Content-Type: multipart/mixed; boundary="-------------0806040741140" This is a multi-part message in MIME format. ---------------0806040741140 Content-Type: text/plain; name="08-103.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="08-103.keywords" Glauber dynamics, Kawasaki dynamics, critical droplet, metastable transition time, last-exit biased distribution, Dirichlet principle, Berman-Konsowa principle, capacity, flow, cluster expansion. ---------------0806040741140 Content-Type: application/postscript; name="BHS.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="BHS.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.94b Copyright 2004 Radical Eye Software %%Title: BHSfinal.dvi %%CreationDate: Wed Jun 04 14:30:38 2008 %%Pages: 45 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 CMR12 CMR8 CMBX10 CMR10 CMMI10 CMMI7 CMSY10 %%+ MSBM10 CMR7 CMSY7 CMTI10 CMR6 CMR9 CMBX12 CMMI8 CMMI6 CMEX10 CMSY6 %%+ CMSY8 MSAM10 MSBM7 CMMI5 MSAM7 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips.exe -P pdf -I c -t A4 BHSfinal.dvi %DVIPSParameters: dpi=8000 %DVIPSSource: TeX output 2008.06.04:1427 %%BeginProcSet: tex.pro 0 0 %! /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/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]{Ci}imagemask restore}B/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: alt-rule.pro 0 0 %! % Patch by TVZ % Makes dvips files draw rules with stroke rather than fill. % Makes narrow rules more predictable at low resolutions % after distilling to PDF. % May have unknown consequences for very thick rules. % Tested only with dvips 5.85(k). 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All Rights Reserved) readonly def /FullName (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 99 /c put readonly def /FontBBox{37 -250 1349 750}readonly def /UniqueID 5087380 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSAM10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 1 /squareplus put dup 3 /square put dup 4 /squaresolid put dup 12 /squareminus put readonly def /FontBBox{8 -463 1331 1003}readonly def /UniqueID 5031981 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 76 /L put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 121 /y put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 73 /I put dup 80 /P put dup 83 /S put dup 87 /W put dup 88 /X put dup 91 /union put dup 92 /intersection put dup 95 /logicalor put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 110 /backslash put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 17 /equivalence put dup 20 /lessequal put dup 24 /similar put dup 26 /propersubset put dup 33 /arrowright put dup 48 /prime put dup 50 /element put dup 56 /universal put dup 63 /perpendicular put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 80 /P put dup 83 /S put dup 88 /X put dup 91 /union put dup 95 /logicalor put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 110 /backslash put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 2 /bracketleftbig put dup 3 /bracketrightbig put dup 8 /braceleftbig put dup 9 /bracerightbig put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 20 /bracketleftbigg put dup 21 /bracketrightbigg put dup 24 /ceilingleftbigg put dup 25 /ceilingrightbigg put dup 26 /braceleftbigg put dup 27 /bracerightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 34 /bracketleftBigg put dup 35 /bracketrightBigg put dup 40 /braceleftBigg put dup 41 /bracerightBigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 50 /bracketlefttp put dup 51 /bracketrighttp put dup 52 /bracketleftbt put dup 53 /bracketrightbt put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 62 /braceex put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 80 /summationtext put dup 88 /summationdisplay put dup 91 /uniondisplay put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 110 /braceleftBig put dup 111 /bracerightBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CAC6A7BEB5D02276E511FFAF2AE11910 DE076F24311D94D07CACC323F360887F1EA11BDDA7927FF3325986FDB0ABDFC8 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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708F69CDA15CEBDEF286BD0C2325EC6835495DC939718F292270B0067EEC8312 DF97DFB01C58E4F504B95A00CCFB1E7557F698B110FF30E5F6080FC9C1DF7D40 AEDCC7D20A78CF5D48EB65D06B1409126140E801D547ACA41D576B3B5DA307CF 048EE6C383 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 69 /E put dup 78 /N put dup 80 /P put dup 85 /U put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 15 /bullet put dup 18 /reflexsubset put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 26 /propersubset put dup 27 /propersuperset put dup 33 /arrowright put dup 41 /arrowdblright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 56 /universal put dup 57 /existential put dup 59 /emptyset put dup 63 /perpendicular put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 73 /I put dup 80 /P put dup 83 /S put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 91 /union put dup 92 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 22 /mu put dup 23 /nu put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 70 /F put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 85 /U put dup 86 /V put dup 88 /X put dup 90 /Z put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 107 /k put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 115 /s put dup 116 /t put dup 118 /v put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 22 /mu put dup 23 /nu put dup 25 /pi put dup 27 /sigma put dup 28 /tau put dup 58 /period put dup 59 /comma put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 72 /H put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 80 /P put dup 81 /Q put dup 82 /R put dup 85 /U put dup 88 /X put dup 96 /lscript put dup 99 /c put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 114 /r put dup 115 /s put dup 116 /t put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueID 5087383 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 50 /two put dup 58 /colon put dup 61 /equal put dup 65 /A put dup 67 /C put dup 68 /D put dup 70 /F put dup 71 /G put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueID 5000828 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 71 /G put dup 73 /I put dup 75 /K put dup 76 /L put dup 79 /O put dup 80 /P put dup 82 /R put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 3 /Lambda put dup 4 /Xi put dup 6 /Sigma put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 33 /exclam put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 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cleartomark %%EndFont %%BeginFont: CMR9 %!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 53 /five put dup 54 /six put dup 55 /seven put dup 57 /nine put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 71 /G put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 127 /dieresis put readonly def /FontBBox{-39 -250 1036 750}readonly def /UniqueID 5000792 def currentdict end currentfile eexec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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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4921E593A0899C0FBD4F77FEFE59E003F9FD91188B0173DA1BE14A78C3D8B73F F6DEA2458D26C1AF5EC2D38B9DB69E9691940B3174BAD7F2BEA3291F086DDA22 EDA2796FF81BF8EBC5068E70C5EB75DAF9379A20D2EDF555A1A0C147BBD848A4 FFB058C53897D3A9281E36CD0A5184A46677BCE5A1DE1569F246551B5A97CAD6 DF284579AF0EE4FF44A456706E25B980A0661539553E3C9A53E1A18E35911F95 92D1CD7C9DF44D10143A7A6B64F1EFD1D91550A053A6FD37C5609F37B81C3AB7 E1997B9EDE0979C747A3CD04532944919BF82D7187E2C00F3CA66357FBAA62A8 53440F6F496F0BA59E33F5DCFF0D1DC4F8547953229D246AA5AE0DD909B7BE51 C5C591E7103862144178498931496CB820DB2824149D78EFF6CCD1B7DCB1223D 51AEA61B3F447ECEA3E20D1922756E732F7D5F9BDDFB7909FF07548CB12C08D1 E5AD31967277043A4CB591E036BF550FDBC1DE6416106D5A67DC487CDDA6412F 9B4C9CFA1ACEFF0E64641F5D01916BDA19EF54024AC7987CB871685CE928E3FC F241885A34B19E30CAF7998A6155327196A234A34F02808C3957A55F4858A69B FDBE0E3649A65F0A805D4240A54E322308A4876C72F84BBF5FE2E10FD868CE85 D6BBE0ACA725A8A16CDA4D1E6082A13043B960B16316FFB66B85EF07F35C05A1 6BED5C90FC926E713CFC700B07384FD41288D86A30FFE827AB67E8A0D74056D2 594C03F696C8B48481187D44820A6DBCEF2BD1AAE268E8DE55B90A4B4A7689D2 B0DD7E1BB87A944B71D8358E06DB5465726CA8298B3ABCD2CD4FC71288C76849 250E5DB1D6CA0D7A99441157F0E2AE0A7F4BCC697C37FDD38312A5895BA198B4 374DA1522BD581DFE17206D50DD0140758FB85F5AFB62EF9BDE6D1DCB6425CB6 F83DE0F58AD586FA195BFFF947A5F365E125A50E66077FF5943150534F7A7280 AE1DE23D2A0E6F573A6C8D84206DB1F67EA11C9B79FB641CBB3906BF0B9F8101 527746BA787BE3830DC2D820915FD920E2EA48F7872AF07E9B29A1522A4D0810 7E16AC855D1D816966A325F235B4BB1290A5A347857B3561EAAD824830E7EE99 D0AB45DED47B74BC1904347E3180FDC28C92B181B327D10ED1699CD9B7C21C6C 496BD34FEEE6BC1A66664306CD7DD1EBD65EF6D59C3A8AB9EA0F87C0362AD1D5 DF587013CBFAA653EBBD898109CD0988512EBACA008088CB0B0938513611521C FAB5022E837B92E1AA248060B6EF0B4D1B6B51FAE85858FD0988EDEB6C202150 709C1128AAF8DFAE3183F03A35A23BFA8414B08EFEA7D78683297DFBE6DFDB09 EC0B5474700EEADA015DDE4600DA39DDF7AB49FE417FE7CE183B5E93D277F2C9 27F9CE357EA0EA26DFAC47F4FE7B2FBA42C1E18BF7DB59279F1D7BAD511B218A D240330406359D83BE8C67665E8580DA68176F3B94C4C2939DAF6A704E589F13 35066D4AC4DEB00B96F944CB5EE45D6D8CD6FED1EF653279A8E1D7F006021F20 1B574E2834180848F48D76808ABFCA22BFB1A5E1BC9D675CB1CD1EEFA206212B EE655E9642E863C812D46ABA56B76981DD4E854045CCDBF18DF9C3201EA135 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 33 /arrowright put dup 49 /infinity put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueID 5000817 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D251491EBF65A98C9FE2B1CF8D725A70281949 8F4AFFE638BBA6B12386C7F32BA350D62EA218D5B24EE612C2C20F43CD3BFD0D F02B185B692D7B27BEC7290EEFDCF92F95DDEB507068DE0B0B0351E3ECB8E443 E611BE0A41A1F8C89C3BC16B352C3443AB6F665EAC5E0CC4229DECFC58E15765 424C919C273E7FA240BE7B2E951AB789D127625BBCB7033E005050EB2E12B1C8 E5F3AD1F44A71957AD2CC53D917BFD09235601155886EE36D0C3DD6E7AA2EF9C C402C77FF1549E609A711FC3C211E64E8F263D60A57E9F2B47E3480B978AAF63 868AEA25DA3D5413467B76D2F02F8097D2841804B020B210C0470066F33B37E2 05805CEE76C91696F62E34EA09B7B7D6D5A4009030F0FB377D84497D56557DF4 9B39A8B5B98BFC07D37F77324AE22B7B9C462D17C175A20B9F5E818796D45700 1B13FC4967F4CC16F5D3F79E3C52B00CC87F9F95D61C8F048967613999D774AB DC3DDF6FDF058E73B251525C26F7ADF5B0C07F91B5811BEC3F1417952FBA8B17 D2A23BEB89890C9946EED48D6E99344BF26791EEE3D872E4F9004D6BDE27BBA8 B3D2E393F729301C25074ED288D3382FF423F1D983B6556EDA45BB414F94D617 10B6E5F367EDC6EC528A1EB3FD8AB797A8A4BA806A8C95E5ECE33938A26341AA 7A8EAE1EB026141204B442FE9D57668CB305855DFBB1E6D937146064ADE8C887 FD5556AE964181ADFFD656ED68DE1E03EA23FCC04F79C9BFCE1B1B98591C0C87 E4F8E757096D46F6EC2655FFE23E43A76649DB5BC75CB4D15EAE53A76FFA250E 038FDD71C0A8B15469A1D38FA522EAE24FD4EFE1CB3E330C7E01290AE1C3C387 4288B0385D28BA02764AAA6E954F73ED8C904417385DA7B7C02A00997A76906E AAFBC2ADA6A9E693D0AEC89395762E1956880B13DD5642C5D030AC918B0EB3B9 AE6318EC0F7FC5E69E4A09A62D0664119684BEA344B0E775C681F6F78E10A40C 9C06BB8F1F45FF60686D4E6164BCAB289CEE800D560C8CDDA2C86B98975B56CC 066A10DD3A4EF3D6161699F9F7C5083CA1EE40C9EC4DA8EBE39EB3E941 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 3 /Lambda put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 49 /one put dup 50 /two put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueID 5000790 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6CC3F1E9AE32F234EB60FE7D E34995B1ACFF52428EA20C8ED4FD73E3935CEBD40E0EAD70C0887A451E1B1AC8 47AEDE4191CCDB8B61345FD070FD30C4F375D8418DDD454729A251B3F61DAE7C 8882384282FDD6102AE8EEFEDE6447576AFA181F27A48216A9CAD730561469E4 78B286F22328F2AE84EF183DE4119C402771A249AAC1FA5435690A28D1B47486 1060C8000D3FE1BF45133CF847A24B4F8464A63CEA01EC84AA22FD005E74847E 01426B6890951A7DD1F50A5F3285E1F958F11FC7F00EE26FEE7C63998EA1328B C9841C57C80946D2C2FC81346249A664ECFB08A2CE075036CEA7359FCA1E90C0 F686C3BB27EEFA45D548F7BD074CE60E626A4F83C69FE93A5324133A78362F30 8E8DCC80DD0C49E137CDC9AC08BAE39282E26A7A4D8C159B95F227BDA2A281AF A9DAEBF31F504380B20812A211CF9FEB112EC29A3FB3BD3E81809FC6293487A7 455EB3B879D2B4BD46942BB1243896264722CB59146C3F65BD59B96A74B12BB2 9A1354AF174932210C6E19FE584B1B14C00E746089CBB17E68845D7B3EA05105 EEE461E3697FCF835CBE6D46C75523478E766832751CF6D96EC338BDAD57D53B 52F5340FAC9FE0456AD13101824234B262AC0CABA43B62EBDA39795BAE6CFE97 563A50AAE1F195888739F2676086A9811E5C9A4A7E0BF34F3E25568930ADF80F 0BDDAC3B634AD4BA6A59720EA4749236CF0F79ABA4716C340F98517F6F06D9AB 7ED8F46FC1868B5F3D3678DF71AA772CF1F7DD222C6BF19D8EF0CFB7A76FC6D1 0AD323C176134907AB375F20CFCD667AB094E2C7CB2179C4283329C9E435E7A4 1E042AD0BAA059B3F862236180B34D3FCED833472577BACD472A47CD4FFE5D78 D8A2086B7832F31FAFBCE0747E0B7F76063E08B889BD206283B8A22EB49056DF 4AE3154E631457AB9BF8E20CEAF41E6962856CA88AEBC6509DEF888C6DC8EC09 8A5297DD0C41A55B44BF2EF9461ECB205972A9E0798FC286104C3C5D70AEB70B A33EE2C389AF73B30E63D28077BCED58BA53D2177A6318EA6BCDD7B3C556B737 22C0B6AC5AE0B3A6714D8B3C50D89638C09DB2F2152C9953908F007A4FF53A79 ABC96DE7FEC5FADC2CEBFD250F13DE67915C3467C29E13C6FCF30139A2708421 4DD9969B44A1AC89DF6DD770C4715BCA6AB5E6662E91080B062CA514A1ECAE65 5A376C72FBCB1A5A6E13B118053F2084E2B5FA819E841DA39361DC5EF5AF4B4F F7A169C9CAF8D454AC5972ECC0E82699BDFBD0D4D083AA8FE012A2E985E9A161 22048526C16276FF1ECD8FA6F2F8A82B8F48131EEDFFC692150175C293411C03 F056BE75166EAEB9700A12DDF161F5C9B9A104EC86A66D3FE9472F9CBCAB3C85 A74C1F423B5F0CA8B0181CC1D787E8A7468C2D0EE5E1F69E657A1934EC6B3DBC FABA948A7B89F05A3EEAE3ECA9657959C9D4165C8C7FA6599B01838A29BE8D13 1A7CB44027AC95CABB1541F66B772D0B6E765C959F939F039743170DA183AE4B DD5D76D9A76E0CA58A86FA2F7FB6690C859A75F2300D0434F7A2083A4D179F66 EBF0AE161D4BDEE9CFCA45218C51E45C936E27C4F7CE62765B25EFDEDE138B60 8859F275858DBB7EECE81DCF688B0507B35725CCD40AF0E3F89483088E263253 5E2B78E9DB06007C2958A593A2B34A08236FA1350030265E672487AAEDDD3304 42EFB37E17F6AC65C35EF0DDF5B310F2C9864CEDBB8259A908EE9C4616928838 190F170BAB4E31AF05ED591C6A78282C87CAFBFB8D5C172DA7A3867E8836E1FA 336488BA2ACB23D63908737F41ADE8FCA04083EABFF8C1BA6D74BD8A211F8A6F 163F066CF215CBC19AFB1D89328ACB4C18EC4D40CFCD5C3EEE6FD46E0BBCA01D 3E145305EA660C31667B19A09CFC192B5D24486FB0AE2B84BC66E18FF84F3F7C 728F042469CAAE3EADA1082A5B7595BAABF35B6949229A46687F4A8BD911706F 2440360D514E15F0A9F532B2A8FF35848A454355C316409CE40E2409A4B12BA5 F7AF56DC60995BD6EC5769A8F1EC0200BC6671D999E3091BC5FD5B2D127F23D6 E187E72B2782B00D02B52378110467E962105BA97E4E4E78BF9DCF6BBA904F5F EC0A6ECE614CBDBD0EE8374BAB51810EA745F2A96049EFD70E9AD462A221EF0A 9727EB5FA30A95D0138AD902561F26B6DCAF240C096ECECBD7053FC8D03665DD A28BBACBC3DE3C98B69C84F52C79163E92FA72470E4A2004A7308BE06012E110 0815D892334A56B5DF28D8069D80363B6E721AE9A36FC4958E8F60FC5FE8108A AE353EABFEA212DB25BE03750039DD22C1865D0C2083A77194C7DA4AA6DF8A1F 5500556631BBAD4ABA914E 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /beta put dup 59 /comma put dup 76 /L put dup 96 /lscript put dup 99 /c put dup 121 /y put readonly def /FontBBox{0 -250 1171 750}readonly def /UniqueID 5087382 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 3 /Lambda put dup 4 /Xi put dup 20 /caron put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 58 /colon put dup 61 /equal put dup 91 /bracketleft put dup 93 /bracketright put dup 94 /circumflex put dup 97 /a put dup 99 /c put dup 101 /e put dup 102 /f put dup 103 /g put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 120 /x put dup 126 /tilde put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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78[{}2 885.568 /MSBM7 rf /Fh 243[861 7[861 861 1[861 1[{}4 1106.96 /MSAM10 rf /Fi 134[736 1[1007 736 774 542 550 574 736 774 697 774 1162 387 736 1[387 774 697 426 639 774 620 774 678 12[970 774 1045 1[953 1047 2[838 5[877 916 1069 1007 992 1054 6[387 2[697 697 697 697 697 697 697 2[387 465 32[774 12[{}46 1212.12 /CMBX10 rf /Fj 243[943 7[943 943 1[943 1[{}4 1212.12 /MSAM10 rf /Fk 145[470 3[261 2[470 470 6[627 2[627 627 2[670 934 3[570 2[658 6[518 4[728 498 615 752 14[627 941 255 14[941 6[732 4[732 732 16[470 2[732{}25 885.568 /CMSY8 rf /Fl 145[424 3[252 2[424 424 6[553 3[553 2[587 4[506 2[583 11[638 451 540 657 1[640 6[467 5[553 1[243 14[812 6[640 1[640 3[640 2[640 13[424 2[640{}24 664.176 /CMSY6 rf /Fm 144[808 808 4[572 572 12[1347 2[1751 7[1279 14[1061 1061 1[1077 1[1077 1[1077 1[1077 2[808 808 808 808 1061 1061 6[976 976 4[707 707 960 960 4[909 909 707 707 2[640 640 892 892 724 724 6[707 707 4[505 505 556 556{}41 1212.12 /CMEX10 rf /Fn 134[420 464 4[388 4[511 726 1[439 340 296 482 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80dc57414f97f477988a4878e379ec123b4a5b3f3aaa6d7a91010ab480a1310c 4889ceab90387904f58f98fcd1f88ef2fee7db51e208a32b7a2d562639aa5ed4 3c85e0df2ede76d99bf77561615d 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndResource /F17 /DBZWMW+CMMI12 [ /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef 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Fn(\014)22594 34761 y(x)p Fl(\030)p Fn(y)25431 32628 y Fq(\033)43 b Fv(\()p Fq(x)p Fv(\))p Fq(\033)g Fv(\()p Fq(y)g Fv(\))p Fq(;)2629 b(\033)380 b Fp(2)336 b(X)36165 32825 y Fr(\014)36794 32628 y Fq(;)10171 b Fv(\(1.20\))1200 36684 y(where)487 b Fp(\000)p Fq(U)607 b(<)475 b Fv(0)487 b(is)g(the)h(binding)g(energy)f(and)h Fq(x)475 b Fp(\030)g Fq(y)531 b Fv(means)487 b(that)i Fq(x)e Fv(and)h Fq(y)531 b Fv(are)486 b(neigh)-34 b(b)34 b(oring)488 b(sites.)1200 38189 y(Th)-34 b(us,)367 b(w)-34 b(e)356 b(are)f(w)-34 b(orking)356 b(in)g(the)g Fs(c)-62 b(anonic)g(al)387 b(ensemble)p Fv(,)363 b(i.e.,)h(there)356 b(is)f(no)h(term)g(analogous) g(to)g(the)g(second)1200 39695 y(term)404 b(in)g(\(1.1\).)539 b(The)405 b(n)-34 b(um)g(b)34 b(er)405 b(of)f(particles)g(in)g(\003) 24281 39892 y Fr(\014)25314 39695 y Fv(is)21705 42418 y Fq(n)22433 42615 y Fr(\014)23398 42418 y Fv(=)336 b Fp(d)202 b Fq(\032)26045 42615 y Fr(\014)26674 42418 y Fp(j)p Fv(\003)27853 42615 y Fr(\014)28481 42418 y Fp(j)g(e)p 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /SVVVSG+CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 85 /U put readonly def /FontBBox{-30 -250 1026 750}readonly def currentdict end currentfile eexec d9d66f633b846a97b686a97e45a3d0aa0529731c99a784ccbe85b4993b2eebde 3b12d472b7cf54651ef21185116a69ab1096ed4bad2f646635e019b6417cc77b 532f85d811c70d1429a19a5307ef63eb5c5e02c89fc6c20f6d9d89e7d91fe470 b72befda23f5df76be05af4ce93137a219ed8a04a9d7d6fdf37e6b7fcde0d90b 986423e5960a5d9fbb4c956556e8df90cbfaec476fa36fd9a5c8175c9af513fe d919c2ddd26bdc0d99398b9f4d03d6a8f05b47af95ef28a9c561dbdc98c47cf5 5250011d19e9366eb6fd153d3a100caa6212e3d5d93990737f8d326d347b7edc 4391c9df440285b8fc159d0e98d4258fc57892dcc57f7903449e07914fbe9e67 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Fv(Hamiltonian)516 b(giv)-34 b(en)515 b(b)-34 b(y)515 b(\(1.24\))h(\(see)f(Figs.)g(3)g (and)h(4\).)872 b(Put)516 b Fq(N)653 b Fv(=)34155 34127 y FH(1)p 34155 34325 471 49 v 34155 35022 a(3)34758 34604 y Fq(`)35263 34165 y FH(2)35263 34904 y Fr(c)35789 34604 y Fv(\()p Fq(`)36765 34165 y FH(2)36765 34904 y Fr(c)37635 34604 y Fp(\000)343 b Fv(1\).)872 b(F)-101 b(or)515 b Fq(M)654 b Fp(2)521 b Fo(N)516 b Fv(with)1200 36110 y Fq(M)469 b Fp(\025)336 b Fq(`)4629 36292 y Fr(c)5092 36110 y Fv(,)404 b(de\014ne)9944 38833 y Fp(D)10879 39021 y Fr(M)12269 38833 y Fv(=)13549 37851 y Fm(\010)14256 38833 y Fq(\033)380 b Fp(2)336 b(X)17338 39030 y Fr(\014)18102 38833 y Fv(:)605 b Fp(9)202 b Fq(x)337 b Fp(2)g Fv(\003)22936 39030 y Fr(\014)23969 38833 y Fv(suc)-34 b(h)405 b(that)g(supp)q([\()p Fq(\033)43 b Fv(\)])337 b Fp(\033)g Fq(R)36716 39021 y Fr(M)42 b(;M)38976 38833 y Fv(\()p Fq(x)p Fv(\))40611 37851 y Fm(\011)41319 38833 y Fq(;)5646 b Fv(\(1.34\))1200 41556 y(i.e.,)563 b(the)533 b(set)g(of)g(con\014gurations)h(con)-34 b(taining)533 b(a)g(sup)34 b(ercritical)531 b(droplet)i(of)g(size)f Fq(M)132 b Fv(.)924 b(F)-101 b(or)532 b(our)h(results)1200 43062 y(b)34 b(elo)-34 b(w)404 b(to)h(b)34 b(e)404 b(v)-67 b(alid)403 b(w)-34 b(e)404 b(need)h(to)f(assume)h(that)15123 45785 y(lim)14737 46602 y Fr(\014)46 b Fk(!1)17395 45785 y Fp(j)p Fv(\003)18574 45982 y Fr(\014)19202 45785 y Fp(j)202 b Fq(\032)20368 45982 y Fr(\014)21333 45785 y Fv(=)336 b Fp(1)p Fq(;)3012 b Fv(lim)26788 46602 y Fr(\014)46 b Fk(!1)29445 45785 y Fp(j)p Fv(\003)30624 45982 y Fr(\014)31252 45785 y Fp(j)202 b Fq(e)32355 45284 y Fk(\000)p Fr(\014)46 b FH(\000)34640 45785 y Fv(=)337 b(0)p Fq(:)10439 b Fv(\(1.35\))1200 49065 y(This)445 b(\014rst)g(condition)g(sa)-34 b(ys)445 b(that)g(the)g(n)-34 b(um)g(b)34 b(er)445 b(of)g(particles)f(tends)h(to)g(in\014nit)-34 b(y)-101 b(,)455 b(and)445 b(ensures)g(that)g(the)1200 50571 y(formation)405 b(of)f(a)g(critical)f(droplet)i(somewhere)f(do)34 b(es)404 b(not)h(globally)e(deplete)h(the)h(surrounding)g(gas.)1200 53737 y Fi(Theorem)464 b(1.4)606 b Fs(In)435 b(the)d(r)-62 b(e)g(gime)433 b Fv(\(1.27\))p Fs(,)h(subje)-62 b(ct)431 b(to)i Fv(\(1.32\))h Fs(and)f Fv(\(1.35\))p Fs(,)h(the)f(fol)62 b(lowing)432 b(hold:)1200 55242 y(\(a\))13536 57257 y Fv(lim)13150 58074 y Fr(\014)46 b Fk(!1)15808 57257 y Fp(j)p Fv(\003)16987 57454 y Fr(\014)17615 57257 y Fp(j)18496 56437 y Fv(4)p Fq(\031)p 18287 56978 1760 49 v 18287 58088 a(\014)64 b Fv(\001)20381 57257 y Fq(e)20945 56756 y Fk(\000)p Fr(\014)46 b FH(\000)23096 57257 y Fo(E)23904 57927 y Fr(\027)24425 57456 y Fx(\()p Fl(S)25298 57265 y Fn(c)25724 57456 y Fl(n)26293 57288 y Fx(~)26148 57456 y Fl(C)30 b Fx(\))p Fl([C)27983 57265 y Fx(+)24367 58235 y Fl(S)24873 58460 y Fn(L)28770 56275 y Fm(\000)29325 57257 y Fq(\034)29855 57657 y FH(\()p Fk(S)30856 57405 y Fn(c)31283 57657 y Fk(n)31921 57433 y FH(~)31753 57657 y Fk(C)48 b FH(\))p Fk([C)33837 57405 y Fx(+)34569 56275 y Fm(\001)35461 57257 y Fv(=)37124 56437 y(1)p 36874 56978 1107 49 v 36874 58088 a Fq(N)38113 57257 y(:)8852 b Fv(\(1.36\))1200 60200 y Fs(\(b\))9854 62214 y Fv(lim)9468 63031 y Fr(\014)46 b Fk(!1)12125 62214 y Fp(j)p Fv(\003)13304 62411 y Fr(\014)13932 62214 y Fp(j)14814 61394 y Fv(4)p Fq(\031)p 14604 61936 1760 49 v 14604 63046 a(\014)64 b Fv(\001)16698 62214 y Fq(e)17262 61714 y Fk(\000)p Fr(\014)46 b FH(\000)19413 62214 y Fo(E)20221 62781 y Fr(\027)20742 62251 y Fl(D)21380 62476 y Fn(M)20684 63089 y Fl(S)21190 63314 y Fn(L)22591 62214 y Fv(\()q Fq(\034)23593 62402 y Fk(D)24321 62551 y Fn(M)25274 62214 y Fv(\))337 b(=)27744 61394 y(1)p 27494 61936 1107 49 v 27494 63046 a Fq(N)28733 62214 y(;)2680 b Fp(8)202 b Fq(`)33130 62396 y Fr(c)33930 62214 y Fp(\024)337 b Fq(M)468 b Fp(\024)337 b Fv(2)p Fq(`)39245 62396 y Fr(c)39977 62214 y Fp(\000)269 b Fv(1)p Fq(:)5170 b Fv(\(1.37\))1200 65938 y(The)498 b(pro)34 b(of)498 b(of)h(Theorem)f(1.4,)520 b(whic)-34 b(h)499 b(is)f(the)g(analog)g(of)h(Theorem)f(1.2,)520 b(will)498 b(b)34 b(e)497 b(giv)-34 b(en)498 b(in)g(Section)g(4.)1200 67443 y(P)-34 b(art)386 b(\(a\))g(sa)-34 b(ys)386 b(that)h(the)f(a)-34 b(v)g(erage)386 b(time)f(to)h(create)f(a)h(critical)e(droplet)i(is)g ([1)232 b(+)g Fq(o)p Fv(\(1\)]\()p Fq(\014)64 b Fv(\001)p Fq(=)p Fv(4)p Fq(\031)43 b Fv(\))p Fq(e)45595 67003 y Fr(\014)j FH(\000)46813 67443 y Fq(N)132 b Fp(j)p Fv(\003)49098 67640 y Fr(\014)49726 67443 y Fp(j)p Fv(.)1200 68949 y(The)393 b(factor)g Fq(\014)64 b Fv(\001)p Fq(=)p Fv(4)p Fq(\031)436 b Fv(comes)392 b(from)h(the)g(simple)f(random)h(w)-34 b(alk)393 b(that)h(is)e(p)34 b(erformed)392 b(b)-34 b(y)393 b(the)g(free)f(particle)1200 70454 y(\\from)470 b(the)g(gas)g(to)h(the) f(proto-critical)f(droplet")h(\(i.e.,)486 b(the)470 b(dynamics)g(go)34 b(es)469 b(from)h Fp(C)41741 70014 y Fk(\000)42998 70454 y Fv(to)h Fp(C)45255 70014 y FH(+)46042 70454 y Fv(\),)486 b(whic)-34 b(h)1200 71960 y(slo)g(ws)472 b(do)-34 b(wn)473 b(the)f(n)-34 b(ucleation.)741 b(The)472 b(factor)g Fq(N)603 b Fv(coun)-34 b(ts)473 b(the)f(n)-34 b(um)g(b)34 b(er)472 b(of)g(shap)34 b(es)472 b(of)g(the)f(proto-critical)25194 75417 y(10)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 1200 1313 a Fv(droplet)323 b(\(see)g(Bo)-34 b(vier,)338 b(den)324 b(Hollander,)338 b(and)324 b(Nardi)f([7]\).)511 b(P)-34 b(art)324 b(\(b\))g(sa)-34 b(ys)323 b(that,)340 b(once)323 b(the)h(critical)d(droplet)1200 2819 y(is)404 b(created,)f(it)h(rapidly)g(gro)-34 b(ws)405 b(to)f(a)h(droplet)f(that) h(has)g(t)-34 b(wice)404 b(the)h(size.)1200 5321 y Fi(Remarks:)510 b Fv(\(1\))348 b(As)f(for)h(Theorem)f(1.2\(c\),)359 b(w)-34 b(e)348 b(exp)34 b(ect)346 b(Theorem)i(1.4\(b\))g(to)g(hold)f(for)h(v) -67 b(alues)347 b(of)g Fq(M)480 b Fv(that)1200 6826 y(gro)-34 b(w)405 b(with)g Fq(\014)467 b Fv(as)404 b Fq(M)469 b Fv(=)337 b Fq(e)13024 6386 y Fr(o)p FH(\()p Fr(\014)46 b FH(\))14839 6826 y Fv(.)539 b(See)404 b(Section)g(4.2)g(for)g(more)g (details.)1200 8332 y(\(2\))d(In)f(App)34 b(endix)401 b(D)f(w)-34 b(e)400 b(will)g(sho)-34 b(w)401 b(that)h(the)e Fs(aver)-62 b(age)489 b Fv(probabilit)-34 b(y)401 b(under)g(the)f (Gibbs)h(measure)f Fq(\026)48394 8529 y Fr(\014)49423 8332 y Fv(of)1200 9837 y(destro)-34 b(ying)328 b(a)f(sup)34 b(ercritical)326 b(droplet)h(and)h(returning)g(to)f(a)g (con\014guration)i(in)e Fp(S)38070 10025 y Fr(L)39093 9837 y Fv(is)f(exp)34 b(onen)-34 b(tially)327 b(small)1200 11342 y(in)382 b Fq(\014)64 b Fv(.)531 b(Hence,)386 b(the)d(crosso)-34 b(v)g(er)382 b(from)h Fp(S)18926 11530 y Fr(L)20004 11342 y Fv(to)g Fp(S)22289 10903 y Fr(c)22751 11342 y Fp(n)23577 11036 y Fv(~)23357 11342 y Fp(C)297 b([)227 b(C)26036 10903 y FH(+)27205 11342 y Fv(represen)-34 b(ts)383 b(the)g(true)f (threshold)i(for)e(n)-34 b(ucleation,)1200 12848 y(and)405 b(Theorem)f(1.4\(a\))h(represen)-34 b(ts)404 b(the)h(true)f(n)-34 b(ucleation)405 b(time.)1200 14353 y(\(3\))387 b(It)g(w)-34 b(as)387 b(sho)-34 b(wn)388 b(in)f(Bo)-34 b(vier,)389 b(den)e(Hollander,)i(and)e(Nardi)g([7])f(that)i(the)f(a)-34 b(v)g(erage)386 b(crosso)-34 b(v)g(er)386 b(time)g(in)h(a)1200 15859 y(\014nite)405 b(b)34 b(o)-34 b(x)404 b(\003)g(equals)9684 19152 y Fq(K)87 b(e)11364 18652 y Fr(\014)46 b FH(\000)12581 19152 y Fv([1)269 b(+)g Fq(o)p Fv(\(1\)])p Fq(;)606 b Fv(with)405 b Fq(K)424 b Fv(=)336 b Fq(K)87 b Fv(\(\003)p Fq(;)202 b(`)27387 19334 y Fr(c)27850 19152 y Fv(\))337 b Fp(\030)30071 18332 y Fv(log)218 b Fp(j)p Fv(\003)p Fp(j)p 30071 18873 3283 49 v 31042 19983 a Fv(4)p Fq(\031)34829 18332 y Fv(1)p 33821 18873 2622 49 v 33821 19983 a Fq(N)132 b Fp(j)p Fv(\003)p Fp(j)36575 19152 y Fq(;)606 b Fv(\003)336 b Fp(!)h Fo(Z)41053 18652 y FH(2)41579 19152 y Fq(:)5386 b Fv(\(1.38\))1200 22524 y(This)450 b(matc)-34 b(hes)450 b(the)g Fp(j)p Fv(\003)12057 22721 y Fr(\014)12685 22524 y Fp(j)p Fv(-dep)34 b(endence)450 b(in)f(Theorem)h(1.4,)460 b(with)451 b(the)f(logarithmic)f(factor)g(in)h(\(1.38\))g(ac-)1200 24029 y(coun)-34 b(ting)463 b(for)f(the)h(extra)e(factor)h Fq(\014)64 b Fv(\001)462 b(in)g(Theorem)g(1.4)f(compared)h(to)h (Theorem)f(1.2.)711 b(Note)462 b(that)h(this)1200 25535 y(factor)488 b(is)f(particularly)g(in)-34 b(teresting,)508 b(since)487 b(it)h(sa)-34 b(ys)487 b(that)i(the)f Fs(e\013e)-62 b(ctive)508 b(b)-62 b(ox)510 b(size)577 b Fv(resp)34 b(onsible)487 b(for)g(the)1200 27040 y(formation)405 b(of)f(a)g(critical)f(droplet)i(is)f Fq(L)19416 27237 y Fr(\014)20044 27040 y Fv(.)1200 30618 y Ft(1.4)1495 b(Outline)1200 33239 y Fv(The)403 b(remainder)g(of)g(this)h(pap)34 b(er)403 b(is)f(organized)h(as)g(follo)-34 b(ws.)539 b(In)403 b(Section)g(2)g(w)-34 b(e)403 b(presen)-34 b(t)404 b(a)f(brief)g(sk)-34 b(etc)g(h)403 b(of)1200 34744 y(the)481 b(basic)g(ingredien)-34 b(ts)481 b(of)g(the)g(p)34 b(oten)-34 b(tial-theoretic)481 b(approac)-34 b(h)482 b(to)f(metastabilit)-34 b(y)-101 b(.)769 b(In)481 b(particular,)499 b(w)-34 b(e)1200 36250 y(exhibit)317 b(a)g Fs(r)-62 b(elation)408 b Fv(b)34 b(et)-34 b(w)g(een)318 b(a)-34 b(v)g(erage)317 b(crosso)-34 b(v)g(er)316 b(times)h(and)h(capacities,)333 b(and)318 b(w)-34 b(e)317 b(state)h Fs(two)353 b(variational)1200 37755 y(r)-62 b(epr)g(esentations)469 b Fv(for)374 b(capacities,)k(the) c(\014rst)g(of)g(whic)-34 b(h)374 b(is)f(suitable)g(for)h(deriving)e (upp)34 b(er)374 b(b)34 b(ounds)374 b(and)g(the)1200 39261 y(second)446 b(for)g(deriving)f(lo)-34 b(w)g(er)447 b(b)34 b(ounds.)665 b(Section)446 b(3)g(con)-34 b(tains)447 b(the)f(pro)34 b(of)446 b(of)h(our)f(results)g(for)g(the)g(case)f(of) 1200 40766 y(Glaub)34 b(er)276 b(dynamics.)497 b(This)277 b(will)f(b)34 b(e)276 b(tec)-34 b(hnically)276 b(relativ)-34 b(ely)275 b(easy)-101 b(,)302 b(and)277 b(will)f(giv)-34 b(e)276 b(a)h(\014rst)g(\015a)-34 b(v)g(or)277 b(of)g(ho)-34 b(w)278 b(our)1200 42271 y(metho)34 b(d)378 b(w)-34 b(orks.)530 b(In)378 b(Section)g(4)f(w)-34 b(e)379 b(deal)e(with)h(Ka)-34 b(w)g(asaki)379 b(dynamics.)529 b(Here)377 b(w)-34 b(e)378 b(will)f(encoun)-34 b(ter)379 b(sev)-34 b(eral)1200 43777 y(rather)385 b(more)g(di\016cult)g(issues,)k(all)384 b(coming)h(from)g(the)h(fact)f(that)h(Ka)-34 b(w)g(asaki)386 b(dynamics)f(is)g Fs(c)-62 b(onservative)p Fv(.)1200 45282 y(The)409 b(\014rst)g(is)f(to)h(understand)h(wh)-34 b(y)409 b(the)g(constan)-34 b(t)410 b(\000,)f(represen)-34 b(ting)409 b(the)f(lo)34 b(cal)408 b(energetic)f(cost)i(to)g(create) 1200 46788 y(a)457 b(critical)e(droplet,)470 b(in)-34 b(v)g(olv)g(es)457 b(the)g(grand-canonical)g(Hamiltonian,)470 b(ev)-34 b(en)457 b(though)h(w)-34 b(e)457 b(are)g(w)-34 b(orking)457 b(in)1200 48293 y(the)396 b(canonical)f(ensem)-34 b(ble.)536 b(This)395 b(m)-34 b(ystery)396 b(will,)g(of)g(course,)g(b) 34 b(e)395 b(resolv)-34 b(ed)395 b(b)-34 b(y)396 b(the)g(observ)-67 b(ation)395 b(that)h(the)1200 49799 y(formation)413 b(of)h(a)e (critical)g(droplet)h(reduces)f(the)h(en)-34 b(trop)g(y)414 b(of)f(the)g(system:)556 b(the)413 b(precise)f(computation)i(of)1200 51304 y(this)i(en)-34 b(trop)g(y)416 b(loss)f(yields)g(\000)g(via)g Fs(e)-62 b(quivalenc)g(e)440 b(of)j(ensembles)p Fv(.)570 b(The)416 b(second)g(problem)f(is)g(to)h(con)-34 b(trol)415 b(the)1200 52810 y(probabilit)-34 b(y)494 b(of)h(a)f(particle)f(mo)-34 b(ving)494 b(from)h(the)f(gas)g(to)h(the)f(proto-critical)f(droplet)i (at)f(the)g(last)h(stage)1200 54315 y(of)476 b(the)g(n)-34 b(ucleation.)754 b(This)477 b(non-lo)34 b(calit)-34 b(y)475 b(issue)h(will)f(b)34 b(e)476 b(dealt)g(with)g(via)f(upp)34 b(er)476 b(and)h(lo)-34 b(w)g(er)476 b(estimates.)1200 55821 y(App)34 b(endices)404 b(A{D)g(collect)g(some)g(tec)-34 b(hnical)404 b(lemmas)g(that)h(are)e(needed)i(in)f(Sections)g(3{4.)3082 57658 y(The)533 b(extension)g(of)h(our)f(results)g(to)g(higher)g (dimensions)h(is)e(limited)h(only)g(b)-34 b(y)533 b(the)g(com)-34 b(binatorial)1200 59164 y(problems)380 b(in)-34 b(v)g(olv)g(ed)380 b(in)g(the)g(computation)h(of)f(the)g(n)-34 b(um)g(b)34 b(er)381 b(of)f(critical)f(droplets)h(\(whic)-34 b(h)381 b(is)e(hard)i(in)e(the)1200 60669 y(case)473 b(of)h(Ka)-34 b(w)g(asaki)475 b(dynamics\))f(and)h(of)f(the)g(probabilit)-34 b(y)474 b(for)g(simple)g(random)g(w)-34 b(alk)474 b(to)g(hit)g(a)g (critical)1200 62175 y(droplet)428 b(of)h(a)f(giv)-34 b(en)427 b(shap)34 b(e)429 b(when)f(coming)g(from)h(far.)610 b(W)-101 b(e)427 b(will)g(not)i(pursue)g(this)f(generalization)g(here.) 1200 63680 y(The)468 b(relev)-67 b(an)-34 b(t)466 b(results)i(on)f(a)g Fq(\014)64 b Fv(-indep)34 b(enden)-34 b(t)469 b(b)34 b(o)-34 b(x)467 b(in)g Fo(Z)27895 63240 y FH(3)28888 63680 y Fv(can)h(b)34 b(e)467 b(found)h(in)g(Ben)e(Arous)i(and)g(Cerf)f ([1])1200 65186 y(\(Glaub)34 b(er\))592 b(and)h(den)f(Hollander,)639 b(Nardi,)f(Olivieri,)f(and)592 b(Scopp)34 b(ola)593 b([17)o(])f(\(Ka) -34 b(w)g(asaki\).)1103 b(F)-101 b(or)592 b(recen)-34 b(t)1200 66691 y(o)g(v)g(erviews)491 b(on)h(droplet)g(gro)-34 b(wth)493 b(in)f(metastabilit)-34 b(y)-101 b(,)514 b(w)-34 b(e)492 b(refer)e(the)j(reader)d(to)i(den)h(Hollander)e([15,)g(16])1200 68196 y(and)414 b(Bo)-34 b(vier)413 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Fv(\))36769 25317 y Fr(c)37233 25757 y Fq(;)20677 27262 y Fv(1)p Fq(;)6987 b Fv(for)404 b Fq(\033)380 b Fp(2)336 b(A)p Fq(;)20677 28768 y Fv(0)p Fq(;)6987 b Fv(for)404 b Fq(\033)380 b Fp(2)336 b(B)37 b Fq(;)47908 27264 y Fv(\(2.1\))1200 30835 y(where)16789 32340 y Fq(\034)17319 32528 y Fk(A)18463 32340 y Fv(=)336 b(inf)95 b Fp(f)p Fq(t)337 b(>)f Fv(0)135 b(:)606 b Fq(\033)26254 32522 y Fr(t)26985 32340 y Fp(2)337 b(A)p Fq(;)202 b(\033)30330 32599 y Fr(t)30670 32347 y Fl(\000)31890 32340 y Fq(=)-740 b Fp(2)336 b(Ag)p Fq(;)13097 b Fv(\(2.2\))1200 34509 y(\()p Fq(\033)2364 34691 y Fr(t)2759 34509 y Fv(\))3230 34691 y Fr(t)p Fk(\025)p FH(0)5319 34509 y Fv(is)491 b(the)g(con)-34 b(tin)g(uous-time)493 b(Mark)-34 b(o)g(v)492 b(c)-34 b(hain)491 b(with)h(state)g(space)f Fp(X)177 b Fv(,)512 b(and)492 b Fo(P)39808 34691 y Fr(\033)40924 34509 y Fv(is)e(its)i(la)-34 b(w)491 b(starting)1200 36014 y(from)404 b Fq(\033)43 b Fv(.)539 b(This)404 b(function)i(is)e Fs(harmonic)471 b Fv(and)405 b(is)f(the)g(unique)h(solution)g(of)f(the) h Fs(Dirichlet)431 b(pr)-62 b(oblem)16716 38519 y Fv(\()p Fq(Lh)18710 38707 y Fk(A)p Fr(;)p Fk(B)20419 38519 y Fv(\)\()p Fq(\033)43 b Fv(\))1108 b(=)f(0)p Fq(;)g(\033)380 b Fp(2)336 b Fv(\()p Fp(A)270 b([)f(B)37 b Fv(\))34083 38079 y Fr(c)34547 38519 y Fq(;)16716 40025 y(h)17414 40213 y Fk(A)p Fr(;)p Fk(B)19123 40025 y Fv(\()p Fq(\033)43 b Fv(\))2875 b(=)1107 b(1)p Fq(;)g(\033)380 b Fp(2)336 b(A)p Fq(;)16716 41530 y(h)17414 41718 y Fk(A)p Fr(;)p Fk(B)19123 41530 y Fv(\()p Fq(\033)43 b Fv(\))2875 b(=)1107 b(0)p Fq(;)g(\033)380 b Fp(2)336 b(B)37 b Fq(;)47908 40027 y Fv(\(2.3\))1200 44039 y(where)404 b(the)h(generator)e(is)h(the) h(matrix)f(with)h(en)-34 b(tries)13389 46649 y Fq(L)p Fv(\()p Fq(\033)-24 b(;)202 b(\033)16629 46149 y Fk(0)16939 46649 y Fv(\))338 b(=)e Fq(c)19552 46846 y Fr(\014)20180 46649 y Fv(\()p Fq(\033)-24 b(;)202 b(\033)22595 46149 y Fk(0)22906 46649 y Fv(\))270 b Fp(\000)f Fq(\016)25398 46853 y Fr(\033)-21 b(;\033)26746 46601 y Fl(0)27301 46649 y Fq(c)27826 46846 y Fr(\014)28455 46649 y Fv(\()p Fq(\033)43 b Fv(\))p Fq(;)2626 b(\033)-24 b(;)202 b(\033)35040 46149 y Fk(0)35687 46649 y Fp(2)337 b(X)177 b Fq(;)9697 b Fv(\(2.4\))1200 49259 y(with)405 b Fq(c)4486 49456 y Fr(\014)5114 49259 y Fv(\()p Fq(\033)43 b Fv(\))405 b(the)g(total)f(rate)g(at)h(whic)-34 b(h)405 b(the)f(dynamics)h(lea)-34 b(v)g(es)403 b Fq(\033)43 b Fv(,)15882 52117 y Fq(c)16407 52314 y Fr(\014)17036 52117 y Fv(\()p Fq(\033)g Fv(\))337 b(=)21593 50966 y Fm(X)20331 53650 y Fr(\033)20901 53398 y Fl(0)21198 53650 y Fk(2X)130 b(nf)p Fr(\033)32 b Fk(g)24808 52117 y Fq(c)25333 52314 y Fr(\014)25961 52117 y Fv(\()p Fq(\033)-24 b(;)202 b(\033)28376 51617 y Fk(0)28687 52117 y Fv(\))p Fq(;)2626 b(\033)380 b Fp(2)337 b(X)177 b Fq(:)12190 b Fv(\(2.5\))1200 56050 y(A)404 b(related)g(quan)-34 b(tit)g(y)405 b(is)f(the)h Fs(e)-62 b(quilibrium)430 b(me)-62 b(asur)g(e)494 b Fv(on)404 b Fp(A)p Fv(,)g(whic)-34 b(h)405 b(is)f(de\014ned)h(as)16376 58660 y Fq(e)16940 58848 y Fk(A)p Fr(;)p Fk(B)18648 58660 y Fv(\()p Fq(\033)43 b Fv(\))338 b(=)e Fp(\000)p Fv(\()p Fq(Lh)24880 58848 y Fk(A)p Fr(;)p Fk(B)26589 58660 y Fv(\)\()p Fq(\033)43 b Fv(\))p Fq(;)2627 b(\033)380 b Fp(2)337 b(A)p Fq(:)12683 b Fv(\(2.6\))1200 61270 y(The)405 b(equilibrium)e(measure)h(also)g(has) g(a)g(probabilistic)h(meaning,)f(namely)-101 b(,)16398 64463 y Fo(P)17139 64645 y Fr(\033)17763 64463 y Fv(\()p Fq(\034)18764 64651 y Fk(B)19796 64463 y Fq(<)336 b(\034)21605 64651 y Fk(A)22413 64463 y Fv(\))h(=)24633 63630 y Fq(e)25197 63818 y Fk(A)p Fr(;)p Fk(B)26905 63630 y Fv(\()p Fq(\033)43 b Fv(\))p 24633 64184 3952 49 v 25192 65294 a Fq(c)25717 65491 y Fr(\014)26346 65294 y Fv(\()p Fq(\033)g Fv(\))28717 64463 y Fq(;)2626 b(\033)380 b Fp(2)337 b(A)p Fq(:)12705 b Fv(\(2.7\))1200 67721 y(The)405 b(k)-34 b(ey)403 b(ob)67 b(ject)405 b(w)-34 b(e)405 b(will)e(w)-34 b(ork)405 b(with)f(is)g(the)h Fs(c)-62 b(ap)g(acity)p Fv(,)400 b(whic)-34 b(h)405 b(is)f(de\014ned)h (as)17510 70445 y Fw(CAP)o Fv(\()p Fp(A)p Fq(;)202 b Fp(B)37 b Fv(\))337 b(=)24712 69293 y Fm(X)24613 71919 y Fr(\033)32 b Fk(2A)26764 70445 y Fq(\026)27494 70642 y Fr(\014)28123 70445 y Fv(\()p Fq(\033)43 b Fv(\))p Fq(e)30365 70633 y Fk(A)p Fr(;)p Fk(B)32074 70445 y Fv(\()p Fq(\033)g Fv(\))p Fq(:)13819 b Fv(\(2.8\))25194 75417 y(12)p eop end %%Page: 13 13 TeXDict begin 13 12 bop 1200 1313 a Ft(2.2)1495 b(Relation)499 b(b)42 b(et)-42 b(w)g(een)499 b(crosso)-42 b(v)g(er)501 b(time)d(and)h(capacit)-42 b(y)1200 3919 y Fv(The)306 b(\014rst)h(imp)34 b(ortan)-34 b(t)307 b(ingredien)-34 b(t)307 b(of)f(the)h(p)34 b(oten)-34 b(tial-theoretic)306 b(approac)-34 b(h)307 b(to)g(metastabilit)-34 b(y)306 b(is)g(a)g(form)-34 b(ula)1200 5425 y(for)430 b(the)h(a)-34 b(v)g(erage)430 b(crosso)-34 b(v)g(er)430 b(time)h(from)f Fp(A)g Fv(to)h Fp(B)37 b Fv(.)618 b(T)-101 b(o)430 b(state)h(this)g (form)-34 b(ula,)438 b(w)-34 b(e)430 b(de\014ne)h(the)g(probabilit)-34 b(y)1200 6930 y(measure)404 b Fq(\027)6600 6491 y Fk(B)6523 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36536 y Fm(X)20573 39161 y Fr(\033)32 b Fk(2B)22409 38909 y Fn(c)23037 37687 y Fq(\026)23767 37884 y Fr(\014)24396 37687 y Fv(\()p Fq(\033)43 b Fv(\))202 b Fq(h)26974 37875 y Fk(A)p Fr(;)p Fk(B)28684 37687 y Fv(\()p Fq(\033)43 b Fv(\))337 b Fp(\024)g Fq(\026)32709 37884 y Fr(\014)33338 37687 y Fv(\()p Fp(B)34642 37187 y Fr(c)35105 37687 y Fv(\))p Fq(:)11389 b Fv(\(2.12\))1200 41490 y(W)-101 b(e)529 b(will)g(see)g(that)i(in)f(our)g(setting)g Fq(\026)19324 41687 y Fr(\014)19953 41490 y Fv(\()p Fp(B)21257 41050 y Fr(c)21720 41490 y Fp(nA)p Fv(\))546 b(=)g Fq(o)p Fv(\()p Fq(\026)27589 41687 y Fr(\014)28218 41490 y Fv(\()p Fp(A)p Fv(\)\))531 b(as)f Fq(\014)609 b Fp(!)546 b(1)p Fv(,)561 b(so)530 b(that)g(the)h(sum)f(in)f(the)1200 42995 y(righ)-34 b(t-hand)363 b(side)d(of)i(\(2.10\))f(is)g Fp(\030)336 b Fq(\026)17488 43192 y Fr(\014)18117 42995 y Fv(\()p Fp(A)p Fv(\))362 b(and)f(the)h(computation)g(of)f(the)g(crosso)-34 b(v)g(er)361 b(time)g(reduces)f(to)h(the)1200 44501 y(estimation)405 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Fv(\)])31737 70421 y FH(2)32263 70921 y Fq(;)2626 b(h)135 b Fv(:)605 b Fp(X)514 b(!)337 b Fv([0)p Fq(;)202 b Fv(1])p Fq(;)4611 b Fv(\(2.13\))25194 75417 y(13)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 1200 1313 a Fv(it)404 b(follo)-34 b(ws,)405 b(via)e(\(2.1\))i(and)g(\(2.7{2.8\),)f(that)20032 4037 y Fw(CAP)o Fv(\()p Fp(A)p Fq(;)202 b Fp(B)37 b Fv(\))337 b(=)f Fp(E)108 b Fv(\()p Fq(h)29051 4225 y Fk(A)p Fr(;)p Fk(B)30760 4037 y Fv(\))p Fq(:)15734 b Fv(\(2.14\))1200 6760 y(Elemen)-34 b(tary)404 b(v)-67 b(ariational)403 b(calculus)h(sho)-34 b(ws)405 b(that)h(the)e(capacit)-34 b(y)404 b(satis\014es)h(the)g Fs(Dirichlet)431 b(principle)91 b Fv(:)1200 9926 y Fi(Prop)39 b(osition)465 b(2.2)606 b Fs(F)-93 b(or)434 b(any)f(two)h(non-empty)e(disjoint)g(sets)g Fp(A)p Fq(;)202 b Fp(B)373 b(\032)336 b(X)177 b Fs(,)18085 12649 y Fw(CAP)o Fv(\()p Fp(A)p Fq(;)202 b Fp(B)37 b Fv(\))337 b(=)2026 b(min)25756 13349 y Fn(h)94 b Fx(:)425 b Fl(X)97 b(!)p Fx([0)p Fn(;)p Fx(1])25320 14133 y Fn(h)p 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35907 y Fr(\014)37016 35710 y Fv(\()p Fq(\033)-24 b(;)202 b(\033)39431 35270 y Fk(0)39741 35710 y Fv(\))337 b Fq(>)g Fv(0.)1200 38876 y Fi(De\014nition)466 b(2.3)606 b Fs(Given)498 b(two)g(non-empty)g(disjoint)e(sets)h Fp(A)p Fq(;)202 b Fp(B)491 b(\032)454 b(X)177 b Fs(,)515 b(a)498 b(lo)-62 b(op-fr)g(e)g(e)496 b(non-ne)-62 b(gative)497 b(unit)1200 40382 y(\015ow,)434 b Fq(f)130 b Fs(,)434 b(fr)-62 b(om)433 b Fp(A)h Fs(to)f Fp(B)470 b Fs(is)433 b(a)h(function)e Fq(f)265 b Fv(:)606 b Fp(E)445 b(!)337 b Fv([0)p Fq(;)202 b Fp(1)p Fv(\))434 b Fs(such)d(that:)1200 41887 y(\(a\))h Fv(\()p Fq(f)130 b Fv(\()p Fq(e)p Fv(\))340 b Fq(>)c Fv(0)h(=)-202 b Fp(\))336 b Fq(f)130 b Fv(\()p Fp(\000)p Fq(e)p Fv(\))339 b(=)d(0\))434 b Fp(8)202 b Fq(e)338 b Fp(2)f(E)108 b Fs(.)1200 43392 y(\(b\))432 b Fq(f)565 b Fs(satis\014es)432 b(Kir)-62 b(cho\013)126 b('s)432 b(law:)12471 45145 y Fm(X)12199 47786 y Fr(\033)12769 47534 y Fl(0)13067 47786 y Fk(2X)14696 46297 y Fq(f)130 b Fv(\()p Fq(\033)-24 b(;)202 b(\033)17834 45796 y Fk(0)18145 46297 y Fv(\))337 b(=)20626 45145 y Fm(X)20233 47786 y Fr(\033)20803 47534 y Fl(00)21343 47786 y Fk(2X)22972 46297 y Fq(f)130 b Fv(\()p Fq(\033)24902 45796 y Fk(00)25469 46297 y Fq(;)202 b(\033)43 b Fv(\))p Fq(;)2680 b Fp(8)202 b Fq(\033)381 b Fp(2)336 b(X)177 b(n)p Fv(\()p Fp(A)270 b([)g(B)37 b Fv(\))p Fq(:)7901 b Fv(\(2.16\))1200 50201 y Fs(\(c\))432 b Fq(f)565 b Fs(is)433 b(normalize)-62 b(d:)15017 50736 y Fm(X)14918 53361 y Fr(\033)32 b Fk(2A)17341 50736 y Fm(X)17069 53377 y Fr(\033)17639 53125 y Fl(0)17937 53377 y Fk(2X)19565 51887 y Fq(f)130 b Fv(\()p Fq(\033)-24 b(;)202 b(\033)22703 51387 y Fk(0)23015 51887 y Fv(\))337 b(=)f(1)h(=)27718 50736 y Fm(X)27325 53377 y Fr(\033)27895 53125 y Fl(00)28435 53377 y Fk(2X)30107 50736 y Fm(X)30064 53361 y Fr(\033)32 b Fk(2B)32102 51887 y Fq(f)130 b Fv(\()p Fq(\033)34032 51387 y Fk(00)34599 51887 y Fq(;)202 b(\033)43 b Fv(\))p Fq(:)10620 b Fv(\(2.17\))1200 55294 y Fs(\(d\))432 b(A)-31 b(ny)434 b(p)-62 b(ath)432 b(fr)-62 b(om)433 b Fp(A)h Fs(to)f Fp(B)470 b Fs(along)433 b(e)-62 b(dges)432 b Fq(e)i Fs(such)d(that)i Fq(f)130 b Fv(\()p Fq(e)p Fv(\))338 b Fq(>)f Fv(0)434 b Fs(is)f(self-avoiding.)1200 56799 y(The)g(sp)-62 b(ac)g(e)432 b(of)h(al)62 b(l)432 b(lo)-62 b(op-fr)g(e)g(e)431 b(non-ne)-62 b(gative)432 b(unit)h(\015ows)g(fr)-62 b(om)434 b Fp(A)f Fs(to)g Fp(B)471 b Fs(is)433 b(denote)-62 b(d)431 b(by)i Fo(U)42720 56987 y Fk(A)p Fr(;)p Fk(B)44428 56799 y Fs(.)3082 59965 y Fv(A)428 b(natural)g(\015o)-34 b(w)429 b(is)f(the)h Fs(harmonic)454 b(\015ow)p Fv(,)434 b(whic)-34 b(h)429 b(is)e(constructed)i(from)f(the)h(equilibrium)e(p)34 b(oten)-34 b(tial)1200 61471 y Fq(h)1898 61659 y Fk(A)p Fr(;)p Fk(B)4010 61471 y Fv(as)3902 64760 y Fq(f)4495 64948 y Fk(A)p Fr(;)p Fk(B)6203 64760 y Fv(\()p Fq(\033)-24 b(;)202 b(\033)8618 64259 y Fk(0)8929 64760 y Fv(\))337 b(=)13590 63940 y(1)p 11149 64481 5487 49 v 11149 65591 a Fw(CAP)o Fv(\()p Fp(A)p Fq(;)202 b Fp(B)37 b Fv(\))16971 64760 y Fq(\026)17701 64957 y 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y Fv(\()p Fq(\033)-24 b(;)202 b(\033)41858 32407 y Fk(0)42168 32847 y Fv(\))156 b Fp(_)g Fq(c)44284 33044 y Fr(\014)44913 32847 y Fv(\()p Fq(\033)46120 32407 y Fk(0)46431 32847 y Fq(;)202 b(\033)43 b Fv(\))337 b(=)g(1)1200 34352 y(b)-34 b(y)404 b(\(1.3\).)539 b(Th)-34 b(us,)406 b(it)e(su\016ces)g(to)h(sho)-34 b(w)405 b(that)13406 36592 y Fq(\026)14136 36789 y Fr(\014)14766 36592 y Fv(\()p Fp(C)71 b Fv(\))337 b Fp(\024)f Fq(N)19007 36774 y FH(1)19735 36592 y Fp(j)p Fv(\003)20914 36789 y Fr(\014)21542 36592 y Fp(j)202 b Fq(e)22645 36092 y Fk(\000)p Fr(\014)46 b FH(\000)24796 36592 y Fv([1)269 b(+)g Fq(o)p Fv(\(1\)])2828 b(as)405 b Fq(\014)400 b Fp(!)337 b(1)p Fq(:)9714 b Fv(\(3.5\))1200 38832 y(F)-101 b(or)331 b(ev)-34 b(ery)329 b Fq(\033)380 b Fp(2)337 b(P)430 b Fv(there)330 b(are)g(one)h(or)g(more)f(rectangles) g Fq(R)27932 39029 y Fr(`)28316 39140 y Fn(c)28743 39029 y Fk(\000)p FH(1)p Fr(;`)30590 39140 y Fn(c)31073 38832 y Fv(\()p Fq(x)p Fv(\),)346 b Fq(x)336 b Fv(=)h Fq(x)p Fv(\()p 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All Rights Reserved) readonly def /FullName (CMR17) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /GNGDCB+CMR17 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 43 /plus put dup 49 /one put dup 50 /two put readonly def /FontBBox{-33 -250 945 749}readonly def currentdict end currentfile eexec d9d66f633b846a97b686a97e45a3d0aa052a014267b7904eb3c0d3bd0b83d891 016ca6ca4b712adeb258faab9a130ee605e61f77fc1b738abc7c51cd46ef8171 9098d5fee67660e69a7ab91b58f29a4d79e57022f783eb0fbbb6d4f4ec35014f d2decba99459a4c59df0c6eba150284454e707dc2100c15b76b4c19b84363758 469a6c558785b226332152109871a9883487dd7710949204ddcf837e6a8708b8 2bdbf16fbc7512faa308a093fe5f075ea0a10a15b0ed05d5039da41b32b16e95 a3ce9725a429b35bad796912fc328e3a28f96fcada20a598e247755e7e7ff801 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndResource /F10 /CAYSSX+CMMI7 [ /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/K/L/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /EZHZJE+CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 27 /sigma put dup 81 /Q put readonly def /FontBBox{-32 -250 1048 750}readonly def currentdict end currentfile eexec d9d66f633b846a97b686a97e45a3d0aa0529731c99a784ccbe85b4993b2eebde 3b12d472b7cf54651ef21185116a69ab1096ed4bad2f646635e019b6417cc77b 532f85d811c70d1429a19a5307ef63eb5c5e02c89fc6c20f6d9d89e7d91fe470 b72befda23f5df76be05af4ce93137a219ed8a04a9d7d6fdf37e6b7fcde0d90b 986423e5960a5d9fbb4c956556e8df90cbfaec476fa36fd9a5c8175c9af513fe d919c2ddd26bdc0d99398b9f4d03d5993dfc0930297866e1cd0a319b6b1fd958 9e394a533a081c36d456a09920001a3d2199583eb9b84b4dee08e3d12939e321 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndResource /F15 /WDKTMF+CMR17 [ /.notdef/.notdef/.notdef/Lambda/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef ] ipeMakeFont %%BeginResource: font FWGWBO+CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /FWGWBO+CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put readonly def /FontBBox{-34 -251 988 750}readonly def currentdict end currentfile eexec d9d66f633b846a97b686a97e45a3d0aa052a014267b7904eb3c0d3bd0b83d891 016ca6ca4b712adeb258faab9a130ee605e61f77fc1b738abc7c51cd46ef8171 9098d5fee67660e69a7ab91b58f29a4d79e57022f783eb0fbbb6d4f4ec35014f d2decba99459a4c59df0c6eba150284454e707dc2100c15b76b4c19b84363758 469a6c558785b226332152109871a9883487dd7710949204ddcf837e6a8708b8 2bdbf16fbc7512faa308a093fe5cf4e9d2405b169cd5365d6eced5d768d66d6c 68618b8c482b341f8ca38e9bb9bafcfaad9c2f3fd033b62690986ed43d9c9361 3645b82392d5cae11a7cb49d7e2e82dcd485cba04c77322eb2e6a79d73dc194e 59c120a2dabb9bf72e2cf256dd6eb54eecba588101abd933b57ce8a3a0d16b28 51d7494f73096df53bdc66bbf896b587df9643317d5f610cd9088f9849126f23 dde030f7b277dd99055c8b119cae9c99158ac4e150cdfc2c66ed92ebb4cc092a aa078ce16247a1335ad332daa950d20395a7384c33ff72eaa31a5b89766e635f 45c4c068ad7ee867398f0381b07cb94d29ff097d59ff9961d195a948e3d87c31 821e9295a56d21875b41988f7a16a1587050c3c71b4e4355bb37f255d6b237ce 96f25467f70fa19e0f85785ff49068949ccc79f2f8ae57d5f79bb9c5cf5eed5d 9857b9967d9b96cdcf73d5d65ff75afabb66734018bae264597220c89fd17379 26764a9302d078b4eb0e29178c878fd61007eea2ddb119ae88c57ecfef4b71e4 140a34951ddc3568a84cc92371a789021a103a1a347050fda6ecf7903f67d213 1d0c7c474a9053866e9c88e65e6932ba87a73686eab0019389f84d159809c498 1e7a30ed942eb211b00dbff5bcc720f4e276c3339b31b6eabbb078430e6a09bb 377d3061a20b1eb98796b8607eecbc699445eaa866c38e02df59f5edd378303a 0733b90e7835c0aaf32ba04f1566d8161ea89cd4d14ddb953f8b910bfc8a7f03 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68057b88356a6fa2cbd311cb4f4608c44b89d08bf6 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndResource /F17 /FWGWBO+CMR12 [ /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /parenleft/parenright/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef ] ipeMakeFont %%BeginResource: font WVEFLG+CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /WVEFLG+CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /beta put dup 80 /P put dup 120 /x put readonly def /FontBBox{-30 -250 1026 750}readonly def currentdict end currentfile eexec d9d66f633b846a97b686a97e45a3d0aa0529731c99a784ccbe85b4993b2eebde 3b12d472b7cf54651ef21185116a69ab1096ed4bad2f646635e019b6417cc77b 532f85d811c70d1429a19a5307ef63eb5c5e02c89fc6c20f6d9d89e7d91fe470 b72befda23f5df76be05af4ce93137a219ed8a04a9d7d6fdf37e6b7fcde0d90b 986423e5960a5d9fbb4c956556e8df90cbfaec476fa36fd9a5c8175c9af513fe d919c2ddd26bdc0d99398b9f4d03d6a8f05b47af95ef28a9c561dbdc98c47cf5 5250011d19e9366eb6fd153d3a100caa6212e3d5d93990737f8d326d347b7edc 4391c9df440285b8fc159d0e98d4258fc57892dcc57f7903449e07914fbe9e67 3c15c2153c061eb541f66c11e7ee77d5d77c0b11e1ac55101da976ccacab6993 eed1406fbb7ff30eac9e90b90b2af4ec7c273ca32f11a5c1426ff641b4a2fb2f 4e68635c93db835737567faf8471cbc05078dcd4e40e25a2f4e5af46c234cf59 2a1ce8f39e1ba1b2a594355637e474167ead4d97d51af0a899b44387e1fd933a 323afda6ba740534a510b4705c0a15647afbf3e53a82bf320dd96753639be49c 2f79a1988863ef977b800c9db5b42039c23eb86953713f730e03ea22ff7bb2c1 d97d33fd77b1bdcc2a60b12cf7805cfc90c5b914c0f30a673df9587f93e47cea 5932dd1930560c4f0d97547bcd805d6d854455b13a4d7382a22f562d7c55041f 0fd294bdaa1834820f894265a667e5c97d95ff152531ef97258f56374502865d a1e7c0c5fb7c6fb7d3c43feb3431095a59fbf6f61cec6d6dee09f4eb0fd70d77 2a8b0a4984c6120293f6b947944be23259f6eb64303d627353163b6505fc8a60 00681f7a3968b6cbb49e0420a691258f5e7b07b417157803fcbe9b9fb1f80fd8 ca0e6fa3322f94a8847c0a16abb8a1bfa3919d1af65af3953125ece3cbef2287 6e4ba0b1b4b229ffb653dd37745649c933a94c7cc05eee11fe2a407495138c0e 3d757e8dbeaf91ef4f35814a77a13789e5aa8d6779032a2ca2e5fca74dbee713 b7e40f46880751e8b01dcae309e0afc3d9d83ecd6f51caba8fc42a3b1575419b c6c535fd31c2cdf058fe624edc940674c5a3aa2977a1232a7256608f7fcfc3e7 acc91c46d3467f8080f7beb07a387ad8eb57e35599fad0b8a2871316b7aeabf1 4f78dae26ae9383911859598abbac7ba2fefa415364287bd01df601fd627bea9 30b19643e50bfdcaf7d905453c16f6e38f0a2e2e66f8c498a739125371cbfdf5 bcfa2f78dc434d6dcb3385baa21cb69bed8f72692dc15fcde1c2f9ec7914da66 a02049e2bd2c4e583addc31e9fe7e6a53806a350d8a6c671e7c16a3433eddbd4 02cfd0f3113ebec9d645faf2eb5e6511e8e0b20fb846a8f8391943d86bb72ac2 d003c3410d0dabee440c128bc427f5cbe2af19a5237ce11407744b68a87f7aa1 8fb9651d5dd5860efe05be6d5b258cba784c504261bfaf0c3a1a1060198c46cc e88ee5e605379534669f6f5bafa09fd8b6c09c28853ff9b7b0a601342971450d 11ce8b1bc19fd9a436ed9f39f4370467100490d3424f60a2a837457988812fc8 0c22b56d914260f106337b1d6faa035375baa520e25483c95dbb503d5b36aa3f 1af460586bacf5d077d9bfaffcdff65584bb29dcf547ce761e9bd4cfb8003dd3 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a7262844b3e5ec6f5dc89e4f59c321ba8579c9e3340e4648a987ee7f819e4017 e422cf284a8c784e6800b9d9f96e7f037ed0168e703290e8cfc5dc9798a88645 7a4970258c768f75419891452e0d08fb5058fa6eee84491b 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndResource /F19 /WVEFLG+CMMI12 [ /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/beta/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /P/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /x/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef ] ipeMakeFont %%BeginResource: font JBYIYY+CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /JBYIYY+CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put readonly def /FontBBox{-36 -250 1070 750}readonly def currentdict end currentfile eexec d9d66f633b846a97b686a97e45a3d0aa052a014267b7904eb3c0d3bd0b83d891 016ca6ca4b712adeb258faab9a130ee605e61f77fc1b738abc7c51cd46ef8171 9098d5fee67660e69a7ab91b58f29a4d79e57022f783eb0fbbb6d4f4ec35014f d2decba99459a4c59df0c6eba150284454e707dc2100c15b76b4c19b84363758 469a6c558785b226332152109871a9883487dd7710949204ddcf837e6a8708b8 2bdbf16fbc7512faa308a093fe5cf4e9d2405b169cd5365d6eced5d768d66d6c 68618b8c482b341f8ca38e9bb9bafcfaad9c2f3fd033b62690986ed43d9c9361 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /VPLQRQ+CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 121 /y put readonly def /FontBBox{-24 -250 1110 750}readonly def currentdict end currentfile eexec d9d66f633b846a97b686a97e45a3d0aa0529731c99a784ccbe85b4993b2eebde 3b12d472b7cf54651ef21185116a69ab1096ed4bad2f646635e019b6417cc77b 532f85d811c70d1429a19a5307ef63eb5c5e02c89fc6c20f6d9d89e7d91fe470 b72befda23f5df76be05af4ce93137a219ed8a04a9d7d6fdf37e6b7fcde0d90b 986423e5960a5d9fbb4c956556e8df90cbfaec476fa36fd9a5c8175c9af513fe d919c2ddd26bdc0d99398b9f4d03d6a8f05b47af95ef28a9c561dbdc98c47cf5 5250011d19e9366eb6fd153d3a100caa6212e3d5d93990737f8d326d347b7edc 4391c9df440285b8fc159d0e98d4258fc57892ddf753642cd526a96aceda4120 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494e1677876e900e3cabf4daf3ca34312cc537b5e363e0d5f5df321a1d0b2b1b 1132afb30d07e5fb136972eb36c97474d049e3e91d5c82287a139a9954e29fe7 3b9e5f5f09694e959ebac2859694494c8251c31a89ad7a6c4cc63c1be0f6722b 0a0a54ff9929e7e1da545da058 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndResource /F15 /UZLCHX+CMR9 [ /.notdef/.notdef/.notdef/.notdef/Xi/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/two/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef ] ipeMakeFont %%BeginResource: font GHXRYQ+CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /GHXRYQ+CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 48 /zero put readonly def /FontBBox{-20 -250 1193 750}readonly def currentdict end currentfile eexec d9d66f633b846a97b686a97e45a3d0aa052a014267b7904eb3c0d3bd0b83d891 016ca6ca4b712adeb258faab9a130ee605e61f77fc1b738abc7c51cd46ef8171 9098d5fee67660e69a7ab91b58f29a4d79e57022f783eb0fbbb6d4f4ec35014f d2decba99459a4c59df0c6eba150284454e707dc2100c15b76b4c19b84363758 469a6c558785b226332152109871a9883487dd7710949204ddcf837e6a8708b8 2bdbf16fbc7512faa308a093fe5cf4e9d2405b169cd5365d6eced5d768d66d6c 68618b8c482b341f8ca38e9bb9bafcfaad9c2f3fd033b62690986ed43d9c9361 3645b82392d5cae11a7cb49d7e2e82dcd485cba17d1afff95f4224cf7ecee45c bfb7c8c77c22a01c345078d28d3ecbf804cdc2fe5025fa0d05ccc5efc0c4f87e cbed13dddf8f34e404f471c6dd2e43331d73e89bbc71e7bf889f6293793fef5a c9dd3792f032e37a364c70914843f7aa314413d022ae3238730b420a7e9d0cf5 d0e24f501451f9cdece10af7e14ff15c4f12f3fca47dd9cd3c7aea8d1551017d 23131c09ed104c052054520268a4fa3c6338ba6cf14c3de3baf2ea35296ee3d8 d6496277e11dff6076fe64c8a8c3419fa774473d63223ffa41cbae609c3d976b 93dfb4079adc7c4ef07303f93808dda9f651f61bccf79555059a44cbaf84a711 6d98083cef58230d54ad486c74c4a257fc703acf918219d0a597a5f680b606e4 ef94adf8bf91a5096a806db64ec96636a98397d22a74932eb7346a9c4b5ee953 cb3c80aa634bfc28aa938c704bda8dc4d13551ccfe2b2784be8bf54502eba9af d49b79237b9c56310550bc30e9108bb06eac755d6aa4e688efe2a0aab17f20fe 00cd0bff1b9cb6bda0fa3a29a3117388b6686657a150ce6421fd5d420f4f7fb5 b0daa1ba19d638676e9cf159ac7325ef17b9f74e082bef75e10a31c7011c0ffa 99b797ce549b5c45238dd0fadd6b99d233ac69282df0d91ea2dbd08ce0083904 a6d968d5ae3bd159d01bdff42d16111bc0a517c66b43972080d9dd4f3b9ae7fb 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndResource /F16 /GHXRYQ+CMR6 [ /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /zero/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef ] ipeMakeFont %%BeginResource: font LKWDSE+CMMI9 %!PS-AdobeFont-1.1: CMMI9 1.100 %%CreationDate: 1996 Jul 23 07:53:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /LKWDSE+CMMI9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 27 /sigma put dup 74 /J put dup 104 /h put readonly def /FontBBox{-29 -250 1075 750}readonly def currentdict end currentfile eexec d9d66f633b846a97b686a97e45a3d0aa0529731c99a784ccbe85b4993b2eebde 3b12d472b7cf54651ef21185116a69ab1096ed4bad2f646635e019b6417cc77b 532f85d811c70d1429a19a5307ef63eb5c5e02c89fc6c20f6d9d89e7d91fe470 b72befda23f5df76be05af4ce93137a219ed8a04a9d7d6fdf37e6b7fcde0d90b 986423e5960a5d9fbb4c956556e8df90cbfaec476fa36fd9a5c8175c9af513fe d919c2ddd26bdc0d99398b9f4d03d5993dfc0930297866e1cd0a319b6b1fd958 9e394a533a081c36d6f5ca5fed4f9ac9ade41e04f9fc52e758c9f45a92bed935 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5a79b6d614b4979f10988bc76ed53a5f45315cd7da216221f842fd0f3e050dd2 bac23c984d506d8f7d614bcb6b244f5f41321549bb0bd041fbf3053307168680 3435e9c942fd1c1c0d9a6b89e1493f97e7aa44f25589105817439757f2c361ff a8dd0bcdaf62c69f4061b0c080b02e077d0698cb6cebcb7d1079abbc85ba83aa 6347c572967519b287c66d37c193ab6f994a14cf773c22bcc4d1b9d37842ae0f 8f93d9cb4a47d2a8e4d89ddcdc030b51847a0a2215efba8ef73b0f1b9321263c 16034c224cbf2d15aeaa6587d6c4b446b4ff7d5817a9cccf3de67725b03d1dc9 14e2f47d3c4985fa9efc373be0ee7206ea43f172c0bd55ae16aae28292b7c464 4058a9b5adbd0b0200b2880389a48717e9e2de6faa97738167ecfd9a47e1e7bd 29cf38bae6d2140e263a9e4ec7ccfded74f9030a53348507b33c9ce2fefef502 3835a7a9e4584864b1a3e60e71df17253648c6cbe14b0648402602842e5165fc 806586fe9b7bb652de935110d65c954a6e4a5349c661cfb8e81efcd9f2f21864 9cbe097ee74c24ffb814f97f71e15d04bb27d3bf1876fc146eb808a8c471a131 4ba95c0a763c26575a4f80fef0beaea44231a120829be3a6743ef92524fd197c a4072459ac84036b307c0da763b2da645dc6c6c4c7fb1197595bb3df6c0989c1 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndResource /F17 /LKWDSE+CMMI9 [ /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/sigma/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/J/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /h/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef ] ipeMakeFont %%BeginResource: font RBWCJZ+CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /RBWCJZ+CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 49 /one put dup 50 /two put readonly def /FontBBox{-251 -250 1009 969}readonly def currentdict end currentfile eexec d9d66f633b846a97b686a97e45a3d0aa052a014267b7904eb3c0d3bd0b83d891 016ca6ca4b712adeb258faab9a130ee605e61f77fc1b738abc7c51cd46ef8171 9098d5fee67660e69a7ab91b58f29a4d79e57022f783eb0fbbb6d4f4ec35014f d2decba99459a4c59df0c6eba150284454e707dc2100c15b76b4c19b84363758 469a6c558785b226332152109871a9883487dd7710949204ddcf837e6a8708b8 2bdbf16fbc7512faa308a093fe5cf7158f1163bc1f3352e22a1452e73feca8a4 87100fb1ffc4c8af409b2067537220e605da0852ca49839e1386af9d7a1a455f 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndResource /F8 /RBWCJZ+CMR10 [ /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/one/two/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef ] ipeMakeFont %%BeginResource: font XKFKBM+CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /XKFKBM+CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 75 /K put dup 76 /L put readonly def /FontBBox{0 -250 1171 750}readonly def currentdict end currentfile eexec d9d66f633b846a97b686a97e45a3d0aa0529731c99a784ccbe85b4993b2eebde 3b12d472b7cf54651ef21185116a69ab1096ed4bad2f646635e019b6417cc77b 532f85d811c70d1429a19a5307ef63eb5c5e02c89fc6c20f6d9d89e7d91fe470 b72befda23f5df76be05af4ce93137a219ed8a04a9d7d6fdf37e6b7fcde0d90b 986423e5960a5d9fbb4c956556e8df90cbfaec476fa36fd9a5c8175c9af513fe d919c2ddd26bdc0d99398b9f4d03d77639df1232a4d6233a9caf69b151dfd33f c0962eac6e3ebfb8ad256a3c654eaaf9a50c51bc6fa90b61b60401c235afab7b 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndResource /F10 /XKFKBM+CMMI7 [ /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/K/L/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef ] ipeMakeFont %%BeginResource: font KSZWWH+CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /KSZWWH+CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 27 /sigma put dup 61 /slash put dup 81 /Q put readonly def /FontBBox{-32 -250 1048 750}readonly def currentdict end currentfile eexec d9d66f633b846a97b686a97e45a3d0aa0529731c99a784ccbe85b4993b2eebde 3b12d472b7cf54651ef21185116a69ab1096ed4bad2f646635e019b6417cc77b 532f85d811c70d1429a19a5307ef63eb5c5e02c89fc6c20f6d9d89e7d91fe470 b72befda23f5df76be05af4ce93137a219ed8a04a9d7d6fdf37e6b7fcde0d90b 986423e5960a5d9fbb4c956556e8df90cbfaec476fa36fd9a5c8175c9af513fe d919c2ddd26bdc0d99398b9f4d03d5993dfc0930297866e1cd0a319b6b1fd958 9e394a533a081c36d456a09920001a3d2199583eb9b84b4dee08e3d12939e321 990cd249827d9648574955f61baaa11263a91b6c3d47a5190165b0c25abf6d3e 6ec187e4b05182126bb0d0323d943170b795255260f9fd25f2248d04f45dfbfb def7ff8b19bfef637b210018ae02572b389b3f76282beb29cc301905d388c721 59616893e774413f48de0b408bc66dce3fe17cb9f84d205839d58014d6a88823 d9320ae93af96d97a02c4d5a2bb2b8c7925c4578003959c46e3ce1a2f0eac4bf 8b9b325e46435bde60bc54d72bc8acb5c0a34413ac87045dc7b84646a324b808 6fd8e34217213e131c3b1510415ce45420688ed9c1d27890ec68bd7c1235faf9 1dab3a369dd2fc3be5cf9655c7b7eda7361d7e05e5831b6b8e2eec542a7b38ee 03be4bac6079d038acb3c7c916279764547c2d51976baba94ba9866d79f13909 95aa39b0f03103a07cbdf441b8c5669f729020af284b7ff52a29c6255fcaacf1 74109050fba2602e72593fbcbfc26e726ee4aef97b7632bc4f5f353b5c67fed2 3ea752a4a57b8f7feff1d7341d895f0a3a0be1d8e3391970457a967eff84f6d8 47750b1145b8cc5bd96ee7aa99ddc9e06939e383bda41175233d58ad263ebf19 afc0e2f840512d321166547b306c592b8a01e1fa2564b9a26dac14256414e4c8 42616728d918c74d13c349f4186ec7b9708b86467425a6fdb3a396562f7ee4d8 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b(a)h(con)-34 b(tribution)356 b(that)g(is)f(a)f(factor)h Fq(O)34 b Fv(\()p Fq(e)30736 19293 y Fk(\000)p Fr(\016)f(\014)32547 19733 y Fv(\))355 b(smaller.)521 b(Indeed,)364 b(in)355 b(analogy)g(with)1200 21239 y(\(3.17\))405 b(w)-34 b(e)404 b(obtain,)h(writing)f Fq(\033)380 b Fp(\030)337 b Fq(\033)17861 20799 y Fk(0)18575 21239 y Fv(when)405 b Fq(c)22265 21436 y Fr(\014)22894 21239 y Fv(\()p Fq(\033)24101 20799 y Fk(0)24412 21239 y Fq(;)202 b(\033)43 b Fv(\))337 b Fq(>)f Fv(0,)3273 23991 y Fw(CAP)o Fv(\()p Fp(S)6682 24179 y Fr(L)7378 23991 y Fq(;)202 b Fp(S)8742 23490 y Fr(c)9204 23991 y Fp(nC)71 b Fv(\))337 b(=)g Fw(CAP)o Fv(\()p Fp(S)16107 23490 y Fr(c)16570 23991 y Fp(nC)71 b Fq(;)202 b Fp(S)19158 24179 y Fr(L)19853 23991 y Fv(\))337 b Fp(\025)f Fw(CAP)o Fv(\()p Fp(D)25550 24179 y Fr(L)26246 23991 y Fq(;)202 b Fp(S)27519 24179 y Fr(L)28214 23991 y Fv(\))3610 27347 y Fp(\025)5434 26195 y Fm(X)4889 28836 y Fr(\033)5459 28584 y Fl(0)5757 28836 y Fk(2D)7112 28985 y Fn(L)8064 26869 y FH(1)p 8064 27067 V 8064 27765 a(2)9187 26195 y Fm(X)9002 28669 y Fn(\033)24 b Fl(2C)10506 28894 y Fn(L)9094 29573 y(\033)g Fl(\030)p Fn(\033)10734 29382 y Fl(0)11458 25274 y Fm(")12298 26527 y Fq(f)130 b Fv(\()p Fq(\033)14228 26087 y Fk(0)14540 26527 y Fq(;)202 b(\033)43 b Fv(\))p 12298 27068 3989 49 v 12773 28178 a Fq(\026)13503 28375 y Fr(\014)14132 28178 y Fv(\()p Fq(\033)g Fv(\))16689 27347 y(+)18034 26527 y Fq(f)130 b Fv(\()p Fq(\033)-24 b(;)202 b(\015)21064 26709 y Fr(\033)21689 26527 y Fv(\(1\)\))p 18034 27068 5676 49 v 19352 28178 a Fq(\026)20082 28375 y Fr(\014)20711 28178 y Fv(\()p Fq(\033)43 b Fv(\))24111 27347 y(+)25323 25831 y Fr(K)63 b Fk(\000)p FH(1)25477 26195 y Fm(X)25494 28830 y Fr(k)24 b FH(=1)30514 26527 y Fq(f)130 b Fv(\()p Fq(\015)32336 26709 y Fr(\033)32962 26527 y Fv(\()p Fq(k)38 b Fv(\))p Fq(;)202 b(\015)35740 26709 y Fr(\033)36365 26527 y Fv(\()p Fq(k)308 b Fv(+)269 b(1\)\))p 27716 27068 15618 49 v 27716 28178 a Fq(\026)28446 28375 y Fr(\014)29076 28178 y Fv(\()p Fq(\015)30175 28360 y Fr(\033)30799 28178 y Fv(\()p Fq(k)38 b Fv(\)\))p Fq(c)33406 28375 y Fr(\014)34036 28178 y Fv(\()p Fq(\015)35135 28360 y Fr(\033)35760 28178 y Fv(\()p Fq(k)g Fv(\))p Fq(;)202 b(\015)38538 28360 y Fr(\033)39163 28178 y Fv(\()p Fq(k)308 b Fv(+)269 b(1\)\))43467 25274 y Fm(#)44174 25456 y Fk(\000)p FH(1)3610 31752 y Fp(\025)5434 30600 y Fm(X)4889 33241 y Fr(\033)5459 32989 y Fl(0)5757 33241 y Fk(2D)7112 33390 y Fn(L)8064 31274 y FH(1)p 8064 31472 471 49 v 8064 32170 a(2)9187 30600 y Fm(X)9002 33074 y Fn(\033)24 b Fl(2C)10506 33299 y Fn(L)9094 33978 y(\033)g Fl(\030)p Fn(\033)10734 33787 y Fl(0)11458 31752 y Fq(\026)12188 31949 y Fr(\014)12817 31752 y Fv(\()p Fq(\033)43 b Fv(\))14697 30406 y Fm(h)15271 31752 y Fq(f)130 b Fv(\()p Fq(\033)17201 31251 y Fk(0)17513 31752 y Fq(;)202 b(\033)43 b Fv(\))269 b(+)g Fq(f)130 b Fv(\()p Fq(\033)-24 b(;)202 b(\015)23770 31934 y Fr(\033)24396 31752 y Fv(\(1\)\))270 b(+)f Fq(C)87 b(K)g(e)30530 31251 y Fk(\000)p Fr(\014)46 b(\016)32341 30406 y Fm(i)32913 30708 y Fk(\000)p FH(1)3610 36664 y Fv(=)336 b([1)269 b(+)g Fq(o)p Fv(\(1\)])202 b Fq(\026)10718 36861 y Fr(\014)11348 36664 y Fv(\()p Fp(C)12457 36852 y Fr(L)13152 36664 y Fv(\))13825 34591 y Fm( )14918 35844 y Fv(2)p Fq(`)16029 36026 y Fr(c)16762 35844 y Fp(\000)269 b Fv(4)p 14918 36385 3662 49 v 15962 37495 a(2)p Fq(`)17073 37677 y Fr(c)20156 35844 y Fv(1)p 19047 36385 2824 49 v 19047 37578 a(1)h(+)21268 37101 y FH(1)p 21268 37299 471 49 v 21268 37996 a(2)22273 36664 y Fv(+)23618 35844 y(1)p 23618 36385 607 49 v 23618 37495 a(2)25176 35844 y(4)p 24692 36385 1574 49 v 24692 37495 a(2)p Fq(`)25803 37677 y Fr(c)27908 35844 y Fv(1)p 26734 36385 2954 49 v 26867 37101 a FH(1)p 26867 37299 471 49 v 26867 37996 a(2)27739 37578 y Fv(+)29084 37101 y FH(1)p 29084 37299 V 29084 37996 a(2)29820 34591 y Fm(!)3610 40270 y Fv(=)336 b([1)269 b(+)g Fq(o)p Fv(\(1\)])202 b Fq(\026)10718 40467 y Fr(\014)11348 40270 y Fv(\()p Fp(C)12457 40458 y Fr(L)13152 40270 y Fv(\))13959 39450 y Fq(N)14933 39632 y FH(2)p 13959 39991 1500 49 v 13959 41101 a Fq(N)14933 41283 y FH(1)15591 40270 y Fq(:)47302 32413 y Fv(\(3.33\))1200 43521 y(Using)404 b(\(3.21\))h(and)g(the)f(remarks)g(follo)-34 b(wing)405 b(it,)e(w)-34 b(e)405 b(get)f(the)h(desired)f(lo)-34 b(w)g(er)404 b(b)34 b(ound.)p 49878 43521 522 731 v 1200 47763 a Ft(3.3)1495 b(Pro)42 b(of)498 b(of)g(Theorem)h(1.2\(c\))1200 50383 y Fv(W)-101 b(rite)5374 51833 y Fm(X)4838 54458 y Fr(\033)32 b Fk(2D)6789 54153 y Fn(c)6763 54766 y(M)7863 52984 y Fq(\026)8593 53181 y Fr(\014)9222 52984 y Fv(\()p Fq(\033)43 b Fv(\))p Fq(h)11598 53172 y Fk(S)12168 53321 y Fn(L)12787 53172 y Fr(;)p Fk(D)13776 53321 y Fn(M)14730 52984 y Fv(\()p Fq(\033)g Fv(\))337 b(=)18341 51833 y Fm(X)18025 54458 y Fr(\033)32 b Fk(2S)19792 54607 y Fn(L)20611 52984 y Fq(\026)21341 53181 y Fr(\014)21970 52984 y Fv(\()p Fq(\033)43 b Fv(\))p Fq(h)24346 53172 y Fk(S)24916 53321 y Fn(L)25535 53172 y Fr(;)p Fk(D)26524 53321 y Fn(M)27477 52984 y Fv(\()p Fq(\033)g Fv(\))271 b(+)32003 51833 y Fm(X)30638 54517 y Fr(\033)32 b Fk(2D)32589 54212 y Fn(c)32563 54825 y(M)33461 54517 y Fk(nS)34501 54666 y Fn(L)35320 52984 y Fq(\026)36050 53181 y Fr(\014)36680 52984 y Fv(\()p Fq(\033)43 b Fv(\))p Fq(h)39056 53172 y Fk(S)39626 53321 y Fn(L)40244 53172 y Fr(;)p Fk(D)41233 53321 y Fn(M)42187 52984 y Fv(\()p Fq(\033)g Fv(\))16745 56662 y(=)337 b Fq(\026)18755 56859 y Fr(\014)19384 56662 y Fv(\()p Fp(S)20589 56850 y Fr(L)21285 56662 y Fv(\))269 b(+)24602 55510 y Fm(X)23238 58195 y Fr(\033)32 b Fk(2D)25189 57890 y Fn(c)25163 58503 y(M)26061 58195 y Fk(nS)27101 58344 y Fn(L)27920 56662 y Fq(\026)28650 56859 y Fr(\014)29279 56662 y Fv(\()p Fq(\033)43 b Fv(\))p Fo(P)31698 56844 y Fr(\033)32324 56662 y Fv(\()p Fq(\034)33325 56850 y Fk(S)33895 56999 y Fn(L)34904 56662 y Fq(<)337 b(\034)36714 56850 y Fk(D)37442 56999 y Fn(M)38395 56662 y Fv(\))p Fq(:)47302 55471 y Fv(\(3.34\))1200 60862 y(The)524 b(last)f(sum)h(is)g (b)34 b(ounded)524 b(ab)34 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Fp(\024)336 b Fq(M)132 b Fv(2)p Fq(`)11958 64055 y Fr(c)12592 63873 y Fp(\000)170 b Fv(1.)522 b(Indeed,)364 b(under)356 b(that)f(restriction)g(the)g(energy)f(of)h(a)f(square)h (droplet)g(of)g(size)1200 65378 y Fq(M)536 b Fv(is)404 b(strictly)f(larger)h(than)h(the)f(energy)g(of)g(a)g(critical)f (droplet.)1200 67216 y Fs(Pr)-62 b(o)g(of)p Fv(.)403 b(The)366 b(pro)34 b(of)367 b(of)f(Theorem)g(1.2\(c\))h(follo)-34 b(ws)366 b(along)h(the)f(same)g(lines)g(as)g(that)h(of)g(Theorems)f (1.2\(a{b\))1200 68721 y(in)j(Sections)h(3.1{3.2.)526 b(The)369 b(main)h(p)34 b(oin)-34 b(t)370 b(is)e(to)i(pro)-34 b(v)g(e)369 b(that)h Fw(CAP)o Fv(\()p Fp(S)32278 68909 y Fr(L)32974 68721 y Fq(;)202 b Fp(D)34448 68909 y Fr(M)35501 68721 y Fv(\))337 b(=)g([1)199 b(+)g Fq(o)p Fv(\(1\)])p Fw(CAP)o Fv(\()p Fp(S)45755 68909 y Fr(L)46450 68721 y Fq(;)j Fp(S)47814 68281 y Fr(c)48277 68721 y Fp(nC)71 b Fv(\).)1200 70227 y(Since)321 b Fw(CAP)o Fv(\()p Fp(S)7691 70415 y Fr(L)8387 70227 y Fq(;)202 b Fp(D)9861 70415 y Fr(M)10914 70227 y Fv(\))337 b Fp(\024)g Fw(CAP)o Fv(\()p Fp(S)16411 70415 y Fr(L)17106 70227 y Fq(;)202 b Fp(S)18470 69787 y Fr(c)18933 70227 y Fp(nC)71 b Fv(\),)338 b(whic)-34 b(h)322 b(w)-34 b(as)323 b(estimated)f(in)g(Section)643 b(3.2,)337 b(w)-34 b(e)322 b(need)g(only)g(pro)-34 b(v)g(e)1200 71732 y(a)452 b(lo)-34 b(w)g(er)452 b(b)34 b(ound)454 b(on)e Fw(CAP)o Fv(\()p Fp(S)14400 71920 y Fr(L)15096 71732 y Fq(;)202 b Fp(D)16570 71920 y Fr(M)17623 71732 y Fv(\).)683 b(This)453 b(is)f(done)g(b)-34 b(y)453 b(using)f(a)h (\015o)-34 b(w)453 b(that)g(breaks)f(do)-34 b(wn)454 b(an)e Fq(M)434 b Fp(\002)301 b Fq(M)25194 75417 y Fv(22)p eop end %%Page: 23 23 TeXDict begin 23 22 bop 1200 1313 a Fv(droplet)420 b(to)g(a)f(square)g (or)h(quasi-square)f(droplet)h Fq(Q)25367 1501 y Fr(L)26482 1313 y Fv(in)f(the)h(canonical)f(w)-34 b(a)g(y)-101 b(,)424 b(whic)-34 b(h)421 b(tak)-34 b(es)419 b Fq(M)45973 874 y FH(2)46779 1313 y Fp(\000)279 b Fq(v)43 b Fv(\()p Fq(L)p Fv(\))1200 2819 y(steps)519 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9084 y Fv(almost)496 b(ev)-34 b(erywhere)494 b(in)i(\003)44785 9281 y Fr(\014)45413 9084 y Fv(,)518 b(and)496 b(the)1200 10590 y(factor)404 b Fq(M)5979 10150 y FH(2)6505 10590 y Fq(e)7069 10150 y Fk(\000)p Fr(\016)33 b(\014)9284 10590 y Fv(replacing)403 b Fq(K)87 b(e)16152 10150 y Fk(\000)p Fr(\016)33 b(\014)18367 10590 y Fv(in)404 b(\(3.20\))h(still) f(is)g Fq(o)p Fv(\(1\).)p 49878 10590 522 731 v 1200 15055 a Fu(4)1793 b(Pro)50 b(of)597 b(of)g(Theorem)j(1.4)1200 18049 y Ft(4.1)1495 b(Pro)42 b(of)498 b(of)g(Theorem)h(1.4\(a\))1200 20581 y Fi(4.1.1)1394 b(Estimate)464 b(of)12926 19672 y Fm(P)14206 20981 y Fr(\033)32 b Fk(2S)65 b([)p FH(\()17199 20757 y(~)17031 20981 y Fk(C)47 b(nC)18591 20729 y Fx(+)19269 20981 y FH(\))19892 20581 y Fq(\026)20622 20778 y Fr(\014)21251 20581 y Fv(\()p Fq(\033)c Fv(\))p Fq(h)23627 20981 y Fk(S)24197 21130 y Fn(L)24816 20981 y Fr(;)p FH(\()p Fk(S)26078 20729 y Fn(c)26504 20981 y Fk(n)27143 20757 y FH(~)26974 20981 y Fk(C)48 b FH(\))p Fk([C)29058 20729 y 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47454 y Fi(2.)465 b(Decomp)39 b(osition)464 b(of)h(con\014gurations.)539 b Fv(De\014ne)405 b(\(compare)f(with)h(\(3.6\)\))17071 49927 y(^)16851 50234 y Fp(C)71 b Fv(\(0\))338 b(=)20725 49252 y Fm(\010)21432 50234 y Fq(\033)22899 50480 y Fr(B)23609 50629 y Fn(L)24171 50862 y(\014)24724 50629 y(;L)25538 50862 y(\014)26147 50480 y FH(\(0\))27539 50234 y Fv(:)606 b Fq(\033)380 b Fp(2)30919 49927 y Fv(~)30699 50234 y Fp(C)71 b Fv(\(0\))32956 49252 y Fm(\011)33664 50234 y Fq(;)17071 52243 y Fv(\024)16851 52550 y Fp(C)g Fv(\(0\))338 b(=)20725 51568 y Fm(\010)21432 52550 y Fq(\033)22899 52796 y FH([)p Fr(B)23870 52945 y Fn(L)24432 53178 y(\014)24985 52945 y(;L)25799 53178 y(\014)26408 52796 y FH(\(0\)])27871 52544 y Fn(c)28489 52550 y Fv(:)605 b Fq(\033)380 b Fp(2)31869 52243 y Fv(~)31649 52550 y Fp(C)71 b Fv(\(0\))33906 51568 y Fm(\011)34614 52550 y Fq(:)47908 51499 y Fv(\(4.6\))1200 55783 y(Then)310 b(ev)-34 b(ery)308 b Fq(\033)380 b Fp(2)9815 55477 y Fv(~)9595 55783 y Fp(C)71 b Fv(\(0\))310 b(can)f(b)34 b(e)309 b(uniquely)g(decomp)34 b(osed)310 b(as)f Fq(\033)380 b Fv(=)401 b(^)-671 b Fq(\033)123 b Fp(_)145 b Fv(\024)-671 b Fq(\033)352 b Fv(for)309 b(some)374 b(^)-671 b Fq(\033)380 b Fp(2)40979 55477 y Fv(^)40759 55783 y Fp(C)71 b Fv(\(0\))310 b(and)375 b(\024)-671 b Fq(\033)380 b Fp(2)48025 55477 y Fv(\024)47805 55783 y Fp(C)71 b Fv(\(0\).)1200 57289 y(Note)354 b(that)6875 56982 y(^)6656 57289 y Fp(C)71 b Fv(\(0\))355 b(has)f Fq(K)424 b Fv(=)337 b Fq(`)14617 57471 y Fr(c)15080 57289 y Fv(\()p Fq(`)16056 57471 y Fr(c)16688 57289 y Fp(\000)170 b Fv(1\))g(+)g(2)353 b(particles)h(and) 28456 56982 y(\024)28236 57289 y Fp(C)71 b Fv(\(0\))355 b(has)g Fq(n)33688 57486 y Fr(\014)34485 57289 y Fp(\000)170 b Fq(K)441 b Fv(particles)354 b(\(and)h(recall)e(that,)1200 58794 y(b)-34 b(y)404 b(the)h(\014rst)g(half)f(of)h(\(1.35\),)f Fq(n)15810 58991 y Fr(\014)16775 58794 y Fp(!)337 b(1)404 b Fv(as)g Fq(\014)d Fp(!)336 b(1)p Fv(\).)539 b(De\014ne)13257 61517 y Fp(C)13966 61017 y FH(fp)14831 61517 y Fv(\(0\))338 b(=)17996 60535 y Fm(\010)18703 61517 y Fq(\033)380 b Fp(2)21141 61211 y Fv(~)20921 61517 y Fp(C)71 b Fv(\(0\))135 b(:)606 b Fq(H)25264 61714 y Fr(\014)25892 61517 y Fv(\()p Fq(\033)43 b Fv(\))338 b(=)e Fq(H)30195 61714 y Fr(\014)30824 61517 y Fv(\()65 b(^)-671 b Fq(\033)43 b Fv(\))270 b(+)f Fq(H)34992 61714 y Fr(\014)35621 61517 y Fv(\()65 b(\024)-671 b Fq(\033)43 b Fv(\))37299 60535 y Fm(\011)38006 61517 y Fq(;)9565 b Fv(\(4.7\))1200 64240 y(i.e.,)505 b(the)486 b(set)g(of)h(con\014gurations)g(consisting)g(of)f(a)g(proto-critical)f (droplet)h(and)h(a)f(free)g(particle)f(inside)1200 65746 y Fq(B)2119 65934 y Fr(L)2759 66090 y Fn(\014)3312 65934 y Fr(;L)4213 66090 y Fn(\014)4822 65746 y Fv(\(0\))369 b(not)g(in)-34 b(teracting)368 b(with)h(the)g(particles)e(outside)i Fq(B)29583 65934 y Fr(L)30223 66090 y Fn(\014)30776 65934 y Fr(;L)31677 66090 y Fn(\014)32286 65746 y Fv(\(0\).)527 b(W)-101 b(rite)368 b Fp(C)38742 65306 y FH(fp)o Fr(;)p Fk(\000)40600 65746 y Fv(\(0\))h(and)g Fp(C)45547 65306 y FH(fp)o Fr(;)p FH(+)47406 65746 y Fv(\(0\))f(to)1200 67384 y(denoting)456 b(the)g(subsets)h(of)e Fp(C)14760 66944 y FH(fp)15626 67384 y Fv(\(0\))h(where)f(the)h(free)f(particle)g (is)g(at)h(distance)g Fq(L)38535 67581 y Fr(\014)39163 67384 y Fv(,)468 b(resp)34 b(ectiv)-34 b(ely)-101 b(,)466 b(2)456 b(from)1200 68890 y(the)439 b(proto-critical)f(droplet.)643 b(Split)439 b(the)h(righ)-34 b(t-hand)440 b(side)f(of)g(\(4.5\))g(in) -34 b(to)440 b(a)e(con)-34 b(tribution)441 b(coming)e(from)1200 70395 y Fq(\033)-24 b(;)202 b(\033)3144 69955 y Fk(0)3791 70395 y Fp(2)337 b(C)5645 69955 y FH(fp)6510 70395 y Fv(\(0\))405 b(and)g(the)f(rest:)17958 71901 y(r.h.s.\(4)p Fq(:)p Fv(5\))337 b(=)f Fp(j)p Fv(\003)25881 72098 y Fr(\014)26510 71901 y Fp(j)202 b Fv([)p Fq(I)95 b(I)363 b Fv(+)269 b Fq(\015)30750 72083 y FH(2)31276 71901 y Fv(\()p Fq(\014)64 b Fv(\)])p Fq(;)14266 b Fv(\(4.8\))25194 75417 y(24)p eop end %%Page: 25 25 TeXDict begin 25 24 bop 1200 1313 a Fv(where)8302 2819 y Fq(I)95 b(I)431 b Fv(=)4659 b(min)13575 3683 y Fn(h)94 b Fx(:)424 b Fl(C)15290 3482 y Fx(fp)16049 3683 y(\(0\))p Fl(!)p Fx([0)p Fn(;)p Fx(1])11307 4466 y Fn(h)p Fl(j)12041 4900 y(C)12522 4699 y Fx(fp)o Fn(;)p Fl(\000)14172 4900 y Fx(\(0\))15273 4466 y Fl(\021)p Fx(1)p Fn(;)141 b(h)p Fl(j)17446 4900 y(C)17927 4699 y Fx(fp)p Fn(;)p Fx(+)19560 4900 y(\(0\))20661 4466 y Fl(\021)p Fx(0)22174 2342 y FH(1)p 22174 2540 471 49 v 22174 3237 a(2)24494 1667 y Fm(X)22979 4401 y Fr(\033)-21 b(;\033)24327 4149 y Fl(0)24625 4401 y Fk(2C)25797 4149 y Fx(fp)26558 4401 y FH(\(0\))27962 2819 y Fq(\026)28692 3016 y Fr(\014)29321 2819 y Fv(\()p Fq(\033)43 b Fv(\))p Fq(c)31524 3016 y Fr(\014)32154 2819 y Fv(\()p Fq(\033)-24 b(;)202 b(\033)34569 2318 y Fk(0)34879 2819 y Fv(\))g([)p Fq(h)p Fv(\()p Fq(\033)43 b Fv(\))271 b Fp(\000)e Fq(h)p Fv(\()p Fq(\033)41653 2318 y Fk(0)41964 2819 y Fv(\)])42772 2318 y FH(2)47908 2819 y Fv(\(4.9\))1200 6926 y(and)478 b Fq(\015)4258 7108 y FH(2)4783 6926 y Fv(\()p Fq(\014)64 b Fv(\))478 b(is)e(an)i(error)e(term)h(that)h(will)f(b)34 b(e)477 b(estimated)g(in)g(Step)h(6.)758 b(This)477 b(term)g(will)g(turn)h(out) g(to)f(b)34 b(e)1200 8432 y(small)404 b(b)34 b(ecause)404 b(of)g(loss)g(of)g(en)-34 b(trop)g(y)406 b(when)f(the)f(particle)g(is)g (at)g(the)h(b)34 b(oundary)-101 b(.)1200 10933 y Fi(3.)465 b(Reduction)h(to)e(capacit)-39 b(y)464 b(of)h(simple)f(random)h(w)-39 b(alk.)538 b Fv(Estimate)3291 13551 y Fq(I)95 b(I)431 b Fv(=)4659 b(min)8564 14415 y Fn(h)94 b Fx(:)424 b Fl(C)10279 14214 y Fx(fp)11038 14415 y(\(0\))p Fl(!)p Fx([0)p Fn(;)p Fx(1])6295 15199 y Fn(h)p Fl(j)7029 15632 y(C)7510 15431 y Fx(fp)p Fn(;)p Fl(\000)9161 15632 y Fx(\(0\))10262 15199 y Fl(\021)p Fx(1)p Fn(;)141 b(h)p Fl(j)12435 15632 y(C)12916 15431 y Fx(fp)p Fn(;)p Fx(+)14549 15632 y(\(0\))15650 15199 y Fl(\021)p Fx(0)9528 17157 y FH(1)p 9528 17355 V 9528 18053 a(2)11495 16483 y Fm(X)10383 19284 y FH(\024)-520 b Fr(\033)32 b(;)50 b FH(\024)-520 b Fr(\033)11734 19032 y Fl(0)12032 19284 y Fk(2)12827 19060 y FH(\024)12659 19284 y Fk(C)48 b FH(\(0\))17263 16483 y Fm(X)16254 19133 y Fx(^)-453 b Fn(\033)24 b(;)47 b Fx(^)-453 b Fn(\033)17459 18942 y Fl(0)17757 19133 y(2)18455 18965 y Fx(^)18310 19133 y Fl(C)30 b Fx(\(0\):)14789 20080 y(^)-453 b Fn(\033)23 b Fl(_)47 b Fx(\024)-453 b Fn(\033)25 b(;)47 b Fx(^)-453 b Fn(\033)17047 19889 y Fl(0)17345 20080 y(_)47 b Fx(\024)-453 b Fn(\033)18398 19889 y Fl(0)18696 20080 y(2C)19730 19879 y Fx(fp)20490 20080 y(\(0\))21870 17635 y Fq(\026)22600 17832 y Fr(\014)23229 17635 y Fv(\()65 b(^)-671 b Fq(\033)313 b Fp(_)335 b Fv(\024)-671 b Fq(\033)43 b Fv(\))202 b Fq(c)27718 17832 y Fr(\014)28347 17635 y Fv(\()65 b(^)-671 b Fq(\033)312 b Fp(_)335 b Fv(\024)-671 b Fq(\033)43 b(;)267 b Fv(^)-671 b Fq(\033)32912 17134 y Fk(0)33492 17635 y Fp(_)334 b Fv(\024)-671 b Fq(\033)35305 17134 y Fk(0)35616 17635 y Fv(\))202 b([)p Fq(h)p Fv(\()65 b(^)-671 b Fq(\033)313 b Fp(_)335 b Fv(\024)-671 b Fq(\033)43 b Fv(\))269 b Fp(\000)h Fq(h)p Fv(\()65 b(^)-671 b Fq(\033)44473 17134 y Fk(0)45053 17635 y Fp(_)335 b Fv(\024)-671 b Fq(\033)46867 17134 y Fk(0)47177 17635 y Fv(\)])47985 17134 y FH(2)4883 21864 y Fp(\024)3648 b Fv(min)7960 22695 y Fn(g)118 b Fx(:)9285 22527 y(^)9140 22695 y Fl(C)29 b Fx(\(0\))p Fl(!)p Fx([0)p Fn(;)p Fx(1])6295 23479 y Fn(g)24 b Fl(j)7120 23711 y Fx(^)6975 23879 y Fl(C)7456 23688 y(\000)8151 23879 y Fx(\(0\))9251 23479 y Fl(\021)p Fx(1)p Fn(;)141 b(g)24 b Fl(j)11515 23711 y Fx(^)11370 23879 y Fl(C)11851 23688 y Fx(+)12527 23879 y(\(0\))13628 23479 y Fl(\021)p Fx(0)9528 25405 y FH(1)p 9528 25603 V 9528 26300 a(2)10930 24730 y Fm(X)10383 27531 y FH(\024)-520 b Fr(\033)32 b Fk(2)11698 27307 y FH(\024)11530 27531 y Fk(C)47 b FH(\(0\))15985 24730 y Fm(X)14976 27380 y Fx(^)-453 b Fn(\033)23 b(;)47 b Fx(^)-453 b Fn(\033)16180 27189 y Fl(0)16479 27380 y(2)17177 27212 y Fx(^)17032 27380 y Fl(C)30 b Fx(\(0\):)13660 28327 y(^)-453 b Fn(\033)23 b Fl(_)47 b Fx(\024)-453 b Fn(\033)25 b(;)47 b Fx(^)-453 b Fn(\033)15918 28136 y Fl(0)16216 28327 y(_)47 b Fx(\024)-453 b Fn(\033)24 b Fl(2C)18303 28127 y Fx(fp)19063 28327 y(\(0\))20443 25882 y Fq(\026)21173 26079 y Fr(\014)21802 25882 y Fv(\()65 b(^)-671 b Fq(\033)313 b Fp(_)334 b Fv(\024)-671 b Fq(\033)43 b Fv(\))202 b Fq(c)26290 26079 y Fr(\014)26919 25882 y Fv(\()65 b(^)-671 b Fq(\033)313 b Fp(_)334 b Fv(\024)-671 b Fq(\033)44 b(;)266 b Fv(^)-670 b Fq(\033)31485 25381 y Fk(0)32064 25882 y Fp(_)335 b Fv(\024)-671 b Fq(\033)43 b Fv(\))202 b([)p Fq(g)43 b Fv(\()65 b(^)-671 b Fq(\033)44 b Fv(\))270 b Fp(\000)f Fq(g)43 b Fv(\()65 b(^)-671 b Fq(\033)40498 25381 y Fk(0)40810 25882 y Fv(\)])41618 25381 y FH(2)42144 25882 y Fq(;)47302 29634 y Fv(\(4.10\))1200 31140 y(where)5103 30833 y(^)4883 31140 y Fp(C)5592 30700 y Fk(\000)6379 31140 y Fv(\(0\),)9112 30833 y(^)8892 31140 y Fp(C)71 b Fv(\(0\))11149 30700 y FH(+)12519 31140 y Fv(denote)583 b(the)g(subsets)g(of)25021 30833 y(^)24801 31140 y Fp(C)71 b Fv(\(0\))583 b(where)f(the)h(free)f (particle)g(is)g(at)h(distance)f Fq(L)49434 31337 y Fr(\014)50063 31140 y Fv(,)1200 32645 y(resp)34 b(ectiv)-34 b(ely)-101 b(,)402 b(2)i(from)g(the)h(proto-critical)f(droplet,)g(and)h(the)f (inequalit)-34 b(y)404 b(comes)g(from)g(substituting)15116 35368 y Fq(h)p Fv(\()65 b(^)-671 b Fq(\033)313 b Fp(_)335 b Fv(\024)-671 b Fq(\033)43 b Fv(\))337 b(=)f Fq(g)43 b Fv(\()65 b(^)-671 b Fq(\033)44 b Fv(\))p Fq(;)2692 b Fv(^)-671 b Fq(\033)380 b Fp(2)28893 35062 y Fv(^)28673 35368 y Fp(C)71 b Fv(\(0\))p Fq(;)470 b Fv(\024)-671 b Fq(\033)380 b Fp(2)34109 35062 y Fv(\024)33889 35368 y Fp(C)71 b Fv(\(0\))p Fq(;)10819 b Fv(\(4.11\))1200 38091 y(and)472 b(afterw)-34 b(ards)472 b(replacing)e(the)i(double)f (sum)h(o)-34 b(v)g(er)535 b(\024)-671 b Fq(\033)43 b(;)267 b Fv(\024)-671 b Fq(\033)28318 37652 y Fk(0)29077 38091 y Fp(2)30553 37785 y Fv(\024)30333 38091 y Fp(C)71 b Fv(\(0\))472 b(b)-34 b(y)471 b(the)g(single)g(sum)g(o)-34 b(v)g(er)536 b(\024)-671 b Fq(\033)491 b Fp(2)48362 37785 y Fv(\024)48142 38091 y Fp(C)71 b Fv(\(0\))1200 39597 y(b)34 b(ecause)419 b Fq(c)6225 39794 y Fr(\014)6853 39597 y Fv(\()65 b(^)-671 b Fq(\033)323 b Fp(_)345 b Fv(\024)-671 b Fq(\033)43 b(;)267 b Fv(^)-671 b Fq(\033)11439 39157 y Fk(0)12029 39597 y Fp(_)345 b Fv(\024)-671 b Fq(\033)13853 39157 y Fk(0)14163 39597 y Fv(\))363 b Fq(>)f Fv(0)419 b(only)g(if)h(either)484 b(^)-671 b Fq(\033)405 b Fv(=)427 b(^)-671 b Fq(\033)27721 39157 y Fk(0)28451 39597 y Fv(or)484 b(\024)-671 b Fq(\033)405 b Fv(=)427 b(\024)-671 b Fq(\033)33090 39157 y Fk(0)33820 39597 y Fv(\(the)420 b(dynamics)g(up)34 b(dates)420 b(one)g(site)1200 41102 y(at)404 b(a)h(time\).)538 b(Next,)404 b(estimate)2471 43788 y(r.h.s.)o(\(4)p Fq(:)p Fv(10\))2808 46283 y Fp(\024)4684 45132 y Fm(X)4136 47932 y FH(\024)-519 b Fr(\033)31 b Fk(2)5452 47708 y FH(\024)5283 47932 y Fk(C)48 b FH(\(0\))8476 45463 y Fv(1)p 7366 46004 2826 49 v 7366 47517 a Fq(Z)8280 46814 y FH(\()p Fr(n)9217 46970 y Fn(\014)9771 46814 y FH(\))8193 47908 y Fr(\014)10527 46283 y Fq(e)11091 45783 y Fk(\000)p Fr(\014)e(H)13170 45939 y Fn(\014)13722 45783 y FH(\()k(\024)-520 b Fr(\033)32 b FH(\))18593 46283 y Fv(min)17078 47114 y Fn(g)118 b Fx(:)18403 46946 y(^)18258 47114 y Fl(C)30 b Fx(\(0\))p Fl(!)p Fx([0)p Fn(;)p Fx(1])15414 47898 y Fn(g)24 b Fl(j)16239 48130 y Fx(^)16094 48298 y Fl(C)16575 48107 y(\000)17269 48298 y Fx(\(0\))18370 47898 y Fl(\021)p Fx(1)p Fn(;)141 b(g)24 b Fl(j)20634 48130 y Fx(^)20489 48298 y Fl(C)20970 48107 y Fx(+)21646 48298 y(\(0\))22746 47898 y Fl(\021)p Fx(0)24260 45806 y FH(1)p 24260 46004 471 49 v 24260 46701 a(2)27570 45132 y Fm(X)26678 47781 y Fx(^)-453 b Fn(\033)23 b(;)47 b Fx(^)-453 b Fn(\033)27882 47590 y Fl(0)28181 47781 y(2)28879 47613 y Fx(^)28734 47781 y Fl(C)30 b Fx(\(0\))25245 48728 y(^)-453 b Fn(\033)23 b Fl(_)47 b Fx(\024)-453 b Fn(\033)25 b(;)47 b Fx(^)-453 b Fn(\033)27503 48537 y Fl(0)27801 48728 y(_)47 b Fx(\024)-453 b Fn(\033)24 b Fl(2C)29888 48528 y Fx(fp)30648 48728 y(\(0\))32028 46283 y Fq(e)32592 45783 y Fk(\000)p Fr(\014)46 b(H)34671 45939 y Fn(\014)35224 45783 y FH(\()k(^)-520 b Fr(\033)31 b FH(\))36782 46283 y Fq(c)37307 46480 y Fr(\014)37936 46283 y Fv(\()65 b(^)-671 b Fq(\033)43 b(;)267 b Fv(^)-671 b Fq(\033)40418 45783 y Fk(0)40729 46283 y Fv(\))202 b([)p Fq(g)43 b Fv(\()65 b(^)-671 b Fq(\033)44 b Fv(\))270 b Fp(\000)f Fq(g)43 b Fv(\()65 b(^)-671 b Fq(\033)47349 45783 y Fk(0)47660 46283 y Fv(\)])48468 45783 y FH(2)48994 46283 y Fq(;)47302 50035 y Fv(\(4.12\))1200 51541 y(where)345 b(w)-34 b(e)345 b(used)g Fq(H)10087 51738 y Fr(\014)10716 51541 y Fv(\()p Fq(\033)43 b Fv(\))337 b(=)g Fq(H)15019 51738 y Fr(\014)15647 51541 y Fv(\()65 b(^)-671 b Fq(\033)44 b Fv(\))151 b(+)g Fq(H)19579 51738 y Fr(\014)20207 51541 y Fv(\()65 b(\024)-671 b Fq(\033)43 b Fv(\))346 b(from)f(\(4.7\))g(and) h(write)f Fq(c)33738 51738 y Fr(\014)34366 51541 y Fv(\()65 b(^)-671 b Fq(\033)43 b(;)267 b Fv(^)-671 b Fq(\033)36848 51101 y Fk(0)37159 51541 y Fv(\))345 b(to)g(denote)h(the)f(transition) 1200 53046 y(rate)572 b(asso)34 b(ciated)572 b(with)g(the)h(Ka)-34 b(w)g(asaki)572 b(dynamics)g(restricted)g(to)g Fq(B)34246 53234 y Fr(L)34886 53390 y Fn(\014)35439 53234 y Fr(;L)36340 53390 y Fn(\014)36949 53046 y Fv(\(0\),)614 b(whic)-34 b(h)573 b(clearly)d(equals)1200 54698 y Fq(c)1725 54895 y Fr(\014)2353 54698 y Fv(\()65 b(^)-671 b Fq(\033)414 b Fp(_)436 b Fv(\024)-671 b Fq(\033)43 b(;)267 b Fv(^)-671 b Fq(\033)7121 54258 y Fk(0)7802 54698 y Fp(_)435 b Fv(\024)-671 b Fq(\033)43 b Fv(\))556 b(for)g(ev)-34 b(ery)620 b(\024)-671 b Fq(\033)632 b Fp(2)19046 54392 y Fv(\024)18827 54698 y Fp(C)71 b Fv(\(0\))556 b(suc)-34 b(h)557 b(that)622 b(^)-671 b Fq(\033)413 b Fp(_)436 b Fv(\024)-671 b Fq(\033)43 b(;)266 b Fv(^)-670 b Fq(\033)31600 54258 y Fk(0)32281 54698 y Fp(_)435 b Fv(\024)-671 b Fq(\033)632 b Fp(2)590 b(C)36891 54258 y FH(fp)37756 54698 y Fv(\(0\))557 b(b)34 b(ecause)555 b(there)h(is)f(no)1200 56204 y(in)-34 b(teraction)406 b(b)34 b(et)-34 b(w)g(een)406 b(the)g(particles)f(inside)g(and)h (outside)g Fq(B)29831 56392 y Fr(L)30471 56548 y Fn(\014)31025 56392 y Fr(;L)31926 56548 y Fn(\014)32535 56204 y Fv(\(0\).)543 b(The)405 b(minim)-34 b(um)407 b(in)e(the)h(r.h.s.)e(of)1200 57709 y(\(4.12\))h(can)f(b)34 b(e)404 b(estimated)g(from)h(ab)34 b(o)-34 b(v)g(e)404 b(b)-34 b(y)23298 59462 y Fm(X)22608 62146 y Fr(\033)32 b Fk(2P)74 b FH(\(0\))25941 60613 y Fp(V)26684 60810 y Fr(\014)27313 60613 y Fv(\()p Fq(\033)43 b Fv(\))18311 b(\(4.13\))1200 64726 y(with)405 b Fp(P)100 b Fv(\(0\))405 b(the)f(set)g(of)h(proto-critical)e(droplets)i(with)g (lo)-34 b(w)g(er-left)404 b(corner)g(at)g(0,)g(and)11509 67630 y Fp(V)12252 67827 y Fr(\014)12881 67630 y Fv(\()p Fq(\033)43 b Fv(\))338 b(=)5089 b(min)19702 68451 y Fn(f)171 b Fx(:)424 b Fc(Z)21373 68260 y Fx(2)21834 68451 y Fl(!)p Fx([0)p Fn(;)p Fx(1])16309 69235 y Fn(f)76 b Fl(j)17036 69504 y Fn(P)17554 69615 y(\033)18109 69504 y Fx(\(0\))19210 69235 y Fl(\021)p Fx(1)p Fn(;)141 b(f)76 b Fl(j)21376 69504 y Fx([)p Fn(B)22225 69729 y(L)22787 69962 y(\014)23341 69729 y(;L)24155 69962 y(\014)24763 69504 y Fx(\(0\)])26043 69313 y Fn(c)26524 69235 y Fl(\021)p Fx(0)28038 67153 y FH(1)p 28038 67351 V 28038 68048 a(2)29568 66479 y Fm(X)28976 69118 y Fn(x;x)30156 68927 y Fl(0)30453 69118 y(2)p Fc(Z)31449 68927 y Fx(2)29511 69926 y Fn(x)p Fl(\030)p Fn(x)31079 69735 y Fl(0)32044 67630 y Fv([)p Fq(f)130 b Fv(\()p Fq(x)p Fv(\))270 b Fp(\000)f Fq(f)130 b Fv(\()p Fq(x)38108 67130 y Fk(0)38420 67630 y Fv(\)])39228 67130 y FH(2)39754 67630 y Fq(;)7211 b Fv(\(4.14\))25194 75417 y(25)p eop end %%Page: 26 26 TeXDict begin 26 25 bop 1200 1313 a Fv(where)469 b Fq(P)5548 1495 y Fr(\033)6173 1313 y Fv(\(0\))g(is)g(the)g(supp)34 b(ort)470 b(of)f(the)h(proto-critical)e(droplet)h(in)g Fq(\033)43 b Fv(,)485 b(and)469 b Fq(x)445 b Fp(\030)f Fq(x)40040 874 y Fk(0)40819 1313 y Fv(means)469 b(that)h Fq(x)f Fv(and)1200 2819 y Fq(x)1893 2379 y Fk(0)2599 2819 y Fv(are)396 b(neigh)-34 b(b)34 b(oring)397 b(sites.)536 b(Indeed,)398 b(\(4.13\))f(is)f(obtained)h(from)f(the)h(expression)f (in)g(\(4.12\))h(b)-34 b(y)397 b(dropping)1200 4324 y(the)404 b(restriction)469 b(^)-671 b Fq(\033)313 b Fp(_)334 b Fv(\024)-671 b Fq(\033)43 b(;)267 b Fv(^)-671 b Fq(\033)13186 3884 y Fk(0)13766 4324 y Fp(_)334 b Fv(\024)-671 b Fq(\033)380 b Fp(2)337 b(C)17770 3884 y FH(fp)18636 4324 y Fv(\(0\),)404 b(substituting)10102 6992 y Fq(g)43 b Fv(\()p Fq(P)11972 7174 y Fr(\033)12598 6992 y Fv(\(0\))270 b Fp([)g(f)p Fq(x)p Fp(g)p Fv(\))336 b(=)h Fq(f)130 b Fv(\()p Fq(x)p Fv(\))p Fq(;)2627 b(\033)380 b Fp(2)337 b(P)100 b Fv(\(0\))p Fq(;)404 b(x)337 b Fp(2)f Fq(B)33351 7180 y Fr(L)33991 7336 y Fn(\014)34545 7180 y Fr(;L)35446 7336 y Fn(\014)36055 6992 y Fv(\(0\))p Fp(n)p Fq(P)38987 7174 y Fr(\033)39612 6992 y Fv(\(0\))p Fq(;)5805 b Fv(\(4.15\))1200 9660 y(and)493 b(noting)h(that)g Fq(c)10745 9857 y Fr(\014)11373 9660 y Fv(\()p Fq(P)12622 9842 y Fr(\033)13248 9660 y Fv(\(0\))329 b Fp([)f(f)p Fq(x)p Fp(g)p Fq(;)202 b(P)19483 9842 y Fr(\033)20108 9660 y Fv(\(0\))329 b Fp([)g(f)p Fq(x)24421 9221 y Fk(0)24731 9660 y Fp(g)p Fv(\))485 b(=)f(1)493 b(when)g Fq(x)484 b Fp(\030)h Fq(x)35370 9221 y Fk(0)36172 9660 y Fv(and)494 b(zero)e(otherwise.)804 b(What)1200 11166 y(\(4.13\))405 b(sa)-34 b(ys)404 b(is)g(that)16994 12671 y Fp(V)17737 12868 y Fr(\014)18366 12671 y Fv(\()p Fq(\033)43 b Fv(\))337 b(=)g Fw(CAP)o Fv(\()p Fq(P)25114 12853 y Fr(\033)25739 12671 y Fv(\(0\))p Fq(;)202 b Fv([)p Fq(B)29082 12859 y Fr(L)29722 13015 y Fn(\014)30276 12859 y Fr(:L)31177 13015 y Fn(\014)31786 12671 y Fv(\(0\)])33671 12171 y Fr(c)34134 12671 y Fv(\))12697 b(\(4.16\))1200 14869 y(is)437 b(the)h(capacit)-34 b(y)438 b(of)g(simple)f(random)h(w) -34 b(alk)438 b(b)34 b(et)-34 b(w)g(een)439 b(the)f(proto-critical)f (droplet)g Fq(P)41093 15051 y Fr(\033)41719 14869 y Fv(\(0\))h(in)f Fq(\033)481 b Fv(and)438 b(the)1200 16374 y(exterior)403 b(of)h Fq(B)7984 16562 y Fr(L)8624 16718 y Fn(\014)9178 16562 y Fr(:L)10079 16718 y Fn(\014)10687 16374 y Fv(\(0\).)540 b(No)-34 b(w,)404 b(de\014ne)18525 19117 y(\024)18269 19424 y Fq(Z)19183 18785 y FH(\()p Fr(n)p Fk(\000)p Fr(K)63 b FH(\))19096 19815 y Fr(\014)22130 19424 y Fv(\(0\))337 b(=)25891 18272 y Fm(X)25344 21073 y FH(\024)-519 b Fr(\033)31 b Fk(2)26660 20849 y FH(\024)26491 21073 y Fk(C)48 b FH(\(0\))28441 19424 y Fq(e)29005 18923 y Fk(\000)p Fr(\014)e(H)31084 19079 y Fn(\014)31637 18923 y FH(\()k(\024)-520 b Fr(\033)31 b FH(\))32994 19424 y Fq(:)13971 b Fv(\(4.17\))1200 23598 y(Then)405 b(w)-34 b(e)404 b(obtain)h(via)f(\(4.13\))h(that)14229 27513 y(r.h.s.\(4)p Fq(:)p Fv(12\))337 b Fp(\024)f Fq(e)22143 27013 y Fk(\000)p Fr(\014)46 b FH(\000)24037 26700 y Fl(\003)25162 26171 y Fv(\024)24907 26477 y Fq(Z)25821 25838 y FH(\()p Fr(n)p Fk(\000)p Fr(K)63 b FH(\))25734 26868 y Fr(\014)28767 26477 y Fv(\(0\))p 24907 27234 5409 49 v 26198 28747 a Fq(Z)27112 28044 y FH(\()p Fr(n)28049 28200 y Fn(\014)28603 28044 y FH(\))27025 29138 y Fr(\014)31341 26362 y Fm(X)30650 29046 y Fr(\033)32 b Fk(2P)74 b FH(\(0\))33983 27513 y Fp(V)34726 27710 y Fr(\014)35355 27513 y Fv(\()p Fq(\033)43 b Fv(\))p Fq(;)9932 b Fv(\(4.18\))1200 31673 y(where)442 b(\000)5501 31233 y Fk(\003)6427 31673 y Fv(=)400 b Fp(\000)p Fq(U)132 b Fv([\()p Fq(`)10986 31855 y Fr(c)11744 31673 y Fp(\000)294 b Fv(1\))14058 31233 y FH(2)14879 31673 y Fv(+)h Fq(`)16622 31855 y Fr(c)17085 31673 y Fv(\()p Fq(`)18061 31855 y Fr(c)18819 31673 y Fp(\000)g Fv(1\))g(+)f(1])442 b(is)h(the)f(binding)i(energy)d(of)i(the) g(proto-critical)e(droplet)1200 33178 y(\(compare)404 b(with)h(\(1.33\)\).)1200 35652 y Fi(4.)k(Capacit)-39 b(y)408 b(estimate.)522 b Fv(F)-101 b(or)356 b(future)h(reference)e(w) -34 b(e)356 b(state)g(the)h(follo)-34 b(wing)356 b(estimate)g(on)g (capacities)g(for)1200 37158 y(simple)404 b(random)h(w)-34 b(alk.)1200 40259 y Fi(Lemma)464 b(4.3)606 b Fs(L)-62 b(et)315 b Fq(U)468 b Fp(\032)337 b Fo(Z)13888 39819 y FH(2)14729 40259 y Fs(b)-62 b(e)315 b(any)f(set)h(such)e(that)h Fp(f)p Fv(0)p Fp(g)337 b(\032)f Fq(U)469 b Fp(\032)336 b Fq(B)32001 40456 y Fr(k)24 b(;k)33345 40259 y Fv(\(0\))p Fs(,)340 b(with)315 b Fq(k)375 b Fp(2)336 b Fo(N)12 b Fp([)g(f)p Fv(0)p Fp(g)317 b Fs(indep)-62 b(endent)1200 41765 y(of)550 b Fq(\014)64 b Fs(.)910 b(L)-62 b(et)551 b Fq(V)819 b Fp(\032)550 b Fo(Z)10811 41325 y FH(2)11888 41765 y Fs(b)-62 b(e)550 b(any)h(set)f(such)g(that)f Fv([)p Fq(B)24666 41953 y Fr(K)63 b(L)26162 42109 y Fn(\014)26716 41953 y Fr(;K)g(L)28473 42109 y Fn(\014)29081 41765 y Fv(\(0\)])30966 41325 y Fr(c)31979 41765 y Fp(\032)550 b Fq(V)819 b Fp(\032)550 b Fv([)p Fq(B)37747 41953 y Fr(L)38387 42109 y Fn(\014)38940 41953 y Fr(;L)39841 42109 y Fn(\014)40449 41765 y Fv(\(0\)])42334 41325 y Fr(c)42798 41765 y Fs(,)580 b(with)551 b Fq(K)637 b Fp(2)549 b Fo(N)1200 43270 y Fs(indep)-62 b(endent)432 b(of)h Fq(\014)64 b Fs(.)557 b(Then)5487 45938 y Fw(CAP)7893 44956 y Fm(\000)8448 45938 y Fp(f)p Fv(0)p Fp(g)p Fq(;)202 b Fv([)p Fq(B)12061 46126 y Fr(K)63 b(L)13557 46282 y Fn(\014)14110 46126 y Fr(;K)g(L)15867 46282 y Fn(\014)16476 45938 y Fv(\(0\)])18361 45438 y Fr(c)18825 44956 y Fm(\001)19717 45938 y Fp(\024)336 b Fw(CAP)201 b Fv(\()q Fq(U)m(;)h(V)269 b Fv(\))337 b Fp(\024)g Fw(CAP)30708 44956 y Fm(\000)31263 45938 y Fq(B)32182 46135 y Fr(k)24 b(;k)33526 45938 y Fv(\(0\))p Fq(;)202 b Fv([)p Fq(B)36869 46126 y Fr(L)37509 46282 y Fn(\014)38063 46126 y Fr(;L)38964 46282 y Fn(\014)39573 45938 y Fv(\(0\)])41458 45438 y Fr(c)41921 44956 y Fm(\001)42679 45938 y Fq(:)4286 b Fv(\(4.19\))1200 48606 y Fs(Mor)-62 b(e)g(over,)431 b(via)i Fv(\(1.29{1.30\))p Fs(,)1897 51784 y Fw(CAP)4303 50802 y Fm(\000)4858 51784 y Fq(B)5777 51981 y Fr(k)24 b(;k)7121 51784 y Fv(\(0\))p Fq(;)202 b Fv([)p Fq(B)10464 51972 y Fr(K)63 b(L)11960 52128 y Fn(\014)12514 51972 y Fr(;K)g(L)14271 52128 y Fn(\014)14880 51784 y Fv(\(0\)])16765 51283 y Fr(c)17228 50802 y Fm(\001)18121 51784 y Fv(=)336 b([1)269 b(+)g Fq(o)p Fv(\(1\)])28460 50964 y(2)p Fq(\031)p 24632 51505 8997 49 v 24632 52615 a Fv(log)17 b(\()p Fq(K)87 b(L)28610 52812 y Fr(\014)29239 52615 y Fv(\))270 b Fp(\000)f Fv(log)219 b Fq(k)34098 51784 y Fv(=)337 b([1)269 b(+)g Fq(o)p Fv(\(1\)])40820 50964 y(4)p Fq(\031)p 40610 51505 1760 49 v 40610 52615 a(\014)64 b Fv(\001)42502 51784 y Fq(;)2680 b(\014)401 b Fp(!)336 b(1)p Fq(:)47302 54005 y Fv(\(4.20\))1200 57106 y Fs(Pr)-62 b(o)g(of)p Fv(.)403 b(The)459 b(inequalities)g(in)f (\(4.19\))i(follo)-34 b(w)459 b(from)h(standard)g(monotonicit)-34 b(y)460 b(prop)34 b(erties)458 b(of)i(capacities.)1200 58612 y(The)520 b(asymptotic)g(estimate)g(in)g(\(4.20\))g(for)g (capacities)f(of)h(concen)-34 b(tric)520 b(b)34 b(o)-34 b(xes)520 b(are)f(standard)i(\(see)f(e.g.)1200 60117 y(La)-34 b(wler)404 b([20)o(],)g(Section)g(2.3\),)g(and)h(also)f(follo) -34 b(w)405 b(b)-34 b(y)404 b(comparison)h(to)f(Bro)-34 b(wnian)405 b(motion.)p 49878 60117 522 731 v 3082 62591 a(W)-101 b(e)366 b(can)g(apply)h(Lemma)f(4.3)g(to)h(estimate)f Fp(V)23820 62788 y Fr(\014)24449 62591 y Fv(\()p Fq(\033)43 b Fv(\))367 b(in)f(\(4.16\),)375 b(since)366 b(the)g(proto-critical)g (droplet)h(with)1200 64097 y(lo)-34 b(w)g(er-left)405 b(corner)e(in)h(0)g(\014ts)h(inside)f(the)h(b)34 b(o)-34 b(x)404 b Fq(B)23422 64294 y FH(2)p Fr(`)24276 64405 y Fn(c)24704 64294 y Fr(;)p FH(2)p Fr(`)25819 64405 y Fn(c)26302 64097 y Fv(\(0\).)539 b(This)405 b(giv)-34 b(es)13168 67341 y Fp(V)13911 67538 y Fr(\014)14540 67341 y Fv(\()p Fq(\033)43 b Fv(\))337 b(=)18177 66521 y(4)p Fq(\031)p 17968 67062 1760 49 v 17968 68172 a(\014)64 b Fv(\001)20062 67341 y([1)269 b(+)g Fq(o)p Fv(\(1\)])p Fq(;)2626 b Fp(8)202 b Fq(\033)381 b Fp(2)336 b(P)100 b Fv(\(0\))p Fq(;)405 b(\014)400 b Fp(!)337 b(1)p Fq(:)8870 b Fv(\(4.21\))1200 70590 y(Moro)-34 b(v)g(er,)403 b(from)g(Bo)-34 b(vier,)401 b(den)j(Hollander,)f(and)g(Nardi)g([7],)f(Lemmas)h (3.4.2{3.4.3,)f(w)-34 b(e)403 b(kno)-34 b(w)404 b(that)g Fq(N)469 b Fv(=)1200 72096 y Fp(jP)100 b Fv(\(0\))p Fp(j)p Fv(,)404 b(the)g(n)-34 b(um)g(b)34 b(er)405 b(of)g(shap)34 b(es)404 b(of)h(the)f(proto-critical)g(droplet,)g(equals)g Fq(N)469 b Fv(=)38668 71619 y FH(1)p 38668 71817 471 49 v 38668 72514 a(3)39271 72096 y Fq(`)39776 71656 y FH(2)39776 72395 y Fr(c)40302 72096 y Fv(\()p Fq(`)41278 71656 y FH(2)41278 72395 y Fr(c)42074 72096 y Fp(\000)269 b Fv(1\).)25194 75417 y(26)p eop end %%Page: 27 27 TeXDict begin 27 26 bop 1200 1313 a Fi(5.)465 b(Equiv)-77 b(alence)463 b(of)i(ensem)-39 b(bles.)539 b Fv(According)404 b(to)g(Lemma)g(B.1)f(in)i(App)34 b(endix)404 b(B,)f(w)-34 b(e)405 b(ha)-34 b(v)g(e)12924 3938 y(\024)12668 4244 y Fq(Z)13582 3542 y FH(\()p Fr(n)14519 3698 y Fn(\014)15073 3542 y Fk(\000)p Fr(K)63 b FH(\))13495 4636 y Fr(\014)17082 4244 y Fv(\(0\))p 12668 5002 5963 49 v 14237 6515 a Fq(Z)15151 5812 y FH(\()p Fr(n)16088 5968 y Fn(\014)16641 5812 y FH(\))15064 6906 y Fr(\014)19100 5281 y Fv(=)336 b(\()p Fq(\032)21477 5478 y Fr(\014)22106 5281 y Fv(\))22577 4780 y Fr(K)23691 5281 y Fq(\026)24421 5478 y Fr(\014)25050 5281 y Fv(\()p Fp(S)91 b Fv(\))202 b([1)270 b(+)f Fq(o)p Fv(\(1\)])p Fq(;)2626 b(\014)401 b Fp(!)336 b(1)p Fq(:)8238 b Fv(\(4.22\))1200 9248 y(This)575 b(is)f(an)h(\\equiv)-67 b(alence)572 b(of)j(ensem)-34 b(bles")575 b(prop)34 b(ert)-34 b(y)574 b(relating)g(the)h(probabilities)f(to)h(\014nd)g Fq(n)46610 9445 y Fr(\014)47621 9248 y Fp(\000)383 b Fq(K)87 b Fv(,)1200 10754 y(resp)34 b(ectiv)-34 b(ely)-101 b(,)444 b Fq(n)8815 10951 y Fr(\014)9881 10754 y Fv(particles)437 b(inside)h([)p Fq(B)19504 10942 y Fr(L)20144 11098 y Fn(\014)20697 10942 y Fr(;L)21598 11098 y Fn(\014)22207 10754 y Fv(\(0\)])24092 10314 y Fr(c)24993 10754 y Fv(\(recall)f (\(4.6\)\).)640 b(Com)-34 b(bining)439 b(\(4.2{4.3\),)446 b(\(4.5\),)h(\(4.8\),)1200 12259 y(\(4.10\),)404 b(\(4.12\),)h (\(4.18\))f(and)h(\(4.21{4.22\),)g(w)-34 b(e)404 b(get)3056 15492 y Fw(CAP)o Fv(\()p Fp(S)91 b Fq(;)202 b Fp(C)7804 14991 y FH(+)8591 15492 y Fv(\))337 b Fp(\024)g Fq(\015)11307 15674 y FH(1)11832 15492 y Fv(\()p Fq(\014)64 b Fv(\))270 b(+)f Fp(j)p Fv(\003)16185 15689 y Fr(\014)16813 15492 y Fp(j)p Fq(\015)17778 15674 y FH(2)18303 15492 y Fv(\()p Fq(\014)64 b Fv(\))270 b(+)f Fq(N)334 b Fp(j)p Fv(\003)23964 15689 y Fr(\014)24592 15492 y Fp(j)25474 14672 y Fv(4)p Fq(\031)p 25264 15213 1760 49 v 25264 16323 a(\014)64 b Fv(\001)27358 15492 y Fq(e)27922 14991 y Fk(\000)p Fr(\014)46 b FH(\000)30073 15492 y Fq(\026)30803 15689 y Fr(\014)31432 15492 y Fv(\()p Fp(S)91 b Fv(\))202 b([1)270 b(+)f Fq(o)p Fv(\(1\)])p Fq(;)2626 b(\014)401 b Fp(!)336 b(1)p Fq(;)1856 b Fv(\(4.23\))1200 18796 y(where)491 b(w)-34 b(e)492 b(use)g(that)g(\000)12317 18356 y Fk(\003)13170 18796 y Fv(+)327 b(\001)p Fq(K)570 b Fv(=)482 b(\000)491 b(de\014ned)h(in)f(\(1.33\).)801 b(This)492 b(completes)g(the)f(pro)34 b(of)492 b(of)g(the)g(upp)34 b(er)1200 20302 y(b)g(ound,)405 b(pro)-34 b(vided)405 b(that)g(the)f(error)f(terms)i Fq(\015)21953 20484 y FH(1)22478 20302 y Fv(\()p Fq(\014)64 b Fv(\))404 b(and)h Fq(\015)27559 20484 y FH(2)28085 20302 y Fv(\()p Fq(\014)64 b Fv(\))404 b(are)g(negligible.)1200 22803 y Fi(6.)542 b(Second)g(error)g(term.)739 b Fv(T)-101 b(o)471 b(estimate)h(the)f(error)f(term)h Fq(\015)30851 22985 y FH(2)31376 22803 y Fv(\()p Fq(\014)64 b Fv(\),)488 b(note)472 b(that)g(the)f(con\014gurations)i(in)1420 24003 y(~)1200 24309 y Fp(C)71 b Fv(\(0\))p Fp(nC)4772 23869 y FH(fp)5638 24309 y Fv(\(0\))385 b(are)f(those)h(for)f(whic)-34 b(h)386 b(inside)e Fq(B)22351 24497 y Fr(L)22991 24653 y Fn(\014)23544 24497 y Fr(;L)24445 24653 y Fn(\014)25054 24309 y Fv(\(0\))h(there)f(is)g(a)g(proto-critical)g(droplet)h(whose)g (lo)-34 b(w)g(er-)1200 25814 y(left)358 b(corner)g(is)g(at)g(0,)367 b(and)359 b(a)g(particle)e(that)j(is)e(at)g(the)h(b)34 b(oundary)359 b(and)g(attac)-34 b(hed)359 b(to)g(some)f(cluster)g (outside)1200 27320 y Fq(B)2119 27508 y Fr(L)2759 27664 y Fn(\014)3312 27508 y Fr(;L)4213 27664 y Fn(\014)4822 27320 y Fv(\(0\).)539 b(Recalling)404 b(\(4.5{4.9\),)g(w)-34 b(e)404 b(therefore)g(ha)-34 b(v)g(e)3941 30394 y Fq(\015)4569 30576 y FH(2)5095 30394 y Fv(\()p Fq(\014)64 b Fv(\))337 b Fp(\024)10489 29242 y Fm(X)8403 32080 y Fr(\033)32 b Fk(2)9768 31857 y FH(~)9600 32080 y Fk(C)48 b FH(\(0\))p Fk(nC)12363 31828 y Fx(fp)13123 32080 y FH(\(0\))15678 29242 y Fm(X)14932 32080 y Fr(\033)15502 31828 y Fl(0)15800 32080 y Fk(2)16595 31857 y FH(~)16427 32080 y Fk(C)f FH(\(0\))18377 30394 y Fq(\026)19107 30591 y Fr(\014)19736 30394 y Fv(\()p Fq(\033)c Fv(\))p Fq(c)21939 30591 y Fr(\014)22568 30394 y Fv(\()p Fq(\033)-24 b(;)202 b(\033)24983 29893 y Fk(0)25294 30394 y Fv(\))g([)p Fq(h)p Fv(\()p Fq(\033)43 b Fv(\))270 b Fp(\000)f Fq(h)p Fv(\()p Fq(\033)32067 29893 y Fk(0)32379 30394 y Fv(\)])33187 29893 y FH(2)34050 30394 y Fp(\024)336 b Fv(6)p Fq(\026)36665 30591 y Fr(\014)37294 30394 y Fv(\()37985 30088 y(~)37765 30394 y Fp(C)72 b Fv(\(0\))p Fp(nC)41338 29893 y FH(fp)42204 30394 y Fv(\(0\)\))p Fq(;)2742 b Fv(\(4.24\))1200 34867 y(where)414 b(w)-34 b(e)414 b(use)g(that)h Fq(h)135 b Fv(:)13046 34561 y(~)12826 34867 y Fp(C)71 b Fv(\(0\))353 b Fp(!)g Fv([0)p Fq(;)202 b Fv(1],)416 b Fq(\026)20909 35064 y Fr(\014)21538 34867 y Fv(\()p Fq(\033)43 b Fv(\))p Fq(c)23741 35064 y Fr(\014)24370 34867 y Fv(\()p Fq(\033)-24 b(;)202 b(\033)26785 34427 y Fk(0)27096 34867 y Fv(\))353 b(=)f Fq(\026)29945 35064 y Fr(\014)30575 34867 y Fv(\()p Fq(\033)43 b Fv(\))276 b Fp(^)g Fq(\026)34343 35064 y Fr(\014)34972 34867 y Fv(\()p Fq(\033)36179 34427 y Fk(0)36490 34867 y Fv(\),)417 b(and)d(there)g(are)f(6)h(p)34 b(ossible)1200 36373 y(transitions)494 b(from)10474 36066 y(~)10254 36373 y Fp(C)71 b Fv(\(0\))p Fp(nC)13826 35933 y FH(fp)14692 36373 y Fv(\(0\))495 b(to)18525 36066 y(~)18306 36373 y Fp(C)70 b Fv(\(0\):)718 b(3)494 b(through)h(a)e(mo)-34 b(v)g(e)494 b(b)-34 b(y)494 b(the)g(particle)e(at)i(the)g(b)34 b(oundary)494 b(of)1200 37878 y Fq(B)2119 38066 y Fr(L)2759 38222 y Fn(\014)3312 38066 y Fr(;L)4213 38222 y Fn(\014)4822 37878 y Fv(\(0\))405 b(and)g(3)f(through)h(a)g(mo)-34 b(v)g(e)404 b(b)-34 b(y)404 b(a)g(particle)g(in)g(the)h(cluster)e(outside)i Fq(B)38458 38066 y Fr(L)39098 38222 y Fn(\014)39652 38066 y Fr(;L)40553 38222 y Fn(\014)41161 37878 y Fv(\(0\).)540 b(Since)12373 40795 y Fq(H)13381 40992 y Fr(\014)14009 40795 y Fv(\()p Fq(\033)43 b Fv(\))338 b Fp(\025)e Fq(H)18312 40992 y Fr(\014)18941 40795 y Fv(\()65 b(^)-671 b Fq(\033)43 b Fv(\))270 b(+)f Fq(H)23109 40992 y Fr(\014)23737 40795 y Fv(\()65 b(\024)-671 b Fq(\033)44 b Fv(\))270 b Fp(\000)f Fq(U)m(;)2626 b(\033)379 b Fp(2)33123 40489 y Fv(~)32903 40795 y Fp(C)71 b Fv(\(0\))p Fp(nC)36475 40295 y FH(fp)37342 40795 y Fv(\(0\))p Fq(;)8075 b Fv(\(4.25\))1200 43519 y(it)404 b(follo)-34 b(ws)405 b(from)f(the)h(same)f(argumen)-34 b(t)405 b(as)f(in)g(Steps)h(3)g(and)f(5)h(that)8517 46242 y Fq(\026)9247 46439 y Fr(\014)9877 46242 y Fv(\()10568 45935 y(~)10348 46242 y Fp(C)71 b Fv(\(0\))p Fp(nC)13920 45741 y FH(fp)14786 46242 y Fv(\(0\)\))338 b Fp(\024)f Fq(N)d(e)20295 45741 y Fk(\000)p Fr(\014)46 b FH(\000)22189 45429 y Fl(\003)22925 46242 y Fv(\()p Fq(\032)24023 46439 y Fr(\014)24652 46242 y Fv(\))25123 45741 y Fr(K)63 b FH(+1)27439 46242 y Fq(\026)28169 46439 y Fr(\014)28798 46242 y Fv(\()p Fp(S)91 b Fv(\))202 b Fq(e)31331 45741 y Fr(\014)46 b(U)32894 46242 y Fv(4\()p Fq(K)357 b Fp(\000)269 b Fv(1\))202 b([1)270 b(+)f Fq(o)p Fv(\(1\)])p Fq(;)4219 b Fv(\(4.26\))1200 48965 y(where)313 b(\()p Fq(\032)5712 49162 y Fr(\014)6341 48965 y Fv(\))6812 48525 y Fr(K)63 b FH(+1)9238 48965 y Fv(comes)313 b(from)g(the)g(fact)g(that)h Fq(n)23055 49162 y Fr(\014)23770 48965 y Fp(\000)87 b Fv(\()p Fq(K)173 b Fv(+)87 b(1\))313 b(particles)g(are)f(outside)h Fq(B)40602 49153 y Fr(L)41242 49309 y Fn(\014)41796 49153 y Fk(\000)p FH(1)p Fr(;L)43899 49309 y Fn(\014)44452 49153 y Fk(\000)p FH(1)45710 48965 y Fv(\(0\))h(\(once)1200 50603 y(more)362 b(use)g(Lemma)g(B.1)g(in)g(App)34 b(endix)363 b(B\),)370 b Fq(e)22180 50163 y Fr(\014)46 b(U)23902 50603 y Fv(comes)362 b(from)g(the)h(gap)g(in)f(\(4.25\),)371 b(and)363 b(4\()p Fq(K)273 b Fp(\000)186 b Fv(1\))362 b(coun)-34 b(ts)1200 52109 y(the)443 b(maximal)f(n)-34 b(um)g(b)34 b(er)443 b(of)g(places)f(at)h(the)g(b)34 b(oundary)443 b(of)g Fq(B)29279 52297 y Fr(L)29919 52453 y Fn(\014)30473 52297 y Fr(;L)31374 52453 y Fn(\014)31982 52109 y Fv(\(0\))h(where)e(the)h(particle)f(can)g(in)-34 b(teract)1200 53614 y(with)305 b(particles)f(outside)i Fq(B)13622 53802 y Fr(L)14262 53958 y Fn(\014)14815 53802 y Fr(;L)15716 53958 y Fn(\014)16325 53614 y Fv(\(0\))f(due)g(to)g(the)g (constrain)-34 b(t)306 b(that)f(de\014nes)h Fp(S)395 b Fv(\(recall)304 b(De\014nition)h(1.3\)\(a\)\).)1200 55253 y(Since)492 b Fq(\032)5080 55450 y Fr(\014)5708 55253 y Fq(e)6272 54813 y Fr(\014)46 b(U)8116 55253 y Fv(=)483 b Fq(o)p Fv(\(1\))492 b(b)-34 b(y)493 b(\(1.27\),)514 b(w)-34 b(e)493 b(therefore)e(see)h(that)h Fq(\015)30331 55435 y FH(2)30857 55253 y Fv(\()p Fq(\014)64 b Fv(\))492 b(indeed)g(is)g(small)g(compared)g(to)g(the)1200 56758 y(main)404 b(term)g(of)h(\(4.23\).)1200 59260 y Fi(7.)688 b(First)g(error)g(term.)1121 b Fv(T)-101 b(o)599 b(estimate)f(the)h (error)e(term)i Fq(\015)30960 59442 y FH(1)31485 59260 y Fv(\()p Fq(\014)64 b Fv(\),)647 b(w)-34 b(e)598 b(de\014ne)h(the)g (sets)g(of)f(pairs)g(of)1200 60765 y(con\014gurations)14817 62479 y Fp(I)15477 62661 y FH(1)16339 62479 y Fv(=)337 b Fp(f)p Fv(\()p Fq(\033)-24 b(;)202 b(\021)43 b Fv(\))337 b Fp(2)g Fv([)p Fp(X)23881 61776 y FH(\()p Fr(n)24818 61932 y Fn(\014)25370 61776 y FH(\))23704 62870 y Fr(\014)25792 62479 y Fv(])26129 61978 y FH(2)26789 62479 y Fv(:)606 b Fq(\033)379 b Fp(2)337 b(S)91 b Fq(;)202 b(\021)380 b Fp(2)337 b(S)34265 61978 y Fr(c)34727 62479 y Fp(n)35553 62172 y Fv(~)35333 62479 y Fp(C)71 b(g)p Fq(;)14817 64852 y Fp(I)15477 65034 y FH(2)16339 64852 y Fv(=)337 b Fp(f)p Fv(\()p Fq(\033)-24 b(;)202 b(\021)43 b Fv(\))337 b Fp(2)g Fv([)p Fp(X)23881 64149 y FH(\()p Fr(n)24818 64305 y Fn(\014)25370 64149 y FH(\))23704 65244 y Fr(\014)25792 64852 y Fv(])26129 64352 y FH(2)26789 64852 y Fv(:)606 b Fq(\033)379 b Fp(2)30169 64546 y Fv(~)29949 64852 y Fp(C)71 b Fq(;)202 b(\021)380 b Fp(2)337 b(S)34149 64352 y Fr(c)34611 64852 y Fp(n)35437 64546 y Fv(~)35217 64852 y Fp(C)71 b(g)p Fq(;)47302 63567 y Fv(\(4.27\))1200 67088 y(and)405 b(estimate)11215 69406 y Fq(\015)11843 69588 y FH(1)12368 69406 y Fv(\()p Fq(\014)64 b Fv(\))337 b Fp(\024)15810 68928 y FH(1)p 15810 69126 471 49 v 15810 69824 a(2)17255 67890 y(2)16615 68254 y Fm(X)16729 70860 y Fr(i)p FH(=1)19448 68254 y Fm(X)18568 70938 y FH(\()p Fr(\033)-21 b(;\021)32 b FH(\))p Fk(2I)21727 71073 y Fn(i)22281 69406 y Fq(\026)23011 69603 y Fr(\014)23640 69406 y Fv(\()p Fq(\033)43 b Fv(\))202 b Fq(c)26045 69603 y Fr(\014)26675 69406 y Fv(\()p Fq(\033)-24 b(;)202 b(\021)43 b Fv(\))337 b(=)31219 68928 y FH(1)p 31219 69126 V 31219 69824 a(2)31823 69406 y Fv(\006\()p Fp(I)33829 69588 y FH(1)34355 69406 y Fv(\))270 b(+)36441 68928 y FH(1)p 36441 69126 V 36441 69824 a(2)37044 69406 y Fv(\006\()p Fp(I)39050 69588 y FH(2)39577 69406 y Fv(\))p Fq(:)6917 b Fv(\(4.28\))25194 75417 y(27)p eop end %%Page: 28 28 TeXDict begin 28 27 bop 1200 1313 a Fv(The)405 b(sum)f(\006\()p Fp(I)8263 1495 y FH(1)8790 1313 y Fv(\))g(can)g(b)34 b(e)404 b(written)h(as)3748 4239 y(\006\()p Fp(I)5754 4421 y FH(1)6280 4239 y Fv(\))337 b(=)g Fp(j)p Fv(\003)9547 4436 y Fr(\014)10175 4239 y Fp(j)10803 3087 y Fm(X)10714 5713 y Fr(\033)32 b Fk(2P)13573 3087 y Fm(X)12844 5925 y Fr(\021)g Fk(2S)14610 5673 y Fn(c)15037 5925 y Fk(n)15676 5701 y FH(~)15507 5925 y Fk(C)16255 4239 y Fq(c)16780 4436 y Fr(\014)17408 4239 y Fv(\()p Fq(\021)43 b(;)202 b(\033)43 b Fv(\))202 b(1)21078 2893 y Fm(n)21887 4239 y Fp(j)p Fv(supp)q([)p Fq(\021)43 b Fv(])270 b Fp(\\)f Fq(B)28307 4427 y Fr(L)28947 4583 y Fn(\014)29500 4427 y Fr(;L)30401 4583 y Fn(\014)31010 4239 y Fv(\(0\))p Fp(j)337 b Fv(=)g Fq(K)35628 2893 y Fm(o)37881 3419 y Fv(1)p 36771 3960 2826 49 v 36771 5473 a Fq(Z)37685 4770 y FH(\()p Fr(n)38622 4926 y Fn(\014)39175 4770 y FH(\))37598 5864 y Fr(\014)39931 4239 y Fq(e)40495 3738 y Fk(\000)p Fr(\014)46 b(H)42574 3894 y Fn(\014)43127 3738 y FH(\()p Fr(\021)32 b FH(\))44418 4239 y Fq(;)2547 b Fv(\(4.29\))1200 8656 y(where)470 b(w)-34 b(e)470 b(use)g(that)h Fq(\026)12203 8853 y Fr(\014)12832 8656 y Fv(\()p Fq(\033)43 b Fv(\))p Fq(c)15035 8853 y Fr(\014)15664 8656 y Fv(\()p Fq(\033)-24 b(;)202 b(\021)43 b Fv(\))447 b(=)f Fq(\026)21025 8853 y Fr(\014)21654 8656 y Fv(\()p Fq(\021)43 b Fv(\))p Fq(c)23766 8853 y Fr(\014)24396 8656 y Fv(\()p Fq(\021)g(;)202 b(\033)43 b Fv(\),)487 b Fq(\033)-24 b(;)202 b(\021)489 b Fp(2)446 b(X)32677 7953 y FH(\()p Fr(n)33614 8109 y Fn(\014)34167 7953 y FH(\))32500 9048 y Fr(\014)34588 8656 y Fv(,)486 b(and)471 b Fq(c)38359 8853 y Fr(\014)38987 8656 y Fv(\()p Fq(\021)43 b(;)202 b(\033)43 b Fv(\))448 b(=)e(0,)486 b Fq(\021)j Fp(2)446 b(S)48285 8216 y Fr(c)48748 8656 y Fp(n)49574 8350 y Fv(~)49354 8656 y Fp(C)71 b Fv(,)1200 10162 y Fq(\033)514 b(=)-740 b Fp(2)337 b(P)503 b Fv(\(recall)403 b(De\014nition)i(1.3\(b\)\).)540 b(W)-101 b(e)403 b(ha)-34 b(v)g(e)13912 12578 y Fq(H)14920 12775 y Fr(\014)15548 12578 y Fv(\()p Fq(\021)43 b Fv(\))338 b Fp(\025)f Fq(H)19761 12775 y Fr(\014)20389 12578 y Fv(\()87 b(^)-693 b Fq(\021)44 b Fv(\))269 b(+)g Fq(H)24466 12775 y Fr(\014)25095 12578 y Fv(\()87 b(\024)-693 b Fq(\021)44 b Fv(\))269 b Fp(\000)g Fq(k)38 b(U)m(;)2627 b(\021)380 b Fp(2)337 b(S)35574 12077 y Fr(c)36036 12578 y Fp(n)36862 12271 y Fv(~)36642 12578 y Fp(C)71 b Fq(;)9614 b Fv(\(4.30\))1200 14994 y(where)375 b Fq(k)414 b Fv(coun)-34 b(ts)377 b(the)f(n)-34 b(um)g(b)34 b(er)376 b(of)g(pairs)f(of)h(particles)f(in)-34 b(teracting)376 b(across)f(the)h(b)34 b(oundary)376 b(of)g Fq(B)45811 15182 y Fr(L)46451 15338 y Fn(\014)47005 15182 y Fr(;L)47906 15338 y Fn(\014)48514 14994 y Fv(\(0\).)1200 16693 y(Moreo)-34 b(v)g(er,)403 b(since)h Fq(\021)515 b(=)-741 b Fp(2)12234 16387 y Fv(~)11960 16693 y Fq(C)87 b Fv(,)404 b(w)-34 b(e)404 b(ha)-34 b(v)g(e)21416 18199 y Fq(H)22424 18396 y Fr(\014)23053 18199 y Fv(\()87 b(^)-693 b Fq(\021)43 b Fv(\))337 b Fp(\025)g Fv(\000)27015 17698 y Fk(\003)27810 18199 y Fv(+)269 b Fq(U)m(:)17118 b Fv(\(4.31\))1200 20270 y(Inserting)404 b(\(4.30{4.31\))h(in)-34 b(to)405 b(\(4.29\),)f(w)-34 b(e)405 b(obtain)8168 23836 y(\006\()p Fp(I)10174 24018 y FH(1)10701 23836 y Fv(\))337 b Fp(\024)g(j)p Fv(\003)13968 24033 y Fr(\014)14596 23836 y Fp(j)202 b Fq(e)15699 23336 y Fk(\000)p Fr(\014)46 b FH(\000)17593 23023 y Fl(\003)18329 23836 y Fq(\026)19059 24033 y Fr(\014)19688 23836 y Fv(\()p Fp(S)91 b Fv(\))202 b([1)270 b(+)f Fq(o)p Fv(\(1\)])27204 22321 y Fr(K)26757 22684 y Fm(X)26775 25320 y Fr(k)24 b FH(=0)28508 23836 y Fv(\()p Fq(\032)29606 24033 y Fr(\014)30235 23836 y Fv(\))30706 23336 y Fr(K)63 b FH(+)p Fr(k)33065 23836 y Fv([4\()p Fq(K)357 b Fp(\000)269 b Fv(1\)])38491 23336 y Fr(k)39262 23836 y Fq(e)39826 23336 y Fr(\014)46 b FH(\()p Fr(k)24 b Fk(\000)p FH(1\))p Fr(U)11509 26999 y Fv(=)337 b Fp(j)p Fv(\003)13968 27196 y Fr(\014)14596 26999 y Fp(j)202 b Fq(e)15699 26499 y Fk(\000)p Fr(\014)46 b FH(\000)17647 26999 y Fq(\026)18377 27196 y Fr(\014)19006 26999 y Fv(\()p Fp(S)91 b Fv(\))202 b([1)270 b(+)f Fq(o)p Fv(\(1\)])202 b Fq(e)26639 26499 y Fk(\000)p Fr(\014)46 b(U)28731 26999 y Fq(;)47302 24826 y Fv(\(4.32\))1200 29527 y(where)499 b(\()p Fq(\032)5898 29724 y Fr(\014)6527 29527 y Fv(\))6998 29087 y Fr(K)63 b FH(+)p Fr(k)9654 29527 y Fv(comes)499 b(from)g(the)h(fact)g(that)g Fq(n)24403 29724 y Fr(\014)25364 29527 y Fp(\000)333 b Fv(\()p Fq(K)420 b Fv(+)332 b Fq(k)38 b Fv(\))500 b(particles)f(are)f (outside)i Fq(B)43743 29715 y Fr(L)44383 29871 y Fn(\014)44937 29715 y Fk(\000)p FH(1)p Fr(;L)47040 29871 y Fn(\014)47593 29715 y Fk(\000)p FH(1)48851 29527 y Fv(\(0\))1200 31032 y(\(once)317 b(more)f(use)h(Lemma)g(B.1)f(in)h(App)34 b(endix)317 b(B\),)334 b(and)318 b(the)f(inequalit)-34 b(y)316 b(again)h(uses)h(an)f(argumen)-34 b(t)318 b(similar)1200 32538 y(as)397 b(in)h(Steps)g(3)f(and)h(5.)536 b(Therefore)398 b(\006\()p Fp(I)19648 32720 y FH(1)20174 32538 y Fv(\))g(is)f(small)g (compared)g(to)h(the)f(main)h(term)f(of)h(\(4.23\).)537 b(The)397 b(sum)1200 34043 y(\006\()p Fp(I)3206 34225 y FH(2)3733 34043 y Fv(\))404 b(can)g(b)34 b(e)404 b(estimated)h(as) 12235 36640 y(\006\()p Fp(I)14241 36822 y FH(2)14767 36640 y Fv(\))337 b(=)16855 35489 y Fm(X)16859 38327 y Fr(\033)32 b Fk(2)18224 38103 y FH(~)18056 38327 y Fk(C)19537 35489 y Fm(X)18808 38327 y Fr(\021)g Fk(2S)20574 38075 y Fn(c)21000 38327 y Fk(n)21639 38103 y FH(~)21470 38327 y Fk(C)22218 36640 y Fq(\026)22948 36837 y Fr(\014)23578 36640 y Fv(\()p Fq(\033)43 b Fv(\))202 b Fq(c)25983 36837 y Fr(\014)26612 36640 y Fv(\()p Fq(\033)-24 b(;)202 b(\021)43 b Fv(\))15575 40385 y(=)337 b Fp(j)p Fv(\003)18034 40582 y Fr(\014)18662 40385 y Fp(j)19798 39233 y Fm(X)19201 42071 y Fr(\033)32 b Fk(2)20566 41847 y FH(~)20398 42071 y Fk(C)47 b FH(\(0\))22348 40385 y Fq(\026)23078 40582 y Fr(\014)23707 40385 y Fv(\()p Fq(\033)c Fv(\))26918 39233 y Fm(X)25587 42071 y Fr(\021)32 b Fk(2S)27353 41819 y Fn(c)27780 42071 y Fk(n)28419 41847 y FH(~)28250 42071 y Fk(C)48 b FH(\(0\))30201 40385 y Fq(c)30726 40582 y Fr(\014)31354 40385 y Fv(\()p Fq(\033)-24 b(;)202 b(\021)43 b Fv(\))15575 43973 y Fp(\024)337 b(j)p Fv(\003)18034 44170 y Fr(\014)18662 43973 y Fp(j)202 b Fq(\026)19931 44170 y Fr(\014)20560 43973 y Fv(\()21251 43666 y(~)21031 43973 y Fp(C)71 b Fv(\(0\)\))23961 42991 y Fm(\010)24670 43973 y Fq(e)25234 43472 y Fk(\000)p Fr(\014)202 b(U)27752 43973 y Fv(+)269 b(\(4)p Fq(L)30866 44170 y Fr(\014)31495 43973 y Fv(\))202 b Fq(\032)32795 44170 y Fr(\014)33626 43973 y Fv([1)269 b(+)g Fq(o)p Fv(\(1\)])38523 42991 y Fm(\011)39230 43973 y Fq(;)47302 40199 y Fv(\(4.33\))1200 46287 y(where)388 b(the)g(\014rst)h(term)f(comes)g(from)g(detac)-34 b(hing)389 b(a)f(particle)f(from)i(the)f(critical)f(droplet)h(and)h (the)f(second)1200 47793 y(term)457 b(from)h(a)f(extra)g(particle)g(en) -34 b(tering)458 b Fq(B)21412 47981 y Fr(L)22052 48137 y Fn(\014)22605 47981 y Fr(;L)23506 48137 y Fn(\014)24115 47793 y Fv(\(0\).)699 b(The)458 b(term)f(b)34 b(et)-34 b(w)g(een)459 b(braces)e(is)g Fq(o)p Fv(\(1\).)699 b(Moreo)-34 b(v)g(er,)1200 49492 y Fq(\026)1930 49689 y Fr(\014)2559 49492 y Fv(\()3250 49186 y(~)3030 49492 y Fp(C)71 b Fv(\(0\)\))375 b(=)f Fq(\026)8180 49689 y Fr(\014)8809 49492 y Fv(\()p Fp(C)9989 49052 y FH(fp)10855 49492 y Fv(\(0\)\))285 b(+)f Fq(\026)15116 49689 y Fr(\014)15745 49492 y Fv(\()16437 49186 y(~)16216 49492 y Fp(C)72 b Fv(\(0\))p Fp(nC)19789 49052 y FH(fp)20655 49492 y Fv(\(0\)\).)606 b(The)427 b(second)g(term)f(w)-34 b(as)427 b(estimated)g(in)f(\(4.26\),)433 b(the)426 b(\014rst)1200 50998 y(term)404 b(can)g(again)g(b)34 b(e)404 b(estimated)h(as)f(in)g(Steps)h(3)f(and)h(5:)2441 54658 y Fq(\026)3171 54855 y Fr(\014)3801 54658 y Fv(\()p Fp(C)4981 54158 y FH(fp)5847 54658 y Fv(\(0\)\))337 b(=)10080 53507 y Fm(X)9532 56366 y FH(^)-519 b Fr(\033)31 b Fk(2)10848 56142 y FH(^)10679 56366 y Fk(C)48 b FH(\(0\))14083 53507 y Fm(X)13716 56112 y Fx(\024)-453 b Fn(\033)24 b Fl(2)14867 55944 y Fx(\024)14722 56112 y Fl(C)30 b Fx(\(0\))12809 57060 y(^)-453 b Fn(\033)24 b Fl(_)47 b Fx(\024)-453 b Fn(\033)24 b Fl(2C)15349 56859 y Fx(fp)16109 57060 y(\(0\))17489 54658 y Fq(\026)18219 54855 y Fr(\014)18848 54658 y Fv(\()65 b(^)-671 b Fq(\033)313 b Fp(_)335 b Fv(\024)-671 b Fq(\033)43 b Fv(\))337 b(=)f Fq(N)e(e)26098 54158 y Fk(\000)p Fr(\014)46 b FH(\000)27992 53845 y Fl(\003)29117 53315 y Fv(\024)28862 53622 y Fq(Z)29776 52919 y FH(\()p Fr(n)30713 53075 y Fn(\014)31266 52919 y Fk(\000)p Fr(K)63 b FH(\))29689 54013 y Fr(\014)33275 53622 y Fv(\(0\))p 28862 54379 5963 49 v 30430 55892 a Fq(Z)31344 55189 y FH(\()p Fr(n)32281 55345 y Fn(\014)32834 55189 y FH(\))31257 56283 y Fr(\014)35293 54658 y Fv(=)337 b Fq(N)d(e)38445 54158 y Fk(\000)p Fr(\014)46 b FH(\000)40595 54658 y Fq(\026)41325 54855 y Fr(\014)41955 54658 y Fv(\()p Fp(S)91 b Fv(\))202 b([1)269 b(+)h Fq(o)p Fv(\(1\)])p Fq(:)47302 58367 y Fv(\(4.34\))1200 59872 y(Therefore)404 b(also)g(\006\()p Fp(I)11132 60054 y FH(2)11659 59872 y Fv(\))g(is)g(small)g(compared)g(to)g(the)h(main)f(term)g(of)h (\(4.23\).)1200 62782 y(Lo)-34 b(w)g(er)404 b(b)34 b(ound)p 1200 62976 6923 49 v 2 w(:)522 b(The)373 b(pro)34 b(of)372 b(of)h(the)g(lo)-34 b(w)g(er)372 b(b)34 b(ound)374 b(follo)-34 b(ws)373 b(the)f(same)h(line)e(of)i(argumen)-34 b(t)373 b(as)g(for)f(Glaub)34 b(er)1200 64288 y(dynamics)391 b(in)f(that)i(it)f(relies)e(on)i(the)g(construction)h(of)f(a)f (suitable)h(unit)h(\015o)-34 b(w.)535 b(This)391 b(\015o)-34 b(w)392 b(will,)g(ho)-34 b(w)g(ev)g(er,)1200 65793 y(b)34 b(e)559 b(considerably)g(more)g(di\016cult.)1005 b(In)559 b(particular,)598 b(w)-34 b(e)559 b(will)g(no)h(longer)f(b)34 b(e)559 b(able)g(to)h(get)f(a)-34 b(w)g(a)g(y)561 b(with)1200 67298 y(c)-34 b(ho)34 b(osing)341 b(a)f(deterministic)g(\015o)-34 b(w,)354 b(and)341 b(the)g(full)f(p)34 b(o)-34 b(w)g(er)341 b(of)g(the)f(Berman-Konso)-34 b(w)g(a)342 b(v)-67 b(ariational)339 b(principle)1200 68804 y(has)405 b(to)f(b)34 b(e)404 b(brough)-34 b(t)406 b(to)e(b)34 b(ear.)538 b(The)405 b(pro)34 b(of)404 b(comes)g(in)g(5)g(steps.)3082 70590 y(F)-101 b(or)463 b(future)g(reference)f(w)-34 b(e)463 b(state)g(the)h(follo)-34 b(wing)463 b(prop)34 b(ert)-34 b(y)463 b(of)g(the)g(harmonic)g(function)h(for)f(simple)1200 72096 y(random)405 b(w)-34 b(alk)404 b(on)h Fo(Z)10968 71656 y FH(2)11493 72096 y Fv(.)25194 75417 y(28)p eop end %%Page: 29 29 TeXDict begin 29 28 bop 1200 1313 a Fi(Lemma)464 b(4.4)606 b Fs(L)-62 b(et)406 b Fq(g)451 b Fs(b)-62 b(e)406 b(the)g(harmonic)f (function)h(of)g(simple)f(r)-62 b(andom)407 b(walk)f(on)h Fq(B)40057 1501 y FH(2)p Fr(L)41167 1657 y Fn(\014)41720 1501 y Fr(;)p FH(2)p Fr(L)43091 1657 y Fn(\014)43701 1313 y Fv(\(0\))g Fs(\(which)f(is)1200 2819 y(e)-62 b(qual)432 b(to)g Fv(1)i Fs(on)g Fp(f)p Fv(0)p Fp(g)g Fs(and)f Fv(0)h Fs(on)f Fv([)p Fq(B)17133 3007 y FH(2)p Fr(L)18243 3163 y Fn(\014)18797 3007 y Fr(;)p FH(2)p Fr(L)20168 3163 y Fn(\014)20777 2819 y Fv(\(0\)])22662 2379 y Fr(c)23125 2819 y Fs(\).)557 b(Then)434 b(ther)-62 b(e)432 b(exists)g(a)h(c)-62 b(onstant)433 b Fq(C)423 b(<)337 b Fp(1)434 b Fs(such)e(that)11904 4380 y Fm(X)12561 6925 y Fr(e)13654 5531 y Fv([)p Fq(g)43 b Fv(\()p Fq(z)53 b Fv(\))271 b Fp(\000)e Fq(g)43 b Fv(\()p Fq(z)323 b Fv(+)269 b Fq(e)p Fv(\)])22217 5713 y FH(+)23342 5531 y Fp(\024)336 b Fq(C)53 b(=L)26971 5728 y Fr(\014)30079 5531 y Fp(8)202 b Fq(z)390 b Fp(2)336 b Fv([)p Fq(B)34308 5719 y Fr(L)34948 5875 y Fn(\014)35502 5719 y Fr(;L)36403 5875 y Fn(\014)37011 5531 y Fv(\(0\)])38896 5031 y Fr(c)39360 5531 y Fq(:)7605 b Fv(\(4.35\))1200 9304 y Fs(Pr)-62 b(o)g(of)p Fv(.)403 b(See)351 b(e.g.)f(La)-34 b(wler,)360 b(Sc)-34 b(hramm)352 b(and)g(W)-101 b(erner)350 b([21)o(],)361 b(Lemma)351 b(5.1.)520 b(The)352 b(pro)34 b(of)351 b(can)g(b)34 b(e)350 b(giv)-34 b(en)351 b(via)f(the)1200 10810 y(estimates)404 b(in)g(La)-34 b(wler)404 b([20],)f(Section)i(1.7,)e(or)h(via)g(a)g (coupling)g(argumen)-34 b(t.)p 49878 10810 522 731 v 1200 13674 a Fi(1.)452 b(Starting)h(con\014gurations.)536 b Fv(W)-101 b(e)393 b(start)g(our)h(\015o)-34 b(w)394 b(on)f(a)h(subset)g(of)f(the)h(con\014gurations)g(in)g Fp(C)46998 13234 y FH(+)48178 13674 y Fv(that)1200 15179 y(is)415 b(su\016cien)-34 b(tly)415 b(large)g(and)g(su\016cien)-34 b(tly)416 b(con)-34 b(v)g(enien)g(t.)572 b(Let)415 b Fp(C)29321 14670 y FH(+)29250 15527 y(2)30463 15179 y Fp(\032)355 b(C)32470 14740 y FH(+)33672 15179 y Fv(denote)416 b(the)g(set)f(of)g(con\014gurations)1200 16685 y(ha)-34 b(ving)377 b(a)f(proto-critical)f(with)i(lo)-34 b(w)g(er-left)377 b(corner)e(at)i(some)f(site)g Fq(x)336 b Fp(2)h Fv(\003)34668 16882 y Fr(\014)35296 16685 y Fv(,)382 b(a)376 b(free)f(particle)h(at)g (distance)h(2)1200 18190 y(from)424 b(this)g(proto-critical)e(droplet,) 429 b(no)423 b(other)h(particles)f(in)g(the)h(b)34 b(o)-34 b(x)424 b Fq(B)34876 18378 y FH(2)p Fr(L)35986 18534 y Fn(\014)36540 18378 y Fr(;)p FH(2)p Fr(L)37911 18534 y Fn(\014)38520 18190 y Fv(\()p Fq(x)p Fv(\),)k(and)c(satisfying)g(the) 1200 19696 y(constrain)-34 b(ts)412 b(in)f Fp(S)9540 19884 y Fr(L)10236 19696 y Fv(,)h(i.e.,)f(all)g(other)g(b)34 b(o)-34 b(xes)411 b(of)g(size)g(2)p Fq(L)26618 19893 y Fr(\014)27657 19696 y Fv(carry)f(no)i(more)f(particles)f(than)i (there)f(are)g(in)g(a)1200 21201 y(proto-critical)435 b(droplet.)635 b(This)437 b(is)e(the)i(same)f(as)g Fp(C)24725 20761 y FH(+)25512 21201 y Fv(,)444 b(except)435 b(that)i(the)g(b)34 b(o)-34 b(x)436 b(around)h(the)f(proto-critical)1200 22707 y(droplet)404 b(has)h(size)e(2)p Fq(L)11267 22904 y Fr(\014)12300 22707 y Fv(rather)h(than)i Fq(L)19597 22904 y Fr(\014)20225 22707 y Fv(.)3082 24826 y(Let)551 b Fq(K)671 b Fv(=)583 b Fq(`)9132 25008 y Fr(c)9595 24826 y Fv(\()p Fq(`)10571 25008 y Fr(c)11402 24826 y Fp(\000)368 b Fv(1\))g(+)g(2)552 b(b)34 b(e)552 b(the)g(v)-34 b(olume)552 b(of)h(the)f(critical)f(droplet,)589 b(and)553 b(let)f Fp(S)42868 24123 y FH(\()p Fr(n)43805 24279 y Fn(\014)44357 24123 y Fk(\000)p Fr(K)63 b FH(\))42777 25173 y(2)46919 24826 y Fv(b)34 b(e)551 b(the)1200 26331 y(analogue)394 b(of)g Fp(S)485 b Fv(when)394 b(the)g(total)h(n)-34 b(um)g(b)34 b(er)394 b(of)g(particles)g(is)f Fq(n)29504 26528 y Fr(\014)30381 26331 y Fp(\000)248 b Fq(K)481 b Fv(and)394 b(the)h(b)34 b(o)-34 b(xes)393 b(in)h(whic)-34 b(h)395 b(w)-34 b(e)394 b(coun)-34 b(t)1200 27837 y(particles)474 b(ha)-34 b(v)g(e)475 b(size)f(2)p Fq(L)12792 28034 y Fr(\014)13896 27837 y Fv(\(compare)g(with)i(De\014nition)f(1.3\).)751 b(Similarly)473 b(as)i(in)g(\(3.17\),)492 b(our)475 b(task)g(is)f(to)1200 29342 y(deriv)-34 b(e)355 b(a)g(lo)-34 b(w)g(er)356 b(b)34 b(ound)356 b(for)g Fw(CAP)o Fv(\()p Fp(S)17709 29530 y Fr(L)18405 29342 y Fq(;)202 b Fv(\()p Fp(S)20240 28902 y Fr(c)20702 29342 y Fp(n)21528 29036 y Fv(~)21308 29342 y Fp(C)71 b Fv(\))172 b Fp([)g(C)24349 28902 y FH(+)25137 29342 y Fv(\))337 b(=)f Fw(CAP)o Fv(\(\()p Fp(S)31195 28902 y Fr(c)31659 29342 y Fp(n)32485 29036 y Fv(~)32265 29342 y Fp(C)71 b Fv(\))172 b Fp([)g(C)35306 28902 y FH(+)36093 29342 y Fq(;)202 b Fp(S)37366 29530 y Fr(L)38061 29342 y Fv(\))337 b Fp(\025)g Fw(CAP)o Fv(\()p Fp(C)43462 29530 y Fr(L)44158 29342 y Fq(;)202 b Fp(S)45431 29530 y Fr(L)46126 29342 y Fv(\),)365 b(where)1200 30848 y Fp(C)1838 31036 y Fr(L)2870 30848 y Fp(\032)337 b(C)4859 30338 y FH(+)4788 31195 y(2)5982 30848 y Fp(\032)g(C)7971 30408 y FH(+)9162 30848 y Fv(de\014ned)405 b(b)-34 b(y)10467 33528 y Fp(C)11105 33716 y Fr(L)12137 33528 y Fv(=)336 b Fp(f)p Fq(\033)313 b Fp([)269 b Fq(P)16883 33775 y FH(\()p Fr(y)32 b FH(\))18167 33528 y Fv(\()p Fq(x;)202 b(z)53 b Fv(\))135 b(:)606 b Fq(\033)380 b Fp(2)337 b(S)25079 32825 y FH(\()p Fr(n)26016 32981 y Fn(\014)26569 32825 y Fk(\000)p Fr(K)63 b FH(\))24988 33875 y(2)28578 33528 y Fq(;)403 b(x;)202 b(y)380 b Fp(2)337 b Fv(\003)33511 33725 y Fr(\014)34140 33528 y Fq(;)404 b Fv(\()p Fq(x;)202 b(y)43 b(;)202 b(z)53 b Fv(\))p Fp(?)p Fq(\033)43 b Fp(g)6169 b Fv(\(4.36\))1200 36330 y(is)428 b(the)g(analog)h(of)f(\(3.13\),)434 b(namely)-101 b(,)434 b(the)428 b(set)g(of)h(con\014gurations)g (obtained)g(from)f Fp(S)40162 35627 y FH(\()p Fr(n)41099 35783 y Fn(\014)41652 35627 y Fk(\000)p Fr(K)63 b FH(\))40071 36677 y(2)44089 36330 y Fv(b)-34 b(y)429 b(adding)g(a)1200 37835 y(critical)379 b(droplet)h(somewhere)g(in)g(\003)17720 38032 y Fr(\014)18728 37835 y Fv(\(lo)-34 b(w)g(er-left)381 b(corner)e(at)h Fq(x)p Fv(,)385 b(protub)34 b(erance)380 b(at)g Fq(y)43 b Fv(,)385 b(free)380 b(particle)f(at)i Fq(z)53 b Fv(\))1200 39341 y(suc)-34 b(h)348 b(that)g(it)g(do)34 b(es)347 b(not)h(in)-34 b(teract)347 b(with)h(the)g(particles)f(in)g Fq(\033)390 b Fv(and)348 b(has)g(an)g(empt)-34 b(y)348 b(b)34 b(o)-34 b(x)347 b(of)g(size)g(2)p Fq(L)45717 39538 y Fr(\014)46693 39341 y Fv(around)1200 40846 y(it.)532 b(Note)386 b(that)h(the)g Fq(n)11194 41043 y Fr(\014)12055 40846 y Fp(\000)233 b Fq(K)472 b Fv(particles)386 b(can)g(blo)34 b(c)-34 b(k)385 b(at)h(most)h Fq(n)30100 41043 y Fr(\014)30728 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b(su\016ces)f(to)h(consider)f(the)h(case)f(where)g(the)h (critical)e(droplet)h(is)g(added)i(at)e Fq(x)372 b Fv(=)g(0,)430 b(b)34 b(ecause)1200 48645 y(the)404 b(union)i(o)-34 b(v)g(er)403 b Fq(x)h Fv(trivially)f(pro)34 b(duces)404 b(a)g(factor)h Fp(j)p Fv(\003)25685 48842 y Fr(\014)26313 48645 y Fp(j)p Fv(.)1200 50966 y Fi(2.)422 b(Ov)-39 b(erall)423 b(strategy)-116 b(.)526 b Fv(Starting)368 b(from)g(a)f(con\014guration) i(in)e Fp(C)31097 51154 y Fr(L)31792 50966 y Fv(\(0\),)376 b(w)-34 b(e)367 b(will)g(successiv)-34 b(ely)366 b(pic)-34 b(k)367 b Fq(K)283 b Fp(\000)196 b Fq(L)1200 52472 y Fv(particles)569 b(from)g(the)h(critical)e(droplet)i(\(starting)g(with) h(the)e(free)g(particle)g(at)h Fq(z)622 b Fv(at)570 b(distance)f(2\))h (and)1200 53977 y(mo)-34 b(v)g(e)625 b(them)h(out)g(of)f(the)h(b)34 b(o)-34 b(x)625 b Fq(B)17620 54165 y Fr(L)18260 54321 y Fn(\014)18813 54165 y Fr(;L)19714 54321 y Fn(\014)20323 53977 y Fv(\(0\),)681 b(placing)625 b(them)h(essen)-34 b(tially)624 b(uniformly)h(in)h(the)f(ann)-34 b(ulus)1200 55483 y Fq(B)2119 55671 y FH(2)p Fr(L)3229 55827 y Fn(\014)3783 55671 y Fr(;)p FH(2)p Fr(L)5154 55827 y Fn(\014)5763 55483 y Fv(\(0\))p Fp(n)p Fq(B)8836 55671 y Fr(L)9476 55827 y Fn(\014)10030 55671 y Fr(;L)10931 55827 y Fn(\014)11540 55483 y Fv(\(0\).)727 b(Once)467 b(this)g(has)h(b)34 b(een)466 b(ac)-34 b(hiev)g(ed,)483 b(the)467 b(con\014guration)h(is)f (in)g Fp(S)43235 55671 y Fr(L)43930 55483 y Fv(.)727 b(Eac)-34 b(h)467 b(suc)-34 b(h)1200 56988 y(mo)g(v)g(e)456 b(will)f(pro)34 b(duce)456 b(an)g(en)-34 b(trop)g(y)457 b(of)f(order)g Fq(L)23074 56548 y FH(2)23074 57357 y Fr(\014)23703 56988 y Fv(,)468 b(whic)-34 b(h)457 b(will)e(b)34 b(e)456 b(enough)g(to)h(comp)34 b(ensate)456 b(for)g(the)g(loss)1200 58493 y(of)499 b(energy)g(in)g(tearing)g(do)-34 b(wn)500 b(the)g(droplet)f(\(recall)f(Fig.)h(4\).)823 b(The)500 b(order)e(in)i(whic)-34 b(h)499 b(the)h(particles)e(are)1200 59999 y(remo)-34 b(v)g(ed)316 b(follo)-34 b(ws)316 b(the)g(canonical)g (order)f(emplo)-34 b(y)g(ed)316 b(in)g(the)g(lo)-34 b(w)g(er)316 b(b)34 b(ound)318 b(for)d(Glaub)34 b(er)316 b(dynamics)g(\(recall)1200 61504 y(Fig.)404 b(6\).)539 b(As)404 b(for)g(Glaub)34 b(er,)404 b(w)-34 b(e)404 b(will)g(use)g(Prop)34 b(osition)404 b(2.4)g(to)h(estimate)7972 65188 y Fw(CAP)o Fv(\()p Fp(C)11285 65376 y Fr(L)11980 65188 y Fq(;)202 b Fp(S)13253 65376 y Fr(L)13948 65188 y Fv(\))338 b Fp(\025)e(j)p Fv(\003)17215 65385 y Fr(\014)17843 65188 y Fp(j)19264 64037 y Fm(X)18382 66721 y Fr(\033)32 b Fk(2C)20077 66870 y Fn(L)20694 66721 y FH(\(0\))23034 64037 y Fm(X)22099 66582 y Fr(\015)156 b FH(:)471 b Fr(\015)23957 66705 y Fx(0)24418 66582 y FH(=)p Fr(\033)25921 65188 y Fo(P)26662 64688 y Fr(f)27267 65188 y Fv(\()p Fq(\015)67 b Fv(\))29106 63559 y Fr(\034)102 b FH(\()p Fr(\015)52 b FH(\))29125 64037 y Fm(X)29143 66672 y Fr(k)24 b FH(=0)31097 63479 y Fm(\024)33834 64368 y Fq(f)130 b Fv(\()p Fq(\015)35656 64565 y Fr(k)36225 64368 y Fq(;)202 b(\015)37392 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Fv(particles)g(exits)g(the)g(b)34 b(o)-34 b(x)404 b Fq(B)25935 72284 y Fr(L)26575 72440 y Fn(\014)27129 72284 y Fr(;L)28030 72440 y Fn(\014)28639 72096 y Fv(\(0\))g(.)25194 75417 y(29)p eop end %%Page: 30 30 TeXDict begin 30 29 bop 3082 1313 a Fv(The)336 b(di\013erence)f(b)34 b(et)-34 b(w)g(een)337 b(Glaub)34 b(er)336 b(and)h(Ka)-34 b(w)g(asaki)336 b(is)g(that,)350 b(while)336 b(in)g(Glaub)34 b(er)335 b(the)i(droplet)f(can)g(b)34 b(e)1200 2819 y(torn)341 b(do)-34 b(wn)342 b(via)d(single)h(spin-\015ips,)354 b(in)340 b(Ka)-34 b(w)g(asaki)341 b(after)g(w)-34 b(e)340 b(ha)-34 b(v)g(e)341 b(detac)-34 b(hed)341 b(a)g(particle)e(from)i(the) f(droplet)1200 4324 y(w)-34 b(e)365 b(need)g(to)g(mo)-34 b(v)g(e)365 b(it)g(out)g(of)h(the)f(b)34 b(o)-34 b(x)365 b Fq(B)20149 4512 y Fr(L)20789 4668 y Fn(\014)21342 4512 y Fr(;L)22243 4668 y Fn(\014)22852 4324 y Fv(\(0\),)373 b(whic)-34 b(h)366 b(tak)-34 b(es)365 b(a)f(large)g(n)-34 b(um)g(b)34 b(er)366 b(of)f(steps.)526 b(Th)-34 b(us,)374 b Fq(\034)137 b Fv(\()p Fq(\015)67 b Fv(\))1200 5830 y(is)438 b(the)h(sum)g(of)f Fq(K)380 b Fp(\000)291 b Fq(L)439 b Fv(stopping)g(times,)447 b(eac)-34 b(h)438 b(except)g(the)h(\014rst)g(of)g(whic)-34 b(h)439 b(is)f(a)g(sum)h(of)g (t)-34 b(w)g(o)439 b(stopping)1200 7335 y(times)334 b(itself,)347 b(one)334 b(to)g(detac)-34 b(h)334 b(the)h(particle)e(and)h(one)g(to)g (mo)-34 b(v)g(e)334 b(it)g(out)g(of)g(the)g(b)34 b(o)-34 b(x)334 b Fq(B)39912 7523 y Fr(L)40552 7679 y Fn(\014)41105 7523 y Fr(;L)42006 7679 y Fn(\014)42615 7335 y Fv(\(0\).)516 b(With)334 b(eac)-34 b(h)1200 8841 y(motion)389 b(of)g(a)f(single)g (particle)f(w)-34 b(e)389 b(need)f(to)h(gain)g(an)f(en)-34 b(trop)g(y)390 b(factor)e(of)h(order)f(close)f(to)i(1)p Fq(=\032)44264 9038 y Fr(\014)44892 8841 y Fv(.)534 b(This)388 b(will)1200 10346 y(b)34 b(e)402 b(done)g(b)-34 b(y)403 b(constructing)g(a)f(\015o)-34 b(w)403 b(that)g(in)-34 b(v)g(olv)g(es)402 b(only)g(the)h(motion)g(of)f(this)h(single)e (particle,)h(based)g(on)1200 11852 y(the)392 b(harmonic)f(function)h (of)g(the)f(simple)g(random)h(w)-34 b(alk)391 b(in)g(the)h(b)34 b(o)-34 b(x)391 b Fq(B)34481 12040 y FH(2)p Fr(L)35591 12196 y Fn(\014)36145 12040 y Fr(;)p FH(2)p Fr(L)37516 12196 y Fn(\014)38125 11852 y Fv(\(0\))h(up)g(to)f(the)h(b)34 b(oundary)1200 13357 y(of)493 b(the)g(b)34 b(o)-34 b(x)492 b Fq(B)8175 13545 y Fr(L)8815 13701 y Fn(\014)9369 13545 y Fr(;L)10270 13701 y Fn(\014)10878 13357 y Fv(\(0\).)804 b(Outside)493 b Fq(B)19093 13545 y Fr(L)19733 13701 y Fn(\014)20287 13545 y Fr(;L)21188 13701 y Fn(\014)21796 13357 y Fv(\(0\))g(the)g(\015o)-34 b(w)494 b(b)34 b(ecomes)492 b(more)g(complex:)714 b(w)-34 b(e)493 b(mo)34 b(dify)492 b(it)h(in)1200 15021 y(suc)-34 b(h)438 b(a)f(w)-34 b(a)g(y)438 b(that)g(a)f(small)g(fraction)g(of)g(the)h(\015o)-34 b(w,)446 b(of)437 b(order)g Fq(L)30802 14512 y Fk(\000)p FH(1+)p Fr(\017)30802 15413 y(\014)33609 15021 y Fv(for)g(some)g Fq(\017)391 b(>)h Fv(0)437 b(small)f(enough,)446 b(is)1200 16527 y(going)454 b(in)-34 b(to)455 b(the)f(direction)g(of)g(remo)-34 b(ving)454 b(the)g(next)g(particle)f(from)h(the)h(droplet.)688 b(The)454 b(reason)g(for)g(this)1200 18032 y(c)-34 b(hoice)408 b(is)g(that)i(w)-34 b(e)408 b(w)-34 b(an)g(t)411 b(to)d(mak)-34 b(e)408 b(sure)h(that)g(the)g(\015o)-34 b(w)410 b(b)34 b(ecomes)407 b(su\016cien)-34 b(tly)409 b(small,)g(of)f(order)g Fq(L)47693 17522 y Fk(\000)p FH(2+)p Fr(\017)47693 18424 y(\014)50063 18032 y Fv(,)1200 19538 y(so)329 b(that)g(this)h(can)e (comp)34 b(ensate)329 b(for)g(the)g(fact)g(that)h(the)f(Gibbs)g(w)-34 b(eigh)g(t)330 b(in)f(the)g(denominator)h(of)f(the)g(lo)-34 b(w)g(er)1200 21043 y(b)34 b(ound)424 b(in)e(\(2.20\))h(is)f(reduced)h (b)-34 b(y)423 b(a)f(factor)h Fq(e)22463 20603 y Fk(\000)p Fr(\014)46 b(U)24977 21043 y Fv(when)423 b(the)g(protub)34 b(erance)423 b(is)f(detac)-34 b(hed.)595 b(The)423 b(reason)1200 22549 y(for)408 b(the)h(extra)f Fq(\017)g Fv(is)g(that)i(w)-34 b(e)409 b(w)-34 b(an)g(t)410 b(to)f(mak)-34 b(e)408 b(sure)g(that,)i (along)f(most)g(of)g(the)f(paths,)j(the)d(protub)34 b(erance)1200 24054 y(is)404 b(detac)-34 b(hed)405 b(b)34 b(efore)404 b(the)g(\014rst)h(particle)e(lea)-34 b(v)g(es)404 b(the)h(b)34 b(o)-34 b(x)404 b Fq(B)28917 24242 y FH(2)p Fr(L)30027 24398 y Fn(\014)30581 24242 y Fr(;)p FH(2)p Fr(L)31952 24398 y Fn(\014)32561 24054 y Fv(\(0\).)3082 25885 y(Once)430 b(the)h(protub)34 b(erance)431 b(detac)-34 b(hes)431 b(itself)g(from)g(the)g(proto-critical,)436 b(the)431 b(\014rst)g(particle)f(stops)i(and)1200 27391 y(the)583 b(second)g(particle)g(mo)-34 b(v)g(es)583 b(in)g(the)g(same)g(w)-34 b(a)g(y)583 b(as)g(the)h(\014rst)f(particle)f(did)i(when)f(it)g(mo)-34 b(v)g(ed)584 b(a)-34 b(w)g(a)g(y)1200 28896 y(from)487 b(the)g(proto-critical)f(droplet,)507 b(and)487 b(so)g(on.)786 b(This)487 b(is)f(rep)34 b(eated)487 b(un)-34 b(til)487 b(no)g(more)f(than)i Fq(L)e Fv(particles)1200 30402 y(remain)423 b(in)g Fq(B)7615 30590 y Fr(L)8255 30746 y Fn(\014)8808 30590 y Fr(;L)9709 30746 y Fn(\014)10318 30402 y Fv(\(0\),)428 b(b)-34 b(y)424 b(whic)-34 b(h)424 b(time)e(w)-34 b(e)424 b(ha)-34 b(v)g(e)423 b(reac)-34 b(hed)424 b Fp(S)30385 30590 y Fr(L)31080 30402 y Fv(.)595 b(As)423 b(w)-34 b(e)424 b(will)e(see,)427 b(the)d(only)f(signi\014can)-34 b(t)1200 31907 y(con)g(tribution)400 b(to)g(the)f(lo)-34 b(w)g(er)399 b(b)34 b(ound)400 b(comes)f(from)g(the)h(motion)f(of)g (the)h(\014rst)f(particle)f(\(as)i(for)f(Glaub)34 b(er\),)1200 33413 y(and)442 b(this)g(coincides)f(with)h(the)g(upp)34 b(er)442 b(b)34 b(ound)443 b(established)f(earlier.)649 b(The)442 b(details)f(of)h(the)g(construction)1200 34918 y(are)404 b(to)g(some)g(exten)-34 b(t)405 b(arbitrary)e(and)i(there)f (are)g(man)-34 b(y)405 b(other)f(c)-34 b(hoices)404 b(imaginable.)1200 37401 y Fi(3.)586 b(First)g(particle.)854 b Fv(W)-101 b(e)509 b(\014rst)h(construct)g(the)g(\015o)-34 b(w)510 b(that)h(mo)-34 b(v)g(es)509 b(the)h(particle)f(at)h(distance)f(2)h (from)1200 38906 y(the)545 b(proto-critical)f(droplet)g(to)h(the)g(b)34 b(oundary)545 b(of)g(the)g(b)34 b(o)-34 b(x)544 b Fq(B)31745 39094 y Fr(L)32385 39250 y Fn(\014)32939 39094 y Fr(;L)33840 39250 y Fn(\014)34448 38906 y Fv(\(0\).)960 b(This)545 b(\015o)-34 b(w)546 b(will)e(consist)g(of)1200 40412 y(indep)34 b(enden)-34 b(t)429 b(\015o)-34 b(ws)429 b(for)f(eac)-34 b(h)428 b(\014xed)g(shap)34 b(e)428 b(and)h(lo)34 b(cation)427 b(of)h(the)g(critical)f(droplet.)610 b(This)428 b(\014rst)g(part)g(of) 1200 41917 y(the)404 b(\015o)-34 b(w)406 b(will)d(b)34 b(e)404 b(seen)g(to)h(pro)34 b(duce)404 b(the)g(essen)-34 b(tial)405 b(con)-34 b(tribution)405 b(to)g(the)f(lo)-34 b(w)g(er)405 b(b)34 b(ound.)3082 43748 y(W)-101 b(e)416 b(lab)34 b(el)415 b(the)i(con\014gurations)h(in)f Fp(C)20122 43936 y Fr(L)20817 43748 y Fv(\(0\))g(b)-34 b(y)417 b Fq(\033)43 b Fv(,)419 b(describing)e(the)g(shap)34 b(e)416 b(of)h(the)g(critical)e(droplet,)420 b(as)1200 45254 y(w)-34 b(ell)483 b(as)h(the)g(con\014guration)g(outside)g(the)g(b)34 b(o)-34 b(x)484 b Fq(B)24644 45442 y FH(2)p Fr(L)25754 45598 y Fn(\014)26308 45442 y Fr(;)p FH(2)p Fr(L)27679 45598 y Fn(\014)28288 45254 y Fv(\(0\),)503 b(and)485 b(w)-34 b(e)483 b(lab)34 b(el)483 b(the)h(p)34 b(osition)484 b(of)f(the)h(free)1200 46759 y(particle)403 b(in)i Fq(\033)447 b Fv(b)-34 b(y)404 b Fq(z)10382 46941 y FH(1)10908 46759 y Fv(\()p Fq(\033)43 b Fv(\).)3082 48591 y(Let)482 b Fq(g)526 b Fv(b)34 b(e)482 b(the)h(harmonic)g(function)g(for)g(simple)f (random)h(w)-34 b(alk)483 b(with)g(b)34 b(oundary)484 b(conditions)f(0)f(on)1200 50096 y([)p Fq(B)2456 50284 y FH(2)p Fr(L)3566 50440 y Fn(\014)4120 50284 y Fr(;)p FH(2)p Fr(L)5491 50440 y Fn(\014)6100 50096 y Fv(\(0\)])7985 49656 y Fr(c)8852 50096 y Fv(and)405 b(1)f(on)h(the)f(critical)f (droplet.)539 b(Then)405 b(w)-34 b(e)404 b(c)-34 b(ho)34 b(ose)404 b(our)h(\015o)-34 b(w)405 b(to)f(b)34 b(e)9041 53980 y Fq(f)130 b Fv(\()p Fq(\033)43 b Fv(\()p Fq(z)53 b Fv(\))p Fq(;)202 b(\033)43 b Fv(\()p Fq(z)14893 53480 y Fk(0)15206 53980 y Fv(\)\))337 b(=)17764 51908 y Fm(\()18741 53135 y Fq(C)19607 53317 y FH(1)20335 53135 y Fv([)p Fq(g)43 b Fv(\()p Fq(z)53 b Fv(\))270 b Fp(\000)g Fq(g)43 b Fv(\()p Fq(z)323 b Fv(+)269 b Fq(e)p Fv(\)])28898 53317 y FH(+)29686 53135 y Fq(;)1212 b Fv(if)404 b Fq(z)32963 52696 y Fk(0)33610 53135 y Fv(=)336 b Fq(z)323 b Fv(+)269 b Fq(e;)202 b Fp(k)p Fq(e)p Fp(k)337 b Fv(=)g(1)p Fq(;)18741 54942 y Fv(0)p Fq(;)11551 b Fv(otherwise)p Fq(;)47302 53980 y Fv(\(4.38\))1200 57695 y(where)473 b Fq(\033)43 b Fv(\()p Fq(z)53 b Fv(\))473 b(is)g(the)g(con\014guration)i(obtained)f (from)f Fq(\033)516 b Fv(b)-34 b(y)473 b(placing)g(the)g(\014rst)g (particle)g(at)g(site)g Fq(z)53 b Fv(.)744 b(The)1200 59200 y(constan)-34 b(t)406 b Fq(C)6955 59382 y FH(1)7885 59200 y Fv(is)e(c)-34 b(hosen)405 b(to)f(ensure)g(that)i Fq(f)535 b Fv(de\014nes)404 b(a)g(unit)h(\015o)-34 b(w)406 b(in)e(the)g(sense)g(of)h(De\014nition)g(2.3,)e(i.e.,)2688 60848 y Fm(X)1806 63532 y Fr(\033)32 b Fk(2C)3501 63681 y Fn(L)4118 63532 y FH(\(0\))5523 61999 y Fq(C)6389 62181 y FH(1)7691 60848 y Fm(X)7117 63532 y Fr(z)7555 63655 y Fx(1)8016 63532 y FH(\()p Fr(\033)g FH(\))p Fr(;e)10015 61999 y Fv([)p Fq(g)43 b Fv(\()p Fq(z)12008 62181 y FH(1)12534 61999 y Fv(\()p Fq(\033)g Fv(\)\))187 b Fp(\000)g Fq(g)43 b Fv(\()p Fq(z)17656 62181 y FH(1)18183 61999 y Fv(\()p Fq(\033)g Fv(\))187 b(+)g Fq(e)p Fv(\)])337 b(=)f Fq(C)25032 62181 y FH(1)26845 60848 y Fm(X)25963 63532 y Fr(\033)32 b Fk(2C)27658 63681 y Fn(L)28275 63532 y FH(\(0\))29881 61999 y Fw(CAP)32287 61017 y Fm(\000)32843 61999 y Fq(P)33621 62181 y Fr(\033)34246 61999 y Fv(\(0\))p Fq(;)202 b Fv([)p Fq(B)37589 62187 y FH(2)p Fr(L)38699 62343 y Fn(\014)39253 62187 y Fr(;)p FH(2)p Fr(L)40624 62343 y Fn(\014)41233 61999 y Fv(\(0\)])43118 61499 y Fr(c)43582 61017 y Fm(\001)44474 61999 y Fv(=)336 b(1)p Fq(;)606 b Fv(\(4.39\))1200 66074 y(where)403 b Fq(P)5482 66256 y Fr(\033)6107 66074 y Fv(\(0\))g(denotes)h(the)f(supp)34 b(ort)404 b(of)f(the)g (proto-critical)f(droplet)h(in)g Fq(\033)43 b Fv(,)403 b(and)g(the)h(capacit)-34 b(y)402 b(refers)g(to)1200 67579 y(the)i(simple)g(random)h(w)-34 b(alk.)3082 69411 y(No)g(w,)404 b(let)g Fq(z)8547 68971 y FH(1)9073 69411 y Fv(\()p Fq(k)38 b Fv(\))405 b(b)34 b(e)404 b(the)g(lo)34 b(cation)404 b(of)h(the)f(\014rst)h(particle)e(at)i(time)f Fq(k)38 b Fv(,)404 b(and)15728 72096 y Fq(\034)16395 71595 y FH(1)17258 72096 y Fv(=)337 b(inf)94 b Fp(f)p Fq(k)375 b Fp(2)337 b Fo(N)135 b Fv(:)606 b Fq(z)25339 71595 y FH(1)25865 72096 y Fv(\()p Fq(k)38 b Fv(\))337 b Fp(2)g Fv([)p Fq(B)30214 72284 y Fr(L)30854 72440 y Fn(\014)31407 72284 y Fr(;L)32308 72440 y Fn(\014)32917 72096 y Fv(\(0\)])34802 71595 y Fr(c)35265 72096 y Fp(g)11431 b Fv(\(4.40\))25194 75417 y(30)p eop end %%Page: 31 31 TeXDict begin 31 30 bop 1200 1313 a Fv(b)34 b(e)443 b(the)g(\014rst)h (time)f(when,)453 b(under)443 b(the)h(Mark)-34 b(o)g(v)443 b(c)-34 b(hain)443 b(asso)34 b(ciated)443 b(to)h(the)f(\015o)-34 b(w)444 b Fq(f)130 b Fv(,)454 b(it)443 b(exits)f Fq(B)45811 1501 y Fr(L)46451 1657 y Fn(\014)47005 1501 y Fr(;L)47906 1657 y Fn(\014)48514 1313 y Fv(\(0\).)1200 2819 y(Let)404 b Fq(\015)471 b Fv(b)34 b(e)403 b(a)i(path)g(of)f(this)h(Mark)-34 b(o)g(v)404 b(c)-34 b(hain.)539 b(Then,)405 b(b)-34 b(y)404 b(\(4.38{4.39\),)h(w)-34 b(e)404 b(ha)-34 b(v)g(e)12948 5312 y Fr(\034)13468 4999 y Fx(1)12563 5676 y Fm(X)12580 8311 y Fr(k)24 b FH(=0)16612 6007 y Fq(f)130 b Fv(\()p Fq(\015)18434 6204 y Fr(k)19004 6007 y Fq(;)202 b(\015)20171 6204 y Fr(k)24 b FH(+1)21941 6007 y Fv(\))p 14648 6548 9729 49 v 14648 7659 a Fq(\026)15378 7856 y Fr(\014)16007 7659 y Fv(\()p Fq(\015)17106 7856 y Fr(k)17675 7659 y Fv(\))p Fq(c)18671 7856 y Fr(\014)19300 7659 y Fv(\()p Fq(\015)20399 7856 y Fr(k)20968 7659 y Fq(;)202 b(\015)22135 7856 y Fr(k)24 b FH(+1)23905 7659 y Fv(\))24846 6827 y(=)26258 6007 y Fq(C)27124 6189 y FH(1)27651 6007 y Fv([)p Fq(g)43 b Fv(\()p Fq(z)29697 5567 y FH(1)30223 6007 y Fv(\(0\)\))271 b Fp(\000)e Fq(g)43 b Fv(\()p Fq(z)35434 5567 y FH(1)35961 6007 y Fv(\()p Fq(\034)37099 5567 y FH(1)37625 6007 y Fv(\)\)])p 26258 6548 12646 49 v 30854 7659 a Fq(\026)31584 7856 y Fr(\014)32213 7659 y Fv(\()p Fq(\015)33312 7841 y FH(0)33838 7659 y Fv(\))47302 6827 y(\(4.41\))1200 10602 y(where)582 b(the)g(sum)g(o)-34 b(v)g(er)582 b(the)h Fq(g)43 b Fv('s)582 b(is)f(telescoping)h(b)34 b(ecause)582 b(only)f(paths)j(along)e(whic)-34 b(h)582 b(the)h Fq(g)43 b Fv(-function)1200 12108 y(decreases)420 b(carry)f(p)34 b(ositiv)-34 b(e)420 b(probabilit)-34 b(y)-101 b(,)425 b(and)c Fq(c)23494 12305 y Fr(\014)24123 12108 y Fv(\()p Fq(\015)25222 12305 y Fr(k)25790 12108 y Fq(;)202 b(\015)26957 12305 y Fr(k)24 b FH(+1)28728 12108 y Fv(\))364 b(=)g(1)420 b(for)h(all)f(0)364 b Fp(\024)f Fq(k)402 b Fp(\024)364 b Fq(\034)40751 11668 y FH(1)41697 12108 y Fv(b)34 b(ecause)420 b(the)h(\014rst)1200 13613 y(particle)403 b(is)h(free.)538 b(W)-101 b(e)404 b(ha)-34 b(v)g(e)404 b Fq(g)43 b Fv(\()p Fq(z)16188 13173 y FH(1)16715 13613 y Fv(\(0\)\))338 b(=)f(1,)403 b(while,)h(b)-34 b(y)405 b(Lemma)e(4.4,)h(there)g(exists)g(a)g Fq(C)423 b(<)337 b Fp(1)404 b Fv(suc)-34 b(h)405 b(that)15498 16336 y Fq(g)43 b Fv(\()p Fq(x)p Fv(\))338 b Fp(\024)f Fq(C)53 b(=)202 b Fv(log)219 b Fq(L)23692 16533 y Fr(\014)24320 16336 y Fq(;)2626 b(x)337 b Fp(2)g Fv([)p Fq(B)30714 16524 y Fr(L)31354 16680 y Fn(\014)31907 16524 y Fr(;L)32808 16680 y Fn(\014)33416 16336 y Fv(\(0\)])35301 15836 y Fr(c)35765 16336 y Fq(:)11200 b Fv(\(4.42\))1200 19059 y(Therefore)14825 20032 y Fr(\034)15345 19719 y Fx(1)14440 20395 y Fm(X)14457 23031 y Fr(k)24 b FH(=0)18490 20727 y Fq(f)130 b Fv(\()p Fq(\015)20312 20924 y Fr(k)20881 20727 y Fq(;)202 b(\015)22048 20924 y Fr(k)24 b FH(+1)23819 20727 y Fv(\))p 16526 21268 9729 49 v 16526 22378 a Fq(\026)17256 22575 y Fr(\014)17885 22378 y Fv(\()p Fq(\015)18984 22575 y Fr(k)19553 22378 y Fv(\))p Fq(c)20549 22575 y Fr(\014)21177 22378 y Fv(\()p Fq(\015)22276 22575 y Fr(k)22845 22378 y Fq(;)202 b(\015)24012 22575 y Fr(k)24 b FH(+1)25783 22378 y Fv(\))26723 21547 y(=)29167 20727 y Fq(C)30033 20909 y FH(1)p 28136 21268 3456 49 v 28136 22378 a Fq(\026)28866 22575 y Fr(\014)29495 22378 y Fv(\()p Fq(\015)30594 22560 y FH(0)31120 22378 y Fv(\))31926 21547 y([1)269 b(+)g Fq(o)p Fv(\(1\)])p Fq(:)10142 b Fv(\(4.43\))1200 24824 y(Next,)404 b(b)-34 b(y)404 b(Lemma)g(4.3,)f(w)-34 b(e)405 b(ha)-34 b(v)g(e)7909 27989 y Fw(CAP)10113 27007 y Fm(\000)10668 27989 y Fq(P)11446 28171 y Fr(\033)12071 27989 y Fv(\(0\))p Fq(;)202 b Fv([)p Fq(B)15414 28177 y FH(2)p Fr(L)16524 28333 y Fn(\014)17079 28177 y Fr(;)p FH(2)p Fr(L)18450 28333 y Fn(\014)19059 27989 y Fv(\(0\)])20944 27489 y Fr(c)21407 27007 y Fm(\001)22299 27989 y Fv(=)23921 27169 y(4)p Fq(\031)p 23712 27710 1760 49 v 23712 28820 a(\014)64 b Fv(\001)25806 27989 y([1)269 b(+)g Fq(o)p Fv(\(1\)])p Fq(;)2627 b(\033)379 b Fp(2)337 b(C)36522 28177 y Fr(L)37218 27989 y Fv(\(0\))p Fq(;)404 b(\014)c Fp(!)337 b(1)p Fq(;)3611 b Fv(\(4.44\))1200 31294 y(\(b)34 b(ecause)513 b Fp(f)p Fv(0)p Fp(g)k(\032)h Fq(P)10839 31476 y Fr(\033)11464 31294 y Fv(\(0\))g Fp(\032)f Fq(B)15909 31491 y FH(2)p Fr(`)16763 31602 y Fn(c)17191 31491 y Fr(:)p FH(2)p Fr(`)18306 31602 y Fn(c)18790 31294 y Fv(\(0\))c(for)g(all)f Fq(\033)560 b Fp(2)518 b(C)27824 31482 y Fr(L)28519 31294 y Fv(\(0\)\).)865 b(Since)513 b Fq(N)649 b Fv(=)518 b Fp(jC)39073 31482 y Fr(L)39768 31294 y Fv(\(0\))p Fp(j)p Fv(,)540 b(it)512 b(follo)-34 b(ws)514 b(from)1200 32799 y(\(4.39\))405 b(that)20304 33994 y(1)p 19911 34535 1393 49 v 19911 35645 a Fq(C)20777 35827 y FH(1)21773 34814 y Fv(=)336 b Fq(N)24703 33994 y Fv(4)p Fq(\031)p 24493 34535 1760 49 v 24493 35645 a(\014)64 b Fv(\001)26588 34814 y([1)269 b(+)g Fq(o)p Fv(\(1\)])p Fq(;)15480 b Fv(\(4.45\))1200 37620 y(and)405 b(so)f(\(4.43\))h(b)34 b(ecomes)8064 39520 y Fm(2)8064 41702 y(4)9257 40442 y Fr(\034)9777 40129 y Fx(1)8872 40805 y Fm(X)8890 43440 y Fr(k)24 b FH(=0)12922 41137 y Fq(f)130 b Fv(\()p Fq(\015)14744 41334 y Fr(k)15313 41137 y Fq(;)202 b(\015)16480 41334 y Fr(k)24 b FH(+1)18251 41137 y Fv(\))p 10958 41678 9729 49 v 10958 42788 a Fq(\026)11688 42985 y Fr(\014)12317 42788 y Fv(\()p Fq(\015)13416 42985 y Fr(k)13985 42788 y Fv(\))p Fq(c)14981 42985 y Fr(\014)15610 42788 y Fv(\()p Fq(\015)16709 42985 y Fr(k)17277 42788 y Fq(;)202 b(\015)18444 42985 y Fr(k)24 b FH(+1)20215 42788 y Fv(\))20819 39520 y Fm(3)20819 41702 y(5)21627 39822 y Fk(\000)p FH(1)23221 41957 y Fv(=)337 b Fq(\026)25231 42154 y Fr(\014)25860 41957 y Fv(\()p Fq(\015)26959 42139 y FH(0)27485 41957 y Fv(\))202 b Fq(N)29809 41137 y Fv(4)p Fq(\031)p 29599 41678 1760 49 v 29599 42788 a(\014)64 b Fv(\001)31694 41957 y([1)269 b(+)g Fq(o)p Fv(\(1\)])p Fq(:)2424 b(\014)400 b Fp(!)337 b(1)p Fq(;)3766 b Fv(\(4.46\))1200 46073 y(This)478 b(is)f(the)h(con)-34 b(tribution)479 b(w)-34 b(e)478 b(w)-34 b(an)g(t,)498 b(b)34 b(ecause)477 b(when)h(w)-34 b(e)478 b(sum)g(\(4.46\))g(o)-34 b(v)g(er)478 b Fq(\015)38889 46255 y FH(0)39873 46073 y Fv(=)459 b Fq(\033)502 b Fp(2)459 b(C)44375 46261 y Fr(L)45071 46073 y Fv(\(0\))478 b(\(recall)1200 47579 y(\(4.37\)\),)405 b(w)-34 b(e)404 b(get)g(a)h(factor)16761 49084 y Fq(\026)17491 49281 y Fr(\014)18120 49084 y Fv(\()p Fp(C)19229 49272 y Fr(L)19925 49084 y Fv(\(0\)\))338 b(=)e Fq(e)24125 48584 y Fk(\000)p Fr(\014)46 b FH(\000)26276 49084 y Fq(\026)27006 49281 y Fr(\014)27635 49084 y Fv(\()p Fp(S)91 b Fv(\))202 b([1)270 b(+)f Fq(o)p Fv(\(1\)])p Fq(:)12463 b Fv(\(4.47\))1200 51651 y(T)-101 b(o)448 b(see)g(wh)-34 b(y)449 b(\(4.47\))f(is)g(true,)459 b(recall)447 b(from)h(\(4.36\))h(that)g Fp(C)28440 51839 y Fr(L)29135 51651 y Fv(\(0\))g(is)f(obtained)h(from)f Fp(S)41156 50948 y FH(\()p Fr(n)42093 51104 y Fn(\014)42646 50948 y Fk(\000)p Fr(K)63 b FH(\))41065 51998 y(2)45103 51651 y Fv(b)-34 b(y)449 b(adding)1200 53156 y(a)429 b(critical)e(droplet)i (with)g(lo)-34 b(w)g(er-left)429 b(corner)f(at)h(the)h(origin)e(that)i (do)34 b(es)428 b(not)i(in)-34 b(teract)429 b(with)g(the)g Fq(n)47141 53353 y Fr(\014)48055 53156 y Fp(\000)286 b Fq(K)1200 54662 y Fv(particles)404 b(elsewhere)f(in)h(\003)13712 54859 y Fr(\014)14341 54662 y Fv(.)538 b(Hence)17520 58763 y Fq(\026)18250 58960 y Fr(\014)18880 58763 y Fv(\()p Fp(C)19989 58951 y Fr(L)20684 58763 y Fv(\(0\)\))338 b(=)f Fq(e)24885 58263 y Fk(\000)p Fr(\014)46 b FH(\000)26779 57950 y Fl(\003)27903 57420 y Fv(~)27648 57727 y Fq(Z)28562 57024 y FH(\()p Fr(n)29499 57180 y Fn(\014)30052 57024 y Fk(\000)p Fr(K)63 b FH(\))28475 58118 y Fr(\014)32061 57727 y Fv(\(0\))p 27648 58484 5963 49 v 29216 59997 a Fq(Z)30130 59294 y FH(\()p Fr(n)31067 59450 y Fn(\014)31620 59294 y FH(\))30043 60388 y Fr(\014)33743 58763 y Fq(;)13222 b Fv(\(4.48\))1200 63256 y(where)4987 62949 y(~)4732 63256 y Fq(Z)5646 62553 y FH(\()p Fr(n)6583 62709 y Fn(\014)7136 62553 y Fk(\000)p Fr(K)63 b FH(\))5559 63647 y Fr(\014)9146 63256 y Fv(\(0\))432 b(is)f(the)h(analog)g(of)20016 62949 y(\024)19761 63256 y Fq(Z)20675 62553 y FH(\()p Fr(n)21612 62709 y Fn(\014)22165 62553 y Fk(\000)p Fr(K)63 b FH(\))20588 63647 y Fr(\014)24174 63256 y Fv(\(0\))432 b(\(de\014ned)h(in)f (\(4.17\)\))g(obtained)h(b)-34 b(y)432 b(requiring)f(that)1200 64761 y(the)469 b Fq(n)4080 64958 y Fr(\014)5022 64761 y Fp(\000)312 b Fq(K)556 b Fv(particles)469 b(are)f(in)i([)p Fq(R)17611 64958 y Fr(`)17995 65069 y Fn(c)18422 64958 y Fr(;`)19067 65069 y Fn(c)19549 64761 y Fv(\(0\)])21434 64321 y Fr(c)22367 64761 y Fv(instead)g(of)f([)p Fq(B)29315 64949 y Fr(L)29955 65105 y Fn(\014)30508 64949 y Fr(;L)31409 65105 y Fn(\014)32018 64761 y Fv(\(0\)])33903 64321 y Fr(c)34366 64761 y Fv(.)734 b(Ho)-34 b(w)g(ev)g(er,)485 b(it)469 b(will)g(follo)-34 b(w)470 b(from)1200 66267 y(the)404 b(pro)34 b(ofs)405 b(of)f(Lemmas)g(B.1{B.2)f(in)h(App)34 b(endix)405 b(B)e(that,)i(just)h(as)e(in)g(\(4.22\),)13533 68959 y(~)13278 69265 y Fq(Z)14192 68562 y FH(\()p Fr(n)15129 68718 y Fn(\014)15682 68562 y Fk(\000)p Fr(K)63 b FH(\))14105 69657 y Fr(\014)17691 69265 y Fv(\(0\))p 13278 70023 V 14846 71535 a Fq(Z)15760 70832 y FH(\()p Fr(n)16697 70988 y Fn(\014)17250 70832 y FH(\))15673 71927 y Fr(\014)19709 70302 y Fv(=)337 b(\()p Fq(\032)22087 70499 y Fr(\014)22716 70302 y Fv(\))23187 69801 y Fr(K)24300 70302 y Fq(\026)25030 70499 y Fr(\014)25660 70302 y Fv(\()p Fp(S)91 b Fv(\))202 b([1)269 b(+)g Fq(o)p Fv(\(1\)])p Fq(;)2627 b(\014)64 b(\034)137 b Fp(1)p Fq(;)8846 b Fv(\(4.49\))25194 75417 y(31)p eop end %%Page: 32 32 TeXDict begin 32 31 bop 1200 1313 a Fv(whic)-34 b(h)413 b(yields)f(\(4.47\))i(b)34 b(ecause)412 b(\000)350 b(=)h(\000)19256 874 y Fk(\003)20056 1313 y Fv(+)275 b Fq(K)87 b Fv(\001.)564 b(F)-101 b(or)413 b(the)g(remaining)g(part)g(of)g(the)g(construction)g (of)g(the)1200 2819 y(\015o)-34 b(w)405 b(it)f(therefore)g(su\016ces)g (to)h(ensure)f(that)h(the)g(sum)g(b)34 b(ey)-34 b(ond)404 b Fq(\034)31567 2379 y FH(1)32497 2819 y Fv(giv)-34 b(es)404 b(a)g(smaller)f(con)-34 b(tribution.)1200 5269 y Fi(4.)392 b(Second)g(particle.)518 b Fv(Once)340 b(the)i(\014rst)f(particle)f (\(i.e.,)352 b(the)341 b(free)g(particle\))f(has)h(left)g(the)g(b)34 b(o)-34 b(x)341 b Fq(B)45811 5457 y Fr(L)46451 5613 y Fn(\014)47005 5457 y Fr(;L)47906 5613 y Fn(\014)48514 5269 y Fv(\(0\),)1200 6775 y(w)-34 b(e)432 b(need)f(to)h(allo)-34 b(w)431 b(the)h(second)g(particle)f(\(i.e.,)437 b(the)431 b(protub)34 b(erance\))433 b(to)e(detac)-34 b(h)433 b(itself)e(from)g (the)h(proto-)1200 8280 y(critical)535 b(droplet)i(and)h(to)f(mo)-34 b(v)g(e)537 b(out)h(of)f Fq(B)21775 8468 y Fr(L)22415 8624 y Fn(\014)22968 8468 y Fr(;L)23869 8624 y Fn(\014)24478 8280 y Fv(\(0\))g(as)g(w)-34 b(ell.)936 b(The)537 b(problem)g(is)g (that)h(detac)-34 b(hing)537 b(the)1200 9919 y(second)481 b(particle)f(reduces)g(the)h(Gibbs)h(w)-34 b(eigh)g(t)482 b(app)34 b(earing)480 b(in)h(the)g(denominator)h(b)-34 b(y)481 b Fq(e)42547 9479 y Fk(\000)p Fr(U)94 b(\014)44639 9919 y Fv(,)499 b(while)480 b(the)1200 11424 y(incremen)-34 b(ts)395 b(of)g(the)g(\015o)-34 b(w)396 b(are)f(reduced)f(only)h(to)g (ab)34 b(out)396 b(1)p Fq(=L)29399 11621 y Fr(\014)30028 11424 y Fv(.)535 b(Th)-34 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Fr(;)p FH(2)p Fr(L)29402 18105 y Fn(\014)30011 17761 y Fv(\(0\))p Fp(n)p Fq(B)33084 17949 y Fr(L)33724 18105 y Fn(\014)34279 17949 y Fr(;L)35180 18105 y Fn(\014)35788 17761 y Fv(\(0\))h(for)g (simple)f(random)h(w)-34 b(alk.)1200 19267 y(This)405 b(go)34 b(es)403 b(as)i(follo)-34 b(ws.)3082 21087 y(Let)508 b Fq(p)p Fv(\()p Fq(z)53 b(;)202 b(z)392 b Fv(+)338 b Fq(e)p Fv(\))510 b(denote)f(the)g(transition)g(probabilities)f(of)h (simple)f(random)h(w)-34 b(alk)509 b(driv)-34 b(en)509 b(b)-34 b(y)508 b(the)1200 22593 y(harmonic)404 b(function)i Fq(g)447 b Fv(on)405 b Fq(B)14899 22781 y FH(2)p Fr(L)16009 22937 y Fn(\014)16563 22781 y Fr(;)p FH(2)p Fr(L)17934 22937 y Fn(\014)18543 22593 y Fv(\(0\).)539 b(Put)5447 26180 y(~)-709 b Fq(p)p Fv(\()p Fq(z)53 b(;)202 b(z)323 b Fv(+)269 b Fq(e)p Fv(\))337 b(=)12332 24471 y Fm(\032)13794 25345 y Fq(p)p Fv(\()p Fq(z)53 b(;)202 b(z)323 b Fv(+)269 b Fq(e)p Fv(\))p Fq(;)7534 b Fv(if)404 b Fq(z)390 b Fp(2)337 b Fq(B)31165 25533 y Fr(L)31805 25689 y Fn(\014)32358 25533 y Fr(;L)33259 25689 y Fn(\014)33868 25345 y Fv(\(0\))p Fq(;)13794 26899 y Fv(\(1)270 b Fp(\000)f Fq(L)17178 26389 y Fk(\000)p FH(1+)p Fr(\017)17178 27290 y(\014)19548 26899 y Fv(\))202 b Fq(p)p Fv(\()p Fq(z)53 b(;)202 b(z)323 b Fv(+)269 b Fq(e)p Fv(\))p Fq(;)1107 b Fv(if)404 b Fq(z)390 b Fp(2)337 b Fq(B)31165 27087 y FH(2)p Fr(L)32275 27243 y Fn(\014)32829 27087 y Fr(;)p FH(2)p Fr(L)34200 27243 y Fn(\014)34809 26899 y Fv(\(0\))p Fp(n)p Fq(B)37882 27087 y Fr(L)38522 27243 y Fn(\014)39076 27087 y Fr(;L)39977 27243 y Fn(\014)40586 26899 y Fv(\(0\))p Fq(:)47302 26180 y Fv(\(4.50\))1200 29804 y(Use)480 b(the)h(transition)g(probabilities) 584 b(~)-709 b Fq(p)p Fv(\()p Fq(z)53 b(;)202 b(z)373 b Fv(+)320 b Fq(e)p Fv(\))481 b(to)g(de\014ne)g(a)f(path)i(measure) 38552 29498 y(~)38281 29804 y Fq(P)168 b Fv(.)768 b(This)480 b(path)i(measure)1200 31310 y(describ)34 b(es)518 b(simple)h(random)h (w)-34 b(alk)519 b(driv)-34 b(en)519 b(b)-34 b(y)520 b Fq(g)43 b Fv(,)548 b(but)520 b(with)g(a)g(killing)e(probabilit)-34 b(y)519 b Fq(L)42271 30800 y Fk(\000)p FH(1+)p Fr(\017)42271 31701 y(\014)45160 31310 y Fv(inside)g(the)1200 32815 y(ann)-34 b(ulus)406 b Fq(B)6604 33003 y FH(2)p Fr(L)7714 33159 y Fn(\014)8268 33003 y Fr(;)p FH(2)p Fr(L)9639 33159 y Fn(\014)10248 32815 y Fv(\(0\))p Fp(n)p Fq(B)13321 33003 y Fr(L)13961 33159 y Fn(\014)14515 33003 y Fr(;L)15416 33159 y Fn(\014)16025 32815 y Fv(\(0\).)539 b(Put)11722 35787 y Fq(k)38 b Fv(\()p Fq(z)53 b(;)202 b(z)323 b Fv(+)269 b Fq(e)p Fv(\))337 b(=)18769 34635 y Fm(X)19376 37181 y Fr(\015)20993 35480 y Fv(~)20722 35787 y Fq(P)168 b Fv(\()p Fq(\015)67 b Fv(\))24036 36034 y FH(\()p Fr(z)37 b(;z)g FH(+)p Fr(e)p FH(\))p Fk(2)p Fr(\015)28368 35787 y Fq(;)2626 b(z)389 b Fp(2)337 b Fq(B)34348 35975 y FH(2)p Fr(L)35458 36131 y Fn(\014)36012 35975 y Fr(;)p FH(2)p Fr(L)37383 36131 y Fn(\014)37992 35787 y Fv(\(0\))p Fq(:)7425 b Fv(\(4.51\))1200 39542 y(This)405 b(edge)f(function)h(satis\014es)g (the)f(follo)-34 b(wing)405 b(equations:)5383 42057 y Fp(\017)269 b Fq(k)38 b Fv(\()p Fq(z)53 b(;)202 b(z)323 b Fv(+)269 b Fq(e)p Fv(\))337 b(=)g([)p Fq(g)43 b Fv(\()p Fq(z)53 b Fv(\))270 b Fp(\000)f Fq(g)43 b Fv(\()p Fq(z)324 b Fv(+)269 b Fq(e)p Fv(\)])21868 42239 y FH(+)22656 42057 y Fq(;)7538 43895 y Fv(if)404 b Fq(z)389 b Fp(2)337 b Fq(B)11666 44083 y Fr(L)12306 44239 y Fn(\014)12860 44083 y Fr(;L)13761 44239 y Fn(\014)14369 43895 y Fv(\(0\))p Fq(;)5383 45865 y Fp(\017)269 b Fq(k)38 b Fv(\()p Fq(z)53 b(;)202 b(z)323 b Fv(+)269 b Fq(e)p Fv(\))337 b(=)g(0)p Fq(;)7538 47702 y Fv(if)404 b Fq(z)389 b Fp(2)337 b Fq(B)11666 47890 y FH(2)p Fr(L)12776 48046 y Fn(\014)13330 47890 y Fr(;)p FH(2)p Fr(L)14701 48046 y Fn(\014)15310 47702 y Fv(\(0\))p Fp(n)p Fq(B)18383 47890 y Fr(L)19023 48046 y Fn(\014)19577 47890 y Fr(;L)20478 48046 y Fn(\014)21087 47702 y Fv(\(0\))405 b(and)g([)p Fq(g)43 b Fv(\()p Fq(z)53 b Fv(\))270 b Fp(\000)g Fq(g)43 b Fv(\()p Fq(z)323 b Fv(+)269 b Fq(e)p Fv(\)])33960 47884 y FH(+)35084 47702 y Fv(=)337 b(0)p Fq(;)5383 49891 y Fp(\017)269 b Fv(\(1)h Fp(\000)f Fq(L)9642 49381 y Fk(\000)p FH(1+)p Fr(\017)9642 50282 y(\014)12012 49891 y Fv(\))12685 48739 y Fm(X)13342 51285 y Fr(e)14638 49891 y Fq(k)38 b Fv(\()p Fq(z)323 b Fv(+)269 b Fq(e;)202 b(z)53 b Fv(\))20799 50137 y Fr(g)32 b FH(\()p Fr(z)37 b FH(+)p Fr(e)p FH(\))p Fk(\000)p Fr(g)32 b FH(\()p Fr(z)37 b FH(\))p Fr(>)p FH(0)27667 49891 y Fv(=)28946 48739 y Fm(X)29603 51285 y Fr(e)30899 49891 y Fq(k)h Fv(\()p Fq(z)53 b(;)202 b(z)323 b Fv(+)269 b Fq(e)p Fv(\))37060 50137 y Fr(g)32 b FH(\()p Fr(z)37 b FH(\))p Fk(\000)p Fr(g)32 b FH(\()p Fr(z)37 b FH(+)p Fr(e)p FH(\))p Fr(>)p FH(0)7538 52902 y Fv(if)808 b Fq(z)389 b Fp(2)337 b Fq(B)12070 53090 y FH(2)p Fr(L)13180 53246 y Fn(\014)13734 53090 y Fr(;)p FH(2)p Fr(L)15105 53246 y Fn(\014)15714 52902 y Fv(\(0\))p Fp(n)p Fq(B)18787 53090 y Fr(L)19427 53246 y Fn(\014)19982 53090 y Fr(;L)20883 53246 y Fn(\014)21491 52902 y Fv(\(0\))p Fq(:)47302 47492 y Fv(\(4.52\))1200 55510 y(Note)415 b(that)i(inside)e(the)g(ann)-34 b(ulus)417 b Fq(B)17743 55698 y FH(2)p Fr(L)18853 55854 y Fn(\014)19407 55698 y Fr(;)p FH(2)p Fr(L)20778 55854 y Fn(\014)21387 55510 y Fv(\(0\))p Fp(n)p Fq(B)24460 55698 y Fr(L)25100 55854 y Fn(\014)25655 55698 y Fr(;L)26556 55854 y Fn(\014)27164 55510 y Fv(\(0\))f(at)g(eac)-34 b(h)415 b(site)g(the)h(\015o)-34 b(w)416 b(out)g(is)f(less)g(than)h (the)1200 57175 y(\015o)-34 b(w)405 b(in)f(b)-34 b(y)405 b(a)f(leaking)f(factor)i(1)269 b Fp(\000)g Fq(L)18358 56665 y Fk(\000)p FH(1+)p Fr(\017)18358 57566 y(\014)20728 57175 y Fv(.)538 b(W)-101 b(e)404 b(pic)-34 b(k)404 b Fq(\017)337 b(>)f Fv(0)404 b(so)g(small)g(that)14099 60080 y Fq(e)14663 59579 y Fr(\014)46 b(U)16427 60080 y Fv(is)404 b(exp)34 b(onen)-34 b(tially)403 b(smaller)h(in)g Fq(\014)467 b Fv(than)406 b Fq(L)35527 59570 y FH(2)p Fk(\000)p Fr(\017)35527 60471 y(\014)37165 60080 y Fq(;)9800 b Fv(\(4.53\))1200 62700 y(\(whic)-34 b(h)354 b(is)f(p)34 b(ossible)354 b(b)-34 b(y)353 b(\(1.27\))h(and)g(\(1.29{1.30\)\).)522 b(The)354 b(imp)34 b(ortan)-34 b(t)355 b(fact)e(for)h(us)f(is)g(that)i (this)f(leaky)e(\015o)-34 b(w)1200 64206 y(is)404 b(dominated)h(b)-34 b(y)405 b(the)f(harmonic)g(\015o)-34 b(w)406 b(asso)34 b(ciated)404 b(with)h Fq(g)43 b Fv(,)404 b(in)g(particular,)g(the)g (\015o)-34 b(w)406 b(in)e(satis\014es)9826 65789 y Fm(X)10483 68334 y Fr(e)11779 66940 y Fq(k)38 b Fv(\()p Fq(z)323 b Fv(+)269 b Fq(e;)202 b(z)53 b Fv(\))337 b Fp(\024)18826 65789 y Fm(X)19483 68334 y Fr(e)20576 66940 y Fv([)p Fq(g)43 b Fv(\()p Fq(z)323 b Fv(+)270 b Fq(e)p Fv(\))g Fp(\000)f Fq(g)43 b Fv(\()p Fq(z)53 b Fv(\)])29139 67122 y FH(+)32351 66940 y Fp(8)202 b Fq(z)391 b Fp(2)336 b Fq(B)36244 67128 y FH(2)p Fr(L)37354 67284 y Fn(\014)37908 67128 y Fr(;)p FH(2)p Fr(L)39279 67284 y Fn(\014)39888 66940 y Fv(\(0\))p Fq(;)5529 b Fv(\(4.54\))1200 70590 y(\(and)486 b(the)g(same)f(applies)g(for)g(the)g(\015o)-34 b(w)487 b(out\).)782 b(This)486 b(inequalit)-34 b(y)485 b(holds)g(b)34 b(ecause)485 b Fq(g)529 b Fv(satis\014es)485 b(the)h(same)1200 72096 y(equations)405 b(as)f(in)g(\(4.50{4.51\))h (but)g(without)h(the)e(leaking)g(factor)g(1)269 b Fp(\000)g Fq(L)35139 71586 y Fk(\000)p FH(1+)p Fr(\017)35139 72487 y(\014)37509 72096 y Fv(.)25194 75417 y(32)p eop end %%Page: 33 33 TeXDict begin 33 32 bop 3082 1313 a Fv(Using)519 b(this)g(leaky)f (\015o)-34 b(w,)549 b(w)-34 b(e)519 b(can)g(no)-34 b(w)520 b(construct)g(a)f(\015o)-34 b(w)520 b(in)-34 b(v)g(olving)519 b(the)g(\014rst)h(t)-34 b(w)g(o)520 b(particles,)547 b(as)1200 2819 y(follo)-34 b(ws:)3766 5201 y Fp(\017)270 b Fq(f)130 b Fv(\()p Fq(\033)43 b Fv(\()p Fq(z)7607 5383 y FH(1)8134 5201 y Fq(;)202 b(a)p Fv(\))p Fq(;)g(\033)43 b Fv(\()p Fq(z)12095 5383 y FH(1)12890 5201 y Fv(+)269 b Fq(e;)202 b(a)p Fv(\)\))337 b(=)g Fq(C)19271 5383 y FH(1)19797 5201 y Fq(k)38 b Fv(\()p Fq(z)21501 5383 y FH(1)22027 5201 y Fq(;)202 b(z)23130 5383 y FH(1)23925 5201 y Fv(+)269 b Fq(e)p Fv(\))p Fq(;)5921 7039 y Fv(if)404 b Fq(z)7596 7221 y FH(1)8459 7039 y Fp(2)336 b Fq(B)10522 7227 y FH(2)p Fr(L)11632 7383 y Fn(\014)12186 7227 y Fr(;)p FH(2)p Fr(L)13557 7383 y Fn(\014)14166 7039 y Fv(\(0\))p Fq(;)3766 9227 y Fp(\017)270 b Fq(f)130 b Fv(\()p Fq(\033)43 b Fv(\()p Fq(z)7607 9409 y FH(1)8134 9227 y Fq(;)202 b(a)p Fv(\))p Fq(;)g(\033)43 b Fv(\()p Fq(z)12095 9409 y FH(1)12621 9227 y Fq(;)202 b(b)p Fv(\)\))337 b(=)g Fq(C)17105 9409 y FH(1)17631 9227 y Fq(L)18456 8717 y Fk(\000)p FH(1+)p Fr(\017)18456 9619 y(\014)21027 8076 y Fm(X)21685 10621 y Fr(e)22980 9227 y Fq(k)38 b Fv(\()p Fq(z)24684 9409 y FH(1)25210 9227 y Fq(;)202 b(z)26313 9409 y FH(1)27108 9227 y Fv(+)269 b Fq(e)p Fv(\))p Fq(;)5921 12239 y Fv(if)404 b Fq(z)7596 12421 y FH(1)8459 12239 y Fp(2)336 b Fq(B)10522 12427 y FH(2)p Fr(L)11632 12583 y Fn(\014)12186 12427 y Fr(;)p FH(2)p Fr(L)13557 12583 y Fn(\014)14166 12239 y Fv(\(0\))p Fp(n)p Fq(B)17239 12427 y Fr(L)17879 12583 y Fn(\014)18434 12427 y Fr(;L)19335 12583 y Fn(\014)19943 12239 y Fv(\(0\))p Fq(;)3766 15276 y Fp(\017)270 b Fq(f)130 b Fv(\()p Fq(\033)43 b Fv(\()p Fq(z)7607 15458 y FH(1)8134 15276 y Fq(;)202 b(z)9237 15458 y FH(2)9762 15276 y Fv(\))p Fq(;)g(\033)43 b Fv(\()p Fq(z)12543 15458 y FH(1)13069 15276 y Fq(;)202 b(z)14172 15458 y FH(2)14967 15276 y Fv(+)269 b Fq(e)p Fv(\)\))338 b(=)19302 13203 y Fm(\()20279 15276 y Fq(C)21145 15458 y FH(1)21671 15276 y Fq(L)22496 14766 y Fk(\000)p FH(1+)p Fr(\017)22496 15667 y(\014)25068 14124 y Fm(X)25725 16670 y Fr(e)27021 15276 y Fq(k)38 b Fv(\()p Fq(z)28725 15458 y FH(1)29251 15276 y Fq(;)202 b(z)30354 15458 y FH(1)31148 15276 y Fv(+)269 b Fq(e)p Fv(\))33395 13203 y Fm(\))34575 15276 y Fv([)p Fq(g)43 b Fv(\()p Fq(z)36568 15458 y FH(2)37094 15276 y Fv(\))270 b Fp(\000)f Fq(g)43 b Fv(\()p Fq(z)40703 15458 y FH(2)41499 15276 y Fv(+)269 b Fq(e)p Fv(\)])44083 15458 y FH(+)44870 15276 y Fq(;)5921 18288 y Fv(if)404 b Fq(z)7596 18470 y FH(1)8459 18288 y Fp(2)336 b Fq(B)10522 18476 y FH(2)p Fr(L)11632 18632 y Fn(\014)12186 18476 y Fr(;)p FH(2)p Fr(L)13557 18632 y Fn(\014)14166 18288 y Fv(\(0\))p Fp(n)p Fq(B)17239 18476 y Fr(L)17879 18632 y Fn(\014)18434 18476 y Fr(;L)19335 18632 y Fn(\014)19943 18288 y Fv(\(0\))p Fq(;)202 b(z)22594 18470 y FH(2)23457 18288 y Fp(2)337 b Fq(B)25521 18476 y Fr(L)26161 18632 y Fn(\014)26714 18476 y Fr(;L)27615 18632 y Fn(\014)28224 18288 y Fv(\(0\))p Fp(n)p Fq(P)31156 18470 y Fr(\033)31782 18288 y Fv(\(0\))p Fq(:)47302 11757 y Fv(\(4.55\))1200 20931 y(Here,)366 b(w)-34 b(e)357 b(write)g Fq(a)g Fv(and)i Fq(b)e Fv(for)g(the)h(lo)34 b(cations)357 b(of)h(the)g(second)f(particle)g(prior)g(and)h(after)f (it)h(detac)-34 b(hes)358 b(itself)1200 22436 y(from)370 b(the)h(proto-critical)e(droplet,)377 b(and)371 b Fq(\033)43 b Fv(\()p Fq(z)21912 22618 y FH(1)22438 22436 y Fq(;)202 b(z)23541 22618 y FH(2)24067 22436 y Fv(\))370 b(for)g(the)h (con\014guration)g(obtained)h(from)e Fq(\033)413 b Fv(b)-34 b(y)371 b(placing)1200 23942 y(the)496 b(\014rst)g(particle)f(\(that)i (w)-34 b(as)497 b(at)f(distance)g(2)f(from)h(the)g(proto-critical)f (droplet\))i(at)e(site)h Fq(z)45247 24124 y FH(1)46268 23942 y Fv(and)g(the)1200 25447 y(second)349 b(particle)g(\(that)i(w) -34 b(as)349 b(the)h(protub)34 b(erance\))350 b(at)f(site)g Fq(z)28694 25629 y FH(2)29220 25447 y Fv(.)520 b(The)350 b(\015o)-34 b(w)350 b(for)f(other)g(motions)h(is)f(zero,)359 b(and)1200 26953 y(the)404 b(constan)-34 b(t)406 b Fq(C)9042 27135 y FH(1)9973 26953 y Fv(is)d(the)i Fs(same)494 b Fv(as)404 b(in)g(\(4.38{4.39\))3082 28790 y(W)-101 b(e)403 b(next)i(de\014ne)f(t)-34 b(w)g(o)406 b(further)f(stopping)g(times,)f (namely)-101 b(,)18146 31513 y Fq(\020)18765 31013 y FH(2)19629 31513 y Fv(=)336 b(inf)95 b Fp(f)p Fq(k)375 b Fp(2)337 b Fo(N)135 b Fv(:)605 b Fq(z)27709 31013 y FH(2)28235 31513 y Fv(\()p Fq(\015)29334 31710 y Fr(k)29903 31513 y Fv(\))337 b(=)g Fq(b)p Fp(g)p Fq(;)13848 b Fv(\(4.56\))1200 34236 y(i.e.,)347 b(the)336 b(\014rst)f(time)g(the)h(second)f(particle) f(\(the)i(protub)34 b(erance\))336 b(detac)-34 b(hes)336 b(itself)f(from)g(the)g(proto-critical)1200 35742 y(droplet,)404 b(and)15297 37247 y Fq(\034)15964 36747 y FH(2)16826 37247 y Fv(=)337 b(inf)94 b Fp(f)p Fq(k)375 b Fp(2)337 b Fo(N)135 b Fv(:)606 b Fq(z)24907 36747 y FH(2)25433 37247 y Fv(\()p Fq(\015)26532 37444 y Fr(k)27101 37247 y Fv(\))337 b Fp(2)f Fv([)p Fq(B)30309 37435 y Fr(L)30949 37591 y Fn(\014)31502 37435 y Fr(;L)32403 37591 y Fn(\014)33012 37247 y Fv(\(0\)])34897 36747 y Fr(c)35361 37247 y Fp(g)p Fq(;)10998 b Fv(\(4.57\))1200 39472 y(i.e.,)385 b(the)e(\014rst)h(time) e(the)h(second)g(particle)f(exits)g(the)h(b)34 b(o)-34 b(x)383 b Fq(B)29208 39660 y Fr(L)29848 39816 y Fn(\014)30401 39660 y Fr(;L)31302 39816 y Fn(\014)31911 39472 y Fv(\(0\).)532 b(Note)383 b(that,)388 b(since)382 b(w)-34 b(e)383 b(c)-34 b(ho)34 b(ose)382 b(the)1200 40978 y(leaking)422 b(probabilit)-34 b(y)424 b(to)f(b)34 b(e)423 b Fq(L)15572 40538 y Fk(\000)p FH(1+)p Fr(\017)17942 40978 y Fv(,)k(the)d(probabilit)-34 b(y)423 b(that)h Fq(\020)30294 40538 y FH(2)31244 40978 y Fv(is)f(larger)f(than)i(the)f(\014rst)h(time)f(the)g(\014rst)1200 42483 y(particle)417 b(exits)f Fq(B)9394 42671 y FH(2)p Fr(L)10504 42827 y Fn(\014)11058 42671 y Fr(;)p FH(2)p Fr(L)12429 42827 y Fn(\014)13038 42483 y Fv(\(0\))i(is)f(of)h(order)f (exp[)p Fp(\000)p Fq(L)24772 42043 y Fr(\017)24772 42852 y(\014)25400 42483 y Fv(])g(and)h(hence)f(is)g(negligible.)577 b(W)-101 b(e)417 b(will)f(disregard)h(the)1200 43989 y(con)-34 b(tributions)406 b(of)e(suc)-34 b(h)405 b(paths)h(in)e(the)g (lo)-34 b(w)g(er)405 b(b)34 b(ound.)539 b(P)-34 b(aths)406 b(with)f(this)f(prop)34 b(ert)-34 b(y)405 b(will)e(b)34 b(e)404 b(called)f Fs(go)-62 b(o)g(d)p Fv(.)3082 45826 y(W)-101 b(e)414 b(will)g(next)h(sho)-34 b(w)416 b(that)g(\(4.41\))f (also)g(holds)g(if)g(w)-34 b(e)415 b(extend)g(the)g(sum)g(along)g(an) -34 b(y)415 b(path)h(of)f(p)34 b(ositiv)-34 b(e)1200 47332 y(probabilit)g(y)419 b(up)g(to)g Fq(\020)11293 46892 y FH(2)11820 47332 y Fv(.)582 b(The)419 b(reason)g(for)f(this)i (lies)d(in)i(Lemma)f(\015o)-34 b(w-lb.11.)583 b(Let)418 b Fq(\015)486 b Fv(b)34 b(e)418 b(a)h(path)h(that)f(has)1200 48837 y(a)460 b(p)34 b(ositiv)-34 b(e)459 b(probabilit)-34 b(y)460 b(under)g(the)h(path)f(measure)g Fo(P)27101 48397 y Fr(f)28165 48837 y Fv(asso)34 b(ciated)460 b(with)g Fq(f)591 b Fv(stopp)34 b(ed)460 b(at)g Fq(\034)44803 48397 y FH(2)45329 48837 y Fv(.)705 b(W)-101 b(e)459 b(will)1200 50343 y(assume)405 b(that)g(this)f(path)i(is)e(go)34 b(o)g(d)404 b(in)g(the)g(sense)g(describ)34 b(ed)404 b(ab)34 b(o)-34 b(v)g(e.)539 b(T)-101 b(o)404 b(that)h(end)g(w)-34 b(e)405 b(decomp)34 b(ose)3074 52768 y Fr(\034)3594 52456 y Fx(2)2689 53132 y Fm(X)2707 55767 y Fr(k)24 b FH(=0)6739 53464 y Fq(f)130 b Fv(\()p Fq(\015)8561 53661 y Fr(k)9130 53464 y Fq(;)202 b(\015)10297 53661 y Fr(k)24 b FH(+1)12068 53464 y Fv(\))p 4775 54005 9729 49 v 4775 55115 a Fq(\026)5505 55312 y Fr(\014)6134 55115 y Fv(\()p Fq(\015)7233 55312 y Fr(k)7802 55115 y Fv(\))p Fq(c)8798 55312 y Fr(\014)9427 55115 y Fv(\()p Fq(\015)10526 55312 y Fr(k)11094 55115 y Fq(;)202 b(\015)12261 55312 y Fr(k)24 b FH(+1)14032 55115 y Fv(\))2824 58773 y(=)4488 57258 y Fr(\034)5008 56945 y Fx(1)4103 57621 y Fm(X)4121 60256 y Fr(k)g FH(=0)8153 57953 y Fq(f)130 b Fv(\()p Fq(\015)9975 58150 y Fr(k)10545 57953 y Fq(;)202 b(\015)11712 58150 y Fr(k)24 b FH(+1)13482 57953 y Fv(\))p 6189 58494 V 6189 59604 a Fq(\026)6919 59801 y Fr(\014)7548 59604 y Fv(\()p Fq(\015)8647 59801 y Fr(k)9216 59604 y Fv(\))p Fq(c)10212 59801 y Fr(\014)10841 59604 y Fv(\()p Fq(\015)11940 59801 y Fr(k)12509 59604 y Fq(;)202 b(\015)13676 59801 y Fr(k)24 b FH(+1)15446 59604 y Fv(\))16320 58773 y(+)18177 57193 y Fr(\020)18652 56881 y Fx(2)19113 57193 y Fk(\000)p FH(2)18371 57621 y Fm(X)17532 60322 y Fr(k)g FH(=)p Fr(\034)19298 60069 y Fx(1)19758 60322 y FH(+1)23259 57953 y Fq(f)130 b Fv(\()p Fq(\015)25081 58150 y Fr(k)25651 57953 y Fq(;)202 b(\015)26818 58150 y Fr(k)24 b FH(+1)28588 57953 y Fv(\))p 21295 58494 V 21295 59604 a Fq(\026)22025 59801 y Fr(\014)22654 59604 y Fv(\()p Fq(\015)23753 59801 y Fr(k)24322 59604 y Fv(\))p Fq(c)25318 59801 y Fr(\014)25947 59604 y Fv(\()p Fq(\015)27046 59801 y Fr(k)27615 59604 y Fq(;)202 b(\015)28782 59801 y Fr(k)24 b FH(+1)30552 59604 y Fv(\))31426 58773 y(+)33839 57258 y Fr(\034)34359 56945 y Fx(2)33454 57621 y Fm(X)32638 60322 y Fr(k)g FH(=)p Fr(\020)34359 60069 y Fx(2)34819 60322 y Fk(\000)p FH(1)38320 57953 y Fq(f)130 b Fv(\()p Fq(\015)40142 58150 y Fr(k)40711 57953 y Fq(;)202 b(\015)41878 58150 y Fr(k)24 b FH(+1)43649 57953 y Fv(\))p 36356 58494 V 36356 59604 a Fq(\026)37086 59801 y Fr(\014)37715 59604 y Fv(\()p Fq(\015)38814 59801 y Fr(k)39383 59604 y Fv(\))p Fq(c)40379 59801 y Fr(\014)41008 59604 y Fv(\()p Fq(\015)42107 59801 y Fr(k)42675 59604 y Fq(;)202 b(\015)43842 59801 y Fr(k)24 b FH(+1)45613 59604 y Fv(\))2824 62112 y(=)336 b Fq(I)365 b Fv(+)269 b Fq(I)95 b(I)364 b Fv(+)269 b Fq(I)95 b(I)g(I)g(:)47302 57538 y Fv(\(4.58\))1200 64801 y(The)281 b(term)g Fq(I)376 b Fv(w)-34 b(as)282 b(already)e(estimated)h (in)g(\(4.41{4.47\).)498 b(T)-101 b(o)282 b(estimate)f Fq(I)95 b(I)g Fv(,)305 b(w)-34 b(e)281 b(use)g(\(4.42\))h(and)f (\(4.54{4.55\))1200 66306 y(to)404 b(b)34 b(ound)406 b(\(compare)e(with)h(\(4.41\)\))13157 69768 y Fq(I)95 b(I)431 b Fp(\024)337 b Fq(C)16895 69950 y FH(1)17756 68948 y Fq(g)43 b Fv(\()p Fq(z)19465 68508 y FH(1)19991 68948 y Fv(\()p Fq(\020)21081 68508 y FH(2)21608 68948 y Fv(\)\))271 b Fp(\000)e Fq(g)43 b Fv(\()p Fq(z)25742 68508 y FH(1)26268 68948 y Fv(\()p Fq(\034)27406 68508 y FH(1)27933 68948 y Fv(\)\))p 17756 69489 11120 49 v 21588 70599 a Fq(\026)22318 70796 y Fr(\014)22947 70599 y Fv(\()p Fq(\015)24046 70781 y FH(0)24572 70599 y Fv(\))29345 69768 y Fp(\024)336 b Fq(C)31490 69950 y FH(1)32351 68926 y Fv([)p Fq(C)53 b(=)202 b Fv(log)219 b Fq(L)37008 69123 y Fr(\014)37637 68926 y Fv(])p 32351 69489 5623 49 v 33435 70599 a Fq(\026)34165 70796 y Fr(\014)34794 70599 y Fv(\()p Fq(\015)35893 70781 y FH(0)36419 70599 y Fv(\))38107 69768 y Fq(;)8858 b Fv(\(4.59\))25194 75417 y(33)p eop end %%Page: 34 34 TeXDict begin 34 33 bop 1200 1313 a Fv(whic)-34 b(h)496 b(is)e(negligible)g(compared)h(to)g Fq(I)590 b Fv(due)495 b(to)g(the)g(factor)g Fq(C)53 b(=)202 b Fv(log)219 b Fq(L)33869 1510 y Fr(\014)34498 1313 y Fv(.)810 b(It)495 b(remains)f(to)i(estimate)e Fq(I)95 b(I)g(I)g Fv(.)1200 2819 y(Note)404 b(that)10721 5306 y Fq(I)95 b(I)g(I)432 b Fv(=)17130 4399 y Fq(f)130 b Fv(\()p Fq(\015)18952 4662 y Fr(\020)19427 4409 y Fx(2)19888 4662 y Fk(\000)p FH(1)21146 4399 y Fq(;)202 b(\015)22313 4662 y Fr(\020)22788 4409 y Fx(2)23303 4399 y Fv(\))p 14354 5028 12197 49 v 14354 6138 a Fq(\026)15084 6335 y Fr(\014)15713 6138 y Fv(\()p Fq(\015)16812 6401 y Fr(\020)17287 6148 y Fx(2)17748 6401 y Fk(\000)p FH(1)19005 6138 y Fv(\))p Fq(c)20001 6335 y Fr(\014)20630 6138 y Fv(\()p Fq(\015)21729 6401 y Fr(\020)22204 6148 y Fx(2)22665 6401 y Fk(\000)p FH(1)23922 6138 y Fq(;)g(\015)25089 6401 y Fr(\020)25564 6148 y Fx(2)26079 6138 y Fv(\))26953 5306 y(+)28765 3791 y Fr(\034)29285 3479 y Fx(2)28380 4155 y Fm(X)28165 6855 y Fr(k)24 b FH(=)p Fr(\020)29886 6603 y Fx(2)32645 4486 y Fq(f)130 b Fv(\()p Fq(\015)34467 4683 y Fr(k)35036 4486 y Fq(;)202 b(\015)36203 4683 y Fr(k)24 b FH(+1)37974 4486 y Fv(\))p 30681 5028 9729 49 v 30681 6138 a Fq(\026)31411 6335 y Fr(\014)32040 6138 y Fv(\()p Fq(\015)33139 6335 y Fr(k)33708 6138 y Fv(\))p Fq(c)34704 6335 y Fr(\014)35333 6138 y Fv(\()p Fq(\015)36432 6335 y Fr(k)37000 6138 y Fq(;)202 b(\015)38167 6335 y Fr(k)24 b FH(+1)39938 6138 y Fv(\))40542 5306 y Fq(:)6423 b Fv(\(4.60\))1200 8521 y(The)355 b(\014rst)g(term)g (corresp)34 b(onds)355 b(to)g(the)g(mo)-34 b(v)g(e)355 b(when)g(the)g(protub)34 b(erance)355 b(detac)-34 b(hes)356 b(itself)e(from)h(the)g(proto-)1200 10027 y(critical)510 b(droplet.)862 b(Its)512 b(n)-34 b(umerator)513 b(is)e(giv)-34 b(en)512 b(b)-34 b(y)512 b Fq(f)130 b Fv(\()p Fq(\033)43 b Fv(\()p Fq(z)27585 10209 y FH(1)28113 10027 y Fq(;)202 b(a)p Fv(\))p Fq(;)g(\033)43 b Fv(\()p Fq(z)32074 10209 y FH(1)32599 10027 y Fq(;)202 b(b)p Fv(\)\))512 b(\(for)h(some)e Fq(z)41255 10209 y FH(1)42297 10027 y Fp(2)516 b Fv([)p Fq(B)44877 10215 y Fr(L)45517 10371 y Fn(\014)46070 10215 y Fr(;L)46971 10371 y Fn(\014)47580 10027 y Fv(\(0\)])49465 9587 y Fr(c)49929 10027 y Fv(\))1200 11691 y(whic)-34 b(h,)470 b(b)-34 b(y)458 b(Lemma)e(4.4)g(and)i(\(4.54{4.55\),)470 b(is)456 b(smaller)g(than)i Fq(C)31522 11873 y FH(1)32048 11691 y Fq(L)32873 11181 y Fk(\000)p FH(1+)p Fr(\017)32873 12082 y(\014)35243 11691 y Fq(C)87 b(L)37021 11181 y Fk(\000)p FH(1)37021 12082 y Fr(\014)38703 11691 y Fv(=)424 b Fq(C)40936 11873 y FH(1)41462 11691 y Fq(C)87 b(L)43240 11181 y Fk(\000)p FH(2+)p Fr(\017)43240 12082 y(\014)45610 11691 y Fv(.)697 b(On)456 b(the)1200 13196 y(other)404 b(hand,)h(its)g(denominator)g(is)e(giv)-34 b(en)404 b(b)-34 b(y)15984 15321 y Fq(\026)p Fv(\()p Fq(\015)17813 15584 y Fr(\020)18288 15331 y Fx(2)18749 15584 y Fk(\000)p FH(1)20006 15321 y Fv(\))p Fq(c)21002 15518 y Fr(\014)21631 15321 y Fv(\()p Fq(\015)22730 15584 y Fr(\020)23205 15331 y Fx(2)23666 15584 y Fk(\000)p FH(1)24923 15321 y Fq(;)202 b(\015)26090 15584 y Fr(\020)26565 15331 y Fx(2)27080 15321 y Fv(\))338 b(=)e Fq(\026)29898 15518 y Fr(\014)30527 15321 y Fv(\()p Fq(\015)31626 15503 y FH(0)32152 15321 y Fv(\))p Fq(e)33187 14820 y Fk(\000)p Fr(U)94 b(\014)35279 15321 y Fq(:)11686 b Fv(\(4.61\))1200 17445 y(The)441 b(same)g(holds)g(for)g(the)g(denominators)h(in)f(all)f(the)h(other)g (terms)g(in)g Fq(I)95 b(I)g(I)g Fv(,)449 b(while)441 b(the)g(n)-34 b(umerators)442 b(in)1200 18951 y(these)404 b(terms)g(satisfy)h(the)g(b)34 b(ound)12258 21075 y Fq(f)130 b Fv(\()p Fq(\015)14080 21272 y Fr(k)14650 21075 y Fq(;)202 b(\015)15817 21272 y Fr(k)24 b FH(+1)17587 21075 y Fv(\))337 b Fp(\024)g Fq(C)20541 21257 y FH(1)21269 21075 y Fq(C)289 b(L)23249 20565 y Fk(\000)p FH(2+)p Fr(\017)23249 21467 y(\014)25619 20093 y Fm(\002)26124 21075 y Fq(g)43 b Fv(\()p Fq(z)27833 20575 y FH(2)28359 21075 y Fv(\()p Fq(\015)29458 21272 y Fr(k)30027 21075 y Fv(\)\))270 b Fp(\000)f Fq(g)43 b Fv(\()p Fq(z)34160 20575 y FH(2)34687 21075 y Fv(\()p Fq(\015)35786 21272 y Fr(k)24 b FH(+1)37557 21075 y Fv(\)\))38499 20093 y Fm(\003)39005 21075 y Fq(:)7960 b Fv(\(4.62\))1200 23200 y(Adding)405 b(up)g(the)f(v)-67 b(arious)404 b(terms,)g(w)-34 b(e)404 b(get)g(that)5256 25813 y Fq(I)95 b(I)g(I)432 b Fp(\024)9921 24993 y Fq(C)10787 25175 y FH(1)p 8889 25534 3456 49 v 8889 26645 a Fq(\026)9619 26842 y Fr(\014)10248 26645 y Fv(\()p Fq(\015)11347 26827 y FH(0)11873 26645 y Fv(\))12477 25813 y Fq(L)13302 25303 y Fk(\000)p FH(2+)p Fr(\017)13302 26205 y(\014)15672 25813 y Fq(e)16236 25313 y Fr(\014)46 b(U)17596 24468 y Fm(\020)18320 25813 y Fv(1)269 b(+)g([)p Fq(g)43 b Fv(\()p Fq(z)22453 25313 y FH(2)22980 25813 y Fv(\()p Fq(\020)24070 25313 y FH(2)24597 25813 y Fv(\)\))270 b Fp(\000)f Fq(g)43 b Fv(\()p Fq(z)28730 25313 y FH(2)29257 25813 y Fv(\()p Fq(\034)30395 25313 y FH(2)30921 25813 y Fv(\)])31729 24468 y Fm(\021)32790 25813 y Fp(\024)34931 24993 y Fv(2)p Fq(C)36403 25175 y FH(1)p 34202 25534 V 34202 26645 a Fq(\026)34932 26842 y Fr(\014)35561 26645 y Fv(\()p Fq(\015)36660 26827 y FH(0)37186 26645 y Fv(\))37790 25813 y Fq(L)38615 25303 y Fk(\000)p FH(2+)p Fr(\017)38615 26205 y(\014)40985 25813 y Fq(e)41549 25313 y Fr(\014)j(U)42909 25813 y Fq(:)4056 b Fv(\(4.63\))1200 28824 y(The)364 b(righ)-34 b(t-hand)364 b(side)f(is)g(smaller)f(than)j Fq(I)458 b Fv(b)-34 b(y)363 b(a)g(factor)h Fq(L)27930 28315 y Fk(\000)p FH(2+)p Fr(\017)27930 29216 y(\014)30299 28824 y Fq(e)30863 28385 y Fr(\014)46 b(U)32223 28824 y Fv(,)371 b(whic)-34 b(h,)372 b(b)-34 b(y)364 b(\(4.53\),)371 b(is)363 b(exp)34 b(onen)-34 b(tially)1200 30330 y(small)404 b(in)g Fq(\014)64 b Fv(.)1200 32532 y Fi(5.)602 b(Remaining)h (particles.)898 b Fv(The)524 b(lesson)g(from)g(the)g(previous)g(steps)g (is)f(that)i(w)-34 b(e)524 b(can)g(construct)h(a)1200 34038 y(\015o)-34 b(w)394 b(with)h(the)e(prop)34 b(ert)-34 b(y)394 b(that)g(eac)-34 b(h)394 b(time)f(w)-34 b(e)393 b(remo)-34 b(v)g(e)393 b(a)h(particle)e(from)i(the)g(droplet)f(w)-34 b(e)394 b(gain)f(a)h(factor)1200 35543 y Fq(L)2025 35033 y Fk(\000)p FH(2+)p Fr(\017)2025 35935 y(\014)4395 35543 y Fv(,)403 b(i.e.,)f(almost)h Fq(e)11899 35103 y Fk(\000)p FH(\001)p Fr(\014)14045 35543 y Fv(.)538 b(\(This)404 b(en)-34 b(trop)g(y)404 b(gain)g(corresp)34 b(onds)403 b(to)h(the)g(gain)f(from)h(the)f(magnetic)h(\014eld)1200 37049 y(in)302 b(Glaub)34 b(er)302 b(dynamics,)322 b(or)302 b(from)g(the)g(activit)-34 b(y)302 b(in)g(Ka)-34 b(w)g(asaki)302 b(dynamics)g(on)h(a)f(\014nite)g(op)34 b(en)302 b(b)34 b(o)-34 b(x.\))505 b(W)-101 b(e)302 b(can)1200 38554 y(con)-34 b(tin)g(ue)356 b(our)e(\015o)-34 b(w)355 b(b)-34 b(y)355 b(tearing)f(do)-34 b(wn)356 b(the)e(critical)f(droplet)i(in)f (the)g(same)g(order)g(as)h(w)-34 b(e)354 b(did)h(for)f(Glaub)34 b(er)1200 40060 y(dynamics.)634 b(Eac)-34 b(h)436 b(remo)-34 b(v)-67 b(al)435 b(corresp)34 b(onds)436 b(to)g(a)f(\015o)-34 b(w)437 b(that)g(is)f(built)g(in)g(the)g(same)g(w)-34 b(a)g(y)437 b(as)e(describ)34 b(ed)436 b(in)1200 41565 y(Step)412 b(4)f(for)g(the)g(second)g(particle.)559 b(There)410 b(will)h(b)34 b(e)410 b(some)h(minor)g(mo)34 b(di\014cations)412 b(in)-34 b(v)g(olving)411 b(a)g(negligible)1200 43070 y(fraction)402 b(of)g(paths)h(where)e(a)h(particle)e(hits)j(a)e (particle)g(that)i(w)-34 b(as)402 b(mo)-34 b(v)g(ed)402 b(out)g(earlier,)f(but)h(this)g(is)g(of)f(no)1200 44576 y(consequence.)706 b(As)460 b(a)h(result)f(of)g(the)h(construction,)475 b(the)460 b(sums)h(along)f(the)h(remainders)f(of)g(these)h(paths)1200 46081 y(will)404 b(giv)-34 b(e)403 b(only)h(negligible)f(con)-34 b(tributions.)3082 47819 y(Th)g(us,)511 b(w)-34 b(e)490 b(ha)-34 b(v)g(e)489 b(sho)-34 b(wn)491 b(that)f(the)g(lo)-34 b(w)g(er)489 b(b)34 b(ound)490 b(coincides,)510 b(up)490 b(to)f(a)g(factor)g(1)326 b(+)g Fq(o)p Fv(\(1\),)511 b(with)489 b(the)1200 49325 y(upp)34 b(er)404 b(b)34 b(ound)406 b(and)f(the)f(lemma)g(is)g(pro)-34 b(v)g(en.)p 49878 49325 522 731 v 1200 53173 a Ft(4.2)1495 b(Pro)42 b(of)498 b(of)g(Theorem)h(1.4\(b\))1200 55694 y Fv(The)405 b(same)f(observ)-67 b(ation)404 b(holds)g(as)h(in)f(\(3.34\).)1200 57432 y Fs(Pr)-62 b(o)g(of)p Fv(.)403 b(The)458 b(pro)34 b(of)459 b(of)f(Theorem)g(1.4\(b\))h(follo)-34 b(ws)458 b(along)g(the)h(same)f(lines)f(as)h(that)h(of)g(Theorem)f(1.4\(a\).) 1200 58937 y(The)345 b(main)g(p)34 b(oin)-34 b(t)345 b(is)g(to)g(pro)-34 b(v)g(e)345 b(that)h Fw(CAP)o Fv(\()p Fp(D)21677 59125 y Fr(M)22730 58937 y Fq(;)202 b Fp(S)24003 59125 y Fr(L)24698 58937 y Fv(\))337 b(=)g([1)150 b(+)g Fq(o)p Fv(\(1\)])p Fw(CAP)o Fv(\()p Fp(C)34829 58497 y FH(+)35617 58937 y Fq(;)202 b Fp(S)36890 59125 y Fr(L)37585 58937 y Fv(\).)519 b(Since)344 b Fw(CAP)o Fv(\()p Fp(S)45426 59125 y Fr(L)46122 58937 y Fq(;)202 b Fp(D)47596 59125 y Fr(M)48649 58937 y Fv(\))337 b Fp(\024)1200 60443 y Fw(CAP)o Fv(\()p Fp(S)4609 60631 y Fr(L)5305 60443 y Fq(;)202 b Fp(C)6553 60003 y FH(+)7339 60443 y Fv(\),)553 b(w)-34 b(e)522 b(need)h(only)f(pro)-34 b(v)g(e)523 b(a)g(lo)-34 b(w)g(er)522 b(b)34 b(ound)524 b(on)f Fw(CAP)o Fv(\()p Fp(D)33398 60631 y Fr(M)34452 60443 y Fq(;)202 b Fp(S)35725 60631 y Fr(L)36420 60443 y Fv(\).)894 b(This)523 b(is)f(done)h(in)f (almost)1200 61948 y(exactly)418 b(the)i(same)f(w)-34 b(a)g(y)420 b(as)f(for)h(Glaub)34 b(er,)422 b(b)-34 b(y)420 b(using)f(the)h(construction)g(giv)-34 b(en)419 b(there)g(and)h (substituting)1200 63454 y(eac)-34 b(h)404 b(Glaub)34 b(er)404 b(mo)-34 b(v)g(e)404 b(b)-34 b(y)405 b(a)f(\015o)-34 b(w)405 b(in)-34 b(v)g(olving)405 b(the)f(motion)h(of)f(just)i(t)-34 b(w)g(o)405 b(particles.)3082 65191 y(Note)316 b(that,)335 b(as)316 b(long)h(as)f Fq(M)469 b Fv(=)336 b Fq(e)17643 64751 y Fr(o)p FH(\()p Fr(\014)46 b FH(\))19459 65191 y Fv(,)333 b(an)317 b Fq(M)226 b Fp(\002)94 b Fq(M)447 b Fv(droplet)317 b(can)f(b)34 b(e)316 b(added)h(at)g Fp(j)p Fv(\003)39628 65388 y Fr(\014)40256 65191 y Fp(j)94 b(\000)g Fq(o)p Fv(\()p Fp(j)p Fv(\003)43962 65388 y Fr(\014)44589 65191 y Fp(j)p Fv(\))316 b(lo)34 b(cations)1200 66697 y(to)316 b(a)g(con\014guration)i Fq(\033)379 b Fp(2)337 b(S)407 b Fv(\(compare)316 b(with)h(\(4.36\)\).)510 b(The)316 b(only)g(no)-34 b(v)g(elt)g(y)317 b(is)e(that)j(w)-34 b(e)316 b(ha)-34 b(v)g(e)316 b(to)h(ev)-34 b(en)g(tually)1200 68202 y(remo)g(v)g(e)406 b(the)g(cloud)g(of)h(particles)e(that)i(is)f (pro)34 b(duced)406 b(in)g(the)h(ann)-34 b(ulus)408 b Fq(B)34965 68390 y FH(2)p Fr(L)36075 68546 y Fn(\014)36628 68390 y Fr(;)p FH(2)p Fr(L)37999 68546 y Fn(\014)38609 68202 y Fv(\(0\))p Fp(n)p Fq(B)41682 68390 y Fr(L)42322 68546 y Fn(\014)42876 68390 y Fr(;L)43777 68546 y Fn(\014)44386 68202 y Fv(\(0\).)545 b(This)406 b(is)1200 69890 y(done)488 b(in)g(m)-34 b(uc)g(h)489 b(the)g(same)e(w)-34 b(a)g(y)489 b(as)f(b)34 b(efore.)789 b(As)488 b(long)g(as)g(only)g Fq(e)32226 69450 y Fr(o)p FH(\()p Fr(\014)46 b FH(\))34530 69890 y Fv(particles)487 b(ha)-34 b(v)g(e)488 b(to)g(b)34 b(e)488 b(remo)-34 b(v)g(ed,)1200 71395 y(p)34 b(oten)-34 b(tial)405 b(collisions)e(b)34 b(et)-34 b(w)g(een)405 b(particles)f(can)g(b)34 b(e)404 b(ignored)g(as)g(they)h(are)e (su\016cien)-34 b(tly)404 b(unlik)-34 b(ely)-101 b(.)p 49878 71395 V 25194 75417 a(34)p eop end %%Page: 35 35 TeXDict begin 35 34 bop 1200 1313 a Fu(A)1793 b(App)50 b(endix:)799 b(sparseness)601 b(of)c(sub)50 b(critical)599 b(droplets)1200 4331 y Fv(Recall)403 b(De\014nition)i(1.1\(a\))g(and)f (\(3.11{3.12\).)539 b(In)405 b(this)f(section)g(w)-34 b(e)405 b(pro)-34 b(v)g(e)404 b(\(3.22\).)1200 7590 y Fi(Lemma)464 b(A.1)606 b Fv(lim)10570 7787 y Fr(\014)46 b Fk(!1)13467 7112 y FH(1)p 13416 7310 574 49 v 13416 8008 a Fr(\014)14324 7590 y Fv(log)16225 6929 y Fr(\026)16791 7085 y Fn(\014)17344 6929 y FH(\()p Fk(S)65 b(nW)74 b FH(\))p 16225 7311 3964 49 v 16964 8024 a Fr(\026)17530 8180 y Fn(\014)18083 8024 y FH(\()p Fk(S)65 b FH(\))20658 7590 y Fv(=)336 b Fp(\0001)p Fs(.)1200 10987 y(Pr)-62 b(o)g(of)p Fv(.)403 b(W)-101 b(e)399 b(will)f(pro)-34 b(v)g(e)400 b(that)g(lim)16698 11184 y Fr(\014)46 b Fk(!1)19595 10510 y FH(1)p 19543 10708 574 49 v 19543 11405 a Fr(\014)20452 10987 y Fv(log)218 b Fq(\026)22949 11184 y Fr(\014)23579 10987 y Fv(\()p Fp(S)91 b(nW)100 b Fv(\))p Fq(=\026)28585 11184 y Fr(\014)29215 10987 y Fv(\()p Fj(\014)p Fv(\))337 b(=)f Fp(\0001)p Fv(.)537 b(Since)399 b Fj(\014)336 b Fp(2)h(S)91 b Fv(,)400 b(this)g(will)f(pro)-34 b(v)g(e)1200 12493 y(the)404 b(claim.)3082 14310 y(Let)g Fq(f)130 b Fv(\()p Fq(\014)64 b Fv(\))405 b(b)34 b(e)404 b(the)g(function)i (satisfying)e(\(3.11\).)539 b(W)-101 b(e)404 b(b)34 b(egin)404 b(b)-34 b(y)404 b(noting)h(that)7149 16912 y Fq(\026)7879 17109 y Fr(\014)8508 16912 y Fv(\()p Fp(S)91 b(nW)100 b Fv(\))337 b Fp(\024)g Fq(\026)14525 17109 y Fr(\014)15154 16912 y Fv(\()p Fp(I)89 b Fv(\))1617 b(with)g Fp(I)427 b Fv(=)24801 15930 y Fm(\010)25508 16912 y Fq(\033)380 b Fp(2)337 b(S)225 b Fv(:)404 b Fp(j)p Fv(supp)q([)p Fq(C)33464 17100 y Fr(B)34273 16912 y Fv(\()p Fq(\033)43 b Fv(\)])p Fp(j)337 b Fq(>)f Fp(j)p Fv(\003)39420 17109 y Fr(\014)40048 16912 y Fp(j)p Fq(=f)130 b Fv(\()p Fq(\014)64 b Fv(\))43406 15930 y Fm(\011)44115 16912 y Fq(;)3153 b Fv(\(A.1\))1200 19515 y(b)34 b(ecause)410 b(the)g(b)34 b(o)g(otstrap)411 b(p)34 b(ercolation)410 b(map)h(increases)e(the)h(n) -34 b(um)g(b)34 b(er)412 b(of)e(\(+1\)-spins.)558 b(Let)410 b Fp(D)34 b Fv(\()p Fq(k)k Fv(\))410 b(denote)1200 21020 y(the)475 b(set)g(of)g(con\014gurations)h(whose)f(supp)34 b(ort)476 b(consists)g(on)f Fq(k)512 b Fv(non-in)-34 b(teracting)477 b(sub)34 b(critical)474 b(rectangles.)1200 22525 y(Put)366 b Fq(C)4401 22707 y FH(1)5264 22525 y Fv(=)337 b(\()p Fq(`)7520 22707 y Fr(c)8176 22525 y Fv(+)193 b(2\)\()p Fq(`)11365 22707 y Fr(c)12021 22525 y Fv(+)g(1\).)525 b(Since)366 b(the)g(union)h(of)f(a)g(sub)34 b(critical)365 b(rectangle)g(and)i(its)e(exterior)g(b)34 b(oundary)1200 24031 y(has)595 b(at)g(most)h Fq(C)9251 24213 y FH(1)10371 24031 y Fv(sites,)643 b(it)594 b(follo)-34 b(ws)596 b(that)g(in)e Fp(I)685 b Fv(there)594 b(are)g(at)i(least)e Fp(j)p Fv(\003)36375 24228 y Fr(\014)37004 24031 y Fp(j)p Fq(=C)38813 24213 y FH(1)39338 24031 y Fq(f)130 b Fv(\()p Fq(\014)64 b Fv(\))596 b(non-in)-34 b(teracting)1200 25536 y(rectangles.)538 b(Th)-34 b(us,)405 b(w)-34 b(e)405 b(ha)-34 b(v)g(e)9359 29224 y Fq(\026)10089 29421 y Fr(\014)10718 29224 y Fv(\()p Fp(I)89 b Fv(\))338 b Fp(\024)14939 27706 y Fr(K)15732 27817 y Fx(max)15237 28073 y Fm(X)14027 31234 y Fr(k)24 b FH(=)15930 30591 y Fl(j)p Fx(\003)16734 30823 y Fn(\014)17288 30591 y Fl(j)p 15405 31032 2661 40 v 15405 31581 a Fn(C)15992 31780 y Fx(1)16453 31581 y Fn(f)76 b Fx(\()p Fn(\014)34 b Fx(\))18400 29224 y Fq(F)168 b Fv(\()p Fq(k)38 b Fv(\))1618 b(with)f Fq(F)168 b Fv(\()p Fq(k)38 b Fv(\))339 b(=)31283 28404 y(1)p 30858 28945 1457 49 v 30858 30056 a Fq(Z)31685 30253 y Fr(\014)33946 28073 y Fm(X)33419 30547 y Fn(\033)24 b Fl(2X)35059 30779 y Fn(\014)35706 30547 y Fx(:)32782 31526 y Fn(C)46 b Fx(\()p Fn(\033)24 b Fx(\))p Fl(2D)e Fx(\()p Fn(k)14 b Fx(\))37195 29224 y Fq(e)37759 28724 y Fk(\000)p Fr(\014)202 b(H)39994 28880 y Fn(\014)40547 28724 y FH(\()p Fr(\033)32 b FH(\))41904 29224 y Fq(;)5364 b Fv(\(A.2\))1200 34018 y(where)404 b Fq(K)5734 34200 y FH(max)7878 34018 y Fp(\024)336 b(j)p Fv(\003)10336 34215 y Fr(\014)10964 34018 y Fp(j)p Fv(.)3082 35836 y(Next,)403 b(note)i(that)16752 37783 y Fq(F)168 b Fv(\()p Fq(k)38 b Fv(\))338 b Fp(\024)f Fv(\(2)22005 37283 y Fr(C)22676 37406 y Fx(1)23193 37783 y Fv(\))23664 37283 y Fr(k)24993 36963 y Fv(1)p 24568 37504 V 24568 38615 a Fq(Z)25395 38812 y Fr(\014)27081 36632 y Fm(X)26359 39316 y Fr(\033)32 b Fk(2D)26 b FH(\()p Fr(k)e FH(\))29959 37783 y Fq(e)30523 37283 y Fk(\000)p Fr(\014)46 b(H)32602 37439 y Fn(\014)33154 37283 y FH(\()p Fr(\033)32 b FH(\))34511 37783 y Fq(:)12757 b Fv(\(A.3\))1200 41270 y(Since)610 b(the)g(b)34 b(o)g(otstrap)610 b(p)34 b(ercolation)610 b(map)g(is)g(do)-34 b(wnhill,)662 b(the)610 b(energy)f(of)h(a)g(sub)34 b(critical)609 b(rectangle)g(is)1200 42776 y(b)34 b(ounded)408 b(b)34 b(elo)-34 b(w)407 b(b)-34 b(y)408 b Fq(C)12142 42958 y FH(2)13009 42776 y Fv(=)341 b(2)p Fq(J)388 b Fp(\000)271 b Fq(h)407 b Fv(\(recall)f(Fig.)h(9\),)h(and)f(the)h(n)-34 b(um)g(b)34 b(er)407 b(of)h(w)-34 b(a)g(ys)408 b(to)f(place)f Fq(k)445 b Fv(rectangles)1200 44436 y(in)404 b(\003)3456 44633 y Fr(\014)4489 44436 y Fv(is)g(at)g(most)10159 43454 y Fm(\000)10714 43853 y Fk(j)p FH(\003)11628 44009 y Fn(\014)12181 43853 y Fk(j)11322 44899 y Fr(k)12443 43454 y Fm(\001)12998 44436 y Fv(,)g(it)g(follo)-34 b(ws)405 b(that)1860 47864 y Fq(F)168 b Fv(\()p Fq(k)38 b Fv(\))339 b Fp(\024)d Fv(2)6642 47364 y Fr(C)7313 47487 y Fx(1)7774 47364 y Fr(k)8545 46155 y Fm(\022)9437 47044 y Fp(j)p Fv(\003)10616 47241 y Fr(\014)11244 47044 y Fp(j)10175 48696 y Fq(k)11581 46155 y Fm(\023)12675 47864 y Fq(\026)13405 48061 y Fr(\014)14035 47864 y Fv(\()p Fj(\014)p Fv(\))202 b Fq(e)16686 47364 y Fk(\000)p Fr(C)18089 47487 y Fx(2)18550 47364 y Fr(\014)46 b(k)20029 47864 y Fp(\024)337 b Fv(2)21915 47364 y Fr(C)22586 47487 y Fx(1)23047 47364 y Fr(k)23817 47864 y Fv(\()p Fq(C)25154 48046 y FH(1)25681 47864 y Fq(ef)130 b Fv(\()p Fq(\014)64 b Fv(\)\))29131 47364 y Fr(k)29904 47864 y Fq(\026)30634 48061 y Fr(\014)31263 47864 y Fv(\()p Fj(\014)p Fv(\))202 b Fq(e)33914 47364 y Fk(\000)p Fr(C)35317 47487 y Fx(2)35779 47364 y Fr(\014)46 b(k)37258 47864 y Fp(\024)336 b Fq(\026)39267 48061 y Fr(\014)39896 47864 y Fv(\()p Fj(\014)p Fv(\))405 b(exp[)p Fp(\000)45451 47387 y FH(1)p 45451 47585 471 49 v 45451 48282 a(2)46053 47864 y Fq(C)46919 48046 y FH(2)47648 47864 y Fq(\014)64 b(k)38 b Fv(])p Fq(;)47605 50036 y Fv(\(A.4\))1200 51541 y(where)441 b(the)g(second)h(inequalit)-34 b(y)441 b(uses)g(that)h Fq(k)38 b Fv(!)398 b Fp(\025)g Fq(k)25162 51101 y Fr(k)25731 51541 y Fq(e)26295 51101 y Fk(\000)p Fr(k)27597 51541 y Fv(,)449 b Fq(k)437 b Fp(2)398 b Fo(N)p Fv(,)451 b(and)441 b(the)h(third)g(inequalit)-34 b(y)440 b(uses)i(that)1200 53047 y Fq(f)130 b Fv(\()p Fq(\014)64 b Fv(\))338 b(=)e Fq(e)5796 52607 y Fr(o)p FH(\()p Fr(\014)46 b FH(\))7612 53047 y Fv(.)539 b(W)-101 b(e)403 b(th)-34 b(us)406 b(ha)-34 b(v)g(e)11734 55284 y Fr(K)12527 55395 y Fx(max)12032 55650 y Fm(X)10821 58811 y Fr(k)24 b FH(=)12725 58169 y Fl(j)p Fx(\003)13529 58401 y Fn(\014)14083 58169 y Fl(j)p 12200 58610 2661 40 v 12200 59159 a Fn(C)12787 59358 y Fx(1)13248 59159 y Fn(f)76 b Fx(\()p Fn(\014)34 b Fx(\))15397 56802 y Fq(F)168 b Fv(\()p Fq(k)38 b Fv(\))743 b Fp(\024)336 b Fv(2)p Fq(\026)21313 56999 y Fr(\014)21942 56802 y Fv(\()p Fj(\014)p Fv(\))202 b Fq(f)130 b Fv(\()p Fq(\014)64 b Fv(\))26917 55959 y Fp(j)p Fv(\003)28096 56156 y Fr(\014)28724 55959 y Fp(j)p 26781 56523 2417 49 v 26781 57633 a Fq(f)130 b Fv(\()p Fq(\014)64 b Fv(\))29734 56802 y(exp)31788 55093 y Fm(\024)32428 56802 y Fp(\000)33504 56324 y FH(1)p 33504 56522 471 49 v 33504 57220 a(2)34239 55982 y Fq(C)35105 56164 y FH(2)p 34239 56523 1393 49 v 34239 57633 a Fq(C)35105 57815 y FH(1)35966 56802 y Fq(\014)37187 55959 y Fp(j)p Fv(\003)38366 56156 y Fr(\014)38994 55959 y Fp(j)p 37051 56523 2417 49 v 37051 57633 a Fq(f)130 b Fv(\()p Fq(\014)64 b Fv(\))39600 55093 y Fm(\025)40442 56802 y Fq(;)6826 b Fv(\(A.5\))1200 61596 y(whic)-34 b(h)405 b(is)f(the)g(desired)g (estimate)g(b)34 b(ecause)404 b Fp(j)p Fv(\003)22611 61793 y Fr(\014)23240 61596 y Fp(j)p Fq(=f)130 b Fv(\()p Fq(\014)64 b Fv(\))405 b(tends)g(to)f(in\014nit)-34 b(y)-101 b(.)p 49878 61596 522 731 v 1200 66332 a Fu(B)1793 b(App)50 b(endix:)789 b(equiv)-100 b(alence)580 b(of)e(ensem)-50 b(bles)580 b(and)f(t)-50 b(ypicalit)g(y)580 b(of)d(holes)1200 69349 y Fv(F)-101 b(or)404 b Fq(m)337 b Fp(2)g Fo(N)p Fv(,)404 b(let)7694 72096 y Fp(S)8519 71595 y FH(\()p Fr(m)p FH(\))10476 72096 y Fv(=)11755 71114 y Fm(\010)12462 72096 y Fq(\033)380 b Fp(2)337 b(X)15722 71457 y FH(\()p Fr(m)p FH(\))15545 72487 y Fr(\014)17476 72096 y Fv(:)606 b Fp(j)p Fv(supp)q([)p Fq(\033)43 b Fv(])268 b Fp(\\)i Fq(B)24929 72284 y Fr(L)25569 72440 y Fn(\014)26122 72284 y Fr(;L)27023 72440 y Fn(\014)27632 72096 y Fv(\()p Fq(x)p Fv(\))p Fp(j)337 b(\024)f Fq(`)31725 72278 y Fr(c)32188 72096 y Fv(\()p Fq(`)33164 72278 y Fr(c)33897 72096 y Fp(\000)269 b Fv(1\))h(+)f(1)404 b Fp(8)202 b Fq(x)337 b Fp(2)g Fv(\003)42570 72293 y Fr(\014)43198 71114 y Fm(\011)47656 72096 y Fv(\(B.1\))25194 75417 y(35)p eop end %%Page: 36 36 TeXDict begin 36 35 bop 1200 1313 a Fv(and)16262 2282 y(\024)15987 2589 y Fq(C)16940 2088 y FH(\()p Fr(m)p FH(\))18560 2589 y Fv(\(0\))337 b(=)21725 1607 y Fm(\010)22432 2589 y Fq(\033)23899 2835 y Fk(2)p Fr(B)25236 2984 y Fn(L)25798 3217 y(\014)26351 2984 y(;L)27165 3217 y(\014)27774 2835 y FH(\(0\))29166 2589 y Fv(:)606 b Fq(\033)379 b Fp(2)337 b(S)33151 2088 y FH(\()p Fr(m)p FH(\))34771 1607 y Fm(\011)35478 2589 y Fq(;)16281 4785 y Fv(\024)16026 5092 y Fq(Z)16940 4453 y FH(\()p Fr(m)p FH(\))16853 5483 y Fr(\014)18560 5092 y Fv(\(0\))g(=)23126 3940 y Fm(X)21725 6741 y Fr(\033)32 b Fk(2)23131 6517 y FH(\024)22922 6741 y Fr(C)23656 6479 y Fx(\()p Fn(m)p Fx(\))25076 6741 y FH(\(0\))26481 5092 y Fq(e)27045 4591 y Fk(\000)p Fr(\014)202 b(H)68 b FH(\()p Fr(\033)32 b FH(\))30705 5092 y Fq(:)47656 4557 y Fv(\(B.2\))1200 8823 y(The)375 b(latter)f(is)g(the)h(partition)g (sum)g(restricted)f(to)h Fq(B)25484 9011 y Fr(L)26124 9167 y Fn(\014)26677 9011 y Fr(;L)27578 9167 y Fn(\014)28187 8823 y Fv(\(0\))g(when)h(it)e(carries)f Fq(m)i Fv(particles.)528 b(In)374 b(App)34 b(en-)1200 10328 y(dix)349 b(B.1)f(w)-34 b(e)349 b(pro)-34 b(v)g(e)349 b(a)g(lemma)g(ab)34 b(out)350 b(ratios)f(of)g(partition)h(sums)f(that)h(w)-34 b(as)350 b(used)g(in\(4.22\),)360 b(\(4.26\),)g(\(4.32\))1200 11834 y(and)428 b(\(4.49\).)609 b(In)427 b(App)34 b(endix)428 b(B.2)f(w)-34 b(e)427 b(pro)-34 b(v)g(e)428 b(that)h(lim)26402 12031 y Fr(\014)46 b Fk(!1)29115 11834 y Fq(\026)29845 12031 y Fr(\014)30474 11834 y Fv(\()31223 11527 y(\024)30945 11834 y Fp(S)91 b Fv(\(0\)\))p Fq(=\026)35125 12031 y Fr(\014)35756 11834 y Fv(\()p Fp(S)g Fv(\))376 b(=)f(1,)433 b(whic)-34 b(h)428 b(is)f(needed)h(in)1200 13339 y(the)404 b(pro)34 b(of)405 b(of)f(this)h(lemma.)1200 16917 y Ft(B.1)1495 b(Equiv)-83 b(alence)498 b(of)g(ensem)-42 b(bles)1200 19537 y Fv(Recall)403 b(\(1.22\),)i(\(4.6\))f(and)h(\(4.17\).)1200 22979 y Fi(Lemma)464 b(B.1)9080 22672 y Fv(\024)8824 22979 y Fq(Z)9738 22276 y FH(\()p Fr(n)10675 22432 y Fn(\014)11229 22276 y Fk(\000)p Fr(s)p FH(\))9651 23370 y Fr(\014)12817 22979 y Fv(\(0\))p Fq(=)-67 b(Z)15818 22276 y FH(\()p Fr(n)16755 22432 y Fn(\014)17308 22276 y FH(\))15731 23370 y Fr(\014)18066 22979 y Fv(=)337 b(\()p Fq(\032)20444 23176 y Fr(\014)21073 22979 y Fv(\))21544 22539 y Fr(s)22237 22979 y Fq(\026)22967 23176 y Fr(\014)23596 22979 y Fv(\()p Fp(S)91 b Fv(\))202 b([1)270 b(+)f Fq(o)p Fv(\(1\)])433 b Fs(as)g Fq(\014)400 b Fp(!)337 b(1)434 b Fs(for)f(al)62 b(l)433 b Fq(s)337 b Fp(2)f Fo(N)p Fs(.)1200 26145 y(Pr)-62 b(o)g(of)p Fv(.)403 b(The)i(pro)34 b(of)404 b(pro)34 b(ceeds)403 b(via)h(upp)34 b(er)404 b(and)h(lo)-34 b(w)g(er)405 b(b)34 b(ounds.)1200 28646 y(Upp)g(er)404 b(b)34 b(ound:)p 1200 29076 7378 49 v 405 w(Let)14796 29997 y(\024)14518 30303 y Fp(S)91 b Fv(\(0\))337 b(=)18508 29321 y Fm(\010)19215 30303 y Fq(\033)380 b Fp(2)337 b(S)281 b Fv(:)605 b(supp)r([)p Fq(\033)43 b Fv(])268 b Fp(\\)i Fq(B)29564 30491 y Fr(L)30204 30647 y Fn(\014)30757 30491 y Fr(;L)31658 30647 y Fn(\014)32267 30303 y Fv(\(0\))338 b(=)e Fp(;)36038 29321 y Fm(\011)36745 30303 y Fq(:)10574 b Fv(\(B.3\))1200 32528 y(W)-101 b(rite)4893 35834 y Fq(\026)5623 36031 y Fr(\014)6253 35834 y Fv(\()7002 35528 y(\024)6724 35834 y Fp(S)91 b Fv(\(0\)\))338 b(=)12428 35014 y(1)p 11318 35556 2826 49 v 11318 37068 a Fq(Z)12232 36365 y FH(\()p Fr(n)13169 36521 y Fn(\014)13722 36365 y FH(\))12145 37459 y Fr(\014)15075 34683 y Fm(X)14528 37483 y FH(\024)-520 b Fr(\033)32 b Fk(2)15843 37260 y FH(\024)15675 37483 y Fk(C)48 b FH(\(0\))21489 34683 y Fm(X)17758 37201 y Fn(\020)40 b Fl(\032)p Fx([)p Fn(B)19667 37426 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42436 y Fq(n)1928 42633 y Fr(\014)2869 42436 y Fp(\000)312 b Fq(s)469 b Fv(particles)f(and)h (then)h(placing)e(another)i Fq(s)f Fv(particles.)731 b(Because)468 b(the)h(in)-34 b(teraction)470 b(is)e(attractiv)-34 b(e,)1200 43941 y(w)g(e)404 b(ha)-34 b(v)g(e)11227 45447 y Fq(H)12235 45644 y Fr(\014)12864 45447 y Fv(\()65 b(\024)-671 b Fq(\033)313 b Fp(_)269 b Fq(\020)89 b Fv(\))338 b Fp(\024)e Fq(H)19133 45644 y Fr(\014)19762 45447 y Fv(\()65 b(\024)-671 b Fq(\033)43 b Fv(\))270 b(+)f Fq(H)23930 45644 y Fr(\014)24558 45447 y Fv(\()p Fq(\020)89 b Fv(\))406 b(and)f Fq(H)29890 45644 y Fr(\014)30518 45447 y Fv(\()p Fq(\020)89 b Fv(\))338 b Fp(\024)f Fv(0)p Fq(;)2626 b Fp(8)267 b Fv(\024)-671 b Fq(\033)43 b(;)202 b(\020)89 b(:)7284 b Fv(\(B.5\))1200 47672 y(Consequen)-34 b(tly)-101 b(,)6816 51213 y Fq(\026)7546 51410 y Fr(\014)8175 51213 y Fv(\()8925 50907 y(\024)8646 51213 y Fp(S)92 b Fv(\(0\)\))338 b Fp(\025)13108 49504 y Fm(\022)14000 50393 y Fq(n)14728 50590 y Fr(\014)14394 52045 y Fq(s)15357 49504 y Fm(\023)16249 49806 y Fk(\000)p FH(1)18951 50393 y Fv(1)p 17841 50935 V 17841 52447 a Fq(Z)18755 51744 y FH(\()p Fr(n)19692 51900 y Fn(\014)20246 51744 y FH(\))18668 52839 y Fr(\014)21599 50062 y Fm(X)21051 52863 y FH(\024)-519 b Fr(\033)31 b Fk(2)22367 52639 y FH(\024)22198 52863 y Fk(C)48 b FH(\(0\))24148 51213 y Fq(e)24712 50713 y Fk(\000)p Fr(\014)e(H)26791 50869 y Fn(\014)27344 50713 y FH(\()k(\024)-520 b Fr(\033)31 b FH(\))32767 50062 y Fm(X)29036 52580 y Fn(\020)40 b Fl(\032)p Fx([)p Fn(B)30945 52805 y(L)31507 53038 y(\014)32059 52805 y(;L)32873 53038 y(\014)33482 52580 y Fx(\(0\)])34762 52389 y Fn(c)35187 52580 y Fl(n)p Fx(supp)q([)47 b(\024)-453 b Fn(\033)23 b Fx(])32676 53785 y Fl(j)p Fn(\020)40 b Fl(j)p Fx(=)p Fn(s)39316 51576 y Fk(f)50 b FH(\024)-520 b Fr(\033)31 b Fk(_)p Fr(\020)61 b Fk(2)42297 51353 y FH(\024)42084 51576 y Fk(S)k FH(\(0\))p Fk(g)44447 51213 y Fq(:)2872 b Fv(\(B.6\))3082 56574 y(W)-101 b(e)395 b(next)h(estimate)h(the)f(second)g(sum,)i(uniformly)e(in)461 b(\024)-671 b Fq(\033)44 b Fv(.)535 b(When)397 b(w)-34 b(e)396 b(add)h(the)f Fq(s)g Fv(particles,)h(w)-34 b(e)397 b(m)-34 b(ust)1200 58080 y(mak)g(e)466 b(sure)g(not)h(to)f(violate)g (the)g(requiremen)-34 b(t)466 b(that)h(all)f(b)34 b(o)-34 b(xes)466 b Fq(B)32527 58268 y Fr(L)33167 58424 y Fn(\014)33720 58268 y Fr(;L)34621 58424 y Fn(\014)35230 58080 y Fv(\()p Fq(x)p Fv(\),)482 b Fq(x)439 b Fp(2)i Fv(\003)40907 58277 y Fr(\014)41535 58080 y Fv(,)481 b(carry)465 b(not)i(more)1200 59731 y(than)460 b Fq(K)545 b Fv(particles)458 b(\(note)h(that)16748 59425 y(\024)16470 59731 y Fp(S)91 b Fv(\(0\))428 b Fp(\032)f(S)549 b Fv(and)459 b(recall)e(De\014nition)i(1.3\(a\)\).)702 b(P)-34 b(artition)459 b(\003)43994 59928 y Fr(\014)44623 59731 y Fp(n)p Fq(B)46148 59919 y Fr(L)46788 60075 y Fn(\014)47341 59919 y Fr(;L)48242 60075 y Fn(\014)48851 59731 y Fv(\(0\))1200 61237 y(in)-34 b(to)489 b(b)34 b(o)-34 b(xes)489 b(of)f(size)g Fq(L)11837 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Fk(g)17841 67407 y Fp(\025)19120 65698 y Fm(\022)20013 66587 y Fp(j)p Fv(\003)21192 66784 y Fr(\014)21820 66587 y Fp(n)p Fq(B)23345 66775 y Fr(L)23985 66931 y Fn(\014)24538 66775 y Fr(;L)25439 66931 y Fn(\014)26048 66587 y Fv(\(0\))p Fp(j)270 b(\000)f Fq(n)30143 66784 y Fr(\014)31041 66587 y Fp(\000)g Fv(\()p Fq(n)33452 66784 y Fr(\014)34081 66587 y Fq(=K)87 b Fv(\)\(3)p Fq(L)38176 66784 y Fr(\014)38805 66587 y Fv(\))39276 66147 y FH(2)29624 68238 y Fq(s)39803 65698 y Fm(\023)40695 67407 y Fq(;)2626 b Fp(8)267 b Fv(\024)-671 b Fq(\033)43 b(:)2050 b Fv(\(B.7\))25194 75417 y(36)p eop end %%Page: 37 37 TeXDict begin 37 36 bop 1200 1313 a Fv(But)437 b Fq(n)4368 1510 y Fr(\014)4997 1313 y Fq(L)5822 874 y FH(2)5822 1682 y Fr(\014)6842 1313 y Fv(=)391 b Fq(o)p Fv(\()p Fq(n)9963 1510 y Fr(\014)10591 1313 y Fq(=\032)11824 1510 y Fr(\014)12453 1313 y Fv(\))h(=)f Fq(o)p Fv(\()p Fp(j)p Fv(\003)16888 1510 y Fr(\014)17516 1313 y Fp(j)p Fv(\))437 b(and)h Fq(L)21976 874 y FH(2)21976 1682 y Fr(\014)22997 1313 y Fv(=)391 b Fq(o)p Fv(\(1)p Fq(=\032)27229 1510 y Fr(\014)27857 1313 y Fv(\))h(=)g Fq(o)p Fv(\()p Fp(j)p Fv(\003)32293 1510 y Fr(\014)32921 1313 y Fp(j)p Fv(\))437 b(b)-34 b(y)437 b(\(1.22\))h(and)g(\(1.29{1.30\),)445 b(and)1200 2819 y(so)502 b(the)g(righ)-34 b(t-hand)503 b(side)f(of)g(\(B.7\))g(equals)f([1)334 b(+)h Fq(o)p Fv(\(1\)])202 b Fp(j)p Fv(\003)28467 3016 y Fr(\014)29095 2819 y Fp(j)29432 2379 y Fr(s)29922 2819 y Fq(=s)p Fv(!)502 b(as)g Fq(\014)562 b Fp(!)500 b(1)p Fv(.)831 b(Since)502 b(the)g(binomial)g(in)1200 4324 y(\(B.6\))404 b(equals)g([1)269 b(+)g Fq(o)p Fv(\(1\)])202 b(\()p Fq(n)14323 4521 y Fr(\014)14952 4324 y Fv(\))15423 3884 y Fr(s)15914 4324 y Fq(=s)p Fv(!)404 b(with)h Fq(n)21318 4521 y Fr(\014)22283 4324 y Fv(=)336 b Fp(d)p Fq(\032)24728 4521 y Fr(\014)25356 4324 y Fp(j)p Fv(\003)26535 4521 y Fr(\014)27164 4324 y Fp(je)p Fv(,)403 b(w)-34 b(e)404 b(end)h(up)g(with)g(\(recall)e(\(4.17\)\))15221 8426 y Fq(\026)15951 8623 y Fr(\014)16581 8426 y Fv(\()17330 8119 y(\024)17052 8426 y Fp(S)91 b Fv(\(0\)\))338 b Fp(\025)21901 7083 y Fv(\024)21646 7389 y Fq(Z)22560 6687 y FH(\()p Fr(n)23497 6843 y Fn(\014)24050 6687 y Fk(\000)p Fr(s)p FH(\))22473 7781 y Fr(\014)25638 7389 y Fv(\(0\))p 21646 8147 5542 49 v 23004 9659 a Fq(Z)23918 8957 y FH(\()p Fr(n)24855 9113 y Fn(\014)25408 8957 y FH(\))23831 10051 y Fr(\014)27522 8426 y Fv(\()p Fq(\032)28620 8623 y Fr(\014)29249 8426 y Fv(\))29720 7925 y Fk(\000)p Fr(s)31145 8426 y Fv([1)269 b(+)g Fq(o)p Fv(\(1\)])p Fq(;)11277 b Fv(\(B.8\))1200 12073 y(or)15975 13245 y(\024)15720 13551 y Fq(Z)16634 12848 y FH(\()p Fr(n)17571 13004 y Fn(\014)18124 12848 y Fk(\000)p Fr(s)p FH(\))16547 13942 y Fr(\014)19712 13551 y Fv(\(0\))p 15720 14309 V 17078 15821 a Fq(Z)17992 15118 y FH(\()p Fr(n)18929 15274 y Fn(\014)19482 15118 y FH(\))17905 16212 y Fr(\014)21731 14587 y Fp(\024)336 b Fv(\()p Fq(\032)24108 14784 y Fr(\014)24737 14587 y Fv(\))25208 14087 y Fr(s)25901 14587 y Fq(\026)26631 14784 y Fr(\014)27260 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33639 y(f)50 b FH(\024)-520 b Fr(\033)32 b Fk(_)p Fr(\020)61 b Fk(2)46362 33415 y FH(\024)46150 33639 y Fk(S)j FH(\(0\))p Fk(g)3289 38980 y Fp(\024)337 b Fv([1)269 b(+)g Fq(o)p Fv(\(1\)])10056 37638 y(\024)9801 37944 y Fq(Z)10715 37241 y FH(\()p Fr(n)11652 37397 y Fn(\014)12205 37241 y Fk(\000)p Fr(s)p FH(\))10628 38335 y Fr(\014)13793 37944 y Fv(\(0\))p 9801 38702 5542 49 v 11159 40214 a Fq(Z)12073 39511 y FH(\()p Fr(n)13010 39667 y Fn(\014)13563 39511 y FH(\))11986 40606 y Fr(\014)15677 38980 y Fv(\()p Fq(\032)16775 39177 y Fr(\014)17404 38980 y Fv(\))17875 38480 y Fk(\000)p Fr(s)8071 43198 y Fv(+)10525 42378 y(1)p 9416 42919 2826 49 v 9416 44432 a Fq(Z)10330 43729 y FH(\()p Fr(n)11267 43885 y Fn(\014)11820 43729 y FH(\))10243 44823 y Fr(\014)13173 42047 y Fm(X)12625 44847 y FH(\024)-519 b Fr(\033)31 b Fk(2)13941 44623 y FH(\024)13772 44847 y Fk(C)48 b FH(\(0\))15722 41489 y Fm(\022)16615 42378 y Fq(n)17343 42575 y Fr(\014)17009 44030 y Fq(s)17971 41489 y Fm(\023)18863 41791 y Fk(\000)p FH(1)20525 43198 y Fq(e)21089 42698 y Fk(\000)p Fr(\014)e(H)23168 42854 y Fn(\014)23721 42698 y FH(\()k(\024)-520 b Fr(\033)31 b FH(\))26733 42047 y Fm(X)26633 44353 y Fn(s)27021 44552 y Fx(1)27483 44353 y Fn(;s)28123 44552 y Fx(2)25412 45138 y(1)p Fl(\024)p Fn(s)26846 45337 y Fx(1)27307 45138 y(+)p Fn(s)28316 45337 y Fx(2)28778 45138 y Fl(\024)p Fn(s)8071 47174 y Fp(\002)13147 46022 y Fm(X)9416 48541 y Fn(\020)40 b Fl(\032)p Fx([)p Fn(B)11325 48766 y(L)11887 48998 y(\014)12439 48766 y(;L)13253 48998 y(\014)13862 48541 y Fx(\(0\)])15142 48350 y Fn(c)15567 48541 y Fl(n)p Fx(supp)q([)47 b(\024)-453 b Fn(\033)23 b Fx(])13056 49745 y Fl(j)p Fn(\020)40 b Fl(j)p Fx(=)p Fn(s)18965 47174 y Fq(e)19529 46674 y Fk(\000)p Fr(\014)46 b(H)68 b FH(\()p Fr(\020)61 b FH(\))23870 47421 y Fk(fj)p Fr(\020)g Fk(\\)p Fr(@)102 b FH(\024)-520 b Fr(\033)32 b Fk(j)p FH(=)p Fr(s)28249 47544 y Fx(1)28710 47421 y Fk(g)989 b(f)p Fr(s)31074 47544 y Fx(2)31849 47421 y FH(in)-26 b(teracting)315 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y Fl(n)p Fx(supp[)47 b(\024)-453 b Fn(\033)23 b Fx(])8384 61376 y Fl(j)p Fn(\020)40 b Fl(j)p Fx(=)p Fn(s)14495 58805 y Fq(e)15059 58304 y Fk(\000)p Fr(\014)46 b(H)68 b FH(\()p Fr(\020)61 b FH(\))19400 59052 y Fk(fj)p Fr(\020)g Fk(\\)p Fr(@)102 b FH(\024)-520 b Fr(\033)32 b Fk(j)p FH(=)p Fr(s)23779 59175 y Fx(1)24240 59052 y Fk(g)989 b(f)p Fr(s)26604 59175 y Fx(2)27379 59052 y FH(in)-26 b(teracting)315 b(particles)g(in)e Fr(\020)61 b Fk(g)38911 59168 y(f)50 b FH(\024)-520 b Fr(\033)32 b Fk(_)p Fr(\020)61 b Fk(2)41893 58944 y FH(\024)41680 59168 y Fk(S)k FH(\(0\))p Fk(g)9594 63379 y Fp(\024)337 b(j)p Fv(\003)12053 63576 y Fr(\014)12681 63379 y Fp(j)13018 62878 y Fr(s)p Fk(\000)p Fr(s)14620 63001 y Fx(1)15081 62878 y Fk(\000)p Fr(s)16248 63001 y Fx(2)16966 63379 y Fv(\(4)p Fq(n)18771 63576 y Fr(\014)19400 63379 y Fv(\))19871 62878 y Fr(s)20306 63001 y Fx(1)22015 62227 y Fm(X)21227 65040 y Fr(\033)21765 65151 y Fl(2)22374 65040 y Fk(S)23009 64746 y Fx(\()p Fn(s)23717 64945 y Fx(2)24178 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b(with)f(the)h Fq(n)47573 69336 y Fr(\014)48545 69139 y Fp(\000)344 b Fq(s)25194 75417 y Fv(37)p eop end %%Page: 38 38 TeXDict begin 38 37 bop 1200 1313 a Fv(particles)404 b(already)f(presen)-34 b(t.)539 b(Next,)404 b(w)-34 b(e)404 b(write)12461 2461 y Fm(X)11674 5274 y Fr(\033)12212 5385 y Fl(2)12821 5274 y Fk(S)13456 4979 y Fx(\()p Fn(s)14164 5178 y Fx(2)14624 4979 y(\))15202 3613 y Fq(e)15766 3112 y Fk(\000)p Fr(\014)46 b(H)68 b FH(\()p Fr(\033)32 b FH(\))20202 3859 y Fk(f)p Fr(s)21107 3982 y Fx(2)21882 3859 y FH(in)-26 b(teracting)315 b(particles)f(in)g Fr(\033)32 b Fk(g)16657 8040 y Fv(=)18506 6510 y Fr(s)18941 6633 y Fx(2)18079 6889 y Fm(X)17937 9479 y Fr(m)p FH(=1)20633 6525 y Fr(m)20173 6889 y Fm(X)20232 9494 y Fr(j)51 b FH(=1)24106 6889 y Fm(X)22259 9375 y Fx(2)p Fl(\024)p Fn(k)23744 9574 y Fx(1)24205 9375 y Fn(;:::;k)25904 9586 y(j)26338 9375 y Fl(\024)p Fn(K)22754 10135 y Fb(P)23552 10275 y Fn(j)23552 10973 y(i)p Fx(=1)25071 10633 y Fn(k)25510 10844 y(i)25862 10633 y Fx(=)p 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Fb(P)20460 42951 y Fn(j)20460 43649 y(i)p Fx(=1)21980 43309 y Fn(k)22419 43520 y(i)22770 43309 y Fx(=)p Fn(s)23779 43508 y Fx(2)25009 40716 y Fq(e)25573 40216 y Fk(\000)p Fr(\014)27192 39552 y Fa(P)28186 39782 y Fn(j)28186 40510 y(i)p Fx(=1)29564 40216 y FH([)p Fr(H)30598 40372 y Fn(k)31037 40583 y(i)31444 40216 y FH(+\()p Fr(k)33032 40351 y Fn(i)33383 40216 y Fk(\000)p FH(1\)\001])36052 40716 y Fq(;)47050 31357 y Fv(\(B.13\))1200 45572 y(where)383 b(in)h(the)g(last)f(inequalit)-34 b(y)384 b(w)-34 b(e)384 b(insert)f(the)h(b)34 b(ound)385 b Fq(\014)27677 45132 y Fk(\000)p FH(1)29136 45572 y Fv(log)219 b Fp(j)p Fv(\003)32083 45769 y Fr(\014)32711 45572 y Fp(j)336 b(\025)h Fv(\001,)387 b(whic)-34 b(h)385 b(is)e(a)g(immediate)h(from)1200 47078 y(\(1.22\))405 b(and)g(\(1.35\).)3082 48825 y(No)-34 b(w,)349 b Fq(H)7132 49022 y Fr(k)7622 49157 y Fn(i)8160 48825 y Fv(+)131 b Fq(k)9865 49007 y Fr(i)10241 48825 y Fv(\001)335 b(is)g(the)h Fs(gr)-62 b(and-c)g(anonic)g(al)457 b Fv(energy)334 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65300 y Fk(\000)p Fr(s)12511 70444 y Fp(\024)337 b Fq(C)289 b Fv([1)269 b(+)g Fq(o)p Fv(\(1\)])20433 69102 y(\024)20178 69408 y Fq(Z)21092 68705 y FH(\()p Fr(n)22029 68861 y Fn(\014)22582 68705 y Fk(\000)p Fr(s)p FH(\))21005 69799 y Fr(\014)24170 69408 y Fv(\(0\))p 20178 70166 V 21536 71678 a Fq(Z)22450 70975 y FH(\()p Fr(n)23387 71131 y Fn(\014)23940 70975 y FH(\))22363 72069 y Fr(\014)25852 70444 y Fv(\()p Fq(\032)26950 70641 y Fr(\014)27579 70444 y Fv(\))28050 69944 y Fk(\000)p Fr(s)30928 69293 y Fm(X)30828 71599 y Fn(s)31216 71798 y Fx(1)31678 71599 y Fn(;s)32318 71798 y Fx(2)29607 72384 y(1)p Fl(\024)p Fn(s)31041 72583 y Fx(1)31502 72384 y(+)p Fn(s)32511 72583 y Fx(2)32973 72384 y Fl(\024)p Fn(s)34133 70444 y Fv(\()p Fq(e)35168 69944 y Fr(U)94 b(\014)36529 70444 y Fq(\032)37156 70641 y Fr(\014)37784 70444 y Fv(\))38255 69944 y Fr(s)38690 70067 y Fx(1)39409 70444 y Fq(e)39973 69944 y Fk(\000)p Fr(\014)46 b(\017)41715 70444 y Fq(:)47050 68354 y Fv(\(B.15\))25194 75417 y(38)p eop end %%Page: 39 39 TeXDict begin 39 38 bop 1200 1313 a Fv(Since)404 b(\001)337 b Fq(>)f(U)132 b Fv(,)404 b(w)-34 b(e)404 b(ha)-34 b(v)g(e)405 b Fq(e)13835 874 y Fr(U)94 b(\014)15195 1313 y Fq(\032)15822 1510 y Fr(\014)16787 1313 y Fp(\024)337 b Fv(1)404 b(and)h(so)f(the)g (sum)h(is)f Fq(o)p Fv(\(1\).)539 b(Hence)15975 4072 y(\024)15720 4378 y Fq(Z)16634 3676 y FH(\()p Fr(n)17571 3832 y Fn(\014)18124 3676 y Fk(\000)p Fr(s)p FH(\))16547 4770 y Fr(\014)19712 4378 y Fv(\(0\))p 15720 5136 5542 49 v 17078 6648 a Fq(Z)17992 5946 y FH(\()p Fr(n)18929 6102 y Fn(\014)19482 5946 y FH(\))17905 7040 y Fr(\014)21731 5415 y Fp(\025)336 b Fv(\()p Fq(\032)24108 5612 y Fr(\014)24737 5415 y Fv(\))25208 4914 y Fr(s)25901 5415 y Fq(\026)26631 5612 y Fr(\014)27260 5415 y Fv(\()28009 5108 y(\024)27731 5415 y Fp(S)91 b Fv(\(0\)\))202 b([1)271 b(+)e Fq(o)p Fv(\(1\)])p Fq(:)11037 b Fv(\(B.16\))1200 9382 y(The)405 b(claim)e(no)-34 b(w)405 b(follo)-34 b(ws)405 b(b)-34 b(y)404 b(using)h(Lemma)f(B.2)f(b)34 b(elo)-34 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Fv(\()10517 23006 y(\024)10239 23313 y Fp(S)91 b Fv(\(0\)\))9486 26449 y(+)11288 24933 y Fr(K)10840 25297 y Fm(X)10698 27888 y Fr(m)p FH(=1)13735 25297 y Fm(X)12935 28286 y Fr(\021)32 b Fk(2X)14866 27815 y Fx(\()p Fn(m)p Fx(\))14736 28602 y Fn(\014)18138 25297 y Fm(X)16622 28406 y Fn(\020)40 b Fl(2X)18279 27815 y Fx(\()p Fn(n)19110 28048 y(\014)19663 27815 y Fl(\000)p Fn(m)p Fx(\))18182 28767 y Fn(\014)17748 29477 y(\021)24 b Fl(_)p Fn(\020)40 b Fl(2S)21872 25629 y Fq(e)22436 25189 y Fk(\000)p Fr(\014)202 b(H)68 b FH(\()p Fr(\021)32 b Fk(_)p Fr(\020)61 b FH(\))p 21872 26170 5261 49 v 23089 27682 a Fq(Z)24003 26979 y FH(\()p Fr(n)24940 27135 y Fn(\014)25493 26979 y FH(\))23916 28074 y Fr(\014)28198 26695 y Fk(f)p FH(supp[)p Fr(\021)32 b FH(])p Fk(\032)p Fr(B)33076 26844 y Fn(L)33638 27076 y(\014)34191 26844 y(;L)35005 27076 y(\014)35614 26695 y FH(\(0\))p Fk(g)787 b(f)p FH(supp)o([)p Fr(\020)61 b FH(])p Fk(\032)p FH([)p Fr(B)43182 26844 y Fn(L)43744 27076 y(\014)44297 26844 y(;L)45111 27076 y(\014)45720 26695 y FH(\(0\)])47183 26443 y Fn(c)47610 26695 y Fk(g)7129 31238 y Fp(\024)337 b Fq(\026)9139 31435 y Fr(\014)9768 31238 y Fv(\()10517 30932 y(\024)10239 31238 y Fp(S)91 b Fv(\(0\)\))271 b(+)e Fq(\015)15194 31420 y FH(1)15719 31238 y Fv(\()p Fq(\014)64 b Fv(\))270 b(+)f Fq(\015)19521 31420 y FH(2)20046 31238 y Fv(\()p Fq(\014)64 b Fv(\))p Fq(;)47050 32710 y Fv(\(B.17\))1200 34215 y(where)2351 37785 y Fq(\015)2979 37967 y FH(1)3505 37785 y Fv(\()p Fq(\014)g Fv(\))337 b(=)7403 36270 y Fr(K)6955 36634 y Fm(X)6813 39225 y Fr(m)p FH(=1)9850 36634 y Fm(X)9050 39623 y Fr(\021)32 b Fk(2X)10981 39152 y Fx(\()p Fn(m)p Fx(\))10851 39939 y Fn(\014)13976 36634 y Fm(X)12737 39547 y Fn(\020)40 b Fl(2X)14394 39152 y Fx(\()p Fn(n)p Fl(\000)p Fn(m)p Fx(\))14297 39908 y Fn(\014)13586 40618 y(\021)24 b Fl(_)p Fn(\020)40 b Fl(2S)17434 36965 y Fq(e)17998 36525 y Fk(\000)p Fr(\014)202 b FH([)p Fr(H)68 b FH(\()p Fr(\021)32 b FH(\)+)p Fr(H)68 b FH(\()p Fr(\020)61 b FH(\)])p 17434 37506 7461 49 v 19751 39019 a Fq(Z)20665 38316 y FH(\()p Fr(n)21602 38472 y Fn(\014)22155 38316 y FH(\))20578 39410 y Fr(\014)25960 38032 y Fk(f)p FH(supp[)p Fr(\021)32 b FH(])p Fk(\032)p Fr(B)30838 38181 y Fn(L)31400 38413 y(\014)31953 38181 y(;L)32767 38413 y(\014)33376 38032 y FH(\(0\))p Fk(g)787 b(f)p FH(supp)o([)p Fr(\020)61 b FH(])p Fk(\032)p FH([)p Fr(B)40944 38181 y Fn(L)41506 38413 y(\014)42059 38181 y(;L)42873 38413 y(\014)43482 38032 y FH(\(0\)])44945 37780 y Fn(c)45372 38032 y Fk(g)47050 37785 y Fv(\(B.18\))1200 43106 y(and)415 b Fq(\015)4195 43288 y FH(2)4720 43106 y Fv(\()p Fq(\014)64 b Fv(\))415 b(is)f(a)g(term)h(that)g(arises)f(from)g(particles)g(in)-34 b(teracting)415 b(accross)f(the)g(b)34 b(oundary)416 b(of)e Fq(B)45811 43294 y Fr(L)46451 43450 y Fn(\014)47005 43294 y Fr(;L)47906 43450 y Fn(\014)48514 43106 y Fv(\(0\).)1200 44612 y(W)-101 b(e)403 b(will)h(sho)-34 b(w)405 b(that)h(b)34 b(oth)405 b Fq(\015)14697 44794 y FH(1)15222 44612 y Fv(\()p Fq(\014)64 b Fv(\))404 b(and)h Fq(\015)20303 44794 y FH(2)20829 44612 y Fv(\()p Fq(\014)64 b Fv(\))404 b(are)g(negligible.)3082 46449 y(Estimate,)g(with)h(the)f(help)h(of)f (\(B.9\))g(\(and)i(recalling)c(\(B.1{B.2\)\),)4267 50484 y Fq(\015)4895 50666 y FH(1)5420 50484 y Fv(\()p Fq(\014)64 b Fv(\))337 b Fp(\024)9318 48969 y Fr(K)8871 49332 y Fm(X)8729 51923 y Fr(m)p FH(=1)11353 49141 y Fv(\024)11098 49448 y Fq(Z)12012 48745 y FH(\()p Fr(n)12949 48901 y Fn(\014)13502 48745 y Fk(\000)p Fr(m)p FH(\))11925 49839 y Fr(\014)p 11098 50205 4390 49 v 11880 51718 a Fq(Z)12794 51015 y FH(\()p Fr(n)13731 51171 y Fn(\014)14284 51015 y FH(\))12707 52109 y Fr(\014)16541 49332 y Fm(X)15822 52112 y Fr(\021)32 b Fk(2S)17588 51851 y Fx(\()p Fn(m)p Fx(\))19212 50484 y Fq(e)19776 49983 y Fk(\000)p Fr(\014)46 b(H)68 b FH(\()p Fr(\021)32 b FH(\))24147 50731 y Fk(f)p FH(supp)o([)p Fr(\021)g FH(])p Fk(\032)p Fr(B)29024 50880 y Fn(L)29586 51112 y(\014)30140 50880 y(;L)30954 51112 y(\014)31562 50731 y FH(\(0\))p Fk(g)7449 55090 y Fv(=)337 b([1)269 b(+)g Fq(o)p Fv(\(1\)])202 b Fq(\026)14558 55287 y Fr(\014)15187 55090 y Fv(\()15936 54783 y(\024)15658 55090 y Fp(S)92 b Fv(\(0\)\))19295 53575 y Fr(K)18848 53938 y Fm(X)18705 56529 y Fr(m)p FH(=1)20740 55090 y Fv(\()p Fq(\032)21838 55287 y Fr(\014)22467 55090 y Fv(\))22938 54589 y Fr(m)24746 53938 y Fm(X)24028 56718 y Fr(\021)32 b Fk(2S)25794 56457 y Fx(\()p Fn(m)p Fx(\))27418 55090 y Fq(e)27982 54589 y Fk(\000)p Fr(\014)202 b(H)68 b FH(\()p Fr(\021)32 b FH(\))32509 55336 y Fk(f)p FH(supp[)p Fr(\021)g FH(])p Fk(\032)p Fr(B)37387 55485 y Fn(L)37949 55718 y(\014)38502 55485 y(;L)39316 55718 y(\014)39925 55336 y FH(\(0\))p Fk(g)7449 59696 y Fv(=)337 b([1)269 b(+)g Fq(o)p Fv(\(1\)])202 b Fq(\026)14558 59893 y Fr(\014)15187 59696 y Fv(\()15936 59389 y(\024)15658 59696 y Fp(S)92 b Fv(\(0\)\))19295 58180 y Fr(K)18848 58544 y Fm(X)18705 61135 y Fr(m)p FH(=1)20740 59696 y Fv(\()p Fq(\032)21838 59893 y Fr(\014)22467 59696 y Fv(\))22938 59195 y Fr(m)24488 58180 y(m)24028 58544 y Fm(X)24087 61150 y Fr(j)51 b FH(=1)27961 58544 y Fm(X)26114 61031 y Fx(2)p Fl(\024)p Fn(k)27599 61230 y Fx(1)28060 61031 y Fn(;:::;k)29759 61242 y(j)30193 61031 y Fl(\024)p Fn(K)26609 61790 y Fb(P)27407 61931 y Fn(j)27407 62629 y(i)p Fx(=1)28926 62289 y Fn(k)29365 62500 y(i)29717 62289 y Fx(=)p Fn(m)33214 58544 y Fm(X)32027 61362 y Fn(C)46 b Fx(=)p Fl([)33834 61004 y Fn(j)33834 61703 y(i)p Fx(=1)35213 61362 y Fn(C)35800 61574 y(i)32139 62320 y Fl(j)p Fn(C)32978 62532 y(i)33330 62320 y Fl(j)p Fx(=)p Fn(k)34642 62532 y(i)35135 62320 y Fl(8)141 b Fn(i)36486 59696 y Fq(e)37050 59195 y Fk(\000)p Fr(\014)38670 58531 y Fa(P)39663 58761 y Fn(j)39663 59490 y(i)p Fx(=1)41198 59195 y Fr(H)68 b FH(\()p Fr(C)43076 59330 y Fn(i)43427 59195 y FH(\))43848 59696 y Fq(;)47050 55718 y Fv(\(B.19\))25194 75417 y(39)p eop end %%Page: 40 40 TeXDict begin 40 39 bop 1200 1313 a Fv(where)380 b(the)h(last)f (equalit)-34 b(y)380 b(is)g(a)g(cluster)g(expansion)h(as)f(in)h (\(B.12\).)530 b(Using)381 b(once)f(more)g(the)g(isop)34 b(erimetric)1200 2819 y(inequalit)-34 b(y)-101 b(,)404 b(w)-34 b(e)404 b(get)g(\(recall)f(\(1.29\)\))3038 6247 y Fq(\015)3666 6429 y FH(1)4192 6247 y Fv(\()p Fq(\014)64 b Fv(\))337 b Fp(\024)f Fv([1)269 b(+)g Fq(o)p Fv(\(1\)])202 b Fq(\026)13329 6444 y Fr(\014)13959 6247 y Fv(\()14708 5940 y(\024)14430 6247 y Fp(S)91 b Fv(\(0\)\))18268 4731 y Fr(K)17821 5095 y Fm(X)17679 7686 y Fr(m)p FH(=1)19714 6247 y Fv(\()p Fq(\032)20812 6444 y Fr(\014)21441 6247 y Fv(\))21912 5746 y Fr(m)23461 4731 y(m)23002 5095 y Fm(X)23060 7701 y Fr(j)51 b FH(=1)26935 5095 y Fm(X)25087 7582 y Fx(2)p Fl(\024)p Fn(k)26572 7781 y Fx(1)27033 7582 y Fn(;:::)q(;k)28733 7793 y(j)29166 7582 y Fl(\024)p Fn(K)25583 8341 y Fb(P)26380 8482 y Fn(j)26380 9180 y(i)p Fx(=1)27900 8839 y Fn(k)28339 9051 y(i)28690 8839 y Fx(=)p Fn(m)30868 6247 y Fq(e)31432 5746 y Fk(\000)p Fr(\014)32894 5082 y Fa(P)33888 5312 y Fn(j)33888 6040 y(i)p Fx(=1)35423 5746 y Fr(H)68 b FH(\()p Fr(k)37120 5881 y Fn(i)37471 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14354 y Fn(k)30566 14566 y(i)30917 14354 y Fx(=)p Fn(m)33095 11761 y Fq(e)33659 11261 y Fk(\000)p Fr(\014)35121 10597 y Fa(P)36114 10827 y Fn(j)36114 11555 y(i)p Fx(=1)37650 11261 y Fr(H)38423 11417 y Fn(k)38862 11628 y(i)6221 17500 y Fv(=)336 b Fq(C)289 b(\026)9385 17697 y Fr(\014)10014 17500 y Fv(\()10764 17194 y(\024)10485 17500 y Fp(S)92 b Fv(\(0\)\))14122 15985 y Fr(K)13675 16349 y Fm(X)13532 18940 y Fr(m)p FH(=1)16431 15985 y Fr(m)15972 16349 y Fm(X)16030 18954 y Fr(j)51 b FH(=1)19905 16349 y Fm(X)18057 18835 y Fx(2)p Fl(\024)p Fn(k)19542 19034 y Fx(1)20003 18835 y Fn(;:::;k)21702 19046 y(j)22136 18835 y Fl(\024)p Fn(K)18553 19595 y Fb(P)19350 19735 y Fn(j)19350 20433 y(i)p Fx(=1)20870 20093 y Fn(k)21309 20304 y(i)21660 20093 y Fx(=)p Fn(m)23838 17500 y Fq(e)24402 17000 y Fk(\000)p Fr(\014)25864 16335 y Fa(P)26857 16565 y Fn(j)26857 17294 y(i)p Fx(=1)28236 17000 y FH([)p Fr(H)29270 17156 y Fn(k)29709 17367 y(i)30116 17000 y FH(+)p Fr(k)31338 17135 y Fn(i)31689 17000 y FH(\001)p Fk(\000)p FH(\(\001)p Fk(\000)p Fr(\016)35503 17156 y Fn(\014)36056 17000 y FH(\)])6221 22113 y Fp(\024)336 b Fq(C)8453 21612 y Fk(0)8966 22113 y Fq(\026)9696 22310 y Fr(\014)10325 22113 y Fv(\()11074 21806 y(\024)10796 22113 y Fp(S)91 b Fv(\(0\)\))202 b Fq(e)14406 21612 y Fk(\000)p Fr(\014)46 b(\017)47050 13588 y Fv(\(B.20\))1200 24379 y(for)404 b(some)g Fq(\017)337 b(>)f Fv(0)404 b(and)h(constan)-34 b(ts)406 b Fq(C)19 b(;)202 b(C)19311 23940 y Fk(0)19959 24379 y Fq(<)337 b Fp(1)404 b Fv(that)h(are)f(indep)34 b(enden)-34 b(t)405 b(of)g Fq(\014)64 b Fv(.)3082 26143 y(Estimate,)404 b(with)h(the)f(help)h(of)f(\(B.9\),)3433 29732 y Fq(\015)4061 29914 y FH(2)4586 29732 y Fv(\()p Fq(\014)64 b Fv(\))337 b Fp(\024)8484 28217 y Fr(K)8037 28581 y Fm(X)7895 31172 y Fr(m)p FH(=1)10850 28581 y Fm(X)10131 31360 y Fr(\021)32 b Fk(2S)11897 31099 y Fx(\()p Fn(m)p Fx(\))13521 29732 y Fq(e)14085 29232 y Fk(\000)p Fr(\014)46 b(H)68 b FH(\()p Fr(\021)32 b FH(\))18184 28217 y Fr(m)17725 28581 y Fm(X)17742 31216 y Fr(k)24 b FH(=1)19677 29732 y Fq(e)20241 29232 y Fr(\014)46 b(k)24 b(U)23048 29979 y Fk(f)p FH(supp[)p Fr(\021)32 b FH(])p Fk(\032)p Fr(B)27926 30128 y Fn(L)28488 30360 y(\014)29041 30128 y(;L)29855 30360 y(\014)30464 29979 y FH(\(0\))p Fk(g)32984 28390 y Fv(\024)32729 28696 y Fq(Z)33643 27993 y FH(\()p Fr(n)34580 28149 y Fn(\014)35133 27993 y Fk(\000)p FH(\()p Fr(m)p FH(+)p Fr(k)24 b FH(\)\))33556 29087 y Fr(\014)p 32729 29453 6367 49 v 34499 30966 a Fq(Z)35413 30263 y FH(\()p Fr(n)36350 30419 y Fn(\014)36903 30263 y FH(\))35326 31357 y Fr(\014)6615 34338 y Fp(\024)8484 32823 y Fr(K)8037 33187 y Fm(X)7895 35778 y Fr(m)p FH(=1)10850 33187 y Fm(X)10131 35966 y Fr(\021)32 b Fk(2S)11897 35705 y Fx(\()p Fn(m)p Fx(\))13521 34338 y Fq(e)14085 33838 y Fk(\000)p Fr(\014)46 b(H)68 b FH(\()p Fr(\021)32 b FH(\))18184 32823 y Fr(m)17725 33187 y Fm(X)17742 35822 y Fr(k)24 b FH(=1)19677 34338 y Fq(e)20241 33838 y Fr(\014)46 b(k)24 b(U)23048 34585 y Fk(f)p FH(supp[)p Fr(\021)32 b FH(])p Fk(\032)p Fr(B)27926 34734 y Fn(L)28488 34966 y(\014)29041 34734 y(;L)29855 34966 y(\014)30464 34585 y FH(\(0\))p Fk(g)32596 34338 y Fv(\()p Fq(\032)33694 34535 y Fr(\014)34323 34338 y Fv(\))34794 33838 y Fr(m)p FH(+)p Fr(k)37129 34338 y Fq(\026)37859 34535 y Fr(\014)38488 34338 y Fv(\()39237 34032 y(\024)38959 34338 y Fp(S)91 b Fv(\(0\)\))202 b([1)271 b(+)e Fq(o)p Fv(\(1\)])6615 38944 y Fp(\024)337 b Fv([1)269 b(+)g Fq(o)p Fv(\(1\)])202 b Fq(\026)13724 39141 y Fr(\014)14353 38944 y Fv(\()15102 38638 y(\024)14824 38944 y Fp(S)92 b Fv(\(0\)\))18461 37429 y Fr(K)18014 37793 y Fm(X)17871 40383 y Fr(m)p FH(=1)19906 38944 y Fv(\()p Fq(\032)21004 39141 y Fr(\014)21633 38944 y Fv(\))22104 38444 y Fr(m)23913 37793 y Fm(X)23194 40572 y Fr(\021)32 b Fk(2S)24960 40311 y Fx(\()p Fn(m)p Fx(\))26584 38944 y Fq(e)27148 38444 y Fk(\000)p Fr(\014)46 b(H)68 b FH(\()p Fr(\021)32 b FH(\))31247 37429 y Fr(m)30788 37793 y Fm(X)30805 40428 y Fr(k)24 b FH(=1)32740 38944 y Fq(e)33304 38444 y Fk(\000)p Fr(\014)46 b(k)24 b FH(\(\001)p Fk(\000)p Fr(U)94 b FH(\))38889 39191 y Fk(f)p FH(supp)o([)p Fr(\021)32 b FH(])p Fk(\032)p Fr(B)43766 39340 y Fn(L)44328 39572 y(\014)44881 39340 y(;L)45695 39572 y(\014)46304 39191 y FH(\(0\))p Fk(g)48032 38944 y Fq(;)47050 41885 y Fv(\(B.21\))1200 43391 y(and)405 b(w)-34 b(e)404 b(can)h(pro)34 b(ceed)403 b(as)h(\(B.19{B.20\))g(to)g(sho)-34 b(w)406 b(that)f(this)g(term)f(is)g(negligible.)p 49878 43391 522 731 v 1200 47340 a Ft(B.3)1495 b(A)-42 b(t)g(ypicalit)g(y)501 b(of)d(critical)i(droplets)1200 49887 y Fv(The)405 b(follo)-34 b(wing)404 b(lemma)g(w)-34 b(as)405 b(used)g(in)f(Section)g(4.1.1.)1200 52533 y Fi(Lemma)464 b(B.3)605 b Fv(lim)10508 52730 y Fr(\014)46 b Fk(!1)13221 52533 y Fq(\026)13951 52730 y Fr(\014)14580 52533 y Fv(\()15271 52227 y(~)15051 52533 y Fp(C)71 b(nC)17075 52093 y FH(+)17862 52533 y Fv(\))p Fq(=\026)19669 52730 y Fr(\014)20299 52533 y Fv(\()p Fp(S)91 b Fv(\))337 b(=)g(0)p Fs(.)1200 55180 y(Pr)-62 b(o)g(of)p Fv(.)403 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y Fk(\003)37749 67765 y Fv(+)269 b Fq(K)87 b Fv(\001\))11229 70243 y(r.h.s.\(B.22\))336 b Fp(\024)g Fq(N)e Fp(j)p Fv(\003)21318 70440 y Fr(\014)21947 70243 y Fp(j)202 b Fq(e)23050 69743 y Fk(\000)p Fr(\014)46 b FH(\000)24944 69430 y Fl(\003)25680 70243 y Fv(\()p Fq(\032)26778 70440 y Fr(\014)27407 70243 y Fv(\))27878 69743 y Fr(K)28991 70243 y Fq(\026)29721 70440 y Fr(\014)30351 70243 y Fv(\()31100 69937 y(\024)30822 70243 y Fp(S)91 b Fv(\(0\)\))202 b([1)270 b(+)f Fq(o)p Fv(\(1\)])17552 72253 y(=)336 b Fq(N)e Fp(j)p Fv(\003)21318 72450 y Fr(\014)21947 72253 y Fp(j)202 b Fq(e)23050 71752 y Fk(\000)p Fr(\014)46 b FH(\000)25200 72253 y Fq(\026)25930 72450 y Fr(\014)26559 72253 y Fv(\()p Fp(S)91 b Fv(\))202 b([1)270 b(+)f Fq(o)p Fv(\(1\)])p Fq(;)2626 b(\014)401 b Fp(!)336 b(1)p Fq(;)47050 71216 y Fv(\(B.23\))25194 75417 y(40)p eop end %%Page: 41 41 TeXDict begin 41 40 bop 1200 1313 a Fv(whic)-34 b(h)405 b(is)f Fq(o)p Fv(\()p Fq(\026)7676 1510 y Fr(\014)8305 1313 y Fv(\()p Fp(S)91 b Fv(\)\))405 b(b)-34 b(y)405 b(\(1.35\).)p 49878 1313 522 731 v 1200 6129 a Fu(C)1793 b(App)50 b(endix:)799 b(T)-50 b(ypicalit)g(y)600 b(of)e(starting)h (con\014gurations)1200 9167 y Fv(In)342 b(Sections)h(C.1{C.2)g(w)-34 b(e)342 b(pro)-34 b(v)g(e)343 b(the)g(claims)e(made)i(in)f(the)h (remarks)e(b)34 b(elo)-34 b(w)343 b(\(1.9\),)355 b(resp)34 b(ectiv)-34 b(ely)-101 b(,)353 b(\(1.32\).)1200 12745 y Ft(C.1)1495 b(Glaub)42 b(er)1200 15365 y Fs(Pr)-62 b(o)g(of)p Fv(.)403 b(Split)17563 16870 y Fp(S)428 b Fv(=)336 b Fp(S)20738 17058 y Fr(L)21703 16870 y Fp([)269 b Fv(\()p Fp(S)361 b(n)269 b(S)25955 17058 y Fr(L)26651 16870 y Fv(\))337 b(=)f Fp(S)29472 17058 y Fr(L)30437 16870 y Fp([)269 b(U)32273 17058 y Fr(>L)33700 16870 y Fq(;)13602 b Fv(\(C.1\))1200 19095 y(where)409 b Fp(U)5469 19283 y Fr(>L)7241 19095 y Fp(\032)345 b(S)501 b Fv(are)408 b(those)i(con\014gurations)h Fq(\033)453 b Fv(for)409 b(whic)-34 b(h)410 b Fq(C)30141 19283 y Fr(B)30950 19095 y Fv(\()p Fq(\033)43 b Fv(\))411 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44860 y Fv(\))337 b Fp(\024)8949 43709 y Fm(X)8643 46334 y Fr(x)p Fk(2)p FH(\003)10453 46490 y Fn(\014)11248 43709 y Fm(X)11208 46334 y Fr(\033)32 b Fk(2S)13972 45149 y(f)p Fr(x)p Fk(2)p Fr(X)16442 44896 y Fn(>L)17699 45149 y FH(\()p Fr(\033)g FH(\))p Fk(g)19728 44860 y Fq(\026)20458 45057 y Fr(\014)21088 44860 y Fv(\()p Fq(\033)43 b Fv(\))7364 48784 y(=)8949 47633 y Fm(X)8643 50258 y Fr(x)p Fk(2)p FH(\003)10453 50414 y Fn(\014)11248 47633 y Fm(X)11208 50258 y Fr(\033)32 b Fk(2S)13972 49073 y(f)p Fr(x)p Fk(2)p Fr(X)16442 48821 y Fn(>L)17699 49073 y FH(\()p Fr(\033)g FH(\))p Fk(g)20286 47964 y Fv(1)p 19861 48506 1457 49 v 19861 49616 a Fq(Z)20688 49813 y Fr(\014)21854 48784 y Fv(exp)23908 47439 y Fm(n)24986 48784 y Fp(\000)269 b Fq(\014)26948 47439 y Fm(h)27520 48784 y Fq(H)28626 47803 y Fm(\000)29181 48784 y Fq(\033)43 b Fp(j)30254 49031 y Fr(R)30963 49224 y Fn(`)31307 49423 y Fx(1)31768 49224 y(\()p Fn(x)p Fx(\))p Fn(;`)33468 49423 y Fx(2)33929 49224 y(\()p Fn(x)p Fx(\))35088 49031 y FH(\()p Fr(x)p FH(\))36405 47803 y Fm(\001)37229 48784 y Fv(+)270 b Fq(H)39548 47803 y Fm(\000)40103 48784 y Fq(\033)43 b Fp(j)41176 49031 y Fr(R)41890 48726 y Fn(c)41885 49384 y(`)42229 49583 y Fx(1)42690 49384 y(\()p Fn(x)p Fx(\))p Fn(;`)44390 49583 y Fx(2)44851 49384 y(\()p Fn(x)p Fx(\))46010 49031 y FH(\()p Fr(x)p FH(\))47326 47803 y Fm(\001)47882 47439 y(io)7364 53076 y Fp(\024)336 b Fq(e)9207 52575 y Fk(\000)p Fr(\014)46 b FH(\000)11101 52724 y Fn(L)p Fx(+1)13308 51924 y Fm(X)13002 54549 y Fr(x)p Fk(2)p FH(\003)14812 54705 y Fn(\014)15607 51924 y Fm(X)15567 54549 y Fr(\033)32 b Fk(2S)18331 53364 y(f)p Fr(x)p Fk(2)p Fr(X)20801 53112 y Fn(>L)22058 53364 y FH(\()p Fr(\033)g FH(\))p Fk(g)24645 52256 y Fv(1)p 24220 52797 V 24220 53907 a Fq(Z)25047 54104 y Fr(\014)25809 53076 y Fq(e)26373 52138 y Fk(\000)p Fr(\014)46 b(H)28520 51156 y Fm(\000)29076 52138 y Fr(\033)32 b Fk(j)29907 52348 y Fn(R)30521 52119 y(c)30519 52746 y(`)30863 52945 y Fx(1)31325 52746 y(\()p Fn(x)p Fx(\))p Fn(;`)33025 52945 y Fx(2)33486 52746 y(\()p Fn(x)p Fx(\))34644 52348 y(\()p Fn(x)p Fx(\))35802 51156 y Fm(\001)36413 53076 y Fq(;)47639 55976 y Fv(\(C.5\))1200 57481 y(where)371 b(\000)5430 57669 y Fr(L)p FH(+1)7699 57481 y Fv(is)g(the)i(energy)d (of)i Fq(Q)17089 57669 y Fr(L)p FH(+1)18987 57481 y Fv(\(0\).)529 b(In)371 b(the)h(last)g(step)g(w)-34 b(e)372 b(use)g(the)g(fact)g(that) h(the)f(b)34 b(o)g(otstrap)372 b(map)1200 58987 y(is)330 b(do)-34 b(wnhill)332 b(and)g(that)g(the)f(energy)f(of)h Fq(Q)20072 59175 y Fr(L)20768 58987 y Fv(\(0\))g(is)g(increasing)f (with)i Fq(L)p Fv(.)514 b(Since)330 b(the)i(energy)e(of)h(a)g(sub)34 b(critical)1200 60492 y(rectangle)404 b(is)f(non-negativ)-34 b(e,)405 b(w)-34 b(e)405 b(get)16295 63215 y Fq(\026)17025 63412 y Fr(\014)17655 63215 y Fv(\()p Fp(U)18885 63403 y Fr(>L)20312 63215 y Fv(\))337 b Fp(\024)f Fq(N)23373 63403 y Fr(L)p FH(+1)25473 63215 y Fq(e)26037 62715 y Fk(\000)p Fr(\014)46 b FH(\000)27931 62864 y Fn(L)p Fx(+1)29832 63215 y Fp(j)p Fv(\003)31011 63412 y Fr(\014)31639 63215 y Fp(j)202 b Fq(\026)32908 63412 y Fr(\014)33537 63215 y Fv(\()p Fp(S)91 b Fv(\))12335 b(\(C.6\))1200 65938 y(with)405 b Fq(N)4935 66126 y Fr(L)p FH(+1)7236 65938 y Fv(coun)-34 b(ting)406 b(the)e(n)-34 b(um)g(b)34 b(er)406 b(of)e(con\014gurations)i(with)f(supp)34 b(ort)405 b(in)f Fq(Q)37511 66126 y Fr(L)p FH(+1)39409 65938 y Fv(\(0\).)3082 67776 y(On)415 b(the)h(other)f(hand,)k(b)-34 b(y)416 b(considering)f(only)g(those)h(con\014gurations)h(in)e Fp(U)37689 67964 y Fr(>L)39531 67776 y Fv(that)h(ha)-34 b(v)g(e)416 b(a)f Fq(Q)46953 67964 y Fr(L)p FH(+1)48851 67776 y Fv(\(0\))1200 69282 y(droplet,)404 b(w)-34 b(e)405 b(get)14146 70951 y Fq(\026)14876 71148 y Fr(\014)15505 70951 y Fv(\()p Fp(U)16735 71139 y Fr(>L)18162 70951 y Fv(\))337 b Fp(\025)f Fq(N)21223 71139 y Fr(L)p FH(+1)23323 70951 y Fq(e)23887 70450 y Fk(\000)p Fr(\014)46 b FH(\000)25781 70599 y Fn(L)p Fx(+1)27682 70951 y Fp(j)p Fv(\003)28861 71148 y Fr(\014)29489 70951 y Fp(j)202 b Fq(\026)30758 70292 y FH([)p Fr(Q)31759 70441 y Fn(L)p Fx(+1)33404 70292 y FH(\(0\)])34867 69980 y Fn(c)30758 71342 y Fr(\014)35350 70951 y Fv(\()p Fp(S)91 b Fv(\))p Fq(;)10185 b Fv(\(C.7\))25194 75417 y(41)p eop end %%Page: 42 42 TeXDict begin 42 41 bop 1200 1313 a Fv(where)552 b(the)g(last)h(factor) f(is)g(the)g(Gibbs)h(w)-34 b(eigh)g(t)553 b(of)f(the)h (con\014gurations)g(in)f Fp(S)643 b Fv(with)553 b(supp)34 b(ort)553 b(outside)1200 2983 y([)p Fq(Q)2495 3171 y Fr(L)p FH(+1)4392 2983 y Fv(\(0\)])6277 2543 y Fr(c)6741 2983 y Fv(.)538 b(It)404 b(easy)g(to)h(sho)-34 b(w)405 b(that)g Fq(\026)19437 2324 y FH([)p Fr(Q)20438 2473 y Fn(L)p Fx(+1)22083 2324 y FH(\(0\)])23546 2011 y Fn(c)19437 3374 y Fr(\014)24029 2983 y Fv(\()p Fp(S)91 b Fv(\))338 b(=)e Fq(\026)28143 3180 y Fr(\014)28772 2983 y Fv(\()p Fp(S)91 b Fv(\)[1)270 b(+)f Fq(o)p Fv(\(1\)])404 b(as)g Fq(\014)d Fp(!)336 b(1)405 b Fv(and)g(so)10172 5829 y Fq(\026)10902 6026 y Fr(\014)11531 5829 y Fv(\()p Fp(U)12761 6017 y Fr(>L)14188 5829 y Fv(\))338 b Fp(\025)e Fq(N)17250 6017 y Fr(L)p FH(+1)19349 5829 y Fq(e)19913 5328 y Fk(\000)p Fr(\014)46 b FH(\000)21807 5477 y Fn(L)p Fx(+1)23709 5829 y Fp(j)p Fv(\003)24888 6026 y Fr(\014)25516 5829 y Fp(j)202 b Fq(\026)26785 6026 y Fr(\014)27414 5829 y Fv(\()p Fp(S)91 b Fv(\))202 b([1)270 b(+)f Fq(o)p Fv(\(1\)])p Fq(;)2626 b(\014)400 b Fp(!)337 b(1)p Fq(:)6211 b Fv(\(C.8\))1200 8391 y(Com)-34 b(bining)406 b(\(C.6{C.7\),)f(w)-34 b(e)404 b(conclude)g(that)h(lim)24365 8588 y Fr(\014)46 b Fk(!1)27078 8391 y Fq(\026)27808 8588 y Fr(\014)28437 8391 y Fv(\()p Fp(U)29667 8579 y Fr(>L)31094 8391 y Fv(\))p Fq(=\026)32901 8588 y Fr(\014)33531 8391 y Fv(\()p Fp(S)91 b Fv(\))337 b(=)f(0)404 b(if)h(and)f(only)g(if)20613 10953 y(lim)20227 11770 y Fr(\014)46 b Fk(!1)22884 10953 y Fp(j)p Fv(\003)24063 11150 y Fr(\014)24692 10953 y Fp(j)202 b Fq(e)25795 10452 y Fk(\000)p FH(\000)27115 10601 y Fn(L)p Fx(+1)29151 10953 y Fv(=)336 b(0)p Fq(:)16266 b Fv(\(C.9\))p 49878 13961 522 731 v 1200 18097 a Ft(C.2)1495 b(Ka)-42 b(w)g(asaki)1200 20691 y Fs(Pr)-62 b(o)g(of)p Fv(.)403 b(Split)17563 22196 y Fp(S)428 b Fv(=)336 b Fp(S)20738 22384 y Fr(L)21703 22196 y Fp([)269 b Fv(\()p Fp(S)361 b(n)269 b(S)25955 22384 y Fr(L)26651 22196 y Fv(\))337 b(=)f Fp(S)29472 22384 y Fr(L)30437 22196 y Fp([)269 b(U)32273 22384 y Fr(>L)33700 22196 y Fq(;)12996 b Fv(\(C.10\))1200 24340 y(where)475 b Fp(U)5535 24528 y Fr(>L)7416 24340 y Fp(\032)454 b(S)566 b Fv(are)474 b(those)i(con\014gurations)g Fq(\033)518 b Fv(for)475 b(whic)-34 b(h)476 b(there)f(exists)f(an)h Fq(x)g Fv(suc)-34 b(h)476 b(that)g Fp(j)p Fv(supp)q([)p Fq(\033)43 b Fv(])316 b Fp(\\)1200 25846 y Fq(B)2119 26034 y Fr(L)2759 26190 y Fn(\014)3312 26034 y Fr(;L)4213 26190 y Fn(\014)4822 25846 y Fv(\()p Fq(x)p Fv(\))p Fp(j)337 b Fq(>)f(L)p Fv(.)539 b(Then)3687 29728 y Fq(\026)4417 29925 y Fr(\014)5047 29728 y Fv(\()p Fp(U)6277 29916 y Fr(>L)7704 29728 y Fv(\))337 b Fp(\024)10097 28576 y Fm(X)9791 31201 y Fr(x)p Fk(2)p FH(\003)11601 31357 y Fn(\014)12396 28576 y Fm(X)12356 31201 y Fr(\033)32 b Fk(2S)15664 28212 y Fr(K)15217 28576 y Fm(X)14389 31201 y Fr(m)p FH(=)p Fr(L)p FH(+1)17997 29728 y Fq(\026)18727 29925 y Fr(\014)19357 29728 y Fv(\()p Fq(\033)43 b Fv(\))21968 29974 y Fk(fj)p FH(supp[)p Fr(\033)32 b FH(])p Fk(\\)p Fr(B)27068 30123 y Fn(L)27630 30356 y(\014)28184 30123 y(;L)28998 30356 y(\014)29606 29974 y FH(\()p Fr(x)p FH(\))p Fk(j)p FH(=)p Fr(m)p Fk(g)33556 29728 y Fv(=)336 b Fp(j)p Fv(\003)36014 29925 y Fr(\014)36643 29728 y Fp(j)202 b Fv([)p Fq(')p Fv(\()p Fq(\014)64 b Fv(\))269 b(+)g Fq(\015)67 b Fv(\()p Fq(\014)d Fv(\)])p Fq(;)2487 b Fv(\(C.11\))1200 33617 y(where)3222 37026 y Fq(')p Fv(\()p Fq(\014)64 b Fv(\))337 b(=)8598 35510 y Fr(K)8151 35874 y Fm(X)7323 38499 y Fr(m)p FH(=)p Fr(L)p FH(+1)11732 35874 y Fm(X)10931 38863 y Fr(\021)32 b Fk(2X)12862 38392 y Fx(\()p Fn(m)p Fx(\))12732 39179 y Fn(\014)16134 35874 y Fm(X)14618 38983 y Fn(\020)40 b Fl(2X)16275 38392 y Fx(\()p Fn(n)17106 38625 y(\014)17660 38392 y Fl(\000)p Fn(m)p Fx(\))16178 39344 y Fn(\014)15744 40054 y(\021)24 b Fl(_)p Fn(\020)40 b Fl(2S)19868 36206 y Fq(e)20432 35766 y Fk(\000)p Fr(\014)46 b FH([)p Fr(H)68 b FH(\()p Fr(\021)32 b FH(\)+)p Fr(H)68 b FH(\()p Fr(\020)61 b FH(\)])p 19868 36747 7304 49 v 22108 38259 a Fq(Z)23022 37556 y FH(\()p Fr(n)23959 37712 y Fn(\014)24512 37556 y FH(\))22935 38651 y Fr(\014)28238 37272 y Fk(f)p FH(supp[)p Fr(\021)32 b FH(])p Fk(\032)p Fr(B)33116 37421 y Fn(L)33678 37654 y(\014)34231 37421 y(;L)35045 37654 y(\014)35654 37272 y FH(\(0\))p Fk(g)989 b(f)p FH(supp)o([)p Fr(\020)61 b FH(])p Fk(\032)p FH([)p Fr(B)43424 37421 y Fn(L)43986 37654 y(\014)44539 37421 y(;L)45353 37654 y(\014)45962 37272 y FH(\(0\)])47425 37020 y Fn(c)47852 37272 y Fk(g)47033 41324 y Fv(\(C.12\))1200 42830 y(and)332 b Fq(\015)67 b Fv(\()p Fq(\014)d Fv(\))332 b(is)f(an)h(error)e(term)h(arising)g (from)h(particles)f(in)-34 b(teracting)332 b(accross)f(the)g(b)34 b(oundary)333 b(of)e Fq(B)45811 43018 y Fr(L)46451 43174 y Fn(\014)47005 43018 y Fr(;L)47906 43174 y Fn(\014)48514 42830 y Fv(\(0\).)1200 44335 y(By)403 b(the)i(same)f(argumen)-34 b(t)405 b(as)f(in)h(\(B.21\),)e(this)i(term)f(is)g(negligible.)537 b(Moreo)-34 b(v)g(er,)4710 48209 y Fq(')p Fv(\()p Fq(\014)64 b Fv(\))337 b Fp(\024)10087 46694 y Fr(K)9639 47057 y Fm(X)8811 49683 y Fr(m)p FH(=)p Fr(L)p FH(+1)12808 46866 y Fv(\024)12553 47172 y Fq(Z)13467 46470 y FH(\()p Fr(n)14404 46626 y Fn(\014)14957 46470 y Fk(\000)p Fr(m)p FH(\))13380 47564 y Fr(\014)p 12553 47930 4390 49 v 13335 49443 a Fq(Z)14249 48740 y FH(\()p Fr(n)15186 48896 y Fn(\014)15739 48740 y FH(\))14162 49834 y Fr(\014)17075 46500 y Fm(\022)18887 47057 y(X)18169 49837 y Fr(\021)32 b Fk(2S)19935 49576 y Fx(\()p Fn(m)p Fx(\))21558 48209 y Fq(e)22122 47708 y Fk(\000)p Fr(\014)203 b(H)68 b FH(\()p Fr(\021)32 b FH(\))26650 48455 y Fk(f)p FH(supp[)p Fr(\021)g FH(])p Fk(\032)p Fr(B)31528 48604 y Fn(L)32090 48837 y(\014)32643 48604 y(;L)33457 48837 y(\014)34066 48455 y FH(\(0\))p Fk(g)35794 46500 y Fm(\023)7532 52815 y Fp(\024)336 b Fv([1)270 b(+)f Fq(o)p Fv(\(1\)])202 b Fq(\026)14641 53012 y Fr(\014)15270 52815 y Fv(\()p Fp(S)91 b Fv(\))18515 51300 y Fr(K)18067 51663 y Fm(X)17239 54289 y Fr(m)p FH(=)p Fr(L)p FH(+1)20646 52815 y Fv(\()p Fq(\032)21744 53012 y Fr(\014)22373 52815 y Fv(\))22844 52314 y Fr(m)23934 51106 y Fm(\022)25746 51663 y(X)25028 54443 y Fr(\021)32 b Fk(2S)26794 54182 y Fx(\()p Fn(m)p Fx(\))28417 52815 y Fq(e)28981 52314 y Fk(\000)p Fr(\014)46 b(H)68 b FH(\()p Fr(\021)32 b FH(\))33352 53061 y Fk(f)p FH(supp[)p Fr(\021)g FH(])p Fk(\032)p Fr(B)38230 53210 y Fn(L)38792 53443 y(\014)39345 53210 y(;L)40159 53443 y(\014)40768 53061 y FH(\(0\))p Fk(g)42496 51106 y Fm(\023)43388 52815 y Fq(;)47033 50574 y Fv(\(C.13\))1200 56813 y(where)375 b(in)g(the)g(last)g(inequalit)-34 b(y)375 b(w)-34 b(e)375 b(use)g(Lemmas)g(B.1{B.2.)527 b(No)-34 b(w)376 b(pro)34 b(ceed)374 b(as)h(in)g(\(B.19{B.20\),)380 b(via)374 b(the)1200 58318 y(cluster)404 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FH([\000)24392 66273 y Fn(L)p Fx(+1)26037 66124 y Fk(\000)p FH(\(\001)p Fk(\000)p Fr(\016)29067 66280 y Fn(\014)29620 66124 y FH(\)])30303 66624 y Fq(;)47033 63701 y Fv(\(C.14\))1200 69085 y(where)343 b Fq(H)5652 69282 y Fr(k)6563 69085 y Fv(is)g(the)g(energy)f(of)h(a)g(droplet)g(with)h Fq(k)381 b Fv(particles)342 b(that)i(is)f(closest)f(to)h(a)g(square)g(or)f (quasi-square,)1200 70590 y(\000)1958 70778 y Fr(L)p FH(+1)4192 70590 y Fv(=)336 b Fq(H)6479 70778 y Fr(L)p FH(+1)8531 70590 y Fv(+)155 b(\()p Fq(L)g Fv(+)g(1\)\001,)359 b(and)348 b(the)f(second)h(inequalit)-34 b(y)347 b(uses)g(the)g(isop)34 b(erimetric)346 b(inequalit)-34 b(y)346 b(together)1200 72096 y(with)405 b(the)f(fact)h(that)g Fq(H)12073 72293 y Fr(k)12911 72096 y Fv(+)269 b Fq(k)38 b Fv(\001)405 b(is)e(increasing)h(in)g Fq(k)442 b Fv(for)405 b(sub)34 b(critical)403 b(droplets.)25194 75417 y(42)p eop end %%Page: 43 43 TeXDict begin 43 42 bop 3082 1313 a Fv(On)459 b(the)h(other)g(hand,)474 b(b)-34 b(y)460 b(considering)g(only)f(those)h(con\014gurations)h(in)e Fp(U)38142 1501 y Fr(>L)40028 1313 y Fv(that)i(ha)-34 b(v)g(e)460 b(a)f(droplet)1200 2819 y(with)405 b Fq(L)269 b Fv(+)g(1)404 b(paticles,)g(w)-34 b(e)404 b(get)14789 5542 y Fq(')p Fv(\()p Fq(\014)64 b Fv(\))337 b Fp(\025)g Fv([1)269 b(+)g Fq(o)p Fv(\(1\)])202 b Fq(C)289 b(\026)p Fv(\()p Fp(S)91 b Fv(\))202 b Fq(e)28408 5042 y Fk(\000)p Fr(\014)46 b FH([\000)30563 5191 y Fn(L)p Fx(+1)32208 5042 y Fk(\000)p FH(\(\001)p Fk(\000)p Fr(\016)35238 5198 y Fn(\014)35791 5042 y FH(\)])36474 5542 y Fq(:)10222 b Fv(\(C.15\))1200 8265 y(Com)-34 b(bining)346 b(\(C.11\))f(and)f (\(C.14{C.15\),)357 b(w)-34 b(e)345 b(conclude)f(that)h(lim)31297 8462 y Fr(\014)46 b Fk(!1)34009 8265 y Fq(\026)34739 8462 y Fr(\014)35369 8265 y Fv(\()p Fp(U)36599 8453 y Fr(>L)38026 8265 y Fv(\))p Fq(=\026)39833 8462 y Fr(\014)40462 8265 y Fv(\()p Fp(S)91 b Fv(\))338 b(=)e(0)344 b(if)g(and)h(only)1200 9771 y(if)17907 11276 y(lim)17521 12093 y Fr(\014)46 b Fk(!1)20179 11276 y Fp(j)p Fv(\003)21358 11473 y Fr(\014)21986 11276 y Fp(j)202 b Fq(e)23089 10776 y Fk(\000)p Fr(\014)g FH(\(\000)25505 10925 y Fn(L)p Fx(+1)27149 10776 y Fk(\000)p FH(\(\001)p Fk(\000)p Fr(\016)30179 10932 y Fn(\014)30732 10776 y FH(\)\))31856 11276 y Fv(=)337 b(0)p Fq(:)12954 b Fv(\(C.16\))p 49878 13947 522 731 v 1200 18763 a Fu(D)1794 b(App)50 b(endix:)799 b(The)598 b(critical)h(droplet)g(is)f(the)h (threshold)1200 21801 y Fv(In)378 b(this)g(app)34 b(endix)378 b(w)-34 b(e)378 b(sho)-34 b(w)378 b(that)h(our)f(estimates)f(on)h (capacities)f(imply)h(that)g(the)g Fs(aver)-62 b(age)466 b Fv(probabilit)-34 b(y)1200 23306 y(under)402 b(the)g(Gibbs)h(measure) f Fq(\026)15687 23503 y Fr(\014)16717 23306 y Fv(of)h(destro)-34 b(ying)402 b(a)g(sup)34 b(ercritical)400 b(droplet)i(and)h(returning)f (to)g(a)g(con\014gu-)1200 24812 y(ration)442 b(in)g Fp(S)6996 25000 y Fr(L)8133 24812 y Fv(is)f(exp)34 b(onen)-34 b(tially)442 b(small)f(in)h Fq(\014)64 b Fv(.)651 b(W)-101 b(e)441 b(will)h(giv)-34 b(e)441 b(the)h(pro)34 b(of)442 b(for)g(Ka)-34 b(w)g(asaki)442 b(dynamics,)451 b(the)1200 26317 y(pro)34 b(of)404 b(for)h(Glaub)34 b(er)403 b(dynamics)i(b)34 b(eing)404 b(simpler.)3082 28155 y(Pic)-34 b(k)316 b Fq(M)468 b Fp(\025)337 b Fq(`)9134 28337 y Fr(c)9597 28155 y Fv(.)509 b(Recall)316 b(from)g(\(2.7\))h(that)h Fq(e)22697 28343 y Fk(D)23425 28492 y Fn(M)24324 28343 y Fr(;)p Fk(S)25155 28492 y Fn(L)25827 28155 y Fv(\()p Fq(\033)43 b Fv(\))338 b(=)e Fq(c)29647 28352 y Fr(\014)30276 28155 y Fv(\()p Fq(\033)43 b Fv(\))p Fo(P)32695 28337 y Fr(\033)33522 28155 y Fv(\()q Fq(\034)34524 28343 y Fk(S)35094 28492 y Fn(L)36103 28155 y Fq(<)336 b(\034)37912 28343 y Fk(D)38640 28492 y Fn(M)39594 28155 y Fv(\))317 b(for)f Fq(\033)380 b Fp(2)337 b(D)45302 28343 y Fr(M)46355 28155 y Fv(.)509 b(Hence)1200 29660 y(summing)405 b(o)-34 b(v)g(er)404 b Fq(\033)380 b Fp(2)336 b Fq(@)67 b Fp(D)12851 29848 y Fr(M)13905 29660 y Fv(,)403 b(the)i(in)-34 b(ternal)404 b(b)34 b(oundary)405 b(of)g Fp(D)28988 29848 y Fr(M)30041 29660 y Fv(,)f(w)-34 b(e)404 b(get)h(using)f(\(2.8\))h(that)9857 31291 y Fm(P)11137 32559 y Fr(\033)32 b Fk(2)p Fr(@)52 b Fk(D)13610 32708 y Fn(M)14765 32200 y Fq(\026)15495 32397 y Fr(\014)16124 32200 y Fv(\()p Fq(\033)43 b Fv(\))p Fq(c)18327 32397 y Fr(\014)18957 32200 y Fv(\()p Fq(\033)g Fv(\))p Fo(P)21376 32382 y Fr(\033)22203 32200 y Fv(\()p Fq(\034)23204 32388 y Fk(S)23774 32537 y Fn(L)24784 32200 y Fq(<)336 b(\034)26593 32388 y Fk(D)27321 32537 y Fn(M)28274 32200 y Fv(\))p 9857 32902 18889 49 v 13913 33103 a Fm(P)15192 34370 y Fr(\033)32 b Fk(2)p Fr(@)52 b Fk(D)17665 34519 y Fn(M)18820 34012 y Fq(\026)19550 34209 y Fr(\014)20180 34012 y Fv(\()p Fq(\033)43 b Fv(\))p Fq(c)22383 34209 y Fr(\014)23012 34012 y Fv(\()p Fq(\033)g Fv(\))29215 33180 y(=)32465 32360 y Fw(CAP)o Fv(\()p Fp(S)35874 32548 y Fr(L)36570 32360 y Fq(;)202 b Fp(D)38044 32548 y Fr(M)39097 32360 y Fv(\))p 30628 32902 10779 49 v 30628 33103 a Fm(P)31907 34370 y Fr(\033)32 b Fk(2)p Fr(@)52 b Fk(D)34380 34519 y Fn(M)35535 34012 y Fq(\026)36265 34209 y Fr(\014)36895 34012 y Fv(\()p Fq(\033)43 b Fv(\))p Fq(c)39098 34209 y Fr(\014)39727 34012 y Fv(\()p Fq(\033)g Fv(\))41539 33180 y Fq(:)5712 b Fv(\(D.1\))1200 36757 y(Clearly)-101 b(,)446 b(the)439 b(left-hand)h(side)e(of)h(\(D.1\))g(is)f(the)h(escap) 34 b(e)437 b(probabilit)-34 b(y)440 b(to)e Fp(S)35832 36945 y Fr(L)36966 36757 y Fv(from)h Fq(@)67 b Fp(D)41512 36945 y Fr(M)43003 36757 y Fs(aver)-62 b(age)g(d)561 b Fv(with)1200 38262 y(resp)34 b(ect)514 b(to)h(the)g(canonical)f (Gibbs)i(measure)e Fq(\026)23874 38459 y Fr(\014)25018 38262 y Fv(conditioned)h(on)g Fq(@)67 b Fp(D)35100 38450 y Fr(M)36668 38262 y Fv(w)-34 b(eigh)g(ted)516 b(b)-34 b(y)515 b(the)g(outgoing)1200 39767 y(rate)400 b Fq(c)4216 39964 y Fr(\014)4844 39767 y Fv(.)537 b(T)-101 b(o)401 b(sho)-34 b(w)401 b(that)g(this)g(quan)-34 b(tit)g(y)402 b(is)d(small,)i(it)f(su\016ces)g(to)h(sho)-34 b(w)401 b(that)g(the)g(denominator)g(is)f(large)1200 41273 y(compared)k(to)h (the)f(n)-34 b(umerator.)3082 43110 y(By)403 b(Lemma)h(4.2,)4727 46276 y Fw(CAP)o Fv(\()p Fp(S)8136 46464 y Fr(L)8832 46276 y Fq(;)202 b Fp(D)10306 46464 y Fr(M)11359 46276 y Fv(\))337 b Fp(\024)f Fw(CAP)o Fv(\()p Fp(S)16855 46464 y Fr(L)17551 46276 y Fq(;)202 b Fv(\()p Fp(S)19386 45775 y Fr(c)20118 46276 y Fp(n)21213 45969 y Fv(~)20994 46276 y Fp(C)71 b Fv(\))269 b Fp([)h(C)24230 45775 y FH(+)25017 46276 y Fv(\))337 b(=)f Fq(N)e Fp(j)p Fv(\003)29591 46473 y Fr(\014)30220 46276 y Fp(j)31101 45456 y Fv(4)p Fq(\031)p 30891 45997 1760 49 v 30891 47107 a Fv(\001)p Fq(\014)32986 46276 y(e)33550 45775 y Fk(\000)p Fr(\014)46 b FH(\000)35701 46276 y Fq(\026)36431 46473 y Fr(\014)37060 46276 y Fv(\()p Fp(S)91 b Fv(\)[1)269 b(+)h Fq(o)p Fv(\(1\)])p Fq(:)3526 b Fv(\(D.2\))1200 49513 y(On)369 b(the)h(other)g(hand,)377 b(note)369 b(that)i Fq(@)67 b Fp(D)18607 49701 y Fr(M)20029 49513 y Fv(con)-34 b(tains)371 b(all)d(con\014gurations)j Fq(\033)412 b Fv(for)370 b(whic)-34 b(h)370 b(there)f(is)g(an)h Fq(M)331 b Fp(\002)200 b Fq(M)1200 51018 y Fv(droplet)396 b(somewhere)g(in)g(\003)13748 51215 y Fr(\014)14377 51018 y Fv(,)h(all)e Fq(L)17611 51215 y Fr(\014)18240 51018 y Fv(-b)34 b(o)-34 b(xes)396 b(not)g(con)-34 b(taining)397 b(this)g(droplet)f(carry)f(at)h(most)h Fq(K)483 b Fv(particles,)1200 52524 y(and)535 b(there)g(is)f(a)h(free)f(particle)g(somewhere)h(in)f (\003)25032 52721 y Fr(\014)25661 52524 y Fv(.)930 b(The)535 b(last)g(condition)g(ensures)g(that)h Fq(c)45099 52721 y Fr(\014)45727 52524 y Fv(\()p Fq(\033)43 b Fv(\))555 b Fp(\025)f Fv(1.)1200 54029 y(Therefore)404 b(w)-34 b(e)404 b(can)h(use)f(Lemma)g(B.1)f(to)h(estimate)3381 56686 y Fm(X)2845 59311 y Fr(\033)32 b Fk(2D)4770 59460 y Fn(M)5870 57837 y Fq(\026)6600 58034 y Fr(\014)7229 57837 y Fv(\()p Fq(\033)43 b Fv(\))p Fq(c)9432 58034 y Fr(\014)10061 57837 y Fv(\()p Fq(\033)g Fv(\))338 b Fp(\025)e(j)p Fv(\003)14535 58034 y Fr(\014)15164 57837 y Fp(j)202 b Fq(e)16267 57337 y Fk(\000)p Fr(\014)46 b(H)18346 57651 y Fn(M)19189 57460 y Fx(2)20350 56495 y Fv(\024)20095 56801 y Fq(Z)21009 56098 y FH(\()p Fr(n)21946 56254 y Fn(\014)22499 56098 y Fk(\000)p Fr(M)24229 55786 y Fx(2)24690 56098 y FH(\))20922 57192 y Fr(\014)p 20095 57559 5017 49 v 21190 59071 a Fq(Z)22104 58368 y FH(\()p Fr(n)23041 58524 y Fn(\014)23594 58368 y FH(\))22017 59463 y Fr(\014)25581 57837 y Fv(=)336 b Fp(j)p Fv(\003)28039 58034 y Fr(\014)28667 57837 y Fp(j)202 b Fq(e)29770 57337 y Fk(\000)p Fr(\014)46 b(H)31849 57651 y Fn(M)32692 57460 y Fx(2)33466 57837 y Fv(\()p Fq(\032)34564 58034 y Fr(\014)35193 57837 y Fv(\))35664 57337 y Fr(M)36662 57024 y Fx(2)37381 57837 y Fq(\026)38111 58034 y Fr(\014)38740 57837 y Fv(\()p Fp(S)91 b Fv(\))202 b([1)270 b(+)f Fq(o)p Fv(\(1\)])p Fq(;)1644 b Fv(\(D.3\))1200 61872 y(where)577 b Fq(H)5886 62135 y Fr(M)6884 61883 y Fx(2)7977 61872 y Fv(is)g(the)h(energy)f(of)g (an)h Fq(M)517 b Fp(\002)384 b Fq(M)710 b Fv(droplet.)1058 b(Com)-34 b(bining)579 b(\(D.2{D.3\))e(w)-34 b(e)578 b(\014nd)g(that)g(the)1200 63378 y(left-hand)405 b(side)f(of)h(\(D.1\)) f(is)g(b)34 b(ounded)406 b(from)e(ab)34 b(o)-34 b(v)g(e)404 b(b)-34 b(y)14184 65057 y Fm(\022)15076 66767 y Fq(N)16525 65947 y Fv(4)p Fq(\031)p 16315 66488 1760 49 v 16315 67598 a Fv(\001)p Fq(\014)18208 65057 y Fm(\023)23153 65947 y Fv(exp)202 b([)p Fp(\000)p Fq(\014)64 b Fv(\000])p 19637 66488 12210 49 v 19637 67598 a(exp)202 b([)o Fp(\000)p Fq(\014)64 b Fv(\()p Fq(H)25199 67861 y Fr(M)26197 67609 y Fx(2)26983 67598 y Fv(+)269 b(\001)p Fq(M)30513 67248 y FH(2)31039 67598 y Fv(\)])32182 66767 y([1)g(+)g Fq(o)p Fv(\(1\)])p Fq(;)10172 b Fv(\(D.4\))1200 70257 y(whic)-34 b(h)405 b(is)f(exp)34 b(onen)-34 b(tially)404 b(small)f(in)h Fq(\014)468 b Fv(b)34 b(ecause)404 b(\000)336 b Fq(>)h(H)26968 70520 y Fr(M)27966 70268 y Fx(2)28751 70257 y Fv(+)269 b(\001)p Fq(M)32281 69817 y FH(2)33211 70257 y Fv(for)404 b(all)g Fq(M)469 b Fp(\025)336 b Fq(`)40179 70439 y Fr(c)40642 70257 y Fv(.)25194 75417 y(43)p eop end %%Page: 44 44 TeXDict begin 44 43 bop 1200 1313 a Fu(References)1806 4351 y Fv([1])606 b(G.)359 b(Ben)f(Arous)i(and)g(R.)f(Cerf,)368 b(Metastabilit)-34 b(y)360 b(of)f(the)h(three-dimensional)g(Ising)f(mo) 34 b(del)358 b(on)i(a)f(torus)3692 5857 y(at)404 b(v)-34 b(ery)403 b(lo)-34 b(w)405 b(temp)34 b(erature,)404 b(Electron.)f(J.)h (Probab.)g(1)g(\(1996\))h(Researc)-34 b(h)404 b(pap)34 b(er)404 b(10.)1806 8358 y([2])606 b(K.A.)e(Berman)g(and)i(M.H.)f (Konso)-34 b(w)g(a,)656 b(Random)607 b(paths)f(and)g(cuts,)656 b(electrical)603 b(net)-34 b(w)g(orks,)656 b(and)3692 9864 y(rev)-34 b(ersible)402 b(Mark)-34 b(o)g(v)404 b(c)-34 b(hains,)405 b(SIAM)f(J.)g(Discrete)f(Math.)i(3)f(\(1990\))h(311{319.) 1806 12366 y([3])606 b(A.)439 b(Bianc)-34 b(hi,)449 b(A.)440 b(Bo)-34 b(vier,)448 b(and)441 b(D.)f(Io\013e,)448 b(Sharp)441 b(asymptotics)g(for)f(metastabilit)-34 b(y)441 b(in)f(the)h(random)3692 13871 y(\014eld)404 b(Curie-W)-101 b(eiss)403 b(mo)34 b(del,)404 b(preprin)-34 b(t)405 b(June)f(2008.)1806 16373 y([4])606 b(A.)356 b(Bo)-34 b(vier,)364 b(Metastabilit)-34 b(y:)515 b(a)356 b(p)34 b(oten)-34 b(tial)358 b(theoretic)e(approac)-34 b(h,)366 b(Pro)34 b(ceedings)356 b(of)g(ICM)h(2006,)365 b(EMS)3692 17878 y(Publishing)404 b(House,)h(2006,)e(pp.)i(499{518.) 1806 20380 y([5])606 b(A.)408 b(Bo)-34 b(vier,)409 b(Metastabilit)-34 b(y)-101 b(,)410 b(in:)549 b(Pro)34 b(ceedings)408 b(of)h(the)g(2006)h (Prague)e(Summer)i(Sc)-34 b(ho)34 b(ol)409 b(on)g(Mathe-)3692 21886 y(matical)403 b(Statistical)i(Mec)-34 b(hanics)404 b(\(ed.)h(R.)f(Kotec)-34 b(k)17 b(\023)-623 b(y\),)403 b(to)i(b)34 b(e)403 b(published)j(b)-34 b(y)404 b(Springer,)g(2008.) 1806 24387 y([6])606 b(A.)371 b(Bo)-34 b(vier,)376 b(M.)371 b(Ec)-34 b(kho\013,)378 b(V.)371 b(Ga)-34 b(yrard,)378 b(and)372 b(M.)g(Klein,)k(Metastabilit)-34 b(y)373 b(and)f(lo)-34 b(w)372 b(lying)f(sp)34 b(ectra)371 b(in)3692 25893 y(rev)-34 b(ersible)402 b(Mark)-34 b(o)g(v)404 b(c)-34 b(hains,)405 b(Comm)-34 b(un.)405 b(Math.)g(Ph)-34 b(ys.)404 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