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Kawasaki dynamics, spectral gap, equivalence of ensembles
---------------9907220853309
Content-Type: application/x-tex; name="microsoft.tex"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="microsoft.tex"

\magnification=\magstephalf
\tolerance=10000
%%  -------------------------------------------------------------------------
%%
%%  Equazioni con nomi simbolici
%%
%%      $$   x=1	\Eq(ciccio)  $$
%%      By \equ(ciccio) we get ...
%% 
%%  Dentro \eqalignno invece di \Eq si usa \eq.
%%  Per far riferimento ad una formula definita nel futuro: \eqf
%%  -------------------------------------------------------------------------
%%
%%  Teoremi con nomi simbolici
%%
%%      \nproclaim Proposition[peppe]. 
%%      If bla bla, then blu blu.
%%
%%      {\it Proof.} It is easy to check that ...
%%
%%      Because of Proposition \thm[peppe], we know that ...
%% 
%%  Per far riferimento ad un teorema definito nel futuro: \thf
%%  Per far riferimento a formule o teoremi definiti in altri file
%%  di cui si dispone il .aux, includere lo statement
%%      \include{file}
%%  e usare \eqf o \thf
%%
%%  Se e' presente il comando \BOZZA, viene stampato sul margine
%%  sinistro il nome simbolico della formula (o del teorema).
%%  -------------------------------------------------------------------------
%%
%%  All'inizio di ogni sezione includere
%%
%%      \expandafter\ifx\csname sezioniseparate\endcsname\relax%
%%         \input macro \fi
%%      \numsec=n                
%%      \numfor=1\numtheo=1\pgn=1
%%
%%  dove n e' il numero della sezione
%%  Le Appendici hanno numeri negativi (\numsec=-1, -2, ecc...)
%%  -------------------------------------------------------------------------
%%
%%  Fonti
%%
%%  Vengono caricate le fonti msam, msbm, eufm. 
%%  -------------------------------------------------------------------------
%%%%%%%%%%%%%%% FORMATO
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\baselineskip=14pt plus0.1pt minus0.1pt 
\parindent=25pt
\lineskip=4pt\lineskiplimit=0.1pt      
\parskip=0.1pt plus1pt
%
\let\ds=\displaystyle
\let\txt=\textstyle
\let\st=\scriptstyle
\let\sst=\scriptscriptstyle
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%  FONTS
\font\twelverm=cmr12 
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%
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%
\font\sixrm=cmr6
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\font\sixsy=cmsy6
\font\sixbf=cmbx6
%
\font\caps=cmcsc10
\font\bigcaps=cmcsc10 scaled \magstep1
%
%%%%%%%%%%%%%%%%%%%%%%%%% GRECO
%
\let\a=\alpha \let\b=\beta  \let\c=\chi \let\d=\delta  \let\e=\varepsilon
\let\f=\varphi \let\g=\gamma \let\h=\eta    \let\k=\kappa  \let\l=\lambda
\let\m=\mu   \let\n=\nu   \let\o=\omega    \let\p=\pi  \let\ph=\varphi
\let\r=\rho  \let\s=\sigma \let\t=\tau   \let\th=\vartheta
\let\y=\upsilon \let\x=\xi \let\z=\zeta
\let\D=\Delta \let\F=\Phi  \let\G=\Gamma  \let\L=\Lambda \let\Th=\Theta
\let\O=\Omega \let\P=\Pi   \let\Ps=\Psi \let\Si=\Sigma \let\X=\Xi
\let\Y=\Upsilon
%
%%%%%%%%%%%%%%%%%%%%%%% CALLIGRAFICHE
%
\def\cA{{\cal A}} \def\cB{{\cal B}} \def\cC{{\cal C}} \def\cD{{\cal D}}
\def\cE{{\cal E}} \def\cF{{\cal F}} \def\cG{{\cal G}} \def\cH{{\cal H}}
\def\cI{{\cal I}} \def\cJ{{\cal J}} \def\cK{{\cal K}} \def\cL{{\cal L}}
\def\cM{{\cal M}} \def\cN{{\cal N}} \def\cO{{\cal O}} \def\cP{{\cal P}}
\def\cQ{{\cal Q}} \def\cR{{\cal R}} \def\cS{{\cal S}} \def\cT{{\cal T}}
\def\cU{{\cal U}} \def\cV{{\cal V}} \def\cW{{\cal W}} \def\cX{{\cal X}}
\def\cY{{\cal Y}} \def\cZ{{\cal Z}} 
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figure
%
\newdimen\xshift 
\newdimen\yshift
\newdimen\xwidth 
%
\def\eqfig#1#2#3#4#5#6{
  \par\xwidth=#1 \xshift=\hsize \advance\xshift 
  by-\xwidth \divide\xshift by 2
  \yshift=#2 \divide\yshift by 2
  \vbox{
  \line{\hglue\xshift \vbox to #2{
    \smallskip
    \vfil#3 
    \special{psfile=#4.ps}
    }
    \hfill\raise\yshift\hbox{#5}
  }
  \smallskip
  \centerline{#6}
  }
  \smallskip}
%
\def\figini#1{%
  \def\8{\write13}%
  \catcode`\%=12\catcode`\{=12\catcode`\}=12
  \catcode`\<=1\catcode`\>=2
  \openout13=#1.ps}
%
\def\figfin{%
  \closeout13
  \catcode`\%=14\catcode`\{=1
  \catcode`\}=2\catcode`\<=12\catcode`\>=12}
%
%
%%%%%%%%%%%%%%%%%%%%%  Numerazione pagine
%
\def\data{\number\day/\ifcase\month\or gennaio \or febbraio \or marzo \or
aprile \or maggio \or giugno \or luglio \or agosto \or settembre
\or ottobre \or novembre \or dicembre \fi/\number\year}
%
%%\newcount\tempo
%%\tempo=\number\time\divide\tempo by 60}
%
\setbox200\hbox{$\scriptscriptstyle \data $}
%
\newcount\pgn 
\pgn=1
\def\foglio{%
  \numsection\pgn
  \global\advance\pgn by 1}
%
%
%%%%%%%%%%%%%%%%% EQUAZIONI E TEOREMI CON NOMI SIMBOLICI
%
\def\begintex{%
  \openin14=\jobname.aux 
  \ifeof14 
    \relax 
  \else
    \input \jobname.aux \closein14 
  \fi 
  \openout15=\jobname.aux
}
\def\endtex{}
%
\global\newcount\numsec
\global\newcount\numfor
\global\newcount\numfig
\global\newcount\numtheo
%
\gdef\profonditastruttura{\dp\strutbox}
%
\def\senondefinito#1{\expandafter\ifx\csname#1\endcsname\relax}
%
\def\SIA #1,#2,#3 {\senondefinito{#1#2}%
   \expandafter\xdef\csname #1#2\endcsname{#3}\else
   \write16{???? ma #1,#2 e' gia' stato definito !!!!}\fi}
%
\def\etichetta(#1){(\numsection\numfor)
   \SIA e,#1,(\numsection\numfor)
   \global\advance\numfor by 1
   \write15{\string\FU (#1){\equ(#1)}}
   \write16{ EQ \equ(#1) == #1  }}
%
\def\oldetichetta(#1){
   \senondefinito{fu#1}\clubsuit(#1)\else
   \csname fu#1\endcsname\fi}
%
\def\FU(#1)#2{\SIA fu,#1,#2 }
%
\def\tetichetta(#1){{\numsection\numtheo}%
   \SIA theo,#1,{\numsection\numtheo}
   \global\advance\numtheo by 1%
   \write15{\string\FUth (#1){\thm[#1]}}%
   \write16{ TH \thm[#1] == #1  }}
%

\def\oldtetichetta(#1){%----------------------- mnemonic label
   \senondefinito{futh#1}\clubsuit(#1)\else
   \csname futh#1\endcsname\fi}
%
%
\def\FUth(#1)#2{\SIA futh,#1,#2 }
%
\def\getichetta(#1){Fig. \number\numfig
 \SIA e,#1,{\number\numfig}
 \global\advance\numfig by 1
 \write15{\string\FU (#1){\equ(#1)}}
 \write16{ Fig. \equ(#1) ha simbolo  #1  }}
%
\newdimen\gwidth
%
\def\BOZZA{
 \def\alato(##1){
 {\vtop to \profonditastruttura{\baselineskip
 \profonditastruttura\vss
 \rlap{\kern-\hsize\kern-1.3truecm{$\scriptstyle##1$}}}}}
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 {\vtop to \profonditastruttura{\baselineskip
 \profonditastruttura\vss
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 \def\talato(##1){\rlap{\sixrm\kern -1.3truecm ##1}}
 \def\thm{\teo}\def\thf{\teo}
}
%
\def\alato(#1){}
\def\galato(#1){}
\def\talato(#1){}
%
%
\def\numsection#1{%
  \ifnum\numsec=-100%
    \relax\number#1%
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\fi
\fi
\fi
}
%
%\def\geq(#1){\getichetta(#1)\galato(#1)}
%
\def\Thm[#1]{\tetichetta(#1)}
\def\thf[#1]{\senondefinito{futh#1}$\clubsuit$[#1]\else
   \csname futh#1\endcsname\fi}
\def\thm[#1]{\senondefinito{theo#1}$\spadesuit$[#1]\else
   \csname theo#1\endcsname\fi}
%
\def\Eq(#1){\eqno{\etichetta(#1)\alato(#1)}}
\def\eq(#1){\etichetta(#1)\alato(#1)}
\def\eqv(#1){\senondefinito{fu#1}$\clubsuit$(#1)\else
   \csname fu#1\endcsname\fi}
\def\equ(#1){\senondefinito{e#1}$\spadesuit$(#1)\else
   \csname e#1\endcsname\fi}
\let\eqf=\eqv
%
\def\nonumeration{%--------------- used in partial printings
  \let\etichetta=\oldetichetta
  \let\tetichetta=\oldtetichetta
  \let\equ=\eqf
  \let\thm=\thf
}
%
%
% -------------------------------------------------------------------------
%
%  Numerazione verso il futuro ed eventuali paragrafi
%  precedenti non inseriti nel file da compilare
%
\def\include#1{%
\openin13=#1.aux \ifeof13 \relax \else
\input #1.aux \closein13 \fi
}
%
%
% -------------------------------------------------------------------------
%
% 
\def\fine{\vfill\eject}
\def\sezioniseparate{%
   \def\fine{\par \vfill \supereject \end }}
%
% -------------------------------------------------------------------------
%
\footline={\rlap{\hbox{$\sst \data$}\ $\st[\number\pageno]$}\hss\tenrm
\foglio\hss}

%
%------------------------- Altre macro da chiamare ------------
%
\def\page{\vfill\eject}
\def\smallno{\smallskip\noindent}
\def\medno{\medskip\noindent}
\def\bigno{\bigskip\noindent}
\def\\{\hfill\break}
\def\acapo{\hfill\break\noindent}
\def\thsp{\thinspace}
\def\x{\thinspace}
\def\tthsp{\kern .083333 em}
\def\mathindent{\parindent=50pt}
\def\club{$\clubsuit$}
\def\cclub{\club\club\club}
\def\?{\mskip -10mu}
%
%------------------------ itemizing
%
\def\itm#1{\item{(#1)}}
\let\itemm=\itemitem
\def\bu{\smallskip\item{$\bullet$}}
\def\bul{\medskip\item{$\bullet$}}
\def\indbox#1{\hbox to \parindent{\hfil\ #1\hfil} }
\def\citem#1{\item{\indbox{#1}}}
\def\citemitem#1{\itemitem{\indbox{#1}}}
\def\litem#1{\item{\indbox{#1\hfill}}}
%
\def\ref[#1]{[#1]}
%
\def\beginsubsection#1\par{\bigskip\leftline{\it #1}\nobreak\smallskip
	    \noindent}
%
\newfam\msafam
\newfam\msbfam
\newfam\eufmfam
%
% -------------------------------------------------- math macros --------
%
%
\def\hexnumber#1{%
  \ifcase#1 0\or 1\or 2\or 3\or 4\or 5\or 6\or 7\or 8\or
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\font\sevenmsa=msam7
\font\fivemsa=msam5
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\scriptscriptfont\msafam=\fivemsa        
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\edef\msafamhexnumber{\hexnumber\msafam}%
%
%   \mathchardef\restriction"1\msafamhexnumber16
%   "class, family, position (found on amstex guide)
%
\mathchardef\restriction"1\msafamhexnumber16
\mathchardef\ssim"0218
\mathchardef\square"0\msafamhexnumber03
\mathchardef\eqd"3\msafamhexnumber2C
\def\QED{\ifhmode\unskip\nobreak\fi\quad
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\font\fivemsb=msbm5
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\scriptscriptfont\msbfam=\fivemsb
\def\Bbb#1{\fam\msbfam\relax#1}    
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\font\seveneufm=eufm7
\font\fiveeufm=eufm5
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\scriptfont\eufmfam=\seveneufm
\scriptscriptfont\eufmfam=\fiveeufm
\def\frak#1{{\fam\eufmfam\relax#1}}
\let\goth\frak
%
\def\bZ{{\Bbb Z}}
\def\bF{{\Bbb F}}
\def\bR{{\Bbb R}}
\def\bC{{\Bbb C}}
\def\bE{{\Bbb E}}
\def\bP{{\Bbb P}}
\def\bI{{\Bbb I}}
\def\bN{{\Bbb N}}
\def\bL{{\Bbb L}}
\def\bV{{\Bbb V}}

\def\Fg{{\frak g}}
\def\({\left(}
\def\){\right)}
%
%-------------------------------------------------------------------
%
% ------- Per compatibilita'
%
\let\integer=\bZ
\let\real=\bR
\let\complex=\bC
\let\Ee=\bE
\let\Pp=\bP
\let\Dir=\cE
\let\Z=\integer
\let\uline=\underline
\def\Zp{{\integer_+}}
\def\ZpN{{\integer_+^N}}
\def\ZZ{{\integer^2}}
\def\ZZt{\integer^2_*}
\def\ee#1{{\vec {\bf e}_{#1}}}
%
\let\neper=e
\let\ii=i
\let\mmin=\wedge
\let\mmax=\vee
\def\lefkg{ \le }
\def\lefkgstrong{ \preceq }
\def\gefkg{ \ge }
\def\gefkgstrong{ \succeq }
\def\identity{ {1 \mskip -5mu {\rm I}}  }
\def\ie{\hbox{\it i.e.\ }}
\let\id=\identity
\let\emp=\emptyset
\let\sset=\subset
\def\ssset{\subset\subset}
\let\setm=\backslash
\def\nep#1{ \neper^{#1}}
\let\uu=\underline
\def\ov#1{{1\over#1}}
\let\nea=\nearrow
\let\dnar=\downarrow
\let\imp=\Rightarrow
\let\de=\partial
\def\dep{\partial^+}
\def\deb{\bar\partial}
\def\tc{\thsp | \thsp}
\let\<=\langle
\let\>=\rangle
\def\tpl{{| \mskip -1.5mu | \mskip -1.5mu |}}
\def\tnorm#1{\tpl #1 \tpl}
\def\uno{{\uu 1}}
\def\mno{{- \uu 1}}
%
\def\xx{ {\{x\}} }
\def\xy{ { \{x,y\} } }
\def\XY{ {\< x, y \>} }

\def\pmu{\{-1,1\}}
%
\def\Pro{\noindent{\it Proof.}}
%
\def\sump{\mathop{{\sum}'}}
\def\tr{ \mathop{\rm tr}\nolimits }
\def\intt{ \mathop{\rm int}\nolimits }
\def\ext{ \mathop{\rm ext}\nolimits }
\def\Tr{ \mathop{\rm Tr}\nolimits }
\def\ad{ \mathop{\rm ad}\nolimits }
\def\Ad{ \mathop{\rm Ad}\nolimits }
\def\dim{ \mathop{\rm dim}\nolimits }
\def\weight{ \mathop{\rm weight}\nolimits }
\def\Orb{ \mathop{\rm Orb} }
\def\Var{ \mathop{\rm Var}\nolimits }
\def\Cov{ \mathop{\rm Cov}\nolimits }
\def\mean{ \mathop{\bf E}\nolimits }
\def\EE{ \mathop\Ee\nolimits }
\def\PP{ \mathop\Pp\nolimits }
\def\diam{\mathop{\rm diam}\nolimits}
\def\sgn{\mathop{\rm sgn}\nolimits}
\def\prob{\mathop{\rm Prob}\nolimits}
\def\gap{\mathop{\rm gap}\nolimits}
\def\osc{\mathop{\rm osc}\nolimits}
\def\supp{\mathop{\rm supp}\nolimits}
\def\Dom{\mathop{\rm Dom}\nolimits}
%
\def\tto#1{\buildrel #1 \over \longrightarrow}
\def\con#1{{\buildrel #1 \over \longleftrightarrow}}
%
\def\norm#1{ | #1 | }
\def\ninf#1{ \| #1 \|_\infty }
\def\scalprod#1#2{ \thsp<#1, \thsp #2>\thsp }
\def\inte#1{\lfloor #1 \rfloor}
\def\ceil#1{\lceil #1 \rceil}
\def\intl{\int\limits}
%
\outer\def\nproclaim#1 [#2]#3. #4\par{\medbreak \noindent
   \talato(#2){\bf #1 \Thm[#2]#3.\enspace }%
   {\sl #4\par }\ifdim \lastskip <\medskipamount 
   \removelastskip \penalty 55\medskip \fi}
%
\def\thmm[#1]{#1}
\def\teo[#1]{#1}
%
%------------------------------ tilde
%
\def\sttilde#1{%
\dimen2=\fontdimen5\textfont0
\setbox0=\hbox{$\mathchar"7E$}
\setbox1=\hbox{$\scriptstyle #1$}
\dimen0=\wd0
\dimen1=\wd1
\advance\dimen1 by -\dimen0
\divide\dimen1 by 2
\vbox{\offinterlineskip%
   \moveright\dimen1 \box0 \kern - \dimen2\box1}
}
%
\def\ntilde#1{\mathchoice{\widetilde #1}{\widetilde #1}%
   {\sttilde #1}{\sttilde #1}}

%
%-------------------------------------------------------------------
%
%\sezioniseparate %----------- togliere quando si stampa tutto insieme
%\let\g=\o %      %------------------ per il Mac 
%



\def\bye{%
\par\vfill\supereject
\message{******** Run TeX twice to resolve cross-references *****************}%
\message{******** Run TeX twice to resolve cross-references *****************}%
\message{******** Run TeX twice to resolve cross-references *****************}%
\message{******** Run TeX twice to resolve cross-references *****************}%
\end}


\long\def\newsection#1\par{%
\advance\numsec by 1%
\numfor=1%
\numtheo=1%
\pgn=1%
\vskip 0pt plus.3\vsize \penalty -250 \vskip 0pt plus-.3\vsize \bigskip 
\vskip \parskip \message {#1}\leftline {\bf \number\numsec. #1}%
\nobreak \smallskip \noindent}

\long\def\appendix#1\par{%
\numsec=-101%
\numfor=1%
\numtheo=1%
\pgn=1%
\vskip 0pt plus.3\vsize \penalty -250 \vskip 0pt plus-.3\vsize \bigskip 
\vskip \parskip \message {#1}\leftline {\bf Appendix. #1}%
\nobreak \smallskip \noindent}


\def\parsk{%
   \baselineskip=11pt plus0.1pt minus0.1pt 
   \lineskip=4pt\lineskiplimit=0.1pt      
   \parskip=2pt plus1pt
}


%-------------------------------------------- amstex
\def\frac#1#2{{#1\over#2}}

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\def\]{\right]}

\mathchardef\sqsubset"3\msafamhexnumber40
\mathchardef\sqsupset"3\msafamhexnumber41
\mathchardef\Cdot"0\msafamhexnumber05

% like eqfig, but needs full filename (so it doesn't have to be .ps)
%
\def\insertfig#1#2#3#4#5#6{
  \par\xwidth=#1 \xshift=\hsize \advance\xshift 
  by-\xwidth \divide\xshift by 2
  \yshift=#2 \divide\yshift by 2
  \vbox{
  \line{\hglue\xshift \vbox to #2{
    \smallskip
    \vfil#3 
    \special{psfile=#4}
    }
    \hfill\raise\yshift\hbox{#5}
  }
  \smallskip
  \centerline{#6}
  }
  \smallskip}
%

\def\beginsubsection#1\par{\bigskip\leftline{\caps #1}\nobreak\smallskip
	    \noindent}

\def\BOZZA{
 \def\alato(##1){
 {\vtop to \profonditastruttura{\baselineskip
 \profonditastruttura\vss
 \rlap{\kern-\hsize\kern-1.3truecm{$\scriptstyle##1$}}}}}
 \def\galato(##1){ \gwidth=\hsize \divide\gwidth by 2
 {\vtop to \profonditastruttura{\baselineskip
 \profonditastruttura\vss
 \rlap{\kern-\gwidth\kern-1.3truecm{$\scriptstyle##1$}}}}}
 \def\talato(##1){\rlap{\sixrm\kern -1.3truecm ##1}}
% \def\thm{\teo}\def\thf{\teo}
}
%
\def\qq{^{J,\emp}_{\L,N}}
\def\nuj{{\nu\qq}}
\def\hT{{\tilde h}}
%
\def\TT{{\ntilde T}}
\def\thT{{\ntilde \th}}
\def\dd{ {d-1\over d} }
\def\tL{{\ntilde L}}
\def\phb{{\bar\ph}}
\def\psb{{\bar\psi}}
\def\Sh{{\hat S}}
\def\ug{{\uu\g}}
\def\ul{{\uu\l}}
\def\uth{{\uu\th}}
\def\Lb{{\bar\L}}
\def\Db{{\bar\D}}

\def\Ltop{{\L_{\rm top}}}
\def\Lbot{{\L_{\rm bottom}}}

\def\nuh{{\hat\nu}}
\def\nub{{\bar\nu}}
\def\eh{{\hat e}}
\def\Sh{{\hat S}}

\def\Zt{{\ntilde Z}}
\def\depl{\dep \mskip -5mu \L}

\def\indh{ {\st x,y  \in V \atop\st |x-y|=1 } }
\def\indhb{ {\st  x\in V, \, y\in V^c \atop\st |x-y|=1 } }
\def\bJt{{\b,J,\t}}
\def\Et{{\ntilde\cE}}
\def\EL{{\Et_\L}}
\def\ELh{{\hat\cE_\L}}
\def\d{\delta}
\def\s{\sigma}
\def\l{\lambda}
%\def\La{\Lambda}
\def\a{\alpha}
\def\b{\beta}
\def\e{\epsilon}
\def\g{\gamma}
\def\xt{\tilde{x}}
\def\pp{\partial}
\def\12{{1\over 2}}

%----------------------------------------- local macros -------------------
%
\def\qq{^{J,\emp}_{\L,N}}
\def\nuj{{\nu\qq}}
%
\def\dd{ {d-1\over d} }
%
\def\hT{{\tilde h}}
\def\TT{{\ntilde T}}
\def\TE{{\ntilde T}}
\def\thT{{\ntilde \th}}
\def\tL{{\ntilde L}}
\def\Ob{{\bar\O}}
\def\phb{{\bar\ph}}
\def\psb{{\bar\psi}}
\def\ug{{\uu\g}}
\def\Sh{{\hat S}}

\def\ug{{\uu\g}}
\def\ul{{\uu\l}}
\def\uth{{\uu\th}}
\def\Lb{{\bar\L}}

\def\Ltop{{\L_{\rm top}}}
\def\Lbot{{\L_{\rm bottom}}}

\def\nuh{{\hat\nu}}
\def\nub{{\bar\nu}}
\def\eh{{\hat e}}
\def\Sh{{\hat S}}

\def\Zt{{\ntilde Z}}
\def\depl{\dep \mskip -5mu \L}

\def\indh{ {\st x,y  \in V \atop\st |x-y|=1 } }
\def\indhb{ {\st  x\in V, \, y\in V^c \atop\st |x-y|=1 } }
\def\bJt{{\b,J,\t}}
\def\Et{{\ntilde\cE}}
\def\EL{{\Et_\L}}
\def\ELh{{\hat\cE_\L}}

\def\Zar{Zahradn\'\i k}

\def\mum[[#1,#2,#3,#4,#5,#6]]{
  \left[ \, {#1\atop#4} \, {#2\atop#5} \, {#3\atop#6} \, \right]}

\let\sss=\sset
\def\dz{d}

\def\xz{{ (x,z) \in V\times W }}

\def\pp{\partial}
\def\12{{1\over 2}}

\def\mHp{{\hat m^+}}
\def\mHm{{\hat m^-}}
\def\mHUp{{\hat m_U^+}}
\def\mHUm{{\hat m_U^-}}
\def\ul{{\underline \l}}
\def\Ag{{A_\ug}}
\def\Alz{{A_\l^\circ}}
\def\Bg{{B_\ug}}
\def\Agz{{A^\circ_\ug}}
\def\Bgz{{B^\circ_\ug}}
\def\Agc{{A^\circ_\g}}
\def\Bgc{{B^\circ_\g}}
\def\Ags{{A_\g^{\circ\circ}}}
\def\Bgs{{B_\g^{\circ\circ}}}


\def\CsLe{{C^*_{\L,\emp}}}
\def\Ednat{{E^{\d,nat}_\ug}}
\def\npt{{\cN^m_L}}

\def\vpha{\vphantom{\bigl[}}
\def\vphb{\vphantom{\bigl[_1^1}}

\def\nud{{\dot\nu}}
\def\cEd{{\dot\cE}}












%\input /home/martin/tex/optex-2.1
%\input /home/martin/tex/macros_to_add
%\input /home/cancrini/tex/optex-2.1
%\input /home/cancrini/tex/macros_to_add
%\input optex-2.1
%\input macros_to_add

%\BOZZA

\begintex
\numsec=0


%----------------------------------------- local macros -------------------
%
\def\qq{^{J,\emp}_{\L,N}}
\def\nuj{{\nu\qq}}
%
\def\dd{ {d-1\over d} }
%
\def\hT{{\tilde h}}
\def\TT{{\ntilde T}}
\def\TE{{\ntilde T}}
\def\thT{{\ntilde \th}}
\def\tL{{\ntilde L}}
\def\Ob{{\bar\O}}
\def\phb{{\bar\ph}}
\def\psb{{\bar\psi}}
\def\ug{{\uu\g}}
\def\Sh{{\hat S}}

\def\ug{{\uu\g}}
\def\ul{{\uu\l}}
\def\uth{{\uu\th}}
\def\Lb{{\bar\L}}

\def\Ltop{{\L_{\rm top}}}
\def\Lbot{{\L_{\rm bottom}}}

\def\nuh{{\hat\nu}}
\def\nub{{\bar\nu}}
\def\eh{{\hat e}}
\def\Sh{{\hat S}}

\def\Zt{{\ntilde Z}}
\def\depl{\dep \mskip -5mu \L}

\def\indh{ {\st x,y  \in V \atop\st |x-y|=1 } }
\def\indhb{ {\st  x\in V, \, y\in V^c \atop\st |x-y|=1 } }
\def\bJt{{\b,J,\t}}
\def\Et{{\ntilde\cE}}
\def\EL{{\Et_\L}}
\def\ELh{{\hat\cE_\L}}

\def\Zar{Zahradn\'\i k}

\def\mum[[#1,#2,#3,#4,#5,#6]]{
  \left[ \, {#1\atop#4} \, {#2\atop#5} \, {#3\atop#6} \, \right]}

\let\sss=\sset
\def\dz{d}

\def\xz{{ (x,z) \in V\times W }}

\def\pp{\partial}
\def\12{{1\over 2}}

\def\mHp{{\hat m^+}}
\def\mHm{{\hat m^-}}
\def\mHUp{{\hat m_U^+}}
\def\mHUm{{\hat m_U^-}}

\def\Ag{{A_\ug}}
\def\Alz{{A_\l^\circ}}
\def\Bg{{B_\ug}}
\def\Agz{{A^\circ_\ug}}
\def\Bgz{{B^\circ_\ug}}
\def\Agc{{A^\circ_\g}}
\def\Bgc{{B^\circ_\g}}
\def\Ags{{A_\g^{\circ\circ}}}
\def\Bgs{{B_\g^{\circ\circ}}}


\def\CsLe{{C^*_{\L,\emp}}}
\def\Ednat{{E^{\d,nat}_\ug}}
\def\npt{{\cN^m_L}}

\def\vpha{\vphantom{\bigl[}}
\def\vphb{\vphantom{\bigl[_1^1}}

\def\nud{{\dot\nu}}
\def\cEd{{\dot\cE}}
%
%--------------------------------- Figures
%
%\def\captionone{The set $U$}
%\def\captiontwo{A collection of edges (left) is split into a collection
%  of contours (right)}
%\def\captionthr{ An example of how to construct the $\nea$--path of Proposition
%  \thm[prop2]}


%\def\figureone{\insertfig{116pt}{116pt}{}{set_u.eps}{}{Fig. 1. \captionone}}
%\def\figuretwo{\insertfig{420pt}{126pt}{}{fig3.psc}{}{Fig. 2. \captiontwo}}
%\def\figurethr{\insertfig{380pt}{269pt}{}{contour4.eps}{}%
%   {Fig. 3. \captionthr}}


%--------------------------------- INIZIO
%
%
\font\ttlfnt=cmss12 scaled 1200 %small caps
\font\bit=cmbxti10 %bold italic text mode
%
\begingroup
\nopagenumbers
\footline={}
%
%
% Author. Initials then last name in upper and lower case
% Point after initials
%
\def\author#1
{\vskip 18pt\tolerance=10000
\noindent\centerline{\caps #1}\vskip 0.8truecm}
%
% Address
%
\def\address#1
{\vskip 4pt\tolerance=10000
\noindent #1\par\vskip 0.5truecm}
%
% Abstract
%
\def\abstract#1{\noindent{\bf Abstract.\ }#1\par}
%
\vskip 1cm
%
%
%  \line{\hfill
%    \special{psfile=sail.eps}
%    }
%
\centerline{\ttlfnt 
  On the spectral gap of Kawasaki dynamics}
\medno
\centerline{\ttlfnt under a mixing condition revisited }
%
%
%--------
\vskip 0.4truecm
%--------
\vskip 0.5truecm
\author{
N. Cancrini $^{1}$, and
F. Martinelli $^{2}$
}
%
\address{%
{\ninerm 
\item{$^1$}
  Dipartimento di Energetica, 
  Universit\`a dell'Aquila, Italy 
  and INFM Unit\`a di Roma ``La Sapienza'' 
\item{}
  e-mail: nicoletta.cancrini@roma1.infn.it 
\item{$^2$}
  Dipartimento di Matematica, Universit\`a di Roma Tre, Italy 
\item{}
  e-mail: martin@mat.uniroma3.it 
}}
%


\bigno
\abstract{We consider a conservative stochastic 
spin exchange dynamics which is reversible with respect the 
canonical Gibbs measure of 
a lattice gas model. We assume that the corresponding 
grand canonical measure satisfies a suitable 
strong mixing condition. We give an 
alternative and quite natural, from the physical point of view, 
proof of the famous Lu--Yau result which states
that the relaxation time in a box of side 
$L$ scales like $L^2$. We then show how to use such an estimate to 
prove a decay to equilibrium for local functions 
of the form ${1\over t^{\a -\e}}$ where $\e$ is positive and 
arbitrarily small and $\a = \ov2$ for $d=1$, $\a=1$ for 
$d\ge 2$. } 
%
\vskip 1cm
\noindent 
{\bf Key Words:}
Kawasaki dynamics, spectral gap, equivalence of ensembles.


{\parindent=0pt
\footnote{}{\ninerm
Mathematics Subject Classification: 82B44, 82C22, 82C44, 60K35}
}

\footline={\sixrm \hfil v1.02}


\vfill\eject
\endgroup



%--------------------------------- 

\newsection Introduction

The problem of computing the relaxation time of stochastic Monte Carlo
algorithms for models of classical spin models in $\Z^d$ has attracted 
in the last years considerable attention and many new rigorous 
techniques have been developed giving rise to 
nice progresses in  
probability theory and statistical mechanics. If for simplicity we 
confine ourselves to $\pm 1$ (or $0$-$1$ in the lattice gas picture) spins, 
the two most studied random 
dynamics have been non-- conservative Glauber type algorithms, in which 
a spin at a time flips its value with a rate satisfying 
the detailed balance condition w.r.t. the grand canonical 
Gibbs measure, and conservative dynamics 
in which nearest neighbors 
spins exchange theirs values with a rate 
satisfying 
the detailed balance condition w.r.t. to the canonical 
Gibbs measure. 
\smallno
It turns out that the conservation of the particle number (in 
the lattice gas picture,
or of the magnetization in the usual $\pm 1$ spin variables) 
makes the analysis of the relaxational properties of 
conservative dynamics much more difficult than in the 
non--conservative case with interesting analogies with the 
problem of the Goldstone mode in quantum mechanics \ref[A]. 
\acapo
For Glauber dynamics the general
picture is relatively clear 
for a wide class of models both in the one phase 
and in the phase coexistence region with the notable exception of 
the critical point (see e.g. \ref[M1] and references therein). In particular,
for the two dimensional Ising case with zero external field, 
the spectral gap of the generator of a Glauber
dynamics does not go to zero in the thermodynamic limit for any temperature 
above the critical 
one, while 
below the critical temperature the spectral gap in a box of side 
$L$ and free boundary conditions 
becomes exponentially small in $L$ with a precise rate related to the 
surface tension. 
\smallno
In the conservative case, instead, the fundamental 
results of \ref[LY], \ref[Y] 
on the spectral gap and logarithmic Sobolev inequality of 
Kawasaki dynamics in the 
one phase region state that, under a suitable mixing condition on the 
grand canonical Gibbs measure which for the two dimensional Ising model 
holds for any temperature
above the critical one, the spectral gap in a box of side $L$ 
shrinks like $L^{-2}$, while   
in the phase coexistence region, and at least 
for the two dimensional Ising model with periodic or 
free boundary condition, 
it becomes exponentially small in the side of the box 
\ref[CCM]. 
The diffusive scaling $L^{2}$ for the relaxation time (in what follows 
identified with 
the inverse of the spectral gap of the generator) of 
Kawasaki dynamics   
proved in \ref[LY] is a key stone in the study of the 
hydrodynamical limit of the Ising model \ref[VY], 
and its proof required the development of a rather 
sophisticated (and intricate) 
technology which posed new, non trivial, problems on 
the theory of canonical Gibbs measures and their detailed equivalence 
to grand canonical ones (see also \ref[BZ1], 
\ref[BZ2], \ref[CM1], \ref[BCO] and \ref[CZ]). 
\smallno
Unfortunately \ref[LY] and particularily 
\ref[Y] are quite difficult to study and the application of 
their techniques to other related problems, for example 
lattice gases with random interactions in the so--called Griffiths 
phase, seems to require a considerable effort. 
With this motivation we decided to 
reprove the result of \ref[LY] by different means and in way that looks,
at least to us, 
intuitively more appealing and natural to apply to other contexts 
\ref[CM2]. 
Although our proof would have never 
found its way without some very nice  
ideas that we found 
in \ref[LY] and \ref[Y], we have tried to be completely 
selfcontained apart the necessary results on finite volume 
canonical Gibbs measures and on the  
equivalence of ensembles that we have developed in a separate paper
\ref[CM1]. We have also added to the original result a simple proof 
of the 
power law relaxation to equilibrium of strictly local (\ie whose 
support is independent of the total volume) observables 
which, at least in one and two dimensions,
is arbitrarily close to the expected $t^{-d/2}$ result (see also \ref[BZ2]). 
 
   
\smallno
Let us now explain in simple terms the strategy behind our proof by firstly 
reviewing the way in which one can prove that the spectral gap for 
non--conservative Glauber dynamics does not shrink to zero with the volume 
in the high temperature phase.
\acapo 
Take a cube $Q_{2L}$ of side $2L$ and 
divide it into two equal  
rectangles $R_i(L,2L)$. Then, if truncated correlations 
in the grand canonical Gibbs measure decay fast enough 
(typically exponentially fast), it is not difficult to see that 
the relaxation time for the original square $Q_{2L}$ is not larger 
than $(1+\e_L)\times$(relaxation time in each rectangle), 
$\e_L = O(L^{-\ov2})$, in strict analogy with what happens at 
infinite temperature where the Glauber evolution becomes a product dynamics. 
Thus the relaxation time only increases by a factor
$(1+\e_L)^2$ when we double the scale and the result follows 
(see section 4 of \ref[M1] for details).
\acapo
For a conservative dynamics like the Kawasaki dynamics 
such a reasoning does not apply because the 
conservation of the number of particles introduces a global constraint in 
the system and even at infinite temperature the dynamics does not factorize 
into independent components. More precisely, the relaxation time in 
$Q_{2L}$ is related to the relaxation time of the modified dynamics in which 
the two rectangles do not exchange particles but feel each other 
only through the transition rates {\it and} 
to the relaxation time 
of the process of exchange of particles between the two halves of $Q_{2L}$. 
Such a simple observation suggests to try to separate the two effects 
which are, apriori, strongly interlaced and to analyze them separately. 
Technically this is possible and it can be achieved if one plugs into 
the variational characterization of the spectral gap, 
$ \gap = \inf_{f} {\Dir(f,f)\over \Var(f)}$,  
the formula of ``conditional variance''
$$
\Var(f) = E \bigl(\Var(f \tc n)\bigr) + \Var\bigl( E(f \tc n)\bigr)
\Eq(intro2)
$$
where $\Var(\cdot)$, $\Dir(\cdot,\cdot)$ and $E(\cdot)$ denote respectively 
the variance, the Dirichlet form (quadratic form of the 
generator) and expectation 
w.r.t. the canonical Gibbs measure in $Q_{2L}$ with e.g. $N$ particles, 
while $\Var(\cdot \tc n)$ denotes the variance 
conditioned to have $n$ of particles in e.g. 
the upper half of $Q_{2L}$.
The latter 
should now be controled by an argument similar to that used in 
the non--conservative case although special care needs to be taken because 
covariances in a multicanonical Gibbs measure 
are quite different from their 
grand canonical analogues.\acapo 
In order to control the effects produced by the 
extra conservation law, \ie to be able to 
bound the last term in the r.h.s. of 
\equ(intro2),
one is led naturally to study the distribution of the number of particles 
in half cube under the canonical Gibbs distribution in $Q_{2L}$ 
and in particular to prove a Poincar\'e inequality for it of the 
form $\Var(g(n)) \le k(N) E\bigl( ({d\over dn}g)^2\bigr)$ for any $g$, 
$k(N)=O(N)$, where ${d\over dn}$
is the discrete derivative. Notice that ${d\over dn}g$ 
measure the effects on the function $g$ of the exchange of one particle 
between the two rectangles. Once such a step has been carried out 
it is not too difficult to complete the scale reduction from $2L$ to $L$.
The final result is that 
the relaxation time for the original square $Q_{2L}$ is not larger 
than $(1+\e_L)\times$(relaxation time in each rectangle) plus 
const.$\times L^2$. This last term, which was absent 
in the analysis of the Glauber dynamics, comes precisely from the second 
term in the r.h.s. of \equ(intro2). The diffusive limit now follows
at once.
\smallno
Let us conclude with a short roadmap of the paper. 
\bul In section 2 we define the setting, the mixing condition we need 
and state the two main results.
\bul In section 3 we collect several technical results that will be needed
later on. A detailed description of these results 
is given at the beginning of the section.   
\bul In section 4 we study in some details the distribution of the number 
of particles in an atom of a given 
partition
of a finite set $\L$ under a multicanonical measure.
\bul In section 5 we prove recursively theorem 2.3 on the diffusive scaling 
of the spectral gap.
\bul In section 6 we prove theorem 2.4 on 
the power law relaxation to equilibrium of local observables.
\bul Finally in the appendix we reprove with some simplifications the 
famous two--blocks estimate of \ref[LY].

\bigno
{\bf Acknowledgements}. 
This work was begun while F.M.
was a guest of the Institut Henri Poincar\'e for the special semester
on ``Large deviations, logarithmic Sobolev inequalities and 
statistical mechanics'' in the spring 1998. F.M. warmly acknowledges
the organizers, and in particular F.Comets and L.Saloff--Coste, 
for the excellent hospitality there and the stimulating 
scientific atmosphere. N.C. acknoledges S.Olla for the warm
hospitality at the Department of Mathematics at the University
of Cergy--Pontoise. During this work 
we have benefit of several interesting and instructing 
discussion with L.Bertini and F.Cesi.





\vskip 1truecm
\newsection Notation and results

\bigno
In this section we first define the setting in which we will 
work (spin model, Gibbs measure, 
dynamics), then we define the basic mixing condition on the 
Gibbs measure and subsequently state the main theorem on this work.
 
\vskip 0.5cm \beginsubsection \number\numsec.1 The lattice and the 
configuration space

\medno
{\it The lattice.}
We consider the $d$ dimensional lattice $\Z^d$ with {\it sites\/} 
$x = (x_1, \ldots, x_d )$
and norms
$$
  |x|_p = (\sum_{i=1}^d  |x_i|^p )^{1/p} \quad p\ge 1 \quad {\rm and}\qquad
  |x| = |x|_\infty = \max_{i \in \{1, \ldots, d\} } |x_i| \,.
$$
The associated distance functions are denoted by 
$d_p(\cdot, \cdot)$ and 
$d(\cdot, \cdot)$.
By $Q_L$ we denote the cube of all $x=(x_1,\ldots, x_d) \in \Z^d$ such that
$x_i \in \{ 0, \ldots, L-1 \}$.
If $x\in \Z^d$, $Q_L(x)$ stands for $Q_L + x$.
We also let $B_L$ be the
ball (w.r.t $d(\cdot, \cdot)$) of radius $L$ centered at the origin, \ie
$B_L = Q_{2L+1}( (-L, \ldots, -L))$.
If $\L$ is a finite subset of $\Z^d$ we write $\L \ssset \Z^d$. 
The cardinality of $\L$ is denoted by $|\L|$.
$\bF$ is the set of all nonempty finite subsets of $\Z^d$.
$[x,y]$ is the {\it closed segment} with endpoints $x$ and $y$. 
The {\it edges} of $\Z^d$ are those $e=[x,y]$ with
$x,y$ nearest neighbors in $\Z^d$.
The {\it boundary of an edge} $e=[x,y]$ is $\d e = \{x,y\}$
The {\it boundary of a set of edges} $\a$ is the set $\d \a$ of 
all sites that
belong to an odd number of edges of $\a$.
A set of edges is called {\it closed} if its boundary is empty.
We denote by $\cE_\L$ the set of all edges such that both endpoints are
in $\L$ and by $\bar\cE_\L$ the set of all edges with at least one
endpoint in $\L$. Viceversa, for a set of edges $X$, $\cV(X)$ stands
for the set of all sites which are endpoints of at least one edge in $X$.
\smallno
Given $\L \sset\Z^d$ we define its interior and exterior 
boundaries as respectively,
$\de^- \L = \{ x \in \L : \, d(x, \L^c)\le 1 \}$ and
$\dep \L = \{ x \in \L^c : \, d(x, \L)\le 1 \}$, and more generally
we define the boundaries of width $n$ as
$\de_n \L = \{ x \in \L : \, d(x, \L^c)\le n \}$,
$\dep_n \L = \{ x \in \L^c : \, d(x, \L)\le n \}$.
For $\L\sset \Z^d$ we also let
$$
  \d\L = \{ \, e^* : e=[x,y],\ x\in\L,\ y\in \L^c,\ |x-y|_1 = 1 \, \}
  \Eq(dL) \,.
$$

\medno
{\it Regular sets.}
A finite subset $\L$ of $\Z^d$ is said to be 
{\it $l$--regular}, $l\in \Z_+$, 
if 
%  Attenzione: questa e' la definizione genrale. A noi serve con x=0
%  there exists $x\in \Z^d$ such that  
%$ \L$ is the union of a
%  finite number of cubes $Q_l(x^i+x)$ where $x^i \in l \Z^d$.
$\L$ is the union of a
finite number of cubes $Q_l(x^i)$ where $x^i \in l \Z^d$.
We denote the class of all such sets by $\bF_l$. 
Notice that any set is $1$--regular \ie $\bF_{l=1}=\bF$.

\medno
{\it The configuration space.}
Our {\it configuration space} is
$\O = S^{\Z^d}$, where $S=\{0,1\}$, or
$\O_V = S^V$ for some $V\subset \Z^d$. 
The single spin space $S$ is endowed with the discrete topology
and $\O$ with the corresponding product topology.
Given $\s\in \O$ and $\L \sset \Z^d$ we denote by $\s_\L$
the natural projection over $\O_\L$.
If $U$, $V$ are disjoint, $\s_U \t_V$ is the configuration on $U\cup V$ which
is equal to  $\s$ on $U$ and $\t$ on $V$.
Given $V\in\bF$ we define the {\it number of particles\/}
$N_V: \O \mapsto \bN$ as 
$$
  N_V(\s) = \sum_{x\in V} \s(x)
  \Eq(mag)
$$
while the {\it density} is given by $\rho_V = N_V/ |V|$.

If $f$ is a function on $\O$, $\D_f$ denotes the smallest subset
of $\Z^d$ such that $f(\s)$ depends only on $\s_{\D_f}$.
$f$ is called {\it local} if $\D_f$ is finite. 
The {\it $l$--support} of a
function $\D_f^{(l)}$, $l\in \Z_+$, 
is the smallest $l$--regular set $V$ such that $\D_f \sset  V$. 
$\cF_\L$ stands for the $\s-$algebra generated by the set of projections
$\{ \p_x \}$,  $x\in\L$,
 from $\O$ to $\pmu$,
where $\pi_x : \s \mapsto \s(x)$. 
When $\L=\Z^d$ we set $\cF = \cF_{\Z^d}$ and $\cF$ coincides
with the Borel $\s-$algebra on $\O$ with respect to the topology
introduced above.
By $\| f\|_\infty$ we mean the 
supremum norm of $f$.
The {\it gradient} of a function $f$ is defined as
$$
  (\nabla_x f)(\s) = f(\s^x) - f(\s)
$$
where $\s^x \in \O$ is the configuration obtained from $\s$, by flipping the
spin at the site $x$. Finally ${\rm Osc}(f) = \sup_{\s,\h}|f(\s)-f(\h)|$. 

\medno
\vskip 0.5cm \beginsubsection \number\numsec.2 The interaction and the Gibbs Measures.

\medno


\nproclaim Definition [potential]. 
A finite range, translation--invariant potential 
$\{\Phi_\L\}_{\L\in \bF}$ is a collection of real, 
local functions 
on $\O$ 
with the following properties
\item{(1)} $\Phi_\L = \Phi_{\L+x}$ for all 
  $\L \in \bF$ and all $x\in \Z^d$
\item{(2)} For each $\L$ the support of $\Phi_\L$ coincides with $\L$
\item{(3)} There exists $r>0$ such that
  $\Phi_\L = 0$ if $\diam \L > r$. $r$ is called the {\it range}
  of the interaction.
\item{(4)} $\|\Phi\| := \sum_{\L\ni 0} \|\Phi_\L\|_\infty \, < \, \infty$

\medno
Given a collection of real numbers 
$\ul = \{\l_x\}_{x\in \Z^d}$ and a {\it potential\/} $\Phi$ we define 
$\Phi^{\sst \ul}$ as 
$$
     \Phi^{\sst \ul}_\L(\s) = 
              \cases{ (h + \l_x)\s(x) & if $\L = \{x\}$ \cr
                      \Phi_\L(\s)  & otherwise
	}
$$
where $h$ is the chemical potential  
(one body part of $\Phi$).\acapo 
Given a {\it potential\/} $\Phi$ ($\Phi^{\sst \ul}$)  
and $V \in \bF$, we define
the Hamiltonian $H^\Phi_V : \O \mapsto \bR$ by
$$
  H_V^\Phi(\s) = - \sum_{\L: \, \L\cap V \ne \emp} \Phi_\L(\s)
$$
For $\s, \t \in \O$ we also let
$H_V^{\Phi,\t}(\s) = H_V^\Phi (\s_V \t_{V^c} )$
and $\t$ is called the {\it boundary condition}.
For each $V\in \bF$, $\t\in \O$ the
(finite volume) conditional Gibbs measure on $(\O, \cF)$, are given by
$$
    d\mu^{\Phi,\t}_V(\s) = 
    \cases{ \bigl(Z^{\Phi,\t}_V\bigr)^{-1}
    \exp[ \,-  H^{\Phi,\t}_V(\s) \,] 
                                & if $\s(x) = \t(x)$ for all $x\in V^c$ \cr
    \vphantom{\Bigl(}
    0 & otherwise. \cr }
    \Eq(finvolmea)
$$
where $Z^{\Phi,\t}_V$ is the proper normalization factor called partition
function. 
Notice that in \equ(finvolmea) we have adsorbed in the interaction 
$\Phi$ the usual
inverse temperature factor $\beta$ in front of the Hamiltonian.
In most notation we will drop the superscript $\Phi$ if that does not generate
confusion. Moreover, whenever we consider $\Phi^{\sst \ul}$ instead of $\Phi$, 
we will write $H_V^{\t,\ul}$ for the finite volume Hamiltonian and $\mu_V^{\t,\ul}$ for the corresponding finite volume Gibbs 
measure. 


Given a measurable bounded function $f$ on $\O$,
$\mu_V (f)$ denotes the {\it function} $\s \mapsto \mu^{\s}_V(f)$ where 
$\mu^{\s}_V(f)$ is just the average of $f$ w.r.t. $\mu^{\s}_V$.
Analogously, for any event $X$,  $\mu^{\t}_V (X) := \mu^{\t}_V (\id_X)$, 
where
$\id_X$ is the characteristic function of $X$. $\mu^{\t}_V(f,g)$ stands
for the covariance or {\it truncated correlation} 
(with respect to $\mu_V^{\t}$) of $f$ and $g$.
The set of measures \equ(finvolmea) satisfies the DLR compatibility
conditions
$$
   \mu^{\t}_\L( \mu_V (X) ) = \mu^{\t}_\L (X) \qquad
   \forall\, X \in \cF
   \qquad
   \forall\,
   V\sset \L\ssset\Z^d  
   \Eq(DLR)
$$


\nproclaim Definition [Gibbs].
A probability measure $\mu$ on $(\O, \cF)$ is called a 
{\it Gibbs measure\/}
for $\Phi$ if
$$
   \mu( \mu_V (X) ) = \mu(X) \qquad
   \forall\, X \in \cF
   \qquad
   \forall\,
   V\in \bF  
   \Eq(DLRi)
$$
see e.g. \ref[G].

\smallno
We introduce the {\it canonical Gibbs measures\/} on
$(\O, \cF)$ defined as
$$
  \nu_{\L, N}^{\t} = \mu_\L^{\t}( \cdot \tc N_\L = N ) \qquad 
  N \in \{ 0, 1, \ldots, |\L| \} 
  \Eq(cano)
$$
where $N_\L$ is the number of particles (\ie spins equal to $+1$)
in $\L$.

\medno
\vskip 0.5cm \beginsubsection \number\numsec.3 The dynamics

\medno
We consider the so--called Kawasaki dynamics in which particles 
(spins with $\s(x)=+1$) can
jump to nearest neighbor empty ($\s(x)=0$) locations, keeping
the total number of particles constant.
For $\s\in\O$, let $\s^{xy}$ be the configuration obtained from $\s$
by exchanging the spins $\s(x)$ and $\s(y)$. Let $t_{xy} \s  = \s^{xy}$
and define $(T_{xy} f)(\s) = f ( t_{xy} \s )$
The stochastic dynamics we want to study is determined by the Markov generators
$L_V$, $V\ssset \Z^d$, defined by 
$$
  (L_V f)(\s) = \sum_{ [x,y] \in \cE_V } c_{xy}(\s) \, (\nabla_{xy} f)(\s)
  \qquad
  \s\in\O\,,  \quad f:\O\mapsto \bR
  \Eq(gnrt)
$$
where $\nabla_{xy} = T_{xy} - \id$.
The nonnegative real quantities
$c_{xy}(\s )$
are the {\it transition rates\/} for the process.
\smallno
The general assumptions on the transition rates are
\item{(1)} {\it Finite range.}
   $c_{xy}(\s)$ depends only on the spins $\s(z)$ with $d(\{x,y\},z)\le r$ 
\item{(2)} {\it Detailed balance.} For all $\s\in\O$ and $[x,y]\in \cE_{\Z^d}$
$$
   \exp\bigl[ - \,H_\xy(\s)   \bigr]
   c_{xy}(\s) =
   \exp\bigl[ - \, H_\xy(\s^{xy}) \bigr]
   c_{xy}(\s^{xy})
   \Eq(dbal)
$$
\item{(3)} {\it Positivity and boundedness.} 
There exist positive real numbers $c_m(\b)$
$c_M(\b)$ such that
$$
   c_m \le
   c_{xy}(\s) \le
   c_M 
   \qquad
   \forall x,y \in \Z^d \, ,\s\in\O \,.
   \Eq(bounded)
$$

\smallno
We denote by $L_{V,N}^{\t}$ the operator $L_V$ acting on 
$L^2(\O, \nu_{V,N}^{\t})$ (this amounts to choosing $\t$ as the boundary 
condition and $N$ as the number of particles).
Assumptions (1), (2) and (3) guarantee that there exists a
unique Markov process whose generator is $L^{\t}_{V,N}$, and whose semigroup
we denote by $(T^{V,N,\t}_t)_{t\ge 0}$.
$L^{\t}_{V,N}$ is a bounded operator on $L^2(\O, \nu_{V,N}^{\t})$
and $\nu_{V,N}^{\t}$ is its
unique invariant measure.
Moreover $\nu_{V,N}^{\t}$ is {\it reversible} with respect to the
process, \ie $L_{V,N}^{\t}$ is self--adjoint on $L^2(\O,\nu_{V,N}^{\t})$.
\smallno
A fundamental quantity associated with the dynamics of
a reversible system is the gap of the generator, \ie
$$
   \gap( L^{\t}_{V,N} ) =
   \inf {\rm spec}\, 
   (- L^{\t}_{V,N} \restriction \identity^\perp  )
$$
where $\identity^\perp$ is the subspace of
$L^2(\O, \nu_{V,N}^{\t})$ orthogonal to the constant functions.
We let $\Dir$ be the Dirichlet form associated with the generator 
$L_{V,N}^\t$,
$$
   \Dir^{\t}_{V,N} (f,f) = 
   \< f ,\, -L_{V,N}^\t f \>_{L^2(\O, \nu_{V,N}^\t)}=
   {1\over 2} \sum_{[x,y]\in \cE_V}
   \nu^{\t}_{V,N}  
   \[\,
    c_{xy} \,
     (\nabla_{xy} f) ^2 
  \,\]
   \Eq(dir)
$$
and $\Var^{\t}_{V,N}$ is the variance relative to the probability
measure $\nu^{\t}_{V,N}$. 
The gap can also be characterized as
$$
   \gap( L^{\t}_{V,N} )
   = \inf_{\st f\in L^2(\O, \nu^{\t}_{V,N}), \atop \st 
             \Var^{\t}_{V,N}(f) \ne 0 } \
   {\Dir^{\t}_{V,N} (f,f) \over \Var^{\t}_{V,N}(f) } \,.
   \Eq(gap)
$$

\medno
\vskip 0.5cm \beginsubsection \number\numsec.4 Definition of the 
mixing condition and main results.


\medno
In order to formulate our basic mixing condition on the interaction $\Phi$ 
we fix positive numbers $C,m,l$ with $l\in {\bf N}$. We then say that 
a collection 
of real numbers $\ul := \{\l_x\}_{x\in \Z^d}$ 
is $l$--regular if, for all $i\in \Z^d$  
and all $x\in Q_l(x^i)$, $x^i\in l\Z^d$, $\l_x =\l_{x^i}$.  

\smallno
\proclaim Definition of property $USMT(C,m,l)$.
For any 
$l$--regular set $\L$ and any pair of bounded local functions $f$ and $g$  
$$
\sup_{\ul \atop \st \ul \,{\rm is}\, l-{\rm regular}}
\sup_\t|\mu_\L^{\t,\ul}(f,g)| \le 
     C\sup_\t\mu_\L^{\t,\ul}(|f|)\,
             \mu_{\L\setminus \D^{(l)}_f}^{\t,\ul}(|g|)
           \sum_{x\in \partial_{r}^- \D^{(l)}_f}
           \sum_{y\in \partial^{-}_r \D^{(l)}_g} \nep{-m|x-y|}
$$
provided that $d(\D^{(l)}_f,\D^{(l)}_g) \ge l$.

\bigno
{\it Remark.} The expert reader may have noticed that our condition
is different, and in principle stronger, than the one used in \ref[LY]
and [Y] because we require the exponential decay of cavariances 
uniformly in the chemical potential even when the latter {\it varies} 
over the 
atoms of a partition of $\L$ while in the above mentioned papers 
the chemical potential is assumed to be constant over $\L$. 
In two dimension, followig the ideas of \ref[MOS], one 
can prove \ref[BCO] that the two conditions are equivalent. 
In higher dimension, contrary to what claimed in 
section 8 of \ref[Y], one can construct examples 
in which a kind of phase transition occurs 
along the interface between two different 
atoms even if for all $l$--regular sets $\L$ 
the covariances decay exponentially 
fast uniformly w.r.t. to constant chemical potentials.  
We remark that the above mixing condition plays an important role also in other contexts like 
in the analysis of renormalization group pathologies \ref[BCO]. 
\bigno
We are finally in a position to formulate the main results of this paper  
on the spectral
gap of the generator of Kawasaki dynamics in a finite volume.

\nproclaim Theorem [main].
Assume that there exist positive numbers $C,m,l$, with $l\in {\bf N}$, such 
that property $USMT(C,m,l)$ holds. Then there exists positive constants 
$c_1, c_2$ such that
$$
   c_1 L^{-2} \le \min_{N,\t} \gap( L^{\t}_{Q_L, N} ) \le 
                  \max_{N,\t} \gap( L^{\t}_{Q_L, N} ) \le c_2 L^{-2}
   \Eq(main.lb)
$$

\noindent
A nice consequence of the above estimate is an inverse polynomial 
bound on the time decay to 
equilibrium in $L^2\(d\nu_{\L,N}^\t\)$ of local observables. 

\nproclaim Theorem [local].
Assume that there exist positive numbers $C,m,l$, with $l\in {\bf N}$, such 
that property $USMT(C,m,l)$ holds. Then for any $\e\in (0,1)$ and any 
local function $f$ with $0\in \D_f$ 
there exists a positive constant 
$C_{f,\e}$ such that 
for any integer $L$ multiple of $\,l$ and any integer $N\in \{1,\dots, (2L)^d\}$ 
$$
  \Var_{\L,N}^\t\( \nep{tL_{\L,N}^\t}f\) \le 
   C_{f,\e}\, \,{1\over t^{\a-\e}}
$$
where $\L := B_L$ and $\a = \ov2$
in $d=1$, $\a=1$ for $d > 1$.

\bigno
{\it Remark.} In the infinite temperature case 
(simple exclusion model) (see \ref[BZ1]) the time 
evolution of a local linear function of the form 
$f(\s) = \sum_x a_x \s(x)$, where $\{a_x\}_{x\in \Z^d} \in l^2\(\Z^d\)$, 
can be computed exactly: 
$\nep{tL_{\L}}f =  \sum_x \(\nep{t\D_\L}a\)_x \s(x)$ where 
$\D_\L$ is the discrete Laplacian on $\L$ with the appropriate 
boundary conditions. In particular one gets in this case that
$\|\nep{tL_{\L}}f\|_2^2 \le C_f t^{-{d/2}}$. Our exponent is thus arbitrarily 
close to the correct one only for $d=1,2$. 
\bigno

\vskip 1truecm
\newsection Preliminary results

\bigno
Here we collect several preliminary results that will be
used in the future. Although the reader may skip this section  
during a first
reading and come back when these results are needed, we still think useful to
give a short roadmap of the section.

\bul In paragraphs $\S 3.1$ and $\S 3.2$  
we collect the necessary results on the 
comparison between finite volume multicanonical Gibbs measures, namely 
grand canonical Gibbs measures conditioned to have a specified number 
of particles in the atoms of a given partition of a finite set $\L$,   
and the corresponding unconditioned measures. 

\bul In paragraph $\S 3.3$, in strict analogy with a similar result 
for non conservative dynamics (see section 4 of \ref[M1]),  
we prove a simple but important 
result whose physical meaning is roughly the following. Consider 
the Kawasaki dynamics in a a box $\L$ that is covered by two rectangles, 
$\L_1$ and $\L_2$, each of size $\approx {|\L|\over 2}$ but such that
their overlap is a long and thin strip whose size also scales like 
$|\L|$. Besides the usual constraint that the total number of particles
is conserved impose the additional constraint that the number of particles
in each rectangle {\it and} in their overlap does not change with time. 
Then the relaxation time of the dynamics is not larger than a constant 
close to one times the largest among the relaxation times of the simple 
Kawasaki dynamics in each of the two rectangles.  

\bul In paragraph $\S 3.4$ we begin to discuss an important topic of our approach
to the computation of the spectral gap of Kawasaki dynamics, namely 
the distribution of the number of particles in an atom of a given 
partition
of a finite set $\L$ under a multicanonical measure. In this paragraph 
we only give two simple results and we postpone a more detailed analysis
to section 4.

\bul In paragraph $\S 3.5$ we show how to prove a sharp Poincar\`e inequality 
for 
a symmetric, finite volume, one 
dimensional random walk whose invariant measure is kind of bell--shaped
around the mean. The reason of our interest 
in such a topic is that, as we will show in section 4, 
the distribution of the number of particles in an atom of a given 
partition
of a finite set $\L$ under a multicanonical measure has exactly the 
above property and the Poincar\`e inequality is an effective tool 
to bound from above the variance of a random variable.        

\bul In paragraph $\S 3.6$ we recall and slightly extend a key result 
of \ref[LY] related to the so--called ``two block estimate''. 
More precisely the result states that the covariance between an arbitrary 
function $f$ and the spatial average of translationally covariant local 
functions can be bounded from above by the inverse of the volume times 
an arbitrarily small constant times the variance of $f$ plus a large
constant times the Dirichlet form of $f$ (see the appendix for more details).

\bul Finally, in paragraphs $\S 3.7, \, 3.8,\, 3.9$,  
we first show how to compute and then 
how to estimate quantities like 
$\nu_{\L,{\bf N}}^\t\( 
        \[{d\over dn}\nu_{\L,{\bf N}}^\t\(f\tc N_{V}=n\)\]^2
                     \)$,  
where $\nu_{\L,{\bf N}}^\t$ is a multi canonical measure over the atoms 
$\{\L_i\}_{i=1}^k$ 
of a partition of a finite set $\L$, $V$ is a subset of a given 
atom $\L_j$ and ${d\over dn}$ is the 
discrete derivative. As explained in section 5, terms like the above one 
naturally appear in the recursive bound of the spectral gap 
and, roughly speaking, they measure the influence on the relaxation time 
of Kawasaki dynamics 
due to the 
exchange of particles between different atoms of the partition.


\vskip 0.5cm \beginsubsection \number\numsec.1 Existence of the tilting fields

\medno
We begin by recalling the following quite general result on the relationship
between particle numbers and the chemical potential (see the appendix 
in \ref[CM1]).
\smallno
Let $\L= \cup_{i=1}^k \L_i$, where the atoms $\L_1,\dots ,\L_k$ are
pairwise disjoint and 
such that there exists $x_1, \dots,x_k$, 
$x_i\in \L_i \quad i=1,\dots,k$, with  
$\min_{i\neq j} d(x_i,x_j) \ge 2r$. Then, for any possible 
values $N_1,\dots, N_k$ of the particle numbers in $\L_1,\dots,\L_k$ and for 
any boundary condition $\t$, there
exists a unique choice of $\(\l_1,\dots,\l_k\)$ such that
$\mu_\L^{\t,\ul}\(N_{\L_i}\) = N_i,\quad i=1,\dots,k$, 
where the chemical potential $\ul$ is constant and equal to $\l_i$ inside 
$\L_i$. \acapo
Next we give a simple result on the dependence on the average particle 
number and on the boundary condition of the chemical potential.
 

\nproclaim Lemma [der]. 
Assume property $USMT(C,m,l)$. 
Let $f$ be a bounded local function
and $\l =\l (\t,n)$ 
the chemical potential 
such that the grand canonical Gibbs state on $\L=Q_L$, $L$ a multiple of the 
basic scale $l$, satisfies
$\mu_\L^{\t,\l}(N_\L)=n $
with $n=(0,|\L|)$. Let $\a\in (0,1)$. Then there exists 
a constant $k$ independent on $L$ such that for any $L$ large enough 
\itm1 For any function $f$ such that $\D_f \sset \L$ and
$d(\D_f,\partial^+_r\L)\ge L^\a$
$$
 \eqalign{
  &i)\phantom{i} \qquad 
  \sup_{y \in \dep_r \L}\ninf{\nabla_y\mu_\L^{\t,\l}(f)}
                    \le\,k\ninf{f}\,{|\D_f|\over|\L|}  \cr
  &ii) \qquad \sup_{y \in \dep_r\L} \ninf{\nabla_y {d\over dn} 
               \mu_\L^{\t,\l}(f)}
             \le k\ninf{f}\,{1\over n}\,{|\D_f|\over|\L|}  \cr
	}	
$$ 
\itm2  For any function $f$ such that $\D_f \sset \L$ 
$$
  \eqalign{
   &i)\phantom{ii} \qquad \ninf{{d\over dn}\mu_\L^{\t,\l}\(f\)}
  \le\, k\ninf{f}{|\D_f|\over|\L|} \cr
  &ii) \qquad 
     \ninf{{d^2\over dn^2}
  \mu_\L^{\t,\l}\(f\)}\le\, k\ninf{f}\,{1\over n}{|\D_f|\over |\L|} \cr
	}
$$
        
\Pro\ 
Parts $i)$ of $1)$ and $2)$
follow immediately from the bounds 
$$
 \eqalign{
  &\sup_{y \in \dep_r \L}|\l(\t^{y},n) - 
  \l(\t,n)| \le {k_1 \over n}  \cr
  &|{d\l(\t,n)\over dn}|  =|{1\over\mu_{\L}^{\t,\l(\t,n)}(N_\L,N_\L)}|
  \le\, {k_1 \over n}  \cr
	}
\Eq(d/dn)
$$ 
together with 
$$
 |{d\over d\l} \mu_{\L}^{\t,\l(\t,n)}(f)| =
 |\mu_{\L}^{\t,\l(\t,n)}\bigl( N_{\L}, f\,\bigr) | 
            \le \,k_2\ninf{f}\,{n |\D_f|\over|\L|}
$$
for suitable constants $k_1,k_2$ independent of $L$
(see proposition 3.1 and section 7.1 in 
\ref[CM1]).\acapo
Part $ii)$ of $1)$ is a little bit tedious but it follows 
without problems from the bound 
$|\nabla_y \mu_\L^{\t,\l}(N_\L,N_\L)| \le k_3$
which, together with \equ(d/dn), 
implies that $|\nabla_y {d\over dn}\l(\t,n)| \le {k_4\over n^2}$.
Part $ii)$ of $2)$ is straightforward if we observe that
$$
  |{d^2\l(\t,n)\over dn^2}| =|{\mu_{\L}^{\t,\l(\t,n)}
  \bigl(N_\L, N_\L, N_\L\bigr)
  \over\mu_{\L}^{\t,\l(\t,n)}\bigl(N_\L, N_\L\bigr)}|\,
  \big({d\l(\t,n)\over dn}\big)^2 \le k_5 {1\over n^2}
 $$
and 
$$
  {d^2\over dn^2}\mu_{\L}^{\t,\l(\t,n)}(f)=
  \mu_{\L}^{\t,\l(\t,n)}(f,N_\L)\,{d^2\l(\t,n)\over dn^2}+
  \mu_{\L}^{\t,\l(\t,n)}(f,N_\L,N_\L)\,\big({d\l(\t,n)\over dn}\big)^2
$$
(see again proposition 3.1 in \ref[CM1]). \QED



\vskip 0.5cm \beginsubsection \number\numsec.2 Equivalence of ensembles

\medno
Here we recall some fine results on the finite volume 
equivalence of ensembles that will be crucial in most of our 
future arguments. We refer the reader to sections 5, 7.2 and 7.3 of 
\ref[CM1].

\smallno
Fix $\d,\e\in (0,1)$, $\e \ll 1$, 
and two integers $k,l$ such that $\d k <1$. 
Let $L_1,\dots, L_k$ be large multiples of the basic length scale 
$l$, let $L=\sum_i L_i$  and assume that $L_j \ge \d L$ for any $j$. 
We then choose  
one coordinate direction, e.g. the $d$ direction, and we take 
$\L=Q_L$, $\L_1$ equal to the first slice of $\L$  orthogonal to 
$d$--direction of width $L_1$, \ie  
$\L_1 = \{x\in \L~:~ 0 \le x_d < L_1\}$, $\L_2$ equal to the 
slice of $\L$ on top of $\L_1$ of width $L_2$ and so on. Let also  
${\bf N}=\{N_i\}_{i=1}^k$ be a set of possible values of 
${\bf N}_\L := \{N_{\L_i}\}_{i=1}^k$ 
and let us assume, for a given boundary condition $\t$, 
that $\ul$ is constant on each set $\L_i$ and such that 
$\mu_\L^{\t,\ul}(N_{\L_i}) = N_i$, $i=1,\dots,k$. We denote by 
$\nu_{\L,{\bf N}}^\t$ the multi canonical Gibbs measure
$\mu_\L^{\t,\ul}\bigl(\cdot\,|\, N_{\L_i} = N_i, \,i=1,\dots,k\,\bigr)$ and
by $\O_\t$ the set of configurations $\t'$ that coincide with $\t$ in the 
half space $\{x\in \Z^d~:~ x_d < L\}$. 

\smallno
Next, given $M>0$ and $\D \sset \L$, we say that
$\D$ is good if either 
\item{a)} $\D$ together with its $M\log |\L|$ neighborhood is entirely 
contained in some
atom $\L_i$ with \hbox{$\rho_i:= {N_i\over |\L_i|} \ge  |\L|^{-\e}$} 

\noindent
or 
\item{b)} $\D$ together with its $M$ neighborhood is entirely contained in some
atom $\L_i$ with \hbox{$\rho_i \le  |\L|^{-\e}$}. 

\noindent
A set is bad if it is not good. 
For good sets $\D \sset \L_i$, $i=1,\dots,k$, we define 
$$
{\bar \D} = \cases{ 
    \{x\in \L: \; d(x,\D)\le M\log|\L| \,\} & if $\rho_i \ge  |\L|^{-\e}$ \cr
    \{x\in \L: \; d(x,\D)\le M\,\} & if $\rho_i \le  |\L|^{-\e}$ \cr     
                  }
$$
while for bad sets $\D$ 
$$
{\bar \D} = \{x\in \L: \; d(x,\D)\le M\log|\L| \,\} 
$$
With these notations the results on the finite volume equivalence of 
ensembles that will be essential for the rest of this paper read as follows.

\nproclaim  Proposition [EQ].
Assume condition $USMT(C,m,l)$.
Then, for any 
$l$, $M$ large enough and $\e$ small enough independent 
of $\{\rho_i\}_{i=1}^k$, 
there exist  
constants $C'=C'(C,m,\|\Phi\|,l,\d,M,\e)$, 
$L_0=L_0(C,m,\|\Phi\|,l,M,\d,\e)$ such that, if $L\ge L_0$,
%
\itm1 Assume $k=1$. Then 
for all local functions $f,g$ with $l$--support contained 
in $\L$ such that $|\D_f^{(l)}| \le |\L|^{1-\e}$ and similarly for $g$
$$
  |\,\nu_{\L,N}^\t(f,g)\,| \le 
  C(f,g)\rho\,\Bigl[ 
  {1\over |\L|}+ \nep{-md(\D^{(l)}_f,\D^{(l)}_g)}\Bigr] 
$$
where $C(f,g) = C'\ninf{f}\,\ninf{g}|\D^{(l)}_f|^2\,|\D^{(l)}_g|^2$.
Moreover
$$
  |\,\nu_{\L,N}^\t(f) - \mu_\L^{\t,\l}(f)\,| \le C' \ninf{f} {|\D_f|\over |\L|}
$$            %
\itm2 Assume $k \ge 2$. Then
for all local functions $f,g$ with $l$--support contained 
in $\L$ such that $|\D_f^{(l)}| \le |\L|^{1-\e}$ and similarly for $g$
$$
  |\,\nu_{\L,{\bf N}}^\t(f,g)\,| \le 
  C(f,g)A_\L(f,g)
  \cases{\min\{\nu_{\L,{\bf N}}^\t(|f|),\nu_{\L,{\bf N}}^\t(|g|)\}\, 
  & if $\D^{(l)}_f$ and $\D^{(l)}_g$ are both good \cr 
   & or bad \cr
  \noalign{\vskip3pt}
  \nu_{\L,{\bf N}}^\t(|f|) & if $\D^{(l)}_f$ is bad\cr
  \noalign{\vskip3pt}
  \nu_{\L,{\bf N}}^\t(|g|) & if $\D^{(l)}_g$ is bad\cr
  } 
$$                           
where $C(f,g) = C'\ninf{f}\,\ninf{g}|\D^{(l)}_f|\,|\D^{(l)}_g|$ and
$$
   A_\L(f,g) = \cases{
              {1\over |\L|}+ \nep{-md(\D^{(l)}_f,\D^{(l)}_g)} & 
                if $\D^{(l)}_f$ or $\D^{(l)}_g$ is good \cr
             \noalign{\vskip3pt}
              {1\over |\L|}\({|\bar \D^{(l)}_g|\over |\D^{(l)}_g|}\)^2 
                                   \({|\bar \D^{(l)}_f|\over |\D^{(l)}_f|}\)^2
                                 + \nep{-md(\D^{(l)}_f,\D^{(l)}_g)} & 
	        otherwise\cr
	}                           
$$
%
\itm3 
Assume $k \ge 2$ and 
$\D \equiv \D_f^{(l)}\sset \L_n$, $n \le k$. Then
$$
\sup_{\t'\in \O_\t} |\nu_{\L,{\bf N}}^\t(f) -\nu_{\L,{\bf N}}^{\t'}(f)| 
     \le C'\ninf{f}\cases{
%               
            |\D|\,\Bigl[{1\over |\L|} + 
 {1\over L^{k-n+1}}\Bigr]
%                 
          & if $\D$ is good \cr
                 &  \cr
              {|\D|\over |\L|}\bigl({|\bar \D|\over |\D|}\bigr)^2 +
        \max_{j=n,n\pm 1}~ \Bigl[{ 
        |{\bar \D}\cap \L_j|^\12\over L^{k-j+1}}~\Bigr]
%			
	  & if $\D$ is bad \cr
		}    
$$
%
\itm4
Assume $k \ge 2$ and 
$\D_f^{(l)} \sset \partial_r\L_n \cap \dep_r\L_{n-1}$. 
Let
${\bar n} = \inte{\,|{d-1\over 2}-1|\,} +1$ and 
$\g(L) = \left[{(\log L)^2\over L} + 
{(\log L)^{\ov2}\over L^{{\bar n}+1-(d -1)/2}}\right]$.  
Then
$$
\sup_{\t'\in \O_\t} |\nu_{\L,{\bf N}}^\t(f) -\nu_{\L,{\bf N}}^{\t'}(f)| \le 
C'{\rm Osc}(f)\, \g(L)^{\inte{{k-n+1\over {\bar n}+1}}}
$$
where $Osc(f) := \max_{\s,\h}|f(\s)-f(\h)|$.

\bigno
{\it Remark.} Actually the first part of the proposition holds in 
a much more general geometric context (see $\S 7.3$ of \ref[CM1]).

\bigno
For future purposes the next result 
is stated in a slighlty more general form.

\nproclaim Proposition [EQ1].
Assume condition $USMT(C,m,l)$. Fix $\d \in (0,1)$, $\L\in \bF_l$ 
and a partition of $\L$ into $l$--regular sets $\L_1,\dots,\L_k$. Let 
$f$ be such 
that $|\L_j \setminus \D_f| \ge \d |\L_j|$ for any $j=1,\dots k$. 
Then for any $l$ large enough there 
exists a constant $A$ depending only on $C,m,\|\Phi\|,l,k,\d$  
such that 
$$
\nu_{\L,{\bf N}}^\t(|f|) \le A\, \mu_\L^{\t,\ul}(|f|)
$$
where $\bf N$ and $\ul$ are as in the beginning of this paragraph. 
In particular
$$
\nu_{\L,{\bf N}}^\t(f,f) \le A \,\mu_\L^{\t,\ul}(f,f)
$$

\Pro\ Following \ref[CM1] (see formula (5.3) there) we write
$$
\nu_{\L,{\bf N}}^\t(|f|) = {\int\limits_{-T_1}^{T_1}dt_1\dots 
                 \int\limits_{-T_k}^{T_k}dt_k \, 
      \mu_\L^{\t,\ul}\(\nep{i\sum_j{t_j\over \s_i}(N_{\L_j}(\h)-N_j)} |f|\)
                \over 
                \int\limits_{-T_1}^{T_1}dt_1\dots 
        \int\limits_{-T_k}^{T_k}dt_k \,\, F(t_1,\dots,t_k)}
\Eq(e1)
$$
where $\s_j^2 := \mu_\L^{\t,\ul}\(N_{\L_j},N_{\L_j}\)$, $T_j := \pi \s_j$ 
and  
$F(t_1,\dots,t_k) := 
\mu\bigl(\nep{i\sum_j{t_j\over \s_j}(N_{\L_j}(\h)-N_j)}\bigr)$.\acapo
With our assumptions the denominator of \equ(e1) is bounded from below 
by a suitable constant depending only on $C,m,\|\Phi\|,l,k$ (see lemma 5.2
in [CM1]). In order 
to bound from above the numerator we observe
that
$$
 \mu_\L^{\t,\ul}\(\nep{i\sum_j{t_j\over \s_j}(N_{\L_j}(\h)-N_j)} |f|\) =
  \mu_\L^{\t,\ul}\(
       \mu_\L^{\t,\ul}\(
        \nep{i\sum_i{t_j\over \s_j}(N_{\L_j}(\h)-N_j)} 
                       \tc \cF_{\D_f}\)
                         |f|\)
\Eq(e2)
$$
and that, because of proposition 3.1, 3.3 
of \ref[CM1], for any choice of 
$t_j \in [-T_j,T_j], \; j=1,\dots,k$ we have $\ninf{\mu_\L^{\t,\ul}\(
\nep{i\sum_j{t_j\over \s_j}(N_{\L_j}(\h)-N_j)} \tc \cF_{\D_f}\)} \le 
\nep{-\a \sum_{j=1}^k t_j^2}$ for a suitable constant 
$\a$ depending only on $C,m,\|\Phi\|,l,k,\d$. 
Notice that it is here that the hypothesis 
$|\L_j \setminus \D_f| \ge \d |\L_j|$ for any $j=1,\dots k$ is being used.
The first statement of the proposition now follows at once.
To prove the statement about the variance we simply observe that
$$
 \nu_{\L,{\bf N}}^\t(f,f) \le 
 \nu_{\L,{\bf N}}^\t\( (\,f-\mu_{\L}^{\t,\ul}(f)\,)^2 \)
$$
\QED


\vskip 0.5cm \beginsubsection \number\numsec.3 A block dynamics bound 

\medno
Here we give a result that is a key step in our recursive bound of the 
spectral gap of Kawasaki dynamics. For simplicity we discuss our estimate in 
two dimensions and at the end we explain how to generalize it to 
higher dimensions.
\smallno 
Fix $\d\in (0,\ov{4})$ and an integer $l$. Let $L\in \Zp$ be a multiple of $l$ and 
let $\L$ denote the rectangle
$$
	\L = \bigl\{(x_1,x_2)\in \Z^2;\ 0\,\le \, x_1\,\le \, l_1-1, \ 
	               0\,\le \, x_2\,\le \, l_2-1\ \bigr\} \ ,
			\qquad l_1,l_2\in \Zp
$$ 
with 
$\min\{l_1, l_2\} \ge 0.1\max\{l_1,l_2\}$ and
$\max\{l_1,l_2\}\,\le \, L$. \acapo
Let $\L_1 = \bigl\{(x_1,x_2)\in \L;\, 
	               0\,\le \, x_2\,\le \, (\ov2 +2\d)\,l_2\ \bigr\}$, 
$\L_2 = \bigl\{(x_1,x_2)\in \L ;\, 
	               (\ov2 +\d)\,l_2\,\le \, x_2\,\le \, l_2\ \bigr\}$, 
and let $\L_3 = \L_1\cap \L_2$.\acapo
Let finally $N_i$ be possible values of the number of particles in 
$\L_i$, $i=1,2,3$, and let $\nu_{\L,{\bf N}}^\t$ be the multi canonical 
Gibbs measure
$\mu_\L^{\t}\bigl(\cdot\,|\, N_{\L_i} = N_i, \,i=1,\dots,3\,\bigr)$. 
Then we have 

\nproclaim Proposition [Block].
Assume condition $USMT(C,m,l)$.
Then, for any $\e > 0$ 
there exist  
$L_0=L_0(\e,C,m,\|\Phi\|,l,\d)$ such that, if $L\ge L_0$,  
$$
   \nu_{\L,{\bf N}}^\t\(f,f\) \le 
   (1+\e)\nu_{\L,{\bf N}}^\t\( \Var^\h_{\L_1,{\bf N}_1}(f) + \Var^\h_{\L_2,{\bf N}_2}(f)\)
$$
where $\Var^\h_{\L_i,{\bf N}_i}(f)$, $i=1,2$, denotes the variance of $f$ w.r.t. the 
multicanonical measure on $\L_i$ with $N_i$ particles, $N_3$ of which
are in $\L_3$, and boundary condition $\h$ on $\dep_r \L_i$.

\Pro\ Fix $\e \in (0,1)$ and
consider the continuous time reversible w.r.t. $\nu_{\L,{\bf N}}^\t$ 
Markov chain on $\O_\L$ ({\it block dynamics}) 
in which, with rate 
one, either $\L_1$ or $\L_2$ is chosen and there the ``old'' configuration 
$\s$ is replaced by a ``new'' one $\s'$ 
distributed according to the multicanonical 
Gibbs measure of the chosen block 
with total number of particles equal to either $N_1$ or $N_2$, 
$N_3$ particles in $\L_3$ and boundary condition $\s$ outside. 
The above proposition simply says that 
the spectral gap of our chain is larger than ${1\over 1+\e}$ provided that
$L$ is large enough. \acapo
It is in fact easy to check that the associated Dirichlet form is 
given by 
$$
\cE_{\rm block}\(f,f\) = 
\nu_{\L,{\bf N}}^\t\( \Var^\h_{\L_1,{\bf N}_1}(f) + \Var^\h_{\L_2,{\bf N}_2}(f)\)      
$$
and that the action of the corresponding semigroup $T_{\rm block}(t)$ 
is given by (see e.g. section 3 of \ref[M1])
$$
  T_{\rm block}(t) f =
  \sum_{n=0}^\infty 
  {(2 t)^n \over n!} 
  \nep{-2t} \ov{2^n}
  \sum_{X \in \{ 1,2 \}^n } \nu_{X_1} \cdots \nu_{X_n} (f)
  \Eq(GP1.4)
$$
where $(\nu_{i}f)(\s) := \nu_{\L_i,N_i}^\s\(f \tc N_{\L_3}=N_3\)$, $i=1,2$.
Since $(\nu_i)^2 = \nu_i$ the last summation
(over $X$)
in \equ(GP1.4) can be written as
$$
  \sum_{k=0}^{n-1}
  {n-1 \choose k}
  ( {\hat A}_{k+1} + {\hat B}_{k+1} ) f
  \Eq(GP1.6)
$$
where 
$$
  {\hat A}_k = ( \nu_1 \circ \nu_2 )^{ \inte{k/2} } \circ 
    \nu_1^{ k - 2 \inte{k/2} }
  \qquad
  {\hat B}_k = ( \nu_2 \circ \nu_1 )^{ \inte{k/2} } \circ 
    \nu_2^{ k - 2 \inte{k/2} }
$$
If now $f$ is an arbitrary
bounded measurable function on $\O_\L$, such that 
$\nu_{\L,{\bf N}}^\t(f) = 0$, we get
$$
  \ninf{ \nu_1\,\nu_2 \,\nu_1 f} \le
  \ninf{ \nu_{\L,{\bf N}}^\t\, \nu_2 \,\nu_1 f } +
  \ninf{ \nu_{\L,{\bf N}}^\t\, \nu_2\, \nu_1 \,f - \nu_1\, \nu_2 \,\nu_1\, f} 
  \Eq(GP1.8)
$$
By construction the first term on the RHS of \equ(GP1.8)
is equal to $\nu_{\L,{\bf N}}^\t(f)= 0$. 
Furthermore, since the interaction has
range $r$ and the number of particles is fixed in each 
set $\L_i$, $i=1,2,3$, the function $h \equiv \nu_2\, \nu_1 f$ is 
$\cF_{\L^c \cup \dep_r \L_2}$ measurable. This fact 
implies that
$$
\ninf{\nu_{\L,{\bf N}}^\t\, \nu_2\, \nu_1 \,f - \nu_1\, \nu_2\, \nu_1\, f } \le
  \sup_{\s,\h \in \O_{\dep_r \L_1} }|\nu_1(h)(\s) - \nu_1(h)(\h)|
  \Eq(GP1.10)
$$   
Thanks to (2) of proposition \thf[EQ], the r.h.s. of \equ(GP1.10) 
is smaller than $C' \ninf{h} \,\({\log L\over L}\)^{\ov2}$ which is smaller 
than $\({\e\over 1+\e}\)^3 \ninf{f}$ if $L$ is large enough. 
Iterating this inequality we get
$$
  \ninf{ {\hat A}_k f } \le \({\e\over 1+\e}\)^{ 3\inte{{k\over 3}} } \ninf{f}
  \qquad
  \ninf{ {\hat B}_k f } \le \({\e\over 1+\e}\)^{ 3\inte{ {k\over 3} }} \ninf{f}
  \Eq(GP1.12)
$$
Thus, the sup norm of \equ(GP1.6) is not greater than
$$
  \ninf{f}
%  {2 \over \({\e\over 1+\e}\)^3 }
  2( 1 + {\e\over 1+\e})^{n-1}
$$
which, inserted back into \equ(GP1.4) yields
$$
  \ninf{ T_{\rm block}(t) f } \le
  \ninf{f}
  4 %\({\e\over 1+\e}\)^{ -3 }
  \nep{- (1 - {\e\over 1+\e}) t} 
$$
In other words the spectral gap of the block dynamics is larger 
than $(1 - {\e\over 1+\e})$ so that 
$$
   \nu_{\L,{\bf N}}^\t\(f,f\) \le 
                (1 - {\e\over 1+\e})^{-1}\cE_{\rm block}(f,f)
                               = 
   (1+\e)\nu_{\L,{\bf N}}^\t\( \Var^\h_{\L_1,{\bf N}_1}(f) + \Var^\h_{\L_2,{\bf N}_2}(f)\)
\QED
$$

\bigno
{\it Remark.} The restriction of $d=2$ comes from (2) of 
proposition \thf[EQ]. In fact, in e.g. three dimensions,  
the r.h.s. of \equ(GP1.10) would have been bounded by only 
$\ninf{h}\sqrt{\log L}$ which is a completely useless bound ! 
The way out is to divide the ``safety belt'' $\L_3$ 
into several layers (just two in $d=3$) each one 
of width proportional to $\d L$, fix the number of particles in each one 
of them  
and then use (3) of 
proposition \thf[EQ]. In other words, in higher dimensions 
the block dynamics has a fast relaxation only if we prevent the exchange
of particles also inside a certain number of layers  of $\L_3$.
   
\vskip 0.5cm \beginsubsection \number\numsec.4 On the distribution of the particle number 

\medno
Here we provide some simple results on the distribution of the particle numbers
in the atoms of a partition of a given set $\L$. Throughout 
this subsection the setting will be as follows. 

\smallno
Let $l\in \integer_+$, $\d, \e \in (0,1)$, 
and let us consider $\L\in \bF_l$, 
$\L = \cup_{i\in I}Q_l(x^i)$, such that 
$|\partial^-_{l-1}\L | \le |\L|^{d-1+\e\over d}$. 
Let $I_1\dots I_k$ be a partition 
of $I$, let $\L_i = \cup_{j\in I_i}Q_l(x^j)$ and assume that 
$0 < \d \le {|\L_i|\over |\L|}$ and 
$|\partial^-_{l-1}\L_i | \le |\L|^{d-1+\e\over d}$ for all $i=1,\dots,k$. 
\acapo
Let also  
${\bf N}=\{N_i\}_{i=1}^k$ be a set of possible values of 
${\bf N}_\L := \{N_{\L_i}\}_{i=1}^k$. Given a boundary condition $\t$, 
let $\ul$ 
be the chemical potential, constant on each atom, 
such that $\mu_\L^{\t,\ul}\({\bf N_{\L}}\)={\bf N}$ 
(see above for the existence of $\ul$). 
Then we have (see Lemma 5.2 in \ref[CM1])

%  Lower and Upper bounds su Prob(N_1,N_2,....,N_K)            %

\nproclaim Proposition [N].
Assume property $USMT\(C,m,l\)$. Let $\s^2_i := \mu_\L^{\t,\ul}\(N_{\L_i},N_{\L_i}\)$ 
$i=1,\dots,k$. Then 
$$
 {1\over C'}  {1\over \prod_i\s_i} \le  \mu_\L^{\t,\ul}\({\bf N}_\L = {\bf N}\) 
                          \le C'  {1\over \prod_i\s_i}
$$
for a suitable constant $C'=C'\(C,m,l,\ninf{\Phi}\) >1 $. 


%  Bound rozzo su Prob(k+1)/Prob(k)		%
\noindent
The next result concerns the way particles distribute inside one 
block of the partition. More precise results in this direction 
are given in the next section.
\smallno
Pick $j\in [1,\dots,k]$ and divide $\L_j$ into two subsets $V,\,W$. 
Then we have
 
\nproclaim Proposition [mui].
For all 
$n \in [0, |V|-1]$
$$
   {(|V|-n)(N_j -n)\over (n+1)(|W|-N_j+n+1)} \, \nep{ -2\|\Phi\| }\le
   { \nu_{\L,{\bf N}}^{\t} \( N_V = n+1\) 
   \over \nu_{\L,{\bf N}}^{\t} \( N_V = n \)
	} 
   \le
   \nep{ 2\|\Phi\|}\, {(|V|-n)(N_j -n)\over (n+1)(|W|-N_j+n+1)} \, 
$$
where $\nu_{\L,{\bf N}}^\t(\cdot ) := 
\mu_\L^{\t,\ul}\bigl( \cdot \tc {\bf N}_\L = {\bf N}\bigr)$.

\bigno
{\it Remark.} It is important that the error is $\nep{ 2\|\Phi\|}$
and not $\nep{ 2\|\Phi^\ul \|}$.
\bigno
\Pro\
We write
$$
   \eqalign{
	&\mu_\L^{\t,\ul} \( N_V = n+1\,;\, {\bf N}_\L = {\bf N}  \) \cr
        &=
	{1 \over (n+1)(|W|-N_j +n +1) } \sum_{x\in V \atop \sst y \in W}
        \mu_\L^{\t,\ul} 
        \( \id_{ \{ {\bf N}_\L = {\bf N} \} }\id_{ \{N_V=n+1\} } 
                \s(x)\(1-\s(y)\) \)\cr
  	&={1\over (n+1)(|W|-N_j + n +1)}\sum_{x\in V \atop \sst y\in W } 
    	\mu_\L^{\t,\ul} \( e^{-(\nabla_{xy} H^{\t,\ul}_{\L} )(\s)} 
             \(1-\s(x)\)\s(y)
    	\id_{\{{\bf N}_\L={\bf N}\}}\id_{ \{N_V=n\} }\) \cr
    	&\le {(|V|-n)(N_j -n)\over (n+1)(|W|-N_j+n+1)} e^{2\|\Phi\|}
               \mu_\L^{\t,\ul} \(N_V = n\,;\, {\bf N}_\L = {\bf N} \)\cr
     	}  
$$
where we used the change of variable $\s\mapsto t_{xy}\s$ to obtain the second
equality and \hfill
$\|e^{-\nabla_{xy} H^{\t,\ul}_{\L} } \|_\infty\le e^{2\|\Phi\|}$
for the last inequality. The lower bound is analogous.
\QED

%\vfill\eject


%             Stima di Cheeger			%

\vskip 0.5cm \beginsubsection \number\numsec.5 Spectral gap of one dimensional discrete 
random walks.

\medno
\nproclaim Proposition [ON].
Let $\g$ be a positive 
probability measure on the integers $\O=\{N_1,N_1+1,\ldots,N_2\}$ and let 
$N^*\in \O$ be the largest integer such that $\sum_{n \le N^*}\g(n) \le \ov2$.
Then, for all functions $f$ on $\O$ we have
$$
  \Var(f) \le C_\g \, \sum_{n=N_1+1}^{N_2} (\g(n) \mmin \g(n-1)) \,
  \bigl[ f(n) - f(n-1) \bigr]^2
$$
where
$$
  C_\g = 4\,\max \[ \( \sup_{ n\le N^*+1}  
  \sum_{j\le n}{\g(j)\over \g(n)}\)^2 \;,\; 
    \(\sup_{ n\ge N^*}  
  \sum_{j\ge n}{\g(j)\over \g(n)}\)^2 \]
$$

\Pro\ 
We consider a continuous time Markov chain with transition rates
$$
  c(n,j)= \cases{ {\g(j)\over\g(n)}\mmin 1 & if $j = n\pm 1$ \cr
                  0 & otherwise. \cr
  }
  \Eq(tr)
$$
Since the rates satisfy the detailed balance condition
$$
  \g(n)\, c(n,j)=\g(j)\, c(j,n)
  \Eq(db)
$$
the probability measure $\g$ is reversible
with respect to the chain.
The associated Dirichlet form is given by
$$ 
  \eqalign{
  \Dir_\g(f,f) &=\sum_{n=1}^N \g(n)c(n,n-1)\[f(n-1)-f(n)\]^2 =\cr
  &=\sum_{n=1}^N (\g(n) \mmin \g(n-1)) \,
  \bigl[ f(n) - f(n-1) \bigr]^2 \,. \cr
  }
$$
If we denote by $\l$ the spectral gap of the generator of the chain, we have 
$$
  \Var_\g(f) \le {1\over \l}~ \Dir_\g (f,f) = 
  {1\over \l}\sum_{i=1}^N (\g(n) \mmin \g(n-1)) \,
  \bigl[ f(n) - f(n-1) \bigr]^2 \,.
$$
To conclude the proof we need a lower bound for $\l$.
Cheeger's inequality (see Theorem 2.1 in \ref[LS]) states that
$$
  \l \geq {I^2\over 8 M}
$$
where
$M=\sup_{i}~(c(n,n+1)+c(n,n-1))$
and
$$
  I=\min_{A\subset\O}{\sum_{(j,k)\in A \times A^c}
  c(j,k)\g(j) \over \g(A)\(1-\g(A)\)}
$$
With the choice \equ(tr) $M\le 2$. 
As the state space $\O$ is countable and connected, by Corollary 4.4 in 
\ref[LS], 
the minimum can be taken over all
subsets $A\subset\O$ such that $A$ and $A^c$ are connected.
Using the symmetry between $A$ and $A^c$ we can also impose 
$\g(A)\leq 1/2$. We can thus write, using 
\equ(db) and \equ(tr),
$$
\eqalign{
  {2\over I} &\leq \max \[\sup_{n\leq N^*}{\sum_{j\leq n}\g(j)
  \over\g(n)\mmin\g(n+1)}\;,\; \sup_{n\geq N^*+1}{\sum_{j\geq n}\g(j)
  \over\g(n)\mmin\g(n-1)} \] \cr
    &\leq \max \[ \( \sup_{ n\le N^*+1} \sum_{j\le n}{\g(j)\over \g(n)}\)
         \;,\; 
         \( \sup_{ n\ge N^*} \sum_{j\ge n}{\g(j)\over \g(n)}\) \] \cr
	}
  \Eq(gi) 
$$
\QED
\medno
\nproclaim Proposition [ON1].
In the same hypotheses of proposition \thf[ON] assume that 
for some $N_1 \le \bar N \le N_2$ and $\a >0$ 
${\g(j-1)\over \g(j)} \le \nep{-\a(\bar N-j)}$ for all $j \le \bar N$ and 
${\g(j+1)\over \g(j)} \le \nep{-\a(j-\bar N)}$ for all $j \ge \bar N$.
Then for all functions $f$ on $\O$ we have
$$
  \Var(f) \le C\max \{{1\over \a},1\}\, \sum_{n=N_1+1}^{N_2} (\g(n) 
  \mmin \g(n-1)) \,
  \bigl[ f(n) - f(n-1) \bigr]^2
$$ 
for a suitable numerical constant $C$ independent of $f$ and all the other
parameters.

\Pro\ Without loss of generality we can assume $N^* \le \bar N$. 
Using the hypothesis, for any $n \le \bar N$ we have 
$$
\eqalign{
\sum_{j\le n}{\g(j)\over \g(n)} &= 
1+ \sum_{j < n}\, \prod^{n}_{z=j+1} \,{\g(z-1)\over \g(z)}      \cr 
&\le 1 + \sum_{j < n}\, \prod^{n}_{z=j+1}\,\nep{-\a(\bar N-z)}   \cr
& \le 1+  \sum_{j < n} \nep{-{\a\over 2} (i-j)^2}                           \cr
&\le 1+ {c \over \sqrt{\a}}                                                \cr
	}
$$
for a suitable numerical constant $c$. 
Similarly for $\sum_{j\ge n}{\g(j)\over \g(n)}$, $n \ge \bar N$. \acapo
Therefore, 
for any \hbox{$n\in \O\cap [N^*+1, \bar N]$}, 
$\g(n) \ge {\sqrt{\a}\over 2\(c+\sqrt{\a}\)}$ so that  
$$
\sup_{n\leq N^*+1}\sum_{j\leq n}{\g(j)
  \over\g(n) }\le c'\max\{ {1\over \sqrt{\a}},1\}
$$
for a suitable numerical constant $c'$.
Analogously 
$$
\eqalign{
\sup_{n\geq N^*} \sum_{j\geq n} {\g(j)\over \g(n)} &\le
 1+ {c \over \sqrt{\a}}+\, 
        \sup_{n\in \O\cap [N^*+1, \bar N] } {1\over \g(n) } \cr  
     &\le c''\max\{ {1\over \sqrt{\a}},1\} \cr
	}
$$
The thesis now follows 
from proposition \thf[ON].
\QED

\vskip 0.5cm 
\beginsubsection \number\numsec.6 A key bound on special covariances.

\medno
Assume the same setting of paragraph $\S 3.4$. For any local function 
$f$ and for any 
$x\in \Z^d$, let 
$\D_f(x) := \D_f+x$ and 
$f_x(\s) := f\(\s_{\D_f(x)}\)$. Then one 
has the following interesting result.

\nproclaim Lemma [Yaug].
Assume condition $USMT(C,m,l)$ and fix $i,j \in \{1,\dots,k\}$ with $i\neq j$. 
Let $g,h$ be two local functions
with support containing the origin and of diameter smaller than $2r$, 
$r$ being the range of the interaction. Then for any 
$\e > 0$ there exists $C_\e$ such that for any $f$
$$
  \nu_{\L,{\bf N}}^\t\bigl(f,{1\over |\L_i||\L_j|}
    \sum_{x\in \L_i\atop z\in \L_j}g_{x}h_{z}\bigr)^2     \leq
  \,{C_\e\over |\L|}\, \nu_{\L,{\bf N}}^\t\(\sum_{[x,z]\in \cE_\L}
  c_{xz}(\nabla_{xz}f)^2 \)+
  {\e\over |\L|}\, \Var_{\L,{\bf N}}^\t(f) 
$$ 
provided that $|\L|$ is larger than $C_\e$.

\bigno
{\it Remark.} When $k=1$, namely the partition of $\L$ is the trivial one,  
and $g(\h) \equiv 1$
the above result is nothing but the so called ``two--blocks estimate'' of 
lemma 4.4. of \ref[LY].  
\bigno
\Pro\ 
Fix $\e > 0$. In view of the above remark we can assume, 
without loss of generality, that $\nu_{\L,{\bf N}}^\t(g_x) =0$ for any 
$x\in \L_i$.\acapo
Let for notation simplicity $G := {1\over |\L_i|} \sum_{x\in \L_i} g_x$, 
$\cH = {1\over |\L_j|}\sum_{z\in \L_j } h_z$ and let us write 
$\cH = \cH^{\rm in} + \cH^{\rm ext}$ where $\cH^{\rm in}$ is the sum over 
those $z$'s in $\L_j$ such that $\D_h(z) \sset \L_j$ and $\cH^{\rm ext}$ 
the rest. Then, 
using the formula relating the
covariance of two functions $f$ and $g$ w.r.t. the measure 
$\nu_{\L,{\bf N}}^\t$
to the covariance w.r.t. the same measure conditioned to a sub $\s$-algebra,
we get
$$
  \eqalign{
  \nu_{\L,{\bf N}}^\t\bigl(f,{1\over |\L_i||\L_j|}
           &\sum_{x\in \L_i\atop z\in \L_j} g_x h_z \bigr)^2 
             = \nu_{\L,{\bf N}}^\t\bigl(f,G\cH \bigr)^2 \cr
        &\le \;
        2 \ninf{\cH^{\rm ext}}^2 \Var_{\L,{\bf N}}^\t(G) 
                                    \Var_{\L,{\bf N}}^\t(f)
                   + 4 \ninf{\cH^{\rm in}}^2 \nu_{\L,{\bf N}}^\t\( \[ 
           \nu_{\L,{\bf N}}^\t \Bigl(f,G 
         \tc \cF_{\L_i^c} \Bigr) \]^2 
			\) 
			\cr 
  		&\phantom{aaaa} + 
      4 \[ \nu_{\L,{\bf N}}^\t \(f,
      \,    
           \nu_{\L,{\bf N}}^\t \Bigl(G \tc \cF_{\L_i^c} \Bigr)
                      \cH^{\rm in} 
                             \)
				\]^2       \cr
    & \le \;
     {1\over |\L|}\[ C' \({|\dep_r\L_j|\over |\L_j|}\)^2 + 
        {\e\over 2}\, \] \Var_{\L,{\bf N}}^\t(f) +
   	\,{C_\e\over |\L|}\, \nu_{\L,{\bf N}}^\t\(
             \sum_{[x,z]\in \cE_\L} c_{xz}(\nabla_{xz}f)^2 \) \cr
      &\phantom{aaaa} +  
        4 \[ \nu_{\L,{\bf N}}^\t \(f,\,    
           \nu_{\L,{\bf N}}^\t\bigl(G\tc \cF_{\L_i^c} \bigr)
                      \cH^{\rm in} \)
				\]^2       \cr
  }
  \Eq(covz)
$$
where we have used the hypothesis $ {|\L_i|\over |\L|} \ge \d$ together with 
lemma 4.4. of \ref[LY] (see proposition \thf[ly] for a simpler proof) 
to bound the term
$\[ 
           \nu_{\L,{\bf N}}^\t \Bigl(f, G 
         \tc \cF_{\L_i^c} \Bigr) \]^2 $
and proposition \thf[EQ1] to get 
$ \Var_{\L,{\bf N}}^\t(G) \le C'' /|\L|$.
\acapo
The third term in the.r.h.s of \equ(covz) can be bounded from 
above by 
$$
\ninf{ \nu_{\L,{\bf N}}^\t (G \tc \cF_{\L_i^c} )}^2 \, 
\Var_{\L,{\bf N}}^\t(\cH^{\rm in} )\,
\Var_{\L,{\bf N}}^\t(f)
\Eq(second)  
$$
In turn, the second factor in the r.h.s. of \equ(second), using proposition 
\thf[EQ1] together with the mixing condition, is bounded from above 
by ${C_1 \over |\L|}$. The first factor in the r.h.s. of \equ(second), 
thanks to the hypothesis 
$\nu_{\L,{\bf N}}^\t(g_x) =0$, to a simple telescopic argument and to 
part $1)$ of proposition \thf[EQ],  
is bounded from above by
$$
\eqalign{
\Bigl[\, \sup_{\t,\t' \in \O_{\dep_r\L_i}} {1\over |\L_i|} \sum_{x\in \L_i} 
       &|\nu_{\L_i,N_i}^\t (g_x)-\nu_{\L_i,N_i}^{\t'} (g_x)| \,\Bigr]^2 \cr
  &\le \[\sup_{\t \in \O_{\dep_r \L_i}} {1\over |\L_i|} \sum_{x\in \L_i}
     \sum_{y\in \dep_r\L_i}\nep{2\|\Phi\|} 
     |\nu_{\L_i,N_i}^\t (g_x,\nep{-\nabla_y H_{\L_i}^\t} )| \]^2 
     \le C' \[{|\dep_r \L_i|\over |\L_i|}\]^2
			\cr
		}
$$
In conclusion, for any $\e >0$, 
the first and third term in the.r.h.s of \equ(covz) can be bounded from 
above by 
$
{\e\over 2|\L|} 
\Var_{\L,{\bf N}}^\t(f)  
$
provided that $|\L|$ is large enough. The proof is complete. \QED

%    Calcolo del gradiente di E(f | n)		%

\vskip 0.5cm \beginsubsection \number\numsec.7 Computing the gradient 
with respect to the particle number.

\medno
Assume the same setting of paragraph $\S 3.4$ and 
let $V$ and $W$ be such that $V \cap W = \emp$ and $\L_j = V \cup W$ for some 
$j\in \{1,\dots,k\}$.
Here we give a result that allows us to compute for any function $f$
the gradient w.r.t. to $n$ of $\nu_{\L,{\bf N}}^{\t}( f \tc N_{V} = n )$
and to bound it in terms 
of the Dirichlet form and the variance of $f$. 

\smallno
For $x,z\in \Z^d$, we define the events
$$
  E_{xz} = \{ \s\in \O: \s(x) = 1, \ \s(z) = 0 \} \,.
  \Eq(ED)
$$
Then we have

\nproclaim Proposition [diff].
Let $V$ and $W$ be such that $V \cap W = \emp$ and $\L_j = V \cup W$ for some 
$j\in \{1,\dots,k\}$.
Let $\g(n) = \nu_{\L,{\bf N}}^{\t}\{ N_V=n\}$. 
Let also $c_n=n(|W|-N_j+n)$, that is number of particles in $V\times$ number 
of holes in $W$, and let $c'_n = n(|V|-N_j+n)$. 
Then, for all functions $f$ on $\O$ we have 
\itm1
$$
  \eqalign{
  \nu_{\L,{\bf N}}^{\t}( f \tc N_{V} = n ) &- 
  \nu_{\L,{\bf N}}^{\t}( f \tc N_{V} = n-1 ) = \cr
  &
  \ov{c_n}  { \g(n-1) \over \g(n)}
  \sum_{ \st x\in V \atop
    \st z\in W }
  \nu_{\L,{\bf N}}^{\t} \[ (\nabla_{zx} f) \id_{E_{zx}} \nep{ - \nabla_{xz} H_\L }
    \tc N_{V} = n-1 \]  \cr
  + 
    &\ov{c_n}  { \g(n-1) \over \g(n)} 
   \sum_{ \st x\in V \atop
    \st z\in W }
  \nu_{\L,{\bf N}}^{\t} \[ \(\nep{ - \nabla_{xz} H_\L }-1\) \id_{E_{zx}} ,f
    \tc N_{V} = n-1 \] \cr
	}
$$
\itm2
$$
\eqalign{
  \nu_{\L,{\bf N}}^{\t}( f \tc N_{V} = n ) &- 
  \nu_{\L,{\bf N}}^{\t}( f \tc N_{V} = n-1 ) = \cr
  &
  -{1\over c'_{N-n+1}}{\g(n)\over \g(n-1)}
  \sum_{ \st x\in V \atop
    \st z\in W }
  \nu_{\L,{\bf N}}^{\t} \[  (\nabla_{xz} f) \id_{E_{xz}}\nep{ - \nabla_{xz} H_\L }
    \tc N_{V} = n \] - \cr
  &- 
  {1\over c'_{N-n+1}}{\g(n)\over \g(n-1)}
  \sum_{ \st x\in V \atop
    \st z\in W }
  \nu_{\L,{\bf N}}^{\t} \[ \(\nep{ - \nabla_{xz} H_\L }-1\) \id_{E_{xz}},f
    \tc N_{V} = n \] 
  \cr
}
$$

\noindent
{\it Remark}. A similar statement is contained in Lemma \teo[3.1] in \ref[LY].
\bigno
Before the proof we note that given $\L_j,\, V,\, W, \, N_j$ as above, 
let $n_{\rm max},\,n_{\rm min}$ be the maximum and minimum 
value of the particle number in $V$ under the constraint that 
$N_{\L_i} = N_i,\quad \forall \,i=1,\dots,k$. Let 
$ u = \inte{ \rho_j |V| }$ where $\rho_j ={N_j\over |\L_j|}$
and let, for $n\in \[n_{\rm min},n_{\rm max}\]$,
$$
   \eqalign{
	A(n) &= \cases{
   \ov{c_n}  { \g(n-1) \over \g(n)}\sum_{ \st x\in V \atop \st z\in W }
  \nu_{\L,{\bf N}}^{\t} \[ (\nabla_{zx} f) \id_{E_{zx}} 
                       \nep{ - \nabla_{xz} H_\L } \tc N_{V} = n-1 \]
                        & if $n \le u$ \cr
              \noalign{\vskip8pt}
 {1\over c'_{N-n+1}}{\g(n)\over \g(n-1)}\sum_{ \st x\in V \atop\st z\in W }
  \nu_{\L,{\bf N}}^{\t} \[  (\nabla_{xz} f) \id_{E_{xz}}
                       \nep{ - \nabla_{xz} H_\L }\tc N_{V} = n \] 
                        & otherwise    
	} \cr
              \noalign{\vskip12pt}
	B(n) &= \cases{
   \ov{c_n}  { \g(n-1) \over \g(n)} 
   \sum_{ \st x\in V \atop \st z\in W }
     \nu_{\L,{\bf N}}^{\t} \[ \(\nep{ - \nabla_{xz} H_\L }-1\) \id_{E_{zx}} ,f
   	 	\tc N_{V} = n-1 \]
	& if $n \le u$ \cr
	\noalign{\vskip8pt}
   {1\over c'_{N-n+1}}{\g(n)\over \g(n-1)}
  \sum_{ \st x\in V \atop
    \st z\in W }
  \nu_{\L,{\bf N}}^{\t} \[ \(\nep{ - \nabla_{xz} H_\L }-1\) \id_{E_{xz}},f
    \tc N_{V} = n \]
	& otherwise
	}
	}
\Eq(aa3)
$$
With this definition we have immediately 

\nproclaim Corollary [Dif].
In the same setting of proposition \thf[diff] 
$$
\eqalign{
\sum_{i} \(\g(n)\mmin \g(n-1)\) &\[\nu_{\L,{\bf N}}^{\t}( f \tc N_{V} = n ) - 
  \nu_{\L,{\bf N}}^{\t}( f \tc N_{V} = n-1 )\]^2 \cr 
  &\le
\sum_{i} \(\g(n)\mmin \g(n-1)\) \[ A(n)^2 + B(n)^2\]
		\cr
	}
$$


\noindent
{\it Proof of the proposition.} \  
Adding and subtracting $T_{xz}f$ we can write
$$
\eqalign{
  &
  \nu_{\L,{\bf N}}^{\t}( f \tc N_{V} = n )=
  \cr
  &
  {1\over c_n}\sum_{\st x\in V\atop\st  z\in W}
  \nu_{\L,{\bf N}}^{\t}\[ (f- T_{xz}f)\id_{E_{xz}}\tc N_{V} = n \]~+
  {1\over c_n}\sum_{\st x\in V\atop \st z\in W}
  \nu_{\L,{\bf N}}^{\t}\[ (T_{xz} f) \id_{E_{xz}} \tc N_{V} = n \]
  \cr
  }
  \Eq(aa)
$$
After the change of variable $\s\mapsto\s^{xz}$ and using the
equality $\nu_{\L,{\bf N}}^{\t}(fg) = \nu_{\L,{\bf N}}^{\t}(f,g) + \nu_{\L,{\bf N}}^{\t}(f) \nu_{\L,{\bf N}}^{\t}(g)$,
we can write the first term in the r.h.s. of \equ(aa) as
$$
\ov{c_n}  { \g(n-1) \over \g(n)}
  \sum_{ \st x\in V \atop
    \st z\in W }
  \nu_{\L,{\bf N}}^{\t} \[ (\nabla_{zx} f) \id_{E_{zx}} \nep{ - \nabla_{xz} H_\L }
    \tc N_{V} = n-1 \]
$$
while the second term becomes
$$
  \eqalign{
  &
  {1\over c_n}~{\g(n-1)\over \g(n)}
  \sum_{\st x\in V\atop \st z\in W}\nu_{\L,{\bf N}}^{\t}
  \[\(e^{-\nabla_{xz}H_\L}-1\)\id_{E_{zx}}, f\tc N_{V} = n-1 \]~+
  \cr
  &  
  {1\over c_n}~{\g(n-1) \over \g(n)}
  \sum_{\st x\in V\atop \st z\in W}\nu_{\L,{\bf N}}^{\t}
  \[e^{-\nabla_{xz}H_\L}\id_{E_{zx}}\tc N_{V} = n-1 \]~
  \nu_{\L,{\bf N}}^{\t}\(f\tc N_{V} = n-1 \)
  \cr
  }
  \Eq(aa2)
$$
where we have exploit the fact that 
$\nu_{\L,{\bf N}}^{\t}\(\cdot \tc N_{V} = n-1 \)$--almost surely
$\sum_{\st x\in V\atop \st z\in W}\id_{E_{zx}} = c'_{N-n+1}$ 
in order 
to substract $1$ from the $e^{-\nabla_{xz}H_\L}$ term.
Taking $f=1$ in equation \equ(aa) we obtain that the term multiplying
$\nu_{\L,{\bf N}}^{\t}\(f\tc N_{V} = n-1 \)$ in \equ(aa2) is equal
to one 
and the result is obtained.
\smallno
The second equality follows from the first by interchanging $V$ and $W$
\QED

% Stima con i cammini del primo termine di d/di E(f | N_V= n)		%

\vskip 0.5cm
\beginsubsection \number\numsec.8 Bound on $\sum_{i} \(\g(n)\mmin \g(n-1)\) A(n)^2$.

\medno
We now show how to bound from above the term
$\sum_{i} \(\g(n)\mmin \g(n-1)\) A(n)^2$ of corollary \thf[Dif]. 
We first need
the following definition.

\nproclaim Definition [cho].
{\rm 
Given a finite connected subset $\L$ of $\Z^d$
a {\it path choice in\/} $\L$ is a collection 
$\l = \{ \l_{xz} : (x,z) \in \L \times \L \}$ such that
$\l_{xz}$ is a self--avoiding path from
$x$ to $z$ inside $\L$. 
}

\smallno
Given a path choice $\l$ in $\L_j$ and $V,W$ as above, we let 
$$
  \eqalign{
  \cG_V(\l) &=   
  \max_{ e \in \cE_\L} \# \{ (x,z) \in V\times W
  : \l_{xz} \sqsupset e \} \cr
  \cD_V(\l) &= \max_\xz |\l_{xz}|
  }
$$
With this notation we have the following result

\nproclaim Proposition [wdir].
There exists
some $C$ depending only on $\|\Phi\|$ such that
$$
  \sum_{n=n_{\rm min}}^{n_{\rm max}} (\g(n) \mmin \g(n-1)) \,A(n)^2
  \le
  C \, \[ \,{\cG_V(\l)\cD_V(\l)\over 
         \rho_j (1-\rho_j)\,|V|\,|W|} 
     \sum_{e \sqsubset \cE_\L }\nu^\t_{\L,{\bf N}} [ (\nabla_e f)^2 ]
  \, \]
$$

\bigno
{\it Proof of Proposition \thm[wdir].}
Assume for simplicity that $n \le u$. Then we observe that
$$
{1\over c_n}~{\g(n-1) \over \g(n)}
  \sum_{\st x\in V\atop \st z\in W}\nu_{\L,{\bf N}}^{\t}
  \[e^{-\nabla_{xz}H_\L}\id_{E_{zx}}\tc N_{V} = n-1 \] = 1
$$
(see right after \equ(aa2)) so that  
$$
        \nep{-2\|\Phi\|}~ {1\over c'_{N_j-n+1}} \le 
               {1\over c_n}~{\g(n-1) \over \g(n)}
               \le \nep{2\|\Phi\|}~ {1\over c'_{N_j-n+1}}
$$
Moreover 

$$
  \ov{c'_{N_j-n+1}}\,
  \sum_\xz
  \nu^\t_{\L,{\bf N}} \[ \, E_{zx}
    \tc N_{V} = n-1 \, \] = 1
$$
We can thus use  the Schwarz inequality and obtain
$$
  \eqalign{
  &\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}A(n)^2 =  \cr
  &= 
  \Bigl[
  \ov{ c_n }{ \g(n-1) \over \g(n)}
  \sum_\xz
  \nu^\t_{\L,{\bf N}} \[ \, \nep{-\nabla_{xz}H}\nabla_{xz} f 
    \tc N_{V} = n-1, E_{zx} \, \] 
  \nu^\t_{\L,{\bf N}} \[ \, E_{zx}
    \tc N_{V} = n-1 \, \] 
  \Bigr]^2
  \cr
  &\le
  C\ov{ c'_{N_j-u+1} }
  \sum_\xz
  \nu^\t_{\L,{\bf N}} \[ \, (\nabla_{xz} f)^2
    \tc N_{V} = n-1, E_{zx} \, \] 
  \nu^\t_{\L,{\bf N}} \[ \, E_{zx}
    \tc N_{V} = n-1 \, \] 
  \cr 
 &\le C\ov{ c'_{N_j-u+1} }
  \sum_\xz
  \nu^\t_{\L,{\bf N}} \[ \, (\nabla_{xz} f)^2
    \tc N_{V} = n-1\, \]}
  \Eq(a12)
$$
since $c'_n$ is 
increasing in $n$.
Similarily, for $n > u$, we obtain
$$
A(n)^2 \le 
C\ov{ c_{u} }
  \sum_\xz
  \nu^\t_{\L,{\bf N}} \[ \, (\nabla_{xz} f)^2
    \tc N_{V} = n\, \]
  \Eq(a12bis)
$$
Notice that 
$c_u \ge c' \rho_j (1-\rho_j)\,|V|\,|W|$ and similarily for 
$c'_{N_j-u+1} \ge C' \rho_j |W|\((1-\rho_j)|V|\)$ where $c'$ is some suitable 
constant.
Thus, by \equ(a12) and \equ(a12bis), we obtain
$$
  \sum_{n=n_{\rm min}}^{n_{\rm max}} (\g(n) \mmin \g(n-1)) \,A(n)^2 \le 
  C''{1\over 
         \rho_j (1-\rho_j)\,|V|\,|W|}
    \sum_\xz \nu^\t_{\L,{\bf N}} \[ \, (\nabla_{xz} f)^2\]
  \Eq(AY)
$$
Let now  $\l$ be  any path choice. 
Thanks to Lemma \teo[4.3] in \ref[Y] we get that there exists $C_3(\|\Phi\|)$
such that
$$
  \nu^\t_{\L,{\bf N}}\[ \,
  (\nabla_{xz} f)^2 
  \,\] \le
  C_3(\b) \,
  |\l_{xz}| \, 
  \sum_{e \sqsubset \l_{xz} }
  \nu^\t_{\L,{\bf N}}[ (\nabla_e f)^2 ] 
$$
which, together with \equ(AY) and the definition of $\cG_V(\l)$
and $\cD_V(\l)$, finishes the proof.
\QED

\vskip 0.5cm 
\beginsubsection \number\numsec.9 
Bound on $\sum_{n} \(\g(n)\mmin \g(n-1)\) B(n)^2$.

\medno
Here we give an estimate of the $B$--term in corollary
\thm[Dif] under a regularity assumption on the sets 
$V$ and $W$. We refer to $\S 3.7$ for all the necessary notation.

\nproclaim Proposition [Bi].
Assume property $USMT(C,m,l)$ and, without loss of generality, 
$\rho_j\le 1/2$. Fix $\e >0$ and assume that $|\dep_r V \cap \L_j | 
\le \e |\L|$.
\itm1
There exists a constant $K$ independent of $f$ such that 
$$
  \sum_{n=n_{min}}^{n_{max}} (\g(n)\wedge\g(n-1))\, B(n)^2
  \le\, {K\over|\L|}\Var^\t_{\L,{\bf N}}(f)
$$
\itm2
There exists $C_{\e}$ independent of $f$ and $C_0$ independent of $\e$ and $f$
such that  
$$
  \sum_{n=n_{min}}^{n_{max}} (\g(n)\wedge\g(n-1))\, B(n)^2
  \le\, {C_{\e}\over \rho_j^2 |\L|}\, 
     \sum_{e \sqsubset \cE_\L }\nu^\t_{\L,{\bf N}} [ (\nabla_e f)^2 ] +
  {C_0\e\over \rho_j^2 |\L|}\,\Var^\t_{\L,{\bf N}}(f) 
$$

\Pro\  \acapo
$(1)$ Let $n\le u$ where $u = \inte{\rho_j |V|}$. 
Then, as in the proof of proposition \thf[wdir],
$$
  B(n)^2\le 
  \,e^{4\ninf{\Phi}}\,
  \nu^\t_{\L,{\bf N}}\Bigl(f,{1\over c_{N_j-u+1}' }
  \sum_{x\in V\atop z\in W}
  \bigl(\nep{ - \nabla_{xz} H_\L }-1\bigr) \id_{E_{zx}} 
  \tc N_V = n-1 \Bigr)^2
  \Eq(ipic)
$$ 
Notice that $(\nep{ - \nabla_{xz} H_\L }-1) \id_{E_{zx}} = 0$ unless 
there are at least two particles in an $r$--neighborhood of 
$\{x\}\cup\{z\}$. Therefore $\nu^\t_{\L,{\bf N}}
\bigl(|(\nep{ - \nabla_{xz} H_\L }-1) \id_{E_{zx}}| 
\tc N_V = n-1\bigr)\le\,c\, \rho_j^2$ for a suitable constant $c$.
Schwartz inequality, part $2)$ of proposition \thm[EQ] together with the 
bound 
$c_{N_j-u+1} \ge c'(\d) \rho_j (1-\rho_j)\,|\L_j|^2$ 
(see the proof of proposition \thf[wdir])
 show that the r.h.s. of 
\equ(ipic) can be bounded from above by
$
  {K'\over |\L_j|}\,\nu^\t_{\L,{\bf N}}\bigl(f,f \tc N_V= n-1 \bigr)
$.
Thus
$$
	\sum_{n\le u}(\g(n)\wedge\g(n-1))\, B(n)^2 \le 
	{K'\over |\L_j|}\nu_{\L,{\bf N}}(f,f) \le 
	{K'\over \d |\L|}\nu_{\L,{\bf N}}(f,f)
$$ 
because of the assumption 
$|\L_j| \ge \d |\L|$. Similarly one proceeds for $n > u$.



\medno
$(2)$ Let $g_x(\s) := \nep{-\nabla_x H_\L^\t(\s)}(1-\s(x))$,  
$h_z(\s) := \nep{-\nabla_z H_\L^\t(\s)}\s(z)$. Notice that 
$\nep{ - \nabla_{xz} H_\L }\id_{E_{zx}} - g_x \,h_z =0$ unless $d(x,z) \le r$.
Therefore
$$
  \eqalign{
   	B(n)^2  &\le 
     			\,e^{4\ninf{\Phi}}\,
    	 \nu^\t_{\L,{\bf N}} \Bigl(f,{1\over \rho_j |V||W|}  
      \sum_{x\in V\atop z\in W} \bigl(\nep{ - \nabla_{xz} H_\L } - 1 \bigr) 
        \id_{E_{zx}}  \tc N_V = n-1 \Bigr)^2  \cr
	   &\le 2\,e^{4\ninf{\Phi}}\, 
     \nu^\t_{\L,{\bf N}}\Bigl(f,{1\over \rho_j |V||W|}
   \sum_{x\in V\atop z\in W}
  \bigl(\nep{ - \nabla_{xz} H_\L }\id_{E_{zx}} - g_x h_z \bigr) 
      \tc N_V = n-1 \Bigr)^2 
    \cr &\phantom{\le} +
    2\,e^{4\ninf{\Phi}}\,
     \nu^\t_{\L,{\bf N}}\Bigl(f,{1\over \rho_j |V||W|}
   \sum_{x\in V\atop z\in W}  g_x h_z  
      \tc N_V = n-1 \Bigr)^2   \cr
  	}
\Eq(xxx)
$$
where we have dropped the minus one from 
$\bigl(\nep{ - \nabla_{xz} H_\L } - 1 \bigr)$ 
because $\sum_{\sst x\in V, \atop z\in W}\id_{E_{zx}}$ is constant under 
$ \nu^\t_{\L,{\bf N}}(\cdot \tc N_V= n-1)$.\acapo
The first term in the r.h.s. of \equ(xxx),  
using Schwartz inequality together with proposition \thf[EQ1],
condition $USMT$ and the assumption 
$|\partial^-_r V \cap \L_j | \le \e |\L|$, 
is bounded from above by
$$
      {C''\e \over |\L|} \nu^\t_{\L,{\bf N}} \Bigl(f,f \tc N_V= n-1\Bigr) 
$$ 
for a suitable constant $C''$.\acapo
The second term in the r.h.s. of \equ(xxx),  
using lemma \thf[Yaug], is bounded from above by
$$
{C_{\e}\over \rho_j^2 |\L|}\, 
     \sum_{e \sqsubset \cE_\L }
      \nu^\t_{\L,{\bf N}}\bigl( [\nabla_e f]^2 \tc N_V= n-1 \bigr) +
  {\e\over \rho_j^2 |\L|}\, \nu^\t_{\L,{\bf N}} \Bigl(f,f \tc N_V= n-1\Bigr)
$$
Thus
$$
\sum_{n\le u}(\g(n)\wedge\g(n-1))\, B(n)^2 \le
{C_{\e}\over \rho_j^2 |\L|}\, 
     \sum_{e \sqsubset \cE_\L }
      \nu^\t_{\L,{\bf N}}\bigl( [\nabla_e f ]^2 \bigr) +
  {C_0\e\over \rho_j^2 |\L|}\, \nu^\t_{\L,{\bf N}}(f,f)
$$
for a suitable constants $C_0$ independent of $\e$. 
Similarly one proceeds for the case $n > u$. \QED

\vskip 1truecm

\newsection More on the distribution of the number of 
particles inside one block.

\bigno
Fix $\d\in (0,1)$ and an integer $k \ge 2$ with $\d k <1$. 
Let $L_1,\dots, L_k$ be large multiples of the basic length scale 
$l$, let $L=\sum_i L_i$  and assume that $L_j \ge \d L$ for any $j$. 
We then choose  
one coordinate direction, e.g. the $d$ direction, and we take 
$\L=Q_L$, $\L_1$ equal to the first slice of $\L$  orthogonal to 
the $d$--axis of width $L_1$, \ie  
$\L_1 = \{x\in \L~:~ 0 \le x_d < L_1\}$, $\L_2$ equal to the 
slice of $\L$ on top of $\L_1$ of width $L_2$ and so on.
Let also  
${\bf N}=\{N_i\}_{i=1}^k$ be a set of possible values of 
${\bf N}_\L := \{N_{\L_i}\}_{i=1}^k$ 
and let us assume, for a given boundary condition $\t$, 
that $\ul$ is constant on each block $\L_i$ and such that 
$\mu_\L^{\t,\ul}(N_{\L_i}) = N_i$, $i=1,\dots,k$. We denote by 
$\nu_{\L,{\bf N}}^\t$ the multi canonical Gibbs measure
$\mu_\L^{\t,\ul}\bigl(\cdot\,|\, N_{\L_i} = N_i, \,i=1,\dots,k\,\bigr)$.
\smallno 
Pick now $j \in \{1,\dots,k\}$ and consider the new finer partition of $\L$ 
obtained from the previous one by splitting $\L_j$ into 
two slices orthogonal to the $d$--direction, $V$ and $W$, in such a way that  
$\d \le {|V|/|W|} \le 1-\d$ and $V$ and $W$ are still multiples 
of the basic length scale $l$. Assume without loss of generality that 
${N_j\over |\L_j|} \le \ov 2$ and denote by $N^*$ the average number of 
particles in $V$ according to $\mu_\L^{\t,\ul}$. Let also  
$n_{\rm min}=\max\{0,N_j-|W|\}$ and $n_{\rm max}=\min\{|V|,N_j\}$ be 
the smallest and the largest value of $N_V(\s)$ under the constraint that 
$N_{\L_j}(\s) = N_j$. It is easy to check that 
$c(\d)^{-1}\le {n_{\rm max}-N^*\over N^*} \le c(\d)$ 
for a suitable constant $c(\d) \ge 1$ and 
similarily for ${N^*-n_{\rm min} \over N^*}$.  
\smallno
In what follows we will consider the distribution of the number 
of particles in $V$ under the measure $\nu_{\L,{\bf N}}^\t$. More precisely
we define $\g=\{\g(n)\}$ to be the probability measure on 
$I= \{n \in [n_{\rm min}, n_{\rm max}]:\; n \hbox{ is an integer }\}$, 
given by 
$$
  \g(n) := \nu_{\L,{\bf N}}^\t \(N_{V}=n\) 
$$
In order to obtain sharp bounds on $\g(n)$, $n\in I$, 
we modify the chemical potential
$\ul$ in an $n$--dependent way 
in such a way that the value $n$ becomes equal to the 
average of $N_V$. More precisely, given $n\in [n_{\rm min}, n_{\rm max}]$,  
let $\ul(n) = \{ \l_1,\dots, \l_{j-1}, \l_V, \l_W, \l_{j+1},\dots ,\l_k\}$  
be a new chemical potential constant on the atoms of the new partition and 
such that 
$$
\eqalign{
  \mu_\L^{\t,\ul(n)}(N_{\L_i}) &= N_i, \quad i=1,\cdots,k \cr
  \mu_\L^{\t,\ul(n)}(N_{V})&= n \cr
  \mu_\L^{\t,\ul(n)}(N_{W})&= N_j -n \cr
	}
  \Eq(eqln)
$$
It is then easy to check that $\g$ can be written in the Gibbsian form, 
$\g(n) = e^{-H(n)}\varphi(n)$, where 
$$
\eqalign{
  H(n) &:=\sum_{i\neq j}\(\l_i(n)-\l_i\)N_i + \l_V\, n + \l_W(N_j-n) -
          \l_j N_j -\log\( {Z_\L^{\t,\ul(n)}\over Z_\L^{\t,\ul}}\) \cr
  \varphi(n) &:= 
 {\mu_\L^{\t,\ul(n)}\({\bf N}_\L= {\bf N}\,;\, 
                       N_{V}=n\)
  \over\mu_\L^{\t,\ul}\({\bf N}_\L= {\bf N}\)} \cr
	}
  \Eq(Wphi)
$$
Finally, given $\e\in (0,1)$, we consider for technical reasons 
the ``$\epsilon$--regularization'' 
of 
$\g$ defined by  
$$
  \tilde{\g}(n):=\cases{
  {1\over Z}e^{-H(n)} & if $n\in I_\e \equiv 
                [N^*+\e(n_{\rm min}-N^*),\,N^*+\e (n_{\rm max}-N^*)] \cap I$
  \cr
  \g(n) & otherwise\cr}
  \Eq(nut)
$$ 
where $  Z:={\sum_{n\in I_\e}e^{-H(n)}\over\sum_{n\in I_\e}\g(n)}$.
\acapo
The following lemma shows that the relative density between $\g$ and its
regularization is bounded uniformly in the size of $I$.

\nproclaim Lemma [nux].
Assume property $USMT(C,m,l)$.   
Then there exists $C'=C'(C,m,l,\d,\e,\ninf{\Phi}) \ge 1$ such that 
$$
  (C')^{-1}\le \inf_{n\in I}{\g(n)\over\tilde{\g}(n)}
            \le \sup_{n\in I}{\g(n)\over\tilde{\g}(n)}\le C'
$$

\Pro\ By the definition \equ(nut) of $\tilde \g$ we immediately have
$$
  {\g(n)\over\tilde{\g}(n)}   
  \le \max\[\sup_{n,n'\in I_\e}{\varphi(n')\over\varphi(n)},\, 1 \]
$$
Thus we must prove
$$
  \sup_{n,n'\in I_\e}{\varphi(n')\over\varphi(n)} \le C'
  \Eq(phiny)
$$
Thanks to proposition \thf[N] and 
by the definition of $\varphi(n)$ 
we have
$$
  {\varphi(n')\over\varphi(n)}\le C_1\, 
 {\s_V(n)\over \s_V(n')}{\s_W(n)\over \s_W(n')}
  \prod_{i\neq j}{\s_i(n)\over \s_i(n')} 
$$
for a suitable constant $C_1$, where 
$\s^2_i(n) := \mu_\L^{\t,\ul(n)}\(N_{\L_i},N_{\L_i}\)$ and similarily for 
$\s^2_V(n),\,\s^2_W(n)$.
\acapo
It is one of the result of proposition 3.1 of \ref[CM1] 
that, under our mixing assumption, the variance of the number of particles in 
each atom can be bounded from above and below by a suitable constant times 
its average. Therefore
$$
   \sup_{n,n'\in I_\e} {\varphi(n')\over\varphi(n)}\le 
  C_2\,\[ \sup_{n,n'\in I_\e} 
          {n(N_j-n)\over n'(N_j-n')}\]^{\ov2} \le C' 
$$
for a suitable constant $C'$ because of the definition of $I$ and 
the assumption on the ratio $|V|\over |W|$. Similarily 
one proves the lower bound. \QED

\bigno
The next lemma proves that the tails of $\g$ are at least exponential.

\nproclaim Lemma [Lxx1].
There exists a positive constant $\e_0 =\e_0 (\d,\ninf{\Phi})$ such that
$\forall \e \in (\e_0 ,1)$
$$
  \eqalign{
  {\g(n+1)\over\g(n)}\le \ov2 & ~~~{\rm if}~~~ 
  n \in [N^*+\e\, (n_{\rm max}-N^*), n_{\rm max}]  \cr
  {\g(n-1)\over\g(n)}\le \ov2 & ~~~{\rm if}~~~ 
  n \in [n_{\rm min}, N^*+\e\, (n_{\rm min}-N^*)]\cr
  }
$$

\Pro\ Using the definition of the canonical measure 
and proposition \thf[mui] we have 
$$
  {\g(n+1)\over \g(n)} 
  \le \nep{2\ninf{\Phi}} {(|V|-n)(N_j-n)\over
  (n+1)(|W|-N_j+n+1)} 
  \Eq(muxx1)
$$
Thus the r.h.s. of \equ(muxx1) is smaller 
than $\nep{2\ninf{\Phi}}\,{1\over\d}\,{1-\e\over \e} \le \ov2$
for any $n \in [N^*+\e\, (n_{\rm max}-N^*), n_{\rm max}] $, 
provided that $\e$ is 
sufficiently close to $1$, because of our assumption on 
$V,\,W$ and the definition of $n_{\rm max}$.  
A similar reasoning applies to 
${\g(n-1)\over\g(n)}$ for $n\in [n_{\rm min}, N^*+\e\, (n_{\rm min}-N^*)]$ \QED

\bigno
Our last result shows that the 
$\e$-regularized version of $\g$ is bell--shaped 
around $n=N^*$ with width proportional to $\sqrt{N^*}$.

\nproclaim Lemma [Wx].
Assume property $USMT(C,m,l)$. Then 
there exists a positive constant $C''=C''(C,m,l,\e,\ninf{\Phi})$ such that
$$
  \eqalign{
  H(n+1)-H(n) &\ge  C''\,{n-N^*\over N^*}  ~~~~~{\rm if}~~~
  n\in[N^*,N^*+\e\, (n_{\rm max}-N^*)] \cr
  H(n-1)-H(n)  &\ge  C''\,{|n-N^*|\over N^*} ~~ ~~~{\rm if}~~~
  n\in[N^*+\e\, (n_{\rm min}-N^*),N^*)\cr
  }
$$

\Pro\  By the definition \equ(Wphi) we have 
$$
  {dH\over dn}=\l_V(n)-\l_{W}(n)
$$
so that we can write
$$
  \eqalign{
  H(n+1)-H(n)&=\int_n^{n+1}\, dn' \int_0^{n'}\, dn'' \,
                   {d\over dn''}\(\l_V(n'')-\l_{W}(n'')\)
  \cr
  H(n-1)-H(n)&=\int_{n-1}^{n}\, dn'\int_0^{N'}\, dn''\, 
      {d\over dn''}\(\l_V(n')-\l_{W}(n'')\)
  \cr
  }
\Eq(Wx0)
$$
The integrand in \equ(Wx0) can be computed thanks to the following identities 
valid for any $i\neq j$ that can be easily 
obtained by taking the derivative w.r.t. $n$ of  both sides 
of \equ(eqln) 

$$
  \cases{
  \sum_{k\neq j}\mu_\L^{\t,\ul(n)}\(N_{\L_i},N_{\L_k}\){d\over dn}\l_k(n)
   + \mu_\L^{\t,\ul(n)}(N_{V},N_{\L_i}){d\over dn}\l_V(n) + 
   \mu_\L^{\t,\ul(n)}(N_{W},N_{\L_i}){d\over dn}\l_W(n)   =0 
  \cr 
  \cr
  \sum_{k\neq j}\mu_\L^{\t,\ul(n)}(N_{\L_k},N_{V}){d\over dn}\l_k(n) 
  + \mu_\L^{\t,\ul(n)}(N_{V},N_{V}){d\over dn}\l_V(n)  
  + \mu_\L^{\t,\ul(n)}(N_{V},N_{W}){d\over dn}\l_W(n) = 1  \cr
  \cr
  \sum_{k\neq j}\mu_\L^{\t,\ul(n)}(N_{\L_k},N_{W}){d\over dn}\l_j(n) 
  + \mu_\L^{\t,\ul(n)}(N_{V},N_{W}){d\over dn}\l_V(n)  
  + \mu_\L^{\t,\ul(n)}(N_{W},N_{W}){d\over dn}\l_W(n) = -1\cr
  }
\Eq(Wx1)
$$
In order to solve the above linear system we can use 
the following general bounds for the covariances of 
the number of particles in the atoms of a partition of $\L$ 
(see proposition 3.1 of \ref[CM1])
$$
  \eqalign{
  \mu_\L^{\t,\ul}\(N_{\L_i},N_{\L_i}\) &\ge  A^{-1}\rho_i |\L_i|       \cr
  |\mu_\L^{\t,\ul}\(N_{\L_i},N_{\L_j}\)| &\le 
        A \rho_i \rho_j \( |\partial_r^-\L_i||\partial_r^-\L_j| \)^{1/2} 
                 \cr
  |\mu_\L^{\t,\ul}\(N_{\L_i},N_{\L_j}\)| &\le 
        A |\L_i|\,|\L_j|\nep{-m d(\L_i,\L_j)} \cr
  }
\Eq(Wx2)
$$
where $\rho_i = {\mu_\L^{\t,\ul}\(N_{\L_i}\)\over |\L_i|}$. \acapo
If we apply the above bounds in our case we get that, 
for $L$ large enough , we can solve \equ(Wx1) by Neumann series and obtain
$$
\eqalign{
  c_1^{-1}{1\over \s_V^2(n)} &\le {d\over dn}\l_V(n) \le 
  c_1 {1\over \s^2_V(n)} \cr
  c_1^{-1}{1\over \s^2_W(n)} &\le -{d\over dn}\l_W(n) \le 
  c_1 {1\over \s^2_W(n)} \cr
	}
$$
where $c_1=c_1(C,l,m,r,\ninf{\Phi})>0$ is a suitable constant. \acapo
In 
particular ${d\over dn}\(\l_V(n)-\l_{W}(n)\) \ge c_1^{-1}{1\over \s^2_V(n)}$.
Take now $n \in [N^*,N^*+\e\, (n_{\rm max}-N^*)] $. 
If we use once more the fact 
that $\s_V^2(n)$ is comparable to the average number of particles in $V$, 
$n$ (see the proof of lemma \thf[nux]), we get from \equ(Wx0), the fact 
that $\d \le {|V|\over|W|} \le 1-\d$ and 
some simple considerations on the size of $n_{\rm max}$ that
$$
  H(n+1)-H(n)\ge c_2 \int_n^{n+1}\, dn'\int_0^{n'}\, dn''\, 
                            {1\over N^* + \e\,(n_{\rm max}- N^*)}
  \ge c_3 {n\over N^*}
$$
A similar reasoning applies to $H(n-1)-H(n)$ when
$n\in[N^*+\e\, (n_{\rm min}-N^*),N^*)$. \QED 

\bigno
We are finally in a position to state the main result of this section.

\nproclaim Theorem [Wx1].
Assume property $USMT(C,m,l)$. Then 
there exists $c_0 = c_0\( C,m, \d, \|\Phi\|\)$ such 
that for all $f:\O_\L \mapsto \real$ that depend only on $N_V(\s)$ 
the following Poincar\`e inequality holds
$$
\nu_{\L,{\bf N}}^\t\(f,f\) \le 
   c_0 N^* \sum_{n\in I} \(\g(n) \mmin \g(n-1)\)\[f(n)-f(n-1)\]^2
$$

\Pro\ Using lemma \thf[nux] it is sufficient to prove the Poincare' 
inequality for the regularized measure $\tilde \g(n)$. Pick $\e$ sufficiently 
close to one in such a way that lemma \thf[Lxx1] holds. Then, thanks 
to lemma \thf[Wx] and the fact that ${n_{\rm max}-N^*\over N^*} 
\le c(\d)$, there 
exists positive constants $c_1,\,c_2$ depending on 
$C,m,\|\Phi\|,\d$ such that 

$$
{ \tilde \g(n+1)\over \tilde \g(n)} \le \cases{
             \exp\(-c_1 {n-N^*\over N^*}\) & if $n \in I$, $n \ge N^*$ and 
                              $n \neq \inte{N^*+\e\,(n_{\rm max}-N^*)} $ \cr
             \cr
              c_2 & if ~~$n = \inte{N^*+\e\,(n_{\rm max}-N^*)} $ \cr
	}
\Eq(Wx3)
$$
and similarily for $ { \tilde \g(n-1)\over \tilde \g(n)}$, $n \le 0$.
We can therefore apply proposition \thf[ON1] and get the result. \QED

\bigno
{\it Remark.} The reader may worry about the fact that \equ(Wx3) is 
not, strictly speaking, identical to the condition in 
proposition \thf[ON1], because of the ``spurious'' point 
$n = \inte{N^*+\e\,(n_{\rm max}-N^*)} $ where the original distribution 
$\g$ starts to agree with its regularization $\tilde \g$. 
However, as one can readily check, 
the presence of this single point only affects the final result by a 
multiplicative constant depending on $C,m,\|\Phi\|,\d$.

\bigno
{\it Remark.} It is interesting to notice that, in the same hypotheses 
of theorem \thf[Wx1] and using the discrete version of the arguments 
described in section 7 of \ref[Le],  
one can prove \ref[M2] a logarithmic Sobolev 
inequality of the form
$$
\nu_{\L,{\bf N}}^\t\(f^2\log f\) \le 
   c'_0 N^* \sum_{n\in I} \(\g(n) \mmin \g(n-1)\)\[f(n)-f(n-1)\]^2
$$
for any non--negative function $f$ that depends only on $N_V(\s)$ and 
such that $\nu_{\L,{\bf N}}^\t\(f^2\)=1$. This observation is useful 
if one wants to bound from above the logarithmic Sobolev constant 
of Kawasaki dynamics \ref[CM3] following the same approach described in 
this paper.







\vskip 1truecm  
	

\newsection Recursive estimate of the spectral gap.

\bigno
In this section we prove the main result via a recursive analysis 
on the behaviour of the spectral gap when the linear size of 
the volume under consideration is doubled.\acapo
For simplicity
we carry out our analysis in two dimensions but the extension to higher 
dimension is straightforward.
\smallno
Let $R(l_1,l_2)$ denote the rectangle
$$
	R(l_1,l_2) = \bigl\{(x_1,x_2)\in \Z^2;\ 0\,\le \, x_1\,\le \, l_1-1, \ 
	               0\,\le \, x_2\,\le \, l_2-1\ \bigr\} \ ;
			\qquad l_1,l_2\in \Zp
$$ 
and let $\cR_L$ be the family of ``fat'' rectangles  
with ``size'' smaller than $L$, namely 
those rectangles $R(l_1,l_2,x) \equiv R(l_1,l_2) +x\,$, $x\in \Z^2$, with 
$l_1\mmin l_2 \ge 0.1\,(l_1\mmax l_2)$ and
$l_1\mmax l_2 \,\le \, L$. \acapo
Let also 
$$
        g(L) = \min_{R\in \cR_L}\min_{N,\t}
		\,\gap( L^{\t}_{R,N} ) 
$$ 
where $\gap( L^{\t}_{R,N} )$ has  
been defined in \equ(gap).
\acapo
With the above notation  we will prove the following recursive bound.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Stima del gap(2L) con gap(L)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\nproclaim Theorem [recur].
Assume $USMT(C,m,l)$. 
Then there exists a positive 
constant $k=k(d,r,\|\Phi\|)$ such that 
$$
	g(2L)^{-1} \le  {3\over 2}\,g(L)^{-1} + k L^2
$$ 
for all $L$ large enough. 
In particular ${\ds \inf_{L}}\, g(L)\,L^2 > 0$.

\Pro\
Let us consider a rectangle 
$\L \equiv R(l_1,l_2)\in \cR_{2L}$ and assume, without loss of 
generality, that the longest side is $l_2$. If $l_2 \le L$ then, 
using the definition of $g(L)$, 
$\min_{N,\t}\,\gap( L^{\t}_{R,N} ) \ge g(L)$. Thus we assume
$L < l_2 \le 2L$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%    Definisco i due blocchi               \%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\noindent
We fix a small number $\d\in (0,10^{-2})$ and we set $d = \inte{\d L}$.  
Given an integer $j\in [1,\inte{{1\over 10\d}}]$ we partition $\L$ into three 
atoms $\{\L_i\}_{i=1}^3$ as follows (we omit the index $j$ for simplicity)

$$
  \eqalign{
	\L_1&= \{x\in \L;\ 0\,\le \, x_2 \,\le \, {l_2/2} + jd\} \cr 
	\L_2&= \{x\in \L;\ {l_2/2} + (j-1)d < x_2 \,\le \, l_2-1\} \cr 
	\L_3&= \L_1\cap \L_2 \cr
          }
\Eq(blocks)
$$
Fix now a boundary condition $\t$ outside $\L$ and a number of particles 
$N\in [0,\dots,|\L|]$. We will then use twice 
the formula relating the variance of a function $f$ w.r.t. the measure 
$\nu_{\L,N}^\t$ to the variance of $f$ w.r.t. the measure 
$\nu_{\L,N}^\t$ conditioned to a sub $\s$--algebra $\cF_0$
$$
    \nu_{\L,N}^\t \(f,f\) = \nu_{\L,N}^\t\( \nu_{\L,N}^\t \(f,f \tc \cF_0 \)\) 
+ \nu_{\L,N}^\t \(\nu_{\L,N}^\t \(f\tc \cF_0\),\nu_{\L,N}^\t \(f\tc \cF_0\)\)  
\Eq(cvar)
$$
to write
$$
\eqalign{
\nu_{\L,N}^\t \(f,f\) &= 
\nu_{\L,N}^\t\( \nu_{\L,N}^\t \(f,f \tc \cF_1 \) \) +
\nu_{\L,N}^\t \(f_1,f_1\) \cr
    &= \nu_{\L,N}^\t\( 
       \nu_{\L,N}^\t \(f,f \tc \cF_{1,3}\) 
	             \) + 
     \nu_{\L,N}^\t \( \nu_{\L,N}^\t \(f_{1,3},f_{1,3} \tc \cF_1\) 
                    \) + 
     \nu_{\L,N}^\t \(f_1,f_1\)   \cr
	}
\Eq(start)
$$
where $\cF_1$ and $\cF_{1,3}$ are the $\s$--algebras generated by $N_{\L_1}$ 
and $\{N_{\L_1}, \,N_{\L_3}\}$ respectively,  
and \hbox{$f_1 := \nu_{\L,N}^\t \(f\tc \cF_1 \)$}, 
$f_{1,3}:= \nu_{\L,N}^\t \(f\tc \cF_{1,3} \)$. 
Formula \equ(start) will 
represent our basic starting point. We will now 
examine separately each term in the r.h.s. of \equ(start).

\vskip 0.5cm \beginsubsection \number\numsec.1 Analysis of the first term in 
the r.h.s. of \equ(start).

\medno
For any small $\e$ and large enough $L$, 
the first term in the r.h.s. of \equ(start) 
can be bounded from above using proposition \thf[Block]
by
$$
 \nu_{\L,N}^\t\( \nu_{\L,N}^\t \(f,f \tc \cF_{1,3} \) \) 
                           \le 
   (1+\e)\nu_{\L,N}^\t\( \nu_{\L_1,N_1}^\h \(f,f\) +
                         \nu_{\L_2,N_2}^\h \(f,f\) \)
\Eq(1term)
$$
where the average is over the random variables $\h$, $N_{\L_1}$ and 
$N_{\L_2}$. Above we have used the bound
$\nu_{\L,N}^\t\( \nu_{\L_1,N_1}^\h \(f,f \tc N_{\L_3}\)\) \le 
\nu_{\L,N}^\t\( \nu_{\L_1,N_1}^\h \(f,f\)\)$ which follows 
at once from \equ(cvar).
\acapo
Let us now examine the spectral 
gap of the bottom rectangle $\L_1$, the reasoning being 
similar for the top one.\acapo
There are two cases to analyze:
\smallno
\item{a)} $l_1 \,\le \, {3\over 2}L$. In this case 
one easily verifies that 
$\L_1 \in \cR_{{3\over 2}L}$.

\item{b)} $l_1 > {3\over 2}L$. In this case 
$\L_1\in \cR_{2L}$ 
but now the {\it longest} side is $l_1$ and 
the {\it shortest} one is smaller  than 
${l_2\over 2}+jd + 1$ which in turn is smaller than 
$1.2L$ since $l_2\,\le \, 2L$ and $jd \,\le \, {L\over 10}$
\smallno
Therefore
$
{\ds \min_{\h,N_1}}\, \gap(L_{\L_1,N_1}^\h) \ge 
\min\{g({3\over 2}L),\, \hat g(2L)\} 
$
where
$$
        \hat g(2L) = \min_{R\,\in \,\cR_{2L} \atop{ \sst 
       l_1 \,\le \, 1.2L, \atop \sst l_2 \ge {3\over2}L }}\min_{\t, N}
        \gap( L^{\t}_{R(l_1,l_2),N} )
$$ 
In conclusion we obtain that the r.h.s. of \equ(1term) is smaller 
than
$$
(1+\e)\max \{\,g({3\over 2}L)^{-1},\, \hat g(2L)^{-1}\,\}
     \[ \Dir_{\L,N}^\t(f,f) + {1\over 2} \sum_{[x,y]\in \cE_{\L_3}}
   \nu^{\t}_{\L,N}\[\,c_{xy} \,(\nabla_{xy} f) ^2 \,\] \]
\Eq(1term.1)
$$
uniformly in $j\in [1,{1\over 10\d}]$. 
Notice that the ``spurious'' term ${1\over 2} \sum_{[x,y]\in \cE_{\L_3}}
   \nu^{\t}_{\L,N}\[\,c_{xy} \,(\nabla_{xy} f) ^2 \,\]$ comes from the 
fact that $\L_1$ and $\L_2$ overlap. 

\vskip 0.5cm \beginsubsection \number\numsec.2  Analysis of the second and third term in 
the r.h.s. of \equ(start).

\medno
Here we bound from above the second and third term in \equ(start).
The necessary steps are identical in both cases and therefore, 
for shortness, we treat only the third term which is 
(notationally speaking) also the simplest. 
Later on we will state without further comments the 
analogous result for the second one. 

\medno
Let $\rho := {N\over |\L|}$ and assume, 
without loss of generality, that $\rho \le \ov2$.
Let also $N_1^* = \mu_\L^{\t,\ul}\(N_{\L_1}\)$, $\mu_\L^{\t,\ul}$ being the 
grand canonical ensemble with average particle number equal to $N$,  and let 
$\g(n) := \nu_{\L,N}^\t\(N_{\L_1} = n\)$. Then, using 
theorem \thf[Wx1], we can write
$$
  \eqalign{
	\nu_{\L,N}^\t \(f_1,f_1\) &\le 
	c_0 N_1^* \sum_{n} \(\g(n) \mmin \g(n-1)\)
             \[\nu_{\L,N}^\t \(f\tc N_{\L_1}=n\) - 
                        \nu_{\L,N}^\t \(f\tc N_{\L_1}=n-1\)\]^2  \cr
        &\le c_0 N_1^* \sum_{n} \(\g(n) \mmin \g(n-1)\)[A(n)^2 + B(n)^2] \cr
	}
\Eq(3term)
$$
where $A(n)$ and $B(n)$ have been defined in \equ(aa3). \acapo
In order to estimate the first term in the 
r.h.s of \equ(3term) we first need to fix a 
{\it path choice} (see proposition \thf[wdir] and definition \thf[cho] 
for the necessary notation). \acapo
Our choice 
is the following. Given $x\in \L_1$
and $z\in \L_2\setminus \L_1$, 
start increasing (or decreasing) the first coordinate
of $x$ until it is equal to the first coordinate of $z$. Then adjust
the second coordinate until you get to $z$. With this
particular path choice it is easy to see that $\cG_V(\l) \le (2L)^{3}$.
Assume in fact that the path $\l_{xz}$ contains the edge $e=[u,v]$ 
where  $u$ and $v$ differ in the $j^{th}$ coordinate. This means that
$x_i = u_i$ for all $i>j$ and $z_i = u_i$ for all $i<j$. So the
number of possible pairs $(x,z)$ is not greater than 
$(2L)^j \, (2L)^{2-j+1} = (2L)^{3}$. Moreover with this geometry 
$\cD_V(\l) \le 4L$. \acapo
Thanks to proposition 
\thf[wdir] first term in the r.h.s. of \equ(3term) is smaller than
$$
C N_1^* {L^4\over \rho(1-\rho) |\L_1| |\L_2|}\, \Dir_{\L,N}^\t(f,f) 
\Eq(3term.1)
$$
Notice that $N_1^* \le  C' \rho |\L_1|$ 
(see proposition 3.1 of \ref[CM1]) so that 
$$
	CN_1^* {L^4\over \rho(1-\rho) |\L_1| |\L_2|}\, \Dir_{\L,N}^\t(f,f) 
	 \le 
        C' L^2 \Dir_{\L,N}^\t(f,f)
\Eq(3term.2)
$$
We now turn to the estimate of the second term in the r.h.s. of \equ(3term).  
Using proposition \thf[Bi] we get that for any $\e>0$ 
there exists a constant $C(\e)$ 
such that the second term 
in the r.h.s. of \equ(3term) is smaller than
$$
  C(\e)\,\Dir_{\L,N}^\t(f,f) + \e\, \nu_{\L,N}^\t\(f,f\)
\Eq(3term.3)
$$
In conclusion, for any $\e>0$ there exists a constant $C(\e)$ such that 
the third term in the r.h.s. of \equ(start) is smaller than
$$
	\nu_{\L,N}^\t \(f_1,f_1\) \le \(C'' L^2 + C(\e)\) \,
            \Dir_{\L,N}^\t(f,f) + 
         \e\, \nu_{\L,N}^\t\(f,f\) 
\Eq(3term.4)
$$
for a suitable constant $C''$.
A similar bound holds for the second term in the r.h.s. of \equ(start).

\vskip 0.5cm \beginsubsection \number\numsec.3  The recursion completed.

\medno
We are finally in  a position to complete the proof of theorem \thf[recur].
If we put together \equ(3term.4) and \equ(1term.1) 
we get that, for any $\e \in (0,\ov2)$
$$
\eqalign{ 
    \hbox{ r.h.s. of \equ(start) } &\le 
        (1+\e)\max \{\,g({3\over 2}L)^{-1},\, \hat g(2L)^{-1}\,\}
      \Bigl[ \Dir_{\L,N}^\t(f,f) 
        + 
      {1\over 2} \sum_{[x,y]\in \cE_{\L_3}}
      \nu^{\t}_{\L,N}\[\,c_{xy} \,(\nabla_{xy} f) ^2 \,\] \Bigr] \cr
       &\phantom{\le}  +
       2\(C'' L^2 + C(\e)\) \,\Dir_{\L,N}^\t(f,f) + 
         2\e\, \nu_{\L,N}^\t\(f,f\) \cr
	} 
\Eq(fina1)
$$
that is 
$$
\eqalign{
\nu_{\L,N}^\t\(f,f\) &\le  
         \({1+\e\over 1-2\e}\)\,
        \max \Bigl\{\,g({3\over 2}L)^{-1},\, \hat g(2L)^{-1}\,\Bigr\}
         \Bigr[ \Dir_{\L,N}^\t(f,f) 
         + 
         {1\over 2} \sum_{[x,y]\in \cE_{\L_3}}
      \nu^{\t}_{\L,N}\[\,c_{xy} \,(\nabla_{xy} f) ^2 \,\] \Bigr] 
         \cr 
         &\phantom{aaa} + 
         kL^2\,\Dir_{\L,N}^\t(f,f) \cr
	}
\Eq(fina2)
$$ 
for a suitable constant $k$.\acapo
Finally we average w.r.t. to the index $j$ (see \equ(blocks)) and use 
the observation that, as $j$ varies in $[1,{1\over 10\d}]$, 
the strips $\L_3 := \L_3^{(j)}$ 
are disjoint. In particular
$$
          \ov2 \sum_{j\in [1,{1\over 10\d}] } 
         \sum_{[x,y]\in \cE_{\L^{(j)}_3}}
      \nu^{\t}_{\L,N}\[\,c_{xy} \,(\nabla_{xy} f) ^2 \,\] \le \Dir_{\L,N}^\t(f,f)
$$
so that 
$$
\eqalign{
          &\nu_{\L,N}^\t\(f,f\) \le \cr 
         \({1+\e\over 1-2\e}\)\(1+ \inte{10\d}\)
        &\max \Bigl\{\,g({3\over 2}L)^{-1},\, \hat g(2L)^{-1}\,\Bigr\}
                 \, \Dir_{\L,N}^\t(f,f) + 
         kL^2\,\Dir_{\L,N}^\t(f,f) \cr
	}
\Eq(fina3)
$$ 
In other words 
$$
               \gap(L_{\L,N}^\t)^{-1} \le 
 \({1+\e\over 1-2\e}\)\(1+ \inte{10\d}\)
        \max \Bigl\{\,g({3\over 2}L)^{-1},\, \hat g(2L)^{-1}\,\Bigr\} + kL^2
\Eq(fina4)
$$
Notice that if the original rectangle $\L$ was such that $l_1 \le 1.2 L$ 
while $l_2 \ge {3\over 2}L$, \ie $\L$ was chosen in the sub--class of 
$\cR_{2L}$ entering in the definition of $\hat g(2L)$, then we would have 
obtained the inequality \equ(fina4) with the factor 
$  \max \{\,g({3\over 2}L)^{-1},\, \hat g(2L)^{-1}\,\}$ replaced 
by $g({3\over 2}L)^{-1}$ simply because case {\it b)} right after \equ(1term)
would have been impossible. Thus
$$
\hat g(2L)^{-1} \le \({1+\e\over 1-2\e}\)\(1+ \inte{10\d}\)
        g({3\over 2}L)^{-1}+ kL^2
\Eq(fina5)
$$
If we combine \equ(fina4) with \equ(fina5) we finally get
$$
               \gap(L_{\L,N}^\t)^{-1} \le 
 \({1+\e\over 1-2\e}\)^2\(1+ \inte{10\d}\)^2
        \,g({3\over 2}L)^{-1} + k'L^2
\Eq(fina6)
$$
for another constant $k'$. Thus
$$
g(2L)^{-1} \le \({1+\e\over 1-2\e}\)^2\(1+ \inte{10\d}\)^2
        \,g({3\over 2}L)^{-1} + k'L^2
$$
and two more iterations prove the recursive inequality of the theorem provided 
that the two parameters $\e,\d$ were chosen small enough.\acapo
Finally 
the fact that ${\ds \min_L} \,\(g(L)\,L^2\) > 0$ is a trivial consequence 
of the recursive bound. \QED

\vskip 1truecm


\newsection Proof of Theorem \thf[local].

\bigno
Fix a local function $f$ with $0\in \D_f$ and 
denote by $E_\l$ the spectral projection 
associated to the interval $[0,\l]$ 
for the selfadjoint operator $-L^\t_{\L,N}$ 
on $L^2\(\O_\L, d\nu_{\L,N}^\t\)$. Assume that 
$\nu_{\L,N}^\t\(f\)=0$. Then we will prove that for any $\e \in (0,1)$ 
there exists a constant $C_{f,\e}$ independent of $\L$ and $N$ such that
$$
\|E_\l f\|^2_2 \le C_{f,\e} \l^{\a -\e}
\Eq(loc.1)
$$ 
where $\|\cdot\|_2$ denote the $L^2\(\O_\L, d\nu_{\L,N}^\t\)$--norm  
and $\a = \a(d)$ is as in the theorem. It is clear that once such an estimate
is available then the result follows by the simple formula
$$
\eqalign{
\|\nep{tL_{\L,N}^\t}f \|_2^2  &\le
\sum_{j=0}^\infty \nep{-j} \| E_{{j+1\over t}}f - E_{{j\over t}}f\|_2^2 
\cr
&\le C_{f,\e}\, {1\over t^{\a-\e}}\,
\sum_{j=0}^\infty \nep{-j} (j+1)^{\a -\e} 
\cr
&\le C'_{f,\e}\,{1\over t^{\a-\e}} \cr
	}
$$
Let us prove \equ(loc.1). For any integer $L' \le \ov2 L$ 
multiple of $l$ let 
$\cF_{L'} := \cF_{\L \setminus B_{L'}}$ and let 
\hbox{ $f_{L'} := \nu_{\L,N}^\t(f\tc \cF_{L'})$}. 
If we use proposition \thf[EQ1] together with the Glauber spectral 
gap inequality for the grand canonical Gibbs measure $\mu_\L^{\t,\l}$ 
( the chemical potential $\l$ here is, as always, 
such that $\mu_\L^{\t,\l}(N_\L)=N$ ), 
$$
\mu_\L^{\t,\l}(g,g) \le C_1\, \mu_\L^{\t,\l}( \sum_{x\in \D_g} [\nabla_x g]^2)
$$
for any $g$ and a suitable constant $C_1$ depending on $C,m,l, \|\Phi\|$ 
(see \ref[M1]), we get
$$
 \Var_{\L,N}^\t\(f_{L'}\) \le 
 C_1\, \mu_\L^{\t,\l}( \sum_{x\in B_{L'}} [\nabla_x f_{L'}]^2)
\Eq(loc.2.0)
$$
It is not difficult to check at this point, using 
the hypothesis $USMT(C,m,l)$ together with 
the results on the equivalence of ensembles of \ref[CM1] and lemma 
\thm[der], 
that uniformly in $L$, the r.h.s. of \equ(loc.2.0) is bounded from above 
by $C_f {1\over (L')^d}$ for some constant $C_f$.\acapo
Observe now that, for any function $g$ and for any integer 
$l_0 \le \ov2 L$ multiple of the basic length scale $l$, 
the formula of the conditional covariance (see e.g. \equ(cvar) ) together with 
the definition of spectral gap and the result of theorem \thf[main] 
give the following 
inequality
$$
\eqalign{
 \nu_{\L,N}^\t(g,f)^2   &\le 
2\, \nu_{\L,N}^\t\(\nu_{\L,N}^\t\(g,f\tc \cF_{l_0}\)^2\) 
+ 2\, \Var_{\L,N}^\t(g) \,\Var_{\L,N}^\t\(f_{l_0}\) \cr
&\le 
2\, \nu_{\L,N}^\t\(\nu_{\L,N}^\t\(g,f\tc \cF_{l_0}\)^2\) 
+ 2C_f\, {1\over l_0^d}\, \Var_{\L,N}^\t(g)\cr
&\le 
  C'_f \, \,\[ l_0^2 \,\Dir_{\L,N}^\t(g,g)
+ {1\over l_0^d}\, \Var_{\L,N}^\t(g) \]\cr
	}
\Eq(loc.3)
$$
Notice that, if we take $g:= E_\l f$, then \equ(loc.3) gives
$\|E_\l f\|_2^2 \le C'_f \, \,\[ l_0^2 \,\l + {1\over l_0^d}\]
$
which, if we optimize over the free parameter $l_0$, becomes 
$\|E_\l f\|_2^2 \le C''_f \l^{d\over d+2}$. \acapo
It is important to
observe that the optimal 
choice is $l_0 = O\(\l^{-{1\over d+2}}\) \ll L$ since $\l \ge C L^{-2}$
because of theorem \thf[main].
\smallno 
We will now use \equ(loc.3) 
as the starting point of a recursive
procedure whose final result will be a bound like \equ(loc.3) but with 
the factor $l_0^2$ replaced by $l_0^\g$ with $\g \le \e$ if $d \ge 2$ and 
$\g \le 1 + \e$ in $d=1$. Clearly such a bound will suffice to prove 
\equ(loc.1) because of the little argument given above.

\nproclaim Lemma [loc1]. 
Let $\b_d = 0$ if $d\ge 2$ and $\b_d=1$ if $d=1$. 
In the same hypotheses of theorem \thf[local] 
assume that for some $\b \in [\b_d,2)$, some constant $C(f,\b)$,
all pairs $l_1 \le \ov2 \, l_2$ 
multiples of $l$ and all $N$ the following inequality 
holds 
$$
 \nu_{B_{l_2},N}^\t(g,f)^2   \le 
   C(f,\b) \, \,\[ l_1^\b \,\Dir_{B_{l_2},N}^\t(g,g)
+ {1\over l_1^d}\, \Var_{B_{l_2},N}^\t(g) \]
\Eq(loc.4)
$$
Then there exists a new constant $C'(f,\b)$ such that 
the same inequality holds with $\b$ replaced by 
$\b' =  {2\b\over d+\b} $.

\Pro\ Pick $l_1 \le \ov2 \, l_2$ 
and apply \equ(loc.3) to $\L := B_{l_2}$ 
and $l_0 :=l_1$ to get 
$$
   \nu_{\L,N}^\t(g,f)^2   \le  
    2\, \nu_{\L,N}^\t\(\nu_{\L,N}^\t\(g,f\tc \cF_{l_1}\)^2\) 
    + 2C_f\, {1\over l_1^d}\, \Var_{\L,N}^\t(g)
\Eq(loc.5)
$$
For any $l_3 \le \ov2 \, l_1$
multiple of $l$ we can bound the first term in the 
r.h.s. of \equ(loc.5) by 
$$
  C(f,\b) \, \, \nu_{\L,N}^\t\( \[ l_3^\b \,
  \Dir_{\L,N}^\t\(g,g \tc \cF_{l_1}\)
  + {1\over l_3^d}\, \Var_{\L,N}^\t\(g \tc \cF_{l_1}\) \] \) 
  \le C_1(f,\b) \, \, \[ l_3^\b + {l_1^2\over l_3^d}\] \,
  \Dir_{\L,N}^\t(g,g)
  \Eq(loc.6)
$$
for a suitable constant $C_1(f,\b)$,
where we have used theorem \thf[main] to bound 
$\Var_{\L,N}^\t\(g \tc \cF_{l_1}\)$ in terms of the Dirichlet form 
$\Dir_{\L,N}^\t\(g,g \tc \cF_{l_1}\)$. \acapo
We now chose 
$l_3 = \ov2 \, l_1^{2\over \b + d} \le \ov2 \, l_1$ and 
get that the r.h.s. of \equ(loc.6) is 
bounded from above by 
$$
 C'(f,\b)\,l_1^{2\b\over d+\b}\,\Dir_{\L,N}^\t(g,g) = 
 C'(f,\b)\,l_1^{\b'}\,\Dir_{\L,N}^\t(g,g) 
$$
and the result follows. \QED
\medno
Notice that the inequality of the lemma holds for $\b=2$ because of 
\equ(loc.3). Moreover it is simple to convince
oneself that the sequence $\b_{n+1} =  {2\b_n\over d+\b_n}$, $\b_0=2$ 
converges to $\b_d$ from above. 
Therefore, for any $\e \in (0,1)$ there exists a constant $C_{f,\e}$ such 
that, for 
all pairs $l_1 \le \ov2\, l_2$ multiples of $l$ and all $N$ 
$$
 \nu_{B_{l_2},N}^\t(g,f)^2   \le 
   C_{f,\e}\cases{ l_1^{1+\e} \,\Dir_{B_{l_2},N}^\t(g,g)
            + {1\over l_1}\, \Var_{B_{l_2},N}^\t(g) & if $d=1$ \cr
                  l_1^{\e} \,\Dir_{B_{l_2},N}^\t(g,g)
            + {1\over l^d_1}\, \Var_{B_{l_2},N}^\t(g)  & if $d \ge 2$ \cr
	}
\Eq(loc.7)
$$
The bound \equ(loc.1) on $\|E_l f\|_2^2$ then follows and, as a consequence,
the statement of the theorem. \QED

\vskip 1truecm
\appendix  Proof of Lu--Yau's two block estimate.

\noindent
Here we reprove, for completeness, lemma 4.4 of \ref[LY] 
which played a key role in the estimates in $\S 3.9$. Our proof is 
close to the original one with some simplification due to the fact that
we have at our disposal proposition \thf[EQ1]. 
\smallno
The setting is the following. We fix $\L = Q_L$, $L$ a multiple of 
the basic length scale $l$, an integer $N \in [0,\dots,|\L|]$ and a local 
function $g$ such that $0\in \D_g$ and ${\rm diam}(\D_g) \le 2r$. As in 
lemma \thf[Yaug] we let $G = {1\over |\L|}\sum_{x\in \L} g_x$ 
where $g_x$ is  the translate of $g$ by $x \in \L$. Let also 
$\rho := {N\over |\L|}$.

\nproclaim Proposition [ly].
Assume property $USMT(C,m,l)$. 
Then for any 
$\e > 0$ there exists $C_\e$ such that for any $f$
$$
  \nu_{\L,N}^\t\bigl(f,G\bigr)^2     \leq
  \,{C_\e\over |\L|}\, \Dir_{\L,N}^\t(f,f) +
  {\e\over |\L|}\, \Var_{\L,N}^\t(f) 
$$ 
provided that $|\L|$ is larger than $C_\e$.

\Pro\ Fix $ \e >0$. 
If $\rho \ll \e$ or $1-\rho \ll \e$ 
the statement follows at once from the Schwartz 
inequality together with part $1)$ of proposition \thf[EQ]. 
We will thus assume, without further notice, that 
$\rho \in (\e, 1-\e)$.
\smallno
We define $\{C_\a\}_{\a \in I}$ to be a collection of cubes of side 
$l_0$ multiple of $l$ such that for any 
$\a \neq \b$ ${\rm dist}(C_\a, C_\b) \ge l_0^{\ov2}$, 
${\rm dist}(C_\a, \partial \L) \ge L^{\ov2}$ and 
$|\L \setminus \cup_{\a} C_\a | \le l_0^{-\ov2} |\L|$. 
Clearly such a 
construction is always possible. 
\acapo
Next we observe that, without loss of generality, we 
can replace $g_x$ by $g_x - \d \s(x)$, $\d$ being an arbitrary constant
independent of $x$, because $\sum_{x\in \L}\s(x) = N$ almost surely 
w.r.t. $\nu_{\L,N}^\t$. Accordingly 
we define $G_\d := G -\d\, N_\L$. Our choice of $\d$ will be made later
but we anticipate that it will be almost independent of $l_0$.
We then set 
$$
    G_\d^{\rm int} := 
    {1\over |\L|}\sum_{x\in \cup_\a C^{\rm int}_\a}\bigl(g_x - \d\s(x)\bigr) 
    \quad \hbox{and} \quad 
     G_\d^{\rm ext} := G_\d - G_\d^{\rm in}   
$$   
where  
$C_\a^{\rm int} = \{x\in C_\a \,:\; d(x, C_\a^c) \ge l_0^{1/4}\,\}$.
Notice that 
$$
\Var_{\L,N}^\t(G_\d^{\rm ext}) \le 
 C' \mu_\L^{\t,\l}(G_\d^{\rm ext},G_\d^{\rm ext}) 
  \le C'' {|\L \setminus \cup_{\a} C_\a | + 
      \sum_{\a} |C_\a\setminus C_\a^{\rm int}| \over |\L|^2} 
   \le C'' l_0^{-\ov2} {1\over |\L|}
\Eq(A1)
$$
because of proposition \thf[EQ1], the mixing condition $USMT$ and 
the definition of $\{C_\a\}_{\a \in I}$.
In particular, for any given $\e >0$,  
$$
\nu_{\L,N}^\t\bigl(f,G_\d^{\rm ext}\bigr)^2 \le {\e\over |\L|} 
\Var_{\L,N}^\t(f)
$$ 
provided that 
$l_0 \gg \e^{-2}$. 
\smallno
We now turn to bound the relevant part 
$\nu_{\L,N}^\t\bigl(f,G_\d^{\rm int}\bigr)^2$. Let 
$\cF_0$ be the $\s$--algebra generated by the the random variables 
$\{\s(x)\}_{x\in \L\setminus \cup_\a C_\a},\, \{ N_\a\}_{\a\in I}$, 
where $N_\a(\s) := \sum_{x\in C_\a}\s(x)$. Then we write

$$
\eqalign{
\nu_{\L,N}^\t\bigl(f,G_\d^{\rm int}\bigr)^2 &\le
2\,\nu_{\L,N}^\t\Bigl(  
     \nu_{\L,N}^\t\bigl(f,G_\d^{\rm int} \tc \cF_0 \bigr)\,\Bigr)^2 +
2\,\nu_{\L,N}^\t\Bigl(
     f,
           \nu_{\L,N}^\t\bigl( G_\d^{\rm int}\tc \cF_0 \bigr)\,\Bigr)^2 \cr
 &\le 2\, \Var_{\L,N}^\t(G_\d^{\rm int})
          \nu_{\L,N}^\t\Bigl( 
                \Var_{\L,N}^\t\bigl(f \tc \cF_0 \bigr)\,\Bigr) +
 2\,\nu_{\L,N}^\t\Bigl(
     f,
           \nu_{\L,N}^\t\bigl( G_\d^{\rm int}\tc \cF_0 \bigr)\,\Bigr)^2 \cr
  &\le {C(l_0)\over |\L|} \Dir_{\L,N}^\t(f,f) +
  2\, \Var_{\L,N}^\t(f)\,
       \Var_{\L,N}^\t\( \nu_{\L,N}^\t\bigl( G_\d^{\rm int}\tc \cF_0 \bigr) \)	
	}
\Eq(A2)
$$
where we have used \equ(A1) to bound $\Var_{\L,N}^\t(G_\d^{\rm int})$ 
by ${C'\over |\L|}$ and the estimate 
$$
    \nu_{\L,N}^\t\Bigl(\, \Var_{\L,N}^\t(f \tc \cF_0 )\,\Bigr) \le 
    C(l_0)\, \Dir_{\L,N}^\t(f,f)
$$ 
for some constant $C(l_0)$, valid since  
$\nu_{\L,N}^\t(\cdot\tc \cF_0 )$ is the  product of canonical Gibbs measures 
over the cubes $C_\a$. Actually, using theorem 2.1 of \ref[CCM], the 
constant $C(l_0)$ is not larger than $\exp(c\,l_0^{d-1})$.
\smallno
The key point is now to prove that, for any $\e >0$, 
$\Var_{\L,N}^\t\( \nu_{\L,N}^\t\bigl( G_\d^{\rm int}\tc \cF_0 \bigr) \)	$ 
is smaller than ${\e\over |\L|}$ provided that $l_0$ is large enough.\acapo
Notice that 
$\nu_{\L,N}^\t\bigl( G_\d^{\rm int}\tc \cF_0 \bigr)(\h) $ 
is the sum of local functions
$$
\nu_{\L,N}^\t\bigl( G_\d^{\rm int}\tc \cF_0 \bigr)(\h) = 
 {1\over |\L|}\, \sum_{\a\in I} 
\nu_{C_\a,N_\a(\h)}^\h 
\Bigl( \sum_{x\in C_\a^{\rm int}} g_x -\d \s(x)\,\Bigr)  
\equiv {1\over |\L|}\, \sum_{\a\in I} G^\d_\a(\h)
$$
Thus, if we order in an arbitrary way the set $I$, we can split the 
above sum into the sum of even and odd $\a$'s and apply proposition 
\thf[EQ1] to each term and get
$$
 \Var_{\L,N}^\t\( \nu_{\L,N}^\t\bigl( G_\d^{\rm int}\tc \cF_0 \bigr) \)
     \le C' {1\over |\L|^2 } \, 
      \mu_\L^{\t,\l}\( \sum_\a G^\d_\a,
                  \sum_\a G^\d_\a \)
$$
for some constant $C'$ independent of $\L$ and $l_0$. 
\smallno
Let now 
$\xi^\d_\a(\h):=\mu_{C_\a}^{\h,\l(\h)}\Bigl( 
       \sum_{x\in C_\a^{\rm int}} [g_x -\d \s(x)]\,\Bigr)$, 
where the chemical potential $\l(\h)$ is such that
$\mu_{C_\a}^{\h,\l(\h)}(N_\a) = N_\a(\h)$. In the (rare) case in which 
$N_\a(\h)=0$ ($N_\a(\h)= l_0^d$) the measure $\mu_{C_\a}^{\h,\l(\h)}$
will simply be the Dirac measure on the constant configuration identically 
equal to $0$ ($1$). Thanks to {\it 1)} of proposition \thf[EQ]
we have 
$\sup_{\h} |G^\d_\a(\h) - \xi^\d_\a(\h)| \le C'$. \acapo
In particular, using to the mixing condition together with the fact that
${\rm dist}(C_\a, C_\b) \ge l_0^{\ov2}$ for any $\a\neq \b$, we get
$$
\eqalign{
 {1\over |\L|^2} \mu_\L^{\t,\l}\Bigl( \sum_\a G^\d_\a -\xi^\d_\a,
                  \sum_\a G^\d_\a -\xi^\d_\a \Bigr) &\le 
 {C\over |\L|^2} \sum_\a \mu_\L^{\t,\l}\( G^\d_\a -\xi^\d_\a, G^\d_\a -\xi^\d_\a \) \cr
	&\le
   C' {1\over |\L| l_0^d} \le {\e\over |\L|}
	} 
$$
for $l_0$ large enough.
It is therefore enough to bound
$\mu_\L^{\t,\l}\Bigl( \sum_\a \xi^\d_\a,\sum_\a \xi^\d_\a \Bigr)$ 
(this is the second point where our version of the two blocks estimate differs 
from the original one). 
\smallno
If we use the Poincar\`e inequality 
$\mu_\L^{\t,\l}(f,f) \le C' \mu_\L^{\t,\l}\(\sum_{x\in \L} (\nabla_x f)^2\)$
valid because of the mixing condition $USMT(C,m,l)$ (see e.g. \ref[M1]) we get
$$
\mu_\L^{\t,\l}\Bigl( \sum_\a \xi^\d_\a,\sum_\a \xi^\d_\a \Bigr) \le 
\mu_\L^{\t,\l}\Bigl( 
\sum_{y\in \L}\, \bigl[\, \nabla_y \, \sum_{\a\in I} \xi^\d_\a \, \bigr]^2 
                                       \Bigr)
\Eq(A3)
$$
Observe now that, by construction,
$\nabla_y \, \xi^\d_\a =0 $ unless ${\rm dist}(y,C_\a) \le r$. Thus 
$$
\mu_\L^{\t,\l}\Bigl( 
\sum_{y\in \L}\, \bigl[\, \nabla_y \, \sum_{\a\in I} \xi^\d_\a \, \bigr]^2 
                \Bigr)
= \sum_{\a\in I} \sum_{y\in \L \atop \st {\rm dist}(y,C_\a) \le r}\, 
         \mu_\L^{\t,\l}\Bigl( \bigl[\, \nabla_y \,  \xi^\d_\a \, \bigr]^2\Bigr)
\Eq(A4)
$$
Let us estimate a generic term 
$\mu_\L^{\t,\l}\(\bigl[\, \nabla_y \,  \xi^\d_\a \, \bigr]^2\)$. It is at this 
stage that the subtraction with the free parameter $\d$ made 
at the beginning becomes important. It is in fact clear that without such a
subtraction generically one expects 
$ \Bigl[\,\nabla_y \, \xi^\d_\a \,\Bigr]^2 = O(1)$
(take for instance the trivial case $g_x(\s)= \s(x)$) which would imply 
that the r.h.s. of \equ(A3) is of the order of $|\L|$. 
The way out in order to gain a factor of $\e$  
is to choose $\d$ in such a way that 
$\mu_\L^{\t,\l}\(\bigl[\, \nabla_y \,  \xi^\d_\a \, \bigr]^2\)$ becomes itself 
a variance which again can be bounded using the 
Poincar\`e inequality above (see \ref[LY]). 
\acapo
Let 
$\tilde \nabla_y f(\s) = (1-\s(y))\nabla_y \, f  - 
\s(y) \nabla_y f$ and notice that  
$$
\bigl[\, \nabla_y \,  \xi^{\d}_\a \, \bigr]^2 = 
\bigl[\, \tilde \nabla_y\, \xi^{\d}_\a\, \bigr]^2 =  
\bigl[\, \tilde \nabla_y \, \mu_{C_\a}^{\h,\l(\h)}\bigl( 
                  \sum_{x\in C_\a^{\rm int}} g_x \Bigr) -\d\,\bigr]^2 
$$
 
\nproclaim Lemma [ly1].
Assume $USMT(C,m,l)$. Let $C_0$ be the cube of side $l_0$ centered at 
$y^*$, the 
center of $\L$, and define 
$$
   \d^* := \mu_\L^{\t,\l}\Bigl( 
             \tilde \nabla_{y^*}\, 
               \mu_{C_0}^{\h,\l(\h)}\bigl( 
                \sum_{x\in C_0^{\rm int}} g_x\,\bigr)\,\Bigr)
$$
Then there exists a constants $k_1, k_2$ independent of $L$ and $l_0$, and
$\a>0$ such that
$\d^* \le k_1$ and
$$
\mu_\L^{\t,\l}\(\bigl[\, \nabla_y \,  \xi^{\d^*}_\a \, \bigr]^2\) \le 
     \cases{ k_2 & if $y \in \dep_r C_\a$ \cr
              \noalign{\vskip3pt}
               {k_2\over l_0^d} & if $y \in C_\a$ \cr
	}
$$

\Pro\ The fact that $\d^*$ is bounded from above uniformly in $L,l_0$
as well as the result 
for $y \in \dep_r C_\a$  
follow immediately from $1)$ of lemma \thf[der]. 
\smallno
Let us now consider $y\in C_\a$. \acapo
In this case, under the flip of the variable 
$\h(y)$, the value of $\xi_\a^{\d^*}$ changes only because 
the number of particles of $\h$ varies by $\pm 1$. 
Notice that
$$
|\,\mu_{\L}^{\t,\l}\Bigl(\, \tilde \nabla_y \, \mu_{C_\a}^{\h,\l(\h)}\bigl( 
                  \sum_{x\in C_\a^{\rm int}} g_x \bigr)
   \,\Bigr) -\d^* | \le k_6 \nep{-m \sqrt{L}}
$$
because of translation invariance, the mixing condition and the 
assumption $\hbox{\rm dist}(C_\a, \dep\L)\ge L^{\ov2}$. Thus 
$$
\eqalign{
\mu_\L^{\t,\l}\(\bigl[\, \nabla_y \,  \xi^{\d^*}_\a \, \bigr]^2\) &\le
\mu_\L^{\t,\l}\(
    \tilde \nabla_y \,\xi^{\d^*}_\a, \tilde \nabla_y \,\xi^{\d^*}_\a \) 
 + k_7 \nep{-m \sqrt{L}} \cr 
           &\le
  \sum_{z\in C_\a\cup \dep_r C_\a}\, 
  k_8\, \mu_\L^{\t,\l}\Bigl( 
\bigl[\, \tilde \nabla_z \tilde \nabla_y \,  \xi^{\d^*}_\a \, \bigr]^2\Bigr) 
   + k_7 \nep{-m \sqrt{L}} \cr
	}
\Eq(z)
$$
where we have used once more the Poincar\`e inequality 
for the Glauber dynamics. \acapo
We are left with the estimate of 
$\tilde \nabla_z \tilde \nabla_y \,\xi^{\d^*}_\a(\h)$. \acapo
Suppose for definiteness 
that $\h(z)=\h(y)= 0$, call $N_\a(\h)=n$ and let $\l(s) = \l(\h,s)$ the 
chemical potential such that 
$\mu_{C_\a}^{\h,\l(\h,s)}\bigl(N_\a \bigr)=s$
with $s\in [0,l_0^d]$. 
In what follows we will assume without further notice 
than $0 < n < \ov2 l_0^d$. If instead e.g. $n=0$ then we will simply 
bound $|\tilde \nabla_z \tilde \nabla_y\,  \xi^{\d^*}_\a|$ by $Cl_0^d$
for a suitable constant $C$. 
\acapo
Let us first consider $z\in C_\a$. 
Then a little computation shows that 
$$
  |\tilde \nabla_z \tilde \nabla_y \, \xi^{\d^*}_\a(\h)| \le 
  \int\nolimits_{n}^{n+1} ds \int\nolimits_s^{s+1}dt \,
  |\,{d^2\over dt^2}\,
  \mu_{C_\a}^{\h,\l(t)}\bigl(\sum_{x\in C_\a^{\rm int}} g_x\bigr)\,|
  \le k_9 {1\over n }
  \Eq(dsec)
$$
Let us now take $z\in \dep_r C_\a$. 
Then lemma \thf[der] gives 
$$
  |\tilde \nabla_z \tilde \nabla_y \, \xi_\a(\h)| 
  \le\,\int\nolimits_{n}^{n+1} ds\, 
  |\tilde \nabla_z 
    {d\over ds}\,
     \mu_{C_\a}^{\h,\l(s)}\bigl(\sum_{x\in C_\a^{\rm int}} g_x\bigr)\,|
    \le {k_{10}\over n}
\Eq(ddsec)
$$
If we now remember 
that the density $\rho$ belongs to the interval $(\e,1-\e)$,
standard large deviations bounds 
for $\mu_\L^{\t,\l}$ together with \equ(dsec) and \equ(ddsec) imply that 
$\mu_\L^{\t,\l}\( \Bigl[\, \tilde \nabla_z \tilde \nabla_y \,  
              \xi^{\d^*}_\a \, \Bigr]^2\) \le k_{11}(\e) {1\over l_0^{2d}}$. 
Thus the r.h.s. of \equ(z) is smaller than 
$k_{12}\, {1\over l_0^d}$ and the lemma follows.   \QED


\beginsection References

\frenchspacing\magnification=\magstephalf

\medno
\item{[A]} %%
  C.~Albanese:
  {\it A Goldstone mode inthe Kawasaki--Ising model}. Journal of Stat. Phys.
  {\bf 77} No. 1/2, 77-87 (1994).

\item{[BCO]} %%
   L.~Bertini, E.~N.~M.~Cirillo and E.~Olivieri:
   {\it Renormalization--Group transformations under strong mixing conditions:
   Gibbsianness and convergence of renormalized interactions}.
   Preprint (1999).

\item{[BZ1]} %%
   L.~Bertini and B.~Zegarlinski:
   {\it Coercive inequalities for Gibbs measures.}
   J. Funct. Anal. {\bf 162}, 257-289, (1999).

\item{[BZ2]} %%
   L.~Bertini and B.~Zegarlinski:
   {\it Coercive inequalities for Kawasaki dynamics. The product case.}
   \hbox{http://mpej.unige.ch/mp\_arc/c/96/96-561.ps.gz.uu} .

\item{[CCM]} %%
   N.~Cancrini, F.~Cesi and F.~Martinelli:
   {\it Kawasaki dynamics at low temperature.}
   Journal of Stat. Phy. {\bf 95} Nos 1/2, 219-275 (1999).

\item{[CM1]} %%
   N.~Cancrini and F.~Martinelli:
   {\it Comparison of finite volume canonical and grand canonical
        Gibbs measures under a mixing condition.}
   Preprint Roma april 1999.

\item{[CM2]} %%
   N.~Cancrini and F.~Martinelli:
   {\it Spectral gap of Kawaski dynamics for the dilute Ising model 
   in the Griffiths phase.}
   In preparation.

\item{[CM3]} %%
   N.~Cancrini and F.~Martinelli:
   {\it Logarithmic Sobolev of Kawasaki dynamics in the one phase 
   region revisited.}
   In preparation.

\item{[CZ]} %%
   F.~Comets and O.~Zeitouni:
   {\it Information estimates and Markov random fields}.
   Preprint (1999).

\item{[G]} %%
   H.~O.~Georgii:
   {\it Gibbs measures and phase transitions.}
   De Gruyter Series in Mathematics {\bf 9} Berlin: Walter de Gruyter (1988).

\item{[Le]} %%
   M.~Ledoux:
   {\it Concentration of measure and logarithmic Sobolev inequalities.}\acapo
   \hbox{http://www-sv.cict.fr/lsp/ledoux/index} .

\item{[LS]} %%-
   G. F. Lawler and A. D. Sokal:
   Bounds on the $L^2$ spectrum for Markov chains and Markov processes:
   A generalization of Cheeger's inequality.
   Trans. Amer. Math. Soc. {\bf 309}, No 2, 557 (1988).
   
\item{[LY]} %%-
   S.~T.~Lu and H--T.~Yau:
   {\it Spectral gap and logarithmic Sobolev inequality for Kawasaki
   and Glauber dynamics}.
   Commun. Math. Phys. {\bf 156}, 399-433 (1993).

\item{[M1]} %%-
   F.~Martinelli:
   {\it Lectures on Glauber dynamics for discrete spin models}.
   Proceedings of the Saint Flour summer school in probability theory
   1997. To appear in Lecture Notes in Mathematics.

\item{[M2]} %%-
   F.~Martinelli:
   {\it Logarithmic Sobolev constant for discrete random walks}.
   In preparation

\item{[MOS]} %%
   F. Martinelli, E. Olivieri and R.H.Schonmann: 
   {\it For Gibbs state of 2D lattice spin systems 
   weak mixing implies strong mixing.}
   Commun. Math. Phys. {\bf 165}, 33 (1994).
 
\item{[VY]} %%
   S.R.S.~Varadhan, H--T.~Yau:
   {\it Diffusive limit of lattice gases with mixing conditions}.
   Preprint 1999.

\item{[Y]} %%
   H--T.~Yau:
   {\it Logarithmic Sobolev Inequality for Lattice Gases with 
   Mixing Conditions.}
   Comm.Math.Phys. {\bf 181}, 367-408, (1996).

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