This is a postscript file generated with "dvips" from a silicon-graphics ".dvi" file. The original LaTeX files (three files) are available from R. Fernandez [Email: fernandez@eldpa.epfl.ch]. However, these files include diagrams drawn using the "picture" environment that have been observed not to be totally machine-independent. The postscript file, on the other hand, seems to print well. BODY %!PS-Adobe-2.0 %%Creator: dvips 5.47 Copyright 1986-91 Radical Eye Software %%Title: notes.dvi %%Pages: 95 1 %%BoundingBox: 0 0 596 843 %%EndComments %%BeginProcSet: tex.pro /TeXDict 200 dict def TeXDict begin /N /def load def /B{bind def}N /S /exch load def /X{S N}B /TR /translate load N /isls false N /vsize 10 N /@rigin{ isls{[0 1 -1 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale Resolution VResolution vsize neg mul TR matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}N /@letter{/vsize 10 N}B /@landscape{/isls true N /vsize -1 N}B /@a4{/vsize 10.6929133858 N}B /@a3{ /vsize 15.5531 N}B /@ledger{/vsize 16 N}B /@legal{/vsize 13 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 /IE 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 IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /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 dup definefont setfont}B /ch-width{ch-data dup length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{128 ch-data dup length 3 sub get sub}B /ch-yoff{ ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]{ch-image} imagemask restore}B /D{/cc X dup type /stringtype ne{]}if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 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}N /eop{clear SI restore showpage userdict /eop-hook known{eop-hook}if}N /@start{userdict /start-hook known{start-hook}if /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for}N /p /show load N /RMat[1 0 0 -1 0 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval(Display)eq}{pop false}ifelse}{false}ifelse end{{gsave TR -.1 -.1 TR 1 1 scale rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 -.1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /a{ moveto}B /delta 0 N /tail{dup /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{clear SS restore}B end %%EndProcSet TeXDict begin 1000 300 300 @start /Fa 1 70 df<000007F000003FFC0000FFFE000181FE 0006007E000C003E0018003C003800380070003000F0004000F0000001F0000001F0000001F800 0001F8000001FC000000FF000000FFC000007FFC00001FF0000007E000001C00000020000000C0 00000180000003000000060000000E0000001C0000003C000000380000007800000078000020F8 0000C0F8000180F8000300FC000600FE000C00FF0018007FC060003FFFC0001FFF000007F80000 1F2B7EA920>69 D E /Fb 2 65 df<00000000001800000000003C00000000007C0000000000F8 0000000001F00000000003E00000000007C0000000000F80000000001F00000000003E00000000 007C0000000000F80000000001F00000000003E00000000007C0000000000F80000000001F0000 0000003E00000000007C0000000000F80000000001F00000000003E00000000007C0000000000F 80000000001F00000000003E00000000007C0000000000F80000000001F00000000003E0000000 0007C0000000000F80000000001F00000000003E00000000007C0000000000F80000000001F000 00000003E00000000007C0000000000F80000000001F00000000003E00000000007C0000000000 F80000000000F000000000006000000000002E2E82AB2A>0 D<600000000000F00000000000F8 00000000007C00000000003E00000000001F00000000000F800000000007C00000000003E00000 000001F00000000000F800000000007C00000000003E00000000001F00000000000F8000000000 07C00000000003E00000000001F00000000000F800000000007C00000000003E00000000001F00 000000000F800000000007C00000000003E00000000001F00000000000F800000000007C000000 00003E00000000001F00000000000F800000000007C00000000003E00000000001F00000000000 F800000000007C00000000003E00000000001F00000000000F800000000007C00000000003E000 00000001F00000000000F800000000007C00000000003C0000000000182E2E82AB2A>64 D E /Fc 1 122 df<0000FFC000000007FFF80000001FFFFE0000003FFFFF000000FFFFFFC000 01FFFFFFE00003FFFFFFF00007FFFFFFF8000FFFFFFFFC001FFFFFFFFE001FFFFFFFFE003FFFFF FFFF003FFFFFFFFF007FFFFFFFFF807FFFFFFFFF807FFFFFFFFF80FFFFFFFFFFC0FFFFFFFFFFC0 FFFFFFFFFFC0FFFFFFFFFFC0FFFFFFFFFFC0FFFFFFFFFFC0FFFFFFFFFFC0FFFFFFFFFFC0FFFFFF FFFFC0FFFFFFFFFFC07FFFFFFFFF807FFFFFFFFF807FFFFFFFFF803FFFFFFFFF003FFFFFFFFF00 1FFFFFFFFE001FFFFFFFFE000FFFFFFFFC0007FFFFFFF80003FFFFFFF00001FFFFFFE00000FFFF FFC000003FFFFF0000001FFFFE00000007FFF800000000FFC000002A2A95942A>121 D E /Fd 11 122 df<7E000000FFC00000FFF800007FFE000003FF0000003FC000000FE0000003 F0000001F80000007C0000003E0000001F0000000F0000000F80000007C0000003C0000003E000 0001E0000001E0000001F0000000F0000000F0000000F800000078000000780000007800000078 00000078000000301D1D998332>8 D<0000003000000078000000780000007800000078000000 78000000F8000000F0000000F0000000F0000001F0000001E0000003E0000003C0000007C00000 0F8000000F0000001F0000003E0000007C000001F8000003F000000FE000003FC00003FF00007F FE0000FFF80000FFE000007E0000001D1D999C32>I<60000000F0000000F0000000F0000000F0 000000F0000000F80000007800000078000000780000007C0000003C0000003E0000001E000000 1F0000000F8000000780000007C0000003E0000001F0000000FC0000007E0000003F8000001FE0 000007FE000003FFF00000FFF800003FF8000003F01D1D809C32>I<000003F000003FF80000FF F80003FFF00007FE00001FE000003F8000007E000000FC000001F0000003E0000007C000000780 00000F8000001F0000001E0000003E0000003C0000007C0000007C0000007800000078000000F8 000000F0000000F0000000F0000000F0000000F0000000600000001D1D808332>I<7F00000000 FFF0000000FFFE0000007FFF80000001FFE00000000FF800000003FE00000000FF000000003F80 0000000FC000000003E000000001F000000000F8000000007C000000003E000000001E00000000 1F000000000F000000000F80000000078000000007C000000003C000000003E000000001E00000 0001E000000001F000000000F000000000F000000000F000000000F80000000078000000007800 0000007800000000780000000078000000007800000000302525A18342>I<0000000030000000 00780000000078000000007800000000780000000078000000007800000000F800000000F00000 0000F000000000F000000000F000000001F000000001E000000003E000000003C000000007C000 000007800000000F800000000F000000001F000000001E000000003E000000007C00000000F800 000001F000000003E00000000FC00000003F80000000FF00000003FE0000000FF8000001FFE000 007FFF800000FFFE000000FFF80000007F000000002525A1A442>I<6000000000F000000000F0 00000000F000000000F000000000F000000000F000000000F80000000078000000007800000000 780000000078000000007C000000003C000000003E000000001E000000001F000000000F000000 000F80000000078000000007C000000003C000000003E000000001F000000000F8000000007C00 0000003E000000001F800000000FE000000007F800000003FE00000000FF800000003FFC000000 0FFFF0000003FFF8000000FFF800000007F0252580A442>I<00000007F0000000FFF8000003FF F800000FFFF000003FFC000000FF80000003FE00000007F80000000FE00000001F800000003E00 0000007C00000000F800000001F000000003E000000003C000000007C000000007800000000F80 0000000F000000001F000000001E000000003E000000003C000000007C000000007C0000000078 0000000078000000007800000000F800000000F000000000F000000000F000000000F000000000 F000000000F00000000060000000002525808342>I<00003FF000000003FFFF0000000FFFFFC0 00001FFFFFE000007FE01FF80000FE0001FC0001F800007E0003F000003F0007C000000F800F80 000007C00F00000003C01F00000003E03E00000001F03C00000000F07C00000000F87800000000 78780000000078780000000078F8000000007CF0000000003CF0000000003CF0000000003CF000 0000003CF0000000003CF0000000003CF0000000003CF0000000003CF8000000007C7800000000 787800000000787800000000787C00000000F83C00000000F03E00000001F01F00000003E00F00 000003C00F80000007C007C000000F8003F000003F0001F800007E0000FE0001FC00007FE01FF8 00001FFFFFE000000FFFFFC0000003FFFF000000003FF000002E2E97962A>105 D<0000001FFE00000000000001FFFFE000000000000FFFFFFC00000000003FFFFFFF0000000000 FFF003FFC000000001FF00003FE000000007F8000007F80000000FE0000001FC0000001F800000 007E0000003F000000003F0000007C000000000F800000F80000000007C00001F00000000003E0 0003E00000000001F00003C00000000000F00007C00000000000F8000F8000000000007C000F00 00000000003C001F0000000000003E001E0000000000001E003E0000000000001F003C00000000 00000F003C0000000000000F007C0000000000000F807800000000000007807800000000000007 80780000000000000780F800000000000007C0F000000000000003C0F000000000000003C0F000 000000000003C0F000000000000003C0F000000000000003C0F000000000000003C0F000000000 000003C0F000000000000003C0F000000000000003C0F000000000000003C0F800000000000007 C07800000000000007807800000000000007807800000000000007807C0000000000000F803C00 00000000000F003C0000000000000F003E0000000000001F001E0000000000001E001F00000000 00003E000F0000000000003C000F8000000000007C0007C00000000000F80003C00000000000F0 0003E00000000001F00001F00000000003E00000F80000000007C000007C000000000F8000003F 000000003F0000001F800000007E0000000FE0000001FC00000007F8000007F800000001FF0000 3FE000000000FFF003FFC0000000003FFFFFFF00000000000FFFFFFC000000000001FFFFE00000 000000001FFE000000004242A1A03E>110 D<0000FFC000000007FFF80000001FFFFE0000003F FFFF000000FFFFFFC00001FFFFFFE00003FFFFFFF00007FFFFFFF8000FFFFFFFFC001FFFFFFFFE 001FFFFFFFFE003FFFFFFFFF003FFFFFFFFF007FFFFFFFFF807FFFFFFFFF807FFFFFFFFF80FFFF FFFFFFC0FFFFFFFFFFC0FFFFFFFFFFC0FFFFFFFFFFC0FFFFFFFFFFC0FFFFFFFFFFC0FFFFFFFFFF C0FFFFFFFFFFC0FFFFFFFFFFC0FFFFFFFFFFC07FFFFFFFFF807FFFFFFFFF807FFFFFFFFF803FFF FFFFFF003FFFFFFFFF001FFFFFFFFE001FFFFFFFFE000FFFFFFFFC0007FFFFFFF80003FFFFFFF0 0001FFFFFFE00000FFFFFFC000003FFFFF0000001FFFFE00000007FFF800000000FFC000002A2A 95942A>121 D E /Fe 1 79 df<1E01F807004005804005804008C08008608008608008308010 1900101900100D00100700300600F80200150E7E8D16>78 D E /Ff 58 123 df<0001FF81FE00001FFFEFFF80007F80FF87C000FC00FE0FE001F801FE0FE003F801FC0F E007F001FC0FE007F001FC07C007F001FC000007F001FC000007F001FC000007F001FC000007F0 01FC000007F001FC000007F001FC0000FFFFFFFFF800FFFFFFFFF800FFFFFFFFF80007F001FC00 0007F001FC000007F001FC000007F001FC000007F001FC000007F001FC000007F001FC000007F0 01FC000007F001FC000007F001FC000007F001FC000007F001FC000007F001FC000007F001FC00 0007F001FC000007F001FC000007F001FC000007F001FC000007F001FC000007F001FC000007F0 01FC00007FFF1FFFE0007FFF1FFFE0007FFF1FFFE0002B2A7FA928>11 D<0001FF803FE000001F FFE3FFF800007F80FFF01E0000FE007F801F0001F800FF003F0003F801FF007F8007F001FE007F 8007F001FE007F8007F001FE007F8007F000FE003F0007F000FE001E0007F000FE00000007F000 FE00000007F000FE00000007F000FE000000FFFFFFFFFFFF80FFFFFFFFFFFF80FFFFFFFFFFFF80 07F000FE003F8007F000FE003F8007F000FE003F8007F000FE003F8007F000FE003F8007F000FE 003F8007F000FE003F8007F000FE003F8007F000FE003F8007F000FE003F8007F000FE003F8007 F000FE003F8007F000FE003F8007F000FE003F8007F000FE003F8007F000FE003F8007F000FE00 3F8007F000FE003F8007F000FE003F8007F000FE003F8007F000FE003F807FFF0FFFE3FFF87FFF 0FFFE3FFF87FFF0FFFE3FFF8352A7FA939>14 D<00030007000E001C0038007000F001E003C003 C007C007800F800F001F001F003F003E003E007E007E007E007C007C00FC00FC00FC00FC00FC00 FC00FC00FC00FC00FC00FC00FC007C007C007E007E007E003E003E003F001F001F000F000F8007 8007C003C003C001E000F000700038001C000E00070003103C7AAC1B>40 D<8000C000E000700038001C001E000F000780078007C003C003E001E001F001F001F800F800F8 00FC00FC00FC007C007C007E007E007E007E007E007E007E007E007E007E007E007E007C007C00 FC00FC00FC00F800F801F801F001F001E003E003C007C0078007800F001E001C0038007000E000 C00080000F3C7BAC1B>I<1C003E007F00FF80FF80FF807F003E001C0009097B8813>46 D<000E00001E00007E0007FE00FFFE00FFFE00F8FE0000FE0000FE0000FE0000FE0000FE0000FE 0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE 0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE007FFFFE7FFFFE7FFF FE17277BA622>49 D<00FF800003FFF0000FFFFC001F03FE003800FF007C007F80FE003FC0FF00 3FC0FF003FE0FF001FE0FF001FE07E001FE03C003FE000003FE000003FC000003FC000007F8000 007F000000FE000000FC000001F8000003F0000003E00000078000000F0000001E0000003C00E0 007000E000E000E001C001C0038001C0070001C00FFFFFC01FFFFFC03FFFFFC07FFFFFC0FFFFFF 80FFFFFF80FFFFFF801B277DA622>I<007F800003FFF00007FFFC000F81FE001F00FF003F80FF 003F807F803F807F803F807F801F807F800F007F800000FF000000FF000000FE000001FC000001 F8000007F00000FFC00000FFF0000001FC0000007E0000007F0000007F8000003FC000003FC000 003FE000003FE03C003FE07E003FE0FF003FE0FF003FE0FF003FC0FF007FC07E007F807C007F00 3F01FE001FFFFC0007FFF00000FF80001B277DA622>I<00000E0000001E0000003E0000007E00 0000FE000000FE000001FE000003FE0000077E00000E7E00000E7E00001C7E0000387E0000707E 0000E07E0000E07E0001C07E0003807E0007007E000E007E000E007E001C007E0038007E007000 7E00E0007E00FFFFFFF8FFFFFFF8FFFFFFF80000FE000000FE000000FE000000FE000000FE0000 00FE000000FE000000FE00007FFFF8007FFFF8007FFFF81D277EA622>I<0C0003000F803F000F FFFE000FFFFC000FFFF8000FFFF0000FFFE0000FFFC0000FFE00000E0000000E0000000E000000 0E0000000E0000000E0000000E7FC0000FFFF8000F80FC000E003E000C003F0000001F8000001F C000001FC000001FE000001FE018001FE07C001FE0FE001FE0FE001FE0FE001FE0FE001FC0FC00 1FC078003F8078003F803C007F001F01FE000FFFF80003FFF00000FF80001B277DA622>I<0007 F000003FFC0000FFFE0001FC0F0003F01F8007E03F800FC03F801FC03F801F803F803F801F003F 8000007F0000007F0000007F000000FF000000FF0FC000FF3FF800FF707C00FFC03E00FFC03F00 FF801F80FF801FC0FF001FC0FF001FE0FF001FE0FF001FE07F001FE07F001FE07F001FE07F001F E03F001FE03F001FC01F801FC01F803F800FC03F0007E07E0003FFFC0000FFF000003FC0001B27 7DA622>I<380000003E0000003FFFFFF03FFFFFF03FFFFFF07FFFFFE07FFFFFC07FFFFF807FFF FF0070000E0070000E0070001C00E0003800E0007000E000E0000000E0000001C0000003800000 07800000078000000F0000000F0000001F0000001F0000003F0000003E0000003E0000007E0000 007E0000007E0000007E000000FE000000FE000000FE000000FE000000FE000000FE000000FE00 0000FE0000007C0000003800001C297CA822>I<003FC00001FFF00003FFFC0007C07E000F003F 001E001F001E000F803E000F803E000F803F000F803F000F803FC00F003FF01F001FFC1E001FFE 3C000FFFF80007FFE00003FFF80001FFFC0001FFFE0007FFFF000F0FFF801E03FFC03C01FFC07C 007FE078001FE0F80007E0F80007E0F80003E0F80003E0F80003E0F80003C07C0003C07C000780 3F000F001FC03E000FFFFC0003FFF800007FC0001B277DA622>I<1C003E007F00FF80FF80FF80 7F003E001C000000000000000000000000000000000000001C003E007F00FF80FF80FF807F003E 001C00091B7B9A13>58 D<000003800000000007C00000000007C0000000000FE0000000000FE0 000000000FE0000000001FF0000000001FF0000000003FF8000000003FF8000000003FF8000000 0073FC0000000073FC00000000F3FE00000000E1FE00000000E1FE00000001C0FF00000001C0FF 00000003C0FF80000003807F80000007807FC0000007003FC0000007003FC000000E003FE00000 0E001FE000001E001FF000001C000FF000001FFFFFF000003FFFFFF800003FFFFFF80000780007 FC0000700003FC0000700003FC0000E00001FE0000E00001FE0001E00001FF0001C00000FF0001 C00000FF00FFFE001FFFFEFFFE001FFFFEFFFE001FFFFE2F297EA834>65 DI<00003FF001800003FFFE0380000FFFFF878000 3FF007DF8000FF8001FF8001FE00007F8003FC00003F8007F000001F800FF000000F801FE00000 07801FE0000007803FC0000007803FC0000003807FC0000003807F80000003807F8000000000FF 8000000000FF8000000000FF8000000000FF8000000000FF8000000000FF8000000000FF800000 0000FF8000000000FF80000000007F80000000007F80000000007FC0000003803FC0000003803F C0000003801FE0000003801FE0000007000FF00000070007F000000E0003FC00001E0001FE0000 3C0000FF8000F800003FF007E000000FFFFFC0000003FFFF000000003FF8000029297CA832>I< FFFFFFF80000FFFFFFFF8000FFFFFFFFE00003FC001FF80003FC0007FC0003FC0001FE0003FC00 00FF0003FC00007F8003FC00003FC003FC00001FC003FC00001FE003FC00001FE003FC00000FF0 03FC00000FF003FC00000FF003FC00000FF003FC00000FF803FC00000FF803FC00000FF803FC00 000FF803FC00000FF803FC00000FF803FC00000FF803FC00000FF803FC00000FF803FC00000FF8 03FC00000FF003FC00000FF003FC00000FF003FC00001FE003FC00001FE003FC00001FC003FC00 003FC003FC00007F8003FC00007F0003FC0001FE0003FC0003FC0003FC001FF800FFFFFFFFE000 FFFFFFFF8000FFFFFFFC00002D297DA835>III<00007FE003000003FFFC0700001FFFFF0F00003FF00FFF0000 FF8001FF0001FE0000FF0003F800003F0007F000003F000FF000001F001FE000000F001FE00000 0F003FC000000F003FC0000007007FC0000007007F80000007007F8000000000FF8000000000FF 8000000000FF8000000000FF8000000000FF8000000000FF8000000000FF8000000000FF800000 0000FF8001FFFFF87F8001FFFFF87F8001FFFFF87FC00000FF003FC00000FF003FC00000FF001F E00000FF001FE00000FF000FF00000FF0007F00000FF0003F80000FF0001FE0000FF0000FF8001 FF00003FF007BF00001FFFFF1F000003FFFE0F0000007FF003002D297CA836>I76 DII<0000FFE000000007FFFC0000003FC07F8000007F001FC000 01FC0007F00003F80003F80007F00001FC000FF00001FE001FE00000FF001FE00000FF003FC000 007F803FC000007F807FC000007FC07F8000003FC07F8000003FC07F8000003FC0FF8000003FE0 FF8000003FE0FF8000003FE0FF8000003FE0FF8000003FE0FF8000003FE0FF8000003FE0FF8000 003FE0FF8000003FE0FF8000003FE07F8000003FC07FC000007FC07FC000007FC03FC000007F80 3FC000007F801FE00000FF001FE00000FF000FF00001FE0007F00001FC0003F80003F80001FC00 07F00000FF001FE000003FC07F8000000FFFFE00000000FFE000002B297CA834>II<0000FFE000000007FFFC0000003FC07F8000007F001FC00001FC00 07F00003F80003F80007F00001FC000FF00001FE001FE00000FF001FE00000FF003FC000007F80 3FC000007F807FC000007FC07FC000007FC07F8000003FC07F8000003FC0FF8000003FE0FF8000 003FE0FF8000003FE0FF8000003FE0FF8000003FE0FF8000003FE0FF8000003FE0FF8000003FE0 FF8000003FE0FF8000003FE07F8000003FC07F8000003FC07FC000007FC03FC000007F803FC000 007F801FE00000FF001FE01F00FF000FF03F81FE0007F060C1FC0003F8C063F80001FCC077F000 00FFC07FE000003FE07F8000000FFFFE00000000FFFE00600000003E00600000003F0060000000 1F81E00000001FFFE00000001FFFC00000000FFFC00000000FFFC000000007FF8000000007FF80 00000003FF0000000001FE000000000078002B357CA834>II<007F806003FFF0E007FFF9E00F807FE01F001FE03E0007E07C0003E07C0001E0FC00 01E0FC0001E0FC0000E0FE0000E0FE0000E0FF000000FFC000007FFE00007FFFE0003FFFFC001F FFFE000FFFFF8007FFFFC003FFFFE000FFFFE00007FFF000007FF000000FF8000007F8000003F8 600001F8E00001F8E00001F8E00001F8F00001F0F00001F0F80003F0FC0003E0FF0007C0FFE01F 80F3FFFF00E0FFFE00C01FF0001D297CA826>I<7FFFFFFFFFC07FFFFFFFFFC07FFFFFFFFFC07F 803FC03FC07E003FC007C078003FC003C078003FC003C070003FC001C0F0003FC001E0F0003FC0 01E0E0003FC000E0E0003FC000E0E0003FC000E0E0003FC000E0E0003FC000E000003FC0000000 003FC0000000003FC0000000003FC0000000003FC0000000003FC0000000003FC0000000003FC0 000000003FC0000000003FC0000000003FC0000000003FC0000000003FC0000000003FC0000000 003FC0000000003FC0000000003FC0000000003FC0000000003FC0000000003FC0000000003FC0 000000003FC00000007FFFFFE000007FFFFFE000007FFFFFE0002B287EA730>II87 D89 D<01FF800007FFF0000F81F8001FC07E00 1FC07E001FC03F000F803F8007003F8000003F8000003F8000003F80000FFF8000FFFF8007FC3F 800FE03F803F803F803F003F807F003F80FE003F80FE003F80FE003F80FE003F807E007F807F00 DF803F839FFC0FFF0FFC01FC03FC1E1B7E9A21>97 DI<001FF80000FFFE0003F01F0007E03F800FC03F801F803F803F801F007F800E007F00 00007F000000FF000000FF000000FF000000FF000000FF000000FF000000FF0000007F0000007F 0000007F8000003F8001C01F8001C00FC0038007E0070003F01E0000FFFC00001FE0001A1B7E9A 1F>I<00003FF80000003FF80000003FF800000003F800000003F800000003F800000003F80000 0003F800000003F800000003F800000003F800000003F800000003F800000003F800000003F800 001FE3F80000FFFBF80003F03FF80007E00FF8000FC007F8001F8003F8003F8003F8007F0003F8 007F0003F8007F0003F800FF0003F800FF0003F800FF0003F800FF0003F800FF0003F800FF0003 F800FF0003F8007F0003F8007F0003F8007F0003F8003F8003F8001F8003F8000F8007F80007C0 0FF80003F03BFF8000FFF3FF80003FC3FF80212A7EA926>I<003FE00001FFF80003F07E0007C0 1F000F801F801F800F803F800FC07F000FC07F0007C07F0007E0FF0007E0FF0007E0FFFFFFE0FF FFFFE0FF000000FF000000FF0000007F0000007F0000007F0000003F8000E01F8000E00FC001C0 07E0038003F81F0000FFFE00001FF0001B1B7E9A20>I<0007F0003FFC00FE3E01F87F03F87F03 F07F07F07F07F03E07F00007F00007F00007F00007F00007F00007F000FFFFC0FFFFC0FFFFC007 F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007 F00007F00007F00007F00007F00007F00007F00007F0007FFF807FFF807FFF80182A7EA915>I< 00FF81F003FFE7F80FC1FE7C1F80FC7C1F007C383F007E107F007F007F007F007F007F007F007F 007F007F007F007F003F007E001F007C001F80FC000FC1F8001FFFE00018FF8000380000003800 00003C0000003E0000003FFFF8001FFFFF001FFFFF800FFFFFC007FFFFE01FFFFFF03E0007F07C 0001F8F80000F8F80000F8F80000F8F80000F87C0001F03C0001E01F0007C00FC01F8003FFFE00 007FF0001E287E9A22>II<07000F801FC0 3FE03FE03FE01FC00F8007000000000000000000000000000000FFE0FFE0FFE00FE00FE00FE00F E00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE0FFFEFFFE FFFE0F2B7DAA14>I107 DIII<003FE00001FFFC0003 F07E000FC01F801F800FC03F800FE03F0007E07F0007F07F0007F07F0007F0FF0007F8FF0007F8 FF0007F8FF0007F8FF0007F8FF0007F8FF0007F8FF0007F87F0007F07F0007F03F800FE03F800F E01F800FC00FC01F8007F07F0001FFFC00003FE0001D1B7E9A22>II<001FC038 0000FFF0780003F838F80007E00DF8000FC007F8001FC007F8003F8003F8007F8003F8007F8003 F8007F0003F800FF0003F800FF0003F800FF0003F800FF0003F800FF0003F800FF0003F800FF00 03F8007F0003F8007F0003F8007F8003F8003F8003F8001F8007F8000FC007F80007E01FF80003 F07BF80000FFF3F800003FC3F800000003F800000003F800000003F800000003F800000003F800 000003F800000003F800000003F800000003F80000003FFF8000003FFF8000003FFF8021277E9A 24>II<03FE300FFFF01E03F03800F0700070F00070F00070F80070FC00 00FFE0007FFE007FFF803FFFE01FFFF007FFF800FFF80003FC0000FC60007CE0003CF0003CF000 38F80038FC0070FF01E0F7FFC0C1FF00161B7E9A1B>I<00700000700000700000700000F00000 F00000F00001F00003F00003F00007F0001FFFF0FFFFF0FFFFF007F00007F00007F00007F00007 F00007F00007F00007F00007F00007F00007F00007F00007F00007F03807F03807F03807F03807 F03807F03803F03803F87001F86000FFC0001F8015267FA51B>IIIIII<3FFFFF803FFFFF803F007F003C00FE003801FE007803FC007803F8007007F800700F F000700FE000001FC000003FC000007F8000007F000000FF000001FE038001FC038003F8038007 F803800FF007800FE007801FE007003FC00F003F801F007F007F00FFFFFF00FFFFFF00191B7E9A 1F>I E /Fg 1 51 df<7FFFFFC0FFFFFFE0C0000060C0000060C0000060C0000060C0000060C0 000060C0000060C0000060C0000060C0000060C0000060C0000060C0000060C0000060C0000060 C0000060C0000060C0000060C0000060C0000060C0000060C0000060C0000060FFFFFFE0FFFFFF E01B1B7B9E25>50 D E /Fh 6 117 df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i 15 121 df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j 12 58 df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k 4 107 df0 D<081C1C3838383070706060C0C0060D7E8D09>48 D<1F00F833818440C2028064018038018018 01801C018026014043022181CC1F00F8180B7D8A1E>I106 D E /Fl 19 121 df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m 32 122 df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n 76 124 df<00000FC0F8000030718E0000E0F31E00 00C0F71E0001C0660C0001800E000003800E000003800E000003800E000007001C000007001C00 0007001C000007001C000007001C0000FFFFFFC0000E003800000E003800000E003800000E0038 00001C007000001C007000001C007000001C007000001C007000001C00E000003800E000003800 E000003800E000003800E000003801C000007001C000007001C000007001C000007001C0000060 03800000E003800000E003800000E003000000C003000001C0070000718E060000F19E0C0000F3 1E180000620C3000003C07C00000272D82A21E>11 D<00000FE0000030180000E01C0001C03C00 01803C0003803800038000000380000007000000070000000700000007000000070000000E0000 00FFFFE0000E00E0000E00E0000E01C0001C01C0001C01C0001C01C0001C0380001C0380003803 80003803800038070000380700003807000070070800700E1000700E1000700E1000700E2000E0 062000E003C000E0000000E0000000C0000001C0000001C0000071800000F1800000F300000062 0000003C0000001E2D82A21B>I<00000FF01FE000003838601800006079C01C0000C07B803C00 01C033003C0001C00700380003800700000003800700000003800E00000003800E00000007000E 00000007000E00000007000E00000007001C000000FFFFFFFFE0000E001C00E0000E001C00E000 0E001C01C0000E003801C0000E003801C0001C003801C0001C00380380001C00380380001C0070 0380001C00700380001C00700700003800700700003800700700003800E00708003800E00E1000 3800E00E10007000E00E10007000E00E20007001C00620007001C003C0006001C0000000E00180 000000E00380000000C00380000000C00300000071C703000000F18F06000000F10F0C00000062 06180000003C03E00000002E2D82A22B>14 D<0180038007800E000C00180030006000C0000909 6FA219>19 D<007000F001F001F001F001E001E001E003C003C003C00380038003800700070007 000600060006000C000C000C00080008000800000000000000000000007000F800F800F000E000 0C247AA30F>33 D<0C06001E0F003F1F803F1F801D0E8002010002010002010004020004020008 0400100800201000402000804000110F78A219>I<0C1E3F3F1D02020204040810204080080F75 A20F>39 D<00008000010000020000040000080000100000300000600000C00000C00001800003 00000300000600000600000E00000C00001C000018000018000038000030000030000070000070 0000600000600000E00000E00000E00000C00000C00000C00000C00000C00000C00000C00000C0 0000C00000C00000C00000C00000C0000040000060000060000020000030000010000008000011 3278A414>I<000800000400000600000200000300000300000100000180000180000180000180 000180000180000180000180000180000180000180000180000180000380000380000380000300 000300000700000700000600000600000E00000C00000C00001C00001800003800003000003000 00600000600000C000018000018000030000060000040000080000100000200000400000800000 113280A414>I<0E1E1E1E1E02020404080810204080070F7D840F>44 DI<70F8F8F0E005057A840F>I<0001000300030006001E002E03CE001C001C001C001C0038 003800380038007000700070007000E000E000E000E001C001C001C001C0038003800380038007 80FFFC10217AA019>49 D<000FC000106000603800801800801C01001C02201E02101E04101E04 101E04101E08203C08203C0840380840780880F00700E00001C000030000060000180000200000 C0000100000200000400100800301000202000605F80C063FFC040FF80807F00801E0017227CA0 19>I<000FC000307000C01801001C02001C04000C04401C08201C08201C08201C08403808C038 0700700000600001C000070000FC000007000003800003800001C00001C00001C00003C06003C0 F003C0F00380E00780800700800E00801C0040380020F0001F800016227BA019>I<0000180000 380000380000700000700000700000E00000E00000E00000C00001C00001800003800003000003 00000600000600000C00000C000018000010000031800061C00043800083800183800303800207 000407000807003FC700403E00800FF0000E00000E00001C00001C00001C00001C000038000038 00003800003000152B7EA019>I<00400400703800FFF000FFE000BF8000800001000001000001 0000010000020000020000023E0002C3000501800601C00401C00001E00001E00001E00001E000 01E00001E07003C0F003C0F003C0E00780800700800F00800E00401C0040380030E0000F800016 227BA019>I<0003E0000C1000380800603800C07801C0780380300700000700000E00001E0000 1E00001C7C003C86003D03007A03807C03807801C07803C0F803C0F003C0F003C0F003C0E00780 E00780E00780E00700E00F00E00E00E01C0060180060300030E0000F800015227AA019>I<0278 0204FC0407FC040FFC080F0C181E04701803A03000602000406000C04000808001800003800003 00000700000600000E00000E00001C00001C00003C0000380000780000780000F00000F00000F0 0001E00001E00001E00003E00003C00003C000018000172279A019>I<000FC000306000401000 801801000803000C03000C06001807001807001007003007C06007E0C003F18001FE0000FC0000 FE00033F00061F800C07C01803C03001C06001C06000C0C000C0C000C0C00080C00180C00100C0 0200C006006008003030000FC00016227BA019>I<07000F800F800F000E000000000000000000 00000000000000000000000000007000F800F800F000E00009157A940F>58 D<00E001F001F001E001C0000000000000000000000000000000000000000000000E001E001E00 1E001E0002000200040004000800080010002000400080000C1F7D940F>I<0000030000000300 000007000000070000000F0000000F0000001F0000002F0000002F0000004F0000004F80000087 800000878000010780000207800002078000040780000407800008078000080780001007800030 078000200780007FFF80004007C0008007C0008003C0010003C0030003C0020003C0040003C004 0003C00C0003C03C0007C0FF003FFC1E237DA224>65 D<00FFFFE0000F0038000F001C000F001E 001E000E001E000F001E000F001E000F003C000E003C001E003C001E003C003C00780078007800 F0007801E00078078000FFFF8000F001E000F000F000F0007801E0007801E0003801E0003C01E0 003C03C0007803C0007803C0007803C000F0078000F0078001E0078003C0078007000F801E00FF FFF00020227DA122>I<00007F00800003808100000E00630000380027000070001F0000E0000E 0001C0000E000380000E000700000E000F000004000E000004001E000004003C000004003C0000 0800780000000078000000007800000000F000000000F000000000F000000000F000000000F000 000000E000000000E000002000E000002000E000004000E000004000F000008000700000800070 00010000380002000018000400001C0008000006003000000381C0000000FE000000212479A223 >I<00FFFFF000000F003C00000F000E00000F000700001E000380001E000380001E0001C0001E 0001C0003C0001C0003C0001E0003C0001E0003C0001E000780001E000780001E000780001E000 780001E000F00003C000F00003C000F00003C000F00003C001E000078001E000078001E0000700 01E0000F0003C0000E0003C0001C0003C0003C0003C00038000780007000078000E000078001C0 0007800700000F801C0000FFFFF0000023227DA125>I<00FFFFFF80000F000780000F00018000 0F000180001E000180001E000180001E000100001E000100003C000100003C000100003C010100 003C01000000780200000078020000007806000000780E000000FFFC000000F00C000000F00C00 0000F00C000001E008000001E008000001E008040001E000080003C000080003C000080003C000 100003C000100007800020000780006000078000C000078001C0000F8007C000FFFFFF80002122 7DA121>I<00FFFFFF000F000F000F0003000F0003001E0003001E0003001E0002001E0002003C 0002003C0002003C0102003C010000780200007802000078060000780E0000FFFC0000F00C0000 F00C0000F00C0001E0080001E0080001E0080001E0000003C0000003C0000003C0000003C00000 078000000780000007800000078000000F800000FFFC000020227DA120>I<00007F0080000380 8100000E00630000380027000070001F0000E0000E0001C0000E000380000E000700000E000F00 0004000E000004001E000004003C000004003C00000800780000000078000000007800000000F0 00000000F000000000F000000000F000000000F0003FFC00E00001E000E00001E000E00001E000 E00003C000E00003C000F00003C000700003C0007000078000380007800018000F80001C001380 0006002300000381C1000000FE000000212479A226>I<00FFF87FFC000F000780000F00078000 0F000780001E000F00001E000F00001E000F00001E000F00003C001E00003C001E00003C001E00 003C001E000078003C000078003C000078003C000078003C0000FFFFF80000F000780000F00078 0000F000780001E000F00001E000F00001E000F00001E000F00003C001E00003C001E00003C001 E00003C001E000078003C000078003C000078003C000078003C0000F8007C000FFF87FFC002622 7DA124>I<00FFF8000F00000F00000F00001E00001E00001E00001E00003C00003C00003C0000 3C0000780000780000780000780000F00000F00000F00000F00001E00001E00001E00001E00003 C00003C00003C00003C0000780000780000780000780000F8000FFF80015227DA113>I<0007FF C000003C0000003C0000003C00000078000000780000007800000078000000F0000000F0000000 F0000000F0000001E0000001E0000001E0000001E0000003C0000003C0000003C0000003C00000 078000000780000007800000078000000F0000000F0000380F0000780F0000F81E0000F81E0000 F03C0000403800004070000021E000001F8000001A237CA11A>I<00FFF807FC000F0001E0000F 0001C0000F000100001E000200001E000400001E001000001E002000003C004000003C00800000 3C010000003C04000000780800000078180000007838000000787C000000F0BC000000F23C0000 00F41E000000F81E000001F01F000001E00F000001E00F000001E00F800003C007800003C00780 0003C003C00003C003C000078003C000078001E000078001E000078001E0000F8001F000FFF80F FE0026227DA125>I<00FFFC00000F8000000F0000000F0000001E0000001E0000001E0000001E 0000003C0000003C0000003C0000003C00000078000000780000007800000078000000F0000000 F0000000F0000000F0000001E0000001E0000001E0002001E0002003C0004003C0004003C00080 03C0008007800180078001000780030007800F000F803E00FFFFFE001B227DA11F>I<00FF8000 07FC000F80000F80000F80001780000F80001780001780002F000013C0002F000013C0004F0000 13C0008F000023C0009E000023C0011E000023C0011E000023C0021E000043C0043C000043C004 3C000043C0083C000041E0083C000081E01078000081E02078000081E02078000081E040780001 01E040F0000101E080F0000101E100F0000101E100F0000200F201E0000200F201E0000200F401 E0000200F801E0000400F803C0000400F003C0000400F003C0000C00E003C0001E00C007C000FF C0C07FFC002E227DA12C>I<00FF000FFC000F8001E0000F800180000FC000800013C001000013 C001000011E001000011E001000021E002000020F002000020F002000020F00200004078040000 40780400004078040000403C040000803C080000803E080000801E080000801E080001001F1000 01000F100001000F10000100079000020007A000020007A000020003E000020003E000040003C0 00040001C000040001C0000C0001C0001E00008000FFC000800026227DA124>I<0000FE000007 8380000C00E0003800700070003800E0003801C0001C0380001C0700001C0F00001E1E00001E1C 00001E3C00001E3C00001E7800001E7800001E7800001EF000003CF000003CF000003CF0000078 F0000078E0000078E00000F0E00000F0E00001E0E00001C0F00003C0F00007807000070078000E 0038001C001C0038000E00E0000703800001FC00001F2479A225>I<00FFFFE0000F0038000F00 1E000F000E001E0007001E0007001E0007001E0007003C000F003C000F003C000F003C001E0078 001E0078003C00780078007800E000F003C000FFFE0000F0000000F0000001E0000001E0000001 E0000001E0000003C0000003C0000003C0000003C0000007800000078000000780000007800000 0F800000FFF8000020227DA121>I<0000FE0000078380000C00E0003800700070003800E00038 01C0003C0380001C0700001C0F00001E1E00001E1E00001E3C00001E3C00001E7800001E780000 1E7800001EF000003CF000003CF000003CF0000038F0000078E0000078E00000F0E00000F0E000 01E0E00001C0E00003C0F01E03807061070070810E0038811C001C8138000E81E0000783804001 FD004000010040000180800001818000018100000387000001FE000001FE000001FC000000F000 1F2D79A225>I<00FFFFC0000F0070000F003C000F001C001E000E001E000E001E000F001E000F 003C001E003C001E003C001E003C003C0078003800780070007801E00078078000FFFC0000F00E 0000F0070000F0038001E003C001E003C001E003C001E003C003C0078003C0078003C0078003C0 078007800F0007800F0107800F01078007020F800702FFF8038C000000F020237DA124>I<0001 F020000E0C40001802C0003001C0006001C000E0018000C0018001C0018001C0018003C0010003 C0010003C0000003C0000003E0000001F8000001FF000000FFE000007FF000001FF8000003FC00 00007C0000003C0000001E0000001E0000001E0020001C0020001C0020001C0020001800600038 0060003000700060007000C000C8018000C607000081FC00001B247DA21B>I<1FFFFFF81E03C0 381803C0183003C01820078018200780184007801040078010400F0010800F0010800F0010000F 0000001E0000001E0000001E0000001E0000003C0000003C0000003C0000003C00000078000000 780000007800000078000000F0000000F0000000F0000000F0000001E0000001E0000001E00000 01E0000003E00000FFFF00001D2277A123>I<3FFE03FF03C0007803C0006003C0002007800040 0780004007800040078000400F0000800F0000800F0000800F0000801E0001001E0001001E0001 001E0001003C0002003C0002003C0002003C000200780004007800040078000400780004007000 0800F0000800F00010007000100070002000700040003000400038018000180200000E0C000003 F00000202377A124>I II<007FFFFE007E001E0078003C00600078 00C000F000C001E0008003E0008003C00100078001000F0001001E0000003C0000007C00000078 000000F0000001E0000003C00000078000000F8000000F0000001E0000003C00400078004000F0 008001F0008001E0008003C00100078001000F0003001E0006001E0006003C001E0078007C00FF FFFC001F227DA11E>90 D<000FF0000FF0000C0000180000180000180000180000300000300000 300000300000600000600000600000600000C00000C00000C00000C00001800001800001800001 80000300000300000300000300000600000600000600000600000C00000C00000C00000C000018 0000180000180000180000300000300000300000300000600000600000600000600000FF0000FF 000014317EA40F>I<010080020100040200080400100800201000201000402000402000804000 B85C00FC7E00FC7E00F87C00703800110F73A219>I<000FF0000FF00000300000600000600000 600000600000C00000C00000C00000C00001800001800001800001800003000003000003000003 00000600000600000600000600000C00000C00000C00000C000018000018000018000018000030 0000300000300000300000600000600000600000600000C00000C00000C00000C0000180000180 00018000018000FF0000FF0000143182A40F>I<00F8C00185C00705C00E03800E03801C03803C 0380380700780700780700780700F00E00F00E00F00E00F00E10F01C20701C20703C20305C4030 8C400F078014157B9419>97 D<03C03F8003800380038007000700070007000E000E000E000E00 1C001CF81D0C1E0E3C0638073807380F700F700F700F700FE01EE01EE01EE03CE038E038607060 E031C01F0010237BA216>I<007E0001C1000301800703800E07801C07803C0000380000780000 780000780000F00000F00000F00000F00000F00100700100700200300C001830000FC00011157B 9416>I<00003C0003F80000380000380000380000700000700000700000700000E00000E00000 E00000E00001C000F9C00185C00705C00E03800E03801C03803C03803807007807007807007807 00F00E00F00E00F00E00F00E10F01C20701C20703C20305C40308C400F078016237BA219>I<00 F803840E021C023C0238027804F018FFE0F000F000E000E000E000E000E002E002600470183060 0F800F157A9416>I<00003E0000470000CF00018F000186000380000380000380000700000700 000700000700000700000E0000FFF0000E00000E00000E00001C00001C00001C00001C00001C00 00380000380000380000380000380000700000700000700000700000700000E00000E00000E000 00E00000C00001C00001C000718000F18000F300006200003C0000182D82A20F>I<001F180030 B800E0B801C07001C0700380700780700700E00F00E00F00E00F00E01E01C01E01C01E01C01E01 C01E03800E03800E0780060B8006170001E700000700000700000E00000E00000E00701C00F018 00F0300060E0003F8000151F7E9416>I<00F0000FE00000E00000E00000E00001C00001C00001 C00001C000038000038000038000038000070000071F0007218007C0C00F00E00F00E00E00E00E 00E01C01C01C01C01C01C01C01C0380380380380380380380704700708700E08700E10700610E0 06206003C016237DA219>I<00C001E001C001C0000000000000000000000000000000001C0023 00430043008700870087000E000E001C001C001C00380038003840708070807080710032001C00 0B217BA00F>I<0000E00001E00001E00000C00000000000000000000000000000000000000000 00000000001E00002300004380008380008380010380010380000700000700000700000700000E 00000E00000E00000E00001C00001C00001C00001C000038000038000038000038000070000070 0000700070E000F0C000F180006300003C0000132B82A00F>I<00F0000FE00000E00000E00000 E00001C00001C00001C00001C0000380000380000380000380000700000701E0070210070C700E 10F00E10F00E20600E40001D80001E00001FC0001C7000383800383800381C00381C2070384070 3840703840701880E01880600F0014237DA216>I<01E01FC001C001C001C00380038003800380 07000700070007000E000E000E000E001C001C001C001C00380038003800380070007000700071 00E200E200E200E200640038000B237CA20C>I<1C0F80F8002610C10C00476066060087807807 008780780700870070070087007007000E00E00E000E00E00E000E00E00E000E00E00E001C01C0 1C001C01C01C001C01C01C001C01C0382038038038403803807040380380708038038030807007 0031003003001E0023157B9428>I<1C0F002631C04740C08780E08780E08700E08700E00E01C0 0E01C00E01C00E01C01C03801C03801C03801C0704380708380E08380E103806107006203003C0 16157B941B>I<007E0001C3000381800701C00E01C01C01E03C01E03801E07801E07801E07801 E0F003C0F003C0F00380F00780700700700E00700C0030180018700007C00013157B9419>I<01 C1F002621804741C08780C08700E08700E08701E00E01E00E01E00E01E00E01E01C03C01C03C01 C03C01C07803807003807003C0E003C1C0072380071E000700000700000E00000E00000E00000E 00001C00001C00001C0000FFC000171F7F9419>I<00F8400184C00705C00E03800E03801C0380 3C0380380700780700780700780700F00E00F00E00F00E00F00E00F01C00701C00703C00305C00 30B8000F380000380000380000700000700000700000700000E00000E00000E0000FFE00121F7B 9416>I<1C1F002620804741C08783C08703C08701808700000E00000E00000E00000E00001C00 001C00001C00001C000038000038000038000038000070000030000012157B9415>I<00FC0001 83000200800401800C03800C03000C00000F00000FF00007FC0003FE00003E00000F0000070070 0700F00600F00600E004004008002030001FC00011157D9414>I<00C001C001C001C001C00380 0380038003800700FFF8070007000E000E000E000E001C001C001C001C00380038003800381070 2070207040708031001E000D1F7C9E10>I<1E00602300E04380E04381C08381C08701C08701C0 0703800E03800E03800E03801C07001C07001C07001C07081C0E10180E101C0E101C1E200C2620 07C3C015157B941A>I<1E03802307C04387C04383C08381C08700C08700C00700800E00800E00 800E00801C01001C01001C01001C02001C02001C04001C08001C08000C300003C00012157B9416 >I<1E0060E02300E1F04380E1F04381C0F08381C0708701C0308701C030070380200E0380200E 0380200E0380201C0700401C0700401C0700401C0700801C0700801C0701001C0F01000C0F0200 06138C0003E0F0001C157B9420>I<03C1E0046210083470103CF02038F0203860203800007000 00700000700000700000E00000E00000E00000E02061C040F1C040F1C080E2C100446200383C00 14157D9416>I<1E00302300704380704380E08380E08700E08700E00701C00E01C00E01C00E01 C01C03801C03801C03801C03801C07001C07001C07001C0F000C3E0003CE00000E00000E00001C 00601C00F03800F03000E0600080C0004380003E0000141F7B9418>I<01E02003F06007F8C004 1F8008010008020000040000080000100000200000400000800001000002000004008008010010 03003F060061FC0040F80080700013157D9414>II E /Fo 43 127 df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p 22 115 df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q 68 123 df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r 43 116 df12 D<0006000C001800300070006000C001C00180038003000700 06000E000C001C001C0018003800380038003000700070007000700070007000E000E000E000E0 00E000E000E000E000E000E000E000E000E000E000E000E000E000E00070007000700070007000 7000300038003800380018001C001C000C000E000600070003000380018001C000C00060007000 300018000C00060F4A788119>16 DI<0000300000600000C0000180000300000700000E00000C0000 180000380000300000700000E00000C00001C0000180000380000380000300000700000600000E 00000E00000C00001C00001C00001C000018000038000038000038000038000070000070000070 0000700000700000700000700000E00000E00000E00000E00000E00000E00000E00000E00000E0 0000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E0000070 00007000007000007000007000007000007000003800003800003800003800001800001C00001C 00001C00000C00000E00000E000006000007000003000003800003800001800001C00000C00000 E000007000003000003800001800000C00000E000007000003000001800000C000006000003014 6377811F>IIII<0000700001F00003C0000780000E00001C00003800 00700000700000F00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E000 00E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E000 00E00000E00000E00000E00000E00000E00000E00001C00001C00001C000038000070000060000 0E0000380000700000C000007000003800000E000006000007000003800001C00001C00001C000 00E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E000 00E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E000 00E00000E00000E00000E00000F000007000007000003800001C00000E000007800003C00001F0 00007014637B811F>26 DI<0000180000300000600000E00000C0000180000380000700000600000E00000C00001C00 00380000380000700000700000E00000E00001E00001C00001C000038000038000038000078000 0700000700000F00000E00000E00001E00001E00001E00001C00001C00003C00003C00003C0000 3C0000380000780000780000780000780000780000780000780000780000700000F00000F00000 F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000 F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000700000780000 7800007800007800007800007800007800007800003800003C00003C00003C00003C00001C0000 1C00001E00001E00001E00000E00000E00000F0000070000070000078000038000038000038000 01C00001C00001E00000E00000E000007000007000003800003800001C00000C00000E00000600 0007000003800001800000C00000E0000060000030000018157C768121>32 DIII<00001C00003C00 00F80001E00003C0000780000F00000E00001E00003C00003C00003C0000780000780000780000 780000780000780000780000780000780000780000780000780000780000780000780000780000 780000780000780000780000780000780000780000780000780000780000780000780000780000 780000780000780000780000780000780000780000780000780000F00000F00000F00001E00001 E00003C0000380000700000E00001C0000780000E00000E000007800001C00000E000007000003 800003C00001E00001E00000F00000F00000F00000780000780000780000780000780000780000 780000780000780000780000780000780000780000780000780000780000780000780000780000 780000780000780000780000780000780000780000780000780000780000780000780000780000 7800007800007800007800007800007800003C00003C00003C00001E00000E00000F0000078000 03C00001E00000F800003C00001C167C7B8121>40 DI<00000C0000180000380000300000600000E00000C00001C000038000 0380000700000700000E00000E00001C00001C0000380000380000780000700000F00000F00000 E00001E00001E00003C00003C00003C00003C0000780000780000780000F80000F00000F00000F 00001F00001F00001E00001E00001E00003E00003E00003E00003E00003C00003C00003C00007C 00007C00007C00007C00007C00007C00007C00007C0000780000780000F80000F80000F80000F8 0000F80000F80000F80000F80000F80000F80000F80000F80000F80000F80000F80000F80000F8 0000164B748024>48 DIIII<001C001C001C001C001C001C001C001C001C001C001C001C001C001C001C001C001C 001C001C001C001C001C001C001C001C001C001C001C001C001C001C001C001C001C001C001C00 1C001C001C001C001C001C001C001C001C001C001C001C001C001C001C001C001C001C001C001C 001C001C001C001C001C001C001C001C001C001C001C001C001C001C001CFFFCFFFCFFFC0E4A80 811C>III<0018007800F001 E003C007800F001F001E003E003C007C007C007800F800F800F800F800F800F800F800F800F800 F800F800F800F800F800F800F800F800F800F800F800F800F800F8000D25707E25>III<007C007C007C007C007C007C007C007C007C007C007C007C007C007C007C007C007C007C00 7C007C007C007C007C00F800F800F800F001F001E003C007C007800F001E007C00F000C0000E25 798025>I<007C007C007C007C007C007C007C007C007C007C007C007C007C007C007C007C007C 007C007C007C007C007C007C007C00F800F800F800F001F001E003E003C0078007000E001C0038 00F000C000F00038001C000E000700078003C003E001E001F000F000F800F800F8007C007C007C 007C007C007C007C007C007C007C007C007C007C007C007C007C007C007C007C007C007C007C00 7C007C0E4D798025>III64 D<00007C00007C00007C00007C00007C00007C00007C00007C00007C00007C00007C00007C0000 7C00007C00007C00007C00007C0000780000780000F80000F80000F80000F80000F80000F80000 F80000F80000F00000F00000F00001F00001F00001F00001F00001E00001E00001E00003E00003 E00003C00003C00003C00007C0000780000780000780000F00000F00000F00000F00001E00001E 00001C00003C00003C0000380000780000700000700000E00000E00001C00001C0000380000380 000700000700000E00000C00001C0000180000300000700000600000C00000164B7F8224>I80 DI<0000F800018400030600060E000604000E00000E00000E00000C00001C00001C0000 1C00001C00001C00001C00001C00001C00001C0000380000380000380000380000380000380000 380000380000380000380000700000700000700000700000700000700000700000700000700000 600000E00000E00000E00040C000E0C000C180004300003E0000172E7E7F14>I88 DI<000000038000000006600000000C700000 000CF00000000CF00000001C600000001800000000380000000038000000003800000000700000 0000700000000070000000007000000000F000000000E000000000E000000000E000000001E000 000001E000000001C000000001C000000003C000000003C000000003C000000003C00000000780 0000000780000000078000000007800000000F800000000F800000000F000000000F000000001F 000000001F000000001F000000001F000000001E000000003E000000003E000000003E00000000 3E000000003C000000007C000000007C000000007C000000007C000000007800000000F8000000 00F800000000F800000000F800000000F000000001F000000001F000000001F000000001F00000 0001E000000001E000000003E000000003E000000003C000000003C000000003C000000003C000 00000780000000078000000007800000000780000000070000000007000000000F000000000F00 0000000E000000000E000000000E000000001E000000001C000000001C000000001C0000000018 00000000380000000038000000003000000000700000006060000000F060000000F0C0000000E1 8000000063000000001E00000000245C7E7F17>I104 DI<0000E00003E0000F8000 1E00003C0000700000700000E00000E00000E00000E00000E00000E00000E00000E00000E00000 E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000 E00001C00001C0000380000700000E00003C0000F00000F000003C00000E000007000003800001 C00001C00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000 E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000700000 7000003C00001E00000F800003E00000E0134A7C811C>110 DI<000000000200000000060000000006000000 0006000000000C000000000C000000000C000000000C000000000C000000001800000000180000 000018000000001800000000180000000030000000003000000000300000000030000000003000 00000060000000006000000000600000000060000000006000000000C000000000C000000000C0 00000000C000000000C00000000180000000018000000001800000000180000000018000000003 000000000300000000030000000003000000000300000000060000000006000000000600000000 0600000000060000000006000000000C000000000C000000000C000000000C000000000C000000 001800000000180000000018000000001800000000180000000030000000003000000000300000 000030000000003000000000600000000060000000006000080000600008000060001C0000C000 1C0000C0003C0000C0003C0000C0005C0000C0005C000180008E000180000E000180000E000180 000E000180000E000300000E000300000700030000070003000007000300000700060000070006 0000038006000003800600000380060000038006000003800C000003800C000001C00C000001C0 0C000001C00C000001C018000001C018000001C018000000E018000000E018000000E030000000 E030000000E0300000007030000000703000000070600000007060000000706000000070600000 00386000000038C000000038C000000038C000000038C000000038C00000001D800000001D8000 00001D800000001D800000001D800000000F000000000F000000000F000000000F000000000F00 0000000E000000000600000000060000000006000000277D7C812A>115 D E /Fs 51 115 df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t 78 123 df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u 59 123 df<000FF83F00007FFDFFC001F81F E3E003E03F87E007C03F87E00F803F07E00F803F03C00F801F00000F801F00000F801F00000F80 1F00000F801F00000F801F0000FFFFFFFC00FFFFFFFC000F801F00000F801F00000F801F00000F 801F00000F801F00000F801F00000F801F00000F801F00000F801F00000F801F00000F801F0000 0F801F00000F801F00000F801F00000F801F00000F801F00000F801F00000F801F00007FF0FFF0 007FF0FFF00023237FA221>11 D<000FF000007FFC0001F80E0003E01F0007C03F000F803F000F 803F000F801E000F800C000F8000000F8000000F8000000F800000FFFFFF00FFFFFF000F801F00 0F801F000F801F000F801F000F801F000F801F000F801F000F801F000F801F000F801F000F801F 000F801F000F801F000F801F000F801F000F801F000F801F000F801F007FF0FFE07FF0FFE01B23 7FA21F>I<00180030006000C001C00380070007000E001E001C003C003C003C00780078007800 7800F800F000F000F000F000F000F000F000F000F000F000F000F80078007800780078003C003C 003C001C001E000E0007000700038001C000C00060003000180D317BA416>40 DI<387CFEFFFF7F3B03030706 060C1C18702008117C8610>44 DI<387CFEFEFE7C3807077C 8610>I<00FE0007FFC00F83E01F01F03E00F83E00F87C007C7C007C7C007CFC007CFC007EFC00 7EFC007EFC007EFC007EFC007EFC007EFC007EFC007EFC007EFC007EFC007EFC007E7C007C7C00 7C7C007C3E00F83E00F81F01F00F83E007FFC000FE0017207E9F1C>48 D<00180000780001F800 FFF800FFF80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F800 01F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F800 01F8007FFFE07FFFE013207C9F1C>I<03FC000FFF003C1FC07007E07C07F0FE03F0FE03F8FE03 F8FE01F87C01F83803F80003F80003F00003F00007E00007C0000F80001F00003E000038000070 0000E01801C0180380180700180E00380FFFF01FFFF03FFFF07FFFF0FFFFF0FFFFF015207D9F1C >I<00FE0007FFC00F07E01E03F03F03F03F81F83F81F83F81F81F03F81F03F00003F00003E000 07C0001F8001FE0001FF000007C00001F00001F80000FC0000FC3C00FE7E00FEFF00FEFF00FEFF 00FEFF00FC7E01FC7801F81E07F00FFFC001FE0017207E9F1C>I<0000E00001E00003E00003E0 0007E0000FE0001FE0001FE00037E00077E000E7E001C7E00187E00307E00707E00E07E00C07E0 1807E03807E07007E0E007E0FFFFFEFFFFFE0007E00007E00007E00007E00007E00007E00007E0 00FFFE00FFFE17207E9F1C>I<1000201E01E01FFFC01FFF801FFF001FFE001FF8001BC0001800 0018000018000018000019FC001FFF001E0FC01807E01803E00003F00003F00003F80003F83803 F87C03F8FE03F8FE03F8FC03F0FC03F07007E03007C01C1F800FFF0003F80015207D9F1C>I<00 1F8000FFE003F07007C0F00F01F81F01F83E01F83E01F87E00F07C00007C0000FC0800FC7FC0FC FFE0FD80F0FF00F8FE007CFE007CFC007EFC007EFC007EFC007E7C007E7C007E7C007E3C007C3E 007C1E00F80F00F00783E003FFC000FF0017207E9F1C>I<6000007800007FFFFE7FFFFE7FFFFC 7FFFF87FFFF87FFFF0E00060E000C0C00180C00300C00300000600000C00001C00001800003800 00780000780000F00000F00000F00001F00001F00001F00003F00003F00003F00003F00003F000 03F00003F00001E00017227DA11C>I<00FE0003FFC00703E00E00F01C00F01C00783C00783E00 783F00783F80783FE0F01FF9E01FFFC00FFF8007FFC003FFE007FFF01E7FF83C1FFC7807FC7801 FEF000FEF0003EF0001EF0001EF0001CF8001C7800383C00381F01F00FFFC001FF0017207E9F1C >I<387CFEFEFE7C380000000000000000387CFEFEFE7C3807167C9510>58 D<000070000000007000000000F800000000F800000000F800000001FC00000001FC00000003FE 00000003FE00000003FE00000006FF000000067F0000000E7F8000000C3F8000000C3F80000018 3FC00000181FC00000381FE00000300FE00000300FE00000600FF000006007F00000E007F80000 FFFFF80000FFFFF800018001FC00018001FC00038001FE00030000FE00030000FE000600007F00 0600007F00FFE00FFFF8FFE00FFFF825227EA12A>65 DI<0003FE0080001FFF818000FF01E38001F8003F8003E0001F8007C0000F800F800007 801F800007803F000003803F000003807F000001807E000001807E00000180FE00000000FE0000 0000FE00000000FE00000000FE00000000FE00000000FE00000000FE000000007E000000007E00 0001807F000001803F000001803F000003801F800003000F8000030007C000060003F0000C0001 F800380000FF00F000001FFFC0000003FE000021227DA128>IIII<0003FE0040001FFFC0C0007F00F1C001F800 3FC003F0000FC007C00007C00FC00003C01F800003C03F000001C03F000001C07F000000C07E00 0000C07E000000C0FE00000000FE00000000FE00000000FE00000000FE00000000FE00000000FE 00000000FE000FFFFC7E000FFFFC7F00001FC07F00001FC03F00001FC03F00001FC01F80001FC0 0FC0001FC007E0001FC003F0001FC001FC003FC0007F80E7C0001FFFC3C00003FF00C026227DA1 2C>I73 D76 DI<0007FC0000003FFF800000FC07E00003F001F80007E000FC000FC0007E001F80003F 001F80003F003F00001F803F00001F807F00001FC07E00000FC07E00000FC0FE00000FE0FE0000 0FE0FE00000FE0FE00000FE0FE00000FE0FE00000FE0FE00000FE0FE00000FE0FE00000FE07E00 000FC07F00001FC07F00001FC03F00001F803F80003F801F80003F000FC0007E0007E000FC0003 F001F80000FC07E000003FFF80000007FC000023227DA12A>79 DI<0007FC0000003FFF800000FC07E00003F001F80007E000FC00 0FC0007E001F80003F001F80003F003F00001F803F00001F807F00001FC07E00000FC07E00000F C0FE00000FE0FE00000FE0FE00000FE0FE00000FE0FE00000FE0FE00000FE0FE00000FE0FE0000 0FE0FE00000FE07E00000FC07F00001FC07F00001FC03F00001F803F81F03F801F83F83F000FC7 0C7E0007E606FC0003F607F80000FF07E000003FFF80000007FF80200000038020000001C02000 0001E0E0000001FFE0000001FFC0000000FFC0000000FFC00000007F800000007F000000001E00 232C7DA12A>II<01FC0407FF8C1F03FC3C00 7C7C003C78001C78001CF8000CF8000CFC000CFC0000FF0000FFE0007FFF007FFFC03FFFF01FFF F80FFFFC03FFFE003FFE0003FF00007F00003F00003FC0001FC0001FC0001FE0001EE0001EF000 3CFC003CFF00F8C7FFE080FF8018227DA11F>I<7FFFFFFF807FFFFFFF807E03F80F807803F807 807003F803806003F80180E003F801C0E003F801C0C003F800C0C003F800C0C003F800C0C003F8 00C00003F800000003F800000003F800000003F800000003F800000003F800000003F800000003 F800000003F800000003F800000003F800000003F800000003F800000003F800000003F8000000 03F800000003F800000003F800000003F800000003F8000003FFFFF80003FFFFF80022227EA127 >I<3FFFFFE03FFFFFE03F801FC03E003FC03C003F8038007F007000FF007000FE007001FE0060 03FC006003F8006007F8000007F000000FE000001FE000001FC000003FC000007F8000007F0000 00FF000000FE006001FC006003FC006003F8006007F800E00FF000E00FE000E01FE001C01FC001 C03F8003C07F8007C07F003FC0FFFFFFC0FFFFFFC01B227DA122>90 D<07FC001FFF803F07C03F 03E03F01E03F01F01E01F00001F00001F0003FF003FDF01FC1F03F01F07E01F0FC01F0FC01F0FC 01F0FC01F07E02F07E0CF81FF87F07E03F18167E951B>97 DI<00FF8007FFE00F83F01F03F03E03F07E03F07C01E07C00 00FC0000FC0000FC0000FC0000FC0000FC00007C00007E00007E00003E00301F00600FC0E007FF 8000FE0014167E9519>I<0001FE000001FE0000003E0000003E0000003E0000003E0000003E00 00003E0000003E0000003E0000003E0000003E0000003E0001FC3E0007FFBE000F81FE001F007E 003E003E007E003E007C003E00FC003E00FC003E00FC003E00FC003E00FC003E00FC003E00FC00 3E00FC003E007C003E007C003E003E007E001E00FE000F83BE0007FF3FC001FC3FC01A237EA21F >I<00FE0007FF800F87C01E01E03E01F07C00F07C00F8FC00F8FC00F8FFFFF8FFFFF8FC0000FC 0000FC00007C00007C00007E00003E00181F00300FC07003FFC000FF0015167E951A>I<003F80 00FFC001E3E003C7E007C7E00F87E00F83C00F80000F80000F80000F80000F80000F8000FFFC00 FFFC000F80000F80000F80000F80000F80000F80000F80000F80000F80000F80000F80000F8000 0F80000F80000F80000F80000F80000F80007FF8007FF80013237FA211>I<03FC1E0FFF7F1F0F 8F3E07CF3C03C07C03E07C03E07C03E07C03E07C03E03C03C03E07C01F0F801FFF0013FC003000 003000003800003FFF801FFFF00FFFF81FFFFC3800FC70003EF0001EF0001EF0001EF0001E7800 3C7C007C3F01F80FFFE001FF0018217E951C>II<1C003F007F007F007F003F001C000000000000000000000000000000 FF00FF001F001F001F001F001F001F001F001F001F001F001F001F001F001F001F001F001F001F 00FFE0FFE00B247EA310>I107 DIII<00FE0007FFC00F83E01E00F03E00F87C007C7C007C7C007CFC007EFC007EFC007EFC00 7EFC007EFC007EFC007E7C007C7C007C3E00F81F01F00F83E007FFC000FE0017167E951C>II<00FE030007FF87000FC1C7001F006F003F003F007E00 3F007E001F007C001F00FC001F00FC001F00FC001F00FC001F00FC001F00FC001F00FC001F007E 001F007E001F003E003F001F007F000FC1DF0007FF9F0001FC1F0000001F0000001F0000001F00 00001F0000001F0000001F0000001F0000001F000000FFE00000FFE01B207E951E>II<0FF3003FFF00781F00 600700E00300E00300F00300FC00007FE0007FF8003FFE000FFF0001FF00000F80C00780C00380 E00380E00380F00700FC0E00EFFC00C7F00011167E9516>I<0180000180000180000180000380 000380000780000780000F80003F8000FFFF00FFFF000F80000F80000F80000F80000F80000F80 000F80000F80000F80000F80000F80000F81800F81800F81800F81800F81800F830007C30003FE 0000F80011207F9F16>IIIIII<7FFFF07FFFF07C03E07007C0600FC0E01F80C01F00C03E00C07E0000FC0000F80001F00003 F03007E03007C0300F80701F80703F00603E00E07C03E0FFFFE0FFFFE014167E9519>I E /Fv 48 123 df<00001FF801FE000001FFFE1FFF80000FFFFF7FFFC0001FF00FFF0FE0007F80 1FFC1FF000FF003FF81FF001FE003FF81FF001FE003FF01FF003FC003FF00FE003FC001FF007C0 03FC000FF0038003FC000FF0000003FC000FF0000003FC000FF0000003FC000FF0000003FC000F F0000003FC000FF0000003FC000FF00000FFFFFFFFFFF000FFFFFFFFFFF000FFFFFFFFFFF00003 FC000FF0000003FC000FF0000003FC000FF0000003FC000FF0000003FC000FF0000003FC000FF0 000003FC000FF0000003FC000FF0000003FC000FF0000003FC000FF0000003FC000FF0000003FC 000FF0000003FC000FF0000003FC000FF0000003FC000FF0000003FC000FF0000003FC000FF000 0003FC000FF0000003FC000FF0000003FC000FF0000003FC000FF0000003FC000FF0000003FC00 0FF0000003FC000FF0000003FC000FF0000003FC000FF000007FFFE1FFFFC0007FFFE1FFFFC000 7FFFE1FFFFC00034327FB130>11 D<0000700000F00001C0000380000780000F00001E00003C00 007C0000F80000F80001F00003F00003E00007E00007C0000FC0000FC0001F80001F80001F8000 3F00003F00003F00007F00007F00007E00007E00007E0000FE0000FE0000FE0000FE0000FE0000 FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE00007E00007E00007E0000 7F00007F00003F00003F00003F00001F80001F80001F80000FC0000FC00007C00007E00003E000 03F00001F00000F80000F800007C00003C00001E00000F000007800003800001C00000F0000070 14497AB520>40 DI< 1F003F807FC0FFE0FFE0FFF0FFF0FFF07FF03FF01F700070007000E000E000E001C001C0038003 8007000E001C00380010000C197A8A17>44 DI<0001E0000003E000000FE000007FE0001FFFE000FFFFE000FFBFE000E03FE0 00003FE000003FE000003FE000003FE000003FE000003FE000003FE000003FE000003FE000003F E000003FE000003FE000003FE000003FE000003FE000003FE000003FE000003FE000003FE00000 3FE000003FE000003FE000003FE000003FE000003FE000003FE000003FE000003FE000003FE000 003FE000003FE000003FE000003FE000003FE000003FE0007FFFFFF07FFFFFF07FFFFFF01C2E7A AD29>49 D<003FF00001FFFE0007FFFF800FC07FE01E001FF03C000FF87F0007FC7F8007FEFFC0 07FEFFC003FEFFC003FFFFC003FF7F8003FF7F8003FF3F0003FF000003FF000003FE000003FE00 0007FC000007FC00000FF800000FF000001FE000001FC000003F8000007F000000FE000001F800 0001F0000003E00000078007000F0007001E0007003C000F0078000E00F0000E01C0001E03FFFF FE07FFFFFE0FFFFFFE1FFFFFFE3FFFFFFE7FFFFFFCFFFFFFFCFFFFFFFCFFFFFFFC202E7CAD29> I<000FFC0000007FFF800001F01FE00003C00FF000070007F8000FE007FC000FF007FC001FF007 FE001FF807FE001FF807FE001FF807FE001FF807FE000FF007FC0007E007FC00018007FC000000 0FF80000000FF00000001FE00000001FC00000007F8000001FFE0000001FFC0000001FFF800000 001FF000000007F800000003FC00000003FE00000003FF00000001FF80000001FF800E0001FFC0 3F8001FFC07FC001FFC07FC001FFC0FFE001FFC0FFE001FFC0FFE001FF80FFE001FF80FFC003FF 007F8003FF003F0003FE001F0007FC000FE01FF80007FFFFE00001FFFF8000001FFC0000222E7D AD29>I<0000007800000000F800000001F800000003F800000007F800000007F80000000FF800 00001FF80000003FF80000007FF800000077F8000000F7F8000001E7F8000003C7F800000787F8 00000707F800000F07F800001E07F800003C07F800007807F800007007F80000F007F80001E007 F80003C007F800078007F8000F0007F8000F0007F8001E0007F8003C0007F800780007F800F000 07F800FFFFFFFFF0FFFFFFFFF0FFFFFFFFF000000FF80000000FF80000000FF80000000FF80000 000FF80000000FF80000000FF80000000FF80000000FF800000FFFFFF0000FFFFFF0000FFFFFF0 242E7EAD29>I<0C0000380FC003F80FFFFFF80FFFFFF00FFFFFE00FFFFFC00FFFFF800FFFFE00 0FFFFC000FFFF0000FFF00000F0000000F0000000F0000000F0000000F0000000F0000000F0000 000F0FF8000F7FFF000FFFFFC00FF01FE00F800FF00F0007F80E0007FC000003FC000003FE0000 03FE000003FF000003FF1E0003FF3F0003FF7F8003FFFF8003FFFFC003FFFFC003FEFF8003FEFF 8003FE7F0007FC7C0007F83C000FF01E001FE00FC07FC007FFFF8001FFFE00003FE000202E7CAD 29>I<00007F80000007FFF000001FC07800007F001C0000FC001E0001F8007E0003F800FF0007 F001FF000FF001FF000FE001FF001FE001FF003FE000FE003FE0007C003FC00000007FC0000000 7FC00000007FC0000000FFC3FF8000FFC7FFE000FFCFBFF000FFDC03F800FFF801FC00FFF001FE 00FFF000FF00FFE000FF80FFE000FF80FFE000FF80FFC000FFC0FFC000FFC0FFC000FFC07FC000 FFC07FC000FFC07FC000FFC07FC000FFC03FC000FFC03FC000FF803FC000FF801FE000FF801FE0 00FF000FE001FE0007F001FC0003F803F80001FC0FF00000FFFFE000003FFF80000007FC000022 2E7DAD29>I<38000000003E000000003FFFFFFFC03FFFFFFFC03FFFFFFFC03FFFFFFF807FFFFF FF007FFFFFFE007FFFFFFC007FFFFFF8007FFFFFF000780001F000700003E000700007C000F000 0F8000E0000F0000E0001E0000E0003E000000007C00000000F800000000F800000001F0000000 01F000000003F000000003E000000007E000000007E00000000FE00000000FE00000001FC00000 001FC00000001FC00000003FC00000003FC00000003FC00000003FC00000003FC00000007FC000 00007FC00000007FC00000007FC00000007FC00000007FC00000007FC00000007FC00000003F80 0000003F800000000E00000022307BAF29>I<0007FC0000003FFF800000F80FE00001E003F000 03C001F800078000FC000F8000FC000F80007E000F80007E001F80007E001FC0007E001FC0007E 001FE0007E001FF800FC001FFE00FC000FFF81F8000FFFE1F0000FFFF7E00007FFFFC00003FFFF 800001FFFFC00000FFFFE000007FFFF80001FFFFFC0003F7FFFE000FE1FFFF001F807FFF003F00 3FFF803F000FFF807E0003FFC07E0000FFC0FC00007FC0FC00001FC0FC00001FC0FC00000FC0FC 00000FC0FC00000F807E00000F807E00000F007F00001F003F00001E001FC0007C000FF803F800 03FFFFE00000FFFF8000001FFC0000222E7DAD29>I<0000007C0000000000007C000000000000 FE000000000000FE000000000000FE000000000001FF000000000001FF000000000003FF800000 000003FF800000000007FFC00000000007FFC00000000007FFC0000000000FFFE0000000000F7F E0000000001F7FF0000000001E3FF0000000001E3FF0000000003E3FF8000000003C1FF8000000 007C1FFC00000000780FFC00000000780FFC00000000F80FFE00000000F007FE00000001F007FF 00000001E003FF00000001E003FF00000003E003FF80000003C001FF80000007C001FFC0000007 8000FFC00000078000FFC000000FFFFFFFE000000FFFFFFFE000001FFFFFFFF000001E00003FF0 00001E00003FF000003C00003FF800003C00001FF800007C00001FFC00007800000FFC00007800 000FFC0000F0000007FE0000F0000007FE0001F0000007FF0003F8000003FF00FFFFC001FFFFFE FFFFC001FFFFFEFFFFC001FFFFFE37317DB03E>65 D<000003FF80018000003FFFF003800001FF FFFC0F800007FF007F1F80001FF8000FBF80003FE00003FF8000FF800000FF8001FF0000007F80 03FE0000003F8007FC0000003F8007FC0000001F800FF80000001F801FF80000000F801FF00000 000F803FF000000007803FF000000007807FF000000007807FE000000007807FE000000000007F E00000000000FFE00000000000FFE00000000000FFE00000000000FFE00000000000FFE0000000 0000FFE00000000000FFE00000000000FFE00000000000FFE000000000007FE000000000007FE0 00000000007FE000000000007FF000000003803FF000000003803FF000000003801FF000000003 801FF800000007800FF8000000070007FC000000070007FC0000000E0003FE0000001E0001FF00 00003C0000FF8000007800003FE00000F000001FF80003E0000007FF003F80000001FFFFFE0000 00003FFFF80000000003FF80000031317BB03C>67 D69 DI<000003FF80018000003FFFF003800001FFFFFC0F800007FF007F1F80 001FF8000FBF80003FE00003FF8000FF800000FF8001FF0000007F8003FE0000003F8007FC0000 003F8007FC0000001F800FF80000001F801FF80000000F801FF00000000F803FF000000007803F F000000007807FF000000007807FE000000007807FE000000000007FE00000000000FFE0000000 0000FFE00000000000FFE00000000000FFE00000000000FFE00000000000FFE00000000000FFE0 0000000000FFE00000000000FFE00007FFFFFE7FE00007FFFFFE7FE00007FFFFFE7FE0000001FF 807FF0000001FF803FF0000001FF803FF0000001FF801FF0000001FF801FF8000001FF800FF800 0001FF8007FC000001FF8007FC000001FF8003FE000001FF8001FF000001FF8000FF800001FF80 003FE00003FF80001FF80007FF800007FF803F3F800001FFFFFE1F8000003FFFF80780000003FF C0018037317BB041>I73 D76 DI80 D82 D<001FF0018000FFFF038003FFFFC78007F00FFF800F8001FF801F00007F803F00001F803E0000 0F807E00000F807E00000780FE00000780FE00000780FE00000380FF00000380FF00000380FF80 000000FFE00000007FFC0000007FFFE000007FFFFE00003FFFFFC0001FFFFFF0001FFFFFF8000F FFFFFC0003FFFFFE0001FFFFFF00007FFFFF80001FFFFF800000FFFFC0000007FFC0000000FFE0 0000003FE00000003FE00000001FE06000001FE0E000000FE0E000000FE0E000000FE0E000000F C0F000000FC0F000000FC0F800001F80FC00001F80FF00003F00FFC0007E00FFFC01FC00F1FFFF F800E03FFFE000C007FF000023317BB02E>I<3FFFFFFFFFFF003FFFFFFFFFFF003FFFFFFFFFFF 003FE00FFC01FF007F000FFC003F807E000FFC001F807C000FFC000F8078000FFC00078078000F FC00078070000FFC00038070000FFC00038070000FFC00038070000FFC000380E0000FFC0001C0 E0000FFC0001C0E0000FFC0001C0E0000FFC0001C000000FFC00000000000FFC00000000000FFC 00000000000FFC00000000000FFC00000000000FFC00000000000FFC00000000000FFC00000000 000FFC00000000000FFC00000000000FFC00000000000FFC00000000000FFC00000000000FFC00 000000000FFC00000000000FFC00000000000FFC00000000000FFC00000000000FFC0000000000 0FFC00000000000FFC00000000000FFC00000000000FFC00000000000FFC00000000000FFC0000 0000000FFC00000000000FFC00000000000FFC000000007FFFFFFF8000007FFFFFFF8000007FFF FFFF800032307DAF39>I<007FF8000003FFFF000007FFFFC0000FE01FE0001FF007F0001FF003 F8001FF003FC001FF001FE000FE001FE0007C001FE00010001FE00000001FE00000001FE000001 FFFE00003FFFFE0001FFF1FE0007FE01FE000FF001FE001FC001FE003F8001FE007F8001FE00FF 0001FE00FF0001FE00FF0001FE00FF0001FE00FF0003FE007F8003FE007FC00EFE003FF03CFF00 0FFFF87FF807FFF03FF800FF800FF825207E9F28>97 D<0007FF00007FFFE000FFFFF003FC03F8 07F007FC0FE007FC1FE007FC3FC007FC3FC003F87FC001F07F8000407F800000FF800000FF8000 00FF800000FF800000FF800000FF800000FF800000FF8000007F8000007FC000007FC000003FC0 000E3FE0000E1FE0001C0FF0001C07F8007803FF01F000FFFFE0007FFF800007FC001F207D9F25 >99 D<00000007E0000003FFE0000003FFE0000003FFE00000003FE00000001FE00000001FE000 00001FE00000001FE00000001FE00000001FE00000001FE00000001FE00000001FE00000001FE0 0000001FE00000001FE00000001FE0000FF81FE0007FFF1FE001FFFFDFE003FE03FFE007F800FF E00FE0003FE01FE0001FE03FC0001FE03FC0001FE07F80001FE07F80001FE07F80001FE0FF8000 1FE0FF80001FE0FF80001FE0FF80001FE0FF80001FE0FF80001FE0FF80001FE0FF80001FE07F80 001FE07F80001FE07F80001FE03FC0001FE03FC0001FE01FC0003FE00FE0007FE007F001FFE003 FC07DFF001FFFF9FFF007FFE1FFF000FF01FFF28327DB12E>I<0007FC0000003FFF800000FFFF E00003FC07F00007F801F8000FE000FC001FE0007E003FC0007E003FC0003F007FC0003F007F80 003F007F80003F80FF80003F80FF80003F80FFFFFFFF80FFFFFFFF80FFFFFFFF80FF80000000FF 80000000FF800000007F800000007F800000003FC00000003FC00003801FC00003801FE0000780 0FF0000F0007F8001E0003FE00FC0000FFFFF800003FFFE0000003FF000021207E9F26>I<0000 FF000007FFC0001FFFE0003FC7F0007F0FF800FE0FF801FE0FF801FC0FF803FC07F003FC03E003 FC01C003FC000003FC000003FC000003FC000003FC000003FC000003FC0000FFFFF800FFFFF800 FFFFF80003FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC00 0003FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC 000003FC000003FC000003FC000003FC000003FC000003FC000003FC00007FFFF0007FFFF0007F FFF0001D327EB119>I<001FF007E000FFFE3FF001FFFF7FF807F83FF1F80FE00FE1F80FE00FE0 F01FC007F0601FC007F0003FC007F8003FC007F8003FC007F8003FC007F8003FC007F8001FC007 F0001FC007F0000FE00FE0000FE00FE00007F83FC00007FFFF000006FFFE00000E1FF000000E00 0000001E000000001E000000001F000000001F800000001FFFFFC0000FFFFFF8000FFFFFFE0007 FFFFFF0003FFFFFF8007FFFFFFC01FFFFFFFE03F00007FE07E00000FF0FC000007F0FC000003F0 FC000003F0FC000003F0FC000003F07E000007E03F00000FC01FC0003F800FF801FF0007FFFFFE 0000FFFFF000001FFF8000252F7E9F29>I<01F800000000FFF800000000FFF800000000FFF800 0000000FF80000000007F80000000007F80000000007F80000000007F80000000007F800000000 07F80000000007F80000000007F80000000007F80000000007F80000000007F80000000007F800 00000007F80000000007F807F8000007F83FFF000007F87FFF800007F8F03FC00007F9C01FE000 07FB000FE00007FE000FF00007FE000FF00007FC000FF00007FC000FF00007F8000FF00007F800 0FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF000 07F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F800 0FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF000FFFFC1FFFF80FFFFC1FFFF80 FFFFC1FFFF8029327DB12E>I<01C00007F0000FF8000FF8001FFC001FFC001FFC000FF8000FF8 0007F00001C00000000000000000000000000000000000000000000000000001F800FFF800FFF8 00FFF8000FF80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F8 0007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F8 00FFFF80FFFF80FFFF8011337DB217>I<01F800FFF800FFF800FFF8000FF80007F80007F80007 F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007 F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007 F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007 F800FFFFC0FFFFC0FFFFC012327DB117>108 D<03F007F8000FF000FFF03FFF007FFE00FFF07F FF80FFFF00FFF0F03FC1E07F800FF1C01FE3803FC007F3000FE6001FC007F6000FFC001FE007FE 000FFC001FE007FC000FF8001FE007FC000FF8001FE007F8000FF0001FE007F8000FF0001FE007 F8000FF0001FE007F8000FF0001FE007F8000FF0001FE007F8000FF0001FE007F8000FF0001FE0 07F8000FF0001FE007F8000FF0001FE007F8000FF0001FE007F8000FF0001FE007F8000FF0001F E007F8000FF0001FE007F8000FF0001FE007F8000FF0001FE007F8000FF0001FE007F8000FF000 1FE007F8000FF0001FE007F8000FF0001FE0FFFFC1FFFF83FFFFFFFFC1FFFF83FFFFFFFFC1FFFF 83FFFF40207D9F45>I<03F007F80000FFF03FFF0000FFF07FFF8000FFF0F03FC0000FF1C01FE0 0007F3000FE00007F6000FF00007FE000FF00007FC000FF00007FC000FF00007F8000FF00007F8 000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF0 0007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8 000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF000FFFFC1FFFF80FFFFC1FFFF 80FFFFC1FFFF8029207D9F2E>I<0007FE0000003FFFC00000FFFFF00003FC03FC0007F000FE00 0FE0007F001FC0003F803FC0003FC03FC0003FC07F80001FE07F80001FE07F80001FE0FF80001F F0FF80001FF0FF80001FF0FF80001FF0FF80001FF0FF80001FF0FF80001FF0FF80001FF07F8000 1FE07F80001FE07F80001FE03FC0003FC03FC0003FC01FE0007F800FE0007F0007F801FE0003FE 07FC0001FFFFF800003FFFC0000007FE000024207E9F29>I<01F80FF000FFF87FFE00FFF9FFFF 80FFFFE07FC00FFF001FE007FE000FF007F80007F807F80007FC07F80003FC07F80003FE07F800 03FE07F80001FE07F80001FF07F80001FF07F80001FF07F80001FF07F80001FF07F80001FF07F8 0001FF07F80001FF07F80001FE07F80003FE07F80003FE07F80003FC07F80007FC07FC0007F807 FE000FF007FF001FE007FBE07FC007F9FFFF0007F87FFE0007F81FE00007F800000007F8000000 07F800000007F800000007F800000007F800000007F800000007F800000007F800000007F80000 0007F8000000FFFFC00000FFFFC00000FFFFC00000282E7E9F2E>I<0007F800E0007FFE01E000 FFFF83E003FE07C3E007F801E7E00FF0007FE01FE0007FE03FE0003FE03FC0001FE07FC0001FE0 7FC0001FE07F80001FE0FF80001FE0FF80001FE0FF80001FE0FF80001FE0FF80001FE0FF80001F E0FF80001FE0FF80001FE07F80001FE07FC0001FE07FC0001FE03FC0001FE03FE0003FE01FE000 7FE00FF000FFE007F801FFE003FE07DFE001FFFF1FE0007FFE1FE0000FF01FE00000001FE00000 001FE00000001FE00000001FE00000001FE00000001FE00000001FE00000001FE00000001FE000 00001FE00000001FE0000003FFFF000003FFFF000003FFFF282E7D9F2C>I<03F03F00FFF07FC0 FFF1FFE0FFF3C7F00FF38FF807F70FF807F60FF807FE0FF807FC07F007FC03E007FC008007F800 0007F8000007F8000007F8000007F8000007F8000007F8000007F8000007F8000007F8000007F8 000007F8000007F8000007F8000007F8000007F8000007F8000007F80000FFFFE000FFFFE000FF FFE0001D207E9F22>I<00FF870007FFEF001FFFFF003F007F003C001F0078000F00F8000700F8 000700F8000700FC000700FF000000FFF800007FFFC0003FFFF0003FFFFC000FFFFE0007FFFF00 01FFFF80001FFF800000FFC000001FC060000FC0E00007C0E00007C0F00007C0F8000780F8000F 80FE000F00FF803E00FFFFFC00F3FFF800C07FC0001A207D9F21>I<0038000038000038000038 0000380000780000780000780000F80000F80001F80003F80007F8001FF800FFFFFEFFFFFEFFFF FE07F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F8 0007F80007F80007F80007F80707F80707F80707F80707F80707F80707F80703F80E03FC0E01FE 1C00FFF8007FF0000FE0182E7EAD20>I<01F80003F000FFF801FFF000FFF801FFF000FFF801FF F0000FF8001FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007 F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000F F00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007 F8000FF00007F8001FF00007F8001FF00003F8003FF00003F8006FF00001FE03CFF80000FFFF8F FF80007FFF0FFF80000FFC0FFF8029207D9F2E>III II<3FFFFFFC3FFFFFFC3FFFFFFC3FC00FF83E001FF03C003FF038003FE078007FC07800FF 807001FF807001FF007003FE007007FC00000FF800000FF800001FF000003FE000007FC00E007F C00E00FF800E01FF000E03FE000E07FE001E07FC001E0FF8001C1FF0003C3FF0007C3FE000FC7F C007FCFFFFFFFCFFFFFFFCFFFFFFFC1F207E9F25>I E /Fw 47 125 df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x 7 117 df<0000E000000000E000000001F000000001F000000001F000000003F80000 0003F800000006FC00000006FC0000000EFE0000000C7E0000000C7E000000183F000000183F00 0000303F800000301F800000701FC00000600FC00000600FC00000C007E00000FFFFE00001FFFF F000018003F000018003F000030001F800030001F800060001FC00060000FC000E0000FE00FFE0 0FFFE0FFE00FFFE0231F7E9E28>65 D<07FC001FFF003F0F803F07C03F03E03F03E00C03E00003 E0007FE007FBE01F03E03C03E07C03E0F803E0F803E0F803E0FC05E07E0DE03FF8FE0FE07E1714 7F9319>97 DI<01FE0007FF801F0F C03E0FC03E0FC07C0FC07C0300FC0000FC0000FC0000FC0000FC0000FC00007C00007E00003E00 603F00C01F81C007FF0001FC0013147E9317>I114 D<0FE63FFE701E600EE006E006F800FFC07FF83FFC1FFE03FE001FC007 C007E007F006F81EFFFCC7F010147E9315>I<01800180018003800380038007800F803F80FFFC FFFC0F800F800F800F800F800F800F800F800F800F800F860F860F860F860F8607CC03F801F00F 1D7F9C14>I E /Fy 62 123 df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z 3 122 df<040004000400C460E4E03F800E003F80E4E0C4600400040004000B0D7E8D11> 3 D<040E0E1C1C1C38383070706060C0C0070F7F8F0A>48 D<0C000C000C000C000C000C00FFC0 FFC00C000C000C000C000C000C000C000C000C000C000C000C000C000C000C000C000C000C000A 1A7E9310>121 D E /FA 10 122 df0 D<400020C000606000C03001801803000C0600060C0003180001B00000E00000E00001B0000318 00060C000C06001803003001806000C0C0006040002013147A9320>2 D<018001800180018041 82F18F399C0FF003C003C00FF0399CF18F4182018001800180018010127E9215>I17 D<000000040000000002000000000200000000010000000000800000000040FFFFFFFF F8FFFFFFFFF8000000004000000000800000000100000000020000000002000000000400250E7E 902A>33 D<0000600000600000C00000C000018000018000018000030000030000060000060000 0C00000C0000180000180000180000300000300000600000600000C00000C00001800001800003 00000300000300000600000600000C00000C000018000018000030000030000030000060000060 0000C0000040000013287A9D00>54 D<003C00E001C00180038003800380038003800380038003 8003800380038003800380030007001C00F0001C00070003000380038003800380038003800380 038003800380038003800380018001C000E0003C0E297D9E15>102 D I106 D<02000700070007000700070007000200020002007270FFF8727002 000200070007000700070007000700070007000700070007000700070007000700070007000200 020002000200020002000D267E9C12>121 D E /FB 51 123 df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df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df<0000000C000000000000001E000000000000001E000000000000001E00 0000000000003F000000000000003F000000000000003F000000000000007F800000000000007F 800000000000007F80000000000000FFC0000000000000FFC0000000000000FFC0000000000001 9FE00000000000019FE00000000000039FF00000000000030FF00000000000030FF00000000000 070FF800000000000607F800000000000607F800000000000C07FC00000000000C03FC00000000 000C03FC00000000001803FE00000000001801FE00000000001801FE00000000003000FF000000 00003000FF00000000003000FF000000000060007F800000000060007F8000000000E0007F8000 000000C0003FC000000000C0003FC000000001C0003FE00000000180001FE00000000180001FE0 0000000380001FF000000003FFFFFFF000000003FFFFFFF000000007FFFFFFF800000006000007 F800000006000007F80000000C000007FC0000000C000003FC0000000C000003FC000000180000 01FE00000018000001FE00000018000001FE00000030000000FF00000030000000FF0000007000 0000FF800000600000007F800000600000007F800000F00000007FC00003F80000007FC0000FFC 000001FFE000FFFFC0003FFFFFC0FFFFC0003FFFFFC0FFFFC0003FFFFFC03A3D7DBC41>65 D70 D73 D<003FC0000001FFF8000007C07E00000E001F00001E000F80001F8007 C0003FC003E0003FC003F0003FC003F0003FC001F0001F8001F8000F0001F800000001F8000000 01F800000001F800000001F80000001FF8000007FFF800003FF9F80000FF01F80003F801F80007 F001F8000FC001F8001F8001F8003F8001F8007F0001F8007F0001F8007E0001F80CFE0001F80C FE0001F80CFE0001F80CFE0003F80CFE0003F80C7E0006F80C7F0006F80C3F000C7C181F80187E 380FE0703FF003FFE01FE0007F800F8026287CA62B>97 D<0003FE00001FFFC0003E01F000F800 3801F0003C03E000FC07C001FE0FC001FE0F8001FE1F8001FE3F0000FC3F0000787F0000007F00 00007E0000007E000000FE000000FE000000FE000000FE000000FE000000FE000000FE000000FE 000000FE0000007E0000007F0000007F0000003F0000033F0000031F8000031F8000060FC00006 07C0000C03E0001801F0003000F800E0007E03C0001FFF000003FC0020287DA626>99 D<00000007E000000003FFE000000003FFE000000003FFE0000000001FE00000000007E0000000 0007E00000000007E00000000007E00000000007E00000000007E00000000007E00000000007E0 0000000007E00000000007E00000000007E00000000007E00000000007E00000000007E0000000 0007E00000000007E00000000007E0000001FC07E000000FFF87E000003F03C7E00000FC0067E0 0001F00037E00003E0001FE00007C0000FE0000FC0000FE0000F800007E0001F800007E0003F00 0007E0003F000007E0007F000007E0007E000007E0007E000007E000FE000007E000FE000007E0 00FE000007E000FE000007E000FE000007E000FE000007E000FE000007E000FE000007E000FE00 0007E0007E000007E0007E000007E0007F000007E0003F000007E0003F000007E0001F000007E0 001F80000FE0000F80000FE00007C0001FE00003E0001FE00001F00077F80000F800E7FFC0003E 0387FFC0001FFF07FFC00003FC07E0002A3D7DBB30>I<0003FC0000001FFF0000007E07C00000 F801E00001E000F00003E000780007C0007C000F80003E001F80003E001F00003F003F00001F00 3F00001F007F00001F007E00001F807E00001F80FE00001F80FE00001F80FFFFFFFF80FFFFFFFF 80FE00000000FE00000000FE00000000FE00000000FE000000007E000000007E000000007F0000 00007F000000003F000001803F000001801F800001800F800003000FC000030007C000060003E0 000C0001F00018000078007000003F01E000000FFF80000001FE000021287EA626>I<0007F001 F8003FFE0FFC007C1F1E3E00F007F03E01E003E01C03C001E00807C001F0000FC001F8000F8000 F8001F8000FC001F8000FC001F8000FC001F8000FC001F8000FC001F8000FC001F8000FC000F80 00F8000FC001F80007C001F00003C001E00001E003C00001F0078000037C1F0000073FFE000006 07F0000006000000000E000000000E000000000E000000000F000000000F0000000007C0000000 07FFFFC00003FFFFFC0001FFFFFF0000FFFFFF8003FFFFFFC00780003FE01F000007F03E000001 F83C000000F87C000000F8780000007CF80000007CF80000007CF80000007CF80000007CF80000 007C7C000000F87C000000F83E000001F01F000003E00F800007C003E0001F0001FC00FE00003F FFF0000007FF800027397EA52B>103 D<01F800000000FFF800000000FFF800000000FFF80000 000007F80000000001F80000000001F80000000001F80000000001F80000000001F80000000001 F80000000001F80000000001F80000000001F80000000001F80000000001F80000000001F80000 000001F80000000001F80000000001F80000000001F80000000001F80000000001F807F8000001 F81FFE000001F8781F000001F8E00F800001F98007C00001FB0007E00001FB0003E00001FE0003 F00001FC0003F00001FC0003F00001FC0003F00001F80003F00001F80003F00001F80003F00001 F80003F00001F80003F00001F80003F00001F80003F00001F80003F00001F80003F00001F80003 F00001F80003F00001F80003F00001F80003F00001F80003F00001F80003F00001F80003F00001 F80003F00001F80003F00001F80003F00001F80003F00001F80003F00001F80003F00001F80003 F00003FC0007F800FFFFF1FFFFE0FFFFF1FFFFE0FFFFF1FFFFE02B3C7EBB30>I<01C00003F000 07F8000FF8000FF8000FF80007F80003F00001C000000000000000000000000000000000000000 00000000000000000000000000000001F8007FF8007FF8007FF80007F80001F80001F80001F800 01F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F800 01F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F800 03FC00FFFFE0FFFFE0FFFFE0133A7FB917>I<01F800FFF800FFF800FFF80007F80001F80001F8 0001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F8 0001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F8 0001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F8 0001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80003FC00FFFFF0FFFF F0FFFFF0143C7FBB17>108 D<01F807F80000FFF81FFE0000FFF8781F0000FFF8E00F800007F9 8007C00001FB0007E00001FB0003E00001FE0003F00001FC0003F00001FC0003F00001FC0003F0 0001F80003F00001F80003F00001F80003F00001F80003F00001F80003F00001F80003F00001F8 0003F00001F80003F00001F80003F00001F80003F00001F80003F00001F80003F00001F80003F0 0001F80003F00001F80003F00001F80003F00001F80003F00001F80003F00001F80003F00001F8 0003F00001F80003F00001F80003F00001F80003F00003FC0007F800FFFFF1FFFFE0FFFFF1FFFF E0FFFFF1FFFFE02B267EA530>110 D<0001FE0000000FFFC000003F03F00000F8007C0001F000 3E0003E0001F0007C0000F800F800007C00F800007C01F000003E03F000003F03F000003F03F00 0003F07E000001F87E000001F87E000001F8FE000001FCFE000001FCFE000001FCFE000001FCFE 000001FCFE000001FCFE000001FCFE000001FCFE000001FC7E000001F87E000001F87F000003F8 3F000003F03F000003F01F000003E01F800007E00F800007C007C0000F8003E0001F0001F0003E 0000F8007C00003F03F000000FFFC0000001FE000026287EA62B>I<01F80FF00000FFF83FFE00 00FFF8F01F0000FFF9C007C00003FB0003E00001FE0001F00001FC0000F80001FC0000FC0001F8 00007E0001F800007E0001F800007F0001F800003F0001F800003F8001F800003F8001F800001F 8001F800001FC001F800001FC001F800001FC001F800001FC001F800001FC001F800001FC001F8 00001FC001F800001FC001F800001FC001F800001F8001F800003F8001F800003F8001F800003F 0001F800007F0001F800007E0001F80000FC0001FC0000FC0001FC0001F80001FE0003F00001FB 0007E00001F9C00FC00001F8F03F000001F83FFC000001F80FE0000001F80000000001F8000000 0001F80000000001F80000000001F80000000001F80000000001F80000000001F80000000001F8 0000000001F80000000001F80000000001F80000000003FC00000000FFFFF0000000FFFFF00000 00FFFFF00000002A377EA530>I<03F00F80FFF03FE0FFF070F0FFF0C1F807F183F801F303F801 F603F801F601F001F6004001FC000001FC000001FC000001FC000001F8000001F8000001F80000 01F8000001F8000001F8000001F8000001F8000001F8000001F8000001F8000001F8000001F800 0001F8000001F8000001F8000001F8000001F8000001F8000001F8000001F8000003FC0000FFFF F800FFFFF800FFFFF8001D267EA522>114 D<007F818003FFE3800F807F801E001F8038000780 380007807000038070000380F0000180F0000180F0000180F8000180F8000180FE0000007F8000 007FFC00003FFFC0001FFFF0000FFFFC0003FFFE0001FFFF00001FFF800000FF8000001FC04000 0FC0C00007E0C00003E0E00001E0E00001E0E00001E0E00001E0F00001E0F00001C0F80003C0FC 000380FC000700F7000E00E3C03C00C0FFF800C03FC0001B287DA622>I<001800000018000000 180000001800000018000000380000003800000038000000380000007800000078000000F80000 00F8000001F8000003F8000007F800001FFFFF00FFFFFF00FFFFFF0001F8000001F8000001F800 0001F8000001F8000001F8000001F8000001F8000001F8000001F8000001F8000001F8000001F8 000001F8000001F8000001F8000001F8000001F8000001F8018001F8018001F8018001F8018001 F8018001F8018001F8018001F8018001F8018000F8018000F8030000FC0300007C0300003E0600 001F0C00000FF8000003F00019367EB421>I<01F80003F000FFF801FFF000FFF801FFF000FFF8 01FFF00007F8000FF00001F80003F00001F80003F00001F80003F00001F80003F00001F80003F0 0001F80003F00001F80003F00001F80003F00001F80003F00001F80003F00001F80003F00001F8 0003F00001F80003F00001F80003F00001F80003F00001F80003F00001F80003F00001F80003F0 0001F80003F00001F80003F00001F80003F00001F80003F00001F80003F00001F80007F00001F8 0007F00001F80007F00000F8000FF00000F8000FF00000FC001BF000007C001BFC00003E0073FF E0001F81E3FFE0000FFF83FFE00001FE03F0002B277EA530>I E end %%EndProlog %%BeginSetup %%Feature: *Resolution 300 TeXDict begin @a4 %%EndSetup %%Page: 1 1 bop 181 509 a FD(F)-7 b(unctional)30 b(In)n(tegrals)g(and)e(their)i (Applications)143 616 y FC(Notes)16 b(for)g(a)h(course)f(for)h(the)f(\\T)l (roisi)o(\022)-23 b(eme)14 b(Cycle)h(de)i(la)f(Ph)o(ysique)f(en)h(Suisse)g (Romande")358 672 y(giv)o(en)g(in)g(Lausanne,)h(Switzerland,)e(during)h(the)g (summer)e(of)j(1992)761 820 y FB(Da)n(vid)i(C.)h(Brydges)1227 799 y FA(\003)624 895 y FB(Departmen)n(t)g(of)g(Mathematics)710 970 y(Univ)n(ersit)n(y)f(of)h(Virginia)659 1045 y(Charlottesville,)d(V)-7 b(A)20 b(22903)932 1119 y(USA)878 1224 y(Notes)f(b)n(y)745 1299 y(Rob)r(erto)g(F)-5 b(ern\023)-30 b(andez)1243 1277 y FA(y)585 1373 y FB(Institut)20 b(de)g(Ph)n(ysique)e(Theorique)439 1448 y(Ecole)g(P)n(olytec)n(hnique)g(F)-5 b(ederale)19 b(de)h(Lausanne)759 1523 y(Ch-1015)g(Ecublens)767 1597 y(SWITZERLAND)p 75 2378 735 2 v 131 2409 a Fz(\003)150 2424 y Fy(This)14 b(w)o(ork)f(is)h(partially)e (supp)q(orted)j(b)o(y)f(NSF)g(gran)o(t)g(DMS)f(9102564)132 2459 y Fz(y)150 2474 y Fy(W)m(ork)g(supp)q(orted)i(b)o(y)f(the)g(F)m(onds)g (National)e(Suisse)p eop %%Page: 2 2 bop 894 307 a Fx(Abstract)265 391 y Fw(These)15 b(lectures)h(are)e(concerned) i(with)f(the)g(analysis)g(and)g(applications)h(of)e(functional)197 448 y(in)o(tegrals)g(de\014ned)h(b)o(y)e(small)i(p)q(erturbations)f(of)f (Gaussian)g(measures.)20 b(The)13 b(cen)o(tral)h(topic)197 504 y(is)j(the)g(renormalization-group.)26 b(F)l(ollo)o(wing)18 b(Wilson)f(and)h(P)o(olc)o(hinski,)g(an)f(e\013ectiv)o(e)g(p)q(o-)197 561 y(ten)o(tial)11 b(is)g(studied)h(as)e(a)g(function)i(of)e(an)h (ultra-violet)g(cuto\013.)18 b(By)11 b(c)o(hanging)g(the)f(cuto\013)g(in)i(a) 197 617 y(con)o(tin)o(uous)i(manner,)f(one)h(obtains)f(a)h(di\013eren)o(tial) g(equation)g(for)f(the)g(e\013ectiv)o(e)h(p)q(oten)o(tial.)197 674 y(It)g(is)g(sho)o(wn)g(that)f(b)o(y)h(con)o(v)o(erting)g(this)g(equation) h(to)e(an)h(in)o(tegral)g(equation)h(and)f(generat-)197 730 y(ing)j(an)f(iterativ)o(e)h(solution)g(one)g(obtains)f(the)h(Ma)o(y)o(er)e (expansion)i(of)f(classical)i(statistical)197 787 y(mec)o(hanics.)26 b(Some)17 b(results)g(on)g(the)g(con)o(v)o(ergence)g(of)g(suc)o(h)g(an)g (expansion)h(are)e(deduced,)197 843 y(with)g(applications)h(to)e(Coulom)o(b)g (and)h(Y)l(uk)m(a)o(w)o(a)f(gases.)20 b(Ho)o(w)o(ev)o(er,)14 b(this)i(metho)q(d)g(of)f(solv-)197 900 y(ing)h(for)f(the)g(e\013ectiv)o(e)h (p)q(oten)o(tial)g(turns)f(out)h(to)e(b)q(e)i(of)f(limited)j(v)m(alue)e(due)g (to)f(\\large-\014eld)197 956 y(problems",)g(whic)o(h)h(w)o(e)e(explain.)22 b(T)l(o)14 b(ac)o(hiev)o(e)i(b)q(etter)e(results)i(w)o(e)e(abandon)h(the)g (e\013ectiv)o(e)197 1012 y(action)i(and)g(represen)o(t)h(the)f(partition)g (function)h(as)f(a)f(p)q(olymer)i(gas.)25 b(The)17 b(p)q(olymer)h(gas)197 1069 y(represen)o(tation)12 b(has)f(enough)h(in)g(common)f(with)h(the)g (e\013ectiv)o(e)g(action)f(that)g(w)o(e)h(are)f(able)h(to)197 1125 y(exactly)j(rep)q(eat)g(the)g(previous)h(metho)q(d)f(of)f(iterating)h (an)g(in)o(tegral)g(equation,)g(concluding)197 1182 y(with)d(the)g(\015o)o(w) g(of)f(the)h(activities)h(of)f(the)g(p)q(olymers)g(giv)o(en)g(b)o(y)g (\\cluster)h(expansions".)19 b(This)197 1238 y(is)g(exp)q(ounded)g(in)g (considerable)h(detail,)g(using)e(as)g(illustration)i(the)e(example)h(of)f (dip)q(ole)197 1295 y(gases.)g(In)c(addition,)g(there)e(are)h(t)o(w)o(o)e(in) o(tro)q(ductory)i(sections|indep)q(ende)q(n)o(t)i(of)d(the)h(other)197 1351 y(lectures|in)22 b(whic)o(h,)f(as)e(a)g(motiv)m(ation,)i(problems)f(in)h (the)e(theory)g(of)g(Coulom)o(b)h(gases)197 1408 y(and)d(p)q(olymer)g(ph)o (ysics)h(are)e(sho)o(wn)g(to)g(b)q(e)i(related)f(to)f(functional)i(in)o (tegrals)e(of)h(the)f(t)o(yp)q(e)197 1464 y(studied)i(here.)24 b(In)18 b(particular,)f(a)f(heuristic)i(discussion)h(is)e(presen)o(ted)g(on)g (ho)o(w)f(quan)o(tum)197 1521 y(e\013ects)f(can)g(destro)o(y)g(Deb)o(y)o(e)g (screening.)p eop %%Page: 3 3 bop 75 315 a Fv(Con)n(ten)n(ts)75 424 y Fu(1)45 b(In)n(tro)r(duction)1427 b(1)75 533 y(2)45 b(The)19 b(Sine-Gordon)f(T)-5 b(ransformation)918 b(4)148 594 y FC(2.1)50 b(Coulom)o(b)16 b(Systems)22 b Ft(:)j(:)f(:)h(:)f(:)h (:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f (:)h(:)f(:)h(:)f(:)g(:)95 b FC(6)148 654 y(2.2)50 b(Energy)17 b(and)g(Normal)d(Ordering)46 b Ft(:)25 b(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f (:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)95 b FC(8)148 714 y(2.3)50 b(The)17 b(Quan)o(tum)e(Statistical)g(Coulom)o(b)h(System)31 b Ft(:)24 b(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)95 b FC(8)148 774 y(2.4)50 b(Problems)12 b Ft(:)24 b(:)h(:)f(:)h(:)f(:)h(:)f(:)h (:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f (:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)71 b FC(15)75 883 y Fu(3)45 b(P)n(olymers,)17 b(Levy)g(Pro)r(cesses,)g(F)-5 b(unctional)19 b(In)n(tegrals)552 b(17)148 943 y FC(3.1)50 b(Levy)16 b(Pro)q(cesses)46 b Ft(:)24 b(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)h (:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)71 b FC(19)148 1003 y(3.2)50 b(F)l(urther)16 b(Dev)o(elopmen)o(ts)44 b Ft(:)24 b(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f (:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)71 b FC(22)75 1112 y Fu(4)45 b(Flo)n(w)20 b(Equations)1327 b(24)148 1173 y FC(4.1)50 b(P)o(erturbations)17 b(of)f(Gaussian)i(Measures)g Ft(:)24 b(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:) 71 b FC(24)148 1233 y(4.2)50 b(Wilson)17 b(Renormalization)41 b Ft(:)25 b(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h (:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)71 b FC(25)148 1293 y(4.3)50 b(Problems)12 b Ft(:)24 b(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f (:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f (:)g(:)71 b FC(29)75 1402 y Fu(5)45 b(The)19 b(E\013ectiv)n(e)d(Action)1200 b(31)148 1462 y FC(5.1)50 b(F)l(unctional)16 b(Equations)26 b Ft(:)f(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f (:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)71 b FC(31)148 1522 y(5.2)50 b(F)l(ormal)15 b(Series)h(for)g(the)g(Classical)g(Action)k Ft(:)25 b(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g (:)71 b FC(32)148 1583 y(5.3)50 b(F)l(ormal)15 b(Series)h(for)g(the)g(Quan)o (tum)f(Action)42 b Ft(:)24 b(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h (:)f(:)h(:)f(:)g(:)71 b FC(37)148 1643 y(5.4)50 b(Con)o(v)o(ergence)16 b(of)g(the)g(F)l(ormal)f(Series)38 b Ft(:)25 b(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:) g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)71 b FC(41)148 1703 y(5.5)50 b(Problems)12 b Ft(:)24 b(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)h (:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f (:)h(:)f(:)h(:)f(:)g(:)71 b FC(43)75 1812 y Fu(6)45 b(The)19 b(Ma)n(y)n(er)g(Expansion)1166 b(46)148 1872 y FC(6.1)50 b(The)17 b(Ursell)d(Co)q(e\016cien)o(ts)35 b Ft(:)24 b(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h (:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)71 b FC(46)260 1932 y(6.1.1)56 b(Equiv)m(alen)o(t)16 b(De\014nitions)43 b Ft(:)25 b(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f (:)h(:)f(:)h(:)f(:)g(:)71 b FC(47)260 1992 y(6.1.2)56 b(T)l(ree-Graph)17 b(Iden)o(tit)o(y)47 b Ft(:)24 b(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g (:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)71 b FC(50)260 2053 y(6.1.3)56 b(T)l(ree-Graph)17 b(Domination)42 b Ft(:)24 b(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f (:)g(:)71 b FC(52)148 2113 y(6.2)50 b(Con)o(v)o(ergence)16 b(of)g(Ma)o(y)o(er)f(Expansions)22 b Ft(:)j(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g (:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)71 b FC(53)148 2173 y(6.3)50 b(Application:)27 b(Con)o(v)o(ergence)19 b(of)h(the)f(Ma)o(y)o (er)f(Expansion)j(of)e(the)h(Y)l(uk)m(a)o(w)o(a)f(Gas)260 2233 y(in)d(Tw)o(o)h(Dimensions)31 b Ft(:)24 b(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f (:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)71 b FC(55)148 2293 y(6.4)50 b(E\013ectiv)o(e)15 b(P)o(oten)o(tials)h(as)h(Ma)o (y)o(er)e(Expansions)26 b Ft(:)f(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h (:)f(:)h(:)f(:)g(:)71 b FC(57)148 2354 y(6.5)50 b(Problems)12 b Ft(:)24 b(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h (:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)71 b FC(58)973 2599 y(iii)p eop %%Page: 4 4 bop 75 307 a Fu(7)45 b(Cluster)18 b(Expansions)1235 b(60)148 367 y FC(7.1)50 b(Manipulations)17 b(with)f Fs(E)t FC(xp)42 b Ft(:)25 b(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h (:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)71 b FC(67)75 476 y Fu(8)45 b(The)19 b(Renormalization)c(Group)k(\(R)n(G\))868 b(70)148 537 y FC(8.1)50 b(The)17 b(Choice)e(of)i(Large)g(Field)e(Regulator)42 b Ft(:)25 b(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f (:)g(:)71 b FC(72)148 597 y(8.2)50 b(Reblo)q(c)o(king)16 b(B)40 b Ft(:)25 b(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f (:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)71 b FC(74)148 657 y(8.3)50 b(Renormalization)15 b(\(Dip)q(ole)h(Systems\))g Ft(:)25 b(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)h (:)f(:)g(:)71 b FC(76)75 766 y Fu(References)1517 b(87)974 2599 y FC(iv)p eop %%Page: 1 5 bop 75 315 a Fv(1)83 b(In)n(tro)r(duction)75 425 y FC(These)16 b(lectures)f(are)h(concerned)g(with)g(the)g(analysis)g(and)h(applications)f (of)g(functional)g(in)o(tegrals)75 485 y(whic)o(h)g(lo)q(ok)g(lik)o(e)805 487 y Fr(Z)855 504 y(Y)872 591 y Fq(x)917 545 y Ft(d\036)p FC(\()p Ft(x)p FC(\))8 b Ft(e)1068 525 y Fp(\000)p Fq(S)r Fo(\()p Fq(\036)p Fo(\))1169 545 y FC(;)629 b(\(1)p Ft(:)p FC(1\))75 664 y(where)24 b(\\)248 630 y Fr(Q)288 674 y Fq(x)318 664 y Ft(d\036)p FC(\()p Ft(x)p FC(\)")h(denotes)g(a)g(v)o(ery)f(m)o(ysterious)f (\\measure"|often)h(denoted)h Fs(D)q Ft(\036)g FC(in)g(the)75 724 y(literature|and)580 794 y Ft(S)s FC(\()p Ft(\036)p FC(\))13 b(=)745 735 y Fr(Z)795 794 y Ft(dx)856 733 y Fr(\024)883 760 y FC(1)p 883 782 25 2 v 883 828 a(2)912 794 y(\()p Fs(r)p Ft(\036)p FC(\))1021 773 y Fo(2)1052 794 y FC(+)e Ft(v)r FC(\()p Ft(\036;)d Fs(r)p Ft(\036;)g(:)g(:)g(:)m FC(\))1364 733 y Fr(\025)1394 794 y Ft(:)404 b FC(\(1)p Ft(:)p FC(2\))75 897 y(In)17 b(the)f(ph)o(ysics)g (literature,)g(these)g(formal)g(expressions)h(are)g(mostly)e(used)i(to)g (generate)g(p)q(ertur-)75 957 y(bation)f(series,)e(semiclassical)e(appro)o (ximations)i(and)i(more)d(recen)o(tly)h(Mon)o(te-Carlo)h(algorithms.)75 1017 y(Since)c(Wilson's)g(in)o(v)o(en)o(tion)f(of)i(the)f(renormalization)f (group)i(\(R)o(G\))g([60])f(the)g(p)q(erturbation)i(theory)75 1077 y(of)k(these)f(in)o(tegrals)h(has)g(dev)o(elop)q(ed)f(in)o(to)g(a)h (deep)g(sub)s(ject)f(whic)o(h)g(at)h(least)g(for)g(small)e(p)q(erturba-)75 1138 y(tions)h Ft(v)i FC(is)e(b)o(y)g(no)o(w)h(w)o(ell)d(understo)q(o)q(d)k (and)f(w)o(e)f(will)f(ha)o(v)o(e)h(nothing)g(to)h(add)g(to)g(this)f(topic.) 148 1198 y(Instead,)i(w)o(e)f(w)o(ould)h(lik)o(e)e(to)i(concen)o(trate)f(on)h Fn(analytic)h FC(asp)q(ects.)27 b(W)l(e)17 b(shall)h(try)f(to)h(explain)75 1258 y(ho)o(w)k(one)h(can)f(con)o(trol)g(the)g(errors)g(b)q(ey)o(ond)g(p)q (erturbation)h(theory)f(so)g(that)h(the)f(Wilson)g(R)o(G)75 1318 y(b)q(ecomes)16 b(the)i(basis)g(for)g(a)g(mathematicall)o(y)c(complete)i (discussion)h(of)h(this)g(t)o(yp)q(e)f(of)h(functional)75 1378 y(in)o(tegral.)k(In)17 b(Wilson's)f(language,)i(w)o(e)e(will)g(b)q(e)h (concerned)f(with)g(the)h(\\irrelev)m(an)o(t")f(terms.)21 b(The)75 1439 y(question)d(is:)26 b(are)19 b(these)g(irrelev)m(an)o(t)e(terms)g Fn(r)n(e)n(al)r(ly)i FC(irrelev)m(an)o(t,)e(or)i(are)g(they)f(so)i(n)o (umerous)d(that)75 1499 y(their)h(o)o(v)o(erall)f(con)o(tribution)i(b)q (ecomes)e(comparable)h(to)h(that)g(of)g(the)f(\\relev)m(an)o(t")h(terms?)28 b(Here)75 1559 y(are)22 b(some)e(problems)h(that)h(re\015ect)e(this)i (question:)31 b(Is)22 b(there)f(Deb)o(y)o(e)f(screening)h(in)h(Quan)o(tum)75 1619 y(Coulom)o(b)13 b(systems?)20 b(Do)q(es)15 b(the)f(expansion)g(of)g(the) g(pressure)g(in)g(p)q(o)o(w)o(ers)g(of)h(activit)o(y)d(\(the)i(Ma)o(y)o(er)75 1679 y(expansion\))22 b(con)o(v)o(erge)e(for)i(a)g(gas)g(of)g(dip)q(oles)f (or)h(for)g(the)f(Kosterlitz-Thouless)g(phase)h(of)g(the)75 1740 y(classical)15 b(t)o(w)o(o)h(dimensional)e(Coulom)o(b)h(gas?)23 b(What)16 b(is)g(the)f(end-to-end)i(distance)f(of)g(a)g(p)q(olymer)75 1800 y(on)h(a)f(four)h(dimensional)e(lattice?)148 1860 y(A)f(program)h(lik)o (e)e(this)h(w)o(as,)h(after)g(Wilson,)f(\014rst)h(tak)o(en)f(up)h(b)o(y)f (Galla)o(v)o(otti)g(et)g(al.)21 b([7])14 b(in)g(their)75 1920 y(construction)j(of)g(sup)q(er-renormalizable)e(\014eld)h(theories)g(suc)o(h) h(as)g Ft(\036)1359 1902 y Fo(4)1359 1932 y(3)1378 1920 y FC(.)23 b(Since)16 b(then)g(a)h(n)o(um)o(b)q(er)e(of)75 1980 y(problems,)f(whic)o(h)g (as)i(an)g(optimistic)d(y)o(outh)j(I)f(w)o(ould)g(ha)o(v)o(e)g(considered)g (to)g(b)q(e)h(a)g(h)o(undred)f(y)o(ears)75 2041 y(in)k(the)g(future,)g(ha)o (v)o(e)f(b)q(een)h(successfully)f(attac)o(k)o(ed)g([19)q(,)g(3)q(,)g(31)q(,)h (40)q(].)29 b(Ho)o(w)o(ev)o(er)17 b(in)i(our)h(zeal)e(to)75 2101 y(go)e(ev)o(er)e(further)h(in)o(to)g(new)h(territory)e(w)o(e)h(ha)o(v)o (e)g(sometimes)d(neglected)i(to)i(explain)e(what)j(w)o(e)d(are)75 2161 y(doing)i(in)f(language)h(that)g(can)g(easily)e(b)q(e)i(understo)q(o)q (d)h(b)o(y)d(outsiders.)22 b(I)15 b(consider)g(these)g(lectures)75 2221 y(to)i(b)q(e)f(a)h(time)d(to)i Fn(b)n(e)n(gin\(!\))22 b FC(this)17 b(task.)k(The)16 b(plan)h(of)f(campaign)g(is)g(as)h(follo)o(ws.) 148 2281 y(T)l(o)i(displa)o(y)e(the)h(surprisingly)f(ric)o(h)g(range)i(of)f (applications)g(of)g(these)g(functional)f(in)o(tegrals,)75 2342 y(in)c(Sections)h(2)g(and)g(3)g(I)f(review)g(ho)o(w)g(problems)g(in)g (statistical)g(mec)o(hanics)e(suc)o(h)j(as)g(the)f(Coulom)o(b)75 2402 y(gas)k(and)g(p)q(olymer)d(ph)o(ysics)h(can)i(b)q(e)f(transformed)f(in)o (to)h(these)f(in)o(tegrals.)21 b(One)16 b(of)g(the)g(problems)75 2462 y(that)g(I)f(discuss)h(here)f(\(heuristically\))f(is)i(to)g(pro)o(v)o(e) e(that)j(quan)o(tum)d(e\013ects)h(destro)o(y)h(screening)f(in)982 2599 y(1)p eop %%Page: 2 6 bop 75 307 a FC(the)14 b(quan)o(tum)e(Coulom)o(b)h(gas.)21 b(These)14 b(in)o(tro)q(ductory)g(sections)f(are)h(purely)f(for)h(motiv)m (ation.)19 b(The)75 367 y(main)c(topics)h(are)g(indep)q(enden)o(t)g(of)g (these)g(sections.)148 428 y(In)h(Section)g(4)h(I)f(b)q(egin)h(the)f(main)f (topic|the)g(renormalization)g(group)j(\(R)o(G\).)d(A)h(standard)75 488 y(approac)o(h)h(is)f(to)h(in)o(tro)q(duce)f(an)g(ultra-violet)g(cuto\013) h(at)f(a)h(length)f(scale)g Ft(l)q FC(\()p Ft(t)p FC(\))f(in)o(to)h(the)g (Gaussian)75 548 y(part)f Ft(d\026)p FC(\()p Ft(\036)p FC(\))g(of)g(\(1.1\),) f(so)i(that)f(it)f(b)q(ecomes)907 512 y Fr(R)943 548 y Ft(d\026)997 555 y Fq(t)1012 548 y FC(\()p Ft(\036)p FC(\),)h(and)g(then)f(to)h(de\014ne)f (an)h(ev)o(olution)f(of)h(the)75 608 y(e\013ectiv)o(e)h(p)q(oten)o(tial)i Ft(V)503 615 y Fq(t)538 608 y FC(b)o(y)f(insisting)h(that)913 573 y Fr(R)949 608 y Ft(d\026)1003 615 y Fq(t)1018 608 y FC(\()p Ft(\036)p FC(\))p Ft(e)1108 590 y Fq(V)1129 594 y Fm(t)1164 608 y FC(remain)e(unc)o(hanged)j(when)f(the)g(cuto\013)75 668 y Ft(l)q FC(\()p Ft(t)p FC(\))f(is)g(c)o(hanged)h(b)o(y)f(v)m(arying)h(the)g (parameter)e Ft(t)p FC(.)29 b(In)18 b(Section)g(5)h(I)g(describ)q(e)f(ho)o(w) h(this)f(leads)h(to)75 729 y(a)i(di\013eren)o(tial)f(equation)g(for)h Ft(V)679 736 y Fq(t)715 729 y FC(and)h(attempt)d(to)i(solv)o(e)f(it)h(b)o(y)f (con)o(v)o(erting)g(it)g(to)h(an)g(in)o(tegral)75 789 y(equation)c(and)h (generating)f(a)h(series)e(b)o(y)h(iteration.)23 b(The)17 b(resulting)g (solution)g(is)g(closely)f(related)75 849 y(to)k(the)f(Ma)o(y)o(er)f (expansion)i(of)g(classical)e(statistical)h(mec)o(hanics)e(and)j(in)f (Section)g(6)h(I)f(pause)h(to)75 909 y(dev)o(elop)g(this)h(idea.)35 b(This)21 b(leads)g(in)o(to)f(some)g(prett)o(y)g(results)h(on)g(the)g(Y)l(uk) m(a)o(w)o(a)g(and)g(Coulom)o(b)75 969 y(gases)15 b(in)f(t)o(w)o(o)g (dimensions)e(but)i(is)g(of)h(limited)c(use)j(in)f(the)h(con)o(trol)g(of)g (the)g(R)o(G)g(b)q(ecause)g(this)g(is)g(an)75 1029 y(expansion)f(in)g(p)q(o)o (w)o(ers)g(of)g(the)g(initial)f(p)q(oten)o(tial)h Ft(V)1012 1036 y Fq(t)p Fo(=0)1072 1029 y FC(.)20 b(An)o(y)12 b(metho)q(d)g(based)i(on) f(the)g(con)o(v)o(ergence)75 1090 y(of)k(suc)o(h)g(an)g(expansion)h(is)e (unable)h(to)g(distinguish)g(b)q(et)o(w)o(een)f Ft(V)1270 1097 y Fq(t)p Fo(=0)1348 1090 y FC(and)h Fs(\000)p Ft(V)1510 1097 y Fq(t)p Fo(=0)1570 1090 y FC(,)g(y)o(et)f(the)h(c)o(hange)75 1150 y(from)g(e.g.,)h Ft(\036)327 1132 y Fo(4)366 1150 y FC(to)g Fs(\000)p Ft(\036)495 1132 y Fo(4)533 1150 y FC(is)h(a)g(c)o(hange)f(from)g (a)h(stable)f(to)h(an)g(unstable)g(in)o(teraction.)27 b(This)19 b(is)f(an)75 1210 y(instance)e(of)h(a)f(\\large)h(\014eld)e(problem".)148 1270 y(T)l(o)i(get)f(around)i(this)e(large)g(\014eld)g(problem,)e(in)i (Section)g(7)h(I)f(partly)g(abandon)i(the)e(represen-)75 1330 y(tation)h Ft(e)242 1312 y Fq(V)263 1316 y Fm(t)294 1330 y FC(in)f(fa)o(v)o(or)g(of)457 1457 y Ft(e)480 1436 y Fq(V)501 1440 y Fm(t)530 1457 y Fs(\000)-9 b(!)645 1403 y Fp(1)632 1415 y Fr(X)624 1507 y Fq(N)t Fo(=0)731 1423 y FC(1)p 714 1445 58 2 v 714 1491 a Ft(N)5 b FC(!)832 1415 y Fr(X)785 1507 y Fq(X)814 1512 y Fl(1)831 1507 y Fq(;:::;X)910 1513 y Fm(N)947 1415 y Fr(Y)965 1506 y Fq(j)1008 1457 y Ft(K)t FC(\()p Ft(X)1112 1464 y Fq(j)1131 1457 y Ft(;)j(\036)p FC(\))p Ft(e)1224 1431 y Fp(\000)1256 1400 y Fr(R)1276 1448 y Fk(\000)p Fm(U)1331 1431 y Fq(dx)e(v)1392 1435 y Fm(t)1406 1431 y Fo(\()p Fq(\036;)p Fp(r)p Fq(\036)p Fo(\))1517 1457 y Ft(:)75 1611 y FC(In)12 b(these)f(notes)i(w)o(e)e(only)h (ha)o(v)o(e)g(time)d(for)k(those)f(systems)f(where)h(the)g Ft(v)1373 1618 y Fq(t)1399 1611 y FC(can)g(b)q(e)g(tak)o(en)g(to)g(b)q(e)h (zero)1880 1593 y Fo(1)1899 1611 y FC(.)75 1671 y(The)20 b(function)f Ft(v)397 1678 y Fq(t)431 1671 y FC(con)o(tains)g(parts)i(of)e(the)h(original) f(e\013ectiv)o(e)f(action)h Ft(V)1448 1678 y Fq(t)1483 1671 y FC(whic)o(h)g(are)h(dominan)o(t)75 1732 y(and)i(pla)o(y)f(an)h(essen)o (tial)e(role)h(in)g(main)o(taining)f(stabilit)o(y|for)g(the)i Ft(\036)1416 1714 y Fo(4)1416 1744 y(4)1457 1732 y FC(theory)f(it)g(w)o(ould) g(ha)o(v)o(e)75 1801 y(the)f(form)286 1781 y Fq(\025)307 1785 y Fm(t)p 286 1789 35 2 v 290 1818 a Fo(4!)346 1801 y FC(:)f Ft(\036)408 1783 y Fo(4)447 1801 y FC(:)h(+)524 1776 y Fq(\026)545 1764 y Fl(2)545 1786 y Fm(t)p 524 1789 39 2 v 534 1818 a Fo(2)587 1801 y FC(:)f Ft(\036)649 1783 y Fo(2)689 1801 y FC(:)g(+)765 1781 y Fq(z)781 1785 y Fm(t)p 765 1789 31 2 v 771 1818 a Fo(2)820 1801 y FC(:)g(\()p Fs(r)p Ft(\036)p FC(\))962 1783 y Fo(2)1001 1801 y FC(:|whilst)g(the)h(functionals)f Ft(K)1591 1808 y Fq(t)1607 1801 y FC(\()p Ft(X)q(;)8 b(\036)p FC(\))20 b(con)o(tain)75 1861 y(corrections)g(whic)o(h)g(are)g(higher)g(order)g(in)g Ft(\025)p FC(.)34 b(The)21 b(argumen)o(t)e Ft(X)25 b FC(signi\014es)20 b(that)h Ft(K)1705 1868 y Fq(t)1740 1861 y FC(dep)q(ends)75 1921 y(on)h(the)f(\014eld)f Ft(\036)p FC(\()p Ft(x)p FC(\))h(only)g(for)h Ft(x)g Fs(2)h Ft(X)t FC(.)36 b(Eac)o(h)21 b(set)h Ft(X)j FC(is)c(called)g(a)g (p)q(olymer.)35 b(The)21 b(set)g Fs(\000)p Ft(U)27 b FC(is)75 1982 y(the)21 b(complemen)n(t)e(of)i(the)h(union)f(of)h(the)f(p)q(olymers)f Ft(X)1136 1989 y Fo(1)1156 1982 y Ft(;)8 b(:)g(:)g(:)g(;)g(X)1306 1989 y Fq(N)1340 1982 y FC(.)37 b(Under)20 b(the)i(action)f(of)h(the)75 2042 y(renormalisation)d(group)h(the)g(p)q(olymer)e(activities)g Ft(K)1100 2049 y Fq(t)1115 2042 y FC(\()p Ft(X)t FC(\))i(ev)o(olv)o(e)e(b)o (y)h(equations)h(that)g(can)g(b)q(e)75 2102 y(studied)d(in)h(v)o(ery)e(close) i(analogy)g(to)g(the)g(ones)g(satis\014ed)g(b)o(y)f(the)g(e\013ectiv)o(e)f (action)i Ft(V)1680 2109 y Fq(t)1695 2102 y FC(.)26 b(One)17 b(can)75 2162 y(write)h(b)q(eautiful)h(series)f(solutions)i(\(called)d (cluster)h(expansions\))i(for)f(these)f(activities)g(but)h(the)75 2222 y(main)g(ob)s(jectiv)o(e)e(in)j(Section)f(7)h(is)g(to)g(devise)f(a)h (metho)q(d)f(of)i(measuring)d(the)i(size)f(\(a)h(norm\))f(of)75 2283 y(the)d(functionals)f Ft(K)t FC(\()p Ft(X)q(;)8 b(\036)p FC(\).)22 b(This)16 b(norm)f(captures)h(the)f(information)g(necessary)g(to)i (ensure)e(that)p 75 2329 735 2 v 131 2360 a Fj(1)150 2375 y Fy(I)d(will)e(discuss)j(the)f(the)h(case)g Fi(\036)640 2360 y Fj(4)640 2385 y(4)670 2375 y Fy(in)e(whic)o(h)h Fi(v)853 2381 y Fh(t)879 2375 y FA(6)p Fy(=)g(0)f(in)h(an)f(expanded)i(v)o(ersion)f (of)f(these)j(notes)e(to)g(b)q(e)g(published)75 2425 y(elsewhere)982 2599 y FC(2)p eop %%Page: 3 7 bop 75 307 a FC(under)18 b(the)g(R)o(G)g(ev)o(olution,)f(whilst)g Ft(\025)792 314 y Fq(t)826 307 y FC(remains)f(small,)h(these)g(p)q(olymers)g (will)g(remain)f(a)j(dilute)75 378 y(gas,)e(so)g(that)f(the)g(factor)h Ft(e)585 352 y Fp(\000)618 321 y Fr(R)637 369 y Fk(\000)p Fm(U)693 352 y Fq(dx)6 b(v)754 356 y Fm(t)768 352 y Fo(\()p Fq(\036;)p Fp(r)p Fq(\036)p Fo(\))895 378 y FC(is)16 b(the)g(dominan)o(t)f(part.)148 438 y(There)k(is)f(some)g(clumsiness)f(in)i(the)g(construction)g(of)g(this)g (norm.)28 b(P)o(erhaps)19 b(someone)f(else)75 498 y(will)d(b)q(e)h(able)g(to) h(do)f(b)q(etter.)21 b(The)16 b(less)g(information)f(one)h(needs)g(to)h(hold) f(within)g(this)g(norm)f(the)75 559 y(more)e(e\016cien)o(t)g(the)h(sc)o(heme) e(will)h(b)q(e.)21 b(The)15 b(norm)e(k)o(eeps)h(trac)o(k)g(of)g(v)m (ariational)h(deriv)m(ativ)o(es)e(of)i(all)75 619 y(orders,)h(in)g(fact)g(it) g(enforces)g(a)h(kind)e(of)i(analyticit)o(y)d(in)i Ft(\036)p FC(,)g(whic)o(h)f(probably)i(is)f(not)h(necessary)l(.)148 679 y(In)i(Section)f(8)h(I)f(discuss)h(in)f(considerable)g(detail)g(the)g(case)h (where)f Ft(v)r FC(\()p Ft(\036)p FC(\))g(actually)g(dep)q(ends)75 747 y(only)e(on)h Fs(r)p Ft(\036)p FC(.)22 b(The)17 b(dip)q(ole)f(gas)i(or)f Ft(e)771 696 y Fr(R)804 727 y Fq(dx)6 b Fo(\()p Fp(r)p Fq(\036)p Fo(\))926 715 y Fl(4)943 727 y Fo(\()p Fq(x)p Fo(\))1009 747 y FC(are)16 b(examples)f(of)i(suc)o(h)f(a)h(functional)g(in)o(tegral.)75 807 y(With)g(some)e(further)i(w)o(ork,)f(this)h(leads)g(to)g(a)g(pro)q(of)h (that)f(the)g(pressure)g(for)g(the)f(dip)q(ole)h(gas)h(has)75 867 y(a)g(con)o(v)o(ergen)o(t)f(expansion)h(in)f(p)q(o)o(w)o(ers)h(of)g (activit)o(y)l(,)e(a)i(result)g(\014rst)g(obtained)g(b)o(y)f(Ga)o(w)o(edzki)g (and)75 928 y(Kupiainen,)k(who)h(w)o(ere)e(the)h(\014rst)g(to)g(really)f (understand)h(the)g(dip)q(ole)g(gas.)36 b(The)21 b(metho)q(ds)f(of)75 988 y(this)d(section)g(ha)o(v)o(e)g(also)h(b)q(een)g(used)f(in)g(the)h (course)f(of)h(w)o(ork)g(on)g(Coulom)o(b)e(and)i(Y)l(uk)m(a)o(w)o(a)g(gases) 75 1048 y(in)h(t)o(w)o(o)h(dimensions)e(b)o(y)h(Dimo)q(c)o(k)e(and)j(Hurd)g ([43].)31 b(They)19 b(ha)o(v)o(e)f(extended)h(these)g(metho)q(ds)g(to)75 1108 y(calculate)13 b(the)g(correlations)g(as)h(opp)q(osed)h(to)f(just)g(the) f(e\013ectiv)o(e)f(action.)20 b(They)13 b(also)i(ha)o(v)o(e)d(pro)o(v)o(ed)75 1168 y(that)21 b(the)f(t)o(w)o(o)g(dimensional)f(Coulom)o(b)g(gas)j(in)e(a)h (\014nite)f(v)o(olume)e(without)i(ultra-violet)g(cuto\013)75 1229 y(has)f(pressure)g(analytic)f(in)h(the)f(activit)o(y)f(for)i(all)g(in)o (v)o(erse)e(temp)q(eratures)g(up)i(to)g(the)g(Kosterlitz)75 1289 y(Thouless)k(transition)f(at)h Ft(\014)j FC(=)e(8)p Ft(\031)g FC(pro)o(vided)d(at)h(eac)o(h)g(of)h(the)e(thresholds)i Ft(\014)j FC(=)e(8)p Ft(\031)r FC(\(1)15 b Fs(\000)1847 1269 y Fo(1)p 1836 1277 40 2 v 1836 1306 a(2)p Fq(n)1880 1289 y FC(\),)75 1349 y Ft(n)f FC(=)g(1)p Ft(;)8 b FC(2)p Ft(;)g Fs(\001)g(\001)g(\001)17 b FC(an)f(in\014nite)g(v)m(acuum)f(energy)h(renormalization)e(is)i(p)q (erformed.)148 1409 y(I)k(thank)g(m)o(y)e(colab)q(orators)k(P)o(aul)e(F)l (ederbush,)g(T)l(om)f(Kennedy)l(,)h(H.T.)f(Y)l(au,)h(John)h(Im)o(brie,)75 1469 y(Pronob)12 b(Mitter)e(and)h(Georg)h(Keller)d(for)i(m)o(uc)o(h)d(of)k (the)e(w)o(ork)h(presen)o(ted)f(here,)h(Rob)q(erto)g(F)l(ern\023)-24 b(andez)75 1530 y(for)16 b(generously)g(agreeing)g(to)g(help)f(in)h (preparing)g(these)f(notes)h(and)h(Andreas)f(P)o(ordt)g(for)g(correc-)75 1590 y(tions.)41 b(Of)23 b(course)g(I)g(will)e(ha)o(v)o(e)i(only)f(m)o(yself) f(to)i(thank)g(for)g(an)o(y)g(errors.)41 b(I)23 b(wish)g(to)g(thank)75 1650 y(Galla)o(v)o(otti)e(and)h(Benfatto)g(for)g(their)e(hospitalit)o(y)h(in) h(Rome)e(whilst)h(some)g(of)h(this)f(w)o(ork)h(w)o(as)75 1710 y(b)q(eing)17 b(prepared.)23 b(Ideas)17 b(close)f(to)i(the)e(ones)h(presen)o (ted)f(here)h(ha)o(v)o(e)f(b)q(een)g(separately)h(adv)m(anced)75 1770 y(b)o(y)12 b(Mac)o(k)h(and)g(P)o(ordt)g([45)q(,)f(46)q(,)h(54])g(and)g (indeed)g(it)f(should)i(b)q(e)f(realized)e(that)j(the)e(w)o(ork)h(presen)o (ted)75 1831 y(here)f(is)g(a)g(cleaned)g(up)g(v)o(ersion)g(of)h(ideas)f(that) h(are)f(implicit)d(in)j(all)g(the)g(w)o(ork)g(on)h(phase)f(cell)f(expan-)75 1891 y(sions)16 b([25)q(,)f(24)q(].)21 b(Finally)14 b(I)i(am)f(v)o(ery)f (grateful)i(to)h(Philipp)q(e)e(Martin)g(and)i(Prof.)k(L.)16 b(Sc)o(hellen)o(b)q(erg)75 1951 y(and)e(the)g(T)l(roisi)o(\022)-23 b(eme)11 b(Cycle)i(de)g(la)h(Ph)o(ysique)e(en)i(Suisse)f(Romande,)g(for)h (the)f(opp)q(ortunit)o(y)h(to)g(giv)o(e)75 2011 y(these)i(lectures.)982 2599 y(3)p eop %%Page: 4 8 bop 75 315 a Fv(2)83 b(The)27 b(Sine-Gordon)h(T)-7 b(ransformation)75 425 y FC(This)18 b(iden)o(tit)o(y)d(rewrites)i(the)h(partition)f(function)h (for)g(a)g(system)e(with)h(t)o(w)o(o-b)q(o)q(dy)i(forces)e(in)h(the)75 485 y(Grand)d(Canonical)g(Ensem)o(ble)d(as)j(a)f(sup)q(erp)q(osition)h(of)g (partition)f(functions)g(of)h(ideal)e(gases)j(with)75 545 y(\(imaginary\))f (external)g(\014elds.)21 b(The)16 b(sup)q(erp)q(osition)i(is)e(o)o(v)o(er)f (these)h(external)f(\014elds.)148 606 y(Supp)q(ose)d(our)g(particles)e(are)h (describ)q(ed)f(b)o(y)g(a)i(co)q(ordinate)f Ft(\030)17 b FC(=)c(\()p Ft(x;)8 b(\017)p FC(\),)j(where)g Ft(x)f FC(is)h(the)g(p)q(osition)75 666 y(and)18 b Ft(\017)g FC(an)g(in)o(ternal)e(co)q(ordinate)j(suc)o(h)e(as)h (c)o(harge,)g(or)g(orien)o(tation)f(in)g(the)h(case)f(of)h(dip)q(oles.)26 b(W)l(e)75 726 y(consider)15 b(particles)g(in)g(a)h(\014nite)f(con)o(tainer,) g(and)h(with)f(a)h(\014nite)f(\(or)h(compact\))e(space)i(of)g(in)o(ternal)75 786 y(co)q(ordinates,)24 b(so)e Ft(\030)j FC(ranges)d(o)o(v)o(er)f(a)i (compact)d(set)i(that)h(w)o(e)e(shall)h(denote)g(\003.)38 b(The)22 b(partition)75 846 y(function)16 b(for)h(particles)e(in)o(teracting)g(with)h (a)h(t)o(w)o(o-b)q(o)q(dy)g(p)q(oten)o(tial)f Ft(u)g FC(and)h(external)e (\014eld)h Ft(f)22 b FC(is)627 982 y Ft(Z)17 b FC(=)750 928 y Fp(1)738 941 y Fr(X)729 1033 y Fq(N)t Fo(=0)836 949 y FC(1)p 819 971 58 2 v 819 1017 a Ft(N)5 b FC(!)890 924 y Fr(Z)943 982 y FC(~)-27 b Ft(z)965 962 y Fq(N)1007 982 y Ft(d)1032 962 y Fq(N)1066 982 y Ft(\030)19 b(e)1129 962 y Fp(\000)p Fq(\014)r(U)t Fo(\()p Fq(\030)1236 967 y Fl(1)1252 962 y Fq(;:::)o(;\030)1317 968 y Fm(N)1345 962 y Fo(\))1812 982 y FC(\(2)p Ft(:)p FC(1\))75 1125 y(where)671 1210 y Ft(U)g FC(=)780 1177 y(1)p 780 1199 25 2 v 780 1245 a(2)844 1156 y Fq(N)829 1169 y Fr(X)818 1260 y Fq(i;j)r Fo(=1)909 1210 y Ft(u)p FC(\()p Ft(\030)977 1217 y Fq(i)991 1210 y Ft(;)8 b(\030)1034 1217 y Fq(j)1053 1210 y FC(\))j(+)1146 1156 y Fq(N)1132 1169 y Fr(X)1133 1260 y Fq(i)p Fo(=1)1200 1210 y Ft(f)5 b FC(\()p Ft(\030)1269 1217 y Fq(i)1284 1210 y FC(\))p Ft(:)495 b FC(\(2)p Ft(:)p FC(2\))75 1337 y(Here)194 1302 y Fr(R)230 1337 y Ft(d\030)24 b FC(represen)o(ts)c(the)g(in)o(tegral)g (b)q(oth)h(o)o(v)o(er)f(co)q(ordinates)h(and)g(in)o(ternal)e(space)i(v)m (ariables,)75 1398 y(and)16 b(it)g(ranges)g(o)o(v)o(er)f(the)g(compact)g(set) h(\003.)21 b(In)15 b(particular,)g(the)h(partition)f(function)h(for)g(an)g (ideal)75 1458 y(gas)h(in)f(an)h(external)e(\014eld)h Ft(\036)g FC(is)560 1587 y Ft(Z)593 1594 y Fo(ideal)668 1587 y FC(\()p Ft(\036)p FC(\))42 b(=)856 1546 y Fr(X)946 1553 y FC(1)p 929 1576 58 2 v 929 1621 a Ft(N)5 b FC(!)1000 1529 y Fr(Z)1053 1587 y FC(~)-27 b Ft(z)1075 1567 y Fq(N)1117 1587 y Ft(d)1142 1567 y Fq(N)1176 1587 y Ft(\030)19 b(e)1239 1567 y Fq(\014)1266 1538 y Fr(P)1310 1548 y Fm(N)1310 1581 y Fl(1)1345 1567 y Fq(\036)p Fo(\()p Fq(\030)1396 1572 y Fm(j)1411 1567 y Fo(\))777 1700 y FC(=)41 b(exp)939 1640 y Fr(\024)961 1642 y(Z)1014 1700 y FC(~)-27 b Ft(z)r(d\030)11 b(e)1116 1680 y Fq(\014)r(\036)p Fo(\()p Fq(\030)q Fo(\))1204 1640 y Fr(\025)1248 1700 y Ft(:)550 b FC(\(2.3\))148 1826 y(Let)13 b(us)h(\014rst)f(consider)g(the)g(case)g(of)h (particles)e(on)i(a)f Fn(lattic)n(e)t FC(.)21 b(F)l(or)13 b(simplicit)o(y)d (w)o(e)i(consider)h(\014rst)75 1886 y Ft(f)24 b FC(=)18 b(0,)h(the)g(reader)g (can)g(easily)f(furnish)h(the)f(small)g(c)o(hanges)h(needed)f(to)h(deal)g (with)f(a)i(general)75 1946 y Ft(f)5 b FC(.)21 b(W)l(e)16 b(write)e(the)i(in) o(teraction)e(energy)h Ft(U)21 b FC(as)16 b(a)g(sum)e(of)i(t)o(w)o(o-b)q(o)q (dy)h(energies)d Ft(u)p FC(\()p Ft(\030)1605 1953 y Fq(i)1619 1946 y Ft(;)8 b(\030)1662 1953 y Fq(j)1681 1946 y FC(\))15 b(times)f(the)75 2007 y(\\particle)i(densities")f(at)i Ft(\030)579 2014 y Fq(i)610 2007 y FC(and)f Ft(\030)725 2014 y Fq(j)744 2007 y FC(:)709 2140 y Ft(U)k FC(=)818 2107 y(1)p 818 2129 25 2 v 818 2175 a(2)856 2099 y Fr(X)859 2191 y Fq(\030)q(;\030)903 2182 y Fk(0)924 2140 y Ft(\032)p FC(\()p Ft(\030)r FC(\))p Ft(u)p FC(\()p Ft(\030)r(;)8 b(\030)1125 2120 y Fp(0)1139 2140 y FC(\))p Ft(\032)p FC(\()p Ft(\030)1225 2120 y Fp(0)1237 2140 y FC(\))p Ft(;)542 b FC(\(2)p Ft(:)p FC(4\))75 2291 y(with)782 2375 y Ft(\032)p FC(\()p Ft(\030)r FC(\))14 b(=)948 2322 y Fq(N)934 2334 y Fr(X)936 2425 y Fq(i)p Fo(=1)1002 2375 y Ft(\016)r FC(\()p Ft(\030)g Fs(\000)c Ft(\030)1150 2382 y Fq(i)1165 2375 y FC(\);)614 b(\(2)p Ft(:)p FC(5\))982 2599 y(4)p eop %%Page: 5 9 bop 75 307 a FC(where)20 b(the)h Ft(\016)h FC(refers)e(to)h(the)f(Kronec)o(k) o(er)f(delta.)34 b(Then)21 b(exp)o(\()p Fs(\000)p Ft(\014)s(U)5 b FC(\))21 b(=)g(exp[\()p Fs(\000)p Ft(\014)s(=)p FC(2\))1716 274 y Fr(P)1767 307 y Ft(\032u\032)p FC(])g(is)75 367 y(a)h(Gaussian)g (function)f(of)g(the)g(v)o(ector)f Ft(\032)p FC(.)37 b(Pro)o(vided)20 b Ft(u)h FC(is)g(p)q(ositiv)o(e)f(de\014nite)h(w)o(e)f(can)i(use)f(the)75 428 y(iden)o(tit)o(y)14 b(\(F)l(ourier)i(transform)g(of)g(a)h(Gaussian)g(is)f (a)h(Gaussian\))209 562 y(exp)291 489 y Fr(")316 562 y Fs(\000)360 528 y Ft(\014)p 360 550 31 2 v 363 596 a FC(2)395 562 y(\()p Ft(\032;)8 b(u\032)p FC(\))532 489 y Fr(#)571 562 y FC(=)13 b(Const.)769 503 y Fr(Z)819 520 y(Y)837 613 y Fq(\030)880 562 y Ft(d\036)p FC(\()p Ft(\030)r FC(\))26 b(exp)1104 501 y Fr(\024)1126 562 y Fs(\000)1170 528 y FC(1)p 1170 550 25 2 v 1170 596 a(2)1199 562 y Ft(\014)s FC(\()p Ft(\036;)8 b(u)1328 541 y Fp(\000)p Fo(1)1374 562 y Ft(\036)p FC(\))j Fs(\000)f Ft(i\014)s FC(\()p Ft(\036;)e(\032)p FC(\))1643 501 y Fr(\025)1673 562 y Ft(:)125 b FC(\(2)p Ft(:)p FC(6\))75 710 y(Here)17 b(\()p Ft(\032;)8 b(u\032)p FC(\))17 b(=)402 677 y Fr(P)446 720 y Fq(\030)q(;\030)490 711 y Fk(0)501 720 y Fp(2)p Fo(\003)559 710 y Ft(\032)p FC(\()p Ft(\030)r FC(\))p Ft(u)p FC(\()p Ft(\030)r(;)8 b(\030)760 692 y Fp(0)773 710 y FC(\))p Ft(\032)p FC(\()p Ft(\030)859 692 y Fp(0)872 710 y FC(\))18 b(and)h Ft(u)1034 692 y Fp(\000)p Fo(1)1099 710 y FC(is)f(the)g(in)o(v)o(erse)e(of)j(the)f(matrix)e(with)i(ro)o (ws)75 770 y(and)h(columns)f(lab)q(elled)f(b)o(y)h(the)h(particle)e(co)q (ordinates)j(in)e(\003)h(and)g(en)o(tries)e Ft(u)p FC(\()p Ft(\030)r(;)8 b(\030)1626 752 y Fp(0)1639 770 y FC(\).)28 b(In)19 b(a)g(more)75 830 y(compact)c(w)o(a)o(y)l(,)g(this)i(form)o(ula)d(can)j(b)q (e)f(written)g(as)692 945 y Ft(e)715 925 y Fp(\000)747 909 y Fm(\014)p 747 917 19 2 v 749 938 a Fl(2)771 925 y Fo(\()p Fq(\032;u\032)p Fo(\))880 945 y FC(=)932 887 y Fr(Z)982 945 y Ft(d\026)p FC(\()p Ft(\036)p FC(\))8 b Ft(e)1134 925 y Fp(\000)p Fq(i\014)r Fo(\()p Fq(\036;\032)p Fo(\))1274 945 y Ft(;)524 b FC(\(2)p Ft(:)p FC(7\))75 1061 y(where)19 b Ft(d\026)i FC(is)f(a)g (zero-mean)e(Gaussian)k(probabilit)o(y)c(measure)h(completely)e(de\014ned)i (b)o(y)h(its)f(co-)75 1121 y(v)m(ariance)d(\(a)h(matrix\))660 1138 y Fr(Z)710 1196 y Ft(d\026)p FC(\()p Ft(\036)p FC(\))8 b Ft(\036)p FC(\()p Ft(\030)r FC(\))p Ft(\036)p FC(\()p Ft(\030)1000 1176 y Fp(0)1013 1196 y FC(\))14 b(=)g Ft(\014)1129 1176 y Fp(\000)p Fo(1)1175 1196 y Ft(u)p FC(\()p Ft(\030)r(;)8 b(\030)1290 1176 y Fp(0)1303 1196 y FC(\))p Ft(:)476 b FC(\(2)p Ft(:)p FC(8\))75 1295 y(If)16 b(w)o(e)g(substitute)g(\(2.4\))g(and)h(\(2.7\))g(in)o (to)f(\(2.1\),)g(w)o(e)f(obtain)519 1427 y Ft(Z)i FC(=)621 1369 y Fr(Z)671 1427 y Ft(d\026)p FC(\()p Ft(\036)p FC(\))822 1373 y Fp(1)809 1386 y Fr(X)801 1478 y Fq(N)t Fo(=0)907 1394 y FC(1)p 891 1416 58 2 v 891 1461 a Ft(N)5 b FC(!)957 1427 y(~)-27 b Ft(z)979 1407 y Fq(N)1077 1386 y Fr(X)1020 1478 y Fq(\030)1036 1483 y Fl(1)1053 1478 y Fq(;:::)o(;\030)1118 1484 y Fm(N)1147 1478 y Fp(2)p Fo(\003)1203 1427 y Ft(e)1226 1403 y Fq(i)1243 1374 y Fr(P)1287 1417 y Fm(j)1309 1403 y Fq(\036)p Fo(\()p Fq(\030)1360 1408 y Fm(j)1376 1403 y Fo(\))p Fq(\032)p Fo(\()p Fq(\030)1438 1408 y Fm(j)1453 1403 y Fo(\))1812 1427 y FC(\(2)p Ft(:)p FC(9\))75 1573 y(and,)16 b(using)h(\(2.5\),)530 1705 y Ft(Z)45 b FC(=)688 1646 y Fr(Z)737 1705 y Ft(d\026)p FC(\()p Ft(\036)p FC(\))888 1651 y Fp(1)876 1664 y Fr(X)867 1755 y Fq(N)t Fo(=0)974 1671 y FC(1)p 957 1693 V 957 1739 a Ft(N)5 b FC(!)1023 1705 y(~)-27 b Ft(z)1045 1684 y Fq(N)1144 1664 y Fr(X)1087 1756 y Fq(\030)1103 1761 y Fl(1)1119 1756 y Fq(;:::;\030)1185 1762 y Fm(N)1213 1756 y Fp(2)p Fo(\003)1269 1705 y Ft(e)1292 1680 y Fq(i)1310 1651 y Fr(P)1353 1695 y Fm(j)1376 1680 y Fq(\036)p Fo(\()p Fq(\030)1427 1685 y Fm(j)1442 1680 y Fo(\))608 1842 y FC(=)688 1783 y Fr(Z)737 1842 y Ft(d\026)p FC(\()p Ft(\036)p FC(\))8 b Ft(Z)899 1849 y Fo(ideal)975 1842 y FC(\()p Ft(i\036)p FC(\))p Ft(:)715 b FC(\(2.10\))75 1957 y(This)19 b(last)g(form)o(ula)f(is)h(the)g Fn(Sine-Gor)n(don)h(tr)n (ansformation)p FC(.)29 b(W)l(e)18 b(notice)h(that)g(in)g(order)g(to)g(use)75 2017 y(\(2.6\),)12 b(the)g(diagonal)g(terms)f Ft(u)p FC(\()p Ft(\030)r(;)d(\030)r FC(\))k(\(\\self-energy"\))g Fn(must)g FC(b)q(e)g(included.)18 b(This)12 b(will)f(need)g(further)75 2077 y(atten)o(tion)24 b(when)g(aplying)g(the)h(Sine-Gordon)g(transformation) f(to)h(forces)f(singular)g(at)h(short)75 2137 y(distances|for)16 b(example)e(Coulom)o(b)i(or)g(Y)l(uk)m(a)o(w)o(a.)148 2198 y(Siegert)e([57])g(and)h(Kac)f([44])g(disco)o(v)o(ered)f(the)h(Sine-Gordon)h (transformation)f(and)h(used)f(it)g(for)75 2258 y(con)o(tin)o(uum)g(systems.) 20 b(A)c(cautious)h(statemen)o(t)d(of)j(the)f(result)f(is:)75 2354 y Fu(Prop)r(osition)j(2.1)g(\(The)g(Sine-Gordon)g(T)-5 b(ransformation\))24 b Fn(L)n(et)12 b Ft(u)p FC(\()p Ft(\030)r(;)c(\030)1547 2336 y Fp(0)1559 2354 y FC(\))13 b Fn(b)n(e)g(p)n(ositive)f(semi-)75 2414 y(de\014nite)j(and)f(c)n(ontinuously)h(di\013er)n(entiable)g(in)f(the)g (p)n(osition)f(c)n(o)n(or)n(dinates,)g(then)i(ther)n(e)e(is)h(a)f(Gaus-)75 2474 y(sian)18 b(me)n(asur)n(e)g Ft(d\026)p FC(\()p Ft(\036)p FC(\))g Fn(on)h(the)g(sp)n(ac)n(e)e(of)h(c)n(ontinuous)h(\014elds)h Ft(\036)15 b Fs(2)g(C)s FC(\(\003\))p Fn(,)j(which)h(is)f(char)n(acterize)n (d)982 2599 y FC(5)p eop %%Page: 6 10 bop 75 307 a Fn(by)15 b(\(1\))216 272 y Fr(R)252 307 y Ft(d\026)p FC(\()p Ft(\036)p FC(\))p Ft(\036)p FC(\()p Ft(\030)r FC(\))f(=)g(0)i Fn(for)e(al)r(l)i Ft(\030)h Fs(2)d FC(\003)p Fn(;)h(and)g(\(2\))g(c)n (ovarianc)n(e)g Ft(\014)1300 289 y Fp(\000)p Fo(1)1347 307 y Ft(u)p Fn(,)g(i.e.,)1502 272 y Fr(R)1538 307 y Ft(d\026)p FC(\()p Ft(\036)p FC(\))8 b Ft(\036)p FC(\()p Ft(\030)r FC(\))p Ft(\036)p FC(\()p Ft(\030)1828 289 y Fp(0)1842 307 y FC(\))14 b(=)75 367 y Ft(\014)106 349 y Fp(\000)p Fo(1)152 367 y Ft(u)p FC(\()p Ft(\030)r(;)8 b(\030)267 349 y Fp(0)280 367 y FC(\))p Fn(;)17 b(such)h(that)705 443 y Ft(Z)g FC(=)808 384 y Fr(Z)858 443 y Ft(d\026)p FC(\()p Ft(\036)p FC(\))8 b Ft(Z)1020 450 y Fo(ideal)1096 443 y FC(\()p Ft(i\036)i FC(+)h Ft(f)5 b FC(\))p Ft(:)506 b FC(\(2)p Ft(:)p FC(11\))75 580 y(The)23 b(reader)f(migh)o(t)g(w)o (onder)g(what)i(is)f(mean)o(t)e(b)o(y)h(a)h Fn(Gaussian)g FC(measure)f(in)g (this)h(functional)75 640 y(con)o(text.)d(This)d(is)f(de\014ned)g(b)o(y)g (requiring)f(that)532 702 y Fr(Z)581 761 y Ft(d\026)p FC(\()p Ft(\036)p FC(\))8 b Ft(e)733 740 y Fq(i)752 709 y Fr(R)785 740 y Fq(d\032)p Fo(\()p Fq(\030)q Fo(\))p Fq(\036)p Fo(\()p Fq(\030)q Fo(\))947 761 y FC(=)14 b Ft(e)1022 738 y Fp(\000)1063 724 y Fl(1)p 1054 730 34 2 v 1054 751 a(2)p Fm(\014)1098 706 y Fr(R)g(R)1166 738 y Fq(d\032)p Fo(\()p Fq(\030)q Fo(\))p Fq(u)p Fo(\()p Fq(\030)q(;\030)1325 726 y Fk(0)1335 738 y Fo(\))p Fq(d\032)p Fo(\()p Fq(\030)1416 726 y Fk(0)1426 738 y Fo(\))1442 761 y Ft(;)332 b FC(\(2)p Ft(:)p FC(12\))75 884 y(where)14 b Ft(d\032)g FC(is)g(an)o(y)g(\014nite)f(Borel)g(measure)g(on)i(\003.)20 b(This)14 b(is)g(a)g(natural)h(generalization)e(of)i(\(2.7\))f(\(the)75 945 y(F)l(ourier)i(transform)f(of)i(a)g(Gaussian)g(is)f(a)h(Gaussian\).)75 1029 y Fu(Pro)r(of.)27 b FC(The)13 b(pro)q(of)g(is)f(basically)f(a)i (transcription)f(of)g(the)g(ab)q(o)o(v)o(e)g(discussion.)20 b(Starting)13 b(from)e(the)75 1089 y(form)o(ula)18 b(ab)q(o)o(v)o(e,)h(the)g (pro)q(of)h(go)q(es)g(exactly)e(as)i(ab)q(o)o(v)o(e,)f(except)f(that)h (\(2.5\))h(turns)f(in)o(to)g Ft(d\032)p FC(\()p Ft(\030)r FC(\))h(=)75 1150 y Ft(\014)114 1116 y Fr(P)157 1130 y Fq(N)157 1162 y Fo(1)199 1150 y Ft(\016)r FC(\()p Ft(\030)e Fs(\000)c Ft(\030)355 1157 y Fq(j)374 1150 y FC(\))p Ft(d\030)r FC(,)24 b(with)e Ft(\016)i FC(the)d(Dirac)h(delta.)39 b(The)22 b(remaining)e(\(di\016cult\))h(question)h (is)g(the)75 1210 y(existence)14 b(of)j(a)f(Gaussian)h(measure)e(on)h(a)h (space)f(of)g(con)o(tin)o(uous)g(functions.)21 b(This)16 b(is)g(discussed,)75 1270 y(for)k(instance,)g(in)f([59)q(,)g(33)q(].)31 b(In)19 b(these)h(references)e(the)i(measure)e(is)i(constructed)f(on)i(a)f(Hilb)q (ert)75 1330 y(space)c(of)g(functions.)21 b(Our)15 b(h)o(yp)q(otheses)h(on)g Ft(u)g FC(guaran)o(tee)g(that)g(the)f(Hilb)q(ert)f(space)i(is)f(a)i(space)e (of)75 1390 y Fn(c)n(ontinuous)i FC(functions.)1348 b Fg(2)75 1535 y Ff(2.1)70 b(Coulom)n(b)22 b(Systems)75 1627 y FC(F)l(or)c(Coulom)o(b)g (forces)g(b)q(et)o(w)o(een)g(c)o(harges)g Ft(\017)f FC(=)g Fs(\006)p FC(1,)i(one)f(w)o(ould)h(lik)o(e)d(to)j(c)o(ho)q(ose)g Ft(\030)h FC(=)d(\()p Ft(x;)8 b(\017)p FC(\))18 b(and)75 1687 y Ft(u)p FC(\()p Ft(\030)r(;)8 b(\030)190 1669 y Fp(0)203 1687 y FC(\))16 b(=)g Ft(\017\017)332 1669 y Fp(0)343 1687 y Ft(G)p FC(\()p Ft(x)c Fs(\000)g Ft(x)519 1669 y Fp(0)530 1687 y FC(\))18 b(where)f Ft(G)i FC(satis\014es)f Fs(\000)p FC(\001)p Ft(G)p FC(\()p Ft(x;)8 b(y)r FC(\))15 b(=)h Ft(\016)r FC(\()p Ft(x)11 b Fs(\000)h Ft(y)r FC(\),)17 b(that)h(is,)g(with)f Ft(G)h FC(b)q(eing)75 1747 y(the)g(k)o(ernel)e(of)j(a)f(Green)g(function)f(\()p Fs(\000)p FC(\001\))862 1729 y Fp(\000)p Fo(1)909 1747 y FC(\()p Ft(x;)8 b(y)r FC(\).)26 b(In)17 b(more)g(than)i(one)f(dimension)e(this)i(k)o(ernel)75 1808 y(is)d(singular)h(at)f Ft(x)f FC(=)g Ft(y)i FC(so)g(there)f(are)g (problems)f(with)h(the)g(Sine-Gordon)h(transformation)g(whic)o(h,)75 1868 y(as)j(w)o(e)e(ha)o(v)o(e)h(describ)q(ed)f(it,)h(requires)f(all)h(the)g (in)o(tegrals)f(\(2.8\))i(to)f(b)q(e)g(\014nite,)g(and)h(in)e(particular,)75 1928 y(as)g(emphasized)d(ab)q(o)o(v)o(e,)i Fn(\014nite)j(diagonal)f(terms)t FC(,)e(i.e.)688 1990 y Fr(Z)738 2049 y Ft(d\026)p FC(\()p Ft(\036)p FC(\))8 b Ft(\036)896 2028 y Fo(2)916 2049 y FC(\()p Ft(\030)r FC(\))15 b(=)f Ft(u)p FC(\()p Ft(\030)r(;)8 b(\030)r FC(\))14 b Ft(<)g Fs(1)p Ft(:)480 b FC(\(2)p Ft(:)p FC(13\))75 2169 y(F)l(ourier-transforming)16 b(the)g(co)q(ordinates:)490 2290 y Ft(u)p FC(\(\()p Ft(x;)8 b(\017)p FC(\))p Ft(;)g FC(\()p Ft(y)r(;)g(\017)754 2270 y Fp(0)764 2290 y FC(\)\))14 b(=)f(Const)p Ft(:)1022 2232 y Fr(Z)1075 2290 y FC(^)-27 b Ft(u)p FC(\()p Ft(\017;)8 b(\017)1181 2270 y Fp(0)1192 2290 y Ft(;)g(k)r FC(\))p Ft(e)1283 2270 y Fq(ik)q Fo(\()p Fq(x)p Fp(\000)p Fq(y)q Fo(\))1417 2290 y Ft(dk)17 b(;)75 2414 y FC(w)o(e)i(see)f(that)i(condition)e(\(2.13\))i (is)f(equiv)m(alen)o(t)e(to)i(requiring)f(the)h(in)o(tegrabilit)o(y)e(of)22 b(^)-27 b Ft(u)p FC(\()p Ft(k)r FC(\).)29 b(This)75 2474 y(is)22 b(not)h(satis\014ed)g(for)g(the)f(Coulom)o(b)g(\()s(^)-27 b Ft(u)24 b Fs(/)g FC(1)p Ft(=k)1011 2456 y Fo(2)1032 2474 y FC(\))e(or)h(Y)l(uk)m(a)o(w)o(a)g(\()s(^)-27 b Ft(u)24 b Fs(/)g FC(1)p Ft(=)p FC(\(1)17 b(+)e Ft(k)1648 2456 y Fo(2)1668 2474 y FC(\)\))22 b(forces)h(in)982 2599 y(6)p eop %%Page: 7 11 bop 75 307 a FC(dimensions)14 b Ft(d)g(>)g FC(1,)h(b)q(ecause)g(of)h(the)e (slo)o(w)i(deca)o(y)e(at)h(high)h(w)o(a)o(v)o(e)e(n)o(um)o(b)q(ers)f Ft(k)k FC(\(for)f(the)e(Coulom)o(b)75 367 y(p)q(oten)o(tial)i(there)g(is)g (also)g(trouble)g(on)h(the)f(infrared)g(end)g(for)h Ft(d)d Fs(\024)g FC(2\).)148 428 y(The)j(ph)o(ysical)f(in)o(terpretation)h(of)g (these)g(problems)e(is)i(the)g(p)q(ossible)h(existence)d(of)i(an)h(insta-)75 488 y(bilit)o(y)c(with)h(resp)q(ect)h(to)g(collapse.)k(Repulsiv)o(e)14 b(short-range)j(forces)f(or)g(quan)o(tum)e(mec)o(hanics)f(are)75 548 y(necessary)k(to)i(prev)o(en)o(t)d(suc)o(h)h(a)i(collapse)e(in)g(three)g (or)i(more)d(dimensions.)24 b(A)18 b(standard)h(w)o(a)o(y)e(to)75 608 y(mo)q(del)11 b(suc)o(h)h(forces)g(is)g(to)g(supp)q(ose)i(that)e(the)g(p) q(oin)o(t)g(c)o(harges)h(are)f(spread)g(out)h(in)o(to)f(little)e(spherical)75 668 y(c)o(harge-distributions)16 b(on)h(a)g(length)f(scale)f Ft(`)p FC(.)22 b(Then)16 b Ft(u)g FC(is)g(the)g(F)l(ourier)g(transform)g(of) 747 799 y(^)-27 b Ft(u)p FC(\()p Ft(\017;)8 b(\017)853 778 y Fp(0)864 799 y Ft(;)g(k)r FC(\))13 b(=)h Ft(\017\017)1037 778 y Fp(0)1053 765 y Ft(F)7 b FC(\()p Ft(`)1131 747 y Fo(2)1151 765 y Ft(k)1178 747 y Fo(2)1198 765 y FC(\))p 1053 787 164 2 v 1112 833 a Ft(k)1139 818 y Fo(2)1221 799 y Ft(;)553 b FC(\(2)p Ft(:)p FC(14\))75 926 y(where)216 884 y Fs(p)p 257 884 39 2 v 257 926 a Ft(F)23 b FC(is)16 b(the)g(F)l(ourier)f(transform)h(of)h(the)f(c) o(harge)g(distribution,)f(and)i Ft(F)22 b FC(is)16 b(assumed)g(to)g(b)q(e)75 986 y(a)j(su\016cien)o(tly)d(smo)q(oth)i(function)g(that)g(tak)o(es)g(the)g (v)m(alue)g(1)g(at)h(the)f(origin)g(and)h(deca)o(ys)e(rapidly)75 1047 y(as)k Ft(k)h Fs(!)f(1)p FC(.)32 b(The)21 b(length)f Ft(`)g FC(corresp)q(onds)i(to)e(an)h(ultra)o(violet)d(cuto\013;)23 b(roughly)d(sp)q(eaking,)h(for)75 1107 y(distances)h(greater)g(than)h Ft(`)g FC(the)e(original)h(p)q(oten)o(tial)g(di\013ers)g(little)f(from)g(the) h(one)g(de\014ned)g(b)o(y)75 1167 y(\(2.14\).)28 b(In)18 b(the)g(case)g(that) h(the)f(particles)f(are)i(in)e(a)i(p)q(erio)q(dic)f(b)q(o)o(x)h Ft(u)f FC(is)g(obtained)g(from)i(^)-27 b Ft(u)18 b FC(b)o(y)g(a)75 1227 y(F)l(ourier)e(series)f(in)h(whic)o(h)g(the)g Ft(k)g FC(=)d(0)k(term)e (is)h(omitted.)148 1287 y(When)f(applying)g(the)f(Sine-Gordon)i (transformation)e(to)i(systems)d(with)i(in)o(teractions)f(of)h(the)75 1348 y(form)g(\(2.14\))i(it)f(is)g(useful)g(to)g(recognise)g(that)822 1451 y Ft(\036)p FC(\()p Ft(x;)8 b(\017)p FC(\))13 b(=)h Ft(\017\036)p FC(\()p Ft(x)p FC(\))f Ft(:)622 b FC(\(2)p Ft(:)p FC(15\))75 1555 y(where)12 b(on)h(the)f(left)g(hand)h(side)f Ft(\036)p FC(\()p Ft(x;)c(\017)p FC(\))j(is)h(distributed)g(according)h(to)g(a)f (Gaussian)i(measure)d(whic)o(h)75 1615 y(has)16 b(a)g(co)o(v)m(ariance)f(of)h (the)f(form)695 1595 y Fo(1)p 694 1603 22 2 v 694 1632 a Fq(\014)720 1615 y Ft(\017\017)760 1597 y Fp(0)771 1615 y Ft(u)p FC(\()p Ft(x;)8 b(x)896 1597 y Fp(0)907 1615 y FC(\))15 b(and)h(on)g(the)g(righ)o(t)f (hand)h(side)f Ft(\036)p FC(\()p Ft(x)p FC(\))g(is)g(distributed)75 1684 y(according)h(to)g(a)g(Gaussian)h(measure)e(whic)o(h)g(has)i(co)o(v)m (ariance)1260 1664 y Fo(1)p 1258 1672 V 1258 1701 a Fq(\014)1284 1684 y Ft(u)p FC(\()p Ft(x;)8 b(x)1409 1665 y Fp(0)1420 1684 y FC(\).)21 b(This)16 b(is)g(b)q(ecause)g(b)q(oth)75 1744 y(sides)c(are)g (Gaussian)h(v)m(ariables)e(with)h(the)g(same)e(co)o(v)m(ariance)i(and)g (therefore)f(are)h(equal)f(as)i(random)75 1804 y(v)m(ariables.)26 b(In)17 b(particular,)h(if)i(~)-27 b Ft(z)r(d\030)21 b FC(giv)o(es)c(equal)g (w)o(eigh)o(t)g(to)h Ft(\017)e FC(=)h Fs(\006)p FC(1)h(\(c)o(harge)f (symmetry\),)e(then)75 1864 y(the)h(Sine-Gordon)h(transformation)f(rewrites)g (the)g(partition)g(function)g(as:)508 1983 y Ft(Z)i FC(=)610 1925 y Fr(Z)660 1983 y Ft(d\026)p FC(\()p Ft(\036)p FC(\))8 b(exp)873 1923 y Fr(\024)895 1983 y FC(2)927 1925 y Fr(Z)977 1983 y FC(cos)h Ft(\014)s FC([)p Ft(\036)p FC(\()p Ft(x)p FC(\))g(+)i Ft(f)5 b FC(\()p Ft(x)p FC(\)])s(~)-27 b Ft(z)r(dx)1436 1923 y Fr(\025)1466 1983 y Ft(:)308 b FC(\(2)p Ft(:)p FC(16\))75 2103 y(where)16 b Ft(f)22 b FC(is)16 b(the)g(external)f(\014eld.)148 2186 y(A)h(particular)g(case)g(of)h(in)o(terest)e(here,)g(is)h(that)h(of)g(a) f(gas)i(of)e Fn(dip)n(oles)t FC(.)22 b(In)16 b(this)g(case)g Ft(\030)h FC(=)d(\()p Ft(x;)d FC(^)-27 b Ft(p)p FC(\))75 2246 y(where)20 b(^)-28 b Ft(p)17 b FC(is)f(a)h(unit)f(v)o(ector)f(in)h Fu(R)695 2228 y Fq(d)715 2246 y FC(,)g(the)h(direction)e(of)i(the)f(dip)q (ole.)21 b Ft(d\030)c FC(=)d Ft(dx)8 b(dS)s FC(\()t(^)-28 b Ft(p)q FC(\),)16 b(where)g Ft(dS)k FC(is)75 2307 y(the)e(Leb)q(esgue)h (measure)d(on)j(the)f(surface)g(of)g(the)g(unit)g(sphere.)26 b(In)18 b Ft(k)r FC(-space)h(the)f(co)o(v)m(ariance)f(of)75 2367 y Ft(d\026)p FC(\()p Ft(\036)p FC(\))g(is)641 2456 y(^)-27 b Ft(u)p FC(\()p Ft(k)r(;)12 b FC(^)-28 b Ft(p;)12 b FC(^)-28 b Ft(p)804 2436 y Fp(0)816 2456 y FC(\))14 b(=)f(\()t(^)-28 b Ft(p)12 b Fs(\001)f Ft(k)r FC(\))1031 2423 y Ft(F)c FC(\()p Ft(`)1109 2405 y Fo(2)1128 2423 y Ft(k)1155 2405 y Fo(2)1175 2423 y FC(\))p 1031 2445 164 2 v 1089 2491 a Ft(k)1116 2476 y Fo(2)1199 2456 y FC(\()t(^)-28 b Ft(p)1242 2436 y Fp(0)1265 2456 y Fs(\001)11 b Ft(k)r FC(\))p Ft(:)438 b FC(\(2)p Ft(:)p FC(17\))982 2599 y(7)p eop %%Page: 8 12 bop 75 307 a FC(By)16 b(the)g(argumen)o(t)f(w)o(e)g(used)i(for)f(\(2.15\))h Ft(\036)p FC(\()p Ft(\030)r FC(\))e Fs(!)e FC(\()t(^)-28 b Ft(p)12 b Fs(\001)f(r)p FC(\))p Ft(\036)p FC(\()p Ft(x)p FC(\))k(so)361 436 y Ft(Z)j FC(=)464 377 y Fr(Z)513 436 y Ft(d\026)p FC(\()p Ft(\036)p FC(\))8 b(exp)726 375 y Fr(\024)748 436 y FC(2)780 377 y Fr(Z)833 436 y FC(~)-27 b Ft(z)r(dx)8 b(dS)s FC(\()t(^)-28 b Ft(p)q FC(\))8 b(cos)h Ft(\014)s FC([)t(^)-28 b Ft(p)11 b Fs(\001)f(r)p Ft(\036)p FC(\()p Ft(x)p FC(\))h(+)g Ft(f)5 b FC(\()p Ft(x;)12 b FC(^)-28 b Ft(p)p FC(\)])1575 375 y Fr(\025)1605 436 y Ft(;)169 b FC(\(2)p Ft(:)p FC(18\))75 565 y(where)16 b Ft(d\026)h FC(is)f(the)g(same)f(as)i(in)f(\(2.16\).)75 709 y Ff(2.2)70 b(Energy)23 b(and)h(Normal)e(Ordering)75 801 y FC(It)15 b(is)h(more)e(standard)j(to)g(exclude)d(self)h(energies)g(in)g (de\014ning)h(the)g(in)o(teraction)e(\(2.2\).)22 b(These)15 b(are)75 862 y(the)21 b(terms)e Ft(i)j FC(=)g Ft(j)i FC(in)d Ft(U)5 b FC(.)35 b(T)l(o)22 b(ac)o(hiev)o(e)d(this)i(w)o(e)g(put)g(the)g (self)f(energies)g(in)o(to)h(the)g(activit)o(y)e(b)o(y)75 922 y(de\014ning)830 982 y Ft(z)d FC(=)e Ft(e)944 961 y Fp(\000)976 948 y Fl(1)p 976 954 16 2 v 976 974 a(2)996 961 y Fq(\014)r(u)p Fo(\()p Fq(\030)q(;\030)q Fo(\))1114 982 y FC(~)-27 b Ft(z)r(;)638 b FC(\(2)p Ft(:)p FC(19\))75 1069 y(and)16 b(tak)o(e)g(them)e(out)i(of)g(the) g(in)o(teraction.)k(No)o(w)c(one)g(writes)f(the)h(Sine-Gordon)h (transformation)75 1129 y(in)f(the)g(form)608 1196 y Ft(Z)i FC(=)710 1137 y Fr(Z)760 1196 y Ft(d\026)p FC(\()p Ft(\036)p FC(\))8 b(exp)973 1135 y Fr(\022)1003 1137 y(Z)1053 1196 y Ft(z)r(d\030)17 b FC(:)c Ft(e)1191 1175 y Fq(i\014)r(\036)p Fo(\()p Fq(\030)q Fo(\))1306 1196 y FC(:)1320 1135 y Fr(\023)1358 1196 y Ft(;)416 b FC(\(2)p Ft(:)p FC(20\))75 1305 y(where)16 b(the)g Fn(normal)h(or)n(der)n(e)n(d)e FC(:)e Ft(e)687 1287 y Fq(i\036)736 1305 y FC(:)j(is)g(de\014ned)g(b)o(y)736 1415 y(:)d Ft(e)786 1395 y Fq(i\014)r(\036)p Fo(\()p Fq(\030)q Fo(\))901 1415 y FC(:=)g Ft(e)994 1381 y Fl(1)p 993 1387 V 993 1408 a(2)1014 1395 y Fq(\014)r(u)p Fo(\()p Fq(\030)q(;\030)q Fo(\))1128 1415 y Ft(e)1151 1395 y Fq(i\014)r(\036)p Fo(\()p Fq(\030)q Fo(\))1252 1415 y Ft(:)522 b FC(\(2)p Ft(:)p FC(21\))148 1525 y(It)17 b(will)f(turn)h(out)h(that)g(that)f Ft(z)j FC(is)d(the)g(\\righ)o(t")g (activit)o(y)f(from)g(the)h(ph)o(ysical)f(p)q(oin)o(t)h(of)h(view.)75 1586 y(F)l(or)23 b(example,)e(if)h(w)o(e)g(remo)o(v)o(e)d(the)j(ultra-violet) g(cuto\013)h(b)o(y)f(taking)g Ft(`)h FC(to)g(zero)f(in)g(the)g Ft(d)j FC(=)f(2)75 1646 y(Y)l(uk)m(a)o(w)o(a)c(gas)h(than)f(the)g(limit)d(is) j(a)h(gas)g(at)f(zero)g(densit)o(y)f(if)j(~)-27 b Ft(z)22 b FC(is)e(held)f(\014xed)h(but)g(has)g(densit)o(y)75 1706 y(appro)o(ximately)12 b(equal)i(to)h Ft(z)i FC(if)d Ft(z)i FC(is)e(held)g(\014xed)g(and)h(small.)k (W)l(e)c(will)e(b)q(e)i(discussing)f(this)h(further)75 1766 y(in)h(Section)g(5)75 1911 y Ff(2.3)70 b(The)22 b(Quan)n(tum)h(Statistical)d (Coulom)n(b)i(System)75 2003 y FC(This)f(section)f(is)h(con)o(tains)f(some)g (heuristic)p 734 2010 182 2 v 20 w(argumen)o(ts)f(dra)o(wn)j(from)d(a)i(pap)q (er)h(I)e(am)g(writing)75 2063 y(with)d(Georg)g(Keller)e([15].)23 b(W)l(e)16 b(thank)h(Erhard)h(Seiler)d(for)i(his)g(help)f(in)g(man)o(y)g(con) o(v)o(ersations)g(on)75 2123 y(these)j(topics.)32 b(W)l(e)20 b(should)g(also)g(men)o(tion)e(that)i(the)g(original)f(observ)m(ation)i(that) f(there)f(migh)o(t)75 2183 y(not)e(b)q(e)f(screening)g(in)g(Quan)o(tum)f (Coulom)o(b)g(plasma)h(w)o(as)g(due)h(to)f(P)o(aul)g(F)l(ederbush.)148 2244 y(If)e(w)o(e)g(include)f(momen)o(ta)f(and)k(kinetic)c(energy)i(then)h (the)f(partition)g(function)h(for)g(a)f(\(c)o(harge-)75 2304 y(symmetri)o(c\))f Fn(classic)n(al)k FC(Coulom)o(b)f(gas)h(is)676 2436 y Ft(Z)h FC(=)779 2395 y Fr(X)869 2402 y FC(1)p 852 2424 58 2 v 852 2470 a Ft(N)5 b FC(!)923 2377 y Fr(Z)973 2436 y Ft(d)998 2415 y Fq(N)1032 2436 y Ft(p)s FC(~)-27 b Ft(z)1081 2415 y Fq(N)1115 2436 y Ft(d)1140 2415 y Fq(N)1174 2436 y Ft(\030)11 b(e)1229 2415 y Fp(\000)p Fq(\014)r(H)1788 2436 y FC(\(2)p Ft(:)p FC(22\))982 2599 y(8)p eop %%Page: 9 13 bop 75 307 a FC(with)751 385 y Ft(H)18 b FC(=)861 343 y Fr(X)945 351 y Ft(p)969 333 y Fo(2)969 363 y Fq(i)p 934 373 68 2 v 934 419 a FC(2)p Ft(m)1017 385 y FC(+)1071 351 y(1)p 1071 373 25 2 v 1071 419 a(2)1109 326 y Fr(Z)1159 385 y Ft(\032u\032)551 b FC(\(2)p Ft(:)p FC(23\))75 482 y(where)769 542 y Ft(\032)p FC(\()p Ft(x)p FC(\))13 b(=)925 500 y Fr(X)993 542 y Ft(\017)1013 549 y Fq(i)1027 542 y Ft(\016)r FC(\()p Ft(x)d Fs(\000)h Ft(x)1186 549 y Fq(i)1200 542 y FC(\))569 b(\(2)p Ft(:)p FC(24\))75 626 y(so)17 b(that)710 628 y Fr(Z)760 686 y Ft(\032u\032)d FC(=)904 645 y Fr(X)972 686 y Ft(\017)992 693 y Fq(i)1006 686 y Ft(\017)1026 693 y Fq(j)1044 686 y Ft(u)p FC(\()p Ft(x)1119 693 y Fq(i)1143 686 y Fs(\000)d Ft(x)1221 693 y Fq(j)1239 686 y FC(\))j Ft(:)502 b FC(\(2)p Ft(:)p FC(25\))75 786 y Ft(u)p FC(\()p Ft(x)9 b Fs(\000)f Ft(y)r FC(\))14 b(=)g(\\1)p Ft(=r)412 768 y Fp(00)434 786 y FC(.)21 b(The)15 b(quotes)h(around)g(the)f(1)p Ft(=r)j FC(remind)13 b(us)j(that,)f(as)h(discussed)f(in)g(Example)75 846 y(1,)g(it)g(is)f(necesary)h(to)g(cuto\013)g(the)g(singularit)o(y)f(of)h (the)g(Coulom)o(b)f(p)q(oten)o(tial)g(at)h(short)h(distances)f(in)75 906 y(order)h(to)h(ha)o(v)o(e)e(a)i(stable)f(in)o(teraction.)k(This)c (cuto\013)h(will)e(b)q(e)h(c)o(haracterised)g(b)o(y)f(a)i(length)f Ft(l)q FC(.)k(The)75 966 y(natural)f(c)o(hoice)f(for)h(this)g(length)g(is)f (the)h(size)f(of)h(the)g(t)o(ypical)f(one-particle)g(w)o(a)o(v)o(efunction)g (in)g(a)75 1027 y(corresp)q(onding)f(quan)o(tum)e(ideal)h(gas)875 1173 y Ft(l)e FC(=)956 1087 y Fr(s)p 998 1087 88 2 v 1003 1140 a Ft(\014)s FC(\026)-25 b Ft(h)1060 1118 y Fo(2)p 1003 1162 78 2 v 1020 1208 a Ft(m)1099 1173 y FC(;)675 b(\(2)p Ft(:)p FC(26\))75 1292 y(since)12 b(it)g(is)g(the)h(P)o(auli)f(exclusion)f (principle)g(and)i(quan)o(tum)f(mec)o(hanics)e(that)j(giv)o(e)f(rise)g(to)h (a)g(stable)75 1352 y(system)i(whic)o(h)g(w)o(e)h(are)g(appro)o(ximating)g (classically)l(.)148 1413 y(The)g(other)h(lengths)f(whic)o(h)g(naturally)g (arise)g(are)g Ft(\014)j FC(and)d(the)g(Deb)o(y)o(e)f(length)858 1536 y Ft(l)873 1543 y Fq(D)919 1536 y FC(=)1024 1502 y(1)p 976 1524 122 2 v 976 1533 a Fs(p)p 1017 1533 80 2 v 1017 1570 a FC(2)p Ft(z)r(\014)1115 1536 y(;)659 b FC(\(2)p Ft(:)p FC(27\))75 1663 y(where)16 b Ft(z)i FC(includes)d(also)i(the)f(con)o(tribution)g(from)f (the)h(momen)o(tum)c(in)o(tegral:)635 1779 y Ft(z)k FC(=)g(~)-27 b Ft(z)759 1720 y Fr(Z)808 1779 y Ft(dpe)880 1758 y Fp(\000)925 1743 y Fm(\014)p 913 1751 43 2 v 913 1771 a Fl(2)p Fm(m)961 1758 y Fq(p)979 1747 y Fl(2)998 1779 y Ft(e)1021 1758 y Fp(\000)p Fq(\014)r Fp(\001)p Fo(\(self)s Fp(\000)p Fo(energy)o(\))p Fq(=)o Fo(2)1339 1779 y Ft(:)435 b FC(\(2)p Ft(:)p FC(28\))75 1897 y(In)16 b(general,)g(for)g(reasonable)h(cuto\013)g(prescriptions)e(self) f Fs(\000)d FC(energy)j(=)g(O\(1)p Ft(=`)p FC(\).)148 1958 y(F)l(or)j(this)f(system)e(the)i(follo)o(wing)g(theorem)f(has)i(b)q(een)f (pro)o(v)o(ed)g([12,)g(13)q(,)f(38)q(,)h(39)q(].)75 2064 y Fu(Theorem)h(2.2)24 b Fn(F)l(or)743 2125 y Ft(z)r(l)784 2104 y Fo(3)817 2125 y Fs(\034)14 b Ft(e)904 2104 y Fp(\000)939 2088 y Fm(\014)p 936 2097 26 2 v 936 2117 a Fl(2)p Fm(l)968 2125 y Ft(;)67 b(z)r(l)1090 2104 y Fo(3)1089 2137 y Fq(D)1134 2125 y Fs(\035)14 b FC(1)p Ft(;)552 b FC(\(2)p Ft(:)p FC(29\))75 2209 y Fn(al)r(l)14 b(char)n(ge-char)n(ge)g(c)n(orr)n(elations)e(de)n(c)n(ay) g(exp)n(onential)r(ly,)k(i.e.,)e(ther)n(e)f(ar)n(e)f(c)n(onstants)i Ft(C)1667 2216 y Fo(1)1699 2209 y Fn(and)f Ft(L)h(>)g FC(0)75 2269 y Fn(such)k(that)743 2329 y Fs(jh)p Ft(\032)p FC(\()p Ft(x)p FC(\))p Ft(\032)p FC(\()p Ft(y)r FC(\))p Fs(ij)13 b(\024)h Ft(C)1090 2336 y Fo(1)1109 2329 y Ft(e)1137 2292 y Fk(\000j)p Fm(x)p Fk(\000)p Fm(y)q Fk(j)p 1137 2301 101 2 v 1177 2322 a Fm(L)1788 2329 y FC(\(2)p Ft(:)p FC(30\))75 2414 y Fn(and)f(higher)h(trunc) n(ate)n(d)f(char)n(ge)g(c)n(orr)n(elations)g(de)n(c)n(ay)f(exp)n(onential)r (ly)k(with)d(the)h(length)h(of)e(the)h(short-)75 2474 y(est)k(tr)n(e)n(e)f (joining)h(the)g(p)n(ositions)f(of)h(the)g(observables.)982 2599 y FC(9)p eop %%Page: 10 14 bop 148 307 a FC(Also)18 b Ft(L)f Fs(')g Ft(l)381 314 y Fq(D)430 307 y FC(when)i Ft(z)r(l)601 289 y Fo(3)638 307 y FC(and)g Ft(z)r(l)776 289 y Fo(3)775 319 y Fq(D)824 307 y FC(are)f(as)h(in)f(the)f (theorem.)26 b(As)18 b Ft(z)h FC(is)f(roughly)h(the)e(densit)o(y)75 367 y(of)e(particles,)f(the)h(in)o(terpretation)f(of)i(the)f(h)o(yp)q (othesis)g(\(2.29\))h(is)f(that)g(there)g(are)g(man)o(y)e(particles)75 428 y(within)j(a)h(Deb)o(y)o(e)e(sphere)i(\(righ)o(tmost)e(condition\),)h (whic)o(h)g(constitute)g(an)h(appro)o(ximately)d(ideal)75 488 y(gas)23 b(\(leftmost)e(condition\).)39 b(An)22 b(exp)q(onen)o(tial)f(deca)o (y)g(of)i(the)f(kind)g(pro)o(v)o(ed)f(in)h(this)g(theorem)75 548 y(is)g(generally)e(referred)h(to)h(as)g(\\Deb)o(y)o(e)f(screening".)38 b(The)21 b(reason)i(for)f(this)f(screening)g(can)h(b)q(e)75 608 y(understo)q(o)q(d)d(b)o(y)d(the)h(follo)o(wing)g(simpli\014ed)e(argumen) o(t.)22 b(By)17 b(the)g(Sine-Gordon)h(represen)o(tation,)75 668 y(the)g(partition)h(function)f(for)h(the)f(in)o(teracting)g(gas)h(is)g(a) g(sup)q(erp)q(osition)g(of)g(partition)f(functions)75 729 y(for)f(ideal)e (gases)i(of)g(particles)e(in)h(external)f(\014elds:)743 851 y Ft(Z)j FC(=)846 792 y Fr(Z)896 851 y Ft(d\026)p FC(\()p Ft(\036)p FC(\))8 b Ft(Z)1058 858 y Fo(ideal)1133 851 y FC(\()p Ft(i\036)p FC(\))14 b Ft(;)543 b FC(\(2)p Ft(:)p FC(31\))75 973 y(with)722 1033 y Ft(Z)755 1040 y Fo(ideal)830 1033 y FC(\()p Ft(i\036)p FC(\))14 b(=)f Ft(e)1002 981 y Fr(R)1036 1013 y Fq(dp)r Fo(~)-20 b Fq(z)q(d\030)7 b(e)1146 1001 y Fk(\000)p Fm(\014)q(h)p Fl(\()p Fm(i\036)p Fl(\))1788 1033 y FC(\(2)p Ft(:)p FC(32\))75 1120 y(and)17 b(putting)f(the)g(mass,)g Ft(m)d FC(=)h(1,)802 1257 y Ft(h)p FC(\()p Ft(i\036)p FC(\))f(=)984 1223 y Ft(p)1008 1205 y Fo(2)p 984 1246 45 2 v 994 1291 a FC(2)1044 1257 y(+)e Ft(i\017\036)i(:)602 b FC(\(2)p Ft(:)p FC(33\))75 1383 y(As)16 b(there)g(is)g(c)o(harge)g(symmetry)d(\(c.f.)20 b(\(2.16\)\))549 1511 y Ft(Z)e FC(=)652 1453 y Fr(Z)702 1511 y Ft(d\026)p FC(\()p Ft(\036)p FC(\))f(exp)923 1451 y Fr(\024)945 1511 y FC(2)977 1453 y Fr(Z)1027 1511 y Ft(z)f FC(:)d(cos)c Ft(\014)s(\036)p FC(\()p Ft(x)p FC(\))k(:)g Ft(dx)1386 1451 y Fr(\025)1416 1511 y Ft(;)358 b FC(\(2)p Ft(:)p FC(34\))75 1640 y(where)17 b Ft(z)i FC(is)e(as)h(in)f(\(2.28\).)25 b(In)17 b(the)g(regime)e(\(2.29\))j(only)f (small)f(v)m(alues)h(of)h Ft(\014)s(\036)p FC(\()p Ft(x)p FC(\))e(con)o (tribute,)g(so)75 1700 y(appro)o(ximating)f(cos\(\))f Fs(')g FC(1)d Fs(\000)660 1681 y Fo(1)p 660 1689 18 2 v 660 1717 a(2)683 1700 y FC(\(\))721 1682 y Fo(2)757 1700 y FC(w)o(e)k(obtain)508 1840 y Ft(Z)i Fs(')611 1781 y Fr(Z)661 1840 y Ft(d\026)p FC(\()p Ft(\036)p FC(\))g(exp)8 b Fs(\000)946 1806 y Ft(\014)p 926 1828 71 2 v 926 1874 a FC(2)p Ft(l)966 1857 y Fo(2)965 1886 y Fq(D)1010 1781 y Fr(Z)1060 1840 y Ft(dx)g FC(\(:)13 b Ft(\036)1196 1819 y Fo(2)1230 1840 y FC(:)g(+Const.\))h Ft(;)308 b FC(\(2)p Ft(:)p FC(35\))75 1975 y(where)20 b(\\Const.")36 b(is)21 b(indep)q(enden)o(t) e(of)i Ft(\036)p FC(.)35 b(Hence)19 b(the)h(partition)h(function)f(is)h (appro)o(ximately)75 2035 y(determined)14 b(b)o(y)h(a)i(new)f(Gaussian)i (measure)938 2017 y Fo(2)520 2167 y Ft(d\026)574 2146 y Fp(0)586 2167 y FC(\()p Ft(\036)p FC(\))c(=)779 2133 y(1)p 724 2155 135 2 v 724 2201 a(Norm.)863 2167 y Ft(d\026)p FC(\()p Ft(\036)p FC(\))j(exp)8 b Fs(\000)1148 2133 y Ft(\014)p 1128 2155 71 2 v 1128 2201 a FC(2)p Ft(l)1168 2184 y Fo(2)1167 2213 y Fq(D)1212 2108 y Fr(Z)1262 2167 y Ft(dx)22 b FC(:)13 b Ft(\036)1393 2146 y Fo(2)1427 2167 y FC(:)g Ft(;)320 b FC(\(2)p Ft(:)p FC(36\))75 2306 y(of)12 b(co)o(v)m(ariance)f(prop)q(ortional)h(to)g(\()p Fs(\000)p FC(\001)q(+)q Ft(l)841 2313 y Fq(D)872 2306 y FC(\))891 2288 y Fp(\000)p Fo(1)938 2306 y FC(,)g(and)g(hence)f(exhibiting)f(exp)q (onen)o(tial)g(damping)h(\(it)75 2367 y(is)h(the)g(same)f(co)o(v)m(ariance)g (matrix)g(as)i(in)e(the)h(Y)l(uk)m(a)o(w)o(a)g(in)o(teraction\).)19 b(Within)12 b(this)g(appro)o(ximation)p 75 2413 735 2 v 131 2454 a Fj(2)150 2469 y Fy(:)f Fi(\036)198 2454 y Fh(N)241 2469 y Fy(:)i(is)h(de\014ned)h(b)o(y)e(:)e Fi(e)550 2454 y Fh(i\014)q(\036)616 2469 y Fy(:=)672 2438 y Fr(P)727 2451 y Fh(\014)747 2438 y Fe(N)p 727 2460 47 2 v 731 2483 a Fh(N)s Fj(!)791 2469 y Fy(:)g Fi(\036)839 2454 y Fh(N)882 2469 y Fy(:)969 2599 y FC(10)p eop %%Page: 11 15 bop 75 307 a FC(this)14 b(co)o(v)m(ariance)f(predicts)h(exp)q(onen)o(tial)f (deca)o(y)g(for)h(the)g(c)o(harge-c)o(harge)g(correlations)g(\(Problems)75 367 y(2.4.1-2.4.2)23 b(b)q(elo)o(w\).)37 b(So,)24 b(once)d(w)o(e)h(do)g(a)g (substan)o(tial)g(amoun)o(t)f(of)h(w)o(ork)g(to)g(sho)o(w)g(that)h(this)75 428 y(appro)o(ximation)15 b(is)h(v)m(alid,)g(\(2.30\))h(follo)o(ws.)148 513 y(F)l(or)i(the)f Fn(quantum-me)n(chanic)n(al)j FC(Coulom)o(b)c(gas,)j (there)e(is)g(go)q(o)q(d)i(evidence)d(that)i(the)f(\(exp)q(o-)75 573 y(nen)o(tial\))d(screening)h(is)g(lost.)22 b(Rather)17 b(it)f(seems)f(that)h(after)h(a)g(n)o(um)o(b)q(er)d(of)j(Deb)o(y)o(e)e (lengths,)h(there)75 633 y(is)f(a)i(cross-o)o(v)o(er)e(from)g(exp)q(onen)o (tial)f(to)j(p)q(o)o(w)o(er-la)o(w)e(deca)o(y)l(.)20 b(This)c(p)q(ossibilit)o (y)f(w)o(as)h(\014rst)g(noted)g(in)75 693 y([13],)d(p428.)21 b(Since)12 b(then)h(sev)o(eral)f(pap)q(ers)h([10)q(,)f(1,)h(2])g(ha)o(v)o(e)f (left)g(us)h(in)f(no)i(doubt)f(that)g(this)g(actually)75 753 y(o)q(ccurs,)j(but)g(there)f(is)g(as)h(y)o(et)f(no)h(pro)q(of.)22 b(In)16 b(particular)f(the)g(detailed)g(calculations)g(of)h(Alastuey)75 814 y(and)i(Martin)g([1,)f(2)q(])g(sho)o(w)i(within)e(p)q(erturbation)h (theory)g(\(the)g(Wigner-Kirkw)o(o)q(o)q(d)g(expansion\))75 874 y(that)h(screening)f(is)g(destro)o(y)o(ed)g(b)o(y)g(e\013ects)g(due)g(to) h(diagrams)f(with)g(p)q(o)o(w)o(er-la)o(w)h(deca)o(y)f(at)g(order)75 934 y(\026)-25 b Ft(h)103 913 y Fo(4)140 934 y FC(and)18 b(higher.)25 b(They)17 b(sho)o(w,)h(for)f(example,)f(that)i(for)f(NaCl)g(ions)h(in)f(w)o (ater)h(at)g(ro)q(om)f(temp)q(er-)75 994 y(ature)h(there)f(will)g(b)q(e)h (screening)f(out)h(to)g(ab)q(out)h(60)g(Deb)o(y)o(e)d(lengths)i(at)g(whic)o (h)f(p)q(oin)o(t)h(there)g(is)f(a)75 1054 y(cross-o)o(v)o(er)f(to)h(a)f(p)q (o)o(w)o(er)g(la)o(w)g(tail.)21 b(This)16 b(is)g(a)h(tin)o(y)e(e\013ect)h (but)g(they)g(note)g(that)h(it)f(ma)o(y)e(b)q(e)j(m)o(uc)o(h)75 1115 y(larger)i(in)f(electron)g(gases.)30 b(According)19 b(to)g(their)f (analysis)h(the)g(t)o(ypical)e(p)q(o)o(w)o(er-la)o(w)i(is)g Ft(r)1773 1096 y Fp(\000)p Fo(6)1839 1115 y FC(but)75 1175 y(the)13 b(p)q(o)o(w)o(er)h(can)g(b)q(e)f(higher)h(dep)q(ending)f(on)h(the)g (correlation)f(and)h(the)f(system.)19 b(This)14 b(violation)f(of)75 1235 y(screening)g(has)h(nothing)g(to)g(do)g(with)f(statistics.)20 b(It)13 b(is)g(a)h(mec)o(hanism)c(similar)h(to)j(that)g(of)f(the)h(V)l(an)75 1295 y(der)i(W)l(aals)i(forces,)e(but)h(it)f(o)q(ccurs)h([2])g(ev)o(en)e(for) i(one)g(comp)q(onen)o(t)f(plasmas)g(in)g(whic)o(h)g(there)g(are)75 1355 y(no)k(atoms)e(or)i(molecules.)27 b(P)o(ossible)18 b(implications)f(for) i(the)g(binding)g(energy)f(of)i(Noble)e(Metals)75 1416 y(ha)o(v)o(e)g(b)q (een)g(discussed)g(b)o(y)g(Maggs)h(and)g(Ascroft)f(in)g([47)q(])g(who)h(ha)o (v)o(e)e(indep)q(enden)o(tly)g(noted)i(the)75 1476 y(failure)c(of)i (screening.)148 1536 y(Let)k(us)h(presen)o(t)f(a)g(simple)e(mo)q(del)h(to)i (explain)e(wh)o(y)h(quan)o(tum)f(\015uctuations)h(can)h(destro)o(y)75 1596 y(the)f(screening)f(in)g(the)h(sense)g(of)g(the)f(theorem)g(ab)q(o)o(v)o (e.)35 b(F)l(or)21 b(simplici)o(t)o(y)d(w)o(e)i(discuss)h(the)g(case)75 1656 y(of)h(Boltzmann)e(statistics,)j(but)f(there)f(are)h(similar)e(represen) o(tations)h(for)h(F)l(ermion)e(and)j(Bose)75 1716 y(statistics.)29 b(This)18 b(simpli\014cation)f(is)i(reasonable)g(since)f(w)o(e)g(are)h (discussing)g(a)g(regime)e(in)h(whic)o(h)75 1777 y(the)23 b(gas)i(is)e(v)o (ery)f(far)i(from)e(degenerate;)k(b)o(y)d(the)g(h)o(yp)q(otheses)h(\(2.29\))g (of)g(the)f(theorem)f Ft(l)27 b Fs(\034)75 1837 y FC(in)o(terparticle)14 b(distance.)148 1897 y(W)l(e)f(\014rst)h(m)o(ust)d(\014nd)j(the)f(analogous)i (of)f(the)f(Sine-Gordon)h(represen)o(tation)e(for)i(the)f(quan)o(tum)75 1957 y(case.)21 b(The)16 b(starting)h(expression)f(is)738 2095 y Ft(Z)i FC(=)841 2054 y Fr(X)917 2062 y FC(~)-27 b Ft(z)939 2044 y Fq(N)p 914 2084 59 2 v 914 2130 a Ft(N)5 b FC(!)986 2095 y(tr)1032 2047 y Fr(\020)1057 2095 y Ft(e)1080 2075 y Fp(\000)p Fq(\014)r(H)1158 2081 y Fm(N)1189 2047 y Fr(\021)1235 2095 y Ft(:)539 b FC(\(2)p Ft(:)p FC(37\))75 2221 y(T)l(o)18 b(deal)g(with)f(the)h(failure)f(of)h(comm)o(utativ)o(it)o(y)o(|[)p Ft(p;)8 b(x)p FC(])13 b Fs(6)p FC(=)j(0|w)o(e)h(pro)q(ceed)h(as)g(in)g(the)f (standard)75 2281 y(deriv)m(ations)d(of)h(the)f(F)l(eynman-Kac)f(path-in)o (tegral)h(form)o(ula:)19 b(w)o(e)14 b(decomp)q(ose)f Ft(H)1605 2288 y Fq(N)1653 2281 y FC(=)h Ft(K)1746 2288 y Fq(N)1787 2281 y FC(+)7 b Ft(U)1865 2288 y Fq(N)1899 2281 y FC(,)75 2341 y(where)16 b Ft(K)257 2348 y Fq(N)307 2341 y FC(is)g(the)g(kinetic)f(term,)f(and)j(use)f (T)l(rotter's)g(form)o(ula:)275 2460 y(tr)9 b Ft(e)345 2439 y Fp(\000)p Fq(\014)r(H)423 2445 y Fm(N)495 2460 y FC(=)58 b(lim)574 2484 y Fq(m)p Fp(!1)684 2460 y FC(tr)730 2411 y Fr(\020)755 2460 y Ft(e)778 2439 y Fp(\000)815 2423 y Fm(\014)p 810 2432 28 2 v 810 2452 a(m)842 2439 y Fq(K)872 2445 y Fm(N)903 2460 y Ft(e)926 2439 y Fp(\000)962 2423 y Fm(\014)p 958 2432 V 958 2452 a(m)990 2439 y Fq(U)1014 2445 y Fm(N)1045 2411 y Fr(\021)1070 2419 y Fq(m)969 2599 y FC(11)p eop %%Page: 12 16 bop 495 324 a FC(=)58 b(lim)574 349 y Fq(m)p Fp(!1)684 266 y Fr(Z)734 324 y Ft(dX)p 759 331 45 2 v 804 335 a Fo(0)832 324 y Ft(dX)p 857 331 V 902 335 a Fo(1)938 324 y Ft(:)8 b(:)g(:)f(;)h(dX)p 1050 331 V 1095 335 a Fq(m)1137 324 y Ft(\016)r FC(\()p Ft(X)p 1180 331 V 1223 335 a Fq(m)1268 324 y Fs(\000)j Ft(X)p 1318 331 V 1362 335 a Fo(0)1382 324 y FC(\))574 449 y Fs(\002)638 395 y Fq(m)627 408 y Fr(Y)621 500 y Fq(k)q Fo(=1)694 449 y FC(exp)777 376 y Fr(")801 449 y FC(\()p Fs(\000)p FC(1)p Ft(=)p FC(2\))p Fs(j)p Ft(X)p 964 456 V 1009 460 a Fq(k)q Fp(\000)p Fo(1)1086 449 y Fs(\000)g Ft(X)p 1136 456 V 1181 460 a Fq(k)1202 449 y Fs(j)1216 429 y Fo(2)1236 389 y Fr(\022)1277 416 y Ft(\014)p 1271 438 43 2 v 1271 483 a(m)1319 389 y Fr(\023)1349 400 y Fp(\000)p Fo(1)1407 449 y FC(+)g Ft(U)1489 456 y Fq(N)1523 449 y FC(\()p Ft(X)p 1542 456 45 2 v 1587 460 a Fq(k)1608 449 y FC(\))1638 416 y Ft(\014)p 1632 438 43 2 v 1632 483 a(m)1680 376 y Fr(#)1788 449 y FC(\(2.38\))75 585 y(No)o(w,)17 b(as)h(in)f(the)g (Sine-Gordon)h(transformation)f(for)h(classical)e(systems,)g(w)o(e)h(write)g Fn(e)n(ach)g FC(factor)75 650 y Ft(e)98 632 y Fp(\000)134 616 y Fm(\014)p 130 624 28 2 v 130 645 a(m)162 632 y Fq(U)186 638 y Fm(N)215 632 y Fo(\()p Fq(X)p 229 639 32 2 v 260 642 a Fm(i)273 632 y Fo(\))302 650 y FC(as)c(a)h(F)l(ourier)e(transform)g(of)h(a)g (Gaussian,)i(using)e(the)f(iden)o(tit)o(y)f(\(2.6\).)21 b(This)12 b(generates)75 710 y(a)j(family)d(of)j(external)e(\014elds)h Ft(\036)653 717 y Fq(k)675 710 y FC(\()p Ft(x)p FC(\),)g(eac)o(h)g(one)g(in)o (tegrated)g(with)g(a)h(Gaussian)h(measure)d Ft(d\026)1792 717 y Fq(k)1814 710 y FC(.)21 b(All)75 770 y(these)16 b(Gaussian)h(measures)f (are)g(in)g(fact)g(iden)o(tical,)e(with)i(zero)g(mean)f(and)i(co)o(v)m (ariance)1743 751 y Fq(m)p 1743 759 V 1748 788 a(\014)1779 770 y Ft(u)p FC(.)429 915 y Ft(Z)45 b FC(=)587 873 y Fr(X)601 965 y Fq(N)663 881 y FC(~)-27 b Ft(z)685 863 y Fq(N)p 660 903 59 2 v 660 949 a Ft(N)5 b FC(!)749 915 y(lim)732 940 y Fq(m)p Fp(!1)842 856 y Fr(Z)892 915 y Ft(dX)p 917 922 45 2 v 962 925 a Fo(0)990 915 y Ft(dX)p 1015 922 V 1059 925 a Fo(1)1096 915 y Ft(:)j(:)g(:)f(;)h(dX)p 1208 922 V 1253 925 a Fq(m)1294 915 y Ft(\016)r FC(\()p Ft(X)p 1337 922 V 1381 925 a Fq(m)1426 915 y Fs(\000)i Ft(X)p 1475 922 V 1520 925 a Fo(0)1540 915 y FC(\))587 1060 y Fs(\002)651 1007 y Fq(m)640 1019 y Fr(Y)634 1111 y Fq(k)q Fo(=1)707 1060 y FC(exp)789 987 y Fr(")814 1060 y FC(\()p Fs(\000)p FC(1)p Ft(=)p FC(2\))p Fs(j)p Ft(X)p 977 1067 V 1022 1071 a Fq(k)q Fp(\000)p Fo(1)1099 1060 y Fs(\000)h Ft(X)p 1149 1067 V 1193 1071 a Fq(k)1215 1060 y Fs(j)1229 1040 y Fo(2)1248 1000 y Fr(\022)1290 1027 y Ft(\014)p 1284 1049 43 2 v 1284 1095 a(m)1331 1000 y Fr(\023)1362 1011 y Fp(\000)p Fo(1)1409 987 y Fr(#)587 1207 y Fs(\002)634 1148 y Fr(Z)684 1207 y Ft(d\026)738 1214 y Fq(k)760 1207 y FC(\()p Ft(\036)808 1214 y Fq(k)829 1207 y FC(\))d(exp)939 1134 y Fr(")963 1207 y Fs(\000)p Ft(i)p FC(\()p Ft(\036)1067 1214 y Fq(k)1088 1207 y Ft(;)g(\032)1135 1214 y Fq(k)1156 1207 y FC(\))1186 1173 y Ft(\014)p 1180 1195 V 1180 1241 a(m)1228 1134 y Fr(#)1274 1207 y Ft(:)500 b FC(\(2.39\))75 1340 y(In)17 b(the)g(limit)d Ft(m)h Fs(!)g(1)p FC(,)i(the)g(pro)q(duct)g(o)o(v)o(er)g Ft(k)i FC(with)e(incremen)o(t)d Ft(\014)s(=m)i FC(b)q(ecomes)g(\(formally!\))22 b(an)75 1400 y(in)o(tegral)15 b(in)h(the)g(exp)q(onen)o(t.)21 b(That)c(is)f Ft(k)i FC(is)e(replaced)f(b)o(y)h(a)g(con)o(tin)o(uum)e (\\time")h Ft(\034)6 b FC(;)16 b(the)g(sequence)75 1460 y(of)21 b(\014elds)g Ft(\036)294 1467 y Fq(k)315 1460 y FC(\()p Ft(x)p FC(\))g(b)q(ecomes)e(a)j(time-dep)q(enden)o(t)c(external)i(\014eld)h Ft(\036)p FC(\()p Ft(\034)s(;)8 b(x)p FC(\);)22 b(and)g(the)e(sequence)g(of) 75 1521 y(Gaussian)i(measures)e Ft(d\026)557 1528 y Fq(k)601 1521 y FC(assem)o(ble)f(a)i(\\space-time")g(Gaussian)h(measure)e Ft(d\026)1612 1528 y Fq(u)p Fp(\012)p Fq(I)1701 1521 y FC(whose)i(co-)75 1581 y(v)m(ariance)17 b(is)f Ft(u)p FC(\()p Ft(x)11 b Fs(\000)g Ft(y)r FC(\))p Ft(\016)r FC(\()p Ft(\034)16 b Fs(\000)c Ft(\033)r FC(\).)22 b(P)o(erm)o(uting)15 b(this)h(measure)g(with)g(the)h(sum)f(o)o(v)o (er)g Ft(N)22 b FC(w)o(e)16 b(obtain)75 1641 y(again)h(the)f(Sine-Gordon)h (transformation)f(\(2.31\))741 1758 y Ft(Z)i FC(=)844 1700 y Fr(Z)894 1758 y Ft(d\026)948 1765 y Fq(u)p Fp(\012)p Fq(I)1025 1758 y Ft(Z)1058 1765 y Fo(ideal)1133 1758 y FC(\()p Ft(i\036)p FC(\))8 b Ft(;)549 b FC(\(2)p Ft(:)p FC(40\))75 1873 y(but)16 b(with)533 1937 y Ft(Z)566 1944 y Fo(ideal)654 1937 y FC(=)e(exp)789 1896 y Fr(X)812 1983 y Fq(\017)857 1879 y Fr(Z)907 1937 y Ft(dW)985 1917 y Fq(\014)1009 1937 y FC(\()p Ft(X)t FC(\))p Ft(e)1114 1915 y Fp(\000)p Fq(i\017)1173 1884 y Fr(R)1201 1894 y Fm(\014)1193 1932 y Fl(0)1228 1915 y Fq(d\034)9 b(\036)p Fo(\()p Fq(\034)r(;X)s Fo(\()p Fq(\034)t Fo(\)\))1441 1937 y Ft(:)333 b FC(\(2)p Ft(:)p FC(41\))75 2065 y(Here)13 b Ft(dW)266 2047 y Fq(\014)304 2065 y FC(is)h(the)g(Wiener)g(measure|formall)o(y)d(prop)q(ortional)16 b(to)f(exp)1432 2017 y Fr(\020)1457 2065 y Fs(\000)1504 2029 y Fr(R)1532 2042 y Fq(\014)1524 2077 y Fo(0)1564 2065 y Ft(d\034)1629 2045 y Fo(1)p 1629 2053 18 2 v 1629 2082 a(2)1652 2065 y FC(\()p Ft(dx=d\034)6 b FC(\))1819 2047 y Fo(2)1839 2017 y Fr(\021)1864 2065 y FC(|)75 2131 y(supp)q(orted)14 b(o)o(v)o(er)d(con)o(tin)o(uous)i (closed)f(paths)i Ft(X)t FC(\()p Ft(\034)6 b FC(\))p Ft(;)i(\034)19 b Fs(2)14 b FC([0)p Ft(;)8 b(\014)s FC(].)19 b(The)13 b(com)o(bination)e (\(2.40\){\(2.41\))75 2191 y(is)16 b(a)h(represen)o(tation)e(for)i(the)e (quan)o(tum)g(partition)i(function)e(whic)o(h)h(app)q(ears)h(in)f([28].)21 b(It)16 b(is)g(also)75 2251 y(deriv)o(ed)e(and)i(used)f(in)g([2].)20 b(The)c(expression)f(of)g Ft(Z)1012 2258 y Fo(ideal)1103 2251 y FC(in)g(terms)e(of)j(the)f(Wiener)g(measure)f(is)h(due)75 2312 y(to)i(Ginibre)e([34].)148 2372 y(Notice)g(that)i(there)f(is)g(a)g (Goldstone)h(mo)q(de:)738 2474 y Ft(\036)p FC(\()p Ft(\034)s(;)8 b(x)p FC(\))13 b Fs(!)g Ft(\036)p FC(\()p Ft(\034)s(;)8 b(x)p FC(\))j(+)g Ft(f)5 b FC(\()p Ft(\034)h FC(\))969 2599 y(12)p eop %%Page: 13 17 bop 75 311 a FC(where)11 b Ft(f)16 b FC(is)11 b(an)o(y)f(function)h(suc)o(h)g (that)772 276 y Fr(R)800 289 y Fq(\014)792 324 y Fo(0)832 311 y Ft(d\034)6 b(f)f FC(\()p Ft(\034)h FC(\))14 b(=)f(0.)20 b(This)11 b(will)f(b)q(e)h(the)g(origin)g(of)g(the)g(long)g(range)75 371 y(forces.)20 b(The)14 b(in)o(tuition)e(is)h(that)h(the)f(F)l(eynman-Kac)f (form)o(ula)g(represen)o(ts)h(the)g(quan)o(tum)g(gas)h(as)g(a)75 431 y(classical)e(gas)i(of)f(closed)g(c)o(harge)f(lo)q(ops)i(with)f(instan)o (taneous)h(Coulom)o(b)e(in)o(teractions.)19 b(Eac)o(h)13 b(lo)q(op)75 492 y(represen)o(ts)k(the)g(quan)o(tum)f(uncertain)o(t)o(y)g(around)i(a)g (classical)f(p)q(osition.)25 b(This)17 b(leads)h(to)g(a)f(time-)75 552 y(dep)q(enden)o(t)e(dip)q(ole)g(force)g(sup)q(erimp)q(osed)g(on)i(the)e (Coulom)o(b)f(force)i(for)f(the)h(classical)e(system.)20 b(A)75 612 y(dip)q(ole)11 b(can)h(p)q(olarise)g(other)g(dip)q(oles)g(leading)f(to)i (induced)e(dip)q(ole-dip)q(ole)g(or)h(m)o(ultip)q(ole-m)n(ulti)o(p)q(ole)75 672 y(forces)17 b(whic)o(h)f(are)i(p)q(o)o(w)o(er)f(la)o(ws.)23 b(The)18 b(standard)g(textb)q(o)q(ok)g(discussions)f(do)h(not)f(see)g(this)g (e\013ect)75 732 y(b)q(ecause)j(they)g(mak)o(e)e(a)i(static)g(appro)o (ximation)e(whic)o(h)i(loses)g(these)f(time)f(dep)q(endences.)31 b(The)75 793 y(mec)o(hanism)18 b(is)i(v)o(ery)g(similar)f(to)i(that)g(of)g (the)g(V)l(an)g(der)f(W)l(aals)i(forces,)f(except)f(that)h(it)g(tak)o(es)75 853 y(place)16 b(without)g(an)o(y)g(need)g(for)h(neutral)f(ob)s(jects)g(suc)o (h)g(as)g(atoms)g(or)h(molecules.)148 913 y(Unfortunately)e(the)f(delta)h (function)g(in)g(the)f(co)o(v)m(ariance)h Ft(u)p FC(\()p Ft(x)8 b Fs(\000)g Ft(y)r FC(\))p Ft(\016)r FC(\()p Ft(\034)13 b Fs(\000)c Ft(\033)r FC(\))14 b(means)g(that)i(the)75 973 y(time-dep)q(enden)o(t)10 b(external)h(\014elds)g Ft(\036)h FC(are)g(as)h(singular)f(as)h(white)e (noise)h(in)g(their)f Ft(t)p FC(-dep)q(endence)g(and)75 1033 y(so)k(\(2.40\))g(is)f(formal.)20 b(W)l(e)14 b(are)g(going)h(to)g(fail)f(to)h (face)f(these)g(questions)g(and)h(instead)f(concen)o(trate)75 1094 y(on)j(a)f(simple)e(mo)q(del)h(obtained)i(b)o(y)f(making)f(the)h(follo)o (wing)g(simpli\014cations:)135 1190 y(1.)24 b(W)l(e)16 b(replace)f(the)h(con) o(tin)o(uum)e(of)j(p)q(ossible)g(particle)e(p)q(ositions)i(b)o(y)f(a)g (simple)e(cubic)i(lattice)197 1251 y(of)i(spacing)h(l)f(with)f(p)q(erio)q (dic)h(b)q(oundary)i(conditions.)26 b(If)18 b(this)g(is)g(done)g(then)g(the)g (natural)197 1311 y(ob)s(ject)k(to)h(stand)g(in)f(for)h(the)g(Wiener)e (measure)g Ft(dW)1240 1293 y Fq(\014)1287 1311 y FC(is)h(the)g(measure)g(on)h (all)f(simple)197 1371 y(random)16 b(w)o(alks)g(that)h(form)e(lo)q(ops)i Ft(X)t FC(\()p Ft(\034)6 b FC(\))16 b(parametrised)f(b)o(y)h Ft(\034)k Fs(2)14 b FC([0)p Ft(;)8 b(\014)s FC(].)135 1471 y(2.)24 b(W)l(e)f(appro)o(ximate)f(the)i(measure)e Ft(dW)944 1453 y Fq(\014)991 1471 y FC(b)o(y)h(a)h(new)g(measure)e(supp)q(orted)i(only) g(on)g(the)197 1531 y(follo)o(wing)16 b(t)o(w)o(o)g(t)o(yp)q(es)g(of)h(w)o (alk:)218 1631 y(\(a\))24 b(The)17 b(paths)g(that)f(sta)o(y)g(at)h(some)e (site)h Ft(x)g FC(for)g(all)g Ft(\034)k Fs(2)14 b FC([0)p Ft(;)8 b(\014)s FC(].)215 1711 y(\(b\))24 b(The)c(paths)g(that)f(jump)f(b)q(et)o(w)o (een)h(t)o(w)o(o)g(neigh)o(b)q(oring)g(sites,)h(that)f(is)g(w)o(alks)g(c)o (harac-)304 1771 y(terized)c(b)o(y)h(t)o(w)o(o)g(nearest)g(neigh)o(b)q(our)h (lattice)e(sites)h Ft(x;)8 b(y)r FC(,)15 b(suc)o(h)h(that)809 1904 y Ft(X)t FC(\()p Ft(\034)6 b FC(\))13 b(=)983 1831 y Fr(\()1037 1873 y Ft(x)42 b FC(if)15 b Ft(\034)20 b Fs(2)14 b FC([0)p Ft(;)1304 1852 y Fq(\014)p 1304 1862 22 2 v 1306 1890 a Fo(2)1330 1873 y FC(\))1037 1936 y Ft(y)46 b FC(if)15 b Ft(\034)20 b Fs(2)14 b FC([)1258 1914 y Fq(\014)p 1258 1924 V 1260 1953 a Fo(2)1284 1936 y Ft(;)8 b(\014)s FC(\))13 b Ft(:)75 2053 y FC(W)l(e)k(assume)f(that)h Ft(dW)513 2035 y Fq(\014)553 2053 y FC(assigns)h(probabilit)o(y)e(1)c Fs(\000)f Ft(q)18 b FC(to)f(the)g(paths)g (of)g(t)o(yp)q(e)g(\(a\),)f(where)h Ft(q)f Fs(\034)f FC(1;)75 2113 y(while)20 b(paths)h(of)g(t)o(yp)q(e)f(\(b\))g(o)q(ccur)h(with)f (remaining)f(probabilit)o(y)l(,)h Ft(q)r FC(.)34 b(This)20 b(parameter)g Ft(q)i FC(do)q(es)75 2173 y(not)e(dep)q(end)g(on)g(the)g (parameters)e Ft(\014)s(;)8 b(z)r(;)g(:)g(:)g(:)18 b FC(b)q(ecause)i(w)o(e)f (ha)o(v)o(e)g(set)h(the)f(lattice)g(spacing)h(equal)75 2233 y(to)d(the)g(natural)h(length)e(scale)h Ft(l)q FC(.)23 b(Although)17 b(the)g(mass)f Ft(m)h FC(w)o(as)g(set)g(equal)g(to)g(one,)g(our)g(analysis)75 2293 y(dep)q(ends)d(on)h Ft(m)f FC(only)f(through)i Ft(l)g FC(so)g(this)f(normalization)e(is)i(no)h(longer)f(in)f(e\013ect.)20 b(The)14 b(t)o(w)o(o-b)q(o)q(dy)75 2354 y(Coulom)o(b)j(in)o(teraction)g (energy)h(is)g(replaced)f(b)o(y)h Ft(u)1032 2361 y Fo(lattice)1130 2354 y FC(,)g(the)g(in)o(v)o(erse)f(of)h(the)g(lattice)f(Laplacian)75 2414 y(with)12 b(p)q(erio)q(dic)g(b)q(oundary)i(conditions)e(and)h(the)f (zero-mo)q(de)f(pro)s(jected)h(out.)20 b(W)l(e)12 b(call)g(this)g(mo)q(del)75 2474 y(the)k Fn(lattic)n(e)i(Semi-Quantum)i(Mo)n(del)p FC(.)969 2599 y(13)p eop %%Page: 14 18 bop 148 307 a FC(The)17 b(consequence)f(of)h(these)f(c)o(hanges)h(is)g(that)g (the)g(dep)q(endence)f(of)h Ft(Z)1479 314 y Fo(ideal)1554 307 y FC(\()p Ft(i\036)p FC(\))f(on)h Ft(\036)p FC(\()p Ft(\034)s(;)8 b(x)p FC(\))17 b(is)75 367 y(only)f(through)747 450 y Ft(\036)776 457 y Fo(1)795 450 y FC(\()p Ft(x)p FC(\))e(=)934 416 y(1)p 931 438 31 2 v 931 484 a Ft(\014)975 391 y Fr(Z)1017 404 y Fq(\014)998 485 y Fo(0)1048 450 y Ft(d\034)6 b(\036)p FC(\()p Ft(\034)s(;)i(x)p FC(\))75 559 y(and)689 632 y Ft(\036)718 639 y Fo(2)737 632 y FC(\()p Ft(x)p FC(\))14 b(=)876 598 y(1)p 873 620 V 873 666 a Ft(\014)917 573 y Fr(Z)958 586 y Fq(\014)940 668 y Fo(0)990 632 y Ft(d\034)6 b(\036)p FC(\()p Ft(\034)s(;)i(x)p FC(\))p Ft(\022)q FC(\()p Ft(\034)e FC(\))13 b Ft(;)75 742 y FC(where)682 821 y Ft(\022)q FC(\()p Ft(\034)6 b FC(\))14 b(=)837 748 y Fr(\()891 790 y FC(1)80 b(if)16 b Ft(\034)k Fs(2)14 b FC([0)p Ft(;)1193 769 y Fq(\014)p 1193 779 22 2 v 1195 807 a Fo(2)1219 790 y FC(\))891 853 y Fs(\000)p FC(1)41 b(if)16 b Ft(\034)k Fs(2)14 b FC([)1147 831 y Fq(\014)p 1147 841 V 1149 870 a Fo(2)1173 853 y Ft(;)8 b(\014)s FC(\))1292 821 y Ft(:)75 936 y FC(These)15 b(\014elds)g(are)g(indep)q(enden)o(tly)e (distributed)i(according)g(to)g(the)g(p)q(erio)q(dic)g(massless)f(Gaussian)75 997 y(measure)h Ft(d\026)i FC(with)f(co)o(v)m(ariance)g Ft(\014)714 979 y Fp(\000)p Fo(1)760 997 y Ft(u)788 1004 y Fo(lattice)902 997 y FC(on)h(\014elds)f(whose)h(spatial)f(a)o(v)o(erage)g(v)m(anishes.)148 1057 y(After)g(a)g(brief)g(calculation)f(the)h(partition)g(function)g(b)q (ecomes)198 1195 y Ft(Z)46 b FC(=)356 1137 y Fr(Z)406 1195 y Ft(d\026)p FC(\()p Ft(\036)508 1202 y Fo(1)528 1195 y FC(\))555 1137 y Fr(Z)605 1195 y Ft(d\026)p FC(\()p Ft(\036)707 1202 y Fo(2)727 1195 y FC(\)exp)829 1122 y Fr(")853 1195 y FC(2)p Ft(z)r FC(\(1)12 b Fs(\000)f Ft(q)r FC(\))1058 1154 y Fr(X)1078 1241 y Fq(x)1126 1195 y Ft(l)1142 1175 y Fo(3)1175 1195 y FC(:)i(cos)q(\()p Ft(\014)s(\036)1347 1202 y Fo(1)1366 1195 y FC(\()p Ft(x)p FC(\)\))g(:)1478 1122 y Fr(#)356 1336 y FC(exp)439 1263 y Fr(")463 1336 y FC(2)p Ft(z)518 1303 y(q)p 517 1325 25 2 v 517 1370 a FC(6)555 1295 y Fr(X)566 1382 y Fq(xy)623 1336 y Ft(l)639 1316 y Fo(3)672 1336 y FC(:)h(cos)765 1276 y Fr(\022)801 1303 y Ft(\014)p 801 1325 31 2 v 804 1370 a FC(2)836 1336 y([)p Ft(\036)879 1343 y Fo(1)898 1336 y FC(\()p Ft(x)p FC(\))d(+)g Ft(\036)1053 1343 y Fo(1)1072 1336 y FC(\()p Ft(y)r FC(\)])1150 1276 y Fr(\023)1194 1336 y FC(:)i(:)h(cos)1314 1276 y Fr(\022)1349 1303 y Ft(\014)p 1349 1325 V 1352 1370 a FC(2)1385 1336 y([)p Ft(\036)1428 1343 y Fo(2)1447 1336 y FC(\()p Ft(x)p FC(\))d Fs(\000)f Ft(\036)1602 1343 y Fo(2)1622 1336 y FC(\()p Ft(y)r FC(\)])1700 1276 y Fr(\023)1743 1336 y FC(:)1757 1263 y Fr(#)1788 1336 y FC(\(2.42\))75 1481 y(If)18 b Ft(q)h FC(is)f(set)g(to)h(zero)f(w)o(e)g (rev)o(ert)f(to)h(the)g(classical)g(Coulom)o(b)f(gas.)28 b(If)18 b Ft(\036)1415 1488 y Fo(1)1452 1481 y FC(is)g(set)g(to)h(zero)f(then)g(b)o (y)75 1541 y(rev)o(ersing)d(the)h(Sine-Gordon)h(transformation)g(w)o(e)e (obtain)491 1679 y Ft(Z)j Fs(!)606 1638 y Fr(X)679 1646 y FC(\()p Ft(z)r(q)r(=)p FC(3\))814 1628 y Fq(N)p 679 1668 169 2 v 734 1714 a Ft(N)5 b FC(!)939 1638 y Fr(X)861 1725 y Fq(x)881 1730 y Fl(1)898 1725 y Fq(y)915 1730 y Fl(1)933 1725 y Fq(;:::)o(;x)1002 1731 y Fm(N)1030 1725 y Fq(y)1047 1731 y Fm(N)1085 1679 y Ft(l)1101 1659 y Fo(3)p Fq(N)1152 1679 y Ft(e)1175 1659 y Fp(\000)p Fq(\014)r(U)t Fo(\()p Fq(x)1286 1664 y Fl(1)1302 1659 y Fq(y)1319 1664 y Fl(1)1336 1659 y Fq(;:::;x)1406 1665 y Fm(N)1434 1659 y Fq(y)1451 1665 y Fm(N)1480 1659 y Fo(\))75 1827 y FC(where)22 b Ft(U)5 b FC(\()p Ft(x)307 1834 y Fo(1)326 1827 y Ft(y)350 1834 y Fo(1)370 1827 y Ft(;)j(:)g(:)g(:)f(;)h(x)507 1834 y Fq(N)540 1827 y Ft(y)564 1834 y Fq(N)598 1827 y FC(\))22 b(is)f(the)h(Coulom)o(b)f(energy)l (,)i(without)f(self)f(energies,)h(of)h Ft(N)k FC(lattice)75 1887 y(dip)q(oles)c(comp)q(osed)f(of)h(p)q(ositiv)o(e)f(and)h(negativ)o(e)f (c)o(harges)h(at)g(\()p Ft(x)1317 1894 y Fo(1)1337 1887 y Ft(;)8 b(y)1383 1894 y Fo(1)1402 1887 y FC(\))p Ft(;)g(:)g(:)g(:)f(;)h FC(\()p Ft(x)1577 1894 y Fq(N)1611 1887 y Ft(;)g(y)1657 1894 y Fq(N)1690 1887 y FC(\);)25 b(p)q(ositiv)o(e)75 1947 y(c)o(harges)16 b(at)h Ft(x)p FC(,)e(negativ)o(e)h(c)o(harges)g(at)h Ft(y)h FC(co)q(ordinates.)148 2008 y(W)l(e)e(conjecture)g(that)h(this)f(mo)q(del)f (will)g(ha)o(v)o(e)h(the)g(follo)o(wing)g(un)o(usual)h(b)q(eha)o(vior:)k(for) c Ft(z)r FC(,)f Ft(\014)s FC(,)f Ft(l)75 2068 y FC(as)i(in)f(the)g(screening) f(theorem)g(ab)q(o)o(v)o(e)h(there)g(is)g(a)h Ft(L)d Fs(\031)f Ft(l)1144 2075 y Fq(D)1192 2068 y FC(suc)o(h)j(that)h(as)g Ft(r)e Fs(\021)1562 2046 y Fq(x)p Fp(\000)p Fq(y)p 1562 2056 66 2 v 1583 2085 a(L)1647 2068 y Fs(!)f(1)730 2211 y(h)p Ft(\036)778 2218 y Fo(1)798 2211 y FC(\()p Ft(x)p FC(\))8 b(;)g Ft(\036)923 2218 y Fo(1)942 2211 y FC(\()p Ft(y)r FC(\))p Fs(i)28 b(\030)1122 2177 y Ft(c)1143 2184 y Fo(1)p 1110 2199 64 2 v 1110 2245 a Ft(\014)s(L)1184 2177 y(e)1207 2159 y Fp(\000)p Fq(r)p 1184 2199 70 2 v 1207 2245 a Ft(r)1788 2211 y FC(\(2)p Ft(:)p FC(43\))535 2392 y Fs(h)p Ft(\036)583 2371 y Fo(2)583 2404 y(1)603 2392 y FC(\()p Ft(x)p FC(\))8 b(;)g Ft(\036)728 2371 y Fo(2)728 2404 y(1)747 2392 y FC(\()p Ft(y)r FC(\))p Fs(i)28 b(\030)911 2319 y Fr(")951 2358 y Ft(c)972 2365 y Fo(1)p 940 2380 64 2 v 940 2426 a Ft(\014)s(L)1013 2358 y(e)1036 2340 y Fp(\000)p Fq(r)p 1013 2380 70 2 v 1036 2426 a Ft(r)1087 2319 y Fr(#)1111 2330 y Fo(2)1142 2392 y FC(+)1202 2358 y Ft(c)1223 2365 y Fo(2)p 1196 2380 53 2 v 1196 2426 a Ft(L)1229 2411 y Fo(4)1262 2319 y Fr(")1300 2358 y Ft(l)p 1292 2380 34 2 v 1292 2426 a(L)1330 2319 y Fr(#)1354 2330 y Fo(4)1382 2392 y Ft(r)1405 2371 y Fp(\000)p Fo(6)1788 2392 y FC(\(2)p Ft(:)p FC(44\))969 2599 y(14)p eop %%Page: 15 19 bop 75 307 a Fs(h)p Ft(A)8 b FC(;)g Ft(B)s Fs(i)23 b(\021)g(h)p Ft(AB)s Fs(i)15 b(\000)f(h)p Ft(A)p Fs(ih)p Ft(B)s Fs(i)p FC(.)39 b(W)l(e)21 b(ha)o(v)o(e)g(put)h(in)g(the)f(exp)q(onen)o(tially)f(deca)o(ying) h(term)f(in)i(the)75 367 y(last)17 b(equation)f(to)h(emphasise)f(that)h (since)f Ft(l)h FC(is)g(m)o(uc)o(h)d(smaller)h(than)i Ft(L)g FC(there)f(is)h(a)g(in)o(termedi)o(ate)75 428 y(region)g(where)g(exp)q(onen)o (tial)g(deca)o(y)f(is)h(dominan)o(t.)23 b(More)17 b(generally)g(t)o(w)o(o-p)q (oin)o(t)g(correlations)g(of)75 488 y(o)q(dd)c(functions)e(of)h Ft(\036)452 495 y Fo(1)483 488 y FC(will)f(deca)o(y)g(exp)q(onen)o(tially)f (and)i(t)o(w)o(o-p)q(oin)o(t)g(correlations)f(of)h(ev)o(en)e(functions)75 548 y(will)k(deca)o(y)g(b)o(y)g(a)h(p)q(o)o(w)o(er)g(la)o(w.)21 b(Static)14 b(c)o(harge)h(densities)1127 530 y Fo(3)1161 548 y Ft(\032)1186 555 y Fo(stat)1262 548 y FC(are)g(o)q(dd)h(functions)e(so)i(w) o(e)e(exp)q(ect)75 608 y Fs(h)p Ft(\032)119 615 y Fo(stat)181 608 y Ft(\032)206 615 y Fo(stat)267 608 y Fs(i)e FC(to)h(ha)o(v)o(e)e(exp)q (onen)o(tial)h(deca)o(y)f(in)h(this)g(mo)q(del)f(but)h Fs(h)p Ft(\032)1265 590 y Fo(2)1265 620 y(stat)1327 608 y FC(;)c Ft(\032)1374 590 y Fo(2)1374 620 y(stat)1435 608 y Fs(i)13 b FC(will)e(deca)o(y)g(b)o(y)h (a)g(p)q(o)o(w)o(er)75 668 y(la)o(w.)37 b(This)21 b(b)q(eha)o(vior)h(has)g (come)e(ab)q(out)j(b)q(ecause)e(our)h(appro)o(ximations)f(ha)o(v)o(e)g(inadv) o(erten)o(tly)75 729 y(created)c(a)h(mo)q(del)f(whic)o(h)f(is)i(ev)o(en)e (separately)i(in)f Ft(\036)1060 736 y Fo(1)1097 729 y FC(and)h Ft(\036)1222 736 y Fo(2)1260 729 y FC(so)g(ev)o(ery)e(F)l(eynman)g(diagram)h (for)75 789 y Fs(h)p Ft(\036)123 796 y Fo(1)143 789 y FC(\()p Ft(x)p FC(\))8 b(;)g Ft(\036)268 796 y Fo(2)287 789 y FC(\()p Ft(y)r FC(\))p Fs(i)14 b FC(has)g(a)h(path)f(of)g(massiv)o(e)e Ft(\036)877 796 y Fo(1)910 789 y FC(propagators)k(connecting)d(the)h(t)o(w)o (o)g(p)q(oin)o(ts,)g(but)g(this)75 849 y(need)k(not)g(b)q(e)g(the)g(case)g (with)g Fs(h)p Ft(\036)699 831 y Fo(2)699 861 y(1)719 849 y FC(\()p Ft(x)p FC(\))8 b(;)g Ft(\036)844 831 y Fo(2)844 861 y(2)864 849 y FC(\()p Ft(y)r FC(\))p Fs(i)p FC(.)26 b(The)18 b Ft(\036)1118 856 y Fo(1)1156 849 y FC(propagators)j(are)d(massiv)o(e)e(b)q (ecause)i(of)75 909 y(screening)e(from)f(the)h(cos)8 b Ft(\036)588 916 y Fo(1)622 909 y Fs(\031)13 b FC(1)f Fs(\000)765 889 y Fo(1)p 765 897 18 2 v 765 926 a(2)787 909 y Ft(\036)816 891 y Fo(2)816 921 y(1)836 909 y FC(.)148 969 y(T)l(o)k(obtain)f(\(2.43\))h(and)f (\(2.44\))h(w)o(e)e(rewite)g(the)h(action)g(in)g(\(2.42\))g(in)g(terms)e(of)i (dimensionless)75 1029 y(v)m(ariables)f Ft(x)f FC(=)h Ft(l)383 1036 y Fq(D)415 1029 y Ft(x)443 1011 y Fp(0)468 1029 y FC(and)h Ft(\036)590 1036 y Fq(i)604 1029 y FC(\()p Ft(x)p FC(\))e(=)782 1010 y Fo(1)p 740 1018 102 2 v 740 1023 a Fs(p)p 782 1023 60 2 v 34 x Fq(\014)r(l)815 1063 y Fm(D)846 1029 y Ft(\036)875 1011 y Fp(0)875 1042 y Fq(i)889 1029 y FC(\()p Ft(x)936 1011 y Fp(0)948 1029 y FC(\))h(with)g(the)f(appro)o(ximations)1514 996 y Fr(P)1557 1040 y Fq(x)1588 1029 y Ft(l)1604 1011 y Fo(3)1637 1029 y Fs(\000)-9 b(!)1731 994 y Fr(R)1767 1029 y Ft(dx)14 b FC(and)75 1116 y(cos\(\))g Fs(\030)g FC(1)d Fs(\000)335 1097 y Fo(1)p 335 1105 18 2 v 335 1134 a(2)357 1116 y FC(\(\))395 1098 y Fo(2)415 1116 y FC(.)21 b(Omitting)15 b(constan)o(ts)i(that)f(are)h (indep)q(enden)o(t)e(of)h(the)g(parameters)g Ft(z)r FC(,)f Ft(\014)k FC(and)75 1177 y Ft(l)e FC(the)f(action)g(b)q(ecomes)226 1265 y(1)p 226 1287 25 2 v 226 1333 a(2)263 1240 y Fr(Z)305 1299 y FC(\()p Fs(r)p Ft(\036)395 1278 y Fp(0)395 1311 y Fo(1)414 1299 y FC(\))433 1278 y Fo(2)464 1299 y FC(+)518 1265 y(1)p 518 1287 V 518 1333 a(2)555 1240 y Fr(Z)597 1299 y FC(\()p Fs(r)p Ft(\036)687 1278 y Fp(0)687 1311 y Fo(2)706 1299 y FC(\))725 1278 y Fo(2)756 1299 y FC(+)810 1265 y(1)p 810 1287 V 810 1333 a(2)847 1240 y Fr(Z)889 1299 y FC(\()p Ft(\036)937 1278 y Fp(0)937 1311 y Fo(1)956 1299 y FC(\))975 1278 y Fo(2)1006 1299 y FC(+)1068 1265 y Ft(\014)p 1060 1287 47 2 v 1060 1333 a(l)1075 1340 y Fq(D)1112 1299 y FC(\()1151 1265 y Ft(l)p 1136 1287 V 1136 1333 a(l)1151 1340 y Fq(D)1187 1299 y FC(\))1206 1278 y Fo(2)1234 1240 y Fr(Z)1289 1299 y FC(:)e Ft(\036)1346 1278 y Fp(0)1346 1311 y Fo(1)1365 1273 y(2)1399 1299 y FC(::)f(\()p Fs(r)p Ft(\036)1530 1278 y Fp(0)1530 1311 y Fo(2)1549 1299 y FC(\))1568 1278 y Fo(2)1601 1299 y FC(:)h Ft(;)145 b FC(\(2)p Ft(:)p FC(45\))75 1432 y(and)17 b(p)q(erturbation)g(theory)f(for)g(this)g(action)h(leads)f(to)h (\(2.43\))f(and)h(\(2.44\).)75 1576 y Ff(2.4)70 b(Problems)75 1669 y Fu(2.4.1)24 b FC(Find)14 b(out)i(what)g(the)f(c)o(harge-densit)o(y)f (correlations)h(b)q(ecome)f(in)h(Sine-Gordon)h(language:)723 1779 y Fs(h)p Ft(\032)p FC(\()p Ft(\030)r FC(\))8 b Ft(\032)p FC(\()p Ft(\030)903 1758 y Fp(0)917 1779 y FC(\))p Fs(i)28 b FC(=)22 b(?)195 b Ft(\030)17 b Fs(6)p FC(=)d Ft(\030)1374 1758 y Fp(0)197 1889 y FC(Hin)o(t:)20 b(Di\013eren)o(tiate)15 b(with)h(resp)q(ect)g(to)h(an)f(external)g(\014eld.)75 1990 y Fu(2.4.2)24 b FC(Let)15 b Ft(U)5 b FC(\()p Ft(x)386 1997 y Fo(1)414 1990 y Ft(y)438 1997 y Fo(1)458 1990 y Ft(;)j(:)g(:)g(:)f(;)h(x) 595 1997 y Fq(N)637 1990 y Ft(y)661 1997 y Fq(N)694 1990 y FC(\))15 b(b)q(e)h(the)f(Coulom)o(b)g(energy)g(including)f(se\013-energies)h (for)h(a)g(sys-)197 2051 y(tem)e(of)h(c)o(harges)h(of)f(+)p Ft(\017)g FC(at)h(the)f(sites)g Ft(x)925 2058 y Fo(1)945 2051 y Ft(;)8 b(:)g(:)g(:)f(;)h(x)1082 2058 y Fq(N)1131 2051 y FC(and)16 b Fs(\000)p Ft(\017)f FC(at)g(the)h(sites)f Ft(y)1573 2058 y Fo(1)1592 2051 y Ft(;)8 b(:)g(:)g(:)f(;)h(y)1725 2058 y Fq(N)1759 2051 y FC(,)15 b(where)197 2111 y Ft(x)225 2118 y Fq(i)247 2111 y Ft(y)271 2118 y Fq(i)307 2111 y FC(are)22 b(nearest)g(neigh)o(b)q(ors) h(and)f(are)g(in)g(a)g(\014nite)g(subset)g(\003)g(of)g(a)h(lattice.)37 b(Consider)197 2171 y(Diric)o(hlet)18 b(b)q(oundary)k(conditions.)35 b(\(That)21 b(is,)g(the)f(t)o(w)o(o-b)q(o)q(dy)i(in)o(teraction)e Ft(u)g FC(is)g(the)h(so-)197 2231 y(lution)e(of)g Fs(\000)p FC(\001)p Ft(u)p FC(\()p Ft(x;)8 b(y)r FC(\))17 b(=)h Ft(\016)r FC(\()p Ft(x)12 b Fs(\000)h Ft(y)r FC(\),)19 b(where)g(\001)f(is)h(the)g (\014nite-di\013erence)f(laplacian,)h(with)p 75 2278 735 2 v 131 2308 a Fj(3)150 2323 y Fy(These)c(are)g(observ)n(ables)f(obtained)g(b)o (y)g(di\013eren)o(tiating)g(the)g(\(logarithm)e(of)s(\))i(the)g(partition)g (function)f(\(2.37\))75 2373 y(with)19 b(resp)q(ect)j(to)d(a)g(time)f(indep)q (enden)o(t)j(external)f(\014eld)g Fi(f)t Fy(\()p Fi(x)p Fy(\))g(inserted)h (in)o(to)d(the)i(Hamiltonian)d(b)o(y)i Fi(H)1794 2379 y Fh(N)1846 2373 y FA(\000)-7 b(!)75 2423 y Fi(H)110 2429 y Fh(N)149 2423 y Fy(+)189 2389 y Fr(R)224 2423 y Fi(dx)7 b(\032)p Fy(\()p Fi(x)p Fy(\))p Fi(f)t Fy(\()p Fi(x)p Fy(\).)18 b(This)13 b(corresp)q(onds)i (to)e Fi(\036)860 2429 y Fj(1)890 2423 y FA(\000)-6 b(!)11 b Fi(\036)994 2429 y Fj(1)1020 2423 y Fy(+)d Fi(f)18 b Fy(in)13 b(the)h(Sine-Gordon)e(in)o(tegrand.)18 b(They)c(measure)75 2473 y(the)g(resp)q(onse)i(to)e(an)g(in\014nitesmal)e(external)i(\014eld.)969 2599 y FC(15)p eop %%Page: 16 20 bop 197 307 a FC(b)q(oundary)17 b(conditions)g Ft(u)p FC(\()p Ft(x;)8 b(y)r FC(\))k(=)i(0)j(if)f Ft(x)g FC(or)g Ft(y)i FC(is)e(not)h(in)f (\003.\))21 b(Sho)o(w)16 b(that)585 407 y Fr(X)577 498 y Fq(N)t Fp(\025)p Fo(0)667 414 y Ft(z)692 396 y Fq(N)p 667 436 59 2 v 667 482 a Ft(N)5 b FC(!)747 407 y Fr(X)739 494 y Fq(x)759 499 y Fl(1)782 494 y Fq(y)799 499 y Fl(1)825 448 y Fs(\001)j(\001)g(\001)912 407 y Fr(X)891 494 y Fq(x)911 500 y Fm(N)946 494 y Fq(y)963 500 y Fm(N)1000 448 y FC(exp)g([)p Fs(\000)p Ft(U)d FC(\()p Ft(x)1221 455 y Fo(1)1248 448 y Ft(y)1272 455 y Fo(1)1292 448 y Ft(;)j(:)g(:)g(:)f(;)h(x)1429 455 y Fq(N)1471 448 y Ft(y)1495 455 y Fq(N)1528 448 y FC(\)])660 600 y(=)712 541 y Fr(Z)762 600 y Ft(d\026)p FC(\()p Ft(\036)p FC(\))17 b(exp)982 526 y Fr(")1007 600 y FC(2)p Ft(z)1064 558 y Fr(X)1072 645 y Fq(x)6 b(y)1132 600 y FC(cos)q([)p Ft(\036)p FC(\()p Ft(x)p FC(\))k Fs(\000)h Ft(\036)p FC(\()p Ft(y)r FC(\)])1474 526 y Fr(#)1519 600 y Ft(;)197 744 y FC(where)18 b(the)g(sums)f(run)h(o)o(v)o(er)f Fn(al)r(l)j FC(nearest-neigh)o(b)q(or)e(pairs)h Ft(x)1338 751 y Fq(i)1352 744 y Ft(;)8 b(y)1398 751 y Fq(i)1411 744 y FC(,)18 b Ft(x;)8 b(y)r FC(,)18 b(ev)o(en)f(if)g(one)h(p)q(oin)o(t)197 805 y(is)e(outside)g(the)g(lattice,)f(but)h Ft(\036)p FC(\()p Ft(x)p FC(\))e(=)f(0)k(if)f Ft(x)d Fs(62)h FC(\003.)21 b(The)c(measure)e Ft(d\026)i FC(is)f(de\014ned)g(b)o(y)469 958 y Ft(d\026)p FC(\()p Ft(\036)p FC(\))e(=)716 924 y(1)p 661 946 135 2 v 661 992 a(Norm)p Ft(:)808 958 y(d)833 938 y Fo(\003)860 958 y Ft(\036)22 b FC(exp)994 873 y Fr(2)994 947 y(4)1022 958 y Fs(\000)p FC(\(1)p Ft(=)p FC(2\))1196 917 y Fr(X)1180 1008 y Fq(x)5 b(y)q Fp(2)p Fo(\003)1272 958 y FC(\()p Ft(\036)p FC(\()p Ft(x)p FC(\))11 b Fs(\000)f Ft(\036)p FC(\()p Ft(y)r FC(\)\))1558 934 y Fo(2)1577 873 y Fr(3)1577 947 y(5)1627 958 y Ft(:)969 2599 y FC(16)p eop %%Page: 17 21 bop 75 315 a Fv(3)83 b(P)n(olymers,)26 b(Levy)g(Pro)r(cesses,)g(F)-7 b(unctional)27 b(In)n(tegrals)75 425 y FC(In)15 b(this)h(section)f(w)o(e)g (review)g(the)g(transformation)h(of)g(problems)e(in)h(p)q(olymer)f(ph)o (ysics)h(in)o(to)g(func-)75 485 y(tional)d(in)o(tegrals)f(of)h(the)g(t)o(yp)q (e)f(\(1.2\),)i(but)f(sometime)o(s)d(including)i(F)l(ermionic)e(\014elds.)20 b(These)12 b(topics)75 545 y(are)k(discussed)g(in)g(great)h(detail)e(in)h (the)g(b)q(o)q(ok)i([27].)148 606 y(A)d(p)q(olymer)e Ft(!)j FC(=)e(\()p Ft(x)533 613 y Fo(0)552 606 y Ft(;)8 b(x)602 613 y Fo(1)622 606 y Ft(;)g(:)g(:)g(:)f(;)h(x)759 613 y Fq(N)792 606 y FC(\))15 b(is)g(a)h(sequence)e(of)h(monomers)e(describ)q(ed)h(b)o(y)h (the)g(co)q(ordi-)75 666 y(nate)f Ft(x)f FC(b)q(elonging)h(to)g(a)g(\014nite) f(set)g(\003.)20 b(The)14 b(p)q(osition)g Ft(x)f FC(of)h(a)g(monomer)d(in)i (\003)g(unites)h(information)75 726 y(ab)q(out)i(the)f(sp)q(ecies)g(of)h(the) f(p)q(olymer)e(and)j(its)f(p)q(osition)h(in)f(a)g(discrete)f(appro)o (ximation)g(to)i(space.)75 786 y(The)h(t)o(ypical)e(\003)i(will)f(b)q(e)h(a)g (\014nite)f(subset)i(of)f Fu(Z)954 768 y Fq(d)991 786 y FC(with)g Ft(d)g FC(the)g(dimension)e(of)i(space.)23 b(In)17 b(this)g(case)75 846 y(all)f(monomers)f(w)o(ould)h(b)q(e)h(of)g(the)f(same)g(sp)q(ecies.)22 b(T)l(o)17 b(a)g(\014rst)g(appro)o(ximation)e(the)h(activit)o(y)f(of)i(a)75 907 y(p)q(olymer)d Ft(!)19 b FC(w)o(ould)d(b)q(e)g(exp)q(ected)f(to)h(b)q(e)h (a)f(pro)q(duct)h(of)f(Gibbs)g(factors,)h(one)f(for)g(eac)o(h)g(monomer)75 967 y(and)h(one)f(for)h(eac)o(h)f(link,)e(th)o(us)577 1096 y Ft(z)r FC(\()p Ft(!)r FC(\))g(=)751 1042 y Fq(N)740 1054 y Fr(Y)738 1145 y Fq(i)p Fo(=0)886 1062 y FC(1)p 808 1084 180 2 v 808 1130 a Ft(A)p FC(\()p Ft(x)892 1137 y Fq(i)906 1130 y Ft(;)8 b(x)956 1137 y Fq(i)969 1130 y FC(\))1002 1042 y Fq(N)t Fp(\000)p Fo(1)1013 1054 y Fr(Y)1009 1145 y Fq(j)r Fo(=0)1078 1096 y FC(\()p Fs(\000)p Ft(A)p FC(\()p Ft(x)1220 1103 y Fq(j)1237 1096 y Ft(;)g(x)1287 1103 y Fq(j)r Fo(+1)1350 1096 y FC(\)\))p Ft(;)410 b FC(\(3)p Ft(:)p FC(1\))75 1232 y(where)17 b(w)o(e)f(ha)o(v)o(e)g (enco)q(ded)h(the)g(link)f(activities)f(as)i(o\013-diagonal)i(elemen)o(ts)14 b(of)j(a)h(matrix)d Ft(A)p FC(\()p Ft(x;)8 b(y)r FC(\))75 1292 y(and)24 b(monomer)d(activities)g(as)j(in)o(v)o(erses)e(of)h(the)g(diagonal)h (en)o(tries.)41 b(The)24 b(matrix)d Ft(A)i FC(has)h(the)75 1352 y(prop)q(erties)702 1446 y Ft(A)p FC(\()p Ft(x;)8 b(y)r FC(\))k Fs(\024)i FC(0)41 b Fs(8)p Ft(x)13 b Fs(6)p FC(=)g Ft(y)j Fs(2)e FC(\003;)699 1519 y Ft(A)p FC(\()p Ft(x;)8 b(x)p FC(\))13 b Fs(\025)h FC(0)87 b Fs(8)p Ft(x)12 b Fs(2)i FC(\003)p Ft(;)619 b FC(\(3.2\))75 1613 y(and)17 b(in)f(order)g(that)460 1580 y Fr(P)512 1613 y Ft(z)r FC(\()p Ft(!)r FC(\))g(b)q(e)h(\014nite)e(if)h (one)g(end)g(of)h Ft(!)h FC(is)e(\014xed,)829 1666 y Fr(X)849 1753 y Fq(y)897 1708 y Ft(A)p FC(\()p Ft(x;)8 b(y)r FC(\))k Fs(\025)i FC(0)p Ft(;)106 b Fs(8)p Ft(x)p FC(;)570 1822 y Ft(A)11 b FC(+)g Ft(A)704 1801 y Fq(t)735 1822 y FC(is)16 b(a)g(p)q(ositiv)o(e)g (de\014nite)f(matrix)o Ft(:)477 b FC(\(3.3\))75 1916 y(W)l(e)13 b(use)g Ft(!)j FC(:)d Ft(u)g Fs(!)h Ft(v)g FC(to)g(denote)e(the)h(ev)o(en)o (t)e(that)j(there)e(is)h(a)g(p)q(olymer)e Ft(!)k FC(with)e Ft(x)1543 1923 y Fo(0)1576 1916 y FC(=)h Ft(u)f FC(and)g Ft(x)1788 1923 y Fq(N)1835 1916 y FC(=)h Ft(v)75 1976 y FC(for)j(some)e Ft(N)5 b FC(.)21 b(Then)c(for)f(a)h(non-in)o(teracting)f(and)h(non-self-in)o (teracting)e(p)q(olymer)g(system,)643 2095 y Ft(P)674 2102 y Fq(u)697 2095 y FC(\()p Ft(!)h FC(:)e Ft(u)f Fs(!)h Ft(v)r FC(\))f(=)1046 2028 y Fr(P)1090 2072 y Fq(!)q Fo(:)p Fq(u)p Fp(!)p Fq(v)1208 2062 y Ft(z)r FC(\()p Ft(!)r FC(\))p 1010 2084 330 2 v 1010 2096 a Fr(P)1054 2140 y Fq(y)1083 2096 y Fr(P)1127 2140 y Fq(!)q Fo(:)p Fq(u)p Fp(!)p Fq(y)1244 2129 y Ft(z)r FC(\()p Ft(!)r FC(\))1812 2095 y(\(3)p Ft(:)p FC(4\))75 2219 y(where)j Ft(P)247 2226 y Fq(u)286 2219 y FC(is)g(the)g(probabilit)o(y)f (conditioned)h(on)h(there)f(b)q(eing)g(a)h(p)q(olymer)d(whic)o(h)i(starts)h (at)g Ft(u)p FC(.)148 2279 y(The)e(connection)g(b)q(et)o(w)o(een)g(this)g (non-in)o(teracting)g(system)f(and)h(in)o(tegrals)g(of)h(t)o(yp)q(e)f (\(1.1\))g(is)g(a)75 2340 y(trivialit)o(y:)k(de\014ne)d(a)g(Gaussian)i (measure)453 2434 y Ft(d\026)p FC(\()p Ft(\036)p FC(\))d(=)648 2393 y Fr(Y)641 2484 y Fq(x)p Fp(2)p Fo(\003)708 2434 y FC(\()p Ft(d)758 2421 y FC(\026)752 2434 y Ft(\036)q FC(\()p Ft(x)p FC(\))10 b Fs(^)i Ft(d\036)p FC(\()p Ft(x)p FC(\)\))c Ft(e)1073 2413 y Fp(\000)p Fo(\()p Fq(\036;A)1176 2405 y Fo(\026)1171 2413 y Fq(\036)p Fo(\))1208 2434 y Ft(=)p FC(Normalization)276 b(\(3)p Ft(:)p FC(5\))969 2599 y(17)p eop %%Page: 18 22 bop 75 307 a FC(for)15 b(the)g Fn(c)n(omplex)g FC(\014eld)f Ft(\036)g FC(=)g Fs(f)p Ft(\036)p FC(\()p Ft(x)p FC(\))f Fs(2)h Fu(C)g FC(:)g Ft(x)f Fs(2)h FC(\003)p Fs(g)p FC(,)h(with)f(\()p Ft(\036;)8 b(A)1271 294 y FC(\026)1265 307 y Ft(\036)o FC(\))14 b(=)1378 274 y Fr(P)1430 307 y Ft(\036)p FC(\()p Ft(x)p FC(\))p Ft(A)p FC(\()p Ft(x;)8 b(y)r FC(\))1682 294 y(\026)1676 307 y Ft(\036)n FC(\()p Ft(y)r FC(\).)20 b(Then)622 446 y Ft(P)653 453 y Fq(u)676 446 y FC(\()p Ft(!)c FC(:)d Ft(u)h Fs(!)f Ft(v)r FC(\))h(=)1025 377 y Fr(R)1061 413 y Ft(d\026)1121 399 y FC(\026)1115 413 y Ft(\036)p FC(\()p Ft(u)p FC(\))p Ft(\036)p FC(\()p Ft(v)r FC(\))p 988 435 351 2 v 988 448 a Fr(P)1032 491 y Fq(y)1061 445 y Fr(R)1097 481 y Ft(d\026)1157 468 y FC(\026)1151 481 y Ft(\036)q FC(\()p Ft(u)p FC(\))p Ft(\036)p FC(\()p Ft(y)r FC(\))1352 446 y Ft(:)446 b FC(\(3)p Ft(:)p FC(6\))75 584 y(The)16 b(probabilit)o(y)g(that)h(giv)o(en)e(a)i(p)q(olymer)e(starting)i(at)g Ft(a)f FC(and)h(another)g(p)q(olymer)d(starting)j(at)g Ft(b)p FC(,)75 644 y(one)f(of)h(them)e(ends)h(at)h Ft(c)f FC(and)h(the)f(other)g(at) h Ft(d)f FC(is)g(lik)o(ewise)755 702 y Fr(R)791 737 y Ft(d\026)851 724 y FC(\026)845 737 y Ft(\036)p FC(\()p Ft(a)p FC(\))944 724 y(\026)938 737 y Ft(\036)p FC(\()p Ft(b)p FC(\))p Ft(\036)p FC(\()p Ft(c)p FC(\))p Ft(\036)p FC(\()p Ft(d)p FC(\))p 700 759 560 2 v 700 772 a Fr(P)744 816 y Fq(x;y)803 770 y Fr(R)839 805 y Ft(d\026)899 792 y FC(\026)893 805 y Ft(\036)p FC(\()p Ft(a)p FC(\))992 792 y(\026)986 805 y Ft(\036)o FC(\()p Ft(b)p FC(\))p Ft(\036)p FC(\()p Ft(x)p FC(\))p Ft(\036)p FC(\()p Ft(y)r FC(\))1279 771 y(;)519 b(\(3)p Ft(:)p FC(7\))75 909 y(and)21 b(so)h(on)f(for)g(higher)g(order)g(correlations.)35 b(T)l(o)21 b(pro)o(v)o(e)f(these)h(statemen)o(ts)e(w)o(e)h(ha)o(v)o(e)h(only) f(to)75 969 y(note)i(that)h(the)f(righ)o(t)g(hand)h(side)f(of)g(\(3.6\))h (can)f(b)q(e)h(expressed)e(in)h(terms)f(of)h(the)g(propagator)75 1029 y(\()p Ft(A)131 1011 y Fp(\000)p Fo(1)178 1029 y FC(\)\()p Ft(u;)8 b(v)r FC(\))19 b(and)429 996 y Fr(P)481 1029 y Ft(z)r FC(\()p Ft(!)r FC(\))h(is)g(a)g(neat)h(w)o(a)o(y)f(of)g(writing)g(the)g (expansion)h Ft(A)1496 1011 y Fp(\000)p Fo(1)1563 1029 y Fs(\021)f FC(\()p Ft(D)c Fs(\000)d Ft(O)q FC(\))1806 1011 y Fp(\000)p Fo(1)1875 1029 y FC(=)75 1089 y Ft(D)116 1071 y Fp(\000)p Fo(1)174 1089 y FC(+)d Ft(D)263 1071 y Fp(\000)p Fo(1)311 1089 y Ft(O)q(D)390 1071 y Fp(\000)p Fo(1)448 1089 y FC(+)g Ft(D)537 1071 y Fp(\000)p Fo(1)585 1089 y Ft(O)q(D)664 1071 y Fp(\000)p Fo(1)713 1089 y Ft(O)q(D)792 1071 y Fp(\000)p Fo(1)851 1089 y FC(+)g Fs(\001)e(\001)g(\001) g Ft(;)15 b FC(where)h Ft(D)h FC(is)f(the)f(digaonal)i(part)f(of)g Ft(A)f FC(and)h Ft(O)i FC(is)75 1149 y(the)e(o\013-diagonal)i(part)f(of)f Ft(A)p FC(.)148 1210 y(In)i(summary:)23 b(w)o(e)18 b(ha)o(v)o(e)f(seen)h (that)g(the)g(correlations)g(of)h(Gaussian)g(in)o(tegrals)f(can)g(b)q(e)g (rep-)75 1270 y(resen)o(ted)e(in)h(terms)f(of)i(probabilities)e(of)i(p)q(oin) o(ts)f(b)q(eing)h(connected)e(b)o(y)h(p)q(olymers)f(pro)o(vided)h(the)75 1330 y(in)o(v)o(erse)c(propagator)j(ob)q(eys)f(conditions)g(\(3.2\),)g (\(3.3\),)g(i.e.,)d(is)j Fn(ferr)n(omagnetic)p FC(.)21 b(Let)14 b(us)h(note)g(that)75 1390 y(if)h(in)g(\(1.1\))294 1355 y Fr(R)322 1390 y FC(\()p Fs(r)p Ft(\036)p FC(\))431 1372 y Fo(2)466 1390 y FC(w)o(ere)g(replaced)g(b)o(y)g(the)g(\014nite)g(di\013erence)f(appro)o (ximation)h(for)h(a)g(lattice)e Fu(Z)1892 1372 y Fq(d)75 1450 y FC(with)k Ft(\036)f FC(=)g(0)h(outside)g(some)e(set)i(\003)f Fs(\032)g Fu(Z)852 1432 y Fq(d)891 1450 y FC(then,)h(with)f Ft(v)i FC(=)e(0,)h(\(1.1\))g(w)o(ould)g(describ)q(e)f(nearest-)75 1510 y(neigh)o(b)q(or)g(p)q(olymers)d(inside)i(\003.)24 b(The)17 b(real)g(\014eld)g(in)g(\(1.1\))g(as)h(opp)q(osed)h(to)e(the)g(complex)e (\014eld)i(in)75 1571 y(our)g(discussion)f(has)h(to)g(do)f(with)h(whether)f (the)g(p)q(olymers)e(are)j(orien)o(ted)e(or)i(not.)75 1655 y Fu(In)n(teractions)p FC(.)29 b(Supp)q(ose)20 b(all)e(the)h(monomers)e(whic) o(h)h(comprise)f(all)h(the)h(p)q(olymers)e(are)i(in)o(ter-)75 1715 y(acting)d(b)o(y)g(t)o(w)o(o-b)q(o)q(dy)h(p)q(oten)o(tials)g Ft(u)p FC(\()p Ft(x;)8 b(y)r FC(\).)20 b(This)c(w)o(ould)g(in)o(tro)q(duce)g (a)h(Gibbs)f(factor)518 1854 y Ft(G)p FC(\()p Ft(n)p 575 1861 30 2 v 1 w FC(\))e(=)f(exp)772 1781 y Fr(\()806 1854 y Fs(\000)858 1820 y Ft(\014)p 858 1843 31 2 v 861 1888 a FC(2)901 1813 y Fr(X)907 1900 y Fq(x;y)969 1854 y Ft(n)p FC(\()p Ft(x;)8 b(!)p 1067 1861 33 2 v 2 w FC(\))p Ft(u)p FC(\()p Ft(x;)g(y)r FC(\))p Ft(n)p FC(\()p Ft(y)r(;)g(!)p 1356 1861 V 1 w FC(\))1406 1781 y Fr(\))1447 1854 y Ft(;)351 b FC(\(3)p Ft(:)p FC(8\))75 1999 y Ft(n)p FC(\()p Ft(x;)8 b(!)p 173 2006 V 2 w FC(\))19 b(is)g(the)h(total)f (n)o(um)o(b)q(er)f(of)i(times)d(the)i(collection)f(of)i(p)q(olymers)e Ft(!)p 1453 2006 V 21 w FC(=)i Fs(f)p Ft(!)1617 2006 y Fo(1)1637 1999 y Ft(;)8 b(!)1689 2006 y Fo(2)1708 1999 y Ft(;)g(:)g(:)g(:)g(;)g(!)1848 2006 y Fq(M)1888 1999 y Fs(g)75 2060 y FC(con)o(tains)18 b(a)g(monomer)e(of)j Ft(x)p FC(.)26 b(F)l(or)18 b(a)g(p)q(olymer)f Ft(!)r FC(,)h Ft(n)p FC(\()p Ft(x;)8 b(!)r FC(\))18 b(is)f(the)h(n)o(um)o(b)q(er)e(of)j (times)d Ft(x)h FC(o)q(ccures)75 2120 y(in)f(the)g(list)f Ft(!)h FC(=)e(\()p Ft(x)442 2127 y Fo(0)461 2120 y Ft(;)8 b(x)511 2127 y Fo(1)531 2120 y Ft(;)g(:)g(:)g(:)f(;)h(x)668 2127 y Fq(N)701 2120 y FC(\),)16 b(and)h(if)f(there)f(are)h(man)o(y)f(p)q(olymers)g Fs(f)p Ft(!)1492 2127 y Fo(1)1512 2120 y Ft(;)8 b(!)1564 2127 y Fo(2)1584 2120 y Ft(;)g(:)g(:)g(:)f(;)h(!)1723 2127 y Fq(M)1763 2120 y Fs(g)p FC(,)16 b(then)75 2180 y Ft(n)p FC(\()p Ft(x;)8 b(!)p 173 2187 V 2 w FC(\))14 b(=)289 2147 y Fr(P)342 2180 y Ft(n)p FC(\()p Ft(x;)8 b(!)470 2187 y Fq(i)484 2180 y FC(\).)75 2264 y Fu(Question)p FC(.)20 b(Do)q(es)c Ft(\036)475 2246 y Fo(2)495 2264 y FC(\()p Ft(x)p FC(\))d Fs(\021)h Ft(\036)p FC(\()p Ft(x)p FC(\))728 2251 y(\026)722 2264 y Ft(\036)o FC(\()p Ft(x)p FC(\))h(corresp)q(ond)g(to)h Ft(n)p FC(\()p Ft(!)r(;)8 b(x)p FC(\))14 b(in)h(the)g(sense)f(that)i(correlations)75 2325 y(of)k Ft(d\026)p FC(\()p Ft(\036)p FC(\))p Ft(G)p FC(\()p Ft(\036)p 312 2341 29 2 v 342 2303 a Fo(2)361 2325 y FC(\))p Ft(=)412 2289 y Fr(R)449 2325 y Ft(d\026G)p FC(\()p Ft(\036)p 560 2341 V 590 2303 a Fo(2)610 2325 y FC(\))f(describ)q(e)g(p)q(olymers)f(in) o(teracting)g(with)i(the)f(Gibbs)g(factor)h(\(3.8\)?)75 2390 y(W)l(e)i(ha)o(v)o(e)f(set)h Ft(\036)p 365 2406 V -22 x Fo(2)438 2390 y Fs(\021)h FC(\()p Ft(\036)p FC(\()p Ft(x)p FC(\))620 2376 y(\026)614 2390 y Ft(\036)p FC(\()p Ft(x)p FC(\)\))728 2397 y Fq(x)p Fp(2)p Fo(\003)819 2390 y FC(The)f(answ)o(er)h(to)f(this)g (equations)h(is)f(actually)f Fn(NO)p FC(,)h(but)75 2474 y(follo)o(wing)d (Symanzik)f([58],)i(w)o(e)f(will)g(see)g(that)h(if)f(p)q(olymers)g(are)h (replaced)e(b)o(y)i(Levy)f(pro)q(cesses)969 2599 y(18)p eop %%Page: 19 23 bop 75 307 a FC(whic)o(h)14 b(are)h(functions)h Ft(X)t FC(\()p Ft(s)p FC(\))f(of)g(a)h(con)o(tin)o(uous)f(parameter)f(s,)h(and)g(if)g(the)g (quan)o(tities)f Ft(n)p FC(\()p Ft(x;)8 b(!)r FC(\))14 b(are)75 367 y(replaced)h(b)o(y)h Fn(o)n(c)n(cup)n(ation)h(times)742 479 y Ft(\034)769 458 y Fq(t)784 479 y FC(\()p Ft(x)p FC(\))c(=)915 420 y Fr(Z)957 433 y Fq(t)938 515 y Fo(0)980 479 y Ft(ds)8 b(\016)1058 486 y Fq(x)1080 479 y FC(\()p Ft(X)t FC(\()p Ft(s)p FC(\)\))p Ft(;)575 b FC(\(3)p Ft(:)p FC(9\))75 585 y(then)16 b(the)g(rule)703 645 y Ft(\036)732 625 y Fo(2)752 645 y FC(\()p Ft(x)p FC(\))d Fs(\021)h Ft(\036)p FC(\()p Ft(x)p FC(\))985 632 y(\026)979 645 y Ft(\036)o FC(\()p Ft(x)p FC(\))g Fs( )-8 b(!)13 b Ft(\034)6 b FC(\()p Ft(x)p FC(\))503 b(\(3)p Ft(:)p FC(10\))75 725 y(b)q(ecomes)11 b(v)m(alid)g(so)i(that)f(the)f(measure)g Ft(d\026)p FC(\()p Ft(\036)p FC(\))p Ft(G)p FC(\()p Ft(\036)1007 707 y Fo(2)1028 725 y FC(\))g(do)q(es)i(indeed)e(describ)q(e)g(a)h(\(Grand)h (Canonical)75 785 y(Ensem)o(ble\))g(of)j(ob)s(jects)g Ft(X)p 534 792 45 2 v 18 w FC(=)e Fs(f)p Ft(X)709 792 y Fo(1)729 785 y FC(\()p Ft(s)771 792 y Fo(1)791 785 y FC(\))p Ft(;)8 b(:)g(:)g(:)f(;)h(X) 959 792 y Fq(M)999 785 y FC(\()p Ft(s)1041 792 y Fq(M)1080 785 y FC(\))p Fs(g)16 b FC(with)f(in)o(teraction)g Ft(G)p FC(\()1552 752 y Fr(P)1605 785 y Ft(\034)1632 767 y Fq(t)1626 797 y(j)1646 785 y FC(\))h(as)g(opp)q(osed)75 845 y(to)i Ft(G)p FC(\()p Ft(n)p 193 852 30 2 v 1 w FC(\).)26 b(It)17 b(is)h(p)q(ossible)g(to)g (descend)g(to)g(p)q(olymers)f(b)o(y)g(in)o(tegrating)h(o)o(v)o(er)f(all)g Ft(X)t FC(\()p Ft(s)p FC(\))h(whic)o(h)f(are)75 905 y(di\013eren)o(t)j (parameterizations)g(of)i(the)f(same)f(p)q(olymer)g Ft(!)r FC(.)36 b(Then)22 b Ft(G)p FC(\()1417 872 y Fr(P)1469 905 y Ft(\034)1496 887 y Fq(t)1490 918 y(j)1511 905 y FC(\))f(b)q(ecomes)f(another) 75 974 y(\(more)15 b(complicated\))f(function)708 961 y(~)697 974 y Ft(G)p FC(\()p Ft(n)p 754 981 V FC(\).)75 1116 y Ff(3.1)70 b(Levy)23 b(Pro)r(cesses)75 1208 y FC(A)17 b(Levy)f(pro)q(cess)i Ft(X)t FC(\()p Ft(t)p FC(\))f(is)f(a)i(random)e(function)h(of)g(a)g(con)o (tin)o(uous)g(time)e Ft(t)g Fs(\025)f FC(0)k(whic)o(h)e(describ)q(es)75 1268 y(the)h Fn(history)g FC(of)h(a)g(drunk)m(ard)g(who,)g(giv)o(en)f(that)h (he)f(is)h(at)g(a)g(site)f Ft(x)p FC(,)g(w)o(aits)g(for)h(a)g(random)f(time)f Ft(t)75 1329 y FC(distributed)g(according)g(to)705 1389 y Ft(dp)754 1396 y Fq(x)777 1389 y FC(\()p Ft(t)p FC(\))d(=)h Ft(A)p FC(\()p Ft(x;)8 b(x)p FC(\))p Ft(e)1074 1368 y Fp(\000)p Fq(A)p Fo(\()p Fq(x;x)p Fo(\))p Fq(t)1218 1389 y Ft(dt;)513 b FC(\(3)p Ft(:)p FC(11\))75 1468 y(and)19 b(then,)g(with)f(probabilit)o(y)g Fs(\000)p Ft(A)p FC(\()p Ft(x;)8 b(y)r FC(\))p Ft(=)m(A)p FC(\()p Ft(x;)g(x)p FC(\))16 b(jumps)h(to)i(site)f Ft(y)r FC(,)g(and)i(with)e(the)g (remaining)75 1528 y(probabilit)o(y)j(1)16 b(+)423 1495 y Fr(P)466 1539 y Fq(y)q Fp(6)p Fo(=)p Fq(x)543 1528 y Ft(A)p FC(\()p Ft(x;)8 b(y)r FC(\))p Ft(=)m(A)p FC(\()p Ft(x;)g(x)p FC(\))20 b(jumps)h(to)h(a)h(state)f(called)f Fn(the)i(gr)n(aveyar)n(d)e FC(|and)i(is)75 1589 y(nev)o(er)15 b(again)i(seen)f(amongst)g(the)g(states)h (of)g(the)f(living,)e(whic)o(h)i(are)g(the)g(sites)g(in)g(\003.)148 1649 y(This)g(pro)q(cess)h(is)f(asso)q(ciated)h(with)f Ft(e)844 1631 y Fp(\000)p Fq(tA)929 1649 y FC(as)h(w)o(e)e(can)i(see)e(formally)g(b)o (y)g(decomp)q(osing)h Ft(A)g FC(in)o(to)75 1709 y(its)g(diagonal)h(part)g(\() p Ft(D)q FC(\))g(and)g(its)f(o\013-diagonal)i(part)f(\()p Fs(\000)p Ft(O)q FC(\))f(and)h(writing)482 1817 y Ft(e)505 1796 y Fp(\000)p Fq(tA)588 1817 y FC(=)c Ft(e)662 1796 y Fp(\000)p Fq(t)p Fo([)p Fq(D)q Fp(\000)p Fq(O)q Fo(])850 1817 y FC(=)59 b(lim)930 1846 y Fq(N)t Fp(!1)1040 1769 y Fr(h)1060 1817 y Ft(e)1083 1796 y Fp(\000)1122 1783 y Fm(t)p 1115 1789 27 2 v 1115 1809 a(N)1147 1796 y Fq(D)1179 1817 y Ft(e)1214 1783 y Fm(t)p 1206 1789 V 1206 1809 a(N)1238 1796 y Fq(O)1268 1769 y Fr(i)1287 1780 y Fq(N)850 1941 y FC(=)g(lim)930 1971 y Fq(N)t Fp(!1)1040 1880 y Fr(\024)1062 1941 y Ft(e)1085 1920 y Fp(\000)1125 1907 y Fm(t)p 1117 1913 V 1117 1934 a(N)1149 1920 y Fq(D)1189 1880 y Fr(\022)1220 1941 y FC(1)11 b(+)1323 1907 y Ft(t)p 1309 1929 45 2 v 1309 1975 a(N)1359 1941 y(O)1397 1880 y Fr(\023\025)1450 1890 y Fq(N)1492 1941 y Ft(:)282 b FC(\(3.12\))75 2049 y(Expand)12 b(the)f(righ)o(t)g(hand)g(side)g(as)h(a)f(matrix)f(pro)q(duct)i(and)g(observ) o(e)e(that)i(the)f(resulting)f(sequences)75 2109 y(of)15 b(matrix)e(indices)h (can)h(b)q(e)g(in)o(terpreted)f(as)h(histories)g(of)g(the)g(t)o(yp)q(e)f (describ)q(ed)g(in)h(the)g(paragraph)75 2169 y(ab)q(o)o(v)o(e)h(sampled)f(at) i(in)o(terv)m(als)e(of)i(time)d(of)i(length)1044 2150 y Fq(t)p 1034 2158 32 2 v 1034 2187 a(N)1071 2169 y FC(.)148 2230 y(A)h(more)f (accurate)h(w)o(a)o(y)g(of)g(putting)h(this)f(is)g(that)g(to)h(the)f(exp)q (onen)o(tial)f Ft(e)1535 2212 y Fp(\000)p Fq(tA)1621 2230 y FC(of)h(an)o(y)g(matrix)75 2290 y(satisfying)e(the)f(conditions)h(\(3.2\))g (is)f(asso)q(ciated)i(a)f(probabilit)o(y)f(measure)f Ft(dP)1529 2297 y Fq(u)1553 2290 y FC(\()p Ft(X)t FC(\))i(on)g(the)f(space)75 2350 y(of)j(all)e(histories)h Fs(f)p Ft(X)t FC(\()p Ft(t)p FC(\))e(:)f Ft(X)t FC(\(0\))i(=)f Ft(u)p Fs(g)i FC(with)g(the)g(prop)q (erties:)629 2457 y(\()p Ft(e)671 2436 y Fp(\000)p Fq(tA)739 2457 y FC(\)\()p Ft(u;)8 b(v)r FC(\))13 b(=)937 2398 y Fr(Z)987 2457 y Ft(dP)1043 2464 y Fq(u)1066 2457 y FC(\()p Ft(X)t FC(\))8 b Ft(\016)1178 2464 y Fq(v)1199 2457 y FC(\()p Ft(X)t FC(\()p Ft(t)p FC(\)\))p Ft(;)437 b FC(\(3)p Ft(:)p FC(13\))969 2599 y(19)p eop %%Page: 20 24 bop 75 307 a FC(where)16 b Ft(\016)238 314 y Fq(x)259 307 y FC(\()p Ft(y)r FC(\))e(=)g(1)i(if)g Ft(x)d FC(=)h Ft(y)r FC(,)i(0)g (otherwise,)g(and)h(more)e(generally)l(,)f(the)i Fn(F)l(eynman-Kac)j FC(form)o(ula)508 383 y Fr(\020)533 432 y Ft(e)556 411 y Fp(\000)p Fq(t)p Fo(\()p Fq(A)p Fo(+)p Fq(V)8 b Fo(\))707 383 y Fr(\021)740 432 y FC(\()p Ft(u;)g(v)r FC(\))13 b(=)919 373 y Fr(Z)969 432 y Ft(dP)1025 439 y Fq(u)1056 432 y Ft(e)1079 378 y Fr(R)1107 388 y Fm(t)1099 426 y Fl(0)1127 409 y Fq(V)7 b Fo(\()p Fq(X)s Fo(\()p Fq(s)p Fo(\)\))p Fq(ds)1294 432 y Ft(\016)1316 439 y Fq(v)1336 432 y FC(\()p Ft(X)t FC(\()p Ft(t)p FC(\)\))p Ft(:)300 b FC(\(3)p Ft(:)p FC(14\))75 556 y(On)16 b(the)g(left-hand)h(side,)e Ft(V)c FC(\()p Ft(x)p FC(\))16 b(is)g(acting)h(as)f(a)h(diagonal)g(matrix)e Ft(V)c FC(\()p Ft(x)p FC(\))p Ft(\016)1457 563 y Fq(x)1478 556 y FC(\()p Ft(y)r FC(\).)148 616 y(If)i(w)o(e)g(presen)o(t)g Ft(dP)486 623 y Fq(u)523 616 y FC(this)g(w)o(a)o(y)l(,)g(then)g(the)h (question)f(arises:)20 b(What)14 b(do)q(es)g(this)f(ha)o(v)o(e)g(to)h(do)g (with)75 676 y(our)k(\014rst)g(description)g(of)g Ft(X)t FC(\()p Ft(t)p FC(\))f(as)i(a)f(dilatory)g(drunk)m(ard)g(and)h(what)f(do)q(es)h(it)e (ha)o(v)o(e)g(to)h(do)h(with)75 737 y(the)d(p)q(olymer?)k(T)l(o)d(answ)o(er)f (this,)g(de\014ne,)g(in)f(terms)g(of)i Ft(dp)1171 744 y Fq(x)1210 737 y FC(giv)o(en)e(in)h(\(3.11\),)461 846 y Ft(dp)510 825 y Fq(n)510 858 y(x)534 846 y FC(\()p Ft(t)p FC(\))41 b(=)h(\()p Ft(dp)779 853 y Fq(x)813 846 y Fs(\003)11 b Ft(dp)898 853 y Fq(x)931 846 y Fs(\003)g(\001)d(\001)g(\001)k(\003)f Ft(dp)1122 853 y Fq(x)1144 846 y FC(\)\()p Ft(t)p FC(\))711 950 y Fs(\021)763 891 y Fr(Z)813 950 y Ft(dp)862 957 y Fq(x)885 950 y FC(\()p Ft(t)922 957 y Fo(1)941 950 y FC(\))d Fs(\001)g(\001)g(\001)h Ft(dp)1084 957 y Fq(x)1107 950 y FC(\()p Ft(t)1144 957 y Fq(n)1167 950 y FC(\))p Ft(\016)1218 877 y Fr( )1250 950 y Ft(t)i Fs(\000)1348 896 y Fq(n)1329 908 y Fr(X)1330 999 y Fq(i)p Fo(=1)1397 950 y Ft(t)1415 957 y Fq(i)1429 877 y Fr(!)1470 950 y Ft(dt:)261 b FC(\(3.15\))75 1092 y(If)20 b(a)i(drunk)m(ard)g(is)e(conditioned)h(to)g (visit)f Ft(n)h FC(times)e(the)i(p)q(oin)o(t)g Ft(x)p FC(,)h(then)e(this)h (is)g(ho)o(w)g(his)g(total)75 1152 y(time)16 b(sp)q(en)o(t)h(at)h Ft(x)g FC(will)e(b)q(e)i(distributed.)25 b(When)18 b Ft(n)e FC(=)g(0)i(de\014ne)g Ft(dp)1337 1134 y Fo(0)1357 1152 y FC(\()p Ft(t)p FC(\))e Fs(\021)g Ft(\016)r FC(\()p Ft(t)p FC(\))p Ft(dt)p FC(.)25 b(De\014ne,)17 b(for)h(a)75 1212 y(p)q(olymer)d Ft(!)r FC(,)h(the)g(probabilit)o(y)f(measure)720 1321 y Ft(d)745 1301 y Fq(!)771 1321 y Ft(p)p FC(\()p Ft(t)p 814 1328 18 2 v FC(\))f(=)924 1280 y Fr(Y)916 1372 y Fq(x)p Fp(2)p Fo(\003)993 1321 y Ft(dp)1042 1301 y Fq(n)p Fo(\()p Fq(x;!)q Fo(\))1042 1334 y Fq(x)1146 1321 y FC(\()p Ft(t)p FC(\()p Ft(x)p FC(\)\))520 b(\(3)p Ft(:)p FC(16\))75 1469 y(on)16 b(the)f(collection)f Ft(t)p 441 1476 V 14 w FC(=)f Fs(f)p Ft(t)p FC(\()p Ft(x)p FC(\))g(:)h Ft(x)f Fs(2)h FC(\003)p Fs(g)p FC(.)21 b(Then)16 b(one)f(can)h(pass)g(from)f(the)g (measure)f Ft(dP)1708 1476 y Fq(u)1731 1469 y FC(\()p Ft(X)t FC(\))i(to)g(a)75 1529 y(sum)f(o)o(v)o(er)h(p)q(olymers)e(b)o(y)75 1642 y Fu(Prop)r(osition)k(3.1)388 1707 y Fr(Z)430 1720 y Fp(1)411 1801 y Fo(0)475 1765 y Ft(dt)535 1707 y Fr(Z)585 1765 y Ft(dP)641 1772 y Fq(u)664 1765 y FC(\()p Ft(X)t FC(\))p Ft(F)7 b FC(\()p Ft(\034)p 804 1772 27 2 v 830 1745 a Fq(t)845 1765 y FC(\))p Ft(\016)886 1772 y Fq(v)906 1765 y FC(\()p Ft(x)p FC(\()p Ft(t)p FC(\)\))13 b(=)1117 1724 y Fr(X)1093 1811 y Fq(!)q Fo(:)p Fq(u)p Fp(!)p Fq(v)1208 1765 y Ft(z)r FC(\()p Ft(!)r FC(\))1311 1707 y Fr(Z)1361 1765 y Ft(d)1386 1745 y Fq(!)1412 1765 y Ft(p)p FC(\()p Ft(t)p 1455 1772 18 2 v FC(\))p Ft(F)7 b FC(\()p Ft(t)p 1550 1772 V -1 w FC(\))p Ft(;)188 b FC(\(3)p Ft(:)p FC(17\))75 1907 y Fn(wher)n(e)18 b Ft(\034)p 213 1914 27 2 v 239 1889 a Fq(t)268 1907 y FC(=)c Fs(f)p Ft(\034)372 1889 y Fq(t)386 1907 y FC(\()p Ft(x)p FC(\))g(:)f Ft(x)h Fs(2)g FC(\003)p Fs(g)j Fn(is)g(the)h(c)n(ol)r(le)n(ction)i(of)d(r)n(andom)f(variables)746 2039 y Ft(\034)773 2018 y Fq(t)788 2039 y FC(\()p Ft(x)p FC(\))d Fs(\021)920 1980 y Fr(Z)961 1993 y Fq(t)943 2074 y Fo(0)984 2039 y Ft(\016)1006 2046 y Fq(x)1028 2039 y FC(\()p Ft(X)t FC(\()p Ft(s)p FC(\)\))p Ft(ds;)555 b FC(\(3)p Ft(:)p FC(18\))75 2166 y Fn(and)18 b Ft(F)25 b Fn(is)17 b(any)h(b)n(ounde)n(d)g(smo)n(oth)g (function.)25 b(We)18 b(se)n(e)g(that)g Ft(\034)1235 2148 y Fq(t)1250 2166 y FC(\()p Ft(x)p FC(\))g Fn(is)g(the)g(total)h(time)f(that)g Ft(X)t FC(\()p Ft(s)p FC(\))p Fn(,)75 2226 y FC(0)c Fs(\024)g Ft(s)g Fs(\024)f Ft(t)p Fn(,)k(sp)n(ends)h(at)f Ft(x)p Fn(.)75 2365 y Fu(Pro)r(of)i(of)f(Prop)r(osition)g(3.1.)j FC(V)l(erify)15 b(it)g(for)i(the)f(sp)q(ecial)g(case)777 2474 y Ft(F)7 b FC(\()p Ft(t)p 835 2481 18 2 v -2 w FC(\))14 b(=)g Ft(e)959 2453 y Fp(\000)p Fq(i)1004 2424 y Fr(P)1048 2468 y Fm(x)1073 2453 y Fq(V)8 b Fo(\()p Fq(x)p Fo(\))p Fq(t)p Fo(\()p Fq(x)p Fo(\))1788 2474 y FC(\(3)p Ft(:)p FC(19\))969 2599 y(20)p eop %%Page: 21 25 bop 75 307 a FC(b)o(y)22 b(ev)m(aluating)g(the)g(righ)o(t)g(hand)g(side)g (explicitly)d(and)k(recognizing)f(that)g(it)g(is)g(the)g(resolv)o(en)o(t)75 367 y(expansion)c(for)g(\()p Ft(A)12 b FC(+)g Ft(iV)f FC(\))572 349 y Fp(\000)p Fo(1)619 367 y FC(\()p Ft(u;)d(v)r FC(\))17 b(when)h Ft(A)12 b FC(+)g Ft(iV)29 b FC(is)18 b(split)f(in)o(to)g(its)h (diagonal)h(part)f Ft(D)i FC(and)e(o\013-)75 428 y(diagonal)i(part)g(0)f(so)h (that)g(\()p Ft(A)12 b FC(+)h Ft(iV)e FC(\))791 409 y Fp(\000)p Fo(1)857 428 y FC(=)19 b(\()p Ft(D)14 b FC(+)f Ft(O)q FC(\))1095 409 y Fp(\000)p Fo(1)1162 428 y FC(=)19 b Ft(D)1260 409 y Fp(\000)p Fo(1)1321 428 y FC(+)13 b Ft(D)1413 409 y Fp(\000)p Fo(1)1461 428 y Ft(O)q(D)1540 409 y Fp(\000)p Fo(1)1601 428 y FC(+)g Fs(\001)8 b(\001)g(\001)q FC(.)29 b(F)l(or)20 b(the)75 488 y(left)c(hand)i(side)f(of)h(\(3.17\))g(w)o(e)f(notice)g(that)906 455 y Fr(P)950 498 y Fq(x)972 488 y Ft(V)11 b FC(\()p Ft(x)p FC(\))p Ft(\034)1104 470 y Fq(t)1119 488 y FC(\()p Ft(x)p FC(\))k(=)1254 452 y Fr(R)1281 465 y Fq(t)1273 500 y Fo(0)1304 488 y Ft(ds)8 b(V)k FC(\()p Ft(X)t FC(\()p Ft(s)p FC(\)\),)18 b(so)g(it)e(to)q(o)j(can)e(b) q(e)75 548 y(ev)m(aluated)f(b)o(y)g(the)g(F)l(eynman-Kac)f(form)o(ula)g (\(3.14\).)22 b(It)15 b(is)677 608 y Fr(Z)718 621 y Fp(1)700 703 y Fo(0)764 667 y Ft(dt)8 b(e)838 646 y Fp(\000)p Fq(t)p Fo(\()p Fq(A)p Fo(+)p Fq(iV)g Fo(\))1016 667 y FC(=)14 b(\()p Ft(A)c FC(+)h Ft(iV)g FC(\))1258 646 y Fp(\000)p Fo(1)1305 667 y Ft(:)469 b FC(\(3)p Ft(:)p FC(20\))75 784 y(No)o(w)13 b(the)g(general)h(case)f(follo)o(ws)g(b)o(y)g(appro)o(ximating)g Ft(F)19 b FC(with)14 b(linear)e(sup)q(erp)q(osition)j(of)e(functions)75 844 y(of)k(the)f(form)f(\(3.19\),)h(e.g.,)554 962 y Ft(F)7 b FC(\()p Ft(t)p 612 969 18 2 v -2 w FC(\))14 b(=)713 903 y Fr(Z)763 962 y Ft(dV)p 788 969 40 2 v 847 949 a FC(^)836 962 y Ft(F)6 b FC(\()p Ft(V)p 893 969 V 11 w FC(\))i(exp)1042 913 y Fr(\020)1067 962 y Fs(\000)p Ft(i)1131 920 y Fr(X)1199 962 y Ft(V)j FC(\()p Ft(x)p FC(\))p Ft(t)p FC(\()p Ft(x)p FC(\))1388 913 y Fr(\021)1420 962 y Ft(:)354 b FC(\(3)p Ft(:)p FC(21\))1875 1064 y Fg(2)148 1124 y FC(No)o(w)18 b(w)o(e)g(return)h(to)f(our)h(discussion) g(of)g(in)o(teractions)e(and)i(mak)o(e)e(the)h Ft(\036)1534 1106 y Fo(2)1571 1124 y Fs( )-8 b(!)17 b Ft(\034)24 b FC(assertion)75 1185 y(more)15 b(precise)g(with)75 1290 y Fu(Theorem)i(3.2)24 b Fn([14)o(])18 b(F)l(or)f(any)g(b)n(ounde)n(d)h(smo)n(oth)e(function)j Ft(G)p FC(\()p Ft(t)p 1290 1297 18 2 v FC(\))489 1351 y Fr(Z)539 1409 y Ft(d\026)8 b(G)p FC(\()p Ft(\036)p 658 1426 29 2 v 688 1388 a Fo(2)708 1409 y FC(\))733 1396 y(\026)727 1409 y Ft(\036)p FC(\()p Ft(u)p FC(\))p Ft(\036)p FC(\()p Ft(v)r FC(\))40 b(=)1034 1351 y Fr(Z)1076 1364 y Fp(1)1058 1445 y Fo(0)1122 1409 y Ft(dt)1181 1351 y Fr(Z)1231 1409 y Ft(dP)1287 1416 y Fq(u)1319 1409 y Ft(\016)1341 1416 y Fq(v)1361 1409 y FC(\()p Ft(X)t FC(\()p Ft(t)p FC(\)\))1034 1521 y Fs(\002)1081 1462 y Fr(Z)1131 1521 y Ft(d\026)8 b(G)p FC(\()p Ft(\036)p 1250 1537 V 1281 1499 a Fo(2)1311 1521 y FC(+)j Ft(\034)p 1360 1528 27 2 v 1387 1500 a Fq(t)1402 1521 y FC(\))367 b(\(3.22\))75 1644 y Fn(wher)n(e)18 b Ft(\036)p 213 1660 29 2 v 241 1622 a Fo(2)275 1644 y Fs(\021)c FC(\()p Ft(\036)p FC(\()p Ft(x)p FC(\))448 1631 y(\026)442 1644 y Ft(\036)o FC(\()p Ft(x)p FC(\)\))555 1651 y Fq(x)p Fp(2)p Fo(\003)625 1644 y Ft(:)75 1893 y Fu(Pro)r(of.)27 b FC(V)l(erify)15 b(b)q(oth)i(sides)f(for)h(the)f(sp)q(ecial)g(case)776 1996 y Ft(G)p FC(\()p Ft(t)p 833 2003 18 2 v 1 w FC(\))d(=)h Ft(e)959 1975 y Fp(\000)p Fq(i)1004 1946 y Fr(P)1048 1989 y Fm(x)1073 1975 y Fq(V)8 b Fo(\()p Fq(x)p Fo(\))p Fq(t)p Fo(\()p Fq(x)p Fo(\))75 2098 y FC(using)17 b(the)f(same)g(reasoning)h(as)g(in)f(the)g(pro)q (of)i(of)f(Prop)q(osition)g(3.1)g(to)g(ev)m(aluate)f(the)h(righ)o(t)f(hand)75 2158 y(side)11 b(and)g(using)h(the)e(fact)h(that)672 2137 y Fq(d\026)6 b(G)p 655 2147 107 2 v 655 2148 a Fr(R)688 2180 y Fq(d\026)h(G)777 2158 y FC(is)k(a)g(Gaussian)i(measure)c(whose)j(co)o(v)m (ariance)f(is)f(\()p Ft(A)p FC(+)p Ft(V)i FC(\))1811 2140 y Fp(\000)p Fo(1)1869 2158 y FC(to)75 2233 y(ev)m(aluate)i(the)g(left)g(hand)h (side.)20 b(Then)15 b(pass)g(to)g(more)e(general)h(functions)h Ft(G)g FC(using)f(sup)q(erp)q(osition)75 2293 y(as)j(in)f(\(3.11\).)1545 b Fg(2)148 2354 y FC(A)o(t)13 b(this)g(p)q(oin)o(t)h(one)g(can)f(use)h(the)f (Prop)q(osition)i(3.1)f(to)g(rewrite)e(the)i(righ)o(t)f(hand)h(side)f(of)h (\(3.22\))75 2414 y(as)23 b(a)f(sum)f(o)o(v)o(er)g(p)q(olymers,)h(ho)o(w)o (ev)o(er)e(the)i(form)o(ulas)f(with)h(the)g(Levy)f(pro)q(cesses)i(are)f (simpler)75 2474 y([21,)16 b(22)q(].)969 2599 y(21)p eop %%Page: 22 26 bop 148 307 a FC(Symanzik)11 b(sho)o(w)o(ed)i([58)q(])f(that)i Ft(\036)753 289 y Fo(2)786 307 y FC(can)f(b)q(e)g(in)o(terpreted)f(as)h(the)g (sum)f(of)i(the)f(o)q(ccupation)g(times)75 367 y(for)k(a)f(Grand)h(Canonical) g(Ensem)o(ble)d(of)i(L)o(\023)-23 b(e)q(vy)15 b(pro)q(cesses)i(whic)o(h)f (are)g(closed)g(lo)q(ops,)h(e.g.,)110 454 y Fr(Z)160 512 y Ft(d\026G)p FC(\()p Ft(\036)p 271 529 29 2 v 301 491 a Fo(2)332 512 y FC(+)11 b Ft(\034)p 381 519 27 2 v 6 w FC(\))j(=)f(lim)493 541 y Fq(\017)p Fp(!)p Fo(0)560 512 y FC(\(1)p Ft(=)p Fs(N)668 519 y Fq(\017)685 512 y FC(\))712 471 y Fr(X)806 479 y FC(1)p 786 501 66 2 v 786 546 a Ft(M)5 b FC(!)865 454 y Fr(Z)915 512 y Ft(dP)971 519 y Fq(\017)988 512 y FC(\()p Ft(X)1047 519 y Fo(1)1067 512 y FC(\))j Fs(\001)g(\001)g(\001)1161 454 y Fr(Z)1210 512 y Ft(dP)1266 519 y Fq(\017)1284 512 y FC(\()p Ft(X)1343 519 y Fq(M)1382 512 y FC(\))g Ft(G)1455 427 y Fr(0)1457 502 y(@)1505 458 y Fq(M)1493 471 y Fr(X)1493 562 y Fq(j)r Fo(=1)1562 512 y Ft(\034)p 1562 519 27 2 v 1589 523 a Fq(j)1618 512 y FC(+)j Ft(\034)p 1667 519 V 1694 427 a Fr(1)1694 502 y(A)1739 512 y Ft(;)35 b FC(\(3)p Ft(:)p FC(23\))75 657 y(where)17 b Fs(N)258 664 y Fq(\017)291 657 y FC(is)g(the)g(normalization)e (factor|obtained)j(b)o(y)e(replacing)g Ft(G)g Fs(!)f FC(1)i(in)g(the)g(righ)o (t-hand)75 717 y(side|and)551 731 y Fr(Z)601 790 y Ft(dP)657 797 y Fq(\017)674 790 y FC(\()p Ft(X)t FC(\))d Fs(\021)822 731 y Fr(Z)864 744 y Fp(1)846 826 y Fq(\017)915 756 y Ft(dt)p 915 778 43 2 v 927 824 a(t)971 748 y Fr(X)991 836 y Fq(x)1039 731 y Fr(Z)1089 790 y Ft(dP)1145 797 y Fq(x)1168 790 y FC(\()p Ft(X)t FC(\))p Ft(\016)1272 797 y Fq(x)1294 790 y FC(\()p Ft(X)t FC(\()p Ft(t)p FC(\)\))p Ft(:)342 b FC(\(3)p Ft(:)p FC(24\))75 954 y Ff(3.2)70 b(F)-6 b(urther)23 b(Dev)n(elopmen)n(ts)75 1047 y FC(De)14 b(Gennes)h([20])f(considered)f(the)i(limit)c(where)j(the)g(n) o(um)o(b)q(er)f(of)h(comp)q(onen)o(ts)g(\(t)o(w)o(o)g(in)g(our)h(case\))75 1107 y(of)22 b Ft(\036)f FC(tends)g(to)h(zero.)36 b(This)21 b(has)h(the)f(e\013ect)g(of)h(driving)f(the)g(activit)o(y)e(of)j(all)f (closed)g(lo)q(ops)h(in)75 1167 y(Symanzik's)e(form)o(ula)i(to)h(zero)f(so)h (that)g(quan)o(tities)f(suc)o(h)g(as)h(the)g(end-to-end)g(distance)f(of)h(a) 75 1227 y(single)16 b(self-in)o(teracting)f(p)q(olymer)f(can)j(b)q(e)f (studied)g(b)o(y)g(Theorem)f(3.2.)148 1287 y(W)l(e)e(prefer)f(another)i (metho)q(d)e(in)o(tro)q(duced)h(b)o(y)f(P)o(arisi-Sourlas)i([51])f(and)g (separately)g(McKane)75 1348 y([49])h(to)h(ac)o(hiev)o(e)d(the)i(same)f (result.)20 b(They)14 b(in)o(tro)q(duce)f(an)o(ti-comm)o(uting)f(v)m (ariables)i Ft( )r FC(\()p Ft(x)p FC(\))p Ft(;)1766 1334 y FC(\026)1757 1348 y Ft( )p FC(\()p Ft(x)p FC(\))g(as)75 1408 y(w)o(ell)h(as)i Ft(\036)p FC(\()p Ft(x)p FC(\).)k(F)l(ollo)o(wing)15 b(the)h(usual)h(rules)f(of)g(Berezin)f(calculus)g(there)h(is)g(a)h(partner) 457 1517 y Ft(d\026)p FC(\()p Ft( )r FC(\))d(=)656 1475 y Fr(Y)649 1567 y Fq(x)p Fp(2)p Fo(\003)717 1517 y FC(\()p Ft(d)771 1504 y FC(\026)761 1517 y Ft( )q FC(\()p Ft(x)p FC(\))p Ft(d )r FC(\()p Ft(x)p FC(\)\))d Fs(\001)g Ft(e)1063 1496 y Fp(\000)p Fo(\()p Fq( )q(;A)1171 1487 y Fo(\026)1164 1496 y Fq( )q Fo(\))1204 1517 y Ft(=)p FC(Normalization)256 b(\(3)p Ft(:)p FC(25\))75 1659 y(to)17 b Ft(d\026)p FC(\()p Ft(\036)p FC(\),)f(and)h(w)o(e)f(can)g (obtain)h(the)f(P)o(arisi-Sourlas-McKane)g(form)o(ula)75 1766 y Fu(Theorem)h(3.3)24 b Fn(F)l(or)16 b(any)i FC(smo)q(oth)f Fn(b)n(ounde)n(d)h Ft(G)p FC(\()p Ft(t)p 1009 1773 18 2 v FC(\))p Fn(,)215 1828 y Fr(Z)265 1886 y Ft(d\026)p FC(\()p Ft(\036)p FC(\))394 1828 y Fr(Z)444 1886 y Ft(d\026)p FC(\()p Ft( )r FC(\))8 b Ft(G)p FC(\()p Ft(\036)p 635 1903 29 2 v 665 1865 a Fo(2)696 1886 y FC(+)j Ft( )p 745 1903 34 2 v 778 1865 a Fo(2)798 1886 y FC(\))823 1873 y(\026)817 1886 y Ft(\036)p FC(\()p Ft(u)p FC(\))p Ft(\036)p FC(\()p Ft(v)r FC(\))h(=)1069 1828 y Fr(Z)1111 1841 y Fp(1)1092 1922 y Fo(0)1156 1886 y Ft(dt)1216 1828 y Fr(Z)1266 1886 y Ft(dP)1322 1893 y Fq(u)1345 1886 y Ft(\016)1367 1893 y Fq(v)1387 1886 y FC(\()p Ft(X)t FC(\()p Ft(t)p FC(\)\))p Ft(G)p FC(\()p Ft(\034)p 1582 1893 27 2 v 1609 1866 a Fq(t)1624 1886 y FC(\))p Ft(:)131 b FC(\(3)p Ft(:)p FC(26\))75 2013 y Fn(wher)n(e)18 b Ft(\036)p 213 2029 29 2 v 241 1992 a Fo(2)272 2013 y FC(+)11 b Ft( )p 321 2029 34 2 v 355 1992 a Fo(2)388 2013 y Fs(\021)j FC(\()p Ft(\036)p FC(\()p Ft(x)p FC(\))561 2000 y(\026)555 2013 y Ft(\036)o FC(\()p Ft(x)p FC(\))d(+)g Ft( )r FC(\()p Ft(x)p FC(\))818 2000 y(\026)809 2013 y Ft( )q FC(\()p Ft(x)p FC(\)\))927 2020 y Fq(x)p Fp(2)p Fo(\003)996 2013 y Fn(.)75 2264 y Fu(Pro)r(of.)27 b FC(W)l(e)15 b(can)f(exactly)g(rep)q(eat)g(the)h(pro)q(of)g(of)g(Theorem)e(3.2.)21 b(The)15 b(F)l(ermions)e(pla)o(y)h(no)h(role)f(in)75 2324 y(the)h(algebra.)21 b(W)l(e)15 b(obtain)h(the)f(same)f(righ)o(t)g(hand)i(side)f(except)f(that) 1375 2289 y Fr(R)1411 2324 y Ft(d\026G)p FC(\()p Ft(\036)p 1522 2341 29 2 v 1552 2303 a Fo(2)1581 2324 y FC(+)9 b Ft(\034)p 1628 2331 27 2 v 5 w FC(\))15 b(is)g(replaced)75 2385 y(b)o(y)649 2398 y Fr(Z)699 2457 y Ft(d\026)p FC(\()p Ft(\036)p FC(\))837 2398 y Fr(Z)887 2457 y Ft(d\026)p FC(\()p Ft( )r FC(\))8 b Ft(G)p FC(\()p Ft(\036)p 1078 2473 29 2 v 1108 2436 a Fo(2)1139 2457 y FC(+)j Ft( )p 1188 2473 34 2 v 1221 2436 a Fo(2)1252 2457 y FC(+)g Ft(\034)p 1301 2464 27 2 v 6 w FC(\))441 b(\(3)p Ft(:)p FC(27\))969 2599 y(22)p eop %%Page: 23 27 bop 75 307 a FC(whic)o(h)27 b(is)h(equal)f(to)h Ft(G)p FC(\()p Ft(\034)p 555 314 27 2 v 6 w FC(\)!)g(This)g(claim)d(follo)o(ws)j(b)o(y)f(c)o (hec)o(king)f(the)i(sp)q(ecial)f(case)h Ft(G)p FC(\()p Ft(t)p 1804 314 18 2 v FC(\))34 b(=)75 367 y(exp\()p Fs(\000)p Ft(i)233 334 y Fr(P)284 367 y Ft(V)12 b FC(\()p Ft(x)p FC(\))p Ft(t)p FC(\()p Ft(x)p FC(\)\).)36 b(In)21 b(this)g(case)h(the)f(in)o(tegral)g (\(3.27\))h(factors)g(in)o(to)f(t)o(w)o(o)h(Gaussian)g(in)o(te-)75 428 y(grals,)d(one)g(o)o(v)o(er)e Ft(\036)i FC(giving)f(a)h(determinan)o(t,)d (and)j(the)g(other)f(o)o(v)o(er)g Ft( )i FC(giving)e(an)h(in)o(v)o(erse)e (deter-)75 488 y(minan)o(t)c(whic)o(h)i(exactly)f(cancel.)19 b(As)c(usual,)h(more)d(general)i(functions)g Ft(G)h FC(are)f(then)g(obtained) g(b)o(y)75 548 y(sup)q(erp)q(osition)i(as)g(in)f(\(3.21\).)1244 b Fg(2)969 2599 y FC(23)p eop %%Page: 24 28 bop 75 315 a Fv(4)83 b(Flo)n(w)27 b(Equations)75 440 y Ff(4.1)70 b(P)n(erturbations)23 b(of)g(Gaussian)h(Measures)75 532 y FC(The)16 b(ob)s(jectiv)o(e)e(of)i(the)g(rest)g(of)g(these)f(lectures)g(will)g(b)q(e)h (the)g(accurate)f(ev)m(aluation)i(of)f(functional)75 592 y(in)o(tegrals)g(of) g Fn(smal)r(l)i FC(p)q(erturbations)f(of)f(Gaussian)i(measures,)d(of)h(the)g (form)819 645 y Fr(Z)869 704 y Ft(d\026)p FC(\()p Ft(\036)p FC(\))e Ft(Z)t FC(\(\003)p Ft(;)8 b(\036)p FC(\))p Ft(:)634 b FC(\(4)p Ft(:)p FC(1\))75 819 y(Here)13 b Ft(d\026)i FC(is)e(a)i(Gaussian)g (measure)e(on)h(the)g(space)g Fs(C)s FC(\(\003\))g(of)g(con)o(tin)o(uous)g (functions)g(on)g(a)h(compact)75 879 y(set)i(\003)f Fs(\032)g FC(R)293 861 y Fq(d)325 879 y Fs(\002)11 b FC(\012,)18 b(where)f(\012)h(is)f (a)h(space)f(for)h Fn(internal)h FC(indices)d(suc)o(h)i(as)g(c)o(harge)f(or)h (direction)e(of)75 939 y(a)j(dip)q(ole.)28 b(The)19 b(measure)e Ft(d\026)j FC(has)f(mean)e(zero)i(and)g(is)f(sp)q(eci\014ed)h(uniquely)e(b)o (y)h(its)g(co)o(v)m(ariance)1893 921 y Fo(4)75 999 y Ft(C)t FC(\()p Ft(\030)r(;)8 b(\030)201 981 y Fp(0)213 999 y FC(\),)683 1016 y Fr(Z)733 1074 y Ft(d\026)787 1081 y Fq(C)817 1074 y FC(\()p Ft(\036)p FC(\))13 b Ft(\036)p FC(\()p Ft(\030)r FC(\))p Ft(\036)p FC(\()p Ft(\030)1058 1054 y Fp(0)1071 1074 y FC(\))h(=)g Ft(C)t FC(\()p Ft(\030)r(;)8 b(\030)1282 1054 y Fp(0)1294 1074 y FC(\))499 b(\(4)p Ft(:)p FC(2\))148 1172 y(The)18 b(in)o(tegrand)g Ft(Z)t FC(\()p Ft(\036)p FC(\))f Fs(\021)f Ft(Z)t FC(\(\003)p Ft(;)8 b(\036)p FC(\))18 b(is)g(a)g(functional)g(de\014ned)g(on)g Fs(C)s FC(\(\003\),)g(whic)o(h)f(is)h(a)h(Banac)o(h)75 1232 y(space)e(under)g(the)g(suprem)o(um)d(norm.)22 b(The)17 b(reader)g(could)g(k) o(eep)f(the)h(follo)o(wing)f(applications)h(of)75 1292 y(the)f(in)o(tegrals)g (\(4.1\))g(in)g(mind:)121 1385 y(\(i\))24 b(If)d(w)o(e)g(write)f Ft(Z)t FC(\(\003)p Ft(;)8 b(\036)p FC(\))22 b(=)h Ft(e)723 1367 y Fp(\000)p Fq(V)7 b Fo(\()p Fq(\036)p Fo(\))850 1385 y FC(and)22 b Ft(V)11 b FC(\()p Ft(\036)p FC(\))21 b(is)g(an)h(in)o(tegral)f (of)g(a)h(lo)q(cal)f(function)g(of)h Ft(\036)p FC(,)197 1445 y(then)15 b(\(4.1\))h(can)f(b)q(e)g(though)o(t)h(as)g(a)g(F)l(eynman)d(in)o (tegral)i(for)g(a)h(quan)o(tum)e(\014eld)h(theory)g(with)197 1506 y(ultra)o(violet)g(and)i(infrared)e(cuto\013.)108 1604 y(\(ii\))23 b(By)f(the)g(Sine-Gordon)h(represen)o(tation,)g(the)g (grand-canonical)g(partition)g(functions)f(of)197 1665 y(statistical)d(mec)o (hanics)f(systems)h(with)h(t)o(w)o(o-b)q(o)q(dy)h(in)o(teractions)e(can)h(b)q (e)h(written)e(in)h(the)197 1725 y(form)14 b(\(4.1\).)21 b(In)15 b(this)f(case)h Ft(Z)k FC(corresp)q(onds)d(to)g(the)e(\(grand-canonical\))i (partiton)g(function)197 1785 y(of)g(an)h(ideal)f(gas.)94 1884 y(\(iii\))23 b(In)17 b(the)g(ab)q(o)o(v)o(e)g(cases,)h(the)f(log)h(of)g (\(4.1\))f(corresp)q(onds)i(to)f(a)f(free)g(energy)l(.)24 b(Ho)o(w)o(ev)o (er,)15 b(if)i(w)o(e)197 1944 y(in)o(tro)q(duce)f(an)g(extra)g(\014eld)g Ft( )r FC(,)f(the)h(corresp)q(onding)i(log)710 2056 y Ft(G)p FC(\(\011\))c(=)g(log)961 1997 y Fr(Z)1011 2056 y Ft(d\026)p FC(\()p Ft(\036)p FC(\))g Ft(Z)t FC(\(\003)p Ft(;)8 b(\036)j FC(+)g Ft( )r FC(\))412 b(\(4)p Ft(:)p FC(3\))197 2168 y(is)21 b(the)f(generating)h(functional)g(for)g(the)g(amputated)f(Greens)h(functions) f(of)i(the)e(theory)l(.)197 2228 y(The)i(understanding)i(of)e(in)o(tegrals)g (of)h(the)f(form)f(\(4.1\))i(allo)o(ws)f(us)h(to)g(tak)o(e)f(con)o(tin)o(uum) 197 2288 y(limits)14 b(or)i(disco)o(v)o(er)f(the)h(long)h(distance)f(prop)q (erties)g(of)h(a)f(statistical-mec)o(hanical)e(theory)l(.)p 75 2329 735 2 v 131 2359 a Fj(4)150 2374 y Fy(Not)19 b(all)g(functions)g Fi(C)s Fy(\()p Fi(\030)r(;)7 b(\030)593 2359 y Fz(0)605 2374 y Fy(\))19 b(can)h(arise)g(as)g(co)o(v)n(ariances)g(of)f(Gaussian)g (measures:)30 b(su\016cien)o(t)20 b(conditions)75 2424 y(are)f(discussed)h (in)e([59)o(].)30 b(In)19 b(particular)f(it)g(is)g(su\016cien)o(t)h(if)e Fi(C)s Fy(\()p Fi(\030)r(;)7 b(\030)1168 2409 y Fz(0)1179 2424 y Fy(\))18 b(is)h(a)f(p)q(ositiv)o(e-semide\014nite)f(con)o(tin)o(uously)75 2474 y(di\013eren)o(tiable)d(function)g(on)f(\003)c FA(\002)h Fy(\003.)969 2599 y FC(24)p eop %%Page: 25 29 bop 148 307 a FC(But)16 b(in)f(the)g(sequel,)g(w)o(e)g(shall)h(consider)f (general)h(expressions)g(of)g(the)f(form)g(\(4.1\),)g(requiring)75 367 y(only)d(that)h Ft(Z)t FC(\()p Ft(\036)p FC(\))f(not)g(gro)o(w)h(more)e (rapidly)h(as)h Fs(j)p Ft(\036)p Fs(j)g(!)h(1)e FC(than)h(what)f(is)h (required)e(to)h(b)q(e)g(in)o(tegrable)75 428 y(with)20 b(resp)q(ect)f(to)h (a)g(Gaussian)h(measure)e(and)h(that)g Ft(Z)t FC(\()p Ft(\036)p FC(\))g(has)g(v)m(ariational)g(deriv)m(ativ)o(es)f(\(same)75 488 y(as)e(F)l(r)o(\023)-23 b(e)o(c)o(het)16 b(deriv)m(ativ)o(es\))e(of)j (all)f(orders)g(and)h(that)g(these)f(b)q(e)g(measures.)k(Let)c(us)h(recall)e (that)i(the)75 548 y(v)m(ariational)g(deriv)m(ativ)o(es)e(of)i(a)h (functional)e(are)h(de\014ned)f(via)h(usual)g(deriv)m(ativ)o(es)e(with)i (resp)q(ect)f(to)75 608 y(linear)f(incremen)o(ts)d(of)k(the)f(argumen)o(ts.) 20 b(F)l(or)15 b(instance,)g(deriv)m(ativ)o(es)f(of)h(the)g(form)f Ft(\016)1654 590 y Fq(N)1688 608 y Ft(Z)q(=\016)r(\036)1799 590 y Fq(N)1847 608 y FC(are)75 668 y(functionals)i(with)g(the)g(de\014ning)h (prop)q(ert)o(y)f(that)g(for)h(an)o(y)f(functions)g Ft(f)1408 675 y Fo(1)1428 668 y Ft(;)8 b(:)g(:)g(:)f(;)h(f)1561 675 y Fq(N)1611 668 y FC(in)16 b Fs(C)s FC(\(\003\),)468 785 y Ft(@)p 444 807 77 2 v 444 853 a(@)s(\025)501 860 y Fo(1)534 819 y Fs(\001)8 b(\001)g(\001)637 785 y Ft(@)p 606 807 91 2 v 606 853 a(@)s(\025)663 860 y Fq(N)701 819 y Ft(Z)751 734 y Fr(0)751 808 y(@)788 819 y Ft(\036)j FC(+)891 765 y Fq(N)877 778 y Fr(X)877 869 y Fq(j)r Fo(=1)946 819 y Ft(\025)974 826 y Fq(j)993 819 y Ft(f)1017 826 y Fq(j)1035 734 y Fr(1)1035 808 y(A)1072 732 y(\014)1072 757 y(\014)1072 782 y(\014)1072 807 y(\014)1072 832 y(\014)1072 856 y(\014)1085 883 y Fq(\025)p 1085 890 21 2 v Fo(=0)1167 819 y FC(=)522 924 y Fr(Z)580 983 y Ft(d\030)626 990 y Fo(1)655 983 y Fs(\001)d(\001)g(\001)722 924 y Fr(Z)780 983 y Ft(d\030)826 990 y Fq(N)968 949 y Ft(\016)992 931 y Fq(N)1025 949 y Ft(Z)t FC(\()p Ft(\036)p FC(\))p 873 971 352 2 v 873 1017 a Ft(\016)r(\036)p FC(\()p Ft(\030)966 1024 y Fo(1)986 1017 y FC(\))g Fs(\001)g(\001)g(\001)g Ft(\016)r(\036)p FC(\()p Ft(\030)1172 1024 y Fq(N)1205 1017 y FC(\))1229 983 y Ft(f)d FC(\()p Ft(\030)1298 990 y Fo(1)1319 983 y FC(\))j Fs(\001)g(\001)g(\001)g Ft(f)d FC(\()p Ft(\030)1481 990 y Fq(N)1516 983 y FC(\))p Ft(:)263 b FC(\(4.4\))75 1121 y(In)15 b(man)o(y)f(cases,)i(for)f(eac)o(h)g (\014eld-con\014guration)i Ft(\036)e FC(these)g(functionals)h(are)f(measures) g(on)h(\003)1792 1103 y Fq(N)1825 1121 y FC(,)f(for)75 1182 y(example:)661 1230 y Ft(\016)p 614 1252 118 2 v 614 1298 a(\016)r(\036)p FC(\()p Ft(x)p FC(\))788 1230 y Ft(\016)p 742 1252 116 2 v 742 1298 a(\016)r(\036)p FC(\()p Ft(y)r FC(\))871 1205 y Fr(Z)921 1264 y Ft(dz)c(\036)1009 1243 y Fo(2)1028 1264 y FC(\()p Ft(z)r FC(\))j(=)g(2)p Ft(\016)r FC(\()p Ft(x)c Fs(\000)h Ft(y)r FC(\))p Ft(;)441 b FC(\(4)p Ft(:)p FC(5\))75 1375 y(but)19 b(they)f(could)g(b)q(e)h (more)e(singular)i(ob)s(jects,)f(lik)o(e)e(deriv)m(ativ)o(es)i(of)g(delta)g (functions.)28 b(W)l(e)19 b(shall)75 1435 y(require)k(that)h(\()p Ft(\016)403 1417 y Fq(N)436 1435 y Ft(Z)q(=\016)r(\036)547 1417 y Fq(N)581 1435 y FC(\)\()p Ft(\036)p FC(\))f(exists)h(for)g(eac)o(h)g Ft(N)29 b FC(and)24 b Ft(\036)g Fn(as)h(a)f(signe)n(d)h(Bor)n(el)g(me)n(asur) n(e)e FC(on)75 1495 y(\003)6 b Fs(\002)g(\001)i(\001)g(\001)e(\002)g FC(\003.)20 b(Those)14 b(cases)g(pro)q(ducing)g(deriv)m(ativ)o(es)f(of)h (deltas,)f(for)h(instance)g(functionals)g(of)g Fs(r)p Ft(\036)p FC(,)75 1555 y(will)e(b)q(e)i(handled)f(b)o(y)g(in)o(tro)q(ducing)g (additional)g(\014elds)g(plus)g(additional)h(constrain)o(ts,)f(for)h (instance)75 1616 y(the)i(requiremen)o(ts)d(that)k(these)f(extra)g(\014elds)g (coincide)f(with)h(the)g(gradien)o(ts.)75 1727 y Fu(De\014nition)i(4.1)24 b Fn(A)17 b(functional)h Ft(A)p FC(\()p Ft(\036)p FC(\))f Fn(de\014ne)n(d)h (on)f Fs(C)s FC(\(\003\))g Fn(is)g(smo)n(oth)f(if)h(it)g(p)n(osesses)f (derivatives)75 1787 y(of)g(al)r(l)i(or)n(ders)d(which)h(exist)h(as)f(Bor)n (el)h(me)n(asur)n(es)e(as)h(describ)n(e)n(d)g(ab)n(ove.)23 b(It)16 b(is)g(of)g(limite)n(d)h(gr)n(owth)f(if)75 1847 y(these)i (derivatives)h(ar)n(e)e(al)r(l)h(b)n(ounde)n(d)g(ac)n(c)n(or)n(ding)f(to)391 1982 y Fs(k)424 1924 y Fr(Z)474 1982 y Ft(d)499 1962 y Fq(N)533 1982 y Ft(\030)11 b(f)5 b FC(\()p Ft(\030)634 1989 y Fo(1)654 1982 y Ft(;)j(:)g(:)g(:)f(;)h(\030)784 1989 y Fq(N)818 1982 y FC(\))938 1948 y Ft(\016)962 1930 y Fq(N)995 1948 y Ft(A)p FC(\()p Ft(\036)p FC(\))p 842 1971 352 2 v 842 2016 a Ft(\016)r(\036)p FC(\()p Ft(\030)935 2023 y Fo(1)955 2016 y FC(\))g Fs(\001)g(\001)g(\001)g Ft(\016)r(\036)p FC(\()p Ft(\030)1141 2023 y Fq(N)1174 2016 y FC(\))1198 1982 y Fs(k)14 b(\024)g Ft(C)1325 1989 y Fq(N)r(;\017)1380 1982 y Ft(e)1403 1960 y Fq(\017)1423 1929 y Fr(R)1443 1977 y Fl(\003)1471 1960 y Fq(d\030)7 b(\036)1533 1948 y Fl(2)1550 1960 y Fo(\()p Fq(\030)q Fo(\))1812 1982 y FC(\(4)p Ft(:)p FC(6\))75 2117 y Fn(for)17 b(any)g Ft(N)r(;)8 b(f)s(;)g(\017)14 b(>)f FC(0)18 b Fn(with)g Ft(f)23 b Fn(c)n(ontinuous)18 b(and)g Fs(j)p Ft(f)5 b Fs(j)14 b(\024)f FC(1)p Fn(.)75 2261 y Ff(4.2)70 b(Wilson)21 b(Renormalization)75 2354 y FC(Wilson)e(has)i(taugh)o(t)f(us)g (that)g(an)g(e\013ectiv)o(e)d(w)o(a)o(y)j(to)g(study)f(in)o(tegrals)g(of)h (the)f(form)g(\(4.1\))h(is)f(to)75 2414 y(successiv)o(ely)13 b(eliminate)f(the)i(\\degrees)h(of)h(freedom")d(presen)o(t)h(in)h(the)g (Gaussian)h(measure)d Ft(d\026)j FC(b)o(y)75 2474 y(p)q(erforming)j(the)h (in\014nite)g(n)o(um)o(b)q(er)e(of)j(in)o(tegrations)f(on)g(the)g(in)o (tegrand)h Ft(Z)j FC(in)c(small)e(bunc)o(hes.)969 2599 y(25)p eop %%Page: 26 30 bop 75 307 a FC(In)17 b(this)g(w)o(a)o(y)g(one)g(obtains)h(a)f(sequence)f(of) i(smo)q(other)e(and)i(smo)q(other)f(Gaussian)h(measures)e(but)75 367 y(applied)g(to)g(in)o(tegrands)h(whic)o(h)e(ha)o(v)o(e)g(p)q(o)q(orer)j (lo)q(calit)o(y)d(prop)q(erties.)21 b(If)16 b(one)g(manages)g(to)h(con)o (trol)75 428 y(the)g(loss)h(of)g(lo)q(calit)o(y)e(of)h(the)g(in)o(tegrand)h (then)f(at)h(the)f(end)g(one)g(is)h(left)e(with)h(an)h(in)o(tegral)f(o)o(v)o (er)f(a)75 488 y(trivial)g(delta-function)h(measure)f(and)h(the)g(problem)f (is)h(solv)o(ed.)23 b(T)l(o)18 b(dev)o(elop)e(this)h(program)g(w)o(e)75 548 y(decomp)q(ose)f(the)g(co)o(v)m(ariance)f(of)i Ft(\036)f FC(in)g(the)g(form)682 633 y Ft(C)t FC(\()p Ft(\030)r(;)8 b(\030)808 613 y Fp(0)820 633 y FC(\))14 b(=)g Ft(C)944 613 y Fp(0)955 633 y FC(\()p Ft(\030)r(;)8 b(\030)1042 613 y Fp(0)1055 633 y FC(\))j(+)g Ft(B)s FC(\()p Ft(\030)r(;)d(\030)1261 613 y Fp(0)1273 633 y FC(\))p Ft(:)75 719 y FC(Then)465 746 y Fr(Z)515 804 y Ft(d\026)569 811 y Fq(C)599 804 y FC(\()p Ft(\036)p FC(\))p Ft(Z)t FC(\()p Ft(\036)p FC(\))42 b(=)892 746 y Fr(Z)942 804 y Ft(d\026)996 811 y Fq(C)1023 802 y Fk(0)1037 804 y FC(\()p Ft(\036)1085 784 y Fp(0)1096 804 y FC(\))1123 746 y Fr(Z)1174 804 y Ft(d\026)1228 811 y Fq(B)1259 804 y FC(\()p Ft(\020)t FC(\))p Ft(Z)t FC(\()p Ft(\036)1407 784 y Fp(0)1429 804 y FC(+)11 b Ft(\020)t FC(\))290 b(\(4.7\))812 913 y Fs(\021)892 854 y Fr(Z)942 913 y Ft(d\026)996 920 y Fq(C)1023 911 y Fk(0)1037 913 y FC(\()p Ft(\036)1085 892 y Fp(0)1096 913 y FC(\))8 b(\()p Ft(\026)1171 920 y Fq(B)1213 913 y Fs(\003)j Ft(Z)t FC(\)\()p Ft(\036)1353 892 y Fp(0)1365 913 y FC(\))j Ft(:)75 1013 y FC(T)l(o)h(pro)o(v) o(e)f(this)g(it)g(su\016ces,)g(b)o(y)g(using)h(the)f(de\014ning)h(prop)q(ert) o(y)f(of)h(Gaussian)h(measures)d(\(2.12\),)i(to)75 1074 y(pro)o(v)o(e)f(it)g (in)g(the)g(sp)q(ecial)g(case)g(where)h Ft(Z)t FC(\()p Ft(\036)p FC(\))e(=)h(exp)8 b Ft(i)p FC(\()p Ft(\036;)g(f)d FC(\))o(,)15 b(i.e.,)1292 1038 y Fr(R)1328 1074 y Ft(d\026)1382 1081 y Fq(C)1409 1072 y Fk(0)1424 1074 y FC(\()p Ft(\036)1472 1055 y Fp(0)1483 1074 y FC(\))1510 1038 y Fr(R)1547 1074 y Ft(d\026)1601 1081 y Fq(B)1632 1074 y FC(\()p Ft(\020)t FC(\))8 b(exp)o([)p Ft(i)p FC(\()p Ft(\036)1856 1055 y Fp(0)1875 1074 y FC(+)75 1134 y Ft(\020)t(;)g(f)d FC(\)])14 b(=)f(exp[\()p Fs(\000)p FC(1)p Ft(=)p FC(2\)\()p Ft(f)s(;)8 b(C)t(f)d FC(\)].)20 b(This)d(in)f(turn)g(is)g (a)h(consequence)e(of)i(\(2.12\).)148 1194 y(F)l(orm)o(ula)c(\(4.7\))i(is)g (the)f(formalization)g(of)h(Wilson's)f(idea:)20 b(the)15 b(RHS)f(in)o(v)o (olv)o(es)f(a)i(\\smo)q(other")75 1254 y(Gaussian)20 b(measure|)p Ft(d\026)565 1261 y Fq(C)592 1252 y Fk(0)605 1254 y FC(|and)f(a)f(less)h(lo)q (cal)f(in)o(tegrand|the)g(con)o(v)o(olution)g(of)g Ft(d\026)1715 1261 y Fq(B)1765 1254 y FC(and)h Ft(Z)t FC(.)75 1314 y(Heuristically)l(,)12 b Ft(B)17 b FC(is)d(c)o(hosen)g(so)i(to)e(include)g(all)g(\015uctuations)h (up)f(to)h(a)g(certain)f(space)g(scale)g(c)o(har-)75 1374 y(acterized)h(b)o (y)i(a)g(\\length")g Ft(`)p FC(\()p Ft(t)p FC(\))g(so)g(that)h(the)e(random)h (\014eld)f(aso)q(ciated)h(to)g(the)g(measure)e Ft(d\026)1821 1381 y Fq(C)1848 1372 y Fk(0)1880 1374 y FC(is)75 1435 y(appro)o(ximately)i (constan)o(t)j(on)f(cub)q(es)g(of)h(side)f(of)g(order)g Ft(`)p FC(\()p Ft(t)p FC(\).)30 b(In)19 b(man)o(y)f(R)o(G)h(approac)o(hes,)h(this)75 1495 y(remo)o(v)m(al)15 b(of)h(\015uctuations)h(is)g(done)f(in)g(discrete)g (steps,)g(for)h(instance)f(increasing)g Ft(`)p FC(\()p Ft(t)p FC(\))g(b)o(y)g(a)h(\014xed)75 1555 y(factor)i(in)g(eac)o(h)g(iteration.)29 b(W)l(e)19 b(ha)o(v)o(e)f(found,)i(follo)o(wing)e([53)q(],)h(that)g(it)g(is)f (more)g(con)o(v)o(enien)o(t)f(to)75 1615 y(pro)q(ceed)e(in)f(a)h Fn(c)n(ontinuum)g FC(of)g(steps,)g(that)g(is,)f(in)o(tegrating)h(out)g (\015uctuations)g(o)o(v)o(er)e(an)j(\\in\014nites-)75 1675 y(imal")h(in)o(terv)m(al)h(so)h(to)g(obtain)g(a)g(\\\015o)o(w")g(of)g (Gaussian)h(measures)d(with)i(asso)q(ciated)g(in)o(tegrands)75 1736 y(describ)q(ed)e(b)o(y)g Fn(di\013er)n(ential)j(e)n(quations)t FC(.)26 b(Let)18 b(us)g(presen)o(t)f(an)i(heuristic)d(discussion)i(of)g(this) g(pro-)75 1796 y(cedure.)i(The)15 b(\\in\014nitesmal")f(in)o(tegration)g(of)i (degrees)e(of)i(freedom)d(corresp)q(onds)j(to)g(letting)e(the)75 1856 y(co)o(v)m(ariance)i Ft(B)j FC(b)q(e)d(\\in\014nitesmal")f(in)h(the)g (sense)g(that)527 1941 y(sup)555 1981 y Fq(\030)608 1941 y Ft(B)s FC(\()p Ft(\030)r(;)8 b(\030)735 1921 y Fp(0)748 1941 y FC(\))14 b Fs(\034)f FC(1)i(;)56 b(sup)982 1981 y Fq(\030)1035 1883 y Fr(Z)1085 1941 y Fs(j)p Ft(B)s FC(\()p Ft(\030)r(;)8 b(\030)1226 1921 y Fp(0)1238 1941 y FC(\))p Fs(j)g Ft(d\030)1327 1921 y Fp(0)1354 1941 y Fs(\034)13 b FC(1)i Ft(:)342 b FC(\(4)p Ft(:)p FC(8\))75 2058 y(In)16 b(this)g(case,)g(expanding)g Ft(Z)t FC(\()p Ft(\036)667 2040 y Fp(0)690 2058 y FC(+)11 b Ft(\020)t FC(\))16 b(up)h(to)f(fourth)h(order)f(in)g Ft(\020)k FC(w)o(e)c(obtain:)226 2171 y(\()p Ft(\026)274 2178 y Fq(B)316 2171 y Fs(\003)11 b Ft(Z)t FC(\)\()p Ft(\036)456 2151 y Fp(0)467 2171 y FC(\))42 b(=)607 2113 y Fr(Z)657 2171 y Ft(d\026)711 2178 y Fq(B)742 2171 y FC(\()p Ft(\030)r FC(\))811 2111 y Fr(\032)843 2171 y Ft(Z)t FC(\()p Ft(\036)928 2151 y Fp(0)940 2171 y FC(\))11 b(+)1019 2113 y Fr(Z)1068 2171 y Ft(d\030)1131 2138 y(\016)r(Z)t FC(\()p Ft(\036)1240 2119 y Fp(0)1251 2138 y FC(\))p 1131 2160 139 2 v 1138 2205 a Ft(\016)r(\036)1191 2191 y Fp(0)1201 2205 y FC(\()p Ft(\030)r FC(\))1275 2171 y Ft(\020)t FC(\()p Ft(\030)r FC(\))p Ft(d\030)656 2308 y FC(+)713 2274 y(1)p 713 2296 25 2 v 713 2342 a(2)750 2249 y Fr(Z)800 2308 y Ft(d\030)858 2249 y Fr(Z)907 2308 y Ft(d\030)955 2287 y Fp(0)1034 2274 y Ft(\016)1058 2256 y Fo(2)1077 2274 y Ft(Z)t FC(\()p Ft(\036)1162 2256 y Fp(0)1173 2274 y FC(\))p 981 2296 263 2 v 981 2342 a Ft(\016)r(\036)1034 2327 y Fp(0)1045 2342 y FC(\()p Ft(\030)r FC(\))p Ft(\016)r(\036)1159 2327 y Fp(0)1171 2342 y FC(\()p Ft(\030)1213 2327 y Fp(0)1225 2342 y FC(\))1249 2308 y Ft(\020)t FC(\()p Ft(\030)r FC(\))p Ft(\020)t FC(\()p Ft(\030)1402 2287 y Fp(0)1415 2308 y FC(\))25 b(+)g Ft(O)q FC(\()p Ft(\020)1604 2287 y Fo(4)1624 2308 y FC(\))1643 2247 y Fr(\033)528 2444 y FC(=)41 b Ft(Z)t FC(\()p Ft(\036)692 2423 y Fp(0)704 2444 y FC(\))25 b(+)815 2410 y(1)p 815 2432 25 2 v 815 2478 a(2)853 2385 y Fr(Z)903 2444 y Ft(d\030)960 2385 y Fr(Z)1010 2444 y Ft(d\030)1058 2423 y Fp(0)1136 2410 y Ft(\016)1160 2392 y Fo(2)1179 2410 y Ft(Z)t FC(\()p Ft(\036)1264 2392 y Fp(0)1276 2410 y FC(\))p 1084 2432 263 2 v 1084 2478 a Ft(\016)r(\036)1137 2464 y Fp(0)1148 2478 y FC(\()p Ft(\030)r FC(\))p Ft(\016)r(\036)1262 2464 y Fp(0)1274 2478 y FC(\()p Ft(\030)1316 2464 y Fp(0)1328 2478 y FC(\))1352 2444 y Ft(B)s FC(\()p Ft(\030)r(;)8 b(\030)1479 2423 y Fp(0)1491 2444 y FC(\))25 b(+)g Ft(O)q FC(\()p Ft(B)1695 2423 y Fo(2)1715 2444 y FC(\))14 b Ft(:)50 b FC(\(4.9\))969 2599 y(26)p eop %%Page: 27 31 bop 75 307 a FC(\(As)15 b Ft(\026)194 314 y Fq(B)240 307 y FC(has)h(zero)f(mean,)f(the)h(exp)q(ectation)g(of)h(o)q(dd-degree)g (monomials)d(in)i(the)g(\014eld)g Ft(\020)k FC(is)c(zero.\))75 367 y(Therefore,)g(to)i(leading)f(order)g(in)g Ft(B)s FC(,)g(form)o(ula)e (\(4.9\))j(implies)d(that)615 434 y Fr(Z)665 492 y Ft(d\026)719 499 y Fq(C)749 492 y FC(\()p Ft(\036)p FC(\))8 b Ft(Z)t FC(\()p Ft(\036)p FC(\))28 b(=)1021 434 y Fr(Z)1071 492 y Ft(d\026)1125 499 y Fq(C)1152 490 y Fk(0)1167 492 y FC(\()p Ft(\036)1215 472 y Fp(0)1226 492 y FC(\))8 b Ft(Z)1290 472 y Fp(0)1302 492 y FC(\()p Ft(\036)1350 472 y Fp(0)1362 492 y FC(\))407 b(\(4)p Ft(:)p FC(10\))75 615 y(with)438 692 y Ft(Z)475 671 y Fp(0)486 692 y FC(\()p Ft(\036)534 671 y Fp(0)546 692 y FC(\))11 b Fs(\000)g Ft(Z)t FC(\()p Ft(\036)p FC(\))27 b(=)828 658 y(1)p 828 680 25 2 v 828 726 a(2)865 633 y Fr(Z)915 692 y Ft(d\030)972 633 y Fr(Z)1022 692 y Ft(d\030)1070 671 y Fp(0)1148 658 y Ft(\016)1172 640 y Fo(2)1192 658 y Ft(Z)t FC(\()p Ft(\036)1277 640 y Fp(0)1288 658 y FC(\))p 1096 680 263 2 v 1096 726 a Ft(\016)r(\036)1149 712 y Fp(0)1160 726 y FC(\()p Ft(\030)r FC(\))p Ft(\016)r(\036)1274 712 y Fp(0)1286 726 y FC(\()p Ft(\030)1328 712 y Fp(0)1340 726 y FC(\))1364 692 y Ft(B)s FC(\()p Ft(\030)r(;)8 b(\030)1491 671 y Fp(0)1504 692 y FC(\))13 b Ft(:)238 b FC(\(4)p Ft(:)p FC(11\))75 806 y(In)14 b(the)f(limit)e(where)j Ft(C)507 788 y Fp(0)532 806 y FC(b)q(ecomes)f(\\in\014nitesimally")e(close)i(to)i Ft(C)i FC(this)d(\014nite-diference)e(equation)75 866 y(b)q(ecomes)19 b(a)h(di\013eren)o(tial)f(equation,)h(and)h(the)f(neglect)f(of)h(the)g(term)f Ft(O)q FC(\()p Ft(B)1503 848 y Fo(2)1522 866 y FC(\))i(is)e(fully)g (justi\014ed.)75 927 y(W)l(e)d(can)g(then)g(iterate)f(\(4.10\))i(un)o(til)e (w)o(e)g(reac)o(h)h(a)g(zero-co)o(v)m(ariance)g(Gaussian)h(measure,)e(that)h (is,)75 987 y(a)h Ft(\016)r FC(-function.)148 1047 y(The)i(formalization)d (of)j(this)f(idea)g(corresp)q(onds)i(to)f(constructing)f(a)h(family)d(of)j (co)o(v)m(ariances)75 1107 y Ft(C)110 1114 y Fp(1)p Fq(t)181 1107 y FC(parametrized)f(b)o(y)j(a)g(con)o(tin)o(uous)f(parameter)g Ft(t)p FC(,)h(suc)o(h)f(that)h Ft(C)1396 1114 y Fp(1)p Fq(t)1468 1107 y Fs(!)g FC(0)g(as)g Ft(t)g Fs(!)h(1)p FC(.)34 b(T)l(o)75 1167 y(repro)q(duce)15 b(the)f(argumen)o(t)g(sk)o(etc)o(hed)f(ab)q(o)o(v)o (e,)h(w)o(e)h(need)f(some)g(additional)g(conditions)h(related)f(to)75 1228 y(\(4.8\),)i(whic)o(h)f(ensures)h(a)g(su\016cien)o(tly)e(smo)q(oth)h(v)m (ariation)i(of)f(the)f(co)o(v)m(ariances)h(with)f Ft(t)p FC(.)21 b(W)l(e)16 b(shall)75 1288 y(call)k(suc)o(h)h(a)g(family)d(a)k Fn(deformation)t FC(.)34 b(In)21 b(this)f(and)i(what)f(follo)o(ws,)h(a)f (double)g(subindex)f(\\)p Ft(ts)p FC(")75 1348 y(where)13 b Ft(t)h Fs(\025)f Ft(s)h FC(in)f(a)h(co)o(v)m(ariance)f(or)h(measure)f(is)g (in)o(tended)f(to)i(emphasize)e(the)h(in)o(terpretation)g(that)75 1408 y(the)j(corresp)q(onding)h(random)f(\014eld)g(has)h(\015uctuations)g (only)f(on)g(the)g(scales)g(\\b)q(et)o(w)o(een)g Ft(s)g FC(and)h Ft(t)p FC(".)75 1522 y Fu(De\014nition)h(4.2)24 b Fn(A)14 b FC(deformation)f Fn(of)g Ft(C)18 b Fn(is)c(a)g(family)f Ft(C)1126 1529 y Fp(1)p Fq(t)1190 1522 y Fn(of)h(c)n(ovarianc)n(es)g(such)g(that)h Ft(C)1739 1529 y Fp(1)p Fo(0)1807 1522 y FC(=)f Ft(C)t Fn(;)75 1582 y FC(lim)143 1589 y Fq(t)p Fp(!1)236 1582 y Ft(C)271 1589 y Fp(1)p Fq(t)321 1582 y FC(\()p Ft(\030)r(;)8 b(\030)r FC(\))20 b(=)e(0)i Fn(uniformly)f(in)i Ft(\030)r Fn(;)g(and)1007 1570 y FC(_)990 1582 y Ft(C)1025 1589 y Fq(t)1058 1582 y Fs(\021)d Ft(dC)1175 1589 y Fp(1)p Fq(t)1225 1582 y Ft(=dt)i Fn(exists)h(and)f (satis\014es:)27 b(\(i\))20 b(it)g(is)75 1643 y(c)n(ontinuously)h(di\013er)n (entiable)g(in)f Ft(\030)r(;)8 b(\030)782 1625 y Fp(0)795 1643 y Fn(;)20 b(\(ii\))g(it)g(is)f(ne)n(gative-de\014nit)q(e;)24 b(and)c(\(iii\))f(it)h(is)f(inte)n(gr)n(able)75 1703 y(in)f(the)g(sense)g (that)g FC(sup)518 1715 y Fq(\030)545 1667 y Fr(R)581 1703 y Fs(j)612 1690 y FC(_)595 1703 y Ft(C)630 1710 y Fq(t)645 1703 y FC(\()p Ft(\030)r(;)8 b(\030)732 1685 y Fp(0)745 1703 y FC(\))p Ft(d\030)812 1685 y Fp(0)825 1703 y Fs(j)17 b Fn(is)g(\014nite.)75 1817 y(R)n(emark:)24 b FC(The)19 b(range)g([0)p Ft(;)8 b Fs(1)p FC(\))18 b(for)h(the)g(parameter)e Ft(t)i FC(is,)f(of)i(course,)e(arbitrary)l (.)29 b(In)19 b(some)e(cases)75 1877 y(w)o(e)d(shall)g(use)g(instead)g Ft(t)f Fs(2)h FC([0)p Ft(;)8 b FC(1])14 b(or)g(some)f(other)i(range.)21 b(W)l(e)13 b(put)i(a)f(single)g(index)f(on)1701 1865 y(_)1685 1877 y Ft(C)1720 1884 y Fq(t)1748 1877 y FC(b)q(ecause)75 1937 y(di\013eren)o(tiation)i(with)h(resp)q(ect)g(to)h Ft(t)f FC(mak)o(es)e(this)j (a)f(single)g(scale)g(ob)s(ject.)148 1998 y(No)o(w)23 b(w)o(e)f(can)h (formalize)d(the)i(steps)h(\(4.9\){\(4.11\).)42 b(F)l(orm)o(ula)21 b(\(4.11\))i(b)q(ecomes)f(a)h(partial)75 2058 y(\(functional\))14 b(di\013eren)o(tial)f(equation)i(for)g(a)g(family)d(of)j(functionals)g Ft(Z)1373 2065 y Fq(t)p Fo(0)1406 2058 y FC(.)20 b(As)15 b(Gaussian)g (functions)75 2118 y(are)k(fundamen)o(tal)f(solutions)i(of)g(the)f(heat)g (equation,)h(it)f(is)g(no)h(surprise)f(to)g(\014nd)h(out)g(that)g Ft(Z)1880 2125 y Fq(t)p Fo(0)75 2178 y FC(is)j(the)g(solution)h(of)f(a)h (suitably)f(form)o(ulated)f(heat)h(equation.)42 b(The)24 b(result)f(is)g(the) g(follo)o(wing)75 2238 y(lemma.)18 b(A)d(commen)n(t)e(ab)q(out)j(notation:)22 b(in)15 b(what)h(follo)o(ws,)f(a)h(functional)f(will)f(b)q(e)h(lab)q(elled)g (with)75 2299 y(subindexes)j(\\)p Ft(t)p FC(0")h(to)g(help)f(us)h(remem)o(b)q (er)c(that)k(it)f(w)o(as)h(constructed)g(b)o(y)f(in)o(tegration)g(o)o(v)o(er) g(the)75 2359 y(scales)e(b)q(et)o(w)o(een)g(0)h(and)g Ft(t)p FC(.)k(Th)o(us)c Ft(Z)746 2366 y Fq(t)p Fo(0)796 2359 y FC(is)f(alw)o(a)o(ys) g(to)h(b)q(e)g(in)o(tegrated)f(with)g(resp)q(ect)g(to)h(a)g(measure)75 2419 y Ft(d\026)129 2426 y Fp(1)p Fq(t)196 2419 y FC(whic)o(h)f(in)o (tegrates)f(o)o(v)o(er)h(all)f(scales)i(larger)f(than)h Ft(t)p FC(.)969 2599 y(27)p eop %%Page: 28 32 bop 75 307 a Fu(Lemma)16 b(4.3)24 b Fn(L)n(et)18 b Ft(d\026)509 314 y Fp(1)p Fq(t)579 307 y Fn(b)n(e)i(the)f(Gaussian)g(me)n(asur)n(es)g (whose)g(c)n(ovarianc)n(es)h(ar)n(e)e Ft(C)1673 314 y Fp(1)p Fq(t)1742 307 y Fn(in)i(a)f(de-)75 367 y(formation)e(of)g Ft(C)t Fn(,)g(then)i(for)d(any)i Ft(t)13 b Fs(\025)h FC(0)p Fn(,)752 431 y Fr(Z)802 490 y Ft(d\026Z)46 b FC(=)1015 431 y Fr(Z)1065 490 y Ft(d\026)1119 497 y Fp(1)p Fq(t)1169 490 y Ft(Z)1202 497 y Fq(t)p Fo(0)1788 490 y FC(\(4.12\))935 577 y(=)k(lim)1015 605 y Fq(t)p Fp(!1)1106 577 y Ft(Z)1139 584 y Fq(t)p Fo(0)1172 577 y FC(\(0\))554 b(\(4.13\))75 697 y Fn(if)17 b Ft(Z)155 704 y Fq(t)p Fo(0)206 697 y Fn(ob)n(eys)g(the)h(functional)h(he)n(at)e(e)n (quation)804 796 y Ft(@)s(Z)866 803 y Fq(t)p Fo(0)p 804 818 95 2 v 828 863 a Ft(@)s(t)945 829 y FC(=)1029 796 y(1)p 1029 818 25 2 v 1029 863 a(2)1067 829 y(\001)1108 836 y Fq(t)1122 829 y Ft(Z)1155 836 y Fq(t)p Fo(0)1788 829 y FC(\(4.14\))833 914 y Ft(Z)866 921 y Fo(00)945 914 y FC(=)41 b Ft(Z)18 b(:)75 1024 y Fn(Her)n(e)529 1101 y FC(\001)570 1108 y Fq(t)584 1101 y Ft(Z)617 1108 y Fq(t)p Fo(0)664 1101 y FC(=)716 1043 y Fr(Z)766 1101 y Ft(d\030)823 1043 y Fr(Z)873 1101 y Ft(d\030)921 1081 y Fp(0)1012 1068 y Ft(\016)1036 1050 y Fo(2)1055 1068 y Ft(Z)1088 1075 y Fq(t)p Fo(0)p 947 1090 240 2 v 947 1136 a Ft(\016)r(\036)p FC(\()p Ft(\030)r FC(\))p Ft(\016)r(\036)p FC(\()p Ft(\030)1156 1121 y Fp(0)1167 1136 y FC(\))1200 1101 y(\()p Fs(\000)1274 1089 y FC(_)1258 1101 y Ft(C)1293 1108 y Fq(t)1307 1101 y FC(\()p Ft(\030)r(;)8 b(\030)1394 1081 y Fp(0)1407 1101 y FC(\)\))p Ft(:)329 b FC(\(4)p Ft(:)p FC(15\))75 1214 y Fn(A)o(lso,)18 b(a)f(solution)h(for)f(the)h(he)n(at)f(e)n(quation)i(is)689 1336 y Ft(Z)722 1343 y Fq(t)p Fo(0)755 1336 y FC(\()p Ft(\036)p FC(\))14 b(=)888 1277 y Fr(Z)937 1336 y Ft(d\026)991 1343 y Fq(t)p Fo(0)1024 1336 y FC(\()p Ft(\020)t FC(\))p Ft(Z)t FC(\()p Ft(\036)d FC(+)g Ft(\020)t FC(\))p Ft(;)498 b FC(\(4)p Ft(:)p FC(16\))75 1458 y Fn(wher)n(e)18 b Ft(d\026)267 1465 y Fq(t)p Fo(0)317 1458 y Fn(has)f(c)n(ovarianc)n(e)h Ft(C)679 1465 y Fp(1)p Fo(0)745 1458 y Fs(\000)11 b Ft(C)830 1465 y Fp(1)p Fq(t)880 1458 y Fn(.)75 1597 y Fu(Pro)r(of.)27 b FC(It)13 b(is)g(a)g (transcription)g(of)g(the)g(previous)f(steps)i(\(4.8\){\(4.11\).)21 b(W)l(rite)12 b Ft(C)1567 1604 y Fp(1)p Fq(t)1631 1597 y FC(=)i Ft(C)1718 1604 y Fp(1)p Fq(t)1766 1595 y Fk(0)1783 1597 y FC(+)t Ft(C)1860 1604 y Fq(t)1873 1595 y Fk(0)1884 1604 y Fq(t)1899 1597 y FC(.)75 1657 y(As)i(in)g(\(4.7\),)g(for)h Fn(any)f FC(smo)q(oth)g (functional)g Ft(Z)935 1664 y Fq(t)p Fo(0)422 1724 y Fr(Z)472 1782 y Ft(d\026)526 1789 y Fp(1)p Fq(t)577 1782 y FC(\()p Ft(\036)p FC(\))8 b Ft(Z)685 1789 y Fq(t)p Fo(0)718 1782 y FC(\()p Ft(\036)p FC(\))13 b(=)850 1724 y Fr(Z)900 1782 y Ft(d\026)954 1789 y Fp(1)p Fq(t)1002 1780 y Fk(0)1016 1782 y FC(\()p Ft(\036)1064 1762 y Fp(0)1075 1782 y FC(\))1102 1724 y Fr(Z)1152 1782 y Ft(d\026)1206 1789 y Fq(t)1219 1780 y Fk(0)1231 1789 y Fq(t)1254 1782 y FC(\()p Ft(\020)t FC(\))p Ft(Z)1350 1789 y Fq(t)p Fo(0)1383 1782 y FC(\()p Ft(\036)1431 1762 y Fp(0)1453 1782 y FC(+)e Ft(\020)t FC(\))j Ft(:)214 b FC(\(4)p Ft(:)p FC(17\))75 1907 y(A)16 b(T)l(a)o(ylor)g(expansion)g(of)h Ft(Z)596 1914 y Fq(t)p Fo(0)629 1907 y FC(\()p Ft(\036)677 1889 y Fp(0)699 1907 y FC(+)11 b Ft(\020)t FC(\))17 b(yields)e(the)h(analogue)h(of)g(\(4.9\):)100 2047 y(\()p Ft(\026)148 2054 y Fq(t)161 2045 y Fk(0)172 2054 y Fq(t)193 2047 y Fs(\003)6 b Ft(Z)257 2054 y Fq(t)p Fo(0)290 2047 y FC(\)\()p Ft(\036)357 2026 y Fp(0)369 2047 y FC(\))28 b(=)f Ft(Z)514 2054 y Fq(t)p Fo(0)547 2047 y FC(\()p Ft(\036)595 2026 y Fp(0)606 2047 y FC(\))21 b(+)709 2013 y(1)p 709 2035 25 2 v 709 2081 a(2)746 1988 y Fr(Z)796 2047 y Ft(d\030)854 1988 y Fr(Z)903 2047 y Ft(d\030)951 2026 y Fp(0)1015 2013 y Ft(\016)1039 1995 y Fo(2)1058 2013 y Ft(Z)1091 2020 y Fq(t)p Fo(0)1124 2013 y FC(\()p Ft(\036)1172 1995 y Fp(0)1184 2013 y FC(\))p 977 2035 263 2 v 977 2081 a Ft(\016)r(\036)1030 2067 y Fp(0)1041 2081 y FC(\()p Ft(\030)r FC(\))p Ft(\016)r(\036)1155 2067 y Fp(0)1167 2081 y FC(\()p Ft(\030)1209 2067 y Fp(0)1221 2081 y FC(\))1245 2047 y Ft(C)1280 2054 y Fq(t)1293 2045 y Fk(0)1304 2054 y Fq(t)1319 2047 y FC(\()p Ft(\030)r(;)8 b(\030)1406 2026 y Fp(0)1419 2047 y FC(\))20 b(+)g Ft(O)1554 1999 y Fr(\020)1579 2047 y FC(\()p Ft(C)1633 2054 y Fq(t)1646 2045 y Fk(0)1657 2054 y Fq(t)1672 2047 y FC(\))1691 2026 y Fo(2)1711 1999 y Fr(\021)1749 2047 y Ft(:)25 b FC(\(4)p Ft(:)p FC(18\))75 2185 y(Expand)18 b(the)g(\014rst)g(term)e(in)h(this)h(iden)o(tit)o(y)e(in)h(p)q(o) o(w)o(ers)h(of)g(\()p Ft(t)12 b Fs(\000)f Ft(t)1295 2167 y Fp(0)1307 2185 y FC(\))17 b(and)i(use)f(the)f(h)o(yp)q(otheses)h(on)92 2233 y(_)75 2245 y Ft(C)110 2252 y Fq(t)141 2245 y FC(and)f Ft(Z)t FC(:)169 2381 y Ft(\026)198 2388 y Fq(t)211 2379 y Fk(0)222 2388 y Fq(t)248 2381 y Fs(\003)11 b Ft(Z)317 2388 y Fq(t)p Fo(0)378 2381 y FC(=)27 b Ft(Z)476 2388 y Fq(t)489 2379 y Fk(0)501 2388 y Fo(0)531 2381 y FC(+)580 2308 y Fr(")610 2348 y Ft(@)s(Z)672 2355 y Fq(t)p Fo(0)p 610 2370 95 2 v 636 2415 a Ft(\016)r(t)709 2308 y Fr(#)733 2433 y Fq(t)p Fo(=)p Fq(t)786 2424 y Fk(0)808 2381 y FC(\()p Ft(t)10 b Fs(\000)h Ft(t)923 2361 y Fp(0)934 2381 y FC(\))g Fs(\000)1019 2348 y FC(1)p 1019 2370 25 2 v 1019 2415 a(2)1057 2381 y([)o(\001)1111 2388 y Fq(t)1126 2381 y Ft(Z)1159 2388 y Fq(t)p Fo(0)1192 2381 y FC(])1205 2396 y Fq(t)p Fo(=)p Fq(t)1258 2386 y Fk(0)1280 2381 y FC(\()p Ft(t)f Fs(\000)h Ft(t)1395 2361 y Fp(0)1407 2381 y FC(\))g(+)g Ft(o)1509 2333 y Fr(\020)1534 2381 y Ft(t)f Fs(\000)h Ft(t)1630 2361 y Fp(0)1642 2333 y Fr(\021)1680 2381 y Ft(:)94 b FC(\(4)p Ft(:)p FC(19\))969 2599 y(28)p eop %%Page: 29 33 bop 75 313 a FC(\(W)l(e)23 b(ha)o(v)o(e)g(used)h(the)f(facts)h(that)756 289 y Fq(@)r(C)802 298 y Fm(t)814 291 y Fk(0)825 298 y Fm(t)p 756 302 83 2 v 781 330 a Fq(@)r(t)870 313 y FC(=)946 294 y Fq(@)p 939 302 34 2 v 939 330 a(@)r(t)977 313 y FC(\()p Ft(C)1031 320 y Fp(1)p Fq(t)1079 311 y Fk(0)1109 313 y Fs(\000)15 b Ft(C)1198 320 y Fp(1)p Fq(t)1248 313 y FC(\))27 b(=)f Fs(\000)1413 301 y FC(_)1397 313 y Ft(C)1432 320 y Fq(t)1447 313 y FC(;)g(and)f(that)f Ft(C)1738 320 y Fq(tt)1791 313 y FC(=)j(0.\))75 373 y(Therefore,)15 b(from)h(\(4.17\))g(and)h(\(4.19\))499 465 y Ft(d)p 490 487 43 2 v 490 533 a(dt)546 440 y Fr(Z)596 499 y Ft(d\026)650 506 y Fp(1)p Fq(t)701 499 y Ft(Z)734 506 y Fq(t)p Fo(0)808 499 y FC(=)44 b(lim)888 529 y Fq(t)901 519 y Fk(0)912 529 y Fp(!)p Fq(t)973 429 y Fr(R)1009 465 y Ft(d\026)1063 472 y Fp(1)p Fq(t)1114 465 y Ft(Z)1147 472 y Fq(t)p Fo(0)1191 465 y Fs(\000)1240 429 y Fr(R)1276 465 y Ft(d\026)1330 472 y Fp(1)p Fq(t)1378 463 y Fk(0)1392 465 y Ft(Z)1425 472 y Fq(t)1438 463 y Fk(0)1450 472 y Fo(0)p 973 487 497 2 v 1167 533 a Ft(t)1185 518 y Fp(0)1208 533 y Fs(\000)10 b Ft(t)808 628 y FC(=)42 b Fs(\000)935 570 y Fr(Z)984 628 y Ft(d\026)1038 635 y Fp(1)p Fq(t)1098 555 y Fr(")1127 594 y Ft(@)s(Z)1189 601 y Fq(t)p Fo(0)p 1127 617 95 2 v 1151 662 a Ft(@)s(t)1237 628 y Fs(\000)1292 594 y FC(1)p 1292 617 25 2 v 1292 662 a(2)1321 628 y(\001)1362 635 y Fq(t)1377 628 y Ft(Z)1410 635 y Fq(t)p Fo(0)1443 555 y Fr(#)1489 628 y FC(;)75 759 y(whic)o(h)17 b(sho)o(ws)i(that)f(\(4.14\))h(is)f(su\016cien)o (t)e(for)942 723 y Fr(R)978 759 y Ft(d\026)1032 766 y Fp(1)p Fq(t)1082 759 y Ft(Z)1115 766 y Fq(t)p Fo(0)1166 759 y FC(to)j(b)q(e)f (constan)o(t)g(with)g Ft(t)p FC(.)26 b(This)18 b(pro)o(v)o(es)75 819 y(\(4.12\).)148 879 y(The)h(iden)o(tit)o(y)e(\(4.13\))i(follo)o(ws)g (from)f(the)h(fact)f(that)i Ft(d\026)1206 886 y Fp(1)p Fq(t)1256 879 y FC(\()p Ft(\036)p FC(\))f Fs(!)f Ft(\016)r FC(\()p Ft(\036)p FC(\))g(w)o(eakly)g(as)h Ft(t)f Fs(!)g(1)p FC(,)75 939 y(b)o(y)e(the)g (condition)g Ft(C)476 946 y Fp(1)p Fq(t)526 939 y FC(\()p Ft(\030)r(;)8 b(\030)r FC(\))15 b Fs(!)f FC(0)i(in)g(the)g(de\014nition)g(of)g(a)h (deformation)e(of)i Ft(C)t FC(.)148 1000 y(The)f(pro)q(of)i(of)e(\(4.16\))h (is)f(left)g(as)h(an)f(exercise)f(for)h(the)g(reader)g(\(Problem)f(4.3.1\).) 205 b Fg(2)75 1060 y Fn(R)n(emark:)24 b FC(The)16 b(standard)h(Wilson)f(pro)q (cedure)f(corresp)q(onds)i(to)f(the)g(iden)o(tit)o(y)e(\(4.13\).)21 b(W)l(e)16 b(ha)o(v)o(e,)75 1120 y(ho)o(w)o(ev)o(er,)22 b(stated)h(the)f (lemma)e(so)j(to)g(highligh)o(t)e(also)i(the)g(in)o(termedi)o(ate)d(step)i (\(4.12\).)41 b(This,)75 1180 y(in)20 b(a)h(sense,)f(is)g(a)h(sligh)o(t)f (generalization)f(of)i(the)f(usual)g(prescription)g(that)g(pro)o(vides)g(an)h (extra)75 1240 y(\015exibilit)o(y:)j(it)18 b(sho)o(ws)i(ho)o(w)f(the)f(ideas) h(can)g(b)q(e)g(used)g(ev)o(en)f(in)g(those)i(cases)f(in)f(whic)o(h)g(for)i (some)75 1301 y(reason)d(w)o(e)f(are)g(unable)g(to)h(iterate)e(the)h(pro)q (cedure)h(all)e(the)h(w)o(a)o(y)g(to)h Ft(t)c FC(=)h Fs(1)p FC(.)75 1443 y Ff(4.3)70 b(Problems)75 1536 y Fu(4.3.1)24 b FC(Pro)o(v)o(e)16 b(that)i(\(4.16\))g(solv)o(es)e(\(4.14\).)25 b(Notice)16 b(that)i(this)f(means)f(that)i(the)f(solution)g(of)h(the)197 1596 y Fn(functional)d FC(heat)f(equation)g(is)g(the)f(con)o(v)o(olution)g (of)h(a)g(Gaussian)h(measure)e(with)g(the)h(initial)197 1656 y(condition.)21 b(This)15 b(is)g(exactly)f(analogous)j(to)f(the)f(familiar)e (PDE)j(case.)21 b([Hin)o(t:)e(Pro)q(ceed)c(as)197 1716 y(in)h(the)g(previous) g(pro)q(of)h(to)g(sho)o(w)g(that)957 1697 y Fq(@)p 950 1705 34 2 v 950 1734 a(@)r(t)989 1716 y Ft(\026)1018 1723 y Fq(t)p Fo(0)1061 1716 y Fs(\003)11 b Ft(Z)18 b FC(=)c(\(1)p Ft(=)p FC(2\)\001)1351 1723 y Fq(t)1367 1716 y FC(\()p Ft(\026)1415 1723 y Fq(t)p Fo(0)1458 1716 y Fs(\003)d Ft(Z)t FC(\).])75 1815 y Fu(4.3.2)24 b FC(More)16 b(generally)l(,)e(sho)o(w)j(that)g(a)g (functional)e Ft(A)1090 1822 y Fq(t)1121 1815 y FC(solv)o(es)h(the)g (di\013eren)o(tial)f(equation)807 1903 y Ft(@)s(A)873 1910 y Fq(t)p 807 1925 80 2 v 824 1971 a Ft(@)s(t)933 1937 y FC(=)42 b(\001)1054 1944 y Fq(t)1068 1937 y Ft(A)1105 1944 y Fq(t)1130 1937 y FC(+)11 b Ft(f)1203 1944 y Fq(t)1218 1937 y FC(\()p Ft(A)1274 1944 y Fq(t)1289 1937 y FC(\))835 2021 y Ft(A)872 2028 y Fo(0)933 2021 y FC(=)42 b Ft(A)13 b(;)197 2121 y FC(where,)j(for)h (eac)o(h)g Ft(t)p FC(,)f Ft(f)610 2128 y Fq(t)642 2121 y FC(is)g(a)h (reasonable)h(function)e(\(e.g.)g(in)o(v)o(olving)g(p)q(o)o(w)o(ers\),)g(if)h Ft(A)1759 2128 y Fq(t)1790 2121 y FC(solv)o(es)197 2181 y(the)f(in)o(tegral)f (equation)670 2300 y Ft(A)707 2307 y Fq(t)749 2300 y FC(=)28 b Ft(\026)844 2307 y Fq(t)p Fo(0)888 2300 y Fs(\003)11 b Ft(A)24 b FC(+)1048 2241 y Fr(Z)1089 2255 y Fq(t)1071 2336 y Fo(0)1113 2300 y Ft(\026)1142 2307 y Fq(ts)1184 2300 y Fs(\003)11 b Ft(f)1244 2307 y Fq(s)1262 2300 y FC(\()p Ft(A)1318 2307 y Fq(s)1336 2300 y FC(\))d Ft(ds)15 b(:)197 2414 y FC(This)e(is)g(the)g (functional-equation)g(analogous)i(of)f(the)f(w)o(ell-kno)o(wn)f(pro)q(cess)i (of)g(replacing)e(a)197 2474 y(non-linear)i(PDE)g(b)o(y)f(the)g(in)o(tegral)g (equation)g(obtained)h(b)o(y)f(con)o(v)o(olving)f(the)i(non-linearit)o(y)969 2599 y(29)p eop %%Page: 30 34 bop 197 307 a FC(with)18 b(the)f(fundamen)o(tal)g(solution)h(of)g(the)g (linear)f(part.)26 b(This)18 b(in)o(tegral)f(equation)h(is)g(then)197 367 y(used)e(to)h(generate)f(a)h(diagramatic)e(expansion.)75 469 y Fu(4.3.3)24 b Fn(A)17 b(form)f(of)h(Wick's)h(the)n(or)n(em)t FC(.)i(Using)15 b(Exercise)g(4.3.1)h(pro)o(v)o(e)f(that)h(if)g Ft(V)27 b FC(is)15 b(a)i(p)q(olynomial)197 529 y(in)f Ft(\036)g FC(then)726 659 y Ft(\026)755 666 y Fq(C)796 659 y Fs(\003)11 b Ft(V)53 b FC(=)42 b(exp)o([)1086 625 y(1)p 1086 647 25 2 v 1086 693 a(2)1115 659 y(\001)1156 666 y Fq(C)1185 659 y FC(])p Ft(V)912 785 y Fs(\021)993 743 y Fr(X)1083 751 y FC(1)p 1066 773 58 2 v 1066 819 a Ft(N)5 b FC(!)1137 724 y Fr(\022)1172 751 y FC(1)p 1172 773 25 2 v 1172 819 a(2)1202 785 y(\001)1243 792 y Fq(C)1272 724 y Fr(\023)1303 732 y Fq(N)1345 785 y Ft(V)415 b FC(\(4.20\))197 911 y(where)617 983 y(\001)658 990 y Fq(C)715 983 y Fs(\021)782 925 y Fr(Z)805 1019 y Fo(\003)840 983 y Ft(d\030)897 925 y Fr(Z)920 1019 y Fo(\003)955 983 y Ft(d\030)1003 963 y Fp(0)1073 949 y Ft(\016)p 1020 972 129 2 v 1020 1017 a(\016)r(\036)1073 1024 y Fq(i)1087 1017 y FC(\()p Ft(\030)r FC(\))1154 983 y Ft(C)t FC(\()p Ft(\030)r(;)8 b(\030)1280 963 y Fp(0)1292 983 y FC(\))1376 949 y Ft(\016)p 1316 972 144 2 v 1316 1017 a(\016)r(\036)1369 1024 y Fq(j)1386 1017 y FC(\()p Ft(\030)1428 1003 y Fp(0)1441 1017 y FC(\))1479 983 y Ft(:)295 b FC(\(4)p Ft(:)p FC(21\))197 1097 y(Pro)o(v)o(e)15 b(that)i(the)f(Wic)o(k)f(p)q(o)o(w)o(ers)i(de\014ned)f (b)o(y)810 1207 y(:)d Ft(\036)866 1186 y Fq(n)903 1207 y FC(:)h(\()p Ft(x)p FC(\))f Fs(\021)h Ft(e)1086 1186 y Fp(\000)1118 1173 y Fl(1)p 1118 1179 16 2 v 1118 1199 a(2)1138 1186 y Fo(\001)1167 1192 y Fm(C)1195 1207 y Ft(\036)1224 1186 y Fq(n)1248 1207 y FC(\()p Ft(x)p FC(\))474 b(\(4)p Ft(:)p FC(22\))197 1317 y(are)19 b(orthogonal,)543 1282 y Fr(R)579 1317 y Ft(d\026)633 1324 y Fq(C)682 1317 y FC(:)g Ft(\036)744 1299 y Fq(n)787 1317 y FC(:)g(\()p Ft(x)p FC(\))g(:)g Ft(\036)967 1299 y Fq(m)1019 1317 y FC(:)g(\()p Ft(y)r FC(\))g(=)g(0)h(if)f Ft(n)g Fs(6)p FC(=)g Ft(m)p FC(,)h(and)g(that)g(this)f(w)o(a)o(y)g(of)197 1382 y(de\014ning)e(:)d Ft(\036)439 1364 y Fq(n)477 1382 y FC(:)i(agrees)h(with)g(another)g(standard)h(de\014nition:)k(:)14 b Ft(e)1437 1364 y Fq(i\036)p Fo(\()p Fq(x)p Fo(\))1534 1382 y FC(:)p Fs(\021)g Ft(e)1629 1350 y Fl(1)p 1628 1356 V 1628 1376 a(2)1648 1363 y Fq(C)r Fo(\()p Fq(x;x)p Fo(\))1755 1382 y Ft(e)1778 1364 y Fq(i\036)p Fo(\()p Fq(x)p Fo(\))1875 1382 y FC(=)197 1409 y Fr(P)254 1420 y Fq(i)266 1409 y Fm(N)295 1420 y Fo(:)p Fq(\036)326 1409 y Fm(N)355 1420 y Fo(:)p 254 1430 111 2 v 289 1459 a Fq(N)t Fo(!)369 1442 y FC(,)i(c.f.)k(Exercise)15 b(5.5.2)i(b)q(elo)o(w.)969 2599 y(30)p eop %%Page: 31 35 bop 75 315 a Fv(5)83 b(The)27 b(E\013ectiv)n(e)g(Action)75 425 y FC(In)14 b(ph)o(ysical)f(applications,)h(one)h(exp)q(ects)e(that)i(the) f(functional)g Ft(Z)k FC(will)13 b(ha)o(v)o(e)h Fn(lo)n(c)n(ality)h(pr)n(op)n (erties)t FC(.)75 485 y(By)23 b(this)h(w)o(e)f(mean)g(that)h(it)g(appro)o (ximately)d(factorizes)j(when)g(ev)m(aluated)f(on)i(a)f(function)g Ft(\036)75 545 y FC(whic)o(h)16 b(is)g(the)h(sum)e(of)i(t)o(w)o(o)g (functions)f(with)h(distan)o(t)f(supp)q(orts.)24 b(One)17 b(w)o(a)o(y)f(to)h (see)f(the)g(presence)75 606 y(of)21 b(lo)q(calit)o(y)f(prop)q(erties)h(is)f (to)h(de\014ne)g(the)f Fn(e\013e)n(ctive)k(action)d Ft(V)32 b FC(b)o(y)21 b Ft(Z)k FC(=)d Ft(e)1515 587 y Fq(V)1545 606 y FC(.)35 b(In)20 b(terms)g(of)h Ft(V)11 b FC(,)75 666 y(lo)q(calit)o(y)16 b(is)h(the)g(prop)q(ert)o(y)g(that)h Ft(V)29 b FC(is)17 b(\(appro)o (ximately\))e(of)j(the)f(form)1415 630 y Fr(R)1451 666 y Ft(d\030)11 b(f)5 b FC(\()p Ft(\036)p FC(\()p Ft(\030)r FC(\))p Ft(;)j Fs(r)p Ft(\036)p FC(\()p Ft(\030)r FC(\))p Ft(;)g(:)g(:)g(:)p FC(\).)75 726 y(When)16 b(applying)f(Lemma)f(4.3)i(w)o(e)f(are)h(in)o (terested)e(in)h(the)h(exten)o(t)e(to)i(whic)o(h)f(lo)q(calit)o(y)g(prop)q (erties)75 786 y(of)h Ft(Z)163 793 y Fq(t)p Fo(0)212 786 y FC(are)g(preserv)o(ed)f(under)h(ev)o(olution)g(b)o(y)f(the)h(heat)g(equation) g(\(4.14\).)22 b(This)16 b(can)g(b)q(e)h(done)f(b)o(y)75 846 y(de\014ning)g Ft(V)287 853 y Fq(t)p Fo(0)336 846 y FC(b)o(y)g Ft(Z)437 853 y Fq(t)p Fo(0)484 846 y FC(=)e(exp)o(\()p Ft(V)657 853 y Fq(t)p Fo(0)690 846 y FC(\))i(and)h(studying)f(the)g(resulting)g (equation)g(for)h(it.)75 990 y Ff(5.1)70 b(F)-6 b(unctional)22 b(Equations)75 1083 y FC(The)16 b(heat)h(equation)f(\(4.14\))h(for)f Ft(Z)731 1090 y Fq(t)p Fo(0)780 1083 y FC(implies)e(that)j Ft(V)1080 1090 y Fq(t)p Fo(0)1129 1083 y FC(solv)o(es)630 1181 y Ft(@)s(V)687 1188 y Fq(t)p Fo(0)p 630 1203 90 2 v 652 1249 a Ft(@)s(t)738 1214 y FC(=)795 1181 y(1)p 795 1203 25 2 v 795 1249 a(2)833 1214 y([)o(\001)887 1221 y Fq(t)902 1214 y Ft(V)930 1221 y Fq(t)p Fo(0)974 1214 y FC(+)11 b(\()p Fs(r)p Ft(V)1112 1221 y Fq(t)p Fo(0)1152 1214 y Ft(;)16 b Fs(r)p Ft(V)1252 1221 y Fq(t)p Fo(0)1285 1214 y FC(\))1304 1221 y Fq(t)1319 1214 y FC(])8 b Ft(;)457 b FC(\(5)p Ft(:)p FC(1\))75 1342 y(where)15 b Fs(r)p Ft(V)26 b FC(denotes)15 b(the)g(measure)781 1323 y Fq(@)r(V)p 763 1331 86 2 v 763 1359 a(@)r(\036)p Fo(\()p Fq(\030)q Fo(\))853 1342 y Ft(d\030)j FC(and,)e(giv)o(en)e(a)i(suitable)e(family)f(of)j (co)o(v)m(ariances)f Ft(C)1863 1349 y Fp(1)p Fq(t)75 1409 y FC(\(De\014nition)h(4.2\),)g(\()g Ft(;)24 b FC(\))524 1416 y Fq(t)555 1409 y FC(is)16 b(the)g(bilinear)g(form)f(\(\\dot)i(pro)q(duct"\)) g(de\014ned)f(b)o(y)513 1531 y(\()p Ft(\026)561 1538 y Fo(1)589 1531 y Ft(;)h(\026)649 1538 y Fo(2)669 1531 y FC(\))688 1538 y Fq(t)716 1531 y Fs(\021)769 1473 y Fr(Z)819 1531 y Ft(d\026)873 1538 y Fo(1)893 1531 y FC(\()p Ft(\030)933 1538 y Fo(1)953 1531 y FC(\))980 1473 y Fr(Z)1030 1531 y Ft(d\026)1084 1538 y Fo(2)1104 1531 y FC(\()p Ft(d\030)1169 1538 y Fo(2)1190 1531 y FC(\))8 b([)p Fs(\000)1286 1519 y FC(_)1270 1531 y Ft(C)1305 1538 y Fq(t)1319 1531 y FC(\()p Ft(\030)1359 1538 y Fo(1)1379 1531 y Ft(;)g(\030)1422 1538 y Fo(2)1442 1531 y FC(\)])75 1650 y(for)17 b(an)o(y)f(t)o(w)o(o)g(measures)f Ft(\026)572 1657 y Fo(1)592 1650 y FC(,)h Ft(\026)651 1657 y Fo(2)687 1650 y FC(in)g(\003.)21 b(Explicitly)390 1779 y(\()p Fs(r)p Ft(V)479 1786 y Fq(t)p Fo(0)520 1779 y Ft(;)16 b Fs(r)p Ft(V)620 1786 y Fq(t)p Fo(0)653 1779 y FC(\))672 1786 y Fq(t)714 1779 y FC(=)780 1721 y Fr(Z)829 1779 y Ft(d\030)875 1786 y Fo(1)904 1721 y Fr(Z)954 1779 y Ft(d\030)1000 1786 y Fo(2)1057 1746 y Ft(\016)r(V)1109 1753 y Fq(t)p Fo(0)p 1034 1768 132 2 v 1034 1814 a Ft(\016)r(\036)p FC(\()p Ft(\030)1127 1821 y Fo(1)1146 1814 y FC(\))1178 1731 y Fr(h)1198 1779 y Fs(\000)1253 1767 y FC(_)1237 1779 y Ft(C)1272 1786 y Fq(t)1286 1779 y FC(\()p Ft(\030)1326 1786 y Fo(1)1346 1779 y Ft(;)8 b(\030)1389 1786 y Fo(2)1409 1779 y FC(\))1428 1731 y Fr(i)1484 1746 y Ft(\016)r(V)1536 1753 y Fq(t)p Fo(0)p 1461 1768 V 1461 1814 a Ft(\016)r(\036)p FC(\()p Ft(\030)1554 1821 y Fo(2)1573 1814 y FC(\))1812 1779 y(\(5)p Ft(:)p FC(2\))75 1914 y(\(\\dot)18 b(pro)q(duct")h(b)q(et)o(w)o(een)d(t)o(w)o(o)h(\\gradien)o (ts"\).)26 b(The)17 b(equation)g(\(5.1\))h(w)o(as)g(already)f(in)o(tro)q (duced)75 1975 y(b)o(y)f(Wilson)g([60],)g(and)g(its)g(imp)q(ortance)g(w)o(as) g(emphasized)f(b)o(y)g(P)o(olc)o(hinsky)g([53)q(].)148 2035 y(A)i(solution)g(to)g(\(5.1\))g(can)g(sometimes)d(b)q(e)j(found)h(b)o(y)e (rewriting)g(it)h(as)g(an)g(in)o(tegral)g(equation)75 2095 y(\(Exercise)e(4.3.2\))524 2177 y Ft(V)552 2184 y Fq(t)p Fo(0)599 2177 y FC(=)f Ft(\026)680 2184 y Fq(t)p Fo(0)723 2177 y Fs(\003)d Ft(V)787 2184 y Fo(00)836 2177 y FC(+)890 2144 y(1)p 890 2166 25 2 v 890 2212 a(2)928 2119 y Fr(Z)969 2132 y Fq(t)951 2213 y Fo(0)993 2177 y Ft(ds)d(\026)1078 2184 y Fq(ts)1121 2177 y Fs(\003)j FC(\()p Fs(r)p Ft(V)1246 2184 y Fq(s)p Fo(0)1290 2177 y Ft(;)16 b Fs(r)p Ft(V)1390 2184 y Fq(s)p Fo(0)1426 2177 y FC(\))1445 2184 y Fq(s)1812 2177 y FC(\(5)p Ft(:)p FC(3\))75 2278 y(and)h(generating)g(a)g(series)e(b)o(y)h(iteration.)22 b(W)l(e)16 b(observ)o(e)g(that)h(the)f(\014rst)h(term)d(in)i(the)h(righ)o (t-hand-)75 2338 y(side)f(corresp)q(onds)h(to)g(the)f(solution)g(of)h(the)f (linear)f(part.)22 b(The)16 b(notation)h(is)641 2457 y(\()p Ft(\026)689 2464 y Fq(ts)731 2457 y Fs(\003)11 b Ft(F)c FC(\)\()p Ft(\036)p FC(\))13 b Fs(\021)958 2398 y Fr(Z)1008 2457 y Ft(d\026)1062 2464 y Fq(ts)1093 2457 y FC(\()p Ft(\020)t FC(\))p Ft(F)7 b FC(\()p Ft(\036)k FC(+)g Ft(\020)t FC(\))969 2599 y(31)p eop %%Page: 32 36 bop 457 307 a Ft(d\026)511 314 y Fq(ts)559 307 y FC(is)16 b(Gaussian)h(with)f (co)o(v)m(ariance)g Ft(C)1199 314 y Fq(ts)1244 307 y Fs(\021)d Ft(C)1331 314 y Fp(1)p Fq(s)1396 307 y Fs(\000)e Ft(C)1481 314 y Fp(1)p Fq(t)1812 307 y FC(\(5)p Ft(:)p FC(4\))75 394 y(\(see)16 b(commen)o(t)d(b)q(efore)j(De\014nition)g(4.2\).)148 455 y(F)l(rom)f(\(4.13\))i(w)o(e)e(see)h(that)g(to)g(compute)f(the)h(free)f (energy)g(\(or)i(the)e(generating)i(functionals\))75 515 y(w)o(e)f(only)g (need)g(the)g(v)m(alue)g(of)g Ft(V)661 522 y Fp(1)p Fo(0)717 515 y FC(:)729 637 y(log)801 578 y Fr(Z)850 637 y Ft(d\026)8 b(Z)19 b FC(=)j(lim)1016 664 y Fq(t)p Fp(!1)1108 637 y Ft(V)1136 644 y Fq(t)p Fo(0)1168 637 y FC(\(0\))15 b Ft(;)553 b FC(\(5)p Ft(:)p FC(5\))75 759 y(but,)18 b(as)g(commen)o(te)o(d)d(ab)q(o)o(v)o(e,)i(w)o (e)h(wish)f(to)h(b)q(e)g(sligh)o(t)o(y)e(more)h(general)g(and)h(consider)g Ft(V)1735 766 y Fq(t)p Fo(0)1785 759 y FC(for)g(all)75 819 y(v)m(alues)e(of)h Ft(t)p FC(.)148 880 y(W)l(e)11 b(turn)h(to)f(the)g (iterativ)o(e)f(solution)h(of)h(the)f(in)o(tegral)f(\015o)o(w)i(\(5.3\),)g (whic)o(h)f(will)f(b)q(e)h(an)h(expansion)75 940 y(in)20 b(terms)e(of)i(tree) f(diagrams.)32 b(W)l(e)20 b(p)q(oin)o(t)g(out)g(that)g(this)g(in)o(tegral)f (\015o)o(w)h(generates)g(essen)o(tially)75 1000 y(the)d(same)f(expansion)i (as)g(Galla)o(v)o(otti)e Fn(et)j(al.)25 b FC(ha)o(v)o(e)17 b(b)q(een)g(using)h([8,)f(29],)g(but)h(they)e(lik)o(e)g(to)i(w)o(ork)75 1060 y(in)o(tegrating)23 b(out)g(the)g(degrees)f(of)i(freedom)d(in)i Fn(\014nite)i(steps)e FC(instead)g(of)g Fn(in\014nitesimal)i(steps)t FC(.)75 1120 y(They)19 b(also)g(ha)o(v)o(e)f(tree)g(diagrams)h(but)g(their)f (trees)h(are)g(not)g(to)g(b)q(e)g(confused)g(with)g(ours:)27 b(their)75 1181 y(trees)14 b(lab)q(el)h(the)f(hierarc)o(h)o(y)f(of)i(scales)g (in)f(their)g(expansion)h(whereas)g(out)g(trees)f(are)h(con)o(v)o(en)o (tional)75 1241 y(F)l(eynman)g(diagrams)h(or)g(sometimes)d(Ma)o(y)o(er)j (diagrams.)75 1385 y Ff(5.2)70 b(F)-6 b(ormal)22 b(Series)g(for)i(the)e (Classical)f(Action)75 1478 y FC(T)l(o)c(familiarize)d(ourselv)o(es)i(with)h (the)f(pro)q(cess)i(of)f(generating)g(a)g(series)g(solution)g(from)e (\(5.3\),)i(let)75 1538 y(us)g(\014rst)f(see)g(what)i(happ)q(ens)f(if)f(all)g (the)g(con)o(v)o(olutions)g(with)g(resp)q(ect)g(to)h Ft(\026)g FC(are)f(left)g(out;)g(that)h(is,)75 1598 y(if)f(w)o(e)g(replace)f(\(5.3\))h (b)o(y)613 1730 y Ft(V)652 1710 y Fo(cl)641 1742 y Fq(t)p Fo(0)694 1730 y FC(=)e Ft(V)774 1737 y Fo(00)823 1730 y FC(+)877 1696 y(1)p 877 1719 25 2 v 877 1764 a(2)914 1672 y Fr(Z)956 1685 y Fq(t)937 1766 y Fo(0)979 1730 y Ft(ds)8 b FC(\()p Fs(r)p Ft(V)1135 1710 y Fo(cl)1124 1742 y Fq(s)p Fo(0)1171 1730 y Ft(;)16 b Fs(r)p Ft(V)1282 1710 y Fo(cl)1271 1742 y Fq(s)p Fo(0)1310 1730 y FC(\))1329 1737 y Fq(s)1361 1730 y Ft(:)437 b FC(\(5)p Ft(:)p FC(6\))75 1854 y(W)l(e)16 b(notice)g(that)g(this)g(is)g (the)g(in)o(tegral)g(equation)g(asso)q(ciated)h(to)g(the)f(functional)g(PDE) 715 1959 y Ft(@)s(V)782 1941 y Fo(cl)772 1972 y Fq(t)p Fo(0)p 715 1981 96 2 v 739 2027 a Ft(@)s(t)856 1993 y FC(=)941 1959 y(1)p 941 1981 25 2 v 941 2027 a(2)970 1993 y(\()p Fs(r)p Ft(V)1070 1972 y Fo(cl)1059 2005 y Fq(t)p Fo(0)1106 1993 y Ft(;)g Fs(r)p Ft(V)1217 1972 y Fo(cl)1206 2005 y Fq(t)p Fo(0)1244 1993 y FC(\))1263 2000 y Fq(t)1812 1993 y FC(\(5.7\))748 2087 y Ft(V)787 2067 y Fo(cl)776 2100 y(00)856 2087 y FC(=)42 b Ft(V)964 2094 y Fo(00)75 2197 y FC(whic)o(h)16 b(is)g(the)g(Hamilton-Jacobi)e(equation)j (obtained)f(b)o(y)g(lea)o(ving)f(out)i(the)f(Laplacian)h(in)f(\(5.1\).)75 2258 y(So)c(our)g(w)o(arm-up)f(study)h(of)g(\(5.6\))g(yields)f(in)g(fact)h (an)g(expression)f(for)h(what,)h(in)f(the)f(quan)o(tum-\014eld)75 2318 y(in)o(terpretation)k(\(application)h(\(i\))g(in)g(Section)g(4.1\),)g (corresp)q(onds)h(to)g(the)f(classical)f(action.)148 2378 y(Equation)c (\(5.6\))h(lends)e(itself)g(to)h(a)g(natural)g(pro)q(cess)h(of)f(iteration:) 18 b(to)11 b(zeroth)g(order)g Ft(V)1739 2360 y Fo(cl)1728 2390 y Fq(t)p Fo(0)1781 2378 y Fs(\031)i Ft(V)1861 2385 y Fo(00)1899 2378 y FC(;)75 2438 y(inserting)h(this)g(appro)o(ximation)f(in)g(the)h(RHS)g (\()p Ft(V)998 2420 y Fo(cl)987 2450 y Fq(s)p Fo(0)1039 2438 y Fs(\031)g Ft(V)1120 2445 y Fo(00)1158 2438 y FC(\))g(w)o(e)f(get)h(a)h (\014rst-order)g(appro)o(ximation)969 2599 y(32)p eop %%Page: 33 37 bop 75 307 a FC(for)14 b Ft(V)175 314 y Fq(t)p Fo(0)207 307 y FC(,)g(whic)o(h)f(can)g(b)q(e)g(substituted)h(in)f(the)g(RHS)g(to)g (generate)g(a)h(second-order)g(appro)o(ximation,)75 367 y(and)j(so)g(on.)k (The)c(\014rst)f(few)g(terms)f(of)i(the)f(series)f(generated)h(in)g(this)g(w) o(a)o(y)g(are:)238 499 y Ft(V)277 478 y Fo(cl)266 511 y Fq(t)p Fo(0)346 499 y FC(=)42 b Ft(V)454 506 y Fo(00)511 499 y FC(+)573 465 y(1)p 573 487 25 2 v 573 533 a(2)611 440 y Fr(Z)652 453 y Fq(t)634 534 y Fo(0)676 499 y Ft(ds)8 b FC(\()p Fs(r)p Ft(V)821 506 y Fo(00)867 499 y Ft(;)16 b Fs(r)p Ft(V)967 506 y Fo(00)1004 499 y FC(\))1023 506 y Fq(s)1053 499 y FC(+)1102 438 y Fr(\022)1137 465 y FC(1)p 1137 487 V 1137 533 a(2)1167 438 y Fr(\023)1197 446 y Fo(2)1225 440 y Fr(Z)1267 453 y Fq(t)1248 534 y Fo(0)1290 499 y Ft(ds)1338 506 y Fo(1)1366 440 y Fr(Z)1408 453 y Fq(s)1424 458 y Fl(1)1389 534 y Fo(0)1452 499 y Ft(ds)1500 506 y Fo(2)437 615 y Fs(\002)487 554 y Fr(\024)509 567 y(\020)533 615 y Fs(r)p Ft(V)603 622 y Fo(00)649 615 y Ft(;)g FC(\()p Fs(r)740 594 y Fo(2)759 615 y Ft(V)787 622 y Fo(00)833 615 y Ft(;)h Fs(r)p Ft(V)934 622 y Fo(00)971 615 y FC(\))990 622 y Fq(s)1006 627 y Fl(2)1026 567 y Fr(\021)1050 642 y Fq(s)1066 647 y Fl(1)1097 615 y FC(+)1146 567 y Fr(\020)1171 615 y Fs(r)p Ft(V)1241 622 y Fo(00)1287 615 y Ft(;)f FC(\()p Fs(r)p Ft(V)1406 622 y Fo(00)1451 615 y Ft(;)g Fs(r)1523 594 y Fo(2)1543 615 y Ft(V)1571 622 y Fo(00)1608 615 y FC(\))1627 622 y Fq(s)1643 627 y Fl(2)1663 567 y Fr(\021)1688 642 y Fq(s)1704 647 y Fl(1)475 731 y FC(+)513 683 y Fr(\020)537 731 y FC(\()p Fs(r)598 711 y Fo(2)618 731 y Ft(V)646 738 y Fo(00)692 731 y Ft(;)g Fs(r)p Ft(V)792 738 y Fo(00)829 731 y FC(\))848 738 y Fq(s)864 743 y Fl(2)892 731 y Ft(;)g Fs(r)p Ft(V)992 738 y Fo(00)1030 683 y Fr(\021)1054 758 y Fq(s)1070 763 y Fl(1)1101 731 y FC(+)1150 683 y Fr(\020)1175 731 y FC(\()p Fs(r)p Ft(V)1264 738 y Fo(00)1309 731 y Ft(;)h Fs(r)1382 711 y Fo(2)1401 731 y Ft(V)1429 738 y Fo(00)1467 731 y FC(\))1486 738 y Fq(s)1502 743 y Fl(2)1530 731 y Ft(;)f Fs(r)p Ft(V)1630 738 y Fo(00)1667 683 y Fr(\021)1692 758 y Fq(s)1708 763 y Fl(1)1728 671 y Fr(\025)1812 822 y FC(\(5.8\))346 895 y(=)42 b Ft(g)449 902 y Fo(0)469 895 y FC(\()p Ft(t)p FC(\))11 b(+)g Ft(g)608 902 y Fo(1)628 895 y FC(\()p Ft(t)p FC(\))f(+)h Ft(g)766 902 y Fo(2)786 895 y FC(\()p Ft(t)p FC(\))g(+)g Fs(\001)d(\001)g (\001)22 b Ft(:)75 1004 y FC(Here)16 b Fs(r)233 986 y Fq(k)254 1004 y Ft(V)28 b FC(denotes)17 b(the)f(measure)848 984 y Fq(\016)865 972 y Fm(k)884 984 y Fq(V)p 767 992 228 2 v 767 1021 a(\016)q(\036)p Fo(\()p Fq(\030)835 1026 y Fl(1)851 1021 y Fo(\))p Fp(\001\001\001)o Fq(\016)q(\036)p Fo(\()p Fq(\030)962 1027 y Fm(k)981 1021 y Fo(\))1008 1004 y Ft(d\030)1054 1011 y Fo(1)1082 1004 y Fs(\001)8 b(\001)g(\001)h Ft(d\030)1195 1011 y Fq(k)1217 1004 y FC(.)23 b(W)l(e)16 b(notice)h(that)g(in)f(the)h(iteration)75 1071 y(of)i(\(5.6\))h (eac)o(h)e(additional)h(lev)o(el)e(of)i(brac)o(k)o(ets)f(\()h Ft(;)27 b FC(\))1062 1078 y Fq(C)1111 1071 y FC(comes)17 b(together)i(with)g (a)g(\\)p Fs(r)p FC(")h(op)q(erator;)75 1131 y(hence,)d(a)i(term)d(of)i(the)g (form)f Fs(r)689 1113 y Fq(k)709 1131 y Ft(V)737 1138 y Fo(00)793 1131 y FC(app)q(ears)i(only)f(inside)f Ft(k)j FC(nested)e(lev)o(els)e(of)i (suc)o(h)g(brac)o(k)o(ets.)75 1191 y(The)f(b)q(o)q(ok)o(eeping)g(of)h(this)f (t)o(yp)q(e)f(of)h(series)g(is)g(usually)f(done)h(with)g(the)g(aid)g(of)g (diagrams:)23 b(T)l(erms)75 1251 y(are)c(asso)q(ciated)h(to)f(graphs)h(whose) f(edges)g(represen)o(t)f(co)o(v)m(ariances)g Fs(\000)1421 1239 y FC(_)1404 1251 y Ft(C)1439 1258 y Fq(s)1476 1251 y FC(\(\\propagators"\))k (and)75 1312 y(where)16 b(a)h(v)o(ertex)d(with)i Ft(k)j FC(impinging)14 b(lines)i(corresp)q(onds)h(to)g(a)f(measure)f Fs(r)1497 1293 y Fq(k)1518 1312 y FC(.)21 b(F)l(or)c(instance)519 1528 y Fr(\020)544 1576 y Fs(r)p Ft(V)614 1583 y Fo(00)660 1576 y Ft(;)f FC(\()p Fs(r)751 1556 y Fo(2)770 1576 y Ft(V)798 1583 y Fo(00)844 1576 y Ft(;)g Fs(r)p Ft(V)944 1583 y Fo(00)982 1576 y FC(\))1001 1583 y Fq(s)1017 1588 y Fl(2)1036 1528 y Fr(\021)1061 1603 y Fq(s)1077 1608 y Fl(1)1125 1576 y Fs(\021)1250 1566 y Fd(y)p 1271 1568 125 4 v 1316 1549 a Fq(s)1332 1554 y Fl(1)1416 1566 y Fd(y)p 1415 1545 4 125 v 1430 1488 a Fq(s)1446 1493 y Fl(2)1416 1400 y Fd(y)1812 1576 y FC(\(5)p Ft(:)p FC(9\))215 1729 y(=)267 1670 y Fr(Z)317 1729 y Ft(d\030)363 1736 y Fo(1)392 1729 y Ft(d\030)438 1736 y Fo(2)475 1729 y Ft(d\030)521 1736 y Fo(3)549 1729 y Ft(d\030)595 1736 y Fo(4)650 1695 y Ft(\016)r(V)702 1702 y Fo(00)p 629 1717 132 2 v 629 1763 a Ft(\016)r(\036)p FC(\()p Ft(\030)722 1770 y Fo(1)741 1763 y FC(\))774 1681 y Fr(h)793 1729 y Fs(\000)848 1716 y FC(_)832 1729 y Ft(C)867 1736 y Fq(s)885 1729 y FC(\()p Ft(\030)925 1736 y Fo(1)945 1729 y Ft(;)8 b(\030)988 1736 y Fo(2)1008 1729 y FC(\))1027 1681 y Fr(i)1137 1695 y Ft(\016)1161 1677 y Fo(2)1180 1695 y Ft(V)1208 1702 y Fo(00)p 1060 1717 263 2 v 1060 1763 a Ft(\016)r(\036)p FC(\()p Ft(\030)1153 1770 y Fo(2)1172 1763 y FC(\))p Ft(\016)r(\036)p FC(\()p Ft(\030)1284 1770 y Fo(3)1304 1763 y FC(\))1336 1681 y Fr(h)1355 1729 y Fs(\000)1411 1716 y FC(_)1394 1729 y Ft(C)1429 1736 y Fq(s)1447 1729 y FC(\()p Ft(\030)1487 1736 y Fo(3)1507 1729 y Ft(;)g(\030)1550 1736 y Fo(4)1570 1729 y FC(\))1589 1681 y Fr(i)1643 1695 y Ft(\016)r(V)1695 1702 y Fo(00)p 1622 1717 132 2 v 1622 1763 a Ft(\016)r(\036)p FC(\()p Ft(\030)1715 1770 y Fo(4)1735 1763 y FC(\))1772 1729 y Ft(:)75 1841 y FC(In)16 b(this)g(w)o(a)o(y)l(,)f(\(5.8\))i(can)f(b)q(e)h(depicted)e(as)190 2109 y Ft(V)229 2088 y Fo(cl)315 2109 y FC(=)439 2088 y Fc(y)496 2109 y FC(+)574 2075 y(1)p 574 2097 25 2 v 574 2143 a(2)612 2050 y Fr(Z)654 2063 y Fq(t)635 2145 y Fo(0)677 2109 y Ft(ds)792 2088 y Fd(y)p 813 2090 125 4 v 867 2074 a Fq(s)958 2088 y Fd(y)1025 2109 y FC(+)1104 2075 y(1)p 1104 2097 25 2 v 1104 2143 a(4)1141 2050 y Fr(Z)1183 2063 y Fq(t)1164 2145 y Fo(0)1206 2109 y Ft(ds)1254 2116 y Fo(1)1282 2050 y Fr(Z)1324 2063 y Fq(s)1340 2068 y Fl(1)1305 2145 y Fo(0)1368 2109 y Ft(ds)1416 2116 y Fo(2)1436 2036 y Fr(")1519 2098 y Fd(y)p 1540 2100 125 4 v 1586 2082 a Fq(s)1602 2087 y Fl(1)1685 2098 y Fd(y)p 1684 2078 4 125 v 1700 2020 a Fq(s)1716 2025 y Fl(2)1685 1932 y Fd(y)1748 2109 y FC(+)551 2430 y Fd(y)566 2416 y Fb(\000)607 2374 y(\000)649 2333 y(\000)661 2321 y(\000)596 2332 y Fq(s)612 2337 y Fl(1)717 2430 y Fd(y)p 715 2410 V 731 2352 a Fq(s)747 2357 y Fl(2)717 2264 y Fd(y)780 2441 y FC(+)888 2430 y Fd(y)p 909 2432 125 4 v 954 2414 a Fq(s)970 2419 y Fl(2)1054 2430 y Fd(y)p 1052 2410 4 125 v 1068 2352 a Fq(s)1084 2357 y Fl(1)1054 2264 y Fd(y)1117 2441 y FC(+)1225 2430 y Fd(y)p 1246 2432 125 4 v 1291 2414 a Fq(s)1307 2419 y Fl(1)1391 2430 y Fd(y)1240 2416 y Fb(\000)1281 2374 y(\000)1323 2333 y(\000)1335 2321 y(\000)1271 2332 y Fq(s)1287 2337 y Fl(2)1391 2264 y Fd(y)1443 2368 y Fr(#)1495 2441 y FC(+)35 b Fs(\001)8 b(\001)g(\001)23 b Ft(:)125 b FC(\(5.10\))969 2599 y(33)p eop %%Page: 34 38 bop 148 307 a FC(Let)19 b(us)f(analyze)g(a)h(bit)f(more)f(the)h(double-in)o (tegral)g(term)f(on)i(the)f(righ)o(t,)g(whic)o(h)g(w)o(e)g(denote)75 367 y(as)23 b Ft(g)164 374 y Fo(2)184 367 y FC(\()p Ft(t)p FC(\).)39 b(Of)22 b(course,)h(the)f(four)g(diagrams)g(giv)o(e)f(the)h(same)g (con)o(tribution,)g(but)h(it)e(is)h(con)o(v)o(e-)75 428 y(nien)o(t)15 b(to)h(k)o(eep)f(the)g(distinction)g(and)i(do)f(some)f(juggling)h(to)g (obtain)g(an)h(expression)e(with)h(b)q(etter)75 488 y(mnemonics.)i(The)f (\014rst)f(t)o(w)o(o)g(diagrams)g(represen)o(t)f(the)h(term)326 711 y Fr(\020)351 759 y Fs(r)p Ft(V)421 766 y Fo(00)467 759 y Ft(;)g Fs(r)p FC(\()p Fs(r)p Ft(V)628 766 y Fo(00)673 759 y Ft(;)g Fs(r)p Ft(V)773 766 y Fo(00)810 759 y FC(\))829 766 y Fq(s)845 771 y Fl(2)865 711 y Fr(\021)890 786 y Fq(s)906 791 y Fl(1)953 759 y Fs(\021)1079 749 y Fd(y)p 1100 750 125 4 v 1145 732 a Fq(s)1161 737 y Fl(1)1245 749 y Fd(y)p 1243 728 4 125 v 1259 670 a Fq(s)1275 675 y Fl(2)1245 583 y Fd(y)1308 759 y FC(+)1416 749 y Fd(y)1431 734 y Fb(\000)1472 692 y(\000)1514 651 y(\000)1526 639 y(\000)1461 650 y Fq(s)1477 655 y Fl(1)1582 749 y Fd(y)p 1580 728 V 1596 670 a Fq(s)1612 675 y Fl(2)1582 583 y Fd(y)1648 759 y Ft(:)126 b FC(\(5)p Ft(:)p FC(11\))75 948 y(This)24 b(corresp)q(onds)h(to)g(a)f(partition)g(of)h(the)f(three)f(v)o (ertices)1289 928 y Fd(i)1280 936 y Fo(1)1351 928 y Fd(i)1342 936 y Fo(2)1351 865 y Fd(i)1342 874 y Fo(3)1396 948 y FC(in)o(to)h(t)o(w)o(o) g(sets:)37 b(a)24 b(v)o(ertex)75 1008 y(represen)o(ting)e(the)h(left)f (factor)h Ft(V)714 1015 y Fo(00)775 1008 y FC(in)f(the)h(dot)g(pro)q(duct)h (\(5.11\))g(and)f(a)g(set)g(of)h(t)o(w)o(o)e(v)o(ertices)75 1069 y(represen)o(ting)16 b(the)h(righ)o(t)f(factor)h(\()p Fs(r)p Ft(V)783 1076 y Fo(00)829 1069 y Ft(;)f Fs(r)p Ft(V)929 1076 y Fo(00)966 1069 y FC(\))985 1076 y Fq(s)1001 1081 y Fl(2)1021 1069 y FC(.)23 b(W)l(e)17 b(obtain)g(a)g(more)f(attractiv)o(e)g(form)o(ula)f (b)o(y)75 1129 y Fn(symmetrization)20 b FC(o)o(v)o(er)15 b(the)h(lab)q(els)g (of)h(the)f Ft(V)27 b FC(factors.)22 b(That)17 b(is,)e(one)i(writes)f (instead)173 1352 y Fr(\020)198 1400 y Fs(r)p Ft(V)268 1407 y Fo(00)313 1400 y Ft(;)h Fs(r)p FC(\()p Fs(r)p Ft(V)475 1407 y Fo(00)520 1400 y Ft(;)f Fs(r)p Ft(V)620 1407 y Fo(00)657 1400 y FC(\))676 1407 y Fq(s)692 1412 y Fl(2)712 1352 y Fr(\021)737 1427 y Fq(s)753 1432 y Fl(1)800 1400 y FC(=)871 1366 y(1)p 871 1388 25 2 v 871 1434 a(3)900 1327 y Fr(" )1016 1390 y Fd(y)p 1037 1391 125 4 v 1082 1373 a Fq(s)1098 1378 y Fl(1)1182 1390 y Fd(y)p 1181 1369 4 125 v 1197 1312 a Fq(s)1213 1317 y Fl(2)1182 1224 y Fd(y)1245 1400 y FC(+)1353 1390 y Fd(y)1368 1375 y Fb(\000)1410 1334 y(\000)1451 1292 y(\000)1463 1280 y(\000)1399 1291 y Fq(s)1415 1296 y Fl(1)1519 1390 y Fd(y)p 1518 1369 V 1534 1312 a Fq(s)1550 1317 y Fl(2)1519 1224 y Fd(y)1571 1327 y Fr(!)256 1732 y FC(+)310 1659 y Fr( )402 1722 y Fd(y)p 423 1723 125 4 v 469 1705 a Fq(s)485 1710 y Fl(2)568 1722 y Fd(y)p 567 1701 4 125 v 583 1644 a Fq(s)599 1649 y Fl(1)568 1556 y Fd(y)631 1732 y FC(+)739 1722 y Fd(y)p 760 1723 125 4 v 806 1705 a Fq(s)822 1710 y Fl(2)906 1722 y Fd(y)754 1707 y Fb(\000)796 1666 y(\000)837 1624 y(\000)849 1612 y(\000)785 1623 y Fq(s)801 1628 y Fl(1)906 1556 y Fd(y)957 1659 y Fr(!)1018 1732 y FC(+)1083 1659 y Fr( )1175 1722 y Fd(y)p 1196 1723 V 1241 1705 a Fq(s)1257 1710 y Fl(1)1341 1722 y Fd(y)1190 1707 y Fb(\000)1231 1666 y(\000)1273 1624 y(\000)1285 1612 y(\000)1220 1623 y Fq(s)1236 1628 y Fl(2)1341 1556 y Fd(y)1404 1732 y FC(+)1512 1722 y Fd(y)1527 1707 y Fb(\000)1568 1666 y(\000)1610 1624 y(\000)1622 1612 y(\000)1557 1623 y Fq(s)1573 1628 y Fl(2)1678 1722 y Fd(y)p 1676 1701 4 125 v 1692 1644 a Fq(s)1708 1649 y Fl(1)1678 1556 y Fd(y)1730 1659 y Fr(!#)1801 1732 y Ft(:)1788 1836 y FC(\(5.12\))75 1946 y(The)k(same)g(RHS)g(is)g (obtained)h(from)e(the)h(analogous)j(symmet)o(rization)17 b(of)k(the)f(p)q (erm)o(uted)f(dot)75 2006 y(pro)q(duct)258 1958 y Fr(\020)283 2006 y FC(\()p Fs(r)344 1988 y Fo(2)363 2006 y Ft(V)391 2013 y Fo(00)437 2006 y Ft(;)d Fs(r)p Ft(V)537 2013 y Fo(00)574 2006 y FC(\))593 2013 y Fq(s)609 2018 y Fl(2)637 2006 y Ft(;)h Fs(r)p Ft(V)738 2013 y Fo(00)775 1958 y Fr(\021)800 2033 y Fq(s)816 2038 y Fl(1)835 2006 y FC(.)22 b(Adding)16 b(b)q(oth)h(con)o (tributions)f(one)g(obtains)189 2299 y Ft(g)212 2306 y Fo(2)232 2299 y FC(\()p Ft(t)p FC(\))58 b(=)453 2266 y(1)p 446 2288 38 2 v 446 2333 a(3!)497 2241 y Fr(Z)539 2254 y Fq(t)521 2335 y Fo(0)562 2299 y Ft(ds)610 2306 y Fo(1)638 2241 y Fr(Z)680 2254 y Fq(s)696 2259 y Fl(1)662 2335 y Fo(0)724 2299 y Ft(ds)772 2306 y Fo(2)792 2226 y Fr(" )908 2289 y Fd(y)p 929 2291 125 4 v 974 2272 a Fq(s)990 2277 y Fl(1)1074 2289 y Fd(y)p 1073 2268 4 125 v 1089 2211 a Fq(s)1105 2216 y Fl(2)1074 2123 y Fd(y)1137 2299 y FC(+)1245 2289 y Fd(y)p 1266 2291 125 4 v 1312 2272 a Fq(s)1328 2277 y Fl(2)1411 2289 y Fd(y)p 1410 2268 4 125 v 1426 2211 a Fq(s)1442 2216 y Fl(1)1411 2123 y Fd(y)1463 2226 y Fr(!)969 2599 y FC(34)p eop %%Page: 35 39 bop 831 473 a FC(+)886 400 y Fr( )978 463 y Fd(y)992 448 y Fb(\000)1034 407 y(\000)1075 365 y(\000)1088 353 y(\000)1023 364 y Fq(s)1039 369 y Fl(1)1144 463 y Fd(y)p 1142 442 4 125 v 1158 385 a Fq(s)1174 390 y Fl(2)1144 297 y Fd(y)1207 473 y FC(+)1315 463 y Fd(y)1330 448 y Fb(\000)1371 407 y(\000)1413 365 y(\000)1425 353 y(\000)1360 364 y Fq(s)1376 369 y Fl(2)1481 463 y Fd(y)p 1479 442 V 1495 385 a Fq(s)1511 390 y Fl(1)1481 297 y Fd(y)1533 400 y Fr(!)831 805 y FC(+)886 732 y Fr( )978 795 y Fd(y)p 998 797 125 4 v 1044 778 a Fq(s)1060 783 y Fl(2)1144 795 y Fd(y)992 780 y Fb(\000)1034 739 y(\000)1075 697 y(\000)1088 685 y(\000)1023 696 y Fq(s)1039 701 y Fl(1)1144 629 y Fd(y)1207 805 y FC(+)1315 795 y Fd(y)p 1336 797 V 1381 778 a Fq(s)1397 783 y Fl(1)1481 795 y Fd(y)1330 780 y Fb(\000)1371 739 y(\000)1413 697 y(\000)1425 685 y(\000)1360 696 y Fq(s)1376 701 y Fl(2)1481 629 y Fd(y)1533 732 y Fr(!#)1788 909 y FC(\(5.13\))346 1204 y(=)453 1170 y(1)p 446 1192 38 2 v 446 1238 a(3!)497 1145 y Fr(Z)539 1158 y Fq(t)521 1240 y Fo(0)562 1204 y Ft(ds)610 1211 y Fo(1)638 1145 y Fr(Z)680 1158 y Fq(t)662 1240 y Fo(0)703 1204 y Ft(ds)751 1211 y Fo(2)771 1131 y Fr(")854 1193 y Fd(y)p 875 1195 125 4 v 921 1177 a Fq(s)937 1182 y Fl(1)1021 1193 y Fd(y)p 1019 1173 4 125 v 1035 1115 a Fq(s)1051 1120 y Fl(2)1021 1027 y Fd(y)1083 1204 y FC(+)1192 1193 y Fd(y)p 1212 1195 125 4 v 1258 1177 a Fq(s)1274 1182 y Fl(2)1358 1193 y Fd(y)1206 1179 y Fb(\000)1248 1137 y(\000)1289 1096 y(\000)1301 1084 y(\000)1237 1095 y Fq(s)1253 1100 y Fl(1)1358 1027 y Fd(y)1421 1204 y FC(+)1529 1193 y Fd(y)1543 1179 y Fb(\000)1585 1137 y(\000)1626 1096 y(\000)1639 1084 y(\000)1574 1095 y Fq(s)1590 1100 y Fl(1)1695 1193 y Fd(y)p 1693 1173 4 125 v 1709 1115 a Fq(s)1725 1120 y Fl(2)1695 1027 y Fd(y)1747 1131 y Fr(#)1785 1204 y Ft(:)1788 1308 y FC(\(5.14\))148 1404 y(In)17 b(fact,)f(as)h(w)o(e)f (shall)h(pro)o(v)o(e,)e(the)i(same)e(t)o(yp)q(e)h(of)h(symmetriz)o(ation)d (can)j(b)q(e)g(applied)f(at)h(eac)o(h)75 1464 y(order)k Ft(N)26 b FC(of)21 b(the)g(iteration.)35 b(W)l(e)20 b(obtain)i(a)f(sum)f(o)o(v)o(er)g (all)g(trees)h Ft(T)27 b FC(of)21 b Ft(N)27 b FC(v)o(ertices.)33 b(A)o(t)20 b(eac)o(h)75 1525 y(v)o(ertex)15 b(sits)i(a)g(cop)o(y)g Ft(\036)496 1532 y Fq(i)510 1525 y FC(\()p Ft(\030)r FC(\))g(of)h(the)e (\014eld)h Ft(\036)p FC(\()p Ft(\030)r FC(\))g(and)g(to)h(eac)o(h)e(line)g (\()p Ft(ij)s FC(\))e Fs(2)h Ft(T)24 b FC(there)16 b(corresp)q(onds)i(a)75 1585 y(parameter)d Ft(s)331 1592 y Fq(ij)378 1585 y FC(and)h(an)h(op)q (erator)474 1707 y(\001)515 1687 y Fo(\()p Fq(ij)r Fo(\))515 1720 y Fq(s)531 1725 y Fm(ij)600 1707 y Fs(\021)666 1649 y Fr(Z)689 1743 y Fo(\003)724 1707 y Ft(d\030)782 1649 y Fr(Z)805 1743 y Fo(\003)840 1707 y Ft(d\030)888 1687 y Fp(0)958 1674 y Ft(\016)p 905 1696 129 2 v 905 1742 a(\016)r(\036)958 1749 y Fq(i)972 1742 y FC(\()p Ft(\030)r FC(\))1047 1659 y Fr(h)1066 1707 y Fs(\000)1121 1695 y FC(_)1105 1707 y Ft(C)1140 1714 y Fq(s)1156 1719 y Fm(ij)1186 1707 y FC(\()p Ft(\030)r(;)8 b(\030)1273 1687 y Fp(0)1285 1707 y FC(\))1304 1659 y Fr(i)1398 1674 y Ft(\016)p 1337 1696 144 2 v 1337 1742 a(\016)r(\036)1390 1749 y Fq(j)1408 1742 y FC(\()p Ft(\030)1450 1727 y Fp(0)1462 1742 y FC(\))1500 1707 y Ft(:)274 b FC(\(5)p Ft(:)p FC(15\))75 1831 y(Therefore)16 b(the)g(whole)g(tree)g(represen)o(ts)f(the)h(op)q(erator) 807 1927 y(\001)848 1907 y Fq(T)848 1939 y(s)p 848 1946 19 2 v 903 1927 a Fs(\021)995 1886 y Fr(Y)969 1980 y Fo(\()p Fq(ij)r Fo(\))p Fp(2)p Fq(T)1082 1927 y FC(\001)1123 1907 y Fo(\()p Fq(ij)r Fo(\))1123 1939 y Fq(s)1139 1944 y Fm(ij)1788 1927 y FC(\(5)p Ft(:)p FC(16\))75 2075 y(acting)j(on)g(the)f Ft(N)5 b FC(-\014eld)19 b(functional)780 2042 y Fr(Q)819 2055 y Fq(N)819 2088 y(i)p Fo(=1)887 2075 y Ft(V)915 2082 y Fo(00)953 2075 y FC(\()p Ft(\036)1001 2082 y Fq(i)1015 2075 y FC(\).)28 b(The)19 b(parameters)e Ft(s)1456 2082 y Fq(ij)1505 2075 y FC(m)o(ust)g(b)q(e)i(in)o (tegrated)75 2136 y(from)c(0)i(to)f Ft(t)p FC(,)g(a)g(fact)h(w)o(e)e(shall)i (represen)o(t)e(with)h(the)g(notation)1261 2100 y Fr(R)1289 2113 y Fq(t)1281 2148 y Fo(0)1312 2136 y Ft(d)1337 2118 y Fq(T)1365 2136 y Ft(s)p FC(.)21 b(The)16 b(\014nal)h(result)e(is)75 2234 y Fu(Prop)r(osition)j(5.1)24 b Fn(L)n(et)18 b Ft(V)587 2241 y Fo(00)644 2234 y Fn(b)n(e)i(a)f(smo)n(oth)f(functional.)29 b(Then)19 b(in)h(the)f(sense)h(of)f(formal)g(p)n(ower)75 2295 y(series)f(in)f(p)n(owers)g(of)h Ft(V)518 2302 y Fo(00)555 2295 y Fn(,)g(the)g(unique)h(solution)f(to)g(\(5.6\),)f(or)f(\(5.7\),)h(is) 502 2428 y Ft(V)541 2408 y Fo(cl)530 2441 y Fq(t)p Fo(0)583 2428 y FC(=)634 2387 y Fr(X)649 2479 y Fq(N)724 2395 y FC(1)p 708 2417 58 2 v 708 2463 a Ft(N)5 b FC(!)806 2387 y Fr(X)779 2479 y Fq(T)14 b Fo(on)9 b Fq(N)901 2370 y Fr(Z)942 2383 y Fq(t)924 2464 y Fo(0)965 2428 y Ft(d)990 2408 y Fq(T)1018 2428 y Ft(s)f FC(\001)1090 2408 y Fq(T)1090 2441 y(s)p 1090 2448 19 2 v 1144 2374 a(N)1133 2387 y Fr(Y)1131 2478 y Fq(i)p Fo(=1)1196 2428 y Ft(V)1224 2435 y Fo(00)1262 2428 y FC(\()p Ft(\036)1310 2435 y Fq(i)1324 2428 y FC(\))1343 2354 y Fr(\014)1343 2379 y(\014)1343 2403 y(\014)1343 2428 y(\014)1343 2453 y(\014)1357 2484 y Fq(\036)1378 2489 y Fm(i)1391 2484 y Fo(=)p Fq(\036)1464 2428 y Ft(;)310 b FC(\(5)p Ft(:)p FC(17\))969 2599 y(35)p eop %%Page: 36 40 bop 75 307 a Fn(wher)n(e)24 b Ft(T)31 b Fn(is)24 b(summe)n(d)g(over)h(al)r(l) g(c)n(onne)n(cte)n(d)h(tr)n(e)n(e)d(gr)n(aphs)h(on)g Ft(N)30 b Fn(vertic)n(es)25 b(lab)n(el)r(le)n(d)h(by)f Ft(i;)8 b(j)29 b FC(=)75 367 y(1)p Ft(;)8 b(:)g(:)g(:)g(;)g(N)21 b Fn(and)15 b(ther)n(e)h(is)f(one)i(inte)n(gr)n(al)f(over)g(a)f(p)n(ar)n(ameter)f Ft(s)1190 374 y Fq(ij)1236 367 y Fn(for)h(e)n(ach)h(line)h Ft(ij)h Fn(in)e(the)g(tr)n(e)n(e)f(gr)n(aph)75 428 y Ft(T)7 b Fn(.)148 542 y(R)n(emarks:)24 b FC(This)14 b(is)f(the)g(statemen)o(t)f(of)i (the)f(kno)o(wn)h(fact)f(that)h(tree)f(graphs)i(giv)o(e)d(the)h(classical)75 602 y(action)j(i.e.)f(the)h(solution)g(to)g(the)g(Hamilton-Jacobi)f (equation.)21 b(If)16 b(the)g(functional)f Ft(V)1686 609 y Fo(00)1740 602 y FC(is)h(of)h(the)75 662 y(form)190 627 y Fr(R)226 662 y Ft(d\030)11 b(\036)312 644 y Fq(n)336 662 y FC(\()p Ft(\030)r FC(\),)16 b(the)g(v)o(ertex)e(functions)j(are)f(represen)o(ted)f(b)o(y)g(a)i (set)f(of)g Ft(n)g FC(\\legs")h(asso)q(ciated)g(to)75 722 y(the)h(dumm)o(y)e (index)i Ft(\030)j FC(that)e(is)g(in)o(tegrated)f(o)o(v)o(er.)28 b(With)18 b(suc)o(h)g(represen)o(tation,)h(the)f(successiv)o(e)75 782 y(terms)d(of)h(\(5.8\))h(corresp)q(ond)g(to)g(the)f Ft(\036)804 764 y Fq(n)827 782 y FC(-)p Fn(tr)n(e)n(e)k FC(F)l(eynman)15 b(diagrams)g(\(Exercise)g(5.5.2\).)148 843 y(An)h(elegan)o(t)g(w)o(a)o(y)g (to)h(pro)o(v)o(e)e(\(5.17\))i(is)g(to)f(c)o(hec)o(k)f(that)i(it)f (satis\014es)g(the)h(di\013eren)o(tial)e(equation)75 903 y(\(5.7\).)33 b(This)20 b(is)g(sk)o(etc)o(hed)e(in)i(Exercise)e(5.5.4)j(for)f(the)g(fully)f (quan)o(tum)f(case.)33 b(Here)19 b(is)g(a)i(more)75 963 y(constructiv)o(e)15 b(pro)q(of.)75 1048 y Fu(Pro)r(of.)27 b FC(W)l(rite)754 1118 y Ft(g)777 1125 y Fq(N)811 1118 y FC(\()p Ft(t)p FC(\))g(=)981 1084 y(1)p 964 1106 58 2 v 964 1152 a Ft(N)5 b FC(!)1060 1077 y Fr(X)1035 1168 y Fq(T)15 b Fo(on)5 b Fq(N)1154 1118 y Ft(I)1180 1098 y Fq(T)1176 1130 y(t)1220 1118 y Ft(:)75 1241 y FC(W)l(e)20 b(shall)g(pro)q(of)i(b)o(y)e(induction)g(on)h Ft(N)k FC(that)c(it)f(tak)o(es) g(the)h(form)e(\(5.17\).)34 b(F)l(rom)20 b(the)g(previous)75 1301 y(discussion)13 b(w)o(e)g(kno)o(w)g(that)g Ft(g)618 1308 y Fo(1)652 1301 y FC(\(in)f(fact)h(also)h Ft(g)935 1308 y Fo(2)968 1301 y FC(and)g Ft(g)1083 1308 y Fo(3)1103 1301 y FC(\))f(has)h(the)e(form)g (giv)o(en)h(in)f(\(5.17\).)21 b(Assume)75 1361 y(it)16 b(is)g(so)h(for)f(all) g Ft(g)398 1368 y Fq(M)454 1361 y FC(with)g Ft(M)j Fs(\024)14 b Ft(N)i Fs(\000)11 b FC(1.)22 b(F)l(rom)15 b(the)h(in)o(tegral)f(equation)h (\(5.6\))h(w)o(e)f(ha)o(v)o(e:)342 1509 y Ft(g)365 1516 y Fq(N)399 1509 y FC(\()p Ft(t)p FC(\))41 b(=)576 1455 y Fq(N)t Fp(\000)p Fo(1)584 1467 y Fr(X)576 1559 y Fq(K)r Fo(=1)666 1475 y FC(1)p 666 1497 25 2 v 666 1543 a(2)704 1450 y Fr(Z)745 1463 y Fq(t)727 1544 y Fo(0)768 1509 y Ft(ds)824 1461 y Fr(\020)850 1509 y Fs(r)p Ft(g)915 1516 y Fq(K)949 1509 y FC(\()p Ft(s)p FC(\))8 b Ft(;)16 b Fs(r)p Ft(g)1113 1516 y Fq(N)t Fp(\000)p Fq(K)1206 1509 y FC(\()p Ft(s)p FC(\))1267 1461 y Fr(\021)1292 1536 y Fq(s)1788 1509 y FC(\(5.18\))496 1661 y(=)581 1627 y(1)p 581 1649 V 581 1695 a(2)619 1607 y Fq(N)t Fp(\000)p Fo(1)627 1619 y Fr(X)618 1711 y Fq(K)r Fo(=1)827 1627 y FC(1)p 709 1649 261 2 v 709 1695 a Ft(K)t FC(!\()p Ft(N)g Fs(\000)10 b Ft(K)t FC(\)!)1045 1619 y Fr(X)1012 1711 y Fq(T)1033 1716 y Fl(1)1060 1711 y Fo(on)f Fq(K)987 1746 y(T)1008 1751 y Fl(2)1035 1746 y Fo(on)g Fq(N)t Fp(n)p Fq(K)1177 1602 y Fr(Z)1218 1615 y Fq(t)1200 1696 y Fo(0)1241 1661 y Ft(ds)1297 1613 y Fr(\020)1323 1661 y Fs(r)p Ft(I)1391 1640 y Fq(T)1412 1645 y Fl(1)1387 1673 y Fq(s)1438 1661 y Ft(;)16 b Fs(r)p Ft(I)1536 1640 y Fq(T)1557 1645 y Fl(2)1532 1673 y Fq(s)1575 1613 y Fr(\021)1600 1688 y Fq(s)1632 1661 y Ft(:)142 b FC(\(5.19\))75 1847 y(A)o(t)14 b(this)g(p)q(oin)o(t)g(w)o(e)g(symmetriz)o (e,)d(as)k(ab)q(o)o(v)o(e,)f(among)h(all)e(p)q(ossible)i(w)o(a)o(ys)f(of)h (dividing)e(the)h(v)o(ertices)75 1907 y Ft(N)20 b Fs(\021)15 b(f)p FC(1)p Ft(;)8 b(:)g(:)g(:)g(;)g(N)d Fs(g)16 b FC(in)o(to)h(t)o(w)o(o)g (sets.)23 b(F)l(or)17 b(a)g(giv)o(en)f(in)o(teger)g Ft(K)t FC(,)h(there)f(are)1443 1859 y Fr(\020)1473 1885 y Fq(N)1473 1924 y(K)1510 1859 y Fr(\021)1552 1907 y FC(w)o(a)o(ys)g(to)i(c)o(ho)q(osing) 75 1973 y(a)f(set)f Ft(X)i Fs(\032)c(f)p FC(1)p Ft(;)8 b(:)g(:)g(:)f(;)h(N)d Fs(g)17 b FC(of)f(cardinalit)o(y)f Fs(j)p Ft(X)t Fs(j)f FC(=)g Ft(K)t FC(;)i(hence)455 2105 y Ft(g)478 2112 y Fq(N)512 2105 y FC(\()p Ft(t)p FC(\))27 b(=)666 2072 y(1)p 666 2094 25 2 v 666 2140 a(2)717 2072 y(1)p 700 2094 58 2 v 700 2140 a Ft(N)5 b FC(!)786 2064 y Fr(X)771 2156 y Fq(X)s Fp(\032)p Fq(N)933 2064 y Fr(X)900 2156 y Fq(T)921 2161 y Fl(1)947 2156 y Fo(on)k Fq(X)875 2190 y(T)896 2195 y Fl(2)923 2190 y Fo(on)g Fq(N)t Fp(n)p Fq(X)1064 2047 y Fr(Z)1105 2060 y Fq(t)1087 2141 y Fo(0)1128 2105 y Ft(ds)1184 2057 y Fr(\020)1210 2105 y Fs(r)p Ft(I)1278 2085 y Fq(T)1299 2090 y Fl(1)1274 2118 y Fq(s)1325 2105 y Ft(;)16 b Fs(r)p Ft(I)1423 2085 y Fq(T)1444 2090 y Fl(2)1419 2118 y Fq(s)1462 2057 y Fr(\021)1487 2132 y Fq(s)1519 2105 y Ft(:)255 b FC(\(5)p Ft(:)p FC(20\))75 2294 y(The)14 b(t)o(w)o(o)g(trees)g Ft(T)407 2301 y Fo(1)440 2294 y FC(and)h Ft(T)562 2301 y Fo(2)595 2294 y FC(and)g(the)e(connecting)h(b)q(ond)h(\(parametrized)d(b)o(y)i Ft(s)p FC(\))g(coming)f(from)g(the)75 2354 y(dot)j(pro)q(duct,)h(assem)o(ble) d(a)i(tree)f Ft(T)22 b FC(on)17 b Ft(N)5 b FC(.)21 b(Ho)o(w)o(ev)o(er,)14 b(the)i(same)e(tree)i Ft(T)22 b FC(app)q(ears)17 b(exactly)e Fn(twic)n(e)75 2414 y FC(in)i(the)f(sum)g(\(5.20\),)h(namely)e(in)i(the)f (partitions)h(\()p Ft(X)q(;)8 b(N)18 b Fs(n)11 b Ft(X)t FC(\))17 b(and)h(\()p Ft(N)e Fs(n)c Ft(X)q(;)c(X)t FC(\).)24 b(Therefore,)16 b(in)969 2599 y(36)p eop %%Page: 37 41 bop 75 307 a FC(passing)17 b(to)g(a)g(sum)e(o)o(v)o(er)g(trees)h Ft(T)k Fs(\032)14 b Ft(N)21 b FC(w)o(e)16 b(m)o(ust)f(cancel)g(the)h(factor)h (1)p Ft(=)p FC(2.)22 b(The)16 b(result)g(is)698 436 y Ft(g)721 443 y Fq(N)755 436 y FC(\()p Ft(t)p FC(\))27 b(=)925 402 y(1)p 908 424 58 2 v 908 470 a Ft(N)5 b FC(!)1004 394 y Fr(X)979 486 y Fq(T)15 b Fo(on)5 b Fq(N)1108 394 y Fr(X)1106 486 y Fq(b)p Fp(2)p Fq(T)1178 436 y Ft(I)1204 415 y Fq(T)1200 448 y(t)1231 436 y FC(\()p Ft(b)p FC(\))75 577 y(where)504 662 y Ft(I)530 642 y Fq(T)526 674 y(t)557 662 y FC(\()p Ft(b)p FC(\))27 b(=)709 604 y Fr(Z)750 617 y Fq(t)732 698 y Fo(0)773 662 y Ft(ds)821 669 y Fq(b)855 604 y Fr(Z)897 617 y Fq(s)913 623 y Fm(b)878 698 y Fo(0)939 662 y Ft(d)964 642 y Fq(T)5 b Fp(n)p Fq(b)1025 662 y Ft(s)j FC(\001)1097 642 y Fq(T)1097 674 y(s)p 1097 681 19 2 v 1150 608 a(N)1140 621 y Fr(Y)1138 712 y Fq(i)p Fo(=1)1203 662 y Ft(V)1231 669 y Fo(00)1269 662 y FC(\()p Ft(\036)1317 669 y Fq(i)1331 662 y FC(\))1350 587 y Fr(\014)1350 612 y(\014)1350 637 y(\014)1350 662 y(\014)1350 687 y(\014)1363 718 y Fq(\036)1384 723 y Fm(i)1398 718 y Fo(=)p Fq(\036)1470 662 y Ft(:)75 797 y FC(The)20 b(sum)f(o)o(v)o(er)h(all)f Ft(b)i Fs(2)f Ft(T)27 b FC(yields)19 b(full)g(in)o(tegrals)1048 761 y Fr(R)1076 774 y Fq(t)1067 809 y Fo(0)1099 797 y Ft(ds)1147 804 y Fq(ij)1197 797 y FC(for)i(all)f(\()p Ft(ij)s FC(\))g Fs(6)p FC(=)g Ft(b)g FC(b)q(ecause)g([0)p Ft(;)8 b(t)p FC(])1821 779 y Fq(N)1875 797 y FC(=)75 857 y Fs([)108 839 y Fq(N)108 869 y(i)p Fo(=1)167 857 y Fs(f)p Ft(s)215 864 y Fo(1)235 857 y Ft(;)g(:)g(:)g(:)f(;)h(s)367 864 y Fq(N)401 857 y Fs(j)p Ft(s)438 864 y Fq(j)474 857 y Fs(\024)17 b Ft(s)553 864 y Fq(i)583 857 y FC(for)g Ft(j)j FC(=)e(1)p Ft(;)8 b(:)g(:)g(:)g(;)g(N)d Fs(g)18 b FC(\(same)g(step)g(as)h(the)g (transition)f(b)q(et)o(w)o(een)g(\(5.13\))75 917 y(and)f(\(5.14\))g(ab)q(o)o (v)o(e\).)1393 b Fg(2)148 1001 y FC(W)l(e)20 b(ha)o(v)o(e)e(stated)i(Prop)q (osition)h(5.1)f(in)g(a)g(form)e(suitable)h(for)h(comparisons)g(with)f(the)h (fully)75 1061 y(quan)o(tum)d(case)g(\(Prop)q(osition)i(5.2)f(b)q(elo)o(w\).) 26 b(Ho)o(w)o(ev)o(er,)16 b(for)i(the)g(presen)o(t)f(classical)g(case)h(the)g (in-)75 1121 y(tegrals)j(o)o(v)o(er)e(the)h(parameters)f Ft(s)712 1128 y Fq(ij)763 1121 y FC(can)i(trivially)d(b)q(e)i(p)q(erformed)g(as)h(the) f(in)o(tegrand)g(in)g(\(5.17\))75 1182 y(factors)g(in)o(to)f(indep)q(enden)o (t)g(in)o(tegrals)818 1146 y Fr(R)846 1159 y Fq(t)838 1194 y Fo(0)869 1182 y Ft(ds)917 1189 y Fq(ij)956 1182 y FC([)p Fs(\000)1025 1169 y FC(_)1009 1182 y Ft(C)1044 1189 y Fq(s)1060 1194 y Fm(ij)1089 1182 y FC(])g(=)g Ft(C)1214 1189 y Fq(t)p Fo(0)1246 1182 y FC(.)31 b(Therefore,)20 b(adopting)g(the)f(con-)75 1242 y(v)o(en)o(tion)c(that)i(unlab)q(eled)f(lines)f(corresp)q(ond)i(to)g(co) o(v)m(ariances)f Ft(C)1283 1249 y Fq(t)p Fo(0)1315 1242 y FC(:)500 1347 y Fd(y)p 521 1348 125 4 v 124 w(y)725 1367 y Fs(\021)792 1309 y Fr(Z)833 1322 y Fq(t)815 1403 y Fo(0)856 1367 y Ft(ds)972 1347 y Fd(y)p 993 1348 V 1047 1333 a Fq(s)1138 1347 y Fd(y)1197 1367 y FC(=)1322 1347 y Fd(y)p 1342 1348 V 1390 1337 a Fq(t)p Fo(0)1488 1347 y Fd(y)1533 1367 y Ft(;)241 b FC(\(5)p Ft(:)p FC(21\))75 1494 y(w)o(e)16 b(can)g(write)g Ft(V)400 1476 y Fo(cl)389 1507 y Fq(t)p Fo(0)444 1494 y FC(as)h(a)f(sum)g(o)o(v)o(er)f(unlab) q(elled)g(tree-diagrams:)215 1638 y Ft(V)254 1617 y Fo(cl)243 1650 y Fq(t)p Fo(0)339 1638 y FC(=)435 1596 y Fr(X)449 1688 y Fq(N)525 1604 y FC(1)p 508 1626 58 2 v 508 1672 a Ft(N)5 b FC(!)606 1596 y Fr(X)580 1688 y Fq(T)14 b Fo(on)9 b Fq(N)702 1638 y FC(\001)743 1617 y Fq(T)743 1650 y(t)p Fo(0)801 1584 y Fq(N)790 1596 y Fr(Y)788 1687 y Fq(i)p Fo(=1)853 1638 y Ft(V)881 1645 y Fo(00)919 1638 y FC(\()p Ft(\036)967 1645 y Fq(i)981 1638 y FC(\))1000 1563 y Fr(\014)1000 1588 y(\014)1000 1613 y(\014)1000 1638 y(\014)1000 1663 y(\014)1014 1693 y Fq(\036)1035 1698 y Fm(i)1048 1693 y Fo(=)p Fq(\036)1121 1638 y Ft(;)339 1840 y FC(=)464 1820 y Fc(y)520 1840 y FC(+)599 1807 y(1)p 599 1829 25 2 v 599 1875 a(2)688 1820 y Fd(y)p 708 1821 125 4 v 124 w(y)468 2157 y FC(+)542 2123 y(1)p 535 2145 38 2 v 535 2191 a(3!)578 2084 y Fr(")661 2147 y Fd(y)p 682 2148 125 4 v 125 w(y)p 826 2126 4 125 v 828 1980 a(y)890 2157 y FC(+)999 2147 y Fd(y)1013 2132 y Fb(\000)1055 2090 y(\000)1096 2049 y(\000)1108 2037 y(\000)1165 2147 y Fd(y)p 1163 2126 V 1165 1980 a(y)1228 2157 y FC(+)1336 2147 y Fd(y)p 1357 2148 125 4 v 124 w(y)1350 2132 y Fb(\000)1392 2090 y(\000)1433 2049 y(\000)1446 2037 y(\000)1502 1980 y Fd(y)1554 2084 y Fr(#)1605 2157 y FC(+)36 b Fs(\001)8 b(\001)g(\001)22 b Ft(:)15 b FC(\(5.22\))75 2322 y Ff(5.3)70 b(F)-6 b(ormal)22 b(Series)g(for)i(the)e(Quan)n(tum)h (Action)75 2414 y FC(Let)h(us)g(turn)g(no)o(w)g(to)g(the)f(series)g(for)h (the)g(fully)e(quan)o(tum)g(action)i(de\014ned)g(b)o(y)f(\(5.3\).)43 b(The)75 2474 y(additional)15 b(ingredien)o(t)g(is)g(the)g(con)o(v)o(olution) g(b)o(y)g(Gaussian)h(measures.)k(T)l(o)c(understand)g(its)f(role,)969 2599 y(37)p eop %%Page: 38 42 bop 75 307 a FC(consider)21 b(\014rst)g(the)f(term)f Ft(\026)613 314 y Fq(t)p Fo(0)660 307 y Fs(\003)14 b Ft(V)727 314 y Fo(00)786 307 y FC(\(solution)21 b(to)g(the)g(linear)f(part)h(of)g(\(5.1\)\).)36 b(Assuming)19 b(for)75 367 y(simplicit)n(y)e(that)j Ft(V)438 374 y Fo(00)496 367 y FC(is)g(a)g(p)q(olynomial)f(in)g Ft(\036)h FC(w)o(e)f(can)h(ev)m(aluate)g Ft(\026)1350 374 y Fq(t)p Fo(0)1396 367 y Fs(\003)14 b Ft(V)1463 374 y Fo(00)1520 367 y FC(with)20 b(the)g(follo)o(wing)75 428 y(handy)d(form)o(ulas.)j(See)15 b(Exercise)g(4.3.3.)635 544 y Ft(\026)664 551 y Fq(t)p Fo(0)708 544 y Fs(\003)c Ft(V)772 551 y Fo(00)852 544 y FC(=)42 b(exp)o([)1025 510 y(1)p 1025 532 25 2 v 1025 578 a(2)1054 544 y(\001)1095 551 y Fq(t)p Fo(0)1127 544 y FC(])8 b Ft(V)1177 551 y Fo(00)851 670 y Fs(\021)932 629 y Fr(X)1022 636 y FC(1)p 1005 659 58 2 v 1005 704 a Ft(N)d FC(!)1076 610 y Fr(\022)1112 636 y FC(1)p 1112 659 25 2 v 1112 704 a(2)1141 670 y(\001)1182 677 y Fq(t)p Fo(0)1214 610 y Fr(\023)1245 618 y Fq(N)1286 670 y Ft(V)1314 677 y Fo(00)1788 670 y FC(\(5.23\))75 783 y(where)540 856 y(\001)581 863 y Fq(t)p Fo(0)641 856 y Fs(\021)708 797 y Fr(Z)731 891 y Fo(\003)766 856 y Ft(d\030)823 797 y Fr(Z)846 891 y Fo(\003)881 856 y Ft(d\030)929 835 y Fp(0)999 822 y Ft(\016)p 947 844 129 2 v 947 890 a(\016)r(\036)1000 897 y Fq(i)1013 890 y FC(\()p Ft(\030)r FC(\))1080 856 y Ft(C)1115 863 y Fq(t)p Fo(0)1147 856 y FC(\()p Ft(\030)r(;)j(\030)1234 835 y Fp(0)1247 856 y FC(\))1331 822 y Ft(\016)p 1271 844 144 2 v 1271 890 a(\016)r(\036)1324 897 y Fq(j)1341 890 y FC(\()p Ft(\030)1383 875 y Fp(0)1396 890 y FC(\))1433 856 y Ft(:)341 b FC(\(5)p Ft(:)p FC(24\))75 967 y(F)l(ollo)o(wing)17 b(the)g(ab)q(o)o(v)o(e)h(con)o(v)o(en)o(tion)e(of)i (using)g(lines)f(for)h(2-p)q(oin)o(t)g(correlations)g(\(\\propagators"\))75 1027 y(w)o(e)e(can)g(diagramatically)f(depict)439 1235 y Ft(\026)468 1242 y Fq(t)p Fo(0)512 1235 y Fs(\003)c Ft(V)576 1242 y Fo(00)677 1235 y FC(=)790 1215 y Fc(y)846 1235 y FC(+)925 1202 y(1)p 925 1224 25 2 v 925 1269 a(2)1010 1215 y Fd(y)977 1083 y(\013)p 977 1181 4 75 v 977 1208 a(\012)1039 1083 y(\010)p 1039 1181 V 1039 1208 a(\011)p 1003 1208 13 4 v 1003 1083 V 1052 1155 a Fq(t)p Fo(0)1109 1235 y FC(+)1188 1202 y(1)p 1188 1224 25 2 v 1188 1269 a(8)1289 1223 y Fd(y)1256 1092 y(\013)p 1256 1190 4 75 v 1256 1216 a(\012)1319 1092 y(\010)p 1319 1190 V 1319 1216 a(\011)p 1283 1216 13 4 v 1283 1092 V 1332 1163 a Fq(t)p Fo(0)1256 1233 y Fd(\013)p 1256 1331 4 75 v 1256 1357 a(\012)1319 1233 y(\010)p 1319 1331 V 1319 1357 a(\011)p 1283 1357 13 4 v 1283 1233 V 1332 1304 a Fq(t)p Fo(0)1389 1235 y FC(+)27 b Fs(\001)8 b(\001)g(\001)23 b Ft(:)239 b FC(\(5)p Ft(:)p FC(25\))75 1444 y(The)15 b(v)o(ertices)e(represen)o(t)h(functional)g (deriv)m(ativ)o(es)g(of)h Ft(V)1109 1451 y Fo(00)1147 1444 y FC(.)21 b(The)15 b(n)o(um)o(b)q(er)e(of)i(deriv)m(ativ)o(es)e(is)i(equal)75 1504 y(to)k(the)g(n)o(um)o(b)q(er)e(of)i(ends)h(of)f(lines)f(at)h(the)g(v)o (ertex.)28 b(The)19 b(next)f(term)f(in)i(the)f(iteration)h(of)g(\(5.3\))75 1564 y(in)o(v)o(olv)o(es)14 b(the)i(in)o(tegral)g(o)o(v)o(er)f Ft(s)h FC(of)152 1661 y Ft(\026)181 1668 y Fq(ts)223 1661 y Fs(\003)11 b FC(\()p Fs(r)p FC(\()p Ft(\026)368 1668 y Fq(s)p Fo(0)415 1661 y Fs(\003)g Ft(V)479 1668 y Fo(00)517 1661 y FC(\))d Ft(;)17 b Fs(r)p FC(\()p Ft(\026)665 1668 y Fq(s)p Fo(0)711 1661 y Fs(\003)11 b Ft(V)775 1668 y Fo(00)813 1661 y FC(\)\))851 1676 y Fq(s)897 1661 y FC(=)28 b([)o Ft(\026)1005 1668 y Fq(ts)1048 1661 y Fs(\003)11 b FC(\()p Ft(\026)1132 1668 y Fq(s)p Fo(0)1179 1661 y Fs(\012)g Ft(\026)1258 1668 y Fq(s)p Fo(0)1295 1661 y FC(\)])f Fs(\003)h FC(\()p Fs(r)p Ft(V)1463 1668 y Fo(00)1509 1661 y Ft(;)16 b Fs(r)p Ft(V)1609 1668 y Fo(00)1646 1661 y FC(\))1665 1668 y Fq(s)1697 1661 y Ft(:)77 b FC(\(5)p Ft(:)p FC(26\))75 1758 y(As)12 b(the)h(con)o(v)o(olution)e (of)i(Gaussians)h(is)f(again)g(a)g(Gaussian)h(whose)f(co)o(v)m(ariance)f (matrix)f(is)h(the)h(sum)75 1818 y(of)20 b(the)g(co)o(v)m(ariance)f(matrices) f(and)j(the)e(tensor)i(pro)q(duct)f(of)g(Gaussians)i(is)d(a)i(Gaussian)g (whose)75 1879 y(co)o(v)m(ariance)14 b(matrix)e(is)i(the)g(direct)f(sum)g(of) h(the)g(co)o(v)m(ariance)g(matrices,)e(w)o(e)i(see)f(that)i(the)f(measure)701 1976 y Ft(\026)730 1983 y Fq(ts)759 1987 y Fl(max)847 1976 y Fs(\021)28 b Ft(\026)943 1983 y Fq(ts)985 1976 y Fs(\003)11 b FC(\()p Ft(\026)1069 1983 y Fq(s)p Fo(0)1117 1976 y Fs(\012)g Ft(\026)1196 1983 y Fq(s)p Fo(0)1232 1976 y FC(\))22 b Ft(:)501 b FC(\(5)p Ft(:)p FC(27\))75 2073 y(is)15 b(in)f(fact)h(a)g(new)g(Gaussian)h (measure)d(\(the)h(reasons)i(for)f(the)g(subscript)f Ft(s)1464 2080 y Fo(max)1547 2073 y FC(will)f(b)q(e)i(seen)g(more)75 2133 y(clearly)g(b)q(elo)o(w\).)23 b(This)17 b(measure)e(in)o(tegrates)h(o)o (v)o(er)g(v)m(ariables)h(\()p Ft(\036)1291 2140 y Fq(i)1305 2133 y FC(\()p Ft(\030)r FC(\)\))1385 2140 y Fq(i)p Fo(=1)p Fq(;)p Fo(2)1489 2133 y FC(where)f(the)g(index)g Ft(i)h FC(is)75 2193 y(there)f(b)q(ecause)h(there)e(are)i(t)o(w)o(o)f(factors)h(of)g Ft(V)922 2200 y Fo(00)977 2193 y FC(and)g(eac)o(h)f(of)g(them)f(is)i(n)o(um)o (b)q(ered)d(b)o(y)i(a)h(v)m(alue)f(of)75 2253 y Ft(i)i FC(indicating)f(that)h (the)g Ft(i)549 2235 y Fo(th)602 2253 y FC(factor)g(is)g(ev)m(aluated)g(on)g (the)g(\014eld)g Ft(\036)1306 2260 y Fq(i)1320 2253 y FC(\()p Ft(\030)r FC(\).)27 b(F)l(rom)16 b(\(5.27\))j(the)f(reader)75 2313 y(ma)o(y)d(c)o(hec)o(k)f(that)j(the)f(co)o(v)m(ariance)g(of)g Ft(\026)822 2320 y Fq(ts)851 2324 y Fl(max)928 2313 y FC(is)349 2382 y Fr(Z)399 2441 y Ft(d\026)453 2448 y Fq(ts)482 2452 y Fl(max)551 2441 y Ft(\036)580 2448 y Fq(i)594 2441 y FC(\()p Ft(\030)r FC(\))p Ft(\036)684 2448 y Fq(j)703 2441 y FC(\()p Ft(\030)745 2420 y Fp(0)757 2441 y FC(\))e(=)842 2368 y Fr(\()896 2410 y Ft(C)931 2417 y Fq(t)p Fo(0)963 2410 y FC(\()p Ft(\030)r(;)8 b(\030)1050 2392 y Fp(0)1063 2410 y FC(\))42 b Fs(f)p Ft(i;)8 b(j)s Fs(g)13 b FC(=)h Fs(f)p FC(1)p Ft(;)8 b FC(1)p Fs(g)14 b FC(or)g Fs(f)p FC(2)p Ft(;)8 b FC(2)p Fs(g)896 2471 y Ft(C)931 2478 y Fq(ts)962 2471 y FC(\()p Ft(\030)r(;)g(\030)1049 2452 y Fp(0)1062 2471 y FC(\))43 b Fs(f)p Ft(i;)8 b(j)s Fs(g)13 b FC(=)h Fs(f)p FC(1)p Ft(;)8 b FC(2)p Fs(g)14 b Ft(:)1788 2441 y FC(\(5)p Ft(:)p FC(28\))969 2599 y(38)p eop %%Page: 39 43 bop 148 307 a FC(If)15 b(w)o(e)f(no)o(w)i(pro)q(ceed)f(as)g(in)g(\(5.25\),)g (w)o(e)g(obtain)h(that)f(the)g(second-order)g(term)f(is)g(the)h(in)o(tegral) 75 367 y(o)o(v)o(er)g Ft(s)i FC(of)250 466 y Ft(\026)279 473 y Fq(ts)308 477 y Fl(max)380 466 y Fs(\003)11 b FC(\()p Fs(r)p Ft(V)505 473 y Fo(00)551 466 y Ft(;)16 b Fs(r)p Ft(V)651 473 y Fo(00)688 466 y FC(\))707 473 y Fq(s)333 603 y FC(=)63 b Ft(\026)463 610 y Fq(ts)492 614 y Fl(max)564 603 y Fs(\003)605 555 y Fr(n)632 603 y FC(\001)673 610 y Fq(s)691 603 y FC([)p Ft(V)733 610 y Fo(00)771 603 y FC(\()p Ft(\036)819 610 y Fo(1)838 603 y FC(\))p Ft(V)885 610 y Fo(00)923 603 y FC(\()p Ft(\036)971 610 y Fo(2)990 603 y FC(\)])1023 555 y Fr(o)1051 554 y(\014)1051 579 y(\014)1051 603 y(\014)1064 630 y Fq(\036)1085 635 y Fm(i)1099 630 y Fo(=)p Fq(\036)333 774 y FC(=)g Ft(\026)463 781 y Fq(ts)492 785 y Fl(max)588 774 y Fs(\003)683 753 y Fd(y)p 704 755 125 4 v 758 739 a Fq(s)849 753 y Fd(y)333 926 y FC(=)434 865 y Fr(\024)456 926 y FC(1)11 b(+)545 892 y(1)p 545 914 25 2 v 545 960 a(2)574 926 y(\001)615 933 y Fq(ts)644 937 y Fl(max)716 926 y FC(+)g Ft(:)d(:)g(:)822 865 y Fr(\025)879 926 y Fs(\003)974 905 y Fd(y)p 995 907 125 4 v 1049 892 a Fq(s)1140 905 y Fd(y)333 1183 y FC(=)460 1163 y Fd(y)p 481 1164 V 535 1149 a Fq(s)626 1163 y Fd(y)685 1183 y FC(+)35 b(2)12 b Fs(\001)824 1150 y FC(1)p 824 1172 25 2 v 824 1218 a(2)909 1163 y Fd(y)876 1031 y(\013)p 876 1129 4 75 v 876 1156 a(\012)938 1031 y(\010)p 938 1129 V 938 1156 a(\011)p 902 1156 13 4 v 902 1031 V 952 1094 a Fq(t)p Fo(0)p 929 1164 125 4 v 984 1149 a Fq(s)1075 1163 y Fd(y)1126 1183 y FC(+)36 b(2)11 b Fs(\001)1265 1150 y FC(1)p 1265 1172 25 2 v 1265 1218 a(2)1354 1163 y Fd(y)1360 1127 y(\017)p 1360 1167 4 9 v 1360 1202 a(\016)1510 1127 y(\014)p 1510 1167 V 1510 1202 a(\015)p 1395 1202 84 4 v 1395 1127 V 1422 1115 a Fq(ts)1429 1228 y(s)1520 1163 y Fd(y)1578 1183 y FC(+)27 b Fs(\001)8 b(\001)g(\001)23 b Ft(:)1788 1271 y FC(\(5.29\))75 1371 y(Tw)o(o)d(remarks)e(are)h(in)g(order.)30 b(First,)20 b(note)f(the)g(extra)g(factors)h(2)g(in)f(fron)o(t)g(of)g(the)g(second)h(and) 75 1431 y(third)e(diagram.)29 b(F)l(or)18 b(the)h(second)g(diagram,)f(this)h (corresp)q(onds)h(to)f(the)f(functional)h(Laplacian)75 1491 y(acting)d(either)f(on)i(the)f(\014rst)g(or)h(on)g(the)f(second)g(v)o(ertex.) k(F)l(or)c(the)g(third)g(diagram,)f(the)h(action)g(of)75 1551 y(the)j(Laplacian)g(has)h(also)g(t)o(w)o(o)f(p)q(ossibilities)f(corresp)q (onding)i(to)f(the)g(\014rst)g Ft(\016)r(=\016)r(\036)f FC(acting)h(on)h(the) 75 1611 y(\014rst)15 b(or)h(the)f(second)g(v)o(ertex.)k(The)d(second)f (remark)e(concerns)i(the)g(lab)q(els)g(of)h(the)f(di\013eren)o(t)f(lines:)75 1671 y(they)19 b(refer)g(to)h(the)f(relev)m(an)o(t)f(scales)i(of)f(the)h (\015uctuations.)31 b(The)20 b(lab)q(el)f(\\)p Ft(ts)p FC(")h(corresp)q(onds) g(to)g(a)75 1732 y(co)o(v)m(ariance)15 b Ft(C)345 1739 y Fq(ts)389 1732 y FC(=)f Fs(\000)p FC([)p Ft(C)529 1739 y Fp(1)p Fq(t)587 1732 y Fs(\000)8 b Ft(C)669 1739 y Fp(1)p Fq(s)723 1732 y FC(])14 b(in)o(v)o(olving)g(scales)h(b)q(et)o(w)o(een)f Ft(s)h FC(and)g Ft(t)p FC(.)21 b(On)15 b(the)g(other)g(hand,)g(a)75 1792 y(line)f(with)h(a)g (single)g(lab)q(el)g Ft(s)g FC(represen)o(ts)f(a)i(co)o(v)m(ariance)e Fs(\000)1162 1779 y FC(_)1146 1792 y Ft(C)1181 1799 y Fp(1)p Fq(s)1250 1792 y FC(that)h(is,)g(in)o(v)o(olving)e(only)i(the)g(scale)75 1852 y Ft(s)p FC(.)148 1912 y(One)h(conclusion)h(from)e(what)j(w)o(e)e(ha)o (v)o(e)f(done)i(up)g(to)g(no)o(w)g(is)f(that)h(the)g(role)f(of)h(the)f(con)o (v)o(olu-)75 1972 y(tions)g(with)f(resp)q(ect)h(to)g(Gaussian)h(measures)d (is)i(to)g(put)g(in)f(the)h(lo)q(ops)g(of)g(the)g(F)l(eynman)e(graphs.)75 2033 y(In)k(addition,)g(w)o(e)g(start)h(to)f(see)g(that)g(ev)o(en)f(when)i (the)e(iteration)h(of)h(\(5.3\))f(yields)f(man)o(y)g(succes-)75 2093 y(siv)o(e)g(con)o(v)o(olutions)h(with)h(resp)q(ect)f(to)h(di\013eren)o (t)f(Gaussian)h(measures,)f(the)g(magic)f(of)i(Gaussian)75 2153 y(calculus)c(allo)o(ws)g(us)h(to)g(com)o(bine)d(them)h(in)o(to)h(an)h(o) o(v)o(erall)e(con)o(v)o(olution)g(with)i(resp)q(ect)f(to)h Fn(one)j FC(big)75 2213 y(Gaussian)f(measure.)j(T)l(o)c(understand)g(this)g (b)q(etter)f(it)g(is)h(p)q(erhaps)g(con)o(v)o(enien)o(t)e(to)i(con)o(tin)o (ue)e(our)75 2273 y(p)q(edestrian)k(analysis)g(to)g(one)g(further)g(order.)29 b(Tw)o(o)20 b(iterations)e(of)i(the)e(in)o(tegral)h(term)e(in)h(\(5.3\))75 2334 y(yield)d(a)i(term)d(corresp)q(onding)j(to)g(\(5.11\))g(whic)o(h)e(is) 191 2396 y Fr(Z)232 2409 y Fq(t)214 2491 y Fo(0)256 2455 y Ft(ds)304 2462 y Fo(1)340 2396 y Fr(Z)382 2409 y Fq(s)398 2414 y Fl(1)363 2491 y Fo(0)426 2455 y Ft(ds)474 2462 y Fo(2)502 2455 y Ft(\026)531 2462 y Fq(ts)560 2467 y Fl(1)591 2455 y Fs(\003)627 2407 y Fr(\020)652 2455 y Fs(r)p Ft(\026)723 2462 y Fq(s)739 2467 y Fl(1)756 2462 y Fo(0)787 2455 y Fs(\003)c Ft(V)851 2462 y Fo(00)897 2455 y Ft(;)16 b Fs(r)p Ft(\026)998 2462 y Fq(s)1014 2467 y Fl(1)1032 2462 y Fq(s)1048 2467 y Fl(2)1078 2455 y Fs(\003)11 b FC(\()p Fs(r)p Ft(\026)1204 2462 y Fq(s)1220 2467 y Fl(2)1238 2462 y Fo(0)1269 2455 y Fs(\003)g Ft(V)1333 2462 y Fo(00)1379 2455 y Ft(;)16 b Fs(r)p Ft(\026)1480 2462 y Fq(s)1496 2467 y Fl(2)1513 2462 y Fo(0)1544 2455 y Fs(\003)11 b Ft(V)1608 2462 y Fo(00)1646 2455 y FC(\))1665 2469 y Fq(s)1681 2474 y Fl(2)1700 2407 y Fr(\021)1725 2482 y Fq(s)1741 2487 y Fl(1)1783 2455 y Ft(:)969 2599 y FC(39)p eop %%Page: 40 44 bop 75 307 a FC(The)23 b(deriv)m(ativ)o(es)f(with)h(resp)q(ect)g(to)h(\010)g (can)f(b)q(e)h(comm)o(ute)o(d)d(through)j(the)f(con)o(v)o(olutions)g(and)75 367 y(p)q(erformed)15 b(on)i Ft(V)404 374 y Fo(00)459 367 y FC(to)g(obtain)g(sev)o(eral)e(terms,)g(lab)q(elled)g(as)j(b)q(efore)e(b)o(y)g (tree)g(graphs.)24 b(The)16 b(term)75 428 y(that)h(corresp)q(onds)g(to)g(the) f(\014rst)g(tree)g(graph)h(in)f(\(5.11\))h(is)481 496 y Fr(Z)522 509 y Fq(t)504 590 y Fo(0)545 555 y Ft(ds)593 562 y Fo(1)630 496 y Fr(Z)672 509 y Fq(s)688 514 y Fl(1)653 590 y Fo(0)716 555 y Ft(ds)764 562 y Fo(2)800 555 y FC([)p Ft(\026)843 562 y Fq(ts)872 567 y Fl(1)903 555 y Fs(\003)11 b FC(\()p Ft(\026)987 562 y Fq(s)1003 567 y Fl(1)1021 562 y Fo(0)1051 555 y Fs(\012)g FC(\()p Ft(\026)1149 562 y Fq(s)1165 567 y Fl(1)1183 562 y Fq(s)1199 567 y Fl(2)1230 555 y Fs(\003)g FC(\()p Ft(\026)1314 562 y Fq(s)1330 567 y Fl(2)1348 562 y Fo(0)1378 555 y Fs(\012)g Ft(\026)1457 562 y Fq(s)1473 567 y Fl(2)1491 562 y Fo(0)1511 555 y FC(\)\)\)])486 669 y Fs(\003)519 621 y Fr(\020)544 669 y Fs(r)p Ft(V)614 676 y Fo(00)659 669 y Ft(;)17 b FC(\()p Fs(rr)p Ft(V)821 676 y Fo(00)866 669 y Ft(;)f Fs(r)p Ft(V)966 676 y Fo(00)1003 669 y FC(\))1022 684 y Fq(s)1038 689 y Fl(2)1058 621 y Fr(\021)1083 696 y Fq(s)1099 701 y Fl(1)1118 607 y Fr(\014)1118 632 y(\014)1118 657 y(\014)1118 682 y(\014)1132 709 y Fq(\036)1153 714 y Fl(1)1170 709 y Fo(=)p Fq(\036)1218 714 y Fl(2)1236 709 y Fo(=)p Fq(\036)1284 714 y Fl(3)1302 709 y Fo(=)p Fq(\036)1374 669 y Ft(:)400 b FC(\(5.30\))75 801 y(As)12 b(for)h(the)f(second-order)h (term,)e(w)o(e)g(see)h(that)h Ft(\026)975 808 y Fq(ts)1004 812 y Fl(max)1079 801 y Fs(\021)g FC([)p Ft(\026)1174 808 y Fq(ts)1203 813 y Fl(1)1234 801 y Fs(\003)e FC(\()p Ft(\026)1318 808 y Fq(s)1334 813 y Fl(1)1352 808 y Fo(0)1382 801 y Fs(\012)g FC(\()p Ft(\026)1480 808 y Fq(s)1496 813 y Fl(1)1514 808 y Fq(s)1530 813 y Fl(2)1561 801 y Fs(\003)g FC(\()p Ft(\026)1645 808 y Fq(s)1661 813 y Fl(2)1679 808 y Fo(0)1710 801 y Fs(\012)f Ft(\026)1788 808 y Fq(s)1804 813 y Fl(2)1822 808 y Fo(0)1842 801 y FC(\)\)\)])75 862 y(is)15 b(a)g(new)g(Gaussian)h(measure|whic)o(h)d(in) o(tegrates)i(o)o(v)o(er)f(v)m(ariables)g(\()p Ft(\036)1404 869 y Fq(i)1418 862 y FC(\()p Ft(\030)r FC(\)\))1498 869 y Fq(i)p Fo(=1)p Fq(;)p Fo(2)p Fq(;)p Fo(3)1613 862 y FC(|with)h(co)o(v)m(ari-) 75 922 y(ance)403 1045 y Fr(Z)453 1104 y Ft(d\026)507 1111 y Fq(ts)536 1115 y Fl(max)605 1104 y Ft(\036)634 1111 y Fq(i)648 1104 y FC(\()p Ft(\030)r FC(\))p Ft(\036)738 1111 y Fq(j)757 1104 y FC(\()p Ft(\030)799 1083 y Fp(0)811 1104 y FC(\))42 b(=)952 979 y Fr(8)952 1016 y(>)952 1029 y(>)952 1041 y(>)952 1054 y(<)952 1128 y(>)952 1141 y(>)952 1153 y(>)952 1166 y(:)1010 1013 y Ft(C)1045 1020 y Fq(t)p Fo(0)1077 1013 y FC(\()p Ft(\030)r(;)8 b(\030)1164 995 y Fp(0)1177 1013 y FC(\))57 b Ft(i)14 b FC(=)f Ft(j)1010 1073 y(C)1045 1080 y Fq(ts)1074 1085 y Fl(2)1093 1073 y FC(\()p Ft(\030)r(;)8 b(\030)1180 1055 y Fp(0)1193 1073 y FC(\))41 b Fs(f)p Ft(i;)8 b(j)s Fs(g)13 b FC(=)h Fs(f)p FC(2)p Ft(;)8 b FC(3)p Fs(g)1010 1133 y Ft(C)1045 1140 y Fq(ts)1074 1145 y Fl(1)1093 1133 y FC(\()p Ft(\030)r(;)g(\030)1180 1115 y Fp(0)1193 1133 y FC(\))41 b Fs(f)p Ft(i;)8 b(j)s Fs(g)13 b FC(=)h Fs(f)p FC(1)p Ft(;)8 b FC(2)p Fs(g)1010 1193 y Ft(C)1045 1200 y Fq(ts)1074 1205 y Fl(1)1093 1193 y FC(\()p Ft(\030)r(;)g(\030)1180 1175 y Fp(0)1193 1193 y FC(\))41 b Fs(f)p Ft(i;)8 b(j)s Fs(g)13 b FC(=)h Fs(f)p FC(1)p Ft(;)8 b FC(3)p Fs(g)1788 1104 y FC(\(5.31\))872 1325 y Fs(\021)41 b Ft(C)987 1332 y Fq(ts)1016 1336 y Fl(max)1076 1325 y FC(\()p Ft(\030)r(;)8 b(\030)1163 1305 y Fp(0)1176 1325 y FC(\))14 b Ft(;)565 b FC(\(5.32\))75 1430 y(where)19 b Ft(s)242 1437 y Fo(max)328 1430 y FC(is)g(the)g(maxim)n(um)c(v)m(alue)k(of)g(the)g(b)q (ond)g(parameter)f(\(in)h(this)g(case)f(the)h(maxim)o(um)75 1490 y(of)h Ft(s)157 1497 y Fo(1)196 1490 y FC(and)g Ft(s)317 1497 y Fo(2)337 1490 y FC(\))f(found)h(along)h(an)o(y)e(path)h(connecting)f (the)h(v)o(ertices)d Ft(i)p FC(,)j Ft(j)i FC(with)e Ft(i)f Fs(6)p FC(=)g Ft(j)s FC(.)31 b(If)19 b Ft(i)g FC(=)h Ft(j)75 1551 y(s)98 1558 y Fo(max)180 1551 y Fs(\021)13 b FC(0.)148 1611 y(This)k(discussion)f(can)g(b)q(e)h(extended)e(to)i(all)e(orders)i(to)g (obtain)75 1719 y Fu(Prop)r(osition)h(5.2)24 b Fn(L)n(et)16 b Ft(V)585 1726 y Fo(00)639 1719 y Fn(b)n(e)g(smo)n(oth)f(functional)j(of)e (limite)n(d)h(gr)n(owth.)k(Then)c(in)g(the)f(sense)i(of)75 1779 y(formal)f(p)n(ower)g(series)h(in)g(p)n(owers)f(of)g Ft(V)813 1786 y Fo(00)851 1779 y Fn(,)g(a)g(solution)i(to)e(\(5.1\),)g(or)g(\(5.3\),)g (is)389 1921 y Ft(V)417 1928 y Fq(t)p Fo(0)463 1921 y FC(=)515 1880 y Fr(X)529 1972 y Fq(N)605 1888 y FC(1)p 588 1910 58 2 v 588 1956 a Ft(N)5 b FC(!)686 1880 y Fr(X)659 1972 y Fq(T)15 b Fo(on)9 b Fq(N)782 1863 y Fr(Z)823 1876 y Fq(t)805 1957 y Fo(0)846 1921 y Ft(d)871 1901 y Fq(T)899 1921 y Ft(s)f(\026)959 1928 y Fq(ts)988 1932 y Fl(max)1060 1921 y Fs(\003)1101 1848 y Fr(\()1134 1921 y FC(\001)1175 1901 y Fq(T)1175 1934 y(s)p 1175 1941 19 2 v 1224 1867 a(N)1213 1880 y Fr(Y)1211 1971 y Fq(i)p Fo(=1)1276 1921 y Ft(V)1304 1928 y Fo(00)1342 1921 y FC(\()p Ft(\036)1390 1928 y Fq(i)1404 1921 y FC(\))1423 1848 y Fr(\))1456 1847 y(\014)1456 1872 y(\014)1456 1896 y(\014)1456 1921 y(\014)1456 1946 y(\014)1470 1977 y Fq(\036)1491 1982 y Fm(i)1504 1977 y Fo(=)p Fq(\036)1577 1921 y Ft(;)197 b FC(\(5)p Ft(:)p FC(33\))75 2070 y Fn(wher)n(e)18 b Ft(\026)242 2077 y Fq(ts)271 2081 y Fl(max)349 2070 y Fn(is)f(the)h(Gaussian)f(me)n(asur)n(e)g (on)h(the)g(\014elds)g Fs(f)p Ft(\036)1213 2077 y Fq(i)1227 2070 y FC(\()p Ft(\030)r FC(\))p Fs(g)1344 2076 y Fq(\030)q Fp(2)p Fo(\003)1319 2110 y(1)p Fp(\024)p Fq(i)p Fp(\024)p Fq(N)1459 2070 y Fn(with)g(c)n(ovarianc)n(e)530 2230 y Ft(C)565 2237 y Fq(ts)594 2241 y Fl(max)655 2230 y FC(\(\()p Ft(\030)r(;)8 b(i)p FC(\))p Ft(;)g FC(\()p Ft(\030)838 2210 y Fp(0)850 2230 y Ft(;)g(j)s FC(\)\))14 b(=)998 2172 y Fr(Z)1040 2185 y Fq(t)1021 2266 y(s)1037 2270 y Fl(max)1106 2230 y Ft(ds)1171 2182 y Fr(h)1191 2230 y Fs(\000)1246 2218 y FC(_)1230 2230 y Ft(C)1265 2237 y Fq(s)1283 2230 y FC(\()p Ft(\030)r(;)8 b(\030)1370 2210 y Fp(0)1383 2230 y FC(\))1402 2182 y Fr(i)1444 2230 y Ft(:)330 b FC(\(5)p Ft(:)p FC(34\))75 2354 y Fn(Her)n(e)17 b Ft(s)214 2361 y Fo(max)299 2354 y Fn(is)g(the)g(lar)n(gest)h(p)n(ar)n(ameter)d Ft(s)836 2361 y Fq(k)q(l)885 2354 y Fn(enc)n(ounter)n(e)n(d)j(on)f(the)h (unique)g(p)n(ath)f(in)g(the)h(tr)n(e)n(e)e(gr)n(aph)75 2414 y Ft(T)28 b Fn(which)23 b(le)n(ads)f(fr)n(om)e(vertex)k Ft(i)d Fn(to)h(vertex)i Ft(j)s Fn(.)35 b(This)22 b(solution)h(is)f(unique)h(within)g (the)f(class)h(of)75 2474 y(formal)17 b(p)n(ower)g(series)h(whose)g(c)n(o)n (e\016cients)g(ar)n(e)f(smo)n(oth)g(and)g(have)h(limite)n(d)g(gr)n(owth.)969 2599 y FC(40)p eop %%Page: 41 45 bop 75 307 a Fu(Pro)r(of.)27 b FC(A)19 b(direct)f(v)o(eri\014cation)g(that)h (\(5.33\))h(is)e(the)h(unique)f(formal-series)g(solution)h(with)g(the)75 367 y(prescrib)q(ed)d(initial)e(v)m(alue)i Ft(V)604 374 y Fo(00)658 367 y FC(is)g(left)g(as)h(an)f(exercise)f(\(Problem)f(5.5.4\).)22 b(W)l(e)16 b(will)f(b)q(e)h(pro)o(ving)g(a)75 428 y(closely)f(related)h (result)f(when)i(w)o(e)f(come)e(to)j(Prop)q(osition)g(6.2.)608 b Fg(2)148 511 y FC(The)17 b(in)o(tegration)h(o)o(v)o(er)e Ft(s)p 604 518 23 2 v 17 w FC(in)h(\(5.33\))h(is)f(not)h(as)g(simple)d(as)j (for)g(the)f(classical)f(case)i(\(see)f(com-)75 572 y(men)o(t)k(at)i(the)g (end)g(of)g(Prop)q(osition)h(5.1)f(ab)q(o)o(v)o(e\).)41 b(The)23 b(reason)h(is)e(that)i(the)e(measure)g Ft(\026)1823 579 y Fq(ts)1852 583 y Fl(max)75 632 y FC(com)o(bines)16 b(the)i(di\013eren)o(t)g(parameters)f Ft(s)845 639 y Fq(ij)893 632 y FC(in)h(a)h(non-factorizable)f(manner.)26 b(The)18 b(net)g(result)g(is)75 692 y(that)c(in)f(passing)h(to)g(unlab)q (elled)f(F)l(eynman)e(diagrams)i(\(that)h(is)f(to)h(diagrams)f(where)g(all)g (the)g(lines)75 752 y(corresp)q(onds)k(to)e(co)o(v)m(ariances)g Ft(C)689 759 y Fq(t)p Fo(0)722 752 y FC(\),)g(one)g(obtains)h(extra)g(com)o (binatorial)d(factors.)22 b(F)l(or)15 b(example,)75 812 y(the)h Ft(s)p FC(-in)o(tegration)g(of)h(\(5.29\))g(yields)337 1020 y Ft(V)365 1027 y Fq(t)p Fo(0)428 1020 y FC(=)556 999 y Fd(y)p 576 1000 125 4 v 124 w(y)780 1020 y FC(+)909 999 y Fd(y)876 868 y(\013)p 876 966 4 75 v 876 992 a(\012)939 868 y(\010)p 939 966 V 939 992 a(\011)p 903 992 13 4 v 903 868 V 930 1000 125 4 v 1075 999 a(y)1127 1020 y FC(+)1205 986 y(1)p 1205 1008 25 2 v 1205 1054 a(2)1294 999 y Fd(y)1301 963 y(\017)p 1301 1003 4 9 v 1301 1038 a(\016)1450 963 y(\014)p 1450 1003 V 1450 1038 a(\015)p 1335 1038 84 4 v 1335 963 V 1460 999 a(y)1518 1020 y FC(+)28 b Fs(\001)8 b(\001)g(\001)146 b FC(\(5)p Ft(:)p FC(35\))75 1141 y(\(Exercise)15 b(5.5.1\).)75 1284 y Ff(5.4)70 b(Con)n(v)n(ergence)22 b(of)h(the)g(F)-6 b(ormal)23 b(Series)75 1377 y FC(W)l(e)d(\014rst)h(emphasize)e(that)i(the)f(ab)q(o)o(v)o(e)h(formal) e(series)h(are)h(expansions)g(in)f(p)q(o)o(w)o(ers)h(of)g Ft(V)1790 1384 y Fo(00)1828 1377 y FC(.)34 b(A)75 1437 y(more)16 b(precise)h(w)o(a)o(y) g(to)h(state)f(this)h(is)f(to)h(replace)f(throughout)i Ft(V)1300 1444 y Fo(00)1353 1437 y Fs(!)d Ft(z)r(V)1472 1444 y Fo(00)1510 1437 y FC(,)h(in)g(whic)o(h)g(case)h(the)75 1497 y Ft(N)5 b FC(-tree)17 b(term)e(b)q(ecomes)g(the)i(co)q(e\016cien)o(t)e(of)i Ft(z)939 1479 y Fq(N)989 1497 y FC(in)f(the)h(resulting)f(expansion.)23 b(When)17 b(are)g(these)75 1557 y(series)f(con)o(v)o(ergen)o(t?)148 1617 y(\(i\))h(If)g Ft(V)295 1624 y Fo(00)350 1617 y FC(is)g(a)g(lo)q(cal)g (p)q(olynomial)f(in)h Ft(\036)g FC(with)g(constan)o(t)h(or)f(globally)g(b)q (ounded)h(co)q(e\016cien)o(ts,)75 1678 y(then)d(the)f(tree)g(expansion)h (\(5.17\))g(con)o(v)o(erges)930 1660 y Fo(5)964 1678 y FC(for)g Ft(t)f FC(small)f(dep)q(ending)i(on)g(sup)9 b Fs(j)p Ft(\036)p FC(\()p Ft(\030)r FC(\))p Fs(j)p FC(.)20 b(One)15 b(w)o(a)o(y)75 1738 y(to)g(understand)h(this)f(is)g(to)g(note)g(that)h(there)e(is)h(a)1026 1718 y Fo(1)p 1014 1726 42 2 v 1014 1755 a Fq(N)t Fo(!)1076 1738 y FC(in)f(\(5.17\))i(whic)o(h)e(is)h(enough)h(to)f(con)o(trol)g(the)75 1798 y(n)o(um)o(b)q(er)h(of)i(tree)g(graphs)h(at)f(order)g Ft(N)23 b FC(\(Ca)o(yley's)17 b(Theorem)g(sa)o(ys)h(the)g(n)o(um)o(b)q(er)e (of)i(tree)f(graphs)75 1858 y(on)h Ft(N)j(>)15 b FC(2)j(v)o(erices)e(is)h Ft(N)552 1840 y Fq(N)t Fp(\000)p Fo(2)631 1858 y FC(.)24 b(Th)o(us)18 b(the)f(problem)f(reduces)g(to)i(sho)o(wing)g(that)g(an)o(y)f(tree)g(graph)75 1918 y(on)22 b Ft(N)k FC(v)o(ertices)20 b(is)h(b)q(ounded)h(b)o(y)f Ft(O)q FC(\()p Ft(t)801 1900 y Fq(N)t Fp(\000)p Fo(1)879 1918 y FC(\),)h(whic)o(h)f(is)g(an)h(easy)f(exercise)e(or)j(see)f(the)g(pro)q(of)h (of)75 1979 y(Prop)q(osition)16 b(6.4.)21 b(More)14 b(generally)l(,)f(the)h (tree)g(expansion)h(is)f(con)o(v)o(ergen)o(t)f(for)i Ft(t)f FC(small)e(dep)q(ending)75 2039 y(on)19 b(sup)8 b Fs(j)p Ft(\036)p FC(\()p Ft(\030)r FC(\))p Fs(j)19 b FC(whenev)o(er)e Ft(V)607 2046 y Fo(00)663 2039 y FC(is)h(analytic)f(in)h Ft(\036)g FC(in)g(the)g (sense)g(that)h(there)f(exists)f(a)i(constan)o(t)g Ft(c)75 2099 y FC(suc)o(h)d(that)218 2223 y(sup)158 2265 y Fq(\030)174 2270 y Fl(1)191 2265 y Fp(2)p Fo(\003)6 b Fq(;)f Fp(j)p Fq(F)g Fp(j\024)p Fo(1)382 2189 y FC(1)p 366 2212 58 2 v 366 2257 a Ft(N)g FC(!)437 2148 y Fr(\014)437 2173 y(\014)437 2198 y(\014)437 2223 y(\014)437 2248 y(\014)451 2165 y(Z)500 2223 y Ft(d\030)546 2230 y Fo(2)583 2223 y Fs(\001)j(\001)g(\001)650 2165 y Fr(Z)700 2223 y Ft(d\030)746 2230 y Fq(M)945 2189 y Ft(\016)969 2171 y Fq(M)1008 2189 y Ft(V)1036 2196 y Fo(00)1074 2189 y FC(\()p Ft(\036)p FC(\))p 799 2212 489 2 v 799 2257 a Ft(\016)r(\036)p FC(\()p Ft(\030)892 2264 y Fo(1)911 2257 y FC(\))p Ft(\016)r(\036)p FC(\()p Ft(\030)1023 2264 y Fo(2)1043 2257 y FC(\))g Fs(\001)g(\001)g(\001)g Ft(\016)r(\036)p FC(\()p Ft(\030)1229 2264 y Fq(M)1268 2257 y FC(\))1292 2223 y Ft(F)f FC(\()p Ft(\030)1371 2230 y Fo(2)1391 2223 y Ft(;)h Fs(\001)g(\001)g(\001)g Ft(;)g(\030)1522 2230 y Fq(M)1562 2223 y FC(\))1581 2148 y Fr(\014)1581 2173 y(\014)1581 2198 y(\014)1581 2223 y(\014)1581 2248 y(\014)1594 2275 y Fq(\036)p Fo(=0)1690 2223 y Ft(<)28 b(c)1777 2203 y Fq(N)1824 2223 y Ft(:)1788 2323 y FC(\(5)p Ft(:)p FC(36\))75 2383 y(This)16 b(t)o(yp)q(e)g(of)h(result)e(is)h(standard.)23 b(There)16 b(is)g(some)f (discussion)i(of)f(it)g(in)g([17].)p 75 2428 735 2 v 131 2459 a Fj(5)150 2474 y Fy(uniformly)11 b(in)i(\003)h(after)g(division)f(b)o(y)g (the)i(v)o(olume)d FA(j)p Fy(\003)p FA(j)e(\021)1061 2441 y Fr(R)1081 2489 y Fj(\003)1112 2474 y Fi(d\030)969 2599 y FC(41)p eop %%Page: 42 46 bop 148 307 a FC(\(ii\))15 b(If)h Ft(V)306 314 y Fo(00)344 307 y FC(\()p Ft(\036)p FC(\))g(has)h Fn(b)n(ounde)n(d)h(derivatives)j FC(as)c(in)f(e.g.,)f(functionals)h(of)h Ft(\036)f FC(suc)o(h)g(as)1685 272 y Fr(R)1704 320 y Fo(\003)1739 307 y Ft(d\030)75 367 y(p)p FC(\()p Ft(e)141 349 y Fq(i\036)p Fo(\()p Fq(\030)q Fo(\))220 367 y Ft(;)8 b(e)265 349 y Fp(\000)p Fq(i\036)p Fo(\()p Fq(\030)q Fo(\))372 367 y FC(\),)25 b(where)e Ft(p)h FC(is)g(a)g(p)q(olynomial,)g(then) f(the)h(series)f(whose)h(terms)e(are)i(giv)o(en)f(in)75 428 y(Prop)q(osition)17 b(5.2)g(is)f(con)o(v)o(ergen)o(t)f(for)h Ft(t)g FC(small)f(b)o(y)h(reduction)f(to)i(the)f(case)g(just)h(considered)f (with)595 538 y(sup)677 488 y Fr(\014)677 513 y(\014)677 538 y(\014)p Ft(\026)11 b Fs(\003)g FC(\001)808 517 y Fq(T)844 496 y Fr(Y)905 538 y Ft(V)933 545 y Fo(00)971 488 y Fr(\014)971 513 y(\014)971 538 y(\014)28 b Fs(\024)f FC(sup)1161 488 y Fr(\014)1161 513 y(\014)1161 538 y(\014)p FC(\001)1216 517 y Fq(T)1251 496 y Fr(Y)1312 538 y Ft(V)1340 545 y Fo(00)1378 488 y Fr(\014)1378 513 y(\014)1378 538 y(\014)396 b FC(\(5)p Ft(:)p FC(37\))75 648 y(whenev)o(er)15 b Ft(\026)i FC(is)f(a)g(probabilit)o (y)g(measure.)148 708 y(The)23 b(dominance)f(b)o(y)h(tree)f(expansions)i(can) f(not)h(b)q(e)f(exp)q(ected)f(to)h(hold)h(more)d(generally)75 768 y(when)g(the)f(deriv)m(ativ)o(es)f(are)i(not)g(b)q(ounded.)34 b(F)l(or)21 b(example,)e(for)i Ft(V)1358 775 y Fo(00)1417 768 y FC(=)g Ft(\025)1512 732 y Fr(R)1549 768 y Ft(d\030)r(\036)1626 750 y Fo(4)1646 768 y FC(\()p Ft(\030)r FC(\),)h(the)e(tree)75 828 y(expansion)12 b(con)o(v)o(erges)g(and)h(for)f Fs(j)p Ft(\036)p FC(\()p Ft(\030)r FC(\))p Fs(j)h FC(b)q(ounded)g(uniformly)d(in)i Ft(\030)j FC(it)c(is)h(analytic)g(in)g Ft(\025)g FC(for)h Fs(j)p Ft(\025)p Fs(j)h(\034)g FC(1.)75 888 y(Suc)o(h)23 b(analyticit)o(y)f(ob)o (viously)h(do)q(es)i(not)f(extend)f(to)h(the)f(full)g(series)g(with)g(lo)q (ops)i(b)q(ecause)f(it)75 949 y(w)o(ould)18 b(imply)d(analogous)k(prop)q (erties)f(for)g Ft(Z)923 956 y Fq(t)p Fo(0)972 949 y FC(=)e(exp)7 b Ft(V)1136 956 y Fq(t)p Fo(0)1169 949 y FC(.)25 b(Ho)o(w)o(ev)o(er,)16 b(b)o(y)h(\(4.16\))h Ft(Z)1664 956 y Fq(t)p Fo(0)1697 949 y FC(\()p Ft(\036)e FC(=)g(0\))h(=)75 973 y Fr(R)111 1009 y Ft(d\026)165 1016 y Fq(t)p Fo(0)206 1009 y FC(exp)8 b Ft(\025)325 973 y Fr(R)362 1009 y Ft(d\030)j(\036)448 991 y Fo(4)468 1009 y FC(\()p Ft(\030)r FC(\))24 b(whic)o(h)e(is)h(div)o(ergen)o(t)f(and)i(forces)f(the)g (series)f(in)h(p)q(o)o(w)o(ers)h(of)f Ft(\025)h FC(to)f(b)q(e)75 1069 y(div)o(ergen)o(t)15 b(b)o(y)g(the)h(monotone)g(con)o(v)o(ergence)f (theorem.)148 1129 y(On)20 b(the)f(other)g(hand)h(when)f Ft(V)733 1136 y Fo(00)790 1129 y FC(is)h(a)f(trigonometric)f(p)q(olynomial)g(there)h (is)g(an)g(imp)q(ortan)o(t)75 1189 y(smo)q(othing)i(e\013ect)f(of)g(the)h (con)o(v)o(olution)f(in)g(\(5.3\).)34 b(The)21 b(con)o(v)o(olution)e(with)i (resp)q(ect)f(to)h Ft(\026)1823 1196 y Fq(ts)1852 1200 y Fl(max)75 1249 y FC(can)16 b(b)q(e)f(p)q(erformed)f(explicitly)f(using)i(\(2.12\))h (whic)o(h)f(implies,)d(for)k(an)o(y)f(Gaussian)i(measure)d(with)75 1310 y(co)o(v)m(ariance)i Ft(C)t FC(:)432 1451 y Ft(\026)461 1458 y Fq(C)502 1451 y Fs(\003)551 1397 y Fq(N)540 1409 y Fr(Y)538 1500 y Fq(i)p Fo(=1)603 1451 y Ft(e)626 1430 y Fq(i\014)r(\036)681 1435 y Fm(i)694 1430 y Fo(\()p Fq(\030)724 1435 y Fm(i)736 1430 y Fo(\))779 1451 y FC(=)28 b(exp)928 1365 y Fr(2)928 1440 y(4)956 1451 y Fs(\000)1000 1417 y FC(1)p 1000 1439 25 2 v 1000 1485 a(2)1037 1409 y Fr(X)1048 1500 y Fq(i;j)1105 1451 y Ft(C)t FC(\()p Ft(\030)1184 1458 y Fq(i)1198 1451 y Ft(;)8 b(\030)1243 1430 y Fp(0)1241 1463 y Fq(j)1259 1451 y FC(\))1278 1365 y Fr(3)1278 1440 y(5)1327 1397 y Fq(N)1316 1409 y Fr(Y)1314 1500 y Fq(i)p Fo(=1)1380 1451 y Ft(e)1403 1430 y Fq(i\014)r(\036)1458 1435 y Fm(i)1470 1430 y Fo(\()p Fq(\030)1500 1435 y Fm(i)1512 1430 y Fo(\))1542 1451 y Ft(:)232 b FC(\(5)p Ft(:)p FC(38\))75 1607 y(W)l(e)19 b(set)g Ft(C)k FC(=)c Ft(C)391 1614 y Fq(ts)420 1618 y Fl(max)499 1607 y FC(whic)o(h)g(is)g(p)q(ositiv)o(e-de\014nite)1033 1588 y Fo(6)1051 1607 y FC(,)h(so)g(that)g(the)f(exp)q(onen)o(t)g(is)g(p)q (ositiv)o(e.)29 b(This)75 1667 y(sho)o(ws)15 b(that)g(the)g(con)o(v)o (olution)e(has)j(a)f(damping)e(e\013ect.)21 b(Corresp)q(ondingly)l(,)15 b(b)q(ounds)h(b)q(etter)e(than)75 1727 y(\(5.37\))j(can)g(b)q(e)f(obtained.) 23 b(In)16 b(fact,)g(form)o(ula)f(\(5.38\))i(applies)f(exactly)f(when)i(the)f (initial)f(data)j(is)75 1787 y(the)13 b(result)g(of)g(the)g(Sine-Gordon)h (transformation)g(from)e(classical)g(statistical)h(mec)o(hanics)e(\()p Ft(V)1823 1794 y Fo(00)1875 1787 y FC(=)75 1812 y Fr(R)119 1847 y Ft(z)r(d\030)17 b FC(:)c(exp)8 b Ft(i\036)p FC(\()p Ft(\030)r FC(\))14 b(:\).)21 b(In)15 b(this)g(case,)g(the)g(expansion)h (\(5.33\))g(b)q(ecomes)f(an)g(expansion)h(in)f(p)q(o)o(w)o(ers)75 1907 y(of)h(the)e(activit)o(y)g Ft(z)j FC(whic)o(h)e(m)o(ust)e(therefore)i(b) q(e)g(the)g(Ma)o(y)o(er)f(expansion)i(for)f(the)g(logarithm)f(of)i(the)75 1968 y(partition)i(function)f(if)h Ft(t)e Fs(!)g(1)p FC(.)26 b(Ho)o(w)o(ev)o(er)16 b(in)h(the)h(statistical)f(mec)o(hanical)e(framew)o (ork,)h(rather)75 2028 y(than)c(using)h(the)e(Sine-Gordon)i(transformation)f (to)g(\\imp)q(ort")f(results)h(from)f(the)g(e\013ectiv)o(e-action)75 2088 y(formalism,)18 b(it)h(is)h(b)q(etter)f(to)i(form)o(ulate)d(the)h(\015o) o(w)i(equations)f(and)g(these)g(expansions)g(directly)75 2148 y(in)d(the)h(statistical)f(mec)o(hanics)f(systems.)25 b(This)18 b(is)f(b)q(ecause)h(the)g(Sine-Gordon)h(transformation)75 2208 y(requires)c(p)q(ositiv)o(e)g(de\014niteness)h(of)g(the)g(p)q(oten)o(tial)g (whereas)g(stabilit)o(y)f(in)g(statistical)h(mec)o(hanics)75 2269 y(is)i(more)f(general,)i(for)f(example)e(an)o(y)j(repulsiv)o(e)d(p)q (oten)o(tial)j(ough)o(t)g(to)f(b)q(e)h(allo)o(w)o(ed.)27 b(This)18 b(study)75 2329 y(of)f(the)f(Ma)o(y)o(er)f(co)q(e\016cien)o(ts)f(is)i(the)g (sub)s(ject)g(of)h(the)f(next)f(section.)p 75 2373 735 2 v 131 2403 a Fj(6)150 2418 y Fy(It)f(is)f(not)h(completely)f(ob)o(vious)g(that) h(this)g(co)o(v)n(ariance)g(is)f(p)q(ositiv)o(e-de\014nite.)19 b(See)c(Prop)q(osition)e(6.18)969 2599 y FC(42)p eop %%Page: 43 47 bop 75 307 a Ff(5.5)70 b(Problems)75 400 y Fu(5.5.1)24 b FC(Sho)o(w)16 b(that)h(the)f Ft(s)p FC(-in)o(tegration)g(of)h(\(5.29\))g(pro)q(duces)f (\(5.35\),)h(that)g(is,)e(sho)o(w)i(that)669 469 y Fr(Z)710 482 y Fq(t)692 563 y Fo(0)733 527 y Ft(ds)841 507 y Fd(y)847 471 y(\017)p 847 511 4 9 v 847 546 a(\016)997 471 y(\014)p 997 511 V 997 546 a(\015)p 882 546 84 4 v 882 471 V 909 459 a Fq(ts)916 572 y(s)1007 507 y Fd(y)1066 527 y FC(=)1136 494 y(1)p 1136 516 25 2 v 1136 562 a(2)1225 507 y Fd(y)1231 471 y(\017)p 1231 511 4 9 v 1231 546 a(\016)1381 471 y(\014)p 1381 511 V 1381 546 a(\015)p 1266 546 84 4 v 1266 471 V 1391 507 a(y)1436 527 y Ft(:)338 b FC(\(5)p Ft(:)p FC(39\))75 667 y Fu(5.5.2)24 b FC(F)l(ollo)o(w)17 b(the)i(discussion)f(of)h(Sections)f (5.2{5.3)h(through)g(for)g(the)f(particular)g(example)e(of)197 728 y Ft(V)225 735 y Fo(00)287 728 y FC(=)349 692 y Fr(R)385 728 y Ft(d\030)11 b(\036)471 709 y Fo(4)491 728 y FC(\()p Ft(\030)r FC(\).)40 b(Obtain)22 b(the)g(\014rst)h(few)f Fn(unlab)n(el)r(le)n(d)30 b FC(\(i.e.)38 b Ft(s)p 1422 735 23 2 v FC(-in)o(tegrated\))22 b(F)l(eynman)197 788 y(diagrams)16 b(using)g(in)g(addition)h(the)f(standard)h (graphic)g(con)o(v)o(en)o(tion)903 893 y Ft(V)931 900 y Fo(00)997 893 y Fs(\021)1079 926 y Fb(\000)1121 885 y(\000)-84 b(@)1121 926 y(@)1193 893 y Ft(:)197 1011 y FC(F)l(or)16 b(the)g(classical)g(part,)g (the)g(answ)o(er)h(is)327 1244 y Ft(V)366 1224 y Fo(cl)355 1256 y Fq(t)p Fo(0)422 1244 y Fs(\021)504 1277 y Fb(\000)546 1236 y(\000)-84 b(@)546 1277 y(@)615 1244 y FC(+)669 1210 y(4)11 b Fs(\001)g FC(4)p 669 1233 85 2 v 692 1278 a(2!)p 775 1237 84 4 v 814 1277 4 84 v 816 1237 125 4 v 899 1237 84 4 v 939 1277 4 84 v 1009 1244 a(+)g(3)h Fs(\001)1124 1210 y FC(4)f Fs(\001)g FC(\(4)h Fs(\001)e FC(3\))i Fs(\001)f FC(4)p 1124 1233 244 2 v 1227 1278 a(3!)p 1388 1237 84 4 v 1428 1277 4 84 v 1430 1237 125 4 v 1513 1237 84 4 v 1553 1277 4 84 v 1553 1236 4 125 v 1513 1113 84 4 v 1553 1153 4 84 v 1623 1244 a(+)28 b Fs(\001)8 b(\001)g(\001)22 b Ft(:)197 1365 y FC(In)17 b(the)f(last)i(term,) d(the)h(factor)i(3)f(is)g(due)g(to)g(the)g(m)o(ultiplic)o(it)o(y)c(of)18 b(the)e(diagram)h(\(all)f(the)h(3)197 1425 y(p)q(ossible)d(trees)g(with)g (three)g(v)o(ertices)f(giv)o(e)g(the)h(same)f(con)o(tribution\).)20 b(The)15 b(fully)e(quan)o(tum)197 1486 y(solution)j(requires)g(more)f(com)o (binatorics.)k(One)d(should)h(obtain)321 1615 y Ft(V)349 1622 y Fo(00)428 1615 y Fs(\021)41 b Ft(V)548 1595 y Fo(cl)536 1628 y(0)p Fq(t)586 1615 y FC(+)640 1582 y(4)12 b Fs(\001)f FC(3)p 640 1604 85 2 v 671 1650 a(2)p 746 1608 125 4 v 809 1576 a Fd(n)898 1615 y FC(+)992 1582 y(4!)p 952 1604 119 2 v 952 1650 a(2!)g Fs(\001)g FC(2)1050 1635 y Fo(2)1123 1601 y Fd(nn)520 1737 y FC(+)574 1703 y(4)g Fs(\001)g FC(4)p 574 1726 85 2 v 597 1771 a(2!)674 1737 y Fs(\001)704 1703 y FC(3)h Fs(\001)f FC(3)g Fs(\001)g FC(\(2\))p 704 1726 184 2 v 784 1771 a(2)903 1737 y Fs(\001)933 1703 y FC(1)p 933 1726 25 2 v 933 1771 a(2)1008 1693 y Fd(\013)p 1008 1729 4 13 v 1008 1755 a(\012)1133 1693 y(\010)p 1133 1729 V 1133 1755 a(\011)p 1035 1755 75 4 v 1035 1693 V 1008 1693 a(\010)p 1008 1729 4 13 v 1008 1755 a(\011)p 979 1755 7 4 v 979 1693 V 1133 1693 a(\013)p 1133 1729 4 13 v 1133 1755 a(\012)p 1159 1755 7 4 v 1159 1693 V 1193 1737 a FC(+)g(2)g Fs(\001)1307 1703 y FC(4)h Fs(\001)f FC(4)p 1307 1726 85 2 v 1331 1771 a(2!)1408 1737 y Fs(\001)1438 1703 y FC(3)g Fs(\001)g FC(2)p 1438 1726 V 1468 1771 a(2)p 1544 1730 187 4 v 1606 1698 a Fd(n)p 1689 1730 84 4 v 1729 1770 4 84 v 520 1859 a FC(+)574 1825 y(4)g Fs(\001)g FC(4)p 574 1847 85 2 v 597 1893 a(2!)674 1859 y Fs(\001)704 1825 y FC(3)h Fs(\001)f FC(2)g Fs(\001)g FC(3)g Fs(\001)g FC(2)h Fs(\001)f FC(\(3)g Fs(\001)g FC(2)g Fs(\001)g FC(2\))p 704 1847 425 2 v 857 1893 a(3!)g Fs(\001)g FC(2)955 1879 y Fo(3)1145 1859 y Fs(\001)1175 1825 y FC(1)p 1175 1847 25 2 v 1175 1893 a(2)1281 1815 y Fd(\013)p 1281 1851 4 13 v 1281 1877 a(\012)1405 1815 y(\010)p 1405 1851 V 1405 1877 a(\011)p 1307 1877 75 4 v 1307 1815 V 1251 1844 a(n)125 b(n)1497 1859 y FC(+)27 b Fs(\001)8 b(\001)g(\001)22 b Ft(:)1788 1947 y FC(\(5.40\))197 2053 y(Note)12 b(the)h(di\013eren)o(t)f (sources)h(of)g(the)g(com)o(binatoric)e(factors:)20 b(\(i\))12 b(factors)i(due)f(to)g(m)o(ultiplic)o(-)197 2113 y(it)o(y)i(of)h(diagrams)g (\(the)g(factor)h(2)f(in)g(the)g(\014fth)g(term\);)e(\(ii\))h(factors)i(due)f (to)g(the)g(underlying)197 2173 y(tree)d(expansion)h(\(e.g.)20 b(the)14 b(factor)g(4)6 b Fs(\001)g FC(4)p Ft(=)p FC(2!)15 b(in)e(all)h(the)f(t)o(w)o(o-v)o(ertex)g(diagrams\);)h(\(iii\))e(factors)197 2233 y(due)j(to)g(the)f(application)h(of)g(Wic)o(k's)e(theorem)h(\(in)g(the)g (form)g(of)h(Exercise)f(4.3.3\);)h(and)g(\(iv\))197 2293 y(factors)20 b(due)e(to)i(the)f Ft(s)p FC(-in)o(tegration)g(\(the)f(factors)i(1/2)g(in)f (the)g(diagrams)f(with)h(a)h(cen)o(tral)197 2354 y(lo)q(op,)i(see)e(Exercise) f(5.5.1\).)35 b(The)20 b(factors)h(\(iii\))e(include)h(a)g(denominator)g Ft(N)5 b FC(!)14 b Fs(\001)g FC(2)1766 2336 y Fq(N)1820 2354 y FC(and,)197 2414 y(in)k(the)g(n)o(umerator,)f(w)o(e)h(ha)o(v)o(e)g (surrounded)h(b)o(y)e(brac)o(k)o(ets)h(the)g(con)o(tributions)g(due)g(to)h (the)197 2474 y(di\013eren)o(t)f(w)o(a)o(ys)h(in)f(whic)o(h)g(the)h (Laplacians)h(can)f(act)g(o)o(v)o(er)f(the)g(v)o(ertices)f(to)j(pro)q(duce)f (the)969 2599 y(43)p eop %%Page: 44 48 bop 197 307 a FC(giv)o(en)18 b(diagram.)29 b(F)l(or)19 b(instance,)f(in)h (the)g(factor)g(\(3)13 b Fs(\001)g FC(2)g Fs(\001)g FC(2\))19 b(of)g(the)g(last)g(diagram)f(\(whic)o(h)197 367 y(comes)j(from)g(the)g (action)h(of)h(\001)812 349 y Fo(3)831 367 y FC(\),)g(3)f(coun)o(ts)h(whic)o (h)e(Laplacian)h(go)q(es)i(to)e(link)f(the)h(t)o(w)o(o)197 428 y(v)o(ertices,)14 b(one)j(of)g(the)f(2s)h(coun)o(ts)f(the)h(t)o(w)o(o)f (p)q(ossibilities)g(for)g(this)h(link)e(\(\014rst)i Ft(@)s(=@)s(\036)e FC(acting)197 488 y(on)h(the)g(\014rst)f(or)h(the)g(second)g(v)o(ertex\),)d (and)k(the)e(remaining)f(2)i(coun)o(ts)g(whic)o(h)f(of)h(the)f(other)197 548 y(Laplacians)i(form)e(a)i(tadp)q(ole)g(on)f(v)o(ertex)f(1.)197 629 y(Alternativ)o(ely)l(,)e(the)j(formal)f(series)g(\(5.40\))i(can)g(b)q(e)f (written)g(in)g(the)g(form)412 766 y Ft(V)440 773 y Fo(00)520 766 y FC(=)41 b Ft(V)639 745 y Fo(cl)627 778 y(0)p Fq(t)677 766 y FC(+)11 b(6)d Ft(C)793 773 y Fq(t)p Fo(0)826 766 y FC(\(0)p Ft(;)g FC(0\))p 951 759 125 4 v 169 w(+)j(3)d Ft(C)1223 745 y Fo(2)1219 778 y Fq(t)p Fo(0)1252 766 y FC(\(0)p Ft(;)g FC(0\))k(+)f(36)1523 722 y Fd(\013)p 1523 758 4 13 v 1523 784 a(\012)1648 722 y(\010)p 1648 758 V 1648 784 a(\011)p 1550 784 75 4 v 1550 722 V 1523 722 a(\010)p 1523 758 4 13 v 1523 784 a(\011)p 1494 784 7 4 v 1494 722 V 1648 722 a(\013)p 1648 758 4 13 v 1648 784 a(\012)p 1674 784 7 4 v 1674 722 V 599 849 a FC(48)d Ft(C)690 856 y Fq(t)p Fo(0)724 849 y FC(\(0)p Ft(;)g FC(0\))848 882 y Fb(\000)890 841 y(\000)-84 b(@)890 882 y(@)959 849 y FC(+)11 b(36)d Ft(C)1103 828 y Fo(2)1099 861 y Fq(t)p Fo(0)1132 849 y FC(\(0)p Ft(;)g FC(0\))1255 805 y Fd(\013)p 1255 841 4 13 v 1255 867 a(\012)1380 805 y(\010)p 1380 841 V 1380 867 a(\011)p 1282 867 75 4 v 1282 805 V 1409 849 a FC(+)27 b Fs(\001)8 b(\001)g(\001)22 b Ft(;)197 973 y FC(where)12 b(the)h(factors)h Ft(C)606 980 y Fq(t)p Fo(0)638 973 y FC(\(0)p Ft(;)8 b FC(0\))13 b(come)f(when)h(a)g(v)o(ertex)e(is)i (joined)g(b)o(y)f(a)h(line)f(to)h(itself)f(\(tadp)q(ole)197 1034 y(graphs\).)21 b(This)12 b(t)o(yp)q(e)g(of)g(con)o(tributions)g(is)g (absen)o(t)g(if)g(w)o(e)g(start)h(with)f Ft(V)1499 1041 y Fo(00)1550 1034 y FC(=)1602 998 y Fr(R)1638 1034 y Ft(d\030)f FC(:)d Ft(\036)1746 1016 y Fo(4)1766 1034 y FC(\()p Ft(\030)r FC(\):)1841 1041 y Fq(C)1866 1045 y Fk(1)p Fm(t)197 1102 y FC(where)22 b(normal)f(ordering)i (:)8 b(:)33 b(is)22 b(de\014ned)g(b)o(y)g(:)8 b Ft(e)1131 1084 y Fq(i\036)p Fo(\()p Fq(x)p Fo(\))1213 1102 y FC(:)1227 1109 y Fq(C)1264 1102 y Fs(\021)24 b Ft(e)1355 1070 y Fl(1)p 1355 1076 16 2 v 1355 1097 a(2)1375 1084 y Fq(C)r Fo(\()p Fq(x;x)p Fo(\))1481 1102 y Ft(e)1504 1084 y Fq(i\036)1563 1102 y Fs(\021)1626 1069 y Fr(P)1683 1079 y Fo(\()p Fq(i)p Fo(\))1723 1067 y Fm(N)1751 1079 y Fo(:)p Fq(\036)1782 1067 y Fm(N)1811 1079 y Fo(:)1821 1085 y Fm(C)1847 1079 y Fo(\()p Fq(x)p Fo(\))p 1683 1091 211 2 v 1768 1119 a Fq(N)t Fo(!)1899 1102 y FC(.)197 1162 y(Sho)o(w)11 b(that)g Ft(\026)449 1169 y Fq(t;)p Fo(0)492 1162 y Fs(\003)p FC(:)d Ft(e)562 1144 y Fq(i\036)p Fo(\()p Fq(x)p Fo(\))644 1162 y FC(:)658 1169 y Fq(C)683 1174 y Fk(1)p Fm(;)p Fl(0)750 1162 y FC(=:)g Ft(e)833 1144 y Fq(i\036)p Fo(\()p Fq(x)p Fo(\))915 1162 y FC(:)929 1169 y Fq(C)954 1173 y Fk(1)p Fm(;t)1020 1162 y FC(and)k(deduce)e(that)h Ft(\026)1396 1169 y Fq(t;)p Fo(0)1439 1162 y Fs(\003)p FC(:)d Ft(\036)p FC(\()p Ft(x)p FC(\))1581 1144 y Fq(N)1614 1162 y FC(:)1628 1169 y Fq(C)1653 1174 y Fk(1)p Fm(;)p Fl(0)1720 1162 y FC(=:)g Ft(\036)p FC(\()p Ft(x)p FC(\))1875 1144 y Fq(N)1908 1162 y FC(:)1922 1169 y Fq(C)1947 1173 y Fk(1)p Fm(;t)2002 1162 y FC(.)75 1264 y Fu(5.5.3)24 b FC(No)o(w)13 b(let)g(us)h(review)f(the)g(standard)i(\(m)o(uc)o(h)d(easier)h(metho)q(d\))g (to)h(obtain)h(the)e(series)g(\(5.40\).)197 1324 y(F)l(rom)18 b(the)h(discussion)g(of)h(Section)e(4)i(\(in)f(particular)g(Lemma)e(4.3)i (and)h(Exercise)e(4.3.3\),)197 1384 y(one)e(concludes)g(that)731 1454 y(exp)8 b Ft(V)842 1461 y Fq(t)p Fo(0)903 1454 y FC(=)27 b(exp)1051 1394 y Fr(\024)1078 1421 y FC(1)p 1078 1443 25 2 v 1078 1489 a(2)1107 1454 y(\001)1148 1461 y Fq(t)p Fo(0)1180 1394 y Fr(\025)1211 1454 y FC(exp)7 b Ft(V)1321 1461 y Fo(00)1373 1454 y Ft(:)401 b FC(\(5)p Ft(:)p FC(41\))197 1558 y(The)16 b(material)e(of)i(the)g(presen)o(t)f(section)h(should)g(su\016ce)f(to)h (compute)f(the)h(diagrammatics)197 1618 y(asso)q(ciated)j(to)g(the)f(\014rst) h(few)f(p)q(o)o(w)o(ers)g(\(\001)p Ft(=)p FC(2\))1062 1600 y Fq(N)1096 1618 y Ft(V)1135 1600 y Fq(M)1193 1618 y FC(in)g(the)g(righ)o (t-hand)h(side.)27 b(T)l(o)19 b(obtain)197 1678 y Ft(V)225 1685 y Fq(t)p Fo(0)281 1678 y FC(one)k(m)o(ust)e(tak)o(e)i(the)g (logarithm|in)e(the)i(sense)f(of)i(formal)d(p)q(o)o(w)o(er)i(series|of)g(the) 197 1738 y(resulting)c(expression;)i(this)e(is)g(easy:)28 b(it)19 b(corresp)q(onds)i(to)f(k)o(eeping)e(only)i(the)f(connected)197 1798 y(diagrams.)k(In)17 b(particular)g(con)o(vince)f(y)o(ourself)g(that)h (in)g(this)g(metho)q(d)f(the)h(com)o(binatorics)197 1858 y(of)d(eac)o(h)f (diagram)h(is)f(the)h(same)f(as)h(coun)o(ting)g(ho)o(w)g(man)o(y)f(w)o(a)o (ys)g(there)h(are)f(to)i(pair)f(lab)q(elled)197 1919 y(legs)i(on)h(lab)q (elled)e(v)o(ertices)f(in)i(order)h(to)f(pro)q(duce)h(a)g(diagram)e(with)h(a) h(giv)o(en)e(top)q(ology)l(.)75 2020 y Fu(5.5.4)24 b FC(Pro)o(v)o(e)16 b(Prop)q(osition)j(5.2)f(b)o(y)f(sho)o(wing)h(that)g(\(5.33\))g(solv)o(es)f (equation)g(\(5.1\))h(in)f(the)h(sense)197 2081 y(of)f(formal)e(p)q(o)o(w)o (er)h(series,)f(and)i(moreo)o(v)o(er)d(that)j(it)f(is)g(the)g(only)g(formal)f (series)h(with)g Ft(V)1815 2088 y Fo(00)1869 2081 y FC(as)197 2141 y(initial)f(condition.)21 b(Some)15 b(guidelines:)197 2222 y(\(i\))21 b(By)g(substituting)h(in)f(\(5.1\))h(a)g(series)e(of)i(the)g (form)e Ft(V)1293 2229 y Fq(t)p Fo(0)1349 2222 y FC(=)1410 2188 y Fr(P)1453 2232 y Fq(N)1495 2222 y Ft(g)1518 2229 y Fq(N)1552 2222 y FC(\()p Ft(t)p FC(\))h(and)i(equating)197 2282 y(p)q(o)o(w)o(ers)16 b(of)h Ft(V)11 b FC(,)16 b(sho)o(w)h(that)g(the)f Ft(g)817 2289 y Fq(N)851 2282 y FC('s)g(m)o(ust)e(satisfy)j(the)f(coupled)g(system)e (of)j(equations)506 2393 y Ft(@)s(g)558 2400 y Fq(N)591 2393 y FC(\()p Ft(t)p FC(\))p 506 2415 141 2 v 553 2461 a Ft(@)s(t)666 2426 y FC(=)722 2393 y(1)p 722 2415 25 2 v 722 2461 a(2)760 2353 y Fr(")784 2426 y FC(\001)825 2433 y Fq(t)840 2426 y Ft(g)863 2433 y Fq(N)897 2426 y FC(\()p Ft(t)p FC(\))10 b(+)1035 2372 y Fq(N)1021 2385 y Fr(X)1012 2477 y Fq(K)r Fo(=1)1089 2426 y FC(\()p Fs(r)p Ft(g)1173 2433 y Fq(K)1207 2426 y FC(\()p Ft(t)p FC(\))e Ft(;)16 b Fs(r)p Ft(g)1366 2433 y Fq(N)t Fp(\000)p Fq(K)1459 2426 y FC(\()p Ft(t)p FC(\)\))1534 2433 y Fq(t)1549 2353 y Fr(#)1595 2426 y Ft(:)179 b FC(\(5)p Ft(:)p FC(42\))969 2599 y(44)p eop %%Page: 45 49 bop 197 315 a FC(\(ii\))23 b(Sho)o(w)h(that)g Ft(g)558 322 y Fq(N)592 315 y FC(\()p Ft(t)p FC(\))i(=)756 296 y Fo(1)p 744 304 42 2 v 744 333 a Fq(N)t Fo(!)799 282 y Fr(P)842 326 y Fq(T)15 b Fo(on)9 b Fq(N)967 280 y Fr(R)994 293 y Fq(t)986 328 y Fo(0)1017 315 y Ft(d)1042 297 y Fq(T)1070 315 y Ft(s)f(\026)1130 322 y Fq(ts)1159 326 y Fl(max)1236 315 y Fs(\003)1283 267 y Fr(n)1310 315 y FC(\001)1351 297 y Fq(T)1351 328 y(s)p 1351 335 19 2 v 1387 282 a Fr(Q)1426 295 y Fq(N)1426 328 y(i)p Fo(=1)1493 315 y Ft(V)1521 322 y Fo(00)1559 315 y FC(\()p Ft(\036)1607 322 y Fq(i)1621 315 y FC(\))1640 267 y Fr(o)1668 266 y(\014)1668 291 y(\014)1668 315 y(\014)1682 342 y Fq(\036)1703 347 y Fm(i)1716 342 y Fo(=)p Fq(\036)1790 315 y FC(solv)o(es)197 390 y(\(5.42\).)38 b(Note)21 b(that)h(when)f Ft(d=dt)h FC(acts)g(on)g Ft(\026)1061 397 y Fq(ts)1090 401 y Fl(max)1172 390 y FC(one)g(obtains)g(the)f(linear)g (part)h(of)g(the)197 451 y(equation,)13 b(b)q(ecause)g(\()p Ft(d=dt)p FC(\)\()p Ft(\026)762 458 y Fq(ts)791 462 y Fl(max)856 451 y Fs(\003)t Ft(F)7 b FC(\))13 b(=)h(\(1)p Ft(=)p FC(2\)\001)1159 458 y Fq(C)1184 462 y Fm(t)1200 451 y FC(\()p Ft(\026)1248 458 y Fq(ts)1277 462 y Fl(max)1342 451 y Fs(\003)t Ft(F)7 b FC(\))12 b(\(Exercise)f(4.3.1\).)20 b(Then)197 511 y(sho)o(w)15 b(that)g(the)f(second)h(term)e(in)h(\(5.42\))i(sho)o(ws)f(up)g(when)g Ft(d=dt)g FC(acts)g(on)g(the)f(upp)q(er)h(limits)197 571 y(of)g(the)g(in)o (tegrals)530 535 y Fr(R)558 549 y Fq(t)550 584 y Fo(0)581 571 y Ft(d)606 553 y Fq(T)634 571 y Ft(s)p FC(.)21 b(F)l(or)15 b(an)h(answ)o(er:)21 b(either)14 b(\(a\))h(lo)q(ok)h(to)g(the)e(pro)q(of)j (of)e(Prop)q(osition)197 631 y(6.2)k(do)o(wn)h(the)f(road,)h(where)f(similar) e(considerations)i(pla)o(y)f(a)i(role;)f(or)h(\(b\))f(lo)q(ok)g(up)h(the)197 691 y(pro)q(of)d(in)f(reference)f([17,)h(Theorem)f(3.4].)197 772 y(\(iii\))e(Sho)o(w)j(uniqueness)e(of)h(the)g(solution)g(pro)q(ceding)g (b)o(y)f(induction)h(on)g Ft(N)5 b FC(.)21 b([Hin)o(t:)e(As)c(the)197 833 y(non-linear)g(part)h(of)f(equation)g(\(5.42\))h(dep)q(ends)g(on)f Ft(g)1200 840 y Fq(L)1227 833 y FC('s)g(with)g Ft(L)f(<)g(N)5 b FC(,)15 b(b)o(y)g(induction)f(one)197 893 y(concludes)i(that)h(t)o(w)o(o)g (solutions)g Ft(f)844 900 y Fq(N)894 893 y FC(and)g Ft(g)1012 900 y Fq(N)1063 893 y FC(ob)q(eying)g(the)f(same)g(initial)f(condition)i(m)o (ust)197 953 y(satisfy)f(the)g(linear)g(equation)g(\()p Ft(d=dt)p FC(\)\()p Ft(f)941 960 y Fq(N)986 953 y Fs(\000)11 b Ft(g)1059 960 y Fq(N)1093 953 y FC(\))j(=)g(\(1)p Ft(=)p FC(2\)\001)1329 960 y Fq(t)1344 953 y FC(\()p Ft(f)1387 960 y Fq(N)1432 953 y Fs(\000)d Ft(g)1505 960 y Fq(N)1539 953 y FC(\).])969 2599 y(45)p eop %%Page: 46 50 bop 75 315 a Fv(6)83 b(The)27 b(Ma)n(y)n(er)g(Expansion)75 425 y FC(This)d(section)g(is)g(an)g(in)o(terlude)f(in)h(the)f(main)g(dev)o (elopmen)o(t.)42 b(F)l(ollo)o(wing)24 b([17,)g(16])g(w)o(e)g(shall)75 485 y(form)o(ulate)c(\015o)o(w)i(equations)g(directly)e(in)i(the)g(con)o (text)f(of)h(the)g(grand)g(canonical)g(ensem)o(ble)d(of)75 545 y(statistical)e(mec)o(hanics)e(and)i(the)g(co)q(e\016cien)o(ts)f(of)i (the)e(Ma)o(y)o(er)g(expansion)i(and)g(conclude)e(with)h(a)75 606 y(brief)f(discussion)h(of)f(the)h(relationship)f(with)g(the)h(e\013ectiv) o(e)d(p)q(oten)o(tial)i(and)i(Gaussian)f(in)o(tegrals.)75 666 y(The)23 b(resulting)g(form)o(ulas)f(for)h(Ursell)f(co)q(e\016cien)o(ts)g (are)h(impro)o(v)o(em)o(en)o(t)o(s)e(o)o(v)o(er)h(the)h(tree)g(graph)75 726 y(form)o(ulas)14 b(discussed)i(in)f(m)o(y)e(Les)j(Houc)o(hes)f(Lectures)g ([9])g(and)h(in)f(particular)g(replace)f(the)h(rather)75 786 y(ugly)h(app)q(endix)g(on)h(tree)e(graph)i(form)o(ulas)e(in)h(those)h(notes.) k(These)16 b(t)o(yp)q(es)g(of)h(form)o(ulas)e(ha)o(v)o(e)g(an)75 846 y(ev)o(olution)g([35)q(,)g(11)q(,)h(4,)g(5)q(,)f(17)q(,)h(23)q(])g(in)g (whic)o(h)f(they)h(ha)o(v)o(e)f(b)q(ecome)g(progressiv)o(ely)g(neater.)148 907 y(First,)k(w)o(e)g(recall)f(the)g(Ma)o(y)o(er)g(expansion)i(for)f(a)h (system)d(of)j(particles)e(in)o(teracting)g(b)o(y)g(t)o(w)o(o-)75 967 y(b)q(o)q(dy)f(forces)f Ft(u)p FC(\()p Ft(\030)r(;)8 b(\030)450 949 y Fp(0)463 967 y FC(\).)21 b(Let)c Ft(Z)j FC(b)q(e)c(the)g(Grand)h (Canonical)g(partition)f(function:)652 1099 y Ft(Z)i Fs(\021)763 1057 y Fr(X)755 1149 y Fq(N)t Fp(\025)p Fo(0)862 1065 y FC(1)p 845 1087 58 2 v 845 1133 a Ft(N)5 b FC(!)908 1099 y Ft(z)933 1078 y Fq(N)975 1040 y Fr(Z)1024 1099 y Ft(d)1049 1078 y Fq(N)1083 1099 y Ft(\030)11 b(e)1138 1078 y Fp(\000)p Fq(\014)r(U)t Fo(\()p Fq(N)r(;\030)q Fo(\))1314 1099 y FC(;)484 b(\(6)p Ft(:)p FC(1\))75 1248 y(with)589 1351 y Ft(U)5 b FC(\()p Ft(N)r(;)j(\030)p 709 1368 24 2 v 3 w FC(\))14 b Fs(\021)819 1252 y Fr(8)819 1289 y(>)819 1302 y(<)819 1376 y(>)819 1389 y(:)903 1258 y(X)876 1352 y Fo(\()p Fq(ij)r Fo(\))p Fp(\032)p Fq(N)999 1299 y Ft(u)p FC(\()p Ft(\030)1067 1306 y Fq(i)1081 1299 y Ft(;)8 b(\030)1124 1306 y Fq(j)1143 1299 y FC(\))41 b Ft(N)19 b Fs(\025)14 b FC(2)876 1406 y(0)303 b Ft(N)19 b FC(=)14 b(1)g Ft(;)1812 1351 y FC(\(6)p Ft(:)p FC(2\))75 1495 y(where)k(\()p Ft(ij)s FC(\))g Fs(\032)f Ft(N)24 b FC(means)18 b(that)h Ft(i)f FC(and)h Ft(j)j FC(are)d(summed)d(o)o (v)o(er)i(pairs)g Ft(i;)8 b(j)21 b FC(with)e(1)f Fs(\024)g Ft(i)f(<)h(j)j Fs(\024)d Ft(N)5 b FC(.)75 1555 y(\()p Ft(N)23 b FC(is)17 b(iden)o(ti\014ed)g(with)g Fs(f)p FC(1)p Ft(;)8 b FC(2)p Ft(;)g(:)g(:)g(:)g(;)g(N)d Fs(g)p FC(.)26 b(More)17 b(generally)l(,)g(w)o(e)g(will)g(de\014ne)g Ft(U)5 b FC(\()p Ft(X)q(;)j(\030)p 1622 1571 V 3 w FC(\))18 b(for)g(an)o(y)g(set)75 1615 y Ft(X)g Fs(\032)c(f)p FC(1)p Ft(;)8 b FC(2)p Ft(;)g(:)g(:)g(:)o Fs(g)p FC(\).)21 b(The)15 b(Ma)o(y)o(er)e(expansion)i(is)g(the)f Fn(formal)20 b FC(expansion)14 b(for)h(log)10 b Ft(Z)18 b FC(in)d(p)q(o)o(w)o (ers)g(of)g Ft(z)784 1720 y FC(1)p 766 1742 62 2 v 766 1788 a Fs(j)p FC(\003)p Fs(j)841 1754 y FC(log)9 b Ft(Z)18 b Fs(\030)1028 1700 y Fp(1)1015 1712 y Fr(X)1037 1803 y Fo(1)1092 1754 y Ft(b)1113 1761 y Fq(N)1146 1754 y Ft(z)1171 1733 y Fq(N)1205 1754 y Ft(;)593 b FC(\(6)p Ft(:)p FC(3\))75 1898 y(where)16 b Fs(j)p FC(\003)p Fs(j)d FC(=)343 1863 y Fr(R)363 1911 y Fo(\003)397 1898 y Ft(d\030)r FC(.)75 2043 y Ff(6.1)70 b(The)22 b(Ursell)f(Co)r(e\016cien)n(ts)75 2135 y FC(The)15 b(Ma)o(y)o(er)e(co)q(e\016cien)o(ts)h Ft(b)588 2142 y Fq(N)636 2135 y FC(in)g(\(6.3\))h(will)f(ultimately)e(b)q(e)j (obtained)g(b)o(y)f(a)i(form)o(ula)d(of)i(the)g(form)638 2267 y Ft(b)659 2274 y Fq(N)706 2267 y FC(=)781 2234 y(1)p 763 2256 V 763 2302 a Fs(j)p FC(\003)p Fs(j)859 2234 y FC(1)p 843 2256 58 2 v 843 2302 a Ft(N)5 b FC(!)914 2209 y Fr(Z)964 2267 y Ft(d)989 2247 y Fq(N)1023 2267 y Ft(\030)r FC(\()p Ft(e)1088 2247 y Fp(\000)p Fq(\014)r(U)1166 2267 y FC(\))1185 2274 y Fq(c)1203 2267 y FC(\()p Ft(N)r(;)j(\030)p 1285 2284 24 2 v 3 w FC(\))p Ft(;)470 b FC(\(6)p Ft(:)p FC(4\))75 2409 y(where)17 b(the)g(co)q(e\016cien)o(ts)e(\()p Ft(e)591 2391 y Fp(\000)p Fq(\014)r(U)669 2409 y FC(\))688 2416 y Fq(c)723 2409 y FC(are)i(called)f Fn(Ursel)r(l)j(c)n(o)n(e\016cients)t FC(.)25 b(Their)17 b(existence)e(and)j (prop-)75 2469 y(erties)d(will)f(b)q(e)i(the)f(sub)s(ject)g(of)h(the)f(next)g (sections.)21 b(Th)o(us)16 b(these)f(co)q(e\016cien)o(ts)f(are)i(to)g(b)q(e)g (de\014ned)969 2599 y(46)p eop %%Page: 47 51 bop 75 307 a FC(so)17 b(that)663 386 y(log)10 b Ft(Z)17 b Fs(\030)846 344 y Fr(X)838 436 y Fq(N)t Fp(\025)p Fo(1)928 352 y Ft(z)953 334 y Fq(N)p 928 374 59 2 v 928 420 a Ft(N)5 b FC(!)1000 327 y Fr(Z)1049 386 y Ft(d)1074 365 y Fq(N)1108 386 y Ft(\030)11 b FC(\()p Ft(e)1182 365 y Fp(\000)p Fq(\014)r(U)1260 386 y FC(\))1279 393 y Fq(c)1311 386 y Ft(;)487 b FC(\(6)p Ft(:)p FC(5\))75 514 y(\(in)16 b(notational)h(corresp)q(ondence)f(with)g(\(6.1\)\).) 75 644 y Fu(6.1.1)55 b(Equiv)m(alen)n(t)17 b(De\014nitions)75 737 y FC(P)o(erhaps)12 b(the)g(b)q(est)g(w)o(a)o(y)g([56])g(to)g(get)g(at)h (the)e(Ursell)g(co)q(e\016cien)o(ts)f(is)i(to)g(de\014ne)g(them)e(b)o(y)i (recursiv)o(ely)75 797 y(solving:)21 b(F)l(or)c(eac)o(h)e(set)h Ft(X)j Fs(\032)13 b(f)p FC(1)p Ft(;)8 b FC(2)p Ft(;)g(:)g(:)g(:)p Fs(g)544 951 y Ft(e)567 930 y Fp(\000)p Fq(\014)r(U)t Fo(\()p Fq(X)s Fo(\))746 951 y FC(=)833 892 y Fp(j)p Fq(X)s Fp(j)828 909 y Fr(X)826 1002 y Fq(k)q Fo(=1)926 917 y FC(1)p 918 939 41 2 v 918 985 a Ft(k)r FC(!)1013 909 y Fr(X)972 1001 y Fq(X)1001 1006 y Fl(1)1018 1001 y Fq(;:::)o(;X)1096 1007 y Fm(k)1156 897 y Fq(k)1139 909 y Fr(Y)1137 1001 y Fq(i)p Fo(=1)1194 951 y FC(\()p Ft(e)1236 930 y Fp(\000)p Fq(\014)r(U)1314 951 y FC(\))1333 958 y Fq(c)1351 951 y FC(\()p Ft(X)1410 958 y Fq(i)1424 951 y FC(\))746 1110 y Fs(\021)886 1068 y Fr(X)826 1162 y Fp(f)p Fq(X)873 1167 y Fl(1)890 1162 y Fq(;:::)o(;X)968 1168 y Fm(k)987 1162 y Fp(g)1046 1056 y Fq(k)1029 1068 y Fr(Y)1027 1159 y Fq(i)p Fo(=1)1084 1110 y FC(\()p Ft(e)1126 1089 y Fp(\000)p Fq(\014)r(U)1204 1110 y FC(\))1223 1117 y Fq(c)1241 1110 y FC(\()p Ft(X)1300 1117 y Fq(i)1314 1110 y FC(\))479 b(\(6.6\))75 1263 y(where)14 b(in)h(the)f(\014rst)h(equation)f(the)h(inner)f(sum)f(ranges)j(o)o(v)o(er)e (all)g(k-tuples)g(of)h(disjoin)o(t)f(non-empt)o(y)75 1323 y(subsets)f Ft(X)280 1330 y Fo(1)300 1323 y Ft(;)8 b(:)g(:)g(:)g(;)g(X)450 1330 y Fq(k)485 1323 y Fs(\032)14 b(f)p FC(1)p Ft(;)8 b(:)g(:)g(:)g(;)g(N)d Fs(g)12 b FC(forming)g(a)h(partition)f(of)h(the)f(set)h Ft(X)t FC(.)20 b(W)l(e)12 b(ha)o(v)o(e)g(delib)q(erately)75 1384 y(omitted)h(the)h Ft(\030)p 337 1400 24 2 v 17 w FC(argumen)o(t)g(b)q(ecause)g(it)g(is)h (useful)f(to)g(consider)h(\(6.6\))f(as)h(a)g(pro)q(cedure)g(for)g(passing)75 1444 y(from)j(one)h(set)g(of)g(co)q(e\016cien)o(ts)e Ft(\036)p FC(\()p Ft(X)t FC(\))i(lab)q(eled)g(b)o(y)f(subsets)h Ft(X)k Fs(\032)18 b(f)p FC(1)p Ft(;)8 b FC(2)p Ft(;)g(:)g(:)g(:)p Fs(g)19 b FC(to)g(another)g(set)g(of)75 1504 y(co)q(e\016cien)o(ts)c Ft(\036)p FC(\()p Ft(X)t FC(\))h(also)h(lab)q(eled)f(b)o(y)g(subsets.)148 1564 y(The)j(pro)q(of)h(that)g(\(6.6\))f(implies,)e(as)j(a)f(formal)f(p)q(o)o (w)o(er)h(series,)g(\(6.5\))g(is)g(standard)h(and)g(v)o(ery)75 1624 y(simple)13 b(\(Exercise)h(6.5.1\);)h(the)g(fact)g(that)h(it)f(uniquely) f(determines)f(\()p Ft(e)1400 1606 y Fp(\000)p Fq(\014)r(U)1478 1624 y FC(\))1497 1631 y Fq(c)1529 1624 y FC(can)j(b)q(e)f(seen)g(recur-)75 1685 y(siv)o(ely)h(starting)i(with)g Ft(N)k FC(=)16 b(1.)27 b(The)18 b(iden)o(tit)o(y)d(\(6.6\))j(implies)e(another)i(p)q(opular)h (expression)e(for)75 1745 y(the)f(Ursell)f(co)q(e\016cien)o(ts.)20 b(W)l(e)c(\014rst)g(notice)g(that)553 1855 y Ft(e)576 1834 y Fp(\000)p Fq(\014)r(U)t Fo(\()p Fq(N)t Fo(\))755 1855 y FC(=)865 1813 y Fr(Y)834 1907 y Fo(\()p Fq(ij)r Fo(\))p Fp(\032)p Fq(N)957 1807 y Fr(h\020)1001 1855 y Ft(e)1024 1834 y Fp(\000)p Fq(\014)r(u)p Fo(\()p Fq(\030)1123 1839 y Fm(i)1136 1834 y Fq(;\030)1162 1839 y Fm(j)1177 1834 y Fo(\))1204 1855 y Fs(\000)11 b FC(1)1278 1807 y Fr(\021)1314 1855 y FC(+)g(1)1387 1807 y Fr(i)1812 1855 y FC(\(6.7\))755 1986 y(=)868 1945 y Fr(X)834 2037 y Fq(G)17 b Fo(on)e Fq(N)998 1945 y Fr(Y)971 2039 y Fo(\()p Fq(ij)r Fo(\))p Fp(2)p Fq(G)1086 1938 y Fr(\020)1111 1986 y Ft(e)1134 1966 y Fp(\000)p Fq(\014)r(u)p Fo(\()p Fq(\030)1233 1971 y Fm(i)1245 1966 y Fq(;\030)1271 1971 y Fm(j)1287 1966 y Fo(\))1314 1986 y Fs(\000)10 b FC(1)1387 1938 y Fr(\021)1421 1986 y Ft(:)377 b FC(\(6.8\))75 2143 y(The)18 b(sum)e(is)i(o)o(v)o(er)f(all)g(graphs)i(with)e (v)o(ertices)f(on)i Ft(N)k Fs(\021)16 b(f)p FC(1)p Ft(;)8 b(:)g(:)g(:)g(;)g (N)d Fs(g)17 b FC(\(the)h(graphs)h(ma)o(y)d(con)o(tain)75 2203 y(isolated)c(v)o(ertices)e(not)i(b)q(elonging)g(to)g(an)o(y)g(b)q(ond)h(\()p Ft(ij)s FC(\)\).)19 b(W)l(e)11 b(p)q(erform)g(this)h(sum)f(b)o(y)g(\014rst)h (summing)75 2263 y(o)o(v)o(er)18 b(all)h(graphs)h Ft(G)g FC(compatible)d (with)i(a)g(partition)g Fs(f)p Ft(X)1151 2270 y Fo(1)1172 2263 y Ft(;)8 b(:)g(:)g(:)f(;)h(X)1321 2270 y Fq(k)1343 2263 y Fs(g)p FC(,)19 b(and)h(then)f(summing)d(o)o(v)o(er)75 2323 y(partitions.)37 b(A)21 b(graph)h Ft(G)g FC(is)f(said)h Fn(c)n(omp)n(atible)g FC(with)f(the)g(partition)g Fs(f)p Ft(X)1473 2330 y Fo(1)1493 2323 y Ft(;)8 b(:)g(:)g(:)g(;)g(X)1643 2330 y Fq(k)1664 2323 y Fs(g)22 b FC(i\013)f Ft(G)h FC(is)f(a)75 2383 y(union)15 b(of)f(maximal)d(connected)j(comp)q(onen)o(ts)g Ft(G)990 2390 y Fo(1)1010 2383 y Ft(;)8 b(:)g(:)g(:)f(;)h(G)1157 2390 y Fq(k)1179 2383 y FC(,)14 b(where)g Ft(G)1384 2390 y Fq(i)1413 2383 y FC(is)g(a)h(connected)f(graph)h(on)969 2599 y(47)p eop %%Page: 48 52 bop 75 307 a Ft(X)115 314 y Fq(i)130 307 y Ft(;)8 b(i)13 b FC(=)h(1)p Ft(;)8 b(:)g(:)g(:)f(;)h(k)r FC(.)22 b(In)16 b(this)g(w)o(a)o(y)g (w)o(e)f(obtain)i(\(for)g Ft(N)i Fs(\025)13 b FC(2\):)418 473 y Ft(e)441 453 y Fp(\000)p Fq(\014)r(U)t Fo(\()p Fq(X)q(;\030)p 572 466 17 2 v -1 w Fo(\))616 473 y FC(=)727 432 y Fr(X)668 526 y Fp(f)p Fq(X)715 531 y Fl(1)732 526 y Fq(;:::)o(;X)810 532 y Fm(k)829 526 y Fp(g)874 419 y Fq(k)857 432 y Fr(Y)855 523 y Fq(i)p Fo(=1)921 375 y Fr(2)921 448 y(6)921 475 y(4)989 432 y(X)963 518 y Fm(G)c Fl(on)g Fm(X)1062 526 y(i)953 548 y Fl(connected)1125 432 y Fr(Y)1099 526 y Fo(\()p Fq(ij)r Fo(\))p Fp(2)p Fq(G)1205 473 y FC(\()p Ft(e)1247 453 y Fp(\000)p Fq(\014)r(u)p Fo(\()p Fq(\030)1346 458 y Fm(i)1359 453 y Fq(;\030)1385 458 y Fm(j)1400 453 y Fo(\))1427 473 y Fs(\000)i FC(1\))1520 375 y Fr(3)1520 448 y(7)1520 475 y(5)1556 473 y Ft(:)242 b FC(\(6)p Ft(:)p FC(9\))75 644 y(Comparing)16 b(with)g(\(6.6\))h(w)o(e)e(see)h(that)350 865 y(\()p Ft(e)392 844 y Fp(\000)p Fq(\014)r(U)470 865 y FC(\))489 872 y Fq(c)506 865 y FC(\()p Ft(N)r(;)8 b(\030)p 588 881 24 2 v 3 w FC(\))14 b(=)697 715 y Fr(8)697 753 y(>)697 765 y(>)697 778 y(>)697 790 y(>)697 803 y(>)697 815 y(<)697 890 y(>)697 902 y(>)697 915 y(>)697 927 y(>)697 939 y(>)697 952 y(:)795 714 y(X)775 800 y Fm(G)9 b Fl(on)g Fm(N)759 829 y Fl(connected)931 714 y Fr(Y)905 808 y Fo(\()p Fq(ij)r Fo(\))p Fp(2)p Fq(G)1011 756 y FC(\()p Ft(e)1053 735 y Fp(\000)p Fq(\014)r(u)p Fo(\()p Fq(\030)1152 740 y Fm(i)1165 735 y Fq(;\030)1191 740 y Fm(j)1206 735 y Fo(\))1233 756 y Fs(\000)i FC(1\))42 b(for)16 b Ft(N)j Fs(\025)14 b FC(2)754 875 y(1)590 b(for)16 b Ft(N)j FC(=)14 b(1)754 977 y(0)590 b(for)16 b Ft(N)j FC(=)14 b(0)g Ft(;)1788 865 y FC(\(6)p Ft(:)p FC(10\))75 1083 y(This)g(expression)g(constitutes)g (another)g(complete)e(c)o(haracterization)h(of)h(the)g(Ursell)e(co)q (e\016cien)o(ts,)75 1143 y(equiv)m(alen)o(t)19 b(to)i(\(6.6\).)35 b(In)20 b(con)o(trast)h(with)g(\(6.8\),)g(the)g(sum)e(in)i(\(6.10\))g(is)f (only)h(o)o(v)o(er)f Fn(c)n(onne)n(cte)n(d)75 1204 y FC(graphs)j(on)f Fs(f)p FC(1)p Ft(;)8 b FC(2)p Ft(;)g(:)g(:)g(:)g(;)g(N)d Fs(g)p FC(.)38 b(Expression)22 b(\(6.10\))g(and)g(all)g(of)g(our)g(form)o(ulas)e(un) o(til)h(the)h(end)f(of)75 1264 y(Section)f(6.1)h(only)g(require)e(that)i Ft(U)27 b FC(b)q(e)20 b(of)i(the)e(form)g Ft(U)5 b FC(\()p Ft(N)g FC(\))22 b(=)1323 1231 y Fr(P)1367 1274 y Fo(\()p Fq(ij)r Fo(\))p Fp(\032)p Fq(N)1492 1264 y Ft(u)p FC(\()p Ft(ij)s FC(\);)g(the)e (fact)h(that)75 1324 y Ft(u)p FC(\()p Ft(ij)s FC(\))16 b(migh)o(t)e(ha)o(v)o (e)i(the)g(structure)f Ft(u)p FC(\()p Ft(\030)809 1331 y Fq(i)824 1324 y Ft(;)8 b(\030)867 1331 y Fq(j)885 1324 y FC(\))16 b(pla)o(ys)g(no)h (part)g(in)f(their)f(dev)o(elopmen)o(t.)148 1384 y(W)l(e)e(shall)f(no)o(w)h (describ)q(e)f(another)i(w)o(a)o(y)l(,)e(tak)o(en)h(from)e([17)q(],)h(to)h (obtain)h(the)e(Ursell)f(co)q(e\016cien)o(ts.)75 1444 y(It)19 b(follo)o(ws)g(the)g(general)h(philosoph)o(y)f(adv)o(o)q(cated)h(in)f(these)g (notes,)h(of)f(considering)h(con)o(tin)o(uous)75 1505 y(\015o)o(ws)j(and)g(c) o(haracterizing)e(ob)s(jects)g(as)i(solutions)g(of)g(di\013eren)o(tial)d (equations.)40 b(Moreo)o(v)o(er,)22 b(in)75 1565 y(Section)12 b(6.4)h(w)o(e)e(shall)i(see)f(that)g(it)g(is)g(in)h(direct)e(corresp)q (ondence)h(with)g(solutions)h(of)g(the)f(e\013ectiv)o(e-)75 1625 y(action)k(equation)g(\(5.3\).)148 1685 y(As)h(a)h(motiv)m(ation,)e(w)o (e)h(notice)g(that)h(expressions)f(of)h(the)f(form)f(\(6.10\))i(are)g (seldomly)d(useful)75 1745 y(to)20 b(pro)o(v)o(e)g(uniform)e(con)o(v)o (ergence)h(of)h(the)g(Ma)o(y)o(er)f(expansion.)33 b(In)20 b(man)o(y)e (situations)j(there)e(are)75 1805 y(to)q(o)f(man)o(y)e(connected)h(graphs)h Ft(G)g FC(in)o(v)o(olv)o(ed:)j(the)c(con)o(v)o(ergence)f(dep)q(ends)i (crucially)d(of)j(delicate)75 1866 y(cancellations)h(whic)o(h)h(are)g(lost)g (if)g(absolute)g(v)m(alues)h(are)f(used)g(to)g(b)q(ound)i(the)e(con)o (tribution)f(of)75 1926 y(eac)o(h)14 b(graph.)22 b(There)14 b(is,)h(therefore,)f(the)g(need)h(for)g(expressions)g(in)o(v)o(olving)e (less,)h(simpler)f(graphs.)75 1986 y(These)18 b(are)g(the)g(tree-graph)g (iden)o(tities.)25 b(Previous)18 b(w)o(a)o(ys)f(to)i(obtain)f(the)g (tree-graph)g(iden)o(tities)75 2046 y([9,)d(and)g(references)f(therein])g(w)o (ere)g(based)h(in)g(the)g(use)g(of)g(decoupling)g(parmeters)e(to)j(estimate|) 75 2106 y(via)g(in)o(tegration)g(b)o(y)g(parts|the)g(e\013ect)g(of)g (succesiv)o(e)f(b)q(ond)i(remo)o(v)m(als)e(that)i(partition)f(the)g(sites)75 2167 y(in)o(to)g(decoupled)f(clusters.)20 b(Here,)15 b(instead,)h(a)g(single) f(parameter)g Ft(t)h FC(will)f(b)q(e)h(in)o(tro)q(duced)f(and)i(the)75 2227 y(e\013ect)f(of)g(in\014nitesimal)e(v)m(ariations)j(\(di\013eren)o(tial) e(equations\))h(will)f(b)q(e)h(estimated.)148 2287 y(Supp)q(ose,)f(then,)e (that)h(to)g(eac)o(h)f(pair)h(\()p Ft(ij)s FC(\))f Fs(\032)h(f)p FC(1)p Ft(;)8 b FC(2)p Ft(;)g(:)g(:)g(:)p Fs(g)13 b FC(w)o(e)g(are)h(giv)o (en)f(a)h(con)o(tin)o(uously)e(di\013er-)75 2347 y(en)o(tiable)f(function)i Ft(\014)s(u)501 2354 y Fq(t)p Fo(0)532 2347 y FC(\()p Ft(ij)s FC(\))g(of)g Ft(t)p FC(.)19 b(If)13 b(w)o(e)f(let)g Ft(F)939 2354 y Fq(t)953 2347 y FC(\()p Ft(X)t FC(\))i(=)g(\()p Ft(e)1143 2329 y Fp(\000)p Fq(\014)r(U)1216 2334 y Fm(t)p Fl(0)1247 2347 y FC(\))1266 2354 y Fq(c)1283 2347 y FC(\()p Ft(X)t FC(\))f(then)g(a)g(short) g(computation)75 2407 y(whic)o(h)18 b(will)f(b)q(e)i(carried)f(out)g(shortly) h(sho)o(ws)g(that)g Ft(F)1082 2414 y Fq(t)1115 2407 y FC(satis\014es)g(the)f (system,)f(one)i(equation)f(for)969 2599 y(48)p eop %%Page: 49 53 bop 75 307 a FC(eac)o(h)16 b Ft(X)t FC(,)445 396 y Ft(d)p 436 418 43 2 v 436 464 a(dt)492 430 y(F)524 437 y Fq(t)539 430 y FC(\()p Ft(X)t FC(\))42 b(=)747 396 y(1)p 747 418 25 2 v 747 464 a(2)801 389 y Fr(X)785 480 y Fq(i;j)r Fp(2)p Fq(X)878 430 y FC(\()p Fs(\000)p Ft(\014)10 b FC(_)-21 b Ft(u)995 437 y Fq(t)1009 430 y FC(\()p Ft(ij)s FC(\)\))750 508 y Fr(8)750 545 y(>)750 557 y(>)750 570 y(<)750 645 y(>)750 657 y(>)750 670 y(:)787 620 y Ft(F)819 627 y Fq(t)834 620 y FC(\()p Ft(X)t FC(\))11 b(+)985 578 y Fr(X)976 670 y Fq(I)s Fp(\032)p Fq(X)1090 578 y Fr(X)1093 670 y Fq(i)p Fp(3)p Fq(I)1066 702 y(j)r Fp(2)p Fq(X)s Fp(n)p Fq(I)1187 620 y Ft(F)1219 627 y Fq(t)1233 620 y FC(\()p Ft(I)t FC(\))p Ft(F)1329 627 y Fq(t)1343 620 y FC(\()p Ft(X)k Fs(n)c Ft(I)t FC(\))1498 508 y Fr(9)1498 545 y(>)1498 557 y(>)1498 570 y(=)1498 645 y(>)1498 657 y(>)1498 670 y(;)1543 620 y Ft(:)231 b FC(\(6.11\))75 794 y(with)16 b(the)g(initial)f(condition)678 914 y Ft(F)710 921 y Fq(t)p Fo(=0)770 914 y FC(\()p Ft(X)t FC(\))f(=)918 841 y Fr(\()972 883 y FC(1)42 b(if)19 b Fs(j)p Ft(X)t Fs(j)14 b FC(=)g(1)972 943 y(0)42 b(if)19 b Fs(j)p Ft(X)t Fs(j)14 b(6)p FC(=)g(1)g Ft(:)1788 914 y FC(\(6)p Ft(:)p FC(12\))75 1051 y(If)i Ft(X)i FC(=)c Fs(;)i FC(or)h Fs(j)p Ft(X)t Fs(j)d FC(=)g(1,)j(equation)f(\(6.11\))h(is)f(to)h(b)q(e)f(understo)q(o)q(d)i(as) 1370 1031 y Fq(d)p 1363 1039 31 2 v 1363 1068 a(dt)1399 1051 y Ft(F)1431 1058 y Fq(t)1446 1051 y FC(\()p Ft(X)t FC(\))c(=)g(0)j(and)f (from)g(the)75 1111 y(initial)f(conditions)h(one)h(concludes)e(that)694 1231 y Ft(F)726 1238 y Fq(t)741 1231 y FC(\()p Ft(X)t FC(\))f(=)889 1158 y Fr(\()943 1200 y FC(0)58 b(if)16 b Ft(X)i FC(=)c Fs(;)943 1261 y FC(1)58 b(if)16 b Fs(j)p Ft(X)t Fs(j)e FC(=)f(1)i Ft(:)1788 1231 y FC(\(6)p Ft(:)p FC(13\))75 1363 y(The)21 b(system)e(\(6.11\){\(6.12\)) j(c)o(haracterizes)d(the)i(Ursell)e(co)q(e\016cien)o(ts)g(for)i(all)f Ft(X)26 b Fs(\032)21 b FC(\003)f(b)q(ecause)75 1423 y(the)g(in)o(tegrating)f (b)q(oth)i(sides)f(of)g(the)g(equations)g(w)o(e)g(obtain)g(a)h(set)e(of)i (equations)f(that)g(can)g(b)q(e)75 1483 y(recursiv)o(ely)10 b(and)k(uniquely)e(solv)o(ed)g(.)20 b(All)12 b(this)g(discussion)h(can)h(b)q (e)f(summarize)o(d)e(in)h(the)h(follo)o(wing)75 1587 y Fu(Prop)r(osition)18 b(6.1)24 b Fn(The)18 b(e)n(quations)g(\(6.11\){\(6.12\))e(have)j(a)e(unique)i (solution)g(which)f(for)f(every)75 1647 y(\014nite)i(subset)f Ft(X)g Fs(\032)c(f)p FC(1)p Ft(;)8 b FC(2)p Ft(;)g(:)g(:)g(:)p Fs(g)17 b Fn(is)g(the)h(Ursel)r(l)h(c)n(o)n(e\016cient)g(given)g(by:)385 1755 y Ft(F)417 1762 y Fq(t)431 1755 y FC(\()p Ft(X)t FC(\))42 b(=)634 1706 y Fr(\020)659 1755 y Ft(e)682 1734 y Fp(\000)p Fq(\014)r(U)755 1739 y Fm(t)p Fl(0)786 1706 y Fr(\021)811 1782 y Fq(c)836 1755 y FC(\()p Ft(X)t FC(\))555 1907 y(=)634 1807 y Fr(8)634 1845 y(>)634 1857 y(<)634 1932 y(>)634 1944 y(:)744 1806 y(X)709 1898 y Fq(G)17 b Fo(on)e Fq(X)697 1927 y Fo(connected)890 1806 y Fr(Y)863 1901 y Fo(\()p Fq(ij)r Fo(\))p Fp(2)p Fq(G)978 1800 y Fr(\020)1003 1848 y Ft(e)1026 1827 y Fp(\000)p Fq(\014)r(u)1095 1832 y Fm(t)p Fl(0)1124 1827 y Fo(\()p Fq(ij)r Fo(\))1193 1848 y Fs(\000)c FC(1)1267 1800 y Fr(\021)1333 1848 y FC(if)17 b Fs(j)p Ft(X)t Fs(j)d Ft(>)g FC(1)692 1973 y(1)617 b(if)17 b Fs(j)p Ft(X)t Fs(j)d FC(=)g(1)g(;)1788 1907 y(\(6.14\))75 2070 y Fn(or)j(by)g(r)n(e)n(cursive)h(solution)g(of)661 2206 y Ft(e)684 2186 y Fp(\000)p Fq(\014)r(U)757 2191 y Fm(t)p Fl(0)786 2186 y Fo(\()p Fq(X)s Fo(\))861 2206 y FC(=)972 2165 y Fr(X)913 2259 y Fp(f)p Fq(X)960 2264 y Fl(1)977 2259 y Fq(;:::)o(;X)1055 2265 y Fm(k)1074 2259 y Fp(g)1117 2152 y Fq(k)1100 2165 y Fr(Y)1117 2255 y Fo(1)1161 2206 y Ft(F)1193 2213 y Fq(t)1208 2206 y FC(\()p Ft(X)1267 2213 y Fq(j)1285 2206 y FC(\))p Ft(;)470 b FC(\(6)p Ft(:)p FC(15\))75 2362 y Fn(wher)n(e)13 b Ft(U)241 2369 y Fq(t)p Fo(0)273 2362 y FC(\()p Ft(X)t FC(\))h Fs(\021)449 2320 y Fr(X)422 2414 y Fo(\()p Fq(ij)r Fo(\))p Fp(\032)p Fq(X)545 2362 y Ft(u)573 2369 y Fq(t)p Fo(0)605 2362 y FC(\()p Ft(ij)s FC(\))p Fn(,)f(and)g(the)g(sum) f(of)g(the)h(RHS)g(is)f(over)h(al)r(l)h(p)n(artitions)e Fs(f)p Ft(X)1710 2369 y Fo(1)1730 2362 y Ft(;)c(:)g(:)g(:)f(;)h(X)1879 2369 y Fq(k)1901 2362 y Fs(g)75 2474 y Fn(of)17 b Ft(X)t Fn(,)h(with)g Ft(k)h Fn(arbitr)n(ary.)i(If)c Ft(k)f FC(=)e(0)p Fn(,)783 2441 y Fr(Q)823 2454 y Fq(k)823 2486 y Fo(1)852 2474 y Ft(F)884 2481 y Fq(t)899 2474 y FC(\()p Ft(\036)p FC(\))f Fs(\021)h FC(1)p Fn(.)969 2599 y FC(49)p eop %%Page: 50 54 bop 75 307 a Fu(Pro)r(of.)27 b FC(The)16 b(pro)q(of)g(is)e(complete)f(except) h(for)h(sho)o(wing)h(that)f(the)g(Ursell)f(co)q(e\016cien)o(t)f(de\014ned)i (b)o(y)75 367 y(\(6.15\))i(solv)o(es)f(\(6.11\),)g(\(6.12\).)148 428 y(Indeed,)d(if)h(w)o(e)g(denote)g(b)o(y)f Ft(f)5 b FC(\()p Ft(t)p FC(\))14 b(the)g(righ)o(t-hand)h(side)e(of)i(\(6.15\),)g(the)e(\015o)o (w)i(equation)f(\(\(6.11\)\))75 488 y(implies)568 549 y Ft(d)-8 b(f)p 568 571 47 2 v 570 617 a(dt)634 583 y FC(=)745 541 y Fr(X)686 635 y Fp(f)p Fq(X)733 640 y Fl(1)750 635 y Fq(;:::)o(;X)828 641 y Fm(k)847 635 y Fp(g)893 529 y Fq(k)876 541 y Fr(Y)873 634 y Fq(`)p Fo(=1)940 583 y Ft(F)972 590 y Fq(t)987 583 y FC(\()p Ft(X)1046 590 y Fq(`)1063 583 y FC(\))1117 541 y Fr(X)1090 635 y Fo(\()p Fq(ij)r Fo(\))p Fp(\032)p Fq(X)1213 583 y Fs(\000)p Ft(\014)11 b FC(_)-22 b Ft(u)1311 590 y Fq(t)1325 583 y FC(\()p Ft(ij)s FC(\))p Ft(;)371 b FC(\(6)p Ft(:)p FC(16\))75 715 y(where)18 b(the)h(terms)e(for)h(whic)o(h)g(\()p Ft(ij)s FC(\))g Fs(\032)f Ft(X)853 722 y Fq(`)889 715 y FC(for)i(some)e Ft(`)i FC(come)e(from)h(the)g (\014rst)h(term)e(in)h(the)g(\015o)o(w)75 776 y(\(6.11\))g(whilst)g(the)f (terms)g(with)g Ft(i)g Fs(2)f Ft(X)821 783 y Fq(`)838 776 y Ft(;)8 b(j)20 b Fs(2)c Ft(X)989 783 y Fq(`)1003 774 y Fk(0)1017 776 y Ft(;)8 b(`)17 b Fs(6)p FC(=)g Ft(`)1151 758 y Fp(0)1181 776 y FC(come)f(from)h(the)g(second)h(term)e(in)i(the)75 836 y(\015o)o(w)f(\(6.11\).)k(This)c(equation)f(\(6.16\))h(can)f(b)q(e)h (rewritten)e(as)722 954 y Ft(d)-8 b(f)p 722 976 V 724 1022 a(dt)788 987 y FC(=)839 902 y Fr(0)839 977 y(@)889 946 y(X)876 1038 y Fq(ij)r Fp(\032)p Fq(X)971 987 y Fs(\000)p Ft(\014)11 b FC(_)-22 b Ft(u)1069 994 y Fq(t)1083 987 y FC(\()p Ft(ij)s FC(\))1161 902 y Fr(1)1161 977 y(A)1205 987 y Ft(f)19 b(;)526 b FC(\(6)p Ft(:)p FC(17\))75 1139 y(and)24 b(the)f(initial)f(conditions)i (\(6.12\))g(imply)d Ft(f)5 b FC(\(0\))26 b(=)g(1.)43 b(The)23 b(only)h(solution)f(to)h(this)f(linear)75 1199 y(ODE)18 b(problem)f(is)g Ft(f)5 b FC(\()p Ft(t)p FC(\))17 b(=)f Ft(e)622 1181 y Fp(\000)p Fq(\014)r(U)695 1186 y Fm(t)p Fl(0)724 1181 y Fo(\()p Fq(X)s Fo(\))785 1199 y FC(.)26 b(By)17 b(the)h(equiv)m(alence)e(b)q(et)o(w)o(een)h (\(6.15\))i(and)f(\(6.14\),)h(this)75 1259 y(concludes)d(the)g(pro)q(of.)1372 b Fg(2)75 1389 y Fu(6.1.2)55 b(T)-5 b(ree-Graph)18 b(Iden)n(tit)n(y)75 1481 y FC(The)g(next)g(task)g(is)g(to)h(solv)o(e)e(\(6.11\){\(6.12\))j (again,)f(but)f(this)g(time)e(using)j(the)f(tree)f(graph)i(gen-)75 1542 y(erating)h(expansion)g(analogous)i(to)e(the)g(one)g(w)o(e)g(ha)o(v)o(e) f(already)h(seen)f(for)i(the)e(e\013ectiv)o(e)f(action)75 1602 y(when)e(w)o(e)g(solv)o(ed)g(\(5.42\).)75 1714 y Fu(Prop)r(osition)i(6.2)g (\(T)-5 b(ree-Graph)18 b(Iden)n(tit)n(y\))23 b Fn(F)l(or)j(any)h(\014nite)g (set)g Ft(X)35 b Fs(\032)30 b(f)p FC(1)p Ft(;)8 b FC(2)p Ft(;)g(:)g(:)g(:)p Fs(g)26 b Fn(with)75 1774 y Fs(j)p Ft(X)t Fs(j)14 b Ft(>)g FC(1)p Fn(,)243 1856 y Fr(\020)267 1904 y Ft(e)290 1884 y Fp(\000)p Fq(\014)r(U)363 1889 y Fm(t)p Fl(0)394 1856 y Fr(\021)419 1931 y Fq(c)445 1904 y FC(\()p Ft(X)t FC(\))41 b(=)681 1863 y Fr(X)648 1955 y Fq(T)21 b Fo(on)15 b Fq(X)783 1846 y Fr(Z)824 1859 y Fq(t)806 1940 y Fo(0)848 1904 y Ft(d)873 1884 y Fq(T)900 1904 y Ft(s)957 1863 y Fr(Y)932 1957 y Fo(\()p Fq(ij)r Fo(\))p Fp(2)p Fq(T)1036 1904 y FC(\()p Fs(\000)p Ft(\014)c FC(_)-22 b Ft(u)1153 1912 y Fq(s)p Fo(\()p Fq(ij)r Fo(\))1226 1904 y FC(\()p Ft(ij)s FC(\)\))8 b(exp)g([)o Fs(\000)p Ft(\014)s(U)1530 1911 y Fq(ts)1559 1915 y Fl(max)1619 1904 y FC(\()p Ft(X)t FC(\)])g Ft(;)322 2052 y(U)355 2059 y Fq(ts)384 2063 y Fl(max)445 2052 y FC(\()p Ft(X)t FC(\))41 b(=)675 2010 y Fr(X)648 2104 y Fo(\()p Fq(ij)r Fo(\))p Fp(\032)p Fq(X)771 1993 y Fr(Z)812 2006 y Fq(t)794 2088 y(s)810 2092 y Fl(max)879 2052 y Ft(ds)17 b FC(_)-23 b Ft(u)963 2059 y Fq(s)982 2052 y FC(\()p Ft(ij)s FC(\))568 2199 y(=)648 2141 y Fr(Z)689 2154 y Fq(t)671 2235 y Fo(0)712 2199 y Ft(d\034)840 2158 y Fr(X)781 2252 y Fq(C)r Fp(2)p Fq(\015)r Fo(\()p Fq(\034)r(;s)p 894 2259 17 2 v(;T)5 b Fo(\))975 2199 y FC(_)-22 b Ft(u)995 2206 y Fq(\034)1016 2199 y FC(\()p Ft(C)t FC(\))p Ft(;)681 b FC(\(6.18\))75 2354 y Fn(wher)n(e)21 b Ft(s)239 2361 y Fo(max)326 2354 y FC(=)e Ft(s)406 2361 y Fo(max)473 2354 y FC(\()p Ft(ij)s FC(\))h Fn(is)h(the)g(lar)n(gest)f(p)n(ar)n(ameter)f Ft(s)1122 2361 y Fq(k)q(l)1175 2354 y Fn(enc)n(ounter)n(e)n(d)i(on)g(the)g (unique)g(p)n(ath)f(in)75 2414 y(the)j(tr)n(e)n(e)g(gr)n(aph)e Ft(T)29 b Fn(which)24 b(le)n(ads)f(fr)n(om)e(vertex)j Ft(i)e Fn(to)h(vertex)h Ft(j)s Fn(.)39 b Ft(T)29 b Fn(is)22 b(summe)n(d)g(over)h(al) r(l)h(tr)n(e)n(e-)75 2474 y(gr)n(aphs)18 b(on)i(the)f(set)h(of)f(vertic)n(es) h Ft(X)t Fn(,)g Ft(s)p 777 2481 23 2 v 17 w Fs(\021)c FC(\()p Ft(s)p FC(\()p Ft(ij)s FC(\))h(:)f(\()p Ft(ij)s FC(\))h Fs(2)g Ft(T)7 b FC(\))18 b Fn(is)h(a)h(set)f(of)g(p)n(ar)n(ameters,)f(one)i(for)969 2599 y FC(50)p eop %%Page: 51 55 bop 75 308 a Fn(e)n(ach)21 b(b)n(ond)g FC(\()p Ft(ij)s FC(\))f Fn(in)h Ft(T)7 b Fn(,)21 b(which)g(ar)n(e)f(inte)n(gr)n(ate)n(d)h(fr)n(om)e FC(0)j Fn(to)e Ft(t)h Fn(by)1322 273 y Fr(R)1349 286 y Fq(t)1341 321 y Fo(0)1372 308 y Ft(d)1397 290 y Fq(T)1425 308 y Ft(s)p Fn(,)h(the)f(family)f Ft(\015)s FC(\()p Ft(\034)s(;)8 b(s)p 1813 315 23 2 v(;)g(T)f FC(\))75 369 y Fn(is)19 b(the)g(p)n(artition)f(of)h Ft(X)k Fn(into)c(clusters)h Fs(f)p Ft(C)875 376 y Fo(1)895 369 y Ft(;)8 b(C)952 376 y Fo(2)971 369 y Ft(;)g(:)g(:)g(:)f(;)h(C)1115 376 y Fq(k)1137 369 y Fs(g)p Fn(,)18 b(wher)n(e)h(e)n(ach)g Ft(C)1480 376 y Fq(i)1513 369 y Fn(c)n(onsists)g(of)g(vertic)n(es)75 429 y(c)n(onne)n(cte)n(d)f(by)g Ft(T)f Fs(n)11 b(f)p FC(\()p Ft(ij)s FC(\))j Fs(2)g Ft(T)20 b FC(:)13 b Ft(s)p FC(\()p Ft(ij)s FC(\))h Fs(\025)f Ft(\034)6 b Fs(g)p Fn(.)148 528 y FC(The)17 b(second)f(form)o(ula)f(for)i Ft(U)693 535 y Fq(ts)722 539 y Fl(max)799 528 y FC(follo)o(ws)f(from)f(the)h(\014rst)h(b)o(y)f(merely)d (unra)o(v)o(elling)i(the)h(de\014-)75 588 y(nitions)e(although)h(it)f(migh)o (t)f(giv)o(e)g(y)o(ou)h(a)h(headac)o(he)f(trying)g(to)h(see)f(it.)20 b(This)14 b(form)o(ula)f(sho)o(ws)i(that)75 648 y Ft(U)108 655 y Fq(ts)137 659 y Fl(max)214 648 y FC(is)h(a)g(stable)h(in)o(teraction)e (when)25 b(_)-23 b Ft(u)845 655 y Fq(s)880 648 y FC(is)16 b(stable.)75 730 y Fu(Pro)r(of.)27 b FC(By)19 b(the)g(last)h(prop)q(osition)h(it)e (su\016ces)g(to)h(sho)o(w)g(that)g(the)f(righ)o(t-hand)h(side)1736 717 y(~)1725 730 y Ft(F)1757 737 y Fq(t)1771 730 y FC(\()p Ft(X)t FC(\))g(of)75 790 y(\(6.18\))i(solv)o(es)f(the)g(\015o)o(w)g(\(6.11\)) h(and)g(satis\014es)f(the)g(initial)f(condition)h(\(6.12\).)37 b(This)21 b(last)h(fact)75 850 y(is)g(ob)o(vious)h(\()343 837 y(~)332 850 y Ft(F)364 857 y Fq(t)378 850 y FC(\()p Ft(X)t FC(\))h(=)g(1)f(b)o(y)f(de\014nition)g(when)g Fs(j)p Ft(X)t Fs(j)i FC(=)g(1\).)40 b(T)l(o)23 b(c)o(hec)o(k)d(Equation)j(\(6.11\))g(w)o(e) 75 910 y(notice)d(that)i(when)477 891 y Fq(d)p 470 899 31 2 v 470 927 a(dt)527 910 y FC(acts)f(on)716 898 y(~)705 910 y Ft(F)27 b FC(it)20 b(can)h(di\013eren)o(tiate)f(the)h(upp)q(er)g(limit)d(of)j (the)g(in)o(tegral)f(in)75 970 y(the)e(exp)q(onen)o(t)h(resulting)f(in)g(a)h (term)d(whic)o(h)i(matc)o(hes)f(the)h(linear)g(term)f(in)h(\(6.11\).)29 b(The)18 b(other)75 1031 y(p)q(ossibilit)o(y)13 b(is)h(for)436 1011 y Fq(d)p 430 1019 V 430 1048 a(dt)480 1031 y FC(to)g(act)g(on)h(the)f (upp)q(er)g(limit)e(of)i(the)1151 995 y Fr(R)1179 1008 y Fq(t)1171 1043 y Fo(0)1202 1031 y Ft(ds)p FC(\()p Ft(ij)s FC(\))g(inside)1478 995 y Fr(R)1506 1008 y Fq(t)1498 1043 y Fo(0)1529 1031 y Ft(d)1554 1013 y Fq(T)1582 1031 y Ft(s)p FC(,)g(in)g(whic)o(h)g(case)75 1091 y(w)o(e)i(obtain)h(a)f(sum)f(o)o(v)o(er)h Ft(ij)g Fs(\032)e Ft(X)20 b FC(of)d(terms)538 1166 y Fr(X)528 1258 y Fq(T)549 1263 y Fm(i)561 1258 y Fq(;T)592 1263 y Fm(j)616 1149 y Fr(Z)658 1162 y Fq(t)639 1243 y Fo(0)681 1207 y Ft(d)706 1187 y Fq(T)727 1192 y Fm(i)742 1207 y Ft(s)773 1149 y Fr(Z)815 1162 y Fq(t)796 1243 y Fo(0)838 1207 y Ft(d)863 1187 y Fq(T)884 1192 y Fm(j)902 1207 y Ft(s)8 b(I)t FC(\()p Ft(T)1007 1214 y Fq(i)1021 1207 y FC(\))p Ft(I)t FC(\()p Ft(T)1114 1214 y Fq(j)1131 1207 y FC(\)\()p Fs(\000)p Ft(\014)i FC(_)-21 b Ft(u)1267 1214 y Fq(t)1280 1207 y FC(\()p Ft(ij)s FC(\)\))735 1375 y(exp)818 1290 y Fr(2)818 1365 y(4)846 1375 y Fs(\000)p Ft(\014)938 1334 y Fr(X)923 1426 y Fq(k)q(l)p Fp(\032)p Fq(X)1021 1317 y Fr(Z)1062 1330 y Fq(t)1044 1411 y(s)1060 1415 y Fl(max)1129 1375 y Ft(d\034)23 b FC(_)-23 b Ft(u)1217 1382 y Fq(\034)1238 1375 y FC(\()p Ft(k)r(l)q FC(\))1319 1290 y Fr(3)1319 1365 y(5)1355 1375 y Ft(;)419 b FC(\(6.19\))75 1513 y(where)21 b(the)g(sum)f(o)o(v)o(er)h Ft(T)28 b FC(b)q(ecomes)20 b(a)h(sum)g(o)o(v)o(er)f Ft(T)1076 1520 y Fq(i)1111 1513 y FC(and)i Ft(T)1240 1520 y Fq(j)1279 1513 y FC(whic)o(h)f(are)g(the)h(comp)q (onen)o(ts)e(of)75 1573 y Ft(T)d Fs(n)11 b(f)p FC(\()p Ft(ij)s FC(\))p Fs(g)16 b FC(that)h(con)o(tain)f Ft(i)g FC(and)h Ft(j)i FC(resp)q(ectiv)o(ely)14 b(and)688 1676 y Ft(I)t FC(\()p Ft(T)762 1683 y Fq(i)775 1676 y FC(\))f(=)878 1635 y Fr(Y)859 1727 y Fq(k)q(`)p Fp(2)p Fq(T)937 1732 y Fm(i)958 1628 y Fr(\020)983 1676 y Fs(\000)p Ft(\014)e FC(_)-22 b Ft(u)1081 1684 y Fq(s)p Fo(\()p Fq(k)q(`)p Fo(\))1160 1676 y FC(\()p Ft(k)r(`)p FC(\))1245 1628 y Fr(\021)1278 1676 y Ft(;)496 b FC(\(6)p Ft(:)p FC(20\))75 1814 y(and)13 b(lik)o(ewise)d(for)i Ft(I)t FC(\()p Ft(T)485 1821 y Fq(j)503 1814 y FC(\).)20 b(The)12 b(main)f(p)q(oin)o(t)h(is)g(that)h Ft(s)p FC(\()p Ft(ij)s FC(\))f(is)g(no)o(w)g(set)h(equal)e(to)i Ft(t)f FC(and)h(b)o(y)e(referring)75 1875 y(to)i(the)f(de\014nition)g(of)g Ft(s)498 1882 y Fo(max)578 1875 y FC(w)o(e)g(see)g(that)h(the)f(exp)q(onen)o (tial)f(in)h(\(6.19\),)i(call)d(it)h(exp)o(\()p Fs(\000)p Ft(\014)s(W)7 b FC(\()p Ft(T)t(;)h(s)p 1785 1882 23 2 v -1 w FC(\()p Ft(T)f FC(\)\),)75 1935 y(factors)17 b(and)g(\(6.19\))g(b)q(ecomes)218 2054 y Fs(\000)p Ft(\014)11 b FC(_)-22 b Ft(u)316 2061 y Fq(t)330 2054 y FC(\()p Ft(ij)s FC(\))426 2012 y Fr(X)415 2104 y Fq(T)436 2109 y Fm(i)449 2104 y Fq(;T)480 2109 y Fm(j)504 1993 y Fr(\022)535 1995 y(Z)576 2009 y Fq(t)558 2090 y Fo(0)599 2054 y Ft(d)624 2033 y Fq(T)645 2038 y Fm(i)660 2054 y Ft(sI)t FC(\()p Ft(T)757 2061 y Fq(i)770 2054 y FC(\))p Ft(e)812 2033 y Fp(\000)p Fq(\014)r(W)5 b Fo(\()p Fq(T)934 2038 y Fm(i)947 2033 y Fq(;s)p Fo(\()p Fq(T)1008 2038 y Fm(i)1020 2033 y Fo(\)\))1050 1993 y Fr(\023)j(\022)1119 1995 y(Z)1161 2009 y Fq(t)1142 2090 y Fo(0)1184 2054 y Ft(d)1209 2033 y Fq(T)1230 2038 y Fm(j)1248 2054 y Ft(sI)t FC(\()p Ft(T)1345 2061 y Fq(j)1362 2054 y FC(\))p Ft(e)1404 2033 y Fp(\000)p Fq(\014)r(W)d Fo(\()p Fq(T)1526 2038 y Fm(j)1541 2033 y Fq(;s)p Fo(\()p Fq(T)1602 2038 y Fm(j)1618 2033 y Fo(\)\))1648 1993 y Fr(\023)974 2181 y FC(=)14 b Fs(\000)p Ft(\014)c FC(_)-21 b Ft(u)1124 2188 y Fq(t)1137 2181 y FC(\()p Ft(ij)s FC(\))1251 2139 y Fr(X)1255 2231 y Fq(I)s Fp(3)p Fq(i)1228 2263 y(X)s Fp(n)p Fq(I)s Fp(3)p Fq(j)1359 2168 y FC(~)1348 2181 y Ft(F)1380 2188 y Fq(t)1395 2181 y FC(\()p Ft(I)t FC(\))1469 2168 y(~)1459 2181 y Ft(F)1491 2188 y Fq(t)1504 2181 y FC(\()p Ft(X)16 b Fs(n)11 b Ft(I)t FC(\))i Ft(:)101 b FC(\(6.21\))75 2354 y(On)22 b(recalling)f(that)i(w)o(e)e(ha)o(v)o(e)h(to)g(sum)f(o)o(v)o(er)g(the)h(b)q (ond)h(\()p Ft(ij)s FC(\))h(=)g(\()p Ft(j)s(i)p FC(\))d(w)o(e)h(see)g(that)g (this)g(term)75 2414 y(matc)o(hes)14 b(the)i(second)g(term)e(in)h(the)h (\015o)o(w)g(\(6.11\))h(so)f(w)o(e)g(ha)o(v)o(e)f(pro)o(v)o(ed)g(that)h(the)g (righ)o(t-hand)g(side)86 2461 y(~)75 2474 y Ft(F)6 b FC(\()p Ft(X)t FC(\))17 b(of)f(\(6.18\))h(satis\014es)g(the)f(\015o)o(w.)1104 b Fg(2)969 2599 y FC(51)p eop %%Page: 52 56 bop 75 307 a Fu(6.1.3)55 b(T)-5 b(ree-Graph)18 b(Domination)75 400 y FC(The)c(preceding)e(tree-graph)i(iden)o(tit)o(y)e(yields)g(a)i(basic)f (to)q(ol)i(for)e(the)h(study)f(of)h(the)f(con)o(v)o(ergence)f(of)75 460 y(Ma)o(y)o(er)e(expansions)i(and)f(man)o(y)f(related)h(expansions:)19 b(the)11 b(tree-graph)h(domination)e(inequalities.)75 574 y Fu(Prop)r(osition)18 b(6.3)g(\(T)-5 b(ree-Graph)18 b(domination\))118 676 y Fn(\(i\))24 b(If)17 b Ft(u)p FC(\()p Ft(ij)s FC(\))c Fs(\025)h FC(0)k Fn(for)e(al)r(l)j FC(\()p Ft(ij)s FC(\))p Fn(,)e(then)646 786 y Fs(j)p FC(\()p Ft(e)702 765 y Fp(\000)p Fq(\014)r(U)780 786 y FC(\))799 793 y Fq(c)816 786 y FC(\()p Ft(X)t FC(\))p Fs(j)28 b(\024)1006 744 y Fr(X)1024 836 y Fq(T)1100 744 y Fr(Y)1075 838 y Fo(\()p Fq(ij)r Fo(\))p Fp(2)p Fq(T)1187 786 y Fs(j)p Ft(e)1224 765 y Fp(\000)p Fq(\014)r(u)p Fo(\()p Fq(ij)r Fo(\))1362 786 y Fs(\000)11 b FC(1)p Fs(j)p Ft(:)324 b FC(\(6)p Ft(:)p FC(22\))197 939 y Fn(wher)n(e)18 b Ft(T)23 b Fn(is)18 b(summe)n(d)f(over)g(al)r(l)i(tr)n(e)n(e-gr)n(aphs)e(on)g(the)h (set)g(of)g(vertic)n(es)g Ft(X)t Fn(.)103 1041 y(\(ii\))24 b(If)18 b Ft(u)277 1048 y Fq(t)p Fo(0)309 1041 y FC(\()p Ft(ij)s FC(\))c(=)454 1005 y Fr(R)482 1019 y Fq(t)473 1054 y Fo(0)505 1041 y Ft(d\034)g FC(_)-22 b Ft(u)585 1048 y Fq(\034)606 1041 y FC(\()p Ft(ij)s FC(\))p Fn(,)18 b(wher)n(e)26 b FC(_)-22 b Ft(u)883 1048 y Fq(t)915 1041 y Fn(is)18 b FC(stable)p Fn(,)g(that)g(is,)g (ther)n(e)g(exists)1567 1028 y FC(_)1550 1041 y Ft(B)1587 1048 y Fq(\034)1623 1041 y Fs(\025)d FC(0)j Fn(such)g(that)197 1101 y(for)f(al)r(l)i(\014nite)f(sets)g Ft(C)t Fn(,)878 1120 y Fr(X)853 1214 y Fo(\()p Fq(ij)r Fo(\))p Fp(\032)p Fq(C)980 1161 y FC(_)-22 b Ft(u)1000 1168 y Fq(\034)1035 1161 y Fs(\025)14 b(\000)1143 1149 y FC(_)1127 1161 y Ft(B)1164 1168 y Fq(\034)1185 1161 y Fs(j)p Ft(C)t Fs(j)p Ft(;)522 b FC(\(6)p Ft(:)p FC(23\))197 1292 y Fn(then,)18 b(for)f(any)g(\014nite)i(set)f Ft(X)t Fn(,)396 1422 y FC(\()p Ft(a)p FC(\))501 1372 y Fr(\014)501 1397 y(\014)501 1422 y(\014)515 1373 y(\020)540 1422 y Ft(e)563 1403 y Fp(\000)p Fq(\014)r(U)636 1408 y Fm(t)p Fl(0)666 1373 y Fr(\021)691 1449 y Fq(c)717 1422 y FC(\()p Ft(X)t FC(\))799 1372 y Fr(\014)799 1397 y(\014)799 1422 y(\014)42 b Fs(\024)941 1380 y Fr(X)907 1472 y Fq(T)21 b Fo(on)15 b Fq(X)1118 1380 y Fr(Y)1092 1474 y Fo(\()p Fq(ij)r Fo(\))p Fp(2)p Fq(T)1205 1361 y Fr(\024)1227 1363 y(Z)1268 1376 y Fq(t)1250 1457 y Fo(0)1292 1422 y Ft(ds)8 b Fs(j)p Ft(\014)j FC(_)-22 b Ft(u)1421 1429 y Fq(s)1438 1422 y FC(\()p Ft(ij)s FC(\))p Fs(j)1530 1361 y Fr(\025)1560 1422 y Ft(e)1583 1401 y Fq(\014)r Fp(j)p Fq(X)s Fp(j)p Fq(B)1684 1406 y Fm(t)p Fl(0)1788 1422 y FC(\(6.24\))401 1569 y(\()p Ft(b)p FC(\))501 1519 y Fr(\014)501 1544 y(\014)501 1569 y(\014)515 1521 y(\020)540 1569 y Ft(e)563 1551 y Fp(\000)p Fq(\014)r(U)636 1556 y Fm(t)p Fl(0)666 1521 y Fr(\021)691 1596 y Fq(c)717 1569 y FC(\()p Ft(X)t FC(\))799 1519 y Fr(\014)799 1544 y(\014)799 1569 y(\014)42 b Fs(\024)938 1528 y Fr(X)907 1619 y Fq(T)21 b Fo(on)15 b(X)1113 1528 y Fr(Y)1087 1622 y Fo(\()p Fq(ij)r Fo(\))p Fp(2)p Fq(T)1200 1508 y Fr(\024)1222 1510 y(Z)1263 1524 y Fq(t)1245 1605 y Fo(0)1286 1569 y Ft(ds)8 b Fs(j)p Ft(\014)k FC(_)-23 b Ft(u)1415 1576 y Fq(s)1433 1569 y FC(\()p Ft(ij)s FC(\))p Fs(j)p Ft(e)1548 1548 y Fo(2)p Fq(\014)r(B)1615 1552 y Fm(ts)1644 1508 y Fr(\025)1674 1569 y Ft(;)100 b FC(\(6.25\))197 1734 y Fn(wher)n(e)18 b Ft(B)372 1741 y Fq(ts)417 1734 y Fs(\021)469 1698 y Fr(R)497 1711 y Fq(t)489 1746 y(s)537 1721 y FC(_)520 1734 y Ft(B)557 1741 y Fq(\034)579 1734 y Ft(d\034)6 b Fn(.)148 1848 y(R)n(emarks:)24 b FC(Inequalit)o(y)19 b(\(6.22\))i(w)o(as)f(the)g (\014rst)h(one)g(pro)o(v)o(ed)e([55)q(,)g(48)q(,)h(52)q(].)33 b(It)20 b(can)g(b)q(e)h(more)75 1908 y(dramatically)14 b(stated)j(in)f(the)g (form)491 1972 y Fr(\014)491 1997 y(\014)491 2021 y(\014)491 2046 y(\014)491 2071 y(\014)491 2096 y(\014)505 2017 y(X)521 2109 y Fq(G)600 2017 y Fr(Y)573 2111 y Fo(\()p Fq(ij)r Fo(\))p Fp(2)p Fq(G)680 2059 y FC(\()p Ft(e)722 2038 y Fp(\000)p Fq(\014)r(u)p Fo(\()p Fq(ij)r Fo(\))860 2059 y Fs(\000)11 b FC(1\))953 1972 y Fr(\014)953 1997 y(\014)953 2021 y(\014)953 2046 y(\014)953 2071 y(\014)953 2096 y(\014)981 2059 y Fs(\024)1033 2017 y Fr(X)1051 2109 y Fq(T)1127 2017 y Fr(Y)1102 2111 y Fo(\()p Fq(ij)r Fo(\))p Fp(2)p Fq(T)1214 2059 y Fs(j)p Ft(e)1251 2038 y Fp(\000)p Fq(\014)r(u)p Fo(\()p Fq(ij)r Fo(\))1389 2059 y Fs(\000)g FC(1)p Fs(j)j Ft(;)283 b FC(\(6)p Ft(:)p FC(26\))75 2212 y(whic)o(h)23 b(sho)o(ws)h(the)f(large)h(amoun)o(t)f(of)h(cancellations) e(presen)o(ted)h(in)g(the)g(original)h(expression)75 2273 y(\(6.10\))e(for)g (the)f(Ursell)f(co)q(e\016cien)o(ts.)36 b(Inequalit)o(y)20 b(\(6.24\))i(yields)e(b)q(ounds)j(similar)c(to)j(Ruelle's)75 2333 y(criterium)11 b([56)q(,)j(page)g(84,)h(Theorem)e(4.3.1].)20 b(Inequalit)o(y)13 b(\(6.25\))h(is)g(an)h(impro)o(v)o(em)o(en)o(t)c(o)o(v)o (er)i(\(6.24\))75 2393 y(that)k(allo)o(ws,)e(for)i(instance,)e(to)i(treat)f (the)g(Y)l(uk)m(a)o(w)o(a)g(gas)h(b)q(ey)o(ond)g(the)f(region)g(of)g (stabilit)o(y)f(of)i(the)75 2453 y(in)o(teraction.)969 2599 y(52)p eop %%Page: 53 57 bop 75 307 a Fu(Pro)r(of.)27 b FC(W)l(e)17 b(obtain)f(part)h(\(i\))f(b)o(y)g (c)o(ho)q(osing)h Ft(u)958 314 y Fq(t)p Fo(0)990 307 y FC(\()p Ft(ij)s FC(\))d(=)g Ft(t)8 b(u)p FC(\()p Ft(ij)s FC(\))16 b(with)g(0)e Fs(\024)g Ft(t)g Fs(\024)g FC(1)i(in)g(Prop)q(osition)75 367 y(6.2.)21 b(Then)c(w)o(e)e(b)q(ound)i(the)f(exp)q(onen)o(t)g(in)f(the)h(prop) q(osition)h(b)o(y)f(dropping)25 b(_)-23 b Ft(u)1512 374 y Fq(k)q(`)1564 367 y FC(if)15 b Ft(k)r(`)i FC(is)f(not)g(in)g Ft(T)7 b FC(,)582 477 y(exp)g([)p Fs(\000)p Ft(\014)s(U)781 484 y Fq(ts)810 488 y Fl(max)869 477 y FC(])41 b Fs(\025)1027 436 y Fr(X)1005 530 y Fo(\()p Fq(ij)r Fo(\))p Fp(2)p Fq(T)1118 419 y Fr(Z)1159 432 y Fo(1)1141 513 y Fq(s)1157 517 y Fl(max)1226 477 y Ft(ds)17 b FC(_)-23 b Ft(u)1310 484 y Fq(s)1328 477 y FC(\()p Ft(ij)s FC(\))925 626 y(=)1013 584 y Fr(X)1005 676 y Fq(ij)r Fp(2)p Fq(T)1090 567 y Fr(Z)1132 580 y Fo(1)1113 661 y Fq(s)p Fo(\()p Fq(ij)r Fo(\))1196 626 y Ft(ds)17 b FC(_)-23 b Ft(u)1280 633 y Fq(s)1298 626 y FC(\()p Ft(ij)s FC(\))412 b(\(6.27\))75 755 y(b)o(y)18 b(de\014nition)g(of)h Ft(s)445 762 y Fo(max)513 755 y FC(.)27 b(When)19 b(this)f(is)h(substituted)f(in)o(to)g(Prop)q(osition) i(6.2)f(w)o(e)f(can)g(explicitly)75 815 y(ev)m(aluate)266 779 y Fr(R)302 815 y Ft(d)327 797 y Fq(T)355 815 y Ft(s)e FC(and)h(the)f(result)g (is)g(part)g(\(a\).)148 875 y(P)o(art)h(\(iia\):)j(Substitute)c(the)g (stabilit)o(y)f(b)q(ound)j(in)o(to)d(Prop)q(osition)j(6.2.)148 935 y(P)o(art)13 b(\(iib\):)18 b(W)l(e)12 b(tak)o(e)g(adv)m(an)o(tage)h(of)g (the)f(fact)g(that)g(when)h Ft(\034)18 b FC(in)12 b(the)g(exp)q(onen)o(t)g (of)g(Prop)q(osition)75 995 y(6.2)21 b(is)g(small)e(compared)g(with)i(most)f (of)h(the)f(parameters)g Ft(s)p FC(\()p Ft(ij)s FC(\),)h(then)f(man)o(y)g(of) h(the)f(clusters)75 1055 y Ft(C)i Fs(2)e Ft(\015)s FC(\()p Ft(\034)s(;)8 b(s)p 278 1062 23 2 v -1 w(;)g(T)f FC(\))19 b(ha)o(v)o(e)f (only)h(one)h(particle)e Fs(j)p Ft(C)t Fs(j)g FC(=)h(1)h(and)g(w)o(e)f(can)g (replace)f Fs(\000)1568 1043 y FC(_)1551 1055 y Ft(B)1588 1062 y Fq(\034)1610 1055 y Fs(j)p Ft(C)t Fs(j)g FC(b)o(y)h(zero)g(in)75 1116 y(the)g(stabilit)o(y)g(b)q(ound.)33 b(This)19 b(idea)h(app)q(eared)g(in) g([36])f(and)i([6)o(].)32 b(Th)o(us)20 b(in)f(Prop)q(osition)i(6.2)f(w)o(e)75 1176 y(substitute)451 1224 y Fr(X)392 1318 y Fq(C)r Fp(2)p Fq(\015)r Fo(\()p Fq(\034)r(;s)p 505 1325 17 2 v -1 w(;T)5 b Fo(\))586 1265 y FC(_)-22 b Ft(u)606 1272 y Fq(\034)627 1265 y FC(\()p Ft(C)t FC(\))41 b Fs(\025)g(\000)881 1253 y FC(_)864 1265 y Ft(B)901 1272 y Fq(\034)922 1265 y Fs(jf)p Ft(i)14 b FC(:)f Ft(i)h Fs(2)30 b FC(some)15 b Ft(C)20 b FC(with)c Fs(j)p Ft(C)t Fs(j)d Ft(>)h FC(1)p Fs(gj)745 1392 y(\025)41 b(\000)881 1380 y FC(_)864 1392 y Ft(B)901 1399 y Fq(\034)922 1392 y Fs(jf)p Ft(i)14 b FC(:)f Ft(s)p FC(\()p Ft(ij)s FC(\))g Fs(\024)h Ft(\034)22 b FC(for)17 b(some)e Ft(j)s Fs(gj)745 1516 y(\025)41 b(\000)881 1503 y FC(_)864 1516 y Ft(B)901 1523 y Fq(\034)934 1474 y Fr(X)931 1566 y Fq(i)p Fp(2)p Fq(X)1013 1474 y Fr(X)1011 1567 y Fq(b)p Fp(2)p Fq(T)1018 1597 y(b)p Fp(3)p Fq(i)1089 1516 y Ft(\022)q FC(\()p Ft(\034)16 b Fs(\000)11 b Ft(s)p FC(\()p Ft(b)p FC(\)\))p Ft(;)454 b FC(\(6.28\))75 1674 y(where)16 b Ft(\022)q FC(\()p Ft(t)p FC(\))f(is)h(the)g(step)g(function,)f Ft(\022)q FC(\()p Ft(t)p FC(\))f(=)f(1)k(if)e Ft(t)f Fs(\025)f FC(0)p Ft(;)8 b(\022)q FC(\()p Ft(t)p FC(\))14 b(=)g(0)i(if)f Ft(t)f(<)g FC(0)i(and)h Ft(b)e FC(is)h(a)g(b)q(ond)h Ft(k)r(`)g FC(in)75 1734 y Ft(T)7 b FC(.)21 b(Since)15 b(ev)o(ery)g(b)q(ond)i(con)o(tains)f(t)o (w)o(o)g(v)o(ertices,)644 1782 y Fr(X)585 1876 y Fq(C)r Fp(2)p Fq(\015)r Fo(\()p Fq(\034)r(;s)p 698 1883 V(;T)5 b Fo(\))780 1823 y FC(_)-23 b Ft(u)799 1830 y Fq(\034)820 1823 y FC(\()p Ft(C)t FC(\))14 b Fs(\025)f(\000)p FC(2)1043 1811 y(_)1026 1823 y Ft(B)1063 1830 y Fq(\034)1095 1782 y Fr(X)1093 1874 y Fq(b)p Fp(2)p Fq(T)1166 1823 y Ft(\022)q FC(\()p Ft(\034)k Fs(\000)10 b Ft(s)p FC(\()p Ft(b)p FC(\)\))p Ft(:)377 b FC(\(6)p Ft(:)p FC(29\))75 1959 y(P)o(art)16 b(\(iib\))g(follo)o(ws)g(imme)o(diately)d (on)j(putting)h(this)f(b)q(ound)h(in)o(to)f(Prop)q(osition)i(6.2.)219 b Fg(2)75 2100 y Ff(6.2)70 b(Con)n(v)n(ergence)22 b(of)h(Ma)n(y)n(er)h (Expansions)75 2193 y FC(The)g(tree-graph)g(b)q(ounds)h(for)f(the)f(Ursell)f (co)q(e\016cien)o(ts)g(immedi)o(ately)e(yield)j(criteria)f(for)i(the)75 2253 y(absolute)e(con)o(v)o(ergence)e(of)i(Ma)o(y)o(er)f(expansions.)38 b(Indeed,)22 b(as)h(review)o(ed)d(in)h(Section)h(6.1,)h(this)75 2313 y(expansion)17 b(has)g(the)f(form)558 2397 y(1)p 539 2419 62 2 v 539 2465 a Fs(j)p FC(\003)p Fs(j)614 2431 y FC(log)9 b Ft(Z)718 2438 y Fq(t)p Fo(0)765 2431 y FC(=)825 2389 y Fr(X)817 2481 y Fq(N)t Fp(\025)p Fo(0)906 2397 y Ft(z)931 2379 y Fq(N)p 906 2419 59 2 v 906 2465 a Ft(N)c FC(!)978 2372 y Fr(Z)1028 2431 y Ft(d)1053 2410 y Fq(N)1087 2431 y Ft(\030)11 b FC(\()p Ft(e)1161 2410 y Fp(\000)p Fq(\014)r(U)1234 2415 y Fm(t)p Fl(0)1265 2431 y FC(\))1284 2438 y Fq(c)1301 2431 y FC(\()p Ft(N)r(;)d(\030)r FC(\))15 b Ft(;)334 b FC(\(6)p Ft(:)p FC(30\))969 2599 y(53)p eop %%Page: 54 58 bop 75 307 a FC(and)23 b(one)g(is)f(in)o(terested)f(in)h(the)g(range)h(of)g (fugacities)f(for)h(whic)o(h)f(the)g(series)g(is)g(analytic)g(and)75 367 y(moreo)o(v)o(er)17 b(con)o(v)o(erges)i(term-b)o(y-term)d(to)k(an)g (analytic)f(series)f(as)j Fs(j)p FC(\003)p Fs(j)d(!)i(1)p FC(.)30 b(Suc)o(h)20 b(prop)q(erties)75 428 y(hold)c(at)h(least)f(if)g(the)g(series) 564 513 y Fr(X)555 605 y Fq(N)t Fp(\025)p Fo(0)645 521 y Fs(j)p Ft(z)r Fs(j)698 503 y Fq(N)p 645 543 87 2 v 659 589 a Ft(N)5 b FC(!)745 555 y(sup)765 594 y Fq(\030)781 599 y Fl(1)826 496 y Fr(Z)876 555 y Ft(d\030)922 562 y Fo(2)951 555 y Fs(\001)j(\001)g(\001)g Ft(d\030)1063 562 y Fq(N)1106 555 y Fs(j)p FC(\()p Ft(e)1162 534 y Fp(\000)p Fq(\014)r(U)1235 539 y Fm(t)p Fl(0)1265 555 y FC(\))1284 562 y Fq(c)1302 555 y FC(\()p Ft(N)r(;)g(\030)r FC(\))p Fs(j)348 b FC(\(6)p Ft(:)p FC(31\))75 702 y(con)o(v)o(erges.)20 b(If)15 b(w)o(e)g(no)o(w)h(use)f(the)h(tree-graph)g(b)q(ounds)h(of)e(Prop)q (osition)i(6.3)f(w)o(e)f(readily)f(conclude)75 762 y(the)i(follo)o(wing.)75 875 y Fu(Prop)r(osition)i(6.4)24 b Fn(If)17 b(either)118 976 y(\(i\))24 b Ft(u)p FC(\()p Ft(\030)265 983 y Fq(i)279 976 y Ft(;)8 b(\030)322 983 y Fq(j)340 976 y FC(\))14 b Fs(\025)g FC(0)k Fn(for)f(al)r(l)h FC(\()p Ft(ij)s FC(\))p Fn(,)f(and)702 1101 y FC(sup)724 1141 y Fq(\030)740 1146 y Fm(i)783 1042 y Fr(Z)833 1101 y Ft(d\030)879 1108 y Fq(j)906 1101 y Fs(j)p Ft(e)943 1080 y Fp(\000)p Fq(\014)r(u)p Fo(\()p Fq(\030)1042 1085 y Fm(i)1054 1080 y Fq(;\030)1080 1085 y Fm(j)1096 1080 y Fo(\))1123 1101 y Fs(\000)11 b FC(1)p Fs(j)j(\021)f Ft(q)j(<)d Fs(1)372 b FC(\(6)p Ft(:)p FC(32\))75 1246 y Fn(or)103 1347 y(\(ii\))24 b Ft(u)225 1354 y Fq(t)p Fo(0)257 1347 y FC(\()p Ft(\030)297 1354 y Fq(i)312 1347 y Ft(;)8 b(\030)355 1354 y Fq(j)373 1347 y FC(\))17 b Fn(is)h(as)f(in)h(\(ii\))f(of)g(Pr)n(op)n(osition) f(6.3,)h(and)570 1479 y FC(sup)593 1519 y Fq(\030)609 1524 y Fm(i)652 1421 y Fr(Z)694 1434 y Fq(t)675 1515 y Fo(0)717 1479 y Ft(ds)782 1418 y Fr(\024)804 1421 y(Z)853 1479 y Ft(d\030)899 1486 y Fq(j)918 1479 y Fs(j)p Ft(\014)11 b FC(_)-22 b Ft(u)991 1486 y Fq(s)1009 1479 y FC(\()p Ft(\030)1049 1486 y Fq(i)1063 1479 y Ft(;)8 b(\030)1106 1486 y Fq(j)1124 1479 y FC(\))p Fs(j)1157 1418 y Fr(\025)1187 1479 y Ft(e)1210 1459 y Fo(2)p Fq(\014)r(B)1277 1463 y Fm(ts)1320 1479 y Fs(\021)14 b Ft(q)h(<)f Fs(1)f FC(;)249 b(\(6)p Ft(:)p FC(33\))75 1640 y Fn(then)18 b(the)g(Mayer)f(exp)n(ansion)h (\(in)g(\014nite)h(or)d(in\014nite)k(volume\))e(is)f(c)n(onver)n(gent)i(for) 886 1749 y Ft(q)r Fs(j)p Ft(z)r Fs(j)p Ft(e)12 b(<)i FC(1)g Ft(:)686 b FC(\(6)p Ft(:)p FC(34\))75 1886 y Fu(Pro)r(of.)27 b FC(The)14 b(tric)o(k)d(is)i(to)h(p)q(erform)e(the)h(in)o(tegral)f(in)h (\(6.18\))h(o)o(v)o(er)e(the)h(v)o(ertices)e Ft(\030)1569 1893 y Fo(1)1589 1886 y Ft(;)d(:)g(:)g(:)g(;)g(\030)1720 1893 y Fq(N)1767 1886 y FC(of)13 b(eac)o(h)75 1946 y(tree)18 b(graph,)h(b)o(y)f (starting)h(from)f(the)g(outermost)g(branc)o(hes)g(\(\\lea)o(v)o(es"\))g(and) h(w)o(orking)f(in)o(w)o(ards.)75 2006 y(In)i(this)f(fashion)i(eac)o(h)e(in)o (tegral)h(factors)g(in)o(to)g(a)g(sequence)f(of)h(indep)q(enden)o(t)f(in)o (tegrals)g(and)i(w)o(e)75 2067 y(obtain)c(that)f(the)g(series)g(\(6.31\))h (is)f(ma)s(jorized|b)q(oth)f(in)h(cases)g(\(i\))g(and)h(\(ii\)|b)o(y)550 2165 y Fr(X)542 2257 y Fq(N)t Fp(\025)p Fo(1)631 2173 y Fs(j)p Ft(z)r Fs(j)684 2154 y Fq(N)p 631 2195 V 652 2240 a Ft(N)758 2165 y Fr(X)731 2257 y Fq(T)d Fo(on)9 b Fq(N)853 2206 y Ft(q)877 2186 y Fq(N)t Fp(\000)p Fo(1)969 2206 y FC(=)1029 2165 y Fr(X)1021 2257 y Fq(N)t Fp(\025)p Fo(1)1111 2173 y Fs(j)p Ft(z)r Fs(j)1164 2154 y Fq(N)p 1111 2195 V 1132 2240 a Ft(N)1202 2206 y(N)1246 2186 y Fq(N)t Fp(\000)p Fo(2)1325 2206 y Ft(q)1349 2186 y Fq(N)t Fp(\000)p Fo(1)1441 2206 y Ft(:)333 b FC(\(6)p Ft(:)p FC(35\))75 2354 y(The)20 b(last)g(expression)f(follo)o(ws)h(from)f(the)h(fact)f(that)i (a)f(tree)f(of)h Ft(N)25 b FC(v)o(ertices)18 b(has)j Ft(N)e Fs(\000)13 b FC(1)20 b(b)q(onds)75 2414 y(and)15 b(of)f(Ca)o(yley's)f (theorem)f(\(there)h(are)h(exactly)f Ft(N)1030 2396 y Fq(N)t Fp(\000)p Fo(2)1123 2414 y FC(trees)g(of)i Ft(N)k FC(v)o(ertices\).)g (Condition)14 b(\(6.34\))75 2474 y(follo)o(ws)i(from)f(Stirling's)h(theorem.) 1129 b Fg(2)969 2599 y FC(54)p eop %%Page: 55 59 bop 148 307 a FC(An)14 b(imp)q(ortan)o(t)g(particular)g(case)g(of)h(\(6.33\)) g(is)f(when)h(the)f(functions)g Fs(f)8 b FC(_)-22 b Ft(u)1488 314 y Fq(s)1506 307 y FC(\()p Ft(\030)1546 314 y Fq(i)1561 307 y Ft(;)8 b(\030)1604 314 y Fq(j)1622 307 y FC(\))p Fs(g)1666 323 y Fq(i;j)r Fp(2)p Fq(C)1771 307 y FC(consti-)75 367 y(tute)16 b(a)h(p)q(ositiv)o(e-de\014nite)e(form)g(for)h(all)g(\014nite)g(sets)g Ft(C)t FC(.)21 b(In)16 b(this)g(case)433 454 y Fr(X)419 546 y Fq(i;j)r Fp(2)p Fq(C)524 496 y FC(_)-22 b Ft(u)544 503 y Fq(s)562 496 y FC(\()p Ft(\030)602 503 y Fq(i)616 496 y Ft(;)8 b(\030)659 503 y Fq(j)678 496 y FC(\))27 b(=)795 462 y(1)p 795 484 25 2 v 795 530 a(2)852 454 y Fr(X)838 546 y Fq(i;j)r Fp(2)p Fq(C)854 580 y(i)p Fp(6)p Fo(=)p Fq(j)948 496 y FC(_)-22 b Ft(u)968 503 y Fq(s)986 496 y FC(\()p Ft(\030)1026 503 y Fq(i)1040 496 y Ft(;)8 b(\030)1083 503 y Fq(j)1102 496 y FC(\))28 b Fs(\025)c(\000)1256 462 y FC(1)p 1256 484 V 1256 530 a(2)1303 454 y Fr(X)1302 546 y Fq(i)p Fp(2)p Fq(C)1382 496 y FC(_)-23 b Ft(u)1401 503 y Fq(s)1419 496 y FC(\()p Ft(\030)1459 503 y Fq(i)1474 496 y Ft(;)8 b(\030)1517 503 y Fq(i)1531 496 y FC(\))14 b Ft(;)210 b FC(\(6)p Ft(:)p FC(36\))75 678 y(hence)15 b(w)o(e)h(can)h(c)o(ho)q(ose)763 736 y(_)746 748 y Ft(B)783 755 y Fq(\034)832 748 y FC(=)903 715 y(1)p 903 737 V 903 783 a(2)940 748 y(sup)969 788 y Fq(\030)1022 748 y Fs(j)8 b FC(_)-22 b Ft(u)1064 755 y Fq(\034)1085 748 y FC(\()p Ft(\030)r(;)8 b(\030)r FC(\))p Fs(j)23 b Ft(:)546 b FC(\(6)p Ft(:)p FC(37\))75 881 y Fu(Corollary)18 b(6.5)24 b Fn(If)i Fs(f)8 b FC(_)-22 b Ft(u)533 888 y Fq(t)548 881 y FC(\()p Ft(\030)588 888 y Fq(i)602 881 y Ft(;)8 b(\030)645 888 y Fq(j)663 881 y FC(\))p Fs(g)707 897 y Fq(i;j)r Fp(2)p Fq(C)825 881 y Fn(is)26 b(p)n(ositive-de\014nite)j(for) d(al)r(l)i(\014nite)f(sets)g Ft(C)t Fn(,)i(then)e(the)75 941 y(Mayer)17 b(exp)n(ansion)h(\(in)g(\014nite)g(or)f(in\014nite)i(volume\))g (is)e(analytic)h(if)585 1072 y Fs(j)p Ft(z)r Fs(j)p Ft(\014)676 1013 y Fr(Z)717 1026 y Fq(t)699 1108 y Fo(0)740 1072 y Ft(ds)815 1059 y FC(_)796 1072 y Ft(A)q FC(\()p Ft(s)p FC(\))8 b(exp)g Ft(\014)1024 1013 y Fr(Z)1066 1026 y Fq(t)1047 1108 y(s)1089 1072 y Ft(d\034)20 b FC(_)-20 b Ft(a)p FC(\()p Ft(\034)6 b FC(\))27 b Ft(<)1338 1038 y FC(1)p 1338 1060 V 1339 1106 a Ft(e)1381 1072 y(;)393 b FC(\(6)p Ft(:)p FC(38\))75 1195 y Fn(wher)n(e)720 1243 y FC(_)702 1255 y Ft(A)o FC(\()p Ft(s)p FC(\))28 b(=)g(sup)921 1295 y Fq(\030)974 1197 y Fr(Z)1024 1255 y Ft(d\030)1072 1235 y Fp(0)1093 1255 y Fs(j)8 b FC(_)-22 b Ft(u)1135 1262 y Fq(s)1153 1255 y FC(\()p Ft(\030)r(;)8 b(\030)1240 1235 y Fp(0)1253 1255 y FC(\))p Fs(j)502 b FC(\(6)p Ft(:)p FC(39\))75 1371 y Fn(and)753 1431 y FC(_)-20 b Ft(a)p FC(\()p Ft(s)p FC(\))27 b(=)h(sup)955 1470 y Fq(\030)1009 1431 y Fs(j)8 b FC(_)-22 b Ft(u)1051 1438 y Fq(\034)1072 1431 y FC(\()p Ft(\030)r(;)8 b(\030)1159 1410 y Fp(0)1172 1431 y FC(\))p Fs(j)22 b Ft(:)547 b FC(\(6)p Ft(:)p FC(40\))75 1596 y Ff(6.3)70 b(Application:)25 b(Con)n(v)n(ergence)16 b(of)h(the)g(Ma)n(y)n(er)g(Expansion)h(of)f(the)232 1671 y(Y)-6 b(uk)l(a)n(w)n(a)24 b(Gas)g(in)e(Tw)n(o)h(Dimensions)75 1763 y FC(W)l(e)h(will)f(deriv)o(e)g(the)g(result,)j(\014rst)e(obtained)h(b)o (y)e(Benfatto)h([6],)i(that)e(the)g(t)o(w)o(o-dimensional)75 1824 y(Y)l(uk)m(a)o(w)o(a)18 b(gas)h(has)f(a)h(con)o(v)o(ergen)o(t)d(Ma)o(y)o (er)h(expansion)h(for)g Ft(\014)h(<)d FC(4)p Ft(\031)r FC(.)26 b(This)18 b(is)g(a)g(system)f(whic)o(h)g(is)75 1884 y(not)e(stable)f(in)g (the)h(sense)f(of)h(Ruelle)d(so)k(this)e(result)g(cannot)h(b)q(e)f(directly)f (deduced)h(from)f(Ruelle's)75 1944 y(criterion)i([56,)h(page)h(84,)f(Theorem) f(4.3.1].)21 b(The)16 b(reason)h(wh)o(y)f(the)f(estimates)g(here)g(do)i(b)q (etter)f(is)75 2004 y(that)h(within)f(the)g(\015o)o(w)h(equation)g(approac)o (h)g(one)g(can)g(separate)g(out)g(the)f(simplest)f(parts)i(of)g(the)75 2064 y(Ursell)c(co)q(e\016cien)o(ts)g(and)j(solv)o(e)d(the)i(\015o)o(w)g (equation)f(for)h(them)e(exactly)l(.)20 b(This)14 b(is)h(in)f(essence)g(what) 75 2125 y(w)o(as)19 b(done)f(ab)q(o)o(v)o(e)g(in)g(the)g(pro)q(of)h(of)f (part)h(\(iib\))e(of)h(Prop)q(osition)i(6.3)e(when)g(w)o(e)g(recognized)f (that)75 2185 y(there)g(is)h(no)g(need)g(for)g(a)g(stabilit)o(y)f(b)q(ound)i (for)f(clusters)f(consisting)h(of)h(a)f(single)f(particle.)25 b(This)75 2245 y(idea)c(will)f(recurr)g(in)h(the)g(coming)e(section)i(on)h (cluster)e(expansions,)i(but)f(there)g(p)q(erturbation)75 2305 y(theory)16 b(will)f(b)q(e)i(used)f(to)h(calculate)e(the)h(relev)m(an)o(t)f (parts)i(more)e(accurately)l(.)148 2365 y(The)h(particles)g(are)g(describ)q (ed)g(b)o(y)g Ft(\030)g FC(=)e(\()p Ft(x;)8 b(\017)p FC(\))15 b(with)h Ft(\017)e FC(=)g Fs(\006)p FC(1)p Ft(:)i FC(The)g(t)o(w)o(o-b)q(o)q (dy)h(p)q(oten)o(tial)f(is)765 2474 y Ft(u)p FC(\()p Ft(\030)r(;)8 b(\030)880 2453 y Fp(0)893 2474 y FC(\))13 b Fs(\021)h Ft(\017\017)1018 2453 y Fp(0)1029 2474 y Ft(u)p FC(\()p Ft(x)c Fs(\000)h Ft(x)1192 2453 y Fp(0)1204 2474 y FC(\))565 b(\(6)p Ft(:)p FC(41\))969 2599 y(55)p eop %%Page: 56 60 bop 75 307 a FC(with)656 377 y Ft(u)p FC(\()p Ft(x)p FC(\))14 b(=)864 343 y(1)p 820 366 112 2 v 820 411 a(\(2)p Ft(\031)r FC(\))912 397 y Fo(2)945 319 y Fr(Z)995 377 y Ft(d)1020 357 y Fo(2)1040 377 y Ft(k)5 b FC(^)-27 b Ft(u)p FC(\()p Ft(k)r FC(\))p Ft(e)1183 357 y Fq(ik)q Fp(\001)p Fo(\()p Fq(x)p Fp(\000)p Fq(x)1305 345 y Fk(0)1315 357 y Fo(\))1788 377 y FC(\(6)p Ft(:)p FC(42\))75 484 y(for)833 554 y(^)g Ft(u)p FC(\()p Ft(k)r FC(\))14 b(=)1047 521 y(1)p 994 543 132 2 v 994 589 a(1)d(+)g Ft(k)1105 574 y Fo(2)1144 554 y Ft(:)630 b FC(\(6)p Ft(:)p FC(43\))75 656 y(Since)14 b Ft(u)p FC(\()p Ft(x)p FC(\))h(is)g(singular)g(at)h Ft(x)d FC(=)h(0)h(w)o(e)g(in)o(tro)q(duce)g(a)g(cuto\013)h(at)f(length)g (scale)g Ft(`)f Fs(\034)g FC(1)h(b)o(y)g(replacing)78 716 y(^)-27 b Ft(u)16 b FC(with)701 786 y(^)-28 b Ft(u)725 793 y Fo(1)p Fq(`)759 786 y FC(\()p Ft(k)r FC(\))14 b(=)948 752 y(1)p 895 774 V 895 820 a(1)d(+)g Ft(k)1006 806 y Fo(2)1042 786 y Fs(\000)1172 752 y FC(1)p 1097 774 175 2 v 1097 820 a Ft(`)1117 806 y Fp(\000)p Fo(2)1176 820 y FC(+)g Ft(k)1252 806 y Fo(2)1276 786 y Ft(:)498 b FC(\(6)p Ft(:)p FC(44\))75 888 y(\(W)l(e)21 b(substract)h(the)f(length)g (scales)h(\\up)f(to)h(scale)f Ft(l)q FC(".\))37 b(W)l(e)21 b(no)o(w)h(in)o(tro)q(duce)e(the)i(con)o(tin)o(uum)75 948 y(parameter)15 b Ft(t)h FC(v)m(arying)g(from)f Ft(`)i FC(to)g(1,)f(de\014ning)683 1065 y(^)-27 b Ft(u)708 1072 y Fq(t`)737 1065 y FC(\()p Ft(k)r FC(\))14 b(=)946 1032 y(1)p 872 1054 172 2 v 872 1100 a Ft(t)890 1085 y Fp(\000)p Fo(2)948 1100 y FC(+)d Ft(k)1024 1085 y Fo(2)1060 1065 y Fs(\000)1190 1032 y FC(1)p 1115 1054 175 2 v 1115 1100 a Ft(`)1135 1085 y Fp(\000)p Fo(2)1193 1100 y FC(+)g Ft(k)1269 1085 y Fo(2)1294 1065 y Ft(:)480 b FC(\(6)p Ft(:)p FC(45\))75 1180 y(W)l(e)17 b(are)g(in)o(terested)f(in)h(the)g(case)g Ft(t)e FC(=)h(1.)25 b(F)l(rom)16 b(the)h(results)g(of)g(the)g(previous)g(section)g (w)o(e)g(learn)75 1241 y(that,)h(to)f(sho)o(w)i(analyticit)o(y)c(of)j(the)f (Ma)o(y)o(er)f(expansion)i(uniformly)d(in)j(the)f(cuto\013,)h(it)f(is)g (enough)75 1301 y(to)g(obtain)f(a)h(b)q(ound)g(of)g(the)f(t)o(yp)q(e)g (\(6.33\):)595 1422 y(sup)623 1462 y Fq(\030)677 1362 y Fr(\024)699 1364 y(Z)740 1377 y Fo(1)722 1458 y(0)768 1422 y Ft(ds)824 1364 y Fr(Z)875 1422 y Ft(d\030)923 1402 y Fp(0)944 1422 y Fs(j)p Ft(\014)10 b FC(_)-21 b Ft(u)1017 1429 y Fq(s)1034 1422 y FC(\()p Ft(\030)r(;)8 b(\030)1121 1402 y Fp(0)1134 1422 y FC(\))p Fs(j)p Ft(e)1190 1402 y Fo(2)p Fq(\014)r(B)1257 1407 y Fl(1)p Fm(s)1289 1362 y Fr(\025)1325 1422 y Fs(\024)13 b Ft(q)389 b FC(\(6)p Ft(:)p FC(46\))75 1549 y(with)16 b(a)g(constan)o(t)g Ft(q)i FC(indep)q(enden)o(t)d(of)h Ft(`)p FC(.)21 b(In)16 b(particular,)f(as) 25 b(_)-22 b Ft(u)1235 1556 y Fq(s)1269 1549 y FC(is)15 b(p)q(ositiv)o(e)g (de\014nite,)g(w)o(e)g(can)h(use)75 1610 y(Corollary)g(6.5)h(with)284 1670 y Fr(Z)325 1683 y Fo(1)307 1764 y Fq(s)353 1728 y Ft(d\034)k FC(_)-20 b Ft(a)o FC(\()p Ft(\034)6 b FC(\))41 b(=)h Ft(u)653 1735 y Fo(1)p Fq(s)689 1728 y FC(\(0\))28 b(=)844 1670 y Fr(Z)894 1728 y Ft(d)919 1708 y Fo(2)939 1728 y Ft(k)5 b FC(^)-27 b Ft(u)994 1735 y Fo(1)p Fq(s)1030 1728 y FC(\()p Ft(k)r FC(\))545 1855 y(=)625 1794 y Fr(\022)675 1821 y FC(1)p 660 1843 54 2 v 660 1889 a(2)p Ft(\031)719 1794 y Fr(\023)750 1802 y Fo(2)778 1796 y Fr(Z)828 1855 y Ft(d)853 1834 y Fo(2)873 1855 y Ft(k)908 1794 y Fr(\032)998 1821 y FC(1)p 944 1843 132 2 v 944 1889 a(1)12 b(+)f Ft(k)1056 1875 y Fo(2)1091 1855 y Fs(\000)1223 1821 y FC(1)p 1146 1843 177 2 v 1146 1889 a Ft(s)1169 1875 y Fp(\000)p Fo(2)1227 1889 y FC(+)g Ft(k)1303 1875 y Fo(2)1328 1794 y Fr(\033)1373 1855 y FC(=)1445 1821 y(1)p 1430 1843 54 2 v 1430 1889 a(4)p Ft(\031)1497 1855 y FC(log)q(\()p Ft(s)1602 1834 y Fp(\000)p Fo(2)1649 1855 y FC(\))j Ft(;)92 b FC(\(6.47\))75 1972 y(and,)16 b(from)g(\(6.45\),)514 2073 y(_)496 2086 y Ft(A)o FC(\()p Ft(s)p FC(\))e(=)g(sup)687 2125 y Fq(\030)741 2027 y Fr(Z)790 2086 y Ft(d\030)838 2065 y Fp(0)868 2086 y FC(_)-23 b Ft(u)887 2093 y Fq(s)906 2086 y FC(\()p Ft(\030)r(;)8 b(\030)993 2065 y Fp(0)1006 2086 y FC(\))13 b(=)h(2)1122 2027 y Fr(Z)1173 2086 y Ft(dx)p Fs(j)8 b FC(_)-22 b Ft(u)1268 2093 y Fq(s)1286 2086 y FC(\()p Ft(x)p FC(\))p Fs(j)13 b FC(=)h(4)p Ft(s:)296 b FC(\(6)p Ft(:)p FC(48\))75 2216 y(\(The)22 b(factor)g(2)g(in)g(the)g (middle)d(term)h(is)i(due)g(to)g(the)g(t)o(w)o(o)g(v)m(alues)g(of)g(the)g(c)o (harge,)g Ft(\017)h FC(=)h Fs(\006)p FC(1.\))75 2276 y(Therefore,)13 b(from)f(\(6.38\))i(w)o(e)f(see)f(that)i(the)f(Ma)o(y)o(er)f(expansion)h (\(6.3\){\(6.4\))h(con)o(v)o(erges)f Fn(uniformly)75 2336 y FC(as)k(the)f(cuto\013)h Ft(`)f FC(decreases)g(to)h(zero)f(pro)o(vided)688 2455 y Fs(j)p Ft(z)r Fs(j)p FC(\(4)p Ft(\014)s FC(\))842 2396 y Fr(Z)883 2409 y Fo(1)864 2491 y Fq(`)911 2455 y Ft(ds)967 2407 y Fr(\020)992 2455 y Ft(ss)1038 2434 y Fp(\000)1070 2419 y Fl(2)p Fm(\014)p 1070 2427 35 2 v 1070 2447 a Fl(4)p Fm(\031)1112 2407 y Fr(\021)1164 2455 y Ft(<)1235 2421 y FC(1)p 1235 2443 25 2 v 1236 2489 a Ft(e)1278 2455 y(;)496 b FC(\(6)p Ft(:)p FC(49\))969 2599 y(56)p eop %%Page: 57 61 bop 75 307 a FC(that)17 b(is,)e(if)h Ft(\014)g(<)e FC(4)p Ft(\031)k FC(and)f Ft(z)h FC(is)e(in)g(the)g(disk)746 439 y Ft(e)p FC(\(2)p Ft(\014)s FC(\)[1)11 b Fs(\000)977 405 y Ft(\014)p 965 427 54 2 v 965 473 a FC(4)p Ft(\031)1024 439 y FC(])1038 419 y Fp(\000)p Fo(1)1085 439 y Fs(j)p Ft(z)r Fs(j)i Ft(<)h FC(1)p Ft(:)547 b FC(\(6)p Ft(:)p FC(50\))148 562 y(By)11 b(using)h(the)f(form)o (ula)f(in)h(Prop)q(osition)i(6.2)e(it)g(is)h(not)g(di\016cult)e(to)h(v)o (erify)l(,)f(with)i(these)f(uniform)75 622 y(b)q(ounds,)17 b(that)f(the)g(connected)f(parts)h(\(i.e.)e(the)i(deriv)m(ativ)o(es)e(of)i (\(4.3\)\))g(also)g(ha)o(v)o(e)f(limits)f(as)i Ft(`)e Fs(#)g FC(0)75 682 y(for)h Ft(\014)j FC(and)d Ft(z)i FC(as)f(ab)q(o)o(v)o(e.)21 b(Th)o(us)15 b(the)g(mo)q(del)f(without)h(cuto\013s)h(has)g(a)f(con)o(v)o (ergen)o(t)f(expansion)h(near)75 742 y Ft(z)h FC(=)d(0.)22 b(This)16 b(is)g(in)g(fact)g(a)h(w)o(a)o(y)f(to)o(w)o(ards)g(the)g (construction)g(of)h(the)f(massiv)o(e)e(Sine-Gordon)j(\014eld)75 803 y(theory)f(for)h Ft(\014)f(<)e FC(4)p Ft(\031)k FC(and)e Fs(j)p Ft(z)r Fs(j)g FC(small.)75 888 y Fu(Op)r(en)k(Problems)p FC(.)j(By)16 b(recen)o(t)g(results)i([43],)f(whic)o(h)g(I)g(shall)g(discuss)g (later,)g(it)g(is)g(kno)o(wn)h(that)75 948 y(this)f(mo)q(del)f(actually)h (has)h(a)g(con)o(v)o(ergen)o(t)e(Ma)o(y)o(er)g(expansion)i(for)f Ft(\014)i(<)c FC(8)p Ft(\031)k FC(pro)o(vided)e(one)h(omits)75 1008 y(the)f(\014rst)g Ft(n)h FC(terms)e(whic)o(h)g(div)o(erge)g(successiv)o (ely)f(of)j(thresholds)f Ft(\014)1339 1015 y Fq(n)1378 1008 y FC(=)e(8)p Ft(\031)r FC(\(1)d Fs(\000)1606 988 y Fo(1)p 1596 996 40 2 v 1596 1025 a(2)p Fq(n)1640 1008 y FC(\))17 b(whic)o(h)f(w)o(ere)75 1068 y(in)o(v)o(estigated)f(in)h([8,)g(50].)148 1128 y(W)l(e)g(w)o(ould)h (lik)o(e)d(to)j(see)f(a)g(direct)g(pro)q(of)h(of)g(this)f(based)h(on)g (further)f(insigh)o(ts)g(in)o(to)g(the)g(Ursell)75 1189 y(co)q(e\016cien)o (ts.)j(In)c([17])f(this)h(w)o(as)g(ac)o(hiev)o(ed)e(for)i(4)p Ft(\031)h Fs(\024)d Ft(\014)j(<)e FC(16)p Ft(\031)r(=)p FC(3,)i(but)f(it)f (seems)f(to)i(b)q(e)g(a)g(di\016cult)75 1249 y(problem)e(for)i Ft(\014)i Fs(\025)c FC(6)p Ft(\031)r FC(.)21 b(The)15 b(same)e(situation)i (holds)h(for)f(the)f(dip)q(ole)h(gas)h(\(or)f(Coulom)o(b)f(gas)i(with)75 1309 y Ft(\014)22 b(>)d FC(8)p Ft(\031)r FC(\))g(and)h(cuto\013s)g(at)g (short)g(distance)f(to)h(remo)o(v)o(e)d(ultra)o(violet)h(div)o(ergences:)26 b(it)19 b(is)g(kno)o(wn)75 1369 y(that)e(for)g(the)f(dip)q(ole)g(gas)h(the)f (pressure)h(is)f(analytic)g([30,)g(18)q(],)f(but)i(not)g(b)o(y)f(direct)f (estimates)g(on)75 1429 y(the)f(Ursell)f(co)q(e\016cien)o(ts.)20 b(W)l(e)14 b(shall)g(b)q(e)h(discussing)g(this)f(pro)q(of)i(later)e(as)h(w)o (ell.)k(F)l(or)c(the)f(Coulom)o(b)75 1490 y(gas)j(at)f Ft(\014)g(>)e FC(8)p Ft(\031)j FC(in)e(t)o(w)o(o-dimensions,)f(w)o(e)i(are)f(sure)h(that)g (the)f(pressure)h(is)f(analytic)g(but)h(Dimo)q(c)o(k)75 1550 y(and)h(Hurd)f(ha)o(v)o(e)f(not)i(y)o(et)e(pro)o(v)o(ed)h(it.)75 1694 y Ff(6.4)70 b(E\013ectiv)n(e)21 b(P)n(oten)n(tials)g(as)i(Ma)n(y)n(er)h (Expansions)75 1787 y FC(What)18 b(is)f(the)g(relation)g(b)q(et)o(w)o(een)f (the)h(last)g(section)g(on)h(the)f(Ma)o(y)o(er)f(expansion)h(and)h(section)f (5.3)75 1847 y(on)g(the)f(formal)f(expansion)i(for)f(the)g(action?)22 b(In)17 b(the)f(case)g(where)g(the)g(t)o(w)o(o-b)q(o)q(dy)i(p)q(oten)o(tial)e Ft(u)g FC(is)75 1907 y(p)q(ositiv)o(e)11 b(de\014nite)g(and)i(regular)f(on)h (the)e(diagonal)i(w)o(e)f(can)g(use)g(the)g(Sine-Gordon)h(transformation)75 1967 y(discussed)j(in)g(Section)g(2)g(to)h(rewrite)e(the)h(partition)h (function)f(\(6.1\),)623 2092 y Ft(Z)i FC(=)726 2034 y Fr(Z)775 2092 y Ft(d\026)829 2100 y Fq(u=\014)892 2092 y FC(\()p Ft(\036)p FC(\))8 b(exp)o(\()1060 2034 y Fr(Z)1110 2092 y Ft(z)r(d\030)17 b FC(:)c Ft(e)1248 2072 y Fq(i\014)r(\036)1318 2092 y FC(:\))p Ft(:)423 b FC(\(6)p Ft(:)p FC(51\))75 2228 y(where)16 b(:)d Ft(e)266 2210 y Fq(i\014)r(\036)336 2228 y FC(:)p Fs(\021)g Ft(e)430 2196 y Fl(1)p 430 2202 16 2 v 430 2222 a(2)450 2209 y Fq(\014)r(u)p Fo(\()p Fq(\030)q(;\030)q Fo(\))565 2228 y Ft(e)588 2210 y Fq(i\014)r(\036)660 2228 y FC(inserts)j(factors)h(whic)o(h)e (cancel)g(self-energies)g(in)h(the)g(in)o(teraction.)75 2288 y(Then)g(w)o(e)g(c)o(ho)q(ose)h(a)g(deformation)e Ft(C)777 2295 y Fp(1)p Fq(t)841 2288 y Fs(\021)e Ft(u)921 2295 y Fp(1)p Fq(t)971 2288 y Ft(=\014)19 b FC(and)e(con)o(tin)o(ue)e(with)796 2413 y(=)848 2354 y Fr(Z)897 2413 y Ft(d\026)951 2420 y Fp(1)p Fq(t)1010 2413 y FC(exp\()p Ft(V)1132 2420 y Fq(t)p Fo(0)1165 2413 y FC(\))p Ft(;)590 b FC(\(6)p Ft(:)p FC(52\))969 2599 y(57)p eop %%Page: 58 62 bop 75 307 a FC(where)16 b Ft(V)244 314 y Fq(t)p Fo(0)293 307 y FC(solv)o(es)g(the)g(\015o)o(w)g(\(5.1\))h(with)f(initial)f(data)747 424 y Ft(V)775 431 y Fo(00)827 424 y FC(=)878 366 y Fr(Z)928 424 y Ft(z)r(d\030)i FC(:)c Ft(e)1066 404 y Fq(i\014)r(\036)1136 424 y FC(:)1150 431 y Fp(1)p Fo(0)1219 424 y Ft(;)555 b FC(\(6)p Ft(:)p FC(53\))75 552 y(where)19 b(:)f Ft(e)274 534 y Fq(i\014)r(\036)348 552 y FC(:)362 559 y Fp(1)p Fq(t)412 552 y Fs(\021)g Ft(e)497 520 y Fl(1)p 497 526 16 2 v 497 547 a(2)517 534 y Fq(\014)r(C)564 538 y Fk(1)p Fm(t)608 534 y Fo(\()p Fq(\030)q(;\030)q Fo(\))681 552 y Ft(e)704 534 y Fq(i\014)r(\036)760 552 y FC(.)29 b(The)19 b(series)g(solution)g(\(5.33\))h(obtained)f(b)o(y)f(iterating)h(the)75 612 y(in)o(tegral)14 b(\015o)o(w)g(\(5.3\))h(is)f(clearly)f(an)i(expansion)f (in)g(p)q(o)o(w)o(ers)h(of)f Ft(z)r FC(,)h(as)g(is)f(the)g(Ma)o(y)o(er)f (expansion.)21 b(Its)75 673 y(limit)12 b Ft(t)i Fs(!)f(1)i FC(yields)f(an)h(expansion)h(for)f(log)9 b Ft(Z)19 b FC(that)d(m)o(ust)d (therefore)h(coincide)g(with)h(the)g(Ma)o(y)o(er)75 733 y(expansion.)148 793 y(If)h(w)o(e)g(substitute)g(in)o(to)g(\(5.1\))g(the)g(functional)g Fn(F)l(ourier)g FC(series)452 933 y Ft(V)480 940 y Fq(t)p Fo(0)513 933 y FC(\()p Ft(\036)p FC(\))e(=)646 891 y Fr(X)736 899 y FC(1)p 719 921 58 2 v 719 967 a Ft(N)5 b FC(!)790 874 y Fr(Z)840 933 y Ft(z)865 912 y Fq(N)898 933 y Ft(d)923 912 y Fq(N)957 933 y Ft(\030)11 b(V)1017 940 y Fq(t)p Fo(0)1050 933 y FC(\()p Ft(N)r(;)d(\030)p 1132 949 24 2 v 3 w FC(\))1194 879 y Fq(N)1184 891 y Fr(Y)1201 982 y Fo(1)1251 933 y FC(:)13 b Ft(e)1301 912 y Fq(i\014)r(\036)p Fo(\()p Fq(\030)1386 917 y Fm(j)1401 912 y Fo(\))1430 933 y FC(:)1444 940 y Fp(1)p Fq(t)1521 933 y Ft(:)253 b FC(\(6)p Ft(:)p FC(54\))75 1069 y(W)l(e)12 b(obtain)h(a)g(system)e(of)h (\015o)o(w)h(equations)g(and)g(initial)e(conditions)h(for)h(the)f(co)q (e\016cien)o(ts)f Ft(V)1741 1076 y Fq(t)p Fo(0)1774 1069 y FC(\()p Ft(N)r(;)d(\030)p 1856 1086 V 3 w FC(\),)75 1129 y(whic)o(h)19 b(is)g(the)g(same)g(as)h(the)f(equations)g(\(6.11\){\(6.12\))j(that)d(c)o (haracterized)f(the)i(Ursell)d(co)q(e\016-)75 1196 y(cien)o(ts)212 1148 y Fr(\020)237 1196 y Ft(e)260 1178 y Fp(\000)p Fq(\014)r(U)333 1183 y Fm(t)p Fl(0)364 1148 y Fr(\021)388 1223 y Fq(c)414 1196 y FC(\()p Ft(N)r(;)8 b(\030)r FC(\).)22 b(Th)o(us)691 1319 y Ft(V)719 1326 y Fq(t)p Fo(0)752 1319 y FC(\()p Ft(N)r(;)8 b(\030)r FC(\))15 b(=)942 1271 y Fr(\020)967 1319 y Ft(e)990 1298 y Fp(\000)p Fq(\014)r(U)1063 1303 y Fm(t)p Fl(0)1094 1271 y Fr(\021)1119 1346 y Fq(c)1144 1319 y FC(\()p Ft(N)r(;)8 b(\030)p 1226 1335 V 3 w FC(\))14 b Ft(:)491 b FC(\(6)p Ft(:)p FC(55\))75 1462 y Ff(6.5)70 b(Problems)75 1555 y Fu(6.5.1)24 b FC(Sho)o(w)16 b(that)h(\(6.6\))f(implies)e(that,)i(as)h(a)g(formal)e(p)q(o)o(w)o(er)h (series,)353 1700 y(exp)436 1615 y Fr(2)436 1690 y(4)472 1659 y(X)464 1751 y Fq(N)t Fp(\025)p Fo(1)549 1700 y Ft(z)574 1680 y Fq(N)615 1642 y Fr(Z)665 1700 y Ft(d)690 1680 y Fq(N)724 1700 y Ft(\030)k FC(\()p Ft(e)807 1680 y Fp(\000)p Fq(\014)r(U)885 1700 y FC(\))904 1707 y Fq(c)921 1700 y FC(\()p Ft(N)r(;)8 b(\030)p 1003 1717 V 3 w FC(\))1046 1615 y Fr(3)1046 1690 y(5)1087 1700 y FC(=)1148 1659 y Fr(X)1139 1751 y Fq(N)t Fp(\025)p Fo(0)1229 1667 y Ft(z)1254 1649 y Fq(N)p 1229 1689 59 2 v 1229 1735 a Ft(N)d FC(!)1301 1642 y Fr(Z)1351 1700 y Ft(d)1376 1680 y Fq(N)1410 1700 y Ft(\030)11 b(e)1465 1680 y Fp(\000)p Fq(\014)r(U)t Fo(\()p Fq(N)r(;\030)p 1596 1693 17 2 v -1 w Fo(\))1626 1700 y Ft(:)148 b FC(\(6)p Ft(:)p FC(56\))197 1846 y(F)l(or)16 b(an)h(answ)o(er)g(see)e([9],) h(page)h(176)g(or)g([56].)75 1952 y Fu(6.5.2)24 b FC(Chec)o(k)17 b(directly)h(that)h(if)699 1904 y Fr(\020)724 1952 y Ft(e)747 1934 y Fp(\000)p Fq(\014)r(U)820 1939 y Fm(t)p Fl(0)851 1904 y Fr(\021)875 1979 y Fq(c)912 1952 y FC(is)f(de\014ned)h(b)o(y)f(\(6.10\))i (then)e(it)h(is)f(the)h(unique)f(solu-)197 2020 y(tion)h(to)g(the)f(system)f (\(6.11\){\(6.12\).)31 b(Hin)o(ts:)25 b(T)l(o)19 b(c)o(hec)o(k)e(that)i(it)f (is)h(a)g(solution,)g(a)g(useful)197 2081 y(in)o(termediate)13 b(expression)j(is:)312 2171 y Ft(d)p 303 2193 43 2 v 303 2239 a(dt)351 2205 y FC(\()p Ft(e)393 2184 y Fp(\000)p Fq(\014)r(U)466 2189 y Fm(t)p Fl(0)497 2205 y FC(\))516 2212 y Fq(c)533 2205 y FC(\()p Ft(X)t FC(\))42 b(=)741 2171 y(1)p 741 2193 25 2 v 741 2239 a(2)796 2163 y Fr(X)779 2255 y Fq(i;j)r Fp(2)p Fq(X)872 2205 y FC(\()p Fs(\000)p Ft(\014)11 b FC(_)-22 b Ft(u)989 2212 y Fq(t)1003 2205 y FC(\()p Ft(ij)s FC(\)\))1159 2163 y Fr(X)1125 2255 y Fq(G)16 b Fo(on)g Fq(X)1113 2283 y Fo(connected)1308 2163 y Fr(Y)1279 2257 y Fo(\()p Fq(k)q(`)p Fo(\))p Fp(2)p Fq(G)1399 2157 y Fr(\020)1424 2205 y Ft(e)1447 2184 y Fp(\000)p Fq(\014)r(u)1516 2189 y Fm(t)p Fl(0)1545 2184 y Fo(\()p Fq(k)q(`)p Fo(\))1619 2205 y Fs(\000)11 b FC(1)1693 2157 y Fr(\021)748 2370 y Fs(\000)802 2336 y FC(1)p 802 2358 V 802 2404 a(2)857 2328 y Fr(X)840 2420 y Fq(i;j)r Fp(2)p Fq(X)933 2370 y FC(\()p Fs(\000)p Ft(\014)g FC(_)-22 b Ft(u)1050 2377 y Fq(t)1064 2370 y FC(\()p Ft(ij)s FC(\)\))1161 2309 y Fr(\024)1239 2328 y(X)1204 2420 y Fq(G)17 b Fo(on)e Fq(X)1192 2448 y Fo(connected)1215 2478 y Fq(G)p Fp(63)p Fo(\()p Fq(ij)r Fo(\))1393 2328 y Fr(Y)1363 2422 y Fo(\()p Fq(k)q(`)p Fo(\))p Fp(2)p Fq(G)1484 2322 y Fr(\020)1508 2370 y Ft(e)1531 2349 y Fp(\000)p Fq(\014)r(u)1600 2354 y Fm(t)p Fl(0)1630 2349 y Fo(\()p Fq(k)q(`)p Fo(\))1704 2370 y Fs(\000)c FC(1)1778 2322 y Fr(\021)969 2599 y FC(58)p eop %%Page: 59 63 bop 943 328 a Fs(\000)1049 286 y Fr(X)1014 378 y Fq(G)17 b Fo(on)f Fq(X)1002 406 y Fo(connected)1025 436 y Fq(G)p Fp(3)p Fo(\()p Fq(ij)r Fo(\))1224 286 y Fr(Y)1195 381 y Fo(\()p Fq(k)q(`)p Fo(\))p Fp(2)p Fq(G)1179 419 y Fo(\()p Fq(k)q(`)p Fo(\))p Fp(6)p Fo(=\()p Fq(ij)r Fo(\))1336 280 y Fr(\020)1361 328 y Ft(e)1384 307 y Fp(\000)p Fq(\014)r(u)1453 312 y Fm(t)p Fl(0)1482 307 y Fo(\()p Fq(k)q(`)p Fo(\))1556 328 y Fs(\000)11 b FC(1)1630 280 y Fr(\021)1655 267 y(\025)1691 328 y Ft(:)83 b FC(\(6.57\))75 558 y Fu(6.5.3)24 b FC(This)13 b(exercise)f(obtains)i(a)g(useful)f (expression)g(for)h(truncated)f(exp)q(ectations.)20 b(Let)14 b Ft(\026)1781 565 y Fq(C)1824 558 y FC(b)q(e)g(a)197 618 y(Gaussian)k (measure)e(with)g(regular)h(co)o(v)m(ariance)g Ft(C)t FC(.)22 b(De\014ne)17 b(the)g(truncated)g(exp)q(ectations)197 678 y Fs(h)p Ft(F)248 685 y Fo(1)268 678 y FC(;)8 b Ft(F)322 685 y Fo(2)341 678 y FC(;)g Ft(:)g(:)g(:)f FC(;)h Ft(F)482 685 y Fq(N)515 678 y Fs(i)17 b FC(of)f(smo)q(oth)h(functionals)f(\()p Ft(F)1078 685 y Fq(i)1092 678 y FC(\()p Ft(\036)p FC(\)\))1178 685 y Fq(i)p Fo(=1)p Fq(;)p Fo(2)p Fq(;:::)n(;N)1361 678 y FC(b)o(y)468 821 y Fs(h)p Ft(F)519 828 y Fo(1)539 821 y FC(;)8 b Ft(F)593 828 y Fo(2)612 821 y FC(;)g Ft(:)g(:)g(:)g FC(;)g Ft(F)754 828 y Fq(N)787 821 y Fs(i)14 b(\021)873 748 y Fr(")991 787 y Ft(@)1020 769 y Fq(N)p 902 809 242 2 v 902 855 a Ft(@)s(\025)959 862 y Fo(1)987 855 y Ft(:)8 b(:)g(:)f(@)s(\025)1109 862 y Fq(N)1156 821 y FC(log)1227 762 y Fr(Z)1277 821 y Ft(d\026)1331 828 y Fq(C)1370 821 y Ft(e)1393 771 y Fr(P)1442 800 y Fq(\025)1463 805 y Fm(i)1476 800 y Fq(F)1498 805 y Fm(i)1513 748 y Fr(#)1538 873 y Fq(\025)p 1538 880 21 2 v -1 w Fo(=0)1627 821 y Ft(:)147 b FC(\(6)p Ft(:)p FC(58\))197 970 y(W)l(ork)19 b(out)h(explicitly)d(the)i (\014rst)h(few)f(truncated)g(exp)q(ectations,)h Ft(N)k FC(=)19 b(1)p Ft(;)8 b FC(2)p Ft(;)g FC(3)20 b(and)g(pro)o(v)o(e)197 1030 y(that)429 1165 y Fs(h)p Ft(F)480 1172 y Fo(1)499 1165 y FC(;)8 b Ft(F)553 1172 y Fo(2)573 1165 y FC(;)g Ft(:)g(:)g(:)f FC(;)h Ft(F)714 1172 y Fq(N)747 1165 y Fs(i)42 b FC(=)921 1123 y Fr(X)887 1215 y Fq(T)21 b Fo(on)16 b Fq(N)1022 1106 y Fr(Z)1064 1119 y Fp(1)1045 1200 y Fo(0)1110 1165 y Ft(d)1135 1144 y Fq(T)1162 1165 y Ft(s)1193 1106 y Fr(Z)1244 1165 y Ft(d\026)1298 1172 y Fq(C)1323 1176 y Fm(ts)1350 1180 y Fl(max)1412 1165 y FC(\001)1453 1144 y Fq(T)1453 1177 y(s)p 1453 1184 19 2 v 1501 1111 a(N)1491 1123 y Fr(Y)1489 1214 y Fq(i)p Fo(=1)1554 1165 y Ft(F)1586 1172 y Fq(i)1600 1165 y FC(\()p Ft(\036)1648 1172 y Fq(i)1662 1165 y FC(\))197 1308 y(where)g(the)g(notation)h(is)f(as)h(in)f(Prop)q (osition)h(5.2.)969 2599 y(59)p eop %%Page: 60 64 bop 75 315 a Fv(7)83 b(Cluster)26 b(Expansions)75 425 y FC(In)17 b(Section)g(5)h(w)o(e)f(studied)g(the)g(solution)h(of)g(the)f(functional)g (heat)h(equation)f(of)h(Lemma)d(4.3)j(b)o(y)75 485 y(writing)j Ft(Z)281 492 y Fq(t)318 485 y FC(=)g Ft(e)400 467 y Fq(V)421 471 y Fm(t)457 485 y FC(and)h(\014nding)f(the)g(\015o)o(w)g(of)g Ft(V)1007 492 y Fq(t)1043 485 y FC(b)o(y)g(iterating)f(an)i(in)o(tegral)e (equation)h(for)g Ft(V)1884 492 y Fq(t)1899 485 y FC(.)75 545 y(This)c(metho)q(d)f(is)g(e\013ectiv)o(e)f(in)h(trac)o(king)g(the)h(lo)q (calit)o(y)e(prop)q(erties)i(of)g Ft(Z)1426 552 y Fq(t)1441 545 y FC(:)22 b(the)17 b Ft(L)1595 552 y Fo(1)1631 545 y FC(norms)f Fs(k)p Ft(V)1831 552 y Fq(t;N)1888 545 y Fs(k)75 613 y FC(app)q(earing)i(in)f (\(6.31\))h(are)g(small)d(if)i Ft(Z)792 620 y Fq(t)825 613 y FC(resem)o(bles)e(exp)1127 565 y Fr(h)1147 613 y Ft(z)1180 578 y Fr(R)1216 613 y Ft(d\030)r(e)1287 595 y Fq(i\014)r(\036)p Fo(\()p Fq(\030)q Fo(\))1389 565 y Fr(i)1408 613 y FC(.)25 b(Ho)o(w)o(ev)o(er,)15 b(an)o(y)i(metho)q(d)75 673 y(based)k(on)f(the)g (e\013ectiv)o(e)e(p)q(oten)o(tial)i Ft(V)32 b FC(will)19 b(encoun)o(ter)g (di\016culties)f(if)i(the)g(initial)e Ft(V)1705 680 y Fo(0)1746 673 y FC(has)j(an)o(y)75 734 y(kind)16 b(of)h(gro)o(wth)h(as)f Fs(j)p Ft(\036)p Fs(j)d(!)g(1)j FC(b)q(ecause)g(a)g(norm)f(of)h Ft(V)28 b FC(will)15 b(not)j(kno)o(w)e(the)h(di\013erence)e(b)q(et)o(w)o(een) 75 794 y Fs(\006)p Ft(V)27 b FC(and)17 b(one)g(of)f(them)f(migh)o(t)f(b)q(e)j (unstable,)f(e.g.,)f Fs(\006)p Ft(\025)1127 758 y Fr(R)1163 794 y Ft(\036)1192 776 y Fo(4)1212 794 y Ft(:)148 854 y FC(F)l(or)21 b(these)g(reasons)g(another)h(kind)e(of)h(expansion,)h(called)e(a)h(cluster)f (expansion)h([35)q(],)g(has)75 914 y(b)q(een)c(in)o(tro)q(duced)f(in)h(quan)o (tum)e(\014eld)i(theory)l(.)23 b(These)16 b(expansions)i(still)d(trac)o(k)i (lo)q(calit)o(y)e(but)i(are)75 974 y(able)j(to)g(distinguish)h(stable)f(and)g (unstable)h(p)q(oten)o(tials.)33 b(In)20 b(this)g(section)f(w)o(e)h(will)f (sho)o(w)i(that)75 1035 y(they)16 b(can)g(b)q(e)g(understo)q(o)q(d)i(in)d(v)o (ery)g(close)h(analogy)h(to)f(what)h(w)o(e)e(ha)o(v)o(e)h(done)g(so)h(far,)e (one)i(simply)75 1095 y(c)o(hanges)f(what)h(is)f(mean)o(t)f(b)o(y)h(exp)q (onen)o(tial!)148 1155 y(In)e(this)h(section,)f Ft(\030)i FC(=)e(\()p Ft(x;)8 b(\013)p FC(\))15 b(where)f Ft(\013)h FC(is)f(an)h(index)f(whic)o(h)g (tak)o(es)g(on)h(discrete)e(v)m(alues)i(from)e(a)75 1215 y(\014nite)j(set)h (\012.)24 b(It)17 b(lab)q(els)g(prop)q(erties)g(suc)o(h)f(as)i(c)o(harge)f (or)g(comp)q(onen)o(ts)g(of)g(a)h(direction)e(for)h(\014elds)75 1275 y(connected)c(with)h(dip)q(ole)f(gases,)i(and)f Ft(x)g Fs(2)g FC(R)905 1257 y Fq(d)939 1275 y FC(is)g(the)f(particle)g(co)q (ordinate.)21 b(The)13 b(term)g Fn(unit)i(blo)n(ck)75 1336 y FC(\(\001\))i(will)e(mean)h(a)h Fn(close)n(d)h FC(cub)q(e)f(of)g(side)f (one)h(cen)o(tered)f(on)h(one)g(of)h(the)e(p)q(oin)o(ts)i(of)f(a)g(unit)g (lattice)75 1396 y Fu(Z)109 1378 y Fq(d)150 1396 y Fs(\032)j FC(R)245 1378 y Fq(d)265 1396 y FC(.)33 b(A)19 b Fn(p)n(olymer)535 1378 y Fo(7)574 1396 y Ft(X)25 b FC(is)20 b(a)g(\014nite,)g(non-empt)o(y)f (union)h(of)g(closed)g(unit)g(blo)q(c)o(ks.)32 b(It)20 b(need)75 1456 y(not)f(b)q(e)h(connected.)28 b Fn(The)20 b(length)i(sc)n(ale)f(on)f (which)g(the)h(blo)n(cks)g(ar)n(e)e(\\unit")i(is)e(chosen)i(to)f(b)n(e)h(of) 75 1516 y(the)g(same)f(or)n(der)f(of)h(magnitude)h(as)f(the)g(length)i(sc)n (ale)f(given)h(by)e(an)g(ultr)n(a-violet)i(cuto\013)f(in)f(the)75 1576 y(me)n(asur)n(e)c Ft(d\026)317 1583 y Fq(t)332 1576 y Fn(.)21 b FC(This)15 b(is)g(b)q(ecause)h(at)f(this)g(scale)f(the)h(\014elds)g (b)q(ecome)e(appro)o(ximately)g(constan)o(t)j(so)75 1637 y(there)g(are)g(no)h (signi\014can)o(t)f(\015uctuations)g(inside)g(suc)o(h)g(a)h(unit)f(blo)q(c)o (k.)148 1697 y(W)l(e)g(will)f(represen)o(t)g(solutions)h Ft(Z)770 1704 y Fq(t)802 1697 y FC(to)g(the)g(functional)f(heat)i(equation)f(of)g (Lemma)e(4.3)i(in)g(the)75 1757 y(form)c(of)i(a)g Fn(p)n(olymer)g(gas)p FC(.)21 b(These)13 b(represen)o(tations)h(w)o(ere)e(\014rst)i(studied)f(and)i (used)e(systematically)75 1817 y(in)j([37]:)527 1915 y Ft(Z)t FC(\(\003)p Ft(;)8 b(\036)p FC(\))13 b(=)772 1861 y Fp(1)760 1873 y Fr(X)752 1965 y Fq(N)t Fo(=0)858 1881 y FC(1)p 842 1903 58 2 v 842 1949 a Ft(N)5 b FC(!)1015 1873 y Fr(X)942 1965 y Fq(X)971 1970 y Fl(1)989 1965 y Fq(;:::)o(;X)1067 1971 y Fm(N)1096 1965 y Fp(\032)p Fo(\003)918 2001 y Fq(X)947 2006 y Fm(i)960 2001 y Fp(\\)p Fq(X)1013 2006 y Fm(j)1029 2001 y Fo(=)p Fp(;)k Fo(if)j Fq(i)p Fp(6)p Fo(=)p Fq(j)1198 1861 y(N)1187 1873 y Fr(Y)1185 1964 y Fq(i)p Fo(=1)1251 1915 y Ft(K)t FC(\()p Ft(X)1355 1922 y Fq(i)1369 1915 y Ft(;)c(\036)p FC(\))p Ft(;)359 b FC(\(7)p Ft(:)p FC(1\))75 2098 y(The)11 b(quan)o(tit)o(y)f Ft(K)t FC(\()p Ft(X)q(;)e(\036)p FC(\))j(is)f(called)g(a)h Fn(p)n(olymer)h(activity)p FC(,)g(and)g(v)m(ariational)f(deriv)m(ativ)o(es)1722 2075 y Fq(\016)1739 2063 y Fm(N)1768 2075 y Fq(K)r Fo(\()p Fq(X)q(;\036)p Fo(\))p 1686 2087 238 2 v 1686 2116 a Fq(\016)q(\036)p Fo(\()p Fq(\030)1754 2121 y Fl(1)1770 2116 y Fo(\))p Fp(\001\001\001)p Fq(\016)q(\036)p Fo(\()p Fq(\030)1882 2122 y Fm(N)1910 2116 y Fo(\))75 2165 y FC(v)m(anish)i(unless)g(eac)o(h)g Ft(\030)491 2172 y Fq(i)519 2165 y Fs(2)h Ft(X)t FC(.)20 b(In)13 b(this)g(sense)f Ft(K)t FC(\()p Ft(X)q(;)c(\036)p FC(\))14 b(dep)q(ends)f(on)g Ft(\036)p FC(\()p Ft(\030)r FC(\))h(only)f(for)g Ft(\030)j Fs(2)e Ft(X)t FC(,)g(where)75 2226 y(w)o(e)d(denote)h(in)f(this)h(w)o(a)o(y)f (the)h(prop)q(ert)o(y)f(that)h Ft(\030)17 b FC(=)d(\()p Ft(x;)8 b(\013)p FC(\))j(and)i Ft(x)g Fs(2)h Ft(X)t FC(.)20 b(F)l(rom)11 b(no)o(w)h(on)g(it)f(should)h(b)q(e)75 2286 y(assumed)j(that)g(all)g(sums)g (o)o(v)o(er)f(sets)i(extend)e(o)o(v)o(er)g(the)i(class)f(of)h Fn(al)r(lowe)n(d)h(sets)t FC(,)f(whic)o(h)f(are)g(unions)75 2346 y(of)i Fn(c)n(el)r(ls)g FC(whic)o(h)f(are)g Fn(op)n(en)h FC(unit)e(blo)q(c)o(ks,)h Fn(op)n(en)g FC(faces)g(of)h(unit)f(blo)q(c)o(ks,)f Ft(:)8 b(:)g(:)g(;)16 b Fn(op)n(en)g FC(edges,)g(v)o(ertices.)p 75 2390 735 2 v 131 2420 a Fj(7)150 2435 y Fy(These)f(p)q(olymers)e(ha)o(v)o (e)g(no)h(connection)h(with)e(the)i(p)q(olymers)e(of)g(Section)h(3)969 2599 y FC(60)p eop %%Page: 61 65 bop 75 307 a FC(The)18 b(n)o(ull)g(set)g(is)g(not)h(a)f(cell)f(but)i(is)f (also)h(allo)o(w)o(ed.)26 b(W)l(e)18 b(extend)g(the)g(de\014nition)g(of)g Ft(K)t FC(\()p Ft(X)q(;)8 b(\036)p FC(\))19 b(to)75 367 y(arbitrary)d(allo)o (w)o(ed)g(sets)g(with)g(the)g(con)o(v)o(en)o(tion)f Ft(K)t FC(\()p Ft(X)q(;)8 b(\036)p FC(\))14 b(=)g(0)j(if)e Ft(X)21 b FC(is)16 b(not)h(a)g(p)q(olymer.)148 428 y(The)c(expansion)h(\(7.1\))f(lo)q (oks)h(lik)o(e)d(an)j(exp)q(onen)o(tial,)f(and)g(in)g(fact,)h(it)e(is)h(one)h (but)f(in)g(a)g(di\013eren)o(t)75 488 y(pro)q(duct)595 608 y(\()p Ft(A)d Fs(\016)h Ft(B)s FC(\)\()p Ft(X)t FC(\))j(=)1012 575 y Fr(P)989 668 y Ft(Y)s(;)8 b(Z)925 729 y(Y)974 723 y FC(_)964 729 y Fs([)p Ft(Z)18 b FC(=)13 b Ft(X)1165 608 y(A)p FC(\()p Ft(Y)d FC(\))p Ft(B)s FC(\()p Ft(Z)t FC(\))419 b(\(7)p Ft(:)p FC(2\))75 810 y(de\014ned)21 b(on)h(functions)f(of)h(allo)o(w)o(ed)e(sets.)36 b(Here)21 b Ft(Z)1071 804 y FC(_)1062 810 y Fs([)p Ft(Y)33 b FC(=)23 b Ft(X)i FC(means)c(that)g Ft(Z)e Fs([)14 b Ft(Y)34 b FC(=)22 b Ft(X)k FC(and)75 870 y Ft(Z)17 b Fs(\\)d Ft(Y)31 b FC(=)19 b Fs(;)p Ft(:)g FC(F)l(unctions)h(on)g(allo)o(w)o(ed)f(sets)g(are)h (a)g(comm)o(utativ)n(e)d(algebra)j(under)f(the)h(pro)q(duct)75 931 y(\(7.2\))d(whic)o(h)e(has)i(the)f(iden)o(tit)o(y)736 1070 y(1\()p Ft(X)t FC(\))f(=)908 997 y Fr(\()963 1039 y FC(1)h(if)g Ft(X)i FC(=)c Fs(;)963 1099 y FC(0)i(otherwise.)1812 1070 y(\(7)p Ft(:)p FC(3\))75 1179 y(If)g(w)o(e)g(also)g(de\014ne)671 1257 y Fg(2)p FC(\()p Ft(X)t FC(\))f(=)856 1184 y Fr(\()911 1227 y FC(1)h(if)g Ft(X)i FC(=)30 b(is)16 b(a)h(cell)911 1287 y(0)f(otherwise,) 1812 1257 y(\(7)p Ft(:)p FC(4\))75 1369 y(\()p Fg(2)g FC(=)h(\\the)f (space"\))h(then)f(w)o(e)g(can)g(rewrite)f(the)h(p)q(olymer)f(gas)i (compactly)e(as)683 1463 y Ft(Z)t FC(\(\003)p Ft(;)8 b(\036)p FC(\))13 b(=)h Fs(E)t FC(xp[)p Fg(2)d FC(+)g Ft(K)t FC(]\(\003)p Ft(;)d(\036)p FC(\))p Ft(;)514 b FC(\(7)p Ft(:)p FC(5\))75 1558 y(where)16 b Fs(E)t FC(xp)h(is)f(the)g(exp)q(onen)o(tial)f(de\014ned)h (for)h Ft(A)f FC(suc)o(h)g(that)g Ft(A)p FC(\()p Fs(;)p FC(\))e(=)f(0)k (using)g(a)f(pro)q(duct)626 1674 y Fs(E)t FC(xp[)p Ft(K)t FC(])d(=)h(1)e(+)f Ft(K)k FC(+)1048 1640 y(1)p 1042 1663 38 2 v 1042 1708 a(2!)1085 1674 y Ft(K)g Fs(\016)c Ft(K)k FC(+)c Fs(\001)d(\001)g(\001)g Ft(:)450 b FC(\(7)p Ft(:)p FC(6\))148 1788 y(There)12 b(is)g(also)h(a)g (logarithm)f Fs(L)p FC(n[1)s(+)s Ft(A)p FC(])h(=)h Ft(A)s Fs(\000)1026 1768 y Fo(1)p 1026 1776 18 2 v 1026 1805 a(2)1049 1788 y Ft(A)s Fs(\016)s Ft(A)s FC(+)1203 1768 y Fo(1)p 1203 1776 V 1203 1805 a(3)1225 1788 y Ft(A)s Fs(\016)s Ft(A)s Fs(\016)s Ft(A)8 b Fs(\001)g(\001)g(\001)13 b FC(de\014ned)f(on)h(all)f Ft(A)g FC(with)75 1848 y Ft(A)p FC(\()p Fs(;)p FC(\))17 b(=)g(0.)28 b(It)17 b(is)h(imp)q(ortan)o(t)g(to)g(realize)f(that,)i(when)f(ev)m(aluated)g (on)h(a)g(giv)o(en)e(set)h Ft(X)t FC(,)h(all)f(these)75 1908 y(series)h(are)h(\014nite)g(b)q(ecause)g(no)g(allo)o(w)o(ed)f(set)h Ft(X)25 b FC(can)20 b(b)q(e)g(inde\014nitely)e(sub)q(divided)h(in)o(to)h (prop)q(er)75 1968 y(subsets)g(whic)o(h)f(are)h(allo)o(w)o(ed.)31 b(Th)o(us)21 b(there)e(will)f(b)q(e)i(no)h(analysis)f(problems)e(in)i (manipulating)75 2028 y(these)c(functions.)408 2010 y Fo(8)148 2089 y FC(If)h(w)o(e)f(substitute)h Ft(Z)532 2096 y Fq(t)562 2089 y FC(=)e Fs(E)t FC(xp[)p Fg(2)d FC(+)f Ft(K)851 2096 y Fq(t)866 2089 y FC(])17 b(in)o(to)f(the)h(functional)f(heat)h(equation)g(of)g (Lemma)e(4.3)75 2149 y(w)o(e)h(\014nd)g(that)h Ft(Z)383 2156 y Fq(t)414 2149 y FC(solv)o(es)f(the)g(heat)h(equation)f Fn(i\013)g FC(the)g(p)q(olymer)f(activit)o(y)f Ft(K)1498 2156 y Fq(t)1530 2149 y FC(solv)o(es)692 2231 y Ft(@)s(K)762 2238 y Fq(t)p 692 2253 85 2 v 711 2299 a Ft(@)s(t)795 2265 y FC(=)852 2231 y(1)p 852 2253 25 2 v 852 2299 a(2)881 2265 y(\()p Ft(K)941 2272 y Fq(t\036\036)1010 2265 y FC(+)d Ft(K)1100 2272 y Fq(t\036)1147 2265 y Fs(\016)g Ft(K)1224 2272 y Fq(t\036)1260 2265 y FC(\))p Ft(;)519 b FC(\(7)p Ft(:)p FC(7\))p 75 2329 735 2 v 131 2359 a Fj(8)150 2374 y Fy(This)13 b(pro)q(duct)h(is)f(reminiscen)o(t)f(of)h(a)f (pro)q(duct)i(on)f(functions)g(of)g(sequences)i(in)o(v)o(en)o(ted)f(b)o(y)e (Ruelle)h([56)o(].)k(There)75 2424 y(is)e(a)h(di\013erence)h(whic)o(h)f(is)f (imp)q(ortan)o(t:)20 b(our)15 b(pro)q(duct)i(is)e(designed)h(to)g(suppress)h (m)o(ultiple)d(o)o(v)o(erlap)g(of)h(p)q(olymers)75 2474 y(whic)o(h)f(leads)g (to)g(arti\014cial)e(large)i(\014eld)g(problems.)969 2599 y FC(61)p eop %%Page: 62 66 bop 75 307 a FC(where)450 380 y Ft(K)491 387 y Fq(\036)525 380 y Fs(\016)11 b Ft(K)602 387 y Fq(\036)640 380 y FC(=)691 321 y Fr(Z)741 380 y Ft(d\030)787 387 y Fo(1)824 321 y Fr(Z)874 380 y Ft(d\030)920 387 y Fo(2)985 346 y Ft(\016)r(K)p 954 368 132 2 v 954 414 a(\016)r(\036)p FC(\()p Ft(\030)1047 421 y Fo(1)1066 414 y FC(\))1101 380 y Fs(\016)1173 346 y Ft(\016)r(K)p 1142 368 V 1142 414 a(\016)r(\036)p FC(\()p Ft(\030)1235 421 y Fo(2)1254 414 y FC(\))1278 380 y([)p Fs(\000)1347 367 y FC(_)1331 380 y Ft(C)s FC(\()p Ft(\030)1409 387 y Fo(1)1429 380 y Ft(;)d(\030)1472 387 y Fo(2)1492 380 y FC(\)])p Ft(:)273 b FC(\(7)p Ft(:)p FC(8\))148 489 y(As)16 b(in)g(Section)g(5)g(w)o(e)g(can)h(con)o(v)o(ert)e(this)h(to)h (an)f(equiv)m(alen)o(t)f(in)o(tegral)g(equation)545 613 y Ft(K)586 620 y Fq(t)615 613 y FC(=)e Ft(\026)695 620 y Fq(t)p Fo(0)739 613 y Fs(\003)e Ft(K)816 620 y Fo(0)848 613 y FC(+)902 580 y(1)p 902 602 25 2 v 902 647 a(2)939 555 y Fr(Z)981 568 y Fo(1)962 649 y(0)1009 613 y Ft(ds)d(\026)1094 620 y Fq(ts)1137 613 y Fs(\003)j FC(\()p Ft(K)1233 620 y Fq(s;\036)1294 613 y Fs(\016)g Ft(K)1371 620 y Fq(s\036)1410 613 y FC(\))p Ft(:)369 b FC(\(7)p Ft(:)p FC(9\))75 731 y(Iteration)17 b(of)h(this)g(in)o(tegral)f(equation)g (yields)g(a)h Fn(\014nite)h FC(series)e(for)g Ft(K)1366 738 y Fq(t)1399 731 y FC(whic)o(h)g(is)h(called)e(a)i Fn(cluster)75 791 y(exp)n(ansion)p FC(.)j(This)15 b(expansion)f(w)o(as)h(really)e(in)o(v)o (en)o(ted)f(b)o(y)h(Glimm,)e(Ja\013e)k(and)g(Sp)q(encer)e(in)h([35)q(])f(but) 75 852 y(this)18 b(presen)o(tation)g(based)h(on)g([18)q(])f(rev)o(eals)f(the) h(structure)g(more)f(clearly)g(than)i(they)f(probably)75 912 y(understo)q(o)q(d)g(at)f(the)f(time.)148 972 y(W)l(e)j(can)g(obtain)h(a)g(b) q(eautiful)e(explicit)f(form)o(ula)h(for)i(the)e(result)h(of)h(iterating)e (this)h(in)o(tegral)75 1032 y(equation)d(b)o(y)g(follo)o(wing)g(the)g(pro)q (cedure)g(of)h(Section)e(5.3)i(and)g(the)f(result)g(is)231 1155 y Ft(K)272 1162 y Fq(t)288 1155 y FC(\()p Ft(\036)p FC(\))41 b(=)475 1114 y Fr(X)490 1206 y Fq(N)565 1121 y FC(1)p 549 1144 58 2 v 549 1189 a Ft(N)5 b FC(!)647 1114 y Fr(X)620 1206 y Fq(T)14 b Fo(on)9 b Fq(N)742 1097 y Fr(Z)783 1110 y Fq(t)765 1191 y Fo(0)807 1155 y Ft(d)832 1135 y Fq(T)859 1155 y Ft(s)f(\026)919 1162 y Fq(ts)948 1166 y Fl(max)1020 1155 y Fs(\003)j FC(\001)1097 1135 y Fq(T)1097 1167 y(s)p 1097 1174 19 2 v 1138 1155 a Fs(f)p Ft(K)1204 1162 y Fo(0)1224 1155 y FC(\()p Ft(\036)1272 1162 y Fo(1)1292 1155 y FC(\))g Fs(\016)g(\001)d(\001)g(\001)j(\016)g Ft(K)1504 1162 y Fo(0)1524 1155 y FC(\()p Ft(\036)1572 1162 y Fq(N)1605 1155 y FC(\))p Fs(gj)1663 1170 y Fq(\036)1684 1175 y Fm(i)1697 1170 y Fo(=)p Fq(\036)1788 1155 y FC(\(7.10\))75 1292 y(The)17 b(notation)g(is)g(as)g(in)g(Prop)q(osition)h(5.33.)23 b(Ho)o(w)o(ev)o(er)15 b(if)h(all)h(w)o(e)f(w)o(an)o(t)h(is)f(estimates)f (then)i(there)75 1352 y(is)f(a)h(simpler)d(pro)q(cedure)i(whic)o(h)f(w)o(e)h (no)o(w)h(describ)q(e.)148 1412 y(Since)c(a)h(cluster)g(expansion)g(is)f (\014nite,)h(con)o(v)o(ergence)e(is)i(not)g(an)g(issue,)g(but)g(one)g(still)f (w)o(an)o(ts)h(to)75 1472 y(measure)j(the)g(exten)o(t)g(to)i(whic)o(h)e(lo)q (calit)o(y)g(is)h(preserv)o(ed)e(under)i(the)g(ev)o(olution.)26 b(In)17 b(the)h(presen)o(t)75 1533 y(con)o(text)i(the)g(lo)q(calit)o(y)g (question)g(means:)29 b(to)21 b(what)h(exten)o(t)d Ft(K)1278 1540 y Fq(t)1293 1533 y FC(\()p Ft(X)t FC(\))i(remains)f(small)f(whenev)o(er) 75 1593 y(the)g(p)q(olymer)f(is)h(spread)h(out)f(or)h(large,)g(if)e(it)h(is)g (true)g(at)h Ft(t)f FC(=)g(0?)31 b(Another)19 b(p)q(oin)o(t)g(to)h(notice)f (is)75 1653 y(that)d(the)f Fn(p)n(artition)h(function)g Ft(Z)t FC(\(\003\))f(loses)g(its)g(meaning)f(as)i(\003)f(b)q(ecomes)f(an)i (in\014nite)e(set)h(but)h(the)75 1713 y(collection)c(\()p Ft(K)349 1720 y Fq(t)364 1713 y FC(\()p Ft(X)t FC(\))p Ft(;)c(X)19 b FC(ranges)14 b(o)o(v)o(er)e(p)q(olymers\))h(remains)f(a)i(go)q(o)q(d)h (description)e(of)h(the)f(ev)o(olution)75 1773 y(in)j(the)g(in\014nite)f(v)o (olume)f(limit.)148 1833 y(These)i(considerations)h(are)f(em)o(b)q(o)q(died)f (in)h(the)g(de\014nition)g(of)g(the)g(follo)o(wing)g(norms:)152 1965 y Fs(k)p Ft(h)205 1945 y Fq(N)239 1965 y Ft(K)280 1972 y Fq(N)314 1965 y FC(\()p Ft(X)q(;)8 b(\036)p FC(\))p Fs(k)469 1972 y Fo(\001)p 469 1979 32 2 v 515 1965 a Fs(\021)41 b FC(sup)567 2007 y Fq(F)5 b Fo(:)p Fp(j)p Fq(F)g Fp(j\024)p Fo(1)705 1907 y Fr(Z)728 2001 y Fo(\001)757 2006 y Fl(1)785 1965 y Ft(hd\030)859 1972 y Fo(1)896 1965 y Fs(\001)j(\001)g(\001)962 1907 y Fr(Z)985 2001 y Fo(\001)1014 2007 y Fm(N)1054 1965 y Ft(hd\030)1128 1972 y Fq(N)1235 1931 y Ft(\016)1259 1913 y Fq(N)1292 1931 y Ft(K)t FC(\()p Ft(X)q(;)g(\036)p FC(\))p 1176 1954 352 2 v 1176 1999 a Ft(\016)r(\036)p FC(\()p Ft(\030)1269 2006 y Fo(1)1288 1999 y FC(\))g Fs(\001)g(\001)g(\001)h Ft(\016)r(\036)p FC(\()p Ft(\030)1475 2006 y Fq(N)1508 1999 y FC(\))1532 1965 y Ft(F)e FC(\()p Ft(\030)1611 1972 y Fo(1)1630 1965 y Ft(;)h(:)g(:)g(:)f(;)h (\030)1760 1972 y Fq(N)1794 1965 y FC(\))p Ft(;)1788 2061 y FC(\(7)p Ft(:)p FC(11\))75 2121 y(where)15 b(\001)256 2128 y Fo(1)275 2121 y Ft(;)8 b(:)g(:)g(:)f(;)h FC(\001)425 2128 y Fq(N)474 2121 y FC(is)14 b(an)i Ft(N)5 b FC(-tuple)15 b(of)g(closed)g(unit) g(blo)q(c)o(ks)f(and)i Ft(F)21 b FC(ranges)16 b(o)o(v)o(er)e(all)h(con)o(tin) o(uous)75 2181 y(functions)j(of)g Ft(N)23 b FC(argumen)o(ts)17 b Ft(\030)666 2188 y Fo(1)686 2181 y Ft(;)8 b(:)g(:)g(:)g(;)g(\030)817 2188 y Fq(N)851 2181 y FC(.)26 b(Note)17 b(that)i(v)m(ariational)f(deriv)m (ativ)o(es)e(of)j Ft(K)j FC(are)c(mea-)75 2241 y(sures.)31 b(No)o(w)20 b(w)o(e)f(w)o(an)o(t)g(to)h(imp)q(ose)f(analyticit)o(y)e(in)j Ft(\036)f FC(in)g(a)h(strip)f(of)h(radius)g Ft(h)g(>)f FC(0)h(ab)q(out)h(the) 75 2301 y(real-)p Ft(\036)16 b FC(axis.)21 b(Roughly)c(sp)q(eaking,)f(for)g (all)g Ft(N)5 b FC(,)634 2388 y Fr(X)587 2480 y Fo(\001)616 2485 y Fl(1)634 2480 y Fq(;:::)o(;)p Fo(\001)712 2486 y Fm(N)750 2430 y Fs(k)p Ft(K)816 2437 y Fq(N)850 2430 y FC(\()p Ft(X)q(;)j(\036)p FC(\))p Fs(k)1005 2437 y Fo(\001)p 1005 2444 32 2 v 1050 2430 a Fs(\024)14 b Ft(C)1148 2396 y(N)5 b FC(!)p 1146 2418 62 2 v 1146 2464 a Ft(h)1174 2449 y Fq(N)1226 2396 y Ft(g)r FC(\()p Ft(X)q(;)j(\036)p FC(\))p 1226 2418 156 2 v 1248 2464 a(\000\()p Ft(X)t FC(\))1387 2430 y Ft(;)387 b FC(\(7)p Ft(:)p FC(12\))969 2599 y(62)p eop %%Page: 63 67 bop 75 307 a FC(where)13 b(\000\()p Ft(X)t FC(\))h(is)g(a)f(w)o(eigh)o(t)g (on)h(sets)f(whic)o(h)g(quan)o(ti\014es)g(lo)q(calit)o(y)l(.)19 b(W)l(e)13 b(call)g(it)g(a)h Fn(lar)n(ge)h(set)g(r)n(e)n(gulator.)75 367 y FC(W)l(e)h(will)f(c)o(ho)q(ose)i(it)f(later,)f(but)h(for)h(the)f (presen)o(t)g(section)f(it)h(can)h(b)q(e)f(an)o(ything)g(that)h(satis\014es) 599 465 y(\000\()p Ft(X)t FC(\))d Fs(\025)g FC(1)p Ft(;)57 b FC(\000\()p Ft(X)16 b Fs([)11 b Ft(Y)g FC(\))j Fs(\024)g FC(\000\()p Ft(X)t FC(\)\000\()p Ft(Y)e FC(\))p Ft(;)407 b FC(\(7)p Ft(:)p FC(13\))75 563 y(e.g.,)19 b(\000\()p Ft(X)t FC(\))i(=)e(1.)32 b(The)20 b(function)g Ft(g)r FC(\()p Ft(X)q(;)8 b(\036)p FC(\))20 b(quan)o(ti\014es)f(the)g(gro)o(wth)i(in)e Ft(K)24 b FC(as)c Fs(j)p Ft(\036)p Fs(j)g(!)f(1)p FC(.)31 b(It)20 b(is)75 623 y(called)15 b(a)i Fn(lar)n(ge-\014eld)i(r)n(e)n(gulator)c FC(and)i(can)g(b)q(e)f(an)o(y)g(function)g(that)g(is)g(in)o(tegrable)f(with)h (resp)q(ect)g(to)75 683 y Ft(d\026)p FC(\()p Ft(\036)p FC(\))h(and)g (satis\014es)534 781 y Ft(g)r FC(\()p Ft(X)q(;)8 b(\036)p FC(\))14 b Fs(\025)g FC(1)p Ft(;)57 b(g)r FC(\()p Ft(X)949 775 y FC(_)939 781 y Fs([)q Ft(Y)s(;)8 b(\036)p FC(\))13 b Fs(\025)h Ft(g)r FC(\()p Ft(X)q(;)8 b(\036)p FC(\))p Ft(g)r FC(\()p Ft(Y)s(;)g(\036)p FC(\))p Ft(:)334 b FC(\(7)p Ft(:)p FC(14\))75 878 y(The)20 b(large-\014eld)f(regulator)h(will)e(b)q(e)i(explained)f(further)g(at)h(the)f (end)h(of)g(this)f(section.)31 b(W)l(e)20 b(in-)75 938 y(corp)q(orate)i(all)f (the)g(inequalities)e(\(7.12\))j(in)o(to)f(one)g(norm)f(with)h(the)g(follo)o (wing)g(de\014nitions)g(for)75 999 y Fs(k)c(k)142 1006 y Fq(g)q(;)p Fo(\000)p Fq(;h)223 999 y FC(,)450 1096 y Fs(k)p Ft(h)503 1076 y Fq(N)536 1096 y Ft(K)577 1103 y Fq(N)611 1096 y FC(\()p Ft(X)t FC(\))p Fs(k)718 1103 y Fq(g)780 1096 y Fs(\021)860 1055 y Fr(X)876 1147 y Fo(\001)p 876 1154 30 2 v 929 1096 a FC(sup)955 1136 y Fq(\036)1010 1096 y Fs(k)p Ft(h)1063 1076 y Fq(N)1097 1096 y Ft(K)1138 1103 y Fq(N)1172 1096 y FC(\()p Ft(X)q(;)8 b(\036)p FC(\))p Fs(k)1327 1103 y Fo(\001)p 1327 1110 32 2 v 1367 1096 a Ft(g)1392 1076 y Fp(\000)p Fo(1)1439 1096 y FC(\()p Ft(X)q(;)g(\036)p FC(\))418 1221 y Fs(k)p Ft(h)471 1201 y Fq(N)504 1221 y Ft(K)545 1228 y Fq(N)579 1221 y FC(\()p Ft(X)t FC(\))p Fs(k)686 1228 y Fq(g)q(;)p Fo(\000)780 1221 y Fs(\021)41 b FC(sup)887 1256 y Fq(x)950 1180 y Fr(X)942 1272 y Fq(X)s Fp(3)p Fq(x)1025 1221 y Fs(k)p Ft(h)1078 1201 y Fq(N)1112 1221 y Ft(K)1153 1228 y Fq(N)1187 1221 y FC(\()p Ft(X)t FC(\))p Fs(k)1294 1228 y Fq(g)1314 1221 y FC(\000\()p Ft(X)t FC(\))562 1367 y Fs(k)p Ft(K)t Fs(k)657 1374 y Fq(g)q(;)p Fo(\000)p Fq(;h)780 1367 y Fs(\021)873 1313 y Fp(1)860 1325 y Fr(X)882 1416 y Fo(0)950 1333 y FC(1)p 934 1355 58 2 v 934 1401 a Ft(N)5 b FC(!)996 1367 y Fs(k)p Ft(h)1049 1346 y Fq(N)1083 1367 y Ft(K)1124 1374 y Fq(N)1158 1367 y Fs(k)1183 1374 y Fq(g)q(;)p Fo(\000)1235 1367 y Ft(:)539 b FC(\(7.15\))75 1496 y(The)16 b(co)o(v)m(ariance)g Ft(C)k FC(will)15 b(b)q(e)h(measured)f(b)o(y)601 1594 y Fs(k)p Ft(C)t Fs(k)690 1601 y Fo(\000)727 1594 y Fs(\021)e FC(sup)801 1633 y Fo(\001)861 1552 y Fr(X)871 1645 y Fo(\001)900 1635 y Fk(0)929 1594 y Fs(j)p Ft(C)t FC(\(\001)p Ft(;)8 b FC(\001)1105 1573 y Fp(0)1116 1594 y FC(\))p Fs(j)p FC(\000\(\001)i Fs([)i FC(\001)1335 1573 y Fp(0)1346 1594 y FC(\))p Ft(;)409 b FC(\(7)p Ft(:)p FC(16\))75 1725 y(where)676 1785 y Ft(C)t FC(\(\001)p Ft(;)8 b FC(\001)838 1765 y Fp(0)848 1785 y FC(\))14 b Fs(\021)62 b FC(sup)933 1825 y Fq(\030)q Fp(2)p Fo(\001)p Fq(;\030)1030 1816 y Fk(0)1041 1825 y Fp(2)p Fo(\001)1094 1816 y Fk(0)1113 1785 y Fs(j)p Ft(C)t FC(\()p Ft(\030)r(;)8 b(\030)1253 1765 y Fp(0)1265 1785 y FC(\))p Fs(j)p Ft(:)476 b FC(\(7)p Ft(:)p FC(17\))75 1899 y(No)o(w)19 b(w)o(e)f(obtain)h(the)g(follo)o(wing)f(theorem)f (b)o(y)h(taking)h(the)g Fs(k)8 b(k)1272 1906 y Fq(g)q(;)p Fo(\000)p Fq(;h)1373 1899 y FC(norm)18 b(of)h(the)f(in)o(tegral)g(\015o)o(w)75 1959 y(equation)d(\(7.9\))g(whic)o(h)f(reduces)h(it)f(to)i(a)f(one-v)m (ariable)g(inequalit)o(y)e(for)i(the)g(norm,)f(whic)o(h)g(can)h(b)q(e)75 2019 y(solv)o(ed)h(explicitly)l(.)75 2119 y Fu(Theorem)h(7.1)24 b Fn(If)17 b Ft(g)483 2126 y Fq(t)515 2119 y Fn(is)g(a)g(family)g(of)h(lar)n (ge)f(\014eld)i(r)n(e)n(gulators)d(that)i(ar)n(e)f(inte)n(gr)n(able)h(in)g (the)f(sense)725 2217 y Ft(\026)754 2224 y Fq(ts)797 2217 y Fs(\003)11 b Ft(g)856 2224 y Fq(s)888 2217 y Fs(\024)j Ft(g)964 2224 y Fq(t)979 2217 y Ft(;)58 b FC(0)14 b Fs(\024)g Ft(s)f Fs(\024)h Ft(t;)75 2315 y Fn(and)k(at)f Ft(t)d FC(=)f(0)p Fn(,)18 b(for)f(some)h Ft(h)13 b(>)h(h)692 2296 y Fp(0)718 2315 y Ft(>)g FC(0)p Fn(,)682 2439 y Fs(k)p Ft(K)748 2446 y Fq(t)p Fo(=0)809 2439 y Fs(k)834 2446 y Fq(g)851 2451 y Fl(0)868 2446 y Fq(;)p Fo(\000)p Fq(;h)946 2439 y Fs(\024)1052 2405 y FC(\()p Ft(h)d Fs(\000)g Ft(h)1188 2387 y Fp(0)1199 2405 y FC(\))1218 2387 y Fo(2)p 1003 2427 284 2 v 1003 2478 a FC(16)1059 2442 y Fr(R)1088 2456 y Fq(t)1080 2491 y Fo(0)1111 2478 y Fs(k)1152 2465 y FC(_)1136 2478 y Ft(C)1171 2485 y Fq(s)1189 2478 y Fs(k)1214 2485 y Fo(\000)1238 2478 y Ft(ds)1292 2439 y(;)969 2599 y FC(63)p eop %%Page: 64 68 bop 75 307 a Fn(then)595 367 y Ft(\026)624 374 y Fo(0)655 367 y Fs(\003)11 b(E)t FC(xp)q([)p Fg(2)g FC(+)g Ft(K)927 374 y Fo(0)947 367 y FC(])i(=)h Ft(\026)1055 374 y Fq(t)1081 367 y Fs(\003)d(E)t FC(xp)q(\()p Fg(2)g FC(+)g Ft(K)1358 374 y Fq(t)1373 367 y FC(\))75 451 y Fn(with)723 511 y Fs(k)p Ft(K)789 518 y Fq(t)804 511 y Fs(k)829 518 y Fq(g)846 522 y Fm(t)860 518 y Fq(;)p Fo(\000)p Fq(;h)922 510 y Fk(0)949 511 y Fs(\024)j(k)p Ft(K)1068 518 y Fq(t)p Fo(=0)1128 511 y Fs(k)1153 518 y Fq(g)1170 523 y Fl(0)1187 518 y Fq(;)p Fo(\000)p Fq(;h)1251 511 y Ft(:)75 640 y Fu(Pro)r(of.)27 b FC(Rename)22 b(the)h Ft(h)g FC(and)g Ft(h)720 622 y Fp(0)755 640 y FC(in)g(the)f(theorem)g Ft(h)1134 647 y Fq(t)1149 640 y Ft(;)8 b(h)1199 647 y Fo(0)1241 640 y FC(resp)q(ectiv)o(ely)l(.)39 b(F)l(or)23 b Ft(p)j FC(=)f(0)p Ft(;)8 b FC(1)p Ft(;)g(:)g(:)g(:)p FC(,)90 680 y Fq(@)111 669 y Fm(p)p 80 688 59 2 v 80 717 a Fq(@)r(h)121 708 y Fm(p)144 700 y Fs(k)p Ft(K)210 707 y Fq(t)225 700 y Fs(k)250 707 y Fq(h)286 700 y Fs(\021)354 680 y Fq(@)375 669 y Fm(p)p 344 688 V 344 717 a Fq(@)r(h)385 708 y Fm(p)408 700 y Fs(k)p Ft(K)474 707 y Fq(t)489 700 y Fs(k)514 707 y Fq(g)531 711 y Fm(s)547 707 y Fq(;)p Fo(\000)p Fq(;h)626 700 y FC(are)15 b(\(semi\)norms.)j(By)d(taking)g (\(semi\)norms)d(of)j(b)q(oth)h(sides)f(of)g(the)75 760 y(in)o(tegral)g (equation)492 882 y Ft(K)533 889 y Fq(t)590 882 y FC(=)41 b Ft(\026)698 889 y Fq(t)p Fo(0)742 882 y Fs(\003)11 b Ft(K)819 889 y Fo(0)850 882 y FC(+)904 848 y(1)p 904 870 25 2 v 904 916 a(2)942 824 y Fr(Z)983 837 y Fq(t)965 918 y Fo(0)1007 882 y Ft(ds)1080 870 y FC(_)1063 882 y Ft(C)t(\026)1131 889 y Fq(ts)1173 882 y Fs(\003)g(f)p Ft(K)1275 889 y Fq(s;\036)1336 882 y Fs(\016)g Ft(K)1413 889 y Fq(s;\036)1462 882 y Fs(g)301 b FC(\(7.18\))75 1002 y(that)17 b(corresp)q(onds)g(to)g(the)f(\015o)o(w)g(\(7.9\),)g(w)o(e)g (obtain,)h(for)f(an)o(y)g Ft(h)p FC(,)471 1143 y Fs(k)p Ft(K)537 1150 y Fq(t)552 1143 y Fs(k)577 1150 y Fq(h)641 1143 y Fs(\024\024)41 b(k)p Ft(K)826 1150 y Fo(0)846 1143 y Fs(k)871 1150 y Fq(h)905 1143 y FC(+)954 1085 y Fr(Z)995 1098 y Fq(t)977 1179 y Fo(0)1018 1143 y Ft(ds)8 b Fs(k)1116 1131 y FC(_)1099 1143 y Ft(C)1134 1150 y Fq(s)1153 1143 y Fs(k)1178 1150 y Fo(\000)1210 1070 y Fr( )1262 1110 y Ft(@)p 1248 1132 57 2 v 1248 1178 a(@)s(h)1310 1143 y Fs(k)p Ft(K)1376 1150 y Fq(s)1395 1143 y Fs(k)1420 1150 y Fq(h)1442 1070 y Fr(!)1475 1082 y Fo(2)1503 1143 y Ft(:)271 b FC(\(7.19\))75 1282 y(The)15 b(notation)h Fs(\024\024)f FC(means)f(that)i (the)f(b)q(ound)h(holds)g(with)1198 1263 y Fq(@)1219 1251 y Fm(p)p 1188 1271 59 2 v 1188 1300 a Fq(@)r(h)1229 1290 y Fm(p)1267 1282 y FC(applied)f(to)g(eac)o(h)g(side.)20 b(F)l(or)c(more)75 1343 y(details)f(on)h(this)f(step,)g(see)g(p.)21 b(380)c(of)f([18].)k (Iterating)15 b(\(7.18\))h(generates)g(a)g(\014nite)f(series)f(b)q(ecause)75 1403 y(a)k(p)q(olymer)e Ft(X)22 b FC(can)c(only)g(b)q(e)f(sub)q(divided)h(in) o(to)f(\014nitely)f(man)o(y)h(prop)q(er)h(subsets.)26 b(This)18 b(series)f(is)75 1463 y Fs(\024\024)c FC(b)q(ounded)h(b)o(y)f(the)g(series)g (that)h(results)f(from)f(iterating)h(\(7.19\))h(\(whic)o(h)f(is)g(an)h (in\014nite)f(series\).)75 1523 y(But)j(this)g(series)g(in)f(its)i(turn)f(is) g Fs(\024\024)g FC(dominated)f(b)o(y)h(the)g(series)f(generated)h(b)o(y)578 1662 y Ft(k)603 1669 y Fq(t)618 1662 y FC(\()p Ft(h)p FC(\))41 b(=)h Ft(k)830 1669 y Fo(0)850 1662 y FC(\()p Ft(h)p FC(\))11 b(+)976 1604 y Fr(Z)1017 1617 y Fq(t)999 1698 y Fo(0)1041 1662 y Ft(ds)d Fs(k)1139 1650 y FC(_)1122 1662 y Ft(C)1157 1669 y Fq(s)1175 1662 y Fs(k)1200 1669 y Fo(\000)1233 1589 y Fr( )1271 1628 y Ft(@)s(k)p 1271 1651 57 2 v 1271 1696 a(@)s(h)1332 1589 y Fr(!)1365 1601 y Fo(2)1393 1662 y FC(;)381 b(\(7.20\))573 1766 y Ft(k)598 1773 y Fo(0)618 1766 y FC(\()p Ft(h)p FC(\))41 b(=)h Fs(k)p Ft(K)871 1773 y Fo(0)891 1766 y Fs(k)916 1773 y Fq(g)933 1778 y Fl(0)950 1773 y Fq(;)p Fo(\000)p Fq(;h)1014 1766 y Ft(;)760 b FC(\(7.21\))75 1868 y(This)16 b(in)o(tegral)g(equation)g (generates)g(a)h(solution,)f(analytic)g(near)g Ft(h)e FC(=)g(0)p Ft(;)i FC(to)842 1974 y Ft(@)s(k)p 842 1996 56 2 v 842 2042 a(@)s(\034)916 2007 y FC(=)968 1934 y Fr( )1007 1974 y Ft(@)s(k)p 1006 1996 57 2 v 1006 2042 a(@)s(h)1068 1934 y Fr(!)1100 1946 y Fo(2)1128 2007 y Ft(;)646 b FC(\(7)p Ft(:)p FC(22\))75 2138 y(where)817 2208 y Ft(\034)19 b FC(=)909 2150 y Fr(Z)950 2163 y Fq(t)932 2244 y Fo(0)974 2208 y Ft(ds)8 b Fs(k)1072 2196 y FC(_)1055 2208 y Ft(C)1090 2215 y Fq(s)1108 2208 y Fs(k)1133 2215 y Fo(\000)1157 2208 y Ft(:)617 b FC(\(7)p Ft(:)p FC(23\))75 2309 y(But)16 b(for)g Ft(\034)22 b FC(small,)14 b(the)i(reader)f(can)i(c)o (hec)o(k)d(that)i(\(7.22\))h(has)g(a)f(solution)h(whic)o(h)e(is)h(analytic)f (in)h Ft(h)p FC(,)75 2369 y(namely)614 2456 y Ft(k)r FC(\()p Ft(\034)s(;)8 b(h)p FC(\))13 b(=)h Ft(k)r FC(\()p Ft(\034)20 b FC(=)14 b(0)p Ft(;)8 b(h)1031 2463 y Fq(cl)1059 2456 y FC(\))j Fs(\000)1144 2423 y FC(\()p Ft(h)g Fs(\000)g Ft(h)1280 2430 y Fq(cl)1308 2423 y FC(\))1327 2405 y Fo(2)p 1144 2445 203 2 v 1220 2491 a FC(4)p Ft(\034)1352 2456 y(;)422 b FC(\(7)p Ft(:)p FC(24\))969 2599 y(64)p eop %%Page: 65 69 bop 75 307 a FC(where)15 b Ft(h)243 314 y Fq(cl)286 307 y FC(is)g(determined) e(b)o(y)h(requiring)g(that)i(the)f(righ)o(t)f(hand)i(side)f(of)g(\(7.24\))h (b)q(e)f(stationary)h(in)75 367 y Ft(h)103 374 y Fq(cl)131 367 y FC(.)k([This)12 b(solution)g(w)o(as)g(obtained)g(b)o(y)f(applying)h (the)f(action)h(principle)e(to)i(the)g(Hamilton-Jacobi)75 428 y(equation)18 b(\(7.22\)].)27 b(Using)18 b(the)g(h)o(yp)q(otheses)g(on)g Ft(h)1027 435 y Fo(0)1047 428 y Ft(;)8 b(h)1097 435 y Fq(t)1112 428 y FC(,)18 b(Rouc)o(h)o(\023)-23 b(es)18 b(theorem)e(implies)g(that)i (there)75 488 y(is)c(a)h(unique)e(solution)i Ft(h)528 495 y Fq(cl)556 488 y FC(\()p Ft(\034)s(;)8 b(h)p FC(\))14 b(suc)o(h)g(that)h Ft(h)922 495 y Fq(cl)951 488 y FC(\()p Ft(\034)k FC(=)14 b(0)1086 470 y Fo(+)1116 488 y Ft(;)8 b(h)p FC(\))13 b(=)h Ft(h)p FC(,)h(whic)o(h)e (is)h(analytic)g(for)g Fs(j)p Ft(h)p Fs(j)g(\024)g Ft(h)1898 495 y Fq(t)75 548 y FC(and)j(real)g(for)g Ft(h)g FC(real.)22 b(Also)17 b Fs(j)p Ft(h)654 555 y Fq(cl)682 548 y Fs(j)e(\024)f Ft(h)792 555 y Fo(0)812 548 y FC(.)23 b(Therefore)16 b(\(7.24\))i(implies)c Ft(k)r FC(\()p Ft(\034)s(;)8 b(h)1498 555 y Fq(t)1512 548 y FC(\))15 b Fs(\024)g Ft(k)r FC(\(0)p Ft(;)8 b(h)1720 555 y Fo(0)1740 548 y FC(\),)16 b(whic)o(h)75 608 y(is)g(the)f(b)q(ound)i(claimed)c (in)j(the)f(theorem.)20 b(The)15 b(solution)h(w)o(e)g(ha)o(v)o(e)f(found)h (is)g(unique)f(within)g(the)75 668 y(class)f(of)h(functions)f(analytic)g(in)g (h)h(ab)q(out)g Ft(h)f FC(=)g(0)h(b)q(ecause)f(\(7.22\))h(determines)d(the)i (p)q(o)o(w)o(er)h(series.)75 729 y(More)h(details)g(can)g(b)q(e)h(found)f(on) h(page)g(381)g(of)g([18].)75 807 y Fu(A)j(c)n(hoice)g(for)g(the)f(large)g (\014eld)h(regulator)f Ft(g)r Fu(:)k FC(In)17 b(order)g(to)h(apply)f(Theorem) f(7.1)i(w)o(e)f(m)o(ust)75 867 y(c)o(ho)q(ose)23 b(a)f(large)g(\014eld)g (regulator)g Ft(g)j FC(and)e(a)f(large)g(set)g(regulator)h(\000.)39 b(Go)q(o)q(d)24 b(c)o(hoices)d(of)i(these)75 927 y(ob)s(jects)16 b(will)g(b)q(e)g(discussed)h(in)f(the)g(next)g(section)g(but)h(here)f(w)o(e)g (will)f(illustrate)h(the)g(role)g(of)h Ft(g)i FC(in)75 988 y(b)q(ounding)14 b(a)e(\014eld)g(p)q(oin)o(t)o(wise)g(in)g Ft(x)h FC(with)f(the)g(aid)h(of)f(stabilit)o(y)g(in)g(a)h(mo)q(del)e(with)h (a)h Ft(\036)1638 970 y Fo(4)1670 988 y FC(in)o(teraction.)75 1048 y(F)l(or)j(simplicit)o(y)d(w)o(e)j(tak)o(e)f(\000)g(=)e(1.)148 1108 y(Consider)733 1168 y Ft(E)769 1175 y Fo(0)789 1168 y FC(\()p Ft(X)t FC(\))h Fs(\021)f Ft(e)960 1146 y Fp(\000)995 1132 y Fm(\025)p 992 1138 24 2 v 992 1159 a Fl(4!)1027 1115 y Fr(R)1047 1163 y Fm(X)1082 1146 y Fq(\036)1103 1134 y Fl(4)1120 1146 y Fq(dx)1160 1168 y Ft(\036)p FC(\()p Ft(x)p FC(\))533 b(\(7)p Ft(:)p FC(25\))75 1246 y(if)17 b(X)g(is)h(a)g(unit)f(blo)q(c)o(k,)g (otherwise)h Ft(E)s FC(\()p Ft(X)t FC(\))e(=)h(0.)25 b Ft(x)16 b Fs(2)h Ft(X)22 b FC(or)c(more)e(generally)h Ft(x)g FC(is)h(near)g Ft(X)t FC(.)25 b(The)75 1306 y(norm)18 b Fs(k)p Ft(E)s Fs(k)294 1313 y Fq(g)q(;)p Fo(\000)p Fq(;h)394 1306 y FC(in)o(v)o(olv)o(es)e(all)i (deriv)m(ativ)o(es)f(of)i Ft(E)j FC(but)c(in)g(particular)g(w)o(e)g(\014rst)h (need)f(to)h(see)f(that)75 1366 y(\(7.25\))13 b(is)g(small)e(in)i(suprem)o (um)d(norm.)19 b(Ho)o(w)o(ev)o(er)11 b Ft(\036)i FC(can)g(b)q(e)g(large)g (just)g(in)f(a)h(small)f(neigh)o(b)q(orho)q(o)q(d)75 1426 y(of)18 b(one)g(p)q(oin)o(t)g Ft(x)f Fs(2)g Ft(X)22 b FC(without)d(the)e(factor)920 1391 y Fr(R)956 1426 y Ft(\036)985 1408 y Fo(4)1022 1426 y FC(in)h(the)f(exp)q(onen)o(t)h(b)q(eing)g(large.)27 b(The)18 b(remedy)75 1486 y(is)f(to)g(recognize)e(that)j(the)e(\014elds)g(whic)o(h)g (are)h(t)o(ypical)e(for)i(the)g(measure)e Ft(d\026)1498 1493 y Fq(t)1531 1486 y FC(cannot)i(mak)o(e)e(suc)o(h)75 1546 y(\015uctuations)j (b)q(ecause)g(the)f(ultra-violet)f(cuto\013)i(in)g Ft(d\026)1114 1553 y Fq(t)1146 1546 y FC(mak)o(es)e(them)g Fn(sti\013)p FC(.)25 b(This)17 b(is)h(where)f(the)75 1607 y(large)c(\014eld)g(regulator)g Ft(g)r FC(\()p Ft(X)q(;)8 b(\036)p FC(\))14 b(whic)o(h)e(is)h(really)f(a)i (large)f(gradien)o(t)g(regulator)h(comes)e(in.)19 b Fn(Supp)n(ose)75 1667 y(the)f(dimension)g(of)f(sp)n(ac)n(e-time)h Ft(d)c FC(=)g(1)j(and)g(let) 681 1790 y Ft(g)r FC(\()p Ft(X)q(;)8 b(\036)p FC(\))14 b(=)g(exp)989 1757 y Ft(\024)p 989 1779 29 2 v 991 1825 a FC(2)1031 1732 y Fr(Z)1054 1826 y Fq(X)1096 1717 y Fr( )1134 1757 y Ft(d\036)p 1134 1779 55 2 v 1135 1825 a(dx)1193 1717 y Fr(!)1226 1729 y Fo(2)1254 1790 y Ft(dx)481 b FC(\(7)p Ft(:)p FC(26\))75 1906 y(Then)16 b(from)448 2003 y Fs(j)p Ft(\036)p FC(\()p Ft(x)p FC(\))p Fs(j)40 b(\024)721 1969 y FC(1)p 697 1991 73 2 v 697 2037 a Fs(j)p Ft(X)t Fs(j)782 1944 y Fr(Z)805 2038 y Fq(X)847 2003 y Ft(dy)10 b Fs(j)p Ft(\036)p FC(\()p Ft(y)r FC(\))h(+)g(\()p Ft(\036)p FC(\()p Ft(x)p FC(\))f Fs(\000)h Ft(\036)p FC(\()p Ft(y)r FC(\)\))p Fs(j)611 2134 y(\024)721 2100 y FC(1)p 697 2123 V 697 2168 a Fs(j)p Ft(X)t Fs(j)782 2076 y Fr(Z)805 2170 y Fq(X)847 2134 y Ft(dy)f Fs(j)p Ft(\036)p FC(\()p Ft(y)r FC(\))p Fs(j)g FC(+)1115 2100 y(1)p 1091 2123 V 1091 2168 a Fs(j)p Ft(X)t Fs(j)1177 2076 y Fr(Z)1200 2170 y Fq(X)1242 2134 y Ft(dy)1309 2076 y Fr(Z)1351 2089 y Fq(x)1332 2170 y(y)1381 2134 y Ft(dz)h Fs(j)1459 2100 y Ft(d\036)p 1459 2123 55 2 v 1461 2168 a(dz)1518 2134 y Fs(j)611 2289 y(\024)692 2216 y Fr(")745 2255 y FC(1)p 721 2277 73 2 v 721 2323 a Fs(j)p Ft(X)t Fs(j)806 2230 y Fr(Z)829 2324 y Fq(X)871 2289 y Ft(dy)g Fs(j)p Ft(\036)p Fs(j)988 2268 y Fo(4)1007 2216 y Fr(#)1036 2213 y Fl(1)p 1036 2219 16 2 v 1036 2240 a(4)1069 2289 y FC(+)1118 2216 y Fr(")1143 2289 y Fs(j)p Ft(X)t Fs(j)1223 2230 y Fr(Z)1246 2324 y Fq(X)1288 2289 y Ft(dz)g Fs(j)1366 2255 y Ft(d\036)p 1366 2277 55 2 v 1368 2323 a(dz)1425 2289 y Fs(j)1439 2268 y Fo(2)1458 2216 y Fr(#)1487 2213 y Fl(1)p 1487 2219 16 2 v 1487 2240 a(2)1518 2289 y Ft(;)256 b FC(\(7.27\))75 2409 y(w)o(e)21 b(\014nd)h(that)g(if)f Ft(\024)i(>)g FC(0)f(is)f(held)g(\014xed)g(and)h Ft(\025)g FC(is)g(made)e(small)g(then)h Fs(j)p Ft(E)s FC(\()p Ft(X)q(;)8 b(\036)p FC(\))p Fs(j)p Ft(g)1673 2391 y Fp(\000)p Fo(1)1721 2409 y FC(\()p Ft(X)q(;)g(\036)p FC(\))23 b Fs(\024)75 2473 y Ft(O)q FC(\()p Ft(\025)160 2455 y Fp(\000)192 2442 y Fl(1)p 193 2448 V 193 2468 a(4)216 2473 y FC(\).)d(In)14 b(other)g(w)o(ords)g(the)g(factor)g(of)g Ft(\036)g FC(is)g(of)g(order)g Ft(\025)1158 2455 y Fp(\000)1190 2442 y Fl(1)p 1191 2448 V 1191 2468 a(4)1227 2473 y FC(when)g(the)g(exp)q(onen)o (t)g Ft(exp)p FC([)p Fs(\000)1777 2454 y Fq(\025)p 1775 2462 28 2 v 1775 2491 a Fo(4!)1814 2438 y Fr(R)1850 2473 y Ft(\036)1879 2455 y Fo(4)1899 2473 y FC(])969 2599 y(65)p eop %%Page: 66 70 bop 75 307 a FC(and)16 b(the)f(regulator)h Ft(g)i FC(are)d(tak)o(en)g(in)o (to)h(accoun)o(t.)21 b(\(7.27\))16 b(is)f(an)h(example)d(of)j(a)g Fn(Sob)n(olev)i(estimate)p FC(.)75 367 y(Sob)q(olev)12 b(estimates)f(hold)h (for)g Ft(d)i(>)g FC(1)e(but)g(one)g(needs)g(one)g(extra)g(deriv)m(ativ)o(e)e (for)i(eac)o(h)g(dimension)1881 349 y Fo(9)1899 367 y FC(.)75 428 y(Therefore)k(for)g(dimension)f Ft(d)p FC(,)h(w)o(e)g(c)o(ho)q(ose)497 541 y Ft(g)r FC(\()p Ft(X)q(;)8 b(\036)p FC(\))100 b(=)42 b(exp)915 493 y Fr(h)935 541 y Fs(\000)979 521 y Fq(\024)p 979 529 21 2 v 980 558 a Fo(2)1004 541 y Fs(kr)p Ft(\036)p Fs(k)1125 523 y Fo(2)1125 553 y Fq(q)q(;X)1195 541 y Fs(\000)11 b Ft(C)t(onst:)p Fs(j)p Ft(X)t Fs(j)1462 493 y Fr(i)497 625 y Fs(k)p Ft(\036)p Fs(k)576 606 y Fo(2)576 637 y Fq(q)q(;X)752 625 y Fs(\021)868 583 y Fr(X)832 677 y Fo(1)p Fp(\024j)p Fq(\013)p Fp(j\024)p Fq(q)972 566 y Fr(Z)995 660 y Fq(X)1037 625 y Fs(j)p Ft(@)1080 604 y Fq(\013)1104 625 y Ft(\036)p FC(\()p Ft(x)p FC(\))p Fs(j)1213 604 y Fo(2)1232 625 y Ft(dx;)497 732 y(q)k Fs(\025)f Ft(d)d FC(+)g(1)p Ft(;)1788 633 y FC(\(7)p Ft(:)p FC(28\))75 852 y(where)k Ft(@)244 834 y Fq(\013)268 852 y Ft(\036)f FC(denotes)i(a)f(partial)g(deriv)m (ativ)o(e)e(of)i Ft(\036)g FC(of)g(order)g Fs(j)p Ft(\013)p Fs(j)p FC(.)21 b(Then)15 b(for)g Ft(h)f FC(=)g Ft(O)q FC(\()p Ft(\025)1654 833 y Fp(\000)1686 820 y Fl(1)p 1687 826 16 2 v 1687 846 a(4)1709 852 y FC(\))h(and)h(an)o(y)75 912 y Ft(\024)e(>)g FC(0)i(held)g(\014xed.)664 972 y Fs(k)p Ft(E)s Fs(k)753 979 y Fq(g)q(;)p Fo(\000)p Fq(;h)849 972 y FC(=)e Ft(O)q FC(\()p Ft(\025)991 938 y Fl(1)p 992 944 V 992 965 a(4)1014 972 y FC(\))p Ft(;)73 b FC(as)17 b Ft(\025)d Fs(!)g FC(0)p Ft(:)464 b FC(\(7)p Ft(:)p FC(29\))75 1059 y(The)11 b Ft(\036)g FC(deriv)m(ativ)o(es)e(whic)o(h)h (are)h(part)g(of)h(the)e(de\014nition)h(\(7.15\))g(of)g(the)g(norm)f(ha)o(v)o (e)g(to)h(b)q(e)g(estimated)75 1132 y(in)f(v)m(ariation)i(norm)e(and)h(this)g (is)f(done)h(using)h(e.g.,)e(1)p Ft(=)p Fs(j)p FC(\001)p Fs(j)1128 1096 y Fr(R)1147 1145 y Fo(\001)1187 1132 y Ft(d\030)h(f)5 b FC(\()p Ft(\030)r FC(\))p Ft(\036)1363 1114 y Fo(3)1384 1132 y FC(\()p Ft(\030)r FC(\))14 b Fs(\024)g(f)p FC(1)p Ft(=)p Fs(j)p FC(\001)p Fs(j)1662 1096 y Fr(R)1682 1145 y Fo(\001)1722 1132 y Ft(d\030)d(\036)1808 1114 y Fo(4)1828 1132 y FC(\()p Ft(\030)r FC(\))p Fs(g)1919 1092 y Fl(3)p 1919 1098 V 1919 1119 a(4)75 1192 y FC(for)18 b Fs(j)p Ft(f)5 b Fs(j)18 b(\024)e FC(1,)j(to)g(estimate)d(factors)j(resulting)e(from)g(deriv)m(ativ)o(es)g(of)i (the)f(exp)q(onen)o(tial)f(in)h(\(7.25\))75 1252 y(in)e(terms)f(of)h(the)g (exp)q(onen)o(t.)148 1312 y(The)d(fact)h(that)f(the)g(\014elds)g Ft(\036)g FC(actually)g(are)g Fn(sti\013)g FC(is)h(captured)f(b)o(y)g(the)g (h)o(yp)q(othesis)g Ft(\026)1700 1319 y Fq(ts)1736 1312 y Fs(\003)5 b Ft(g)1789 1319 y Fq(s)1822 1312 y Fs(\024)13 b Ft(g)1897 1319 y Fq(t)75 1373 y FC(in)k(Theorem)f(7.1.)25 b(Let)18 b(us)g(tak)o(e)e Ft(g)723 1380 y Fq(s)760 1373 y FC(as)i(in)f(\(7.28\),)h(and)g(let)e Ft(t)g FC(=)g Ft(s)11 b FC(+)h Ft(ds)18 b FC(so)g(that)g(the)f(co)o(v)m (ariance)75 1433 y(of)g Ft(\026)160 1440 y Fq(ts)207 1433 y FC(is)f(in\014nitesimal.)j(Then,)725 1562 y Ft(\026)754 1569 y Fq(ts)797 1562 y Fs(\003)11 b Ft(g)856 1569 y Fq(s)889 1562 y FC(=)i Ft(g)963 1569 y Fq(s)993 1562 y FC(+)1047 1529 y(1)p 1047 1551 25 2 v 1047 1596 a(2)1076 1562 y(\()p Ft(g)1118 1569 y Fq(s)1137 1562 y FC(\))1156 1569 y Fq(\036\036)1200 1562 y Ft(ds:)526 b FC(\(7)p Ft(:)p FC(30\))75 1688 y(W)l(e)15 b(\014nd)h(that)g (if)e(w)o(e)h(let)g Ft(\024)g FC(in)g(\(7.28\))h(and)g(the)f(Const.)22 b Fs(\021)14 b Ft(v)i FC(dep)q(end)g(on)g Ft(s;)8 b(\024)14 b FC(=)f Ft(\024)1615 1695 y Fq(s)1634 1688 y Ft(;)8 b(v)15 b FC(=)f Ft(v)1771 1695 y Fq(s)1789 1688 y FC(,)h(then)75 1748 y Ft(\026)104 1755 y Fq(ts)147 1748 y Fs(\003)c Ft(g)206 1755 y Fq(s)238 1748 y Fs(\024)j Ft(g)314 1755 y Fq(t)345 1748 y FC(holds)j(for)f Ft(t)e FC(=)f Ft(s)f FC(+)f Ft(ds)16 b FC(pro)o(vided)693 1846 y Ft(d\024)746 1853 y Fq(s)p 693 1868 72 2 v 705 1914 a Ft(ds)784 1880 y Fs(\025)e Ft(a)863 1887 y Fq(s)881 1880 y Ft(\024)909 1859 y Fo(2)909 1892 y Fq(s)977 1880 y FC(;)1053 1846 y Ft(dv)1102 1853 y Fq(s)p 1053 1868 68 2 v 1062 1914 a Ft(ds)1139 1880 y Fs(\025)g Ft(b)1213 1887 y Fq(s)1231 1880 y Ft(\024)1259 1887 y Fq(s)1277 1880 y Ft(;)497 b FC(\(7)p Ft(:)p FC(31\))75 2014 y(where)22 b(the)g Ft(a)338 2021 y Fq(s)379 2014 y FC(and)h Ft(b)501 2021 y Fq(s)541 2014 y FC(are)f(determined)e(b)o(y)i (the)g(partial)h(deriv)m(ativ)o(es)e(of)h(the)g(co)o(v)m(ariance)1876 2001 y(_)1859 2014 y Ft(C)1894 2021 y Fq(s)75 2074 y FC(up)e(to)g(order)g Ft(s)g FC(in)g Ft(x)f FC(and)i(in)e Ft(y)r FC(,)h(e.g.,)g Ft(@)850 2056 y Fq(\013)847 2086 y(x)874 2074 y Ft(@)903 2056 y Fq(\014)900 2086 y(y)942 2061 y FC(_)926 2074 y Ft(C)961 2081 y Fq(s)979 2074 y FC(\()p Ft(x;)8 b(y)r FC(\))p Ft(;)g Fs(j)p Ft(\013)p Fs(j)19 b(\024)h Ft(s;)8 b Fs(j)p Ft(\014)s Fs(j)19 b(\024)h Ft(s)p FC(,)g(whic)o(h)f(ha)o(v)o(e)g(to)h(exist)75 2134 y(as)h(con)o(tin)o (uous)f(functions)h(for)f Ft(x)h Fs(')g Ft(y)h FC(and)f(to)g(b)q(e)f(in)o (tegrable)g(in)g Ft(x)13 b Fs(\000)h Ft(y)r FC(.)33 b(These)21 b(are)f(natural)75 2194 y(conditions)f(b)q(ecause,)h(for)g Ft(t)e FC(=)h Ft(s)13 b FC(+)g Ft(ds)p FC(,)20 b Ft(@)878 2176 y Fq(\013)875 2207 y(x)902 2194 y Ft(@)931 2176 y Fq(\013)928 2207 y(y)972 2182 y FC(_)956 2194 y Ft(C)991 2201 y Fq(s)1009 2194 y FC(\()p Ft(x;)8 b(y)r FC(\))p Ft(ds)19 b FC(is)g(the)g(co)o(v)m (ariance)f(of)i Ft(@)1655 2176 y Fq(\013)1652 2207 y(x)1679 2194 y Ft(\020)t FC(\()p Ft(x)p FC(\))f(whic)o(h)75 2263 y(o)q(ccurs)e(in)f (\()p Ft(\026)331 2270 y Fq(ts)374 2263 y Fs(\003)c Ft(g)434 2270 y Fq(s)452 2263 y FC(\)\()p Ft(\036)p FC(\))j(=)605 2227 y Fr(R)641 2263 y Ft(d\026)695 2270 y Fq(ts)727 2263 y FC(\()p Ft(\020)t FC(\))p Ft(g)813 2270 y Fq(s)832 2263 y FC(\()p Ft(\036)c FC(+)g Ft(\020)t FC(\).)23 b(The)16 b(regularit)o(y)g(of)h(these)f(deriv)m (ativ)o(es)g(of)1841 2250 y(_)1825 2263 y Ft(C)k FC(is)75 2323 y(the)c(condition)g(that)h(guaran)o(tees)g(that)f Ft(d\026)881 2330 y Fq(ts)929 2323 y FC(liv)o(es)f(only)h(on)h Fn(sti\013)f FC(\014elds)g Ft(\036)p FC(.)p 75 2367 735 2 v 131 2398 a Fj(9)150 2413 y Fy(F)m(or)d(this)h(crude)h(argumen)o(t.)i(It)d(is)g(p)q(ossible)g(to)g (get)g(a)o(w)o(a)o(y)f(with)1186 2396 y Fj(1)p 1186 2403 17 2 v 1186 2427 a(2)1217 2413 y Fy(+)d Fi(\017)j Fy(deriv)n(ativ)o(es)h(p)q(er) h(dimension)969 2599 y FC(66)p eop %%Page: 67 71 bop 75 307 a Ff(7.1)70 b(Manipulations)22 b(with)f Fa(E)5 b FB(xp)75 400 y FC(W)l(e)15 b(illustrate)g(some)g(of)h(the)f(prop)q(erties)h (of)g Fs(E)t FC(xp)g(whic)o(h)f(w)o(e)h(will)e(need)i(in)f(the)g(next)h (section)f(and)75 460 y(also)i(sho)o(w)g(ho)o(w)g(to)g(rewrite)f(Gibbs)g (factors)h(in)g(the)f(form)f Fs(E)t FC(xp)q([)p Fg(2)c FC(+)g Ft(K)t FC(])16 b(along)i(with)e(b)q(ounds)i(on)75 520 y Ft(K)t FC(.)148 580 y(Let)f Ft(\036)p FC(\()p Ft(\030)r FC(\))d(=)417 561 y Fq(@)p 397 568 62 2 v 397 597 a(@)r(x)438 601 y Fm(\026)463 580 y Ft(\036)p FC(\()p Ft(x)p FC(\))p Ft(;)723 708 y(Z)756 715 y Fo(0)776 708 y FC(\(\003)p Ft(;)8 b(\036)p FC(\))41 b(=)h Ft(e)1043 686 y Fp(\000)1076 654 y Fr(R)1096 703 y Fl(\003)1124 686 y Fq(\025\036)1166 674 y Fl(4)1183 686 y Fo(\()p Fq(\030)q Fo(\))p Fq(d\030)748 760 y Fr(Z)798 818 y Ft(d\030)r FC(\()p Fs(\001)p FC(\))g Fs(\021)1044 764 y Fq(d)1023 777 y Fr(X)1020 867 y Fq(\026)p Fo(=1)1095 760 y Fr(Z)1145 818 y Ft(dx)p FC(\()p Fs(\001)p FC(\))p Ft(:)524 b FC(\(7.32\))75 960 y(W)l(e)16 b(can)g(rewrite)g(this)g(in)g(terms)e(of)j Fs(E)t FC(xp)g(b)o(y)f(de\014ning) 492 1096 y Ft(F)7 b FC(\()p Ft(X)t FC(\))41 b(=)733 1023 y Fr(\()787 1072 y Ft(e)810 1049 y Fp(\000)p Fq(\025)864 1018 y Fr(R)884 1066 y Fm(X)918 1049 y Fq(\036)939 1038 y Fl(4)957 1049 y Fq(d\030)1051 1072 y FC(if)16 b Ft(X)21 b FC(is)16 b(a)g(unit)g(blo)q (c)o(k)787 1132 y(0)240 b(otherwise)p Ft(;)1788 1096 y FC(\(7.33\))492 1212 y Ft(R)p FC(\()p Ft(X)t FC(\))43 b(=)e Ft(e)756 1191 y Fq(F)5 b Fo(\()p Fq(X)s Fo(\))855 1212 y Fs(\000)11 b FC(1)p Ft(;)845 b FC(\(7.34\))75 1316 y(then)687 1420 y Ft(Z)720 1427 y Fo(0)740 1420 y FC(\(\003\))42 b(=)969 1379 y Fr(Y)954 1470 y Fq(X)s Fp(\032)p Fo(\003)1058 1420 y Ft(e)1081 1400 y Fq(F)5 b Fo(\()p Fq(X)s Fo(\))854 1539 y FC(=)969 1497 y Fr(Y)954 1589 y Fq(X)s Fp(\032)p Fo(\003)1058 1539 y FC(\(1)11 b(+)g Ft(R)p FC(\()p Ft(X)t FC(\)\))489 b(\(7.35\))75 1681 y(\(whic)o(h)16 b(de\014nes)g Ft(R)p FC(\).)148 1741 y(No)o(w)f(w)o(e)g(expand)g(the)f(pro)q (duct.)22 b(It)14 b(do)q(es)i(not)g(imm)o(ediatel)o(y)c(giv)o(e)i Fs(E)t FC(xp)q([)p Fg(2)8 b FC(+)g Ft(R)p FC(])15 b(b)q(ecause)h(t)o(w)o(o)75 1801 y(di\013eren)o(t)i(sets)i Ft(X)j FC(can)d(touc)o(h,)f(therefore)g(w)o(e) f(group)i(them)e(together)h(whenev)o(er)f(this)h(happ)q(ens)75 1862 y(b)o(y)75 1945 y Fu(De\014nition)f(6.1.1.)i FC(Giv)o(en)15 b(an)o(y)h(function)g Ft(K)t FC(\()p Ft(X)t FC(\))h(on)g(p)q(olymers,)d(let) 700 2049 y Ft(K)745 2028 y Fo(+)775 2049 y FC(\()p Ft(X)t FC(\))g(=)965 2008 y Fr(X)922 2102 y Fp(f)p Fq(X)969 2107 y Fm(i)982 2102 y Fp(g!)p Fq(X)1075 2008 y Fr(Y)1096 2099 y Fq(i)1136 2049 y Ft(K)t FC(\()p Ft(X)1240 2056 y Fq(i)1255 2049 y FC(\))p Ft(;)75 2202 y FC(where)20 b Fs(f)p Ft(X)285 2209 y Fq(i)299 2202 y Fs(g)h(!)g Ft(X)j FC(means)c(that)h Fs(f)p Ft(X)809 2209 y Fq(i)823 2202 y Fs(g)f FC(is)g(summed)e(o)o(v)o(er)h(all)h Fn(sets)1384 2183 y Fo(10)1443 2202 y Fs(f)p Ft(X)1508 2209 y Fo(1)1528 2202 y Ft(;)8 b(:)g(:)g(:)f(;)h(X)1677 2209 y Fq(N)1732 2202 y FC(:)20 b Fs([)p Ft(X)1839 2209 y Fq(i)1875 2202 y FC(=)75 2262 y Ft(X)q(;)8 b(N)24 b(>)17 b FC(1)g(arbitrary)p Fs(g)h FC(suc)o(h)h(that)g Ft(X)792 2269 y Fo(1)812 2262 y Ft(;)8 b(:)g(:)g(:)f(;)h(X)961 2269 y Fq(N)1014 2262 y FC(are)18 b(distinct)g(and)h (the)g(o)o(v)o(erlap)e(graph)j(is)e(con-)75 2322 y(nected.)25 b(The)18 b Fn(overlap)i(gr)n(aph)d FC(is)g(the)h(graph)h(on)f(v)o(ertices)e Fs(f)p FC(1)p Ft(;)8 b(:)g(:)g(:)g(;)g(N)d Fs(g)18 b FC(whose)g(b)q(onds)h (are)f(those)75 2382 y(pairs)e(\()p Ft(ij)s FC(\))g(suc)o(h)g(that)h Ft(X)544 2389 y Fq(i)570 2382 y Fs(\\)11 b Ft(X)654 2389 y Fq(j)686 2382 y Fs(6)p FC(=)j Fs(;)p FC(.)p 75 2428 735 2 v 115 2459 a Fj(10)150 2474 y Fy(So)f(the)i(order)f(of)g Fi(X)469 2480 y Fj(1)488 2474 y Fi(;)7 b(:)g(:)g(:)e(;)i(X)615 2480 y Fh(N)660 2474 y Fy(do)q(es)14 b(not)g(distinguish)f FA(f)p Fi(X)1092 2480 y Fj(1)1111 2474 y Fi(;)7 b(:)g(:)g(:)e(;)i(X)1238 2480 y Fh(N)1269 2474 y FA(g)969 2599 y FC(67)p eop %%Page: 68 72 bop 148 307 a FC(In)19 b(other)g(w)o(ords)h Ft(K)530 289 y Fo(+)559 307 y FC(\()p Ft(X)t FC(\))g(is)f(a)g(sum)f(o)o(v)o(er)g(all)h(w)o (a)o(ys)g(of)g(\014lling)f(a)i(set)f Ft(X)k FC(with)c(p)q(olymers)f(so)75 367 y(that)f(the)g(p)q(olymers)e(o)o(v)o(erlap)h(to)i(form)d(a)j(connected)e (clump.)21 b(Under)16 b(these)h(circumstances)e(the)75 428 y(norm)k(of)h(the)g(clump)e(of)i(p)q(olymers)f(is)h(roughly)g(sp)q(eaking)h (the)e(pro)q(duct)i(of)f(the)g(norms)f(of)i(the)75 488 y(individual)14 b(p)q(olymers)g(except)f(that)j(if)e(sev)o(eral)g(p)q(olymers)g(o)o(v)o (erlap)g(then)h(the)g(resulting)g(function)75 548 y(of)h Ft(\036)f FC(will)g(gro)o(w)h(as)g(the)g(pro)q(duct)g(of)g(the)f(gro)o(wths)i(in)e Ft(\036)g FC(of)h(the)g(p)q(olymers)e(that)i(o)o(v)o(erlap.)k(This)c(is)75 608 y(what)h(w)o(e)f(see)g(in)75 722 y Fu(Prop)r(osition)i(7.2)24 b Fn(F)l(or)17 b(any)g(functional)i Ft(K)118 884 y Fn(\(i\))644 903 y Fr(Y)626 995 y Fq(Y)9 b Fp(\032)p Fq(X)714 944 y FC([1)i(+)g Ft(K)t FC(\()p Ft(Y)g FC(\)])i(=)h Fs(E)t FC(xp)q([)p Fg(2)c FC(+)h Ft(K)16 b FC(+)11 b Ft(K)1358 924 y Fo(+)1387 944 y FC(]\()p Ft(X)t FC(\))103 1090 y Fn(\(ii\))24 b(F)l(or)17 b Ft(\015)g(>)d FC(1)j Fn(ther)n(e)h(is)f Ft(C)632 1097 y Fq(\015)672 1090 y Fn(such)g(that)h(for)f(any)g FC(\000)732 1200 y Fs(k)p Ft(K)802 1180 y Fo(+)831 1200 y Fs(k)856 1207 y Fq(g)874 1198 y Fm(\034)893 1207 y Fq(;)p Fo(\000)p Fq(;h)971 1200 y Fs(\024)1032 1159 y Fr(X)1024 1251 y Fq(N)t Fp(\025)p Fo(2)1109 1200 y Ft(C)1148 1180 y Fq(N)1144 1213 y(\015)1181 1200 y Fs(k)p Ft(K)t Fs(k)1276 1180 y Fq(N)1276 1213 y(g)q(;\015)r Fo(\000)p Fq(;h)197 1357 y Fn(wher)n(e)g FC(\()p Ft(\015)s FC(\000\)\()p Ft(X)t FC(\))d(=)g Ft(\015)606 1339 y Fp(j)p Fq(X)s Fp(j)659 1357 y FC(\000\()p Ft(X)t FC(\))p Fn(,)k(and)f Ft(\034)22 b Fn(is)17 b(the)g(maximum)g(overlap)g (of)f(sets)h(in)g(the)g(supp)n(ort)197 1417 y(of)g Ft(R)p Fn(:)645 1478 y Ft(\034)j FC(=)14 b(sup)765 1512 y Fq(x)819 1478 y FC(#)p Fs(f)p Ft(X)k FC(:)c Ft(X)k Fs(3)c Ft(x)j Fn(and)h Ft(R)p FC(\()p Ft(X)t FC(\))c Fs(6)p FC(=)g(0)p Fs(g)p Ft(:)75 1637 y Fu(Pro)r(of.)27 b FC(P)o(art)17 b(\(i\))f(is)g(left)f(to)i(the)f(reader,)f(and)i(for)g(part)f (\(ii\),)f(see)h([18],)g(Lemma)e(5.1.)181 b Fg(2)148 1722 y FC(Th)o(us)614 1782 y Ft(e)637 1760 y Fp(\000)670 1729 y Fr(R)689 1777 y Fl(\003)718 1760 y Fq(\025\036)760 1748 y Fl(4)777 1760 y Fo(\()p Fq(\030)q Fo(\))5 b Fq(d\030)878 1782 y FC(=)13 b Fs(E)t FC(xp)q([)p Fg(2)e FC(+)g Ft(R)g FC(+)g Ft(R)1258 1762 y Fo(+)1289 1782 y FC(]\(\003\))413 b(\(7)p Ft(:)p FC(36\))75 1869 y(and)14 b(the)f(prop)q(osition)h(is)f(the)g(estimate)e(w)o(e)i(need)g (to)g(sho)o(w)h(that)g Fs(k)p Ft(R)1339 1851 y Fo(+)1369 1869 y Fs(k)1394 1876 y Fq(g)q(;)p Fo(\000)p Fq(;h)1489 1869 y FC(is)f(\(v)o (ery\))f(small)f(when)75 1929 y Fs(k)p Ft(R)p Fs(k)162 1936 y Fq(g)q(;)p Fo(\000)p Fq(;h)261 1929 y FC(is)16 b(small.)148 2015 y(There)j(is)h(another)g(prop)q(osition)g(whic)o(h)f(is)h(not)g(of)f (immediate)d(relev)m(ance)i(but)i(whic)o(h)f(is)g(of)75 2075 y(the)13 b(same)g(nature)h(and)g(is)f(needed)g(later,)g(so)i(w)o(e)e(include) f(it)h(no)o(w.)21 b(By)13 b(de\014nition)g(of)h Fs(E)t FC(xp)g(w)o(e)f(ha)o (v)o(e)75 2135 y Fs(E)t FC(xp)q([)p Ft(A)e FC(+)h Ft(B)s FC(])17 b(=)g Fs(E)t FC(xp[)p Ft(A)p FC(])11 b Fs(\016)i(E)t FC(xp[)p Ft(B)s FC(],)k(but)h(when)h(w)o(e)f(w)o(an)o(t)g(an)g(ordinary)h(pro)q(duct)g (of)f(t)o(w)o(o)g Fs(E)t FC(xp)q('s)75 2195 y(then)e(the)g(result)g(is)75 2309 y Fu(Prop)r(osition)i(7.3)67 b Fn(\(i\))501 2419 y Fs(E)t FC(xp)q([)p Fg(2)11 b FC(+)g Ft(A)p FC(])p Fs(E)t FC(xp)o([)p Fg(2)g FC(+)g Ft(B)s FC(])i(=)h Fs(E)t FC(xp)q([)p Fg(2)c FC(+)h Ft(A)g FC(+)g Ft(B)j FC(+)d Ft(A)f Fs(_)i Ft(B)s FC(])p Ft(;)969 2599 y FC(68)p eop %%Page: 69 73 bop 197 307 a Fn(wher)n(e)580 367 y FC(\()p Ft(A)11 b Fs(_)g Ft(B)s FC(\)\()p Ft(X)t FC(\))j(=)980 326 y Fr(X)897 420 y Fp(f)p Fq(X)944 425 y Fm(i)957 420 y Fp(g)p Fq(;)p Fp(f)p Fq(Y)1023 425 y Fm(j)1039 420 y Fp(g!)p Fq(X)1132 326 y Fr(Y)1152 417 y Fq(i)1193 367 y Ft(A)p FC(\()p Ft(X)1289 374 y Fq(i)1303 367 y FC(\))1330 326 y Fr(Y)1349 417 y Fq(j)1392 367 y Ft(B)s FC(\()p Ft(Y)1479 374 y Fq(j)1497 367 y FC(\))p Ft(;)197 503 y Fn(wher)n(e)k Fs(f)p Ft(X)400 510 y Fq(i)414 503 y Fs(g)p Ft(;)8 b Fs(f)p Ft(Y)514 510 y Fq(j)532 503 y Fs(g)14 b(!)g Ft(X)21 b Fn(me)n(ans)d(that)g(the)f(sum)h(is)f(over)h(families)g(of)f(sets)h Fs(f)p Ft(X)1690 510 y Fo(1)1710 503 y Ft(;)8 b(:)g(:)g(:)f(;)h(X)1859 510 y Fq(N)1893 503 y Ft(;)197 563 y(Y)225 570 y Fo(1)245 563 y Ft(;)g(:)g(:)g(:)f(;)h(Y)382 570 y Fq(M)422 563 y Fs(g)22 b Fn(with)h Ft(N)k Fs(\025)c FC(1)p Fn(,)g Ft(M)28 b Fs(\025)22 b FC(1)p Fn(,)i(such)f(that)f FC(\()p Fs([)p Ft(X)1281 570 y Fq(i)1296 563 y FC(\))14 b Fs([)h FC(\()p Fs([)p Ft(Y)1457 570 y Fq(j)1476 563 y FC(\))22 b(=)g Ft(X)t Fn(,)i(the)f Ft(X)t Fn('s)f(ar)n(e)197 623 y(disjoint,)e(the)g Ft(Y)11 b Fn('s)19 b(ar)n(e)g(disjoint,)h(and)g(the)g(overlap)g(gr)n(aph)e(on)i Fs(f)p FC(1)p Ft(;)8 b(:)g(:)g(:)g(;)g(N)r(;)g FC(1)1648 605 y Fp(0)1660 623 y Ft(;)g(:)g(:)g(:)f(;)h(M)1821 605 y Fp(0)1833 623 y Fs(g)20 b Fn(is)197 683 y(c)n(onne)n(cte)n(d.)103 785 y(\(ii\))k(F)l(or)17 b Ft(\015)g(>)d FC(1)j Fn(ther)n(e)h(is)f Ft(C)632 792 y Fq(\015)672 785 y Fn(such)g(that)h(for)f(any)g FC(\000)p Fn(,)501 895 y Fs(k)p Ft(A)11 b Fs(_)g Ft(B)s Fs(k)683 902 y Fq(g)700 908 y Fm(A)725 902 y Fq(g)742 908 y Fm(B)768 902 y Fq(;)p Fo(\000)p Fq(;h)846 895 y Fs(\024)929 853 y Fr(X)898 945 y Fq(N)r(;M)t Fp(\025)p Fo(1)1028 895 y Ft(C)1067 874 y Fq(N)t Fo(+)p Fq(M)1063 907 y(\015)1165 895 y Fs(k)p Ft(A)p Fs(k)1252 874 y Fq(N)1252 907 y(g)1269 913 y Fm(A)1294 907 y Fq(;\015)r Fo(\000)p Fq(;h)1378 895 y Fs(k)p Ft(B)s Fs(k)1468 874 y Fq(M)1468 907 y(g)1485 913 y Fm(B)1510 907 y Fq(;\015)r Fo(\000)p Fq(;h)1594 895 y Ft(;)197 1053 y Fn(wher)n(e)18 b FC(\()p Ft(\015)s FC(\000\)\()p Ft(X)t FC(\))c(=)g Ft(\015)607 1035 y Fp(j)p Fq(X)s Fp(j)660 1053 y FC(\000\()p Ft(X)t FC(\))p Fn(.)75 1192 y Fu(Pro)r(of.)27 b FC(A)16 b(simple)e(v)m(ariation)j(on)g(the)f (pro)q(of)h(of)g(Prop)q(osition)g(7.2.)538 b Fg(2)969 2599 y FC(69)p eop %%Page: 70 74 bop 75 319 a Fv(8)83 b(The)27 b(Renormalization)g(Group)g(\(R)n(G\))75 429 y FC(This)16 b(section)g(is)g(based)h(on)g([18,)f(42)q(].)148 489 y(The)c(R)o(G)f(mak)o(es)f(its)h(app)q(earance)i(when)e(the)h(parameter)e Ft(t)h FC(that)h(en)o(ters)f(in)o(to)g(the)g(deformation)75 549 y(of)24 b(the)g(co)o(v)m(ariance)g(is)g(tied)f(to)h(the)g(c)o(hange)g(of) h(a)f(length)g(scale)g Ft(`)g FC(whic)o(h)f(parametrizes)g(the)75 610 y(smo)q(othness)17 b(of)f(the)g(\014eld)g Ft(\036)p FC(.)21 b(E.g.,)15 b(w)o(e)h(tak)o(e)g Ft(`)e FC(=)g Ft(`)p FC(\()p Ft(t)p FC(\))g(=)g Ft(`)1184 617 y Fo(0)1204 610 y Ft(e)1227 591 y Fq(t)1257 610 y FC(and)j(consider)809 735 y Ft(Z)842 742 y Fq(t)871 735 y FC(=)923 676 y Fr(Z)972 735 y Ft(d\026)1026 742 y Fq(t)1042 735 y Ft(Z)1075 742 y Fq(t)1090 735 y FC(\()p Ft(\036)p FC(\))p Ft(;)641 b FC(\(8)p Ft(:)p FC(1\))75 857 y(where)16 b Ft(d\026)270 864 y Fq(t)302 857 y FC(has)h(co)o(v)m(ariance)644 920 y Fr(Z)694 979 y Ft(d\026)748 986 y Fq(t)764 979 y Ft(\036)p FC(\()p Ft(x)p FC(\))p Ft(\036)p FC(\()p Ft(y)r FC(\))c(=)g Ft(C)1051 987 y Fq(`)p Fo(\()p Fq(t)p Fo(\))1108 979 y FC(\()p Ft(x)e Fs(\000)f Ft(y)r FC(\))745 1092 y Ft(C)780 1099 y Fq(`)796 1092 y FC(\()p Ft(x)p FC(\))k(=)f Ft(`)947 1071 y Fp(\000)p Fo(\()p Fq(d)p Fp(\000)p Fo(2\))1068 1092 y Ft(C)1103 1099 y Fo(1)1131 1031 y Fr(\022)1167 1058 y Ft(x)p 1167 1080 28 2 v 1171 1126 a(`)1199 1031 y Fr(\023)1238 1092 y Ft(;)560 b FC(\(8.2\))75 1217 y(and)17 b(the)f(F)l(ourier)g(transform)f(of)i Ft(C)737 1224 y Fq(`)770 1217 y FC(is)816 1339 y(^)805 1352 y Ft(C)840 1359 y Fq(`)857 1352 y FC(\()p Ft(k)r FC(\))c(=)992 1318 y Ft(F)7 b FC(\()p Ft(`)1070 1300 y Fo(2)1090 1318 y Ft(k)1117 1300 y Fo(2)1136 1318 y FC(\))p 992 1340 164 2 v 1036 1386 a Ft(\033)r(k)1093 1372 y Fo(2)1160 1352 y Ft(;)638 b FC(\(8)p Ft(:)p FC(3\))75 1478 y(where)18 b Ft(F)7 b FC(\()p Ft(t)p FC(\))17 b(is)h(a)g(smo)q(oth)h(decreasing)f(function)g(of)g Ft(t)f Fs(\025)g FC(0)h(whic)o(h)g(deca)o(ys)f(rapidly)h(as)h Ft(t)d Fs(!)h(1)p FC(,)75 1538 y(together)g(with)h(all)e(deriv)m(ativ)o(es)g (and)i Ft(F)7 b FC(\(0\))15 b(=)h Ft(F)998 1520 y Fp(0)1009 1538 y FC(\(0\))g(=)f(1)p Ft(;)8 b(\033)17 b(>)f FC(0)i(is)f(the)g (dielectric)d(constan)o(t)k(or)75 1598 y(Field)d(Strength)h(Renormalization.) 148 1658 y(The)j(function)g Ft(F)26 b FC(imp)q(oses)18 b(an)h(ultra-violet)g (cuto\013)g(in)g(suc)o(h)g(a)g(w)o(a)o(y)g(that)h(\014elds)e Ft(\036)p FC(\()p Ft(x)p FC(\))h(that)75 1718 y(are)f(t)o(ypical)e(for)i(the) f(measure)g Ft(d\026)726 1725 y Fq(t)759 1718 y FC(are)h(functions)f(whic)o (h)g(are)h(self-similar)d(on)j(all)g(scales)f(do)o(wn)75 1779 y(to)g Ft(`)p FC(\()p Ft(t)p FC(\))f(and)h(at)g(this)f(scale)g(they)g(b)q (ecome)f(appro)o(ximately)g(constan)o(t.)22 b(This)17 b(picture)e(motiv)m (ates,)75 1839 y(for)h(example,)d(the)j(c)o(hoice)e(of)i(blo)q(c)o(k)g(size)f (in)g(the)h(cluster)f(expansion)h(of)g(the)f(last)h(section.)21 b(\\Unit)75 1899 y(blo)q(c)o(k")14 b(should)h(ha)o(v)o(e)f(side)g(1)h(in)g(a) g(unit)f(of)h(length)f(comparable)g(to)h Ft(`)p FC(\()p Ft(t)p FC(\))f(b)q(ecause)h(a)g(blo)q(c)o(k)f(should)75 1959 y(b)q(e)19 b(an)h(ob)s(ject)f(with)g(the)g(prop)q(ert)o(y)g(that)h(the)f(\014eld)f(is)h (essen)o(tially)f(constan)o(t)i(within)e(it.)30 b(If)19 b(the)75 2019 y(blo)q(c)o(k)f(size)f(is)i(alw)o(a)o(ys)f(k)o(ept)g(comparable)f(with)h Ft(`)p FC(\()p Ft(t)p FC(\),)h(then)f(the)g(cluster)g(expansion)g(of)h(the)f (last)75 2079 y(section)13 b(will)g(pro)o(vide)g(error)h(estimates)e(that)i (scale)f(according)h(to)g(simple)e(dimensional)g(analysis.)148 2140 y(The)f(collection)e(of)i(\014elds)g(\()p Ft(\036)p FC(\()p Ft(x)p FC(\))p Ft(;)d Fs(r)802 2147 y Fq(\026)824 2140 y Ft(\036)p FC(\()p Ft(x)p FC(\))p Ft(;)g Fs(r)983 2147 y Fq(\026)1005 2140 y Fs(r)1047 2147 y Fq(\027)1068 2140 y Ft(\036)p FC(\()p Ft(x)p FC(\))p Ft(;)g(:)g(:)g(:)o FC(\))14 b Fs(\021)f FC(\()p Ft(\036)p FC(\()p Ft(x)p FC(\))p Ft(;)8 b(\036)p FC(\()p Ft(x;)g(\026)p FC(\))p Ft(;)g(\036)p FC(\()p Ft(x;)g(\026;)g(\027)s FC(\))p Ft(;)g(:)g(:)g(:)n FC(\))75 2200 y(will)18 b(b)q(e)i(denoted)f Ft(\036)p FC(\()p Ft(\030)r FC(\))h(so)g(in)f(this)g(section)g Ft(\030)j FC(=)d(\()p Ft(x;)8 b(\013)p FC(\))20 b(with)f Ft(\013)g FC(=)g Fs(;)p FC(,)h Ft(\026)p FC(,)g(\()p Ft(\026;)8 b(\027)s FC(\),)20 b(or)g Ft(:)8 b(:)g(:)19 b FC(up)g(to)75 2260 y(some)c(p)q(ermanen) o(tly)f(\014xed)i(\014nite)g(n)o(um)o(b)q(er)e(of)j(deriv)m(ativ)o(es.)148 2320 y(F)l(or)g(the)f(rest)g(of)g(this)g(section,)922 2380 y Ft(`)942 2387 y Fo(0)976 2380 y FC(=)e(1)p Ft(:)75 2468 y FC(I)i(will)f(write)h Ft(d\026)380 2475 y Fq(`)413 2468 y FC(sometimes)d (instead)k(of)f Ft(d\026)927 2475 y Fq(t)959 2468 y FC(where)g Ft(`)e FC(=)g Ft(e)1209 2450 y Fq(t)1223 2468 y FC(.)969 2599 y(70)p eop %%Page: 71 75 bop 75 307 a Fu(Con)n(v)n(en)n(tions.)32 b Ft(a)19 b Fs(\035)g FC(1)h(means)e Ft(a)h FC(su\016cien)o(tly)f(large)h(dep)q(ending)h(only)f(on) h(the)f(dimension)f Ft(d)75 367 y FC(unless)i(I)g(explicitly)d(men)o(tion)h (dep)q(endence)h(on)i(other)f(parameters.)31 b(Similarly)18 b(for)i Ft(a)g Fs(\034)g FC(1.)33 b(I)75 428 y(will)16 b(use)i Ft(C)q(;)8 b(C)347 435 y Fo(1)366 428 y Ft(;)g(C)423 435 y Fo(2)443 428 y Ft(;)g(:)g(:)g(:)f(;)17 b FC(for)h(constan)o(ts)g(whic)o(h)f (dep)q(end)h(only)f(on)h(dimension)e Ft(d)p FC(.)26 b(If)17 b(a)h(constan)o(t)75 488 y(dep)q(ends)h(on)g(a)g(parameter,)e(e.g.,)h Ft(\025)p FC(,)h(I)f(will)g(indicate)f(it)h(b)o(y)g Ft(C)t FC(\()p Ft(\025)p FC(\))h(unless)f Ft(\025)h FC(has)g(already)g(b)q(een)75 548 y(\014xed)d(once)g(and)h(for)f(all.)75 658 y Fu(P)n(erio)r(dic)j (Boundary)i(Conditions.)26 b FC(F)l(or)18 b(simplicit)n(y)d(I)i(shall)h (assume)f(\003)h(is)f(a)h(torus,)h(whose)75 718 y(sides)e(are)g(equal)f(and)h (equal)f(to)i(a)f(p)q(o)o(w)o(er)g(of)g Ft(L)g FC(where)f Ft(L)h FC(is)g(an)g(in)o(teger)f(that)h(will)f(b)q(e)h(pro)q(duced)75 778 y(b)q(elo)o(w.)36 b(The)21 b(F)l(ourier)f(transform)h(b)q(ecomes)f(a)h(F) l(ourier)g(series)f(and)i(in)f(the)f(series)h(for)g Ft(C)1807 785 y Fq(`)1845 778 y FC(the)75 839 y Ft(k)16 b FC(=)e(0)i(term)f(is)h (omitted.)148 949 y(F)l(or)d(the)g(rest)f(of)h(this)g(section)f(I)h(will)e(b) q(e)i(describing)f(ho)o(w)h(to)g(c)o(ho)q(ose)h(the)e(large)h(\014eld)f (regulator)75 1009 y Ft(g)22 b FC(and)e(the)f(large)g(set)g(regulator)h (\000.)31 b(The)20 b(c)o(hoice)e(of)i Ft(g)h FC(will)e(ha)o(v)o(e)f(the)h(go) q(o)q(d)j(prop)q(ert)o(y)d(that)h(it)75 1069 y(is)c(\(almost\))g(in)o(v)m (arian)o(t)g(under)g(the)h(\015o)o(w.)22 b(I)16 b(will)g(concen)o(trate)f(on) i(the)g(simplest)d(case)j(where)f(the)75 1129 y(p)q(erturbation)21 b Ft(K)k FC(dep)q(ends)d(on)f Ft(\036)g FC(only)f(through)i Fs(r)p Ft(\036)e FC(but)h(these)g(c)o(hoices)f(of)h Ft(g)i FC(and)e(\000)g(are)g(of)75 1189 y(m)o(uc)o(h)14 b(more)h(general)h(use.)148 1249 y(The)j(basic)h(example)d(of)i(a)h(system)e(where)h(the)g(in)o (teraction)f(is)h(a)h(functional)f(of)g Fs(r)p Ft(\036)g FC(is)g(the)75 1310 y(dip)q(ole)d(gas)h(w)o(e)e(discussed)h(in)g(Section)f(2.1.)22 b(W)l(e)16 b(ha)o(v)o(e)f(seen)g(in)h(Section)g(7.1)g(ho)o(w)g(to)h(rewrite)e (the)75 1370 y(in)o(teraction)g(in)h(\(2.18\))h(with)f(the)f(external)h (\014eld)f Ft(f)k FC(=)14 b(0)j(in)e(the)h(form)f Fs(E)t FC(xp)q([)p Fg(2)10 b FC(+)h Ft(K)t FC(]\(\003\))16 b(required)75 1430 y(for)k(this)f(section)g(and)i(Prop)q(osition)f(7.2)g(sho)o(ws)h(that)f(the)f (norm)g(of)h(the)f(resulting)g Ft(K)24 b FC(is)19 b(small)75 1490 y(when)13 b(the)g(activit)o(y)h(~)-27 b Ft(z)15 b FC(is)e(small.)19 b(Giv)o(en)12 b(suc)o(h)h(an)g(initial)f Ft(K)1172 1497 y Fq(`)p Fo(=1)1247 1490 y FC(w)o(e)h(ma)o(y)f(consider)g(a)i(p)q(erturbation)75 1550 y Ft(K)t FC(\()p Ft(\017)p 139 1557 20 2 v FC(\))g(=)h Ft(K)286 1557 y Fo(1)317 1550 y FC(+)366 1517 y Fr(P)419 1550 y Ft(\017)439 1557 y Fq(i)452 1550 y Ft(Q)491 1557 y Fq(i)522 1550 y FC(b)o(y)h(some)f(non-translation)j(in)o(v)m(arian)o(t)e(functionals)h Ft(Q)1547 1557 y Fq(i)1561 1550 y FC(.)22 b(Eac)o(h)17 b Ft(Q)f FC(should)75 1611 y(b)q(e)j(a)g(lo)q(cal)f(observ)m(able)h(b)o(y)f(whic)o(h)g (I)h(mean)e(that)i(there)f(is)h(a)g(set)f Ft(Y)30 b FC(\(where)19 b(the)f(observ)m(able)h(is)75 1671 y(lo)q(cated\))i(and)h Ft(Q)p FC(\()p Ft(X)t FC(\))f(v)m(anishes)h(unless)f Ft(X)27 b Fs(\033)22 b Ft(Y)11 b FC(.)36 b(The)21 b(exp)q(ectation)g(of)g(observ)m(ables)h(can)f (b)q(e)75 1731 y(obtained)f(b)o(y)g(di\013eren)o(tiating)678 1711 y Fo(1)p 674 1719 27 2 v 674 1748 a Fq(Z)713 1695 y Fr(R)758 1731 y Ft(d\026)812 1738 y Fq(`)p Fo(=1)874 1731 y Fs(E)t FC(xp[)p Fg(2)14 b FC(+)f Ft(K)t FC(\()p Ft(\017)p FC(\)]\(\003\))19 b(with)h(resp)q(ect)g(to)g Ft(\017)p 1630 1738 20 2 v 20 w FC(and)h(setting)75 1791 y Ft(\017)p 75 1798 V 17 w FC(=)d(0.)28 b(I)18 b(sa)o(y)g(\\can")i(but)e(no)h(one)g(has)g(quite)e(done)i(it)f(this)g (w)o(a)o(y)g(y)o(et)g(using)h(the)f(metho)q(d)f(I)h(am)75 1851 y(ab)q(out)e(to)f(describ)q(e)e(and)i(I)f(will)g(b)q(e)g(discussing)h(the)f (tra)s(jectory)g(only)g(at)h Ft(\017)p 1435 1858 V 14 w FC(=)e(0)i(whereas)g (I)f(should)75 1912 y(b)q(e)h(considering)g(an)g(in\014nitesmal)e(neigh)o(b)q (orho)q(o)q(d.)23 b(W)l(ork)15 b(on)g(this)g(is)g(no)o(w)g(in)g(progress.)21 b(Dimo)q(c)o(k)75 1972 y(and)15 b(Hurd)g([41])g(ha)o(v)o(e)f(included)g(an)h (external)f(\014eld)1046 1936 y Fr(R)1091 1972 y Ft(dxJ)5 b FC(\()p Ft(x)p FC(\))p Ft(\036)p FC(\()p Ft(x)p FC(\))13 b(in)i(the)f(in)o (teraction)g(in)h(\(2.18\))75 2032 y(and)i(follo)o(w)o(ed)f(the)g(ev)o (olution)g(of)h Ft(K)k FC(with)16 b(a)i(norm)d(that)i(includes)f(deriv)m (ativ)o(es)f(with)i(resp)q(ect)f(to)75 2092 y Ft(J)25 b FC(at)20 b Ft(J)25 b FC(=)20 b(0.)33 b(F)l(rom)19 b(this)h(they)g(ha)o(v)o(e)f (obtained)h(the)g(asymptotics)f(of)h(correlation)g(functions.)75 2152 y(The)c(other)g(t)o(yp)q(e)f(of)i(result)e(that)i(comes)d(out)j(of)f (the)f(metho)q(ds)h(I)f(am)g(ab)q(out)j(to)e(describ)q(e)f(is)h(that)75 2213 y(for)22 b(the)g(dip)q(ole)g(gas)i(the)e(pressure)g(is)g(analytic)g(in)f (the)h(activit)o(y)f(at)h(small)f(activit)o(y)g([30,)h(18].)75 2273 y(Results)16 b(on)i(correlations)e(ha)o(v)o(e)g(already)h(b)q(een)f (obtained)h(b)o(y)f(di\013eren)o(t)g(metho)q(ds)g(b)o(y)h(Ga)o(w)o(edzki)75 2333 y(and)e(Kupiainen)e(e.g.)21 b([32])14 b(and)g(also)h(b)o(y)f(F)l(eldman) e(et)i(al.)21 b([26].)f(In)14 b(the)g(mean)o(tim)o(e)d(I)j(am)f(reduced)75 2393 y(to)18 b(asking)g(y)o(ou)g(to)g(trust)f(me)f(that)i(the)g(forthcoming)e (sc)o(heme)g(is)h(go)q(o)q(d)j(for)e(more)e(than)i(just)g(an)75 2453 y(elab)q(orate)f(calculation)f(of)h(an)h(ob)s(ject)e(without)h(ph)o (ysical)f(signi\014cance,)f(the)i(partition)g(function)969 2599 y(71)p eop %%Page: 72 76 bop 75 307 a FC(without)17 b(an)o(y)f(external)f(\014elds!)148 367 y(I)h(will)f(pro)o(v)o(e,)g(for)i(an)f(initial)f Ft(K)21 b FC(sub)s(ject)15 b(to)i(certain)e(symmetries,)e(that)135 469 y(1.)24 b(that)d(there)e(is)h(a)h(tra)s(jectory)e Ft(\033)792 476 y Fq(`)829 469 y FC(for)h(the)g Ft(\033)i FC(in)e(the)g(co)o(v)m(ariance) f(\(8.3\))i(in)f Ft(d\026)p FC(,)h(a)g(v)m(acuum)197 529 y(energy)16 b Ft(E)389 536 y Fq(`)422 529 y FC(and)g(a)h(\015o)o(w)g Ft(K)700 536 y Fq(`)733 529 y FC(suc)o(h)f(that)766 651 y Ft(e)789 631 y Fq(E)815 637 y Fm(`)840 593 y Fr(Z)890 651 y Ft(d\026)944 658 y Fq(`;\033)988 664 y Fm(`)1006 651 y Fs(E)t FC(xp[)p Fg(2)11 b FC(+)g Ft(K)1241 658 y Fq(`)1258 651 y FC(]\(\003\))197 774 y(is)16 b(\014xed)g(and)909 834 y Fs(k)p Ft(K)975 841 y Fq(`)992 834 y Fs(k)1017 841 y Fq(g)q(;)p Fo(\000)p Fq(;h)1113 834 y Fs(!)d FC(0)197 921 y(.)135 1023 y(2.)24 b(that)18 b(if)f Ft(`)370 1005 y Fp(0)398 1023 y Ft(>)f(`)i FC(then)g Ft(K)644 1030 y Fq(`)658 1021 y Fk(0)672 1023 y FC(\()p Ft(X)t FC(\))g(is)f(a)h(functional)f (only)h(of)g Ft(K)1301 1030 y Fq(`)1318 1023 y FC(\()p Ft(Y)11 b FC(\))17 b(for)h Ft(Y)27 b Fs(\032)16 b Ft(X)22 b FC(in)17 b(the)h(sense)197 1083 y(of)e Ft(\017)p 252 1090 20 2 v 16 w FC(dep)q(endence.)75 1185 y Ft(g)i FC(app)q(earing)e(in)g(the)f(norm)g(of)h Ft(K)j FC(has)e(a)f(sligh)o(t)f(dep)q(endence)f(on)i Ft(`)h FC(whic)o(h)d(stabilizes)h(as)h Ft(`)e Fs(!)g(1)p FC(.)75 1245 y(This)f(is)f(natural)h(and)g(comes)f(ab)q(out)i(b)q(ecause)e(part)h(of)g Ft(K)k FC(is)12 b(b)q(eing)h(withdra)o(wn)g(in)o(to)f(the)h(measure)75 1305 y Ft(d\026)129 1312 y Fq(`;\033)173 1318 y Fm(`)191 1305 y FC(.)27 b(This)18 b(withdra)o(w)o(al)g(of)h(a)f(factor)h(of)f(the)g(form)f Ft(e)1124 1287 y Fp(\000)1156 1273 y Fl(1)p 1156 1279 16 2 v 1156 1300 a(2)1177 1287 y Fq(\016)q(\033)1214 1293 y Fm(`)1229 1287 y Fo(\()p Fp(r)p Fq(\036)p Fo(\))1307 1275 y Fl(2)1344 1305 y FC(a\013ects)i(the)f(gro)o(wth)g(of)h Ft(K)j FC(at)75 1365 y(large)16 b Fs(r)p Ft(\036)p FC(.)21 b(Item)14 b(\(2\))j(w)o(ould)f(b)q (e)g(imp)q(ortan)o(t)g(when)g(observ)m(ables)h(are)f(analysed.)75 1510 y Ff(8.1)70 b(The)22 b(Choice)f(of)j(Large)f(Field)e(Regulator)75 1602 y FC(The)d(c)o(hoice)f(of)i Ft(g)g FC(=)e Ft(g)r FC(\()p Ft(X)q(;)8 b(\036)p FC(\))19 b(in)f(the)g(norm)f Fs(k)8 b(k)985 1609 y Fq(g)q(;)p Fo(\000)p Fq(;h)1086 1602 y FC(is)18 b(made)f(in)h(suc)o(h) g(a)g(w)o(a)o(y)g(as)h(to)g(\(i\))e(satisfy)75 1662 y(the)j(requiremen)n(ts)d (\(7.14\),)k(\(ii\))e(\(essen)o(tially\))f(satisfy)i(the)g(requiremen)o(t)c Ft(\026)1522 1669 y Fq(ts)1567 1662 y Fs(\003)e Ft(g)1629 1669 y Fq(s)1668 1662 y Fs(\024)19 b Ft(g)1749 1669 y Fq(t)1784 1662 y FC(in)h(the)75 1722 y(cluster)14 b(expansion)i(Theorem)e(7.1)i(in)e (suc)o(h)h(a)h(w)o(a)o(y)f(that)h(under)f(rescaling)g Ft(g)1507 1729 y Fq(t)1537 1722 y FC(returns)h(to)f Ft(g)1786 1729 y Fq(s)1805 1722 y FC(,)g(\(iii\))75 1783 y Fs(j)p Ft(@)118 1765 y Fq(\014)141 1783 y Ft(\036)p FC(\()p Ft(x)p FC(\))p Fs(j)p FC(,)g(1)f Fs(\024)g(j)p Ft(\014)s Fs(j)f(\024)g Ft(d)f FC(+)f(2,)16 b(are)g(dominated)f(b)o(y)h Ft(g)r FC(\()p Ft(X)q(;)8 b(\036)p FC(\))17 b(if)e Ft(x)h FC(is)h(in)e Ft(X)t FC(.)148 1843 y(Let)h Ft(@)264 1825 y Fq(\014)287 1843 y Ft(\036)p FC(\()p Ft(x)p FC(\))f(denote)h(an)o(y)g(partial)f(deriv)m(ativ)o(e)f(of)i Ft(\036)p FC(\()p Ft(x)p FC(\))g(of)g(order)g Fs(j)p Ft(\014)s Fs(j)p FC(.)k(The)15 b(Sob)q(olev)h(norm)f(of)75 1903 y(order)h Ft(s)h FC(is)f(de\014ned)g(b)o(y)596 2029 y Fs(k)p Ft(\036)p Fs(k)675 2036 y Fq(s;X)q(;`)771 2029 y Fs(\021)858 1987 y Fr(X)823 2081 y Fo(1)p Fp(\024j)p Fq(\014)r Fp(j\024)p Fq(s)961 1968 y Fr(\022)992 1970 y(Z)1015 2065 y Fq(X)1057 2029 y Fs(j)p Ft(@)1100 2008 y Fq(\014)1123 2029 y Ft(\036)p Fs(j)1166 2008 y Fo(2)1185 1968 y Fr(\023)1224 2029 y Ft(`)1244 2008 y Fo(2\()p Fp(j)p Fq(\014)r Fp(j\000)p Fo(1\))1378 2029 y Ft(:)420 b FC(\(8)p Ft(:)p FC(4\))75 2182 y(It)15 b(has)i(the)e(prop)q(ert)o(y)g(that)h(if)f Ft(s)h FC(is)f(large)h(enough)g(dep)q(ending)g(on)g(the)g(dimension,)d(then)j (there)f(is)75 2243 y Ft(C)110 2250 y Fq(s)145 2243 y FC(suc)o(h)h(that)g (for)h Ft(`)d Fs(\025)g FC(1,)i(for)g(all)g(m)o(ulti-indice)o(s)e(1)g Fs(\024)g(j)p Ft(\014)s Fs(j)f(\024)g Ft(d)f FC(+)f(2,)766 2353 y Fs(j)p Ft(@)809 2332 y Fq(\014)832 2353 y Ft(\036)p FC(\()p Ft(x)p FC(\))p Fs(j)i(\024)h Ft(C)1042 2360 y Fq(s)1060 2353 y Fs(k)p Ft(\036)p Fs(k)1139 2360 y Fq(s;X)q(;`)1812 2353 y FC(\(8)p Ft(:)p FC(5\))148 2463 y(pro)o(vided)i Ft(x)g FC(is)g(in)g(the)g (p)q(olymer)e Ft(X)t FC(.)969 2599 y(72)p eop %%Page: 73 77 bop 148 307 a FC(The)16 b(regulator)h(is)f(de\014ned)g(to)h(b)q(e)509 444 y Ft(g)532 451 y Fq(\017;`)573 444 y FC(\()p Ft(X)q(;)8 b(\036)p FC(\))14 b(=)g(exp)852 371 y Fr( )892 410 y Ft(\017)p 890 432 25 2 v 890 478 a FC(2)919 444 y Fs(k)p Ft(\036)p Fs(k)998 451 y Fq(s;X)q(;`)1099 444 y FC(+)1164 410 y Ft(\017`)p 1162 432 46 2 v 1162 478 a FC(2)p Ft(c)1220 386 y Fr(Z)1243 480 y Fq(@)r(X)1306 444 y Fs(jr)p Ft(\036)p Fs(j)1405 424 y Fo(2)1424 371 y Fr(!)1465 444 y Ft(:)333 b FC(\(8)p Ft(:)p FC(6\))75 581 y Ft(c)16 b FC(dep)q(ends)h(only)f(on)h Ft(d)f FC(and)h(is)f(\014xed)g(b) o(y)g(the)g(follo)o(wing)g(lemm)o(a.)75 742 y Fu(Lemma)g(8.1)56 b Fn(\(a\))798 827 y Fs(j)p Ft(@)841 806 y Fq(\014)863 827 y Ft(\036)p FC(\()p Ft(x)p FC(\))p Fs(j)14 b(\024)1047 793 y Ft(C)1082 800 y Fq(s)p 1043 815 62 2 v 1043 825 a Fs(p)p 1085 825 20 2 v 36 x Ft(\017)1118 827 y(g)1141 834 y Fq(\017;`)1182 827 y FC(\()p Ft(X)q(;)8 b(\036)p FC(\))500 b(\(8)p Ft(:)p FC(7\))197 941 y Fn(for)17 b(any)g Ft(x)g Fn(in)h Ft(X)t Fn(,)g Ft(`)c Fs(\025)g FC(1)p Fn(,)k Ft(\017)13 b(>)h FC(0)p Fn(,)k FC(1)c Fs(\024)f(j)p Ft(\014)s Fs(j)g(\024)h Ft(d)d FC(+)g(2)p Fn(.)110 1042 y(\(b\))25 b(If)f Fs(f)p Ft(X)320 1049 y Fq(i)334 1042 y Fs(g)g Fn(is)g(any)g(set)h(of)e(p)n(olymers)h(which)g(have)h(at)f (most)g Ft(\034)32 b Fs(\021)25 b FC(sup)1542 1054 y Fq(x)1572 1042 y FC(#)p Fs(f)p Ft(i)g FC(:)h Ft(X)1760 1049 y Fq(i)1801 1042 y Fs(3)g Ft(x)p Fs(g)197 1102 y Fn(overlapping)19 b(at)e(any)h(p)n (oint,)f(then)h(if)f Ft(c)h Fn(is)f(su\016ciently)i(lar)n(ge)f(dep)n(ending)g (only)g(on)g Ft(d)p Fn(,)743 1211 y Ft(g)766 1218 y Fq(\017;`)807 1211 y FC(\()p Fs([)p Ft(X)899 1218 y Fq(i)913 1211 y Ft(;)8 b(\036)p FC(\))14 b Fs(\025)1049 1169 y Fr(Y)1070 1261 y Fq(i)1111 1211 y Ft(g)1134 1219 y Fq(\017=\034)r(;`)1210 1211 y FC(\()p Ft(X)1269 1218 y Fq(i)1283 1211 y Ft(;)8 b(\036)p FC(\))p Ft(:)445 b FC(\(8)p Ft(:)p FC(8\))110 1375 y Fn(\(c\))25 b(Fix)16 b(any)f Ft(L)f(>)g FC(1)p Fn(.)22 b(F)l(or)16 b Ft(\017)d(>)h FC(0)i Fn(smal)r(l)h(dep)n(ending)g(on)f Ft(L)p Fn(,)g(ther)n(e)g(exists)g Ft(v)r FC(\()p Ft(\017)p FC(\))d(=)h(0\()p Ft(\017)p FC(\))h Fn(as)h Ft(\017)d Fs(!)h FC(0)197 1435 y Fn(such)k(that)774 1526 y Ft(\026)803 1533 y Fq(`)817 1524 y Fk(0)829 1533 y Fq(`)856 1526 y Fs(\003)11 b Ft(g)915 1533 y Fq(\017;`)970 1526 y Fs(\024)j Ft(g)1046 1533 y Fq(\017;`)1084 1524 y Fk(0)1106 1453 y Fr( )1144 1492 y Ft(`)1164 1474 y Fp(0)p 1144 1514 33 2 v 1150 1560 a Ft(`)1181 1453 y Fr(!)1214 1464 y Fq(v)q Fo(\()p Fq(\017)p Fo(\))p Fp(j)p Fq(X)s Fp(j)1812 1526 y FC(\(8)p Ft(:)p FC(9\))197 1647 y Fn(for)j FC(1)d Fs(\024)g Ft(`)g Fs(\024)f Ft(`)472 1629 y Fp(0)498 1647 y Fs(\024)h Ft(L)p Fn(,)k(and)f(al)r(l)i Ft(\033)c Fs(\025)882 1628 y Fo(1)p 882 1636 18 2 v 882 1664 a(2)922 1647 y Fn(\()p Ft(\033)k Fn(app)n(e)n(ars)d(in)i(\(8.3\)\).)148 1760 y FC(F)l(rom)d(part)h(\(c\))f(w)o(e)g(see)g(that)h Ft(g)i FC(do)q(es)e(not)h(quite)d(satisfy)i(the)f(righ)o(t)g(h)o(yp)q(othesis)h Ft(\026)1690 1767 y Fq(ts)1731 1760 y Fs(\003)10 b Ft(g)1789 1767 y Fq(s)1822 1760 y Fs(\024)j Ft(g)1897 1767 y Fq(t)75 1820 y FC(to)k(apply)f(Theorem)f(7.1)h(but)749 1880 y(\026)-26 b Ft(g)770 1887 y Fq(\017;`)811 1880 y FC(\()p Ft(X)t FC(\))14 b Fs(\021)g Ft(g)983 1887 y Fq(\017;`)1024 1880 y FC(\()p Ft(X)t FC(\))p Ft(`)1126 1860 y Fq(v)q Fo(\()p Fq(\017)p Fo(\))p Fp(j)p Fq(X)s Fp(j)1788 1880 y FC(\(8)p Ft(:)p FC(10\))75 1967 y(do)q(es.)148 2027 y(F)l(rom)h(no)o(w)i(on,)f(I)g(will)f(write)772 2135 y Fs(k)p Ft(K)t Fs(k)867 2142 y Fq(\017;)p Fo(\000)p Fq(;h)959 2135 y Fs(\021)e(k)p Ft(K)t Fs(k)1106 2142 y Fq(g)1123 2146 y Fm(\017)1138 2142 y Fq(;)p Fo(\000)p Fq(;h)1202 2135 y Ft(:)572 b FC(\(8)p Ft(:)p FC(11\))75 2353 y Fu(Pro)r(of)19 b(of)h(Lemm)o(a)d(8.1.)22 b FC(See)16 b([42)q(,)g(App)q(endix)g(A].)g(There)g(are)h(three)f(ideas,)h (one)f(is)h(describ)q(ed)75 2413 y(in)e(the)g(discussion)h(of)g(large)f (\014eld)g(regulators)i(in)e(Section)g(6,)g(namely)f(that)i Ft(\024)f FC(ev)o(olv)o(es)f(according)75 2473 y(to)i(an)h(equation)f(of)g (the)g(form)659 2454 y Fq(d\024)p 659 2462 39 2 v 663 2491 a(dt)717 2473 y FC(=)d(Const.)23 b Ft(\024)957 2455 y Fo(2)992 2473 y FC(so)17 b Ft(\024)f FC(can)g(b)q(e)g(dominated)f(b)o(y)g Ft(\024)1587 2480 y Fo(1)1607 2473 y FC(\()p Ft(`=`)1690 2480 y Fo(1)1711 2473 y FC(\))1730 2455 y Fq(\017)1762 2473 y FC(for)i(an)o(y)969 2599 y(73)p eop %%Page: 74 78 bop 75 307 a Ft(\017)19 b(>)h FC(0)g(for)g(1)g Fs(\024)g Ft(`)g Fs(\024)g Ft(L)g FC(if)f Ft(\024)625 314 y Fo(1)664 307 y FC(is)h(small.)30 b(The)20 b(second)g(idea)f(is)h(that)g(the)f(co)q(e\016cien)o(t)f(of)i(\()p Fs(r)p Ft(\036)p FC(\))1893 289 y Fo(2)75 367 y FC(need)f(not)h(increase)e (at)i(all)f(b)q(ecause)h(w)o(e)e(can)i(transfer)g(its)f(gro)o(wth)h(to)f(a)h (higher)f(deriv)m(ativ)o(e)f(b)o(y)75 428 y(in)o(tegration)h(b)o(y)h(parts.) 32 b(This)19 b(generates)h(the)g(b)q(oundary)h(in)o(tegral.)30 b(The)20 b(third)f(idea)h(whic)o(h)f(is)75 488 y(needed)c(in)f(part)i(\(b\))f (is)g(to)h(dominate)e(b)q(oundary)j(in)o(tegrals)d(b)o(y)h(v)o(olume)e(in)o (tegrals)i(b)o(y)g(a)g(Sob)q(olev)75 548 y(inequalit)o(y)l(.)1562 b Fg(2)75 648 y Fu(Commen)n(ts.)148 708 y FC(\(1\))15 b(W)l(e)f(shall)g (later)g(sligh)o(tly)f(c)o(hange)i(the)f(deformation)f(so)i(w)o(e)f(will)g (need)g(the)g(fact)g(that)h(part)75 768 y(\(c\))h(holds)g(for)h(0)d Ft(<)g(\017)g Fs(\034)f FC(1)k(dep)q(ending)f(on)h Ft(L)f FC(and)h(constan)o (ts)g Ft(A;)8 b(B)s FC(:)619 861 y Ft(A)41 b FC(=)73 b(sup)777 902 y Fq(\013;)p Fp(j)p Fq(\013)p Fp(j\024)p Fo(2)p Fq(s)948 861 y FC(sup)929 900 y Fo(1)p Fp(\024)p Fq(`)p Fp(\024)p Fq(L)1049 861 y Fs(j)p Ft(@)1092 840 y Fq(\013)1132 848 y FC(_)1116 861 y Ft(C)1151 868 y Fq(`)1167 861 y FC(\()p Ft(x)p FC(\))p Fs(j)616 990 y Ft(B)44 b FC(=)73 b(sup)777 1031 y Fq(\013;)p Fp(j)p Fq(\013)p Fp(j\024)p Fo(2)p Fq(s)948 990 y FC(sup)929 1029 y Fo(1)p Fp(\024)p Fq(`)p Fp(\024)p Fq(L)1049 931 y Fr(Z)1098 990 y Fs(j)p Ft(@)1141 969 y Fq(\013)1182 977 y FC(_)1166 990 y Ft(C)1201 997 y Fq(`)1217 990 y FC(\()p Ft(x)p FC(\))p Fs(j)p Ft(dx;)424 b FC(\(8.12\))75 1115 y(regardless)14 b(of)g(what)h(co)o(v)m (ariance)e Ft(C)739 1122 y Fq(`)769 1115 y FC(is)h(b)q(eing)g(deformed)e(and) i(regardless)g(of)g(what)h(the)e(deforma-)75 1175 y(tion)k(is.)23 b(In)17 b(particular,)f(these)h(conditions)g(hold)g(for)g(the)f(standard)j (deformation)d(with)g Ft(\033)j FC(\014xed)75 1235 y(discussed)d(ab)q(o)o(v)o (e.)148 1295 y(\(2\))j(The)f Ft(`)h FC(dep)q(endence)f(in)g(\(8.4\))h(is)f(c) o(hosen)h(to)f(mak)o(e)f(the)h(norm)g(rescale)g(to)g(itself)g(after)g(a)75 1355 y(R)o(G)d(step)h(\(de\014ned)f(b)q(elo)o(w\).)21 b(Ho)o(w)o(ev)o(er)13 b(Lemma)g(8.1)j(is)f(v)m(alid)g(with)h(an)o(y)f(p)q(ositiv)o(e)g(p)q(o)o(w)o (ers)g(of)h Ft(`)g FC(in)75 1416 y(the)g(Sob)q(olev)h(norm.)75 1557 y Ff(8.2)70 b(Reblo)r(c)n(king)21 b(B)75 1650 y FC(In)i(order)g(to)g(k)o (eep)f(the)g(blo)q(c)o(k)h(size)f(comparable)g(with)h(the)f(scale)h Ft(`)p FC(\()p Ft(t)p FC(\),)h(whic)o(h)e(is)h(increasing)75 1710 y(as)18 b(the)g(co)o(v)m(ariance)f(is)h(deformed,)e(w)o(e)i(m)o(ust)e(p) q(erio)q(dically)h(stop)i(the)e(deformation)g(and)h(rewrite)75 1770 y Fs(E)t FC(xp)c Fs(\021)g(E)t FC(xp)309 1782 y Fo(1)343 1770 y FC(on)h(a)g(new)g(scale)f(\()p Fs(E)t FC(xp)765 1782 y Fq(L)791 1770 y FC(\))h(where)f Ft(L)h FC(is)f(an)i(in)o(teger)d Ft(>)h FC(1)h(so)h(that)f(the)f(new)h(p)q(olymers)75 1830 y(\()p Ft(L)p FC(-p)q(olymers\))f(are)h(based)g(on)h(blo)q(c)o(ks)f Ft(L)g FC(times)e(larger.)21 b(Supp)q(ose)16 b Ft(X)j FC(is)c(a)g(1-p)q (olymer.)20 b(Let)1817 1818 y(\026)1803 1830 y Ft(X)f FC(b)q(e)75 1890 y(the)d(smallest)f Ft(L)p FC(-p)q(olymer)g(that)h(con)o(tains)h Ft(X)t FC(.)k(Then)646 1983 y Fs(E)t FC(xp)730 1995 y Fo(1)749 1983 y FC(\()p Fg(2)12 b FC(+)f Ft(K)t FC(\))j(=)f Fs(E)t FC(xp)1079 1995 y Fq(L)1105 1983 y FC(\()p Fg(2)e FC(+)g Fs(B)r Ft(K)t FC(\))p Ft(;)454 b FC(\(8)p Ft(:)p FC(13\))75 2075 y(where)18 b(on)g(the)g(righ)o(t)g(hand)g(side)g Fg(2)g FC(and)h Fs(B)r Ft(K)i FC(are)d(de\014ned)g(on)g(sets)h(asso)q(ciated)g(with)e Ft(L)p FC(-blo)q(c)o(ks)75 2135 y(and)g(for)f(an)o(y)g Ft(L)p FC(-p)q(olymer)g Ft(U)729 2196 y Fs(B)r Ft(K)t FC(\()p Ft(U)5 b FC(\))14 b Fs(\021)960 2154 y Fr(X)951 2248 y Fp(f)p Fq(X)998 2253 y Fm(i)1011 2248 y Fp(g)1037 2154 y Fr(Y)1058 2245 y Fq(i)1099 2196 y Ft(K)t FC(\()p Ft(X)1203 2203 y Fq(i)1217 2196 y FC(\))p Ft(;)538 b FC(\(8)p Ft(:)p FC(14\))75 2322 y(and)17 b Fs(f)p Ft(X)235 2329 y Fq(i)249 2322 y Fs(g)f FC(is)g(summed)d(o)o(v)o(er)i(all)h (sets)g Fs(f)p Ft(X)863 2329 y Fo(1)883 2322 y Ft(;)8 b(:)g(:)g(:)g(;)g(X) 1033 2329 y Fq(N)1080 2322 y FC(:)22 b Ft(N)g FC(arbitrary)p Fs(g)16 b FC(of)g(1-p)q(olymers)f(suc)o(h)h(that)519 2412 y(\(i\))41 b Fs([)659 2400 y FC(\026)645 2412 y Ft(X)685 2419 y Fq(i)714 2412 y FC(=)13 b Ft(U)5 b FC(;)505 2473 y(\(ii\))57 b(the)16 b(o)o(v)o(erlap)g(graph)h(on)g Fs(f)p Ft(X)1153 2480 y Fq(i)1167 2473 y Fs(g)f FC(is)g(connected.)1788 2443 y(\(8)p Ft(:)p FC(15\))969 2599 y(74)p eop %%Page: 75 79 bop 75 307 a FC(The)16 b Fn(overlap)i(gr)n(aph)e FC(is)g(all)f(pairs)i Fs(f)p FC(\()p Ft(ij)s FC(\))c(:)871 295 y(\026)857 307 y Ft(X)897 314 y Fq(i)922 307 y Fs(\\)981 295 y FC(\026)967 307 y Ft(X)1007 314 y Fq(j)1039 307 y Fs(6)p FC(=)h Ft(\036)p Fs(g)p FC(.)148 367 y(This)i Fs(B)i FC(at)e(\014rst)h(sigh)o(t)f(is)g(not)h(v)o(ery)e(prep)q (ossessing)i(but)f(with)g(the)g(righ)o(t)g(c)o(hoice)f(of)h(large)h(set)75 428 y(regulator)j(\000)h(it)f(has)g(a)h(go)q(o)q(d)h(prop)q(ert)o(y)e(that)g (captures)h(Wilson's)f(observ)m(ation)g(that)h(only)f(the)75 488 y(lo)q(cal)c(parts)h(of)g(an)f(in)o(teraction)g(are)g(relev)m(an)o(t.)75 584 y Fu(Choice)j(of)g FC(\000)p Fu(.)j FC(W)l(e)16 b(w)o(an)o(t)h(to)f(c)o (ho)q(ose)h(\000\()p Ft(X)t FC(\))h(in)e(suc)o(h)g(a)h(w)o(a)o(y)f(that)h (the)f(activities)f Ft(K)t FC(\()p Ft(X)q(;)8 b(\036)p FC(\))17 b(will)75 644 y(ha)o(v)o(e)d(to)h(b)q(e)g(v)o(ery)e(small)g(if)h Ft(X)19 b FC(is)c(not)g(close)f(to)h(b)q(eing)g(a)g(unit)f(blo)q(c)o(k,)g (i.e.,)f(lo)q(cal.)21 b(T)l(o)15 b(quan)o(tify)f(the)75 704 y(suppression)k(of)g(p)q(olymers)e(with)h(man)o(y)f(blo)q(c)o(ks)h(w)o(e)f (include)g(a)i(factor)g Ft(A)1460 686 y Fp(j)p Fq(X)s Fp(j)1530 704 y FC(where)f Ft(A)e Fs(\025)g FC(1,)j(and)75 765 y Fs(j)p Ft(X)t Fs(j)c FC(is)g(the)g(n)o(um)o(b)q(er)e(of)i(blo)q(c)o(ks)g(in)f(the)h (p)q(olymer)f Ft(X)t FC(.)20 b(T)l(o)15 b(quan)o(tify)e(the)h(suppression)h (of)f(p)q(olymers)75 825 y(with)i(widely)f(separated)i(blo)q(c)o(ks)f(w)o(e)g (include)f(a)h(factor)765 911 y(\002\()p Ft(X)t FC(\))e(=)g(inf)967 941 y Fq(T)1024 869 y Fr(Y)1018 962 y Fq(b)p Fp(2)p Fq(T)1090 911 y Ft(\022)q FC(\()p Fs(j)p Ft(b)p Fs(j)p FC(\))p Ft(;)573 b FC(\(8)p Ft(:)p FC(16\))75 1033 y(where)18 b Ft(T)25 b FC(runs)20 b(o)o(v)o(er)d(tree)h(graphs)i(connecting)f(the)f(unit)g(blo)q(c)o(ks)h(that) g(comprise)e Ft(X)23 b FC(b)o(y)18 b(b)q(onds)75 1093 y Ft(b)26 b Fs(2)g Ft(T)t(;)8 b Fs(j)p Ft(b)p Fs(j)22 b FC(is)h(the)g(length)610 1075 y Fo(11)671 1093 y FC(of)h(suc)o(h)f(a)g(b)q(ond.)44 b Ft(\022)q FC(\()p Ft(s)p FC(\))23 b(is)g(an)h(increasing)f(function)g(of)h Ft(s;)8 b(s)23 b FC(a)75 1153 y(non-negativ)o(e)16 b(in)o(teger,)f(suc)o(h)h (that)h Ft(\022)q FC(\(0\))d(=)g Ft(\022)q FC(\(1\))g(=)g(1)i(and)674 1258 y Ft(\022)706 1197 y Fr(\022\032)778 1224 y Ft(s)p 773 1246 34 2 v 773 1292 a(L)811 1197 y Fr(\033\023)886 1258 y Fs(\024)e Ft(A)976 1237 y Fp(\000)p Fo(1)1023 1258 y Ft(\022)q FC(\()p Ft(s)p FC(\))p Ft(;)56 b(s)14 b Fs(\025)g FC(2)p Ft(;)482 b FC(\(8)p Ft(:)p FC(17\))75 1363 y(where)18 b Fs(f)p Ft(x)p Fs(g)h FC(is)f(the)h(smallest)e(in)o(teger)h Fs(\025)f Ft(x)p FC(.)29 b(\(As)18 b Ft(s)g Fs(!)g(1)p Ft(;)8 b(\022)q FC(\()p Ft(s)p FC(\))18 b Fs(\030)g Ft(s)1391 1345 y Fq(p)1430 1363 y FC(with)g Ft(p)h FC(=)f(log)1704 1374 y Fq(L)1730 1363 y Ft(A)p FC(\).)28 b(The)75 1423 y(p)q(oin)o(t)17 b(of)f(this)h(requiremen)n(t) d(is)i(that)h(whenev)o(er)e(a)i(p)q(olymer)e(consisting)i(of)f(t)o(w)o(o)h (or)g(more)e(blo)q(c)o(ks)75 1483 y(whic)o(h)h(are)h(not)g(neigh)o(b)q(ors)g (is)f(reblo)q(c)o(k)o(ed,)f(then)i(on)g(the)f(scale)g(of)h(the)f(new)h(blo)q (c)o(ks)f(the)h(distance)75 1543 y(b)q(et)o(w)o(een)d(them)e(will)i(b)q(e)g (smaller)f(and)i(so)g(the)f(reblo)q(c)o(k)o(ed)g(p)q(olymer)e(will)i(b)q(e)g (smaller)f(in)h(the)g(norm)75 1603 y(appropriate)21 b(to)g(the)f(new)g(blo)q (c)o(ks.)33 b(In)20 b(this)h(w)o(a)o(y)f(w)o(e)g(capture)g(Wilson's)g(idea)g (that)h(non-lo)q(cal)75 1664 y(parts)16 b(of)h(the)e(in)o(teraction)g(are)h (irrelev)m(an)o(t.)j(If)c(a)i(p)q(olymer)d(consists)i(of)g(man)o(y)e(blo)q(c) o(ks)i(then)f(when)75 1724 y(reblo)q(c)o(k)o(ed)k(it)i(will)f(ha)o(v)o(e)g (few)o(er)g(blo)q(c)o(ks,)h(therefore)g(there)f(will)g(b)q(e)h(few)o(er)f (factors)h(of)h Ft(A)e FC(in)h(the)75 1784 y(norm)16 b(of)g(the)g(new)h(p)q (olymer)e(so)i(again)g(it)f(will)f(lo)q(ok)i(smaller.)j(These)c(ideas)g(are)h (summarised)d(in)75 1844 y(the)i(follo)o(wing)g(prop)q(osition.)148 1904 y(Giv)o(en)f Ft(K)21 b FC(de\014ne)16 b Ft(K)533 1911 y Fq(S)559 1904 y FC(,)f(the)h(part)h(of)g Ft(K)j FC(that)d(liv)o(es)d(on)j (s)p 1177 1911 20 2 v(mall)e(sets,)h(b)o(y)444 2019 y Ft(K)485 2026 y Fq(S)511 2019 y FC(\()p Ft(X)t FC(\))e Fs(\021)659 1946 y Fr(\()714 1988 y Ft(K)t FC(\()p Ft(X)t FC(\))58 b(if)15 b Ft(X)21 b FC(connected)16 b(and)h Fs(j)p Ft(X)t Fs(j)c(\024)h FC(2)1489 1970 y Fq(d)714 2049 y FC(0)161 b(otherwise)p Ft(;)1788 2019 y FC(\(8)p Ft(:)p FC(18\))75 2138 y(and)19 b(let)e Ft(K)285 2145 y Fq(B)334 2138 y FC(b)q(e)i(de\014ned)e(b)o(y)h Ft(K)j FC(=)c Ft(K)800 2145 y Fq(S)839 2138 y FC(+)12 b Ft(K)930 2145 y Fq(B)961 2138 y FC(.)27 b(Let)18 b(\000)h(b)q(e)f(constructed)g(as)h(ab)q (o)o(v)o(e.)27 b(Fix)1770 2120 y Fo(12)1825 2138 y Ft(\021)20 b FC(so)75 2198 y(that)d(1)d Ft(<)g(\021)297 2180 y Fo(2\(2)347 2169 y Fm(d)364 2180 y Fo(\))393 2198 y Ft(<)450 2179 y Fo(3)p 450 2187 18 2 v 450 2216 a(2)473 2198 y FC(,)h(and)i(set)f(\()p Ft(\021)r FC(\000\)\()p Ft(X)t FC(\))f(=)f Ft(\021)942 2180 y Fp(j)p Fq(X)s Fp(j)995 2198 y FC(\000\()p Ft(X)t FC(\))p Ft(;)8 b FC(\()p Ft(\021)1174 2180 y Fp(\000)p Fo(1)1221 2198 y FC(\000\)\()p Ft(X)t FC(\))15 b(=)f Ft(\021)1445 2180 y Fp(\000j)p Fq(X)s Fp(j)1525 2198 y FC(\000\()p Ft(X)t FC(\))p Ft(:)j FC(Then)p 75 2279 735 2 v 115 2310 a Fj(11)150 2325 y Fy(All)c(lengths)h(are)g (measured)g(b)o(y)g FA(j)p Fy(\()p Fi(x)725 2331 y Fj(1)743 2325 y Fi(;)7 b(:)g(:)g(:)e(;)i(x)860 2331 y Fh(d)878 2325 y Fy(\))p FA(j)k Fy(=)h(sup)1024 2335 y Fh(i)1044 2325 y FA(j)p Fi(x)1080 2331 y Fh(i)1093 2325 y FA(j)i Fy(where)h Fi(x)1263 2331 y Fh(i)1290 2325 y Fy(is)f(measured)g(in)f(units)h(of)g(blo)q(c)o(k)f (size.)115 2359 y Fj(12)150 2374 y Fy(This)f(m)o(ysterious)g Fi(\021)i Fy(is)e(b)q(est)i(ignored,)e(i.e.,)f(set)j(equal)e(to)h(one,)f(the) h(\014rst)h(time)d(through)i(this)f(argumen)o(t.)17 b(It)c(is)75 2424 y(there)h(mostly)d(b)q(ecause)k(I)e(w)o(an)o(t)f(to)h(reserv)o(e)i(some) c(deca)o(y)j(in)e(the)i(size)f(of)f(the)i(p)q(olymer)d(to)i(cancel)h(a)e (small)f(gro)o(wth)75 2474 y(in)i(the)i(v)p 195 2481 22 2 v(acuum)e(energy)h (part)g Fi(v)q Fy(\()p Fi(\017)p Fy(\))h(of)e(the)i(large)e(\014eld)h (regulator.)k(See)d(\(8.9\))969 2599 y FC(75)p eop %%Page: 76 80 bop 75 307 a Fu(Prop)r(osition)18 b(8.2)24 b Fn(F)l(or)17 b(al)r(l)h Ft(\017)c Fs(\025)g FC(0)p Ft(;)8 b(h)p 767 314 29 2 v 14 w Fs(\025)13 b FC(0)p Ft(;)18 b Fn(if)f Fs(k)p Ft(K)t Fs(k)1059 316 y Fq(\017;\021)1102 306 y Fk(\000)p Fl(1)1143 316 y Fo(\000)p Fq(;h)p 1175 323 21 2 v 1211 307 a Fs(\034)d FC(1)k Fn(dep)n(ending)g(only)g (on)g Ft(\021)r Fn(,)436 420 y Fs(kB)r Ft(K)t Fs(k)566 427 y Fq(\017;\021)q Fo(\000)p Fq(;h)p 641 434 V 676 420 a Fs(\024)c FC(2)p Ft(L)786 399 y Fq(d)815 372 y Fr(n)843 420 y Fs(k)p Ft(K)909 427 y Fq(S)934 420 y Fs(k)959 428 y Fq(\017;\021)1002 419 y Fk(\000)p Fl(1)1044 428 y Fo(\000)p Fq(;h)p 1076 435 V 1109 420 a FC(+)d Ft(A)1195 399 y Fp(\000)p Fo(1)1242 420 y Fs(k)p Ft(K)1308 427 y Fq(B)1338 420 y Fs(k)1363 428 y Fq(\017;\021)1406 419 y Fk(\000)p Fl(1)1448 428 y Fo(\000)p Fq(;h)p 1480 435 V 1502 372 a Fr(o)1538 420 y Ft(:)236 b FC(\(8)p Ft(:)p FC(19\))75 532 y Fu(Pro)r(of:)20 b FC(See)13 b(Lemmas)f(3.1)i(and)g(3.2)g(in)g([18].)20 b(The)14 b(main)e(p)q(oin)o(t)i(is)g(that)g(the)f(small)f(sets)i(are)g(those) 75 592 y(sets)19 b(that)h(do)g(not)f(b)q(ecome)f(smaller)f(as)j(measured)e(b) o(y)h(\000)h(when)f(they)g(are)g(minimal)o(ly)d(co)o(v)o(ered)75 652 y(b)o(y)g(L-blo)q(c)o(ks.)23 b(T)l(o)17 b(understand)h(the)e(gro)o(wth)h (b)o(y)f Ft(L)1035 634 y Fq(d)1072 652 y FC(of)h(the)g(small)e(set)h(part,)h (note)g(that)g(a)g(single)75 712 y(blo)q(c)o(k)f(can)g(o)q(ccup)o(y)g(an)o(y) g(of)h Ft(L)638 694 y Fq(d)674 712 y FC(p)q(ositions)h(and)e(still)g(b)q(e)g (co)o(v)o(ered)f(b)o(y)g(the)h(same)g(L-blo)q(c)o(k.)101 b Fg(2)148 818 y FC(No)o(w)16 b(w)o(e)g(c)o(ho)q(ose)h Ft(\021)h Fn(onc)n(e)g(and)g(for)e(al)r(l)p FC(,)i(and)e(let)887 920 y Ft(A)d FC(=)h Ft(L)1022 900 y Fp(\000)p Fo(2)p Fq(d)1087 920 y Ft(;)75 1023 y FC(so)j(that)g(the)f Ft(K)366 1030 y Fq(B)413 1023 y FC(con)o(tribution)f(to)i Fs(B)r Ft(K)j FC(is)c(suppressed)g(b)o(y)g (a)h(factor)f Ft(L)1420 1004 y Fp(\000)p Fq(d)1468 1023 y FC(.)75 1166 y Ff(8.3)70 b(Renormalizati)o(on)21 b(\(Dip)r(ole)f(Systems\))75 1258 y FC(W)l(e)g(ha)o(v)o(e)f(seen)h(in)g(Prop)q(osition)h(8.2)g(that)g(the) e(reblo)q(c)o(king)h(op)q(eration)h Fs(B)g FC(automatically)e(sup-)75 1318 y(presses)14 b(the)g(part)g Ft(K)462 1325 y Fq(B)507 1318 y FC(of)g Ft(K)k FC(that)d(liv)o(es)d(on)i(big)g(sets)g(but)h(the)e(lo)q(cal) h(parts)h Ft(K)1512 1325 y Fq(s)1545 1318 y FC(that)f(liv)o(e)e(on)i Fn(smal)r(l)75 1379 y(sets)j FC(de\014ned)f(b)o(y)565 1439 y Ft(X)j Fs(2)14 b(S)j(\()-25 b(\))14 b Ft(X)21 b FC(is)16 b(connected)f(and)i Fs(j)p Ft(X)t Fs(j)d(\024)g FC(2)1402 1418 y Fq(d)1788 1439 y FC(\(8)p Ft(:)p FC(20\))75 1523 y(gro)o(w)j(b)o(y)e(a)i(v) o(olume)d(factor)i Ft(L)641 1505 y Fq(d)678 1523 y FC(under)g Fs(B)r FC(.)k(Th)o(us)c(the)g(c)o(hoice)f(of)h(\000)h(has)g(captured)f (Wilson's)g(idea)75 1583 y(that)h(only)f(lo)q(cal)g(parts)h(of)f(the)g(in)o (teraction)g(can)g(b)q(e)g(relev)m(an)o(t.)148 1643 y(T)l(o)24 b(examine)d(the)i(lo)q(cal)g(parts)g(carefully)f(w)o(e)h(de\014ne)f(a)i(lo)q (calization)e(op)q(erator)j Fs(L)p FC(.)42 b(I)22 b(am)75 1703 y(b)q(orro)o(wing)13 b(this)e(terminolgy)f(from)g(an)i(analogous)i (constuction)e(whic)o(h)e(app)q(ears)j(in)f(man)o(y)e(pap)q(ers)75 1763 y(b)o(y)16 b(Galla)o(v)o(otti)f(et)h(al.,)f(e.g.,)g(see)h(section)g(8)h (in)e([29)q(].)21 b(Let)131 1934 y(\()p Fs(L)p Ft(K)t FC(\)\()p Ft(X)t FC(\))15 b(=)397 1822 y Fr(8)397 1860 y(>)397 1872 y(>)397 1884 y(<)397 1959 y(>)397 1972 y(>)397 1984 y(:)454 1865 y FC(0)1146 b(if)16 b Ft(X)25 b Fs(2)1744 1867 y(j)1788 1865 y(S)479 1912 y Fq(P)463 1925 y Fr(X)454 2017 y Fq(N)t Fo(=0)561 1932 y FC(1)p 544 1955 58 2 v 544 2000 a Ft(N)5 b FC(!)615 1893 y Fr(")640 1908 y(Z)689 1966 y Fs(\001)j(\001)g(\001)756 1908 y Fr(Z)825 1932 y Ft(\016)849 1914 y Fq(N)882 1932 y Ft(K)t FC(\()p Ft(X)q(;)g(\036)14 b FC(=)g(0\))p 811 1955 352 2 v 811 2000 a Ft(\016)r(\036)p FC(\()p Ft(\030)904 2007 y Fo(1)923 2000 y FC(\))8 b Fs(\001)g(\001)g(\001)h Ft(\016)r(\036)p FC(\()p Ft(\030)1110 2007 y Fq(N)1143 2000 y FC(\))1175 1966 y Ft(\036)p FC(\()p Ft(\030)1244 1973 y Fo(1)1264 1966 y FC(\))f Fs(\001)g(\001)g(\001)h Ft(\036)p FC(\()p Ft(\030)1427 1973 y Fq(N)1460 1966 y FC(\))1479 1893 y Fr(#)1504 2018 y FC(lo)q(c)1624 1966 y(if)16 b Ft(X)i Fs(2)c(S)1788 2063 y FC(\(8)p Ft(:)p FC(21\))75 2123 y([)p Ft(M)5 b FC(])155 2141 y(lo)q(c)234 2123 y(means)15 b(that)i(the)f(monomial)e Ft(M)21 b FC(is)16 b(replaced)g(b)o(y)g(a)g(lo)q(cal)g(p)q(olynomial)f(of)i (the)f(form)563 2245 y([)p Ft(M)5 b FC(])643 2263 y(lo)q(c)720 2245 y(=)771 2204 y Fr(X)790 2291 y Fq(\013)p 790 2298 23 2 v 840 2245 a Ft(c)861 2252 y Fq(\013)p 861 2259 25 2 v 894 2187 a Fr(Z)917 2281 y Fq(X)959 2245 y Ft(@)988 2225 y Fq(\013)1011 2230 y Fl(1)1029 2245 y Ft(\036)p FC(\()p Ft(\030)r FC(\))j Fs(\001)g(\001)g(\001)h Ft(@)1223 2225 y Fq(\013)1246 2231 y Fm(N)1276 2245 y Ft(\036)p FC(\()p Ft(\030)r FC(\))f Ft(d\030)368 b FC(\(8)p Ft(:)p FC(22\))75 2403 y(suc)o(h)12 b(that)h Ft(M)8 b Fs(\000)s FC([)p Ft(M)d FC(])460 2420 y(lo)q(c)534 2403 y(has)13 b(dimension)e(more)g(negativ)o(e)g(than)i(the)f(\()p Ft(p)s FC(+)s(1\))1470 2384 y(th)1530 2403 y(term)f(in)h(the)g(T)l(a)o(ylor)75 2474 y(expansion)18 b(of)g Ft(K)k FC(in)17 b Ft(\036)h FC(ab)q(out)h Ft(\036)d FC(=)h(0.)26 b(The)18 b(dimension)e(is)h(computed)g(b)o(y)g (assigning)i Ft(L)1757 2456 y Fp(\000)1789 2441 y Fm(d)p Fk(\000)p Fl(2)p 1789 2448 56 2 v 1809 2469 a(2)1869 2474 y FC(to)969 2599 y(76)p eop %%Page: 77 81 bop 75 307 a FC(eac)o(h)15 b Ft(\036)p FC(\()p Ft(x)p FC(\))g(and)h Ft(L)421 289 y Fp(\000)p Fo(1)483 307 y FC(to)g(eac)o(h)f(gradien)o(t.)20 b(F)l(or)c(this)f(computation)f(do)i(not)g(assign)g(dimensions)e(to)75 367 y(the)i(in)o(tegrals)g(in)g Ft(\030)j FC(or)d(k)o(ernels)f(in)h Ft(\030)r FC(.)22 b(T)l(o)17 b(see)f(that)g([)p Ft(M)5 b FC(])1127 385 y(lo)q(c)1206 367 y(exists)16 b(insert)598 498 y Ft(\036)p FC(\()p Ft(\030)r FC(\))e Fs(\021)g Ft(@)784 477 y Fq(\013)808 498 y Ft(\036)p FC(\()p Ft(x)p FC(\))f(=)h Ft(@)997 477 y Fq(\013)1021 498 y Ft(\036)p FC(\()p Ft(z)r FC(\))d(+)1173 439 y Fr(Z)1214 452 y Fq(x)1196 534 y(z)1245 498 y Ft(d)p FC(\()p Ft(@)1318 477 y Fq(\013)1342 498 y Ft(\036)p FC(\))398 b(\(8)p Ft(:)p FC(23\))75 628 y(for)18 b(eac)o(h)f Ft(\036)p FC(\()p Ft(\030)r FC(\))h(in)f Ft(M)23 b FC(and)18 b(consider)g Ft(M)j FC(=)930 608 y Fo(1)p 913 616 52 2 v 913 645 a Fp(j)p Fq(X)s Fp(j)978 592 y Fr(R)997 641 y Fq(X)1039 628 y Ft(M)14 b(dz)r FC(.)26 b(This)17 b(con)o(tains)h(a)g(part)g(whic)o(h)f(comes)75 695 y(solely)h(from)f(the)h Ft(@)444 677 y Fq(\013)468 695 y Ft(\036)p FC(\()p Ft(z)r FC(\))h(terms)e(in)h(\(8.23\))h(and)g(this)f(part)h(is)f(lo)q (cal)h(as)g(in)f(\(8.22\).)28 b(Ev)o(erything)75 755 y(else)12 b(in)h(this)g(expression)g(for)g Ft(M)19 b FC(is)13 b(of)g(dimension)f(impro) o(v)o(ed)e(b)o(y)j(one)g(b)q(ecause)g(it)g(has)h(at)f(least)g(one)75 815 y(more)e(gradien)o(t)h(arising)g(from)f(the)h Ft(d)h FC(in)e(\(8.23\).)21 b(W)l(e)12 b(can)g(tak)o(e)g(all)f(these)h(terms)f(and)h(use)h(the)f(same)75 876 y(metho)q(d)g(to)g(write)g(them)f(as)i(lo)q(cal)g(p)q(olynomials)e(plus)h (remainders)f(of)i(impro)o(v)o(ed)d(dimension)h(un)o(til)75 936 y(all)i(remainders)g(ha)o(v)o(e)g(su\016cien)o(tly)f(negativ)o(e)h (dimension.)19 b(This)14 b(pro)q(cess)g(generates)g(dep)q(endence)75 996 y(on)20 b(\014elds)g Ft(@)304 978 y Fq(\014)327 996 y Ft(\036)p FC(\()p Ft(x)p FC(\))f(where)g Ft(\014)j FC(b)q(elongs)f(to)f(a)g(range,)h (Range\()p Ft(\014)s FC(\),)f(determined)d(b)o(y)i(the)h(c)o(hoice)e(of)75 1056 y Ft(p)p FC(.)30 b(If,)18 b(for)h(example,)e Ft(p)i FC(=)f(3,)i(then)f Ft(M)f Fs(\000)13 b FC([)p Ft(M)5 b FC(])961 1074 y(lo)q(c)1042 1056 y(con)o(tains)19 b Fs(r)p Ft(\036@)1336 1038 y Fq(\014)1358 1056 y Ft(\036)g FC(with)g Fs(j)p Ft(\014)s Fs(j)e FC(=)i Ft(d)13 b FC(+)g(2)19 b(whose)75 1121 y(dimensions)14 b(are)i Ft(L)438 1103 y Fp(\000)p Fo(\(2)p Fq(d)p Fo(+1\))592 1121 y FC(whic)o(h)f(is)h(one)g (more)e(negativ)o(e)h(than)i(\()p Fs(r)p Ft(\036)p FC(\))1405 1103 y Fo(4)1439 1121 y FC(whic)o(h)e(is)h(the)g(dominan)o(t)75 1190 y(term)e(in)i(the)f(\()p Ft(p)c FC(+)f(1\))476 1172 y(th)540 1190 y(term)k(in)i(the)f(T)l(a)o(ylor)h(expansion)g(of)g Ft(K)k FC(in)15 b Ft(\036)h FC(ab)q(out)h Ft(\036)d FC(=)g(0.)21 b(This)16 b(is)g(the)75 1250 y(reason)h(for)g(the)f(h)o(yp)q(otheses)g(w)o(e)g(will)f (imp)q(ose)g(on)i Ft(K)j FC(dep)q(endence)c(on)g(\014elds)g Ft(@)1574 1232 y Fq(\014)1597 1250 y Ft(\036)p FC(.)148 1310 y(I)g(mak)o(e)e(the)i(follo)o(wing)g(assumptions)h(on)f Ft(K)t FC(\()p Ft(X)q(;)8 b(\036)p FC(\):)121 1411 y(\(i\))24 b Ft(K)t FC(\()p Ft(X)q(;)8 b(\036)p FC(\))17 b(dep)q(ends)f(on)h Ft(\036)f FC(only)g(through)h Ft(@)1010 1393 y Fq(\014)1033 1411 y Ft(\036)p FC(\()p Ft(x)p FC(\),)f(1)e Fs(\024)g(j)p Ft(\014)s Fs(j)e(\024)i Ft(d)d FC(+)g(2;)108 1513 y(\(ii\))23 b Ft(K)t FC(\()p Ft(X)q(;)8 b(\036)p FC(\))14 b(=)g Ft(K)t FC(\()p Ft(X)q(;)8 b Fs(\000)p Ft(\036)p FC(\);)94 1614 y(\(iii\))23 b Ft(K)e FC(is)c(in)o(v)m(arian)o(t)f (under)h(rotations)g(and)h(parit)o(y)l(,)e(i.e.,)f(if)h Ft(r)i FC(is)f(a)g(re\015ection)f(in)h(a)g(co)q(ordinate)197 1675 y(h)o(yp)q(erplane)f(or)g(a)h(rotation)g(b)o(y)f(90)851 1657 y Fp(\016)871 1675 y FC(,)g(then)816 1784 y Ft(K)t FC(\()p Ft(r)q(X)q(;)8 b(r)q(\036)p FC(\))15 b(=)f Ft(K)t FC(\()p Ft(X)q(;)8 b(\036)p FC(\))p Ft(:)75 1914 y FC(These)14 b(assumptions)g(hold)g(for)h(the) f(dip)q(ole)f(gas,)i(for)g(example.)j(If)c(w)o(e)f(ha)o(v)o(e)h(\(i\))f(and)i (\(ii\),)e(then)h(w)o(e)75 1974 y(can)j(and)g(will)e(de\014ne)h Fs(L)h FC(with)f Ft(p)f FC(=)f(3)i(\(whic)o(h)g(will)f(b)q(e)i(the)f(same)g (as)h Fs(L)f FC(with)h Ft(p)d FC(=)g(2)j(b)q(ecause)g(the)75 2034 y(3)99 2016 y Fq(d)138 2034 y FC(v)m(ariational)h(deriv)m(ativ)o(e)e(of) i Ft(K)k FC(v)m(anishes)c(at)g Ft(\036)f FC(=)f(0\).)27 b(\(iii\))16 b(is)i(useful)f(b)q(ecause)i(it)e(guaran)o(tees)75 2095 y(that)g Fs(L)p Ft(K)k FC(will)16 b(lead)g(to)h(a)g(renormalization)e(of)i Ft(\033)r FC(.)22 b(Without)16 b(it)g(one)h(w)o(ould)g(ha)o(v)o(e)f(to)h (allo)o(w)f Ft(\033)i FC(to)75 2155 y(b)q(ecome)d(a)h(bilinear)g(form)f Ft(\033)610 2162 y Fq(\026\027)652 2155 y Ft(k)677 2162 y Fq(\026)701 2155 y Ft(k)726 2162 y Fq(\027)764 2155 y FC(instead)h(of)h Ft(\033)r(k)1045 2137 y Fo(2)1064 2155 y FC(.)148 2215 y(No)o(w)d(I)g(will)g (summarize)d(a)k(sequence)e(of)i(op)q(erations)g(that)g(will)e(constitute)h (one)h(R.)e(G.)i(trans-)75 2275 y(formation.)40 b(Supp)q(ose)24 b(this)f(op)q(eration)h(has)f(already)g(b)q(een)f(accomplished)f(some)h(n)o (um)o(b)q(er)f(of)75 2335 y(times,)14 b(and)j(as)g(a)f(result)g(w)o(e)g(ha)o (v)o(e)f(written)h(the)g(original)g(partition)g(function)g(in)g(the)g(form) 597 2457 y Ft(e)620 2436 y Fp(\000)p Fq(E)675 2425 y Fm(j)702 2398 y Fr(Z)752 2457 y Ft(d\026)806 2468 y Fq(L;\033)861 2459 y Fl(\()p Fm(j)q Fl(\))904 2457 y Fs(E)t FC(xp)987 2469 y Fo(1)1015 2409 y Fr(h)1035 2457 y Fg(2)11 b FC(+)g Ft(K)1177 2432 y Fo(\()p Fq(j)r Fo(\))1173 2469 y Fq(L)1223 2409 y Fr(i)1251 2457 y FC(\(\003)1304 2436 y Fo(\()p Fq(j)r Fo(\))1349 2457 y FC(\))p Ft(;)406 b FC(\(8)p Ft(:)p FC(24\))969 2599 y(77)p eop %%Page: 78 82 bop 75 309 a FC(where)15 b Ft(d\026)269 320 y Fq(L;\033)324 310 y Fl(\()p Fm(j)q Fl(\))383 309 y FC(has)h(co)o(v)m(ariance)e Ft(F)7 b FC(\()p Ft(L)794 291 y Fo(2)814 309 y Ft(k)841 291 y Fo(2)860 309 y FC(\))p Ft(=)p FC(\()p Ft(\033)952 291 y Fo(\()p Fq(j)r Fo(\))998 309 y Ft(k)1025 291 y Fo(2)1045 309 y FC(\).)21 b(The)15 b(subscript)g Ft(L)h FC(on)g Ft(d\026)g FC(means)e(that)i(the)75 369 y(length)g(scale)g(of)h(the)f(ultra-violet)f(cuto\013)i(in)f(the)g(co)o (v)m(ariance)g(of)g Ft(d\026)h FC(is)g Ft(`)d FC(=)g Ft(L)p FC(.)22 b(The)16 b(subscript)g(1)75 429 y(on)h Fs(E)t FC(xp)f(means)f(that)i (it)f(is)f(de\014ned)h(on)h Fn(unit)g FC(blo)q(c)o(ks.)k(\003)1132 411 y Fo(\()p Fq(j)r Fo(\))1193 429 y FC(is)16 b(the)g(original)g(torus)h (\003)f(rescaled)f(b)o(y)75 489 y Ft(L)108 471 y Fp(\000)p Fq(j)154 489 y FC(.)21 b Ft(L)c FC(will)e(b)q(e)h(sp)q(eci\014ed)g(b)q(elo)o (w.)21 b(An)16 b(R.)g(G.)g(transformation)g(will)f(consist)h(of:)148 549 y Fu(\(1\))j(Renormalization)14 b Fs(E)21 b FC(in)c(whic)o(h)f(the)g Fs(L)p Ft(K)21 b FC(parts)c(of)g Ft(K)k FC(are)c(e)p 1409 556 22 2 v(xtracted)f(and)h(cancelled)75 610 y(b)o(y)f(a)g(shift)g(in)g(the)g(co) o(v)m(ariance)g(of)h Ft(d\026)780 621 y Fq(L;\033)835 611 y Fl(\()p Fm(j)q Fl(\))878 610 y FC(.)k(T)l(o)c(do)g(this,)f(let)f Ft(X)21 b FC(b)q(e)16 b(an)o(y)g(allo)o(w)o(ed)g(set)g(as)h(de\014ned)75 670 y(b)q(elo)o(w)f(\(7.2\))h(and)g(set)758 730 y(\012\()p Ft(X)t FC(\))d Fs(\021)955 688 y Fr(X)942 780 y Fq(Y)8 b Fp(\032)p Fq(X)1037 730 y Fs(L)p Ft(K)1112 737 y Fq(L)1139 730 y FC(\()p Ft(Y)j FC(\))p Ft(:)558 b FC(\(8)p Ft(:)p FC(25\))75 861 y(\012)16 b(is)h(quadratic)f(in)g Fs(r)p Ft(\036)f FC(b)o(y)h(our)h(c)o(hoice)e(of)h Fs(L)p FC(.)22 b(Then,)16 b(dropping)h(indices)e Ft(j)k FC(and)e Ft(\033)1633 843 y Fo(\()p Fq(j)r Fo(\))1678 861 y FC(,)219 983 y Ft(e)242 962 y Fp(\000)p Fq(E)307 924 y Fr(Z)357 983 y Ft(d\026)411 990 y Fq(L)438 983 y Fs(E)t FC(xp)521 995 y Fo(1)541 983 y FC([)p Fg(2)11 b FC(+)g Ft(K)693 990 y Fq(L)720 983 y FC(]\(\003\))i(=)h Ft(e)894 962 y Fp(\000)p Fo(\()p Fq(E)r Fo(+)p Fq(\016)q(E)r Fo(\))1058 924 y Fr(Z)1108 983 y Ft(d\026)1162 990 y Fq(L;)p Fo(\012)1224 983 y Fs(E)t FC(xp)1308 995 y Fo(1)1327 983 y FC([)p Fg(2)d FC(+)g Fs(E)t Ft(K)1509 990 y Fq(L)1536 983 y FC(]\(\003\))p Ft(;)152 b FC(\(8)p Ft(:)p FC(26\))75 1105 y(where)16 b Ft(d\026)270 1112 y Fq(L;)p Fo(\012)348 1105 y FC(is)g(the)g(new)h(Gaussian)g(measure)e(de\014ned)h(b)o(y)748 1215 y Ft(d\026)802 1222 y Fq(L;)p Fo(\012)906 1215 y FC(=)41 b Ft(e)1008 1195 y Fq(\016)q(E)1054 1215 y Ft(d\026)1108 1222 y Fq(L)1135 1215 y Ft(e)1158 1195 y Fo(\012\(\003\))767 1304 y Ft(e)790 1284 y Fp(\000)p Fq(\016)q(E)906 1304 y FC(=)985 1246 y Fr(Z)1035 1304 y Ft(d\026)1089 1311 y Fq(L)1116 1304 y Ft(e)1139 1284 y Fo(\012\(\003\))1218 1304 y Ft(;)556 b FC(\(8.27\))75 1427 y(so)14 b(that)g Ft(d\026)289 1434 y Fq(L;)p Fo(\012)364 1427 y FC(is)f(normalized)e(and)j Fs(E)t Ft(K)k FC(is)13 b(a)h(new)f(set)g (of)h(p)q(olymer)d(activities)h(de\014ned)h(b)o(y)g(solving)620 1537 y Fs(E)t FC(xp)q([)p Fg(2)e FC(+)g Fs(E)t Ft(K)886 1544 y Fq(L)913 1537 y FC(])i(=)h Ft(e)1015 1516 y Fp(\000)p Fo(\012)1070 1537 y Fs(E)t FC(xp[)p Fg(2)d FC(+)g Ft(K)1305 1544 y Fq(L)1332 1537 y FC(];)428 b(\(8)p Ft(:)p FC(28\))75 1647 y Ft(E)114 1628 y Fo(\()p Fq(j)r Fo(+1\))219 1647 y FC(=)13 b Ft(E)309 1628 y Fo(\()p Fq(j)r Fo(\))366 1647 y FC(+)e Ft(\016)r(E)s FC(.)148 1707 y Fu(\(2\))19 b(Reblo)r(c)n(king)c Fs(B)i FC(in)g(whic)o(h)e(w) o(e)i(c)o(hange)f(the)g(blo)q(c)o(ks)h(from)e(unit)h(to)h Ft(L)p FC(-blo)q(c)o(ks)g(using)g(the)75 1767 y(op)q(eration)g Fs(B)h FC(de\014ned)e(in)g(Section)f(7.2,)608 1877 y Fs(E)t FC(xp)691 1889 y Fo(1)711 1877 y FC([)p Fg(2)c FC(+)g Fs(E)t Ft(K)893 1884 y Fq(L)920 1877 y FC(])j(=)f Fs(E)t FC(xp)1083 1889 y Fq(L)1109 1877 y FC([)p Fg(2)e FC(+)g Fs(B)r(E)t Ft(K)1326 1884 y Fq(L)1353 1877 y FC(])p Ft(:)407 b FC(\(8)p Ft(:)p FC(29\))148 1987 y Fu(\(3\))23 b(Rescaling)d Fs(R)h FC(in)f(whic)o(h)g(the)g(c)o(hange)g (of)h(v)m(ariables)g Ft(x)g FC(=)g Ft(Lx)1453 1969 y Fp(0)1464 1987 y Ft(;)8 b(\036)p FC(\()p Ft(x)p FC(\))21 b(=)g Ft(L)1694 1969 y Fo(1)p Fp(\000)p Fq(d=)p Fo(2)1795 1987 y Ft(\036)1824 1969 y Fp(0)1835 1987 y FC(\()p Ft(x)1882 1969 y Fp(0)1894 1987 y FC(\))75 2047 y(brings)c Ft(L)p FC(-blo)q(c)o(ks)f(bac)o(k)g(to)g (unit)g(blo)q(c)o(ks)g(and)h(rescales)f(the)g(Gaussian)h(measure)e(so)i(that) g Ft(L)d Fs(!)g FC(1:)470 2169 y Ft(e)493 2149 y Fp(\000)p Fq(E)548 2137 y Fl(\()p Fm(j)q Fl(+1\))637 2111 y Fr(Z)686 2169 y Ft(d\026)740 2176 y Fq(L;)p Fo(\012)802 2169 y Fs(E)t FC(xp)886 2181 y Fq(L)920 2121 y Fr(h)940 2169 y Fg(2)d FC(+)g Fs(B)r(E)t Ft(K)1147 2144 y Fo(\()p Fq(j)r Fo(\))1143 2182 y Fq(L)1193 2121 y Fr(i)1221 2169 y FC(\(\003)1274 2149 y Fo(\()p Fq(j)r Fo(\))1319 2169 y FC(\))553 2278 y(=)i Ft(e)627 2258 y Fp(\000)p Fq(E)682 2246 y Fl(\()p Fm(j)q Fl(+1\))771 2220 y Fr(Z)821 2278 y Ft(d\026)875 2285 y Fo(1)p Fq(;)p Fp(R)p Fo(\012)961 2278 y Fs(E)t FC(xp)1053 2230 y Fr(h)1072 2278 y Fg(2)f FC(+)f Ft(K)1215 2253 y Fo(\()p Fq(j)r Fo(+1\))1211 2289 y(1)1306 2230 y Fr(i)1333 2278 y FC(\(\003)1386 2258 y Fo(\()p Fq(j)r Fo(+1\))1477 2278 y FC(\))p Ft(;)278 b FC(\(8.30\))75 2400 y(where)787 2461 y Ft(K)832 2435 y Fo(\()p Fq(j)r Fo(+1\))828 2471 y(1)937 2461 y FC(=)14 b Fs(RB)r(E)t Ft(K)1141 2435 y Fo(\()p Fq(j)r Fo(\))1137 2473 y Fq(L)1187 2461 y Ft(:)587 b FC(\(8)p Ft(:)p FC(31\))969 2599 y(78)p eop %%Page: 79 83 bop 148 307 a Fu(\(4\))23 b(In)n(tegrating)g(Out)d FC(in)g(whic)o(h)f (Theorem)g(7.1)i(is)f(used)g(to)h(adv)m(ance)f Ft(`)h FC(=)g(1)f(to)h Ft(`)g FC(=)f Ft(L)75 367 y FC(b)o(y)d(deforming)f(the)h(co)o(v)m(ariance)g (of)h Ft(d\026)807 374 y Fo(1)p Fq(;)p Fp(R)p Fo(\012)910 367 y FC(so)g(that)g(it)e(b)q(ecomes)h Ft(d\026)1377 379 y Fq(L;\033)1432 369 y Fl(\()p Fm(j)q Fl(+1\))1531 367 y FC(whic)o(h)f(is)h(the)h(same)75 428 y(Gaussian)j(measure)e(whic)o(h)g(w)o(e)g(started)i(with)e(at)i(the)e (outset)h(of)h(this)e(R)o(G)h(transformation)g(in)75 488 y(\(8.24\))d(except) e(that)i Ft(\033)505 470 y Fo(\()p Fq(j)r Fo(\))566 488 y FC(has)g(b)q(ecome) e Ft(\033)858 470 y Fo(\()p Fq(j)r Fo(+1\))948 488 y FC(.)122 554 y Fr(Z)172 613 y Ft(d\026)226 620 y Fo(1)p Fq(;)p Fp(R)p Fo(\012)312 613 y Fs(E)t FC(xp)395 625 y Fo(1)423 565 y Fr(h)443 613 y Fg(2)c FC(+)g Ft(K)585 587 y Fo(\()p Fq(j)r Fo(+1\))581 624 y(1)676 565 y Fr(i)704 613 y FC(\(\003)757 592 y Fo(\()p Fq(j)r Fo(+1\))847 613 y FC(\))j(=)932 554 y Fr(Z)982 613 y Ft(d\026)1036 624 y Fq(L;\033)1091 614 y Fl(\()p Fm(j)q Fl(+1\))1173 613 y Fs(E)t FC(xp)1256 625 y Fo(1)1284 565 y Fr(h)1304 613 y Fg(2)d FC(+)g Ft(K)1446 587 y Fo(\()p Fq(j)r Fo(+1\))1442 625 y Fq(L)1537 565 y Fr(i)1565 613 y FC(\(\003)1618 592 y Fo(\()p Fq(j)r Fo(+1\))1708 613 y FC(\))p Ft(;)47 b FC(\(8)p Ft(:)p FC(32\))75 738 y(and)17 b(so)g(\(8.24\))g(with)f Ft(j)j FC(has)e(b)q(ecome)e(\(8.24\))i(with)f Ft(j)e FC(+)d(1.)148 798 y(This)17 b(completes)d(the)i(R)o(G)g(transformation)g(\(steps)g (\(1\)-\(4\)\).)148 858 y(The)23 b(in)o(tegration)f(out)h(in)o(v)o(olv)o(es)d (a)j(deformation)f(of)h(the)f(co)o(v)m(ariance)g(whic)o(h)g(is)g(not)h(quite) 75 918 y(rescaling)c(b)q(ecause)i(the)e(initial)g(co)o(v)m(ariance,)h(that)g (of)g Ft(d\026)1172 925 y Fo(1)p Fq(;)p Fp(R)p Fo(\012)1258 918 y FC(,)g(is)g(not)g(quite)f(equal)h(to)g Ft(C)1793 925 y Fq(`)p Fo(=1)1875 918 y FC(=)75 979 y Fs(R)p Ft(C)152 986 y Fq(`)p Fo(=)p Fq(L)220 979 y FC(.)i(This)16 b(is)h(of)f(course)h(the)f(w)o (a)o(y)g(in)g(whic)o(h)g(anomalous)g(dimensions)f(can)i(app)q(ear)g(in)f (other)75 1039 y(more)g(complicated)g(mo)q(dels.)25 b(In)17 b(this)h(mo)q(del)f(there)g(is)g(no)i(need)e(to)h(use)g(a)g(deformation)f (whic)o(h)75 1099 y(is)e(not)g(quite)f(canonical)h(rescaling)g(but)g(it)f(is) h(attractiv)o(e)f(b)q(ecause)h(it)g(restores)g(the)g(co)o(v)m(ariance)f(to)75 1159 y(a)i(simple)e(form)h(from)g(whic)o(h)g(the)h(role)g(of)g Ft(\033)h FC(as)g(a)f(dielectric)e(constan)o(t)i(or)g(eqiv)m(alen)o(tly)e(as) j(a)f(\014eld)75 1219 y(strength)f(renormalisation)e(is)h(immedi)o(ately)d (apparen)o(t.)21 b(If)14 b(w)o(e)g(simply)e(rescaled)i(the)g(co)o(v)m (ariance)75 1279 y(of)20 b Ft(d\026)188 1286 y Fo(1)p Fq(;)p Fp(R)p Fo(\012)273 1261 y(13)330 1279 y FC(then)f Ft(\033)i FC(w)o(ould)e(b)q(e)h(em)o(b)q(edded)d(in)i(the)g(co)o(v)m(ariance)g(in)g(a)h (more)e(complicated)f(w)o(a)o(y)75 1340 y(and)e(one)g(w)o(ould)g(need)f(to)h (pro)o(v)o(e)f(a)h(small)e(lemma)f(to)j(sho)o(w)g(that)g(the)g(asymptotic)e (deca)o(y)h(of)h(suc)o(h)75 1400 y(a)h(co)o(v)m(ariance)f(is)g(of)h(the)f (form)657 1380 y Fo(1)p 656 1388 22 2 v 656 1417 a Fq(\033)682 1400 y Fs(k)p FC(\()p Ft(x)9 b Fs(\000)g Ft(y)r FC(\))p Fs(k)881 1382 y Fo(2)p Fp(\000)p Fq(d)961 1400 y FC(as)17 b(b)q(e\014ts)e(a)h (dielectric)d(constan)o(t.)22 b(I)15 b(am)f(indebted)75 1460 y(to)j(Pronob)g(Mitter)e(for)i(education)f(on)h(this)f(p)q(oin)o(t.)148 1520 y(No)o(w)d(let)g(us)h(consider)f(ho)o(w)h(to)g(c)o(ho)q(ose)g(the)f (parameters)g Ft(L;)8 b(\024)13 b FC(in)g(the)g(large)h(\014eld)f(regulator)h Ft(g)1890 1527 y Fq(\024)75 1580 y FC(and)f Ft(h)g FC(in)g(the)g(norm)f Fs(k)c(k)524 1587 y Fq(g)541 1591 y Fm(\024)561 1587 y Fq(;)p Fo(\000)p Fq(;h)639 1580 y FC(=)13 b Fs(k)8 b(k)748 1587 y Fq(\024;)p Fo(\000)p Fq(;h)833 1580 y FC(.)20 b(W)l(e)13 b(will)e(construct)i (a)h(constan)o(t)f Ft(C)1513 1587 y Fo(1)1546 1580 y FC(whic)o(h)f (determines)75 1641 y(these)k(parameters)f(and)i(some)e(others)i(according)f (to:)75 1751 y Fu(P)n(arameters)98 1852 y FC(\(A\))24 b(Cho)q(ose)18 b Ft(L)e FC(large)g(so)h(that)706 1833 y Fq(C)731 1838 y Fl(1)p 706 1841 43 2 v 706 1869 a Fq(L)730 1860 y Fm(d)767 1852 y Ft(<)824 1833 y Fo(1)p 824 1841 18 2 v 824 1869 a(2)846 1852 y FC(.)100 1998 y(\(B\))24 b(Let)15 b Ft(\024)311 2005 y Fq(j)343 1998 y FC(=)f Ft(\024)423 2005 y Fo(0)473 1941 y Fq(j)451 1956 y Fr(X)472 2047 y Fo(0)519 1998 y FC(2)543 1977 y Fp(\000)p Fq(n)595 1998 y FC(.)20 b(This)15 b(will)f(b)q(e)h(a)g(sequence)f(of)h(n)o (um)o(b)q(ers)e(that)i(limits)e(the)h(large)h(\014eld)197 2090 y(b)q(eha)o(vior)j(of)h(the)f(p)q(olymer)f(activities)g(pro)q(duced)i(after)g Ft(j)i FC(R)o(G)d(steps,)h(roughly)g(sp)q(eaking)197 2155 y(b)o(y)d Fs(j)p Ft(K)324 2137 y Fo(\()p Fq(j)r Fo(\))369 2155 y Fs(j)e(\024)g Ft(e)478 2122 y Fl(1)p 477 2128 16 2 v 477 2148 a(2)497 2135 y Fq(\024)517 2140 y Fm(j)540 2104 y Fr(R)567 2135 y Fo(\()p Fp(r)p Fq(\036)p Fo(\))645 2123 y Fl(2)197 2236 y FC(Cho)q(ose)k Ft(\024)395 2243 y Fo(0)431 2236 y FC(so)f(small)d(that)j(for)f(all)g Ft(j)h Fs(\025)c FC(0)943 2346 y Ft(L)976 2326 y Fq(v)q Fo(\()p Fq(\024)1028 2331 y Fm(j)1045 2326 y Fo(\))1074 2346 y Fs(\024)h Ft(\021)r(;)p 75 2390 735 2 v 115 2421 a Fj(13)150 2436 y Fy(as)g(w)o(as)f (done)i(in)e([18)o(])969 2599 y FC(79)p eop %%Page: 80 84 bop 197 307 a FC(where)16 b Ft(v)r FC(\()p Ft(\024)p FC(\))f(app)q(ears)j(in) e(Lemma)e(8.1,)i(part)h(\(c\).)197 388 y(De\014ne)g Ft(\017)369 395 y Fq(j)403 388 y FC(=)461 363 y Fq(\024)481 368 y Fm(j)q Fl(+1)536 363 y Fp(\000)p Fq(\024)583 368 y Fm(j)p 461 377 140 2 v 521 405 a Fq(\034)623 388 y FC(where)f Ft(\034)23 b FC(is)18 b(the)f(largest)g(n)o(um)o(b)q(er)f(of)h(small)f(sets)h(that)h(can)f (o)o(v)o(erlap)197 448 y(at)h(a)g(p)q(oin)o(t,)g Ft(\034)j FC(=)c(sup)613 460 y Fq(x)643 448 y FC(#)p Fs(f)p Ft(X)j Fs(2)d(S)j FC(:)k Ft(X)c Fs(3)d Ft(x)p Fs(g)p FC(.)25 b(It)17 b(is)g(v)o(ery)f(imp)q (ortan)o(t)h(to)h(this)g(argumen)o(t)197 509 y(that)f(this)f(n)o(um)o(b)q(er) e Ft(\034)22 b FC(is)16 b(\014nite.)99 610 y(\(C\))25 b(Cho)q(ose)18 b Ft(h)e FC(large)g(so)h(that)883 670 y Ft(\024)911 677 y Fo(0)931 670 y Ft(h)959 650 y Fo(2)979 670 y Ft(L)1012 650 y Fp(\000)p Fo(3)p Fq(d)p Fp(\000)p Fo(2)1136 670 y Fs(\025)d FC(1)p Ft(:)197 758 y FC(Note)k(that)h(if)f Ft(j)j Fs(\035)d FC(1,)h(then)g Ft(\017)772 765 y Fq(j)790 758 y Ft(h)818 739 y Fo(2)856 758 y Fs(\024)e FC(1.)29 b(Recall)18 b(that)h Ft(h)f FC(is)h(the)f(radius)h(of)g (analyticit)o(y)e(of)197 818 y(the)f(p)q(olymer)f(activities)148 919 y(Ha)o(ving)h(set)g(these)g(parameters,)f(I)h(mak)o(e)e(the)i(follo)o (wing)75 1029 y Fu(Inductiv)n(e)i(Assumptions)d FC(\()p Ft(I)699 1011 y Fo(\()p Fq(j)r Fo(\))744 1029 y FC(\))121 1146 y(\(i\))24 b Fs(k)p Ft(K)267 1121 y Fo(\()p Fq(j)r Fo(\))263 1158 y Fq(L)313 1146 y Fs(k)338 1153 y Fq(\024)358 1158 y Fm(j)374 1153 y Fq(;)p Fo(\000)p Fq(;h)452 1146 y Fs(\024)505 1098 y Fr(\020)534 1127 y Fq(C)559 1132 y Fl(1)p 534 1135 43 2 v 534 1163 a Fq(L)558 1154 y Fm(d)582 1098 y Fr(\021)607 1110 y Fq(j)633 1146 y FC(;)108 1300 y(\(ii\))f Ft(\033)227 1282 y Fo(\()p Fq(j)r Fo(\))286 1300 y Fs(\025)14 b FC(1)9 b Fs(\000)426 1280 y Fo(1)p 426 1288 18 2 v 426 1317 a(8)479 1243 y Fq(j)457 1258 y Fr(X)478 1349 y Fo(0)525 1300 y FC(2)549 1279 y Fp(\000)p Fq(n)616 1300 y FC(where)15 b Ft(\033)786 1282 y Fo(\()p Fq(j)r Fo(\))847 1300 y FC(app)q(ears)h(in)f(the)h(Gaussian)h(measure)d(co)o(v)m(ariance)h (\(8.3\).)197 1400 y(Note)h(that)g(for)h Ft(j)g Fs(\035)c FC(1)p Ft(;)8 b FC(\()p Ft(\017)680 1407 y Fq(j)699 1400 y Ft(h)727 1382 y Fo(2)746 1400 y FC(\))p Fs(k)p Ft(K)835 1382 y Fo(\()p Fq(j)r Fo(\))881 1400 y Fs(k)906 1407 y Fq(\024)926 1412 y Fm(j)943 1407 y Fq(;)p Fo(\000)p Fq(;h)1020 1400 y Fs(\024)14 b FC(1,)i(indeed,)f(as)i(small)e(as)i(y)o(ou)f(lik)o(e.)75 1597 y Fu(Theorem)h(8.3)24 b Fn(If)15 b Ft(j)i Fs(\035)d FC(1)i Fn(dep)n(ending)h(on)f(these)h(choic)n(es)g(of)e Ft(L;)8 b(\024)1318 1604 y Fq(j)1337 1597 y Ft(;)g(h)p Fn(,)16 b(then)h Ft(I)1551 1579 y Fo(\()p Fq(j)r Fo(\))1610 1597 y FC(=)-8 b Fs(\))13 b Ft(I)1729 1579 y Fo(\()p Fq(j)r Fo(+1\))1835 1597 y Fn(and)147 1665 y Fr(Z)197 1724 y Ft(d\026)251 1735 y Fq(L;\033)306 1725 y Fl(\()p Fm(j)q Fl(\))349 1724 y Fs(E)t FC(xp)433 1736 y Fo(1)453 1724 y FC([)p Fg(2)e FC(+)g Ft(K)609 1698 y Fo(\()p Fq(j)r Fo(\))605 1736 y Fq(L)654 1724 y FC(]\(\003)721 1703 y Fo(\()p Fq(j)r Fo(\))766 1724 y FC(\))42 b(=)f Ft(e)929 1703 y Fp(\000)p Fq(\016)q(E)1001 1692 y Fl(\()p Fm(j)q Fl(\))1052 1665 y Fr(Z)1101 1724 y Ft(d\026)1155 1735 y Fq(L;\033)1210 1725 y Fl(\()p Fm(j)q Fl(+1\))1292 1724 y Fs(E)t FC(xp)1376 1736 y Fo(1)1395 1724 y FC([)p Fg(2)11 b FC(+)g Ft(K)1551 1698 y Fo(\()p Fq(j)r Fo(+1\))1547 1736 y Fq(L)1642 1724 y FC(]\(\003)1709 1703 y Fo(\()p Fq(j)r Fo(+1\))1799 1724 y FC(\);)677 1861 y Ft(\016)r(E)740 1841 y Fo(\()p Fq(j)r Fo(\))827 1861 y FC(=)906 1803 y Fr(Z)956 1861 y Ft(d\026)1010 1873 y Fq(L;\033)1065 1863 y Fl(\()p Fm(j)q Fl(\))1117 1861 y FC(exp)1199 1776 y Fr(0)1199 1850 y(@)1266 1820 y(X)1236 1917 y Fq(Y)d Fp(\032)p Fo(\003)1315 1908 y Fl(\()p Fm(j)q Fl(\))1365 1861 y Fs(L)p Ft(K)1444 1841 y Fo(\()p Fq(j)r Fo(\))1490 1861 y FC(\()p Ft(Y)j FC(\))1567 1776 y Fr(1)1567 1850 y(A)1611 1861 y Ft(:)148 2021 y FC(T)l(o)18 b(use)g(this)g(theorem)e(w)o (e)h(pic)o(k)g Ft(j)793 2028 y Fo(0)829 2021 y Fs(\035)g FC(1)h(so)g(that)g Ft(I)1132 2003 y Fo(\()p Fq(j)r Fo(\))1194 2021 y FC(=)-8 b Fs(\))16 b Ft(I)1316 2003 y Fo(\()p Fq(j)r Fo(+1\))1424 2021 y FC(for)i(all)f Ft(j)i Fs(\025)e Ft(j)1684 2028 y Fo(0)1721 2021 y FC(and)i(then)75 2081 y(rename)f(the)h(initial)f Ft(K)t FC(\()p Ft(X)t FC(\))i(and)g(initial)e(\003,)i Ft(K)982 2063 y Fq(j)996 2068 y Fl(0)1035 2081 y FC(and)g(\003)1167 2063 y Fq(j)1181 2068 y Fl(0)1220 2081 y FC(resp)q(ectiv)o(ely)l(.)28 b(Then)20 b(pro)o(vided)f(the)75 2141 y(norm)h(of)i(the)f(initial)e Ft(K)26 b FC(is)21 b(small,)f(as)i(required)e(b)o(y)g Ft(I)1129 2123 y Fq(j)1143 2128 y Fl(0)1162 2141 y FC(,)i(the)f(theorem)f(implies)e (that)k(w)o(e)e(can)75 2201 y(rep)q(eat)15 b(R)o(G)g(steps)g(un)o(til)f(\003) 581 2183 y Fo(\()p Fq(j)r Fo(\))641 2201 y FC(b)q(ecomes)g(a)h(single)g(unit) g(blo)q(c)o(k)f(\001.)20 b(In)15 b([18,)g(43)q(,)f(41)q(])h(this)f(is)h(used) g(to)75 2261 y(calculate)g(the)g(logarithm)f(of)i(the)g(partition)f(function) h(b)o(y)f(reducing)g(it)g(to)h(the)f(logarithm)g(of)h(the)75 2322 y(partition)g(function)g(in)g(a)h(single)f(blo)q(c)o(k.)148 2382 y(The)i(conclusions)g(follo)o(w)g(from)f(our)h(discussion)h(ab)q(o)o(v)o (e)f(except)e(for)j Ft(I)1483 2364 y Fo(\()p Fq(j)r Fo(\))1545 2382 y FC(=)-8 b Fs(\))16 b Ft(I)1667 2364 y Fo(\()p Fq(j)r Fo(+1\))1757 2382 y FC(,)i(whic)o(h)75 2442 y(will)d(require)g(the)h(follo)o (wing)g(Lemma)e(8.4)j(and)g(Prop)q(osition)g(8.5.)969 2599 y(80)p eop %%Page: 81 85 bop 75 307 a Fu(Lemma)16 b(8.4)24 b Fn(L)n(et)16 b Ft(R)f FC(=)e Ft(e)579 289 y Fp(\000L)p Fq(K)675 307 y Fs(\000)d FC(1)p Fn(.)23 b(Ther)n(e)17 b(is)g Ft(C)1013 314 y Fo(2)1050 307 y Fn(such)g(that)g(for)g (al)r(l)h Ft(\017;)8 b(h;)g(\024)14 b Fs(\025)f FC(0)18 b Fn(with)f Ft(\017h)1787 289 y Fo(2)1821 307 y Fs(\024)c FC(1)p Fn(,)75 367 y(and)18 b(any)f FC(\000)118 469 y Fn(\(i\))24 b Fs(kL)p Ft(K)t Fs(k)326 476 y Fq(\017;)p Fo(\000)p Fq(;h)418 469 y Fs(\024)14 b Ft(C)506 476 y Fo(2)548 449 y(1)p 531 457 52 2 v 531 486 a Fq(\017h)565 477 y Fl(2)587 469 y Fs(k)p Ft(K)t Fs(k)682 476 y Fq(\024;)p Fo(\000)p Fq(;h)767 469 y Ft(:)75 570 y Fn(If,)j(in)h(addition,)f FC(\()p Ft(\017h)473 552 y Fo(2)493 570 y FC(\))512 552 y Fp(\000)p Fo(1)559 570 y Fs(k)p Ft(K)t Fs(k)654 577 y Fq(\024;)p Fo(\000)p Fq(;h)752 570 y Fs(\034)d FC(1)p Fn(,)j(then)103 672 y(\(ii\))24 b Fs(k)p Ft(R)p Fs(k)284 679 y Fq(\017;)p Fo(\000)p Fq(;h)376 672 y Fs(\024)14 b Ft(C)464 679 y Fo(2)506 652 y(1)p 489 660 V 489 689 a Fq(\017h)523 680 y Fl(2)545 672 y Fs(k)p Ft(K)t Fs(k)640 679 y Fq(\024;)p Fo(\000)p Fq(;h)88 793 y Fn(\(iii\))24 b Fs(k)p Ft(R)11 b FC(+)g Fs(L)p Ft(K)t Fs(k)423 800 y Fq(\017;)p Fo(\000)p Fq(;h)516 793 y Fs(\024)j Ft(C)604 800 y Fo(2)631 744 y Fr(\020)678 773 y Fo(1)p 661 781 V 661 810 a Fq(\017h)695 800 y Fl(2)718 793 y Fs(k)p Ft(K)t Fs(k)813 800 y Fq(\024;)p Fo(\000)p Fq(;h)897 744 y Fr(\021)922 756 y Fo(2)950 793 y Ft(:)148 913 y FC(F)l(or)f(a)h (detailed)e(pro)q(of,)i(see)e([18)q(,)h(Prop)q(osition)h(4.2].)20 b(Ho)o(w)o(ev)o(er,)11 b(notice)i(that)g(\(i\))g(follo)o(ws)f(easily)75 973 y(from)735 1033 y Fs(j)p Ft(\036)p FC(\()p Ft(\030)r FC(\))p Fs(j)853 1013 y Fo(2)887 1033 y Fs(\024)h Ft(O)q FC(\()p Ft(\017)1016 1013 y Fp(\000)p Fo(1)1064 1033 y FC(\))p Ft(g)1106 1040 y Fq(\017)1122 1033 y FC(\()p Ft(X)q(;)8 b(\036)p FC(\))75 1120 y(if)23 b Ft(\030)29 b FC(=)d(\()p Ft(x;)8 b(\026)p FC(\))p Ft(;)g(x)25 b Fs(2)h Ft(X)t FC(.)43 b(T)l(o)24 b(understand)h(\(ii\),)e (notice)g(that)h(\(i\))f(sa)o(ys)h(that)f(for)h Ft(\036)p FC(\()p Ft(\030)r FC(\))g(large,)75 1187 y Ft(R)14 b Fs(\030)g FC(exp)262 1138 y Fr(\020)292 1167 y Fo(1)p 292 1175 18 2 v 292 1204 a(2)314 1187 y Fs(k)p Ft(K)t Fs(k)414 1165 y Fq(\036)435 1153 y Fl(2)p 414 1175 39 2 v 414 1204 a Fq(h)434 1194 y Fl(2)457 1138 y Fr(\021)482 1187 y FC(,)i(so)h(w)o(e)e(need)h(\()p Ft(\017h)824 1169 y Fo(2)844 1187 y FC(\))863 1169 y Fp(\000)p Fo(1)910 1187 y Fs(k)p Ft(K)t Fs(k)e(\024)f FC(1)k(to)f(dominate)f Ft(R)i FC(b)o(y)f Ft(g)1530 1194 y Fq(\017)1547 1187 y FC(.)21 b(\(iii\))15 b(is)h(exp)q(ected)75 1255 y(since)f Ft(R)d Fs(\000)f(L)p Ft(K)18 b FC(=)c Ft(O)q FC(\()p Fs(L)p Ft(K)t FC(\))593 1237 y Fo(2)614 1255 y FC(.)148 1365 y(The)22 b(next)f(prop)q(osition)i(is)e(related)g(to)h (the)f(fact)h(that)g(if)f(a)h(function)g Ft(f)5 b FC(\()p Ft(z)r FC(\))21 b(is)h(analytic)f(in)75 1425 y(a)h(strip)g(of)g(width)f Ft(h)h FC(then)g Ft(f)641 1407 y Fo(\()p Fq(n)p Fo(\))692 1425 y FC(\()p Ft(z)r FC(\))g(is,)g(in)g(a)g(strip)f(of)h(width)g Ft(\013h)p FC(,)h(do)o(wn)f(in)g(size)f(b)o(y)g(a)h(factor)75 1485 y Ft(\013)106 1467 y Fq(n)130 1485 y FC(.)27 b(The)18 b(generalised)g(norm)f Fs(k)p FC(\()p Fs(\001)p FC(\))p Fs(k)756 1492 y Fq(\024;)p Fo(\000)p Fq(;h)p 818 1499 21 2 v 858 1485 a FC(where)h(eac)o(h)g(of)g(the)g(\014elds)g Fs(r)p Ft(\036;)8 b Fs(rr)p Ft(\036;)g(:)g(:)g(:)16 b FC(is)i(assigned)75 1546 y(a)k(separate)g(radius)g(of)g(analyticit)o(y)e(mak)o(es)g(an)j(app)q (earance)f(here.)37 b Ft(h)p 1408 1553 29 2 v 23 w Fs(\021)23 b FC(\()p Ft(h)1568 1553 y Fp(r)p Fq(\036)1621 1546 y Ft(;)8 b(h)1671 1553 y Fp(rr)p Fq(\036)1753 1546 y Ft(;)g(:)g(:)g(:)o FC(\))23 b Fs(\021)75 1606 y FC(\()p Ft(h)122 1615 y Fq(@)143 1606 y Fm(\014)163 1615 y Fq(\036)186 1606 y FC(\))205 1614 y Fq(\014)r Fp(2)p Fo(Range)n(\()p Fq(\014)r Fo(\))413 1606 y Fs(\021)16 b FC(\()p Ft(h)515 1613 y Fq(\014)538 1606 y FC(\))557 1614 y Fq(\014)r Fp(2)p Fo(Range)n(\()p Fq(\014)r Fo(\))750 1606 y FC(.)24 b(It)17 b(is)g(de\014ned,)g(b)o(y)f(replacing)h(in)g(equation) g(\(7.11\),)h Ft(hd\030)i FC(b)o(y)75 1666 y Ft(h)103 1673 y Fq(\014)127 1666 y Ft(d\030)r FC(.)38 b(Recall)21 b(that)h Ft(\030)k FC(=)d(\()p Ft(x;)8 b(\014)s FC(\),)753 1630 y Fr(R)789 1666 y Ft(d\030)25 b FC(in)o(tegrates)c(o)o(v)o(er)g Ft(x)g FC(and)h Ft(\014)i FC(and)f(that)f(Range\()p Ft(\014)s FC(\))f(w)o(as)75 1726 y(discussed)16 b(when)h(the)f(lo)q(calisation)g(op)q(erator)h Fs(L)g FC(w)o(as)g(de\014ned.)75 1828 y Fu(Prop)r(osition)h(8.5)24 b Fn(L)n(et)16 b FC([\(1)11 b Fs(\000)e(L)p FC(\))p Ft(K)t FC(])785 1835 y Fq(S)828 1828 y Fn(denote)18 b(the)g(smal)r(l)g(set)f(p)n (art)f(of)h FC(\(1)11 b Fs(\000)f(L)p FC(\))p Ft(K)21 b Fn(as)c(de\014ne)n(d) h(in)75 1888 y(Pr)n(op)n(osition)f(7.2.1.)24 b(L)n(et)18 b Ft(\013h)p 564 1895 60 2 v 18 w Fn(b)n(e)h(given)h(by)e FC(\()p Ft(\013h)p 913 1895 V FC(\))991 1895 y Fq(\014)1030 1888 y FC(=)d Ft(\013L)1147 1870 y Fp(\000j)p Fq(\014)r Fp(j)p Fo(+1)1263 1888 y Ft(h)p Fn(.)25 b(F)l(or)17 b(e)n(ach)i Ft(p)c FC(=)h(0)p Ft(;)8 b FC(1)p Ft(;)g(:)g(:)g(:)o Fn(,)19 b(ther)n(e)75 1948 y(is)i Ft(C)166 1955 y Fo(3)185 1948 y FC(\()p Ft(p)p FC(\))h Fn(such)f(that)g(for)g(al)r(l)h Ft(\013;)8 b(h;)g(\024)21 b(>)f FC(0)i Fn(with)f FC(\(\()p Ft(\013h)p 1044 1955 V FC(\))1122 1955 y Fq(\014)1146 1948 y FC(\))1165 1930 y Fo(2)1185 1948 y Ft(\024)f Fs(\025)g FC(1)i Fn(for)e(al)r(l)i Ft(\014)h Fs(2)e FC(Range\()p Ft(\014)s FC(\))p Fn(,)g(and)75 2015 y Ft(L)108 1997 y Fp(\000)140 1983 y Fm(d)p 141 1989 17 2 v 141 2010 a Fl(2)178 2015 y Fs(\024)13 b Ft(\013)h Fs(\024)g FC(1)p Fn(,)k(and)f(any)h FC(\000)p Fn(,)545 2125 y Fs(k)p FC([\(1)10 b Fs(\000)h(L)p FC(\))p Ft(K)t FC(])799 2132 y Fq(S)825 2125 y Fs(k)850 2132 y Fq(\024;)p Fo(\000)p Fq(;\013h)p 912 2139 43 2 v 970 2125 a Fs(\024)j Ft(C)1058 2132 y Fo(3)1078 2125 y FC(\()p Ft(p)p FC(\))p Ft(\013)1171 2105 y Fq(p)p Fo(+1)1236 2125 y Fs(k)p Ft(K)t Fs(k)1331 2132 y Fq(\024;)p Fo(\000)p Fq(;h)1429 2125 y Ft(:)148 2244 y FC(In)j(this)f(section,)g Ft(p)f FC(=)f(3,)j(so)g(I)g (write)f Ft(C)883 2251 y Fo(3)917 2244 y FC(=)e Ft(C)1004 2251 y Fo(3)1024 2244 y FC(\()p Ft(p)p FC(\).)23 b(Also)16 b(I)g(\014x)h Ft(\013)e FC(=)f(2)p Ft(L)1492 2225 y Fp(\000)1524 2212 y Fm(d)p 1525 2218 17 2 v 1525 2238 a Fl(2)1548 2244 y FC(,)j(where)f Ft(L)h FC(will)e(b)q(e)75 2304 y(c)o(hosen)h(b)q(elo)o(w.)75 2414 y Fu(Sk)n(etc)n(h)e(of)g(pro)r(of.)20 b FC(\(A)12 b(detailed)f(pro)q(of) i(of)g(a)f(v)o(ery)f(similar)f(prop)q(osition)k(is)e(giv)o(en)f(in)h([18,)h (Lemma)75 2474 y(4.3].\))29 b(First)18 b(note)h(that)h(v)m(ariational)f (deriv)m(ativ)o(es)e(of)i(order)g Ft(n)g Fs(\025)f Ft(p)13 b FC(+)g(1)19 b(for)g Ft(K)k FC(and)c(\(1)14 b Fs(\000)e(L)p FC(\))p Ft(K)969 2599 y FC(81)p eop %%Page: 82 86 bop 75 307 a FC(coincide.)21 b(Because)c Ft(h)f FC(is)h(scaled)f(b)o(y)h(a)g (factor)g(of)g Ft(\013)g FC(in)f(the)h(left)f(hand)h(side)f(the)h(con)o (tribution)f(of)75 367 y(all)e(deriv)m(ativ)o(es)g(of)h(order)g Ft(n)f Fs(\025)f FC(4)j(is)e(do)o(wn)i(b)o(y)e(a)h(factor)h(of)f Ft(\013)1202 349 y Fq(n)1239 367 y Fs(\024)f Ft(\013)1323 349 y Fq(p)1343 367 y FC(.)21 b(This)15 b(is)g(part)g(of)g(the)g(reason)75 428 y(for)k Ft(\013)183 409 y Fq(p)221 428 y FC(on)g(the)f(righ)o(t)g(hand)h (side.)27 b(The)18 b(other)h(part)f(comes)f(from)h(the)g(fact)g(that)h(w)o(e) f(can)g(write)75 488 y(lo)o(w)o(er)d(deriv)m(ativ)o(es)f(in)h(terms)f(of)i (higher)g(ones.)21 b(F)l(or)16 b(example)d(consider)j(the)f Ft(n)f FC(=)g(0)i(con)o(tribution)75 548 y(to)g(the)f(norm)g(of)h(\(1)10 b Fs(\000)g(L)p FC(\))p Ft(K)t FC(:)21 b(w)o(e)15 b(can)h(express)f(it)h(in)f (terms)f(of)i(the)f Ft(p)c FC(+)e(1)16 b(v)m(ariational)g(deriv)m(ativ)o(e)75 608 y(using)h(\(for)f Ft(p)e FC(=)g(3\),)420 738 y(\(1)e Fs(\000)f(L)p FC(\))p Ft(K)t FC(\()p Ft(\036)p FC(\))j(=)755 680 y Fr(Z)797 693 y Fo(1)779 774 y(0)825 738 y Ft(dt)881 704 y FC(\(1)e Fs(\000)e Ft(t)p FC(\))1022 686 y Fo(3)p 881 727 161 2 v 943 772 a FC(3!)1055 680 y Fr(Z)1110 704 y Ft(\016)1134 686 y Fo(4)1153 704 y Ft(K)p 1110 727 89 2 v 1118 772 a(\016)r(\036)1171 758 y Fo(4)1211 738 y FC(\()p Ft(t\036)p FC(\))e Ft(\036\036\036\036)j FC(+)g Ft(:)d(:)g(:)f(;)229 b FC(\(8)p Ft(:)p FC(33\))75 864 y(where)16 b Ft(:)8 b(:)g(:)17 b FC(is)f(the)h(di\013erence)e Ft(M)i Fs(\000)11 b FC([)p Ft(M)5 b FC(])837 871 y Fq(loc)899 864 y FC(discussed)17 b(in)f(\(8.22\),)h(\(8.23\).)24 b(I)16 b(will)g(drop)h(this)g(part)75 924 y(b)q(ecause)f(it)g(is)g(easy)h(to)f(estimate)f(b)o(y)g Ft(C)t(\013)853 906 y Fq(p)p Fo(+1)918 924 y Fs(k)p Ft(K)t Fs(k)h FC(Therefore)101 1056 y Fs(j)p FC(\(1)c Fs(\000)e(L)p FC(\))p Ft(K)t FC(\()p Ft(\036)p FC(\))p Ft(g)409 1035 y Fp(\000)p Fo(1)457 1056 y FC(\()p Ft(\036)p FC(\))p Fs(j)41 b(\024)659 981 y Fr(\014)659 1006 y(\014)659 1031 y(\014)659 1056 y(\014)659 1080 y(\014)673 997 y(Z)715 1010 y Fo(1)696 1091 y(0)743 1056 y Ft(dt)799 1022 y FC(\(1)11 b Fs(\000)g Ft(t)p FC(\))940 1004 y Fo(3)p 799 1044 161 2 v 860 1090 a FC(3!)973 997 y Fr(Z)1027 1022 y Ft(\016)1051 1004 y Fo(4)1071 1022 y Ft(K)p 1027 1044 89 2 v 1035 1090 a(\016)r(\036)1088 1075 y Fo(4)1121 1056 y FC(\()p Ft(t\036)p FC(\))p Ft(g)1231 1035 y Fp(\000)p Fo(1)1277 1056 y FC(\()p Ft(t\036)p FC(\))1370 1007 y Fr(n)1398 1056 y Ft(\036\036\036\036g)1539 1035 y Fp(\000)p Fo(1)1594 1007 y Fr(\020)1619 1012 y Fs(p)p 1660 1012 123 2 v 1660 1056 a FC(1)h Fs(\000)e Ft(t)1763 1041 y Fo(2)1783 1056 y Ft(\036)1812 1007 y Fr(\021)o(o)1864 981 y(\014)1864 1006 y(\014)1864 1031 y(\014)1864 1056 y(\014)1864 1080 y(\014)579 1195 y Fs(\024)669 1161 y FC(4!)p 664 1183 48 2 v 664 1229 a Ft(h)692 1215 y Fo(4)717 1195 y Fs(k)p Ft(K)t Fs(k)812 1202 y Fq(\024;)p Fo(\000)p Fq(;h)905 1137 y Fr(Z)946 1150 y Fo(1)928 1231 y(0)974 1195 y Ft(dt)1030 1161 y FC(\(1)i Fs(\000)f Ft(t)p FC(\))1172 1143 y Fo(3)p 1030 1183 161 2 v 1092 1229 a FC(3!)1255 1161 y(Const.)p 1201 1183 247 2 v 1201 1229 a(\()p Ft(\024)p FC(\(1)g Fs(\000)g Ft(t)1370 1215 y Fo(2)1389 1229 y FC(\)\))1427 1215 y Fo(2)579 1300 y Fs(\024)41 b FC(Const.)22 b Ft(\013)850 1280 y Fo(4)870 1300 y Fs(k)p Ft(K)t Fs(k)965 1307 y Fq(\024;)p Fo(\000)p Fq(;h)1050 1300 y Ft(;)724 b FC(\(8.34\))75 1401 y(using)24 b(con)o(v)o(ergence)e(of)h (the)g Ft(t)g FC(in)o(tegral)g(and)h(\()p Ft(\013h)p FC(\))1065 1383 y Fo(2)1085 1401 y Ft(\024)i Fs(\025)g FC(1.)43 b(W)l(e)23 b(can)h(put)f(together)h(similar)75 1461 y(estimates)c(on)h(the)g Ft(n)h FC(=)g(1)p Ft(;)8 b FC(2)p Ft(;)g(:)g(:)g(:)g(;)g(p)22 b FC(parts)g(of)f(the)g Fs(k)p FC([\(1)14 b Fs(\000)g(L)p FC(\))p Ft(K)t FC(])1333 1468 y Fq(S)1359 1461 y Fs(k)21 b FC(norm)f(to)i(conclude)e (as)i(in)75 1521 y(Lemma)14 b(8.4.)1534 b Fg(2)75 1626 y Fu(Pro)r(of)19 b(of)f(Theorem)f(8.3.)k FC(In)16 b(particular)g(this)g(pro)o(v)o(es)f(that)i Ft(I)1303 1608 y Fo(\()p Fq(j)r Fo(\))1362 1626 y FC(=)-8 b Fs(\))13 b Ft(I)1481 1608 y Fo(\()p Fq(j)r Fo(+1\))1588 1626 y FC(if)j Ft(j)g Fs(\035)e FC(1.)148 1686 y Fu(\(1\))19 b(Renormalization.)h FC(The)d(task)g(is)f(to)i(solv)o(e)e(\(8.28\))h(for)g Fs(E)t Ft(K)22 b FC(and)17 b(estimate)e(the)h(norm)75 1746 y(of)h Fs(E)t Ft(K)t FC(.)22 b(I)15 b(b)q(egin)i(b)o(y)f(pro)o(ving)g(that)760 1846 y Fs(E)t Ft(K)j FC(=)13 b(\(1)f Fs(\000)f(L)p FC(\))p Ft(K)k FC(+)c Ft(H)q(;)569 b FC(\(8)p Ft(:)p FC(35\))75 1954 y(\()p Ft(K)18 b FC(=)c Ft(K)250 1928 y Fo(\()p Fq(j)r Fo(\))246 1966 y Fq(L)295 1954 y FC(\))j(where)f Ft(H)t FC(,)g(for)g(h)p 620 1961 28 2 v(igher)g(order)h(in)e Ft(K)t FC(,)h(is)g(giv)o(en)g(b)o(y)577 2054 y Ft(H)i FC(=)c(\()p Ft(R)d FC(+)g Fs(L)p Ft(K)t FC(\))h(+)f Ft(R)999 2034 y Fo(+)1040 2054 y FC(+)g(\()p Ft(R)g FC(+)g Ft(R)1242 2034 y Fo(+)1272 2054 y FC(\))g Fs(_)h Ft(K)q(;)385 b FC(\(8)p Ft(:)p FC(36\))75 2155 y Ft(R)18 b Fs(\021)f Ft(e)209 2136 y Fp(\000L)p Fq(K)307 2155 y Fs(\000)c FC(1,)19 b Ft(R)453 2136 y Fo(+)501 2155 y FC(w)o(as)g(de\014ned)f(in)g(De\014nition)g(6.1.1,)h Ft(A)12 b Fs(_)h Ft(B)21 b FC(w)o(as)e(de\014ned)f(in)g(Prop)q(osition)75 2215 y(7.3.)k(T)l(o)16 b(obtain)h(this)f(form)o(ula,)f(\014rst)h(note)h(that) 442 2315 y Ft(e)465 2295 y Fp(\000)p Fo(\012\()p Fq(X)s Fo(\))621 2315 y FC(=)717 2274 y Fr(Y)700 2366 y Fq(Y)8 b Fp(\032)p Fq(X)796 2267 y Fr(\020)821 2315 y Ft(e)844 2295 y Fp(\000L)p Fq(K)r Fo(\()p Fq(Y)g Fo(\))996 2315 y Fs(\000)j FC(1)g(+)g(1)1154 2267 y Fr(\021)621 2434 y FC(=)717 2393 y Fr(Y)700 2484 y Fq(Y)d Fp(\032)p Fq(X)787 2434 y FC(\(1)k(+)f Ft(R)p FC(\()p Ft(Y)h FC(\)\))h(=)h Fs(E)t FC(xp)q([)p Fg(2)c FC(+)h Ft(R)h FC(+)f Ft(R)1419 2413 y Fo(+)1449 2434 y FC(]\()p Ft(X)t FC(\))243 b(\(8.37\))969 2599 y(82)p eop %%Page: 83 87 bop 75 307 a FC(b)o(y)16 b(Prop)q(osition)h(7.2.)22 b(Then,)16 b(b)o(y)f(de\014nition)h(of)h Fs(E)t Ft(K)t FC(,)371 402 y Fs(E)t FC(xp)q([)p Fg(2)10 b FC(+)h Fs(E)t Ft(K)t FC(])42 b(=)g Ft(e)799 381 y Fp(\000)p Fo(\012)853 402 y Fs(E)t FC(xp)q([)p Fg(2)11 b FC(+)g Ft(K)t FC(])696 475 y(=)42 b Fs(E)t FC(xp[)p Fg(2)11 b FC(+)g Ft(R)h FC(+)f Ft(R)1105 454 y Fo(+)1135 475 y FC(])p Fs(E)t FC(xp[)p Fg(2)g FC(+)g Ft(K)t FC(])696 547 y(=)42 b Fs(E)t FC(xp[)p Fg(2)11 b FC(+)g Ft(K)k FC(+)c Ft(R)h FC(+)f Ft(R)1210 527 y Fo(+)1251 547 y FC(+)g Ft(K)k Fs(_)c FC(\()p Ft(R)h FC(+)f Ft(R)1554 527 y Fo(+)1584 547 y FC(\)])75 642 y(\(b)o(y)16 b(Prop)q(osition)h(7.3\))743 702 y(=)c Fs(E)t FC(xp)q([)p Fg(2)e FC(+)g Ft(K)k Fs(\000)c(L)p Ft(K)k FC(+)c Ft(H)t FC(])496 b(\(8)p Ft(:)p FC(38\))75 783 y(whic)o(h)16 b(pro)o(v)o(es)f(the)h(form)o(ula)f(for)i Fs(E)t Ft(K)t FC(.)148 843 y Ft(H)23 b FC(lo)q(oks)d(complicated)c(and)k(it)e(is,)h(but)g(all)f(w)o (e)h(need)f(is)h(that)g(there)f(is)h(a)g(constan)o(t)g Ft(C)k FC(suc)o(h)75 903 y(that)587 990 y Fs(k)p Ft(H)t Fs(k)681 999 y Fq(\024)701 1004 y Fm(j)q Fl(+1)756 999 y Fq(;\021)785 989 y Fk(\000)p Fl(1)826 999 y Fo(\000)p Fq(;h)894 990 y Fs(\024)14 b Ft(C)994 917 y Fr( )1062 956 y FC(1)p 1031 979 86 2 v 1031 1024 a Ft(\017)1051 1031 y Fq(j)1069 1024 y Ft(h)1097 1010 y Fo(2)1122 990 y Fs(k)p Ft(K)t Fs(k)1217 997 y Fq(\024)1237 1002 y Fm(j)1254 997 y Fq(;)p Fo(\000)p Fq(;h)1318 917 y Fr(!)1351 929 y Fo(2)1379 990 y Ft(;)395 b FC(\(8)p Ft(:)p FC(39\))75 1099 y(where)16 b Ft(\021)g(>)d FC(1)k(w)o(as)g(\014xed)f(once)g(and)h(for)f (all)g(just)g(b)q(elo)o(w)h(Prop)q(osition)g(8.2.)148 1159 y(I)22 b(shall)f(write)g Fs(k)p Ft(K)t Fs(k)h FC(instead)g(of)g Fs(k)p Ft(K)t Fs(k)884 1166 y Fq(\024)904 1171 y Fm(j)920 1166 y Fq(;)p Fo(\000)p Fq(;h)1006 1159 y FC(in)f(this)h(pro)q(of.)39 b Ft(H)26 b FC(consists)c(of)g(three)f(pieces,)75 1220 y Ft(R)12 b FC(+)f Fs(L)p Ft(K)q(;)d(R)308 1201 y Fo(+)354 1220 y FC(and)17 b(\()p Ft(R)12 b FC(+)f Ft(R)603 1201 y Fo(+)633 1220 y FC(\))g Fs(_)g Ft(K)t FC(.)22 b(It)15 b(is)h(enough)h(to)g(pro)o(v)o(e)e(\(8.39\))i (for)g(eac)o(h)f(piece.)148 1280 y Fu(Piece)i(1:)j FC(This)16 b(is)g(small)f(b)q(ecause)i Ft(R)d Fs(\030)f(\000L)p Ft(K)t FC(.)22 b(Th)o(us)546 1374 y Fs(k)p Ft(R)11 b FC(+)g Fs(L)p Ft(K)t Fs(k)772 1383 y Fq(\024)792 1388 y Fm(j)q Fl(+1)848 1383 y Fq(;\021)877 1373 y Fk(\000)p Fl(1)918 1383 y Fo(\000)p Fq(;h)986 1374 y Fs(\024)i(k)p Ft(R)f FC(+)f Fs(L)p Ft(K)t Fs(k)1265 1383 y Fq(\017)1279 1388 y Fm(j)1296 1383 y Fq(;\021)1325 1373 y Fk(\000)p Fl(1)1366 1383 y Fo(\000)p Fq(;h)1420 1374 y Ft(;)354 b FC(\(8)p Ft(:)p FC(40\))75 1469 y(b)q(ecause)16 b Ft(\024)283 1476 y Fq(j)r Fo(+1)361 1469 y Fs(\025)d Ft(\017)433 1476 y Fq(j)989 1529 y Fs(\024)g(k)p Ft(R)f FC(+)f Fs(L)p Ft(K)t Fs(k)1268 1536 y Fq(\017)1282 1541 y Fm(j)1298 1536 y Fq(;)p Fo(\000)p Fq(;h)1788 1529 y FC(\(8)p Ft(:)p FC(41\))75 1610 y(b)q(ecause)16 b Ft(\021)281 1592 y Fp(\000)p Fo(1)328 1610 y FC(\000)f Fs(\024)e FC(\000.)22 b(No)o(w)16 b(apply)g(Lemma)e(8.4)j(and)g (conclude)e(the)h(b)q(ound)i(on)e(piece)f(1.)148 1670 y Fu(Piece)g(2:)k Ft(R)394 1652 y Fo(+)438 1670 y FC(is)13 b(small)f(for)i(the)f(same)f(reason) i(that)g(all)f(\()p Fs(\001)p FC(\))1250 1652 y Fo(+)1293 1670 y FC(quan)o(tities)f(are)i(small)e(|)h(it)g(is)g(of)75 1730 y(order)j Ft(K)247 1712 y Fo(2)266 1730 y FC(.)21 b(In)16 b(Prop)q(osition)g (7.2,)g(replace)e(\000)i(b)o(y)g Ft(\015)1021 1712 y Fp(\000)p Fo(1)1068 1730 y FC(\000,)f(set)h Ft(\015)h FC(=)1296 1699 y Fs(p)p 1338 1699 26 2 v 31 x Ft(\021)r(;)8 b(\017)13 b FC(=)h Ft(\034)6 b(\017)1518 1737 y Fq(j)1551 1730 y FC(and)16 b Ft(K)i FC(=)c Ft(R)p FC(.)21 b(The)75 1790 y(result)15 b(is)g(a)h(b)q(ound)g(for)g Ft(R)558 1772 y Fo(+)603 1790 y FC(in)f(terms)f(of)h Ft(R)h FC(whic)o(h)f(w)o(e)g(estimate)e(using)j(Lemma)d(8.4)i(in)g(terms)f(of)75 1851 y Ft(K)20 b FC(obtaining:)i(there)16 b(is)g Ft(C)k FC(suc)o(h)c(that)g (for)h Ft(j)g Fs(\035)c FC(1)k(\(to)f(mak)o(e)f Fs(k)p Ft(K)t Fs(k)h FC(small\))634 1985 y Fs(k)p Ft(R)696 1964 y Fo(+)725 1985 y Fs(k)750 2009 y Fq(\034)t(\017)784 2014 y Fm(j)800 2009 y Fq(;\021)829 1993 y Fk(\000)858 1984 y Fl(1)p 858 1988 16 2 v 858 2006 a(2)880 2009 y Fo(\000)p Fq(;h)948 1985 y Fs(\024)e Ft(C)1048 1912 y Fr( )1116 1951 y FC(1)p 1086 1973 86 2 v 1086 2019 a Ft(\017)1106 2026 y Fq(j)1124 2019 y Ft(h)1152 2005 y Fo(2)1176 1985 y Fs(k)p Ft(K)t Fs(k)1271 1912 y Fr(!)1304 1923 y Fo(2)1332 1985 y Ft(;)442 b FC(\(8)p Ft(:)p FC(42\))75 2121 y(and)19 b(since)f Ft(\021)320 2103 y Fp(\000)352 2089 y Fl(1)p 353 2095 16 2 v 353 2116 a(2)375 2121 y FC(\000)h Fs(\025)e Ft(\021)506 2103 y Fp(\000)p Fo(1)553 2121 y FC(\000)j(this)e(is)h (also)g(a)g(b)q(ound)h(for)f(the)g(norm)f Fs(k)p Ft(R)1404 2103 y Fo(+)1434 2121 y Fs(k)1459 2130 y Fq(\034)t(\017)1493 2135 y Fm(j)1509 2130 y Fq(;\021)1538 2120 y Fk(\000)p Fl(1)1579 2130 y Fo(\000)p Fq(;h)1633 2121 y FC(.)29 b(Finally)l(,)17 b(for)75 2181 y Ft(j)g Fs(\035)c FC(1)k(w)o(e)f(ha)o(v)o(e)f Ft(\034)6 b(\017)447 2188 y Fq(j)479 2181 y Fs(\024)13 b Ft(\024)559 2188 y Fq(j)r Fo(+1)623 2181 y FC(,)j(so)h(this)f(is)g(a)g(b)q(ound)i(for)e Fs(k)p Ft(R)1184 2163 y Fo(+)1214 2181 y Fs(k)1239 2190 y Fq(\024)1259 2195 y Fm(j)q Fl(+1)1314 2190 y Fq(;\021)1343 2180 y Fk(\000)p Fl(1)1384 2190 y Fo(\000)p Fq(;h)1455 2181 y FC(as)h(required.)148 2242 y Fu(Piece)d(3:)19 b FC(This)13 b(is)f(small)f(b)q(ecause)i(it)f(also)h (is)g(of)g(order)g Ft(K)1216 2224 y Fo(2)1248 2242 y FC(since)f(it)g(is)h (formed)e(b)o(y)h(the)g(o)o(v)o(erlap)75 2302 y(of)19 b(t)o(w)o(o)f(or)g (more)f(p)q(olymers.)26 b(First)18 b(w)o(e)g(com)o(bine)e(\(8.42\))j(and)g (Lemma)d(8.4)i(to)h(get:)25 b(there)18 b(is)g Ft(C)75 2362 y FC(suc)o(h)e(that)599 2441 y Fs(k)p Ft(R)11 b FC(+)g Ft(R)758 2420 y Fo(+)788 2441 y Fs(k)813 2465 y Fq(\034)t(\017)847 2470 y Fm(j)863 2465 y Fq(;\021)892 2450 y Fk(\000)921 2440 y Fl(1)p 921 2444 V 921 2462 a(2)943 2465 y Fo(\000)p Fq(;h)1011 2441 y Fs(\024)j Ft(C)1111 2368 y Fr( )1179 2407 y FC(1)p 1148 2429 86 2 v 1148 2475 a Ft(\017)1168 2482 y Fq(j)1186 2475 y Ft(h)1214 2461 y Fo(2)1239 2441 y Fs(k)p Ft(K)t Fs(k)1334 2368 y Fr(!)1375 2441 y Ft(:)399 b FC(\(8)p Ft(:)p FC(43\))969 2599 y(83)p eop %%Page: 84 88 bop 148 307 a FC(No)o(w)16 b(w)o(e)f(estimate)g(\()p Ft(R)10 b FC(+)h Ft(R)678 289 y Fo(+)708 307 y FC(\))f Fs(_)h Ft(K)20 b FC(b)o(y)15 b(taking)i Ft(A)c FC(=)h Ft(R)d FC(+)f Ft(R)1295 289 y Fo(+)1325 307 y Ft(;)e(B)16 b FC(=)e Ft(K)q(;)8 b(g)1539 314 y Fq(A)1582 307 y FC(=)14 b Ft(g)1657 314 y Fq(\034)t(\017)1691 319 y Fm(j)1709 307 y Ft(;)8 b(g)1754 314 y Fq(B)1799 307 y FC(=)13 b Ft(g)1873 314 y Fq(\024)1893 319 y Fm(j)75 367 y FC(in)j(Prop)q(osition)h(7.34,)g(with)f Ft(\015)h FC(=)716 337 y Fs(p)p 757 337 26 2 v 757 367 a Ft(\021)h FC(and)f(\000)g(replaced)e(b) o(y)h Ft(\021)1227 349 y Fp(\000)p Fo(1)1274 367 y FC(\000.)22 b(The)16 b(result)g(is)487 462 y Fs(k)p FC(\()p Ft(R)c FC(+)f Ft(R)666 441 y Fo(+)696 462 y FC(\))g Fs(_)g Ft(K)t Fs(k)840 470 y Fq(\024)860 475 y Fm(j)q Fl(+1)915 470 y Fq(;\021)944 461 y Fk(\000)p Fl(1)985 470 y Fo(\000)p Fq(;h)570 542 y Fs(\024)681 501 y Fr(X)623 595 y Fq(N)r(;M)t Fp(\025)p Fo(\(1)p Fq(;)p Fo(1\))808 542 y Ft(C)847 522 y Fq(N)t Fo(+)p Fq(M)843 554 y(\015)945 542 y Fs(k)p Ft(R)h FC(+)f Ft(R)1105 522 y Fo(+)1135 542 y Fs(k)1160 522 y Fq(N)1160 570 y(\034)t(\017)1194 575 y Fm(j)1210 570 y Fq(;\021)1239 554 y Fk(\000)1267 544 y Fl(1)p 1267 548 16 2 v 1267 566 a(2)1290 570 y Fo(\000)p Fq(;h)1344 542 y Fs(k)p Ft(K)t Fs(k)1439 522 y Fq(M)1478 542 y Ft(;)296 b FC(\(8.44\))75 693 y(where)17 b(again)g(I)g(used)g Ft(\021)520 675 y Fp(\000)552 661 y Fl(1)p 552 667 V 552 688 a(2)575 693 y FC(\000)e Fs(\024)g FC(\000)j(in)e(the)h Ft(K)k FC(norm.)i(W)l(e)17 b(conclude)f(the)h(b)q(ound)h(\(8.39\))f(on)h(piece)75 754 y(3)f(using)f(\(8.43\))h(and)g Ft(j)g Fs(\035)c FC(1.)22 b(This)16 b(concludes)g(the)g(pro)q(of)i(of)e Ft(H)21 b FC(b)q(ound)c(\(8.39\).)148 814 y Fu(\(2\))24 b(Reblo)r(c)n(king.)35 b Fs(E)t Ft(K)28 b FC(=)22 b(\(1)15 b Fs(\000)g(L)p FC(\))p Ft(K)k FC(+)14 b Ft(H)26 b FC(as)c(in)f(\(8.35\).)37 b(F)l(rom)21 b(Prop)q(osition)h(8.5)g(w)o(e)75 874 y(estimate)14 b(the)i(small)f(set)h(part)h(of)f(\(1)c Fs(\000)f(L)p FC(\))p Ft(K)455 968 y Fs(k)p FC([\(1)g Fs(\000)g(L)p FC(\))p Ft(K)t FC(])710 975 y Fq(S)735 968 y Fs(k)760 977 y Fq(\024)780 982 y Fm(j)q Fl(+1)835 977 y Fq(;\021)864 967 y Fk(\000)p Fl(1)905 977 y Fo(\000)p Fq(;\013h)p 937 984 43 2 v 1024 968 a Fs(\024)41 b(k)p FC([\(1)11 b Fs(\000)g(L)p FC(\))p Ft(K)t FC(])1359 975 y Fq(S)1384 968 y Fs(k)1409 975 y Fq(\024)1429 980 y Fm(j)1446 975 y Fq(;)p Fo(\000)p Fq(;\013h)p 1488 982 V 1024 1047 a Fs(\024)41 b FC(16)p Ft(C)1187 1054 y Fo(3)1207 1047 y Ft(L)1240 1026 y Fp(\000)p Fo(2)p Fq(d)1306 1047 y Fs(k)p Ft(K)t Fs(k)p Ft(;)373 b FC(\(8.45\))75 1141 y(where)16 b Ft(\013)e FC(=)g(2)p Ft(L)370 1123 y Fp(\000)p Fq(d=)p Fo(2)453 1141 y FC(.)22 b(Using)16 b Ft(H)i FC(=)c Ft(O)q FC(\()p Ft(K)838 1123 y Fo(2)858 1141 y FC(\),)i(\(8.39\),)g(w)o(e)g(can)g(tak)o(e)g Ft(j)h Fs(\035)d FC(1)i(so)h(that)612 1235 y Fs(k)p FC([)p Fs(E)t Ft(K)t FC(])740 1242 y Fq(S)765 1235 y Fs(k)790 1243 y Fq(\024)810 1248 y Fm(j)q Fl(+1)865 1243 y Fq(;\021)894 1234 y Fk(\000)p Fl(1)935 1243 y Fo(\000)p Fq(;\013h)p 967 1250 V 1026 1235 a Fs(\024)d FC(17)p Ft(C)1162 1242 y Fo(3)1182 1235 y Ft(L)1215 1215 y Fp(\000)p Fo(2)p Fq(d)1281 1235 y Fs(k)p Ft(K)t Fs(k)412 b FC(\(8)p Ft(:)p FC(46\))75 1329 y(\(write)15 b([)p Fs(E)t Ft(K)t FC(])321 1336 y Fq(S)360 1329 y FC(=)f([\(1)d Fs(\000)f(L)p FC(\))p Ft(K)t FC(])641 1336 y Fq(S)677 1329 y FC(+)g([)p Ft(H)t FC(])797 1336 y Fq(S)838 1329 y FC(and)16 b(estimate)f(eac)o(h)g(small)f(set)i(part)h (separately)e(b)o(y)h(the)75 1390 y(triangle)i(inequalit)o(y\).)26 b(No)o(w)19 b(w)o(e)f(apply)g(the)h(reblo)q(c)o(king)e(Prop)q(osition)j(8.2)f (to)g(conclude,)f(again)75 1450 y(using)f Ft(H)h FC(=)13 b Ft(O)q FC(\()p Ft(K)414 1432 y Fo(2)435 1450 y FC(\),)j(that)g(for)h Ft(j)g Fs(\035)c FC(1,)589 1544 y Fs(kB)r(E)t Ft(K)t Fs(k)749 1551 y Fq(\024)769 1556 y Fm(j)q Fl(+1)824 1551 y Fq(;\021)q Fo(\000)p Fq(;\013h)p 885 1558 V 944 1544 a Fs(\024)g FC(\(34)p Ft(C)1098 1551 y Fo(3)1130 1544 y FC(+)e(2\))p Ft(L)1255 1523 y Fp(\000)p Fq(d)1303 1544 y Fs(k)p Ft(K)t Fs(k)390 b FC(\(8)p Ft(:)p FC(47\))75 1638 y(whic)o(h)16 b(b)o(y)f(the)h(inductiv)o(e)f(h)o(yp)q (othesis)h(is)g(less)g(than)932 1732 y Fs(\024)e FC(\()p Ft(C)1039 1739 y Fo(1)1058 1732 y Ft(L)1091 1712 y Fp(\000)p Fq(d)1139 1732 y FC(\))1158 1712 y Fo(\()p Fq(j)r Fo(+1\))1788 1732 y FC(\(8)p Ft(:)p FC(48\))75 1827 y(pro)o(vided)833 1887 y Ft(C)868 1894 y Fo(1)901 1887 y FC(=)g(34)p Ft(C)1036 1894 y Fo(3)1068 1887 y FC(+)d(2)p Ft(:)633 b FC(\(8)p Ft(:)p FC(49\))75 1967 y Fn(Thus,)17 b(the)h Ft(C)329 1974 y Fo(1)366 1967 y Fn(in)g(the)g (inductive)h(hyp)n(othesis)e(is)g(determine)n(d)h(by)g Ft(C)1350 1974 y Fo(3)1387 1967 y Fn(in)f(Pr)n(op)n(osition)f(8.5.)148 2028 y Fu(\(3\))24 b(Rescaling.)34 b FC(Letting)21 b Ft(x)h FC(=)f Ft(Lx)873 2009 y Fp(0)885 2028 y Ft(;)8 b(\036)p FC(\()p Ft(x)p FC(\))21 b(=)h Ft(L)1116 2009 y Fo(1)p Fp(\000)p Fq(d=)p Fo(2)1217 2028 y Ft(\036)1246 2009 y Fp(0)1257 2028 y FC(\()p Ft(x)1304 2009 y Fp(0)1315 2028 y FC(\))p Ft(;)8 b FC(\()p Fs(r)p Ft(\036)p FC(\)\()p Ft(x)p FC(\))21 b(=)h Ft(L)1645 2009 y Fp(\000)p Fq(d=)p Fo(2)1728 2028 y Fs(r)1770 2009 y Fp(0)1781 2028 y Ft(\036)1810 2009 y Fp(0)1822 2028 y FC(\()p Ft(x)1869 2009 y Fp(0)1880 2028 y FC(\),)75 2088 y(whic)o(h)16 b(implies)f Ft(h)410 2095 y Fp(r)p Fq(\036)478 2088 y FC(=)g Ft(L)564 2070 y Fq(d=)p Fo(2)620 2088 y Ft(h)648 2070 y Fp(0)648 2100 y(r)p Fq(\036)701 2088 y FC(,)i Ft(h)760 2095 y Fp(rr)p Fq(\036)857 2088 y FC(=)e Ft(L)943 2070 y Fq(d=)p Fo(2+1)1044 2088 y Ft(h)1072 2070 y Fp(0)1072 2100 y(rr)p Fq(\036)1154 2088 y FC(,)i Ft(:)8 b(:)g(:)17 b FC(and,)g(referring)f(to)i(the)f (de\014nition)75 2148 y(of)g Ft(g)h FC(in)e(\(8.6\))647 2208 y Ft(g)670 2215 y Fq(\024)690 2220 y Fm(j)q Fl(+1)745 2215 y Fq(;L)781 2208 y FC(\()p Ft(X)q(;)8 b(\036)p FC(\))14 b(=)g Ft(g)1000 2215 y Fq(\024)1020 2220 y Fm(j)q Fl(+1)1075 2215 y Fq(;)p Fo(1)1105 2208 y FC(\()p Ft(L)1157 2188 y Fp(\000)p Fo(1)1204 2208 y Ft(X)q(;)8 b(\036)1296 2188 y Fp(0)1308 2208 y FC(\))p Ft(:)447 b FC(\(8)p Ft(:)p FC(50\))75 2293 y(Th)o(us,)16 b(recalling)f Ft(\013)f FC(=)g(2)p Ft(L)561 2275 y Fp(\000)p Fq(d=)p Fo(2)661 2293 y FC(and)j(\(8.48\),)410 2394 y Fs(k)p Ft(K)480 2368 y Fo(\()p Fq(j)r Fo(+1\))476 2405 y(1)571 2394 y Fs(k)596 2401 y Fq(\024)616 2406 y Fm(j)q Fl(+1)671 2401 y Fq(;\021)q Fo(\000)p Fq(;)p Fo(2)p Fq(h)813 2394 y Fs(\021)41 b(kRB)r(E)t Ft(K)t Fs(k)1095 2401 y Fq(\024)1115 2406 y Fm(j)q Fl(+1)1170 2401 y Fq(;\021)q Fo(\000)p Fq(;)p Fo(2)p Fq(h)813 2473 y FC(=)h Fs(kB)r(E)t Ft(K)t Fs(k)1053 2480 y Fq(\024)1073 2485 y Fm(j)q Fl(+1)1128 2480 y Fq(;\021)q Fo(\000)p Fq(;\013h)p 1189 2487 V 1247 2473 a Fs(\024)14 b FC(\()p Ft(C)1354 2480 y Fo(1)1374 2473 y Ft(L)1407 2453 y Fp(\000)p Fq(d)1454 2473 y FC(\))1473 2453 y Fo(\()p Fq(j)r Fo(+1\))1564 2473 y Ft(:)210 b FC(\(8.51\))969 2599 y(84)p eop %%Page: 85 89 bop 148 307 a Fu(\(4\))18 b(In)n(tegrating)f(out.)k FC(First)15 b(w)o(e)g(c)o(ho)q(ose)h(the)f(deformation)g(of)h(co)o(v)m(ariance)f(that)g (w)o(e)h(w)o(an)o(t)75 367 y(to)f(use)f(in)g(Theorem)f(7.1.)21 b(W)l(e)14 b(w)o(an)o(t)g(to)h(deform)e(the)h(co)o(v)m(ariance)g(of)h Ft(d\026)1414 374 y Fo(1)p Fq(;)p Fp(R)p Fo(\012)1514 367 y FC(in)o(to)f(the)g(co)o(v)m(ariance)75 428 y(of)j Ft(d\026)185 439 y Fq(L;\033)240 429 y Fl(\()p Fm(j)q Fl(+1\))337 428 y FC(whic)o(h)f(is)849 510 y(^)838 523 y Ft(C)873 530 y Fq(L)912 523 y FC(=)969 489 y Ft(F)7 b FC(\()p Ft(L)1060 471 y Fo(2)1079 489 y Ft(k)1106 471 y Fo(2)1126 489 y FC(\))p 969 511 176 2 v 973 557 a Ft(\033)1003 542 y Fo(\()p Fq(j)r Fo(+1\))1094 557 y Ft(k)1121 542 y Fo(2)1788 523 y FC(\(8)p Ft(:)p FC(52\))75 622 y(\012)16 b(has)h(the)f(form)g(\(up)g(to)h(a)f Ft(\036)g FC(indep)q(enden)o(t)g(part\))521 739 y(\012\(\003)609 718 y Fo(\()p Fq(j)r Fo(\))655 739 y FC(\))e(=)770 697 y Fr(X)740 795 y Fq(Y)8 b Fp(\032)p Fo(\003)819 785 y Fl(\()p Fm(j)q Fl(\))869 739 y Ft(c)890 746 y Fq(\013)p 890 753 25 2 v 914 739 a FC(\()p Ft(Y)j FC(\))999 680 y Fr(Z)1023 774 y Fq(Y)1070 739 y Ft(dz)r(@)1149 718 y Fq(\013)1172 723 y Fl(1)1190 739 y Ft(\036)p FC(\()p Ft(z)r FC(\))p Ft(@)1311 718 y Fq(\013)1334 723 y Fl(2)1353 739 y Ft(\036)p FC(\()p Ft(z)r FC(\))p Ft(;)329 b FC(\(8)p Ft(:)p FC(53\))75 882 y(whic)o(h)13 b(is)h(a)g(translation)h(in)o(v)m(arian)o (t)e(functional)h(of)g Ft(\036)p FC(,)g(so)g(passing)h(to)f(the)g(F)l(ourier) f(transform)h(and)75 942 y(remem)n(b)q(ering)g(the)i(parit)o(y)f(assumptions) h(on)h Ft(K)k FC(there)15 b(are)h Ft(\016)r(\033)i FC(and)f Ft(!)h FC(suc)o(h)e(that)558 1063 y Fs(R)p FC(\012\(\003)688 1042 y Fo(\()p Fq(j)r Fo(\))734 1063 y FC(\))e(=)f Fs(\000)862 1029 y FC(1)p 862 1051 V 862 1097 a(2)900 1022 y Fr(X)920 1114 y Fq(k)960 1063 y Fs(f)p Ft(\016)r(\033)r(k)1066 1042 y Fo(2)1095 1063 y FC(+)e Ft(!)r FC(\()p Ft(k)r FC(\))p Fs(gj)1286 1050 y FC(^)1280 1063 y Ft(\036)p FC(\()p Ft(k)r FC(\))p Fs(j)1388 1042 y Fo(2)1408 1063 y Ft(;)366 b FC(\(8)p Ft(:)p FC(54\))75 1203 y(where)16 b(the)g(p)q(olynomial)f Ft(!)r FC(\()p Ft(k)r FC(\))f(=)g Ft(O)q FC(\(\()p Ft(k)818 1185 y Fo(2)838 1203 y FC(\))857 1185 y Fo(2)877 1203 y FC(\))i(as)h Ft(k)f Fs(!)e FC(0.)22 b(The)16 b(quadratic)g(parts)h(of)g(\012)f(and)h(conse-)75 1263 y(quen)o(tly)11 b Ft(!)j FC(are)f(formed)e(from)g(the)h(second)h(deriv)m (ativ)o(es)e(of)h Ft(K)t FC(.)20 b(Therefore)12 b(b)o(y)g(their)g (construction,)75 1324 y(c.f.)20 b(\(8.22\),)d(there)e(is)i Ft(C)i FC(suc)o(h)d(that)746 1450 y Fs(j)p Ft(@)789 1429 y Fq(\013)786 1462 y(k)813 1450 y Ft(!)r FC(\()p Ft(k)g FC(=)e(0\))p Fs(j)g(\024)1090 1416 y Ft(C)p 1085 1438 48 2 v 1085 1484 a(h)1113 1469 y Fo(2)1147 1450 y Fs(k)p Ft(K)t Fs(k)546 b FC(\(8)p Ft(:)p FC(55\))75 1564 y(and)845 1636 y Fs(j)p Ft(\016)r(\033)r Fs(j)13 b(\024)1002 1602 y Ft(C)p 997 1624 V 997 1670 a(h)1025 1656 y Fo(2)1059 1636 y Fs(k)p Ft(K)t Fs(k)p Ft(:)620 b FC(\(8)p Ft(:)p FC(56\))75 1733 y(By)16 b(de\014nition)f(the)h(co)o(v)m(ariance)g(of)h Ft(d\026)798 1740 y Fo(1)p Fq(;)p Fp(R)p Fo(\012)900 1733 y FC(is)488 1804 y Fr( )526 1843 y Ft(\033)556 1825 y Fo(\()p Fq(j)r Fo(\))601 1843 y Ft(k)628 1825 y Fo(2)p 525 1866 124 2 v 525 1911 a Ft(F)7 b FC(\()p Ft(k)610 1897 y Fo(2)630 1911 y FC(\))665 1877 y(+)k Ft(\016)r(\033)r(k)795 1857 y Fo(2)824 1877 y FC(+)g Ft(!)r FC(\()p Ft(k)r FC(\))970 1804 y Fr(!)1003 1814 y Fp(\000)p Fo(1)1064 1877 y FC(=)1251 1843 y Ft(F)c FC(\()p Ft(k)1336 1825 y Fo(2)1355 1843 y FC(\))p 1121 1866 383 2 v 1121 1911 a Ft(\033)1151 1897 y Fo(\()p Fq(j)r Fo(+1\))1241 1911 y Ft(k)1268 1897 y Fo(2)1288 1911 y FC([1)k(+)g Ft(G)p FC(\()p Ft(k)r FC(\)])1788 1877 y(\(8)p Ft(:)p FC(57\))75 2007 y(where)572 2109 y Ft(\033)602 2089 y Fo(\()p Fq(j)r Fo(+1\))733 2109 y FC(=)42 b Ft(\033)843 2089 y Fo(\()p Fq(j)r Fo(\))899 2109 y FC(+)11 b Ft(\016)r(\033)588 2201 y(G)p FC(\()p Ft(k)r FC(\))42 b(=)g Ft(\016)r(\033)r FC(\()p Ft(F)7 b FC(\()p Ft(k)971 2180 y Fo(2)989 2201 y FC(\))k Fs(\000)g FC(1\))g(+)1177 2167 y Ft(!)r FC(\()p Ft(k)r FC(\))p 1177 2189 98 2 v 1202 2235 a Ft(k)1229 2221 y Fo(2)1279 2201 y Ft(F)c FC(\()p Ft(k)1364 2180 y Fo(2)1383 2201 y FC(\))p Ft(:)372 b FC(\(8.58\))75 2315 y(De\014ne)16 b(the)g(deformation)f(to)i(b)q(e,)f(for)g(1)f Fs(\024)e Ft(`)i Fs(\024)e Ft(L)p FC(,)697 2431 y(^)686 2444 y Ft(C)725 2423 y Fp(0)721 2456 y Fq(`)752 2444 y FC(=)960 2410 y Ft(F)7 b FC(\()p Ft(`)1038 2392 y Fo(2)1057 2410 y Ft(k)1084 2392 y Fo(2)1104 2410 y FC(\))p 808 2432 467 2 v 808 2478 a Ft(\033)838 2464 y Fo(\()p Fq(j)r Fo(+1\))929 2478 y Ft(k)956 2464 y Fo(2)975 2478 y FC([1)k(+)g Ft(a)p FC(\()p Ft(`)p FC(\))p Ft(G)p FC(\()p Ft(k)r FC(\)])1279 2444 y Ft(;)495 b FC(\(8)p Ft(:)p FC(59\))969 2599 y(85)p eop %%Page: 86 90 bop 75 313 a FC(where)19 b Ft(a)p FC(\()p Ft(`)p FC(\))h(is)g(a)g(linear)e (function)i(suc)o(h)f(that)i Ft(a)p FC(\(1\))e(=)h(1)p Ft(;)8 b(a)p FC(\()p Ft(L)p FC(\))19 b(=)h(0,)g(so)h(that)1603 300 y(^)1592 313 y Ft(C)1631 295 y Fp(0)1627 325 y Fq(`)1663 313 y FC(in)o(terp)q(olates)75 373 y(b)q(et)o(w)o(een)277 360 y(^)266 373 y Ft(C)301 380 y Fq(L)345 373 y FC(and)e(the)g(co)o(v)m(ariance)f(of)h Ft(d\026)879 380 y Fo(1)p Fq(;)p Fp(R)p Fo(\012)965 373 y FC(.)28 b(The)19 b Fn(crucial)h(p)n(oint)e FC(is)h(that)g(the)f(co)q(e\016cien)o(t)f (of)75 433 y Ft(k)102 415 y Fo(2)142 433 y FC(in)j(the)g(denominator,)g (i.e.,)f(the)h(strength)g(of)h(the)f(singularit)o(y)l(,)g(do)q(es)g(not)h(c)o (hange)f(b)q(ecause)75 493 y Ft(G)p FC(\(0\))f(=)f(0.)29 b(Pro)o(vided)18 b Ft(\033)556 475 y Fo(\()p Fq(j)r Fo(+1\))665 493 y FC(remains)f(b)q(ounded) i(a)o(w)o(a)o(y)g(from)e(zero)i(uniformly)d(in)j Ft(j)i Fs(\035)d FC(1)h(this)75 554 y(deformation)c(has)i(the)f(follo)o(wing)g(prop)q(erties:) 21 b(for)c Ft(j)g Fs(\035)c FC(1,)121 651 y(\(i\))24 b(The)18 b(constan)o(ts)h Ft(A)f FC(and)h Ft(B)h FC(in)e(\(8.12\),)h(the)f(h)o(yp)q (othesis)g(for)h(the)e(Lemma)f(8.1)j(on)g Ft(g)1782 658 y Fq(\024)1804 651 y FC(,)g(will)197 711 y(b)q(e)d(b)q(ounded)h(b)o(y)f(a)h(constan)o(t)g Ft(C)t FC(\()p Ft(L)p FC(\))c(=)h Ft(C)20 b FC(since)15 b Ft(L)h FC(is)h(c)o(hosen;)108 811 y(\(ii\))23 b Fs(k)238 798 y FC(_)222 811 y Ft(C)261 793 y Fp(0)257 823 y Fq(`)273 811 y Fs(k)298 818 y Fq(\021)q Fo(\000)357 811 y FC(is)16 b(b)q(ounded)i(b)o(y)d(a)i (constan)o(t)g Ft(C)t FC(\()p Ft(L)p FC(\))c(=)h Ft(C)20 b FC(since)15 b Ft(L)i FC(is)f(c)o(hosen.)75 953 y(F)l(rom)i(\(8.56\))h(w)o(e)g (see)f(that)h(if)g Ft(\033)691 935 y Fo(\()p Fq(j)r Fo(\))754 953 y Fs(\025)f FC(1)c Fs(\000)905 934 y Fo(1)p 905 942 18 2 v 905 971 a(8)944 895 y(\()p Fq(j)r Fo(\))936 912 y Fr(X)957 1003 y Fo(0)1004 953 y FC(2)1028 933 y Fp(\000)p Fq(n)1080 953 y FC(,)19 b(then)g Ft(\033)1257 935 y Fo(\()p Fq(j)r Fo(+1\))1365 953 y Fs(\025)f FC(1)13 b Fs(\000)1516 934 y Fo(1)p 1516 942 V 1516 971 a(8)1547 895 y(\()p Fq(j)r Fo(+1\))1561 912 y Fr(X)1583 1003 y Fo(0)1644 953 y FC(2)1668 933 y Fp(\000)p Fq(n)1738 953 y Fn(which)21 b(is)75 1054 y(p)n(art)16 b(\(ii\))i(of)f(the)h(inductive)h (hyp)n(othesis)d Ft(I)868 1036 y Fo(\()p Fq(j)r Fo(+1\))958 1054 y FC(.)21 b(In)16 b(particular,)f Ft(\033)1322 1036 y Fo(\()p Fq(j)r Fo(\))1381 1054 y Fs(\025)1439 1034 y Fo(3)p 1439 1042 V 1439 1071 a(4)1478 1054 y FC(for)h(an)o(y)h Ft(j)f Fs(\035)e FC(1.)148 1114 y(No)o(w)21 b(w)o(e)g(apply)g(Theorem)e(7.1)j(with)f (the)g(c)o(hoices:)29 b(\()p Ft(h)17 b FC(in)f(Theorem)f(7.1\))22 b(=)g(2)p Ft(h)p FC(,)g Ft(h)1777 1096 y Fp(0)1811 1114 y FC(=)g Ft(h)p FC(,)75 1174 y(\(\000)17 b(in)f(Theorem)f(\(6.1\))f(=)f Ft(\021)r FC(\000,)k(\()p Ft(g)698 1181 y Fq(`)731 1174 y FC(in)f(Theorem)e (7.1)q(\))g(=)h(\026)-26 b Ft(g)1163 1181 y Fq(`)1180 1174 y FC(,)16 b(where)g(\(c.f.,)e(\(8.10\),)653 1279 y(\026)-26 b Ft(g)674 1286 y Fq(`)691 1279 y FC(\()p Ft(X)q(;)8 b(\036)p FC(\))14 b(=)g Ft(g)910 1286 y Fq(\024)930 1291 y Fm(j)q Fl(+1)985 1286 y Fq(;`)1012 1279 y FC(\()p Ft(X)q(;)8 b(\036)p FC(\))p Ft(`)1162 1258 y Fq(v)q Fo(\()p Fq(\024)1214 1263 y Fm(j)q Fl(+1)1270 1258 y Fp(j)p Fq(X)s Fp(j)1323 1279 y Ft(:)451 b FC(\(8)p Ft(:)p FC(60\))75 1383 y(By)16 b(Lemma)e(8.1,)i(this)g(satis\014es)h Ft(\026)722 1390 y Fq(`)736 1381 y Fk(0)747 1390 y Fq(`)775 1383 y Fs(\003)c FC(\026)-26 b Ft(g)834 1390 y Fq(`)865 1383 y Fs(\024)13 b Ft(g)940 1390 y Fq(`)954 1381 y Fk(0)985 1383 y FC(whic)o(h)i(is)h(one)h(h)o(yp)q(othesis)f(of)g(Theorem)f(7.1.)148 1443 y(Cho)q(ose)j Ft(j)f Fs(\035)c FC(1,)j(so)h(that)g(b)o(y)f(the)g(b)q (ound)h(\(8.51\))g(on)g Ft(K)1194 1418 y Fo(\()p Fq(j)r Fo(+1\))1190 1454 y(1)620 1577 y Fs(k)p Ft(K)690 1552 y Fo(\()p Fq(j)r Fo(+1\))686 1588 y(1)781 1577 y Fs(k)806 1584 y Fq(\024)826 1589 y Fm(j)q Fl(+1)881 1584 y Fq(;\021)q Fo(\000)p Fq(;)p Fo(2)p Fq(h)996 1577 y Fs(\024)1184 1544 y Ft(h)1212 1526 y Fo(2)p 1053 1566 310 2 v 1053 1616 a FC(16)1109 1581 y Fr(R)1138 1594 y Fq(L)1130 1629 y Fo(1)1172 1616 y Fs(k)1214 1604 y FC(_)1197 1616 y Ft(C)1236 1599 y Fp(0)1232 1629 y Fq(`)1248 1616 y Fs(k)1273 1623 y Fq(\021)q Fo(\000)1316 1616 y Ft(d`)1788 1577 y FC(\(8)p Ft(:)p FC(61\))75 1713 y(whic)o(h)e(is)h(the)g(other)g(h)o(yp)q(othesis)g(of)h(Theorem)d(7.1.) 22 b(Then)16 b(b)o(y)f(the)h(c)o(hoice)f(of)h Ft(\024)1579 1720 y Fo(0)1615 1713 y FC(in)g(P)o(arameters)75 1774 y(part)h(\(B\),)644 1841 y Fs(k)p Ft(K)714 1815 y Fo(\()p Fq(j)r Fo(+1\))710 1853 y Fq(L)805 1841 y Fs(k)830 1848 y Fq(\024)850 1853 y Fm(j)q Fl(+1)905 1848 y Fq(;)p Fo(\000)p Fq(;h)983 1841 y Fs(\024)c(k)p Ft(K)1105 1815 y Fo(\()p Fq(j)r Fo(+1\))1101 1853 y Fq(L)1196 1841 y Fs(k)1222 1848 y Fo(\026)-19 b Fq(g)1238 1854 y Fm(L)1261 1848 y Fq(;\021)q Fo(\000)p Fq(;h)1788 1841 y FC(\(8)p Ft(:)p FC(62\))75 1930 y(b)q(ecause)16 b(the)g(factor)g Ft(L)511 1912 y Fq(v)q Fo(\()p Fq(\024)p Fo(\))p Fp(j)p Fq(X)s Fp(j)646 1930 y FC(in)h(\026)-25 b Ft(g)18 b FC(is)d(cancelled)f(b)o(y)i(the)f(factor)h Ft(\021)1316 1912 y Fp(j)p Fq(X)s Fp(j)1385 1930 y FC(in)g Ft(\021)r FC(\000.)21 b(By)15 b(the)h(conclusion)75 1990 y(of)h(Theorem)d (7.1,)963 2050 y Fs(\024)f(k)p Ft(K)1081 2057 y Fo(1)1101 2050 y Fs(j)1115 2057 y Fq(\024)1135 2062 y Fm(j)q Fl(+1)1190 2057 y Fq(;\021)q Fo(\000)p Fq(;)p Fo(2)p Fq(h)1788 2050 y FC(\(8)p Ft(:)p FC(63\))75 2135 y(whic)o(h)j(b)o(y)f(the)h(b)q(ound)i(\(8.51\))f(is) 977 2275 y Fs(\024)1030 2215 y Fr(\022)1065 2242 y Ft(C)1100 2249 y Fo(1)p 1065 2264 55 2 v 1066 2310 a Ft(L)1099 2295 y Fq(d)1125 2215 y Fr(\023)1155 2221 y Fo(\()p Fq(j)r Fo(+1\))1255 2275 y Ft(:)519 b FC(\(8)p Ft(:)p FC(64\))75 2404 y(Th)o(us,)19 b(w)o(e)e(ha)o(v)o(e)h(pro)o(v)o(ed)f(the)h(Inductiv)o(e)e(Assumption)h Ft(I)1164 2386 y Fo(\()p Fq(j)r Fo(+1\))1273 2404 y FC(and)i(the)f(pro)q(of)h (of)f(Theorem)f(8.3)75 2464 y(is)f(complete.)1545 b Fg(2)969 2599 y FC(86)p eop %%Page: 87 91 bop 75 315 a Fv(References)99 425 y FC([1])24 b(A.)18 b(Alastuey)f(and)i(Ph.) f(A.)f(Martin.)27 b(Absence)17 b(of)h(exp)q(onen)o(tial)g(clustering)f(for)i (the)f(static)175 485 y(quan)o(tum)d(correlations)h(and)g(the)g (time-displaced)e(correlations)h(in)h(c)o(harged)g(\015uids.)21 b Fn(Eur)n(o-)175 545 y(phys.)c(L)n(ett.)p FC(,)f(6:385{390,)i(1988.)99 647 y([2])24 b(A.)c(Alastuey)g(and)h(Ph.)f(A.)g(Martin.)34 b(Absence)19 b(of)i(exp)q(onen)o(tial)f(clustering)g(in)g(quan)o(tum)175 707 y(Coulom)o(b)c(\015uids.)21 b Fn(Phys.)c(R)n(ev.)g(A)p FC(,)f(40:6485{6520)q(,)j(1989.)99 809 y([3])24 b(T.)h(Balaban.)47 b(Large)26 b(\014eld)e(renormalization,)h(I)q(I.)f(Lo)q(calization,)j(exp)q (onen)o(tiation)e(and)175 869 y(b)q(ounds)d(for)f(the)g(R)f(op)q(eration.)35 b Fn(Comm.)21 b(in)h(Math.)f(Physics)p FC(,)g(122:355{392)q(,)j(1989.)36 b(See)175 929 y(also)17 b(previous)f(pap)q(ers.)99 1031 y([4])24 b(G.)d(Battle)f(and)i(P)l(.)e(F)l(ederbush.)35 b(A)20 b(phase)i(cell)d (cluster)h(expansion)i(for)f(Euclidean)f(\014eld)175 1091 y(theories.)h Fn(A)o(nnals)d(of)g(Physics)p FC(,)d(142:95{139,)k(1982.)99 1193 y([5])24 b(G.)18 b(Battle)f(and)i(P)l(.)f(F)l(ederbush.)26 b(A)17 b(note)h(on)h(cluster)e(expansions,)i(tree)e(graph)i(iden)o(tities,) 175 1253 y(extra)318 1234 y Fo(1)p 306 1242 42 2 v 306 1270 a Fq(N)t Fo(!)369 1253 y FC(factors!!!)26 b Fn(L)n(etters)17 b(in)h(Mathematic)n(al)g(Physics)p FC(,)d(8:55{57,)j(1984.)99 1355 y([6])24 b(G.)f(Benfatto.)41 b(An)23 b(iterated)f(Ma)o(y)o(er)g (expansion)h(for)g(the)g(Yuk)m(a)o(w)o(a)g(gas.)42 b Fn(J.)24 b(Statistic)n(al)175 1415 y(Physics)p FC(,)16 b(41:671{684,)j(1985.)99 1517 y([7])24 b(G.)17 b(Benfatto,)f(N.)g(Cassandro,)j(G.)d(Galla)o(v)o(otti,) g(F.)g(Nicolo,)g(E.)g(Olivieri,)e(E.)j(Presutti,)f(and)175 1577 y(E.)21 b(Scacciatelli.)34 b(Some)20 b(probabilistic)g(tec)o(hniques)g (in)h(\014eld)f(theory)l(.)36 b Fn(Comm.)22 b(in)g(Math.)175 1637 y(Physics)p FC(,)16 b(59:143{166,)j(1978.)99 1739 y([8])24 b(G.)16 b(Benfatto,)g(G.)g(Galla)o(v)o(otti,)e(and)j(F.)e(Nicolo.)20 b(On)c(the)g(massiv)o(e)e(Sine-Gordon)j(theory)f(in)175 1799 y(the)g(\014rst)h(few)f(regions)g(of)h(collapse.)k Fn(Comm.)c(in)g(Math.)g (Physics)p FC(,)f(83:387{410,)j(1982.)99 1901 y([9])24 b(D.)f(C.)f(Brydges.) 40 b(A)22 b(short)h(course)g(on)g(cluster)f(expansions.)41 b(In)22 b(K.)g(Osterw)o(alder)g(and)175 1961 y(R.)c(Stora,)i(editors,)f Fn(Critic)n(al)g(Phenomena,)j(R)n(andom)d(Systems,)h(Gauge)g(The)n(ories)p FC(,)f(pages)175 2021 y(131{183,)i(Amsterdam,)15 b(New)i(Y)l(ork,)g(1984.)i (Les)f(Houc)o(hes)f(Summer)e(Sc)o(ho)q(ol,)j(North)g(Hol-)175 2081 y(land.)75 2183 y([10])24 b(D.)d(C.)h(Brydges)f(and)h(Seiler)e(E.)37 b(Absence)20 b(of)i(screening)f(in)g(certain)f(lattice)h(gauge)h(and)175 2243 y(plasma)16 b(mo)q(dels.)k Fn(J.)d(Statistic)n(al)h(Physics)p FC(,)e(42:405{424,)j(1986.)75 2345 y([11])24 b(D.)16 b(C.)g(Brydges)g(and)h (P)l(.)f(F)l(ederbush.)21 b(A)15 b(new)h(form)g(of)g(the)g(Ma)o(y)o(er)f (expansion)h(in)g(classical)175 2405 y(statistical)g(mec)o(hanics.)j Fn(J.)e(Math.)g(Phys.)p FC(,)e(19:2064{206)q(7,)k(1978.)969 2599 y(87)p eop %%Page: 88 92 bop 75 307 a FC([12])24 b(D.)e(C.)g(Brydges)g(and)h(P)l(.)f(F)l(ederbush.)38 b(Deb)o(y)o(e)21 b(screening.)38 b Fn(Comm.)22 b(in)i(Math.)e(Physics)p FC(,)175 367 y(73:197{246,)d(1980.)75 468 y([13])24 b(D.)18 b(C.)g(Brydges)f(and)i(P)l(.)e(F)l(ederbush.)26 b(Deb)o(y)o(e)17 b(screening)g(in)h(classical)f(Coulom)o(b)g(systems.)175 528 y(In)g(G.)f(V)l(elo)g(and)h(A.S.)e(Wigh)o(tman,)g(editors,)i Fn(R)o(igor)n(ous)f(A)o(tomic)i(and)g(Mole)n(cular)g(Physics)p FC(,)175 588 y(pages)f(371{439,)i(Oxford,)d(1981.)h(Erice)e(Summer)f(Sc)o(ho) q(ol,)i(Plen)o(um)e(Press.)75 688 y([14])24 b(D.)f(C.)g(Brydges,)g(J.)g(F)l (r\177)-24 b(ohlic)o(h,)23 b(and)g(T.)g(Sp)q(encer.)40 b(The)23 b(random)g(w)o(alk)f(represen)o(tation)175 749 y(of)f(classical)f(spin)g (systems)g(and)h(correlation)f(inequalities.)32 b Fn(Comm.)20 b(in)i(Math.)f(Physics)p FC(,)175 809 y(83:123{150,)e(1982.)75 909 y([15])24 b(D.)12 b(C.)h(Brydges)e(and)i(G.)g(Keller.)g(Deb)o(y)o(e)e (screening)g(in)h(quan)o(tum)f(Coulom)o(b)h(systems.)h(W)l(ork)175 969 y(in)j(progress.)75 1070 y([16])24 b(D.)h(C.)f(Brydges)h(and)g(T.)g (Kennedy)l(.)45 b(Con)o(v)o(ergence)24 b(of)h(m)o(ultiscale)d(Ma)o(y)o(er)h (series.)46 b(In)175 1130 y(M.)11 b(Mebkhout)g(and)i(R.)e(S)o(\023)-23 b(en)o(\023)g(eor,)12 b(editors,)g Fn(VIII)g(International)j(Congr)n(ess)e (on)g(Mathematic)n(al)175 1190 y(Physics)p FC(,)j(pages)h(311{317,)h(Singap)q (ore,)f(1986.)h(W)l(orld)e(Scien)o(ti\014c)e(Publishing)i(Co.)75 1290 y([17])24 b(D.)15 b(C.)f(Brydges)g(and)i(T.)e(Kennedy)l(.)k(Ma)o(y)o(er) 13 b(expansions)i(and)h(the)e(Hamilton-Jacobi)f(equa-)175 1350 y(tion.)21 b Fn(J.)c(Statistic)n(al)i(Physics)p FC(,)c(48:19{49,)j(1987.)75 1451 y([18])24 b(D.)19 b(C.)f(Brydges)g(and)h(H.T.)f(Y)l(au.)28 b(Grad)19 b Ft(\036)f FC(p)q(erturbations)i(of)f(massless)e(Gaussian)j (\014elds.)175 1511 y Fn(Comm.)d(in)h(Math.)f(Physics)p FC(,)f(129:351{392,)j (1990.)75 1611 y([19])24 b(P)l(.A.F)11 b(da)h(V)l(eiga.)i Fn(Construction)g (de)g(mo)n(deles)g(non-r)n(enormalisables)h(en)g(th)o(\023)-24 b(eorie)14 b(quantique)175 1672 y(des)k(champs)p FC(.)j(PhD)c(thesis,)e(Univ) o(ersit)o(\023)-23 b(e)14 b(de)i(P)o(aris-Sud)h(Cen)o(tre)f(Orsa)o(y)l(,)f (1991.)75 1772 y([20])24 b(P)l(.)18 b(G.)f(De)g(Gennes.)26 b(Exp)q(onen)o(ts)18 b(for)g(the)f(excluded)f(v)o(olume)f(problem)h(as)j (deriv)o(ed)d(b)o(y)h(the)175 1832 y(Wilson)f(metho)q(d.)21 b Fn(Physics)c(L)n(ett.)p FC(,)f(38A:339{340,)i(1972.)75 1932 y([21])24 b(E.B.)f(Dynkin.)42 b(Mark)o(o)o(v)22 b(pro)q(cesses)i(as)h(a)e(to) q(ol)h(in)g(\014eld)f(theory)l(.)42 b Fn(J.)24 b(F)l(unct.)h(A)o(nalysis)p FC(,)175 1993 y(50:167{187,)19 b(1983.)75 2093 y([22])24 b(E.B.)18 b(Dynkin.)30 b(P)o(olynomials)18 b(of)i(the)f(o)q(ccupation)h(\014eld)e(and)i (related)f(random)g(\014elds.)30 b Fn(J.)175 2153 y(F)l(unct.)19 b(A)o(nalysis)p FC(,)d(58:20{52,)i(1984.)75 2253 y([23])24 b(P)l(.)13 b(F)l(ederbush.)i(A)e(note)g(on)g(the)g(ultimate)d(tree)j(graph)g (inequalit)o(y)l(.)h(Univ)o(ersit)o(y)c(of)j(Mic)o(higan)175 2314 y(preprin)o(t,)i(1990.)75 2414 y([24])24 b(P)l(.)19 b(F)l(ederbush.)30 b(A)19 b(phase)g(cell)f(approac)o(h)i(to)g(Yang)f(Mills)f(theory)h(V.)30 b Fn(Comm.)19 b(in)i(Math.)175 2474 y(Physics)p FC(,)16 b(127:433{457,)j (1990.)k(See)16 b(also)g(previous)g(pap)q(ers.)969 2599 y(88)p eop %%Page: 89 93 bop 75 307 a FC([25])24 b(J.)18 b(F)l(eldman,)e(V.)i(Magnen,)g(J.and)h(Riv)m (asseau,)f(and)h(R.)f(Seneor.)27 b(A)17 b(renormalizable)f(\014eld)175 367 y(theory:)36 b(the)23 b(massiv)o(e)e(Gross)k(Nev)o(eu)d(mo)q(del)g(in)h (t)o(w)o(o)g(dimensions.)41 b Fn(Comm.)24 b(in)g(Math.)175 428 y(Physics)p FC(,)16 b(103:67{103,)j(1986.)75 526 y([26])24 b(J.)16 b(F)l(eldman,)d(V.)i(Magnen,)h(J.and)g(Riv)m(asseau,)g(and)h(R.)e (Seneor.)20 b(Construction)d(and)f(Borel)175 586 y(summabilit)o(y)c(of)j (infrared)g Ft(\036)719 568 y Fo(4)719 599 y(4)754 586 y FC(b)o(y)g(a)h (phase)g(space)f(expansion.)20 b Fn(Comm.)c(in)h(Math.)f(Physics)p FC(,)175 646 y(109:437{480)q(,)i(1987.)75 745 y([27])24 b(R.)15 b(F)l(ern\023)-24 b(andez,)14 b(J.)h(F)l(r\177)-24 b(ohlic)o(h,)13 b(and)j(A.)e(Sok)m(al.)20 b Fn(R)n(andom)15 b(Walks,)i(Critic)n(al)f (Phenomena)h(and)175 805 y(T)l(riviality)h(in)g(Quantum)g(Field)g(The)n(ory)p FC(.)i(Springer-V)l(erlag,)15 b(New)h(Y)l(ork,)g(1992.)75 904 y([28])24 b(J.)16 b(F)l(r\177)-24 b(ohlic)o(h)14 b(and)j(Y.M.)d(P)o(ark.)20 b(Correlation)c(inequalities)e(and)j(the)e(thermo)q(dynamic)e(limit)175 964 y(for)k(classical)g(and)g(quan)o(tum)f(con)o(tin)o(uous)h(systems.)22 b Fn(Comm.)17 b(in)h(Math.)g(Physics)p FC(,)f(59:235{)175 1024 y(266,)g(1978.)75 1122 y([29])24 b(G.)d(Galla)o(v)o(otti.)34 b(Renormalization)19 b(group.)36 b(T)l(ec)o(hnical)19 b(rep)q(ort,)j(T)l (roisi)o(\022)-23 b(eme)19 b(Cycle)h(de)h(la)175 1183 y(Ph)o(ysique)15 b(en)i(Suisse)f(Romande,)f(1990.)75 1281 y([30])24 b(K.)17 b(Ga)o(w)o(edzki)g(and)h(A.)f(Kupiainen.)25 b(Blo)q(c)o(k)16 b(spin)i(renormalization)e(group)i(for)g(dip)q(ole)f(gas)175 1341 y(and)g(\()p Fs(5)p Ft(\036)p FC(\))382 1323 y Fo(4)401 1341 y FC(.)k Fn(A)o(nn.)d(Phys.)p FC(,)d(147:198,)j(1983.)75 1440 y([31])24 b(K.)16 b(Ga)o(w)o(edzki)f(and)i(A.)f(Kupiainen.)k(Massless)d (lattice)e Ft(\036)1256 1422 y Fo(4)1256 1452 y(4)1292 1440 y FC(theory:)21 b(Rigorous)c(con)o(trol)f(of)h(a)175 1500 y(renormalizable)f (asymptotically)f(free)i(mo)q(del.)25 b Fn(Comm.)18 b(in)h(Math.)f(Physics)p FC(,)g(99:197{252,)175 1560 y(1985.)75 1659 y([32])24 b(K.)14 b(Ga)o(w)o(edzki)f(and)i(A.)f(Kupiainen.)k(Asymptotic)12 b(freedom)h(b)q(ey)o (ond)h(p)q(erturbation)h(theory)l(.)175 1719 y(In)k(K.)f(Osterw)o(alder)g (and)h(R.)f(Stora,)i(editors,)f Fn(Critic)n(al)g(Phenomena,)j(R)n(andom)c (Systems,)175 1779 y(Gauge)g(The)n(ories)p FC(.)e(Les)g(Houc)o(hes,)f(North)i (Holland,)e(1986.)75 1878 y([33])24 b(I.M.)18 b(Gel'fand,)g(G.E.)h(Shilo)o (v,)f(N.)g(Y)l(a.)g(Vilenkin,)f(and)j(M.I.)d(Graev.)29 b Fn(Gener)n(alise)n (d)20 b(F)l(unc-)175 1938 y(tions)p FC(,)c(v)o(olume)e(4.)22 b(Academic)14 b(Press,)i(New)f(Y)l(ork,)h(1964{1968.)75 2036 y([34])24 b(J.)14 b(Ginibre.)j(Some)c(applications)i(of)f(functional)g(in)o (tegration)g(in)g(statistical)g(mec)o(hanics.)i(In)175 2096 y(C.)c(DeWitt)g(and)g(R.)g(Stora,)h(editors,)g Fn(Statistic)n(al)h(Me)n (chanics)g(and)g(Quantum)h(Field)f(The)n(ory)p FC(.)175 2157 y(Les)j(Houc)o(hes,)e(Gordon)j(and)e(Breac)o(h,)f(1971.)75 2255 y([35])24 b(J.)14 b(Glimm,)e(A.)h(Ja\013e,)i(and)g(T.)g(Sp)q(encer.)j (The)c(Wigh)o(tman)g(axioms)f(and)i(particle)f(structure)175 2315 y(in)i(the)g Ft(P)7 b FC(\()p Ft(\036)p FC(\))421 2322 y Fo(2)457 2315 y FC(quan)o(tum)15 b(\014eld)h(mo)q(del.)k Fn(A)o(nn.)e(Math.)p FC(,)d(100:585,)j(1974.)75 2414 y([36])24 b(M.)18 b(G\177)-24 b(opfert)18 b(and)h(G.)f(Mac)o(k.)26 b(Iterated)17 b(Ma)o(y)o(er)g(expansions)i(for)f(classical)f(gases.)28 b Fn(Comm.)175 2474 y(in)18 b(Math.)f(Physics)p FC(,)f(81:97{126,)i(1981.)969 2599 y(89)p eop %%Page: 90 94 bop 75 307 a FC([37])24 b(C.)15 b(Grub)q(er)g(and)g(H.)f(Kunz.)k(General)d (prop)q(erties)g(of)g(p)q(olymer)e(systems.)k Fn(Comm.)f(in)g(Math.)175 367 y(Physics)p FC(,)g(22:133{161,)j(1971.)75 468 y([38])24 b(J.)c(Im)o(brie.)31 b(Deb)o(y)o(e)19 b(screening)h(for)g(Jellium)e(and)j (other)f(Coulom)o(b)f(systems.)33 b Fn(Comm.)20 b(in)175 528 y(Math.)d(Physics)p FC(,)f(87:515{565,)j(1983.)75 628 y([39])24 b(J.)18 b(Im)o(brie.)23 b(Iterated)17 b(Ma)o(y)o(er)g(expansions)i(and)f (their)f(application)h(to)g(Coulom)o(b)f(systems.)175 688 y(In)h(J.)g(F)l (r\177)-24 b(ohlic)o(h,)17 b(editor,)h Fn(Sc)n(aling)j(and)e(Self)h (Similarity)g(in)f(Physics.)g(R)n(enormalization)g(in)175 749 y(Statistic)n(al)g(Me)n(chanics)f(and)f(Dynamics)p FC(.)f(Birkh\177)-24 b(auser,)15 b(Boston,)h(1983.)75 849 y([40])24 b(J.)18 b(Im)o(brie.)24 b(Renormalization)17 b(group)i(metho)q(ds)e(in)h(gauge)h(theories.)27 b(In)18 b(K.)f(Osterw)o(alder)175 909 y(and)24 b(R.)e(Stora,)j(editors,)e Fn(Critic)n(al)h(Phenomena,)i(R)n(andom)c(Systems,)j(Gauge)g(The)n(ories)p FC(,)175 969 y(pages)16 b(297{358,)i(Amsterdam,)12 b(New)j(Y)l(ork,)f(1984.)j (Les)e(Houc)o(hes)g(Summer)d(Sc)o(ho)q(ol,)k(North)175 1029 y(Holland.)75 1130 y([41])24 b(Dimo)q(c)o(k)15 b(J.)h(and)h(T.)g(R.)f(Hurd.) 22 b(A)16 b(renormalisation)f(group)j(analysis)e(of)h(correlation)f(func-)175 1190 y(tions)h(for)f(the)g(Dip)q(ole)g(gas.)23 b(McMaster)15 b(Univ)o(ersit)o(y)f(preprin)o(t.)75 1290 y([42])24 b(Dimo)q(c)o(k)18 b(J.)h(and)h(T.)g(R.)f(Hurd.)30 b(A)20 b(renormalisation)e(group)i(analysis)g (of)g(the)f(Kosterlitz-)175 1350 y(Thouless)e(phase.)22 b(McMaster)15 b(Univ)o(ersit)o(y)f(preprin)o(t.)75 1451 y([43])24 b(Dimo)q(c)o(k)c(J.)i (and)g(T.)g(R.)f(Hurd.)38 b(Construction)23 b(of)f(the)g(t)o(w)o(o)f (dimensional)g(Sine-Gordon)175 1511 y(mo)q(del)15 b(for)i Ft(\014)f(<)e FC(8)p Ft(\031)r FC(.)21 b(McMaster)15 b(Univ)o(ersit)o(y)f(Preprin)o(t,)h (1992.)75 1611 y([44])24 b(M.)17 b(Kac.)26 b(On)18 b(the)f(partition)h (function)g(of)g(one)g(dimensional)e(gas.)26 b Fn(Phys.)19 b(Fluids)p FC(,)e(2:8{12,)175 1672 y(1959.)75 1772 y([45])24 b(G.)12 b(Mac)o(k)g(and)h(A.)e(P)o(ordt.)16 b(Con)o(v)o(ergen)o(t)11 b(p)q(erturbation)i(expansions)g(in)f(Euclidean)g(quan)o(tum)175 1832 y(\014eld)k(theory)l(.)21 b Fn(Comm.)c(in)h(Math.)f(Physics)p FC(,)e(97:267{298,)k(1985.)75 1932 y([46])24 b(G.)f(Mac)o(k)f(and)i(A.)e(P)o (ordt.)41 b(Con)o(v)o(ergen)o(t)22 b(w)o(eak)h(coupling)g(expansions)g(for)g (lattice)f(\014eld)175 1993 y(theories)16 b(that)h(lo)q(ok)f(lik)o(e)f(p)q (erturbation)i(series.)j Fn(R)n(ev.)e(in)f(Math.)g(Phys.)p FC(,)f(1:47{87,)h(1989.)75 2093 y([47])24 b(A.C.)c(Maggs)h(and)g(N.W.)e (Ashcroft.)34 b(Electronic)19 b(\015uctuation)i(and)g(cohesion)f(in)g (metals.)175 2153 y Fn(Phys.)d(R)n(ev.)g(L)n(ett.)p FC(,)f(59:113{116,)j (1987.)75 2253 y([48])24 b(V.A.)19 b(Malyshev.)32 b(Uniform)19 b(cluster)g(estimates)g(for)h(lattice)g(mo)q(dels.)32 b Fn(Comm.)20 b(in)i(Math.)175 2314 y(Physics)p FC(,)16 b(64:131{157,)j(1979.)75 2414 y([49])24 b(A.J.)16 b(McKane.)22 b(Reform)o(ulation)15 b(of)i Ft(n)e Fs(!)g FC(0)i(mo)q(dels)f(using)h(an)o(ticomm)n(uting)d(scalar) j(\014elds.)175 2474 y Fn(Physics)h(L)n(ett.)f(A)p FC(,)f(76:22,)h(1980.)969 2599 y(90)p eop %%Page: 91 95 bop 75 307 a FC([50])24 b(F.)15 b(Nicolo,)f(J.)g(Renn,)h(and)h(A.)e (Steinman.)j(On)e(the)g(massiv)o(e)e(Sine-Gordon)j(equation)f(in)g(all)175 367 y(regions)i(of)f(collapse.)21 b Fn(Comm.)c(in)h(Math.)f(Physics)p FC(,)e(105:291{326)q(,)k(1986.)75 469 y([51])24 b(G.)18 b(P)o(arisi)f(and)i (N.)e(Sourlas.)26 b(Self-a)o(v)o(oiding)17 b(w)o(alk)g(and)i(sup)q (ersymmetry)l(.)j Fn(J.)d(Phys.)f(L)n(ett.)p FC(,)175 529 y(41:1403{140)q(6,) h(1980.)75 631 y([52])24 b(O.)c(P)o(enrose.)33 b(Con)o(v)o(ergence)20 b(of)g(fugacit)o(y)g(expansions)h(for)f(classical)g(systems.)32 b(In)20 b(T.)g(A.)175 691 y(Bak,)d(editor,)g Fn(Statistic)n(al)i(Me)n (chanics,)h(F)l(oundations)f(and)f(Applic)n(ations)p FC(,)g(pages)g(101{109,) 175 751 y(New)e(Y)l(ork,)f(1967.)j(I.U.P)l(.A.P)l(.)13 b(Meeting)i(\(Cop)q (enhagen)j(1966\),)f(Benjamin)d(Inc.)75 853 y([53])24 b(J.)f(P)o(olc)o (hinski.)41 b(Renormalization)21 b(and)j(e\013ectiv)o(e)e(Lagrangians.)44 b Fn(Nucle)n(ar)25 b(Physics)f(B)p FC(,)175 913 y(231:269,)18 b(1984.)75 1015 y([54])24 b(A.)17 b(P)o(ordt.)26 b(Con)o(v)o(ergen)o(t)17 b(m)o(ultigrid)e(p)q(olymer)h(expansions)i(and)g(renormalisation)f(for)h(Eu-) 175 1075 y(clidean)e(\014eld)f(theory)l(.)21 b(DESY)c(preprin)o(t,)d(1992.)75 1177 y([55])24 b(G.)f(C.)f(Rota.)42 b(On)22 b(the)h(foundations)h(of)f(com)o (binatorial)e(theory)i(I.)f(Theory)g(of)h(Mobius)175 1237 y(functions.)e Fn(Wahrsch.)c(Z.)g(V)l(erw.)i(Geb.)p FC(,)d(2:340{368,)i(1964.)75 1339 y([56])24 b(D.)16 b(Ruelle.)k Fn(Statistic)n(al)e(Me)n(chanics)p FC(.)k(W.A.)15 b(Benjamin)f(Inc.,)h(London,)i(1969.)75 1440 y([57])24 b(A.J.F.)15 b(Siegert.)21 b(P)o(artition)16 b(functions)g(as)h(a)o (v)o(erages)f(of)h(functionals)g(of)f(Gaussian)i(random)175 1501 y(functions.)j Fn(Physic)n(a)p FC(,)16 b(26:530{535,)i(1960.)75 1602 y([58])24 b(K.)f(Symanzik.)42 b(Euclidean)23 b(quan)o(tum)f(theory)l(.) 43 b(In)24 b(R.)f(Jost,)j(editor,)e Fn(L)n(o)n(c)n(al)f(Quantum)175 1663 y(The)n(ory)p FC(.)15 b(Academic)e(Press,)k(New)e(Y)l(ork,)h(London,)h (1969.)75 1764 y([59])24 b(S.R.S.)d(V)l(aradhan.)41 b Fn(Sto)n(chastic)24 b(Pr)n(o)n(c)n(esses)p FC(.)39 b(Couran)o(t)23 b(Institute)f(Lecture)g (Notes,)h(New)175 1824 y(Y)l(ork,)16 b(1968.)75 1926 y([60])24 b(K.)13 b(G.)g(Wilson)g(and)g(J.)g(Kogut.)k(The)c(renormalization)e(group)j (and)g(the)f Ft(\017)g FC(expansion.)j Fn(Phys.)175 1986 y(R)n(ep.)h(\(Se)n (ct)h(C)g(of)f(Phys)g(L)n(ett.\))p FC(,)e(12:75{200,)k(1974.)969 2599 y(91)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ENDBODY