This is a multi-part message in MIME format. ---------------0212110511206 Content-Type: text/plain; name="02-514.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-514.comments" 21 pages, 1 figure ---------------0212110511206 Content-Type: text/plain; name="02-514.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-514.keywords" Contact Matrices, Information, Coarse Graining ---------------0212110511206 Content-Type: application/postscript; name="ContMatr.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="ContMatr.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: ContMatr.dvi %%Pages: 21 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -ta4 -f ContMatr.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.12.11:1128 %%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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Fv(F)-8 b(or)32 b(SR)-11 b(W)33 b(on)g Fq(Z)885 5046 y Fn(2)921 5082 y Fv(,)g Ft(R)1055 5097 y Fz(N)1150 5082 y Fs(\030)1279 5043 y Fz(\031)r(N)p 1265 5059 134 4 v 1265 5116 a Fn(ln)12 b Fz(N)1409 5082 y Fv(,)32 b(more)g(precisely)h(\(see)h(e.g.)f([15)o (]\),)1151 5320 y Fq(E)13 b Fv(\()p Ft(R)1324 5335 y Fz(N)1397 5320 y Fv(\))28 b(=)1620 5252 y Ft(\031)t(N)p 1576 5297 236 4 v 1576 5388 a Fv(ln)16 b(8)p Ft(N)1838 5320 y Fv(\(1)22 b(+)g Fs(O)d Fv(\(1)p Ft(=)e Fv(ln)e Ft(N)10 b Fv(\))q(\))16 b Ft(:)694 b Fv(\(4.1\))1828 5637 y(10)p eop %%Page: 11 11 11 10 bop 1396 1418 a currentpoint currentpoint translate 0.96912 0.96912 scale neg exch neg exch translate 1396 1418 a 1396 1418 a gsave currentpoint currentpoint translate 0 neg rotate neg exch neg exch translate 1396 1418 a @beginspecial 176 @llx 0 @lly 295 @urx 117 @ury 1190 @rwi @setspecial %%BeginDocument: p2d.eps %!PS-Adobe-3.0 EPSF-3.0 %%Title: D:\PATRIK\ARTDIPL\P2D.EPS %%Creator: Micrografx Graphics Engine %%CreationDate: Fri Jul 20 20:09:00 2001 %%BoundingBox: 176 0 295 117 %%Pages: (atend) %%LanguageLevel: 1 %%EndComments %%BeginProlog %%BeginResource: procset RangerDict 1 1 /RangerDict 150 dict def RangerDict begin /bd{bind def}bind def /ld{load def}bd /p/closepath ld /cc/concat ld /v/curveto ld /e/eofill ld /f/fill ld /gr/grestore ld /gs/gsave ld /l/lineto ld /m/moveto ld /n/newpath ld /r/rlineto ld /sc/scale ld /sd/setdash ld /g/setgray ld /lc/setlinecap ld /lj/setlinejoin ld /lw/setlinewidth ld /c/setrgbcolor ld /sp/showpage ld /s/stroke ld /tf/transform ld /tr/translate ld /xd{exch def}bd /min{2 copy gt{exch} if pop}bd /arct where {pop} {/arct{arcto pop pop pop pop}bd} ifelse /setcmykcolor where {pop /k/setcmykcolor ld} {/k{4 1 roll 3 {3 index add 1 min 1 exch sub 3 1 roll} repeat c pop}bd} ifelse /MGX_CustomColor where {pop /x/MGX_CustomColor ld} {/x{exch pop dup 1 ne {4 {dup 6 -1 roll mul exch} repeat} if pop k}bd} ifelse /op{/MGX_Overprint exch 1 eq {true} {false} ifelse def}bd /_mx matrix def /smx{/_mx _mx currentmatrix def}bd /rmx{_mx setmatrix}bd /_ctm matrix def /sctm{/_ctm _ctm currentmatrix def /_res 72 72 matrix defaultmatrix dtransform _ctm idtransform pop abs def}bd /rctm{_ctm setmatrix}bd /el{smx tr sc 1 eq {0 -1 m} if 0 0 1 -90 270 arc rmx}bd /ea{smx tr sc 1 eq {arcn} {arc} ifelse 1 eq {0 0 l} if rmx}bd /rt{1 eq {m} {l} ifelse 2 copy 0 r 0 exch r neg 0 r neg 0 exch r}bd /rr{smx tr /_wy exch 3 index div def /_wx exch 3 index div def sc _wx 2 div 0 3 -1 roll 1 eq {m} {l} ifelse 0 0 0 _wy 0.5 arct 0 _wy _wx _wy 0.5 arct _wx _wy _wx 0 0.5 arct _wx 0 0 0 0.5 arct _wx 2 div 0 l rmx}bd /clp{/_ct xd {{gs _ct 1 eq {eoclip} {clip} ifelse}stopped {gr currentflat 1 add setflat} {exit} ifelse} loop}bd /horz{_y1 _w 2 div add _w _y2 {dup _x1 exch m _x2 exch l s} for}bd /vert{_x1 _w 2 div add _w _x2 {dup _y1 m _y2 l s} for}bd /fdiag{_x1 _w _x2 _y2 _y1 sub add{_y2 m _y2 _y1 sub neg dup r s} for}bd /bdiag{_x1 _y2 _y1 sub sub _w _x2{_y2 m _y2 _y1 sub dup neg r s} for}bd /hf{/_ht xd clp rctm /_w _res 12 div def pathbbox _w div cvi 1 add _w mul /_y2 xd _w div cvi 1 add _w mul /_x2 xd _w div cvi _w mul /_y1 xd _w div cvi _w mul /_x1 xd 1 lw [] 0 sd n _ht 0 eq {horz} if _ht 1 eq {vert} if _ht 2 eq {fdiag} if _ht 3 eq {bdiag} if _ht 4 eq {horz vert} if _ht 5 eq {fdiag bdiag} if gr}bd /bf{/_im xd clp rctm /_w _res 12 div round def pathbbox _w div cvi 1 add _w mul /_y2 xd _w div cvi 1 add _w mul /_x2 xd _w div cvi _w mul /_y1 xd _w div cvi _w mul /_x1 xd /_wx _x2 _x1 sub def /_wy _y2 _y1 sub def /_swx _wx _w div cvi 8 mul def /_swy _wy _w div cvi 8 mul def /_ic 0 def _x1 _y1 tr _wx _wy sc _swx _swy false [_swx 0 0 _swy 0 0] /_swx _swx 8 div def {_im _ic _swx div cvi 8 mod 1 getinterval /_ic _ic 1 add def} imagemask gr}bd /_rgb 3 string def /_cmyk 4 string def /_g 1 string def /rgbtograyx{_g 0 currentfile _rgb readhexstring pop 0 get .3 mul _rgb 1 get .59 mul add _rgb 2 get .11 mul add cvi put _g}bd /rgbtogray{_g 0 currentfile _rgb readstring pop 0 get .3 mul _rgb 1 get .59 mul add _rgb 2 get .11 mul add cvi put _g}bd /cmyktograyx{_g 0 currentfile _cmyk readhexstring pop 0 get .3 mul _cmyk 1 get .59 mul add _cmyk 2 get .11 mul add _cmyk 3 get add cvi 255 min 255 exch sub put _g}bd /cmyktogray{_g 0 currentfile _cmyk readstring pop 0 get .3 mul _cmyk 1 get .59 mul add _cmyk 2 get .11 mul add _cmyk 3 get add cvi 255 min 255 exch sub put _g}bd /readimg{currentfile _imgstr readstring pop}bd /readimgx{currentfile _imgstr readhexstring pop}bd /i{gs tr sc /_imgstr exch string def [4 index 0 0 6 index 0 0] exch 1 eq {{readimg}} {{readimgx}} ifelse image gr}bd /colorimage where {pop /ic{gs tr sc /_imgstr exch string def /_p xd [4 index 0 0 6 index 0 0] exch 1 eq {{readimg}} {{readimgx}} ifelse false _p colorimage gr}bd} {/ic{gs tr sc /_imgstr exch string def /_p xd [4 index 0 0 6 index 0 0] exch 1 eq {_p 3 eq {{rgbtogray}} {{cmyktogray}} ifelse} {_p 3 eq {{rgbtograyx}} {{cmyktograyx}} ifelse} ifelse image gr}bd} ifelse /im{gs tr sc /_imgstr exch string def 1 eq {true} {false} ifelse [3 index 0 0 5 index 0 0] {readimgx} imagemask gr}bd /bullets{{/bullet}repeat} bd /WE[/grave/acute/circumflex/tilde/macron /breve/dotaccent/dieresis/ring/cedilla/hungarumlaut/ogonek/caron/dotlessi 18 bullets StandardEncoding 32 95 getinterval aload pop 3 bullets /quotesinglbase/florin/quotedblbase/ellipsis/dagger/daggerdbl/circumflex /perthousand/Scaron/guilsinglleft/OE 4 bullets/quoteleft/quoteright /quotedblleft/quotedblright/bullet/endash/emdash/tilde/trademark/scaron /guilsinglright/oe 2 bullets /Ydieresis/space/exclamdown/cent/sterling /currency/yen/brokenbar/section/dieresis/copyright/ordfeminine /guillemotleft/logicalnot/hyphen/registered/macron/degree/plusminus /twosuperior/threesuperior/acute/mu/paragraph/periodcentered/cedilla /onesuperior/ordmasculine/guillemotright/onequarter/onehalf/threequarters /questiondown/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/AE/Ccedilla /Egrave/Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis 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/FontInfo known {/FontInfo get} if /UnderlinePosition known not {pop /Courier findfont} if /FontInfo get /UnderlinePosition get 1000 div 0 exch _mxf tf /_ulpos xd pop 0 .3 _mxf tf /_sopos xd pop currentfont dup dup /FontInfo known {/FontInfo get} if /UnderlineThickness known not {pop /Courier findfont} if /FontInfo get /UnderlineThickness get 1000 div _ys mul /_ulsiz xd } bd /to {m show n} bd /tp {m false charpath} bd /ul {1 eq {/_pos _ulpos def} {/_pos _sopos def} ifelse 1 eq {gs [] 0 sd 0 lc _ulsiz lw _pos add m 0 r s gr} {_pos add _ulsiz 2 div add m 0 _ulsiz neg r 0 r 0 _ulsiz r p} ifelse} bd end %%EndResource %%EndProlog %%BeginSetup RangerDict begin 0 306 tr 0.036 -0.036 sc sctm %%EndSetup /#copies 1 def %%Page: 1 1 %%BeginPageSetup %%EndPageSetup 6281 8377 m 0 -3053 r s 6156 5574 m 125 -250 r 125 250 r -125 -125 r p gs e gr s 4882 7105 m 3307 0 r s 7939 6980 m 250 125 r -250 125 r 125 -125 r p gs e gr s 0 127 127 6281 7105 el gs 0.129 g e gr s 0 127 127 5009 7105 el gs 0.129 g e 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All Rights Reserved.) readonly def /FullName (TimesNewRomanITALIC) readonly def /FamilyName (TimesNewRomanITALIC) readonly def /Weight (Converted) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -109 def /UnderlineThickness 49 def end readonly def /FontName /TimesNewRomanITALIC def /Encoding 256 array 0 1 255{1 index exch /.notdef put}for dup 32 /C32 put dup 33 /C33 put dup 34 /C34 put dup 35 /C35 put dup 36 /C36 put dup 37 /C37 put dup 38 /C38 put dup 39 /C39 put dup 40 /C40 put dup 41 /C41 put dup 42 /C42 put dup 43 /C43 put dup 44 /C44 put dup 45 /C45 put dup 46 /C46 put dup 47 /C47 put dup 48 /C48 put dup 49 /C49 put dup 50 /C50 put dup 51 /C51 put dup 52 /C52 put dup 53 /C53 put dup 54 /C54 put dup 55 /C55 put dup 56 /C56 put dup 57 /C57 put dup 58 /C58 put dup 59 /C59 put dup 60 /C60 put dup 61 /C61 put dup 62 /C62 put dup 63 /C63 put dup 64 /C64 put dup 65 /C65 put dup 66 /C66 put dup 67 /C67 put dup 68 /C68 put dup 69 /C69 put dup 70 /C70 put dup 71 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put dup 187 /C187 put dup 133 /C133 put dup 192 /C192 put dup 195 /C195 put dup 213 /C213 put dup 140 /C140 put dup 156 /C156 put dup 150 /C150 put dup 151 /C151 put dup 147 /C147 put dup 148 /C148 put dup 145 /C145 put dup 146 /C146 put dup 247 /C247 put dup 255 /C255 put dup 159 /C159 put dup 164 /C164 put dup 139 /C139 put dup 155 /C155 put dup 135 /C135 put dup 183 /C183 put dup 130 /C130 put dup 132 /C132 put dup 137 /C137 put dup 194 /C194 put dup 202 /C202 put dup 193 /C193 put dup 203 /C203 put dup 200 /C200 put dup 205 /C205 put dup 206 /C206 put dup 207 /C207 put dup 204 /C204 put dup 211 /C211 put dup 212 /C212 put dup 210 /C210 put dup 218 /C218 put dup 219 /C219 put dup 217 /C217 put dup 136 /C136 put dup 152 /C152 put dup 184 /C184 put dup 138 /C138 put dup 154 /C154 put dup 166 /C166 put dup 208 /C208 put dup 240 /C240 put dup 221 /C221 put dup 253 /C253 put dup 222 /C222 put dup 254 /C254 put dup 215 /C215 put dup 185 /C185 put dup 178 /C178 put dup 179 /C179 put dup 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cleartomark %%EndResource /TimesNewRomanITALIC 0 397 445 445 fnt (y) 6495 5169 to /TimesNewRomanITALIC 0 384 431 431 fnt (x) 7901 7236 to %%PageTrailer %%Trailer end %%Pages: 1 %%EOF %%EndDocument @endspecial 2387 1418 a currentpoint grestore moveto 2387 1418 a 1396 1418 a currentpoint currentpoint translate 1 0.96912 div 1 0.96912 div scale neg exch neg exch translate 1396 1418 a 634 1790 a Fv(Figure)31 b(1:)43 b(The)34 b(pattern)f Ft(Q)p Fv(.)44 b(The)33 b(visited)f(sites)h(of)f Ft(Q)h Fv(are)g(blac)m(k.)404 2059 y(Next)i(w)m(e)g(apply)f(a)g(result)g(of)f (v)-5 b(an)34 b(Wijland,)f(Caser)j(and)e(Hilhorst)e([15].)48 b(Let)35 b(the)257 2240 y(supp)s(ort)j(of)f Ft(!)k Fv(b)s(e)c (de\014ned)i(to)e(b)s(e)h(the)g(set)g(of)f(p)s(oin)m(ts)g(visited)g(b)m (y)h Ft(!)t Fv(.)58 b(Consider)37 b(t)m(w)m(o)257 2420 y(\014nite)j(disjoin)m(t)f(sets)i(of)f(lattice)e(p)s(oin)m(ts)i Ft(A)1865 2435 y Fz(u)1950 2420 y Fv(and)g Ft(A)2220 2435 y Fz(v)2261 2420 y Fv(.)66 b(The)41 b(\\pattern")f(cen)m(tered)i (at)257 2601 y Fp(x)35 b Fv(asso)s(ciated)g(with)g(the)g(sets)h Ft(A)1479 2616 y Fz(u)1559 2601 y Fv(and)f Ft(A)1824 2616 y Fz(v)1900 2601 y Fv(is)g(a)f(con\014guration)h(of)f Fs(j)p Ft(A)2892 2616 y Fz(v)2933 2601 y Fs(j)g Fv(visited)h(sites)257 2781 y Fp(x)23 b Fv(+)f Fp(z)o Fv(,)40 b Fp(z)c Fs(2)h Ft(A)815 2796 y Fz(v)893 2781 y Fv(and)h(of)f Fs(j)p Ft(A)1305 2796 y Fz(u)1350 2781 y Fs(j)h Fv(un)m(visited)g(sites)g Fp(x)22 b Fv(+)g Fp(z)2291 2745 y Fo(0)2314 2781 y Fv(,)39 b Fp(z)2430 2745 y Fo(0)2490 2781 y Fs(2)d Ft(A)2665 2796 y Fz(u)2711 2781 y Fv(.)58 b(W)-8 b(e)39 b(sa)m(y)f(that)g(the)257 2962 y(pattern)c(app)s(ears)g(in)f(the)h(supp)s(ort)g(of)f Ft(!)k Fv(at)c Fp(x)h Fv(if)f(the)h(lattice)e(p)s(oin)m(ts)h Fp(x)22 b Fv(+)g Fp(z)p Fv(,)34 b Fp(z)29 b Fs(2)h Ft(A)3427 2977 y Fz(v)3468 2962 y Fv(,)257 3142 y(are)j(in)f(the)i(supp)s(ort)f (of)f Ft(!)k Fv(and)d(the)h(lattice)d(p)s(oin)m(ts)i Fp(x)22 b Fv(+)g Fp(z)2452 3106 y Fo(0)2475 3142 y Fv(,)33 b Fp(z)2585 3106 y Fo(0)2636 3142 y Fs(2)c Ft(A)2804 3157 y Fz(u)2849 3142 y Fv(,)k(are)g(not)g(in)f(the)257 3323 y(supp)s(ort)37 b(of)f Ft(!)t Fv(.)53 b(The)38 b(n)m(um)m(b)s(ers) f(of)e(times)h(that)g(a)g(pattern)g(app)s(ears)h(in)e(the)i(supp)s(ort) 257 3504 y(of)32 b Ft(!)k Fv(is)c(then)i(the)f(n)m(um)m(b)s(er)g(of)f (di\013eren)m(t)g Fp(x)c Fs(2)g Fq(Z)2054 3467 y Fz(n)2131 3504 y Fv(suc)m(h)34 b(that)f(it)e(app)s(ears)i(at)f Fp(x)p Fv(.)404 3684 y(Let)51 b(us)g(consider)g(the)h(\\pattern)f Ft(Q)p Fv(")f(de\014ned)j(as)e(the)g(set)g(comp)s(osed)g(of)g(the)257 3865 y(follo)m(wing)65 b(sets)k Ft(A)1003 3880 y Fz(v)1112 3865 y Fv(and)f Ft(A)1410 3880 y Fz(u)1455 3865 y Fv(:)113 b Ft(A)1668 3880 y Fz(v)1709 3865 y Fv(\()p Ft(Q)p Fv(\))88 b(=)f Fs(f)p Fv(\(0)p Ft(;)17 b Fv(0\))p Ft(;)g Fv(\()p Fs(\000)p Fv(1)p Ft(;)g Fv(0\))p Fs(g)66 b Fv(and)i Ft(A)3134 3880 y Fz(u)3179 3865 y Fv(\()p Ft(Q)p Fv(\))87 b(=)257 4045 y Fs(f)p Fv(\(1)p Ft(;)17 b Fv(0\))p Ft(;)g Fv(\(0)p Ft(;)g Fs(\000)p Fv(1\))p Ft(;)g Fv(\(0)p Ft(;)g Fv(1\))p Ft(;)g Fv(\()p Fs(\000)p Fv(1)p Ft(;)g Fv(1\))p Fs(g)29 b Fv(\(see)34 b(Figure)d(1\).)404 4226 y(Let)i Ft(Q)656 4241 y Fz(N)751 4226 y Fv(=)28 b Ft(Q)932 4241 y Fz(N)1000 4226 y Fv(\()p Ft(!)t Fv(\))k(b)s(e)h(the)g(n)m(um)m(b)s(er)g(of)g (times)f(that)g Ft(Q)h Fv(app)s(ears)g(in)g(the)g(supp)s(ort)257 4406 y(of)e Ft(!)t Fv(.)42 b(Then)33 b(using)d([15],)i Fq(E)12 b Fv(\()p Ft(Q)1394 4421 y Fz(N)1467 4406 y Fv(\))28 b(=)1705 4367 y Fz(\031)1748 4344 y Fe(2)1783 4367 y Fz(N)p 1647 4384 259 4 v 1647 4441 a Fn(\(ln)12 b(8)p Fz(N)7 b Fn(\))1870 4422 y Fe(2)1915 4406 y Ft(m)2000 4421 y Fn(1)2059 4406 y Fv(+)19 b Fs(O)2253 4296 y Fr(\020)2420 4367 y Fz(N)p 2322 4384 V 2322 4441 a Fn(\(ln)12 b(8)p Fz(N)7 b Fn(\))2545 4422 y Fe(3)2590 4296 y Fr(\021)2681 4406 y Fv(where)32 b Ft(m)3046 4421 y Fn(1)3114 4406 y Fv(=)27 b Ft(m)3302 4421 y Fn(1)3342 4406 y Fv(\()p Ft(Q)p Fv(\))257 4587 y(is)32 b(a)g(constan)m(t,)h(and)f Ft(Q)1122 4602 y Fz(N)1211 4587 y Fs(\000)22 b Fq(E)12 b Fv(\()p Ft(Q)1485 4602 y Fz(N)1559 4587 y Fv(\))27 b Fs(')1778 4548 y Fn(2)p Fo(A)p 1739 4564 169 4 v 1739 4621 a Fn(ln)12 b(8)p Fz(N)1918 4587 y Fq(E)h Fv(\()p Ft(Q)2094 4602 y Fz(N)2167 4587 y Fv(\))p Ft(\015)5 b Fv(\()p Ft(N)10 b Fv(\))33 b(where)g Ft(\015)5 b Fv(\()p Ft(N)10 b Fv(\))33 b(is)e(a)h(random)257 4768 y(v)-5 b(ariable)26 b(\(V)-8 b(aradhan's)28 b(renormalized)e(lo)s(cal)f(time)h (of)h(self-in)m(tersections\))g(with)h(mean)257 4948 y(0)33 b(and)g(v)-5 b(ariance)33 b(1.)44 b Fs(A)32 b Fv(is)h(a)g(constan)m(t)g(giv)m(en)h(in)e([15)o(])h(whose)i(v)-5 b(alue)32 b(is)g Fs(\030)d Fv(1)p Ft(:)p Fv(3034.)44 b(W)-8 b(e)257 5129 y(computed)33 b Ft(m)792 5144 y Fn(1)864 5129 y Fv(\014nding)g Ft(m)1275 5144 y Fn(1)1342 5101 y Fs(\030)1343 5133 y Fv(=)1447 5129 y(2)p Ft(:)p Fv(78)22 b Fs(\001)g Fv(10)1791 5093 y Fo(\000)p Fn(3)1884 5129 y Ft(:)404 5309 y Fv(These)34 b(results)f(imply)e(the)i(follo)m(wing)d (prop)s(osition.)1828 5637 y(11)p eop %%Page: 12 12 12 11 bop 257 573 a Fp(Prop)s(osition)36 b(4.1.)49 b Fk(F)-7 b(or)29 b(simple)f(r)-5 b(andom)29 b(walks)g(on)g Fq(Z)2368 537 y Fn(2)2405 573 y Fk(,)i(ther)-5 b(e)29 b(exists)h(a)f Ft(\027)34 b(>)28 b Fv(0)h Fk(such)257 753 y(that)1198 1002 y Fv(lim)1163 1064 y Fz(N)7 b Fo(!1)1384 1002 y Fq(P)1460 892 y Fr(\020)1521 1002 y Fv(deg)18 b Fs(C)6 b Fv(\()p Ft(!)t Fv(\))27 b Fs(\025)i 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y Fs(\000)1059 3603 y Fn(1)p 1045 3620 64 4 v 1045 3677 a Fz(N)1135 3643 y Fv(ln)16 b Fq(P)p Fv(\()p Ft(R)1404 3658 y Fz(N)1473 3643 y Ft(=)-5 b(N)38 b Fs(\025)28 b Ft(k)s Fv(\))36 b([6],)h(w)m(e)g(conclude)f(that,)h(if)e Ft(\014)2975 3658 y Fz(J)3058 3643 y Fv(=)e(0,)k(for)f Ft(\013)257 3823 y Fv(small)23 b(enough)j(the)g(n)m(um)m(b)s(er)g(of)f(constructed) i(R)-11 b(W)26 b(exceeds)i(the)d(total)f(n)m(um)m(b)s(er)i(of)f(R)-11 b(W)257 4004 y(with)33 b Ft(n)27 b Fs(2)i Fv([)p Ft(N)10 b Fv(\(1)22 b(+)g(20)p Ft(\013)q Fv(\))p Ft(;)17 b(N)10 b Fv(\(1)21 b(+)h Ft(c)1561 4019 y Fn(2)1601 4004 y Ft(\013)q Fv(\)])32 b(and)h Ft(I)1994 4019 y Fz(n)2041 4004 y Ft(=n)27 b Fs(\024)h Ft(k)2331 4019 y Fn(1)2393 4004 y Fv(+)22 b Ft(c)2533 4019 y Fn(2)2572 4004 y Ft(\013)q Fv(.)43 b(Therefore)34 b Ft(\014)3202 4019 y Fz(J)3279 4004 y Ft(>)27 b Fv(0.)257 4184 y Fp(Outline)37 b(of)h(the)f(pro)s(of)h(of)g (prop)s(osition)f(3.2:)44 b Fq(P)p Fv(\()p Ft(R)2387 4199 y Fz(N)2456 4184 y Ft(=)-5 b(N)38 b Fs(\024)29 b Ft(")p Fv(\))j(is)h(not)f(exp)s(onen-)257 4365 y(tially)k(small)g(in)h Ft(N)49 b Fv(\(see)39 b(e.g.)f([6])g(and)h(pro)s(of)e(of)h(\(2.1a\)\).) 59 b(Let)38 b Ft(J)2729 4380 y Fn(1)2806 4365 y Fv(=)f([0)p Ft(;)17 b Fv(1)25 b Fs(\000)h Ft(")p Fv(\))38 b(and)257 4545 y Ft(J)311 4560 y Fn(2)390 4545 y Fv(=)h([1)27 b Fs(\000)g Ft(";)17 b Fv(1].)63 b(Since)40 b Ft(d)p Fv(\()p Ft(J)1373 4560 y Fn(1)1412 4545 y Fv(\))f Ft(>)g Fv(0,)i Ft(W)14 b Fv(\()p Ft(N)c Fv(\))1991 4560 y Fz(J)2030 4569 y Fe(1)2108 4545 y Fv(is)39 b(exp)s(onen)m(tially)f(small)f (compared)257 4726 y(with)24 b Ft(W)14 b Fv(\()p Ft(N)c Fv(\))741 4741 y Fz(J)780 4750 y Fe(2)842 4726 y Fv(for)23 b Ft(N)35 b Fv(large)23 b(enough.)41 b(Therefore)25 b(for)e(large)g Ft(N)10 b Fv(,)26 b Ft(W)14 b Fv(\()p Ft(N)c Fv(\))28 b Fs(')g Ft(W)14 b Fv(\()p Ft(N)c Fv(\))3313 4741 y Fz(J)3352 4750 y Fe(2)3418 4726 y Fs(\024)257 4907 y(j)p Fv(\012)355 4922 y Fz(N)423 4907 y Fs(j)p Fq(P)p Fv(\()p Ft(R)622 4922 y Fz(N)691 4907 y Ft(=)-5 b(N)37 b Fs(\024)28 b Ft(")p Fv(\),)33 b(from)e(whic)m(h)i(follo)m(ws)e(\(3.13\).)404 5087 y(The)61 b(complete)f(pro)s(of)f(can)i(b)s(e)g(found)f(at)g Fk(http://www-m5.mathematik.tu-)257 5268 y(muenchen.de/p)-5 b(ers/ferr)g(ari/homep)g(age/downlo)g(ad/app)g(en)o(dix.ps)o(.gz)p Fv(.)1828 5637 y(19)p eop %%Page: 20 20 20 19 bop 257 573 a Fu(References)306 852 y Fv([1])49 b(I.)33 b(Bahar,)f(A.R.)h(A)m(tilgan,)e(and)h(B.)h(Erman,)f(F)-8 b(olding)30 b(Des.)j Fp(2)p Fv(,)g(173)f(\(1997\).)306 1116 y([2])49 b(P)-8 b(.)43 b(Billingsley)-8 b(,)42 b Fk(Er)-5 b(go)g(dic)43 b(The)-5 b(ory)44 b(and)g(Information)p Fv(,)g(New)f(Y)-8 b(ork,)46 b(Wiley)c(ed.)458 1296 y(\(1965\).)306 1560 y([3])49 b(P)-8 b(.L.)66 b(F)-8 b(errari,)72 b Fk(Contact)64 b(matric)-5 b(es)65 b(for)g(r)-5 b(andom)64 b(walks)p Fv(,)72 b(EPFL-Lausanne)458 1740 y(diploma)30 b(thesis.)306 2004 y([4])49 b(K.F.)37 b(F)-8 b(reed,)40 b Fk(Polymers)f(as)g (self-avoiding)f(walks)p Fv(,)g(Ann.)h(Probab.)f Fp(9)p Fv(,)h(837{556)458 2185 y(\(1981\).)306 2448 y([5])49 b(T.)44 b(Haliloglu,)f(I.)h(Bahar,)j(and)d(B.)g(Erman,)j Fk(Gaussian)e(Dynamics)f(of)h(F)-7 b(olde)i(d)458 2629 y(Pr)g(oteins)p Fv(,)32 b(Ph)m(ys.)j(Rev.)e(Lett.)g Fp(79)p Fv(,)f(3090{3093)f(\(1997\).)306 2892 y([6])49 b(Y.)d(Hamana)e(and)h (H.)h(Kesten,)k Fk(A)d(lar)-5 b(ge)46 b(deviation)g(r)-5 b(esult)47 b(for)g(the)f(r)-5 b(ange)47 b(of)458 3073 y(r)-5 b(andom)29 b(walk)g(and)g(for)h(the)g(Wiener)g(sausage)p Fv(,)d(Probab.)h(Theory)g(Relat.)e(Fields,)458 3253 y Fp(120)33 b Fv(\(2\),)f(183{208)f(\(2001\).)306 3517 y([7])49 b(T.F.)29 b(Ha)m(v)m(el,)h(G.M.)e(Cripp)s(en,)i(and)e(I.D.)h (Kun)m(tz,)h(Biop)s(olymers,)e Fp(18)p Fv(,)h(73)f(\(1979\).)306 3781 y([8])49 b(H.)40 b(Kesten,)j Fk(On)e(the)h(Numb)-5 b(er)41 b(of)h(Self-A)n(voiding)e(Walks)p Fv(,)h(J.)f(Math.)g(Ph)m(ys.) i Fp(4)p Fv(,)458 3961 y(960)32 b(\(1963\).)306 4225 y([9])49 b(K.F.)32 b(Lau)h(and)f(K.A.)h(Dill,)c(Macromolecules)j Fp(22)p Fv(,)h(3986)e(\(1989\).)257 4488 y([10])49 b(S.)33 b(Lifson)f(and)g(C.)h(Sander,)h(Nature)e(\(London\),)h Fp(282)p Fv(,)g(109)f(\(1979\).)257 4752 y([11])49 b(N.)38 b(Madras)g(and)h(G.)e(Slade,)i Fk(The)g(Self-A)n(voiding)f(Walk)p Fv(,)h(Probabilit)m(y)e(and)h(Its)458 4933 y(Applications,)31 b(Birk\177)-49 b(auser)33 b(Boston)f(ed.)i(\(1996\).)257 5196 y([12])49 b(Th.)h(M.)g(Nieu)m(w)m(enh)m(uizen,)56 b Fk(T)-7 b(r)i(apping)49 b(and)g(Lifshitz)h(T)-7 b(ails)50 b(in)g(R)-5 b(andom)49 b(Me-)458 5377 y(dia,)34 b(Self-A)n(ttr)-5 b(acting)34 b(Polymers)h(and)f(the)h(Numb)-5 b(er)35 b(of)f(Distinct)h(Sites)f(Visite)-5 b(d:)1828 5637 y Fv(20)p eop %%Page: 21 21 21 20 bop 458 573 a Fk(A)40 b(R)-5 b(enormalize)g(d)38 b(Instanton)g(Appr)-5 b(o)g(ach)39 b(in)g(Thr)-5 b(e)g(e)39 b(Dimensions)p Fv(,)e(Ph)m(ys.)i(Rev.)458 753 y(Lett.)33 b Fp(62)f Fv(\(4\),)h(357{360)d(\(1989\).)257 1017 y([13])49 b(M.)55 b(V)-8 b(endruscolo,)61 b(B.)55 b(Subramanian,)k(I.)c(Kan)m (ter,)61 b(E.)55 b(Doman)m(y)-8 b(,)60 b(and)55 b(J.L.)458 1198 y(Leb)s(o)m(witz,)e Fk(Statistic)-5 b(al)49 b(Pr)-5 b(op)g(erties)50 b(of)f(Contact)h(Maps)p Fv(,)i(Ph)m(ys.)f(Rev.)e(E,)g Fp(59)p Fv(,)458 1378 y(977{984)31 b(\(1999\).)257 1642 y([14])49 b(M.)27 b(V)-8 b(endruscolo,)29 b(E.)e(Kussell,)h(and)f(E.)g (Doman)m(y)-8 b(,)28 b(F)-8 b(olding)24 b(Des.)k Fp(2)p Fv(,)g(295)e(\(1997\).)257 1905 y([15])49 b(F.)33 b(v)-5 b(an)33 b(Wijland,)f(S.)i(Caser,)h(and)e(H.J.)h(Hilhorst,)e Fk(T)-7 b(op)i(olo)g(gy)35 b(of)g(the)h(supp)-5 b(ort)36 b(of)458 2086 y(the)f(two-dimensional)e(r)-5 b(andom)33 b(walk)p Fv(,)f(J.)h(Ph)m(ys.)i(A)d Fp(30)p Fv(,)h(507)f(\(1997\).)257 2350 y([16])49 b(F.)27 b(v)-5 b(an)27 b(Wijland)f(and)h(H.J.)g (Hilhorst,)g Fk(Universal)j(Fluctuations)g(in)f(the)h(Supp)-5 b(ort)458 2530 y(of)34 b(the)h(R)-5 b(andom)34 b(Walk)p Fv(,)f(J.)f(Stat.)h(Ph)m(ys.)h Fp(89)p Fv(,)f(119)f(\(1997\).)257 2794 y([17])49 b(F.)g(v)-5 b(an)50 b(Wijland,)i Fk(Th)n(\022)-47 b(ese)p Fv(,)52 b(Univ)m(ersit)m(\023)-46 b(e)49 b(de)h(P)m(aris-Sud,)k (U.F.R.)49 b(Scien)m(ti\014que)458 2974 y(d'Orsa)m(y)34 b(\(1998\).)1828 5637 y(21)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0212110511206--