Content-Type: multipart/mixed; boundary="-------------9907220853309" This is a multi-part message in MIME format. ---------------9907220853309 Content-Type: text/plain; name="99-275.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-275.keywords" 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 %\hoffset=0.5truecm %\voffset=0.5truecm %\hsize=16.5truecm %\vsize=22.0truecm \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 \font\twelvei=cmmi12 \font\twelvesy=cmsy10 \font\twelvebf=cmbx12 \font\twelvett=cmtt12 \font\twelveit=cmti12 \font\twelvesl=cmsl12 % \font\ninerm=cmr9 \font\ninei=cmmi9 \font\ninesy=cmsy9 \font\ninebf=cmbx9 \font\ninett=cmtt9 \font\nineit=cmti9 \font\ninesl=cmsl9 % \font\eightrm=cmr8 \font\eighti=cmmi8 \font\eightsy=cmsy8 \font\eightbf=cmbx8 \font\eighttt=cmtt8 \font\eightit=cmti8 \font\eightsl=cmsl8 % \font\seven=cmr7 % \font\sixrm=cmr6 \font\sixi=cmmi6 \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$}}}}} \def\galato(##1){ \gwidth=\hsize \divide\gwidth by 2 {\vtop to \profonditastruttura{\baselineskip \profonditastruttura\vss \rlap{\kern-\gwidth\kern-1.3truecm{$\scriptstyle##1$}}}}} \def\talato(##1){\rlap{\sixrm\kern -1.3truecm ##1}} \def\thm{\teo}\def\thf{\teo} } % \def\alato(#1){} \def\galato(#1){} \def\talato(#1){} % % \def\numsection#1{% \ifnum\numsec=-100% \relax\number#1% \else \ifnum\numsec=-101% \relax A.\number#1% \else \ifnum\numsec<0% A\number-\numsec.\number#1% \else \number\numsec.\number#1% \fi \fi \fi } % %\def\geq(#1){\getichetta(#1)\galato(#1)} % \def\Thm[#1]{\tetichetta(#1)} \def\thf[#1]{\senondefinito{futh#1}$\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 9\or A\or B\or C\or D\or E\or F\fi} % \font\tenmsa=msam10 \font\sevenmsa=msam7 \font\fivemsa=msam5 \textfont\msafam=\tenmsa \scriptfont\msafam=\sevenmsa \scriptscriptfont\msafam=\fivemsa % \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 \ifmmode\square\else$\square$\fi} % \font\tenmsb=msbm10 \font\sevenmsb=msbm7 \font\fivemsb=msbm5 \textfont\msbfam=\tenmsb \scriptfont\msbfam=\sevenmsb \scriptscriptfont\msbfam=\fivemsb \def\Bbb#1{\fam\msbfam\relax#1} % \font\teneufm=eufm10 \font\seveneufm=eufm7 \font\fiveeufm=eufm5 \textfont\eufmfam=\teneufm \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}} \def\[{\left[} \def\]{\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) $$ where $A(n)$ and $B(n)$ have been defined in \equ(aa3). \acapo In order to estimate the first term in the r.h.s of \equ(3term) we first need to fix a {\it path choice} (see proposition \thf[wdir] and definition \thf[cho] for the necessary notation). \acapo Our choice is the following. Given $x\in \L_1$ and $z\in \L_2\setminus \L_1$, start increasing (or decreasing) the first coordinate of $x$ until it is equal to the first coordinate of $z$. Then adjust the second coordinate until you get to $z$. With this particular path choice it is easy to see that $\cG_V(\l) \le (2L)^{3}$. Assume in fact that the path $\l_{xz}$ contains the edge $e=[u,v]$ where $u$ and $v$ differ in the $j^{th}$ coordinate. This means that $x_i = u_i$ for all $i>j$ and $z_i = u_i$ for all $i0$ there exists a constant $C(\e)$ such that the second term in the r.h.s. of \equ(3term) is smaller than $$ 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). \end ---------------9907220853309--