 603 y Fv(\011)2707 670 y Fz(:)456 823 y FG(More)j(precisely)-7 b(,)27 b(let)h(us)g(de\014ne)f(p)r(ositiv)n(e) h(lattice)f(semi-axes)g Fx(P)2491 788 y FD(+)2484 846 y(0)2573 823 y FG(and)g Fx(L)2791 788 y FD(+)2791 846 y(0)2874 823 y FG(as)775 970 y Fx(P)840 935 y FD(+)833 992 y(0)918 970 y FG(=)1005 903 y Fv(\010)1054 970 y FG(\(0)p Fz(;)14 b(k)s FG(\))22 b Fx(2)i FE(Z)1405 936 y FD(2)1460 900 y Fv(\014)1460 949 y(\014)1510 970 y Fz(k)i(>)d FG(0)1709 903 y Fv(\011)1881 970 y Fu(and)109 b Fx(L)2171 935 y FD(+)2171 992 y(0)2249 970 y FG(=)2337 903 y Fv(\010)2385 970 y FG(\()p Fz(k)s(;)14 b FG(0\))23 b Fx(2)g FE(Z)2737 936 y FD(2)2791 900 y Fv(\014)2791 949 y(\014)2842 970 y Fz(k)j(>)c FG(0)3040 903 y Fv(\011)3102 970 y Fz(:)456 1114 y FG(Then)27 b(w)n(e)h(sa)n(y)e(that)i Fz(!)e Fx(2)d(Z)1334 1135 y FD(+)1417 1114 y FG(i\013)793 1269 y Fz(!)s FG(\(0\))f Fx(2)i(P)1120 1234 y FD(+)1113 1291 y(0)1174 1269 y Fz(;)42 b(!)s FG(\()p Fz(n)p FG(\))23 b Fx(2)h(L)1567 1234 y FD(+)1567 1291 y(0)1733 1269 y Fu(and)108 b Fz(!)s FG(\()p Fz(l)r FG(\))23 b Fx(2)g Fw(Quad)2399 1284 y FD(+)2472 1269 y Fx(\\)c FE(Z)2608 1235 y FD(2)81 b Fy(8)2764 1269 y Fz(l)24 b FG(=)f(1)p Fz(;)14 b(::;)g(n)j Fx(\000)h FG(1)p Fz(:)-2822 b FG(\(1.1\))456 1418 y(Notice)25 b(that)h(in)f(the)h(ab)r(o)n(v)n(e)e(de\014nition)i (the)f(length)h Fx(j)p Fz(!)s Fx(j)d FG(=)f Fz(n)k FG(of)f(a)g(path)g Fz(!)j FG(w)n(as)d(not)g(assumed)456 1518 y(to)i(b)r(e)i(\014xed.)38 b(Eac)n(h)27 b(path)h Fz(!)e Fx(2)f(Z)1524 1538 y FD(+)1607 1518 y FG(splits)j Fw(Quad)2012 1533 y FD(+)2095 1518 y FG(in)n(to)g(t)n(w)n(o)f(comp)r(onen)n(ts)h(|)g(the)g(b)r(ounded)456 1617 y(one)e(\(con)n(taining)g(the)i(origin\))e(and)g(the)i(un)n(b)r (ounded)f(one.)36 b(Let)27 b FO(A)2588 1629 y FD(+)2643 1617 y FG(\()p Fz(!)s FG(\))g(denote)g(the)g(area)f(of)456 1717 y(the)i(b)r(ounded)g(comp)r(onen)n(t.)456 1872 y FO(De\014nition)i(1.1.)40 b FG(Giv)n(en)28 b Fz(Q)23 b Fx(2)g FE(N)38 b FG(de\014ne)27 b(the)h(canonical)f(set)g(of)h(paths) f Fx(Z)2844 1895 y FC(Q;)p FD(+)2999 1872 y FG(via)1350 2030 y Fx(Z)1410 2050 y FC(Q;)p FD(+)1560 2030 y FG(=)1648 1962 y Fv(\010)1697 2030 y Fz(!)e Fx(2)f(Z)1913 2050 y FD(+)1991 1959 y Fv(\014)1991 2009 y(\014)2042 2030 y FO(A)2114 2042 y FD(+)2169 2030 y FG(\()p Fz(!)s FG(\))f(=)g Fz(Q)2465 1962 y Fv(\011)2527 2030 y Fz(:)-2094 b FG(\(1.2\))555 2201 y(Giv)n(en)29 b Fz(\014)g(>)24 b FG(0)k(w)n(e)g(asso)r(ciate)g (with)h(an)n(y)f(self-a)n(v)n(oiding)e(path)j Fz(!)i FG(the)e(w)n(eigh)n(t)f Fz(e)3055 2171 y Fy(\000)p FC(\014)s Fy(j)p FC(!)r Fy(j)3235 2201 y FG(.)40 b(It)29 b(is)456 2301 y(kno)n(wn)e([37)o(])g(that)h(there)g(exists)f(the)h(critical)f(v) -5 b(alue)27 b Fz(\014)2178 2313 y FC(c)2235 2301 y Fx(2)d FG(\(0)p Fz(;)14 b Fx(1)p FG(\),)27 b(suc)n(h)h(that)985 2470 y Fz(\014)g(>)22 b(\014)1194 2482 y FC(c)1334 2470 y Fx(\()-14 b(\))1609 2391 y Fv(X)1592 2573 y FC(x)p Fy(2)p Ft(Z)1719 2556 y Fs(2)1760 2470 y Fz(g)1800 2482 y FC(\014)1844 2470 y FG(\()p Fz(x)p FG(\))1979 2423 y Fr(def)1988 2470 y FG(=)2102 2391 y Fv(X)2085 2573 y FC(x)p Fy(2)p Ft(Z)2212 2556 y Fs(2)2292 2391 y Fv(X)2252 2567 y 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8250 4200 m 8250 4500 l gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P9 [16 0 0 -8 550.00 280.00] PATmp PATsp ef gr PATusp gs col0 s gr % Polyline n 11475 4500 m 11550 4500 l gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P10 [8 0 0 -16 765.00 300.00] PATmp PATsp ef gr PATusp gs col0 s gr % Polyline [60] 0 sd n 11550 1200 m 11550 4575 l gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P10 [8 0 0 -16 770.00 80.00] PATmp PATsp ef gr PATusp gs col0 s gr [] 0 sd % Polyline n 11550 4500 m 11550 7200 l gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P10 [8 0 0 -16 770.00 300.00] PATmp PATsp ef gr PATusp gs col0 s gr % Polyline n 11550 4500 m 11550 7200 l gs col7 0.90 shd ef gr gs col0 s gr % Polyline n 8250 4200 m 8250 4500 l gs col7 0.90 shd ef gr gs col0 s gr % Polyline [60] 0 sd n 10950 3300 m 10950 3975 l gs col7 0.90 shd ef gr gs col0 s gr [] 0 sd % Polyline n 8250 1125 m 8250 7425 l gs col0 s gr % Polyline 15.000 slw n 11543 7200 m 11558 7200 l gs col0 s gr % Polyline 7.500 slw n 7950 7200 m 11550 7200 l gs col0 s gr % Polyline n 14550 7200 m 15450 7200 l gs col0 s gr % Polyline n 8250 2175 m 8250 2177 l 8250 2182 l 8250 2190 l 8250 2204 l 8250 2223 l 8250 2249 l 8250 2281 l 8250 2320 l 8250 2366 l 8250 2417 l 8250 2474 l 8250 2536 l 8250 2602 l 8250 2671 l 8250 2741 l 8250 2813 l 8250 2886 l 8250 2957 l 8250 3028 l 8250 3097 l 8250 3163 l 8250 3227 l 8250 3289 l 8250 3347 l 8250 3403 l 8250 3456 l 8250 3505 l 8250 3552 l 8250 3596 l 8250 3637 l 8250 3676 l 8250 3712 l 8250 3746 l 8250 3778 l 8250 3809 l 8250 3837 l 8250 3863 l 8250 3889 l 8250 3912 l 8250 3913 l 8250 3967 l 8250 4013 l 8250 4053 l 8250 4086 l 8250 4114 l 8250 4137 l 8250 4155 l 8250 4170 l 8250 4182 l 8250 4190 l 8250 4195 l 8250 4198 l 8250 4200 l gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P9 [16 0 0 -8 550.00 145.00] PATmp PATsp ef gr PATusp gs col0 s gr % Ellipse n 8250 2100 38 38 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr % Ellipse n 11550 4500 38 38 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr % Ellipse n 14550 7200 38 38 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr % Polyline n 300 1200 m 300 7500 l gs col0 s gr % Polyline n 300 7200 m 6000 7200 l gs col0 s gr % Polyline 15.000 slw gs clippath 1530 1881 m 1500 2025 l 1470 1881 l 1470 2055 l 1530 2055 l cp clip n 1500 1200 m 1500 2025 l gs col0 s gr gr % arrowhead 7.500 slw n 1530 1881 m 1500 2025 l 1470 1881 l 1500 1905 l 1530 1881 l cp gs 0.00 setgray ef gr col0 s % Polyline n 300 1650 m 600 1650 l 600 2025 l 900 2025 l 900 1725 l 1200 1725 l 1200 1950 l 1050 1950 l 1050 2100 l 1650 2100 l 1650 1950 l 1950 1950 l 1950 2100 l 2250 2100 l 2250 2250 l 1950 2250 l 1950 2400 l 2400 2400 l 2400 2250 l 2550 2250 l 2550 2625 l 2850 2625 l 2850 2400 l 2700 2400 l 2700 2250 l 2850 2250 l 3000 2325 l 3000 2550 l 3150 2550 l 3150 2475 l 3300 2475 l 3300 2700 l 3000 2700 l 3000 2850 l 3450 2850 l 3450 2625 l 3675 2625 l 3750 2925 l 3525 2925 l 3525 3225 l 4050 3225 l 4050 3075 l 4200 3075 l 4200 3450 l 3975 3450 l 3975 3600 l 4350 3600 l 4350 3825 l 4875 3825 l 4875 4050 l 5025 4050 l 5025 4350 l 5100 4350 l 5100 4500 l 4875 4500 l 4875 4125 l 4650 4125 l 4650 4500 l 4800 4500 l 4800 4725 l 5250 4725 l 5250 4950 l 4950 4950 l 4950 5175 l 5100 5175 l 5100 5025 l 5400 5025 l 5400 5475 l 5625 5475 l 5625 5850 l 5475 5850 l 5475 5625 l 5250 5625 l 5250 6000 l 5700 6000 l 5700 6300 l 5400 6300 l 5400 6600 l 5550 6600 l 5550 6450 l 5850 6450 l 5850 6750 l 5475 6750 l 5475 6900 l 5850 6900 l 5850 7200 l gs col0 s gr % Polyline n 6000 7200 m 6450 7200 l gs col0 s gr % Polyline 15.000 slw n 300 1800 m 302 1800 l 306 1800 l 313 1800 l 325 1801 l 341 1802 l 362 1802 l 387 1804 l 416 1805 l 449 1807 l 484 1809 l 520 1811 l 558 1814 l 596 1817 l 634 1821 l 671 1824 l 708 1828 l 744 1832 l 780 1837 l 815 1842 l 849 1847 l 883 1853 l 918 1859 l 953 1866 l 988 1874 l 1024 1882 l 1061 1891 l 1100 1900 l 1129 1907 l 1159 1915 l 1191 1923 l 1222 1932 l 1255 1941 l 1289 1951 l 1324 1961 l 1360 1971 l 1397 1982 l 1435 1993 l 1474 2005 l 1514 2018 l 1555 2030 l 1597 2043 l 1639 2057 l 1682 2071 l 1726 2085 l 1770 2099 l 1814 2114 l 1858 2129 l 1903 2144 l 1948 2159 l 1992 2174 l 2036 2190 l 2080 2205 l 2124 2221 l 2167 2236 l 2210 2252 l 2252 2267 l 2293 2282 l 2334 2297 l 2373 2312 l 2413 2327 l 2451 2342 l 2489 2356 l 2527 2371 l 2564 2386 l 2600 2400 l 2638 2415 l 2676 2431 l 2713 2446 l 2751 2462 l 2788 2477 l 2826 2493 l 2863 2510 l 2901 2526 l 2938 2543 l 2976 2559 l 3013 2576 l 3051 2594 l 3088 2611 l 3126 2629 l 3163 2646 l 3200 2664 l 3236 2682 l 3272 2700 l 3308 2718 l 3343 2736 l 3377 2754 l 3411 2771 l 3444 2789 l 3476 2806 l 3508 2824 l 3538 2841 l 3568 2858 l 3597 2874 l 3625 2890 l 3652 2907 l 3679 2923 l 3704 2938 l 3729 2954 l 3753 2969 l 3777 2985 l 3800 3000 l 3827 3018 l 3854 3037 l 3880 3056 l 3907 3074 l 3933 3094 l 3959 3114 l 3984 3134 l 4010 3155 l 4036 3176 l 4061 3198 l 4087 3220 l 4112 3243 l 4137 3266 l 4162 3290 l 4186 3314 l 4210 3338 l 4234 3363 l 4257 3388 l 4280 3413 l 4302 3439 l 4324 3464 l 4345 3490 l 4366 3516 l 4386 3541 l 4406 3567 l 4426 3593 l 4444 3620 l 4463 3646 l 4482 3673 l 4500 3700 l 4515 3723 l 4531 3747 l 4546 3771 l 4562 3796 l 4577 3821 l 4593 3848 l 4610 3875 l 4626 3903 l 4643 3932 l 4659 3962 l 4676 3992 l 4694 4024 l 4711 4056 l 4729 4089 l 4746 4123 l 4764 4157 l 4782 4192 l 4800 4228 l 4818 4264 l 4836 4300 l 4854 4337 l 4871 4374 l 4889 4412 l 4906 4449 l 4924 4487 l 4941 4524 l 4958 4562 l 4974 4599 l 4990 4637 l 5007 4674 l 5023 4712 l 5038 4749 l 5054 4787 l 5069 4824 l 5085 4862 l 5100 4900 l 5114 4936 l 5129 4973 l 5144 5011 l 5158 5049 l 5173 5087 l 5188 5127 l 5203 5166 l 5218 5207 l 5233 5248 l 5248 5290 l 5264 5333 l 5279 5376 l 5295 5420 l 5310 5464 l 5326 5508 l 5341 5552 l 5356 5597 l 5371 5642 l 5386 5686 l 5401 5730 l 5415 5774 l 5429 5818 l 5443 5861 l 5457 5903 l 5470 5945 l 5482 5986 l 5495 6026 l 5507 6065 l 5518 6103 l 5529 6140 l 5539 6176 l 5549 6211 l 5559 6245 l 5568 6278 l 5577 6309 l 5585 6341 l 5593 6371 l 5600 6400 l 5609 6439 l 5618 6476 l 5626 6512 l 5634 6547 l 5641 6582 l 5647 6617 l 5653 6651 l 5658 6685 l 5663 6720 l 5668 6756 l 5672 6792 l 5676 6829 l 5679 6866 l 5683 6904 l 5686 6942 l 5689 6980 l 5691 7016 l 5693 7051 l 5695 7084 l 5696 7113 l 5698 7138 l 5698 7159 l 5699 7175 l 5700 7187 l 5700 7194 l 5700 7198 l 5700 7200 l gs col0 s gr % Polyline 7.500 slw [60] 0 sd n 300 1500 m 302 1500 l 307 1501 l 315 1503 l 329 1505 l 347 1508 l 372 1512 l 401 1518 l 435 1524 l 473 1530 l 514 1538 l 558 1545 l 602 1553 l 647 1562 l 692 1570 l 736 1578 l 779 1586 l 820 1593 l 859 1601 l 896 1608 l 932 1615 l 966 1622 l 998 1629 l 1029 1635 l 1059 1642 l 1089 1648 l 1117 1655 l 1145 1662 l 1172 1668 l 1200 1675 l 1228 1682 l 1255 1689 l 1283 1696 l 1311 1704 l 1340 1712 l 1369 1720 l 1399 1728 l 1430 1737 l 1461 1746 l 1492 1755 l 1524 1765 l 1556 1775 l 1589 1785 l 1622 1795 l 1655 1805 l 1689 1816 l 1722 1826 l 1755 1837 l 1789 1848 l 1822 1858 l 1855 1869 l 1887 1880 l 1919 1890 l 1951 1901 l 1983 1911 l 2014 1921 l 2045 1932 l 2076 1942 l 2107 1952 l 2138 1963 l 2167 1972 l 2196 1982 l 2226 1992 l 2256 2002 l 2287 2013 l 2319 2023 l 2351 2034 l 2384 2045 l 2417 2057 l 2451 2068 l 2485 2080 l 2520 2092 l 2555 2105 l 2591 2117 l 2627 2130 l 2663 2143 l 2698 2156 l 2734 2169 l 2770 2182 l 2805 2195 l 2840 2208 l 2874 2221 l 2908 2234 l 2941 2247 l 2974 2260 l 3006 2273 l 3038 2286 l 3069 2299 l 3099 2312 l 3129 2324 l 3158 2337 l 3188 2350 l 3216 2363 l 3245 2376 l 3274 2390 l 3303 2403 l 3332 2417 l 3362 2432 l 3392 2447 l 3422 2462 l 3452 2478 l 3482 2494 l 3513 2511 l 3544 2528 l 3575 2545 l 3606 2563 l 3637 2581 l 3668 2600 l 3699 2618 l 3729 2637 l 3760 2657 l 3790 2676 l 3819 2695 l 3848 2715 l 3877 2734 l 3905 2754 l 3932 2774 l 3959 2793 l 3986 2813 l 4012 2833 l 4038 2852 l 4063 2872 l 4088 2892 l 4113 2913 l 4136 2932 l 4159 2952 l 4182 2972 l 4206 2992 l 4230 3014 l 4254 3036 l 4278 3058 l 4302 3081 l 4327 3105 l 4352 3130 l 4377 3155 l 4403 3180 l 4428 3207 l 4454 3234 l 4479 3261 l 4505 3289 l 4531 3317 l 4556 3346 l 4581 3375 l 4606 3404 l 4631 3433 l 4655 3463 l 4679 3492 l 4703 3522 l 4726 3551 l 4748 3580 l 4770 3609 l 4792 3639 l 4813 3668 l 4834 3696 l 4854 3725 l 4874 3754 l 4893 3783 l 4913 3813 l 4931 3840 l 4949 3869 l 4967 3897 l 4984 3926 l 5002 3956 l 5020 3987 l 5039 4018 l 5057 4050 l 5075 4082 l 5094 4115 l 5112 4149 l 5130 4184 l 5149 4220 l 5168 4256 l 5186 4292 l 5205 4329 l 5223 4367 l 5241 4405 l 5259 4443 l 5277 4482 l 5295 4520 l 5313 4559 l 5330 4598 l 5347 4636 l 5363 4675 l 5380 4713 l 5396 4752 l 5411 4790 l 5426 4827 l 5441 4865 l 5456 4902 l 5470 4939 l 5484 4976 l 5498 5013 l 5512 5050 l 5525 5088 l 5538 5123 l 5550 5159 l 5563 5195 l 5575 5231 l 5588 5268 l 5600 5306 l 5613 5344 l 5625 5383 l 5638 5422 l 5651 5462 l 5663 5502 l 5676 5543 l 5688 5584 l 5701 5625 l 5713 5667 l 5725 5708 l 5737 5750 l 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l 5021 5466 l 5033 5499 l 5045 5532 l 5057 5565 l 5069 5599 l 5081 5633 l 5093 5668 l 5105 5702 l 5117 5737 l 5129 5772 l 5140 5806 l 5152 5841 l 5163 5875 l 5173 5909 l 5184 5942 l 5194 5975 l 5204 6007 l 5213 6039 l 5222 6071 l 5231 6101 l 5239 6131 l 5247 6161 l 5254 6190 l 5262 6219 l 5268 6247 l 5275 6275 l 5282 6305 l 5288 6334 l 5294 6364 l 5300 6394 l 5306 6425 l 5311 6457 l 5317 6489 l 5322 6523 l 5327 6558 l 5333 6595 l 5338 6633 l 5343 6674 l 5348 6715 l 5354 6759 l 5359 6803 l 5364 6849 l 5369 6895 l 5374 6940 l 5378 6984 l 5383 7026 l 5387 7064 l 5390 7098 l 5393 7128 l 5395 7152 l 5397 7171 l 5399 7184 l 5399 7193 l 5400 7198 l 5400 7200 l gs col0 s gr [] 0 sd % Polyline n 4800 3600 m 4350 4050 l gs col0 s gr % Polyline 15.000 slw gs clippath 4227 4131 m 4350 4050 l 4269 4173 l 4392 4050 l 4350 4008 l cp clip n 4350 4050 m 3900 4500 l gs col0 s gr gr % arrowhead 7.500 slw n 4227 4131 m 4350 4050 l 4269 4173 l 4265 4135 l 4227 4131 l cp gs 0.00 setgray ef gr col0 s % Polyline 15.000 slw gs clippath 4923 3519 m 4800 3600 l 4881 3477 l 4758 3600 l 4800 3642 l cp clip n 4800 3600 m 5400 3000 l gs col0 s gr gr % arrowhead 7.500 slw n 4923 3519 m 4800 3600 l 4881 3477 l 4885 3515 l 4923 3519 l cp gs 0.00 setgray ef gr col0 s /Times-Italic ff 450.00 scf sf 2250 8025 m gs 1 -1 sc (a\)) col0 sh gr /Times-Italic ff 450.00 scf sf 9450 8100 m gs 1 -1 sc (b\)) col0 sh gr $F2psEnd rs end showpage %%EndDocument @endspecial 854 395 a FA(j)891 355 y FC(Q)879 420 y(\014)s(;)p FD(+)1113 773 y Fz(!)1397 654 y(Q)1463 624 y FC(\027)t(=)p FD(2)1594 654 y Fx(\030)23 b Fz(M)1772 624 y FC(\027)2176 536 y Fz(!)2228 548 y FD(1)2129 843 y Fz(Q)2195 855 y FD(1)1940 1032 y Fz(R)2082 1316 y(R)q FG(\()p Fz(M)k FG(+)18 b Fz(l)r FG(\))2389 1576 y Fz(M)108 b(M)27 b FG(+)18 b Fz(l)2743 1316 y(Q)2809 1328 y FD(2)2861 1009 y Fz(!)2913 1021 y FD(2)754 1828 y FF(Figure)36 b(1.3.)k FG(a\))31 b(T)-7 b(ub)r(e)31 b(path)g Fz(!)i FG(b\))e(Basic)f (splitting)h Fz(!)g FG(=)c Fz(!)2753 1840 y FD(1)2811 1828 y Fx([)21 b Fz(!)2939 1840 y FD(2)3006 1828 y FG(of)31 b(a)754 1928 y(path)d(from)f Fx(Z)1204 1950 y FC(Q;)p FD(+)456 2214 y FO(Lemma)i(1.4.)40 b Fn(F)-7 b(or)27 b(ev)n(ery)f Fz(")d(>)g FG(0)k Fn(there)g(exists)h Fz(c)2049 2226 y FD(1)2109 2214 y Fz(>)23 b FG(0)p Fn(,)k(suc)n(h)g(that)883 2449 y FE(Z)944 2461 y FC(\014)s(;)o FD(+)1053 2381 y Fv(\000)1091 2449 y Fx(Z)1170 2414 y Fr(tub)r(e)1151 2469 y FC(Q;)p FD(+)1317 2449 y FG(;)41 b Fz(l)r FG(\()p Fz(!)s FG(\))23 b Fx(\025)g Fz(M)1728 2414 y FC(")1763 2381 y Fv(\001)1824 2449 y Fx(\024)f FG(exp)2052 2307 y Fv(\()2119 2449 y Fx(\000)2184 2304 y Fv(s)p 2267 2304 173 4 v 2320 2393 a Fz(Q)p 2277 2430 153 4 v 2277 2506 a(q)2314 2518 y FC(\014)s(;)p FD(+)2439 2449 y Fz(w)2498 2461 y FC(\014)s(;)p FD(+)2632 2449 y Fx(\000)c Fz(c)2751 2461 y FD(1)2789 2449 y Fz(M)2879 2414 y FC(")2914 2307 y Fv(\))2994 2449 y Fz(:)-2561 b FG(\(1.16\))555 2695 y(Th)n(us,)23 b(w)n(e)e(can)g(restrict)g(atten)n(tion)h(only)f(to)h (bridges)e Fz(!)26 b Fx(2)d(Z)2466 2664 y Fr(tub)r(e)2447 2717 y FC(Q;)p FD(+)2607 2695 y FG(satisfying)e(the)h(condition)456 2808 y Fz(l)r FG(\()p Fz(!)s FG(\))g Fx(\024)h Fz(M)802 2777 y FC(")837 2808 y FG(.)36 b(In)25 b(the)g(latter)f(case)g(the)h (corresp)r(onding)d(break)i(p)r(oin)n(t)h Fz(x)p FG(\()p Fz(!)s FG(\))e(=)2920 2740 y Fv(\000)2958 2808 y Fz(M)e FG(+)12 b Fz(l)r FG(\()p Fz(!)s FG(\))p Fz(;)i(R)3384 2740 y Fv(\001)3421 2808 y FG(,)456 2907 y Fz(R)23 b FG(=)g Fz(R)q FG(\()p Fz(!)s FG(\),)j(splits)g(the)g(bridge)f Fz(!)j FG(in)n(to)d(t)n(w)n(o)g(pieces,)h Fz(!)g FG(=)c Fz(!)2351 2919 y FD(1)2403 2907 y Fx([)15 b Fz(!)2525 2919 y FD(2)2562 2907 y FG(,)26 b(as)f(sho)n(wn)g(in)g(Fig.)h(1.3)13 b(b\).)456 3007 y(Notice)24 b(that)g Fz(!)941 3019 y FD(2)1003 3007 y FG(is)g(a)g(corner)f(path,)i Fz(!)1666 3019 y FD(2)1726 3007 y Fx(2)e(A)1870 3019 y FC(R)1925 3007 y FG(,)i(whereas)e Fz(!)2336 3019 y FD(1)2397 3007 y FG(is)h(a)g(bridge)g(from)g Fx(B)3042 3030 y FC(M)6 b FD(+)p FC(l)3211 3007 y FG(whic)n(h,)456 3108 y(in)24 b(addition,)h(has)f(no)g(break)f(lines)i(in)f(the)h(in)n(terv)-5 b(al)24 b([)p Fz(M)t(;)14 b(M)20 b FG(+)12 b Fz(l)r FG(].)35 b(Let)25 b(us)f(use)g Fx(B)2971 3131 y FC(M)s(;l)3106 3108 y FG(to)h(denote)456 3209 y(the)g(set)g(of)g(all)f(suc)n(h)h (bridges)f(and)g FE(B)1604 3221 y FC(M)s(;l)1746 3209 y FG(to)g(denote)h(the)g(corresp)r(onding)e(partition)i(function.)456 3308 y(W)-7 b(e)28 b(obtain:)886 3462 y FE(Z)947 3474 y FC(\014)s(;)p FD(+)1071 3462 y FG(\()p Fz(Q)p FG(\))23 b(=)1436 3383 y Fv(X)1312 3565 y Fy(j)p FC(R)p Fy(\000)p FC(M)6 b Fy(j\024)p FC(M)1644 3549 y Fq(\027)1722 3383 y Fv(X)1695 3562 y FC(l)p Fy(\024)p FC(M)1837 3545 y Fq(")2133 3383 y Fv(X)1883 3565 y FC(Q)1935 3573 y Fs(1)1968 3565 y FD(+)p FC(Q)2071 3573 y Fs(2)2104 3565 y FD(+)p FC(R)p FD(\()p FC(M)g FD(+)p FC(l)p FD(\)=)p FC(Q)2516 3462 y FE(B)2565 3474 y FC(M)t(;l)2696 3462 y FG(\()q Fz(Q)2795 3474 y FD(1)2831 3462 y FG(\))15 b FE(A)2943 3474 y FC(R)3017 3462 y FG(\()p Fz(Q)3115 3474 y FD(2)3152 3462 y FG(\))1219 3773 y(+)j Fz(O)1367 3656 y Fv(\022)1429 3773 y FG(exp)1556 3656 y Fv(\032)1618 3773 y Fx(\000)1683 3628 y Fv(s)p 1766 3628 173 4 v 1819 3717 a Fz(Q)p 1776 3754 153 4 v 1776 3830 a(q)1813 3842 y FC(\014)s(;)p FD(+)1938 3773 y Fz(w)1997 3785 y FC(\014)s(;)p FD(+)2132 3773 y Fx(\000)g Fz(c)2251 3785 y FD(2)2288 3773 y Fz(M)2378 3739 y FC(\027)2437 3773 y Fx(\000)g Fz(c)2556 3785 y FD(3)2593 3773 y Fz(M)2683 3739 y FC(")2719 3656 y Fv(\033\023)2842 3773 y Fz(:)456 3645 y FG(\(1.17\))555 4003 y(This)k(is)f(our)g(basic)f (decomp)r(osition)h(of)h(the)f(canonical)g(partition)g(function)g FE(Z)3004 4015 y FC(\014)r(;)p FD(+)3113 4003 y FG(\()p Fz(Q)p FG(\).)35 b(The)456 4102 y(target)c(asymptotic)h(form)n(ula)g (\(1.9\))g(follo)n(ws)g(from)g(the)h(lo)r(cal)f(limit)h(results)f(on)g (the)h(bridge)456 4202 y(and)21 b(corner)g(canonical)f(partition)i (functions)g(\(see)g(\(1.32\))f(and)h(\(1.34\))f(b)r(elo)n(w\),)i(whic) n(h)f(enable)456 4301 y(an)27 b(asymptotically)f(sharp)h(re-summation)g (in)h(\(1.17\),)e(as)h(it)h(is)g(carried)e(in)i(Sect.)g(2.4.)456 4498 y(1.6.)46 b FO(Conjugate)k(quan)m(tities)f(and)h(the)g (parametrised)e(represen)m(tation.)41 b FG(De\014ne)456 4598 y Fz(m)529 4610 y FC(\014)573 4598 y FG(\()p Fx(\001)p FG(\))29 b(to)f(b)r(e)g(the)g(con)n(v)n(ex)f(conjugate)g(of)h Fz(\034)1827 4610 y FC(\014)1872 4598 y FG(\(1)p Fz(;)14 b Fx(\001)p FG(\);)28 b(th)n(us,)g(for)g(an)n(y)f Fx(j)p Fz(t)p Fx(j)c Fz(<)g(\034)2803 4610 y FC(\014)2848 4598 y FG(\(1)p Fz(;)14 b FG(0\))28 b(or,)f(equiv)-5 b(a-)456 4697 y(len)n(tly)e(,)27 b(for)g(an)n(y)g Fz(t)c Fx(2)h(D)1182 4709 y FC(\014)1227 4697 y FG(,)1381 4858 y Fz(\034)1417 4870 y FC(\014)1462 4791 y Fv(\000)1500 4858 y FG(1)p Fz(;)14 b(m)1652 4824 y Fy(0)1652 4879 y FC(\014)1696 4858 y FG(\()p Fz(t)p FG(\))1790 4791 y Fv(\001)1848 4858 y FG(+)k Fz(m)2004 4870 y FC(\014)2048 4858 y FG(\()p Fz(t)p FG(\))24 b(=)e Fz(tm)2356 4824 y Fy(0)2356 4879 y FC(\014)2401 4858 y FG(\()p Fz(t)p FG(\))p Fz(:)456 5016 y FG(The)j(function)h Fx(\000)p Fz(m)1085 5028 y FC(\014)1155 5016 y FG(can)f(b)r(e)h(though)n(t)f(as)g(the)h(mass)f(of) g(the)h(tilted)g(bridge)f(grand-canonical)456 5116 y(partition)30 b(function.)47 b(In)31 b(the)g(sequel)f(w)n(e)h(shall)f(frequen)n(tly)h (rely)f(on)g(the)i(results)e(from)g([33)o(],)456 5216 y(whic)n(h)d(w)n(e,)g(for)g(the)h(sak)n(e)f(of)g(con)n(v)n(enience,)g (collect)g(in)h(the)g(prop)r(erties)f FO(P1)p FG(-)p FO(P5)g FG(b)r(elo)n(w:)p eop %%Page: 9 9 9 8 bop 1488 226 a FD(SELF-A)-7 b(V)n(OIDING)29 b(POL)-5 b(YGONS)999 b(9)562 425 y FO(P1:)41 b FG(The)27 b(in)n(terior)g(of)g (the)h(e\013ectiv)n(e)g(domain)f(of)h Fz(m)2228 437 y FC(\014)2300 425 y FG(is)1043 612 y Fx(D)1107 624 y FC(\014)1175 564 y Fr(def)1184 612 y FG(=)46 b(in)n(t)1408 544 y Fv(\010)1456 612 y Fz(t)1509 541 y Fv(\014)1509 591 y(\014)1560 612 y Fz(m)1633 624 y FC(\014)1677 612 y FG(\()p Fz(t)p FG(\))24 b Fz(<)f Fx(1)1966 544 y Fv(\011)2037 612 y FG(=)2125 544 y Fv(\000)2163 612 y Fx(\000)p Fz(\034)2264 624 y FC(\014)2309 612 y FG(\(1)p Fz(;)14 b FG(0\))p Fz(;)g(\034)2567 624 y FC(\014)2611 612 y FG(\(1)p Fz(;)g FG(0\))2796 544 y Fv(\001)2834 612 y Fz(:)-2401 b FG(\(1.18\))562 770 y FO(P2:)41 b Fz(m)816 782 y FC(\014)888 770 y FG(is)28 b(analytic)f(on)g Fx(D)1467 782 y FC(\014)1512 770 y FG(.)562 870 y FO(P3:)41 b Fz(m)816 882 y FC(\014)888 870 y FG(is)28 b(strictly)f(con)n(v)n(ex;)f(moreo)n(v)n(er,)f(for)i(ev) n(ery)f([)p Fz(a;)14 b(b)p FG(])23 b Fx(\032)g(D)2616 882 y FC(\014)1680 1028 y FG(min)1653 1085 y FC(t)p Fy(2)p FD([)p FC(a;b)p FD(])1859 1028 y Fz(m)1932 994 y Fy(00)1932 1048 y FC(\014)1977 1028 y FG(\()p Fz(t)p FG(\))h Fz(>)e FG(0)p Fz(:)-1791 b FG(\(1.19\))562 1237 y FO(P4:)41 b FG(F)-7 b(or)27 b(ev)n(ery)f(closed)h(in)n(terv)-5 b(al)27 b([)p Fz(a;)14 b(b)p FG(])23 b Fx(\032)f(D)1993 1249 y FC(\014)2038 1237 y FG(,)1382 1406 y FE(B)1432 1418 y FC(n)1483 1406 y FG(\()p Fz(t)p FG(\))i(=)e Fz(\026)p FG(\()p Fz(t)p FG(\))p Fz(e)1871 1371 y FC(nm)1971 1380 y Fq(\014)2011 1371 y FD(\()p FC(t)p FD(\))2105 1406 y FG(\()q(1)c(+)g FA(o)h FG(\()q(1\))o(\))14 b Fz(;)661 1564 y FG(uniformly)28 b(in)g Fz(t)23 b Fx(2)g FG([)p Fz(a;)14 b(b)p FG(].)37 b(The)27 b(co)r(e\016cien)n(t)h Fz(\026)f FG(ab)r(o)n(v)n(e)g(is)g(giv)n(en)g(b)n(y:)1381 1764 y Fz(\026)p FG(\()p Fz(t)p FG(\))c(=)1636 1672 y Fv(n)1691 1685 y(X)1733 1864 y FC(k)1825 1764 y Fz(k)s FE(F)1927 1776 y FC(k)1974 1764 y FG(\()p Fz(t)p FG(\))p Fz(e)2107 1729 y Fy(\000)p FC(k)q(m)2254 1738 y Fq(\014)2294 1729 y FD(\()p FC(t)p FD(\))2375 1672 y Fv(o)2430 1689 y Fy(\000)p FD(1)456 1764 y FG(\(1.20\))661 1996 y(is)35 b(also)e(an)h(analytic)g(function)g(on)g Fx(D)1887 2008 y FC(\014)1932 1996 y FG(.)57 b(F)-7 b(urthermore,)35 b(there)g(exists)f(a)f(con)n(tin)n(uous)661 2096 y(p)r(ositiv)n(e)27 b(function)i Fz(d)1337 2108 y FC(\014)1382 2096 y FG(\()p Fx(\001)p FG(\))f(on)f Fx(D)1676 2108 y FC(\014)1721 2096 y FG(,)h(suc)n(h)f(that,)h(uniformly)f(in)h Fz(n)g FG(and)f Fz(t)c Fx(2)h FG([)p Fz(a;)14 b(b)p FG(],)1541 2264 y FE(F)1598 2276 y FC(n)1649 2264 y FG(\()p Fz(t)p FG(\))23 b Fx(\024)g Fz(e)1893 2230 y Fy(\000)p FC(nd)2021 2239 y Fq(\014)2059 2230 y FD(\()p FC(t)p FD(\))2140 2264 y FE(B)2190 2276 y FC(n)2241 2264 y FG(\()p Fz(t)p FG(\))p Fz(:)-1902 b FG(\(1.21\))562 2423 y FO(P5:)41 b FG(The)27 b(map)1433 2581 y Fz(t)c Fx(7!)g FG(\()p Fz(t;)14 b Fx(\000)p Fz(m)1829 2593 y FC(\014)1874 2581 y FG(\()p Fz(t)p FG(\)\))p Fz(;)180 b(t)24 b Fx(2)f(D)2399 2593 y FC(\014)2444 2581 y Fz(;)661 2742 y FG(giv)n(es)k(a)g (parametrised)f(represen)n(tation)g(the)i(b)r(oundary)f Fz(@)5 b FO(K)2617 2754 y FC(\014)2679 2742 y Fx(\\)2753 2675 y Fv(\010)2802 2742 y FG(\()p Fz(x;)14 b(y)s FG(\))3017 2672 y Fv(\014)3017 2722 y(\014)3068 2742 y Fz(y)26 b(>)c FG(0)3264 2675 y Fv(\011)3312 2742 y FG(.)456 2939 y(1.7.)46 b FO(Separation)26 b(of)e(masses.)39 b FG(The)22 b(form)n(ula)e (\(1.21\))h(is)g(an)g(instance)g(of)h(the)g(separation)e(of)456 3039 y(the)30 b(deca)n(y)f(rates)h(t)n(yp)r(e)g(phenomenon,)h(whic)n(h) f(lies)g(in)h(the)f(heart)g(of)g(the)h(Ornstein-Zernik)n(e)456 3138 y(theory)-7 b(.)75 b(In)41 b(particular,)i(the)e(corresp)r(onding) e(deca)n(y)g(estimates)i(are)e(ubiquitous)i(in)g(the)456 3238 y(lo)r(cal)26 b(limit)j(approac)n(h)c(w)n(e)i(pursue)g(in)h(this)g (w)n(ork.)35 b(Let)28 b(us)f(form)n(ulate)g(a)g(general)f(p)r(oin)n (t-wise)456 3338 y(statemen)n(t)h(of)h(this)g(sort,)e(whic)n(h)i(will)g (b)r(e)g(rep)r(eatedly)f(referred)f(to)i(in)g(the)g(sequel:)456 3500 y FO(Prop)s(osition)i(1.5.)40 b Fn(Fix)28 b(t)n(w)n(o)e(n)n(um)n (b)r(ers)h Fz(r)n(;)14 b(\016)27 b(>)22 b FG(0)p Fn(,)28 b(and)f(consider)f(the)i(lattice)g(cone)1439 3659 y Fx(C)1483 3671 y FC(r)1543 3659 y FG(=)1630 3591 y Fv(\010)1679 3659 y FG(\()p Fz(x)1758 3671 y FD(1)1796 3659 y Fz(;)14 b(x)1880 3671 y FD(2)1917 3659 y FG(\))1973 3588 y Fv(\014)1973 3638 y(\014)2023 3659 y Fx(j)p Fz(x)2093 3671 y FD(2)2131 3659 y Fx(j)23 b(\024)g Fz(r)r(x)2351 3671 y FD(1)2389 3591 y Fv(\011)2438 3659 y Fz(:)-2005 b FG(\(1.22\))456 3822 y Fn(Then)32 b(there)f(exist)h(p)r(ositiv)n(e)f(constan)n(ts)g Fz(c)1813 3834 y FD(1)1880 3822 y FG(=)f Fz(c)2011 3834 y FD(1)2048 3822 y FG(\()p Fz(r)n(;)14 b(\016)s FG(\))33 b Fn(and)e Fz(c)2458 3834 y FD(2)2525 3822 y FG(=)f Fz(c)2656 3834 y FD(2)2693 3822 y FG(\()p Fz(r)n(;)14 b(\016)s FG(\))p Fn(,)34 b(suc)n(h)d(that)h(uni-)456 3921 y(formly)d(in)h Fz(x)d Fx(2)g(C)1020 3933 y FC(r)1087 3921 y Fn(and)i(in)h(all)g (sub-in)n(terv)-5 b(als)28 b FG([)p Fz(a;)14 b(b)p FG(])27 b Fx(\032)f FG([0)p Fz(;)14 b(x)2384 3933 y FD(1)2421 3921 y FG(])30 b Fn(of)g(the)g(length)g Fx(j)p Fz(b)19 b Fx(\000)h Fz(a)p Fx(j)26 b(\025)h Fz(\016)s(x)3407 3933 y FD(1)456 4021 y Fn(one)g(has)g(the)h(follo)n(wing)e(inequalit)n (y)1009 4179 y Fz(g)1049 4191 y FC(\014)1107 4179 y FG(\()p Fz(x)p FG(;)42 b Fk(!)28 b Fj(has)e(no)g(break)f(lines)i(on)e Fu([)p Fk(a;)14 b(b)p Fu(])p FG(\))23 b Fx(\024)g Fz(c)2394 4191 y FD(1)2431 4179 y Fz(e)2470 4145 y Fy(\000)p FC(c)2552 4153 y Fs(2)2584 4145 y FC(x)2622 4153 y Fs(1)2658 4179 y Fz(g)2698 4191 y FC(\014)2742 4179 y FG(\()p Fz(x)p FG(\);)-2420 b(\(1.23\))456 4337 y Fn(here,)27 b(giv)n(en)g(a)g(family) g Fx(E)35 b Fn(of)28 b(self-a)n(v)n(oiding)d(paths)j Fz(!)i Fn(w)n(e)d(de\014ne:)1518 4507 y Fz(g)1558 4519 y FC(\014)1616 4507 y FG(\()p Fz(x)p FG(;)14 b Fx(E)7 b FG(\))24 b(=)1967 4429 y Fv(X)1927 4604 y FC(!)r FD(:0)p Fy(!)p FC(x)1962 4670 y(!)r Fy(2E)2140 4507 y Fz(e)2179 4473 y Fy(\000)p FC(\014)s Fy(j)p FC(!)r Fy(j)2359 4507 y Fz(:)555 4817 y FG(In)34 b(the)g(case)e(of)i(the)f(self-a)n(v)n (oiding)f(w)n(alks)g(the)i(estimate)f(\(1.23\))g(has)g(b)r(een)h (established)456 4917 y(in)j([33)o(])g(follo)n(wing)f(the)i(earlier)e (w)n(ork)f(of)i([20].)65 b(The)37 b(in)n(trinsic)g(renormalisation)e (pro)r(of)i(of)456 5016 y(the)30 b(mass-gap)f(t)n(yp)r(e)h(statemen)n (ts)g(has)g(b)r(een)h(recen)n(tly)e(dev)n(elop)r(ed)h(in)g(a)g(more)g (complicated)456 5116 y(con)n(text)40 b(of)g(the)h(nearest)e(neigh)n(b) r(our)h(Bernoulli)g(b)r(ond)g(p)r(ercolation)g(in)g([12)o(])h(and)f(in) h(the)456 5216 y(con)n(text)27 b(of)g(high)h(temp)r(erature)f(Ising)g (mo)r(dels)h(in)g([13)o(].)p eop %%Page: 10 10 10 9 bop 456 226 a FD(10)814 b(OST)-5 b(AP)28 b(HR)-5 b(YNIV)29 b(AND)g(DMITR)-5 b(Y)29 b(IOFFE)555 425 y FG(The)21 b(inequalities)f(\(1.21\))f(and)h(\(1.23\))g(imply)h(the)f(follo)n (wing)g(relation)f(b)r(et)n(w)n(een)h(the)h(bridge)456 525 y(and)e(full)h(connectivit)n(y)f(functions)h(\(c.f.)h([20)o(])f (and)f([33)o(]\):)33 b(Giv)n(en)20 b Fz(x)j Fx(2)h FE(Z)2679 495 y FD(2)2730 525 y FG(\(with)c Fz(x)2990 537 y FD(1)3051 525 y Fx(\025)j FG(0\))c(de\014ne)456 624 y Fz(h)504 636 y FC(\014)548 624 y FG(\()p Fz(x)p FG(\))29 b(and)e Fz(f)890 636 y FC(\014)935 624 y FG(\()p Fz(x)p FG(\))h(as)731 781 y Fz(h)779 793 y FC(\014)824 781 y FG(\()p Fz(x)p FG(\))c(=)1151 702 y Fv(X)1111 878 y FC(!)r FD(:0)p Fy(!)p FC(x)1047 956 y(!)r Fy(\000)e Fu(bridge)1388 781 y Fz(e)1427 746 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Fz(g)2310 1346 y FC(\014)2355 1334 y FG(\()p Fz(x)p FG(\))24 b Fx(\024)e Fz(c)2613 1346 y FD(4)2651 1334 y Fz(h)2699 1346 y FC(\014)2743 1334 y FG(\()p Fz(x)p FG(\))p Fz(:)-2421 b FG(\(1.24\))456 1487 y(1.8.)46 b FO(Bridge)38 b(and)h(corner)f (partition)g(functions.)j FG(The)33 b(leading)g(asymptotics)f(of)h(the) 456 1587 y(bridge)h(and)h(corner)f(partition)g(functions)i FE(B)1916 1599 y FC(M)s(;l)2069 1587 y FG(and)e FE(A)2303 1599 y FC(R)2363 1587 y FG(,)j(whic)n(h)e(sho)n(w)g(up)g(in)h(the)f (basic)456 1686 y(decomp)r(osition)19 b(form)n(ula)f(\(1.17\),)j(are)d (related)h(to)h(the)g(follo)n(wing)e(v)-5 b(ariational)19 b(problem:)32 b(Find)977 1882 y Fz( )s FG(\()p Fz(q)s FG(\))1162 1835 y Fr(def)1171 1882 y FG(=)f(min)1406 1765 y Fv(\032)1468 1769 y(Z)1551 1790 y FD(1)1514 1958 y(0)1602 1882 y Fz(\034)1638 1894 y FC(\014)1683 1882 y FG(\(1)p Fz(;)14 b(u)1842 1848 y Fy(0)1865 1882 y FG(\()p Fz(\030)t FG(\)\))g Fz(d\030)2122 1812 y Fv(\014)2122 1861 y(\014)2173 1882 y Fz(u)p 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y Fv(Z)1612 2547 y FD(1)1575 2716 y(0)1649 2640 y FG(\(1)18 b Fx(\000)g Fz(\030)t FG(\))p Fz(m)1969 2606 y Fy(0)1969 2660 y FC(\014)2014 2573 y Fv(\000)2052 2640 y FG(\(1)h Fx(\000)f Fz(\030)t FG(\))p Fz(t)2330 2573 y Fv(\001)2382 2640 y Fz(d\030)2488 2593 y Fr(def)2497 2640 y FG(=)32 b Fz(q)s FG(\()p Fz(t)p FG(\))24 b(=)e Fz(q)s(:)-2446 b FG(\(1.27\))456 2823 y(By)31 b(prop)r(erties)g FO(P2)g FG(and)h FO(P3)f FG(the)h(function)g Fz(q)s FG(\()p Fz(t)p FG(\))h(is)e(analytic)g(on)g Fx(D)2672 2835 y FC(\014)2749 2823 y FG(and,)h(moreo)n(v)n(er,)e(has)456 2923 y(the)e(analytic)f(in)n(v)n(erse)f Fz(t)d FG(=)f Fz(t)p FG(\()p Fz(q)s FG(\))29 b(on)e(the)h(latter)f(domain.)555 3022 y(Notice)f(that)g(b)n(y)g(the)h(con)n(v)n(ex)d(dualit)n(y)i(b)r (et)n(w)n(een)g Fz(m)2190 3034 y FC(\014)2234 3022 y FG(\()p Fx(\001)p FG(\))h(and)f Fz(\034)2544 3034 y FC(\014)2589 3022 y FG(\(1)p Fz(;)14 b Fx(\001)g FG(\))26 b(the)g(minimal)h(v)-5 b(alue)456 3122 y Fz( )s FG(\()p Fz(q)s FG(\))28 b(can)f(b)r(e)h 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FG(+)18 b Fz(l)r FG(\))1307 1226 y FD(2)1344 1256 y Fn(,)28 b Fz(t)23 b FG(=)g Fz(t)p FG(\()p Fz(q)1635 1268 y FD(1)1672 1256 y FG(\))p Fn(,)28 b(and)1472 1427 y Fz(\026)1522 1439 y FC(l)1548 1427 y FG(\()p Fz(t)p FG(\))23 b(=)1753 1348 y Fv(X)1754 1527 y FC(l)1775 1510 y Fl(0)1798 1527 y Fy(\025)p FC(l)1887 1427 y Fz(e)1926 1393 y Fy(\000)p FC(l)1999 1368 y Fl(0)2021 1393 y FC(m)2080 1402 y Fq(\014)2119 1393 y FD(\()p FC(t)p FD(\))2200 1427 y FE(F)2256 1439 y FC(l)2278 1423 y Fl(0)2310 1427 y FG(\()p Fz(t)p FG(\))p Fz(:)-1971 b FG(\(1.33\))456 1673 y FO(Prop)s(osition)30 b(1.7.)40 b Fn(As)c Fz(R)h Fx(!)g(1)p Fn(,)h(the)e(asymptotics)f(of)h(the)g(corner)f(partition)g (function)456 1773 y FE(A)521 1785 y FC(R)581 1773 y FG(\()p Fz(Q)679 1785 y FD(2)717 1773 y FG(\))27 b Fn(is)h(giv)n(en)f (b)n(y)1152 2013 y FE(A)1217 2025 y FC(R)1277 2013 y FG(\()p Fz(Q)1375 2025 y FD(2)1413 2013 y FG(\))c(=)1556 1864 y Fv(s)p 1639 1864 367 4 v 1668 1957 a Fz(\024)p FG(\()p Fz(s)p FG(\))p Fz(\026)p 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b(shall)f(pro)n(v)n(e)f(the)i(ab)r(o)n(v)n(e)f(t)n (w)n(o)f(statemen)n(ts)i(in)g(Sect.)f(3)h(and)f(Sect.)h(4)f(resp)r (ectiv)n(ely)-7 b(.)1429 2910 y(2.)45 b FF(V)-10 b(aria)k(tional)31 b(Pr)n(oblems)555 3059 y FG(In)j(this)g(section,)g(assuming)f(v)-5 b(alidit)n(y)33 b(of)g(Prop)r(ositions)f(1.6)g(and)i(1.7,)g(w)n(e)f (deriv)n(e)f(sharp)456 3159 y(estimates)d(on)g(the)h(v)-5 b(ariational)28 b(sum)i(\()p Fz(M)f FG(+)19 b Fz(l)r FG(\))p Fz( )s FG(\()p Fz(q)2089 3171 y FD(1)2126 3159 y FG(\))h(+)g Fz(R)q( )s FG(\()p Fz(q)2453 3171 y FD(2)2490 3159 y FG(\))30 b(that)g(en)n(ters)f(the)h(basic)f(de-)456 3258 y(comp)r(osition)h(form)n(ula)g(\(1.17\))g(through)g(the)i (relations)d(\(1.32\))h(and)h(\(1.34\).)46 b(The)31 b(ev)n(en)n(tual) 456 3358 y(computation)18 b(leads)g(to)h(the)g(main)g(assertion)e (\(1.9\))h(of)h(Theorem)f(1,)i(and)f(it)g(will)g(b)r(e)g(p)r(erformed) 456 3457 y(in)29 b(the)g(concluding)f(Sect.)h(2.4.)40 b(The)29 b(tec)n(hniques)g(emplo)n(y)n(ed)f(in)h(the)g(preparatory)d 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rx ry 2 copy rmoveto % cx cy cchar rx ry } forall pop pop pop pop pop % - currentpoint newpath moveto } bind def /PATcg { 7 dict dup begin /lw currentlinewidth def /lc currentlinecap def /lj currentlinejoin def /ml currentmiterlimit def /ds [ currentdash ] def /cc [ currentrgbcolor ] def /cm matrix currentmatrix def end } bind def % PATdraw - calculates the boundaries of the object and % fills it with the current pattern /PATdraw { % proc save exch PATpcalc % proc nw nh px py 5 -1 roll exec % nw nh px py newpath PATfill % - restore } bind def % PATfill - performs the tiling for the shape /PATfill { % nw nh px py PATfill - PATDict /CurrentPattern get dup begin setfont % Set the coordinate system to Pattern Space PatternGState PATsg % Set the color for uncolored pattezns PaintType 2 eq { PATDict /PColor get PATsc } if % Create the string for showing 3 index string % nw nh px py str % Loop for each of the pattern sources 0 1 Multi 1 sub { % nw nh px py str source % Move to the starting location 3 index 3 index % nw nh px py str source px py moveto % nw nh px py str source % For multiple sources, set the appropriate color Multi 1 ne { dup PC exch get PATsc } if % Set the appropriate string for the source 0 1 7 index 1 sub { 2 index exch 2 index put } for pop % Loop over the number of vertical cells 3 index % nw nh px py str nh { % nw nh px py str currentpoint % nw nh px py str cx cy 2 index show % nw nh px py str cx cy YStep add moveto % nw nh px py str } repeat % nw nh px py str } for 5 { pop } repeat end } bind def % PATkshow - kshow with the current pattezn /PATkshow { % proc string exch bind % string proc 1 index 0 get % string proc char % Loop over all but the last character in the string 0 1 4 index length 2 sub { % string proc char idx % Find the n+1th character in the string 3 index exch 1 add get % string proe char char+1 exch 2 copy % strinq proc char+1 char char+1 char % Now show the nth character PATsstr dup 0 4 -1 roll put % string proc chr+1 chr chr+1 (chr) false charpath % string proc char+1 char char+1 /clip load PATdraw % Move past the character (charpath modified the current point) currentpoint newpath moveto % Execute the user proc (should consume char and char+1) mark 3 1 roll % string proc char+1 mark char char+1 4 index exec % string proc char+1 mark... cleartomark % string proc char+1 } for % Now display the last character PATsstr dup 0 4 -1 roll put % string proc (char+1) false charpath % string proc /clip load PATdraw neewath pop pop % - } bind def % PATmp - the makepattern equivalent /PATmp { % patdict patmtx PATmp patinstance exch dup length 7 add % We will add 6 new entries plus 1 FID dict copy % Create a new dictionary begin % Matrix to install when painting the pattern TilingType PATtcalc /PatternGState PATcg def PatternGState /cm 3 -1 roll put % Check for multi pattern sources (Level 1 fast color patterns) currentdict /Multi known not { /Multi 1 def } if % Font dictionary definitions /FontType 3 def % Create a dummy encoding vector /Encoding 256 array def 3 string 0 1 255 { Encoding exch dup 3 index cvs cvn put } for pop /FontMatrix matrix def /FontBBox BBox def /BuildChar { mark 3 1 roll % mark dict char exch begin Multi 1 ne {PaintData exch get}{pop} ifelse % mark [paintdata] PaintType 2 eq Multi 1 ne or { XStep 0 FontBBox aload pop setcachedevice } { XStep 0 setcharwidth } ifelse currentdict % mark [paintdata] dict /PaintProc load % mark [paintdata] dict paintproc end gsave false PATredef exec true PATredef grestore cleartomark % - } bind def currentdict end % newdict /foo exch % /foo newlict definefont % newfont } bind def % PATpcalc - calculates the starting point and width/height % of the tile fill for the shape /PATpcalc { % - PATpcalc nw nh px py PATDict /CurrentPattern get begin gsave % Set up the coordinate system to Pattern Space % and lock down pattern PatternGState /cm get setmatrix BBox aload pop pop pop translate % Determine the bounding box of the shape pathbbox % llx lly urx ury grestore % Determine (nw, nh) the # of cells to paint width and height PatHeight div ceiling % llx lly urx qh 4 1 roll % qh llx lly urx PatWidth div ceiling % qh llx lly qw 4 1 roll % qw qh llx lly PatHeight div floor % qw qh llx ph 4 1 roll % ph qw qh llx PatWidth div floor % ph qw qh pw 4 1 roll % pw ph qw qh 2 index sub cvi abs % pw ph qs qh-ph exch 3 index sub cvi abs exch % pw ph nw=qw-pw nh=qh-ph % Determine the starting point of the pattern fill %(px, py) 4 2 roll % nw nh pw ph PatHeight mul % nw nh pw py exch % nw nh py pw PatWidth mul exch % nw nh px py end } bind def % Save the original routines so that we can use them later on /oldfill /fill load def /oldeofill /eofill load def /oldstroke /stroke load def /oldshow /show load def /oldashow /ashow load def /oldwidthshow /widthshow load def /oldawidthshow /awidthshow load def /oldkshow /kshow load def % These defs are necessary so that subsequent procs don't bind in % the originals /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def /PATredef { MyAppDict begin { /fill { /clip load PATdraw newpath } bind def /eofill { /eoclip load PATdraw newpath } bind def /stroke { PATstroke } bind def /show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def /ashow { 0 0 null 6 3 roll PATawidthshow } bind def /widthshow { 0 0 3 -1 roll PATawidthshow } bind def /awidthshow { PATawidthshow } bind def /kshow { PATkshow } bind def } { /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def } ifelse end } bind def false PATredef % Conditionally define setcmykcolor if not available /setcmykcolor where { pop } { /setcmykcolor { 1 sub 4 1 roll 3 { 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat setrgbcolor - pop } bind def } ifelse /PATsc { % colorarray aload length % c1 ... cn length dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor } ifelse } ifelse } bind def /PATsg { % dict begin lw setlinewidth lc setlinecap lj setlinejoin ml setmiterlimit ds aload pop setdash cc aload pop setrgbcolor cm setmatrix end } bind def /PATDict 3 dict def /PATsp { true PATredef PATDict begin /CurrentPattern exch def % If it's an uncolored pattern, save the color CurrentPattern /PaintType get 2 eq { /PColor exch def } if /CColor [ currentrgbcolor ] def end } bind def % PATstroke - stroke with the current pattern /PATstroke { countdictstack save mark { currentpoint strokepath moveto PATpcalc % proc nw nh px py clip newpath PATfill } stopped { (*** PATstroke Warning: Path is too complex, stroking with gray) = cleartomark restore countdictstack exch sub dup 0 gt { { end } repeat } { pop } ifelse gsave 0.5 setgray oldstroke grestore } { pop restore pop } ifelse newpath } bind def /PATtcalc { % modmtx tilingtype PATtcalc tilematrix % Note: tiling types 2 and 3 are not supported gsave exch concat % tilingtype matrix currentmatrix exch % cmtx tilingtype % Tiling type 1 and 3: constant spacing 2 ne { % Distort the pattern so that it occupies % an integral number of device pixels dup 4 get exch dup 5 get exch % tx ty cmtx XStep 0 dtransform round exch round exch % tx ty cmtx dx.x dx.y XStep div exch XStep div exch % tx ty cmtx a b 0 YStep dtransform round exch round exch % tx ty cmtx a b dy.x dy.y YStep div exch YStep div exch % tx ty cmtx a b c d 7 -3 roll astore % { a b c d tx ty } } if grestore } bind def /PATusp { false PATredef PATDict begin CColor PATsc end } bind def % this is the pattern fill program from the Second edition Reference Manual % with changes to call the above pattern fill % left30 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def /BBox [0 0 1 1] def /XStep 1 def /YStep 1 def /PatWidth 1 def /PatHeight 1 def /Multi 2 def /PaintData [ { clippath } bind { 32 16 true [ 32 0 0 -16 0 16 ] {} imagemask } bind ] def /PaintProc { pop exec fill } def currentdict end /P1 exch def % right30 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def /BBox [0 0 1 1] def /XStep 1 def /YStep 1 def /PatWidth 1 def /PatHeight 1 def /Multi 2 def /PaintData [ { clippath } bind { 32 16 true [ 32 0 0 -16 0 16 ] {<00030003000c000c0030003000c000c0030003000c000c00 30003000c000c00000030003000c000c0030003000c000c0 030003000c000c0030003000c000c000>} imagemask } bind ] def /PaintProc { pop exec fill } def currentdict end /P2 exch def 1.1111 1.1111 scale %restore scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def $F2psBegin %%Page: 1 1 10 setmiterlimit 0.06000 0.06000 sc % Polyline 7.500 slw n 975 5100 m 2775 5100 l gs col0 s gr % Polyline 15.000 slw n 975 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l 5543 3040 l 5548 3056 l 5549 3073 l 5547 3090 l 5542 3108 l 5537 3125 l 5533 3143 l 5529 3161 l 5528 3180 l 5530 3199 l 5537 3220 l 5550 3241 l 5569 3262 l 5594 3282 l 5625 3300 l 5656 3313 l 5689 3324 l 5721 3331 l 5753 3336 l 5785 3338 l 5816 3339 l 5846 3338 l 5876 3335 l 5906 3332 l 5934 3328 l 5962 3324 l 5988 3319 l 6011 3314 l 6031 3310 l 6048 3306 l 6060 3304 l 6068 3302 l 6073 3301 l 6075 3300 l gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P2 [16 0 0 -8 245.00 200.00] PATmp PATsp ef gr PATusp gs col0 s gr 7.500 slw % Ellipse n 7687 5587 266 266 0 360 DrawEllipse gs col7 0.50 shd ef gr gs col0 s gr % Ellipse n 7687 3675 252 252 0 360 DrawEllipse gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P2 [16 0 0 -8 495.67 228.20] PATmp PATsp ef gr PATusp gs col0 s gr % Polyline n 900 5100 m 3600 5100 l gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P2 [16 0 0 -8 60.00 340.00] PATmp PATsp ef gr PATusp gs col0 s gr % Polyline n 2700 5100 m 3675 5100 l gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 180.00 340.00] PATmp PATsp ef gr PATusp gs col0 s gr % Polyline n 2700 5100 m 3675 5100 l gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 180.00 340.00] PATmp PATsp ef gr PATusp gs col0 s gr % Polyline n 675 5100 m 900 5100 l gs col0 s gr % Polyline n 6075 5100 m 6450 5100 l gs col7 0.50 shd ef gr gs col0 s gr % Polyline n 6075 5100 m 6075 6300 l gs col0 s gr % Polyline n 975 2700 m 975 6300 l gs col0 s gr % Polyline n 6075 2700 m 6075 3300 l gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P2 [16 0 0 -8 405.00 180.00] PATmp PATsp ef gr PATusp gs col0 s gr $F2psEnd rs end %%EndDocument @endspecial 2399 4164 a Fu(area)e(computed)f(with)g(sign)i(+)2399 4625 y(area)f(computed)f(with)g(sign)i Fi(\000)1241 4152 y FC(j)1070 5071 y FF(Figure)32 b(2.4.)40 b FG(A)28 b(curv)n(e)f FA(j)35 b Fx(2)24 b(J)2079 5083 y FC(a)2147 5071 y FG(and)j(its)h (signed)f(area)p eop 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l 10339 6958 l 10355 6978 l 10372 7001 l 10390 7026 l 10408 7052 l 10425 7079 l 10442 7105 l 10457 7130 l 10471 7151 l 10482 7169 l 10490 7183 l 10495 7192 l 10499 7198 l 10500 7200 l gs col0 s gr $F2psEnd rs showpage %%EndDocument @endspecial 1501 3579 a Fz(q)1538 3591 y FD(1)1525 3909 y Fz(xy)2021 3957 y(q)2058 3969 y FD(2)1903 4169 y Fz(x)1265 3697 y(y)2257 3838 y(j)2291 3850 y FD(2)1714 3484 y Fz(j)1748 3496 y FD(1)1209 4438 y FF(Figure)32 b(2.6.)40 b FG(Splitting)28 b(of)g(a)f(path)h Fz(j)g Fx(2)23 b(C)2594 4450 y FC(x;y)555 4640 y FG(Consequen)n(tly)-7 b(,)27 b(the)h(minimisation)g(problem)f (\(2.5\))g(can)g(b)r(e)h(rewritten)g(as)922 4800 y(min)749 4850 y FC(q)779 4858 y Fs(1)812 4850 y FD(+)p FC(q)893 4858 y Fs(2)926 4850 y FD(=)p FC(q)1007 4859 y Fq(\014)r(;)p Fs(+)1108 4850 y Fy(\000)p FC(xy)1247 4800 y FG(min)1385 4708 y Fv(n)1441 4800 y Fx(W)1523 4812 y FC(\014)1581 4800 y FG(\()q FA(j)1639 4812 y FD(1)1676 4800 y FG(\))19 b(+)f Fx(W)1892 4812 y FC(\014)1951 4800 y FG(\()p 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5181 y FD(2)2319 5216 y Fz(q)2356 5228 y FC(\014)s(;)p FD(+)2472 5216 y Fz(=\034)2550 5228 y FC(\014)2594 5216 y FG(\(1)p Fz(;)14 b FG(0\))2779 5181 y FD(2)p eop %%Page: 14 14 14 13 bop 456 226 a FD(14)814 b(OST)-5 b(AP)28 b(HR)-5 b(YNIV)29 b(AND)g(DMITR)-5 b(Y)29 b(IOFFE)456 425 y FG(In)k(the)h (latter)e(case,)i(in)g(view)f(of)g(the)g(relations)f(\(1.27\))h(and)g (\(2.2\),)h(the)g(v)-5 b(ariational)31 b(prob-)456 525 y(lem)37 b(\(2.6\))g(is)g(equiv)-5 b(alen)n(t)37 b(to)g(the)g(follo)n (wing)f(t)n(w)n(o-dimensional)g(constrain)n(t)f(minimisation)456 624 y(problem:)1043 793 y(min)1181 700 y Fv(n)1236 793 y Fz(xe)p FG(\()p Fz(t)p FG(\))19 b(+)f Fz(y)s(e)p FG(\()p Fz(s)p FG(\))1727 722 y Fv(\014)1727 772 y(\014)1778 793 y Fz(x)1825 758 y FD(2)1863 793 y Fz(q)s FG(\()p Fz(t)p FG(\))h(+)f Fz(y)2143 758 y FD(2)2180 793 y Fz(q)s FG(\()p Fz(s)p FG(\))23 b(=)g Fz(q)2471 805 y FC(\014)s(;)p FD(+)2605 793 y Fx(\000)18 b Fz(xy)2779 700 y Fv(o)2834 793 y Fz(:)-2401 b FG(\(2.7\))456 983 y(Let)634 961 y(^)617 983 y Fz( )t FG(\()p Fz(x;)14 b(y)s FG(\))41 b(denote)g(the)g(v)-5 b(alue)40 b(of)h(the)g(ab)r(o)n(v)n(e)e(minim)n(um,)2496 967 y(^)2495 983 y Fz(t)p FG(\()p Fz(x;)14 b(y)s FG(\))41 b(and)j(^)-45 b Fz(s)p FG(\()p Fz(x;)14 b(y)s FG(\))41 b(|)g(the)456 1082 y(corresp)r(onding)25 b(minimisers.)456 1244 y FO(Lemma)k(2.2.)40 b Fn(There)g(exists)f Fz(\016)47 b(>)c FG(0)p Fn(,)f(suc)n(h)e(that)2186 1222 y FG(^)2169 1244 y Fz( )s Fn(,)2293 1228 y FG(^)2292 1244 y Fz(t)g Fn(and)j FG(^)-45 b Fz(s)40 b Fn(are)f(analytic)g(on)g(the)i Fz(\016)s Fn(-)456 1343 y(neigh)n(b)r(ourho)r(o)r(d)d(\(2.4\))g(of)h FG(\()p Fz(x)1429 1355 y FD(0)1467 1343 y Fz(;)14 b(y)1545 1355 y FD(0)1582 1343 y FG(\))p Fn(.)72 b(Moreo)n(v)n(er,)39 b(for)g(ev)n(ery)e FG(\()p Fz(x;)14 b(y)s FG(\))40 b Fn(satisfying)e(\(2.4\),)k(the)456 1445 y(functions)815 1430 y FG(^)814 1445 y Fz(t)p FG(\()p Fz(x;)14 b(y)s FG(\))28 b Fn(and)j FG(^)-46 b Fz(s)p FG(\()p Fz(x;)14 b(y)s FG(\))28 b Fn(v)n(erify)f(the)h(follo)n(wing)f(transv)n(ersalit)n (y)e(condition:)1647 1580 y FG(^)1646 1595 y Fz(t)p FG(\()p Fz(x;)14 b(y)s FG(\))p 1646 1632 223 4 v 1734 1709 a Fz(x)1902 1652 y FG(=)2003 1595 y(^)-46 b Fz(s)p FG(\()p Fz(x;)14 b(y)s FG(\))p 1999 1632 232 4 v 2093 1709 a Fz(y)2241 1652 y(:)-1808 b FG(\(2.8\))456 1848 y FA(Pr)l(o)l(of)33 b(of)g(L)l(emma)f(2.2.)46 b FG(Applying)30 b(the)g(metho)r(d)h(of)f (Lagrange)e(m)n(ultipliers)i(and)f(using)h(the)456 1948 y(dualit)n(y)19 b(relation)f(\(recall)h(that)h Fz(m)1516 1960 y FC(\014)1561 1948 y FG(\()p Fx(\001)p FG(\))g(is)f(the)h(con)n (v)n(ex)e(conjugate)h(of)h Fz(\034)2633 1960 y FC(\014)2678 1948 y FG(\(1)p Fz(;)14 b Fx(\001)p FG(\)\),)21 b Fz(\034)2965 1918 y Fy(0)2956 1971 y FC(\014)3002 1948 y FG(\(1)p Fz(;)14 b(m)3186 1918 y Fy(0)3186 1971 y FC(\014)3230 1948 y FG(\()p Fz(t)p FG(\)\))24 b(=)456 2048 y Fz(t)p FG(,)j(w)n(e)h(obtain:)1107 2139 y Fv(8)1107 2214 y(>)1107 2239 y(>)1107 2264 y(>)1107 2289 y(>)1107 2314 y(>)1107 2339 y(<)1107 2488 y(>)1107 2513 y(>)1107 2538 y(>)1107 2563 y(>)1107 2588 y(>)1107 2613 y(:)1194 2184 y(\000)1233 2236 y FG(^)1232 2251 y Fz(tx)19 b Fx(\000)f Fz(\025x)1506 2217 y FD(2)1544 2184 y Fv(\001)1596 2138 y(Z)1679 2159 y FD(1)1642 2327 y(0)1717 2251 y FG(\(1)g Fx(\000)g Fz(\030)t FG(\))1964 2217 y FD(2)2001 2251 y Fz(m)2074 2217 y Fy(00)2074 2272 y FC(\014)2119 2184 y Fv(\000)2157 2251 y FG(\(1)g Fx(\000)g Fz(\030)t FG(\))2405 2236 y(^)2404 2251 y Fz(t)2435 2184 y Fv(\001)2487 2251 y Fz(d\030)27 b FG(=)c(0)p Fz(;)1194 2423 y Fv(\000)1236 2490 y FG(^)-46 b Fz(sy)21 b Fx(\000)d Fz(\025y)1508 2455 y FD(2)1546 2423 y Fv(\001)1598 2377 y(Z)1681 2397 y FD(1)1644 2565 y(0)1718 2490 y FG(\(1)g Fx(\000)g Fz(\030)t FG(\))1965 2455 y FD(2)2003 2490 y Fz(m)2076 2455 y Fy(00)2076 2510 y FC(\014)2121 2423 y Fv(\000)2159 2490 y FG(\(1)g Fx(\000)g Fz(\030)t FG(\))s(^)-45 b Fz(s)2445 2423 y Fv(\001)2497 2490 y Fz(d\030)28 b FG(=)22 b(0)p Fz(;)1194 2672 y(x)1241 2637 y FD(2)1279 2672 y Fz(q)s 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3287 y Fz(t;)17 b FG(^)-45 b Fz(s;)14 b(\025;)g(x;)g(y)s FG(\))23 b(=)g FO(0)p Fz(:)-2425 b FG(\(2.10\))456 3571 y(Recalling)27 b(no)n(w)f(the)i(de\014nitions)g(\(1.27\))f(and)g (\(1.29\),)g(w)n(e)h(see)f(that)h(the)g(matrix)1152 3764 y Fz(@)5 b(F)12 b FG(\()1299 3749 y(^)1298 3764 y Fz(t;)18 b FG(^)-46 b Fz(s;)14 b(\025;)g(x;)g(y)s FG(\))p 1152 3801 535 4 v 1267 3884 a Fz(@)5 b FG(\()1349 3869 y(^)1348 3884 y Fz(t;)18 b FG(^)-46 b Fz(s;)14 b(\025)p FG(\))1720 3820 y(=)1807 3653 y Fv(2)1807 3803 y(4)2040 3718 y FG(1)273 b(0)179 b Fx(\000)p Fz(x)2040 3818 y FG(0)273 b(1)181 b Fx(\000)p Fz(y)1946 3920 y(x)1993 3890 y FD(2)2030 3920 y Fz(\033)s FG(\()2113 3905 y(^)2112 3920 y Fz(t)q FG(\))83 b Fz(y)2302 3890 y FD(2)2339 3920 y Fz(\033)s FG(\()s(^)-45 b Fz(s)q FG(\))119 b(0)2688 3653 y Fv(3)2688 3803 y(5)456 4064 y FG(is)42 b(non-degenerate.)81 b(By)42 b(analyticit)n(y)g(of)h Fz(m)1965 4076 y FC(\014)2053 4064 y FG(this)g(implies)g(analyticit)n(y)f(of)h(the)g(v)n(ector)456 4176 y(\()489 4161 y(^)488 4176 y Fz(t;)17 b FG(^)-45 b Fz(s;)14 b(\025)p FG(\)\()p Fz(x;)g(y)s FG(\))25 b(in)g(a)e(neigh)n (b)r(ourho)r(o)r(d)g(of)h(\()p Fz(x)1821 4188 y FD(0)1859 4176 y Fz(;)14 b(x)1943 4188 y FD(0)1981 4176 y FG(\).)36 b(The)24 b(analyticit)n(y)g(of)2757 4154 y(^)2740 4176 y Fz( )j FG(follo)n(ws)c(then)i(from)456 4301 y(the)j(relation)922 4279 y(^)905 4301 y Fz( )s FG(\()p Fz(x;)14 b(y)s FG(\))24 b(=)g Fz(xe)p FG(\()1386 4285 y(^)1385 4301 y Fz(t)p FG(\))c(+)e Fz(y)s(e)p FG(\()s(^)-45 b Fz(s)p FG(\).)39 b(Finally)-7 b(,)28 b(notice)g(that)h Fz(t)2553 4313 y FD(0)2614 4254 y Fr(def)2623 4301 y FG(=)2722 4285 y(^)2721 4301 y Fz(t)p FG(\()p Fz(x)2830 4313 y FD(0)2868 4301 y Fz(;)14 b(x)2952 4313 y FD(0)2989 4301 y FG(\))25 b(=)i(^)-45 b Fz(s)p FG(\()p Fz(x)3253 4313 y FD(0)3291 4301 y Fz(;)14 b(x)3375 4313 y FD(0)3412 4301 y FG(\))456 4400 y(satis\014es)29 b Fz(m)837 4370 y Fy(0)837 4424 y FC(\014)881 4400 y FG(\()p Fz(t)943 4412 y FD(0)981 4400 y FG(\))e(=)g(1)i(.)44 b(Indeed,)31 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753 y Fz(\033)s FG(\()p Fz(t)2038 765 y FD(0)2076 753 y FG(\))2119 629 y Fv(\000)2157 696 y FG(\()p Fz(x)h Fx(\000)f Fz(x)2385 708 y FD(0)2423 696 y FG(\))g(+)g(\()p Fz(y)k Fx(\000)c Fz(y)2775 708 y FD(0)2812 696 y FG(\))2844 629 y Fv(\001)2882 646 y FD(2)1687 880 y FG(+)g Fz(o)1824 813 y Fv(\000)1863 880 y FG(\()p Fz(x)h Fx(\000)f Fz(x)2091 892 y FD(0)2128 880 y FG(\))2160 846 y FD(2)2216 880 y FG(+)g(\()p Fz(y)k Fx(\000)c Fz(y)2518 892 y FD(0)2555 880 y FG(\))2587 846 y FD(2)2624 813 y Fv(\001)2676 880 y Fz(:)456 771 y FG(\(2.17\))456 1031 y(2.4.)46 b FO(Pro)s(of)24 b(of)g(Theorem)f(1.)41 b FG(In)21 b(the)g(notation)f(of)h(Sect.)g(2.2)f (the)i(scaling)d(b)r(et)n(w)n(een)i(the)g(mi-)456 1131 y(croscopic)c(quan)n(tities)i(asso)r(ciated)f(with)i(a)f(path)h Fz(!)26 b Fx(2)d(Z)2227 1101 y Fr(tub)r(e)2209 1154 y FC(Q;)p FD(+)2366 1131 y FG(\(Fig.)d(1.3\))e(and)i(the)g(macroscopic) 456 1231 y(quan)n(tities)27 b(asso)r(ciated)f(with)i(the)g(optimal)g (quarter)e(shap)r(e)i(curv)n(e)e FA(j)2644 1243 y FC(\014)s(;)p FD(+)2783 1231 y FG(=)d Fz(@)5 b FO(K)2995 1243 y FC(\014)s(;)p FD(+)3129 1231 y Fx(\\)18 b Fw(Quad)3389 1246 y FD(+)456 1331 y FG(is)28 b(giv)n(en)g(b)n(y)g(\(recall)g(that)h Fz(M)37 b FG(is)29 b(the)g FA(inte)l(ger)g FG(part)f(of)g(the)i(middle) f(p)r(oin)n(t)f(of)h(the)g(horizon)n(tal)456 1439 y(pro)5 b(jection)26 b(of)i FA(j)982 1399 y FC(Q)970 1464 y(\014)s(;)p FD(+)1086 1439 y FG(\))1535 1527 y Fv(\014)1535 1577 y(\014)1535 1627 y(\014)1535 1677 y(\014)1535 1727 y(\014)1563 1528 y(s)p 1646 1528 173 4 v 1699 1617 a Fz(Q)p 1656 1654 153 4 v 1656 1730 a(q)1693 1742 y FC(\014)s(;)p FD(+)1837 1673 y Fx(\000)1930 1617 y Fz(M)p 1930 1654 90 4 v 1932 1730 a(x)1979 1742 y FD(0)2029 1527 y Fv(\014)2029 1577 y(\014)2029 1627 y(\014)2029 1677 y(\014)2029 1727 y(\014)2108 1673 y Fx(\024)2255 1617 y FG(1)p 2233 1654 85 4 v 2233 1730 a Fz(x)2280 1742 y FD(0)2328 1673 y Fz(;)-1895 b FG(\(2.18\))456 1884 y(where,)27 b(as)g(in)h(Sect.)f(2.2,) g(\()p Fz(x)1356 1896 y FD(0)1394 1884 y Fz(;)14 b(x)1478 1896 y FD(0)1516 1884 y FG(\))28 b(is)f(the)h(middle)g(p)r(oin)n(t)g (of)g FA(j)2411 1896 y FC(\014)s(;)p FD(+)2527 1884 y FG(.)555 1984 y(The)d(tub)r(e)h(condition)e(\(1.14\))g(and)h(the)g (estimate)g(\(1.16\))f(enable)g(to)h(restrict)f(the)h(range)f(of)456 2084 y(summation)j(in)h(the)g(righ)n(t)f(hand)g(side)h(of)f(\(1.17\))g (to,)879 2237 y Fz(l)d Fx(\024)f Fz(c)1052 2249 y FD(1)1089 2237 y Fz(M)1179 2203 y FC(")1214 2237 y Fz(;)97 b Fx(j)p Fz(R)19 b Fx(\000)g Fz(M)9 b Fx(j)22 b(\024)h Fz(M)1836 2203 y FC(\027)1877 2237 y Fz(;)124 b Fu(and)2271 2167 y Fv(\014)2271 2217 y(\014)2298 2237 y Fz(Q)2364 2249 y FC(i)2410 2237 y Fx(\000)18 b Fz(M)2583 2203 y FD(2)2620 2237 y Fz(q)s FG(\()p Fz(t)2722 2249 y FD(0)2759 2237 y FG(\))2791 2167 y Fv(\014)2791 2217 y(\014)2842 2237 y Fx(\024)23 b Fz(c)2966 2249 y FD(3)3003 2237 y Fz(Q)3079 2177 y Fs(1+)p Fq(\027)p 3079 2190 104 3 v 3116 2223 a Fs(2)3196 2237 y Fz(;)-2763 b FG(\(2.19\))456 2379 y(with)30 b(an)n(y)e(\014xed)i(v)-5 b(alues)29 b(of)g Fz(")d(>)g FG(0)j(and)g Fz(\027)j Fx(2)27 b FG(\(1)p Fz(=)p FG(2)p Fz(;)14 b FG(1\).)41 b(In)29 b(what)h(follo)n(ws)e(w)n(e) h(shall)g(c)n(ho)r(ose)g Fz(")456 2479 y FG(and)e Fz(\027)33 b FG(in)28 b(suc)n(h)f(a)g(w)n(a)n(y)g(that)1405 2650 y Fz(\027)d FG(+)18 b Fz(")23 b(<)g FG(1)193 b Fu(and)e Fz(\027)28 b(<)2420 2594 y FG(2)p 2420 2631 42 4 v 2420 2707 a(3)2472 2650 y Fz(:)-2039 b FG(\(2.20\))456 2815 y(In)27 b(the)h(asymptotic)f(expressions)f(\(1.32\))h(and)h(\(1.34\))e (the)i(areas)e Fz(q)2596 2827 y FD(1)2661 2815 y FG(and)i Fz(q)2860 2827 y FD(2)2925 2815 y FG(scale)e(as)1268 2994 y Fz(q)1305 3006 y FD(1)1365 2994 y FG(=)1571 2938 y Fz(Q)1637 2950 y FD(1)p 1463 2975 320 4 v 1463 3051 a FG(\()p Fz(M)h FG(+)18 b Fz(l)r FG(\))1745 3027 y FD(2)1986 2994 y Fu(and)191 b Fz(q)2338 3006 y FD(2)2399 2994 y FG(=)2496 2938 y Fz(Q)2562 2950 y FD(2)p 2496 2975 103 4 v 2497 3051 a Fz(R)2561 3027 y FD(2)2609 2994 y Fz(:)456 3191 y FG(Therefore,)29 b(b)n(y)g(the)h(last)f(of)h(the)g(relations)f (\(2.19\),)g Fx(j)p Fz(q)2186 3203 y FC(i)2234 3191 y Fx(\000)19 b Fz(q)s FG(\()p Fz(t)2420 3203 y FD(0)2458 3191 y FG(\))p Fx(j)26 b(\024)h Fz(c)2667 3203 y FD(1)2704 3191 y Fz(Q)2770 3160 y FD(\()p FC(\027)t Fy(\000)p FD(1\))p FC(=)p FD(2)3041 3191 y Fx(\024)f Fz(c)3168 3203 y FD(2)3205 3191 y Fz(M)3295 3160 y FC(\027)t Fy(\000)p FD(1)3421 3191 y FG(,)456 3290 y(and,)h(consequen)n(tly)-7 b(,)967 3427 y Fz(t)p FG(\()p Fz(q)1066 3439 y FD(1)1103 3427 y FG(\))24 b(=)e Fz(t)1276 3439 y FD(0)1332 3427 y FG(+)c Fz(O)1494 3360 y Fv(\000)1532 3427 y Fz(M)1622 3393 y FC(\027)t Fy(\000)p FD(1)1748 3360 y Fv(\001)1966 3427 y FG(and)166 b Fz(s)p FG(\()p Fz(q)2374 3439 y FD(2)2412 3427 y FG(\))23 b(=)g Fz(t)2585 3439 y FD(0)2640 3427 y FG(+)18 b Fz(O)2803 3360 y Fv(\000)2841 3427 y Fz(M)2931 3393 y FC(\027)t Fy(\000)p FD(1)3057 3360 y Fv(\001)3109 3427 y Fz(;)-2676 b FG(\(2.21\))456 3569 y(where)35 b Fz(t)h FG(=)g Fz(t)p FG(\()p Fz(q)970 3581 y FD(1)1008 3569 y FG(\))g(and)g Fz(s)g FG(=)g Fz(s)p FG(\()p Fz(q)1530 3581 y FD(2)1568 3569 y FG(\))g(are)f(reco)n(v)n(ered)e(from)i Fz(q)s FG(\()p Fz(t)p FG(\))i(=)g Fz(q)2672 3581 y FD(1)2745 3569 y FG(and)e Fz(q)s FG(\()p Fz(s)p FG(\))i(=)f Fz(q)3232 3581 y FD(2)3306 3569 y FG(\(see)456 3668 y(\(1.27\)\))27 b(resp)r(ectiv)n(ely)-7 b(.)555 3768 y(Let)30 b(us)g(\014x)g Fz(l)i FG(and)d Fz(R)i FG(in)f(a)g(compliance)f(with)i(the)f(\(2.19\))f (and)h(sum)g(o)n(v)n(er)e(all)i(admissible)456 3868 y Fz(Q)522 3880 y FD(1)581 3868 y FG(and)22 b Fz(Q)803 3880 y FD(2)862 3868 y FG(\(the)h(third)f(sum)h(in)f(\(1.17\)\).)35 b(By)22 b(the)g(scaling)g(relation)f(\(2.18\),)h(the)h(microscopic)456 3967 y(break)32 b(p)r(oin)n(t)j(with)f(co)r(ordinates)e(\()p Fz(M)g FG(+)22 b Fz(l)r(;)14 b(R)q FG(\))33 b(corresp)r(onds,)h(in)g (terms)g(of)g(the)g(v)-5 b(ariational)456 4067 y(problem)27 b(\(2.5\),)g(to)h(the)g(macroscopic)d(p)r(oin)n(t)j(\()p Fz(x;)14 b(y)s FG(\))28 b(with)g(the)g(co)r(ordinates)e(\(see)i(Fig.)f (1.3\))1270 4247 y Fz(x)c FG(=)1438 4191 y Fz(M)k FG(+)18 b Fz(l)p 1438 4228 218 4 v 1502 4304 a(M)1665 4247 y(x)1712 4259 y FD(0)1943 4247 y Fu(and)192 b Fz(y)26 b FG(=)2436 4191 y Fz(R)p 2423 4228 90 4 v 2423 4304 a(M)2523 4247 y(x)2570 4259 y FD(0)2607 4247 y Fz(;)456 4422 y FG(optimal)h(slop)r (es)1004 4407 y(^)1003 4422 y Fz(t)c FG(=)1145 4407 y(^)1144 4422 y Fz(t)p FG(\()p Fz(x;)14 b(y)s FG(\),)31 b(^)-45 b Fz(s)23 b FG(=)j(^)-45 b Fz(s)p FG(\()p Fz(x;)14 b(y)s FG(\),)28 b(and)f(the)h(areas)1140 4546 y(^)1121 4567 y Fz(Q)1187 4579 y FD(1)1247 4567 y FG(=)23 b(\()p Fz(M)k FG(+)18 b Fz(l)r FG(\))1617 4533 y FD(2)1654 4567 y Fz(q)s FG(\()1727 4552 y(^)1726 4567 y Fz(t)p FG(\))194 b Fu(and)2317 4546 y FG(^)2298 4567 y Fz(Q)2364 4579 y FD(2)2424 4567 y FG(=)22 b Fz(R)2575 4533 y FD(2)2612 4567 y Fz(q)s FG(\()s(^)-45 b Fz(s)p FG(\))p Fz(:)456 4704 y FG(Notice,)27 b(that)h(the)g(transv)n(ersalit)n(y)d(condition)i(\(2.8\))h(reads)e(in) i(the)g(ab)r(o)n(v)n(e)e(notation)h(as)1827 4828 y(^)1826 4843 y Fz(t)p 1733 4880 218 4 v 1733 4956 a(M)g FG(+)18 b Fz(l)1983 4899 y FG(=)2096 4843 y(^)-45 b Fz(s)p 2081 4880 64 4 v 2081 4956 a(R)2154 4899 y(:)-1721 b FG(\(2.22\))456 5071 y(By)27 b(\(2.19\),)g(the)h(range)e(of)i(admissible)f(areas)f Fz(Q)1994 5083 y FD(1)2031 5071 y Fz(;)14 b(Q)2134 5083 y FD(2)2198 5071 y FG(\(with)28 b Fz(R)q FG(,)g Fz(l)h FG(\014xed\))f(is)f(included)h(in)1368 5216 y Fx(f)p Fz(Q)1476 5228 y FD(1)1512 5216 y Fz(;)14 b(Q)1615 5228 y FD(2)1679 5145 y Fv(\014)1679 5195 y(\014)1735 5216 y Fx(j)p Fz(Q)1824 5228 y FC(i)1870 5216 y Fx(\000)1972 5195 y FG(^)1953 5216 y Fz(Q)2019 5228 y FC(i)2046 5216 y Fx(j)23 b Fz(<)g(c)2216 5228 y FD(4)2253 5216 y Fz(M)2343 5181 y FD(1+)p FC(\027)2468 5216 y Fx(g)p Fz(:)p eop %%Page: 17 17 17 16 bop 1488 226 a FD(SELF-A)-7 b(V)n(OIDING)29 b(POL)-5 b(YGONS)966 b(17)456 425 y FG(Th)n(us,)23 b(using)f(the)h(relations)e (\(1.32\),)i(\(1.34\),)g(and)f(\(2.21\))g(as)g(w)n(ell)g(as)g(the)h (analytic)e(prop)r(erties)456 525 y(of)27 b(the)h(functions)g Fz(\026)p FG(,)g Fz(\024)p FG(,)f Fz(\033)k FG(and)d Fz(\026)1540 537 y FC(l)1565 525 y FG(,)g(w)n(e)f(rewrite)g(the)h (third)g(sum)g(in)g(\(1.17\))e(as:)800 677 y Fz(\026)p FG(\(0\))p Fz(\026)1006 689 y FC(l)1031 677 y FG(\()p Fz(t)1093 689 y FD(0)1131 677 y FG(\))1163 606 y Fv(p)p 1246 606 362 4 v 71 x Fz(\026)p FG(\()p Fz(t)1358 689 y FD(0)1396 677 y FG(\))p Fz(\024)p FG(\()p Fz(t)1538 689 y FD(0)1575 677 y FG(\))p 800 715 808 4 v 918 805 a(2)p Fz(\031)s(\033)s FG(\()p Fz(t)1122 817 y FD(0)1160 805 y FG(\))1192 732 y Fx(p)p 1261 732 228 4 v 73 x Fz(M)1351 781 y FD(3)1388 805 y Fz(R)1452 781 y FD(3)1841 656 y Fv(X)1631 849 y Fy(j)p FC(Q)1703 857 y Fq(i)1730 849 y Fy(\000)1796 834 y FD(^)1782 849 y FC(Q)1834 857 y Fq(i)1860 849 y Fy(j\024)p FC(c)1962 857 y Fs(4)1994 849 y FC(M)2063 824 y Fs(1+)p Fq(\027)1655 945 y FC(Q)1707 953 y Fs(1)1740 945 y FD(+)p FC(Q)1843 953 y Fs(2)1876 945 y FD(=)1941 930 y(^)1927 945 y FC(Q)1979 953 y Fs(1)2011 945 y FD(+)2077 930 y(^)2062 945 y FC(Q)2114 953 y Fs(2)2185 734 y Fz(e)2224 700 y Fy(\000)p FD(\()p FC(M)6 b FD(+)p FC(l)p FD(\))p FC( )r FD(\()p FC(q)2571 708 y Fs(1)2604 700 y FD(\))p Fy(\000)p FC(R )r FD(\()p FC(q)2834 708 y Fs(2)2867 700 y FD(\))2911 734 y FG(\(1)18 b(+)g FA(o)h FG(\()q(1\))o(\))c Fz(:)-2867 b FG(\(2.23\))456 1080 y(No)n(w)27 b(\(recall)g(the)h(de\014nition)g(of)f Fz(e)p FG(\()p Fz(t)p FG(\))h(in)g(\(2.2\)\),)759 1267 y(\()p Fz(M)g FG(+)18 b Fz(l)r FG(\))p Fz( )s FG(\()p Fz(q)1168 1279 y FD(1)1205 1267 y FG(\))h(+)f Fz(R)q( )s FG(\()p Fz(q)1529 1279 y FD(2)1566 1267 y FG(\))h Fx(\000)1710 1211 y Fz(M)p 1710 1248 90 4 v 1712 1324 a(x)1759 1336 y FD(0)1826 1245 y FG(^)1809 1267 y Fz( )s FG(\()p Fz(x;)14 b(y)s FG(\))24 b(=)f(\()p Fz(M)k FG(+)18 b Fz(l)r FG(\))p Fz(e)p FG(\()p Fz(t)p FG(\))g(+)g Fz(R)q(e)p FG(\()p Fz(s)p FG(\))h Fx(\000)3004 1211 y Fz(M)p 3004 1248 V 3006 1324 a(x)3053 1336 y FD(0)3120 1245 y FG(^)3103 1267 y Fz( )s FG(\()p Fz(x;)14 b(y)s FG(\))1260 1443 y(=)23 b(\()p Fz(M)k FG(+)18 b Fz(l)r FG(\))1644 1376 y Fv(\000)1682 1443 y Fz(e)p FG(\()p Fz(t)p FG(\))h Fx(\000)f Fz(e)p FG(\()1989 1428 y(^)1988 1443 y Fz(t)o FG(\))2049 1376 y Fv(\001)2106 1443 y FG(+)g Fz(R)d FG(\()p Fz(e)p FG(\()s(^)-45 b Fz(s)p FG(\))19 b Fx(\000)f Fz(e)p FG(\()p Fz(s)p FG(\)\))c Fz(:)456 1336 y FG(\(2.24\))555 1593 y(Next,)44 b(using)c(the)g(con)n(v)n(ex)f(dualit)n(y)h(b)r(et)n(w)n (een)g Fz(\034)2130 1605 y FC(\014)2175 1593 y FG(\(1)p Fz(;)14 b Fx(\001)g FG(\))40 b(and)g Fz(m)2642 1605 y FC(\014)2687 1593 y FG(\()p Fx(\001)p FG(\),)k(relations)39 b(\(2.21\){)456 1692 y(\(2.20\),)27 b(\(1.27\){\(1.29\),)e(and)j(p)r (ositivit)n(y)f(of)h Fz(\033)s FG(\()p Fx(\001)p FG(\),)g(w)n(e)g (obtain)786 1897 y Fz(e)p FG(\()p Fz(t)p FG(\))18 b Fx(\000)g Fz(e)p FG(\()1092 1881 y(^)1091 1897 y Fz(t)p FG(\))24 b(=)1256 1881 y(^)1255 1897 y Fz(t)1299 1784 y Fv(Z)1382 1804 y FD(1)1345 1972 y(0)1419 1897 y FG(\(1)18 b Fx(\000)g Fz(\030)t FG(\))1666 1829 y Fv(\000)1705 1897 y Fz(m)1778 1862 y Fy(0)1778 1917 y FC(\014)1823 1897 y FG(\(\(1)g Fx(\000)g Fz(\030)t FG(\))p Fz(t)p FG(\))h Fx(\000)f Fz(m)2339 1862 y Fy(0)2339 1917 y FC(\014)2384 1897 y FG(\(\(1)h Fx(\000)f Fz(\030)t FG(\))2665 1881 y(^)2664 1897 y Fz(t)p FG(\))2726 1829 y Fv(\001)2778 1897 y Fz(d\030)1274 2160 y FG(+)1367 2104 y(1)p 1367 2141 42 4 v 1367 2217 a(2)1432 2047 y Fv(Z)1515 2068 y FD(1)1478 2236 y(0)1576 2026 y Fv(\000)1614 2093 y Fz(m)1687 2063 y Fy(0)1687 2116 y FC(\014)1732 2093 y FG(\(\(1)g Fx(\000)g Fz(\030)t FG(\))p Fz(t)p FG(\))h Fx(\000)f Fz(m)2248 2063 y Fy(0)2248 2116 y FC(\014)2293 2093 y FG(\(\(1)h Fx(\000)f Fz(\030)t FG(\))2574 2078 y(^)2573 2093 y Fz(t)p FG(\))2635 2026 y Fv(\001)2673 2043 y FD(2)p 1576 2141 1135 4 v 1913 2224 a Fz(m)1986 2195 y Fy(00)1986 2249 y FC(\014)2031 2224 y FG(\(\(1)g Fx(\000)g Fz(\030)t FG(\))2311 2209 y(^)2310 2224 y Fz(t)q FG(\))2734 2160 y Fz(d\030)23 b FG(+)18 b Fz(o)2973 2093 y Fv(\000)3011 2160 y FG(\()p Fz(t)h Fx(\000)3176 2145 y FG(^)3175 2160 y Fz(t)p FG(\))3237 2126 y FD(2)3275 2093 y Fv(\001)1177 2423 y FG(=)1256 2407 y(^)1255 2423 y Fz(t)1285 2355 y Fv(\000)1323 2423 y Fz(q)s FG(\()p Fz(t)p FG(\))g Fx(\000)f Fz(q)s FG(\()1632 2407 y(^)1631 2423 y Fz(t)p FG(\))1693 2355 y Fv(\001)1750 2423 y FG(+)1843 2366 y(\()p Fz(q)s FG(\()p Fz(t)p FG(\))i Fx(\000)e Fz(q)s FG(\()2185 2351 y(^)2184 2366 y Fz(t)p FG(\)\))2278 2336 y FD(2)p 1843 2404 473 4 v 1986 2486 a FG(2)p Fz(\033)s FG(\()2111 2471 y(^)2110 2486 y Fz(t)p FG(\))2344 2423 y(+)g Fz(o)2481 2355 y Fv(\000)2519 2423 y FG(\()p Fz(t)h Fx(\000)2684 2407 y FG(^)2683 2423 y Fz(t)p FG(\))2745 2388 y FD(2)2783 2355 y Fv(\001)2834 2423 y Fz(:)456 2623 y FG(No)n(w,)j(the)h(scaling)e(relation)h(for)f (the)i(area)e Fz(q)s FG(\()p Fx(\001)p FG(\))h(together)g(with)h(the)f (transv)n(ersalit)n(y)e(condition)456 2731 y(\(2.22\))26 b(and)i(the)g(iden)n(tit)n(y)g Fz(Q)1373 2743 y FD(1)1428 2731 y FG(+)18 b Fz(Q)1577 2743 y FD(2)1637 2731 y FG(=)1743 2710 y(^)1725 2731 y Fz(Q)1791 2743 y FD(1)1846 2731 y FG(+)1948 2710 y(^)1929 2731 y Fz(Q)1995 2743 y FD(2)2059 2731 y FG(imply)1149 2878 y(\()p Fz(M)27 b FG(+)18 b Fz(l)r FG(\))1432 2863 y(^)1431 2878 y Fz(t)1461 2811 y Fv(\000)1499 2878 y Fz(q)s FG(\()p Fz(t)p FG(\))h Fx(\000)f Fz(q)s FG(\()1808 2863 y(^)1807 2878 y Fz(t)p FG(\))1869 2811 y Fv(\001)1926 2878 y FG(+)g Fz(R)t FG(^)-45 b Fz(s)2111 2811 y Fv(\000)2150 2878 y Fz(q)s FG(\()p Fz(s)p FG(\))19 b Fx(\000)f Fz(q)s FG(\()s(^)-45 b Fz(s)p FG(\))2538 2811 y Fv(\001)2599 2878 y FG(=)23 b(0)p Fz(:)456 3023 y FG(As)28 b(a)f(result,)h(the)g(RHS)h(of)f(\(2.24\))f(reads,)g(due)h (to)g(relations)f(\(2.19\),)g(\(2.21\))g(and)h(the)g(c)n(hoice)456 3122 y(of)f Fz(\027)33 b FG(in)28 b(\(2.20\),)774 3326 y(\()p Fz(M)g FG(+)18 b Fz(l)r FG(\))1066 3270 y(\()p Fz(q)s FG(\()p Fz(t)p FG(\))i Fx(\000)e Fz(q)s FG(\()1408 3254 y(^)1407 3270 y Fz(t)p FG(\)\))1501 3239 y FD(2)p 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(and)h(the)f(scaling)g(relation)g(\(2.18\),)964 1983 y Fz(\022)r(x)1052 1995 y FD(0)1090 1983 y Fz(M)1180 1948 y FD(2)1217 1983 y Fz(\033)s FG(\()p Fz(t)1329 1995 y FD(0)1367 1983 y FG(\))c(=)1577 1927 y Fz(q)1614 1939 y FC(\014)s(;)p FD(+)p 1520 1964 267 4 v 1520 2040 a Fz(x)1567 2011 y FD(3)1567 2062 y(0)1605 2040 y Fz(\033)s FG(\()p Fz(t)1717 2052 y FD(0)1754 2040 y FG(\))1797 1983 y Fz(x)1844 1995 y FD(0)1881 1983 y Fz(M)1971 1948 y FD(2)2008 1983 y Fz(\033)s FG(\()p Fz(t)2120 1995 y FD(0)2158 1983 y FG(\))g(=)g Fz(q)2338 1995 y FC(\014)s(;)p FD(+)2467 1866 y Fv(\022)2539 1927 y Fz(M)p 2539 1964 90 4 v 2541 2040 a(x)2588 2052 y FD(0)2638 1866 y Fv(\023)2699 1883 y FD(2)2760 1983 y FG(=)f Fz(Q:)456 2169 y FG(Finally)-7 b(,)39 b(using)d(the)h(relation)f Fz(\024)p FG(\()p Fz(t)1568 2181 y FD(0)1605 2169 y FG(\))i(=)g Fz(\026)p FG(\()p Fz(t)1890 2181 y FD(0)1928 2169 y FG(\),)h(v)n(eri\014ed)d(in)h (Theorem)f(4.2)g(of)g(Sect.)h(4,)i(w)n(e)456 2268 y(deduce)27 b(the)h(claim)g(of)f(Theorem)g(1)g(with)h(\(cf.)h(\(1.20\)\))1258 2446 y Fz(c)1294 2458 y FC(\014)1362 2446 y FG(=)22 b Fz(\026)p FG(\(0\))i Fx(\021)1716 2354 y Fv(n)1772 2367 y(X)1813 2546 y FC(k)1905 2446 y Fz(k)s FE(F)2007 2458 y FC(k)2054 2446 y FG(\(0\))p Fz(e)2199 2412 y Fy(\000)p FC(k)q(m)2346 2421 y Fq(\014)2385 2412 y FD(\(0\))2475 2354 y Fv(o)2530 2371 y Fy(\000)p FD(1)2619 2446 y Fz(;)456 2661 y FG(the)29 b(in)n(v)n(erse)f(of)h(mean)g(length)g(of)h(\\horizon) n(tal")c(irreducible)j(bridges)f(\(i.e.,)i(with)g(v)-5 b(anishing)456 2761 y(tilt)28 b Fz(t)p FG(\).)p 3384 2761 4 57 v 3388 2708 50 4 v 3388 2761 V 3437 2761 4 57 v 982 2934 a(3.)46 b FF(Asymptotics)31 b(of)h(bridge)g(p)-6 b(ar)g(tition)32 b(functions)555 3084 y FG(In)25 b(this)f(section)g(w)n (e)g(dev)n(elop)f(our)h(basic)f(lo)r(cal)h(limit)h(tec)n(hniques)f(and) g(use)g(them)h(to)f(pro)n(v)n(e)456 3183 y(Prop)r(osition)i(1.6.)456 3336 y(3.1.)46 b FO(The)33 b(notation)e(and)h(the)g(results.)456 3488 y FG(3.1.1.)41 b FA(Asymptotics)31 b(of)g FE(B)1312 3500 y FC(n)1363 3421 y Fv(\000)1401 3488 y Fz(n)1451 3458 y FD(2)1488 3488 y Fz(q)1525 3500 y FC(n)1570 3421 y Fv(\001)1608 3488 y FA(.)42 b FG(Giv)n(en)29 b(a)f(bridge)g Fz(!)g FG(=)2411 3421 y Fv(\000)2449 3488 y Fz(!)s FG(\(0\))p Fz(;)14 b(:::;)g(!)s FG(\()p Fz(m)p FG(\))2945 3421 y Fv(\001)2983 3488 y FG(,)29 b(the)g(cen)n(tred)456 3588 y(signed)24 b(area)f Fz(a)p FG(\()p Fz(!)s FG(\))i(is)f(de\014ned)h(as) f(the)h(algebraic)e(sum)i(of)g(the)g(comp)r(onen)n(ts)f(b)r(ounded)h(b) n(y)f(the)456 3687 y(tra)5 b(jectory)18 b(of)i Fz(!)j FG(and)c(the)i(segmen)n(t)e(connecting)h(the)g(end-p)r(oin)n(ts)g Fz(!)s FG(\(0\))g(and)g Fz(!)s FG(\()p Fz(m)p FG(\))g(\(Fig.)g(3.7\).) 643 4941 y @beginspecial 0 @llx 0 @lly 921 @urx 358 @ury 3135 @rwi @setspecial %%BeginDocument: sap_area.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: sap_area.eps %%Creator: fig2dev Version 3.2 Patchlevel 3a %%CreationDate: Sat Nov 24 16:26:58 2001 %%For: ostap@authors (Ostap Hryniv) %%BoundingBox: 0 0 921 358 %%Magnification: 1.0000 %%EndComments /MyAppDict 100 dict dup begin def /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save newpath 0 358 moveto 0 0 lineto 921 0 lineto 921 358 lineto closepath clip newpath -17.0 501.0 translate 1 -1 scale % This junk string is used by the show operators /PATsstr 1 string def /PATawidthshow { % cx cy cchar rx ry string % Loop over each character in the string { % cx cy cchar rx ry char % Show the character dup % cx cy cchar rx ry char char PATsstr dup 0 4 -1 roll put % cx cy cchar rx ry char (char) false charpath % cx cy cchar rx ry char /clip load PATdraw % Move past the character (charpath modified the % current point) currentpoint % cx cy cchar rx ry char x y newpath moveto % cx cy cchar rx ry char % Reposition by cx,cy if the character in the string is cchar 3 index eq { % cx cy cchar rx ry 4 index 4 index rmoveto } if % Reposition all characters by rx ry 2 copy rmoveto % cx cy cchar rx ry } forall pop pop pop pop pop % - currentpoint newpath moveto } bind def /PATcg { 7 dict dup begin /lw currentlinewidth def /lc currentlinecap def /lj currentlinejoin def /ml currentmiterlimit def /ds [ currentdash ] def /cc [ currentrgbcolor ] def /cm matrix currentmatrix def end } bind def % PATdraw - calculates the boundaries of the object and % fills it with the current pattern /PATdraw { % proc save exch PATpcalc % proc nw nh px py 5 -1 roll exec % nw nh px py newpath PATfill % - restore } bind def % PATfill - performs the tiling for the shape /PATfill { % nw nh px py PATfill - PATDict /CurrentPattern get dup begin setfont % Set the coordinate system to Pattern Space PatternGState PATsg % Set the color for uncolored pattezns PaintType 2 eq { PATDict /PColor get PATsc } if % Create the string for showing 3 index string % nw nh px py str % Loop for each of the pattern sources 0 1 Multi 1 sub { % nw nh px py str source % Move to the starting location 3 index 3 index % nw nh px py str source px py moveto % nw nh px py str source % For multiple sources, set the appropriate color Multi 1 ne { dup PC exch get PATsc } if % Set the appropriate string for the source 0 1 7 index 1 sub { 2 index exch 2 index put } for pop % Loop over the number of vertical cells 3 index % nw nh px py str nh { % nw nh px py str currentpoint % nw nh px py str cx cy 2 index show % nw nh px py str cx cy YStep add moveto % nw nh px py str } repeat % nw nh px py str } for 5 { pop } repeat end } bind def % PATkshow - kshow with the current pattezn /PATkshow { % proc string exch bind % string proc 1 index 0 get % string proc char % Loop over all but the last character in the string 0 1 4 index length 2 sub { % string proc char idx % Find the n+1th character in the string 3 index exch 1 add get % string proe char char+1 exch 2 copy % strinq proc char+1 char char+1 char % Now show the nth character PATsstr dup 0 4 -1 roll put % string proc chr+1 chr chr+1 (chr) false charpath % string proc char+1 char char+1 /clip load PATdraw % Move past the character (charpath modified the current point) currentpoint newpath moveto % Execute the user proc (should consume char and char+1) mark 3 1 roll % string proc char+1 mark char char+1 4 index exec % string proc char+1 mark... cleartomark % string proc char+1 } for % Now display the last character PATsstr dup 0 4 -1 roll put % string proc (char+1) false charpath % string proc /clip load PATdraw neewath pop pop % - } bind def % PATmp - the makepattern equivalent /PATmp { % patdict patmtx PATmp patinstance exch dup length 7 add % We will add 6 new entries plus 1 FID dict copy % Create a new dictionary begin % Matrix to install when painting the pattern TilingType PATtcalc /PatternGState PATcg def PatternGState /cm 3 -1 roll put % Check for multi pattern sources (Level 1 fast color patterns) currentdict /Multi known not { /Multi 1 def } if % Font dictionary definitions /FontType 3 def % Create a dummy encoding vector /Encoding 256 array def 3 string 0 1 255 { Encoding exch dup 3 index cvs cvn put } for pop /FontMatrix matrix def /FontBBox BBox def /BuildChar { mark 3 1 roll % mark dict char exch begin Multi 1 ne {PaintData exch get}{pop} ifelse % mark [paintdata] PaintType 2 eq Multi 1 ne or { XStep 0 FontBBox aload pop setcachedevice } { XStep 0 setcharwidth } ifelse currentdict % mark [paintdata] dict /PaintProc load % mark [paintdata] dict paintproc end gsave false PATredef exec true PATredef grestore cleartomark % - } bind def currentdict end % newdict /foo exch % /foo newlict definefont % newfont } bind def % PATpcalc - calculates the starting point and width/height % of the tile fill for the shape /PATpcalc { % - PATpcalc nw nh px py PATDict /CurrentPattern get begin gsave % Set up the coordinate system to Pattern Space % and lock down pattern PatternGState /cm get setmatrix BBox aload pop pop pop translate % Determine the bounding box of the shape pathbbox % llx lly urx ury grestore % Determine (nw, nh) the # of cells to paint width and height PatHeight div ceiling % llx lly urx qh 4 1 roll % qh llx lly urx PatWidth div ceiling % qh llx lly qw 4 1 roll % qw qh llx lly PatHeight div floor % qw qh llx ph 4 1 roll % ph qw qh llx PatWidth div floor % ph qw qh pw 4 1 roll % pw ph qw qh 2 index sub cvi abs % pw ph qs qh-ph exch 3 index sub cvi abs exch % pw ph nw=qw-pw nh=qh-ph % Determine the starting point of the pattern fill %(px, py) 4 2 roll % nw nh pw ph PatHeight mul % nw nh pw py exch % nw nh py pw PatWidth mul exch % nw nh px py end } bind def % Save the original routines so that we can use them later on /oldfill /fill load def /oldeofill /eofill load def /oldstroke /stroke load def /oldshow /show load def /oldashow /ashow load def /oldwidthshow /widthshow load def /oldawidthshow /awidthshow load def /oldkshow /kshow load def % These defs are necessary so that subsequent procs don't bind in % the originals /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def /PATredef { MyAppDict begin { /fill { /clip load PATdraw newpath } bind def /eofill { /eoclip load PATdraw newpath } bind def /stroke { PATstroke } bind def /show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def /ashow { 0 0 null 6 3 roll PATawidthshow } bind def /widthshow { 0 0 3 -1 roll PATawidthshow } bind def /awidthshow { PATawidthshow } bind def /kshow { PATkshow } bind def } { /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def } ifelse end } bind def false PATredef % Conditionally define setcmykcolor if not available /setcmykcolor where { pop } { /setcmykcolor { 1 sub 4 1 roll 3 { 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat setrgbcolor - pop } bind def } ifelse /PATsc { % colorarray aload length % c1 ... cn length dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor } ifelse } ifelse } bind def /PATsg { % dict begin lw setlinewidth lc setlinecap lj setlinejoin ml setmiterlimit ds aload pop setdash cc aload pop setrgbcolor cm setmatrix end } bind def /PATDict 3 dict def /PATsp { true PATredef PATDict begin /CurrentPattern exch def % If it's an uncolored pattern, save the color CurrentPattern /PaintType get 2 eq { /PColor exch def } if /CColor [ currentrgbcolor ] def end } bind def % PATstroke - stroke with the current pattern /PATstroke { countdictstack save mark { currentpoint strokepath moveto PATpcalc % proc nw nh px py clip newpath PATfill } stopped { (*** PATstroke Warning: Path is too complex, stroking with gray) = cleartomark restore countdictstack exch sub dup 0 gt { { end } repeat } { pop } ifelse gsave 0.5 setgray oldstroke grestore } { pop restore pop } ifelse newpath } bind def /PATtcalc { % modmtx tilingtype PATtcalc tilematrix % Note: tiling types 2 and 3 are not supported gsave exch concat % tilingtype matrix currentmatrix exch % cmtx tilingtype % Tiling type 1 and 3: constant spacing 2 ne { % Distort the pattern so that it occupies % an integral number of device pixels dup 4 get exch dup 5 get exch % tx ty cmtx XStep 0 dtransform round exch round exch % tx ty cmtx dx.x dx.y XStep div exch XStep div exch % tx ty cmtx a b 0 YStep dtransform round exch round exch % tx ty cmtx a b dy.x dy.y YStep div exch YStep div exch % tx ty cmtx a b c d 7 -3 roll astore % { a b c d tx ty } } if grestore } bind def /PATusp { false PATredef PATDict begin CColor PATsc end } bind def % crosshatch lines 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def /BBox [0 0 1 1] def /XStep 1 def /YStep 1 def /PatWidth 1 def /PatHeight 1 def /Multi 2 def /PaintData [ { clippath } bind { 16 16 true [ 16 0 0 -16 0 16 ] {} imagemask } bind ] def /PaintProc { pop exec fill } def currentdict end /P11 exch def /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def $F2psBegin %%Page: 1 1 10 setmiterlimit 0.06000 0.06000 sc 30.000 slw % Ellipse n 1425 8100 225 225 0 360 DrawEllipse gs col7 0.85 shd ef gr gs col0 s gr % Ellipse n 7725 8100 225 225 0 360 DrawEllipse gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P11 [16 0 0 -16 500.00 525.00] PATmp PATsp ef gr PATusp gs col0 s gr % Polyline 0.000 slw n 600 5700 m 900 5700 l 900 6300 l 1500 6300 l 1500 6600 l 2700 6600 l 2700 6000 l 2100 6000 l 2100 5400 l 2400 5400 l 2400 5025 l gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P11 [16 0 0 -16 40.00 335.00] PATmp PATsp ef gr PATusp % Polyline n 3900 4425 m 3900 4800 l 5100 4800 l 5100 4500 l 4500 4500 l 4500 4200 l gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P11 [16 0 0 -16 260.00 280.00] PATmp PATsp ef gr PATusp % Polyline 30.000 slw n 8400 5700 m 15600 5700 l gs col0 s gr % Polyline 0.000 slw n 10200 5700 m 10200 5400 l 10500 5400 l 10500 4500 l 10200 4500 l 10200 3900 l 11400 3900 l 11400 4200 l 11700 4200 l 11700 3300 l 12300 3300 l 12300 3900 l 12000 3900 l 12000 4800 l 13200 4800 l 13200 4500 l 12600 4500 l 12600 3600 l 13200 3600 l 13200 3000 l 14100 3000 l 14100 3300 l 15000 3300 l 15000 5700 l gs col7 0.85 shd ef gr % Polyline 30.000 slw n 10200 5700 m 10200 5400 l 10500 5400 l 10500 4500 l 10200 4500 l 10200 3900 l gs col0 s gr % Polyline n 2400 5025 m 2400 4500 l 2100 4500 l 2100 3900 l 3300 3900 l 3300 4200 l 3600 4200 l 3600 3300 l 4200 3300 l 4200 3900 l 3900 3900 l 3900 4425 l gs col7 0.85 shd ef gr gs col0 s gr % Polyline n 4500 4200 m 4500 3600 l 5100 3600 l 5100 3000 l 6000 3000 l 6000 3300 l 6900 3300 l gs col7 0.85 shd ef gr gs col0 s gr 7.500 slw % Ellipse n 600 5700 38 38 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr % Ellipse n 6900 3300 38 38 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr % Ellipse n 15000 3300 38 38 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr 30.000 slw % Ellipse n 8700 5700 38 38 0 360 DrawEllipse gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P11 [16 0 0 -16 577.47 377.47] PATmp PATsp ef gr PATusp gs col0 s gr % Polyline 7.500 slw n 600 2400 m 600 7500 l gs col0 s gr % Polyline n 300 5700 m 7500 5700 l gs col0 s gr % Polyline n 600 5700 m 6900 3300 l gs col0 s gr % Polyline n 3900 4425 m 4500 4200 l gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P11 [16 0 0 -16 260.00 280.00] PATmp PATsp ef gr PATusp gs col0 s gr % Polyline n 8700 5700 m 15000 3300 l gs col0 s gr % Polyline n 12000 4425 m 12600 4200 l gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P11 [16 0 0 -16 800.00 280.00] PATmp PATsp ef gr PATusp gs col0 s gr % Polyline [60] 0 sd n 6900 2400 m 6900 7500 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 15000 2400 m 15000 7500 l gs col0 s gr [] 0 sd % Polyline 0.000 slw n 9000 5700 m 9000 6300 l 9600 6300 l 9600 6600 l 10800 6600 l 10800 6000 l 10200 6000 l 10200 5700 l 9000 5700 l cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P11 [16 0 0 -16 600.00 380.00] PATmp PATsp ef gr PATusp % Polyline n 10500 5025 m 10500 5025 l gs col0 s gr % Polyline 15.000 slw n 10500 5025 m 10500 4500 l 10200 4500 l 10200 3900 l 11400 3900 l 11400 4200 l 11700 4200 l 11700 3300 l 12300 3300 l 12300 3900 l 12000 3900 l 12000 4425 l gs col0 s gr % Polyline n 12000 4425 m 12000 4800 l 13200 4800 l 13200 4500 l 12600 4500 l 12600 4200 l gs col0 s gr % Polyline n 12600 4200 m 12600 3600 l 13200 3600 l 13200 3000 l 14100 3000 l 14100 3300 l 15000 3300 l gs col0 s gr % Polyline 0.000 slw n 8700 5700 m 10200 5700 l 10200 5400 l 10500 5400 l 10500 5025 l % Polyline 7.500 slw n 8700 2400 m 8700 7500 l gs col0 s gr % Polyline [60] 0 sd n 15000 3300 m 15600 3300 l gs col0 s gr [] 0 sd % Polyline gs clippath 15015 7230 m 15015 7170 l 14830 7170 l 14980 7200 l 14830 7230 l cp 8685 7170 m 8685 7230 l 8870 7230 l 8720 7200 l 8870 7170 l cp eoclip n 8700 7200 m 15000 7200 l gs col0 s gr gr % arrowhead n 8870 7170 m 8720 7200 l 8870 7230 l 8840 7200 l 8870 7170 l cp gs 0.00 setgray ef gr col0 s % arrowhead n 14830 7230 m 14980 7200 l 14830 7170 l 14860 7200 l 14830 7230 l cp gs 0.00 setgray ef gr col0 s % Polyline gs clippath 15270 5715 m 15330 5715 l 15330 5530 l 15300 5680 l 15270 5530 l cp 15330 3285 m 15270 3285 l 15270 3470 l 15300 3320 l 15330 3470 l cp eoclip n 15300 3300 m 15300 5700 l gs col0 s gr gr % arrowhead n 15330 3470 m 15300 3320 l 15270 3470 l 15300 3440 l 15330 3470 l cp gs 0.00 setgray ef gr col0 s % arrowhead n 15270 5530 m 15300 5680 l 15330 5530 l 15300 5560 l 15270 5530 l cp gs 0.00 setgray ef gr col0 s % Polyline 30.000 slw n 8700 5700 m 9000 5700 l 9000 6300 l 9600 6300 l 9600 6600 l 10800 6600 l 10800 6000 l 10200 6000 l 10200 5700 l gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P11 [16 0 0 -16 580.00 380.00] PATmp PATsp ef gr PATusp gs col0 s gr % Polyline n 600 5700 m 900 5700 l 900 6300 l 1500 6300 l 1500 6600 l 2700 6600 l 2700 6000 l 2100 6000 l 2100 5400 l 2400 5400 l 2400 5025 l gs col0 s gr % Polyline n 3900 4425 m 3900 4800 l 5100 4800 l 5100 4500 l 4500 4500 l 4500 4200 l gs col0 s gr % Polyline n 10200 3900 m 11400 3900 l 11400 4200 l 11700 4200 l 11700 3300 l 12300 3300 l 12300 3900 l 12000 3900 l 12000 4800 l 13200 4800 l 13200 4500 l 12600 4500 l 12600 3600 l 13200 3600 l 13200 3000 l 14100 3000 l 14100 3300 l 15000 3300 l gs col0 s gr % Polyline 7.500 slw n 8700 5700 m 15000 3300 l gs col0 s gr % Polyline n 8700 5700 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Fz(C)6 b(e)1643 1964 y Fy(\000)p FC(cn)1766 1939 y Fq(\020)1822 1998 y FG(+)1905 1928 y Fv(\014)1905 1978 y(\014)1932 1998 y FE(E)1982 1959 y Fr(reg)1982 2019 y FC(t;n)2078 1931 y Fv(\000)2116 1998 y Fz(e)2155 1964 y FC(t)p FD(\001)p FC(=n)2332 1998 y Fx(\000)18 b FG(1)c(;)g Fx(j)p FG(\001)p Fx(j)22 b(\024)h Fz(n)2783 1964 y FD(1)p Fy(\000)p FC(\021)2908 1998 y FG(;)28 b Fz(n)3009 1964 y FD(2)3046 1998 y Fz(q)3083 2010 y FC(n)3151 1928 y Fv(\014)3151 1978 y(\014)3202 1998 y Fx(M)3302 1931 y Fv(\001)3340 1928 y(\014)3340 1978 y(\014)1409 2154 y Fx(\024)23 b FG(2)p Fz(C)6 b(e)1643 2120 y Fy(\000)p FC(cn)1766 2095 y Fq(\020)1822 2154 y FG(+)18 b Fz(O)1970 2087 y Fv(\000)2008 2154 y Fz(n)2058 2120 y Fy(\000)p FC(\021)2151 2087 y Fv(\001)2189 2154 y FE(P)2241 2114 y Fr(reg)2241 2174 y FC(t;n)2329 2087 y Fv(\000)2367 2154 y Fz(A)2429 2120 y FC(R)2507 2154 y FG(=)23 b Fz(n)2645 2120 y FD(2)2682 2154 y Fz(q)2719 2166 y FC(n)2787 2083 y Fv(\014)2787 2133 y(\014)2838 2154 y Fx(M)2938 2087 y Fv(\001)456 2293 y FG(uniformly)k(in)h Fx(M)f FG(under)h(consideration.)35 b(As)28 b(a)f(result,)896 2445 y FE(B)945 2411 y Fr(re)q(g)945 2466 y FC(n;\013;")1110 2378 y Fv(\000)1149 2445 y Fz(e)1188 2411 y FC(tA)1263 2386 y Fq(R)1309 2411 y FC(=n)1402 2445 y FG(;)h Fz(n)1503 2411 y FD(2)1540 2445 y Fz(q)1577 2457 y FC(n)1622 2378 y Fv(\001)1683 2445 y FG(=)23 b FE(B)1820 2411 y Fr(re)q(g)1820 2466 y FC(n;\013;")1986 2378 y Fv(\000)2024 2445 y Fz(e)2063 2411 y FC(tA)2138 2386 y Fq(I)2171 2411 y FC(=n)2250 2378 y Fv(\001)2288 2445 y Fz(O)2353 2378 y Fv(\000)2392 2445 y Fz(e)2431 2411 y Fy(\000)p FC(cn)2554 2386 y Fq(\020)2591 2378 y Fv(\001)1679 2600 y FG(+)18 b FE(B)1811 2566 y Fr(re)q(g)1811 2621 y FC(n;\013;")1976 2533 y Fv(\000)2014 2600 y Fz(e)2053 2566 y FC(tA)2128 2541 y Fq(I)2161 2566 y FC(=n)2254 2600 y FG(;)28 b Fz(n)2355 2566 y FD(2)2392 2600 y Fz(q)2429 2612 y FC(n)2475 2533 y Fv(\001\000)2551 2600 y FG(1)18 b(+)g Fz(O)r FG(\()p Fz(n)2841 2566 y Fy(\000)p FC(\021)2934 2600 y 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Fr(r)q(eg)891 3564 y FC(n;\013;")1057 3476 y Fv(\000)1095 3543 y Fz(e)1134 3509 y FC(z)r(A)1218 3484 y Fq(I)1251 3509 y FC(=n)1330 3476 y Fv(\001)1391 3543 y FG(=)1479 3468 y Fv(p)p 1562 3468 313 4 v 75 x Fz(\026)p FG(\(0\))p Fz(\026)p FG(\()p Fz(z)t FG(\))13 b(exp)2015 3451 y Fv(n)2071 3543 y Fz(n)2135 3430 y Fv(Z)2217 3451 y FD(1)2180 3619 y(0)2268 3543 y Fz(m)2341 3555 y FC(\014)2386 3476 y Fv(\000)2424 3543 y FG(\(1)18 b Fx(\000)g Fz(\030)t FG(\))p Fz(z)2714 3476 y Fv(\001)2766 3543 y Fz(d\030)2849 3451 y Fv(o)2905 3476 y(\000)2943 3543 y FG(1)g(+)g Fz(o)p FG(\(1\))3232 3476 y Fv(\001)456 3543 y FG(\(3.18\))456 3724 y Fn(holds,)27 b(as)g Fz(n)c Fx(!)g(1)p Fn(,)28 b(uniformly)f(in)h(complex)f Fz(z)k Fn(satisfying)c(the)h(condition)1408 3863 y Fx(<)p Fz(z)f Fx(2)c FG([)p Fz(a;)14 b(b)p FG(])p Fz(;)1978 3793 y Fv(\014)1978 3843 y(\014)2005 3863 y Fx(=)p Fz(z)2108 3793 y Fv(\014)2108 3843 y(\014)2158 3863 y Fx(\024)23 b Fz(A=)2350 3799 y Fx(p)p 2419 3799 50 4 v 64 x Fz(n:)-2036 b FG(\(3.19\))555 4016 y(The)28 b(pro)r(of)f(of)g(the)h(lemma)g(is)f (giv)n(en)g(in)h(Sect.)g(3.4)f(b)r(elo)n(w.)456 4174 y(3.2.6.)41 b FA(L)l(o)l(c)l(al)32 b(limit)g(asymptotics.)43 b FG(Let)29 b(the)h(segmen)n(t)f([)p Fz(a;)14 b(b)p FG(])25 b Fx(\032)h(D)2556 4186 y FC(\014)2630 4174 y FG(b)r(e)j(as)g(\014xed)g (ab)r(o)n(v)n(e.)41 b(F)-7 b(or)456 4274 y(an)n(y)26 b Fz(t)d Fx(2)h FG([)p Fz(a;)14 b(b)p FG(],)27 b(in)n(tro)r(duce)g(the) h(tilted)h(probabilit)n(y)d(distribution)1436 4490 y FE(P)1488 4450 y Fr(reg)1488 4511 y FC(t;n)1577 4490 y FG(\()14 b Fx(\001)g FG(\))23 b(=)1813 4426 y FE(B)1862 4396 y Fr(re)q(g)1862 4447 y FC(n;\013;")2027 4359 y Fv(\000)2079 4426 y Fx(\001)14 b FG(;)g Fz(e)2192 4396 y FC(tA)2267 4371 y Fq(I)2300 4396 y FC(=n)2393 4359 y Fv(\001)p 1813 4471 619 4 v 1843 4552 a FE(B)1892 4512 y Fr(re)q(g)1892 4561 y FC(n;\013;")2071 4484 y Fv(\000)2123 4552 y Fz(e)2162 4528 y FC(tA)2237 4511 y Fq(I)2270 4528 y FC(=n)2363 4484 y Fv(\001)2441 4490 y Fz(:)-2008 b 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Fn(satisfying)h(\(3.1\).)p eop %%Page: 25 25 25 24 bop 1488 226 a FD(SELF-A)-7 b(V)n(OIDING)29 b(POL)-5 b(YGONS)966 b(25)555 425 y FG(Consequen)n(tly)-7 b(,)38 b(in)e(view)g(of)g(the)g(c)n(hange)f(of)h(measure)f(form)n(ula)g (\(3.13\),)j(it)e(remains)g(to)456 525 y(insert)23 b(the)h(asymptotics) f(\(3.18\))g(and)g(\(3.21\))g(in)n(to)g(the)h(decomp)r(osition)f (\(3.17\))g(and)g(to)h(apply)456 624 y(the)k(dualit)n(y)f(relation)g (\(1.28\).)36 b(The)27 b(conclusion)g(\(3.11\))g(of)g(Prop)r(osition)f (3.3)h(follo)n(ws.)p 3384 624 4 57 v 3388 572 50 4 v 3388 624 V 3437 624 4 57 v 555 724 a(W)-7 b(e)28 b(turn)g(no)n(w)f(to)g (the)h(pro)r(of)f(of)h(Lemmas)f(3.5{3.7.)456 880 y(3.3.)46 b FO(Pro)s(of)32 b(of)g(Lemma)d(3.5.)41 b FG(Giv)n(en)27 b(a)g(collection)g(of)h(in)n(teger)e(heigh)n(ts)1417 1018 y Fx(H)e FG(=)f Fx(f)p Fz(h)1689 1030 y FD(1)1725 1018 y Fz(;)14 b(g)1802 1030 y FD(1)1839 1018 y Fz(;)g(h)1924 1030 y FD(2)1961 1018 y Fz(;)g(:::;)g(g)2144 1030 y FC(K)t Fy(\000)p FD(1)2293 1018 y Fz(;)g(h)2378 1030 y FC(K)2441 1018 y Fx(g)456 1155 y FG(w)n(e)19 b(sa)n(y)h(that)g(a)g(regular)e (bridge)i Fz(!)26 b Fx(2)d(B)1673 1167 y FC(n)1738 1155 y FG(is)d Fx(H)q FG(-compatible)g(if)h(its)f(mesoscopic)f(decomp)r (osition)456 1255 y(\(3.8\))27 b(satis\014es:)617 1362 y Fy(8)672 1397 y Fz(j)h FG(=)23 b(1)p Fz(;)14 b(:::;)g(K)q(;)96 b(H)7 b FG(\()p Fz(!)1358 1409 y FC(j)1393 1397 y FG(\))23 b(=)g Fz(h)1584 1409 y FC(j)1812 1397 y Fu(and)2128 1362 y Fy(8)2183 1397 y Fz(j)28 b FG(=)23 b(1)p Fz(;)14 b(:::;)g(K)23 b Fx(\000)18 b FG(1)p Fz(;)97 b(H)7 b FG(\()p Fz(\015)3008 1409 y FC(j)3043 1397 y FG(\))23 b(=)g Fz(g)3226 1409 y FC(j)3260 1397 y Fz(:)456 1534 y FG(A)28 b(collection)f Fx(H)h FG(is)g(called)f FA(r)l(e)l(gular)37 b FG(i\013)627 1638 y Fy(8)683 1672 y Fz(j)28 b FG(=)22 b(1)p Fz(;)14 b(:::;)g(K)q(;)97 b Fx(j)p Fz(h)1280 1684 y FC(j)1315 1672 y Fx(j)23 b(\024)f Fz(r)1485 1684 y FD(3)1523 1672 y Fz(n)1573 1638 y FC(\013)1814 1672 y Fu(and)2129 1638 y Fy(8)2185 1672 y Fz(j)28 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Fx(H)1988 5011 y Fv(\001)2048 5078 y Fx(\024)23 b Fz(c)2172 5090 y FD(3)2223 5078 y FG(exp)2350 5011 y Fv(\010)2398 5078 y Fx(\000)p Fz(c)2499 5090 y FD(4)2536 5078 y Fz(n)2586 5044 y FD(1)p Fy(\000)p FD(2\()p FC(\013)p FD(+)p FC(")p FD(+)p FC(\021)r FD(\))2973 5011 y Fv(\011)456 5216 y FG(uniformly)k(in)h Fz(t)23 b Fx(2)g FG([)p Fz(a;)14 b(b)p FG(],)28 b(\014xed)f (collections)g Fx(M)g FG(and)h Fx(H)q FG(,)f(and)h(all)f Fz(n)h FG(large)e(enough.)p eop %%Page: 26 26 26 25 bop 456 226 a FD(26)814 b(OST)-5 b(AP)28 b(HR)-5 b(YNIV)29 b(AND)g(DMITR)-5 b(Y)29 b(IOFFE)555 425 y FG(Finally)-7 b(,)28 b(\(3.16\))f(follo)n(ws)f(directly)i(from)f(\(3.22\))g(and)g (the)h(last)f(b)r(ound.)p 3384 425 4 57 v 3388 372 50 4 v 3388 425 V 3437 425 4 57 v 456 589 a(3.4.)46 b FO(Pro)s(of)28 b(of)f(Lemma)e(3.6.)40 b FG(Fix)24 b(an)n(y)f Fz(t)g Fx(2)g FG([)p Fz(a;)14 b(b)p FG(])24 b(and)f(a)h(regular)e(collection)h Fx(M)g FG(of)h(break-)456 702 y(ing)g(p)r(oin)n(ts.)36 b(On)25 b(the)g(ev)n(en)n(t)f Fx(fMg)p FG(,)g(the)h(partition)g (function)g FE(B)2450 714 y FC(n)2502 702 y FG(\()p Fz(e)2573 672 y FC(tA)2648 647 y Fq(I)2681 672 y FC(=n)2760 702 y FG(;)14 b Fx(M)p FG(\))25 b(\(cf.)g(\(1.3\)\))g(has)456 801 y(the)j(follo)n(wing)e(factorisation)g(prop)r(ert)n(y:)967 1020 y FE(B)1016 1032 y FC(n)1068 1020 y FG(\()p Fz(e)1139 985 y FC(tA)1214 960 y Fq(I)1247 985 y FC(=n)1326 1020 y FG(;)14 b Fx(M)p FG(\))23 b(=)1633 916 y FC(K)1610 941 y Fv(Y)1606 1118 y FC(j)s FD(=1)1734 1020 y FE(B)1784 1035 y FC(W)9 b FD(\()p FC(!)1923 1043 y Fq(j)1960 1035 y FD(\))1990 952 y Fv(\000)2028 1020 y Fz(t)p FG(\()p Fz(!)2142 1032 y FC(j)2177 1020 y FG(\))2209 952 y Fv(\001)2261 916 y FC(K)t Fy(\000)p FD(1)2281 941 y Fv(Y)2281 1120 y FC(l)p FD(=1)2420 1020 y FE(F)2465 1035 y FC(W)g FD(\()p FC(\015)2597 1044 y Fq(l)2627 1035 y FD(\))2657 952 y Fv(\000)2695 1020 y Fz(t)p FG(\()p Fz(\015)2800 1032 y FC(l)2826 1020 y FG(\))2858 952 y Fv(\001)2896 1020 y Fz(;)-2463 b FG(\(3.23\))456 1250 y(with)28 b(\(cf.)g(\(3.14\)\))1139 1434 y Fz(t)p FG(\()p Fz(!)1253 1446 y FC(j)1288 1434 y FG(\))1343 1387 y Fr(def)1352 1434 y FG(=)1459 1378 y Fz(N)1526 1390 y FC(j)p 1459 1415 102 4 v 1485 1491 a Fz(n)1571 1434 y(t)23 b FG(=)1711 1342 y Fv(\020)1761 1434 y FG(1)18 b Fx(\000)1914 1378 y FG(2)p Fz(j)23 b Fx(\000)18 b FG(1)p 1914 1415 224 4 v 1966 1491 a(2)p Fz(K)2166 1434 y FG(+)2259 1378 y Fz(L)2316 1390 y FC(j)2368 1378 y Fx(\000)g Fz(R)2514 1390 y FC(j)s Fy(\000)p FD(1)p 2259 1415 376 4 v 2401 1491 a FG(2)p Fz(n)2644 1342 y Fv(\021)2694 1434 y Fz(t;)1139 1638 y(t)p FG(\()p Fz(\015)1244 1650 y FC(l)1270 1638 y FG(\))1325 1591 y Fr(def)1334 1638 y FG(=)1441 1582 y Fz(M)1522 1594 y FC(l)p 1441 1619 107 4 v 1469 1695 a Fz(n)1557 1638 y(t)23 b FG(=)1697 1546 y Fv(\020)1747 1638 y FG(1)18 b Fx(\000)1925 1582 y Fz(l)p 1900 1619 77 4 v 1900 1695 a(K)2005 1638 y FG(+)2098 1582 y Fz(L)2155 1594 y FC(l)2198 1582 y Fx(\000)g Fz(R)2344 1594 y FC(l)p 2098 1619 272 4 v 2188 1695 a FG(2)p Fz(n)2380 1546 y Fv(\021)2429 1638 y Fz(t:)456 1529 y FG(\(3.24\))456 1803 y(Next,)28 b(w)n(e)f(de\014ne)1412 1965 y FE(B)1462 1931 y FC(\013;")1462 1986 y(n)1566 1965 y FG(\()p Fz(t)p FG(\))1684 1918 y Fr(def)1693 1965 y FG(=)1789 1886 y Fv(X)1810 2065 y Fy(M)1923 1965 y FE(B)1972 1977 y FC(n)2023 1965 y FG(\()p Fz(e)2094 1931 y FC(tA)2169 1906 y Fq(I)2203 1931 y FC(=n)2282 1965 y FG(;)14 b Fx(M)p FG(\))p Fz(;)-2018 b FG(\(3.25\))456 2181 y(where)25 b(the)i(sum)f(runs)g(o)n(v)n(er)e (all)i(regular)e(collections)i Fx(M)p FG(.)36 b(Let)26 b Fx(B)2526 2193 y FC(n;\013;")2711 2181 y FG(denote)g(the)h(ensem)n (ble)456 2281 y(of)g(all)f(bridges)g Fz(!)k FG(whose)c(mesoscopic)g (collections)g Fx(M)p FG(\()p Fz(!)s FG(\))h(of)g(breaking)e(p)r(oin)n (ts)i(are)f(regular.)555 2380 y(As)32 b(w)n(e)f(shall)g(see)g(b)r(elo)n (w,)h(in)g(some)f(complex)g(neigh)n(b)r(ourho)r(o)r(d)f Fx(U)40 b FG(of)31 b(the)h(segmen)n(t)f([)p Fz(a;)14 b(b)p FG(])456 2493 y(the)33 b(partition)f(function)h FE(B)1334 2463 y Fr(re)q(g)1334 2513 y FC(n;\013;")1499 2426 y Fv(\000)1537 2493 y Fz(e)1576 2463 y FC(z)r(A)1660 2438 y Fq(I)1693 2463 y FC(=n)1773 2426 y Fv(\001)1843 2493 y FG(is)g(w)n(ell)f(appro)n(ximated)f(b)n(y)h FE(B)2805 2463 y FC(\013)q(;")2805 2513 y(n)2910 2493 y FG(\()p Fz(z)t FG(\);)j(the)e(latter,)456 2592 y(ho)n(w)n(ev)n(er,)25 b(p)r(ossesses)h(the)i(follo)n(wing)f(prop)r(ert)n(y)-7 b(.)456 2746 y FO(Lemma)29 b(3.8.)40 b Fn(Assume)24 b(that)h Fz(\013)g Fn(and)f Fz(")g Fn(satisfy)f(\(3.10\))h(with)h Fz(")f Fn(small)f(enough,)i FG(0)d Fz(<)h FG(4)p Fz(")f(<)h(\013)p Fn(.)456 2846 y(Then)31 b(there)g(exists)g(a)g(complex)g(neigh)n(b)r (ourho)r(o)r(d)g Fx(U)39 b Fn(of)32 b(the)g(segmen)n(t)e FG([)p Fz(a;)14 b(b)p FG(])31 b Fn(suc)n(h)g(that)h(the)456 2945 y(relation)907 3126 y FE(B)956 3091 y FC(\013;")956 3146 y(n)1061 3126 y FG(\()p Fz(z)t FG(\))22 b(=)1278 3051 y Fv(p)p 1361 3051 313 4 v 75 x Fz(\026)p FG(\(0\))p Fz(\026)p FG(\()p Fz(z)t FG(\))14 b(exp)1815 3033 y Fv(n)1870 3126 y Fz(n)1934 3013 y Fv(Z)2017 3033 y FD(1)1980 3201 y(0)2068 3126 y Fz(m)2141 3138 y FC(\014)2186 3058 y Fv(\000)2224 3126 y FG(\(1)k Fx(\000)g Fz(x)p FG(\))p Fz(z)2521 3058 y Fv(\001)2573 3126 y Fz(dx)2663 3033 y Fv(o)2719 3058 y(\000)2757 3126 y FG(1)g(+)g Fz(O)r FG(\()p Fz(n)3047 3091 y Fy(\000)p FC(")3135 3126 y FG(\))3167 3058 y Fv(\001)456 3126 y FG(\(3.26\))456 3309 y Fn(holds)27 b(true,)g(as)g Fz(n)c Fx(!)g(1)p Fn(,)28 b(uniformly)g(in)f Fz(z)g Fx(2)c(U)8 b Fn(.)456 3463 y FA(Pr)l(o)l(of.)43 b FG(Let)27 b Fz(t)894 3475 y FC(j)957 3463 y FG(denote)h(the)g (statistically)f(a)n(v)n(eraged)d(v)-5 b(alue)28 b(of)g Fz(t)p FG(\()p Fz(!)2581 3475 y FC(j)2615 3463 y FG(\),)1268 3648 y Fz(t)1298 3660 y FC(j)1356 3648 y Fx(\021)1443 3556 y Fv(\020)1493 3648 y FG(1)18 b Fx(\000)1646 3592 y FG(2)p Fz(j)23 b Fx(\000)18 b FG(1)p 1646 3629 224 4 v 1699 3705 a(2)p Fz(K)1879 3556 y Fv(\021)1929 3648 y Fz(t;)180 b(j)28 b FG(=)22 b(1)p Fz(;)14 b(:)g(:)g(:)g(;)g(K)q(:) -2177 b FG(\(3.27\))555 3818 y(Our)39 b(argumen)n(t)g(b)r(elo)n(w)g(is) h(based)f(on)h(the)g(follo)n(wing)e(prop)r(erties)h([33)o(]:)62 b(there)39 b(exist)h(a)456 3917 y(complex)27 b(neigh)n(b)r(ourho)r(o)r (d)f Fx(U)36 b FG(of)27 b([)p Fz(a;)14 b(b)p FG(])27 b(and)h(a)f(p)r(ositiv)n(e)g(constan)n(t)35 b(~)-51 b Fz(\013)24 b FG(=)31 b(~)-51 b Fz(\013)q FG(\()p Fx(U)8 b FG(\))28 b(suc)n(h)f(that)h(the)456 4017 y(relations)1294 4101 y Fy(1)1267 4126 y Fv(X)1267 4304 y FC(k)q Fy(\025)p FD(2)1402 4204 y Fz(e)1448 4170 y FD(~)-40 b FC(\013k)1525 4134 y Fv(\014)1525 4184 y(\014)1552 4204 y FE(F)1597 4216 y FC(k)1644 4204 y FG(\()p Fz(z)t FG(\))14 b(exp)1892 4137 y Fv(\010)1940 4204 y Fx(\000)p Fz(k)s(m)2124 4216 y FC(\014)2168 4204 y FG(\()p Fz(z)t FG(\))2275 4137 y Fv(\011)2323 4134 y(\014)2323 4184 y(\014)2374 4204 y Fz(<)23 b Fx(1)p Fz(;)911 4476 y FG(1)18 b(+)1078 4372 y FC(n)1119 4347 y Fq(")1054 4397 y Fv(X)1054 4576 y FC(k)q FD(=2)1175 4476 y FG(\()p Fz(k)j Fx(\000)d FG(1\))p FE(F)1473 4488 y FC(k)1520 4476 y FG(\()p Fz(z)t FG(\))c(exp)1767 4409 y Fv(\010)1816 4476 y Fx(\000)p Fz(k)s(m)2000 4488 y FC(\014)2044 4476 y FG(\()p Fz(z)t FG(\))2151 4409 y Fv(\011)2222 4476 y FG(=)2377 4420 y(1)p 2320 4457 157 4 v 2320 4533 a Fz(\026)p FG(\()p Fz(z)t FG(\))2505 4476 y(+)k Fz(o)p FG(\()p Fz(n)2710 4442 y FC(")2746 4476 y Fz(e)2785 4442 y Fy(\000)7 b FD(~)-40 b FC(\013)o(n)2920 4417 y Fq(")2957 4476 y FG(\))1267 4691 y(log)14 b FE(B)1437 4703 y FC(n)1488 4691 y FG(\()p Fz(z)t FG(\))19 b Fx(\000)f Fz(nm)1820 4703 y FC(\014)1864 4691 y FG(\()p Fz(z)t FG(\))g Fx(\000)g FG(log)c Fz(\026)p FG(\()p Fz(z)t FG(\))23 b(=)g Fz(o)p FG(\()p Fz(e)2572 4656 y Fy(\000)7 b FD(~)-40 b FC(\013n)2712 4691 y FG(\))p Fz(;)456 4417 y FG(\(3.28\))456 4833 y(hold)31 b(uniformly)g(in)g Fz(z)i Fx(2)c(U)39 b FG(and)32 b(all)e Fz(n)i FG(su\016cien)n(tly)f(large.)46 b(Here)31 b(w)n(e)g(ha)n(v)n(e)f(used)h(the)h(iden-)456 4932 y(tit)n(y)27 b([33])1595 5037 y Fy(1)1568 5062 y Fv(X)1567 5240 y FC(k)q FD(=1)1702 5141 y FE(F)1747 5153 y FC(k)1794 5141 y FG(\()p Fx(\001)p FG(\)e)1918 5106 y Fy(\000)p FC(k)q(m)2065 5115 y Fq(\014)2105 5106 y FD(\()p Fy(\001)p FD(\))2203 5141 y Fx(\021)c FG(1)p eop %%Page: 27 27 27 26 bop 1488 226 a FD(SELF-A)-7 b(V)n(OIDING)29 b(POL)-5 b(YGONS)966 b(27)456 425 y FG(v)-5 b(alid)40 b(ev)n(erywhere)f(on)h Fx(U)8 b FG(.)76 b(Note)40 b(that)h(the)g(\014rst)f(t)n(w)n(o)g(prop)r (erties)g(in)g(\(3.28\))g(imply)h(the)456 525 y(follo)n(wing)26 b(useful)i(estimate:)865 743 y(1)18 b(+)1093 639 y FC(n)1134 614 y Fq(")1069 664 y Fv(X)1008 843 y FC(l)p Fy(\025)p FD(1)p FC(;r)r Fy(\025)p FD(1)1265 743 y FE(F)1310 755 y FC(l)p FD(+)p FC(r)1425 651 y Fv(\020)1474 743 y Fz(z)1513 755 y FD(0)1569 743 y FG(+)g(\()p Fz(r)j Fx(\000)d Fz(l)r FG(\))1898 687 y Fz(z)p 1894 724 50 4 v 1894 800 a(n)1954 651 y Fv(\021)2017 743 y FG(exp)2144 651 y Fv(n)2199 743 y Fx(\000)p FG(\()p Fz(l)i FG(+)e Fz(r)r FG(\))p Fz(m)2568 755 y FC(\014)2614 651 y Fv(\020)2663 743 y Fz(z)2702 755 y FD(0)2758 743 y FG(+)g(\()p Fz(r)j Fx(\000)d Fz(l)r FG(\))3087 687 y Fz(z)p 3083 724 V 3083 800 a(n)3143 651 y Fv(\021)o(o)1573 983 y FG(=)1661 916 y Fv(\000)1699 983 y Fz(\026)p FG(\()p Fz(z)1820 995 y FD(0)1857 983 y FG(\))1889 916 y Fv(\001)1928 933 y Fy(\000)p FD(1)2035 983 y FG(+)g Fz(o)p FG(\()p Fz(n)2240 949 y FD(3)p FC(")p Fy(\000)p FD(2)2394 983 y FG(\))456 823 y(\(3.29\))456 1126 y(v)-5 b(alid)30 b(uniformly)h(in)g Fz(z)1180 1138 y FD(0)1248 1126 y FG(and)f Fz(z)k FG(b)r(elonging)c(to)h Fx(U)39 b FG(and)30 b(all)h Fz(n)f FG(large)g(enough)g(\(thanks)g(to)h (the)456 1226 y(term-b)n(y-term)26 b(cancellation)h(in)h(the)f(linear)g (part)836 1340 y FD(2)p FC(n)910 1315 y Fq(")829 1365 y Fv(X)829 1544 y FC(k)q FD(=2)950 1352 y Fv(h)989 1444 y FE(F)1034 1410 y Fy(0)1034 1465 y FC(k)1080 1444 y FG(\()p Fz(z)1151 1456 y FD(0)1189 1444 y FG(\))18 b(+)g FE(F)1367 1456 y FC(k)1414 1444 y FG(\()p Fz(z)1485 1456 y FD(0)1522 1444 y FG(\))p Fz(m)1627 1410 y Fy(0)1627 1465 y FC(\014)1672 1444 y FG(\()p Fz(z)1743 1456 y FD(0)1780 1444 y FG(\))1812 1352 y Fv(i)1866 1444 y FG(exp)1992 1352 y Fv(n)2048 1444 y Fx(\000)p Fz(k)s(m)2232 1456 y FC(\014)2276 1444 y FG(\()p Fz(z)2347 1456 y FD(0)2384 1444 y FG(\))2416 1352 y Fv(o)2567 1365 y(X)2485 1547 y FD(1)p Fy(\024)p FC(l;r)r Fy(\024)p FC(n)2737 1522 y Fq(")2531 1625 y FC(l)p FD(+)p FC(r)r FD(=)p FC(k)2769 1444 y FG(\()p Fz(r)k Fx(\000)c Fz(l)r FG(\))3016 1388 y Fz(z)p 3012 1425 V 3012 1501 a(n)456 1749 y FG(of)27 b(the)h(T)-7 b(a)n(ylor)26 b(expansion)h(of)g(the)h(LHS)g(in)g (\(3.29\))f(around)f Fz(z)2422 1761 y FD(0)2459 1749 y FG(\).)555 1849 y(In)i(addition,)g(w)n(e)f(shall)g(use)g(the)h(follo) n(wing)f(simple)h(observ)-5 b(ation)26 b(to)h(b)r(e)h(c)n(hec)n(k)n(ed) f(b)r(elo)n(w.)661 1968 y Fn(Let)f Fz(z)t FG(\()p Fz(!)935 1980 y FC(j)969 1968 y FG(\))f Fn(and)g Fz(z)1224 1980 y FC(j)1284 1968 y Fn(b)r(e)h(the)f(complex)g(coun)n(terparts)e(of)i Fz(t)p FG(\()p Fz(!)2544 1980 y FC(j)2579 1968 y FG(\))h Fn(and)f Fz(t)2826 1980 y FC(j)2886 1968 y Fn(de\014ned)h(in)661 2067 y(\(3.24\))g(and)h(\(3.27\))f(corresp)r(ondingly)-7 b(.)35 b(Then,)27 b(uniformly)g(in)g Fz(!)f Fx(2)d(B)2850 2079 y FC(n;\013;")3036 2067 y Fn(and)j Fz(z)661 2167 y Fn(under)i(consideration,)e(w)n(e)h(ha)n(v)n(e)860 2382 y FG(log)1008 2278 y FC(K)985 2303 y Fv(Y)981 2480 y FC(j)s FD(=1)1109 2382 y FE(B)1159 2397 y FC(W)9 b FD(\()p FC(!)1298 2405 y Fq(j)1335 2397 y FD(\))1365 2315 y Fv(\000)1403 2382 y Fz(z)t FG(\()p Fz(!)1530 2394 y FC(j)1564 2382 y FG(\))1596 2315 y Fv(\001)1658 2382 y FG(=)1775 2278 y FC(K)1745 2303 y Fv(X)1748 2480 y FC(j)s FD(=1)1879 2382 y FG(log)14 b Fz(\026)p FG(\()p Fz(z)2121 2394 y FC(j)2156 2382 y FG(\))19 b(+)f Fz(n)2340 2348 y FC(\013)2431 2278 y(K)2401 2303 y Fv(X)2403 2480 y FC(j)s FD(=1)2535 2382 y Fz(m)2608 2394 y FC(\014)2652 2382 y FG(\()p Fz(z)2723 2394 y FC(j)2758 2382 y FG(\))1653 2684 y Fx(\000)1736 2581 y FC(K)t Fy(\000)p FD(1)1749 2606 y Fv(X)1751 2782 y FC(j)s FD(=1)1881 2617 y Fv(\000)1919 2684 y Fz(L)1976 2696 y 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FG(log)c Fz(\026)p FG(\()5 b(\026)-47 b Fz(z)2106 653 y FC(l)2131 641 y FG(\))2163 549 y Fv(o)2233 641 y FG(exp)2360 549 y Fv(n)2415 641 y Fz(O)r FG(\()p Fz(K)6 b(n)2639 607 y FD(3)p FC(")p Fy(\000)p FD(1)2793 641 y FG(\))19 b(+)f Fz(O)r FG(\()p Fz(n)3074 607 y FC(")p Fy(\000)p FC(\013)3205 641 y FG(\))3237 549 y Fv(o)891 873 y FG(=)979 798 y Fv(p)p 1062 798 313 4 v 75 x Fz(\026)p FG(\(0\))p Fz(\026)p FG(\()p Fz(z)t FG(\))1375 806 y Fv(\000)1413 873 y FG(1)g(+)g Fz(O)r FG(\()p Fz(n)1703 839 y FD(3)p FC(")p Fy(\000)p FC(\013)1867 873 y FG(\))1899 806 y Fv(\001)1937 873 y Fz(:)456 1013 y FG(It)28 b(remains)e(to)i(observ)n(e)e(that)h(\(thanks)h(to)g(smo)r (othness)e(of)i Fz(m)2446 1025 y FC(\014)2491 1013 y FG(\()p Fx(\001)p FG(\)\))861 1232 y Fz(n)911 1197 y FC(\013)1002 1128 y(K)972 1153 y Fv(X)975 1330 y FC(j)s FD(=1)1106 1232 y Fz(m)1179 1244 y FC(\014)1224 1232 y FG(\()p Fz(z)1295 1244 y FC(j)1330 1232 y FG(\))18 b Fx(\000)g Fz(n)1527 1119 y Fv(Z)1610 1139 y FD(1)1573 1307 y(0)1661 1232 y Fz(m)1734 1244 y FC(\014)1779 1164 y Fv(\000)1817 1232 y Fz(z)t FG(\()p Fz(x)p FG(\))1971 1164 y Fv(\001)2023 1232 y Fz(dx)24 b FG(=)e Fz(O)r FG(\()p Fz(n=K)2490 1197 y FD(2)2527 1232 y FG(\))i(=)e Fz(O)r FG(\()p Fz(n)2817 1197 y FD(2)p FC(\013)p Fy(\000)p FD(1)2983 1232 y FG(\))p Fz(:)555 1455 y FG(Finally)-7 b(,)38 b(w)n(e)d(justify)i (\(3.30\).)60 b(T)-7 b(o)35 b(this)h(end,)i(\014x)e(an)n(y)f Fz(!)k Fx(2)e(B)2574 1467 y FC(n;\013;")2733 1455 y FG(.)61 b(According)35 b(to)g(the)456 1554 y(de\014nitions)28 b(\(3.24\))g(and)g(\(3.27\),)g(w)n(e)g(ha)n(v)n(e)f Fz(z)t FG(\()p Fz(!)1967 1566 y FC(j)2002 1554 y FG(\))19 b Fx(\000)g Fz(z)2176 1566 y FC(j)2235 1554 y FG(=)24 b Fz(O)r FG(\()p Fz(n)2471 1524 y FC(")2507 1554 y Fz(=n)p FG(\))k(and)g(therefore)g(the)h(last)456 1654 y(relation)d(in)i (\(3.28\))f(implies)h(\(recall)f(that)h Fz(K)g FG(=)23 b Fz(n)2050 1624 y FD(1)p Fy(\000)p FC(\013)2210 1654 y FG(and)k Fz(\026)h FG(is)f(an)g(analytic)g(function\))670 1766 y FC(K)639 1791 y 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FC(K)1583 2749 y FG(=)g(0;)k(therefore,)f(using)f(again)f(the)i(T)-7 b(a)n(ylor)30 b(form)n(ula,)j(w)n(e)456 2849 y(get)27 b(the)h(relations)850 2956 y FC(K)820 2981 y Fv(X)822 3158 y FC(j)s FD(=1)940 3060 y FG(\()p Fz(L)1029 3072 y FC(j)1082 3060 y FG(+)18 b Fz(R)1228 3072 y FC(j)s Fy(\000)p FD(1)1348 3060 y FG(\))p Fz(m)1453 3072 y FC(\014)1498 3060 y FG(\()p Fz(z)1569 3072 y FC(j)1604 3060 y FG(\))23 b(=)1747 2956 y FC(K)t Fy(\000)p FD(1)1759 2981 y Fv(X)1762 3158 y FC(j)s FD(=1)1891 3060 y FG(\()p Fz(L)1980 3072 y FC(j)2034 3060 y FG(+)18 b Fz(R)2180 3072 y FC(j)2215 3060 y FG(\))p Fz(m)2320 3072 y FC(\014)2365 3060 y FG(\()5 b(\026)-47 b Fz(z)2436 3072 y FC(j)2471 3060 y FG(\))1654 3362 y(+)1785 3306 y Fz(z)p 1747 3343 V 1747 3419 a FG(2)p Fz(K)1889 3258 y FC(K)t Fy(\000)p FD(1)1902 3283 y Fv(X)1904 3460 y FC(j)s FD(=1)2034 3362 y FG(\()p Fz(L)2123 3374 y FC(j)2176 3362 y Fx(\000)18 b Fz(R)2322 3374 y FC(j)2357 3362 y FG(\))p Fz(m)2462 3328 y Fy(0)2462 3383 y FC(\014)2507 3362 y 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Fz(dx)167 b FG(as)27 b Fz(n)c Fx(!)g(1)p Fz(:)456 4607 y FG(Recalling)h(the)i(non-degeneracy)c(of)j (the)h(function)g Fz(\033)s FG(\()p Fz(t)p FG(\))g(from)f(\(1.29\))f (and)h(using)g(the)g(T)-7 b(a)n(ylor)456 4706 y(form)n(ula,)26 b(w)n(e)i(immediately)f(deduce)h(the)g(follo)n(wing)f(fact.)456 4855 y FO(Corollary)33 b(3.9.)40 b Fn(There)27 b(exist)g Fz(\016)f(>)d FG(0)k Fn(and)h Fz(\021)e(>)d FG(0)k Fn(suc)n(h)g(that)h (the)g(estimate)1427 4976 y Fx(j)p FE(B)1499 4946 y FC(\013)q(;")1499 4997 y(n)1604 4976 y FG(\()p Fz(t)18 b FG(+)g Fz(i\034)9 b FG(\))p Fx(j)p 1427 5013 471 4 v 1538 5089 a FE(B)1587 5049 y FC(\013;")1587 5099 y(n)1692 5089 y FG(\()p Fz(t)p FG(\))1930 5032 y Fx(\024)23 b FG(exp)2145 4965 y Fv(\010)2193 5032 y Fx(\000)p Fz(\021)s(n\034)2397 4998 y FD(2)2435 4965 y Fv(\011)456 5216 y Fn(holds)k(uniformly)g(in)h Fz(t)23 b Fx(2)h FG([)p Fz(a;)14 b(b)p FG(])27 b Fn(and)g Fx(j)p Fz(\034)9 b Fx(j)24 b(\024)f Fz(\016)s Fn(.)p eop %%Page: 29 29 29 28 bop 1488 226 a FD(SELF-A)-7 b(V)n(OIDING)29 b(POL)-5 b(YGONS)966 b(29)555 425 y FG(W)-7 b(e)22 b(turn)f(no)n(w)g(to)g(the)h (pro)r(of)f(of)g(\(3.18\).)34 b(In)22 b(a)e(w)n(a)n(y)-7 b(,)22 b(similar)e(to)i(\(3.25\)+\(3.23\),)e(w)n(e)h(rewrite)779 659 y FE(B)828 625 y Fr(re)q(g)828 679 y FC(n;\013;")993 659 y FG(\()p Fz(z)t FG(\))i Fx(\021)g FE(B)1260 625 y Fr(re)q(g)1260 679 y FC(n;\013;")1426 592 y Fv(\000)1464 659 y Fz(e)1503 625 y FC(z)r(A)1587 600 y Fq(I)1620 625 y FC(=n)1699 592 y Fv(\001)1760 659 y FG(=)1848 580 y Fv(X)1869 758 y Fy(M)2009 555 y FC(K)1986 580 y Fv(Y)1981 757 y FC(j)s FD(=1)2110 659 y FE(B)2159 619 y Fr(re)q(g)2159 687 y FC(W)10 b FD(\()p FC(!)2299 695 y Fq(j)2336 687 y FD(\))2366 592 y Fv(\000)2404 659 y Fz(z)t FG(\()p Fz(!)2531 671 y FC(j)2565 659 y FG(\))2597 592 y Fv(\001)2649 555 y FC(K)t Fy(\000)p FD(1)2668 580 y Fv(Y)2669 759 y FC(l)p FD(=1)2808 659 y FE(F)2864 619 y Fr(reg)2853 687 y FC(W)f FD(\()p FC(\015)2985 696 y Fq(l)3015 687 y FD(\))3045 592 y Fv(\000)3083 659 y Fz(z)t FG(\()p Fz(\015)3201 671 y FC(l)3226 659 y FG(\))3258 592 y Fv(\001)3296 659 y Fz(;)-2863 b FG(\(3.32\))456 900 y(where)26 b(the)h(functions)g FE(B)1243 870 y Fr(re)q(g)1243 920 y FC(m)1334 900 y FG(\()p Fx(\001)p FG(\))g(and)g FE(F)1665 870 y Fr(reg)1654 920 y FC(m)1755 900 y FG(\()p Fx(\001)p FG(\))h(are)e(de\014ned)h(in)g (an)f(ob)n(vious)g(w)n(a)n(y)f(to)i(satisfy)f(the)456 1000 y(conditions)e FO(\(R1\){\(R3\))g FG(of)h(De\014nition)g(3.2.)35 b(As)25 b(w)n(e)g(shall)f(see)g(b)r(elo)n(w,)h(these)g(functions)g(are) 456 1099 y(w)n(ell)i(appro)n(ximated)f(b)n(y)h(their)h(non-restricted)e (analogues:)661 1225 y Fn(Let)g(the)h(p)r(ositiv)n(e)e(constan)n(ts)g Fz(\013)p Fn(,)i Fz(")f Fn(and)f(the)i(segmen)n(t)e FG([)p Fz(a;)14 b(b)p FG(])26 b Fn(b)r(e)g(as)f(\014xed)h(ab)r(o)n(v)n(e.)661 1325 y(Then)h(there)f(exist)h(large)e(enough)g Fz(r)1807 1337 y FD(2)1845 1325 y Fn(,)i Fz(r)1932 1337 y FD(3)1996 1325 y Fn(\(recall)f(De\014nition)h(3.2\))f(and)g(p)r(ositiv)n(e)661 1424 y Fz(c)p Fn(,)i Fz(\020)34 b Fn(suc)n(h)27 b(that)h(the)g (relations)970 1519 y Fv(\014)970 1569 y(\014)998 1590 y FE(B)1047 1602 y FC(m)1116 1590 y FG(\()p Fz(z)t FG(\))19 b Fx(\000)f FE(B)1374 1556 y Fr(re)q(g)1374 1610 y FC(m)1465 1590 y FG(\()p Fz(z)t FG(\))1572 1519 y Fv(\014)1572 1569 y(\014)1622 1590 y Fx(\024)23 b FE(B)1759 1602 y FC(m)1828 1590 y FG(\()p Fz(t)p FG(\))c Fx(\000)f FE(B)2074 1556 y Fr(reg)2074 1610 y FC(m)2164 1590 y FG(\()p Fz(t)p FG(\))24 b(=)e Fz(O)r FG(\()p Fz(e)2505 1556 y Fy(\000)p FC(cm)2646 1531 y Fq(\020)2684 1590 y FE(B)2734 1602 y FC(m)2803 1590 y FG(\()p Fz(t)p FG(\)\))971 1675 y Fv(\014)971 1725 y(\014)998 1745 y FE(F)1043 1757 y FC(m)1112 1745 y FG(\()p Fz(z)t FG(\))d Fx(\000)f FE(F)1377 1711 y Fr(reg)1365 1766 y FC(m)1467 1745 y FG(\()p Fz(z)t FG(\))1574 1675 y Fv(\014)1574 1725 y(\014)1625 1745 y Fx(\024)k FE(F)1757 1757 y FC(m)1826 1745 y FG(\()p Fz(t)p FG(\))d Fx(\000)f FE(F)2078 1711 y Fr(re)q(g)2067 1766 y FC(m)2169 1745 y FG(\()p Fz(t)p FG(\))23 b(=)g Fz(O)r FG(\()p Fz(e)2510 1711 y Fy(\000)p FC(cm)2651 1686 y Fq(\020)2689 1745 y FE(F)2734 1757 y FC(m)2803 1745 y FG(\()p Fz(t)p FG(\)\))456 1661 y(\(3.33\))661 1903 y Fn(hold)28 b(uniformly)f(in)h Fz(m)23 b Fx(2)g FE(N)38 b Fn(and)28 b Fz(z)e Fx(2)d FE(C)49 b Fn(satisfying)27 b Fx(<)p Fz(z)f Fx(2)e FG([)p Fz(a;)14 b(b)p FG(])p Fn(.)555 2029 y FG(W)-7 b(e)22 b(p)r(ostp)r(one)g(the)g(pro)r(of)f(of)g(the)h(b) r(ounds)g(\(3.33\))f(till)h(the)g(end)g(of)g(the)g(section)f(and)g (deduce)456 2128 y(v)-5 b(alidit)n(y)27 b(of)h(the)g(asymptotics)e (\(3.18\))h(\014rst.)555 2228 y(Our)g(k)n(ey)g(observ)-5 b(ation)26 b(is)i(the)g(next)f(estimate:)37 b(if)28 b Fz(z)f Fx(2)c FE(C)49 b FG(is)27 b(suc)n(h)h(that)f Fx(<)p Fz(z)g Fx(2)c FG([)p Fz(a;)14 b(b)p FG(],)27 b(then)776 2328 y Fv(\014)776 2377 y(\014)803 2398 y FE(B)853 2364 y FC(\013;")853 2419 y(n)957 2398 y FG(\()p Fz(t)19 b FG(+)f Fz(i\034)9 b FG(\))19 b Fx(\000)f FE(B)1378 2364 y Fr(re)q(g)1378 2419 y FC(n;\013;")1543 2398 y FG(\()p Fz(t)h FG(+)f Fz(i\034)9 b FG(\))1813 2328 y Fv(\014)1813 2377 y(\014)1864 2398 y Fx(\024)23 b FE(B)2001 2364 y FC(\013)q(;")2001 2419 y(n)2106 2398 y FG(\()p Fz(t)p FG(\))c Fx(\000)f FE(B)2351 2364 y Fr(re)q(g)2351 2419 y FC(n;\013;")2516 2398 y FG(\()p Fz(t)p FG(\))24 b(=)f Fz(O)r FG(\()p Fz(e)2858 2364 y Fy(\000)p FC(cn)2981 2339 y Fq(\020)3019 2398 y FG(\))p FE(B)3101 2364 y FC(\013;")3101 2419 y(n)3205 2398 y FG(\()p Fz(t)p FG(\))p Fz(;)-2866 b FG(\(3.34\))456 2551 y(with)24 b(p)r(ossibly)g(di\013eren)n(t)g (constan)n(t)f Fz(c)g(>)f FG(0.)36 b(Indeed,)25 b(the)f(inequalit)n(y)f (ab)r(o)n(v)n(e)g(follo)n(ws)g(directly)456 2651 y(from)h(the)i (mesoscopic)d(represen)n(tations)g(of)i(the)h(partition)f(functions)g FE(B)2766 2621 y FC(\013;")2766 2671 y(n)2871 2651 y FG(\()p Fx(\001)p FG(\))g(and)g FE(B)3192 2621 y Fr(reg)3192 2671 y FC(n;\013;")3357 2651 y FG(\()p Fx(\001)p FG(\))456 2753 y(\(recall)39 b(\(3.32\))h(and)g(its)h(non-restricted)e(analogue)g (\(3.25\)+\(3.23\)\))g(com)n(bined)h(with)h(the)456 2853 y(apriori)26 b(b)r(ounds)1073 2930 y Fv(\014)1073 2980 y(\014)1100 3001 y FE(B)1150 3013 y FC(m)1219 3001 y FG(\()p Fz(t)19 b FG(+)f Fz(is)p FG(\))1483 2930 y Fv(\014)1483 2980 y(\014)1533 3001 y Fx(\024)23 b FE(B)1670 3013 y FC(m)1739 3001 y FG(\()p Fz(t)p FG(\))p Fz(;)2149 2930 y Fv(\014)2149 2980 y(\014)2176 3001 y FE(F)2221 3013 y FC(m)2290 3001 y FG(\()p Fz(t)c FG(+)f Fz(is)p FG(\))2554 2930 y Fv(\014)2554 2980 y(\014)2604 3001 y Fx(\024)23 b FE(F)2737 3013 y FC(m)2806 3001 y FG(\()p Fz(t)p FG(\))p Fz(;)1045 3077 y Fv(\014)1045 3127 y(\014)1073 3147 y FE(B)1122 3117 y Fr(re)q(g)1122 3168 y FC(m)1212 3147 y FG(\()p Fz(t)c FG(+)f Fz(is)p FG(\))1476 3077 y Fv(\014)1476 3127 y(\014)1527 3147 y Fx(\024)k FE(B)1664 3117 y Fr(reg)1664 3168 y FC(m)1754 3147 y FG(\()p Fz(t)p FG(\))p Fz(;)2121 3077 y Fv(\014)2121 3127 y(\014)2149 3147 y FE(F)2205 3117 y Fr(reg)2193 3168 y FC(m)2295 3147 y FG(\()p Fz(t)d FG(+)f Fz(is)p FG(\))2559 3077 y Fv(\014)2559 3127 y(\014)2609 3147 y Fx(\024)23 b FE(F)2753 3117 y Fr(re)q(g)2742 3168 y FC(m)2843 3147 y FG(\()p Fz(t)p FG(\))456 3074 y(\(3.35\))456 3301 y(v)-5 b(alid)34 b(uniformly)g(in)h Fz(t)f Fx(2)h FG([)p Fz(a;)14 b(b)p FG(],)35 b(real)f Fz(s)p FG(,)i(and)e(in)n(teger) g Fz(m)g Fx(\025)g FG(2.)56 b(On)35 b(the)f(other)g(hand,)i(the)456 3400 y(equalit)n(y)27 b(is)g(a)g(simple)h(corollary)d(of)j(the)g (estimate)767 3535 y Fv(\014)767 3584 y(\014)767 3634 y(\014)838 3526 y FC(l)796 3551 y Fv(Y)795 3728 y FC(i)p FD(=1)916 3630 y Fz(a)960 3642 y FC(i)1006 3630 y Fx(\000)1133 3526 y FC(l)1090 3551 y Fv(Y)1089 3728 y FC(i)p FD(=1)1211 3630 y Fz(b)1247 3642 y FC(i)1274 3535 y Fv(\014)1274 3584 y(\014)1274 3634 y(\014)1325 3630 y Fx(\024)22 b Fz(l)16 b FG(max)1515 3682 y FC(j)1607 3535 y Fv(\014)1607 3584 y(\014)1607 3634 y(\014)1645 3574 y Fz(a)1689 3586 y FC(j)1742 3574 y Fx(\000)i Fz(b)1861 3586 y FC(j)p 1645 3611 251 4 v 1735 3687 a Fz(b)1771 3699 y FC(j)1906 3535 y Fv(\014)1906 3584 y(\014)1906 3634 y(\014)1990 3526 y FC(l)1948 3551 y Fv(Y)1947 3728 y FC(i)p FD(=1)2069 3630 y Fz(c)2105 3642 y FC(i)2326 3630 y Fu(with)26 b Fk(c)2534 3638 y FH(i)2582 3630 y Fu(=)21 b(max)o(\()p Fi(j)p Fk(a)2898 3638 y FH(i)2924 3630 y Fi(j)p Fk(;)13 b Fi(j)p Fk(b)3033 3638 y FH(i)3060 3630 y Fi(j)p Fu(\),)456 3893 y FG(the)28 b(uniform)g(b)r(ounds)g(\(3.33\))f(and)g(the)i(observ) -5 b(ation)26 b(that)i Fz(K)6 b(O)r FG(\()p Fz(e)2579 3862 y Fy(\000)p FC(c)p FD(\()p FC(n)2728 3837 y Fq(")2760 3862 y FD(\))2786 3837 y Fq(\020)2824 3893 y FG(\))24 b(=)f Fz(O)r FG(\()p Fz(e)3104 3862 y Fy(\000)p FC(c)3186 3837 y Fl(0)3208 3862 y FC(n)3249 3837 y Fq(\020)3279 3821 y Fl(0)3310 3893 y FG(\))28 b(as)456 3992 y Fz(n)22 b Fx(!)i(1)p FG(.)555 4092 y(No)n(w)36 b(the)g(target)f(b)r(ound)h (\(3.18\))f(follo)n(ws)g(directly)g(from)h(\(3.34\))f(and)h(a)f(simple) h(obser-)456 4191 y(v)-5 b(ation)36 b(that)g(the)h(ratio)e FE(B)1312 4161 y FC(\013)q(;")1312 4212 y(n)1417 4191 y FG(\()p Fz(z)t FG(\))p Fz(=)p FE(B)1614 4161 y FC(\013)q(;")1614 4212 y(n)1719 4191 y FG(\()p Fz(t)p FG(\))i(is)f(uniformly)g(separated) f(from)h(zero)f(for)h(an)n(y)f Fz(z)456 4291 y FG(satisfying)27 b(\(3.19\).)555 4391 y(Finally)-7 b(,)31 b(w)n(e)f(pro)n(v)n(e)f(the)h (b)r(ounds)h(\(3.33\).)44 b(Since)30 b(the)h(inequalities)f(in)h (\(3.33\))e(are)g(simple)456 4490 y(term-b)n(y-term)j(estimates)i(com)n (bined)f(with)h(the)g(apriori)f(b)r(ounds)g(of)h(the)g(t)n(yp)r(e)g (\(3.35\),)h(w)n(e)456 4590 y(shall)25 b(concen)n(trate)f(ourselv)n(es) f(on)j(the)f(pro)r(of)g(of)h(the)g(equalities)f(only)-7 b(.)35 b(The)26 b(latter,)g(ho)n(w)n(ev)n(er,)456 4690 y(ha)n(v)n(e)g(a)i(natural)f(probabilistic)f(in)n(terpretation.)37 b(Namely)-7 b(,)28 b(in)g(terms)f(of)h(the)g(\(tilted\))h(bridge)456 4789 y(and)e(irreducible)g(bridge)g(distributions)937 5002 y FE(P)1000 4967 y FC(t)989 5022 y(m)1050 5002 y FG(\()p Fx(\001)p FG(\))1161 4955 y Fr(def)1170 5002 y FG(=)1277 4940 y FE(B)1326 4952 y FC(m)1395 4873 y Fv(\000)1433 4940 y Fx(\001)p FG(;)14 b Fz(e)1532 4910 y FC(tH)t FD(\()p FC(!)r FD(\))1716 4873 y Fv(\001)p 1277 4983 477 4 v 1307 5063 a FE(B)1356 5075 y FC(m)1425 4996 y Fv(\000)1463 5063 y Fz(e)1502 5039 y FC(tH)t FD(\()p FC(!)r FD(\))1686 4996 y Fv(\001)1764 5002 y Fz(;)346 b FE(Q)2203 4967 y FC(t)2191 5022 y(m)2260 5002 y FG(\()p Fx(\001)p FG(\))2371 4955 y Fr(def)2380 5002 y FG(=)2486 4940 y FE(F)2531 4952 y FC(m)2600 4873 y Fv(\000)2638 4940 y Fx(\001)p FG(;)14 b Fz(e)2737 4910 y FC(tH)t FD(\()p FC(\015)t FD(\))2915 4873 y Fv(\001)p 2486 4983 467 4 v 2516 5063 a FE(F)2561 5075 y FC(m)2630 4996 y Fv(\000)2668 5063 y Fz(e)2707 5039 y FC(tH)t FD(\()p FC(\015)t FD(\))2885 4996 y Fv(\001)456 5002 y FG(\(3.36\))456 5216 y(the)28 b(equalities)f(in)h(\(3.33\))e(follo)n(w)h(directly)h(from)f(the)h (prop)r(erties:)p eop %%Page: 30 30 30 29 bop 456 226 a FD(30)814 b(OST)-5 b(AP)28 b(HR)-5 b(YNIV)29 b(AND)g(DMITR)-5 b(Y)29 b(IOFFE)661 425 y Fn(Fix)f Fz(")23 b(>)f FG(0)28 b Fn(small)f(enough;)g(then,)h(for)f(some)g Fz(c)c(>)g FG(0)p Fn(,)k Fz(\021)f(>)d FG(0)p Fn(,)k(the)h(estimates) 958 639 y FE(P)1021 605 y FC(t)1010 660 y(m)1085 497 y Fv( )1241 565 y Fx(j)p Fz(H)7 b FG(\()p Fz(!)s FG(\))p Fx(j)23 b(\024)g Fz(r)1630 577 y FD(3)1667 565 y Fz(m;)97 b Fx(j)p Fz(a)p FG(\()p Fz(!)s FG(\))p Fx(j)23 b(\024)g Fz(r)2217 577 y FD(3)2255 565 y Fz(m)2328 535 y FD(3)p FC(=)p FD(2+)p FC(")1193 685 y Fy(8)1234 715 y Fz(\015)1323 654 y FD(e)1305 715 y Fx(\032)g Fz(!)s(;)41 b Fk(\015)t Fj(-irreducible)49 b FG(=)-14 b Fx(\))22 b Fz(W)12 b FG(\()p Fz(\015)5 b FG(\))24 b Fz(<)e(m)2527 685 y FC(")2604 497 y Fv(!)2693 639 y Fx(\025)h FG(1)17 b Fx(\000)i Fz(e)2963 605 y Fy(\000)p FC(cm)3104 580 y Fq(\020)3141 639 y Fz(;)1199 909 y FE(Q)1269 875 y FC(t)1258 930 y(m)1327 817 y Fv(\020)1377 909 y Fx(j)p Fz(H)7 b FG(\()p Fz(\015)e FG(\))p Fx(j)23 b(\024)f Fz(r)1758 921 y FD(2)1796 909 y Fz(m;)14 b Fx(j)p Fz(a)p FG(\()p Fz(\015)5 b FG(\))p Fx(j)23 b(\024)g Fz(r)2256 921 y FD(2)2293 909 y Fz(m)2366 875 y FD(2)2403 817 y Fv(\021)2476 909 y Fx(\025)g FG(1)p Fx(\000)o Fz(e)2709 875 y Fy(\000)p FC(cm)2850 850 y Fq(\020)456 748 y FG(\(3.33)637 718 y Fy(0)659 748 y FG(\))661 1076 y Fn(hold,)28 b(as)f Fz(m)c Fx(!)g(1)p Fn(,)28 b(uniformly)f(in)h Fz(t)23 b Fx(2)g FG([)p Fz(a;)14 b(b)p FG(])p Fn(.)555 1197 y FG(T)-7 b(o)31 b(v)n(erify)f(the)h(\014rst)g(b)r(ound)h(ab)r(o)n(v)n (e,)e(it)h(is)g(enough)g(to)f(c)n(hec)n(k)h(that)g(for)f(an)n(y)g Fz(")f(>)f FG(0)j(there)456 1296 y(are)26 b Fz(c)d(>)g FG(0)k(and)g Fz(\020)j(>)23 b FG(0)k(suc)n(h)g(that)h(the)g (inequalities)1021 1457 y FE(P)1084 1423 y FC(t)1073 1478 y(m)1135 1390 y Fv(\000)1187 1423 y Fy(9)1228 1457 y Fz(\015)1317 1396 y FD(e)1299 1457 y Fx(\032)23 b Fz(!)s(;)41 b Fk(\015)t Fu(-irreducible)26 b(&)f Fz(W)12 b FG(\()p Fz(\015)5 b FG(\))23 b Fx(\025)g Fz(m)2427 1423 y FC(")2476 1390 y Fv(\001)2537 1457 y Fx(\024)g Fz(e)2664 1423 y Fy(\000)p FC(cm)2805 1398 y Fq(\020)2842 1457 y Fz(;)1409 1610 y FE(P)1472 1576 y FC(t)1461 1631 y(m)1523 1543 y Fv(\000)1575 1610 y Fx(j)p Fz(H)7 b FG(\()p Fz(!)s FG(\))p Fx(j)23 b Fz(>)g(r)1964 1622 y FD(3)2001 1610 y Fz(m)2088 1543 y Fv(\001)2149 1610 y Fx(\024)g Fz(e)2276 1576 y Fy(\000)p FC(cm)2417 1551 y Fq(\020)2454 1610 y Fz(;)1064 1763 y FE(P)1127 1729 y FC(t)1116 1784 y(m)1177 1696 y Fv(\000)1229 1763 y Fx(j)p Fz(a)p FG(\()p Fz(!)s FG(\))p Fx(j)g Fz(>)g(r)1586 1775 y FD(3)1624 1763 y Fz(m)1697 1729 y FD(3)p FC(=)p FD(2+)p FC(")1906 1693 y Fv(\014)1906 1743 y(\014)1957 1763 y Fx(j)p Fz(H)7 b FG(\()p Fz(!)s FG(\))p Fx(j)23 b(\024)g Fz(r)2346 1775 y FD(3)2384 1763 y Fz(m)2471 1696 y Fv(\001)2532 1763 y Fx(\024)f Fz(e)2658 1729 y Fy(\000)p FC(cm)2799 1704 y Fq(\020)456 1907 y FG(hold,)g(uniformly)f(in)h Fz(t)h Fx(2)g FG([)p Fz(a;)14 b(b)p FG(],)22 b(for)f(all)g Fz(m)g FG(large)f(enough.)34 b(Ho)n(w)n(ev)n(er,)20 b(the)i(\014rst)f(one)g (is)g(a)g(direct)456 2007 y(corollary)26 b(of)i(\(1.24\),)g(the)h (second)f(one)g(is)g(ob)n(vious)f(pro)n(vided)h Fz(r)2491 2019 y FD(3)2553 2007 y Fz(>)c FG(0)k(is)g(large)g(enough,)g(and)456 2107 y(the)g(last)g(b)r(ound)h(follo)n(ws)e(directly)h(from)g(the)h (strict)f(triangle)g(inequalit)n(y)g(\(1.8\))g(for)g Fz(m)g FG(large)456 2206 y(enough.)555 2306 y(T)-7 b(o)30 b(v)n(erify)f(the)i(second)e(b)r(ound)i(in)f(\(3.33)1868 2276 y Fy(0)1891 2306 y FG(\),)h(w)n(e)e(observ)n(e)g(\014rst)h(that)g (the)h(irreducible)e(t)n(w)n(o)456 2406 y(p)r(oin)n(t)e(function)i Fz(f)1039 2418 y FC(\014)1111 2406 y FG(eviden)n(tly)e(satis\014es:) 1033 2558 y Fz(f)1074 2570 y FC(\014)1119 2558 y FG(\()p Fz(m;)14 b(H)7 b FG(\))23 b Fz(<)g(g)1520 2570 y FC(\014)1564 2558 y FG(\()p Fz(m;)14 b(H)7 b FG(\))23 b(=)g Fz(g)1965 2570 y FC(\014)2009 2558 y FG(\()p Fz(H)r(;)14 b(m)p FG(\))24 b Fz(<)f FG(e)2403 2523 y Fy(\000)p FC(H)t(\034)2544 2532 y Fq(\014)2583 2523 y FD(\(1)p FC(;m=H)t FD(\))2843 2558 y Fz(:)456 2699 y FG(Since,)k Fz(\034)731 2711 y FC(\014)777 2699 y FG(\(1)p Fz(;)14 b Fx(\001)p FG(\))27 b(is)h(con)n(tin)n(uous)e(at)i(zero,)e(the)i(large)e(deviation)i (estimate)1441 2844 y FE(Q)1511 2810 y FC(t)1500 2865 y(m)1569 2777 y Fv(\000)1621 2844 y Fx(j)p Fz(H)7 b FG(\()p Fz(\015)e FG(\))p Fx(j)23 b Fz(>)f(r)2002 2856 y FD(2)2040 2844 y Fz(m)2127 2777 y Fv(\001)2188 2844 y Fx(\024)h Fz(e)2315 2810 y Fy(\000)p FC(cm)456 2986 y FG(holds)g(uniformly)g(in)h Fz(t)f Fx(2)h FG([)p Fz(a;)14 b(b)p FG(])22 b Fx(\032)h(D)1605 2998 y FC(\014)1673 2986 y FG(for)g(an)n(y)g Fz(r)1986 2998 y FD(2)2048 2986 y FG(satisfying)g(the)h(follo)n(wing)e(t)n(w)n(o) h(conditions:)526 3169 y Fz(\016)589 3122 y Fr(def)598 3169 y FG(=)31 b Fz(\034)730 3181 y FC(\014)776 3169 y FG(\(1)p Fz(;)14 b FG(1)p Fz(=r)1008 3181 y FD(2)1044 3169 y FG(\))19 b Fx(\000)f FG(max)13 b Fx(fj)p Fz(a)p Fx(j)p Fz(;)h Fx(j)p Fz(b)p Fx(jg)22 b Fz(>)g FG(0)83 b(and)g Fz(\016)21 b Fx(\001)e Fz(r)2227 3181 y FD(2)2288 3169 y Fz(>)j Fx(\000)14 b FG(lim)g(inf)2539 3219 y FC(m)2725 3169 y FG(min)2698 3227 y FC(t)p Fy(2)p FD([)p FC(a;b)p FD(])2929 3113 y FG(1)p 2914 3150 73 4 v 2914 3226 a Fz(m)3010 3169 y FG(log)g FE(F)3188 3181 y FC(m)3257 3169 y FG(\()p Fz(t)p FG(\))p Fz(:)456 3362 y FG(Similarly)-7 b(,)26 b(for)f(an)n(y)h(\014xed)g Fz(r)r FG(,)h(the)f(strict)g (triangle)g(inequalit)n(y)f(\(1.8\))h(implies)g(that)h(the)f(upp)r(er) 456 3462 y(b)r(ound)1161 3604 y FE(Q)1231 3569 y FC(t)1220 3624 y(m)1289 3536 y Fv(\000)1341 3604 y Fx(j)p Fz(a)p FG(\()p Fz(\015)5 b FG(\))p Fx(j)23 b Fz(>)g(r)1691 3616 y FD(2)1728 3604 y Fz(m)1801 3569 y FD(2)1862 3533 y Fv(\014)1862 3583 y(\014)1912 3604 y Fx(j)p Fz(H)7 b FG(\()p Fz(\015)e FG(\))p Fx(j)23 b(\024)g Fz(r)r(m)2383 3536 y Fv(\001)2445 3604 y Fx(\024)f Fz(e)2571 3569 y Fy(\000)p FC(cm)2716 3604 y Fz(;)456 3745 y FG(also)34 b(holds)i(uniformly)f(in)h Fz(t)h Fx(2)g FG([)p Fz(a;)14 b(b)p FG(])36 b Fx(\032)g(D)1870 3757 y FC(\014)1951 3745 y FG(and)g(for)f(an)n(y)g Fz(r)2458 3757 y FD(2)2531 3745 y FG(large)g(enough.)60 b(Th)n(us,)38 b(the)456 3845 y(second)20 b(of)h(the)h(inequalities)f(in)g(\(3.33)1643 3815 y Fy(0)1666 3845 y FG(\))g(follo)n(ws)f(once)h Fz(r)2205 3857 y FD(2)2264 3845 y FG(has)g(b)r(een)g(c)n(hosen)g(to)g(b)r(e)g (su\016cien)n(tly)456 3945 y(large.)p 3384 3945 4 57 v 3388 3892 50 4 v 3388 3945 V 3437 3945 4 57 v 456 4108 a(3.5.)46 b FO(Pro)s(of)35 b(of)g(Lemma)e(3.7.)40 b FG(F)-7 b(rom)30 b(the)g(p)r(oin)n(t)h(of)f(view)g(of)h(the)f(tilted)i (probabilit)n(y)d(dis-)456 4208 y(tribution)h FE(P)861 4168 y Fr(reg)861 4229 y FC(t;n)950 4208 y FG(\()14 b Fx(\001)g FG(\))31 b(de\014ned)g(in)f(\(3.20\),)h(Lemma)f(3.6)g (implies)g(the)h(\(uniform\))g(cen)n(tral)f(limit)456 4318 y(asymptotics)d(for)h(the)h(prop)r(erly)e(rescaled)h(ideal)g(area) f Fz(A)2279 4288 y FC(I)2317 4318 y FG(.)40 b(In)28 b(particular,)g (the)g(mean)h(v)-5 b(alue)456 4418 y(satis\014es)1145 4598 y FE(E)1195 4558 y Fr(reg)1195 4618 y FC(t;n)1291 4531 y Fv(\000)1329 4598 y Fz(A)1391 4564 y FC(I)1429 4598 y Fz(=n)1521 4531 y Fv(\001)1581 4598 y FG(=)23 b Fz(n)1748 4542 y(d)p 1729 4579 83 4 v 1729 4655 a(ds)1835 4485 y Fv(Z)1918 4505 y FD(1)1881 4674 y(0)1969 4598 y Fz(m)2042 4610 y FC(\014)2087 4531 y Fv(\000)2125 4598 y FG(\(1)18 b Fx(\000)g Fz(\030)t FG(\))p Fz(s)2411 4531 y Fv(\001)2463 4598 y Fz(d\030)2569 4478 y Fv(\014)2569 4527 y(\014)2569 4577 y(\014)2569 4627 y(\014)2597 4681 y FC(s)p FD(=)p FC(t)2732 4598 y Fz(:)456 4794 y FG(T)-7 b(aking)37 b(no)n(w)h Fz(t)k FG(=)f Fz(t)1135 4806 y FC(n)1218 4794 y FG(to)e(b)r(e)g(the)g(conjugate)f(v)-5 b(alue)38 b(to)h Fz(q)2371 4806 y FC(n)2455 4794 y FG(in)g(\(1.27\))e (and)i(recalling)e(the)456 4893 y(self-a)n(v)n(eraging)30 b(prop)r(ert)n(y)k(\(3.16\))f(of)h(the)h(area)d(correction)h(\001\()p Fz(!)s FG(\))h Fx(\021)g Fz(A)2787 4863 y FC(R)2842 4893 y FG(\()p Fz(!)s FG(\))23 b Fx(\000)f Fz(A)3133 4863 y FC(I)3172 4893 y FG(\()p Fz(!)s FG(\),)36 b(w)n(e)456 4993 y(obtain)1004 5173 y FE(E)1053 5133 y Fr(re)q(g)1053 5194 y FC(t)1078 5202 y Fq(n)1125 5194 y FC(;n)1190 5106 y Fv(\000)1228 5173 y Fz(A)1290 5139 y FC(R)1345 5173 y Fz(=n)1437 5106 y Fv(\001)1497 5173 y FG(=)23 b Fz(n)1649 5060 y Fv(Z)1731 5081 y FD(1)1695 5249 y(0)1769 5173 y FG(\(1)18 b Fx(\000)g Fz(\030)t FG(\))p Fz(m)2089 5139 y Fy(0)2089 5194 y FC(\014)2134 5106 y Fv(\000)2172 5173 y FG(\(1)g Fx(\000)g Fz(\030)t FG(\))p Fz(t)2449 5185 y FC(n)2495 5106 y Fv(\001)2547 5173 y Fz(d\030)27 b FG(=)c Fz(nq)2828 5185 y FC(n)2873 5173 y Fz(:)p eop %%Page: 31 31 31 30 bop 1488 226 a FD(SELF-A)-7 b(V)n(OIDING)29 b(POL)-5 b(YGONS)966 b(31)456 425 y FG(In)30 b(other)f(w)n(ords,)g Fz(t)1075 437 y FC(n)1151 425 y FG(is)h(the)g(prop)r(er)f(tilt)i(in)f (the)h(study)f(the)g(asymptotics)f(of)h(the)h(probabil-)456 525 y(it)n(y)c FE(P)632 495 y Fr(reg)632 545 y FC(n)715 458 y Fv(\000)753 525 y Fz(A)815 495 y FC(R)893 525 y FG(=)c Fz(n)1031 495 y FD(2)1068 525 y Fz(q)1105 537 y FC(n)1150 458 y Fv(\001)1188 525 y FG(.)555 624 y(Let)28 b Fz(\037)756 636 y FC(n)802 624 y FG(\()p Fz(\034)9 b FG(\))29 b(b)r(e)f(the)h Fz(t)1227 636 y FC(n)1272 624 y FG(-tilted)f(c)n(haracteristic)e(function)j(of)f(the)g(cen)n (tred)g(normalised)e(real)456 724 y(area,)974 894 y Fz(\037)1026 906 y FC(n)1071 894 y FG(\()p Fz(\034)9 b FG(\))1204 847 y Fr(def)1213 894 y FG(=)32 b FE(E)1359 854 y Fr(re)q(g)1359 915 y FC(t)1384 923 y Fq(n)1431 915 y FC(;n)1510 894 y FG(exp)1636 802 y Fv(n)1742 838 y Fz(i\034)p 1702 875 155 4 v 1702 953 a(n)1752 929 y FD(3)p FC(=)p FD(2)1866 827 y Fv(\000)1904 894 y Fz(A)1966 860 y FC(R)2039 894 y Fx(\000)18 b Fz(n)2172 860 y FD(2)2209 894 y Fz(q)2246 906 y FC(n)2291 827 y Fv(\001)2330 802 y(o)1218 1172 y FG(=)1316 1108 y FE(B)1365 1078 y Fr(reg)1365 1129 y FC(n;\013;")1530 1041 y Fv(\000)1568 1108 y Fz(e)1607 1078 y FC(i\034)7 b FD(\001)p FC(=n)1798 1053 y Fs(3)p Fq(=)p Fs(2)1893 1108 y FG(;)14 b Fz(e)1969 1078 y FD(\()p FC(t)2020 1086 y Fq(n)2060 1078 y FD(+)p FC(i\034)7 b(=)2206 1036 y Fy(p)p 2260 1036 42 3 v 2260 1078 a FC(n)p FD(\))p FC(A)2377 1053 y Fq(I)2411 1078 y FC(=n)2490 1041 y Fv(\001)p 1316 1153 1213 4 v 1643 1234 a FE(B)1692 1194 y Fr(re)q(g)1692 1243 y FC(n;\013;")1857 1166 y Fv(\000)1895 1234 y Fz(e)1934 1210 y FC(t)1959 1218 y Fq(n)2000 1210 y FC(A)2050 1193 y Fq(I)2083 1210 y FC(=n)2162 1166 y Fv(\001)2552 1172 y FG(exp)2678 1105 y Fv(\010)2727 1172 y Fx(\000)p Fz(i\034)2866 1108 y Fx(p)p 2935 1108 50 4 v 64 x Fz(nq)3022 1184 y FC(n)3067 1105 y Fv(\011)3115 1172 y Fz(:)456 1044 y FG(\(3.37\))456 1380 y(The)36 b(function)h Fz(\037)1021 1392 y FC(n)1067 1380 y FG(\()p Fz(\034)9 b FG(\))38 b(b)r(eing)e(2)p Fz(\031)s(n)1588 1349 y FD(3)p FC(=)p FD(2)1692 1380 y FG(-p)r(erio)r(dic,)j(the)e(in)n(v)n(ersion)e(form)n (ula)g(for)h(the)h(F)-7 b(ourier)456 1479 y(transform)26 b(implies)1119 1687 y FE(P)1171 1647 y Fr(reg)1171 1708 y FC(t)1196 1716 y Fq(n)1235 1708 y FC(;n)1300 1620 y Fv(\000)1338 1687 y Fz(A)1400 1653 y FC(R)1478 1687 y FG(=)d Fz(n)1616 1653 y FD(2)1653 1687 y Fz(q)1690 1699 y FC(n)1735 1620 y Fv(\001)1796 1687 y FG(=)1996 1631 y(1)p 1894 1668 247 4 v 1894 1746 a(2)p Fz(\031)s(n)2036 1722 y FD(3)p FC(=)p FD(2)2164 1574 y Fv(Z)2247 1595 y FC(\031)r(n)2329 1570 y Fs(3)p Fq(=)p Fs(2)2210 1763 y Fy(\000)p FC(\031)r(n)2344 1746 y Fs(3)p Fq(=)p Fs(2)2453 1687 y Fz(\037)2505 1699 y FC(n)2550 1687 y FG(\()p Fz(\034)9 b FG(\))14 b Fz(d\034)5 b(:)456 1875 y FG(Th)n(us,)29 b(the)h(lo)r(cal)f(limit)h(result)g(\(3.20\))e(is)i(an)f(immediate)h (corollary)d(of)i(the)h(follo)n(wing)f(three)456 1975 y(facts.)456 2115 y FO(Claim)g(3.10.)40 b Fn(Fix)28 b(an)n(y)f Fz(A)c(>)g FG(0)p Fn(.)36 b(Then)28 b(the)g(con)n(v)n(ergence)1457 2284 y FG(log)14 b Fz(\037)1630 2296 y FC(n)1675 2284 y FG(\()p Fz(\034)9 b FG(\))20 b(+)1897 2228 y(1)p 1897 2265 42 4 v 1897 2341 a(2)1948 2284 y Fz(\033)s FG(\()p Fz(t)2060 2296 y FC(n)2106 2284 y FG(\))p Fz(\034)2183 2249 y FD(2)2245 2284 y Fx(\000)-15 b(!)23 b FG(0)-1987 b(\(3.38\))456 2446 y Fn(holds,)27 b(as)g Fz(n)c Fx(!)g(1)p Fn(,)28 b(uniformly)f(in)h Fx(j)p Fz(\034)9 b Fx(j)24 b(\024)e Fz(A)p Fn(.)456 2586 y FO(Claim)29 b(3.11.)40 b Fn(There)27 b(exist)h(p)r(ositiv)n(e)f(constan)n(ts)g Fz(\016)s Fn(,)g Fz(\021)s Fn(,)h Fz(C)6 b Fn(,)28 b Fz(c)p Fn(,)g(and)f Fz(\020)34 b Fn(suc)n(h)28 b(that)g(the)g(b)r(ound) 1345 2663 y Fv(\014)1345 2713 y(\014)1372 2733 y Fz(\037)1424 2745 y FC(n)1470 2733 y FG(\()p Fz(\034)9 b FG(\))1579 2663 y Fv(\014)1579 2713 y(\014)1631 2733 y Fx(\024)22 b FG(2)14 b(exp)1900 2666 y Fv(\010)1949 2733 y Fx(\000)p Fz(\021)s(\034)2103 2699 y FD(2)2141 2666 y Fv(\011)2207 2733 y FG(+)k Fz(C)6 b(e)2394 2699 y Fy(\000)p FC(cn)2517 2674 y Fq(\020)456 2733 y FG(\(3.39\))456 2875 y Fn(is)27 b(true)h(uniformly)f(in)h Fx(j)p Fz(\034)9 b Fx(j)24 b(\024)e Fz(\016)1432 2815 y Fx(p)p 1502 2815 50 4 v 1502 2875 a Fz(n)27 b Fn(and)h(all)f Fz(n)g Fn(large)g(enough.)456 3014 y FO(Claim)i(3.12.)40 b Fn(Let)33 b Fz(\016)i(>)c FG(0)h Fn(b)r(e)h(as)f(\014xed)h(ab)r(o)n(v)n(e.)51 b(Then)32 b(there)h(exist)g(p)r(ositiv)n(e)f(constan)n(ts)f Fz(c)456 3114 y Fn(and)c Fz(\020)34 b Fn(suc)n(h)27 b(that)h(the)g(inequalit)n (y)1557 3178 y Fv(\014)1557 3228 y(\014)1585 3249 y Fz(\037)1637 3261 y FC(n)1682 3249 y FG(\()p Fz(\034)9 b FG(\))1791 3178 y Fv(\014)1791 3228 y(\014)1843 3249 y Fx(\024)23 b FG(exp)2057 3181 y Fv(\010)2106 3249 y Fx(\000)p Fz(cn)2257 3214 y FC(\020)2294 3181 y Fv(\011)456 3249 y FG(\(3.40\))456 3397 y Fn(is)k(satis\014ed)g(uniformly)h(in)g Fz(\016)1374 3337 y Fx(p)p 1443 3337 V 60 x Fz(n)23 b Fx(\024)f(j)p Fz(\034)9 b Fx(j)24 b(\024)f Fz(\031)s(n)1906 3367 y FD(3)p FC(=)p FD(2)2038 3397 y Fn(and)k(all)h Fz(n)f Fn(large)f(enough.)555 3537 y FG(The)j(rest)f(of)h(this)f(section)h(is) f(dev)n(oted)g(to)h(the)g(pro)r(of)f(of)g(these)h(facts.)39 b(The)29 b(cen)n(tral)f(limit)456 3637 y(theorem)g(con)n(v)n(ergence)f (\(3.38\))h(and)g(the)i(Gaussian)e(upp)r(er)h(b)r(ound)g(\(3.39\))f(in) i(the)f(region)f(of)456 3736 y(small)33 b Fz(\034)9 b FG(,)36 b Fx(j)p Fz(\034)9 b Fx(j)34 b(\024)f Fz(\016)1046 3676 y Fx(p)p 1115 3676 V 60 x Fz(n)p 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Fv(\000)1680 5217 y Fz(e)1719 5183 y Fy(\000)p FC(cn)1842 5158 y Fq(\020)1879 5150 y Fv(\001)1936 5217 y FG(+)2019 5147 y Fv(\014)2019 5196 y(\014)2046 5217 y Fz(\037)2098 5183 y FC(I)2098 5238 y(n)2143 5217 y FG(\()p Fz(\034)9 b FG(\))2252 5147 y Fv(\014)2252 5196 y(\014)2281 5217 y Fz(O)2346 5150 y Fv(\000)2385 5217 y Fz(n)2435 5183 y Fy(\000)p FC(\021)2527 5150 y Fv(\001)p eop %%Page: 32 32 32 31 bop 456 226 a FD(32)814 b(OST)-5 b(AP)28 b(HR)-5 b(YNIV)29 b(AND)g(DMITR)-5 b(Y)29 b(IOFFE)456 425 y FG(uniformly)e(in)h Fx(j)p Fz(\034)9 b Fx(j)24 b Fz(<)e(\016)1173 365 y Fx(p)p 1243 365 50 4 v 1243 425 a Fz(n)o FG(.)555 525 y(No)n(w,)30 b(let)g Fz(\034)40 b FG(satisfy)29 b Fx(j)p Fz(\034)9 b Fx(j)28 b(\024)e Fz(A)k FG(with)g Fz(A)g FG(\014xed)g(ab)r(o)n(v)n (e.)42 b(Then)30 b(a)f(direct)h(calculation)f(based)456 624 y(on)e(the)h(asymptotics)f(\(3.18\))g(sho)n(ws)f(that)578 857 y Fz(\037)630 823 y FC(I)630 878 y(n)676 857 y FG(\()p Fz(\034)9 b FG(\))24 b Fx(\021)906 793 y FE(B)956 763 y 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(teger\))f(partition)h Fx(D)e FG(=)2403 4097 y Fv(\010)2451 4164 y Fz(d)2494 4176 y FD(1)2532 4164 y Fz(;)14 b(d)2612 4176 y FD(2)2649 4164 y Fz(;)g(:)g(:)g(:)g(;)g(d)2877 4176 y FC(N)2940 4097 y Fv(\011)3016 4164 y Fn(of)29 b(a)e(natural)456 4264 y(n)n(um)n(b)r(er)g Fz(n)g Fn(is)h(called)f (regular,)f(if)546 4392 y FG(1\))744 4362 y Fy(8)786 4392 y Fz(j)i FG(=)23 b(1)p Fz(;)14 b FG(2)p Fz(;)g(:)g(:)g(:)e(;)i(N)t (;)180 b(d)1557 4404 y FC(j)1615 4392 y Fz(<)23 b(n)1753 4362 y FC(")1788 4392 y Fn(,)546 4578 y FG(2\))744 4548 y Fy(8)786 4578 y Fz(k)j FG(=)d(1)p Fz(;)14 b FG(2)p Fz(;)g(:)g(:)g(:)e(;)i(n)1297 4548 y FD(1)p Fy(\000)p FC(")1417 4578 y Fn(,)1468 4548 y Fy(9)1510 4578 y Fz(j)32 b Fn(suc)n(h)c(that)g Fz(d)1987 4590 y FC(j)2045 4578 y Fx(\025)22 b FG(3)28 b Fn(and)2365 4470 y FC(j)s Fy(\000)p FD(1)2363 4499 y Fv(X)2370 4678 y FC(l)p FD(=1)2497 4578 y Fz(d)2540 4590 y FC(l)2589 4578 y Fx(2)2667 4511 y Fv(\002)2702 4578 y FG(\()p Fz(k)21 b Fx(\000)d FG(1\))p Fz(n)3005 4544 y FC(")3040 4578 y Fz(;)c(k)s(n)3173 4544 y FC(")3208 4511 y Fv(\001)3246 4578 y Fn(.)555 4814 y FG(Note)36 b(that)f(for)g(an)n(y)g(regular)f(partition)h Fx(D)j FG(=)e Fx(D)2162 4826 y FC(n)2242 4814 y FG(of)g(an)f(in)n (teger)f(n)n(um)n(b)r(er)i Fz(n)f FG(and)g(an)n(y)456 4914 y Fx(D)r FG(-compatible)26 b(bridge)f Fz(!)h Fx(2)d(B)1439 4884 y Fr(reg)1436 4934 y FC(n;\013;")1595 4914 y FG(,)k(eac)n(h)e (segmen)n(t)2150 4847 y Fv(\002)2184 4914 y FG(\()p Fz(k)19 b Fx(\000)d FG(1\))p Fz(n)2483 4884 y FC(")2518 4914 y Fz(;)e(k)s(n)2651 4884 y FC(")2686 4847 y Fv(\001)2750 4914 y FG(in)n(tersects)26 b(a)g(span)g(of)456 5016 y(an)h(irreducible) g(comp)r(onen)n(t)g Fz(\015)1449 5020 y Fl(\017)1512 5016 y FG(of)g(width)h Fz(W)12 b FG(\()p Fz(\015)2006 5020 y Fl(\017)2041 5016 y FG(\))24 b Fx(\025)e FG(3.)555 5116 y(Due)34 b(to)f(the)h(massgap)d(condition)j(\(1.21\),)g(the)f (function)h Fx(D)r FG(\()p Fz(!)s FG(\))g(though)n(t)f(as)g(a)f(random) 456 5216 y(v)-5 b(ariable)26 b(in)i Fz(!)e Fx(2)d(B)1078 5185 y Fr(reg)1075 5236 y FC(n;\013;")1262 5216 y FG(has)k(the)h(follo) n(wing)e(imp)r(ortan)n(t)h(prop)r(ert)n(y:)p eop %%Page: 33 33 33 32 bop 1488 226 a FD(SELF-A)-7 b(V)n(OIDING)29 b(POL)-5 b(YGONS)966 b(33)661 425 y Fn(Fix)29 b(an)n(y)f Fz(")d(>)g FG(0)j Fn(small)h(enough.)40 b(Then)29 b(there)f(exist)h(p)r(ositiv)n (e)f(constan)n(ts)g Fz(c)h Fn(and)661 525 y Fz(\020)34 b Fn(suc)n(h)28 b(that)g(the)g(estimate)1336 689 y FE(P)1388 650 y Fr(reg)1388 710 y FC(t;n)1477 622 y Fv(\000)1528 689 y Fx(D)r FG(\()p Fz(!)s FG(\))h Fj(is)d(not)f(regular)2216 622 y Fv(\001)2277 689 y Fz(<)e(e)2404 655 y Fy(\000)p FC(cn)2527 630 y Fq(\020)456 689 y FG(\(3.42\))661 844 y Fn(holds,)28 b(as)f Fz(n)22 b Fx(!)i(1)p Fn(,)j(uniformly)h(in)g Fz(t)23 b Fx(2)g FG([)p Fz(a;)14 b(b)p FG(])p Fn(.)555 969 y FG(Let)27 b FE(B)752 939 y Fr(reg)752 990 y FC(n;\013;")917 902 y Fv(\000)955 969 y Fx(\001)1001 899 y Fv(\014)1001 949 y(\014)1052 969 y Fx(D)1118 902 y Fv(\001)1183 969 y FG(denote)f(the)g(restriction)g(of)g(the)g(bridge)g(partition)f (function)i FE(B)3192 939 y Fr(reg)3192 990 y FC(n;\013;")3357 969 y FG(\()p Fx(\001)p FG(\))456 1072 y(to)d(the)i(ensem)n(ble)f(of)f Fx(D)r FG(-compatible)h(bridges.)35 b(Thanks)25 b(to)g(the)g(uniform)g (b)r(ound)g(\(3.42\),)g(the)456 1171 y(target)h(inequalit)n(y)h (\(3.40\))g(follo)n(ws)g(directly)g(from)g(the)h(follo)n(wing)f(fact.) 456 1331 y FO(Lemma)i(3.14.)40 b Fn(Supp)r(ose)28 b(that)g Fz(")f Fn(satis\014es)g FG(0)c Fz(<)g FG(12)p Fz(")f(<)g FG(1)28 b Fn(and)f(let)h FG([)p Fz(a;)14 b(b)p FG(])23 b Fx(\032)g(D)2984 1343 y FC(\014)3056 1331 y Fn(b)r(e)28 b(as)f(\014xed)456 1430 y(ab)r(o)n(v)n(e.)35 b(Then)28 b(there)f(exist)h(p)r(ositiv)n(e)f(constan)n(ts)g Fz(c)g Fn(and)h Fz(\020)34 b Fn(suc)n(h)27 b(that)h(the)g(estimate)918 1507 y Fv(\014)918 1557 y(\014)918 1606 y(\014)946 1602 y FE(B)995 1572 y Fr(re)q(g)995 1623 y FC(n;\013;")1174 1510 y Fv(\020)1224 1602 y FG(exp)1364 1510 y Fv(n)1420 1540 y(P)1507 1561 y FC(N)1507 1627 y FD(1)1584 1602 y Fz(t)1614 1614 y FC(k)1655 1602 y Fz(H)7 b FG(\()p Fz(\015)1806 1614 y FC(k)1847 1602 y FG(\))18 b(+)2026 1570 y FC(i\034)p 1990 1584 133 4 v 1990 1634 a(n)2031 1618 y Fs(3)p Fq(=)p Fs(2)2133 1602 y Fz(A)2195 1572 y FC(R)2249 1602 y FG(\()p Fz(!)s FG(\))2368 1510 y Fv(o)2447 1532 y(\014)2447 1582 y(\014)2498 1602 y Fx(D)2564 1510 y Fv(\021)2614 1507 y(\014)2614 1557 y(\014)2614 1606 y(\014)p 918 1670 1724 4 v 1190 1775 a FE(B)1240 1735 y Fr(reg)1240 1785 y FC(n;\013;")1419 1683 y Fv(\020)1468 1775 y FG(exp)1609 1683 y Fv(n)1664 1713 y(P)1752 1733 y FC(N)1752 1800 y FD(1)1829 1775 y Fz(t)1859 1787 y FC(k)1900 1775 y Fz(H)7 b FG(\()p Fz(\015)2051 1787 y FC(k)2092 1775 y FG(\))2124 1683 y Fv(o)2202 1704 y(\014)2202 1754 y(\014)2253 1775 y Fx(D)2319 1683 y Fv(\021)2674 1689 y Fx(\024)23 b Fz(e)2801 1654 y Fy(\000)p FC(c)2883 1662 y Fs(1)2915 1654 y FC(n)2956 1629 y Fq(")456 1689 y FG(\(3.43\))456 1963 y Fn(is)40 b(true,)k(for)d(all)f Fz(n)h Fn(large)e(enough,)k(uniformly)e(in)g Fz(n)2220 1933 y FD(1)p FC(=)p FD(2)p Fy(\000)p FC(")2453 1963 y Fx(\024)j(j)p Fz(\034)9 b Fx(j)46 b(\024)f Fz(\031)s(n)2909 1933 y FD(3)p FC(=)p FD(2)3013 1963 y Fn(,)f(in)d(regular)456 2063 y(partitions)27 b Fx(D)e(\021)e(D)1076 2075 y FC(n)1148 2063 y Fn(of)28 b Fz(n)p Fn(,)g(and)f(in)h(all)f(collections)g(of)g (tilts)i Fz(t)2415 2075 y FD(1)2452 2063 y Fn(,)f Fz(:)14 b(:)g(:)27 b Fn(,)h Fz(t)2708 2075 y FC(N)2794 2063 y Fx(2)23 b FG([)p Fz(a;)14 b(b)p FG(])p Fn(.)555 2222 y FG(The)20 b(rest)g(of)g(this)g(section)g(is)f(dev)n(oted)h(to)g(the)g (pro)r(of)g(of)f(this)i(lemma.)34 b(It)20 b(will)h(b)r(e)f(p)r (erformed)456 2322 y(in)27 b(sev)n(eral)f(steps,)i(the)g(k)n(ey)f(idea) g(b)r(eing)h(as)f(follo)n(ws.)555 2421 y(Let)h Fz(!)j FG(b)r(e)d(an)n(y)f Fx(D)r FG(-compatible)g(bridge.)37 b(Using)28 b(the)g(microscopic)e(decomp)r(osition)h(of)h(the)456 2521 y(real)e(area,)821 2749 y Fz(A)883 2715 y FC(R)938 2749 y FG(\()p Fz(!)s FG(\))d(=)1198 2645 y FC(N)1168 2670 y Fv(X)1211 2846 y FD(1)1301 2749 y Fz(n)1351 2761 y FC(j)1386 2749 y Fz(H)7 b FG(\()p Fz(\015)1537 2761 y FC(j)1572 2749 y FG(\))19 b(+)1737 2645 y FC(N)1706 2670 y Fv(X)1749 2846 y FD(1)1840 2749 y Fz(a)p FG(\()p Fz(\015)1959 2761 y FC(j)1994 2749 y FG(\))p Fz(;)208 b Fu(with)26 b Fk(n)2478 2757 y FH(j)2532 2749 y Fu(=)21 b Fk(d)2653 2757 y FH(j)2685 2749 y Fk(=)p Fu(2)d(+)2885 2656 y FH(N)2856 2678 y Ff(X)2857 2837 y FH(k)q FK(=)p FH(j)2980 2749 y Fk(d)3020 2758 y FH(k)3058 2749 y Fu(,)456 2978 y FG(w)n(e)27 b(rewrite)g(the)h(LHS)g(of)f(\(3.43\))g(as)1244 3107 y FC(N)1220 3132 y Fv(Y)1257 3308 y FD(1)1340 3211 y FE(E)1389 3166 y FC(t)1414 3174 y Fq(j)1389 3236 y FC(d)1424 3244 y Fq(j)1465 3119 y Fv(\020)1515 3211 y FG(exp)1642 3119 y Fv(n)1747 3155 y Fz(i\034)p 1707 3192 155 4 v 1707 3269 a(n)1757 3245 y FD(3)p FC(=)p FD(2)1871 3143 y Fv(\000)1909 3211 y Fz(n)1959 3223 y FC(j)1994 3211 y Fz(H)7 b FG(\()p Fz(\015)2145 3223 y FC(j)2180 3211 y FG(\))19 b(+)f Fz(a)p FG(\()p Fz(\015)2433 3223 y FC(j)2468 3211 y FG(\))2500 3143 y Fv(\001)2538 3119 y(o\021)2643 3211 y Fz(;)-2210 b FG(\(3.44\))456 3442 y(where)29 b FE(E)747 3412 y FC(t)747 3466 y(d)792 3442 y FG(\()p Fx(\001)p FG(\))h(denotes)g(the)g(exp)r(ectation)f(corresp)r (onding)f(to)h(the)h(tilted)h(irreducible)e(distri-)456 3542 y(bution)f FE(Q)777 3512 y FC(t)777 3565 y(d)822 3542 y FG(,)g(cf.)g(\(3.36\).)555 3641 y(Let)i Fz(")d(>)f FG(0)k(b)r(e)g(as)f(\014xed)h(ab)r(o)n(v)n(e.)42 b(W)-7 b(e)30 b(claim)g(that)g(the)g(follo)n(wing)f(three)h(prop)r(erties)f (hold)456 3741 y(true)e(uniformly)h(in)f(3)c Fx(\024)g Fz(d)g Fx(\024)g Fz(n)1463 3711 y FC(")1498 3741 y FG(.)661 3867 y Fn(a\))29 b(There)g(exists)f(a)h(p)r(ositiv)n(e)f(constan)n(t)g Fz(c)1987 3879 y FD(1)2053 3867 y Fn(suc)n(h)h(that)g(uniformly)g(in)g Fx(j)p Fz(s)p Fx(j)d(\024)e Fz(n)3151 3837 y Fy(\000)p FC(")661 3966 y Fn(and)k Fz(t)23 b Fx(2)g FG([)p Fz(a;)14 b(b)p FG(])p Fn(,)1494 4055 y Fv(\014)1494 4104 y(\014)1522 4125 y FE(E)1571 4091 y FC(t)1571 4146 y(d)1616 4058 y Fv(\000)1654 4125 y Fz(e)1693 4091 y FC(isH)t FD(\()p FC(\015)t FD(\))1900 4058 y Fv(\001)1939 4055 y(\014)1939 4104 y(\014)1989 4125 y Fx(\024)23 b FG(1)18 b Fx(\000)g Fz(c)2256 4137 y FD(1)2293 4125 y Fz(s)2332 4091 y FD(2)2369 4125 y FG(;)-1936 b(\(3.45\))661 4285 y Fn(b\))27 b(There)e(exists)h(a) f(p)r(ositiv)n(e)h(constan)n(t)f Fz(c)1974 4297 y FD(2)2037 4285 y Fn(suc)n(h)h(that)g(uniformly)f(in)i Fx(j)p Fz(s)p Fx(j)c(\024)f Fz(n)3118 4255 y Fy(\000)p FD(2)p FC(")661 4384 y Fn(and)28 b Fx(j)p Fz(h)p Fx(j)23 b(\024)f Fz(r)1064 4396 y FD(2)1102 4384 y Fz(n)1152 4354 y FC(")1187 4384 y Fn(,)1264 4543 y FE(E)1313 4564 y FC(d)1358 4476 y Fv(\000)1396 4543 y Fz(e)1435 4509 y FC(isa)p FD(\()p FC(\015)t FD(\))1643 4473 y Fv(\014)1643 4523 y(\014)1694 4543 y Fz(H)7 b FG(\()p Fz(\015)e FG(\))23 b(=)f Fz(h)2040 4476 y Fv(\001)2101 4543 y Fx(\025)h FG(1)18 b Fx(\000)g Fz(c)2368 4555 y FD(2)2405 4543 y Fz(s)2444 4509 y FD(2)2481 4543 y Fz(n)2531 4509 y FD(4)p FC(")2599 4543 y FG(;)-2166 b(\(3.46\))661 4697 y Fn(c\))24 b(There)f(exists)g(a)g(p)r(ositiv)n(e)g (constan)n(t)f Fz(c)1949 4709 y FD(3)2010 4697 y Fn(suc)n(h)h(that)h (uniformly)f(in)g Fx(j)p Fz(s)p Fx(j)h(\024)e Fz(\031)27 b Fn(and)661 4797 y Fx(j)p Fz(h)p Fx(j)c(\024)g Fz(r)903 4809 y FD(2)941 4797 y Fz(n)991 4767 y FC(")1026 4797 y Fn(,)1366 4964 y FE(E)1415 4985 y FC(d)1460 4897 y Fv(\000)1498 4964 y Fz(e)1537 4930 y FC(isa)p FD(\()p FC(\015)t FD(\))1745 4894 y Fv(\014)1745 4944 y(\014)1796 4964 y Fz(H)7 b FG(\()p Fz(\015)e FG(\))23 b(=)f Fz(h)2142 4897 y Fv(\001)2203 4964 y Fx(\024)h Fz(e)2330 4930 y Fy(\000)p FC(c)2412 4938 y Fs(3)2443 4930 y FC(s)2474 4905 y Fs(2)2511 4964 y Fz(:)-2078 b FG(\(3.47\))555 5116 y(P)n(ostp)r(oning)30 b(the)i(pro)r(of)f(of)h(these)g(prop)r (erties)e(for)h(a)h(while,)g(w)n(e)g(deduce)f(\014rst)h(the)g(claim)456 5216 y(\(3.43\))26 b(of)i(the)g(lemma.)p eop %%Page: 34 34 34 33 bop 456 226 a FD(34)814 b(OST)-5 b(AP)28 b(HR)-5 b(YNIV)29 b(AND)g(DMITR)-5 b(Y)29 b(IOFFE)555 425 y FG(T)-7 b(o)25 b(b)r(egin)h(with,)h(supp)r(ose)e(that)h Fz(n)1646 395 y FD(1+)p FC(")1788 425 y Fz(<)d Fx(j)p Fz(\034)9 b Fx(j)24 b(\024)f Fz(\031)s(n)2179 395 y FD(3)p FC(=)p FD(2)2283 425 y FG(.)36 b(Then)26 b(the)g(absolute)f(v)-5 b(alue)26 b(of)f(the)456 525 y(pro)r(duct)i(\(3.44\))g(is)g(b)r(ounded) i(ab)r(o)n(v)n(e,)d(thanks)h(to)h(estimate)f(\(3.47\),)g(b)n(y)880 656 y FC(N)847 681 y Fv(O)893 857 y FD(1)986 760 y FE(Q)1045 715 y FC(t)1070 723 y Fq(j)1045 785 y FC(d)1080 793 y Fq(j)1121 693 y Fv(\000)1159 760 y FG(max)1220 812 y FC(j)1327 760 y Fx(j)p Fz(H)7 b FG(\()p Fz(\015)1501 772 y FC(j)1536 760 y FG(\))p Fx(j)23 b Fz(>)g(r)1739 772 y FD(2)1777 760 y Fz(n)1827 726 y FC(")1862 693 y Fv(\001)1184 1001 y FG(+)1267 909 y Fv(\020)1375 1001 y FG(max)1330 1055 y FD(3)p Fy(\024)p FC(d)e(r)3063 1863 y FD(2)3101 1851 y Fz(n)3151 1817 y FC(")3186 1784 y Fv(\001)1291 2103 y FG(+)74 b(max)1374 2161 y Fy(j)p FC(h)p Fy(j\024)p FC(r)1536 2169 y Fs(2)1567 2161 y FC(n)1608 2144 y Fq(")1654 2103 y FE(E)1704 2124 y FC(d)1739 2132 y Fq(j)1779 2011 y Fv(\020)1829 2103 y FG(1)18 b Fx(\000)g FG(exp)2099 2011 y Fv(n)2204 2047 y Fz(i\034)p 2164 2084 V 2164 2162 a(n)2214 2138 y FD(3)p FC(=)p FD(2)2328 2103 y Fz(a)p FG(\()p Fz(\015)2447 2115 y FC(j)2482 2103 y FG(\))2514 2011 y Fv(o)2593 2033 y(\014)2593 2083 y(\014)2644 2103 y Fz(H)7 b FG(\()p Fz(\015)2795 2115 y FC(j)2830 2103 y FG(\))23 b(=)g Fz(h)3021 2011 y Fv(\021)3070 2103 y Fz(:)456 1856 y FG(\(3.48\))456 2304 y(No)n(w,)k(in)h(eac)n(h)e(in)n (terv)-5 b(al)1189 2457 y Fz(n)1239 2426 y FD(3)p FC(=)p FD(2)p 1081 2494 372 4 v 1081 2570 a FG(\()p Fz(k)21 b FG(+)d(1\))p Fz(n)1384 2546 y FD(2)p FC(")1485 2513 y Fz(<)23 b Fx(j)p Fz(\034)9 b Fx(j)24 b(\024)1791 2457 y Fz(n)1841 2426 y FD(3)p FC(=)p FD(2)p 1786 2494 165 4 v 1786 2570 a Fz(k)s(n)1882 2546 y FD(2)p FC(")1960 2513 y Fz(;)180 b(k)26 b FG(=)c Fz(n)2369 2478 y FD(3)p FC(")2438 2513 y Fz(;)14 b(:)g(:)g(:)f(;)h(n)2672 2478 y 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Fx(2)d(B)1281 4074 y Fr(lo)r(c)1267 4127 y FC(n;n)1369 4111 y Fq(\027)1405 4127 y FC(;k)1492 4104 y FG(whose)i(irreducible)h (sub-bridges)e(ha)n(v)n(e)h(spans)h(shorter)e(than)456 4209 y Fz(n)506 4179 y FC(")541 4209 y FG(.)37 b(W)-7 b(e)28 b(claim)f(that)h(uniformly)g(in)f Fz(m)c FG(=)g(1)p Fz(;)14 b(:::;)g(n)2039 4179 y FC(")2102 4209 y FG(and)27 b Fz(t)2293 4221 y FD(1)2330 4209 y Fz(;)14 b(t)2397 4221 y FD(2)2457 4209 y Fx(2)24 b FG([)p Fz(a;)14 b(b)p FG(],)894 4370 y FE(F)939 4336 y Fr(lo)r(c)939 4391 y FC(m;n)1059 4374 y Fq(\027)1101 4391 y FC(;k)1161 4278 y Fv(\020)1211 4370 y Fz(e)1260 4305 y Fq(t)1283 4317 y Fs(1)p 1260 4323 56 3 v 1269 4356 a Fq(n)1325 4336 y FC(A)1375 4311 y Fq(R)1422 4336 y FD(+)p FC(t)1498 4344 y Fs(2)1531 4336 y FC(H)1594 4370 y FG(;)g Fx(j)p Fz(H)7 b FG(\()p Fz(\015)e FG(\))p Fx(j)23 b Fz(>)f(r)2012 4382 y FD(2)2050 4370 y Fz(n)2100 4336 y FC(")2163 4370 y Fu(or)k Fx(j)p Fz(a)p FG(\()p Fz(\015)5 b FG(\))p Fx(j)23 b Fz(>)g(r)2607 4382 y FD(2)2645 4370 y Fz(n)2695 4336 y FD(2)p FC(")2763 4278 y Fv(\021)1888 4546 y Fx(\024)f Fz(e)2014 4512 y Fy(\000)p FC(c)2096 4520 y Fs(18)2156 4512 y FC(n)2197 4486 y Fq(")2234 4546 y FE(F)2278 4512 y Fr(l)q(o)r(c)2278 4566 y FC(m;n)2398 4550 y Fq(\027)2441 4566 y FC(;k)2501 4454 y Fv(\020)2551 4546 y Fz(e)2600 4480 y Fq(t)2623 4492 y Fs(1)p 2599 4498 V 2609 4532 a Fq(n)2665 4512 y FC(A)2715 4486 y Fq(R)2762 4512 y FD(+)p FC(t)2838 4520 y Fs(2)2871 4512 y FC(H)2934 4454 y Fv(\021)2983 4546 y Fz(;)456 4702 y FG(as)33 b(so)r(on)f(as)h Fz(r)906 4714 y FD(2)978 4702 y FG(is)g(c)n(hosen)g(to)g(b)r(e)h (su\016cien)n(tly)g(large.)53 b(Indeed,)35 b(assuming)e(that)h Fz(")f FG(is)h(small)456 4802 y(enough)28 b(to)g(satisfy)g Fz(")19 b FG(+)g Fz(\027)30 b(<)25 b FG(1,)j(w)n(e)h(obtain)f Fz(A)1946 4772 y FC(R)2001 4802 y FG(\()p Fz(\015)5 b FG(\))p Fz(=n)24 b FG(=)g FA(o)c FG(\(1\))29 b(uniformly)f(in)h Fz(\015)h Fx(2)25 b(F)3224 4772 y Fr(lo)r(c)3205 4825 y FC(m;n)3325 4809 y Fq(\027)3361 4825 y FC(;k)3421 4802 y FG(.)456 4916 y(Since)687 4883 y FD(1)p 674 4897 59 4 v 674 4944 a FC(m)757 4916 y FG(log)14 b FE(F)922 4885 y Fr(l)q(o)r(c)922 4939 y FC(m;n)1042 4922 y Fq(\027)1085 4939 y FC(;k)1145 4916 y FG(\()p Fz(t)p FG(\))20 b(is)f(b)r(ounded)h (ab)r(o)n(v)n(e)d(and)i(b)r(elo)n(w)g(uniformly)g(in)g Fz(m)g FG(and)g(in)h Fz(t)j Fx(2)g FG([)p Fz(a;)14 b(b)p FG(])23 b Fx(\032)456 5016 y(D)520 5028 y FC(\014)564 5016 y FG(,)35 b(the)e(last)g(estimate)g(follo)n(ws)f(from)g(the)i (H\177)-42 b(older)32 b(inequalit)n(y)h(and)f(the)i(strict)f(p)r (ositivit)n(y)456 5116 y(of)38 b(the)g(connectivit)n(y)g(deca)n(y)f (rates)h Fz(\034)1685 5128 y FC(\014)1730 5116 y FG(,)j(once)c(the)i (constan)n(t)e Fz(r)2529 5128 y FD(2)2605 5116 y FG(has)h(b)r(een)h(c)n (hosen)e(to)h(b)r(e)456 5216 y(su\016cien)n(tly)27 b(large.)p eop %%Page: 42 42 42 41 bop 456 226 a FD(42)814 b(OST)-5 b(AP)28 b(HR)-5 b(YNIV)29 b(AND)g(DMITR)-5 b(Y)29 b(IOFFE)555 425 y FG(Going)c(bac)n(k) f(to)g(the)i(splitting)f(\(3.57\))f(and,)i(accordingly)-7 b(,)23 b(adjusting)i(the)h(analysis)d(of)i(the)456 525 y(decomp)r(osition)i(form)n(ula)f(\(3.59\))h(with)h(resp)r(ect)g(to)f (the)h(ev)n(en)n(t)1331 675 y Fx(fj)p Fz(H)7 b FG(\()p Fz(\015)e FG(\))p Fx(j)22 b Fz(>)h(r)1754 687 y FD(2)1792 675 y Fz(n)1842 641 y FC(")1905 675 y Fu(or)j Fx(j)p Fz(a)p FG(\()p Fz(\015)5 b FG(\))p Fx(j)23 b Fz(>)g(r)2349 687 y FD(2)2386 675 y Fz(n)2436 641 y FD(2)p FC(")2505 675 y Fx(g)p Fz(;)456 824 y FG(w)n(e)k(conclude)g(that)765 978 y FE(B)815 943 y Fr(lo)r(c)832 998 y FC(n;n)934 981 y Fq(\027)970 998 y FC(;k)1031 885 y Fv(\020)1081 978 y Fz(e)1136 921 y Fq(t)p 1129 930 37 3 v 1129 963 a(n)1176 943 y FC(A)1226 918 y Fq(R)1277 978 y FG(;)1327 943 y Fy(9)1369 978 y Fz(\015)1456 916 y FC(e)1440 978 y Fx(\032)22 b Fz(!)31 b Fu(irreducible)26 b(with)g Fx(j)p Fz(H)7 b FG(\()p Fz(\015)e FG(\))p Fx(j)23 b Fz(>)g(r)2547 990 y FD(2)2584 978 y Fz(n)2634 943 y FC(")2697 978 y Fu(or)j Fx(j)p Fz(a)p FG(\()p 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Fy(\000)p FC(\013)1515 2282 y FG(and)34 b(the)g(bridges)f Fz(!)2177 2294 y FC(j)2246 2282 y FG(in)h(the)g(decomp)r(osition)g (\(3.8\).)55 b(The)456 2382 y(length)27 b(of)f(the)h(span)g(of)g Fz(!)1284 2394 y FC(j)1345 2382 y FG(is)g(at)g(most)f Fz(n)1781 2352 y FC(\013)1828 2382 y FG(.)37 b(Because)26 b(of)h(\(3.62\))f(and)g(the)h(lo)r(calit)n(y)g(prop)r(ert)n(y)456 2481 y(\(3.52\))f(it)i(remains)f(to)h(sho)n(w)e(that)i(uniformly)g(in)g Fz(m)23 b Fx(\024)f Fz(n)2284 2451 y FC(\013)2359 2481 y FG(and)27 b Fz(t)c Fx(2)h FG([)p Fz(a;)14 b(b)p FG(])22 b Fx(\032)h(D)2989 2493 y FC(\014)3034 2481 y FG(,)647 2658 y FE(B)696 2624 y Fr(l)q(o)r(c)714 2679 y FC(m;n)834 2662 y Fq(\027)870 2679 y FC(k)925 2566 y Fv(\020)974 2658 y Fz(e)1013 2624 y FC(tH)t FD(\()p FC(!)r FD(\))1197 2658 y FG(;)14 b Fx(j)p Fz(H)7 b FG(\()p Fz(!)s FG(\))p Fx(j)23 b Fz(>)f(r)1622 2670 y FD(3)1660 2658 y Fz(n)1710 2624 y FC(\013)1785 2658 y Fu(or)k Fx(j)p Fz(a)p FG(\()p Fz(!)s FG(\))p Fx(j)d Fz(>)g(r)2236 2670 y FD(3)2274 2658 y Fz(n)2324 2624 y FD(3)p FC(=)p FD(2)p FC(\013)p FD(+)p FC(")2553 2566 y Fv(\021)2271 2834 y Fx(\024)g Fz(e)2398 2800 y Fy(\000)p FC(c)2480 2808 y Fs(19)2539 2800 y FC(n)2580 2775 y Fq(")2617 2834 y FE(B)2666 2800 y Fr(l)q(o)r(c)2683 2855 y FC(m;n)2803 2838 y Fq(\027)2840 2855 y FC(k)2894 2742 y Fv(\020)2944 2834 y Fz(e)2983 2800 y FC(tH)t FD(\()p FC(!)r FD(\))3167 2742 y Fv(\021)3230 2834 y Fz(:)456 3003 y FG(Ho)n(w)n(ev)n(er,)i(as)i(in)g(the)h (preceding)f(step,)g(this)h(already)e(follo)n(ws)g(in)i(a)f(standard)f (w)n(a)n(y)g(from)h(the)456 3103 y(H\177)-42 b(older)31 b(inequalit)n(y)h(and)g(the)g(strict)g(triangle)f(inequalit)n(y)h (\(1.8\),)h(as)e(so)r(on)g(as)h(the)g(constan)n(t)456 3203 y Fz(r)493 3215 y FD(3)558 3203 y FG(is)27 b(large)g(enough.)36 b(The)28 b(pro)r(of)f(of)g(Lemma)g(3.17)g(is)g(completed.)p 3384 3203 4 57 v 3388 3150 50 4 v 3388 3203 V 3437 3203 4 57 v 970 3414 a(4.)46 b FF(Asymptotics)31 b(of)h(corner)f(p)-6 b(ar)g(tition)33 b(functions)555 3563 y FG(The)h(pro)r(of)f(of)g(Prop)r (osition)f(1.7)g(closely)h(follo)n(ws)f(the)i(line)f(of)h(reasoning)d (dev)n(elop)r(ed)i(in)456 3663 y(Sect.)k(3.)66 b(As)38 b(in)f(the)h(pro)r(of)f(of)h(Theorem)e(3.1)h(w)n(e)g(emplo)n(y)g(the)g Fz(K)45 b FG(=)39 b Fz(n)2877 3633 y FD(1)p Fy(\000)p FC(\013)3047 3663 y FG(mesoscopic)456 3763 y(splitting)d(of)g(a)f(path) h Fz(!)j Fx(2)e(A)1423 3775 y FC(n)1505 3763 y FG(with)f(resp)r(ect)g (to)f(the)h(irreducible)g(animals)f Fz(\015)3031 3775 y FD(1)3068 3763 y Fz(;)14 b(:)g(:)g(:)g(;)g(\015)3296 3775 y FC(K)t Fy(\000)p FD(1)456 3862 y FG(whic)n(h)31 b(happ)r(en)h(to)g(sit)g(on)f(the)h(p)r(oles)g Fx(P)1759 3874 y FC(j)s(n)1830 3858 y Fq(\013)1877 3862 y FG(.)49 b(Exactly)31 b(as)g(it)h(has)f(b)r(een)h(done)g(in)g(the)g(pro)r(of)456 3962 y(of)d(\(3.60\))f(it)h(is)g(p)r(ossible)g(to)g(rule)f(out)h(the)h (case)e(of)h(max)2267 3974 y FC(j)2316 3962 y Fz(W)12 b FG(\()p Fz(\015)2481 3974 y FC(j)2516 3962 y FG(\))26 b Fz(>)f(n)2714 3932 y FC(")2750 3962 y FG(,)k(and)g(consider)f(only) 456 4061 y(those)f(paths)g Fz(!)f Fx(2)d(A)1121 4073 y FC(n)1195 4061 y FG(whic)n(h)k(admit)h(the)g(disjoin)n(t)g (mesoscopic)e(decomp)r(osition)1280 4211 y Fz(!)53 b FG(=)23 b Fz(!)1525 4223 y FD(1)1580 4211 y Fx(_)c Fz(\015)1697 4223 y FD(1)1753 4211 y Fx(_)g Fz(!)1879 4223 y FD(2)1934 4211 y Fx(_)g(\001)14 b(\001)g(\001)k(_)h Fz(\015)2240 4223 y FC(K)t Fy(\000)p FD(1)2407 4211 y Fx(_)g Fz(!)2533 4223 y FC(K)2597 4211 y Fz(:)-2164 b FG(\(4.1\))555 4362 y(The)33 b(decomp)r(osition)g(\(4.1\))f(di\013ers)h(from)g(\(3.8\))g (in)g(t)n(w)n(o)f(resp)r(ects:)47 b(\014rst)33 b(of)g(all,)h Fz(!)3244 4374 y FD(1)3314 4362 y FG(is)f(a)456 4462 y(corner)c(path;)k(in)f(addition,)g(all)f(the)g(bridges)f Fz(\015)1989 4474 y FD(1)2027 4462 y Fz(;)14 b(:)g(:)g(:)f(;)h(!)2263 4474 y FC(K)2358 4462 y FG(are)30 b(sub)5 b(ject)32 b(to)f(the)g (global)f(con-)456 4561 y(strain)n(t)e(to)h(sta)n(y)g(ab)r(o)n(v)n(e)e (the)j(horizon)n(tal)e(semi-axis)g Fx(L)2194 4526 y Fy(\000)2194 4584 y FD(0)2250 4561 y FG(.)42 b(The)29 b(latter)g(constrain)n(t,)g (ho)n(w)n(ev)n(er,)456 4662 y(has)c(little)i(impact)f(on)f(the)i (asymptotic)e(prop)r(erties)g(of)h FE(A)2301 4674 y FC(n)2352 4595 y Fv(\000)2390 4662 y Fz(n)2440 4632 y FD(2)2477 4662 y Fz(q)2514 4674 y FC(n)2560 4595 y Fv(\001)2624 4662 y FG(whenev)n(er)f(the)h(sequence)456 4762 y(of)h(rescaled)f (areas)g Fx(f)p Fz(q)1156 4774 y FC(n)1201 4762 y Fx(g)h FG(is)h(b)r(ounded)g(a)n(w)n(a)n(y)d(from)j(zero.)555 4862 y(Indeed,)35 b(using)e Fx(A)1144 4832 y Fr(reg)1144 4882 y FC(n;\013;")1336 4862 y FG(to)g(denote)g(the)h(ensem)n(ble)e(of) h(regular)f(corner)f(paths)i(satisfying)456 4964 y(\(R1\)-\(R3\))28 b(of)f(De\014nition)i(3.2)e(and)g FE(A)1676 4934 y Fr(re)q(g)1676 4985 y FC(n;\013;")1869 4964 y FG(to)h(denote)g(the)g(corresp)r(onding) e(partition)h(func-)456 5064 y(tion,)g(w)n(e)h(follo)n(w)e(literally)h (the)h(pro)r(of)f(of)h(Prop)r(osition)e(3.4)h(to)g(deduce:)1060 5216 y FE(A)1125 5228 y FC(n)1176 5148 y Fv(\000)1214 5216 y Fz(n)1264 5181 y FD(2)1301 5216 y Fz(q)1338 5228 y FC(n)1384 5148 y Fv(\001)1472 5216 y FG(=)51 b FE(A)1642 5181 y Fr(reg)1642 5236 y FC(n;\013;")1807 5148 y Fv(\000)1845 5216 y Fz(n)1895 5181 y FD(2)1932 5216 y Fz(q)1969 5228 y FC(n)2014 5148 y Fv(\001)2098 5216 y FG(+)46 b Fz(o)2249 5148 y Fv(\000)2287 5216 y Fz(e)2326 5181 y Fy(\000)p FC(n )r FD(\()p FC(q)2521 5189 y Fq(n)2562 5181 y FD(\))p Fy(\000)p FC(c)2670 5189 y Fs(1)2702 5181 y FC(n)2743 5156 y Fq(")2779 5148 y Fv(\001)2817 5216 y Fz(:)p eop %%Page: 43 43 43 42 bop 1488 226 a FD(SELF-A)-7 b(V)n(OIDING)29 b(POL)-5 b(YGONS)966 b(43)456 425 y FG(Next,)28 b(tilting)g(the)g(area)e Fz(A)1316 395 y FC(R)1398 425 y FG(b)n(y)h(the)h(conjugate)f(v)-5 b(alue)28 b Fz(t)23 b FG(=)g Fz(t)p FG(\()p Fz(q)2487 437 y FC(n)2532 425 y FG(\),)28 b(w)n(e)f(obtain)h(\(cf.)g(\(3.13\)\):) 1130 588 y FE(A)1184 554 y Fr(reg)1184 609 y FC(n;\013;")1349 521 y Fv(\000)1387 588 y Fz(n)1437 554 y FD(2)1474 588 y Fz(q)1511 600 y FC(n)1556 521 y Fv(\001)1645 588 y FG(=)50 b Fz(e)1799 554 y Fy(\000)p FC(ntq)1947 562 y Fq(n)1992 588 y FE(A)2046 554 y Fr(re)q(g)2046 609 y FC(n;\013;")2211 521 y Fv(\000)2249 588 y Fz(e)2288 554 y FC(tA)2363 529 y Fq(R)2410 554 y FC(=n)2489 588 y FG(;)28 b Fz(n)2590 554 y FD(2)2627 588 y Fz(q)2664 600 y FC(n)2709 521 y Fv(\001)2747 588 y Fz(:)456 734 y FG(F)-7 b(urther,)25 b(using)g(an)h(argumen)n(t)e(similar)h(to)g(the)h(one)f(emplo)n(y)n(ed) f(in)i(the)g(pro)r(of)f(of)g(Lemma)g(3.5)456 833 y(w)n(e)i(get)g(the)h (relation)925 992 y FE(A)979 957 y Fr(re)q(g)979 1012 y FC(n;\013;")1144 924 y Fv(\000)1182 992 y Fz(e)1221 957 y FC(tA)1296 932 y Fq(R)1343 957 y FC(=n)1422 992 y FG(;)g Fz(n)1523 957 y FD(2)1560 992 y Fz(q)1597 1004 y FC(n)1642 924 y Fv(\001)1731 992 y FG(=)50 b FE(A)1900 957 y Fr(r)q(eg)1900 1012 y FC(n;\013;")2066 924 y Fv(\000)2104 992 y Fz(e)2143 957 y FC(tA)2218 932 y Fq(I)2251 957 y FC(=n)2330 992 y FG(;)28 b Fz(n)2431 957 y FD(2)2468 992 y Fz(q)2505 1004 y FC(n)2550 924 y Fv(\001\000)2626 992 y FG(1)18 b(+)g FA(o)i FG(\(1\))2937 924 y Fv(\001)456 1137 y FG(and)27 b(factorise)g(the)g(partition)h(function)g(in)g(the)g (RHS)g(as)1031 1296 y FE(A)1085 1261 y Fr(reg)1085 1316 y FC(n;\013;")1250 1228 y Fv(\000)1288 1296 y Fz(e)1327 1261 y FC(tA)1402 1236 y Fq(I)1435 1261 y FC(=n)1514 1296 y FG(;)g Fz(n)1615 1261 y FD(2)1652 1296 y Fz(q)1689 1308 y FC(n)1734 1228 y Fv(\001)1823 1296 y FG(=)50 b FE(A)1992 1261 y Fr(re)q(g)1992 1316 y FC(n;\013;")2158 1228 y Fv(\000)2196 1296 y Fz(e)2235 1261 y FC(tA)2310 1236 y Fq(I)2343 1261 y FC(=n)2422 1228 y Fv(\001)2460 1296 y FE(P)2512 1256 y Fr(reg)2512 1316 y FC(t;n)2601 1228 y Fv(\000)2639 1296 y Fz(n)2689 1261 y FD(2)2726 1296 y Fz(q)2763 1308 y FC(n)2808 1228 y Fv(\001)2846 1296 y Fz(:)-2413 b FG(\(4.2\))456 1441 y(Th)n(us,)32 b(as)f(in)h(the)f(case)g(of)h(bridge)e(partition)i(functions,)g(w)n(e)g (ha)n(v)n(e)e(split)i(the)g(problem)f(in)n(to)456 1549 y(t)n(w)n(o:)58 b(\014nding)38 b(the)h(sharp)f(asymptotics)g(of)g FE(A)1976 1519 y Fr(re)q(g)1976 1570 y FC(n;\013;")2141 1482 y Fv(\000)2180 1549 y Fz(e)2219 1519 y FC(tA)2294 1494 y Fq(I)2327 1519 y FC(=n)2406 1482 y Fv(\001)2482 1549 y FG(and)h(deriving)f(a)g(lo)r(cal)g(limit)456 1655 y(result)27 b(for)g(the)h(tilted)g(probabilit)n(y)f(measure)f FE(P)1978 1615 y Fr(reg)1978 1676 y FC(t;n)2067 1655 y FG(\()p Fx(\001)p FG(\))j(in)e(the)h(ensem)n(ble)g(of)f(corner)f (paths.)456 1814 y FO(Remark)k(4.0.1.)40 b FG(If)26 b(the)g(sequence)f (of)h(rescaled)e(areas)g Fx(f)p Fz(q)2342 1826 y FC(n)2387 1814 y Fx(g)i FG(is)f(b)r(ounded)h(a)n(w)n(a)n(y)e(from)i(zero,)456 1913 y(then)32 b(so)g(is)g(the)h(sequence)f(of)g(conjugate)g(tilts)h Fx(f)p Fz(t)2069 1925 y FC(n)2145 1913 y FG(=)d Fz(t)p FG(\()p Fz(q)2339 1925 y FC(n)2385 1913 y FG(\))p Fx(g)p FG(.)51 b(Th)n(us,)33 b(since)f(the)h(a)n(v)n(eraged)456 2013 y(v)-5 b(alue)40 b(of)g(the)h(heigh)n(t)f Fz(H)7 b FG(\()p Fz(!)1370 2025 y FD(1)1407 2013 y FG(\))41 b(is)f(of)g(the)h(order)e Fz(n)2119 1983 y FC(\013)2166 2013 y FG(,)44 b(w)n(e)c(infer)g(from)g(the)h(strict)f(triangle)456 2113 y(inequalit)n(y)25 b(\(1.8\))g(that)h(those)f(remaining)g(parts)f (of)i(the)g(tra)5 b(jectories)24 b Fz(\015)2736 2125 y FD(1)2787 2113 y Fx(_)15 b(\001)f(\001)g(\001)g(_)h Fz(!)3090 2125 y FC(K)3179 2113 y FG(in)26 b(\(4.1\))456 2213 y(whic)n(h)f(violate)f(the)h(global)f(constrain)n(t)g(to)h(sta)n (y)g(ab)r(o)n(v)n(e)e(the)j(semi-axis)e Fx(L)2787 2178 y Fy(\000)2787 2235 y FD(0)2868 2213 y FG(giv)n(e)g(a)h(negligible)456 2313 y(con)n(tribution)i(to)g(either)h(of)f(the)h(terms)f(on)h(the)g (righ)n(t)f(hand)g(side)h(of)f(\(4.2\).)555 2482 y(As)22 b(far)f(as)f(the)i FE(A)1079 2451 y Fr(re)q(g)1079 2502 y FC(n;\013;")1244 2414 y Fv(\000)1282 2482 y Fz(e)1321 2451 y FC(tA)1396 2426 y Fq(I)1429 2451 y FC(=n)1508 2414 y Fv(\001)1568 2482 y FG(term)f(is)h(considered,)f(let)h(us)f (remo)n(v)n(e)f(the)i(aforemen)n(tioned)456 2599 y(constrain)n(t)k(and) h(denote)h(the)g(mo)r(di\014ed)g(partition)f(function)2438 2577 y(~)2428 2599 y FE(A)2482 2568 y Fr(re)q(g)2482 2619 y FC(n;\013;")2647 2599 y FG(.)37 b(By)28 b(\(1.8\),)1008 2772 y FE(A)1062 2738 y Fr(reg)1062 2793 y FC(n;\013;")1227 2705 y Fv(\000)1265 2772 y Fz(e)1304 2738 y FC(tA)1379 2713 y Fq(I)1412 2738 y FC(=n)1491 2705 y Fv(\001)1580 2772 y FG(=)1705 2751 y(~)1695 2772 y FE(A)1749 2738 y Fr(re)q(g)1749 2793 y FC(n;\013;")1914 2705 y Fv(\000)1953 2772 y Fz(e)1992 2738 y FC(tA)2067 2713 y Fq(I)2100 2738 y FC(=n)2179 2705 y Fv(\001\000)2255 2772 y FG(1)18 b(+)g Fz(o)2438 2705 y Fv(\000)2476 2772 y Fz(e)2515 2738 y Fy(\000)p FC(c)2597 2746 y Fs(2)2629 2738 y FD(\()p FC(t)p FD(\))p FC(n)2747 2713 y Fq(\013)2793 2705 y Fv(\001\001)2869 2772 y Fz(;)456 2942 y FG(for)36 b(ev)n(ery)g Fz(t)k(>)f FG(0)e(\014xed.)66 b(In)37 b(its)h(turn,)i(the)d(partition)g(function) 2647 2921 y(~)2638 2942 y FE(A)2692 2912 y Fr(re)q(g)2692 2963 y FC(n;\013;")2857 2875 y Fv(\000)2895 2942 y Fz(e)2934 2912 y FC(tA)3009 2887 y Fq(I)3042 2912 y FC(=n)3122 2875 y Fv(\001)3197 2942 y FG(can)g(b)r(e)456 3042 y(studied)30 b(along)e(the)j(lines)e(of)h(the)g(pro)r(of)g(of)f(Lemma)h(3.6,)g (except)f(that)i(in)f(order)e(to)i(con)n(trol)456 3141 y(the)f(con)n(tribution)f(of)h Fz(!)1222 3153 y FD(1)1288 3141 y FG(one)f(needs)h(to)g(dev)n(elop)f(a)g(sharp)g(asymptotic)g (expression)g(for)g(the)456 3241 y(tilted)g(corner)e(generation)g (function)i FE(A)1728 3253 y FC(n)1779 3241 y FG(\()p Fz(t)p FG(\).)555 3341 y(As)c(in)g(the)g(case)f(of)g(bridges,)h(the)g (lo)r(cal)f(limit)i(result)e(for)g FE(P)2403 3301 y Fr(reg)2403 3361 y FC(t;n)2492 3273 y Fv(\000)2530 3341 y Fz(n)2580 3311 y FD(2)2617 3341 y Fz(q)2654 3353 y FC(n)2699 3273 y Fv(\001)2761 3341 y FG(requires)f(some)i(care,)456 3440 y(but)i(an)f(ob)n(vious)f(and)h(straigh)n(tforw)n(ard)e(mo)r (di\014cation)i(of)h(the)g(argumen)n(ts)e(emplo)n(y)n(ed)g(in)i(the)456 3540 y(pro)r(of)h(of)g(Lemma)h(3.7)e(yields:)456 3696 y FO(Claim)j(4.1.)40 b Fn(Fix)28 b(an)n(y)f Fz(\016)f(>)d FG(0)p Fn(.)36 b(As)28 b Fz(n)23 b Fx(!)g(1)p Fn(,)1119 3876 y FE(P)1171 3836 y Fr(reg)1171 3896 y FC(t;n)1260 3809 y Fv(\000)1298 3876 y Fz(A)1360 3842 y FC(R)1438 3876 y FG(=)g Fz(n)1576 3842 y FD(2)1613 3876 y Fz(q)1650 3888 y FC(n)1695 3809 y Fv(\001)1784 3876 y FG(=)2114 3820 y(1)p 1909 3857 453 4 v 1909 3874 a Fv(p)p 1992 3874 370 4 v 71 x FG(2)p Fz(\031)s(n)2134 3921 y FD(3)2171 3945 y Fz(\033)s FG(\()p Fz(t)2283 3957 y FC(n)2329 3945 y FG(\))2371 3809 y Fv(\000)2409 3876 y FG(1)18 b(+)g FA(o)h FG(\()q(1\))2720 3809 y Fv(\001)2758 3876 y Fz(;)456 4086 y Fn(uniformly)27 b(in)h(the)g(rescaled)e(areas)g Fx(f)p Fz(q)1680 4098 y FC(n)1725 4086 y Fx(g)h Fn(satisfying)1426 4254 y Fz(\016)54 b Fx(\024)c Fz(q)1669 4266 y FC(n)1765 4254 y Fx(\024)1966 4197 y Fz(q)2003 4209 y FC(\014)s(;)p FD(+)p 1890 4234 303 4 v 1890 4310 a Fz(\034)1926 4322 y FC(\014)1972 4310 y FG(\(0)p Fz(;)14 b FG(1\))2157 4287 y FD(2)2203 4254 y FG(\(1)k Fx(\000)g Fz(\016)s FG(\))p Fz(:)-2017 b FG(\(4.3\))456 4461 y FO(Remark)30 b(4.1.1.)40 b FG(The)25 b(second)f(inequalit)n(y)g(ab)r(o)n(v)n(e)g (pla)n(ys)g(the)h(same)g(role)e(as)i(\(3.1\))f(of)h(Theo-)456 4561 y(rem)f(3.1,)g(whereas)f(the)i(\014rst)f(constrain)n(t)f(in)h (\(4.3\))h(suppresses)e(the)h(e\013ect)h(of)f(en)n(tropic)g(repul-)456 4661 y(sion)f(of)g(the)h(comp)r(onen)n(ts)f Fz(\015)1348 4673 y FD(1)1386 4661 y FG(,)g Fz(:)14 b(:)g(:)28 b FG(,)d Fz(!)1657 4673 y FC(K)1744 4661 y FG(from)e Fx(L)1993 4626 y Fy(\000)1993 4683 y FD(0)2073 4661 y FG(\(this)h(repulsion)f(b)r (ecomes)h(non-negligible)456 4761 y(for)j Fz(q)620 4773 y FC(n)688 4761 y Fx(\030)c FG(0\).)555 4917 y(T)-7 b(o)25 b(summarise)f(the)h(ab)r(o)n(v)n(e)f(discussion:)34 b(essen)n(tially)24 b(the)i(only)e(new)h(ingredien)n(t)f(required)456 5016 y(for)39 b(the)h(pro)r(of)g(of)g(Prop)r(osition)e(1.7)h(is)h(an)g (asymptotically)e(sharp)i(computation)f(of)h(the)456 5116 y(corner)28 b(generating)h(functions)h FE(A)1546 5128 y FC(n)1598 5116 y FG(\()p Fz(t)p FG(\))d(=)g FE(A)1876 5128 y FC(n)1927 5116 y FG(\()p Fz(e)1998 5086 y FC(tH)t FD(\()p FC(!)r FD(\))2182 5116 y FG(\).)45 b(More)29 b(precisely)-7 b(,)30 b(w)n(e)f(should)h(pro)n(v)n(e)456 5216 y(the)e(follo)n(wing)e(corner)g(coun)n(terpart)g(of)i(\(3.28\):)p eop %%Page: 44 44 44 43 bop 456 226 a FD(44)814 b(OST)-5 b(AP)28 b(HR)-5 b(YNIV)29 b(AND)g(DMITR)-5 b(Y)29 b(IOFFE)456 438 y FO(Theorem)h(4.2.) 40 b Fn(Let)29 b FG([)p Fz(a;)14 b(b)p FG(])26 b Fx(\032)f(D)1546 402 y FD(+)1544 463 y FC(\014)1628 391 y Fr(def)1636 438 y FG(=)35 b Fx(D)1800 450 y FC(\014)1864 438 y Fx(\\)20 b FG(\(0)p Fz(;)14 b(\034)2086 450 y FC(\014)2131 438 y FG(\(0)p Fz(;)g FG(1\)\))p Fn(.)42 b(Then)29 b(there)g(exists)g(a)g (complex)456 545 y(neigh)n(b)r(ourho)r(o)r(d)f Fx(U)39 b Fn(of)30 b FG([)p Fz(a;)14 b(b)p FG(])29 b Fn(in)h FE(C)15 b Fn(,)37 b(a)29 b(non-v)-5 b(anishing)29 b(analytic)g (function)i Fz(\024)p FG(\()p Fx(\001)p FG(\))f Fn(on)g Fx(U)38 b Fn(and)30 b(a)456 644 y(p)r(ositiv)n(e)d(constan)n(t)35 b FG(~)-50 b Fz(\013)23 b(>)g FG(0)p Fn(,)k(suc)n(h)g(that)1323 745 y Fv(\014)1323 795 y(\014)1323 845 y(\014)1350 841 y Fz(e)1389 807 y Fy(\000)p FC(nm)1541 816 y Fq(\014)1580 807 y FD(\()p FC(z)r FD(\))1670 841 y FE(A)1736 853 y FC(n)1787 841 y FG(\()p Fz(z)t FG(\))18 b Fx(\000)g Fz(\024)p FG(\()p Fz(z)t FG(\))2150 745 y Fv(\014)2150 795 y(\014)2150 845 y(\014)2200 841 y Fx(\024)2304 785 y FG(1)p 2298 822 54 4 v 2306 898 a(~)-50 b Fz(\013)2361 841 y(e)2400 807 y Fy(\000)7 b FD(~)-40 b FC(\013n)2540 841 y Fz(;)-2107 b FG(\(4.4\))456 1026 y Fn(uniformly)27 b(in)h Fz(n)23 b Fx(2)g FE(N)38 b Fn(and)27 b(in)h Fz(z)e Fx(2)e(U)8 b Fn(.)456 1126 y(F)-7 b(urthermore,)23 b(the)h(function)h Fz(\024)e Fn(ab)r(o)n(v)n(e)g(is)h(related)f(to)g(the)i(bridge)e (prefactor)f(function)j Fz(\026)f Fn(\(see)456 1226 y(\(1.20\)\))j(as)g (follo)n(ws:)36 b(F)-7 b(or)27 b(eac)n(h)f Fz(t)d Fx(2)h(D)1668 1191 y FD(+)1666 1251 y FC(\014)1751 1226 y Fn(de\014ne)k Fz(t)2021 1196 y Fy(\003)2082 1226 y Fx(2)23 b(D)2226 1191 y FD(+)2224 1251 y FC(\014)2310 1226 y Fn(via)1603 1400 y Fz(m)1676 1365 y Fy(0)1676 1420 y FC(\014)1721 1400 y FG(\()p Fz(t)p FG(\))p Fz(m)1888 1365 y Fy(0)1888 1420 y FC(\014)1933 1400 y FG(\()p Fz(t)1995 1365 y Fy(\003)2034 1400 y FG(\))51 b(=)f(1)p Fz(:)-1841 b FG(\(4.5\))456 1561 y Fn(Then,)1530 1723 y Fz(\024)p FG(\()p Fz(t)p FG(\))p Fz(\024)p FG(\()p Fz(t)1782 1689 y Fy(\003)1821 1723 y FG(\))51 b(=)f Fz(\026)p FG(\()p Fz(t)p FG(\))p Fz(\026)p FG(\()p Fz(t)2275 1689 y Fy(\003)2314 1723 y FG(\))p Fz(:)-1913 b FG(\(4.6\))456 1888 y Fn(In)27 b(particular,)g(if)h Fz(t)1072 1900 y FD(0)1137 1888 y Fn(satis\014es)f Fz(m)1516 1858 y Fy(0)1516 1911 y FC(\014)1560 1888 y FG(\()p Fz(t)1622 1900 y FD(0)1660 1888 y FG(\))c(=)g(1)p Fn(,)k(then)h Fz(\024)p FG(\()p Fz(t)2194 1900 y FD(0)2231 1888 y FG(\))c(=)e Fz(\026)p FG(\()p Fz(t)2486 1900 y FD(0)2524 1888 y FG(\))p Fn(.)555 2054 y FG(The)28 b(rest)f(of)h(the)g(section)f(is)g(dev)n(oted)g(to)h (the)g(pro)r(of)f(of)g(Theorem)g(4.2.)456 2258 y(4.1.)46 b FO(Asymptotic)31 b(b)s(eha)m(viour)h(of)g FE(A)1767 2270 y FC(n)1818 2258 y FG(\()p Fz(z)t FG(\))p FO(.)41 b FG(Due)28 b(to)g(Remark)f(4.0.1,)f(for)h(ev)n(ery)f Fz(t)d Fx(2)h(D)3347 2223 y FD(+)3345 2283 y FC(\014)3402 2258 y FG(,)1477 2475 y(lim)1448 2525 y FC(n)p Fy(!1)1649 2419 y FG(1)p 1645 2456 50 4 v 1645 2532 a Fz(n)1719 2475 y FG(log)14 b FE(A)1905 2487 y FC(n)1956 2475 y FG(\()p Fz(t)p FG(\))52 b(=)e Fz(m)2290 2487 y FC(\014)2335 2475 y FG(\()p Fz(t)p FG(\))p Fz(:)-1996 b FG(\(4.7\))456 2666 y(In)40 b(order)g(to)g(deriv)n(e)g(sharp)r(er)f(asymptotics)h(as)g (asserted)g(in)h(\(4.4\),)i(it)f(is)e(con)n(v)n(enien)n(t)g(to)456 2765 y(in)n(v)n(estigate)30 b(corner)g(generating)h(functions)h(of)f (the)i(form)e FE(A)2415 2777 y FC(n)p FD(+)p FC(k)2554 2765 y FG(\()p Fz(z)t FG(\))h(enabling,)g(th)n(us,)h(some)456 2865 y(degree)c(of)i(freedom)f(in)h(c)n(ho)r(osing)e Fz(n)h FG(and)h Fz(k)s FG(.)45 b(Namely)-7 b(,)31 b(w)n(e)g(shall)f (consider)f Fz(n)i FG(and)f Fz(m)h FG(to)f(b)r(e)456 2965 y(of)d(the)h(same)f(order)f(and,)i(moreo)n(v)n(er,)d(request)i (that)h(they)g(satisfy)1592 3109 y(1)p 1592 3146 42 4 v 1592 3222 a(4)1666 3165 y Fz(<)1766 3109 y(k)p 1764 3146 50 4 v 1764 3222 a(n)1847 3165 y(<)1944 3109 y(k)22 b FG(+)c(1)p 1944 3146 189 4 v 2014 3222 a Fz(n)2166 3165 y(<)23 b FG(4)p Fz(:)-1863 b FG(\(4.8\))456 3355 y(Recall)27 b(that)h(the)g(starting)e(p)r(oin)n(t)i(of)g(corner)e (paths)i(from)f Fx(A)2393 3367 y FC(n)p FD(+)p FC(k)2554 3355 y FG(is)g(\()p Fx(\000)p Fz(n)18 b Fx(\000)g Fz(k)s(;)c FG(0\).)37 b(By)27 b(\(1.23\))456 3455 y(most)f(of)g(these)g(paths)g (should)g(ha)n(v)n(e)f(break)h(lines)g(in)g(the)h(in)n(terv)-5 b(al)26 b(\()p Fx(\000)p Fz(n)15 b Fx(\000)h Fz("n;)e Fx(\000)p Fz(n)g FG(+)h Fz("n)p FG(\).)37 b(T)-7 b(o)456 3554 y(b)r(e)27 b(precise,)g(giv)n(en)f Fz(!)g Fx(2)d(A)1302 3566 y FC(n)p FD(+)p FC(k)1463 3554 y FG(let)k(us)h(use)f Fz(\015)g FG(=)c Fz(\015)5 b FG(\()p Fz(!)s FG(\))27 b(to)g(denote)g(the)h(irreducible)f(animal)g(of)456 3655 y Fz(!)j FG(whic)n(h)e(sits)g(on)g(the)h(p)r(ole)f Fx(P)1421 3667 y Fy(\000)p FC(n)1518 3655 y FG(.)38 b(Then)28 b(\(1.23\))g (implies)g(that)g(for)g(ev)n(ery)f(\014xed)h([)p Fz(a;)14 b(b)p FG(])23 b Fx(\032)h(D)3389 3619 y FD(+)3387 3680 y FC(\014)456 3757 y FG(there)j(exists)g Fz(c)933 3769 y FD(1)993 3757 y Fz(>)c FG(0,)k(suc)n(h)h(that)814 3922 y FE(A)879 3934 y FC(n)p FD(+)p FC(k)1018 3854 y Fv(\000)1056 3922 y Fz(e)1095 3887 y FC(tH)t FD(\()p FC(!)r FD(\))1293 3922 y FG(;)f Fw(Span)p FG(\()p Fz(\015)5 b FG(\))23 b Fx(6\032)g FG(\()p Fx(\000)p Fz(n)18 b Fx(\000)g Fz("n;)c Fx(\000)p Fz(n)j FG(+)h Fz("n)p FG(\))2448 3854 y Fv(\001)2537 3922 y Fx(\024)50 b Fz(e)2691 3887 y Fy(\000)p FC(c)2773 3895 y Fs(1)2805 3887 y FC(n)p FD(+\()p FC(n)p FD(+)p FC(k)q FD(\))p FC(m)3136 3896 y Fq(\014)3176 3887 y FD(\()p FC(t)p FD(\))456 3922 y FG(\(4.9\))456 4086 y(uniformly)27 b(in)h Fz(t)23 b Fx(2)g FG([)p Fz(a;)14 b(b)p FG(])28 b(and)f Fz(n)h FG(su\016cien)n(tly)f(large.)555 4187 y(Let)40 b(us)g(remo)n(v)n(e)e(the)i(condition)f(to)h(sta)n(y)f(ab)r(o) n(v)n(e)f(the)i(semi-axis)e Fx(L)2793 4151 y Fy(\000)2793 4209 y FD(0)2890 4187 y FG(on)h(the)h(in)n(terv)-5 b(al)456 4286 y(\()p Fx(\000)p Fz(n)19 b Fx(\000)g Fz("n;)14 b FG(0\))28 b(and)h(denote)g(the)g(mo)r(di\014ed)h(partition)e(function)i FE(A)2590 4298 y FC(n)p FD(+)p FC(k)r(;")2780 4286 y FG(\()p Fz(t)p FG(\).)42 b(Since)29 b(for)f(eac)n(h)456 4390 y Fz(t)23 b Fx(2)g FG([)p Fz(a;)14 b(b)p FG(])23 b Fx(\032)f(D)926 4354 y FD(+)924 4415 y FC(\014)1008 4390 y FG(the)27 b(a)n(v)n(erage)c(slop)r(e)j(of)g(the)h(paths)f(con)n (tributing)f(to)h FE(A)2752 4402 y FC(n)p FD(+)p FC(k)q(;")2942 4390 y FG(\()p Fz(t)p FG(\))g(is)h Fz(m)3218 4360 y Fy(0)3218 4413 y FC(\014)3262 4390 y FG(\()p Fz(t)p FG(\))d Fz(>)456 4497 y FG(0,)33 b(w)n(e)f(infer)g(from)g(the)h(strict)f(triangle)g (inequalit)n(y)g(\(1.8\))g(that)h(there)f(exists)g Fz(c)3023 4509 y FD(2)3091 4497 y Fz(>)f FG(0,)i(suc)n(h)456 4596 y(that)1171 4758 y FE(A)1237 4770 y FC(n)p FD(+)p FC(k)q(;")1427 4758 y FG(\()p Fz(t)p FG(\))19 b Fx(\000)f FE(A)1688 4770 y FC(n)p FD(+)p FC(k)1827 4758 y FG(\()p Fz(t)p FG(\))51 b Fx(\024)f Fz(e)2126 4724 y Fy(\000)p FC(c)2208 4732 y Fs(2)2240 4724 y FC(n)p FD(+\()p FC(n)p FD(+)p FC(k)q FD(\))p FC(m)2571 4733 y Fq(\014)2611 4724 y FD(\()p FC(t)p FD(\))2692 4758 y Fz(;)-2259 b FG(\(4.10\))456 4923 y(uniformly)27 b(in)h Fz(t)23 b Fx(2)g FG([)p Fz(a;)14 b(b)p FG(])28 b(and)f Fz(n)h FG(large)e(enough.)555 5022 y(Since)i Fz(m)845 5034 y FC(\014)917 5022 y FG(is)g(analytic)f(in)h(a) f(complex)g(neigh)n(b)r(ourho)r(o)r(d)g(of)g([)p Fz(a;)14 b(b)p FG(])27 b(and)h(since,)f(eviden)n(tly)-7 b(,)991 5184 y Fx(j)p FE(A)1079 5196 y FC(n)p FD(+)p FC(k)r(;")1269 5184 y FG(\()p Fz(z)t FG(\))19 b Fx(\000)f FE(A)1543 5196 y FC(n)p FD(+)p FC(k)1682 5184 y FG(\()p Fz(z)t FG(\))p Fx(j)50 b(\024)h FE(A)2043 5196 y FC(n)p FD(+)p FC(k)r(;")2233 5184 y FG(\()p Fx(<)p Fz(z)t FG(\))18 b Fx(\000)g FE(A)2566 5196 y FC(n)q FD(+)p FC(k)2705 5184 y FG(\()p Fx(<)p Fz(z)t FG(\))p Fz(;)-2439 b FG(\(4.11\))p eop %%Page: 45 45 45 44 bop 1488 226 a FD(SELF-A)-7 b(V)n(OIDING)29 b(POL)-5 b(YGONS)966 b(45)456 425 y FG(the)32 b(inequalities)g(\(4.9\))g(and)g (\(4.10\))g(actually)g(hold)g(in)g(some)g(complex)g(neigh)n(b)r(ourho)r (o)r(d)f Fx(U)456 525 y FG(of)c([)p Fz(a;)14 b(b)p FG(]:)37 b(There)27 b(exists)g Fz(c)1278 537 y FD(3)1338 525 y Fz(>)c FG(0,)k(suc)n(h)g(that)h(the)g(relations)777 711 y FE(A)842 723 y FC(n)p FD(+)p FC(k)981 644 y Fv(\000)1019 711 y Fz(e)1058 677 y FC(z)r(H)t FD(\()p FC(!)r FD(\))1264 711 y FG(;)14 b Fw(Span)p FG(\()p Fz(\015)5 b FG(\))23 b Fx(6\032)g FG(\()p Fx(\000)p Fz(n)18 b Fx(\000)g Fz("n;)c Fx(\000)p Fz(n)j FG(+)h Fz("n)p FG(\))2406 644 y Fv(\001)2467 711 y FG(=)23 b Fz(o)2609 619 y Fv(\020)2659 711 y Fz(e)2698 677 y Fy(\000)p FC(c)2780 685 y Fs(3)2811 677 y FC(n)p FD(+\()p FC(n)p FD(+)p FC(k)q FD(\))p FC(m)3142 686 y Fq(\014)3182 677 y FD(\()p FC(z)r FD(\))3272 619 y Fv(\021)1287 894 y FE(A)1353 906 y FC(n)p FD(+)p FC(k)q(;")1543 894 y FG(\()p Fz(z)t FG(\))18 b Fx(\000)g FE(A)1816 906 y FC(n)p FD(+)p FC(k)1955 894 y FG(\()p Fz(z)t FG(\))23 b(=)g Fz(o)2227 802 y Fv(\020)2276 894 y Fz(e)2315 860 y Fy(\000)p FC(c)2397 868 y Fs(3)2429 860 y FC(n)p FD(+\()p FC(n)p FD(+)p FC(k)q FD(\))p FC(m)2760 869 y Fq(\014)2800 860 y FD(\()p FC(z)r FD(\))2890 802 y Fv(\021)-2484 b FG(\(4.12\))456 1076 y(hold)27 b(uniformly)h(in)f Fz(z)g Fx(2)c(U)8 b FG(.)37 b(Therefore,)762 1299 y FE(A)827 1311 y FC(n)p FD(+)p FC(k)966 1299 y FG(\()p Fz(z)t FG(\))51 b(=)1263 1195 y FC("n)1239 1220 y Fv(X)1246 1399 y FC(l)p FD(=0)1397 1195 y FC("n)1373 1220 y Fv(X)1374 1396 y FC(r)r FD(=0)1507 1299 y FE(A)1572 1311 y FC(n)p Fy(\000)p FC(l)1696 1299 y FG(\()p Fz(z)t FG(\))p FE(F)1859 1311 y FC(l)p FD(+)p FC(r)1974 1299 y FG(\()p Fz(z)t FG(\))p FE(B)2130 1311 y FC(k)r Fy(\000)p FC(r)2262 1299 y FG(\()p Fz(z)t FG(\))46 b(+)f FA(o)2587 1207 y Fv(\020)2636 1299 y Fz(e)2675 1265 y Fy(\000)p FC(c)2757 1273 y Fs(3)2789 1265 y FC(n)p FD(+\()p FC(n)p FD(+)p FC(k)q FD(\))p FC(m)3120 1274 y Fq(\014)3160 1265 y FD(\()p FC(z)r FD(\))3250 1207 y Fv(\021)3313 1299 y Fz(;)-2880 b FG(\(4.13\))456 1535 y(also)26 b(uniformly)i(in)f Fz(z)g Fx(2)c(U)8 b FG(.)555 1634 y(De\014ne)28 b(no)n(w)1512 1796 y Fz(\024)1560 1808 y FC(n)1605 1796 y FG(\()p Fz(z)t FG(\))23 b(=)f Fz(e)1861 1762 y Fy(\000)p FC(nm)2013 1771 y Fq(\014)2052 1762 y FD(\()p FC(z)r FD(\))2142 1796 y FE(A)2207 1808 y FC(n)2258 1796 y FG(\()p Fz(z)t FG(\))p Fz(:)456 1960 y FG(Multiplying)28 b(b)r(oth)g(sides)f(of)g(\(4.13\))g(b)n(y)h Fz(e)1792 1930 y Fy(\000)p FD(\()p FC(n)p FD(+)p FC(k)q FD(\))p FC(m)2083 1939 y Fq(\014)2122 1930 y FD(\()p FC(z)r FD(\))2240 1960 y FG(and)f(using)g(the)h(relation)1281 2129 y Fz(e)1320 2095 y Fy(\000)p FC(nm)1472 2104 y Fq(\014)1511 2095 y FD(\()p FC(z)r FD(\))1601 2129 y FE(B)1650 2141 y FC(n)1701 2129 y FG(\()p Fz(z)t FG(\))23 b(=)g Fz(\026)p FG(\()p Fz(z)t FG(\))2090 2062 y Fv(\000)2128 2129 y FG(1)18 b(+)g FA(o)5 b FG(\()p Fz(e)2389 2095 y Fy(\000)p FC(c)2471 2103 y Fs(4)2503 2095 y FC(n)2548 2129 y FG(\))2580 2062 y Fv(\001)456 2291 y FG(v)-5 b(alid)22 b(uniformly)g(in)g Fz(n)g FG(large)f(enough)g(and)h(in)g Fz(z)k FG(from)21 b(\(p)r(ossibly)h(further)g(shrink)n(ed\))g(complex)456 2390 y(neigh)n(b)r(ourho)r(o)r(d)j Fx(U)34 b FG(of)27 b([)p Fz(a;)14 b(b)p FG(],)26 b(w)n(e)g(arriv)n(e)e(to)i(the)h(follo)n (wing)e(recursion)g(t)n(yp)r(e)h(relation)f(for)h(the)456 2490 y(functions)i Fx(f)p Fz(\024)904 2502 y FC(n)p FD(+)p FC(k)1036 2490 y Fx(g)p FG(:)565 2820 y Fz(\024)613 2832 y FC(n)p FD(+)p FC(k)746 2820 y FG(\()p Fz(z)t FG(\))23 b(=)963 2753 y Fv(\000)1001 2820 y Fz(\026)p FG(\()p Fz(z)t FG(\))c(+)f Fz(o)p FG(\()p Fz(e)1371 2786 y Fy(\000)p FC(c)1453 2794 y Fs(5)1485 2786 y FC(n)1530 2820 y FG(\))1562 2753 y Fv(\001)1638 2716 y FC("n)1614 2741 y Fv(X)1622 2920 y FC(l)p FD(=0)1772 2716 y FC("n)1748 2741 y Fv(X)1750 2917 y FC(r)r FD(=0)1882 2820 y Fz(\024)1930 2832 y FC(n)p Fy(\000)p FC(l)2048 2820 y FG(\()p Fz(z)t FG(\))p Fz(e)2194 2786 y Fy(\000)p FD(\()p FC(l)p FD(+)p FC(r)r FD(\))p FC(m)2462 2795 y Fq(\014)2500 2786 y FD(\()p FC(z)r FD(\))2590 2820 y FE(F)2647 2832 y FC(l)p FD(+)p FC(r)2762 2820 y FG(\()p Fz(z)t FG(\))g(+)g Fz(o)3024 2753 y Fv(\000)3062 2820 y Fz(e)3101 2786 y Fy(\000)p FC(c)3183 2794 y Fs(3)3215 2786 y FC(n)3260 2753 y Fv(\001)3312 2820 y Fz(;)456 2643 y FG(\(4.14\))456 3060 y(whic)n(h)27 b(holds)g(uniformly)g(in)g Fz(z)f Fx(2)e(U)35 b FG(and)27 b(in)h Fz(n;)14 b(k)30 b FG(satisfying)c Fz(n=)p FG(4)c Fz(<)h(k)j(<)c FG(4)p Fz(n)p FG(.)36 b(Th)n(us,)27 b(giv)n(en)g(a)456 3160 y(couple)g(\()p Fz(n;)14 b(k)s FG(\))28 b(as)f(in)g(\(4.8\),)519 3388 y Fz(\024)567 3400 y FC(n)p FD(+)p FC(k)q FD(+1)784 3388 y FG(\()p Fz(z)t FG(\))18 b Fx(\000)g Fz(\024)1040 3400 y FC(n)p FD(+)p FC(k)1173 3388 y FG(\()p Fz(z)t FG(\))23 b(=)g FA(o)5 b FG(\()p Fz(e)1509 3354 y Fy(\000)p FC(c)1591 3362 y Fs(6)1623 3354 y FC(n)1668 3388 y FG(\))1738 3284 y FC("n)1714 3309 y Fv(X)1722 3488 y FC(l)p FD(=0)1872 3284 y FC("n)1848 3309 y Fv(X)1850 3485 y FC(r)r FD(=0)1982 3388 y Fz(\024)2030 3400 y FC(n)p Fy(\000)p FC(l)2148 3388 y FG(\()p Fz(z)t FG(\))p Fz(e)2294 3354 y Fy(\000)p FD(\()p FC(l)p FD(+)p FC(r)r FD(\))p FC(m)2562 3363 y Fq(\014)2600 3354 y FD(\()p FC(z)r FD(\))2691 3388 y FE(F)2747 3400 y FC(l)p FD(+)p FC(r)2862 3388 y FG(\()p Fz(z)t FG(\))18 b(+)3070 3321 y Fv(\000)3108 3388 y Fz(e)3147 3354 y Fy(\000)p FC(c)3229 3362 y Fs(3)3261 3354 y FC(n)3306 3321 y Fv(\001)3358 3388 y Fz(:)456 3628 y FG(By)29 b(the)h(separation) e(of)h(masses)g(\(1.21\),)g(the)h(neigh)n(b)r(ourho)r(o)r(d)e Fx(U)38 b FG(can)29 b(b)r(e)h(c)n(hosen)f(in)h(suc)n(h)f(a)456 3728 y(w)n(a)n(y)d(that)i(the)g(estimate)1469 3892 y Fx(j)p Fz(e)1531 3858 y Fy(\000)p FC(nm)1683 3867 y Fq(\014)1722 3858 y FD(\()p FC(z)r FD(\))1812 3892 y FE(F)1868 3904 y FC(n)1919 3892 y FG(\()p Fz(z)t FG(\))p Fx(j)23 b(\024)g Fz(c)2196 3904 y FD(8)2233 3892 y Fz(e)2272 3858 y Fy(\000)p FC(c)2354 3866 y Fs(7)2386 3858 y FC(n)456 4054 y FG(is)k(v)-5 b(alid)28 b(uniformly)f(in)h Fz(n)23 b Fx(2)g FE(N)38 b FG(and)27 b(in)h Fz(z)f Fx(2)c(U)8 b FG(.)37 b(It)28 b(follo)n(ws)f(that)h(the)g(limit)1587 4236 y Fz(\024)p FG(\()p Fz(z)t FG(\))1793 4189 y Fr(def)1801 4236 y FG(=)89 b(lim)1926 4285 y FC(n)p Fy(!1)2113 4236 y Fz(\024)2161 4248 y FC(n)2206 4236 y FG(\()p Fz(z)t FG(\))456 4426 y(exists)35 b(on)h Fx(U)44 b FG(and,)37 b(furthermore,)g(the)f(uniform) g(exp)r(onen)n(tial)g(rate)f(of)g(con)n(v)n(ergence)f(\(4.4\))456 4526 y(holds.)555 4625 y(In)28 b(order)e(to)i(c)n(hec)n(k)f(that)h Fz(\024)p FG(\()p Fx(\001)p FG(\))g(do)r(es)f(not)h(v)-5 b(anish)28 b(in)g(a)f(\(p)r(ossibly)g(smaller)g(than)h Fx(U)36 b FG(neigh-)456 4725 y(b)r(ourho)r(o)r(d)d(of)g([)p Fz(a;)14 b(b)p FG(])34 b(assume,)g(to)g(the)g(con)n(trary)-7 b(,)33 b(that)h(this)g(is)f(not)h(the)g(case.)54 b(Then)34 b(there)456 4825 y(exists)27 b Fz(t)c Fx(2)g FG([)p Fz(a;)14 b(b)p FG(],)28 b(suc)n(h)f(that)h Fz(\024)p FG(\()p Fz(t)p FG(\))23 b(=)g(0.)36 b(By)28 b(\(4.4\))f(this)h(w)n(ould)f(imply)h (that)1496 5025 y(lim)1467 5075 y FC(n)p Fy(!1)1668 4969 y FG(1)p 1664 5006 50 4 v 1664 5082 a Fz(n)1738 5025 y FG(log)14 b Fz(\024)1907 5037 y FC(n)1952 5025 y FG(\()p Fz(t)p FG(\))24 b Fz(<)e Fx(\000)r FG(~)-44 b Fz(c)23 b(<)f FG(0)p Fz(;)456 5216 y FG(in)27 b(a)h(clear)e(con)n(tradiction)g (to)i(the)g(principal)f(rate)g(of)h(deca)n(y)e(form)n(ula)h(\(4.7\).)p eop %%Page: 46 46 46 45 bop 456 226 a FD(46)814 b(OST)-5 b(AP)28 b(HR)-5 b(YNIV)29 b(AND)g(DMITR)-5 b(Y)29 b(IOFFE)456 425 y FG(4.2.)46 b FO(Lo)s(cal)41 b(limit)c(b)s(eha)m(viour)j(of)g(the)f(corner)i (connectivities)e Fz(a)2833 437 y FC(\014)2878 425 y FG(\()p Fz(x)p FG(\))p FO(.)j FG(Exactly)33 b(as)456 525 y(in)j(the)h(case)e(of)h(bridge)f(partition)h(functions)h([33)o(])f (the)g(analytic)g(con)n(trol)f(\(4.4\))h(o)n(v)n(er)e(mo-)456 624 y(men)n(t)d(generating)e(functions)i(sets)f(up)h(the)g(stage)f(for) g(a)h(lo)r(cal)f(limit)h(description)f(of)h(corner)456 724 y(connectivities)c Fz(a)1019 736 y FC(\014)1064 724 y FG(\()p Fz(x)p FG(\),)1365 920 y Fz(a)1409 932 y FC(\014)1453 920 y FG(\()p Fz(x)p FG(\))1616 873 y Fr(def)1625 920 y FG(=)1955 841 y Fv(X)1749 1023 y FC(!)r FD(:\()p Fy(\000)p FC(x)1928 1031 y Fs(1)1960 1023 y FC(;)p FD(0\))p Fy(7!)p FD(\(0)p FC(;x)2222 1031 y Fs(2)2254 1023 y FD(\))1909 1104 y FC(!)r Fy(2A)2051 1112 y Fq(x)2084 1124 y Fs(1)2294 920 y Fz(e)2333 886 y Fy(\000)p FC(\014)s Fy(j)p FC(!)r Fy(j)2512 920 y Fz(:)456 1267 y FG(In)20 b(view)g(of)g(the)h (conditions)e(of)h(Theorem)g(4.2)f(these)h(estimates)g(should)g(hold)g (in)h(the)f(language)456 1367 y(of)29 b(conjugate)f(tilts)i(uniformly)f (in)h Fz(t)c Fx(2)g FG([)p Fz(a;)14 b(b)p FG(])26 b Fx(\032)f(D)2063 1331 y FD(+)2061 1392 y FC(\014)2118 1367 y FG(,)30 b(whic)n(h)g (translates)e(to)h(uniform)g(results)456 1474 y(for)e Fz(a)627 1486 y FC(\014)671 1474 y FG(\()p Fz(x)p FG(\))i(for)e (directions)1253 1670 y Fz(x)c Fx(2)1417 1649 y FG(~)1402 1670 y Fx(C)1446 1682 y FC(r)1505 1623 y Fr(def)1514 1670 y FG(=)32 b Fx(f)p FG(\()p Fz(y)1726 1682 y FD(1)1762 1670 y Fz(;)14 b(y)1840 1682 y FD(2)1877 1670 y FG(\))37 b(:)g Fz(y)2047 1682 y FD(2)2084 1670 y Fz(=r)26 b(<)c(y)2317 1682 y FD(1)2377 1670 y Fz(<)h(y)2506 1682 y FD(2)2543 1670 y Fz(r)r Fx(g)p Fz(:)456 1838 y FG(As)33 b(in)g([33)o(],)i(the)e (only)f(remaining)h(prop)r(ert)n(y)e(to)i(b)r(e)h(c)n(hec)n(k)n(ed)d (is)i(an)g(exp)r(onen)n(tial)g(deca)n(y)f(of)456 1937 y FE(A)521 1949 y FC(n)572 1937 y FG(\()p Fz(t)17 b FG(+)f Fz(is)p FG(\))26 b(for)g(large)f(v)-5 b(alues)26 b(of)h Fz(s)p FG(,)g(that)f(is)h(in)g(the)g(case)e(when)i Fz(t)16 b FG(+)g Fz(is)26 b FG(do)r(es)h(not)f(necessarily)456 2037 y(b)r(elongs)h(to)g(the)h(complex)f(neigh)n(b)r(ourho)r(o)r(d)g Fx(U)36 b FG(describ)r(ed)27 b(in)h(Theorem)f(4.2.)456 2205 y FO(Lemma)i(4.3.)40 b Fn(F)-7 b(or)34 b(an)n(y)g Fz(t)i Fx(2)f(D)1526 2170 y FD(+)1524 2230 y FC(\014)1616 2205 y Fn(and)g(for)f(an)n(y)g(complex)h(neigh)n(b)r(ourho)r(o)r(d)f Fx(U)43 b Fn(of)35 b Fz(t)g Fn(\(suc)n(h)456 2321 y(that)e Fx()c FG(0)p Fn(,)i(suc)n(h)f(that)h(uniformly)f(in)h Fz(n)456 2424 y Fn(su\016cien)n(tly)27 b(large)f(one)i(has)f(the)h (follo)n(wing)e(inequalit)n(y)1474 2637 y FG(sup)1334 2710 y FC(t)p FD(+)p FC(is)p Fy(62U)6 b FC(;)p Fy(j)p FC(s)p Fy(j)p FC(<\031)1754 2516 y Fv(\014)1754 2566 y(\014)1754 2616 y(\014)1754 2666 y(\014)1792 2581 y FE(A)1857 2593 y FC(n)1908 2581 y FG(\()p Fz(t)19 b FG(+)f Fz(is)p FG(\))p 1792 2618 381 4 v 1876 2694 a FE(A)1941 2706 y FC(n)1993 2694 y FG(\()p Fz(t)p FG(\))2182 2516 y Fv(\014)2182 2566 y(\014)2182 2616 y(\014)2182 2666 y(\014)2260 2637 y Fz(<)50 b(e)2414 2603 y Fy(\000)p FC(\016)r(n)2543 2637 y Fz(:)-2110 b FG(\(4.15\))555 2873 y(In)36 b(view)f(of)g(the)g(corresp)r(onding)f(result)h(for)f(the) i(bridge)e(partition)h(functions)h([33)o(])f(the)456 2972 y(pro)r(of)30 b(of)h(the)h(lemma)e(is)h(essen)n(tially)f(con)n (tained)h(in)g(\(4.13\).)47 b(Indeed,)32 b(b)n(y)f(\(4.11\),)g(a)f(w)n (eak)n(er)456 3072 y(v)n(ersion)c(of)h(\(4.13\),)699 3306 y FE(A)764 3318 y FC(n)p FD(+)p FC(k)903 3306 y FG(\()p Fz(z)t FG(\))50 b(=)1200 3203 y FC("n)1176 3228 y Fv(X)1183 3406 y FC(l)p FD(=0)1333 3203 y FC("n)1310 3228 y Fv(X)1311 3403 y FC(r)r FD(=0)1443 3306 y FE(A)1509 3318 y FC(n)p Fy(\000)p FC(l)1633 3306 y FG(\()p Fz(z)t FG(\))p FE(F)1796 3318 y FC(l)p FD(+)p FC(r)1911 3306 y FG(\()p Fz(z)t FG(\))p FE(B)2067 3318 y FC(k)r Fy(\000)p FC(r)2198 3306 y FG(\()p Fz(z)t FG(\))19 b(+)f Fz(o)2461 3214 y Fv(\020)2510 3306 y Fz(e)2549 3272 y Fy(\000)p FC(c)2631 3280 y Fs(3)2663 3272 y FC(n)p FD(+\()p FC(n)p FD(+)p FC(k)q FD(\))p FC(m)2994 3281 y Fq(\014)3034 3272 y FD(\()p FC(t)p FD(\))3115 3214 y Fv(\021)3178 3306 y Fz(;)456 3553 y FG(holds)23 b(for)h(an)n(y)f(v)-5 b(alue)24 b(of)g Fz(s)g FG(in)h Fz(z)h FG(=)d Fz(t)11 b FG(+)g Fz(is)p FG(.)35 b(Since)25 b Fz(\024)p FG(\()p Fx(\001)p FG(\))f(do)r(esn't)h(v)-5 b(anish)24 b(in)g Fx(U)8 b FG(,)25 b(w)n(e)f(divide)g(b)r(oth)456 3653 y(sides)j(of)h(the)h 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Fy(00)1867 4895 y FC(\014)1912 4870 y FG(\()p Fz(t)p FG(\))2030 4784 y Fz(e)2069 4750 y Fy(\000)p FC(\034)2152 4759 y Fq(\014)2190 4750 y FD(\()p FC(x)p FD(\))2284 4717 y Fv(\000)2322 4784 y FG(1)18 b(+)g FA(o)h FG(\()q(1\))2633 4717 y Fv(\001)2671 4784 y Fz(;)-2238 b FG(\(4.16\))456 5081 y Fn(uniformly)35 b(in)h Fz(x)i FG(=)e(\()p Fz(n;)14 b(k)s FG(\))37 b Fx(2)1474 5060 y FG(~)1459 5081 y Fx(C)1503 5093 y FC(r)1540 5081 y Fn(.)61 b(The)36 b(ab)r(o)n(v)n(e)f(tilt)h Fz(t)g Fn(is)g(related)f (to)h(the)g(p)r(oin)n(t)g Fz(x)h Fn(b)n(y)e(the)456 5181 y(dualit)n(y)27 b(relation)g Fz(m)1115 5151 y Fy(0)1115 5204 y FC(\014)1159 5181 y FG(\()p Fz(t)p FG(\))d(=)f Fz(k)s(=n)p Fn(.)p eop %%Page: 47 47 47 46 bop 1488 226 a FD(SELF-A)-7 b(V)n(OIDING)29 b(POL)-5 b(YGONS)966 b(47)456 425 y FO(Remark)30 b(4.4.1.)40 b FG(Recall)35 b([33)o(])h(that)g(the)g(bridge)f(t)n(w)n(o-p)r(oin)n(t)g (functions)h(satisfy)f(a)g(similar)456 525 y(asymptotic)27 b(form)n(ula:)36 b(F)-7 b(or)27 b(ev)n(ery)f Fz(r)g(<)c 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b(Comm)n(un.)24 b(Math.)i(Ph)n(ys.)g Fc(131)p FN(,)601 4136 y(1,)d(1{50)i(\(1990\))491 4219 y([3])35 b(K.)30 b(Azuma:)47 b FB(Weighte)l(d)33 b(sums)h(of)f(c)l(ertain)g(dep)l(endent)g(r)l(andom)i(variables)5 b FN(,)34 b(T^)-35 b(ohoku)33 b(Math.)e(J.)g Fc(19)p FN(,)601 4302 y(357{367)25 b(\(1967\))491 4385 y([4])35 b(O.)25 b(Benois,)i(T.)f(Bo)r(dineau,)i(P)-6 b(.)26 b(Butta,)i(E.)e (Presutti:)37 b FB(On)28 b(the)g(validity)g(of)g(van)h(der)f(Waals)i (the)l(ory)e(of)601 4468 y(surfac)l(e)e(tension)5 b FN(,)24 b(Mark)n(o)n(v)f(Pro)r(c.)h(Relat.)f(Fields)g Fc(3)p FN(,)g(175{198)j(\(1997\))491 4551 y([5])35 b(O.)28 b(Benois,)i(T.)e (Bo)r(dineau,)j(E.)e(Presutti:)42 b FB(L)l(ar)l(ge)30 b(deviations)h(in)g(the)f(van)h(der)g(Waals)g(limit)6 b FN(,)30 b(Sto)r(c)n(h.)601 4634 y(Pro)r(c.)23 b(Appl.)g Fc(75)p FN(,)f(89{104)k(\(1998\))491 4717 y([6])35 b(T.)21 b(Bo)r(dineau:)32 b FB(The)24 b(Wul\013)h(c)l(onstruction)g(in)f(thr)l (e)l(e)h(and)g(mor)l(e)h(dimensions)p FN(,)d(Comm)n(un.)d(Math.)i(Ph)n (ys.)601 4800 y Fc(207)p FN(,)g(1,)h(197-229)i(\(1999\))491 4883 y([7])35 b(T.)23 b(Bo)r(dineau:)32 b FB(Phase)26 b(c)l(o)l(existenc)l(e)g(for)g(Kac)f(Ising)h(mo)l(dels)p FN(,)f(preprin)n(t)f(\(2000\))491 4967 y([8])35 b(T.)g(Bo)r(dineau,)40 b(D.)35 b(Io\013e,)40 b(Y.)c(V)-6 b(elenik:)56 b FB(R)n(igor)l(ous)38 b(pr)l(ob)l(abilistic)h(analysis)f(of)f(e)l(quilibrium)g(crystal)601 5050 y(shap)l(es)p FN(,)25 b(J.)e(Math.)h(Ph)n(ys.)f Fc(41)p FN(,3,)f(1033-1098)k(\(2000\))491 5133 y([9])35 b(T.)16 b(Bo)r(dineau,)j(D.)d(Io\013e,)k(Y.)c(V)-6 b(elenik:)28 b FB(A)19 b(micr)l(osc)l(opic)i(derivation)f(of)g(the)f(Winterb)l (ottom)h(c)l(onstruction)601 5216 y(|)25 b(an)h Fc(L)850 5225 y FK(1)910 5216 y FB(appr)l(o)l(ach)p FN(,)g(to)f(app)r(ear)f(in)f FB(J.)j(Statist.)f(Phys.)f FN(\(2001\))p eop %%Page: 49 49 49 48 bop 1488 226 a FD(SELF-A)-7 b(V)n(OIDING)29 b(POL)-5 b(YGONS)966 b(49)456 425 y FN([10])35 b(T.)23 b(Bo)r(dineau,)h(E.)f (Presutti:)31 b(priv)l(ate)25 b(comm)n(unication)e(\(2001\))456 508 y([11])35 b(A.)23 b(Bo)n(vier,)h(M.)g(Zahradnik:)33 b FB(The)26 b(low-temp)l(er)l(atur)l(e)i(phase)f(of)g(Kac-Ising)e(mo)l (dels)5 b FN(,)26 b(J.)e(Statist.)h(Ph)n(ys.)601 591 y Fc(87)p FN(,)d(No.1-2,)h(311{332)j(\(1997\))456 674 y([12])35 b(M.)25 b(Campanino,)i(D.)e(Io\013e:)38 b FB (Ornstein-Zernike)26 b(b)l(ehaviour)k(the)l(ory)e(the)g(Bernoul)t(li)h (b)l(ond)g(p)l(er)l(c)l(olation)601 757 y(on)d Fc(Z)753 734 y FH(d)789 757 y FN(,)d(to)h(app)r(ear)g(in)47 b(Annals)24 b(Prob.)f(\(2002\))456 840 y([13])35 b(M.)26 b(Campanino,)h(D.)f (Io\013e,)j(Y.)d(V)-6 b(elenik:)39 b FB(Ornstein-Zernike)26 b(the)l(ory)j(for)g(\014nite)g(r)l(ange)g(Ising)g(mo)l(dels)601 923 y(ab)l(ove)d FJ(T)835 931 y FH(c)867 923 y FN(,)d(preprin)n(t)h (\(2001\))456 1006 y([14])35 b(J.)23 b(Cardy:)31 b FB(Me)l(an)26 b(ar)l(e)l(a)h(of)f(self-avoiding)g(lo)l(ops)p FN(,)f(Ph)n(ys.)e(Rev.)h (Lett.)g Fc(72)p FN(,)f(11,)h(158-1583)h(\(1994\))456 1089 y([15])35 b(M.)29 b(Cassandro,)j(E.)d(Presutti:)45 b FB(Phase)33 b(tr)l(ansitions)f(in)f(Ising)h(systems)g(with)g(long)g (but)f(\014nite)g(r)l(ange)5 b FN(,)601 1172 y(Mark)n(o)n(v)23 b(Pro)r(c.)h(Relat.)f(Fields)h Fc(2)p FN(,)e(241{262)k(\(1996\))456 1255 y([16])35 b(R.)30 b(Cerf:)45 b FB(L)l(ar)l(ge)33 b(deviations)g(for)g(thr)l(e)l(e)g(dimensional)h(sup)l(er)l(critic)l (al)f(p)l(er)l(c)l(olation)5 b FN(,)36 b(Ast)n(\023)-33 b(erisque)31 b Fc(267)p FN(,)601 1338 y(1{177)25 b(\(2000\))456 1421 y([17])35 b(R.)21 b(Cerf,)g(A.)f(Pisztora:)31 b FB(On)23 b(the)h(Wul\013)g(crystal)f(in)h(the)f(Ising)h(mo)l(del)7 b FN(,)23 b(Ann.)e(Probab.)g Fc(28)p FN(,)g(3,)h(947{1017)601 1504 y(\(2000\))456 1588 y([18])35 b(R.)d(Cerf,)j(A.)d(Pisztora:)51 b FB(Phase)36 b(c)l(o)l(existenc)l(e)e(in)g(Ising,)j(Potts)e(and)g(p)l (er)l(c)l(olation)i(mo)l(dels)5 b FN(,)37 b(preprin)n(t)601 1671 y(\(2000\))456 1754 y([19])e(F.)28 b(Cesi,)h(G.)g(Guadagni,)i(F.)d (Martinelli,)h(R.H.)e(Sc)n(honmann:)42 b FB(On)30 b(the)h (two-dimensional)h(sto)l(chastic)601 1837 y(Ising)26 b(mo)l(del)h(in)f(the)f(phase)i(c)l(o)l(existenc)l(e)f(r)l(e)l(gion)h (ne)l(ar)f(the)g(critic)l(al)f(p)l(oint)6 b FN(,)25 b(J.)f(Statist.)g (Ph)n(ys.)g Fc(85)p FN(,)f(1-2,)601 1920 y(55-102)h(\(1996\))456 2003 y([20])35 b(J.T.)20 b(Cha)n(y)n(es)h(and)g(L.)f(Cha)n(y)n(es:)31 b FB(Ornstein-Zernike)21 b(b)l(ehavior)i(for)g(self-avoiding)g(walks)h (at)f(al)t(l)g(noncrit-)601 2086 y(ic)l(al)j(temp)l(er)l(atur)l(es)5 b FN(,)25 b(Comm)n(un.)c(Math.)j(Ph)n(ys.)f Fc(105)p FN(,)g(221-238)h(\(1986\))456 2169 y([21])35 b(R.L.)27 b(Dobrushin:)41 b FB(A)30 b(statistic)l(al)g(b)l(ehaviour)h(of)f(shap)l (es)i(of)e(b)l(oundaries)i(of)e(phases)p FN(,)h(In)e(Kotec)n(k)q(\023) -36 b(y,)31 b(R.)601 2252 y(\(ed.\))17 b(Phase)g(T)-6 b(ransitions:)27 b(Mathematics,)18 b(Ph)n(ysics,)f(Biology)g(...,)f (60{70;)j(Singap)r(ore:)29 b(W)-6 b(orld)16 b(Scien)n(ti\014c)601 2335 y(\(1993\))456 2418 y([22])35 b(R.)18 b(Dobrushin,)i(O.)f(Hryniv:) 28 b FB(Fluctuations)23 b(of)f(shap)l(es)h(of)f(lar)l(ge)h(ar)l(e)l(as) g(under)f(p)l(aths)h(of)f(r)l(andom)h(walks)5 b FN(,)601 2501 y(Probab.)23 b(Theory)h(Relat.)g(Fields)f Fc(105)p FN(,)f(423{458)k(\(1996\))456 2584 y([23])35 b(R.)24 b(Dobrushin,)g(O.)g(Hryniv:)33 b FB(Fluctuations)28 b(of)f(the)g(phase) g(b)l(oundary)i(in)d(the)g(2D)h(Ising)g(ferr)l(omagnet)p FN(,)601 2667 y(Comm)n(un.)21 b(Math.)j(Ph)n(ys.)f Fc(189)p FN(,)f(395{445)k(\(1997\))456 2750 y([24])35 b(R.)29 b(Dobrushin,)i(R.)f(Kotec)n(k)q(\023)-36 b(y,)33 b(S.)d(Shlosman:)43 b Fb(W)-6 b(ul\013)31 b(Construction:)45 b(a)30 b(Global)g(Shap)r(e)i (from)c(Lo)r(cal)601 2833 y(In)n(teraction)p FN(.)22 b(\(T)-6 b(ranslations)20 b(of)g(mathematical)g(monographs,)g Fc(104)p FN(\))g(Pro)n(vidence,)h(R.I.:)29 b(Amer.)18 b(Math.)601 2916 y(So)r(c.)24 b(\(1992\))456 2999 y([25])35 b(R.L.)24 b(Dobrushin,)h(S.)g(Shlosman:)34 b FB(L)l(ar)l(ge)28 b(and)g(mo)l(der)l(ate)h(deviations)f(in)f(the)g(Ising)g(mo)l(del)p FN(,)g(Adv)l(ances)601 3082 y(So)n(v.)d(Math.,)f Fc(20)p FN(,)f(91-219)j(\(1994\))456 3165 y([26])35 b(M.E.)23 b(Fisher,)i(A.J.)f(Guttmann,)i(S.G.)f(Whittington:)36 b FB(Two-dimensional)29 b(lattic)l(e)e(vesicles)f(and)i(p)l(oly-)601 3248 y(gons)p FN(,)23 b(J.)h(Ph)n(ys.)f(A)g Fc(24)p FN(,)g(13,)h (3095{3106)i(\(1991\))456 3331 y([27])35 b(G.)f(Galla)n(v)n(otti:)53 b FB(The)35 b(phase)i(sep)l(ar)l(ation)g(line)e(in)h(the)f (two-dimensional)i(Ising)e(mo)l(del)7 b FN(,)37 b(Comm)n(un.)601 3414 y(Math.)23 b(Ph)n(ys.)h Fc(27)p FN(,)e(103-136)j(\(1972\))456 3497 y([28])35 b(Y.)15 b(Higuc)n(hi:)27 b FB(On)18 b(some)h(limit)f (the)l(or)l(em)i(r)l(elate)l(d)f(to)f(the)g(phase)i(sep)l(ar)l(ation)g (line)e(in)h(the)f(two-dimensional)601 3580 y(Ising)25 b(mo)l(del)7 b FN(,)25 b(Z.)e(W)-6 b(ahrsc)n(h.)24 b(V)-6 b(erw.)23 b(Gebiete)i Fc(50)p FN(,)e(3,)g(287-315)i(\(1979\))456 3663 y([29])35 b(W.)i(Ho)r(e\013ding:)59 b FB(Pr)l(ob)l(ability)40 b(ine)l(qualities)e(for)h(sums)g(of)g(b)l(ounde)l(d)h(r)l(andom)g (variables)p FN(,)h(J.)c(Amer.)601 3746 y(Statist.)24 b(Asso)r(c.)f Fc(58)p FN(,)g(13{30)i(\(1963\))456 3829 y([30])35 b(O.)25 b(Hryniv:)36 b FB(On)27 b(lo)l(c)l(al)j(b)l(ehaviour) f(of)f(the)g(phase)h(sep)l(ar)l(ation)h(line)e(in)g(the)f FN(2)p FB(D)h(Ising)h(mo)l(del)p FN(,)e(Probab.)601 3912 y(Theory)d(Relat.)f(Fields)h Fc(110)p FN(,)e(1,)h(91{107)i(\(1998\))456 3995 y([31])35 b(O.)e(Hryniv,)i(R.)f(Kotec)n(k)q(\023)-36 b(y:)53 b FB(Surfac)l(e)36 b(tension)f(and)h(the)f(Ornstein-Zernike)e (b)l(ehaviour)j(for)f(the)g(2D)601 4078 y(Blume-Cap)l(el)27 b(mo)l(del)p FN(,)d(to)h(app)r(ear)f(in)f(J.)h(Statist.)g(Ph)n(ys.)f (\(2002\))456 4161 y([32])35 b(O.)23 b(Hryniv,)f(Y.)h(V)-6 b(elenik:)32 b(in)23 b(preparation)456 4244 y([33])35 b(D.)20 b(Io\013e:)31 b FB(Ornstein-Zernike)21 b(b)l(ehaviour)k(and)f (analyticity)f(of)g(shap)l(es)i(for)f(self-avoiding)f(walks)i(on)e FL(Z)3395 4221 y FH(d)3425 4244 y FN(,)601 4327 y(Mark)n(o)n(v)g(Pro)r (c.)h(Relat.)f(Fields)h Fc(4)p FN(,)e(323-350)j(\(1998\))456 4410 y([34])35 b(D.)21 b(Io\013e:)31 b FB(L)l(ar)l(ge)25 b(deviations)g(for)f(the)g FN(2)p FB(D)g(Ising)g(mo)l(del:)34 b(a)24 b(lower)h(b)l(ound)g(without)g(cluster)f(exp)l(ansions)p FN(,)601 4493 y(J.)f(Statist.)h(Ph)n(ys.)g Fc(74)p FN(,)e(1-2,)i (411{432)h(\(1994\))456 4576 y([35])35 b(D.)24 b(Io\013e,)i(R.H.)e(Sc)n (honmann:)34 b FB(Dobrushin-Kote)l(ck)o(\023)-36 b(y-Shlosman)29 b(the)l(or)l(em)f(up)f(to)g(the)g(critic)l(al)f(temp)l(er-)601 4659 y(atur)l(e)p FN(,)d(Comm)n(un.)f(Math.)i(Ph)n(ys.)f Fc(199)p FN(,)f(1,)i(117-167)g(\(1998\))456 4742 y([36])35 b(J.L.)21 b(Leb)r(o)n(witz,)j(A.)d(Mazel,)h(E.)g(Presutti:)31 b FB(Liquid-vap)l(or)25 b(phase)g(tr)l(ansitions)g(for)g(systems)f (with)g(\014nite-)601 4825 y(r)l(ange)i(inter)l(actions)p FN(,)d(J.)g(Statist.)i(Ph)n(ys.)e Fc(94)p FN(,)g(5-6,)g(955-1025)i (\(1999\))456 4908 y([37])35 b(N.)23 b(Madras)g(and)h(G.)g(Slade:)31 b Fb(The)24 b(Self-Av)n(oiding)f(Random)h(W)-6 b(alk)t FN(,)23 b(Boston,)i(Birkh\177)-35 b(auser)23 b(\(1993\))456 4991 y([38])35 b(F.)17 b(Martinelli:)27 b FB(L)l(e)l(ctur)l(es)20 b(on)h(Glaub)l(er)g(dynamics)f(for)h(discr)l(ete)f(spin)g(mo)l(dels)5 b FN(.)19 b(Lectures)g(on)f(probabilit)n(y)601 5074 y(theory)29 b(and)f(statistics)h(\(Sain)n(t-Flour,)f(1997\),)i(93{191,)h(Lecture)e (Notes)f(in)g(Math.,)h Fc(1717)p FN(,)e(Springer,)601 5157 y(Berlin)22 b(\(1999\))p eop %%Page: 50 50 50 49 bop 456 226 a FD(50)814 b(OST)-5 b(AP)28 b(HR)-5 b(YNIV)29 b(AND)g(DMITR)-5 b(Y)29 b(IOFFE)456 425 y FN([39])35 b(R.A.)28 b(Minlos,)j(Y)-6 b(a.G.)29 b(Sinai:)44 b FB(The)31 b(phenomenon)i(of)f(\\phase)g(sep)l(ar)l(ation)-7 b(")33 b(at)e(low)i(temp)l(er)l(atur)l(es)f(in)601 508 y(some)26 b(lattic)l(e)f(mo)l(dels)i(of)f(a)g(gas)g(I,)e FN(Math.)f(USSR-Sb.)h Fc(2)p FN(,)f(335{395)i(\(1967\))456 591 y([40])35 b(R.A.)28 b(Minlos,)j(Y)-6 b(a.G.)29 b(Sinai:)44 b FB(The)31 b(phenomenon)i(of)f (\\phase)g(sep)l(ar)l(ation)-7 b(")33 b(at)e(low)i(temp)l(er)l(atur)l (es)f(in)601 674 y(some)26 b(lattic)l(e)f(mo)l(dels)i(of)f(a)g(gas)g (II,)e FN(T)-6 b(rans.)23 b(Mosco)n(w)h(Math.)g(So)r(c.)g Fc(19)p FN(,)e(121{196)k(\(1968\))456 757 y([41])35 b(C.-E.)28 b(P\014ster:)45 b FB(L)l(ar)l(ge)32 b(deviations)g(and)g(phase)h(sep)l (ar)l(ation)h(in)d(the)h(two)g(dimensional)h(Ising)f(mo)l(del)7 b FN(,)601 840 y(Helv.)23 b(Ph)n(ys.)g(Acta)i Fc(64)p FN(,)d(953{1054)k(\(1991\))456 923 y([42])35 b(C.-E.)c(P\014ster,)k(Y.) d(V)-6 b(elenik:)49 b FB(L)l(ar)l(ge)34 b(deviations)h(and)f(c)l (ontinuum)h(limit)f(in)g(the)g FN(2)p FB(D)g(Ising)g(mo)l(del)g FN(,)601 1006 y(Probab.)23 b(Theory)h(Relat.)g(Fields)f Fc(109)p FN(,)f(4,)i(435-506)h(\(1997\))456 1089 y([43])35 b(C.-E.)23 b(P\014ster,)j(Y.)e(V)-6 b(elenik:)35 b FB(Interfac)l(e,)27 b(surfac)l(e)h(tension)f(and)h(r)l(e)l(entr)l(ant)g(pinning)f(tr)l (ansition)h(in)e(the)601 1172 y(2D)f(Ising)h(mo)l(del)p FN(,)f(Comm)n(un.)c(Math.)j(Ph)n(ys.)f Fc(204)p FN(,)g(269-312)h (\(1999\))456 1255 y([44])35 b(A.)25 b(Pisztora:)35 b FB(Surfac)l(e)28 b(or)l(der)g(lar)l(ge)g(deviations)g(of)g(Ising,)g (Potts)g(and)g(p)l(er)l(c)l(olation)i(mo)l(dels)5 b FN(,)27 b(Probab.)601 1338 y(Theory)d(Relat.)f(Fields)h Fc(104)p FN(,)e(427{466)j(\(1996\))456 1421 y([45])35 b(R.H.)28 b(Sc)n(honmann,)j(S.B.)d(Shlosman:)42 b FB(Complete)31 b(analyticity)f(for)h FN(2)p FB(D)g(Ising)g(c)l(omplete)l(d)7 b FN(.)29 b(Comm)n(un.)601 1504 y(Math.)23 b(Ph)n(ys.)h Fc(170)p FN(,)e(453{482)j(\(1995\))456 1588 y([46])35 b(R.H.)30 b(Sc)n(honmann,)k(S.B.)c(Shlosman:)46 b FB(Wul\013)34 b(dr)l(oplets)g(and)g(the)e(metastable)h(r)l(elaxation)h(of)g(kinetic) 601 1671 y(Ising)25 b(mo)l(dels)5 b FN(.)26 b(Comm)n(un.)21 b(Math.)j(Ph)n(ys.)f Fc(194)p FN(,)f(no.)i(2,)f(389{462)j(\(1998\))555 1826 y FM(DMA-EPFL,)h(CH-1015)d(La)o(usanne,)g(Switzerland)555 1909 y FB(Curr)l(ent)i(addr)l(ess)5 b FN(:)33 b(Statistical)25 b(Lab)r(oratory,)f(Cam)n(bridge)f(Univ)n(ersit)n(y,)g(Cam)n(bridge,)f (CB3)i(0WB,)g(UK)555 1992 y FB(E-mail)i(addr)l(ess)5 b FN(:)33 b Fa(o.hryniv@statslab.cam.)q(ac.)q(uk)555 2133 y FM(F)-8 b(a)o(cul)l(ty)25 b(of)g(Industrial)g(Engineering,)f (Technion,)i(Haif)-6 b(a)24 b(3200,)g(Israel)555 2216 y FB(E-mail)i(addr)l(ess)5 b FN(:)33 b Fa(ieioffe@ie.technion.ac)q(.il) p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0111261105364--