Content-Type: multipart/mixed; boundary="-------------0211160833296" This is a multi-part message in MIME format. ---------------0211160833296 Content-Type: text/plain; name="02-466.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-466.keywords" Gibbs vs non-Gibbs, generalized Gibbs measures, variational principle, renormalization group, disordered systems, random field Ising model, Morita approach. ---------------0211160833296 Content-Type: application/postscript; name="ggmAP.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="ggmAP.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: ggmAP.dvi %%CreationDate: Sat Nov 16 15:15:23 2002 %%Pages: 32 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMBX12 CMR12 CMSY8 CMBX10 CMR10 CMBX9 CMR9 CMSY6 CMMI10 %%+ CMSY10 CMR8 CMTI10 MSBM7 CMMI6 CMEX10 MSBM10 CMMI8 CMR6 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips.exe -P pdf -G0 ggmAP %DVIPSParameters: dpi=8000, compressed %DVIPSSource: TeX output 2002.11.16:1515 %%BeginProcSet: tex.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/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: alt-rule.pro %! % Patch by TVZ % Makes dvips files draw rules with stroke rather than fill. % Makes narrow rules more predictable at low resolutions % after distilling to PDF. % May have unknown consequences for very thick rules. % Tested only with dvips 5.85(k). 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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. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /Lambda put dup 8 /Phi put dup 22 /macron put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 58 /colon put dup 61 /equal put dup 100 /d put dup 103 /g put dup 108 /l put dup 111 /o put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueXX 5000789 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 17 /eta put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 27 /sigma put dup 33 /omega put dup 59 /comma put dup 60 /less put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 72 /H put dup 75 /K put dup 76 /L put dup 78 /N put dup 82 /R put dup 86 /V put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 103 /g 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 112 /p put dup 118 /v put dup 120 /x put dup 121 /y put readonly def /FontBBox{-24 -250 1110 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 76 /L put dup 78 /N put dup 80 /P put dup 82 /R put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueXX 5031982 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d56491bdecdcafdc988206c5a457a19270b 37d0ab776e03eaa7eb568eeab6b5e79dec03b0dcbf923a2aa8e4f4deda2cb043 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818455cc8dd8d40939f3ba8aa5abc54590c7286b172193c23adb1b4f9931a9f7 73b960b1f5b760438028478e4442dc93a567b07e20017c18fca2a1daa2649dee 5a6ec51fd6c3f4b27367c31e95a56ba4bbeda0493bc02306037e2ba69add9b94 373364db316691e73d6811cd76fbc92b0c67ca18dcdf05807f17d1b309456ea9 d113ea0fccd6eb73bd8dfdbd5905a5e38bbf61cfe58fb01ec95a4bdd9565433d 4a7abb8dca6f76c45b254635c3352c04b4ac72a87290d2af4efd37444af8e454 8e3db7d1d177ebc3950d14988805c6bea0f48129b71531be0ea4f2fa62e7842a 6661249ec9e0a2be7a4c8822d06fbd1bf74e8dee8403bea1ee0ecc4352dd18ae cad4ccf99102f544e53befc4935e09818dcd7079ceb4b8a92bc7f261c0b5ef69 bf404c8b72bcba626a24be5ac5811ec354e74f141b53aba4bc9c9e9a48dff459 39ddd8c6158c809dd570d0bbeca1cdbe6a1630ad3495a233191952ebf398d885 b90baacc0c3c32c48fba9638b512bc93c3647e361427402454d4dba1ac7adaa4 15a2c75aea3959f4a711eb4577568ad4043a1b0b9c58ade9f21bd68d41809f07 4893b58def29908d781d27625e22b5d2ba0036ea5a6e5643f86a08d003a8933f 09defead9383482ebe236dac98e348c8c6c64a1f264e95573beb69e424c40143 d080258b8fbdd159f9c84f4d6f20d18593567c2e6500f56e393a41c29cdb9040 7bb67255cb97dd70328b6762e0d1800afe1f854cb905f3938a6aaa41d7ba1ad7 1b565df5db985d2019e97cf445159fdfca550ac86aa8330bca42b2d6e1ed95b8 af440c11e1219d1035e734ab49ef298347a4ddfe04ec5c9c82d4a8a265ce331d d26004e3fffba6c52284a1d3808f3ac71af5b53ccf20558442f66153a069c6f7 d6027895b2e10d177b09d5ca7ad3305091b4e7d4362a8ec7ae2f9382d43039d5 8b9da11d11a43b957eec779f93c1c7dca89bfc788e332049a017e8c676030e7d 5c8a3fca937f4d8188bdebd56186525a159388dac60dbaed0eb8e727e1c1cf40 efb3e9cf63b1bea041da6936d118ad15e46b233aa9cc6756c9778f99600dd3a8 67b10931ba6705a3c13871d93e22c4943e45f16ec07a9917597e33da26c01c4f 4911eebbbe90ceeb7f09026c980b23bfb1a554e6ad7c935d52b0cc93f1ff294d 23af5db0927699cf35fbc0d6770a91d728adbaaa0d580bbf5ee1ff1c89c91ccb 0647c2a60725e04d2f4e80e61346bdf3dc53719437f2941acd2034451c5e2e4d 406e45a51acc45e7df20752376121251abcea8ec848ed02bb8b06e5b7ad746c6 b0bd9ba058bc7d539303a6d6560ec99d92e07c8d5240ae6601f08d50076ccc9a abf89a669388b2ba7a4ceb3b4f13f49ed806dfe6f7e7f2fcac039010a43e3955 59ec386b4486e1db89aebaf6902a7246999b824e42a90a68b32232eade51244a 61d9354bdf1669d44fd322597f5dcb9185fffb60e80251f582aac40558e2327d 54a3a2862ef119c53f51dcb3d637ed65a0319a0460981e7ea9c050152476620b 209ea9bc7093be4f6042644cef29d86c99935cb2b987bc945086f7af019edfbc ae76d627a489927be3d388b11a37e55830fe3f856321110b113789fa0c73e080 2362753b67829ac2b6ef38eca3c92e17e42dcc4875646e5a4efe402b3ac5d6e0 75c769b58a3d19eeda4763d3f8e80c51cdc0fb2f0e265aca7fe0a3610a551aa5 6e35528651997ca6d2bc8b564cc7788d728cb61dd31c1ce2fea4d71832039329 026d82c4a29f15fca1b1e1410e1c9026f5d44cdd97d5c1d77c1b670ba2e82f41 e4bc69b81694a0d88bca9a4061fecfbb175071c9b087a50fca577456a05f4cd9 fa07a17741f24a9bad50cf91ddd6a330f31cb8ebef5cc9e54de657ff2da4743c 532b7493f0ad167aabdf5b12029fab39ccf40d1399af0624ec688f0defcc2d6c 0bfb3517bfe525add1f936557ed0400b57ff82c0ca5017616a85fda58e91e7a9 0d491d72d1cd269d37582af13ba2be560c276089ad51be3486460c596fc0a30f 524c58895ba36b6d3edb6ad193440e939dced5944e46b5357f22bc98e6c16b53 4cb28f0ca38b274fce017d551c851dca792286951035c3f780c1966d0d6344b2 94ef8e53e3b594 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 17 /eta put dup 22 /mu put dup 27 /sigma put dup 33 /omega put dup 59 /comma put dup 64 /partialdiff put dup 65 /A put dup 67 /C put dup 72 /H put dup 76 /L put dup 78 /N put dup 86 /V put dup 99 /c put dup 100 /d put dup 103 /g put dup 107 /k put dup 110 /n put dup 114 /r put dup 120 /x put dup 121 /y put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueXX 5087381 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 58 /colon put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 73 /I put dup 74 /J put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 86 /V 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 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 dup 122 /z put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueXX 5000769 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be79011d1f5bfae5c1f476ee6f05eb1d2caeb 269958b194521197b312fcced4867f3c8fbd030bd715d8ffda1dcd454b174e7a 1a97b59fe770e67702519d9d9b23d61ac08424d555242a8ca08c49aef300945d 99b999a79ce74804ae6bfde623f4463371442f6523a5f6ce19c839a708c02513 2e22c696c8ccade45680e5197189d0f98e7f0d5f955e353970b392cf530a68cc 56b0035ddfbf206c3074beeb0739dcbca272a6e629fb7aea2c5ba7bae50c7b4c a595df78200c352997ec3ee564df229fbb5473f5e8ccb1cc0153e9a7e299a8ea 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 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 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 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 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36f7e7b2182d5956a3e931bd9dccdb8070719b39e805c66fce1dcc4592ca273a f4b4fe0e997ccbf05f8887993ed1066e688216962937dbeea651b702 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 1 /Delta put dup 3 /Lambda put dup 8 /Phi put dup 10 /Omega put dup 22 /macron put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 52 /four put dup 56 /eight put dup 58 /colon put dup 61 /equal put dup 65 /A put dup 82 /R put dup 94 /circumflex put dup 97 /a put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 114 /r put dup 116 /t put dup 118 /v put dup 126 /tilde put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueXX 5000791 def currentdict end currentfile eexec 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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 10 /circlemultiply put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 22 /precedesequal put dup 24 /similar put dup 26 /propersubset put dup 27 /propersuperset put dup 28 /lessmuch put dup 33 /arrowright put dup 34 /arrowup put dup 40 /arrowdblleft put dup 41 /arrowdblright put dup 44 /arrowdblboth put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 59 /emptyset put dup 70 /F put dup 71 /G put dup 72 /H put dup 76 /L put dup 77 /M put dup 83 /S put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueXX 5000820 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f67052788e6703bb5ee49533870bca1f113ad8 3750d597b842d8d96c423ba1273ddd32f3a54a912a443fcd44f7c3a6fe3956b0 aa1e784aaec6fce08dae0c76da9d0a3eba57b98a6233d9e9f0c3f00fcc6b2c6a 9ba23af389e6dfff4efec3de05d6276c6be417703ce508377f25960ef4ed83b4 9b01b873f3a639ce00f356229b6477a081933fef3bb80e2b9dffa7f75567b1fa 4d739b772f8d674e567534c6c5bbf1cf615372be20b18472f7aa58be8c216dbd df81cc0a86b6d8318ca68fe22c8af13b54d7576fe4ca5a7af9005ea5cc4edb79 c0ab668e4fec4b7f5a9eb5f0e4c088cd818ecc4feb4b40ec8bd2981bf2336074 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 33 /omega put dup 58 /period put dup 59 /comma put dup 60 /less put dup 62 /greater put dup 64 /partialdiff 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 75 /K put dup 76 /L put dup 78 /N put dup 80 /P put dup 81 /Q put dup 82 /R put dup 84 /T put dup 85 /U put dup 86 /V put dup 90 /Z put dup 97 /a 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9feb428e1369feb1ab16e6d375485b80a68670c3d06df48e4119b2fba9154210 e2c24ccd2e310fad39de1547f76669fe252b5976c024a891ae01eb9dc805409b 87370c78ec91eee920275227bf7f9a4a1ef18bd69620e8166ef5509d8d691b82 ab504c9d0115b9ca02666b5ce16e134469e55b8bc4ad528cc4315fe5e1e49609 bbf47cce2105ede7ab0f44cd5a0d8c01e0053a1a8904b24888ec56b7e0031602 27f2131c569aa50f16cb8d19ad9a3665491a6b80d24f790381bdc03fae7cae65 4fb090a94063ab462fffbe78a2c1fd5cbdbad2229b0fbbf0fbc836dfd96725fc 81d3bb4b6cb7d553db96461ff28798f539c7741d9a809581da860339761a3bc0 6c89ca8103924aa120097102adbe12fa477ee2008edfc28d64f4416b037ea437 b3967419e85d936340ecc77224c0e356564899a311da6d3aa62c40151d3bdb69 3058fd49808b2a5343dbec8ab4202d181df0a0b7f9490f68a1dfa7799874dae9 8008664dee1a4d5d612cd30484dc7fb0a9c17c4d3be5b2 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /asteriskmath put dup 26 /propersubset put dup 48 /prime put dup 51 /owner put dup 54 /negationslash put dup 59 /emptyset put dup 92 /intersection put dup 104 /angbracketleft put dup 105 /angbracketright put dup 110 /backslash put dup 121 /dagger put dup 122 /daggerdbl put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueXX 5000816 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b6a18979f258b8c6ed3 ccafbc21aca420c9c83eea371adc20e038b4d7b8ac303004b0aa205f04135140 76407216032fdd22e6219da8f16b28ca12524deb7bca073cc5eba65c102a5e85 fd48e6d062cd4283ee570a7774597e5bf0e3400b6be72db0115f3cb12db70ce0 83722870cddfadee715f10f1fcaf20e06f3c54afe5ca238539bfe2b596116e83 f5371ff18fa5003d8543226cfd4025f9940365b392a858d27f078d3abcffe4a1 54e78c7692d1a32bf935967c64f01b24788ff8325d61145e2d4a489fd986fb77 38e6b254522c77ca2797a504a9ce4676a77ebacb026eca94dde5922c936f8e90 c43e285195b1e87c21cc7609dfc41c3c5fd0853f8388e76e0cb774a5085e1f77 293f52d9322ee262d1d60b17eb07f50a5167b7b7c97efc76180b8f50a8281171 4df7cf9c12d47f8dede69d4aa142e4e47f478ce521614fe6e66b325b33f3c426 f514543b06c9e6960860b23a735e205e1af752398ccf0bc9a9fa94de00170c3c acc90627b2626ae90f3e099f5c3d9491f2e5a58ff710a7dfe46af39cb1caaf77 44a6e965a8a09c6b0c6a267f265303e264ae422f395691fb01e7fbd41d914687 70375d5eba3d17c8033305fc521b19fc670ae8eac591a8db1789e3a7cff5cfd9 d56e4c0dddd5b9c5db9aba6f3861d7acb48bca7df5dcf6c40da46f8d769c2bc8 d5c917ae2d330aab648b8ff6de5a75de87c0376efe5da021ffab7bcc30bc254b fffbaa32f5d6d6220e8b33c186e704c1c0f5f9e335d2e1e52a1374f89da1d0b0 f027047c6e42c1847df8b0df6f274046b28909f1d633a80a0f891db6864f7f04 819865a4af6eac0b1a6b84c1712415dc7554917cb91fdded57631e969144767e 6842a10b50722bf440ddb6c6e193a33b4b265ffeb3553d6483844ef808946db3 9d9221cfb31b30e6b58d0772f9e905a3d8c2381ba15f778b5c6bdda074a6ec39 af3fe47ee6e429cb8345368129cb2aedd40cd0c5d53e26c49f4ea04c184c293c da55584bc7265178c6ca6384159672449a5fd7c338ea43813c7489f7c0dcf546 892dc6b9dd11acea6a6995fea41b56577a19a14b79221a14a02ec6a6aea11cbd 58daf9583fbd6d5848fb13bd032c1b7ce5d057e97183dcb44a31e8735fa144fa e18a4358148bd896a43005c340808336ff0c54a811c0826c432e333f5f333d1b b5d5b8aca80f056533fb08e7132cb38c70a6ced33f5f801bf0df2584f994b550 b7b9c69363f3ba3e9dda6e504799648fafdbc50c9f4c96b212e1ec03d8713d17 085d3c6353812ec853ace4a7d12fea970b93ff5f4b7f405b2465fd754e696f69 d15863999495d717c1fbc09e1a5ebf7ea3cd0484d851391c4ec2299d71d1700f fee59ccda4588557035be6bcc864296cfb4653264bdfe3bdc8e1a17b9512ce34 f0021084af190c8a2b39de5afaabb0663ae3f613ca3d54a44cb7f7f8385f038d 1beff42694bdd81e01eb8cae41d05aedd7891cf50821f736676cfe9f81174cf0 6c6b1609fbb2fdc13fdd87a5da74ee5b6d171e237172e94ba0640cea7a2fbe24 31b023731da11585104ca11ec957976fbf96d56dbc74a5d134afb0764f9f3fe8 400d15d225e64011de60e53cdf8f5b9a6f8a518cdccbad6aae90e635743bd6b7 e3d70e801618159468be1f9ac2804e05156a2495d7423854ade698c85afdc202 2e58f8b4744ac56d79fcfca2753804343acc6d9b5029c35b664bab39b9e32c9b ab55633b5384b7f469fb498504e61aca2263a2c0f33f9b4aec72eb9675c996de f1497ca52c7664fe9785a80ef3c5ef56d7293fc19b85765a765910e97de0cbfb 4258bde5aa2fa838480b4a344a863b7f54bd4c790b557fbb136b01d865f0de2a 7f3ac386ee9f526006d28c36c3e37f8030a62f0c6b48f459be6a09b8260dd0ef 0a43be7087027db62da5bb4e46a73b62bf6acf76448478e1cc7dbfb54a33183e 398abbd9b60537a49101b9aa484be904569c51143086d9ad35fa4c22fb6d8a2b 1964e24b33b500d6b8beb00b2b932f732c9e3bb7dfeac23c196e548744192a98 71f8a9fd4dfef03d95c3a557da370d7adc8701944a5ca9dd625e3f7aac5e30cc 6ba63ea17ebb886a56850a04d5c2e702c2979371a2abbf96b47888dc4df14619 5d82209ced837609cee009122a523d9670e86afad31249b825da82495c985f41 f3851c4d84943caa66f9c0ea8c6901375494cb3840764eb211748afb7800d424 3acb2fb44a3dc95a1af81628105ae9d92d336efd9d483455221c73f42f366197 3b4edcafe3fbe578549f41d38a37ba7d49d321ee07e92634530b736fb4e77671 a1d1cc60f3f1f0e278e0127dc78fbba139a26a0dbe6cbab186787673163acf3a 36baedbebc4e1242c60d61848dd390300ef3fc5acc2fa749570fbeabcf462ac7 13b5f0d1ce732bb07fbf8f24ed7394efdb0b2fcaf1341fcfa9fd5312cf8e0b49 545de59f99bc42f093ade47fbf0a586913be26be8f9d325f08c6aacd77536da1 f6df63037c27e05bce087401b97c1aaa9d841de37f25758d8ae38748810b1828 f5362a7c286c867ce3bb73291d319ccaa9523e65dfdfa67112e6b36192cce05d d4777cfcf89586799fc483d30bd2a407ef8d8223e2730868690287468768b9f1 25dfc3ee34feacde92860031888f63b422b41a45afebaefd4b36be4779c724f5 ede47cfbcfa2a45062e2ab4aae8b7e55dd7d837b22bd9b0a04ac5a1fe46ce197 b8a9d5467dcf35ec10f71552192b296b22a28676ab4a09768cf641d38aee040a f25657bd631ad8e1bd65ab9a1255cc6af2bbcde1019b53e738d618a8589b2c3f bf0a786beae952ac56247ea0949865345d242ee214659d4951e1e086767c6356 153b99442f611ed697721dc737683211e3234f136e4c2c4e998af27f8147d516 b1adaee39f10ef678168f004f850edfcd7c01fd437cd1f0cdaeaa7f949d7fe55 b1094a7c7591c0a4d6b0bc2d4eab889583f235e4671455e8ed5ac794c81a2a5b a225611f136f80ea358bbf80760476f348a0b735d73123f66023d9b48dad785f 336150e88e929f0fc1b1409c29b7a623ee0b8e1a0aa347434a644bce20242074 7431b3e935de377e15abc736b07451573abc3b90d10775b0e9ee42edf0249314 a7ad3ea382ba2188ed8c421899a553732f1efd11dd8b29c57086095909655442 82d94e2f8f1526fd95292a54a4a15d42e443a29e25fdd441d9793848ed056ad2 a032f3b80283d4a5ce459f3f4e9f7bae7fe6559e173e7a81ce6945e45223557f 645eb54be73034e6d4861f1e9b958898ff2c293d01dc0eef8dbad46da02eadb1 d8dc3682f87907a6dfd8a7f7b00aff78563882 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 12 /fi put dup 34 /quotedblright 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 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 64 /at put dup 65 /A put dup 66 /B put dup 69 /E put dup 70 /F put dup 71 /G put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-39 -250 1036 750}readonly def /UniqueXX 5000792 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b448685bdd20eb05d5e7c2126132e33a59a7170609 dcf4871a5d023c9ef57d3362d9f2d7a440bb69bf653364105f16f4d0f03582f9 aced3d05cc76489b16e3fa8a446094d30038b06ecceda269f2eab9d19a99c7f9 39f9548f206c5a457a19270b2b82c43b091dfc5573468eaa3e7a4a32f8042891 d85e4b180fcbcb3091d2800e54c87d84ce9cad6869b5aabbbe47f40c68799893 d22b765295e1e69e33aa048b7ed98ba480ceca91f3ebf8ef85fe9a3976909626 b95ac5940d53f9b02215d84a44837ba25ed15cce0d504f1d335065594f3bc824 5405407591cccb11cfd4645da60d960c0b93f187b0cf7b105543c0b70f89af5d 264b6c026e3ae646acf145950202ec73282111e3e601cf2bcde22ce3edf6db23 516481420f26552ff4472d749811f27768150450d0d0ebe3c79f999e99b5c0f2 2eaebb12d97782b1bd91b2a1f62a76412548ad53c0dd411d4a08c0f071c2c218 63d9adb75a4621803ecb84c2bb235b620b658984b2d8e0c4637e2811bc8f0d04 6c8935afc70141e1b2d9c23bdb251d304b3378faf8928bd09686aa0340fb0cb1 dc48c996ef91530ff078666fad227a3589f50b605267212d3a65ebc1019a8eec 9a0739a00279471a01e1505c17658c10030fae32f274fdc8c8774b0d5406c384 a1d17068dcb0c2575c562fdc5a2176609bad9a2c255e426a4325a4fc3053c7b3 2f35f7bbc9aa50135f30223360fa2fbf019294b3af98224c5c05d6038db08bc1 4103010094dc215ef9ccfb96b237851332c8ed81a9184cced4edbda26b381f0c 73a2bc4a92533b0e667e82ce49ddd8ac9e34d9d40fd4b5b9910eab3b1234bbf3 af6608d66eaa424c158eddb75fbe39462708da8f5c18d92fbdb2f534598a844a e076f96a741f8002af3b38d7e4bfa363f5f5d1193014f71cbc45f23794bfc025 aa27bc25b5e771671713e5dacf67fbb40f32a7fa51f4c34b4228b7df03749f96 22a890409263e6eec7fb172a2d92ac1a51777954f5765a672e574c4f86fd3350 3b58e737c3405c8007da023cf73125dafc276a1921043afb2b36fe8d2aa9ed18 e680c433a340ee107d2ebdf0188377f336874342c61f2ff06b5eccef31bf5ff8 6aa08b5182ab9a4d3d9daab961ed902001d6016295526e499fedb5bb0e4693d0 e72680a45cefe422b1c74f7b691ae020241e8bb5713603a8eff1175da7443225 c3253a897659bbb2b0888949ba1534f92c57b0359799b1b44a65084f13ff81cd a8b71659f8d115539a584a17a1aebb4209da3af3306ac6e9adec840a14a6260c 79bb10e85cd07a8f95c85306e2e36732791ed47c2a9d2b20362ce2c055c1ee37 fd98d7d31ae18ac09450d4ffa94d784ffb750fe0d1fc95abfdc07189c6b44110 dd7dfc04222dfb84383a4e46c2ead20157766255080664eb2d1b85f77bd09319 5d2698e3a99b7253f138faff0abc9dba139db769eb4fb926ddcc7cbbc1bf73d8 6f6edf77366cda67c3f124965a402a772fbb3424ed7ec11a0de29cd69028ed61 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29e62720339b7fb9588eb1f1e3018ee960f273a49877264af091c2f42b90d635 291f6d793e8ae694247d25c31e287b9d662304a61c15f45aeaad7f705347794d c899e92b003ace8ff21a63afcd8867bd261836d02c6edb4bc4d7ab71710994f2 4471f747455996e1e80574764d52d026ed803129145fbcd9268fc39538cd5614 8e04ba6a660a3f93aeff05bb81ab79cb0632fbcf32b38b2823e45e4f310931f1 73b94301ead044ef678a9f13ed467ecc031929d42f90d8d53f9d471c74e79e13 b4cde40b2372ee2285f7db7f5c47c9dac79b28ff9d47aad588cda5c0c2771485 3f2ed2fe4d0124e32323c7b940a42ce02b1c185efbc745271a3db80345ca3a6a c11c7a9bb640ffd51fe5b05cf75940d14c4582ed9db49690bb42976145d29aef b9b8 0000000000000000000000000000000000000000000000000000000000000000 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 58 /colon put dup 65 /A put dup 97 /a put dup 98 /b put dup 99 /c put dup 114 /r put dup 115 /s put dup 116 /t put readonly def /FontBBox{-58 -250 1195 750}readonly def /UniqueXX 5000767 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be79011d1f5bfae5c1f476ee6f05eb1d2caeb 269958b194521197b312fcced4867f3c8fbd030bd715d8ffda1dcd454b174e7a 1a97b59fe770e67702519d9d9b23d61ac08424d55fd4dda249cff0b56b9f3aff e9d0de215c02a52a6bc77155ff6b8cba5cd6646ba331254ac58ace650a967d3b 272331b87b6df06d5ab9d80fabe9f9ceac10139b61244814dd9fc295ed42d1b5 cd11c2e4beab318b20f51ca2c2930657e5343ab7bcf0e8870a0d12de4ffacb6c b3fdbdda481c2fcb84408d3d902e9a32070b2af6cd9317a33a42a43857c114b3 f4c3005cef9401f1c2bad3e69150d7145b79f95c9cfaf7a335b277e6435ab374 f6a3e78e124ac1e4615511f743ad65c5d778403a840310ac4902985f107fd33c 0049623e4c496cd353d2ae2a4804110c6a420a38ece5d5b235c5e35886a0987d 284e6110527b653b1b2c68c636ef423536180acadb954843eff4d9a82568a713 b682c574e30f793f093aef755c650e0e7175c2e3138695212d9fea7bd4b5805c a90985b134c6bc15a19afae0bfb3066363ca3e2568c2ac4559a69fc32feb443b 61243b4508b7524d974636855c83c8149d489b68fe18bd497116a953431251df bd09fa1d2ef66c85b997ab2f6b7c6d9d096cb43931379ff9782fa00c5f3cbd06 04265d16891ffc4f66e4a0307765d384c32aa83a4e4f89ac79ef8e76eed08450 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7c8af995d6e612a769e062c6c4edc78895454b9d6bcce4d133d43949020cfa9f 52e23a3bf8ee81211de2f7cf8eaf99b375a448f606f8f91618c7e72aec2a92c1 099fd0cdc68f47c915679a25c402bf96aded0e4d267d92cb3895a71e07930d64 92cd86507b1aaaca28997036606c508d6b6a01da72492e629c6c29ff767786a3 a2537ce023287e869fca3d53b2d56fbeb677497dbe920059a179f769ca243b64 493945905de1f0c4bcb8f806a89f6f6aca7b910f293c62e95a1132338f7aca79 0f3dbf410cad86acf55f4981f4571889257d0e3b8cbc5c3c466f26e89e810fdd f34f35a21168d0d95ec0073658f414aeef08ba16f22a599a4ea9e903181e760a 4c32a9b31ec7e2a771821d8c3b93c7c28e66803462c66052778d364e88addf84 98a0bb93f64b46e315005fa89c1fbf894bcb74cd0e029206ee1aad624e06ac17 4077988559933bfa21b29ea1ac98f20e61d4dfca294edad1e34a496307914429 339b69470a3f8fa24067709850bfdc37bd7fd56f914a08451c0245275e709da3 f8be3d44eeece3285f48d6053c41954e33cc2b5c09e734590d1f89b5ead523fc 35c678968a88fb7b2c08dc98541cb4d541d5b396c69d079b1f190160e6c99c79 ba1643a47fd22ca57dae71202b831c6f752b6f1f6cfba8f1d64a41865456eb36 841c32a536daddd88a5e9f1e3bd359b7565c1311646095a09d7e3aa70de67621 719fa49207b75bf8aeb0cd6ae93224ce2e0c3feebe1bb3f41340ebc782cb07cd e34dde5e3180f58ff42dc6830c1f718f8d650d13f33a3dd6151a8673a5360798 9cd669482d244f646794ede03d7e7815ba3349643a61c28257b736cab39c6f8d d4897a456b88f2811a1e2c333823402b980035bd7e73d962338623cd982f6d61 c57ee8939977406fd2decd1db47f5e60bb4f3621e138647af703d5037aef83bd 9753f8acbed8c751c80d3204e718d0954e09268dff2345c7098ebd120cb477c6 718469560dd7ae19240956eb195ec3968ba01c4edf477b11972363a1720073f5 34473d38c15ff1deaa1e249567e80d84bac5941b3827f385ec116dd788f81c09 d23994338abbce90e70b77d4cbd6433bf01d0c493e3218c2e893deb39d1a4cf7 47b7d5b3f56cf3741995c6f7075067aef76d200ab70db5955b576888f1ad71bf 057b0b6128cfeaed6935972f26f4ff643ddfbd4439dcd5bb4d77bfbb1ea555ec f7871e1148791fbcbed72e0fc5d208c6aee24dab5516a7a63e8f47f1a9d88c07 da8c72e3f6dc84016bf73d879cd6b2ea2f40b2241f435b9e2068e8ac9ef83634 58887d71b29c892327a7a6ea707a33d7ff6ff1364d65daf7f18495d7dce87222 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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. 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 1 /Delta put dup 3 /Lambda put dup 8 /Phi put dup 10 /Omega 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 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 64 /at 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 88 /X put dup 89 /Y put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 96 /quoteleft 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 /UniqueXX 5000793 def currentdict end currentfile eexec 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 67 /C put dup 68 /D put dup 70 /F put dup 75 /K 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 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put 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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 2 /multiply put dup 3 /asteriskmath put dup 21 /greaterequal put dup 26 /propersubset put dup 33 /arrowright put dup 34 /arrowup 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 91 /union put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 110 /backslash put dup 121 /dagger put dup 122 /daggerdbl put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueXX 5000818 def currentdict end currentfile eexec 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9eecaee8fb296390e86a576d5d41c81f5a288ca5afbea6a172d9a42871f114d7 4bf2aa67114ff476925663cfe76a309eb9e433093897 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 44 /comma put dup 48 /zero put dup 49 /one put dup 50 /two put dup 54 /six put dup 65 /A put dup 67 /C put dup 70 /F put dup 75 /K put dup 76 /L put dup 78 /N put dup 82 /R put dup 97 /a put dup 98 /b put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 121 /y put dup 127 /dieresis put readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueXX 5000794 def currentdict end currentfile eexec 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(b)34 b(e)421 b(view)-34 b(ed)421 b(as)g(a)g(transformation)i(on)e(the) h(lev)-34 b(el)419 b(of)j(Hamilto-)1200 54940 y(nians.)681 b(Later)451 b(on,)464 b(man)-34 b(y)452 b(other)g(examples)f(of)h (\\non-Gibbsian")h(measures)f(app)34 b(eared)452 b(in)f(the)h(con)-34 b(text)1200 56445 y(of)457 b(join)-34 b(t)458 b(measures)e(of)h (disordered)g(spin)g(systems)f([17],)469 b(time)457 b(ev)-34 b(olution)457 b(of)g(Gibbs)g(measures)f([5],)469 b(and)1200 57950 y(dynamical)541 b(systems)h([22)o(],)575 b(pro)-34 b(viding)542 b(further)g(motiv)-67 b(ation)542 b(for)f(the)h (construction)g(of)g(a)f(generalized)1200 59456 y(Gibbs)405 b(formalism.)3082 61293 y(As)503 b(so)34 b(on)503 b(as)h(the)f(\014rst) h(examples)f(of)g(non-Gibbsian)i(measures)e(app)34 b(eared,)528 b(Dobrushin)504 b(prop)34 b(osed)1200 62799 y(a)528 b(program)g(of)h (\\Gibbsian)g(restoration)f(of)h(non-Gibbsian)h(\014elds",)559 b(arguing)528 b(that)i(the)e(phenomenon)1200 64304 y(of)550 b(non-Gibbsianness)i(is)d(caused)h(b)-34 b(y)550 b(\\exceptional")f (con\014gurations,)587 b(whic)-34 b(h)550 b(are)f(negligible)g(in)g (the)p 1200 65359 19680 45 v 2563 66076 a Fn(\003)3043 66499 y Fp(WIAS,)341 b(Mohrenstrasse)g(39,)h(10117)f(Berlin,)i(German) -28 b(y)-85 b(.)455 b(E-Mail:)i(kuelsk)-28 b(e@wias-b)28 b(erlin.de)2606 67326 y Fn(y)3043 67749 y Fp(Eurandom,)566 b(L.G.)522 b(1.48,)567 b(TU)521 b(Eindho)-28 b(v)g(en,)566 b(P)-28 b(ostbus)520 b(513,)566 b(5600)522 b(MB)f(Eindho)-28 b(v)g(en,)566 b(The)522 b(Netherlands.)995 b(E-Mail:)1200 68967 y(len)-28 b(y@eurandom.tue.nl)2606 69793 y Fn(z)3043 70217 y Fp(F)-85 b(aculteit)379 b(Wiskunde)f(En)e(Informatica)h(TU)f (Eindho)-28 b(v)g(en,)386 b(P)-28 b(ostbus)376 b(513,)386 b(5600)377 b(MB)h(Eindho)-28 b(v)g(en,)385 b(The)377 b(Netherlands.)1200 71434 y(E-Mail:f.h.j.redig@tue.nl)25497 75417 y Fr(1)p eop %%Page: 2 2 2 1 bop 1200 1313 a Fr(measure-theoretic)350 b(sense.)520 b(He)350 b(th)-34 b(us)351 b(prop)34 b(osed)351 b(the)f(notion)h(of)f (a)g(\\w)-34 b(eakly)350 b(Gibbsian")h(measure,)360 b(where)1200 2819 y(the)574 b(existence)f(of)h(the)g(\014nite-v)-34 b(olume)574 b(Hamiltonian)h(is)e(not)i(required)d(uniformly)i(in)g(the) g(b)34 b(oundary)1200 4324 y(condition,)463 b(but)451 b(only)g(for)g(b)34 b(oundary)451 b(conditions)h(in)e(a)h(set)g(of)g (measure)f(one.)678 b(This)452 b(is)e(clearly)f(enough)1200 5830 y(to)583 b(de\014ne)h(the)g(Gibbsian)g(form)f(of)g(the)h (conditional)g(probabilities,)627 b(and)584 b(Gibbs)g(measures)f(via)f (the)1200 7335 y(DLR)602 b(equations.)1135 b(Since)602 b(Dobrushin)i(and)g(Shlosman)g(\(1997\),)652 b(man)-34 b(y)604 b(pap)34 b(ers)602 b(ha)-34 b(v)g(e)604 b(b)34 b(een)602 b(written)1200 8841 y(sho)-34 b(wing)416 b(the)g(\\w)-34 b(eak)415 b(Gibbs")h(prop)34 b(ert)-34 b(y)415 b(of)h(renormalized)e (lo)-34 b(w)415 b(temp)34 b(erature)415 b(phases,)j(see)d(e.g.,)i([2)o (,)e(23,)1200 10346 y(21,)389 b(25)q(],)j(and)e(of)g(join)-34 b(t)391 b(measures)f(of)g(disordered)f(spin)i(systems,)h([17,)d(18)q (].)533 b(P)-34 b(arallel)389 b(to)h(this,)i(F)-101 b(ern\023)-606 b(andez)1200 11852 y(and)428 b(P\014ster)f(\(1997\))h(dev)-34 b(elop)34 b(ed)427 b(ideas)h(ab)34 b(out)428 b(generalized)e(regularit) -34 b(y)426 b(prop)34 b(erties)427 b(of)h(the)f(conditional)1200 13357 y(probabilities.)521 b(They)351 b(pro)-34 b(v)g(ed)352 b(that)g(the)g(decimation)f(of)h(the)f(lo)-34 b(w)352 b(temp)34 b(erature)351 b(plus)h(phase)g(of)f(the)h(Ising)1200 14863 y(mo)34 b(del)588 b(is)f(consisten)-34 b(t)590 b(with)f(a)f(monotone)h(righ)-34 b(t-con)g(tin)g(uous)591 b(system)d(of)h(conditional)f(probabilities.)1200 16368 y(In)548 b(the)g(framew)-34 b(ork)549 b(of)f(in)-34 b(v)g(estigating) 549 b(regularit)-34 b(y)547 b(of)i(the)f(conditional)h(probabilities,) 583 b(the)549 b(notion)f(of)1200 17874 y(\\almost)409 b(Gibbs")g(w)-34 b(as)410 b(in)-34 b(tro)34 b(duced)409 b(in)g([23].)551 b(A)409 b(measure)f Fm(\026)h Fr(is)f(called)g(almost) h(Gibbs)g(if)g(its)f(conditional)1200 19379 y(probabilities)279 b(ha)-34 b(v)g(e)280 b(a)f(v)-34 b(ersion)279 b(whic)-34 b(h)281 b(is)e(con)-34 b(tin)g(uous)281 b(on)f(a)f(set)g(of)h Fm(\026)p Fr(-measure)g(one.)497 b(If)279 b(one)g(do)34 b(es)279 b(not)h(insist)1200 20884 y(on)398 b(\\absolute")g(con)-34 b(v)g(ergence)396 b(of)i(the)f(sums)h(of)g(p)34 b(oten)-34 b(tials)398 b(constituting)h(\014nite-v)-34 b(olume)397 b(Hamiltonians,)1200 22390 y(then)439 b(almost)g(Gibbs)g(implies)f(w) -34 b(eak)439 b(Gibbs,)447 b(but)440 b(the)f(con)-34 b(v)g(erse)438 b(is)h(not)g(true,)447 b(see)438 b([20,)g(23].)641 b(In)439 b([9)o(])g(it)f(is)1200 23895 y(pro)-34 b(v)g(ed)323 b(that)h(e.g.)511 b(the)323 b(decimation)g(of)g(the)g(plus)h(phase)f (of)g(the)g(lo)-34 b(w)323 b(temp)34 b(erature)323 b(Ising)g(mo)34 b(del)322 b(is)g(almost)1200 25401 y(Gibbs,)429 b(and)c(the)f (criterion)f(to)i(c)-34 b(haracterize)422 b(an)j(essen)-34 b(tial)424 b(p)34 b(oin)-34 b(t)425 b(of)f(discon)-34 b(tin)g(uit)g(y)426 b(of)e(the)g(conditional)1200 26906 y(probabilities)560 b(giv)-34 b(en)560 b(in)g([6)o(])g(strongly)g (suggests)h(that)g(man)-34 b(y)561 b(other)f(examples)f(of)h (renormalized)f(lo)-34 b(w)1200 28412 y(temp)34 b(erature)486 b(phases)h(are)f(almost)h(Gibbs.)786 b(The)486 b(in)-34 b(v)g(estigation)488 b(of)e(generalized)g(Gibbs)h(prop)34 b(erties)485 b(of)1200 29917 y(the)363 b(non-Gibbsian)h(measures)f (whic)-34 b(h)364 b(app)34 b(ears)362 b(e.g.)524 b(as)363 b(transformations)h(of)f(Gibbs)h(measures)e(is)h(called)1200 31423 y(the)404 b(\\\014rst)h(part)g(of)f(the)h(Dobrushin)g(program".) 3082 33260 y(The)419 b(\\second)h(part)g(of)f(the)h(Dobrushin)g (program")f(then)h(consists)g(in)f(building)h(a)f(thermo)34 b(dynamic)1200 34766 y(formalism)344 b(within)i(the)f(new)g(class)f(of) h(\\generalized)f(Gibbs)h(measures".)518 b(The)346 b(question)f (whether)g(in)f(the)1200 36271 y(con)-34 b(text)396 b(of)f(w)-34 b(eakly)395 b(Gibbsian)h(measures)e(there)h(is)g(a)g(reasonable)g (notion)h(of)f(\\ph)-34 b(ysical)396 b(equiv)-67 b(alence")393 b({)1200 37777 y(i.e.,)358 b(if)348 b(t)-34 b(w)g(o)349 b(systems)f(of)g(conditional)h(probabilities)f(share)g(a)g(Gibbs)h (measure,)358 b(then)349 b(they)f(are)g(equal)f({)h(is)1200 39282 y(already)300 b(raised)h(in)g([2)o(].)504 b(In)301 b(the)g(classical)f(Gibbs)h(formalism,)322 b(ph)-34 b(ysical)300 b(equiv)-67 b(alence)299 b(corresp)34 b(onds)301 b(to)h(zero)1200 40788 y(relativ)-34 b(e)450 b(en)-34 b(trop)g(y)452 b(densit)-34 b(y)-101 b(,)463 b(or)451 b(zero)f(\\information)i(distance".)679 b(Generally)450 b(sp)34 b(eaking,)462 b(one)452 b(w)-34 b(ould)452 b(lik)-34 b(e)1200 42293 y(to)408 b(obtain)h(a)f(relation)g (b)34 b(et)-34 b(w)g(een)408 b(v)-67 b(anishing)409 b(relativ)-34 b(e)406 b(en)-34 b(trop)g(y)409 b(densit)-34 b(y)409 b(and)g(conditional)f(probabilities.)1200 43799 y(F)-101 b(or)495 b(Gibbs)g(measures)f(with)i(a)f(translation)g(in)-34 b(v)-67 b(arian)-34 b(t)495 b(uniformly)g(absolutely)g(con)-34 b(v)g(ergen)g(t)495 b(p)34 b(oten)-34 b(tial,)1200 45304 y(a)498 b(translation)h(in)-34 b(v)-67 b(arian)-34 b(t)499 b(probabilit)-34 b(y)499 b(measure)f Fm(\026)g Fr(has)h(zero)e(relativ) -34 b(e)498 b(en)-34 b(trop)g(y)499 b(densit)-34 b(y)499 b Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))499 b(with)1200 46809 y(resp)34 b(ect)471 b(to)g(a)g(Gibbs)h(measure)f Fm(\027)548 b Fr(if)472 b(and)g(only)f(if)g Fm(\026)g Fr(is)g(Gibbs)h(with)g(the)g(same)f(p)34 b(oten)-34 b(tial.)741 b(Ph)-34 b(ysically)1200 48315 y(sp)34 b(eaking,)429 b(this)c(means)f(that)i(the)e(only)g(minimizers)g(of)g(the)h(free)f (energy)f(are)h(the)h(equilibrium)e(phases.)1200 49820 y(In)530 b(complete)g(generalit)-34 b(y)-101 b(,)561 b(i.e.)916 b(without)532 b(an)-34 b(y)530 b(lo)34 b(calit)-34 b(y)529 b(requiremen)-34 b(ts,)562 b Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))547 b(=)g(0)530 b(do)34 b(es)530 b(not)h(imply)1200 51326 y(that)483 b Fm(\026)f Fr(and)g Fm(\027)558 b Fr(ha)-34 b(v)g(e)482 b(an)-34 b(ything)483 b(in)f(common,)501 b(see)481 b(e.g.)771 b(the)482 b(example)e(in)i ([31])f(where)h(a)f(measure)h Fm(\027)558 b Fr(is)1200 52831 y(constructed)405 b(suc)-34 b(h)405 b(that)g(for)g(an)-34 b(y)404 b(translation)h(in)-34 b(v)-67 b(arian)-34 b(t)405 b(probabilit)-34 b(y)404 b(measure)g Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))338 b(=)f(0.)3082 54669 y(In)382 b(this)h(pap)34 b(er)383 b(w)-34 b(e)383 b(in)-34 b(v)g(estigate)383 b(the)g(relation)f(b)34 b(et)-34 b(w)g(een)384 b Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))338 b(=)e(0)383 b(and)g(the)g(prop)34 b(ert)-34 b(y)383 b(of)g(ha)-34 b(ving)383 b(a)1200 56174 y(common)398 b(system)f(of)g(conditional)h (probabilities)f(for)h(general)e(quasilo)34 b(cal)396 b(measures,)i(almost)g(Gibbsian)1200 57680 y(measures)h(and)g(w)-34 b(eakly)399 b(Gibbsian)g(measures.)537 b(W)-101 b(e)398 b(will)g(w)-34 b(ork)399 b(in)g(the)g(con)-34 b(text)400 b(of)f(lattice)f(spin)i(systems)1200 59185 y(with)436 b(a)g(single-site)f(spin)h(taking)f(a)h(\014nite)g(n)-34 b(um)g(b)34 b(er)436 b(of)g(v)-67 b(alues.)631 b(Let)435 b Fm(\015)503 b Fr(denote)436 b(a)f(translation)h(in)-34 b(v)-67 b(arian)-34 b(t)1200 60691 y(system)383 b(of)g(conditional)g (probabilities,)k(and)c Fl(G)23121 60873 y Fk(in)-26 b(v)24444 60691 y Fr(\()p Fm(\015)67 b Fr(\))383 b(the)h(set)e(of)h (all)g(translation)g(in)-34 b(v)-67 b(arian)-34 b(t)383 b(probabilit)-34 b(y)1200 62196 y(measures)420 b(ha)-34 b(ving)420 b Fm(\015)487 b Fr(as)420 b(a)g(v)-34 b(ersion)419 b(of)h(their)g(conditional)h(probabilities.)585 b(If)420 b Fm(\015)487 b Fr(is)419 b(con)-34 b(tin)g(uous)422 b(then,)j(for)1200 63702 y Fm(\027)599 b Fl(2)522 b(G)4449 63884 y Fk(in)-26 b(v)5772 63702 y Fr(\()p Fm(\015)67 b Fr(\),)543 b(w)-34 b(e)516 b(obtain)g Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))523 b(=)f(0)515 b(if)g(and)h(only)f (if)h Fm(\026)522 b Fl(2)g(G)31546 63884 y Fk(in)-26 b(v)32869 63702 y Fr(\()p Fm(\015)67 b Fr(\).)872 b(If)516 b Fm(\015)582 b Fr(is)515 b(con)-34 b(tin)g(uous)517 b Fm(\026)p Fr(-almost)1200 65207 y(ev)-34 b(erywhere,)593 b(then)558 b(w)-34 b(e)557 b(obtain)g(that)h Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))592 b(=)f(0)556 b(and)i Fm(\027)667 b Fl(2)591 b(G)32150 65389 y Fk(in)-26 b(v)33473 65207 y Fr(\()p Fm(\015)67 b Fr(\))558 b(implies)e Fm(\026)591 b Fl(2)g(G)43376 65389 y Fk(in)-26 b(v)44699 65207 y Fr(\()p Fm(\015)67 b Fr(\).)996 b(More)1200 66713 y(generally)-101 b(,)613 b(for)573 b Fm(\027)695 b Fl(2)617 b(G)12265 66895 y Fk(in)-26 b(v)13588 66713 y Fr(\()p Fm(\015)67 b Fr(\))574 b(and)f Fm(\026)618 b Fl(2)g(M)22554 66203 y Fk(+)22554 67060 y(1)23914 66713 y Fr(concen)-34 b(trating)573 b(on)g(a)g(set)g(of)g(\\go)34 b(o)g(d)573 b(con\014gurations",)1200 68218 y(w)-34 b(e)474 b(obtain)g(the)g(existence)e(of)i Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\),)491 b(an)474 b(explicit)e(expression)h(for)h(it)f(where)g Fm(\027)551 b Fr(en)-34 b(ters)473 b(only)g(through)1200 69724 y(its)557 b(conditional)h(probabilities)f(and)h(the)g(relation)f Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))593 b(=)f(0)557 b(implies)f Fm(\026)593 b Fl(2)f(G)40295 69906 y Fk(in)-26 b(v)41618 69724 y Fr(\()p Fm(\015)67 b Fr(\).)998 b(The)558 b(\\go)34 b(o)g(d)1200 71229 y (con\014gurations")449 b(here)e(are)h(de\014ned)g(suc)-34 b(h)449 b(that)g(a)e(telescoping)h(pro)34 b(cedure)447 b(-)h(inspired)g(b)-34 b(y)448 b(the)g(metho)34 b(d)25497 75417 y(2)p eop %%Page: 3 3 3 2 bop 1200 1313 a Fr(of)559 b(Sulliv)-67 b(an)559 b([30])g(-)g(con) -34 b(v)g(erges)559 b(almost)h(surely)-101 b(.)1003 b(These)559 b(results,)598 b(together)560 b(with)g(some)f(examples)f(of)1200 2819 y(non-Gibbsian)484 b(measures)f(to)g(whic)-34 b(h)483 b(they)g(apply)f(sho)-34 b(w)484 b(that)g(almost)f(Gibbsian)g(measures) f(exhibit)h(a)1200 4324 y(reasonable)535 b(thermo)34 b(dynamic)536 b(formalism.)933 b(The)536 b(fact)g(that)g(some)g(concen) -34 b(tration)536 b(prop)34 b(erties)536 b(of)f(the)1200 5830 y(measures)419 b(are)g(required)f(is)h(reminiscen)-34 b(t)419 b(of)h(the)f(situation)i(in)e(un)-34 b(b)34 b(ounded)422 b(spin)d(systems,)k(an)d(analogy)1200 7335 y(already)403 b(p)34 b(oin)-34 b(ted)406 b(out)e(b)-34 b(y)405 b(Dobrushin,)g([28].) 3082 9132 y(The)501 b(con)-34 b(text)501 b(of)g(join)-34 b(t)501 b(measures)f(of)h(disordered)g(spin)g(systems)f(pro)-34 b(vides)501 b(a)f(go)34 b(o)g(d)501 b(source)f(of)g(ex-)1200 10637 y(amples)478 b(for)h(v)-67 b(alidit)-34 b(y)477 b(and)i(failure)f(of)h(the)f(v)-67 b(ariational)478 b(principle.)760 b(Here)478 b(b)-34 b(y)478 b(join)-34 b(t)480 b(measure)e(w)-34 b(e)479 b(mean)1200 12143 y(the)498 b(join)-34 b(t)499 b(distribution)g(b)34 b(oth)498 b(of)g(the)g(spins)h(and)f(the)g (disorder.)819 b(In)498 b(these)f(examples)h(\(esp)34 b(ecially)496 b(for)1200 13648 y(the)383 b(random)g(\014eld)g(Ising)f (mo)34 b(del\))382 b(there)h(is)f(a)g(precise)g(criterion)f(separating) i(the)g(almost)g(Gibbsian)g(case)1200 15153 y(from)539 b(the)g(w)-34 b(eakly)538 b(Gibbsian)h(case.)942 b(In)538 b(particular,)572 b(for)539 b(the)g(random)g(\014eld)g(Ising)f(mo)34 b(del,)572 b(the)539 b(join)-34 b(t)1200 16659 y(measure)420 b(is)g(alw)-34 b(a)g(ys)421 b(w)-34 b(eakly)420 b(Gibbs,)k(and)d(at)g (lo)-34 b(w)420 b(temp)34 b(eratures)421 b(w)-34 b(e)420 b(pro)-34 b(v)g(e)421 b(here)f(that)h(it)f(ev)-34 b(en)420 b(admits)1200 18164 y(a)395 b(translation)h(in)-34 b(v)-67 b(arian)-34 b(t)395 b(p)34 b(oten)-34 b(tial)396 b(whic)-34 b(h)395 b(deca)-34 b(ys)395 b(almost)h(surely)e(as)h(a)g(stretc)-34 b(hed)395 b(exp)34 b(onen)-34 b(tial)395 b(\(so)h(in)1200 19670 y(particular)476 b(con)-34 b(v)g(erges)476 b Fj(absolutely)c Fr(a.s.\).)755 b(If)476 b(there)g(is)g(no)g(phase)h(transition,)495 b(then)477 b(the)g(join)-34 b(t)477 b(measure)1200 21175 y(for)525 b(the)g(random)h(\014eld)e(Ising)h(mo)34 b(del)524 b(is)h(almost)g(Gibbs)h(\(but)g(not)f(Gibbs)h(in)f(dimension)g(t)-34 b(w)g(o)526 b(at)f(lo)-34 b(w)1200 22681 y(temp)34 b(erature\).)653 b(In)442 b(the)g(almost)h(Gibbsian)g(regime)e(w)-34 b(e)442 b(obtain)h(the)g(v)-67 b(ariational)441 b(principle)h(whereas)g(in)1200 24186 y(the)340 b(w)-34 b(eakly)339 b(but)i(not-almost)g(Gibbsian)f (regime)f(w)-34 b(e)340 b(sho)-34 b(w)341 b(the)f(in)-34 b(v)-67 b(alidit)-34 b(y)339 b(of)h(the)g(v)-67 b(ariational)339 b(principle.)1200 25692 y(More)350 b(precisely)-101 b(,)359 b(in)350 b(that)i(case)d(the)i(join)-34 b(t)351 b(measure)f(for)h(the)f (min)-34 b(us)351 b(phase)g(\()p Fm(K)37585 25252 y Ft(\000)38373 25692 y Fr(\))g(is)f(not)h(consisten)-34 b(t)351 b(with)1200 27197 y(the)509 b(\(w)-34 b(eakly)509 b(Gibbsian\))h(system)f(of)g (conditional)g(probabilities)g(of)g(the)g(plus)g(phase)g(\()p Fm(K)44124 26757 y Fk(+)44912 27197 y Fr(\),)535 b(but)510 b(one)1200 28703 y(easily)498 b(obtains)i(that)g(the)f(relativ)-34 b(e)498 b(en)-34 b(trop)g(y)500 b(densities)f Fm(h)p Fr(\()p Fm(K)30108 28263 y Ft(\000)30896 28703 y Fl(j)p Fm(K)32349 28263 y Fk(+)33136 28703 y Fr(\))d(=)e Fm(h)p Fr(\()p Fm(K)37825 28263 y Fk(+)38613 28703 y Fl(j)p Fm(K)40066 28263 y Ft(\000)40853 28703 y Fr(\))h(=)g(0.)823 b(Ph)-34 b(ysically)1200 30208 y(sp)34 b(eaking,)440 b(this)433 b(means)g(that)h(w)-34 b(e)434 b(are)e(in)h(the)g (pathological)g(situation)h(where)f(a)g(minimizer)f(of)h(the)g(free) 1200 31714 y(energy)439 b(is)g(not)i(a)e(\\phase")h(\(in)g(the)h(DLR)e (sense\).)645 b(A)-34 b(t)440 b(the)g(same)g(time,)448 b(w)-34 b(e)440 b(also)f(treat)h(the)g(join)-34 b(t)441 b(mea-)1200 33219 y(sures)377 b(in)f(a)h(v)-34 b(ery)376 b(broad)i(sense,)j(i.e.,)g(for)c(p)34 b(ossibly)377 b(non-i.i.d.)529 b(disorder,)382 b(w)-34 b(e)377 b(pro)-34 b(v)g(e)377 b(existence)f(of)h(relativ)-34 b(e)1200 34725 y(en)g(trop)g(y)342 b(densit)-34 b(y)-101 b(,)354 b(giv)-34 b(e)341 b(an)g(explicit)g (represen)-34 b(tation)342 b(in)f(terms)g(of)g(the)h(de\014ning)g(p)34 b(oten)-34 b(tials,)354 b(and)342 b(discuss)1200 36230 y(implications)404 b(of)h(our)f(results)g(for)g(the)h(Morita)f(approac) -34 b(h)405 b([26].)3082 38026 y(Our)288 b(pap)34 b(er)289 b(is)g(organized)f(as)h(follo)-34 b(ws:)481 b(in)289 b(Section)g(2)g(w)-34 b(e)289 b(in)-34 b(tro)34 b(duce)289 b(basic)g(de\014nitions)h(and)g(notations,)1200 39532 y(discuss)272 b(the)f(di\013eren)-34 b(t)272 b(generalized)e(Gibbs)i (measures)f(and)i(de\014ne)e(the)h(v)-67 b(ariational)271 b(principle.)493 b(In)272 b(Section)1200 41037 y(3)508 b(w)-34 b(e)508 b(pro)-34 b(v)g(e)508 b(the)g(v)-67 b(ariational)507 b(principle)g(for)h(some)g(class)g(of)g(almost)g(Gibbsian)h(measures,) 533 b(using)508 b(the)1200 42543 y(tec)-34 b(hnique)328 b(of)f(\\relativ)-34 b(e)326 b(energies")h(of)g(Sulliv)-67 b(an)327 b([30].)512 b(In)328 b(Section)f(4)g(w)-34 b(e)328 b(pro)-34 b(v)g(e)327 b(the)h(v)-67 b(ariational)326 b(principle)1200 44048 y(for)562 b(measures)g(with)h(translation)g(in) -34 b(v)-67 b(arian)-34 b(t)563 b(con)-34 b(tin)g(uous)565 b(system)d(of)g(conditional)h(probabilities.)1013 b(In)1200 45554 y(Section)501 b(5)f(w)-34 b(e)501 b(giv)-34 b(e)500 b(the)h(example)f(of)h(the)f(GriSing)h(random)g(\014eld)g(and)g(the)g (decimation)g(of)f(the)h(lo)-34 b(w-)1200 47059 y(temp)34 b(erature)530 b(plus)h(phase)g(of)f(the)h(Ising)f(mo)34 b(del.)916 b(In)530 b(Section)h(6)f(w)-34 b(e)530 b(discuss)h(the)g (examples)e(of)i(join)-34 b(t)1200 48565 y(measures)404 b(of)g(disordered)h(spin)f(systems.)1200 52636 y Fo(2)1793 b(Preliminaries)1200 55677 y Fi(2.1)1495 b(Con\014guration)501 b(space)1200 58256 y Fr(The)376 b(con\014guration)h(space)e(is)g(an)h (in\014nite)g(pro)34 b(duct)376 b(space)f(\012)337 b(=)g Fm(E)32299 57816 y Fh(Z)32889 57503 y Fg(d)33802 58256 y Fr(with)376 b Fm(E)445 b Fr(a)375 b(\014nite)h(set.)529 b(Its)376 b(Borel-)p Fm(\033)43 b Fr(-)1200 59761 y(\014eld)348 b(is)f(denoted)i(b)-34 b(y)347 b Fl(F)120 b Fr(.)521 b(W)-101 b(e)347 b(denote)h(b)-34 b(y)348 b Fl(S)427 b Fr(=)22878 58779 y Ff(\010)23585 59761 y Fr(\003)336 b Fl(\032)h Fe(Z)26851 59321 y Fd(d)27390 59761 y Fm(;)202 b Fl(j)p Fr(\003)p Fl(j)336 b Fm(<)g Fl(1)32272 58779 y Ff(\011)33327 59761 y Fr(the)348 b(set)g(of)g(the)g(\014nite)g (subsets)g(of)g Fe(Z)49860 59321 y Fd(d)1200 61267 y Fr(and)407 b(for)g(an)-34 b(y)407 b(\003)340 b Fl(2)h(S)91 b Fr(,)407 b(\012)12484 61455 y Fk(\003)13533 61267 y Fr(=)340 b Fm(E)15781 60827 y Fk(\003)16489 61267 y Fr(.)545 b Fl(F)18242 61455 y Fk(\003)19357 61267 y Fr(denotes)407 b(the)g Fm(\033)43 b Fr(-algebra)406 b(generated)h(b)-34 b(y)407 b Fl(f)p Fm(\033)43 b Fr(\()p Fm(x)p Fr(\))341 b(:)f Fm(x)h Fl(2)g Fr(\003)p Fl(g)p Fr(.)545 b(F)-101 b(or)406 b(all)1200 62772 y Fm(\033)-24 b(;)202 b(!)393 b Fl(2)350 b Fr(\012,)414 b(w)-34 b(e)413 b(denote)f Fm(\033)12738 62960 y Fk(\003)13446 62772 y Fm(;)202 b(!)14739 62960 y Fk(\003)15859 62772 y Fr(the)412 b(pro)67 b(jections)413 b(on)g(\012)26768 62960 y Fk(\003)27888 62772 y Fr(and)g(also)f(write)f Fm(\033)36493 62960 y Fk(\003)37201 62772 y Fm(!)37955 62960 y Fk(\003)38608 62708 y Fg(c)39502 62772 y Fr(for)h(the)g(con\014guration)1200 64278 y(whic)-34 b(h)515 b(agrees)e(with)i Fm(\033)556 b Fr(in)514 b(\003)g(and)g(with)h Fm(!)558 b Fr(in)514 b(\003)24549 63838 y Fd(c)25011 64278 y Fr(.)868 b(The)514 b(set)g(of)g(probabilit)-34 b(y)515 b(measures)f(on)g(\(\012)p Fm(;)202 b Fl(F)120 b Fr(\))515 b(is)1200 65783 y(denoted)487 b(b)-34 b(y)486 b Fl(M)9082 65273 y Fk(+)9082 66130 y(1)9869 65783 y Fr(.)783 b(A)485 b(function)i Fm(f)617 b Fr(is)485 b(said)h(to)g(b)34 b(e)485 b Fj(lo)-62 b(c)g(al)484 b Fr(if)i(there)g(exists)f(\001)473 b Fl(2)f(S)577 b Fr(suc)-34 b(h)486 b(that)h Fm(f)617 b Fr(is)485 b Fl(F)49156 65971 y Fk(\001)49996 65783 y Fr(-)1200 67289 y(measurable.)535 b(W)-101 b(e)393 b(denote)g(b)-34 b(y)394 b Fl(L)f Fr(the)h(set)g(of)f (all)g(lo)34 b(cal)392 b(functions.)537 b(The)393 b(uniform)h(closure)f (of)h Fl(L)f Fr(is)g Fm(C)87 b Fr(\(\012\),)1200 68794 y(the)404 b(set)h(of)f(con)-34 b(tin)g(uous)407 b(functions)e(on)g (\012.)3082 70590 y(On)481 b(\012,)500 b(translations)482 b Fl(f)p Fm(\034)14720 70772 y Fd(x)15769 70590 y 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b(eci\014cation)500 b(and)f(quasilo)42 b(calit)-42 b(y)1200 12335 y Fs(De\014nition)466 b(2.1)606 b Fj(A)466 b Fr(sp)34 b(eci\014cation)467 b Fj(on)f Fr(\(\012)p Fm(;)202 b Fl(F)120 b Fr(\))467 b Fj(is)f(a)f(family)g Fm(\015)d Fr(=)395 b Fl(f)p Fm(\015)33816 12523 y Fk(\003)34524 12335 y Fm(;)202 b Fr(\003)395 b Fl(2)g(S)91 b(g)466 b Fj(of)g(pr)-62 b(ob)g(ability)462 b(kernels)1200 13841 y(fr)-62 b(om)433 b Fr(\012)4940 14029 y Fk(\003)5593 13776 y Fg(c)6509 13841 y Fj(to)g Fl(F)554 b Fj(that)432 b(ar)-62 b(e)2633 16328 y(1.)605 b Fr(Prop)34 b(er)p Fj(:)557 b(F)-93 b(or)433 b(al)62 b(l)433 b Fm(B)398 b Fl(2)337 b(F)16116 16516 y Fk(\003)16769 16264 y Fg(c)17251 16328 y Fj(,)433 b Fm(\015)18684 16516 y Fk(\003)19391 16328 y Fr(\()p Fm(B)61 b Fl(j)p Fm(!)43 b Fr(\))339 b(=)d Fs(1)24762 16516 y Fd(B)25571 16328 y Fr(\()p Fm(!)43 b Fr(\))p Fj(.)2633 18714 y(2.)605 b Fr(Consisten)-34 b(t)p Fj(:)560 b(If)433 b Fr(\003)337 b Fl(\032)f Fr(\003)15301 18274 y Ft(0)16045 18714 y Fj(ar)-62 b(e)433 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b(the)348 b(consistency)g(condition)h(\(2\))g(is)f(required)f(to)i (hold)f(for)h Fj(every)1200 40551 y Fr(con\014guration)514 b Fm(!)561 b Fl(2)517 b Fr(\012,)540 b(and)513 b(not)g(only)g(for)f (almost)h(ev)-34 b(ery)511 b Fm(!)562 b Fl(2)517 b Fr(\012.)864 b(This)513 b(is)f(b)34 b(ecause)512 b(the)h(notion)g(of)1200 42057 y(sp)34 b(eci\014cation)510 b(is)f(de\014ned)i(without)h(an)-34 b(y)510 b(reference)f(to)h(a)g(particular)f(measure.)856 b(A)510 b(sp)34 b(eci\014cation)509 b Fm(\015)577 b Fr(is)1200 43562 y(translation)405 b(in)-34 b(v)-67 b(arian)-34 b(t)405 b(if)f(for)g(all)f Fm(A)337 b Fl(2)g(F)120 b Fr(,)404 b(\003)337 b Fl(2)f(S)91 b Fr(,)404 b Fm(!)381 b Fl(2)337 b Fr(\012:)18924 45939 y Fm(\015)19552 46127 y Fk(\003+)p Fd(x)21521 45939 y Fr(\()p Fm(A)p Fl(j)p Fm(!)43 b Fr(\))338 b(=)f Fm(\015)26752 46127 y Fk(\003)27459 45939 y Fr(\()p Fm(\034)28460 46121 y Fd(x)29046 45939 y Fm(A)p Fl(j)p Fm(\034)30822 46121 y Fd(x)31406 45939 y Fm(!)43 b Fr(\))1200 48315 y(In)398 b(this)h(pap)34 b(er)398 b(w)-34 b(e)398 b(will)g Fj(always)e Fr(restrict)i(to)g(the)h (case)e(of)i(non-n)-34 b(ull)399 b(sp)34 b(eci\014cations,)400 b(i.e.,)d(for)i(an)-34 b(y)398 b(\003)337 b Fl(2)f(S)91 b Fr(,)1200 49821 y(there)404 b(exist)g(0)336 b Fm(<)h(a)10033 50009 y Fk(\003)11077 49821 y Fm(<)g(b)12877 50009 y Fk(\003)13921 49821 y Fm(<)g Fr(1)404 b(suc)-34 b(h)405 b(that)16071 52198 y Fm(a)16712 52386 y Fk(\003)17756 52198 y Fm(<)337 b Fr(inf)19132 52925 y Fd(\033)-21 b(;\021)20712 52198 y Fm(\015)21340 52386 y Fk(\003)22048 52198 y Fr(\()p Fm(\033)43 b Fl(j)p Fm(\021)g Fr(\))337 b Fl(\024)g Fr(sup)26596 53161 y Fd(\033)-21 b(;\021)28352 52198 y Fm(\015)28980 52386 y Fk(\003)29687 52198 y Fr(\()p Fm(\033)43 b Fl(j)p Fm(\021)g Fr(\))338 b Fm(<)e(b)34484 52386 y Fk(\003)35192 52198 y Fm(:)1200 55730 y Fs(De\014nition)466 b(2.2)606 b Fj(A)371 b(pr)-62 b(ob)g(ability)367 b(me)-62 b(asur)g(e)368 b Fm(\026)j Fj(on)g Fr(\(\012)p Fm(;)202 b Fl(F)120 b Fr(\))372 b Fj(is)e(said)f(to)g(b)-62 b(e)461 b Fr(consisten)-34 b(t)372 b Fj(with)e(a)g(sp)-62 b(e)g(ci\014c)g(ation)1200 57235 y Fm(\015)516 b Fj(\(or)448 b(sp)-62 b(e)g(ci\014e)g(d)447 b(by)h Fm(\015)67 b Fj(\))449 b(if)f(the)g(latter)f(is)i(a)f(r)-62 b(e)g(alization)447 b(of)h(its)g(\014nite-volume)f(c)-62 b(onditional)446 b(pr)-62 b(ob)g(abilities,)1200 58741 y(that)432 b(is,)h(if)g(for)g(al)62 b(l)433 b Fm(A)336 b Fl(2)h(F)554 b Fj(and)434 b Fr(\003)336 b Fl(2)h(S)91 b Fj(,)433 b(and)g(for)g Fm(\026)p Fj(-a.e.)556 b Fm(!)43 b Fj(,)19060 62026 y Fm(\026)p Fr([)p Fm(A)p Fl(jF)22244 62214 y Fk(\003)22897 61962 y Fg(c)23379 62026 y Fr(]\()202 b Fm(!)246 b Fr(\))337 b(=)g Fm(\015)28105 62214 y Fk(\003)28812 62026 y Fr(\()p Fm(A)p Fl(j)202 b Fm(!)246 b Fr(\))p Fm(:)15369 b Fr(\(2.3\))1200 64788 y(Equiv)-67 b(alen)-34 b(tly)-101 b(,)403 b Fm(\026)h Fr(is)g(consisten)-34 b(t)405 b(with)g Fm(\015)471 b Fr(if)20454 66006 y Ff(Z)21667 67655 y Fr(\()p Fm(\015)22766 67843 y Fk(\003)23473 67655 y Fm(f)130 b Fr(\))p Fm(d\026)339 b Fr(=)27646 66006 y Ff(Z)29060 67655 y Fm(f)130 b(d\026)1200 70590 y Fr(for)394 b(all)f Fm(f)468 b Fl(2)336 b Fm(C)87 b Fr(\(\012\).)537 b(W)-101 b(e)393 b(denote)h(b)-34 b(y)394 b Fl(G)72 b Fr(\()p Fm(\015)67 b Fr(\))395 b(the)f(set)g(of)g(measures)g(consisten) -34 b(t)395 b(with)f Fm(\015)67 b Fr(.)535 b(F)-101 b(or)394 b(a)g(translation)1200 72096 y(in)-34 b(v)-67 b(arian)-34 b(t)404 b(sp)34 b(eci\014cation,)404 b Fl(G)14254 72278 y Fk(in)-26 b(v)15577 72096 y Fr(\()p Fm(\015)67 b Fr(\))405 b(is)f(the)g(set)h(of)f(translation)h(in)-34 b(v)-67 b(arian)-34 b(t)405 b(elemen)-34 b(ts)404 b(of)g Fl(G)72 b Fr(\()p Fm(\015)67 b Fr(\))405 b(.)25497 75417 y(4)p eop %%Page: 5 5 5 4 bop 1200 1313 a Fs(De\014nition)466 b(2.4)2039 b Fj(1.)606 b(A)f(sp)-62 b(e)g(ci\014c)g(ation)602 b Fm(\015)672 b Fj(is)704 b Fr(quasilo)34 b(cal)604 b Fj(if)g(for)h(e)-62 b(ach)603 b Fr(\003)647 b Fl(2)g(S)696 b Fj(and)605 b(e)-62 b(ach)603 b Fm(f)736 b Fj(lo)-62 b(c)g(al,)4230 2819 y Fm(\015)4858 3007 y Fk(\003)5566 2819 y Fm(f)467 b Fl(2)337 b Fm(C)87 b Fr(\(\012\))p Fj(.)2633 5278 y(2.)605 b(A)350 b(pr)-62 b(ob)g(ability)347 b(me)-62 b(asur)g(e)349 b Fm(\026)h Fj(is)449 b Fr(quasilo)34 b(cal)350 b Fj(if)f(it)h(is)g(c) -62 b(onsistent)348 b(with)h(some)h(quasilo)-62 b(c)g(al)346 b(sp)-62 b(e)g(ci\014c)g(ation.)1200 8815 y Fi(2.3)1495 b(P)-42 b(oten)g(tials)501 b(and)e(Gibbs)f(measures)1200 11415 y Fr(Examples)404 b(of)g(quasilo)34 b(cal)403 b(measures)h(are)g Fj(Gibbs)432 b(me)-62 b(asur)g(es)402 b Fr(de\014ned)j(via)e(p)34 b(oten)-34 b(tials.)1200 14433 y Fs(De\014nition)466 b(2.5)2039 b Fj(1.)606 b(A)459 b Fr(p)34 b(oten)-34 b(tial)460 b Fj(is)f(a)f(family)g Fr(\010)384 b(=)e Fl(f)p Fr(\010)29842 14621 y Fd(A)30986 14433 y Fr(:)h Fm(A)g Fl(2)g(S)91 b(g)459 b Fj(of)g(lo)-62 b(c)g(al)457 b(functions)h(such)f(that)4230 15938 y(for)433 b(al)62 b(l)433 b Fm(A)337 b Fl(2)g(S)91 b Fj(,)433 b Fr(\010)12797 16126 y Fd(A)13991 15938 y Fj(is)g Fl(F)16163 16126 y Fd(A)16923 15938 y Fj(-me)-62 b(asur)g(able.)2633 18398 y(2.)605 b(A)434 b(p)-62 b(otential)431 b(is)i(tr)-62 b(anslation)432 b(invariant)g(if)i(for)f(al)62 b(l)432 b Fm(A)337 b Fl(2)g(S)91 b Fj(,)433 b Fm(x)337 b Fl(2)f Fe(Z)35010 17958 y Fd(d)35984 18398 y Fj(and)433 b Fm(!)380 b Fl(2)337 b Fr(\012)p Fj(:)21943 20994 y Fr(\010)22818 21182 y Fd(A)p Fk(+)p Fd(x)24840 20994 y Fr(\()p Fm(!)43 b Fr(\))338 b(=)e(\010)29071 21182 y Fd(A)29832 20994 y Fr(\()p Fm(\034)30833 21176 y Fd(x)31418 20994 y Fm(!)43 b Fr(\))1200 24013 y Fs(De\014nition)466 b(2.6)606 b Fj(A)434 b(p)-62 b(otential)431 b(is)i(said)f(to)h(b)-62 b(e)2633 26720 y(1.)605 b Fr(Con)-34 b(v)g(ergen)g(t)406 b(at)434 b Fj(the)e(c)-62 b(on\014gur)g(ation)432 b Fm(!)478 b Fj(if)433 b(for)g(al)62 b(l)433 b Fr(\003)336 b Fl(2)h(S)525 b Fj(the)432 b(sum)24651 28279 y Ff(X)23932 30963 y Fd(A)p Ft(\\)p Fk(\003)p Ft(6)p Fk(=)p Ft(;)27321 29430 y Fr(\010)28196 29618 y Fd(A)28957 29430 y Fr(\()p Fm(!)43 b Fr(\))17212 b(\(2.7\))4230 33252 y Fj(is)433 b(c)-62 b(onver)g(gent.)2633 35712 y(2.)605 b Fr(Uniformly)404 b(con)-34 b(v)g(ergen)g(t)435 b Fj(if)e(c)-62 b(onver)g(genc)g(e)431 b(in)j(\(2.7\))e(is)h(uniform)g (in)h Fm(!)43 b Fj(.)2633 38172 y(3.)605 b Fr(Uniformly)404 b(absolutely)h(con)-34 b(v)g(ergen)g(t)434 b Fj(\(UA)-31 b(C\))433 b(if)h(for)f(al)62 b(l)432 b Fr(\003)337 b Fl(2)f(S)21381 39798 y Ff(X)20663 42482 y Fd(A)p Ft(\\)p Fk(\003)p Ft(6)p Fk(=)p Ft(;)24052 40949 y Fr(sup)24656 41912 y Fd(!)26079 40949 y Fl(j)p Fr(\010)27291 41137 y Fd(A)28051 40949 y Fr(\()p Fm(!)43 b Fr(\))p Fl(j)675 b Fm(<)e Fl(1)p Fm(:)1200 45296 y Fr(F)-101 b(or)404 b(a)g(general)g(p)34 b(oten)-34 b(tial)404 b(\010,)h(w)-34 b(e)404 b(de\014ne)h(the)f(measurable)g(set)h(of)f(its)g(p)34 b(oin)-34 b(ts)405 b(of)g(con)-34 b(v)g(ergence:)15741 47892 y(\012)16616 48080 y Fk(\010)17688 47892 y Fr(=)336 b Fl(f)p Fm(!)381 b Fl(2)337 b Fr(\012)g(:)740 b(\010)337 b(is)404 b(con)-34 b(v)g(ergen)g(t)405 b(at)g Fm(!)43 b Fl(g)p Fm(:)3082 50800 y Fr(In)297 b(order)g(to)h(de\014ne)g(Gibbs)h (measures,)318 b(w)-34 b(e)298 b(consider)f(a)h(UA)-34 b(C)298 b(p)34 b(oten)-34 b(tial)298 b(and)g(de\014ne)h(its)e Fj(\014nite-volume)1200 52305 y(Hamiltonian)403 b Fr(for)h(\003)336 b Fl(2)h(S)495 b Fr(and)405 b(b)34 b(oundary)405 b(condition)g Fm(!)380 b Fl(2)337 b Fr(\012)405 b(b)-34 b(y)17868 55016 y Fm(H)18974 54515 y Fk(\010)18876 55315 y(\003)19709 55016 y Fr(\()p Fm(\033)43 b Fl(j)p Fm(!)g Fr(\))338 b(=)24857 53864 y Ff(X)24139 56548 y Fd(A)p Ft(\\)p Fk(\003)p Ft(6)p Fk(=)p Ft(;)27527 55016 y Fr(\010)28402 55204 y Fd(A)29163 55016 y Fr(\()p Fm(\033)30327 55204 y Fk(\003)31035 55016 y Fm(!)31789 55204 y Fk(\003)32442 54951 y Fg(c)32924 55016 y Fr(\))p Fm(:)1200 59510 y Fs(De\014nition)466 b(2.8)606 b Fj(L)-62 b(et)434 b Fr(\010)g Fj(b)-62 b(e)433 b(UA)-31 b(C.)433 b(The)g Fr(Gibbs)405 b(sp)34 b(eci\014cation)434 b Fm(\015)32057 59070 y Fk(\010)33225 59510 y Fj(with)f(p)-62 b(otential)432 b Fr(\010)i Fj(is)f(de\014ne)-62 b(d)433 b(by)18540 62549 y Fm(\015)19235 62048 y Fk(\010)19168 62849 y(\003)19969 62549 y Fr(\()p Fm(\033)43 b Fl(j)p Fm(!)g Fr(\))339 b(=)25924 61729 y(1)p 24532 62270 3390 49 v 24532 63438 a Fm(Z)25446 63021 y Fk(\010)25359 63820 y(\003)26181 63438 y Fr(\()p Fm(!)43 b Fr(\))28055 62549 y Fm(e)28619 62048 y Ft(\000)p Fd(H)30192 61736 y Fc(\010)30124 62349 y(\003)30825 62048 y Fk(\()p Fd(\033)32 b Ft(j)p Fd(!)g Fk(\))1200 66067 y Fj(wher)-62 b(e)433 b(the)f(p)-62 b(artition)432 b(function)g Fm(Z)17292 65627 y Fk(\010)17205 66426 y(\003)18027 66067 y Fr(\()p Fm(!)43 b Fr(\))436 b Fj(is)d(the)f(normalizing)h(c)-62 b(onstant.)1200 69085 y Fr(A)344 b(measure)g Fm(\026)h Fr(is)f(a)g Fj(Gibbs)377 b(me)-62 b(asur)g(e)342 b Fr(if)i(there)g(exists)g(a)g(UA)-34 b(C)345 b(p)34 b(oten)-34 b(tial)345 b(\010)g(suc)-34 b(h)345 b(that)g Fm(\026)337 b Fl(2)g(G)72 b Fr(\()p Fm(\015)45225 68645 y Fk(\010)45960 69085 y Fr(\).)519 b(Gibbs)1200 70590 y(measures)441 b(are)f(quasilo)34 b(cal)440 b(and)i(con)-34 b(v)g(ersely)-101 b(,)449 b(an)-34 b(y)442 b(non-n)-34 b(ull)442 b(quasilo)34 b(cal)440 b(measure)h(can)g(b)34 b(e)441 b(written)h(in)f(a)1200 72096 y(Gibbsian)405 b(w)-34 b(a)g(y)405 b(\(see)f([14])f(and)i(more)f (details)g(in)g(Section)h(4\).)25497 75417 y(5)p eop %%Page: 6 6 6 5 bop 1200 1313 a Fi(2.4)1495 b(Generalized)499 b(Gibbs)g(measures) 1200 3869 y Fs(De\014nition)466 b(2.9)606 b Fj(A)348 b(me)-62 b(asur)g(e)346 b Fm(\027)425 b Fj(is)347 b Fr(w)-34 b(eakly)311 b(Gibbs)348 b Fj(if)g(ther)-62 b(e)346 b(exists)g(a)i(p)-62 b(otential)345 b Fr(\010)j Fj(such)e(that)h Fm(\027)77 b Fr(\(\012)46971 4057 y Fk(\010)47706 3869 y Fr(\))337 b(=)g(1)1200 5374 y Fj(and)18738 7511 y Fm(\027)279 b Fr([)o Fm(\033)20645 7699 y Fk(\003)21353 7511 y Fl(jF)22561 7699 y Fk(\003)23214 7446 y Fg(c)23695 7511 y Fr(])202 b(\()p Fm(!)43 b Fr(\))338 b(=)27724 6691 y Fm(e)28288 6251 y Ft(\000)p Fd(H)29861 5939 y Fc(\010)29793 6552 y(\003)30494 6251 y Fk(\()p Fd(\033)32 b Ft(j)p Fd(!)g Fk(\))p 27724 7232 5006 49 v 28532 8400 a Fm(Z)29446 7983 y Fk(\010)29359 8782 y(\003)30181 8400 y Fr(\()p Fm(!)43 b Fr(\))1200 10258 y Fj(for)433 b Fm(\027)77 b Fj(-almost)432 b(every)g Fm(!)43 b Fj(.)1200 12967 y Fs(Remark)464 b(2.10)606 b Fj(Some)551 b(authors)e(insist)g(on)j(the) d(almost)h(sur)-62 b(ely)549 b Fr(absolute)j Fj(c)-62 b(onver)g(genc)g(e)549 b(of)h(the)g(sums)1200 14473 y(de\014ning)596 b Fm(H)7053 14033 y Fk(\010)6955 14832 y(\003)7788 14473 y Fj(.)1045 b(However,)635 b(for)596 b(the)f(de\014nition)g(of)h(the)f (we)-62 b(akly)594 b(Gibbsian)g(sp)-62 b(e)g(ci\014c)g(ation)594 b(ther)-62 b(e)594 b(is)i(no)1200 15978 y(r)-62 b(e)g(ason)433 b(to)f(pr)-62 b(efer)433 b(absolute)d(c)-62 b(onver)g(genc)g(e.)1200 18687 y Fs(De\014nition)466 b(2.11)606 b Fj(L)-62 b(et)374 b Fm(\015)440 b Fj(b)-62 b(e)373 b(a)h(sp)-62 b(e)g(ci\014c)g(ation.) 534 b(A)374 b(c)-62 b(on\014gur)g(ation)372 b Fm(!)418 b Fj(is)373 b(said)f(to)h(b)-62 b(e)373 b(a)g(p)-62 b(oint)373 b(of)g(c)-62 b(ontinuity)1200 20193 y(for)433 b Fm(\015)501 b Fj(if)433 b(for)g(al)62 b(l)432 b Fr(\003)337 b Fl(2)g(S)91 b Fj(,)433 b Fm(f)467 b Fl(2)337 b(L)p Fj(,)434 b Fm(\015)17541 20381 y Fk(\003)18248 20193 y Fm(f)565 b Fj(is)433 b(c)-62 b(ontinuous)431 b(at)i Fm(!)43 b Fj(.)1200 22902 y Fr(F)-101 b(or)404 b(a)g(giv)-34 b(en)404 b Fm(\015)67 b Fr(,)404 b(\012)9861 23084 y Fd(\015)10857 22902 y Fr(denotes)h(its)g (measurable)f(set)g(of)g(p)34 b(oin)-34 b(ts)405 b(of)g(con)-34 b(tin)g(uit)g(y)-101 b(.)1200 25344 y Fs(De\014nition)466 b(2.12)606 b Fj(A)369 b(me)-62 b(asur)g(e)366 b Fm(\027)446 b Fj(is)367 b(c)-62 b(al)62 b(le)-62 b(d)367 b Fr(almost)333 b(Gibbs)369 b Fj(if)f(ther)-62 b(e)367 b(exists)g(a)h(sp)-62 b(e)g(ci\014c)g(ation)366 b Fm(\015)435 b Fj(such)366 b(that)1200 26850 y Fm(\027)414 b Fl(2)336 b(G)72 b Fr(\()p Fm(\015)67 b Fr(\))435 b Fj(and)e Fm(\027)77 b Fr(\(\012)10598 27032 y Fd(\015)11191 26850 y Fr(\))337 b(=)f(1)p Fj(.)1200 29292 y Fr(If)545 b Fm(\027)623 b Fr(is)545 b(almost)h(Gibbs,)581 b(then)546 b(there)g(exists)f(an)g(almost)h(surely)f(con)-34 b(v)g(ergen)g(t)546 b(p)34 b(oten)-34 b(tial)547 b(\010)e(suc)-34 b(h)547 b(that)1200 30798 y Fm(\027)j Fr(is)472 b(w)-34 b(eakly)473 b(Gibbsian)g(for)g(\010,)490 b(and)474 b(th)-34 b(us)474 b(almost)f(Gibbsianness)h(implies)f(w)-34 b(eak)473 b(Gibbsianness.)745 b(The)1200 32303 y(con)-34 b(v)g(erse)399 b(is)h(not)g(true:)537 b(a)399 b(measure)h(can)g(b)34 b(e)399 b(w)-34 b(eakly)399 b(Gibbs)i(and)f(for)g(the)g(asso)34 b(ciated)400 b(p)34 b(oten)-34 b(tial)400 b(\010,)h(\012)49174 32591 y Fd(\015)49711 32339 y Fc(\010)1200 33808 y Fr(is)439 b(of)h(measure)f(zero)f([20,)h(23].)644 b(If)439 b(a)h(measure)f(is)g (almost)h(Gibbs)g(and)g(translation)g(in)-34 b(v)-67 b(arian)-34 b(t,)448 b(then)440 b(the)1200 35314 y(corresp)34 b(onding)404 b(p)34 b(oten)-34 b(tial)405 b(can)f(b)34 b(e)404 b(c)-34 b(hosen)405 b(to)f(b)34 b(e)404 b(translation)h(in)-34 b(v)-67 b(arian)-34 b(t.)1200 38765 y Fi(2.5)1495 b(Relativ)-42 b(e)498 b(en)-42 b(trop)g(y)500 b(and)f(v)-83 b(ariational)501 b(principle)1200 41320 y Fr(F)-101 b(or)496 b Fm(\026;)202 b(\027)567 b Fl(2)490 b(M)8656 40810 y Fk(+)8656 41691 y(1)p Fd(;)p Fk(in)-26 b(v)10711 41320 y Fr(,)518 b(the)497 b Fj(\014nite-volume)516 b(r)-62 b(elative)516 b(entr)-62 b(opy)601 b Fr(at)497 b(v)-34 b(olume)496 b(\003)490 b Fl(2)g(S)587 b Fr(of)496 b Fm(\026)h Fr(relativ)-34 b(e)495 b(to)h Fm(\027)573 b Fr(is)1200 42826 y(de\014ned)405 b(as)12893 45879 y Fm(h)13591 46067 y Fk(\003)14299 45879 y Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))674 b(=)19274 43031 y Ff(8)19274 44121 y(>)19274 44485 y(<)19274 46667 y(>)19274 47031 y(:)21242 42948 y(Z)21915 45697 y Fk(\012)22985 43778 y Fm(d\026)24346 43966 y Fk(\003)p 22985 44319 2070 49 v 23051 45430 a Fm(d\027)24281 45618 y Fk(\003)25389 44598 y Fr(log)27289 43778 y Fm(d\026)28650 43966 y Fk(\003)p 27289 44319 V 27355 45430 a Fm(d\027)28585 45618 y Fk(\003)29693 44598 y Fm(d\027)1087 b Fr(if)674 b Fm(\026)34121 44786 y Fk(\003)35166 44598 y Fl(\034)336 b Fm(\027)37313 44786 y Fk(\003)20905 47764 y Fr(+)p Fl(1)9091 b Fr(otherwise)p Fm(:)47302 45879 y Fr(\(2.13\))1200 49692 y(The)487 b(notation)g Fm(\026)9434 49880 y Fk(\003)10629 49692 y Fr(refers)e(to)i(the)f (distribution)i(of)f Fm(!)26632 49880 y Fk(\003)27826 49692 y Fr(when)g Fm(!)530 b Fr(is)486 b(distributed)i(according)e(to)g Fm(\026)p Fr(.)786 b(By)1200 51197 y(Jensen's)404 b(inequalit)-34 b(y)-101 b(,)404 b Fm(h)12501 51385 y Fk(\003)13209 51197 y Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))337 b Fl(\025)g Fr(0.)538 b(The)405 b Fj(r)-62 b(elative)431 b(entr)-62 b(opy)510 b Fr(of)404 b Fm(\026)h Fr(relativ)-34 b(e)403 b(to)h Fm(\027)481 b Fr(is)404 b(the)h(limit)18103 54038 y Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))674 b(=)1058 b(lim)23777 54766 y Fd(n)p Ft(!1)27332 53218 y Fr(1)p 26564 53760 2142 49 v 26564 54870 a Fl(j)p Fr(\003)27743 55052 y Fd(n)28369 54870 y Fl(j)28838 54038 y Fm(h)29536 54226 y Fk(\003)30189 54337 y Fg(n)30811 54038 y Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))13806 b(\(2.14\))1200 57165 y(where)362 b(\003)5505 57347 y Fd(n)6467 57165 y Fr(=)337 b([)p Fm(n;)202 b(n)p Fr(])10416 56725 y Fd(d)11139 57165 y Fl(\\)185 b Fe(Z)12940 56725 y Fd(d)13841 57165 y Fr(is)362 b(a)g(sequence)f(of)h(cub)34 b(es)362 b(\(this)h(can)f(b)34 b(e)361 b(replaced)h(b)-34 b(y)362 b(a)g(V)-101 b(an)362 b(Ho)-34 b(v)g(e)362 b(sequence\).)1200 58670 y(In)450 b(what)h(follo)-34 b(ws,)461 b(if)450 b(w)-34 b(e)450 b(write)g(lim)18027 58966 y Fk(\003)p Ft(")p Fh(Z)19740 58714 y Fg(d)20480 58670 y Fm(f)130 b Fr(\(\003\))451 b(w)-34 b(e)450 b(mean)g(that)h(the)g(limit)e(is)g(tak)-34 b(en)451 b(along)f(a)f(V)-101 b(an)451 b(Ho)-34 b(v)g(e)1200 60294 y(sequence.)630 b(The)436 b(de\014ning)g(limit)e(\(2.14\))i(is)e (kno)-34 b(wn)437 b(to)e(exist)f(if)h Fm(\027)465 b Fl(2)389 b(M)34768 59784 y Fk(+)34768 60664 y(1)p Fd(;)p Fk(in)-26 b(v)37257 60294 y Fr(is)435 b(a)g(translation)g(in)-34 b(v)-67 b(arian)-34 b(t)1200 61992 y(Gibbs)503 b(measure)f(with)h(a)g Fj(tr)-62 b(anslation)522 b(invariant)501 b Fr(UA)-34 b(C)503 b(p)34 b(oten)-34 b(tial)503 b(and)g Fm(\026)e Fl(2)f(M)39929 61482 y Fk(+)39929 62362 y(1)p Fd(;)p Fk(in)-26 b(v)42486 61992 y Fr(arbitrary)-101 b(.)832 b(The)1200 63690 y(Kolmogoro)-34 b(v-Sinai)404 b(en)-34 b(trop)g(y)405 b Fm(h)p Fr(\()p Fm(\026)p Fr(\))h(is)e(de\014ned)h(for) f Fm(\026)337 b Fl(2)g(M)29142 63180 y Fk(+)29142 64060 y(1)p Fd(;)p Fk(in)-26 b(v)31196 63690 y Fr(:)14630 66770 y Fm(h)p Fr(\()p Fm(\026)p Fr(\))338 b(=)f Fl(\000)586 b Fr(lim)19763 67497 y Fd(n)p Ft(!1)23318 65950 y Fr(1)p 22550 66491 V 22550 67602 a Fl(j)p Fr(\003)23729 67784 y Fd(n)24355 67602 y Fl(j)25026 65619 y Ff(X)25046 68164 y Fd(\033)25584 68313 y Fc(\003)26136 68424 y Fg(n)26979 66770 y Fm(\026)p Fr(\()p Fm(\033)28873 66958 y Fk(\003)29526 67069 y Fg(n)30148 66770 y Fr(\))202 b(log)219 b Fm(\026)202 b Fr(\()q Fm(\033)34686 66958 y Fk(\003)35339 67069 y Fg(n)35960 66770 y Fr(\))g Fm(:)10332 b Fr(\(2.15\))3082 70590 y(W)-101 b(e)371 b(are)g(no)-34 b(w)373 b(ready)f(to)g(state)h (the)f(v)-67 b(ariational)371 b(principle)g(for)h(sp)34 b(eci\014cations)372 b(and)h(measures,)378 b(whic)-34 b(h)1200 72096 y(giv)g(es)404 b(a)g(relation)g(b)34 b(et)-34 b(w)g(een)405 b(zero)e(relativ)-34 b(e)403 b(en)-34 b(trop)g(y)405 b(and)g(equalit)-34 b(y)404 b(of)g(conditional)h(probabilities.)25497 75417 y(6)p eop %%Page: 7 7 7 6 bop 1200 1313 a Fs(De\014nition)466 b(2.16)606 b Fj(L)-62 b(et)586 b Fm(\015)652 b Fj(b)-62 b(e)585 b(a)g(sp)-62 b(e)g(ci\014c)g(ation,)620 b Fm(\027)689 b Fl(2)612 b(G)27936 1495 y Fk(in)-26 b(v)29260 1313 y Fr(\()p Fm(\015)67 b Fr(\))586 b Fj(and)f Fl(M)612 b(\032)f(M)39067 804 y Fk(+)39067 1684 y(1)p Fd(;)p Fk(in)-26 b(v)41122 1313 y Fj(.)1013 b(We)585 b(say)f(that)g(a)1200 2819 y Fr(v)-67 b(ariational)403 b(principle)h(holds)h(for)f(the)g(triple)g(\()p Fm(\015)67 b(;)202 b(\027)10 b(;)202 b Fl(M)p Fr(\))433 b Fj(if)1200 5653 y Fs(\(0\))606 b Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))435 b Fj(exists)d(for)h(al)62 b(l)433 b Fm(\026)337 b Fl(2)g(M)p Fj(.)1200 8154 y Fs(\(1\))606 b Fm(\026)337 b Fl(2)g(G)6520 8336 y Fk(in)-26 b(v)7843 8154 y Fr(\()p Fm(\015)67 b Fr(\))270 b Fl(\\)f(M)434 b Fj(implies)e Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))337 b(=)g(0)p Fj(.)1200 10656 y Fs(\(2\))606 b Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))338 b(=)e(0)434 b Fj(and)f Fm(\026)337 b Fl(2)g(M)433 b Fj(implies)f Fm(\026)337 b Fl(2)g(G)23166 10838 y Fk(in)-26 b(v)24489 10656 y Fr(\()p Fm(\015)67 b Fr(\))p Fj(.)1200 13822 y Fr(Items)384 b(\(1\))h(and)g(\(2\))g(are)f(called)f(the)i(\014rst)g (and)g(second)f(part)h(of)f(the)h(v)-67 b(ariational)384 b(principle.)531 b(The)384 b(second)1200 15328 y(part)431 b(is)f(true)g(for)h(an)-34 b(y)431 b(translation)g(in)-34 b(v)-67 b(arian)-34 b(t)431 b(quasilo)34 b(cal)429 b(measure)h Fm(\027)507 b Fr([11])430 b(\(with)h Fl(M)380 b Fr(=)g Fl(M)44832 14818 y Fk(+)44832 15698 y(1)p Fd(;)p Fk(in)-26 b(v)46887 15328 y Fr(\).)617 b(The)1200 16833 y(\014rst)557 b(part)g(is)g(pro)-34 b(v)g(ed)557 b(for)g(translation)h(in)-34 b(v)-67 b(arian)-34 b(t)557 b(Gibbs)g(measures)g(asso)34 b(ciated)557 b(with)g(a)g(translation)1200 18338 y(in)-34 b(v)-67 b(arian)-34 b(t)466 b(UA)-34 b(C)466 b(p)34 b(oten)-34 b(tial)467 b(\(with)f Fl(M)439 b Fr(=)g Fl(M)22680 17829 y Fk(+)22680 18709 y(1)p Fd(;)p Fk(in)-26 b(v)25200 18338 y Fr(also\).)723 b(W)-101 b(e)465 b(extend)h(this)g(result)g(to)g(an) -34 b(y)466 b(translation)1200 19844 y(in)-34 b(v)-67 b(arian)-34 b(t)464 b(quasilo)34 b(cal)463 b(measure)h(in)g(Section)h (4.)718 b(In)465 b([9)o(],)479 b(the)464 b(second)h(part)f(has)h(b)34 b(een)464 b(pro)-34 b(v)g(ed)465 b(for)f(some)1200 21349 y(renormalized)503 b(non-Gibbsian)j(FK)-34 b(G)505 b(measures.)838 b(In)505 b(general,)528 b(the)505 b(set)f Fl(M)g Fr(will)f(b)34 b(e)504 b(a)g(set)h(of)f(transla-)1200 22855 y(tion)532 b(in)-34 b(v)-67 b(arian)-34 b(t)533 b(probabilit)-34 b(y)532 b(measures)g(concen)-34 b(trating)533 b(on)g(\\go)34 b(o)g(d)532 b(con\014gurations")h(\(e.g.,)563 b(p)34 b(oin)-34 b(ts)533 b(of)1200 24360 y(con)-34 b(tin)g(uit)g(y)406 b(of)e(conditional)h(probabilities\).)1200 28512 y Fo(3)1793 b(V)-149 b(ariational)598 b(principle)h(for)f(generalized)h(Gibbs)f (measures)1200 31549 y Fr(W)-101 b(e)396 b(study)h(the)f(v)-67 b(ariational)396 b(principle)f(for)h(generalized)g(Gibbs)g(measures.) 536 b(W)-101 b(e)396 b(\014rst)g(pro)-34 b(v)g(e)397 b(the)f(second)1200 33055 y(part)473 b(for)f(almost)h(Gibbsian)g (measures,)489 b(whic)-34 b(h)473 b(is)f(a)g(rather)g(straigh)-34 b(tforw)g(ard)475 b(tec)-34 b(hnical)472 b(extension)g(of)1200 34560 y(Georgii)403 b(\(1988\),)i(Chapter)g(15.)1200 38138 y Fi(3.1)1495 b(Second)499 b(part)g(of)f(the)h(v)-83 b(ariational)500 b(principle)f(for)g(almost)h(Gibbsian)f(measures)1200 40759 y Fs(Theorem)464 b(3.1)606 b Fj(L)-62 b(et)466 b Fm(\015)533 b Fj(b)-62 b(e)466 b(a)g(tr)-62 b(anslation)464 b(invariant)i(sp)-62 b(e)g(ci\014c)g(ation)463 b(on)j Fr(\(\012)p Fm(;)202 b Fl(F)120 b Fr(\))468 b Fj(and)e Fm(\027)473 b Fl(2)396 b(G)44583 40941 y Fk(in)-26 b(v)45906 40759 y Fr(\()p Fm(\015)67 b Fr(\))p Fj(.)656 b(F)-93 b(or)1200 42264 y(al)62 b(l)433 b Fm(\026)337 b Fl(2)f(M)6602 41754 y Fk(+)6602 42634 y(1)p Fd(;)p Fk(in)-26 b(v)8657 42264 y Fj(,)18333 43971 y Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))338 b(=)e(0)18333 45477 y Fm(\026)p Fr(\(\012)20409 45659 y Fd(\015)21002 45477 y Fr(\))h(=)g(1)24493 43380 y Ff(o)25301 44726 y Fr(=)-202 b Fl(\))673 b Fm(\026)337 b Fl(2)g(G)30860 44908 y Fk(in)-26 b(v)32183 44726 y Fr(\()p Fm(\015)67 b Fr(\))1200 47600 y Fj(and)433 b(thus)f(such)g(a)h (me)-62 b(asur)g(e)432 b Fm(\026)i Fj(is)f(almost)f(Gibbs)g(w.r.t.)557 b Fm(\015)67 b Fj(.)1200 50766 y(Pr)-62 b(o)g(of)p Fr(.)605 b(Cho)34 b(ose)573 b Fm(\027)694 b Fl(2)617 b(G)12786 50948 y Fk(in)-26 b(v)14109 50766 y Fr(\()p Fm(\015)67 b Fr(\))573 b(and)g Fm(\026)g Fr(suc)-34 b(h)573 b(that)g Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))619 b(=)d(0.)1044 b(W)-101 b(e)571 b(ha)-34 b(v)g(e)573 b(to)g(pro)-34 b(v)g(e)572 b(that)i(for)e(an)-34 b(y)1200 52272 y Fm(g)380 b Fl(2)337 b(L)p Fm(;)202 b Fr(\003)336 b Fl(2)h(S)91 b Fr(:)21654 53777 y Fm(\026)p Fr(\()p Fm(\015)23483 53965 y Fk(\003)24191 53777 y Fm(g)313 b Fl(\000)269 b Fm(g)43 b Fr(\))338 b(=)e(0)p Fm(:)17962 b Fr(\(3.2\))1200 56002 y(Fix)404 b Fm(g)380 b Fl(2)337 b(L)404 b Fr(and)h(\001)337 b Fl(2)f(S)495 b Fr(suc)-34 b(h)405 b(that)h Fm(g)447 b Fr(is)404 b Fl(F)21268 56190 y Fk(\001)22108 56002 y Fr(-measurable.)538 b(The)405 b(h)-34 b(yp)34 b(othesis)18407 59167 y Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))338 b(=)593 b(lim)23408 60083 y Fk(\003)p Ft(")p Fh(Z)25121 59831 y Fg(d)26393 58347 y Fr(1)p 25938 58889 1516 49 v 25938 59999 a Fl(j)p Fr(\003)p Fl(j)27586 59167 y Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))338 b(=)e(0)14716 b(\(3.3\))1200 62867 y(implies)530 b(that)i(for)f(ev)-34 b(ery)529 b(\003)548 b Fl(2)g(S)91 b Fr(,)562 b(the)531 b(densit)-34 b(y)531 b Fm(f)25089 63055 y Fk(\003)26345 62867 y Fr(=)27969 62319 y Fd(d\026)29019 62468 y Fc(\003)p 27969 62589 1658 49 v 28021 63285 a Fd(d\027)28968 63434 y Fc(\003)30290 62867 y Fr(exists)f(and)i(is)e(a)h(b)34 b(ounded)532 b(p)34 b(ositiv)-34 b(e)531 b Fl(F)49288 63055 y Fk(\003)49996 62867 y Fr(-)1200 64373 y(measurable)404 b(function.)540 b(In)-34 b(tro)34 b(duce)404 b(lo)34 b(cal)403 b(appro)-34 b(ximations)406 b(of)e Fm(\015)31683 64561 y Fk(\003)32391 64373 y Fm(g)43 b Fr(:)17999 67096 y Fm(g)18620 66595 y Ft(\000)18577 67396 y Fd(n)19408 67096 y Fr(\()p Fm(\033)g Fr(\))1107 b(=)1332 b(inf)24243 67903 y Fd(!)32 b Ft(2)p Fk(\012)26369 67096 y Fm(\015)26997 67284 y Fk(\003)27704 67096 y Fm(g)43 b Fr(\()p Fm(\033)29489 67284 y Fk(\003)30142 67395 y Fg(n)30764 67096 y Fm(!)31518 67284 y Fk(\003)32171 67031 y Fg(c)32171 67505 y(n)32793 67096 y Fr(\))17999 69559 y Fm(g)18620 69058 y Fk(+)18577 69858 y Fd(n)19408 69559 y Fr(\()p Fm(\033)g Fr(\))1107 b(=)1156 b(sup)24243 70601 y Fd(!)32 b Ft(2)p Fk(\012)26369 69559 y Fm(\015)26997 69747 y Fk(\003)27704 69559 y Fm(g)43 b Fr(\()p Fm(\033)29489 69747 y Fk(\003)30142 69858 y Fg(n)30764 69559 y Fm(!)31518 69747 y Fk(\003)32171 69494 y Fg(c)32171 69968 y(n)32793 69559 y Fr(\))p Fm(:)25497 75417 y Fr(7)p eop %%Page: 8 8 8 7 bop 1200 1313 a Fr(In)411 b(the)g(quasilo)34 b(cal)410 b(case,)i(w)-34 b(e)411 b(ha)-34 b(v)g(e)411 b Fm(g)18543 874 y Fk(+)18500 1613 y Fd(n)19605 1313 y Fl(\000)274 b Fm(g)21443 874 y Ft(\000)21400 1613 y Fd(n)22578 1313 y Fl(!)348 b Fr(0)411 b(uniformly)g(when)h Fm(n)e Fr(go)34 b(es)411 b(to)g(in\014nit)-34 b(y)-101 b(,)413 b(whereas)e(here)1200 2819 y(w)-34 b(e)407 b(ha)-34 b(v)g(e)408 b Fm(g)6406 2379 y Fk(+)6363 3119 y Fd(n)7465 2819 y Fl(\000)271 b Fm(g)9300 2379 y Ft(\000)9257 3119 y Fd(n)10430 2819 y Fl(!)341 b Fr(0)407 b(on)h(the)f(set)g(\012)19543 3001 y Fd(\015)20543 2819 y Fr(of)h Fm(\026)p Fr(-measure)f(one,)g(and)h (hence,)g(b)-34 b(y)407 b(dominated)h(con)-34 b(v)g(ergence)1200 4324 y(in)404 b Fm(L)3439 3884 y Fk(1)3965 4324 y Fr(\()p Fm(\026)p Fr(\).)539 b(T)-101 b(o)405 b(obtain)g(\(3.2\))g(decomp)34 b(ose:)16801 6707 y Fm(\026)p Fr(\()p Fm(\015)18630 6895 y Fk(\003)19338 6707 y Fm(g)313 b Fl(\000)270 b Fm(g)43 b Fr(\))337 b(=)g Fm(A)25060 6889 y Fd(n)25955 6707 y Fr(+)270 b Fm(B)28087 6889 y Fd(n)28983 6707 y Fr(+)f Fm(C)31061 6889 y Fd(n)31957 6707 y Fr(+)g Fm(D)34173 6889 y Fd(n)47908 6707 y Fr(\(3.4\))1200 9089 y(where)18301 11471 y Fm(A)19210 11653 y Fd(n)20944 11471 y Fr(=)1106 b Fm(\026)p Fr(\()p Fm(\015)24822 11659 y Fk(\003)25531 11471 y Fm(g)313 b Fl(\000)269 b Fm(g)28255 10970 y Ft(\000)28212 11771 y Fd(n)29042 11471 y Fr(\))18291 13308 y Fm(B)19210 13490 y Fd(n)20944 13308 y Fr(=)1106 b Fm(\027)77 b Fr(\(\()p Fm(g)25232 12808 y Ft(\000)25189 13608 y Fd(n)26290 13308 y Fl(\000)269 b Fm(\015)28130 13496 y Fk(\003)28838 13308 y Fm(g)43 b Fr(\))p Fm(f)30523 13555 y Fk(\003)31176 13666 y Fg(n)31743 13555 y Ft(n)p Fk(\003)32922 13308 y Fr(\))18344 15146 y Fm(C)19210 15328 y Fd(n)20944 15146 y Fr(=)1106 b Fm(\027)77 b Fr(\()p Fm(f)24733 15393 y Fk(\003)25386 15504 y Fg(n)25953 15393 y Ft(n)p Fk(\003)27131 15146 y Fr(\()p Fm(\015)28230 15334 y Fk(\003)28938 15146 y Fm(g)313 b Fl(\000)269 b Fm(g)43 b Fr(\)\))18207 16984 y Fm(D)19211 17166 y Fd(n)20944 16984 y Fr(=)1106 b Fm(\027)77 b Fr(\(\()p Fm(f)25204 17230 y Fk(\003)25857 17341 y Fg(n)26425 17230 y Ft(n)p Fk(\003)27872 16984 y Fl(\000)269 b Fm(f)29677 17172 y Fk(\003)30330 17283 y Fg(n)30952 16984 y Fr(\))p Fm(g)43 b Fr(\))p Fm(:)1200 19366 y Fr(Using)19308 20871 y(0)336 b Fl(\024)h Fm(\015)22158 21059 y Fk(\003)22865 20871 y Fm(g)313 b Fl(\000)269 b Fm(g)25589 20371 y Ft(\000)25546 21171 y Fd(n)26714 20871 y Fl(\024)336 b Fm(g)28614 20371 y Fk(+)28571 21171 y Fd(n)29671 20871 y Fl(\000)270 b Fm(g)31505 20371 y Ft(\000)31462 21171 y Fd(n)1200 22926 y Fm(A)2109 23108 y Fd(n)3072 22926 y Fl(!)337 b Fr(0)404 b(as)g Fm(n)g Fr(go)34 b(es)403 b(to)i(in\014nit)-34 b(y)-101 b(.)539 b(F)-101 b(or)404 b Fm(B)20038 23108 y Fd(n)20665 22926 y Fr(,)g(use)7690 25308 y(0)337 b Fl(\024)f(j)p Fm(B)11168 25490 y Fd(n)11795 25308 y Fl(j)g Fr(=)g Fm(\027)14625 24326 y Ff(\000)15181 25308 y Fr(\()p Fm(\015)16280 25496 y Fk(\003)16988 25308 y Fm(g)313 b Fl(\000)269 b Fm(g)19712 24807 y Ft(\000)19669 25608 y Fd(n)20500 25308 y Fr(\))p Fm(f)21564 25555 y Fk(\003)22217 25666 y Fg(n)22783 25555 y Ft(n)p Fk(\003)23962 24326 y Ff(\001)24854 25308 y Fl(\024)336 b Fm(\027)77 b Fr(\()p Fm(f)27873 25555 y Fk(\003)28526 25666 y Fg(n)29093 25555 y Ft(n)p Fk(\003)30272 25308 y Fr(\()p Fm(g)31364 24807 y Fk(+)31321 25608 y Fd(n)32421 25308 y Fl(\000)269 b Fm(g)34254 24807 y Ft(\000)34211 25608 y Fd(n)35042 25308 y Fr(\)\))338 b(=)e Fm(\026)p Fr(\()p Fm(g)39423 24807 y Fk(+)39380 25608 y Fd(n)40481 25308 y Fl(\000)269 b Fm(g)42314 24807 y Ft(\000)42271 25608 y Fd(n)43102 25308 y Fr(\))p Fm(;)1200 27690 y Fr(to)404 b(obtain)h Fm(B)7371 27872 y Fd(n)8335 27690 y Fl(!)337 b Fr(0)404 b(as)g Fm(n)336 b Fl(!)h(1)p Fr(.)3082 29471 y(Since)556 b Fm(\027)668 b Fl(2)592 b(G)72 b Fr(\()p Fm(\015)67 b Fr(\),)595 b(and)557 b Fm(f)15530 29718 y Fk(\003)16183 29829 y Fg(n)16750 29718 y Ft(n)p Fk(\003)18519 29471 y Fl(2)591 b(F)20789 29659 y Fk(\003)21442 29406 y Fg(c)21924 29471 y Fr(,)k Fm(C)23722 29653 y Fd(n)24939 29471 y Fr(=)c(0.)997 b(The)557 b(fact)g(that)h Fm(D)37383 29653 y Fd(n)38600 29471 y Fl(!)591 b Fr(0)557 b(follo)-34 b(ws)557 b(from)g(the)1200 30976 y(assumption)406 b(of)e(zero)f(relativ)-34 b(e)404 b(en)-34 b(trop)g(y)405 b(densit)-34 b(y:)539 b(see)404 b(Georgii)f(\(1988\),)i(p)f(324.)p 49878 32757 522 731 v 1200 35800 a Fs(Remark)464 b(3.5)2039 b Fj(1.)605 b(The)518 b(r)-62 b(ole)518 b(of)g Fl(M)g Fj(in)h(De\014nition)f(2.16)f(is)h (playe)-62 b(d)516 b(her)-62 b(e)517 b(by)h(the)f(set)g(of)h(me)-62 b(asur)g(es)4230 37306 y(c)g(onc)g(entr)g(ating)432 b(on)i(the)e(p)-62 b(oints)432 b(of)h(c)-62 b(ontinuity)432 b(of)h Fm(\015)500 b Fj(\()p Fm(\026)337 b Fl(2)g(M)433 b Fj(if)g(and)g(only)g(if)h Fm(\026)p Fr(\(\012)42683 37488 y Fd(\015)43276 37306 y Fr(\))337 b(=)f(1)p Fj(\).)2633 39694 y(2.)605 b(R)-62 b(emark)433 b(that)f(in)i(The)-62 b(or)g(em)433 b(3.1,)f(we)i(do)e(not) i(ask)e(any)h(c)-62 b(onc)g(entr)g(ation)432 b(pr)-62 b(op)g(erties)431 b(of)i Fm(\027)77 b Fj(.)1200 43162 y Fi(3.2)1495 b(First)569 b(part)g(of)f(the)g(v)-83 b(ariational)570 b(principle)f(for)f(some)h(almost)g(Gibbsian)g(mea-)4604 44711 y(sures)1200 47275 y Fr(T)-101 b(o)462 b(obtain)g(the)h(\014rst)f (part)g(of)g(the)g(v)-67 b(ariational)461 b(principle,)475 b(it)461 b(will)g(turn)i(out)f(that)h(concen)-34 b(tration)463 b(of)f Fm(\026)1200 48780 y Fr(on)442 b(the)g(set)g(\012)7851 48962 y Fd(\015)8886 48780 y Fr(is)f(not)i(the)f(righ)-34 b(t)442 b(condition.)653 b(W)-101 b(e)441 b(need)h(that)h(some)f (particular)f(class)h(of)g(\\telescoping)1200 50286 y (con\014gurations")348 b(are)f(p)34 b(oin)-34 b(ts)348 b(of)f(con)-34 b(tin)g(uit)g(y)348 b(of)g(the)f(sp)34 b(eci\014cation.)519 b(This)348 b(reminds)f(of)g(asking)g(con)-34 b(tin)g(uit)g(y)1200 51791 y(prop)34 b(erties)507 b(of)g(the)h (one-sided)g(conditional)g(probabilities.)848 b(In)507 b(the)h(case)f(of)g(\(uniformly\))i(con)-34 b(tin)g(uous)1200 53297 y(sp)34 b(eci\014cations,)404 b(this)h(distinction)g(b)34 b(et)-34 b(w)g(een)405 b(one-sided)f(and)h(t)-34 b(w)g(o-sided)406 b(is)e(of)g(course)g(not)h(visible.)3082 55077 y(W)-101 b(e)540 b(c)-34 b(ho)34 b(ose)540 b(a)h(particular)f(v)-67 b(alue)540 b(written)h(+1)f(in)g(the)h(state)g(space)g Fm(E)610 b Fr(and)541 b(denote)g(b)-34 b(y)541 b("+")f(the)1200 56583 y(con\014guration)484 b(whose)g(v)-67 b(alue)482 b(is)h(+1)f(ev)-34 b(erywhere.)774 b(T)-101 b(o)484 b(an)-34 b(y)483 b(con\014guration)h Fm(\033)512 b Fl(2)468 b Fr(\012,)503 b(w)-34 b(e)483 b(asso)34 b(ciate)482 b(the)1200 58088 y(con\014guration)405 b Fm(\033)9279 57649 y Fk(+)10471 58088 y Fr(de\014ned)g(b)-34 b(y)17964 61873 y Fm(\033)18700 61372 y Fk(+)19488 61873 y Fr(\()p Fm(x)p Fr(\))337 b(=)22739 59388 y Ff(8)22739 60479 y(<)22739 62661 y(:)24370 60365 y Fm(\033)43 b Fr(\()p Fm(x)p Fr(\))674 b(if)1105 b(x)673 b Fl(\024)g Fr(0)24370 63376 y(+1)1347 b(if)1104 b(x)673 b Fm(>)g Fr(0)p Fm(:)1200 65826 y Fr(Here,)419 b(the)f(order)f Fl(\024)g Fr(is)g(lexicographic.)576 b(W)-101 b(e)417 b(de\014ne)h(then)g(\012)29381 65386 y Fd(<)p Fk(0)29381 66126 y Fd(\015)31056 65826 y Fr(to)g(b)34 b(e)416 b(the)i(subset)g(of) g(\012)g(of)f(the)h(con\014gu-)1200 67331 y(rations)404 b Fm(\033)448 b Fr(suc)-34 b(h)405 b(that)g(the)f(new)h (con\014guration)g Fm(\033)24409 66891 y Fk(+)25601 67331 y Fr(is)f(a)g(go)34 b(o)g(d)404 b(con\014guration)h(for)f Fm(\015)67 b Fr(:)18829 69714 y(\012)19704 69213 y Fd(<)p Fk(0)19704 70013 y Fd(\015)21299 69714 y Fr(=)22579 68732 y Ff(\010)23286 69714 y Fm(\033)380 b Fl(2)336 b Fr(\012)p Fm(;)539 b(\033)27990 69213 y Fk(+)29114 69714 y Fl(2)337 b Fr(\012)31134 69896 y Fd(\015)31727 68732 y Ff(\011)32434 69714 y Fm(:)1200 72096 y Fr(This)405 b(set)f(will)f(b)34 b(e)404 b(describ)34 b(ed)404 b(in)g(di\013eren)-34 b(t)405 b(examples)e(in)h(Section)h(5.)25497 75417 y(8)p eop %%Page: 9 9 9 8 bop 1200 1313 a Fs(3.2.1)1394 b(Results)1200 3851 y Fr(W)-101 b(e)406 b(consider)f(a)i(pair)f(\()p Fm(\015)67 b(;)202 b(\027)77 b Fr(\))406 b(with)h Fm(\027)417 b Fl(2)340 b(G)20431 4033 y Fk(in)-26 b(v)21754 3851 y Fr(\()p Fm(\015)67 b Fr(\))407 b(and)g(a)f(measure)g Fm(\026)g Fr(whic)-34 b(h)408 b(satis\014es)e(the)h(follo)-34 b(wing)407 b(con-)1200 5357 y(dition:)3082 8617 y Fs(Condition)466 b(C1)22617 10123 y Fm(\026)p Fr(\(\012)24693 9622 y Fd(<)p Fk(0)24693 10423 y Fd(\015)25952 10123 y Fr(\))337 b(=)g(1)p Fm(:)1200 12101 y Fr(W)-101 b(e)403 b(also)h(in)-34 b(tro)34 b(duce)18867 14312 y Fm(e)19431 13811 y Fk(+)19431 14611 y Fd(\027)20555 14312 y Fr(:=)336 b Fl(\000)458 b Fr(lim)23316 15227 y Fk(\003)p Ft(")p Fh(Z)25029 14975 y Fg(d)26301 13492 y Fr(1)p 25846 14033 1516 49 v 25846 15143 a Fl(j)p Fr(\003)p Fl(j)27696 14312 y Fr(log)219 b Fm(\027)77 b Fr(\(+)31554 14500 y Fk(\003)32262 14312 y Fr(\))1200 16945 y(whenev)-34 b(er)404 b(it)g(exists.)1200 19285 y Fs(Theorem)464 b(3.6)606 b Fj(Under)434 b(the)e(c)-62 b(ondition)432 b(C1:)2633 21624 y(1.)605 b Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))435 b Fj(exists)d(if)h(and)h(only)e (if)i Fm(e)19273 21184 y Fk(+)19273 21924 y Fd(\027)20494 21624 y Fj(exists)e(and)h(then)15222 24556 y Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))338 b(=)e Fm(e)20786 24055 y Fk(+)20786 24855 y Fd(\027)21843 24556 y Fl(\000)269 b Fm(h)p Fr(\()p Fm(\026)p Fr(\))i Fl(\000)26908 22906 y Ff(Z)27582 25655 y Fk(\012)28519 24556 y Fr(log)30419 23736 y Fm(\015)31047 23918 y Fk(0)31572 23736 y Fr(\()p Fm(\033)32779 23296 y Fk(+)33567 23736 y Fl(j)p Fm(\033)34640 23296 y Fk(+)35427 23736 y Fr(\))p 30419 24277 5480 49 v 30709 25387 a Fm(\015)31337 25569 y Fk(0)31863 25387 y Fr(\(+)p Fl(j)p Fm(\033)34350 25037 y Fk(+)35137 25387 y Fr(\))36031 24556 y Fm(\026)p Fr(\()p Fm(d\033)43 b Fr(\))p Fm(:)8501 b Fr(\(3.7\))4230 27433 y Fj(wher)-62 b(e)433 b Fm(h)p Fr(\()p Fm(\026)p Fr(\))i Fj(is)e(the)g(Kolmo)-62 b(gor)g(ov-Sinai)430 b(entr)-62 b(opy)433 b(of)f Fm(\026)p Fj(.)2633 29770 y(2.)605 b(If)434 b(mor)-62 b(e)g(over)432 b Fm(\026)337 b Fl(2)g(G)13671 29952 y Fk(in)-26 b(v)14994 29770 y Fr(\()p Fm(\015)67 b Fr(\))434 b Fj(and)f Fm(e)19983 29330 y Fk(+)19983 30070 y Fd(\027)21205 29770 y Fj(exists,)e(then) 19778 32636 y Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))338 b(=)592 b(lim)24778 33551 y Fk(\003)p Ft(")p Fh(Z)26491 33299 y Fg(d)27763 31816 y Fr(1)p 27308 32357 1516 49 v 27308 33467 a Fl(j)p Fr(\003)p Fl(j)29158 32636 y Fr(log)31059 31816 y Fm(\026)p Fr(\(+)33203 32004 y Fk(\003)33911 31816 y Fr(\))p 31059 32357 3324 49 v 31086 33467 a Fm(\027)77 b Fr(\(+)33176 33655 y Fk(\003)33884 33467 y Fr(\))34516 32636 y Fm(:)13055 b Fr(\(3.8\))1200 35936 y(T)-101 b(o)475 b(get)h(the)f(more)g(usual)g(expression)g(of)h (the)f(v)-67 b(ariational)474 b(principle,)492 b(w)-34 b(e)476 b(add)g(an)f(extra)g(condition)g(to)1200 37442 y(the)404 b(condition)h(C1:)3082 40702 y Fs(Condition)466 b(C2)13236 43265 y Fm(\026)337 b Fl(2)g(G)16169 43447 y Fk(in)-26 b(v)17492 43265 y Fr(\()p Fm(\015)67 b Fr(\))337 b(is)g(suc)-34 b(h)337 b(that)807 b(lim)26045 44180 y Fk(\003)p Ft(")p Fh(Z)27758 43928 y Fc(d)29052 42445 y Fr(1)p 28598 42986 1516 49 v 28598 44097 a Fl(j)p Fr(\003)p Fl(j)30448 43265 y Fr(log)32348 42445 y Fm(\026)p Fr(\(+)34492 42633 y Fk(\003)35201 42445 y Fr(\))p 32348 42986 3324 49 v 32375 44097 a Fm(\027)77 b Fr(\(+)34465 44285 y Fk(\003)35173 44097 y Fr(\))36142 43265 y(=)336 b(0)p Fm(:)1200 46510 y Fs(Theorem)464 b(3.9)606 b Fj(Assume)433 b(that)f(c)-62 b(onditions)431 b(C1)j(and)f(C2)g(ar)-62 b(e)433 b(ful\014l)62 b(le)-62 b(d.)556 b(Then)2633 48849 y(1.)605 b Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))338 b(=)f(0)p Fj(.)2633 51186 y(2.)605 b Fm(e)4794 50746 y Fk(+)4794 51486 y Fd(\027)6016 51186 y Fj(exists)432 b(and)h Fm(e)12253 50746 y Fk(+)12253 51486 y Fd(\027)13377 51186 y Fr(=)337 b Fm(e)15221 50746 y Fk(+)15221 51486 y Fd(\026)16008 51186 y Fj(.)2633 53645 y(3.)605 b Fm(h)p Fr(\()p Fm(\013)t Fl(j)p Fm(\027)77 b Fr(\))436 b Fj(exists)c(for)h(al) 62 b(l)432 b Fm(\013)342 b Fl(2)337 b(M)18805 53135 y Fk(+)18805 54015 y(1)p Fd(;)p Fk(in)-26 b(v)21293 53645 y Fj(satisfying)431 b(C1.)1200 56234 y Fr(As)404 b(a)g(corollary)f(of)h (these)h(theorems,)f(w)-34 b(e)404 b(obtain)h(the)g(usual)g(\014rst)f (part)h(of)f(the)h(v)-67 b(ariational)403 b(principle.)1200 58573 y Fs(Theorem)464 b(3.10)606 b Fj(L)-62 b(et)416 b Fm(\026)337 b Fl(2)g(M)15946 58063 y Fk(+)15946 58943 y(1)p Fd(;)p Fk(in)-26 b(v)18417 58573 y Fj(and)415 b Fm(\027)f Fl(2)337 b(G)23632 58755 y Fk(in)-26 b(v)24955 58573 y Fr(\()p Fm(\015)67 b Fr(\))417 b Fj(such)d(that)h(c)-62 b(onditions)414 b(C1)i(and)g(C2)g(hold)f(and)g Fm(e)49612 58133 y Fk(+)49612 58873 y Fd(\027)1200 60079 y Fj(exists.)556 b(Then)2633 62418 y(1.)605 b Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))435 b Fj(exists.)2633 64755 y(2.)605 b Fm(\026)337 b Fl(2)g(G)7163 64937 y Fk(in)-26 b(v)8486 64755 y Fr(\()p Fm(\015)67 b Fr(\))434 b Fj(implies)e Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))338 b(=)f(0)p Fj(.)1200 67344 y Fs(Remark)464 b(3.11)606 b Fj(The)476 b(existenc)-62 b(e)474 b(of)i(the)f(limit)h(de\014ning)g Fm(e)28900 66904 y Fk(+)28900 67643 y Fd(\027)30164 67344 y Fj(is)g(guar)-62 b(ante)g(e)g(d)473 b(for)j(e.g.)684 b(r)-62 b(enormalization)1200 68849 y(gr)g(oup)439 b(tr)-62 b(ansformations)438 b(of)i(Gibbs)e(me)-62 b(asur)g(es,)440 b(and)g(for)g Fm(\027)518 b Fj(with)440 b(p)-62 b(ositive)437 b(c)-62 b(orr)g(elations)438 b(\(by)h(sub)-62 b(additiv-)1200 70355 y(ity\).)716 b(Mor)-62 b(e)g(over,)498 b(in)487 b(the)f(c)-62 b(ase)486 b(of)g(tr)-62 b(ansformations)485 b(of)h(Gibbs)f(me)-62 b(asur)g(es,)498 b(c)-62 b(ondition)486 b(C2)h(is)f(also)g(e)-62 b(asy)1200 71860 y(to)433 b(verify.)555 b(Se)-62 b(e)434 b(Se)-62 b(ction)432 b(5)i(b)-62 b(elow.)25497 75417 y Fr(9)p eop %%Page: 10 10 10 9 bop 1200 1313 a Fi(3.3)1495 b(Pro)42 b(ofs)1200 3934 y Fs(Pro)d(of)464 b(of)h(Theorem)f(3.6)1200 7100 y Fr(First)404 b(w)-34 b(e)405 b(need)f(the)h(follo)-34 b(wing)1200 9934 y Fs(Lemma)464 b(3.12)606 b Fj(If)434 b Fm(\026)p Fr(\(\012)12576 9494 y Fd(<)p Fk(0)12576 10233 y Fd(\015)13834 9934 y Fr(\))337 b(=)g(1)p Fj(,)433 b(then)2633 12767 y(1.)605 b(Uniformly)433 b(in)h Fm(!)381 b Fl(2)336 b Fr(\012)p Fj(,)12207 16126 y Fr(lim)11822 16854 y Fd(n)p Ft(!1)15377 15306 y Fr(1)p 14610 15847 2142 49 v 14610 16958 a Fl(j)p Fr(\003)15789 17140 y Fd(n)16414 16958 y Fl(j)17086 14476 y Ff(Z)17759 17225 y Fk(\012)18696 16126 y Fr(log)20700 15306 y Fm(\015)21328 15494 y Fk(\003)21981 15605 y Fg(n)22602 15306 y Fr(\()p Fm(\033)43 b Fl(j)p Fm(!)g Fr(\))p 20597 15847 4923 49 v 20597 16958 a Fm(\015)21225 17146 y Fk(\003)21878 17257 y Fg(n)22499 16958 y Fr(\(+)p Fl(j)p Fm(!)g Fr(\))25652 16126 y Fm(\026)p Fr(\()p Fm(d\033)g Fr(\))338 b(=)30308 14476 y Ff(Z)30982 17225 y Fk(\012)31918 16126 y Fr(log)33819 15306 y Fm(\015)34447 15488 y Fk(0)34972 15306 y Fr(\()p Fm(\033)36179 14866 y Fk(+)36967 15306 y Fl(j)p Fm(\033)38040 14866 y Fk(+)38827 15306 y Fr(\))p 33819 15847 5480 49 v 34109 16958 a Fm(\015)34737 17140 y Fk(0)35263 16958 y Fr(\(+)p Fl(j)p Fm(\033)37750 16608 y Fk(+)38537 16958 y Fr(\))39431 16126 y Fm(\026)p Fr(\()p Fm(d\033)43 b Fr(\))p Fm(:)2633 19996 y Fj(2.)605 b(F)-93 b(or)434 b Fm(\027)413 b Fl(2)337 b(G)72 b Fr(\()p Fm(\015)67 b Fr(\))p Fj(,)12750 23423 y Fr(lim)12365 24150 y Fd(n)p Ft(!1)15921 22603 y Fr(1)p 15153 23144 2142 49 v 15153 24254 a Fl(j)p Fr(\003)16332 24436 y Fd(n)16958 24254 y Fl(j)17629 21773 y Ff(Z)18303 24522 y Fk(\012)19239 23423 y Fr(log)21265 22603 y Fm(\027)77 b Fr(\()p Fm(\033)23105 22791 y Fk(\003)23758 22902 y Fg(n)24379 22603 y Fr(\))p 21140 23144 3836 49 v 21140 24254 a Fm(\027)g Fr(\(+)23230 24442 y Fk(\003)23883 24553 y Fg(n)24504 24254 y Fr(\))25109 23423 y Fm(\026)p Fr(\()p Fm(d\033)43 b Fr(\))338 b(=)29765 21773 y Ff(Z)30438 24522 y Fk(\012)31375 23423 y Fr(log)33276 22603 y Fm(\015)33904 22785 y Fk(0)34429 22603 y Fr(\()p Fm(\033)35636 22163 y Fk(+)36424 22603 y Fl(j)p Fm(\033)37497 22163 y Fk(+)38284 22603 y Fr(\))p 33276 23144 5480 49 v 33566 24254 a Fm(\015)34194 24436 y Fk(0)34719 24254 y Fr(\(+)p Fl(j)p Fm(\033)37206 23904 y Fk(+)37994 24254 y Fr(\))38888 23423 y Fm(\026)p Fr(\()p Fm(d\033)43 b Fr(\))p Fm(:)4230 26795 y Fj(In)435 b(p)-62 b(articular,)430 b(the)j(limit)f(dep)-62 b(ends)432 b(only)h(on)h(the)e(p)-62 b(air)433 b Fr(\()p Fm(\015)67 b(;)202 b(\026)p Fr(\))p Fj(.)1200 29960 y Fs(Remark)464 b(3.13)606 b Fj(If)522 b Fm(\026)h Fj(is)f(er)-62 b(go)g(dic)520 b(under)h(tr)-62 b(anslations,)542 b(we)522 b(have)f(a)h(slightly)d(str)-62 b(onger)521 b(statement)g(for)1200 31641 y(item)576 b(1)g(:)7280 31164 y Fk(1)p 6645 31362 1742 49 v 6645 32075 a Ft(j)p Fk(\003)7559 32186 y Fg(n)8125 32075 y Ft(j)8721 30664 y Ff(R)9294 32067 y Fk(\012)10231 31641 y Fr(log)12212 31044 y Fd(\015)12697 31193 y Fc(\003)13305 31044 y Fk(\()p Fd(\033)32 b Ft(j)p Fd(!)g Fk(\))p 12131 31362 3435 49 v 12131 32075 a Fd(\015)12616 32224 y Fc(\003)13223 32075 y Fk(\(+)p Ft(j)p Fd(!)g Fk(\))15698 31641 y Fm(\026)p Fr(\()p Fm(d\033)43 b Fr(\))578 b Fj(c)-62 b(onver)g(ges)574 b(in)j Fe(L)27301 31201 y Fk(1)27827 31641 y Fr(\()p Fm(\026)p Fr(\))h Fj(to)31676 30664 y Ff(R)32249 32067 y Fk(\012)33186 31641 y Fr(log)35086 31044 y Fd(\015)35571 31167 y Fc(0)36032 31044 y Fk(\()p Fd(\033)36968 30731 y Fc(+)37644 31044 y Ft(j)p Fd(\033)38475 30731 y Fc(+)39151 31044 y Fk(\))p 35086 31362 4431 49 v 35343 32087 a Fd(\015)35828 32210 y Fc(0)36289 32087 y Fk(\(+)p Ft(j)p Fd(\033)38218 31835 y Fc(+)38894 32087 y Fk(\))39650 31641 y Fm(\026)p Fr(\()p Fm(d\033)43 b Fr(\))p Fj(,)613 b(uniformly)576 b(in)1200 33146 y Fm(!)381 b Fl(2)336 b Fr(\012)p Fj(.)1200 36312 y(Pr)-62 b(o)g(of)p Fr(.)2681 39146 y(1.)606 b(The)356 b(pro)34 b(of)356 b(uses)g(relativ)-34 b(e)354 b(energies)h(as)g(in)h (Sulliv)-67 b(an)355 b(\(1973\).)523 b(F)-101 b(or)356 b(all)f(\003)336 b Fl(2)h(S)91 b Fm(;)202 b(\033)-24 b(;)202 b(!)380 b Fl(2)336 b Fr(\012,)366 b(w)-34 b(e)356 b(de\014ne,)11101 42315 y Fm(E)12066 41806 y Fk(+)11996 42697 y(\003)12853 42315 y Fr(\()p Fm(\033)43 b Fl(j)p Fm(!)g Fr(\))338 b(=)e(log)19264 41718 y Fd(\015)19749 41867 y Fc(\003)20356 41718 y Fk(\()p Fd(\033)32 b Ft(j)p Fd(!)g Fk(\))p 19183 42037 3435 49 v 19183 42749 a Fd(\015)19668 42898 y Fc(\003)20275 42749 y Fk(\(+)p Ft(j)p Fd(!)g Fk(\))23827 42315 y Fr(and)1078 b Fm(D)34 b Fr(\()p Fm(\033)43 b Fr(\))337 b(=)g Fm(E)32155 41806 y Fk(+)32085 42756 y Ft(f)p Fk(0)p Ft(g)33551 42315 y Fr(\()p Fm(\033)43 b Fl(j)p Fm(\033)g Fr(\))338 b(=)e(log)39901 41718 y Fd(\015)40386 41841 y Fc(0)40847 41718 y Fk(\()p Fd(\033)32 b Ft(j)p Fd(\033)g Fk(\))p 39820 42037 3241 49 v 39820 42749 a Fd(\015)40305 42872 y Fc(0)40765 42749 y Fk(\(+)p Ft(j)p Fd(\033)g Fk(\))43193 42315 y Fr(.)4230 45557 y(W)-101 b(e)374 b(consider)h(an)g(appro)-34 b(ximation)376 b(of)f Fm(\033)22779 45118 y Fk(+)23941 45557 y Fr(at)g(\014nite)h(v) -34 b(olume)374 b(\003)h(with)g(b)34 b(oundary)376 b(condition)f Fm(!)419 b Fr(and)4230 47063 y(de\014ne)405 b(the)f Fj(telesc)-62 b(oping)431 b(c)-62 b(on\014gur)g(ation)402 b Fm(T)24032 46623 y Fd(!)23864 47422 y Fk(\003)24705 47063 y Fr([)p Fm(x;)202 b(\033)-24 b(;)202 b Fr(+]:)14826 53570 y Fm(T)15702 53069 y Fd(!)15534 53870 y Fk(\003)16375 53570 y Fr([)p Fm(x;)g(\033)-24 b(;)202 b Fr(+]\()p Fm(y)43 b Fr(\))336 b(=)23627 48903 y Ff(8)23627 49994 y(>)23627 50358 y(>)23627 50721 y(>)23627 51085 y(>)23627 51449 y(>)23627 51812 y(>)23627 52176 y(<)23627 54358 y(>)23627 54721 y(>)23627 55085 y(>)23627 55449 y(>)23627 55812 y(>)23627 56176 y(>)23627 56540 y(:)25258 49804 y Fm(!)43 b Fr(\()p Fm(y)g Fr(\))1446 b(if)1538 b Fm(y)380 b Fl(2)337 b Fr(\003)34286 49364 y Fd(c)25258 52815 y Fm(\033)43 b Fr(\()p Fm(y)g Fr(\))1507 b(if)1538 b Fm(y)717 b Fl(\024)673 b Fm(x;)538 b(y)381 b Fl(2)336 b Fr(\003)25258 55826 y(+1)2273 b(if)1538 b Fm(y)717 b(>)673 b(x;)538 b(y)381 b Fl(2)336 b Fr(\003)p Fm(:)4230 59953 y Fr(Using)405 b(the)f(consistency)g(prop)34 b(ert)-34 b(y)405 b(of)f Fm(\015)67 b Fr(,)404 b(w)-34 b(e)404 b(ha)-34 b(v)g(e)405 b(b)-34 b(y)404 b(telescoping,)18365 62790 y Fm(E)19330 62280 y Fk(+)19260 63172 y(\003)20116 62790 y Fr(\()p Fm(\033)43 b Fl(j)p Fm(!)g Fr(\))339 b(=)24575 61639 y Ff(X)24546 64264 y Fd(x)p Ft(2)p Fk(\003)26558 62790 y Fm(E)27523 62290 y Fk(+)27453 63090 y Fd(x)28309 62790 y Fr(\()p Fm(\033)43 b Fl(j)p Fm(T)30729 62290 y Fd(!)30561 63090 y Fk(\003)31402 62790 y Fr([)p Fm(x;)202 b(\033)-24 b(;)202 b Fr(+]\))p Fm(:)4230 66611 y Fr(By)404 b(translation)h(in)-34 b(v)-67 b(ariance)403 b(of)h Fm(\015)67 b Fr(,)18335 69448 y Fm(E)19300 68938 y Fk(+)19230 69830 y(\003)20087 69448 y Fr(\()p Fm(\033)43 b Fl(j)p Fm(!)g Fr(\))338 b(=)24546 68296 y Ff(X)24516 70922 y Fd(x)p Ft(2)p Fk(\003)26528 69448 y Fm(D)34 b Fr(\()p Fm(\034)28567 69630 y Ft(\000)p Fd(x)29883 69448 y Fm(T)30759 68948 y Fd(!)30591 69748 y Fk(\003)31432 69448 y Fr([)p Fm(x;)202 b(\033)-24 b(;)202 b Fr(+]\))p Fm(:)25194 75417 y Fr(10)p eop %%Page: 11 11 11 10 bop 4230 1313 a Fr(By)404 b(translation)h(in)-34 b(v)-67 b(ariance)403 b(of)h Fm(\026)p Fr(,)11865 2780 y Ff(Z)12538 5529 y Fk(\012)13475 4429 y Fm(E)14440 3920 y Fk(+)14370 4811 y(\003)15023 4922 y Fg(n)15644 4429 y Fr(\()p Fm(\033)43 b Fl(j)p Fm(!)g Fr(\))p Fm(\026)p Fr(\()p Fm(d\033)g Fr(\))1110 b(=)24967 3278 y Ff(X)24655 5903 y Fd(x)p Ft(2)p Fk(\003)26465 6014 y Fg(n)27233 2780 y Ff(Z)27906 5529 y Fk(\012)28843 4429 y Fm(D)34 b Fr(\()p Fm(\034)30882 4611 y Ft(\000)p Fd(x)32198 4429 y Fm(T)33074 3929 y Fd(!)32906 4729 y Fk(\003)33747 4429 y Fr([)p Fm(x;)202 b(\034)35846 4611 y Fd(x)36430 4429 y Fm(\033)-24 b(;)202 b Fr(+]\))p Fm(\026)p Fr(\()p Fm(d\033)43 b Fr(\))p Fm(:)4230 8207 y Fr(Therefore,)404 b(w)-34 b(e)404 b(ha)-34 b(v)g(e)405 b(to)g(pro)-34 b(v)g(e)404 b(that,)h(uniformly)f(in)g Fm(!)43 b Fr(,)10761 11795 y(lim)10376 12522 y Fd(n)p Ft(!1)13932 10975 y Fr(1)p 13164 11516 2142 49 v 13164 12626 a Fl(j)p Fr(\003)14343 12808 y Fd(n)14969 12626 y Fl(j)15640 9722 y Ff( )16912 10643 y(X)16600 13269 y Fd(x)p Ft(2)p Fk(\003)18410 13380 y Fg(n)19178 10145 y Ff(Z)19851 12894 y Fk(\012)20586 10449 y Ff(h)21159 11795 y Fm(D)34 b Fr(\()p Fm(\034)23198 11977 y Ft(\000)p Fd(x)24514 11795 y Fm(T)25390 11294 y Fd(!)25222 12095 y Fk(\003)25875 12206 y Fg(n)26497 11795 y Fr([)p Fm(x;)202 b(\034)28596 11977 y Fd(x)29180 11795 y Fm(\033)-24 b(;)202 b Fr(+]\))268 b Fl(\000)h Fm(D)34 b Fr(\()p Fm(\033)35864 11294 y Fk(+)36651 11795 y Fr(\))37122 10449 y Ff(i)37695 11795 y Fm(\026)p Fr(\()p Fm(d\033)43 b Fr(\))40734 9722 y Ff(!)42032 11795 y Fr(=)336 b(0)p Fm(:)4230 15573 y Fr(By)404 b(de\014nition,)11807 21106 y Fm(\034)12337 21288 y Ft(\000)p Fd(x)13653 21106 y Fm(T)14529 20605 y Fd(!)14361 21405 y Fk(\003)15014 21516 y Fg(n)15636 21106 y Fr([)p Fm(x;)202 b(\034)17735 21288 y Fd(x)18319 21106 y Fm(\033)-24 b(;)202 b Fr(+])336 b(=)22422 17166 y Ff(8)22422 18257 y(>)22422 18621 y(>)22422 18985 y(>)22422 19348 y(>)22422 19712 y(<)22422 21894 y(>)22422 22257 y(>)22422 22621 y(>)22422 22985 y(>)22422 23348 y(:)24053 18093 y Fm(\034)24583 18275 y Ft(\000)p Fd(x)25900 18093 y Fm(!)43 b Fr(\()p Fm(y)g Fr(\))1446 b(if)1538 b Fm(y)313 b Fr(+)269 b Fm(x)337 b Fl(2)f Fr(\003)37102 17653 y Fd(c)37102 18393 y(n)24053 21104 y Fr(+)4726 b(if)1538 b(0)673 b Fm(<)337 b(y)380 b(;)202 b(y)313 b Fr(+)269 b Fm(x)337 b Fl(2)g Fr(\003)41175 21286 y Fd(n)24053 24115 y Fm(\033)43 b Fr(\()p Fm(y)g Fr(\))3354 b(if)1538 b Fm(y)380 b Fl(\024)674 b Fr(0)p Fm(;)538 b(y)313 b Fr(+)269 b Fm(x)337 b Fl(2)g Fr(\003)41175 24297 y Fd(n)41801 24115 y Fm(:)4230 27289 y Fr(No)-34 b(w,)468 b(pic)-34 b(k)454 b Fm(\017)420 b(>)h Fr(0,)466 b Fm(!)f Fl(2)420 b Fr(\012)455 b(and)g Fm(\033)464 b Fl(2)421 b Fr(\012)23130 26849 y Fd(<)p Fk(0)23130 27589 y Fd(\015)24388 27289 y Fr(.)689 b(Using)455 b(the)f(fact)h(that)h Fm(\033)36863 26849 y Fk(+)38104 27289 y Fr(is)f(a)f(p)34 b(oin)-34 b(t)455 b(of)g(con)-34 b(tin)g(uit)g(y)4230 28795 y(of)533 b Fm(D)34 b Fr(,)564 b(w)-34 b(e)533 b(c)-34 b(ho)34 b(ose)533 b Fm(n)14293 28977 y Fk(0)15351 28795 y Fr(suc)-34 b(h)534 b(that)g Fm(\030)56 b Fl(j)21892 28983 y Fk(\003)22545 29094 y Fg(n)23056 29293 y Fc(0)24179 28795 y Fr(=)550 b Fm(\033)26408 28355 y Fk(+)27196 28795 y Fl(j)27533 28983 y Fk(\003)28186 29094 y Fg(n)28697 29293 y Fc(0)29801 28795 y Fr(implies)532 b Fl(j)p Fm(D)34 b Fr(\()p Fm(\030)56 b Fr(\))355 b Fl(\000)g Fm(D)34 b Fr(\()p Fm(\033)40845 28355 y Fk(+)41632 28795 y Fr(\))p Fl(j)551 b(\024)g Fm(\017)p Fr(.)924 b(Remark)4230 30458 y(that)433 b Fm(\034)7414 30640 y Ft(\000)p Fd(x)8731 30458 y Fm(T)9607 30018 y Fd(!)9439 30817 y Fk(\003)10092 30928 y Fg(n)10714 30458 y Fr([)p Fm(x;)202 b(\034)12813 30640 y Fd(x)13397 30458 y Fm(\033)-24 b(;)202 b Fr(+])430 b(and)j Fm(\033)19436 30018 y Fk(+)20655 30458 y Fr(di\013er)e(only)h (on)g(the)g(set)g Fl(f)p Fm(y)426 b Fl(2)383 b Fe(Z)35876 30018 y Fd(d)36798 30458 y Fr(:)g Fm(x)287 b Fr(+)h Fm(y)426 b Fl(2)383 b Fr(\003)42782 30018 y Fd(c)42782 30758 y(n)43408 30458 y Fl(g)p Fr(.)621 b(Therefore,)4230 31963 y(the)393 b(di\013erence)f Fl(j)p Fm(D)34 b Fr(\()p Fm(\033)14301 31523 y Fk(+)15087 31963 y Fr(\))246 b Fl(\000)g Fm(D)34 b Fr(\()p Fm(\034)19032 32145 y Ft(\000)p Fd(x)20348 31963 y Fm(T)21224 31523 y Fd(!)21056 32323 y Fk(\003)21709 32434 y Fg(n)22331 31963 y Fr([)p Fm(x;)202 b(\034)24430 32145 y Fd(x)25014 31963 y Fm(\033)-24 b(;)202 b Fr(+]\))p Fl(j)391 b Fr(can)i(only)f(b)34 b(e)392 b(bigger)g(than)i Fm(\017)e Fr(for)g Fm(x)g Fr(suc)-34 b(h)394 b(that)4230 33469 y(\(\003)5543 33651 y Fd(n)6114 33774 y Fc(0)6900 33469 y Fl(\000)269 b Fm(x)p Fr(\))h Fl(\\)f Fr(\003)11465 33029 y Fd(c)11465 33768 y(n)12428 33469 y Fl(6)p Fr(=)336 b Fl(;)p Fr(.)4230 35456 y(Therefore,)18552 37703 y(1)p 17785 38245 V 17785 39355 a Fl(j)p Fr(\003)18964 39537 y Fd(n)19589 39355 y Fl(j)20059 37130 y Ff(\014)20059 37857 y(\014)20059 38584 y(\014)21112 37372 y(X)20800 39997 y Fd(x)p Ft(2)p Fk(\003)22610 40108 y Fg(n)23378 37542 y Ff(\002)23883 38523 y Fm(D)34 b Fr(\()p Fm(\034)25922 38705 y Ft(\000)p Fd(x)27238 38523 y Fm(T)28114 38023 y Fd(!)27946 38823 y Fk(\003)28599 38934 y Fg(n)29221 38523 y Fr([)p Fm(x;)202 b(\034)31320 38705 y Fd(x)31904 38523 y Fm(\033)-24 b(;)202 b Fr(+]\))269 b Fl(\000)g Fm(D)34 b Fr(\()p Fm(\033)38589 38023 y Fk(+)39376 37542 y Ff(\003)40217 37130 y(\014)40217 37857 y(\014)40217 38584 y(\014)11795 42273 y Fl(\024)336 b Fm(\017)270 b Fr(+)16285 41453 y(2)p 15517 41994 V 15517 43105 a Fl(j)p Fr(\003)16696 43287 y Fd(n)17322 43105 y Fl(j)18128 42273 y(jj)336 b Fm(D)370 b Fl(jj)21186 42455 y Ft(1)22855 40879 y Ff(\014)22855 41606 y(\014)22855 42334 y(\014)23259 42273 y Fl(f)p Fm(x)337 b Fl(2)g Fr(\003)26882 42455 y Fd(n)28181 42273 y Fr(:)673 b(\(\003)30504 42455 y Fd(n)31075 42578 y Fc(0)31861 42273 y Fl(\000)269 b Fm(x)p Fr(\))h Fl(\\)f Fr(\003)36426 41773 y Fd(c)36426 42573 y(n)37389 42273 y Fl(6)p Fr(=)336 b Fl(;)h(g)40217 40879 y Ff(\014)40217 41606 y(\014)40217 42334 y(\014)4230 45480 y Fr(and)405 b(this)g(is)f(less)f(than)j(2)p Fm(\017)e Fr(for)g Fm(n)g Fr(big)g(enough.)539 b(So)405 b(w)-34 b(e)404 b(obtain)h(that)16495 47727 y(1)p 15728 48268 V 15728 49379 a Fl(j)p Fr(\003)16907 49561 y Fd(n)17532 49379 y Fl(j)18002 47153 y Ff(\014)18002 47881 y(\014)18002 48608 y(\014)19055 47396 y(X)18742 50021 y Fd(x)p Ft(2)p Fk(\003)20552 50132 y Fg(n)21320 47565 y Ff(\002)21826 48547 y Fm(D)34 b Fr(\()p Fm(\034)23865 48729 y Ft(\000)p Fd(x)25181 48547 y Fm(T)26057 48047 y Fd(!)25889 48847 y Fk(\003)26542 48958 y Fg(n)27164 48547 y Fr([)p Fm(x;)202 b(\034)29263 48729 y Fd(x)29847 48547 y Fm(\033)-24 b(;)202 b Fr(+]\))268 b Fl(\000)h Fm(D)34 b Fr(\()p Fm(\033)36531 48047 y Fk(+)37318 48547 y Fr(\))37789 47565 y Ff(\003)38631 47153 y(\014)38631 47881 y(\014)38631 48608 y(\014)4230 52494 y Fr(con)-34 b(v)g(erges)480 b(to)h(zero)e(on)i(the)g(set)f(of)h (\012)22157 52054 y Fd(<)p Fk(0)22157 52794 y Fd(\015)23895 52494 y Fr(of)f(full)h Fm(\026)p Fr(-measure,)499 b(uniformly)480 b(in)g Fm(!)43 b Fr(.)768 b(By)480 b(dominated)4230 54000 y(con)-34 b(v)g(ergence,)404 b(w)-34 b(e)404 b(then)h(obtain)10554 57116 y(lim)10169 57843 y Fd(n)p Ft(!1)12824 57116 y Fr(sup)13428 58079 y Fd(!)15751 56296 y Fr(1)p 14984 56837 V 14984 57947 a Fl(j)p Fr(\003)16163 58129 y Fd(n)16788 57947 y Fl(j)17460 55466 y Ff(Z)18133 58215 y Fk(\012)19070 55722 y Ff(\014)19070 56449 y(\014)19070 57176 y(\014)19989 55964 y(X)19676 58589 y Fd(x)p Ft(2)p Fk(\003)21486 58700 y Fg(n)22254 56134 y Ff(\002)22759 57116 y Fm(D)34 b Fr(\()p Fm(\034)24798 57298 y Ft(\000)p Fd(x)26115 57116 y Fm(T)26991 56615 y Fd(!)26823 57415 y Fk(\003)27664 57116 y Fr([)p Fm(x;)202 b(\034)29763 57298 y Fd(x)30347 57116 y Fm(\033)-24 b(;)202 b Fr(+]\))268 b Fl(\000)h Fm(D)34 b Fr(\()p Fm(\033)37031 56615 y Fk(+)37818 57116 y Fr(\))38289 56134 y Ff(\003)38795 55722 y(\014)38795 56449 y(\014)38795 57176 y(\014)39199 57116 y Fm(\026)p Fr(\()p Fm(d\033)43 b Fr(\))338 b(=)e(0)4230 60894 y(whic)-34 b(h)405 b(implies)f(statemen)-34 b(t)405 b(1)f(of)h(the)g(lemma.)2681 63363 y(2.)606 b(Denote)17303 65202 y Fm(F)18082 65390 y Fk(\003)18735 65501 y Fg(n)19357 65202 y Fr(\()p Fm(\026;)202 b(\027)77 b Fr(\))338 b(=)24762 64382 y(1)p 23994 64924 V 23994 66034 a Fl(j)p Fr(\003)25173 66216 y Fd(n)25799 66034 y Fl(j)26470 63553 y Ff(Z)27144 66302 y Fk(\012)28081 65202 y Fr(log)30106 64382 y Fm(\027)77 b Fr(\()p Fm(\033)31946 64570 y Fk(\003)32599 64681 y Fg(n)33221 64382 y Fr(\))p 29981 64924 3836 49 v 29981 66034 a Fm(\027)g Fr(\(+)32071 66222 y Fk(\003)32724 66333 y Fg(n)33346 66034 y Fr(\))33950 65202 y Fm(\026)p Fr(\()p Fm(d\033)43 b Fr(\))p Fm(:)4230 68027 y Fr(Using)405 b Fm(\027)413 b Fl(2)337 b(G)72 b Fr(\()p Fm(\015)67 b Fr(\),)404 b(w)-34 b(e)405 b(obtain)14482 71436 y Fm(F)15261 71624 y Fk(\003)15914 71735 y Fg(n)16536 71436 y Fr(\()p Fm(\026;)202 b(\027)77 b Fr(\))337 b(=)21940 70616 y(1)p 21172 71157 2142 49 v 21172 72267 a Fl(j)p Fr(\003)22351 72449 y Fd(n)22977 72267 y Fl(j)23649 69786 y Ff(Z)24322 72535 y Fk(\012)25259 71436 y Fr(log)27263 69561 y Ff(R)27835 70963 y Fk(\012)28772 70537 y Fm(\015)29400 70725 y Fk(\003)30053 70836 y Fg(n)30674 70537 y Fr(\()p Fm(\033)43 b Fl(j)p Fm(!)g Fr(\))p Fm(\027)77 b Fr(\()p Fm(d!)43 b Fr(\))p 27160 71157 9480 49 v 27160 71302 a Ff(R)27732 72705 y Fk(\012)28669 72279 y Fm(\015)29297 72467 y Fk(\003)29950 72578 y Fg(n)30571 72279 y Fr(\(+)p Fl(j)p Fm(!)g Fr(\))p Fm(\027)77 b Fr(\()p Fm(d!)43 b Fr(\))36771 71436 y Fm(\026)p Fr(\()p Fm(d\033)g Fr(\))p Fm(:)25194 75417 y Fr(11)p eop %%Page: 12 12 12 11 bop 4230 1313 a Fr(Use)12896 3299 y(inf)12671 4106 y Fd(!)32 b Ft(2)p Fk(\012)15033 2479 y Fm(\015)15661 2667 y Fk(\003)16314 2778 y Fg(n)16935 2479 y Fr(\()p Fm(\033)43 b Fl(j)p Fm(!)g Fr(\))p 14930 3020 4923 49 v 14930 4130 a Fm(\015)15558 4318 y Fk(\003)16211 4429 y Fg(n)16832 4130 y Fr(\(+)p Fl(j)p Fm(!)g Fr(\))20658 3299 y Fl(\024)22511 1424 y Ff(R)23083 2826 y Fk(\012)24020 2401 y Fm(\015)24648 2589 y Fk(\003)25301 2700 y Fg(n)25922 2401 y Fr(\()p Fm(\033)g Fl(j)p Fm(!)g Fr(\))p Fm(\027)77 b Fr(\()p Fm(d!)43 b Fr(\))p 22407 3020 9480 49 v 22407 3166 a Ff(R)22980 4568 y Fk(\012)23917 4142 y Fm(\015)24545 4330 y Fk(\003)25198 4441 y Fg(n)25818 4142 y Fr(\(+)p Fl(j)p Fm(!)g Fr(\))p Fm(\027)77 b Fr(\()p Fm(d!)43 b Fr(\))32692 3299 y Fl(\024)723 b Fr(sup)34309 4342 y Fd(!)32 b Ft(2)p Fk(\012)36670 2479 y Fm(\015)37298 2667 y Fk(\003)37951 2778 y Fg(n)38572 2479 y Fr(\()p Fm(\033)43 b Fl(j)p Fm(!)g Fr(\))p 36567 3020 4923 49 v 36567 4130 a Fm(\015)37195 4318 y Fk(\003)37848 4429 y Fg(n)38469 4130 y Fr(\(+)p Fl(j)p Fm(!)g Fr(\))41622 3299 y Fm(:)4230 6090 y Fr(Let)404 b Fm(\017)337 b(>)f Fr(0)404 b(b)34 b(e)404 b(giv)-34 b(en)404 b(and)h Fm(!)381 b Fr(=)336 b Fm(!)43 b Fr(\()p Fm(n;)202 b(\033)-24 b(;)202 b(\017)p Fr(\),)404 b Fm(!)25350 5650 y Ft(0)25999 6090 y Fr(=)336 b Fm(!)28075 5650 y Ft(0)28387 6090 y Fr(\()p Fm(n;)202 b(\033)-24 b(;)202 b(\017)p Fr(\))403 b(suc)-34 b(h)405 b(that)12417 7319 y Ff(Z)13091 10068 y Fk(\012)14252 8969 y Fr(inf)14028 9776 y Fd(!)32 b Ft(2)p Fk(\012)16153 8969 y Fr(log)18157 8149 y Fm(\015)18785 8337 y Fk(\003)19438 8448 y Fg(n)20059 8149 y Fr(\()p Fm(\033)43 b Fl(j)p Fm(!)g Fr(\))p 18054 8690 V 18054 9801 a Fm(\015)18682 9989 y Fk(\003)19335 10100 y Fg(n)19956 9801 y Fr(\(+)p Fl(j)p Fm(!)g Fr(\))23109 8969 y Fm(\026)p Fr(\()p Fm(d\033)g Fr(\))338 b Fl(\025)27765 7319 y Ff(Z)28438 10068 y Fk(\012)29375 8969 y Fr(log)31379 8149 y Fm(\015)32007 8337 y Fk(\003)32660 8448 y Fg(n)33281 8149 y Fr(\()p Fm(\033)43 b Fl(j)p Fm(!)g Fr(\()p Fm(n;)202 b(\033)-24 b(;)202 b(\017)p Fr(\)\))p 31276 8690 8831 49 v 31276 9801 a Fm(\015)31904 9989 y Fk(\003)32557 10100 y Fg(n)33178 9801 y Fr(\(+)p Fl(j)p Fm(!)43 b Fr(\()p Fm(n;)202 b(\033)-24 b(;)202 b(\017)p Fr(\)\))40509 8969 y Fl(\000)269 b Fm(\017)4230 11793 y Fr(and)12094 12006 y Ff(Z)12767 14755 y Fk(\012)13753 13656 y Fr(sup)13704 14698 y Fd(!)32 b Ft(2)p Fk(\012)15830 13656 y Fr(log)17833 12836 y Fm(\015)18461 13024 y Fk(\003)19114 13135 y Fg(n)19735 12836 y Fr(\()p Fm(\033)43 b Fl(j)p Fm(!)g Fr(\))p 17730 13377 4923 49 v 17730 14487 a Fm(\015)18358 14675 y Fk(\003)19011 14786 y Fg(n)19632 14487 y Fr(\(+)p Fl(j)p Fm(!)g Fr(\))22785 13656 y Fm(\026)p Fr(\()p Fm(d\033)g Fr(\))338 b Fl(\024)27441 12006 y Ff(Z)28115 14755 y Fk(\012)29052 13656 y Fr(log)31056 12836 y Fm(\015)31684 13024 y Fk(\003)32337 13135 y Fg(n)32958 12836 y Fr(\()p Fm(\033)43 b Fl(j)p Fm(!)35299 12396 y Ft(0)35610 12836 y Fr(\()p Fm(n;)202 b(\033)-24 b(;)202 b(\017)p Fr(\)\))p 30952 13377 9142 49 v 30952 14487 a Fm(\015)31580 14675 y Fk(\003)32233 14786 y Fg(n)32854 14487 y Fr(\(+)p Fl(j)p Fm(!)35402 14137 y Ft(0)35714 14487 y Fr(\()p Fm(n;)g(\033)-24 b(;)202 b(\017)p Fr(\)\))40496 13656 y(+)269 b Fm(\017:)4230 16307 y Fr(No)-34 b(w)405 b(use)f(the)h(\014rst)g(item)f(of)g(the)h (lemma)e(and)i(c)-34 b(ho)34 b(ose)404 b Fm(N)537 b Fr(suc)-34 b(h)405 b(that)g(for)f(all)g Fm(n)336 b Fl(\025)g Fm(N)132 b Fr(,)12750 20070 y(sup)13354 21033 y Fd(!)14777 18676 y Ff(\014)14777 19404 y(\014)14777 20131 y(\014)16082 19250 y Fr(1)p 15314 19792 2142 49 v 15314 20902 a Fl(j)p Fr(\003)16493 21084 y Fd(n)17119 20902 y Fl(j)17790 18421 y Ff(Z)18463 21170 y Fk(\012)19400 20070 y Fr(log)21404 19250 y Fm(\015)22032 19438 y Fk(\003)22685 19549 y Fg(n)23306 19250 y Fr(\()p Fm(\033)43 b Fl(j)p Fm(!)g Fr(\))p 21301 19792 4923 49 v 21301 20902 a Fm(\015)21929 21090 y Fk(\003)22582 21201 y Fg(n)23203 20902 y Fr(\(+)p Fl(j)p Fm(!)g Fr(\))26356 20070 y Fm(\026)p Fr(\()p Fm(d\033)g Fr(\))270 b Fl(\000)30878 18421 y Ff(Z)31551 21170 y Fk(\012)32488 20070 y Fm(D)34 b Fr(\()p Fm(\033)34733 19570 y Fk(+)35520 20070 y Fr(\))p Fm(\026)p Fr(\()p Fm(d\033)43 b Fr(\))39030 18676 y Ff(\014)39030 19404 y(\014)39030 20131 y(\014)39772 20070 y Fl(\024)336 b Fm(\017:)4230 22637 y Fr(F)-101 b(or)404 b Fm(n)337 b Fl(\025)f Fm(N)132 b Fr(,)404 b(w)-34 b(e)404 b(obtain)11754 23720 y Ff(Z)12428 26469 y Fk(\012)13364 25370 y Fm(D)34 b Fr(\()p Fm(\033)15609 24869 y Fk(+)16396 25370 y Fr(\))p Fm(\026)p Fr(\()p Fm(d\033)43 b Fr(\))271 b Fl(\000)e Fr(2)p Fm(\017)674 b Fl(\024)f Fm(F)25556 25558 y Fk(\003)26209 25669 y Fg(n)26831 25370 y Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))674 b Fl(\024)31806 23720 y Ff(Z)32479 26469 y Fk(\012)33416 25370 y Fm(D)34 b Fr(\()p Fm(\033)35661 24869 y Fk(+)36448 25370 y Fr(\))p Fm(\026)p Fr(\()p Fm(d\033)43 b Fr(\))271 b(+)e(2)p Fm(\017:)p 49878 28465 522 731 v 3082 30726 a Fs(Pro)39 b(of)464 b(of)g(Theorem)h(3.6)2681 33331 y Fr(1.)606 b(Denote)17126 35171 y Fm(h)17824 35353 y Fd(n)18451 35171 y Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))337 b(:=)23990 34351 y(1)p 23222 34892 2142 49 v 23222 36003 a Fl(j)p Fr(\003)24401 36185 y Fd(n)25027 36003 y Fl(j)25699 34020 y Ff(X)25718 36565 y Fd(\033)26256 36714 y Fc(\003)26808 36825 y Fg(n)27652 35171 y Fm(\026)p Fr(\()p Fm(\033)29546 35359 y Fk(\003)30199 35470 y Fg(n)30820 35171 y Fr(\))202 b(log)33394 34351 y Fm(\026)p Fr(\()p Fm(\033)35288 34539 y Fk(\003)35941 34650 y Fg(n)36563 34351 y Fr(\))p 33394 34892 3641 49 v 33421 36003 a Fm(\027)77 b Fr(\()p Fm(\033)35261 36191 y Fk(\003)35914 36302 y Fg(n)36536 36003 y Fr(\))37167 35171 y Fm(:)4230 38584 y Fr(W)-101 b(e)392 b(recall)g(that)i(for)e Fm(\026)337 b Fl(2)g(M)17657 38074 y Fk(+)17657 38954 y(1)p Fd(;)p Fk(in)-26 b(v)19712 38584 y Fr(\(\012\),)395 b(the)f(limit)e(of)h Fm(h)29289 38766 y Fd(n)29916 38584 y Fr(\()p Fm(\026)p Fr(\))337 b(:=)g Fl(\000)35253 38106 y Fk(1)p 34618 38304 1742 49 v 34618 39018 a Ft(j)p Fk(\003)35532 39129 y Fg(n)36098 39018 y Ft(j)36694 37675 y Ff(P)37973 38942 y Fd(\033)38511 39091 y Fc(\003)39063 39202 y Fg(n)39942 38584 y Fm(\026)p Fr(\()p Fm(\033)41836 38772 y Fk(\003)42489 38883 y Fg(n)43111 38584 y Fr(\))202 b(log)219 b Fm(\026)p Fr(\()p Fm(\033)47446 38772 y Fk(\003)48099 38883 y Fg(n)48721 38584 y Fr(\))393 b(is)4230 40259 y(the)405 b Fj(Kolmo)-62 b(gor)g(ov-Sinai)430 b(entr)-62 b(opy)403 b Fr(of)h Fm(\026)h Fr(denoted)g Fm(h)p Fr(\()p Fm(\026)p Fr(\).)540 b(W)-101 b(e)403 b(write)9235 43138 y Fm(h)9933 43320 y Fd(n)10559 43138 y Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))338 b(=)538 b Fl(\000)p Fm(h)16704 43320 y Fd(n)17331 43138 y Fr(\()p Fm(\026)p Fr(\))270 b Fl(\000)21588 42318 y Fr(1)p 20820 42860 2142 49 v 20820 43970 a Fl(j)p Fr(\003)21999 44152 y Fd(n)22625 43970 y Fl(j)23296 41987 y Ff(X)23316 44532 y Fd(\033)23854 44681 y Fc(\003)24406 44792 y Fg(n)25249 43138 y Fm(\026)p Fr(\()p Fm(\033)27143 43326 y Fk(\003)27796 43437 y Fg(n)28418 43138 y Fr(\))202 b(log)31117 42318 y Fm(\027)77 b Fr(\()p Fm(\033)32957 42506 y Fk(\003)33610 42617 y Fg(n)34231 42318 y Fr(\))p 30992 42860 3836 49 v 30992 43970 a Fm(\027)g Fr(\(+)33082 44158 y Fk(\003)33735 44269 y Fg(n)34356 43970 y Fr(\))35432 43138 y Fl(\000)37747 42318 y Fr(1)p 36979 42860 2142 49 v 36979 43970 a Fl(j)p Fr(\003)38158 44152 y Fd(n)38784 43970 y Fl(j)39455 43138 y Fr(log)219 b Fm(\027)77 b Fr(\(+)43313 43326 y Fk(\003)43966 43437 y Fg(n)44587 43138 y Fr(\))p Fm(:)4230 46602 y Fr(When)460 b(condition)h(C1)f(holds,)475 b(the)460 b(asymptotic)h(b)34 b(eha)-34 b(vior)459 b(of)h(the)h(second)f(term)g(of)g(the)h(r.h.s.)705 b(is)4230 48108 y(giv)-34 b(en)405 b(b)-34 b(y)406 b(Lemma)f(3.12.)541 b(Hence,)405 b(the)h(relativ)-34 b(e)404 b(en)-34 b(trop)g(y)406 b(exists)f(if)g(and)h(only)f(if)g Fm(e)42356 47668 y Fk(+)42356 48408 y Fd(\027)43549 48108 y Fr(exists,)g(and)h(it)4230 49613 y(is)e(giv)-34 b(en)404 b(b)-34 b(y)405 b(\(3.7\).)2681 51955 y(2.)606 b(W)-101 b(e)441 b(consider)f Fm(\026)398 b Fl(2)g(G)14170 52137 y Fk(in)-26 b(v)15493 51955 y Fr(\()p Fm(\015)67 b Fr(\))442 b(suc)-34 b(h)441 b(that)h Fm(\026)p Fr(\(\012)25081 51515 y Fd(<)p Fk(0)25081 52255 y Fd(\015)26340 51955 y Fr(\))398 b(=)g(1)441 b(and)g(use)g(the)h (follo)-34 b(wing)441 b(decomp)34 b(osition)441 b(of)4230 53460 y(the)405 b(\014nite-v)-34 b(olume)405 b(relativ)-34 b(e)403 b(en)-34 b(trop)g(y:)8205 57626 y Fm(h)8903 57808 y Fd(n)9530 57626 y Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))337 b(=)33807 b(\(3.14\))9106 59281 y(1)p 8338 59822 V 8338 60933 a Fl(j)p Fr(\003)9517 61115 y Fd(n)10143 60933 y Fl(j)10814 58950 y Ff(X)10834 61495 y Fd(\033)11372 61644 y Fc(\003)11924 61755 y Fg(n)12767 60101 y Fm(\026)p Fr(\()p Fm(\033)14661 60289 y Fk(\003)15314 60400 y Fg(n)15936 60101 y Fr(\))202 b(log)18635 59281 y Fm(\026)p Fr(\()p Fm(\033)20529 59469 y Fk(\003)21182 59580 y Fg(n)21804 59281 y Fr(\))p 18510 59822 3891 49 v 18510 60933 a Fm(\026)p Fr(\(+)20654 61121 y Fk(\003)21307 61232 y Fg(n)21929 60933 y Fr(\))22802 60101 y Fl(\000)24915 59281 y Fr(1)p 24147 59822 2142 49 v 24147 60933 a Fl(j)p Fr(\003)25326 61115 y Fd(n)25952 60933 y Fl(j)26624 58950 y Ff(X)26643 61495 y Fd(\033)27181 61644 y Fc(\003)27733 61755 y Fg(n)28577 60101 y Fm(\026)p Fr(\()p Fm(\033)30471 60289 y Fk(\003)31124 60400 y Fg(n)31745 60101 y Fr(\))g(log)34444 59281 y Fm(\027)77 b Fr(\()p Fm(\033)36284 59469 y Fk(\003)36937 59580 y Fg(n)37559 59281 y Fr(\))p 34319 59822 3836 49 v 34319 60933 a Fm(\027)g Fr(\(+)36409 61121 y Fk(\003)37062 61232 y Fg(n)37684 60933 y Fr(\))38557 60101 y(+)40670 59281 y(1)p 39902 59822 2142 49 v 39902 60933 a Fl(j)p Fr(\003)41081 61115 y Fd(n)41707 60933 y Fl(j)42378 60101 y Fr(log)44279 59281 y Fm(\026)p Fr(\(+)46423 59469 y Fk(\003)47076 59580 y Fg(n)47698 59281 y Fr(\))p 44279 59822 3891 49 v 44306 60933 a Fm(\027)g Fr(\(+)46396 61121 y Fk(\003)47049 61232 y Fg(n)47671 60933 y Fr(\))48302 60101 y Fm(:)4230 63565 y Fr(By)395 b(Lemma)f(3.12,)j(in)e(the)h(limit) f Fm(n)336 b Fl(!)h(1)p Fr(,)396 b(the)g(\014rst)g(t)-34 b(w)g(o)397 b(terms)e(of)g(the)h(r.h.s.)535 b(are)395 b(functions)h(of)g Fm(\015)4230 65071 y Fr(rather)338 b(than)h(functions)h(of)e Fm(\026;)202 b(\027)414 b Fl(2)337 b(G)21191 65253 y Fk(in)-26 b(v)22514 65071 y Fr(\()p Fm(\015)67 b Fr(\))338 b(and)h(cancel)e(out.)517 b(Hence,)351 b(the)338 b(relativ)-34 b(e)337 b(en)-34 b(trop)g(y)339 b(exists)4230 66576 y(if)390 b(and)h(only)f(if)g(the)h(third)g(term)f (con)-34 b(v)g(erges.)533 b(Using)391 b(Item)f(1)g(\(existence)g(of)g (relativ)-34 b(e)389 b(en)-34 b(trop)g(y\),)394 b(w)-34 b(e)4230 68082 y(obtain)405 b(the)g(existence)e(of)i(the)f(limit)g (\(3.8\))g(and)h(the)g(equalit)-34 b(y)18851 70961 y Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))338 b(=)721 b(lim)23851 71689 y Fd(n)p Ft(!1)27608 70141 y Fr(1)p 26639 70682 2546 49 v 26639 71793 a Fl(j)202 b Fr(\003)28020 71975 y Fd(n)28847 71793 y Fl(j)29519 70961 y Fr(log)31420 70141 y Fm(\026)p Fr(\(+)33564 70329 y Fk(\003)34217 70440 y Fg(n)34838 70141 y Fr(\))p 31420 70682 3891 49 v 31447 71793 a Fm(\027)77 b Fr(\(+)33537 71981 y Fk(\003)34190 72092 y Fg(n)34811 71793 y Fr(\))35443 70961 y Fm(:)25194 75417 y Fr(12)p eop %%Page: 13 13 13 12 bop 1200 1313 a Fs(Pro)39 b(of)464 b(of)h(Theorem)f(3.9)2681 4406 y Fr(1.)606 b(This)417 b(is)g(direct)f(consequence)g(of)h(Theorem) f(3.6)h(and)g(\(3.8\):)564 b(under)417 b(the)g(conditions)g Fm(C)87 b Fr(1)417 b(and)g Fm(C)87 b Fr(2,)4230 5912 y Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))338 b(=)f(0.)2681 8392 y(2.)606 b(The)405 b(existence)e(of)i(the)f(relativ) -34 b(e)403 b(en)-34 b(trop)g(y)405 b(pro)-34 b(v)g(es)405 b(that)g Fm(e)31013 7953 y Fk(+)31013 8692 y Fd(\027)32205 8392 y Fr(exists)f(and)g(is)g(giv)-34 b(en)404 b(b)-34 b(y)17753 11753 y Fm(e)18317 11252 y Fk(+)18317 12052 y Fd(\027)19441 11753 y Fr(=)337 b Fm(h)p Fr(\()p Fm(\026)p Fr(\))270 b(+)24574 10103 y Ff(Z)25988 11753 y Fr(log)27888 10933 y Fm(\015)28516 11115 y Fk(0)29042 10933 y Fr(\()p Fm(\033)30249 10493 y Fk(+)31036 10933 y Fl(j)p Fm(\033)32109 10493 y Fk(+)32896 10933 y Fr(\))p 27888 11474 5480 49 v 28178 12584 a Fm(\015)28806 12766 y Fk(0)29332 12584 y Fr(\(+)p Fl(j)p Fm(\033)31819 12234 y Fk(+)32606 12584 y Fr(\))33500 11753 y Fm(\026)p Fr(\()p Fm(d\033)43 b Fr(\))p Fm(:)4230 15128 y Fr(Com)-34 b(bined)406 b(with)f(C2)f(this)h (pro)-34 b(v)g(es)404 b Fm(e)21309 14688 y Fk(+)21309 15427 y Fd(\026)22434 15128 y Fr(=)336 b Fm(e)24277 14688 y Fk(+)24277 15427 y Fd(\027)25065 15128 y Fr(.)2681 17730 y(3.)606 b(Consider)310 b(an)-34 b(y)311 b(other)f(measure)f Fm(\013)342 b Fl(2)336 b(M)22812 17221 y Fk(+)22812 18101 y(1)p Fd(;)p Fk(in)-26 b(v)25176 17730 y Fr(suc)-34 b(h)311 b(that)g(C1)f(holds.)508 b(The)310 b(existence)f(of)h(the)g(relativ)-34 b(e)4230 19359 y(en)g(trop)g(y)405 b Fm(h)p Fr(\()p Fm(\013)t Fl(j)p Fm(\026)p Fr(\))i(follo)-34 b(ws)404 b(b)-34 b(y)405 b(com)-34 b(bining)405 b(the)f(existence)g(of)g Fm(e)33215 18919 y Fk(+)33215 19658 y Fd(\027)34407 19359 y Fr(with)h(Theorem)f (3.6,)f(and)15246 22722 y Fm(h)p Fr(\()p Fm(\013)t Fl(j)p Fm(\027)77 b Fr(\))338 b(=)f Fm(e)20860 22222 y Fk(+)20860 23022 y Fd(\027)21917 22722 y Fl(\000)269 b Fm(h)p Fr(\()p Fm(\013)t Fr(\))i Fl(\000)27031 21072 y Ff(Z)28445 22722 y Fr(log)30346 21902 y Fm(\015)30974 22084 y Fk(0)31499 21902 y Fr(\()p Fm(\033)32706 21462 y Fk(+)33494 21902 y Fl(j)p Fm(\033)34567 21462 y Fk(+)35354 21902 y Fr(\))p 30346 22443 V 30636 23554 a Fm(\015)31264 23736 y Fk(0)31789 23554 y Fr(\(+)p Fl(j)p Fm(\033)34276 23203 y Fk(+)35064 23554 y Fr(\))35958 22722 y Fm(\013)t Fr(\()p Fm(d\033)43 b Fr(\))p Fm(:)4230 26097 y Fr(If)404 b(moreo)-34 b(v)g(er)404 b Fm(\013)409 b Fr(satis\014es)c(C2,)f(w)-34 b(e)404 b(also)g(obtain)h(that)h Fm(e)29716 25657 y Fk(+)29716 26397 y Fd(\013)30907 26097 y Fr(exists)e(and)h(equals)f Fm(e)40852 25657 y Fk(+)40852 26397 y Fd(\027)41639 26097 y Fr(.)1200 29655 y Fi(3.4)1495 b(Generalization)1200 32265 y Fr(In)566 b(the)h(h)-34 b(yp)34 b(othesis)567 b(of)f(the)h(theorems)f(ab)34 b(o)-34 b(v)g(e,)606 b(the)567 b(plus-con\014guration)h(pla)-34 b(ys)566 b(a)g(particular)g(role)f(of) 1200 33770 y(telescoping)505 b(reference)e(con\014guration.)842 b(Without)506 b(to)34 b(o)505 b(m)-34 b(uc)g(h)506 b(e\013ort,)530 b(w)-34 b(e)505 b(obtain)h(the)f(follo)-34 b(wing)506 b(gen-)1200 35276 y(eralization)455 b(where)g(w)-34 b(e)456 b(telescop)34 b(e)455 b(w.r.t)g(a)h(random)g(con\014guration)h Fm(\030)511 b Fr(c)-34 b(hosen)457 b(from)e(some)h(translation)1200 36781 y(in)-34 b(v)-67 b(arian)-34 b(t)434 b(measure)f Fm(\025)p Fr(.)625 b(Results)434 b(of)g(the)f(previous)g(section)g(are) g(reco)-34 b(v)g(ered)432 b(b)-34 b(y)434 b(c)-34 b(ho)34 b(osing)434 b Fm(\025)385 b Fr(=)g Fm(\016)46563 36963 y Fk(+)47350 36781 y Fr(.)625 b(The)1200 38287 y(generalization)493 b(to)h(a)g(random)h(telescoping)e(con\014guration)j(will)d(b)34 b(e)493 b(natural)i(in)f(the)g(con)-34 b(text)494 b(of)h(join)-34 b(t)1200 39792 y(measures)404 b(of)g(disordered)h(spin)f(systems)g(in)h (Section)f(6.)3082 41619 y(F)-101 b(or)404 b(an)-34 b(y)404 b Fm(\030)56 b(;)202 b(\033)380 b Fl(2)337 b Fr(\012,)404 b(w)-34 b(e)404 b(de\014ne)h(the)g(concatenated)g(con\014guration)g Fm(\033)35298 41179 y Fd(\030)35802 41619 y Fr(:)15739 45682 y Fl(8)p Fm(x)337 b Fl(2)g Fe(Z)19395 45181 y Fd(d)19935 45682 y Fm(;)202 b(\033)21210 45181 y Fd(\030)21713 45682 y Fr(\()p Fm(x)p Fr(\))337 b(=)24965 43197 y Ff(8)24965 44288 y(<)24965 46470 y(:)26596 44174 y Fm(\033)43 b Fr(\()p Fm(x)p Fr(\))674 b(if)1104 b(x)673 b Fl(\024)g Fr(0)26596 47185 y Fm(\030)56 b Fr(\()p Fm(x)p Fr(\))674 b(if)1104 b(x)673 b Fm(>)g Fr(0)p Fm(:)1200 50107 y Fr(and)497 b(the)h(set)f(\012)8690 49517 y Fd(\030)35 b(;<)p Fk(0)8690 50265 y Fd(\015)11154 50107 y Fr(to)497 b(b)34 b(e)496 b(the)i(subset)f(of)h(\012)331 b Fl(\002)g Fr(\012)497 b(of)g(the)g(con\014gurations)i(\()p Fm(\033)-24 b(;)202 b(\030)56 b Fr(\))497 b(suc)-34 b(h)497 b(that)h(the)f(new)1200 51613 y(con\014guration)405 b Fm(\033)9279 51173 y Fd(\030)10187 51613 y Fr(is)f(a)g(go)34 b(o)g(d)404 b(con\014guration)h(for)g Fm(\015)67 b Fr(:)16438 54273 y(\012)17313 53773 y Fd(\030)35 b(;<)p Fk(0)17313 54573 y Fd(\015)19617 54273 y Fr(=)20897 53291 y Ff(\010)21604 54273 y Fr(\()p Fm(\033)-24 b(;)202 b(\030)56 b Fr(\))337 b Fl(2)g Fr(\012)269 b Fl(\002)g Fr(\012)p Fm(;)539 b(\033)30665 53773 y Fd(\030)31506 54273 y Fl(2)336 b Fr(\012)33525 54455 y Fd(\015)34118 53291 y Ff(\011)34825 54273 y Fm(:)1200 56933 y Fr(W)-101 b(e)403 b(also)h(generalize)f Fm(e)11875 56494 y Fk(+)11875 57233 y Fd(\027)13067 56933 y Fr(and)i(denotes)17094 60149 y Fm(e)17658 59648 y Fd(\025)17658 60448 y(\027)18600 60149 y Fr(=)336 b Fl(\000)458 b Fr(lim)21024 61064 y Fk(\003)p Ft(")p Fh(Z)22737 60812 y Fg(d)24009 59329 y Fr(1)p 23554 59870 1516 49 v 23554 60980 a Fl(j)p Fr(\003)p Fl(j)25404 58499 y Ff(Z)26077 61248 y Fk(\012)27014 60149 y Fr(log)219 b Fm(\027)77 b Fr(\()p Fm(\030)30459 60337 y Fk(\003)31167 60149 y Fr(\))p Fm(\025)p Fr(\()p Fm(d\030)56 b Fr(\))12798 b(\(3.15\))1200 63465 y(pro)-34 b(vided)404 b(this)h(limit)f(exists.)3082 65292 y(W)-101 b(e)547 b(consider)h(a)f(sp)34 b(eci\014cation)548 b Fm(\015)67 b Fr(,)584 b(measures)547 b Fm(\027)653 b Fl(2)577 b(G)28706 65474 y Fk(in)-26 b(v)30029 65292 y Fr(\()p Fm(\015)67 b Fr(\),)584 b Fm(\026;)202 b(\025)576 b Fl(2)g(M)37979 64782 y Fk(+)37979 65662 y(1)p Fd(;)p Fk(in)-26 b(v)40034 65292 y Fr(,)583 b(and)549 b(the)f(follo)-34 b(wing)1200 66797 y(conditions:)1200 69568 y Fs(C'1)606 b Fm(\025)269 b Fl(\012)g Fm(\026)p Fr(\(\012)8161 68979 y Fd(\030)35 b(;<)p Fk(0)8161 69726 y Fd(\015)10130 69568 y Fr(\))337 b(=)f(1.)1200 72049 y Fs(C'2)606 b Fr(lim)5580 72345 y Fk(\003)p Ft(")p Fh(Z)7293 72093 y Fg(d)8518 71572 y Fk(1)p 8166 71770 1176 49 v 8166 72483 a Ft(j)p Fk(\003)p Ft(j)9676 71073 y Ff(R)10248 72475 y Fk(\012)11185 71067 y Ff(\000)11943 72049 y Fr(log)13844 71501 y Fd(d\026)14894 71650 y Fc(\003)p 13844 71771 1658 49 v 13896 72467 a Fd(d\027)14843 72616 y Fc(\003)15634 71067 y Ff(\001)16190 72049 y Fr(\()p Fm(\030)17191 72237 y Fk(\003)17899 72049 y Fr(\))p Fm(\025)p Fr(\()p Fm(d\030)20709 72237 y Fk(\003)21418 72049 y Fr(\))337 b(=)g(0.)25194 75417 y(13)p eop %%Page: 14 14 14 13 bop 1200 1313 a Fr(The)393 b(follo)-34 b(wing)393 b(theorems)g(are)f(the)h(straigh)-34 b(tforw)g(ard)395 b(generalizations)d(of)g(Theorem)h(3.6)f(and)i(3.10,)g(and)1200 2819 y(their)404 b(pro)34 b(ofs)404 b(follo)-34 b(w)405 b(the)g(same)f(lines.)1200 5985 y Fs(Theorem)464 b(3.16)606 b Fj(Under)434 b(the)e(c)-62 b(ondition)432 b(C'1,)2633 8819 y(1.)605 b Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))435 b Fj(exists)d(if)h(and)h(only)e(if)i Fm(e)19273 8379 y Fd(\025)19273 9118 y(\027)20311 8819 y Fj(exists)e(and)h(then) 13458 12325 y Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))338 b(=)e Fm(e)19022 11824 y Fd(\025)19022 12624 y(\027)19896 12325 y Fl(\000)269 b Fm(h)p Fr(\()p Fm(\026)p Fr(\))i Fl(\000)24961 10675 y Ff(Z)25635 13424 y Fk(\012)p Ft(\002)p Fk(\012)27983 12325 y Fr(log)29884 11505 y Fm(\015)30512 11687 y Fk(0)31037 11505 y Fr(\()p Fm(\033)32244 11065 y Fd(\030)32748 11505 y Fl(j)p Fm(\033)33821 11065 y Fd(\030)34324 11505 y Fr(\))p 29884 12046 4913 49 v 30210 13157 a Fm(\015)30838 13339 y Fk(0)31364 13157 y Fr(\()p Fm(\030)56 b Fl(j)p Fm(\033)33494 12806 y Fd(\030)33998 13157 y Fr(\))34929 12325 y Fm(\026)p Fr(\()p Fm(d\033)43 b Fr(\))p Fm(\025)p Fr(\()p Fm(d\030)56 b Fr(\))p Fm(:)6131 b Fr(\(3.17\))2633 16398 y Fj(2.)605 b(If)434 b(mor)-62 b(e)g(over)432 b Fm(\026)337 b Fl(2)g(G)13671 16580 y Fk(in)-26 b(v)14994 16398 y Fr(\()p Fm(\015)67 b Fr(\))434 b Fj(and)f Fm(e)19983 15958 y Fd(\025)19983 16697 y(\027)21022 16398 y Fj(exists,)f(then)16094 19691 y Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))337 b(=)593 b(lim)21094 20606 y Fk(\003)p Ft(")p Fh(Z)22807 20354 y Fg(d)24079 18871 y Fr(1)p 23624 19412 1516 49 v 23624 20522 a Fl(j)p Fr(\003)p Fl(j)25474 18041 y Ff(Z)26148 20790 y Fk(\012)27084 18709 y Ff(\000)27842 19691 y Fr(log)29743 18871 y Fm(d\026)31104 19059 y Fk(\003)p 29743 19412 2070 49 v 29809 20522 a Fm(d\027)31039 20710 y Fk(\003)31944 18709 y Ff(\001)32500 19691 y Fr(\()p Fm(\030)33501 19879 y Fk(\003)34210 19691 y Fr(\))p Fm(\025)p Fr(\()p Fm(d\030)37020 19879 y Fk(\003)37728 19691 y Fr(\))p Fm(:)1200 23725 y Fs(Theorem)464 b(3.18)606 b Fj(Consider)499 b Fm(\026)457 b Fl(2)f(M)19334 23216 y Fk(+)19334 24096 y(1)p Fd(;)p Fk(in)-26 b(v)21389 23725 y Fm(;)202 b(\015)566 b Fj(a)500 b(sp)-62 b(e)g(ci\014c)g(ation,)513 b Fm(\027)533 b Fl(2)457 b(G)34598 23907 y Fk(in)-26 b(v)35921 23725 y Fr(\()p Fm(\015)67 b Fr(\))500 b Fj(such)e(that)h Fm(e)43990 23286 y Fd(\025)43990 24025 y(\027)45094 23725 y Fj(exists)f(and)1200 25231 y(c)-62 b(onditions)432 b(C'1)h(and)g(C'2)g(ar)-62 b(e)433 b(true.)556 b(Then)2633 28065 y(1.)605 b Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))435 b Fj(exists)d(and)h(is)g(given)g(by)g(\(3.17\).)2633 30566 y(2.)605 b Fm(\026)337 b Fl(2)g(G)7163 30748 y Fk(in)-26 b(v)8486 30566 y Fr(\()p Fm(\015)67 b Fr(\))434 b Fj(implies)e Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))338 b(=)f(0)p Fj(.)1200 34718 y Fo(4)1793 b(V)-149 b(ariational)598 b(principle)h(for)f(quasilo)50 b(cal)598 b(measures)1200 37756 y Fr(The)527 b(usual)g(w)-34 b(a)g(y)527 b(to)g(pro)-34 b(v)g(e)526 b Fm(\026)541 b Fl(2)g(G)18024 37938 y Fk(in)-26 b(v)19347 37756 y Fr(\()p Fm(\015)67 b Fr(\))541 b Fl(\()-202 b(\))541 b Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))541 b(=)g(0)526 b(in)h(the)f(Gibbsian)i(con) -34 b(text)527 b(uses)f(that)i Fm(\015)1200 39261 y Fr(is)441 b(a)h(sp)34 b(eci\014cation)441 b(asso)34 b(ciated)442 b(with)g(a)g(translation)g(in)-34 b(v)-67 b(arian)-34 b(t)442 b(and)g(UA)-34 b(C)443 b(p)34 b(oten)-34 b(tial)442 b(\010,)451 b(and)442 b(go)34 b(es)441 b(via)1200 40767 y(existence)533 b(and)i(b)34 b(oundary)534 b(condition)h(indep)34 b(endence)534 b(of)g(pressure)g(\(see)g([11)o(]\).)928 b(Since)534 b(for)g(a)f(general)1200 42272 y(quasilo)34 b(cal)510 b(sp)34 b(eci\014cation)510 b Fm(\015)578 b Fr(w)-34 b(e)511 b(cannot)h(rely)d(on)i(the)g(existence)f(of)h(suc)-34 b(h)512 b(a)f(p)34 b(oten)-34 b(tial)511 b(\(see)g([14])f(and)1200 43777 y(op)34 b(en)460 b(problem)g(in)g([6]\),)474 b(w)-34 b(e)460 b(sho)-34 b(w)462 b(here)d(that)i(the)g(w)-34 b(eak)g(er)460 b(prop)34 b(ert)-34 b(y)460 b(of)h(uniform)f(con)-34 b(v)g(ergence)460 b(of)g(the)1200 45283 y(v)-67 b(acuum)371 b(p)34 b(oten)-34 b(tial)372 b(whic)-34 b(h)372 b(can)f(b)34 b(e)371 b(asso)34 b(ciated)371 b(to)h(the)f(quasilo)34 b(cal)370 b Fm(\015)438 b Fr(\(see)371 b([14]\))g(su\016ces)g(to)h (obtain)g(zero)1200 46788 y(relativ)-34 b(e)403 b(en)-34 b(trop)g(y)-101 b(.)1200 49954 y Fs(Theorem)464 b(4.1)606 b Fj(L)-62 b(et)479 b Fm(\015)547 b Fj(b)-62 b(e)478 b(a)h(tr)-62 b(anslation)478 b(invariant)g(quasilo)-62 b(c)g(al)475 b(sp)-62 b(e)g(ci\014c)g(ation,)488 b Fm(\027)496 b Fl(2)420 b(G)42602 50136 y Fk(in)-26 b(v)43925 49954 y Fr(\()p Fm(\015)67 b Fr(\))480 b Fj(and)f Fm(\026)420 b Fl(2)1200 51460 y(M)2656 50950 y Fk(+)2656 51830 y(1)p Fd(;)p Fk(in)-26 b(v)4710 51460 y Fj(.)558 b(Then)433 b Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))435 b Fj(exists)d(for)h(al)62 b(l)433 b Fm(\026)337 b Fl(2)g(M)23275 50950 y Fk(+)23275 51830 y(1)p Fd(;)p Fk(in)-26 b(v)25763 51460 y Fj(and)18232 54183 y Fm(\026)337 b Fl(2)g(G)21165 54365 y Fk(in)-26 b(v)22488 54183 y Fr(\()p Fm(\015)67 b Fr(\))539 b Fl(\()-202 b(\))539 b Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))338 b(=)e(0)p Fm(:)1200 57349 y Fj(Pr)-62 b(o)g(of)p Fr(.)605 b(The)415 b(implication)g(of)g(the)h (left)f(\()p Fj(the)442 b(se)-62 b(c)g(ond)442 b(p)-62 b(art)p Fr(\))414 b(is)h(pro)-34 b(v)g(ed)415 b(in)g([11].)571 b(T)-101 b(o)415 b(pro)-34 b(v)g(e)415 b(the)g(\014rst)h(part,)1200 58854 y(w)-34 b(e)422 b(need)g(the)h(follo)-34 b(wing)422 b(lemma)g(to)g(c)-34 b(hec)g(k)422 b(h)-34 b(yp)34 b(othesis)422 b(of)h(Theorem)f(3.10.)591 b(Condition)424 b(C2)e(is)f(trivially)1200 60360 y(true)404 b(when)h Fm(\015)471 b Fr(is)404 b(quasilo)34 b(cal)403 b(\(\012)16187 59920 y Fd(<)p Fk(0)16187 60659 y Fd(\015)17782 60360 y Fr(=)337 b(\012\).)1200 63526 y Fs(Lemma)464 b(4.2)606 b Fj(F)-93 b(or)507 b(al)62 b(l)506 b Fm(\026;)202 b(\027)547 b Fl(2)470 b(G)17088 63708 y Fk(in)-26 b(v)18411 63526 y Fr(\()p Fm(\015)67 b Fr(\))508 b Fj(with)e Fm(\015)574 b Fj(tr)-62 b(anslation)506 b(invariant)g(and)h(quasilo)-62 b(c)g(al,)521 b Fm(e)44825 63086 y Fk(+)44825 63825 y Fd(\027)45613 63526 y Fm(;)202 b(e)46716 63086 y Fk(+)46716 63825 y Fd(\026)48010 63526 y Fj(exist)1200 65031 y(and)19311 66871 y Fr(lim)18927 67598 y Fd(n)p Ft(!1)22482 66051 y Fr(1)p 21714 66592 2142 49 v 21714 67702 a Fl(j)p Fr(\003)22893 67884 y Fd(n)23519 67702 y Fl(j)24191 66871 y Fr(log)26091 66051 y Fm(\026)p Fr(\(+)28235 66239 y Fk(\003)28888 66350 y Fg(n)29510 66051 y Fr(\))p 26091 66592 3891 49 v 26118 67702 a Fm(\027)77 b Fr(\(+)28208 67890 y Fk(\003)28861 68001 y Fg(n)29483 67702 y Fr(\))30451 66871 y(=)336 b(0)p Fm(:)25194 75417 y Fr(14)p eop %%Page: 15 15 15 14 bop 1200 1313 a Fj(Pr)-62 b(o)g(of)p Fr(.)605 b(Kozlo)-34 b(v)399 b(\(1974\))i(pro)-34 b(v)g(es)400 b(that)i(to)e(an)-34 b(y)400 b(translation)h(in)-34 b(v)-67 b(arian)-34 b(t)400 b(quasilo)34 b(cal)399 b(sp)34 b(eci\014cation)400 b Fm(\015)467 b Fr(there)1200 2819 y(corresp)34 b(onds)445 b(a)f(translation)i(in)-34 b(v)-67 b(arian)-34 b(t)444 b(uniformly)h(con)-34 b(v)g(ergen)g(t)446 b(v)-67 b(acuum)444 b(p)34 b(oten)-34 b(tial)445 b(\010)g(suc)-34 b(h)446 b(that)f Fm(\015)471 b Fr(=)1200 4324 y Fm(\015)1895 3884 y Fk(\010)2630 4324 y Fr(.)3082 6144 y(By)403 b(uniform)i(con)-34 b(v)g(ergence,)403 b(w)-34 b(e)405 b(ha)-34 b(v)g(e)17328 8992 y(lim)17072 9907 y Fk(\003)p Ft(")p Fh(Z)18785 9655 y Fg(d)19469 8992 y Fr(sup)20097 9955 y Fd(\033)21631 7598 y Ff(\014)21631 8325 y(\014)21631 9052 y(\014)23998 7840 y(X)22035 10525 y Fd(A)p Ft(3)p Fk(0)p Fd(;A)p Ft(\\)p Fk(\003)26083 10272 y Fg(c)26510 10525 y Ft(6)p Fk(=)p Ft(;)27914 8992 y Fr(\010)28789 9180 y Fd(A)29550 8992 y Fr(\()p Fm(\033)43 b Fr(\))31228 7598 y Ff(\014)31228 8325 y(\014)31228 9052 y(\014)32306 8992 y Fr(=)336 b(0)p Fm(:)13380 b Fr(\(4.3\))1200 12946 y(Remark)432 b(that)i(in)f(\(4.3\),) 440 b(the)433 b(absolute)h(v)-67 b(alue)432 b(is)g Fj(outside)521 b Fr(the)433 b(sum,)440 b(i.e.,)e(\(4.3\))c(means)f(that)h(the)f (series)1200 13542 y Ff(P)2479 14809 y Fd(A)p Ft(3)p Fk(0)4539 14451 y Fr(\010)5414 14639 y Fd(A)6175 14451 y Fr(\()p Fm(\033)43 b Fr(\))425 b(is)f(con)-34 b(v)g(ergen)g(t)426 b(in)e(the)h(sup-norm)h(top)34 b(ology)424 b(on)h Fm(C)87 b Fr(\(\012\),)431 b(but)425 b(not)h(necessarily)d Fj(absolutely)1200 15957 y(c)-62 b(onver)g(gent)p Fr(.)533 b(W)-101 b(e)393 b(can)h(de\014ne)g(a)f(Hamiltonian)h(and)g(a)g(partition)g(function)h (for)f(an)-34 b(y)394 b(\003)336 b Fl(2)h(S)91 b Fm(;)202 b(\021)43 b(;)202 b(\033)379 b Fl(2)337 b Fr(\012,)396 b(as)1200 17462 y(usual:)10720 18967 y Fm(H)11826 18378 y Fd(\021)11728 19349 y Fk(\003)12436 18967 y Fr(\()p Fm(\033)43 b Fr(\))337 b(=)16449 17816 y Ff(X)15731 20500 y Fd(A)p Ft(\\)p Fk(\003)p Ft(6)p Fk(=)p Ft(;)19120 18967 y Fr(\010)19995 19155 y Fd(A)20755 18967 y Fr(\()p Fm(\033)21919 19155 y Fk(\003)22627 18967 y Fm(\021)23229 19155 y Fk(\003)23882 18903 y Fg(c)24363 18967 y Fr(\))1348 b(and)f Fm(Z)30308 19155 y Fk(\003)31016 18967 y Fr(\()p Fm(!)43 b Fr(\))339 b(=)34436 17816 y Ff(X)34373 20441 y Fd(\033)32 b Ft(2)p Fk(\012)36452 18967 y Fm(e)37016 18467 y Ft(\000)p Fd(H)38589 18154 y Fg(!)38521 18768 y Fc(\003)39186 18467 y Fk(\()p Fd(\033)g Fk(\))40543 18967 y Fm(:)7028 b Fr(\(4.4\))1200 22411 y(Lemma)404 b(4.2)f(is)h(no)-34 b(w)406 b(a)e(direct)f (consequence)h(of)h(the)f(follo)-34 b(wing)1200 25449 y Fs(Lemma)464 b(4.5)2039 b Fj(1.)17576 27160 y Fr(lim)17192 27888 y Fd(n)p Ft(!1)20040 27160 y Fr(sup)19846 28123 y Fd(!)32 b(;\021)g(;\033)23162 26340 y Fr(1)p 22394 26882 2142 49 v 22394 27992 a Fl(j)p Fr(\003)23573 28174 y Fd(n)24199 27992 y Fl(j)24668 26130 y Ff(\014)24668 26857 y(\014)25072 27160 y Fm(H)26178 26571 y Fd(\021)26080 27542 y Fk(\003)26733 27653 y Fg(n)27354 27160 y Fr(\()p Fm(\033)43 b Fr(\))271 b Fl(\000)e Fm(H)31621 26660 y Fd(!)31523 27460 y Fk(\003)32176 27571 y Fg(n)32797 27160 y Fr(\()p Fm(\033)43 b Fr(\))34475 26130 y Ff(\014)34475 26857 y(\014)35216 27160 y Fr(=)337 b(0)p Fm(:)10469 b Fr(\(4.6\))2633 30453 y Fj(2.)19837 32293 y Fr(lim)19452 33020 y Fd(n)p Ft(!1)22107 32293 y Fr(sup)22328 33256 y Fd(!)32 b(;\021)25034 31473 y Fr(1)p 24267 32014 V 24267 33124 a Fl(j)p Fr(\003)25446 33306 y Fd(n)26072 33124 y Fl(j)26743 32293 y Fr(log)28644 31473 y Fm(Z)29471 31661 y Fk(\003)30124 31772 y Fg(n)30745 31473 y Fr(\()p Fm(!)43 b Fr(\))p 28644 32014 3843 49 v 28720 33124 a Fm(Z)29547 33312 y Fk(\003)30200 33423 y Fg(n)30822 33124 y Fr(\()p Fm(\021)g Fr(\))32956 32293 y(=)336 b(0)p Fm(:)12730 b Fr(\(4.7\))1200 36101 y Fj(Pr)-62 b(o)g(of)p Fr(.)1143 b(W)-101 b(e)402 b(follo)-34 b(w)403 b(the)g(standard)h(line)e(of)g (the)h(argumen)-34 b(t)404 b(used)f(b)-34 b(y)403 b(Israel)e(\(1986\))i (to)g(pro)-34 b(v)g(e)403 b(existence)1200 37607 y(and)460 b(b)34 b(oundary)461 b(condition)f(indep)34 b(endence)460 b(of)f(the)h(pressure)g(for)f(a)h(UA)-34 b(C)460 b(p)34 b(oten)-34 b(tial,)474 b(but)461 b(w)-34 b(e)459 b(detail)h(it)1200 39112 y(b)34 b(ecause)457 b(the)h(v)-67 b(acuum)458 b(p)34 b(oten)-34 b(tial)458 b(is)g(only)f(uniformly)h(con)-34 b(v)g(ergen)g(t.)700 b(Clearly)-101 b(,)470 b(\(4.6\))458 b(implies)f(\(4.7\):)646 b(for)1200 40618 y(all)403 b Fm(n)337 b Fl(2)f Fe(N)p Fr(,)4012 43801 y(exp)6066 42455 y Ff(n)7143 43801 y Fl(\000)463 b Fr(sup)8355 44764 y Fd(!)32 b(;\021)g(;\033)10568 42407 y Ff(\014)10568 43134 y(\014)10568 43862 y(\014)10972 43801 y Fm(H)12078 43211 y Fd(\021)11980 44182 y Fk(\003)12633 44293 y Fg(n)13254 43801 y Fr(\()p Fm(\033)43 b Fr(\))270 b Fl(\000)g Fm(H)17521 43300 y Fd(!)17423 44101 y Fk(\003)18076 44212 y Fg(n)18697 43801 y Fr(\()p Fm(\033)43 b Fr(\))20375 42407 y Ff(\014)20375 43134 y(\014)20375 43862 y(\014)20779 42455 y(o)21924 43801 y Fl(\024)337 b Fr(sup)23425 44764 y Fd(!)32 b(;\021)25363 42981 y Fm(Z)26190 43169 y Fk(\003)26843 43280 y Fg(n)27465 42981 y Fr(\()p Fm(!)43 b Fr(\))p 25363 43522 V 25439 44632 a Fm(Z)26266 44820 y Fk(\003)26919 44931 y Fg(n)27542 44632 y Fr(\()p Fm(\021)g Fr(\))29676 43801 y Fl(\024)336 b Fr(exp)33009 42455 y Ff(n)34213 43801 y Fr(sup)34019 44764 y Fd(!)32 b(;\021)g(;\033)36232 42407 y Ff(\014)36232 43134 y(\014)36232 43862 y(\014)36636 43801 y Fm(H)37742 43211 y Fd(\021)37644 44182 y Fk(\003)38297 44293 y Fg(n)38918 43801 y Fr(\()p Fm(\033)43 b Fr(\))270 b Fl(\000)f Fm(H)43184 43300 y Fd(!)43086 44101 y Fk(\003)43739 44212 y Fg(n)44360 43801 y Fr(\()p Fm(\033)43 b Fr(\))46038 42407 y Ff(\014)46038 43134 y(\014)46038 43862 y(\014)46443 42455 y(o)47251 43801 y Fm(:)1200 47185 y Fr(T)-101 b(o)404 b(pro)-34 b(v)g(e)405 b(\(4.6\),)f(w)-34 b(e)405 b(write)8427 50100 y Fm(H)9533 49511 y Fd(\021)9435 50482 y Fk(\003)10088 50593 y Fg(n)10709 50100 y Fr(\()p Fm(\033)43 b Fr(\))270 b Fl(\000)f Fm(H)14975 49600 y Fd(!)14877 50400 y Fk(\003)15530 50511 y Fg(n)16151 50100 y Fr(\()p Fm(\033)43 b Fr(\))338 b(=)22455 48949 y Ff(X)19446 51633 y Fd(A)p Ft(\\)p Fk(\003)21431 51744 y Fg(n)21997 51633 y Ft(6)p Fk(=)p Ft(;)p Fd(;A)p Ft(\\)p Fk(\003)25445 51381 y Fg(c)25445 51855 y(n)26012 51633 y Ft(6)p Fk(=)p Ft(;)27214 48755 y Ff(h)27787 50100 y Fr(\010)28662 50288 y Fd(A)29422 50100 y Fr(\()p Fm(\033)30586 50288 y Fk(\003)31239 50399 y Fg(n)31861 50100 y Fm(\021)32463 50288 y Fk(\003)33116 50036 y Fg(c)33116 50509 y(n)33737 50100 y Fr(\))270 b Fl(\000)f Fr(\010)36565 50288 y Fd(A)37325 50100 y Fr(\()p Fm(\033)38489 50288 y Fk(\003)39142 50399 y Fg(n)39764 50100 y Fm(!)40518 50288 y Fk(\003)41171 50036 y Fg(c)41171 50509 y(n)41793 50100 y Fr(\))42264 48755 y Ff(i)42836 50100 y Fm(:)1200 54036 y Fr(and)405 b(w)-34 b(e)404 b(\014rst)h(remark:)3815 56036 y(1)p 3047 56577 2142 49 v 3047 57687 a Fl(j)p Fr(\003)4226 57869 y Fd(n)4852 57687 y Fl(j)5321 55462 y Ff(\014)5321 56189 y(\014)5321 56916 y(\014)8734 55704 y(X)5725 58389 y Fd(A)p Ft(\\)p Fk(\003)7710 58500 y Fg(n)8277 58389 y Ft(6)p Fk(=)p 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Fg(c)42660 58610 y(n)43226 58389 y Ft(6)p Fk(=)p Ft(;)44630 56856 y Fr(\010)45505 57044 y Fd(A)46266 56856 y Fr(\()p Fm(\033)43 b Fr(\))47944 55462 y Ff(\014)47944 56189 y(\014)47944 56916 y(\014)48349 56856 y Fm(:)1200 60791 y Fr(W)-101 b(e)403 b(obtain)8006 63405 y(sup)8634 64368 y Fd(\033)9831 62011 y Ff(\014)9831 62738 y(\014)9831 63465 y(\014)12298 62253 y(X)10235 64938 y Fd(A)p Ft(3)p Fd(x;A)p Ft(\\)p Fk(\003)14343 64685 y Fg(c)14343 65159 y(n)14910 64938 y Ft(6)p Fk(=)p Ft(;)16314 63405 y Fr(\010)17189 63593 y Fd(A)17949 63405 y Fr(\()p Fm(\033)43 b Fr(\))19627 62011 y Ff(\014)19627 62738 y(\014)19627 63465 y(\014)21139 63405 y Fr(=)23189 62011 y Ff(\014)23189 62738 y(\014)23189 63465 y(\014)23649 62253 y(X)23593 64879 y Fd(A)p Ft(3)p Fd(x)25657 63405 y Fr(\010)26532 63593 y Fd(A)27292 63405 y Fr(\()p Fm(\033)g Fr(\))607 b Fl(\000)32303 62253 y Ff(X)30789 64879 y Fd(A)p Ft(3)p Fd(x;A)p Ft(\032)p Fk(\003)35002 64990 y Fg(n)35770 63405 y Fr(\010)36645 63593 y Fd(A)37406 63405 y Fr(\()p Fm(\033)43 b Fr(\))39084 62011 y Ff(\014)39084 62738 y(\014)39084 63465 y(\014)21139 67117 y Fr(=)23189 65723 y Ff(\014)23189 66450 y(\014)23189 67178 y(\014)23619 65965 y(X)23593 68591 y Fd(A)p Ft(3)p Fk(0)25598 67117 y Fr(\010)26473 67305 y Fd(A)27233 67117 y Fr(\()p Fm(\034)28234 67299 y Fd(x)28819 67117 y Fm(\033)g Fr(\))607 b Fl(\000)34326 65965 y Ff(X)31845 68650 y Fd(A)p Ft(3)p Fk(0)p Fd(;A)p Ft(\032)p Fk(\(\003)36364 68761 y Fg(n)36931 68650 y Ft(\000)p Fd(x)p Fk(\))38760 67117 y Fr(\010)39635 67305 y Fd(A)40396 67117 y Fr(\()p Fm(\034)41397 67299 y Fd(x)41982 67117 y Fm(\033)43 b Fr(\))43189 65723 y Ff(\014)43189 66450 y(\014)43189 67178 y(\014)21139 70829 y Fl(\024)1107 b Fr(sup)23877 71882 y Fd(\030)25014 69435 y Ff(\014)25014 70162 y(\014)25014 70890 y(\014)28661 69678 y(X)25418 72362 y Fd(A)p Ft(3)p Fk(0)p Fd(;A)p Ft(\\)p Fk(\(\003)29832 72473 y Fg(n)30399 72362 y Ft(\000)p Fd(x)p Fk(\))32027 72110 y Fg(c)32453 72362 y Ft(6)p Fk(=)p Ft(;)33857 70829 y Fr(\010)34732 71017 y Fd(A)35493 70829 y Fr(\()p Fm(\030)56 b Fr(\))37021 69435 y Ff(\014)37021 70162 y(\014)37021 70890 y(\014)37426 70829 y Fm(:)25194 75417 y Fr(15)p eop %%Page: 16 16 16 15 bop 1200 1313 a Fr(Pic)-34 b(k)403 b Fm(\017)337 b(>)g Fr(0)404 b(and)h(c)-34 b(ho)34 b(ose)404 b(\001)g(suc)-34 b(h)405 b(that)18842 4037 y(sup)19530 5089 y Fd(\030)20667 2643 y Ff(\014)20667 3370 y(\014)20667 4097 y(\014)23100 2885 y(X)21071 5569 y Fd(A)p Ft(3)p Fk(0)p Fd(;A)p Ft(\\)p Fk(\001)25250 5317 y Fg(c)25677 5569 y Ft(6)p Fk(=)p Ft(;)27082 4037 y Fr(\010)27957 4225 y Fd(A)28717 4037 y Fr(\()p Fm(\030)56 b Fr(\))30245 2643 y Ff(\014)30245 3370 y(\014)30245 4097 y(\014)30987 4037 y Fl(\024)336 b Fm(\017)1200 8033 y Fr(then)9820 9311 y Ff(\014)9820 10038 y(\014)9820 10765 y(\014)13467 9553 y(X)10224 12238 y Fd(A)p Ft(3)p Fk(0)p Fd(;A)p Ft(\\)p Fk(\(\003)14638 12349 y Fg(n)15205 12238 y Ft(\000)p Fd(x)p Fk(\))16833 11986 y Fg(c)17259 12238 y Ft(6)p Fk(=)p Ft(;)18663 10705 y Fr(\010)19538 10893 y Fd(A)20299 10705 y Fr(\()p Fm(\030)56 b Fr(\))21827 9311 y Ff(\014)21827 10038 y(\014)21827 10765 y(\014)22905 10705 y Fl(\024)24521 8220 y Ff(8)24521 9311 y(<)24521 11493 y(:)26152 9197 y Fm(\017)2241 b Fr(if)1875 b(\(\003)32780 9379 y Fd(n)33676 9197 y Fl(\000)269 b Fm(x)p Fr(\))337 b Fl(\033)f Fr(\001)26152 12208 y Fm(C)1867 b Fr(if)1875 b(\(\003)32780 12390 y Fd(n)33676 12208 y Fl(\000)269 b Fm(x)p Fr(\))g Fl(\\)h Fr(\001)38409 11768 y Fd(c)39208 12208 y Fl(6)p Fr(=)337 b Fl(;)1200 14332 y Fr(where)18695 15837 y Fm(C)424 b Fr(=)336 b(sup)21953 16890 y Fd(\030)23089 14443 y Ff(\014)23089 15171 y(\014)23089 15898 y(\014)23519 14686 y(X)23493 17311 y Fd(A)p Ft(3)p Fk(0)25498 15837 y Fr(\010)26373 16025 y Fd(A)27133 15837 y Fr(\()p Fm(\030)56 b Fr(\))28661 14443 y Ff(\014)28661 15171 y(\014)28661 15898 y(\014)29740 15837 y Fm(<)673 b Fl(1)p Fm(:)1200 19373 y Fr(Since)404 b(for)g(an)-34 b(y)405 b(\001)337 b Fl(\032)f Fe(Z)11944 18934 y Fd(d)12888 19373 y Fr(\014nite,)17294 22667 y(lim)16909 23394 y Fd(n)p Ft(!1)19564 22667 y Fm(\017)20526 21847 y Fl(jf)p Fm(x)g Fr(:)g(\001)270 b(+)f Fm(x)g Fl(\\)g Fr(\003)28544 21407 y Fd(c)28544 22146 y(n)29507 21847 y Fl(6)p Fr(=)337 b Fl(;gj)p 20526 22388 11810 49 v 25360 23498 a(j)p Fr(\003)26539 23680 y Fd(n)27165 23498 y Fl(j)32805 22667 y Fr(=)f(0)1200 25971 y(w)-34 b(e)404 b(obtain)13841 27683 y(lim)202 b(sup)15410 28646 y Fd(n)18654 26863 y Fr(1)p 17886 27404 2142 49 v 17886 28514 a Fl(j)p Fr(\003)19065 28696 y Fd(n)19691 28514 y Fl(j)21011 26532 y Ff(X)20699 29157 y Fd(x)p Ft(2)p Fk(\003)22509 29268 y Fg(n)23277 27683 y Fr(sup)23965 28736 y Fd(\030)25102 26289 y Ff(\014)25102 27016 y(\014)25102 27744 y(\014)27481 26532 y(X)25506 29216 y Fd(x)p Ft(3)p Fd(x;A)p Ft(\\)p Fk(\003)29439 28964 y Fg(c)29439 29437 y(n)30005 29216 y Ft(6)p Fk(=)p Ft(;)31409 27683 y Fr(\010)32284 27871 y Fd(A)33045 27683 y Fr(\()p Fm(\030)56 b Fr(\))34573 26289 y Ff(\014)34573 27016 y(\014)34573 27744 y(\014)35651 27683 y Fl(\024)673 b Fm(\017)1200 31298 y Fr(whic)-34 b(h)405 b(b)-34 b(y)404 b(the)h(arbitrary)f(c)-34 b(hoice)403 b(of)i Fm(\017)337 b(>)f Fr(0)404 b(pro)-34 b(v)g(es)405 b(\(4.6\))f(and)h(the)g(statemen) -34 b(t)405 b(of)g(the)f(lemma.)p 49878 31298 522 731 v 3082 33800 a(T)-101 b(o)410 b(deriv)-34 b(e)409 b(Lemma)h(4.2)f(from) h(Lemma)g(4.5,)g(w)-34 b(e)411 b(only)e(ha)-34 b(v)g(e)411 b(to)f(pro)-34 b(v)g(e)410 b(that)h(for)f(all)f Fm(\027)423 b Fl(2)347 b(G)45340 33982 y Fk(in)-26 b(v)46663 33800 y Fr(\()p Fm(\015)67 b Fr(\),)411 b Fm(e)49612 33360 y Fk(+)49612 34099 y Fd(\027)1200 35305 y Fr(exists)404 b(and)h(is)f(indep)34 b(enden)-34 b(t)405 b(of)g Fm(\015)67 b Fr(.)538 b(F)-101 b(or)404 b(suc)-34 b(h)405 b(a)f(measure)g Fm(\027)77 b Fr(,)404 b(write)18591 38959 y Fm(\027)77 b Fr(\(+)20681 39147 y Fk(\003)21388 38959 y Fr(\))338 b(=)23476 37309 y Ff(Z)24149 40058 y Fk(\012)25219 38139 y Fm(e)25783 37589 y Ft(\000)p Fd(H)27356 37155 y Fg(\021)27288 37897 y Fc(\003)27840 38008 y Fg(n)28463 37589 y Fk(\(+\))p 25219 38680 4763 49 v 25520 39790 a Fm(Z)26347 39978 y Fk(\003)27000 40089 y Fg(n)27622 39790 y Fr(\()p Fm(\021)43 b Fr(\)\))30114 38959 y Fm(\027)77 b Fr(\()p Fm(d\021)43 b Fr(\))1200 42392 y(where)404 b Fm(H)5811 41803 y Fd(\021)5713 42774 y Fk(\003)6366 42885 y Fg(n)7391 42392 y Fr(is)g(de\014ned)h Fj(via)496 b Fr(the)404 b(v)-67 b(acuum)404 b(p)34 b(oten)-34 b(tial)405 b(of)g Fm(\015)471 b Fr(in)404 b(\(4.4\).)539 b(W)-101 b(e)403 b(use)i(Lemma)e(4.5)h(to)h(write)18805 46295 y Fm(\027)77 b Fr(\(+)20895 46483 y Fk(\003)21603 46295 y Fr(\))22411 45961 y Fl(\030)22411 46350 y Fr(=)23691 44645 y Ff(Z)24364 47394 y Fk(\012)25434 45474 y Fm(e)25998 45035 y Ft(\000)p Fd(H)27571 44637 y Fc(+)27503 45343 y(\003)28248 45035 y Fk(\(+\))p 25434 46016 4333 49 v 26750 47206 a Fm(Z)27664 46696 y Fk(+)27577 47588 y(\003)29900 46295 y Fm(\027)g Fr(\()p Fm(d\021)43 b Fr(\))1200 49964 y(where)509 b Fm(a)5451 50152 y Fk(\003)6669 49630 y Fl(\030)6669 50019 y Fr(=)8123 49964 y Fm(b)8643 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Fk(\()p Fd(R)5 b Fk(\))20456 7424 y Fd(A)21958 7043 y Fr(\()p Fm(\033)43 b Fr(\)\))25685 5873 y Fl(\024)1443 b Fr(sup)28699 6836 y Fd(\033)30098 4479 y Ff(\014)30098 5206 y(\014)30098 5934 y(\014)33453 4721 y(X)30704 7406 y Fd(A)p Ft(\032)p Fk(\003)p Fd(;)p Fk(diam\(A\))p Fd(>)p Fk(R)38156 5873 y Fr(\010)39031 6061 y Fd(A)39791 5873 y Fr(\()p Fm(\033)43 b Fr(\))41469 4479 y Ff(\014)41469 5206 y(\014)41469 5934 y(\014)25685 9585 y Fl(\024)1106 b Fr(sup)28362 10548 y Fd(\033)29791 8434 y Ff(X)29761 11059 y Fd(x)p Ft(2)p Fk(\003)31773 8191 y Ff(\014)31773 8919 y(\014)31773 9646 y(\014)35014 8434 y(X)32379 11118 y Fd(A)p Ft(3)p Fd(x;)p Fk(diam\(A\))p Fd(>)p Fk(R)39603 9585 y Fr(\010)40478 9773 y Fd(A)41239 9585 y Fr(\()p Fm(\033)43 b Fr(\))42917 8191 y Ff(\014)42917 8919 y(\014)42917 9646 y(\014)25685 13297 y Fl(\024)27764 12146 y Ff(X)27734 14771 y Fd(x)p Ft(2)p Fk(\003)29746 13297 y Fr(sup)30373 14260 y Fd(\033)31773 11903 y Ff(\014)31773 12631 y(\014)31773 13358 y(\014)35014 12146 y(X)32379 14830 y Fd(A)p Ft(3)p Fd(x;)p Fk(diam\(A\))p Fd(>)p Fk(R)39603 13297 y Fr(\010)40478 13485 y Fd(A)41239 13297 y Fr(\()p Fm(\033)g Fr(\))42917 11903 y Ff(\014)42917 12631 y(\014)42917 13358 y(\014)25685 17010 y Fr(=)1106 b Fl(j)p Fr(\003)p Fl(j)202 b Fr(sup)30079 17973 y Fd(\033)31478 15616 y Ff(\014)31478 16343 y(\014)31478 17070 y(\014)34691 15858 y(X)32084 18542 y Fd(A)p Ft(3)p Fk(0)p Fd(;)p Fk(diam)r(\(A\))p Fd(>)p Fk(R)39250 17010 y Fr(\010)40125 17198 y Fd(A)40885 17010 y Fr(\()p Fm(\033)43 b Fr(\))42563 15616 y Ff(\014)42563 16343 y(\014)42563 17070 y(\014)1200 20710 y Fr(and)9366 21215 y Ff(P)10646 22482 y Fd(\033)11473 22124 y Fr(exp)201 b(\()p Fl(\000)15142 21215 y Ff(P)16422 22482 y Fd(A)p Ft(\032)p Fk(\003)18768 22124 y Fr(\010)19643 21485 y Fk(\()p Fd(R)5 b Fk(\))19643 22506 y Fd(A)21145 22124 y Fr(\()p Fm(\033)43 b Fr(\)\))p 9217 22732 14228 49 v 9217 23272 a Ff(P)10497 24539 y Fd(\033)11323 24181 y Fr(exp)202 b(\()p Fl(\000)14993 23272 y Ff(P)16273 24539 y Fd(A)p Ft(\032)p Fk(\003)18619 24181 y Fr(\010)19494 23542 y Fk(\()p Fd(R)20574 23290 y Fn(0)20873 23542 y Fk(\))19494 24562 y Fd(A)21294 24181 y Fr(\()p Fm(\033)43 b Fr(\)\))23914 23011 y Fl(\024)337 b(j)p Fr(\003)p Fl(j)202 b Fr(sup)27539 23974 y Fd(\033)28938 21617 y Ff(\014)28938 22344 y(\014)28938 23072 y(\014)33194 21859 y(X)29544 24544 y Fd(A)p Ft(3)p Fk(0)p Fd(;)p Fk(diam)q(\(A\))p Fd(>)p Fk(R)36664 23880 y Fb(V)37605 24544 y Fk(R)38297 24292 y Fn(0)38797 23011 y Fr(\010)39672 23199 y Fd(A)40433 23011 y Fr(\()p Fm(\033)43 b Fr(\))42111 21617 y Ff(\014)42111 22344 y(\014)42111 23072 y(\014)1200 26648 y Fr(to)404 b(conclude)h(that)g Fl(f)p Fm(P)168 b Fr(\(\010)13189 26208 y Fk(\()p Fd(R)5 b Fk(\))14692 26648 y Fr(\))p Fm(;)202 b(R)346 b(>)337 b Fr(0)p Fl(g)404 b Fr(is)g(a)g(Cauc)-34 b(h)g(y)405 b(net)g(with)g(limit)16128 29536 y(lim)15672 30343 y Fd(R)5 b Ft(!1)18470 29536 y Fm(P)168 b Fr(\(\010)20762 29035 y Fk(\()p Fd(R)5 b Fk(\))22265 29536 y Fr(\))337 b(=)592 b(lim)24352 30451 y Fk(\003)p Ft(")p Fh(Z)26065 30199 y Fg(d)27337 28716 y Fr(1)p 26882 29257 1516 49 v 26882 30367 a Fl(j)p Fr(\003)p Fl(j)28732 29536 y Fr(log)219 b Fm(Z)31414 29035 y Fk(free)31327 29836 y(\003)33297 29536 y Fr(=)336 b Fm(e)35140 29035 y Fk(+)35140 29836 y Fd(\027)1200 32637 y Fr(whic)-34 b(h)384 b(dep)34 b(ends)384 b(only)f(on)g(the)h(v)-67 b(acuum)383 b(p)34 b(oten)-34 b(tial)384 b(\(hence)f(on)h(the)g(sp)34 b(eci\014cation)383 b Fm(\015)67 b Fr(\).)531 b(This)384 b(pro)-34 b(v)g(es)384 b(that)1200 34143 y Fm(e)1764 33703 y Fk(+)1764 34442 y Fd(\027)2956 34143 y Fr(and)404 b Fm(e)5876 33703 y Fk(+)5876 34442 y Fd(\026)7068 34143 y Fr(exist)g(for)g(all)g Fm(\026;)202 b(\027)413 b Fl(2)337 b(G)17623 34325 y Fk(in)-26 b(v)18946 34143 y Fr(\()p Fm(\015)67 b Fr(\),)404 b(and)h(dep)34 b(ends)405 b(of)g Fm(\015)471 b Fr(only)-101 b(.)538 b(Therefore,)17176 37327 y(lim)16920 38242 y Fk(\003)p Ft(")p Fh(Z)18633 37990 y Fg(d)19905 36507 y Fr(1)p 19450 37048 V 19450 38158 a Fl(j)p 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b(GriSing)g(random)g(\014eld)g(is)g(an)g(example)f(of)h(join)-34 b(t)493 b(measure)f(of)g(disordered)f(systems,)514 b(studied)493 b(in)1200 55877 y(more)436 b(exten)-34 b(t)437 b(in)g(Section)f(6.)636 b(It)437 b(has)g(b)34 b(een)436 b(studied)i(in)e([8])g(and)h(pro)-34 b(vides)437 b(an)g(easy)f(example)g(of)h(a)f(non-)1200 57383 y(Gibbsian)528 b(random)g(\014elds)f(whic)-34 b(h)528 b(\014ts)f(in)g(the)h(framew)-34 b(ork)527 b(of)g(our)h(theorems.)907 b(The)527 b(random)h(\014eld)f(is)1200 58888 y(constructed)477 b(as)g(follo)-34 b(ws.)756 b(Sites)476 b(are)g(empt)-34 b(y)477 b(or)f(o)34 b(ccupied)476 b(according)g(a)h(Bernoulli)e(pro)34 b(duct)477 b(measure)1200 60394 y(of)407 b(parameter)f Fm(p)341 b(<)g(p)11228 60576 y Fd(c)12097 60394 y Fr(where)406 b Fm(p)16214 60576 y Fd(c)17083 60394 y Fr(is)h(the)g(p)34 b(ercolation)406 b(threshold)h(for)g(site)f(p)34 b(ercolation)407 b(on)f Fe(Z)44914 59954 y Fd(d)45454 60394 y Fr(.)546 b(F)-101 b(or)406 b(an)-34 b(y)1200 61899 y(realization)377 b Fm(\021)422 b Fr(of)378 b(o)34 b(ccupancies)378 b(where)g(all)f(o)34 b(ccupied)378 b(clusters)g(are)f(\014nite,)384 b(w)-34 b(e)378 b(ha)-34 b(v)g(e)379 b(the)f(Gibbs)h(measure)1200 63405 y(on)404 b(con\014gurations)i Fm(\033)380 b Fl(2)337 b(f\000)p Fr(1)p Fm(;)202 b Fr(+1)p Fl(g)17772 62965 y Fh(Z)18362 62652 y Fg(d)23965 64910 y Fm(\026)24695 64320 y Fd(\021)24695 65302 y(\014)25325 64910 y Fr(\()p Fm(d\033)43 b Fr(\))1200 66963 y(whic)-34 b(h)320 b(is)f(the)h(pro)34 b(duct)320 b(of)g(free)f(b)34 b(oundary)320 b(condition)g(Ising)f (measures)h(on)f(the)h(o)34 b(ccupied)319 b(clusters.)510 b(More)1200 68468 y(precisely)-101 b(,)349 b(under)339 b Fm(\026)10444 67878 y Fd(\021)10444 68860 y(\014)11411 68468 y Fr(spin)f(con\014gurations)h(of)f(o)34 b(ccupied)338 b(clusters)g Fm(C)424 b Fr(are)338 b(indep)34 b(enden)-34 b(t)339 b(and)g(distributed)1200 69974 y(as:)16648 71685 y Fm(\026)17378 71882 y Fd(\014)46 b(;C)19001 71685 y Fr(\()p Fm(\033)20165 71873 y Fd(C)20954 71685 y Fr(\))337 b(=)23639 70865 y(1)p 23174 71407 1536 49 v 23174 72517 a Fm(Z)24001 72705 y Fk(\003)25179 71685 y Fm(e)25743 71112 y Ft(\000)p Fd(\014)27206 70448 y Fb(P)28199 71389 y Fn(h)p Fg(xy)24 b Fn(i\032)p Fg(C)31268 71112 y Fd(\033)32 b Fk(\()p Fd(x)p Fk(\))p Fd(\033)g Fk(\()p Fd(y)g Fk(\))25194 75417 y Fr(17)p eop %%Page: 18 18 18 17 bop 1200 1313 a Fr(The)405 b(GriSing)f(random)h(\014eld)f(is)g (then)h(de\014ned)g(as:)21386 3981 y Fm(\030)56 b Fr(\()p Fm(x)p Fr(\))338 b(=)e Fm(\033)43 b Fr(\()p Fm(x)p Fr(\))p Fm(\021)g Fr(\()p Fm(x)p Fr(\))p Fm(:)1200 6649 y Fr(In)404 b(w)-34 b(ords,)405 b Fm(\030)56 b Fr(\()p Fm(x)p Fr(\))337 b(=)g(0)404 b(for)g(uno)34 b(ccupied)405 b(sites)f(and)h(equal)e(to)i (the)f(spin)h Fm(\033)43 b Fr(\()p Fm(x)p Fr(\))405 b(at)f(o)34 b(ccupied)404 b(sites.)3082 8477 y(W)-101 b(e)403 b(denote)i(b)-34 b(y)405 b Fm(K)11788 8674 y Fd(p;\014)13556 8477 y Fr(the)f(la)-34 b(w)405 b(of)f(the)h(random)g(\014eld)f Fm(\030)56 b Fr(.)3082 10306 y(It)446 b(is)h(kno)-34 b(wn)448 b(that)g(for)f(an)-34 b(y)447 b Fm(p)407 b Fl(2)h Fr(\(0)p Fm(;)202 b Fr(1\),)457 b Fm(\014)511 b Fr(large)446 b(enough,)458 b Fm(K)31999 10503 y Fd(p;\014)33809 10306 y Fr(is)447 b(not)g(a)g(Gibbs)g(measure)g (\(see)f([8])1200 11811 y(for)392 b Fm(p)336 b(<)h(p)5879 11993 y Fd(c)6733 11811 y Fr(and)393 b([17])e(for)h(an)-34 b(y)393 b Fm(p)336 b Fl(2)h Fr(\(0)p Fm(;)202 b Fr(1\)\).)535 b(The)393 b(p)34 b(oin)-34 b(ts)393 b(of)f(essen)-34 b(tial)392 b(discon)-34 b(tin)g(uit)g(y)394 b(of)e(the)g(conditional) 1200 13317 y(probabilities)404 b Fm(K)9175 13514 y Fd(p;\014)10539 13317 y Fr(\()p Fm(\033)43 b Fr(\(0\))p Fl(j)p Fm(\030)14161 13612 y Fh(Z)14751 13360 y Fg(d)15235 13612 y Ft(nf)p Fk(0)p Ft(g)17172 13317 y Fr(\))405 b(are)e(a)h(subset)i(of)10825 15984 y Fm(D)370 b Fr(=)336 b Fl(f)p Fm(\030)393 b Fr(:)336 b Fm(\030)461 b Fr(con)-34 b(tains)405 b(an)f(in\014nite)h(cluster)f (of)g(o)34 b(ccupied)404 b(sites)p Fl(g)p Fm(:)1200 18652 y Fr(Since)493 b Fm(p)484 b(<)h(p)7586 18834 y Fd(c)8048 18652 y Fr(,)515 b(there)493 b(exists)g(a)f(sp)34 b(eci\014cation)493 b Fm(\015)560 b Fr(suc)-34 b(h)494 b(that)g Fl(f)p Fm(K)31983 18849 y Fd(p;\014)33347 18652 y Fl(g)485 b Fr(=)f Fl(G)72 b Fr(\()p Fm(\015)67 b Fr(\))494 b(and)g(suc)-34 b(h)493 b(that)h(for)f(the)1200 20158 y(con)-34 b(tin)g(uit)g(y)332 b(p)34 b(oin)-34 b(ts)331 b(\012)11294 20340 y Fd(\015)11887 20158 y Fr(,)344 b(w)-34 b(e)331 b(ha)-34 b(v)g(e)330 b Fm(K)18028 20355 y Fd(p;\014)19392 20158 y Fr(\(\012)20738 20340 y Fd(\015)21332 20158 y Fr(\))337 b(=)f(1,)345 b(i.e.,)e Fm(K)27966 20355 y Fd(p;\014)29660 20158 y Fr(is)330 b(almost)h(Gibbs.)514 b(Moreo)-34 b(v)g(er,)344 b(if)330 b(w)-34 b(e)331 b(c)-34 b(ho)34 b(ose)1200 21663 y Fm(\030)1730 21845 y Fk(0)2740 21663 y Fl(\021)484 b Fr(0)492 b(as)h(a)f(telescoping)h(reference)e(con\014guration,)515 b(then)494 b(clearly)c Fm(\033)528 b Fl(2)484 b Fm(D)37564 21223 y Fd(c)38518 21663 y Fr(implies)492 b Fm(\033)43457 21223 y Fd(\030)43871 21346 y Fc(0)44871 21663 y Fl(2)484 b Fm(D)47201 21223 y Fd(c)47663 21663 y Fr(,)514 b(i.e.,)1200 23169 y(in)541 b(this)g(case)g(\012)8829 23351 y Fd(\015)9986 23169 y Fl(\032)564 b Fr(\012)12368 22729 y Fd(<)p Fk(0)12368 23468 y Fd(\015)13626 23169 y Fr(.)949 b(Therefore,)575 b(in)541 b(this)g(example)f(condition)i(C1)f(is)g(satis\014ed)g(as)g (so)34 b(on)542 b(as)f Fm(\026)1200 24674 y Fr(concen)-34 b(trates)405 b(on)f Fm(D)10901 24234 y Fd(c)11363 24674 y Fr(.)539 b(Using)404 b Fl(f)p Fm(K)17281 24871 y Fd(p;\014)18646 24674 y Fl(g)336 b Fr(=)h Fl(G)72 b Fr(\()p Fm(\015)67 b Fr(\),)404 b(and)17246 28214 y(lim)16990 29130 y Fk(\003)p Ft(")p Fh(Z)18703 28877 y Fg(d)19975 27394 y Fr(1)p 19520 27935 1516 49 v 19520 29046 a Fl(j)p Fr(\003)p Fl(j)21370 28214 y Fr(log)219 b Fm(K)24167 28411 y Fd(p;\014)25531 28214 y Fr(\(0)26608 28402 y Fk(\003)27317 28214 y Fr(\))337 b(=)f(log)17 b(\(1)270 b Fl(\000)f Fm(p)p Fr(\))1200 31067 y(w)-34 b(e)404 b(obtain)i(the)e(follo)-34 b(wing)405 b(prop)34 b(osition:)1200 34169 y Fs(Prop)39 b(osition)465 b(5.1)606 b Fj(If)434 b Fm(\026)p Fr(\()p Fm(D)34 b Fr(\))337 b(=)g(0)433 b Fj(then)g Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(K)23722 34366 y Fd(p;\014)25087 34169 y Fr(\))h Fj(exists)f(and)g (is)g(zer)-62 b(o)432 b(if)h(and)h(only)f(if)g Fm(\026)337 b Fr(=)f Fm(K)46239 34366 y Fd(p;\014)47604 34169 y Fj(.)1200 37729 y Fi(5.2)1495 b(Decimation)1200 40340 y Fr(Let)499 b Fm(\026)4197 39830 y Fk(+)4197 40731 y Fd(\014)5485 40340 y Fr(\(resp.)825 b Fm(\026)10013 39830 y Ft(\000)10013 40731 y Fd(\014)10801 40340 y Fr(\))500 b(b)34 b(e)500 b(the)g(lo)-34 b(w-temp)34 b(erature)500 b(\()p Fm(\014)560 b(>)496 b(\014)28736 40522 y Fd(c)29199 40340 y Fr(\))k(plus)g(\(resp.) 826 b(min)-34 b(us\))501 b(phase)g(of)f(the)g(Ising)1200 42069 y(mo)34 b(del)420 b(on)h Fe(Z)7327 41629 y Fd(d)7866 42069 y Fr(.)588 b(F)-101 b(or)420 b Fm(b)365 b Fl(2)f Fe(N)p Fr(,)424 b Fm(\027)15351 41559 y Fk(+)15274 42460 y Fd(\014)16559 42069 y Fr(denotes)d(its)g(decimation,)j(i.e.,)f(the)e (distribution)h(of)f Fl(f)p Fm(\033)43 b Fr(\()p Fm(bx)p Fr(\))365 b(:)e Fm(x)h Fl(2)g Fe(Z)49254 41629 y Fd(d)49794 42069 y Fl(g)1200 43788 y Fr(when)354 b Fm(\033)396 b Fr(is)352 b(distributed)i(according)f(to)g Fm(\026)20291 43278 y Fk(+)20291 44179 y Fd(\014)21079 43788 y Fr(.)521 b(It)353 b(is)f(kno)-34 b(wn)355 b(that)f Fm(\027)31405 43278 y Fk(+)31328 44179 y Fd(\014)32544 43788 y Fr(is)f(not)h(a)e (Gibbs)i(measure)f([6)o(].)521 b(In)353 b([10])1200 45507 y(it)438 b(is)h(pro)-34 b(v)g(ed)439 b(that)h(there)e(exists)g(a)h (monotone)g(sp)34 b(eci\014cation)439 b Fm(\015)31161 45067 y Fk(+)32386 45507 y Fr(suc)-34 b(h)440 b(that)f Fm(\027)38491 44997 y Fk(+)38414 45899 y Fd(\014)39673 45507 y Fl(2)394 b(G)72 b Fr(\()p Fm(\015)42834 45067 y Fk(+)43621 45507 y Fr(\).)642 b(In)439 b([9)o(])g(it)f(is)1200 47226 y(pro)-34 b(v)g(ed)434 b(that)h(the)f(p)34 b(oin)-34 b(ts)435 b(of)f(con)-34 b(tin)g(uit)g(y)435 b(\012)21656 47485 y Fd(\015)22193 47233 y Fc(+)23359 47226 y Fr(satisfy)f Fm(\027)27849 46716 y Fk(+)27772 47618 y Fd(\014)28636 47226 y Fr(\(\012)29982 47485 y Fd(\015)30519 47233 y Fc(+)31252 47226 y Fr(\))386 b(=)g(1,)441 b(i.e.,)e Fm(\027)37824 46716 y Fk(+)37747 47618 y Fd(\014)39045 47226 y Fr(is)433 b(almost)h(Gibbs.)628 b(The)1200 48875 y(p)34 b(oin)-34 b(t)458 b(of)g(con)-34 b(tin)g(uit)g(y)460 b(of)e Fm(\015)13759 48436 y Fk(+)15004 48875 y Fr(can)f(b)34 b(e)458 b(describ)34 b(ed)457 b(as)h(those)g(con\014gurations)h Fm(\021)501 b Fr(for)458 b(whic)-34 b(h)458 b(the)g(\\in)-34 b(ternal)1200 50381 y(spins")374 b(do)f(not)h(exhibit)f(a)h(phase)f(transition)h (when)g(the)g(decimated)f(spins)h(are)f(\014xed)g(to)h(b)34 b(e)373 b Fm(\021)43 b Fr(.)528 b(E.g.,)378 b(the)1200 51886 y(all)314 b(plus)h(and)h(the)f(all)f(min)-34 b(us)316 b(con\014gurations)g(are)e(elemen)-34 b(ts)315 b(of)g(\012)31282 52145 y Fd(\015)31819 51893 y Fc(+)32551 51886 y Fr(,)332 b(but)316 b(the)f(alternating)g(con\014guration)1200 53392 y(is)404 b(not.)3082 55220 y(The)447 b(\014rst)f(part)h(of)g(the) g(v)-67 b(ariational)445 b(principle)h(for)g(\()p Fm(\015)28548 54780 y Fk(+)29336 55220 y Fm(;)202 b(\027)30551 54710 y Fk(+)30474 55612 y Fd(\014)31338 55220 y Fm(;)g Fl(M)p Fr(\))446 b(has)g(already)g(b)34 b(een)446 b(pro)-34 b(v)g(ed)447 b(in)f([9])1200 56726 y(\(and)374 b(is)e(direct)g(b)-34 b(y)373 b(Theorem)f(3.1\),)379 b(with)373 b(a)g(set)g Fl(M)f Fr(consisting)h(of)g(the)g(translation)g(in)-34 b(v)-67 b(arian)-34 b(t)373 b(measures)1200 58231 y(whic)-34 b(h)405 b(concen)-34 b(trate)405 b(on)f(\012)13728 58490 y Fd(\015)14265 58238 y Fc(+)15401 58231 y Fr(.)539 b(Here)403 b(w)-34 b(e)405 b(complete)e(this)i(result)f(b)-34 b(y)405 b(adding)g(a)f(second)g(part:)1200 61010 y Fs(Theorem)464 b(5.2)606 b Fj(F)-93 b(or)434 b(any)f Fm(\026)337 b Fl(2)g(M)17760 60500 y Fk(+)17760 61380 y(1)p Fd(;)p Fk(in)-26 b(v)20248 61010 y Fj(satisfying)431 b(C1)i(for)g Fm(\015)30189 60570 y Fk(+)30977 61010 y Fj(,)2633 63788 y(1.)605 b Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(\027)77 b Fr(\))435 b Fj(exists.)2633 66271 y(2.)605 b(We)433 b(have)f(the)h(e)-62 b(quivalenc)g(e)18488 68939 y Fm(\026)337 b Fl(2)g(G)21421 69121 y Fk(in)-26 b(v)22744 68939 y Fr(\()p Fm(\015)23910 68439 y Fk(+)24698 68939 y Fr(\))1010 b Fl(\()-202 b(\))1011 b Fm(h)p Fr(\()p Fm(\026)p Fl(j)p Fm(\027)32324 68439 y Fk(+)33111 68939 y Fr(\))338 b(=)e(0)p Fm(:)3082 72096 y Fr(W)-101 b(e)403 b(\014rst)i(use)f(a)g(lemma.)25194 75417 y(18)p eop %%Page: 19 19 19 18 bop 1200 1313 a Fs(Lemma)464 b(5.3)606 b Fm(\026)337 b Fl(2)g(G)72 b Fr(\()p Fm(\015)12701 874 y Fk(+)13488 1313 y Fr(\))434 b Fj(and)f Fm(\026)p Fr(\(\012)18823 1572 y Fd(\015)19360 1320 y Fc(+)20093 1313 y Fr(\))337 b(=)g(1)433 b Fj(implies)21514 3709 y Fm(\027)22190 3199 y Fk(+)22113 4100 y Fd(\014)23650 3709 y Fl(\026)673 b Fm(\026)h Fl(\026)f Fm(\027)28962 3199 y Fk(+)28885 4100 y Fd(\014)29749 3709 y Fm(:)17822 b Fr(\(5.4\))1200 6526 y Fj(Pr)-62 b(o)g(of)p Fr(.)605 b(Consider)405 b Fm(f)535 b Fr(monotone.)k(By)404 b(monotonicit)-34 b(y)405 b(of)f Fm(\015)28530 6086 y Fk(+)29721 6526 y Fr([10],)f(for)i(all)e (\003)337 b Fl(2)f(S)91 b Fr(,)9468 7830 y Ff(Z)10883 9480 y Fm(f)130 b(d\026)337 b Fr(=)14584 7830 y Ff(Z)15257 10579 y Fk(\012)15992 9480 y Fr(\()p Fm(\015)17158 8970 y Fk(+)17091 9861 y(\003)17946 9480 y Fm(f)130 b Fr(\)\()p Fm(!)43 b Fr(\))p Fm(\026)p Fr(\()p Fm(d!)g Fr(\))341 b Fl(\024)25600 7830 y Ff(Z)26273 10579 y Fk(\012)27008 9480 y Fr(\()p Fm(\015)28174 8970 y Fk(+)28107 9861 y(\003)28962 9480 y Fm(f)130 b Fr(\)\(+\))p Fm(\026)p Fr(\()p Fm(d!)43 b Fr(\))340 b(=)d(\()p Fm(\015)37927 8970 y Fk(+)37860 9861 y(\003)38714 9480 y Fm(f)130 b Fr(\)\(+\))p Fm(:)1200 12634 y Fr(T)-101 b(aking)404 b(the)h(limit)f(\003)336 b Fl(")h Fe(Z)13153 12195 y Fd(d)14096 12634 y Fr(giv)-34 b(es,)404 b(and)h(using)f Fm(\015)23626 12125 y Fk(+)23559 13016 y(\003)24413 12634 y Fr(\()p Fl(\001j)p Fr(+\))g(go)34 b(es)404 b(to)h Fm(\027)32201 12125 y Fk(+)32124 13026 y Fd(\014)32988 12634 y Fr(,)20958 14199 y Ff(Z)22372 15849 y Fm(f)130 b(d\026)338 b Fl(\024)26073 14199 y Ff(Z)27487 15849 y Fm(f)130 b(d\027)29517 15339 y Fk(+)29440 16240 y Fd(\014)30305 15849 y Fm(:)1200 18868 y Fr(Similarly)-101 b(,)453 b(using)445 b Fm(\026)p Fr(\(\012)11929 19050 y Fd(\015)12522 18868 y Fr(\))404 b(=)f(1,)454 b(and)445 b(the)g(expression)f(of)h(\012)28843 19050 y Fd(\015)29879 18868 y Fr(in)g([10],)453 b(w)-34 b(e)445 b(ha)-34 b(v)g(e)445 b Fm(\015)39365 18428 y Fk(+)40152 18868 y Fr(\()p Fm(f)130 b Fr(\))405 b(=)e Fm(\015)44263 18428 y Ft(\000)45050 18868 y Fr(\()p Fm(f)130 b Fr(\),)456 b Fm(\026)p Fr(-a.s.)1200 20374 y(and)405 b(hence)17090 20484 y Ff(Z)18504 22134 y Fm(f)130 b(d\026)338 b Fr(=)22205 20484 y Ff(Z)23619 22134 y Fm(\015)24314 21624 y Ft(\000)24247 22516 y Fk(\003)25101 22134 y Fr(\()p Fm(f)130 b Fr(\))p Fm(d\026)676 b Fl(\025)d Fm(\015)31114 21624 y Ft(\000)31047 22516 y Fk(\003)31901 22134 y Fm(f)130 b Fr(\()p Fl(\000)p Fr(\))1200 24686 y(whic)-34 b(h)405 b(giv)-34 b(es)20958 25033 y Ff(Z)22372 26682 y Fm(f)130 b(d\026)338 b Fl(\025)26073 25033 y Ff(Z)27487 26682 y Fm(f)130 b(d\027)29517 26173 y Ft(\000)29440 27074 y Fd(\014)30305 26682 y Fm(:)p 49878 29123 522 731 v 3082 31461 a Fr(The)404 b(follo)-34 b(wing)405 b(corollary)e(pro)-34 b(v)g(es)404 b(Theorem)g(5.2)g(using)h(Theorem)f (3.10.)1200 33968 y Fs(Prop)39 b(osition)465 b(5.5)2039 b Fj(1.)606 b Fm(e)14718 33528 y Fk(+)14718 34267 y Fd(\027)15181 34423 y Fg(\014)16126 33968 y Fr(=)336 b Fl(\000)202 b Fr(lim)20234 34263 y Fk(\003)p Ft(")p Fh(Z)21947 34011 y Fg(d)23172 33490 y Fk(1)p 22819 33688 1176 49 v 22819 34401 a Ft(j)p Fk(\003)p Ft(j)24329 33968 y Fr(log)219 b Fm(\027)26773 33458 y Fk(+)26696 34359 y Fd(\014)27560 33968 y Fr(\(+)28974 34156 y Fk(\003)29682 33968 y Fr(\))434 b Fj(exists.)2633 36596 y(2.)605 b(F)-93 b(or)434 b(any)f Fm(\026)337 b Fl(2)g(G)72 b Fr(\()p Fm(\015)12987 36156 y Fk(+)13774 36596 y Fr(\))p Fj(,)21057 38738 y Fr(lim)20801 39654 y Fk(\003)p Ft(")p Fh(Z)22514 39402 y Fg(d)23785 37918 y Fr(1)p 23331 38460 1516 49 v 23331 39570 a Fl(j)p Fr(\003)p Fl(j)25181 38738 y Fr(log)27448 37918 y Fm(\026)p Fr(\(+)29592 38106 y Fk(\003)30300 37918 y Fr(\))p 27081 38460 4057 49 v 27081 39650 a Fm(\027)27757 39140 y Fk(+)27680 40041 y Fd(\014)28544 39650 y Fr(\(+)29958 39838 y Fk(\003)30666 39650 y Fr(\))31607 38738 y(=)337 b(0)p Fm(:)1200 42167 y Fj(Pr)-62 b(o)g(of)p Fr(.)2681 44673 y(1.)606 b(F)-101 b(ollo)-34 b(ws)405 b(from)f(subadditivit)-34 b(y)406 b(and)f(p)34 b(ositiv)-34 b(e)403 b(correlations.)2681 47066 y(2.)606 b(F)-101 b(ollo)-34 b(ws)405 b(from)f(sto)34 b(c)-34 b(hastic)405 b(domination)g(\(5.4\))g(and)14953 50451 y(lim)14697 51366 y Fk(\003)p Ft(")p Fh(Z)16410 51114 y Fg(d)17682 49631 y Fr(1)p 17228 50172 1516 49 v 17228 51282 a Fl(j)p Fr(\003)p Fl(j)19078 50451 y Fr(log)20978 49631 y Fm(\027)21654 49191 y Fk(+)22441 49631 y Fr(\(+)23855 49819 y Fk(\003)24563 49631 y Fr(\))p 20978 50172 4057 49 v 20978 51362 a Fm(\027)21654 50852 y Ft(\000)21577 51753 y Fd(\014)22441 51362 y Fr(\(+)23855 51550 y Fk(\003)24563 51362 y Fr(\))25504 50451 y(=)593 b(lim)26784 51366 y Fk(\003)p Ft(")p Fh(Z)28497 51114 y Fg(d)29768 49631 y Fr(1)p 29314 50172 1516 49 v 29314 51282 a Fl(j)p Fr(\003)p Fl(j)31164 50451 y Fr(log)33064 49414 y Fm(\026)33794 48904 y Fk(+)33794 49806 y Fd(\014)34582 49414 y Fr(\(+)35996 49611 y Fd(b)p Fk(\003)37106 49414 y Fr(\))p 33064 50172 4514 49 v 33064 51362 a Fm(\026)33794 50852 y Ft(\000)33794 51753 y Fd(\014)34582 51362 y Fr(\(+)35996 51559 y Fd(b)p Fk(\003)37106 51362 y Fr(\))38047 50451 y(=)337 b(0)4230 53971 y(where,)531 b(to)507 b(obtain)g(the)f(last)g(equalit)-34 b(y)-101 b(,)531 b(w)-34 b(e)506 b(used)h(that)g Fm(\026)31433 53461 y Fk(+)31433 54363 y Fd(\014)32221 53971 y Fm(;)202 b(\026)33490 53461 y Ft(\000)33490 54363 y Fd(\014)34783 53971 y Fr(are)505 b(the)i(Ising)f(plus)g(and)h(min)-34 b(us)4230 55477 y(phases.)p 49878 57983 522 731 v 1200 61322 a Fs(Remark)464 b(5.6)606 b Fj(We)362 b(c)-62 b(onje)g(ctur)g(e) 360 b(that)h Fm(C)87 b Fr(1)363 b Fj(is)f(satis\014e)-62 b(d)361 b(for)h(any)g(er)-62 b(go)g(dic)361 b(me)-62 b(asur)g(e)360 b Fm(\026)338 b Fl(2)e(G)72 b Fr(\()p Fm(\015)43707 60882 y Fk(+)44495 61322 y Fr(\))363 b Fj(in)g(dimen-)1200 62827 y(sion)443 b Fm(d)354 b Fr(=)f(2)p Fj(.)587 b(This)442 b(amounts)g(of)g(to)h(pr)-62 b(ove)442 b(that)f(the)h(internal)h(spins)f(do)g(not)h(show)g(a)g(phase)e(tr)-62 b(ansition,)1200 64333 y(given)461 b(a)g(\\typic)-62 b(al)459 b(c)-62 b(on\014gur)g(ation)460 b(of)h Fm(\026)p Fj(")h(on)g Fm(b)p Fe(Z)23614 63893 y Fd(d)24616 64333 y Fj(to)f(the)f(left)h(of)g(the)g(origin,)468 b(and)461 b(al)62 b(l)461 b Fr(+)g Fj(on)h Fm(b)p Fe(Z)46334 63893 y Fd(d)47336 64333 y Fj(to)f(the)1200 65838 y(right.)775 b(Fixing)506 b(these)f(de)-62 b(cimate)g(d)503 b(spins)j(acts)f(as)h(a) g(magnetic)f(\014eld,)524 b(pushing)504 b(the)i(spins)f(on)i(the)e (right)1200 67344 y(of)486 b(the)f(origin)h(into)g(a)g(\\plus-like")d (phase)h(and)i(the)g(spins)f(on)i(the)f(left)f(of)h(the)f(origin)h(in)h (a)f(\\plus-like")1200 68849 y(or)418 b(\\minus-like")e(phase,)k(dep) -62 b(ending)416 b(on)j Fm(\026)p Fj(.)553 b(The)418 b(lo)-62 b(c)g(ation)417 b(of)h(the)g(interfac)-62 b(e)416 b(b)-62 b(etwe)g(en)418 b(\\right)f(and)i(left")1200 70355 y(should)466 b(not)i(dep)-62 b(end)466 b(on)i(the)f(b)-62 b(oundary)467 b(c)-62 b(ondition)466 b(in)i Fm(d)400 b Fr(=)f(2)468 b Fj(\(no)g(Basuev)e(tr)-62 b(ansition\).)660 b(However,)475 b(we)1200 71860 y(do)433 b(not)g(have)f(a)h(rigor)-62 b(ous)431 b(pr)-62 b(o)g(of)433 b(of)g(this)f(fact.)25194 75417 y Fr(19)p eop %%Page: 20 20 20 19 bop 1200 1313 a Fo(6)1793 b(Examples)599 b(I)50 b(I:)598 b(Join)-50 b(t)598 b(measure)i(of)d(random)h(spin)h(systems) 1200 4351 y Fr(W)-101 b(e)405 b(consider)g(the)h(join)-34 b(t)407 b(measures)e(of)h(disordered)g(spins-systems)g(on)g(the)g(pro) 34 b(duct)406 b(of)g(spin-space)g(and)1200 5857 y(disorder-space)527 b(de\014ned)h(in)f(terms)g(of)g(a)g(quenc)-34 b(hed)528 b(absolutely)f(con)-34 b(v)g(ergen)g(t)528 b(Gibbs-in)-34 b(teraction)528 b(and)1200 7362 y(an)505 b(a)f(priori-distribution)i (of)f(the)g(disorder)f(v)-67 b(ariables.)839 b(They)505 b(w)-34 b(ere)504 b(treated)h(b)34 b(efore)504 b(in)h([17,)f(18])g(and) 1200 8868 y(pro)-34 b(vide)417 b(a)f(broad)i(class)e(of)h(examples)f (of)h(generalized)f(Gibbs)h(measures.)577 b(A)416 b(sp)34 b(eci\014c)416 b(example)h(of)g(this,)1200 10373 y(the)404 b(GriSing)h(\014eld,)f(w)-34 b(as)405 b(already)e(considered)h(in)g (Section)h(5.1.)3082 12211 y(First)350 b(w)-34 b(e)350 b(pro)-34 b(v)g(e)351 b(that,)361 b(for)350 b(the)h(same)f(quenc)-34 b(hed)350 b(p)34 b(oten)-34 b(tial,)362 b(the)350 b(relativ)-34 b(e)349 b(en)-34 b(trop)g(y)351 b(densit)-34 b(y)351 b(b)34 b(et)-34 b(w)g(een)1200 13716 y(corresp)34 b(onding,)478 b(p)34 b(ossibly)463 b(di\013eren)-34 b(t)464 b(join)-34 b(t)465 b(measures)e(is)h(alw)-34 b(a)g(ys)464 b(zero.)716 b(Next)463 b(w)-34 b(e)464 b(pro)-34 b(v)g(e)464 b(in)f(generalit)-34 b(y)1200 15222 y(that)462 b(these)g(measures)f(are)f(asymptotically)h (decoupling)h(whenev)-34 b(er)461 b(the)h(a-priori)e(distribution)j(of) e(the)1200 16727 y(disorder)396 b(is.)536 b(The)396 b(useful)h(notion)h (of)e(asymptotic)h(decoupled)g(w)-34 b(as)397 b(recen)-34 b(tly)396 b(coined)g(b)-34 b(y)397 b(P\014ster)f(\(2002\),)1200 18233 y(and)483 b(pro)-34 b(vides)482 b(a)g(broad)h(class)f(of)g (measures,)502 b(including)482 b(lo)34 b(cal)481 b(transformations)j (of)e(Gibbs)h(measures,)1200 19738 y(for)524 b(whic)-34 b(h)524 b(the)g(existence)f(of)h(relativ)-34 b(e)523 b(en)-34 b(trop)g(y)524 b(densit)-34 b(y)525 b(and)f(the)g(large)f (deviation)g(principle)h(holds.)1200 21244 y(Using)488 b(results)h(these)f(results,)509 b(w)-34 b(e)488 b(easily)f(obtain)i (existence)f(of)g(the)h(relativ)-34 b(e)487 b(en)-34 b(trop)g(y)489 b(densit)-34 b(y)-101 b(.)791 b(Next)1200 22749 y(w)-34 b(e)562 b(sp)34 b(ecialize)560 b(to)i(the)g(sp)34 b(eci\014c)561 b(example)g(of)h(the)g(random)g(\014eld)g(Ising)g(mo)34 b(del)561 b(in)h(Section)g(6.3.)1010 b(W)-101 b(e)1200 24254 y(fo)34 b(cus)495 b(on)g(the)g(in)-34 b(teresting)495 b(region)g(of)g(the)g(parameter)f(space)h(when)g(there)g(is)f(a)h (phase)g(transition)h(for)1200 25760 y(the)485 b(spin-v)-67 b(ariables,)503 b(for)485 b(almost)f(an)-34 b(y)485 b(con\014guration)h (of)f(disorder)f(v)-67 b(ariables.)778 b(Here)483 b(w)-34 b(e)485 b(sho)-34 b(w)486 b(on)e(the)1200 27265 y(basis)382 b(of)h([18)o(])f(that)h(the)g(join)-34 b(t)383 b(plus)f(and)h(the)g (join)-34 b(t)383 b(min)-34 b(us)383 b(state)f(for)h(the)f(same)g (quenc)-34 b(hed)383 b(p)34 b(oten)-34 b(tial)382 b(are)1200 28771 y(not)446 b(compatible)f(with)g(the)g(same)g(in)-34 b(teraction)445 b(p)34 b(oten)-34 b(tial.)662 b(In)445 b([18)o(])g(it)g(w)-34 b(as)445 b(already)g(sho)-34 b(wn)446 b(that)g(there)1200 30276 y(is)405 b(alw)-34 b(a)g(ys)407 b(a)e(translation-in)-34 b(v)-67 b(arian)-34 b(t)408 b(con)-34 b(v)g(ergen)g(t)406 b(p)34 b(oten)-34 b(tial,)407 b(or)e(a)h(p)34 b(ossibly)405 b(non-translation-in)-34 b(v)-67 b(arian)-34 b(t)1200 31782 y(absolutely)443 b(con)-34 b(v)g(ergen)g(t)444 b(p)34 b(oten)-34 b(tial)444 b(for)f(the)g(corresp) 34 b(onding)443 b(join)-34 b(t)444 b(measure.)655 b(W)-101 b(e)442 b(also)h(discuss)g(this)h(in)1200 33287 y(more)546 b(detail)h(and)g(sk)-34 b(etc)g(h)547 b(a)g(pro)34 b(of)546 b(on)h(the)g(basis)g(of)g([18])f(and)h(the)g(R)-34 b(G-analysis)548 b(of)f(Bricmon)-34 b(t)546 b(and)1200 34793 y(Kupiainen)517 b(\(1988\))h(that)g(sho)-34 b(ws)518 b(that)g(there)e(is)h(a)g (translation-in)-34 b(v)-67 b(arian)-34 b(t)518 b(join)-34 b(t)518 b(p)34 b(oten)-34 b(tial)517 b(that)h(ev)-34 b(en)1200 36298 y(deca)g(ys)455 b(lik)-34 b(e)454 b(a)h(stretc)-34 b(hed)456 b(exp)34 b(onen)-34 b(tial.)692 b(This)456 b(pro)-34 b(vides)455 b(an)h(explicit)e(example)g(of)i(a)f(w)-34 b(eakly)454 b(\(but)j(not)1200 37804 y(almost\))405 b(Gibbsian)g (measure)f(for)g(whic)-34 b(h)405 b(the)f(v)-67 b(ariational)404 b(principle)f(fails.)1200 41382 y Fi(6.1)1495 b(Setup)1200 44802 y Fr(W)-101 b(e)285 b(consider)h(disordered)g(mo)34 b(dels)286 b(of)g(the)h(follo)-34 b(wing)286 b(general)g(t)-34 b(yp)34 b(e.)499 b(W)-101 b(e)285 b(assume)i(that)g(the)f (con\014guration)1200 46307 y(space)293 b(of)h(the)g(quenc)-34 b(hed)294 b(mo)34 b(del)293 b(is)h(again)f(as)h(detailed)f(in)h (Section)g(2.1)f(and)h(w)-34 b(e)294 b(denote)g(the)g(spin)g(v)-67 b(ariables)1200 47813 y(b)-34 b(y)373 b Fm(\033)43 b Fr(.)528 b(Additionally)373 b(w)-34 b(e)373 b(assume)g(that)h(there)f (are)f(also)h(disorder)f(v)-67 b(ariables)372 b Fm(\021)380 b Fr(=)336 b(\()p Fm(\021)40350 47995 y Fd(x)40936 47813 y Fr(\))41407 48109 y Fd(x)p Ft(2)p Fh(Z)43154 47857 y Fg(d)44064 47813 y Fr(en)-34 b(tering)373 b(the)1200 49489 y(game,)504 b(taking)484 b(v)-67 b(alues)484 b(in)g(an)g (in\014nite)h(pro)34 b(duct)485 b(space)f(\()p Fm(E)29138 49049 y Ft(0)29449 49489 y Fr(\))29920 49049 y Fh(Z)30510 48737 y Fg(d)31048 49489 y Fr(,)504 b(where)484 b(again)g Fm(E)39751 49049 y Ft(0)40546 49489 y Fr(is)g(a)g(\014nite)h(set.)778 b(W)-101 b(e)1200 50995 y(denote)418 b(the)f Fj(joint)445 b(variables)414 b Fr(b)-34 b(y)418 b Fm(\030)c Fr(=)358 b(\()p Fm(\030)20072 51177 y Fd(x)20658 50995 y Fr(\))21129 51290 y Fd(x)p Ft(2)p Fh(Z)22876 51038 y Fg(d)23773 50995 y Fr(=)g(\()p Fm(\033)-24 b(;)202 b(\021)43 b Fr(\))359 b(=)f(\()p Fm(\033)30693 51177 y Fd(x)31278 50995 y Fm(;)202 b(\021)32419 51177 y Fd(x)33004 50995 y Fr(\))33475 51290 y Fd(x)p Ft(2)p Fh(Z)35222 51038 y Fg(d)35760 50995 y Fr(.)577 b(It)417 b(will)g(b)34 b(e)417 b(con)-34 b(v)g(enien)g(t)418 b(later)1200 52500 y(also)404 b(to)g(write)g(simply)g(\()p Fm(\033)43 b(\021)g Fr(\))405 b(to)g(denote)g(the)f(pair)g(\()p Fm(\033)-24 b(;)202 b(\021)43 b Fr(\).)3082 54338 y(One)526 b(essen)-34 b(tial)527 b(ingredien)-34 b(t)527 b(of)g(the)f(mo)34 b(del)526 b(is)h(giv)-34 b(en)526 b(b)-34 b(y)527 b(the)g Fj(de\014ning)546 b(p)-62 b(otential)524 b Fr(\010)541 b(=)f(\(\010)46603 54526 y Fd(A)47364 54338 y Fr(\))47835 54633 y Fd(A)p Ft(\032)p Fh(Z)49862 54381 y Fg(d)1200 55843 y Fr(dep)34 b(ending)379 b(on)f(the)g(join)-34 b(t)379 b(v)-67 b(ariables)376 b Fm(\030)393 b Fr(=)337 b(\()p Fm(\033)-24 b(;)202 b(\021)43 b Fr(\).)530 b(\010)25259 56031 y Fd(A)26019 55843 y Fr(\()p Fm(\030)56 b Fr(\))379 b(dep)34 b(ends)378 b(on)h Fm(\030)433 b Fr(only)378 b(through)h Fm(\030)42927 56031 y Fd(A)43688 55843 y Fr(.)529 b(W)-101 b(e)377 b(assume)1200 57348 y(that)f(\010)f(is)f (\014nite)i(range.)528 b(When)375 b(w)-34 b(e)375 b(\014x)g(a)g (realization)f(of)h(the)g(disorder)f Fm(\021)43 b Fr(,)381 b(w)-34 b(e)375 b(ha)-34 b(v)g(e)375 b(a)g(p)34 b(oten)-34 b(tial)376 b(for)e(the)1200 58854 y(spin-v)-67 b(ariables)384 b Fm(\033)427 b Fr(that)385 b(is)f(t)-34 b(ypically)384 b(non-translation)i(in)-34 b(v)-67 b(arian)-34 b(t.)532 b(W)-101 b(e)383 b(then)i(de\014ne)g(the)g(corresp)34 b(onding)1200 60359 y Fj(quenche)-62 b(d)431 b(Gibbs)h(sp)-62 b(e)g(ci\014c)g(ation)401 b Fr(b)-34 b(y)404 b(De\014nition)h(2.8)f (using)h(the)f(notation)9920 63525 y Fm(\026)10700 63024 y Fk(\026)-520 b Fd(\033)10650 63824 y Fk(\003)11358 63525 y Fr([)p Fm(\021)43 b Fr(]\()p Fm(B)61 b Fr(\))337 b(:=)17811 62705 y(1)p 16685 63246 2858 49 v 16685 64356 a Fm(Z)17648 63939 y Fk(\026)-519 b Fd(\033)17512 64738 y Fk(\003)18224 64356 y Fr([)p Fm(\021)43 b Fr(])19878 62373 y Ff(X)20180 64919 y Fd(\033)20718 65068 y Fc(\003)21831 63525 y Fr(1)22437 63713 y Fd(B)23245 63525 y Fr(\()p Fm(\033)24409 63713 y Fk(\003)25182 63525 y Fr(\026)-671 b Fm(\033)25810 63820 y Fh(Z)26400 63568 y Fg(d)26882 63820 y Ft(n)p Fk(\003)28061 63525 y Fr(\))p Fm(e)29096 62881 y Ft(\000)29985 62217 y Fb(P)30978 63158 y Fg(A)p Fc(:)p Fg(A)p Fn(\\)p Fc(\003)p Fn(6)p Fc(=)p Fn(;)34803 62881 y Fk(\010)35483 63030 y Fg(A)36151 62881 y Fk(\()p Fd(\033)37055 63030 y Fc(\003)37712 62881 y Fk(\026)-519 b Fd(\033)38201 63228 y Fa(Z)38644 63037 y Fg(d)39125 63228 y Fn(n)p Fc(\003)40157 62881 y Fd(;\021)32 b Fk(\))41344 63525 y Fm(:)6227 b Fr(\(6.1\))1200 67359 y(Here)578 b(w)-34 b(e)579 b(do)g(not)h(mak)-34 b(e)579 b(the)g(de\014ning)h(p)34 b(oten)-34 b(tial)579 b(explicit)f(an)-34 b(ymore)579 b(in)g(order)f(not)i(to)f(o)-34 b(v)g(erburden)1200 68864 y(notion.)497 b(The)279 b(measures)e(\(6.1\))i(are)e(also)h(called)f (more)g(lo)34 b(osely)277 b Fj(quenche)-62 b(d)314 b(\014nite-volume)i (Gibbs)f(me)-62 b(asur)g(es)p Fr(.)1200 70370 y(Ob)-34 b(viously)-101 b(,)404 b(the)g(\014nite-v)-34 b(olume)405 b(summation)g(is)f(o)-34 b(v)g(er)404 b Fm(\033)27270 70558 y Fk(\003)28314 70370 y Fl(2)337 b Fm(E)30424 69930 y Fk(\003)31132 70370 y Fr(.)25194 75417 y(20)p eop %%Page: 21 21 21 20 bop 3082 1313 a Fr(The)417 b(second)g(ingredien)-34 b(t)417 b(of)g(the)g(quenc)-34 b(hed)418 b(mo)34 b(del)416 b(is)g(the)h(distribution)h(of)f(the)h(disorder)e(v)-67 b(ariables)1200 2819 y Fe(P)p Fr(\()p Fm(d\021)43 b Fr(\).)658 b(Most)444 b(of)g(the)g(times)g(in)f(the)h(theory)g(of)g(disordered)g (systems)f(one)h(considers)g(the)g(case)f(of)h(i.i.d.)1200 4324 y(v)-67 b(ariables,)403 b(but)i(w)-34 b(e)404 b(can)h(and)f(will)g (b)34 b(e)404 b(more)g(general)f(here.)3082 6101 y(The)436 b(ob)67 b(jects)436 b(of)g(in)-34 b(terest)436 b(will)f(then)h(b)34 b(e)436 b(the)g(in\014nite)g(v)-34 b(olume)435 b Fj(joint)462 b(me)-62 b(asur)g(es)433 b Fm(K)42105 5661 y Fk(\026)-519 b Fd(\033)42681 6101 y Fr(\()p Fm(d\030)56 b Fr(\),)444 b(b)-34 b(y)436 b(whic)-34 b(h)1200 7607 y(w)g(e)338 b(understand)i(an)-34 b(y)338 b(limiting)f(measure)g(of)h(lim)23515 7902 y Fk(\003)p Ft(")p Fh(Z)25228 7650 y Fg(d)25968 7607 y Fe(P)p Fr(\()p Fm(d\021)43 b Fr(\))p Fm(\026)29707 7167 y Fk(\026)-520 b Fd(\033)29657 7966 y Fk(\003)30366 7607 y Fr([)p Fm(\021)43 b Fr(]\()p Fm(d\033)g Fr(\))338 b(in)f(the)h(pro)34 b(duct)339 b(top)34 b(ology)337 b(on)h(the)1200 9112 y(space)359 b(of)h(join)-34 b(t)360 b(v)-67 b(ariables.)522 b(Of)360 b(course,)367 b(there)359 b(are)g(examples)f(for)i(di\013eren) -34 b(t)359 b(join)-34 b(t)361 b(measures)e(of)g(the)h(same)1200 10618 y(quenc)-34 b(hed)401 b(Gibbs)h(sp)34 b(eci\014cation)400 b(for)h(di\013eren)-34 b(t)401 b(spin)g(b)34 b(oundary)402 b(conditions)466 b(\026)-671 b Fm(\033)43 b Fr(.)537 b(In)401 b(principle)f(there)h(can)1200 12123 y(ev)-34 b(en)304 b(b)34 b(e)304 b(di\013eren)-34 b(t)305 b(ones)g(for)f(the)h (same)f(spin-b)34 b(oundary)306 b(condition)370 b(\026)-671 b Fm(\033)43 b Fr(,)324 b(dep)34 b(ending)305 b(on)g(the)g (sub-sequence.)3082 13900 y(F)-101 b(or)479 b(all)g(of)h(this)g(the)g (reader)e(migh)-34 b(t)481 b(think)f(of)f(the)h(concrete)f(example)g (of)h(the)g Fj(r)-62 b(andom)502 b(\014eld)g(Ising)1200 15405 y(mo)-62 b(del.)809 b Fr(Here)494 b(the)h(spin)h(v)-67 b(ariables)493 b Fm(\033)19019 15587 y Fd(x)20099 15405 y Fr(tak)-34 b(e)495 b(v)-67 b(alues)494 b(in)h Fl(f\000)p Fr(1)p Fm(;)202 b Fr(1)p Fl(g)p Fr(.)810 b(The)495 b(disorder)g(v)-67 b(ariables)494 b(are)g(giv)-34 b(en)1200 16911 y(b)g(y)479 b(the)h(random)f(\014elds)g Fm(\021)13424 17093 y Fd(x)14488 16911 y Fr(that)h(are)f(i.i.d.)762 b(with)479 b(single-site)g (distribution)h Fe(P)38346 17093 y Fk(0)39351 16911 y Fr(that)g(is)e(supp)34 b(orted)480 b(on)1200 18416 y(a)557 b(\014nite)g(set)g Fl(H)8682 18598 y Fk(0)9764 18416 y Fr(and)h(assumed)f(to)h(b)34 b(e)556 b(symmetric.)996 b(The)557 b(de\014ning)h(p)34 b(oten)-34 b(tial)557 b(\010\()p Fm(\033)-24 b(;)202 b(\021)43 b Fr(\))558 b(is)f(giv)-34 b(en)556 b(b)-34 b(y)1200 19922 y(\010)2075 20168 y Ft(f)p Fd(x;y)32 b Ft(g)4360 19922 y Fr(\()p Fm(\033)-24 b(;)202 b(\021)43 b Fr(\))337 b(=)f Fl(\000)p Fm(\014)64 b(\033)11157 20104 y Fd(x)11741 19922 y Fm(\033)12434 20104 y Fd(y)13332 19922 y Fr(for)346 b(nearest)g(neigh)-34 b(b)34 b(ors)347 b Fm(x;)202 b(y)380 b Fl(2)337 b Fe(Z)28822 19482 y Fd(d)29361 19922 y Fr(,)357 b(\010)30930 20168 y Ft(f)p Fd(x)p Ft(g)32457 19922 y Fr(\()p Fm(\033)-24 b(;)202 b(\021)43 b Fr(\))337 b(=)g Fl(\000)p Fm(h\021)39112 20104 y Fd(x)39697 19922 y Fm(\033)40390 20104 y Fd(x)40974 19922 y Fr(,)357 b(and)347 b(\010)44842 20110 y Fd(A)45939 19922 y Fr(=)337 b(0)346 b(else.)1200 23381 y Fi(6.2)1495 b(Relativ)-42 b(e)498 b(en)-42 b(trop)g(y)500 b(for)f(join)-42 b(t)498 b(measures)1200 26484 y Fr(F)-101 b(or)351 b(the)h(\014rst)f(result)h(w)-34 b(e)351 b(do)h(not)g(need)f(the)h(indep)34 b(endence)351 b(of)h(the)f(disorder)g(\014eld.)521 b(In)352 b(fact,)362 b(without)352 b(an)-34 b(y)1200 27989 y(decoupling)405 b(assumption)g(on)g Fe(P)f Fr(w)-34 b(e)404 b(ha)-34 b(v)g(e)405 b(the)f(follo)-34 b(wing.)1200 30730 y Fs(Theorem)464 b(6.2)606 b Fj(Denote)410 b(by)g Fm(K)16373 30290 y Fk(\026)-519 b Fd(\033)17359 30730 y Fj(and)410 b Fm(K)20855 30290 y Fk(\026)-519 b Fd(\033)21376 29977 y Fn(0)22140 30730 y Fj(two)410 b(joint)g(me)-62 b(asur)g(es)408 b(for)i(the)g(same)f (quenche)-62 b(d)408 b(Gibbs)h(sp)-62 b(e)g(c-)1200 32235 y(i\014c)g(ation)494 b Fm(\026)6669 31795 y Ft(\001)6669 32594 y Fk(\003)7377 32235 y Fr([)p Fm(\021)43 b Fr(]\()p Fm(d\033)g Fr(\))p Fj(,)510 b(obtaine)-62 b(d)493 b(with)h(any)g(two)h (spin)f(b)-62 b(oundary)492 b(c)-62 b(onditions)558 b Fr(\026)-671 b Fm(\033)538 b Fj(\(and)559 b Fr(\026)-671 b Fm(\033)42826 31795 y Ft(0)43632 32235 y Fj(r)-62 b(esp)g(e)g (ctively\),)1200 33741 y(along)497 b(any)g(subse)-62 b(quenc)g(es)494 b Fr(\003)14904 33929 y Fd(N)16299 33741 y Fj(\(and)j Fr(\003)20055 33301 y Ft(0)20055 34100 y Fd(N)21449 33741 y Fj(r)-62 b(esp)g(e)g(ctively\).)746 b(Then)498 b(their)e(r)-62 b(elative)496 b(entr)-62 b(opy)497 b(density)f(van-)1200 35246 y(ishes,)431 b(i.e.,)h Fm(h)p Fr(\()p Fm(K)9357 34806 y Fk(\026)-519 b Fd(\033)9934 35246 y Fl(j)p Fm(K)11436 34806 y Fk(\026)g Fd(\033)11957 34494 y Fn(0)12310 35246 y Fr(\))337 b(=)g(0)p Fj(.)1200 37986 y Fs(Remark)464 b(6.3)606 b Fj(Note)452 b(that)f(we)i(ar)-62 b(e)453 b(mor)-62 b(e)452 b(gener)-62 b(al)452 b(than)g(in)h(the)f (usual)f(set)h(up)g(and)h(we)g(do)f(not)g(ne)-62 b(e)g(d)453 b(to)1200 39492 y(assume)432 b(tr)-62 b(anslation)432 b(invarianc)-62 b(e,)431 b(not)j(even)e(of)h(the)g(de\014ning)g(p)-62 b(otential)431 b Fr(\010)p Fj(.)1200 42232 y Fs(Remark)464 b(6.4)606 b Fj(This)555 b(r)-62 b(esult)554 b(is)h(neither)g(dir)-62 b(e)g(ctly)553 b(r)-62 b(elate)g(d)554 b(to)h(the)g(\014rst)g(p)-62 b(art)555 b(nor)h(to)f(the)g(se)-62 b(c)g(ond)554 b(p)-62 b(art)1200 43738 y(of)481 b(the)f(variational)e(principle.)699 b(It)482 b(do)-62 b(es)479 b(not)i(yield)f(the)g(\014rst)i(p)-62 b(art)480 b(\(which)g(wil)62 b(l)481 b(b)-62 b(e)480 b(pr)-62 b(ove)g(d)480 b(di\013er)-62 b(ently\))1200 45243 y(b)g(e)g(c)g(ause)380 b(it)j(is)g(not)g(cle)-62 b(ar)382 b(that)f(every)h(me)-62 b(asur)g(e)381 b(that)h(is)h(c)-62 b(omp)g(atible)380 b(with)i(the)h(same)f(sp)-62 b(e)g(ci\014c)g(ation) 380 b(as)j Fm(K)49526 44803 y Fk(\026)-519 b Fd(\033)50047 44491 y Fn(0)1200 46749 y Fj(c)-62 b(an)484 b(b)-62 b(e)483 b(written)h(in)h(terms)e(of)h Fm(K)16856 46309 y Fk(\026)-519 b Fd(\033)17432 46749 y Fj(.)709 b(Applie)-62 b(d)482 b(to)h(the)g(r)-62 b(andom)484 b(\014eld)f(Ising)h(mo)-62 b(del)483 b(in)i(Se)-62 b(ction)483 b(6.3,)495 b(this)1200 48254 y(r)-62 b(esult)378 b(wil)62 b(l)380 b(dispr)-62 b(ove)377 b(the)i(se)-62 b(c)g(ond)379 b(p)-62 b(art)378 b(of)i(the)f(variational)e(principle)h(for)h(we)-62 b(akly)379 b(but)g(not)g(almost)g(Gibbs)1200 49760 y(me)-62 b(asur)g(es.)1200 52500 y(Pr)g(o)g(of)p Fr(.)605 b(W)-101 b(e)283 b(ha)-34 b(v)g(e)284 b(from)g(the)g(de\014nition)g(of)g(the)g(join)-34 b(t)285 b(measures)f(as)f(limit)g(p)34 b(oin)-34 b(ts)285 b(with)f(suitable)g(sequences)1200 54006 y(of)404 b(v)-34 b(olumes)9569 55254 y Fm(K)10734 54814 y Fk(\026)-519 b Fd(\033)11310 55254 y Fr(\()p Fm(\033)12474 55442 y Fk(\003)13182 55254 y Fm(\021)13784 55442 y Fk(\003)14492 55254 y Fr(\))p 9420 55795 5693 49 v 9420 56911 a Fm(K)10585 56561 y Fk(\026)g Fd(\033)11106 56309 y Fn(0)11459 56911 y Fr(\()p Fm(\033)12623 57099 y Fk(\003)13331 56911 y Fm(\021)13933 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27817 55795 14363 49 v 27817 56979 a(lim)29501 57167 y Fd(N)30599 56002 y Ff(R)31609 56979 y Fe(P)p Fr(\()p Fm(d)87 b Fr(~)-693 b Fm(\021)44 b Fr(\)1)35175 57161 y Fd(\021)35647 57310 y Fc(\003)36310 56979 y Fm(\026)37090 56561 y Fk(\026)-520 b Fd(\033)37610 56309 y Fn(0)37040 57360 y Fk(\003)37693 57509 y Fg(N)38518 56979 y Fr([)87 b(~)-693 b Fm(\021)43 b Fr(]\()p Fm(\033)41001 57167 y Fk(\003)41709 56979 y Fr(\))47908 56074 y(\(6.5\))1200 59227 y(Here)551 b(and)i(later)f(w) -34 b(e)552 b(will)g(write)g(in)g(short)h(1)22741 59409 y Fd(\021)23213 59558 y Fc(\003)24428 59227 y Fr(for)f(the)h(indicator) f(function)h(of)g(the)f(ev)-34 b(en)g(t)553 b(that)g(the)1200 60733 y(in)-34 b(tegration)417 b(v)-67 b(ariable)503 b(~)-693 b Fm(\021)460 b Fr(coincides)416 b(with)h(the)g(\014xed)g (con\014guration)h Fm(\021)460 b Fr(on)417 b(\003.)575 b(W)-101 b(e)416 b(ha)-34 b(v)g(e)417 b(from)g(the)g(\014nite)1200 62238 y(range)404 b(of)g(the)h(disordered)f(p)34 b(oten)-34 b(tial)405 b(that)15424 64831 y(sup)13229 65890 y Fd(\033)32 b(\021)g Fk(=)p Fd(\033)15605 65638 y Fn(0)15902 65890 y Fd(\021)16406 65638 y Fn(0)17108 65890 y Fr(on)405 b Fk(\003)19444 63437 y Ff(\014)19444 64165 y(\014)19444 64892 y(\014)19848 63680 y(X)20371 66305 y Fd(A)21599 63850 y Ff(\000)22155 64831 y Fr(\010)23030 65019 y Fd(A)23790 64831 y Fr(\()p Fm(\033)43 b(\021)g Fr(\))271 b Fl(\000)e Fr(\010)28471 65019 y Fd(A)29232 64831 y Fr(\()p Fm(\033)30439 64331 y Ft(0)30750 64831 y Fm(\021)31395 64331 y Ft(0)31705 64831 y Fr(\))32176 63850 y Ff(\001)32732 63437 y(\014)32732 64165 y(\014)32732 64892 y(\014)33473 64831 y Fl(\024)337 b Fm(C)35619 65013 y Fk(1)36145 64831 y Fl(j)p Fm(@)67 b Fr(\003)p Fl(j)1200 68232 y Fr(for)455 b(cub)34 b(es)455 b(\003)g(with)h(some)f(\014nite)h(constan)-34 b(t)457 b Fm(C)22649 68414 y Fk(1)23175 68232 y Fr(.)691 b(By)455 b Fm(@)67 b Fr(\003)455 b(w)-34 b(e)455 b(mean)g(the)h Fm(r)34 b Fr(-b)g(oundary)456 b(of)f(\003,)467 b(where)455 b Fm(r)489 b Fr(is)1200 69737 y(the)404 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Fk(^)-520 b Fd(\033)37150 72395 y Fk(\003)37859 72096 y Fr([)p Fm(\021)38798 72284 y Fk(\003)39592 72096 y Fr(^)-693 b Fm(\021)40107 72392 y Fh(Z)40697 72139 y Fg(d)41179 72392 y Ft(n)p Fk(\003)42358 72096 y Fr(]\()p Fm(\033)43859 72284 y Fk(\003)44566 72096 y Fr(\))25194 75417 y(21)p eop %%Page: 22 22 22 21 bop 1200 1313 a Fr(for)416 b(an)-34 b(y)417 b(join)-34 b(t)417 b(reference)e(con\014guration)483 b(^)-671 b Fm(\033)130 b Fr(^)-693 b Fm(\021)43 b Fr(.)575 b(This)417 b(giv)-34 b(es)416 b(the)g(upp)34 b(er)417 b(b)34 b(ound)417 b Fm(e)38619 874 y Fk(4)p Fd(C)39760 997 y Fc(1)40222 874 y Ft(j)p Fd(@)52 b Fk(\003)p Ft(j)42417 1313 y Fr(on)416 b(the)h(r.h.s.)574 b(of)1200 2819 y(\(6.5\),)497 b(b)-34 b(y)479 b(application)g(of)g(the)f(last)h(inequalities)f(on)g(n)-34 b(umerator)480 b(and)f(denominator)g(of)g(\(6.5\))g(for)f(the)1200 4324 y(same)404 b(reference)f(con\014guration.)3082 6142 y(This)540 b(implies)f(for)h(the)g(\014nite-v)-34 b(olume)540 b(relativ)-34 b(e)539 b(en)-34 b(trop)g(y)540 b(an)g(upp)34 b(er)540 b(b)34 b(ound)541 b(of)f(the)g(order)f(of)h(the)1200 7648 y(b)34 b(oundary)-101 b(,)405 b(i.e.,)10796 11344 y Fm(h)11494 11532 y Fk(\003)12202 11344 y Fr(\()p Fm(K)13838 10843 y Fk(\026)-519 b Fd(\033)14415 11344 y Fl(j)p Fm(K)15917 10843 y Fk(\026)g Fd(\033)16438 10531 y Fn(0)16791 11344 y Fr(\))337 b(=)19115 10192 y Ff(X)18878 12738 y Fd(\033)19416 12887 y Fc(\003)20023 12738 y Fd(\021)20495 12887 y Fc(\003)21305 11344 y Fm(K)22470 10843 y Fk(\026)-519 b Fd(\033)23046 11344 y Fr(\()p Fm(\033)24210 11532 y Fk(\003)24918 11344 y Fm(\021)25520 11532 y Fk(\003)26228 11344 y Fr(\))202 b(log)28951 10524 y Fm(K)30116 10084 y Fk(\026)-519 b Fd(\033)30692 10524 y Fr(\()p Fm(\033)31856 10712 y Fk(\003)32564 10524 y Fm(\021)33166 10712 y Fk(\003)33873 10524 y Fr(\))p 28802 11065 5693 49 v 28802 12182 a Fm(K)29967 11832 y Fk(\026)g Fd(\033)30488 11579 y Fn(0)30841 12182 y Fr(\()p Fm(\033)32005 12370 y Fk(\003)32713 12182 y Fm(\021)33315 12370 y Fk(\003)34022 12182 y Fr(\))34963 11344 y Fl(\024)337 b Fr(4)p Fm(C)37715 11526 y Fk(1)38241 11344 y Fl(j)p Fm(@)67 b Fr(\003)p Fl(j)p Fm(:)3082 15235 y Fr(F)-101 b(rom)542 b(that)i(clearly)d(follo)-34 b(ws)543 b(the)f(claim)g Fm(h)p Fr(\()p Fm(K)25232 14796 y Fk(\026)-519 b Fd(\033)25809 15235 y Fl(j)p Fm(K)27311 14796 y Fk(\026)g Fd(\033)27832 14483 y Fn(0)28185 15235 y Fr(\))567 b Fl(\024)g Fr(lim)202 b(sup)34444 15526 y Fd(n)p Ft("1)37452 14758 y Fk(1)p 36816 14956 1742 49 v 36816 15669 a Ft(j)p Fk(\003)37730 15780 y Fg(n)38297 15669 y Ft(j)38691 15235 y Fm(h)39389 15423 y Fk(\003)40042 15534 y Fg(n)40664 15235 y Fr(\()p Fm(K)42300 14796 y Fk(\026)-519 b Fd(\033)42876 15235 y Fl(j)p Fm(K)44378 14796 y Fk(\026)g Fd(\033)44899 14483 y Fn(0)45252 15235 y Fr(\))567 b(=)g(0)543 b(for)1200 16911 y(\(\003)2513 17093 y Fd(n)3139 16911 y Fr(\))3610 17104 y Fd(n)p Ft(2)p Fh(N)5908 16911 y Fr(a)404 b(sequence)g(of)g(cub) 34 b(es.)p 49878 16911 522 731 v 3082 19353 a(Also)468 b(the)h(next)g(theorem)g(can)g(b)34 b(e)468 b(pro)-34 b(v)g(ed)469 b(in)g(a)g(natural)g(w)-34 b(a)g(y)469 b(when)h(w)-34 b(e)469 b(relax)e(the)j(indep)34 b(endence)1200 20858 y(assumption)434 b(of)e(the)g(a)g(priori)f(distribution)i Fe(P)f Fr(of)g(the)h(disorder)e(v)-67 b(ariables.)621 b(It)432 b(sa)-34 b(ys)432 b(that)h(the)g(prop)34 b(ert)-34 b(y)1200 22364 y(of)480 b(b)34 b(eing)479 b Fj(asymptotic)-62 b(al)62 b(ly)499 b(de)-62 b(c)g(ouple)g(d)475 b Fr(carries)j(o)-34 b(v)g(er)480 b(from)g(the)g(distribution)g(of)g(the)g(disorder)g (\014elds)f(to)1200 23869 y(an)-34 b(y)405 b(corresp)34 b(onding)404 b(join)-34 b(t)405 b(distribution.)540 b(F)-101 b(ollo)-34 b(wing)405 b([27)o(],)f(w)-34 b(e)404 b(giv)-34 b(e)404 b(the)g(follo)-34 b(wing)1200 26896 y Fs(De\014nition)466 b(6.6)606 b Fj(A)462 b(pr)-62 b(ob)g(ability)458 b(me)-62 b(asur)g(e)459 b Fe(P)387 b Fl(2)g(M)25825 26386 y Fk(+)25825 27266 y(1)p Fd(;)p Fk(in)-26 b(v)28341 26896 y Fj(is)461 b(c)-62 b(al)62 b(le)-62 b(d)459 b(asymptotic)-62 b(al)62 b(ly)457 b(de)-62 b(c)g(ouple)g(d)458 b(\(AD\))i(if)1200 28402 y(ther)-62 b(e)432 b(exists)g(se)-62 b(quenc)g(es)431 b Fm(g)13538 28584 y Fd(n)14165 28402 y Fj(,)i Fm(c)15495 28584 y Fd(n)16554 28402 y Fj(such)f(that)18763 31188 y Fr(lim)18378 31916 y Fd(n)p Ft(!1)21661 30368 y Fm(c)22186 30550 y Fd(n)p 21166 30910 2142 49 v 21166 32020 a Fl(j)p Fr(\003)22345 32202 y Fd(n)22971 32020 y Fl(j)23777 31188 y Fr(=)336 b(0)p Fm(;)1260 b Fr(lim)26874 31916 y Fd(n)p Ft(!1)29662 30368 y Fm(g)30240 30550 y Fd(n)p 29662 30910 1205 49 v 29901 32020 a Fm(n)31336 31188 y Fr(=)336 b(0)1200 34441 y Fj(and)433 b(for)g(al)62 b(l)433 b Fm(A)337 b Fl(2)f(F)10486 34629 y Fk(\003)11139 34740 y Fg(n)11761 34441 y Fj(,)433 b Fm(B)398 b Fl(2)337 b(F)15899 34629 y Fk(\003)16552 34324 y Fg(c)16552 34908 y(n)p Fc(+)p Fg(g)18088 35019 y(n)19199 34441 y Fj(with)433 b Fe(P)p Fr(\()p Fm(A)p Fr(\))p Fe(P)p Fr(\()p Fm(B)61 b Fr(\))338 b Fl(6)p Fr(=)e(0)p Fj(:)19295 37975 y Fm(e)19859 37475 y Ft(\000)p Fd(c)20999 37586 y Fg(n)21957 37975 y Fl(\024)23538 37155 y Fe(P)p Fr(\()p Fm(A)269 b Fl(\\)h Fm(B)61 b Fr(\))p 23369 37696 5257 49 v 23369 38807 a Fe(P)p Fr(\()p Fm(A)p Fr(\))p Fe(P)p Fr(\()p Fm(B)g Fr(\))29095 37975 y Fl(\024)337 b Fm(e)30939 37475 y Fd(c)31347 37586 y Fg(n)31968 37975 y Fm(:)15603 b Fr(\(6.7\))1200 41584 y Fs(Theorem)464 b(6.8)606 b Fj(Supp)-62 b(ose)429 b Fe(P)i Fj(is)f(asymptotic)-62 b(al)62 b(ly)427 b(de)-62 b(c)g(ouple)g(d)427 b(with)j(functions)g Fm(g)38345 41766 y Fd(n)39402 41584 y Fj(and)h Fm(c)42279 41766 y Fd(n)42904 41584 y Fj(.)557 b(Assume)429 b(that)1200 43089 y Fm(K)2365 42649 y Fk(\026)-519 b Fd(\033)3336 43089 y Fj(is)395 b(a)f(c)-62 b(orr)g(esp)g(onding)393 b(tr)-62 b(anslation)393 b(invariant)h(joint)h(me)-62 b(asur)g(e)393 b(of)i(a)f(quenche)-62 b(d)393 b(r)-62 b(andom)395 b(system,)401 b(with)1200 44595 y(a)419 b(de\014ning)g (\014nite)g(r)-62 b(ange)418 b(p)-62 b(otential.)550 b(Then)419 b Fm(K)22964 44155 y Fk(\026)-519 b 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Fk(\003)19783 52184 y Fg(n)20405 51885 y Fm(\030)20935 52073 y Fd(V)21748 51885 y Fr(\))p 15984 52426 7266 49 v 15984 53537 a Fm(K)g Fr(\()p Fm(\030)18101 53725 y Fk(\003)18754 53836 y Fg(n)19376 53537 y Fr(\))p Fm(K)g Fr(\()p Fm(\030)21964 53725 y Fd(V)22778 53537 y Fr(\))23719 52705 y(=)26160 51885 y Fm(K)g Fr(\()p Fm(\033)28440 52073 y Fk(\003)29093 52184 y Fg(n)29715 51885 y Fm(\021)30317 52073 y Fk(\003)30970 52184 y Fg(n)31591 51885 y Fm(\033)32284 52073 y Fd(V)33096 51885 y Fm(\021)33698 52073 y Fd(V)34510 51885 y Fr(\))p 25131 52426 10881 49 v 25131 53537 a Fm(K)g Fr(\()p Fm(\033)27411 53725 y Fk(\003)28064 53836 y Fg(n)28685 53537 y Fm(\021)29287 53725 y Fk(\003)29940 53836 y Fg(n)30562 53537 y Fr(\))p Fm(K)g Fr(\()p Fm(\033)33313 53725 y Fd(V)34126 53537 y Fm(\021)34728 53725 y Fd(V)35540 53537 y Fr(\))36481 52705 y Fl(\024)336 b Fm(e)38324 52205 y Fd(c)38732 51892 y Fn(0)38732 52426 y Fg(n)39354 52705 y Fm(:)8217 b Fr(\(6.9\))1200 55891 y(W)-101 b(e)403 b(only)h(sho)-34 b(w)406 b(the)e(upp)34 b(er)405 b(b)34 b(ound.)539 b(It)405 b(su\016ces)f(to)g(sho)-34 b(w)15275 59439 y(lim)202 b(sup)16845 60402 y Fd(n)21147 58283 y Fm(K)22312 57844 y Fk(\026)-519 b Fd(\033)22267 58631 y Fk(~)22176 58855 y(\003)22829 58966 y Fg(n)23450 58283 y Fr(\()p Fm(\033)24614 58471 y Fk(\003)25267 58582 y Fg(n)25889 58283 y Fm(\021)26491 58471 y Fk(\003)27144 58582 y Fg(n)27765 58283 y Fm(\033)28458 58471 y Fd(V)29270 58283 y Fm(\021)29872 58471 y Fd(V)30684 58283 y Fr(\))p 19320 59160 13662 49 v 19320 60270 a Fm(K)20485 59853 y Fk(\026)g Fd(\033)20440 60640 y Fk(~)20349 60864 y(\003)21002 61013 y Fg(N)21827 60270 y Fr(\()p Fm(\033)22991 60458 y Fk(\003)23644 60569 y Fg(n)24266 60270 y Fm(\021)24868 60458 y Fk(\003)25521 60569 y Fg(n)26142 60270 y Fr(\))p Fm(K)27778 59853 y Fk(\026)g Fd(\033)27733 60640 y Fk(~)27642 60864 y(\003)28295 61013 y Fg(N)29120 60270 y Fr(\()p Fm(\033)30284 60458 y Fd(V)31096 60270 y Fm(\021)31698 60458 y Fd(V)32510 60270 y Fr(\))33451 59439 y Fl(\024)337 b Fm(e)35295 58938 y Fd(c)35703 59049 y Fg(n)1200 63338 y Fr(for)404 b(an)-34 b(y)405 b(sequence)10486 63032 y(~)10368 63338 y(\003)11210 63526 y Fd(N)12107 63338 y Fr(.)538 b(The)405 b(quan)-34 b(tit)g(y)405 b(under)g(the)f(limsup)h (equals)17599 64783 y Ff(R)18610 65760 y Fe(P)p Fr(\()p Fm(d)87 b Fr(~)-693 b Fm(\021)43 b Fr(\)1)22175 65942 y Fd(\021)22647 66091 y Fc(\003)23199 66202 y Fg(n)23877 65760 y Fr(1)24483 65942 y Fd(\021)24955 66091 y Fg(V)25723 65760 y Fm(\026)26503 65320 y Fk(\026)-520 b Fd(\033)26544 66108 y Fk(~)26453 66332 y(\003)27106 66481 y Fg(N)27930 65760 y Fr([)87 b(~)-693 b Fm(\021)43 b Fr(]\()p Fm(\033)30413 65948 y Fk(\003)31066 66059 y Fg(n)31687 65760 y Fm(\033)32380 65948 y Fd(V)33192 65760 y Fr(\))p 12314 66674 26635 49 v 12314 66820 a Ff(R)13324 67796 y Fe(P)p Fr(\()p Fm(d)87 b Fr(~)-693 b Fm(\021)15769 67978 y Fk(1)16295 67796 y Fr(\)1)17372 67978 y Fd(\021)17844 68127 y Fc(\003)18396 68238 y Fg(n)19074 67796 y Fm(\026)19854 67379 y Fk(\026)-520 b Fd(\033)19895 68167 y Fk(~)19804 68390 y(\003)20457 68539 y Fg(N)21282 67796 y Fr([)87 b(~)-693 b Fm(\021)22221 67978 y Fk(1)22746 67796 y Fr(]\()p Fm(\033)24247 67984 y Fk(\003)24900 68095 y Fg(n)25521 67796 y Fr(\))26194 66820 y Ff(R)27205 67796 y Fe(P)p Fr(\()p Fm(d)87 b Fr(~)-693 b Fm(\021)29650 67978 y Fk(2)30175 67796 y Fr(\)1)31252 67978 y Fd(\021)31724 68127 y Fg(V)32492 67796 y Fm(\026)33272 67379 y Fk(\026)-520 b Fd(\033)33313 68167 y Fk(~)33222 68390 y(\003)33875 68539 y Fg(N)34700 67796 y Fr([)87 b(~)-693 b Fm(\021)35639 67978 y Fk(2)36164 67796 y Fr(]\()p Fm(\033)37665 67984 y Fd(V)38477 67796 y Fr(\))39082 66953 y Fm(:)7883 b Fr(\(6.10\))1200 70590 y(Lo)34 b(ok)370 b(at)h(the)g(term)g(under)h(the)f(disorder-in)-34 b(tegral)371 b(in)g(the)g(n)-34 b(umerator.)528 b(W)-101 b(e)371 b(ha)-34 b(v)g(e)371 b(b)-34 b(y)371 b(the)h(compatibilit)-34 b(y)1200 72096 y(of)404 b(the)h(quenc)-34 b(hed)405 b(k)-34 b(ernels)403 b(that)25194 75417 y(22)p eop %%Page: 23 23 23 22 bop 11297 3455 a Fm(\026)12077 2954 y Fk(\026)-520 b Fd(\033)12118 3742 y Fk(~)12027 3966 y(\003)12680 4115 y Fg(N)13505 3455 y Fr([)p Fm(\021)14444 3643 y Fk(\003)15097 3754 y Fg(n)15718 3455 y Fm(\021)16320 3643 y Fd(V)17219 3455 y Fr(~)-693 b Fm(\021)17734 3751 y Fh(Z)18324 3499 y Fg(d)18806 3751 y Ft(n)p Fk(\(\003)20295 3862 y Fg(n)20862 3751 y Ft([)p Fd(V)209 b Fk(\))22667 3455 y Fr(]\(1)24081 3637 y Fd(\033)24619 3786 y Fc(\003)25171 3897 y Fg(n)25849 3455 y Fr(1)26455 3637 y Fd(\033)26993 3786 y Fg(V)27760 3455 y Fr(\))9248 6207 y(=)11297 4558 y Ff(Z)12711 6207 y Fm(\026)13491 5707 y Fk(\026)-520 b Fd(\033)13532 6495 y Fk(~)13441 6719 y(\003)14094 6868 y Fg(N)14919 6207 y Fr([)p Fm(\021)15858 6395 y Fk(\003)16511 6506 y Fg(n)17132 6207 y Fm(\021)17734 6395 y Fd(V)18633 6207 y Fr(~)-693 b Fm(\021)19148 6503 y Fh(Z)19738 6251 y Fg(d)20220 6503 y Ft(n)p Fk(\(\003)21709 6614 y Fg(n)22276 6503 y Ft([)p Fd(V)209 b Fk(\))24082 6207 y Fr(]\()p Fm(d)65 b Fr(~)-671 b Fm(\033)43 b Fr(\)1)27334 6389 y Fd(\033)27872 6538 y Fg(V)28639 6207 y Fm(\026)29419 5707 y Fk(~)-520 b Fd(\033)29369 6507 y Fk(\003)30022 6618 y Fg(n)30644 6207 y Fr([)p Fm(\021)31583 6395 y Fk(\003)32236 6506 y Fg(n)32857 6207 y Fm(\021)33459 6395 y Fd(V)34358 6207 y Fr(~)-693 b Fm(\021)34873 6503 y Fh(Z)35463 6251 y Fg(d)35945 6503 y Ft(n)p Fk(\(\003)37434 6614 y Fg(n)38001 6503 y Ft([)p Fd(V)209 b Fk(\))39807 6207 y Fr(]\(1)41221 6389 y Fd(\033)41759 6538 y Fc(\003)42311 6649 y Fg(n)42988 6207 y Fr(\))9248 8859 y Fl(\024)1106 b Fm(e)11861 8358 y Fk(2)p Fd(C)13002 8481 y Fc(1)13464 8358 y Ft(j)p Fd(@)52 b Fk(\003)14926 8469 y Fg(n)15492 8358 y Ft(j)15809 8859 y Fm(\026)16589 8358 y Fk(^)-520 b Fd(\033)16539 9158 y Fk(\003)17192 9269 y Fg(n)17814 8859 y Fr([)p Fm(\021)18753 9047 y Fk(\003)19406 9158 y Fg(n)20114 8859 y Fr(^)-693 b Fm(\021)20629 9155 y Fh(Z)21219 8902 y Fg(d)21701 9155 y Ft(n)p Fk(\003)22824 9266 y Fg(n)23446 8859 y Fr(]\()p Fm(\033)24947 9047 y Fk(\003)25600 9158 y Fg(n)26221 8859 y Fr(\))270 b Fl(\002)f Fm(\026)28954 8358 y Fk(\026)-520 b Fd(\033)28995 9146 y Fk(~)28904 9370 y(\003)29557 9519 y Fg(N)30381 8859 y Fr([)p Fm(\021)31320 9047 y Fk(\003)31973 9158 y Fg(n)32594 8859 y Fm(\021)33196 9047 y Fd(V)34096 8859 y Fr(~)-693 b Fm(\021)34611 9155 y Fh(Z)35201 8902 y Fg(d)35683 9155 y Ft(n)p Fk(\(\003)37172 9266 y Fg(n)37738 9155 y Ft([)p Fd(V)210 b Fk(\))39544 8859 y Fr(]\(1)40958 9041 y Fd(\033)41496 9190 y Fg(V)42263 8859 y Fr(\))1200 12838 y(where)335 b(the)h(inequalit)-34 b(y)335 b(follo)-34 b(ws)336 b(from)f(the)h(uniform)g(absolute)f(con)-34 b(v)g(ergence)335 b(of)h(the)f(quenc)-34 b(hed)336 b(p)34 b(oten)-34 b(tial,)1200 14343 y(for)404 b(an)-34 b(y)405 b(reference)e(con\014guration)470 b(^)-671 b Fm(\033)130 b Fr(^)-693 b Fm(\021)44 b Fr(.)3082 16084 y(W)-101 b(e)403 b(use)i(that)10752 18225 y Fm(\026)11532 17725 y Fk(\026)-520 b Fd(\033)11573 18512 y Fk(~)11482 18736 y(\003)12135 18847 y Fg(n)12757 18225 y Fr([)p Fm(\021)13696 18413 y Fk(\003)14349 18524 y Fg(n)14970 18225 y Fr(\()87 b(~)-693 b Fm(\021)16043 18407 y Fk(1)16569 18225 y Fr(\))17040 18521 y Fh(Z)17630 18269 y Fg(d)18113 18521 y Ft(n)p Fk(\003)19236 18632 y Fg(n)19858 18225 y Fr(]\()p Fm(\033)21359 18413 y Fk(\003)22012 18524 y Fg(n)22633 18225 y Fr(\))337 b Fl(\025)f Fm(e)25284 17725 y Ft(\000)p Fk(2)p Fd(C)27157 17848 y Fc(1)27619 17725 y Ft(j)p Fd(@)52 b Fk(\003)29081 17836 y Fg(n)29647 17725 y Ft(j)29964 18225 y Fm(\026)30744 17725 y Fk(^)-520 b Fd(\033)30694 18525 y Fk(\003)31347 18636 y Fg(n)31969 18225 y Fr([)p Fm(\021)32908 18413 y Fk(\003)33561 18524 y Fg(n)34269 18225 y Fr(^)-693 b Fm(\021)34784 18521 y Fh(Z)35374 18269 y Fg(d)35856 18521 y Ft(n)p Fk(\003)36979 18632 y Fg(n)37601 18225 y Fr(]\()p Fm(\033)39102 18413 y Fk(\003)39755 18524 y Fg(n)40376 18225 y Fr(\))1200 20503 y(and)455 b(the)g(similar)f(lo) -34 b(w)g(er)454 b(b)34 b(ound)456 b(on)f(the)g(\014rst)g(disorder-in) -34 b(tegral)454 b(in)h(the)g(denominator)g(of)g(\(6.10\))g(with)1200 22008 y(the)526 b(same)f(reference)f(join)-34 b(t)527 b(reference)d(con\014guration)591 b(^)-671 b Fm(\033)131 b Fr(^)-693 b Fm(\021)43 b Fr(.)902 b(F)-101 b(rom)525 b(this)h(w)-34 b(e)526 b(get)f(an)h(upp)34 b(er)526 b(b)34 b(ound)526 b(on)1200 23514 y(\(6.10\))405 b(in)f(the)g(form)h(of)13385 27535 y Fm(e)13949 27034 y Fk(4)p Fd(C)15090 27157 y Fc(1)15551 27034 y Ft(j)p Fd(@)52 b Fk(\003)17013 27145 y Fg(n)17580 27034 y Ft(j)20839 25365 y Ff(R)21849 26341 y Fe(P)p Fr(\()p Fm(d)87 b Fr(~)-693 b Fm(\021)43 b Fr(\)1)25414 26523 y Fd(\021)25886 26672 y Fc(\003)26438 26783 y Fg(n)27116 26341 y Fr(1)27722 26523 y Fd(\021)28194 26672 y Fg(V)28962 26341 y Fm(\026)29742 25902 y Fk(\026)-520 b Fd(\033)29783 26689 y Fk(~)29692 26913 y(\003)30345 27062 y Fg(N)31170 26341 y Fr([)87 b(~)-693 b Fm(\021)43 b Fr(]\()p Fm(\033)33653 26529 y Fd(V)34465 26341 y Fr(\))p 18029 27256 19717 49 v 18029 27401 a Ff(R)19039 28378 y Fe(P)p Fr(\()p Fm(d)87 b Fr(~)-693 b Fm(\021)21484 28560 y Fk(1)22010 28378 y Fr(\)1)23087 28560 y Fd(\021)23559 28709 y Fc(\003)24111 28820 y Fg(n)24991 27401 y Ff(R)26001 28378 y Fe(P)p Fr(\()p Fm(d)87 b Fr(~)-693 b Fm(\021)28446 28560 y Fk(2)28972 28378 y Fr(\)1)30049 28560 y Fd(\021)30521 28709 y Fg(V)31289 28378 y Fm(\026)32069 27960 y Fk(\026)-520 b Fd(\033)32110 28748 y Fk(~)32019 28972 y(\003)32672 29121 y Fg(N)33497 28378 y Fr([)87 b(~)-693 b Fm(\021)34436 28560 y Fk(2)34961 28378 y Fr(]\()p Fm(\033)36462 28566 y Fd(V)37274 28378 y Fr(\))37878 27535 y Fm(:)9087 b Fr(\(6.11\))3082 30737 y(Last)372 b(w)-34 b(e)373 b(need)g(to)g(con)-34 b(trol)372 b(the)h(in\015uence)g(of)g(the)g(v)-67 b(ariation)372 b(of)g(the)h(random)g(\014elds)g(inside)f(the)h(\014nite)1200 32243 y(v)-34 b(olume)404 b Fm(\021)5977 32431 y Fk(\003)6630 32542 y Fg(n)7655 32243 y Fr(on)h(the)f(Gibbs-exp)34 b(ectation)405 b(outside.)539 b(W)-101 b(e)404 b(ha)-34 b(v)g(e)405 b(that)12098 34591 y Fm(\026)12878 34090 y Fk(\026)-520 b Fd(\033)12919 34878 y Fk(~)12828 35102 y(\003)13481 35251 y Fg(N)14306 34591 y Fr([)p Fm(\021)15245 34779 y Fk(\003)15898 34890 y Fg(n)16606 34591 y Fr(~)-693 b Fm(\021)17121 34886 y Fh(Z)17711 34634 y Fg(d)18193 34886 y Ft(n)p Fk(\003)19316 34997 y Fg(n)19938 34591 y Fr(]\()p Fm(\033)21439 34779 y Fd(V)22251 34591 y Fr(\))337 b Fl(\024)g Fm(e)24903 34090 y Fk(2)p Fd(C)26044 34213 y Fc(1)26505 34090 y Ft(j)p Fd(@)52 b Fk(\003)27967 34201 y Fg(n)28534 34090 y Ft(j)28851 34591 y Fm(\026)29631 34090 y Fk(\026)-520 b Fd(\033)29672 34878 y Fk(~)29581 35102 y(\003)30234 35251 y Fg(N)31058 34591 y Fr([)p Fm(\021)32040 33952 y Fk(\(1\))31997 34972 y(\003)32650 35083 y Fg(n)33385 34591 y Fr(~)-693 b Fm(\021)33900 34886 y Fh(Z)34490 34634 y Fg(d)34972 34886 y Ft(n)p Fk(\003)36095 34997 y Fg(n)36717 34591 y Fr(]\()p Fm(\033)38218 34779 y Fd(V)39030 34591 y Fr(\))1200 37101 y(for)404 b(an)-34 b(y)405 b(con\014gurations)g Fm(\021)448 b Fr(and)405 b Fm(\021)17218 36661 y Fk(\(1\))18880 37101 y Fr(inside)f(\003)23163 37283 y Fd(n)23789 37101 y Fr(.)538 b(This)405 b(giv)-34 b(es)404 b(the)g(follo)-34 b(wing)405 b(upp)34 b(er)404 b(b)34 b(ound)406 b(on)e(\(6.11\))16613 40041 y Fm(e)17177 39541 y Fk(8)p Fd(C)18318 39664 y Fc(1)18780 39541 y Ft(j)p Fd(@)52 b Fk(\003)20242 39652 y Fg(n)20808 39541 y Ft(j)23826 38151 y Ff(R)24836 39127 y Fe(P)p Fr(\()p Fm(d)87 b Fr(~)-693 b Fm(\021)43 b Fr(\)1)28401 39309 y Fd(\021)28873 39458 y Fc(\003)29425 39569 y Fg(n)30103 39127 y Fr(1)30709 39309 y Fd(\021)31181 39458 y Fg(V)p 21258 39763 13260 49 v 21258 39908 a Ff(R)22268 40885 y Fe(P)p Fr(\()p Fm(d)87 b Fr(~)-693 b Fm(\021)24713 41067 y Fk(1)25238 40885 y Fr(\)1)26315 41067 y Fd(\021)26787 41216 y Fc(\003)27339 41327 y Fg(n)28219 39908 y Ff(R)29230 40885 y Fe(P)p Fr(\()p Fm(d)87 b Fr(~)-693 b Fm(\021)31675 41067 y Fk(2)32200 40885 y Fr(\)1)33277 41067 y Fd(\021)33749 41216 y Fg(V)34650 40041 y Fm(:)1200 42914 y Fr(But)622 b(this,)675 b(b)-34 b(y)622 b(the)g(prop)34 b(ert)-34 b(y)621 b(of)h(asymptotic)g(decoupling)g(of)f(the)h(disorder)f (\014eld,)676 b(is)621 b(b)34 b(ounded)622 b(b)-34 b(y)1200 44420 y Fm(e)1764 43980 y Fk(8)p Fd(C)2905 44103 y Fc(1)3367 43980 y Ft(j)p Fd(@)52 b Fk(\003)4829 44091 y Fg(n)5395 43980 y Ft(j)p Fk(+)p Fd(c)6796 44091 y Fg(n)8016 44420 y Fr(and)600 b(the)f(pro)34 b(of)598 b(of)h(the)g(upp)34 b(er)599 b(b)34 b(ound)600 b(in)f(\(6.9\))g(is)f(done.)1123 b(The)599 b(pro)34 b(of)599 b(of)g(the)g(lo)-34 b(w)g(er)1200 45925 y(b)34 b(ound)405 b(is)f(similar.)p 49878 45925 522 731 v 3082 49077 a(Applying)g(P\014ster's)g(theory)h([27)o(],)f(w) -34 b(e)404 b(ha)-34 b(v)g(e)1200 52269 y Fs(Corollary)464 b(6.12)606 b Fj(Supp)-62 b(ose)381 b Fe(P)h Fj(is)h(asymptotic)-62 b(al)62 b(ly)378 b(de)-62 b(c)g(ouple)g(d)379 b(and)j(that)f Fm(K)36506 51829 y Fk(\026)-519 b Fd(\033)37465 52269 y Fj(is)382 b(a)h(c)-62 b(orr)g(esp)g(onding)380 b(tr)-62 b(ans-)1200 53775 y(lation)604 b(invariant)g(joint)g(me)-62 b(asur)g(e)603 b(of)h(a)h(quenche)-62 b(d)602 b(r)-62 b(andom)605 b(system,)646 b(with)604 b(a)h(de\014ning)g(\014nite)g(r) -62 b(ange)1200 55280 y(p)g(otential.)555 b(Then)433 b Fm(h)p Fr(\()p Fm(K)87 b Fl(j)p Fm(K)13598 54840 y Fk(\026)-519 b Fd(\033)14175 55280 y Fr(\))434 b Fj(exists)e(for)h(al) 62 b(l)433 b(tr)-62 b(anslation)432 b(invariant)g(pr)-62 b(ob)g(ability)430 b(me)-62 b(asur)g(es)432 b Fm(K)87 b Fj(.)1200 57532 y Fr(Moreo)-34 b(v)g(er)404 b(w)-34 b(e)404 b(ha)-34 b(v)g(e)405 b(the)f(follo)-34 b(wing)405 b(explicit)e(form)-34 b(ula:)1200 59784 y Fs(Theorem)464 b(6.13)606 b Fj(Supp)-62 b(ose)472 b(that)h(the)g(de\014ning)i(p)-62 b(otential)472 b Fr(\010\()p Fm(\033)-24 b(;)202 b(\021)43 b Fr(\))475 b Fj(is)f(tr)-62 b(anslation)472 b(invariant)i(and)f(that) 1200 61290 y Fe(P)521 b Fj(is)g(asymptotic)-62 b(al)62 b(ly)517 b(de)-62 b(c)g(ouple)g(d.)817 b(Supp)-62 b(ose)519 b(that)h Fm(K)26421 60850 y Fk(\026)-519 b Fd(\033)27518 61290 y Fj(is)521 b(tr)-62 b(anslation)520 b(invariant)g(joint)h(me)-62 b(asur)g(e)520 b(c)-62 b(on-)1200 62795 y(structe)g(d)388 b(with)j(the)e(b)-62 b(oundary)389 b(c)-62 b(ondition)455 b Fr(\026)-671 b Fm(\033)43 b Fj(.)543 b(Supp)-62 b(ose)389 b(that)g Fm(K)478 b Fj(is)390 b(a)h(tr)-62 b(anslation-invariant)388 b(me)-62 b(asur)g(e)389 b(on)1200 64301 y(the)432 b(pr)-62 b(o)g(duct)432 b(sp)-62 b(ac)g(e.)555 b(Denote)432 b(by)h Fm(K)17935 64498 y Fd(d)18909 64301 y Fj(its)g(mar)-62 b(ginal)432 b(on)i(the)e(disor)-62 b(der)432 b(variables)e Fm(\021)43 b Fj(.)558 b(Then)8158 66442 y Fm(h)p Fr(\()p Fm(K)87 b Fl(j)p Fm(K)11945 65942 y Fk(\026)-519 b Fd(\033)12521 66442 y Fr(\))1108 b(=)e Fm(h)p Fr(\()p Fm(K)18347 66639 y Fd(d)18888 66442 y Fl(j)p Fe(P)p Fr(\))269 b Fl(\000)g Fm(h)p Fr(\()p Fm(K)87 b Fr(\))271 b Fl(\000)e Fm(h)p Fr(\()p Fm(K)28355 66639 y Fd(d)28896 66442 y Fr(\))14100 68789 y(+)16175 67638 y Ff(X)16149 70263 y Fd(A)p Ft(3)p Fk(0)18775 67969 y Fr(1)p 18287 68510 1583 49 v 18287 69621 a Fl(j)p Fm(A)p Fl(j)20002 68789 y Fm(K)21118 67444 y Ff(\020)21842 68789 y Fr(\010)22717 68977 y Fd(A)23478 68789 y Fr(\()p Fm(\033)245 b(\021)380 b Fr(=)337 b Fl(\001)p Fr(\))27957 67444 y Ff(\021)28950 68789 y Fr(+)269 b Fm(K)31278 67444 y Ff(\020)32002 68789 y Fr(lim)32518 69596 y Fk(\003)34475 67969 y Fr(1)p 34021 68510 1516 49 v 34021 69621 a Fl(j)p Fr(\003)p Fl(j)35871 68789 y Fr(log)218 b Fm(Z)38601 68289 y Fk(\026)-519 b Fd(\033)38465 69089 y Fk(\003)39177 68789 y Fr(\()p Fm(\021)381 b Fr(=)336 b Fl(\001)p Fr(\))42718 67444 y Ff(\021)1200 72096 y Fj(wher)-62 b(e)433 b Fm(h)p Fr(\()p Fm(K)87 b Fr(\))435 b Fj(is)e(the)f(Kolmo)-62 b(gor)g(ov-Sinai)431 b(entr)-62 b(opy)432 b(\(2.15\).)25194 75417 y Fr(23)p eop %%Page: 24 24 24 23 bop 1200 1313 a Fs(Remark)464 b(6.14)606 b Fj(The)555 b(thir)-62 b(d)553 b(term)i(has)f(the)g(me)-62 b(aning)554 b(of)h(the)e Fm(K)87 b Fj(-exp)-62 b(e)g(ctation)553 b(of)h(the)g(`joint)g(ener)-62 b(gy'.)1200 2819 y(The)458 b(last)g(term)h(is)f(the)g Fm(K)87 b Fj(-me)-62 b(an)459 b(of)f(the)g(\\quenche)-62 b(d)456 b(pr)-62 b(essur)g(e".)631 b(Note)457 b(that)h(it)g(is)h(b)-62 b(oundary)457 b(c)-62 b(ondition)1265 4324 y Fr(\026)-671 b Fm(\033)43 b Fj(-indep)-62 b(endent,)431 b(of)i(c)-62 b(ourse.)1200 7454 y Fs(Remark)464 b(6.15)606 b Fj(In)409 b(the)f(c)-62 b(ase)407 b(that)f Fe(P)j Fj(is)f(a)g(Gibbs)e(distribution,)k(the)e(existenc)-62 b(e)406 b(of)i(the)f(r)-62 b(elative)406 b(entr)-62 b(opy)1200 8960 y(density)432 b(is)h(obtaine)-62 b(d)431 b(dir)-62 b(e)g(ctly,)431 b(i.e.,)h(without)g(r)-62 b(elying)432 b(on)i(P\014ster's)f(the)-62 b(ory.)1200 12090 y(Pr)g(o)g(of)p Fr(.)605 b(W)-101 b(e)404 b(ha)-34 b(v)g(e)4946 14404 y(1)p 4492 14945 1516 49 v 4492 16056 a Fl(j)p Fr(\003)p Fl(j)6140 15224 y Fm(h)6838 15412 y Fk(\003)7546 15224 y Fr(\()p Fm(K)87 b Fl(j)p Fm(K)10635 14724 y Fk(\026)-519 b Fd(\033)11211 15224 y Fr(\))337 b(=)13886 14404 y(1)p 13432 14945 V 13432 16056 a Fl(j)p Fr(\003)p Fl(j)15519 14073 y Ff(X)15282 16618 y Fd(\033)15820 16767 y Fc(\003)16427 16618 y Fd(\021)16899 16767 y Fc(\003)17709 15224 y Fm(K)87 b Fr(\()p Fm(\033)19989 15412 y Fk(\003)20697 15224 y Fm(\021)21299 15412 y Fk(\003)22007 15224 y Fr(\))202 b(log)219 b Fm(K)87 b Fr(\()p Fm(\033)26728 15412 y Fk(\003)27436 15224 y Fm(\021)28038 15412 y Fk(\003)28745 15224 y Fr(\))270 b Fl(\000)31286 14404 y Fr(1)p 30831 14945 V 30831 16056 a Fl(j)p Fr(\003)p Fl(j)32918 14073 y Ff(X)32681 16618 y Fd(\033)33219 16767 y Fc(\003)33826 16618 y Fd(\021)34298 16767 y Fc(\003)35108 15224 y Fm(K)87 b Fr(\()p Fm(\033)37388 15412 y Fk(\003)38096 15224 y Fm(\021)38698 15412 y Fk(\003)39406 15224 y Fr(\))202 b(log)219 b Fm(K)43012 14724 y Fk(\026)-519 b Fd(\033)43588 15224 y Fr(\()p Fm(\033)44752 15412 y Fk(\003)45460 15224 y Fm(\021)46062 15412 y Fk(\003)46769 15224 y Fr(\))1200 19118 y(where)404 b(the)h(\014rst)f(term)g(con)-34 b(v)g(erges)404 b(to)h Fl(\000)p Fm(h)p Fr(\()p Fm(K)87 b Fr(\).)539 b(F)-101 b(or)404 b(the)h(second)f(term)g(w)-34 b(e)405 b(use)f(the)h(appro)-34 b(ximation)13714 22840 y(sup)13594 23893 y Fk(\026)-520 b Fd(\033)32 b(;)50 b Fk(^)-520 b Fd(\033)31 b(;)69 b Fk(^)-539 b Fd(\021)15710 20719 y Ff(\014)15710 21446 y(\014)15710 22173 y(\014)15710 22900 y(\014)15710 23628 y(\014)16114 22840 y Fr(log)17679 20767 y Ff( )21880 22020 y Fm(K)23045 21580 y Fk(\026)-519 b Fd(\033)23621 22020 y Fr(\()p Fm(\033)24785 22208 y Fk(\003)25493 22020 y Fm(\021)26095 22208 y Fk(\003)26803 22020 y Fr(\))p 18772 22561 11611 49 v 18772 23739 a Fe(P)p Fr(\()p Fm(\021)20586 23927 y Fk(\003)21294 23739 y Fr(\))p Fm(\026)22545 23322 y Fk(^)f Fd(\033)22495 24121 y Fk(\003)23203 23739 y Fr([)p Fm(\021)24142 23927 y Fk(\003)24937 23739 y Fr(^)-693 b Fm(\021)25452 24035 y Fh(Z)26042 23783 y Fg(d)26524 24035 y Ft(n)p Fk(\003)27702 23739 y Fr(]\()p Fm(\033)29203 23927 y Fk(\003)29911 23739 y Fr(\))30515 20767 y Ff(!)31475 20719 y(\014)31475 21446 y(\014)31475 22173 y(\014)31475 22900 y(\014)31475 23628 y(\014)32215 22840 y Fl(\024)337 b Fr(2)p Fm(C)34967 23022 y Fk(1)35493 22840 y Fl(j)p Fm(@)67 b Fr(\003)p Fl(j)p Fm(:)1200 26494 y Fr(First)404 b(w)-34 b(e)405 b(ha)-34 b(v)g(e)6378 29393 y Fl(\000)7908 28573 y Fr(1)p 7454 29114 1516 49 v 7454 30224 a Fl(j)p Fr(\003)p Fl(j)9541 28241 y Ff(X)9304 30787 y Fd(\033)9842 30936 y Fc(\003)10449 30787 y Fd(\021)10921 30936 y Fc(\003)11731 29393 y Fm(K)87 b Fr(\()p Fm(\033)14011 29581 y Fk(\003)14719 29393 y Fm(\021)15321 29581 y Fk(\003)16029 29393 y Fr(\))202 b(log)219 b Fe(P)p Fr(\()p Fm(\021)20284 29581 y Fk(\003)20991 29393 y Fr(\))338 b(=)23666 28573 y(1)p 23212 29114 V 23212 30224 a Fl(j)p Fr(\003)p Fl(j)24860 29393 y Fm(h)25558 29581 y Fk(\003)26266 29393 y Fr(\()p Fm(K)27766 29590 y Fd(d)28306 29393 y Fl(j)p Fe(P)p Fr(\))270 b Fl(\000)31924 28573 y Fr(1)p 31470 29114 V 31470 30224 a Fl(j)p Fr(\003)p Fl(j)33320 28241 y Ff(X)33655 30787 y Fd(\021)34127 30936 y Fc(\003)35272 29393 y Fm(K)36301 29590 y Fd(d)36841 29393 y Fr(\()p Fm(\021)37914 29581 y Fk(\003)38622 29393 y Fr(\))202 b(log)220 b Fm(K)42093 29590 y Fd(d)42632 29393 y Fr(\()p Fm(\021)43705 29581 y Fk(\003)44414 29393 y Fr(\))p Fm(:)1200 33287 y Fr(The)332 b(second)g(term)g(con)-34 b(v)g(erges)332 b(to)g Fm(h)p Fr(\()p Fm(K)19252 33484 y Fd(d)19793 33287 y Fr(\).)514 b(The)333 b(\014rst)f(term)g(con)-34 b(v)g(erges)332 b(to)g Fm(h)p Fr(\()p Fm(K)37757 33484 y Fd(d)38297 33287 y Fl(j)p Fe(P)p Fr(\).)515 b(This)332 b(is)g(clear)e(either)1200 34792 y(b)-34 b(y)359 b(the)h(classical)e (theory)h(for)g(the)g(case)g(that)h Fe(P)e Fr(is)h(Gibbs)h(or)f(ev)-34 b(en)358 b(indep)34 b(enden)-34 b(t,)370 b(or)358 b(b)-34 b(y)360 b(P\014ster's)f(theory)1200 36298 y(if)404 b Fe(P)g Fr(is)g(asymptotically)g(decoupled.)539 b(Next,)403 b(b)-34 b(y)405 b(de\014nition)6392 39104 y(log)219 b Fm(\026)8940 38603 y Fk(^)-520 b Fd(\033)8890 39404 y Fk(\003)9598 39104 y Fr([)p Fm(\021)10537 39292 y Fk(\003)11332 39104 y Fr(^)-693 b Fm(\021)11847 39400 y Fh(Z)12437 39148 y Fg(d)12919 39400 y Ft(n)p Fk(\003)14097 39104 y Fr(]\()p Fm(\033)15598 39292 y Fk(\003)16306 39104 y Fr(\))337 b(=)f Fl(\000)20739 37952 y Ff(X)19538 40637 y Fd(A)p Fk(:)p Fd(A)p Ft(\\)p Fk(\003)p Ft(6)p Fk(=)p Ft(;)23893 39104 y Fr(\010)24768 39292 y Fd(A)25529 39104 y Fr(\()p Fm(\033)26693 39292 y Fk(\003)27466 39104 y Fr(^)-671 b Fm(\033)28094 39400 y Fh(Z)28684 39148 y Fg(d)29166 39400 y Ft(n)p Fk(\003)30344 39104 y Fm(\021)30946 39292 y Fk(\003)31741 39104 y Fr(^)-693 b Fm(\021)32256 39400 y Fh(Z)32846 39148 y Fg(d)33328 39400 y Ft(n)p Fk(\003)34506 39104 y Fr(\))270 b Fl(\000)f Fr(log)219 b Fm(Z)39190 38603 y Fk(^)-519 b Fd(\033)39054 39404 y Fk(\003)39766 39104 y Fr(\()p Fm(\021)40839 39292 y Fk(\003)41634 39104 y Fr(^)-693 b Fm(\021)42149 39400 y Fh(Z)42739 39148 y Fg(d)43221 39400 y Ft(n)p Fk(\003)44399 39104 y Fr(\))p Fm(:)1200 43276 y Fr(Using)352 b(translation-in)-34 b(v)-67 b(ariance)352 b(of)g(the)g(measure)f Fm(K)439 b Fr(w)-34 b(e)352 b(get)f(that)i(the)f(application)g(of)42201 42798 y Fk(1)p 41849 42996 1176 49 v 41849 43710 a Ft(j)p Fk(\003)p Ft(j)43359 42299 y Ff(R)44369 43276 y Fm(K)87 b Fr(\()p Fm(d\033)47280 43464 y Fk(\003)47988 43276 y Fm(d\021)49221 43464 y Fk(\003)49929 43276 y Fr(\))1200 45090 y(o)-34 b(v)g(er)657 b(the)h(\014rst)h(sum)f(of)g(the)g(r.h.s.) 1298 b(con)-34 b(v)g(erges)658 b(to)g Fl(\000)28440 44180 y Ff(P)29719 45448 y Fd(A)p Ft(3)p Fk(0)32290 44612 y(1)p 31912 44810 1228 49 v 31912 45523 a Ft(j)p Fd(A)p Ft(j)33272 45090 y Fm(K)87 b Fr(\(\010)35734 45278 y Fd(A)36495 45090 y Fr(\()p Fm(\033)246 b(\021)802 b Fr(=)759 b Fl(\001)p Fr(\)\).)1300 b(T)-101 b(o)658 b(see)f(that)1200 46595 y(the)522 b(a)-34 b(v)g(erage)521 b(o)-34 b(v)g(er)521 b(the)h(last)f(term)h(con)-34 b(v)g(erges)521 b(w)-34 b(e)522 b(use)f(the)h(ergo)34 b(dic)520 b(decomp)34 b(osition)522 b(of)g Fm(K)45043 46792 y Fd(d)46104 46595 y Fr(to)g(write)1200 48100 y Fm(K)2229 48297 y Fd(d)2769 48100 y Fr(\()p Fm(d\021)43 b Fr(\))596 b(=)7120 47124 y Ff(R)8130 48100 y Fm(\032)p Fr(\()p Fm(d\024)p Fr(\))p Fm(\024)p Fr(\()p Fm(d\021)43 b Fr(\))561 b(where)e Fm(\032)p Fr(\()p Fm(d\024)p Fr(\))h(is)e(a)h (probabilit)-34 b(y)560 b(measure)e(that)i(is)f(concen)-34 b(trated)560 b(on)f(the)1200 49606 y(ergo)34 b(dic)469 b(measures)h(on)g Fm(\021)43 b Fr(.)737 b(Fix)470 b(an)-34 b(y)470 b(ergo)34 b(dic)469 b Fm(\024)p Fr(.)737 b(F)-101 b(or)470 b Fm(\024)p Fr(-a.e.)736 b(disorder)470 b(con\014guration)h Fm(\021)513 b Fr(w)-34 b(e)471 b(ha)-34 b(v)g(e)470 b(the)1200 51111 y(existence)436 b(of)h(the)g(limit)g Fl(\000)202 b Fr(lim)15681 51299 y Fk(\003)17076 50634 y(1)p 16724 50832 1176 49 v 16724 51545 a Ft(j)p Fk(\003)p Ft(j)18234 51111 y Fr(log)218 b Fm(Z)20964 50672 y Fk(\026)-519 b Fd(\033)20828 51471 y Fk(\003)21540 51111 y Fr(\()p Fm(\021)435 b Fr(=)391 b Fl(\001)p Fr(\),)445 b(b)-34 b(y)437 b(standard)i(argumen)-34 b(ts)438 b([29].)636 b(The)437 b(con)-34 b(v)g(ergence)1200 52888 y(is)417 b(also)g(in)g Fm(L)7128 52448 y Fk(1)7654 52888 y Fr(,)i(b)-34 b(y)418 b(dominated)g(con)-34 b(v)g(ergence.)577 b(So)418 b(w)-34 b(e)417 b(ma)-34 b(y)417 b(in)-34 b(tegrate)418 b(o)-34 b(v)g(er)417 b Fm(\032)f Fr(to)i(see)e(the)i(statemen)-34 b(t)418 b(of)1200 54393 y(the)404 b(theorem.)p 49878 54393 522 731 v 1200 58615 a Fi(6.3)1495 b(Discussion)337 b(of)d(the)h(\014rst)g(part)h(of)e(the)g(v)-83 b(ariational)337 b(principle)e(for)g(join)-42 b(t)335 b(measures)1200 61231 y Fr(T)-101 b(o)313 b(discuss)g(the)g(\014rst)g(part)g(of)f(the)h (v)-67 b(ariational)312 b(principle)g(w)-34 b(e)313 b(will)e(use)i(an)g (explicit)e(represen)-34 b(tation)314 b(of)e(the)1200 62736 y(conditional)405 b(exp)34 b(ectations)404 b(of)h(the)g(joined)g (measures.)539 b(F)-101 b(or)404 b(this)h(w)-34 b(e)405 b(need)g(to)f(restrict)g(to)h(the)g(case)f(that)1200 64241 y Fe(P)432 b Fr(is)f(a)h(pro)34 b(duct)433 b(measure.)622 b(First,)439 b(in)432 b(the)g(situation)h(detailed)f(b)34 b(elo)-34 b(w,)439 b(w)-34 b(e)432 b(pro)-34 b(v)g(e)433 b(the)f(\014rst)h(part)f(of)g(the)1200 65747 y(v)-67 b(ariational)384 b(principle)g(b)-34 b(y)385 b(direct)g(argumen)-34 b(ts.)533 b(Next,)388 b(w)-34 b(e)385 b(illustrate)g(the)g(criteria)f (giv)-34 b(en)384 b(in)h(the)g(general)1200 67252 y(theory)393 b(of)g(Section)g(3.4)g(b)-34 b(y)393 b(sho)-34 b(wing)395 b(that)f(they)f(can)g(b)34 b(e)392 b(v)-34 b(eri\014ed)393 b(in)g(the)g(con)-34 b(text)394 b(of)f(join)-34 b(t)394 b(measures)f(in)1200 68758 y(the)404 b(almost)h(Gibbsian)g(case,)e (giving)h(then)h(an)f(alternativ)-34 b(e)404 b(pro)34 b(of)405 b(of)f(the)h(v)-67 b(ariational)403 b(principle.)3082 70590 y(W)-101 b(e)403 b(start)i(with)g(the)g(follo)-34 b(wing)404 b(prop)34 b(osition)405 b(of)g([18)o(].)25194 75417 y(24)p eop %%Page: 25 25 25 24 bop 1200 1313 a Fs(Prop)39 b(osition)465 b(6.16)606 b Fj(Assume)468 b(that)g Fe(P)h Fj(is)f(a)h(pr)-62 b(o)g(duct)467 b(me)-62 b(asur)g(e.)662 b(Assume)468 b(that)g(ther)-62 b(e)467 b(is)i(a)g(set)f(of)g(r)-62 b(e)g(al-)1200 2819 y(izations)503 b(of)i Fm(\021)43 b Fj('s)505 b(of)g Fe(P)p Fj(-me)-62 b(asur)g(e)503 b(one)i(such)f(that)g(the)g(quenche)-62 b(d)503 b(in\014nite-volume)g(Gibbs)h(me)-62 b(asur)g(e)504 b Fm(\026)p Fr([)p Fm(\021)43 b Fr(])1200 4324 y Fj(is)435 b(a)h(we)-62 b(ak)435 b(limit)g(of)g(the)g(quenche)-62 b(d)434 b(\014nite-volume)f(me)-62 b(asur)g(es)434 b(\(6.1\).)563 b(Then,)436 b(a)f(version)g(of)g(the)g(in\014nite-)1200 5830 y(volume)332 b(c)-62 b(onditional)331 b(exp)-62 b(e)g(ctation)331 b(of)h(the)h(c)-62 b(orr)g(esp)g(onding)331 b(joint)h(me)-62 b(asur)g(e)332 b Fm(K)36786 5390 y Fd(\026)37408 5830 y Fr(\()p Fm(d\033)-24 b(;)202 b(d\021)43 b Fr(\))337 b(=)f Fe(P)p Fr(\()p Fm(d\021)43 b Fr(\))p Fm(\026)p Fr([)p Fm(\021)g Fr(]\()p Fm(d\033)g Fr(\))1200 7335 y Fj(is)433 b(given)g(by)f(the)h(formula)13306 10710 y Fm(K)14422 10210 y Fd(\026)15246 9728 y Ff(\002)15751 10710 y Fm(\030)16281 10898 y Fk(\003)16989 9680 y Ff(\014)16989 10407 y(\014)17393 10710 y Fm(\030)17923 10898 y Fk(\003)18576 10646 y Fg(c)19058 9728 y Ff(\003)19899 10710 y Fr(=)26189 9856 y Fm(\026)26919 9203 y Fr(ann,)q Fd(\030)29623 9359 y Fg(@)43 b Fc(\003)26919 10237 y Fk(\003)30766 9856 y Fr(\()p Fm(\030)31767 10044 y Fk(\003)32476 9856 y Fr(\))p 21312 10431 16513 49 v 21312 10868 a Ff(R)22322 11845 y Fm(\026)23052 11192 y Fr(ann,)q Fd(\030)25756 11348 y Fg(@)g Fc(\003)23052 12226 y Fk(\003)26899 11845 y Fr(\()p Fm(d)87 b Fr(~)-693 b Fm(\021)28603 12033 y Fk(\003)29311 11845 y Fr(\))p Fm(Q)30740 11255 y Fd(\026)30740 12226 y Fk(\003)31448 11845 y Fr(\()p Fm(\021)32521 12033 y Fk(\003)33229 11845 y Fm(;)289 b Fr(~)-693 b Fm(\021)34370 12033 y Fk(\003)35078 11845 y Fm(;)202 b(\021)36219 12033 y Fk(\003)36872 11780 y Fg(c)37353 11845 y Fr(\))37957 10710 y Fm(:)9008 b Fr(\(6.17\))1200 14511 y Fj(Her)-62 b(e)416 b Fm(\026)4812 13858 y Fr(ann,)q Fd(\030)7516 14014 y Fg(@)43 b Fc(\003)4812 14893 y Fk(\003)8659 14511 y Fr(\()p Fm(\030)9660 14699 y Fk(\003)10369 14511 y Fr(\))417 b Fj(is)f(the)f Fr(trivial)385 b(annealed)h(lo)34 b(cal)385 b(sp)34 b(eci\014cation)416 b Fj(given)g(by)g(in)h(terms)f (of)g(the)f(p)-62 b(otential)1200 16017 y Fm(U)2160 15577 y Fk(triv)2028 16376 y Fd(A)3719 16017 y Fr(\()p Fm(\033)-24 b(;)202 b(\021)43 b Fr(\))337 b(=)f(\010)9005 16205 y Fd(A)9766 16017 y Fr(\()p Fm(\033)-24 b(;)202 b(\021)43 b Fr(\))270 b Fl(\000)f Fr(1)14649 16263 y Fd(A)p Fk(=)p Ft(f)p Fd(x)p Ft(g)17814 16017 y Fr(log)218 b Fe(P)20322 16199 y Fk(0)20848 16017 y Fr(\()p Fm(\021)21921 16199 y Fd(x)22506 16017 y Fr(\))434 b Fj(w.r.t)f(c)-62 b(ounting)433 b(me)-62 b(asur)g(e)432 b(on)h(the)g(pr)-62 b(o)g(duct)431 b(sp)-62 b(ac)g(e.)3082 17802 y(F)-93 b(urther)432 b(we)i(have)e(put) 14029 20214 y Fm(Q)14987 19624 y Fd(\026)14987 20595 y Fk(\003)15695 20214 y Fr(\()p Fm(\021)16811 19713 y Fk(1)16768 20513 y(\003)17476 20214 y Fm(;)202 b(\021)18660 19713 y Fk(2)18617 20513 y(\003)19325 20214 y Fm(;)g(\021)20466 20402 y Fk(\003)21119 20149 y Fg(c)21600 20214 y Fr(\))337 b(=)f Fm(\026)p Fr([)p Fm(\021)25399 19713 y Fk(2)25356 20513 y(\003)26064 20214 y Fm(\021)26666 20402 y Fk(\003)27319 20149 y Fg(c)27801 20214 y Fr(]\()p Fm(e)29173 19713 y Ft(\000)p Fk(\001)p Fd(H)31462 19862 y Fc(\003)32070 19713 y Fk(\()p Fd(\021)32940 19401 y Fc(1)32908 20014 y(\003)33515 19713 y Fd(;\021)34280 19401 y Fc(2)34248 20014 y(\003)34856 19713 y Fd(;\021)35589 19869 y Fg(@)43 b Fc(\003)36678 19713 y Fk(\))37099 20214 y Fr(\))1200 22625 y Fj(wher)-62 b(e)9543 24131 y Fr(\001)p Fm(H)11561 24319 y Fk(\003)12269 24131 y Fr(\()p Fm(\021)13385 23630 y Fk(1)13342 24430 y(\003)14050 24131 y Fm(;)202 b(\021)15234 23630 y Fk(2)15191 24430 y(\003)15898 24131 y Fm(;)g(\021)17039 24319 y Fk(\003)17692 24066 y Fg(c)18173 24131 y Fr(\)\()p Fm(\033)43 b Fr(\))338 b(=)22658 22979 y Ff(X)21940 25663 y Fd(A)p Ft(\\)p Fk(\003)p Ft(6)p Fk(=)p Ft(;)25127 22785 y Ff(\020)25851 24131 y Fr(\010)26726 24319 y Fd(A)27486 24131 y Fr(\()p Fm(\033)-24 b(;)202 b(\021)29810 23630 y Fk(1)29767 24430 y(\003)30475 24131 y Fm(\021)31077 24319 y Fk(\003)31730 24066 y Fg(c)32211 24131 y Fr(\))270 b Fl(\000)f Fr(\010)35039 24319 y Fd(A)35800 24131 y Fr(\()p Fm(\033)-24 b(;)202 b(\021)38124 23630 y Fk(2)38081 24430 y(\003)38788 24131 y Fm(\021)39390 24319 y Fk(\003)40043 24066 y Fg(c)40525 24131 y Fr(\))40996 22785 y Ff(\021)41720 24131 y Fm(:)1200 28335 y Fr(According)452 b(to)h(our)f(assumption)i (on)e(the)h(measurabilit)-34 b(y)452 b(on)h Fm(\026)p Fr([)p Fm(\021)43 b Fr(],)464 b Fm(Q)34211 27746 y Fd(\026)34211 28717 y Fk(\003)35371 28335 y Fr(dep)34 b(ends)453 b(measurably)f(on)h Fm(\021)48929 28523 y Fk(\003)49582 28271 y Fg(c)50063 28335 y Fr(.)1200 29841 y(W)-101 b(e)326 b(\014x)h(a)f(v)-34 b(ersion)327 b(of)g(the)g(map)g(and)g(de\014ne)g(the)h(r.h.s.)512 b(of)327 b(\(6.17\))g(to)g(b)34 b(e)326 b(the)h(sp)34 b(eci\014cation)327 b Fm(\015)43855 29401 y Fd(\026)44476 29841 y Fr(.)513 b(Note)327 b(that)1200 31346 y(for)454 b(the)g(random)g(\014eld)g(Ising)g(mo)34 b(del,)466 b(this)454 b(sp)34 b(eci\014cation)454 b(exists)f(for)h(all)f(random)h(\014eld)g (con\014gurations)1200 32852 y(b)-34 b(y)404 b(monotonicit)-34 b(y)-101 b(.)3082 34637 y(In)396 b(this)h(con)-34 b(text)396 b(w)-34 b(e)397 b(alw)-34 b(a)g(ys)397 b(ha)-34 b(v)g(e)396 b(the)h(\014rst)f(part)h(of)f(the)h(v)-67 b(ariational)395 b(principle.)535 b(Note)397 b(that)g(w)-34 b(e)396 b(do)1200 36143 y(not)405 b(need)f(an)-34 b(y)405 b(further)f(assumption)i(ab)34 b(out)405 b(almost)g(Gibbsianness.)1200 38945 y Fs(Theorem)464 b(6.18)606 b Fj(Assume)497 b(that)g Fe(P)h Fj(is)g(a)g(pr)-62 b(o)g(duct)496 b(me)-62 b(asur)g(e.)749 b(Ther)-62 b(e)498 b(exists)f(a)h(c)-62 b(onstant)497 b Fm(C)585 b Fj(dep)-62 b(ending)1200 40451 y(only)433 b(on)h Fr(\010)p Fj(,)f Fe(P)h Fj(such)e(that)g(for)h(any)g Fm(K)19 b(;)202 b(K)20620 40011 y Ft(0)21268 40451 y Fl(2)337 b(G)72 b Fr(\()p Fm(\015)24372 40011 y Fd(\026)24993 40451 y Fr(\))434 b Fj(one)g(has)18996 43528 y Fr(sup)19684 44580 y Fd(\030)21023 41770 y Ff(\014)21023 42497 y(\014)21023 43225 y(\014)21023 43952 y(\014)21427 43528 y Fr(log)23483 42708 y Fm(K)87 b Fr(\()p Fm(\030)25600 42896 y Fk(\003)26308 42708 y Fr(\))p 23327 43249 3608 49 v 23327 44359 a Fm(K)24443 44009 y Ft(0)24754 44359 y Fr(\()p Fm(\030)25755 44547 y Fk(\003)26464 44359 y Fr(\))27068 41770 y Ff(\014)27068 42497 y(\014)27068 43225 y(\014)27068 43952 y(\014)27809 43528 y Fl(\024)336 b Fm(C)87 b Fl(j)p Fm(@)67 b Fr(\003)p Fl(j)p Fm(:)1200 46822 y Fj(In)434 b(p)-62 b(articular)431 b Fm(h)p Fr(\()p Fm(K)87 b Fl(j)p Fm(K)12065 46382 y Ft(0)12377 46822 y Fr(\))337 b(=)f Fm(h)p Fr(\()p Fm(K)16749 46382 y Ft(0)17061 46822 y Fl(j)p Fm(K)87 b Fr(\))336 b(=)h(0)p Fj(.)1200 49625 y(Pr)-62 b(o)g(of)p Fr(.)605 b(Using)404 b Fm(K)19 b(;)202 b(K)11073 49185 y Ft(0)11721 49625 y Fl(2)337 b(G)72 b Fr(\()p Fm(\015)14825 49185 y Fd(\026)15447 49625 y Fr(\),)404 b(it)g(su\016ces)g(to)h(sho)-34 b(w)405 b(that)g(w)-34 b(e)405 b(ha)-34 b(v)g(e)404 b(the)h(estimate) 20695 51915 y Fm(\015)21390 51326 y Fd(\026)21323 52297 y Fk(\003)22030 51915 y Fr(\()p Fm(\030)23031 52103 y Fk(\003)23740 51915 y Fl(j)p Fm(\030)24607 52103 y Fk(\003)25260 51851 y Fg(c)25741 51915 y Fr(\))p 20695 52491 5518 49 v 20695 53607 a Fm(\015)21390 53017 y Fd(\026)21323 53989 y Fk(\003)22030 53607 y Fr(\()p Fm(\030)23031 53795 y Fk(\003)23740 53607 y Fl(j)p Fm(\030)24663 53190 y Ft(0)24607 53989 y Fk(\003)25260 53737 y Fg(c)25741 53607 y Fr(\))26682 52770 y Fl(\024)337 b Fm(e)28526 52269 y Fd(C)63 b Ft(j)p Fd(@)52 b Fk(\003)p Ft(j)1200 55937 y Fr(where)488 b(the)g(constan)-34 b(t)490 b Fm(C)575 b Fr(is)488 b(indep)34 b(enden)-34 b(t)490 b(of)e(\003)p Fm(;)202 b(\030)56 b(;)202 b(\030)26186 55497 y Ft(0)26496 55937 y Fr(.)790 b(F)-101 b(rom)488 b(the)h(explicit)e(represen)-34 b(tation)489 b(\(6.17\))f(w)-34 b(e)1200 57443 y(obtain)9469 59866 y Fm(\015)10164 59277 y Fd(\026)10097 60248 y Fk(\003)10804 59866 y Fr(\()p Fm(\030)11805 60054 y Fk(\003)12514 59866 y Fl(j)p Fm(\030)13381 60054 y Fk(\003)14034 59802 y Fg(c)14515 59866 y Fr(\))p 9469 60442 V 9469 61558 a Fm(\015)10164 60969 y Fd(\026)10097 61940 y Fk(\003)10804 61558 y Fr(\()p Fm(\030)11805 61746 y Fk(\003)12514 61558 y Fl(j)p Fm(\030)13437 61141 y Ft(0)13381 61940 y Fk(\003)14034 61688 y Fg(c)14515 61558 y Fr(\))16226 60721 y(=)18409 59866 y Fm(\026)19139 59277 y Fr(ann)q Fd(;\030)21767 59433 y Fg(@)43 b Fc(\003)19139 60248 y Fk(\003)22911 59866 y Fr(\()p Fm(\030)23912 60054 y Fk(\003)24620 59866 y Fr(\))p 18409 60442 6683 49 v 18409 61995 a Fm(\026)19139 61261 y Fr(ann)q Fd(;\030)21802 60956 y Fn(0)21767 61577 y Fg(@)g Fc(\003)19139 62376 y Fk(\003)22911 61995 y Fr(\()p Fm(\030)23912 62183 y Fk(\003)24620 61995 y Fr(\))25357 58890 y Ff(R)26367 59866 y Fm(\026)27097 59141 y Fr(ann)r Fd(;\030)29761 58828 y Fn(0)29726 59449 y Fg(@)g Fc(\003)27097 60248 y Fk(\003)30869 59866 y Fr(\()p Fm(d)87 b Fr(~)-693 b Fm(\021)32573 60054 y Fk(\003)33281 59866 y Fr(\))p Fm(Q)34710 59277 y Fd(\026)34710 60248 y Fk(\003)35418 59866 y Fr(\()p Fm(\021)36491 60054 y Fk(\003)37199 59866 y Fm(;)289 b Fr(~)-693 b Fm(\021)38340 60054 y Fk(\003)39048 59866 y Fm(;)202 b(\021)40232 59427 y Ft(0)40189 60226 y Fk(\003)40842 59974 y Fg(c)41323 59866 y Fr(\))p 25357 60442 16438 49 v 25357 60816 a Ff(R)26367 61792 y Fm(\026)27097 61202 y Fr(ann)r Fd(;\030)29726 61358 y Fg(@)43 b Fc(\003)27097 62174 y Fk(\003)30869 61792 y Fr(\()p Fm(d)87 b Fr(~)-693 b Fm(\021)32573 61980 y Fk(\003)33281 61792 y Fr(\))p Fm(Q)34710 61202 y Fd(\026)34710 62174 y Fk(\003)35418 61792 y Fr(\()p Fm(\021)36491 61980 y Fk(\003)37199 61792 y Fm(;)289 b Fr(~)-693 b Fm(\021)38340 61980 y Fk(\003)39048 61792 y Fm(;)202 b(\021)40189 61980 y Fk(\003)40842 61728 y Fg(c)41323 61792 y Fr(\))41927 60721 y Fm(:)5038 b Fr(\(6.19\))1200 64597 y(Using)474 b(the)h(de\014nition)g(of)f Fm(\026)14472 64008 y Fr(ann)q Fd(;\030)17100 64164 y Fg(@)43 b Fc(\003)14472 64979 y Fk(\003)18718 64597 y Fr(and)474 b(using)h(the)f(\014nite)h(range)f(assumption)h(on)g(\010,) 491 b(w)-34 b(e)474 b(obtain)h(the)1200 66103 y(b)34 b(ound)505 b Fm(e)5601 65663 y Fd(c)p Ft(j)p Fd(@)52 b Fk(\003)p Ft(j)8291 66103 y Fr(for)504 b(the)h(\014rst)f(factor)g(on) h(the)f(r.h.s.)837 b(of)504 b(\(6.19\).)839 b(The)504 b(second)g(factor)h(on)f(the)g(r.h.s.)837 b(of)1200 67608 y(\(6.19\))405 b(is)e(b)34 b(ounded)406 b(b)-34 b(y)11052 69117 y Ff( )12012 71189 y Fr(sup)12453 72221 y Fk(~)-539 b Fd(\021)12856 72370 y Fc(\003)14172 70335 y Fm(Q)15130 69745 y Fd(\026)15130 70717 y Fk(\003)15838 70335 y Fr(\()p Fm(\021)16911 70523 y Fk(\003)17619 70335 y Fm(;)288 b Fr(~)-692 b Fm(\021)18760 70523 y Fk(\003)19467 70335 y Fm(;)202 b(\021)20651 69895 y Ft(0)20608 70694 y Fk(\003)21261 70442 y Fg(c)21742 70335 y Fr(\))p 14172 70911 8043 49 v 14172 72027 a Fm(Q)15130 71437 y Fd(\026)15130 72409 y Fk(\003)15838 72027 y Fr(\()p Fm(\021)16911 72215 y Fk(\003)17619 72027 y Fm(;)288 b Fr(~)-692 b Fm(\021)18760 72215 y Fk(\003)19467 72027 y Fm(;)202 b(\021)20608 72215 y Fk(\003)21261 71963 y Fg(c)21742 72027 y Fr(\))22346 69117 y Ff(!)23641 69359 y(R)24651 70335 y Fm(\026)25381 69610 y Fr(ann)q Fd(;\030)28044 69297 y Fn(0)28009 69918 y Fg(@)43 b Fc(\003)25381 70717 y Fk(\003)29153 70335 y Fr(\()p Fm(d)87 b Fr(~)-693 b Fm(\021)30857 70523 y Fk(\003)31565 70335 y Fr(\))p Fm(Q)32994 69745 y Fd(\026)32994 70717 y Fk(\003)33702 70335 y Fr(\()p Fm(\021)34775 70523 y Fk(\003)35483 70335 y Fm(;)289 b Fr(~)-693 b Fm(\021)36624 70523 y Fk(\003)37332 70335 y Fm(;)202 b(\021)38473 70523 y Fk(\003)39126 70271 y Fg(c)39607 70335 y Fr(\))p 23641 70911 16438 49 v 23641 71284 a Ff(R)24651 72261 y Fm(\026)25381 71671 y Fr(ann)q Fd(;\030)28009 71827 y Fg(@)43 b Fc(\003)25381 72642 y Fk(\003)29153 72261 y Fr(\()p Fm(d)87 b Fr(~)-693 b Fm(\021)30857 72449 y Fk(\003)31565 72261 y Fr(\))p Fm(Q)32994 71671 y Fd(\026)32994 72642 y Fk(\003)33702 72261 y Fr(\()p Fm(\021)34775 72449 y Fk(\003)35483 72261 y Fm(;)289 b Fr(~)-693 b Fm(\021)36624 72449 y Fk(\003)37332 72261 y Fm(;)202 b(\021)38473 72449 y Fk(\003)39126 72196 y Fg(c)39607 72261 y Fr(\))40211 71189 y Fm(:)25194 75417 y Fr(25)p eop %%Page: 26 26 26 25 bop 1200 1313 a Fr(Using)522 b(the)g(same)g(argumen)-34 b(t)523 b(on)g Fm(\026)18159 724 y Fr(ann)q Fd(;\030)20787 880 y Fg(@)43 b Fc(\003)18159 1695 y Fk(\003)22452 1313 y Fr(again,)551 b(w)-34 b(e)522 b(see)g(that)h(the)f(second)g(factor)h (is)e(b)34 b(ounded)523 b(b)-34 b(y)1200 2819 y Fm(e)1764 2379 y Fd(C)63 b Ft(j)p Fd(@)52 b Fk(\003)p Ft(j)4276 2819 y Fr(.)538 b(T)-101 b(o)405 b(estimate)f(the)h(\014rst)f(factor,)h (remind)f(the)g(explicit)f(expression)11288 5777 y Fm(Q)12246 5187 y Fd(\026)12246 6158 y Fk(\003)12954 5777 y Fr(\()p Fm(\021)14027 5965 y Fk(\003)14735 5777 y Fm(;)289 b Fr(~)-693 b Fm(\021)15876 5965 y Fk(\003)16583 5777 y Fm(;)202 b(\021)17724 5965 y Fk(\003)18377 5712 y Fg(c)18858 5777 y Fr(\))1108 b(=)e Fm(\026)p Fr([)87 b(~)-693 b Fm(\021)24155 5965 y Fk(\003)24863 5777 y Fm(\021)25465 5965 y Fk(\003)26118 5712 y Fg(c)26600 5777 y Fr(])27139 4431 y Ff(\020)27862 5777 y Fm(e)28426 5276 y Ft(\000)p Fk(\001)p Fd(H)30715 5425 y Fc(\003)31323 5276 y Fk(\()p Fd(\021)32161 5425 y Fc(\003)32769 5276 y Fd(;)69 b Fk(~)-539 b Fd(\021)33502 5425 y Fc(\003)34110 5276 y Fd(;\021)34843 5448 y Fc(\003)35395 5257 y Fg(c)35878 5276 y Fk(\))36299 4431 y Ff(\021)20437 8401 y Fl(\024)1106 b Fm(e)23050 7901 y Fd(c)p Ft(j)p Fd(@)52 b Fk(\003)p Ft(j)25237 8401 y Fm(\026)p Fr([)87 b(~)-693 b Fm(\021)26906 8589 y Fk(\003)27614 8401 y Fm(\021)28216 8589 y Fk(\003)28869 8337 y Fg(c)29350 8401 y Fr(])29889 7056 y Ff(\020)30613 8401 y Fm(e)31177 7901 y Ft(\000)p Fk(\001)p Fd(H)33466 8050 y Fc(\003)34074 7901 y Fk(\()p Fd(\021)34912 8050 y Fc(\003)35519 7901 y Fd(;)69 b Fk(~)-539 b Fd(\021)36252 8050 y Fc(\003)36860 7901 y Fd(;\021)37625 7588 y Fn(0)37593 8224 y Fc(\003)38145 8033 y Fg(c)38628 7901 y Fk(\))39050 7056 y Ff(\021)39975 8401 y Fm(:)1200 11359 y Fr(Here)372 b(the)h(inequalit)-34 b(y)372 b(follo)-34 b(ws)373 b(from)g(the)g(de\014nition)g(of)g Fm(H)28231 11547 y Fk(\003)29311 11359 y Fr(and)g(the)g(\014nite)g (range)f(prop)34 b(ert)-34 b(y)373 b(of)g(\010.)528 b(No)-34 b(w)1200 12865 y(use)379 b(the)g(de\014nition)h(of)f(the)g(quenc)-34 b(hed)380 b(k)-34 b(ernels)378 b(and)h(once)g(again)g(the)g(\014nite)g (range)g(of)g(\010)g(to)g(see)f(that)i(the)1200 14370 y(last)404 b(exp)34 b(ectation)404 b(is)g(b)34 b(ounded)406 b(from)e(ab)34 b(o)-34 b(v)g(e)404 b(b)-34 b(y)12159 17328 y Fm(e)12723 16828 y Fd(c)p Ft(j)p Fd(@)52 b Fk(\003)p Ft(j)14909 17328 y Fm(\026)p Fr([)87 b(~)-693 b Fm(\021)16578 17516 y Fk(\003)17286 17328 y Fm(\021)17931 16828 y Ft(0)17888 17628 y Fk(\003)18541 17376 y Fg(c)19023 17328 y Fr(])19562 15983 y Ff(\020)20285 17328 y Fm(e)20849 16828 y Ft(\000)p Fk(\001)p Fd(H)23138 16977 y Fc(\003)23746 16828 y Fk(\()p Fd(\021)24584 16977 y Fc(\003)25192 16828 y Fd(;)69 b Fk(~)-539 b Fd(\021)25925 16977 y Fc(\003)26533 16828 y Fd(;\021)27298 16515 y Fn(0)27266 17151 y Fc(\003)27818 16960 y Fg(c)28301 16828 y Fk(\))28722 15983 y Ff(\021)29783 17328 y Fr(=)336 b Fm(Q)32020 16739 y Fd(\026)32020 17710 y Fk(\003)32728 17328 y Fr(\()p Fm(\021)33801 17516 y Fk(\003)34509 17328 y Fm(;)289 b Fr(~)-693 b Fm(\021)35650 17516 y Fk(\003)36358 17328 y Fm(;)202 b(\021)37542 16828 y Ft(0)37499 17628 y Fk(\003)38152 17376 y Fg(c)38633 17328 y Fr(\))p Fm(:)1200 20286 y Fr(This)405 b(\014nishes)g(the)f(pro) 34 b(of.)p 49878 20286 522 731 v 3082 24116 a(Let)433 b(us)g(no)-34 b(w)434 b(c)-34 b(hec)g(k)433 b(what)i(can)e(b)34 b(e)433 b(said)g(ab)34 b(out)434 b(the)g(criteria)d(for)j(the)f (\014rst)h(part)g(of)f(the)h(v)-67 b(ariational)1200 25622 y(principle)382 b(for)h(join)-34 b(t)383 b(measures.)532 b(It)382 b(turns)i(out)f(that)h(it)e(is)g(natural)i(to)f(use)f(the)h (criteria)f(giv)-34 b(en)382 b(in)h(Section)1200 27127 y(3.4)310 b(with)h(a)f(measure)g Fm(\025)h Fr(that)g(is)f(not)h(a)f (Dirac)g(measure.)507 b(Instead,)329 b(let)310 b(us)h(tak)-34 b(e)310 b(an)-34 b(y)311 b(translation)g(in)-34 b(v)-67 b(arian)-34 b(t)1200 28633 y(con\014guration)405 b Fm(\033)9279 28193 y Fk(0)10209 28633 y Fr(and)g(put)g Fm(\025)337 b Fr(:=)f Fe(P)270 b Fl(\012)f Fm(\016)20210 28896 y Fd(\033)20780 28643 y Fc(0)21295 28633 y Fr(.)3082 30470 y(First,)352 b(using)340 b(the)g(argumen)-34 b(ts)341 b(giv)-34 b(en)339 b(in)h(the)f(pro)34 b(of)340 b(of)g(Theorem)f(6.13,) 352 b(it)340 b(is)f(simple)g(in)h(this)g(situation)1200 31976 y(to)404 b(see)g(that)h(the)g(limit)e(\(3.15\))i(exists)f(and)h (to)f(giv)-34 b(e)404 b(an)g(explicit)g(expression)f(for)i(it.)1200 34810 y Fs(Prop)39 b(osition)465 b(6.20)606 b Fj(Supp)-62 b(ose)551 b(that)g(the)g(de\014ning)h(p)-62 b(otential)550 b Fr(\010)j Fj(is)f(tr)-62 b(anslation)551 b(invariant.)912 b(Supp)-62 b(ose)1200 36315 y(that)574 b Fm(K)4985 35875 y Fk(\026)-519 b Fd(\033)6136 36315 y Fj(is)574 b(tr)-62 b(anslation)574 b(invariant)g(joint)g(me)-62 b(asur)g(e)574 b(c)-62 b(onstructe)g(d)572 b(with)i(the)h(b)-62 b(oundary)573 b(c)-62 b(ondition)638 b Fr(\026)-671 b Fm(\033)43 b Fj(.)1200 37821 y(Then)8208 39762 y Fm(e)8772 39261 y Fd(\025)8772 40092 y(K)9675 39840 y Fc(\026)-453 b Fg(\033)10575 39762 y Fr(=)337 b Fl(\000)p Fm(h)p Fr(\()p Fe(P)p Fr(\))270 b(+)16687 38610 y Ff(X)16661 41236 y Fd(A)p Ft(3)p Fk(0)18665 38112 y Ff(Z)20079 39762 y Fe(P)p Fr(\()p Fm(d\021)43 b Fr(\))23171 38942 y(\010)24046 39130 y Fd(A)24808 38942 y Fr(\()p Fm(\033)26015 38502 y Fk(0)26541 38942 y Fm(;)202 b(\021)43 b Fr(\))p 23171 39483 5025 49 v 24893 40593 a Fl(j)p Fm(A)p Fl(j)28599 39762 y Fr(+)29811 38112 y Ff(Z)31225 39762 y Fe(P)p Fr(\()p Fm(d\021)g Fr(\))459 b(lim)34386 40677 y Fk(\003)p Ft(")p Fh(Z)36099 40425 y Fg(d)37371 38942 y Fr(1)p 36917 39483 1516 49 v 36917 40593 a Fl(j)p Fr(\003)p Fl(j)38767 39762 y Fr(log)219 b Fm(Z)41498 39261 y Fk(\026)-519 b Fd(\033)41362 40062 y Fk(\003)42073 39762 y Fr([)p Fm(\021)43 b Fr(])1200 43037 y Fj(exists.)3082 45871 y Fr(Put)9505 47377 y Fl(H)10529 47559 y Fd(\026)11487 47377 y Fr(:=)336 b Fl(f)p Fm(\021)380 b Fl(2)337 b(H)12 b Fm(;)202 b(\021)379 b Fl(7!)337 b Fm(Q)20899 46876 y Fd(\026)20899 47676 y(x)21521 47377 y Fr(\()p Fm(\021)22637 46876 y Fk(1)22594 47676 y Fd(x)23179 47377 y Fm(;)202 b(\021)24363 46876 y Fk(2)24320 47676 y Fd(x)24904 47377 y Fm(;)g(\021)26045 47673 y Fh(Z)26635 47420 y Fg(d)27117 47673 y Ft(n)p Fd(x)28173 47377 y Fr(\))404 b(is)g(con)-34 b(tin)g(uous)407 b Fl(8)p Fm(x;)202 b(\021)38885 46876 y Fk(1)38842 47676 y Fd(x)39426 47377 y Fm(;)g(\021)40610 46876 y Fk(2)40567 47676 y Fd(x)41152 47377 y Fl(g)p Fm(;)1200 49602 y Fr(then)429 b(w)-34 b(e)430 b(ha)-34 b(v)g(e)429 b(that)h Fm(\033)43 b(\021)421 b Fl(2)377 b Fr(\012)15084 49784 y Fd(\015)15621 49531 y Fg(\026)16608 49602 y Fl(,)h Fm(\021)421 b Fl(2)378 b(H)21431 49784 y Fd(\026)22052 49602 y Fr(.)612 b(Assume)429 b(that)h Fe(P)p Fr([)p Fl(H)32270 49784 y Fd(\026)32890 49602 y Fr(])378 b(=)f(1.)612 b(Then)430 b(an)-34 b(y)429 b(join)-34 b(t)430 b(measure)e(is)1200 51107 y(almost)358 b(Gibbs.)524 b(This)358 b(w)-34 b(as)359 b(p)34 b(oin)-34 b(ted)358 b(out)h(and)g(discussed)f(in)g(the)g(pap)34 b(ers)358 b([17,)f(18])g(and)i(is)e(apparen)-34 b(t)360 b(from)1200 52613 y(the)404 b(ab)34 b(o)-34 b(v)g(e)405 b(represen)-34 b(tation)405 b(of)f(the)h(conditional)f(exp)34 b(ectation.)3082 54450 y(Let)558 b(us)h(remark)f(that,)598 b(whenev)-34 b(er)558 b Fm(K)646 b Fr(is)558 b(a)h(translation-in)-34 b(v)-67 b(arian)-34 b(t)560 b(probabilit)-34 b(y)559 b(measure)f(on)h(the)1200 55956 y(pro)34 b(duct)545 b(space)g(and)g Fm(K)12931 55516 y Fk(\026)-519 b Fd(\033)14051 55956 y Fr(is)545 b(an)-34 b(y)545 b(join)-34 b(t)545 b(measure)g(with)g (marginal)f Fm(K)34934 55516 y Fk(\026)-519 b Fd(\033)34798 56325 y(d)35510 55956 y Fr(\()p Fm(d\021)43 b Fr(\))572 b(=)f Fe(P)p Fr(\()p Fm(d\021)43 b Fr(\))545 b(w)-34 b(e)545 b(ha)-34 b(v)g(e)545 b(that)1200 57461 y Fm(K)2229 57658 y Fd(d)2769 57461 y Fr(\()p Fm(d\021)43 b Fr(\))341 b Fl(6)p Fr(=)f Fe(P)p Fr(\()p Fm(d\021)43 b Fr(\))341 b Fl(\))g Fm(h)p Fr(\()p Fm(K)87 b Fl(j)p Fm(K)15251 57021 y Fk(\026)-519 b Fd(\033)15828 57461 y Fr(\))340 b Fm(>)g Fr(0.)545 b(This)407 b(is)f(clear)g(from)g(the)h(monotonicit) -34 b(y)407 b(of)g(the)f(relativ)-34 b(e)406 b(en)-34 b(trop)g(y)1200 58967 y(w.r.t.)1092 b(to)589 b(the)g(\014ltration.)1093 b(\([11],)634 b(Prop)34 b(osition)589 b(15.5)f(c\).)1092 b(So,)635 b Fm(h)p Fr(\()p Fm(K)87 b Fl(j)p Fm(K)36399 58527 y Fk(\026)-519 b Fd(\033)36976 58967 y Fr(\))644 b(=)g(0)589 b(w)-34 b(ould)589 b(imply)g(that)1200 60472 y Fm(h)p Fr(\()p Fm(K)3398 60669 y Fd(d)3939 60472 y Fl(j)p Fe(P)p Fr(\))401 b(=)g(0)443 b(whic)-34 b(h)444 b(again)f(w)-34 b(ould)445 b(imply)d Fm(K)23103 60669 y Fd(d)24045 60472 y Fr(=)401 b Fe(P)p Fr(,)452 b(b)-34 b(y)444 b(the)f(classical)f(v)-67 b(ariational)442 b(principle)h (applied)1200 61978 y(to)416 b(the)g(pro)34 b(duct)417 b(measure)f Fe(P)p Fr(.)573 b(So,)419 b(giv)-34 b(en)415 b(a)h(join)-34 b(t)417 b(measure)f Fm(K)30714 61538 y Fk(\026)-519 b Fd(\033)31290 61978 y Fr(,)418 b(the)e(class)g(of)g(in) -34 b(teresting)416 b(measures)g(is)1200 63483 y(reduced)404 b(to)h(the)f(ones)g(ha)-34 b(ving)405 b(the)g(same)f Fm(\021)43 b Fr(-marginal.)1200 66649 y Fs(Prop)c(osition)465 b(6.21)606 b Fj(Supp)-62 b(ose)389 b(that)g Fe(P)i Fj(is)f(a)h(pr)-62 b(o)g(duct)388 b(me)-62 b(asur)g(e)389 b(and)i(that)e Fm(\015)36568 66209 y Fd(\026)37580 66649 y Fj(is)h(the)g(ab)-62 b(ove)389 b(sp)-62 b(e)g(ci\014c)g(ation)1200 68154 y(for)379 b(a)h(tr)-62 b(anslation-invariant)376 b(joint)j(me)-62 b(asur)g(e)378 b Fm(K)23947 67715 y Fd(\026)24569 68154 y Fj(.)539 b(Supp)-62 b(ose)378 b(that)g Fe(P)p Fr(\()p Fl(H)34702 68336 y Fd(\026)35324 68154 y Fr(\))337 b(=)f(1)p Fj(.)540 b(T)-93 b(ake)379 b Fm(K)467 b Fj(a)379 b(tr)-62 b(anslation-)1200 69660 y(invariant)432 b(me)-62 b(asur)g(e)432 b(with)h(mar)-62 b(ginal)433 b Fm(K)19839 69857 y Fd(d)20716 69660 y Fr(=)336 b Fe(P)p Fj(.)3082 71497 y(Then)433 b(c)-62 b(ondition)432 b(C'1)i(holds)d(for)i(the)g(me)-62 b(asur)g(e)432 b Fm(K)87 b Fj(,)433 b(for)g(the)g(ab)-62 b(ove)431 b(choic)-62 b(e)431 b(of)i Fm(\025)p Fj(.)25194 75417 y Fr(26)p eop %%Page: 27 27 27 26 bop 1200 1313 a Fj(Pr)-62 b(o)g(of)p Fr(.)605 b(W)-101 b(e)528 b(ha)-34 b(v)g(e)530 b(to)f(c)-34 b(hec)g(k)529 b(that)h Fm(\025)p Fr(\()p Fm(d\033)20388 874 y Fk(1)20914 1313 y Fm(d\021)22190 874 y Fk(1)22716 1313 y Fr(\))p Fm(K)87 b Fr(\()p Fm(d\033)26141 874 y Fk(2)26668 1313 y Fm(d\021)27944 874 y Fk(2)28470 1313 y Fr(\)-a.s)529 b(a)g(con\014guration)h Fm(\033)40634 874 y Fk(1)40591 1638 y Fd(<)p Fk(0)41848 1313 y Fm(\021)42493 874 y Fk(1)42450 1638 y Fd(<)p Fk(0)43708 1313 y Fm(\033)44444 874 y Fk(2)44401 1660 y Ft(\025)p Fk(0)45658 1313 y Fm(\021)46303 874 y Fk(2)46260 1660 y Ft(\025)p Fk(0)48046 1313 y Fr(is)f(in)1200 2819 y(\012)2075 3001 y Fd(\015)2612 2749 y Fg(\026)3221 2819 y Fr(,)479 b(where)465 b(for)g(a)f(con\014guration)i Fm(\033)507 b Fr(w)-34 b(e)465 b(ha)-34 b(v)g(e)465 b(written)h Fm(\033)28893 3001 y Fd(<)p Fk(0)30587 2819 y Fr(=)438 b(\()p Fm(\033)33132 3001 y Fd(x)33717 2819 y Fr(\))34188 3001 y Fd(x<)p Fk(0)35975 2819 y Fr(,)480 b(etc.)719 b(This)465 b(is)f(equiv)-67 b(alen)-34 b(t)464 b(to)1200 4324 y Fm(\021)1845 3884 y Fk(1)1802 4649 y Fd(<)p Fk(0)3059 4324 y Fm(\021)3704 3884 y Fk(2)3661 4671 y Ft(\025)p Fk(0)5444 4324 y Fl(2)526 b(H)7802 4506 y Fd(\026)8940 4324 y Fr(for)517 b Fe(P)345 b Fl(\012)f Fe(P)p Fr(-a.e.)878 b Fm(\021)17768 3884 y Fk(1)18294 4324 y Fm(;)202 b(\021)19478 3884 y Fk(2)20004 4324 y Fr(,)545 b(since)517 b(b)34 b(oth)518 b Fm(\025)f Fr(and)h Fm(K)605 b Fr(ha)-34 b(v)g(e)517 b(marginal)g Fe(P)p Fr(,)545 b(and)519 b(the)e(later)g(is)1200 5830 y(immediate)404 b(b)34 b(ecause)404 b(it)g(is)g(a)g(pro)34 b(duct)405 b(measure.)p 49878 5830 522 731 v 3082 8332 a(T)-101 b(o)404 b(illustrate)g(the)h(general)e(theory)h(of)h(Section)f (3.4)g(w)-34 b(e)405 b(note)f(the)h(follo)-34 b(wing)1200 11497 y Fs(Corollary)464 b(6.22)606 b Fj(Supp)-62 b(ose)492 b(that)g Fe(P)h Fj(is)g(a)g(pr)-62 b(o)g(duct)491 b(me)-62 b(asur)g(e)492 b(and)h(that)f Fm(\015)36157 11058 y Fd(\026)37272 11497 y Fj(is)h(the)f(ab)-62 b(ove)492 b(sp)-62 b(e)g(ci\014c)g(ation) 1200 13003 y(for)485 b(a)g(tr)-62 b(anslation-invariant)482 b(joint)j(me)-62 b(asur)g(e)484 b Fm(K)24476 12563 y Fd(\026)25097 13003 y Fj(.)713 b(Supp)-62 b(ose)483 b(that)h Fe(P)p Fr(\()p Fl(H)35615 13185 y Fd(\026)36237 13003 y Fr(\))431 b(=)f(1)p Fj(.)713 b(T)-93 b(ake)485 b Fm(K)518 b Fl(2)430 b(G)46704 13185 y Fd(inv)48141 13003 y Fr(\()p Fm(\015)49307 12563 y Fd(\026)49929 13003 y Fr(\))1200 14508 y Fj(with)j(mar)-62 b(ginal)433 b Fm(K)9896 14705 y Fd(d)10772 14508 y Fr(=)337 b Fe(P)p Fj(.)3082 16346 y(Then)433 b(c)-62 b(ondition)432 b(C'2)i(of)e(The)-62 b(or)g(em)433 b(3.18)f(is)h(true)g(and)g(henc)-62 b(e)13531 19752 y Fm(h)p Fr(\()p Fm(K)87 b Fl(j)p Fm(K)17269 19252 y Fd(\026)17891 19752 y Fr(\))337 b(=)g(lim)20494 20560 y Fk(\003)22452 18932 y Fr(1)p 21997 19474 1516 49 v 21997 20584 a Fl(j)p Fr(\003)p Fl(j)23847 18103 y Ff(Z)25261 19752 y Fe(P)p Fr(\()p Fm(d\021)43 b Fr(\))202 b(log)30634 18921 y Fm(K)87 b Fr(\()p Fm(\033)32957 18481 y Fk(0)32914 19280 y(\003)33622 18921 y Fm(\021)34224 19109 y Fk(\003)34932 18921 y Fr(\))p 30323 19474 5391 49 v 30323 20607 a Fm(K)31439 20257 y Fd(\026)32061 20607 y Fr(\()p Fm(\033)33268 20190 y Fk(0)33225 20989 y(\003)33933 20607 y Fm(\021)34535 20795 y Fk(\003)35242 20607 y Fr(\))36183 19752 y(=)337 b(0)1200 23328 y Fj(for)433 b(any)g(tr)-62 b(anslation)432 b(invariant)g(spin-c)-62 b(on\014gur)g(ation)431 b Fm(\033)27398 22888 y Fk(0)27924 23328 y Fj(.)1200 26906 y Fi(6.4)1495 b(Random)467 b(\014eld)g(Ising)f(mo)42 b(del:)649 b(failure)467 b(of)e(the)i(second)g(part)g(of)f(the)h(v)-83 b(ariational)4604 28456 y(principle)1200 32404 y Fr(Let)404 b(us)g(no)-34 b(w)405 b(sp)34 b(ecialize)403 b(to)h(the)h(random)g(\014eld)f(Ising)g (mo)34 b(del.)3082 34242 y(F)-101 b(or)459 b(all)g(what)i(follo)-34 b(ws)460 b(w)-34 b(e)460 b(will)f(denote)h(b)-34 b(y)460 b Fm(K)25173 33802 y Fk(+)25960 34242 y Fr(\()p Fm(d\033)43 b(d\021)g Fr(\))430 b(=)f Fe(P)p Fr(\()p Fm(d\021)43 b Fr(\))p Fm(\026)35036 33802 y Fk(+)35824 34242 y Fr([)p Fm(\021)g Fr(]\()p Fm(d\033)g Fr(\))460 b(the)g(`plus-join)-34 b(t)461 b(mea-)1200 35747 y(sure'.)856 b(Here)509 b(w)-34 b(e)511 b(clearly)d(mean)i(b)-34 b(y)511 b Fm(\026)19596 35307 y Fk(+)20384 35747 y Fr([)p Fm(\021)43 b Fr(]\()p Fm(d\033)g Fr(\))513 b(=)g(lim)27665 36043 y Fk(\003)p Ft(")p Fh(Z)29378 35791 y Fg(d)30118 35747 y Fm(\026)30848 35307 y Fk(+)31635 35747 y Fr([)p Fm(\021)43 b Fr(]\()p Fm(d\033)g Fr(\))511 b(the)f(random)h(in\014nite-v)-34 b(olume)1200 37253 y(Gibbs)371 b(measure)g(on)g(the)g(Ising)f(spins.) 528 b(The)371 b(limit)f(exists)g(for)h(an)-34 b(y)371 b(arbitrary)f(\014xed)h Fm(\021)43 b Fr(,)377 b(b)-34 b(y)371 b(monotonicit)-34 b(y)-101 b(.)1200 38758 y(Similarly)403 b(w)-34 b(e)404 b(write)g Fm(K)12356 38318 y Ft(\000)13144 38758 y Fr(\()p Fm(d\033)43 b(d\021)g Fr(\))337 b(=)g Fe(P)p Fr(\()p Fm(d\021)43 b Fr(\))p Fm(\026)22035 38318 y Ft(\000)22823 38758 y Fr([)p Fm(\021)g Fr(]\()p Fm(d\033)g Fr(\).)539 b(In)404 b(this)h(situation)g(w)-34 b(e)405 b(ha)-34 b(v)g(e)1200 42588 y Fs(Prop)39 b(osition)465 b(6.23)606 b Fj(Assume)363 b(that)g(the)f(quenche)-62 b(d)362 b(r)-62 b(andom)363 b(\014eld)h(Ising)f(mo)-62 b(del)363 b(has)g(a)g(phase)f(tr)-62 b(ansition)1200 44094 y(in)483 b(the)e(sense)g(that)g Fm(\026)11325 43654 y Fk(+)12112 44094 y Fr([)p Fm(\021)43 b Fr(]\()p Fm(\033)14595 44276 y Fd(x)15605 44094 y Fr(=)425 b(+\))g Fm(>)g(\026)20910 43654 y Ft(\000)21697 44094 y Fr([)p Fm(\021)43 b Fr(]\()p Fm(\033)24180 44276 y Fd(x)25190 44094 y Fr(=)425 b(+\))483 b Fj(for)e Fe(P)p Fj(-a.e.)702 b Fm(\021)526 b Fj(and)482 b(for)g(some)f Fm(x)425 b Fl(2)g Fe(Z)46059 43654 y Fd(d)46598 44094 y Fj(.)704 b(Then)1200 45599 y(the)429 b(joint)g(me)-62 b(asur)g(es)428 b Fm(K)12388 45159 y Fk(+)13606 45599 y Fj(and)h Fm(K)17072 45159 y Ft(\000)17860 45599 y Fj(,)h(obtaine)-62 b(d)428 b(with)h(the)g(same)g(de\014ning)h(p)-62 b(otential)428 b(ar)-62 b(e)429 b(not)h(c)-62 b(omp)g(atible)1200 47105 y(with)433 b(the)f(same)h(sp)-62 b(e)g(ci\014c)g(ation.)1200 50271 y Fs(Remark)464 b(6.24)606 b Fj(We)401 b(alr)-62 b(e)g(ady)398 b(know)j(by)g(The)-62 b(or)g(em)400 b(6.2)g(that)f(the)h (r)-62 b(elative)399 b(entr)-62 b(opy)399 b Fm(h)p Fr(\()p Fm(K)42737 49831 y Fk(+)43525 50271 y Fl(j)p Fm(K)44978 49831 y Ft(\000)45765 50271 y Fr(\))j Fj(is)e(zer)-62 b(o,)1200 51776 y(and)417 b(thus)g(we)g(pr)-62 b(ove)417 b(her)-62 b(e)416 b(that)h(the)f(se)-62 b(c)g(ond)416 b(p)-62 b(art)417 b(of)g(the)g(variational)e(principle)h(is)h(not)g (valid)f(in)i(c)-62 b(ase)417 b(of)1200 53282 y(phase)431 b(tr)-62 b(ansition)433 b(for)g(the)f(quenche)-62 b(d)431 b(r)-62 b(andom)433 b(\014eld)g(Ising)h(mo)-62 b(del.)1200 56447 y Fs(Remark)464 b(6.25)606 b Fj(In)422 b(the)d(so-c)-62 b(al)62 b(le)-62 b(d)418 b(\\gr)-62 b(and)420 b(ensemble)f(appr)-62 b(o)g(ach")417 b(to)j(disor)-62 b(der)g(e)g(d)418 b(systems)h(pr)-62 b(op)g(ose)g(d)418 b(in)1200 57953 y(the)514 b(the)-62 b(or)g(etic)g(al)512 b(physics)g(liter)-62 b(atur)g(e)512 b(\(going)i(b)-62 b(ack)514 b(to)g(Morita)g(\(1964\)\))e(it)j(is)f (implicitly)f(assume)-62 b(d)512 b(that)1200 59458 y(the)553 b(p)-62 b(otential)551 b(for)j(the)e(joint)i(me)-62 b(asur)g(e)552 b(always)g(exists)h(and)g(do)-62 b(es)553 b(not)g(dep)-62 b(end)553 b(on)h(the)e(choic)-62 b(e)552 b(of)h(the)1200 60964 y(joint)502 b(me)-62 b(asur)g(e)502 b(for)h(the)f(same)g (de\014ning)h(p)-62 b(otential.)764 b(Her)-62 b(e)503 b(we)g(give)f(a)g(ful)62 b(l)502 b(pr)-62 b(o)g(of)502 b(that)g(non-unicity)g(of)1200 62469 y(the)447 b(joint)g(c)-62 b(onditional)446 b(exp)-62 b(e)g(ctations)445 b(\(and)i(ne)-62 b(c)g(essarily)446 b(of)h(the)g(c)-62 b(orr)g(esp)g(onding)446 b(joint)h(p)-62 b(otential\))446 b(r)-62 b(e)g(al)62 b(ly)1200 63975 y(do)-62 b(es)403 b(happ)-62 b(en,)408 b(despite)402 b(of)j(the)e(fact)g(that)h(the)f(joint)h(me)-62 b(asur)g(es)403 b(ar)-62 b(e)404 b(always)f(we)-62 b(akly)403 b(Gibbs.)546 b(It)405 b(is)f(thus)f(an)1200 65480 y(imp)-62 b(ortant)316 b(example)f(of)h(a)h(p)-62 b(atholo)g(gic)g(al)313 b(b)-62 b(ehavior)314 b(in)j(the)f(Morita)f(appr)-62 b(o)g(ach)314 b(in)k(a)e(wel)62 b(l-known)317 b(disor)-62 b(der)g(e)g(d)1200 66986 y(system,)498 b(in)487 b(a)g(tr)-62 b(anslation-invariant)483 b(situation.)714 b(F)-93 b(or)487 b(a)f(discussion)e(of)i(the)g(pr)-62 b(oblems)485 b(of)h(the)f(Morita) 1200 68491 y(appr)-62 b(o)g(ach)431 b(with)i(the)f(the)-62 b(or)g(etic)g(al)430 b(physics)h(c)-62 b(ommunity,)432 b(se)-62 b(e)432 b([7)o(,)i(15)o(,)f(16)o(].)25194 75417 y Fr(27)p eop %%Page: 28 28 28 27 bop 1200 1313 a Fj(Pr)-62 b(o)g(of)p Fr(.)605 b(The)297 b(pro)34 b(of)296 b(relies)e(on)j(the)f(explicit)f(represen)-34 b(tation)297 b(of)g(prop)34 b(osition)296 b(6.16)g(for)g(the)h (conditional)f(ex-)1200 2819 y(p)34 b(ectations)270 b(of)g Fm(K)9259 2379 y Fk(+)10315 2819 y Fr(\(resp.)494 b Fm(K)14898 2379 y Ft(\000)15686 2819 y Fr(\))269 b(in)h(terms)f(of)h Fm(\026)22924 2379 y Fk(+)23981 2819 y Fr(\(resp.)494 b Fm(\026)28178 2379 y Ft(\000)28965 2819 y Fr(\).)g(W)-101 b(e)269 b(will)g(sho)-34 b(w)271 b(that)39735 1842 y Ff(R)40745 2819 y Fm(K)41861 2379 y Fk(+)42649 2819 y Fr(\()p Fm(d\030)44281 3001 y Fd(x)44811 2749 y Fg(c)45293 2819 y Fr(\))p Fm(K)46880 2379 y Ft(\000)46793 3119 y Fd(x)47668 2819 y Fr(\()p Fl(\001j)p Fm(\030)49343 3001 y Fd(x)49873 2749 y Fg(c)50355 2819 y Fr(\))337 b Fl(6)p Fr(=)1200 4324 y Fm(K)2316 3884 y Fk(+)3103 4324 y Fr(\()p Fl(\001)p Fr(\).)539 b(Let)404 b(us)g(ev)-67 b(aluate)404 b(b)34 b(oth)405 b(sides)f(on)g(the)h(ev)-34 b(en)g(t)405 b Fm(B)398 b Fr(:=)336 b Fl(f)p Fm(\021)30615 4506 y Fd(x)31536 4324 y Fr(=)h(+)p Fm(;)34298 3415 y Ff(P)35577 4683 y Fd(y)32 b Fk(:)p Ft(j)p Fd(y)g Ft(\000)p Fd(x)p Ft(j)p Fk(=1)40076 4324 y Fm(\033)40769 4506 y Fd(y)41657 4324 y Fr(=)337 b(0)p Fl(g)p Fr(.)3082 6162 y(Using)477 b(prop)34 b(osition)477 b(6.16,)494 b(it)476 b(is)h(simple)f(to)h(see)f (that)h(w)-34 b(e)477 b(ha)-34 b(v)g(e)477 b(in)g(particular)f(for)h (the)g(lo)34 b(cal)475 b(ev)-34 b(en)g(t)1200 7667 y Fm(\021)1802 7849 y Fd(x)2724 7667 y Fr(=)336 b(+)404 b(for)g(an)-34 b(y)405 b(con\014guration)g Fm(\033)447 b Fr(with)20739 6758 y Ff(P)22019 8026 y Fd(y)32 b Fk(:)p Ft(j)p Fd(y)g Ft(\000)p Fd(x)p Ft(j)p Fk(=1)26518 7667 y Fm(\033)27211 7849 y Fd(y)28099 7667 y Fr(=)337 b(0)404 b(the)g(form)-34 b(ula)7012 11225 y Fm(K)8128 10725 y Fk(+)8915 11225 y Fr(\()p Fm(\021)9988 11407 y Fd(x)10910 11225 y Fr(=)337 b(+)p Fl(j)p Fm(\033)14163 11407 y Fd(x)14693 11155 y Fg(c)15173 11225 y Fm(\021)15775 11407 y Fd(x)16305 11155 y Fg(c)16787 11225 y Fr(\))g(=)18875 9880 y Ff(\020)19598 11225 y Fr(1)270 b(+)21686 9575 y Ff(Z)23100 11225 y Fm(\026)23830 10725 y Fk(+)24618 11225 y Fr([)p Fm(\021)25557 11407 y Fd(x)26478 11225 y Fr(=)336 b Fl(\000)p Fm(;)202 b(\021)29841 11407 y Fd(x)30371 11155 y Fg(c)30852 11225 y Fr(]\()p Fm(d)65 b Fr(~)-671 b Fm(\033)32984 11407 y Fd(x)33569 11225 y Fr(\))p Fm(e)34604 10725 y Fk(2)p Fd(h)50 b Fk(~)-520 b Fd(\033)36154 10836 y Fg(x)36729 9880 y Ff(\021)37453 10182 y Ft(\000)p Fk(1)39048 11225 y Fr(=:)336 b Fm(r)41245 10725 y Fk(+)42032 11225 y Fr(\()p Fm(\021)43105 11407 y Fd(x)43635 11155 y Fg(c)44117 11225 y Fr(\))1200 14439 y(So)404 b(w)-34 b(e)405 b(get)f(that)12092 16435 y Fm(K)13208 15935 y Fk(+)13995 16435 y Fr(\()p Fm(B)61 b Fr(\))338 b(=)17534 14786 y Ff(Z)18948 16435 y Fe(P)p Fr(\()p Fm(d)87 b Fr(~)-693 b Fm(\021)44 b Fr(\))p Fm(\026)22638 15935 y Fk(+)23426 16435 y Fr([)87 b(~)-693 b Fm(\021)43 b Fr(])24745 15090 y Ff(\020)26714 15284 y(X)25468 17968 y Fd(y)32 b Fk(:)p Ft(j)p Fd(y)g Ft(\000)p Fd(x)p Ft(j)p Fk(=1)29912 16435 y Fm(\033)30605 16617 y Fd(y)31493 16435 y Fr(=)337 b(0)33379 15090 y Ff(\021)34372 16435 y Fl(\002)269 b Fm(r)36165 15935 y Fk(+)36952 16435 y Fr(\()87 b(~)-693 b Fm(\021)38025 16617 y Fd(x)38555 16365 y Fg(c)39037 16435 y Fr(\))1200 20116 y(De\014ne)404 b Fm(r)5535 19676 y Ft(\000)6322 20116 y Fr(\()p Fm(\021)7395 20298 y Fd(x)7925 20046 y Fg(c)8407 20116 y Fr(\))g(as)g(ab)34 b(o)-34 b(v)g(e,)405 b(but)g(with)g(the)f(Gibbs)h(measure)f Fm(\026)30583 19676 y Ft(\000)31371 20116 y Fr(.)538 b(Then)405 b(w)-34 b(e)404 b(ha)-34 b(v)g(e)7128 21747 y Ff(Z)8542 23397 y Fm(K)9658 22897 y Fk(+)10445 23397 y Fr(\()p Fm(d\030)12077 23579 y Fd(x)12607 23327 y Fg(c)13089 23397 y Fr(\))p Fm(K)14676 22897 y Ft(\000)14589 23697 y Fd(x)15464 23397 y Fr(\()p Fl(\001j)p Fm(\030)17139 23579 y Fd(x)17669 23327 y Fg(c)18151 23397 y Fr(\)\()p Fm(B)61 b Fr(\))338 b(=)22161 21747 y Ff(Z)23576 23397 y Fe(P)p Fr(\()p Fm(d)87 b Fr(~)-693 b Fm(\021)43 b Fr(\))p Fm(\026)27265 22897 y Fk(+)28053 23397 y Fr([)87 b(~)-693 b Fm(\021)43 b Fr(])29372 22052 y Ff(\020)31341 22246 y(X)30095 24930 y Fd(y)32 b Fk(:)p Ft(j)p Fd(y)g Ft(\000)p Fd(x)p Ft(j)p Fk(=1)34539 23397 y Fm(\033)35232 23579 y Fd(y)36121 23397 y Fr(=)336 b(0)38006 22052 y Ff(\021)38999 23397 y Fl(\002)270 b Fm(r)40793 22897 y Ft(\000)41579 23397 y Fr(\()87 b(~)-693 b Fm(\021)42652 23579 y Fd(x)43182 23327 y Fg(c)43664 23397 y Fr(\))p Fm(:)1200 27443 y Fr(No)-34 b(w)512 b(it)f(follo)-34 b(ws)512 b(from)f(our)g(assumption)i (that,)538 b(for)511 b Fe(P)p Fr(-a.e.)859 b(con\014guration)599 b(~)-693 b Fm(\021)554 b Fr(w)-34 b(e)512 b(ha)-34 b(v)g(e)511 b(the)h(strict)f(in-)1200 28948 y(equalit)-34 b(y)406 b Fm(r)6396 28508 y Fk(+)7183 28948 y Fr(\()87 b(~)-693 b Fm(\021)8256 29130 y Fd(x)8786 28878 y Fg(c)9268 28948 y Fr(\))341 b Fm(<)g(r)11945 28508 y Ft(\000)12732 28948 y Fr(\()87 b(~)-693 b Fm(\021)13805 29130 y Fd(x)14335 28878 y Fg(c)14817 28948 y Fr(\).)546 b(But)407 b(this)h(sho)-34 b(ws)408 b(that)f(b)34 b(oth)408 b(measures)f(giv)-34 b(e)406 b(di\013eren)-34 b(t)407 b(exp)34 b(ectations)406 b(of)1200 30454 y Fm(B)465 b Fr(and)405 b(\014nishes)g(the)g(claim.)p 49878 30454 522 731 v 3082 34284 a(In)428 b(the)h(follo)-34 b(wing)429 b(w)-34 b(e)429 b(sho)-34 b(w)430 b(from)e(the)h(w)-34 b(eakly)428 b(Gibbsian)h(p)34 b(oin)-34 b(t)430 b(of)e(view)g(that)i Fm(K)42081 33844 y Fk(+)43296 34284 y Fr(and)g Fm(K)46794 33844 y Ft(\000)48009 34284 y Fr(ha)-34 b(v)g(e)1200 35789 y(a)508 b(\\go)34 b(o)g(d")507 b(\(rapidly)h(deca)-34 b(ying\))508 b(almost)h(surely)e(con)-34 b(v)g(ergen)g(t)508 b(translation)h(in)-34 b(v)-67 b(arian)-34 b(t)508 b(p)34 b(oten)-34 b(tial.)850 b(This)1200 37295 y(strengthens)306 b(the)f(results)g(in)f([18],)324 b(where)304 b(the)h(a.s.)506 b(absolutely)304 b(con)-34 b(v)g(ergen)g(t)306 b(p)34 b(oten)-34 b(tial)305 b(is)g(not)g(translation)1200 38800 y(in)-34 b(v)-67 b(arian)-34 b(t.)1200 41966 y Fs(Theorem)464 b(6.26)606 b Fj(Assume)495 b(that)f Fm(d)450 b Fl(\025)f Fr(3)p Fj(,)511 b Fm(\014)560 b Fj(is)495 b(lar)-62 b(ge)495 b(enough,)509 b(the)495 b(r)-62 b(andom)495 b(\014elds)g Fm(\021)41775 42148 y Fd(x)42856 41966 y Fj(ar)-62 b(e)495 b(i.i.d.)743 b(with)1200 43472 y(symmetric)520 b(distribution)e(that)i (is)h(c)-62 b(onc)g(entr)g(ate)g(d)519 b(on)i(\014nitely)g(many)g (values,)540 b(and)521 b(that)f Fm(h)p Fe(P)p Fm(\021)45299 43032 y Fk(2)45256 43771 y Fd(x)46363 43472 y Fj(is)g(su\016-)1200 44977 y(ciently)432 b(smal)62 b(l.)3082 46815 y(Ther)-62 b(e)534 b(exists)g(an)h(absolutely)c(c)-62 b(onver)g(gent)533 b(p)-62 b(otential)533 b(that)g(is)h Fr(translation)516 b(in)-34 b(v)-67 b(arian)-34 b(t)535 b Fj(for)g(the)f(plus)1200 48320 y(joint)446 b(me)-62 b(asur)g(e)446 b Fm(K)9917 47880 y Fk(+)10704 48320 y Fr(\()p Fm(d\033)43 b(d\021)g Fr(\))448 b Fj(for)e(su\016ciently)f(low)i(temp)-62 b(er)g(atur)g(e)444 b(and)j(smal)62 b(l)446 b(disor)-62 b(der.)595 b(It)447 b(de)-62 b(c)g(ays)445 b(like)g(a)1200 49826 y(str)-62 b(etche)g(d)431 b(exp)-62 b(onential.)1200 52992 y(Pr)g(o)g(of)p Fr(.)605 b(Applying)454 b(the)g(remark)f(giv)-34 b(en)453 b(after)h(\(5.5\))g(that)h(rely)d(on)i(Theorem)g(2.4.)686 b(of)454 b([18])f(w)-34 b(e)454 b(ha)-34 b(v)g(e)454 b(the)1200 54497 y(follo)-34 b(wing.)3082 57840 y Fs(F)-116 b(act)464 b(pro)-39 b(v)g(ed)465 b(in)g([18)q(].)3082 59678 y Fr(Assume)379 b(that)h Fm(K)11265 59238 y Fd(\026)11886 59678 y Fr(\()p Fm(d\030)56 b Fr(\))338 b(=)e Fe(P)p Fr(\()p Fm(d\021)43 b Fr(\))p Fm(\026)p Fr([)p Fm(\021)g Fr(]\()p Fm(d\033)g Fr(\))380 b(is)f(a)g(join)-34 b(t)379 b(measure)g(for)g(the)g(random)g(\014eld)g(Ising)g(mo)34 b(del.)1200 61183 y(Denote)404 b(the)h(disorder)f(a)-34 b(v)g(erage)403 b(of)i(the)g(quenc)-34 b(hed)404 b(spin-spin)i (correlation)d(b)-34 b(y)10148 64397 y Fm(c)p Fr(\()p Fm(m)p Fr(\))337 b(:=)2121 b(sup)14633 65499 y Fd(x;y)32 b Fk(:)p Ft(j)p Fd(x)p Ft(\000)p Fd(y)g Ft(j)p Fk(=)p Fd(m)20229 62747 y Ff(Z)21643 64397 y Fe(P)p Fr(\()p Fm(d\021)43 b Fr(\))24602 63003 y Ff(\014)24602 63730 y(\014)24602 64458 y(\014)25007 64397 y Fm(\026)p Fr([)p Fm(\021)g Fr(]\()p Fm(\033)28220 64579 y Fd(x)28805 64397 y Fm(\033)29498 64579 y Fd(y)30050 64397 y Fr(\))270 b Fl(\000)f Fm(\026)p Fr([)p Fm(\021)43 b Fr(]\()p Fm(\033)35216 64579 y Fd(x)35801 64397 y Fr(\))202 b Fm(\026)p Fr([)p Fm(\021)43 b Fr(]\()p Fm(\033)39687 64579 y Fd(y)40240 64397 y Fr(\))40711 63003 y Ff(\014)40711 63730 y(\014)40711 64458 y(\014)41115 64397 y Fm(:)1200 68079 y Fr(Supp)34 b(ose)391 b(w)-34 b(e)390 b(giv)-34 b(e)390 b(ourselv)-34 b(es)390 b(an)-34 b(y)390 b(nonnegativ)-34 b(e)391 b(translation)g(in) -34 b(v)-67 b(arian)-34 b(t)390 b(function)i Fm(w)33 b Fr(\()p Fm(A)p Fr(\))390 b(giving)g(w)-34 b(eigh)g(t)1200 69584 y(to)404 b(a)h(subset)g Fm(A)p Fl(\032)p Fe(Z)10069 69144 y Fd(d)10608 69584 y Fr(.)25194 75417 y(28)p eop %%Page: 29 29 29 28 bop 3082 1313 a Fr(Then)405 b(there)f(is)g(a)g(p)34 b(oten)-34 b(tial)16905 1007 y(\026)16695 1313 y Fm(U)17655 874 y Fd(\026)18276 1313 y Fr(\()p Fm(\021)43 b Fr(\))405 b(on)f(the)h(disorder)f(space)g(satisfying)h(the)f(deca)-34 b(y)404 b(prop)34 b(ert)-34 b(y)10548 3597 y Ff(X)9779 6223 y Fd(A)p Fk(:)p Fd(A)p Ft(3)p Fd(x)12607 6346 y Fc(0)13270 4749 y Fm(w)33 b Fr(\()p Fm(A)p Fr(\))16224 3099 y Ff(Z)17639 4749 y Fe(P)p Fr(\()p Fm(d\021)43 b Fr(\))20800 3719 y Ff(\014)20800 4446 y(\014)21415 4443 y Fr(\026)21205 4749 y Fm(U)22165 4159 y Fd(\026)22033 5130 y(A)22792 4749 y Fr(\()p Fm(\021)g Fr(\))24379 3719 y Ff(\014)24379 4446 y(\014)25121 4749 y Fl(\024)26675 4443 y Fr(\026)26401 4749 y Fm(C)27267 4931 y Fk(1)28062 4749 y Fr(+)29549 4443 y(\026)29274 4749 y Fm(C)30140 4931 y Fk(2)31415 3234 y Ft(1)31010 3597 y Ff(X)30868 6188 y Fd(m)p Fk(=2)33105 4749 y Fm(m)34169 4248 y Fk(2)p Fd(d)p Ft(\000)p Fk(1)36630 4749 y Fr(\026)-854 b Fm(w)32 b Fr(\()p Fm(m)p Fr(\))p Fm(c)p Fr(\()p Fm(m)p Fr(\))1200 8637 y(if)427 b(the)h(r.h.s.)608 b(is)427 b(\014nite.)609 b(Here)674 b(\026)-854 b Fm(w)32 b Fr(\()p Fm(m)p Fr(\))377 b(:=)e Fm(w)21499 7655 y Ff(\000)22055 8637 y Fl(f)p Fm(z)389 b Fl(2)337 b Fe(Z)25567 8197 y Fd(d)26107 8637 y Fr(;)202 b Fm(z)389 b Fl(\025)337 b Fr(0)p Fm(;)202 b Fl(j)p Fm(z)53 b Fl(j)336 b(\024)g Fm(m)p Fl(g)34600 7655 y Ff(\001)35583 8637 y Fr(where)427 b Fl(\025)g Fr(denotes)h(the)g(lexico-)1200 10142 y(graphic)457 b(order.)9643 9836 y(\026)9368 10142 y Fm(C)10234 10324 y Fk(1)10760 10142 y Fm(;)11573 9836 y Fr(\026)11299 10142 y Fm(C)12165 10324 y Fk(2)13148 10142 y Fr(are)g(constan)-34 b(ts,)472 b(dep)34 b(ending)457 b(on)h Fm(\014)64 b(;)202 b(h)p Fr(.)697 b(If)457 b Fm(K)34011 9702 y Fd(\026)35090 10142 y Fr(is)f(translation)j(in)-34 b(v)-67 b(arian)-34 b(t,)470 b(then)1410 11341 y(\026)1200 11648 y Fm(U)2160 11208 y Fd(\026)2781 11648 y Fr(\()p Fm(\021)43 b Fr(\))288 b(is)e(translation)h(in)-34 b(v)-67 b(arian)-34 b(t,)310 b(to)34 b(o.)500 b(The)287 b(total)g(p)34 b(oten)-34 b(tial)287 b Fm(U)30773 11208 y Fk(triv)32332 11648 y Fr(\()p Fm(\033)-24 b(;)202 b(\021)43 b Fr(\))34 b(+)36349 11341 y(\026)36138 11648 y Fm(U)37098 11208 y Fd(\026)37720 11648 y Fr(\()p Fm(\021)43 b Fr(\))287 b(is)f(a)h(p)34 b(oten)-34 b(tial)287 b(for)g Fm(K)49442 11208 y Fd(\026)50063 11648 y Fr(.)1200 13153 y(Here)302 b Fm(U)4924 12713 y Fk(triv)6785 13153 y Fr(is)g(a)h(p)34 b(oten)-34 b(tial)303 b(for)g(the)g(formal)f(Hamiltonian)h Fl(\000)p Fm(\014)30036 12244 y Ff(P)31315 13511 y Fd()34050 13153 y Fm(\033)34743 13335 y Fd(i)35118 13153 y Fm(\033)35811 13335 y Fd(j)36364 13153 y Fl(\000)66 b Fm(h)38273 12244 y Ff(P)39553 13511 y Fd(i)40130 13153 y Fm(\021)40732 13335 y Fd(i)41108 13153 y Fm(\033)41801 13335 y Fd(i)42242 13153 y Fl(\000)43251 12244 y Ff(P)44531 13511 y Fd(i)45109 13153 y Fr(log)218 b Fe(P)47617 13335 y Fk(0)48143 13153 y Fr(\()p Fm(\021)49216 13335 y Fd(i)49592 13153 y Fr(\).)3082 16460 y(It)367 b(w)-34 b(as)367 b(already)f(stated)i(in)f([18])f(that)i (w)-34 b(e)367 b(exp)34 b(ect)366 b(a)h(sup)34 b(erp)g(olynomial)366 b(deca)-34 b(y)367 b(of)g(the)g(quan)-34 b(tit)g(y)368 b Fm(c)p Fr(\()p Fm(m)p Fr(\))1200 17966 y(with)436 b Fm(m)p Fr(,)443 b(when)436 b Fm(m)f Fr(tends)h(to)f(in\014nit)-34 b(y)-101 b(.)633 b(W)-101 b(e)434 b(remark)h(\014rst)g(that)i(it)e(w) -34 b(as)436 b(already)e(stated)i(and)g(pro)-34 b(v)g(ed)436 b(in)1200 19471 y(Bricmon)-34 b(t)510 b(and)h(Kupiainen)g(\(1988\))g (that)h Fl(j)p Fm(\026)p Fr([)p Fm(\021)43 b Fr(]\()p Fm(\033)25234 19653 y Fd(x)25819 19471 y Fm(\033)26512 19653 y Fd(y)27064 19471 y Fr(\))269 b Fl(\000)g Fm(\026)p Fr([)p Fm(\021)43 b Fr(]\()p Fm(\033)32229 19653 y Fd(x)32815 19471 y Fr(\))p Fm(\026)p Fr([)p Fm(\021)g Fr(]\()p Fm(\033)36499 19653 y Fd(y)37051 19471 y Fr(\))p Fl(j)514 b(\024)g Fm(C)87 b Fr(\()p Fm(\021)43 b Fr(\))p Fm(e)42934 19032 y Ft(\000)p Fd(C)63 b(\014)46 b(d)p Fk(\()p Fd(x;y)32 b Fk(\))48043 19471 y Fr(with)1200 20977 y(a)461 b(random)g(constan)-34 b(t)462 b Fm(C)87 b Fr(\()p Fm(\021)43 b Fr(\))462 b(that)g(is)e (\014nite)i(for)e Fe(P)p Fr(-a.e.)708 b Fm(\021)43 b Fr(.)708 b(The)461 b(problem)g(is)g(that)g(in)-34 b(tegrabilit)g(y)461 b(of)g(the)1200 22482 y(constan)-34 b(t)353 b(is)e(not)i(to)f(b)34 b(e)351 b(exp)34 b(ected.)520 b(Unfortunately)-101 b(,)363 b(Bricmon)-34 b(t)352 b(and)g(Kupiainen)g(\(1988\))g(do)g(not)g(con)-34 b(trol)1200 23988 y(explicitly)490 b(in)h(their)g(pap)34 b(er)492 b(the)f(deca)-34 b(y)492 b(of)f(the)h(disorder)f(a)-34 b(v)g(erage)491 b Fm(c)p Fr(\()p Fm(m)p Fr(\).)801 b(No)-34 b(w)492 b(w)-34 b(e)492 b(will)f(reen)-34 b(ter)491 b(their)1200 25493 y(renormalization)347 b(group)h(pro)34 b(of)348 b(and)g(sk)-34 b(etc)g(h)349 b(ho)-34 b(w)348 b(stretc)-34 b(hed)349 b(exp)34 b(onen)-34 b(tial)348 b(deca)-34 b(y)347 b(is)g(obtained)i(for)f Fm(c)p Fr(\()p Fm(m)p Fr(\).)1200 26999 y(Ob)-34 b(viously)-101 b(,)364 b(w)-34 b(e)355 b(cannot)h(rep)34 b(eat)354 b(the)h(details)f(of)h(the)g(R)-34 b(G)356 b(analysis)e(here.)522 b(F)-101 b(or)354 b(a)h(p)34 b(edagogical)354 b(exp)34 b(osition)1200 28504 y(of)493 b(the)h(R)-34 b(G)494 b(for)f(disordered)g(mo)34 b(dels,)515 b(see)493 b(also)g([3])g(where)g(the)g(example)g(of)g(an)h(in)-34 b(terface)493 b(mo)34 b(del)493 b(w)-34 b(as)1200 30010 y(treated.)1200 32926 y Fs(Corollary)464 b(6.27)606 b Fj(\(F)-93 b(r)-62 b(om)398 b([1)o(]\))f(Ther)-62 b(e)397 b(is)f(an)i(exp)-62 b(onent)397 b Fm(\013)341 b(>)c Fr(0)398 b Fj(such)d(that,)403 b(for)397 b(al)62 b(l)397 b Fm(m)h Fj(su\016ciently)c(lar)-62 b(ge)1200 34432 y(we)434 b(have)d(that)22171 35937 y Fm(c)p Fr(\()p Fm(m)p Fr(\))337 b Fl(\024)g Fm(e)26883 35437 y Ft(\000)p Fd(m)28447 35124 y Fg(\013)29092 35937 y Fm(:)17873 b Fr(\(6.28\))3082 38854 y Fj(Sketch)432 b(of)h(pr)-62 b(o)g(of)432 b(b)-62 b(ase)g(d)431 b(on)j(R)-31 b(G:)3082 42162 y Fr(F)-101 b(or)472 b(the)g(\014rst)h(part)g(w)-34 b(e)472 b(follo)-34 b(w)473 b(Bricmon)-34 b(t)472 b(and)h(Kupiainen)f (\(1988\),)490 b(page)472 b(750,)489 b(8.3)472 b(`Exp)34 b(onen)-34 b(tial)1200 43667 y(Deca)g(y)327 b(of)g(Correlations'.)513 b(Fix)327 b Fm(x)g Fr(and)h Fm(y)43 b Fr(.)513 b(W)-101 b(e)327 b(will)g(b)34 b(e)327 b(in)-34 b(terested)327 b(in)h(sending)g(their)f(distance)h(to)f(in\014nit)-34 b(y)-101 b(.)1200 45173 y(Let)427 b(us)h(denote)h(b)-34 b(y)428 b Fm(H)98 b Fl(\032)p Fe(Z)13468 44733 y Fd(d)14435 45173 y Fr(the)428 b(half)g(space)g Fm(H)474 b Fr(:=)375 b Fl(f)p Fm(z)429 b Fl(2)377 b Fe(Z)28952 44733 y Fd(d)29491 45173 y Fm(;)202 b(e)285 b Fl(\001)g Fm(z)429 b Fl(\024)376 b Fm(a)p Fl(g)427 b Fr(for)h Fm(a)376 b(>)g Fr(0,)433 b(where)427 b Fm(e)i Fr(is)e(a)h(\014xed)1200 46678 y(unit)567 b(v)-34 b(ector.)1025 b(Let)566 b(us)g(denote)h(b)-34 b(y)567 b Fm(\026)19214 46866 y Fd(H)20111 46678 y Fr([)p Fm(\021)43 b Fr(])607 b(:=)f(lim)25607 46875 y Fk(\003)p Ft(")p Fd(H)27828 46678 y Fm(\026)28558 46168 y Fk(+)28558 47060 y(\003)29346 46678 y Fr([)p Fm(\021)43 b Fr(].)1025 b(By)565 b(monotonicit)-34 b(y)567 b(w)-34 b(e)567 b(ha)-34 b(v)g(e)567 b(for)f(an)-34 b(y)1200 48183 y(con\014guration)354 b(of)e(random)h(\014elds)g Fm(\021)395 b Fr(that)354 b(the)e(quenc)-34 b(hed)353 b(exp)34 b(ectation)353 b(of)f(the)h(spin)g (at)f(the)h(origin)f(in)g(the)1200 49689 y(measure)404 b Fm(\026)6654 49249 y Fk(+)6654 49989 y Fd(a)7441 49689 y Fr([)p Fm(\021)43 b Fr(])404 b(is)g(bigger)g(than)h(that)g(in)f(the)h (measure)f Fm(\026)28433 49249 y Fk(+)29220 49689 y Fr([)p Fm(\021)43 b Fr(].)3082 51491 y(Rep)34 b(eating)375 b(the)g(FK)-34 b(G-argumen)g(ts)376 b(giv)-34 b(en)375 b(in)f(the)i(\014rst)f(steps)g 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b(in)g(all)g(of)h Fe(Z)9602 60130 y Fd(d)10515 60570 y Fr(that)g(there)f(is)g(no)h(P)-34 b(eierls)372 b(con)-34 b(tour)374 b(around)h(0)e(in)g(that)i(touc)-34 b(hes)374 b(the)g(complemen)-34 b(t)373 b(of)1200 62075 y Fm(H)98 b Fr(.)538 b(Then,)405 b(in)f(the)h(same)f(con\014guration)h Fm(\021)43 b Fr(,)404 b(w)-34 b(e)405 b(ha)-34 b(v)g(e)404 b(that)i(the)e(r.h.s.)538 b(is)404 b(b)34 b(ounded)406 b(b)-34 b(y)16245 64585 y Fm(\026)16975 64075 y Fk(+)16975 64967 y Fd(H)17872 64585 y Fr([)p Fm(\021)43 b Fr(]\()p Fm(\033)20355 64767 y Fk(0)20881 64585 y Fr(\))270 b Fl(\000)f Fm(\026)23564 64084 y Fk(+)24351 64585 y Fr([)p Fm(\021)43 b Fr(]\()p Fm(\033)26834 64767 y Fk(0)27360 64585 y Fr(\))337 b Fl(\024)f Fm(\026)30177 64084 y Fk(+)30965 64585 y Fr([)p Fm(\021)43 b Fr(]\()p Fm(E)33720 64084 y Fd(c)33650 64885 y(H)34546 64585 y Fr(\))p Fm(:)1200 67095 y Fr(No)-34 b(w,)405 b(w)-34 b(e)404 b(can)g(alw)-34 b(a)g(ys)405 b(estimate)f(this)h(exp)34 b(ectation)404 b(as)g(a)g(sum)h(o)-34 b(v)g(er)404 b(probabilities)g(of)h(P)-34 b(eierls)403 b(con)-34 b(tours)17236 69718 y Fm(\026)17966 69217 y Fk(+)18753 69718 y Fr([)p Fm(\021)43 b Fr(]\()p Fm(E)21508 69217 y Fd(c)21438 70017 y(H)22335 69718 y Fr(\))337 b Fl(\024)26011 68566 y Ff(X)25090 71396 y Fg(\015)43 b Fc(:)p Fr(in)-34 b(t)q Fg(\015)43 b Fn(3)p Fc(0)24555 72454 y Fr(in)-34 b(t)q Fg(\015)43 b Fn(\\)p Fg(H)27746 72263 y(c)28173 72454 y Fn(6)p Fc(=)p Fn(;)29553 69718 y Fm(\026)30283 69217 y Fk(+)31071 69718 y Fr([)p Fm(\021)g Fr(]\()p Fm(\015)67 b Fr(\))p Fm(:)25194 75417 y Fr(29)p eop %%Page: 30 30 30 29 bop 1200 1313 a Fr(The)366 b(problem)f(is)g(that)i(there)e(is)g (no)h(uniform)g(P)-34 b(eierls)364 b(estimate)i(for)f(all)g (con\014gurations)i(of)f(the)f(disorder.)1200 2819 y(There)605 b(is)g(ho)-34 b(w)g(ev)g(er)605 b(a)g(\\go)34 b(o)g(d)605 b(ev)-34 b(en)g(t")606 b(in)f(disorder)g(space)g Fm(G)672 b Fr(=)f Fm(G)34271 3007 y Fd(H)35772 2819 y Fr(suc)-34 b(h)606 b(that)h(there)d(really)g(is)h(a)1200 4324 y(P)-34 b(eierls)474 b(estimate)g(for)h(all)f(the)h(\\long")g(con)-34 b(tours)476 b(app)34 b(earing)475 b(in)g(the)g(ab)34 b(o)-34 b(v)g(e)475 b(sum.)750 b(The)476 b Fe(P)p Fr(-probabilit)-34 b(y)1200 5830 y(of)556 b(the)f(complemen)-34 b(t)556 b(of)g(this)g(ev)-34 b(en)g(t)555 b(is)g(small)g(and)h(con)-34 b(trolled)556 b(\(in)f(a)h(v)-34 b(ery-non)g(trivial)554 b(w)-34 b(a)g(y\))557 b(b)-34 b(y)555 b(the)1200 7335 y(renormalization)404 b(group)g(construction.)540 b(F)-101 b(or)404 b Fm(\021)380 b Fl(2)337 b Fm(G)26004 7523 y Fd(H)27304 7335 y Fr(w)-34 b(e)405 b(really)d(ha)-34 b(v)g(e)405 b(that)18673 8741 y Ff(X)17752 11571 y Fg(\015)43 b Fc(:)p Fr(in)-34 b(t)q Fg(\015)43 b Fn(3)p Fc(0)17217 12629 y Fr(in)-34 b(t)q Fg(\015)43 b Fn(\\)p Fg(H)20408 12438 y(c)20835 12629 y Fn(6)p Fc(=)p Fn(;)22215 9893 y Fm(\026)22945 9392 y Fk(+)23733 9893 y Fr([)p Fm(\021)g Fr(]\()p Fm(\015)67 b Fr(\))337 b Fl(\024)f Fm(e)28869 9392 y Ft(\000)p Fd(C)63 b(\014)46 b(d)p Fk(\()p Fd(H)32600 9079 y Fg(c)33026 9392 y Fd(;)p Fk(0\))34179 9893 y Fm(:)1200 14837 y Fr(This)405 b(is)e(stated)j(as)e(\(8.34\))g(in)h([1)o(].)539 b(So)404 b(w)-34 b(e)405 b(ha)-34 b(v)g(e)404 b(that)14792 16189 y Ff(Z)16206 17839 y Fe(P)p Fr(\()p Fm(d\021)43 b Fr(\))p Fm(\026)19895 17338 y Fk(+)20684 17839 y Fr([)p Fm(\021)g Fr(]\()p Fm(E)23439 17338 y Fd(c)23369 18139 y(H)24265 17839 y Fr(\))337 b Fl(\024)g Fe(P)p Fr(\()p Fm(G)28518 17338 y Fd(c)28981 17839 y Fr(\))269 b(+)h Fm(e)31498 17338 y Ft(\000)p Fd(C)63 b(\014)46 b(d)p Fk(\()p Fd(H)35229 17026 y Fg(c)35654 17338 y Fd(;)p Fk(0\))1200 20773 y Fr(F)-101 b(rom)453 b(the)f(construction)i(of)f (the)g(renormalization)f(group)h(in)f(Bricmon)-34 b(t-Kupiainen)453 b(w)-34 b(e)453 b(can)g(see)f(that)1200 22279 y Fm(G)539 b Fr(is)h(expressable)f(in)g(the)h(so-called)f(bad)i(\014elds)f Fs(N)26217 21839 y Fd(k)26217 22578 y(x)26801 22279 y Fr(\()p Fm(\021)43 b Fr(\))541 b(in)f(the)g(form)f Fm(G)563 b Fr(=)f Fl(f)p Fm(\021)43 b(;)202 b Fs(N)41604 21839 y Fd(k)41604 22578 y(x)42189 22279 y Fr(\()p Fm(\021)43 b Fr(\))564 b(=)e(0)404 b Fl(8j)p Fm(x)p Fl(j)562 b Fm(<)1200 23954 y(L;)606 b Fl(8)p Fm(k)460 b(>)6228 23357 y Fk(log)172 b Fd(d)p Fk(\()p Fd(x;H)10083 23044 y Fg(c)10510 23357 y Fk(\))p 6228 23675 4648 49 v 7546 24372 a(log)f Fd(L)11008 23954 y Fl(g)p Fr(.)691 b Fm(L)455 b Fr(is)g(a)g(\014xed)g(\014nite)g (length)h(scale)e(\(the)h(blo)34 b(c)-34 b(k-length)456 b(suitably)f(c)-34 b(hosen)455 b(in)g(the)1200 25709 y(construction)501 b(of)f(the)h(R)-34 b(G\).)501 b(It)f(app)34 b(ears)500 b(here)g(just)h(as)f(a)g(constan)-34 b(t.)828 b Fm(x)496 b Fl(2)h Fe(Z)37872 25270 y Fd(d)38912 25709 y Fr(runs)j(o)-34 b(v)g(er)500 b(sites)g(in)g(the)1200 27215 y(lattice)367 b(and)h Fm(k)406 b Fr(is)367 b(a)h(natural)g(n)-34 b(um)g(b)34 b(er)369 b(denoting)f(the)g Fm(k)38 b Fr(-the)368 b(application)g(of)g(the)g(renormalization)f(group)1200 28720 y(transformation.)540 b(The)405 b(renormalization)e(group)i(giv) -34 b(es)404 b(the)g(probabilistic)g(con)-34 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