Content-Type: multipart/mixed; boundary="-------------0301220745781" This is a multi-part message in MIME format. ---------------0301220745781 Content-Type: text/plain; name="03-29.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-29.keywords" Surface tension, Wulff theory, Pirogov sinai ---------------0301220745781 Content-Type: application/postscript; name="BP.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="BP.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.90a Copyright 2002 Radical Eye Software %%Title: bpf1.dvi %%Pages: 42 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMBX10 CMR9 CMCSC10 CMMI9 CMSY9 CMR10 MSBM10 CMR8 CMTI9 %%+ CMMI10 MSAM10 CMMI8 CMSY10 CMMI6 CMR6 CMEX10 CMSY8 MSBM7 CMSY6 %%+ CMTI10 CMBX8 MSAM7 CMSY5 CMR5 EUFM10 CMBX9 CMTT9 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips bpf1 -o BP.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.01.22:1416 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: texps.pro %! 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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 66 /B put dup 76 /L put dup 78 /N put dup 82 /R put dup 83 /S put dup 84 /T put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6A66A4000A13D5F68BFF326D 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX9 %!PS-AdobeFont-1.1: CMBX9 1.0 %%CreationDate: 1991 Aug 20 16:36:25 % 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 (CMBX9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX9 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 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 readonly def /FontBBox{-58 -250 1195 750}readonly def /UniqueID 5000767 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: EUFM10 %!PS-AdobeFont-1.1: EUFM10 2.1 %%CreationDate: 1992 Nov 20 17:36:20 % Euler fonts were designed by Hermann Zapf. % 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 (EUFM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM10 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 82 /R put readonly def /FontBBox{-26 -224 1055 741}readonly def /UniqueID 5031986 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % 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 (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 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 readonly def /FontBBox{-341 -250 1304 965}readonly def /UniqueID 5000788 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY5 %!PS-AdobeFont-1.1: CMSY5 1.0 %%CreationDate: 1991 Aug 15 07:21:16 % 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 (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY5 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 readonly def /FontBBox{21 -944 1448 791}readonly def /UniqueID 5000815 def currentdict end currentfile eexec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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 8 /braceleftbig put dup 9 /bracerightbig put dup 12 /vextendsingle put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg 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 56 /bracelefttp put dup 57 /bracerighttp put dup 58 /braceleftbt put dup 59 /bracerightbt put dup 60 /braceleftmid put dup 61 /bracerightmid put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 80 /summationtext put dup 82 /integraltext put dup 83 /uniontext put dup 84 /intersectiontext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put dup 91 /uniondisplay put dup 92 /intersectiondisplay put dup 98 /hatwide put dup 110 /braceleftBig put dup 111 /bracerightBig put dup 113 /radicalBig 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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cleartomark %%EndFont %%BeginFont: MSAM7 %!PS-AdobeFont-1.1: MSAM7 2.1 %%CreationDate: 1992 Oct 17 08:30:29 % 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 (MSAM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM7 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 54 /lessorequalslant put dup 62 /greaterorequalslant put readonly def /FontBBox{0 -576 1569 1076}readonly def /UniqueID 5032012 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1C87678CE98C24B934A76220 4DD9B2FF3A49786028E35DDE10AD2C926BD30AD47015FFE9469DE1F793D1C53A C8812CBCD402444EAEA7A50ECC186431EE52EE8453683AEC95E456840BE0C86B C5F5FFF679BB62F2B28D7268A06205508769CDC9C37768A9FE109FF077CE0E17 9831D554030EAE0FCF724C4E634DB32DEC9DD3915E6EA7A11C524241C3AE0E64 D32B4199268E6EAA30BD2D896BF5D3271DCDE73EA7C5B891B83E2B6ECB37A624 27E45D2FAE5E2430B23A077C526652C149C0A74AFC47962A80F4D3CA1C27FDAA C4D9E221DCDE6B7A521F1BC3EB69A04D8EAFF08B20503092DDB12F96197F4053 C3B3CF22AD6C5273505E627A6941022ADBB9A4C468B6E4A9EEE9A28499813AD1 536CF5BBE72A0D586F1E1CAEFC9E4E116AB4195F41D056D7407E121679F9AE3F C1B389368FF1C276CCDABF3CAF54EEB15594E028A92DAA461CEE87D159A5EECE 4C664AFEFCCC72DFEE7D90E9C2B03A4490CFF4C3EEDFC69E0083D898B995D789 48C442FBFFC067A0936B7268123239A73860CA130F18846B2378689E48691C98 0F8110E70AB919F6133A70F964110D67CCC793AD7D0A8909771C20A06342FF94 98938263097E84B970E3197BCE3B659FD9F760C1F3A3D8036EC27B4C040DB93B 5E3C3B033438B0D982F027D524AA72C506FAF860B30A7DF5DF3D73B8E229666C 7B0A7B49D7E6D62C1F79743B266648777FFD457C93E40ADB929E04890CE5F7A3 C37DB82158B3868A62A63D903CB61F15D3F92F2323F2BFB06CCD9188E5BC70F4 2FB0E393984499418AFFE0663AAFD236FDE91DCD8A794DFD4CB9CEDB8A8CB0BD FD4EDF4F2D323E028F0286A4232ACF9860BA7BFAA13537691F5524440AA3E8A2 08B4518909119A58B49ACAD5ACD9DE9B76D97026E59A3388E278EE68807685B3 0430E09F2D9169CE0FBEDA56A05EEA06AFC0D1849975FEA8451827C0E8D7EF2B B2F6438491BEE71BC5D62039FD07381EAE6558C9E0B4FBF78C8043A55C1C0E5A DA8B6E46E7D67D5BFAE7EE3C945DB100426E442933A35D7054AF675BDCD455EF 9EA4460AA15856A66F48525773643CB3AA10E4DB063FB43CD34ED81F0F19B6B1 A88D2102CBED46061536FA18FC3DBD366A6291DA939336DEB0EFE366C8A26C5F 474E614391DE59ED9289A6D1B9A4A09D09EBC8899C42DDA90A16A23E02C4DBC9 1E4FB1DA154EE5155158036E7557A15F00B4F4478BAC117479A15343E5888D9F F8C5A389AD7459F0ABD1A1796CE36D95EE2FE6AC0644CB3CB80D3C8968232652 8D6D4A9A2C4C30330531D95BC52B450B45ECEBF0B88C450E0376A67E3F6EB099 B2C4BFB35D733DCEFEB400BBD4D8CFF7D14F87E1761FD3389F8E4D46E90E9C42 E091266A0A294EF74FE969CEFB3226F308890D7F6EB73853F3A6FCA5222D6361 B2F6A6C2D3831451F8A37D7AB4F4B5885F872F591CBAD1A82776ECA13D013F49 BFA15037B1118FD1ADD66668EDF08C1111D3A709919A4F42269056C31B65A85E 78C269FBF1CF2EC3E2B997E3E2DD178B7F83A64C52B38C9D9054377E1E0C9AB4 F7D24BAE06 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX8 %!PS-AdobeFont-1.1: CMBX8 1.0 %%CreationDate: 1991 Aug 20 16:36:07 % 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 (CMBX8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX8 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 readonly def /FontBBox{-59 -250 1235 750}readonly def /UniqueID 5000766 def currentdict end currentfile eexec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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 12 /fi put dup 13 /fl 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 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 58 /colon put dup 59 /semicolon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 70 /F put dup 71 /G put dup 73 /I 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 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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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 6 /plusminus put dup 48 /prime put dup 67 /C put dup 84 /T put dup 85 /U put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % 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 (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 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 66 /B put dup 83 /S put dup 84 /T put dup 90 /Z put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueID 5032014 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6A66A4000A13D5F68BFF326D 1D432B0D064B56C598F4338C319309181D78E1629A31ECA5DD8536379B03C383 D10F04E2CB7C8461B10646CD63AFEB7608468CA0FCFC4D3458FB43D22879B515 27DD9CCF44C2BFCD95A4DE911E4915FBC02335E9999FD9B546134081D6DA3792 EC4A76DEBA77635BE52E09986268A919CB48B5EFB1A1301EE0683CB5709BC8CE D819D799020CBA673BA39C911075501395B1FD20EAD392C9D5A8C9FD1198C737 D1A614CF0C0432F29DDEB4BF9DB026908DBE89EE522B7D55DE9BF64AFBE6248E 2E10466655EB9083E7D23E3F0EE26154F191BEBC9987930CD4B4CABE1275BDF9 8755EF3D531FDF91D54954FC53F15A38D1E8F8D1E36447484FA2C09D34813615 838B6330FEAE536D08376E4A0FDDF58CDF5647C9F1FF3A7D1ACAD376DB3CADB6 9459F7A5D4F1864863B79E9F93A1EDE8B99C3138D26227C01F6FE0AAC800F2E5 94DD81CF7B1355B642CE45CB532FC5B535D66EDFFEA076C009E87406D9772D71 848C3C53B7496A5D6B58679EF11E114C5F457C6A0D3CDE50278E4A89D5393B1C F877CF4E2142A4D045C4AA9138105D748903BACC28FD43DFEDB341E1FCDBE2EA D412498FBB5374D6836CFBEB13D4C2B7B9625C25B037FDA9DCC42F5679C4B3C1 6340E341F73A921015D6FB8B0D7B8961FB3614B78DEDC0722EC190DA7637E7E8 896BF3D096AA4C870282DDD491740660FFF722FC06DE6FA9763D3FEC37003552 A89D45A917622B6A3693799E9FA88D36959C2229D2A51651DA0D5D4E2333A6BB 929B7D26C143DEDBC05487532A02CB62916A646F58084A79DBE2AE36D0011C73 F97A5912B736C5118E19CAC085B17C9070C59EE00B8CDCEE93DDCDF5BDEFB5F0 8872C77BA070C12886B9AB684FBE46841D2C1DA85E1ECB50200B017DC71FB9D1 7328E842A71AA9C5EFF83D869015AE1160A63F3A1388DF56F8DF2224597766D0 1BD59A87D8CE4861242B4B13E940D92ECD9F39620B65E9EED16251FF33691B65 6B966FE9EC8F3F84CF55F95F2F4AFE08768C448EA17B1B815B966A2F82D27A9D E7D80DC986174F26E407C8962542E2E3021A0CD14BEF6115D2AEE6A1DBBEDDF6 7DD6FF3499B2062A145895C5850DAED28D0503A1D5D855ABF3AB23322140ACC6 FA3F605EEB5EE47BBCCAD61746027527DFBEF5277E7536C0F872A23C26438049 9FA05F5EC3B1C2A236A9F66EFD657FEEE7806E5D141DED9C9A2A46988D5FAAA4 E9FD1FCEDA94FAA63E86897D19DF25318867A9CD4AD110174914AA50A7E026AF 29CC42513310B97169DED9043B002866FFC5A675740CBA7F942CD9837ADAADAD 96DB0480EBD2AB53CD5A360BFBE2B6EBCA99F8540B60A8F3C65EC79B07411021 E769701D79238ABFC11243E333EFFF2847C14AB762CF43FD8AFD7F03330E08EC 54499345598BD37FF80DA8D6F2B6A808EDEEFD3D7D202F28A4E17C15DC89DC9C A5BF859849EB670822FFD07620F0BCBB90ED175F6B0BB54762DE06EBCDD50333 7D2243C1B0E5E86860FA44919AEB59A6FC0C682FD640B83F9285C45579BEBC9F 660370BF55E6492D21986C2F18F851B8580F98F1D51B42CC56DF25CB413135F4 0E668BD93C01FCB0CEC83542275273B41177ADD370795FA85F604CD18E41A8E3 EE0E0CF14B0D8DB47F0E296324DDE60F27A5FDD8B5C31E6E952E3009B5BA3BD6 C26AB1D9E76759639371A9756D028DD7760B3187696EF88053253BC98D5AF9DC 538D014EE5DCB31E8A7168D93292E01714BE4E7ADA789EF48D9070ED8DCE5077 2B167C70A79AD8A20289B5F152CC0DF1F8C7DE330C0EE8DAC0D4BC1E7E27D356 C220D1FF24C464A96E0C937D1681618030576FD9D654C942DD37DFFCA153A0CF 006F5DC6B7B3542501722A01803F04885C05B3213FDD0E190BAE69BFC3FF9544 094E842EFB73DDC22D08DB9CDA217EC09DFE1DC0BB987C0F3F6BD35360BA5B45 F7346C92012327452373BC87C6449CDF36DAB0E11B5893E1CE70B168710F3B48 25E0FD080D141D58E00B5571F2526CE446FDAC8BE0140DAF533E3AC380618F8E 514BD18DE2A301C53B02D82C9234CE11C79B3F81ACF9BE68C2579F0A6F5B5DBD 3C63ACACB54493A5940DB26CB64B41E99EA3287224CDA33109EB346C1B703579 EDF47B68F968A0A673E383B9795BCEC94B7FA36EC6F1222EB793F0B3DA3CB148 4AAF2E05AEC0C89C1737168FE9A5B136F463594BDEF4E8AE1C3FB3E067094B6F E754B4392CD09F03A25B1D3CB179ADB6B8E9D40CBDDF2B3F4744672C54CEA2E1 363A5669742D51690F05169CB7D927A621C1EC6E78CCE3D7134F4C493C323562 3486D7864E849CCAD4E4097ECA34E4ABAE148E798B4D96672FB2BF9A520C8585 C25B82F334330FC011EB9AF93C09EDE74DD19742FFF8C23EAE73222E6DC49693 60C9E58BD9DE3FD078FCC230D881083CE3FB1D5361C6F5E9B3BB1BA3AD18D30D 79F6D6107752B3DDB49A08393AB162D311546E248733D696C58BC434DD0FA66D 1C23069C2F2AC001F7C89861554567A514AA51403A75C1E27F8FC7CE5CD174C9 17E87B513D433C57F782CBA08AD270714F0DC08FCA0D18B967EC6683D69CDBEC 64FA84FE021B77E2774E0EB22CB7B126E18C37B6C78A475663BE94E994FA9F67 7EBFAA01512A37A72E31CAF5D4D196C96836D1CB835251E5E74214386087D48F E5E11EE8EDE5AA591F0F70652A48DD653B6B9FFFEBAB6C8AE6BB8C829F9AE22F 4A29706F71A25405FB56DF74E1C18D3603F6E957D4B466559F5BE55529D81754 8C11875380E23A8F362D1A68B4CF1059F218E7BBF31E23993FCF9A407C2D65B7 B12101B01F1F3B7497D0A9D7DA9C37CE59D1C18A6CC0CB37D718ECF943B95412 EE09B51CA0FDE83DBB60D0333047B17E51724B3173DDF30BD293A55908BE4AD3 6C2BFACBEECFAB2F57CFAE84A0F98E169ABE26D1316E0E 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 1 /periodcentered put dup 3 /asteriskmath put dup 6 /plusminus put dup 7 /minusplus put dup 21 /greaterequal put dup 24 /similar put dup 26 /propersubset put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 54 /negationslash put dup 59 /emptyset put dup 67 /C put dup 68 /D put dup 71 /G put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D5FC1B2109839E5B52DFBB2A7C1B5D8E7E8AA0 5B10EA43D6A8ED61AF5B23D49920D8F79DAB6A59062134D84AC0100187A6CD1F 80F5DDD9D222ACB1C23326A7656A635C4A241CCD32CBFDF8363206B8AA36E107 1477F5496111E055C7491002AFF272E46ECC46422F0380D093284870022523FB DA1716CC4F2E2CCAD5F173FCBE6EDDB874AD255CD5E5C0F86214393FCB5F5C20 9C3C2BB5886E36FC3CCC21483C3AC193485A46E9D22BD7201894E4D45ADD9BF1 CC5CF6A5010B5654AC0BE0DA903DB563B13840BA3015F72E51E3BC80156388BA F83C7D393392BCBC227771CDCB976E93302531916BE0BD8482003329AABB90B5 D88466D4C618DB7CBCC3C244CE78CABA16B5273D9B8F08974A6A648301F019FC 0720D38E0DF2D44B866DF064EC53E1AD7143FB9D68FDA456C5BBD2FC4F57B1AD 1CBD5331934DA29C9499CFC1619CB1C09F518FAFA2B69633DB4ACBDC7551A099 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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 (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 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 /alpha put dup 12 /beta put dup 14 /delta put dup 34 /epsilon put dup 59 /comma put dup 61 /slash put dup 68 /D put dup 75 /K put dup 76 /L put dup 78 /N put dup 82 /R put dup 99 /c put dup 100 /d put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 114 /r put dup 120 /x put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 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16C0EC76630AD3C8C27E5F1D830F52B73756198603D1477C3D2D8E7C05EA66B7 52403C86EDBAC1CEC1281F99EBFA2C798AF250F6822DEF7DC3679B03545188BB 6145B5D0E1365D12A7BC7C6E22472645E9F5C80CB98D2F22A5FC7CC4A05EA2E8 1BC37BE47B857FF3 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 3 /asteriskmath put dup 6 /plusminus put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 26 /propersubset put dup 28 /lessmuch put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 54 /negationslash put dup 56 /universal put dup 57 /existential put dup 59 /emptyset put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 70 /F put dup 71 /G put dup 72 /H put dup 75 /K put dup 77 /M put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 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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 /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 11 /alpha put dup 12 /beta put dup 14 /delta put dup 16 /zeta put dup 24 /xi put dup 25 /pi put dup 27 /sigma put dup 31 /chi put dup 33 /omega put dup 34 /epsilon put dup 58 /period put dup 59 /comma put dup 61 /slash put dup 63 /star put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 68 /D put dup 72 /H put dup 75 /K put dup 76 /L put dup 78 /N put dup 82 /R put dup 86 /V put dup 88 /X put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e 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 112 /p put dup 114 /r put dup 115 /s put dup 118 /v put dup 120 /x put dup 121 /y put dup 126 /vector put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueID 5087383 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 3 /square put dup 32 /squiggleright put dup 54 /lessorequalslant put dup 62 /greaterorequalslant put readonly def /FontBBox{8 -463 1331 1003}readonly def /UniqueID 5031981 def currentdict end currentfile eexec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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 11 /alpha put dup 12 /beta put dup 14 /delta put dup 16 /zeta put dup 17 /eta put dup 22 /mu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 63 /star put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 72 /H put dup 75 /K put dup 76 /L put dup 78 /N put dup 79 /O put dup 80 /P 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 99 /c put dup 100 /d put dup 101 /e put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k 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 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 126 /vector put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 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 AFC27E4A7E07D09FB08355F6EA74E530B0743143F2A871732D62D80F35B19FD2 C7FDF08105847F13D50934419AC647CBA71DF74F4531DC02BBDA22AEEA3FBBBB 407E0ACC52BDC60D01A29407CC4F93EB8BF6D4813E9BA858D54F38918AC82720 4956D50291F0546E50FCAFA6DBD0099123F5ECD4AB338DB310DB4CAE11337A89 8ED99B6F483940C97544F888EAF0CBEB11094A13C073D0061808662A04A82BA0 AD35E8782F854AF66C20C0FEF18D0ECDD1646321B93D327E53D88CA0E825FA95 05AA57BD742586BAB9C3FD5D8D28FCE6E11AA9C75F261015D227281F4CB11551 DA200FB50A0523DEAB4D12A435B240554F52441D759F8E5DD4E48685C8646CD9 94DA4E36637A9CA336D2574780AD5F82AFEC51EFFD575617F571D6F4C19F4F8F 74CA5AAB8BA333F06F5DC213B09FCEC766DCB65B85F2DE22AB5E242209587DA9 E39B2D0149C627B3A5BA992309992CDAF3497C360EB69BFCC89B7CDF57AF589A 96C8823BC779A49BC28831C9E44BB0EFEFA68D7E910A0FFBF4D632BAD5EFE1D0 F6A70C16E482BD774FAC08BAC60BEB90990A6509CED93637461452FE04CD8F90 79355E7F5EE71391BC4C53482F56E34EC9432582045F790239730C1EAFFE8A2B 72015346561DC130735D3E7888947EACE8C74447334C3AF269513F8F9918CA6C 7D4FFC75E7E76C9BE305944CA6110EB36FDFEB519EBC29DD7D35D04D9958D4B8 EEB19B8AEDD26C5350B482E491785AB44AFC868CBE69520DBC12A93AFEFC7E04 D934B6477321C4AE5E60B23255D0758452E02AD274934F046B05411550E7B605 A210C82542CE4B475BFB4656C5E3288A334C6C0E02FF740C3537A0260F64A3B6 32F5E051D842AB795C40F75063EA9D65FEE3BC7278FB78868564F8745ACD19DC 74942BC3FDC730C7323A9E0A6A7A9481A3FEA155AE733848A0BDE9BE7901896E 42737FCED346B74BB3214CB652576C7B16E32FEACB7434D1FC5068A39247424B C7ADCDB5E6D51ABB68AE97DD0264F086403677EE6CD123DB1D67EAAD1DFEA809 177924FFF87DB3622176EA672F6BB470042BFEBDCCF82C6E162F75F453B2554D 1E58CA76D7C9056B8240E2AEBCFFEDCB332753B562F5D76D11933C0BF840D982 587A9C4B7D35F405F962FDE8BBE5C427D524E78355A13F36AEA6EFAC8FF25183 4AF99108B29379BA20E289A9AEAB7BADEFB144C35183A6B34D6C5DF89FFA89E9 9A65865A4468D92B121E481D6445E203F7B0E82F3B8BCD981BC89BFA0E79A45E ACFB486387CF6B6F5AE57FA78E66A03ED013757F423C1D943470570FE1996929 243BAA3807EF933BF943EB189EC38E3124A80441C7DCDB70F20B4F143BDB602B 487495B5BF2BDE75265AE266104B8C5760C345CA9C822CF57932FA71679A1FE1 159ADD9FB615937BA996A0CBD887442F61CFF2CB35E1D1356D0B7D5C5D5BEDB6 4D41B93DF1EC8D26607D64FF22B464DB66A24C24C2C9502846FE5ECC2B20C1C8 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D44B8FBBFAD59E4CC48FF1AFB915545C343CBD4938893346EC5A4B098F3CA041 AABF9F3E722E181BFFA02378AD053AC5F7370E15ED3A25457F5F25962625D703 F0184A2F0840DD131DB7D8255124AD048DE40265E7680D9C6ECF7374AACFB451 E44A1827B7B0BC9CC5D71F16649595EF69D3BFC9B3C9B9942B4CA6936A03C56C F43FAF770E337CD3947CCBB1BB45FD06 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI9 %!PS-AdobeFont-1.1: CMTI9 1.0 %%CreationDate: 1991 Aug 18 21:08:07 % 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 (CMTI9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI9 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 19 /acute put dup 39 /quoteright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 50 /two 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 71 /G put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 87 /W put dup 90 /Z 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 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-35 -250 1148 750}readonly def /UniqueID 5000827 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E3948FFB3DF7BFF10C9BDA4EFE5F68A8CB1526990D1357AE6D2F7C2D2EF8496 4E47B39E6712EB8908A3265E5FAB40567E866C244814449F1E993AAB422C3F1D DFA8C7118584F2E5197FD4BFA3A8AE9E953C6CD4672C0FF51E41C3A919749C1A F06650DF4C5E17492164BDBCDF22609A74BFA7F69960A64B9F949FFC2A807458 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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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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMCSC10 %!PS-AdobeFont-1.1: CMCSC10 1.0 %%CreationDate: 1991 Aug 18 17:46:49 % 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 (CMCSC10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMCSC10 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 18 /grave put dup 19 /acute 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 57 /nine put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N 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 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 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 120 /x put dup 121 /y put readonly def /FontBBox{14 -250 1077 750}readonly def /UniqueID 5000772 def currentdict end currentfile eexec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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 5 /Pi put dup 6 /Sigma put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 19 /acute put dup 22 /macron 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 47 /slash 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 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 90 /Z 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 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY9 %!PS-AdobeFont-1.1: CMSY9 1.0 %%CreationDate: 1991 Aug 15 07:22:27 % 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 (CMSY9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY9 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 21 /greaterequal put dup 66 /B put dup 67 /C put dup 102 /braceleft put dup 103 /braceright put readonly def /FontBBox{-30 -958 1146 777}readonly def /UniqueID 5000819 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: 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 /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 14 /delta put dup 59 /comma put dup 66 /B put dup 76 /L put dup 82 /R put dup 84 /T put dup 85 /U put dup 100 /d put dup 104 /h put dup 110 /n put dup 118 /v put dup 126 /vector put readonly def /FontBBox{-29 -250 1075 750}readonly def /UniqueID 5087384 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 3 /Lambda put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 18 /grave put dup 19 /acute put dup 34 /quotedblright 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 47 /slash 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 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 93 /bracketright 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 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put readonly def /FontBBox{-39 -250 1036 750}readonly def /UniqueID 5000792 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F 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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 11 /ff put dup 12 /fi put dup 13 /fl put dup 40 /parenleft put dup 41 /parenright 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 63 /question put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 72 /H put dup 73 /I 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 89 /Y 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 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 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec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42 46 46 51 74 23 42 28 28 46 42 28 42 46 42 42 46 12[65 51 66 1[62 2[82 57 2[35 1[70 59 1[69 65 64 68 5[28 28 4[46 46 46 46 46 46 1[28 33 28 2[37 37 26[53 51 12[{}54 90.9091 /CMTI10 rf /Fk 157[39 98[{}1 66.4176 /CMEX10 rf /Fl 170[39 35 16[34 18[18 41[48 2[32 2[48{}7 49.8132 /CMSY6 rf /Fm 165[44 5[44 37 16[44 66[{}4 66.4176 /MSBM7 rf /Fn 149[20 2[35 35 9[47 6[44 39 43 60 1[49 1[58 6[42 2[55 37 7[35 4[0 2[47 47 71 19 14[71 6[55 1[55 2[55 13[55 55 2[35 1[20 55{}28 66.4176 /CMSY8 rf /Fo 142[91 1[61 61 11[51 5[101 101 51 116 131 3[76 76 43 1[96 14[80 80 2[81 81 81 81 81 81 6[80 80 6[73 73 4[53 53 72 72 4[68 68 6[67 67 54 54 3[30 2[53 53 6[42 42{}40 90.9091 /CMEX10 rf /Fp 205[30 30 30 4[47 1[24 24 17[30 18[41 1[50 38{}10 49.8132 /CMR6 rf /Fq 135[35 5[29 6[33 25 22 36 3[32 28 16[46 3[48 1[42 51 6[50 6[32 1[19 24[29 19[28 1[35 40 11[{}19 49.8132 /CMMI6 rf /Fr 143[76 1[45 2[45 25 2[45 45 9[61 61 2[65 90 56 57 50 55 77 74 63 2[109 1[69 2[77 54 65 1[70 48 60 73 5[45 1[51 51 1[0 2[61 61 91 15[91 4[91 1[71 1[71 2[71 71 2[71 10[71 2[45 71 25 71{}45 90.9091 /CMSY10 rf /Fs 129[35 4[35 40 1[34 2[33 32 1[36 1[43 62 21 37 29 24 41 2[33 36 31 30 37 29 7[58 1[41 3[53 3[56 1[48 60 2[58 3[58 1[53 53 37 35 1[35 1[20 20 23[33 44 1[44 3[40 1[41 31 7[31 1[31 1[40 45 11[{}45 66.4176 /CMMI8 rf /Ft 193[71 7[71 21[91 28[71 3[{}4 90.9091 /MSAM10 rf /Fu 129[45 3[42 45 52 65 44 52 1[43 41 1[46 44 55 80 1[47 37 31 52 2[42 47 39 1[48 38 5[62 1[75 1[53 62 1[56 69 1[58 69 73 1[62 77 2[76 3[75 65 69 68 48 45 71 45 71 25 25 23[42 57 59 57 54 1[40 52 47 52 40 1[55 4[45 40 1[40 1[51 58 11[{}61 90.9091 /CMMI10 rf /Fv 133[31 37 35 51 35 41 25 31 32 35 39 39 43 63 20 35 1[24 39 35 24 35 39 35 35 39 6[47 2[77 2[55 43 56 1[52 59 1[69 48 59 1[30 57 59 50 52 58 55 54 57 6[24 7[39 3[24 27 24 4[24 19[39 5[45 43 47 11[{}54 74.7198 /CMTI9 rf /Fw 129[35 5[37 3[27 28 2[39 35 39 4[20 3[31 7[20 1[20 1[53 1[72 1[53 51 39 52 1[48 55 53 1[44 2[25 53 1[46 48 54 51 50 53 3[55 1[20 20 35 35 35 35 35 35 35 35 35 35 1[20 2[55 1[27 27 17[35 16[53 1[49 1[59 44{}51 66.4176 /CMR8 rf /Fx 165[61 5[61 51 66 3[66 1[61 9[61 66[{}7 90.9091 /MSBM10 rf /Fy 135[56 2[56 54 42 55 1[51 58 56 68 47 2[27 56 1[49 51 57 54 1[56 9[101 2[71 56 73 1[67 2[90 3[37 7[74 10[50 4[50 2[29 1[29 44[{}29 90.9091 /CMCSC10 rf /Fz 128[45 45 2[45 40 48 48 66 48 51 35 36 36 48 51 45 51 76 25 48 28 25 51 45 28 40 51 40 51 45 2[45 25 45 25 56 2[93 68 68 66 51 67 1[62 71 68 83 57 71 1[33 68 71 59 62 69 66 64 68 3[71 1[25 25 45 45 45 45 45 45 45 45 45 45 45 25 30 25 71 1[35 35 25 4[45 25 10[45 2[45 4[76 51 51 53 4[66 68 61 63 1[76 57{}90 90.9091 /CMR10 rf /FA 152[38 38 34[41 50 44[60 20[60{}6 74.7198 /CMSY9 rf /FB 129[38 7[37 7[46 5[44 3[40 14[52 45 1[58 5[52 9[58 6[21 44[34 14[{}12 74.7198 /CMMI9 rf /FC 134[46 2[46 46 44 34 45 47 42 47 46 56 38 2[23 46 48 1[42 47 44 43 46 10[61 61 59 46 60 1[55 1[61 74 51 1[42 30 2[53 55 62 59 1[61 7[41 1[41 41 41 41 41 41 41 41 1[24 28 24 24[41 41 18[{}50 74.7198 /CMCSC10 rf /FD 132[38 34 41 41 55 41 43 30 30 30 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b(TENSION)g(AND)f(WULFF)h(THEOR)-6 b(Y)983 b(3)118 366 y Fz(\()p Fu(\014)5 b(;)15 b(h)p Fz(\()p Fu(\014)5 b Fz(\)\).)51 b(In)33 b(particular)g(on)g(the)g(phase)g(co)s(existence)i (curv)m(e,)f(one)f(can)g(de\014ne)g(\(see)h(Theorem)e(4.2)118 476 y(b)s(elo)m(w\))i(t)m(w)m(o)h(distinct)f(Gibbs)f(measures)g Fu(\026)1636 438 y Fw(+)1636 509 y Fs(\014)s(;h)p Fw(\()p Fs(\014)s Fw(\))1875 476 y Fz(and)g Fu(\026)2110 438 y Fn(\000)2110 509 y Fs(\014)s(;h)p Fw(\()p Fs(\014)s Fw(\))2348 476 y Fz(whic)m(h)g(are)h(measures)g(on)f(the)h(space)118 625 y Fr(f\006)p Fz(1)p Fr(g)324 592 y Fm(Z)368 568 y Fq(d)410 625 y Fz(.)41 b(They)29 b(are)i(obtained)g(b)m(y)f(taking)i (the)e(thermo)s(dynamic)g(limit)h(of)g Fu(\026)2827 587 y Fw(+)2827 658 y Fs(\014)s(;h)p Fw(\()p Fs(\014)s Fw(\))p Fs(;)p Fw(\003)3131 625 y Fz(and)f Fu(\026)3363 587 y Fn(\000)3363 658 y Fs(\014)s(;h)p Fw(\()p Fs(\014)s Fw(\))p Fs(;)p Fw(\003)3636 625 y Fz(.)118 746 y(Eac)m(h)c(of)f(these)g (measures)g(represen)m(ts)g(a)g(pure)f(state.)41 b(The)24 b(a)m(v)m(eraged)j(magnetization)h(in)d(eac)m(h)h(phase)118 854 y(is)31 b(denoted)f(b)m(y)1023 965 y Fu(m)1103 927 y Fw(+)1103 995 y Fs(\014)1187 965 y Fz(=)25 b Fu(\026)1338 927 y Fw(+)1338 998 y Fs(\014)s(;h)p Fw(\()p Fs(\014)s Fw(\))1543 965 y Fz(\()p Fu(\033)1630 979 y Fw(0)1670 965 y Fz(\))91 b(and)g Fu(m)2114 927 y Fn(\000)2114 995 y Fs(\014)2198 965 y Fz(=)25 b Fu(\026)2349 927 y Fn(\000)2349 998 y Fs(\014)s(;h)p Fw(\()p Fs(\014)s Fw(\))2554 965 y Fz(\()p Fu(\033)2641 979 y Fw(0)2681 965 y Fz(\))15 b Fu(:)719 b Fz(\(2.1\))218 1232 y(Observ)m(e)22 b(that)g(if)g(w)m(e)g (replace)h(plaquettes)g(b)m(y)f(b)s(onds)e(w)m(e)j(reco)m(v)m(er)g (\(mo)s(dulo)f(an)g(additiv)m(e)h(constan)m(t\))118 1340 y(the)f(energy)g(of)f(the)h(nearest)g(neigh)m(b)s(or)g(Ising)f(mo)s (del.)38 b(Our)20 b(system)i(is)f(in)g(our)h(in)m(ten)m(tions)g(the)g (simplest)118 1448 y(mo)s(di\014cation)33 b(of)g(the)g(nearest)g(neigh) m(b)s(or)f(Ising)h(mo)s(del)f(where)g(the)h(spin)f(\015ip)g(symmetry)g (is)g(brok)m(en)118 1556 y(but)g(the)h(ground)e(states)j(are)f(k)m(ept) h(unc)m(hanged.)47 b(This)32 b(c)m(hoice)i(has)e(b)s(een)g(to)i(giv)m (e)g(up)d(an)m(y)i(attempt)118 1664 y(of)38 b(generalit)m(y)i(and)d (instead)h(to)g(fo)s(cus)f(on)h(a)g(particular)g(mo)s(del,)h(where)f (the)f(main)h(ideas)g(are)g(not)118 1771 y(obscured)22 b(b)m(y)i(to)s(o)g(man)m(y)f(tec)m(hnicalities.)41 b(Nev)m(ertheless,) 27 b(w)m(e)c(b)s(eliev)m(e)i(our)e(analysis)g(extends)g(to)h(\014nite) 118 1879 y(range,)31 b(man)m(y)g(b)s(o)s(dy)d(hamiltonians)j(of)g(the)f (form)1661 1956 y Fo(X)1643 2153 y Fs(X)5 b Fn(\032)p Fw(\003)1826 2042 y Fu(V)1879 2056 y Fs(X)1946 2042 y Fz(\()p Fu(\033)2033 2056 y Fs(X)2101 2042 y Fz(\))118 2279 y(pro)m(vided)38 b(they)h(are)g(in)m(to)h(the)e(Pirogo)m(v-Sinai)j (class)e(and)f(under)f(the)i(assumptions)f(that)h(the)g(p)s(o-)118 2386 y(ten)m(tials)i Fu(V)501 2400 y Fs(X)607 2386 y Fz(are)e(symmetric)g(and)g(translational)h(in)m(v)-5 b(arian)m(t,)43 b(with)38 b(ground)g(states)i(the)f(constan)m(t)118 2494 y(con\014gurations.)i(The)30 b(symmetry)g(assumption)g(means)970 2666 y(for)h(all)g Fu(X)r(;)379 b(V)1770 2680 y Fn(R)p Fs(X)1898 2565 y Fo(\020)1952 2666 y Fz(\()p Fr(R)p Fu(\033)s Fz(\))2154 2680 y Fn(R)p Fs(X)2283 2565 y Fo(\021)2362 2666 y Fz(=)25 b Fu(V)2511 2680 y Fs(X)2579 2666 y Fz(\()p Fu(\033)2666 2680 y Fs(X)2733 2666 y Fz(\))15 b Fu(;)667 b Fz(\(2.2\))118 2843 y(where)29 b Fr(R)h Fz(denotes)f(the)h(symmetry)f (wrt)g(the)h(origin)g(and)f(\()p Fr(R)p Fu(\033)s Fz(\))2351 2857 y Fs(j)2414 2843 y Fz(=)24 b Fu(\033)2561 2863 y Fw(\()p Fn(R)p Fw(\))2675 2844 y Fl(\000)p Fp(1)2759 2863 y Fw(\()p Fs(j)t Fw(\))2850 2843 y Fz(.)41 b(W)-8 b(e)31 b(will)e(pursue)f(the)118 2951 y(discussion)i(on)g(p)s(ossible)g (extensions)h(and)f(op)s(en)f(questions)i(in)f(Subsection)g(3.4.)118 3295 y(2.2.)47 b FE(Phase)53 b(co)s(existence.)46 b Fz(The)f (phenomenon)g(of)h(phase)f(segregation)i(will)f(b)s(e)f(describ)s(ed)g (in)118 3403 y(the)e(framew)m(ork)h(of)f(the)g Fx(L)1094 3370 y Fw(1)1133 3403 y Fz(-approac)m(h.)80 b(Let)44 b(us)e(\014rst)g(recall)j(the)e(functional)h(setting.)80 b(On)42 b(the)118 3518 y(macroscopic)31 b(lev)m(el,)h(the)e(system)f (is)h(con\014ned)f(in)g(the)h(torus)2263 3495 y Fo(b)2258 3518 y Fx(T)25 b Fz(=)g([0)p Fu(;)15 b Fz(1])2620 3485 y Fs(d)2692 3518 y Fz(of)29 b Fx(R)2860 3485 y Fs(d)2930 3518 y Fz(and)g(a)h(macroscopic)118 3626 y(con\014guration)40 b(where)g(the)g(pure)f(phases)g(co)s(exist)i(is)f(describ)s(ed)f(b)m(y) g(a)i(function)e Fu(v)k Fz(taking)e(v)-5 b(alues)118 3734 y Fr(f)p Fu(m)243 3696 y Fn(\000)243 3763 y Fs(\014)302 3734 y Fu(;)15 b(m)422 3696 y Fw(+)422 3763 y Fs(\014)482 3734 y Fr(g)p Fz(.)58 b(The)35 b(function)h Fu(v)j Fz(should)c(b)s(e)g (in)m(terpreted)i(as)f(a)g(signed)g(indicator)h(represen)m(ting)f(the) 118 3874 y(lo)s(cal)30 b(order)f(parameter)g(:)40 b(if)29 b Fu(v)1192 3888 y Fs(r)1255 3874 y Fz(=)c Fu(m)1431 3835 y Fw(+)1431 3903 y Fs(\014)1519 3874 y Fz(for)j(some)i Fu(r)d Fr(2)2042 3850 y Fo(b)2037 3874 y Fx(T)p Fz(,)i(then)g(the)g (system)g(should)f(b)s(e)g(lo)s(cally)i(at)g Fu(r)118 3987 y Fz(in)g(equilibrium)g(in)g(the)g(+)g(phase.)218 4095 y(T)-8 b(o)33 b(de\014ne)f(the)h(macroscopic)h(in)m(terfaces,)h (i.e.)49 b(the)33 b(b)s(oundary)e(of)i(the)g(set)g Fr(f)p Fu(v)g Fz(=)c Fu(m)3170 4057 y Fn(\000)3170 4125 y Fs(\014)3229 4095 y Fr(g)p Fz(,)34 b(a)f(con)m(v)m(e-)118 4235 y(nien)m(t)h (functional)f(setting)i(is)e(the)g(space)h(BV)r(\()1755 4212 y Fo(b)1750 4235 y Fx(T)p Fu(;)15 b Fr(f)p Fu(m)1976 4197 y Fn(\000)1976 4264 y Fs(\014)2035 4235 y Fu(;)g(m)2155 4197 y Fw(+)2155 4264 y Fs(\014)2215 4235 y Fr(g)p Fz(\))34 b(of)f(functions)g(of)g(b)s(ounded)e(v)-5 b(ariation)118 4375 y(with)31 b(v)-5 b(alues)31 b Fu(m)677 4336 y Fn(\006)677 4404 y Fs(\014)767 4375 y Fz(in)879 4351 y Fo(b)874 4375 y Fx(T)g Fz(\(see)h([EG])g(for)f(a)g(review\).)44 b(F)-8 b(or)32 b(an)m(y)f Fu(v)f Fr(2)c Fz(BV)r(\()2610 4351 y Fo(b)2605 4375 y Fx(T)p Fu(;)15 b Fr(f)p Fu(m)2831 4336 y Fn(\000)2831 4404 y Fs(\014)2891 4375 y Fu(;)g(m)3011 4336 y Fw(+)3011 4404 y Fs(\014)3070 4375 y Fr(g)p Fz(\))q(,)31 b(there)h(exists)118 4504 y(a)j(generalized)g(notion)g(of)g(the)f(b)s (oundary)e(of)i(the)h(set)g Fr(f)p Fu(v)g Fz(=)c Fu(m)2359 4465 y Fn(\000)2359 4533 y Fs(\014)2418 4504 y Fr(g)k Fz(called)g(reduced)f(b)s(oundary)e(and)118 4633 y(denoted)c(b)m(y)g Fu(@)636 4600 y Fn(\003)675 4633 y Fu(v)s Fz(.)40 b(If)28 b Fr(f)p Fu(v)h Fz(=)c Fu(m)1170 4594 y Fn(\000)1170 4662 y Fs(\014)1229 4633 y Fr(g)j Fz(is)g(a)g(regular)g(set,)h(then)e Fu(@)2192 4600 y Fn(\003)2232 4633 y Fu(v)k Fz(coincides)e(with)e(the)h (usual)f(b)s(oundary)118 4746 y Fu(@)5 b(v)s Fz(.)218 4854 y(The)39 b(in)m(terfacial)j(energy)e(asso)s(ciated)h(to)g(a)f (domain)f(is)h(obtained)g(b)m(y)g(in)m(tegrating)h(the)f(surface)118 4963 y(tension)27 b(along)h(the)f(b)s(oundary)e(of)i(the)h(domain.)39 b(The)27 b(surface)f(tension)i(is)f(a)g(function)g Fu(\034)3187 4978 y Fs(\014)3259 4963 y Fz(:)e Fx(S)3360 4930 y Fw(\()p Fs(d)p Fn(\000)p Fw(1\))3571 4963 y Fr(!)118 5082 y Fx(R)184 5049 y Fw(+)277 5082 y Fz(on)35 b(the)g(set)g(of)g(unit)f(v)m(ectors)i Fx(S)1383 5049 y Fw(\()p Fs(d)p Fn(\000)p Fw(1\))1568 5082 y Fz(,)g(whic)m(h)f(in)f(our)g(mo)s(del)h(has)f(the)h(expression)g (sp)s(eci\014ed)f(in)118 5197 y(Section)d(3.)41 b(The)30 b(W)-8 b(ul\013)31 b(functional)g Fr(W)1496 5212 y Fs(\014)1573 5197 y Fz(is)g(de\014ned)e(in)h Fx(L)2145 5211 y Fw(1)2184 5197 y Fz(\()2224 5173 y Fo(b)2219 5197 y Fx(T)p Fz(\))h(as)f(follo)m (ws)595 5417 y Fr(W)685 5432 y Fs(\014)732 5417 y Fz(\()p Fu(v)s Fz(\))c(=)971 5288 y Fo(\032)1081 5290 y(R)1124 5395 y Fs(@)1165 5376 y Fl(\003)1201 5395 y Fs(v)1257 5363 y Fu(\034)1297 5378 y Fs(\014)1344 5363 y Fz(\()19 b Fu(~)-64 b(n)1434 5377 y Fs(x)1477 5363 y Fz(\))15 b Fu(d)p Fr(H)1651 5377 y Fs(x)1696 5363 y Fu(;)197 b Fz(if)98 b Fu(v)29 b Fr(2)24 b Fz(BV)r(\()2401 5340 y Fo(b)2396 5363 y Fx(T)p Fu(;)15 b Fr(f)p Fu(m)2622 5325 y Fn(\000)2622 5393 y Fs(\014)2682 5363 y Fu(;)g(m)2802 5325 y Fw(+)2802 5393 y Fs(\014)2861 5363 y Fr(g)p Fz(\))q Fu(;)1081 5481 y Fr(1)25 b Fu(;)742 b Fz(otherwise)q Fu(:)3475 5417 y Fz(\(2.3\))p eop end %%Page: 4 4 TeXDict begin 4 3 bop 118 45 a Fw(4)1163 b(T.)23 b(BODINEA)n(U)g(AND)g (E.)g(PRESUTTI)118 373 y Fz(T)-8 b(o)35 b(an)m(y)g(measurable)g(subset) f Fu(A)h Fz(of)1411 349 y Fo(b)1406 373 y Fx(T)o Fz(,)h(w)m(e)f(asso)s (ciate)i(the)e(function)f(1)-25 b(I)2626 387 y Fs(A)2716 373 y Fz(=)32 b Fu(m)2899 334 y Fw(+)2899 402 y Fs(\014)2973 373 y Fz(1)3018 387 y Fs(A)3071 368 y Fq(c)3130 373 y Fz(+)23 b Fu(m)3304 334 y Fn(\000)3304 402 y Fs(\014)3378 373 y Fz(1)3423 387 y Fs(A)3515 373 y Fz(and)118 491 y(simply)30 b(write)h Fr(W)732 506 y Fs(\014)779 491 y Fz(\()p Fu(A)p Fz(\))26 b(=)f Fr(W)1129 506 y Fs(\014)1176 491 y Fz(\(1)-25 b(I)1264 505 y Fs(A)1321 491 y Fz(\).)218 652 y(Fix)31 b(an)g(in)m(terv)-5 b(al)32 b([)p Fu(m)942 666 y Fw(1)981 652 y Fu(;)15 b(m)1101 666 y Fw(2)1141 652 y Fz(])31 b(included)f(in)h(\()p Fu(m)1783 614 y Fn(\000)1783 682 y Fs(\014)1842 652 y Fu(;)15 b(m)1962 614 y Fw(+)1962 682 y Fs(\014)2021 652 y Fz(\).)43 b(The)30 b(equilibrium)g(crystal)i(shap)s(es)e(are)h(the)118 766 y(solutions)j(of)f(the)g(W)-8 b(ul\013)34 b(v)-5 b(ariational)35 b(problem,)e(i.e.)50 b(they)33 b(are)g(the)g(minimizers)h(of)f(the)g (functional)118 874 y Fr(W)208 889 y Fs(\014)285 874 y Fz(under)c(a)i(v)m(olume)g(constrain)m(t)501 1089 y(min)667 960 y Fo(\032)735 1089 y Fr(W)825 1104 y Fs(\014)872 1089 y Fz(\()p Fu(v)s Fz(\))1021 984 y Fo(\014)1021 1039 y(\014)1021 1093 y(\014)1081 1089 y Fu(v)e Fr(2)c Fz(BV)r(\()1414 1065 y Fo(b)1409 1089 y Fx(T)p Fu(;)15 b Fr(f)p Fu(m)1635 1050 y Fn(\000)1635 1118 y Fs(\014)1694 1089 y Fu(;)g(m)1814 1050 y Fw(+)1814 1118 y Fs(\014)1874 1089 y Fr(g)p Fz(\))p Fu(;)198 b(m)2257 1103 y Fw(1)2321 1089 y Ft(6)2432 965 y Fo(Z)2485 1154 y Fk(b)2483 1171 y Fm(T)2561 1089 y Fu(v)2605 1103 y Fs(x)2665 1089 y Fu(dx)25 b Ft(6)g Fu(m)2965 1103 y Fw(2)3004 960 y Fo(\033)3088 1089 y Fu(:)362 b Fz(\(2.4\))118 1298 y(Let)31 b Fr(D)s Fz(\()p Fu(m)469 1312 y Fw(1)508 1298 y Fu(;)15 b(m)628 1312 y Fw(2)668 1298 y Fz(\))31 b(b)s(e)e(the)i(set)g(of)f(minimizers)h(of)37 b(\(2.4\))r(.)118 1478 y(2.3.)47 b FE(Lo)s(cal)31 b(magnetization.)46 b Fz(The)26 b(corresp)s(ondence)g(b)s(et)m(w)m(een)h(the)g(microscopic) h(quan)m(tities)g(and)118 1586 y(the)34 b(functional)h(setting)g (describ)s(ed)e(ab)s(o)m(v)m(e)i(can)f(b)s(e)f(obtained)i(only)f(after) g(some)h(a)m(v)m(eraging)h(pro)s(ce-)118 1694 y(dure,)30 b(as)h(the)g(one)g(w)m(e)h(are)f(going)h(to)f(describ)s(e.)42 b(W)-8 b(e)32 b(\014rst)e(need)h(a)g(few)f(extra)i(notations.)43 b(Let)31 b Fr(B)3513 1661 y Fw(\()p Fs(K)5 b Fw(\))3636 1694 y Fz(,)118 1802 y Fu(K)32 b Fz(=)25 b(2)368 1769 y Fs(k)411 1802 y Fz(,)31 b Fu(k)d Fr(2)d Fx(N)p Fz(,)30 b(b)s(e)g(the)h(partition)g(of)f Fx(Z)1573 1769 y Fs(d)1643 1802 y Fz(in)m(to)i(cub)s(es)d Fx(B)2139 1816 y Fs(K)2207 1802 y Fz(:)41 b(the)31 b(seed)f(of)h(the)f(partition)h(is)1006 1959 y Fx(B)1067 1973 y Fs(K)1135 1959 y Fz(\(0\))c(=)1373 1885 y Fo(\010)1426 1959 y Fu(x)e Fr(2)g Fx(Z)1650 1921 y Fs(d)1715 1959 y Fz(:)h(0)f Ft(6)g Fu(x)1984 1973 y Fs(i)2038 1959 y Fu(<)g(K)q(;)107 b(i)25 b Fz(=)g(1)p Fu(;)15 b(::;)g(d)2718 1885 y Fo(\011)118 2128 y Fz(and)22 b(the)h(other)g(cub)s(es)f(of)h Fr(B)1068 2095 y Fw(\()p Fs(K)5 b Fw(\))1213 2128 y Fz(are)23 b(obtained)g(b)m(y)g(translations) g(b)m(y)g(in)m(teger)h(m)m(ultiples)f(of)g Fu(K)29 b Fz(in)23 b(all)g(co-)118 2237 y(ordinate)k(directions.)40 b(The)25 b(sequence)i Fr(B)1540 2204 y Fw(\()p Fs(K)5 b Fw(\))1663 2237 y Fu(;)15 b(k)29 b Fr(2)24 b Fx(N)p Fz(,)j(is)g(then)f(a)g(compatible)i(sequence)e(of)h(partitions)118 2346 y(of)k Fx(R)288 2313 y Fs(d)328 2346 y Fz(,)f(namely)h(eac)m(h)h (cub)s(e)d Fx(B)1179 2360 y Fs(N)1271 2346 y Fr(2)c(B)1420 2313 y Fw(\()p Fs(N)7 b Fw(\))1572 2346 y Fz(is)30 b(union)g(of)g(cub)s (es)g Fx(B)2331 2360 y Fs(K)2429 2346 y Fz(in)g Fr(B)2598 2313 y Fw(\()p Fs(K)5 b Fw(\))2721 2346 y Fz(,)31 b(if)f Fu(K)i Fz(=)25 b(2)3110 2313 y Fs(k)3178 2346 y Ft(6)g Fu(N)35 b Fz(=)25 b(2)3523 2313 y Fs(n)3571 2346 y Fz(.)218 2562 y(Giv)m(en)33 b Fu(K)j Fz(=)28 b(2)740 2529 y Fs(k)783 2562 y Fz(,)33 b(w)m(e)g(denote)g(b)m(y)g Fx(B)1462 2576 y Fs(K)1530 2562 y Fz(\()p Fu(x)p Fz(\))g(the)f(b)s(o)m(x)h(in)f Fr(B)2191 2529 y Fw(\()p Fs(K)5 b Fw(\))2346 2562 y Fz(whic)m(h)32 b(con)m(tains)i(the)f(p)s(oin)m(t)f Fu(x)d Fr(2)f Fx(Z)3596 2529 y Fs(d)3636 2562 y Fz(.)118 2670 y(The)i(lo)s(cal)i(a)m(v)m (eraged)g(magnetization)h(is)d(de\014ned)g(b)m(y)1295 2862 y Fr(M)1404 2876 y Fs(K)1473 2862 y Fz(\()p Fu(x)p Fz(\))25 b(=)1855 2801 y(1)p 1726 2841 303 4 v 1726 2925 a Fr(j)p Fx(B)1812 2939 y Fs(K)1881 2925 y Fz(\()p Fu(x)p Fz(\))p Fr(j)2129 2776 y Fo(X)2054 2977 y Fs(y)r Fn(2)p Fm(B)2182 2988 y Fq(K)2241 2977 y Fw(\()p Fs(x)p Fw(\))2351 2862 y Fu(\033)2403 2876 y Fs(y)2459 2862 y Fu(:)991 b Fz(\(2.5\))118 3137 y(By)31 b(abuse)f(of)g(notation,)i Fr(M)1115 3151 y Fs(K)1184 3137 y Fz(\()p Fr(\001)p Fz(\))f(can)g(b)s (e)e(view)m(ed)i(also)h(as)e(a)h(piecewise)h(constan)m(t)f(function)f (on)h Fx(R)3578 3104 y Fs(d)3618 3137 y Fz(.)218 3245 y(F)-8 b(or)31 b(simplicit)m(y)h(the)f(microscopic)h(region)g(\003)e (is)h(c)m(hosen)g(as)g Fx(B)2367 3259 y Fs(N)2434 3245 y Fz(\(0\))h(and,)f(imp)s(osing)f(p)s(erio)s(dic)g(b.c.)118 3365 y(it)g(b)s(ecomes)f(the)h(torus)f Fx(T)1017 3379 y Fs(N)1084 3365 y Fz(.)40 b(W)-8 b(e)30 b(call)h Fu( )1529 3379 y Fs(N)1626 3365 y Fz(the)e(map)g(from)g Fx(T)2257 3379 y Fs(N)2353 3365 y Fz(on)m(to)2561 3342 y Fo(b)2556 3365 y Fx(T)p Fz(,)h(obtained)f(b)m(y)g(shrinking)f(b)m(y)118 3473 y(a)j(factor)g(1)p Fu(=)-5 b(N)10 b Fz(.)42 b(W)-8 b(e)31 b(then)f(de\014ne)g(the)h(lo)s(cal)g(magnetization)1159 3640 y Fr(M)1268 3654 y Fs(N)s(;K)1415 3640 y Fz(\()p Fu(r)s Fz(\))26 b(=)f Fr(M)1760 3654 y Fs(K)1828 3566 y Fo(\000)1870 3640 y Fu( )1932 3602 y Fn(\000)p Fw(1)1929 3669 y Fs(N)2027 3640 y Fz(\()p Fu(r)s Fz(\))2141 3566 y Fo(\001)2183 3640 y Fu(;)197 b(r)28 b Fr(2)2565 3616 y Fo(b)2560 3640 y Fx(T)854 b Fz(\(2.6\))118 3816 y(whic)m(h)33 b(is)g(a)g(function)g(on)1046 3792 y Fo(b)1041 3816 y Fx(T)f Fz(piecewise)j(constan)m(t)f(on)f(the)g(b)s(o)m(xes)g Fu( )2501 3830 y Fs(N)2569 3816 y Fz(\()p Fx(B)2665 3830 y Fs(K)2733 3816 y Fz(\),)h Fx(B)2888 3830 y Fs(K)2986 3816 y Fr(\032)29 b Fx(T)3147 3830 y Fs(N)3214 3816 y Fz(.)49 b(The)32 b(lo)s(cal)118 3924 y(order)37 b(parameter)h Fr(M)914 3938 y Fs(N)s(;K)1099 3924 y Fz(c)m(haracterizes)i(the)d(lo)s (cal)i(equilibrium.)61 b(The)37 b(total)j(magnetization)g(in)118 4032 y Fx(T)179 4046 y Fs(N)276 4032 y Fz(is)31 b(simply)f(denoted)g(b) m(y)g Fr(M)1237 4046 y Fs(N)1305 4032 y Fz(.)218 4376 y(W)-8 b(e)31 b(can)g(no)m(w)f(state)i(a)f(result)f(on)g(phase)g(co)s (existence.)118 4543 y FE(Theorem)39 b(2.1.)44 b Fj(Ther)-5 b(e)36 b(exists)g Fu(\014)1339 4557 y Fw(0)1410 4543 y Fu(>)30 b Fz(0)36 b Fj(such)g(that)h(for)f(any)g Fu(\014)g(>)31 b(\014)2556 4557 y Fw(0)2631 4543 y Fj(and)37 b Fz([)p Fu(m)2916 4557 y Fw(1)2955 4543 y Fu(;)15 b(m)3075 4557 y Fw(2)3115 4543 y Fz(])31 b Fr(\032)g Fz(\()p Fu(m)3388 4505 y Fn(\000)3388 4572 y Fs(\014)3447 4543 y Fu(;)15 b(m)3567 4505 y Fw(+)3567 4572 y Fs(\014)3626 4543 y Fz(\))118 4662 y Fj(\(with)35 b Fu(m)434 4676 y Fw(1)500 4662 y Fu(<)26 b(m)677 4676 y Fw(2)717 4662 y Fj(\),)33 b(the)h(fol)5 b(lowing)35 b(holds:)45 b(for)34 b(every)g Fu(\016)c(>)d Fz(0)34 b Fj(ther)-5 b(e)34 b(is)g(a)g(sc)-5 b(ale)34 b Fu(K)2962 4676 y Fw(0)3028 4662 y Fz(=)27 b Fu(K)3203 4676 y Fw(0)3243 4662 y Fz(\()p Fu(\014)5 b(;)15 b(\016)s Fz(\))35 b Fj(such)118 4769 y(that)f(for)f(any)g Fu(K)f Ft(>)25 b Fu(K)905 4783 y Fw(0)434 4974 y Fz(lim)395 5034 y Fs(N)7 b Fn(!1)640 4974 y Fu(\026)695 4992 y Fs(\014)s(;h)p Fw(\()p Fs(\014)s Fw(\))p Fs(;N)998 4846 y Fo(\022)1214 4974 y Fz(inf)1065 5039 y Fs(v)r Fn(2D)r Fw(\()p Fs(m)1294 5048 y Fp(1)1330 5039 y Fs(;m)1412 5048 y Fp(2)1446 5039 y Fw(\))1489 4974 y Fr(kM)1643 4988 y Fs(N)s(;K)1811 4974 y Fr(\000)20 b Fu(v)s Fr(k)1994 4988 y Fw(1)2059 4974 y Ft(6)25 b Fu(\016)2231 4869 y Fo(\014)2231 4924 y(\014)2231 4978 y(\014)2287 4974 y Fu(m)2367 4988 y Fw(1)2431 4974 y Ft(6)g Fr(M)2636 4988 y Fs(N)2729 4974 y Ft(6)g Fu(m)2905 4988 y Fw(2)2944 4846 y Fo(\023)3036 4974 y Fz(=)g(1)15 b Fu(;)118 5188 y Fj(wher)-5 b(e)30 b Fr(D)s Fz(\()p Fu(m)559 5202 y Fw(1)599 5188 y Fu(;)15 b(m)719 5202 y Fw(2)758 5188 y Fz(\))30 b Fj(denotes)g(the)g(set)f(of)h (the)f(e)-5 b(quilibrium)29 b(crystal)i(shap)-5 b(es)39 b Fz(\(2.4\))31 b Fj(\(wher)-5 b(e)30 b(the)g(surfac)-5 b(e)118 5296 y(tension)39 b(is)g(the)f(one)h(de\014ne)-5 b(d)39 b(in)g(Se)-5 b(ction)39 b(3\))g(and)g Fu(\026)1993 5314 y Fs(\014)s(;h)p Fw(\()p Fs(\014)s Fw(\))p Fs(;N)2319 5296 y Fj(is)g(the)f(Gibbs)h(me)-5 b(asur)g(e)39 b(on)g Fx(T)3391 5310 y Fs(N)3496 5296 y Fj(with)118 5404 y(p)-5 b(erio)g(dic)34 b(b)-5 b(oundary)35 b(c)-5 b(onditions.)p eop end %%Page: 5 5 TeXDict begin 5 4 bop 1137 45 a Fw(SURF)-8 b(A)n(CE)31 b(TENSION)g(AND)f(WULFF)h(THEOR)-6 b(Y)983 b(5)1461 366 y Fz(3.)46 b Fy(Surf)-8 b(a)n(ce)32 b(tension)218 528 y Fz(F)-8 b(or)35 b(an)m(y)f(giv)m(en)i(unit)e(v)m(ector)f Fu(~)-42 b(n)31 b Fz(=)h(\()p Fu(n)1554 542 y Fw(1)1593 528 y Fu(;)15 b(:)g(:)g(:)i(;)e(n)1850 543 y Fs(d)1890 528 y Fz(\),)36 b(w)m(e)f(are)f(going)i(to)f(de\014ne)e(the)i(surface)f (tension)118 636 y Fu(\034)158 651 y Fs(\014)205 636 y Fz(\()n Fu(~)-43 b(n)p Fz(\))28 b(in)f(the)g(direction)i(orthogonal)f (to)e Fu(~)-42 b(n)o Fz(.)40 b(Con)m(trary)27 b(to)i(the)e(Ising)g(mo)s (del,)i(the)e(lac)m(k)i(of)f(symmetry)118 744 y(b)s(et)m(w)m(een)k(the) f(t)m(w)m(o)i(pure)d(phases)g(requires)h(a)g(more)g(complex)h (de\014nition)f(of)g(surface)g(tension)h(whic)m(h)118 852 y(relies)h(on)g(t)m(w)m(o)h(new)e(features)h(:)45 b(a)33 b(cuto\013)g(of)g(the)g(in)m(terface)h(\015uctuations)f(and)f (the)g(in)m(tro)s(duction)h(of)118 960 y(p)s(erfect)d(w)m(alls.)118 1135 y(3.1.)47 b FE(In)m(terface)36 b(\015uctuations)h(cuto\013.)46 b Fz(W)-8 b(e)33 b(asso)s(ciate)g(to)g(an)m(y)f(unit)f(v)m(ector)g Fu(~)-43 b(n)27 b Fz(=)h(\()p Fu(n)3213 1149 y Fw(1)3252 1135 y Fu(;)15 b(:)g(:)g(:)i(;)e(n)3509 1150 y Fs(d)3549 1135 y Fz(\))32 b(a)118 1243 y(co)s(ordinate)j(direction)h Fu(j)h Fr(2)32 b(f)p Fz(1)p Fu(;)15 b(::;)g(d)p Fr(g)38 b Fz(in)c(suc)m(h)g(a)h(w)m(a)m(y)h(that)f Fu(n)2314 1257 y Fs(i)2374 1243 y Ft(6)c Fu(n)2531 1257 y Fs(j)2602 1243 y Fz(for)j(all)i Fu(i)e Fz(while)h Fu(n)3238 1257 y Fs(i)3298 1243 y Fu(<)d(n)3456 1257 y Fs(j)3492 1243 y Fz(,)k(for)118 1351 y(an)m(y)d Fu(i)c(>)f(j)5 b Fz(.)48 b(F)-8 b(or)33 b(notational)h(simplicit)m(y)g(supp)s(ose)d Fu(j)j Fz(=)29 b Fu(d)p Fz(,)k(the)g(other)f(cases)i(are)f(treated)g (similarly)-8 b(.)118 1459 y(W)g(e)32 b(set)545 1611 y(\003)608 1626 y Fs(`;m)723 1611 y Fz(\()n Fu(~)-43 b(n)p Fz(\))26 b(=)970 1537 y Fo(\010)1023 1611 y Fu(x)f Fr(2)g Fx(Z)1247 1573 y Fs(d)1287 1611 y Fu(;)197 b Fr(8)p Fu(i)25 b(<)g(d;)107 b Fr(\000)p Fu(`)25 b Ft(6)g Fu(x)2173 1625 y Fs(i)2226 1611 y Ft(6)g Fu(`)p Fz(;)40 b Fr(\000)p Fu(m)25 b Ft(6)g Fz(\()p Fu(x)c Fr(\001)d Fu(~)-43 b(n)p Fz(\))25 b Ft(6)g Fu(m)3141 1537 y Fo(\011)3209 1611 y Fu(:)241 b Fz(\(3.1\))118 1763 y(As)28 b Fu(~)-43 b(n)30 b Fz(is)h(\014xed)e(throughout)h(this)h(section,)g(w)m(e)g(will)g(drop) e(it)i(from)f(the)h(notation.)218 1871 y(The)20 b(surface)g(tension)h Fu(\034)1035 1886 y Fs(\014)1082 1871 y Fz(\()n Fu(~)-43 b(n)p Fz(\))21 b(will)g(b)s(e)f(the)g(thermo)s(dynamic)g(limit)i(of)e (ratios)i(of)f(partition)g(functions)118 1979 y(de\014ned)31 b(on)h(subsets)f(of)h(the)g(slab)g(\003)1392 2007 y Fs(L;)1470 1979 y Fp(11)p Fq(")p 1469 1991 91 3 v 1484 2033 a Fp(10)1570 2007 y Fs(L)1622 1979 y Fz(.)45 b(The)31 b(limit)i(will)f(b)s(e)f(tak)m (en)i(for)f(appropriate)g(sequences)118 2109 y(of)d(the)g(parameters)g (\()p Fu(L;)15 b(")p Fz(\),)31 b(in)d(particular)h(w)m(e)h(require)e Fu(L)h Fz(and)f(\()p Fu("=)p Fz(10\))p Fu(L)j Fz(to)e(b)s(e)f(in)h Fr(f)p Fz(2)3105 2076 y Fs(n)3153 2109 y Fu(;)15 b(n)25 b Fr(2)g Fx(N)p Fr(g)p Fz(.)40 b(W)-8 b(e)118 2217 y(will)31 b(\014rst)e(in)m(tro)s(duce)i(the)f(partition)h(function)f(with)g (mixed)h(b)s(oundary)d(conditions.)218 2433 y(W)-8 b(e)27 b(w)m(an)m(t)f(to)h(imp)s(ose)f(+)f(and)g Fr(\000)h Fz(b)s(oundary)e (conditions)i(on)g(top)g(and)f(b)s(ottom)h(of)g(our)f(domains;)j(it)118 2541 y(will)g(b)s(e)f(con)m(v)m(enien)m(t)j(to)e(lea)m(v)m(e)i(some)e (freedom)g(on)f(their)h(exact)h(lo)s(cation)g(and)e(with)g(this)h(in)f (mind)g(w)m(e)118 2649 y(in)m(tro)s(duce)k(the)g(notion)g(of)g (barriers.)41 b(A)31 b Fj(b)-5 b(arrier)32 b Fz(in)f(a)g(slab)g(\003) 2270 2664 y Fs(`;m)2415 2649 y Fz(is)g(a)g(connected)h(set)f(of)g (plaquettes)118 2757 y(in)f(\003)287 2772 y Fs(`;m)432 2757 y Fz(whic)m(h)g(connects)h(the)g(faces)g(of)f(\003)1604 2772 y Fs(`;m)1750 2757 y Fz(parallel)h(to)d Fu(~)-42 b(e)2226 2772 y Fs(d)2297 2757 y Fz(and)30 b(it)g(is)h(suc)m(h)f(that)g (its)h(complemen)m(t)118 2866 y(in)h(\003)289 2881 y Fs(`;m)435 2866 y Fz(is)g(made)g(of)g(t)m(w)m(o)h(distinct)f(comp)s (onen)m(ts)g(whic)m(h)f(are)h(not)g Fu(?)p Fz(-connected.)46 b(Let)33 b(then)e Fr(C)3424 2833 y Fw(+)3515 2866 y Fz(and)118 2975 y Fr(C)171 2942 y Fn(\000)272 2975 y Fz(b)s(e)40 b(t)m(w)m(o)j(barriers)d(in)h(\003)1114 2992 y Fs(L;)1208 2965 y Fq(")p 1192 2977 61 3 v 1192 3018 a Fp(10)1263 2992 y Fs(L)1342 2975 y Fz(+)27 b Fu("L)l(~)-41 b(e)1586 2990 y Fs(d)1669 2975 y Fz(and)40 b(\003)1919 2992 y Fs(L;)2013 2965 y Fq(")p 1997 2977 V 1997 3018 a Fp(10)2068 2992 y Fs(L)2147 2975 y Fr(\000)28 b Fu("L)l(~)-41 b(e)2392 2990 y Fs(d)2433 2975 y Fz(.)73 b(The)41 b(subset)f(of)i(\003)3196 3003 y Fs(L;)3274 2976 y Fp(11)p Fq(")p 3274 2988 91 3 v 3288 3029 a Fp(10)3374 3003 y Fs(L)3467 2975 y Fz(lying)118 3110 y(b)s(et)m(w)m(een)29 b Fr(C)520 3077 y Fw(+)608 3110 y Fz(and)f Fr(C)836 3077 y Fn(\000)923 3110 y Fz(is)h(denoted)f(b) m(y)g(\003\()p Fr(C)1629 3077 y Fw(+)1689 3110 y Fu(;)15 b Fr(C)1782 3077 y Fn(\000)1842 3110 y Fz(\).)40 b(The)28 b(mixed)g(b)s(oundary)f(conditions)33 b(\026)-50 b Fu(\033)3290 3077 y Fn(\006)3378 3110 y Fz(outside)118 3218 y(\003\()p Fr(C)269 3185 y Fw(+)329 3218 y Fu(;)15 b Fr(C)422 3185 y Fn(\000)481 3218 y Fz(\))31 b(are)g(de\014ned)e(as)i(follo)m(ws)726 3453 y Fr(8)p Fu(x)25 b Fr(62)g Fz(\003\()p Fr(C)1091 3415 y Fw(+)1150 3453 y Fu(;)15 b Fr(C)1243 3415 y Fn(\000)1303 3453 y Fz(\))p Fu(;)202 b Fz(\026)-49 b Fu(\033)1616 3415 y Fn(\006)1613 3475 y Fs(x)1700 3453 y Fz(=)1796 3297 y Fo(\()1869 3391 y Fz(+1)p Fu(;)273 b Fz(if)91 b(\()p Fu(x)21 b Fr(\001)d Fu(~)-43 b(n)p Fz(\))26 b Ft(>)e Fz(0)15 b Fu(;)1869 3521 y Fr(\000)p Fz(1)p Fu(;)273 b Fz(if)91 b(\()p Fu(x)21 b Fr(\001)d Fu(~)-43 b(n)p Fz(\))26 b Fu(<)e Fz(0)15 b Fu(:)118 3692 y Fz(W)-8 b(e)41 b(denote)g(b)m(y)f Fr(S)785 3659 y Fw(+)883 3692 y Fz(\(resp)g Fr(S)1183 3659 y Fn(\000)1242 3692 y Fz(\))g(the)g(set)h(of)f(spin)f (con\014gurations)h(for)g(whic)m(h)f(there)i(is)f(a)g(barrier)118 3804 y(included)32 b(in)g(\003)655 3821 y Fs(L;)748 3794 y Fq(")p 733 3806 61 3 v 733 3847 a Fp(10)804 3821 y Fs(L)877 3804 y Fz(+)980 3768 y Fs("L)p 980 3783 81 4 v 1002 3835 a Fw(2)1067 3804 y Fu(~)-42 b(e)1112 3819 y Fs(d)1185 3804 y Fz(\(resp)32 b(\003)1478 3821 y Fs(L;)1572 3794 y Fq(")p 1556 3806 61 3 v 1556 3847 a Fp(10)1627 3821 y Fs(L)1701 3804 y Fr(\000)1803 3768 y Fs("L)p 1803 3783 81 4 v 1826 3835 a Fw(2)1890 3804 y Fu(~)-42 b(e)1935 3819 y Fs(d)1976 3804 y Fz(\))33 b(where)f(all)h(spins)f(are)h(equal)f (to)i(1)f(\(resp)f Fr(\000)p Fz(1\).)118 3929 y(Finally)-8 b(,)26 b(w)m(e)d(in)m(tro)s(duce)f(the)h(follo)m(wing)h(constrained)f (partition)g(function)f(on)g(\003\()p Fr(C)2952 3896 y Fw(+)3012 3929 y Fu(;)15 b Fr(C)3105 3896 y Fn(\000)3165 3929 y Fz(\))22 b(with)h(mixed)118 4037 y(b)s(oundary)28 b(conditions)j(\(see)h(\014gure)d(1\))527 4217 y Fu(Z)596 4172 y Fn(C)637 4149 y Fp(+)687 4172 y Fs(;)p Fn(C)748 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b(ho)m(w)m(ev)m(er)g(not)f(useful)g(in)f(our)h(con)m(text,)j (b)s(ecause)d(w)m(e)h(w)m(an)m(t)g(to)g(imp)s(ose)f(one)g(of)g(the)g(t) m(w)m(o)i(phases)d(on)p eop end %%Page: 6 6 TeXDict begin 6 5 bop 118 45 a Fw(6)1163 b(T.)23 b(BODINEA)n(U)g(AND)g (E.)g(PRESUTTI)118 366 y Fz(some)29 b(of)f(the)g(w)m(alls;)i(but,)f(as) f(w)m(e)h(are)f(going)h(to)g(see,)h(it)e(is)h(p)s(ossible)e(to)i (de\014ne)f(some)g(sort)h(of)f(p)s(erio)s(dic)118 474 y(conditions)j(on)f(single)h(w)m(alls)h(of)e(the)h(con)m(tainer.)218 582 y(W)-8 b(e)31 b(start)g(b)m(y)g(de\014ning)e(a)i(symmetric)g (partition)g(of)f Fx(Z)2124 549 y Fs(d)2195 582 y Fz(b)m(y)g(the)g(h)m (yp)s(erplane)g(\006)g(orthogonal)i(to)c Fu(~)-42 b(n)118 690 y Fz(and)33 b(con)m(taining)i(0.)49 b(Let)34 b(us)f(\014rst)f(supp) s(ose)g(that)i(the)f(orien)m(tation)g Fu(~)-42 b(n)32 b Fz(is)i(suc)m(h)f(that)g(\006)22 b Fr(\\)g Fx(Z)3329 657 y Fs(d)3399 690 y Fz(=)30 b Fr(f)p Fz(0)p Fr(g)p Fz(.)118 798 y(W)-8 b(e)32 b(then)e(set)363 1014 y Fx(Z)424 976 y Fs(d)424 1036 y Fw(+)508 1014 y Fz(=)604 858 y Fo(\()677 1014 y Fu(x)c Fr(2)e Fx(Z)901 976 y Fs(d)957 909 y Fo(\014)957 964 y(\014)957 1018 y(\014)1169 1014 y Fu(x)1221 1029 y Fs(d)1286 1014 y Ft(>)46 b Fr(\000)1496 900 y Fs(d)p Fn(\000)p Fw(1)1494 927 y Fo(X)1502 1123 y Fs(i)p Fw(=1)1656 952 y Fu(n)1711 966 y Fs(i)p 1650 993 96 4 v 1650 1076 a Fu(n)1705 1091 y Fs(d)1755 1014 y Fu(x)1807 1028 y Fs(i)1835 858 y Fo(\))1929 1014 y Fr(n)20 b(f)p Fz(0)p Fr(g)p Fu(;)199 b Fx(Z)2414 976 y Fs(d)2414 1036 y Fn(\000)2498 1014 y Fz(=)25 b Fx(Z)2655 976 y Fs(d)2715 1014 y Fr(n)c Fz(\()p Fu(Z)2885 976 y Fs(d)2878 1036 y Fw(+)2957 1014 y Fr([)f(f)p Fz(0)p Fr(g)p Fz(\))15 b Fu(:)118 1242 y Fz(Then)1042 1388 y Fx(Z)1103 1350 y Fs(d)1168 1388 y Fz(=)25 b Fx(Z)1325 1350 y Fs(d)1325 1410 y Fw(+)1404 1388 y Fr([)19 b Fx(Z)1545 1350 y Fs(d)1545 1410 y Fn(\000)1624 1388 y Fr([)h(f)p Fz(0)p Fr(g)p Fu(;)199 b Fx(Z)2125 1350 y Fs(d)2125 1410 y Fn(\000)2209 1388 y Fz(=)25 b Fr(R)p Fz(\()p Fx(Z)2478 1350 y Fs(d)2478 1410 y Fw(+)2537 1388 y Fz(\))903 b(\(3.3\))118 1534 y(where)30 b Fr(R)g Fz(is)h(the)f(symmetry)g(wrt)g(0.)218 1642 y(If)40 b(there)h(are)h(sites)f(in)g Fx(Z)1116 1609 y Fs(d)1183 1642 y Fr(\\)27 b Fz(\006)40 b(b)s(esides)h(0,)j(w)m(e)d (split)g(them)g(b)s(et)m(w)m(een)h Fx(Z)2837 1609 y Fs(d)2837 1665 y Fw(+)2936 1642 y Fz(and)e Fx(Z)3184 1609 y Fs(d)3184 1665 y Fn(\000)3284 1642 y Fz(in)g(suc)m(h)h(a)118 1759 y(w)m(a)m(y)36 b(that)g(\(3.3\))h(is)e(preserv)m(ed.)55 b(Notice)37 b(\014rst)d(that)i(if)f Fu(x)e Fr(2)g Fz(\006)23 b Fr(\\)g Fx(Z)2467 1726 y Fs(d)2507 1759 y Fz(,)37 b(also)f Fr(R)p Fu(x)d Fr(2)g Fz(\006)23 b Fr(\\)g Fx(Z)3246 1726 y Fs(d)3286 1759 y Fz(.)55 b(Then)34 b(if)118 1867 y Fu(x)28 b Fz(=)g(\()p Fu(x)384 1881 y Fw(1)424 1867 y Fu(;)15 b(::;)g(x)606 1882 y Fs(d)647 1867 y Fz(\))29 b Fr(6)p Fz(=)f(0,)33 b(w)m(e)f(call)h Fu(i)g Fz(the)f(\014rst)f(in)m (teger)j(so)e(that)h Fu(x)2294 1881 y Fs(i)2350 1867 y Fr(6)p Fz(=)28 b(0)k(\(i.e.)i Fu(x)2762 1881 y Fw(1)2829 1867 y Fz(=)28 b Fu(::)g Fz(=)g Fu(x)3157 1881 y Fs(i)p Fn(\000)p Fw(1)3303 1867 y Fz(=)g(0\))33 b(and)118 1977 y(w)m(e)e(put)f Fu(x)25 b Fr(2)g Fx(Z)643 1944 y Fs(d)643 1999 y Fw(+)732 1977 y Fz(if)30 b Fu(x)867 1991 y Fs(i)920 1977 y Fu(>)25 b Fz(0)31 b(and)f Fu(x)25 b Fr(2)g Fx(Z)1493 1944 y Fs(d)1493 1999 y Fn(\000)1582 1977 y Fz(otherwise.)41 b(Th)m(us)449 2239 y Fx(Z)510 2202 y Fs(d)510 2262 y Fw(+)594 2239 y Fz(=)736 2126 y Fs(d)703 2153 y Fo([)690 2351 y Fs(k)r Fw(=1)901 2138 y Fo(n)1045 2053 y(8)1045 2135 y(<)1045 2298 y(:)1126 2239 y Fu(x)25 b Fr(2)g Fx(Z)1350 2202 y Fs(d)1405 2135 y Fo(\014)1405 2189 y(\014)1405 2244 y(\014)1435 2239 y Fr(8)p Fu(i)g(<)g(k)s(;)46 b(x)1811 2253 y Fs(i)1865 2239 y Fz(=)25 b(0)p Fu(;)46 b(x)2129 2254 y Fs(k)2197 2239 y Fu(>)25 b Fz(0)p Fu(;)46 b(x)2461 2254 y Fs(d)2526 2239 y Ft(>)g Fr(\000)2736 2126 y Fs(d)p Fn(\000)p Fw(1)2733 2153 y Fo(X)2738 2348 y Fs(j)t Fw(=1)2892 2177 y Fu(n)2947 2191 y Fs(j)p 2890 2218 V 2890 2302 a Fu(n)2945 2317 y Fs(d)2995 2239 y Fu(x)3047 2253 y Fs(j)3083 2053 y Fo(9)3083 2135 y(=)3083 2298 y(;)3475 2239 y Fz(\(3.4\))1045 2602 y Fr([)1121 2416 y Fo(8)1121 2497 y(<)1121 2661 y(:)1201 2602 y Fu(x)26 b Fr(2)f Fx(Z)1426 2564 y Fs(d)1481 2497 y Fo(\014)1481 2552 y(\014)1481 2606 y(\014)1511 2602 y Fr(8)p Fu(i)g(<)g(k)s(;)46 b(x)1887 2616 y Fs(i)1940 2602 y Fz(=)25 b(0)p Fu(;)46 b(x)2204 2617 y Fs(k)2272 2602 y Fu(<)25 b Fz(0)p Fu(;)46 b(x)2536 2617 y Fs(d)2602 2602 y Fu(>)25 b Fr(\000)2786 2488 y Fs(d)p Fn(\000)p Fw(1)2784 2516 y Fo(X)2788 2711 y Fs(j)t Fw(=1)2942 2540 y Fu(n)2997 2554 y Fs(j)p 2940 2581 V 2940 2664 a Fu(n)2995 2679 y Fs(d)3045 2602 y Fu(x)3097 2616 y Fs(j)3134 2416 y Fo(9)3134 2497 y(=)3134 2661 y(;)3230 2501 y(o)3306 2602 y Fu(;)118 2854 y Fz(A)37 b(dra)m(wbac)m(k)g(of)g(the)f(de\014nition)h(is)f(that)i(for)c Fu(~)-43 b(n)36 b Fz(orien)m(ted)i(along)f(one)g(of)g(the)g(axis)g(of)g (co)s(ordinates)118 2962 y(the)g(b)s(ottom)g(of)f Fx(Z)778 2929 y Fs(d)778 2985 y Fw(+)873 2962 y Fz(is)h(not)f(\015at.)60 b(This)35 b(could)i(b)s(e)f(a)m(v)m(oided)i(at)f(the)g(price)f(of)h (considering)f(a)h(more)118 3070 y(complicated)32 b(mapping)e(than)g (the)h(simple)f(symmetry)g(wrt)g(0.)218 3286 y(W)-8 b(e)36 b(no)m(w)g(pro)s(ceed)f(in)g(de\014ning)g(the)h(re\015ected)g (hamiltonian)g(in)f Fx(Z)2588 3253 y Fs(d)2628 3286 y Fz(.)56 b(The)35 b(idea)h(is)g(to)g(use)f Fr(R)h Fz(in)118 3394 y(order)k(to)h(glue)g(together)g(di\013eren)m(t)g(regions)g(touc)m (hing)g(the)f(surface)g(\006)g(so)h(that)g(if,)h(for)e(instance,)118 3502 y Fu(x)27 b Fr(2)g Fx(Z)346 3469 y Fs(d)346 3524 y Fw(+)436 3502 y Fz(in)m(teracts)34 b(across)e(\006)f(with)g Fu(y)f Fr(2)d Fx(Z)1609 3469 y Fs(d)1609 3524 y Fn(\000)1699 3502 y Fz(then)k Fu(x)h Fz(will)g(no)m(w)f(in)m(teract)j(with)d Fr(R)p Fz(\()p Fu(y)s Fz(\))d Fr(2)f Fx(Z)3272 3469 y Fs(d)3272 3524 y Fw(+)3330 3502 y Fz(.)45 b(As)31 b(the)118 3610 y(energy)f(is)g(de\014ned)f(in)h(terms)g(of)g(plaquettes,)h(this)f (can)h(b)s(e)e(easily)i(ac)m(hiev)m(ed)h(b)m(y)e(in)m(tro)s(ducing)f(a) i(new)118 3718 y(set)g(of)g(plaquettes)g Fr(f)p Fu(p)p Fr(g)937 3685 y Fs(R)995 3718 y Fz(.)218 3828 y(Plaquettes)d(whic)m(h)f (are)g(en)m(tirely)h(con)m(tained)h(either)e(in)g Fx(Z)2220 3795 y Fs(d)2220 3850 y Fw(+)2278 3828 y Fz(,)h(or)f(in)g Fx(Z)2603 3795 y Fs(d)2603 3850 y Fn(\000)2688 3828 y Fz(or)g(in)g Fu(B)j Fz(=)25 b Fr(f\000)p Fz(1)p Fu(;)15 b Fz(0)p Fu(;)g Fz(1)p Fr(g)3470 3795 y Fs(d)3540 3828 y Fz(are)118 3945 y(unc)m(hanged.)40 b(Instead)30 b(an)m(y)g(plaquette) h Fu(p)e Fz(con)m(taining)i(sites)g(b)s(oth)e(in)g Fx(Z)2597 3912 y Fs(d)2597 3967 y Fw(+)2686 3945 y Fz(and)g(in)g Fx(Z)3028 3912 y Fs(d)3028 3967 y Fn(\000)3116 3945 y Fz(is)h(replaced)g(b)m(y)1095 4180 y Fu(p)25 b Fr(!)1282 4024 y Fo(\()1355 4118 y Fu(p)1401 4085 y Fw(+)1485 4118 y Fz(=)g(\()p Fu(p)20 b Fr(n)h Fx(Z)1809 4085 y Fs(d)1809 4141 y Fn(\000)1868 4118 y Fz(\))f Fr([)g(R)p Fz(\()p Fu(p)g Fr(\\)g Fx(Z)2324 4085 y Fs(d)2324 4141 y Fn(\000)2383 4118 y Fz(\))1355 4248 y Fu(p)1401 4215 y Fn(\000)1485 4248 y Fz(=)25 b(\()p Fu(p)20 b Fr(n)h Fx(Z)1809 4215 y Fs(d)1809 4270 y Fw(+)1868 4248 y Fz(\))f Fr([)g(R)p Fz(\()p Fu(p)g Fr(\\)g Fx(Z)2324 4215 y Fs(d)2324 4270 y Fw(+)2383 4248 y Fz(\))3475 4180 y(\(3.5\))118 4413 y(Notice)42 b(that)e(b)s(oth)e(plaquettes)j Fu(p)e Fz(and)g Fr(R)p Fz(\()p Fu(p)p Fz(\))h(generate)h(the)f(same)g(pair)f Fu(p)2811 4380 y Fn(\006)2870 4413 y Fz(,)j(so)d(that)h(the)g(\\total) 118 4521 y(n)m(um)m(b)s(er")30 b(of)g(old)h(and)e(new)h(plaquettes)i (is)e(the)h(same.)218 4629 y(Extending)22 b(the)h(de\014nition)f(of)h Fu(V)d Fz(\()p Fu(\033)1449 4643 y Fs(p)1489 4629 y Fz(\))j(to)g(the)g (new)f(set)h(of)g(plaquettes,)i(the)e(re\015ected)g(Hamiltonian)118 4737 y(is)31 b(then)1170 4846 y Fu(H)1253 4801 y Fs(R;\033)1366 4814 y Fp(\003)1407 4800 y Fq(c)1246 4875 y Fs(h;)p Fw(\003)1448 4846 y Fz(\()p Fu(\033)s Fz(\))26 b(=)1749 4759 y Fo(X)1695 4968 y Fs(p)p Fn(2f)p Fs(p)p Fn(g)1884 4945 y Fq(R)1704 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5356 y Fr(\000)1646 5295 y Fz(\()n Fu(~)-43 b(n)19 b Fr(\001)e Fu(~)-41 b(e)1843 5310 y Fs(d)1884 5295 y Fz(\))p 1646 5335 V 1686 5419 a Fu(L)1748 5392 y Fs(d)p Fn(\000)p Fw(1)1959 5356 y Fz(log)2227 5292 y Fu(Z)2296 5248 y Fw(+)p Fs(;)p Fn(\000)2289 5321 y Fs(L)p 2102 5335 453 4 v 2102 5354 a Fo(q)p 2193 5354 128 4 v 107 x Fu(Z)2262 5422 y Fw(+)2255 5489 y Fs(L)2335 5354 y Fo(q)p 2426 5354 V 107 x Fu(Z)2495 5422 y Fn(\000)2488 5489 y Fs(L)2579 5356 y Fu(:)825 b Fz(\(3.11\))p eop end %%Page: 9 9 TeXDict begin 9 8 bop 1137 45 a Fw(SURF)-8 b(A)n(CE)31 b(TENSION)g(AND)f(WULFF)h(THEOR)-6 b(Y)983 b(9)118 366 y Fz(Notice)29 b(that)e(this)g(represen)m(tation)h(of)e(the)h(surface)g (tension)g(w)m(ould)g(also)g(require)g(an)f(assumption)g(on)118 474 y(the)21 b(p)s(oten)m(tial)i(similar)e(to)h(\(2.2\))h(in)e(order)f (to)i(pro)s(duce)e(exact)j(cancellations)g(b)s(et)m(w)m(een)f(the)f(n)m (umerator)118 582 y(and)30 b(the)g(denominator.)218 690 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b(the)d(energy)h(in)f(the)h(small)g(regions) g(along)h(the)e(in)m(terface)j(is)d(ev)-5 b(aluated)30 b(b)m(y)e(pasting)h(the)118 798 y(a)h(priori)g(estimates)h(pro)m(vided) f(b)m(y)f(De\014nition)i(3.1.)41 b(The)30 b(iteration)h(is)f(p)s (ossible)f(thanks)g(to)i(the)f(v)m(ery)118 906 y(lo)s(ose)40 b(structure)e(of)h(the)g(de\014nition)g(of)g(the)g(surface)g(tension.) 66 b(The)38 b(limit)i(wrt)e Fu(")h Fz(has)g(no)g(impact)118 1014 y(on)d(the)g(v)-5 b(alue)36 b(of)h(the)f(surface)g(tension,)h(the) g(main)e(motiv)-5 b(ation)38 b(is)e(tec)m(hnical:)54 b(it)36 b(is)g(useful)g(in)f(the)118 1122 y(iteration)d(pro)s(cedure)d (and)h(afterw)m(ards)g(in)h(the)f(completion)i(of)e(Theorem)g(2.1.)218 1373 y(W)-8 b(e)24 b(are)g(going)h(no)m(w)e(to)h(compare)g(the)g (represen)m(tations)g Fu(\034)2204 1388 y Fs(\014)2251 1373 y Fz(\()n Fu(~)-43 b(n)p Fz(\))23 b(and)g Fu(\034)2619 1340 y Fs(?)2609 1401 y(\014)2659 1373 y Fz(\()n Fu(~)-43 b(n)p Fz(\))23 b(of)h(the)f(surface)h(tension.)118 1495 y(According)35 b(to)h(Theorem)e(3.1)i(the)f(con)m(v)m(ergence)i (\(3.9\))f(is)f(uniform)f(o)m(v)m(er)i(the)e(domains)h(\003\()p Fr(C)3414 1462 y Fw(+)3473 1495 y Fu(;)15 b Fr(C)3566 1462 y Fn(\000)3626 1495 y Fz(\))118 1603 y(and)37 b(th)m(us)g(it)g(is) g(enough)g(to)h(de\014ne)f(the)g(surface)g(tension)h(on)f(regular)g (domains)g(of)h(the)f(t)m(yp)s(e)g(\001)3584 1617 y Fs(L)3636 1603 y Fz(.)118 1711 y(F)-8 b(urthermore,)31 b(the)f(p)s(erfect)g(w)m (alls)i(are)f(suc)m(h)f(that)1209 2083 y(lim)1177 2143 y Fs(L)p Fn(!1)1465 2021 y Fz(1)p 1392 2062 193 4 v 1392 2145 a Fu(L)1454 2119 y Fs(d)p Fn(\000)p Fw(1)1624 2083 y Fz(log)1767 1887 y Fo(q)p 1858 1887 128 4 v 107 x Fu(Z)1927 1956 y Fw(+)1920 2023 y Fs(L)2001 1887 y Fo(q)p 2091 1887 V 2091 1994 a Fu(Z)2160 1956 y Fn(\000)2153 2023 y Fs(L)p 1767 2062 453 4 v 1785 2162 a Fu(Z)1854 2118 y Fw(+)p Fs(;R)1847 2191 y(L)2001 2162 y Fu(Z)2070 2118 y Fn(\000)p Fs(;R)2063 2191 y(L)2254 2083 y Fz(=)25 b(0)15 b Fu(:)118 2406 y Fz(It)35 b(remains)h(only)f(to)h(analyze)h(the)e (role)h(of)f(the)h(cuto\013)g(of)f(in)m(terface)i(\015uctuations.)55 b(De\014nition)36 b(3.1)118 2513 y(w)m(ould)30 b(coincide)i(with)e (\(3.11\))i(if)f(the)f(follo)m(wing)i(holds)517 2836 y(lim)486 2897 y Fs(L)p Fn(!1)774 2775 y Fz(1)p 700 2815 193 4 v 700 2898 a Fu(L)762 2872 y Fs(d)p Fn(\000)p Fw(1)933 2836 y Fz(log)1075 2772 y Fu(Z)1144 2728 y Fw(+)p Fs(;)p Fn(\000)1137 2801 y Fs(L)1277 2772 y Fz(\()p Fr(S)1374 2739 y Fw(+)1434 2772 y Fu(;)15 b Fr(S)1536 2739 y Fn(\000)1595 2772 y Fz(\))p 1075 2815 556 4 v 1252 2910 a Fu(Z)1321 2866 y Fw(+)p Fs(;)p Fn(\000)1314 2939 y Fs(L)1666 2836 y Fz(=)56 b(lim)1761 2897 y Fs(L)p Fn(!1)2049 2775 y Fz(1)p 1976 2815 193 4 v 1976 2898 a Fu(L)2038 2872 y Fs(d)p Fn(\000)p Fw(1)2209 2836 y Fz(log)17 b Fu(\026)2396 2792 y Fw(+)p Fs(;)p Fn(\000)2396 2865 y Fs(\014)s(;)p Fw(\001)2518 2876 y Fq(L)2568 2836 y Fz(\()p Fr(S)2665 2799 y Fw(+)2724 2836 y Fu(;)e Fr(S)2826 2799 y Fn(\000)2885 2836 y Fz(\))26 b(=)f(0)15 b 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y(expansion,)c(the)g(extension)g(to)g(more)g(general)h(Pirogo)m (v-Sinai)g(mo)s(dels,)e(as)h(those)g(describ)s(ed)f(at)h(the)118 4055 y(end)f(of)g(Subsection)h(2.1,)g(seems)g(p)s(ossible.)40 b(F)-8 b(or)32 b(more)e(general)i(mo)s(dels)e(sev)m(eral)i(questions)e (remain.)118 4162 y(In)23 b(particular,)j(De\014nition)e(3.1)h(do)s(es) e(not)h(seem)g(appropriate)g(to)g(deal)h(with)e(p)s(erio)s(dic)g (ground)g(states.)118 4270 y(F)-8 b(or)33 b(m)m(ulti-phase)g(mo)s (dels,)f(the)h(solution)g(of)f(the)g(v)-5 b(ariational)34 b(problem)e(is)g(not)h(kno)m(wn)f(and)f(th)m(us)h(a)118 4378 y(macroscopic)42 b(description)f(of)g(phase)g(co)s(existence)i(is) e(a)g(mathematical)i(c)m(hallenge.)74 b(The)41 b(proba-)118 4486 y(bilistic)33 b(p)s(oin)m(t)e(of)h(view)g(is)f(sligh)m(tly)i (di\013eren)m(t)f(since)g(one)g(is)f(in)m(terested)i(to)f(deriv)m(e)g (the)g(macroscopic)118 4594 y(v)-5 b(ariational)27 b(problem)d (\(without)h(solving)h(it\))g(from)e(the)h(microscopic)h(system.)39 b(In)24 b(this)h(case,)j(the)d(dif-)118 4702 y(\014culties)31 b(are)f(of)g(t)m(w)m(o)h(distinct)g(natures:)40 b(geometric)32 b(and)d(probabilistic.)41 b(F)-8 b(or)31 b(a)f(thorough)g(study)f(of) 118 4810 y(the)24 b(geometric)h(problems)e(w)m(e)h(refer)g(the)f (reader)h(to)g(Cerf,)h(Pisztora)g([CePi2].)39 b(F)-8 b(or)24 b(the)g(issues)f(related)118 4918 y(to)39 b(the)f(coarse)h (graining)f(and)f(the)h(surface)g(tension,)i(w)m(e)e(hop)s(e)f(that)i (our)e(approac)m(h)h(can)g(pro)m(vide)118 5026 y(a)33 b(step)g(to)m(w)m(ards)g(the)g(deriv)-5 b(ation)33 b(of)g(phase)f(co)s (existence)i(for)e(m)m(ulti-phase)i(mo)s(dels.)46 b(Nev)m(ertheless,) 118 5134 y(it)40 b(should)f(b)s(e)g(stressed)g(that)h(the)g(in)m (teresting)g(phenomena,)i(as)e(b)s(oundary)d(la)m(y)m(ers,)44 b(o)s(ccurring)39 b(in)118 5242 y(m)m(ulti-phase)e(mo)s(dels)e(cannot)i (b)s(e)e(capture)i(in)e(the)h Fx(L)2030 5209 y Fw(1)2069 5242 y Fz(-framew)m(ork.)58 b(A)37 b(more)f(re\014ned)f(analysis)h(of) 118 5350 y(the)26 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Fp(1)1525 3661 y(\(\001)1599 3647 y Fq(c)1630 3661 y Fp(\))1346 3735 y Fs(h)1663 3705 y Fz(\()p Fu(\033)1750 3725 y Fn(T)1806 3706 y Fl(\000)p Fp(1)1889 3725 y Fw(\(\001\))2006 3705 y Fz(\))p Fu(;)202 b(Z)2337 3661 y Fs(R;\033)2450 3674 y Fp(\001)2500 3660 y Fq(c)2330 3735 y Fs(\014)s(;h;)p Fw(\001)2566 3705 y Fz(=)24 b Fu(Z)2730 3638 y Fs(\033)2770 3661 y Fl(T)2819 3647 y(\000)p Fp(1)2902 3661 y(\(\001)2976 3647 y Fq(c)3008 3661 y Fp(\))2723 3739 y Fs(\014)s(;h;)p Fn(T)2903 3721 y Fl(\000)p Fp(1)2985 3739 y Fw(\(\001\))3475 3705 y Fz(\(4.4\))218 4144 y FE(Pro)s(of.)81 b Fz(Since)44 b Fu(n)j(<)g(n)1107 4111 y Fn(0)1130 4144 y Fz(\()p Fu(x)p Fz(\),)h(it)c(is)g(enough)g(to)g(consider)g(a)g(re\015ection)h(wrt)e(a) h(single)g(surface)118 4252 y(and)34 b(mo)s(dulo)h(a)g(c)m(hange)h(of)f (v)-5 b(ariables)36 b(to)g(w)m(ork)f(in)g(the)g(framew)m(ork)g(of)g (Subsection)g(3.2.)55 b(Supp)s(ose)118 4360 y Fu(x)44 b Fz(is)g(in)g(the)g(upp)s(er)e(part,)48 b Fu(x)g Fr(2)f Fx(Z)1392 4327 y Fs(d)1392 4382 y Fw(+)1451 4360 y Fz(,)h(then,)f(b)m (y)d(induction)g(on)g Fu(k)51 b Ft(6)c Fu(n)d Fz(it)h(is)f(easy)g(to)h (see)g(that)118 4477 y Fu(K)202 4444 y Fs(R)260 4477 y Fz(\()p Fu(x;)15 b(k)s Fz(\))35 b(=)g Fr(T)22 b Fz(\()p Fu(K)7 b Fz(\()p Fu(x;)15 b(k)s Fz(\)\),)40 b(where)35 b Fr(T)59 b Fz(is)36 b(equal)g(to)h(the)f(iden)m(tit)m(y)i(on)e Fu(K)7 b Fz(\()p Fu(x;)15 b(n)p Fz(\))24 b Fr(\\)g Fx(Z)3059 4444 y Fs(d)3059 4499 y Fw(+)3153 4477 y Fz(and)36 b(to)g Fr(R)g Fz(on)118 4594 y Fu(K)7 b Fz(\()p Fu(x;)15 b(n)p Fz(\))20 b Fr(\\)f Fx(Z)580 4561 y Fs(d)580 4616 y Fn(\000)639 4594 y Fz(.)41 b(W)-8 b(e)31 b(next)f(c)m(hec)m(k)i(that)f Fr(T)52 b Fz(is)31 b(one)f(to)h(one.)41 b(If)29 b(it)i(w)m(as)g(not)f (the)g(case,)i(there)e(w)m(ould)g(b)s(e)118 4701 y(t)m(w)m(o)j (distinct)e(sites)h Fu(y)s(;)15 b(z)31 b Fr(2)26 b Fu(K)7 b Fz(\()p Fu(x;)15 b(n)p Fz(\))32 b(suc)m(h)e(that)i Fr(T)23 b Fz(\()p Fu(y)s Fz(\))k(=)f Fr(T)d Fz(\()p Fu(z)t Fz(\).)44 b(This)30 b(w)m(ould)h(mean)g(that)h Fu(z)f Fz(=)26 b Fr(R)p Fz(\()p Fu(y)s Fz(\))118 4809 y(and,)j(since)h Fu(K)7 b Fz(\()p Fu(x;)15 b(n)p Fz(\))30 b(is)f(a)h(con)m(v)m(ex)h (set,)g(then)e(0)h(w)m(ould)f(b)s(e)g(in)g Fu(K)7 b Fz(\()p Fu(x;)15 b(n)p Fz(\),)30 b(whic)m(h)f(is)h(excluded)f(b)s(ecause)118 4919 y Fu(n)39 b Ft(6)g Fu(n)p Fz(\()p Fu(x)p Fz(\).)67 b(Since)39 b Fr(T)61 b Fz(maps)39 b(the)g(plaquettes)h(of)f Fr(f)p Fu(p)p Fr(g)h Fz(in)e Fu(K)7 b Fz(\()p Fu(x;)15 b(n)p Fz(\))39 b(bijectiv)m(ely)i(in)e(those)g Fr(f)p Fu(p)p Fr(g)3488 4886 y Fs(R)3586 4919 y Fz(in)118 5028 y Fu(K)202 4995 y Fs(R)260 5028 y Fz(\()p Fu(x;)15 b(n)p Fz(\),)31 b(\(4.4\))h(follo)m(ws.)42 b(The)30 b(theorem)g(is)h(pro)m(v) m(ed.)1585 b Ft(\003)218 5372 y Fz(The)27 b(previous)h(theorem)h (implies)f(that)h(a)m(w)m(a)m(y)h(from)e(the)h(set)f Fu(B)5 b Fz(,)29 b(the)f(re\015ections)h(ha)m(v)m(e)h(no)e(impact)118 5480 y(on)k(the)h(energy)-8 b(.)47 b(This)31 b(will)i(b)s(e)e(useful)h (to)g(ev)-5 b(aluate)34 b(the)f(corrections)g(to)g(the)g(pressure)d(in) i(presence)p eop end %%Page: 13 13 TeXDict begin 13 12 bop 1137 45 a Fw(SURF)-8 b(A)n(CE)31 b(TENSION)g(AND)f(WULFF)h(THEOR)-6 b(Y)948 b(13)118 366 y Fz(of)29 b(re\015ected)h(b)s(oundaries.)38 b(The)29 b(particular)g(structure)f(of)i(the)f(re\015ecting)g(surfaces)g(will)g (not)g(matter)118 474 y(in)h(the)h(sequel,)g(the)f(analysis)h(only)g (relying)f(on)h(the)f(follo)m(wing)i(assumption:)896 735 y FE(Assumption:)42 b Fz(F)-8 b(or)31 b(all)g Fu(x)25 b Fr(2)g Fx(Z)2024 698 y Fs(d;R)2137 735 y Fz(,)31 b Fu(n)p Fz(\()p Fu(x)p Fz(\))25 b Fu(<)g(n)2546 698 y Fn(0)2569 735 y Fz(\()p Fu(x)p Fz(\).)759 b(\(4.5\))218 888 y(Ha)m(ving)34 b(de\014ned)d(the)i(setup,)g(w)m(e)g(can)g(no)m(w)f (start)h(the)g(analysis)g(whic)m(h)g(b)s(egins)f(b)m(y)g(recalling)i (the)118 996 y(fundamen)m(tal)c(notion)h(of)g(con)m(tours,)g(adapted)f (to)h(the)g(case)g(of)g(re\015ecting)g(surfaces.)118 1222 y(4.1.)47 b FE(Con)m(tours.)f Fz(W)-8 b(e)39 b(will)g(refer)f (explicitly)i(to)f(the)g(case)g(of)g(re\015ections,)i(as)e(underlined)e (b)m(y)h(the)118 1330 y(sup)s(erscript)c Fu(R)q Fz(;)39 b(without)c Fu(R)i Fz(the)f(expressions)f(refer)h(to)g(the)g(case)h (without)f(re\015ections)g(for)g(whic)m(h)118 1438 y(the)e(classical)h (pro)s(ofs)e(apply)g(directly)h(and)f(whic)m(h)g(can)g(an)m(yw)m(a)m(y) i(b)s(e)e(reco)m(v)m(ered)i(from)e(our)g(analysis)118 1546 y(b)m(y)d(replacing)i Fr(f)p Fu(p)p Fr(g)770 1513 y Fs(R)858 1546 y Fz(b)m(y)e Fr(f)p Fu(p)p Fr(g)p Fz(.)218 1655 y(W)-8 b(e)32 b(de\014ne)e(the)g(phase)h(indicator)g(at)h Fu(x)p Fz(,)f Fu(\021)1700 1622 y Fs(R)1697 1678 y(x)1758 1655 y Fz(\()p Fu(\033)s Fz(\),)h(as)f(equal)g(to)g(1)g(\(resp.)g Fr(\000)p Fz(1\))g(if)g Fu(\033)j Fz(is)c(iden)m(tically)j(1)118 1765 y(\(resp.)d Fr(\000)p Fz(1\))h(on)f(all)i Fu(p)25 b Fr(3)g Fu(x)p Fz(,)30 b Fu(p)25 b Fr(2)g(f)p Fu(p)p Fr(g)1363 1732 y Fs(R)1421 1765 y Fz(;)31 b(otherwise)g Fu(\021)1927 1779 y Fs(x)1971 1765 y Fz(\()p Fu(\033)s Fz(\))26 b(=)f(0.)218 1872 y(Calling)45 b Fu(R)q Fz(-connected)g(t)m(w) m(o)h(sites)f Fu(x)f Fz(and)g Fu(y)j Fz(if)d(they)h(b)s(oth)e(b)s (elong)i(to)g(a)f(same)h(plaquette)h(in)118 1980 y Fr(f)p Fu(p)p Fr(g)254 1947 y Fs(R)312 1980 y Fz(,)35 b(the)f(spatial)h(supp)s (orts)c(sp\(\000\))j(of)g(the)g Fu(R)g Fz(con)m(tours)g(\000)f(of)h Fu(\033)j Fz(are)d(the)g(maximal)g Fu(R)q Fz(-connected)118 2090 y(comp)s(onen)m(ts)g(of)g(the)h(set)f Fr(f)p Fu(\021)1128 2057 y Fs(R)1125 2112 y(x)1218 2090 y Fz(=)d(0)p Fr(g)p Fz(.)52 b(W)-8 b(e)35 b(will)g(tacitly)h(supp)s(ose)c(in)i(the)g (sequel)g(that)h(they)f(are)g(all)118 2198 y(b)s(ounded)28 b(sets.)41 b(Let)1497 2290 y(\026)1492 2313 y(\000)25 b(=)1746 2226 y Fo([)1670 2428 y Fs(x)p Fn(2)p Fw(sp\(\000\))1938 2313 y Fu(K)2022 2275 y Fs(R)2079 2313 y Fz(\()p Fu(x;)15 b Fz(2\))1189 b(\(4.6\))118 2562 y(Then)29 b(the)i Fu(R)g Fz(con)m(tours)g(\000)f(of)h Fu(\033)i Fz(are)e(the)g(pairs)e(\000)d(=) f(\()2006 2539 y(\026)2000 2562 y(\000)p Fu(;)15 b(\033)2153 2569 y Fw(\026)2149 2586 y(\000)2197 2562 y Fz(\),)31 b(with)f Fu(\033)2551 2569 y Fw(\026)2547 2586 y(\000)2626 2562 y Fz(the)g(restriction)i(of)e Fu(\033)k Fz(to)3523 2539 y(\026)3518 2562 y(\000.)218 2670 y(Notice)g(that)g(in)e(eac)m(h)i Fu(R)f Fz(connected)g(comp)s(onen)m(t)g(of)2127 2647 y(\026)2121 2670 y(\000)p Fr(n)g Fz(sp\(\000\),)g Fu(\033)2580 2684 y Fs(x)2656 2670 y Fz(is)g(iden)m(tically)i(equal)e(either)118 2778 y(to)i(1)g(or)g(to)g Fr(\000)p Fz(1,)h(while)f(the)f(v)-5 b(alues)35 b(outside)1722 2755 y(\026)1716 2778 y(\000)f(are)h(not)g (determined)f(b)m(y)h(\000)f(and)g(therefore)h(can)g(b)s(e)118 2886 y(arbitrary)-8 b(.)41 b(Let)1599 3001 y Fu(D)28 b Fz(:=)1829 2978 y(\026)1824 3001 y(\000)20 b Fr(n)g Fz(sp\(\000\))1295 b(\(4.7\))118 3136 y(and)26 b(call)i Fu(D)529 3150 y Fw(0)595 3136 y Fz(and)d Fu(D)845 3098 y Fn(\006)842 3164 y Fs(i)930 3136 y Fz(the)i(maximal)g Fu(R)g Fz(connected)g(comp)s(onen)m(ts)g(of)f Fu(D)s Fz(.)40 b Fu(D)2780 3150 y Fw(0)2846 3136 y Fz(is)26 b(the)h(one)f(whic)m(h)g(is)h Fu(R)118 3250 y Fz(connected)32 b(to)f(the)g(un)m(b)s(ounded)c(comp)s(onen)m(t)k(of)1855 3227 y(\026)1849 3250 y(\000)1906 3217 y Fs(c)1940 3250 y Fz(,)g Fu(D)2074 3212 y Fw(+)2071 3277 y Fs(i)2163 3250 y Fz(\(resp.)g Fu(D)2495 3212 y Fn(\000)2492 3277 y Fs(i)2554 3250 y Fz(\))g(are)f(the)h(comp)s(onen)m(ts)g(where)118 3364 y Fu(\033)170 3378 y Fs(x)245 3364 y Fz(\(as)h(sp)s(eci\014ed)f(b) m(y)g(\000\))g(is)h(equal)g(to)g(1)f(\(resp.)h Fr(\000)p Fz(1\).)44 b(W)-8 b(e)32 b(also)g(call)h(in)m(t)2624 3326 y Fn(\006)2624 3391 y Fs(i)2684 3364 y Fz(\(\000\))e(the)h(comp)s (onen)m(t)f(of)3576 3341 y(\026)3570 3364 y(\000)3627 3331 y Fs(c)118 3478 y Fz(whic)m(h)f(is)h Fu(R)g Fz(connected)g(to)g Fu(D)1183 3440 y Fn(\006)1180 3505 y Fs(i)1242 3478 y Fz(.)40 b(Finally)32 b(\000)e(is)g(a)h Fr(\006)f Fz(con)m(tour,)h(if)g Fu(\033)d Fz(=)d Fr(\006)p Fz(1)31 b(on)f Fu(D)2936 3492 y Fw(0)2976 3478 y Fz(.)218 3592 y(The)36 b Fu(R)h Fz(con)m(tours)g(in) f(a)h(b)s(ounded)d(domain)i(\003)g Fr(\032)f Fx(Z)2056 3559 y Fs(d;R)2206 3592 y Fz(with)h(+)g([)p Fr(\000)p Fz(])h(b)s(oundary)d(conditions)j(are)118 3700 y(de\014ned)29 b(as)i(the)f(con)m(tours)h(of)g(the)f(con\014guration)h(\()p Fu(\033)1964 3714 y Fw(\003)2018 3700 y Fu(;)15 b FE(1)2110 3714 y Fw(\003)2159 3696 y Fq(c)2195 3700 y Fz(\))31 b([resp.)f(and)g(of)h(\()p Fu(\033)2872 3714 y Fw(\003)2925 3700 y Fu(;)15 b Fr(\000)p FE(1)3088 3714 y Fw(\003)3137 3696 y Fq(c)3173 3700 y Fz(\)].)218 3810 y(The)30 b(w)m(eigh)m(t)h Fu(w)759 3777 y Fs(R;)p Fw(+)892 3810 y Fz(\(\000\))g(of)g(a)f(+)g Fu(R)h Fz(con)m(tour)g(is)1138 4078 y Fu(w)1205 4040 y Fs(R;)p Fw(+)1337 4078 y Fz(\(\000\))26 b(=)1596 4016 y Fu(e)1638 3983 y Fn(\000)p Fs(\014)s(H)1799 3960 y Fq(R)1794 4006 y(h)1850 3983 y Fw(\()p Fs(\033)1921 3989 y Fp(\026)1917 4002 y(\000)1959 3983 y Fw(\))p 1596 4057 395 4 v 1687 4149 a Fu(e)1729 4123 y Fs(\014)s(h)p Fn(j)1837 4106 y Fw(\026)1833 4123 y(\000)o Fn(j)2052 3964 y Fs(n)2095 3941 y Fl(\000)2041 3991 y Fo(Y)2042 4187 y Fs(i)p Fw(=1)2208 3975 y Fu(Z)2277 3931 y Fs(R;)p Fn(\000)2270 4020 y Fs(\014)s(;h;)p Fw(in)n(t)2477 3990 y Fl(\000)2477 4042 y Fq(i)2529 4020 y Fw(\(\000\))p 2208 4057 425 4 v 2208 4157 a Fu(Z)2277 4113 y Fs(R;)p Fw(+)2270 4202 y Fs(\014)s(;h;)p Fw(in)n(t)2477 4172 y Fl(\000)2477 4224 y Fq(i)2529 4202 y Fw(\(\000\))3475 4078 y Fz(\(4.8\))118 4340 y(The)j(sup)s(erscript)f Fu(R)j Fz(recalls)g(that)f(all)h(quan)m(tities)g(are)f(de\014ned)f(using)g (the)h(collection)i(of)e(plaquettes)118 4454 y Fr(f)p Fu(p)p Fr(g)254 4421 y Fs(R)312 4454 y Fz(.)71 b(The)40 b(term)g Fu(e)874 4421 y Fs(\014)s(h)p Fn(j)982 4405 y Fw(\026)978 4421 y(\000)p Fn(j)1086 4454 y Fz(in)g(the)g(denominator) h(is)f(the)g(Gibbs)g(factor)h(of)g(the)f(con\014guration)h FE(1)3618 4461 y Fw(\026)3613 4478 y(\000)118 4574 y Fz(iden)m(tically)33 b(equal)d(to)i(1)e(in)1096 4551 y(\026)1090 4574 y(\000,)h Fu(e)1245 4541 y Fn(\000)p Fs(\014)s(H)1401 4553 y Fq(h)1441 4541 y Fw(\()p Fi(1)1513 4547 y Fp(\026)1509 4560 y(\000)1551 4541 y Fw(\))1608 4574 y Fz(=)25 b Fu(e)1746 4541 y Fs(\014)s(h)p Fn(j)1854 4524 y Fw(\026)1850 4541 y(\000)p Fn(j)1917 4574 y Fz(.)218 4682 y(The)31 b(w)m(eigh)m(t)i Fu(w)762 4649 y Fs(R;)p Fn(\000)894 4682 y Fz(\(\000\))f(of)g(a)g Fr(\000)f Fu(R)h Fz(con)m(tour)g(is)f(de\014ned)g(symmetrically)h(with)g(the)f(role)h (of)g(+)f(and)118 4790 y Fr(\000)f Fz(in)m(terc)m(hanged.)42 b(With)31 b(these)g(de\014nitions,)f(w)m(e)h(ha)m(v)m(e)g(the)g(iden)m (tit)m(y)1155 4956 y Fu(Z)1224 4912 y Fs(R;)p Fn(\006)1217 4986 y Fs(\014)s(;h;)p Fw(\003)1418 4956 y Fz(=)25 b Fu(e)1556 4919 y Fn(\006)p Fs(\014)s(h)p Fn(j)p Fw(\003)p Fn(j)1858 4870 y Fo(X)1828 5083 y Fn(f)p Fw(\000)1907 5093 y Fq(i)1933 5083 y Fn(g)1968 5053 y Fl(\006)1968 5106 y Fp(\003)2100 4870 y Fo(Y)2061 5083 y Fn(f)p Fw(\000)2140 5093 y Fq(i)2167 5083 y Fn(g)2202 5053 y Fl(\006)2202 5106 y Fp(\003)2269 4956 y Fu(w)2336 4919 y Fs(R;)p Fn(\006)2469 4956 y Fz(\(\000)2561 4970 y Fs(i)2589 4956 y Fz(\))851 b(\(4.9\))118 5247 y(where)34 b Fr(f)p Fz(\000)487 5261 y Fs(i)515 5247 y Fr(g)560 5209 y Fw(+)560 5275 y(\003)654 5247 y Fz([)p Fr(f)p Fz(\000)781 5261 y Fs(i)810 5247 y Fr(g)855 5209 y Fn(\000)855 5275 y Fw(\003)914 5247 y Fz(])g(is)h(a)f(compatible)i(collection)g(of)f(+)f([)p Fr(\000)p Fz(])g Fu(R)h Fz(con)m(tours)f(in)g(\003.)52 b(Tw)m(o)35 b(con)m(tours)118 5355 y(are)c(compatible)h(i\013)e(their)h (spatial)g(supp)s(orts)d(are)j(not)g Fu(R)q Fz({connected.)218 5463 y(F)-8 b(or)25 b(the)g(case)h(without)f(re\015ections)g(w)m(e)h (can)f(apply)f(directly)h(the)g(classical)i(Pirogo)m(v-Sinai)g(theory:) p eop end %%Page: 14 14 TeXDict begin 14 13 bop 118 45 a Fw(14)1128 b(T.)23 b(BODINEA)n(U)g (AND)g(E.)g(PRESUTTI)118 366 y FE(Theorem)43 b(4.2.)i Fj(Ther)-5 b(e)40 b(is)f Fu(c)d(>)g Fz(0)j Fj(and,)i(for)f(any)f Fu(\014)44 b Fj(lar)-5 b(ge)39 b(enough,)i Fu(h)p Fz(\()p Fu(\014)5 b Fz(\))37 b Fr(2)f Fz(\(0)p Fu(;)15 b(ce)3137 333 y Fn(\000)p Fs(\014)s(=)p Fw(2)3312 366 y Fz(\))39 b Fj(so)g(that)118 476 y(the)g(thermo)-5 b(dynamic)42 b(limits)d(of)f Fu(\026)1345 438 y Fn(\006)1345 509 y Fs(\014)s(;h)p Fw(\()p Fs(\014)s Fw(\))p Fs(;)p Fw(\003)1657 476 y Fj(de\014ne)h(distinct)g(DLR)g(me)-5 b(asur)g(es.)61 b(Mor)-5 b(e)g(over,)41 b(for)e(any)118 597 y(c)-5 b(ontour)34 b Fz(\000)p Fj(,)e(the)h(weight)h(without)f(r)-5 b(e\015e)g(ction)35 b(satisfy)1462 754 y Fz(0)26 b Fu(<)f(w)1696 716 y Fn(\006)1756 754 y Fz(\(\000\))g Ft(6)g Fu(e)2046 716 y Fn(\000)p Fs(\014)s(N)2200 727 y Fp(\000)2242 716 y Fs(=)p Fw(2)3429 754 y Fz(\(4.10\))118 905 y Fj(wher)-5 b(e)34 b Fu(N)448 919 y Fw(\000)529 905 y Fj(is)e(the)h(numb)-5 b(er)33 b(of)g(distinct)g(plaquettes)h(which)f(c)-5 b(over)34 b(sp)p Fz(\(\000\))p Fj(.)218 1070 y Fz(In)29 b(the)i(follo)m(wing)h (the)e(b)s(ound)f(\(4.10\))j(will)f(b)s(e)f(refered)g(as)g(a)h(P)m (eierls)h(estimate)g(since)e(it)h(leads)1445 1232 y Fu(\026)1500 1194 y Fn(\006)1500 1265 y Fs(\014)s(;h)p Fw(\()p Fs(\014)s Fw(\))p Fs(;)p Fw(\003)1773 1232 y Fz(\(\000\))26 b Ft(6)f Fu(e)2064 1194 y Fn(\000)p Fs(\014)s(N)2218 1205 y Fp(\000)2260 1194 y Fs(=)p Fw(2)3429 1232 y Fz(\(4.11\))218 1638 y(The)32 b(b)s(ound)f(\(4.10\))k(is)e(actually)h(the)f(crucial)h(p)s(oin)m(t)f (of)g(the)g(theorem,)h(the)f(small)h(w)m(eigh)m(t)g(of)f(the)118 1746 y(con)m(tours)g(is)g(in)f(fact)i(resp)s(onsible)d(for)i(the)g (memory)f(of)h(the)g(b)s(oundary)d(conditions)j(to)h(surviv)m(e)e(the) 118 1854 y(thermo)s(dynamic)k(limit,)j(th)m(us)d(yielding)h(the)f (phase)g(transition.)60 b(Moreo)m(v)m(er,)40 b(as)d(w)m(e)f(will)h (see,)i(if)d Fu(\014)118 1961 y Fz(is)g(large,)j(\(and)c(the)i(w)m (eigh)m(t)g(small\),)i(b)m(y)c(cluster)i(expansion,)g(it)g(is)f(p)s (ossible)f(to)i(exp)s(onen)m(tiate)g(the)118 2069 y(r.h.s.)30 b(of)h([the)g(analogue)h(without)e(re\015ections])i(of)37 b(\(4.9\))c(and)c(th)m(us)i(to)g(compute)g(the)f(\014nite)h(v)m(olume) 118 2177 y(corrections)k(to)g(the)f(pressure.)49 b(This)33 b(is)h(on)f(the)h(other)g(hand)f(also)i(the)f(k)m(ey)g(p)s(oin)m(t)g (in)f(the)h(pro)s(of)f(of)118 2285 y(\(4.10\))r(,)40 b(whic)m(h)d(at)i(\014rst)d(sigh)m(t)j(mak)m(es)f(all)h(the)f(ab)s(o)m (v)m(e)g(to)h(lo)s(ok)f(circular.)63 b(The)37 b(main)g(goal)i(in)e (this)118 2393 y(section)32 b(is)e(to)h(pro)m(v)m(e)g(the)g(b)s(ound)d (\(4.10\))33 b(in)d(case)h(of)g(re\015ections.)118 2974 y(4.2.)47 b FE(Restricted)e(ensem)m(bles.)h Fz(F)-8 b(ollo)m(wing)41 b(Zahradnik,)e(w)m(e)g(construct)g(a)g(m)m(uc)m(h)f(simpler,)j (\014cti-)118 3082 y(tious)35 b(mo)s(del)g(whic)m(h,)h(as)f(a)g (miracle,)j(in)c(the)h(end,)h(turns)e(out)h(to)g(coincide)h(with)f(the) g(real)h(one.)54 b(In)118 3189 y(the)34 b(whole)f(sequel)h Fu(\014)39 b Fz(is)33 b(large)i(enough)e(and)g Fu(h)e Fz(=)f Fu(h)p Fz(\()p Fu(\014)5 b Fz(\),)36 b(see)e(Theorem)f(4.2,)j (will)d(often)h(drop)f(from)118 3297 y(the)e(notation.)218 3405 y(Inspired)e(b)m(y)i(\(4.9\))r(,)f(w)m(e)h(set)g(for)f(an)m(y)h(b) s(ounded)d(region)j(\003)26 b Fr(\032)f Fx(Z)2416 3372 y Fs(d;R)2529 3405 y Fz(,)1177 3575 y(\004)1238 3531 y Fs(R;)p Fn(\006)1238 3605 y Fs(\014)s(;)p Fw(\003)1395 3575 y Fz(=)g Fu(e)1533 3538 y Fn(\006)p Fs(\014)s(h)p Fn(j)p Fw(\003)p Fn(j)1836 3489 y Fo(X)1805 3702 y Fn(f)p Fw(\000)1884 3712 y Fq(i)1911 3702 y Fn(g)1946 3672 y Fl(\006)1946 3725 y Fp(\003)2077 3489 y Fo(Y)2038 3702 y Fn(f)p Fw(\000)2117 3712 y Fq(i)2144 3702 y Fn(g)2179 3672 y Fl(\006)2179 3725 y Fp(\003)2265 3575 y Fz(^)-63 b Fu(w)2314 3538 y Fs(R;)p Fn(\006)2446 3575 y Fz(\(\000)2538 3589 y Fs(i)2567 3575 y Fz(\))827 b(\(4.12\))299 3984 y(^)-63 b Fu(w)348 3946 y Fs(R;)p Fw(+)480 3984 y Fz(\(\000\))26 b(=)f(min)896 3883 y Fo(n)956 3984 y Fu(e)998 3946 y Fn(\000)p Fs(\014)s(N)1159 3923 y Fq(R)1152 3969 y Fp(\000)1210 3946 y Fs(=)p Fw(2)1285 3984 y Fz(;)1335 3922 y Fu(e)1377 3889 y Fn(\000)p Fs(\014)s(H)1538 3866 y Fq(R)1533 3912 y(h)1589 3889 y Fw(\()p Fs(\033)1660 3895 y Fp(\026)1656 3908 y(\000)1698 3889 y Fw(\))p 1335 3963 395 4 v 1426 4055 a Fu(e)1468 4028 y Fs(\014)s(h)p Fn(j)1576 4012 y Fw(\026)1572 4028 y(\000)o Fn(j)1791 3870 y Fs(n)1834 3846 y Fl(\000)1780 3897 y Fo(Y)1781 4093 y Fs(i)p Fw(=1)1946 3881 y Fz(\004)2007 3837 y Fs(R;)p Fn(\000)2007 3926 y Fs(\014)s(;)p Fw(in)n(t)2154 3896 y Fl(\000)2154 3948 y Fq(i)2206 3926 y Fw(\(\000\))p 1946 3963 363 4 v 1946 4063 a Fz(\004)2007 4019 y Fs(R;)p Fw(+)2007 4108 y Fs(\014)s(;)p Fw(in)n(t)2154 4078 y Fl(\000)2154 4130 y Fq(i)2206 4108 y Fw(\(\000\))2319 3883 y Fo(o)2380 3984 y Fu(;)197 b Fz(\000)30 b(a)h(+)f Fu(R)h Fz(con)m(tour)163 b(\(4.13\))300 4361 y(^)-64 b Fu(w)348 4323 y Fs(R;)p Fn(\000)481 4361 y Fz(\(\000\))26 b(=)f(min)896 4260 y Fo(n)957 4361 y Fu(e)999 4323 y Fn(\000)p Fs(\014)s(N)1160 4300 y Fq(R)1153 4346 y Fp(\000)1211 4323 y Fs(=)p Fw(2)1285 4361 y Fz(;)1336 4299 y Fu(e)1378 4266 y Fn(\000)p Fs(\014)s(H)1539 4243 y Fq(R)1534 4289 y(h)1589 4266 y Fw(\()p Fs(\033)1660 4272 y Fp(\026)1656 4285 y(\000)1699 4266 y Fw(\))p 1336 4340 395 4 v 1399 4432 a Fu(e)1441 4405 y Fn(\000)p Fs(\014)s(h)p Fn(j)1604 4389 y Fw(\026)1600 4405 y(\000)o Fn(j)1792 4247 y Fs(n)1835 4223 y Fp(+)1781 4274 y Fo(Y)1782 4470 y Fs(i)p Fw(=1)1947 4258 y Fz(\004)2008 4214 y Fs(R;)p Fw(+)2008 4303 y Fs(\014)s(;)p Fw(in)n(t)2155 4273 y Fp(+)2155 4325 y Fq(i)2205 4303 y Fw(\(\000\))p 1947 4340 362 4 v 1947 4440 a Fz(\004)2008 4396 y Fs(R;)p Fn(\000)2008 4485 y Fs(\014)s(;)p Fw(in)n(t)2155 4455 y Fp(+)2155 4507 y Fq(i)2205 4485 y Fw(\(\000\))2318 4260 y Fo(o)2379 4361 y Fu(;)197 b Fz(\000)30 b(a)h Fr(\000)f Fu(R)h Fz(con)m(tour)164 b(\(4.14\))118 4618 y(where)30 b Fu(N)464 4585 y Fs(R)454 4645 y Fw(\000)553 4618 y Fz(is)g(the)h(n)m(um)m(b)s(er)f(of)g Fu(R)i Fz(plaquettes)f(in)g (sp\(\000\).)41 b(In)30 b(this)h(w)m(a)m(y)h(the)e(w)m(eigh)m(ts)i (automatically)118 4726 y(satisfy)i(the)g(crucial)h(b)s(ound)d (\(4.10\))r(,)j(but)e(\014rst)g(let)h(us)f(c)m(hec)m(k)j(that)e (\(4.12\))r({\(4.14\))j(do)c(really)i(de\014ne)118 4847 y(the)24 b(\\partition)h(functions")e(\004)1177 4803 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b(on)g(the)h(other)g(hand)f(w)m(e)h(kno)m(w)g (\004)1513 5322 y Fs(R;)p Fn(\006)1513 5396 y Fs(\014)s(;)p Fw(\003)1674 5367 y Fz(for)f(all)i(\003)25 b Fr(\032)g Fx(Z)2183 5334 y Fs(d;R)2296 5367 y Fz(,)30 b Fr(j)p Fz(\003)p Fr(j)c Ft(6)f Fu(n)p Fz(,)30 b(w)m(e)g(can)g(use)f(\(4.13\))s ({\(4.14\))118 5480 y(to)h(determine)e(the)h(w)m(eigh)m(ts)h(of)f(all)h (\000)e(if)h(all)h(their)e(in)m(terior)i(parts)e(ha)m(v)m(e)i(v)m (olume)55 b Ft(6)25 b Fu(n)p Fz(;)k(since)g(regions)p eop end %%Page: 15 15 TeXDict begin 15 14 bop 1137 45 a Fw(SURF)-8 b(A)n(CE)31 b(TENSION)g(AND)f(WULFF)h(THEOR)-6 b(Y)948 b(15)118 366 y Fz(\003)33 b(with)f Fr(j)p Fz(\003)p Fr(j)e Fz(=)e Fu(n)21 b Fz(+)h(1)33 b(cannot)g(con)m(tain)h(con)m(tours)f(whose)f(in) m(terior)i(part)e(has)g(v)m(olume)i Fu(>)28 b(n)p Fz(,)33 b(w)m(e)g(can)118 487 y(use)d(\(4.12\))j(to)e(determine)f(\004)1134 443 y Fs(R;)p Fn(\006)1134 516 y Fs(\014)s(;)p Fw(\003)1297 487 y Fz(for)g(suc)m(h)g(a)g(\003,)h(and)f(the)g(induction)h(step)f(is) 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1208 y Fs(R;)p Fn(\006)3565 1241 y Fz(\()p Fr(\001)p Fz(\))118 1349 y(their)31 b(w)m(eigh)m(ts,)i (all)f(suc)m(h)e(notions)i(b)s(eing)e(de\014ned)g(next.)42 b(Analogous)32 b(expressions)f(are)g(v)-5 b(alid)32 b(in)e(the)118 1456 y(absence)h(of)f(re\015ections.)218 1564 y(A)d(+)g Fu(R)g Fz(p)s(olymer)g Fu(\031)h Fz(=)d([\000)1116 1523 y Fs(")1149 1533 y Fq(i)1116 1592 y Fs(i)1179 1564 y Fz(])j(in)e(\005)1402 1520 y Fs(R;)p Fw(+)1402 1593 y(\003)1562 1564 y Fz(\(the)i(de\014nition)e(of)i Fr(\000)e Fz(p)s(olymers)h(is)g (similar)g(and)g(omitted\),)118 1672 y(is)35 b(an)f(unordered,)g (\014nite)g(collection)j(of)e(distinct)g(+)f Fu(R)h Fz(con)m(tours)g (\000)2526 1686 y Fs(i)2588 1672 y Fz(tak)m(en)g(with)g(p)s(ositiv)m(e) g(in)m(teger)118 1780 y(m)m(ultiplicit)m(y)d Fu(")647 1794 y Fs(i)676 1780 y Fz(,)f(and)e(suc)m(h)h(that,)i(setting)1249 1966 y Fu(X)7 b Fz(\()p Fu(\031)s Fz(\))26 b(=)1578 1879 y Fo([)1617 2075 y Fs(i)1720 1966 y Fz(sp)o(\(\000)1898 1980 y Fs(i)1926 1966 y Fz(\))p Fu(;)198 b(\031)28 b Fz(=)d([\000)2442 1924 y Fs(")2475 1934 y Fq(i)2442 1993 y Fs(i)2505 1966 y Fz(])899 b(\(4.16\))118 2225 y Fu(X)7 b Fz(\()p Fu(\031)s Fz(\))32 b(is)e(a)h Fu(R)g Fz(connected)g(subset)f (of)g(\003.)218 2334 y(The)e(w)m(eigh)m(ts)37 b(^)-52 b Fu(!)785 2301 y Fs(R;)p Fn(\006)917 2334 y Fz(\()p Fu(\031)s Fz(\),)30 b Fu(\031)e Fz(=)d([\000)1355 2293 y Fs(")1388 2303 y Fq(i)1355 2361 y Fs(i)1418 2334 y Fz(],)k(are)g(giv)m(en)h(in)e(terms)g(of)h(the)g(w)m(eigh)m(ts)h(of)e (con)m(tours,)49 b(^)-64 b Fu(w)3376 2301 y Fs(R;)p Fn(\006)3509 2334 y Fz(\(\000\):)1310 2529 y(^)-52 b Fu(!)1363 2491 y Fs(R;)p Fn(\006)1495 2529 y Fz(\()p Fu(\031)s Fz(\))26 b(=)f Fu(r)s Fz(\()p Fu(\031)s Fz(\))1926 2443 y Fo(Y)1973 2638 y Fs(i)2077 2529 y Fz(^)-64 b Fu(w)2125 2491 y Fs(R;)p Fn(\006)2258 2529 y Fz(\(\000)2350 2543 y Fs(i)2378 2529 y Fz(\))2413 2491 y Fs(")2446 2501 y Fq(i)3429 2529 y Fz(\(4.17\))118 2783 y(where)1266 2917 y Fu(r)s Fz(\()p Fu(\031)s Fz(\))26 b(=)1557 2831 y Fo(Y)1603 3026 y Fs(i)1673 2917 y Fz(\()p Fu(")1750 2931 y Fs(i)1779 2917 y Fz(!\))1839 2879 y Fn(\000)p Fw(1)2016 2831 y Fo(X)1949 3032 y Fn(G)1995 3013 y Fl(0)2017 3032 y Fn(\032G)t Fw(\()p Fs(\031)r Fw(\))2215 2917 y Fz(\()p Fr(\000)p Fz(1\))2401 2879 y Fn(jG)2467 2856 y Fl(0)2490 2879 y Fn(j)118 3179 y Fz(with)j Fr(G)5 b Fz(\()p Fu(\031)s Fz(\))31 b(the)f(\(abstract\))h (graph)e(of)h Fu(\031)s Fz(,)g(whic)m(h)f(consists)i(of)e(v)m(ertices,) j(lab)s(elled)e(b)m(y)g(the)3255 3111 y Fo(P)3351 3206 y Fs(i)3395 3179 y Fu(")3437 3193 y Fs(i)3495 3179 y Fz(con-)118 3287 y(tours)c(in)g Fu(\031)s Fz(,)h(and)f(of)g(edges,)i (whic)m(h)e(join)g(an)m(y)h(t)m(w)m(o)g(v)m(ertices)h(lab)s(elled)f(b)m (y)f(con)m(tours)h(with)f(in)m(tersecting)118 3395 y(supp)s(orts.)39 b(By)31 b(de\014nition)f Fr(G)5 b Fz(\()p Fu(\031)s Fz(\))31 b(is)g(connected)g(and)f(the)g(sum)g(in)g(\(4.18\))i(is)f(o)m(v)m(er)h (all)f(the)f(connected)118 3503 y(subgraphs)24 b Fr(G)603 3470 y Fn(0)652 3503 y Fz(of)h Fr(G)5 b Fz(\()p Fu(\031)s Fz(\))27 b(whic)m(h)e(con)m(tain)i(all)f(the)1804 3435 y Fo(P)1900 3530 y Fs(i)1944 3503 y Fu(")1986 3517 y Fs(i)2039 3503 y Fz(v)m(ertices;)j Fr(jG)2474 3470 y Fn(0)2498 3503 y Fr(j)d Fz(denotes)g(the)f(n)m(um)m(b)s(er)f(of)i (edges)118 3611 y(in)k Fr(G)283 3578 y Fn(0)307 3611 y Fz(.)218 3719 y(The)38 b(n)m(um)m(b)s(er)g(of)i(connections)g(of)f (eac)m(h)i(site)f(is)f(not)g(increased)h(b)m(y)f(the)g(re\015ection)h (pro)s(cedure.)118 3827 y(Th)m(us,)31 b(for)g Fu(\014)37 b Fz(large)c(enough,)e([KP],)h(the)g(series)g(on)f(the)h(r.h.s.)43 b(of)c(\(4.15\))34 b(is)d(absolutely)i(con)m(v)m(ergen)m(t)118 3935 y(and,)d(giv)m(en)i(an)m(y)e(\014nite)h(sequence)f(\000)1395 3949 y Fw(1)1435 3935 y Fu(;)15 b(::;)g Fz(\000)1622 3949 y Fs(n)1700 3935 y Fz(of)31 b(con)m(tours,)1153 4090 y Fo(X)881 4294 y Fs(\031)r Fn(2)p Fw(\005)1024 4275 y Fq(R;)p Fl(\006)1141 4294 y Fs(;\031)r Fn(3)p Fw(\000)1295 4304 y Fq(i)1321 4294 y Fs(;i)p Fw(=1)p Fs(;::n)1647 4176 y Fr(j)7 b Fz(^)-52 b Fu(!)1732 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5004 y Fs(d;R)2245 5037 y Fz(whic)m(h)f(con)m(tain)i(the)f(origin)h (\(supp)s(osing)118 5151 y(0)j Fr(2)f Fx(Z)350 5118 y Fs(d;R)463 5151 y Fz(\).)54 b Fu(D)37 b Fz(represen)m(t)d(the)h (spatial)h(supp)s(ort)d(of)h(a)h(con)m(tour)h(and)e(2)2634 5118 y Fn(j)p Fs(D)r Fn(j)2772 5151 y Fz(b)s(ounds)e(the)j(n)m(um)m(b)s (er)e(of)118 5264 y(con)m(tours)c(with)f(same)g(spatial)h(supp)s(ort)e Fu(D)s Fz(.)40 b(The)27 b(extra)i(2)g(in)e(2)2321 5231 y Fw(2)p Fn(j)p Fs(D)r Fn(j)2489 5264 y Fz(is)h(for)g(con)m(v)m (enience.)42 b(The)27 b(factor)118 5372 y(2)163 5339 y Fs(d)243 5372 y Fz(in)40 b(the)f(last)i(term)e(of)47 b(\(4.18\))42 b(en)m(ters)e(via)g(the)g(relation)g(\(2)2345 5339 y Fs(d)2387 5372 y Fz(\))p Fu(N)2495 5386 y Fs(D)2599 5372 y Fr(\025)h(j)p Fu(D)s Fr(j)p Fz(,)h Fu(N)2979 5386 y Fs(D)3082 5372 y Fz(the)e(n)m(um)m(b)s(er)e(of)118 5480 y(plaquettes)31 b(needed)g(to)g(co)m(v)m(er)h Fu(D)s Fz(.)p eop end %%Page: 16 16 TeXDict begin 16 15 bop 118 45 a Fw(16)1128 b(T.)23 b(BODINEA)n(U)g (AND)g(E.)g(PRESUTTI)218 366 y Fz(Since)32 b(b)m(y)f(Theorem)h(4.2,)h (the)f(w)m(eigh)m(ts)h Fu(w)1693 333 y Fn(\006)1753 366 y Fz(\(\000\))f(satisfy)g(the)g(same)h(b)s(ounds)c(as)j(the)51 b(^)-64 b Fu(w)3240 333 y Fs(R;)p Fn(\006)3373 366 y Fz(\(\000\),)33 b(w)m(e)118 474 y(ha)m(v)m(e,)f(analogously)g(to)f (\(4.15\))r(,)1199 643 y(log)18 b Fu(Z)1401 605 y Fn(\006)1394 673 y Fs(\014)s(;)p Fw(\003)1535 643 y Fz(=)25 b Fr(\006)p Fu(\014)5 b(h)p Fr(j)p Fz(\003)p Fr(j)46 b Fz(+)2125 557 y Fo(X)2060 775 y Fs(\031)r Fn(2)p Fw(\005)2203 742 y Fq(R;)p Fl(\006)2203 798 y Fp(\003)2335 643 y Fu(!)2395 606 y Fn(\006)2454 643 y Fz(\()p Fu(\031)s Fz(\))850 b(\(4.20\))118 936 y(with)30 b Fu(!)385 903 y Fn(\006)444 936 y Fz(\()p Fu(\031)s Fz(\))h(de\014ned)e(b)m(y)j(\(4.17\))g(ha)m (ving)f Fu(w)1663 903 y Fn(\006)1723 936 y Fz(\(\000\))g(in)f(the)g (place)i(of)49 b(^)-64 b Fu(w)2546 903 y Fs(R;)p Fn(\006)2679 936 y Fz(\(\000\).)41 b(As)31 b(in)f(\(4.18\))r(,)1189 1075 y Fo(X)976 1271 y Fs(\031)r 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Fn(j)118 4497 y Fz(whic)m(h)35 b(pro)m(v)m(es)h(\(4.23\))h(b)s(ecause,)g(as)f(w)m(e)f (are)h(going)g(to)g(see,)h(the)f(sum)e(o)m(v)m(er)i Fu(k)s Fz(,)h(that)f(w)m(e)g(denote)f(b)m(y)118 4605 y Fu(S)5 b Fz(\()p Fu(x)p Fz(\),)31 b(is)g(b)s(ounded)d(b)m(y)i(1.)218 4713 y(Calling)43 b Fu(S)599 4727 y Fs(N)666 4713 y Fz(\()p Fu(x)p Fz(\))g(the)g(sum)f(with)g Fu(k)49 b Fr(\024)c Fu(N)10 b Fz(,)46 b(since)d Fu(S)5 b Fz(\()p Fu(x)p Fz(\))43 b(is)f(the)h(limit)h(as)e Fu(N)56 b Fr(!)45 b(1)e Fz(of)g Fu(S)3447 4727 y Fs(N)3513 4713 y Fz(\()p Fu(x)p Fz(\),)118 4821 y(it)c(su\016ces)f(to)h(pro)m(v)m(e)g(that)g(for)e(all)j Fu(y)h Fz(and)c Fu(N)10 b Fz(,)40 b Fu(S)1866 4835 y Fs(N)1933 4821 y Fz(\()p Fu(y)s Fz(\))f Ft(6)f Fz(1.)65 b(The)37 b(pro)s(of)h(is)g(b)m(y)g(induction)g(on)g Fu(N)10 b Fz(.)118 4928 y Fu(S)174 4942 y Fw(1)213 4928 y Fz(\()p Fu(y)s Fz(\))26 b Fu(<)f Fz(1)31 b(b)m(y)g(\(4.19\))s(.)40 b(Supp)s(ose)29 b Fu(S)1367 4942 y Fs(N)7 b Fn(\000)p Fw(1)1524 4928 y Fz(\()p Fu(x)p Fz(\))26 b 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Fn(j)1025 5372 y Fz(coming)g(from)g(the)f (induction)h(h)m(yp)s(othesis.)39 b(Then,)26 b(b)m(y)i(\(4.19\))s(,)f Fu(S)3305 5386 y Fs(N)3372 5372 y Fz(\()p Fu(x)p Fz(\))f Ft(6)f Fz(1)118 5480 y(for)30 b(an)m(y)h Fu(x)f Fz(and)g(\(4.23\))j(is) d(pro)m(v)m(ed.)41 b(The)30 b(lemma)h(is)f(pro)m(v)m(ed.)1357 b Ft(\003)p eop end %%Page: 17 17 TeXDict begin 17 16 bop 1137 45 a Fw(SURF)-8 b(A)n(CE)31 b(TENSION)g(AND)f(WULFF)h(THEOR)-6 b(Y)948 b(17)218 366 y Fz(By)30 b(the)h(analogue)h(of)37 b(\(4.22\))c(w)m(e)e(conclude)g (con)m(v)m(ergence)h(of)f(the)f(series)h(on)f(the)h(r.h.s.)f(of)1188 588 y Fr(P)1251 602 y Fn(\006)1335 588 y Fz(:=)c Fr(\006)p Fu(h)20 b Fz(+)1706 526 y(1)p 1701 567 57 4 v 1701 650 a Fu(\014)1974 501 y Fo(X)1782 703 y Fs(\031)r Fn(2)p Fw(\005\()p Fn(\006)p Fw(\))p Fs(;X)5 b Fw(\()p Fs(\031)r Fw(\))p Fn(3)p Fw(0)2330 526 y Fu(!)2390 493 y Fn(\006)2449 526 y Fz(\()p Fu(\031)s Fz(\))p 2323 567 259 4 v 2323 650 a Fr(j)p Fu(X)i Fz(\()p Fu(\031)s Fz(\))p Fr(j)3429 588 y Fz(\(4.24\))118 856 y(T)-8 b(o)26 b(study)f(the)h(w)m(eigh)m(ts)h (of)f(the)g(p)s(olymers)f(obtained)h(b)m(y)f(re\015ection)i(w)m(e)f (will)g(use)f(the)h(follo)m(wing)h(three)118 964 y(lemmas,)i(where)f (\003)g(is)g(tacitly)i(supp)s(osed)c(to)j(b)s(e)e(a)h(b)s(ounded)e (region)j(in)e Fx(Z)2691 931 y Fs(d)2731 964 y Fz(.)40 b(They)28 b(refer)f(to)i(the)f(case)118 1071 y(without)38 b(re\015ections)g(and)f(since)h(the)g(magnetic)h(\014eld)e Fu(h)h Fz(is)f(equal)h(to)h Fu(h)p Fz(\()p Fu(\014)5 b Fz(\))38 b(they)g(are)g(part)f(of)h(the)118 1179 y(classical)31 b(Pirogo)m(v-Sinai)g(Theory)-8 b(.)40 b(F)-8 b(or)30 b(con)m(v)m(enience,)h(w)m(e)e(giv)m(e)i(an)e(explicit)h(pro)s(of,)f (consequence)g(of)118 1287 y(Theorem)h(4.2.)118 1459 y FE(Lemma)35 b(4.2.)42 b Fj(F)-7 b(or)34 b Fu(\014)j Fj(lar)-5 b(ge)34 b(enough,)1611 1623 y Fr(P)1674 1637 y Fw(+)1758 1623 y Fz(=)25 b Fr(P)1917 1637 y Fn(\000)2002 1623 y Fz(=)g Fr(P)1268 b Fz(\(4.25\))118 1787 y Fj(wher)-5 b(e)36 b Fr(P)42 b Fj(is)34 b(the)h(thermo)-5 b(dynamic)38 b(pr)-5 b(essur)g(e)36 b(at)f(inverse)f(temp)-5 b(er)g(atur)g(e)38 b Fu(\014)h Fj(and)d(magnetic)f(\014eld)g Fu(h)29 b Fz(=)118 1895 y Fu(h)p Fz(\()p Fu(\014)5 b Fz(\))p Fj(.)43 b(Mor)-5 b(e)g(over,)836 2116 y Fz(log)18 b Fu(Z)1038 2078 y Fn(\006)1031 2145 y Fs(\014)s(;)p Fw(\003)1230 2116 y Fz(=)82 b Fr(\006)p Fu(\014)5 b(h)p Fr(j)p Fz(\003)p Fr(j)22 b Fz(+)1840 2030 y Fo(X)1838 2226 y Fs(x)p Fn(2)p Fw(\003)2099 2030 y Fo(X)1989 2231 y Fs(x)p Fn(2)p Fs(X)5 b Fw(\()p Fs(\031)r Fw(\))p Fn(\032)p Fw(\003)2423 2054 y Fu(!)2483 2021 y Fn(\006)2542 2054 y Fz(\()p Fu(\031)s Fz(\))p 2416 2095 V 2416 2178 a Fr(j)p Fu(X)i Fz(\()p Fu(\031)s Fz(\))p Fr(j)3429 2116 y Fz(\(4.26\))1230 2422 y(=)82 b Fu(\014)5 b Fr(j)p Fz(\003)p Fr(jP)29 b(\000)1910 2335 y Fo(X)1786 2536 y Fs(X)5 b Fw(\()p Fs(\031)r Fw(\))p Fn(\\)p Fw(\003)2042 2518 y Fq(c)2075 2536 y Fn(6)p Fw(=)p Fn(;)2240 2360 y Fr(j)p Fu(X)i Fz(\()p Fu(\031)s Fz(\))22 b Fr(\\)e Fz(\003)p Fr(j)p 2240 2401 423 4 v 2322 2484 a(j)p Fu(X)7 b Fz(\()p Fu(\031)s Fz(\))p Fr(j)2699 2422 y Fu(!)2759 2384 y Fn(\006)2817 2422 y Fz(\()p Fu(\031)s Fz(\))487 b(\(4.27\))218 2821 y FE(Pro)s(of.)69 b Fz(\(4.26\))41 b(is)f(just)f(a)h(rewriting)f(of)47 b(\(4.20\))s(;)d(\(4.25\))e(follo)m (ws)e(from)f(\(4.26\))j(b)m(y)e(taking)g(the)118 2929 y(thermo)s(dynamic)h(limit)h(and)f(using)g(Lemma)g(4.1.)75 b(\(4.27\))43 b(is)f(also)g(a)g(rewriting)f(of)49 b(\(4.26\))r(.)74 b(The)118 3037 y(lemma)31 b(is)f(pro)m(v)m(ed.)2799 b Ft(\003)118 3381 y FE(Lemma)40 b(4.3.)45 b Fj(F)-7 b(or)39 b Fu(\014)j Fj(lar)-5 b(ge)38 b(enough,)g(and)h(c)-5 b(al)5 b(ling)37 b Fu(\016)s Fz(\003)h Fj(the)g(union)f(of)g(al)5 b(l)38 b(sites)f(in)g Fz(\003)3207 3348 y Fs(c)3279 3381 y Fj(which)h(ar)-5 b(e)118 3489 y Fr(\003)p Fj(-c)g(onne)g(cte)g(d)34 b(to)f Fz(\003)p Fj(,)1114 3533 y Fo(\014)1114 3587 y(\014)1114 3642 y(\014)1160 3637 y Fz(log)17 b Fu(Z)1361 3599 y Fn(\006)1354 3667 y Fs(\014)s(;)p Fw(\003)1490 3637 y Fr(\000)j Fu(\014)5 b Fr(j)p Fz(\003)p Fr(jP)1820 3533 y Fo(\014)1820 3587 y(\014)1820 3642 y(\014)1877 3637 y Ft(6)25 b Fu(e)2015 3600 y Fn(\000)p Fs(\014)s(=)p Fw(2+2)2273 3576 y Fq(d)p Fp(+1)2387 3600 y Fs(\013)2437 3637 y Fr(j)p Fu(\016)s Fz(\(\003\))p Fr(j)766 b Fz(\(4.28\))218 3955 y FE(Pro)s(of.)41 b Fz(By)32 b(\(4.27\))1164 4042 y Fo(\014)1164 4096 y(\014)1164 4151 y(\014)1210 4146 y Fz(log)17 b Fu(Z)1411 4108 y Fn(\006)1404 4176 y Fs(\014)s(;)p Fw(\003)1540 4146 y Fr(\000)j Fu(\014)5 b Fr(j)p Fz(\003)p Fr(jP)1870 4042 y Fo(\014)1870 4096 y(\014)1870 4151 y(\014)1927 4146 y Ft(6)2058 4060 y Fo(X)2039 4258 y Fs(x)p Fn(2)p Fs(\016)r Fw(\003)2281 4060 y Fo(X)2223 4261 y Fs(X)g Fw(\()p Fs(\031)r Fw(\))p Fn(3)p Fs(x)2486 4146 y Fr(j)p Fu(!)2571 4109 y Fn(\006)2630 4146 y Fz(\()p Fu(\031)s Fz(\))p Fr(j)118 4419 y Fz(whic)m(h,)30 b(b)m(y)i(\(4.22\))r (,)f(yields)f(\(4.28\))s(.)40 b(The)30 b(lemma)h(is)g(pro)m(v)m(ed.) 1357 b Ft(\003)218 4763 y Fz(The)30 b(\014nal)g(lemma)h(pro)m(v)m(es)g (that)g(the)f(b)s(ound)e(\(4.10\))33 b(w)m(as)e(to)s(o)g(conserv)-5 b(ativ)m(e.)118 4989 y FE(Lemma)35 b(4.4.)42 b Fj(Ther)-5 b(e)33 b(is)g(a)g(c)-5 b(onstant)34 b Fu(c)f Fj(so)g(that,)h(for)f Fu(\014)38 b Fj(lar)-5 b(ge)34 b(enough,)1057 5071 y Fo(\014)1057 5126 y(\014)1057 5180 y(\014)1102 5176 y Fz(log)18 b Fu(Z)1304 5137 y Fw(+)1297 5205 y Fs(\014)s(;)p Fw(\003)1433 5176 y Fr(\000)i Fz(log)d Fu(Z)1725 5137 y Fn(\000)1718 5205 y Fs(\014)s(;)p Fw(\003)1834 5071 y Fo(\014)1834 5126 y(\014)1834 5180 y(\014)1890 5176 y Ft(6)25 b Fz(2)p Fu(e)2073 5138 y Fn(\000)p Fs(\014)s(=)p Fw(2+2)2331 5115 y Fq(d)p Fp(+1)2445 5138 y Fs(\013)2495 5176 y Fr(j)p Fu(\016)s Fz(\(\003\))p Fr(j)708 b Fz(\(4.29\))1160 5421 y Fu(w)1227 5384 y Fn(\006)1287 5421 y Fz(\(\000\))26 b Fr(\024)f Fz(exp)1690 5320 y Fo(n)1771 5421 y Fr(\000)20 b Fu(\014)5 b(N)1991 5435 y Fw(\000)2039 5320 y Fo(\020)2093 5421 y Fz(1)21 b Fr(\000)f Fu(ce)2331 5384 y Fn(\000)p Fs(\014)s(=)p Fw(2)2504 5320 y Fo(\021o)3429 5421 y Fz(\(4.30\))p eop end %%Page: 18 18 TeXDict begin 18 17 bop 118 45 a Fw(18)1128 b(T.)23 b(BODINEA)n(U)g (AND)g(E.)g(PRESUTTI)218 366 y FE(Pro)s(of.)39 b Fz(\(4.29\))28 b(follo)m(ws)e(directly)g(from)f(\(4.28\))s(.)39 b(By)25 b(the)h(analogue)g(of)33 b(\(4.8\))27 b(without)e(re\015ections,)588 558 y Fu(w)655 521 y Fn(\006)714 558 y Fz(\(\000\))h Fr(\024)f Fz(exp)1117 457 y Fo(n)1198 558 y Fr(\000)20 b Fu(\014)5 b(N)1418 572 y Fw(\000)1486 558 y Fz(+)20 b(2)p Fu(\014)5 b Fr(j)p Fu(h)p Fr(jj)1811 535 y Fz(\026)1805 558 y(\000)r Fr(j)20 b Fz(+)g(2)p Fu(e)2087 521 y Fn(\000)p Fs(\014)s(=)p Fw(2+2)2345 497 y Fq(d)p Fp(+1)2460 521 y Fs(\013)2525 472 y Fo(X)2578 667 y Fs(i)2671 558 y Fr(j)p Fu(\016)s Fz(\(in)m(t)2884 520 y Fn(\000)2884 585 y Fs(i)2943 558 y Fz(\(\000\)\))p Fr(j)3130 457 y Fo(o)218 809 y Fz(Notice)36 b(also)g(that)f(the)g(constrain)m(t)h(on)f Fu(h)p Fz(\()p Fu(\014)5 b Fz(\))35 b(can)g(b)s(e)g(easily)g(reco)m(v)m (ered.)56 b(By)35 b(equating)g(the)g(t)m(w)m(o)118 917 y(r.h.s.)30 b(of)38 b(\(4.24\))32 b(and)e(then)g(using)h(\(4.22\))i(in) d(the)h(v)m(ersion)f(without)h(re\015ections,)g(w)m(e)g(get)1518 1099 y Fu(\014)5 b Fr(j)p Fu(h)p Fr(j)26 b Ft(6)f Fu(e)1840 1062 y Fn(\000)p Fs(\014)s(=)p Fw(2+2)2098 1038 y Fq(d)p Fp(+1)2212 1062 y Fs(\013)3429 1099 y Fz(\(4.31\))218 1265 y(Moreo)m(v)m(er,)h(if)21 b Fu(x)k Fr(2)g Fu(\016)s Fz(\(in)m(t)1061 1227 y Fn(\000)1061 1292 y Fs(i)1120 1265 y Fz(\(\000\)\))e(there)f(is)f(a)i(plaquette)f Fu(p)g Fz(suc)m(h)f(that)i Fu(p)s Fr(\\)s Fu(K)7 b Fz(\()p Fu(x;)15 b Fz(2\))25 b Fr(6)p Fz(=)g Fr(;)p Fz(,)f Fu(p)s Fr(\\)s Fz(sp)o(\(\000\))i Fr(6)p Fz(=)f Fr(;)p Fz(,)118 1379 y(so)k(that)h(to)g(eac)m(h)g Fu(x)25 b Fr(2)g([)961 1393 y Fs(i)989 1379 y Fu(\016)s Fz(\(in)m(t)1176 1340 y Fn(\000)1176 1406 y Fs(i)1235 1379 y Fz(\(\000\)\))30 b(w)m(e)g(can)f(asso)s(ciate)i (a)e(plaquette)h(con)m(tributing)g(to)f Fu(N)3279 1393 y Fw(\000)3328 1379 y Fz(,)g(in)g(suc)m(h)118 1487 y(a)i(w)m(a)m(y)g (that)g(the)g(same)g(plaquette)g(is)f(coun)m(ted)h(at)g(most)g Fr(j)p Fu(K)7 b Fz(\(0)p Fu(;)15 b Fz(3\))p Fr(j)33 b Fz(times.)41 b(Th)m(us)401 1678 y Fu(w)468 1641 y Fn(\006)528 1678 y Fz(\(\000\))26 b Fr(\024)f Fz(exp)931 1577 y Fo(n)1012 1678 y Fr(\000)19 b Fu(\014)5 b(N)1231 1692 y Fw(\000)1300 1678 y Fz(+)20 b(2)p Fu(e)1478 1641 y Fn(\000)p Fs(\014)s(=)p Fw(2+2)1736 1617 y Fq(d)p Fp(+1)1851 1641 y Fs(\013)1900 1678 y Fr(j)1931 1655 y Fz(\026)1925 1678 y(\000)p Fr(j)h Fz(+)f(2)p Fu(e)2206 1641 y Fn(\000)p Fs(\014)s(=)p Fw(2+2)2464 1617 y Fq(d)p Fp(+1)2578 1641 y Fs(\013)2628 1678 y Fr(j)p Fu(K)7 b Fz(\(0)p Fu(;)15 b Fz(3\))p Fr(j)p Fu(N)3035 1692 y Fw(\000)3085 1577 y Fo(o)3429 1678 y Fz(\(4.32\))118 1865 y(The)30 b(inequalit)m(y)1143 1993 y Fr(j)1174 1970 y Fz(\026)1168 1993 y(\000)p Fr(j)25 b(\024)g(j)p Fz(sp\(\000\))p Fr(jj)p Fu(K)7 b Fz(\(2)p Fu(;)15 b Fz(0\))p Fr(j)28 b(\024)d Fu(N)2166 2007 y Fw(\000)2215 1993 y Fz(2)2260 1955 y Fs(d)2300 1993 y Fr(j)p Fu(K)7 b Fz(\(2)p Fu(;)15 b Fz(0\))p Fr(j)118 2138 y Fz(concludes)31 b(the)f(pro)s(of)g(of)g(the) h(lemma.)2122 b Ft(\003)218 2482 y Fz(W)-8 b(e)39 b(turn)f(no)m(w)g (bac)m(k)i(to)f(the)g(main)f(goal)i(of)f(the)f(section,)k(namely)d(to)g (pro)m(v)m(e)h(that)f(the)f(b)s(ound)118 2590 y(\(4.10\))33 b(holds)d(also)h(for)f(the)h(w)m(eigh)m(ts)g(with)f(re\015ections.)42 b(The)30 b(pro)s(of)f(is)i(obtained)g(in)f(t)m(w)m(o)h(steps.)118 2818 y FE(Theorem)38 b(4.3.)43 b Fj(F)-7 b(or)36 b(any)f Fu(\014)k Fj(lar)-5 b(ge)36 b(enough)f(the)g(fol)5 b(lowing)36 b(holds.)49 b(L)-5 b(et)35 b Fu(x)29 b Fr(2)f Fx(Z)2947 2785 y Fs(d;R)3095 2818 y Fj(and)35 b Fu(n)29 b Ft(6)f Fu(n)p Fz(\()p Fu(x)p Fz(\))p Fj(;)118 2939 y(then)f(if)g Fz(\003)7 b Fr([)g Fu(@)588 2906 y Fs(R)646 2939 y Fz(\(\003\))26 b Fr(\032)f Fu(K)985 2906 y Fs(R)1042 2939 y Fz(\()p Fu(x;)15 b(n)p Fz(\))p Fj(,)28 b Fz(\004)1376 2895 y Fs(R;)p Fn(\006)1376 2968 y Fs(\014)s(;)p Fw(\003)1533 2939 y Fz(=)d Fu(Z)1698 2895 y Fs(R;)p Fn(\006)1691 2968 y Fs(\014)s(;)p Fw(\003)1857 2939 y Fj(and)i(if)f Fz(\000)h Fj(is)f(a)h Fr(\006)p Fj(,)h Fu(R)f Fj(c)-5 b(ontour)28 b(with)3102 2916 y Fz(\026)3096 2939 y(\000)e Fr(\032)e Fu(K)3358 2906 y Fs(R)3416 2939 y Fz(\()p Fu(x;)15 b(n)p Fz(\))p Fj(,)118 3072 y(then)52 b Fz(^)-64 b Fu(w)387 3039 y Fs(R;)p Fn(\006)520 3072 y Fz(\(\000\))26 b(=)f Fu(w)836 3039 y Fs(R;)p Fn(\006)968 3072 y Fz(\(\000\))h Fu(<)f(e)1259 3039 y Fn(\000)p Fs(\014)s(N)1413 3050 y Fp(\000)1455 3039 y Fs(=)p Fw(2)1530 3072 y Fj(.)218 3363 y FE(Pro)s(of.)56 b Fz(Under)35 b(the)g(assumption)g(that)h Fu(n)d Ft(6)g Fu(n)p Fz(\()p Fu(x)p Fz(\),)k(Theorem)f(4.1)g(applies)g (and)e(therefore)i(the)118 3471 y(pro)s(of)29 b(will)h(follo)m(w)h (from)f(the)g(previous)f(results)g(on)h(the)g(w)m(eigh)m(ts)h(without)f (re\015ection)h(and)e(from)g(the)118 3579 y(one)e(to)g(one)f(corresp)s (ondence)g(b)s(et)m(w)m(een)h Fu(K)7 b Fz(\()p Fu(x;)15 b(n)p Fz(\))27 b(and)e Fu(K)2112 3546 y Fs(R)2170 3579 y Fz(\()p Fu(x;)15 b(n)p Fz(\).)40 b(In)25 b(particular)i(\(4.4\))h (implies)f(that)118 3688 y(for)j(domains)g(strictly)i(con)m(tained)f (in)f Fu(K)1529 3655 y Fs(R)1587 3688 y Fz(\()p Fu(x;)15 b(n)p Fz(\))1543 3865 y Fu(Z)1612 3826 y Fn(\006)1605 3899 y Fs(\014)s(;)p Fn(T)1724 3880 y Fl(\000)p Fp(1)1806 3899 y Fw(\(\003\))1940 3865 y Fz(=)25 b Fu(Z)2105 3820 y Fs(R;)p Fn(\006)2098 3894 y Fs(\014)s(;)p Fw(\003)3429 3865 y Fz(\(4.33\))218 4048 y(In)h(the)h(case)h Fr(j)p Fz(\003)p Fr(j)e Fz(=)e(1,)29 b(an)m(y)e(con)m(tour)g(in)g(\003)g(has)f (no)h(in)m(terior)g(and)g(\(4.12\))r(-\(4.13\))i(allo)m(w)g(to)e (compute)137 4157 y(^)-64 b Fu(w)185 4124 y Fs(R;)p Fn(\006)318 4157 y Fz(\(\000\),)31 b(getting,)h(as)f(in)f(the)h(pro)s(of)e(of)i (Lemma)f(4.4,)1183 4349 y(^)-63 b Fu(w)1232 4312 y Fs(R;)p Fn(\006)1364 4349 y Fz(\(\000\))26 b Ft(6)40 b Fz(exp)1782 4248 y Fo(n)1863 4349 y Fr(\000)20 b Fu(\014)5 b(N)2083 4363 y Fw(\000)2152 4349 y Fz(+)20 b(2)p Fu(\014)5 b Fr(j)p Fu(h)p Fr(jj)2477 4326 y Fz(\026)2471 4349 y(\000)q Fr(j)2554 4248 y Fo(o)118 4555 y Fz(hence,)46 b(for)d Fu(\014)48 b Fz(large)43 b(enough,)65 b(^)-64 b Fu(w)1317 4522 y Fs(R;)p Fn(\006)1450 4555 y Fz(\(\000\))46 b(=)f Fu(w)1806 4522 y Fn(\006)1866 4555 y Fz(\(\000\))h Fu(<)g(e)2198 4522 y Fn(\000)p Fs(\014)s(N)2352 4533 y Fp(\000)2394 4522 y Fs(=)p Fw(2)2468 4555 y Fz(.)78 b(Supp)s(ose)41 b(b)m(y)h(induction)h(that)118 4676 y(for)e(an)m(y)h Fr(j)p Fz(\003)p Fr(j)j Ft(6)e Fu(k)i Fz(\(\003)d(as)g(in)f(the)g(text) i(of)f(the)f(theorem\),)46 b(\004)2335 4632 y Fs(R;)p Fn(\006)2335 4705 y Fs(\014)s(;)p Fw(\003)2510 4676 y Fz(=)e Fu(Z)2694 4638 y Fn(\006)2687 4710 y Fs(\014)s(;)p Fn(T)2806 4691 y Fl(\000)p Fp(1)2888 4710 y Fw(\(\003\))3040 4676 y Fz(=)g Fu(Z)3224 4632 y Fs(R;)p Fn(\006)3217 4705 y Fs(\014)s(;)p Fw(\003)3355 4676 y Fz(.)74 b(Then)118 4798 y(if)44 b(\000)g(is)h(as)f(in)g(the)g(text)i(of)e(the)g(theorem)h (and)f(moreo)m(v)m(er)h(all)h(its)e(in)m(terior)h(parts)f(ha)m(v)m(e)i (v)m(olume)118 4906 y Ft(6)32 b Fu(k)s Fz(,)j(then)f(the)h(second)f (term)h(on)f(the)g(r.h.s.)g(of)42 b(\(4.12\))r(-\(4.13\))37 b(is)e(equal)f(to)i Fu(w)2910 4873 y Fn(\006)2969 4906 y Fz(\()p Fr(T)3077 4873 y Fn(\000)p Fw(1)3171 4906 y Fz(\000\),)g(with)e(the)118 5020 y(ob)m(vious)d(meaning)g(of)g(the)g (notation,)i(whic)m(h)d(b)m(y)h(Lemma)g(4.4)h(is,)f(for)g Fu(\014)36 b Fz(large)c(enough,)f Fu(<)26 b(e)3366 4987 y Fn(\000)p Fs(\014)s(N)3520 4998 y Fp(\000)3561 4987 y Fs(=)p Fw(2)3636 5020 y Fz(.)118 5128 y(Then)37 b(the)h(second)g (term)g(on)g(the)g(r.h.s.)g(of)45 b(\(4.12\))s(-\(4.13\))40 b(is)e(smaller)h(than)e(the)i(\014rst)e(one,)j(hence)137 5237 y(^)-64 b Fu(w)185 5204 y Fs(R;)p Fn(\006)318 5237 y Fz(\(\000\))37 b(=)g Fu(w)657 5204 y Fs(R;)p Fn(\006)790 5237 y Fz(\(\000\).)62 b(Since)38 b(all)g(con)m(tours)g(inside)f(\003)h (ha)m(v)m(e)g(in)m(terior)h(parts)e(with)g(v)m(olume)75 b Ft(6)37 b Fu(k)s Fz(,)118 5358 y(\(4.10\))25 b(sho)m(ws)e(that)h (\004)878 5314 y Fs(R;)p Fn(\006)878 5387 y Fs(\014)s(;)p Fw(\003)1035 5358 y Fz(=)h Fu(Z)1200 5314 y Fn(\006)p Fs(;R)1193 5387 y(\014)s(;)p Fw(\003)1357 5358 y Fz(=)g Fu(Z)1522 5320 y Fn(\006)1515 5392 y Fs(\014)s(;)p Fn(T)1633 5373 y Fl(\000)p Fp(1)1716 5392 y Fw(\(\003\))1824 5358 y Fz(,)g(th)m(us)d(pro)m(ving)h(the)h(induction)e(step.)38 b(The)23 b(theorem)118 5480 y(is)31 b(pro)m(v)m(ed.)3091 b Ft(\003)p eop end %%Page: 19 19 TeXDict begin 19 18 bop 1137 45 a Fw(SURF)-8 b(A)n(CE)31 b(TENSION)g(AND)f(WULFF)h(THEOR)-6 b(Y)948 b(19)218 366 y Fz(Before)31 b(extending)g(the)f(result)h(to)g(general)g(\003,)g(w)m (e)g(state)g(and)f(pro)m(v)m(e)h(the)g(follo)m(wing)h(lemma.)118 593 y FE(Lemma)j(4.5.)42 b Fj(F)-7 b(or)34 b Fu(\014)j Fj(lar)-5 b(ge)34 b(enough,)700 809 y Fz(log)17 b(\004)893 765 y Fs(R;)p Fn(\006)893 838 y Fs(\014)s(;)p Fw(\003)1050 809 y Fz(=)25 b Fu(\014)5 b Fr(j)p Fz(\003)p Fr(jP)29 b Fz(+)1500 723 y Fo(X)1498 919 y Fs(x)p Fn(2)p Fw(\003)1699 708 y Fo(n)2243 723 y(X)1775 927 y Fs(X)5 b Fw(\()p Fs(\031)r Fw(\))p Fn(\\)p Fs(K)2047 908 y Fq(R)2098 927 y Fw(\()p Fs(x)p Fw(;)p Fs(n)p Fw(\()p Fs(x)p Fw(\)\))2349 908 y Fq(c)2381 927 y Fn(6)p Fw(=)p Fn(;)p Fw(;)p Fs(x)p Fn(2)p Fs(X)g Fw(\()p Fs(\031)r Fw(\))p Fn(\032)p Fw(\003)2925 747 y Fz(^)-52 b Fu(!)2978 714 y Fs(R;)p Fn(\006)3110 747 y Fz(\()p Fu(\031)s Fz(\))p 2918 788 318 4 v 2948 871 a Fr(j)p Fu(X)7 b Fz(\()p Fu(\031)s Fz(\))p Fr(j)1644 1123 y(\000)2117 1036 y Fo(X)1730 1238 y Fs(X)e Fw(\()p Fs(\031)r Fw(\))p Fn(\\)p Fs(K)g Fw(\(0;)p Fs(n)p Fw(\()p Fs(x)p Fw(\)\))2248 1219 y Fq(c)2282 1238 y Fn(6)p Fw(=)p Fn(;)p Fw(;0)p Fn(2)p Fs(X)g Fw(\()p Fs(\031)r Fw(\))2718 1061 y Fu(!)2778 1028 y Fn(\006)2836 1061 y Fz(\()p Fu(\031)s Fz(\))p 2711 1102 259 4 v 2711 1185 a Fr(j)p Fu(X)i Fz(\()p Fu(\031)s Fz(\))p Fr(j)2979 1022 y Fo(o)3429 1123 y Fz(\(4.34\))218 1514 y FE(Pro)s(of.)41 b Fz(W)-8 b(e)32 b(write)455 1713 y(log)18 b(\004)649 1669 y Fs(R;)p Fn(\006)649 1742 y Fs(\014)s(;)p Fw(\003)806 1713 y Fz(=)25 b Fr(\006)p Fu(\014)5 b(h)p Fr(j)p Fz(\003)p Fr(j)21 b Fz(+)1308 1627 y Fo(X)1306 1823 y Fs(x)p Fn(2)p Fw(\003)1507 1612 y Fo(n)1852 1627 y(X)1583 1831 y Fs(x)p Fn(2)p Fs(X)5 b Fw(\()p Fs(\031)r Fw(\))p Fn(\032)p Fs(K)1950 1812 y Fq(R)2000 1831 y Fw(\()p Fs(x)p Fw(;)p Fs(n)p Fw(\()p Fs(x)p Fw(\)\))2328 1651 y Fu(!)2388 1618 y Fs(R;)p Fn(\006)2520 1651 y Fz(\()p Fu(\031)s Fz(\))p 2328 1692 318 4 v 2357 1775 a Fr(j)p Fu(X)i Fz(\()p Fu(\031)s Fz(\))p Fr(j)1873 2032 y Fz(+)2427 1945 y Fo(X)1959 2150 y Fs(X)e Fw(\()p Fs(\031)r Fw(\))p Fn(\\)p Fs(K)2231 2131 y Fq(R)2281 2150 y Fw(\()p Fs(x)p Fw(;)p Fs(n)p Fw(\()p Fs(x)p Fw(\)\))2532 2131 y Fq(c)2565 2150 y Fn(6)p Fw(=)p Fn(;)p Fw(;)p Fs(x)p Fn(2)p Fs(X)g Fw(\()p Fs(\031)r Fw(\))p Fn(\032)p Fw(\003)3102 1970 y Fu(!)3162 1937 y Fs(R;)p Fn(\006)3294 1970 y Fz(\()p Fu(\031)s Fz(\))p 3102 2011 V 3131 2094 a Fr(j)p Fu(X)i Fz(\()p Fu(\031)s Fz(\))p Fr(j)3430 1931 y Fo(o)562 2420 y Fu(\014)e Fr(j)p Fz(\003)p Fr(jP)34 b Fz(=)25 b Fr(\006)p Fu(\014)5 b(h)p Fr(j)p Fz(\003)p Fr(j)21 b Fz(+)1328 2333 y Fo(X)1333 2530 y Fs(i)p Fn(2)p Fw(\003)1525 2319 y Fo(n)1934 2333 y(X)1601 2535 y Fs(X)5 b Fw(\()p Fs(\031)r Fw(\))p Fn(\032)p Fs(K)g Fw(\()p Fs(x)p Fw(;)p Fs(n)p Fw(\()p Fs(x)p Fw(\)\);)p Fs(x)p Fn(2)p Fs(X)g Fw(\()p Fs(\031)r Fw(\))2482 2358 y Fu(!)2542 2325 y Fn(\006)2601 2358 y Fz(\()p Fu(\031)s Fz(\))p 2475 2399 259 4 v 2475 2482 a Fr(j)p Fu(X)i Fz(\()p Fu(\031)s Fz(\))p Fr(j)1979 2730 y Fz(+)2456 2644 y Fo(X)2065 2845 y Fs(X)e Fw(\()p Fs(\031)r Fw(\))p Fn(\\)p Fs(K)g Fw(\()p Fs(x)p Fw(;)p Fs(n)p Fw(\()p Fs(x)p Fw(\)\))2588 2826 y Fq(c)2621 2845 y Fn(6)p Fw(=)p Fn(;)p Fw(;)p Fs(x)p Fn(2)p Fs(X)g Fw(\()p Fs(\031)r Fw(\))3061 2669 y Fu(!)3121 2636 y Fn(\006)3180 2669 y Fz(\()p Fu(\031)s Fz(\))p 3054 2709 V 3054 2793 a Fr(j)p Fu(X)i Fz(\()p Fu(\031)s Fz(\))p Fr(j)3323 2629 y Fo(o)118 3019 y Fz(Then)36 b(log)17 b(\004)555 2975 y Fs(R;)p Fn(\006)555 3049 y Fs(\014)s(;)p Fw(\003)712 3019 y Fr(\000)24 b Fu(\014)5 b Fr(j)p Fz(\003)p Fr(jP)45 b Fz(is)37 b(equal)g(to)h(the)f(di\013erence)g(of)g(the)g(r.h.s.)g(of)g (the)g(last)g(t)m(w)m(o)h(equations.)118 3133 y(The)c(\014rst)f(terms)h (in)g(the)h(sum)e(o)m(v)m(er)i Fu(x)g Fz(cancel)g(with)f(eac)m(h)h (other,)h(see)f(the)f(pro)s(of)g(of)g(Theorem)g(4.1,)118 3241 y(and)42 b(\(4.34\))i(follo)m(ws)f(after)g(recalling)g(that)g(the) f(w)m(eigh)m(ts)h(without)f(re\015ections)h(are)g(translational)118 3349 y(in)m(v)-5 b(arian)m(t.)42 b(The)30 b(theorem)h(is)f(pro)m(v)m (ed.)2134 b Ft(\003)118 3701 y FE(Theorem)34 b(4.4.)41 b Fj(F)-7 b(or)32 b(any)g Fu(\014)37 b Fj(lar)-5 b(ge)32 b(enough,)g(for)g(any)g(b)-5 b(ounde)g(d)33 b Fz(\003)26 b Fr(\032)f Fx(Z)2655 3668 y Fs(d;R)2768 3701 y Fj(,)31 b Fz(\004)2888 3657 y Fs(R;)p Fn(\006)2888 3730 y Fs(\014)s(;)p Fw(\003)3045 3701 y Fz(=)25 b Fu(Z)3210 3657 y Fs(R;)p Fn(\006)3203 3730 y Fs(\014)s(;)p Fw(\003)3373 3701 y Fj(and)33 b(for)118 3834 y(any)g(b)-5 b(ounde)g(d,)34 b Fr(\006)p Fj(,)e Fu(R)i Fj(c)-5 b(ontour)34 b Fz(\000)p Fj(,)51 b Fz(^)-64 b Fu(w)1415 3801 y Fs(R;)p Fn(\006)1548 3834 y Fz(\(\000\))26 b(=)f Fu(w)1864 3801 y Fs(R;)p Fn(\006)1996 3834 y Fz(\(\000\))h Fu(<)f(e)2287 3801 y Fn(\000)p Fs(\014)s(N)2441 3812 y Fp(\000)2483 3801 y Fs(=)p Fw(2)2558 3834 y Fj(.)218 4121 y FE(Pro)s(of.)57 b Fz(By)37 b(\(4.34\))h(and)d(\(4.22\))r(,)i(denoting)f(b)m(y)g Fu(n)p Fz(\()p Fu(x;)15 b(y)s Fz(\))36 b(the)g(maximal)g(in)m(teger)h (suc)m(h)f(that)g Fu(y)47 b(=)-55 b Fr(2)118 4231 y Fu(K)202 4198 y Fs(R)260 4231 y Fz(\()p Fu(x)p Fz(;)15 b Fu(n)p Fz(\()p Fu(x;)g(y)s Fz(\)\),)198 4312 y Fo(\014)198 4367 y(\014)198 4422 y(\014)243 4417 y Fz(log)j(\004)437 4373 y Fs(R;)p Fn(\006)437 4446 y Fs(\014)s(;)p Fw(\003)589 4417 y Fr(\000)i Fu(\014)5 b Fr(j)p Fz(\003)p Fr(jP)919 4312 y Fo(\014)919 4367 y(\014)919 4422 y(\014)1059 4417 y Ft(6)108 b Fz(2)1301 4331 y Fo(X)1298 4528 y Fs(x)p Fn(2)p Fw(\003)1500 4417 y Fu(e)1542 4379 y Fn(\000)p Fw([)p Fs(\014)s(=)p Fw(2)p Fn(\000)p Fw(2)1820 4356 y Fq(d)p Fp(+1)1934 4379 y Fs(\013)p Fw(])11 b Fs(n)p Fw(\()p Fs(x)p Fw(\))2177 4417 y Ft(6)25 b Fz(2)2336 4331 y Fo(X)2333 4528 y Fs(x)p Fn(2)p Fw(\003)2605 4331 y Fo(X)2535 4535 y Fs(y)r Fn(2)p Fs(\016)2652 4516 y Fq(R)2703 4535 y Fw(\(\003\))2873 4417 y Fu(e)2915 4379 y Fn(\000)p Fw([)p Fs(\014)s(=)p Fw(2)p Fn(\000)p Fw(2)3193 4356 y Fq(d)p Fp(+1)3307 4379 y Fs(\013)p Fw(])11 b Fs(n)p Fw(\()p Fs(x;y)r Fw(\))1736 4692 y Ft(6)25 b Fz(2)1963 4605 y Fo(X)1892 4810 y Fs(y)r Fn(2)p Fs(\016)2009 4791 y Fq(R)2060 4810 y Fw(\(\003\))2231 4605 y Fo(X)2230 4801 y Fs(n)p Fn(\025)p Fw(1)2429 4692 y Fu(e)2471 4654 y Fn(\000)p Fw(\()p Fs(\014)s(=)p Fw(2)p Fn(\000)p Fw(2)2756 4631 y Fq(d)p Fp(+1)2870 4654 y Fs(\013)p Fw(\))p Fs(n)2990 4692 y Fz(\(2)p Fu(n)20 b Fz(+)g(1\))3316 4654 y Fs(d)118 4957 y Fz(whic)m(h)30 b(yields)1029 4996 y Fo(\014)1029 5051 y(\014)1029 5105 y(\014)1074 5101 y Fz(log)18 b(\004)1268 5057 y Fs(R;)p Fn(\006)1268 5130 y Fs(\014)s(;)p Fw(\003)1420 5101 y Fr(\000)h Fu(\014)5 b Fr(j)p Fz(\003)p Fr(jP)1749 4996 y Fo(\014)1749 5051 y(\014)1749 5105 y(\014)1807 5101 y Ft(6)25 b Fu(c)1942 5115 y Fs(\013)1991 5101 y Fu(e)2033 5063 y Fn(\000)p Fs(\014)s(=)p Fw(2+2)2291 5040 y Fq(d)p Fp(+1)2406 5063 y Fs(\013)2455 5023 y Fo(\014)2455 5078 y(\014)2485 5101 y Fu(\016)2528 5063 y Fs(R)2587 5101 y Fz(\(\003\))2720 5023 y Fo(\014)2720 5078 y(\014)3429 5101 y Fz(\(4.35\))118 5263 y(with)1331 5384 y Fu(c)1370 5398 y Fs(\013)1445 5384 y Fz(=)g(2)p Fu(e)1628 5346 y Fs(\013)1694 5297 y Fo(X)1693 5493 y Fs(n)p Fn(\025)p Fw(1)1892 5384 y Fu(e)1934 5346 y Fn(\000)p Fs(\013n)2081 5384 y Fz(\(2)p Fu(n)c Fz(+)f(1\))2408 5346 y Fs(d)3429 5384 y Fz(\(4.36\))p eop end %%Page: 20 20 TeXDict begin 20 19 bop 118 45 a Fw(20)1128 b(T.)23 b(BODINEA)n(U)g (AND)g(E.)g(PRESUTTI)118 366 y Fz(An)37 b(inductiv)m(e)g(argumen)m(t)g (as)g(in)g(the)g(pro)s(of)f(of)h(Theorem)f(4.3)i(completes)h(then)d (the)h(pro)s(of)f(of)h(the)118 474 y(theorem.)3125 b Ft(\003)218 700 y Fz(Before)31 b(ending)f(this)g(Section,)i(w)m(e)f (collect)h(some)f(estimates)h(used)e(in)g(the)g(next)h(sections.)118 926 y FE(Lemma)k(4.6.)42 b Fj(Given)32 b(any)i(p)-5 b(ositive)33 b(inte)-5 b(ger)33 b Fu(n)p Fj(,)360 1045 y Fo(\014)360 1099 y(\014)360 1154 y(\014)406 1149 y Fz(log)17 b Fu(Z)607 1111 y Fn(\006)600 1179 y Fs(\014)s(;)p Fw(\003)736 1149 y Fr(\000)j(f\006)p Fu(\014)5 b(h)p Fr(j)p Fz(\003)p Fr(j)22 b Fz(+)1279 1063 y Fo(X)1277 1260 y Fs(x)p Fn(2)p Fw(\003)1722 1063 y Fo(X)1478 1264 y Fs(x)p Fn(2)p Fs(X)5 b Fw(\()p Fs(\031)r Fw(\))p Fn(\032)p Fs(K)g Fw(\()p Fs(x)p Fw(;)p Fs(n)p Fw(\))p Fn(\\)p Fw(\003)2130 1088 y Fu(!)2190 1055 y Fn(\006)2249 1088 y Fz(\()p Fu(\031)s Fz(\))p 2123 1128 259 4 v 2123 1212 a Fr(j)p Fu(X)i Fz(\()p Fu(\031)s Fz(\))p Fr(j)2392 1149 y(g)2437 1045 y Fo(\014)2437 1099 y(\014)2437 1154 y(\014)2493 1149 y Ft(6)25 b Fr(j)p Fz(\003)p Fr(j)p Fu(e)2744 1112 y Fn(\000)p Fw(\()p Fs(\014)s(=)p Fw(2)p Fn(\000)p Fw(2)3029 1088 y Fq(d)3067 1112 y Fs(\013)p Fw(\))p Fs(n)3429 1149 y Fz(\(4.37\))231 1420 y Fo(\014)231 1475 y(\014)231 1530 y(\014)276 1525 y Fz(log)18 b Fu(Z)478 1481 y Fs(R;)p Fn(\006)471 1554 y Fs(\014)s(;)p Fw(\003)630 1525 y Fr(\000)i(f\006)p Fu(\014)5 b(h)p Fz(\()p Fu(\014)g Fz(\))p Fr(j)p Fz(\003)p Fr(j)22 b Fz(+)1299 1439 y Fo(X)1297 1636 y Fs(x)p Fn(2)p Fw(\003)1768 1439 y Fo(X)1498 1643 y Fs(x)p Fn(2)p Fs(X)5 b Fw(\()p Fs(\031)r Fw(\))p Fn(\032)p Fs(K)1865 1624 y Fq(R)1916 1643 y Fw(\()p Fs(x;n)p Fw(\))p Fn(\\)p Fw(\003)2194 1464 y Fu(!)2254 1431 y Fs(R;)p Fn(\006)2386 1464 y Fz(\()p Fu(\031)s Fz(\))p 2194 1504 318 4 v 2223 1587 a Fr(j)p Fu(X)i Fz(\()p Fu(\031)s Fz(\))p Fr(j)2522 1525 y(g)2567 1420 y Fo(\014)2567 1475 y(\014)2567 1530 y(\014)2623 1525 y Ft(6)25 b Fr(j)p Fz(\003)p Fr(j)p Fu(e)2874 1487 y Fn(\000)p Fw(\()p Fs(\014)s(=)p Fw(2)p Fn(\000)p Fw(2)3159 1464 y Fq(d)3197 1487 y Fs(\013)p Fw(\))p Fs(n)3429 1525 y Fz(\(4.38\))218 1931 y FE(Pro)s(of.)41 b Fz(\(4.37\))33 b(and)c(\(4.38\))k(follo)m(w)f(from)e(\(4.23\))i(and)e (its)h(analogue)h(without)e(re\015ections.)163 b Ft(\003)218 2157 y Fz(This)32 b(lemma)h(will)g(enable)g(us)f(to)i(estimate)h(the)e (corrections)h(to)f(the)g(pressure.)47 b(Let)33 b(us)f(also)i(ex-)118 2265 y(amine)d(t)m(w)m(o)h(other)e(consequences)h(whic)m(h)f(will)h(b)s (e)f(crucial)h(in)f(the)g(rest)h(of)g(the)f(pap)s(er.)218 2481 y(The)f(\014rst)g(consequence)i(justi\014es)f(the)g(notion)g(of)g (p)s(erfect)g(w)m(alls)h(in)m(tro)s(duced)f(in)f(Subsection)h(3.2.)118 2589 y(W)-8 b(e)25 b(consider)e(the)h(slab)f(\003)1008 2603 y Fs(L;")1136 2589 y Fz(and)g(the)h(re\015ection)g(wrt)f(to)h(the) g(h)m(yp)s(erplane)e(\006)j(=)g Fr(f)p Fu(x)h Fr(2)f Fx(R)3179 2556 y Fs(d)3219 2589 y Fu(;)106 b Fz(\()n Fu(~)-43 b(n)7 b Fr(\001)f Fu(x)p Fz(\))26 b(=)118 2697 y(0)p Fr(g)31 b Fz(whic)m(h)f(splits)h(\003)801 2711 y Fs(L;")936 2697 y Fz(in)m(to)g(t)m(w)m(o)h(non)d(in)m(teracting)k (domains.)118 2873 y FE(Lemma)i(4.7.)42 b Fj(Ther)-5 b(e)33 b(exists)g Fu(c)26 b(>)f Fz(0)33 b Fj(such)g(that)980 2947 y Fo(\014)980 3002 y(\014)980 3056 y(\014)1025 3052 y Fz(log)18 b Fu(Z)1227 3013 y Fw(+)1220 3080 y Fs(L;")1344 3052 y Fr(\000)i Fz(log)e Fu(Z)1637 3007 y Fs(R;)p Fw(+)1630 3080 y Fs(L;")1769 2947 y Fo(\014)1769 3002 y(\014)1769 3056 y(\014)1824 3052 y Ft(6)25 b Fu(ce)2001 3014 y Fn(\000)p Fw(\()p Fs(\014)s(=)p Fw(2)p Fn(\000)p Fw(2)2286 2991 y Fq(d)2324 3014 y Fs(\013)p Fw(\))2401 3052 y Fu(L)2463 3014 y Fs(d)p Fn(\000)p Fw(2)2609 3052 y Fu(;)795 b Fz(\(4.39\))118 3271 y Fj(wher)-5 b(e)34 b Fu(Z)444 3233 y Fw(+)437 3300 y Fs(L;")573 3271 y Fj(denotes)f(the)g(p)-5 b(artition)35 b(function)d(on)h Fz(\003)1974 3285 y Fs(L;"L)2159 3271 y Fj(with)g Fz(+)f Fj(b)-5 b(oundary)34 b(c)-5 b(onditions)34 b(and)g Fu(Z)3530 3227 y Fs(R;)p Fw(+)3523 3300 y Fs(L;")118 3389 y Fj(is)h(the)g(p)-5 b(artition)37 b(function)d(obtaine)-5 b(d)37 b(by)d(r)-5 b(e\015e)g(ction)36 b(\(se)-5 b(e)35 b(Subse)-5 b(ction)35 b(3.2\).)49 b(The)35 b(same)g(statement)118 3497 y(holds)f(with)g Fr(\000)e Fj(b)-5 b(oundary)35 b(c)-5 b(onditions.)218 3673 y FE(Pro)s(of.)41 b Fz(Let)31 b Fu(B)f Fz(=)25 b Fr(f\000)p Fz(1)p Fu(;)15 b Fz(0)p Fu(;)g Fz(1)p Fr(g)1275 3640 y Fs(d)1318 3673 y Fz(.)41 b(F)-8 b(or)31 b(an)m(y)f Fu(x)h Fz(in)f(\003)1970 3687 y Fs(L;")2074 3673 y Fz(,)h(w)m(e)g(set)821 3843 y(\026)-49 b Fu(n)o Fz(\()p Fu(x)p Fz(\))26 b(=)f(min)p Fr(f)p Fu(n;)106 b(K)7 b Fz(\()p Fu(x;)15 b(n)p Fz(\))21 b Fr(\\)e Fu(B)30 b Fr(6)p Fz(=)25 b Fr(;)p Fu(;)15 b(K)7 b Fz(\()p Fu(x;)15 b(n)p Fz(\))21 b Fr(\\)f Fz(\003)2646 3805 y Fs(c)2646 3865 y(L;")2776 3843 y Fr(6)p Fz(=)25 b Fr(;g)118 4013 y Fz(Then)488 4119 y Fo(\014)488 4174 y(\014)488 4228 y(\014)533 4224 y Fz(log)18 b Fu(Z)735 4185 y Fw(+)728 4252 y Fs(L;")852 4224 y Fr(\000)i Fz(log)e Fu(Z)1145 4179 y Fw(+)p Fs(;R)1138 4252 y(L;")1276 4119 y Fo(\014)1276 4174 y(\014)1276 4228 y(\014)1332 4224 y Ft(6)1514 4137 y Fo(X)1443 4334 y Fs(x)p Fn(2)p Fw(\003)1579 4345 y Fq(L;"L)1781 4123 y Fo(\020)2387 4137 y(X)1850 4342 y Fs(X)5 b Fw(\()p Fs(\031)r Fw(\))p Fn(\\)p Fs(K)2122 4323 y Fq(R)2173 4342 y Fw(\()p Fs(x)p Fw(;)t(\026)-39 b Fs(n)p Fw(\()p Fs(x)p Fw(\)\))2424 4323 y Fq(c)2457 4342 y Fn(6)p Fw(=)p Fn(;)p Fw(;)p Fs(x)p Fn(2)p Fs(X)5 b Fw(\()p Fs(\031)r Fw(\))p Fn(\032)p Fw(\003)2918 4353 y Fq(L;"L)3130 4162 y Fu(!)3190 4129 y Fs(R;)p Fw(+)3322 4162 y Fz(\()p Fu(\031)s Fz(\))p 3130 4203 V 3160 4286 a Fr(j)p Fu(X)i Fz(\()p Fu(\031)s Fz(\))p Fr(j)1905 4542 y Fz(+)2382 4455 y Fo(X)1991 4657 y Fs(X)e Fw(\()p Fs(\031)r Fw(\))p Fn(\\)p Fs(K)g Fw(\()p Fs(x)p Fw(;)t(\026)-39 b Fs(n)p Fw(\()p Fs(x)p Fw(\)\))2514 4638 y Fq(c)2547 4657 y Fn(6)p Fw(=)p Fn(;)p Fw(;)p Fs(x)p Fn(2)p Fs(X)5 b Fw(\()p Fs(\031)r Fw(\))2987 4480 y Fu(!)3047 4447 y Fw(+)3106 4480 y Fz(\()p Fu(\031)s Fz(\))p 2980 4521 259 4 v 2980 4604 a Fr(j)p Fu(X)i Fz(\()p Fu(\031)s Fz(\))p Fr(j)3249 4441 y Fo(\021)118 4821 y Fz(The)29 b(con)m(tribution)h(of)g (the)f(p)s(olymers)g(with)g Fu(X)7 b Fz(\()p Fu(\031)s Fz(\))30 b(in)g Fu(K)7 b Fz(\()p Fu(x;)15 b(n)p Fz(\))29 b(and)g(the)h(re\015ected)f(ones)h(in)f Fu(X)7 b Fz(\()p Fu(\031)s Fz(\))31 b(in)118 4930 y Fu(K)202 4897 y Fs(R)260 4930 y Fz(\()p Fu(x;)15 b(n)p Fz(\))30 b(with)h Fu(n)24 b Ft(6)30 b Fz(\026)-50 b Fu(n)p Fz(\()p Fu(x)p Fz(\))31 b(cancel)h(with)e(eac)m(h)h(other)g(b)m(y)f(Theorem)g(4.1.)218 5038 y(Since)j(the)g(w)m(eigh)m(ts)h(of)f(the)g(p)s(olymers)f(are)h (exp)s(onen)m(tially)h(small)g(\(see)f(Lemma)g(4.1\),)i(the)e(result) 118 5146 y(follo)m(ws.)3181 b Ft(\003)218 5372 y Fz(The)30 b(second)i(consequence)g(will)f(b)s(e)g(used)f(in)h(Section)h(6.)43 b(Let)32 b Fx(T)2483 5386 y Fs(N)2581 5372 y Fz(b)s(e)e(the)i(torus)e Fr(f\000)p Fu(N)5 b(;)15 b(:)g(:)g(:)i(;)e(N)10 b Fr(g)3620 5339 y Fs(d)118 5480 y Fz(and)39 b(w)m(e)i(consider)f(a)g(collection)j (of)d(re\015ections)g(inside)g Fx(T)2192 5494 y Fs(N)2299 5480 y Fz(for)f(whic)m(h)h(the)g(assumption)g(\(4.5\))h(is)p eop end %%Page: 21 21 TeXDict begin 21 20 bop 1137 45 a Fw(SURF)-8 b(A)n(CE)31 b(TENSION)g(AND)f(WULFF)h(THEOR)-6 b(Y)948 b(21)118 366 y Fz(satis\014ed.)82 b(Let)44 b Fu(B)49 b Fz(denote)44 b(the)g(b)s(oundary)e(conditions)i(imp)s(osed)g(b)m(y)f(the)i (re\015ections,)j(i.e.)82 b(the)118 474 y(cen)m(ters)31 b(and)f(the)h(b)s(oundaries)e(of)h(eac)m(h)i(re\015ecting)f(surfaces.) 40 b(W)-8 b(e)32 b(ha)m(v)m(e)981 563 y Fo(\014)981 617 y(\014)981 672 y(\014)1026 667 y Fz(log)18 b Fu(Z)1228 630 y Fs(R)1221 690 y(\014)s(;N)1371 667 y Fr(\000)i Fz(log)d Fu(Z)1656 682 y Fs(\014)s(;N)1786 563 y Fo(\014)1786 617 y(\014)1786 672 y(\014)1842 667 y Ft(6)25 b Fu(c)1977 681 y Fs(\013)2027 667 y Fu(e)2069 630 y Fn(\000)p Fw(\()p Fs(\014)s(=)p Fw(2)p Fn(\000)p Fw(2)2354 606 y Fq(d)2391 630 y Fs(\013)p Fw(\))2468 667 y Fr(j)p Fu(B)5 b Fr(j)15 b Fu(;)797 b Fz(\(4.40\))118 875 y(where)36 b Fu(Z)456 842 y Fs(R)449 903 y(\014)s(;N)615 875 y Fz(\(resp.)58 b Fu(Z)958 890 y Fs(\014)s(;N)1088 875 y Fz(\))36 b(denotes)h(the)f (partition)h(functions)f(in)g Fx(T)2612 889 y Fs(N)2715 875 y Fz(with)g(p)s(erio)s(dic)f(b)s(oundary)118 992 y(conditions)c(and)f(with)g(\(resp.)40 b(without\))31 b(re\015ection.)1032 1573 y(5.)46 b Fy(Pr)n(oper)-6 b(ties)33 b(of)h(the)f(surf)-8 b(a)n(ce)33 b(tension)218 1735 y Fz(In)d(the)i(follo)m(wing,)h Fu(\014)j Fz(is)c(\014xed)e(large)j (enough)e(suc)m(h)g(that)h(the)f(results)g(of)h(Section)g(4)g(are)f (satis\014ed)118 1843 y(and)38 b Fu(h)g Fz(refers)g(to)h Fu(h)p Fz(\()p Fu(\014)5 b Fz(\).)65 b(W)-8 b(e)39 b(\014rst)f(deriv)m (e)g(the)h(existence)g(of)f(the)h(thermo)s(dynamic)e(limit)i(for)f(the) 118 1950 y(surface)30 b(tension)h(and)f(then)g(its)h(con)m(v)m(exit)m (y)i(and)c(p)s(ositivit)m(y)-8 b(.)118 2161 y(5.1.)47 b FE(Pro)s(of)32 b(of)f(Theorem)g(3.1.)46 b Fz(The)27 b(pro)s(of)f(can)h(b)s(e)f(split)i(in)m(to)g(three)f(steps.)39 b(First,)28 b(w)m(e)g(are)f(going)118 2269 y(to)33 b(pro)m(v)m(e)h (that)f(the)g(c)m(hoice)h(of)f(the)g(barriers)e(\()p Fr(C)1798 2236 y Fw(+)1858 2269 y Fu(;)15 b Fr(C)1951 2236 y Fn(\000)2011 2269 y Fz(\))33 b(has)f(almost)h(no)g(con)m (tribution)g(on)g(the)f(ratio)118 2377 y(of)g(the)f(partition)h (functions.)44 b(Then,)31 b(an)g(inductiv)m(e)h(pro)s(cedure)e(enables) i(us)e(to)j(impro)m(v)m(e)f(\(3.8\))h(and)118 2485 y(to)e(deriv)m(e)g (the)g(con)m(v)m(ergence)h(\(3.9\))r(.)118 2655 y Fj(Step)h(1)g(:)p 118 2688 309 4 v 218 2763 a Fz(The)d(\014rst)f(step)i(is)f(to)h(pro)m (v)m(e)g(that)647 3028 y Fu(\034)687 3043 y Fs(\014)734 3028 y Fz(\()n Fu(~)-43 b(n)p Fz(\))83 b(=)g(lim)15 b(inf)1153 3086 y Fs(")p Fn(!)p Fw(0)1388 3028 y Fz(lim)g(inf)1420 3088 y Fs(L)p Fn(!1)1714 3028 y Fz(sup)1681 3112 y Fn(C)1722 3093 y Fp(+)1772 3112 y Fs(;)p Fn(C)1833 3093 y Fl(\000)1915 3028 y Fr(\000)1996 2966 y Fz(\()n Fu(~)-43 b(n)20 b Fr(\001)d Fu(~)-42 b(e)2193 2981 y Fs(d)2234 2966 y Fz(\))p 1996 3007 274 4 v 2009 3090 a Fu(\014)5 b(L)2127 3064 y Fs(d)p Fn(\000)p Fw(1)2310 3028 y Fz(log)2452 2951 y Fu(Z)2521 2907 y Fn(C)2562 2883 y Fp(+)2613 2907 y Fs(;)p Fn(C)2674 2883 y Fl(\000)2514 2980 y Fs(L;")2729 2951 y Fz(\()p Fr(S)2826 2918 y Fw(+)2886 2951 y Fu(;)15 b Fr(S)2988 2918 y Fn(\000)3047 2951 y Fz(\))p 2452 3007 630 4 v 2522 3113 a Fu(Z)2591 3069 y Fn(C)2632 3050 y Fp(+)2682 3069 y Fs(;R)2584 3142 y(L;")2774 3113 y Fu(Z)2843 3069 y Fn(C)2884 3050 y Fl(\000)2936 3069 y Fs(;R)2836 3142 y(L;")3107 3028 y Fu(:)343 b Fz(\(5.1\))218 3293 y(This)28 b(b)s(oils)i(do)m(wn)f(to)h(c)m(hec)m(k)h(that)f(there)f(are) h(constan)m(ts)h(\()p Fu(C)2312 3307 y Fw(1)2352 3293 y Fu(;)15 b(C)2457 3307 y Fw(2)2496 3293 y Fz(\))30 b(suc)m(h)f(that)h (for)f(an)m(y)h(\()p Fu(L;)15 b(")p Fz(\))31 b(and)118 3408 y(for)f(an)m(y)h(\()p Fr(C)517 3375 y Fw(+)577 3408 y Fu(;)15 b Fr(C)670 3375 y Fn(\000)729 3408 y Fz(\))31 b(and)f(\()1023 3385 y(~)1007 3408 y Fr(C)1060 3375 y Fw(+)1119 3408 y Fu(;)1176 3385 y Fz(~)1159 3408 y Fr(C)1212 3375 y Fn(\000)1272 3408 y Fz(\))495 3501 y Fo(\014)495 3556 y(\014)495 3610 y(\014)495 3665 y(\014)495 3719 y(\014)495 3774 y(\014)525 3688 y Fz(log)667 3611 y Fu(Z)736 3566 y Fn(C)777 3543 y Fp(+)828 3566 y Fs(;)p Fn(C)889 3543 y Fl(\000)729 3639 y Fs(L;")944 3611 y Fz(\()p Fr(S)1041 3578 y Fw(+)1101 3611 y Fu(;)15 b Fr(S)1203 3578 y Fn(\000)1262 3611 y Fz(\))p 667 3667 V 737 3773 a Fu(Z)806 3728 y Fn(C)847 3709 y Fp(+)897 3728 y Fs(;R)799 3801 y(L;")989 3773 y Fu(Z)1058 3728 y Fn(C)1099 3709 y Fl(\000)1151 3728 y Fs(;R)1051 3801 y(L;")1328 3688 y Fr(\000)k Fz(log)1561 3611 y Fu(Z)1642 3550 y Fw(~)1630 3566 y Fn(C)1671 3543 y Fp(+)1721 3566 y Fs(;)1753 3550 y Fw(~)1741 3566 y Fn(C)1782 3543 y Fl(\000)1623 3639 y Fs(L;")1838 3611 y Fz(\()p Fr(S)1935 3578 y Fw(+)1994 3611 y Fu(;)c Fr(S)2096 3578 y Fn(\000)2156 3611 y Fz(\))p 1561 3667 V 1630 3783 a Fu(Z)1711 3722 y Fw(~)1699 3739 y Fn(C)1740 3720 y Fp(+)1790 3739 y Fs(;R)1692 3812 y(L;")1883 3783 y Fu(Z)1964 3722 y Fw(~)1952 3739 y Fn(C)1993 3720 y Fl(\000)2044 3739 y Fs(;R)1945 3812 y(L;")2201 3501 y Fo(\014)2201 3556 y(\014)2201 3610 y(\014)2201 3665 y(\014)2201 3719 y(\014)2201 3774 y(\014)2272 3688 y Ft(6)25 b Fu(C)2433 3702 y Fw(1)2472 3688 y Fu(L)2534 3650 y Fs(d)2589 3688 y Fz(exp\()p Fr(\000)p Fu(C)2899 3702 y Fw(2)2939 3688 y Fu("L)p Fz(\))15 b Fu(:)357 b Fz(\(5.2\))118 3972 y(The)40 b(ev)m(en)m(ts)j Fr(S)664 3939 y Fw(+)722 3972 y Fu(;)15 b Fr(S)824 3939 y Fn(\000)924 3972 y Fz(decouple)42 b(the)f(in)m (terface)h(from)e(the)h(b)s(oundary)e(e\013ects)j(th)m(us)e(\(5.2\))j (can)e(b)s(e)118 4080 y(deriv)m(ed)31 b(b)m(y)f(using)g(only)g (estimates)i(in)e(a)h(pure)e(phase.)218 4296 y(It)h(is)h(enough)f(to)h (consider)1203 4273 y(~)1186 4296 y Fr(C)1239 4263 y Fn(\000)1324 4296 y Fz(=)25 b Fr(C)1473 4263 y Fn(\000)1532 4296 y Fz(.)40 b(In)30 b(this)g(case,)i(\(5.2\))g(b)s(ecomes)691 4389 y Fo(\014)691 4444 y(\014)691 4498 y(\014)691 4553 y(\014)691 4607 y(\014)691 4662 y(\014)721 4576 y Fz(log)864 4499 y Fu(Z)933 4454 y Fn(C)974 4431 y Fp(+)1024 4454 y Fs(;)p Fn(C)1085 4431 y Fl(\000)926 4527 y Fs(L;")1141 4499 y Fz(\()p Fr(S)1238 4466 y Fw(+)1297 4499 y Fu(;)15 b Fr(S)1399 4466 y Fn(\000)1458 4499 y Fz(\))p 864 4555 V 864 4671 a Fu(Z)945 4610 y Fw(~)933 4627 y Fn(C)974 4608 y Fp(+)1024 4627 y Fs(;)p Fn(C)1085 4608 y Fl(\000)926 4700 y Fs(L;")1141 4671 y Fz(\()p Fr(S)1238 4645 y Fw(+)1297 4671 y Fu(;)g Fr(S)1399 4645 y Fn(\000)1458 4671 y Fz(\))1524 4576 y Fr(\000)20 b Fz(log)1757 4499 y Fu(Z)1826 4454 y Fn(C)1867 4431 y Fp(+)1917 4454 y Fs(;R)1819 4527 y(L;")p 1757 4555 238 4 v 1757 4671 a Fu(Z)1838 4610 y Fw(~)1826 4627 y Fn(C)1867 4608 y Fp(+)1917 4627 y Fs(;R)1819 4700 y(L;")2005 4389 y Fo(\014)2005 4444 y(\014)2005 4498 y(\014)2005 4553 y(\014)2005 4607 y(\014)2005 4662 y(\014)2075 4576 y Ft(6)25 b Fu(C)2236 4590 y Fw(1)2276 4576 y Fu(L)2338 4538 y Fs(d)2393 4576 y Fz(exp\()p Fr(\000)p Fu(C)2703 4590 y Fw(2)2743 4576 y Fu("L)p Fz(\))15 b Fu(:)553 b Fz(\(5.3\))218 4860 y(F)-8 b(or)34 b(an)m(y)g(spin)f(con\014guration)h (in)f Fr(S)1480 4827 y Fw(+)1539 4860 y Fz(,)h(let)h(us)e(denote)h(b)m (y)f Fu(s)2321 4827 y Fw(+)2413 4860 y Fz(the)h(supp)s(ort)e(of)i(the)g (+)f(barrier)g(in)118 4968 y(\003)181 4985 y Fs(L;)275 4958 y Fq(")p 259 4970 61 3 v 259 5011 a Fp(10)330 4985 y Fs(L)397 4968 y Fz(+)492 4932 y Fs("L)p 492 4947 81 4 v 515 4999 a Fw(2)579 4968 y Fu(~)-41 b(e)625 4983 y Fs(d)693 4968 y Fz(whic)m(h)27 b(is)h(the)g(closest)h(to)f(the)g(h)m (yp)s(erplane)f(\006)e(=)g Fr(f)p Fu(x)p Fz(;)43 b(\()p Fu(x)15 b Fr(\001)d Fu(~)-42 b(n)p Fz(\))26 b(=)f(0)p Fr(g)p Fz(.)40 b(This)27 b(particular)118 5093 y(c)m(hoice)h(of)f Fu(s)527 5060 y Fw(+)611 5093 y Fz(will)g(b)s(e)e(stressed)h(b)m(y)h (the)f(notation)h Fu(s)1911 5060 y Fw(+)1995 5093 y Ft( )e Fr(S)2173 5060 y Fw(+)2232 5093 y Fz(.)40 b(The)25 b(constrained)i (partition)g(function)118 5201 y(can)k(b)s(e)f(decomp)s(osed)g(as)g (follo)m(ws)625 5389 y Fu(Z)694 5344 y Fn(C)735 5321 y Fp(+)785 5344 y Fs(;)p Fn(C)846 5321 y Fl(\000)687 5417 y Fs(L;")902 5389 y Fz(\()p Fr(S)999 5351 y Fw(+)1058 5389 y Fu(;)15 b Fr(S)1160 5351 y Fn(\000)1220 5389 y Fz(\))25 b(=)1376 5302 y Fo(X)1400 5504 y Fs(s)1433 5486 y Fp(+)1548 5389 y Fu(e)1590 5351 y Fs(\014)s(h)p Fn(j)p Fs(s)1727 5328 y Fp(+)1777 5351 y Fn(j)1801 5389 y Fu(Z)1870 5344 y Fn(C)1911 5321 y Fp(+)1961 5344 y Fs(;s)2014 5321 y Fp(+)1863 5417 y Fs(L;")2083 5389 y Fu(Z)2152 5344 y Fs(s)2185 5321 y Fp(+)2235 5344 y Fs(;)p Fn(C)2296 5321 y Fl(\000)2145 5417 y Fs(L;")2352 5389 y Fz(\()p Fu(s)2430 5351 y Fw(+)2514 5389 y Ft( )g Fr(S)2692 5351 y Fw(+)2751 5389 y Fu(;)15 b Fr(S)2853 5351 y Fn(\000)2913 5389 y Fz(\))g Fu(;)p eop end %%Page: 22 22 TeXDict begin 22 21 bop 118 45 a Fw(22)1128 b(T.)23 b(BODINEA)n(U)g (AND)g(E.)g(PRESUTTI)118 366 y Fz(with)33 b(the)h(\014rst)f(partition)h (function)f(free)h(of)g(constrain)m(ts)g(so)g(that)g(cluster)g (expansion)f(applies)h(and)118 474 y(the)g(second)f(partition)h (function)g(whic)m(h)f(tak)m(es)i(in)m(to)f(accoun)m(t)h(the)f (constrain)m(t)g(that)g(there)g(is)f(no)h(+)118 582 y(barrier)c(in)g (\003)586 600 y Fs(L;)679 572 y Fq(")p 664 584 61 3 v 664 626 a Fp(10)734 600 y Fs(L)807 582 y Fz(+)908 547 y Fs("L)p 908 562 81 4 v 930 614 a Fw(2)995 582 y Fu(~)-42 b(e)1040 597 y Fs(d)1111 582 y Fz(b)s(elo)m(w)31 b Fu(s)1412 549 y Fw(+)1471 582 y Fz(.)218 697 y(W)-8 b(e)31 b(\014rst)f(write)472 862 y Fu(Z)541 818 y Fn(C)582 795 y Fp(+)632 818 y Fs(;)p Fn(C)693 795 y Fl(\000)534 891 y Fs(L;")749 862 y Fz(\()p Fr(S)846 829 y Fw(+)905 862 y Fu(;)15 b Fr(S)1007 829 y Fn(\000)1066 862 y Fz(\))p 472 918 630 4 v 472 1035 a Fu(Z)553 974 y Fw(~)541 991 y Fn(C)582 972 y Fp(+)632 991 y Fs(;)p Fn(C)693 972 y Fl(\000)534 1063 y Fs(L;")749 1035 y Fz(\()p Fr(S)846 1009 y Fw(+)905 1035 y Fu(;)g Fr(S)1007 1009 y Fn(\000)1066 1035 y Fz(\))1137 939 y(=)1233 853 y Fo(X)1257 1055 y Fs(s)1290 1036 y Fp(+)1389 862 y Fu(Z)1458 818 y Fn(C)1499 795 y Fp(+)1549 818 y Fs(;s)1602 795 y Fp(+)1451 891 y Fs(L;")p 1389 918 268 4 v 1389 1035 a Fu(Z)1470 974 y Fw(~)1458 991 y Fn(C)1499 972 y Fp(+)1549 991 y Fs(;s)1602 972 y Fp(+)1451 1063 y Fs(L;")1692 939 y Fu(e)1734 902 y Fs(\014)s(h)p Fn(j)p Fs(s)1871 878 y Fp(+)1921 902 y Fn(j)1969 862 y Fu(Z)2051 801 y Fw(~)2038 818 y Fn(C)2079 795 y Fp(+)2130 818 y Fs(;s)2183 795 y Fp(+)2031 891 y Fs(L;")2237 862 y Fu(Z)2306 818 y Fs(s)2339 795 y Fp(+)2389 818 y Fs(;)p Fn(C)2450 795 y Fl(\000)2299 891 y Fs(L;")2506 862 y Fz(\()p Fu(s)2584 829 y Fw(+)2668 862 y Ft( )25 b Fr(S)2846 829 y Fw(+)2905 862 y Fu(;)15 b Fr(S)3007 829 y Fn(\000)3066 862 y Fz(\))p 1969 918 1133 4 v 2221 1035 a Fu(Z)2302 974 y Fw(~)2290 991 y Fn(C)2331 972 y Fp(+)2381 991 y Fs(;)p Fn(C)2442 972 y Fl(\000)2283 1063 y Fs(L;")2498 1035 y Fz(\()p Fr(S)2595 1009 y Fw(+)2654 1035 y Fu(;)g Fr(S)2756 1009 y Fn(\000)2815 1035 y Fz(\))3127 939 y Fu(:)323 b Fz(\(5.4\))118 1204 y(Let)31 b Fu(N)k Fz(=)25 b Fu("L=)p Fz(10,)32 b(\(supp)s(ose,)e (for)g(notational)i(simplicit)m(y)-8 b(,)33 b Fu(N)40 b Fz(an)30 b(in)m(teger\),)j(then)600 1469 y(exp)p Fr(f\000)p Fz(4)p Fu(L)962 1432 y Fs(d)1003 1469 y Fu(e)1045 1432 y Fn(\000)p Fw(\()p Fs(\014)s(=)p Fw(2)p Fn(\000)p Fw(2)p Fs(\013)p Fw(\))p Fs(N)1471 1469 y Fr(g)26 b Ft(6)1648 1392 y Fu(Z)1717 1348 y Fn(C)1758 1324 y Fp(+)1808 1348 y Fs(;s)1861 1324 y Fp(+)1710 1421 y Fs(L;")p 1648 1448 268 4 v 1648 1565 a Fu(Z)1729 1504 y Fw(~)1717 1520 y Fn(C)1758 1501 y Fp(+)1808 1520 y Fs(;s)1861 1501 y Fp(+)1710 1593 y Fs(L;")1950 1392 y Fu(Z)2031 1331 y Fw(~)2019 1348 y Fn(C)2060 1324 y Fp(+)2110 1348 y Fs(;R)2012 1421 y(L;")p 1950 1448 238 4 v 1950 1554 a Fu(Z)2019 1510 y Fn(C)2060 1491 y Fp(+)2110 1510 y Fs(;R)2012 1583 y(L;")2223 1469 y Ft(6)40 b Fz(exp)p Fr(f)p Fz(4)p Fu(L)2625 1432 y Fs(d)2666 1469 y Fu(e)2708 1432 y Fn(\000)p Fw(\()p Fs(\014)s(=)p Fw(2)p Fn(\000)p Fw(2)p Fs(\013)p Fw(\))p Fs(N)3134 1469 y Fr(g)296 b Fz(\(5.5\))118 1729 y(follo)m(ws)38 b(from)e(crossed)h(cancellations)i(among)e(the)g(terms)g(in)f(the)h(n)m (umerator)g(and)f(denominator.)118 1837 y(W)-8 b(e)31 b(are)e(going)i(to)f(apply)f(the)h(expansion)f(of)g(the)h(partition)g (function)f(deriv)m(ed)h(in)f(Lemma)g(4.6)i(with)118 1945 y Fu(n)f Fz(=)g Fu(N)10 b Fz(.)49 b(The)33 b(factor)h(4)g(is)f(b)s (ecause)g(there)h(are)f(4)h(partition)g(functions)f(in)m(v)m(olv)m(ed.) 51 b(With)34 b(reference)118 2053 y(to)f(\(4.37\))i(and)d(\(4.38\))r(,) h(the)g(con)m(tribution)g(of)f Fu(x)h Fz(suc)m(h)f(that)h(the)f(scalar) i(pro)s(duct)d(\()p Fu(x)22 b Fr(\001)d Fu(~)-43 b(n)p Fz(\))29 b Ft(>)f Fz(8)p Fu("L=)p Fz(10)118 2189 y(coming)33 b(from)e Fu(Z)718 2145 y Fn(C)759 2122 y Fp(+)809 2145 y Fs(;s)862 2122 y Fp(+)711 2218 y Fs(L;")948 2189 y Fz(and)g Fu(Z)1195 2145 y Fn(C)1236 2122 y Fp(+)1286 2145 y Fs(;R)1188 2218 y(L;")1395 2189 y Fz(cancel)j(with)d(eac)m(h)i (other,)g(as)f(w)m(ell)g(as)g(those)h(from)e Fu(Z)3297 2128 y Fw(~)3285 2145 y Fn(C)3326 2122 y Fp(+)3376 2145 y Fs(;s)3429 2122 y Fp(+)3278 2218 y Fs(L;")3515 2189 y Fz(and)118 2345 y Fu(Z)199 2284 y Fw(~)187 2300 y Fn(C)228 2277 y Fp(+)278 2300 y Fs(;R)180 2373 y(L;")356 2345 y Fz(.)38 b(Symmetrically)-8 b(,)26 b(the)e(con)m(tribution)g(of)g Fu(x)f Fz(suc)m(h)g(that)h(\()p Fu(x)6 b Fr(\001)t Fu(~)-43 b(n)q Fz(\))26 b Fu(<)f Fz(8)p Fu("L=)p Fz(10)g(arising)f(from)f Fu(Z)3463 2300 y Fn(C)3504 2277 y Fp(+)3554 2300 y Fs(;s)3607 2277 y Fp(+)3456 2373 y Fs(L;")118 2500 y Fz(and)30 b Fu(Z)376 2439 y Fw(~)364 2456 y Fn(C)405 2432 y Fp(+)455 2456 y Fs(;s)508 2432 y Fp(+)357 2529 y Fs(L;")593 2500 y Fz(cancel)h(with)f(eac)m(h)i(other,)f(as)f(w)m(ell)i(as)e(those)h (from)f Fu(Z)2481 2439 y Fw(~)2469 2456 y Fn(C)2510 2432 y Fp(+)2560 2456 y Fs(;R)2462 2529 y(L;")2668 2500 y Fz(and)g Fu(Z)2914 2456 y Fn(C)2955 2432 y Fp(+)3005 2456 y Fs(;R)2907 2529 y(L;")3082 2500 y Fz(.)218 2716 y(Finally)-8 b(,)32 b(b)m(y)e(applying)h(\(5.5\))r(,)g(w)m(e)g(get)g (from)f(\(5.4\))r(:)211 2904 y Fu(Z)280 2860 y Fn(C)321 2836 y Fp(+)372 2860 y Fs(;)p Fn(C)433 2836 y Fl(\000)273 2932 y Fs(L;")488 2904 y Fz(\()p Fr(S)585 2871 y Fw(+)645 2904 y Fu(;)15 b Fr(S)747 2871 y Fn(\000)806 2904 y Fz(\))p 211 2960 630 4 v 211 3076 a Fu(Z)293 3015 y Fw(~)280 3032 y Fn(C)321 3013 y Fp(+)372 3032 y Fs(;)p Fn(C)433 3013 y Fl(\000)273 3105 y Fs(L;")488 3076 y Fz(\()p Fr(S)585 3050 y Fw(+)645 3076 y Fu(;)g Fr(S)747 3050 y Fn(\000)806 3076 y Fz(\))960 2981 y Ft(6)1148 2904 y Fu(Z)1217 2860 y Fn(C)1258 2836 y Fp(+)1309 2860 y Fs(;R)1210 2932 y(L;")p 1148 2960 238 4 v 1148 3076 a Fu(Z)1230 3015 y Fw(~)1217 3032 y Fn(C)1258 3013 y Fp(+)1309 3032 y Fs(;R)1210 3105 y(L;")1396 2981 y Fu(e)1438 2943 y Fw(4)p Fs(L)1521 2920 y Fq(d)1558 2943 y Fs(e)1591 2920 y Fl(\000)p Fp(\()p Fq(\014)s(=)p Fp(2)p Fl(\000)p Fp(2)p Fq(\013)p Fp(\))p Fq(N)1981 2894 y Fo(X)2005 3097 y Fs(s)2038 3078 y Fp(+)2153 2904 y Fu(Z)2234 2843 y Fw(~)2222 2860 y Fn(C)2263 2836 y Fp(+)2313 2860 y Fs(;s)2366 2836 y Fp(+)2215 2932 y Fs(L;")2436 2904 y Fu(e)2478 2871 y Fs(\014)s(h)p Fn(j)p Fs(s)2615 2847 y Fp(+)2665 2871 y Fn(j)2703 2904 y Fu(Z)2772 2860 y Fs(s)2805 2836 y Fp(+)2855 2860 y Fs(;)p Fn(C)2916 2836 y Fl(\000)2765 2932 y Fs(L;")2972 2904 y Fz(\()p Fu(s)3050 2871 y Fw(+)3134 2904 y Ft( )26 b Fr(S)3313 2871 y Fw(+)3371 2904 y Fu(;)15 b Fr(S)3473 2871 y Fn(\000)3533 2904 y Fz(\))p 2153 2960 1416 4 v 2546 3076 a Fu(Z)2627 3015 y Fw(~)2615 3032 y Fn(C)2656 3013 y Fp(+)2706 3032 y Fs(;)p Fn(C)2767 3013 y Fl(\000)2608 3105 y Fs(L;")2823 3076 y Fz(\()p Fr(S)2920 3050 y Fw(+)2979 3076 y Fu(;)g Fr(S)3081 3050 y Fn(\000)3140 3076 y Fz(\))960 3330 y Ft(6)1148 3253 y Fu(Z)1217 3208 y Fn(C)1258 3185 y Fp(+)1309 3208 y Fs(;R)1210 3281 y(L;")p 1148 3309 238 4 v 1148 3425 a Fu(Z)1230 3364 y Fw(~)1217 3381 y Fn(C)1258 3362 y Fp(+)1309 3381 y Fs(;R)1210 3454 y(L;")1411 3330 y Fu(e)1453 3292 y Fw(4)p Fs(L)1536 3269 y Fq(d)1573 3292 y Fs(e)1606 3269 y Fl(\000)p Fp(\()p Fq(\014)s(=)p Fp(2)p Fl(\000)p Fp(2)p Fq(\013)p Fp(\))p Fq(N)118 3590 y Fz(In)30 b(the)g(same)h(w)m(a)m(y)g(w)m(e)g(get)982 3766 y Fu(Z)1051 3722 y Fn(C)1092 3699 y Fp(+)1142 3722 y Fs(;)p Fn(C)1203 3699 y Fl(\000)1044 3795 y Fs(L;")1259 3766 y Fz(\()p Fr(S)1356 3733 y Fw(+)1416 3766 y Fu(;)15 b Fr(S)1518 3733 y Fn(\000)1577 3766 y Fz(\))p 982 3822 630 4 v 982 3939 a Fu(Z)1063 3878 y Fw(~)1051 3895 y Fn(C)1092 3876 y Fp(+)1142 3895 y Fs(;)p Fn(C)1203 3876 y Fl(\000)1044 3967 y Fs(L;")1259 3939 y Fz(\()p Fr(S)1356 3913 y Fw(+)1416 3939 y Fu(;)g Fr(S)1518 3913 y Fn(\000)1577 3939 y Fz(\))1730 3843 y Ft(>)1919 3766 y Fu(Z)1988 3722 y Fn(C)2029 3699 y Fp(+)2080 3722 y Fs(;R)1981 3795 y(L;")p 1919 3822 238 4 v 1919 3939 a Fu(Z)2001 3878 y Fw(~)1988 3895 y Fn(C)2029 3876 y Fp(+)2080 3895 y Fs(;R)1981 3967 y(L;")2182 3843 y Fu(e)2224 3806 y Fn(\000)p Fw(4)p Fs(L)2362 3782 y Fq(d)2399 3806 y Fs(e)2432 3782 y Fl(\000)p Fp(\()p Fq(\014)s(=)p Fp(2)p Fl(\000)p Fp(2)p Fq(\013)p Fp(\))p Fq(N)118 4108 y Fz(Recalling)32 b(that)f Fu(N)k Fz(=)25 b Fu("L=)p Fz(10,)33 b(w)m(e)d(ha)m(v)m(e)i(th)m(us)e(completed)h(the)g (pro)s(of)f(of)37 b(\(5.2\))r(.)118 4427 y Fj(Step)c(2)g(:)p 118 4459 309 4 v 218 4535 a Fz(The)d(goal)i(is)e(to)h(deriv)m(e)g(a)g (lo)m(w)m(er)g(b)s(ound)e(for)1198 4788 y Fu(\036)p Fz(\()p Fu(L;)15 b(";)g Fr(C)1524 4751 y Fw(+)1585 4788 y Fu(;)g Fr(C)1678 4751 y Fn(\000)1737 4788 y Fz(\))26 b(=)1950 4711 y Fu(Z)2019 4667 y Fn(C)2060 4644 y Fp(+)2110 4667 y Fs(;)p Fn(C)2171 4644 y Fl(\000)2012 4740 y Fs(L;")2227 4711 y Fz(\()p Fr(S)7 b Fz(\))p 1904 4767 502 4 v 1904 4873 a Fu(Z)1973 4829 y Fn(C)2014 4810 y Fp(+)2064 4829 y Fs(;R)1966 4902 y(L;")2167 4873 y Fu(Z)2236 4829 y Fn(C)2277 4810 y Fl(\000)2328 4829 y Fs(;R)2229 4902 y(L;")118 5048 y Fz(in)30 b(terms)g(of)h Fu(\034)621 5063 y Fs(\014)668 5048 y Fz(\()n Fu(~)-43 b(n)p Fz(\).)41 b(F)-8 b(or)31 b(simplicit)m(y)h Fr(S)g Fz(=)25 b(\()p Fr(S)1717 5015 y Fw(+)1776 5048 y Fu(;)15 b Fr(S)1878 5015 y Fn(\000)1937 5048 y Fz(\).)218 5156 y(The)26 b(previous)g(step)h (\(see)g(\(5.1\))r(\))g(implies)g(that)g(there)g(exists)g(a)g(sequence) g(\()p Fu(")2871 5171 y Fs(k)2915 5156 y Fu(;)15 b(L)3017 5171 y Fs(k)3060 5156 y Fz(\))3095 5171 y Fs(k)21 b Fh(>)e Fw(0)3293 5156 y Fz(suc)m(h)27 b(that)818 5241 y Fo(\014)818 5295 y(\014)818 5350 y(\014)818 5404 y(\014)818 5459 y(\014)859 5338 y Fz(\()n Fu(~)-43 b(n)20 b Fr(\001)d 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1898 y Fo(X)734 2115 y Fw(\()p Fs(c)792 2083 y Fp(+)p Fq(;i)792 2139 y(k)884 2115 y Fs(;c)935 2083 y Fl(\000)p Fq(;i)935 2139 y(k)1028 2115 y Fw(\))1070 1984 y Fu(e)1112 1934 y Fn(\000)p Fs(\014)s(H)1268 1946 y Fq(h)1309 1934 y Fw(\()p Fi(1)1377 1904 y Fl(\006)1377 1957 y(U)1416 1975 y Fq(k)1458 1934 y Fw(\))1489 1984 y Fu(Z)1558 1947 y Fn(C)1599 1923 y Fp(+)1551 2027 y Fw(\001)1610 2008 y Fp(+)1661 2027 y Fw(\()p Fn(f)p Fs(c)1754 1994 y Fp(+)p Fq(;i)1754 2051 y(k)1846 2027 y Fn(g)p Fw(\))1913 1911 y Fo(\000)1955 1984 y Fr(S)2017 1947 y Fw(+)2076 1984 y Fu(;)15 b(c)2155 1940 y Fw(+)p Fs(;i)2155 2014 y(k)2283 1984 y Ft( )25 b Fr(C)2452 1940 y Fw(+)p Fs(;i)2447 2014 y(k)2555 1911 y Fo(\001)2597 1984 y Fu(Z)2666 1947 y Fn(C)2707 1923 y Fl(\000)2659 2027 y Fw(\001)2718 2008 y Fl(\000)2770 2027 y Fw(\()p Fn(f)p Fs(c)2863 1994 y Fl(\000)p Fq(;i)2863 2051 y(k)2956 2027 y Fn(g)p Fw(\))3023 1911 y Fo(\000)3065 1984 y Fr(S)3127 1947 y Fn(\000)3186 1984 y Fu(;)15 b(c)3265 1940 y Fn(\000)p 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b(By)31 b(taking)g(the)g(pro)s(duct)e(o)m(v)m(er)j(all)f Fu(i)p Fz(,)g(w)m(e)g(get)537 3687 y Fo(Y)583 3882 y Fs(i)669 3773 y Fu(Z)738 3718 y Fs(c)769 3686 y Fp(+)p Fq(;i)769 3742 y(k)860 3718 y Fs(;c)911 3686 y Fl(\000)p Fq(;i)911 3742 y(k)731 3802 y Fs(L)779 3814 y Fq(k)817 3802 y Fs(;")870 3814 y Fq(k)1007 3773 y Fz(\()p Fr(S)1097 3788 y Fs(k)1140 3773 y Fz(\))26 b Ft(>)40 b Fz(exp)1466 3645 y Fo(\022)1533 3773 y Fr(\000)1619 3712 y Fu(\014)20 b(L)1752 3679 y Fs(d)p Fn(\000)p Fw(1)p 1614 3752 V 1614 3836 a Fz(\()n Fu(~)-43 b(n)20 b Fr(\001)d Fu(~)-42 b(e)1811 3851 y Fs(d)1852 3836 y Fz(\))1897 3700 y Fo(\000)1939 3773 y Fu(\034)1979 3788 y Fs(\014)2026 3773 y Fz(\()n Fu(~)f(n)p Fz(\))20 b(+)g(1)p Fu(=k)2402 3700 y Fo(\001)2445 3645 y(\023)2552 3687 y(Y)2598 3882 y Fs(i)2684 3773 y Fu(Z)2753 3718 y Fs(c)2784 3686 y Fp(+)p Fq(;i)2784 3742 y(k)2875 3718 y Fs(;R)p Fw(\()p Fs(k)r Fw(\))2746 3802 y Fs(L)2794 3814 y Fq(k)2832 3802 y Fs(;")2885 3814 y Fq(k)3045 3773 y Fu(Z)3114 3718 y Fs(c)3145 3686 y Fl(\000)p Fq(;i)3145 3742 y(k)3238 3718 y Fs(;R)p Fw(\()p Fs(k)r Fw(\))3107 3802 y Fs(L)3155 3814 y Fq(k)3193 3802 y Fs(;")3246 3814 y Fq(k)218 4008 y Fz(W)-8 b(e)22 b(are)g(going)h(to)f (plug)f(the)g(previous)g(inequalit)m(y)i(in)e(\(5.8\))i(in)e(order)g (to)h(reconstruct)g(t)m(w)m(o)h(partition)118 4116 y(function)30 b(on)g(the)h(domains)794 4269 y(\001)870 4231 y Fn(\006)954 4269 y Fz(=)1050 4183 y Fo([)1088 4378 y Fs(i)1191 4168 y Fo(n)1252 4269 y Fz(\001)1328 4231 y Fn(\006)1387 4269 y Fz(\()p Fr(f)p Fu(c)1506 4225 y Fn(\006)p Fs(;i)1506 4298 y(k)1609 4269 y Fr(g)1654 4284 y Fs(k)1698 4269 y Fz(\))k Fr([)g Fz(\003)1927 4231 y Fn(\006)1986 4269 y Fz(\()p Fu(c)2060 4231 y Fn(\006)p Fs(;i)2164 4269 y Fu(;)15 b(R)q Fz(\()p Fu(k)s Fz(\)\))37 b Fr([)d Fu(c)2600 4225 y Fn(\006)p Fs(;i)2600 4298 y(k)2703 4168 y Fo(o)2794 4269 y Fu(:)118 4509 y Fz(Notice)f(that)e(the)g(sets)h(\001)1013 4476 y Fn(\006)1102 4509 y Fz(are)g(sligh)m(tly)g(di\013eren)m(t)f (from)f(\003)2211 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Fr(S)2718 4757 y Fn(\000)2777 4794 y Fu(;)g Fr(C)2870 4750 y Fn(\000)p Fs(;i)2865 4824 y(k)2974 4721 y Fo(\001)3015 4794 y Fu(e)3057 4744 y Fn(\000)p Fs(\014)s(H)3213 4756 y Fq(h)3254 4744 y Fw(\()p Fi(1)3322 4714 y Fl(\006)3322 4767 y(U)3361 4785 y Fq(k)3403 4744 y Fw(\))1733 5020 y Fz(exp)1887 4892 y Fo(\022)1954 5020 y Fr(\000)2040 4959 y Fu(\014)20 b(L)2173 4926 y Fs(d)p Fn(\000)p Fw(1)p 2035 4999 V 2035 5083 a Fz(\()n Fu(~)-43 b(n)20 b Fr(\001)d Fu(~)-42 b(e)2232 5098 y Fs(d)2273 5083 y Fz(\))2318 4947 y Fo(\000)2360 5020 y Fu(\034)2400 5035 y Fs(\014)2447 5020 y Fz(\()n Fu(~)f(n)p Fz(\))20 b(+)g(1)p Fu(=k)2823 4947 y Fo(\001)2866 4892 y(\023)2963 5020 y Fu(;)118 5258 y Fz(where)k Fu(Z)444 5210 y Fn(C)485 5187 y Fp(+)535 5210 y Fs(;R)p Fw(\()p Fs(k)r Fw(\))437 5292 y(\001)496 5273 y Fp(+)705 5185 y Fo(\000)747 5258 y Fr(S)809 5225 y Fw(+)868 5258 y Fu(;)15 b Fr(C)961 5214 y Fw(+)p Fs(;i)956 5288 y(k)1064 5185 y Fo(\001)1130 5258 y Fz(denotes)24 b(the)g(partition)h(function)e (on)h(\001)2520 5225 y Fw(+)2603 5258 y Fz(with)f(a)i(p)s(erfect)e(w)m (all)i(made)f(of)118 5366 y(m)m(ultiple)29 b(re\015ections)g(on)f(the)h (scale)g Fu(L)1456 5381 y Fs(k)1527 5366 y Fz(and)f(taking)h(in)m(to)g (accoun)m(t)h(the)f(o)s(ccurrence)f(of)h(the)f(barriers)118 5480 y Fr(S)180 5447 y Fw(+)269 5480 y Fz(and)i Fr(fC)544 5436 y Fw(+)p Fs(;i)539 5510 y(k)647 5480 y Fr(g)692 5494 y Fs(i)721 5480 y Fz(.)p eop end %%Page: 25 25 TeXDict begin 25 24 bop 1137 45 a Fw(SURF)-8 b(A)n(CE)31 b(TENSION)g(AND)f(WULFF)h(THEOR)-6 b(Y)948 b(25)218 366 y Fz(By)32 b(\(5.7\))1303 496 y Fr(j)p Fu(H)1404 511 y Fs(h)1448 496 y Fz(\()p FE(1)1536 457 y Fn(\006)1536 524 y(U)1580 536 y Fq(k)1622 496 y Fz(\))p Fr(j)26 b Ft(6)f Fu(c)p Fr(jU)1925 511 y Fs(k)1968 496 y Fr(j)h Ft(6)f Fu(c)p Fz(4)p Fu(")2241 511 y Fs(k)2284 496 y Fu(L)2346 458 y Fs(d)p Fn(\000)p Fw(1)118 640 y Fz(so)31 b(that,)g(it)g(only)f(remains)g(to)i(c)m(hec)m(k)g(that)466 891 y(lim)439 952 y Fs(k)r Fn(!1)665 891 y Fz(lim)634 952 y Fs(L)p Fn(!1)900 891 y Fz(inf)853 957 y Fn(C)894 938 y Fp(+)945 957 y Fs(;)p Fn(C)1006 938 y Fl(\000)1181 830 y Fz(1)p 1108 870 193 4 v 1108 954 a Fu(L)1170 927 y Fs(d)p Fn(\000)p Fw(1)1351 891 y Fz(log)1493 822 y Fu(Z)1562 774 y Fn(C)1603 750 y Fp(+)1653 774 y Fs(;R)p Fw(\()p Fs(k)r Fw(\))1555 856 y(\001)1614 837 y Fp(+)1824 748 y Fo(\000)1866 822 y Fr(S)1928 789 y Fw(+)1987 822 y Fu(;)15 b Fr(C)2080 778 y Fw(+)p Fs(;i)2075 851 y(k)2183 748 y Fo(\001)2225 822 y Fu(Z)2294 774 y Fn(C)2335 750 y Fl(\000)2386 774 y Fs(;R)p Fw(\()p Fs(k)r Fw(\))2287 856 y(\001)2346 837 y Fl(\000)2557 748 y Fo(\000)2599 822 y Fr(S)2661 789 y Fn(\000)2720 822 y Fu(;)g Fr(C)2813 778 y Fn(\000)p Fs(;i)2808 851 y(k)2916 748 y Fo(\001)p 1493 870 1465 4 v 1980 976 a Fu(Z)2049 932 y Fn(C)2090 913 y Fp(+)2140 932 y Fs(;R)2042 1005 y(L;")2232 976 y Fu(Z)2301 932 y Fn(C)2342 913 y Fl(\000)2394 932 y Fs(;R)2294 1005 y(L;")2993 891 y Ft(>)25 b Fz(0)15 b Fu(;)301 b Fz(\(5.9\))118 1141 y(b)s(ecause,)31 b(if)f(w)m(e)h(supp)s (ose)e(that)i(the)f(previous)g(inequalit)m(y)i(holds,)e(then)813 1349 y(lim)15 b(inf)844 1410 y Fs(L)p Fn(!1)1090 1287 y Fz(\()n Fu(~)-43 b(n)20 b Fr(\001)d Fu(~)-41 b(e)1288 1302 y Fs(d)1328 1287 y Fz(\))p 1090 1328 274 4 v 1103 1411 a Fu(\014)5 b(L)1221 1385 y Fs(d)p Fn(\000)p Fw(1)1436 1349 y Fz(inf)1389 1415 y Fn(C)1430 1396 y Fp(+)1481 1415 y Fs(;)p Fn(C)1542 1396 y Fl(\000)1608 1349 y Fz(log)18 b Fu(\036)p Fz(\()p Fu(L;)d(";)g Fr(C)2067 1311 y Fw(+)2128 1349 y Fu(;)g Fr(C)2221 1311 y Fn(\000)2280 1349 y Fz(\))26 b Ft(>)45 b Fr(\000)20 b Fu(\034)2588 1364 y Fs(\014)2635 1349 y Fz(\()n Fu(~)-43 b(n)p Fz(\))15 b Fu(:)118 1560 y Fz(whic)m(h)30 b(completes)i(the)e(Theorem)h(3.1.)118 1776 y Fj(Step)i(3)g(:)p 118 1808 309 4 v 218 1889 a Fz(The)k(\014nal)g(step)g(is)h(dev)m(oted)g(to)h(the)e(deriv)-5 b(ation)38 b(of)45 b(\(5.9\))r(.)62 b(This)37 b(amoun)m(ts)h(to)g(pro)m (v)m(e)g(that)g(the)118 1997 y(corrections)43 b(to)f(the)f(pressure)f (for)i(the)f(di\013eren)m(t)h(t)m(yp)s(es)f(of)h(re\015ected)g(b)s (oundary)d(conditions)j(are)118 2104 y(negligible.)218 2324 y(First,)30 b(w)m(e)f(c)m(hec)m(k)i(that)f(the)f(constrained)h (partition)g(function)f Fu(Z)2478 2276 y Fn(C)2519 2253 y Fp(+)2569 2276 y Fs(;R)p Fw(\()p Fs(k)r Fw(\))2471 2358 y(\001)2530 2339 y Fp(+)2740 2250 y Fo(\000)2781 2324 y Fr(S)2843 2291 y Fw(+)2902 2324 y Fu(;)15 b Fr(C)2995 2280 y Fw(+)p Fs(;i)2990 2353 y(k)3098 2250 y Fo(\001)3169 2324 y Fz(is)29 b(asymptot-)118 2469 y(ically)39 b(equiv)-5 b(alen)m(t)38 b(to)g(the)f(non-constrained)g(partition)h(function)e Fu(Z)2570 2421 y Fn(C)2611 2398 y Fp(+)2662 2421 y Fs(;R)p Fw(\()p Fs(k)r Fw(\))2563 2503 y(\001)2622 2484 y Fp(+)2832 2469 y Fz(.)61 b(Let)37 b Fu(\026)3142 2421 y Fw(+)p Fs(;R)p Fw(\()p Fs(k)r Fw(\))3142 2503 y(\001)3201 2484 y Fp(+)3405 2469 y Fz(b)s(e)f(the)118 2577 y(corresp)s(onding)29 b(Gibbs)h(measure.)41 b(Then)29 b(the)i(follo)m(wing)h(holds)421 2753 y Fu(\026)476 2705 y Fw(+)p Fs(;R)p Fw(\()p Fs(k)r Fw(\))476 2787 y(\001)535 2768 y Fp(+)702 2679 y Fo(\000)743 2753 y Fr(S)805 2716 y Fw(+)864 2753 y Fu(;)15 b Fr(C)957 2709 y Fw(+)p Fs(;i)952 2782 y(k)1061 2679 y Fo(\001)1128 2753 y Ft(>)1223 2679 y Fo(\000)1265 2753 y Fz(1)21 b Fr(\000)f Fu(L)1484 2716 y Fs(d)p Fn(\000)p Fw(1)1629 2753 y Fz(exp\()p Fr(\000)p Fu(c"L)p Fz(\))2052 2679 y Fo(\001)q(\000)2137 2753 y Fz(1)g Fr(\000)g Fu(L)2355 2715 y Fs(d)p Fn(\000)p Fw(1)2355 2782 y Fs(k)2501 2753 y Fz(exp\()p Fr(\000)p Fu(c")2827 2768 y Fs(k)2870 2753 y Fu(L)2932 2768 y Fs(k)2975 2753 y Fz(\))3010 2679 y Fo(\001)3052 2702 y Fn(N)3110 2714 y Fq(k)3167 2753 y Fu(:)237 b Fz(\(5.10\))118 2913 y(This)27 b(can)h(b)s(e)e(deriv)m(ed)i (as)g(follo)m(ws.)40 b(The)27 b(o)s(ccurrence)h(of)g(a)g(barrier)e (with)i(blo)s(c)m(ks)f(uniformly)g(lab)s(elled)118 3021 y(b)m(y)e(1)h(in)f(the)h(slab)f(\003)808 3036 y Fs(l)q(;m)942 3021 y Fz(implies)g(that)h(there)g(is)f(no)g(connected)h(set)g(of)g (blo)s(c)m(ks)g(lab)s(elled)g(b)m(y)f Fr(\000)p Fz(1)g(joining)118 3155 y(the)32 b(t)m(w)m(o)g(faces)h(of)e(\003)837 3170 y Fs(l)q(;m)977 3155 y Fz(orthogonal)i(to)c Fu(~)-42 b(n)o Fz(.)44 b(Under)31 b Fu(\026)2000 3107 y Fw(+)p Fs(;R)p Fw(\()p Fs(k)r Fw(\))2000 3189 y(\001)2059 3170 y Fp(+)2225 3155 y Fz(,)h(a)g(P)m(eierls)g(estimate)h(similar)f(to)g (\(4.11\))118 3265 y(\(see)e(theorem)g(4.4\).)42 b(A)29 b(P)m(eierls)h(t)m(yp)s(e)g(argumen)m(t)f(implies)h(then)f(that)g(a)h (connected)g(set)g(of)f Fr(\000)g Fz(blo)s(c)m(ks)118 3384 y(with)37 b(length)g(at)g(least)i Fu(m)d Fz(has)h(a)g(probabilit)m (y)g(smaller)h(than)e(exp\()p Fr(\000)2585 3342 y Fs(\014)p 2585 3363 44 4 v 2589 3415 a Fw(2)2638 3384 y Fu(m)p Fz(\).)60 b(Applying)37 b(recursiv)m(ely)118 3491 y(the)31 b(P)m(eierls)g(argumen)m(t,)g(w)m(e)g(deriv)m(e)g(\(5.10\))r(.)218 3717 y(By)f(h)m(yp)s(othesis)g(on)h(the)f(sequence)h(\()p Fu(")1545 3732 y Fs(k)1588 3717 y Fu(;)15 b(L)1690 3732 y Fs(k)1733 3717 y Fz(\),)31 b(for)f Fu(k)k Fz(large)d(enough)f (\(5.10\))j(implies)999 3898 y Fu(\026)1054 3850 y Fw(+)p Fs(;R)p Fw(\()p Fs(k)r Fw(\))1054 3932 y(\001)1113 3913 y Fp(+)1279 3825 y Fo(\000)1321 3898 y Fr(S)1383 3861 y Fw(+)1441 3898 y Fu(;)15 b Fr(C)1534 3854 y Fw(+)p Fs(;i)1529 3928 y(k)1638 3825 y Fo(\001)1705 3898 y Ft(>)25 b Fz(2)1846 3861 y Fn(\000)p Fw(1)p Fn(\000)p Fs(L)2039 3837 y Fq(d)p Fl(\000)p Fp(1)2165 3861 y Fw(exp)q(\()p Fn(\000)p Fs(c")2419 3873 y Fq(k)2457 3861 y Fs(L)2505 3873 y Fq(k)2543 3861 y Fw(\))2590 3898 y Fu(:)118 4058 y Fz(Therefore)844 4291 y(lim)817 4353 y Fs(k)r Fn(!1)1043 4291 y Fz(lim)1012 4352 y Fs(L)p Fn(!1)1325 4230 y Fz(1)p 1251 4270 193 4 v 1251 4354 a Fu(L)1313 4328 y Fs(d)p Fn(\000)p Fw(1)1484 4291 y Fz(inf)1493 4357 y Fn(C)1534 4338 y Fp(+)1635 4291 y Fz(log)1777 4222 y Fu(Z)1846 4174 y Fn(C)1887 4151 y Fp(+)1938 4174 y Fs(;R)p Fw(\()p Fs(k)r Fw(\))1839 4256 y(\001)1898 4237 y Fp(+)2108 4148 y Fo(\000)2150 4222 y Fr(S)2212 4189 y Fw(+)2271 4222 y Fu(;)15 b Fz(\()p Fr(C)2399 4178 y Fw(+)p Fs(;i)2394 4251 y(k)2503 4222 y Fz(\))2538 4148 y Fo(\001)p 1777 4270 803 4 v 2013 4380 a Fu(Z)2082 4332 y Fn(C)2123 4313 y Fp(+)2173 4332 y Fs(;R)p Fw(\()p Fs(k)r Fw(\))2075 4409 y Fs(L;")2615 4291 y Ft(>)25 b Fz(0)15 b Fu(:)633 b Fz(\(5.11\))118 4550 y(This)30 b(reduces)g(the)g(pro)s(of)g(of)37 b(\(5.9\))c(to)859 4806 y(lim)832 4867 y Fs(k)r Fn(!1)1058 4806 y Fz(lim)1027 4866 y Fs(L)p Fn(!1)1340 4744 y Fz(1)p 1266 4785 193 4 v 1266 4868 a Fu(L)1328 4842 y Fs(d)p Fn(\000)p Fw(1)1546 4806 y Fz(inf)1499 4872 y Fn(C)1540 4853 y Fp(+)1591 4872 y Fs(;)p Fn(C)1652 4853 y Fl(\000)1744 4806 y Fz(log)1886 4736 y Fu(Z)1955 4689 y Fn(C)1996 4665 y Fp(+)2046 4689 y Fs(;R)p Fw(\()p Fs(k)r Fw(\))1948 4770 y(\001)2007 4752 y Fp(+)2232 4736 y Fu(Z)2301 4689 y Fn(C)2342 4665 y Fl(\000)2394 4689 y Fs(;R)p Fw(\()p Fs(k)r Fw(\))2294 4770 y(\001)2353 4752 y Fl(\000)p 1886 4785 679 4 v 1980 4891 a Fu(Z)2049 4847 y Fn(C)2090 4828 y Fp(+)2140 4847 y Fs(;R)2042 4919 y(L;")2232 4891 y Fu(Z)2301 4847 y Fn(C)2342 4828 y Fl(\000)2394 4847 y Fs(;R)2294 4919 y(L;")2600 4806 y Fz(=)25 b(0)15 b Fu(:)648 b Fz(\(5.12\))118 5056 y(Again)30 b(this)e(estimate)i(will)f(follo)m(w) h(from)e(cross)h(cancellations)i(b)s(et)m(w)m(een)e(the)g(4)g (partition)h(functions.)218 5164 y(F)-8 b(ollo)m(wing)25 b(the)f(strategy)g(of)g(step)f(1,)i(the)f(bulk)e(con)m(tribution)i(and) e(the)i(correction)g(to)g(the)g(pressure)118 5272 y(from)30 b(the)h(b)s(oundary)d(terms)i Fr(C)1205 5239 y Fw(+)1264 5272 y Fu(;)15 b Fr(C)1357 5239 y Fn(\000)1447 5272 y Fz(can)31 b(b)s(e)f(estimated)h(b)m(y)g(Lemma)f(4.6;)i(they)e(are)h(of) g(the)f(order)1324 5454 y Fu(L)1386 5417 y Fs(d)1441 5454 y Fz(exp)1595 5354 y Fo(n)1656 5454 y Fz(2)p Fu(L)1763 5417 y Fs(d)1804 5454 y Fu(e)1846 5417 y Fn(\000)p Fw(\()p Fs(\014)s(=)p Fw(2)p Fn(\000)p Fw(2)p Fs(\013)p Fw(\))p Fs("L=)p Fw(10)2395 5354 y Fo(o)p eop end %%Page: 26 26 TeXDict begin 26 25 bop 118 45 a Fw(26)1128 b(T.)23 b(BODINEA)n(U)g (AND)g(E.)g(PRESUTTI)218 366 y Fz(Th)m(us)33 b(it)j(is)e(enough)h(to)g (c)m(hec)m(k)i(that)e(the)g(con)m(tribution)g(of)g(the)g(p)s(erfect)g (w)m(alls)g(in)m(v)m(olv)m(ed)i(in)d(eac)m(h)118 474 y(partition)23 b(function)f(will)g(b)s(e)g(negligible)i(wrt)d(to)i(the) g(surface)f(order.)37 b(W)-8 b(e)24 b(\014rst)d(consider)h(the)h (partition)118 600 y(functions)35 b Fu(Z)584 556 y Fn(C)625 532 y Fp(+)675 556 y Fs(;R)577 629 y(L;")752 600 y Fu(;)15 b(Z)861 556 y Fn(C)902 532 y Fl(\000)954 556 y Fs(;R)854 629 y(L;")1031 600 y Fz(.)55 b(By)35 b(an)f(analogous)j(argumen)m(t)e (of)g(the)g(one)g(used)f(to)i(deriv)m(e)f(\(4.39\))s(,)h(w)m(e)118 729 y(see)24 b(that)g(the)g(corrections)g(to)h(the)e(pressure)f (induced)h(b)m(y)g(the)h(p)s(erfect)f(w)m(all)i(\006)e(are)g(of)h(the)g (order)f Fu(L)3506 696 y Fs(d)p Fn(\000)p Fw(2)3636 729 y Fz(.)218 945 y(W)-8 b(e)42 b(consider)f(no)m(w)g(the)g(partition)h (functions)e(with)h(m)m(ultiple)h(re\015ections.)73 b(The)40 b(p)s(erfect)h(w)m(all)118 1053 y(asso)s(ciated)31 b(to)e(the)h(b)s(o)m (x)f(\003)1051 1020 y Fs(i)1051 1080 y(k)1123 1053 y Fz(is)g(denoted)g(\006)1621 1067 y Fs(i)1678 1053 y Fz(and)f(to)i(eac)m (h)g(re\015ection)g(corresp)s(onds)e(a)h(particular)h(set)118 1161 y Fu(B)187 1175 y Fs(i)245 1161 y Fz(of)h(b)s(oundary)d (conditions.)41 b(The)30 b(set)h Fu(B)1626 1175 y Fs(i)1684 1161 y Fz(comprises)g(the)f(sites)h(around)e(the)i(cen)m(ter)g(of)g (re\015ection)118 1268 y(in)k(\006)295 1282 y Fs(i)358 1268 y Fz(as)g(w)m(ell)i(as)e(the)g(sites)h(outside)g(\003)1531 1235 y Fs(i)1531 1296 y(k)1609 1268 y Fz(whic)m(h)f(are)g(connected)h (to)g(\006)2641 1282 y Fs(i)2669 1268 y Fz(.)55 b(The)35 b(union)f(of)i(the)f Fu(B)3537 1282 y Fs(i)3600 1268 y Fz(is)118 1376 y(denoted)c(b)m(y)f Fu(B)35 b Fz(\(see)c(\014gure)f (2\).)218 1484 y(In)23 b(order)h(to)h(use)f(the)h(estimate)h(of)e (Section)i(4,)g(w)m(e)e(should)g(\014rst)f(c)m(hec)m(k)j(that)f(the)g (assumption)f(\(4.5\))118 1592 y(holds)31 b(for)g(the)g(m)m(ultiple)h (re\015ections)g(at)g(the)f(scale)i Fu(L)1992 1607 y Fs(k)2034 1592 y Fz(.)43 b(Supp)s(ose)30 b(that)h(for)g(some)h Fu(x)p Fz(,)f Fu(n)3189 1559 y Fn(0)3212 1592 y Fz(\()p Fu(x)p Fz(\))d Ft(6)e Fu(n)p Fz(\()p Fu(x)p Fz(\).)118 1702 y(F)-8 b(ollo)m(wing)35 b(the)d(pro)s(of)f(of)h(Theorem)g(4.1)h (there)f(exists)g(a)h(bijectiv)m(e)g(map)f Fr(T)55 b Fz(suc)m(h)31 b(that)i Fu(K)3288 1669 y Fs(R)3345 1702 y Fz(\()p Fu(x;)15 b(n)p Fz(\))29 b(=)118 1811 y Fr(T)23 b Fz(\()p Fu(K)7 b Fz(\()p Fu(x;)15 b(n)p Fz(\)\))27 b(for)e(an)m(y)i Fu(n)e(<)g(n)1122 1778 y Fn(0)1144 1811 y Fz(\()p Fu(x)p Fz(\).)40 b(Th)m(us)25 b Fu(K)1641 1778 y Fs(R)1699 1811 y Fz(\()p Fu(x;)15 b(n)1881 1778 y Fn(0)1904 1811 y Fz(\()p Fu(x)p Fz(\))c Fr(\000)g Fz(1\))28 b(con)m(tains)g(only) e(sites)g(in)g Fu(K)7 b Fz(\()p Fu(x;)15 b(n)3341 1778 y Fn(0)3364 1811 y Fz(\()p Fu(x)p Fz(\))c Fr(\000)g Fz(1\))118 1920 y(or)25 b(in)g(the)h(re\015ection)g(of)f Fu(K)7 b Fz(\()p Fu(x;)15 b(n)1230 1887 y Fn(0)1253 1920 y Fz(\()p Fu(x)p Fz(\))10 b Fr(\000)g Fz(1\))26 b(wrt)f(one)h(p)s(erfect)f(w)m (all.)40 b(By)25 b(construction)h Fu(K)3154 1887 y Fs(R)3212 1920 y Fz(\()p Fu(x;)15 b(n)3394 1887 y Fn(0)3417 1920 y Fz(\()p Fu(x)p Fz(\)\))26 b(is)118 2030 y(obtained)d(b)m(y)g(adding)f (all)h(the)g(plaquettes)h(connected)f(to)g Fu(K)2193 1997 y Fs(R)2251 2030 y Fz(\()p Fu(x;)15 b(n)2433 1997 y Fn(0)2456 2030 y Fz(\()p Fu(x)p Fz(\))5 b Fr(\000)g Fz(1\),)25 b(so)e(that)g(it)g(is)g(imp)s(ossible)118 2139 y(that)f Fu(K)390 2106 y Fs(R)448 2139 y Fz(\()p Fu(x;)15 b(n)630 2106 y Fn(0)653 2139 y Fz(\()p Fu(x)p Fz(\)\))23 b(con)m(tains)g(sites)f(in)f(t)m(w)m(o)i(distinct)g(p)s (erfect)e(w)m(alls)i(\006)2533 2153 y Fs(i)2582 2139 y Fz(and)e(\006)2816 2153 y Fs(j)2874 2139 y Fz(without)h(in)m (tersecting)118 2247 y(the)32 b(b)s(oundaries)e(of)i(\006)911 2261 y Fs(i)970 2247 y Fz(and)f(\006)1214 2261 y Fs(j)1282 2247 y Fz(whic)m(h)g(are)h(included)f(in)g Fu(B)5 b Fz(.)44 b(This)31 b(sho)m(ws)g(that)h Fu(n)p Fz(\()p Fu(x)p Fz(\))c Fu(<)f(n)3338 2214 y Fn(0)3361 2247 y Fz(\()p Fu(x)p Fz(\))32 b(and)118 2355 y(that)f(assumption)f(\(4.5\))i(is)e (satis\014ed.)218 2511 y(In)22 b(eac)m(h)j(partition)f(function)f Fu(Z)1312 2463 y Fn(C)1353 2439 y Fp(+)1404 2463 y Fs(;R)p Fw(\()p Fs(k)r Fw(\))1305 2545 y(\001)1364 2526 y Fp(+)1598 2511 y Fz(or)g Fu(Z)1771 2463 y Fn(C)1812 2439 y Fl(\000)1864 2463 y Fs(;R)p Fw(\()p Fs(k)r Fw(\))1764 2545 y(\001)1823 2526 y Fl(\000)2058 2511 y Fz(there)h(are)2428 2410 y Fo(\020)2512 2475 y Fs(L)p 2493 2490 87 4 v 2493 2542 a(L)2541 2554 y Fq(k)2589 2410 y Fo(\021)2643 2432 y Fs(d)p Fn(\000)p Fw(1)2797 2511 y Fz(re\015ections)g(at)g(the)g(scale) 118 2662 y Fu(L)180 2677 y Fs(k)223 2662 y Fz(.)47 b(Eac)m(h)33 b(re\015ection)h(leads)f(to)g(corrections)h(of)f(the)g(order)f Fu(L)2293 2624 y Fs(d)p Fn(\000)p Fw(2)2293 2692 y Fs(k)2456 2662 y Fz(and)g(o)m(v)m(erall)j(w)m(e)e(get)h(an)e(e\013ect)i(of)118 2794 y(the)27 b(order)516 2758 y Fs(L)564 2735 y Fq(d)p Fl(\000)p Fp(1)p 516 2773 163 4 v 554 2825 a Fs(L)602 2837 y Fq(k)688 2794 y Fz(.)40 b(As)27 b Fu(k)j Fz(div)m(erges)d(this)g (leads)h(to)f(v)-5 b(anishingly)27 b(small)g(con)m(tributions)h(wrt)e (the)h(surface)118 2924 y(order)j Fu(L)418 2891 y Fs(d)p Fn(\000)p Fw(1)548 2924 y Fz(.)218 3032 y(Com)m(bining)g(the)g (previous)g(estimates,)j(w)m(e)d(conclude)h(\(5.12\))r(.)118 3376 y(5.2.)47 b FE(Prop)s(erties.)f Fz(W)-8 b(e)32 b(are)e(going)i(to) f(establish)g(some)f(basic)h(prop)s(erties)f(of)h(the)f(surface)g (tension)118 3547 y FE(Prop)s(osition)46 b(5.1.)h Fj(F)-7 b(or)42 b(any)f Fu(\014)k Fj(lar)-5 b(ge)42 b(enough)e(such)h(that)h (the)f(mo)-5 b(del)42 b(is)e(in)h(the)f(Pir)-5 b(o)g(gov)42 b(Sinai)118 3655 y(r)-5 b(e)g(gime)1625 3778 y Fz(inf)1558 3847 y Fs(~)-34 b(n)p Fn(2)p Fm(S)1686 3828 y Fq(d)p Fl(\000)p Fp(1)1816 3778 y Fu(\034)1856 3793 y Fs(\014)1903 3778 y Fz(\()n Fu(~)-43 b(n)p Fz(\))25 b Fu(>)g Fz(0)p Fu(:)218 3999 y Fz(The)39 b(p)s(ositivit)m(y)j(of)e(the)h(surface)f (tension)g(de\014ned)f(in)h(\(3.11\))i(w)m(as)f(already)g(deriv)m(ed)f (in)g([BKL])118 4107 y(\(nev)m(ertheless)32 b(the)e(existence)i(of)f (the)f(thermo)s(dynamic)g(limit)h(w)m(as)g(an)f(assumption)g(in)g ([BKL]\).)218 4323 y(The)g(homogeneous)h(extension)g(on)f Fx(R)1555 4290 y Fs(d)1625 4323 y Fz(of)h(the)f(surface)h(tension)f(is) h(de\014ned)e(b)m(y)812 4534 y Fr(8)p Fu(x)c Fr(2)f Fx(R)1091 4496 y Fs(d)1132 4534 y Fu(;)197 b(\034)1394 4549 y Fs(\014)1441 4534 y Fz(\()p Fu(x)p Fz(\))26 b(=)f Fr(k)p Fu(x)p Fr(k)1827 4548 y Fw(2)1882 4534 y Fu(\034)1922 4549 y Fs(\014)1984 4405 y Fo(\022)2126 4472 y Fu(x)p 2061 4513 183 4 v 2061 4596 a Fr(k)p Fu(x)p Fr(k)2203 4610 y Fw(2)2253 4405 y Fo(\023)2335 4534 y Fu(;)198 b(\034)2598 4549 y Fs(\014)2644 4534 y Fz(\(0\))27 b(=)e(0)15 b Fu(:)118 4770 y FE(Prop)s(osition)36 b(5.2.)42 b Fj(The)33 b(surfac)-5 b(e)33 b(tension)h Fu(\034)1742 4785 y Fs(\014)1821 4770 y Fj(is)f(c)-5 b(onvex)33 b(on)g Fx(R)2408 4737 y Fs(d)2448 4770 y Fj(.)218 4941 y Fz(As)d(a)h(consequence)g([Am],)g(the)g(functional)f Fr(W)1864 4956 y Fs(\014)1942 4941 y Fz(is)g(lo)m(w)m(er)i(semi-con)m (tin)m(uous.)218 5157 y(The)41 b(de\014nition)g(\(3.8\))i(of)f(the)g (surface)f(tension)h(in)f(the)h(direction)e Fu(~)-43 b(n)41 b Fz(relies)h(on)g(the)g(arbitrary)118 5264 y(c)m(hoice)c(of)e (the)h(orien)m(tation)h(of)e(the)g(slab)h(along)g(one)f(of)g(the)h (axis)f(\(see)h(section)h(3\).)58 b(Nev)m(ertheless,)118 5372 y(since)36 b Fu(\034)386 5387 y Fs(\014)468 5372 y Fz(is)f(con)m(v)m(ex,)j(it)e(is)f(also)h(con)m(tin)m(uous)g(and)e (therefore)i(the)f(v)-5 b(alue)36 b(of)f(the)h(surface)f(tension)g(is) 118 5480 y(indep)s(enden)m(t)29 b(of)i(the)f(arbitrary)g(c)m(hoices)j (in)d(the)g(de\014nition.)p eop end %%Page: 27 27 TeXDict begin 27 26 bop 1137 45 a Fw(SURF)-8 b(A)n(CE)31 b(TENSION)g(AND)f(WULFF)h(THEOR)-6 b(Y)948 b(27)118 366 y Fj(Pr)-5 b(o)g(of)34 b(of)f(Pr)-5 b(op)g(osition)35 b(5.1.)218 474 y Fz(According)24 b(to)h(Theorem)e(3.1,)k(it)d(is)g (enough)f(to)i(pro)m(v)m(e)f(that)h(there)f(is)g Fu(c)2651 489 y Fs(\014)2723 474 y Fu(>)h Fz(0)f(suc)m(h)g(that)g(uniformly)118 582 y(o)m(v)m(er)29 b Fu(~)-42 b(n)30 b Fz(the)g(follo)m(wing)i(holds) 694 874 y Fr(8)p Fu(L)25 b(>)g Fz(0)p Fu(;)15 b Fr(8)p Fu(")26 b(>)1246 812 y Fz(1)p 1238 853 62 4 v 1238 936 a Fu(L)1310 874 y(;)244 b Fz(inf)1532 939 y Fn(C)1573 921 y Fp(+)1623 939 y Fs(;)p Fn(C)1684 921 y Fl(\000)1751 874 y Fz(log)1894 797 y Fu(Z)1963 752 y Fn(C)2004 729 y Fp(+)2054 752 y Fs(;)p Fn(C)2115 729 y Fl(\000)1956 825 y Fs(L;")2171 797 y Fz(\()p Fr(S)2268 764 y Fw(+)2327 797 y Fu(;)15 b Fr(S)2429 764 y Fn(\000)2488 797 y Fz(\))p 1894 853 630 4 v 1958 959 a Fu(Z)2027 914 y Fn(C)2068 895 y Fp(+)2118 914 y Fs(;R)2020 987 y(L;")2221 959 y Fu(Z)2290 914 y Fn(C)2331 895 y Fl(\000)2382 914 y Fs(;R)2283 987 y(L;")2559 874 y Ft(6)45 b Fr(\000)20 b Fu(c)2805 889 y Fs(\014)2852 874 y Fu(L)2914 836 y Fs(d)p Fn(\000)p Fw(1)3060 874 y Fu(:)344 b Fz(\(5.13\))118 1170 y(A)m(t)33 b(this)f(stage)h(the)g(constrain)m(t)g(\()p Fr(S)1349 1137 y Fw(+)1408 1170 y Fu(;)15 b Fr(S)1510 1137 y Fn(\000)1569 1170 y Fz(\))33 b(pla)m(ys)f(no)g(role)h(and)e(can)h(b)s(e)g(dropp)s (ed.)43 b(F)-8 b(urthermore,)33 b(it)118 1278 y(is)e(enough)f(to)h (select)h(the)e(most)h(simple)f(barriers)g Fr(C)1932 1245 y Fw(+)1991 1278 y Fu(;)15 b Fr(C)2084 1245 y Fn(\000)2174 1278 y Fz(and)30 b(to)h(deriv)m(e)905 1554 y Fr(8)p Fu(L)25 b(>)g Fz(0)p Fu(;)15 b Fr(8)p Fu(")26 b(>)1457 1493 y Fz(1)p 1449 1533 62 4 v 1449 1617 a Fu(L)1521 1554 y(;)197 b Fz(log)1998 1477 y Fu(Z)2067 1433 y Fw(+)p Fs(;)p Fn(\000)2060 1506 y Fs(L;")p 1886 1533 427 4 v 1886 1634 a Fu(Z)1955 1590 y Fw(+)p Fs(;R)1948 1662 y(L;")2112 1634 y Fu(Z)2181 1590 y Fn(\000)p Fs(;R)2174 1662 y(L;")2348 1554 y Ft(6)45 b Fr(\000)20 b Fu(c)2594 1569 y Fs(\014)2641 1554 y Fu(L)2703 1517 y Fs(d)p Fn(\000)p Fw(1)2849 1554 y Fu(;)555 b Fz(\(5.14\))118 1857 y(where)26 b Fu(Z)446 1813 y Fw(+)p Fs(;)p Fn(\000)439 1885 y Fs(L;")605 1857 y Fz(denotes)h(the)f(partition)h(function)f (with)g(mixed)g(b)s(oundary)e(conditions)j(in)f(the)g(domain)118 1991 y(\003)181 2005 y Fs(L;"L)334 1991 y Fz(.)40 b(F)-8 b(or)31 b(simplicit)m(y)h(w)m(e)f(supp)s(ose)e(that)i Fu(n)1705 2006 y Fs(d)1770 1991 y Fz(=)25 b(\()n Fu(~)-43 b(n)20 b Fr(\001)d Fu(~)-41 b(e)2064 2006 y Fs(d)2104 1991 y Fz(\))26 b Ft(>)f Fz(1)p Fu(=)2351 1914 y Fr(p)p 2428 1914 48 4 v 2428 1991 a Fu(d)p Fz(.)218 2099 y(As)33 b(explained)g(after)h(the)f(heuristic)g(expansion)g(\(3.10\))r(,)h(the) g(precise)f(deriv)-5 b(ation)34 b(of)f(the)g(surface)118 2209 y(tension)25 b(requires)f(to)i(comp)s(ensate)f(precisely)g(the)g (b)s(oundary)d(surface)j(tensions)g Fu(\034)2970 2170 y Fw(+)2960 2238 y Fs(bd)3055 2209 y Fz(and)f Fu(\034)3276 2170 y Fn(\000)3266 2238 y Fs(bd)3361 2209 y Fz(app)s(ear-)118 2325 y(ing)29 b(in)g(the)g(n)m(umerator)g(and)g(the)g(denominator.)40 b(F)-8 b(or)30 b(\(5.13\))r(,)g(only)f(a)g(crude)g(b)s(ound)d(on)j Fu(\034)3286 2286 y Fw(+)3276 2354 y Fs(bd)3376 2325 y Fz(and)f Fu(\034)3601 2286 y Fn(\000)3591 2354 y Fs(bd)118 2433 y Fz(is)j(necessary)-8 b(.)41 b(More)31 b(precisely)-8 b(,)32 b(b)m(y)f(\(4.38\))s(,)f(there)h(is)f Fu(C)2069 2447 y Fw(1)2134 2433 y Fu(>)25 b Fz(0)31 b(suc)m(h)f(that)864 2571 y Fo(\014)864 2626 y(\014)864 2681 y(\014)895 2676 y Fz(log)17 b Fu(Z)1096 2632 y Fw(+)p Fs(;R)1089 2705 y(L;")1248 2676 y Fz(+)j(log)e Fu(Z)1541 2632 y Fn(\000)p Fs(;R)1534 2705 y(L;")1693 2676 y Fr(\000)h Fu(\014)5 b Fr(P)i(j)p Fz(\003)1997 2690 y Fs(L;"L)2151 2676 y Fr(j)2176 2571 y Fo(\014)2176 2626 y(\014)2176 2681 y(\014)2247 2676 y Ft(6)2353 2615 y Fu(C)2418 2629 y Fw(1)p 2353 2655 105 4 v 2358 2738 a Fu(n)2413 2753 y Fs(d)2467 2676 y Fu(L)2529 2639 y Fs(d)p Fn(\000)p Fw(1)2660 2676 y Fu(e)2702 2639 y Fn(\000)p Fs(\014)s(=)p Fw(2)2890 2676 y Fu(:)514 b Fz(\(5.15\))118 2935 y(Due)26 b(to)g(the)g(mixed)g Fr(\006)f Fz(b.c.)h(the)g(spin)e(con\014gurations)i(whic)m(h)f(con)m (tribute)i(to)f Fu(Z)2875 2890 y Fw(+)p Fs(;)p Fn(\000)2868 2963 y Fs(L;")3034 2935 y Fz(ha)m(v)m(e)g(necessarily)118 3053 y(an)43 b(\\op)s(en")g(con)m(tour)g(\000)f(whose)h(spatial)g(supp) s(ort,)i(sp\(\000\),)g Fr(\003)p Fz(-disconnects)f(the)f(top)g(and)f(b) s(ottom)118 3161 y(faces)27 b(of)f(\003)496 3175 y Fs(L;"L)649 3161 y Fz(.)39 b(The)26 b(complemen)m(t)h(of)1504 3138 y(\026)1499 3161 y(\000)o(,)h(see)e(Subsection)g(4.1)i(for)d (de\014nitions,)i(is)g(made)f(b)m(y)g(a)g(\014nite)118 3268 y(n)m(um)m(b)s(er)38 b(of)g(regions,)k(sa)m(y)d(\001)1155 3282 y Fw(1)1195 3268 y Fu(;)15 b(::;)g Fz(\001)1401 3282 y Fs(n)1449 3268 y Fz(,)41 b(with)d(their)h(b)s(oundaries,)g Fu(\016)s Fz(\001)2573 3282 y Fs(i)2641 3268 y Fz(\(i.e.)h(all)g (plaquettes)f(in)g(\001)3602 3236 y Fs(c)3602 3294 y(i)3636 3268 y Fz(,)118 3376 y Fr(\003)p Fz(-connected)32 b(to)f(\001)805 3390 y Fs(i)833 3376 y Fz(\))g(where)f(the)g(spins)g(ha)m(v)m(e)h(a)g (constan)m(t)h(sign,)e(denoted)h(b)m(y)f Fu(\030)2921 3390 y Fs(i)2949 3376 y Fz(.)40 b(Then)1316 3648 y Fu(Z)1385 3603 y Fw(+)p Fs(;)p Fn(\000)1378 3676 y Fs(L;")1544 3648 y Fz(=)1639 3561 y Fo(X)1683 3758 y Fw(\000)1786 3648 y Fu(e)1828 3610 y Fn(\000)p Fs(\014)s(H)1984 3622 y Fq(h)2024 3610 y Fw(\()p Fs(\033)2095 3616 y Fp(\026)2091 3629 y(\000)2134 3610 y Fw(\))2217 3534 y Fs(n)2181 3561 y Fo(Y)2182 3757 y Fs(i)p Fw(=1)2312 3648 y Fu(Z)2381 3603 y Fs(\030)2412 3613 y Fq(i)2374 3676 y Fw(\001)2433 3686 y Fq(i)118 3932 y Fz(By)32 b(\(4.28\))r(,)f(w)m(e)g(get)825 4204 y Fu(Z)894 4159 y Fw(+)p Fs(;)p Fn(\000)887 4232 y Fs(L;")1052 4204 y Ft(6)25 b Fu(e)1190 4166 y Fs(\014)s Fn(j)p Fw(\003)1302 4177 y Fq(L;"L)1438 4166 y Fn(jP)1532 4117 y Fo(X)1576 4314 y Fw(\000)1679 4204 y Fu(e)1721 4166 y Fn(\000)p Fs(\014)s(H)1877 4178 y Fq(h)1917 4166 y Fw(\()p Fs(\033)1988 4172 y Fp(\026)1984 4185 y(\000)2027 4166 y Fw(\)+)p Fs(\014)s Fn(j)2176 4149 y Fw(\026)2172 4166 y(\000)p Fn(jP)2346 4090 y Fs(n)2309 4117 y Fo(Y)2310 4313 y Fs(i)p Fw(=1)2441 4204 y Fu(e)2483 4166 y Fs(e)2516 4143 y Fl(\000)p Fq(\014)s(=)p Fp(2+2)p Fq(\013)2785 4166 y Fs(N)2841 4178 y Fq(\016)r Fp(\001)2921 4194 y Fq(i)118 4483 y Fz(In)30 b(the)g(last)i(pro)s(duct)d(w)m(e)i(use)f(the) g(inequalit)m(y)1599 4636 y Fs(n)1554 4663 y Fo(X)1563 4858 y Fs(i)p Fw(=1)1701 4749 y Fu(N)1774 4764 y Fs(\016)r Fw(\001)1866 4774 y Fq(i)1922 4749 y Ft(6)25 b Fz(3)2063 4712 y Fs(d)2104 4749 y Fu(N)2177 4763 y Fw(\000)118 5033 y Fz(\(as)j(eac)m(h)g(plaquette)g(in)f Fu(\016)s Fz(\001)1083 5047 y Fs(i)1138 5033 y Fz(is)g Fr(\003)p Fz(-connected)i(to)f(a)f(plaquette)h(of)f(sp\(\000\))g(and)g(the)g (corresp)s(ondence)f(is)118 5143 y(at)h(most)f(3)488 5110 y Fs(d)555 5143 y Fz(to)g(1\).)40 b(Moreo)m(v)m(er,)29 b(b)m(y)d(the)g(de\014nition)g(of)g(con)m(tours)g(and)g(using)f(the)h (fact)h(that)f Fu(h)g Fz(b)s(elongs)118 5265 y(to)31 b(\(0)p Fu(;)15 b(e)391 5232 y Fn(\000)p Fs(\014)s(=)p Fw(2+2)649 5209 y Fq(d)p Fp(+1)765 5232 y Fs(\013)814 5265 y Fz(\))31 b(\(see)g(\(4.31\))r(\))739 5480 y Fr(\000)p Fu(\014)5 b(H)942 5495 y Fs(h)986 5480 y Fz(\()p Fu(\033)1077 5487 y Fw(\026)1073 5504 y(\000)1122 5480 y Fz(\))26 b Ft(6)f Fu(\014)5 b Fr(j)p Fu(h)p Fr(jj)1468 5457 y Fz(\026)1462 5480 y(\000)q Fr(j)20 b(\000)g Fu(\014)5 b(N)1785 5494 y Fw(\000)1859 5480 y Ft(6)45 b Fr(\000)20 b Fu(\014)5 b Fz([1)21 b Fr(\000)f Fu(e)2346 5443 y Fn(\000)p Fs(\014)s(=)p Fw(2+2)2604 5419 y Fq(d)p Fp(+1)2718 5443 y Fs(\013)2768 5480 y Fz(6)2813 5443 y Fs(d)2853 5480 y Fz(])p Fu(N)2951 5494 y Fw(\000)3015 5480 y Fu(:)p eop end %%Page: 28 28 TeXDict begin 28 27 bop 118 45 a Fw(28)1128 b(T.)23 b(BODINEA)n(U)g (AND)g(E.)g(PRESUTTI)218 366 y Fz(The)30 b(previous)f(estimate)k (implies)d(that)h(for)f Fu(\014)36 b Fz(large)31 b(enough,)284 537 y Fu(Z)353 493 y Fw(+)p Fs(;)p Fn(\000)346 566 y Fs(L;")511 537 y Ft(6)25 b Fu(e)649 500 y Fs(\014)s Fn(j)p Fw(\003)761 511 y Fq(L;"L)898 500 y Fn(jP)992 451 y Fo(X)1035 648 y Fw(\000)1138 537 y Fz(exp)1292 437 y Fo(n)1373 537 y Fr(\000)1464 437 y Fo(\020)1518 537 y Fu(\014)5 b Fz([1)21 b Fr(\000)f Fu(e)1798 500 y Fn(\000)p Fs(\014)s(=)p Fw(2+2)2056 476 y Fq(d)p Fp(+1)2170 500 y Fs(\013)2220 537 y Fz(6)2265 500 y Fs(d)2306 537 y Fz(])g Fr(\000)g Fz(6)2487 500 y Fs(d)2528 537 y Fz(2)p Fu(e)2615 500 y Fn(\000)p Fs(\014)s(=)p Fw(2+2)2873 476 y Fq(d)p Fp(+1)2988 500 y Fs(\013)3057 537 y Fr(\000)g Fz(3)3193 500 y Fs(d)3234 537 y Fu(e)3276 500 y Fn(\000)p Fs(\014)s(=)p Fw(2+2)3534 476 y Fq(d)p Fp(+1)3648 500 y Fs(\013)3698 437 y Fo(\021)3752 537 y Fu(N)3825 551 y Fw(\000)3873 437 y Fo(o)782 788 y Ft(6)25 b Fu(e)920 750 y Fs(\014)s Fn(j)p Fw(\003)1032 761 y Fq(L;"L)1168 750 y Fn(jP)1567 702 y Fo(X)1287 907 y Fs(D)r Fn(3)p Fs(x)1434 888 y Fl(\003)1470 907 y Fs(;N)1546 918 y Fq(D)1621 907 y Fh(>)19 b Fw(2)1730 888 y Fl(\000)p Fq(d)1815 907 y Fs(L)1863 888 y Fq(d)p Fl(\000)p Fp(1)2043 788 y Fu(e)2085 750 y Fn(\000)p Fs(\014)s(=)p Fw(2)p Fs(N)2309 761 y Fq(D)2369 788 y Fz(2)2414 750 y Fw(3)2449 727 y Fq(d)2486 750 y Fn(j)p Fs(D)r Fn(j)118 1046 y Fz(where)37 b(the)h(sum)e(is)i(o)m(v)m(er)h(all)f(connected)g(sets)g Fu(D)j Fz(of)c(plaquettes)i(\()p Fu(D)h Fz(standing)e(for)f (sp\(\000\)\))h(whic)m(h)118 1153 y(con)m(tain)45 b Fu(x)504 1120 y Fn(\003)588 1153 y Fz(a)f(p)s(oin)m(t)g(of)g(\003)1108 1167 y Fs(L;"L)1305 1153 y Fr(\003)p Fz(-connected)h(to)g(the)f (surface)g(whic)m(h)g(separates)h(the)f(+)f(and)h Fr(\000)118 1280 y Fz(b)s(oundary)38 b(conditions;)45 b(2)1056 1247 y Fw(3)1091 1223 y Fq(d)1128 1247 y Fn(j)p Fs(D)r Fn(j)1271 1280 y Fz(coun)m(ts)c(the)f(n)m(um)m(b)s(er)f(of)h(con)m(tours)g(with)g (giv)m(en)h(spatial)g(supp)s(ort.)118 1388 y(This)30 b(leads)g(to)1218 1498 y Fu(Z)1287 1454 y Fw(+)p Fs(;)p Fn(\000)1280 1527 y Fs(L;")1445 1498 y Ft(6)25 b Fu(e)1583 1461 y Fs(\014)s Fn(j)p Fw(\003)1695 1472 y Fq(L;"L)1831 1461 y Fn(jP)1910 1498 y Fu(e)1952 1461 y Fn(\000)p Fw(\()p Fs(\014)s(=)p Fw(2)p Fn(\000)p Fs(\013)p Fw(\)2)2309 1437 y Fl(\000)p Fq(d)2395 1461 y Fs(L)2443 1437 y Fq(d)p Fl(\000)p Fp(1)3429 1498 y Fz(\(5.16\))118 1638 y(Inequalities)31 b(\(5.15\))i(and)d(\(5.16\))i(imply)936 1794 y Fu(Z)1005 1750 y Fw(+)p Fs(;)p Fn(\000)998 1822 y Fs(L;")p 823 1850 427 4 v 823 1950 a Fu(Z)892 1906 y Fw(+)p Fs(;R)885 1979 y(L;")1049 1950 y Fu(Z)1118 1906 y Fn(\000)p Fs(;R)1111 1979 y(L;")1285 1871 y Ft(6)40 b Fz(exp)1550 1770 y Fo(n)1631 1871 y Fr(\000)20 b Fu(L)1784 1833 y Fs(d)p Fn(\000)p Fw(1)1915 1797 y Fo(\000)1956 1871 y Fz(2)2001 1833 y Fn(\000)p Fs(d)2097 1871 y Fz(\()2142 1809 y Fu(\014)p 2142 1850 57 4 v 2147 1933 a Fz(2)2229 1871 y Fr(\000)g Fu(\013)p Fz(\))h Fr(\000)2535 1809 y Fu(C)2600 1823 y Fw(1)p 2535 1850 105 4 v 2540 1933 a Fu(n)2595 1948 y Fs(d)2649 1871 y Fu(e)2691 1833 y Fn(\000)p Fs(\014)s(=)p Fw(2)2864 1797 y Fo(\001)2906 1770 y(o)118 2125 y Fz(Since)30 b Fu(n)410 2140 y Fs(d)476 2125 y Ft(>)25 b Fz(1)p Fu(=)662 2048 y Fr(p)p 738 2048 48 4 v 77 x Fu(d)q Fz(,)30 b(for)g Fu(\014)36 b Fz(large)31 b(enough)g(\(5.14\))h(holds.)118 2351 y Fj(Pr)-5 b(o)g(of)34 b(of)f(Pr)-5 b(op)g(osition)35 b(5.2.)218 2459 y Fz(The)g(con)m(v)m(exit)m(y)j(is)e(equiv)-5 b(alen)m(t)37 b(to)g(the)f(p)m(yramidal)f(inequalit)m(y)j(\(see)e(eg)h ([MMR)q(]\).)57 b(T)-8 b(o)36 b(an)m(y)g(col-)118 2567 y(lection)29 b(of)e(unit)g(v)m(ectors)h(\()n Fu(~)-43 b(n)1092 2581 y Fw(1)1131 2567 y Fu(;)15 b(:)g(:)g(:)i(;)c(~)-43 b(n)1388 2582 y Fs(d)p Fw(+1)1519 2567 y Fz(\),)28 b(one)f(asso)s (ciates)i(a)f(p)m(yramid)e(\001\()n Fu(~)-43 b(n)2779 2581 y Fw(1)2818 2567 y Fu(;)15 b(:)g(:)g(:)i(;)c(~)-43 b(n)3075 2582 y Fs(d)p Fw(+1)3205 2567 y Fz(\))28 b(with)f(faces)118 2675 y(\()p Fr(F)218 2689 y Fs(i)247 2675 y Fz(\))282 2689 y Fs(i)336 2675 y Fz(orthogonal)g(to)g(\()n Fu(~)-43 b(n)983 2689 y Fs(i)1011 2675 y Fz(\))1046 2689 y Fs(i)1074 2675 y Fz(.)40 b(Let)26 b Fr(jF)1387 2689 y Fs(i)1416 2675 y Fr(j)g Fz(b)s(e)f(the)h(area)g(of)g Fr(F)2094 2689 y Fs(i)2123 2675 y Fz(.)39 b(Then)24 b(the)i(p)m(yramidal)g (inequalit)m(y)h(means)118 2783 y(that)1317 2963 y Fr(jF)1407 2977 y Fw(1)1447 2963 y Fr(j)15 b Fu(\034)1527 2978 y Fs(\014)1575 2963 y Fz(\()n Fu(~)-43 b(n)1665 2977 y Fw(1)1704 2963 y Fz(\))26 b Ft(6)1878 2849 y Fs(d)p Fw(+1)1876 2877 y Fo(X)1884 3072 y Fs(i)p Fw(=2)2022 2963 y Fr(jF)2112 2977 y Fs(i)2141 2963 y Fr(j)15 b Fu(\034)2221 2978 y Fs(\014)2268 2963 y Fz(\()n Fu(~)-43 b(n)2358 2977 y Fs(i)2386 2963 y Fz(\))15 b Fu(:)968 b Fz(\(5.17\))118 3174 y(The)33 b(deriv)-5 b(ation)35 b(of)e(the)h(p)m(yramidal)g (inequalit)m(y)h(follo)m(ws)g(closely)g(the)f(appro)m(ximation)g(sc)m (heme)h(ex-)118 3282 y(plained)40 b(in)g(the)g(second)g(step)g(of)g (the)g(pro)s(of)f(of)i(Theorem)e(3.1.)71 b(F)-8 b(or)41 b(a)f(giv)m(en)h(\()p Fu(L;)15 b(")p Fz(\),)44 b(instead)c(of)118 3390 y(appro)m(ximating)33 b(the)e(surface)h(tension)f(in)h(the)f(slab) h(\003)2022 3404 y Fs(L;"L)2174 3390 y Fz(\()n Fu(~)-43 b(n)2264 3404 y Fw(1)2304 3390 y Fz(\))31 b(b)m(y)h(lo)s(calizing)h (the)f(in)m(terface)h(in)e(the)118 3498 y(smaller)f(slabs)f(\003)716 3512 y Fs(L)764 3524 y Fq(k)802 3512 y Fs(;")855 3524 y Fq(k)893 3512 y Fs(L)941 3524 y Fq(k)983 3498 y Fz(\()n Fu(~)-43 b(n)1073 3512 y Fw(1)1112 3498 y Fz(\),)30 b(the)g(in)m (terface)g(is)g(constrained)f(to)h(follo)m(w)h(a)e(more)h(complicated)h (p)s(eri-)118 3606 y(o)s(dic)g(pattern.)218 3714 y(More)j(precisely)-8 b(,)35 b(the)e(h)m(yp)s(erplan)f(orthogonal)j(to)51 b Fu(~)-62 b(n)2065 3728 y Fw(1)2137 3714 y Fz(and)33 b(going)h(through)f (0,)h(is)g(pa)m(v)m(ed)g(b)m(y)f(unit)118 3838 y(\()p Fu(d)27 b Fr(\000)e Fz(1\)-dimensional)40 b(cub)s(es)e(denoted)h(b)m(y) g(\()p Fu(C)1796 3805 y Fw(\()p Fs(`)p Fw(\))1884 3838 y Fz(\))1919 3853 y Fs(`)1952 3838 y Fz(.)66 b(F)-8 b(or)40 b(an)m(y)f Fu(`)p Fz(,)i(let)f Fr(F)2713 3790 y Fw(\()p Fs(`)p Fw(\))2704 3864 y(1)2840 3838 y Fz(b)s(e)e(a)i(translate)g(of)f Fr(F)3622 3852 y Fw(1)118 3955 y Fz(rescaled)34 b(appropriately)g(to)g (\014t)f(in)g(the)h(cub)s(e)f Fu(C)1824 3922 y Fw(\()p Fs(`)p Fw(\))1911 3955 y Fz(.)50 b(The)32 b(corresp)s(onding)h(p)m (yramid)g(is)g(denoted)g(b)m(y)118 4063 y(\001)194 4030 y Fw(\()p Fs(`)p Fw(\))282 4063 y Fz(.)40 b(In)30 b(this)g(w)m(a)m(y)-8 b(,)32 b(a)f(p)s(erio)s(dic)f(structure)g(is)g(created)1336 4228 y Fr(Q)c Fz(=)1532 4141 y Fo([)1568 4339 y Fs(`)1648 4154 y Fo(\000)1689 4228 y Fu(C)1761 4190 y Fw(\()p Fs(`)p Fw(\))1869 4228 y Fr([)20 b Fz(\001)2026 4190 y Fw(\()p Fs(`)p Fw(\))2113 4154 y Fo(\001)2175 4228 y Fr(n)h(F)2315 4180 y Fw(\()p Fs(`)p Fw(\))2306 4254 y(1)2418 4228 y Fu(:)118 4465 y Fz(The)h(in)m(terface)i(will)f(b)s(e)f(forced)h(to)g (cross)g(\003)1583 4479 y Fs(L;"L)1735 4465 y Fz(\()n Fu(~)-43 b(n)1825 4479 y Fw(1)1865 4465 y Fz(\))22 b(b)m(y)h(follo)m (wing)h(the)f(p)s(erio)s(dic)f(pattern)g Fu(N)10 b Fr(Q)p Fz(,)25 b(where)118 4576 y Fu(N)35 b Fz(=)25 b Fu(")364 4543 y Fw(2)404 4576 y Fu(L)p Fz(.)40 b(This)27 b(is)h(done)f(b)m(y)h (decomp)s(osing)g(eac)m(h)g(\015at)g(region)h(of)f Fu(N)10 b Fr(Q)28 b Fz(orthogonal)h(to)d Fu(~)-43 b(n)3231 4590 y Fs(i)3287 4576 y Fz(in)m(to)28 b(slabs)118 4684 y(\003)181 4698 y Fs(L)229 4710 y Fq(k)267 4698 y Fs(;")320 4710 y Fq(k)358 4698 y Fs(L)406 4710 y Fq(k)448 4684 y Fz(\()n Fu(~)-43 b(n)538 4698 y Fs(i)566 4684 y Fz(\),)38 b(with)d Fu(L)938 4699 y Fs(k)1015 4684 y Fr(\034)f Fu(N)10 b Fz(.)57 b(The)36 b(in)m(terface)h(is)f(allo)m(w)m(ed)i(to)e (\015uctuate)h(inside)e(eac)m(h)i(slab,)h(th)m(us)118 4792 y(an)30 b(appro)m(ximation)i(of)e(the)h(surface)f(tension)h(in)f (eac)m(h)i(directions)c Fu(~)-42 b(n)2512 4806 y Fs(i)2570 4792 y Fz(can)30 b(b)s(e)g(reco)m(v)m(ered.)43 b(Since)30 b(the)118 4900 y(p)s(ortion)e(of)h(the)f(in)m(terface)j(outside)d(the)h (slabs)f(is)h(small)g(wrt)f(the)g(surface)h(order,)f(its)h(con)m (tribution)g(is)118 5008 y(negligible)j(and)e(w)m(e)h(obtain)688 5182 y Fu(L)750 5149 y Fs(d)p Fn(\000)p Fw(1)p 688 5222 193 4 v 736 5305 a Fu(n)791 5320 y Fs(d)890 5243 y Fu(\034)930 5258 y Fs(\014)977 5243 y Fz(\()n Fu(~)-43 b(n)1067 5257 y Fw(1)1106 5243 y Fz(\))26 b Ft(6)1278 5157 y Fo(X)1329 5354 y Fs(`)1425 5088 y Fo(\()1508 5182 y Fr(j)p Fu(C)1605 5149 y Fw(\()p Fs(`)p Fw(\))1713 5182 y Fr(n)20 b(F)1852 5134 y Fw(\()p Fs(`)p 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1350 l 3044 1350 l 3043 1350 l 3038 1350 l 3024 1350 l 3000 1350 l 2981 1350 l 2964 1350 l 2950 1350 l 2939 1350 l 2931 1350 l 2925 1350 l 2919 1350 l 2911 1350 l 2900 1350 l 2886 1350 l 2869 1350 l 2850 1350 l 2826 1350 l 2812 1350 l 2807 1350 l 2806 1350 l 2803 1350 l 2792 1350 l 2775 1350 l 2758 1350 l 2746 1350 l 2740 1350 l 2738 1350 l 2735 1350 l 2729 1350 l 2717 1350 l 2700 1350 l 2683 1350 l 2671 1350 l 2665 1350 l 2663 1350 l 2660 1350 l 2654 1350 l 2642 1350 l 2625 1350 l 2608 1350 l 2596 1350 l 2590 1350 l 2588 1350 l 2585 1350 l 2579 1350 l 2567 1350 l 2550 1350 l 2533 1350 l 2521 1350 l 2515 1350 l 2513 1350 l 2510 1350 l 2504 1350 l 2492 1350 l 2475 1350 l 2458 1350 l 2446 1350 l 2440 1350 l 2438 1350 l 2435 1350 l 2429 1350 l 2417 1350 l 2400 1350 l 2383 1350 l 2371 1350 l 2365 1350 l 2363 1350 l 2360 1350 l 2354 1350 l 2342 1350 l 2325 1350 l 2307 1350 l 2295 1349 l 2287 1346 l 2281 1344 l 2276 1341 l 2270 1341 l 2260 1343 l 2250 1350 l 2243 1361 l 2242 1371 l 2245 1380 l 2250 1388 l 2255 1395 l 2258 1404 l 2257 1414 l 2250 1425 l 2240 1432 l 2230 1434 l 2224 1434 l 2219 1431 l 2213 1429 l 2205 1426 l 2193 1425 l 2175 1425 l 2157 1425 l 2145 1424 l 2137 1421 l 2131 1419 l 2126 1416 l 2120 1416 l 2110 1418 l 2100 1425 l 2093 1436 l 2092 1446 l 2095 1455 l 2100 1463 l 2105 1470 l 2108 1479 l 2107 1489 l 2100 1500 l 2090 1507 l 2080 1509 l 2074 1509 l 2069 1506 l 2063 1504 l 2055 1501 l 2043 1500 l 2025 1500 l 2008 1500 l 1996 1500 l 1990 1500 l 1988 1500 l 1985 1500 l 1979 1500 l 1967 1500 l 1950 1500 l 1933 1500 l 1921 1500 l 1915 1500 l 1913 1500 l 1910 1500 l 1904 1500 l 1892 1500 l 1875 1500 l 1858 1500 l 1846 1500 l 1840 1500 l 1838 1500 l 1835 1500 l 1829 1500 l 1817 1500 l 1800 1500 l 1783 1500 l 1771 1500 l 1765 1500 l 1763 1500 l 1760 1500 l 1754 1500 l 1742 1500 l 1725 1500 l 1707 1500 l 1695 1499 l 1687 1496 l 1681 1494 l 1676 1491 l 1670 1491 l 1660 1493 l 1650 1500 l 1644 1508 l 1640 1517 l 1639 1525 l 1638 1532 l 1638 1538 l 1638 1543 l 1639 1550 l 1640 1558 l 1644 1567 l 1650 1575 l 1661 1582 l 1671 1584 l 1680 1584 l 1688 1581 l 1695 1579 l 1704 1576 l 1714 1575 l 1725 1575 l 1727 1575 l 1728 1575 l 1730 1575 l 1731 1575 l 1732 1575 l 1733 1575 l 1734 1575 l 1735 1575 l 1736 1575 l 1737 1575 l 1736 1575 l 1735 1575 l 1734 1575 l 1733 1575 l 1732 1575 l 1731 1575 l 1730 1575 l 1728 1575 l 1727 1575 l 1725 1575 l 1715 1575 l 1705 1575 l 1699 1575 l 1694 1575 l 1688 1575 l 1680 1575 l 1668 1575 l cp gs col0 s gr % Polyline 7.500 slw [60] 0 sd n 6000 1200 m 12000 1200 l 12000 7200 l 6000 7200 l cp gs col0 s gr [] 0 sd $F2psEnd rs %%EndDocument endTexFig 385 4182 a Fr(jM)519 4196 y Fs(K)608 4182 y Fr(\000)20 b Fu(m)779 4144 y Fw(+)779 4211 y Fs(\014)837 4182 y Fr(j)26 b Ft(6)f Fu(\020)148 4371 y Fz(+)1684 3993 y Fu(K)1684 4725 y(K)1768 4692 y Fs(\013)2393 4064 y Fu(d)g Ft(>)g Fz(2)656 5146 y FC(Figure)j(4.)38 b FD(Coarse)27 b(grained)g(con\014guration)f(with)g(o)n(v)n(erlapping)g(+)f(go)r(o)r (d)i(blo)r(c)n(ks.)218 5372 y Fz(F)-8 b(or)39 b(large)h(mesoscopic)h(b) s(o)m(xes,)g(the)e(t)m(ypical)i(spin)d(con\014gurations)h(o)s(ccur)g (with)f(o)m(v)m(erwhelming)118 5480 y(probabilit)m(y)-8 b(.)p eop end %%Page: 30 30 TeXDict begin 30 29 bop 118 45 a Fw(30)1128 b(T.)23 b(BODINEA)n(U)g (AND)g(E.)g(PRESUTTI)118 366 y FE(Theorem)35 b(6.1.)42 b Fj(Then)33 b(for)g(any)h Fu(\020)d(>)25 b Fz(0)p Fj(,)33 b(the)g(fol)5 b(lowing)34 b(holds)g(uniformly)g(over)f Fu(N)658 539 y Fr(8f)p Fu(x)806 553 y Fw(1)845 539 y Fu(;)15 b(:)g(:)g(:)i(;)e(x)1099 554 y Fs(`)1132 539 y Fr(g)p Fu(;)109 b(\026)1366 554 y Fs(\014)s(;N)1511 438 y Fo(\020)1565 539 y Fu(u)1617 495 y Fs(\020)1617 568 y(K)1685 539 y Fz(\()p Fu(x)1772 553 y Fw(1)1812 539 y Fz(\))26 b(=)f(0)p Fu(;)15 b(:)g(:)g(:)i(;)e(u)2268 495 y Fs(\020)2268 568 y(K)2337 539 y Fz(\()p Fu(x)2424 554 y Fs(`)2457 539 y Fz(\))26 b(=)e(0)2658 438 y Fo(\021)2754 539 y Ft(6)2850 466 y Fo(\000)2891 539 y Fu(\032)2938 495 y Fs(\020)2938 568 y(K)3007 466 y Fo(\001)3048 488 y Fs(`)3096 539 y Fu(;)354 b Fz(\(6.1\))118 739 y Fj(wher)-5 b(e)34 b(the)f(p)-5 b(ar)g(ameter)35 b Fu(\032)1000 695 y Fs(\020)1000 768 y(K)1101 739 y Fj(vanishes)f(as)f Fu(K)39 b Fj(go)-5 b(es)33 b(to)g(in\014nity.)218 904 y Fz(Despite)38 b(the)g(fact)h(that)f(the)g(mesoscopic)h(phase)e(lab)s (els)h(are)g(not)f(indep)s(enden)m(t,)i(the)f(Theorem)118 1012 y(ab)s(o)m(v)m(e)32 b(ensures)d(that)i(the)f(o)s(ccurence)h(of)f (the)h(bad-blo)s(c)m(ks)f(is)g(dominated)h(b)m(y)f(a)g(Bernoulli)h (measure.)118 1119 y(F)-8 b(or)31 b(the)g(sak)m(e)g(of)g(completness,)g (the)g(pro)s(of)e(of)i(Theorem)f(6.1)h(is)g(recalled)g(in)f(the)h(App)s (endix.)218 1270 y(As)f(in)g(\(2.6\))r(,)h(the)f(macroscopic)i(coun)m (terpart)f(of)f(the)h(phase)f(lab)s(els)h(is)f(de\014ned)f(b)m(y)1183 1433 y Fu(u)1235 1389 y Fs(\020)1235 1462 y(N)s(;K)1383 1433 y Fz(\()p Fu(x)p Fz(\))d(=)f Fu(u)1679 1389 y Fs(\020)1679 1462 y(K)1747 1360 y Fo(\000)1789 1433 y Fu( )1851 1395 y Fn(\000)p Fw(1)1848 1462 y Fs(N)1945 1433 y Fz(\()p Fu(x)p Fz(\))2067 1360 y Fo(\001)2110 1433 y Fu(;)197 b(x)25 b Fr(2)2500 1410 y Fo(b)2495 1433 y Fx(T)15 b Fu(:)118 1614 y Fz(The)30 b(images)h(of)g Fr(B)770 1581 y Fw(\()p Fs(K)5 b Fw(\))923 1614 y Fz(b)s(o)m(xes)31 b(b)m(y)f Fu( )1359 1628 y Fs(N)1457 1614 y Fz(are)g(denoted)h(b)m(y) 2083 1591 y Fo(b)2078 1614 y Fx(B)2139 1628 y Fs(N)s(;K)2286 1614 y Fz(\()p Fu(x)p Fz(\).)218 1725 y(An)m(y)23 b(discrepancy)g(in)g (the)g Fx(L)1200 1692 y Fw(1)1239 1725 y Fz({norm)g(b)s(et)m(w)m(een)h (the)f(coarse)i(graining)e(and)g(the)g(lo)s(cal)i(order)d(param-)118 1841 y(eter)34 b(can)f(b)s(e)g(neglected)i(with)d(sup)s(erexp)s(onen)m (tial)h(probabilit)m(y)-8 b(.)50 b(By)33 b(construction,)i(for)e(an)m (y)h Fu(x)29 b Fr(2)3606 1817 y Fo(b)3601 1841 y Fx(T)118 1957 y Fz(either)e Fr(jM)506 1971 y Fs(N)s(;K)653 1957 y Fz(\()p Fu(x)p Fz(\))12 b Fr(\000)g Fu(u)922 1913 y Fs(\020)922 1985 y(N)s(;K)1070 1957 y Fz(\()p Fu(x)p Fz(\))p Fr(j)27 b Fz(is)f(smaller)h(than)f Fu(\020)33 b Fz(or)26 b(the)h(blo)s(c)m(k)2422 1933 y Fo(b)2417 1957 y Fx(B)2478 1971 y Fs(N)s(;K)2625 1957 y Fz(\()p Fu(x)p Fz(\))g(has)f(lab)s(el)h Fu(u)3200 1913 y Fs(\020)3200 1985 y(N)s(;K)3347 1957 y Fz(\()p Fu(x)p Fz(\))f(=)f(0.)118 2070 y(Using)k(the)g(domination)g(b)m(y)g(Bernoulli)h(p)s(ercolation,)g (the)f(follo)m(wing)i(holds.)39 b(Giv)m(en)30 b(an)m(y)f Fu(\016)g(>)c Fz(0,)30 b(one)118 2178 y(can)36 b(c)m(ho)s(ose)h(the)f (accuracy)h Fu(\020)42 b Fz(of)35 b(the)h(coarse)h(graining)f(and)f(a)h (scale)h Fu(K)2681 2192 y Fw(0)2721 2178 y Fz(\()p Fu(\016)n(;)15 b(\014)5 b Fz(\))38 b(suc)m(h)d(that)i(for)e(an)m(y)118 2286 y(mesoscopic)d Fu(K)g Ft(>)25 b Fu(K)866 2300 y Fw(0)839 2474 y Fz(lim)800 2534 y Fs(N)7 b Fn(!1)1138 2412 y Fz(1)p 1054 2453 214 4 v 1054 2536 a Fu(N)1137 2510 y Fs(d)p Fn(\000)p Fw(1)1293 2474 y Fz(log)18 b Fu(\026)1481 2489 y Fs(\014)s(;N)1625 2373 y Fo(\020)1680 2474 y Fr(kM)1834 2488 y Fs(N)s(;K)2002 2474 y Fr(\000)i Fu(u)2145 2430 y Fs(\020)2145 2503 y(N)s(;K)2292 2474 y 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10200 8260 l 10200 8296 l 10200 8326 l 10200 8351 l 10200 8370 l 10200 8384 l 10200 8393 l 10200 8398 l 10200 8400 l gs col7 0.90 shd ef gr gs col0 s gr % Polyline [60] 0 sd n 1185 5295 m 5100 3825 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 5685 3900 m 9510 4515 l gs col0 s gr [] 0 sd % Polyline gs clippath 3009 3937 m 2952 3953 l 2992 4099 l 2989 3976 l 3050 4083 l cp eoclip n 3150 4545 m 2985 3960 l gs col0 s gr gr % arrowhead n 3050 4083 m 2989 3976 l 2992 4099 l col0 s % Polyline gs clippath 1136 4813 m 1188 4782 l 1111 4652 l 1147 4771 l 1060 4683 l cp eoclip n 480 3645 m 1155 4785 l gs col0 s gr gr % arrowhead n 1060 4683 m 1147 4771 l 1111 4652 l col0 s % Polyline gs clippath 1501 5921 m 1460 5877 l 1350 5982 l 1458 5921 l 1392 6025 l cp eoclip n 435 6900 m 1470 5910 l gs col0 s gr gr % arrowhead n 1392 6025 m 1458 5921 l 1350 5982 l col0 s % Polyline gs clippath 9550 3241 m 9559 3182 l 9409 3161 l 9524 3208 l 9400 3220 l cp 6004 2668 m 5995 2727 l 6145 2748 l 6031 2702 l 6154 2689 l cp eoclip n 6015 2700 m 9540 3210 l gs col0 s gr gr % arrowhead n 6154 2689 m 6031 2702 l 6145 2748 l col0 s % arrowhead n 9400 3220 m 9524 3208 l 9409 3161 l col0 s % Polyline gs clippath 9657 4524 m 9716 4534 l 9743 4386 l 9693 4499 l 9684 4375 l cp 9857 3740 m 9798 3730 l 9771 3878 l 9822 3766 l 9830 3889 l cp eoclip n 9825 3750 m 9690 4515 l gs col0 s gr gr % arrowhead n 9830 3889 m 9822 3766 l 9771 3878 l col0 s % arrowhead n 9684 4375 m 9693 4499 l 9743 4386 l col0 s % Polyline 30.000 slw n 1121 4227 m 4531 3056 l 5111 4758 l 1708 5932 l 1121 4227 l cp gs col0 s gr % Polyline n 5910 3120 m 9481 3626 l 9226 5406 l 5665 4906 l 5910 3120 l cp gs col0 s gr % Polyline 0.000 slw n 825 7575 m 3225 7617 l 3204 8817 l 804 8775 l 825 7575 l cp gs col7 1.00 shd ef gr % Polyline n 906 7001 m 3189 7743 l 2817 8884 l 535 8141 l 906 7001 l cp gs col7 1.00 shd ef gr % Polyline n 3000 7575 m 5400 7575 l 5400 8775 l 3000 8775 l 3000 7575 l cp gs col7 1.00 shd ef gr % Polyline n 5325 7425 m 7725 7425 l 7725 8625 l 5325 8625 l 5325 7425 l cp gs col7 1.00 shd ef gr % Polyline n 9075 7350 m 11475 7350 l 11475 8550 l 9075 8550 l 9075 7350 l cp gs col7 1.00 shd ef gr % Polyline n 9075 7350 m 11475 7350 l 11475 8550 l 9075 8550 l 9075 7350 l cp gs col7 1.00 shd ef gr % Polyline n 9075 7350 m 11475 7350 l 11475 8550 l 9075 8550 l 9075 7350 l cp gs col7 1.00 shd ef gr % Polyline n 9075 7350 m 11475 7350 l 11475 8550 l 9075 8550 l 9075 7350 l cp gs col7 1.00 shd ef gr % Polyline n 9075 7350 m 11475 7350 l 11475 8550 l 9075 8550 l 9075 7350 l cp gs col7 1.00 shd ef gr % Polyline n 9075 7350 m 11475 7350 l 11475 8550 l 9075 8550 l 9075 7350 l cp gs col7 1.00 shd ef gr % Polyline n 9150 7200 m 11550 7200 l 11550 8400 l 9150 8400 l 9150 7200 l cp gs col7 1.00 shd ef gr % Polyline n 9150 7200 m 11550 7200 l 11550 8400 l 9150 8400 l 9150 7200 l cp gs col7 1.00 shd ef gr % Polyline n 8492 7569 m 10609 6441 l 11175 7500 l 9057 8628 l 8492 7569 l cp gs col7 1.00 shd ef gr % Polyline n 7588 7717 m 9971 7430 l 10117 8618 l 7734 8910 l 7588 7717 l cp gs col7 1.00 shd ef gr $F2psEnd rs %%EndDocument endTexFig 2274 2505 a FB(h)2639 2829 y Fp(1)p 2639 2843 31 4 v 2639 2885 a(2)2679 2859 y FB(\016)s(h)2983 3095 y FA(f)p FB(v)g FD(=)c(1)p FA(g)1802 3214 y(f)p FB(v)k FD(=)c FA(\000)p FD(1)p FA(g)1399 2812 y FB(~)-37 b(n)1446 2820 y Fq(i)1771 2722 y Fg(b)1755 2741 y FB(B)1817 2709 y Fq(i)824 2699 y Fg(b)810 2717 y FB(R)869 2686 y Fq(i;)p Fp(+)777 3289 y Fg(b)763 3308 y FB(R)822 3276 y Fq(i;)p Fl(\000)1117 3941 y FC(Figure)28 b(5.)38 b FD(Appro)n(ximation)26 b(b)n(y)e(parallelepip)r(eds.)218 4133 y Fz(According)j(to)g(\(6.2\),)i (the)d(lo)s(cal)i(a)m(v)m(eraged)g(magnetization)h(can)e(b)s(e)f (replaced)g(b)m(y)h(the)f(mesoscopic)118 4241 y(phase)31 b(lab)s(els.)42 b(Therefore)31 b(Prop)s(osition)g(6.2)h(is)f(equiv)-5 b(alen)m(t)32 b(to)g(the)f(follo)m(wing)h(statemen)m(t)h(:)41 b(for)31 b(an)m(y)118 4349 y Fu(\016)f Fz(p)s(ositiv)m(e,)e(there)e (exists)g Fu(K)1097 4363 y Fw(0)1162 4349 y Fz(=)f Fu(K)1335 4363 y Fw(0)1375 4349 y Fz(\()p Fu(\016)n(;)15 b(h)p Fz(\),)29 b Fu(\020)1669 4363 y Fw(0)1733 4349 y Fz(=)c Fu(\020)1869 4363 y Fw(0)1909 4349 y Fz(\()p Fu(\016)n(;)15 b(h)p Fz(\))28 b(suc)m(h)d(that)i(uniformly)d(in)i Fu(K)32 b Ft(>)25 b Fu(K)3323 4363 y Fw(0)3363 4349 y Fz(,)i Fu(\020)k Ft(6)25 b Fu(\020)3622 4363 y Fw(0)280 4588 y Fz(lim)15 b(sup)317 4666 y Fs(N)7 b Fn(!1)668 4527 y Fz(1)p 584 4567 214 4 v 584 4651 a Fu(N)667 4624 y Fs(d)p Fn(\000)p Fw(1)838 4588 y Fz(log)17 b Fu(\026)1025 4603 y Fs(\014)s(;N)1170 4433 y Fo( )1242 4588 y Fu(u)1294 4544 y Fs(\020)1294 4617 y(N)s(;K)1466 4588 y Fr(2)1595 4475 y Fs(`)1559 4502 y Fo(\\)1552 4697 y Fs(i)p Fw(=1)1697 4588 y Fr(V)7 b Fz(\()1812 4565 y Fo(b)1795 4588 y Fu(R)1865 4551 y Fs(i)1893 4588 y Fu(;)15 b(\016)s Fz(v)m(ol)r(\()2146 4565 y Fo(b)2128 4588 y Fu(R)2198 4551 y Fs(i)2227 4588 y Fz(\)\))2297 4433 y Fo(!)2410 4588 y Ft(6)45 b Fr(\000)20 b(W)2707 4603 y Fs(\014)2754 4588 y Fz(\()p Fu(v)s 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b(ation)26 b(of)f(this)h(b)s(oils)f (do)m(wn)g(to)118 5264 y(transfert)33 b(the)g(macroscopic)i Fx(L)1227 5231 y Fw(1)1266 5264 y Fz({b)s(ounds)d(in)m(to)i(a)f (microscopic)i(statemen)m(t)g(on)e(the)g(lo)s(calization)j(of)118 5372 y(an)25 b(in)m(terface)i(inside)e(eac)m(h)i Fu(R)1126 5339 y Fs(i)1125 5399 y(N)1192 5372 y Fz(.)39 b(This)24 b(is)i(a)f(k)m(ey)h(step)g(in)f(the)g Fx(L)2275 5339 y Fw(1)2314 5372 y Fz({approac)m(h)h(and)f(the)h(coarse)g(graining)118 5480 y(will)31 b(pla)m(y)g(a)g(ma)5 b(jor)30 b(role.)p eop end %%Page: 32 32 TeXDict begin 32 31 bop 118 45 a Fw(32)1128 b(T.)23 b(BODINEA)n(U)g (AND)g(E.)g(PRESUTTI)118 366 y Fj(Step)33 b(2)g(:)42 b(Minimal)33 b(se)-5 b(ction)33 b(ar)-5 b(gument.)p 118 399 1417 4 v 218 499 a Fz(The)31 b(microscopic)h(images)h(of)1319 476 y Fo(b)1301 499 y Fu(R)1371 466 y Fs(i;)p Fn(\006)1505 499 y Fz(in)e Fx(T)1673 513 y Fs(N)1771 499 y Fz(are)h(denoted)g Fu(R)2339 455 y Fs(i;)p Fn(\006)2338 528 y Fs(N)2472 499 y Fz(and)f(w)m(e)h(set)g Fu(R)2998 466 y Fs(i)2997 526 y(N)3091 499 y Fz(=)27 b Fu(R)3259 455 y Fs(i;)p Fw(+)3258 528 y Fs(N)3382 499 y Fr([)21 b Fu(R)3534 455 y Fs(i;)p Fn(\000)3533 528 y Fs(N)3636 499 y Fz(.)118 613 y(F)-8 b(or)30 b(simplicit)m(y)-8 b(,)31 b(w)m(e)e(will)g(only)g (pro)m(v)m(e)h(the)f(existence)h(of)f(a)h(+)e(barrier)g Fr(C)2621 580 y Fs(i;)p Fw(+)2753 613 y Fz(lying)h(in)g(the)g(upp)s(er) d(part)118 721 y(of)40 b Fu(R)301 688 y Fs(i)300 748 y(N)407 721 y Fz(and)f(refer)g(to)h([)p FE(?)q Fz(])g(for)f(a)h (complete)h(deriv)-5 b(ation.)69 b(W)-8 b(e)41 b(consider)f Fu(@)2771 688 y Fw(top)2877 721 y Fu(R)2947 688 y Fs(i)2946 748 y(N)3053 721 y Fz(the)g(face)g(of)g Fu(R)3595 688 y Fs(i)3594 748 y(N)118 844 y Fz(orthogonal)28 b(to)e(the)h(v)m(ector)e Fu(~)-43 b(n)1151 858 y Fs(i)1205 844 y Fz(and)26 b(con)m(tiguous)h(to) g Fu(R)2003 800 y Fs(i;)p Fw(+)2002 873 y Fs(N)2105 844 y Fz(.)39 b(Let)27 b Fu(R)2398 800 y Fs(i;)p Fw(top)2397 873 y Fs(N)2573 844 y Fz(b)s(e)f(the)g(set)h(of)f(sites)g(in)g Fu(R)3452 800 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y(bad)h(blo)r(c)n(ks)1802 3541 y FA(f)p FB(v)f FD(=)c FA(\000)p FD(1)p FA(g)2322 3895 y(f)p FB(v)j FD(=)d(1)p FA(g)2770 4297 y(B)2822 4262 y Fq(j)2850 4241 y Ff(\000)2820 4317 y Fq(i)2770 3187 y FA(B)2822 3151 y Fq(j)2850 3130 y Fe(+)2820 3207 y Fq(i)857 4108 y FB(R)916 4073 y Fq(i;)p Fp(+)915 4129 y Fq(N)1008 4060 y Fl(0)857 3399 y FB(R)916 3364 y Fq(i;)p Fl(\000)915 3420 y Fq(N)1009 3351 y Fl(0)1402 4788 y FC(Figure)27 b(6.)39 b FD(Minimal)27 b(sections.)218 5054 y Fz(F)-8 b(or)27 b(an)m(y)g(spin)f(con\014guration)g(suc)m(h)h (that)g Fu(u)1727 5010 y Fs(\020)1727 5083 y(N)s(;K)1900 5054 y Fz(b)s(elongs)g(to)2330 4986 y Fo(T)2406 5012 y Fs(`)2406 5081 y(i)p Fw(=1)2555 5054 y Fr(V)7 b Fz(\()2670 5031 y Fo(b)2653 5054 y Fu(R)2723 5021 y Fs(i)2751 5054 y Fu(;)15 b(\016)s Fz(v)m(ol)r(\()3004 5031 y Fo(b)2986 5054 y Fu(R)3056 5021 y Fs(i)3085 5054 y Fz(\)\),)28 b(the)f(n)m(um)m(b)s(er)118 5167 y(of)k Fj(b)-5 b(ad)31 b Fz(b)s(o)m(xes)g(in)f(a)g(minimal)h(section)g(is)g(b)s(ounded)d(b)m (y)757 5427 y Fu(n)812 5389 y Fw(+)812 5454 y Fs(i)895 5427 y Ft(6)d Fu(\016)s Fz(v)m(ol)r(\()1204 5404 y Fo(b)1186 5427 y Fu(R)1256 5390 y Fs(i)1285 5427 y Fz(\))1330 5366 y(10)p Fu(c)1459 5381 y Fs(d)p 1330 5406 171 4 v 1372 5490 a Fu(\016)s(s)1526 5299 y Fo(\022)1603 5366 y Fu(N)p 1603 5406 84 4 v 1603 5490 a(K)1696 5299 y Fo(\023)1763 5322 y Fs(d)p Fn(\000)p Fw(1)1934 5427 y Ft(6)g Fz(10)p Fu(c)2159 5442 y Fs(d)2201 5427 y Fu(\016)s(s)2287 5390 y Fs(d)p Fn(\000)p Fw(1)2433 5299 y Fo(\022)2510 5366 y Fu(N)p 2510 5406 V 2510 5490 a(K)2604 5299 y Fo(\023)2671 5322 y Fs(d)p Fn(\000)p Fw(1)2832 5427 y Fu(:)p eop end %%Page: 33 33 TeXDict begin 33 32 bop 1137 45 a Fw(SURF)-8 b(A)n(CE)31 b(TENSION)g(AND)f(WULFF)h(THEOR)-6 b(Y)948 b(33)118 372 y Fz(As)249 304 y Fo(P)345 330 y Fs(`)345 399 y(i)p Fw(=1)478 372 y Fr(j)522 349 y Fo(b)503 372 y Fu(B)577 339 y Fs(i)605 372 y Fr(j)26 b Fz(=)e Fu(`s)832 339 y Fs(d)p Fn(\000)p Fw(1)989 372 y Fz(can)j(b)s(e)f(con)m(trolled)i(in)e(terms)h(of)f(the)h (p)s(erimeter)f(of)h Fu(@)2855 339 y Fn(\003)2895 372 y Fu(v)s Fz(,)h(the)e(total)j(n)m(um)m(b)s(er)118 480 y(of)i Fj(b)-5 b(ad)31 b Fz(b)s(o)m(xes)g(is)f(b)s(ounded)e(b)m(y)1324 631 y Fs(`)1272 659 y Fo(X)1281 854 y Fs(i)p Fw(=1)1419 745 y Fu(n)1474 707 y Fw(+)1474 772 y Fs(i)1558 745 y Ft(6)d Fu(\016)19 b(C)7 b Fz(\()p Fu(v)s Fz(\))1917 617 y Fo(\022)1995 684 y Fu(N)p 1994 724 84 4 v 1994 807 a(K)2088 617 y Fo(\023)2155 640 y Fs(d)p Fn(\000)p Fw(1)2316 745 y Fu(:)1134 b Fz(\(6.4\))218 1075 y(F)-8 b(rom)39 b(the)g(v)m(ery)g(construction)g(of)g(the)f(coarse)i(graining,)h(the)e (+)f(spin)g(surfaces)h(asso)s(ciated)h(to)118 1191 y(o)m(v)m(erlapping) f(b)s(o)m(xes)f(with)f Fu(u)1137 1147 y Fs(\020)1137 1220 y(N)s(;K)1322 1191 y Fz(lab)s(els)h(equal)g(to)g(1)g(are)g (connected.)64 b(As)38 b(eac)m(h)g(minimal)g(section)118 1304 y(con)m(tains)27 b(mainly)e(+)g(go)s(o)s(d)h(blo)s(c)m(ks,)h (there)e(exist)h(almost)h(a)f(+)f(barrier)g(in)g(eac)m(h)h(minimal)g (section.)40 b(By)118 1412 y(mo)s(difying)30 b(the)g(spin)g (con\014gurations)g Fu(\033)k Fz(on)c(the)h Fj(b)-5 b(ad)31 b Fz(b)s(o)m(xes,)g(w)m(e)g(will)g(complete)g(these)g(+)f(barriers.)218 1520 y(More)d(precisely)-8 b(,)28 b(w)m(e)e(asso)s(ciate)j(to)d(an)m(y) h(con\014guration)f Fu(\033)k Fz(the)c(con\014guration)32 b(\026)-50 b Fu(\033)29 b Fz(with)d(spins)f(equal)118 1628 y(to)g(+)f(on)g(the)g(b)s(oundary)e(of)i(eac)m(h)h Fj(b)-5 b(ad)25 b Fz(b)s(o)m(x)f(in)g(the)g(minimal)g(section)i Fr(B)2566 1595 y Fs(i)2563 1660 y(j)2596 1641 y Fp(+)2674 1628 y Fz(and)d(equal)i(to)f Fu(\033)k Fz(otherwise.)118 1745 y(The)i(cost)h(of)g(this)f(surgical)h(pro)s(cedure)e(can)i(b)s(e)f (estimated)i(as)e(follo)m(ws.)118 2010 y Fu(\026)173 2025 y Fs(\014)s(;N)318 1854 y Fo( )390 2010 y Fu(u)442 1966 y Fs(\020)442 2038 y(N)s(;K)615 2010 y Fr(2)743 1896 y Fs(`)707 1923 y Fo(\\)700 2119 y Fs(i)p Fw(=1)845 2010 y Fr(V)7 b Fz(\()961 1987 y Fo(b)943 2010 y Fu(R)1013 1972 y Fs(i)1041 2010 y Fu(;)15 b(\016)s Fz(v)m(ol)s(\()1294 1987 y Fo(b)1277 2010 y Fu(R)1347 1972 y Fs(i)1375 2010 y Fz(\)\))1445 1854 y Fo(!)1558 2010 y Ft(6)1740 1923 y Fo(X)1669 2125 y Fw(\()p Fs(i)1720 2134 y Fp(1)1755 2125 y Fs(;:::)o(;i)1878 2137 y Fq(k)1915 2125 y Fw(\))2049 1923 y Fo(X)1973 2125 y Fw(\()p Fs(j)2029 2134 y Fp(1)2064 2125 y Fs(;:::)n(;j)2191 2137 y Fq(k)2228 2125 y Fw(\))2391 1923 y Fo(X)2286 2137 y Fw(\()p Fs(n)2356 2107 y Fp(+)2356 2158 y(1)2407 2137 y Fs(;:::)o(;n)2549 2107 y Fp(+)2549 2160 y Fq(k)2599 2137 y Fw(\))2667 2010 y Fu(\026)2722 2025 y Fs(\014)s(;N)2851 1936 y Fo(\000)2893 2010 y Fr(f)p Fu(n)2993 1972 y Fw(+)2993 2036 y(1)3052 2010 y Fu(;)g(:)g(:)g(:)i(;)e (n)3309 1972 y Fw(+)3309 2039 y Fs(k)3368 2010 y Fr(g)3413 1936 y Fo(\001)3470 2010 y Fu(:)-20 b Fz(\(6.5\))118 2325 y(The)43 b(r.h.s.)81 b(tak)m(es)45 b(in)m(to)g(accoun)m(t)g(the)f (fact)h(that)f(in)g(the)g(domains)f Fu(R)2695 2292 y Fs(i)2719 2301 y Fp(1)2758 2325 y Fu(;)15 b(:)g(:)g(:)h(;)f(R)3029 2292 y Fs(i)3053 2304 y Fq(k)3096 2325 y Fz(,)47 b(the)d(minimal)118 2433 y(sections)28 b(are)g(at)g(heigh)m(ts)g Fu(j)1056 2447 y Fw(1)1096 2433 y Fu(;)15 b(:)g(:)g(:)h(;)f(j)1334 2448 y Fs(k)1403 2433 y Fr(2)25 b Fz([)1524 2397 y Fw(9)p Fs(\016)r(sN)p 1524 2412 165 4 v 1539 2464 a Fw(10)p Fs(K)1699 2433 y Fu(;)1749 2397 y Fs(\016)r(sN)p 1749 2412 130 4 v 1782 2464 a(K)1889 2433 y Fz(])i(and)f(con)m(tain)j Fu(n)2487 2394 y Fw(+)2487 2459 y(1)2546 2433 y Fu(;)15 b(:)g(:)g(:)h(;)f(n)2802 2394 y Fw(+)2802 2462 y Fs(k)2888 2433 y Fj(b)-5 b(ad)29 b Fz(b)s(o)m(xes)e(suc)m(h)g(that)118 2541 y(\(6.4\))i(holds.)39 b(Once)27 b(the)h(lo)s(cation)g(of)g(the)f Fj(b)-5 b(ad)28 b Fz(b)s(o)m(xes)g(is)f(\014xed,)g(the)h(n)m(um)m(b)s (er)d(of)j(spin)e(\015ips)g(to)i(mo)s(dify)118 2673 y Fu(\033)37 b Fz(in)m(to)i(\026)-50 b Fu(\033)37 b Fz(is)c(at)i(most)f Fu(C)7 b Fz(\()p Fu(v)s Fz(\))1122 2599 y Fo(\000)1174 2637 y Fs(N)p 1174 2652 65 4 v 1174 2704 a(K)1248 2599 y Fo(\001)1290 2618 y Fs(d)p Fn(\000)p Fw(1)1451 2673 y Fu(K)1535 2640 y Fs(d)p Fn(\000)p Fw(1)1665 2673 y Fz(.)50 b(By)34 b(construction)39 b(\026)-50 b Fu(\033)37 b Fz(b)s(elongs)c(to)h(the)g(set)g Fr(A)3320 2687 y Fw(1)3393 2673 y Fz(of)f(spin)118 2790 y(con\014gurations)e(whic)m(h)f(con)m (tain)i(a)e(+)g(barrier)g(in)g(the)h(upp)s(er)d(part)i(of)h(eac)m(h)g (domain)g Fu(R)3183 2757 y Fs(i)3182 2817 y(N)292 3064 y Fu(\026)347 3079 y Fs(\014)s(;N)476 2990 y Fo(\000)518 3064 y Fr(f)p Fu(n)618 3026 y Fw(+)618 3090 y(1)677 3064 y Fu(;)15 b(:)g(:)g(:)i(;)e(n)934 3026 y Fw(+)934 3093 y Fs(k)993 3064 y Fr(g)1038 2990 y Fo(\001)1105 3064 y Ft(6)1265 2950 y Fs(k)1226 2978 y Fo(Y)1216 3172 y Fs(\013)p Fw(=1)1367 2936 y Fo(\022)1475 3011 y Fz(\()p Fu(sN)r(=K)7 b Fz(\))1792 2978 y Fs(d)p Fn(\000)p Fw(1)1643 3118 y Fu(n)1698 3085 y Fw(+)1698 3141 y Fs(\013)1965 2936 y Fo(\023)2047 3064 y Fz(exp)2201 2990 y Fo(\000)2243 3064 y Fu(\016)19 b(C)2367 3078 y Fw(2)2406 3064 y Fz(\()p Fu(v)s(;)c(\014)5 b Fz(\))p Fu(N)2702 3026 y Fs(d)p Fn(\000)p Fw(1)2835 2990 y Fo(\001)2901 3064 y Fu(\026)2956 3079 y Fs(\014)s(;N)3086 2990 y Fo(\000)3128 3064 y Fr(A)3201 3078 y Fw(1)3240 2990 y Fo(\001)3297 3064 y Fu(;)118 3339 y Fz(where)26 b(\()p Fu(sN)r(=K)7 b Fz(\))694 3306 y Fs(d)p Fn(\000)p Fw(1)851 3339 y Fz(refers)26 b(to)h(the)f(total)i(n) m(um)m(b)s(er)d(of)h(blo)s(c)m(ks)h(in)f(eac)m(h)h(minimal)f(sections.) 41 b(Summing)118 3446 y(o)m(v)m(er)32 b(all)f(the)g(con\014gurations)f (and)g(using)h(\(6.4\))h(again,)g(w)m(e)f(obtain)577 3557 y Fo(X)472 3770 y Fw(\()p Fs(n)542 3740 y Fp(+)542 3791 y(1)593 3770 y Fs(;:::)o(;n)735 3740 y Fp(+)735 3794 y Fq(k)785 3770 y Fw(\))827 3643 y Fu(\026)882 3658 y Fs(\014)s(;N)1012 3570 y Fo(\000)1054 3643 y Fr(f)p Fu(n)1154 3605 y Fw(+)1154 3669 y(1)1213 3643 y Fu(;)15 b(:)g(:)g(:)h(;)f(n)1469 3605 y Fw(+)1469 3672 y Fs(k)1528 3643 y Fr(g)1573 3570 y Fo(\001)1641 3643 y Ft(6)40 b Fz(exp)1906 3570 y Fo(\000)1948 3643 y Fu(o)p Fz(\()p Fu(\016)s Fz(\))15 b Fu(C)2185 3657 y Fw(3)2226 3643 y Fz(\()p Fu(v)s(;)g(\014)5 b Fz(\))p Fu(N)2522 3606 y Fs(d)p Fn(\000)p Fw(1)2654 3570 y Fo(\001)2721 3643 y Fu(\026)2776 3658 y Fs(\014)s(;N)2906 3570 y Fo(\000)2947 3643 y Fr(A)3020 3657 y Fw(1)3059 3570 y Fo(\001)3116 3643 y Fu(:)334 b Fz(\(6.6\))118 3949 y(Finally)31 b(replacing)i (\(6.6\))f(in)e(\(6.5\))r(,)h(w)m(e)f(get)155 4219 y Fu(\026)210 4234 y Fs(\014)s(;N)354 4064 y Fo( )426 4219 y Fu(u)478 4175 y Fs(\020)478 4248 y(N)s(;K)651 4219 y Fr(2)779 4106 y Fs(`)743 4133 y Fo(\\)737 4328 y Fs(i)p Fw(=1)881 4219 y Fr(V)7 b Fz(\()997 4196 y Fo(b)979 4219 y Fu(R)1049 4182 y Fs(i)1078 4219 y Fu(;)15 b(\016)s Fz(v)m(ol)r(\()1331 4196 y Fo(b)1313 4219 y Fu(R)1383 4182 y Fs(i)1411 4219 y Fz(\)\))1481 4064 y Fo(!)1594 4219 y Ft(6)25 b Fz(2)1735 4182 y Fs(`)1784 4091 y Fo(\022)1861 4158 y Fu(N)p 1861 4199 84 4 v 1861 4282 a(K)1954 4091 y Fo(\023)2021 4114 y Fs(`)2069 4219 y Fz(exp)2223 4146 y Fo(\000)2265 4219 y Fu(o)p Fz(\()p Fu(\016)s Fz(\))15 b Fu(C)2502 4233 y Fw(3)2543 4219 y Fz(\()p Fu(v)s(;)g(\014)5 b Fz(\))p Fu(N)2839 4182 y Fs(d)p Fn(\000)p Fw(1)2972 4146 y Fo(\001)3039 4219 y Fu(\026)3094 4234 y Fs(\014)s(;N)3223 4146 y Fo(\000)3265 4219 y Fr(A)3338 4233 y Fw(1)3377 4146 y Fo(\001)3434 4219 y Fu(:)16 b Fz(\(6.7\))218 4479 y(Rep)s(eating)25 b(the)g(same)h(argumen)m(t,)g(w)m(e)g(can)f(consider) g(instead)g(of)g Fr(A)2564 4493 y Fw(1)2628 4479 y Fz(an)g(ev)m(en)m(t) h Fr(A)f Fz(whic)m(h)f(con)m(tains)118 4587 y(at)32 b(least)g(4)g (barriers)e(in)h(eac)m(h)h Fu(R)1237 4554 y Fs(i)1236 4614 y(N)1303 4587 y Fz(.)43 b(F)-8 b(or)31 b(an)m(y)h(spin)e (con\014guration)h(in)g Fr(A)p Fz(,)g(w)m(e)g(de\014ne)g(the)g(set)h (of)f(sites)118 4711 y Fr(C)171 4678 y Fs(i;)p Fw(+)307 4711 y Fz(as)i(the)f(supp)s(ort)f(of)i(the)g(+)f(barrier)h(in)f Fu(R)1766 4666 y Fs(i;)p Fw(+)1765 4739 y Fs(N)1901 4711 y Fz(whic)m(h)g(is)h(the)g(closest)h(to)g Fu(@)2874 4678 y Fw(top)2980 4711 y Fu(R)3050 4678 y Fs(i)3049 4737 y(N)3116 4711 y Fz(.)48 b(In)32 b(the)h(same)118 4824 y(w)m(a)m(y)-8 b(,)36 b Fr(C)377 4791 y Fs(i;)p Fn(\000)513 4824 y Fz(is)d(the)h(lo)s(cation)g(of)g(the)f Fr(\000)g Fz(barrier)g(in)g(the)g(lo)m(w)m(er)i(part)e(of)h Fu(R)2673 4791 y Fs(i)2672 4851 y(N)2772 4824 y Fz(whic)m(h)f(is)g(the)h(closest) h(to)118 4936 y(\()p Fu(R)223 4903 y Fs(i)222 4963 y(N)290 4936 y Fz(\))325 4903 y Fs(c)360 4936 y Fz(.)58 b(By)37 b(analogy)h(with)e(the)g(notation)i(of)e(Section)h(3,)i(the)d(set)h(of) g(spin)e(con\014gurations)i(whic)m(h)118 5049 y(con)m(tain)32 b(a)f(+)f(and)f(a)i Fr(\000)f Fz(barrier)g(in)g(the)g(domain)h(\003\()p Fr(C)2005 5016 y Fs(i;)p Fw(+)2108 5049 y Fu(;)15 b Fr(C)2201 5016 y Fs(i;)p Fn(\000)2304 5049 y Fz(\))31 b(is)f(denoted)h(b)m(y)f Fr(S)2993 5016 y Fs(i)3046 5049 y Fz(=)25 b(\()p Fr(S)3239 5016 y Fs(i;)p Fw(+)3342 5049 y Fu(;)15 b Fr(S)3444 5016 y Fs(i;)p Fn(\000)3547 5049 y Fz(\).)118 5264 y Fj(Step)33 b(3)g(:)42 b(Surfac)-5 b(e)33 b(tension)g(estimates.)p 118 5297 1389 4 v 218 5372 a Fz(As)h(a)h(consequence)g(of)g(the)g (previous)f(step,)h(for)g(an)m(y)f(spin)g(con\014guration)h(in)f Fr(A)p Fz(,)h(there)g(exists)g(a)118 5480 y(microscopic)40 b(in)m(terface)g(lo)s(calized)h(in)d(eac)m(h)i(cub)s(e)d Fu(R)1993 5447 y Fs(i)1992 5507 y(N)2060 5480 y Fz(.)65 b(Th)m(us)38 b(w)m(e)h(are)g(no)m(w)f(in)g(a)h(go)s(o)s(d)g(shap)s(e)f (to)p eop end %%Page: 34 34 TeXDict begin 34 33 bop 118 45 a Fw(34)1128 b(T.)23 b(BODINEA)n(U)g (AND)g(E.)g(PRESUTTI)118 366 y Fz(c)m(hec)m(k)32 b(that)556 594 y(lim)15 b(sup)593 672 y Fs(N)7 b Fn(!1)969 532 y Fz(1)p 885 573 214 4 v 885 656 a Fu(N)968 630 y Fs(d)p Fn(\000)p Fw(1)1123 594 y Fz(log)18 b Fu(\026)1311 609 y Fs(\014)s(;N)1441 594 y Fz(\()p Fr(A)p Fz(\))25 b Ft(6)45 b Fr(\000)1867 480 y Fs(`)1816 507 y Fo(X)1825 703 y Fs(i)p Fw(=1)1963 470 y Fo(Z)2028 659 y Fk(b)2013 676 y Fs(B)2066 686 y Fq(i)2112 594 y Fu(\034)2152 609 y Fs(\014)2199 594 y Fz(\()n Fu(~)-43 b(n)2289 608 y Fs(i)2317 594 y Fz(\))15 b Fu(d)p Fr(H)2491 608 y Fs(x)2556 594 y Fz(+)20 b Fu(C)7 b Fz(\()p Fu(\014)e(;)15 b(v)s(;)g(\016)s Fz(\))g Fu(;)420 b Fz(\(6.8\))118 838 y(where)33 b Fu(C)7 b Fz(\()p Fu(\014)e(;)15 b(v)s(;)g(\016)s Fz(\))36 b(v)-5 b(anishes)34 b(as)g Fu(\016)j Fz(tends)d(to)g(0.)51 b(Com)m(bining)34 b(the)g(previous)f(inequalit)m(y)i(with)e(\(6.7\))r(,)118 946 y(w)m(e)e(deduce)f(\(6.3\))r(.)41 b(W)-8 b(e)31 b(no)m(w)f(pro)s (ceed)h(in)f(deriving)h(\(6.8\))r(.)218 1104 y(W)-8 b(e)43 b(\014rst)f(pin)g(the)g(in)m(terfaces)i(on)e(the)h(sides)f(of)h(eac)m (h)g Fu(R)2279 1071 y Fs(i)2278 1131 y(N)2388 1104 y Fz(b)m(y)f(imp)s(osing)g(that)h(the)g(b)s(o)m(xes)f(on)118 1228 y(the)35 b(b)s(oundary)e(of)i(eac)m(h)h Fu(R)1080 1183 y Fs(i;)p Fw(+)1079 1256 y Fs(N)1218 1228 y Fz(\(resp)e Fu(R)1520 1183 y Fs(i;)p Fn(\000)1519 1256 y Fs(N)1623 1228 y Fz(\))h(parallel)h(to)d Fu(~)-43 b(n)2191 1242 y Fs(i)2254 1228 y Fz(ha)m(v)m(e)36 b Fu(\021)j Fz(lab)s(els)c(equal)g (to)h(1)f(\(resp)g Fr(\000)p Fz(1\).)118 1344 y(Since)30 b(the)g(heigh)m(t)h(of)f Fu(R)959 1311 y Fs(i)958 1370 y(N)1056 1344 y Fz(is)g Fu(\016)s(s)p Fz(,)g(this)g(pro)s(cedure)f (requires)h(to)g(mo)s(dify)f(at)i(most)g Fu(\016)s(s)3064 1311 y Fs(d)p Fn(\000)p Fw(1)3195 1344 y Fu(N)3278 1311 y Fs(d)p Fn(\000)p Fw(1)3438 1344 y Fz(spins.)118 1451 y(Therefore)38 b(this)g(has)g(no)g(further)f(impact)h(on)g(the)g(ev)-5 b(aluation)40 b(of)e(the)h(statistical)h(w)m(eigh)m(ts)g(of)e(the)118 1559 y(con\014gurations)31 b(b)s(ecause)f(the)h(cost)g(of)g(\015ipping) e(these)i(spins)e(is)h(b)s(ounded)f(b)m(y)h(exp\()p Fu(\016)s(C)7 b Fz(\()p Fu(v)s Fz(\))15 b Fu(N)3371 1526 y Fs(d)p Fn(\000)p Fw(1)3503 1559 y Fz(\).)218 1667 y(In)22 b(this)g(w)m(a)m(y)-8 b(,)26 b(the)d(domain)f Fx(T)1214 1681 y Fs(N)1304 1667 y Fz(is)g(partionned)h(in)m(to)g(the)g(domains)f(\003\()p Fr(C)2667 1634 y Fs(i;)p Fw(+)2771 1667 y Fu(;)15 b Fr(C)2864 1634 y Fs(i;)p Fn(\000)2967 1667 y Fz(\))23 b(and)f(a)h(remainder)118 1775 y(whic)m(h)30 b(will)h(b)s(e)f(denoted)g(b)m(y)g(\001.)867 2110 y Fu(\026)922 2125 y Fs(\014)s(;N)1052 2110 y Fz(\()p Fr(A)p Fz(\))25 b(=)1399 2049 y(1)p 1326 2089 192 4 v 1326 2173 a Fu(Z)1388 2188 y Fs(\014)s(;N)1648 2024 y Fo(X)1543 2227 y Fw(\()p Fn(C)1611 2209 y Fq(i;)p Fp(+)1703 2227 y Fs(;)p Fn(C)1764 2209 y Fq(i;)p Fl(\000)1857 2227 y Fw(\))1900 2110 y Fu(Z)1969 2073 y Fs(!)1962 2133 y Fw(\001)2099 1997 y Fs(`)2055 2024 y Fo(Y)2056 2219 y Fs(i)p Fw(=1)2186 2110 y Fu(Z)2255 2066 y Fn(C)2296 2043 y Fq(i;)p Fp(+)2388 2066 y Fs(;)p Fn(C)2449 2043 y Fq(i;)p Fl(\000)2248 2140 y Fs(N)s(;\016)r(N)2545 2110 y Fz(\()p Fr(S)2642 2073 y Fs(i)2671 2110 y Fz(\))15 b Fu(;)118 2388 y Fz(where)20 b(the)g(b)s(oundary)f(conditions)h Fu(!)j Fz(are)e(imp)s(osed)f(b)m(y)g(the)g(v)-5 b(alues)21 b(of)f(the)h(spins)e(outside)h Fr([)3255 2402 y Fs(i)3283 2388 y Fz(\003\()p Fr(C)3434 2355 y Fs(i;)p Fw(+)3537 2388 y Fu(;)15 b Fr(C)3630 2355 y Fs(i;)p Fn(\000)3734 2388 y Fz(\))218 2496 y(In)m(tro)s(ducing)29 b(b)m(y)i(force)g(the)f (partition)h(functions)f(with)g(the)h(p)s(erfect)f(w)m(alls)h(w)m(e)g (get)460 2751 y Fu(\026)515 2766 y Fs(\014)s(;N)644 2751 y Fz(\()p Fr(A)p 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Fz(are)h(the)f(coun)m(terparts)h(of)f(\003)3303 872 y Fs(N)s(;\016)r(N)3483 857 y Fz(\()n Fu(~)-43 b(n)3573 871 y Fs(i)3601 857 y Fz(\).)118 981 y(Let)30 b Fr(A)353 948 y Fs(i;)p Fw(+)485 981 y Fz(b)s(e)f(the)h(ev)m(en)m(t)i(that)e (there)g(are)g(t)m(w)m(o)h(+)e(barriers)g(in)h Fu(R)2364 937 y Fs(i;)p Fw(+)2363 1009 y Fs(N)2496 981 y Fz(and)f Fr(A)2745 948 y Fs(i;)p Fn(\000)2877 981 y Fz(the)h(analogous)h(ev)m (en)m(t)118 1106 y(with)38 b(t)m(w)m(o)h Fr(\000)f Fz(barriers)f(in)h Fu(R)1147 1062 y Fs(i;)p Fn(\000)1146 1134 y Fs(N)1249 1106 y Fz(.)63 b(The)38 b Fr(\006)g Fz(barrier)f(in)h Fu(R)2131 1062 y Fs(i;)p Fn(\006)2130 1134 y Fs(N)2271 1106 y Fz(whic)m(h)g(is)g(the)g(closest)h(from)f(\()p Fu(R)3426 1073 y Fs(i)3425 1133 y(N)3492 1106 y Fz(\))3527 1073 y Fs(c)3600 1106 y Fz(is)118 1231 y(denoted)29 b(b)m(y)f Fr(C)637 1198 y Fs(i;)p Fn(\006)740 1231 y Fz(.)40 b(W)-8 b(e)29 b(set)g Fr(A)c Fz(=)1293 1163 y Fo(T)1369 1189 y Fs(`)1369 1258 y(i)p Fw(=1)1502 1231 y Fr(A)1575 1198 y Fs(i;)p Fw(+)1693 1231 y Fr(\\)16 b(A)1843 1198 y Fs(i;)p Fn(\000)1945 1231 y Fz(.)40 b(Let)29 b(us)f(also)h(de\014ne)f Fr(D)2800 1198 y Fs(i;)p Fw(+)2930 1231 y Fz(\(resp)g Fr(D)3229 1198 y Fs(i;)p Fn(\000)3332 1231 y Fz(\))g(the)h(set)118 1354 y(of)i(spin)e(con\014gurations)i(suc)m(h)f(that)h(the)g Fu(\021)s Fz(-lab)s(els)h(are)f(equal)g(to)g(1)g(\(resp)f Fr(\000)p Fz(1\))h(on)f(the)h(sides)f(of)h Fu(R)3559 1310 y Fs(i;)p Fw(+)3558 1383 y Fs(N)118 1479 y Fz(\(resp)d Fu(R)414 1435 y Fs(i;)p Fn(\000)413 1508 y Fs(N)517 1479 y Fz(\))g(parallel)i(to)d Fu(~)-43 b(n)1066 1493 y Fs(i)1094 1479 y Fz(.)40 b(In)28 b(order)g(to)h(construct)g(a)g(closed)g(con)m (tour)h(of)e(spins)g(surrounding)e Fu(V)3569 1493 y Fs(N)3636 1479 y Fz(,)118 1593 y(w)m(e)j(de\014ne)f Fr(D)k Fz(as)d(the)f(set)i (of)f(con\014gurations)g(in)f Fr(D)1882 1560 y Fs(i;)p Fw(+)2013 1593 y Fz(and)g Fr(D)2261 1560 y Fs(i;)p Fn(\000)2392 1593 y Fz(suc)m(h)g(that)i(the)f(blo)s(c)m(ks)g(on)f(one)h(side)118 1701 y(of)i Fu(U)294 1668 y Fs(\016)284 1728 y(N)381 1701 y Fz(ha)m(v)m(e)h Fu(\021)s Fz(-lab)s(els)f Fr(\000)f Fz(and)g(+)g(in)g(the)h(other)f(side.)218 1917 y(An)m(y)k(spin)e (con\014guration)i(in)g Fr(A)22 b(\\)g(D)36 b Fz(con)m(tains)f(a)f (microscopic)h(in)m(terface)h(whic)m(h)d(decouples)h Fu(V)3594 1931 y Fs(N)118 2025 y Fz(from)c(its)h(complemen)m(t.)42 b(One)30 b(has)456 2218 y Fu(\026)511 2233 y Fs(\014)s(;N)656 2117 y Fo(\020)710 2218 y Fr(k)p Fu(u)807 2174 y Fs(\020)807 2246 y(N)s(;K)975 2218 y Fr(\000)20 b Fz(1)-25 b(I)1128 2234 y Fk(b)1119 2251 y Fs(V)1180 2218 y Fr(k)1225 2232 y Fw(1)1290 2218 y Ft(6)25 b Fu(\016)1429 2117 y Fo(\021)1525 2218 y Ft(>)f Fu(\026)1675 2233 y Fs(\014)s(;N)1820 2117 y Fo(\020)q(n)1935 2218 y Fr(k)p Fu(u)2032 2174 y Fs(\020)2032 2246 y(N)s(;K)2200 2218 y Fr(\000)c Fz(1)-25 b(I)2353 2234 y Fk(b)2344 2251 y Fs(V)2405 2218 y Fr(k)2450 2232 y Fw(1)2515 2218 y Ft(6)25 b Fu(\016)2654 2117 y Fo(o)2736 2218 y Fr(\\)20 b(A)g(\\)f(D)3063 2117 y Fo(\021)3132 2218 y Fu(:)272 b Fz(\(6.12\))118 2416 y(The)42 b(spin)f (con\014gurations)i(inside)f Fu(V)1443 2430 y Fs(N)1553 2416 y Fz(\(resp)g Fu(V)1866 2383 y Fs(c)1846 2443 y(N)1913 2416 y Fz(\))h(are)g(surrounded)d(b)m(y)i Fr(\000)g Fz(\(resp)g(+\))g (b)s(oundary)118 2524 y(conditions,)35 b(so)f(that)g(they)g(are)f(in)h (equilibrium)e(in)h(the)h Fr(\000)f Fz(\(resp)g(+\))h(pure)e(phase.)50 b(Bulk)34 b(estimate)118 2632 y(imply)c(that)h(one)g(can)f(c)m(ho)s (ose)i Fu(s)e Fz(small)h(enough,)f Fu(\020)1888 2599 y Fn(0)1936 2632 y Fz(=)25 b Fu(\020)2079 2599 y Fn(0)2102 2632 y Fz(\()p Fu(\016)s Fz(\))32 b(and)d Fu(K)2507 2599 y Fn(0)2500 2656 y Fw(0)2565 2632 y Fz(=)c Fu(K)2745 2599 y Fn(0)2738 2656 y Fw(0)2778 2632 y Fz(\()p Fu(\016)s Fz(\))32 b(suc)m(h)e(that)157 2859 y(lim)118 2919 y Fs(N)7 b Fn(!1)378 2859 y Fu(\026)433 2874 y Fs(\014)s(;N)578 2731 y Fo(\022)645 2735 y(Z)704 2925 y Fk(b)695 2941 y Fs(V)752 2922 y Fq(c)803 2859 y Fr(j)p Fu(u)880 2815 y Fs(\020)916 2791 y Fl(0)880 2888 y Fs(N)s(;K)1028 2859 y Fz(\()p Fu(x)p Fz(\))20 b Fr(\000)g Fu(m)1341 2821 y Fw(+)1341 2888 y Fs(\014)1400 2859 y Fr(j)15 b Fu(dx)26 b Ft(>)1672 2797 y Fu(\016)p 1671 2838 46 4 v 1671 2921 a Fz(2)1757 2859 y(or)1883 2735 y Fo(Z)1943 2925 y Fk(b)1934 2941 y Fs(V)2010 2859 y Fr(j)p Fu(u)2087 2815 y Fs(\020)2123 2791 y Fl(0)2087 2888 y Fs(N)s(;K)2234 2859 y Fz(\()p Fu(x)p Fz(\))21 b Fr(\000)f Fu(m)2548 2821 y Fn(\000)2548 2888 y Fs(\014)2607 2859 y Fr(j)15 b Fu(dx)26 b Ft(>)2879 2797 y Fu(\016)p 2878 2838 V 2878 2921 a Fz(2)2964 2754 y Fo(\014)2964 2809 y(\014)2964 2864 y(\014)3024 2859 y Fr(A)20 b(\\)g(D)3271 2731 y Fo(\023)3363 2859 y Fz(=)25 b(0)g Fu(:)118 3084 y Fz(\(This)30 b(limit)h(can)g(b)s(e)f(obtained)g (b)m(y)h(using)e(a)i(pro)s(of)f(similar)h(to)g(the)f(one)h(of)f (Theorem)h(6.4\).)218 3192 y(So)f(that)h(\(6.12\))i(can)d(b)s(e)g (rewritten)g(for)h Fu(N)40 b Fz(large)31 b(enough)f(as)913 3406 y Fu(\026)968 3421 y Fs(\014)s(;N)1113 3305 y Fo(\020)1167 3406 y Fr(k)p Fu(u)1264 3362 y Fs(\020)1300 3338 y Fl(0)1264 3435 y Fs(N)s(;K)1432 3406 y Fr(\000)20 b Fz(1)-25 b(I)1585 3423 y Fk(b)1576 3440 y Fs(V)1637 3406 y Fr(k)1682 3420 y Fw(1)1747 3406 y Ft(6)25 b Fu(\016)1886 3305 y Fo(\021)1982 3406 y Ft(>)2088 3345 y Fz(1)p 2088 3385 V 2088 3468 a(2)2143 3406 y Fu(\026)2198 3421 y Fs(\014)s(;N)2343 3406 y Fz(\()p Fr(A)20 b(\\)g(D)s Fz(\))15 b Fu(:)729 b Fz(\(6.13\))118 3665 y Fj(Step)33 b(3)g(:)42 b(Surfac)-5 b(e)33 b(tension.)p 118 3697 989 4 v 218 3786 a Fz(Let)27 b(\003)f(b)s(e)g(the)g(union)g(of)g(the)h(sets)g(\003)1476 3800 y Fs(i)1529 3786 y Fz(=)e(\003\()p Fr(C)1776 3753 y Fs(i;)p Fw(+)1879 3786 y Fu(;)15 b Fr(C)1972 3753 y Fs(i;)p Fn(\000)2076 3786 y Fz(\).)39 b(The)26 b(con\014gurations)h(in) f(the)g(ev)m(en)m(t)i Fr(A)12 b(\\)g(D)118 3894 y Fz(con)m(tain)37 b(t)m(w)m(o)f(closed)g(surfaces)f(with)g(+)g(and)f Fr(\000)h Fz(blo)s(c)m(ks)h(whic)m(h)f(partition)g(the)h(domain)f Fx(T)3325 3908 y Fs(N)3427 3894 y Fz(in)m(to)h(3)118 4002 y(regions.)1478 4135 y Fx(T)1539 4149 y Fs(N)1631 4135 y Fz(=)25 b(\003)20 b Fr([)g Fz(\001)1967 4097 y Fw(+)2046 4135 y Fr([)g Fz(\001)2203 4097 y Fn(\000)2277 4135 y Fu(;)118 4286 y Fz(where)34 b(\001)461 4253 y Fn(\006)554 4286 y Fz(represen)m(ts)g(the)h(lo)s(cation)g(of)g(the)f Fr(\006)g Fz(pure)g(phases)f(and)h(\003)g(is)h(concen)m(trated)h(along) f(the)118 4394 y(in)m(terface.)42 b(W)-8 b(e)32 b(pro)s(ceed)e(no)m(w)g (to)i(ev)-5 b(aluate)32 b(the)e(RHS)g(of)38 b(\(6.13\))721 4608 y Fu(\026)776 4623 y Fs(\014)s(;N)920 4608 y Fz(\()q Fr(A)20 b(\\)f(D)s Fz(\))41 b Ft(>)1426 4547 y Fz(1)p 1384 4587 130 4 v 1384 4670 a Fu(Z)1446 4684 y Fs(N)1616 4522 y Fo(X)1538 4725 y Fn(C)1579 4706 y Fq(i;)p Fp(+)1671 4725 y Fs(;)p Fn(C)1732 4706 y Fq(i;)p Fl(\000)1840 4608 y Fu(Z)1909 4570 y Fw(+)1902 4642 y(\001)1961 4623 y Fp(+)2016 4608 y Fu(Z)2085 4570 y Fn(\000)2078 4642 y Fw(\001)2137 4623 y Fl(\000)2208 4522 y Fo(Y)2254 4717 y Fs(i)2339 4608 y Fu(Z)2408 4564 y Fn(C)2449 4540 y Fq(i;)p Fp(+)2541 4564 y Fs(;)p Fn(C)2602 4540 y Fq(i;)p Fl(\000)2401 4637 y Fw(\003)2450 4647 y Fq(i)2698 4608 y Fz(\()p Fr(S)2788 4622 y Fs(i)2817 4608 y Fz(\))15 b Fu(;)118 4881 y Fz(where)20 b(w)m(e)h(used)f(analogous)i(notation)g (to)f(Section)g(3)g(for)f(the)h(partition)g(function)g(with)f(mixed)g (b)s(ound-)118 4989 y(ary)28 b(conditions.)40 b(In)m(tro)s(ducing)27 b(the)h(partition)g(functions)f(with)h(re\015ected)g(b)s(oundary)e (conditions)i(w)m(e)118 5097 y(get)381 5360 y Fu(\026)436 5375 y Fs(\014)s(;N)581 5360 y Fz(\()p Fr(A)20 b(\\)g(D)s Fz(\))40 b Ft(>)1086 5299 y Fz(1)p 1044 5340 V 1044 5423 a Fu(Z)1106 5437 y Fs(N)1276 5274 y Fo(X)1198 5478 y Fn(C)1239 5459 y Fq(i;)p Fp(+)1331 5478 y Fs(;)p Fn(C)1392 5459 y Fq(i;)p Fl(\000)1500 5360 y Fu(Z)1569 5322 y Fw(+)1562 5394 y(\001)1621 5376 y Fp(+)1676 5360 y Fu(Z)1745 5322 y Fn(\000)1738 5394 y Fw(\001)1797 5376 y Fl(\000)1853 5360 y Fu(Z)1922 5316 y Fn(C)1963 5293 y Fq(i;)p Fp(+)2054 5316 y Fs(;R)1915 5389 y Fw(\003)1964 5399 y Fq(i)2147 5360 y Fu(Z)2216 5316 y Fn(C)2257 5293 y Fq(i;)p Fl(\000)2349 5316 y Fs(;R)2209 5389 y Fw(\003)2258 5399 y Fq(i)2467 5274 y Fo(Y)2513 5470 y Fs(i)2639 5286 y Fu(Z)2708 5242 y Fn(C)2749 5219 y Fq(i;)p Fp(+)2840 5242 y Fs(;)p Fn(C)2901 5219 y Fq(i;)p Fl(\000)2701 5315 y Fw(\003)2750 5325 y Fq(i)2998 5286 y Fz(\()p Fr(S)3088 5300 y Fs(i)3117 5286 y Fz(\))p 2608 5340 574 4 v 2608 5447 a Fu(Z)2677 5402 y Fn(C)2718 5383 y Fq(i;)p Fp(+)2810 5402 y Fs(;R)2670 5491 y Fw(\003)2719 5462 y Fp(+)2719 5514 y Fq(i)2902 5447 y Fu(Z)2971 5402 y Fn(C)3012 5383 y Fq(i;)p Fl(\000)3105 5402 y Fs(;R)2964 5491 y Fw(\003)3013 5462 y Fl(\000)3013 5514 y Fq(i)3207 5360 y Fu(;)197 b Fz(\(6.14\))p eop end %%Page: 37 37 TeXDict begin 37 36 bop 1137 45 a Fw(SURF)-8 b(A)n(CE)31 b(TENSION)g(AND)f(WULFF)h(THEOR)-6 b(Y)948 b(37)118 366 y Fz(where)35 b(\003)449 328 y Fn(\006)449 394 y Fs(i)543 366 y Fz(refers)g(to)g(the)h(sets)f(\003)1316 333 y Fn(\006)1375 366 y Fz(\()p Fr(C)1463 333 y Fs(i;)p Fn(\006)1567 366 y Fz(\))g(whic)m(h)g(w)m(ere)g(in)m(tro)s(duced)g(in)g(subsection)g (3.2.)56 b(The)34 b(last)118 474 y(term)d(in)g(the)g(RHS)g(is)g(an)g (appro)m(ximation)h(of)f(the)h(surface)f(tension)g(in)g(eac)m(h)h (domain)f(\003)3225 488 y Fs(i)3253 474 y Fz(,)h(therefore)118 582 y(Theorem)e(3.1)i(implies)308 823 y(inf)220 890 y Fn(C)261 871 y Fq(i;)p Fp(+)353 890 y Fs(;)p Fn(C)414 871 y Fq(i;)p Fl(\000)616 761 y Fz(1)p 532 802 214 4 v 532 885 a Fu(N)615 859 y Fs(d)p Fn(\000)p Fw(1)795 737 y Fo(X)849 932 y Fs(i)957 823 y Fz(log)1130 749 y Fu(Z)1199 704 y Fn(C)1240 681 y Fq(i;)p Fp(+)1331 704 y Fs(;)p Fn(C)1392 681 y Fq(i;)p Fl(\000)1192 777 y Fw(\003)1241 787 y Fq(i)1489 749 y Fz(\()p Fr(S)1579 763 y Fs(i)1608 749 y Fz(\))p 1100 802 574 4 v 1100 909 a Fu(Z)1169 865 y Fn(C)1210 846 y Fq(i;)p Fp(+)1301 865 y Fs(;R)1162 954 y Fw(\003)1211 924 y Fp(+)1211 976 y Fq(i)1393 909 y Fu(Z)1462 865 y Fn(C)1503 846 y Fq(i;)p Fl(\000)1596 865 y Fs(;R)1455 954 y Fw(\003)1504 924 y Fl(\000)1504 976 y Fq(i)1709 823 y Ft(>)45 b Fr(\000)1916 737 y Fo(X)1969 932 y Fs(i)2062 699 y Fo(Z)2127 889 y Fk(b)2113 905 y Fs(B)2169 886 y Fq(i)2215 823 y Fu(\034)2255 838 y Fs(\014)2302 823 y Fz(\()n Fu(~)-43 b(n)2392 837 y Fs(i)2420 823 y Fz(\))15 b Fu(d)p Fr(H)2595 785 y Fw(\()p Fs(d)p Fn(\000)p Fw(1\))2594 845 y Fs(x)2801 823 y Fr(\000)20 b Fu(P)13 b Fz(\()p Fu(v)s Fz(\))p Fu(c)p Fz(\()p Fu(\016)n(;)i(N)10 b Fz(\))15 b Fu(;)39 b Fz(\(6.15\))118 1092 y(where)30 b(lim)507 1107 y Fs(\016)r Fn(!)p Fw(0)666 1092 y Fz(lim)792 1106 y Fs(N)7 b Fn(!1)1016 1092 y Fu(c)p Fz(\()p Fu(\016)n(;)15 b(N)10 b Fz(\))27 b(=)e(0)31 b(and)f Fu(P)13 b Fz(\()p Fu(v)s Fz(\))31 b(is)g(the)f(p)s(erimeter)g(of)h Fu(v)s Fz(.)218 1200 y(It)f(remains)g(to)h(c)m(hec)m(k)h(that)626 1434 y(lim)587 1495 y Fs(N)7 b Fn(!1)925 1373 y Fz(1)p 841 1413 214 4 v 841 1497 a Fu(N)924 1471 y Fs(d)p Fn(\000)p Fw(1)1080 1434 y Fz(log)1212 1252 y Fo(0)1212 1415 y(@)1344 1373 y Fz(1)p 1302 1413 130 4 v 1302 1497 a Fu(Z)1364 1511 y Fs(N)1559 1348 y Fo(X)1482 1551 y Fn(C)1523 1533 y Fq(i;)p Fp(+)1614 1551 y Fs(;)p Fn(C)1675 1533 y Fq(i;)p Fl(\000)1783 1434 y Fu(Z)1852 1396 y Fw(+)1845 1468 y(\001)1904 1449 y Fp(+)1974 1434 y Fu(Z)2043 1396 y Fn(\000)2036 1468 y Fw(\001)2095 1449 y Fl(\000)2166 1434 y Fu(Z)2235 1390 y Fn(C)2276 1367 y Fq(i;)p Fp(+)2368 1390 y Fs(;R)2228 1479 y Fw(\003)2277 1449 y Fp(+)2277 1501 y Fq(i)2460 1434 y Fu(Z)2529 1390 y Fn(C)2570 1367 y Fq(i;)p Fl(\000)2663 1390 y Fs(;R)2522 1479 y Fw(\003)2571 1449 y Fl(\000)2571 1501 y Fq(i)2740 1252 y Fo(1)2740 1415 y(A)2845 1434 y Fz(=)25 b(0)15 b Fu(:)403 b Fz(\(6.16\))118 1691 y(Com)m(bining)30 b(inequalities)i(\(6.15\))h(and)c(\(6.16\))k(w)m(e)e(see)g(that)498 1931 y(lim)15 b(inf)522 1992 y Fs(N)7 b Fn(!1)859 1870 y Fz(1)p 775 1910 214 4 v 775 1993 a Fu(N)858 1967 y Fs(d)p Fn(\000)p Fw(1)1029 1931 y Fz(log)17 b Fu(\026)1216 1946 y Fs(\014)s(;N)1361 1931 y Fz(\()p Fr(A)j(\\)g(D)s Fz(\))40 b Ft(>)61 b Fr(\000)1992 1817 y Fs(`)1941 1845 y Fo(X)1949 2040 y Fs(i)p Fw(=1)2087 1807 y Fo(Z)2152 1997 y Fk(b)2138 2013 y Fs(B)2194 1995 y Fq(i)2240 1931 y Fu(\034)2280 1946 y Fs(\014)2327 1931 y Fz(\()n Fu(~)-43 b(n)2417 1945 y Fs(i)2445 1931 y Fz(\))15 b Fu(d)p Fr(H)2620 1894 y Fw(\()p Fs(d)p Fn(\000)p Fw(1\))2619 1954 y Fs(x)2826 1931 y Fr(\000)20 b Fu(o)p Fz(\()p Fu(\016)s Fz(\))15 b Fu(:)118 2160 y Fz(Using)31 b(Theorem)f(6.3)h(and)f(letting)i Fu(\016)i Fz(v)-5 b(anish,)30 b(w)m(e)h(conclude)g(the)f(pro)s(of)g(of) g(Prop)s(osition)h(6.1.)218 2308 y(W)-8 b(e)36 b(turn)e(no)m(w)h(to)g (the)h(deriv)-5 b(ation)35 b(of)42 b(\(6.16\))s(.)54 b(Since)35 b(the)g(re\015ected)h(b)s(oundary)d(conditions)i(de-)118 2416 y(couple)e(the)f(system,)h(the)f(n)m(umerator)g(should)f(b)s(e)g (understo)s(o)s(d)f(as)i(the)h(pro)s(duct)d(of)i(t)m(w)m(o)i (partitions)118 2524 y(functions)e(asso)s(ciated)i(to)g(the)f(sets)1415 2501 y(\026)1400 2524 y(\001)1476 2491 y Fw(+)1564 2524 y Fz(=)c(\001)1740 2491 y Fw(+)1820 2524 y Fr([)1881 2538 y Fs(i)1931 2524 y Fz(\003)1994 2491 y Fw(+)2053 2524 y Fz(\()p Fr(C)2141 2491 y Fs(i;)p Fw(+)2244 2524 y Fz(\))k(and)2506 2501 y(\026)2491 2524 y(\001)2567 2491 y Fn(\000)2655 2524 y Fz(=)c(\001)2831 2491 y Fn(\000)2911 2524 y Fr([)2972 2538 y Fs(i)3022 2524 y Fz(\003)3085 2491 y Fn(\000)3144 2524 y Fz(\()p Fr(C)3232 2491 y Fs(i;)p Fn(\000)3335 2524 y Fz(\),)34 b(where)118 2632 y(\003\()p Fr(C)269 2599 y Fs(i;)p Fn(\006)372 2632 y Fz(\))e(denotes)g(the)f (part)g(of)g(\003)1291 2646 y Fs(i)1320 2632 y Fz(.)43 b(It)31 b(is)g(imp)s(ortan)m(t)h(to)g(note)g(that)f(con)m(trary)h(to)g (\001)3075 2599 y Fn(\006)3134 2632 y Fz(,)g(the)f(sets)3542 2609 y(\026)3527 2632 y(\001)3603 2599 y Fn(\006)118 2740 y Fz(are)f(indep)s(enden)m(t)e(of)h(the)h(c)m(hoice)h(of)f(the)f (surfaces)g Fr(C)1962 2707 y Fs(i;)p Fn(\006)2065 2740 y Fz(.)41 b(In)28 b(particular,)i(follo)m(wing)h(the)f(notation)g(of) 118 2848 y(Section)h(3,)1307 2890 y Fo(X)1307 3093 y Fn(C)1348 3074 y Fq(i;)p Fp(+)1455 2976 y Fu(Z)1524 2938 y Fw(+)1517 3010 y(\001)1576 2991 y Fp(+)1645 2976 y Fu(Z)1714 2932 y Fn(C)1755 2908 y Fq(i;)p Fp(+)1847 2932 y Fs(;R)1707 3021 y Fw(\003)1756 2991 y Fp(+)1756 3043 y Fq(i)1949 2976 y Fz(=)25 b Fu(Z)2114 2939 y Fs(R)2119 2991 y Fw(\026)2107 3008 y(\001)2166 2989 y Fp(+)2221 2976 y Fz(\()p Fr(C)2309 2939 y Fs(i;)p Fw(+)2412 2976 y Fz(\))p Fu(;)118 3206 y Fz(where)35 b(the)i(RHS)e(denotes)h(the)g (partition)h(function)e(on)2159 3183 y(\026)2144 3206 y(\001)2220 3173 y Fw(+)2315 3206 y Fz(under)f(the)i(constrain)m(t)h (that)g(in)e(eac)m(h)118 3320 y Fd(R)193 3276 y Fs(i;)p Fw(+)193 3349 y Fs(N)328 3320 y Fz(there)e(is)f(a)h(+)e(barrier.)46 b(Applying)32 b(the)g(same)h(strategy)g(as)g(for)f(the)g(deriv)-5 b(ation)33 b(of)40 b(\(5.11\))r(,)33 b(w)m(e)118 3428 y(can)e(c)m(hec)m(k)h(that)1218 3626 y(lim)1179 3687 y Fs(N)7 b Fn(!1)1518 3565 y Fz(1)p 1434 3605 V 1434 3689 a Fu(N)1517 3662 y Fs(d)p Fn(\000)p Fw(1)1672 3626 y Fz(log)1815 3554 y Fu(Z)1884 3521 y Fs(R)1889 3574 y Fw(\026)1877 3591 y(\001)1936 3572 y Fp(+)1991 3554 y Fz(\()p Fr(C)2079 3521 y Fs(i;)p Fw(+)2182 3554 y Fz(\))p 1815 3605 403 4 v 1928 3693 a Fu(Z)1997 3662 y Fs(R)2002 3714 y Fw(\026)1990 3731 y(\001)2049 3712 y Fp(+)2253 3626 y Fz(=)24 b(0)15 b Fu(:)118 3847 y Fz(This)30 b(implies)g(that)h (\(6.16\))i(is)d(equiv)-5 b(alen)m(t)32 b(to)1235 4068 y(lim)1196 4129 y Fs(N)7 b Fn(!1)1535 4007 y Fz(1)p 1451 4047 214 4 v 1451 4131 a Fu(N)1534 4104 y Fs(d)p Fn(\000)p Fw(1)1690 4068 y Fz(log)1832 3996 y Fu(Z)1901 3963 y Fs(R)1906 4016 y Fw(\026)1894 4033 y(\001)1953 4014 y Fp(+)2023 3996 y Fu(Z)2092 3963 y Fs(R)2097 4016 y Fw(\026)2085 4033 y(\001)2144 4014 y Fl(\000)p 1832 4047 369 4 v 1951 4131 a Fu(Z)2013 4145 y Fs(N)2235 4068 y Fz(=)25 b(0)15 b Fu(:)1013 b Fz(\(6.17\))218 4256 y(The)24 b(partition)i(functions)e (in)h(the)g(n)m(umerator)g(tak)m(e)i(also)f(in)m(to)g(accoun)m(t)g(the) f(constrain)m(ts)h(imp)s(osed)118 4364 y(b)m(y)32 b(the)h(set)f Fr(D)j Fz(on)d(the)h(spins)e(along)i(the)f(set)h Fu(U)1788 4331 y Fs(\016)1778 4391 y(N)1878 4364 y Fz(and)e(on)h(the)h(sides)f (of)g Fu(R)2738 4331 y Fs(i)2737 4391 y(N)2836 4364 y Fz(parallel)i(to)c Fu(~)-42 b(n)3330 4378 y Fs(i)3357 4364 y Fz(.)47 b(These)118 4472 y(constrain)m(ts)30 b(can)g(b)s(e)e (released)i(up)f(to)g(a)h(small)g(cost)g(wrt)f(the)g(surface)g(order.) 40 b(This)28 b(comes)i(from)f(the)118 4580 y(fact)i(that)f(the)f(ev)m (en)m(t)i Fr(D)h Fz(is)e(supp)s(orted)e(b)m(y)h(at)h(most)g Fu(c)p Fz(\()p Fu(d;)15 b(\016)s Fz(\))p Fu(N)2286 4547 y Fs(d)p Fn(\000)p Fw(1)2449 4580 y Fz(edges)30 b(where)f Fu(c)p Fz(\()p Fu(d;)15 b(\016)s Fz(\))32 b(v)-5 b(anishes)29 b(as)118 4688 y Fu(\016)39 b Fz(go)s(es)e(to)f(0.)56 b(Therefore)35 b(the)h(probabilit)m(y)g(of)g Fr(D)i Fz(is)d(negligible) j(with)d(resp)s(ect)g(to)i(a)e(surface)h(order)118 4795 y(and)30 b(w)m(e)h(get)1200 4861 y Fo(\014)1200 4916 y(\014)1200 4970 y(\014)1200 5025 y(\014)1200 5079 y(\014)1230 5020 y Fz(log)1373 4948 y Fu(Z)1442 4915 y Fs(R)1447 4968 y Fw(\026)1435 4985 y(\001)1494 4966 y Fp(+)1564 4948 y Fu(Z)1633 4915 y Fs(R)1638 4968 y Fw(\026)1626 4985 y(\001)1685 4966 y Fl(\000)p 1373 4999 V 1492 5087 a Fu(Z)1561 5056 y Fs(R)1554 5116 y(N)1751 4861 y Fo(\014)1751 4916 y(\014)1751 4970 y(\014)1751 5025 y(\014)1751 5079 y(\014)1822 5020 y Ft(6)25 b Fu(c)p Fz(\()p Fu(d;)15 b(\016)s Fz(\))p Fu(N)2240 4983 y Fs(d)p Fn(\000)p Fw(1)2388 5020 y Fu(;)1016 b Fz(\(6.18\))118 5260 y(where)29 b Fu(Z)449 5227 y Fs(R)442 5287 y(N)539 5260 y Fz(is)g(the)g (unconstrained)g(partition)h(function)f(on)g Fx(T)2289 5274 y Fs(N)2385 5260 y Fz(for)g(whic)m(h)g(the)h(in)m(teractions)h(in) e(the)118 5372 y(middle)h(of)h(eac)m(h)h Fu(R)795 5339 y Fs(i)794 5399 y(N)892 5372 y Fz(ha)m(v)m(e)g(b)s(een)d(mo)s(di\014ed) h(and)g(replaced)h(b)m(y)f(p)s(erfect)h(w)m(alls.)42 b(Again)32 b(b)m(y)e(the)h(same)118 5480 y(considerations)38 b(as)f(in)g(the)g(last)h(argumen)m(t)f(of)g(the)g(pro)s(of)g(of)g(the)g (upp)s(er)e(b)s(ound)g(\(see)j(Subsection)p eop end %%Page: 38 38 TeXDict begin 38 37 bop 118 45 a Fw(38)1128 b(T.)23 b(BODINEA)n(U)g (AND)g(E.)g(PRESUTTI)118 366 y Fz(6.3\),)33 b(one)f(c)m(hec)m(k)h(that) f(one)g(can)g(\014nd)d(a)j(p)s(olyhedral)f(appro)m(ximation)h(for)g (whic)m(h)f(assumption)g(\(4.5\))118 474 y(is)36 b(satis\014ed.)59 b(The)36 b(corrections)h(to)g(the)g(pressure)e(induced)g(b)m(y)h(the)h (re\015ection)g(are)g(negligible)h(wrt)118 582 y(the)31 b(surface)f(order)g(\(see)h(\(4.40\))s(\))f(so)h(that)1367 792 y(lim)1328 852 y Fs(N)7 b Fn(!1)1642 730 y Fz(1)p 1558 771 214 4 v 1558 854 a Fu(N)1641 828 y Fs(d)p Fn(\000)p Fw(1)1796 792 y Fz(log)1939 730 y Fu(Z)2008 697 y Fs(R)2001 756 y(N)p 1939 771 130 4 v 1939 854 a Fu(Z)2001 868 y Fs(N)2103 792 y Fz(=)25 b(0)15 b Fu(:)118 983 y Fz(This,)30 b(com)m(bined)h(with)f(\(6.18\))i(implies)f(the)f(v)-5 b(alidit)m(y)32 b(of)38 b(\(6.16\))r(.)118 1145 y(6.5.)47 b FE(Exp)s(onen)m(tial)36 b(tigh)m(tness.)46 b Fz(The)31 b(purp)s(ose)f(of)i(this)g(Subsection)f(is)h(to)g(pro)m(v)m(e)h(that)f (phase)f(co-)118 1252 y(existence)h(cannot)e(o)s(ccur)g(b)m(y)g (creation)h(of)g(man)m(y)f(small)g(droplets.)41 b(Rephrased)29 b(in)h(a)g(mathematical)118 1360 y(w)m(a)m(y)-8 b(,)27 b(this)c(means)g(that)i(with)e(an)g(o)m(v)m(erwhelming)i(probabilit)m (y)-8 b(,)26 b(the)e(con\014gurations)f(will)h(concen)m(trate)118 1468 y(close)32 b(to)f(the)f(compact)i(set)763 1621 y Fr(K)832 1635 y Fs(a)899 1621 y Fz(=)995 1548 y Fo(\010)1048 1621 y Fu(v)d Fr(2)c Fz(BV)r(\()1381 1598 y Fo(b)1376 1621 y Fx(T)p Fu(;)15 b Fr(f)p Fu(m)1602 1583 y Fn(\000)1602 1651 y Fs(\014)1661 1621 y Fu(;)g(m)1781 1583 y Fw(+)1781 1651 y Fs(\014)1841 1621 y Fr(g)p Fz(\))26 b Fr(j)91 b Fu(P)13 b Fz(\()p Fr(f)p Fu(v)29 b Fz(=)c Fu(m)2463 1583 y Fn(\000)2463 1651 y Fs(\014)2522 1621 y Fr(g)p Fz(\))h Ft(6)f Fu(a)2772 1548 y Fo(\011)2825 1621 y Fu(;)579 b Fz(\(6.19\))118 1773 y(where)30 b Fu(P)43 b Fz(denotes)31 b(the)g(p)s(erimeter)f(and)g Fu(a)g Fz(will)h(b)s(e)e(c)m(hosen)i (large)h(enough.)118 1928 y FE(Prop)s(osition)40 b(6.3.)j Fj(Ther)-5 b(e)36 b(exists)g(a)g(c)-5 b(onstant)37 b Fu(C)7 b Fz(\()p Fu(\014)e Fz(\))30 b Fu(>)g Fz(0)35 b Fj(such)h(that)g(for)g(al)5 b(l)36 b Fu(\016)j Fj(p)-5 b(ositive)36 b(one)g(c)-5 b(an)118 2036 y(\014nd)33 b Fu(K)376 2050 y Fw(0)416 2036 y Fz(\()p Fu(\016)s Fz(\))h Fj(such)f(that)h(for)f Fu(K)f Ft(>)25 b Fu(K)1383 2050 y Fw(0)460 2224 y Fr(8)p Fu(a)g(>)g Fz(0)p Fu(;)202 b Fz(lim)15 b(sup)989 2303 y Fs(N)7 b Fn(!1)1365 2163 y Fz(1)p 1281 2204 214 4 v 1281 2287 a Fu(N)1364 2261 y Fs(d)p Fn(\000)p Fw(1)1519 2224 y Fz(log)18 b Fu(\026)1707 2239 y Fs(\014)s(;N)1852 2224 y Fz(\()p Fr(M)1996 2238 y Fs(N)s(;K)2168 2224 y Fr(62)25 b(V)7 b Fz(\()p Fr(K)2421 2238 y Fs(a)2464 2224 y Fu(;)15 b(\016)s Fz(\)\))42 b Ft(6)j Fr(\000)20 b Fu(C)7 b Fz(\()p Fu(\014)e Fz(\))15 b Fu(a;)118 2447 y Fj(wher)-5 b(e)34 b Fr(V)7 b Fz(\()p Fr(K)542 2461 y Fs(a)584 2447 y Fu(;)15 b(\016)s Fz(\))34 b Fj(is)f(the)g Fu(\016)s Fj(-neighb)-5 b(orho)g(o)g(d)36 b(of)d Fr(K)1777 2461 y Fs(a)1852 2447 y Fj(in)f Fx(L)2024 2414 y Fw(1)2063 2447 y Fz(\()2103 2423 y Fo(b)2098 2447 y Fx(T)p Fz(\))p Fj(.)218 2602 y Fz(The)22 b(estimate)i(\(6.2\))g(allo) m(ws)f(us)f(to)h(shift)f(our)g(atten)m(tion)i(from)e(the)h(lo)s(cal)g (a)m(v)m(eraged)i(magnetization)118 2710 y(to)31 b(the)g(mesoscopic)g (phase)f(lab)s(els.)41 b(In)30 b(particular)h(Prop)s(osition)f(6.3)i (follo)m(ws)f(from)118 2865 y FE(Theorem)40 b(6.4.)45 b Fj(Fix)36 b Fu(\020)j(>)33 b Fz(0)p Fj(.)54 b(F)-7 b(or)38 b(every)e Fu(a)d(>)f Fz(0)37 b Fj(and)g Fu(\016)g(>)32 b Fz(0)37 b Fj(ther)-5 b(e)37 b(exists)g(a)g(\014nite)g(sc)-5 b(ale)37 b Fu(K)3479 2879 y Fw(0)3519 2865 y Fz(\()p Fu(\016)s Fz(\))p Fj(,)118 2973 y(such)c(that)h(for)f(al)5 b(l)33 b Fu(K)f Ft(>)25 b Fu(K)1068 2987 y Fw(0)758 3157 y Fz(lim)15 b(sup)795 3235 y Fs(N)7 b Fn(!1)1145 3095 y Fz(1)p 1061 3136 V 1061 3219 a Fu(N)1144 3193 y Fs(d)p Fn(\000)p Fw(1)1300 3157 y Fz(log)18 b Fu(\026)1488 3172 y Fs(\014)s(;N)1617 3056 y Fo(\020)1671 3157 y Fu(u)1723 3112 y Fs(\020)1723 3185 y(N)s(;K)1896 3157 y Fr(62)25 b(V)7 b Fz(\()p Fr(K)2149 3171 y Fs(a)2191 3157 y Fu(;)15 b Fz(2)p Fu(\016)s Fz(\))2354 3056 y Fo(\021)2436 3157 y Ft(6)45 b Fr(\000)20 b Fu(c)p Fz(\()p Fu(\014)5 b(;)15 b(K)7 b Fz(\))p Fu(a)15 b(;)409 b Fz(\(6.20\))118 3358 y Fj(wher)-5 b(e)34 b Fu(c)p Fz(\()p Fu(\014)5 b(;)15 b(K)7 b Fz(\))34 b Fj(is)e(a)h(p)-5 b(ositive)34 b(c)-5 b(onstant.)218 3513 y Fz(The)25 b(core)i(of)f(the)g(pro)s(of)f(relies)i (on)f(the)g(con)m(trol)h(of)f(the)g(phase)g(of)g(small)g(con)m(tours)h (b)m(y)f(means)f(of)h(an)118 3621 y(en)m(trop)m(y/energy)34 b(argumen)m(t.)48 b(The)32 b(argumen)m(t)h(is)g(standard)f(and)g(dep)s (ends)f(only)h(on)h(the)g(structure)118 3729 y(of)d(the)h(coarse)g (graining.)41 b(W)-8 b(e)31 b(refer)f(the)h(reader)f(to)g([BIV1)r(])g (\(Theorem)g(2.2.1\),)j(where)c(Prop)s(osition)118 3837 y(6.4)g(w)m(as)g(deriv)m(ed)f(in)f(a)i(complete)g(generalit)m(y)-8 b(.)42 b(Finally)-8 b(,)31 b(notice)e(that)f(similar)h(argumen)m(ts)f (can)g(easily)118 3945 y(b)s(e)i(adapted)g(to)h(m)m(ulti-phase)g(mo)s (dels)f(\(see)i(Remark)e(3.4)i(in)e([BIV2)q(]\).)218 4090 y(Theorem)42 b(2.1)j(can)e(b)s(e)f(obtained)i(b)m(y)e(com)m (bining)i(Prop)s(ositions)f(6.3,)k(6.1,)h(6.2.)79 b(Since)43 b Fr(K)3515 4104 y Fs(a)3600 4090 y Fz(is)118 4198 y(compact)37 b(with)e(resp)s(ect)h(to)g(the)g Fx(L)1355 4165 y Fw(1)1430 4198 y Fz(top)s(ology)h(\(see)g([EG]\),)h(the)e(exp)s(onen)m(tial)g (tigh)m(tness)h(prop)s(ert)m(y)118 4306 y(6.3)32 b(enables)e(us)g(to)h (fo)s(cus)f(only)h(on)f(a)h(\014nite)g(n)m(um)m(b)s(er)e(of)h (con\014gurations)h(close)h(to)f Fr(K)3098 4320 y Fs(a)3140 4306 y Fz(.)41 b(The)30 b(precise)118 4414 y(asymptotic)h(of)e(these)h (con\014gurations)g(is)f(then)h(estimated)g(b)m(y)g(Prop)s(ositions)f (6.1,)i(6.2)f(\(see)h([B1)q(])f(for)118 4522 y(details\).)1076 5103 y Fy(Appendix)j Fz(A.)46 b Fy(Pr)n(oof)34 b(of)g(Theorem)f(6.1)218 5264 y Fz(The)23 b(magnetic)i(\014eld)e(is)h(equal)g(to)g Fu(h)p Fz(\()p Fu(\014)5 b Fz(\))25 b(and)e(omitted)h(from)f(the)h (notation)h(throughout)e(the)h(pro)s(of.)118 5372 y(The)30 b(pro)s(of)g(follo)m(ws)h(the)g(argumen)m(t)f(dev)m(elop)s(ed)h(in)f ([B2)q(].)p eop end %%Page: 39 39 TeXDict begin 39 38 bop 1137 45 a Fw(SURF)-8 b(A)n(CE)31 b(TENSION)g(AND)f(WULFF)h(THEOR)-6 b(Y)948 b(39)118 366 y Fj(Step)33 b(1.)p 118 399 277 4 v 39 w Fz(Let)38 b(us)g(start)h(with) f(a)g(single)h(b)s(o)m(x.)64 b(If)38 b Fx(B)1906 380 y Fs(K)1974 366 y Fz(\()p Fu(x)p Fz(\))h(is)f(not)h(a)f Fj(go)-5 b(o)g(d)40 b Fz(b)s(o)m(x)e(then)g(either)h(there)f(is)118 474 y(a)f(con)m(tour)g(of)g(length)g(at)g(least)h Fu(K)1351 441 y Fs(\013)1437 474 y Fz(crossing)f(the)f(enlarged)i(b)s(oundary)c (or)j(conditionally)h(on)e(the)118 582 y(ev)m(en)m(t)45 b(that)f(the)f(b)s(o)m(x)g Fx(B)998 596 y Fs(K)1066 582 y Fz(\()p Fu(x)p Fz(\))h(is)f(surrounded)e(b)m(y)i(a)h(surface)f(of)g Fu(\021)s Fz(-blo)s(c)m(k)i(spins)d(of)h(sign)g Fu(")3419 596 y Fs(x)3464 582 y Fz(,)j(the)118 690 y(magnetization)c Fr(M)827 704 y Fs(K)895 690 y Fz(\()p Fu(x)p Fz(\))e(is)f(at)m(ypical.) 68 b(These)39 b(t)m(w)m(o)i(o)s(ccurrences)e(can)g(b)s(e)f(estimated)i (separately)-8 b(.)118 798 y(Applying)30 b(the)h(P)m(eierls)g(estimate) h(\(4.11\))s(,)e(w)m(e)h(get)537 970 y Fu(\026)592 985 y Fs(\014)s(;N)722 896 y Fo(\000)763 970 y Fz(there)g(is)f(a)h(con)m (tour)g(crossing)g Fu(@)5 b Fx(B)1953 984 y Fs(K)2021 970 y Fz(\()p Fu(x)p Fz(\))2144 896 y Fo(\001)2211 970 y Ft(6)25 b Fu(K)2391 932 y Fs(d)p Fn(\000)p Fw(1)2536 970 y Fz(exp\()p Fr(\000)p Fu(c)2820 985 y Fs(\014)2868 970 y Fu(K)2952 932 y Fs(\013)3001 970 y Fz(\))15 b Fu(:)376 b Fz(\(A.1\))118 1135 y(Conditionally)33 b(on)f(the)h(o)s(ccurrence)f (of)g(a)h(connected)g(surface)f Fr(S)39 b Fz(of)33 b Fu(\021)s Fz(-blo)s(c)m(k)g(spins)e(of)i(sign)f Fu(")3433 1149 y Fs(x)3509 1135 y Fz(sur-)118 1242 y(rounding)26 b(the)h(b)s(o)m(x)g Fx(B)884 1256 y Fs(K)952 1242 y Fz(\()p Fu(x)p Fz(\),)h(the)f(con\014gurations)g(inside)g Fx(B)2179 1256 y Fs(K)2247 1242 y Fz(\()p Fu(x)p Fz(\))g(are)h(decoupled)f(from)f (the)h(exterior.)118 1350 y(W)-8 b(e)32 b(\014rst)d(use)h(Tc)m(heb)m (yshev)h(inequalit)m(y)284 1620 y Fu(\026)339 1635 y Fs(\014)s(;N)484 1519 y Fo(\020)538 1620 y Fr(fjM)717 1634 y Fs(K)786 1620 y Fz(\()p Fu(x)p Fz(\))21 b Fr(\000)f Fu(m)1100 1580 y Fs(")1133 1588 y Fq(x)1100 1649 y Fs(\014)1176 1620 y Fr(j)25 b Ft(>)g Fu(\020)7 b Fr(g)1444 1542 y Fo(\014)1444 1597 y(\014)1505 1620 y Fr(S)1567 1519 y Fo(\021)1662 1620 y Ft(6)1867 1558 y Fz(1)p 1767 1599 246 4 v 1767 1682 a Fu(\020)1814 1656 y Fw(2)1853 1682 y Fu(K)1937 1656 y Fw(2)p Fs(d)2023 1620 y Fu(\026)2078 1580 y Fs(")2111 1588 y Fq(x)2078 1653 y Fs(\014)s(;)p Fw(in)n(t)o(\()p Fn(S)e Fw(\))2346 1437 y Fo(0)2346 1601 y(@)2426 1546 y(\000)2551 1533 y(X)2483 1735 y Fs(i)p Fn(2)p Fm(B)2598 1746 y Fq(K)2657 1735 y Fw(\()p Fs(x)p Fw(\))2766 1620 y Fu(\033)2818 1634 y Fs(i)2867 1620 y Fr(\000)20 b Fu(m)3038 1580 y Fs(")3071 1588 y Fq(x)3038 1649 y Fs(\014)3113 1546 y Fo(\001)3155 1569 y Fw(2)3194 1437 y Fo(1)3194 1601 y(A)3304 1620 y Fu(:)118 1894 y Fz(where)25 b(in)m(t)q(\()p Fr(S)7 b Fz(\))26 b(is)f(the)h(region)g (surrounded)d(b)m(y)i Fr(S)7 b Fz(.)39 b(As)26 b Fr(S)32 b Fz(has)25 b(b)s(een)g(c)m(hosen)h(as)g(the)f(closest)i(surface)f(to) 118 2002 y(\()p Fx(B)214 2016 y Fs(K)5 b Fw(+)p Fs(K)399 1997 y Fq(\013)446 2002 y Fz(\))481 1969 y Fs(c)516 2002 y Fz(,)32 b(the)h(magnetization)h(inside)e(the)g(b)s(o)m(x)g Fx(B)1979 2016 y Fs(K)2047 2002 y Fz(\()p Fu(x)p Fz(\))g(is)g (measurable)h(after)f(the)g(conditioning.)118 2110 y(Classical)37 b(Pirogo)m(v-Sinai)h(theory)e(ensures)g(also)h(that)f(under)f(the)h (assumptions)f(of)h(Theorem)g(4.2,)118 2218 y(the)31 b(correlations)h(deca)m(y)f(exp)s(onen)m(tially)h(in)e(the)g Fu(")1883 2232 y Fs(x)1928 2218 y Fz(-pure)f(phase,)i(so)f(that)h(w)m (e)g(obtain)898 2421 y Fu(\026)953 2381 y Fs(")986 2389 y Fq(x)953 2454 y Fs(\014)s(;h;)p Fw(in)n(t)o(\()p Fn(S)5 b Fw(\))1282 2320 y Fo(\020)1336 2421 y Fr(fjM)1515 2435 y Fs(K)1585 2421 y Fz(\()p Fu(x)p Fz(\))21 b Fr(\000)e Fu(m)1898 2381 y Fs(")1931 2389 y Fq(x)1898 2450 y Fs(\014)1974 2421 y Fr(j)25 b Ft(>)g Fu(\020)7 b Fr(g)2212 2320 y Fo(\021)2292 2421 y Fr(\024)2480 2359 y Fz(1)p 2398 2400 211 4 v 2398 2483 a Fu(\020)2445 2457 y Fw(2)2484 2483 y Fu(K)2568 2457 y Fs(d)2618 2421 y Fu(\037)15 b(;)737 b Fz(\(A.2\))118 2644 y(where)30 b(the)h(susceptibilit)m(y)g Fu(\037)25 b Fz(=)1267 2576 y Fo(P)1363 2671 y Fs(i)p Fn(2)p Fm(Z)1478 2652 y Fq(d)1534 2644 y Fu(\026)1589 2606 y Fw(+)1589 2674 y Fs(\014)1648 2644 y Fz(\()p Fu(\033)1735 2658 y Fw(0)1774 2644 y Fz(;)15 b Fu(\033)1866 2658 y Fs(i)1895 2644 y Fz(\))31 b(is)f(\014nite.)118 2860 y Fj(Step)j(2.)p 118 2892 277 4 v 31 w Fz(In)d(order)f(to)j(ev)-5 b(aluate)32 b(the)e(probabilit)m(y)h(of)g(the)f(ev)m(en)m(t)1282 2965 y Fo(n)1343 3066 y Fu(u)1395 3021 y Fs(\020)1395 3094 y(K)1463 3066 y Fz(\()p Fu(x)1550 3080 y Fw(1)1590 3066 y Fz(\))c(=)f(0)p Fu(;)15 b(:)g(:)g(:)i(;)e(u)2046 3021 y Fs(\020)2046 3094 y(K)2114 3066 y Fz(\()p Fu(x)2201 3081 y Fs(`)2235 3066 y Fz(\))25 b(=)g(0)2436 2965 y Fo(o)118 3288 y Fz(the)k(partition)g Fr(B)713 3255 y Fw(\()p Fs(K)5 b Fw(\))865 3288 y Fz(is)28 b(sub-divised)g(in)m(to)h Fu(c)1647 3303 y Fs(d)1717 3288 y Fz(sub-partitions)f(\()p Fr(B)2395 3240 y Fw(\()p Fs(K)5 b Fw(\))2392 3315 y Fs(i)2518 3288 y Fz(\))2553 3302 y Fs(i)20 b Fh(6)f Fs(c)2702 3314 y Fq(d)2771 3288 y Fz(suc)m(h)28 b(that)h(t)m(w)m(o)h(cub)s(es)e(of)118 3421 y(size)g Fu(K)21 b Fz(+)14 b Fu(K)554 3388 y Fs(\013)629 3421 y Fz(cen)m(tered)28 b(on)f(t)m(w)m(o)i(sites)f(of)f Fr(B)1642 3373 y Fw(\()p Fs(K)5 b Fw(\))1639 3448 y Fs(i)1792 3421 y Fz(are)27 b(disjoin)m(t.)40 b(By)28 b(applying)f(H\177)-45 b(older)27 b(inequalit)m(y)-8 b(,)30 b(the)118 3529 y(estimate)i (\(6.1\))g(is)f(reduced)e(to)j(cub)s(es)d(whic)m(h)h(are)h(not)g (nearest)f(neigh)m(b)s(ors.)309 3772 y Fu(\026)364 3787 y Fs(\014)s(;N)509 3671 y Fo(\020)564 3772 y Fu(u)616 3727 y Fs(\020)616 3800 y(K)684 3772 y Fz(\()p Fu(x)771 3786 y Fw(1)811 3772 y Fz(\))25 b(=)g(0)p Fu(;)15 b(:)g(:)g(:)i(;)e(u) 1266 3727 y Fs(\020)1266 3800 y(K)1335 3772 y Fz(\()p Fu(x)1422 3787 y Fs(`)1455 3772 y Fz(\))26 b(=)f(0)1657 3671 y Fo(\021)1752 3772 y Ft(6)1888 3654 y Fs(c)1919 3666 y Fq(d)1863 3685 y Fo(Y)1864 3881 y Fs(i)p Fw(=1)1995 3772 y Fu(\026)2050 3787 y Fs(\014)s(;N)2194 3671 y Fo(\020)2249 3772 y Fr(8)p Fu(x)2352 3786 y Fs(j)2413 3772 y Fr(2)g(D)2572 3724 y Fw(\()p Fs(K)5 b Fw(\))2569 3799 y Fs(i)2695 3772 y Fu(;)106 b(u)2878 3727 y Fs(\020)2878 3800 y(K)2946 3772 y Fz(\()p Fu(x)3033 3786 y Fs(j)3070 3772 y Fz(\))26 b(=)f(0)3272 3671 y Fo(\021)3353 3666 y Fp(1)p 3336 3678 65 3 v 3336 3719 a Fq(c)3364 3737 y(d)3445 3772 y Fu(:)118 4059 y Fj(Step)33 b(3.)p 118 4092 277 4 v 38 w Fz(The)38 b(ev)m(en)m(t)874 3959 y Fo(n)935 4059 y Fu(u)987 4015 y Fs(\020)987 4088 y(K)1055 4059 y Fz(\()p Fu(x)1142 4073 y Fw(1)1182 4059 y Fz(\))26 b(=)f(0)p Fu(;)15 b(:)g(:)g(:)i(;)e(u) 1638 4015 y Fs(\020)1638 4088 y(K)1707 4059 y Fz(\()p Fu(x)1794 4074 y Fs(`)1827 4059 y Fz(\))25 b(=)g(0)2028 3959 y Fo(o)2127 4059 y Fz(can)39 b(b)s(e)e(decomp)s(osed)h(in)m(to)g (2)h(terms)f(:)55 b(on)118 4197 y Fu(`)156 4164 y Fn(0)216 4197 y Fz(b)s(o)m(xes)38 b(the)f(densit)m(y)h(is)f(at)m(ypical,)k (whereas)c(there)h(are)g(con)m(tours)f(crossing)h(the)f Fu(`)25 b Fr(\000)f Fu(`)3267 4164 y Fn(0)3328 4197 y Fz(enlarged)118 4305 y(b)s(oundaries)29 b(of)i(the)f(remaining)h(b)s(o) m(xes.)218 4413 y(F)-8 b(or)31 b(a)g(giv)m(en)g(collection)i(of)e Fu(j)36 b Fz(b)s(o)m(xes,)30 b(w)m(e)h(de\014ne)149 4577 y Fr(A)222 4591 y Fs(j)341 4577 y Fz(=)83 b Fr(f)p Fz(The)30 b Fu(j)36 b Fz(b)s(o)m(xes)31 b(are)g(surrounded)c(b)m(y)k Fr(\006)f Fz(surfaces,)g(but)g(their)g(a)m(v)m(eraged)j(magnetizations) 677 4712 y(are)e(non)f(t)m(ypical)q Fr(g)162 4847 y(B)222 4861 y Fs(j)341 4847 y Fz(=)83 b Fr(f)q Fz(There)29 b(are)i(con)m (tours)g(crossing)g(the)f Fu(j)36 b Fz(enlarged)31 b(b)s(oundaries)e (of)i(the)f(b)s(o)m(xes)p Fr(g)h Fu(:)218 5012 y Fz(The)25 b(probabilities)i(of)g(b)s(oth)e(ev)m(en)m(ts)j(can)e(b)s(e)g(ev)-5 b(aluated)27 b(as)f(follo)m(ws.)41 b(As)26 b(the)g Fu(j)32 b Fz(b)s(o)m(xes)26 b(are)h(disjoin)m(t)118 5120 y(and)j(the)g (surfaces)h(of)f(blo)s(c)m(ks)h(decouple)g(the)f(con\014gurations)h (inside)f(eac)m(h)h(b)s(o)m(x)1116 5298 y Fu(\026)1171 5313 y Fs(\014)s(;N)1301 5298 y Fz(\()p Fr(A)1409 5312 y Fs(j)1445 5298 y Fz(\))26 b Ft(6)1602 5224 y Fo(\000)1644 5298 y Fu(\026)1699 5313 y Fs(\014)s(;N)1828 5298 y Fz(\()p Fr(A)1936 5312 y Fw(1)1976 5298 y Fz(\))2011 5224 y Fo(\001)2053 5247 y Fs(j)2114 5298 y Ft(6)2210 5224 y Fo(\000)2252 5298 y Fu(\013)2310 5312 y Fs(K)2379 5224 y Fo(\001)2420 5247 y Fs(j)2472 5298 y Fu(;)118 5464 y Fz(where)k(the)h(constan)m(t)g Fu(\013)962 5478 y Fs(K)1056 5464 y Fz(=)1225 5423 y Fs(\037)p 1162 5443 171 4 v 1162 5501 a(\020)1198 5482 y Fp(2)1232 5501 y Fs(K)1297 5482 y Fq(d)1373 5464 y Fz(w)m(as)f(in)m(tro)s(duced)g(in)g(\(A.2\))r(.)p eop end %%Page: 40 40 TeXDict begin 40 39 bop 118 45 a Fw(40)1128 b(T.)23 b(BODINEA)n(U)g (AND)g(E.)g(PRESUTTI)357 593 y Fu(\026)412 608 y Fs(\014)s(;N)542 593 y Fz(\()p Fr(B)637 607 y Fs(j)674 593 y Fz(\))i(=)880 475 y Fs(j)830 507 y Fo(X)839 702 y Fs(i)p Fw(=1)977 593 y Fu(\026)1032 608 y Fs(\014)s(;N)1161 520 y Fo(\000)1203 593 y Fr(f9)30 b Fu(i)h Fz(con)m(tours)g(crossing)g(the)f Fu(j)36 b Fz(enlarged)31 b(b)s(oundaries)n Fr(g)3173 520 y Fo(\001)3231 593 y Fu(:)118 830 y Fz(W)-8 b(e)32 b(c)m(ho)s(ose)f Fu(i)g Fz(blo)s(c)m(ks)f(as)h(starting)g(p)s(oin)m(ts) f(of)h(these)g(con)m(tours.)41 b(Then)29 b(w)m(e)i(ha)m(v)m(e)h(to)f (ev)-5 b(aluate)1394 912 y Fo(X)1144 1114 y Fn(j)p Fw(\000)1208 1123 y Fp(1)1242 1114 y Fn(j)p Fw(+)p Fn(\001\001\001)o Fw(+)p Fn(j)p Fw(\000)1495 1124 y Fq(i)1521 1114 y Fn(j)19 b Fh(>)g Fs(j)t(K)1732 1095 y Fq(\013)1790 999 y Fu(\026)1845 1014 y Fs(\014)s(;N)1975 999 y Fz(\(\000)2067 1013 y Fw(1)2107 999 y Fu(;)c(:)g(:)g(:)h(;)f Fz(\000)2365 1013 y Fs(i)2394 999 y Fz(\))g Fu(;)118 1263 y Fz(where)30 b(the)h(con)m(tours)f(\(\000)996 1277 y Fw(1)1036 1263 y Fu(;)15 b(:)g(:)g(:)i(;)e Fz(\000)1295 1277 y Fs(i)1323 1263 y Fz(\))31 b(ha)m(v)m(e)g(also)h(to)f(cross)f(eac)m(h)i(b)s (oundaries)d(of)h(the)h Fu(j)36 b Fz(cub)s(es.)218 1371 y(Let)31 b Fu(n)436 1385 y Fs(r)503 1371 y Fz(b)s(e)f(the)h(n)m(um)m(b) s(er)e(of)h(b)s(oundaries)f(crossed)i(b)m(y)f(the)g(con)m(tour)i Fu(r)662 1453 y Fo(X)412 1655 y Fn(j)p Fw(\000)476 1664 y Fp(1)511 1655 y Fn(j)p Fw(+)p Fn(\001\001\001)n Fw(+)p Fn(j)p Fw(\000)763 1665 y Fq(i)789 1655 y Fn(j)19 b Fh(>)h Fs(j)t(K)1001 1636 y Fq(\013)1058 1540 y Fu(\026)1113 1555 y Fs(\014)s(;N)1243 1540 y Fz(\(\000)1335 1554 y Fw(1)1375 1540 y Fu(;)15 b(:)g(:)g(:)h(;)f Fz(\000)1633 1554 y Fs(i)1662 1540 y Fz(\))25 b Ft(6)1969 1453 y Fo(X)1833 1649 y Fs(n)1876 1658 y Fp(1)1911 1649 y Fw(+)p Fn(\001\001\001)o Fw(+)p Fs(n)2123 1659 y Fq(i)2149 1649 y Fw(=)p Fs(j)2326 1453 y Fo(X)2276 1655 y Fw(\(\000)2347 1663 y Fq(r)2382 1655 y Fs(;n)2445 1663 y Fq(r)2480 1655 y Fw(\))2522 1540 y Fu(\026)2577 1555 y Fs(\014)s(;N)2707 1540 y Fz(\(\000)2799 1554 y Fw(1)2838 1540 y Fu(;)15 b(:)g(:)g(:)i(;)e Fz(\000)3097 1554 y Fs(i)3125 1540 y Fz(\))g Fu(:)118 1804 y Fz(If)28 b(a)g(con)m(tour)g(crosses)h Fu(n)961 1818 y Fs(r)1026 1804 y Fz(b)s(oundaries)e(then)g(it)i(has)e(a)i(length)f(at)h(least)g Fu(n)2662 1818 y Fs(r)2699 1804 y Fu(K)2783 1771 y Fs(\013)2848 1804 y Fz(+)15 b(\()p Fu(n)3024 1818 y Fs(r)3077 1804 y Fr(\000)g Fz(1\))p Fu(K)35 b Fz(b)s(ecause)118 1912 y(the)c(distance)g(b)s(et)m(w)m(een)g(the)f(b)s(o)m(xes)h(is)f(at)h (least)h Fu(K)7 b Fz(.)40 b(Th)m(us)478 2048 y Fo(X)229 2249 y Fn(j)p Fw(\000)293 2258 y Fp(1)327 2249 y Fn(j)p Fw(+)p Fn(\001\001\001)n Fw(+)p Fn(j)p Fw(\000)579 2259 y Fq(i)605 2249 y Fn(j)19 b Fh(>)h Fs(j)t(K)817 2230 y Fq(\013)875 2134 y Fu(\026)930 2149 y Fs(\014)s(;N)1059 2134 y Fz(\(\000)1151 2148 y Fw(1)1191 2134 y Fu(;)15 b(:)g(:)g(:)i(;)e Fz(\000)1450 2148 y Fs(i)1478 2134 y Fz(\))109 b Ft(6)1936 2048 y Fo(X)1801 2243 y Fs(n)1844 2252 y Fp(1)1878 2243 y Fw(+)p Fn(\001\001\001)o Fw(+)p Fs(n)2090 2253 y Fq(i)2116 2243 y Fw(=)p Fs(j)2268 2020 y(i)2222 2048 y Fo(Y)2218 2242 y Fs(r)r Fw(=1)2357 2134 y Fz(exp\()p Fr(\000)p Fu(c)2641 2149 y Fs(\014)2689 2134 y Fu(n)2744 2148 y Fs(r)2781 2134 y Fu(K)2865 2097 y Fs(\013)2935 2134 y Fr(\000)20 b Fu(c)3065 2149 y Fs(\014)3112 2134 y Fz(\()p Fu(n)3202 2148 y Fs(r)3260 2134 y Fr(\000)g Fz(1\))p Fu(K)7 b Fz(\))1622 2487 y Ft(6)108 b Fz(exp)o(\()p Fr(\000)p Fu(c)2084 2502 y Fs(\014)2132 2487 y Fu(j)5 b(K)2258 2449 y Fs(\013)2308 2487 y Fz(\))2358 2331 y Fo( )2462 2373 y Fn(1)2431 2400 y Fo(X)2430 2595 y Fs(n)p Fw(=1)2578 2487 y Fz(exp\()p Fr(\000)p Fu(c)2862 2502 y Fs(\014)2910 2487 y Fz(\()p Fu(n)20 b Fr(\000)g Fz(1\))p Fu(K)7 b Fz(\))3310 2331 y Fo(!)3383 2354 y Fs(i)1622 2720 y Ft(6)108 b Fu(C)1873 2683 y Fs(i)1931 2720 y Fz(exp)o(\()p Fr(\000)p Fu(c)2214 2735 y Fs(\014)2262 2720 y Fu(j)5 b(K)2388 2683 y Fs(\013)2438 2720 y Fz(\))15 b Fu(:)118 3103 y(\026)173 3118 y Fs(\014)s(;N)303 3103 y Fz(\()p Fr(B)398 3117 y Fs(j)434 3103 y Fz(\))26 b Ft(6)656 2984 y Fs(j)606 3016 y Fo(X)615 3212 y Fs(i)p Fw(=1)752 2974 y Fo(\022)819 3041 y Fu(j)825 3165 y(i)862 2974 y Fo(\023)929 3103 y Fu(K)1013 3065 y Fw(\()p Fs(d)p Fn(\000)p Fw(1\))p Fs(i)1222 3103 y Fu(C)1294 3065 y Fs(i)1337 3103 y Fz(exp\()p Fr(\000)p Fu(c)1621 3118 y Fs(\014)1668 3103 y Fu(j)5 b(K)1794 3065 y Fs(\013)1844 3103 y Fz(\))26 b Ft(6)40 b Fz(exp\()p Fr(\000)p Fu(c)2300 3118 y Fs(\014)2363 3103 y Fu(j)5 b(K)2489 3065 y Fs(\013)2538 3103 y Fz(\)\(1)22 b(+)e Fu(C)7 b(K)2922 3065 y Fs(d)p Fn(\000)p Fw(1)3052 3103 y Fz(\))3087 3065 y Fs(j)3149 3103 y Fz(=)25 b(\()p Fu(\013)3338 3065 y Fn(0)3338 3125 y Fs(K)3407 3103 y Fz(\))3442 3065 y Fs(j)3494 3103 y Fu(:)118 3346 y Fz(where)30 b(the)h(constan)m(t)g Fu(\013)962 3313 y Fn(0)962 3373 y Fs(K)1061 3346 y Fz(v)-5 b(anishes)30 b(as)h Fu(K)37 b Fz(go)s(es)31 b(to)g(in\014nit)m(y)-8 b(.)218 3454 y(Com)m(bining)30 b(b)s(oth)g(estimates,)i(w)m(e)f(obtain)118 3696 y Fu(\026)173 3711 y Fs(\014)s(;N)318 3595 y Fo(\020)372 3696 y Fu(u)424 3652 y Fs(\020)424 3725 y(K)493 3696 y Fz(\()p Fu(x)580 3710 y Fw(1)619 3696 y Fz(\))26 b(=)f(0)p Fu(;)15 b(:)g(:)g(:)i(u)1035 3652 y Fs(\020)1035 3725 y(K)1103 3696 y Fz(\()p Fu(x)1190 3711 y Fs(`)1224 3696 y Fz(\))25 b(=)g(0)1425 3595 y Fo(\021)1520 3696 y Ft(6)1688 3582 y Fs(`)1637 3610 y Fo(X)1631 3808 y Fs(`)1660 3789 y Fl(0)1683 3808 y Fw(=1)1788 3568 y Fo(\022)1867 3635 y Fu(`)1855 3758 y(`)1893 3732 y Fn(0)1916 3568 y Fo(\023)1998 3696 y Fu(\026)2053 3711 y Fs(\014)s(;N)2183 3696 y Fz(\()p Fr(A)2291 3711 y Fs(`)2320 3692 y Fl(0)2346 3696 y Fz(\))2381 3658 y Fw(1)p Fs(=)p Fw(2)2491 3696 y Fu(\026)2546 3711 y Fs(\014)s(;N)2676 3696 y Fz(\()p Fr(B)2771 3711 y Fs(`)p Fn(\000)p Fs(`)2884 3692 y Fl(0)2910 3696 y Fz(\))2945 3658 y Fw(1)p Fs(=)p Fw(2)3081 3696 y Ft(6)3177 3622 y Fo(\000)3218 3696 y Fu(\013)3276 3710 y Fs(K)3365 3696 y Fz(+)20 b Fu(\013)3514 3658 y Fn(0)3514 3718 y Fs(K)3583 3622 y Fo(\001)3624 3645 y Fs(`)3672 3696 y Fu(:)118 3935 y Fz(This)30 b(completes)h(the)g(pro)s(of.)1625 4146 y Fy(References)118 4291 y FD([AlBe])40 b(G.)22 b(Alb)r(erti,)h(G.)f(Bellettini,)i Fv(Asymptotic)i(b)l(ehavior)f(of)e (a)h(non)h(lo)l(c)l(al)e(anisotr)l(opic)j(mo)l(del)e(for)g(phase)h(tr)l (ansition)6 b FD(,)276 4382 y(J.)26 b(Math.)h(Ann.)d Fc(310)p FD(,)j(No.)f(3,)g(527-560,)i(\(1998\).)118 4476 y([Am])39 b(L.)27 b(Am)n(brosio,)h Fv(On)g(the)i(lower)f(semic)l (ontinuity)g(of)g(quasic)l(onvex)h(inte)l(gr)l(als)g(in)f FD(SBV\(\012)p FB(;)14 b(R)3011 4444 y Fq(k)3048 4476 y FD(\),)28 b(Nonlinear)f(Anal.)276 4567 y Fc(23)p FD(,)f(No.)g(3,)h (405{425,)h(\(1994\).)118 4658 y([ABCP])40 b(G.)32 b(Alb)r(erti,)g(G.)g (Bellettini,)i(M.)e(Cassandro,)j(E.)c(Presutti,)j Fv(Surfac)l(e)f (tension)h(in)e(Ising)g(system)i(with)f(Kac)276 4750 y(p)l(otentials)6 b FD(,)28 b(J.)e(Stat.)g(Ph)n(ys.)f Fc(82)p FD(,)i(743{796)h(\(1996\).)118 4841 y([Al])39 b(K.)27 b(Alexander,)g Fv(Cub)l(e-r)l(o)l(ot)32 b(b)l(oundary)e (\015uctuations)h(for)e(dr)l(oplets)i(in)d(r)l(andom)i(cluster)g(mo)l (dels)6 b FD(,)29 b(Comm.)f(Math.)276 4932 y(Ph)n(ys.)e Fc(224)p FD(,)g(No.)g(3,)g(733{781)j(\(2001\).)118 5024 y([BCP])40 b(G.)27 b(Bellettini,)g(M.)g(Cassandro,)h(E.)e(Presutti,)h Fv(Constr)l(aine)l(d)j(minima)c(of)i(non)g(lo)l(c)l(al)g(fr)l(e)l(e)g (ener)l(gy)h(functionals)6 b FD(,)276 5115 y(J.)26 b(Stat.)g(Ph)n(ys.)g Fc(84)p FD(,)g(1337{1349)j(\(1996\).)118 5206 y([BBBP])41 b(O.)23 b(Benois,)j(T.)e(Bo)r(dineau,)h(P)-6 b(.)24 b(Butta,)g(E.)g (Presutti,)h Fv(On)g(the)i(validity)e(of)h(van)g(der)g(Waals)g(the)l (ory)h(of)e(surfac)l(e)276 5298 y(tension)6 b FD(,)27 b(Mark.)f(Pro)r(c.)h(and)e(Rel.)h(Fields)h Fc(3)p FD(,)f(175{198)i (\(1997\).)118 5389 y([BBP])40 b(O.)24 b(Benois,)h(T.)f(Bo)r(dineau,)h (E.)f(Presutti,)g Fv(L)l(ar)l(ge)j(deviations)f(in)f(the)i(van)e(der)i (Waals)e(limit)7 b FD(,)23 b(Sto)r(c)n(h.)g(Pro)r(c.)i(and)276 5480 y(Appl.)g Fc(75)p FD(,)h(89{104)i(\(1998\).)p eop end %%Page: 41 41 TeXDict begin 41 40 bop 1137 45 a Fw(SURF)-8 b(A)n(CE)31 b(TENSION)g(AND)f(WULFF)h(THEOR)-6 b(Y)948 b(41)118 366 y FD([B1])40 b(T.)27 b(Bo)r(dineau,)g Fv(The)h(Wul\013)g(c)l (onstruction)i(in)e(thr)l(e)l(e)i(and)e(mor)l(e)h(dimensions)6 b FD(,)27 b(Comm.)g(Math.)g(Ph)n(ys.)f Fc(207)p FD(,)h(197{)276 458 y(229)g(\(1999\).)118 549 y([B2])40 b(T.)c(Bo)r(dineau,)j Fv(Phase)f(c)l(o)l(existenc)l(e)h(for)d(the)i(Kac)f(Ising)f(mo)l(dels)6 b FD(,)39 b(75{111,)h(Progr.)d(Probab.,)i(51,)g(Birkhuser)276 640 y(Boston,)27 b(Boston,)g(MA,)f(\(2002\).)118 732 y([BIV1])39 b(T.)29 b(Bo)r(dineau,)g(D.)f(Io\013e,)h(Y.)e(V)-6 b(elenik,)28 b Fv(R)n(igor)l(ous)j(pr)l(ob)l(abilistic)f(analysis)g(of) f(e)l(quilibrium)g(crystal)i(shap)l(es)6 b FD(,)31 b(J.)276 823 y(Math.)26 b(Ph)n(ys.)g Fc(41)p FD(,)g(No.3,)h(1033{1098)i (\(2000\).)118 914 y([BIV2])39 b(T.)23 b(Bo)r(dineau,)h(D.)e(Io\013e,)h (Y.)e(V)-6 b(elenik,)23 b Fv(Winterb)l(ottom)j(c)l(onstruction)g(for)e (\014nite)h(r)l(ange)g(ferr)l(omagnetic)h(mo)l(dels)276 1006 y(:)35 b(a)27 b Fb(L)451 974 y Fp(1)486 1006 y Fv(-appr)l(o)l(ach) p FD(,)h(J.)f(Stat.)e(Ph)n(ys.)h Fc(105)p FD(,)g(No)g(1/2,)h(93{131)h (\(2001\).)118 1097 y([BKL])39 b(J.)e(Bricmon)n(t,)h(K.)e(Kuro)r(da,)i (J.)f(Leb)r(o)n(witz,)i Fv(Surfac)l(e)e(tension)h(and)f(phase)h(c)l(o)l (existenc)l(e)h(for)e(gener)l(al)g(lattic)l(e)276 1188 y(systems)6 b FD(,)28 b(J.)e(Stat.)g(Ph)n(ys.)f Fc(33)p FD(,)i(59{75)g(\(1983\).)118 1280 y([Ce])40 b(R.)25 b(Cerf,)i Fv(L)l(ar)l(ge)i(deviations)g(for)e(thr)l(e)l(e)j(dimensional)d(sup)l (er)l(critic)l(al)i(p)l(er)l(c)l(olation)6 b FD(,)28 b(Ast)n(\023)-36 b(erisque)25 b Fc(267)h FD(\(2000\).)118 1371 y([CePi1])41 b(R.)26 b(Cerf,)i(A.)e(Pisztora,)j Fv(On)f(the)h(Wul\013)f(crystal)h(in)f(the)h(Ising)f(mo)l(del)8 b FD(,)27 b(Ann.)f(Probab.)h Fc(28)p FD(,)g(No.)g(3,)g(947{1017)276 1462 y(\(2000\).)118 1554 y([CePi2])41 b(R.)36 b(Cerf,)41 b(A.)c(Pisztora,)k Fv(Phase)e(c)l(o)l(existenc)l(e)h(in)d(Ising,)j (Potts)f(and)f(p)l(er)l(c)l(olation)h(mo)l(dels)6 b FD(,)40 b(Ann.)c(Inst.)h(H.)276 1645 y(P)n(oincar)27 b(Probab.)f(Statist.,)h Fc(37)p FD(,)f(No.)g(6,)g(643{724)j(\(2001\).)118 1736 y([CIV])39 b(M.Campanino,)29 b(D.Io\013e,)f(Y.V)-6 b(elenik,)27 b Fv(Ornstein-Zernike)k(The)l(ory)f(for)f(the)h(\014nite)f(r)l(ange)h (Ising)f(mo)l(dels)g(ab)l(ove)276 1828 y FB(T)321 1836 y Fq(c)353 1828 y FD(,)d(preprin)n(t)f(\(2001\).)118 1919 y([DKS])38 b(R.L.)21 b(Dobrushin,)g(R.)f(Kotec)n(k)q(\023)-39 b(y,)21 b(S.)f(Shlosman,)i Fv(Wul\013)g(c)l(onstruction:)35 b(a)23 b(glob)l(al)g(shap)l(e)g(fr)l(om)g(lo)l(c)l(al)g(inter)l(action) 6 b FD(,)276 2010 y(AMS)25 b(translations)i(series,)h(v)n(ol)d(104,)i (Pro)n(vidence)f(R.I.)g(\(1992\).)118 2102 y([EG])40 b(L.)21 b(Ev)l(ans,)h(R.)f(Gariep)n(y)-6 b(,)22 b Fv(Me)l(asur)l(e)j (The)l(ory)g(and)f(Fine)f(Pr)l(op)l(erties)i(of)e(F)-6 b(unctions)6 b FD(,)24 b(CR)n(C)e(Press,)h(London)e(\(1992\).)118 2193 y([F])66 b(I.)26 b(F)-6 b(onseca,)26 b Fv(The)i(Wul\013)f(the)l (or)l(em)i(r)l(evisite)l(d)8 b FD(,)27 b(Pro)r(c.Ro)n(y)n(al)g(London)e (So)r(c.)i(Sect.)e(A,)h Fc(432)p FD(,)g(125-145)i(\(1991\).)118 2284 y([FM])40 b(I.)28 b(F)-6 b(onseca,)31 b(S.)d(Mueller,)j Fv(A)f(uniqueness)i(pr)l(o)l(of)e(of)g(the)h(Wul\013)f(The)l(or)l(em)6 b FD(,)30 b(Pro)r(c.)g(Ro)n(y)-6 b(.)28 b(So)r(c.)h(Edin)n(burgh;)h (Sect)276 2376 y(A,)25 b Fc(119)p FD(,)i(125{136)h(\(1991\).)118 2467 y([HK])38 b(O.)24 b(Hryniv,)f(R.)h(Kotec)n(ky)-6 b(,)23 b Fv(Surfac)l(e)k(tension)g(and)f(the)h(Ornstein-Zernike)h(b)l (ehaviour)g(for)d(the)i(2D)f(Blume-Cap)l(el)276 2558 y(mo)l(del)8 b FD(,)26 b(Jour.)g(Stat.)g(Ph)n(ys.,)g Fc(106)p FD(,)g(No.)g(3-4,)h(431{476)h(\(2002\).)118 2650 y([HKZ1])39 b(P)-6 b(.)33 b(Holic)n(ky)-6 b(,)35 b(R.)d(Kotec)n(ky)-6 b(,)35 b(M.)e(Zahradnik,)i Fv(R)n(igid)f(interfac) l(es)i(for)e(lattic)l(e)h(mo)l(dels)g(at)g(low)f(temp)l(er)l(atur)l (es.)p FD(,)276 2741 y(Jour.)26 b(Stat.)g(Ph)n(ys.,)g Fc(50)p FD(,)g(No.)g(3-4,)h(755{812,)h(\(1988\).)118 2832 y([HKZ2])39 b(P)-6 b(.)26 b(Holic)n(ky)-6 b(,)26 b(R.)g(Kotec)n(ky)-6 b(,)26 b(M.)h(Zahradnik,)f Fv(Phase)j(Diagr)l(am)f (of)g(Horizontal)t(ly)h(Invariant)g(Gibbs)f(States)i(for)276 2924 y(L)l(attic)l(e)f(Mo)l(dels)6 b FD(,)26 b(Ann.)f(Henri)h(P)n (oincar)g(ph)n(ys.)f(theo.,)i Fc(3)p FD(,)f(no.)g(2,)g(203{267)i (\(2002\).)118 3015 y([KP])39 b(R.)26 b(Kotec)n(ky)g(and)g(D.)g (Preiss,)i Fv(Cluster)h(Exp)l(ansion)g(for)f(A)n(bstr)l(act)j(Polymer)d (Mo)l(dels)6 b FD(,)28 b(Comm)n(un.)e(Math.)h(Ph)n(ys.,)276 3106 y Fc(103)p FD(,)g(491-498)g(\(1986\).)118 3198 y([I1])50 b(D.)26 b(Io\013e,)g Fv(L)l(ar)l(ge)i(deviations)h(for)f(the)g(2D)g (Ising)f(mo)l(del)h(:)34 b(a)28 b(lower)g(b)l(ound)g(without)h(cluster) g(exp)l(ansions)6 b FD(,)28 b(J.)e(Stat.)276 3289 y(Ph)n(ys.,)g Fc(74)p FD(,)g(411{432)j(\(1994\).)118 3380 y([I2])50 b(D.)22 b(Io\013e,)i Fv(Exact)i(deviation)f(b)l(ounds)h(up)f(to)h FB(T)1598 3388 y Fq(c)1654 3380 y Fv(for)f(the)g(Ising)g(mo)l(del)f(in) g(two)i(dimensions)6 b FD(,)24 b(Prob.)f(Th.)g(Rel.)g(Fields)276 3472 y Fc(102)p FD(,)k(313{330)h(\(1995\).)118 3563 y([ISc])39 b(D.)26 b(Io\013e,)i(R.)e(Sc)n(honmann,)h Fv(Dobrushin-Kote)l(ck)o (\023)-38 b(y-Shlosman)33 b(the)l(ory)d(up)f(to)g(the)g(critic)l(al)g (temp)l(er)l(atur)l(e)6 b FD(,)30 b(Comm.)276 3654 y(Math.)c(Ph)n(ys.)g Fc(199)p FD(,)g(117{167)j(\(1998\).)118 3745 y([LMR])39 b(L.)33 b(Laanait,)j(A.)c(Messager,)37 b(J.)c(Ruiz,)i Fv(Phases)g(c)l(o)l(existenc)l(e)h(and)f(surfac)l(e)g(tensions)g(for)f (the)h(Potts)g(mo)l(del.)p FD(,)276 3837 y(Comm.)27 b(Math.)f(Ph)n(ys.) g Fc(105)p FD(,)g(No.)g(4,)g(527{545)j(\(1986\).)118 3928 y([LMP])40 b(J.)28 b(Leb)r(o)n(witz,)g(Mazel,)h(Presutti,)f Fv(Liquid-vap)l(or)j(phase)f(tr)l(ansitions)g(for)f(systems)i(with)e (\014nite-r)l(ange)i(inter)l(ac-)276 4019 y(tions.)p FD(,)26 b(J.)h(Stat.)e(Ph)n(ys.)h Fc(94)p FD(,)g(No.5-6,)h(955{1025)i (\(1999\).)118 4111 y([MMR])39 b(A.)19 b(Messager,)j(S.)d(Miracle-Sol)n (\023)-36 b(e,)22 b(J.)d(Ruiz,)h Fv(Surfac)l(e)i(tension,)h(step)g(fr)l (e)l(e)e(ener)l(gy)i(and)e(fac)l(ets)h(in)f(the)h(e)l(quilibrium)276 4202 y(crystal)8 b FD(,)27 b(J.)f(Stat.)g(Ph)n(ys.)g Fc(79)p FD(,)g(\(1995\).)118 4293 y([MMRS])39 b(A.)26 b(Messager,)j(S.)d(Miracle-Sol)n(\023)-36 b(e,)28 b(J.)f(Ruiz,)f(S.)g (Shlosman,)h Fv(Interfac)l(es)i(in)f(the)h(Potts)g(mo)l(del.)f(II.)e(A) n(ntonov's)276 4385 y(rule)i(and)g(rigidity)f(of)h(the)g(or)l(der)h (disor)l(der)g(interfac)l(e)p FD(,)e(Comm.)g(Math.)f(Ph)n(ys.)f Fc(140)p FD(,)i(No.2,)g(275{290)h(\(1991\).)118 4476 y([Pf)6 b(])40 b(C.E.)28 b(P\014ster,)h Fv(L)l(ar)l(ge)h(deviations)g (and)g(phase)g(sep)l(ar)l(ation)h(in)e(the)h(two)g(dimensional)f(Ising) f(mo)l(del)8 b FD(,)28 b(Helv.)f(Ph)n(ys.)276 4567 y(Acta)f Fc(64)p FD(,)g(953{1054)j(\(1991\).)118 4659 y([PV2])39 b(C.-E.)30 b(P\014ster,)g(Y)57 b(V)-6 b(elenik,)30 b Fv(L)l(ar)l(ge)h(deviations)h(and)f(c)l(ontinuum)g(limit)e(in)h(the)h (2D)g(Ising)f(mo)l(del)8 b FD(,)30 b(Prob.)f(Th.)276 4750 y(Rel.)d(Fields)g Fc(109)p FD(,)h(435{506)h(\(1997\).)118 4841 y([Pi])43 b(A.)35 b(Pisztora,)k Fv(Surfac)l(e)e(or)l(der)g(lar)l (ge)g(deviations)g(of)f(Ising,)h(Potts)h(and)e(p)l(er)l(c)l(olation)i (mo)l(dels)6 b FD(,)38 b(Prob.)e(Th.)f(Rel.)276 4933 y(Fields)26 b Fc(104)p FD(,)h(427{466)h(\(1996\).)118 5024 y([V)-6 b(e])39 b(Y.)25 b(V)-6 b(elenik,)25 b Fv(Phase)k(sep)l(ar) l(ation)g(as)f(a)f(lar)l(ge)i(deviations)f(pr)l(oblem:)36 b(a)27 b(micr)l(osc)l(opic)i(derivation)f(of)f(surfac)l(e)i(ther-)276 5115 y(mo)l(dynamics)f(for)g(some)g(2D)f(spin)h(systems)6 b FD(,)28 b(Th)n(\022)-36 b(ese)26 b Fc(1712)g FD(EPF-L,)h(\(1997\).)p eop end %%Page: 42 42 TeXDict begin 42 41 bop 118 45 a Fw(42)1128 b(T.)23 b(BODINEA)n(U)g (AND)g(E.)g(PRESUTTI)218 366 y FC(Universit)582 360 y(\023)582 366 y(e)g(P)-7 b(aris)26 b(7)f(and)g(Labora)-5 b(toire)25 b(de)g(Pr)n(obabilit)2117 360 y(\023)2117 366 y(es)h(et)f(Mod)2498 360 y(\022)2498 366 y(eles)g(Al)2778 360 y(\023)2778 366 y(ea)-5 b(toires)25 b(C.N.R.S.)f(UMR)118 458 y(7599,)29 b(U.F.R.)f(Ma)-5 b(th)814 452 y(\023)814 458 y(ema)g(tiques,)27 b(Case)h(7012,)h(2)g(Pla)n(ce)g(Jussieu,)e(F-75251)k(P)-7 b(aris,)28 b(France)218 549 y Fv(E-mail)f(addr)l(ess)6 b FD(:)36 b Fa(bodineau@gauss.math.jussieu.)q(fr)218 704 y FC(Errico)i(Presutti,)h(Dip)-5 b(ar)g(timento)36 b(di)g(Ma)-5 b(tema)g(tica,)39 b(Universit)2447 698 y(\022)2445 704 y(a)c(di)i(R)n(oma)g(Tor)g(Vergat)-5 b(a,)39 b(00133)118 796 y(R)n(oma,)28 b(It)-5 b(al)g(y)218 887 y Fv(E-mail)27 b(addr)l(ess)6 b FD(:)36 b Fa(presutti@mat.uniroma2.it)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0301220745781--