This is a multi-part message in MIME format. ---------------0407091152125 Content-Type: text/plain; name="04-208.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-208.comments" MSC2000: 82B28; 82B44; 60K35. Number of pages 59 ---------------0407091152125 Content-Type: text/plain; name="04-208.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-208.keywords" Lattice systems, Cluster expansion, Disordered systems, Renormalization group. ---------------0407091152125 Content-Type: application/postscript; name="bcobat-preprint.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="bcobat-preprint.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.92b Copyright 2002 Radical Eye Software %%Title: bcobat-preprint.dvi %%Pages: 59 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 CMR12 CMR10 CMBX10 CMTI10 CMTT10 CMBX12 CMTI12 %%+ CMMI12 CMMI8 CMSY10 CMSY8 MSBM10 CMR8 CMEX10 CMSY6 CMMI6 CMR6 %%+ LCIRCLE10 CMSL12 MSBM7 CMTT12 TeX-cmex8 TeX-cmex7 MSAM10 MSAM7 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips bcobat-preprint %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2004.07.09:1834 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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 71 /G put dup 76 /L put dup 84 /T put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueID 5032014 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6A66A4000A13D5F68BFF326D 1D432B0D064B56C598F4338C319309181D78E1629A31ECA5DD8536379B03C383 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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 (CMTT12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch true def end readonly def /FontName /CMTT12 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 /.notdef put readonly def /FontBBox{-1 -234 524 695}readonly def /UniqueID 5000833 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F0364CD5660FE13FF01BC20148F9C480BCD0E C81D5BFC66F04993DD73F0BE0AB13F53B1BA79FE5F618A4F672B16C06BE3251E 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All Rights Reserved) readonly def /FullName (LCIRCLE10) readonly def /FamilyName (LaTeX) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /LCIRCLE10 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 103 /a103 put dup 115 /a115 put dup 116 /a116 put readonly def /FontBBox{-2000 -2000 2040 2040}readonly def /UniqueID 5011942 def currentdict end currentfile eexec D9D66F637A9E5292A4933615152D29EEC26E1BED2E48CAB7AC058698EA30B07E F8BDB66981B14445E1107420FDAF32EDBD5C26E35B334E3AF24373B2A13984D9 1D56801ACCF98612DE2C19685E0F4D52369AD642D169AB57DAB10665C6C01538 497544391FABD08A483766E9B5C7ABA9A537C053A8ADC16DEBD8B5C9E1EDEB21 E4E89B0945C0737B96F0194585BE0D9B958F1030EA84710C31971BA7E93777C2 0658995DC8702442780303E948079822B9EE0C531A5CC49578EB9E683CB77495 DF613FBDC4536BD7A90F044B7EEE6EACA2042539F62789EA39AA16433AC4BD0F 98AAA04894F5487C279D2766B50B23E16D7F1023E387FA4165DF522DB290D390 B669550A43638C3C9CC38C7667A22D27F9DB1542620110A8671B2713C8DB93D3 4C500474AC190831F9C7441DC4A7C27EE88DAEF10D7AEE6FDB22E7625E802B4C FCEE0DC79988734BDA39B9896B1833D4BEC4F91DB321F343F57B3FC5ECA13AEA AB108165934BDE65D2EB5A62BD395AB84402D4A2703E3DC3BEC860D5A4EB785A 1D8D0D1AE7D31BFCD3ABE3E9779B6BC28D93CFCFD8461032D969F1C9A53A6DCA 321062C6750132DD47444F208275E5000794E261F842CA01FD5DCA6F8C92C7EA E2D8D66ADB1194BD8AC59E9191895A2FCA88F32DA2B27717F18C258EE2402C69 5655EE4D15708452687828A90653E9FC2707F1E9ACA413DAD6E5EE607F6DCB97 09B7195203A395C97CCC7390F38D69AEC6323A0AAE0A05958B7987701BBD6497 F95D8653699C39F179820FB8C2DBD7A4E7EDF03E83F7536946A3F532650AC5CD 48C10BA3AB8F67915EAC0053592CA317E0A8EFD1322398DB7D9B32ECE97791A0 0B49839C58E426531833D30F153863E92A79516C2878AEE6916F0FA78F94714E D709C10DB24955FC0DE6769D34E4E1DB2ADED1775C5A5A6313C93B6A0195311B 4A334B4D9B914EA59194239AAD1358BC269B8CBD35A7685ACB29282D46440A38 083B9CB2258BB81E2489B5FBF551AEA1330D4DF70E9E650964A369CA2F77DEBD 0CDBCECDD2DFE4EE609FBBCB1766B2F8A18A4A796E5AAA06B1045AABB36F4551 C3B86D2B605328AAB01B1507282AEF46CD40332C389B259AD1C7FBFFD92C36A7 19BB001822EA510B8B0D6A9E7D96DF6D6F1B1DA30E735A2EE0A5F24386458CCB 23CA23E2AB13300B651EBC58FAD54DE3454D7D285277C6CF4CB0C30AED1615C6 AC6A8AFE29D670F86710F80ED4ACC67AADA61B7DAA1FFA079C3CC09ABE553CC3 2E30B88CF48677C56340E5BFE95AAD1C468903F47B5F8ED16465B1D15F1A7ACC D0117C32C0E43915338B67D1DA6083C96CEF858D0A83027A746EE41DBC129A6F 857D8DA31B6014B1FBD83AF6087BB97E41A3C7E641820A79CA5188ADD64A9A11 887C3138335147D50CFEE1C98C7EC1B47B0FE4851D7B2B5F848CCB712B795B3F C75D146BB854258D31169A3CD7836CC8D970992CCB1A09E885C3D65F5FAF5CCD 025DFA045BA0DF305FBD816EE38E1BA1459E92C1FFB53912C4AACB32CF887954 0C345F495E94E2ED107883DB181F1BF93E9D4D5757B2991A4BFEA9B4B1EF6540 7C252C3685A6966B8A852DFF4F4FCFEC08A05AC2CB6DCDB4BB8EF5EAF92F7BD8 00D1F572 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % 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 (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 0 /.notdef put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 5250011D19E9366EB6FD153D3A100CAA6212E3D5D93990737F8D326D347B7EDC 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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 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 2 /bracketleftbig put dup 3 /bracketrightbig put dup 8 /braceleftbig put dup 9 /bracerightbig put dup 12 /vextendsingle put dup 13 /vextenddouble put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 20 /bracketleftbigg put dup 21 /bracketrightbigg put dup 26 /braceleftbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 40 /braceleftBigg put dup 41 /bracerightBigg put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 62 /braceex put dup 78 /circlemultiplytext put dup 79 /circlemultiplydisplay put dup 80 /summationtext put dup 81 /producttext 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 95 /logicalordisplay put dup 98 /hatwide put dup 99 /hatwider put dup 101 /tildewide put dup 102 /tildewider put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 110 /braceleftBig put dup 111 /bracerightBig put dup 112 /radicalbig 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 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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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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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%%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 /.notdef put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D5FC1B2109839E5B52DFBB2A7C1B5D8E7E8AA0 5B10EA43D6A8ED61AF5B23D49920D8F79DAB6A59062134D84AC0100187A6CD1F 80F5DDD9D222ACB1C23326A7656A635C4A241CCD32CBFDF8363206B8AA36E107 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cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D31FF2B87F97C73D63EECDDA4C49501773468A 27D1663E0B62F461F6E40A5D6676D1D12B51E641C1D4E8E2771864FC104F8CBF 5B78EC1D88228725F1C453A678F58A7E1B7BD7CA700717D288EB8DA1F57C4F09 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI12 %!PS-AdobeFont-1.1: CMTI12 1.0 %%CreationDate: 1991 Aug 18 21:06:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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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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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 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. 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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 0 /.notdef put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR17) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR17 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-33 -250 945 749}readonly def /UniqueID 5000795 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F075EA0A10A15B0ED05D5039DA41B32B16E95 A3CE9725A429B35BAD796912FC328E3A28F96FCADA20A598E247755E7E7FF801 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y(g)g(g)g(g)f(g)h(g)g(g)g(g)1038 473 y(g)g(g)g(g)f(g)h(g)g(g)g(g)1038 174 y(g)g(g)g(g)f(g)h(g)g(g)g(g)1038 1968 y(s)166 b(s)g(s)h(s)f(s)g(s)g(s)h(s)f(s)g(s)1038 1768 y(s)g(s)g(s)h(s)f(s)g(s)g(s)h(s)f(s)g(s)1038 1569 y(s)g(s)g(s)h(s)f(s)g(s)g(s)h(s)f(s)g(s)1038 1370 y(s)g(s)g(s)h(s)f(s)g (s)g(s)h(s)f(s)g(s)1038 1171 y(s)g(s)g(s)h(s)f(s)g(s)g(s)h(s)f(s)g(s) 1038 971 y(s)g(s)g(s)h(s)f(s)g(s)g(s)h(s)f(s)g(s)1038 772 y(s)g(s)g(s)h(s)f(s)g(s)g(s)h(s)f(s)g(s)1038 573 y(s)g(s)g(s)h(s)f(s)g(s)g(s)h(s)f(s)g(s)1038 374 y(s)g(s)g(s)h(s)f(s)g (s)g(s)h(s)f(s)g(s)1038 174 y(s)g(s)g(s)h(s)f(s)g(s)g(s)h(s)f(s)g(s)p 986 2017 4 100 v 1584 2017 V 2182 2017 V 2780 2017 V 986 1420 V 1584 1420 V 2182 1420 V 2780 1420 V 986 822 V 1584 822 V 2182 822 V 2780 822 V 986 224 V 1584 224 V 2182 224 V 2780 224 V 988 1919 100 4 v 1586 1919 V 2183 1919 V 2781 1919 V 988 1322 V 1586 1322 V 2183 1322 V 2781 1322 V 988 724 V 1586 724 V 2183 724 V 2781 724 V 988 126 V 1586 126 V 2183 126 V 2781 126 V 1086 2017 4 100 v 1684 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b(let)33 b Ft(h)27 b Fv(:)h Fp(N)g Fr(!)f Fp(R)1492 422 y Fo(+)1584 407 y Fv(and)32 b(set)1223 641 y Fr(k)p Fv(\010)1343 600 y Fo(\()p Fs(s)p Fo(\))1435 641 y Fr(k)1485 656 y Fo(DM)1635 641 y Fv(:=)1766 547 y Fn(X)1766 758 y Fs(X)5 b Fq(3)p Fo(0)1928 641 y Ft(h)p Fv(\()p Fr(j)p Ft(X)j Fr(j)p Fv(\))17 b Fr(k)p Fv(\010)2342 590 y Fo(\()p Fs(s)p Fo(\))2342 668 y Fs(X)2433 641 y Fr(k)2483 656 y Fq(1)0 955 y Fv(If)33 b(exp)q Fr(f\000)p Ft(\015)5 b(n)p Fr(g)p Ft(h)p Fv(\()p Ft(n)p Fv(\))30 b Fr(!)e Fv(0)33 b(in)h(the)g(limit)f Ft(n)d Fr(!)e(1)33 b Fv(for)g(all)g Ft(\015)h(>)29 b Fv(0,)k(there)h(exist)h(complex)g(in) m(teractions)0 1075 y(with)h(arbitrarily)f(small)h(norm)f Fr(k)24 b(\001)g(k)1387 1090 y Fo(DM)1509 1075 y Fv(,)36 b(giving)g(rise)g(to)f(phase)h(transition)f(in)h(the)g(sense)h(that)e (the)0 1196 y(corresp)s(onding)29 b(partition)f(function)g(v)-5 b(anishes)30 b(for)d(a)h(sequence)i(of)e(cub)s(es)h(\003)2839 1211 y Fs(n)2913 1196 y Fr(!)f(L)3110 1159 y Fo(\()p Fs(s)p Fo(\))3201 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b Fv(\()p Ft(M)3532 4102 y Fo(\()p Fs(`)p Fo(\))3522 4179 y Fs(I)3621 4152 y Fv(\))3659 4116 y Fq(\000)p Fo(1)3753 4152 y Fv(.)0 4273 y(W)-8 b(e)29 b(a)m(v)m(oid)h(the)f(troublesome)h(issue)g(of)e(describing)j(Gibbs)e (measures)h(on)f(non{compact)g(single)h(spin)0 4393 y(space,)k(see)f ([19])g(for)f(a)g(discussion,)j(and)e(consider)h Ft(\027)1968 4357 y Fo(\()p Fs(`)p Fo(\))2088 4393 y Fv(only)g(for)e(\014nite)h Ft(`)p Fv(.)0 4610 y Fu(2.7.)79 b(Main)35 b(r)-5 b(esults)0 4754 y Fv(In)24 b(this)h(subsection)h(w)m(e)g(state)e(the)h(main)f (theorems)i(on)e(the)g(w)m(eak)i(Gibbsianit)m(y)f(of)f(the)h (renormalized)0 4875 y(measure)34 b(and)f(the)g(corresp)s(onding)g(con) m(v)m(ergence)i(as)e Ft(`)28 b Fr(!)f(1)p Fv(.)p Black 0 5073 a Fw(Theorem)32 b(2.1.)p Black 38 w Fu(L)-5 b(et)30 b Ft(U)40 b Fu(b)-5 b(e)30 b(a)g(lattic)-5 b(e)29 b(gas)h(p)-5 b(otential)29 b(satisfying)h(SM\()p Ft(`)2638 5088 y Fo(0)2677 5073 y Ft(;)17 b(b;)g(B)5 b Fu(\).)43 b(Then)29 b(for)h(any)g(lar)-5 b(ge)0 5207 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b(otheses)28 b(of)i(The)-5 b(or)g(em)28 b(3.3)h(b)-5 b(e)29 b(satis\014e)-5 b(d.)42 b(Set)29 b Ft(p)2794 3882 y Fs(k)2865 3867 y Fv(:=)e(sup)3142 3891 y Fs(x)p Fq(2)p Fh(L)3294 3867 y Fp(Q)17 b Fv(\()p Ft(x)28 b Fr(2)g Fp(B)3669 3882 y Fs(k)3712 3867 y Fv(\))p Fu(,)0 3987 y(for)35 b Ft(k)30 b Fv(=)e(0)p Ft(;)17 b Fv(1)p Ft(;)g(:)g(:)g(:)e Fu(;)35 b(set)g(also)f Ft(A)1146 4002 y Fs(k)1189 3987 y Fv(\()p Ft(x)p Fv(\))28 b(:=)f Ft(O)1553 4002 y Fs(\015)1589 4014 y Fl(k)1627 4002 y Fo(+\000)1726 4014 y Fl(k)1768 3987 y Fv(\()p Ft(x)p Fv(\))c Fr(n)f Ft(O)2069 4003 y Fo(\(\000)2140 4015 y Fl(k)2178 4003 y Fq(\000)p Fo(1\))p Fs(=)p Fo(2)2370 3987 y Fv(\()p Ft(x)p Fv(\))35 b Fu(and)g Fr(j)p Ft(A)2827 4002 y Fs(k)2869 3987 y Fr(j)27 b Fv(=)h Fr(j)p Ft(A)3129 4002 y Fs(k)3171 3987 y Fv(\()p Ft(x)p Fv(\))p Fr(j)p Fu(.)45 b(Then)34 b(for)0 4108 y(e)-5 b(ach)34 b Ft(k)d Fv(=)c(0)p Ft(;)17 b Fv(1)p Ft(;)g(:)g(:)g(:)50 b Fu(we)35 b(have)885 4350 y Ft(p)934 4365 y Fs(k)r Fo(+1)1095 4350 y Fr(\024)28 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Fs(d)3000 3117 y Ft(#)3057 3081 y Fs(d)3057 3143 y(k)r Fo(+1)3190 3117 y Fv(,)i(and)e Fr(j)p Ft(O)3555 3132 y Fs(#)3597 3144 y Fl(k)3638 3117 y Fr(j)f(\024)0 3237 y Fv(3)49 3201 y Fs(d)89 3237 y Ft(#)146 3201 y Fs(d)146 3263 y(k)r Fo(+1)280 3237 y Fv(.)p 3709 3237 V 3713 3175 59 4 v 3713 3237 V 3771 3237 4 66 v 0 3547 a Fw(4.)h(The)f(constrained)i(mo)s(dels)0 3691 y Fv(In)j(dealing)f (with)h(the)g(renormalization)g(group)f(transformation)g(it)h(is)g (necessary)h(to)e(pac)m(k)i(spins)0 3812 y(asso)s(ciated)37 b(to)e(di\013eren)m(t)i(sites)g(of)e(the)h(lattice)g(so)g(that)g(a)f (new)i(v)-5 b(ariable,)36 b(often)g(called)h Fu(blo)-5 b(ck)37 b(spin)p Fv(,)0 3932 y(is)c(obtained.)0 4148 y Fu(4.1.)79 b(The)34 b(blo)-5 b(ck)34 b(spin)h(mo)-5 b(dels)0 4292 y Fv(Recall)27 b(the)g(general)g(setup)g(in)m(tro)s (duced)h(in)f(Subsections)h(2.2{2.4)e(for)g(a)g(spin)h(mo)s(del)g(on)g (lattice)g Fr(L)3662 4255 y Fo(\()p Fs(s)p Fo(\))3753 4292 y Fv(,)0 4412 y(with)j Ft(s)e Fr(\025)g 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Fv(the)f(family)g(of)g(all)g(not)g (empt)m(y)h(subsets)h Ft(I)i Fr(\032)30 b(L)3242 852 y Fo(\()p Fs(})p Fo(\))3380 888 y Fv(suc)m(h)35 b(that)0 1009 y(for)c(eac)m(h)h Ft(j)h Fr(2)28 b Ft(I)39 b Fv(w)m(e)33 b(ha)m(v)m(e)f Ft(j)1021 1024 y Fs(k)1092 1009 y Fv(=)27 b Ft(i)1228 1024 y Fs(k)1303 1009 y Fv(and)k Ft(j)1531 1024 y Fs(h)1604 1009 y Fr(2)d(f)p Ft(i)1781 1024 y Fs(h)1845 1009 y Fr(\000)20 b Ft(};)d(i)2081 1024 y Fs(h)2126 1009 y Ft(;)g(i)2203 1024 y Fs(h)2267 1009 y Fv(+)i Ft(})p Fr(g)31 b Fv(for)g(all)g Ft(h)d Fv(=)f(1)p Ft(;)17 b(:)g(:)g(:)f(;)h(d) 30 b Fv(and)h Ft(h)d Fr(6)p Fv(=)g Ft(k)s Fv(.)0 1129 y(W)-8 b(e)33 b(set)1092 1249 y Ft(I)1135 1264 y Fq(\006)1222 1249 y Fv(:=)28 b Ft(@)1409 1208 y Fo(\()p Fs(})p Fo(\))1513 1249 y Ft(I)i Fr(\\)23 b(f)p Ft(j)33 b Fr(2)28 b(L)1961 1208 y Fo(\()p Fs(})p Fo(\))2093 1249 y Fv(:)44 b Ft(j)2204 1264 y Fs(k)2275 1249 y Fv(=)27 b Ft(i)2411 1264 y Fs(k)2476 1249 y Fr(\006)c Ft(})p Fr(g)0 1406 y Fv(and,)49 b(for)d Ft(m)51 b Fr(2)f(M)768 1370 y Fo(\()p Fs(`)p Fo(\))902 1406 y Fv(and)c Ft(\033)54 b Fr(2)d(X)1417 1370 y Fs(`)1450 1406 y Fv(,)e Ft(\033)1581 1421 y Fq(\006)1691 1406 y Fv(:=)h Ft(\033)1899 1425 y Fq(O)1957 1407 y Fl(`)1955 1442 y(})1999 1425 y Fs(I)2030 1434 y Fm(\006)2132 1406 y Fv(and)c Ft(\033)2390 1421 y Fo(0)2480 1406 y Fv(:=)51 b Ft(\033)2689 1425 y Fq(O)2747 1407 y Fl(`)2745 1442 y(})2788 1425 y Fo(\()p Fs(I)2846 1434 y Fk(+)2897 1425 y Fq([)p Fs(I)2975 1434 y Fm(\000)3027 1425 y Fo(\))3054 1407 y Fk(c)3090 1406 y Fv(.)83 b(Recall)47 b Ft(\016)54 b(>)c Fv(0)0 1561 y(has)44 b(b)s(een)g(in)m(tro)s(duced)h(in)e (\(4.8\),)j(giv)m(en)f Ft(J)55 b Fr(\032)47 b(L)1891 1525 y Fo(\()p Fs(`)p Fo(\))2022 1561 y Fv(w)m(e)d(set)h Fr(D)2420 1510 y Fo(\()p Fs(`)p Fo(\))2417 1589 y Fs(\016)2507 1561 y Fv(\()p Ft(J)9 b Fv(\))46 b(:=)g Fr(f)p Ft(m)h Fr(2)f(M)3255 1525 y Fo(\()p Fs(`)p Fo(\))3389 1561 y Fv(:)63 b Fr(j)p Ft(m)3592 1576 y Fs(j)3629 1561 y Fr(j)46 b(\024)0 1681 y Ft(`)41 1645 y Fs(d)p Fo(\(1)p Fs(=)p 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3365 40 3 v 974 3389 a Fv(:=)27 b Fr(f)p Ft(U)1220 3404 y Fs(X)1316 3389 y Fr(2)h(F)1482 3404 y Fs(X)1549 3389 y Ft(;)33 b(X)i Fr(\032\032)29 b(Lg)j Fv(with)522 3626 y Ft(U)598 3577 y Fs(z)p 598 3589 V 588 3653 a(X)656 3626 y Fv(\()p Ft(\021)t Fv(\))27 b(:=)941 3486 y Fn(\032)1058 3565 y Ft(\021)1106 3580 y Fs(x)1166 3565 y Fv(log)18 b Ft(z)1354 3580 y Fs(x)1514 3565 y Fv(if)32 b(there)h(exists)i Ft(x)28 b Fr(2)g(L)k Fv(suc)m(h)i(that)f Ft(X)i Fv(=)28 b Fr(f)p Ft(x)p Fr(g)1058 3686 y Ft(U)1124 3701 y Fs(X)1191 3686 y Fv(\()p Ft(\021)t Fv(\))195 b(if)32 b Fr(j)p Ft(X)8 b Fr(j)27 b Ft(>)g Fv(1)0 3875 y(for)38 b(all)h Ft(\021)j Fr(2)d(X)15 b Fv(;)41 b(w)m(e)f(shall)g(use)g(the)f (notation)f Ft(U)1852 3839 y Fs(z)p 1852 3851 V 1931 3875 a Fv(:=)g(\()p Ft(z)p 2110 3891 50 4 v 5 w(;)17 b(U)2270 3890 y Fs(>)p Fo(1)2364 3875 y Fv(\).)62 b(Giv)m(en)40 b(\003)e Fr(\032\032)h(L)3148 3839 y Fo(\()p Fs(`)p Fo(\))3275 3875 y Fv(and)g Ft(\034)50 b Fr(2)38 b(X)15 b Fv(,)0 3995 y(the)39 b(\014nite)g(v)m(olume)g(Gibbs)g(measure)h(with)f(b)s (oundary)f(condition)h Ft(\034)50 b Fv(asso)s(ciated)39 b(with)g(the)f(lattice)0 4116 y(gas)29 b(p)s(oten)m(tial)g(with)g(not)g (homogeneous)h(activit)m(y)g Ft(U)1981 4079 y Fs(z)p 1981 4091 40 3 v 2050 4116 a Fv(is)f(denoted)h(b)m(y)f Ft(\026)2699 4079 y Fs(\034)2699 4141 y Fo(\003)p Fs(;z)p 2768 4153 36 3 v 2836 4116 a Fv(and)g(the)g(corresp)s(onding)0 4236 y(partition)j(function)i(b)m(y)f Ft(Z)998 4200 y Fs(\034)991 4261 y Fo(\003)p Fs(;z)p 1060 4273 V 1099 4236 a Fv(.)44 b(Namely)-8 b(,)33 b(w)m(e)h(ha)m(v)m(e)828 4458 y Ft(Z)902 4417 y Fs(\034)895 4482 y Fo(\003)p Fs(;z)p 964 4494 V 1031 4458 a Fv(:=)1180 4363 y Fn(X)1161 4574 y Fs(\021)r Fq(2X)1295 4585 y Fk(\003)1359 4458 y Fv(exp)1524 4347 y Fn(n)1662 4363 y(X)1617 4566 y Fl(X)t Fm(\\)p Fk(\003)p Fm(6)p Fk(=)p Fm(;)p Fk(:)1649 4623 y Fm(j)p Fl(X)t Fm(j)p Fl(>)p Fk(1)1876 4458 y Ft(U)1942 4473 y Fs(X)2010 4458 y Fv(\()p Ft(\021)t(\034)2142 4473 y Fo(\003)2191 4454 y Fk(c)2227 4458 y Fv(\))22 b(+)2385 4363 y Fn(X)2389 4574 y Fs(x)p Fq(2)p Fo(\003)2545 4458 y Ft(\021)2593 4473 y Fs(x)2654 4458 y Fv(log)17 b Ft(z)2841 4473 y Fs(x)2886 4347 y Fn(o)3531 4458 y Fv(\(4.10\))146 4787 y(It)23 b(is)h(easy)g(to)f(see)h(that)f(b)m(y)h([13,)t(14],)g(see) h(Remark)e(2)g(in)g([1,)g(p.)h(849],)g(the)g(follo)m(wing)f(lemma,)j (stating)0 4907 y(that)32 b(the)h(strong)g(mixing)g(condition)h (\(2.7\))e(is)h(satis\014ed)h(uniformly)f(in)g(the)g(activities,)h (holds.)p Black 0 5076 a Fw(Lemma)49 b(4.2.)p Black 47 w Fu(L)-5 b(et)43 b(the)g(lattic)-5 b(e)44 b(gas)e(p)-5 b(otential)43 b Ft(U)54 b Fv(=)43 b(\()p Ft(z)t(;)17 b(U)2292 5091 y Fs(>)p Fo(1)2386 5076 y Fv(\))44 b Fu(satisfy)f (Condition)f(SM)p Fv(\()p Ft(`)3465 5091 y Fo(0)3504 5076 y Ft(;)17 b(b;)g(B)5 b Fv(\))p Fu(.)0 5196 y(Then)42 b(ther)-5 b(e)43 b(exist)f Ft(")g(>)h Fv(0)p Fu(,)h Ft(`)1118 5160 y Fq(0)1118 5221 y Fo(0)1200 5196 y Fu(multiple)f(of)f Ft(`)1745 5211 y Fo(0)1785 5196 y Fu(,)i(and)f(two)f(p)-5 b(ositive)43 b(r)-5 b(e)g(als)42 b Ft(b)2880 5160 y Fq(0)2946 5196 y Fv(=)g Ft(b)3105 5160 y Fq(0)3129 5196 y Fv(\()p Ft(";)17 b(b;)g(B)5 b(;)17 b(`)3506 5211 y Fo(0)3545 5196 y Fv(\))43 b Fu(and)0 5316 y Ft(B)79 5280 y Fq(0)134 5316 y Fv(=)32 b Ft(B)321 5280 y Fq(0)344 5316 y Fv(\()p Ft(";)17 b(b;)g(B)5 b(;)17 b(`)721 5331 y Fo(0)760 5316 y Fv(\))31 b Ft(<)h Fr(1)37 b Fu(such)f(that)i(for)f Ft(z)p 1657 5332 50 4 v 36 w Fv(=)32 b Fr(f)p Ft(z)1941 5331 y Fs(x)2016 5316 y Fr(2)g Fv([0)p Ft(;)17 b Fr(1)p Fv(\))31 b(:)48 b Ft(x)33 b Fr(2)f(Lg)k Fu(such)h(that)g Fr(j)p Ft(z)3317 5331 y Fs(x)3385 5316 y Fr(\000)24 b Ft(z)t Fr(j)32 b(\024)g Ft(")p Fu(,)0 5437 y(for)26 b(al)5 b(l)26 b Ft(x)i Fr(2)g(L)p Fu(,)f(the)f(lattic)-5 b(e)27 b(gas)e(p)-5 b(otential)26 b(with)g(not)g(homo)-5 b(gene)g(ous)25 b(activity)h Ft(U)2921 5401 y Fs(z)p 2921 5413 40 3 v 2989 5437 a Fv(=)i(\()p Ft(z)p 3131 5453 50 4 v 4 w(;)17 b(U)3290 5452 y Fs(>)p Fo(1)3385 5437 y Fv(\))26 b Fu(satis\014es)0 5557 y(SM)p Fv(\()p Ft(`)221 5521 y Fq(0)221 5582 y Fo(0)260 5557 y Ft(;)17 b(b)345 5521 y Fq(0)369 5557 y Ft(;)g(B)492 5521 y Fq(0)515 5557 y Fv(\))p Fu(.)p Black 1841 5806 a Fv(24)p Black eop end %%Page: 25 26 TeXDict begin 25 25 bop Black Black 0 407 a Fu(Pr)-5 b(o)g(of)35 b(of)g(The)-5 b(or)g(em)35 b(4.1.)d Fv(Let)i Ft(\026)1210 422 y Fs(z)1282 407 y Fv(b)s(e)g(the)f(unique)i (in\014nite)f(v)m(olume)h(Gibbs)f(measure)g(of)f(the)g(lattice)0 527 y(gas)42 b(with)g(p)s(oten)m(tial)g Ft(U)53 b Fv(=)43 b(\()p Ft(z)t(;)17 b(U)1265 542 y Fs(>)p Fo(1)1360 527 y Fv(\))42 b(and)g(set)g Ft(\045)h Fv(:)g(\(0)p Ft(;)17 b Fv(+)p Fr(1)p Fv(\))42 b Fr(3)i Ft(z)k Fr(\000)-16 b(!)42 b Ft(\045)p Fv(\()p Ft(z)t Fv(\))i(:=)f Ft(\026)3180 542 y Fs(z)3220 527 y Fv(\()p Ft(\021)3306 542 y Fo(0)3345 527 y Fv(\))g Fr(2)h Fv(\(0)p Ft(;)17 b Fv(1\).)0 648 y(Let)40 b Ft(")g(>)g Fv(0)g(b)s(e)g(as)g(in)g(Lemma)h(4.2,)h(b)m(y)e (the)h(con)m(tin)m(uit)m(y)h(of)d Ft(\045)i Fv(w)m(e)g(can)f(c)m(ho)s (ose)h Ft(")3106 611 y Fq(0)3169 648 y Ft(>)f Fv(0)g(suc)m(h)h(that)0 768 y Ft(\045)50 732 y Fq(\000)p Fo(1)161 768 y Fv(\()p Ft(\045)p Fv(\()p Ft(z)t Fv(\))23 b Fr(\000)g Fv(2)p Ft(")592 732 y Fq(0)615 768 y Ft(;)17 b(\045)p Fv(\()p Ft(z)t Fv(\))22 b(+)g(2)p Ft(")1049 732 y Fq(0)1072 768 y Fv(\))28 b Fr(\032)g Fv([)p Ft(z)20 b Fr(\000)15 b Ft(";)i(z)j Fv(+)15 b Ft(")p Fv(].)41 b(The)31 b(thesis)f(follo)m(ws)g (b)m(y)g(Lemma)g(4.2)e(and)i([1,)f(Prop.)0 888 y(5.1].)p 3709 888 4 66 v 3713 826 59 4 v 3713 888 V 3771 888 4 66 v 0 1093 a Fu(4.5.)79 b(On)34 b(the)h(sp)-5 b(arseness)34 b(of)g(b)-5 b(ad)35 b(sites)0 1235 y Fv(In)g(this)h(subsection)h(w)m(e) g(state)e(precisely)j(in)d(whic)m(h)i(sense)g(the)e(bad)h(sites)g(in)f Fr(L)3001 1199 y Fo(\()p Fs(})p Fo(\))3140 1235 y Fv(are)g(sparse.)53 b(W)-8 b(e)0 1355 y(de\014ne)34 b(the)f(map)g Ft(\031)e Fv(:)d Fr(M)928 1319 y Fo(\()p Fs(`)p Fo(\))p Fs(;})1107 1355 y Fr(!)g(f)p Fv(0)p Ft(;)17 b Fv(1)p Fr(g)1477 1319 y Fq(L)1526 1296 y Fk(\()p Fl(})p Fk(\))1652 1355 y Fv(b)m(y)34 b(setting)f(for)f(eac)m(h)h Ft(n)28 b Fr(2)g(M)2776 1319 y Fo(\()p Fs(`)p Fo(\))p Fs(;})2961 1355 y Fv(and)k Ft(t)c Fr(2)g(L)3376 1319 y Fo(\()p Fs(})p Fo(\))1228 1535 y Fn(\000)1273 1615 y Ft(\031)t Fv(\()p Ft(n)p Fv(\))1466 1535 y Fn(\001)1512 1654 y Fs(t)1569 1615 y Fv(:=)1700 1475 y Fn(\032)1816 1563 y Fv(0)83 b(if)32 b Ft(t)c Fr(2)g(L)2263 1512 y Fo(\()p Fs(})p Fo(\))2263 1590 y Fs(\016)2367 1563 y Fv(\()p Ft(n)p Fv(\))1816 1683 y(1)83 b(otherwise)3531 1615 y(\(4.11\))146 1870 y(As)37 b(a)f(\014rst)g(step)h(w)m(e)g(sho)m (w)g(that)f(the)h(probabilit)m(y)g(that)e(a)h(site)h(is)g(bad)f(is)g (exp)s(onen)m(tially)i(small)0 1990 y(in)33 b Ft(`)p Fv(.)p Black 0 2171 a Fw(Theorem)41 b(4.3.)p Black 43 w Fu(L)-5 b(et)38 b(the)f(lattic)-5 b(e)38 b(gas)f(p)-5 b(otential)37 b Ft(U)43 b Fv(=)32 b(\()p Ft(z)t(;)17 b(U)2309 2186 y Fs(>)p Fo(1)2404 2171 y Fv(\))37 b Fu(satisfy)g (Condition)g(SM)p Fv(\()p Ft(`)3465 2186 y Fo(0)3504 2171 y Ft(;)17 b(b;)g(B)5 b Fv(\))p Fu(.)0 2292 y(Then)30 b(ther)-5 b(e)31 b(exists)g(a)g(r)-5 b(e)g(al)30 b Ft(C)35 b Fv(=)28 b Ft(C)7 b Fv(\()p Ft(";)17 b(`)1469 2307 y Fo(0)1507 2292 y Ft(;)g(b;)g(B)5 b Fv(\))28 b Ft(>)f Fv(0)k Fu(such)g(that)g(for)g(any)g(p)-5 b(ositive)30 b(inte)-5 b(ger)31 b Ft(`)g Fu(we)g(have)0 2412 y(that)977 2533 y Fv(sup)944 2625 y Fs(t)p Fq(2L)1065 2606 y Fk(\()p Fl(})p Fk(\))1173 2533 y Ft(\027)1227 2491 y Fo(\()p Fs(`)p Fo(\))p Fs(;})1380 2533 y Fv(\(\()p Ft(\031)t Fv(\()p Ft(n)p Fv(\)\))1687 2548 y Fs(t)1744 2533 y Fv(=)c(1\))h Fr(\024)g Fv(exp)q Fr(f\000)p Ft(C)7 b(`)2461 2491 y Fo(\(1)p Fs(=)p Fo(3)p Fq(\000)p Fo(2)p Fs(\016)r Fo(\))p Fs(d)2786 2533 y Fr(g)695 b Fv(\(4.12\))0 2778 y Fu(Pr)-5 b(o)g(of.)32 b Fv(W)-8 b(e)33 b(ha)m(v)m(e)341 2979 y(sup)308 3072 y Fs(t)p Fq(2L)429 3053 y Fk(\()p Fl(})p Fk(\))537 2979 y Ft(\027)591 2938 y Fo(\()p Fs(`)p Fo(\))p Fs(;})744 2979 y Fv(\(\()p Ft(\031)t Fv(\()p Ft(n)p Fv(\)\))1051 2994 y Fs(t)1108 2979 y Fv(=)27 b(1\))84 b(=)116 b(sup)1542 3072 y Fs(t)p Fq(2L)1663 3053 y Fk(\()p Fl(})p Fk(\))1771 2979 y Ft(\027)1825 2938 y Fo(\()p Fs(`)p Fo(\))1913 2899 y Fn(\000)1959 2979 y Fr(9)p Ft(i)28 b Fr(2)g Ft(Q)2246 2929 y Fo(\()p Fs(`)p Fo(\))2246 3011 y Fs(}=`)2359 2979 y Fv(\()p Ft(t)p Fv(\))g(:)44 b Fr(j)p Ft(m)2682 2994 y Fs(i)2711 2979 y Fr(j)27 b Ft(>)h(`)2911 2938 y Fs(d)p Fo(\(1)p Fs(=)p Fo(6)p Fq(\000)p Fs(\016)r Fo(\))3200 2979 y Ft(\037)3261 2938 y Fq(\000)p Fo(1)p Fs(=)p Fo(2)3426 2899 y Fn(\001)1381 3187 y Fr(\024)84 b Ft(d)1593 3146 y Fs(d)1675 3187 y Fv(sup)1649 3280 y Fs(i)p Fq(2L)1769 3261 y Fk(\()p Fl(`)p Fk(\))1864 3187 y Ft(\026)1923 3107 y Fn(\000)1968 3187 y Fr(j)p Ft(M)2100 3136 y Fo(\()p Fs(`)p Fo(\))2090 3213 y Fs(i)2188 3187 y Fr(j)28 b Ft(>)f(`)2388 3146 y Fs(d)p Fo(\(1)p Fs(=)p Fo(6)p Fq(\000)p Fs(\016)r Fo(\))2678 3187 y Ft(\037)2739 3146 y Fq(\000)p Fo(1)p Fs(=)p Fo(2)2904 3107 y Fn(\001)0 3462 y Fv(W)-8 b(e)34 b(pic)m(k)i Ft(i)30 b Fr(2)h(L)606 3426 y Fo(\()p Fs(`)p Fo(\))693 3462 y Fv(.)48 b(T)-8 b(o)34 b(b)s(ound)g(the)h(righ)m(t)f (hand)g(side)h(of)f(the)g(ab)s(o)m(v)m(e)h(inequalit)m(y)-8 b(,)36 b(w)m(e)f(recall)g(\(2.11\),)0 3582 y(consider)h Ft(L)c(>)f(`)k Fv(in)m(teger,)i(set)e(\001)1251 3597 y Fs(L)1304 3582 y Fv(\()p Ft(i)p Fv(\))c(:=)h Fr(f)p Ft(x)f Fr(2)h(L)g Fv(:)48 b(d\()p Ft(x;)17 b(Q)2257 3597 y Fs(`)2290 3582 y Fv(\()p Ft(i)p Fv(\)\))32 b Fr(\024)g Ft(L)p Fr(g)p Fv(,)k(and)e(use)i(the)g(exp)s(onen)m(tial)0 3703 y(Cheb)m(yshev)g(inequalit)m(y)-8 b(,)34 b(with)g Ft(h)28 b Fr(\025)g Fv(0,)k(as)h(follo)m(ws)85 3948 y Ft(\026)144 3868 y Fn(\000)189 3948 y Ft(M)293 3897 y Fo(\()p Fs(`)p Fo(\))283 3974 y Fs(i)410 3948 y Ft(>)27 b(`)554 3907 y Fs(d)p Fo(\(1)p Fs(=)p Fo(6)p Fq(\000)p Fs(\016)r Fo(\))844 3948 y Ft(\037)905 3907 y Fq(\000)p Fo(1)p Fs(=)p Fo(2)1070 3868 y Fn(\001)1199 3948 y Fv(=)g Ft(\026)1361 3838 y Fn(\020)1491 3854 y(X)1437 4070 y Fs(x)p Fq(2)p Fs(Q)1580 4082 y Fl(`)1609 4070 y Fo(\()p Fs(i)p Fo(\))1688 3948 y Fv(\()p Ft(\021)1774 3963 y Fs(x)1840 3948 y Fr(\000)c Ft(\032)p Fv(\))28 b Ft(>)f(`)2200 3907 y Fo(\(2)p Fs(=)p Fo(3)p Fq(\000)p Fs(\016)r Fo(\))p Fs(d)2490 3838 y Fn(\021)850 4241 y Fr(\024)h Ft(e)1000 4200 y Fq(\000)p Fs(h`)1125 4176 y Fk(\(2)p Fl(=)p Fk(3)p Fm(\000)p Fl(\016)r Fk(\))p Fl(d)1400 4241 y Ft(\026)1459 4130 y Fn(\020)1535 4241 y Fv(exp)1701 4160 y Fn(\010)1759 4241 y Ft(h)1885 4146 y Fn(X)1832 4362 y Fs(x)p Fq(2)p Fs(Q)1975 4374 y Fl(`)2004 4362 y Fo(\()p Fs(i)p Fo(\))2083 4241 y Fv(\()p Ft(\021)2169 4256 y Fs(x)2235 4241 y Fr(\000)22 b Ft(\032)p Fv(\))2422 4160 y Fn(\011)2481 4130 y(\021)850 4533 y Fv(=)27 b Ft(e)998 4492 y Fq(\000)p Fs(h`)1123 4468 y Fk(\(2)p Fl(=)p Fk(3)p Fm(\000)p Fl(\016)r Fk(\))p Fl(d)1415 4397 y Fn(Z)1515 4533 y Ft(\026)p Fv(\()p Ft(d\034)11 b Fv(\))17 b Ft(\026)1830 4492 y Fs(\034)1830 4558 y Fo(\001)1889 4569 y Fl(L)1934 4558 y Fo(\()p Fs(i)p Fo(\))2034 4422 y Fn(\020)2110 4533 y Fv(exp)2275 4452 y Fn(\010)2333 4533 y Ft(h)2460 4438 y Fn(X)2406 4654 y Fs(x)p Fq(2)p Fs(Q)2549 4666 y Fl(`)2578 4654 y Fo(\()p Fs(i)p Fo(\))2657 4533 y Fv(\()p Ft(\021)2743 4548 y Fs(x)2809 4533 y Fr(\000)23 b Ft(\032)p Fv(\))2997 4452 y Fn(\011)3055 4422 y(\021)850 4825 y Fv(=)k Ft(e)998 4784 y Fq(\000)p Fs(h`)1123 4761 y Fk(\(2)p Fl(=)p Fk(3)p Fm(\000)p Fl(\016)r Fk(\))p Fl(d)1415 4690 y Fn(Z)1515 4825 y Ft(\026)p Fv(\()p Ft(d\034)11 b Fv(\))33 b(exp)1952 4744 y Fn(\010)2027 4825 y Fv(log)18 b Ft(Z)2244 4784 y Fs(\034)2237 4850 y Fo(\001)2296 4861 y Fl(L)2342 4850 y Fo(\()p Fs(i)p Fo(\))p Fs(;z)p 2440 4862 36 3 v 3 w Fo(\()p Fs(i;h)p Fo(\))2641 4825 y Fr(\000)23 b Fv(log)17 b Ft(Z)2957 4784 y Fs(\034)2950 4850 y Fo(\001)3009 4861 y Fl(L)3055 4850 y Fo(\()p Fs(i)p Fo(\))p Fs(;z)p 3153 4862 V 3 w Fo(\()p Fs(i;)p Fo(0\))3349 4825 y Fr(\000)23 b Ft(h\032`)3596 4784 y Fs(d)3637 4744 y Fn(\011)3531 5003 y Fv(\(4.13\))0 5123 y(where)33 b(w)m(e)f(used)h(the)f(DLR)f (equations)h(\(2.6\))f(and,)h(for)f Ft(\034)39 b Fr(2)28 b(X)47 b Fv(and)31 b(\003)d Fr(\032\032)g(L)p Fv(,)k(w)m(e)g(ha)m(v)m (e)h(considered)0 5244 y(the)g(partition)f(function)901 5459 y Ft(Z)975 5418 y Fs(\034)968 5484 y Fo(\003)p Fs(;z)p 1037 5496 V 2 w Fo(\()p Fs(i;h)p Fo(\))1243 5459 y Fv(=)1365 5365 y Fn(X)1346 5576 y Fs(\021)r Fq(2X)1480 5587 y Fk(\003)1544 5459 y Fv(exp)1709 5349 y Fn(n)1776 5459 y Ft(H)1857 5474 y Fo(\003)1910 5459 y Fv(\()p Ft(\021)t(\034)2042 5474 y Fo(\003)2091 5455 y Fk(c)2126 5459 y Fv(\))22 b(+)g Ft(h)2459 5365 y Fn(X)2357 5580 y Fs(x)p Fq(2)p Fo(\003)p Fq(\\)p Fs(Q)2596 5592 y Fl(`)2625 5580 y Fo(\()p Fs(i)p Fo(\))2721 5459 y Ft(\021)2769 5474 y Fs(x)2813 5349 y Fn(o)3531 5459 y Fv(\(4.14\))p Black 1841 5806 a(25)p Black eop end %%Page: 26 27 TeXDict begin 26 26 bop Black Black 0 407 a Fv(of)44 b(a)g(lattice)i(gas)e(with)h(not)g(homogeneous)h(activit)m(y)g Ft(z)p 2103 423 50 4 v 4 w Fv(\()p Ft(i;)17 b(h)p Fv(\))45 b(suc)m(h)h(that)e Ft(z)2906 422 y Fs(x)2950 407 y Fv(\()p Ft(i;)17 b(h)p Fv(\))49 b(=)f Ft(z)t(e)3426 371 y Fs(h)3516 407 y Fv(for)c(all)0 527 y Ft(x)39 b Fr(2)g Ft(Q)276 542 y Fs(`)310 527 y Fv(\()p Ft(i)p Fv(\))g(and)g Ft(z)699 542 y Fs(x)743 527 y Fv(\()p Ft(i;)17 b(h)p Fv(\))39 b(=)g Ft(z)k Fv(otherwise;)h(recall)c Ft(z)k Fv(is)39 b(the)h(activit)m(y)g(of)f(the)h(original)e(lattice)i(gas.)0 648 y(Note)33 b(that)f Ft(Z)521 611 y Fs(\034)514 677 y Fo(\003)p Fs(;z)p 583 689 36 3 v 2 w Fo(\()p Fs(i;)p Fo(0\))788 648 y Fv(coincides)j(with)e(the)g(partition)g(function)g Ft(Z)2454 611 y Fs(\034)2447 673 y Fo(\003)2532 648 y Fv(of)f(the)h(original)f(lattice)h(gas.)146 777 y(F)-8 b(rom)32 b(the)h(strong)f(mixing)i(condition)f(SM\()p Ft(`)1841 792 y Fo(0)1881 777 y Ft(;)17 b(b;)g(B)5 b Fv(\))32 b(it)g(follo)m(ws)h(that)g(there)g(exist)g(t)m(w)m(o)h(p)s (ositiv)m(e)0 897 y(reals)g Ft(C)299 912 y Fo(1)338 897 y Fv(\()p Ft(`)417 912 y Fo(0)457 897 y Ft(;)17 b(b;)g(B)5 b Fv(\))29 b Ft(<)f Fr(1)33 b Fv(and)h Ft(C)1230 912 y Fo(2)1269 897 y Fv(\()p Ft(`)1348 912 y Fo(0)1387 897 y Ft(;)17 b(b;)g(B)5 b Fv(\))30 b Ft(>)e Fv(0)34 b(suc)m(h)g(that)g 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Fo(\()p Fs(`)p Fo(\))p Fs(;})2312 439 y(m;)p Fq(f)p Fs(D)2487 451 y Fl(k)2525 439 y Fq(g\\)p Fo(\001)2703 408 y Fv(w)m(e)g(get)311 657 y Ft(\027)359 672 y Fs(C)409 684 y Fl(k)452 657 y Fv(\()p Ft(\015)541 672 y Fs(C)591 684 y Fl(k)633 657 y Fr(j\001)p Ft(;)17 b(\030)5 b Fv(\))26 b Fr(2)i(F)1021 610 y Fs(`;})1011 696 y(@)1052 676 y Fk(\()p Fl(})p Fk(\))1143 696 y Fq(f)p Fs(C)1228 708 y Fl(k)1267 696 y Fq(g\\)p Fo(\()p Fq(A[B)r Fo(\))1703 657 y Fv(and)143 b Ft(\027)2051 672 y Fs(C)2101 684 y Fl(k)2144 657 y Fv(\()p Ft(\015)2233 672 y Fs(C)2283 684 y Fl(k)2325 657 y Fr(j)p Ft(\013)q(\014)6 b(;)17 b Fr(\001)p Fv(\))26 b Fr(2)i(F)2789 610 y Fs(`;})p 2779 662 159 3 v 2779 720 a Fq(f)p Fs(C)2864 732 y Fl(k)2902 720 y Fq(g)2938 676 y Fk(\()p Fl(})p Fk(\))p Fl(;)p Fk(2)3078 720 y Fq(\\)p Fo(\001)3184 702 y Fk(c)3531 657 y Fv(\(5.23\))0 938 y(for)37 b(all)g Ft(\013)g Fr(2)g(X)582 891 y Fs(`;})567 965 y Fq(A)679 938 y Fv(,)i Ft(\014)i Fr(2)c(X)1030 891 y Fs(`;})1015 965 y Fq(B)1127 938 y Fv(,)i 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Fq(D)1936 1909 y Fv(\))17 b Ft(Z)2065 1858 y Fo(\()p Fs(`)p Fo(\))p Fs(;})2058 1964 y(m;)p Fq(f)p Fs(D)2233 1976 y Fl(k)2270 1964 y Fo(+)p Fs(e)2358 1928 y Fk(\(2)p Fl(})p Fk(\))2358 1985 y(2)2480 1964 y Fq(g\\)p Fo(\001)2625 1909 y Fv(\()p Ft(\013)q Fv(\()p Ft(\014)6 b(\015)f Fv(\))2919 1924 y Fo(d)2961 1909 y Fv(\()3009 1883 y(\026)2999 1909 y Ft(\030)3042 1924 y Fq(B)r([C)3182 1909 y Fv(\))3220 1924 y Fo(u)3273 1883 y Fv(\026)3263 1909 y Ft(\030)3306 1924 y Fq(D)3367 1909 y Fv(\))3531 1632 y(\(5.24\))0 2175 y(holds.)44 b(W)-8 b(e)33 b(\014nally)h(remark)f(that)f Ft(\027)1349 2190 y Fs(C)1399 2202 y Fl(k)1442 2175 y Fv(\()p Ft(\015)1531 2190 y Fs(C)1581 2202 y Fl(k)1623 2175 y Fr(j)p Ft(\013)q(\014)6 b(;)1828 2149 y Fv(\026)1819 2175 y Ft(\030)1862 2190 y Fq(D)1921 2175 y Fv(\))28 b(=)f(1)-28 b(I)2146 2197 y Fq(f)p Fs(\015)2217 2208 y Fl(C)2261 2225 y(k)2304 2197 y Fo(=)2366 2179 y(\026)2359 2197 y Fs(\030)2390 2208 y Fl(C)2434 2225 y(k)2477 2197 y Fq(g)2548 2175 y Fv(whenev)m(er)36 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b(inside)g(the)g(v)m(ertices)i(whic)m(h)f(lab)s (el)0 1703 y(the)i(function;)g(more)g(precisely)571 1907 y(\011)647 1856 y Fo(\()p Fs(i;j)t Fo(\))647 1934 y Fs(g)r(;)p Fo(\003)782 1907 y Fv(\()p Fr(\001)p Ft(;)17 b(n)p Fv(\))27 b Fr(2)h(F)1191 1860 y Fs(`;})1181 1938 y Fo(\()p Fs(Y)1249 1947 y Fk(0)1283 1938 y Fo(\()p Fs(g)r Fo(\))p Fq(\\)p Fo(\003)1469 1919 y Fk(c)1502 1938 y Fo(\))p Fq([)p Fs(g)1759 1907 y Fv(and)172 b(\010)2158 1856 y Fo(\()p Fs(i;j)t Fo(\))2158 1934 y Fs(G;s;)p Fo(\003)2338 1907 y Fv(\()p Fr(\001)p Ft(;)17 b(n)p Fv(\))27 b Fr(2)h(F)2747 1860 y Fs(`;})2737 1953 y Fo(\()p Fs(Y)2805 1961 y Fl(s)2838 1953 y Fo(\()p Fs(G)p Fo(\))p Fq(\\)p Fo(\003)3043 1934 y Fk(c)3076 1953 y Fo(\))p Fq([)3164 1937 y Fe(b)3150 1953 y Fs(G)3531 1907 y Fv(\(6.18\))0 2159 y(where)34 b(w)m(e)f(recall)704 2134 y Fn(b)686 2159 y Ft(G)27 b Fv(:=)921 2085 y Fn(S)1004 2188 y Fs(g)r Fq(2)p Fs(G)1163 2159 y Ft(g)t Fv(.)146 2310 y(W)-8 b(e)37 b(can)f(\014nally)i(de\014ne) 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Fn(\011)3531 3616 y Fv(\(6.20\))0 3938 y(W)-8 b(e)46 b(remark)h(that)e(the)i(sums)g(o)m(v)m(er)g Ft(j)52 b Fv(in)46 b(\(6.19\))f(are)h(extended)i(to)d(a)h(\014nite)g(n) m(um)m(b)s(er)i(of)d(terms,)0 4058 y(indeed)33 b(for)e Ft(j)38 b Fv(suc)m(h)33 b(that)f Ft(#)1021 4073 y Fs(j)1086 4058 y Ft(>)27 b Fv(diam)1401 4073 y Fs(})1449 4058 y Fv(\()p Ft(X)8 b Fv(\))32 b(the)g(sets)h Fr(G)2062 4073 y Fs(j)2099 4058 y Fv(\()p Ft(X)8 b Fv(\)\()p Ft(n)p Fv(\))31 b(and)h Fr(R)p 2618 4074 85 4 v 2703 4082 a Fs(j)2739 4058 y Fv(\()p Ft(X)8 b Fv(\)\()p Ft(n)p Fv(\))32 b(are)f(empt)m(y)j(for)d(all)0 4190 y Ft(n)d Fr(2)229 4164 y Fv(\026)180 4190 y Fr(M)300 4153 y Fo(\()p Fs(`)p Fo(\))p Fs(;})452 4190 y Fv(.)43 b(F)-8 b(or)32 b(eac)m(h)i Ft(X)h Fr(\032\032)29 b(L)1285 4153 y Fo(\()p Fs(})p Fo(\))1421 4190 y Fv(and)j Ft(j)i Fr(\025)28 b Fv(1)k(w)m(e)i (\014nally)f(set)1105 4369 y Fn(e)1089 4394 y Fr(G)1148 4409 y Fs(j)1184 4394 y Fv(\()p Ft(X)8 b Fv(\)\()p Ft(n)p Fv(\))28 b(:=)f Fr(f)p Ft(g)k 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Fq(0)3407 5216 y Fv(\))p Fu(;)p Black 115 5437 a(3.)p Black 49 w(we)44 b(have)g Fv(\007)708 5452 y Fo(\003)761 5437 y Fv(\()p Ft(G;)17 b(s)p Fv(\)\()p Ft(n)p Fv(\))45 b(=)h(\007)1381 5452 y Fo(\003)1434 5437 y Fv(\()p Ft(G;)17 b(s)p Fv(\)\()p Ft(n)1773 5401 y Fq(0)1796 5437 y Fv(\))44 b Fu(for)g(any)h Ft(G)h Fr(\032)2502 5412 y Fn(e)2486 5437 y Fr(G)2545 5452 y Fq(\025)p Fo(1)2639 5437 y Fv(\()p Ft(X)8 b Fv(\)\()p Ft(n)p Fv(\))46 b(=)3121 5412 y Fn(e)3105 5437 y Fr(G)3164 5452 y Fq(\025)p Fo(1)3259 5437 y Fv(\()p Ft(X)8 b Fv(\)\()p Ft(n)3520 5401 y Fq(0)3543 5437 y Fv(\))45 b Fu(and)244 5557 y Ft(s)27 b Fr(\025)i Fv(0)34 b Fu(such)h(that)g Ft(Y)984 5572 y Fs(s)1021 5557 y Fv(\()p Ft(G)p Fv(\))27 b Fr(\032)h Ft(X)8 b Fu(;)p Black 1841 5806 a Fv(48)p Black eop end %%Page: 49 50 TeXDict begin 49 49 bop Black Black Black 115 407 a Fu(4.)p Black 49 w(for)29 b(e)-5 b(ach)29 b Ft(j)k Fr(\025)c Fv(1)g Fu(we)g(have)g(that)g Fr(R)p 1416 423 85 4 v 1501 431 a Fs(j)1537 407 y Fv(\()p Ft(G;)17 b(s)p Fv(\)\()p Ft(n)p Fv(\))28 b(=)f Fr(R)p 2045 423 V 2130 431 a Fs(j)2166 407 y Fv(\()p Ft(G;)17 b(s)p Fv(\)\()p Ft(n)2505 371 y Fq(0)2528 407 y Fv(\))29 b Fu(for)h(any)f Ft(G)f Fr(\032)3153 382 y Fn(e)3137 407 y Fr(G)3196 422 y Fq(\025)p Fs(j)t Fo(+1)3378 407 y Fv(\()p Ft(X)8 b Fv(\)\()p Ft(n)p Fv(\))27 b(=)260 524 y Fn(e)244 549 y Fr(G)303 564 y Fq(\025)p Fs(j)t Fo(+1)485 549 y Fv(\()p Ft(X)8 b Fv(\)\()p Ft(n)746 513 y Fq(0)769 549 y Fv(\))35 b Fu(and)f Ft(s)28 b Fr(\025)g Fv(0)34 b Fu(such)h(that)g Ft(Y)1771 564 y Fs(s)1808 549 y Fv(\()p Ft(G)p Fv(\))27 b Fr(\032)h Ft(X)8 b Fu(;)p Black 115 765 a(5.)p Black 49 w(for)46 b(e)-5 b(ach)46 b Ft(j)56 b Fr(\025)50 b Fv(0)p Fu(,)f Ft(i)h Fr(\025)f Ft(j)37 b Fv(+)31 b(1)p Fu(,)49 b Ft(g)k Fr(2)1746 740 y Fn(e)1730 765 y Fr(G)1789 780 y Fs(i)1834 765 y Fv(\()p Ft(X)8 b Fv(\))16 b(\()p Ft(n)p Fv(\))50 b(=)2340 740 y Fn(e)2324 765 y Fr(G)2383 780 y Fs(i)2428 765 y Fv(\()p Ft(X)8 b Fv(\))16 b(\()p Ft(n)2705 729 y Fq(0)2729 765 y Fv(\))p 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Fs(i)1369 1274 y Fv(\()p Ft(X)8 b Fv(\))16 b(\()p Ft(n)1646 1238 y Fq(0)1670 1274 y Fv(\))54 b Fr(6)p Fv(=)f Fr(;)c Fu(and)g Ft(Y)2251 1289 y Fs(s)2287 1274 y Fv(\()p Ft(G)p Fv(\))54 b Fr(\032)h Ft(X)8 b Fu(,)52 b(we)d(have)f Fv(\010)3264 1223 y Fo(\()p Fs(i;j)t Fo(\))3264 1301 y Fs(G;s;)p Fo(\003)3445 1274 y Fv(\()p Fr(\001)p Ft(;)17 b(n)p Fv(\))53 b(=)244 1423 y(\010)314 1372 y Fo(\()p Fs(i;j)t Fo(\))314 1450 y Fs(G;s;)p Fo(\003)494 1423 y Fv(\()p Fr(\001)p Ft(;)17 b(n)662 1387 y Fq(0)685 1423 y Fv(\))p Fu(;)p Black 115 1651 a(7.)p Black 49 w(for)35 b(e)-5 b(ach)34 b Ft(j)f Fr(\025)c Fv(1)p Fu(,)34 b Ft(g)d Fr(2)d(G)1142 1666 y Fs(j)1179 1651 y Fv(\()p Ft(X)8 b Fv(\)\()p Ft(n)p Fv(\))27 b(=)h Fr(G)1668 1666 y Fs(j)1705 1651 y Fv(\()p Ft(X)8 b Fv(\)\()p Ft(n)1966 1615 y Fq(0)1989 1651 y Fv(\))p Fu(,)34 b(we)h(have)f Ft(Z)2534 1600 y Fo(\()p Fs(j)t Fo(\))2527 1678 y Fs(g)r(;)p Fo(\003)2636 1651 y Fv(\()p Fr(\001)p Ft(;)17 b(n)p Fv(\))27 b(=)g Ft(Z)3046 1600 y Fo(\()p Fs(j)t Fo(\))3039 1678 y Fs(g)r(;)p Fo(\003)3148 1651 y Fv(\()p Fr(\001)p Ft(;)17 b(n)3316 1615 y Fq(0)3338 1651 y Fv(\))p Fu(;)p Black 115 1851 a(8.)p Black 49 w(for)35 b(e)-5 b(ach)34 b Ft(j)f Fr(\025)c Fv(1)p Fu(,)34 b Ft(R)p 911 1867 76 4 v 29 w Fr(2)28 b(R)p 1108 1867 85 4 v 24 x Fs(j)1229 1851 y Fv(\()p Ft(X)8 b Fv(\)\()p Ft(n)p Fv(\))27 b(=)h Fr(R)p 1659 1867 V 24 x Fs(j)1780 1851 y Fv(\()p Ft(X)8 b Fv(\)\()p Ft(n)2041 1815 y Fq(0)2064 1851 y Fv(\))p Fu(,)35 b(we)f(have)g Ft(\020)2578 1866 y Fs(R)p 2578 1878 54 3 v(;)p Fo(\003)2704 1851 y Fv(\()p Fr(\001)p Ft(;)17 b(n)p Fv(\))27 b(=)h Ft(\020)3084 1866 y Fs(R)p 3084 1878 V(;)p Fo(\003)3210 1851 y Fv(\()p Fr(\001)p Ft(;)17 b(n)3378 1815 y Fq(0)3400 1851 y Fv(\))p Fu(.)0 2043 y(Pr)-5 b(o)g(of)45 b(of)h(L)-5 b(emma)44 b(6.3.)g Fv(W)-8 b(e)44 b(\014rst)h(pro)m(v)m(e)h(items)f (1{4)e(separately)-8 b(,)49 b(then)c(5)f(and)g(6)g(b)m(y)h(induction.)0 2163 y(Items)34 b(7)e(and)h(8)f(will)h(b)s(e)g(a)f(b)m(ypro)s(duct)i (of)e(the)h(pro)s(of)f(of)g(5)g(and)h(6.)146 2283 y Fu(Item)42 b(1.)d Fv(Let)h Ft(g)k Fr(2)898 2258 y Fn(e)882 2283 y Fr(G)941 2298 y Fs(j)978 2283 y Fv(\()p Ft(X)8 b Fv(\)\()p Ft(n)p Fv(\);)43 b(since)f Ft(X)48 b Fr(\033)41 b Ft(Y)1898 2298 y Fo(0)1937 2283 y Fv(\()p Ft(g)t Fv(\))e Fr(\033)i Ft(B)2300 2232 y Fo(\()p Fs(})p Fo(\))2295 2311 y Fs(#)2337 2321 y Fl(j)2404 2283 y Fv(\()p Ft(g)t Fv(\),)g(item)g(2)f(in)g (Theorem)h(4.4)f(and)0 2436 y Ft(n)58 2451 y Fs(X)180 2436 y Fv(=)54 b Ft(n)368 2400 y Fq(0)368 2462 y Fs(X)484 2436 y Fv(imply)49 b Ft(g)58 b Fr(2)d(G)1058 2451 y Fs(j)1095 2436 y Fv(\()p Ft(n)1191 2400 y Fq(0)1214 2436 y Fv(\).)91 b(No)m(w,)53 b Ft(g)k Fr(2)1880 2411 y Fn(e)1864 2436 y Fr(G)1923 2451 y Fs(j)1960 2436 y Fv(\()p Ft(X)8 b Fv(\)\()p Ft(n)2221 2400 y Fq(0)2244 2436 y Fv(\))48 b(follo)m(ws)h(from)f Ft(g)58 b Fr(2)d(G)3197 2451 y Fs(j)3234 2436 y Fv(\()p Ft(n)3330 2400 y Fq(0)3353 2436 y Fv(\))48 b(and)h(the)0 2570 y(geometrical)31 b(prop)s(ert)m(y)g Ft(Y)971 2585 y Fo(0)1010 2570 y Fv(\()p Ft(g)t Fv(\))c Fr(\032)h Ft(X)8 b Fv(.)43 b(Hence)1732 2544 y Fn(e)1716 2570 y Fr(G)1775 2585 y Fs(j)1811 2570 y Fv(\()p Ft(X)8 b Fv(\)\()p Ft(n)p Fv(\))28 b Fr(\032)2259 2544 y Fn(e)2243 2570 y Fr(G)2302 2585 y Fs(j)2339 2570 y Fv(\()p Ft(X)8 b Fv(\)\()p Ft(n)2600 2533 y Fq(0)2623 2570 y Fv(\))30 b(and,)h(b)m(y)h(in)m(terc)m(hanging)g(the)0 2690 y(role)k(of)f Ft(n)h Fv(and)g Ft(n)652 2654 y Fq(0)675 2690 y Fv(,)h(w)m(e)g(get)e (the)h(equalit)m(y)-8 b(.)55 b(The)37 b(second)g(equalit)m(y)g(follo)m (ws)f(immediately)i(from)e(the)0 2810 y(\014rst)d(one.)146 2931 y Fu(Item)k(2.)f Fv(The)g(pro)s(of)f(of)g(the)h(\014rst)g(equalit) m(y)h(is)f(similar)g(to)g(the)g(pro)s(of)e(of)h(item)h(1.)53 b(Pro)s(of)34 b(of)i(the)0 3051 y(second)f(equalit)m(y)-8 b(.)49 b(Let)34 b Ft(R)p 897 3067 76 4 v 31 w Fr(2)d(R)p 1099 3067 85 4 v 24 x Fs(j)1220 3051 y Fv(\()p Ft(X)8 b Fv(\)\()p Ft(n)p Fv(\))33 b(and)h Ft(G)c Fv(=)g Fr(f)p Ft(g)2053 3066 y Fo(1)2092 3051 y Ft(;)17 b(:)g(:)g(:)f(;)h(g)2358 3066 y Fs(k)2400 3051 y Fr(g)30 b Fv(:=)g Ft(R)p 2613 3067 76 4 v 14 w Fc(\026)2756 3066 y Fs(j)2826 3051 y Fv(b)s(e)k(the)h(collection)g(of)e(all)0 3172 y(the)38 b(v)m(ertices)h(the)f(cluster)h(of)d(p)s(olymers)j Ft(R)p 1561 3187 V 38 w Fv(is)f(built)g(of.)57 b(By)38 b(de\014nition)g Ft(g)2820 3187 y Fs(h)2900 3172 y Fr(2)e(G)3061 3187 y Fs(j)3098 3172 y Fv(\()p Ft(n)p Fv(\))i(for)e(an)m(y)i Ft(h)e Fv(=)0 3292 y(1)p Ft(;)17 b(:)g(:)g(:)f(;)h(k)s Fv(.)44 b(The)33 b(de\014nition)h(of)f(supp)s(ort)g(of)f(a)h(p)s (olymer)g(and)g(supp)18 b Ft(R)p 2480 3308 V 29 w Fv(=)28 b Ft(X)41 b Fv(imply)33 b(that)g Ft(Y)3351 3307 y Fo(0)3390 3292 y Fv(\()p Ft(g)3475 3307 y Fs(h)3520 3292 y Fv(\))28 b Fr(\032)g Ft(X)0 3412 y Fv(for)35 b(an)m(y)h Ft(h)c Fv(=)h(1)p Ft(;)17 b(:)g(:)g(:)e(;)i(k)s Fv(.)52 b(Hence,)37 b Ft(n)1314 3427 y Fs(X)1414 3412 y Fv(=)32 b Ft(n)1580 3376 y Fq(0)1580 3437 y Fs(X)1683 3412 y Fv(and)k(item)g(2)f(in)g (Theorem)i(4.4)e(imply)h Ft(g)3199 3427 y Fs(h)3276 3412 y Fr(2)d(G)3434 3427 y Fs(j)3471 3412 y Fv(\()p Ft(n)3567 3376 y Fq(0)3590 3412 y Fv(\))i(for)0 3533 y(an)m(y)k Ft(h)g Fv(=)e(1)p Ft(;)17 b(:)g(:)g(:)f(;)h(k)s Fv(,)40 b(whic)m(h)g(yields)h Ft(R)p 1353 3549 V 39 w Fr(2)d(R)p 1570 3549 85 4 v 24 x Fs(j)1691 3533 y Fv(\()p Ft(X)8 b Fv(\)\()p Ft(n)1952 3497 y Fq(0)1975 3533 y Fv(\).)62 b(Hence)40 b Fr(R)p 2398 3549 V 24 x Fs(j)2519 3533 y Fv(\()p Ft(X)8 b Fv(\)\()p Ft(n)p Fv(\))38 b Fr(\032)g(R)p 2971 3549 V 24 x Fs(j)3092 3533 y Fv(\()p Ft(X)8 b Fv(\)\()p Ft(n)3353 3497 y Fq(0)3376 3533 y Fv(\))39 b(and,)h(b)m(y)0 3653 y(in)m(terc)m(hanging)34 b(the)f(role)g(of)f Ft(n)h Fv(and)f Ft(n)1414 3617 y Fq(0)1438 3653 y Fv(,)g(w)m(e)i(get)f(the)g (equalit)m(y)-8 b(.)146 3773 y Fu(Item)35 b(3.)d Fv(Recall)h(\(6.8\),)f (let)h Ft(Y)49 b Fr(2)28 b Fv(\007)1459 3788 y Fo(\003)1512 3773 y Fv(\()p Ft(G;)17 b(s)p Fv(\)\()p Ft(n)p Fv(\).)43 b(Then)34 b(w)m(e)f(ha)m(v)m(e)1164 3980 y Ft(Y)49 b Fr(\032)28 b Ft(Y)1432 3995 y Fs(s)1468 3980 y Fv(\()p Ft(G)p Fv(\))115 b(and)h Ft(Y)44 b Fr(\\)22 b Ft(y)2246 3995 y Fs(s)2283 3980 y Fv(\()p Ft(G)p Fv(\))27 b Fr(6)p Fv(=)h Fr(;)0 4187 y Fv(Moreo)m(v)m(er,)54 b Ft(X)62 b Fr(\033)55 b Ft(Y)807 4202 y Fs(s)843 4187 y Fv(\()p Ft(G)p Fv(\))f Fr(\033)h Ft(Y)21 b Fv(,)52 b Ft(n)1397 4202 y Fs(X)1519 4187 y Fv(=)i Ft(n)1707 4150 y Fq(0)1707 4212 y Fs(X)1775 4187 y Fv(,)e(and)c Ft(Y)76 b Fr(2)55 b Fv(\007)2389 4225 y Fo(\003)p Fq(\\L)2534 4190 y Fk(\()p Fl(})p Fk(\))2534 4249 y Fl(\016)2624 4225 y Fo(\()p Fs(n)p Fo(\))2774 4187 y Fv(imply)50 b Ft(Y)75 b Fr(2)55 b Fv(\007)3393 4225 y Fo(\003)p Fq(\\L)3538 4190 y Fk(\()p Fl(})p Fk(\))3538 4249 y Fl(\016)3629 4225 y Fo(\()p Fs(n)3699 4206 y Fm(0)3721 4225 y Fo(\))3753 4187 y Fv(.)0 4332 y(Finally)-8 b(,)45 b Ft(\030)5 b Fv(\()p Ft(Y)20 b Fv(\)\()p Ft(n)p Fv(\))45 b(=)e Ft(G)f Fv(and)h Ft(n)1239 4347 y Fs(Y)1344 4332 y Fv(=)g Ft(n)1521 4296 y Fq(0)1521 4357 y Fs(Y)1625 4332 y Fv(imply)g Ft(\030)5 b Fv(\()p Ft(Y)20 b Fv(\)\()p Ft(n)2205 4296 y Fq(0)2229 4332 y Fv(\))44 b(=)g Ft(G)p Fv(.)72 b(All)42 b(the)h(prop)s(erties)g (ensuring)0 4452 y Ft(Y)69 b Fr(2)49 b Fv(\007\()p Ft(G;)17 b(s)p Fv(\)\()p Ft(n)656 4416 y Fq(0)679 4452 y Fv(\))44 b(ha)m(v)m(e)i(b)s(een)f(v)m(eri\014ed,)50 b(hence)c(w)m(e)f(ha)m(v)m (e)h(\007\()p Ft(G;)17 b(s)p Fv(\)\()p Ft(n)p Fv(\))48 b Fr(\032)g Fv(\007\()p Ft(G;)17 b(s)p Fv(\)\()p Ft(n)3340 4416 y Fq(0)3363 4452 y Fv(\))44 b(and,)k(b)m(y)0 4572 y(in)m(terc)m(hanging)34 b(the)f(role)g(of)f Ft(n)h Fv(and)f Ft(n)1414 4536 y Fq(0)1438 4572 y Fv(,)g(w)m(e)i(get)f(the)g(equalit)m (y)-8 b(.)146 4693 y Fu(Item)45 b(4.)e Fv(Let)h Ft(R)p 695 4709 76 4 v 48 w Fr(2)j(R)p 930 4709 85 4 v 1015 4717 a Fs(j)1051 4693 y Fv(\()p Ft(G;)17 b(s)p Fv(\)\()p Ft(n)p Fv(\),)46 b Ft(F)61 b Fv(=)46 b Fr(f)p Ft(f)1845 4708 y Fo(1)1884 4693 y Ft(;)17 b(:)g(:)g(:)f(;)h(f)2151 4708 y Fs(k)2194 4693 y Fr(g)46 b Fv(:=)h Ft(R)p 2440 4709 76 4 v 31 w Fc(\026)2617 4708 y Fq(\025)p Fs(j)2752 4693 y 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Fv(=)51 b(0.)83 b(F)-8 b(or)46 b(eac)m(h)h Ft(i)k Fr(\025)g Fv(1)45 b(and)i Ft(g)54 b Fr(2)16 863 y Fn(e)0 888 y Fr(G)59 903 y Fs(i)87 888 y Fv(\()p Ft(X)8 b Fv(\)\()p Ft(n)p Fv(\))49 b(=)576 863 y Fn(e)560 888 y Fr(G)619 903 y Fs(i)648 888 y Fv(\()p Ft(X)8 b Fv(\)\()p Ft(n)909 852 y Fq(0)932 888 y Fv(\),)48 b(b)m(y)e(using)g(\(6.10\),)h(item)f(3,)i Ft(Y)70 b Fr(2)50 b Fv(\007)2463 903 y Fo(\003)2516 888 y Fv(\()p Ft(g)t(;)17 b Fv(0\)\()p Ft(n)p Fv(\))48 b(=)h(\007)3119 903 y Fo(\003)3172 888 y Fv(\()p Ft(g)t(;)17 b Fv(0\)\()p Ft(n)3488 852 y Fq(0)3510 888 y Fv(\),)48 b(and)0 1022 y(item)38 b(8)f(in)g(Theorem)h(5.1,)g(w)m(e)g(ha)m(v)m(e)h(that)e(\011) 1707 971 y Fo(\()p Fs(i;)p Fo(0\))1707 1049 y Fs(g)r(;)p Fo(\003)1845 1022 y Fv(\()p Fr(\001)p Ft(;)17 b(n)p Fv(\))35 b(=)g(\011)2273 971 y Fo(\()p Fs(i;)p Fo(0\))2273 1049 y Fs(g)r(;)p Fo(\003)2411 1022 y Fv(\()p Fr(\001)p Ft(;)17 b(n)2579 986 y Fq(0)2601 1022 y Fv(\).)57 b(The)39 b(statemen)m(t)f(in) f(item)h(6)0 1143 y(for)32 b Ft(j)i Fv(=)27 b(0)33 b(is)g(pro)m(v)m(en) h(similarly)-8 b(.)146 1263 y(No)m(w,)29 b(w)m(e)f(\014x)g(the)g(in)m (teger)g Ft(j)33 b Fv(and)27 b(supp)s(ose)h(that)f(the)h(statemen)m(ts) h(in)e(items)h(5)f(and)g(6)g(are)g(v)m(eri\014ed)0 1383 y(for)i(all)h Ft(k)h Fv(=)c(0)p Ft(;)17 b(:)g(:)g(:)f(;)h(j)22 b Fr(\000)16 b Fv(1,)30 b Ft(i)e Fr(\025)h Ft(k)19 b Fv(+)d(1.)42 b(F)-8 b(rom)30 b(the)g(inductiv)m(e)i(h)m(yp)s(otheses)g (and)d(\(6.11\))g(w)m(e)i(ha)m(v)m(e)g(that)0 1504 y(the)i(equalit)m(y) 1423 1624 y(\011)1499 1573 y Fo(\()p Fs(j)t Fo(\))1499 1651 y Fs(g)r(;)p Fo(\003)1607 1624 y Fv(\()p Fr(\001)p Ft(;)17 b(n)p Fv(\))27 b(=)h(\011)2020 1573 y Fo(\()p Fs(j)t Fo(\))2020 1651 y Fs(g)r(;)p Fo(\003)2128 1624 y Fv(\()p Fr(\001)p Ft(;)17 b(n)2296 1583 y Fq(0)2319 1624 y Fv(\))1174 b(\(6.22\))0 1812 y(holds)33 b(for)f(all)h Ft(g)e Fr(2)728 1787 y Fn(e)712 1812 y Fr(G)771 1827 y Fs(j)808 1812 y Fv(\()p Ft(X)8 b Fv(\)\()p Ft(n)p Fv(\))27 b(=)1254 1787 y Fn(e)1238 1812 y Fr(G)1297 1827 y Fs(j)1333 1812 y Fv(\()p Ft(X)8 b Fv(\)\()p Ft(n)1594 1776 y Fq(0)1617 1812 y Fv(\).)44 b(Hence,)34 b(recalling)f(\(6.12\))f(and)h(\(6.13\),)e (w)m(e)j(ha)m(v)m(e)861 2022 y Ft(Z)935 1971 y Fo(\()p Fs(j)t Fo(\))928 2049 y Fs(g)r(;)p Fo(\003)1036 2022 y Fv(\()p Fr(\001)p Ft(;)17 b(n)p Fv(\))27 b(=)h Ft(Z)1447 1971 y Fo(\()p Fs(j)t Fo(\))1440 2049 y Fs(g)r(;)p Fo(\003)1548 2022 y Fv(\()p Fr(\001)p Ft(;)17 b(n)1716 1981 y Fq(0)1739 2022 y Fv(\))143 b(and)h Ft(\027)2275 1971 y Fo(\()p Fs(j)t Fo(\))2269 2049 y Fs(g)r(;)p Fo(\003)p Fs(;n;\030)2521 2022 y Fv(=)27 b Ft(\027)2678 1971 y Fo(\()p Fs(j)t Fo(\))2672 2050 y Fs(g)r(;)p Fo(\003)p Fs(;n)2840 2031 y Fm(0)2862 2050 y Fs(;\030)3531 2022 y Fv(\(6.23\))0 2241 y(for)32 b(an)m(y)h Ft(g)e Fr(2)521 2216 y Fn(e)505 2241 y Fr(G)564 2256 y Fs(j)601 2241 y Fv(\()p Ft(X)8 b Fv(\)\()p Ft(n)p Fv(\))28 b(=)1047 2216 y Fn(e)1031 2241 y Fr(G)1090 2256 y Fs(j)1127 2241 y Fv(\()p Ft(X)8 b Fv(\)\()p Ft(n)1388 2205 y Fq(0)1411 2241 y Fv(\))32 b(and)h Ft(\030)f 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2744 y Fv(\()p Ft(I)8 b Fv(\))44 b Ft(>)f(r)2075 2693 y Fo(\()p Fs(`)p Fo(\))2072 2769 y Fs(i)2163 2744 y Fv(\()p Ft(m)p Fv(\))h Fr(\025)g Ft(d)d Fv(w)m(e)i(ha)m(v)m(e)h(that) d Fr(j)p Ft(I)8 b Fr(j)43 b(\025)i Fv(2.)71 b(If)42 b(it)0 2878 y(do)s(es)j(not)f(exists)i(an)m(y)f Ft(X)55 b Fr(\032\032)49 b(L)1302 2842 y Fo(\()p Fs(`)p Fo(\))p Fs(;})1498 2878 y Fv(suc)m(h)d(that)e Fr(O)2035 2842 y Fs(`)2032 2903 y(})2081 2878 y Ft(X)56 b Fv(=)47 b Ft(I)52 b Fv(w)m(e)46 b(ha)m(v)m(e)f Ft( )2895 2828 y Fo(\()p Fs(`)p Fo(\))2891 2905 y Fs(I)2983 2878 y Fv(\()p Ft(m)p Fv(\))j(=)g(0.)78 b(On)45 b(the)0 3010 y(other)c(hand)h(if)f(there)g(exists)i Ft(X)50 b Fr(\032\032)44 b(L)1547 2974 y Fo(\()p Fs(`)p Fo(\))p Fs(;})1739 3010 y Fv(suc)m(h)f(that)e Fr(O)2270 2974 y Fs(`)2267 3035 y(})2316 3010 y Ft(X)50 b Fv(=)42 b Ft(I)8 b Fv(,)43 b(from)e(\(6.33\))g(w)m(e)h(ha)m(v)m(e)g(that)0 3157 y Ft( )67 3106 y Fo(\()p Fs(`)p Fo(\))63 3184 y Fs(I)155 3157 y Fv(\()p Ft(m)p Fv(\))35 b(=)g(0.)55 b(Indeed)39 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y(Consider,)41 b(no)m(w,)f Ft(m)d Fr(2)1099 3795 y Fv(\026)1049 3821 y Fr(M)1169 3784 y Fo(\()p Fs(`)p Fo(\))1257 3821 y Fv(,)i Ft(i)e Fr(2)h(L)1566 3784 y Fo(\()p Fs(`)p Fo(\))1653 3821 y Fv(,)i(and)e Ft(x)p Fv(\()p Ft(i)p Fv(\))g(as)g(ab)s(o)m(v)m(e.)61 b(By)38 b(using)h(\(6.29\),)f(\(6.33\),)h(and)0 3941 y(item)33 b(3)f(in)h(Theorem)h(6.1,)e(w)m(e)i(then)f(ha)m(v)m(e)h(that) 366 4105 y Fn(X)385 4317 y Fs(I)5 b Fq(3)p Fs(i)527 4115 y Fn(\014)527 4175 y(\014)560 4200 y Ft( )627 4149 y Fo(\()p Fs(`)p Fo(\))623 4227 y Fs(I)715 4200 y Fv(\()p Ft(m)p Fv(\))876 4115 y Fn(\014)876 4175 y(\014)937 4200 y Fv(=)1050 4132 y(1)p 1050 4177 49 4 v 1050 4268 a(2)1109 4200 y Ft(m)1194 4159 y Fo(2)1194 4224 y Fs(i)1256 4200 y Fv(+)1428 4105 y Fn(X)1367 4326 y Fl(I)t Fm(\032\032L)1537 4309 y Fk(\()p Fl(`)p Fk(\))1615 4326 y(:)1364 4383 y Fm(j)p Fl(I)t Fm(j\025)p Fk(2)p Fl(;)11 b(I)t Fm(3)p Fl(i)1662 4115 y Fn(\014)1662 4175 y(\014)1695 4200 y Ft( )1762 4149 y Fo(\()p Fs(`)p Fo(\))1758 4227 y Fs(I)1850 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4852 y Fv(\()p Fr(O)2328 4805 y Fs(})2325 4880 y(`)2377 4852 y Ft(m)p Fv(\))0 5115 y(for)32 b(all)h Ft(m)28 b Fr(2)541 5089 y Fv(\026)492 5115 y Fr(M)612 5078 y Fo(\()p Fs(`)p Fo(\))699 5115 y Fv(.)p Black 1841 5806 a(52)p Black eop end %%Page: 53 54 TeXDict begin 53 53 bop Black Black 146 407 a Fu(Item)35 b(5.)f Fv(By)f(recalling)h(de\014nition)h(\(6.29\))e(and)g(b)m(y)h (using)h(the)e(Mink)m(o)m(wski)k(inequalit)m(y)e(w)m(e)g(ha)m(v)m(e)0 527 y(that)115 718 y(sup)90 811 y Fs(i)p Fq(2L)210 792 y Fk(\()p Fl(`)p Fk(\))304 608 y Fn(h)351 718 y Ft(\027)405 677 y Fo(\()p Fs(`)p Fo(\))494 608 y Fn(\020)553 604 y(\014)553 664 y(\014)553 723 y(\014)603 624 y(X)622 835 y Fs(I)5 b Fq(3)p Fs(i)764 718 y Ft( )831 668 y Fo(\()p Fs(`)p Fo(\))827 745 y Fs(I)918 604 y Fn(\014)918 664 y(\014)918 723 y(\014)952 630 y Fs(q)990 608 y Fn(\021i)1096 630 y Fo(1)p Fs(=q)1272 718 y Fv(=)92 b(sup)1415 811 y Fs(i)p Fq(2L)1535 792 y Fk(\()p Fl(`)p Fk(\))1629 608 y Fn(n)q(h)1743 718 y Ft(\027)1797 677 y Fo(\()p 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Fl(I)t Fm(3)p Fl(i)p Fk(:)579 2047 y Fm(j)p Fl(I)t Fm(j\025)p Fk(2)753 1885 y Ft( )820 1835 y Fo(\()p Fs(`)p Fo(\))816 1912 y Fs(I)908 1771 y Fn(\014)908 1831 y(\014)908 1890 y(\014)941 1797 y Fs(q)979 1775 y Fn(\021)q(i)1086 1797 y Fo(1)p Fs(=q)1222 1885 y Fr(\024)68 b Fv(sup)1327 1978 y Fs(x)p Fq(2L)1463 1959 y Fk(\()p Fl(})p Fk(\))1570 1775 y Fn(h)1617 1885 y Ft(\027)1671 1844 y Fo(\()p Fs(`)p Fo(\))1760 1775 y Fn(\020)1819 1771 y(\014)1819 1831 y(\014)1819 1890 y(\014)1872 1791 y(X)1869 2002 y Fs(X)5 b Fq(3)p Fs(x)2019 1885 y Fv([\011)2122 1835 y Fo(\()p Fs(`)p Fo(\))p Fs(;})2122 1912 y(X)q(;X)2296 1885 y Fr(\016)22 b(O)2450 1838 y Fs(})2447 1913 y(`)2499 1885 y Fv(])2526 1771 y Fn(\014)2526 1831 y(\014)2526 1890 y(\014)2560 1797 y Fs(q)2598 1775 y Fn(\021i)2704 1797 y Fo(1)p Fs(=q)405 2208 y Fr(\024)52 b Fv(sup)493 2301 y Fs(x)p Fq(2L)629 2281 y Fk(\()p Fl(})p Fk(\))737 2097 y Fn(nh)867 2072 y(Z)957 2281 y Fo(\026)922 2298 y Fq(M)1006 2278 y Fk(\()p Fl(`)p Fk(\))1089 2208 y Ft(\027)1143 2167 y Fo(\()p Fs(`)p Fo(\))1231 2208 y Fv(\()p Ft(dm)p Fv(\))1443 2097 y Fn(\020)1522 2113 y(X)1519 2325 y Fs(X)5 b Fq(3)p Fs(x)1685 2208 y Fv(1)-28 b(I)1742 2228 y Fq(f)p Fo(diam)1930 2236 y Fl(})1973 2228 y Fo(\()p Fs(X)5 b Fo(\))p Fq(\024)p Fs(\045;\030)s Fo(\()p Fs(X)g Fo(\)\()p Fq(O)2437 2195 y Fl(})2435 2251 y(`)2482 2228 y Fs(m)p Fo(\)=)p Fq(;g)2702 2208 y Fr(k)p Ft(V)2830 2157 y Fo(\()p Fs(`)p Fo(\))p Fs(;})2809 2235 y(X)q(;X)2982 2208 y Fv(\()p Fr(\001)p Ft(;)17 b Fr(O)3174 2161 y Fs(})3171 2236 y(`)3223 2208 y Ft(m)p Fv(\))p Fr(k)3396 2223 y Fq(1)3471 2097 y Fn(\021)3530 2120 y Fs(q)3568 2097 y Fn(i)3615 2120 y Fo(1)p Fs(=q)834 2477 y Fv(+)932 2366 y Fn(h)996 2341 y(Z)1085 2550 y Fo(\026)1051 2567 y Fq(M)1135 2547 y Fk(\()p Fl(`)p Fk(\))1217 2477 y Ft(\027)1271 2436 y Fo(\()p Fs(`)p Fo(\))1359 2477 y Fv(\()p Ft(dm)p Fv(\))1571 2366 y Fn(\020)1650 2382 y(X)1647 2594 y Fs(X)5 b Fq(3)p Fs(x)1814 2382 y Fn(X)1825 2593 y Fs(j)t Fq(\025)p Fo(1)2149 2382 y Fn(X)1974 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Ft(x)p Fr(g)p Fv(\))2910 5388 y Fn(\003)2951 5410 y Fo(1)p Fs(=q)3531 5468 y Fv(\(6.38\))p Black 1841 5806 a(53)p Black eop end %%Page: 54 55 TeXDict begin 54 54 bop Black Black 0 407 a Fv(with)45 b Ft(c)276 422 y Fo(1)359 407 y Fv(a)f(suitable)h(real)f(dep)s(ending)h (on)f(the)g(norm)g Fr(k)p Ft(U)10 b Fr(k)2276 422 y Fo(0)2360 407 y Fv(of)43 b(the)i(in)m(teraction.)78 b(By)45 b(the)f(same)0 527 y(estimate)34 b(as)f(in)f(\(3.16\),)g(w)m(e)i(ha)m(v)m(e)68 768 y Ft(c)110 783 y Fo(1)149 768 y Ft(`)190 727 y Fs(d)287 768 y Fv(sup)247 861 y Fs(x)p Fq(2L)383 842 y Fk(\()p Fl(})p Fk(\))490 674 y Fn(X)501 884 y Fs(j)t Fq(\025)p Fo(1)651 768 y Ft(#)708 727 y Fo(4)p Fs(d)708 793 y(j)784 688 y Fn(\002)825 768 y Ft(\027)879 727 y Fo(\()p Fs(`)p Fo(\))968 768 y Fv(\()p Fr(f9)p Ft(g)d Fr(2)d(G)1342 783 y Fs(j)1379 768 y Fv(\()p Fr(O)1499 721 y Fs(})1496 796 y(`)1548 768 y Ft(m)p Fv(\))g(:)44 b Ft(Y)1827 783 y Fo(0)1866 768 y Fv(\()p Ft(g)t Fv(\))27 b Fr(3)h Ft(x)p Fr(g)p Fv(\))2257 688 y 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2168 y Fs(i)p Fq(2L)1278 2148 y Fk(\()p Fl(`)p Fk(\))1372 1980 y Fn(X)1390 2192 y Fs(I)5 b Fq(3)p Fs(i)1532 2075 y Ft(e)1577 2034 y Fs(q)r(\013)1656 2046 y Fl(`)1699 2034 y Fo(diam)1851 2046 y Fl(`)1881 2034 y Fo(\()p Fs(I)g Fo(\))p Fs(=)p Fo(\(2)p Fs(d)2104 2010 y Fk(2)2141 2034 y Fo(\))2173 2075 y Fr(j)p Ft(\036)2259 2024 y Fo(\()p Fs(`)p Fo(\))2259 2102 y Fs(I)2346 2075 y Fv(\()p Ft(m)p Fv(\))p Fr(j)765 2320 y(\024)90 b Fv(sup)870 2413 y Fs(m)p Fq(2M)1063 2394 y Fk(\()p Fl(`)p Fk(\))1183 2320 y Fv(sup)1158 2413 y Fs(i)p Fq(2L)1278 2394 y Fk(\()p Fl(`)p Fk(\))1372 2226 y Fn(X)1390 2437 y Fs(I)5 b Fq(3)p Fs(i)1532 2320 y Ft(e)1577 2279 y Fs(q)r(\013)1656 2291 y Fl(`)1699 2279 y Fo(diam)1851 2291 y Fl(`)1881 2279 y Fo(\()p Fs(I)g Fo(\))p Fs(=)p Fo(\(2)p Fs(d)2104 2256 y Fk(2)2141 2279 y Fo(\))2254 2226 y Fn(X)2199 2447 y Fl(X)t Fm(\032L)2344 2429 y Fk(\()p Fl(})p Fk(\))2435 2447 y(:)2214 2512 y Fm(O)2264 2495 y Fl(`)2262 2526 y(})2306 2512 y(X)t Fk(=)p Fl(I)2479 2320 y Fr(j)p 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Fu(Item)38 b(7.)d Fv(W)-8 b(e)37 b(follo)m(w)f(an)f(argumen)m(t)i(analogous)f(to)f(that)h(in)g([1,)g (Section)h(5.3].)53 b(Let)36 b Ft(m)3416 3411 y Fq(0)3473 3447 y Fr(2)d(M)3692 3411 y Fo(\()p Fs(`)p Fo(\))0 3581 y Fv(and)g Ft(J)j Fr(\032\032)29 b(L)532 3545 y Fo(\()p Fs(`)p Fo(\))620 3581 y Fv(,)j(w)m(e)i(de\014ne)g(the)f(follo)m(wing)g (probabilit)m(y)g(k)m(ernel)h(on)f Fr(L)2675 3530 y Fo(\()p Fs(`)p Fo(\))2675 3608 y Fs(J)601 4014 y Ft(q)644 4029 y Fs(J)693 4014 y Fv(\()p Ft(m)816 3973 y Fq(0)839 4014 y Ft(;)17 b(m)p Fv(\))28 b(:=)1319 3831 y(exp)1484 3720 y Fn(n)1604 3736 y(X)1567 3952 y Fs(I)5 b Fq(\\)p Fs(J)h Fq(6)p Fo(=)p Fq(;)1802 3750 y Fn(\002)1843 3831 y Ft( )1910 3780 y Fo(\()p Fs(`)p Fo(\))1906 3858 y Fs(I)1998 3831 y Fv(\()p Ft(mm)2206 3790 y Fq(0)2206 3855 y Fs(J)2251 3837 y Fk(c)2287 3831 y Fv(\))22 b(+)g Ft(\036)2503 3780 y Fo(\()p Fs(`)p Fo(\))2503 3858 y Fs(I)2591 3831 y Fv(\()p Ft(mm)2799 3790 y Fq(0)2799 3855 y Fs(J)2844 3837 y Fk(c)2880 3831 y Fv(\))2918 3750 y Fn(\003)2959 3720 y(o)p 1175 3991 1995 4 v 1238 4023 a(X)1175 4261 y Fs(m)p Fq(2M)1368 4226 y Fk(\()p Fl(`)p Fk(\))1368 4285 y Fl(J)1463 4117 y Fv(exp)1628 4007 y Fn(n)1748 4023 y(X)1711 4239 y Fs(I)5 b Fq(\\)p Fs(J)h Fq(6)p Fo(=)p Fq(;)1946 4037 y Fn(\002)1987 4117 y Ft( )2054 4066 y Fo(\()p Fs(`)p Fo(\))2050 4144 y Fs(I)2142 4117 y Fv(\()p Ft(mm)2350 4076 y Fq(0)2350 4142 y Fs(J)2395 4123 y Fk(c)2431 4117 y Fv(\))22 b(+)g Ft(\036)2647 4066 y Fo(\()p Fs(`)p Fo(\))2647 4144 y Fs(I)2735 4117 y Fv(\()p Ft(mm)2943 4076 y Fq(0)2943 4142 y Fs(J)2988 4123 y Fk(c)3024 4117 y Fv(\))3062 4037 y Fn(\003)3103 4007 y(o)3531 4014 y Fv(\(6.41\))0 4503 y(where)31 b(the)f(functions)h Ft( )929 4452 y Fo(\()p Fs(`)p Fo(\))925 4530 y Fs(I)1047 4503 y Fv(and)f Ft(\036)1292 4452 y Fo(\()p Fs(`)p Fo(\))1292 4530 y Fs(I)1409 4503 y Fv(ha)m(v)m(e)i(b)s(een)e(de\014ned)h(in)g(\(6.29\))e(and)h (\(6.30\).)42 b(Note)30 b(that,)g(giv)m(en)0 4639 y Ft(m)e Fr(2)g(M)327 4588 y Fo(\()p Fs(`)p Fo(\))327 4666 y Fs(J)415 4639 y Fv(,)k(w)m(e)i(ha)m(v)m(e)g Ft(q)886 4654 y Fs(J)935 4639 y Fv(\()p Fr(\001)p Ft(;)17 b(m)p Fv(\))27 b Fr(2)h(B)1357 4588 y Fo(\()p Fs(`)p Fo(\))1354 4666 y Fs(J)1399 4647 y Fk(c)1446 4639 y Fv(.)146 4775 y(Pic)m(k)38 b Ft(J)k Fr(\032\032)34 b(L)721 4739 y Fo(\()p Fs(`)p Fo(\))809 4775 y Fv(,)i Ft(f)44 b Fr(2)33 b(B)1131 4724 y Fo(\()p Fs(`)p Fo(\))1128 4802 y Fs(J)1220 4775 y Fv(,)j(recall)h Ft(M)1651 4739 y Fo(\()p Fs(`)p Fo(\))1775 4775 y Fv(has)f(b)s(een)h (de\014ned)g(in)f(\(2.11\),)g(b)m(y)g(de\014nition)h(of)f(the)0 4896 y(renormalized)d(measure)g Ft(\027)1012 4860 y Fo(\()p Fs(`)p Fo(\))1131 4896 y Fv(and)f(b)m(y)g(standard)g(measure)h(theory)f (w)m(e)g(ha)m(v)m(e)h Ft(\026)p Fv(\()p Ft(f)11 b Fv(\()p Ft(M)3208 4860 y Fo(\()p Fs(`)p Fo(\))3296 4896 y Fv(\)\))28 b(=)f Ft(\027)3557 4860 y Fo(\()p Fs(`)p Fo(\))3645 4896 y Fv(\()p Ft(f)11 b Fv(\))0 5016 y(and)120 5124 y Fn(Z)175 5350 y Fq(M)259 5330 y Fk(\()p Fl(`)p Fk(\))358 5260 y Ft(\027)412 5218 y Fo(\()p Fs(`)p Fo(\))500 5260 y Fv(\()p Ft(dm)674 5218 y Fq(0)697 5260 y Fv(\))832 5165 y Fn(X)768 5404 y Fs(m)p Fq(2M)961 5368 y Fk(\()p Fl(`)p Fk(\))961 5427 y Fl(J)1073 5260 y Ft(q)1116 5275 y Fs(J)1165 5260 y Fv(\()p Ft(m)1288 5218 y Fq(0)1312 5260 y Ft(;)17 b(m)p Fv(\))g Ft(f)11 b Fv(\()p Ft(m)p Fv(\))27 b(=)1846 5124 y Fn(Z)1902 5350 y Fq(X)1982 5260 y Ft(\026)p Fv(\()p Ft(d\021)t Fv(\))2316 5165 y Fn(X)2253 5404 y Fs(m)p Fq(2M)2446 5368 y Fk(\()p Fl(`)p Fk(\))2446 5427 y Fl(J)2557 5260 y Ft(q)2600 5275 y Fs(J)2649 5260 y Fv(\()p Ft(M)2791 5218 y Fo(\()p Fs(`)p Fo(\))2880 5260 y Fv(\()p Ft(\021)t Fv(\))p Ft(;)17 b(m)p Fv(\))g Ft(f)11 b Fv(\()p Ft(m)p Fv(\))119 b(\(6.42\))p Black 1841 5806 a(54)p Black eop end %%Page: 55 56 TeXDict begin 55 55 bop Black Black 0 407 a Fv(The)34 b(equations)f(\(2.15\))f(will)h(th)m(us)h(follo)m(w)f(from)811 675 y Ft(\026)p Fv(\()p Ft(f)11 b Fv(\()p Ft(M)1109 634 y Fo(\()p Fs(`)p Fo(\))1197 675 y Fv(\)\))28 b(=)1404 540 y Fn(Z)1460 765 y Fq(X)1540 675 y Ft(\026)p Fv(\()p Ft(d\021)t Fv(\))1874 581 y Fn(X)1810 819 y Fs(m)p Fq(2M)2003 784 y Fk(\()p Fl(`)p Fk(\))2003 842 y Fl(J)2115 675 y Ft(q)2158 690 y Fs(J)2207 675 y Fv(\()p Ft(M)2349 634 y Fo(\()p Fs(`)p Fo(\))2438 675 y Fv(\()p Ft(\021)t Fv(\))p Ft(;)17 b(m)p Fv(\))g Ft(f)11 b Fv(\()p Ft(m)p Fv(\))561 b(\(6.43\))146 1061 y(F)-8 b(or)35 b Ft(X)40 b Fr(\032)33 b(L)624 1025 y Fo(\()p Fs(`)p Fo(\))712 1061 y Fv(,)j(let)g(us)g(in)m (tro)s(duce)g(the)g(family)g(of)f Ft(\033)t Fv({algebras)h Fr(E)2622 1010 y Fo(\()p Fs(`)p Fo(\))2613 1088 y Fs(X)2742 1061 y Fv(:=)c Ft(\033)t Fr(f)p Ft(M)3090 1010 y Fo(\()p Fs(`)p Fo(\))3080 1087 y Fs(i)3179 1061 y Fv(\()p Fr(\001)p Fv(\))17 b Ft(;)40 b(i)32 b Fr(2)h Ft(X)8 b Fr(g)33 b(\032)0 1181 y(F)72 1196 y Fq(O)128 1208 y Fl(`)157 1196 y Fs(X)261 1181 y Fv(on)j(the)h(con\014guration)g(space)g Fr(X)15 b Fv(.)55 b(No)m(w,)38 b(pic)m(k)g(\003)33 b Fr(\032\032)j(L)2423 1145 y Fo(\()p Fs(})p Fo(\))2562 1181 y Fv(so)h(that)f Ft(V)56 b Fv(:=)34 b Fr(O)3233 1145 y Fs(`)3230 1206 y(})3279 1181 y Fv(\003)g Fr(\033)h Ft(J)9 b Fv(.)55 b(F)-8 b(or)0 1302 y Ft(\021)31 b Fr(2)d(X)15 b Fv(,)33 b(w)m(e)g(set)786 1422 y Ft(G)863 1437 y Fs(V)924 1422 y Fv(\()p Ft(M)1066 1371 y Fo(\()p Fs(`)p Fo(\))1056 1454 y Fs(V)16 b Fq(n)p Fs(J)1197 1422 y Fv(\()p Ft(\021)t Fv(\))p Ft(;)h(\021)1417 1437 y Fq(O)1473 1449 y Fl(`)1502 1437 y Fs(V)1559 1418 y Fk(c)1595 1422 y Fv(\))27 b(:=)h Ft(\026)1850 1342 y Fn(\000)1895 1422 y Ft(f)11 b Fv(\()p Ft(M)2096 1381 y Fo(\()p Fs(`)p Fo(\))2185 1422 y Fv(\))2223 1338 y Fn(\014)2223 1397 y(\014)2256 1422 y Fr(E)2318 1371 y Fo(\()p Fs(`)p Fo(\))2309 1454 y Fs(V)k Fq(n)p Fs(J)2471 1422 y Fr(\012)23 b(F)2643 1437 y Fq(O)2699 1449 y Fl(`)2729 1437 y Fs(V)2785 1418 y Fk(c)2821 1342 y Fn(\001)2867 1422 y Fv(\()p Ft(\021)t Fv(\))536 b(\(6.44\))0 1597 y(W)-8 b(e)33 b(shall)g(pro)m(v)m(e)h(that,)e Ft(\026)p Fv({a.s.,)754 1828 y(lim)713 1902 y Fs(V)15 b Fq("L)853 1882 y Fk(\()p Fl(`)p Fk(\))948 1828 y Ft(G)1025 1843 y Fs(V)1086 1828 y Fv(\()p Ft(M)1228 1778 y Fo(\()p Fs(`)p Fo(\))1218 1860 y Fs(V)h Fq(n)p Fs(J)1359 1828 y Fv(\()p Ft(\021)t Fv(\))p Ft(;)h(\021)1579 1843 y Fq(O)1635 1855 y Fl(`)1664 1843 y Fs(V)1721 1825 y Fk(c)1756 1828 y Fv(\))28 b(=)1989 1734 y Fn(X)1926 1973 y Fs(m)p Fq(2M)2119 1937 y Fk(\()p Fl(`)p Fk(\))2119 1996 y Fl(J)2214 1828 y Ft(q)2257 1843 y Fs(J)2306 1828 y Fv(\()p Ft(M)2448 1787 y Fo(\()p Fs(`)p Fo(\))2536 1828 y Fv(\()p Ft(\021)t Fv(\))p Ft(;)17 b(m)p Fv(\))g Ft(f)11 b Fv(\()p Ft(m)p Fv(\))463 b(\(6.45\))0 2183 y(Therefore,)34 b(b)m(y)f(dominated)h(con)m (v)m(ergence,)h(w)m(e)f(ha)m(v)m(e)71 2446 y Ft(\026)p Fv(\()p Ft(f)11 b Fv(\()p Ft(M)369 2405 y Fo(\()p Fs(`)p Fo(\))457 2446 y Fv(\)\))27 b(=)h Ft(\026)p Fv(\()p Ft(\026)p Fv(\()p Ft(f)11 b Fv(\()p Ft(M)1059 2405 y Fo(\()p Fs(`)p Fo(\))1146 2446 y Fr(jE)1236 2395 y Fo(\()p Fs(`)p Fo(\))1227 2477 y Fs(V)16 b Fq(n)p Fs(J)1390 2446 y Fr(\012)23 b(F)1562 2461 y Fq(O)1618 2473 y Fl(`)1647 2461 y Fs(V)1704 2442 y Fk(c)1739 2446 y Fv(\)\))1871 2385 y Fs(V)15 b Fq("L)2011 2362 y Fk(\()p Fl(`)p Fk(\))1900 2446 y Fr(\000)-17 b(!)2144 2310 y Fn(Z)2200 2536 y Fq(X)2280 2446 y Ft(\026)p Fv(\()p Ft(d\021)t Fv(\))2614 2351 y Fn(X)2551 2590 y Fs(m)p Fq(2M)2744 2554 y Fk(\()p Fl(`)p Fk(\))2744 2613 y Fl(J)2855 2446 y Ft(q)2898 2461 y Fs(J)2947 2446 y Fv(\()p Ft(M)3089 2405 y Fo(\()p Fs(`)p Fo(\))3178 2446 y Fv(\()p Ft(\021)t Fv(\))p Ft(;)17 b(m)p Fv(\))g Ft(f)11 b Fv(\()p Ft(m)p Fv(\))3531 2706 y(\(6.46\))0 2826 y(so)33 b(that)f(\(6.43\))g(holds.) 146 2947 y(W)-8 b(e)49 b(\014nally)h(pro)m(v)m(e)g(\(6.45\))e(in)h(the) g(set)g(of)f(full)h(measure)h(\()p Ft(M)2553 2910 y Fo(\()p Fs(`)p Fo(\))2642 2947 y Fv(\))2680 2910 y Fq(\000)p Fo(1)2774 2947 y Fv(\()2861 2921 y(\026)2812 2947 y Fr(M)2932 2910 y Fo(\()p Fs(`)p Fo(\))3019 2947 y Fv(\).)92 b(By)49 b(the)g(Gibbs)0 3067 y(prop)s(ert)m(y)33 b(of)f(the)h(original)f (measure)h Ft(\026)p Fv(,)f(for)g Ft(\021)f Fr(2)d(X)15 b Fv(,)32 b(suc)m(h)i(that)e Ft(M)2528 3031 y Fo(\()p Fs(`)p Fo(\))2617 3067 y Fv(\()p Ft(\021)t Fv(\))27 b Fr(2)2916 3042 y Fv(\026)2866 3067 y Fr(M)2986 3031 y Fo(\()p Fs(`)p Fo(\))3074 3067 y Fv(,)32 b(and)h Ft(m)3408 3031 y Fq(0)3459 3067 y Fr(2)28 b(M)3673 3082 y Fs(J)3718 3063 y Fk(c)3753 3067 y Fv(,)0 3187 y(suc)m(h)34 b(that)e Ft(m)516 3151 y Fq(0)516 3213 y Fs(J)561 3194 y Fk(c)625 3187 y Fv(=)728 3107 y Fn(\000)774 3187 y Ft(M)878 3151 y Fo(\()p Fs(`)p Fo(\))967 3187 y Fv(\()p Ft(\021)t Fv(\))1095 3107 y Fn(\001)1140 3226 y Fs(J)1185 3207 y Fk(c)1220 3187 y Fv(,)h(w)m(e)g(ha)m(v)m(e)h(that)100 3479 y Ft(G)177 3494 y Fs(V)238 3479 y Fv(\()p Ft(m)361 3438 y Fq(0)361 3503 y Fs(V)16 b Fq(n)p Fs(J)502 3479 y Ft(;)h(\021)594 3494 y Fq(O)650 3506 y Fl(`)680 3494 y Fs(V)737 3475 y Fk(c)772 3479 y Fv(\))28 b(=)f Ft(\026)1000 3432 y Fs(\021)1000 3506 y Fq(O)1056 3518 y Fl(`)1086 3506 y Fs(V)1147 3398 y Fn(\000)1192 3479 y Ft(f)11 b Fv(\()p Ft(M)1393 3438 y Fo(\()p Fs(`)p Fo(\))1482 3479 y Fv(\))1520 3394 y Fn(\014)1520 3454 y(\014)1553 3479 y Ft(M)1657 3428 y Fo(\()p Fs(`)p Fo(\))1647 3510 y Fs(V)16 b Fq(n)p Fs(J)1816 3479 y Fv(=)27 b Ft(m)2004 3438 y Fq(0)2004 3503 y Fs(V)17 b Fq(n)p Fs(J)2146 3398 y Fn(\001)277 3872 y Fv(=)455 3583 y Fn(X)391 3795 y Fs(\033)r Fq(2X)530 3806 y Fm(O)578 3823 y Fl(`)609 3806 y(V)679 3678 y Ft(f)11 b Fv(\()p Ft(M)880 3637 y Fo(\()p Fs(`)p Fo(\))968 3678 y Fv(\()p Ft(\033)t Fv(\)\))17 b Ft(e)1203 3637 y Fs(H)1261 3648 y Fm(O)1309 3666 y Fl(`)1339 3648 y(V)1393 3637 y Fo(\()p Fs(\033)r(\021)1497 3650 y Fm(O)1545 3667 y Fl(`)1577 3650 y(V)1626 3636 y Fk(c)1662 3637 y Fo(\))1710 3678 y Fv(1)-28 b(I)1766 3716 y Fq(f)p Fs(M)1876 3681 y Fk(\()p Fl(`)p Fk(\))1869 3743 y Fl(V)13 b Fm(n)p Fl(J)1993 3716 y Fo(\()p Fs(\033)r Fo(\)=)p Fs(m)2206 3694 y Fm(0)2206 3743 y Fl(V)i Fm(n)p Fl(J)2332 3716 y Fq(g)p 391 3849 1981 4 v 694 3871 a Fn(X)630 4083 y Fs(\033)r Fq(2X)769 4094 y Fm(O)817 4111 y Fl(`)848 4094 y(V)918 3966 y Ft(e)963 3925 y Fs(H)1021 3936 y Fm(O)1069 3953 y Fl(`)1100 3936 y(V)1154 3925 y Fo(\()p Fs(\033)r(\021)1258 3938 y Fm(O)1306 3955 y Fl(`)1338 3938 y(V)1387 3924 y Fk(c)1422 3925 y Fo(\))1470 3966 y Fv(1)-28 b(I)1527 4004 y Fq(f)p Fs(M)1637 3969 y Fk(\()p Fl(`)p Fk(\))1630 4031 y Fl(V)13 b Fm(n)p Fl(J)1754 4004 y Fo(\()p Fs(\033)r Fo(\)=)p Fs(m)1967 3981 y Fm(0)1967 4031 y Fl(V)i Fm(n)p Fl(J)2093 4004 y Fq(g)277 4442 y Fv(=)454 4131 y Fn(X)391 4370 y Fs(m)p Fq(2M)584 4335 y Fk(\()p Fl(`)p Fk(\))584 4393 y Fl(J)679 4226 y Ft(f)c Fv(\()p Ft(m)p Fv(\))996 4131 y Fn(X)932 4343 y Fs(\033)r Fq(2X)1071 4354 y Fm(O)1119 4371 y Fl(`)1150 4354 y(V)1220 4226 y Ft(e)1265 4185 y Fs(H)1323 4196 y Fm(O)1371 4213 y Fl(`)1402 4196 y(V)1455 4185 y Fo(\()p Fs(\033)r(\021)1559 4198 y Fm(O)1607 4215 y Fl(`)1639 4198 y(V)1689 4184 y Fk(c)1724 4185 y Fo(\))1772 4226 y Fv(1)-28 b(I)1829 4264 y Fq(f)p Fs(M)1939 4229 y Fk(\()p Fl(`)p Fk(\))1932 4287 y Fl(V)2017 4264 y Fo(\()p Fs(\033)r Fo(\)=)p Fs(mm)2292 4241 y Fm(0)2292 4291 y Fl(V)15 b Fm(n)p Fl(J)2418 4264 y Fq(g)p 391 4419 2067 4 v 581 4441 a Fn(X)517 4680 y Fs(m)p Fq(2M)710 4644 y Fk(\()p Fl(`)p Fk(\))710 4703 y Fl(J)869 4441 y Fn(X)805 4652 y Fs(\033)r Fq(2X)944 4663 y Fm(O)992 4681 y Fl(`)1024 4663 y(V)1094 4536 y Ft(e)1139 4494 y Fs(H)1197 4505 y Fm(O)1245 4523 y Fl(`)1275 4505 y(V)1329 4494 y Fo(\()p Fs(\033)r(\021)1433 4507 y Fm(O)1481 4525 y Fl(`)1513 4507 y(V)1562 4493 y Fk(c)1597 4494 y Fo(\))1646 4536 y Fv(1)-28 b(I)1702 4574 y Fq(f)p Fs(M)1812 4539 y Fk(\()p Fl(`)p Fk(\))1805 4597 y Fl(V)1890 4574 y Fo(\()p Fs(\033)r Fo(\)=)p Fs(mm)2165 4551 y Fm(0)2165 4600 y Fl(V)16 b Fm(n)p Fl(J)2291 4574 y Fq(g)2495 4442 y Fv(=)2672 4131 y Fn(X)2608 4370 y Fs(m)p Fq(2M)2801 4335 y Fk(\()p Fl(`)p Fk(\))2801 4393 y Fl(J)2896 4226 y Ft(f)11 b Fv(\()p Ft(m)p Fv(\))17 b Ft(Z)3207 4175 y Fo(\()p Fs(`)p Fo(\))3200 4254 y Fs(mm)3324 4235 y Fm(0)3347 4254 y Fs(;V)3427 4226 y Fv(\()p Fr(O)3547 4185 y Fs(`)3580 4226 y Ft(\021)t Fv(\))p 2608 4419 1062 4 v 2790 4441 a Fn(X)2727 4680 y Fs(m)p Fq(2M)2920 4644 y Fk(\()p Fl(`)p 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(ossible)h(failure)f(of)f(the)h(Gibbs)g(prop)s(ert)m(y)g(for)f (measures)i(on)201 4857 y(lattice)26 b(systems.)j(Disordered)e(systems) h(and)e(statistical)h(ph)m(ysics:)43 b(rigorous)26 b(results.")i Fi(Mark)m(o)m(v)201 4977 y(Pro)s(cess.)34 b(Related)f(Fields)g Fw(2)p Fv(,)g(209{224)e(\(1996\).)p Black 0 5181 a([18])p Black 49 w(A.C.D.)36 b(v)-5 b(an)37 b(En)m(ter,)i(R.)d(F)-8 b(ern\023)-49 b(andez,)39 b(\\A)e(remark)g(on)f(di\013eren)m(t)i(norms) g(and)e(analyticit)m(y)i(for)201 5301 y(man)m(y{particle)c(in)m (teractions.")f Fi(J.)g(Statist.)g(Ph)m(ys.)i Fw(56)p Fv(,)d(965{972)f(\(1989\).)p Black 1841 5806 a(57)p Black eop end %%Page: 58 59 TeXDict begin 58 58 bop Black Black Black 0 407 a Fv([19])p Black 49 w(A.C.D.)29 b(v)-5 b(an)30 b(En)m(ter,)h(R.)f(F)-8 b(ern\023)-49 b(andez,)31 b(A.D.)e(Sok)-5 b(al,)30 b(\\Regularit)m(y)g (prop)s(erties)h(and)f(pathologies)201 527 y(of)40 b(p)s(osition{space) i(renormalization{group)f(transformations:)61 b(scop)s(e)42 b(and)f(limitations)g(of)201 648 y(Gibbsian)33 b(theory)-8 b(.")33 b Fi(J.)f(Statist.)h(Ph)m(ys.)i Fw(72)p Fv(,)d(879{1167)f (\(1994\).)p Black 0 851 a([20])p Black 49 w(J.)37 b(F)-8 b(r\177)-49 b(ohlic)m(h,)40 b(J.Z.)d(Im)m(brie,)j(\\Impro)m(v)m(ed)g(p) s(erturbation)d(expansion)i(for)e(disordered)i(systems:)201 971 y(b)s(eating)32 b(Gri\016ths')h(singularities.")h Fi(Comm.)f(Math.)g(Ph)m(ys.)i Fw(96)p Fv(,)e(145{180)d(\(1984\).)p Black 0 1175 a([21])p Black 49 w(G.)i(Galla)m(v)m(otti,)h (\\Instabilities)h(and)f(phase)h(transitions)f(in)g(the)h(Ising)f(mo)s (del.)g(A)g(review.")h Fi(La)201 1295 y(Rivista)f(del)g(Nuo)m(v)m(o)g (Cimen)m(to)h Fw(2)p Fv(,)f(133{169)e(\(1972\).)p Black 0 1499 a([22])p Black 49 w(Galla)m(v)m(otti)h(G.,)f(Martin)i(L\177)-49 b(of)31 b(A.,)i(Miracle)g(Sole)f(S.,)h(in)f(Battelle)h(Seattle)f (\(1971\))f(Rencon)m(tres,)201 1619 y(A.)25 b(Lenard,)i(ed.)f (\(Lecture)h(Notes)f(in)f(Phisics,)k(V)-8 b(ol.)26 b(20,)g(Springer,)i (Berlin,)f(1973\),)f(pp.162-204.)p Black 0 1822 a([23])p Black 49 w(N.M.)33 b(Hugenholtz,)i(\\On)e(the)g(in)m(v)m(erse)j (problem)e(in)f(statistical)h(mec)m(hanics.")i Fi(Comm.)e(Math.)201 1943 y(Ph)m(ys.)g Fw(85)p Fv(,)f(27{38)e(\(1982\).)p Black 0 2146 a([24])p Black 49 w(R.)44 b(Kotec)m(k)q(\023)-50 b(y,)50 b(D.)44 b(Preiss,)49 b(\\Cluster)d(expansion)h(for)d(abstract)h (p)s(olymer)h(mo)s(dels.")f Fi(Comm.)201 2267 y(Math.)32 b(Ph)m(ys.)j Fw(103)p Fv(,)e(491{498)e(\(1986\).)p Black 0 2470 a([25])p Black 49 w(F.)d(Martinelli,)i(E.)f(Olivieri,)i(R.)d(Sc) m(honmann,)j(\\F)-8 b(or)28 b(2{)p Ft(D)i Fv(lattice)g(spin)f(systems)i (w)m(eak)f(mixing)201 2590 y(implies)k(strong)e(mixing.")i Fi(Comm)m(un.)g(Math.)f(Ph)m(ys.)h Fw(165)p Fv(,)f(33{47)e(\(1994\).)p Black 0 2794 a([26])p Black 49 w(C.)25 b(Maes,)j(F.)d(Redig,)i(S.)e (Shlosman,)j(A.)d(V)-8 b(an)25 b(Mo\013aert,)i(\\P)m(ercolation,)g (path)f(large)f(deviations)201 2914 y(and)32 b(w)m(eakly)j(Gibbs)e (states.")g Fi(Comm)m(un.)h(Math.)f(Ph)m(ys.)h Fw(209)p Fv(,)f(517{545)e(\(2000\).)p Black 0 3117 a([27])p Black 49 w(E.)36 b(Olivieri,)i(\\On)e(a)g(cluster)i(expansion)f(for)f (lattice)h(spin)g(systems:)53 b(a)36 b(\014nite)h(size)g(condition)201 3238 y(for)32 b(the)h(con)m(v)m(ergence.")i Fi(J.)d(Statist.)h(Ph)m (ys.)i Fw(50)p Fv(,)d(1179{1200)f(\(1988\).)p Black 0 3441 a([28])p Black 49 w(E.)c(Olivieri,)i(P)-8 b(.)28 b(Picco,)h(\\Cluster)f(expansion)g(for)f Ft(D)s Fv({dimensional)h (lattice)g(systems)h(and)e(\014nite)201 3562 y(v)m(olume)34 b(factorization)e(prop)s(erties.")h Fi(J.)g(Statist.)g(Ph)m(ys.)h Fw(59)p Fv(,)f(221{256)e(\(1990\).)p Black 0 3765 a([29])p Black 49 w(S.B.)56 b(Shlosman,)62 b(\\P)m(ath)56 b(large)f(deviation)i (and)e(other)h(t)m(ypical)h(prop)s(erties)f(of)g(the)g(lo)m(w{)201 3885 y(temp)s(erature)29 b(mo)s(dels,)g(with)g(applications)g(to)f(the) h(w)m(eakly)h(Gibbs)e(states.")h Fi(Mark)m(o)m(v)g(Pro)s(cess.)201 4006 y(Related)k(Fields)g Fw(6)p Fv(,)g(121{133)e(\(2000\).)p Black 0 4209 a([30])p Black 49 w(R.H.)42 b(Sc)m(honmann,)47 b(S.B.)d(Shlosman,)i(\\Complete)e(analyticit)m(y)h(for)d(2D)g(Ising)i (completed.")201 4330 y Fi(Comm.)33 b(Math.)g(Ph)m(ys.)h Fw(170)p Fv(,)f(453{482)e(\(1995\).)p Black 1841 5806 a(58)p Black eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0407091152125--