Content-Type: multipart/mixed; boundary="-------------0809080957901" This is a multi-part message in MIME format. ---------------0809080957901 Content-Type: text/plain; name="08-157.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="08-157.comments" Dipartimento di Matematica, Universit di Roma ``Tor Vergata'', Via della Ricerca Scientifica, I-00133, Roma. e-mail: benfatto@mat.uniroma2.it ---------------0809080957901 Content-Type: text/plain; name="08-157.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="08-157.keywords" Fermion systems, Renormalization group ---------------0809080957901 Content-Type: application/postscript; name="UV_2d_rev.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="UV_2d_rev.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.96dev Copyright 2007 Radical Eye Software %%Title: UV_2d_rev.dvi %%CreationDate: Mon Sep 08 16:47:24 2008 %%Pages: 15 %%PageOrder: Ascend %%BoundingBox: 0 0 595 842 %%DocumentFonts: CMR17 CMR12 CMBX9 CMR9 CMBX12 CMMI7 CMR7 CMR10 CMMI10 %%+ CMSY10 MSBM10 CMBX10 CMTI10 CMEX10 CMSY7 CMBX7 CMMI5 CMR5 CMSY5 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips.exe UV_2d_rev.dvi %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2008.09.08:1646 %%BeginProcSet: tex.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: texps.pro 0 0 %! 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%%EndFont %%BeginFont: CMBX7 %!PS-AdobeFont-1.1: CMBX7 1.0 %%CreationDate: 1991 Aug 20 16:35:49 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX7 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 107 /k put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-55 -250 1289 751}readonly def currentdict end currentfile eexec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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 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 20 /lessequal put dup 21 /greaterequal put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 68 /D put dup 84 /T put dup 106 /bar put dup 110 /backslash put readonly def /FontBBox{-15 -951 1252 782}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 12 /vextendsingle put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 20 /bracketleftbigg put dup 21 /bracketrightbigg put dup 34 /bracketleftBigg put dup 35 /bracketrightBigg put dup 80 /summationtext put dup 81 /producttext put dup 82 /integraltext put dup 83 /uniontext put dup 84 /intersectiontext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put dup 104 /bracketleftBig put dup 105 /bracketrightBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 50 /two put dup 65 /A put dup 66 /B 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 76 /L put dup 77 /M 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 89 /Y put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-163 -250 1146 969}readonly def currentdict end currentfile eexec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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 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 65 /A put dup 82 /R put dup 84 /T put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 103 /g put dup 104 /h 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 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-301 -250 1164 946}readonly def currentdict end currentfile eexec 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 82 /R put dup 84 /T put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly 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 6 /plusminus put dup 17 /equivalence put dup 18 /reflexsubset put dup 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 33 /arrowright put dup 41 /arrowdblright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 59 /emptyset put dup 68 /D put dup 69 /E put dup 84 /T put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 107 /bardbl put dup 110 /backslash put dup 120 /section put readonly def /FontBBox{-29 -960 1116 775}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 18 /theta put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 25 /pi put dup 27 /sigma put dup 28 /tau put dup 31 /chi put dup 32 /psi put dup 34 /epsilon put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 65 /A put dup 67 /C 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 81 /Q put dup 82 /R put dup 84 /T put dup 86 /V put dup 87 /W 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 109 /m put dup 110 /n put dup 112 /p 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 dup 126 /vector put readonly def /FontBBox{-32 -250 1048 750}readonly def currentdict end currentfile eexec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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 1 /Delta put dup 3 /Lambda put dup 11 /ff put dup 12 /fi put dup 14 /ffi put dup 19 /acute put dup 22 /macron put dup 33 /exclam put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 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 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 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 readonly def /FontBBox{-251 -250 1009 969}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /beta put dup 13 /gamma put dup 20 /kappa put dup 22 /mu put dup 25 /pi put dup 27 /sigma put dup 28 /tau put dup 32 /psi put dup 34 /epsilon put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 61 /slash put dup 70 /F put dup 71 /G put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 82 /R put dup 84 /T put dup 86 /V 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 106 /j put dup 107 /k put dup 109 /m put dup 110 /n 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 126 /vector put readonly def /FontBBox{0 -250 1171 750}readonly def currentdict end currentfile eexec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%%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 11 /ff put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 70 /F put dup 84 /T put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 121 /y put readonly def /FontBBox{-39 -250 1036 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX9 %!PS-AdobeFont-1.1: CMBX9 1.0 %%CreationDate: 1991 Aug 20 16:36:25 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 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 currentdict end currentfile eexec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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 12 /fi put dup 18 /grave put dup 34 /quotedblright 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 56 /eight put dup 58 /colon put dup 64 /at put dup 65 /A put dup 66 /B put dup 68 /D put dup 71 /G put dup 73 /I put dup 77 /M put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V 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 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put readonly def /FontBBox{-34 -251 988 750}readonly def currentdict end currentfile eexec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CB4FFDDBDB42CE75AC312535EC1063D5DD85AE9C990AA52057A43043C67A4C3D 9AC4DB9EA0607E20833FEA 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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y(~)1189 3262 y(k)r Fe(\001)o Fn(~)-33 b(x)o Fe(\000)p Fn(x)1373 3270 y Fb(0)1405 3262 y Fn(")1436 3270 y Fb(0)1469 3262 y Fm(\()1492 3247 y Fn(~)1495 3262 y(k)r Fm(\))1575 3180 y Ff(\024)1749 3241 y Fk(\022)r Fl(\()p Fk(x)1869 3253 y Fm(0)1908 3241 y Fl(\))p 1629 3278 431 4 v 1629 3369 a(1)18 b(+)g Fk(e)1811 3345 y Fe(\000)p Fn(\014)s(")1935 3353 y Fb(0)1967 3345 y Fm(\()1990 3329 y Fn(~)1993 3345 y(k)r Fm(\))2088 3297 y Fj(\000)2229 3241 y Fl(1)g Fj(\000)g Fk(\022)r Fl(\()p Fk(x)2492 3253 y Fm(0)2531 3241 y Fl(\))p 2181 3278 430 4 v 2181 3369 a(1)g(+)g Fk(e)2363 3345 y Fm(+)p Fn(\014)s(")2486 3353 y Fb(0)2518 3345 y Fm(\()2541 3329 y Fn(~)2544 3345 y(k)r Fm(\))2621 3180 y Ff(\025)2702 3297 y Fk(;)228 b Fl(\(2.4\))260 3574 y(where)23 b Fk(\022)r Fl(\()p Fk(t)p Fl(\))h(is)f(the)h(step)f(function,)i(equal)e(to)g(1)g(for)g Fk(t)g(>)g Fl(0)g(and)g(equal)g(to)g(0)g(for)g Fk(t)g Fj(\024)g Fl(0)g(\(this)260 3673 y(c)n(hoice)k(ensures)f(that)i Fk(g)s Fl(\(0)p Fk(;)13 b(~)-41 b(x)p Fl(\))23 b(=)g(lim)1437 3689 y Fn(x)1475 3697 y Fb(0)1507 3689 y Fe(!)p Fm(0)1606 3673 y Fa(\000)g Fk(g)s Fl(\()p Fk(x)1796 3685 y Fm(0)1833 3673 y Fk(;)13 b(~)-41 b(x)q Fl(\),)28 b(a)f(required)f(condition)i ([NO]\).)384 3773 y(The)k(function)h Fk(g)s Fl(\()p Fh(x)p Fl(\),)g(whic)n(h)f(represen)n(ts)e(the)i(co)n(v)-5 b(ariance)30 b(of)i(the)g(free)g(fermion)f(gas)260 3873 y(without)24 b(cuto\013s,)g(is)g(an)n(tip)r(erio)r(dic)f(in)g Fk(x)1524 3885 y Fm(0)1586 3873 y Fl(of)g(p)r(erio)r(d)g Fk(\014)28 b Fl(and)c(p)r(erio)r(dic)f(of)g(p)r(erio)r(d)h(2)p Fk(\014)t Fl(;)h(then)260 3983 y(w)n(e)i(shall)g(consider)g(it)h(as)f(de\014ned)g (on)h Fi(T)1552 3935 y Fm(1)1552 4003 y(2)p Fn(\014)1648 3983 y Fj(\002)18 b Fi(T)1798 3935 y Fm(2)1798 4003 y Fn(L)1847 3983 y Fl(.)37 b(This)28 b(function)g(has)f(a)g(singularit)n (y)f(at)260 4082 y(the)31 b(p)r(oin)n(ts)f Fk(x)705 4094 y Fm(0)770 4082 y Fl(=)e(0)p Fk(;)14 b Fj(\006)p Fk(\014)t Fl(,)30 b(whic)n(h)g(can)g(b)r(e)h(describ)r(ed)f(in)h(the)g(follo)n (wing)e(w)n(a)n(y)-7 b(.)44 b(Let)30 b Fk(h)2992 4094 y Fm(0)3030 4082 y Fl(\()p Fk(t)p Fl(\))260 4182 y(b)r(e)c(a)e(smo)r (oth)h(function)h(on)f Fi(R)p Fl(,)h(equal)f(to)g(1)g(for)g Fj(j)p Fk(t)p Fj(j)e(\024)g Fl(1)h(and)i(equal)e(to)i(0)e(for)h Fj(j)p Fk(t)p Fj(j)e(\025)g Fl(2,)i(and)260 4297 y(let)k(us)g(call)643 4275 y(~)642 4297 y Fk(h)690 4309 y Fm(0)728 4297 y Fl(\()p Fk(x)807 4309 y Fm(0)844 4297 y Fl(\))h(the)f(function)h(on)e Fi(T)1560 4248 y Fm(1)1560 4317 y(2)p Fn(\014)1637 4297 y Fl(,)i(whic)n(h)f(is)f(equal)h(to)g Fk(h)2385 4309 y Fm(0)2422 4297 y Fl(\()p Fk(x)2501 4309 y Fm(0)2539 4297 y Fl(\))g(around)f Fk(x)2930 4309 y Fm(0)2993 4297 y Fl(=)d(0)260 4396 y(and)g(equal)f(to)h Fk(h)782 4408 y Fm(0)820 4396 y Fl(\()p Fk(x)899 4408 y Fm(0)950 4396 y Fl(+)14 b Fk(\014)t Fl(\))25 b(around)f Fk(x)1463 4408 y Fm(0)1524 4396 y Fl(=)f Fj(\000)p Fk(\014)t Fl(.)36 b(Let)25 b(us)g(call)g(also)f Fk(h)2399 4408 y Fm(1)2436 4396 y Fl(\()o Fk(~)-41 b(x)q Fl(\))25 b(the)h(function)f(on)260 4511 y Fi(T)326 4463 y Fm(2)326 4531 y Fn(L)376 4511 y Fl(,)30 b(whic)n(h)f(is)g(equal)g(to)g Fk(h)1125 4523 y Fm(0)1162 4511 y Fl(\()o Fk(~)-41 b(x)1241 4481 y Fm(2)1279 4511 y Fl(\))30 b(around)d Fk(~)-41 b(x)26 b Fl(=)g(0,)j(and)g Fk(f)9 b Fl(\()p Fh(x)p Fl(\))27 b(=)2327 4489 y(~)2326 4511 y Fk(h)2374 4523 y Fm(0)2411 4511 y Fl(\()p Fk(x)2490 4523 y Fm(0)2528 4511 y Fl(\))p Fk(h)2608 4523 y Fm(1)2646 4511 y Fl(\()o Fk(~)-41 b(x)p Fl(\).)43 b(One)29 b(can)260 4610 y(easily)e(c)n(hec)n(k)f(that)968 4710 y Fm(2)p Fn(:)p Fm(4)p Fn(a)114 b Fk(f)9 b Fl(\()p Fh(x)p Fl(\))p Fk(g)s Fl(\()p Fh(x)p Fl(\))24 b(=)f Fk(G)p Fl(\()p Fh(x)p Fl(\))d(+)e Fk(R)1982 4722 y Fm(1)2019 4710 y Fl(\()p Fh(x)p Fl(\))24 b Fk(;)773 b Fl(\(2.5\))260 4854 y(where)385 5027 y Fm(2)p Fn(:)p Fm(4)151 b Fk(G)p Fl(\()p Fh(x)p Fl(\))84 b(=)e Fk(h)1080 5039 y Fm(0)1117 5027 y Fl(\()p Fk(x)1196 5039 y Fm(0)1234 5027 y Fl(\))p Fk(h)1314 5039 y Fm(1)1352 5027 y Fl(\()o Fk(~)-41 b(x)q Fl(\))p 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Fm(0)1382 5530 y Fl(\))p Fk(h)1462 5542 y Fm(1)1500 5530 y Fl(\()o Fk(~)-41 b(x)p Fl(\))1648 5474 y(1)p 1621 5511 94 4 v 1621 5587 a Fk(L)1678 5563 y Fm(2)1768 5452 y Ff(X)1736 5632 y Fn(~)1739 5647 y(k)r Fe(2D)1873 5655 y Fc(L)1940 5474 y Fk(e)1979 5444 y Fe(\000)p Fn(i)2051 5429 y(~)2054 5444 y(k)r Fe(\001)o Fn(~)-33 b(x)o Fe(\000)p Fn(x)2238 5452 y Fb(0)2270 5444 y Fn(")2301 5452 y Fb(0)2334 5444 y Fm(\()2357 5429 y Fn(~)2360 5444 y(k)q Fm(\))p 1940 5511 487 4 v 1968 5602 a Fl(1)18 b(+)g Fk(e)2150 5578 y Fm(+)p Fn(\014)s(")2273 5586 y Fb(0)2305 5578 y Fm(\()2328 5563 y Fn(~)2331 5578 y(k)r Fm(\))2455 5530 y Fl(+)433 b(\(2.7\))1671 5871 y(4)p eop end %%Page: 5 5 TeXDict begin 5 4 bop 968 569 a Fl(+)83 b Fk(h)1164 581 y Fm(0)1201 569 y Fl(\()p Fk(x)1280 581 y Fm(0)1336 569 y Fl(+)18 b Fk(\014)t Fl(\))p Fk(h)1550 581 y Fm(1)1588 569 y Fl(\()o Fk(~)-41 b(x)q Fl(\))1736 513 y(1)p 1710 550 94 4 v 1710 626 a Fk(L)1767 602 y Fm(2)1856 490 y Ff(X)1824 670 y Fn(~)1827 685 y(k)r Fe(2D)1961 693 y Fc(L)2028 513 y Fk(e)2067 482 y Fe(\000)p Fn(i)2139 467 y(~)2142 482 y(k)r Fe(\001)o Fn(~)-33 b(x)o Fe(\000)p Fm(\()p Fn(x)2352 490 y Fb(0)2384 482 y Fm(+)p Fn(\014)s Fm(\))p Fn(")2533 490 y Fb(0)2565 482 y Fm(\()2588 467 y Fn(~)2591 482 y(k)r Fm(\))p 2028 550 631 4 v 2128 641 a Fl(1)18 b(+)g Fk(e)2310 617 y Fm(+)p Fn(\014)s(")2433 625 y Fb(0)2465 617 y Fm(\()2488 601 y Fn(~)2491 617 y(k)r Fm(\))2691 569 y Fk(;)260 847 y Fl(and,)27 b(if)h Fk(t)23 b(>)g Fl(0,)k(w)n(e)h(are)e(de\014ning)465 1089 y Fm(2)p Fn(:)p Fm(5)150 b Fk(\016)741 1046 y Fm(\()p Fn(L)p Fm(\))738 1109 y Fn(t)843 1089 y Fl(\()o Fk(~)-41 b(x)q Fl(\))23 b(=)1102 1033 y(1)p 1076 1070 94 4 v 1076 1146 a Fk(L)1133 1122 y Fm(2)1222 1010 y Ff(X)1190 1190 y Fn(~)1193 1205 y(k)r Fe(2D)1327 1213 y Fc(L)1384 1089 y Fk(e)1423 1055 y Fe(\000)p Fn(i)1495 1039 y(~)1498 1055 y(k)r Fe(\001)o Fn(~)-33 b(x)o Fe(\000)p Fn(t)1666 1039 y(~)1669 1055 y(k)1705 1029 y Fb(2)1738 1055 y Fn(=)p Fm(2)1832 1089 y Fl(=)1952 1010 y Ff(X)1919 1223 y Fn(~)g(n)p Fe(2)p Fi(Z)2072 1175 y Fb(2)2128 1033 y Fk(e)2167 1002 y Fe(\000)p Fm(\()o Fn(~)g(x)p Fm(+)o Fn(~)g(nL)p Fm(\))2447 977 y Fb(2)2478 1002 y Fn(=)p Fm(\(2)p Fn(t)p Fm(\))p 2128 1070 499 4 v 2317 1146 a Fl(2)p Fk(\031)s(t)2660 1089 y(:)270 b Fl(\(2.8\))384 1386 y(By)28 b(using)f(\(2.1\))g(and)h (\(2.5\),)f(w)n(e)g(can)h(write:)275 1564 y Fm(2)p Fn(:)p Fm(6)150 b Fk(g)551 1576 y Fn(uv)652 1564 y Fl(=)23 b Fk(f)9 b(g)830 1576 y Fn(uv)926 1564 y Fl(+)18 b(\(1)g Fj(\000)g Fk(f)9 b Fl(\))p Fk(g)1306 1576 y Fn(uv)1408 1564 y Fl(=)22 b Fk(f)9 b(g)21 b Fj(\000)d Fk(f)9 b(g)1779 1576 y Fn(ir)1857 1564 y Fl(+)18 b(\(1)g Fj(\000)g Fk(f)9 b Fl(\))p Fk(g)2237 1576 y Fn(uv)2338 1564 y Fj(\021)23 b Fk(G)c Fl(+)f Fk(R)23 b(;)251 b Fl(\(2.9\))260 1743 y(where)31 b Fk(R)f Fl(=)e Fk(R)753 1755 y Fm(1)812 1743 y Fj(\000)20 b Fk(f)9 b(g)987 1755 y Fn(ir)1068 1743 y Fl(+)20 b(\(1)h Fj(\000)f Fk(f)9 b Fl(\))p Fk(g)1455 1755 y Fn(uv)1534 1743 y Fl(.)48 b(It)31 b(is)h(easy)e(to)h(c)n(hec)n (k)g(that)g Fk(R)q Fl(\()p Fh(x)p Fl(\))h(is)g(a)e(smo)r(oth)260 1858 y(function)22 b(on)g Fi(T)755 1810 y Fm(1)755 1878 y(2)p Fn(\014)840 1858 y Fj(\002)7 b Fi(T)978 1810 y Fm(2)978 1878 y Fn(L)1050 1858 y Fl(and)22 b(that,)h(giv)n(en)e(an)n(y) g(in)n(teger)g Fk(M)32 b(>)23 b Fl(0,)f(there)g(exists)f(a)h(constan)n (t)260 1957 y Fk(C)319 1969 y Fn(M)393 1957 y Fl(,)33 b(indep)r(enden)n(t)h(of)e Fk(L)g Fl(and)g Fk(\014)t Fl(,)i(suc)n(h)e(that,)h(if)g Fk(d)1911 1969 y Fn(\014)1956 1957 y Fl(\()p Fk(x)2035 1969 y Fm(0)2073 1957 y Fk(;)14 b(y)2151 1969 y Fm(0)2188 1957 y Fl(\))33 b(and)f Fk(d)2462 1969 y Fn(L)2512 1957 y Fl(\()o Fk(~)-41 b(x;)13 b(~)-41 b(y)s Fl(\))33 b(denote)f(the)260 2075 y(distances)27 b(on)g Fi(T)798 2027 y Fm(1)798 2096 y Fn(\014)870 2075 y Fl(\(not)h Fi(T)1116 2027 y Fm(1)1116 2096 y(2)p Fn(\014)1194 2075 y Fl(\))g(and)g Fi(T)1482 2027 y Fm(2)1482 2096 y Fn(L)1532 2075 y Fl(,)g(resp)r(ectiv)n(ely)-7 b(,)26 b(then)599 2307 y Fm(2)p Fn(:)p Fm(7)150 b Fj(j)p Fk(R)q Fl(\()p Fh(x)19 b Fj(\000)f Fh(y)q Fl(\))p Fj(j)24 b(\024)1847 2251 y Fk(C)1906 2263 y Fn(M)p 1334 2288 1159 4 v 1334 2364 a Fl(1)18 b(+)g Fk(d)1520 2376 y Fn(\014)1565 2364 y Fl(\()p Fk(x)1644 2376 y Fm(0)1700 2364 y Fj(\000)g Fk(y)1824 2376 y Fm(0)1861 2364 y Fl(\))1893 2340 y Fn(M)1986 2364 y Fl(+)g Fk(d)2112 2376 y Fn(L)2162 2364 y Fl(\()o Fk(~)-41 b(x)19 b Fj(\000)e Fk(~)-41 b(y)s Fl(\))2419 2340 y Fn(M)2526 2307 y Fk(:)363 b Fl(\(2.10\))384 2538 y(As)29 b(it)g(is)f(w)n(ell)h(kno)n(wn,)f(the)h(decomp)r(osition)f (\(2.9\))g(of)h(the)g(co)n(v)-5 b(ariance)26 b(implies)j(that,)260 2637 y(if)k(w)n(e)g(call)g Fk(P)680 2649 y Fn(R)734 2637 y Fl(\()p Fk(d )s Fl(\))h(and)f Fk(P)1152 2649 y Fn(G)1208 2637 y Fl(\()p Fk(d )s Fl(\))h(the)f(Gaussian)g(measures)e(of)i(co)n(v) -5 b(ariance)31 b Fk(R)q Fl(\()p Fh(x)22 b Fj(\000)g Fh(y)q Fl(\))260 2737 y(and)27 b Fk(G)p Fl(\()p Fh(x)20 b Fj(\000)e Fh(y)q Fl(\),)28 b(resp)r(ectiv)n(ely)-7 b(,)27 b(then)h(w)n(e)g(can)f(rewrite)g(\(1.1\))g(in)h(the)g(form)830 2958 y 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b(in)g(this)g(case,)g(there)g(is)g(only)f(one)g(unlab)r (eled)h(tree\);)i(w)n(e)d(call)h Fg(trivial)i(vertic)l(es)f Fl(the)260 2505 y(p)r(oin)n(ts)19 b(where)g(the)g(branc)n(hes)f Fg(c)l(onne)l(cting)k(two)g(non)g(trivial)h(vertic)l(es)d Fl(in)n(tersect)f(the)g(family)260 2604 y(of)34 b(v)n(ertical)e(lines;) 37 b(\014nally)-7 b(,)35 b(w)n(e)f(call)f(v)n(ertices)g(the)h(trivial)f (or)g(non)h(trivial)f(v)n(ertices)f(and)260 2704 y(the)c(endp)r(oin)n (ts)f(\(see)h(the)g(dots)f(in)h(Fig.)37 b(1\).)280 2803 y(6\))20 b(Giv)n(en)g(a)g(v)n(ertex)f Fk(v)s Fl(,)j(w)n(e)e(call)f Fk(h)1308 2815 y Fn(v)1368 2803 y Fl(the)h(frequency)g(index)g(of)g (the)h(v)n(ertical)e(line)h(con)n(taining)260 2903 y(it.)76 b(The)41 b(\014rst)f(v)n(ertex)g(of)g(the)h(tree)f(\(ha)n(ving)g (frequency)g(index)h(0\))f(will)h(b)r(e)g(denoted)260 3003 y Fk(v)300 3015 y Fm(0)337 3003 y Fl(.)52 b(Giv)n(en)32 b(a)g(trivial)g(or)f(non)h(trivial)g(v)n(ertex)f Fk(v)s Fl(,)j Fk(s)1902 3015 y Fn(v)1974 3003 y Fl(will)f(denote)f(the)h(n)n (um)n(b)r(er)f(of)g(lines)260 3102 y(branc)n(hing)j(from)g Fk(v)40 b Fl(\(then)d Fk(s)1207 3114 y Fn(v)1283 3102 y Fl(=)g(1,)h(if)f Fk(v)i Fl(is)d(a)g(trivial)f(v)n(ertex\).)62 b(If)37 b Fk(v)i Fl(is)d(an)g(endp)r(oin)n(t)260 3202 y(and)25 b Fk(v)462 3172 y Fe(0)510 3202 y Fl(is)g(the)h(non)f(trivial) f(v)n(ertex)g(immediately)i(preceding)e(it,)i(there)f(is)g(the)g (constrain)n(t)260 3302 y Fk(h)308 3314 y Fn(v)370 3302 y Fl(=)e Fk(h)506 3314 y Fn(v)541 3297 y Fa(0)586 3302 y Fl(+)18 b(1;)27 b(this)h(follo)n(ws)f(from)g(the)h(fact)g(that)g Fk(G)1944 3314 y Fn(h)1987 3302 y Fl(\(0)p Fk(;)13 b(~)-41 b(x)19 b Fj(\000)e Fk(~)-41 b(y)r Fl(\))24 b(=)e(0.)282 3401 y(7\))g(Giv)n(en)g(a)f(v)n(ertex)g Fk(v)s Fl(,)j(w)n(e)e(shall)f (call)h(the)g Fg(cluster)j(of)g Fk(v)g Fl(the)e(family)f(of)g (space-time)f(p)r(oin)n(ts)260 3501 y(asso)r(ciated)30 b(to)h(all)f(the)i(endp)r(oin)n(ts)f(follo)n(wing)f Fk(v)s Fl(,)j(if)e Fk(v)k Fl(is)c(not)g(an)g(endp)r(oin)n(t,)h(or)e Fk(v)35 b Fl(itself,)260 3600 y(otherwise.)282 3700 y(8\))21 b(Finally)-7 b(,)23 b(w)n(e)e(denote)h Fj(E)1093 3712 y Fn(h)1157 3700 y Fl(and)g Fj(E)1364 3670 y Fn(T)1357 3724 y(h)1438 3700 y Fl(the)g(exp)r(ectation)f(and)h(the)g(truncated)f (exp)r(ectation,)260 3800 y(resp)r(ectiv)n(ely)-7 b(,)27 b(with)h(resp)r(ect)f(to)h(the)g(Gaussian)e(measure)h(with)h(co)n(v)-5 b(ariance)25 b Fk(G)2782 3812 y Fn(h)2826 3800 y Fl(.)384 3999 y(W)-7 b(e)28 b(can)g(expand)f(the)h(functional)g(\(2.19\))e(as) 939 4233 y Fm(3)p Fn(:)p Fm(1)150 b Fk(V)1242 4190 y Fm(\()p Fn(N)6 b Fm(\))1223 4258 y Fn(G)1357 4233 y Fl(\()p Fk(')p Fl(\))24 b(=)1614 4129 y Fe(1)1587 4154 y Ff(X)1586 4333 y Fn(k)q Fm(=0)1724 4154 y Ff(X)1721 4333 y Fn(n)p Fe(\025)p Fm(1)1861 4233 y Fk(V)1928 4190 y Fm(\()p Fn(N)6 b Fm(\))1909 4258 y(2)p Fn(k)q(;n)2044 4233 y Fl(\()p Fk(')p Fl(\))24 b Fk(;)744 b Fl(\(3.1\))260 4515 y(where)37 b Fk(V)577 4472 y Fm(\()p Fn(N)6 b Fm(\))558 4540 y(2)p Fn(k)q(;n)693 4515 y Fl(\()p Fk(')p Fl(\))39 b(is)f(the)h(con)n (tribution)e(of)h(the)h(terms)e(of)h(order)f(2)p Fk(k)j Fl(in)f(the)f(\014eld)g(and)260 4614 y(order)25 b Fk(n)i Fl(in)g(the)g(couplings)f Fk(\025;)14 b(\027)5 b Fl(,)28 b(whic)n(h)f(will)g(further)g(expanded)f(in)h(the)h(follo)n(wing)d(w)n (a)n(y)-7 b(.)260 4726 y(Giv)n(en)24 b(a)h(tree)f Fk(\034)33 b Fj(2)23 b(T)941 4682 y Fm(\()p Fn(N)6 b Fm(\))919 4735 y Fn(n)1055 4726 y Fl(,)26 b(w)n(e)e(asso)r(ciate)g(to)g(it)h(man)n(y)g (di\013eren)n(t)f(terms,)h(eac)n(h)f(term)h(b)r(eing)260 4825 y(c)n(haracterized)j(b)n(y)i(selecting,)h(for)f(an)n(y)f(v)n (ertex)h Fk(v)g Fj(2)e Fk(\034)9 b Fl(,)32 b(a)e(subset)h Fk(P)2430 4837 y Fn(v)2497 4825 y Fj(\032)c Fk(I)2625 4837 y Fn(\034)2667 4825 y Fl(,)k(so)f(that)h(the)260 4925 y(family)d Fj(f)p Fk(P)609 4937 y Fn(v)648 4925 y Fk(;)14 b(v)26 b Fj(2)d Fk(\034)9 b Fj(g)28 b Fl(satis\014es)f(the)h (follo)n(wing)f(conditions.)298 5124 y(1\))39 b(If)f Fk(v)k Fl(is)c(an)h(endp)r(oin)n(t,)i Fk(P)1242 5136 y Fn(v)1320 5124 y Fl(coincides)d(with)h(the)g(set)f(of)h(\014elds)f (app)r(earing)f(in)i(the)260 5224 y(corresp)r(onding)25 b(in)n(teraction)i(term.)290 5323 y(2\))j(If)g Fk(v)k Fl(is)29 b(not)i(an)e(endp)r(oin)n(t)i(\()p Fk(v)g Fl(not)c(e.p.)k(in)f (the)g(follo)n(wing\))g(and)f Fk(v)2501 5293 y Fm(1)2539 5323 y Fk(;)14 b(:)g(:)g(:)f(;)h(v)2766 5293 y Fn(s)2797 5301 y Fc(v)2868 5323 y Fl(are)29 b(the)260 5423 y(v)n(ertices)i (immediately)i(follo)n(wing)e Fk(v)s Fl(,)j(then)f Fk(P)1745 5435 y Fn(v)1816 5423 y Fj(\022)1912 5361 y Ff(S)1981 5381 y Fn(s)2012 5389 y Fc(v)1981 5448 y Fn(i)p Fm(=1)2107 5423 y Fk(P)2160 5435 y Fn(v)2193 5443 y Fc(i)2256 5423 y Fl(and,)h(if)f Fk(v)h(>)g Fl(~)-45 b Fk(v)2738 5435 y Fm(0)2776 5423 y Fl(,)34 b(the)e(\014rst)260 5523 y(non)c(trivial)g (v)n(ertex)f(of)i Fk(\034)9 b Fl(,)29 b Fk(P)1168 5535 y Fn(v)1232 5523 y Fj(6)p Fl(=)24 b Fj(;)p Fl(.)39 b(Moreo)n(v)n(er,)26 b(if)j(w)n(e)f(de\014ne)h Fk(Q)2321 5538 y Fn(v)2356 5522 y Fc(i)2411 5523 y Fl(=)24 b Fk(P)2553 5535 y Fn(v)2606 5460 y Ff(T)2689 5523 y Fk(P)2742 5538 y Fn(v)2777 5522 y Fc(i)2837 5523 y Fl(\(so)k(that)260 5622 y Fk(P)313 5634 y Fn(v)376 5622 y Fl(=)463 5560 y Ff(S)532 5580 y Fn(s)563 5588 y Fc(v)532 5647 y Fn(i)p Fm(=1)658 5622 y Fk(Q)724 5638 y Fn(v)759 5621 y Fc(i)789 5622 y Fl(\),)g(then)h Fk(P)1115 5634 y Fn(v)1148 5642 y Fc(i)1178 5622 y Fj(n)p Fk(Q)1286 5634 y Fn(v)1319 5642 y Fc(i)1372 5622 y Fj(6)p Fl(=)23 b Fj(;)k Fl(for)g(an)n(y)g Fk(i)p Fl(,)g(if)h Fk(s)2007 5634 y Fn(v)2070 5622 y Fk(>)23 b Fl(1.)1671 5871 y(7)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 384 608 a Fl(Let)28 b(us)g(no)n(w)f(de\014ne,)h (for)f(an)n(y)f(set)i Fk(P)35 b Fj(2)23 b Fk(I)1691 620 y Fn(\034)1733 608 y Fl(,)750 803 y Fm(3)p Fn(:)p Fm(1)p Fn(a)115 b Fk(')p Fl(\()p Fk(P)12 b Fl(\))23 b(=)1295 724 y Ff(Y)1281 902 y Fn(f)7 b Fe(2)p Fn(P)1430 803 y Fk(')1484 757 y Fn(\033)1522 766 y Fc(f)1484 813 y Fd(x)1524 822 y Fc(f)1648 803 y Fk(;)97 b( )s Fl(\()p Fk(P)12 b Fl(\))24 b(=)2080 724 y Ff(Y)2065 902 y Fn(f)7 b Fe(2)p Fn(P)2214 803 y Fk( )2271 757 y Fn(\033)2309 766 y Fc(f)2268 813 y Fd(x)2308 822 y Fc(f)2374 803 y Fk(:)556 b Fl(\(3.2\))260 1076 y(One)27 b(can)g(sho)n(w)g(that)h(\(see)f([BGPS])h(for)f (details\))501 1301 y Fm(3)p Fn(:)p Fm(2)150 b Fk(V)804 1258 y Fm(\()p Fn(N)6 b Fm(\))785 1326 y(2)p Fn(k)q(;n)920 1301 y Fl(\()p Fk(')p Fl(\))24 b(=)1206 1222 y Ff(X)1150 1422 y Fn(\034)7 b Fe(2T)1285 1391 y Fb(\()p Fc(N)e Fb(\))1271 1433 y Fc(n)1471 1222 y Ff(X)1397 1404 y Fe(j)p Fn(P)1459 1412 y Fc(v)1489 1424 y Fb(0)1525 1404 y Fe(j)p Fm(=2)p Fn(k)1680 1188 y Ff(Z)1776 1301 y Fk(d)p Fh(x)p 1819 1314 51 4 v 24 w Fk(')p Fl(\()p Fk(P)2032 1313 y Fn(v)2065 1321 y Fb(0)2103 1301 y Fl(\))23 b Fk(W)12 b Fl(\()p Fk(\034)5 b(;)14 b(P)2411 1313 y Fn(v)2444 1321 y Fb(0)2481 1301 y Fk(;)g Fh(x)p 2518 1314 V 1 w Fl(\))23 b Fk(;)306 b Fl(\(3.3\))640 1628 y Fm(3)p Fn(:)p Fm(3)151 b Fk(W)12 b Fl(\()p Fk(\034)5 b(;)14 b(P)1130 1640 y Fn(v)1163 1648 y Fb(0)1200 1628 y Fk(;)g Fh(x)p 1237 1641 V Fl(\))24 b(=)1430 1536 y Ff(h)1578 1549 y(Y)1562 1712 y Fc(v)q(>v)1668 1724 y Fb(0)1493 1771 y Fc(v)g Fm(not)e(e.p.)1793 1549 y Ff(X)1814 1727 y Fn(P)1856 1735 y Fc(v)1926 1536 y Ff(i)1966 1628 y Fk(W)12 b Fl(\()p Fk(\034)5 b(;)14 b Fj(f)p Fk(P)2261 1640 y Fn(v)2300 1628 y Fj(g)p Fk(;)g Fh(x)p 2379 1641 V Fl(\))23 b Fk(;)446 b Fl(\(3.4\))415 2069 y Fm(3)p Fn(:)p Fm(4)316 b Fk(W)12 b Fl(\()p Fk(\034)5 b(;)14 b Fj(f)p Fk(P)1112 2081 y Fn(v)1152 2069 y Fj(g)p Fk(;)g Fh(x)p 1231 2082 V -1 w Fl(\))24 b(=)e(\()p Fj(\000)p Fk(\027)5 b Fl(\))1598 2035 y Fn(n)1639 2043 y Fb(2)1707 1965 y Fn(n)1748 1973 y Fb(4)1691 1990 y Ff(Y)1690 2167 y Fn(i)p Fm(=1)1798 2069 y Fl([)p Fj(\000)p Fk(\025v)s Fl(\()p Fh(x)2059 2081 y Fm(2)p Fn(i)p Fe(\000)p Fm(1)2224 2069 y Fj(\000)18 b Fh(x)2357 2081 y Fm(2)p Fn(i)2418 2069 y Fl(\)])h Fj(\001)438 b Fl(\(3.5\))817 2354 y Fj(\001)854 2212 y Ff(")989 2275 y(Y)902 2450 y Fn(v)25 b Fm(not)d(e.p.)1236 2298 y Fl(1)p 1206 2335 102 4 v 1206 2411 a Fk(s)1245 2423 y 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Fl(.)260 3053 y Fo(4)135 b(Pro)t(of)45 b(of)g(Theorem)g(1.1)24 3235 y Fn(ss)p Fm(4)260 3335 y Fl(By)38 b(the)g(remark)f(follo)n(wing)h (\(2.12\),)i(in)e(order)f(to)h(pro)n(v)n(e)f(the)i(\014rst)f(past)g(of) g(Theorem)260 3434 y(1.1,)31 b(that)h(is)g(the)f(b)r(ounds)h(\(1.7\))f (and)h(\(1.8\),)g(it)g(is)f(su\016cien)n(t)h(to)f(pro)n(v)n(e)f(that,)j (if)e(w)n(e)h(put)260 3534 y Fk(\025)308 3546 y Fm(0)369 3534 y Fl(=)22 b(max)p Fj(fj)p Fk(\025)p Fj(j)p Fk(;)14 b Fj(j)p Fk(\027)5 b Fj(jg)p Fl(,)457 3816 y Fm(4)p Fn(:)p Fm(1)150 b Fl(\()p Fk(\014)t(L)833 3782 y Fm(2)871 3816 y Fl(\))903 3782 y Fe(\000)p Fm(1)1006 3703 y Ff(Z)1103 3816 y Fk(d)p Fh(x)p 1146 3829 V 1210 3646 a Ff(\014)1210 3696 y(\014)1210 3746 y(\014)1210 3795 y(\014)1210 3845 y(\014)1210 3895 y(\014)1294 3737 y(X)1238 3937 y Fn(\034)7 b Fe(2T)1373 3907 y Fb(\()p Fc(N)e Fb(\))1359 3948 y Fc(n)1559 3737 y Ff(X)1485 3919 y Fe(j)p Fn(P)1547 3927 y Fc(v)1577 3939 y Fb(0)1613 3919 y Fe(j)p Fm(=2)p Fn(k)1768 3816 y Fk(W)12 b Fl(\()p Fk(\034)5 b(;)14 b(P)2021 3828 y Fn(v)2054 3836 y Fb(0)2091 3816 y Fk(;)g Fh(x)p 2128 3829 V Fl(\))2210 3646 y Ff(\014)2210 3696 y(\014)2210 3746 y(\014)2210 3795 y(\014)2210 3845 y(\014)2210 3895 y(\014)2261 3816 y Fj(\024)23 b Fl(\()p Fk(C)2440 3828 y Fn(k)2481 3816 y Fk(\025)2529 3828 y Fm(0)2567 3816 y Fl(\))2599 3782 y Fn(n)2668 3816 y Fk(:)262 b Fl(\(4.1\))260 4110 y(T)-7 b(o)29 b(b)r(egin)g(with,)h(w)n(e)f(note)g(that,)g(b)n(y)g (pro)r(ceeding)f(as)h(in)g(App.)42 b(2)29 b(of)g([BGPS)o(])h(\(where)e (an)260 4210 y(essen)n(tial)f(role)g(is)h(pla)n(y)n(ed)g(b)n(y)f(the)i (Gram-Hadamard)d(inequalit)n(y)-7 b(,)28 b(used)g(as)g(in)g([Le]\))g (and)260 4309 y(b)n(y)f(using)g(\(2.16\),)g(w)n(e)h(get)186 4526 y Fm(4)p Fn(:)p Fm(2)422 4430 y Ff(\014)422 4480 y(\014)422 4530 y(\014)449 4526 y Fj(E)500 4491 y Fn(T)493 4546 y(h)553 4458 y Ff(\000)591 4526 y Fk( )s Fl(\()p Fk(P)733 4538 y Fm(1)771 4526 y Fl(\))p Fk(;)14 b(:)g(:)g(:)g(;)g( )s Fl(\()p Fk(P)1130 4538 y Fn(s)1166 4526 y Fl(\))1198 4458 y Ff(\001)1236 4430 y(\014)1236 4480 y(\014)1236 4530 y(\014)1287 4526 y Fj(\024)23 b Fk(C)1440 4422 y Ff(P)1527 4509 y Fc(j)1570 4478 y Fe(j)p Fn(P)1632 4486 y Fc(j)1663 4478 y Fe(j)1687 4526 y Fk(\015)1744 4455 y Fc(h)p 1744 4464 35 4 v 1747 4498 a Fb(2)1800 4422 y Ff(P)1888 4509 y Fc(j)1930 4478 y Fe(j)p Fn(P)1992 4486 y Fc(j)2023 4478 y Fe(j)2061 4447 y Ff(X)2097 4625 y Fn(T)2194 4526 y Fk(e)2233 4491 y Fe(\000)p Fn(\024d)2359 4461 y Fb(\()p Fc(h)p Fb(\))2359 4513 y Fc(T)2442 4491 y Fm(\()p Fn(P)2510 4499 y Fb(1)2542 4491 y Fn(;:::)o(;P)2683 4499 y Fc(s)2715 4491 y Fm(\))2768 4526 y Fk(;)162 b Fl(\(4.2\))260 4782 y(where)36 b Fk(T)49 b Fl(is)37 b(an)f Fg(anchor)l(e)l(d)k(tr)l(e)l(e)e(gr)l(aph)h Fl(b)r(et)n(w)n(een)e(the)g (clusters)g(of)g(space-time)f(p)r(oin)n(ts)260 4882 y(from)30 b(whic)n(h)g(the)h(\014elds)g(lab)r(eled)f(b)n(y)g Fk(P)1518 4894 y Fm(1)1556 4882 y Fk(;)14 b(:)g(:)g(:)g(;)g(P)1794 4894 y Fn(s)1860 4882 y Fl(emerge;)31 b(this)f(means)g(that)h Fk(T)42 b Fl(is)30 b(a)g(set)260 4981 y(of)d(lines)h(connecting)f(t)n (w)n(o)g(p)r(oin)n(ts)g(in)h(di\013eren)n(t)f(clusters,)g(whic)n(h)h(b) r(ecomes)f(a)g(tree)g(graph)260 5081 y(if)40 b(one)f(iden)n(ti\014es)h (all)g(the)g(p)r(oin)n(ts)g(in)g(the)g(same)f(cluster.)73 b(Moreo)n(v)n(er,)40 b(if)g Fh(x)2780 5051 y Fn(i)2848 5081 y Fl(and)g Fh(y)3073 5051 y Fn(i)3101 5081 y Fl(,)260 5181 y Fk(i)27 b Fl(=)g(1)p Fk(;)14 b(:)g(:)g(:)f(;)h(s)30 b Fl(are)f(the)i(space-time)e(co)r(ordinates)g(of)h(the)g(t)n(w)n(o)g (p)r(oin)n(ts)g(connected)g(b)n(y)g(the)260 5280 y(lines)d(b)r (elonging)g(to)h Fk(T)12 b Fl(,)27 b(w)n(e)g(de\014ne:)510 5537 y Fm(4)p Fn(:)p Fm(3)151 b Fk(d)790 5494 y Fm(\()p Fn(h)p Fm(\))790 5562 y Fn(T)885 5537 y Fl(\()p Fk(P)970 5549 y Fm(1)1021 5537 y Fk(:)14 b(:)g(:)g(P)1185 5549 y Fn(s)1221 5537 y Fl(\))23 b(=)1366 5434 y Fn(s)p Fe(\000)p Fm(1)1364 5458 y Ff(X)1370 5635 y Fn(i)p Fm(=1)1484 5537 y Fl(\()p Fk(\015)1564 5503 y Fn(h)1607 5537 y Fk(d)1650 5549 y Fn(\014)1695 5537 y Fl(\()p Fk(x)1774 5503 y Fn(i)1774 5558 y Fm(0)1812 5537 y Fk(;)14 b(y)1893 5503 y Fn(i)1890 5558 y Fm(0)1927 5537 y Fl(\))k(+)g Fk(\015)2108 5503 y Fn(h=)p Fm(2)2218 5537 y Fk(d)2261 5549 y Fn(L)2311 5537 y Fl(\()o Fk(~)-41 b(x)2390 5503 y Fn(i)2418 5537 y Fk(;)13 b(~)-41 b(y)2499 5503 y Fn(i)2527 5537 y Fl(\)\))23 b Fk(:)316 b Fl(\(4.3\))1671 5871 y(8)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 260 508 a Fl(Note)27 b(that,)g(if)h Fk(s)23 b Fl(=)f(1,)27 b(the)h(sum)f(o)n(v)n(er)e Fk(T)38 b Fl(is)27 b(v)n(oid)f(and)h(m)n(ust)g(b)r(e)g(understo)r(o)r(d)g(as)f (a)h(trivial)260 608 y(factor)g(1.)384 708 y(W)-7 b(e)28 b(can)g(use)f(the)h(inequalit)n(y)f(\(4.2\))g(to)h(b)r(ound)g(the)g (r.h.s.)36 b(of)28 b(\(3.5\).)37 b(W)-7 b(e)28 b(get)306 943 y Fm(4)p Fn(:)p Fm(4)316 b Fj(j)p Fk(W)12 b Fl(\()p Fk(\034)5 b(;)14 b Fj(f)p Fk(P)1026 955 y Fn(v)1066 943 y Fj(g)p Fk(;)g Fh(x)p 1145 956 51 4 v Fl(\))p Fj(j)23 b(\024)g Fl(\()p Fk(C)6 b(\025)1506 955 y Fm(0)1544 943 y Fl(\))1576 909 y Fn(n)1635 801 y Ff(")1701 839 y Fn(n)1742 847 y Fb(4)1684 865 y Ff(Y)1684 1041 y Fn(i)p Fm(=1)1805 943 y Fj(j)p Fk(v)s Fl(\()p Fh(x)1953 955 y Fm(2)p Fn(i)p Fe(\000)p Fm(1)2118 943 y Fj(\000)18 b Fh(x)2251 955 y Fm(2)p Fn(i)2312 943 y Fl(\))p Fj(j)2367 801 y Ff(#)2435 943 y Fj(\001)495 b Fl(\(4.4\))708 1228 y Fj(\001)745 1086 y Ff(")880 1149 y(Y)794 1324 y Fn(v)24 b Fm(not)e(e.p.)1127 1172 y Fl(1)p 1097 1209 102 4 v 1097 1285 a Fk(s)1136 1297 y Fn(v)1175 1285 y Fl(!)1208 1228 y Fk(\015)1266 1160 y Fc(h)1300 1168 y(v)p 1266 1175 70 4 v 1287 1208 a Fb(2)1357 1132 y Ff(P)1445 1153 y Fc(s)1473 1161 y(v)1445 1219 y(i)p Fb(=1)1542 1188 y Fm([)p Fe(j)p Fn(P)1623 1196 y Fc(v)1653 1209 y(i)1683 1188 y Fe(j\000j)p Fn(P)1817 1196 y Fc(v)1852 1188 y Fe(j)p Fm(])1909 1149 y Ff(X)1932 1327 y Fn(T)1971 1335 y Fc(v)2043 1228 y Fk(e)2082 1189 y Fe(\000)p Fn(\024d)2208 1159 y Fb(\()p Fc(h)2264 1167 y(v)2299 1159 y Fb(\))2208 1211 y Fc(T)2243 1219 y(v)2326 1189 y Fm(\()p Fn(P)2394 1197 y Fc(v)2424 1209 y Fb(1)2461 1189 y Fn(;:::)n(;P)2601 1197 y Fc(v)2631 1205 y(s)2659 1213 y(v)2703 1189 y Fm(\))2733 1086 y Ff(#)2819 1228 y Fk(;)260 1481 y Fl(where)27 b(w)n(e)g(used)h(the)g(fact)f(that)1298 1418 y Ff(P)1386 1506 y Fn(v)e Fm(not)d(e.p.)1683 1418 y Ff(P)1771 1439 y Fn(s)1802 1447 y Fc(v)1771 1506 y Fn(i)p Fm(=1)1883 1481 y Fl([)p Fj(j)p Fk(P)1982 1493 y Fn(v)2015 1501 y Fc(i)2046 1481 y Fj(j)c(\000)g(j)p Fk(Q)2259 1493 y Fn(v)2292 1501 y Fc(i)2323 1481 y Fj(j)p Fl(])23 b Fj(\024)g Fl(2)p Fk(n)p Fl(.)384 1580 y(Let)41 b(us)f(no)n(w)g(observ)n(e)e(that,)44 b(if)d(w)n(e)f(select)g(one)g (tree)g(graph)f Fk(T)2482 1592 y Fn(v)2561 1580 y Fl(for)h(eac)n(h)f(v) n(ertex)260 1680 y Fk(v)s Fl(,)k(except)d(the)g(endp)r(oin)n(ts,)j(and) d(w)n(e)g(add,)j(for)c(eac)n(h)g(endp)r(oin)n(t)h(of)g(t)n(yp)r(e)g Fk(\025)p Fl(,)k(the)c(line)260 1780 y(connecting)e(the)h(t)n(w)n(o)f (corresp)r(onding)e(space-time)i(p)r(oin)n(ts,)k(w)n(e)c(get)g(a)h (spanning)f(tree)260 1879 y(for)32 b(the)h(all)f(set)g(of)h(p)r(oin)n (ts)f Fh(x)p 1148 1892 51 4 v 33 w Fl(asso)r(ciate)f(to)h Fk(\034)9 b Fl(.)52 b(W)-7 b(e)33 b(can)f(use)g(this)h(spanning)f(tree) g(\(and)260 2009 y(the)27 b(corresp)r(onding)e Fk(e)974 1970 y Fe(\000)p Fn(\024d)1100 1939 y Fb(\()p Fc(h)1156 1947 y(v)1191 1939 y Fb(\))1100 1991 y Fc(T)1135 1999 y(v)1218 1970 y Fm(\()p Fn(P)1286 1978 y Fc(v)1316 1990 y Fb(1)1353 1970 y Fn(;:::)n(;P)1493 1978 y Fc(v)1523 1986 y(s)1551 1994 y(v)1595 1970 y Fm(\))1652 2009 y Fl(or)h Fk(v)s Fl(\()p Fh(x)1878 2021 y Fm(2)p Fn(i)p Fe(\000)p Fm(1)2041 2009 y Fj(\000)17 b Fh(x)2173 2021 y Fm(2)p Fn(i)2234 2009 y Fl(\))27 b(factors\))f(to)h(p)r(erform)f(the) 260 2108 y(in)n(tegrations)f(o)n(v)n(er)h(the)h(space-time)f(p)r(oin)n (ts)h(in)h(the)f(usual)g(w)n(a)n(y)-7 b(,)26 b(b)n(y)g(ordering)g(them) h(in)h(a)260 2208 y(w)n(a)n(y)h(suggested)h(b)n(y)g(the)h(spanning)f (tree)g(so)g(that)h(eac)n(h)f(in)n(tegration)g(in)n(v)n(olv)n(es)e (only)j(one)260 2307 y(di\013erence)f(of)g(co)r(ordinates,)f(except)h (the)h(last)f(one,)g(whic)n(h)g(giv)n(es)f(a)h(v)n(olume)f(factor.)44 b(As)260 2407 y(sho)n(wn)28 b(in)h(App.)42 b(3)28 b(of)h([BGPS],)g(the) h(sum)f(o)n(v)n(er)e(the)i(p)r(ossible)g(c)n(hoices)f(of)g(the)i (spanning)260 2507 y(tree)d(is)h(con)n(trolled,)e(up)i(to)f(a)h Fk(C)1274 2477 y Fn(n)1347 2507 y Fl(constan)n(t,)f(b)n(y)g(the)h(1)p Fk(=s)2086 2519 y Fn(v)2124 2507 y Fl(!)g(factors,)f(hence)g(w)n(e)g (get:)533 2710 y Fm(4)p Fn(:)p Fm(5)316 b Fl(\()p Fk(\014)t(L)1075 2676 y Fm(2)1112 2710 y Fl(\))1144 2676 y Fe(\000)p Fm(1)1248 2597 y Ff(Z)1345 2710 y Fk(d)p Fh(x)p 1388 2723 V Fj(j)p Fk(W)12 b Fl(\()p Fk(\034)5 b(;)14 b Fj(f)p Fk(P)1756 2722 y Fn(v)1796 2710 y Fj(g)p Fk(;)g Fh(x)p 1875 2723 V -1 w Fl(\))p Fj(j)24 b(\024)e Fl(\()p Fk(C)6 b(\025)2235 2722 y Fm(0)2274 2710 y Fl(\))2306 2676 y Fn(n)2351 2710 y Fj(k)s Fl(\026)-45 b Fk(v)s Fj(k)2478 2673 y Fn(n)2519 2681 y Fb(4)2478 2732 y Fm(1)2573 2710 y Fj(\001)357 b Fl(\(4.5\))935 2912 y Fj(\001)1059 2833 y Ff(Y)972 3007 y Fn(v)25 b Fm(not)d(e.p.)1265 2912 y Fk(\015)1323 2844 y Fc(h)1357 2852 y(v)p 1323 2859 70 4 v 1344 2892 a Fb(2)1414 2816 y Ff(P)1502 2836 y Fc(s)1530 2844 y(v)1502 2903 y(i)p Fb(=1)1599 2872 y Fm([)p Fe(j)p Fn(P)1680 2880 y Fc(v)1710 2893 y(i)1740 2872 y Fe(j\000j)p Fn(P)1874 2880 y Fc(v)1909 2872 y Fe(j)p Fm(])1952 2912 y Fk(\015)2000 2877 y Fe(\000)p Fm(2)p Fn(h)2124 2885 y Fc(v)2159 2877 y Fm(\()p Fn(s)2216 2885 y Fc(v)2252 2877 y Fe(\000)p Fm(1\))2390 2912 y Fk(;)260 3164 y Fl(where)31 b Fj(k)s Fl(\026)-45 b Fk(v)r Fj(k)630 3176 y Fm(1)696 3164 y Fl(=)790 3097 y Ff(R)859 3164 y Fk(d)o(~)k(x)36 b Fl(\026)-46 b Fk(v)t Fl(\()o Fk(~)-41 b(x)p Fl(\).)49 b(Note)31 b(that)h Fk(v)s Fl(\()p Fh(x)p Fl(\))g(is)g(not)f(a)g(b)r(ounded)h(function,)g (see)f(\(1.4\);)260 3263 y(hence,)d(in)f(the)h(previous)f(b)r(ound)h (an)f(essen)n(tial)g(role)f(is)i(pla)n(y)n(ed)f(b)n(y)g(the)h(fact)g (that)f(all)h(the)260 3363 y(spanning)f(trees)g(con)n(tain)g(all)g(the) h(lines)g(asso)r(ciated)e(to)h(the)h(endp)r(oin)n(ts)g(of)g(t)n(yp)r(e) f Fk(\025)p Fl(.)384 3462 y(Let)38 b(us)g(no)n(w)f(call)g Fk(n)1055 3474 y Fm(4)p Fn(;v)1185 3462 y Fl(and)h Fk(n)1407 3474 y Fm(2)p Fn(;v)1537 3462 y Fl(the)g(n)n(um)n(b)r(er)f(of)h(endp)r (oin)n(ts)f(of)h(t)n(yp)r(e)g Fk(\025)g Fl(and)g Fk(\027)5 b Fl(,)260 3562 y(resp)r(ectiv)n(ely)-7 b(,)30 b(follo)n(wing)e(the)j (v)n(ertex)e Fk(v)s Fl(,)h(if)h(it)f(is)g(not)g(an)f(endp)r(oin)n(t,)i (and)f Fk(n)2683 3574 y Fn(v)2752 3562 y Fl(their)g(sum.)260 3662 y(Then,)24 b(it)f(is)g(easy)e(to)i(c)n(hec)n(k)f(that,)i(if)f Fk(v)j Fl(is)d(not)f(an)h(endp)r(oin)n(t,)2179 3599 y Ff(P)2269 3687 y Fm(~)-35 b Fn(v)r Fe(\025)p Fn(v)2407 3599 y Ff(P)2495 3620 y Fn(s)2529 3628 y Fb(~)k Fc(v)2495 3687 y Fn(i)p Fm(=1)2606 3662 y Fl([)p Fj(j)p Fk(P)2708 3674 y Fm(~)-36 b Fn(v)2738 3682 y Fc(i)2769 3662 y Fj(j)9 b(\000)g(j)p Fk(P)2954 3674 y Fm(~)-36 b Fn(v)2990 3662 y Fj(j)p Fl(])24 b(=)260 3761 y(4)p Fk(n)352 3773 y Fm(4)p Fn(;v)466 3761 y Fl(+)f(2)p Fk(n)646 3773 y Fm(2)p Fn(;v)761 3761 y Fj(\000)f(j)p Fk(P)924 3773 y Fn(v)964 3761 y Fj(j)35 b Fl(and)f(that)1377 3699 y Ff(P)1467 3786 y Fm(~)-36 b Fn(v)s Fe(\025)p Fn(v)1591 3761 y Fl(\()p Fk(s)1665 3773 y Fm(~)g Fn(v)1725 3761 y Fj(\000)22 b Fl(1\))34 b(=)g Fk(n)2069 3773 y Fm(4)p Fn(;v)2185 3761 y Fl(+)22 b Fk(n)2322 3773 y Fm(2)p Fn(;v)2437 3761 y Fj(\000)h Fl(1.)57 b(Then)34 b(w)n(e)h(can)260 3861 y(rewrite)27 b(\(4.5\))g(as)188 4065 y Fm(4)p Fn(:)p Fm(6)151 b Fl(\()p Fk(\014)t(L)565 4030 y Fm(2)602 4065 y Fl(\))634 4030 y Fe(\000)p Fm(1)737 3952 y Ff(Z)834 4065 y Fk(d)p Fh(x)p 877 4078 51 4 v 1 w Fj(j)p Fk(W)12 b Fl(\()p Fk(\034)5 b(;)14 b Fj(f)p Fk(P)1246 4077 y Fn(v)1285 4065 y Fj(g)p Fk(;)g Fh(x)p 1364 4078 V Fl(\))p Fj(j)23 b(\024)g Fl(\()p Fk(C)6 b(\025)1725 4077 y Fm(0)1763 4065 y Fl(\))1795 4030 y Fn(n)1941 3986 y Ff(Y)1855 4160 y Fn(v)24 b Fm(not)e(e.p.)2148 4065 y Fk(\015)2196 4030 y Fe(\000)p Fn(d)p Fm(\()p Fn(n)2350 4038 y Fb(4)p Fc(;v)2432 4030 y Fn(;n)2493 4038 y Fb(2)p Fc(;v)2575 4030 y Fn(;)p Fe(j)p Fn(P)2657 4038 y Fc(v)2693 4030 y Fe(j)p Fm(\))2766 4065 y Fk(;)164 b Fl(\(4.6\))260 4312 y(where)27 b(the)h Fg(vertex)h(dimension)g Fk(d)p Fl(\()p Fk(n)1410 4324 y Fm(4)1448 4312 y Fk(;)14 b(n)1535 4324 y Fm(2)1572 4312 y Fk(;)g(p)p Fl(\))27 b(is)h(de\014ned)g(as)214 4512 y Fm(4)p Fn(:)p Fm(7)150 b Fk(d)p Fl(\()p Fk(n)575 4524 y Fm(4)613 4512 y Fk(;)14 b(n)700 4524 y Fm(2)737 4512 y Fk(;)g(p)p Fl(\))23 b(=)g(2\()p Fk(n)1083 4524 y Fm(4)1138 4512 y Fl(+)18 b Fk(n)1271 4524 y Fm(2)1327 4512 y Fj(\000)g Fl(1\))g Fj(\000)1595 4456 y Fl(1)p 1595 4493 42 4 v 1595 4569 a(2)1646 4512 y(\(4)p Fk(n)1770 4524 y Fm(4)1826 4512 y Fl(+)g(2)p Fk(n)2001 4524 y Fm(2)2056 4512 y Fj(\000)g Fk(p)p Fl(\))23 b(=)f Fk(n)2373 4524 y Fm(2)2429 4512 y Fj(\000)c Fl(2)g(+)2665 4456 y Fk(p)p 2665 4493 V 2665 4569 a Fl(2)2740 4512 y Fk(:)190 b Fl(\(4.7\))384 4726 y(If,)33 b(giv)n(en)d(the)h(tree)f Fk(\034)39 b Fj(2)29 b(T)1255 4682 y Fm(\()p Fn(N)6 b Fm(\))1234 4735 y Fn(n)1370 4726 y Fl(,)32 b(it)f(turns)g(out)g(that)g Fk(d)p Fl(\()p Fk(n)2191 4738 y Fm(4)p Fn(;v)2283 4726 y Fk(;)14 b(n)2370 4738 y Fm(2)p Fn(;v)2462 4726 y Fk(;)g Fj(j)p Fk(P)2575 4738 y Fn(v)2615 4726 y Fj(j)p Fl(\))29 b Fk(>)f Fl(0)j(for)f(an)n(y)260 4825 y Fk(v)i(>)c(v)465 4837 y Fm(0)533 4825 y Fl(whic)n(h)j(is)g(not)g(an)f(endp)r(oin)n(t,)i (then,)h(b)n(y)d(pro)r(ceeding)g(as)h(in)g Fj(x)p Fl(3)f(of)h([BGPS],)h (one)260 4925 y(can)g(p)r(erform)g(the)h(sum)g(o)n(v)n(er)e(the)i(sets) g Fj(f)p Fk(P)1659 4937 y Fn(v)1698 4925 y Fk(;)14 b(v)35 b(>)c(v)1946 4937 y Fm(0)1983 4925 y Fj(g)i Fl(and)f(all)g(the)i(lab)r (els)e(of)h Fk(\034)42 b Fl(of)33 b(the)260 5024 y(l.h.s.)39 b(of)28 b(\(4.1\),)g(obtaining)g(a)g(\()p Fk(C)6 b(\025)1372 5036 y Fm(0)1410 5024 y Fl(\))1442 4994 y Fn(n)1516 5024 y Fl(b)r(ound.)39 b(Hence,)29 b(if)g(this)f(prop)r(ert)n(y)f(w)n(ere)h (true)g(for)260 5124 y(all)23 b(trees)f(of)h(order)e Fk(n)p 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y(\(b\))29 b Fj(j)p Fk(P)516 683 y Fn(v)556 671 y Fj(j)23 b Fl(=)f(2,)28 b Fk(n)832 683 y Fm(2)p Fn(;v)947 671 y Fl(=)22 b(0)p Fk(;)14 b Fl(1)301 834 y(\(c\))24 b Fj(j)p Fk(P)502 846 y Fn(v)542 834 y Fj(j)f Fl(=)g(0,)h Fk(n)815 846 y Fm(2)p Fn(;v)930 834 y Fl(=)f(0)p Fk(;)14 b Fl(1)p Fk(;)g Fl(2)22 b(\(this)i(can)f(happ)r(en)h(only)f(in)h(the)f(v)n(ertices)g(b)r(et)n (w)n(een)g Fk(v)2929 846 y Fm(0)2990 834 y Fl(and)467 934 y(the)28 b(\014rst)g(non)f(trivial)g(v)n(ertex)g(of)g Fk(\034)9 b Fl(\))384 1108 y(Note)35 b(that)h(this)f(result)f(is)h (strictly)g(related)f(with)i(the)f(space)f(dimension.)59 b(In)35 b(fact)260 1208 y(our)30 b(pro)r(cedure)g(w)n(orks)g(ev)n(en)g (for)h Fk(d)e(>)g Fl(2,)i(the)h(only)f(di\013erence)g(b)r(eing)g(that,) h(instead)f(of)260 1307 y(\(4.7\),)26 b(w)n(e)f(get)g Fk(d)p Fl(\()p Fk(n)861 1319 y Fm(4)899 1307 y Fk(;)14 b(n)986 1319 y Fm(2)1023 1307 y Fk(;)g(p)p Fl(\))23 b(=)g Fj(\000)p Fk(n)1360 1319 y Fm(4)1396 1307 y Fl(\()p Fk(d=)p Fl(2)14 b Fj(\000)g Fl(1\))h(+)f Fk(n)1866 1319 y Fm(2)1917 1307 y Fj(\000)g Fk(d=)p Fl(2)g Fj(\000)g Fl(1)g(+)g Fk(pd=)p Fl(4,)26 b(so)e(that,)j(for)e(an)n(y)260 1407 y(v)-5 b(alue)28 b(of)g Fj(j)p Fk(P)646 1419 y Fn(v)685 1407 y Fj(j)p Fl(,)h(w)n(e)e(ha)n(v)n(e)g(v)n(ertices)g(with)i(negativ) n(e)e(dimension,)h(if)g Fk(n)2435 1419 y Fm(4)p Fn(;v)2555 1407 y Fl(is)g(large)f(enough.)260 1507 y(This)h(mak)n(es)f(completely) i(useless)e(the)i(tree)f(expansion.)38 b(W)-7 b(e)29 b(w)n(an)n(t)e(no)n(w)h(to)g(sho)n(w)g(that,)260 1606 y(if)34 b Fk(d)g Fl(=)e(2,)j(w)n(e)f(can)f(impro)n(v)n(e)f(our)h(b)r (ounds,)j(b)n(y)d(summing)h(the)g(con)n(tributions)f(of)h(some)260 1706 y(trees)27 b(in)h(the)g(l.h.s.)37 b(of)27 b(\(4.1\),)h(b)r(efore)f (exc)n(hanging)f(the)i(absolute)f(v)-5 b(alue)27 b(with)h(the)g(sums.) 384 1805 y(Note)22 b(that,)h(if)e(w)n(e)g(expand,)h(giv)n(en)f(an)n(y)f (v)n(ertex)h Fk(v)s Fl(,)h(all)f(the)h(truncated)f(exp)r(ectations)g (in)260 1905 y(the)28 b(v)n(ertices)h(~)-45 b Fk(v)26 b Fj(\025)d Fk(v)s Fl(,)28 b(w)n(e)f(get)h(a)f(sum)g(of)h(connected)f (F)-7 b(eynmann)28 b(graphs.)35 b(These)28 b(graphs)260 2005 y(can)22 b(b)r(e)g(dra)n(wn)f(in)i(the)f(usual)g(w)n(a)n(y)f(b)n (y)h(represen)n(ting)e(the)j(t)n(w)n(o)e(in)n(teraction)g(terms)h(of)g (\(1.3\))260 2104 y(as)j(in)h(Fig.)36 b(2,)26 b(where)f(the)h(solid)f (lines)h(corresp)r(ond)e(to)h(fermions)g(and)h(the)g(wiggling)e(lines) 260 2204 y(corresp)r(ond)33 b(to)h(the)i(t)n(w)n(o-b)r(o)r(dy)d(p)r (oten)n(tial;)38 b(moreo)n(v)n(er,)c(the)h(arro)n(w)e(on)h(a)h(fermion) f(line)260 2304 y(coming)28 b(in)g(\(out\))h(the)g(space-time)f(p)r (oin)n(t)g Fh(x)h Fl(means)f(that)h(the)g(corresp)r(onding)d(\014eld)j (has)260 2403 y(p)r(ositiv)n(e)23 b(\(negativ)n(e\))f Fk(\033)s Fl(-lab)r(el.)36 b(It)24 b(turns)f(out)g(that)h(the)f(graphs) f(m)n(ust)i(ha)n(v)n(e)e Fj(j)p Fk(P)2745 2415 y Fn(v)2785 2403 y Fj(j)h Fl(external)1055 2640 y Fh(x)1055 3180 y(y)2301 2972 y(x)557 3185 y @beginspecial @setspecial %%BeginDocument: fig3.ps /fx [ 0.00 1.11 2.22 3.33 4.44 5.56 6.67 7.78 8.89 10.00 11.11 12.22 13.33 14.44 15.56 16.67 17.78 18.89 20.00 21.11 22.22 23.33 24.44 25.56 26.67 27.78 28.89 30.00 31.11 32.22 33.33 34.44 35.56 36.67 37.78 38.89 40.00 41.11 42.22 43.33 44.44 45.56 46.67 47.78 48.89 50.00 51.11 52.22 53.33 54.44 55.56 56.67 57.78 58.89 60.00 61.11 62.22 63.33 64.44 65.56 66.67 67.78 68.89 70.00 71.11 72.22 73.33 74.44 75.56 76.67 77.78 78.89 80.00 81.11 82.22 83.33 84.44 85.56 86.67 87.78 88.89 90.00 91.11 92.22 93.33 94.44 95.56 96.67 97.78 98.89 100.00 ] def /fy [ 0.00 2.07 3.98 5.58 6.75 7.39 7.46 6.96 5.91 4.41 2.57 0.52 -1.56 -3.52 -5.21 -6.50 -7.28 -7.50 -7.14 -6.22 -4.82 -3.05 -1.04 1.04 3.05 4.82 6.22 7.14 7.50 7.28 6.50 5.21 3.52 1.56 -0.52 -2.57 -4.41 -5.91 -6.96 -7.46 -7.39 -6.75 -5.58 -3.98 -2.07 0.00 2.07 3.98 5.58 6.75 7.39 7.46 6.96 5.91 4.41 2.57 0.52 -1.56 -3.52 -5.21 -6.50 -7.28 -7.50 -7.14 -6.22 -4.82 -3.05 -1.04 1.04 3.05 4.82 6.22 7.14 7.50 7.28 6.50 5.21 3.52 1.56 -0.52 -2.57 -4.41 -5.91 -6.96 -7.46 -7.39 -6.75 -5.58 -3.98 -2.07 0.00 ] def /cambio_coordinate{ % st: x1 y1 x2 y2 4 copy exch pop exch sub % x1 dy 6 1 roll exch pop sub % dy x1 y1 dx 4 1 roll translate % sposta l'origine; st: dx dy 2 copy exch atan rotate % rotazione di arctan(dy/dx); st: dx dy 2 exp exch 2 exp add sqrt % sqrt(dx^2 +dy^2) } def /nx fx length 1 sub def /normonda { %st: l 100 div dup scale fx 0 get fy 0 get moveto 1 3 nx { dup dup 1 add exch 2 add 3 1 roll exch dup fx exch get 4 1 roll fy exch get 3 1 roll dup fx exch get 3 1 roll fy exch get exch dup fx exch get exch fy exch get curveto } for } def /onda { gsave % uso: x1 y1 x2 y2 onda cambio_coordinate % st: l normonda stroke grestore } def /freccia { gsave % uso: x1 y1 x2 y2 freccia cambio_coordinate % st: l dup 0 0 moveto 0 lineto % st: l 2 div 0 translate 15 rotate 0 0 moveto -5 0 lineto -30 rotate 0 0 moveto -5 0 lineto stroke grestore } def /punto { gsave % uso: x1 y1 punto 2 0 360 newpath arc fill stroke grestore} def 65 60 punto 65 10 punto 30 10 65 10 freccia 65 10 110 10 freccia 30 60 65 60 freccia 65 60 110 60 freccia 65 10 65 60 onda 180 35 215 35 freccia 215 35 250 35 freccia 215 35 punto %%EndDocument @endspecial 1087 3388 a Fl(Figure)k(2:)-200 b Fn(f)7 b(ig)r Fm(3)162 b Fl(The)28 b(in)n(teraction)f(terms)260 3579 y(fermion)19 b(lines,)j(asso)r(ciated)c(to)i(the)g(\014eld)g(v)-5 b(ariables)19 b(in)h Fk(P)2028 3591 y Fn(v)2068 3579 y Fl(,)h(one)f(in)n(ternal)f(wiggling)g(line)h(for)260 3679 y(eac)n(h)k(endp)r(oin)n(t)i(of)f(t)n(yp)r(e)g Fk(\025)h Fl(follo)n(wing)e Fk(v)29 b Fl(and)c Fj(f)1756 3617 y Ff(P)1843 3637 y Fn(s)1877 3645 y Fb(~)-31 b Fc(v)1843 3704 y Fn(i)p Fm(=1)1955 3679 y Fl([)p Fj(j)p Fk(P)2057 3691 y Fm(~)-36 b Fn(v)2087 3699 y Fc(i)2118 3679 y Fj(j)14 b(\000)g(j)p Fk(P)2313 3691 y Fm(~)-36 b Fn(v)2349 3679 y Fj(j)p Fl(])p Fj(g)p Fk(=)p Fl(2)24 b(in)n(ternal)g(fermion)260 3779 y(lines,)30 b(asso)r(ciated)d(to)j(propagators)c(of)j(scales)f Fk(h)1814 3791 y Fm(~)-36 b Fn(v)1850 3779 y Fl(,)30 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(giving)g(a)g(non)h(v)-5 b(anishing)35 b(con)n(tribution)f(are)260 5323 y(those)i(ha)n(ving)g(the)h(structure)g(of)g(a)f Fg(ladder)k(gr)l(aph)p Fl(,)h(that)c(is)f(a)h(graph)f(with)h(t)n(w)n(o) f(op)r(en)260 5423 y(paths)e(with)i(the)f(same)f(orien)n(tation,)h (connected)g(b)n(y)f Fk(n)2076 5435 y Fm(4)p Fn(;v)2203 5423 y Fl(non)g(in)n(tersecting)g(wiggling)260 5523 y(lines,)e(see)f (Fig.)48 b(3.)g(Let)32 b(us)f(no)n(w)g(supp)r(ose)g(that)g Fk(v)k Fl(satis\014es)c(condition)g(\(b\))h(or)e(\(c\))i(and)260 5622 y(\(4.8\);)f(in)g(this)f(case,)g(it)h(is)f(p)r(ossible)g(to)g (build)h(graphs)e(without)i(lo)r(ops)e(only)h(if)h Fk(n)2871 5634 y Fm(4)p Fn(;v)2991 5622 y Fl(=)c(0)1650 5871 y(10)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 260 508 a Fl(and)33 b Fj(j)p Fk(P)503 520 y Fn(v)543 508 y Fj(j)f Fl(=)h(2,)h(so)f(that)g Fk(n)1138 520 y Fn(v)1210 508 y Fl(=)f Fk(n)1357 520 y Fm(2)p Fn(;v)1482 508 y Fl(=)g(1,)i(whic)n(h)f(is)h(imp)r(ossible)f (if)h Fk(v)h(>)e(v)2716 520 y Fm(0)2753 508 y Fl(.)54 b(Hence)34 b(a)260 608 y(tree)d Fk(\034)40 b Fl(ma)n(y)31 b(giv)n(e)f(a)g(non)h(v)-5 b(anishing)31 b(con)n(tribution)f(only)h(if) g(there)g(is)g(no)g(trivial)f(or)g(non)260 708 y(trivial)j(v)n(ertex)g (satisfying)g(\(b\))h(or)f(\(c\),)j(together)c(with)j(\(4.8\),)g (except)e(the)h(trivial)g(tree)260 807 y(with)28 b Fk(n)499 819 y Fn(v)561 807 y Fl(=)23 b Fk(n)699 819 y Fm(2)p Fn(;v)814 807 y Fl(=)g(1.)889 1130 y Fh(z)931 1142 y Fm(1)889 1753 y Fh(y)939 1765 y Fm(1)1305 1130 y Fh(z)1347 1142 y Fm(2)1305 1753 y Fh(y)1355 1765 y Fm(2)1927 1130 y Fh(z)1969 1142 y Fn(m)p Fe(\000)p Fm(1)1927 1753 y Fh(y)1977 1765 y Fn(m)p Fe(\000)p Fm(1)2425 1130 y Fh(z)2467 1142 y Fn(m)2425 1753 y Fh(y)2475 1765 y Fn(m)557 1841 y @beginspecial @setspecial %%BeginDocument: fig2.ps /fx [ 0.00 1.11 2.22 3.33 4.44 5.56 6.67 7.78 8.89 10.00 11.11 12.22 13.33 14.44 15.56 16.67 17.78 18.89 20.00 21.11 22.22 23.33 24.44 25.56 26.67 27.78 28.89 30.00 31.11 32.22 33.33 34.44 35.56 36.67 37.78 38.89 40.00 41.11 42.22 43.33 44.44 45.56 46.67 47.78 48.89 50.00 51.11 52.22 53.33 54.44 55.56 56.67 57.78 58.89 60.00 61.11 62.22 63.33 64.44 65.56 66.67 67.78 68.89 70.00 71.11 72.22 73.33 74.44 75.56 76.67 77.78 78.89 80.00 81.11 82.22 83.33 84.44 85.56 86.67 87.78 88.89 90.00 91.11 92.22 93.33 94.44 95.56 96.67 97.78 98.89 100.00 ] def /fy [ 0.00 2.07 3.98 5.58 6.75 7.39 7.46 6.96 5.91 4.41 2.57 0.52 -1.56 -3.52 -5.21 -6.50 -7.28 -7.50 -7.14 -6.22 -4.82 -3.05 -1.04 1.04 3.05 4.82 6.22 7.14 7.50 7.28 6.50 5.21 3.52 1.56 -0.52 -2.57 -4.41 -5.91 -6.96 -7.46 -7.39 -6.75 -5.58 -3.98 -2.07 0.00 2.07 3.98 5.58 6.75 7.39 7.46 6.96 5.91 4.41 2.57 0.52 -1.56 -3.52 -5.21 -6.50 -7.28 -7.50 -7.14 -6.22 -4.82 -3.05 -1.04 1.04 3.05 4.82 6.22 7.14 7.50 7.28 6.50 5.21 3.52 1.56 -0.52 -2.57 -4.41 -5.91 -6.96 -7.46 -7.39 -6.75 -5.58 -3.98 -2.07 0.00 ] def /cambio_coordinate{ % st: x1 y1 x2 y2 4 copy exch pop exch sub % x1 dy 6 1 roll exch pop sub % dy x1 y1 dx 4 1 roll translate % sposta l'origine; st: dx dy 2 copy exch atan rotate % rotazione di arctan(dy/dx); st: dx dy 2 exp exch 2 exp add sqrt % sqrt(dx^2 +dy^2) } def /nx fx length 1 sub def /normonda { %st: l 100 div dup scale fx 0 get fy 0 get moveto 1 3 nx { dup dup 1 add exch 2 add 3 1 roll exch dup fx exch get 4 1 roll fy exch get 3 1 roll dup fx exch get 3 1 roll fy exch get exch dup fx exch get exch fy exch get curveto } for } def /onda { gsave % uso: x1 y1 x2 y2 onda cambio_coordinate % st: l normonda stroke grestore } def /freccia { gsave % uso: x1 y1 x2 y2 freccia cambio_coordinate % st: l dup 0 0 moveto 0 lineto % st: l 2 div 0 translate 15 rotate 0 0 moveto -5 0 lineto -30 rotate 0 0 moveto -5 0 lineto stroke grestore } def /punto { gsave % uso: x1 y1 punto 2 0 360 newpath arc fill stroke grestore} def /tlinea { gsave % uso: x1 y1 x2 y2 tlinea moveto [4 4] 2 setdash lineto stroke grestore} def -20 0 translate 65 80 punto 65 20 punto 25 20 65 20 freccia 65 20 115 20 freccia 25 80 65 80 freccia 65 80 115 80 freccia 65 20 65 80 onda 115 80 punto 115 20 punto 115 80 200 80 tlinea 115 20 200 20 tlinea 115 20 115 80 onda 200 80 punto 200 20 punto 200 80 250 80 freccia 200 20 250 20 freccia 200 20 200 80 onda 250 80 punto 250 20 punto 250 80 290 80 freccia 250 20 290 20 freccia 250 20 250 80 onda %%EndDocument @endspecial 1207 2044 a Fl(Figure)k(3:)-200 b Fn(f)7 b(ig)r Fm(2)163 b Fl(A)27 b(ladder)g(graph)384 2252 y(W)-7 b(e)28 b(are)f(no)n(w)g(ready)g(to)g(mo)r(dify)i(our)d(expansion.)37 b(Giv)n(en)27 b(a)g(tree)h Fk(\034)33 b Fj(2)23 b(T)2705 2209 y Fm(\()p Fn(N)6 b Fm(\))2683 2261 y Fn(n)2847 2252 y Fl(and)28 b(the)260 2351 y(sets)i Fj(f)p Fk(P)520 2363 y Fn(v)559 2351 y Fk(;)14 b(v)31 b Fj(2)d Fk(\034)9 b Fj(g)p Fl(,)31 b(let)f(us)h(consider)e(the)h(family)h Fk(F)1904 2363 y Fn(\034)1976 2351 y Fl(of)f(all)g(v)n(ertices)f (di\013eren)n(t)i(from)f Fk(v)3064 2363 y Fm(0)3101 2351 y Fl(,)260 2451 y(whic)n(h)23 b(satisfy)f(condition)h(\(a\),)h(\(4.8\)) e(and)h(the)g(further)g(condition)f(that,)j(if)e Fk(v)j Fj(2)e Fk(F)2851 2463 y Fn(\034)2892 2451 y Fl(,)g(there)260 2551 y(is)33 b(no)g(v)n(ertex)i(~)-45 b Fk(v)36 b(<)c(v)s Fl(,)j(except)e(p)r(ossibly)g Fk(v)1632 2563 y Fm(0)1670 2551 y Fl(,)i(whic)n(h)e(also)f(satis\014es)h(\(a\))g(and)g(\(4.8\).)54 b(W)-7 b(e)260 2650 y(w)n(an)n(t)30 b(to)g(rebuild)h(our)f(expansion)g (\(3.3\),)h(b)n(y)f(summing)h(all)f(terms)h(asso)r(ciated)e(to)i(trees) 260 2750 y(whic)n(h)24 b(are)e(obtained)i(from)f Fk(\034)34 b Fl(b)n(y)24 b(substituting,)h(for)e(eac)n(h)g Fk(v)j Fj(2)e Fk(F)2326 2762 y Fn(\034)2368 2750 y Fl(,)g(the)h(subtree)e (starting)260 2849 y(from)32 b Fk(v)j Fl(with)e(another)e(arbitrary)f (subtree)h(with)i(the)f(same)g(n)n(um)n(b)r(er)g Fk(n)2603 2861 y Fn(v)2674 2849 y Fl(of)g(endp)r(oin)n(ts)260 2961 y(of)j(t)n(yp)r(e)g Fk(\025)p Fl(;)40 b(w)n(e)34 b(shall)h(call)g Fj(T)1224 2917 y Fm(\()p Fn(N)6 b Fm(\))1203 2970 y Fn(v)1374 2961 y Fl(the)36 b(family)f(of)g(all)g(suc)n(h)f(subtrees)h(starting)f (from)h Fk(v)s Fl(.)260 3060 y(Moreo)n(v)n(er,)28 b(w)n(e)h(sum)h(also) f(o)n(v)n(er)f(all)i(the)g(c)n(hoices)f(of)h(the)g(sets)g Fk(P)2310 3072 y Fm(~)-36 b Fn(v)2346 3060 y Fl(,)34 b(~)-45 b Fk(v)30 b(>)c(v)s Fl(,)31 b(and)f(w)n(e)g(lea)n(v)n(e)260 3160 y(unc)n(hanged)d(the)i(set)g Fk(P)1001 3172 y Fn(v)1040 3160 y Fl(,)g(again)e(for)h(eac)n(h)g Fk(v)f Fj(2)e Fk(F)1830 3172 y Fn(\034)1872 3160 y Fl(.)39 b(In)29 b(order)e(to)h(describ)r(e)g (the)h(result)f(of)260 3259 y(suc)n(h)f(op)r(eration,)g(w)n(e)g(note)g (that,)h(if)g Fk(v)f Fj(2)c Fk(F)1625 3271 y Fn(\034)1695 3259 y Fl(and)k Fk(m)c Fl(=)g Fk(n)2090 3271 y Fn(v)2129 3259 y Fl(,)120 3469 y Fm(4)p Fn(:)p Fm(9)367 3390 y Ff(Y)286 3590 y Fn(\034)324 3573 y Fa(\003)358 3590 y Fe(2T)456 3559 y Fb(\()p Fc(N)5 b Fb(\))441 3601 y Fc(v)568 3377 y Ff(h)716 3390 y(Y)718 3556 y Fb(~)-31 b Fc(v)q(>v)634 3606 y Fb(~)g Fc(v)24 b Fm(not)e(e.p.)930 3390 y Ff(X)951 3568 y Fn(P)996 3576 y Fb(~)-31 b Fc(v)1064 3377 y Ff(i)1212 3390 y(Y)1214 3558 y Fb(~)g Fc(v)r Fa(\025)p Fc(v)1130 3613 y Fb(~)g Fc(v)24 b Fm(not)e(e.p.)1467 3413 y Fl(1)p 1437 3450 102 4 v 1437 3526 a Fk(s)1479 3538 y Fm(~)-36 b Fn(v)1515 3526 y Fl(!)1548 3469 y Fj(E)1599 3434 y Fn(T)1592 3489 y(h)1634 3497 y Fb(~)-31 b Fc(v)1684 3469 y Fl(\()q Fk( )s Fl(\()p Fk(P)1862 3484 y Fm(~)-36 b Fn(v)1894 3467 y Fb(1)1931 3469 y Fj(n)p Fk(Q)2042 3484 y Fm(~)g Fn(v)2074 3467 y Fb(1)2110 3469 y Fl(\))p Fk(;)14 b(:)g(:)g(:)g(;)g( )s Fl(\()p Fk(P)2472 3481 y Fm(~)-36 b Fn(v)2504 3461 y Fc(s)2535 3475 y Fb(~)-31 b Fc(v)2577 3469 y Fj(n)p Fk(Q)2688 3481 y Fm(~)-36 b Fn(v)2720 3461 y Fc(s)2751 3475 y Fb(~)-31 b Fc(v)2791 3469 y Fl(\)\))23 b(=)10 b(\(4.9\))323 3719 y(1)p 296 3756 96 4 v 296 3832 a Fk(m)p Fl(!)402 3775 y Fj(E)453 3741 y Fn(T)446 3798 y Fm([)p Fn(h)504 3806 y Fc(v)539 3798 y Fn(;N)c Fm(])654 3683 y Ff(\020)704 3775 y Fk( )761 3741 y Fm(+)758 3796 y Fd(z)791 3804 y Fb(1)828 3775 y Fk( )885 3741 y Fm(+)882 3796 y Fd(y)922 3804 y Fb(1)958 3775 y Fl(;)14 b Fk( )1052 3741 y Fm(+)1049 3796 y Fd(z)1082 3804 y Fb(2)1119 3775 y Fk( )1176 3741 y Fm(+)1173 3796 y Fd(y)1213 3804 y Fb(2)1250 3775 y Fk( )1307 3741 y Fe(\000)1304 3796 y Fd(y)1344 3804 y Fb(2)1380 3775 y Fk( )1437 3741 y Fe(\000)1434 3796 y Fd(z)1467 3804 y Fb(2)1504 3775 y Fl(;)g Fk(:)g(:)g(:)g(;)g( ) 1746 3741 y Fm(+)1743 3796 y Fd(z)1776 3804 y Fc(m)p Fa(\000)p Fb(1)1909 3775 y Fk( )1966 3741 y Fm(+)1963 3796 y Fd(y)2003 3804 y Fc(m)p Fa(\000)p Fb(1)2135 3775 y Fk( )2192 3741 y Fe(\000)2189 3796 y Fd(y)2229 3804 y Fc(m)p Fa(\000)p Fb(1)2361 3775 y Fk( )2418 3741 y Fe(\000)2415 3796 y Fd(z)2448 3804 y Fc(m)p Fa(\000)p Fb(1)2581 3775 y Fl(;)g Fk( )2675 3741 y Fe(\000)2672 3796 y Fd(y)2712 3804 y Fc(m)2771 3775 y Fk( )2828 3741 y Fe(\000)2825 3796 y Fd(z)2858 3804 y Fc(m)2918 3683 y Ff(\021)3005 3775 y Fk(;)260 3986 y Fl(where)35 b Fj(E)559 3955 y Fn(T)552 4012 y Fm([)p Fn(h;N)6 b Fm(])746 3986 y Fl(denotes)35 b(the)h(truncated)f(exp)r(ectation)g(with)h(resp)r(ect) f(to)g(the)h(Gaussian)260 4117 y(measure)i(with)i(co)n(v)-5 b(ariance)37 b Fk(G)1274 4132 y Fm([)p Fn(h;N)6 b Fm(])1477 4117 y Fl(=)1584 4054 y Ff(P)1671 4075 y Fn(N)1671 4142 y(h)1710 4125 y Fa(0)1733 4142 y Fm(=)p Fn(h)1841 4117 y Fk(G)1906 4129 y Fn(h)1945 4112 y Fa(0)2011 4117 y Fl(and)39 b(the)h(\014elds)f(in)n(v)n(olv)n(ed)f(in)i(the)260 4216 y(truncated)26 b(exp)r(ectation)g(are)f(those)h(asso)r(ciated)f (to)h(the)h(endp)r(oin)n(ts)f(follo)n(wing)f Fk(v)s Fl(,)i(except)260 4316 y(the)36 b(\014elds)f(asso)r(ciate)g(to)g Fk(P)1151 4328 y Fn(v)1191 4316 y Fl(,)j(that)e(is)f Fk( )1588 4286 y Fe(\000)1585 4336 y Fd(z)1618 4344 y Fb(1)1655 4316 y Fl(,)j Fk( )1773 4286 y Fe(\000)1770 4336 y Fd(y)1810 4344 y Fb(1)1846 4316 y Fl(,)g Fk( )1964 4286 y Fm(+)1961 4336 y Fd(z)1994 4344 y Fc(m)2090 4316 y Fl(and)d Fk( )2316 4286 y Fm(+)2313 4336 y Fd(y)2353 4344 y Fc(m)2412 4316 y Fl(.)61 b(This)36 b(claim)f(easily)260 4416 y(follo)n(ws)f(b)n(y)h (comparing)f(the)i(m)n(ultiscale)f(expansion)g(of)g(log)2243 4349 y Ff(R)2312 4416 y Fk(P)2365 4431 y Fm([)p Fn(h)2423 4439 y Fc(v)2459 4431 y Fn(;N)6 b Fm(])2560 4416 y Fl(\()p Fk(d )s Fl(\))14 b(exp)h Fk(V)k Fl(\()p Fk( )27 b Fl(+)260 4515 y Fk(')p Fl(\))33 b(with)h(the)f(ordinary)e(cum)n(ulan)n(t)h (expansion,)i Fk(P)1900 4530 y Fm([)p Fn(h;N)6 b Fm(])2059 4515 y Fl(\()p Fk(d )s Fl(\))34 b(b)r(eing)f(the)g(measure)f(with)260 4615 y(co)n(v)-5 b(ariance)25 b Fk(G)726 4630 y Fm([)p Fn(h;N)6 b Fm(])886 4615 y Fl(.)384 4739 y(Let)32 b(us)g(no)n(w)f(call) 995 4718 y(~)981 4739 y Fj(T)1047 4696 y Fm(\()p Fn(N)6 b Fm(\))1026 4749 y Fn(n)1194 4739 y Fl(the)32 b(set)f(of)h(trees,)g (whose)f(de\014nition)h(di\013ers)f(from)g(that)h(of)260 4850 y Fj(T)326 4807 y Fm(\()p Fn(N)6 b Fm(\))305 4860 y Fn(n)469 4850 y Fl(for)27 b(the)h(follo)n(wing)e(reasons.)288 5050 y(1\))h(The)h(n)n(um)n(b)r(er)f(of)h(endp)r(oin)n(ts)f(is)h(not)f (\014xed,)h(but)g(is)g(at)f(most)h Fk(n)p Fl(.)287 5149 y(2\))e(Besides)g(the)h(endp)r(oin)n(ts)g(of)f(t)n(yp)r(e)h Fk(\027)32 b Fl(and)27 b Fk(\025)g Fl(\(whose)f(n)n(um)n(b)r(er)h(is)f (still)h(denoted)g(b)n(y)f Fk(n)3087 5161 y Fm(2)260 5249 y Fl(and)k Fk(n)474 5261 y Fm(4)511 5249 y Fl(,)i(resp)r(ectiv)n (ely\),)e(there)h(are)e(also)h Fk(n)1654 5219 y Fe(\003)1654 5270 y Fm(4)1722 5249 y Fl(endp)r(oin)n(ts,)i(to)e(b)r(e)h(called)f(of) g(t)n(yp)r(e)h Fk(\025)2921 5219 y Fe(\003)2990 5249 y Fl(and)260 5349 y Fg(or)l(der)g Fk(n)529 5318 y Fe(\003)529 5369 y Fn(v)595 5349 y Fj(\024)c Fk(n)p Fl(;)k(if)g Fk(F)923 5361 y Fn(\034)995 5349 y Fl(is)f(the)g(set)g(of)g(endp)r(oin)n(ts)g (of)g(t)n(yp)r(e)h Fk(\025)2171 5318 y Fe(\003)2209 5349 y Fl(,)g(the)f(follo)n(wing)f(constrain)n(t)260 5448 y(has)e(to)g(b)r(e)h(satis\014ed:)1033 5548 y Fm(3)p Fn(:)p Fm(2)p Fn(b)121 b Fk(n)1319 5560 y Fm(4)1374 5548 y Fl(+)18 b Fk(n)1507 5560 y Fm(2)1563 5548 y Fl(+)1666 5469 y Ff(X)1646 5647 y Fn(v)r Fe(2)p Fn(F)1768 5655 y Fc(\034)1819 5548 y Fk(n)1869 5514 y Fe(\003)1869 5568 y Fn(v)1932 5548 y Fl(=)k Fk(n)h(:)797 b Fl(\(4.10\))1650 5871 y(11)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 294 508 a Fl(3\))33 b(T)-7 b(o)34 b(eac)n(h)f Fk(v)j Fj(2)e Fk(F)941 520 y Fn(\034)1017 508 y Fl(w)n(e)f(asso)r(ciate)f(a)i(frequency)f(lab)r(el)h Fk(h)2213 520 y Fn(v)2285 508 y Fj(2)g Fl([1)p Fk(;)14 b(N)9 b Fl(],)35 b(satisfying)e(the)260 608 y(constrain)n(t)26 b(that)i Fk(h)878 620 y Fn(v)941 608 y Fl(=)22 b Fk(h)1076 620 y Fn(v)1111 604 y Fa(0)1156 608 y Fl(+)c(1,)28 b(if)g Fk(v)1451 578 y Fe(0)1502 608 y Fl(is)f(the)h(higher)f(v)n(ertex)g (preceding)g Fk(v)k Fl(in)c(the)h(tree.)293 708 y(4\))33 b(T)-7 b(o)32 b(eac)n(h)g Fk(v)j Fj(2)e Fk(F)935 720 y Fn(\034)1010 708 y Fl(w)n(e)f(asso)r(ciate)g Fk(n)1542 677 y Fe(\003)1542 728 y Fn(v)1614 708 y Fl(in)n(teraction)g(terms)g (of)h(t)n(yp)r(e)g Fk(\025)p Fl(;)k(w)n(e)32 b(shall)h(call)260 807 y Fh(x)p 260 820 51 4 v -30 x Fe(\003)310 828 y Fn(v)383 807 y Fl(the)g(set)h(of)f(corresp)r(onding)e(2)p Fk(n)1398 777 y Fe(\003)1398 828 y Fn(v)1470 807 y Fl(space-time)h(p)r(oin)n(ts.) 54 b(Moreo)n(v)n(er,)32 b(if)i(the)f(set)g Fh(x)p 2945 820 V 2996 777 a Fe(\003)2996 828 y Fn(v)3068 807 y Fl(is)260 907 y(written)i(as)g(in)h(the)f(second)g(line)h(of)f(\(4.9\))g(and)g (Fig.)60 b(3,)37 b(with)f Fk(m)f Fl(=)h Fk(n)2601 877 y Fe(\003)2601 927 y Fn(v)2640 907 y Fl(,)h(and)f(w)n(e)f(put)260 1006 y Fh(x)p 260 1019 V 21 x Fn(m)396 1006 y Fl(=)23 b Fh(x)p 484 1019 V -30 x Fe(\003)534 1027 y Fn(v)574 1006 y Fl(,)28 b Fk(h)22 b Fl(=)h Fk(h)831 1018 y Fn(v)870 1006 y Fl(,)28 b(w)n(e)f(shall)h(de\014ne)f(the)h(function)48 1242 y Fm(4)p Fn(:)p Fm(10)287 b Fl(~)-45 b Fk(v)494 1199 y Fm(\()p Fn(N)6 b Fm(\))491 1267 y Fn(m;h)613 1242 y Fl(\()p Fh(x)p 645 1255 V 21 x Fn(m)758 1242 y Fl(\))24 b(=)938 1186 y(1)p 911 1223 96 4 v 911 1299 a Fk(m)p Fl(!)1055 1139 y Fn(m)1032 1163 y Ff(Y)1031 1340 y Fn(i)p Fm(=1)1152 1242 y Fk(v)s Fl(\()p Fh(y)1277 1254 y Fn(i)1324 1242 y Fj(\000)18 b Fh(z)1449 1254 y Fn(i)1477 1242 y Fl(\))h Fj(\001)1361 b Fl(\(4.11\))451 1477 y Fj(\001E)525 1443 y Fn(T)518 1500 y Fm([)p Fn(h)576 1508 y Fc(v)611 1500 y Fn(;N)6 b Fm(])726 1385 y Ff(\020)776 1477 y Fk( )833 1443 y Fm(+)830 1498 y Fd(z)863 1506 y Fb(1)900 1477 y Fk( )957 1443 y Fm(+)954 1498 y Fd(y)994 1506 y Fb(1)1030 1477 y Fl(;)14 b Fk( )1124 1443 y Fm(+)1121 1498 y Fd(z)1154 1506 y Fb(2)1191 1477 y Fk( )1248 1443 y Fm(+)1245 1498 y Fd(y)1285 1506 y Fb(2)1321 1477 y Fk( )1378 1443 y Fe(\000)1375 1498 y Fd(y)1415 1506 y Fb(2)1452 1477 y Fk( )1509 1443 y Fe(\000)1506 1498 y Fd(z)1539 1506 y Fb(2)1576 1477 y Fl(;)g Fk(:)g(:)g(:)f(;)h( )1817 1443 y Fm(+)1814 1498 y Fd(z)1847 1506 y Fc(m)p Fa(\000)p Fb(1)1980 1477 y Fk( )2037 1443 y Fm(+)2034 1498 y Fd(y)2074 1506 y Fc(m)p Fa(\000)p Fb(1)2207 1477 y Fk( )2264 1443 y Fe(\000)2261 1498 y Fd(y)2301 1506 y Fc(m)p Fa(\000)p Fb(1)2433 1477 y Fk( )2490 1443 y Fe(\000)2487 1498 y Fd(z)2520 1506 y Fc(m)p Fa(\000)p Fb(1)2653 1477 y Fl(;)g Fk( )2747 1443 y Fe(\000)2744 1498 y Fd(y)2784 1506 y Fc(m)2843 1477 y Fk( )2900 1443 y Fe(\000)2897 1498 y Fd(z)2930 1506 y Fc(m)2990 1385 y Ff(\021)3076 1477 y Fk(:)384 1746 y Fl(It)28 b(follo)n(ws)f(that)h(w)n(e)f(can)g (substitute)i(\(3.3\))e(with)h(a)f(similar)g(expansion:)337 1958 y Fm(3)p Fn(:)p Fm(2)p Fn(a)114 b Fk(V)640 1915 y Fm(\()p Fn(N)6 b Fm(\))621 1983 y(2)p Fn(k)q(;n)756 1958 y Fl(\()p Fk(')p Fl(\))24 b(=)1043 1879 y Ff(X)986 2079 y Fn(\034)7 b Fe(2)1080 2064 y Fm(~)1069 2079 y Fe(T)1121 2049 y Fb(\()p Fc(N)e Fb(\))1107 2090 y Fc(n)1308 1879 y Ff(X)1233 2061 y Fe(j)p Fn(P)1295 2069 y Fc(v)1325 2081 y Fb(0)1362 2061 y Fe(j)p Fm(=2)p Fn(k)1516 1845 y Ff(Z)1613 1958 y Fk(d)p Fh(x)p 1656 1971 51 4 v 24 w Fk(')p Fl(\()p Fk(P)1869 1970 y Fn(v)1902 1978 y Fb(0)1939 1958 y Fl(\))2019 1937 y(~)1995 1958 y Fk(W)2085 1924 y Fm(\()p Fn(N)h Fm(\))2199 1958 y Fl(\()p Fk(\034)f(;)14 b(P)2362 1970 y Fn(v)2395 1978 y Fb(0)2433 1958 y Fk(;)g Fh(x)p 2470 1971 V Fl(\))23 b Fk(;)314 b Fl(\(4.12\))513 2272 y Fm(3)p Fn(:)p Fm(3)p Fn(a)774 2251 y Fl(~)750 2272 y Fk(W)840 2238 y Fm(\()p Fn(N)6 b Fm(\))954 2272 y Fl(\()p Fk(\034)f(;)14 b(P)1117 2284 y Fn(v)1150 2292 y Fb(0)1188 2272 y Fk(;)g Fh(x)p 1225 2285 V Fl(\))23 b(=)1418 2180 y Ff(h)1566 2193 y(Y)1549 2356 y Fc(v)q(>v)1655 2368 y Fb(0)1481 2414 y Fc(v)h Fm(not)e(e.p.)1781 2193 y Ff(X)1802 2371 y Fn(P)1844 2379 y Fc(v)1900 2205 y Fe(\003)1939 2180 y Ff(i)2002 2251 y Fl(~)1978 2272 y Fk(W)2068 2238 y Fm(\()p Fn(N)6 b Fm(\))2183 2272 y Fl(\()p Fk(\034)f(;)14 b Fj(f)p Fk(P)2388 2284 y Fn(v)2427 2272 y Fj(g)p Fk(;)g Fh(x)p 2506 2285 V Fl(\))23 b Fk(;)278 b Fl(\(4.13\))260 2553 y(where)506 2491 y Ff(P)594 2578 y Fn(P)636 2586 y Fc(v)676 2511 y Fe(\003)748 2553 y Fl(means)33 b(that)h(the)g(sum)g(is)g(constrained)e(b)n(y)i(the)g (condition)g Fj(j)p Fk(P)2782 2565 y Fn(v)2821 2553 y Fj(j)g(\025)f Fl(6,)i(if)260 2653 y Fk(n)310 2665 y Fn(v)372 2653 y Fk(<)23 b(n)510 2622 y Fe(\003)510 2676 y Fn(h)549 2684 y Fc(v)588 2653 y Fl(,)28 b(and)24 2900 y Fm(3)p Fn(:)p Fm(4)p Fn(a)450 2879 y Fl(~)426 2900 y Fk(W)516 2866 y Fm(\()p Fn(N)6 b Fm(\))631 2900 y Fl(\()p Fk(\034)f(;)14 b Fj(f)p Fk(P)836 2912 y Fn(v)875 2900 y Fj(g)p Fk(;)g Fh(x)p 954 2913 V Fl(\))23 b(=)g(\()p Fj(\000)p Fk(\027)5 b Fl(\))1322 2866 y Fn(n)1363 2874 y Fb(2)1431 2796 y Fn(n)1472 2804 y Fb(4)1415 2822 y Ff(Y)1414 2998 y Fn(i)p Fm(=1)1521 2900 y Fl([)p Fj(\000)p Fk(\025v)s Fl(\()p Fh(x)1782 2912 y Fm(2)p Fn(i)p Fe(\000)p Fm(1)1948 2900 y Fj(\000)18 b Fh(x)2081 2912 y Fm(2)p Fn(i)2142 2900 y Fl(\)])2238 2822 y Ff(Y)2211 3000 y Fn(v)r Fe(2)p Fn(F)2333 3008 y Fc(\034)2384 2808 y Ff(h)2423 2900 y Fl(\()p Fj(\000)p Fk(\025)p Fl(\))2600 2866 y Fn(n)2641 2841 y Fa(\003)2641 2883 y Fc(v)2685 2900 y Fl(~)-45 b Fk(v)2725 2857 y Fm(\()p Fn(N)6 b Fm(\))2722 2926 y Fn(n)2763 2909 y Fa(\003)2763 2942 y Fc(v)2799 2926 y Fn(;h)2858 2934 y Fc(v)2897 2900 y Fl(\()p Fh(x)p 2929 2913 V 2980 2866 a Fe(\003)2980 2921 y Fn(v)3019 2900 y Fl(\))3051 2808 y Ff(i)3109 2900 y Fj(\001)426 3195 y(\001)463 3053 y Ff(")598 3116 y(Y)511 3290 y Fn(v)25 b Fm(not)d(e.p.)844 3139 y Fl(1)p 815 3176 102 4 v 815 3252 a Fk(s)854 3264 y 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(o)g(couples)g(of)h(\014elds)f(b)r(elong)h(to)f(t)n(w)n(o)260 3744 y(di\013eren)n(t)e(in)n(teraction)f(terms)h(of)g(t)n(yp)r(e)h Fk(\025)p Fl(,)g(among)e(those)h(asso)r(ciated)f(to)h Fk(v)s Fl(,)h(see)f(item)g(4\))260 3844 y(ab)r(o)n(v)n(e.)35 b(F)-7 b(or)27 b(the)h(other)f(v)n(ertices,)g Fk(P)1413 3856 y Fn(v)1480 3844 y Fl(is)h(de\014ned)g(as)f(b)r(efore.)384 3944 y(It)d(is)g(easy)f(to)g(see)g(that)1156 3923 y(~)1132 3944 y Fk(W)1222 3914 y Fm(\()p Fn(N)6 b Fm(\))1337 3944 y Fl(\()p Fk(\034)f(;)14 b Fj(f)p Fk(P)1542 3956 y Fn(v)1582 3944 y Fj(g)p Fk(;)g Fh(x)p 1661 3957 51 4 v -1 w Fl(\))24 b(satis\014es)f(a)g(b)r(ound)h(similar)f(to)h(\(4.5\).)35 b(The)260 4043 y(only)25 b(di\013erence)h(is)g(that)g(no)n(w)f(w)n(e)g (do)h(not)g(ha)n(v)n(e)e(a)h(spanning)h(tree)f(and)h(a)f(corresp)r (onding)260 4143 y(deca)n(ying)k(factor)h(to)g(p)r(erform)g(the)h(in)n (tegration)e(inside)h(the)h(sets)f Fh(x)p 2386 4156 V -30 x Fe(\003)2436 4164 y Fn(v)2506 4143 y Fl(asso)r(ciated)g(to)g(the) 260 4243 y(endp)r(oin)n(ts)h(of)h(t)n(yp)r(e)f Fk(\025)979 4212 y Fe(\003)1018 4243 y Fl(.)48 b(Ho)n(w)n(ev)n(er,)31 b(if)h(w)n(e)f(shrink)g(ev)n(ery)f(set)h Fh(x)p 2270 4256 V 2321 4212 a Fe(\003)2321 4263 y Fn(v)2391 4243 y Fl(to)h(a)f(p)r(oin)n(t,)h(w)n(e)f(get)h(a)260 4342 y(spanning)26 b(tree)h(b)n(y)g(c)n(ho)r(osing,)e(as)i(b)r(efore,)g(a)f (tree)h(graph)f Fk(T)2142 4354 y Fn(v)2208 4342 y Fl(or)g(the)h(in)n (teraction)f(line)i(for)260 4442 y(the)g(other)f(v)n(ertices;)g(if)h(w) n(e)g(use)g(this)g(spanning)f(tree)g(to)h(p)r(erform)f(in)i(the)f (usual)f(w)n(a)n(y)g(the)260 4541 y(in)n(tegrations)21 b(o)n(v)n(er)g(the)j(di\013erences)e(of)h(co)r(ordinates)f(asso)r (ciated)f(to)i(its)g(lines,)h(w)n(e)f(are)e(left)260 4641 y(with)32 b(the)g(in)n(tegrations)e(o)n(v)n(er)g(the)i(sets)f Fh(x)p 1554 4654 V 1605 4611 a Fe(\003)1605 4662 y Fn(v)1644 4641 y Fl(,)i Fk(v)g Fj(2)d Fk(F)1911 4653 y Fn(\034)1953 4641 y Fl(,)j(with)f(the)g(condition)f(that)h(one)f(of)260 4741 y(the)e(space-time)g(p)r(oin)n(ts)g(from)g(whic)n(h)g(the)h(four)f (external)f(lines)i Fk(P)2392 4753 y Fn(v)2461 4741 y Fl(emerge)e(is)h(\014xed,)h(in)260 4840 y(eac)n(h)g(set.)49 b(It)32 b(follo)n(ws,)f(b)n(y)g(observing)f(that,)j(if)e Fk(v)i Fj(2)d Fk(F)2011 4852 y Fn(\034)2053 4840 y Fl(,)i(the)g(set)f Fk(P)2441 4852 y Fn(v)2513 4840 y Fl(can)g(b)r(e)g(c)n(hosen)g(in)260 4940 y Fk(n)310 4910 y Fe(\003)310 4960 y Fn(v)349 4940 y Fl(\()p Fk(n)431 4910 y Fe(\003)431 4960 y Fn(v)491 4940 y Fj(\000)20 b Fl(1\))31 b(di\013eren)n(t)f(w)n(a)n(ys,)g(and)g(b) n(y)h(using)f(the)h(constrain)n(t)e(\(4.10\),)i(that)g(w)n(e)f(ha)n(v)n (e)f(to)260 5056 y(m)n(ultiply)f(the)g(r.h.s.)36 b(of)28 b(\(4.5\))f(b)n(y)1361 4993 y Ff(Q)1439 5081 y Fn(v)r Fe(2)p Fn(F)1561 5089 y Fc(\034)1616 5056 y Fj(jj)s Fl(~)-45 b Fk(v)1705 5013 y Fm(\()p Fn(N)6 b Fm(\))1702 5081 y Fn(n)1743 5064 y Fa(\003)1743 5097 y Fc(v)1779 5081 y Fn(;h)1838 5089 y Fc(v)1878 5056 y Fj(jj)p Fl(,)28 b(ha)n(ving)e (de\014ned)680 5288 y Fm(4)p Fn(:)p Fm(11)117 b Fj(jj)s Fl(~)-45 b Fk(v)1005 5245 y Fm(\()p Fn(N)6 b Fm(\))1002 5313 y Fn(m;h)1124 5288 y Fj(jj)23 b Fl(=)g(max)1346 5341 y Fn(i)1449 5288 y Fl(sup)1474 5358 y Fd(x)1514 5338 y Fa(\003)1514 5379 y Fc(i)1588 5175 y Ff(Z)1685 5288 y Fk(d)p Fl(\()p Fh(x)p 1760 5301 V 1811 5309 a Fn(m)1874 5288 y Fj(n)p Fh(x)1966 5254 y Fe(\003)1966 5309 y Fn(i)2004 5288 y Fl(\))p Fj(j)s Fl(~)-45 b Fk(v)2102 5245 y Fm(\()p Fn(N)6 b Fm(\))2099 5313 y Fn(m;h)2221 5288 y Fl(\()p Fh(x)p 2253 5301 V 2304 5309 a Fn(m)2367 5288 y Fl(\))p Fj(j)23 b Fk(;)444 b Fl(\(4.15\))260 5523 y(where)38 b Fh(x)561 5492 y Fe(\003)561 5544 y Fn(i)638 5523 y Fl(is)h(one)f(of)g(the)h(four)f(p)r(oin)n(ts)h(from)f(whic)n(h)g (the)h(four)g(external)e(lines)i(of)f Fk(P)3084 5535 y Fn(v)260 5622 y Fl(emerge.)1650 5871 y(12)p eop end %%Page: 13 13 TeXDict begin 13 12 bop 384 508 a Fl(On)41 b(the)g(other)f(hand,)j(the) e(graph)f(expansion)f(of)i(\(4.9\))f(giv)n(es)f(rise)h(to)g(a)h(sum)f (of)260 608 y(2)302 578 y Fn(m)p Fe(\000)p Fm(2)449 608 y Fl(\()p Fk(m)15 b Fj(\000)f Fl(2\)!)25 b(ladder)g(graphs)f(with)i (propagator)d Fk(G)1963 623 y Fm([)p Fn(h;N)6 b Fm(])2123 608 y Fl(;)26 b(hence,)g(b)n(y)f(using)h(\(4.11\))e(and)260 708 y(\(1.4\),)j(w)n(e)g(get)202 955 y Fm(4)p Fn(:)p Fm(12)283 b Fj(jj)s Fl(~)-45 b Fk(v)693 912 y Fm(\()p Fn(N)6 b Fm(\))690 980 y Fn(m;h)812 955 y Fj(jj)24 b(\024)e Fl(2)1011 921 y Fn(m)1088 955 y Fl(sup)1135 1023 y Fn(j)1227 842 y Ff(Z)1323 863 y(h)1463 851 y Fn(m)1440 876 y Ff(Y)1376 1055 y Fn(i)p Fm(=1)p Fn(;i)p Fe(6)p Fm(=)p Fn(j)1623 955 y Fk(dz)1705 967 y Fn(i;)p Fm(0)1785 955 y Fk(d)o(~)-41 b(z)1867 967 y Fn(i)1895 863 y Ff(i)23 b(h)2034 851 y Fn(m)2011 876 y Ff(Y)2010 1053 y Fn(i)p Fm(=1)2131 955 y Fk(d)o(~)-41 b(y)2215 967 y Fn(i)2243 863 y Ff(ih)2359 851 y Fn(m)2336 876 y Ff(Y)2335 1053 y Fn(i)p Fm(=1)2456 955 y Fj(j)s Fl(\026)c Fk(v)t Fl(\()o Fk(~)k(y)2596 967 y Fn(i)2642 955 y Fj(\000)16 b Fk(~)-40 b(z)2764 967 y Fn(i)2791 955 y Fl(\))p Fj(j)2846 863 y Ff(i)2904 955 y Fj(\001)604 1257 y(\001)627 1165 y Ff(h)681 1153 y Fn(m)p Fe(\000)p Fm(1)699 1178 y Ff(Y)699 1355 y Fn(i)p Fm(=1)838 1257 y Fj(j)p Fk(G)926 1272 y Fm([)p Fn(h;N)6 b Fm(])1086 1257 y Fl(\()p Fh(z)1160 1269 y Fn(i)p Fm(+1)1291 1257 y Fj(\000)18 b Fh(z)1416 1269 y Fn(i)1444 1257 y Fl(\))p Fj(jj)p Fk(G)1587 1272 y Fm([)p Fn(h;N)6 b Fm(])1747 1257 y Fl(\()p Fk(z)1818 1269 y Fn(i)p Fm(+1)p Fn(;)p Fm(0)2000 1257 y Fj(\000)18 b Fk(z)2122 1269 y Fn(i;)p Fm(0)2202 1257 y Fk(;)13 b(~)-41 b(y)2280 1269 y Fn(i)p Fm(+1)2410 1257 y Fj(\000)18 b Fk(~)-42 b(y)2534 1269 y Fn(i)2562 1257 y Fl(\))p Fj(j)2617 1165 y Ff(i)2679 1257 y Fk(:)210 b Fl(\(4.16\))260 1515 y(By)27 b(using)g(\(2.15\))g (and)h(\(2.13\),)f(w)n(e)g(can)g(easily)g(see)g(that,)h(uniformly)f(in) h Fk(N)9 b Fl(,)471 1739 y Fm(4)p Fn(:)p Fm(13)721 1626 y Ff(Z)818 1739 y Fk(d)o(~)-41 b(x)p Fj(j)p Fk(G)996 1754 y Fm([)p Fn(h;N)6 b Fm(])1156 1739 y Fl(\()p Fk(x)1235 1751 y Fm(0)1273 1739 y Fk(;)13 b(~)-41 b(x)p Fl(\))p Fj(j)23 b(\024)g Fk(C)6 b Fl([)p Fk(h)1659 1751 y Fm(0)1697 1739 y Fl(\()p Fk(\015)1777 1704 y Fn(h)1819 1739 y Fj(j)p Fk(x)1889 1751 y Fm(0)1927 1739 y Fj(j)p Fl(\))19 b(+)f Fk(h)2132 1751 y Fm(0)2169 1739 y Fl(\()p Fk(\015)2249 1704 y Fn(h)2292 1739 y Fj(j)p Fk(x)2362 1751 y Fm(0)2418 1739 y Fl(+)g Fk(\014)t Fj(j)p Fl(\)])24 b Fk(:)235 b Fl(\(4.17\))260 1962 y(Therefore,)25 b(if)i(w)n(e)f(in)n(tegrate)g(in)g (\(4.16\))g(\014rst)g(the)h(space)e(co)r(ordinates,)h(b)n(y)g(using)g (a)g(span-)260 2062 y(ning)35 b(tree)h(con)n(taining)e(all)i(fermion)f (lines)g(and)h(one)f(in)n(teraction)g(line,)j(then)e(the)g(time)260 2161 y(co)r(ordinates,)25 b(b)n(y)i(using)g(the)g(fact)g(that)g Fk(h)1587 2173 y Fm(0)1624 2161 y Fl(\()p Fk(\015)1704 2131 y Fn(h)1747 2161 y Fj(j)p Fk(x)1817 2173 y Fm(0)1855 2161 y Fj(j)p Fl(\))g(has)f(supp)r(ort)h(in)g(an)f(in)n(terv)-5 b(al)27 b(of)f(size)260 2261 y Fk(\015)308 2231 y Fe(\000)p Fn(h)402 2261 y Fl(,)i(w)n(e)f(get)798 2372 y Fm(4)p Fn(:)p Fm(14)117 b Fj(jj)s Fl(~)-45 b Fk(v)1123 2329 y Fm(\()p Fn(N)6 b Fm(\))1120 2397 y Fn(m;h)1242 2372 y Fj(jj)23 b(\024)g Fk(C)1464 2338 y Fn(m)1527 2372 y Fk(\015)1575 2338 y Fe(\000)p Fn(h)p Fm(\()p Fn(m)p Fe(\000)p Fm(1\))1866 2372 y Fj(k)s Fl(\026)-45 b Fk(v)r Fj(k)1992 2338 y Fn(m)p Fe(\000)p Fm(1)1992 2393 y Fe(1)2140 2372 y Fj(k)s Fl(\026)g Fk(v)s Fj(k)2267 2384 y Fm(1)2327 2372 y Fk(;)562 b Fl(\(4.18\))260 2520 y(where)27 b Fj(k)s Fl(\026)-45 b Fk(v)s Fj(k)627 2532 y Fe(1)720 2520 y Fl(=)22 b(sup)931 2541 y Fn(~)-33 b(x)988 2520 y Fj(j)s Fl(\026)-45 b Fk(v)s Fl(\()o Fk(~)k(x)q Fl(\))p Fj(j)p Fl(.)384 2620 y(The)28 b(b)r(ound)f(\(4.18\))g(implies)g(that)1535 2599 y(~)1510 2620 y Fk(W)1600 2590 y Fm(\()p Fn(N)6 b Fm(\))1715 2620 y Fl(\()p Fk(\034)f(;)14 b Fj(f)p Fk(P)1920 2632 y Fn(v)1960 2620 y Fj(g)p Fk(;)g Fh(x)p 2039 2633 51 4 v Fl(\))27 b(satis\014es)g(a)f(b)r(ound)i(similar)e(to)260 2720 y(\(4.6\).)35 b(W)-7 b(e)22 b(could)f(indeed)i(ev)n(en)e(impro)n (v)n(e)g(the)h(b)r(ound,)h(b)n(y)f(using)f(the)i Fk(\015)2511 2690 y Fe(\000)p Fn(h)p Fm(\()p Fn(m)p Fe(\000)p Fm(1\))2823 2720 y Fl(factor)e(in)260 2819 y(the)i(r.h.s.)35 b(of)24 b(\(4.18\),)f(but)h(w)n(e)f(do)f(not)i(need)f(this)g(impro)n(v)n(emen)n (t.)35 b(It)23 b(will)g(b)r(e)h(su\016cien)n(t)f(to)260 2919 y(use)k(the)g(fact)g(that)g(no)n(w)g(the)g(set)g Fj(f)p Fk(P)1426 2931 y Fn(v)1465 2919 y Fj(g)g Fl(has)f(the)h(imp)r (ortan)n(t)g(prop)r(ert)n(y)f(that,)h(if)h(a)e(v)n(ertex)260 3019 y Fk(v)31 b Fl(has)c(non)h(p)r(ositiv)n(e)f(dimension)g(\(that)i (is)e(it)h(satis\014es)f(either)h(condition)f(\(a\),)h(\(b\))h(or)e (\(c\))260 3118 y(ab)r(o)n(v)n(e\),)e(then)h Fk(n)786 3130 y Fn(v)849 3118 y Fj(\025)c Fk(n)986 3088 y Fe(\003)986 3142 y Fn(h)1025 3150 y Fc(v)1065 3118 y Fl(,)k(that)g(is)g Fk(n)1424 3130 y Fn(v)1489 3118 y Fl(has)g(to)g(b)r(e)g(v)n(ery)f (large,)f(if)j Fk(h)2375 3130 y Fn(v)2440 3118 y Fl(is)f(large.)35 b(T)-7 b(o)25 b(exploit)260 3218 y(this)h(prop)r(ert)n(y)f(in)h(an)f (e\016cien)n(t)h(w)n(a)n(y)-7 b(,)25 b(let)h(us)g(select,)g(giv)n(en)f Fk(\034)9 b Fl(,)27 b(the)f(family)g Fk(R)2691 3230 y Fn(\034)2759 3218 y Fl(of)f(v)n(ertices)260 3317 y(suc)n(h)30 b(that)i Fk(n)684 3329 y Fn(v)751 3317 y Fj(\025)d Fk(n)895 3287 y Fe(\003)895 3341 y Fn(h)934 3349 y Fc(v)1004 3317 y Fl(and)i(there)g(is)f(no)h(v)n(ertex)i(~)-45 b Fk(v)32 b(>)c(v)34 b Fl(with)d(the)h(same)e(prop)r(ert)n(y)-7 b(.)46 b(W)-7 b(e)260 3417 y(note)27 b(that,)h(for)f(an)n(y)g Fk(")c(>)g Fl(0,)695 3613 y Fm(4)p Fn(:)p Fm(15)117 b Fl(1)23 b(=)1114 3535 y Ff(Y)1084 3713 y Fn(v)r Fe(2)p Fn(R)1214 3721 y Fc(\034)1265 3613 y Fk(e)1304 3579 y Fe(\000)p Fn("n)1428 3554 y Fa(\003)1428 3596 y Fc(h)1462 3604 y(v)1506 3613 y Fk(e)1545 3579 y Fn("n)1617 3554 y Fa(\003)1617 3596 y Fc(h)1651 3604 y(v)1718 3613 y Fj(\024)g Fk(e)1845 3579 y Fn("n)1966 3535 y Ff(Y)1935 3713 y Fn(v)r Fe(2)p Fn(R)2065 3721 y Fc(\034)2116 3613 y Fk(e)2155 3579 y Fe(\000)2217 3552 y Fc("\014)p 2217 3567 63 4 v 2234 3599 a Fb(2)2290 3579 y Fn(\015)2329 3554 y Fc(h)2363 3562 y(v)2429 3613 y Fk(;)460 b Fl(\(4.19\))260 3886 y(so)27 b(that,)h(if)g(w)n(e)f(c)n(ho)r(ose)f Fk(")d Fl(=)g(2)p Fk(\014)1269 3856 y Fe(\000)p Fm(1)1358 3886 y Fl(,)28 b(w)n(e)f(get)540 4079 y Fm(4)p Fn(:)p Fm(16)117 b Fl(1)23 b Fj(\024)f Fk(e)967 4045 y Fm(2)p Fn(\014)1041 4020 y Fa(\000)p Fb(1)1118 4045 y Fn(n)1163 4079 y Fl(\(4)p Fk(e)1276 4045 y Fe(\000)p Fm(2)1365 4079 y Fl(\))1397 4045 y Fe(j)p Fn(R)1467 4053 y Fc(\034)1504 4045 y Fe(j)1573 4000 y Ff(Y)1542 4179 y Fn(v)r Fe(2)p Fn(R)1672 4187 y Fc(\034)1723 4079 y Fk(\015)1771 4045 y Fe(\000)p Fm(2)p Fn(h)1895 4053 y Fc(v)1957 4079 y Fj(\024)h Fk(C)2110 4045 y Fn(n)2200 4000 y Ff(Y)2169 4179 y Fn(v)r Fe(2)p Fn(R)2299 4187 y Fc(\034)2350 4079 y Fk(\015)2398 4045 y Fe(\000)p Fm(2)p Fn(h)2522 4053 y Fc(v)2585 4079 y Fk(:)304 b Fl(\(4.20\))384 4342 y(By)28 b(using)f(this)h(b)r(ound,)g(w) n(e)f(get)g(\014nally)482 4561 y Fm(4)p Fn(:)p Fm(6)p Fn(a)280 b Fl(\()p Fk(\014)t(L)1024 4527 y Fm(2)1062 4561 y Fl(\))1094 4527 y Fe(\000)p Fm(1)1197 4448 y Ff(Z)1294 4561 y Fk(d)p Fh(x)p 1337 4574 51 4 v 1 w Fj(j)1435 4540 y Fl(~)1411 4561 y Fk(W)1501 4527 y Fm(\()p Fn(N)6 b Fm(\))1615 4561 y Fl(\()p Fk(\034)f(;)14 b Fj(f)p Fk(P)1820 4573 y Fn(v)1860 4561 y Fj(g)p Fk(;)g Fh(x)p 1939 4574 V Fl(\))p Fj(j)23 b(\024)780 b Fl(\(4.21\))884 4761 y Fj(\024)23 b Fl(\()p Fk(C)6 b(\025)1117 4773 y Fm(0)1155 4761 y Fl(\))1187 4727 y Fn(n)1233 4669 y Ff(h)1317 4682 y(Y)1286 4861 y Fn(v)r Fe(2)p Fn(R)1416 4869 y Fc(\034)1467 4761 y Fk(\015)1515 4727 y Fe(\000)p Fm(2)p Fn(h)1639 4735 y Fc(v)1678 4669 y Ff(i)1818 4682 y(Y)1731 4857 y Fn(v)25 b Fm(not)d(e.p.)2025 4761 y Fk(\015)2073 4727 y Fe(\000)p Fn(d)p Fm(\()p Fn(n)2227 4735 y Fb(4)p Fc(;v)2309 4727 y Fn(;n)2370 4735 y Fb(2)p Fc(;v)2452 4727 y Fn(;)p Fe(j)p Fn(P)2534 4735 y Fc(v)2570 4727 y Fe(j)p Fm(\))2642 4761 y Fk(:)260 5024 y Fl(Let)35 b(us)g(call)g Fk(\034)735 4994 y Fe(\003)809 5024 y Fl(the)g(minimal)g(subtree)g(of)g Fk(\034)45 b Fl(whic)n(h)35 b(con)n(tains)f Fk(v)2389 5036 y Fm(0)2461 5024 y Fl(and)h(the)h(set)f Fk(R)2981 5036 y Fn(\034)3057 5024 y Fl(of)260 5124 y(v)n(ertices.)f(Our)23 b(de\014nitions)h(imply)g(that)f(all)h(the)f(v)n(ertices)g(of)g Fk(\034)34 b Fl(whic)n(h)23 b(ha)n(v)n(e)f(non)i(p)r(ositiv)n(e)260 5224 y(dimension)33 b(b)r(elong)g(to)g Fk(\034)1081 5194 y Fe(\003)1154 5224 y Fl(and)g(that)h(the)g(elemen)n(ts)f(of)g Fk(R)2164 5236 y Fn(\034)2239 5224 y Fl(are)g(the)g(endp)r(oin)n(ts)h (of)f Fk(\034)3062 5194 y Fe(\003)3101 5224 y Fl(.)260 5323 y(Hence,)24 b(w)n(e)e(can)g(distribute)h(the)g Fk(\015)1350 5293 y Fe(\000)p Fm(2)p Fn(h)1474 5301 y Fc(v)1536 5323 y Fl(factors)e(in)i(the)g(r.h.s.)35 b(of)22 b(\(4.21\))g(along)f(the)i (paths)260 5423 y(connecting)28 b(the)g(v)n(ertices)g(in)g Fk(R)1281 5435 y Fn(\034)1351 5423 y Fl(to)h Fk(v)1494 5435 y Fm(0)1531 5423 y Fl(,)g(so)e(to)i(increase)e(the)i(dimension)f (of)g(all)g(v)n(ertices)260 5523 y(of)e Fk(\034)398 5492 y Fe(\003)463 5523 y Fl(of)g(at)g(least)f(t)n(w)n(o)g(units.)37 b(Since)26 b Fk(d)p Fl(\()p Fk(n)1583 5535 y Fm(4)1621 5523 y Fk(;)14 b(n)1708 5535 y Fm(2)1745 5523 y Fk(;)g(p)p Fl(\))26 b(is)g Fj(\025)d(\000)p Fl(1)i(and)h(gro)n(ws)e(linearly)h (with)h Fk(p)p Fl(,)260 5622 y(this)i(is)f(su\016cien)n(t)h(to)f(con)n (trol)g(the)h(sum)f(o)n(v)n(er)f(the)i(set)g Fj(f)p Fk(P)2099 5634 y Fn(v)2138 5622 y Fj(g)p Fl(,)f(see)g(remark)g(after)g(\(4.7\).) 1650 5871 y(13)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 384 508 a Fl(In)26 b(order)d(to)i(complete)g (the)g(pro)r(of)g(of)g(Theorem)f(1.1,)g(w)n(e)h(de\014ne)g(the)h (constan)n(ts)e Fk(p)3001 520 y Fn(n;m)260 627 y Fl(and)18 b(the)h(distributions)f Fk(W)1100 639 y Fm(2)p Fn(k)q(;n;m)1314 627 y Fl(\()p Fh(x)p 1346 640 51 4 v Fk(;)c Fh(y)p 1433 657 52 4 v 2 w Fl(\))19 b(in)f(a)g(w)n(a)n(y)g(analogous)e(to)i Fk(p)2351 584 y Fm(\()p Fn(N)6 b Fm(\))2351 637 y Fn(n;m)2493 627 y Fl(and)18 b Fk(W)2735 584 y Fm(\()p Fn(N)6 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Fl(\()p Fh(x)p 3069 1791 V 3120 1748 a Fe(\003)3120 1799 y Fn(v)3159 1778 y Fl(\))260 1915 y(with)31 b(~)-45 b Fk(v)489 1927 y Fn(n)530 1911 y Fa(\003)530 1944 y Fc(v)566 1927 y Fn(;h)625 1935 y Fc(v)664 1915 y Fl(\()p Fh(x)p 696 1928 V 747 1885 a Fe(\003)747 1936 y Fn(v)786 1915 y Fl(\))24 b(=)h(~)-45 b Fk(v)972 1872 y Fm(\()p Fe(1)p Fm(\))969 1940 y Fn(n)1010 1923 y Fa(\003)1010 1956 y Fc(v)1046 1940 y Fn(;h)1105 1948 y Fc(v)1145 1915 y Fl(\()p Fh(x)p 1177 1928 V -30 x Fe(\003)1227 1936 y Fn(v)1267 1915 y Fl(\).)384 2027 y(In)31 b(order)f(to)h(pro)n(v)n(e)e(the)i(b)r(ounds)g(\(1.9\))f(and)h (\(1.10\),)g(it)g(is)g(su\016cien)n(t)g(to)g(pro)n(v)n(e)e(that)260 2127 y(there)e(exists)g(a)h(constan)n(t)f Fk(\016)1143 2139 y Fn(N)1205 2127 y Fl(,)h(whic)n(h)g(go)r(es)e(to)i(0)f(as)g Fk(N)32 b Fj(!)23 b(1)p Fl(,)28 b(suc)n(h)f(that)512 2393 y Fm(4)p Fn(:)p Fm(1)p Fn(a)281 b Fl(\()p Fk(\014)t(L)1055 2358 y Fm(2)1092 2393 y Fl(\))1124 2358 y Fe(\000)p Fm(1)1227 2280 y Ff(Z)1324 2393 y Fk(d)p 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Fj(j)p Fl(,)24 b(whic)n(h)f(is)f(of)g(the)h(form)f(of)g(the)h (r.h.s.)35 b(in)22 b(\(4.6\),)i(a)d(factor)h Fk(\015)2620 5054 y Fe(\000)p Fn(N)q(=)p Fm(2)2796 5084 y Fl(,)i(without)260 5184 y(a\013ecting)f(the)h(b)r(ound)g(of)g(the)g(sum)g(o)n(v)n(er)e (the)i(trees;)g(in)g(fact)g(this)g(factor)f(can)g(b)r(e)h(comp)r(en-) 260 5284 y(sated)j(b)n(y)g(lo)n(w)n(ering)f(of)i(1)p Fk(=)p Fl(2)e(the)i(v)n(ertex)e(dimension)i(\(whic)n(h)g(is)f Fj(\025)c Fl(1\))k(in)h(all)f(the)h(v)n(ertices)260 5383 y(of)f(the)h(path)g(joining)j(\026)-45 b Fk(v)30 b Fl(with)f(the)f(ro)r (ot.)36 b(It)28 b(follo)n(ws)e(that)929 5592 y(\()p Fk(\014)t(L)1069 5558 y Fm(2)1106 5592 y Fl(\))1138 5558 y Fe(\000)p Fm(1)1242 5479 y Ff(Z)1339 5592 y Fk(d)p Fh(x)p 1382 5605 V Fj(j)p Fl(\001)1524 5604 y Fm(2)p Fn(;N)1640 5592 y Fl(\()p Fh(x)p 1672 5605 V 1 w Fl(\))p Fj(j)d(\024)g Fl(\()p Fk(C)1980 5604 y Fn(k)2021 5592 y Fk(\025)2069 5604 y Fm(0)2107 5592 y Fl(\))2139 5558 y Fn(n)2185 5592 y Fk(\015)2233 5558 y Fe(\000)p 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Fl(,)g(541{664)e(\(1990\).)260 1828 y([BGPS])105 b(G.)38 b(Benfatto,)i(G.)e(Galla)n(v)n(otti,)g(A.)g(Pro)r(cacci,)g(B.)g (Scopp)r(ola:)56 b Fg(Beta)39 b(F)-6 b(unc-)638 1928 y(tions)30 b(and)g(Schwinger)h(F)-6 b(unctions)29 b(for)i(a)f(Many)g(F) -6 b(ermions)30 b(System)g(in)g(One)638 2027 y(Dimension)p Fl(,)e(Comm.)g(Math.)g(Ph)n(ys.)e Fh(160)p Fl(,)h(93{171)e(\(1994\).) 260 2193 y([BGM])132 b(G.)27 b(Benfatto,)f(A.)h(Giuliani,)g(V.)g (Mastropietro:)35 b Fg(L)l(ow)29 b(temp)l(er)l(atur)l(e)f(analysis)638 2293 y(of)40 b(two)g(dimensional)h(F)-6 b(ermi)40 b(systems)f(with)h (symmetric)f(F)-6 b(ermi)40 b(surfac)l(e)g Fl(,)638 2393 y(Ann.)28 b(Henri)g(P)n(oincar)n(\023)-39 b(e)24 b Fh(4)p Fl(,)k(137{193)c(\(2003\).)260 2559 y([BGM1])90 b(G.)32 b(Benfatto,)g(A.)f(Giuliani,)h(V.)g(Mastropietro:)42 b Fg(F)-6 b(ermi)34 b(liquid)g(b)l(ehavior)h(in)638 2658 y(the)g(2D)f(Hubb)l(ar)l(d)h(mo)l(del)g(at)f(low)h(temp)l(er)l(atur)l (es)p Fl(,)f(Ann.)f(Henri)g(P)n(oincar)n(\023)-39 b(e)29 b Fh(7)p Fl(,)638 2758 y(809{898)24 b(\(2006\).)260 2924 y([DR])208 b(G.)27 b(Disertori,)e(V.)i(Riv)-5 b(asseau:)35 b Fg(Inter)l(acting)28 b(F)-6 b(ermi)29 b(Liquid)h(in)e(Two)h(Dimen-) 638 3024 y(sions)j(at)f(Finite)h(T)-6 b(emp)l(er)l(atur)l(e)p Fl(,)29 b(Comm.)h(Math.)f(Ph)n(ys.)f Fh(215)p Fl(,)h(291{341)d(and)638 3123 y(251{390)e(\(2000\).)260 3289 y([FT])218 b(J.)40 b(F)-7 b(eldman,)45 b(E.)40 b(T)-7 b(rub)r(o)n(witz:)62 b Fg(Perturb)l(ation)42 b(the)l(ory)g(for)h(many)f(fermion)638 3389 y(systems)p Fl(,)28 b(Helv.)f(Ph)n(ys.)g(Acta)h Fh(63)p Fl(,)f(156{260)d(\(1990\).)260 3555 y([G])267 b(G.)42 b(Galla)n(v)n(otti:)64 b Fg(R)l(enormalization)44 b(the)l(ory)f(and)g(ultr)l(aviolet)g(stability)h(for)638 3655 y(sc)l(alar)e(\014elds)g(via)h(r)l(enormalization)g(gr)l(oup)e (metho)l(ds)p Fl(,)j(Rev.)d(Mo)r(d.)f(Ph)n(ys.,)638 3754 y Fh(57)p Fl(,)27 b(471{562)e(\(1985\).)260 3920 y([Gi])244 b(J.)31 b(Ginibre:)44 b Fg(Some)33 b(applic)l(ations)j(of)e(functional) f(inte)l(gr)l(ation)g(in)h(statistic)l(al)638 4020 y(me)l(chanics)p Fl(,)41 b(in)d Fg(Statistic)l(al)h(Me)l(chanics)i(and)e(Field)i(The)l (ory)p Fl(,)h(C.)37 b(De)h(Witt)638 4120 y(and)27 b(R.)h(Stora)f(eds.,) h(Gordon)e(and)i(Breac)n(h,)e(New)i(Y)-7 b(ork)26 b(\(1970\).)260 4286 y([Le])243 b(A.)27 b(Lesniewski:)36 b Fg(E\013e)l(ctive)30 b(action)g(for)g(the)g(Y)-6 b(ukawa)29 b(2)h(quantum)e(\014eld)h(The-) 638 4385 y(ory)p Fl(,)g(Comm)n(un.)e(Math.)h(Ph)n(ys.)e Fh(108)p Fl(,)h(437{467)e(\(1987\).)260 4551 y([NO])205 b(J.)26 b(W.)g(Negele,)g(H.)g(Orland:)35 b(Quan)n(tum)25 b(Man)n(y-P)n(article)e(Systems,)j(Addison)638 4651 y(W)-7 b(esley)g(,)28 b(New)g(Y)-7 b(ork)26 b(\(1998\).)260 4817 y([PS])229 b(W.)45 b(P)n(edra,)i(M.)e(Salmhofer:)71 b Fg(Determinat)45 b(b)l(ounds)g(and)h(the)g(Matsub)l(ar)l(a)638 4917 y(UV)31 b(pr)l(oblem)h(of)h(many-fermion)f(systems)p Fl(,)e(arXiv:0707.2810v1)24 b([math-ph])638 5016 y(\(2007\).)260 5182 y([R])271 b(D.)28 b(Ruelle:)37 b Fg(Statistic)l(al)30 b(Me)l(chanics)p Fl(,)g(W.A.)e(Benjamin)g(\(1969\).)1650 5871 y(15)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0809080957901--