Content-Type: multipart/mixed; boundary="-------------0211181121808" This is a multi-part message in MIME format. ---------------0211181121808 Content-Type: text/plain; name="02-469.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-469.comments" 14 pages, 3 figures ---------------0211181121808 Content-Type: text/plain; name="02-469.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-469.keywords" First passage percolation ---------------0211181121808 Content-Type: application/postscript; name="Branching.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Branching.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: Branching.dvi %%Pages: 14 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -ta4 -f Branching.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.11.18:1734 %%BeginProcSet: texc.pro %! 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b(loses)e(the)g(ph)m(ysical)g(in)m(terpretation)f(of)g (the)i(spreading)f(of)f(a)h(\015uid.)f(But)i(directed)f(\014rst)21 5384 y(and)h(last)f(passage)h(p)s(ercolation)e(mo)s(dels)g(are)i(exp)s (ected)h(to)e(b)s(e)h(in)f(the)h(same)f(univ)m(ersalit)m(y)g(class)21 5504 y(under)e(the)f(condition)e(that)i(along)e(the)i(ra)m(y)g(under)h (consideration)d(the)j(macroscopic)e(shap)s(e)h(has)21 5624 y(a)d(non-zero)h(curv)-5 b(ature)33 b([KMH)q(,)f(PS)q(].)1852 5873 y(1)p eop %%Page: 2 2 2 1 bop 21 29 a Fh(1)97 b(INTR)m(ODUCTION)34 b(AND)f(MAIN)g(RESUL)-8 b(T)1742 b Fr(2)21 2859 y currentpoint currentpoint translate 0.56622 0.56622 scale neg exch neg exch translate 21 2859 a 21 2859 a gsave currentpoint currentpoint translate 0 neg rotate neg exch neg exch translate 21 2859 a @beginspecial 18 @llx 23 @lly 819 @urx 577 @ury 8010 @rwi @setspecial %%BeginDocument: Figure1.eps %!PS-Adobe-3.0 EPSF-3.0 %%Title: bigpic %%BoundingBox: 18 23 819 577 %%Creator: ADOBEPS4.DRV Version 4.52 %%CreationDate: 05/21/2002 12:37:41 %%For: Francois %%Pages: 1 %%DocumentNeededResources: (atend) %%DocumentSuppliedResources: (atend) %%LanguageLevel: 2 %%EndComments %%BeginDefaults %%ViewingOrientation: 0 1 -1 0 %%EndDefaults %%BeginProlog %%BeginResource: procset AdobePS_FatalError 4.2 0 userdict begin/FatalErrorIf{{initgraphics findfont 1 index 0 eq{exch pop}{dup length dict begin{1 index/FID ne{def}{pop pop}ifelse}forall/Encoding{ ISOLatin1Encoding}stopped{StandardEncoding}if def currentdict end /ErrFont-Latin1 exch definefont}ifelse exch scalefont setfont counttomark 3 div cvi{moveto show}repeat showpage quit}{cleartomark}ifelse}bind def end %%EndResource %%BeginResource: procset AdobePS_Win_Driver_Incr_L2 4.2 0 userdict /AdobePS_Win_Driver_Incr_L2 250 dict dup begin put [ (Ce travail requiert une imprimante PostScript(R) Niveau 2.) 100 500 (Veuillez envoyer ce fichier vers une imprimante Niveau 2 ou corriger les) 100 485 (paramètres des feuilles de propriétés du pilote, puis recommencez.) 100 470 12 /Times-Roman /languagelevel where {pop languagelevel}{1} ifelse 2 lt FatalErrorIf /VM? {vmstatus exch sub exch pop gt {PrtVMMsg}if} bind def %%BeginResource: procset AdobePS_Win_Utils 4.2 0 /d/def load def/,/load load d/~/exch , d/?/ifelse , d/!/pop , d/`/begin , d/^ /index , d/@/dup , d/+/translate , d/$/roll , d/U/userdict , d/M/moveto , d/- /rlineto , d/&/currentdict , d/:/gsave , d/;/grestore , d/F/false , d/T/true , d/N/newpath , d/E/end , d/Ac/arc , d/An/arcn , d/A/ashow , d/D/awidthshow , d /C/closepath , d/V/div , d/O/eofill , d/L/fill , d/I/lineto , d/-C/rcurveto , d/-M/rmoveto , d/+S/scale , d/Ji/setfont , d/Lc/setlinecap , d/Lj/setlinejoin , d/Lw/setlinewidth , d/S/show , d/LH/showpage , d/K/stroke , d/W/widthshow , d/R/rotate , d/XS/xshow , d/b{bind d}bind d/bd{bind d}bind d/xd{~ d}bd/ld{, d} bd/lw/Lw ld/lc/Lc ld/lj/Lj ld/sg/setgray ld/sgcs{/DeviceGray setcolorspace}bd /bn/bind ld/L2? F/languagelevel where{! languagelevel 2 ge{! T}if}if d/L3? F /languagelevel where{! languagelevel 3 ge{! T}if}if d/g{@ not{U/DefIf_save save put}if U/DefIf_bool 2 ^ put}b/DefIf_El{if U/DefIf_bool get not @{U /DefIf_save get restore}if}b/e{DefIf_El !}b/self & d/reinitialize{[/TextInit /GraphInit/UtilsInit counttomark{@ where{self eq}{F}?{cvx exec}{!}?}repeat cleartomark}b/initialize{`{/Pscript_Win_Data where{!}{U/Pscript_Win_Data & put }?/ADO_mxRot ~ d/TextInitialised? F d reinitialize E}{U/Pscript_Win_Data 230 dict @ ` put/ADO_mxRot ~ d/TextInitialised? F d reinitialize}?}b/terminate{!{& self eq{exit}{E}?}loop E}b/suspend/terminate , d/resume{` Pscript_Win_Data `}b /snap{transform 0.25 sub round 0.25 add ~ 0.25 sub round 0.25 add ~ itransform }b/dsnap{dtransform round ~ round ~ idtransform}b U<04>cvn{}put/setjn{{ statusdict/jobname known{statusdict/jobname 3 -1 $ put}if}stopped cleartomark} b/solid{[]0 setdash}b/setdsh{0 setdash}b/colspRefresh{}b/rp{4 2 $ M 1 ^ 0 - 0 ~ - neg 0 -}b/rr{1 ^ 0 - 0 ~ - neg 0 - C}b/CTMsave{globaldict ` currentglobal T setglobal/SavedCTM matrix currentmatrix d setglobal E}b/CTMrestore{ globaldict ` currentglobal T setglobal/SavedCTM where{! SavedCTM setmatrix}if setglobal E}b/emuxs{! ! currentpoint( )@ 0 6 -1 $ put S ~ 3 ^ 3 ^ get add ~ M 1 add}b/XSE{version cvi 2015 ge{XS}{0/emuxs , 4 -1 $ cshow ! !}?}b/UDF{L2?{ undefinefont}{!}?}b/UDR{L2?{undefineresource}{! !}?}b/freeVM{/Courier findfont [40 0 0 -40 0 0]makefont Ji 2 vmreclaim}b/hfRedefFont{findfont @ length dict ` {1 ^/FID ne{d}{! !}?}forall & E @ ` ~{/CharStrings 1 dict `/.notdef 0 d & E d} if/Encoding 256 array 0 1 255{1 ^ ~/.notdef put}for d E definefont !}bind d /hfMkCIDFont{/CIDFont findresource @ length 2 add dict `{1 ^ @/FID eq ~ @/XUID eq ~/UIDBase eq or or{! !}{d}?}forall/CDevProc ~ d/Metrics2 16 dict d /CIDFontName 1 ^ d & E 1 ^ ~/CIDFont defineresource ![~]composefont !}bind d %%EndResource %%BeginResource: procset AdobePS_Win_Utils_L2 4.2 0 /colspA/DeviceGray d/colspABC/DeviceRGB d/setAorABC{{colspA}{colspABC}? setcolorspace}b/rf/rectfill , d/fx{1 1 dtransform @ 0 ge{1 sub 0.5}{1 add -0.5 }? 3 -1 $ @ 0 ge{1 sub 0.5}{1 add -0.5}? 3 1 $ 4 1 $ idtransform 4 -2 $ idtransform}b/BZ{4 -2 $ snap + +S fx rf}b/rs/rectstroke , d/rc/rectclip , d/sg {@ @ setcolor}b/sco{setcolor}b/colspRefresh{colspABC setcolorspace}b/sgco{{sg }{sco}?}b/UtilsInit{currentglobal{F setglobal}if}b/resourcestatus where{! /ColorRendering/ProcSet resourcestatus{! ! T}{F}?}{F}? not{/ColorRendering<< /GetHalftoneName{currenthalftone @/HalftoneName known{/HalftoneName get}{! /none}?}bn/GetPageDeviceName{currentpagedevice @/PageDeviceName known{ /PageDeviceName get @ null eq{!/none}if}{!/none}?}bn/GetSubstituteCRD{! /DefaultColorRendering/ColorRendering resourcestatus{! !/DefaultColorRendering }{(DefaultColorRendering*){cvn exit}127 string/ColorRendering resourceforall}? }bn>>/defineresource where{!/ProcSet defineresource !}{! !}?}if/buildcrdname{ /ColorRendering/ProcSet findresource ` mark GetHalftoneName @ type @/nametype ne ~/stringtype ne and{!/none}if(.)GetPageDeviceName @ type @/nametype ne ~ /stringtype ne and{!/none}if(.)5 ^ 0 5 -1 1{^ length add}for string 6 1 $ 5 ^ 5{~ 1 ^ cvs length 1 ^ length 1 ^ sub getinterval}repeat ! cvn 3 1 $ ! ! E}b /definecolorrendering{~ buildcrdname ~/ColorRendering defineresource !}b /findcolorrendering where{!}{/findcolorrendering{buildcrdname @/ColorRendering resourcestatus{! ! T}{/ColorRendering/ProcSet findresource ` GetSubstituteCRD E F}?}b}?/selectcolorrendering{findcolorrendering !/ColorRendering findresource setcolorrendering}b/ExecWMForm{execform}b/setpagedevice where{! /realstpgdev/setpagedevice ld}if/SC_topddict 0 d/SC_spdict 0 d/dopgdev{ Pscript_Win_Data/setpagedevice undef SC_topddict @ length 0 gt{realstpgdev}if} bd/stpgdev{SC_topddict @ 3 -1 ${SC_spdict 2 ^ known{SC_spdict 2 ^ get @ 3 -1 $ {put @}forall ! put @}{put @}?}forall ! !}bd/ststpgdev{Pscript_Win_Data /setpagedevice/stpgdev , put/SC_topddict 0 dict store/SC_spdict 3 dict ` /InputAttributes 0 dict d/Policies 0 dict d/OutputAttributes 0 dict d & E store}d/G2UBegin{findresource/FontInfo get/GlyphNames2Unicode get `}bind d /G2CCBegin{findresource/FontInfo get/GlyphNames2HostCode get `}bind d/G2UEnd{E }bind d/AddFontInfoBegin{/FontInfo 8 dict @ `}bind d/AddFontInfo{ /GlyphNames2Unicode 16 dict d/GlyphNames2HostCode 16 dict d}bind d /AddFontInfoEnd{E d}bind d/notdefXPos 0 d/notdefYPos 0 d/SetNotDefCharString{@ findfont/Encoding get 0 get/.notdef eq{@ 1000 selectfont<00>stringwidth /notdefYPos ~ d/notdefXPos ~ d}{/notdefXPos 0 d/notdefYPos 0 d}? findfont /CharStrings get/.notdef{! notdefXPos notdefYPos setcharwidth}put}bind d /T0AddCFFMtx2{/CIDFont findresource/Metrics2 get ` d E}bind d/ %%EndResource %%BeginResource: procset AdobePS_Win_Compat 4.2 0 userdict/AdobePS_Win_Compat 13 dict dup begin/bd{bind def}bind def/ld{load def }bd/CB{pop pop pop pop}bind def/B{pop pop pop pop}bind def/$x matrix def/SS{ /pagesave save def}bind def/RS{/pagesave where{pop pagesave restore}{$x matrix invertmatrix concat}ifelse}bind def/ANSIVec[0/grave 1/acute 2/circumflex 3 /tilde 4/macron 5/breve 6/dotaccent 7/dieresis 8/ring 9/cedilla 10 /hungarumlaut 11/ogonek 12/caron 13/dotlessi 39/quotesingle 96/grave 124/bar 130/quotesinglbase 131/florin 132/quotedblbase 133/ellipsis 134/dagger 135 /daggerdbl 136/circumflex 137/perthousand 138/Scaron 139/guilsinglleft 140/OE 145/quoteleft 146/quoteright 147/quotedblleft 148/quotedblright 149/bullet 150 /endash 151/emdash 152/tilde 153/trademark 154/scaron 155/guilsinglright 156 /oe 159/Ydieresis 160/space 161/exclamdown 164/currency 165/yen 166/brokenbar 167/section 168/dieresis 169/copyright 170/ordfeminine 171/guillemotleft 172 /logicalnot 173/hyphen 174/registered 175/macron 176/degree 177/plusminus 178 /twosuperior 179/threesuperior 180/acute 181/mu 182/paragraph 183 /periodcentered 184/cedilla 185/onesuperior 186/ordmasculine 187 /guillemotright 188/onequarter 189/onehalf 190/threequarters 191/questiondown 192/Agrave 193/Aacute 194/Acircumflex 195/Atilde 196/Adieresis 197/Aring 198 /AE 199/Ccedilla 200/Egrave 201/Eacute 202/Ecircumflex 203/Edieresis 204 /Igrave 205/Iacute 206/Icircumflex 207/Idieresis 208/Eth 209/Ntilde 210/Ograve 211/Oacute 212/Ocircumflex 213/Otilde 214/Odieresis 215/multiply 216/Oslash 217/Ugrave 218/Uacute 219/Ucircumflex 220/Udieresis 221/Yacute 222/Thorn 223 /germandbls 224/agrave 225/aacute 226/acircumflex 227/atilde 228/adieresis 229 /aring 230/ae 231/ccedilla 232/egrave 233/eacute 234/ecircumflex 235/edieresis 236/igrave 237/iacute 238/icircumflex 239/idieresis 240/eth 241/ntilde 242 /ograve 243/oacute 244/ocircumflex 245/otilde 246/odieresis 247/divide 248 /oslash 249/ugrave 250/uacute 251/ucircumflex 252/udieresis 253/yacute 254 /thorn 255/ydieresis]def currentdict{dup type/operatortype eq{[exch]cvx def}{ pop pop}ifelse}forall/initialize{currentdict exch begin begin}bind def /terminate{/@FL where not{pop end end}{pop}ifelse}bind def/suspend/terminate load def/resume/initialize load def/M/moveto load def end put/Courier findfont 10 scalefont setfont %%EndResource end %%EndResource %%EndProlog %%BeginSetup statusdict begin (%%[ ProductName: ) print product print ( ]%%)= flush end /findresource where{pop mark{/CIDParams /ProcSet findresource /SetBuildCompatible get true exch exec}stopped cleartomark}if [ 0 1 -1 0 0 0 ] false AdobePS_Win_Driver_Incr_L2 dup /initialize get exec /mysetup [ .24 0 0 -.24 22.799999 576.960021 ] def %%EndSetup %%Page: 1 1 %%EndPageComments %%BeginPageSetup userdict begin /pagesave save def end mysetup concat [ matrix currentmatrix {dup dup round sub abs .0001 lt{round} if} forall] setmatrix colspRefresh %%EndPageSetup : 130 150 2740 1743 rc AdobePS_Win_Driver_Incr_L2 begin %%BeginResource: procset AdobePS_Win_BW_Images 4.2 0 /iw 0 d/ih 0 d/im_save 0 d/s 0 d/polarity 0 d/smoothflag 0 d/mystring 0 d/bpc 0 d L3?{/itype 4 d}{/itype 1 d}?/imask F d/im0 0 d/im1 0 d/im2 0 d/im3 0 d /setup1asciiproc{[currentfile mystring/readhexstring cvx/! cvx]cvx bind}b /setup1binaryproc{[currentfile mystring/readstring cvx/! cvx]cvx bind}b /setup2asciiproc{currentfile/ASCII85Decode filter/RunLengthDecode filter}b /setup2binaryproc{currentfile/RunLengthDecode filter}b/mycolorspace{colspABC}d /myimagedict{/myimagedict 10 dict d myimagedict @ `/ImageType 1 d /MultipleDataSource F d E}b/imageprocarray[/setup1binaryproc/setup1asciiproc /setup2binaryproc/setup2asciiproc/setup1binarydecodeproc/setup1asciidecodeproc ]d/Q{/im_save save d sgco imageprocarray ~ get/s ~ , d/imask ~ d/polarity ~ d /smoothflag ~ d +/dx 2 ^ d/dy 1 ^ d +S/mystring ~ string d/bpc ~ d/ih ~ d/iw ~ d fx rf}b/X{/im_save save d imageprocarray ~ get/s ~ , d/imask ~ d/polarity ~ d/smoothflag ~ d +/dx 2 ^ d/dy 1 ^ d +S/mystring ~ string d/bpc ~ d/ih ~ d/iw ~ d}b/Z{im_save restore}b/Y{sgco myimagedict @ `/Width iw d/Height ih d/Decode polarity{[1 0]}{[0 1]}? d/ImageMatrix[iw 0 0 ih 0 0]d/DataSource s d /BitsPerComponent 1 d/Interpolate smoothflag d E imagemask}bd %%EndResource %%BeginResource: procset AdobePS_Win_Co_Images_L2 4.2 0 /doclutimage{/rgbclut ~ d ! imask{/im0 ~ d}if bpc @ 8 eq{! 255}{4 eq{15}{3}?}? /hival ~ d[/Indexed colspABC hival rgbclut]setcolorspace myimagedict @ ` imask {/MaskColor[im0]d/ImageType itype d}{/ImageType 1 d}?/Width iw d/Height ih d /Decode[0 hival]d/ImageMatrix[iw 0 0 ih 0 0]d/DataSource s d/BitsPerComponent bpc d/Interpolate smoothflag d E image}b/doCMYKclutimage{/CMYKclut ~ d ! imask {/im0 ~ d}if bpc @ 8 eq{! 255}{4 eq{15}{3}?}?/hival ~ d[/Indexed/DeviceCMYK hival CMYKclut]setcolorspace myimagedict @ ` imask{/MaskColor[im0]d/ImageType itype d}{/ImageType 1 d}?/Width iw d/Height ih d/Decode[0 hival]d/ImageMatrix[ iw 0 0 ih 0 0]d/DataSource s d/BitsPerComponent bpc d/Interpolate smoothflag d E image}b/doNimage{bpc 24 eq{imask{/im2 ~ d/im1 ~ d/im0 ~ d}if colspABC}{imask {/im0 ~ d}if colspA}? setcolorspace myimagedict @ ` imask{bpc 24 eq{/MaskColor [im0 im1 im2]d}{/MaskColor[im0]d}?/ImageType itype d}{/ImageType 1 d}?/Width iw d/Height ih d/Decode bpc 24 eq{[0 1 0 1 0 1]}{[0 1]}? d/ImageMatrix[iw 0 0 ih 0 0]d/DataSource s d/BitsPerComponent bpc 24 eq{8}{bpc}? d/Interpolate smoothflag d E image}b/doCMYKimage{imask{/im3 ~ d/im2 ~ d/im1 ~ d/im0 ~ d}if /DeviceCMYK setcolorspace myimagedict @ ` imask{/MaskColor[im0 im1 im2 im3]d /ImageType itype d}{/ImageType 1 d}?/Width iw d/Height ih d/Decode[0 1 0 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3143 y Fd(2)888 3128 y Fr(+)27 b Fn(x)p Fr(.)40 b(\(4\))f(is)g(the)h(distribution)e (function)h(for)g(a)g(standard)h(T)-8 b(racy-Widom)21 3248 y(random)32 b(v)-5 b(ariable)31 b Fn(\020)788 3263 y Fk(2)859 3248 y Fr([TW)q(].)i(The)g(famous)f(result)h(in)e([BDJ])i (states)g(that)1505 3468 y Fn(L)p Fr(\()p Fn(t)p Fr(\))1711 3441 y Fl(\030)1711 3472 y Fr(=)1816 3468 y(2)p Fn(t)22 b Fr(+)g Fn(t)2055 3427 y Fk(1)p Fj(=)p Fk(3)2165 3468 y Fn(\020)2208 3483 y Fk(2)3607 3468 y Fr(\(5\))21 3688 y(in)44 b(the)g(limit)d(of)j(large)f Fn(t)p Fr(.)i(In)g(brac)m(k)m (ets,)h(w)m(e)f(remark)f(that)h(the)f(pro)s(of)g(in)f([BDJ])h(pro)s (ceeds)i(via)21 3809 y(T)-8 b(o)s(eplitz)32 b(and)g(not,)h(as)f (indicated)g(here,)i(via)e(F)-8 b(redholm)30 b(determinan)m(ts.)167 3929 y(While)36 b(\(5\))h(giv)m(es)h(v)m(ery)g(precise)g(information)c (ab)s(out)i(the)i(t)m(ypical)e(length)h(of)f(directed)i(p)s(oly-)21 4049 y(mer,)30 b(it)f(lea)m(v)m(es)h(un)m(touc)m(hed)i(the)f(issue)f 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y(Figure)e(1)h(whic)m(h)h(displa)m(ys)g(the)f (directed)h(p)s(olymers)e(rotated)i(b)m(y)g Fn(\031)t(=)p Fr(4)e(for)h(b)s(etter)h(visibilit)m(y)-8 b(.)31 b(The)21 5374 y(ro)s(ot)25 b(p)s(oin)m(t)g(is)g(alw)m(a)m(ys)i Fn(S)33 b Fr(=)28 b(\(0)p Fn(;)17 b Fr(0\))25 b(and)h(the)g(end)h(p)s (oin)m(ts)e(lie)f(on)i(the)g(line)f Fn(U)2773 5389 y Fj(t)2831 5374 y Fr(=)i Fl(f)p Fr(\()p Fn(t)8 b Fl(\000)g Fn(x;)17 b(t)8 b Fr(+)g Fn(x)p Fr(\))p Fn(;)17 b Fl(j)p Fn(x)p Fl(j)30 b(\024)21 5494 y Fn(t)p Fl(g)p Fr(.)k(F)-8 b(or)32 b(\014xed)j(realization)c Fn(!)36 b Fr(and)e(for)f(eac)m(h)h (end)g(p)s(oin)m(t)f Fn(E)39 b Fr(w)m(e)c(dra)m(w)f(the)g(set)g(of)f (all)e(maximizers.)21 5614 y(Note)25 b(that,)f(e.g.)g(for)g Fn(E)34 b Fr(=)27 b(\()p Fn(t;)17 b(t)p Fr(\),)25 b(the)f(directed)h(p) s(olymer)e(splits)g(and)i(merges)f(again,)f(whic)m(h)i(re\015ects)p eop %%Page: 4 4 4 3 bop 21 29 a Fh(1)97 b(INTR)m(ODUCTION)34 b(AND)f(MAIN)g(RESUL)-8 b(T)1742 b Fr(4)21 337 y(that)22 b(\005)295 352 y Fk(max)430 337 y Fr(\()p Fn(S;)17 b(E)6 b(;)17 b(!)t Fr(\))k(con)m(tains)h(man)m (y)g(paths,)h(their)e(n)m(um)m(b)s(er)h(presumably)g(gro)m(wing)f(exp)s (onen)m(tially)21 457 y(in)j Fn(t)p Fr(.)h(The)g(resulting)f(net)m(w)m (ork)i(of)e(lines)g(has)h(some)g(resem)m(blance)g(to)f(a)g(riv)m(er)h (net)m(w)m(ork)h(with)f(\(0)p Fn(;)17 b Fr(0\))23 b(as)21 577 y(the)34 b(mouth)f(or)g(to)g(a)g(system)i(of)e(blo)s(o)s(d)e(v)m (essels,)36 b(see)f([Me])f(for)f(related)g(mo)s(dels.)f(T)-8 b(o)34 b(c)m(haracterize)21 698 y(the)44 b(net)m(w)m(ork)h(a)e(natural) f(geometrical)f(ob)5 b(ject)45 b(is)d(the)i Fi(last)h(br)-5 b(anching)51 b Fr(for)43 b(a)g(pair)f(of)h(directed)21 818 y(p)s(olymers)i(with)g(distinct)f(end)i(p)s(oin)m(ts)f([FH].)h(As)f (in)g(Figure)f(1)h(the)h(starting)f(p)s(oin)m(t)f(is)h(alw)m(a)m(ys)21 939 y Fn(S)38 b Fr(=)31 b(\(0)p Fn(;)17 b Fr(0\))34 b(and)h(the)h(end)f (p)s(oin)m(t)f Fn(E)41 b Fr(m)m(ust)36 b(lie)d(on)i(the)g(line)f Fn(U)2335 954 y Fj(t)2365 939 y Fr(.)h(If)g Fn(\031)2582 954 y Fj(i)2645 939 y Fr(is)g(a)f(maximizer)f(with)h(start)21 1059 y(p)s(oin)m(t)j Fn(S)43 b Fr(=)37 b(\(0)p Fn(;)17 b Fr(0\))37 b(and)h(end)g(p)s(oin)m(t)f Fn(E)1468 1074 y Fj(i)1534 1059 y Fl(2)g Fn(U)1703 1074 y Fj(t)1733 1059 y Fr(,)h Fn(i)f Fr(=)f(1)p Fn(;)17 b Fr(2,)38 b(then)g(the)h(last) e(p)s(oin)m(t)g(in)g(whic)m(h)i Fn(\031)3497 1074 y Fk(1)3574 1059 y Fr(and)21 1179 y Fn(\031)76 1194 y Fk(2)147 1179 y Fr(in)m(tersect)33 b(is)d(denoted)j(b)m(y)f Fn(I)8 b Fr(\()p Fn(\031)1282 1194 y Fk(1)1322 1179 y Fn(;)17 b(\031)1421 1194 y Fk(2)1460 1179 y Fr(\).)32 b(W)-8 b(e)31 b(de\014ne)i(the)f Fi(last)i(interse)-5 b(ction)33 b(p)-5 b(oint)40 b Fr(for)31 b(t)m(w)m(o)h(sets)h(of)21 1300 y(maximizers)e(b)m(y)731 1520 y Fn(J)9 b Fr(\()p Fn(E)904 1535 y Fk(1)944 1520 y Fn(;)17 b(E)1060 1535 y Fk(2)1099 1520 y Fr(\))28 b(=)f Fn(I)8 b Fr(\()p Fn(\031)1412 1535 y Fk(1)1452 1520 y Fn(;)17 b(\031)1551 1535 y Fk(2)1590 1520 y Fr(\))33 b(whic)m(h)g(minimizes)d Fn(d)p Fr(\()p Fn(I)8 b Fr(\()p 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(ossibilit)m(y)e(that)i(for)f Fn(\027)k Fr(=)30 b(2)p Fn(=)p Fr(3)k(the)g(distribution)e(of)i(the)21 5623 y(last)29 b(in)m(tersection)h(p)s(oin)m(t)f(is)g(degenerate)i(near)f(the)g (origin.)e(In)i(fact)f(the)i(pro)s(of)d(exploits)i(geometric)p eop %%Page: 5 5 5 4 bop 21 29 a Fh(2)97 b(LAST)34 b(BRANCHING)2609 b Fr(5)21 337 y(asp)s(ects)37 b(for)f(branc)m(hing)g(p)s(oin)m(ts)f (close)h(to)g Fn(U)1695 352 y Fj(t)1724 337 y Fr(,)g(whic)m(h)h(cannot) f(b)s(e)g(used)h(to)f(obtain)f(sharp)h(results)21 457 y(close)d(to)f(the)h(origin.)167 577 y(Another)28 b(w)m(a)m(y)g(to)e(c) m(haracterize)i(the)f(net)m(w)m(ork)i(of)d(Figure)g(1)g(is)h(to)f (consider)h(the)g(line)f(densit)m(y)i(at)21 698 y(the)e(cross-section)f Fn(U)805 713 y Fj(s)842 698 y Fr(,)g(equiv)-5 b(alen)m(tly)24 b(the)i(t)m(ypical)e(distance)h(b)s(et)m(w)m(een)i(maximizers)c(when)j (crossing)21 818 y Fn(U)87 833 y Fj(s)124 818 y Fr(.)31 b(T)-8 b(o)31 b(ha)m(v)m(e)h(a)f(de\014nition,)f(for)g(giv)m(en)i Fn(!)i Fr(let)p 1721 738 105 4 v 30 w Fn(M)1826 833 y Fj(t)1886 818 y Fr(b)s(e)d(all)e(the)j(maximizers)d(with)h(end)i(p)s (oin)m(ts)e(in)h Fn(U)3702 833 y Fj(t)21 939 y Fr(considered)k(as)f(a)f (subset)i(of)e Fl(f)p Fr(\()p Fn(x)1258 954 y Fk(1)1298 939 y Fn(;)17 b(x)1397 954 y Fk(2)1436 939 y Fr(\))p Fn(;)g Fr(0)29 b Fl(\024)h Fn(x)1758 954 y Fk(1)1821 939 y Fr(+)23 b Fn(x)1975 954 y Fk(2)2044 939 y Fl(\024)30 b Fr(2)p Fn(t)p Fl(g)p Fr(.)p 2346 859 V 34 w Fn(M)2451 954 y Fj(t)2514 939 y Fr(consists)k(of)f(straigh)m(t)g(segmen)m(ts)21 1059 y(connecting)41 b(t)m(w)m(o)g(p)s(oin)m(ts)f(of)g Fn(!)j Fr(and)e(straigh)m(t)e(segmen)m(ts)j(connecting)e(0)g(with)g(a)g (p)s(oin)m(t)g(in)f Fn(!)t Fr(.)h(In)21 1179 y(addition)30 b(there)j(is)f(a)f(union)h(of)f(triangles)g(with)g(base)i(con)m(tained) f(in)f Fn(U)2667 1194 y Fj(t)2729 1179 y Fr(and)h(the)h(ap)s(ex)f(a)g (p)s(oin)m(t)f(of)21 1300 y Fn(!)t Fr(.)g(W)-8 b(e)32 b(de\014ne)h Fn(M)686 1315 y Fj(t)748 1300 y Fr(to)e(b)s(e)p 998 1220 V 32 w Fn(M)1103 1315 y Fj(t)1164 1300 y Fr(suc)m(h)i(that)f (in)f(ev)m(ery)i(triangle)d(only)h(the)i(t)m(w)m(o)f(sides)g(emerging)f (from)21 1420 y(the)i(ap)s(ex)g(are)g(retained.)f(Let)971 1613 y Fn(N)1049 1628 y Fj(t)1079 1613 y Fr(\()p Fn(s)p Fr(\))27 b(=)h(#)33 b(of)f(p)s(oin)m(ts)g(of)g Fn(M)2055 1628 y Fj(t)2107 1613 y Fl(\\)23 b Fn(U)2262 1628 y Fj(s)2299 1613 y Fn(;)33 b Fr(0)27 b Fl(\024)i Fn(s)e Fl(\024)h Fn(t:)826 b Fr(\(6\))21 1815 y(If)38 b Fn(s)d Fr(=)h Fn(c)17 b(t)p Fr(,)38 b(0)d Fn(<)h(c)g(<)g Fr(1,)h(then)h(the)g(t)m (ypical)f(distance)h(b)s(et)m(w)m(een)h(lines)e(is)g(of)g(order)g Fn(t)3207 1779 y Fk(2)p Fj(=)p Fk(3)3355 1815 y Fr(and)g(th)m(us)21 1935 y(one)c(exp)s(ects)1593 2056 y Fn(N)1671 2071 y Fj(t)1701 2056 y Fr(\()p Fn(ct)p Fr(\))28 b Fl(')g Fn(t)2022 2015 y Fk(1)p Fj(=)p Fk(3)2132 2056 y Fn(:)21 2218 y Fr(On)k(the)h(other)f(hand)g(for)g(a)g(cross-section)h(closer)f(to)g Fn(U)2099 2233 y Fj(t)2160 2218 y Fr(the)h(n)m(um)m(b)s(er)g(of)e(p)s (oin)m(ts)h(should)g(increase)21 2339 y(faster.)h(In)g(particular)e Fn(N)967 2354 y Fj(t)997 2339 y Fr(\()p Fn(t)p Fr(\))c Fl(')h Fn(t)p Fr(.)33 b(This)g(suggests)h(that)1512 2531 y Fn(N)1590 2546 y Fj(t)1620 2531 y Fr(\()p Fn(t)22 b Fl(\000)h Fn(t)1850 2490 y Fj(\026)1897 2531 y Fr(\))k Fl(')i Fn(t)2103 2490 y Fj(g)r Fk(\()p Fj(\026)p Fk(\))21 2724 y Fr(with)j Fn(g)t Fr(\(0\))27 b(=)g(1)33 b(and)f Fn(g)t Fr(\(1\))27 b(=)h(1)p Fn(=)p Fr(3.)k(In)g(the)h(last)f(section)h (w)m(e)h(pro)m(v)m(e)f(the)g(lo)m(w)m(er)g(b)s(ound)1569 2961 y Fn(g)t Fr(\()p Fn(\026)p Fr(\))27 b Fl(\025)1897 2893 y Fr(5)p 1897 2938 49 4 v 1897 3029 a(6)1978 2961 y Fl(\000)2087 2893 y Fn(\026)p 2087 2938 59 4 v 2092 3029 a Fr(2)2156 2961 y Fn(:)21 3289 y Fm(2)161 b(Last)54 b(branc)l(hing)21 3508 y Fr(W)-8 b(e)30 b(plan)f(to)g(pro)m(v)m(e)h (Theorem)g(1.)f(Before)h(w)m(e)h(in)m(tro)s(duce)e(some)g(notation)f (and)i(state)g(some)f(results)21 3629 y(of)j([BDJ])h(concerning)f (large)g(deviations)g(for)g(the)h(length)f(of)g(maximizers.)167 3749 y(F)-8 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Fr(2)1443 5498 y Fb(p)p 1542 5498 V 1542 5588 a Fn(a)p Fr(\()p Fn(w)s(;)17 b(w)1821 5559 y Ff(0)1843 5588 y Fr(\))22 b Fl(\000)h(j)p Fn(n)p Fl(j)p Fr(\))k Fl(\024)h Fn(c)2329 5603 y Fk(1)2385 5588 y Fr(exp)q(\()p Fl(\000)p Fn(c)2691 5603 y Fk(2)2731 5588 y Fl(j)p Fn(\034)11 b Fl(j)2840 5547 y Fk(3)2879 5588 y Fr(\))p Fn(:)663 b Fr(\(8\))p eop %%Page: 6 6 6 5 bop 21 29 a Fh(2)97 b(LAST)34 b(BRANCHING)2609 b Fr(6)167 337 y(Our)45 b(\014rst)h(step)g(is)f(to)f(pro)m(v)m(e)j(a)e (lemma)d(on)j(the)h(length)e(\(as)i(in)e([Jo],)h(Lemma)f(3.1\))g(and)i (a)21 457 y(geometric)36 b(lemma)f(b)s(ecause)k(b)s(oth)e(will)e(b)s(e) i(used)h(in)f(the)h(pro)s(ofs.)f(Let)g Fl(Z)45 b Fr(b)s(e)38 b(a)f(set)h(of)e(p)s(oin)m(ts)h(in)21 577 y Fg(R)87 541 y Fk(2)87 602 y(+)185 577 y Fr(suc)m(h)e(that)e Fl(jZ)8 b(j)28 b(\024)i Fn(t)924 541 y Fj(m)1024 577 y Fr(for)j(a)g(\014nite)g Fn(m)g Fr(and)g(let)g(for)g(eac)m(h)h Fn(z)g Fl(2)29 b(Z)41 b Fr(b)s(e)34 b Fn(z)2796 541 y Ff(0)2849 577 y Fr(=)28 b Fn(z)g 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3614 y Fd(2)3337 3633 y Fk(\))3375 3496 y Fb(\000)3420 3532 y Ff(\000)p Fk(\(1+)p Fj(k)r Fk(\))3475 3611 y(1)p Ff(\000)p Fj(k)3659 3496 y Fb(\001)3704 3577 y Fr(.)21 3744 y(Then)40 b Fn(z)k Fr(=)38 b Fn(E)33 b Fl(\006)27 b Fn(w)s(e)812 3759 y Fk(2)878 3744 y Fl(\000)g Fn(\025l)r(e)1115 3759 y Fk(1)1194 3744 y Fr(with)38 b Fn(\025)h Fl(\024)g Fr(1)g(suc)m(h)h(that)f Fn(z)44 b Fl(2)39 b Fr([0)p Fn(;)17 b(E)6 b Fr(].)39 b(F)-8 b(or)38 b(the)h(computations)f(w)m(e)21 3865 y(consider)g(the)f("+")g(case,)h(the)g(")p Fl(\000)p Fr(")f(case)h(is)f(obtained)f(replacing)g Fn(w)k Fr(with)d Fl(\000)p Fn(w)j Fr(at)d(the)g(end.)h(Let)21 3985 y Fn(Q)28 b Fr(=)230 3900 y Fb(p)p 329 3900 388 4 v 329 3985 a Fr(2\(1)22 b(+)g Fn(k)639 3956 y Fk(2)678 3985 y Fr(\))33 b(and)g Fn(l)970 3949 y Ff(0)1021 3985 y Fr(=)27 b Fn(\025l)e Fl(\000)d Fr(1.)33 b(Then)917 4246 y Fn(z)f Fr(=)1098 4105 y Fb(\022)1171 4178 y Fn(t)p Fr(\(1)22 b Fl(\000)h Fn(k)s Fr(\))f Fl(\000)h Fn(w)s Fr(\(1)e(+)h Fn(k)s Fr(\))p Fn(=Q)g Fl(\000)h Fn(\025l)r Fr(\(1)f Fl(\000)g Fn(k)s Fr(\))p Fn(=Q)1173 4314 y(t)p Fr(\(1)g(+)g Fn(k)s Fr(\))g(+)g Fn(w)s Fr(\(1)f Fl(\000)i Fn(k)s Fr(\))p Fn(=Q)f Fl(\000)h Fn(\025l)r Fr(\(1)f(+)g Fn(k)s Fr(\))p Fn(=Q)2762 4105 y Fb(\023)21 4495 y Fr(and)908 4648 y Fn(z)957 4607 y Ff(0)1009 4648 y Fr(=)1113 4508 y Fb(\022)1186 4581 y Fn(t)p Fr(\(1)g Fl(\000)h Fn(k)s Fr(\))f Fl(\000)h Fn(w)s Fr(\(1)e(+)h Fn(k)s Fr(\))p Fn(=Q)g Fl(\000)h Fn(l)2294 4545 y Ff(0)2317 4581 y Fr(\(1)f Fl(\000)h Fn(k)s Fr(\))p Fn(=Q)1188 4717 y(t)p Fr(\(1)f(+)g Fn(k)s Fr(\))g(+)g Fn(w)s Fr(\(1)f Fl(\000)i Fn(k)s Fr(\))p Fn(=Q)f Fl(\000)h Fn(l)2294 4688 y Ff(0)2317 4717 y Fr(\(1)f(+)g Fn(k)s Fr(\))p Fn(=Q)2744 4508 y Fb(\023)2817 4648 y Fn(:)21 4865 y Fr(Expansion)33 b(leads)g(to)f(the)h(follo)m(wing)d(results,)539 5044 y Fb(p)p 639 5044 293 4 v 91 x Fn(a)p Fr(\(0)p Fn(;)17 b(z)870 5106 y Ff(0)894 5135 y Fr(\))83 b(=)f Fn(t)1208 5048 y Fl(p)p 1292 5048 265 4 v 1292 5135 a Fr(1)22 b Fl(\000)g Fn(k)1516 5106 y 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Fk(2)p 1435 5559 265 4 v 1436 5650 a Fr(1)f(+)h Fn(k)1658 5621 y Fk(2)1709 5582 y Fn(f)11 b Fr(\()p Fn(k)s(;)17 b(w)s(=\025l)r Fr(\))p Fn(;)p eop %%Page: 8 8 8 7 bop 21 29 a Fh(2)97 b(LAST)34 b(BRANCHING)2609 b Fr(8)21 337 y(where)34 b Fn(f)11 b Fr(\()p Fn(k)s(;)17 b(\020)8 b Fr(\))26 b(=)717 251 y Fb(p)p 816 251 968 4 v 816 337 a Fr(1)c(+)g(4)p Fn(k)s(\020)8 b Fr(\(1)21 b Fl(\000)i Fn(k)1401 308 y Fk(2)1440 337 y Fr(\))1478 308 y Ff(\000)p Fk(1)1595 337 y Fl(\000)f Fn(\020)1745 308 y Fk(2)1784 337 y Fr(.)32 b(It)h(follo)m(ws)e(that)336 571 y Fb(p)p 436 571 293 4 v 90 x Fn(a)p Fr(\(0)p Fn(;)17 b(z)667 632 y Ff(0)691 661 y Fr(\))22 b(+)849 571 y Fb(p)p 948 571 299 4 v 948 661 a Fn(a)p Fr(\()p Fn(z)t(;)17 b(E)6 b Fr(\))23 b Fl(\000)1369 571 y Fb(p)p 1468 571 298 4 v 1468 661 a Fn(a)p Fr(\(0)p Fn(;)17 b(E)6 b Fr(\))28 b(=)f Fl(\000)1974 483 y Fb(s)p 2074 483 408 4 v 2146 593 a Fr(1)22 b Fl(\000)g Fn(k)2370 564 y Fk(2)p 2084 638 388 4 v 2084 729 a Fr(2\(1)g(+)g Fn(k)2394 700 y Fk(2)2433 729 y Fr(\))2481 661 y Fn(h)p Fr(\()p Fn(k)s(;)17 b(w)s(;)g(\025l)r Fr(\))k(+)h Fl(O)s Fr(\()p Fn(w)3228 619 y Fk(2)3267 661 y Fn(=t)p Fr(\))p Fn(;)21 955 y Fr(where)581 1149 y Fn(h)p Fr(\()p Fn(k)s(;)17 b(w)s(;)g(\025l)r Fr(\))26 b(=)i Fn(\025l)1234 1008 y Fb(\022)1307 1149 y Fr(1)22 b Fl(\000)1478 971 y Fb(s)p 1578 971 926 4 v 178 x Fr(1)g(+)1883 1081 y(4)p Fn(k)s(w)p 1757 1126 429 4 v 1757 1217 a(\025l)r Fr(\(1)g Fl(\000)g Fn(k)2107 1188 y Fk(2)2147 1217 y Fr(\))2217 1149 y Fl(\000)2354 1081 y Fn(w)2427 1053 y Fk(2)p 2326 1126 168 4 v 2326 1217 a Fn(\025)2383 1188 y Fk(2)2423 1217 y Fn(l)2454 1188 y Fk(2)2525 1149 y Fr(+)2760 1081 y(2)p Fn(k)s(w)p 2633 1126 429 4 v 2633 1217 a(\025l)r Fr(\(1)g Fl(\000)h Fn(k)2984 1188 y Fk(2)3024 1217 y Fr(\))3071 1008 y Fb(\023)3145 1149 y Fn(:)21 1410 y Fr(It)33 b(is)f(easy)h(to)g(see)g(that)g Fn(h)p Fr(\()p Fn(k)s(;)17 b(w)s(;)g(\025l)r Fr(\))27 b Fl(\025)h Fr(0)k(\(in)g(fact,)g Fn(h)p Fr(\()p Fn(k)s(;)17 b(w)s(;)g(\025l)r Fr(\))27 b(=)h(0)k(only)g(if)2921 1371 y Fk(1)p 2907 1387 64 4 v 2907 1445 a Fj(\025l)3008 1410 y Fr(=)27 b(0\).)33 b(Moreo)m(v)m(er)990 1699 y Fn(h)p Fr(\()p Fn(k)s(;)17 b(w)s(;)g(\025l)r Fr(\))27 b Fl(\030)1580 1632 y Fr(\()p Fn(k)1672 1595 y Fk(2)1734 1632 y Fr(+)22 b(1\))1919 1595 y Fk(2)1958 1632 y Fn(w)2031 1595 y Fk(2)p 1567 1676 517 4 v 1567 1767 a Fr(2\()p Fn(k)1708 1739 y Fk(2)1769 1767 y Fl(\000)h Fr(1\))1956 1739 y Fk(2)1995 1767 y Fn(\025l)2115 1699 y Fr(+)f Fl(O)s Fr(\()p Fn(w)2406 1658 y Fk(3)2445 1699 y Fn(=)p Fr(\()p Fn(\025l)r Fr(\))2658 1658 y Fk(2)2697 1699 y Fr(\))p Fn(;)21 1969 y Fr(and)33 b(the)g(minimal)28 b(v)-5 b(alue)32 b(is)g(obtained)g(for)g Fn(\025)c Fr(=)f(1.)33 b(Consequen)m(tly)-8 b(,)35 b(for)d Fn(l)e Fr(=)d Fn(t)2925 1933 y Fj(\027)3001 1969 y Fr(large)32 b(enough,)353 2153 y Fb(p)p 453 2153 293 4 v 90 x Fn(a)p Fr(\(0)p Fn(;)17 b(z)684 2214 y Ff(0)707 2243 y Fr(\))22 b(+)865 2153 y Fb(p)p 965 2153 299 4 v 90 x Fn(a)p Fr(\()p Fn(z)t(;)17 b(E)6 b Fr(\))23 b 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Fn(z)2013 2751 y Ff(0)2065 2792 y Fr(=)2168 2652 y Fb(\022)2242 2725 y Fn(\013)2305 2689 y Ff(0)2328 2725 y Fn(t)p Fr(\(1)22 b Fl(\000)g Fn(k)s Fr(\))h Fl(\000)f Fn(w)s Fr(\(1)f(+)h Fn(k)s Fr(\))p Fn(=Q)2243 2860 y(\013)2306 2832 y Ff(0)2328 2860 y Fn(t)p Fr(\(1)h(+)f Fn(k)s Fr(\))g(+)g Fn(w)s Fr(\(1)f Fl(\000)i Fn(k)s Fr(\))p Fn(=Q)3282 2652 y Fb(\023)21 3024 y Fr(where)34 b Fn(\013)28 b Fl(2)g Fr(\(0)p Fn(;)17 b Fr(1\))32 b(suc)m(h)i(that)e Fn(z)h Fl(2)28 b Fr([0)p Fn(;)17 b(E)6 b Fr(])32 b(and)h Fn(\013)1850 2988 y Ff(0)1901 3024 y Fr(=)27 b Fn(\013)c Fr(+)f(1)p Fn(=tQ)p Fr(.)33 b(The)g(expansion)g(yields)f(to)101 3213 y Fb(p)p 201 3213 293 4 v 90 x Fn(a)p Fr(\(0)p Fn(;)17 b(z)432 3274 y Ff(0)455 3303 y Fr(\))23 b(+)614 3213 y Fb(p)p 713 3213 299 4 v 713 3303 a Fn(a)p Fr(\()p Fn(z)t(;)17 b(E)6 b Fr(\))23 b Fl(\000)1134 3213 y Fb(p)p 1233 3213 298 4 v 1233 3303 a Fn(a)p Fr(\(0)p Fn(;)17 b(E)6 b Fr(\))84 b(=)f Fl(\000)1868 3162 y Fb(\022)1959 3236 y Fr(1)p 1952 3280 63 4 v 1952 3371 a 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a(6)584 1224 y Fl(\000)693 1180 y Fj(\026)p 693 1201 43 4 v 697 1258 a Fk(2)746 1224 y Fn(:)21 1417 y Fo(Pr)n(oof.)49 b Fr(The)34 b(\014rst)f(part)f(of)g(the)h(pro)s(of)f(is)g(close)h(to)f (the)h(one)g(of)f(Theorem)h(1)f(ii\).)167 1538 y(As)i(in)d(Figure)h(3)g (let)g(us)h(consider)g(t)m(w)m(o)g(\014xed)h(p)s(oin)m(ts)825 1752 y Fn(E)897 1767 y Fk(1)965 1752 y Fr(=)27 b(\()p Fn(t)p Fr(\(1)22 b Fl(\000)h Fn(k)s Fr(\))p Fn(;)17 b(t)p Fr(\(1)22 b(+)g Fn(k)s Fr(\)\))32 b(and)h Fn(E)2152 1767 y Fk(2)2219 1752 y Fr(=)28 b Fn(E)2395 1767 y Fk(1)2456 1752 y Fr(+)22 b Fn(t)2589 1711 y Fj(\027)2633 1752 y Fr(\()p Fl(\000)p Fr(1)p Fn(;)17 b Fr(1\))21 1967 y(with)39 b Fn(k)i Fl(2)e Fr(\()p Fl(\000)p Fr(1)p Fn(;)17 b Fr(1\).)39 b(W)-8 b(e)39 b(lo)s(ok)f(at)g(the)h(region)f(closer)h(than)g Fn(l)i Fr(=)d Fn(t)2529 1931 y Fj(\026)2615 1967 y Fr(from)g(the)h (line)e Fn(U)3282 1982 y Fj(t)3312 1967 y Fr(.)i(W)-8 b(e)40 b(tak)m(e)21 2088 y Fn(w)c Fr(=)e Fn(t)272 2051 y Fj(\027)315 2088 y Fn(=)p 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Fr(Therefore)34 b Fg(P)p Fr(\()p Fn(F)14 b Fr(\))29 b Fl(\024)f Fr(1)p Fn(=t)942 5373 y Fk(2)1014 5410 y Fr(for)k Fn(t)h Fr(large)e(enough.)i (Consequen)m(tly)1217 5624 y Fg(P)p Fr(\(\012)24 b Fl(n)e Fn(D)s Fr(\))27 b Fl(\024)i Fn(t)1770 5583 y Ff(\000)p Fk(2)1886 5624 y Fr(+)22 b Fg(P)p Fr(\()p Fn(F)14 b Fr(\))29 b Fl(\024)f Fr(2)p Fn(t)2414 5583 y Ff(\000)p Fk(2)2509 5624 y Fn(:)p eop %%Page: 13 13 13 12 bop 21 29 a Fh(3)97 b(DENSITY)34 b(OF)e(BRANCHES)2265 b Fr(13)167 337 y(De\014ne)34 b(the)f(set)g(of)f(ev)m(en)m(ts)745 528 y Fn(Q)c Fr(=)f Fl(f)p Fn(!)k Fl(2)d Fr(\012)33 b(s.t.)g Fn(d)p Fr(\()p Fn(J)9 b Fr(\()p Fn(E)1717 543 y Fk(1)1757 528 y Fn(;)17 b(E)1873 543 y Fk(2)1912 528 y Fr(\)\()p Fn(!)t Fr(\))p Fn(;)g(U)2201 543 y Fj(t)2230 528 y Fr(\))27 b Fl(\024)i Fn(l)34 b Fr(with)f Fn(l)c Fr(=)f Fn(t)2884 487 y Fj(\026)2931 528 y Fl(g)p Fn(:)21 719 y Fr(W)-8 b(e)33 b(pro)m(v)m(e)h(that)e(for)g Fn(t)h Fr(large)f(enough)1185 911 y Fg(P)p Fr(\()p Fn(Q)p Fr(\))e Fl(\024)e Fn(t)1567 869 y Ff(\000)p Fk(2)1694 911 y Fr(for)k(all)f Fn(\026)c(<)h Fr(2)p Fn(\027)g Fl(\000)23 b Fr(1)p Fn(=)p Fr(3)p Fn(:)990 b Fr(\(24\))21 1102 y(W)-8 b(e)38 b(consider)g(the)f(ev)m(en)m(t)j Fn(T)49 b Fr(=)36 b Fn(Q)25 b Fl(\\)h Fn(D)40 b Fr(with)d Fn(\026)f(<)f Fr(2)p Fn(\027)d Fl(\000)26 b Fr(1)p Fn(=)p Fr(3.)37 b(As)h(in)e(the)i(previous)g(pro)s(of,)e(for)h Fn(t)21 1222 y Fr(large)i(enough)h(w)m(e)g(ha)m(v)m(e)h Fg(P)p Fr(\()p Fn(Q)p Fr(\))h Fl(\024)e Fg(P)p Fr(\()p Fn(T)14 b Fr(\))28 b(+)f Fg(P)p Fr(\(\012)i Fl(n)e Fn(D)s Fr(\).)39 b(If)g Fn(!)k Fl(2)d Fn(T)53 b Fr(then)41 b(the)f(lo)m(w)m (est)g(maximizer)21 1343 y(from)31 b(0)i(to)f Fn(E)524 1358 y Fk(2)596 1343 y Fr(in)m(tersects)i Fn(@)5 b(C)i Fr(\()p Fn(w)s(;)17 b(l)r Fr(\))1385 1358 y Fk(+)1476 1343 y Fr(at)32 b(some)h(p)s(oin)m(t)e Fn(H)8 b Fr(.)32 b(W)-8 b(e)33 b(de\014ne)h Fn(h)p Fr(\()p Fn(!)t Fr(\))e(and)g Fn(h)3166 1306 y Ff(0)3190 1343 y Fr(\()p Fn(!)t Fr(\))g(as)g(in)g(the) 21 1463 y(previous)h(pro)s(of)f(and)h(for)f Fn(!)f Fl(2)d Fn(T)46 b Fr(w)m(e)34 b(ha)m(v)m(e)821 1654 y Fn(L)p Fr(\(0)p Fn(;)17 b(E)1090 1669 y Fk(2)1130 1654 y Fr(\))27 b Fl(\024)h Fr(2)p Fn(t)1384 1613 y Fj(\016)1444 1654 y Fr(+)22 b(2\()1629 1564 y Fb(p)p 1729 1564 440 4 v 90 x Fn(a)p Fr(\(0)p Fn(;)17 b(h)1967 1626 y Ff(0)1990 1654 y Fr(\()p Fn(!)t Fr(\)\))k(+)2288 1564 y Fb(p)p 2388 1564 479 4 v 90 x Fn(a)p Fr(\()p Fn(h)p Fr(\()p Fn(!)t Fr(\))p Fn(;)c(E)2790 1669 y Fk(2)2828 1654 y Fr(\)\))p Fn(:)21 1846 y Fr(W)-8 b(e)35 b(compute)g(the)g(minimal)30 b(distance)35 b Fn(d)1573 1861 y Fj(m)1674 1846 y Fr(b)s(et)m(w)m(een)h Fn(@)5 b(C)i Fr(\()p Fn(w)s(;)17 b(l)r Fr(\))2409 1861 y Fk(+)2503 1846 y Fr(and)35 b(the)g(segmen)m(t)p 3244 1766 161 4 v 35 w(0)p Fn(E)3365 1861 y Fk(2)3439 1846 y Fr(\014nding)21 1993 y Fn(d)72 2008 y Fj(m)166 1993 y Fr(=)270 1883 y Fb(\020)403 1899 y Ff(p)p 462 1899 36 3 v 55 x Fk(2)p 339 1970 223 4 v 339 1980 a Ff(p)p 398 1980 164 3 v 56 x Fk(1+)p Fj(k)527 2017 y Fd(2)593 1993 y Fl(\000)703 1954 y Fk(1)p 703 1970 36 4 v 703 2028 a(2)748 1883 y Fb(\021)824 1993 y Fn(t)859 1957 y Fj(\027)925 1993 y Fr(+)22 b Fl(O)s Fr(\()p Fn(t)1178 1957 y Fj(\026)p Ff(\000)p Fk(1)1315 1993 y Fr(\).)32 b(Applying)g(Lemma)f(3)i(w)m(e)g(obtain)646 2145 y Fb(p)p 745 2145 440 4 v 745 2235 a Fn(a)p Fr(\(0)p Fn(;)17 b(h)983 2207 y Ff(0)1006 2235 y Fr(\()p Fn(!)t Fr(\)\))k(+)1304 2145 y Fb(p)p 1404 2145 479 4 v 90 x Fn(a)p Fr(\()p Fn(h)p Fr(\()p Fn(!)t Fr(\))p Fn(;)c(E)1806 2250 y Fk(2)1845 2235 y Fr(\))28 b Fl(\024)2016 2145 y Fb(p)p 2115 2145 332 4 v 2115 2235 a Fn(a)p Fr(\(0)p Fn(;)17 b(E)2369 2250 y Fk(2)2409 2235 y Fr(\))22 b Fl(\000)g Fn(C)2645 2194 y Ff(0)2669 2235 y Fr(\()p Fn(k)s Fr(\))p Fn(t)2834 2194 y Fk(2)p Fj(\027)t Ff(\000)p Fj(\026)3010 2235 y Fn(=)p Fr(2)21 2471 y(pro)m(vided)54 b(that)f Fn(\026)62 b(<)h Fr(2)p Fn(\027)42 b Fl(\000)37 b Fr(1)p Fn(=)p Fr(3)52 b(and)h(with)g Fn(C)1917 2435 y Ff(0)1940 2471 y Fr(\()p Fn(k)s Fr(\))63 b(=)f Fn(C)7 b Fr(\()p Fn(k)s Fr(\))2495 2361 y Fb(\020)2628 2377 y 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Ff(0)3499 2821 y Fr(\()p Fn(k)s Fr(\))p Fn(t)3664 2780 y Fj( )3717 2821 y Fr(\))p Fn(:)21 3012 y Fr(Applying)k(\(8\))g (with)g Fn(n)c Fr(=)g Fl(\000)p Fn(C)1167 2976 y Ff(0)1190 3012 y Fr(\()p Fn(k)s Fr(\))p Fn(t)1355 2976 y Fj( )1441 3012 y Fr(w)m(e)33 b(obtain)1234 3204 y Fg(P)p Fr(\()p Fn(T)14 b Fr(\))29 b Fl(\024)f Fn(c)1616 3219 y Fk(1)1672 3204 y Fr(exp)q(\()p Fl(\000)p Fn(c)1978 3219 y Fk(2)2018 3204 y Fn(C)2095 3163 y Ff(00)2137 3204 y Fr(\()p Fn(k)s Fr(\))p Fn(t)2302 3163 y Fk(3)p Fj( )r Ff(\000)p Fk(1)2481 3204 y Fr(\))21 3395 y(with)33 b Fn(C)321 3359 y Ff(00)363 3395 y Fr(\()p Fn(k)s Fr(\))c(=)g Fn(C)704 3359 y Ff(0)727 3395 y Fr(\()p Fn(k)s Fr(\))857 3359 y Fk(3)897 3395 y Fn(=)p Fr(2)995 3313 y Fl(p)p 1077 3313 265 4 v 1077 3395 a Fr(1)22 b Fl(\000)h Fn(k)1302 3366 y Fk(2)1341 3395 y Fr(.)34 b(Therefore)g Fg(P)p Fr(\()p Fn(T)14 b Fr(\))31 b Fl(\024)e Fr(1)p Fn(=t)2320 3359 y Fk(2)2392 3395 y Fr(for)k Fn(t)h Fr(large)e(enough.)i(Finally)c(for)j Fn(t)21 3515 y 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b(for)21 4159 y Fn(j)35 b Fr(=)29 b Fl(\000)p Fn(M)5 b(;)17 b(:)g(:)g(:)34 b(;)17 b(M)10 b Fr(.)34 b(Let)f Fn(W)14 b Fr(\()p Fn(j)6 b Fr(\))33 b(b)s(e)h(the)g(set)g(of)f (all)e(in)m(tersections)j(b)s(et)m(w)m(een)h(the)f(maximizers)e(with)21 4279 y(end)38 b(p)s(oin)m(t)e(at)h Fn(T)650 4294 y Fj(j)724 4279 y Fr(and)g(the)h(ones)f(with)g(end)h(p)s(oin)m(t)e(at)h Fn(T)2168 4294 y Fj(j)t Fk(+1)2295 4279 y Fr(.)g(W)-8 b(e)38 b(de\014ne)g Fn(m)p Fr(\()p Fn(j)6 b Fr(\))37 b(to)g(b)s(e)g(the)h(set)g(of)21 4400 y(p)s(oin)m(ts)32 b(of)g Fn(W)14 b Fr(\()p Fn(j)6 b Fr(\))32 b(whose)i(distance)f(to)f Fn(U)1538 4415 y Fj(t)1601 4400 y Fr(is)g(at)g(most)g Fn(l)e Fr(=)e Fn(t)2255 4364 y Fj(\026)2301 4400 y Fr(.)33 b(Then)100 4684 y Fg(P)p Fr(\()p Fl(9)17 b Fn(j)40 b Fr(s.t.)33 b Fn(m)p Fr(\()p Fn(j)6 b Fr(\))28 b Fl(6)p Fr(=)f Fl(;)p Fr(\))h(=)f Fg(P)1128 4543 y Fb(\022)1246 4559 y Fj(M)7 b Ff(\000)p Fk(1)1273 4589 y Fb([)1220 4801 y Fj(j)t Fk(=)p Ff(\000)p Fj(M)1454 4684 y Fn(m)p Fr(\()p Fn(j)f Fr(\))28 b Fl(6)p Fr(=)f Fl(;)1842 4543 y Fb(\023)1943 4684 y Fl(\024)h Fr(2)p Fn(M)181 b Fr(max)2218 4746 y Fj(j)t Fk(=)p Ff(\000)p Fj(M)s(;:::)10 b(;M)d Ff(\000)p Fk(1)2723 4684 y Fg(P)p Fr(\()p Fn(m)p Fr(\()p Fn(j)f Fr(\))30 b Fl(6)p Fr(=)d Fl(;)p Fr(\))g Fl(\024)i Fr(6)p Fn(t)3465 4643 y Ff(\000)p Fk(1)p Ff(\000)p Fj(\027)21 4977 y Fr(as)f Fn(t)g Fr(go)s(es)g(to)g(in\014nit)m(y)-8 b(.)27 b(Then)i(as)g Fn(t)f Fr(go)s(es)g(to)f(in\014nit)m(y)h(w)m(e)h (ha)m(v)m(e,)g(at)e(distance)i Fn(t)2870 4941 y Fj(\026)2944 4977 y Fr(with)f Fn(\026)f(<)h Fr(2)p Fn(\027)19 b Fl(\000)13 b Fr(1)p Fn(=)p Fr(3,)21 5098 y(at)35 b(least)f(2)p Fn(M)h Fr(+)23 b(1)32 b Fl(\030)g Fn(t)875 5062 y Fk(1)p Ff(\000)p Fj(\027)1043 5098 y Fr(branc)m(hes)37 b(that)e(ha)m(v)m(e)h(not)f(y)m (et)h(merged)e(with)h(probabilit)m(y)e(one.)i(Since)21 5218 y(for)d Fn(k)221 5233 y Fk(0)288 5218 y Fl(\034)27 b Fr(1,)32 b(2)p Fn(M)g Fr(+)22 b(1)27 b(=)h(2[\(1)21 b Fl(\000)h Fn(k)1310 5233 y Fk(0)1349 5218 y Fr(\))p Fn(t)1422 5182 y Fk(1)p Ff(\000)p Fj(\027)1556 5218 y Fr(])f(+)h(1)27 b Fl(\025)h Fn(t)1918 5182 y Fk(1)p Ff(\000)p Fj(\027)2052 5218 y Fr(,)k(for)g(all)e Fn(\033)i Fr(=)27 b(1)22 b Fl(\000)g Fn(\027)34 b(<)27 b Fr(5)p Fn(=)p Fr(6)21 b Fl(\000)h Fn(\026=)p Fr(2)32 b(w)m(e)h(ha)m(v)m(e)1301 5410 y(lim)1286 5469 y Fj(t)p Ff(!1)1469 5410 y Fg(P)17 b Fr(\()q Fn(N)1662 5425 y Fj(t)1692 5410 y Fr(\()p Fn(t)22 b Fl(\000)h Fn(t)1922 5368 y Fj(\026)1969 5410 y Fr(\))28 b Fl(\025)g Fn(t)2175 5368 y Fj(\033)2222 5410 y Fr(\))f(=)h(1)p Fn(:)p 3661 5624 4 66 v 3665 5562 59 4 v 3665 5624 V 3723 5624 4 66 v eop %%Page: 14 14 14 13 bop 21 29 a Fh(REFERENCES)2944 b Fr(14)21 337 y Fm(Ac)l(kno)l(wledgmen)l(ts)21 556 y Fr(W)-8 b(e)33 b(thank)g(Mic)m (hael)f(Pr\177)-49 b(ahofer)33 b(for)f(helping)f(us)j(to)e(generate)h (Figure)f(1.)21 888 y Fm(References)21 1107 y Fr([AD])145 b(D.)21 b(Aldous,)h(P)-8 b(.)22 b(Diaconis,)f Fi(L)-5 b(ongest)24 b(incr)-5 b(e)g(asing)24 b(subse)-5 b(quenc)g(es:)24 b(fr)-5 b(om)24 b(p)-5 b(atienc)g(e)25 b(sorting)f(to)368 1228 y(the)30 b(Baik-Deift-Johansson)c(The)-5 b(or)g(em)p Fr(,)26 b(Bull.)f(Amer.)i(Math.)g(So)s(c.)g Fc(36)p Fr(,)g(413{432)e (\(1999\).)21 1431 y([BDJ])99 b(J.)25 b(Baik,)f(P)-8 b(.A.)26 b(Deift,)d(K.)i(Johansson,)h Fi(On)h(the)h(distribution)g(of)g (the)f(length)h(of)f(the)h(longest)368 1552 y(incr)-5 b(e)g(asing)33 b(subse)-5 b(quenc)g(e)34 b(of)h(r)-5 b(andom)33 b(p)-5 b(ermutations)p Fr(,)32 b(J.)h(Amer.)f(Math.)h(So)s (c.)f Fc(12)h Fr(1119{)368 1672 y(1178)f(\(1999\).)21 1875 y([FH])156 b(D.S.)42 b(Fisher,)g(T.)g(Hw)m(a,)h Fi(A)n(nomalous)f(\015uctuations)i(of)f(dir)-5 b(e)g(cte)g(d)43 b(p)-5 b(olymers)43 b(in)g(r)-5 b(andom)368 1996 y(me)g(dia)p Fr(,)32 b(Ph)m(ys.)i(Rev.)f(B)g Fc(49)p Fr(,)g(3136{3154)d(\(1994\).)21 2199 y([Ha])171 b(J.M.)23 b(Hammersley)-8 b(,)21 b Fi(A)k(few)g(se)-5 b(e)g(d)5 b(lings)24 b(of)h(r)-5 b(ese)g(ar)g(ch)p Fr(,)21 b(Pro)s(c.)h(Sixth)g(Berk)m(eley)i(Symp.)e(Math.)368 2320 y(Statist.)32 b(and)h(Probabilit)m(y)d Fc(1)p Fr(,)j(345{394,)e (Univ)m(ersit)m(y)i(of)f(California)e(Press)k(\(1972\).)21 2523 y([Ke])174 b(H.)42 b(Kesten,)h Fi(Asp)-5 b(e)g(cts)43 b(of)h(\014rst-p)-5 b(assage)42 b(p)-5 b(er)g(c)g(olation)p Fr(,)41 b(Lecture)i(Notes)f(in)f(Math.)h Fc(1180)368 2643 y Fr(125{264)31 b(\(1986\).)21 2847 y([KMH])55 b(J.)38 b(Krug,)g(P)-8 b(.)39 b(Meakin,)f(T.)h(Halpin-Healy)-8 b(,)36 b Fi(A)n(mplitude)k(universality)f(for)h(driven)g(inter-)368 2967 y(fac)-5 b(es)30 b(and)f(dir)-5 b(e)g(cte)g(d)30 b(p)-5 b(olymers)30 b(in)g(r)-5 b(andom)30 b(me)-5 b(dia)p Fr(,)27 b(Ph)m(ys.)i(Rev.)g(A)e Fc(45)p Fr(,)h(638{653)e(\(1992\).)21 3171 y([Jo])194 b(K.)25 b(Johansson,)h Fi(T)-7 b(r)i(ansversal)27 b(\015uctuations)h(for)g(incr)-5 b(e)g(asing)27 b(subse)-5 b(quenc)g(es)27 b(on)h(the)g(plane)p Fr(,)368 3291 y(Probab.)33 b(Theory)h(Relat.)d(Fields)h Fc(116)p Fr(,)g(445{456)f(\(2000\).)21 3494 y([Me])161 b(P)-8 b(.)26 b(Meakin,)f Fi(F)-7 b(r)i(actals,)27 b(Sc)-5 b(aling)27 b(and)h(Gr)-5 b(owth)28 b(F)-7 b(ar)27 b(F)-7 b(r)i(om)27 b(Equilibrium)p Fr(,)e(Cam)m(bridge)g(Uni-)368 3615 y(v)m(ersit)m(y)34 b(Press,)g(Cam)m(bridge,)e(1998.)21 3818 y([Pi])199 b(M.S.T.)29 b(Piza,)e Fi(Dir)-5 b(e)g(cte)g(d)29 b(Polymers)h(in)g(a)g(r)-5 b(andom)29 b(envir)-5 b(onment:)29 b(some)g(r)-5 b(esults)30 b(on)g(\015uc-)368 3939 y(tuations)p Fr(,)j(J.)g(Stat.)f(Ph)m(ys.)i Fc(89)p Fr(,)f(581{603)e(\(1997\).)21 4142 y([PS])173 b(M.)34 b(Pr\177)-49 b(ahofer,)34 b(H.)f(Sp)s(ohn,)h Fi(Sc)-5 b(ale)35 b(invarianc)-5 b(e)34 b(of)h(the)h(PNG)h(dr)-5 b(oplet)35 b(and)g(the)h(A)n(iry)g(pr)-5 b(o-)368 4262 y(c)g(ess)p Fr(,)32 b(J.)h(Stat.)f(Ph)m(ys.)j Fc(108)p Fr(,)d(1071{1106)f(\(2002\).)21 4466 y([TW])123 b(C.A.)31 b(T)-8 b(racy)g(,)31 b(H.)g(Widom,)e Fi(L)-5 b(evel-sp)g(acing)31 b(distributions)i(and)g(the)g(A)n(iry)g(kernel)p Fr(,)d(Comm.)368 4586 y(Math.)j(Ph)m(ys.)h Fc(159)p Fr(,)f(151{174)e(\(1994\).)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0211181121808--