This is a set of macros for the paper.

%%%
%%%   FTMACROS
%%%   Version JF 30/11/92
%%%
\magnification1200
\tolerance=10000
\hsize=17truecm\vsize=23truecm
\parindent=40pt
\multiply\baselineskip by 15\divide \baselineskip by 10
\def\today{\ifcase\month\or January\or February\or March\or April\or
     May\or June\or July\or August\or September\or October\or November\or
     December\fi\space\number\day, \number\year}
\def\header#1{\rm\nopagenumbers \hfil\underbar{#1} \hfil\today\bigskip
     \headline={\rm\ifnum\pageno>1 #1\hfil\today\hfil Page \folio\else\hfil\fi}}
%
%
%
\def\dst{\displaystyle}
\def\sty{\scriptstyle}
\def\sst{\scriptstyle}
\def\tst{\textstyle}     
\def\leftdisplay#1{\leftline{$\displaystyle{#1}$}}
\def\leftrightdisplay#1#2{\line{$\displaystyle{#1}$\hfil$\displaystyle{#2}$}} 
\def\frac#1#2{\dst {#1\over#2}}     % fractions in displaystyle
\def\sfrac#1#2{{\tst{#1\over#2}}}   % fractions in textstyle    
\def\pmb#1{\setbox0=\hbox{#1}       % generate bold face
     \kern-.025em\copy0\kern-\wd0
     \kern.05em\copy0\kern-\wd0
     \kern-.025em\box0}%Knuth puts in \raise.0433em before box0  
\def\shbox#1{\vbox{\hrule\hbox{\vrule\vbox{#1}\vrule}\hrule}}   % box it
%
%  Greek letters
%
\def\al{\alpha}
\def\be{\beta}
\def\ga{\gamma}
\def\de{\delta}
\def\ep{\epsilon}
\def\ze{\zeta}
\def\et{\eta}
\def\th{\theta}
\def\ka{\kappa}
\def\la{\lambda}
\def\rh{\rho}
\def\si{\sigma}
\def\ta{\tau}
\def\up{\upsilon}
\def\ph{\phi}
\def\ch{\chi}
\def\ps{\psi}
\def\om{\omega}
\def\Al{\Alpha}
\def\Be{\Beta}
\def\Ga{\Gamma}
\def\De{\Delta}
\def\Ze{\Zeta}
\def\Et{\Eta}
\def\Th{\Theta}
\def\Ka{\Kappa}
\def\La{\Lambda}
\def\Rh{\Rho}
\def\Si{\Sigma}
\def\Ta{\Tau}
\def\Up{\Upsilon}
\def\Ph{\Phi}
\def\Ch{\Chi}
\def\Ps{\Psi}
\def\Om{\Omega}   
%
% Characters
%
\def\0{{\bf 0}}
\def\a{{\bf a}}
\def\b{{\bf b}}
\def\k{{\bf k}}
\def\el{\hbox{\pmb{$\ell$}}}
\def\m{{\bf m}}
\def\t{{\bf t}}
\def\s{{\bf s}}
\def\S{{\cal S}}
\def\q{{\bf q}}
\def\x{{\bf x}}
\def\y{{\bf y}}
\def\r{{\bf r}}
\def\s{{\bf s}}
\def\L{{\bf L}}
\def\p{{\bf p}}
\def\w{{\bf w}}
\def\z{{\bf z}}
\def\F{{\cal F}}
\def\bpsi{{\bf \Psi}}
%
%  Vincent's capital roman double letters
%
\def\AA{{\it I}\hskip-3pt{\tt A}}
\def\CC{\rlap {\raise 0.4ex \hbox{$\scriptscriptstyle |$}
\hskip -0.1em C}}
\def\FF{\hbox to 8.33887pt{\rm I\hskip-1.8pt F}}
\def\NN{\hbox to 9.3111pt{\rm I\hskip-1.8pt N}}
\def\PP{\hbox to 8.61664pt{\rm I\hskip-1.8pt P}}
\def\QQ{\rlap {\raise 0.4ex \hbox{$\scriptscriptstyle |$}}
{\hskip -0.1em Q}}
\def\RR{\hbox to 9.1722pt{\rm I\hskip-1.8pt R}}
\def\ZZ{\hbox to 8.2222pt{\rm Z\hskip-4pt \rm Z}} %\def\ZZ{Z\!\!\!Z}
%
%
%      Blackboard characters
%
\def\bbbr{{\rm I\!R}}  
\def\bbbn{{\rm I\!N}} 
\def\bbbm{{\rm I\!M}}
\def\bbbh{{\rm I\!H}}
\def\bbbf{{\rm I\!F}}
\def\bbbk{{\rm I\!K}}
\def\bbbp{{\rm I\!P}}
\font \tensans                = cmss10
\font \fivesans               = cmss10 at 5pt
\font \sixsans                = cmss10 at 6pt
\font \sevensans              = cmss10 at 7pt
\font \ninesans               = cmss10 at 9pt
\font \twelvesans             = cmss12
\newfam\sansfam
\textfont\sansfam=\tensans\scriptfont\sansfam=\sevensans
\scriptscriptfont\sansfam=\fivesans
\def\sans{\fam\sansfam\tensans}
\def\bbbone{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l}    %\bbbone
{\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}}
\def\bbbc{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm C$}\hbox{\hbox %\bbbc
to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
{\setbox0=\hbox{$\textstyle\rm C$}\hbox{\hbox
to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
{\setbox0=\hbox{$\scriptstyle\rm C$}\hbox{\hbox
to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
{\setbox0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\hbox
to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}}
\def\bbbe{{\mathchoice {\setbox0=\hbox{\smalletextfont e}\hbox{\raise   %\bbbe
0.1\ht0\hbox to0pt{\kern0.4\wd0\vrule width0.3pt
height0.7\ht0\hss}\box0}}
{\setbox0=\hbox{\smalletextfont e}\hbox{\raise
0.1\ht0\hbox to0pt{\kern0.4\wd0\vrule width0.3pt
height0.7\ht0\hss}\box0}}
{\setbox0=\hbox{\smallescriptfont e}\hbox{\raise
0.1\ht0\hbox to0pt{\kern0.5\wd0\vrule width0.2pt
height0.7\ht0\hss}\box0}}
{\setbox0=\hbox{\smallescriptscriptfont e}\hbox{\raise
0.1\ht0\hbox to0pt{\kern0.4\wd0\vrule width0.2pt
height0.7\ht0\hss}\box0}}}}
\def\bbbq{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm               %\bbbq
Q$}\hbox{\raise
0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}}
{\setbox0=\hbox{$\textstyle\rm Q$}\hbox{\raise
0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}}
{\setbox0=\hbox{$\scriptstyle\rm Q$}\hbox{\raise
0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}}
{\setbox0=\hbox{$\scriptscriptstyle\rm Q$}\hbox{\raise
0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}}}}
\def\bbbt{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm              %\bbbt
T$}\hbox{\hbox to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}
{\setbox0=\hbox{$\textstyle\rm T$}\hbox{\hbox
to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}
{\setbox0=\hbox{$\scriptstyle\rm T$}\hbox{\hbox
to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}
{\setbox0=\hbox{$\scriptscriptstyle\rm T$}\hbox{\hbox
to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}}}
\def\bbbs{{\mathchoice                                               %\bbbs
{\setbox0=\hbox{$\displaystyle     \rm S$}\hbox{\raise0.5\ht0\hbox
to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\hbox
to0pt{\kern0.55\wd0\vrule height0.5\ht0\hss}\box0}}
{\setbox0=\hbox{$\textstyle        \rm S$}\hbox{\raise0.5\ht0\hbox
to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\hbox
to0pt{\kern0.55\wd0\vrule height0.5\ht0\hss}\box0}}
{\setbox0=\hbox{$\scriptstyle      \rm S$}\hbox{\raise0.5\ht0\hbox
to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\raise0.05\ht0\hbox
to0pt{\kern0.5\wd0\vrule height0.45\ht0\hss}\box0}}
{\setbox0=\hbox{$\scriptscriptstyle\rm S$}\hbox{\raise0.5\ht0\hbox
to0pt{\kern0.4\wd0\vrule height0.45\ht0\hss}\raise0.05\ht0\hbox
to0pt{\kern0.55\wd0\vrule height0.45\ht0\hss}\box0}}}}
\def\bbbz{{\mathchoice {\hbox{$\sans\textstyle Z\kern-0.4em Z$}}       %\bbbz
{\hbox{$\sans\textstyle Z\kern-0.4em Z$}}
{\hbox{$\sans\scriptstyle Z\kern-0.3em Z$}}
{\hbox{$\sans\scriptscriptstyle Z\kern-0.2em Z$}}}}
%
%   Symbols
%
\def\const{{\rm const}\,}
\def\sgn{{\rm sgn}}
\def\slim{\mathop{\,\hbox{s-lim}\,}}
\def\cartprod{\mathop{\lower2pt\hbox{{\twelvesans X}}}}
\def\eqdef{{\buildrel \rm def \over =}}
\def\v{\pmb{\big\vert}}
\def\bbar#1{\setbox0=\hbox{$#1$}%
     \copy0
     \raise.0433em\box0}
\def\optbar#1{\vbox{\ialign{##\crcr\hfil${\scriptscriptstyle(}\mkern -1mu
         \vrule height 1.2pt width 3pt depth -.8pt
         {\scriptscriptstyle)}$\hfil\crcr
          \noalign{\kern-1pt\nointerlineskip}$\hfil\displaystyle{#1}\hfil$\crcr}}}
%
% Section heading
%
\font \tafontt                = cmbx10 scaled\magstep2
\font \tbfontss               = cmbx5  scaled\magstep1
\font \tbfonts                = cmbx7  scaled\magstep1
\font \tbfontt                = cmbx10 scaled\magstep1
\font \tasys                  = cmex10 scaled\magstep1
\font \tamsss                 = cmmib10 scaled 833
\font \tamss                  = cmmib10
\font \tams                   = cmmib10 scaled\magstep1
\font \tbst                   = cmsy10 scaled\magstep1
\font \tbsss                  = cmsy5  scaled\magstep1
\font \tbss                   = cmsy7  scaled\magstep1
\font \chfont                 = cmbx12 at 14pt
\font \subchfont              = cmbx12 at 12pt
\def\titlea#1{\centerline{\tafontt #1 }\vskip.75truein}
\def\titleb#1{\removelastskip\vskip.2truein\noindent{\tbfontt #1 }\vskip.4truein}
\def\titlec#1{\removelastskip\vskip\baselineskip\noindent{\bf #1 }
                     \vskip\baselineskip}
\def\sec{\mathhexbox278}
\def\section#1#2{%
      \vskip25pt plus 4pt minus4pt
     \bgroup
 \textfont0=\tbfontt \scriptfont0=\tbfonts \scriptscriptfont0=\tbfontss
 \textfont1=\tams \scriptfont1=\tamss \scriptscriptfont1=\tamsss
 \textfont2=\tbst \scriptfont2=\tbss \scriptscriptfont2=\tbsss
 \textfont3=\tasys \scriptfont3=\tenex \scriptscriptfont3=\tenex
     \baselineskip=16pt
     \lineskip=0pt
     \rightskip 0pt plus 6em
     \pretolerance=10000
     \tbfontt
     \setbox0=\vbox{\vskip25pt
     \noindent
     \if!#1!\ignorespaces#2
     \else\ignorespaces#1\unskip\enspace\ignorespaces#2\fi
     \vskip15pt}%
     \dimen0=\pagetotal
     \ifdim\dimen0<\pagegoal
     \dimen0=\ht0\advance\dimen0 by\dp0\advance\dimen0 by
     4\normalbaselineskip
     \advance\dimen0 by\pagetotal
     \advance\dimen0 by-\pageshrink
     \ifdim\dimen0>\pagegoal\vfill\eject\fi\fi
     \noindent
     \if!#1!\ignorespaces#2
     \else\ignorespaces#1\unskip\enspace\ignorespaces#2\fi
     \egroup
     \vskip12.5pt plus4pt minus4pt
     \ignorespaces}
%
%  Theorem etc.
%
\def\newenvironment#1#2#3#4{\long\def#1##1##2{\removelastskip
\vskip\baselineskip\noindent{#3#2\if!##1!.\else\unskip\ \ignorespaces
##1\unskip\fi\ }{#4\ignorespaces##2\vskip\baselineskip}}}
%  
\newenvironment\lemma{Lemma}{\bf}{\it}
\newenvironment\proposition{Proposition}{\bf}{\it}
\newenvironment\theorem{Theorem}{\bf}{\it}
\newenvironment\corollary{Corollary}{\bf}{\it}
\newenvironment\example{Example}{\it}{\rm}
\newenvironment\problem{Problem}{\bf}{\rm}
\newenvironment\definition{Definition}{\bf}{\rm}
\newenvironment\remark{Remark}{\it}{\rm}
\def \thm#1{\vskip.25truein\noindent{\bf Theorem#1}\par}
\def \lem#1{\vskip.25truein\noindent{\bf Lemma#1}\par}
%
%    Proof, qed, equation
%
\long\def\proof#1{\removelastskip\vskip\baselineskip\noindent{\bf
            Proof\if!#1!\else\ \ignorespaces#1\fi.\ }\ \ \ignorespaces}
\def\sq{\hbox{\rlap{$\sqcap$}$\sqcup$}}
\def\qed{\ifmmode\sq\else{\unskip\nobreak\hfil
           \penalty50\hskip1em\null\nobreak\hfil\sq
           \parfillskip=0pt\finalhyphendemerits=0\endgraf}\fi}
\def \prf{\vskip.25truein\noindent{\bf Proof\ }:\ \ \ }
%\def\endproof{\hfil \break \line{ \hfill\vrule height .75em width .75em depth 0pt}}
\def\endproof{\hfill\vrule height .6em width .6em depth 0pt\vskip.25in }
\def\eqn#1{\eqno{({\rm #1})}}


%      ********   FIG.TEX    ************************
%      (c)  R. A. Adams and R. B. Israel 
%           Dept. of Mathematics
%           The University of British Columbia
%           Vancouver, B.C., Canada V6T 1Y4
%
%           November, 1990.      Version Feldman  09/11/92
%      **********************************************
%
% being macros suitable for the inclusion of figures created by 
% "MG" into TeX documents.
%
%%%%%%%
%%%%%%% SELECT YOUR SYSTEM  NOW!
\newcount\system
%\global\system=1   % for textures 
%\global\system=2   % for msdos
\global\system=3   % for unix(dvips)vincent
%
% In vertical mode use:
%
% \cfig{<filename>}{<caption text>}
%
% to include a centred figure whose PostScript description and 
% associated TeX labels, both produced by "MG", are in the 
% files <filename>.PS and <filename>.LBL respectively. 
% The first line of the .LBL file is the MG version number.  The
% second and third lines are the width and height of the figure 
% (in points). The macro reads them there to provide
% enough vertical space.
%
% example:
%
% \cfig{myplot}{Figure 3}
%
% Similarly, two side-by-side graphs can be inserted by
%
% \twofigs{<filename1>}{<caption1 text>}{<filename2>}{<caption2 text>}
%              
% A graph on the right side of some text can be produced by
% \rfig{<text on the left>}{<filename>}{<caption text>}
%              
%
% Labels on figures will print in the default TeX families 
% (typically 10 point Computer Modern  or whatever families 
% are installed in your version of TeX.)
% If you want to change this use, for example, 
%
%      \def\setlabelsize{\ninepoint}
%
% within your TeX document. To reinstate the defaults use
%
%      \def\setlabelsize{}
%
% macros \ds, \ss, \ts defined below can also be used to control 
% size of labels
%
% \ninepoint and \eightpoint are defined hereunder for CM fonts.  
% If you want any other size or family  you will have to define 
% your own.
%
\font\ninerm=cmr9   \font\sixrm=cmr6   \font\eightrm=cmr8  
\font\ninei=cmmi9   \font\sixi=cmmi6   \font\eighti=cmmi8  
\font\ninesy=cmsy9  \font\sixsy=cmsy6  \font\eightsy=cmsy8 
\font\ninebf=cmbx9  \font\sixbf=cmbx6  \font\eightbf=cmbx8 
\font\nineit=cmti9                     \font\eightit=cmti8 
\font\ninesl=cmsl9                     \font\eightsl=cmsl8 
\font\ninett=cmtt9                     \font\eighttt=cmtt8 

\font\twelverm=cmr12
\font\twelvei=cmmi12
\font\twelvesy=cmsy10 at 12pt
\font\twelvebf=cmbx12
\font\twelveit=cmti12
\font\twelvesl=cmsl12
\font\twelvett=cmtt12
%
\def\twelvepoint{\def\rm{\fam0\twelverm}% switch to 12-point type
 \textfont0=\twelverm \scriptfont0=\eightrm \scriptscriptfont0=\sixrm
 \textfont1=\twelvei \scriptfont1=\eighti \scriptscriptfont1=\sixi
 \textfont2=\twelvesy \scriptfont2=\eightsy \scriptscriptfont2=\sixsy
 \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex
 \textfont\itfam=\twelveit \def\it{\fam\itfam\twelveit}%
 \textfont\slfam=\twelvesl \def\sl{\fam\slfam\twelvesl}%
 \textfont\ttfam=\twelvett \def\tt{\fam\ttfam\twelvett}%
 \textfont\bffam=\twelvebf \scriptfont\bffam=\eightbf
 \scriptscriptfont\bffam=\sixbf \def\bf{\fam\bffam\twelvebf}%
 \normalbaselineskip=14pt
 \setbox\strutbox=\hbox{\vrule height10pt depth4pt width0pt}%
 \let\sc=\tenrm \let\big=\twelvebig \normalbaselines\rm}
%
\def\ninepoint{\def\rm{\fam0\ninerm}% switch to 9-point type
 \textfont0=\ninerm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm
 \textfont1=\ninei \scriptfont1=\sixi \scriptscriptfont1=\fivei
 \textfont2=\ninesy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy
 \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex
 \textfont\itfam=\nineit \def\it{\fam\itfam\nineit}%
 \textfont\slfam=\ninesl \def\sl{\fam\slfam\ninesl}%
 \textfont\ttfam=\ninett \def\tt{\fam\ttfam\ninett}%
 \textfont\bffam=\ninebf \scriptfont\bffam=\sixbf
 \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\ninebf}%
 \normalbaselineskip=11pt
 \setbox\strutbox=\hbox{\vrule height8pt depth3pt width0pt}%
 \let\sc=\sevenrm \let\big=\ninebig \normalbaselines\rm}
%
\def\eightpoint{\def\rm{\fam0\eightrm}% switch to 8-point type
 \textfont0=\eightrm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm
 \textfont1=\eighti \scriptfont1=\sixi \scriptscriptfont1=\fivei
 \textfont2=\eightsy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy
 \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex
 \textfont\itfam=\eightit \def\it{\fam\itfam\eightit}%
 \textfont\slfam=\eightsl \def\sl{\fam\slfam\eightsl}%
 \textfont\ttfam=\eighttt \def\tt{\fam\ttfam\eighttt}%
 \textfont\bffam=\eightbf \scriptfont\bffam=\sixbf
 \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\eightbf}%
 \normalbaselineskip=9pt
 \setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}%
 \let\sc=\sixrm \let\big=\eightbig \normalbaselines\rm}
%
% The next five \TeX\ macros can be used in labels to simplify
% some size choices.  (If they interfere with other macros in
% your system, comment them out.)
%
%
% use the current tex font family for labels
\def\setlabelsize{}
%
% by default, boxes will not be drawn around figures
\newif\ifboxfigures
\boxfiguresfalse
% you can declare \boxfigurestrue in your TeX source if you want
% the boxes drawn
\newdimen\boxrulewidth \newdimen\boxborderwidth
\boxrulewidth=1pt \boxborderwidth=2pt
% boxes drawn around figures will have rules 1 point wide 
% surrounding a 2 point wide border by default.
% You can change these measurements by redefining \boxrulewidth 
% and \boxborderwidth in your TeX source.
%
% macro for drawing boxes
%
\def\boxit#1{\vtop{\hrule height\boxrulewidth%
\hbox{\vrule width\boxrulewidth\kern\boxborderwidth%
\vbox{\kern\boxborderwidth#1\kern\boxborderwidth}\kern\boxborderwidth%
\vrule width\boxrulewidth}%
\hrule height\boxrulewidth}}  
%
% Now for the definitions and main macro for figure inclusion.
%
\newcount\mgversion
\newdimen\pswidth  \newdimen\xleft
\newdimen\psheight \newdimen\ytop \newdimen\ybot
\newcount\justx \newcount\justy
\global\justx=0 \global\justy=0
\newdimen\vpos \newtoks\label 
\newread\labelfile \newdimen\xcoord \newdimen\ycoord
\newif\ifdoit 
\newbox\labox
%
\newcount\temp
\def\readdim#1{\global\read\labelfile to \temp
\global #1=\temp pt}
%
\def\figinsert#1{\par%  #1=filename
\openin\labelfile=#1.lbl                                              
\global\read\labelfile to\mgversion\message{#1}               
\readdim{\pswidth}                                     
\readdim{\psheight}                                    
\ifboxfigures\boxit{\fi\vbox to\psheight {\vfill
%%%
%%% NOTE: next lines may have to be changed for your DVIPS driver %%%
\ifnum\system=1\special{postscript grestore newpath gsave}\fi%%textures
\ifnum\system=2\special{postscript grestore newpath gsave}\fi%%msdos
%\ifnum\system=3\special{" grestore newpath gsave}\fi         %%unix:dvips
\ifnum\system=1\special{postscriptfile  #1.PS }\fi           %%textures
\ifnum\system=2\special{ps: plotfile #1.PS} \fi              %%msdos  
\ifnum\system=3\special{psfile=#1.ps hscale=120 vscale=120} \fi                    %%unix:dvips                                       
%%%
\vskip-\psheight \setlabelsize%                                     
\hbox to\pswidth  {\hss}%                                            
\parindent=0pt\offinterlineskip                                       
\vpos=0 pt%                                                              
\loop\readdim{\xcoord}                                 
\ifnum \temp < -999 \doitfalse\else\doittrue\fi                        
\ifdoit \readdim{\ycoord}                              
\global\read\labelfile to\justx                                       
\global\read\labelfile to\justy                                       
\global\read\labelfile to\label
\global\setbox\labox=\hbox{\label\hskip-0.3em}%    
\advance\vpos by-\ycoord                                              
\vskip-\vpos \vpos=\ycoord                                         
\hbox to\pswidth{\hskip\xcoord %                                 
\hbox to 0pt{\ifnum\justx>0\hss\fi%                                   
\vbox to0pt{%                                                         
\ifnum\justy<2\vss\fi%                                                
\copy\labox\kern0pt%  
\ifnum\justy>0\vss\fi}%                                               
\ifnum\justx<2\hss\fi}%                                               
\hss}%                                                                
\repeat%                                                              
\advance\vpos by-\psheight%                                           
\vskip-\vpos %                                                      
}\ifboxfigures}\fi\closein\labelfile}                                                  
%
% 
%    figcrop{<filename,w/o extension>} treats the first two labels as marking
%    the upper left and lower right corners of the figure. This is for
%    positioning purposes only. The figure may extend beyond the corners.
%    The corner markers are not printed.
%
%
\def\figcrop#1{\par%  #1=filename
\openin\labelfile=#1.lbl                                              
\global\read\labelfile to\mgversion\message{#1}               
\global\read\labelfile to\temp%read overall dimensions                                     
\readdim{\ybot}
\readdim{\xleft}%               read upper left point
\readdim{\ytop}
\global\read\labelfile to\justx%ignore
\global\read\labelfile to\justy%ignore
\global\read\labelfile to\label%ignore
\readdim{\pswidth}%            read lower right point
\global\advance\pswidth by -\xleft
\readdim{\psheight}
\global\advance\ybot by -\psheight
\global\advance\psheight by -\ytop
\global\read\labelfile to\justx%ignore
\global\read\labelfile to\justy%ignore
\global\read\labelfile to\label%ignore                                    
\ifboxfigures\boxit{\fi\vbox to\psheight{\vfill
%%%
%%% NOTE: next line may have to be changed for your DVIPS driver %%%
\ifnum\system=1\special{postscript grestore newpath gsave}\fi  %%textures
\ifnum\system=2\special{postscript grestore newpath gsave}\fi  %%msdos
%\ifnum\system=3\special{" grestore newpath gsave}\fi           %%unix:dvips
\ifnum\system=1
\hbox to \pswidth{\kern-\xleft\special{postscriptfile  #1.PS }\hfil}\fi%textures
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%\input ../ftmacros
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\def\<{\left <}
\def\>{\right >}
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\null\vskip2.5truecm
%  
%   (1) Put title in next two lines
%
%\centerline{The Pauli Exclusion Principle in Many Body Theory}
%\centerline{A Convergent Linked Cluster Expansion for Many Fermion Models}
%\centerline{A Convergent Cluster-Like Expansion for Many Fermion Green's 
%Functions}
\centerline{\subchfont An Infinite Volume Expansion for Many Fermion 
Green's Functions}
%\centerline{}
\vskip20pt
\centerline{Joel Feldman\footnote{$^{*}$}{ Research supported in part by 
the Natural Sciences and Engineering Research Council of Canada and the
Schweizerischer Nationalfonds zur Forderung der wissenschaftlichen
Forschung}$^{\dagger}$}
\centerline{Department of Mathematics}
\centerline{University of British Columbia}
\centerline{Vancouver, B.C. }
\centerline{CANADA\ \   V6T 1Z2}\vskip.5truecm
\centerline{Jacques Magnen$^{\dagger}$, 
Vincent Rivasseau\footnote{$^{\dagger}$}
{ Research supported in part by the Forschungsinstitut f\"ur
Mathematik, Z\"urich}}
\centerline{Centre de Physique Th\'eorique}
\centerline{Ecole Polytechnique}
\centerline{F-91128 Palaiseau Cedex}
\centerline{FRANCE}\vskip.5truecm
\centerline{Eugene Trubowitz}
\centerline{Mathematik}
\centerline{ETH-Zentrum}
\centerline{CH-8092 Z$\ddot{\rm u}$rich}
\centerline{SWITZERLAND}\vskip2truecm

%
%   (2) Dedication (optional)
%
%\vskip1truecm
\noindent
\underbar{Abstract}\ \ \ 
%
%   (3) Put abstract below here
We prove the convergence of a simplified cluster/Mayer expansion at
one energy scale for three space-time dimensional many Fermion
systems. The bounds are uniform in the scale. We iterate them
to show that the sum of all diagrams that contain no
two or four-legged subdiagrams converges.
Our results are suited
to a multiscale construction of the full system. 


\vfill
\eject
\multiply\baselineskip by 15\divide \baselineskip by10



\noindent{\subchfont \sec I Introduction}
\vskip.25truein
 
In this paper we consider many Fermion systems formally characterized
by the effective potential $$ {\cal G}(\psi^e,\bar \psi^e)
=\log\,{1\over Z}\int e^{-\lambda {\cal
V}(\psi+\psi^e,\bar\psi+\bar\psi^e)} d\mu_{C}(\psi,\bar\psi) $$ for
the external fields $\psi^e,\bar\psi^e$. Here, $d\mu_C(\psi,\bar\psi)$
is the fermionic Gaussian measure in the Grassmann variables
$\{\psi(\xi),\bar\psi(\xi)\ |\ \xi\in
\bbbr\times\bbbr^d\times\{\uparrow,\downarrow\}\}$ with propagator $$
C(\xi,\bar\xi)=\delta_{\sigma,\bar\sigma}\int
{d^{d+1}p\over(2\pi)^{d+1}} {e^{i<p,\xi-\bar\xi>_-}
\over ip_0-e(\p)}
$$ 
where 
$$
 <p,\xi>_-=\p\cdot\x-p_0\tau 
$$ 
and 
$$
 e(\p)={\p^2\over2m}-\mu.  
$$ 
The variable $\xi=(\tau,\x,\sigma)$ consists of time,
space and spin components and the $(d+1)$-momentum $p=(p_0,\p)$. The 
interaction is given by 
$$ 
{\cal V}={1\over 2}
\int \prod_{i=1}^4 d\xi_i\ \  
V\!\left(\xi_1,\xi_2,\xi_3,\xi_4\right)
\bar\psi(\xi_1)\bar\psi(\xi_2)\psi(\xi_4)\psi(\xi_3)
$$ 
where the kernel $V\!\left(\xi_1,\xi_2,\xi_3,\xi_4\right)$ is
translation invariant with $V\!\left(0,\xi_2,\xi_3,\xi_4\right)$
integrable and 
$$
\int d\xi=\sum_{\sigma\in\{\uparrow,\downarrow\}}
\int_\bbbr d\tau\int_{\bbbr^d} d\x.
$$ The partition function $$ Z=\int e^{-\lambda {\cal
V}(\psi,\bar\psi)}d\mu_{C}(\psi,\bar\psi) $$ so that ${\cal
G}(0,0)=0.$

The Euclidean Green's functions $$
G_p(\xi_1,\bar\xi_1,\dots,\xi_p,\bar\xi_p)=\prod_{i=1}^p
{\delta^2\hfill\over
\delta\psi^e(\xi_i)\delta\bar\psi^e(\bar\xi_i)}{\cal G} $$ generated
by the effective potential are the connected Green's functions
amputated by the free propagator. By definition, ${\cal G}$ exists
when the norm $$
\norm G_p\norm=\max_{j}\ \sup_{\xi_j}\ \int\prod_{i\ne j}d\xi_i\  
|G_p(\xi_1,\cdots,\xi_{2p})| $$ of each of its moments, $G_p,\ p\ge
1$, is finite. Intuitively, $\norm G_p\norm$ is the supremum in
momentum space of $G_p$. In fact, the supremum in momentum space was
used as the standard norm on vertices in [FT2].

Our long term goal is to give a rigorous proof that the standard model
for an interacting system of electrons and phonons has a
superconducting ground state at sufficiently low temperature.
Perturbation theory and, in particular, the renormalization of the two
point function was controlled in [FT1]. (See [BG] for related results.)
A renormalization group flow
for the four point function was defined and analyzed in [FT2]. Two
additional ingredients are required to complete this program.  First
an infinite volume expansion that combines power counting at fixed
energy with the exclusion principle and second, control of the
Goldstone boson. This paper provides the first ingredient, in three
space-time dimensions. With the exception of Lemma 3 all components of
the expansion apply in all dimensions. We restrict to $d=2$ only when
it is necessary to do so.

  

As in [FT1,2], the model is sliced into energy regimes by decomposing
momentum space into shells around the Fermi surface. The $j^{\rm th}$
slice has covariance $$
C^{(j)}(\xi,\bar\xi)=\delta_{\sigma,\bar\sigma}\int
{d^{d+1}p\over(2\pi)^{d+1}} {e^{i<p,\xi-\bar \xi>_-}
\over ip_0-e(\p)}f_j(p),
$$ where $$ f_j(p)=f\left(M^{-2j}\left(p_0^2+e(\p)^2\right)\right) $$
effectively forces $|ip_0-e(\p)|\sim M^j$.  The function $f\in
C^\infty_0([1,M^4])$. The parameter $M$ is strictly bigger than one so
that the scales near the Fermi surface have $j$ near $-\infty$. The
model is defined in finite volume and at fixed scale by the following
lemma. However, the radius of convergence depends on volume and scale in a
patently unsatisfactory way.$$
\int d\xi=\sum_{\sigma\in\{\uparrow,\downarrow\}}
\int_\bbbr d\tau\int_{\bbbr^d} d\x.
$$\vfill\eject
\lem{\ 1}{\it
$$ {\cal G}_\Lambda^{(j)}(\psi^e,\bar\psi^e) =\log\,{1\over
Z_\Lambda^{(j)}}\int e^{-\lambda {\cal
V}_\Lambda(\psi+\psi^e,\bar\psi+\bar\psi^e)}
d\mu_{C^{(j)}}(\psi,\bar\psi) $$ where $$ {\cal V}_\Lambda={1\over 2}
\int_{\Lambda^4} \prod_{i=1}^4 d\xi_i\ \  
V\!\left(\xi_1,\xi_2,\xi_3,\xi_4\right)
\bar\psi(\xi_1)\bar\psi(\xi_2)\psi(\xi_4)\psi(\xi_3)
$$
 and 
$$ Z_\Lambda^{(j)}=\int e^{-\lambda {\cal V}_\Lambda(\psi,\bar\psi)} 
d\mu_{C^{(j)}}(\psi,\bar\psi) $$ 
is
analytic in $\lambda$ in a neighborhood of the origin that includes at
least the disk of radius $\const\left(M^{2j}|\Lambda|\right)^{-1}.$}
\prf
Expand the exponential 
$$\eqalign{ Z_\Lambda^{(j)}&=\int e^{-\lambda
{\cal V}_\Lambda(\psi,\bar\psi)} d\mu_{C^{(j)}}(\psi,\bar\psi)\cr
&=\sum_{n=0}^\infty {(-\lambda)^n\over n!}\int {\cal
V}_\Lambda(\psi,\bar\psi)^n d\mu_{C^{(j)}}(\psi,\bar\psi)\cr
&=\sum_{n=0}^\infty {(-\lambda)^n\over 2^n n!}\int_{\Lambda^{4n}}
\left(\prod_{j=1}^n\prod_{i=1}^2 d\xi_{j,i}d\bar\xi_{j,i}\right)
\left(\prod_{j=1}^nV(\bar\xi_{j,1},\bar\xi_{j,2},\xi_{j,1},\xi_{j,2})
\right)\det\left[C^{(j)}(\xi_k,\bar\xi_\ell)\right]
}$$ 
where the indices $k$ and $\ell$ run over $\{(j,i)\,|\,1\le j\le
n\,,\,1\le i\le 2\}$.

The $(k,\ell)$ matrix element 
$$\eqalign{ 
C^{(j)}(\xi_k,\bar\xi_\ell)
&=\delta_{\sigma_k,\bar\sigma_\ell}
\int {d^{d+1}p\over(2\pi)^{d+1}} {e^{i<p,\xi_k-\bar\xi_\ell>_-}
\over ip_0-e(\p)}f_j(p)\cr
&=\left<A_k,\bar A_\ell\right> 
}$$ 
where 
$$
 A_k(p,\alpha):=
\delta_{\alpha,\sigma_k}{e^{i<p,\xi_k>_-}
\over\left[ip_0-e(\p)\right]^{1\over 2}}
  f_j(p)^{1/2} 
$$ 
and 
$$
\bar A_\ell(p,\bar \alpha) := \delta_{\bar\alpha,\bar\sigma_\ell}
\left\{{e^{-i<p,\xi_\ell>_-}\over
  {\left[ip_0-e(\p)\right]^{1\over 2}}}\right\}^* f_j(p)^{1/2} 
$$ 
are
in $L^2\left(\bbbr^{d+1}\times\{\uparrow,\downarrow\}\right).$ Here,
any branch of the square root will do, since\ $ip_0-e(\p)$\ does not
vanish on the domain of integration.
 
Consequently, by Gram's inequality, 
$$\eqalign{
\left|\det\left[C^{(j)}(\xi_k,\bar\xi_\ell)\right]\right|
&\le\prod_{k} \|A_k\|_2\ \|\bar A_k\|_2\cr &\le {\rm const}^n\ M^{2jn}
}$$ since 
$$\eqalign{
\|\bar A_k\|_2^2=\|A_k\|_2^2
&=\int{d^{d+1}\p\over(2\pi)^{d+1}}\left|ip_0-e(\p)\right|^{-1}f_j(p)\cr
&\le {\rm const\ }M^{j}.  
}$$ Substituting, 
$$\eqalign{
\left|Z_\Lambda^{(j)}\right|
&\le\sum_{n=0}^\infty {|\lambda|^n\over 2^n n!}\int_{\Lambda^{4n}}
\left(\prod_{j=1}^n\prod_{i=1}^2 d\xi_{j,i}d\bar\xi_{j,i}\right)
\left|\prod_{j=1}^nV(\bar\xi_{j,1},\bar\xi_{j,2},\xi_{j,1},\xi_{j,2})
\right|
\left|\det\left[C^{(j)}(\xi_k,\bar\xi_\ell)\right]\right|\cr
&\le\sum_{n=0}^\infty {|\const\,\lambda|^n\over n!}\int_{\Lambda^{4n}}
\left(\prod_{j=1}^n\prod_{i=1}^2 d\xi_{j,i}d\bar\xi_{j,i}\right)
\left|\prod_{j=1}^nV(\bar\xi_{j,1},\bar\xi_{j,2},\xi_{j,1},\xi_{j,2})
\right|
M^{2jn}\cr &\le\sum_{n=0}^\infty {|\const\,\lambda|^n\over n!}
\left(|\Lambda|M^{2j}\norm V\norm\right)^n\cr
&=\exp\left(\const\,|\lambda||\Lambda|M^{2j}
\norm V\norm\right).
}$$ 
It follows that the partition function is an entire function of
$\lambda$.

Each Taylor coefficient of the expansion of the numerator 
$$
\int e^{-\lambda {\cal V}_\Lambda(\psi+\psi^e,\bar\psi+\bar\psi^e)}
d\mu_{C^{(j)}}(\psi,\bar\psi) 
$$ 
in powers of the external fields is of the form 
$$
\int P(\psi,\bar\psi)e^{-\lambda {\cal V}_\Lambda(\psi,\bar\psi)}
d\mu_{C^{(j)}}(\psi,\bar\psi) 
$$ 
for some polynomial $P$. It is estimated in exactly the same way and is 
also an entire function of $\lambda$.

The Green's functions of ${\cal G}$ are finite sums of finite products
of such numerators divided by a power of $Z_\Lambda^{(j)}$. Thus they
are meromorphic with poles at the zeroes of the partition function.
Therefore the radius of convergence is at least the absolute value of
the smallest root of $Z_\Lambda^{(j)}$.

When $\lambda=0$, the partition function is one. As above 
$$
\left|{d\hfill\over d\lambda}Z_\Lambda^{(j)}\right|
\le \const\,|\Lambda|M^{2j}\norm V\norm
\exp\left(\const\,|\lambda||\Lambda|M^{2j}
\norm V\norm\right)
$$ 
so that 
$$
\left|Z_\Lambda^{(j)}(\lambda)-1\right|\le
\const\,|\lambda||\Lambda|M^{2j}\norm V\norm\exp\left(\const\,|\lambda|
|\Lambda|M^{2j}\norm V\norm\right).
$$
 Finally, for $|\lambda||\Lambda|M^{2j}\norm V\norm\le\const$,
$$
\const\,|\lambda||\Lambda|M^{2j}\norm V\norm\,
\exp\left(\const\,|\lambda||\Lambda|M^{2j}
\norm V\norm\right)\ <\  1\ .
$$ 
\endproof

Let $G^{(j,\Lambda)}_p$ be the $p$-point Green's function generated by
${\cal G}^{(j)}_\Lambda$. By the Lemma above the Taylor series 
$$
G^{(j,\Lambda)}_p=\sum_{n=0}^\infty g_n(p,j,\Lambda)\lambda^n 
$$ 
has a
strictly positive, though possibly $j$ and $\Lambda$ dependent, radius
of convergence. The main result of this paper is
\thm{\ 1}{\it Let $d=2$. There exists a $\ \const\ $, independent of $j$ and 
$\Lambda$, such that $$
\norm g_n(p,j,\Lambda)\norm \le\const^{n+p}M^{(2-5p/2)j}\norm V\norm^n
$$ 
where 
$$
\norm g_n(p,j,\Lambda)\norm=
\max_{k}\ \sup_{\xi_k}\ \int\prod_{i\ne k}d\xi_i\  
|g_n(p,j,\Lambda)(\xi_1,\cdots,\xi_{2p})|.  $$ Furthermore the limits
$$ 
g_n(p,j)=\lim_{\Lambda\rightarrow \bbbr^3}g_n(p,j,\Lambda) 
$$ 
exist and the infinite volume Green's functions at scale $j$ 
$$
G^{(j)}_p=\sum_{n=0}^\infty g_n(p,j)\lambda^n 
$$ 
are analytic in
$|\lambda|<R=(\const\norm V\norm)^{-1}$.}
\vskip .25truein

The reader might have expected the perturbative power counting factor
$M^{{1\over 2}(4-2p)j}$ as in [FT2, Lemma III.1a, (III.16a)] rather
than $M^{(2-5p/2)j}$. However in this theorem we do not try to
optimize the analysis with respect to external legs, and we state the
theorem at the level of ordinary Green's functions, rather than
amputated functions, which appear in a multiscale analysis.  
A finer, more powerful though more complicated bound, which is 
operationally equivalent to the perturbative one is given in Theorem 2 of 
Section III.  By operationally equivalent we mean, for example, that the 
bound
of Theorem 2 is adapted to a multiscale analysis and can be iterated in
a renormalization group flow. In fact Theorem 2 of Section III states
that the sum over all ``completely convergent'' graphs and all scale 
assignments to the lines of the graphs 
is absolutely convergent and analytic for all coupling constants $\lambda$
in a fixed disk around zero. Recall that ``completely convergent'' [Ri]
means that there are neither two nor four point subgraphs with internal
scales higher than those of the external legs. Theorem 2 
shows that our single slice analysis is the correct building block for a 
multiscale analysis.
 
A theorem of this kind is usually proven with a standard cluster/Mayer
expansion [GJS,Br,Ri]. Space-time, $\bbbr^{d+1}$, is paved by cubes
$\Delta$ of side $M^{-j}$ dual to the decay rate $M^j$ of the
propagator. The decay rate is primarily determined by the thickness of
the shell in momentum space. Then one expands in coupling constants
that control the interaction between boxes. One essential prerequisite
for the convergence of such expansions is the $j$ independent estimate
$|Z(\Delta)|\le\const$. For our models, one can see in perturbation
theory that this estimate fails.

The logarithm of the partition function is given perturbatively by the
sum of all connected vacuum graphs. In evaluating a connected vacuum
graph at scale $j$, each propagator contributes 
$(ip_0-e(\p))^{-1}\sim M^{-j}$ and the volume of integration for each 
momentum loop is $\sim  M^{2j}$. Hence the value of a vacuum graph, of 
order $n$, with the
position of one vertex held fixed is 
$\sim M^{-j(2n)}M^{2j(n+1)}\sim M^{2j}$. Integrating the fixed vertex over
 $\Delta$ gives
$|\Delta|M^{2j}=M^{-(d-1)j}$.  This result is a reflection of the
Pauli exclusion principle. The shell in momentum space about the Fermi
surface has volume $M^{2j}$, while the position space volume of the
``dual'' box $\Delta$ is $M^{-(d+1)j}$. The Pauli exclusion principle
now permits $M^{-(d-1)j}$ electrons to be located in $\Delta$ with
momentum restricted to the shell. For $d=1$ there is no true Fermi
surface and consequently one electron in $\Delta$. As $d$ grows the
Pauli exclusion principle becomes progressively weaker and the
estimate on the partition function in $\Delta$ becomes more 
\vfill\eject
\noindent and more $j$ dependent.


There are three naive ways to force the volume in phase space to be
independent of $j$. One either makes the box $\Delta$ smaller, or
decomposes the shell into sufficiently small sectors, or both. In each
case, the number of electrons in such a constrained region would be of
order one, achieving duality in the sense of the exclusion principle.
The first alternative, however, violates duality in the sense of decay
of the propagator.

Let us decompose the shell into $M^{-(d-1)j}$ sectors of side $M^{j}$,
by constructing a smooth partition of unity
$$
1=\sum_{m=1}^{M^{-(d-1)j}}\eta_m(\p),\hskip .25truein
\eta_m(\p)=\eta_m\left({\p\over|\p|}k_F\right)
$$
of the Fermi surface, where $\eta_m$ is supported on the union of the
$m^{th}$ sector, $S_m$, and its neighbors, whose number is at most
$3^{d-1}-1$. There is a corresponding decomposition of the covariance
$$
C^{(j)}=\sum_{m=1}^{M^{-(d-1)j}} C^{(j,m)}
$$ 
where
$$
C^{(j,m)}(\xi,\bar \xi)=\delta_{\sigma,\bar\sigma}
\int {d^{d+1}p\over(2\pi)^{d+1}} {e^{i<p,\xi-\bar\xi>_-}
\over ip_0-e(\p)}f_j(p)\,\eta_m(\p)
$$
and of the fields
$$
\psi=\sum_{m=1}^{M^{-(d-1)j}}\psi^{(m)},\hskip.25truein
\bar\psi=\sum_{m=1}^{M^{-(d-1)j}}\bar\psi^{(m)}.
$$

The standard power counting bound for an individual graph is still
easy to prove when there are sectors. First, one selects a spanning
tree for the graph. To each line not in the tree there is a
corresponding momentum loop, obtained by joining its ends through a
path in the tree. This construction produces a complete set of
independent loops. Ignoring unimportant constants, each propagator is
bounded by its supremum $M^{-j}$.  The volume of integration for each
loop is now $M^{(d+1)j}$. A priori, there is one sector sum with
$M^{-(d-1)j}$ terms for each line. But, by conservation of momentum,
there is only one sector sum per loop. Thus, if there are $n$ vertices
and $E$ external lines, the supremum in momentum space of the graph is
bounded by 
$$\eqalign{
\prod_{\rm lines}M^{-j}\prod_{\rm loops}M^{(d+1)j}M^{-(d-1)j}
&=M^{-j(4n-E)/2}M^{2j\left[(4n-E)/2-(n-1)\right]}\cr &=M^{{1\over
2}(4-E)j}.  
}$$

In the course of a non-perturbative construction, estimates cannot be
made graph by graph because there are too many of them. Rather, the
perturbation series must be blocked and the blocks estimated as units.
The blocks are estimated using the exclusion principle to implement
strong cancellations between the roughly $n!^2$ graphs of order $n$.
However, once the series is blocked, momentum loops can't be defined
and the argument leading to the estimate above cannot be made.
Conservation of momentum has to be implemented at each vertex, rather
than through loops. Even though the volume cutoff $\Lambda$ breaks
exact conservation of momentum, many of the $M^{-2\ell(d-1)j}$ terms
in the sector sums for a general $2\ell$-legged vertex must be zero.
\lem{\ 2}{\it  Fix $\ m\in {\bf Z}^{d+1}\ $ and $\ \ell\ \ge\ 2\ $.  
Then, the number of $2\ell$-tuples 
$$
\left\{S_1,\ \cdots\,S_{2\ell}\right\}
$$ 
of sectors for which there exist $\ \k_i\in \bbbr^d,\ i=1,\ \cdots,\ 
2\ell\ $ satisfying 
$$
\k_i^\prime\in S_i,\ \  |\k_i-\k_i^\prime|\le {\rm const}\ M^j
\ , \ \ \ i=1,\ \cdots,\ 2\ell\ 
$$ 
and 
$$
 |\k_1+\ \cdots\ +\k_{2\ell}|\le{\rm const}\left(1+|m|\right)M^j 
$$ 
is bounded by 
$$
\const^\ell(1+|m|)^dM^{-(2\ell-1)(d-1)j}M^j
\left\{1+|j|\delta_{d,2}\delta_{\ell,2}\right\}.
$$ 
In particular, for a four legged vertex, the number of $4$-tuples is
at most 
$$
\const(1+|n|)^dM^{(-3d+4)j}\left\{1+|j|\delta_{d,2}\right\}.
$$ 
Here, $\ \k^\prime = {\k\over |\k|}\ $ denotes the projection of $\
\k\ $ onto the Fermi surface.}
\vskip.25truein
Lemma 2 is proved in the next section.  Specializing to two space
dimensions, the number of active sector 4-tuples at a vertex is of
order $\ |j|M^{-2j}\ $. Four planar vectors of equal length whose sum
is zero form a parallelogram. The factor $\ M^{-2j}\ $ is natural
since a parallelogram is determined by two of its sides. The
logarithmic factor $\ |j|\ $ is not an artifact of our bounds. It
arises from the degenerate situation in which all four vectors are
roughly collinear. One of the main technical difficulties of the paper
is to overcome the logarithm. Note that in three dimensions, the
parallelogram is hinged and the logarithm $|j|$ jumps to the power
$M^{-j/2}$.

To circumvent the logarithm in two dimensions we divide the Fermi
circle into sectors of length $M^{j/2}$ rather than $M^j$ through a
smooth partition of unity
$$
1=\sum_{\ell=1}^{M^{-j/2}}\zeta_\ell(\p),\hskip .25truein
\zeta_\ell(\p)=\zeta_\ell\left({\p\over|\p|}k_F\right)
$$
 where $\zeta_\ell$ is supported on the $\ell^{th}$ sector,
$\Sigma_\ell$, and its 2 neighbors. We denote by $r_\ell$ the center
of $\Sigma_\ell$. The new sectors are long and skinny since they are
still $M^j$ thick. We shall show in a moment that the sector
propagator $$ C^{(j,\ell)}(\xi,\bar \xi)=\delta_{\sigma,\bar\sigma}
\int {d^{3}p\over(2\pi)^{d+1}} {e^{i<p,\xi-\bar\xi>_-}
\over ip_0-e(\p)}f_j(p)\,\zeta_\ell(\p)
$$ 
decays anisotropically. To accommodate the anisotropy, we will
introduce a different lattice of dual boxes for each sector. The boxes
will be short in the direction perpendicular to $r_\ell$ and long in
the direction of $r_\ell$.

The reason that approximately collinear configurations generate a
logarithm for sectors of length $M^j$, but not for sectors of length
$M^{j/2}$, may be seen in the proof (\sec II) of
\lem{\  3}  {\it  Let $d=2$ and divide the Fermi circle into sectors
$\Sigma_\ell,\ \ell=1,\cdots,M^{-j/2}$ of width $M^{j/2}$. Fix $\ m\in
{\bf Z}^{3}$ and any sector $\Sigma_{\ell_1}$. The number of sector
quadruples $\
\{\Sigma_{\ell_1},\Sigma_{\ell_2},\Sigma_{\ell_3},\Sigma_{\ell_4}\}$
for which there exist $\ \k_i\in \bbbr^2,\ i=1,\ \cdots,\ 4\ $
satisfying 
$$
\k_i^\prime\in \Sigma_{\ell_i},
\ \  |\k_i-\k_i^\prime|\le{\rm const}\, M^j
\ , \ \ \ i=1,\ \cdots,4\ 
$$ 
and 
$$
 |\k_1+\ \cdots\ +\k_{4}|\le{\rm
const}\,\left(1+|m|\right)M^j $$ is bounded by 
$$
\const\,(1+|m|^{2})M^{-j/2}.
$$}

The anisotropic decay of the covariance is given in
\lem{\ 4}
\noindent(a) Let
$$
\rho^{(j,\ell)}(\xi,\bar\xi)\ \eqdef\  1+M^{j}|\xi_{0} -\bar\xi_{0}|
+ M^{j}|\xi_{\|} -\bar\xi_{\|}| + M^{j/2} |\xi_{\bot} -\bar\xi_{\bot}|
$$ 
{\it where $\xi_{\|}$ is the component of $\xi$ parallel to
$r_{\ell}$ and $\xi_{\bot}$ is the component orthogonal to $r_{\ell}$.
The same notation is used for $\bar \xi$. Then, for any $\gamma>0$,}
$$
\vert C^{(j,\ell)} (\xi, \bar \xi) \vert
 \le \const\ M^{3j/2} \rho^{(j,\ell)}(\xi,\bar\xi)^{-\gamma}.  
$$
 

\noindent(b)
$$
\left\vert D_0^{n_0}D_\|^{n_1}D_\bot^{n_2}
\left(e^{-i<\r_\ell,\xi-\bar\xi>_-}C^{(j,\ell)} (\xi, \bar \xi)\right) 
\right\vert
 \le \const\ M^{({3\over 2}+n_0+n_1+{n_2\over 2})j}
\rho^{(j,\ell)}(\xi,\bar\xi)^{-\gamma}. 
$$ 
{\it Here $D_0={\partial\hfill\over \partial\xi_0} ,
D_\|={\r_\ell\over|\r_\ell|}\cdot\nabla_\xi$ and
$D_\bot=\hat\pi_\ell\cdot\nabla_\xi$ where $\hat\pi_\ell$ is any unit
vector perpendicular to $\r_\ell$.}
\prf (a) A pointwise bound on
$$ 
C^{(j,\ell)}(\xi,\bar \xi)=\delta_{\sigma,\bar\sigma}
\int {d^{3}p\over(2\pi)^{3}} {e^{i<p,\xi-\bar\xi>_-}
\over ip_0-e(\p)}f_j(p)\,\zeta_\ell(\p)
$$ 
is obtained by observing that the integrand is bounded by $M^{-j}$
and that the volume of integration is $M^{{5\over 2}j}$. Multiplying
$C^{(j,\ell)}$ by $\rho^{(j,\ell)}(\xi,\bar\xi)^{\gamma}$ is
converted, by integration by parts, into $p$-derivatives acting on 
$$
{1\over ip_0-e(\p)}f_j(p)\,\zeta_\ell(\p)\ \ .  
$$ 
A derivative acting
on $\zeta^{(\ell)}$ produces an $M^{-{1\over 2}j}$ while one acting on
$f$ produces an $M^{-j}$. However, the directional derivative $\hat
\pi_\ell\cdot\nabla_\p$ acting on
$f_j(p)=f\left(M^{-2j}\left(p_0^2+e(\p)^2\right)\right)$ produces 
$$
M^{-2j}2e(\p){1\over 2m}2\p\cdot\hat \pi_\ell 
$$ 
which is bounded by
$\const\ M^{-2j}M^jM^{j/2}|\hat \pi_\ell|=\const\ M^{-j/2}$ on the
support of the integrand.

\noindent(b) Each derivative, with respect to $\xi$ or $\bar\xi$, of
$$ 
e^{-i<\r_\ell,\xi-\bar\xi>_-}C^{(j,\ell)}(\xi,\bar \xi)
=\delta_{\sigma,\bar\sigma}
\int {d^{3}p\over(2\pi)^{3}} {e^{i<p-\r_\ell,\xi-\bar\xi>_-}
\over ip_0-e(\p)}f_j(p)\,\zeta_\ell(\p)
$$ 
brings down a factor of $p-\r_\ell$. On the domain of integration,
the components of this vector\hfill\break
\hfil\figplace{lce1}{1.25in}{0in}\hfil\break
in the time, $\r_\ell$ and $\hat\pi_\ell$ directions are bounded by
$M^j,M^j$ and $M^{j/2}$ respectively. This is obvious except for the
$\r_\ell$ component.  For it we have $$\eqalign{
\left|(p-\r_\ell)_\|\right|&\le k_F-\big(k_F-O(M^{j})\big)\cos \theta\cr
&\le k_F-\big(k_F-O(M^{j})\big)\big(1-O(M^{j/2})^2\big)\cr &\le O(M^j)
}$$ since $\theta\le O(M^{j/2})$.
\endproof
 
We now give a rough description of the expansion. In perturbation
theory, $g_n(p,j,\Lambda)$ is written as ${1\over n!}$ times the sum
of approximately $n!^2$ connected Feynman diagrams of order $n$. Every
connected diagram is spanned by a connected tree. By Cayley's Theorem
there are $n^{n-2}$ labeled trees of order $n$. The number of trees
can be compensated for by the ${1\over n!}$. The expansion starts with
trees and inductively builds graphs from them by joining the remaining
$4n-2p-2(n-1)$ legs.  However, for each fixed tree, these legs may be
joined in $\sim n!$ ways to form connected graphs. As in the standard
cluster expansion, part of this $n!$ will be cancelled using the decay
of the propagator. We now review this procedure.

Pretend that the propagator decays like
$(1+M^j|\xi-\bar\xi|)^{-\gamma}$.  Introduce a lattice, $K_j$, of
cubes of side $M^{-j}$ that paves space-time. If we are in the midst
of the inductive process, some, but not all, legs of the tree have
been joined to form lines. At the next step we select any leg, say in
$\Delta\in K_j$, that has not yet been contracted and sum over all
possible ways of connecting it to another uncontracted leg $\tau$.
Block the sum $$\eqalign{
\sum_{{\rm target\ legs\ }\tau} =
\sum_{\Delta^\prime\in K_j}
\sum_{{\rm target\ legs}\atop {\rm in\ }\Delta^\prime}\ \ .
}$$ Ultimately we must estimate such blocked sums. This is done by
\def\eqone{I.1}
$$\eqalign{
\left|\sum_{\Delta^\prime\in K_j}
\sum_{{\rm target\ legs}\atop {\rm in\ } \Delta^\prime} F(\tau)\right|
&\le\left[\sum_{\Delta^\prime\in K_j}
\big(1+M^j{\rm dist}(\Delta,\Delta^\prime)\big)^{-d-2}\right]\cr
&\hskip.5truein\times\ \sup_{\Delta^\prime\in K_j}
\left|\sum_{{\rm target\ legs}\atop {\rm in\ }\Delta^\prime} 
\big(1+M^j{\rm dist}(\Delta,\Delta^\prime)\big)^{d+2}F(\tau)\right|.
}\eqn{\eqone}$$ The factor $\big(1+M^j{\rm
dist}(\Delta,\Delta^\prime)\big)^{d+2}$ is balanced by the decay of
the propagator. Now the number of terms in the sum $\sum_{{\rm target\
legs}\atop {\rm in\ }\Delta^\prime}$ is the number of target legs in
$\Delta^\prime$ rather than the total number of target legs. When
applied to all contractions, this technique converts ``global $n!$'s''
to ``local $n!$'s''. For a local $n!$ to be large, there must be many
fields of the same momentum slice in a single dual cube. This is
prevented by the Pauli exclusion principle.

To see how the combinatorial analysis suggested in the last two
paragraphs may be carried out in a manner suitable for the proof of
Theorem 1, we develop a complete expansion for a toy model with a
simplified propagator and a local interaction.  All the complications
due to the presence of the Fermi surface are removed, by hand.
The proof of Theorem 1 for the true propagator and
full interaction is presented in \sec III.

Let the dimension $d$ be arbitrary and let $C^{(j)}$ be any propagator
obeying the bounds
\def\eqtwo{I.2}
$$
\left\vert \nabla^n C^{(j)} (\xi, \bar \xi)\right\vert
 \le \const\ M^{({d+1\over 2}+n)j}
\left(1+M^j(|\xi-\bar\xi|\right)^{-\gamma}. 
\eqn{\eqtwo}$$
for some large $\gamma$ and all $n\le N$, which are typical for
strictly renormalizable field theories.  For example the propagator
for the Gross-Neveu model [FMRS], [GK] in two space-time dimensions is of this
type. The interaction of the toy model is $$ {\cal V}_\Lambda={1\over
2}\sum_{\{\uparrow,\downarrow\}}
\int_{\Lambda} d\tau d\x\ \  
\bar\psi(\tau,\x,\sigma)\bar\psi(\tau,\x,\sigma^\prime)
\psi(\tau,\x,\sigma^\prime)\psi(\tau,\x,\sigma).
$$ We now expand $$
\<{\cal A}\>={1\over Z_\Lambda}\int{\cal A} 
e^{-\lambda{\cal V}_\Lambda(\psi,\bar\psi)}
d\mu_{C^{(j)}}(\psi,\bar\psi) $$ where $$ {\cal A}=\int
\prod_{j=1}^{2p} d\xi_j\ A(\xi_1,\cdots,\xi_{2p})
\prod_{j=1}^{2p}\optbar\psi(\xi_j)
$$ is an arbitrary monomial.

Just as in the proof of Lemma 1, Gram's inequality (or, in the
unlikely event that $C^{(j)}$ does not factorize suitably, Hadamard's
inequality) implies that both the numerator and denominator of
$\<{\cal A}\>$ are entire functions of $\lambda$. The denominator $Z$
can have many $j$ and $\Lambda$ dependent zeros. But when $\lambda=0$,
$Z=1$ so that $\<{\cal A}\>$ is meromorphic on all of $\bbbc$ and
analytic at zero. We shall develop a formal power series expansion
for $\<{\cal A}\>$ with the property that for every $N$ 
$$
\<{\cal A}\>=\sum_{n=0}^N a_n(j,\Lambda)\lambda^n+
O\left(\lambda^{N+1}\right).
$$ 
A priori we do not claim that the tail
$O\left(\lambda^{N+1}\right)$ is uniform in $j$ or $\Lambda$.
Nevertheless, since $\<{\cal A}\>$ is analytic at zero we must have
\def\eqthr{I.3}
$$
\<{\cal A}\>=\sum_{n=0}^\infty a_n(j,\Lambda)\lambda^n
\eqn{\eqthr}
$$ in some, possibly $j$ and $\Lambda$ dependent, neighborhood of
zero. Observe that $a_n(j,\Lambda)$ must be the sum of all connected
Feynman diagrams of order $n$ with $2p$ external legs, since
(\eqthr) is an asymptotic expansion.

We shall also show that there exists a $\ \const\ $, independent of
$j$ and $\Lambda$, such that 
$$
|a_n(j,\Lambda)|\le\const^{n+p}M^{(d+1)pj/2}\|A\|_1.  
$$ 
As a
consequence, equation (\eqthr) applies for all
$|\lambda|<R=\const^{-1}$.  Any zeroes of $Z$ that appear in this disk
must be cancelled by zeroes of the numerator. Finally, we shall show
that the limits $a_n(j)=\lim_{\Lambda\rightarrow \bbbr^{d+1}}
a_n(j,\Lambda)$ exist.  This will prove 
$$
\lim_{\Lambda\rightarrow \bbbr^{d+1}}\<{\cal A}\>
=\sum_{n=0}^\infty a_n(j)\lambda^n
$$ 
for all $|\lambda|<R$, the analog of Theorem 1 for the toy model.


The expansion is developed inductively. At the end of step $s$ we have
a sum over a set ${\cal T}_s$ of terms
\def\eqfou{I.4}
$$
\<{\cal A}\>=\sum_{T\in {\cal T}_s} {\int{\cal A}(T,s,\Lambda)
e^{-\lambda{\cal V}_\Lambda(\psi,\bar\psi)}d\mu_{C^{(j)}}
\over
\int e^{-\lambda{\cal V}_\Lambda(\psi,\bar\psi)}d\mu_{C^{(j)}}  }.
\eqn{\eqfou}
$$ Each ${\cal A}(T,s,\Lambda)$ is a monomial in the fields
$\psi,\bar\psi$ of degree $2\pi(T,s)$.  In the event that $\pi(T,s)=0$
the ratio of integrals simplifies to the number ${\cal
A}(T,s,\Lambda)$. Thus
\def\eqfiv{I.5}
$$
\<{\cal A}\>=\sum_{T\in {\cal T}_s\atop \pi(T,s)=0}{\cal A}(T,s,\Lambda)
+\sum_{T\in {\cal T}_s\atop \pi(T,s)\ne 0} {\int{\cal A}(T,s,\Lambda)
e^{-\lambda{\cal V}_\Lambda(\psi,\bar\psi)}d\mu_{C^{(j)}}
\over
\int e^{-\lambda{\cal V}_\Lambda(\psi,\bar\psi)}d\mu_{C^{(j)}} }\ .
\eqn{\eqfiv}
$$ The numbers in the first sum are not touched in subsequent steps of
the expansion. We shall show that each ratio in the second sum is
$O\left(\lambda^s\right)$.

During step $s+1$, two operations are performed. The first is
integration by parts. We apply the integration by parts formulae
\def\eqsix{I.6}
$$\eqalign{
\int \psi(\xi) F(\psi,\bar\psi)d\mu_{C^{(j)}}(\psi,\bar\psi)
&=\int d\bar\xi \ C^{(j)}(\xi,\bar\xi)\int {\delta\hfill\over\delta
\bar\psi(\bar\xi)} F(\psi,\bar\psi)d\mu_{C^{(j)}}(\psi,\bar\psi)\cr
\int \bar\psi(\bar\xi) F(\psi,\bar\psi)d\mu_{C^{(j)}}(\psi,\bar\psi)
&=-\int d\xi \ C^{(j)}(\xi,\bar\xi)\int {\delta\hfill\over\delta
\psi(\xi)} F(\psi,\bar\psi)d\mu_{C^{(j)}}(\psi,\bar\psi)
}\eqn{\eqsix}$$ to each of the fields appearing in the monomial 
${\cal A}(T,s,\Lambda)$.  The order is chosen arbitrarily. Of course if a
derivative ${\dst\delta\hfill\over\dst\delta \optbar\psi(\optbar\xi)}$ 
acts on a field
in ${\cal A}(T,s,\Lambda)$, the field disappears. Thus we need not
apply the integration by parts formulae to eliminate it. If no
derivative acts on the exponential, all fields downstairs disappear,
producing a term of degree zero.  (We use the expression
``downstairs'' to identify fields multiplying the exponential.) Each
time a derivative 
${\dst\delta\hfill\over\dst\delta \optbar\psi(\optbar \xi)}$  acts on the
exponential a new vertex and new fields are brought downstairs. Since
the new vertex comes with a $\lambda$ we see that, by the end of step
$s+1$ all terms with degree $2\pi(T,s,\Lambda)$ different from zero
are $O\left(\lambda^{s+1}\right)$.

We interpret this construction in terms of graphs. Each integration by
parts adds a line, that is, a propagator to the graph. The lines
produced by differentiating the exponential form a spanning tree. The
terms ${\cal A}(T,s,\Lambda)$ of degree zero are values of completely
formed graphs.  The other terms are gestating and will at some later
stage become the values of full graphs.

Fix any $T\in {\cal T}_s$. When all fields of ${\cal A}(T,s,\Lambda)$
have been eliminated by partial integration, we apply the second
operation. It implements the Pauli exclusion principle.  At this point
there are still fields downstairs that were generated by derivatives
acting on the exponential. They will be eliminated by partial
integration in step $s+2$. As in the appendix of [IM], we expand each of 
these fields
in a Taylor polynomial of degree $t$ around $c_\delta$, both to be
determined: $$\eqalign{
\optbar\psi(\xi)=\optbar\psi(c_\delta)+\ \cdots\ 
+&{1\over t!}\big((\xi-c_\delta)\cdot\nabla_\xi\big)^{t}
\ \optbar\psi(c_\delta)\cr
&+{1\over t!}\int_0^1dw (1-w)^{t}
\big((\xi-c_\delta)\cdot\nabla_\xi\big)^{t+1}\ 
\optbar\psi(c_\delta+w(\xi-c_\delta)).
}$$ When, in step $s+2$ a field is eliminated by integration by parts,
we obtain a Taylor expansion of the corresponding propagator. Thus
lines in graphs carry these more general propagators. Note that the
Taylor expansion contains $$
1+(d+1)+(d+1)^2+\cdots+(d+1)^{t}+(d+1)^{t+1}\le 2(d+1)^{t+1} $$ terms.

We now describe how the (field dependent) expansion point $c_\delta$
is determined.  Recall that $K_j$ is a paving of $\bbbr^{d+1}$ by
cubes of side $M^{-j}$. Each interaction vertex downstairs is
rewritten as the sum
\def\eqsev{I.7}
$$ {\cal V}_\Lambda={1\over 2}\sum_{\{\uparrow,\downarrow\}}
\sum_{\Delta\in K_j}\int_{\Lambda\cap\Delta} d\tau d\x\ \  
\bar\psi(\tau,\x,\sigma)\bar\psi(\tau,\x,\sigma^\prime)
\psi(\tau,\x,\sigma^\prime)\psi(\tau,\x,\sigma).
\eqn{\eqsev}$$
In the $s=0$ step the only fields downstairs belong to the
monomial ${\cal A}$. They are also expanded in terms of the paving.
Multiple applications of (\eqsix) and (\eqsev) have generated from
${\cal A}(T,s,\Lambda)$ a sum of terms. Pick a term. Each field of
this term is localized in a cube $\Delta\in K_j$.  Let
$\pi(s+1,\Delta,\sigma,b)$ be the number of fields that have the
specified values of $\Delta,\sigma$ and $b$. Here $s+1$ reminds us
that we are in the midst of step $s+1$ and $b$ distinguishes between
$\psi$'s and $\bar\psi$'s. For each $\sigma,b$ and $\Delta\in K_j$ we
divide $\Delta$ into $\pi(s+1,\Delta,\sigma,b)^\epsilon$ identical
cubes $\delta$ each of side $\pi(s+1,\Delta,\sigma,b)^{-{\epsilon\over
d+1}}M^{-j}$. The value of $\epsilon$ will be picked later. The center
of $\delta$ is called $c_\delta$ and the number of fields in $\delta$
with specified values of $\sigma$ and $b$ is called
$\pi(s+1,\delta,\sigma,b)$. The $\delta$ in the above Taylor expansion
is, of course, that containing $\xi$. In particular the Taylor
expansion must be done inside the $\xi$ integrals. Note that, by the
hypothesis (\eqtwo) on the behavior of the propagator, each
$(\xi-c_\delta)\cdot\nabla_\xi$ that acts on a $\psi$ produces a
factor of
\def\eqeig{I.8}
$$
\pi(s+1,\Delta,\sigma,b)^{-{\epsilon\over d+1}}M^{-j}\cdot M^j.
\eqn{\eqeig}$$

Since the fields anticommute, any nonzero integral may contain at most
one field having any given value of $\sigma,b$ at any $c_\delta$. The
same is true for each derivative of the fields. Thus, for each
$\sigma,b,\delta$, all but $2(d+1)^{t}$ of the fields having this
$\sigma,b$ and located in $\delta$ must be Taylor remainders. That is,
there are at least $\pi(s+1,\delta,\sigma,b)-2(d+1)^{t}$ Taylor
remainders. This completes the description of the expansion. We now
prove
\lem{5} {\it Let the propagator $C^{(j)}$ obey {\rm (\eqtwo)}. Then there 
exists a 
$\ \const\ $, independent of $j$ and $\Lambda$, such that $$
|a_n(j,\Lambda)|\le\const^{n+p}M^{(d+1)pj/2}\|A\|_1.  $$ Furthermore
the limits $$ a_n(j)=\lim_{\Lambda\rightarrow \bbbr^{d+1}}
a_n(j,\Lambda)$$ exist and obey the same bounds. }
\prf
We use the ``method of combinatorial factors'' to keep track of the
many sums in the expansion. This technique uses the elementary
estimate
\def\eqnin{I.9}
$$
\kappa_i>0\ ,\ \sum_i \kappa_i^{-1}\le 1\ \ \     
\Rightarrow\ \ \ \left|\sum_i U_i\right|\le\sup_i
\left|\kappa_i U_i\right|.
\eqn{\eqnin}$$   
to replace each sum by a supremum. To help remember the combinatorial
factor $\kappa_i$ multiplying the value $U_i$ of a given diagram, the
factor is assigned to a specific line or vertex of the diagram.

Here are the combinatorial factors used to control each of the
operations.
\vskip.25truein
\noindent(1){\it Integration by parts.}  
\item{-}A factor of two, assigned to the leg 
initiating the integration by parts suffices to decide whether or not
the leg brings down a new vertex from the exponent.

\noindent When it does, 
\item{-}a factor of two, 
assigned to the target leg, counts the number of possible target legs.


\noindent When it doesn't, we need to work harder. Vertices are 
continuously 
being added downstairs from the exponent during the $(s+1)^{\rm st}$
step.  Thus the set of possible target legs both grows and shrinks as
the stage progresses. On the other hand, all source legs must come
from the set of legs downstairs at the beginning of the stage. So it
is easier to count the number of contractions by labeling each target
leg with the name of the source leg that contracted to it rather than
use the usual procedure, which is to label each source leg with the
name of the target leg to which it contracts.
\item{-}A factor of two per leg suffices 
to decide whether or not the leg is a target leg.

\noindent For each target leg, we organize the sum over source legs 
according 
first, to the cube $\Delta\in K_j$ of the source and second, to which
leg in $\Delta$ is the source.  As in (\eqone)
\item{-}a factor of 
$\ \const\left(1+M^j{\rm dist}(\Delta,\Delta^\prime)\right)^{d+2}\ $,
assigned to the line generated, suffices to control the sum over
$\Delta$.
\item{-}A factor $\pi(s+1,\Delta,\sigma,b)$, assigned to the source leg, 
counts the number of possible source legs within $\Delta$.

\noindent(2){\it Implementation of the Pauli exclusion principle.} 
\item{-} a factor of 
$\ \const\left(1+M^j{\rm dist}(\Delta,\Delta^\prime)\right)^{d+2}\ $
assigned to the propagator that brought a vertex down from the
exponent, will control the sum over localization cubes of the vertex.

\noindent The sum over localization cubes for the fields of ${\cal A}$ is 
not
controlled using combinatorial factors. It will shortly be performed
explicitly.
\item{-}The Taylor expansion splits each leg into at most 
$\ 2(d+1)^{t+1}\ $ 
pieces.  
\vskip .25truein 
It remains only to bound an integral. The
integration variables are the positions of the fields of ${\cal A}$.
The integrand is the supremum, over positions and diagrams, of the
product of
\item{-} the above combinatorial factors
\item{-}$\const\left(1+M^j{\rm dist}
(\Delta,\Delta^\prime)\right)^{-\gamma}$ 
per line of the diagram
\item{-}$|\lambda|\int_{\Delta}d^{d+1}\xi\ 1\ \le |\lambda| M^{-j(d+1)}$ 
per vertex of the diagram
\item{-}$|A(\xi_1,\cdots,\xi_{2p})|$
\item{-}and the factors that come from the Taylor expansions used to 
implement the Pauli exclusion principle.
\vfill\eject

\noindent The latter are 
\def\eqten{I.10}
$$\eqalign{ &\prod_s\prod_{\sigma,b}\ \prod_{\Delta\in
K_j}\prod_{\delta\in \Delta}
\left[\pi(s,\Delta,\sigma,b)^{-{\epsilon\over d+1}}\right]^
{(t+1)\big(\pi(s,\delta,\sigma,b)-2(d+1)^{t}\big)}\cr &\le
\prod_s\prod_{\sigma,b}\ \prod_{\Delta\in K_j}
\left[\pi(s,\Delta,\sigma,b)^{-{\epsilon\over d+1}}\right]^
{(t+1)\big(\pi(s,\Delta,\sigma,b)-2(d+1)^{t}
\pi(s,\Delta,\sigma,b)^\epsilon\big)}.\cr
}\eqn{\eqten}$$ 
Pick any $\zeta>0$. It is possible to choose
$\epsilon$ and $t$, depending only on $d$ and $\zeta$ so that $$
(\eqten)\le\prod_s\prod_{\sigma,b}\ \prod_{\Delta\in K_j}
\const \pi(s,\Delta,\sigma,b)^{-\zeta \pi(s,\Delta,\sigma,b)}.
$$ Altogether $$\eqalign{
\left|a_n(j,\Lambda)\right|&\le \const^{n+p} 
M^{{d+1\over 2}j{4n+2p\over 2}}M^{-(d+1)jn}
\|A\|_1\cr
&=\const^{n+p}M^{{d+1\over 2}pj}\|A\|_1 }$$ The first power of $M^j$
came from covariance bounds and the second from integration over the
positions of the interaction vertices. When the $\Lambda$ constraint
is removed, $a_n$ is expressed as an absolutely convergent series
obeying the same bound.\endproof

Our discussion of the toy model is now complete. The rest of the paper
is devoted to the full model.

%\vfill\eject

\vskip.25truein
\noindent{\subchfont \sec II Sector Counting Lemmas}
\vskip.25truein

In this section we prove Lemmas 2 and 3 formulated in the
introduction.

\lem{\ 2}  {\it Fix $\ m\in \bbbz^{d+1}\ $ and $\ \ell\ge 2\ $.  
Then, the number of $2\ell$-tuples 
$$ 
\left\{S_1,\ \cdots\,S_{2\ell}\right\}
$$ 
of sectors of side $M^j$ on the Fermi sphere
for which there exist $\ \k_i\in \bbbr^d,\ i=1,\ \cdots,\ 2\ell\ $
satisfying 
$$
\k_i^\prime\in S_i,\ \  |\k_i-\k_i^\prime|\le{\rm const}\,M^j
\ , \ \ \ i=1,\ \cdots,\ 2\ell\ 
$$ 
and 
$$
 |\k_1+\ \cdots\ +\k_{2\ell}|\le{\rm const}\left(1+|m|\right)M^j 
$$ 
is bounded by 
$$
\const^\ell(1+|m|)^dM^{-(2\ell-1)(d-1)j}M^j
\left\{1+|j|\delta_{d,2}\delta_{\ell,2}\right\}.
$$ 
Here, $\ \k^\prime = {\k\over |\k|}\ $ denotes the projection of $\
\k\ $ onto the Fermi surface.}

\prf For any fixed 
$\ \k_i,\ i=1,\ \cdots,\ 2\ell-1\ $ there are at most $\
O(1+|m|)^{d-1}\ $ sectors within $\ O\left((1+|m|)M^j\right)\ $ of $\
\k_1+\ \cdots\ +\k_{2\ell-1}\ $.  Thus, the problem is reduced to
determining the number of $(2\ell-1)$-tuples $\ \left\{S_1,\
\cdots\ ,S_{2\ell-1}\right\}\ $ of sectors for which there exist
momenta $\ \k_i\in \bbbr^d,\ i=1,\ \cdots,\ 2\ell-1\ $ with $\
\k_i^\prime\in S_i,\ \ |\k_i-\k_i^\prime|\le {\rm const}\ M^j
\ $,
such that $\ \k_1+\ \cdots\ +\k_{2\ell-1}\ $ is within $\
O\left((1+|m|)M^j\right)\ $ of the Fermi surface.

Since $\ |\k_i-\k_i^\prime|\le {\rm const}\,M^j\ $ there must exist
$\k^\prime_i\in S_i$ such that $\ \k_1^\prime+\ \cdots\
+\k_{2\ell-1}^\prime\ $ is within $\ O\left((1+|m|+\ell)M^j\right)\ $
of the Fermi surface.  As the projections $\ \k_i^\prime\ ,\ 1\le i\le
2\ell-1\ $, vary independently over their sectors, the sum $\
\k_1^\prime+\ \cdots\ +\k_{2\ell-1}^\prime\ $ varies by $\ O(\ell
M^j)\ $.  Thus, the problem is further reduced to counting the number
$\ N_\ell\ $ of $(2\ell-1)$-tuples $\ \left\{S_1,\ \cdots\
,S_{2\ell-1}\right\}\ $ of sectors such that for all $\
\k_i^\prime\in S_i\ $ the sum $\ \k_1^\prime+\ \cdots\
+\k_{2\ell-1}^\prime\ $ is within $\ O\left((1+|m|+\ell)M^j\right)\ $
of the Fermi surface.  Observe that, if the volume of every sector is
at least $\ \const\,M^{(d-1)j}$, 
$$
N_\ell\ \le\ \const^{-(2\ell-1)}M^{-(2\ell-1)(d-1)j}\
\prod_{i=1}^{2\ell-1}\left(\int_{k_FS^{d-1}}d\k_i^\prime\right)
f\left(\k_1^\prime+\ \cdots\ +\k_{2\ell-1}^\prime\right)\ 
$$ 
where $\ f\ $ is a smooth function that is one when 
$\ \k_1^\prime+\ \cdots\ +\k_{2\ell-1}^\prime\ $ is within 
$\ O\left((1+|m|+\ell)M^j\right)\ $
of the Fermi surface and is zero when it is at least a distance $\
2O\left((1+|m|+\ell)M^j\right)\ $ from the Fermi surface.

Rewriting, and recalling that the Fourier transform of the
$(d-1)$-sphere $\ \delta(|\k|-k_F)\ $ is $$\ {\rm const}\
\left(k_F|\p|\right)^{1-{d\over 2}} J_{{d\over 2}-1}(k_F|\p|) $$ we
have $$\leftdisplay{\prod_{i=1}^{2\ell-1}
\left(\int_{k_FS^{d-1}}d\k_i^\prime\right)
f\left(\k_1^\prime+\ \cdots\ +\k_{2\ell-1}^\prime\right)}$$
$$\leftdisplay{\hskip.5truein= {\rm const}^{2\ell-1}\ \int\
\prod_{i=1}^{2\ell-1}d^d\k_i
\ \prod_{i=1}^{2\ell-1}\ \delta(|\k_i|-k_F) 
\ f\left(\k_1+\ \cdots\ +\k_{2\ell-1}\right)}$$
$$\leftdisplay{\hskip.5truein={\rm const}^{2\ell-1}\
\int\ d^d\t\ \prod_{i=1}^{2\ell-1}d^d\k_i\ \prod_{i=1}^{2\ell-1}d^d\p_i\ 
\prod_{i=1}^{2\ell-1}\ \left(k_F|\p_i|\right)^{1-{d\over 2}}
J_{{d\over 2}-1}(k_F|\p_i|)}$$ 
$$\leftdisplay{\hskip 2.1truein\
\prod_{i=1}^{2\ell-1}\ e^{i<\k_i,\p_i>}
\ {\widehat f}(\t)
\ e^{i<\t,\k_1+\ \cdots\ +\k_{2\ell-1}>}}$$
$$\leftdisplay{\hskip.5truein={\rm const}^{2\ell-1}\
\int\ d^d\t\  \prod_{i=1}^{2\ell-1}d^d\p_i
\ \prod_{i=1}^{2\ell-1}\ \left(k_F|\p_i|\right)^{1-{d\over 2}}
J_{{d\over 2}-1}(k_F|\p_i|)
\ \prod_{i=1}^{2\ell-1}\ \delta(\p_i+\t)\ {\widehat f}(\t)}$$
$$\leftdisplay{\hskip.5truein={\rm const}^{2\ell-1}\ \int\
\ d^d\t\ \ \left\{\left(k_F|\t|\right)^{1-{d\over 2}}
J_{{d\over 2}-1}(k_F|\t|)\right\}^{2\ell-1}
\ {\widehat f}(\t)\ .}
$$
\vskip .25truein

The classical estimates %(Stein and Weiss P.158) 
$$J_\alpha(r)\sim \const\,r^\alpha\ ,\ \ r\rightarrow 0$$
$$J_\alpha(r)=O\left(r^{-{1\over 2}}\right)\ ,\ \ r\rightarrow \
\infty\ $$ imply that $$ \left(k_F|\t|\right)^{1-{d\over 2}}
J_{{d\over 2}-1}(k_F|\t|) =\cases{O(1) & ${\rm for\ small}\ |\t| $\cr
O\left( |\t|^{{1-d\over 2}}\right) & $ {\rm for\ large}\ |\t| $\cr} $$
Consequently, $${\widehat f}(\t)={\rm const}\ \int d^d\p\
e^{-i<\p,\t>}\ f(|\p|)\ =\ {\rm const}\ \int_0^\infty dr\ r^{d-1}
\int_{S^{d-1}}d\p^\prime\ e^{-i<r\p^\prime,\t>}\ f(r)$$
$$
\leftdisplay{\hskip .55truein={\rm const}\ \int_0^\infty dr
\ r^{d-1}\ \left(r|\t|\right)^{1-{d\over 2}}J_{{d\over 2}-1}(r|\t|)\ f(r)}
$$ 
$$
\leftdisplay{\hskip .55truein={\rm const}
\ (1+|m|+\ell)\ M^j\ \int_0^\infty dr\ r^{d-1}
\ \left(r|\t|\right)^{1-{d\over 2}}
J_{{d\over 2}-1}(r|\t|)\ {f(r)\over \|f\|_1}} $$ 
$$\leftdisplay{\hskip.55truein=
\cases{O\left((1+|m|+\ell)M^j|\t|^{{1-d\over 2}}\right) & 
${\rm for}\ |\t|\ \ge\ 1 $\cr O\left((1+|m|+\ell)M^j \right) & 
${\rm for}\ |\t|\ \le\ 1$\cr}}$$

We now obtain $$\eqalign{N_\ell\ &\le\ {\rm const}^\ell\
M^{-(2\ell-1)(d-1)j}\ \int d^d\t\ \
\left\{\left(k_F|\t|\right)^{1-{d\over 2}} J_{{d\over
2}-1}(k_F|\t|)\right\}^{2\ell-1}
\ {\widehat f}(\t)\cr
&\le\ {\rm const}^\ell\ (1+|m|)\ M^{-(2\ell-1)(d-1)j}M^j
\ \left\{\int_{|\t|\ \ge\ 1}d^d\t\ |\t|^{-\ell(d-1)}\ +
\ \int_{|\t|\ \le\ 1}d^d\t\right\}\cr
&\le\ \cases{ {\rm const}^\ell\ (1+|m|)\ M^{-(2\ell-1)(d-1)j}M^j &
${\rm for}\ \ell(d-1)\ >\ d $\cr {\rm logarithmically\
divergent}&${\rm for}\ \ell(d-1)= d $\cr}}$$
\vskip .25truein

If $\ \ell(d-1)= d\ $, then $\ \ell=d=2\ $ and we consider the
integral $$\int d^d\t\ \left\{\left(k_F|\t|\right)^{1-{d\over 2}}
J_{{d\over 2}-1}(k_F|\t|)\right\}^3
\ {\widehat f}(\t)$$
Of course $f(t)$ is Schwartz class and hence so is $\widehat f$. The
convergence of the integral is not in question.  However each
derivative of $\ f\ $ produces a factor of $\ M^{-j}\ $.  Thus, we may
gain a power of $\ |\t|^{-2}\ $ in our estimate of $\ {\widehat
f}(\t)\ $ only at the cost of one $\ M^{-2j}\ $.  Taking the geometric
mean between $$|{\widehat f}(\t)|\le{\rm const}\, (1+|m|)\
M^j|\t|^{-{1\over 2}}$$ and $$|{\widehat f}(\t)|\le{\rm const}\,
(1+|m|)\ M^{-j}|\t|^{-{5\over 2}}$$ one obtains for every $\
\epsilon>0\ $ $$|{\widehat f}(\t)|\le{\rm const}\, (1+|m|)\
M^{(1-2\epsilon)j} |\t|^{-({1\over 2}+2\epsilon)}\ .$$ The constant in
the last line is independent of $\ \epsilon\ $. Finally, for $\ d=2\ $
$$N_2\le{\rm const}\ (1+|m|)\ M^{-3j}\ M^j
\ \left\{M^{(-2\epsilon)j}\ \int_{|\t|\ \ge\ 1}d^2\t
\ |\t|^{-(2+2\epsilon)}\ +\ 1\right\}$$
$$=\ {\rm const}\, (1+|m|)\ M^{-3j}\ M^j
\ \left\{M^{(-2\epsilon)j}\ {\rm const}\ \epsilon^{-1} +\ 1\right\}\ .$$
Taking the limit $\ \epsilon \rightarrow 0\ $ gives the desired
result.
\endproof

We now turn to the proof of Lemma 3.  Actually we shall prove the
stronger
\lem{\  $3^\prime$} {\it
Let $d=2$ and $\ell \ge 4$. Let $\Omega_2,\dots,\Omega_\ell$ be 
intervals on the Fermi circle each of length $\Omega\ge M^{j/2}$. 
Let $\Sigma_1$ be a fixed sector. The number of 
$(\ell-1)$-tuples of sectors $\{\Sigma_{2},\cdots,\Sigma_{\ell}\}$ for 
which there exist $\ \k_i\in\bbbr^2,\ i=1,\cdots,\ell\ $ 
satisfying 
$$
\k_{i}^\prime\in \Sigma_{i}\cap\Omega_i,
\ \  |\k_i-\k_i^\prime|\le{\rm const}\, M^j
\ , \ \ \ i=1,\cdots,\ell\ 
$$ 
and 
$$ 
|\k_1+\ \cdots\ +\k_{2\ell}|\le{\rm const}\,\left(1+|m|\right)M^j 
$$ 
is bounded by 
$$
\const^{\ell}\,(1+|m|)^2\left(\Omega M^{-j/2}\right)^{\ell-3}\ .
$$ }
\prf
Denote by $\r_n$ the center of the sector containing $\k_n$. Renumber
$\k_2,\cdots,\k_\ell$ so that $|\r_\ell\cdot\r_{\ell-1}|$ is minimal
amongst all $\left\{|\r_n\cdot\r_p|\ \big|\ n,p\ne 1\right\}$. In
other words $\phi=\angle(\k^\prime_{\ell-1},\k^\prime_\ell)$ is as 
close to $\pi/2$ as possible. All other 
$\angle(\k^\prime_{n},\k^\prime_{p})$'s with $\ n,p\ge 2\ $ 
must be within $\phi+O(M^{j/2})$ of either 0 or $\pi$. 

When $M^i\le\phi\le M^{i+1}$ or $M^i\le\pi-\phi\le M^{i+1}$ the
number of accessible $(\ell-1)$-tuples of sectors is bounded by
$$
N_\ell\prod_{i=2}^{\ell-2}\min\left\{\Omega,M^i\right\}M^{-j/2}
$$
where $N_\ell$ is the number of sectors accessible to the last two
$\k$'s once the sectors for $\k_2,\cdots,\k_{\ell-2}$ have been 
fixed.  We shall shortly show that $N_\ell\le \const^\ell\,(1+|m|)^2$.
The sum over those $i$'s with $M^i\le\Omega$ is bounded by
$$
\const^\ell\,\sum_{i\atop M^i\le\Omega} (1+|m|)^2M^{i(\ell-3)}
M^{-j(\ell-3)/2}
\le \const^\ell\,(1+|m|)^2 \left(\Omega M^{-j/2}\right)^{\ell-3}
$$
provided $\ell\ge 4$. Now consider $M^i>\Omega$. Once the sectors
of all the $\k_i$'s except the $\ell^{\rm th}$ have been selected
there can be at most one $i$ consistent with $\k_\ell$ falling
in $\Omega_\ell$. For this one value of $i$
$$\eqalign{
N_\ell\prod_{i=2}^{\ell-2}\min\left\{ \Omega ,M^i\right\}M^{-j/2}
&\le \const^\ell\,(1+|m|)^2 \left(\Omega M^{-j/2}\right)^{\ell-3}
}$$

It suffices to consider $i$ obeying 
$\ M^i >\const^\ell(1+|m|)M^{j/2}\ $ 
so fix any such $i$ and $\ \Sigma_1,\cdots,\Sigma_{\ell-2}$. We now
compute $\a$ and $\epsilon$, defined by
$$\eqalign{
\a&=-\r_1-\,\cdots\,-\r_{\ell-2}\cr
\a+\epsilon&=\k_{\ell-1}^\prime+\k_\ell^\prime\cr
&=-\k_1-\,\cdots\,-\k_{\ell-2}+O\!\big((1+|m|)M^j\big)\ .
}$$
Chose a coordinate system in which $\ \r_2=(k_F,0)\ $. Then, since
$\ \theta_n=\angle(\r_2,\r_n),\ $ is within 
$\ 2M^{i+1}\ $ of 0 or $\pi$ the $x$ and $y$ coordinates of every
$\r_n$ and $\k_n$, $\ \ 2\le n\le\ell,\ $ obey
$$\eqalign{
x_n&=\pm[k_F+O\!\big(M^j\big)]\cos O\!\big(M^i\big)
=\pm k_F+O\!\big(M^{2i}\big)\cr
y_n&=[k_F+O\!\big(M^j\big)]\sin O\!\big(M^i\big)=O\!\big(M^i\big)
}$$
respectively. Consequently $\ \k_1=-\k_2-\,\cdots\,-\k_{\ell}\ $
and hence $\r_1$ obey
$$\eqalign{
x_1&=\pm k_F+O\!\big((|m|+\ell)M^{2i}\big)\cr
y_1&=O\!\big((|m|+\ell)M^i\big)
}$$
We need to know with greater precision how much $\k_n$ can wiggle.
$$\eqalign{
\k_n-\r_n&=\k_n^\prime-\r_n+O\!\big(M^j\big)\cr
&=k_F\Big(\cos\big(\theta_n+O\!(M^{j/2})\big),
\sin\big(\theta_n+O\!(M^{j/2})\big)\Big)-k_F\Big(\cos\theta_n,
\sin\theta_n\Big)+O\!\big(M^j\big)\cr
&=O\!\big((M^iM^{j/2},M^{j/2})\big)\ .
}$$
Summing the individual wiggles yields
$$\eqalign{
\epsilon&=\sum_{n=1}^{\ell-2}(\r_n-\k_n)+O\!\big((1+|m|)M^j\big)\cr
&=(|m|+\ell)\,O\!\big((M^iM^{j/2},M^{j/2})\big)
}$$
and
$$\eqalign{
\a&=-\r_1-\,\cdots\,-\r_{\ell-2}\cr
&=\k_{\ell-1}+\k_\ell-\sum_{n=1}^{\ell}\k_n
+\sum_{n=1}^{\ell-2}(\k_n-\r_n)\cr
&=N(2k_F,0)+O\!\big((M^{2i},M^{i})\big)
+O\!\big((1+|m|)M^{j})\big)+\ell\, O\!\big((M^iM^{j/2},M^{j/2})\big)\cr
&=N(2k_F,0)+O\!\big((M^{2i},M^{i})\big)
}$$
where $\ N\in\{1,0,-1\}\ $.

We are now in a position to bound $N_\ell$. There are two cases to be 
considered. First, suppose $|N|=1$. Rotate the coordinate system by
$\ \pi\delta_{N,-1}+O\!\big(M^{i}\big)\ $ to make $\a$ run along the 
positive $x$ axis. In the new coordinates,  
$$
\epsilon=(|m|+\ell)\,O\!\big((M^iM^{j/2},M^{j/2})\big)\ .
$$
is still obeyed.\hfill\break
\figplace{lce3}{1.25in}{-.35in}
\vskip.25truein

Then the two components of
$$
k_F\big(\cos\alpha,\sin\alpha\big)
+k_F\big(\cos(\phi-\alpha),-\sin(\phi-\alpha)\big)=\a+\epsilon
$$
are
$$\eqalign{
\cos\alpha+\cos(\phi-\alpha)&={|\a|\over k_F}+(|m|+\ell)\,
O\!\big(M^iM^{j/2}\big)\cr
\sin\alpha-\sin(\phi-\alpha)&=(|m|+\ell)\,O\!\big(M^{j/2}\big)\ .\cr
}$$
The $y$ component implies that
$$
|2\alpha-\phi|=(|m|+\ell)\,O\!\big(M^{j/2}\big)
$$
so that there are $\ \const\,(|m|+\ell)\ $ sectors accessible to
$\k_\ell$ (once $\r_{\ell-1}$ has been fixed) and
$$
\alpha={1\over 2}\phi+(|m|+\ell)\,O\!\big(M^{j/2}\big)\ .
$$
is bounded above and below by $\ \const\,M^i$. Since
$$\eqalign{
\cos\alpha+\cos(\phi-\alpha)
&=\cos\alpha+\cos\alpha\cos(\phi-2\alpha)
-\sin\alpha\sin(\phi-2\alpha)\cr
&=\left[2+(|m|+\ell)^2\,O\!\big(M^{j}\big)\right]\cos\alpha
+(|m|+\ell)\,O\!\big(M^iM^{j/2}\big)\cr
}$$
the $x$ component gives
$$
\alpha=\cos^{-1}\left({|\a|\over 2k_F}\right)
+(|m|+\ell)\,O\!\left({M^iM^{j/2}\over M^i}\right)
$$
Thus, there are $\ \const\,(|m|+\ell)\ $ sectors accessible to
$\k_{\ell-1}$ as well.

Finally, suppose that $N=0$. This time rotate the coordinate system 
by $\ O\!\big(M^{i}\big)\ $ or $\ \pi+O\!\big(M^{i}\big)\ $so that 
$\k_{\ell-1}$ runs along the negative $x$ axis.
\hfill\break
\figplace{lce4}{1.5in}{-.1in}\hfill\break
\noindent The angle $\phi$ is determined by
$$\eqalign{
\sin\left({\pi-\phi\over 2}\right)
&={\left|\a+\epsilon\right|\over 2k_F}\cr
&={|\a|\over 2k_F}+(|m|+\ell)\,O\!\big(M^{j/2}\big)\ .
}$$
Thus
$$
\phi=\pi-2\sin^{-1}\left({|\a|\over 2k_F}\right)
+(|m|+\ell)\,O\!\big(M^{j/2}\big)
$$
and $\k_\ell$ has access to $O\!\big(|m|+\ell\big)$ sectors when 
$\r_{\ell-1}$ is held fixed. The angle $\theta$ is determined by
$$\eqalign{
\left|\sin\left(\theta-{\phi\over 2}\right)\right|
&={\left|\epsilon\cdot\left(\cos\left({\pi-\phi\over 2}\right)
                 ,\sin\left({\pi-\phi\over 2}\right)\right)\right|
           \over|\a|}\cr
&\le{(|m|+\ell)\,O\!\big(M^iM^{j/2}\big)
\over M^i-(|m|+\ell)\,O\!\big(M^{j/2}\big)}\cr
&\le(|m|+\ell)\,O\!\big(M^{j/2}\big)\ .
}$$
In the second last line we used the hypothesis that 
$\ M^i\le\phi\le M^{i+1}\ $. This forces 
$$\eqalign{
\left|\a+\epsilon\right|
&=2k_F\sin\left({\pi-\phi\over 2}\right)\cr
&\ge \const\,M^i\ .
}$$
Once again there are at most $O\!\big(|m|+\ell\big)$ sectors
accessible to $\k_{\ell-1}$.
\endproof 

%\vfill\eject


%\input myfig
\def\eqtwo{I.2}
\def\eqsix{I.6}
\def\eqsev{I.7}

\vskip.25truein
\noindent{\subchfont \sec III The Full Expansion}
\vskip.25truein

In this Section we elaborate on the expansion presented in the
introduction to prove a lemma and the two theorems. All are of
the form of Lemma 5, but treat the true propagator
and interaction. They are restricted to three space-time dimensions.
In Lemma 6, we bound Schwinger functions. Theorem 1 constructs the
effective potential. Finally in Theorem 2, we generalize the expansion
to accommodate $\nu$-body interactions, $\nu\ge 2$, of the type
generated by the renormalization group flow of [FT2].

Throughout this section we expand the true propagator $$
C^{(j)}=\sum_{\ell=1}^{M^{-j/2}} C^{(j,\ell)} $$ where $$
C^{(j,\ell)}(\xi,\bar \xi)=\delta_{\sigma,\bar\sigma}
\int {d^{3}p\over(2\pi)^{d+1}} {e^{i<p,\xi-\bar\xi>_-}
\over ip_0-e(\p)}f_j(p)\,\zeta_\ell(\p)\ .
$$ 
The partition of unity
$$
1=\sum_{\ell=1}^{M^{-j/2}}\zeta_\ell(\p),\hskip .25truein
\zeta_\ell(\p)=\zeta_\ell\left({\p\over|\p|}k_F\right)
$$
divides the Fermi circle into long sectors $\Sigma_\ell$.  We denote
by $r_\ell$ the center of $\Sigma_\ell$. Recall that 
$$
<p,\xi>_-=\p\cdot\x-p_0\tau\ , 
$$ 
$$ 
e(\p)={\p^2\over 2m}-\mu.  
$$ 
and
$$
 f_j(p)=f\left(M^{-2j}\left(p_0^2+e(\p)^2\right)\right).  
$$ 
The
corresponding decomposition of the fields is 
$$
\psi(\xi)=\sum_{\ell=1}^{M^{-j/2}} \psi^{(\ell)}(\xi)\hskip.5in
\bar\psi(\xi)=\sum_{\ell=1}^{M^{-j/2}} \bar\psi^{(\ell)}(\xi)\ .
$$



For each sector $\Sigma_\ell$ we introduce a lattice ${\bf D}_{\ell}$
of rectangular parallelepipeds (called boxes for short) that pave
$\bbbr^{3}$.  These boxes have three axes. One axis is the fixed time
direction and the length of any box in that direction is $M^{-j}$.  In
the orthogonal $\bbbr^{2}$ plane, one of the axes is $r_{\ell}$, the
center of $\Sigma_\ell$, and the length in this direction is $M^{-j}$.
The third axis is orthogonal to $r_{\ell}$ and has length $M^{-j/2}$.


Finally, the two-body interaction in volume $\Lambda\subset \bbbr^3$
is $$ {\cal V}_\Lambda={1\over 2}\hskip-1.5pt
\int \hskip-1.5pt d^3\xi\prod_{i=1}^4d^3\eta_i \ \chi_\Lambda(\xi)
V(\eta_1,\eta_2,\eta_3,\eta_4)
\delta(\eta_1\!+\!\hskip1pt\eta_2\!+\!\hskip1pt\eta_3\!+\hskip1pt\!\eta_4)
{\bar\psi}(\xi+\eta_1)\psi(\xi+\eta_3)
{\bar\psi}(\xi+\eta_2)\psi(\xi+\eta_4).  $$ It is convenient for us to
restrict only the center of mass $\xi$ to $\Lambda$.

\lem{\ 6} {\it Let
$$ {\cal A}=\int \prod_{j=1}^{2p} d\xi_j\ A(\xi_1,\cdots,\xi_{2p})
\prod_{j=1}^{2p}\optbar\psi(\xi_j)
$$ be a monomial of degree $2p$. Define $$
\<{\cal A}\>={1\over Z_\Lambda}\int{\cal A} 
e^{-\lambda{\cal V}_\Lambda(\psi,\bar\psi)}
d\mu_{C^{(j)}}(\psi,\bar\psi) $$ where $$ Z_\Lambda=\int
e^{-\lambda{\cal V}_\Lambda(\psi,\bar\psi)}
d\mu_{C^{(j)}}(\psi,\bar\psi)\ .  $$ Then, there exists a $\ \const\
$, independent of $j$ and $\Lambda$, such that the perturbation series
$$
\<{\cal A}\>=\sum_{n=0}^\infty a_n(j,\Lambda)\lambda^n
$$ converges for all $|\lambda|<R=(\const\norm V\norm )^{-1}$ and the
coefficients satisfy $$ |a_n(j,\Lambda)|\le\const^{n+p}M^{pj/2}\|A\|_1
\norm V\norm^n\ .
$$ Furthermore the limits $$ a_n(j)=\lim_{\Lambda\rightarrow \bbbr^3}
a_n(j,\Lambda) $$ exist. }

In particular the infinite volume single slice Schwinger functions
exist and obey good $j$-dependent bounds.

\prf As in the introduction our expansion is developed inductively. 
Once again, at the end of step $s$ we have a sum over a set ${\cal
T}_s$ of terms
\def\eqIIIone{III.1}
$$\eqalign{
\<{\cal A}\>&=\sum_{T\in {\cal T}_s} {\int{\cal A}(T,s,\Lambda)
e^{-\lambda{\cal V}_\Lambda(\psi,\bar\psi)}d\mu_{C^{(j)}}
\over
\int e^{-\lambda{\cal V}_\Lambda(\psi,\bar\psi)}d\mu_{C^{(j)}}  }\cr
&=\sum_{T\in {\cal T}_s\atop \pi(T,s)=0}{\cal A}(T,s,\Lambda)
+\sum_{T\in {\cal T}_s\atop \pi(T,s)\ne 0} {\int{\cal A}(T,s,\Lambda)
e^{-\lambda{\cal V}_\Lambda(\psi,\bar\psi)}d\mu_{C^{(j)}}
\over
\int e^{-\lambda{\cal V}_\Lambda(\psi,\bar\psi)}d\mu_{C^{(j)}} }\ \ \ .
}\eqn{\eqIIIone}$$ Recall that each ${\cal A}(T,s,\Lambda)$ is a
monomial in the fields $\psi,\bar\psi$ of degree $2\pi(T,s)$.  We used
the fact that the ratio of integrals simplifies to the number ${\cal
A}(T,s,\Lambda)$ when $\pi(T,s)=0$.  Each ratio in the second sum will
be $O\left(\lambda^s\right)$.

We now outline the $(s+1)^{\rm th}$ step in the induction. The details
will be presented shortly. First pick any field of ${\cal
A}(T,s,\Lambda)$.  Apply the integration by parts formula (\eqsix). In
the event that the derivative brings a new vertex down from the
exponent, expand the new vertex as a sum over momentum sectors $\ell$
and position space boxes $\Delta\in {\bf D}_\ell$. Here, this step is
considerably more complicated than (\eqsev).  Next, pick any other
field of the original monomial ${\cal A}(T,s,\Lambda)$ and repeat the
construction. When all the fields of ${\cal A}(T,s,\Lambda)$ have been
exhausted, substitute Taylor expansions much as in \sec I.


The sum over sectors and conjugate boxes is complicated because
conservation of momentum must be exploited to restrict the sums over
sectors at every vertex. We now consider this in more detail. The
volume cutoff interaction is 
$$\eqalign{ {\cal V}_\Lambda&={1\over 2}
\int d^3\xi\prod_{i=1}^4d^3\eta_i \ 
\chi_\Lambda(\xi)V(\eta_1,\eta_2,\eta_3,\eta_4)
\delta(\Sigma \eta_i)
{\bar\psi}(\xi\!+\!\eta_1)\psi(\xi\!+\!\eta_3)
{\bar\psi}(\xi\!+\!\eta_2)\psi(\xi\!+\!\eta_4)\cr &={1\over 2}\int
d^3\xi\prod_{i=1}^4\left(d^3\eta_i{d^3k_i\over (2\pi)^3}\right)
\ \ \chi_\Lambda(\xi)
V(\eta_1,\eta_2,\eta_3,\eta_4)\delta(\Sigma \eta_i)
{\bar\psi}(k_1)\psi(k_3){\bar\psi}(k_2)\psi(k_4)\cr &\hskip2in
e^{-i<k_1,\xi+\eta_1>_-}e^{i<k_3,\xi+\eta_3>_-}e^{-i<k_2,\xi+\eta_2>_-}
e^{i<k_4,\xi+\eta_4>_-}\cr
&={1\over 2}\int
\prod_{i=1}^4{d^3k_i\over (2\pi)^3}\ 
{\widetilde \chi}_\Lambda(-k_1-k_2+k_3+k_4)
<k_1,k_2|V|k_3,k_4>\bar\psi(k_1)\psi(k_3)\bar\psi(k_2)\psi(k_4)\cr
&={1\over 2}\sum_{m\in {\bf Z}^3}\int\prod_{i=1}^4{d^3k_i\over
(2\pi)^3}\
\chi(k_1+k_2-k_3-k_4+M^jm)
{\widetilde \chi}_\Lambda(-k_1-k_2+k_3+k_4)\cr
&\hskip2in<k_1,k_2|V|k_3,k_4>
\bar\psi(k_1)\psi(k_3){\bar\psi}(k_2)\psi(k_4)\ .
}$$ 
Here $$ 1=\sum_{m\in {\bf Z}^3}\chi(M^{-j}k+m) $$ is a partition of
momentum space by $C^\infty_0$ functions supported on cubes of side
$M^j$ and 
$$\eqalign{
 <k_1,k_2|V|k_3,k_4>&=\int\prod_{i=1}^4d^3\eta_i
V(\eta_1,\eta_2,\eta_3,\eta_4)\delta(\eta_1+\eta_2+\eta_3+\eta_4)\cr
&\hskip.5in
e^{-i<k_1,\eta_1>_-}e^{i<k_3,\eta_3>_-}e^{-i<k_2,\eta_2>_-}
e^{i<k_4,\eta_4>_-}.  
}$$ 
Reversing the calculation above expresses
\def\eqIIItwo{III.2}
$$\eqalign{ 
{\cal V}=&\sum_{m\in {\bf Z}^3} {1\over 2}\int
d^3\xi\prod_{i=1}^4d^3\eta_i \
\chi_{m,\Lambda}(\xi)V(\eta_1,\eta_2,\eta_3,\eta_4)
\delta(\Sigma \eta_i){\bar\psi}(\xi\!+\!\eta_1)\psi(\xi\!+\!\eta_3)
{\bar\psi}(\xi\!+\!\eta_2)\psi(\xi\!+\!\eta_4)\cr &=\sum_{m\in {\bf
Z}^3} {1\over 2}\int \prod_{i=1}^4d^3\xi_i \
\chi_{m,\Lambda}\left({\xi_1+\xi_2+\xi_3+\xi_4\over 4}\right)
V(\xi_1,\xi_2,\xi_3,\xi_4)
{\bar\psi}(\xi_1)\psi(\xi_3){\bar\psi}(\xi_2)\psi(\xi_4)
}\eqn{\eqIIItwo}$$ 
where $\chi_{m,\Lambda}$ is the inverse Fourier
transform of $\chi(M^{-j}k+m){\widetilde \chi}_\Lambda(k)$. Even if the
interaction $V$ is of compact support, with respect to the center of
mass, $\chi_{m,\Lambda}V$ is no longer supported in a compact
neighborhood of $\Lambda$ because $\chi$ is of compact support in
momentum space. This is no problem. The vertex is already connected by
propagators to the external generalized vertex ${\cal A}$, so
integration over the center of mass variable is taken care of.

Only two properties of $\chi_{m,\Lambda}$ are required. The first is
\def\eqIIIthr{III.3}
$$\eqalign{
\sup_{\xi\in \bbbr^3} \left|\chi_{m,\Lambda}(\xi)\right|
&\le\sup_\xi \left|M^{3j}\int d^3\eta\ e^{i<\eta,M^jm>_-}
{\widetilde\chi}(M^j\eta) {\chi}_\Lambda(\xi-\eta)\right|\cr &\le 
{\rm const}_N\left(1+|m|\right)^{-N} }
\eqn{\eqIIIthr}$$
independent of $j$, provided derivatives acting on $\chi_\Lambda$ 
produce, at worst, $M^j$'s. In other words $\chi_\Lambda$ must decay 
from 1 down to zero in a distance $O(M^{-j})$. This is a consequence of 
the standard integration by parts trick, as in the proof of Lemma 4, and 
$$
\left|{\nabla_\eta}^b {\widetilde\chi}(\eta)\right|
\le \const^b\  {1\over (1+|\eta|)^{4}}\ .
$$

Once conservation of momentum at a vertex has been implemented by
(\eqIIItwo), each field at the vertex is expanded in sectors.  Recall
that the vertex was produced by an application of the integration by
parts formulae (\eqsix) to a source field. So, one of its legs
inherits its sector number from this source field.  A priori,
decomposing each of the three remaining fields into a sum over sectors
could produce $M^{-3j/2}$ terms. The second property of
$\chi_{m,\Lambda}$ is that there are only $O(|m|^{2} M^{-j/2})$
nonzero terms because of the constraints imposed by conservation of
momentum and the fact that we are in two space dimensions. See Lemma
3.

Finally, each field $\optbar\psi^{(\ell)}$ is expanded in boxes
$\Delta\in{\bf D}_\ell$. Altogether, the ``sum over sectors and
conjugate boxes'' operation replaces (\eqsev) by
\def\eqIIIfou{III.4}
$$\eqalign{ {\cal V} &=\sum_{m\in {\bf
Z}^3}\sum_{\ell_1,\ell_2,\ell_3,\ell_4} {1\over 2}\prod_{i=1}^4
\left(\sum_{\Delta_i\in{\bf D}_{\ell_i}}\int_{\Delta_i} d^3\xi_i\right) 
\ \chi_{m,\Lambda}\left({\xi_1+\xi_2+\xi_3+\xi_4\over 4}\right)\cr
&\hskip2in \times V(\xi_1,\xi_2,\xi_3,\xi_4)
{\bar\psi}^{(\ell_1)}(\xi_1)\psi^{(\ell_3)}(\xi_3)
{\bar\psi}^{(\ell_2)}(\xi_2)\psi^{(\ell_4)}(\xi_4) }\eqn{\eqIIIfou}$$
In step $s=0$ the only fields downstairs are those belonging to the
monomial ${\cal A}$. One decomposes each field in sectors
$\Sigma_\ell$ and then in conjugate boxes $\Delta\in {\bf D}_\ell$.
There is no need for the sum over $m$ associated with conservation of
momentum.


We describe the Taylor expansions in more detail. Now there are
sectors to be taken into account. Let $\pi(s+1,\ell,\Delta,\sigma,b)$
be the number of fields with the specified values of $\sigma, b, \ell$
and $\Delta\in{\bf D}_\ell$.  For each $\ell,\sigma,b$ and $\Delta\in
{\bf D}_\ell$ we divide $\Delta$ into
$\pi(s+1,\ell,\Delta,\sigma,b)^\epsilon$ identical sub-boxes (really
rectangular parallelepipeds) $\delta$ each similar to $\Delta$. Thus
each little box has dimensions $$
\pi(s+1,\ell,\Delta,\sigma,b)^{-{\epsilon\over 3}}M^{-j}\times
\pi(s+1,\ell,\Delta,\sigma,b)^{-{\epsilon\over 3}}M^{-j}\times
\pi(s+1,\ell,\Delta,\sigma,b)^{-{\epsilon\over 3}}M^{-j/2}.
$$The value of $\epsilon$ will be picked later.  As in \sec I the
center of $\delta$ is called $c_\delta$ and the number of fields in
$\delta$ with specified values of $\ell,\sigma$ and $b$ is called
$\pi(s+1,\ell,\delta,\sigma,b)$.

The action of derivatives, given by Lemma 4.b, differs from (\eqtwo)
in several respects. In particular the appearance of the phase $e^{-i
<r_\ell,\xi-\bar\xi>_-}$ forces us to modify the Taylor expansion.
Based on this remark we define $$\eqalign{
\Psi^{(\ell)}(\xi)&=e^{-i<\r_\ell,\xi>}\psi^{(\ell)}(\xi)\cr
\bar\Psi^{(\ell)}(\xi)&=e^{i<\r_\ell,\xi>}\bar\psi^{(\ell)}(\xi).
}$$ We expand each $\Psi^{(\ell)}(\xi),\bar\Psi^{(\ell)}(\xi)$
downstairs in a Taylor polynomial
\def\eqIIIfiv{III.5}
$$\eqalign{
\optbar\Psi^{(\ell)}(\xi)=\optbar\Psi^{(\ell)}(c_\delta)+\ \cdots\ 
+&{1\over t!}\big((\xi-c_\delta)\cdot\nabla_\xi\big)^{t}\
\optbar\Psi^{(\ell)}(c_\delta)\cr
&+{1\over t!}\int_0^1dw (1-w)^{t}
\big((\xi-c_\delta)\cdot\nabla_\xi\big)^{t+1}\ 
\optbar\Psi^{(\ell)}(c_\delta+w(\xi-c_\delta))
}\eqn{\eqIIIfiv}$$ with $\delta$ being the box that contains $\xi$. As
before, the Taylor expansion is done inside the $\xi$ integrals. One
might worry that the restriction to a very small box $\delta$ weakens
conservation of momentum and consequently increases the number of
nonzero terms in the sector sums.  However, the sector sums have
already been cut down, so there is no problem.

Each Taylor expansion contains $$ 1+3+3^2+\cdots+3^t+3^{t+1}\le
2\times3^{t+1} $$ terms. Note that, by Lemma 4.b, each
$(\xi-c_\delta)\cdot\nabla_\xi$ that acts on a $\optbar\Psi^{(\ell)}$
produces a factor of $$
\pi(s+1,\ell,\Delta,\sigma,b)^{-{\epsilon\over 3}}
\left[M^{-j}M^{j}+M^{-j}M^{j}+M^{-j/2}M^{j/2}\right]
=3\pi(s+1,\ell,\Delta,\sigma,b)^{-{\epsilon\over 3}}.  $$ Since the
fields anticommute, any nonzero integral may contain at most one field
having any given value of $\ell,\sigma,b$ at any $c_\delta$.  The same
is true for each derivative of the fields. Thus, for each
$\ell,\sigma,b,\delta$, all but $2\times 3^{t}$ of the fields having
this $\ell,\sigma,b$ and located in $\delta$ must be Taylor
remainders. That is, there are at least
$\pi(s+1,\ell,\delta,\sigma,b)-2\times 3^{t}$ Taylor remainders. This
completes the description of the expansion.

We use the same strategy for performing the estimates as in Lemma 5.
Here are the combinatorial factors used to control each of the three
operations.\vskip.25truein
\noindent(1){\it Integration by parts.}  
\item{-}A factor of two, assigned to the leg 
initiating the integration by parts suffices to decide whether or not
the leg brings down a new vertex from the exponent.

\noindent When it does, 
\item{-}a factor of two, 
assigned to the target leg, counts the number of possible target legs.


\noindent When it doesn't, we count the number of possible source legs for
each target leg by applying the rules of Lemma 5 in each sector
$\ell$.  Precisely, for each target leg, we organize the sum over
source legs according first, to the cube $\Delta\in {\bf D}_\ell$ of
the source and second, to which leg in $\Delta$ is the source.
\item{-}A factor of two per leg suffices to decide whether or not the leg 
is a target leg.
\item{-}A factor of 
$\ \const\rho^{(j,\ell)}(\Delta,\Delta^\prime)^{4}\ $, assigned to the
line generated, suffices to control the sum over $\Delta$.
\item{-}A factor $\pi(s+1,\ell,\Delta,\sigma,b)$, assigned to the source 
leg, counts the number of possible source legs within $\Delta$.

\noindent(2){\it Sums over sectors and conjugate boxes.}\ \  
The sum $\ \sum_{m\in \bbbz^3}\ \cdots\ \chi_{m,\Lambda}\ $ is
controlled by
\item{-} a factor $\ \const|m|^{4}\ $ assigned to the vertex.

\noindent When a vertex first moves downstairs from the exponent the 
sector of one of 
its legs is fixed by the source leg that initiates the process. By
Lemma 3 the number of sectors accessible to the remaining legs is
bounded by
\item{-} a factor $\ \const(1+|m|^{4/3})M^{-j/2}$  assigned to the vertex.
\item{-} If the external vertex does not impose any constraint on the 
sector sums of its $2p$ legs then the number of sectors accessible is
$M^{-pj}$.

\noindent The sums $\sum_{\Delta_i\in D_{\ell}}$ are not controlled by 
combinatorial factors. They will be performed explicitly.


\noindent(3){\it Pauli exclusion principle.} 
\item{-}The Taylor expansion splits each leg into at most 
$\ 2\times 3^{t+1}\ $ pieces.  \vskip .25truein It remains only to
bound an integral. The integration variables are the positions of the
fields of ${\cal A}$ and the interaction vertices ${\cal V}$ that have
been brought down from the exponent. The integrand is the supremum,
over positions and diagrams, of the product of
\item{-} the above combinatorial factors
\item{-}$\const\rho^{(j,\ell)}(\Delta,\Delta^\prime)^{-\gamma}$ 
per line of the diagram
\item{-}$|\lambda|
\left(\sup_{\xi\in \bbbr^3} \left|\chi_{m,\Lambda}(\xi)\right|\right)
|V(\xi_1,\xi_2,\xi_3,\xi_4)|
\le {\rm const}|\lambda|\left(1+|m|\right)^{-N}|V(\xi_1,\xi_2,\xi_3,\xi_4)|
$ 
per vertex of the diagram (see (\eqIIIthr))
\item{-}$|A(\xi_1,\cdots,\xi_{2p})|$
\item{-}and the factors that come from the Taylor expansions used to 
implement the Pauli exclusion principle.

\noindent The latter are 
\def\eqIIIsix{\rm III.6}
$$\eqalign{ &\prod_s\prod_{\ell,\sigma,b}\ 
\prod_{\Delta\in {\bf D}_\ell}
\prod_{\delta\in \Delta} 
\left[3\pi(s,\ell,\Delta,\sigma,b)^{-{\epsilon\over 3}}
\right]^{(t+1)\big(\pi(s,\ell,\delta,\sigma,b)-2\times 3^{t}\big)}\cr 
&\le
\prod_s\prod_{\ell,\sigma,b}\ \prod_{\Delta\in {\bf D}_\ell}
\left[3\pi(s,\ell,\Delta,\sigma,b)^{-{\epsilon\over 3}}
\right]^{(t+1)\big(\pi(s,\ell,\Delta,\sigma,b)-2\times
3^{t}\pi(s,\ell,\Delta,\sigma,b)^\epsilon\big)}.\cr 
}\eqn{\eqIIIsix}$$
Pick any $\zeta>0$. It is possible to choose $\epsilon$ and $t$,
depending only on $\zeta$ so that 
$$
(\eqIIIsix)\le\prod_s\prod_{\ell,\sigma,b}\ \prod_{\Delta\in{\bf D}_\ell}
\const \pi(s,\ell,\Delta,\sigma,b)^{-\zeta \pi(s,\ell,\Delta,\sigma,b)}.
$$ Altogether
\def\eqIIIsev{III.7}
$$\eqalign{
\left|a_n(j,\Lambda)\right|
&\le \const^{n+p}\! \int \prod_i d\xi_i |A|\!
\left(\prod |V|\right)\!
\left(\prod \rho^{(j,\ell)}(\Delta,\Delta^\prime)^{-\gamma+4}\!
\right)\!
M^{{3\over 2}j{4n+2p\over 2}}M^{-jn/2}M^{-pj}\cr }\eqn{\eqIIIsev}$$
where the second product is over vertices and the third is over lines.
The first power of $M^j$ came from covariance bounds and the second
and third from sector sums for fields at interaction vertices and
${\cal A}$ respectively.  After discarding some of the decay factors
$\rho^{(j,\ell)}$, we can view the integrand as a connected tree with
generalized vertices $V$ and $A$ and lines $\rho^{(j,\ell)}$.
Integrate starting at the extremities of the tree and working towards
the root $A$.  Suppose that the extremal vertex $V$ is connected to
the tree by the argument $\xi_1$.  The integral over
$\xi_2,\xi_3,\xi_4$ with $\xi_1$ held fixed produces a $\norm V\norm
$. A decay factor $\rho^{(j,\ell)}$ is enough to fix the box in which
$\xi_1$ lives. The integral over $\xi_1$ within that box costs
$M^{-5j/2}$, the volume of the box.  Repeat for all the other $V$'s.
Finally, the integral over the arguments of ${\cal A}$ gives
$\|A\|_1$. In conclusion $$\eqalign{
\left|a_n(j,\Lambda)\right|&\le \const^{n+p} \|A\|_1\norm V
\norm^n M^{-5jn/2}
M^{3j(n+p/2)}M^{-jn/2}M^{-pj}\cr &=\const^{n+p} \|A\|_1\norm V\norm^n
M^{-pj/2}\ .  }$$


When the finite volume, i.e. $\Lambda$, constraint is removed, $a_n$
is expressed as an absolutely convergent series obeying the same
bound.\endproof

The next order of business is the effective potential $$ {\cal
G}_\Lambda^{(j)}(\psi^e,\bar\psi^e) =\log\,{1\over
Z_\Lambda^{(j)}}\int e^{-\lambda {\cal
V}_\Lambda(\psi+\psi^e,\bar\psi+\bar\psi^e)}
d\mu_{C^{(j)}}(\psi,\bar\psi).  $$ Let $G^{(j,\Lambda)}_p$ be the
$p$-point Green's function generated by ${\cal G}^{(j)}_\Lambda$. We
now prove the
\thm{\ 1}{\it Let $d=2$. There exists a $\ \const\ $, independent of $j$ and 
$\Lambda$, such that 
$$
\norm g_n(p,j,\Lambda)\norm \le\const^{n+p}M^{(2-5p/2)j}\norm V\norm^n
$$ 
where 
$$
\norm g_n(p,j,\Lambda)\norm=
\max_{k}\ \sup_{\xi_k}\ \int\prod_{i\ne k}d\xi_i\  
|g_n(p,j,\Lambda)(\xi_1,\cdots,\xi_{2p})|.  $$ Furthermore the limits
$$ g_n(p,j)=\lim_{\Lambda\rightarrow \bbbr^3}g_n(p,j,\Lambda) $$ exist
and the infinite volume Green's functions at scale $j$ $$
G^{(j)}_p=\sum_{n=0}^\infty g_n(p,j)\lambda^n $$ are analytic in
$|\lambda|<R=(\const\norm V\norm)^{-1}$.}
\vskip.25truein
\prf \ \  
To get started apply a single functional derivative
\def\eqIIIeig{III.8}
$$ {\delta\hfill\over \delta\psi^e}{\cal G}^{(j)}_\Lambda
=-\lambda{\int \left({\delta\hfill\over \delta\psi^e} {\cal
V}_\Lambda(\psi+\psi^e,\bar\psi+\bar\psi^e)\right) 
e^{-\lambda {\cal V}_\Lambda(\psi+\psi^e,\bar\psi+\bar\psi^e)}
d\mu_{C^{(j)}}(\psi,\bar\psi)
\over  
\int e^{-\lambda {\cal V}_\Lambda(\psi+\psi^e,\bar\psi+\bar\psi^e)}
d\mu_{C^{(j)}}(\psi,\bar\psi)}\ \ .
\eqn{\eqIIIeig}$$
Now expand as in Lemma 6. However, before each application of the
integration by parts formulae, decompose the downstairs of the
numerator into a polynomial in $\psi,\psi^e,\bar\psi,\bar\psi^e$. That
is, multiply out products of $\psi+\psi^e$ and $\bar\psi+\bar\psi^e$.
The integration by parts formulae are then applied to the
$\psi,\bar\psi$'s.  The expansion terminates for those monomials that
are independent of $\psi,\bar\psi$, i.e. that are functions of the
external fields $\psi^e,\bar\psi^e$ alone. For these terms the
monomial factors out of the numerator, leaving the quotient of two
identical integrals that cancel.

The rest of the expansion and most of the estimates are the same as
before.  However, the external legs enter the effective potential in a
way different from the Schwinger functions of Lemma 6.

The only combinatorial factors that change are those that count
sectors.  Lemma 3, the main tool for sector counting, does not apply
to vertices containing external legs, since the momentum of the
external leg need not be near the Fermi circle. It is still true that
when a vertex first moves downstairs from the exponent the sector of
one of its legs is fixed by the source leg that initiates the process.
By Lemma 3 the number of sectors accessible to the remaining legs is
bounded by
\item{-} a factor $\ \const(1+|m|^{4/3})M^{-j/2}$,  assigned to the vertex,
when the vertex contains no external legs.
\item{-} If the vertex does contain an external leg, there are at most two
 other internal legs. There are at most $\ M^{-j}$ sectors accessible to
these legs.
\item{-} The first external vertex was created by a functional derivative,
 rather
than integration by parts.  There are at most $\ M^{-3j/2}$ sectors
accessible to its internal legs.

The main bound
\def\eqIIIsevp{III.7'}
$$\eqalign{ &\norm g_n(p,j,\Lambda)\norm\cr &\le \const^{n+p}\! \int
\prod_i d\xi_i\
\left(\prod |V|\right)\!
\left(\prod \rho^{(j,\ell)}(\Delta,\Delta^\prime)^{-\gamma+4}\!
\right)\!
M^{{3\over 2}j{4n-2p\over 2}}M^{-jn/2}M^{-pj}M^{-j/2}
}\eqn{\eqIIIsevp}$$ is obtained from (\eqIIIsev) by deleting $|A|$ and
adjusting the powers of $M^j$. External lines are amputated, so that
the number of lines ${4n+2p\over 2}$ becomes ${4n-2p\over 2}$,
accounting for the first factor of $M^j$. The remaining powers of
$M^j$ come from the sector counts. We have included one sector sum per
vertex, plus one ``extra'' sector sum per external leg, plus an
additional supplement for the first vertex.  Finally, by definition of
the norm $\norm\ \cdot\ \norm$ the position of one argument $\xi_j$ of
one vertex is held fixed, removing one volume integral $M^{-5j/2}$.
Thus
\def\eqIIInin{III.9}
$$\eqalign{
\norm g_n(p,j,\Lambda)\norm
&\le \const^{n+p} \norm V\norm^n
M^{-5j(n-1)/2}M^{3j(n-p/2)}M^{-jn/2}M^{-pj}M^{-j/2}\cr &=\const^{n+p}
\norm V\norm^n M^{(2-5p/2)j}\ .  }\eqn{\eqIIInin}$$
\endproof

Under a renormalization group flow, the effective potential of scale
$j$ becomes the interaction at scale $j-1$. If we are to use an
expansion like that of Theorem 1 in such a setting, we must allow
the interaction to contain monomials of degree $2q$ for all $q\ge 1$,
not just $q=2$. On the other hand we must retain the memory that the
$2q$-legged monomial began life as a bunch of four-legged monomials.

So, let's collect together some additional consequences, not stated in
Theorem 1. First each graph contributing to $g(q,j)$ must have at
least $\max\{1,q-1\}$ (four-legged) vertices so it comes with a power
of at least $\lambda^{\max\{1,q-1\}}$, though the estimates will eat
up a portion of this.  

Second, the magnitude of $g(q,j)$ reflects the the power counting of a 
graph built from four-legged vertices. Consider any graph having $v$ 
four-legged vertices and $2q$ amputated external legs. This graph has 
$2v-q$ propagators. The supremum of a single sector propagator in position
space is $M^{3j/2}$. Each vertex, save one which is held fixed to break
translation invariance, is integrated over all $\bbbr^3$. Each such 
integral gives $M^{-5j/2}$. When $g(q,j)$ acts as an interaction at scale
$j$ or lower, the  momenta of its external legs are restricted to lie
within $M^j$ of the Fermi surface. Then the sector counting Lemma~3 
applies to both internal and external vertices.  Roughly speaking, the 
sector counting Lemma says that the legs of a four-legged vertex are 
paired and that the sector of either leg of the pair determines that of 
the other leg of the pair. As the sector of one leg at each vertex is 
fixed by conservation of momentum, there is one sum over $M^{-j/2}$ 
sectors per vertex. This gives a total power counting factor, including
the sector sums for the external legs, of
$$
M^{(2v-q)3j/2}M^{-(v-1)5j/2}M^{-vj/2}=M^{(5-3q)j/2}\ .
$$
The sector sums for the (amputated) external legs are only performed 
when propagators are later hooked onto them. It is convenient to save up 
the sector sums for external legs in a way that avoids having to 
distinguish between external/internal pairings  and external/external
pairings. So we leave the sum over sectors for each external {\it vertex}
explicit, instead of bounding it by the number of terms times the size
of the maximum term. Each term in the resulting expansion for the
$2q$-point function then has the sectors of all external {\it legs}
fixed. But it also has the sectors of all internal propagators hooked
to the vertex fixed.
 

Third, approximate conservation of momentum increases the number of 
available sectors  by a factor that depends on the degree of 
approximateness in the conservation of momentum. The degree to which 
$g(q,j)$ conserves momentum depends on the degree to which its internal 
vertices conserve momentum. Suppose that the original model is
supported in a volume of size $|\Lambda|=M^{-3J}$ and that the cutoff
function decays smoothly to zero in a distance $M^{-J}$. When we 
apply (\eqIIIfou) each vertex $v$ in $g(q,j)$ is assigned a number $E_v$
with the property that the vertex is zero unless the
sum of the momenta feeding into it is bounded by $E_vM^J$. Consequently,
$g(q,j)$ is zero unless the sum of the momenta feeding into it is bounded
by $\sum_v E_vM^J$.

Based on the motivation of the last paragraph we now consider interactions
of the form
\def\eqIIItena{III.10a}
$$ 
{\cal V}_\Lambda= \sum_{j^\prime>j}\sum_{q=1}^\infty
\int\prod_{i=1}^qd\xi_id\bar\xi_i\ 
V_{\Lambda,q}^{(j^\prime)}(\xi_1,\bar\xi_1,\cdots,\xi_q,\bar\xi_q)
\prod_{i=1}^q\bar\psi(\bar\xi_i)\psi(\xi_i)
\eqn{\eqIIItena}$$ 
when evaluating the effective potential with covariance of scale
$j$. The term ${\cal V}^{(0)}_{\Lambda , 2}$ in this sum is the ordinary 
vertex (\eqIIItwo) with four fields. Each kernel is expressed as a sum
\def\eqIIItenb{III.10b}
$$ V_{\Lambda,q}^{(j^\prime)}=\sum_{m\in{\cal M}_{q}}
V_{\Lambda,q,m}^{(j^\prime)}\ .
\eqn{\eqIIItenb}$$
The index $j^\prime$ gives the scale at which the graphs contributing to
$V_{\Lambda,q,m}^{(j^\prime)}$ were formed.
You should think of $m$ as measuring the extent to which exact
conservation of momentum fails as well as giving the sectors of some
internal lines. In particular,
\def\eqIIItenc{III.10c}
\item{-}there is an $E^{(j^\prime)}_{q,m}\in\bbbz$ such that
$V_{\Lambda,q,m}^{(j^\prime)}$ is zero unless the sum of the momenta 
flowing into it is bounded by  $E^{(j^\prime)}_{q,m}M^J$\hfill
(\eqIIItenc)

\noindent We assume that there is an $0<\alpha<1$ such that, when all
the arguments of $V_{\Lambda,q}^{(j^\prime)}$ have momentum within 
$\ \const M^j\ $ of the Fermi surface and when one argument lies in a
fixed sector of scale $j$, 
\def\eqIIItend{III.10d}
\item{-}the remaining $2q-1$ legs of each nonzero
$V_{\Lambda,q,m}^{(j^\prime)}$ are supported in \hfill\break $C_1^{q}
\exp\left\{\left(E^{(j^\prime)}_{q,m}M^{J-j}\right)^\alpha\right\} 
M^{(2q-3)(j'-j)/2} K_{q,m}^{(j^\prime)}$
sector $(2q-1)$-tuples of scale $j$.\hfill\break\null\hfill
(\eqIIItend)

\noindent That is, $K_{q,m}^{(j^\prime)}$ is the number of sector 
$(2q-1)$-tuples at scale $j'$ when the kernel was first generated.
However, a sector that is introduced with width $M^{j^\prime/2}$ is
subdivided into $M^{(j^\prime-j)/2}$ sectors at scale $j$. The number of 
accessible sectors of scale $j$ for the $2q-1$ remaining external legs 
given their sectors at scale $j'$ is bounded by 
$const^{q}(1+E^{(j^\prime)}_{q,m}M^{J-j})^2 M^{(2q-3)(j'-j)/2}$ 
in Lemma 3. It is convenient for us to use the multiplicative property
of the exponential so we bound 
$(1+EM^{J-j})^2\le\const\exp\left\{\left(EM^{J-j}\right)^\alpha\right\}$.
 
We further assume that the kernel is analytic in the region
$|\lambda|< R$ and obeys the bound
\def\eqIIItene{III.10e}
$$
\sum_{m\in{\cal M}_q}
\exp\left\{\sum_{\imath=J}^{j^\prime-1}
\left(E^{(j^\prime)}_{q,m}M^{J-\imath}\right)^\alpha\right\}
 K_{q,m}^{(j^\prime)}\norm V_{\Lambda,q,m}^{(j^\prime)}\norm 
\le K_1|\lambda|^{q/2}M^{{1\over 2}(5-3q)j^\prime}\ \ .
\eqn{\eqIIItene}$$ 
throughout that region.  Note that $\lambda$ times the vertex
(\eqIIItwo) is of this form, with
$$\eqalign{
V_{\Lambda,q}^{(j^\prime)}&=0 {\rm \ for\ all\ } q\ne 2,\ j^\prime\ne
0\cr {\cal M}_2&=\bbbz^3\cr V_{\Lambda,2,m}^{(0)}&={1\over
2}\chi_{m,\Lambda}V\cr
K_{2,m}^{(0)}&=\const\cr
E^{(0)}_{2,m}&=|m_1|+|m_2|+|m_3|\cr
\norm V_{\Lambda,2,m}^{(0)}\norm&\le
|\lambda|\const\,e^{-\const\,|m|^{1/2}}\norm V\norm\cr
\alpha&={1\over 3}\cr
R&=\infty\cr
}$$
We are making a somewhat stronger hypothesis on the cutoff function
$\chi_{m,\Lambda}$ here than in (\eqIIIthr). The standard $C^\infty$ 
compactly supported functions obey this stronger hypothesis. To be 
precise let
$$\eqalign{
\chi_1(x)&=\cases{0&$x\le 0$\cr
                 e^{-x^{-2}}e^{-(x-1)^{-2}}&$0<x<1$\cr
                 0&$x\ge 1$\cr}\cr
\chi_2(x)&=\left[\int_0^1 \chi_1(t)dt\right]^{-1}\int_0^x \chi_1(t)dt\cr
\chi_3(x)&=\chi_2(x+2)\chi_2(-x-2)\cr
}$$
and finally
\def\eqIIIele{III.11}
$$
\chi_\Lambda(\xi)=\prod_{i=0}^2\chi_3(M^J\xi_i)\ .
\eqn{\eqIIIele}$$
Then
$$
\sup_{x\in\bbbr}\left|
{d^n\hfill\over dx^n}\chi_3(x)\right|\le \const^n\,n^{3n/2}
$$ 
so that the Fourier transform
$$
\left|\widetilde \chi_3(k)\right|\le 4 {\const^{\!n}\,n^{3n/2}\over k^n}
$$
for all even $n\ge 0$. Let $\beta>3/2$. Choosing $n$ to be the even 
integer nearest $|k|^{1/\beta}$ yields
$$\eqalign{
\left|\widetilde \chi_3(k)\right|
&\le\const^{\!n}\,{n^{3n/2}\over n^{\beta n}}\cr
&\le e^{-(\beta-{3\over2})n\ln n+n\ln \const}\cr
&\le \const e^{-(\beta-{3\over2})n}\cr
&\le \const e^{-\const |k|^{1/\beta}}\ .
}$$

We know from the perturbative analysis of [FT2] that two and four point 
interaction vertices  which have internal scales $j'$ higher than the 
external scale $j$ have to be renormalized. This problem will be treated 
in a later paper.  But we can already state a rigorous result if we limit 
ourselves to the part of the theory containing only convergent graphs.  
This part of the model is the sum of all graphs that have no two or four
point subgraphs with the scales assigned to their internal lines higher
than the scales assigned to their external lines. This was called 
completely positive power counting case in [Ri]. To isolate this part
 of the model it suffices to require that, with the exception of 
$q=2,\ j'=0$, all $V_{\Lambda,q}^{(j^\prime)}$ with $q\le 2$, $j'>j$ be
zero.\vfill\eject

\thm{\ 2} {\it Let ${\cal V}$ obey {\rm (III.10)} and let 
$V_{\Lambda,q}^{(j^\prime)}=0$ for $q=1,\ j'>j$ and $q=2,\ 0>j'>j$. Then, 
if $M$ is large enough, there is an $R_0(M,K_1)>0$ such that the effective
potential ${\cal G}$ obeys {\rm (III.10)} for all $|\lambda|<\min(R,R_0)$.}
\prf
The expansion and bounds are the same as for Theorem 1, with the
exception that the vertices are more complicated and that we leave the
sums associated with external vertices explicit. We use a tilde to
designate index sets and constants that refer to the effective
potential. There is, of course the trivial graph in which a
$V_{\Lambda,q}^{(j^\prime)}$ gets fed directly into the effective
potential as a single vertex graph. All nontrivial contributions get
put into $\widetilde V^{(j)}_{\Lambda,p,m}$'s. The new index set 
$$
\widetilde{\cal M}_{p}=\bbbz^{>0}\cartprod_{v=1}^{2p}\ 
{\tst\bigcup\limits_{j_v>j}} \ {\tst\bigcup\limits_{q_v=0}^\infty} \
{\tst\bigcup\limits_{m_v\in{\cal M}_{q_v}\atop
I\subset\{1,\cdots,2q_v\}}}
\{{\rm sector\ assignments\ to \ the\ lines\ }I{\rm\ of\ }
V^{(j_v)}_{\Lambda,q_v,m_v}\}
$$ 
Here $v$ labels the external vertices of terms contributing to
$\widetilde V^{(j)}_{\Lambda,p,m}\,$. In the event that there are
fewer than $2p$ external vertices, the extra $q_v$'s are set to zero.
The set $I$ selects the legs of the vertex $v$ that will be internal
to the $2p$-point function. We shall denote by $\imath_v$ and $e_v$
the number of legs of the vertex $v$ that end up being internal and
external legs, respectively, of $\widetilde V^{(j)}_{\Lambda,p,m}.$
Suppose, for example, that the first seven vertices brought down from
the exponent end up being internal, but that the eighth ends up being
a $V_{\Lambda,q_1,m_1}^{(j_1)}$ with the legs in $I_1$, a proper
subset of $\{1,\cdots,2q_1\}$, being internal and of sectors
$\ell_1,\cdots,\ell_{|I_1|}$. Then this term will contribute to a
$\widetilde V_{\Lambda,p,m}^{(j)}$ with the second component 
(out of $2p+1$ components) of $m$ being 
$(j_1,q_1,m_1,I_1,\ell_1,\cdots,\ell_{|I_1|})$. The new conservation of
momentum index is
$$
E^{(j)}_{p,m}=E+\sum_v E^{(j_v)}_{q_v,m_v}
$$
where $E$, the first of the $2p+1$ components of $m$, is the sum of the
conservation of momentum indices of all internal vertices.
\vfill\eject

The main bound (\eqIIInin) of Theorem 1 is replaced by
\def\eqIIIninp{III.9'}
$$\eqalign{ &\norm \widetilde V_{\Lambda,p,m}^{(j)}\norm
\!\le\! \prod_{\rm internal\atop vertices}\!\!\left(
\sum_{j_v>j}\sum_{q_v}\sum_{m_v\in {\cal M}_{q_v}}
\hskip-5.5pt C_1^{q}\!\exp\!
   \left\{E^{(j_v)^\alpha}_{q_v,m_v}M^{(J-j)\alpha}\!\right\}
M^{(q_v-{3\over 2})(j_v-j)}K_{q_v,m_v}^{(j_v)}
\norm V_{\Lambda,q_v,m_v}^{(j_v)}\norm\!\right)\cr
&\hskip1.5truein\left(\prod_{\rm external\atop vertices}
\norm V_{\Lambda,q_v,m_v}^{(j_v)}\norm\right)
M^{-5j(n-1)/2}M^{{3\over 2}j(\Sigma 2q_v-2p)/2}K_2^{\Sigma 2q_v-2p}
K_3^{\Sigma 2q_v}\cr 
}\eqn{\eqIIIninp}$$
The constant $K_2$ includes the constants arising
from bounding the propagator and summing or taking the supremum of
$\rho^{(j,\ell)}(\Delta,\Delta^\prime)^{-\gamma+4}$. The constant
$K_3$ includes the combinatorial factors associated with propagators
and external legs. The latter include a factor of two for deciding
whether or not a leg contracted to the exponent, a factor of two to
decide which leg was the target leg, in the event that there was
contraction to the exponent (note that $q\le 2^q$) and a factor of
$(2\times 3^{t+1})^2$ from the Taylor expansion. The differences
between $K_2$ and $K_3$ are that the former applies only to
propagators, i.e. internal legs, while the latter is independent of
$M$.

Moving around the powers of $M$
$$\eqalign{
&M^{-5j(n-1)/2}M^{{3\over 2}j(\Sigma 2q_v-2p)/2}
\prod_{\rm internal\atop vertices}M^{(q_v-{3\over 2})(j_v-j)}\cr
&=M^{{1\over 2}(5-3p)j}
\prod_{\rm internal\atop vertices}M^{(q_v-{3\over 2})(j_v-j)}
M^{-{1\over 2}(5-3q_v)j}
\prod_{\rm external\atop vertices}M^{-{1\over 2}(5-3q_v)j}\cr
&=M^{{1\over 2}(5-3p)j}
\prod_{\rm internal\atop vertices}M^{{1\over 2}(2-q_v)(j_v-j)}
M^{-{1\over 2}(5-3q_v)j_v}
\prod_{\rm external\atop vertices}M^{-{1\over 2}(5-3q_v)j}\cr
}$$
we end up with
$$\eqalign{
\norm \widetilde V_{\Lambda,p,m}^{(j)}\norm
&\le \prod_{\rm internal\atop vertices}
\Bigg(\sum_{j_v>j}\sum_{q_v}\sum_{m_v\in {\cal M}_{q_v}}
M^{{1\over 2}(2-q_v)(j_v-j)}M^{-{1\over 2}(5-3q_v)j_v}
\exp\!
   \left\{E^{(j_v)^\alpha}_{q_v,m_v}M^{(J-j)\alpha}\!\right\}\cr
&\hskip2.5truein
(C_1K_2^2K_3^2)^{q_v}
K_{q_v,m_v}^{(j_v)}\norm V_{\Lambda,q_v,m_v}^{(j_v)}\norm\Bigg)\cr
&\hskip.5truein\left(\prod_{\rm external\atop vertices} M^{-{1\over
2}(5-3q_v)j}K_2^{\imath_v}K_3^{2q_v}\norm
V_{\Lambda,q_v,m_v}^{(j_v)}\norm\right) M^{{1\over 2}(5-3p)j}\cr
}$$  

Now summing over $m\in\widetilde{\cal M}_{p}$ entails, for each
external vertex, a factor of two per line (to decide whether it is
internal or not), a sum over $j_v$ and $m_v$ and a sum over sector
assignments to the lines of $V_{\Lambda,q_v,m_v}^{(j_v)}$ that ended
up being internal to $\widetilde V_{\Lambda,p,m}^{(j)}$. Including a
factor $\widetilde K_{p,m}^{(j)}$ accounts for a sum over sector
assignments to the lines that ended up external. So there is a
sum over assignments to all legs and, since
$$
E^{(j)^\alpha}_{q,m}
\le\sum_{{\rm all \atop vertices}}E^{(j_v)^\alpha}_{q_v,m_v}\ ,
$$ 
the left hand side of (\eqIIItena) for $\widetilde V_{\Lambda,p,m}^{(j)}$
is bounded by
$$\eqalign{ 
&M^{-{1\over2}(5-3p)j}\sum_{m\in\widetilde{\cal M}_{p}}
\exp\left\{\sum_{\imath=J}^{j-1}
         \left(E^{(j)^\alpha}_{q,m}M^{J-\imath}\right)\right\}
\widetilde K_{p,m}^{(j)}
\norm \widetilde V_{\Lambda,p,m}^{(j)}\norm\cr
&\hskip.25truein\le
\prod_{\rm all\atop vertices}\Bigg(\sum_{j_v>j}\sum_{q_v}\sum_{m_v\in 
{\cal M}_{q_v}}
M^{{1\over 2}(2-q_v)(j_v-j)}M^{-{1\over 2}(5-3q_v)j_v}
\exp\left\{\sum_{\imath=J}^{j}
         \left(E^{(j_v)^\alpha}_{q_v,m_v}M^{J-\imath}\right)\right\}\cr
&\hskip3.5truein K_2^{\imath_v}{(2C_1K_3^2)}^{q_v}K_{q_v,m_v}^{(j_v)}
\norm V_{\Lambda,q_v,m_v}^{(j_v)}\norm\Bigg)\cr
&\hskip.25truein\le \prod_{\rm all\atop vertices}
\left(\sum_{j_v>j}\sum_{q_v} M^{{1\over 2}(2-q_v)(j_v-j)}K_1K_2^{\imath_v}
{(2C_1K_3^2)}^{q_v}|\lambda|^{q_v/2}\right)\cr
&\hskip.25truein\le |\lambda|^{p/2}\prod_{\rm all\atop vertices}
\left(\sum_{j_v>j}\sum_{q_v}M^{-{1\over 4}(j_v-j)}
M^{{1\over 4}(2-q_v)}K_1K_2^{\imath_v}{(2C_1K_3^2)}^{q_v}
|\lambda|^{\imath_v/4}\right)\cr
&\hskip.25truein\le |\lambda|^{p/2}\prod_{\rm all\atop vertices}
K_1M^{1/2}|\lambda|^{1/8}\left(\sum_{j_v>j}\sum_{q_v} 
M^{-{1\over 4}(j_v-j)}M^{-q_v/4}K_2^{\imath_v}{(2C_1K_3^2)}^{q_v}
|\lambda|^{\imath_v/8}\right)\cr 
}$$ 
For the second inequality we used $j_v>j$. For the third we also used
$q_v\ge 3$. Since $C_1$ and $K_3$ are independent of $M$ we can
choose $M$ so that $ 2C_1K_3^2\le M^{1/8} $. Then, if
$|\lambda|^{1/8}\le K_2^{-1}$, 
$$\eqalign{ 
M^{-{1\over 2}(5-3p)j}\sum_{m\in\widetilde{\cal M}_{p}}
&\exp\left\{\sum_{\imath=J}^{j-1}
         \left(E^{(j)^\alpha}_{q,m}M^{J-\imath}\right)\right\}
\widetilde K_{p,m}^{(j)}
\norm \widetilde V_{\Lambda,p,m}^{(j)}\norm \cr
&\le |\lambda|^{p/2}\prod_{\rm all\atop vertices}
K_1M^{1/2}|\lambda|^{1/8}\left(\sum_{j_v>j}\sum_{q_v} M^{-{1\over
4}(j_v-j)}M^{-q_v/8}\right)\cr &\le |\lambda|^{p/2}\prod_{\rm all\atop
vertices}K_1M^{1/2}\, {M^{-{1\over 4}}\over 1-M^{-{1\over 4}}}\,
{M^{-{5\over 8}}\over 1-M^{-{1\over 8}}}\, |\lambda|^{1/8}\cr &\le
|\lambda|^{p/2} }$$ provided $|\lambda|$ is small enough.
\endproof
Using Theorem 2 inductively, we conclude that the sum over all scales of the 
completely convergent graphs contributing to any given Green's function 
is analytic in $\lambda$ at $\lambda=0$. 




%\vfill\eject
\vskip.25truein
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\vskip.1truein
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\end%