Content-Type: multipart/mixed; boundary="-------------0407201359177" This is a multi-part message in MIME format. ---------------0407201359177 Content-Type: text/plain; name="04-221.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-221.comments" e-mail: Roberto.Fernandez@univ-rouen.fr; pablo@ime.usp.br; gguerber@fing.edu.uy ---------------0407201359177 Content-Type: text/plain; name="04-221.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-221.keywords" birth-and-death processes; random environment; backwards oriented percolation; multiscale analysis; random point processes; random loss networks ---------------0407201359177 Content-Type: application/postscript; name="bdcam.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="bdcam.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.90a Copyright 2002 Radical Eye Software %%Title: bdcam.dvi %%CreationDate: Tue Jul 20 12:54:56 2004 %%Pages: 48 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMBX12 CMR12 CMSY10 CMBX10 CMR10 MSBM10 CMMI8 CMSY7 %%+ CMTT10 CMTI12 CMMI12 CMBX8 CMEX10 CMSY8 CMR8 CMR6 CMBXTI10 CMMI6 %%+ CMSY6 MSBM7 MSAM10 CMBX6 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: DVIPS.EXE bdcam %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2004.07.20:1254 %%BeginProcSet: texc.pro %! 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17 09:05:00 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /square put readonly def /FontBBox{8 -463 1331 1003}readonly def /UniqueID 5031981 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1C87678CE98C24B934A76220 4DD9B2FF3A49786028E35DDE10AD2C926BD30AD47015FFE9469DE1F793D1C53A C8812CBCD402444EAEA7A50EC5FD93D6A04C2783B50EA48059E3E7407537CB8D 4C206846EF0764C05289733920E2399E58AD8F137C229F3CE3E34D2D1EAB2D53 20D44EFAC8EFA4D14A2EFE389D952527F98D0E49BD5BD2C8D58FF9CB9C78D974 75C2AB5467D73D2B5E277A3FDC35909938A9DF0EB91BD9159D3437BE22EE4544 3429AC8E2BFBE34AE54D3BA3AD04BDF3F4F43A2B43992DF88678681B3AB32CFD A23E2C98D1AF00AB206AC95B78BBE6316F7A0AB6BD3236C28C76288B3C25D1EB E9ABB3576C5EC15A71D26177F5883E9B48293D59015615E2EEAF2E9BA04151ED 5497B9A1C41CBA44BAFF13EA218F5EAC11952EE336AD1DBE6CE92F002EAA3B3D 3BE4C3792F3405763C4BD93EFC3B4FC34193439561841BA989DD8D9F9AEE7A7B 24AEB4654B35023C9720B8F31AA9452E29753FB7915CB29977E725611E37C0B7 784BCC26FACF8A7A0EB1E54290D27FFE52B2D87FAD080AD15EE1984C37E0EB30 122C3012D3A16B09C28903D138352AB5462674B6CFB63F1371768D094DDF288C 36FB9B58443F872D61F2CD8CED42FE0EFF3D7E9952A172BB1AFECB60BF79F2B6 04265FDE4F78BC9FD619AA733CD0412F1D9A7C13B271BF827DCBDC8ABAE24FF0 74D3C220621D7FF0EFE62D835A221D0A7C139E2E6681FC2BBA58FA3B80D416EC 3854C63BA040A4262B458340DAA18AA6AEA3BBAC61615CB85982B18664D3D3AF 340C65B969071CF2D0CABEB80E04623D0526F862ECA8280EEE236C535F70561A 854181132E677674AD5E14C6636F57541D3C821F0776D2CB9B8526D4B826791A 0B179B387D47B08982D36BD61C90BA2CDFE045E8BBAE1C5D4DBA9760AC2D9587 67FB0C22BBD0A905E37D82FD03BBFA69003B5FC6EFCA8CCA8516F461AA33D6F3 6F0B583AECD7A2374014C81033772FE29A7CC38DB81E8462044EF083ECA78849 AC11327BA7EB2FE5389E333AB1FEEDCDC90C78C4713BA7769975CD6290D03B5C 6635F5650B5C1898C7108E6A2A09814C43297E4051B12B263C433646D8555F45 C0B380F639A5A2B5E307C67E6B4208970FB2F32CB0D495CBC1A6D78B36524C44 4FDEA1286022EEC44210FC4ABCE3B8E8BD7DC9A57E5CF409205B78672AD1052E 23813AC24D 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 90 /Z put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueID 5032014 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 110 /backslash put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec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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 13 /gamma put dup 14 /delta put dup 17 /eta put dup 18 /theta put dup 20 /kappa put dup 23 /nu put dup 28 /tau put dup 59 /comma put dup 76 /L put dup 78 /N put dup 82 /R put dup 84 /T put dup 86 /V put dup 96 /lscript put dup 98 /b put dup 100 /d put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 114 /r put dup 116 /t put dup 120 /x put dup 121 /y put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 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47813C178EEACAA6220546E23A814F22980C360D86A6565ADD879B4864984D70 A0CCC261B5814ACCB2A2FA5FE72E06FBC2BA706ACA205919095E921857054348 D1A28DD9F53C3B2574EDBF21D908FC351FE6E3FF350F9AD194AEF7110C16635A DC4D5DC068122839DA1A1E90C6F54730A36CC00ED055B6EDFDA27F7D7B85131C 9D99B72EB903FA2F 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBXTI10 %!PS-AdobeFont-1.1: CMBXTI10 1.0 %%CreationDate: 1991 Aug 18 17:46:30 % 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 (CMBXTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMBXTI10 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 73 /I put dup 82 /R put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e 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 115 /s put dup 116 /t put dup 117 /u put readonly def /FontBBox{-29 -250 1274 754}readonly def /UniqueID 5000771 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 59 /semicolon put dup 61 /equal put dup 91 /bracketleft put dup 93 /bracketright put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 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766F532C223094D3A538CD580F85F68B8DA632559B3C59AC662B8AAEA16DC7 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 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 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 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 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 92 /quotedblleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueID 5000769 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 1 /Delta put dup 3 /Lambda put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 83 /S 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 104 /h put dup 105 /i put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 116 /t put dup 117 /u put dup 119 /w put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec 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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 2 /multiply put dup 21 /greaterequal put dup 24 /similar put dup 26 /propersubset 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 107 /bardbl put dup 110 /backslash put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 8 /circleplus put dup 14 /openbullet put dup 18 /reflexsubset put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 26 /propersubset put dup 27 /propersuperset put dup 30 /precedes put dup 32 /arrowleft put dup 33 /arrowright put dup 41 /arrowdblright put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 54 /negationslash put dup 55 /mapsto put dup 57 /existential put dup 59 /emptyset put dup 64 /aleph put dup 65 /A put dup 66 /B put dup 69 /E put dup 71 /G put dup 74 /J put dup 80 /P put dup 82 /R put 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX8 %!PS-AdobeFont-1.1: CMBX8 1.0 %%CreationDate: 1991 Aug 20 16:36:07 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 65 /A put dup 71 /G put dup 74 /J put readonly def /FontBBox{-59 -250 1235 750}readonly def /UniqueID 5000766 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 17 /eta put dup 18 /theta put dup 20 /kappa put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 26 /rho put dup 28 /tau put dup 34 /epsilon put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff 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 82 /R put dup 83 /S put dup 84 /T put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 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 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueID 5087386 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 5250011D19E9366EB6FD153D3A100CAA6212E3D5D93990737F8D326D347B7EDC 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI12 %!PS-AdobeFont-1.1: CMTI12 1.0 %%CreationDate: 1991 Aug 18 21:06:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 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 16 /dotlessi put dup 19 /acute put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 91 /bracketleft put dup 93 /bracketright put dup 94 /circumflex put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put readonly def /FontBBox{-36 -251 1103 750}readonly def /UniqueID 5000829 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (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 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 7 /Upsilon put dup 9 /Psi put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 16 /dotlessi put dup 19 /acute put dup 28 /oslash put dup 33 /exclam put dup 34 /quotedblright put dup 38 /ampersand 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 62 /questiondown 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 81 /Q 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 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 124 /emdash put dup 126 /tilde put dup 127 /dieresis put readonly 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 16 /dotlessi put dup 19 /acute put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 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 65 /A put dup 66 /B put dup 67 /C 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 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 89 /Y put dup 90 /Z put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1F3352E22A1452E73FECA8A4 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cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /asteriskmath put dup 121 /dagger put dup 122 /daggerdbl put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueID 5000817 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D251491EBF65A98C9FE2B1CF8D725A70281949 8F4AFFE638BBA6B12386C7F32BA350D62EA218D5B24EE612C2C20F43CD3BFD0D F02B185B692D7B27BEC7290EEFDCF92F95DDEB507068DE0B0B0351E3ECB8E443 E611BE0A41A1F8C89C3BC16B352C3443AB6F665EAC5E0CC4229DECFC58E15765 424C919C273E7FA240BE7B2E951AB789D127625BBCB7033E005050EB2E12B1C8 E5F3AD1F44A71957AD2CC53D917BFD09235601155886EE36D0C3DD6E7AA2EF9C C402C77FF1549E609A711FC3C211E64E8F263D60A57E9F2B47E3480B978AAF63 868AEA25DA3D5413467B76D2F02F8097D2841804B020B210C0470066F33B37E2 05805CEE7AA960519806B7942CEEDFF4F6FFF13686D71738CF098EF08F6E4D06 61AA847C14F37DAD834CC1DF9F6206E55344F2E6290D454C962DBCE76E035F93 C497E87A91DAD7CEF4D53C65B64BA2A4C167DD3515CDC9EF13B5EDC08B07460D E790D3740251C868DF2D8B22E7ADA0B5A4DF3A7C87988C7A95D909BA5EC93EE4 C2646AE67445DE0A239CF038C9D507D068CC078C9AF68ADFA213EC6DA335032B F3B79C3EBBA5663A3590D89095A519AFF5189BCCD1B2B36992341521D3D9F734 B9E7CA9AC98EAC5B985B3232569A70D721752FC60AF3C4635846553E7D0A741F 859FF2B50B3420DCA9380CCC41D7686F9E70AC9B74FE9680F17C967977FC61D4 794A68ADA7E349BE498D15C8C520654C31FDA9FCD26CDA7937133AC921DBB89C 73D0969FB2258B3B6C98FFAE7546F6EAA97CC055CEAC6257B86F3C279D32A19C 8413A2CB942BF13A0855B598F8037699C0C47A03B29B46F181673B14EC43F37F F6A6E73CDE1AB21F5848AD6C145296E2D10DD2280E006AA68092B4518470992B 181B7F5A301D5123D98B7E129A2FA7F02700E82ED094FCBCCFEAD21AEDA89F6B 4D4C82D6B848C60BB69826A846FE11702B19B7058C089C65B348E6B1F442876F BA345B7431B2195834F01C7D866ED4B752D84B6A14A9AA66E6541A22C073C12E 9A18013CDB8D45867638284875A1D30D8CB7EB2C252BF877E2E5D879FA4DD93A 1F100D31ED14FEBBF95C7D36EDBAFB5FA1523B91883E25420F4016A7FBE65C97 A48FA877019903C3C231CB357BF568636239B7C13F51E5B6DCE524A3636851F0 BADC08A443FC16B4EF636503415F1EB7D16B99B0AE6745A6751BDCAEFD3D9D33 4EFE96A20C5C0E452981124D21FEAA035159C5A09A84F5CE867DBB8DF967B580 52720C3189E4D82178B625F8271612E6736CBDCEDFEB3C874220BB5176C48F82 593801BFFCB1E9B194A3EF81FAEE0833F1741263DBD557A67D76179E70470303 076CA8DAE4FBEA112DCD81180DB09DC9E2FA9F08F666797C77203A4B0BA7B5DE 8EACBB15F6B4D73F4110161817B7509397EE37BA586BB9E72981119762A9F3DA EE71EE6C893C930DB154FC5D9F5EAE7677ABFF79441E8D48B312B02ECB7670B4 FA6DF3E6AC39F4F5BBC8E8812531B9DA94C13FF36F50DA55F6456B3307A957AD 056BDB3F3AE172809F8EA05E6CA2F97113D855B31CA52A0A1775B305AA23B702 F49CC2021F5DD074D66B34B63553BC908D74183C3D6FAACE3C6AEF81B9CF31BA F2B7E846C51A23FEFD033CDACA5872824EC49EC1B0AA72A5EB6BE41A120CEE5B 45DBBF29ACD516EED59B30 0000000000000000000000000000000000000000000000000000000000000000 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 14 /delta put dup 17 /eta put dup 18 /theta put dup 20 /kappa put dup 23 /nu put dup 24 /xi put dup 28 /tau put dup 58 /period put dup 59 /comma put dup 61 /slash put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C 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 82 /R put dup 83 /S put dup 84 /T put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 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 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueID 5087383 def currentdict end currentfile eexec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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 65 /A put dup 68 /D put dup 69 /E put dup 75 /K put dup 79 /O put dup 82 /R put dup 83 /S put dup 87 /W put dup 89 /Y 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{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont TeXDict begin 39158280 55380996 1000 600 600 (bdcam.dvi) @start /Fa 181[35 74[{}1 49.8132 /CMBX6 rf /Fb 154[32 2[32 98[{}2 49.8132 /CMEX10 rf /Fc 252[77 3[{}1 99.6264 /MSAM10 rf /Fd 165[44 90[{}1 66.4176 /MSBM7 rf /Fe 145[32 59[42 61 18 47[48{}5 49.8132 /CMSY6 rf /Ff 134[32 35 3[24 1[29 5[20 33 25 22 4[32 1[27 1[26 9[36 1[36 1[46 3[48 1[42 16[19 30[29 4[31 2[36 1[29 32 2[28 32 1[40 11[{}25 49.8132 /CMMI6 rf /Fg 154[39 2[39 17[74 80[{}3 66.4176 /CMEX10 rf /Fh 138[62 38 49 50 2[59 65 94 30 2[35 1[53 1[53 59 53 1[59 14[86 8[47 73[{}16 99.6264 /CMBXTI10 rf /Fi 162[18 1[18 29[47 1[18 6[30 30 30 30 30 4[47 1[24 24 36[41 3[{}13 49.8132 /CMR6 rf /Fj 133[60 71 71 97 1[75 52 53 55 71 75 67 75 112 37 71 1[37 75 67 41 61 75 60 75 65 4[69 7[94 1[100 1[92 2[128 3[50 1[106 1[88 2[96 102 6[37 4[67 67 67 67 67 2[37 11[69 21[75 12[{}43 119.552 /CMBX12 rf /Fk 136[51 1[39 27 3[39 35 39 59 3[20 39 2[31 39 31 1[35 3[20 1[20 7[39 21[55 1[20 20 5[35 35 35 35 35 4[55 1[27 27 36[49 1[59 1[{}29 66.4176 /CMR8 rf /Fl 145[35 2[35 3[35 35 9[47 47 31[35 4[0 2[47 47 71 19 21[55 1[55 2[55 18[55 1[55{}17 66.4176 /CMSY8 rf /Fm 141[100 2[66 66 4[47 47 1[100 55 2[55 5[111 111 55 127 144 4[83 2[105 52[75 75 4[53 53 73 73 59 59 3[33 12[{}24 99.6264 /CMEX10 rf /Fn 143[83 1[50 2[50 28 2[50 50 9[66 66 7[60 84 1[69 5[68 2[59 1[53 2[65 80 61 4[50 1[55 1[0 0 2[66 66 100 7[100 7[100 100 1[77 2[77 77 1[77 2[77 77 1[77 3[50 5[77 5[77 28 77{}39 99.6264 /CMSY10 rf /Fo 181[42 2[64 5[61 65[{}3 66.4176 /CMBX8 rf /Fp 133[45 48 55 70 1[56 35 46 44 43 49 1[58 85 29 51 40 33 56 47 48 45 51 42 41 51 41 5[67 57 81 92 57 1[57 60 74 3[78 94 66 83 54 43 81 77 63 1[81 70 74 73 51 1[76 49 76 27 27 23[46 5[42 1[50 1[43 48 59 1[56 1[45 48 2[43 51 55 62 11[{}65 99.6264 /CMMI12 rf /Fq 131[100 50 40 47 45 65 45 52 32 40 41 45 50 50 55 80 25 45 30 30 50 45 30 45 50 45 45 50 2[50 30 1[30 3[97 1[72 70 55 71 1[66 75 1[87 61 1[51 38 1[75 64 66 74 70 69 72 5[30 30 50 50 50 50 50 50 50 50 50 50 1[30 35 30 2[40 40 20[50 2[30 3[55 12[{}69 99.6264 /CMTI12 rf /Fr 165[66 7[72 77 61 1[72 8[66 69[{}6 99.6264 /MSBM10 rf /Fs 128[49 49 1[98 49 43 51 51 70 51 54 38 38 38 51 54 49 54 81 27 51 30 27 54 49 30 43 54 43 54 49 2[49 27 49 27 1[73 73 100 73 73 70 54 72 76 66 76 73 89 61 76 50 35 73 77 64 66 75 70 69 73 2[46 76 1[27 27 49 49 49 49 49 49 49 49 49 49 49 27 33 27 76 1[38 38 27 76 3[49 27 4[49 8[49 2[27 1[81 54 54 57 1[76 1[76 2[65 68 76 81 61{}98 99.6264 /CMR12 rf /Ft 133[44 44 2[44 44 44 44 44 1[44 44 44 44 44 2[44 1[44 44 44 44 1[44 44 14[44 11[44 5[44 17[44 44 45[{}24 83.022 /CMTT10 rf /Fu 128[42 42 2[42 1[44 44 1[44 46 32 33 33 44 46 42 46 69 23 2[23 46 42 25 37 46 37 46 42 6[51 62 3[62 1[46 61 1[57 1[62 76 52 1[43 30 62 65 54 57 1[60 59 62 5[23 1[42 42 42 42 42 1[42 42 42 42 1[23 28 23 24[42 2[23 16[{}58 83.022 /CMR10 rf /Fv 133[30 30 117[34 3[{}3 58.1154 /CMSY7 rf /Fw 134[59 59 81 59 62 44 44 46 59 62 56 62 93 31 59 1[31 62 56 34 51 62 50 62 54 11[86 78 62 2[77 2[106 67 88 58 42 88 88 70 74 86 81 80 85 6[31 56 56 56 56 56 56 56 56 56 56 1[31 37 31 2[44 44 27[62 12[{}58 99.6264 /CMBX12 rf /Fx 134[85 85 117 1[90 63 64 66 1[90 81 90 134 45 85 1[45 90 81 49 74 90 72 90 78 12[112 1[120 1[110 3[97 2[60 126 3[124 117 1[122 8[81 2[81 81 81 81 81 2[45 1[45 31[90 12[{}40 143.462 /CMBX12 rf /Fy 133[33 35 40 51 2[25 33 32 32 36 34 43 62 21 37 29 24 41 34 34 33 36 31 30 37 29 5[48 41 58 67 41 1[41 43 53 3[56 68 48 60 39 31 58 4[50 53 53 37 2[35 1[20 20 29[31 3[31 35 2[41 1[33 35 2[31 36 40 45 11[{}57 66.4176 /CMMI8 rf /Fz 165[61 90[{}1 90.9091 /MSBM10 rf /FA 133[40 48 48 66 48 51 35 36 36 48 51 45 51 76 25 48 28 25 51 45 28 40 51 40 51 45 9[93 2[66 8[71 4[59 10[25 25 11[25 30 25 2[35 35 27[51 12[{}38 90.9091 /CMR10 rf /FB 139[41 41 43 14[46 58 51 7[79 1[108 3[58 78 2[79 3[82 5[69 80 2[79 65[{}15 90.9091 /CMBX10 rf /FC 133[37 37 117[42 3[{}3 83.022 /CMSY10 rf /FD 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y(\014)1408 1736 y Fn(\024)56 b(k)p Fp(f)11 b Fn(k)1700 1751 y Fl(1)1791 1736 y Fp(C)1868 1695 y Fo(J)1914 1736 y Fs(\(Supp)17 b(\()p Fp(f)11 b Fs(\)\))33 b(exp)2540 1626 y Fm(h)2587 1736 y Fn(\000)p Fp(m)h Fs(ln)2864 1693 y Fy(q)2902 1736 y Fs(\(1)22 b(+)g Fp(t)p Fs(\))3182 1626 y Fm(i)3365 1736 y Fs(\(2.19\))79 2050 y Fq(with)44 b Fp(C)377 2014 y Fo(J)423 2050 y Fs(\(Supp)18 b(\()p Fp(f)11 b Fs(\)\))46 b(=)1036 1975 y Fm(P)1142 2079 y Fy(x)p Fl(2)p Fk(Supp)12 b(\()p Fy(f)7 b Fk(\))1514 2050 y Fp(C)1591 2014 y Fo(J)1584 2075 y Fy(x)1637 2050 y Fq(.)75 b(The)45 b(supr)-5 b(emum)44 b(is)h(taken)g(over)g(al)5 b(l)44 b(ac)-5 b(c)g(eptable)79 2186 y(c)g(on\014gur)g(ations)33 b(in)i Fr(N)894 2150 y Fo(G)958 2161 y Fi(\003)1008 2186 y Fq(.)-50 2389 y(3.)49 b Fw(Space)30 b(con)m(v)m(ergence.)43 b Fq(As)30 b Fs(\003)d Fn(!)g Fr(Z)1480 2353 y Fy(d)1521 2389 y Fq(,)j Fp(\026)1640 2353 y Fo(J)1640 2414 y Fk(\003)1721 2389 y Fq(c)-5 b(onver)g(ges)28 b(we)-5 b(akly)28 b(to)h Fp(\026)2620 2353 y Fo(J)2694 2389 y Fs(:=)e Fp(\026)2883 2353 y Fo(J)2883 2422 y Fd(Z)2927 2404 y Ff(d)2996 2389 y Fq(at)i(an)g(exp)-5 b(onen-)79 2510 y(tial)39 b(r)-5 b(ate.)58 b(Mor)-5 b(e)39 b(pr)-5 b(e)g(cisely,)40 b(for)f(any)g(function)g Fp(f)50 b Fq(dep)-5 b(ending)37 b(only)i(on)g(animals)f(c)-5 b(ontaine)g(d)79 2630 y(in)34 b(a)h(\014nite)f(set)h Fs(\003)p Fq(:)565 2757 y Fm(\014)565 2817 y(\014)565 2877 y(\014)598 2872 y Fp(\026)657 2831 y Fo(J)703 2872 y Fp(f)d Fn(\000)23 b Fp(\026)942 2831 y Fo(J)942 2896 y Fk(\003)995 2872 y Fp(f)1054 2757 y Fm(\014)1054 2817 y(\014)1054 2877 y(\014)1142 2872 y Fn(\024)56 b(k)p Fp(f)11 b Fn(k)1434 2887 y Fl(1)1628 2777 y Fm(X)1525 2993 y Fy(x)p Fl(2)p Fk(Supp)h(\()p Fy(f)7 b Fk(\))1893 2872 y Fp(C)1970 2831 y Fo(J)1963 2896 y Fy(x)2049 2872 y Fs(exp)2198 2761 y Fm(h)2245 2872 y Fn(\000)p Fp(m)17 b Fs(dist)q(\()p Fp(x;)g Fs(\003)2787 2831 y Fk(c)2822 2872 y Fs(\))2860 2761 y Fm(i)2935 2872 y Fp(:)403 b Fs(\(2.20\))-50 3238 y Fq(4.)49 b Fw(Exp)s(onen)m(tial)36 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b(the)h(time)h(con)m(v)m(ergence)h(due)f (to)e(the)h(sub)s(exp)s(onen)m(tial)i(time)f(dep)s(endence)h(in)e(the)g (righ)m(t-hand)g(side)-165 514 y(of)32 b(\(2.19\).)-19 634 y(W)-8 b(e)34 b(remark)h(that)f(while)h(the)f(size)i(function)e Fp(S)40 b Fs(in)m(v)m(olv)m(ed)c(in)e(h)m(yp)s(othesis)j(\(2.18\))c(is) h(completely)-165 755 y(arbitrary)-8 b(,)42 b(it)f(is)g(clear)g(that)f (it)h(con)m(v)m(eys)i(some)e(idea)g(of)f(the)h(\\mass")g(or)f(\\migh)m (t")h(\(Dobrushin's,)-165 875 y(1996,)47 b(terminology\))e(of)f (animals,)49 b(in)c(the)g(sense)i(that)e(the)g(larger)f Fp(S)6 b Fs(\()p Fp(\015)f Fs(\))45 b(the)g(larger)g(the)g(set)h(of) -165 995 y(animals)39 b(incompatible)h(with)g Fp(\015)5 b Fs(.)62 b(In)40 b(fact,)g(the)f(natural)g(c)m(hoice)h(for)f(the)g (examples)i(presen)m(ted)g(in)-165 1116 y(Section)h(2.2)e(is)i(the)f (size)h(of)f(the)g(relev)-5 b(an)m(t)42 b(set)g(of)f(v)m(ertices)i(of)d (eac)m(h)i(animal)g(\(area)e(of)h(the)g(grain,)-165 1236 y(length)k(of)f(the)h(call,)j(p)s(erimeter)e(of)e(the)h(con)m(tour,)j (etc.\).)80 b(But,)48 b(of)c(course,)49 b(\\natural")44 b(is)h(not)f(a)-165 1357 y(synon)m(ym)34 b(of)e(\\optimal".)-19 1477 y(In)40 b(most)f(cases)i(of)e(in)m(terest,)j(condition)e(\(2.18\)) f(is)h(morally)f(more)h(limitan)m(t)g(than)f(\(2.16\).)63 b(T)-8 b(o)-165 1597 y(formalize)33 b(this)g(fact,)f(let)h(us)g(denote) 1283 1807 y(\007)1359 1766 y Fo(J)1460 1807 y Fs(=)55 b(sup)1645 1886 y Fy(x)1755 1712 y Fm(X)1763 1921 y Fy(\015)t Fl(3)p Fy(x)1915 1807 y Fp(w)1988 1766 y Fo(J)2034 1807 y Fs(\()p Fp(\015)5 b Fs(\))1199 b(\(2.22\))-165 2099 y(and)1008 2239 y(\011)1084 2198 y Fo(J)1186 2239 y Fs(=)57 b(sup)1317 2321 y Fy(\015)t Fl(2)p Fo(G)1586 2172 y Fs(1)p 1511 2216 198 4 v 1511 2307 a Fp(S)6 b Fs(\()p Fp(\015)f Fs(\))1756 2144 y Fm(X)1736 2357 y Fy(\022)r Fk(:)p Fy(\022)r Fl(6\030)p Fy(\015)1937 2239 y Fp(S)h Fs(\()p Fp(\022)s Fs(\))17 b Fp(w)2217 2198 y Fo(J)2262 2239 y Fs(\()p Fp(\022)s Fs(\))27 b Fp(;)925 b Fs(\(2.23\))-165 2514 y(and)40 b(let)h(us)h(in)m(tro)s(duce)f(the)g Fq(halo)47 b Fs(of)40 b(an)g(animal,)j(namely)f(the)f(region)g(of)f Fr(Z)2782 2478 y Fy(d)2864 2514 y Fs(around)g(it)h(b)s(ey)m(ond)-165 2635 y(whic)m(h)34 b(no)e(incompatible)i(is)f(p)s(ossible:)672 2857 y Fp(H)8 b Fs(\()p Fp(\015)d Fs(\))55 b(:=)1107 2762 y Fm(\\)1234 2746 y(n)1300 2857 y Fp(W)42 b Fn(\032)28 b Fr(Z)1605 2816 y Fy(d)1673 2857 y Fs(:)g Fp(V)22 b Fs(\()p Fp(\022)s Fs(\))27 b Fn(\032)i Fp(W)41 b Fs(=)-17 b Fn(\))27 b Fp(\022)k Fn(\030)d Fp(\015)2620 2746 y Fm(o)2687 2768 y Fy(c)2749 2857 y Fp(:)589 b Fs(\(2.24\))-165 3082 y(This)46 b(halo)f(is)h(a)f(\014nite)i(set)f(b)s(ecause)g(of)f (assumption)i(\(2.4\).)81 b(In)46 b(the)g(examples)h(of)e(Section)i (2.2)-165 3202 y Fp(H)8 b Fs(\()p Fp(\015)d Fs(\))27 b(=)g Fp(V)22 b Fs(\()p Fp(\015)5 b Fs(\).)43 b(In)33 b(addition)g(w)m(e)g(denote)785 3455 y Fp(u)841 3470 y Fk(1)935 3455 y Fs(=)147 b(inf)1066 3521 y Fy(\015)t Fl(2)p Fo(G)p Fk(:)p Fy(\015)t Fl(6)p 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Fp(u)2308 4387 y Fk(1)2374 4303 y Fs(\004)2439 4262 y Fo(J)3365 4303 y Fs(\(2.27\))-165 4541 y(whic)m(h)34 b(lead)f(to)f(the)h(follo)m (wing)g(corollary)g(of)f(our)g(main)h(result.)1676 4790 y(13)p eop end %%Page: 14 14 TeXDict begin 14 13 bop -165 153 a Fw(Corollary)37 b(2.28)50 b Fq(Consider)33 b(an)i(animal)e(mo)-5 b(del)34 b(as)h(de\014ne)-5 b(d)34 b(in)g(Se)-5 b(ction)35 b(2.1)f(such)h(that)1220 367 y Fp(u)1276 382 y Fk(1)1342 367 y Fp(>)28 b Fs(0)99 b Fq(and)g Fp(u)1904 382 y Fk(2)1971 367 y Fp(<)28 b Fn(1)f Fp(:)1136 b Fs(\(2.29\))-165 582 y Fq(Then,)37 b(ther)-5 b(e)38 b(exists)i Fm(e)-58 b Fp(")33 b(>)f Fs(0)38 b Fq(such)f(that)h(if)g Fr(E)p Fs(\(\004)1604 546 y Fo(J)1650 582 y Fs(\))33 b Fn(\024)k Fm(e)-58 b Fp(")37 b Fq(ther)-5 b(e)38 b(exist)f(c)-5 b(onstants)37 b Fp(m;)17 b(q)37 b(>)c Fs(0)k Fq(such)h(that)-165 703 y(pr)-5 b(op)g(erties)34 b(1)h(to)g(5)f(of)h(The)-5 b(or)g(em)34 b(2.15)g(hold)g(for)h Fr(P)p Fq(-almost)f(al)5 b(l)34 b(disor)-5 b(der)34 b(c)-5 b(on\014gur)g(ations)34 b Fw(J)p Fq(.)-19 901 y Fs(The)46 b(conditions)g(\(2.29\))e(are)h (trivially)g(satis\014ed)i(b)m(y)e(all)g(the)g(examples)i(in)e(Section) h(2.2,)i(for)-165 1022 y(whic)m(h,)40 b(in)e(fact,)h Fp(u)548 1037 y Fk(1)624 1022 y Fs(=)d Fp(u)792 1037 y Fk(2)867 1022 y Fs(=)h(1.)58 b(In)39 b(our)e(b)s(ounded-diameter)i (setting,)h(they)f(are)e(also)h(satis\014ed)h(if)-165 1142 y(the)33 b(function)g(size)h(is)f(c)m(hosen)h(translation)e(in)m (v)-5 b(arian)m(t.)-165 1287 y Fw(Pro)s(of.)59 b Fs(W)-8 b(e)36 b(\014rst)g(observ)m(e)h(that)e(for)g(an)m(y)h Fp(a)d Fn(\025)g Fs(1)i(there)h(exists)h Fp(C)2347 1302 y Fy(a)2424 1287 y Fs(suc)m(h)g(that)e(ln)2942 1244 y Fy(a)2984 1287 y Fs(\(1)24 b(+)g Fp(x)p Fs(\))33 b Fn(\024)g Fp(C)3501 1302 y Fy(a)3559 1287 y Fp(x)-165 1407 y Fs(for)f(all)g Fp(x)c Fn(\025)g Fs(0.)44 b(Therefore,)1028 1644 y Fr(E)1094 1533 y Fm(h)1141 1644 y Fs(ln)1223 1601 y Fp(a)1278 1644 y Fs(\(1)22 b(+)g(\007)1561 1603 y Fo(J)1607 1644 y Fs(\))1645 1533 y 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Fp(\021)2086 1436 y Fo(J)p Fy(;)p Fk(0)2186 1477 y Fs(\()17 b Fn(\001)g Fp(;)g Fw(A)2415 1436 y Fk(\()p Fy(x;)p Fk(0\))2567 1477 y Fs(\()p Fp(A)p 2605 1493 V 22 w Fn([)23 b Fp(A)p 2789 1493 V -42 x Fl(0)2885 1477 y Fs(\)\))2961 1366 y Fm(\021)3048 1477 y Fp(:)-165 1712 y Fs(Analogously)387 1947 y Fp(Y)e Fs(\()p Fp(A)p 503 1963 V(;)c(A)p 620 1963 V -41 x Fl(0)716 1947 y Fs(\))56 b(=)f Fp(g)992 1837 y Fm(\020)1051 1947 y Fp(\021)1103 1906 y Fo(J)p Fy(;)p Fk(0)1203 1947 y Fs(\()17 b Fn(\001)g Fp(;)g(A)p 1347 1963 V 21 w Fn([)23 b Fp(A)p 1530 1963 V -41 x Fl(0)1626 1947 y Fs(\))1664 1837 y Fm(\021)1779 1947 y Fs(=)55 b Fp(g)1961 1837 y Fm(\020)2020 1947 y Fp(\021)2072 1906 y Fo(J)p Fy(;)p Fk(0)2173 1947 y Fs(\()17 b Fn(\001)g Fp(;)g Fw(A)2402 1906 y Fk(\()p Fy(x;)p Fk(0\))2553 1947 y Fs(\()p Fp(A)p 2591 1963 V 23 w Fn([)22 b Fp(A)p 2775 1963 V -41 x Fl(0)2872 1947 y Fs(\)\))2948 1837 y Fm(\021)3035 1947 y Fp(:)-165 2183 y Fs(With)33 b(these)g(de\014nitions,)i(w)m(e)e(obtain)311 2418 y 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Fn(B)i Fs(is)e Fp(T)14 b Fs(-in)m(v)-5 b(arian)m(t.)43 b(Hence,)798 4501 y Fm(e)792 4526 y Fr(E)858 4416 y Fm(h)905 4526 y Fp(F)14 b Fs(\()p Fp(C)p 1020 4542 V 7 w(;)j(C)p 1141 4542 V 1218 4485 a Fl(0)1241 4526 y Fs(\))g Fw(1)-12 b Fn(fB)s(g)1508 4416 y Fm(i)1610 4526 y Fs(=)1747 4501 y Fm(e)1741 4526 y Fr(E)1808 4416 y Fm(h)1855 4526 y Fp(F)14 b Fs(\()p Fp(T)g(C)p 2041 4542 V 6 w(;)j(T)d(C)p 2232 4542 V 2309 4485 a Fl(0)2332 4526 y Fs(\))27 b Fw(1)-12 b Fn(fB)s(g)2609 4416 y Fm(i)3365 4526 y Fs(\(4.13\))1676 4790 y(21)p eop end %%Page: 22 22 TeXDict begin 22 21 bop -165 153 a Fs(for)28 b(eac)m(h)j(lo)s(cal)e (function)h Fp(F)42 b Fs(on)30 b Fw(G)1129 168 y Fk(\003)1197 153 y Fn(\002)15 b Fw(G)1377 168 y Fk(\003)1431 153 y Fs(.)43 b(But,)30 b(in)g(the)f(presence)j(of)d Fn(B)s Fs(,)h(the)g(function)g Fp(F)43 b Fs(in)m(v)m(olv)m(ed)-165 273 y(in)32 b(\(4.10\))g(is)h(o)s(dd)g(under)g(this)g(transformation.) 44 b(W)-8 b(e)33 b(conclude)h(that)653 495 y Fm(e)648 520 y Fr(E)714 410 y 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Fy(c)2568 1015 y Fs(\))28 b Fp(:)704 b Fs(\(4.15\))-165 1257 y(The)33 b(pro)s(of)f(follo)m(ws)h(from)f(the)h(observ)-5 b(ation)34 b(that)532 1473 y Fm(e)529 1499 y Fr(P)p Fs(\()p Fn(B)696 1458 y Fy(c)731 1499 y Fs(\))55 b Fn(\024)968 1473 y Fm(b)957 1499 y Fr(Q)1034 1458 y Fo(J)1081 1388 y Fm(n)1147 1499 y Fs(SD)17 b(\()p Fw(A)1416 1458 y Fk(\()p Fy(x;)p Fk(0\))1569 1499 y Fs(\))22 b(+)g(SD)16 b(\()p Fw(A)1995 1458 y Fk(\()p Fy(y)r(;)p Fk(0\))2146 1499 y Fs(\))28 b Fn(\025)g(k)p Fp(x)22 b Fn(\000)h Fp(y)t Fn(k)2645 1388 y Fm(o)2739 1499 y Fp(:)p 2863 1448 57 3 v 2863 1500 3 52 v 2917 1500 V 2863 1502 57 3 v -165 1788 a Fj(4.2)135 b(Bounds)44 b(for)h(the)g(size)h(of)f(the)g(clan)-165 1973 y Fs(The)34 b(k)m(ey)h(inequalit)m(y)g(\(3.20\))d(leads)i(to)f (the)h(follo)m(wing)f(b)s(ounds)h(for)f(the)g(probabilities)i(of)d(the) i(time-)-165 2093 y(length)f(and)f(space)i(diameter)g(and)e(size)i(of)e (the)h(clan)g(of)f(a)g(space-time)i(p)s(oin)m(t.)-165 2321 y Fw(Prop)s(osition)j(4.16)49 b 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Fs(while)33 b(\(4.18\))f(follo)m(ws) h(from)358 423 y Fr(Q)435 382 y Fo(J)482 313 y Fm(n)548 423 y Fs(SD)16 b(\()p Fw(A)816 382 y Fk(\()p Fy(x;)p Fk(0\))969 423 y Fs(\))28 b Fp(>)g(L)1205 313 y Fm(o)1327 423 y Fn(\024)1569 329 y Fm(X)1460 545 y Fy(y)r Fk(:)p Fl(k)p Fy(x)p Fl(\000)p Fy(y)r Fl(k\025)p Fy(L)1855 288 y Fm(Z)1955 314 y Fl(1)1910 513 y Fk(0)2046 423 y Fp(G)2123 382 y Fo(J)2169 313 y Fm(\020)2228 423 y Fs(\()p Fp(x;)17 b Fs(0\))28 b Fn(!)f Fs(\()p Fp(y)t(;)17 b Fn(\000)p Fp(t)p Fs(\))2891 313 y Fm(\021)2950 423 y Fp(dt)28 b(:)274 b Fs(\(4.22\))-165 754 y(T)-8 b(o)32 b(obtain)h(\(4.19\))f(w)m(e)h(use) h(\(4.18\))e(to)g(b)s(ound)h(the)g(righ)m(t-hand)f(side)i(of)e(the)h (inequalit)m(y)672 1001 y Fr(Q)749 960 y Fo(J)795 891 y Fm(n)861 1001 y Fs(SS)17 b(\()p Fw(A)1109 960 y Fk(\()p Fy(x;)p Fk(0\))1263 1001 y Fs(\))27 b Fp(>)h(L)1498 891 y Fm(o)1620 1001 y Fn(\024)56 b Fr(Q)1830 960 y Fo(J)1876 891 y Fm(n)1943 1001 y Fs(SD)16 b(\()p Fw(A)2211 960 y Fk(\()p Fy(x;)p Fk(0\))2364 1001 y Fs(\))28 b Fp(>)f(L)2599 960 y Fk(1)p Fy(=d)2711 891 y Fm(o)3365 1001 y Fs(\(4.23\))-165 1243 y(and)32 b(sum)i(o)m(v)m(er)f Fp(L)p Fs(.)44 b Fc(\003)-165 1532 y Fj(4.3)135 b(Pro)t(of)45 b(of)g(Theorem)g(2.15)-165 1717 y Fs(The)d(b)s(ound)f(\(3.20\))f(pro)m(vided)i(b)m(y)g(the)f(k)m (ey)i(Lemma)e(3.18)g(implies,)j(b)m(y)e(Borel-Can)m(telli,)i(the)e(ab-) -165 1837 y(sence)35 b(of)e(bac)m(kw)m(ards)j(orien)m(ted)f(p)s (ercolation.)47 b(Th)m(us,)36 b(w)m(e)e(can)g(apply)h(Theorem)g(4.1)e (for)g(almost-all)-165 1957 y(realizations)g(of)f(the)h(disorder.)-46 2161 y(1.)49 b Fw(Existence)d(and)h(uniqueness.)69 b Fs(Under)41 b(the)f(h)m(yp)s(othesis)j(of)c(no)i(p)s(ercolation)f(in)g (an)m(y)h(re-)79 2281 y(gion)32 b(\003,)h(the)h(pro)s(cess)g(\(3.13\))e (de\014nes)j(a)d(stationary)i(pro)s(cess)g Fp(\021)2468 2245 y Fy(t)2530 2281 y Fs(in)f(\003)g(b)m(y)h(considering)g(those)79 2402 y(cylinders)g(in)f Fw(C)g Fs(with)g(basis)g(con)m(tained)h(in)f (\003.)43 b(Uniqueness)35 b(follo)m(ws)f(from)e(the)h(next)g(item.)-46 2605 y(2.)49 b Fw(Time)30 b(con)m(v)m(ergence.)42 b Fs(W)-8 b(e)27 b(use)g(\(4.17\))f(and)h(\(4.19\))e(to)h(b)s(ound)h(the)g(righ)m (t)f(hand)h(side)h(of)e(\(4.2\))79 2725 y(b)m(y)770 2967 y(2)17 b Fn(k)p Fp(f)11 b Fn(k)995 2982 y Fl(1)1205 2873 y Fm(X)1101 3089 y Fy(x)p Fl(2)p Fk(Supp)i(\()p Fy(f)7 b Fk(\))1453 2857 y Fm(\020)p 1512 2887 77 4 v 110 x Fp(C)1589 2906 y Fo(J)1589 2992 y Fy(x)1652 2967 y Fs(exp)1801 2857 y Fm(n)1867 2967 y Fn(\000)p 1944 2912 86 4 v Fp(m)34 b Fs(ln)2145 2924 y Fy(q)2183 2967 y Fs(\(1)22 b(+)g Fp(bt)p Fs(\))2504 2857 y Fm(o)2593 2967 y Fs(+)2710 2942 y Fm(b)2691 2967 y Fp(C)2768 2926 y Fo(J)2761 2992 y Fy(x)2830 2967 y Fp(e)2875 2926 y Fl(\000)p Fk(\(1)p Fl(\000)p Fy(b)p Fk(\))p Fy(t)3135 2857 y Fm(\021)1033 3311 y Fn(\024)61 b Fs(2)17 b Fn(k)p Fp(f)11 b Fn(k)1396 3326 y Fl(1)1606 3216 y Fm(X)1503 3432 y Fy(x)p Fl(2)p Fk(Supp)h(\()p Fy(f)7 b Fk(\))1871 3311 y Fp(C)1948 3270 y Fo(J)1941 3336 y Fy(x)2027 3311 y 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Fq(with)35 b Fp(t)1832 1574 y Fy(Y)1921 1559 y Fn(\024)28 b Fp(t)2061 1574 y Fy(X)2128 1559 y Fq(,)35 b(we)g(have)f(that,)-127 1774 y Fp(G)-50 1733 y Fo(J)-50 1799 y Fy(W)30 1774 y Fs(\()p Fp(X)r(;)17 b(Y)k Fs(\))83 b Fn(\024)794 1680 y Fm(X)555 1895 y Fy(Z)603 1904 y Fi(1)649 1895 y Fl(2)12 b Fy(@)t(B)802 1906 y Ff(L;T)912 1895 y Fk(\()p Fy(X)5 b Fk(\))13 b Fl(\\)e Fy(W)1193 1774 y Fp(G)1270 1733 y Fo(J)1270 1799 y Fy(B)1323 1811 y Ff(L)p Fi(+)p Ff(\016)n(;T)1507 1799 y Fk(\()p Fy(X)5 b Fk(\))1629 1774 y Fs(\()p Fp(X)r(;)17 b(Z)1861 1789 y Fk(1)1900 1774 y Fs(\))2203 1680 y Fm(X)1955 1895 y Fy(Z)2003 1904 y Fi(2)2049 1895 y Fl(2)12 b Fy(@)t(B)2202 1906 y Ff(L;T)2312 1895 y Fk(\()p Fy(Z)2387 1904 y Fi(1)2422 1895 y Fk(\))g Fl(\\)g Fy(W)2613 1774 y Fp(G)2690 1733 y Fo(J)2690 1799 y Fy(B)2743 1811 y Ff(L)p Fi(+)p Ff(\016)n(;T)2926 1799 y Fk(\()p Fy(Z)3001 1808 y Fi(1)3036 1799 y Fk(\))3067 1774 y Fs(\()p Fp(Z)3172 1789 y Fk(1)3211 1774 y Fp(;)17 b(Z)3322 1789 y Fk(2)3361 1774 y Fs(\))44 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Fp(p)28 b(>)f(\013)q(d)34 b Fq(such)h(that)869 4260 y Fr(P)930 4150 y Fm(n)996 4260 y Fp(x)h Fq(is)e Fs(\()p Fp(m)1314 4275 y Fl(1)1389 4260 y Fp(;)17 b(L)1499 4275 y Fy(k)1542 4260 y Fs(\))22 b Fn(\000)h Fq(r)-5 b(e)g(gular)2026 4150 y Fm(o)2147 4260 y Fn(\025)56 b Fs(1)22 b Fn(\000)2491 4193 y Fs(1)p 2461 4238 110 4 v 2461 4329 a Fp(L)2527 4282 y Fy(p)2527 4357 y(k)3365 4260 y Fs(\(5.28\))-165 4541 y Fq(for)34 b(al)5 b(l)35 b Fp(k)c Fs(=)c(0)p Fp(;)17 b Fs(1)p Fp(;)g Fs(2)p Fp(;)g(:)g(:)g(:)n Fq(,)35 b(for)f(al)5 b(l)35 b Fp(x)28 b Fn(2)g Fr(Z)1310 4505 y Fy(d)1351 4541 y Fq(.)1676 4790 y Fs(29)p eop end %%Page: 30 30 TeXDict begin 30 29 bop -19 153 a Fs(W)-8 b(e)33 b(end)g(this)g (section)h(with)f(the)g(pro)s(of)f(of)g(the)h(\\easy)h(part")e(of)g (the)h(m)m(ultiscale)h(argumen)m(t.)-165 343 y Fw(Theorem)k(5.29)49 b Fq(If)41 b(a)h(sc)-5 b(ale)g(d)41 b(se)-5 b(quenc)g(e)41 b Fs(\()p Fp(L)1552 358 y Fk(0)1592 343 y Fp(;)17 b(\013)q(;)g(T)d Fs(\))41 b Fq(has)g Fp(m)2159 358 y Fl(1)2234 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Fp(B)1072 478 y Fy(n)1115 488 y Ff(i)1141 478 y Fk(\(2\()p Fy(l)q Fk(+)p Fy(\016)r Fk(\)+1\))p Fy(;e)1537 459 y Ff(L)1579 445 y(\027)1628 458 y Fs(\(\()p Fp(y)1752 473 y Fy(i)1780 458 y Fp(;)17 b Fs(0\)\))27 b Fp(;)114 b(i)28 b Fs(=)g(1)p Fp(;)17 b Fs(2)p Fp(;)g(:)g(:)g(:)e(;)i(R)2717 417 y Fl(0)-165 643 y Fs([recall)41 b(the)h(notation)f(in)g(\(5.3\)].) 69 b(W)-8 b(e)41 b(shall)h(con)m(trol)g(the)f(connectivit)m(y)j (function)d(in)h(the)f(spacial)-165 763 y(direction,)33 b(on)g(the)g(\()p Fp(m;)17 b(l)r Fs(\)-regular)32 b(region)1271 1046 y Fp(B)1350 1005 y Fl(0)1429 1046 y Fs(=)55 b Fp(B)1634 1061 y Fy(L)1686 1046 y Fs(\()p Fp(X)8 b Fs(\))22 b Fn(n)1964 922 y Fy(R)2017 898 y Fe(0)1947 952 y Fm([)1945 1162 y Fy(i)p Fk(=1)2076 1046 y Fp(B)2150 1061 y Fy(i)3365 1046 y Fs(\(5.59\))-165 1319 y([w)m(e)33 b(use)h(the)f(notation)f (resulting)i(from)e(\(5.20\))g(and)h(\(5.46\)].)-19 1439 y(If)j Fp(X)41 b Fn(2)34 b Fp(B)383 1403 y Fl(0)406 1439 y Fs(,)j(w)m(e)g(denote)f Fp(@)5 b(B)1064 1454 y Fk(0)1138 1439 y Fs(=)33 b Fn(f)p Fp(X)8 b Fn(g)p Fs(;)37 b(otherwise)h Fp(X)j Fn(2)34 b Fp(B)2234 1454 y Fy(i)2258 1435 y Fe(0)2320 1439 y Fs(for)h(some)i Fp(i)2753 1403 y Fl(0)2812 1439 y Fs(and)f(w)m(e)h(put)g Fp(@)5 b(B)3465 1454 y Fk(0)3538 1439 y Fs(=)-165 1560 y Fp(@)g(B)-35 1575 y Fy(i)-11 1556 y Fe(0)16 1560 y Fs(.)52 b(Similarly)-8 b(,)37 b(if)e Fp(Y)53 b Fn(2)33 b Fp(B)912 1524 y Fl(0)936 1560 y Fs(,)j(w)m(e)g (denote)g Fp(@)5 b(B)1592 1575 y Fy(R)p Fk(+1)1773 1560 y Fs(=)32 b Fn(f)p Fp(Y)21 b Fn(g)p Fs(;)37 b(otherwise)g Fp(Y)53 b Fn(2)33 b Fp(B)2843 1575 y Fy(i)2867 1556 y Fe(00)2947 1560 y Fs(for)i(some)h Fp(i)3379 1524 y Fl(00)3457 1560 y Fs(and)-165 1680 y(w)m(e)g(put)g Fp(@)5 b(B)293 1695 y Fy(R)p Fk(+1)475 1680 y Fs(=)33 b Fp(@)5 b(B)714 1695 y Fy(i)738 1676 y Fe(00)784 1680 y Fs(.)53 b(Ev)m(ery)37 b(connection)g(from)f Fp(X)43 b Fs(to)35 b Fp(Y)57 b Fs(can)36 b(b)s(e)g(decomp)s(osed)h(in)m(to)f(disjoin)m(t)-165 1801 y(connections)e(among)e(some)i(of)e(the)h(cylinders)h Fp(B)1682 1816 y Fy(i)1711 1801 y Fs(.)43 b(Therefore,)10 1968 y Fm(n)76 2078 y Fp(X)35 b Fn(!)292 2094 y Fy(B)345 2105 y Ff(L)392 2094 y Fk(\()p Fy(X)5 b Fk(\))542 2078 y Fp(Y)620 1968 y Fm(o)769 2078 y Fn(\022)954 1954 y Fy(R)1007 1930 y Fe(0)936 1984 y Fm([)930 2193 y Fy(r)r Fk(=1)1419 1984 y Fm([)1098 2199 y Fl(f)p Fy(i)1157 2208 y Fi(1)1192 2199 y Fy(;i)1236 2208 y Fi(2)1270 2199 y Fy(;:::)o(;i)1393 2207 y Ff(r)1427 2199 y Fl(g\032f)p Fk(1)p Fy(;)p Fk(2)p Fy(;:::)o(;R)1794 2181 y Fe(0)1816 2199 y Fl(g)1852 1968 y Fm(n)1918 2078 y Fp(@)g(B)2048 2093 y Fk(0)2116 2078 y Fn(!)2216 2093 y Fy(B)2272 2074 y Fe(0)2327 2078 y Fp(@)g(B)2457 2093 y Fy(i)2481 2102 y Fi(1)2520 1968 y Fm(o)2609 2078 y Fn(\016)2681 1968 y Fm(n)2747 2078 y Fp(@)g(B)2877 2093 y Fy(i)2901 2102 y Fi(1)2969 2078 y Fn(!)3069 2093 y Fy(B)3125 2074 y Fe(0)3179 2078 y Fp(@)g(B)3309 2093 y Fy(i)3333 2102 y Fi(2)3373 1968 y Fm(o)2100 2374 y Fn(\016)17 b(\001)g(\001)g(\001)j (\016)2377 2263 y Fm(n)2444 2374 y Fp(@)5 b(B)2574 2389 y Fy(i)2598 2397 y Ff(r)2665 2374 y Fn(!)2765 2389 y Fy(B)2821 2370 y Fe(0)2875 2374 y Fp(@)g(B)3005 2389 y Fy(R)3058 2370 y Fe(0)3082 2389 y Fk(+1)3177 2263 y Fm(o)3271 2374 y Fp(:)67 b Fs(\(5.60\))-19 2586 y(As)33 b Fp(B)204 2550 y Fl(0)255 2586 y Fn(\032)28 b Fs(\003)428 2550 y Fl(0)473 2586 y Fn(\002)23 b Fr(R)33 b Fs(with)g(\003)968 2550 y Fl(0)1023 2586 y Fs(a)g(\()p Fp(m;)17 b(l)r Fs(\)-regular)32 b(region,)h(w)m(e)g(can)g(apply)g(\(5.24\))f(to)g(obtain)-116 2819 y Fr(Q)-39 2778 y Fo(J)8 2819 y Fn(f)p Fp(@)5 b(B)188 2834 y Fy(j)217 2843 y Fi(1)284 2819 y Fn(!)384 2834 y Fy(B)440 2815 y Fe(0)494 2819 y Fp(@)g(B)624 2834 y Fy(j)653 2843 y Fi(2)692 2819 y Fn(g)56 b(\024)930 2709 y Fm(h)977 2819 y Fs(\()p Fp(R)q Fs([2\()p Fp(l)23 b Fs(+)e Fp(\016)t Fs(\))g(+)f(1]\))1669 2778 y Fy(d)1709 2819 y Fp(e)1754 2778 y Fy(L)1802 2755 y Ff(\027)1845 2709 y Fm(i)1892 2731 y Fk(2)1948 2819 y Fs(exp)2114 2709 y Fm(n)2201 2819 y Fn(\000)h Fp(m)p Fs(\()p Fp(l)j Fs(+)c Fp(\016)t Fs(\))2656 2709 y Fm(\020)2726 2752 y Fp(D)2807 2767 y Fy(j)2836 2776 y Fi(1)2870 2767 y Fy(;j)2919 2776 y Fi(2)p 2726 2796 232 4 v 2742 2887 a Fp(l)k Fs(+)e Fp(\016)2987 2819 y Fn(\000)g Fs(1)3135 2709 y Fm(\021o)3288 2819 y Fp(:)50 b Fs(\(5.61\))-165 3037 y(W)-8 b(e)33 b(ha)m(v)m(e)h(denoted)204 3222 y Fp(D)285 3237 y Fy(j)314 3246 y Fi(1)349 3237 y Fy(;j)398 3246 y Fi(2)491 3222 y Fs(=)55 b(min)785 3111 y Fm(n)851 3222 y Fn(k)p Fp(x)956 3237 y Fk(1)1018 3222 y Fn(\000)22 b Fp(x)1172 3237 y Fk(2)1212 3222 y Fn(k)56 b Fs(:)f(\()p Fp(x)1493 3237 y Fk(1)1533 3222 y Fp(;)17 b(t)p Fs(\))27 b Fn(2)h Fp(@)5 b(B)1901 3237 y Fy(j)1930 3246 y Fi(1)1986 3222 y Fp(;)34 b Fs(\()p Fp(x)2140 3237 y Fk(2)2179 3222 y Fp(;)17 b(s)p Fs(\))28 b Fn(2)g Fp(@)5 b(B)2559 3237 y Fy(j)2588 3246 y Fi(2)2660 3222 y Fs(for)32 b(some)h Fp(t;)17 b(s)3178 3111 y Fm(o)-165 3428 y Fs(and)32 b(used)i(the)f(b)s (ounds)554 3646 y(max)752 3505 y Fm(\032)836 3578 y Fn(k)p Fp(x)941 3593 y Fy(j)970 3602 y Fi(1)1031 3578 y Fn(\000)23 b Fp(x)1186 3593 y Fy(j)1215 3602 y Fi(2)1253 3578 y Fn(k)p 836 3623 467 4 v 971 3714 a Fp(l)h Fs(+)e Fp(\016)1330 3646 y(;)1383 3578 y Fn(j)p Fp(t)g Fn(\000)h Fp(s)p Fn(j)p 1383 3623 259 4 v 1445 3714 a Fp(e)1490 3685 y Fy(L)1538 3666 y Ff(\027)1652 3505 y Fm(\033)1782 3646 y Fn(\025)1925 3578 y(k)o Fp(x)2029 3593 y Fy(j)2058 3602 y Fi(1)2119 3578 y Fn(\000)g Fp(x)2274 3593 y Fy(j)2303 3602 y Fi(2)2342 3578 y Fn(k)p 1925 3623 467 4 v 2059 3714 a Fp(l)h Fs(+)e Fp(\016)2457 3646 y Fn(\025)2600 3578 y Fp(D)2681 3593 y Fy(j)2710 3602 y Fi(1)2744 3593 y Fy(;j)2793 3602 y Fi(2)p 2600 3623 232 4 v 2616 3714 a Fp(l)i Fs(+)e Fp(\016)2868 3646 y(:)-19 3888 y Fs(>F)-8 b(rom)32 b(\(5.60\),)g(\(5.61\))g(and)g (the)h(v)-5 b(an)33 b(den)g(Berg-Kesten)h(inequalit)m(y)g(\(5.13\))e(w) m(e)i(ha)m(v)m(e)-43 4171 y Fp(G)34 4130 y Fo(J)34 4196 y Fy(B)87 4207 y Ff(L)133 4196 y Fk(\()p Fy(X)5 b Fk(\))256 4171 y Fs(\()p Fp(X)r(;)17 b(Y)k Fs(\))83 b Fn(\024)814 4046 y Fy(R)867 4023 y Fe(0)780 4076 y Fm(X)790 4286 y Fy(r)r Fk(=1)1246 4076 y Fm(X)941 4292 y Fl(f)p Fy(i)1000 4301 y Fi(1)1035 4292 y Fy(;i)1079 4301 y Fi(2)1113 4292 y Fy(;:::)n(;i)1235 4300 y Ff(r)1270 4292 y Fl(g\032f)p Fk(1)p Fy(;)p Fk(2)p Fy(;:::)o(;R)1637 4273 y Fe(0)1659 4292 y Fl(g)1711 4060 y Fm(h)1758 4171 y Fs(\()p Fp(R)q Fs([2\()p Fp(l)24 b Fs(+)e Fp(\016)t Fs(\))g(+)h(1]\))2456 4130 y Fy(d)2496 4171 y Fp(e)2541 4130 y Fy(L)2589 4106 y Ff(\027)2632 4060 y Fm(i)2679 4083 y Fk(2\()p Fy(R)2794 4059 y Fe(0)2817 4083 y Fk(+1\))780 4496 y Fn(\002)17 b Fs(exp)1040 4356 y Fm(\032)1115 4496 y Fn(\000)p Fp(m)p Fs(\()p Fp(l)25 b Fs(+)d Fp(\016)t Fs(\))1569 4356 y Fm(\022)1652 4429 y Fp(D)1733 4444 y Fk(0)p Fy(;i)1812 4453 y Fi(1)1873 4429 y Fs(+)g Fp(D)2052 4444 y Fy(i)2076 4453 y Fi(1)2110 4444 y Fy(;i)2154 4453 y Fi(2)2215 4429 y Fs(+)g Fp(:)17 b(:)g(:)k Fs(+)h Fp(D)2628 4444 y Fy(i)2652 4452 y Ff(r)2687 4444 y Fy(;R)2760 4425 y Fe(0)2783 4444 y Fk(+1)p 1652 4474 1226 4 v 2165 4565 a Fp(l)j Fs(+)d Fp(\016)2909 4496 y Fn(\000)h Fs(\()p Fp(r)h Fs(+)e(1\))3300 4356 y Fm(\023\033)3465 4496 y Fp(:)1676 4790 y Fs(35)p eop end %%Page: 36 36 TeXDict begin 36 35 bop -165 153 a Fs(This)31 b(inequalit)m(y)-8 b(,)32 b(together)f(with)f(the)h(b)s(ounds)f Fp(D)1709 168 y Fk(0)p Fy(;i)1788 177 y Fi(1)1844 153 y Fs(+)17 b Fp(D)2018 168 y Fy(i)2042 177 y Fi(1)2076 168 y Fy(;i)2120 177 y Fi(2)2176 153 y Fs(+)g Fp(:)g(:)g(:)f Fs(+)h Fp(D)2574 168 y Fy(i)2598 176 y Ff(r)2632 168 y Fy(;R)2705 149 y Fe(0)2728 168 y Fk(+1)2850 153 y Fn(\025)28 b Fp(L)17 b Fn(\000)h Fs(2[2\()p Fp(l)h Fs(+)d Fp(\016)t Fs(\))h(+)-165 288 y(1][)p Fp(n)-4 303 y Fy(i)20 312 y Fi(1)74 288 y Fs(+)e Fp(n)223 303 y Fy(i)247 312 y Fi(2)301 288 y Fs(+)g Fp(:)i(:)g(:)e Fs(+)g Fp(n)671 303 y Fy(i)695 311 y Ff(r)734 288 y Fs(])28 b Fn(\025)g Fp(L)15 b Fn(\000)g Fs(2[2\()p Fp(l)k Fs(+)c Fp(\016)t Fs(\))g(+)g(1])p Fp(R)30 b Fs(and)1926 214 y Fm(P)2031 240 y Fy(R)2084 217 y Fe(0)2031 317 y Fy(r)r Fk(=1)2176 214 y Fm(P)2281 317 y Fl(f)p Fy(i)2340 326 y Fi(1)2375 317 y Fy(;i)2419 326 y Fi(2)2453 317 y Fy(;:::)o(;i)2576 325 y Ff(r)2610 317 y Fl(g\032f)p Fk(1)p Fy(;)p Fk(2)p Fy(;:::)o(;R)2977 299 y Fe(0)3000 317 y Fl(g)3072 288 y Fs(1)e Fn(\024)g Fs(\()p Fp(R)16 b Fs(+)f(1\)!,)-165 409 y(yield)112 669 y Fp(G)189 628 y Fo(J)189 694 y Fy(B)242 705 y Ff(L)289 694 y Fk(\()p Fy(X)5 b Fk(\))411 669 y Fs(\()p Fp(X)r(;)17 b(Y)k Fs(\))83 b Fn(\024)h Fs(\()p Fp(R)23 b Fs(+)f(1\)!)1283 559 y Fm(h)1330 669 y Fs(\()p Fp(R)q Fs([2\()p Fp(l)i Fs(+)e Fp(\016)t Fs(\))g(+)g(1]\))2027 628 y Fy(d)2067 669 y Fp(e)2112 628 y Fy(L)2160 605 y Ff(\027)2203 559 y Fm(i)2250 581 y Fk(2\()p Fy(R)p Fk(+1\))936 912 y Fn(\002)17 b Fs(exp)1196 771 y Fm(\032)1270 912 y Fn(\000)p Fp(m)p Fs(\()p Fp(l)25 b Fs(+)e Fp(\016)t Fs(\))1725 771 y Fm(\022)1807 844 y Fp(L)g Fn(\000)f Fs(2[2\()p Fp(l)i Fs(+)e Fp(\016)t Fs(\))g(+)g(1])p Fp(R)p 1807 889 859 4 v 2137 980 a(l)j Fs(+)d Fp(\016)2697 912 y 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Fn(\033)56 b Fp(F)1552 2310 y Fy(j)1611 2295 y Fn(\\)22 b Fp(D)1780 2311 y Fl(\000)p Fk(\()p Fy(j)t Fl(\000)p Fk(1)p Fy(=)p Fk(2\))r Fg(b)-41 b Fy(\034)9 b Fk(\()p Fy(l)q Fk(\))2231 2295 y Fp(;)1107 b Fs(\(5.74\))-165 2515 y(where)39 b Fp(F)185 2530 y Fy(j)261 2515 y Fs(is)g(the)g(ev)m(en)m(t)h(that)e(there)i(is)f(no)f (connection)i(inside)g(the)f(slice)g Fp(S)2748 2530 y Fy(j)2823 2515 y Fs(from)g(p)s(oin)m(ts)g(in)g(the)-165 2649 y(de\013ectiv)m(e)34 b(set)g(to)e(regular)h(p)s(oin)m(ts)g (outside)g(the)g(bu\013er)g(zone)2173 2623 y Fm(e)2167 2649 y Fs(\003)o(:)733 2893 y Fp(F)796 2908 y Fy(j)888 2893 y Fs(=)1019 2782 y Fm(n)1085 2893 y Fp(@)5 b(B)1220 2907 y Fg(b)1215 2923 y Fk(\003)1292 2893 y Fn(\\)22 b Fp(S)1440 2908 y Fy(j)1504 2893 y Fn(\000)-16 b(!)1665 2908 y Fy(S)1708 2918 y Ff(j)1740 2908 y Fl(n)p Fy(B)1833 2921 y Fb(b)1828 2934 y Fi(\003)1906 2893 y Fp(B)1980 2923 y Fk(\003[)p Fy(x)p Fk(;)p Fy(L)p Fk(+)p Fy(\016)r Fk(])p Fl(n)2305 2906 y Fg(e)2300 2923 y Fk(\003)2375 2893 y Fn(\\)23 b Fp(S)2524 2908 y Fy(j)2560 2782 y Fm(o)2627 2805 y Fy(c)2689 2893 y Fp(;)649 b Fs(\(5.75\))-165 3135 y(and)37 b Fp(D)110 3150 y Fy(s)184 3135 y Fs(represen)m(ts)i(the)f (lac)m(k)g(of)f(connection)h(within)g(a)f(strip)h(of)f(heigh)m(t)h (\001)f(in)m(v)m(olving)i(de\013ectiv)m(e)-165 3255 y(sites:)735 3375 y Fp(D)816 3390 y Fy(s)909 3375 y Fs(=)1040 3265 y Fm(n)1112 3350 y(e)1106 3375 y Fs(\003)22 b Fn(\002)h(f)p Fp(s)p Fn(g)k(\000)-16 b(!)1635 3389 y Fg(e)1630 3406 y Fk(\003)o Fl(\002)p Fk([)p Fy(s)p Fl(\000)p Fk(\001)p Fy(;s)p Fk(])2009 3350 y Fm(e)2003 3375 y Fs(\003)22 b Fn(\002)g(f)p Fp(s)g Fn(\000)h Fs(\001)p Fn(g)2541 3265 y Fm(o)2607 3287 y Fy(c)2686 3375 y Fp(:)652 b Fs(\(5.76\))-165 3577 y(W)-8 b(e)37 b(are)g(assumming)h(that)e Fp(l)j Fs(is)f(su\016cien)m(tly)h(large)e(so)g(that)h Fm(b)-57 b Fp(\034)12 b Fs(\()p Fp(l)r Fs(\))p Fp(=)p Fs(2)34 b Fp(>)h Fs(1,)i(and)g(hence)j Fm(b)-57 b Fp(\034)12 b Fs(\()p Fp(l)r Fs(\))p Fp(=)p Fs(2)34 b Fp(>)h Fs(\001)-165 3698 y(for)d(all)g(\001)c Fn(2)g Fs(\(0)p Fp(;)17 b Fs(1].)-19 3818 y(Since)35 b(eac)m(h)g Fp(F)535 3782 y Fy(c)521 3843 y(j)603 3818 y Fs(in)m(v)m(olv)m(es)i(connections)f(within)f (regular)f(regions,)h(w)m(e)g(can)f(apply)h(Lemma)g(5.23)-165 3938 y(to)d(obtain)1366 4059 y Fr(Q)1443 4018 y Fo(J)1490 4059 y Fn(f)p Fp(F)1617 4018 y Fy(c)1603 4083 y(j)1651 4059 y Fn(g)c(\024)g Fp(e)1879 4018 y Fl(\000)p Fy(h)p Fk(\()p Fy(l)q Fk(\))2055 4059 y Fp(;)1283 b Fs(\(5.77\))-165 4233 y(with)-116 4453 y Fp(h)p Fs(\()p Fp(l)r Fs(\))56 b(=)f Fp(m)p Fn(f)370 4427 y Fm(b)369 4453 y Fp(\016)t Fs(\()p Fp(l)r Fs(\))21 b Fn(\000)g Fs(\(3)p Fp(l)h Fs(+)e(4)p Fp(\016)k Fs(+)c(1\))p Fn(g)g(\000)h Fs(2)c(ln)h Fm(b)-57 b Fp(\034)12 b Fs(\()p Fp(l)r Fs(\))20 b Fn(\000)h Fp(d)c Fs(ln)1951 4427 y Fm(b)1950 4453 y Fp(\016)t Fs(\()p Fp(l)r Fs(\))j Fn(\000)h Fp(d)c Fs(ln\(2)p Fp(l)22 b Fs(+)e(3)p Fp(\016)k Fs(+)c(1\))g Fn(\000)h Fs(2)c(ln)f Fp(R)29 b(:)50 b Fs(\(5.78\))1676 4790 y(38)p eop end %%Page: 39 39 TeXDict begin 39 38 bop -165 153 a Fs(Hence)768 273 y Fp(h)p Fs(\()p Fp(l)r Fs(\))55 b Fn(\025)h Fp(m)17 b(c)1263 288 y Fk(1)1319 273 y Fp(l)1350 232 y Fy(\024)1417 273 y Fn(\000)23 b Fp(c)1559 288 y Fk(2)1615 273 y Fp(l)1646 232 y Fy(\034)1745 273 y Fn(\025)56 b Fp(c)1920 288 y Fk(3)1976 273 y Fp(l)2007 232 y Fy(\024)p Fl(\000)p Fy(\022)2197 273 y Fn(\025)g Fp(c)2372 288 y Fk(3)2428 273 y Fp(l)2459 232 y Fy(\024)p Fl(\000)p Fy(\022)2588 241 y Fi(0)2654 273 y Fp(;)684 b Fs(\(5.79\))-165 447 y(where)43 b Fp(c)168 462 y Fk(1)207 447 y Fp(;)17 b(c)293 462 y Fk(1)374 447 y Fs(and)42 b Fp(c)615 462 y Fk(3)697 447 y Fs(are)g(p)s(ositiv)m(e)h (constan)m(ts)h(|dep)s(ending)e(on)g Fp(d;)17 b(\016)n(;)g(\024;)g (\034)6 b(;)17 b(\022)45 b Fs(and)d Fp(R)q Fs(|)f(and)h Fp(l)j Fs(is)-165 568 y(large)32 b(enough.)44 b(W)-8 b(e)33 b(ha)m(v)m(e)h(used)g(that)e Fp(m)c Fn(\025)g Fp(l)1523 532 y Fl(\000)p Fy(\022)1617 568 y Fs(,)33 b Fp(\022)e(<)c(\022)1901 583 y Fk(0)1974 568 y Fs(and)32 b Fp(\034)40 b(<)27 b(\024)22 b Fn(\000)h Fp(\022)s Fs(.)-19 688 y(On)33 b(the)g(other)f(hand,)914 920 y Fp(D)995 935 y Fy(s)1087 920 y Fn(\033)1263 825 y Fm(\\)1220 1037 y Fy(\015)t Fl(2)p Fo(G)1376 1050 y Fb(e)1371 1063 y Fi(\003)1433 920 y Fn(f)p Fs([)p Fn(E)1563 935 y Fk(1)1602 920 y Fs(\()p Fp(\015)5 b Fs(\))22 b Fn([)h(E)1898 935 y Fk(2)1937 920 y Fs(\()p Fp(\015)5 b Fs(\)])22 b Fn(\\)g(E)2259 935 y Fk(3)2298 920 y Fs(\()p Fp(\015)5 b Fs(\))p Fn(g)28 b Fp(;)830 b Fs(\(5.80\))-165 1250 y(where)81 1470 y Fn(E)134 1485 y Fk(1)173 1470 y Fs(\()p Fp(\015)5 b Fs(\))82 b(=)h Fn(f)p Fp(\015)38 b Fs(is)33 b(not)f(presen)m(t)i(at)f(time)g Fp(s)22 b Fn(\000)h Fs(\001)p Fn(g)k Fp(;)81 1615 y Fn(E)134 1630 y Fk(2)173 1615 y Fs(\()p Fp(\015)5 b Fs(\))82 b(=)h Fn(f)p Fp(\015)38 b Fs(is)33 b(presen)m(t)h(at)e(time)h Fp(s)23 b Fn(\000)f Fs(\001)33 b(but)g(it)f(do)s(es)h(not)g(surviv)m(e) i(un)m(til)e(time)g Fp(s)p Fn(g)28 b Fp(;)81 1760 y Fn(E)134 1775 y Fk(3)173 1760 y Fs(\()p Fp(\015)5 b Fs(\))82 b(=)h Fn(f)33 b Fs(there)g(is)g(no)g(birth)f(of)h Fp(\015)k Fs(in)c(the)g(in)m(terv)-5 b(al)33 b([)p Fp(s)22 b Fn(\000)h Fs(\001)p Fp(;)17 b(s)p Fs(])p Fn(g)27 b Fp(:)-165 1980 y Fs(W)-8 b(e)33 b(observ)m(e)h(that)107 2200 y Fr(Q)184 2159 y Fo(J)231 2090 y Fm(\020)290 2200 y Fs([)p Fn(E)370 2215 y Fk(1)409 2200 y Fs(\()p Fp(\015)5 b Fs(\))22 b Fn([)h(E)705 2215 y Fk(2)744 2200 y Fs(\()p Fp(\015)5 b Fs(\)])22 b Fn(\\)g(E)1066 2215 y Fk(3)1105 2200 y Fs(\()p Fp(\015)5 b Fs(\))1237 2090 y Fm(\021)1380 2200 y Fs(=)1539 2090 y Fm(\020)1598 2200 y Fr(Q)1675 2159 y Fo(J)1722 2200 y Fs([)p Fn(E)1802 2215 y Fk(1)1841 2200 y Fs(\()p Fp(\015)g Fs(\)])22 b(+)g Fr(Q)2197 2159 y Fo(J)2244 2200 y Fs([)p Fn(E)2324 2215 y Fk(2)2363 2200 y Fs(\()p Fp(\015)5 b Fs(\)])2522 2090 y Fm(\021)2598 2200 y Fr(Q)2675 2159 y Fo(J)2721 2200 y Fs([)p Fn(E)2801 2215 y Fk(3)2840 2200 y Fs(\()p Fp(\015)g Fs(\)])1380 2413 y(=)1539 2302 y Fm(h)1586 2413 y Fp(e)1631 2372 y Fl(\000)p Fy(w)1739 2348 y Fa(J)1778 2372 y Fk(\()p Fy(\015)t Fk(\))1899 2413 y Fs(+)22 b(\(1)g Fn(\000)h Fp(e)2251 2372 y Fl(\000)p Fy(w)2359 2348 y Fa(J)2398 2372 y Fk(\()p Fy(\015)t Fk(\))2497 2413 y Fs(\)\(1)f Fn(\000)h Fp(e)2789 2372 y Fl(\000)p Fk(\001)2907 2413 y Fs(\))2945 2302 y Fm(i)2992 2413 y Fp(e)3037 2372 y Fl(\000)p Fk(\001)p Fy(w)3204 2348 y Fa(J)3243 2372 y Fk(\()p Fy(\015)t Fk(\))1380 2598 y Fs(=)83 b Fp(K)1629 2557 y Fo(J)1622 2622 y Fk(\001)1685 2598 y Fs(\()p Fp(\015)5 b Fs(\))27 b Fp(:)1494 b Fs(\(5.81\))-165 2818 y(This)26 b(result)g(follo)m(ws)f(from)g(w)m(ell-kno)m(wn)i(prop)s(erties)f(on)f (P)m(oisson)h(pro)s(cesses)h(and)e(from)g(the)h(fact)e(that)-165 2938 y(the)j(ev)m(en)m(ts)i(in)e(whic)m(h)h(an)e(animal)h Fp(\015)32 b Fs(is)27 b(activ)-5 b(ated)27 b(at)f(time)i Fn(\000)p Fp(t)f Fs(and)g(surviv)m(es)i(un)m(til)f(time)f(0)f(de\014ne) i(an)-165 3058 y(inhomogeneous)j(P)m(oisson)h(pro)s(cess)g(of)e(rate)g Fp(w)1578 3022 y Fo(J)1623 3058 y Fs(\()p Fp(\015)5 b Fs(\))p Fp(e)1800 3022 y Fl(\000)p Fy(t)1885 3058 y Fs(.)43 b(Therefore,)32 b(due)f(to)f(the)g Fr(Q)2963 3022 y Fo(J)3010 3058 y Fs(-indep)s(endence)-165 3179 y(of)i(ev)m(en)m(ts)j(p)s (ertaining)d(to)h(di\013eren)m(t)g(animals,)1014 3411 y Fr(Q)1091 3370 y Fo(J)1138 3411 y Fn(f)p Fp(D)1269 3426 y Fy(s)1305 3411 y Fn(g)55 b(\025)1578 3316 y Fm(Y)1543 3528 y Fy(\015)t Fl(2)p Fo(G)1699 3541 y Fb(e)1694 3554 y Fi(\003)1757 3411 y Fp(K)1847 3370 y Fo(J)1840 3435 y Fk(\001)1902 3411 y Fs(\()p Fp(\015)5 b Fs(\))56 b Fn(\025)g Fp(e)2268 3370 y Fl(\000)p Fy(l)2345 3346 y Ff(b)2408 3411 y Fp(:)930 b Fs(\(5.82\))-19 3746 y(By)33 b(the)g(FK)m(G-inequalit)m(y)g(w)m(e)h(get,)f(from)f(\(5.74\),)g (\(5.77\))g(and)g(\(5.79\){\(5.81\),)897 3966 y Fr(Q)974 3925 y Fo(J)1021 3966 y Fn(f)p Fp(A)1144 3981 y Fy(j)1181 3966 y Fn(g)82 b(\025)i Fr(Q)1551 3925 y Fo(J)1597 3966 y Fn(f)p Fp(F)1710 3981 y Fy(j)1747 3966 y Fn(g)27 b Fr(Q)1901 3925 y Fo(J)1948 3966 y Fn(f)p Fp(D)2079 3981 y Fl(\000)p Fk(\()p Fy(j)t Fl(\000)p Fk(1)p Fy(=)p Fk(2\))r Fg(b)-41 b Fy(\034)9 b Fk(\()p Fy(l)q Fk(\))2502 3966 y Fn(g)1313 4146 y(\025)1474 4035 y Fm(\020)1533 4146 y Fs(1)22 b Fn(\000)h Fp(e)1749 4105 y Fl(\000)p Fy(c)1835 4114 y Fi(3)1869 4105 y Fy(l)1891 4081 y Ff(\024)p Fe(\000)p Ff(\022)2004 4096 y Fi(0)2047 4035 y Fm(\021)2124 4146 y Fp(e)2169 4105 y Fl(\000)p Fy(l)2246 4081 y Ff(b)1313 4343 y Fn(\025)84 b Fp(e)1519 4302 y Fl(\000)p Fk(2)p Fy(l)1631 4278 y Ff(b)3365 4343 y Fs(\(5.83\))1676 4790 y(39)p eop end %%Page: 40 40 TeXDict begin 40 39 bop -165 153 a Fs(for)32 b Fp(l)j Fs(large)e(enough.)45 b(Th)m(us,)34 b(the)g(indep)s(endence)h(of)e(the) g(ev)m(en)m(ts)i Fp(A)2358 168 y Fy(j)2395 153 y Fs(,)e Fp(j)h Fs(=)28 b(1)p Fp(;)17 b Fs(2)p Fp(;)g(:)g(:)g(:)e(;)i(N)43 b Fs(leads)34 b(to)f(the)-165 273 y(b)s(ound)199 547 y Fr(Q)276 506 y Fo(J)323 547 y Fs(\()p Fp(A)434 506 y Fy(c)469 547 y Fs(\))55 b(=)725 422 y Fy(N)693 452 y Fm(Y)696 662 y Fy(j)t Fk(=1)820 436 y Fm(\020)880 547 y Fs(1)22 b Fn(\000)h Fr(Q)1128 506 y Fo(J)1174 547 y Fs(\()p Fp(A)1285 562 y Fy(j)1322 547 y Fs(\))1360 436 y Fm(\021)1474 547 y Fn(\024)56 b Fs(\(1)22 b Fn(\000)h Fp(e)1861 506 y Fl(\000)p Fk(2)p Fy(l)1973 482 y Ff(b)2009 547 y Fs(\))2047 498 y Fy(N)2169 547 y Fn(\024)56 b Fs(exp)2451 436 y Fm(\020)2510 547 y Fn(\000)p Fp(N)28 b(e)2738 506 y Fl(\000)p Fk(2)p Fy(l)2850 482 y Ff(b)2886 436 y Fm(\021)2973 547 y Fp(:)365 b Fs(\(5.84\))-165 853 y(The)35 b(condition)h Fp(\034)43 b(<)31 b(\013)q(\027)41 b Fs(implies)36 b(that,)f(for)g Fp(l)i Fs(large)d(enough,)i Fp(N)42 b Fn(\025)32 b Fs(exp)q(\()p Fp(l)2657 817 y Fy(\013\027)2746 853 y Fp(=)p Fs(2\),)j(and)g(as,)g(b)s (esides,)-165 974 y Fp(b)28 b(<)f(\013)q(\027)6 b Fs(,)33 b(w)m(e)h(obtain)e(from)h(\(5.84\))e(that)1166 1207 y Fr(Q)1243 1166 y Fo(J)1290 1207 y Fs(\()p Fp(A)1401 1166 y Fy(c)1436 1207 y Fs(\))55 b Fn(\024)h Fs(exp)1811 1096 y Fm(\020)1870 1207 y Fn(\000)p Fp(e)1992 1166 y Fy(l)2014 1142 y Ff(\013\027)2093 1166 y Fy(=)p Fk(4)2168 1096 y Fm(\021)2255 1207 y Fp(;)1083 b Fs(\(5.85\))-165 1435 y(for)32 b Fp(l)i Fs(su\016cien)m(tly)i(large.)-19 1555 y(Finally)-8 b(,)50 b(inequalities)e(\(5.73\))e(and)g(\(5.85\))g (together)g(with)h(the)g(decomp)s(ostion)g(\(5.69\))f(yield)-165 1676 y(\(recall)33 b(that)f Fp(\034)39 b(<)28 b(\013)q(\027)6 b Fs(\))504 1909 y Fp(G)581 1868 y Fo(J)581 1933 y Fy(B)634 1944 y Ff(L)680 1933 y Fk(\()p Fy(X)f Fk(\))803 1909 y Fs(\()p Fp(X)r(;)17 b(Y)k Fs(\))83 b Fn(\024)g Fr(Q)1404 1868 y Fo(J)1451 1798 y Fm(\020)1510 1909 y Fn(f)p Fp(X)36 b Fn(!)1777 1924 y Fy(B)1830 1935 y Ff(L)1876 1924 y Fk(\()p Fy(X)5 b Fk(\))2026 1909 y Fp(Y)21 b Fn(g)h(\\)h Fp(A)2338 1798 y Fm(\021)2419 1909 y Fs(+)f Fr(Q)2594 1868 y Fo(J)2641 1909 y Fs(\()p Fp(A)2752 1868 y Fy(c)2787 1909 y Fs(\))1167 2121 y Fn(\024)83 b Fs(exp)1476 2011 y Fm(n)1543 2121 y Fn(\000)p Fs(\()p Fp(m)23 b Fn(\000)f Fp(e)1910 2080 y Fl(\000)1975 2053 y Ff(l)1995 2032 y(\034)p 1975 2065 59 3 v 1990 2106 a Fi(3)2048 2121 y Fs(\))17 b Fp(e)2158 2053 y Ff(l)2178 2032 y(\034)p 2158 2065 V 2172 2106 a Fi(2)2231 2011 y Fm(o)2320 2121 y Fs(+)22 b(exp)2567 2011 y Fm(n)2633 2121 y Fn(\000)p Fp(e)2765 2053 y Ff(l)2785 2032 y(\013\027)p 2766 2065 99 3 v 2800 2106 a Fi(4)2879 2011 y Fm(o)1167 2334 y Fn(\024)83 b Fs(exp)1476 2223 y Fm(n)1543 2334 y Fn(\000)p Fs(\()p Fp(m)23 b Fn(\000)f Fp(e)1910 2293 y Fl(\000)1975 2266 y Ff(l)1995 2245 y(\034)p 1975 2278 59 3 v 1990 2319 a Fi(3)2048 2334 y Fs(\))p Fp(e)2141 2266 y Ff(l)2161 2245 y(\034)p 2142 2278 V 2156 2319 a Fi(4)2215 2223 y Fm(o)2309 2334 y Fp(;)1029 b Fs(\(5.86\))-165 2578 y(with)34 b Fp(m)23 b Fn(\000)g Fp(e)311 2542 y Fl(\000)p Fy(l)388 2519 y Ff(\034)427 2542 y Fy(=)p Fk(3)532 2578 y Fn(\025)30 b Fs(1)p Fp(=L)803 2542 y Fy(\022)842 2578 y Fs(,)k(for)f Fp(l)j Fs(larger)d(than)h(a)f(certain)h(threshold)h (that)f(dep)s(ends)h(on)e Fp(d)p Fs(,)h Fp(\016)t Fs(,)g Fp(\013)q Fs(,)g Fp(\027)6 b Fs(,)-165 2699 y Fp(m)-80 2714 y Fk(0)-41 2699 y Fs(,)33 b Fp(\022)s Fs(,)g Fp(\024)f Fs(,)p Fp(b)p Fs(,)i Fp(\034)44 b Fs(and)32 b Fp(R)q Fs(.)44 b Fc(\003)-165 2956 y Fw(5.4.3)112 b(Probabilistic)38 b(estimates:)51 b(Go)s(o)s(d)38 b(ev)m(en)m(ts)f(ha)m(v)m(e)i(high)f (probabilit)m(y)-165 3141 y Fs(In)33 b(this)h(section)g(w)m(e)g(sho)m (w)g(that,)f(assuming)i(\(5.47\),)d(the)i(ev)m(en)m(ts)h Fn(A)e Fs(and)g Fn(B)2666 3156 y Fk(\001)2762 3141 y Fs(in)m(tro)s(duced)i(in)e(\(5.51\))-165 3262 y(and)c(\(5.52\))g(ha)m (v)m(e)h(go)s(o)s(d)f(enough)g(probabilit)m(y)i(for)d Fp(l)k Fs(su\016cien)m(tly)g(large.)42 b(Hyp)s(othesis)32 b(\(3.19\))c(is)i(used)-165 3382 y(to)i(b)s(ound)h(the)g(probabilit)m (y)g(of)f(the)h(second)h(ev)m(en)m(t.)-165 3573 y Fw(Lemma)39 b(5.87)49 b Fq(L)-5 b(et)36 b Fp(m)31 b(>)f Fs(0)p Fq(,)36 b Fp(l)c(>)e(\016)t Fq(,)36 b Fp(\013)31 b(>)f Fs(1)36 b Fq(and)g Fp(p)30 b(>)g(\013)q(d)p Fq(.)48 b(Put)37 b Fp(L)30 b Fs(=)g Fp(l)2538 3537 y Fy(\013)2588 3573 y Fq(.)48 b(Pick)36 b(a)g(p)-5 b(ositive)35 b(inte)-5 b(ger)-165 3694 y Fp(R)28 b(>)g(\013)q(p=)p Fs(\()p Fp(p)21 b Fn(\000)i Fp(\013)q(d)p Fs(\))34 b Fq(and)g(de\014ne)g(the)h(event)g Fn(A)f Fq(as)h(in)f(\(5.51\).)44 b(If)964 3944 y Fr(P)1025 3833 y Fm(n)1092 3944 y Fp(x)35 b Fq(is)g Fs(\()p Fp(m;)17 b(l)r Fs(\))22 b Fn(\000)h Fq(r)-5 b(e)g(gular)1968 3833 y Fm(o)2090 3944 y Fn(\025)56 b Fs(1)22 b Fn(\000)2415 3876 y Fs(1)p 2403 3921 71 4 v 2403 4012 a Fp(l)2434 3983 y Fy(p)3365 3944 y Fs(\(5.88\))-165 4188 y Fq(for)34 b(al)5 b(l)35 b Fp(x)28 b Fn(2)g Fr(Z)373 4152 y Fy(d)414 4188 y Fq(,)35 b(then)1324 4348 y Fr(P)p Fn(fAg)55 b(\025)g Fs(1)22 b Fn(\000)1986 4280 y Fs(1)p 1933 4325 155 4 v 1933 4416 a(2)p Fp(L)2048 4387 y Fy(p)2098 4348 y Fp(;)1240 b Fs(\(5.89\))-165 4541 y Fq(assuming)34 b Fp(l)j Fq(lar)-5 b(ge)34 b(exc)-5 b(e)g(e)g(d)34 b(a)h(c)-5 b(ertain)34 b(minimum)g(value)h(that)g(dep)-5 b(ends)33 b(on)i Fp(d)p Fq(,)f Fp(\013)q Fq(,)h Fp(p)g Fq(and)f Fp(R)q Fq(.)1676 4790 y Fs(40)p eop end %%Page: 41 41 TeXDict begin 41 40 bop -165 153 a Fw(Pro)s(of.)48 b Fs(Let)30 b(us)g(sa)m(y)h(that)e(t)m(w)m(o)h(sites)h Fp(x)1305 168 y Fk(1)1345 153 y Fp(;)17 b(x)1444 168 y Fk(2)1511 153 y Fn(2)28 b Fr(Z)1671 117 y Fy(d)1741 153 y Fs(are)i Fp(l)r Fs(-nono)m(v)m(erlapping)h(if)e(the)h(distance)h (b)s(et)m(w)m(een)-165 273 y(the)k(b)s(o)m(xes)h(\003[)p Fp(x)426 288 y Fk(1)465 273 y Fs(;)17 b Fp(l)26 b Fs(+)d Fp(\016)t Fs(])35 b(and)g(\003[)p Fp(x)1114 288 y Fk(2)1154 273 y Fs(;)17 b Fp(l)25 b Fs(+)f Fp(\016)t Fs(])34 b(is)i(strictly)g (larger)e(than)h(one.)50 b(In)35 b(this)g(case,)i(the)e(ev)m(en)m(ts) -165 393 y Fn(f)p Fp(x)-60 408 y Fy(i)1 393 y Fs(is)e(\()p Fp(m;)17 b(l)r Fs(\))22 b Fn(\000)h Fs(regular)p Fn(g)p Fs(,)32 b Fp(i)c Fs(=)g(1)p Fp(;)17 b Fs(2,)31 b(are)i(indep)s(enden)m (t.)46 b(W)-8 b(e)33 b(ha)m(v)m(e)623 715 y Fn(A)703 674 y Fy(c)793 715 y Fn(\032)1223 583 y Fk(\()p Fy(l)q Fk(\))1206 621 y Fm([)926 836 y Fy(x)966 845 y Fi(1)1000 836 y Fy(;:::)o(;x)1139 847 y Ff(R)p Fi(+1)1265 836 y Fl(2)p Fk(\003[)p Fy(x)p Fk(;)p Fy(L)p Fk(+)p Fy(\016)r Fk(])1613 591 y Fy(R)p Fk(+1)1630 621 y Fm(\\)1628 830 y Fy(i)p Fk(=1)1757 605 y Fm(n)1823 715 y Fp(x)1878 730 y Fy(i)1939 715 y Fs(is)g(\()p Fp(m;)17 b(l)r Fs(\))23 b Fn(\000)f Fs(singular)2732 605 y Fm(o)2798 715 y Fp(;)540 b Fs(\(5.90\))-165 1044 y(where)33 b(the)f(index)i(\()p Fp(l)r Fs(\))e(reminds)h(us)g(that)f(the)g(union)h(is)f(o)m(v)m(er)h (collections)h(of)d Fp(R)23 b Fs(+)d(1)32 b(sites)i(whic)m(h)f(are)-165 1164 y Fp(l)r Fs(-nono)m(v)m(erlapping.)44 b(Hence)472 1480 y Fr(P)p Fn(fA)663 1439 y Fy(c)697 1480 y Fn(g)83 b(\024)1287 1348 y Fk(\()p Fy(l)q Fk(\))1254 1386 y Fm(X)990 1602 y Fy(x)1030 1611 y Fi(1)1065 1602 y Fy(;:::)n(;x)1203 1613 y Ff(R)p Fi(+1)1330 1602 y Fl(2)p Fk(\003[)p Fy(x)p Fk(;)p Fy(L)p Fk(+)p Fy(\016)r Fk(])1661 1370 y Fm(h)1708 1480 y Fr(P)1769 1370 y Fm(n)1835 1480 y Fp(x)33 b Fs(is)g(\()p Fp(m;)17 b(l)r Fs(\))22 b Fn(\000)h Fs(singular)2716 1370 y Fm(oi)2829 1392 y Fy(R)p Fk(+1)830 1816 y Fn(\024)1000 1749 y Fs([2\()p Fp(L)g Fs(+)f Fp(\016)t Fs(\)])1413 1713 y Fy(d)p Fk(\()p Fy(R)p Fk(+1\))p 1000 1793 652 4 v 1191 1885 a Fp(l)1222 1856 y Fy(p)p Fk(\()p Fy(R)p Fk(+1\))1717 1816 y Fn(\024)1888 1749 y Fs(1)p 1860 1793 107 4 v 1860 1885 a Fp(L)1926 1856 y Fy(p)1986 1749 y Fs(4)2035 1713 y Fy(d)p Fk(\()p Fy(R)p Fk(+1\))p 1986 1793 288 4 v 1998 1885 a Fp(l)2029 1856 y Fk(\()p Fy(p)p Fl(\000)p Fy(\013d)p Fk(\))2339 1816 y Fn(\024)2535 1749 y Fs(1)p 2482 1793 155 4 v 2482 1885 a(2)p Fp(L)2597 1856 y Fy(p)2674 1816 y Fp(;)664 b Fs(\(5.91\))-165 2059 y(for)32 b Fp(l)i Fs(large)f(enough.)44 b Fc(\003)-165 2283 y Fw(Lemma)39 b(5.92)49 b Fq(L)-5 b(et)28 b Fp(l)i(>)d(\016)t Fq(,)i Fp(\013)f(>)g Fs(1)f Fq(and)g Fp(p)g(>)h Fs(0)p Fq(.)42 b(Put)28 b Fp(L)g Fs(=)g Fp(l)2069 2247 y Fy(\013)2119 2283 y Fq(.)42 b(F)-7 b(or)26 b Fp(\024)i Fq(and)f Fp(b)h Fq(such)f(that)h Fs(0)f Fp(<)h(\024)g(<)f(b=d)-165 2404 y Fq(take)32 b(a)h(c)-5 b(ol)5 b(le)-5 b(ction)32 b(of)g(sites)h Fn(f)p Fp(x)988 2419 y Fy(i)1044 2404 y Fs(:)28 b Fp(i)g Fs(=)f(1)p Fp(;)17 b Fs(2)p Fp(;)g(:)g(:)g(:)e(;)i(R)q Fn(g)33 b Fq(and)f(intr)-5 b(o)g(duc)g(e)33 b(the)g(event)f Fn(B)2865 2419 y Fk(\001)2961 2404 y Fq(as)h(in)f(\(5.52\))g(for)-165 2524 y Fs(\001)c(=)f Fp(e)92 2488 y Fl(\000)p Fy(l)169 2464 y Ff(\021)211 2524 y Fq(,)35 b(with)g Fs(0)27 b Fp(<)h(\021)j(<)d(b)22 b Fn(\000)h Fp(\024d)p Fq(.)44 b(If)35 b Fn(@)27 b Fp(<)h Fn(1)34 b Fq(for)h(some)f Fp(a)28 b(>)g(\013)q Fs(\()p Fp(p)21 b Fs(+)h Fp(d)p Fs(\))p Fp(=\021)t Fq(,)34 b(then)1313 2785 y Fr(P)p Fn(fB)1489 2800 y Fk(\001)1552 2785 y Fn(g)55 b(\025)h Fs(1)22 b Fn(\000)2024 2718 y Fs(1)p 1971 2762 V 1971 2854 a(2)p Fp(L)2086 2825 y Fy(p)3365 2785 y Fs(\(5.93\))-165 3027 y Fq(for)34 b Fp(l)j Fq(lar)-5 b(ger)35 b(than)g(a)f(c)-5 b(ertain)35 b(thr)-5 b(eshold)34 b(that)h(dep)-5 b(ends)34 b(on)g Fp(\013)q Fq(,)h Fp(p)p Fq(,)g Fp(\024)p Fq(,)f Fp(b)p Fq(,)i Fp(d)p Fq(,)e Fp(\021)t Fq(,)h Fn(@)p Fq(.)-165 3252 y Fw(Pro)s(of.)49 b Fs(W)-8 b(e)33 b(start)g(with)g(the)g(follo)m (wing)g(observ)-5 b(ations:)-73 3453 y(\(i\))49 b(F)-8 b(or)31 b(an)m(y)j(collection)f(of)f(sites)i Fn(f)p Fp(x)1305 3468 y Fy(i)1361 3453 y Fs(:)28 b Fp(i)g Fs(=)f(1)p Fp(;)17 b Fs(2)p Fp(;)g(:)g(:)g(:)e(;)i(R)q Fn(g)1212 3762 y Fw(G)1300 3785 y Fl([)1347 3762 y Ff(R)1347 3807 y(i)p Fi(=1)1450 3785 y Fk(\003[)p Fy(x)1559 3795 y Ff(i)1585 3785 y Fy(;l)1627 3766 y Ff(\024)1667 3785 y Fk(])1746 3762 y Fn(\032)1909 3638 y Fy(R)1880 3668 y Fm([)1879 3878 y Fy(i)p Fk(=1)2009 3762 y Fw(G)2097 3778 y Fk(\003[)p Fy(x)2206 3788 y Ff(i)2232 3778 y Fk(;)p Fy(l)2274 3759 y Ff(\024)2314 3778 y Fk(+)p Fy(\016)r Fk(])2454 3762 y Fp(:)884 b Fs(\(5.94\))-100 4108 y(\(ii\))49 b(As)33 b(a)f(consequence)k(of)c(\(i\),)820 4387 y Fn(B)888 4346 y Fy(c)885 4412 y Fk(\001)1004 4387 y Fn(\022)1266 4293 y Fm([)1137 4509 y Fy(y)r Fl(2)p Fk(\003[)p Fy(x)p Fk(;)p Fy(L)p Fk(+)p Fy(\016)r Fk(])1522 4277 y Fm(n)1726 4293 y(X)1588 4504 y Fy(\015)t Fl(2)p Fo(G)1739 4518 y Fi(\003[)p Ff(y)r Fi(;)p Ff(l)1870 4504 y(\024)1909 4518 y Fi(+)p Ff(\016)r Fi(])2024 4387 y Fs(ln)2106 4277 y Fm(h)2277 4320 y Fs(1)p 2163 4364 278 4 v 2163 4456 a Fp(K)2253 4422 y Fo(J)2246 4483 y Fk(\001)2309 4456 y Fs(\()p Fp(\015)5 b Fs(\))2450 4277 y Fm(i)2553 4387 y Fp(>)2699 4320 y(l)2730 4284 y Fy(b)p 2694 4364 76 4 v 2694 4456 a Fp(R)2779 4277 y Fm(o)2845 4387 y Fp(:)493 b Fs(\(5.95\))1676 4790 y(41)p eop end %%Page: 42 42 TeXDict begin 42 41 bop -127 153 a Fs(\(iii\))49 b(F)-8 b(or)31 b(an)m(y)j(\001)28 b Fp(>)f Fs(0,)1094 285 y Fp(K)1184 244 y Fo(J)1177 309 y Fk(\001)1240 285 y Fs(\()p Fp(\015)5 b Fs(\))55 b Fn(\025)h Fs(\(1)22 b Fn(\000)g Fp(e)1813 244 y Fl(\000)p Fk(\001)1932 285 y Fs(\))17 b Fp(e)2032 244 y Fl(\000)p Fk(\001)p Fy(w)2199 220 y Fa(J)2237 244 y Fk(\()p Fy(\015)t Fk(\))2392 285 y Fp(>)55 b Fs(0)p Fp(:)766 b Fs(\(5.96\))-19 488 y(>F)-8 b(rom)32 b(\(5.96\))f(and)h(the)g(inequalit)m(y)i(ln)q(\(1)20 b Fn(\000)i Fp(e)1705 452 y Fl(\000)p Fk(\001)1823 488 y Fs(\))27 b Fn(\025)i Fs(ln\(\001)p Fp(=)p Fs(2\),)j(v)-5 b(alid)32 b(for)f(\001)i(small)f(enough,)h(w)m(e)-165 608 y(obtain)472 732 y Fm(X)334 943 y Fy(\015)t Fl(2)p Fo(G)485 957 y Fi(\003[)p Ff(y)r Fi(;)p Ff(l)616 943 y(\024)656 957 y Fi(+)p Ff(\016)r Fi(])787 826 y Fs(ln)868 716 y Fm(h)1040 759 y Fs(1)p 926 803 278 4 v 926 895 a Fp(K)1016 861 y Fo(J)1009 922 y Fk(\001)1071 895 y Fs(\()p Fp(\015)5 b Fs(\))1213 716 y Fm(i)1343 826 y Fn(\024)1642 732 y Fm(X)1504 943 y Fy(\015)t Fl(2)p Fo(G)1655 957 y Fi(\003[)p Ff(y)r Fi(;)p Ff(l)1786 943 y(\024)1825 957 y Fi(+)p Ff(\016)r Fi(])1940 716 y Fm(h)1987 826 y Fs(\001)p Fp(w)2141 785 y Fo(J)2187 826 y Fs(\()p Fp(\015)g Fs(\))22 b Fn(\000)g Fs(ln)q(\(1)g Fn(\000)g Fp(e)2775 785 y Fl(\000)p Fk(\001)2893 826 y Fs(\))2931 716 y Fm(i)1343 1128 y Fn(\024)1621 1033 y Fm(X)1504 1249 y Fy(z)s Fl(2)p Fk(\003[)p Fy(y)r Fk(;)p Fy(l)1735 1230 y Ff(\024)1774 1249 y Fk(+)p Fy(\016)r Fk(])1916 1033 y Fm(X)1926 1242 y Fy(\015)t Fl(3)p Fy(z)2076 1017 y Fm(h)2123 1128 y Fs(\001)p Fp(w)2277 1087 y Fo(J)2323 1128 y Fs(\()p Fp(\015)5 b Fs(\))22 b Fn(\000)h Fs(ln\(1)f Fn(\000)g 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