%---------------------------------------------------------------------------- % % Filippo Cesi and Fabio Martinelli % % % On the Layering Transition of an SOS % Surface Interacting with a Wall. II. The Glauber Dynamics. % % % Five useful but not necessary figures can be obtained from % the authors via ordinary mail % % INSTRUCTIONS: % 1) If you have any problem with fonts which are not available on % your system, search for the statement % \fnts=1 % and comment it (with a %) % 2) Run TeX twice to resolve cross-references % %---------------------------------------------------------------------------- % %%%%%%%%%%%%%%% FORMATO \magnification=\magstep1 \hoffset=0.cm \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 %%%%%%%%%%%%%%%% 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 %%%%%%%%%%%%%%%%%%%%% 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{\veroparagrafo:\number\pgn \global\advance\pgn by 1} %%%%%%%%%%%%%%%%% EQUAZIONI CON NOMI SIMBOLICI %%% %%% per assegnare un nome simbolico ad una equazione basta %%% scrivere \Eq(...) o, in \eqalignno, \eq(...) o, %%% nelle appendici, \Eqa(...) o \eqa(...): %%% dentro le parentesi e al posto dei ... %%% si puo' scrivere qualsiasi commento; %%% per assegnare un nome simbolico ad una figura, basta scrivere %%% \geq(...); per avere i nomi %%% simbolici segnati a sinistra delle formule e delle figure si deve %%% dichiarare il documento come bozza, iniziando il testo con %%% \BOZZA. Sinonimi \Eq,\EQ,\EQS; \eq,\eqs; \Eqa,\Eqas;\eqa,\eqas. %%% All' inizio di ogni paragrafo si devono definire il %%% numero del paragrafo e della prima formula dichiarando %%% \numsec=... \numfor=... (brevetto Eckmannn); all'inizio del lavoro %%% bisogna porre \numfig=1 (il numero delle figure non contiene la sezione. %%% Si possono citare formule o figure seguenti; le corrispondenze fra nomi %%% simbolici e numeri effettivi sono memorizzate nel file \jobname.aux, che %%% viene letto all'inizio, se gia' presente. E' possibile citare anche %%% formule o figure che appaiono in altri file, purche' sia presente il %%% corrispondente file .aux; basta includere all'inizio l'istruzione %%% \include{nomefile} %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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){(\veroparagrafo.\veraformula) \SIA e,#1,(\veroparagrafo.\veraformula) \global\advance\numfor by 1 \write15{\string\FU (#1){\equ(#1)}} \write16{ EQ \equ(#1) == #1 }} \def\FU(#1)#2{\SIA fu,#1,#2 } %------------------- teoremi ---------------------------- % \def\tetichetta(#1){{\veroparagrafo.\verotheo}% \SIA theo,#1,{\veroparagrafo.\verotheo} \global\advance\numtheo by 1% \write15{\string\FUth (#1){\thm[#1]}}% \write16{ TH \thm[#1] == #1 }} \def\FUth(#1)#2{\SIA futh,#1,#2 } % %-------------------------------------------------------- \def\getichetta(#1){Fig. \verafigura \SIA e,#1,{\verafigura} \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-1.2truecm{$\scriptstyle##1$}}}}} \def\galato(##1){ \gwidth=\hsize \divide\gwidth by 2 {\vtop to \profonditastruttura{\baselineskip \profonditastruttura\vss \rlap{\kern-\gwidth\kern-1.2truecm{$\scriptstyle##1$}}}}} \def\talato(##1){\rlap{\sixrm\kern -1.2truecm ##1}} } \def\alato(#1){} \def\galato(#1){} \def\talato(#1){} \def\veroparagrafo{\ifnum\numsec<0 A\number-\numsec\else \number\numsec\fi} \def\veraformula{\number\numfor} \def\verotheo{\number\numtheo} \def\verafigura{\number\numfig} %\def\geq(#1){\getichetta(#1)\galato(#1)} \def\Thm[#1]{\tetichetta(#1)} \def\thf[#1]{\senondefinito{futh#1}$\clubsuit$#1\else \csname futh#1\endcsname\fi} \def\thm[#1]{\senondefinito{theo#1}thf[#1]\else \csname theo#1\endcsname\fi} \def\Eq(#1){\eqno{\etichetta(#1)\alato(#1)}} \def\eq(#1){\etichetta(#1)\alato(#1)} \def\eqv(#1){\senondefinito{fu#1}$\clubsuit$#1\else \csname fu#1\endcsname\fi} \def\equ(#1){\senondefinito{e#1}eqv(#1)\else \csname e#1\endcsname\fi} \let\EQS=\Eq \let\EQ=\Eq \let\eqs=\eq %%%%%%%%%%%%%%%%%% 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} \openin14=\jobname.aux \ifeof14 \relax \else \input \jobname.aux \closein14 \fi \openout15=\jobname.aux %%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\fine{\vfill\eject} \def\sezioniseparate{% \def\fine{\par \vfill \supereject \end }} \footline={\rlap{\hbox{\copy200}\ $\st[\number\pageno]$}\hss\tenrm \foglio\hss} %---------------- fonti disponibili --------------------------- % \newcount\fnts \fnts=0 \fnts=1 %-----comment if msam, msbm, eufm are not available % % % % \def\oq{\char96} \def\oqq{\oq\oq} \def\page{\vfill\eject} \def\smallno{\smallskip\noindent} \def\medno{\medskip\noindent} \def\bigno{\bigskip\noindent} \def\acapo{\hfill\break} \def\thsp{\thinspace} \def\x{\thinspace} \def\tthsp{\kern .083333 em} \def\mathindent{\parindent=50pt} \def\club{$\clubsuit$} %------------------------ itemizing % \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} % % %------------------------------------------------------------------- %.................. Se non ci sono le fonti % \newfam\msafam \newfam\msbfam \newfam\eufmfam \ifnum\fnts=0 \def\integer{ { {\rm Z} \mskip -6.6mu {\rm Z} } } \def\real{{\rm I\!R}} \def\bb{ \vrule height 6.7pt width 0.5pt depth 0pt } \def\complex{ { {\rm C} \mskip -8mu \bb \mskip 8mu } } \def\Ee{{\rm I\!E}} \def\Pp{{\rm I\!P}} \def\mbox{ \vbox{ \hrule width 6pt \hbox to 6pt{\vrule\vphantom{k} \hfil\vrule} \hrule width 6pt} } \def\QED{\ifhmode\unskip\nobreak\fi\quad \ifmmode\mbox\else$\mbox$\fi} \let\restriction=\lceil % %.................. o se ci sono % \else \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 \mathchardef\square"0\msafamhexnumber03 \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 %-------------------------------------- % %\font\teneusb=eusb10 %\font\seveneusb=eusb7 %\font\fiveeusb=eusb5 %\newfam\eusbfam %\textfont\eusbfam=\teneusb %\scriptfont\eusbfam=\seveneusb %\scriptscriptfont\eusbfam=\fiveeusb %\def\script#1{{\fam\eusbfam\relax#1}} %%%%%%%%%%%%%%%%%%%% \def\integer{{\Bbb Z}} \def\real{{\Bbb R}} \def\complex{{\Bbb C}} \def\Ee{{\Bbb E}} \def\Pp{{\Bbb P}} \def\Iidentity{{\Bbb I}} \fi % %------------------------------------------------------------------- % \def\identity{ {1 \mskip -5mu {\rm I}} } \def\ie{\hbox{\it i.e.\ }} \let\sset=\subset \def\ssset{\subset\subset} \let\neper=e \let\ii=i \let\emp=\emptyset \let\id=\identity \def\ov#1{{1\over#1}} \def\Pro{\noindent{\it Proof.}} \def\sump{\mathop{{\sum}'}} \def\sumh{\mathop{{\sum}^{(h)}}} \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\dim{ \mathop{\rm dim}\nolimits } \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\sign{\mathop{\rm sign}\nolimits} \def\prob{\mathop{\rm Prob}\nolimits} \def\tc{\thsp | \thsp} \let\nea=\nearrow \let\dnar=\downarrow \def\norm#1{ {\Vert #1 \Vert} } \def\inte#1{\lfloor #1 \rfloor} \def\ceil#1{\lceil #1 \rceil} \def\cdotss{$\cdots$} \def\imp{\Rightarrow} \let\de=\partial \def\dep{\partial^+} \def\deb{\bar\partial} \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}} % %------------------------------------------------------------------- \def\papI{{[CM]}} \def\papII{{[CM]}} \def\xx{ {\{x\}} } \def\kmax{{k_{max}}} \def\kmx{\inte{\nep{\b\z\over 20}}} \def\Gb{\bar G} \def\Gt{{\ntilde G}} \def\abar{\bar \a} \def\hb{\bar\h} \def\hba{\bar h} \def\Lb{\bar\L} \def\xit{{\ntilde\xi}} \def\xib{\bar\xi} \def\psit{{\ntilde\psi}} \def\what{\hat w} \def\phh{\hat\ph} \def\Nh{\hat N} \def\Dir{{\cal E}} \def\calD{{\cal D}} \def\Con{{\rm Con}} \let\Z=\integer \def\Zp{{\integer_+}} \def\ZpN{{\integer_+^N}} \def\ZZ{{\integer^2}} \def\ZZt{\integer^2_*} \def\FF{{\bf F}} \def\A{{\cal A}} \def\B{{\cal B}} \def\calZ{{\cal Z}} \def\pmu{\{-1,1\}} \def\conf{\pmu^\ZZ} \def\confV{\pmu^V} \def\pht{{\ntilde \ph}} \def\phb{\bar \ph} \def\Ot{{\ntilde \O}} \def\mx{\setm \{x\}} \def\Jb{\bar J} \def\gap{ {\rm gap} } \def\gtl{ {\ntilde \g} } \def\gtls{\{\gtl\}} \def\gbar{{\bar \g}} \let\gb=\gbar \def\gbarx{{\bar \g\mx}} \def\gdn{\g_\dnar} \def\etl{ {\ntilde \e} } \def\ztl{ {\ntilde z} } \def\wtl{{\ntilde w}} \def\wbar{{\bar w}} \let\wb=\wbar \def\wbb{\overline{\overline w}} \def\wJh{w_{J,h}} \def\wJhtl{{\ntilde \wJh}} \def\wtr{\wJh^{tr}} \def\wt{w_h^{tr}} \def\wmod{w_h^n} \def\wkp{w_h^{k+1}} \def\wkptl{{\ntilde\wkp}} \def\wmodtl{{\ntilde\wmod}} \def\hl{{h_{\l}}} \def\Asc{ {\cors A} } \def\Bsc{ {\cors B} } \def\setm{\setminus} \def\xy{ { \{x,y\} } } \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\indhh{ { x\in V } } \def\indhpsi{{h,\psi}} \def\indJhpsi{{J,h,\psi}} \def\indJhpsip{{J,h,\psi'}} \def\indJh{{J,h}} \def\indJhn{{J,h,n}} \def\indhn{{h,n}} \def\vv{^{\hst,\emp}_\L} \def\hnL{^{h,n}_\L} \def\AWp{_{\A \in W_l^+(\L,k) }} \def\BWp{_{\B \in W_l^+(\L,k) }} \def\GWp{_{\G \in \Con^+(\L,k+1) }} \def\indphOV{{\ph\in\O_V}} \def\gg{ { \g\in\G } } \def\ggp{ { \g\in\G' } } \def\ggint{{\g\in\G\setm\G_{int}}} \def\rraa{{\r\in\A}} \def\GA{{\G\in\A}} \def\Gext{\G_{ext}} \def\Gg{\G\setm \{\g\}} \def\indC(#1){ { \G\in C^*_c(V,#1) } } \def\indCw(#1){ { \G\in C^*_w(V,#1) } } \def\indCl(#1){ { \G\in C^*_{c,l}(V,#1) } } \def\indW(#1){{\A\in W(V,#1)}} \def\indWl(#1){{\A\in W_l(V,#1)}} \def\indWl(#1){{\A\in W_l(V,#1)}} \def\Eg{{E(\g)}} \def\Ig{{I(\g)}} \def\Lg{{L(\g)}} \def\Sg{{S(\g)}} \def\hp{^{h,\psi}} \def\hpL{^{h,\psi}_\L} \def\JhpL{^{J,h,\psi}_\L} \def\HpL{H^{h,\psi}_\L} \def\HJpL{H^{J,h,\psi}_\L} \def\Hn{ { H^{J,h,n} } } \def\HHk{ { H^{J,h,k}_V(\ph) } } \def\HHn{ { H^{J,h,n}_V(\ph) } } \def\HiL{\bar H^{h,\infty}_\L} \def\Hbi{\bar H^{h,\infty}_V(\ph)} \def\HbiL{\HiL(\ph)} \def\dbarp{ \bar D^*_+(V,k,\L) } \def\deltaV{ { \partial V } } \def\dV{\deltaV} \def\deltaL{ \partial \L} \def\deL{\d \L} \def\debL{\bar\partial \L} \def\deV{\d V} \def\debgb{{\deb\gb}} \def\nep#1{ \neper^{#1}} \def\nepm{\nep{-4 \b m}} \def\nepn{\nep{-4 \b n}} \def\nepk{\nep{-4 \b k}} \def\nepmk{\nep{-4 \b (m\wedge k)}} \def\nepnk{\nep{-4 \b (n\wedge k)}} \def\nepml{\nep{-4 \b (m\wedge l)}} \def\nepnl{\nep{-4 \b (n\wedge l)}} \def\nzero{ [ \b \neper^{5 \b k} ] } \def\mmin{\wedge} \def\mmax{\vee} \def\Bigcup{\bigcup} \def\hik{h\in I_k(\b)} \def\hkpm{h_{k+1}^-(\b)} \def\hkpp{h_{k+1}^+(\b)} \def\hkm{h_{k}^-(\b)} \def\hkp{h_{k}^+(\b)} \def\hkst{h_{k}^*(\b)} \def\hkmst{h_{k-1}^*(\b)} \def\hkpst{h_{k+1}^*(\b)} \def\hst{{h^*}} \def\crit{[\hkpp, \hkm]} \def\hcrit{h \in\crit} \def\Zh{\hat Z} \def\Zhe{\hat Z_e} \def\Zhk{\hat Z^k} \def\Zhke{\hat Z^k_e} \def\Zhkp{\hat Z^{k+1}} \def\Zhn{\hat Z^n} \def\Zk{Z^{J,h,k}} \def\Zkp{Z^{J,h,k+1}} \def\Zn{Z^{J,h,n}} \def\Zm{Z^{J,h,m}} \def\mub{\bar \mu} \def\mun{\mu^{J,h,n}} \def\muk{\mu^{J,h,k}} \def\etas{\{\h\}} \def\eetas{{\h\in\etas}} \def\etab{{\bar \h}} \def\zm{\z^{-1}} \def\zer{_{(0)}} \def\Zar{Zahradn\'\i k} \expandafter\ifx\csname sezioniseparate\endcsname%--- non toccare \relax\input macro \fi %--- queste due righe % \font\ttlfnt=cmcsc10 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\vskip 0.5truecm} % % Abstract % \def\abstract#1 { \noindent{\bf Abstract.\ }#1\par} % \vskip 1cm \centerline{\ttlfnt On the Layering Transition of an SOS Surface Interacting} \centerline{\ttlfnt with a Wall. II. \caps The Glauber Dynamics} \vskip 0.5truecm \author{Filippo Cesi $^{\dag}$ {\ninerm and} Fabio Martinelli $^{\ddag}$} % \address{\ninerm \dag Dipartimento di Fisica, Universit\`a \oqq La Sapienza", P.le A. Moro 2, 00185 Roma, Italy \hfill\break \ddag Dipartimento di Matematica, III Universit\`a, Via C. Segre 2, 00146 Roma, Italy \hfill\break \dag e-mail: cesi@vaxrom.roma1.infn.it \hfill\break \ddag e-mail: martin@mat.uniroma3.it} % \abstract{\ninerm %Abstract della II parte We continue our study of the statistical mechanics of a 2D surface above a fixed wall and attracted towards it by means of a very weak positive magnetic field $h$ in the solid on solid (SOS) approximation, when the inverse temperature $\beta$ is very large. In particular we consider a Glauber dynamics for the above model and study the rate of approach to equilibrium in a large cube with arbitrary boundary conditions. Using the results proved in the first paper of this series we show that for all $h\in (h^*_{k+1},h^*_k)$ ($\{h_k^*\}$ being the critical values of the magnetic field found in the previous paper) the gap in the spectrum of the generator of the dynamics is bounded away from zero uniformly in the size of the box and in the boundary conditions. On the contrary, for $h\,=\,h_k^*$ and free boundary conditions, we show that the gap in a cube of side $L$ is bounded from above and from below by a negative exponential of $L$. Our results provide a strong indication that, contrarily to what happens in two dimensions, for the three dimensional dynamical Ising model in a finite cube at low temperature and very small positive external field, with boundary conditions that are opposite to the field on one face of the cube and are absent (free) on the remaining faces, the rate of exponential convergence to equilibrium, which is positive in infinite volume, may go to zero exponentially fast in the side of the cube. } % \vskip 1cm \noindent \vskip 1cm \noindent {\bf Key Words:} SOS model, Layering transition, Pirogov-Sinai theory, Relaxation time {\parindent=0pt \footnote{}{ Work partially supported by grant SC1-CT91-0695 of the Commission of European Communities} \footnote{}{ Mathematics Subject Classification. Primary: 82C24. Secondary: 60K35} } \vfill\eject \endgroup \expandafter\ifx\csname sezioniseparate\endcsname\relax% \input macro \fi \numsec=0 \numfor=1 \numtheo=1 \pgn=1 \beginsection 0. Introduction This paper is the second part of a work, begun in \ref[CM], %------ XXX (which will be also referred to as \papI), about the equilibrium and non equilibrium statistical mechanics of a SOS surface above a fixed wall at low temperature and attracted towards it by a very weak external field. The equilibrium distribution of the model in a finite volume $V\subset \ZZ$ with boundary conditions $\{\psi (y)\}_{y\in \ZZ\setminus V}$ is described by the following Gibbs measure: $$ \mu_V^\psi (\ph ) = {1\over Z^\psi(V)} \exp \Bigl[ - {\beta\over 2} \sum_{x,y\in V; \atop \vert x-y\vert =1} \vert \ph (x)-\ph (y)\vert\,-\,\beta h\sum_{x\in V}\ph (x)\,-\, \sum_{x\in V\,y\in V^c \atop \vert x-y\vert =1} \vert \ph (x)-\psi (y)\vert \Bigr] \Eq(1) $$ where $Z^\psi(V)$ is the associated partition function and the random variable $\ph (x)\in\Zp$ represents the height of the surface at $x\in V$ above the wall. A kinetic version of the model is readily obtained by considering a Glauber dynamics for it, namely a single spin Markov process on the configuration space, reversible with respect to $\mu_V^\psi(\ph)$ and such that each move consists in replacing at some site $x$, $\ph(x)$ with $\ph(x)\pm 1$. We refer the reader to the introduction of \papI{} for a detailed discussion of the main motivation which led us to consider such a model and for a brief description of the main results both for the equilibrium and the non equilibrium case. Here we just recall for the reader's convenience the main result of \papI, since it plays an essential role in our analysis of the non equilibrium case. \proclaim Theorem [CM]. There exists $\b_0$ such that for all $\b\ge\b_0$ there are positive numbers $\{ \hkst \}_{k=1}^{\kmax}$, with $\kmax=\inte{\nep{\b\over 20000}}$, such that the following holds for $k=1,\ldots,\kmax$ \smallno \item{$(i)$} $ {1\over4} \nep{-4\b k} \le \b \hkst \le 4 \nep{-4\b k}$ \item{$(ii)$} if $h^*_{k}(\b) < h < h^*_{k-1}(\b)$ (define $h^*_0(\b) = +\infty$) then \itemm{$(a)$} there exists a unique Gibbs measure for the interaction \equ(1) \itemm{$(b)$} there exist $m(\b,h)>0$, $C(\b,h)>0$ such that for any $N\geq \inte{8/h+1}$ $$ \sup_{\psi,\psi' \in\O } | E_{Q_N}^{h,\psi}\ph (0) - E_{Q_N}^{h,\psi'} \ph (0) | \leq C(\b,h) \nep{ -m(\b,h) \, N} $$ where $ E_{Q_N}^{h,\psi}(\ph (0))$ denotes the expected value of the height of the surface at $x=0$ in a square $Q_N$ of side $N$ and center at the origin, with boundary conditions $\psi$. \item{$(iii)$} if $h = \hkst$ then both partitions functions $Z^k(Q_N)$ and $Z^{k+1}(Q_N)$, with boundary conditions $\psi\equiv k$ and $\psi\equiv k+1$ respectively, admit a convergent cluster expansion. Hence there are at least two distinct extreme Gibbs measures. \smallno In the present paper we turn our attention to the Glauber dynamics of the SOS surface in a square $Q_N$ of side $N$, with boundary conditions $\psi$ and magnetic field $h$; more precisely we analyze the $\gap^{h,\psi}(Q_N)$ in the spectrum of its generator which is related to the relaxation time to equilibrium. Our main result is \proclaim Theorem. In the same setting as in Theorem [CM] we have, for all $k=1, \ldots,\kmax$, \smallno \item{$(i)$} if $h^*_{k}(\b) < h < h^*_{k-1}(\b)$ then there exist $L_0(\b,h)$, $\k(\b,h)>0$ such that $$ \inf_{L\ge L_0} \inf_{\psi\in\O} \gap^{h,\psi}(Q_L) \ge \k(\b,h) $$ \item{$(ii)$} if $h=h^*_k(\b)$, then there exist positive constants $C_1(\b,h)$, $C_2(\b,h)$ such that for all $N>10/h$ $$ C_1(\b,h) \, \nep{ -100 \b k N } \le \gap^{h,\emptyset}(Q_N) \le C_2(\b,h) \, \nep{ -{1\over40} \b N } $$ where $\emptyset$ means free boundary conditions. \smallno The proof of part $(i)$, discussed in Section 2, is organized as follows. Using the methods of \ref[MOS] and some apriori bounds on the moments of the random variable $\ph (x)$ proved in \ref[CM], one easily improve part $(i)$ of Theorem [CM], by showing that a stronger form of weak dependence on the boundary conditions (strong mixing in the language of \ref[MO1], \ref[MO2], \ref[MOS]) holds. Once strong mixing is established, a lower bound of the gap in any large enough cube, uniformly in the volume and in the boundary conditions, follows from the ``block dynamics approach" envisaged in \ref[MO1]. This method requires, however, as an input, a lower bound on the gap in a given fixed block. Such a bound would be trivial were the random variables $\ph (x)$ be bounded. In our case the problem requires some extra work and our solution is based upon the so called \oqq Cheeger inequality" (see \ref[LS]). The proof of the upper bound of part $(ii)$ is discussed in Section 3 and it uses the variationl characterization of the gap. More precisely, using a suitable test function $f$, we show the existence of a bottleneck: the system, in order to relax to equilibrium, has to make an excursion to a region of the configuration space of very small equilibrium measure. Finally Section 4 contains the proof of the lower bound; our method relies upon a novel recursive estimate of the gap that allows us to overcome the problem of the unboundedness of the variables $\ph (x)$. \vskip 0.5cm \noindent {\bf Acknowledgments}\par This work started while one of the authors (F.M) was profiting of a very stimulating reasearch period at the Newton Institute in Cambridge in the framework of the special semester dedicated to "Spatial Structure of Random Fields". He would like to thank the organizers for the kind invitation and the opportunity to discuss with several specialists of the field. We would like also to thank Mazel for interesting discussions about their work. \fine \expandafter\ifx\csname sezioniseparate\endcsname\relax% \input macro \fi %--------------------------------- INIZIO \numsec=1 \numfor=1 \numtheo=1 \pgn=1 \beginsection 1. Preliminaries For the reader's convenience we recall most of the basic definitions that were given in \papI. \beginsubsection 1.1. General definitions We consider the two dimensional lattice $\ZZ$ whose elements are called {\it sites\/} and its dual $\ZZt = \ZZ + (1/2, 1/2)$. For $x,y \in \real^2$ we define two distances $$ d(x,y) = |x-y| = \sum_{i=1}^2 |x_i-y_i| \quad\hbox{and}\quad d_\infty(x,y) = |x-y|_\infty = \max_{i=1,2} |x_i-y_i| \,. $$ $[x,y]$ is the {\it closed segment} with $x,y$ as its endpoints. The {\it edges} of $\ZZ$ ($\ZZt$) are those $e=[x,y]$ with $x,y$ nearest neighbors in $\ZZ$ ($\ZZt$). Given $e$ edge of $\ZZ$, $e^*$ is the unique edge in $\ZZt$ that intersects $e$. The {\it boundary of an edge} $e=[x,y]$ is $\d e = \{x,y\}$ The {\it boundary of a subset of edges} $\a$ is the set of sites $\d \a$ that belong to an odd number of edges of $\a$. A set of edges is called {\it closed} if its boundary is empty. \smallno We will often consider our model on a square $$ Q_N = \cases{ \vphantom{\Bigl(} \bigl\{ \, (x_1,x_2) \in \ZZ : -L \le x_i \le L, \ i=1,2 \, \bigr\} & if $N=2L+1$ \cr \vphantom{\Bigl(} \bigl\{ \, (x_1,x_2) \in \ZZ : -L+1 \le x_i \le L, \ i=1,2 \, \} & if $N=2L$ \cr } $$ $\L$ and $V$ will denote arbitrary subsets of $\ZZ$. If $\L$ is finite we write $\L \ssset \ZZ$. The cardinality of $\L$ is denoted by $|\L|$. We define four kinds of {\it boundaries\/} $$ \eqalign{ \partial \L &= \{ x \in \L : d(x, \L^c)=1 \} \cr \bar\partial \L &= \{ x \in \L : d_\infty(x, \L^c)=1 \} \cr \partial^+\L &= \{ x \in \L^c : d(x, \L)=1 \} \cr \d\L &= \{ e^*=[x,y]^* : \{x,y\}\cap\L\ne\emp,\ \{x,y\}\cap\L^c\ne\emp \} } $$ where $\L^c=\ZZ \setm \L$. $(x_1,\ldots,x_n)$ is called a {\it path} from $x_1$ to $x_n$ if $|x_{i+1}-x_i|=1$ for $i=1,\ldots,n-1$. A $*$--path is the same as a path with $|x_{i+1}-x_i|=1$ replaced by $d_\infty(x_i,x_{i+1})=1$. A ($*$--) path is called {\it self-avoiding\/} if $x_i\ne x_j$ for all $\{i,j\}$ such that $i\ne j$ and $\{i,j\} \ne \{1,n\}$. If $x_1 = x_n$ the ($*$--) path is called {\it closed}. $\L \subset \ZZ$ is said to be {\it connected} ($*$--connected) if for all $x,y$ in $\L$ there exists a path ($*$--path) from $x$ to $y$ which is entirely contained in $\L$. $\L\ssset\ZZ$ is called {\it simply connected} if $\L^c$ is $*$--connected. A set of edges $\a$ is {\it connected} if the union of all its edges is connected in $\real^2$. We denote by $C_B$ the set of all finite closed connected set of edges of $\ZZt$. If $\a \in C_B$ then we define the {\it interior} of $\a$ (see Fig.~1) as the set of all sites $x=(x_1, x_2)\in \ZZ$ such that the half line $$ \{x_1\} \times [x_2, +\infty) $$ intersects $\a$ in an odd number of points. The interior of $\a$ is denoted by $\bar\a$ and is always a (possibly disconnected) simply connected subset of $\ZZ$ for each $\a\in C_B$. $C_B(V)$ is the set of all $\a$ in $C_B$ such that $\bar\a \subset V$. \beginsubsection 1.2 The SOS model The {\it configuration space} of the model is $\O = \Z_+^{\ZZ}$, or $\O_V = \Z_+^V$ for some $V\subset \ZZ$. An element of $\O_V$ will usually be denoted by $\ph = \{ \ph(x), x\in V \}$. If $U\sset V \sset \ZZ$, and $\ph\in \O_V$ we denote by $\ph_U$ the restriction of $\ph$ to the set $U$. Given $V\ssset\ZZ$ and some {\it boundary condition} (b.c.) $\psi\in\O$, one defines the hamiltonian as $$ H^{J,h,\psi}_V(\ph) = \ov2\sum_\indh J(x,y) \,| \ph(x) - \ph(y) | + \sum_\indhb J(x,y) \, | \ph(x) - \psi(y) | + h \sum_\indhh \ph(x) \Eq(H) $$ and we always assume $0\le J(x,y)\le 1$ for all $x,y$. We write $J\in\d V$ if $J(x,y)<1$ only for the boundary terms, \ie if $J(x,y)=1$ unless $[x,y]^*\in\d V$. If we take $J(x,y)=0$ for all boundary terms, then we have {\it free boundary conditions} that we also denote with $$ H^{h,\emp}_V(\ph) = \ov2 \sum_\indh | \ph(x) - \ph(y) | + h \sum_\indhh \ph(x) \,. \Eq(1.20) $$ The partition function is given by $$ Z^{J,h,\psi}(V) = \sum_\indphOV \neper^{-\b H^\indJhpsi_V(\ph) } \,. \Eq(partfun) $$ Given any set $\a$ of dual edges (for instance $\a=\d V$) we define $$ |\a|_J = \sum\nolimits_{e^*=[x,y]^*\in\a} J(x,y) \,. \Eq(lenghtJ) $$ (if $J=1$ everywhere this is just the ordinary {\it length\/} of $\a$). \smallno As in \papI{} we often assume that $J$ satisfies the hypothesis $H(\L,t)$ for some $\L\ssset \ZZ$ and $t\ge 0$, namely that \smallno \item{$(i)$} $J\in\d\L$ \item{$(ii)$} $|\a|_J \ge t |\a|$ for all $\a\in C_B(\L)$ \smallno We also define $$ \D(J) = \{ \hbox{\rm $x\in \ZZ : J(x,y) \ne 1$ for some $y\in\ZZ$} \} \Eq(1.dej) $$ If $H(\L,t)$ holds for $J$ then, clearly, for each $V\sset\L$, $$ | \D(J) \cap V | \le |\de V | \le |\d V| \,. $$ When $J(x,y)=1$ for all $x,y$, we drop the superscript $J$. \smallno For $U\sset \ZZ$, let $\FF_U$ be the $\s$--algebra generated by the collection of sets $$ \{ \ph\in\O : \ph(x)=n \}_{x\in U, n\in\Zp} $$ and let $\FF = \FF_\ZZ$. The (finite volume) conditional Gibbs measure on $(\O, \FF)$ associated with the Hamiltonian \equ(H) is defined as $$ \mu^\indJhpsi_V(\ph) = \cases{ \bigl(Z^\indJhpsi(V)\bigr)^{-1} \neper^{-\b H^\indJhpsi_V(\ph)} & if $\ph(x) = \psi(x)$ for all $x\in V^c$ \cr \vphantom{\Bigl(} 0 & otherwise. \cr } \Eq(finvolmea) $$ We also regard $\mu^{J,h,\psi}_V$ as a measure on $\O_V$ by extending each configuration $\ph\in \O_V$ to the whole space in such a way that it agrees with the boundary conditions outside $V$. The expectation with respect to the measure \equ(finvolmea) is denoted by $\mean^\indJhpsi_V(\cdot)$. The set of probability measures \equ(finvolmea) satisfies the compatibility conditions $$ \mu^\indJhpsi_\L(\ph) = \sum_{\ph'\in\O} \mu^\indJhpsi_\L(\ph') \, \mu^{J,h,\ph'}_V(\ph) \qquad {\rm for\ all\ } V\sset \L\ssset\ZZ \Eq(DLR) $$ We refer to Section 1.2 of \papI\ for the definition of (infinite volume) Gibbs measure, the definition of increasing (decreasing) functions, and the statement of the FKG properties. \begingroup %--------------------------- \def\vv{^{\psi}_V} \def\ms{\mu\vv} \def\vw{_{V,\psi}} \beginsubsection 1.3. The dynamics and our results In the rest of this section, for readability purposes, we assume to have chosen $\b$, $h$ and $J$, so we don't write them explicitly if no confusion arises. The stochastic dynamics we want to study is defined by the Markov generator $$ (L\vv f)(\ph) = \sum_{\indhh, \, s=\pm1} c(x,\ph,s) \left[ f(\ph^{x,s}) - f(\ph) \right] \Eq(gnrt) $$ acting on $L^2(\O, d\mu_V^\psi)$, where $$ \ph^{x,s}(y) = \cases{ \ph(y) & if $y\ne x$ \cr \ph(y)+s & if $y=x$. \cr } $$ In \equ(gnrt) $\ph$ denotes a configuration on the whole lattice $\ZZ$ which, in view of \equ(finvolmea), agrees with the b.c. $\psi$ on $V^c$. In general we identify $L^2(\O, d\mu_V^\psi)$ with $L^2(\O_V, d\mu_V^\psi)$. The nonnegative real quantities $$ c(x,\ph,s)\quad x\in\ZZ\,,\quad \ph\in\O\,, \quad s=\pm1 $$ are the {\it transition rates\/} for the process. We set $c(x,\ph,s)=0$ if $\ph^{x,s} \notin \O$, that is if $\ph(x) = 1$ and $s=-1$. The assumptions on the transition rates are \item{$(H_1)$} {\it Tranlation invariance.} If there exists $y$ such that $\pht(x) = \ph(x+y)$ for all $x\in\ZZ$, then $c(x+y,\ph,s) = c(x,\pht,s)$ for all $x\in\ZZ$, $s=\pm1$ \item{$(H_2)$} {\it Nearest neighbor interactions} If $\ph(y)=\pht(y)$ for all $y$ such that $d(x,y)\le 1$, then $c(x,\ph,s)=c(x,\pht,s)$ \item{$(H_3)$} {\it Attractivity.} If $\ph\le\pht$ and $\ph(x)=\pht(x)$, then $$ \eqalignno{ & c (x,\ph,+) \le c(x,\pht,+) \cr & c (x,\ph,-) \ge c(x,\pht,-) & \eq(attr) \cr } $$ \item{$(H_4)$} {\it Detailed balance.} For all $x$, $\ph$, $s$ such that $\ph^{x,s}\in\O$ $$ \exp\left[ -\b H^{\ph}_\xx(\ph(x)) \right] c(x,\ph,s) = \exp\left[ -\b H^{\ph}_\xx(\ph(x)+s) \right] c(x,\ph^{x,s},-s), \Eq(dbal) $$ \item{$(H_5)$} {\it Positivity and boundedness.} There exist $c_m(\b,h)>0$ and $c_M(\b,h) < \infty$ such that, $$ \eqalign{ & \inf_{\ph\in\O,\, s=\pm1 : \,\ph^{x,s} \in \O} c(x,\ph,s) \ge c_m(\b,h) \cr & \sup_{\ph\in\O} \sum_{s=\pm1} c(x,\ph,s) \le c_M(\b,h) < \infty \cr } \Eq(bounded) $$ \smallno Two cases one may want to keep in mind are ($\c$ is the characteristic function) $$ c(x,\ph,s) = \min\bigl\{ \nep{ -\b \D_{x,s} H(\ph) } \,,\, 1 \bigr\} \, \c \{ \ph^{x,s}\in\O \} $$ and $$ c(x,\ph,s) = \left[ 1 + \exp\bigl( \b \D_{x,s} H(\ph) \bigr) \right]^{-1} \c \{ \ph^{x,s}\in\O \} $$ where $$ \D_{x,s} H(\ph) = H^\ph_\xx(\ph(x)+s) - H^\ph_\xx(\ph(x)) \,. $$ $(H_1)-(H_5)$ garantee that there exists a unique Markov process with semigroup $T\vv(t)$ and generator $L\vv$. $L\vv$ is a bounded operator on $L^2(\O,d\mu\vv)$. The process has a unique invariant measure given by $\mu\vv$. Moreover $\mu\vv$ is {\it reversible} with respect to the process, \ie $L\vv$ is self--adjoint on $L^2(\O,d\mu\vv)$. Given $\ph\in\O$ we denote by $\ph_t$ the random configuration at time $t$ evolving according to the process, so that $$ \EE_V^\ph f(\ph_t) = \int f(\ph_t) \,d\PP_V^\ph = (T\vv(t)) \,f(\ph) \,,\quad\forall\ph\in\O {\rm\ such\ that\ } \ph_{V^c}=\psi_{V^c} \,. $$ $\EE^\ph$ and $\PP^\ph$ stand respectively for the expectation and the probability measure associated with the process starting from $\ph_V$ at time zero and subject to b.c. $\ph_{V^c}$. \smallno Attractivity assumption implies (see for instance \ref[L]) \smallno \item{(1)} If $f$ is an increasing function on $\O_V$ then $T\vv(t) f$ is also increasing for all $t\ge 0$ \item{(2)} If $\r_1$, $\r_2$ are two probability measures on $(\O_V,\FF_V)$ such that $\r_1 \le \r_2$ then \acapo $\r_1 T\vv(t) \le \r_2 T\vv(t)$ for all $t\ge 0$ \item{(3)} For any $\ph$, $\ph'$ in $\O$ such that $\ph\le\ph'$, the {\it standard coupling} \ref[L] $\PP_{V}^{\ph,\ph'}$ of $\ph_t$, $\ph'_t$ is such that $\PP_{V}^{\ph,\ph'} \{ \ph_t \le \ph'_t \} = 1$, for all $t\ge0$. \medno This last property allows us to define a {\it standard\/} coupling of two Gibbs measures which preserve the order of the b.c. Take in fact $\nu^{\psi,\psi'}_V$ as the unique invariant measure of the (standard) coupled process $(\ph_t, \ph'_t)$. Then we have \smallno \item{(1)} $\nu^{\psi,\psi'}_V \{ (\ph,\ph') : \ph = \ph_0 \} = \mu\vv(\ph_0)$ for all $\ph_0\in\O_V$ \item{(2)} $\nu^{\psi,\psi'}_V \{ (\ph,\ph') : \ph' = \ph_0 \} = \mu^{\psi'}_V(\ph_0)$ for all $\ph_0\in\O_V$ \item{(3)} If $\psi\le\psi'$, then $\nu^{\psi,\psi'}_V \{ (\ph,\ph') : \ph\le\ph' \} = 1$ \smallno A fundamental quantity associated with the dynamics of a reversible system is the gap of the generator, \ie $$ \gap^{\psi}(V) = \gap( L\vv ) = \inf {\rm spec}\, (- L\vv \restriction \identity^\perp ) $$ where $\identity^\perp$ is the subspace of $L^2(\O, d\mu\vv)$ orthogonal to the constant functions. The gap can be also characterized as $$ \gap^{\psi}(V) = \inf_{\st f\in L^2(\O, d\mu^{\psi}_V)} {\Dir^{\psi}_V (f,f) \over \Var^{\psi}_V(f) } \Eq(gap) $$ where $\Dir$ is the Dirichlet form associated with the generator $L$, $$ \Dir^{\psi}_V (f,f) = {1\over 2} \sum_{\ph\in \O} \sum_{x\in V,\, s=\pm1} \mu^{\psi}_V(\ph) \, c(x,\ph,s) \left[ f(\ph^{x,s}) - f(\ph) \right]^2 \Eq(var) $$ and $\Var^{h,\psi}_V$ is the variance relative to the probability measure $\mu^{h,\psi}_V$. \smallno Then main result in this paper is \nproclaim Theorem [1.2]. In the same setting as in Theorem \teo[CM] of the Introduction, if the transition rates satisfy $(H_1)$--$(H_5)$, then we have for all $k=1, \cdots, \kmax$ \smallno \item{$(i)$} if $h^*_{k}(\b) < h < h^*_{k-1}(\b)$ then there exist $L_0(\b,h)$, $\k(\b,h)>0$ such that $$ \inf_{L\ge L_0} \inf_{\psi\in\O} \gap^{h,\psi}(Q_L) \ge \k(\b,h) $$ \item{$(ii)$} if $h=h^*_k(\b)$ there exists a positive constant $C_1(\b,h)$ such that for all $N>10/h$ $$ C_1(\b,h) \, \nep{ -100 \b k N } \le \gap^{h,\emptyset}(Q_N) \le c_M(\b,h) \nep{ -{1\over40} \b N } $$ where $c_M$ is given by \equ(bounded). \endgroup %----------------------------------------------------------- \fine \expandafter\ifx\csname sezioniseparate\endcsname\relax% \input macro \fi \begingroup \def\Hbi{\bar H^{h,\infty}_V(\ph)} \def\Z{\integer} \def\ZpN{\Z_+^N} %---------------------------------------------------------- \numsec=2 \numfor=1 \numtheo=1 \pgn=1 \beginsection 2. Lower bound on the gap for $h\ne\hkst$ In this section we will show that, if $h^*_{k}(\b) < h < h^*_{k-1}(\b)$, then the finite volume Gibbs measures show a weak dependence (in a strong sense) on the boundary conditions. As a consequence, we obtain a lower bound for the gap of the dynamics in a finite, large enough square $Q_N$. This lower bound is independent of the boundary conditions and of $N$. Thus we get part $(i)$ of Theorem \thm[1.2]. We will adapt to our model the ideas and techniques developed in \ref[MO1], \ref[MOS], supplemented by results of Section 3 of \papI\ and by Proposition \teo[7.1] of \papI. As in \papI\ we often need a small constant, so we set $$ \z = 1000^{-1} \,. $$ \beginsubsection 2.1. Strong mixing Following the basic result of \ref[MOS], we strengthen the result of Proposition \teo[7.1] of \papI. \nproclaim Theorem [6.2a] {\it (Strong Mixing)}. Let $\b$ be large enough and $h\in(\hkst,\hkmst)$, with $1\le k\le k_{max}= \kmx$. Then there exists $L_0(\b, h)$, $C(\b,h)$ and $m(\b,h)>0$ such that, for any $L\geq L_0$, any $\Delta\sset Q_L$ and any $y\in \de^+ Q_L$, we have $$ \sup_{\st \psi ,\psi'\in\O\atop \st \psi'(x)= \psi(x)\,,\,\forall x\neq y} \sup_{A\in \FF_{\Delta}} | \mu_{Q_L}^{h,\psi}(A) - \mu_{Q_L}^{h,\psi '}(A)| \le C \sum_{x\in \Delta}\nep{-m \, d(x,y) } $$ \noindent {\it Remark.} Following \ref[MO1], we will say that {\it SM(L,C,m)} holds if the above inequality with the prescribed constants is valid for a square $Q_L$. \medno \Pro\ If the random variables were bounded then the result would simply follow from Proposition \teo[7.1] of \papI, thanks to the main theorem of \ref[MOS] which states that, for discrete bounded spin systems with finite range interaction, weak mixing implies strong mixing in two dimensions. Thus we have to take care of the unboundedness of the ${\ph(x)}$'s . If one examines closely the arguments of \ref[MOS] one finds out that the two basic ingredients (besides the dimensionality and the finite range condition) are the following \smallno \item{$(i)$} Weak mixing in the form stated in Proposition \teo[7.1] of \papI{} \item{$(ii)$} let $C$, $m$ be the ``weak mixing'' constants appearing in Proposition \teo[7.1] of \papI, and let $l_0(\b,h)$ be such that $$ C\sum_{j = l_0}^\infty \nep{-m j } \le {1\over 2} \,. $$ Then there exists $\bar N(\b,h) \ge l_0$ such that $$ \inf_{\psi\in\O} \mu_{Q_{\bar N}}^\psi \{\, \ph(x)=\bar\ph(x) \; \forall x\in Q_{\bar N} \,\} >0 \quad {\rm for\ some\ }\phb\in\O \Eq(6.5.1) $$ \smallno While for bounded spins with finite range interaction, \equ(6.5.1) is a trivial consequence of the boundedness of the interaction, in our case \equ(6.5.1) follows from part $(ii)$ of Proposition \teo[3.2] of \papI, if one chooses the reference configuration $\bar \ph$ identically equal to $+1$, and $\bar N = l_0 \mmax \inte{8/h+1}$. In fact from FKG inequality one gets $$ \eqalign{\inf_{\psi\in\O} \mu_{Q_{\bar N}}^\psi\{ \, \ph(x)=+1 \; \forall \, x\in Q_{\bar N} \,\} &\geq \prod_{x\in Q_{\bar N}} \inf _{\psi\in\O} \mu_{Q_{\bar N}}^\psi\{ \ph(x)=+1 \}\geq b_2(\b,h)^{|Q_{\bar N}|} \cr} $$ Thus $(i)$ and $(ii)$ hold also in our case and the theorem follows. \QED \beginsubsection 2.2. Proof of part $(i)$ of Theorem \thm[1.2] We are finally in a position to prove a lower bound for the gap of the dynamics defined in Section 1.3 in any finite, large enough square $Q_N$, independent of the boundary condition and of $N$. We follow very closely Section 4 of \ref[MO1]. Given an even integer $L_0= 2 K_0$ and a cube $Q_N$, $N = n L_0$, we consider its covering by squares $Q^i_{L_0}$ such that if two different squares $Q^i_{L_0}$ and $Q^j_{L_0}$ overlap than necessarily each one of them is the translated by $K_0$ of the other along at least one of the two coordinate axes. Let $$ \calD_n = \{ Q_{L_0}^i \} $$ and consider the block--dynamics defined in the Appendix, determined by $\b,h$, the b.c. $\psi$ and the collection $\calD_n$, through the Markov generator $L_{\calD_n}^{h,\psi}$ given by \eqv(aa3). The same proof of Theorem \teo[4.1] of \ref[MO1] shows that, if $SM(L_0,C,m)$ holds and moreover $L_0$ is so large that $$ C L_0^2\exp (-m\sqrt{L_0} )\leq {1\over L_0^4} \Eq(6.40) $$ then the gap of $L_{\calD_n}^{h,\psi}$ is bounded away from zero uniformly in $\psi$ and in $N = n L_0$. Thus if we choose $L_0 = L_0(\b,h)$ large enough so that both Theorem \thm[6.2a] and \equ(6.40) are satisfied we get $$ \inf_{\psi\in\O} \inf_{n\ge 1} \gap( L_{\calD_n}^{h,\psi} ) \ge \k_1(\b,h) > 0 \,. \Eq(6.20) $$ In Proposition \thf[6.3b], on the other side, we will show that, considering the usual single site dynamics on each block $Q^i_{L_0}$, if $L_0 > \inte{8/h+1}$, we have $$ \inf_{\psi\in\O} \gap (L_{Q_{L_0}}^{h,\psi}) \ge \k_2(\b,h) >0 \,. \Eq(6.7) $$ Finally, Proposition \thf[aa1] gives an estimate the for gap of the single site dynamics in the full volume $Q_N$, in terms of the quantities appearing in \equ(6.20), \equ(6.7). Thus, assuming to have chosen $L_0$ such that $L_0 \ge \inte{8/h+1}$ also holds, we get $$ \inf_{N = n L_0, \, n\in\Zp } \inf_{\psi\in\O} \gap (L_{Q_{N}}^{h,\psi}) \ge \ov4 \k_1 \, \k_2 >0 \,. $$ The factor $\ov4$ is the \oqq overlapping'' factor appearing as the last term in \eqv(aa0). The extension of the theorem to all $N \ge L_0$ is straightforward after one realizes that Theorem \teo[4.1] of \ref[MO1] is valid for any coverings $\{Q_{L_0}^i\}$ of $Q_N$ such that the following holds: for each site $x \in Q_N$ there is a square $Q^i_{L_0}$ $$ d(x, (Q^i_{L_0})^c \setm (Q_N)^c ) \ge \ov{10} L_0 \,. $$ In this way we can ``cover'' any square $Q_N$ with $N$ greater than say $10 L_0$. \QED \smallno We are thus left with the proof of \equ(6.7). \nproclaim Proposition [6.3b]. %------------------------------------- For each $\b,h>0$, $N\ge N_1(h) = \inte{8/h +1}$, we have (remember that we always assume $0\le J(x,y)\le 1$) $$ g_N(\b,h) \equiv \inf_{J\in\d Q_N} \inf_{\psi\in\O} \gap (L_{Q_{N}}^{J,h,\psi}) > 0 \,. $$ \Pro\ We recall that $J\in \d Q_N$ means (see Section 1.2) that $J=1$ everywhere with the possible exception of the boundary terms. Let then $\L=Q_N$. As we did in Section 3 of \papI\ we define $$ \bar H^{h,\infty}_V(\ph) = \ov2 \sum_\indh | \ph(x) - \ph(y) | - \sum_\indhb \ph(x) + h \sum_\indhh \ph(x) \Eq(hinfin) $$ which satisfies $$ (n-1) \,|\deV|_J + H^{J,h,1}_V(\ph) \ge H^{J,h,n}_V(\ph) \ge (n-1) \,|\deV|_J + \bar H^{h,\infty}_V(\ph) \Eq(Hb2) $$ For $X\sset\O_\L$ we define $\de X$ as the set of all $\ph\in X$ such that there exists $\ph'\in X^c$ with $$ \sum_{x\in\L} |\ph(x) - \ph'(x)| = 1. $$ The proof is based on the so called Cheeger inequality given in \ref[LS], which in our case says $$ \gap (L_{\L}^{J,h,\psi}) \ge {q^2 \over 8 M } $$ where, letting $\mu=\mu\JhpL$, $$ \eqalign{ & q = \inf_{\st X\sset\O_\L \atop\st0< \mu(X)\le \ov2} {1\over \mu(X) \, \mu(X^c) } \, \sum_{\ph\in X} \mu(\ph) \sum_{(x,s) : \, \ph^{x,s} \in X^c } c(x,\ph,s) \ge \cr & \ge \inf_{\st X\sset\O_\L \atop\st0< \mu(X)\le \ov2} c_m(\b,h) {\mu(\de X) \over \mu(X) } \,. \cr } $$ and $$ M = \sup_{\ph\in\O_\L} \sum_{x\in\L} \sum_{s=\pm1} c(x,\ph,s) \le |\L| c_M(\b,h) $$ Thanks the boundedness of the transition rates \equ(bounded), the proposition is thus proven if we can show that $$ \inf_{J\in\d V} \inf_{\psi\in\O} \ \inf_{\st X\sset\O_\L \atop\st0< \mu\JhpL(X)\le \ov2} { \mu^{J,h,\psi}_\L( \de X )\over \mu^{J,h,\psi}_\L(X) } > 0 \,. \Eq(XdeX) $$ We denote by $\ph_1$ the configuration on $\L$ which is equal to $1$ everywhere, and we consider two cases depending on whether $\ph_1$ belongs to $X$ or not. \medno \bu $\ph_1 \notin X$ \smallno For each $\ph\in X$ we set $$ \eqalign{ & m(\ph) = \inf \{ i : \ph \mmin i \in X \} \cr & \ph' = \ph \mmin m(\ph) \qquad {\rm and} \qquad \ph'' = \ph\mmin (m(\ph)-1) \cr } $$ So that $\ph'\in X$ and $\ph''\in X^c$. If we define $T \ph = \ph'$, then we claim that $$ { \mu\JhpL(X) \over \mu\JhpL(TX) } \le \sum_{\ph\in\Ot_\L} \nep{ -\b \HbiL} \quad \forall \psi\in\O, \Eq(T) $$ where $\Ot_\L = \{ 0, 1,2,\ldots \}^\L$. To prove \equ(T) one has to observe that, since $\ph(x)\ge\ph(y)$ implies $\ph'(x)\ge\ph'(y)$, then $$ \eqalign{ & |\ph(x) - \ph(y)| = |\ph'(x)-\ph'(y)| + |(\ph-\ph')(x) - (\ph-\ph')(y) | \cr & J(x,y) |\ph(x)-\psi(y)| \ge -(\ph-\ph')(x) + J(x,y) |\ph'(x)-\psi(y)| \,. \cr } $$ This implies $$ H^{J,h,\psi}_\L(\ph) \ge H^{J,h,\psi}_\L(\ph') + \bar H^{h,\infty}_\L(\ph-\ph') \,. \Eq(HH) $$ So, if we let $Z=Z^{J,h,\psi}(\L)$, we have $$ \eqalignno{ & \mu\JhpL(X) = Z^{-1} \sum_{\ph\in\O_\L} \nep{ -\b \HJpL(\ph) } = Z^{-1} \sum_{\pht\in TX} \sum_{\ph\in\O_\L: \, T\ph = \pht} \nep{ -\b \HJpL(\ph) } \le \cr & \le Z^{-1} \sum_{\pht\in TX} \nep{ -\b \HJpL(\pht) } \sum_{\ph\in\O_\L: \, T\ph = \pht} \nep{ -\b \HiL(\ph-\pht) } \le \cr & \le \mu\JhpL(TX) \sum_{\ph\in\Ot_\L} \nep{ -\b \HbiL}. &\eq(XTX) \cr } $$ To complete the proof we are going to use the following facts \medno \item{$(a)$} For each $X$, for each $\pht\in TX$, there exists $\phb\in\de X$ such that $\pht-1\le\phb\le\pht$, and \item{$(b)$} $| \HJpL(\pht) - \HJpL(\phb)| \le 5N^2$ \medno Statement $(a)$ is a trivial consequence of the definition of $T \ph$, while $(b)$ follows easily from $(a)$. Given $(a)$ and $(b)$, we can proceed as follows $$ \eqalign{ & \mu\JhpL(TX) \le Z^{-1} \nep{ 5\b N^2 } \sum_{\pht\in TX} \nep{ -\b \HJpL(\phb) } = \cr & = Z^{-1} \nep{ 5\b N^2 } \sum_{\phb\in\de X} \# \{ \pht\in TX : \pht \to \phb \} \, \nep{ -\b \HJpL(\phb) } \le \nep{ 5\b N^2 } 2^{N^2} \mu\JhpL(\de X) \cr} \Eq(TX) $$ The proof of \equ(XdeX) when $\ph_1\notin X$, follows then from \equ(XTX), \equ(TX) and Proposition \teo[3.1] of \papI{} (the fact that $\O_\L$ should be replaced by $\Ot_\L$ is not essential). \medno \bu $\ph_1 \in X$. \smallno The key ingredient is the following observation \medno \item{$(c)$} there exists $s(\b,h,N)$ such that for all $X\in\O_\L$ with $\mu\JhpL(X)\le 1/2$ for some $\psi\in\O$, there exists $\ph\in X^c$ with $\sup_x \ph(x) \le s(\b,h,N)$. \medno {\it Proof of $(c)$.} By Proposition \teo[3.2] of \papI{} and Chebyshev inequality, we have $$ \mu\JhpL \{ \sup_x \ph(x) > r \} \le \sum_{x\in\L} \mu\JhpL \{ \ph(x) > r \} \le {1\over r} \, |\L|\, b_1(\b,h,1). $$ So, by taking $s(\b,h,N) = [ 4\, |\L|\, b_1(\b,h,1) ]^{-1}$, we get $$ \mu\JhpL \{ \sup_x \ph(x) \le s(\b,h,N) \} \ge {3\over4} $$ which implies $(c)$. \medno Choose now some $\pht\in X^c$ such that $\sup_x \pht(x) \le s$. Since $\ph_1\in X$, there exists $\phb\in\de X$ such that $\ph_1\le \phb\le\pht$. Hence, using the trivial bounds $$ H^{J,h,1}_\L(\ph) \le 5 |\L| \sup_x \ph(x) $$ we obtain (remember \equ(Hb2)), $$ \eqalign{ & \mu\JhpL(\de X) \ge \mu\JhpL(\phb) = (Z^{J,h,\psi}(\L))^{-1} \nep{ -\b \HJpL(\phb) } \ge \cr & \nep{ -\b H^{J,h,1}_\L(\phb) } \Bigl[ \sum_{\ph\in\O_\L} \nep{ -\b \HbiL} \Bigr]^{-1} \ge \Bigl[ \sum_{\ph\in\O_\L} \nep{ -\b \HbiL} \Bigr]^{-1} \nep{ -5 \b |\L| s(\b,h,N) } \,.\cr } $$ This complete the proof of \equ(XdeX) and Proposition \thm[6.3b]. \QED \endgroup \fine \expandafter\ifx\csname sezioniseparate\endcsname\relax% \input macro \fi %---------------------------------------------------------- \numsec=3 \numfor=1 \numtheo=1 \pgn=1 \beginsection 3. Upper bound on the gap at $\hkst$ In this section we prove the upper bound in part $(ii)$ of Theorem \thm[1.2]. More precisely, we have \nproclaim Theorem [8.5]. Let $\b$ be large enough and let $k\in\Zp$ such that $1\le k\le k_{max}=\kmx$. If $N>10/ \hkst$, then $$ \gap \bigl( L^{\hkst,\emp}_{Q_N} \bigr) \le c_M(\b,\hkst) \, \nep{ -{1\over40} \b N } $$ where $c_M(\b,h)$ is defined in \equ(bounded). \smallno We set for simplicity $h^* = \hkst$ and $\L = Q_N$. In order to find an upper bound on $\gap^{\hst,\emp}(\L)$ we are going to use the \oqq look for the bottleneck'' approach, \ie we take advantage of the variational characterization for the gap \equ(gap) and choose an appropriate test function $f_0$ which illustrates how the system, in order to relax to equilibrium, has to make an excursion to a very unlikely region. \smallno Given $\ph\in\O$ we set $$ \s(x) = \sign ( \ph(x) - k - 1/2 ) $$ and, for $U\ssset \ZZ$ $$ \eqalign{ M_U(\s) &= M_U(\s(\ph)) = \sum\nolimits_{x\in U} \s(x) \cr m_U(\s) &= m_U(\s(\ph)) = |U|^{-1} M_U(\s) \,.\cr } $$ In analogy with the solution of the similar problem for the 2D Ising model in the phase coexistence region, we take as a test function $$ f_0(\ph) = \chi \{ M_\L(\ph) > 0 \} - \chi \{ M_\L(\ph) < 0 \} \,. $$ Using \equ(bounded), we find $$ \Dir^{\hst,\emp}_\L(f_0,f_0) \le \mu\vv\{ |M_\L(\ph)| \le 2 \} \, \sup_{\ph\in\O} \sum_{s=\pm} c\vv(x,\ph,s) \le c_M(\b,h) \, \mu\vv\{ |M_\L(\ph)| \le 2 \}, \Eq(dir) $$ while $$ \Var\vv(f_0) = \mu\vv\{ M_\L(\ph) \ne 0 \} - \bigl(\mu\vv\{ M_\L(\ph) > 0 \} - \mu\vv\{ M_\L(\ph) < 0 \} \bigr)^2 \Eq(7.var) $$ The proof of Theorem \thm[8.5] is then a consequence of \equ(gap), \equ(dir), \equ(7.var) and Lemmas \thf[8.3], \thf[8.4] proven below. \beginsubsection 3.1. Bound on the Dirichlet form We want to show that, if $\b$ is large enough and $N\ge 10/\hkst$, then $$ \mu\vv\{ |M_\L(\ph)| \le 2 \} \le \nep{ - cost\, \b |\deL| }. $$ The idea is the following (see also \ref[S]). Take $\L'\sset \L$. Then either there is a \oqq contour" separating the phase $\s=1$ from the phase $\s=-1$ or there is a circuit surrounding $\L'$ where $\s$ is either all plus or all minus. So we need to prove that \smallno \item{$\bullet$} the probabilty of having a large contour of length $l$ with free b.c. goes like \acapo $\exp( - {\rm cost}\, \b\, l)$ \item{$\bullet$} the probability of having $|M_{\L'}| \le 2$ conditioned to the existence of a $(+)$ circuit or a $(-)$ circuit surrounding $\L'$ goes like $\exp( - {\rm cost}\, \b |\d \L'|)$ \medno Before stating the next Lemma we have introduce another definition. Given $\s\in\confV$ let $\s_+\in\conf$ be defined by $$ \s_+(x) = \cases{ \s(x) & if $x\in V$\cr +1 & if $x\in V^c$\cr } $$ The set $\{ e^* : e=[x,y], \ \s_+(x)\ne\s_+(y) \}$ can be written as a union of its connected components $\a_1, \ldots, \a_r$ where $\a_i \in C_B(V)$. A closed set of dual edges $\a$ is called a {\it contour for} $\s$ if $\a=\a_i$, for some $i$. \begingroup \def\muu{\mu\vv} \nproclaim Lemma [8.2]. Let $\b$ be large enough, $1\le k \le\kmax$, $\L=Q_N$. For each $\a\in C_B(\L)$, let $\a_0 = \a \setm \deL$. Then, if $\a\cap\deL\ne\emp$ and $|\a_0|>N/100$, we have $$ \mu^{\hkst,\emp}_\L\{ \ph\in\O_\L : \a \hbox{\rm\ is a contour for } \s(\ph) \} \le \nep{- 0.99 \b |\a_0| } \,. $$ \Pro\ Let $\hst=\hkst$. Given $\a\in C_B(\L)$ we define the event $$ X_\a = \{ \ph\in\O_\L : \hbox{\rm $\a$ is a contour for $\ph$} \}. $$ We also let $\L_1=\abar$, $\L_2=\L\setm\abar$ and $$ \eqalign{ & \D_1 = \{ x\in\L_1 : d(x,\L_2) = 1 \} \cr & \D_2 = \{ x\in\L_2 : d(x,\L_1) = 1 \}. \cr } $$ We observe that both $\L_1$ and $\L_2$ are simply connected. In fact $\L_1$ is the interior of a closed connected set of dual edges, and $\L_2$ is is the interior of $(\a\cup\deL) \setm (\a\cap\deL)$ which is again a closed connected set of dual edges because $\a$ touches $\deL$. So we have $$ X_\a \subset X_\a^1 \cup X_\a^2, $$ where $$ \eqalign{ & X_\a^1 = \{ \ph\in\O_\L : \s(x) = -1\ \forall x\in\D_1, {\rm\ and\ } \s(x) = +1\ \forall x\in\D_2 \} \cr & X_\a^2 = \{ \ph\in\O_\L : \s(x) = +1\ \forall x\in\D_1, {\rm\ and\ } \s(x) = -1\ \forall x\in\D_2 \} \cr } $$ In the following we are going to show how to estimate $\muu(X_\a^1)$. $\muu(X_\a^2)$ can be dealt with in the same way. Choose now the couplings $J(e)$ which correspond to free b.c. on the boundary of $\L$, \ie $$ J(x,y) = \cases{ 1 & if $[x,y]^*\notin\deL$ \cr 0 & if $[x,y]^*\in\deL$ \cr } \Eq(Jfree) $$ In this way, since, for $\ph\in X_\a^1$, $$ H\vv(\ph) = H^{J,\hst,k}_{\L_1}(\ph) + H^{J,\hst,k+1}_{\L_2}(\ph) + |\a_0|, $$ we can write $$ \muu(X_\a^1) = {\nep{ -\b |\a_0| }\, Z^{J,\hst,k}(\L_1)\, Z^{J,\hst,k+1}(\L_2) \over Z^{\hst,\emp}(\L) } \,. \Eq(X1) $$ At this point our strategy depends on whether $|\a\cap\deL|$ is greater than or less than ${1\over2}|\deL|$. In the first case we estimate the denominator as $$ Z^{\hst,\emp}(\L) \ge Z^{J,\hst,k}(\L_1) \, Z^{J,\hst,k}(\L_2) $$ while in the second case we write $$ Z^{\hst,\emp}(\L) \ge Z^{J,\hst,k+1}(\L_1) \, Z^{J,\hst,k+1}(\L_2) \,. $$ We treat only the first case because the second is identical. So, assuming $|\a\cap\deL|\ge {1\over2} |\deL|$, we get $$ \muu(X_\a^1) \le \nep{ -\b |\a_0| } {Z^{J,\hst,k+1}(\L_2) \over Z^{J,\hst,k}(\L_2) }. \Eq(X11) $$ What is left to complete the proof is to convince ourselves that (rememeber the definition of $H(\L,t)$ given before \equ(1.dej) and that $\z = 1000^{-1}$) \medno \item{$(*)$} \hfil {\it $H(\L_2, 10 \z)$ holds for $J$ } \hfil \medno In fact, assuming $(*)$, we apply Corollary \teo[4.2] of \papI{} to estimate the above quotient and obtain $$ \muu(X_\a^1) \le \exp\left[ -\b|\a_0| + |\deL_2| \nep{ -\ov4 \b\z} \right] \le \nep{ -\b|\a_0| + |\a_0| + |\deL| } \le \nep{ |\a_0| ( -\b + 401 ) } $$ where we have used $|\a_0| \ge N/100 $. \medno {\it Proof of $(*)$.} Let $\h\in C_B(\L_2)$. We are going to show that $|\h|_J \ge {1\over3} |\h|$, which is more than enough. We recall that with our choice of $J$, $|\h|_J = |\h\setm\deL|$ and that $\d \L_2 = \a_0 \cup (\deL\setm\a)$, so that $$ |\h\cap\deL| \le | \d \L_2 \cap \deL | = | \deL\setm\a| \le {1\over2} |\deL| \Eq(eta1) $$ because of what we assumed right before \equ(X11). We consider three cases \smallno \item{(1)} If $\h$ does not touch any two opposite sides of $\L$ then $|\h| \ge 2 |\h\cap\deL|$, so \acapo $|\h|_J = |\h| - |\h\cap\deL| \ge {1\over2} |\h|$ \item{(2)} If $\h$ touches exactly 3 sides of $\L$, then, since it is closed, $|\h\setm\deL|\ge{1\over4}|\deL|$, so, by \equ(eta1), $|\h\cap\deL| \le 2 |\h\setm\deL|$ which implies $|\h|_J \ge {1\over3} |\h|$ \item{(3)} If $\h$ touches all 4 sides of $\L$, then $|\h|\ge|\deL|$, so $|\h|_J \ge {1\over2} |\h|$. \smallno This proves $(*)$. \medno Thus we have obtained $$ \muu(X_\a) \le 2 \nep{ |\a_0| ( -\b + 401 ) } \le \nep{ - 0.99 \b |\a_0| } \QED $$ \endgroup \begingroup %---------------------------------------------- \def\LL{{\L'}} \def\ms{\mu\vv} \def\prm#1{^{J,\hst,k}_{\L_#1}} \def\mm#1{\mu\prm#1} \def\Zx{Z^{\hst,\emp}} \def\nume{{3\over8}} \noindent We can now state our first key estimate \nproclaim Lemma [8.3]. Let $\b$ be large enough, $1\le k\le\kmax$ and $\L=Q_N$. Then, if $N\ge 1000$, $$ \mu^{\hkst,\emp}_\L \{ \ph\in\O_\L : |M_\L(\ph)|\le 2 \} \le \nep{ - \ov{12} \b N } $$ \Pro\ We borrow the basic idea from \ref[S]. \smallno Let $\hst=\hkst$. Consider a square $\L'= Q_{N'}$ of side $N' = N - 2 \inte{N/20}$. In this way $d(\d \L, \d \L') = \inte{N/20}$. It is then easy to check that $$ \ms \{ |M_\L(\ph)|\le2 \} \le \ms\{ |m_\LL(\ph)| \le \nume \}. \Eq(7.1) $$ Following \ref[S] we consider the events $$ Y^\pm = \left\{ \ph\in\O_\L : \vcenter{ \hbox{\rm there exists a connected set $R\sset\L\setm\LL$, surrounding $\LL$} \hbox{such that $\s(x) = \pm1$ for all $x\in R$} } \right\}. $$ and, letting $Y=Y^+\cup Y^-$, we write $$ \ms\{ |m_\LL(\ph)| \le \nume \} \le \ms\{ m_\LL(\ph) \le \nume \tc Y^+ \} + \ms\{ m_\LL(\ph) \ge -\nume \tc Y^- \} + \ms( Y^c ). \Eq(tre) $$ The idea is to show that the first two terms are small because of Corollary \teo[4.3] of \papI, while the last term is small because of Lemma \thm[8.2]. The first two terms can be taken care of in the same way, so we will deal only with the first. Consider $\L'$ as a disjoint union of $N'$ horizontal strips $S_j$, each strip being a $N' \times 1$ rectangle. Accordingly write $$ \L' = S_1 \cup S_2 \cup\cdots\cup S_{N'}. $$ By conditioning on the ``most external'' $R$ which surrounds $\L'$, and using the FKG inequality and the Markov property as we did in Proposition \teo[A1.1] of \papI{}, we find $$ \ms\{ m_\LL(\ph) \le \nume \tc Y^+ \} \le \sup_{\st V :\, \L'\sset V \sset \L \atop {V \rm\ connected\ and \atop \rm simply\ connected } } \mu_V^{\hst,k+1} \{ m_\LL(\ph) \le \nume \} \,. $$ Applying Corollary \teo[4.3] of \papI{} to each $S_j$ we get $$ \eqalignno{ & \ms\{ m_\LL(\ph) \le \nume \tc Y^+ \} \le \sup_{\st V :\, \L'\sset V \sset \L \atop {V \rm\ connected\ and \atop \rm simply\ connected } } N' \sup_{1\le j\le N'} \mu_V^{\hst,k+1} \{ m_{S_j}(\ph) \le \nume \} \le \cr & \le N' \nep{ -{4\over 9} \b {5\over8} N' } \le N \nep{ - {1\over 4} \b N } \le \nep{ -{1\over5} \b N } & \eq(uno) \cr } $$ We now complete the proof with an estimate of the last term in \equ(tre). The basic observation here is that if $\ph\in Y^c$, then there is a contour $\a$ for $\ph$ touching both $\deL$ and $\d \L'$, which implies $$ |\a_0| = |\a\setm\deL| \ge 2d(\deL,\d \L') \ge 2\inte{N/20} \,. $$ It is easy to check that for any dual edge $e^*$ $$ \# \{ \a\in C_B(\L) : \a\ni e^*,\, |\a_0| = l \} \le K^{l+4N +1} $$ where $K^l$ is an upper bound for the number of $\a$ of length $l$ containing a fixed edge. Then, by Lemma \thm[8.2], setting $l_0 = 2\inte{N/20}$, $$ \eqalign{ & \ms(Y^c) \le \sum_{\st \a\in C_B(\L) \atop \st \a\cap\deL\ne\emp ,\,\st |\a_0|\ge l_0 } \ms \{ \hbox{\rm $\a$ is a contour for $\ph$ } \} \le \cr & \sum_{e^*\in\deL} \sum_{\st \a\in C_B(\L) \atop \st \a\ni e^* ,\, |\a_0|\ge l_0 } \ms \{ \hbox{\rm $\a$ is a contour for $\ph$ } \} \le \cr & \le |\d\L| \sum_{l=l_0}^\infty K^{l+4N+1} \nep{ -0.99 \b l} \le 2 K^{l_0+4N+1} \nep{ - 0.99 \b l_0 } \,. \cr } $$ Together with \equ(7.1), \equ(tre) and \equ(uno) this proves the theorem (if $N\ge 1000$ then $l_0 \ge \ov{11} N$). \QED \beginsubsection 3.2. Bound on the variance of $f_0$ In order to find a lower bound on the variance of $f_0$, we prove \nproclaim Lemma [8.4]. Let $\b$ be large enough, $1\le k\le k_{max}$. Take $\L = Q_N$ with $N>10/ \hkst$. Then $$ \eqalign{ & \mu^{\hkst,\emp}_\L \{ \ph\in\O_\L : M_\L(\s(\ph)) < 0 \} \ge \nep{ -\ov{20} \b N } \cr & \mu^{\hkst,\emp}_\L \{ \ph\in\O_\L : M_\L(\s(\ph)) > 0 \} \ge \nep{ -\ov{20} \b N } \cr } $$ \Pro\ Let $\hst=\hkst$. We assume, for simplicity that $N=2L+1$ is odd and subdivide $\L$ into two rectangles, so we consider the central horizontal row $$ R = \{ [-L, L] \cap \Z \} \times \{0\} \,. $$ Then we write $$ \L = \L_1 \cup R \cup \L_2 $$ where $\L_1$ is the $N\times L$ rectangle above $R$ and $\L_2$ is the rectangle below $R$. Consider now the events $$ \eqalign{ & X = \{ \ph\in\O_\L : \ph(x) = k, \, \forall x\in R \} \cr & Y_i = \{ \ph\in\O_\L : M_{\L_i}(\s) < 0 \} \qquad i=1,2. \cr } $$ Then $$ \ms\{ M_\L(\s) < 0 \} \ge \ms( Y_1 \cap Y_2 \tc X ) \, \ms(X) = \mm1 (Y_1) \, \mm2(Y_2) \, \ms(X) \, \Eq(min0) $$ where $J$ is taken as in \equ(Jfree). To take care of the first two terms, we observe that $H(\L_1, {1\over4})$ holds for $J$ by an argument very similar to the proof of statement $(*)$ in Lemma \thm[8.2]. So, using the row decomposition as in \equ(uno), and Corollary \teo[4.3] of \papI, we get $$ \mm1 (Y_1) = 1 - \mm1(Y_1^c) \ge 1 - L \nep{ - {1\over9} \b N } \ge {1\over 2} \Eq(Y) $$ (same for $Y_2$). \smallno Let's consider now the last term in \equ(min0). If $\ph\in X$ then $$ H\vv(\ph) = H\prm1(\ph) + H\prm2(\ph) + \hst k N $$ so $$ \ms(X) = {\sum_{\ph\in X} \nep{ - \b H\vv(\ph) } \over \Zx(\L) } = {Z^{J,\hst,k}(\L_1) \, Z^{J,\hst,k}(\L_2) \over \Zx(\L) } \, \nep{ -\b \hst k N} \,. \Eq(msx) $$ The idea is again to use cluster expansion to estimate the quotient of partition functions. For this purpose we introduce a small coupling on the boundary of $\L$ $$ \Jb(x,y) = \cases{ 1 & if $[x,y]^*\notin\deL$ \cr 10 \z & if $[x,y]^*\in\deL$\cr } \Eq(Jsmall) $$ where we have set $\z=1000^{-1}$. Jensen inequality (see Proposition \teo[4.5] of \papI) and Proposition \thf[8.6] (which is given below) yield $$ { Z^{\Jb,\hst,k}(\L) \over \Zx(\L) } \ge \exp\Bigl[ - 10\, \z\b\, |\d\L| \sup_{x\in\de\L} \mean\vv |\ph(x) - k | \Bigr] \ge \nep{ -11 \z\b |\d\L| } \,. \Eq(rZ) $$ Hence, since $H(\L_1, \ov4)$, $H(\L_2, \ov4)$ holds for $J$, and $H(\L, 10\z)$ holds for $\Jb$ for all $V\ssset\ZZ$, all three partition functions $$ Z^{J,\hst,k}(\L_1) \ ,\ Z^{J,\hst,k}(\L_2) \ , \ Z^{\Jb,\hst,k}(\L) $$ have a cluster expansion, thanks to Theorem \teo[4.1] of \papI{}. With the help of Theorem \teo[4.4] of \papI{} is not difficult to show that $$ {Z^{J,\hst,k}(\L_1) \, Z^{J,\hst,k}(\L_2) \over Z^{\Jb,\hst,k}(\L) } \ge \nep{ \b\hst k N - \e(\b) N } \, \Eq(rZ2) $$ where $\e(\b)\to0$ when $\b\to\infty$. Together with \equ(msx), \equ(rZ) this implies $$ \ms(X) \ge \nep{ - 45 \z\b N }, $$ and then, by \equ(min0), \equ(Y) $$ \ms\{ M_\L(\s) < 0 \} \ge \nep{ -50 \z\b N }. $$ The proof of the second statement is identical. \QED In the next Proposition we complete the proof, by showing how to get the second inequality in \equ(rZ). We will actually give a more general statement than the one needed. \nproclaim Proposition [8.6]. Let $\b$ be large enough, $1\le k\le k_{max}$, $\L=Q_N$, $N>10/ \hkst$. Then $$ \sup_{x\in \L} \mean^{\hkst,\emp}_\L |\ph(x) - k | \le 1.1 $$ \Pro\ Let $\hst=\hkst$ and for any positive integer $k$ let, as in \papI, $$ I_k(\b) = [ \hkm, \hkp ] \x, \quad \hkm = {4\over\b} \nep{-4\b k} \x, \quad \hkp = {1\over 4\b} \nep{-4\b (k-1)} \x, \quad h_1^+ = {1\over \b} \nep{ -{\b\over 25}J} \x. $$ By FKG we can write $$ \eqalignno{ & \ms\{ \ph(x) \ge n \} \le \mu_V^{h,\emp} \{ \ph(x) \ge n \} \qquad\forall h\in I_{k+1}(\b) &\eq(2dis) \cr & \ms\{ \ph(x) \le n \} \le \mu_V^{h,\emp} \{ \ph(x) \le n \} \qquad\forall h\in I_{k}(\b) &\eq(2disa) \cr } $$ Choose now $h\in I_{k+1}(\b)$, and a positive integer $j$. We want to show that the RHS of \equ(2dis) is less than $\nep{ -{\z\over3} \b j }$ when $n=k+1+j$. As in \papI{}, for $V\sset\L$, we define the events $$ S_+(\L,j,V) = \left\{ \ph\in\O_\L : \vcenter{\rm \hbox{ there exists a path $(x_1, \ldots, x_s)$ such that } \hbox{$x_1\in V$, $x_s\in\deltaL$ and $\ph(x_i)\ge j$ for each $i$} } \right\}. $$ If $x\in \L$ and $j>0$, trivially we have that $\ph(x)\ge k+j+1$ if and only if $\ph\in S_+(\L,k+1+j, \xx)$. Thus, by Theorem \teo[5.2] of \papI{} we get $$ \mu_V^{h,\emp} \{ \ph(x) \ge k+1+j \} \le \nep{ -\z\b j} \,. $$ If, on the contrary, $x\in\L\setm\de\L$, then we define the event $F$ as $$ F = S_+(\L,k+1+\inte{j/2}+1, \de^+\xx) $$ So $$ \mu_\L^{h,\emp} \{ \ph(x)\ge k+1+j \} \le \mu_\L^{h,\emp} ( \ph(x)\ge k+1+j \tc F^c ) + \mu_\L^{h,\emp} ( F ) \,. \Eq(7.cond) $$ By Proposition \teo[A1.1] of \papI{} one finds $$ \mu_\L^{h,\emp} ( \ph(x)\ge k+1+j \tc F^c ) \le \sup_{\st V\ni x \atop {V \rm\ connected\ and \atop \rm simply\ connected } } \mu_V^{h,k+1+\inte{j/2}} \{ \ph(x)\ge k+1+j \}. $$ Using FKG again and Proposition \teo[3.5] of \papI, and letting $m= \inte{j/2}$, we get $$ \eqalign{ & \mu_V^{h,k+1+m} \{ \ph(x)\ge k+1+j \} \le \mu_V^{h,k+1+m} ( \ph(x)\ge k+1+j \tc \ph(y) > m\ \forall y\in V ) = \cr & = \mu_V^{h,k+1} \{ \ph(x)\ge k+1+j-m \} \le \nep{ -\ov5 \b (j-m) } \le \nep{ -\ov{10} \b j } \cr } $$ On the other side Theorem \teo[5.2] of \papI{} says that $$ \mu_\L^{h,\emp} ( F ) \le \nep{ - \z \b (1+m) } \le \nep{ - \ov2 \z \b j } \,. $$ These last two inequalities, together with \equ(2dis), \equ(7.cond) give $$ \ms\{ \ph(x) \ge k+1+j \} \le \nep{ -{\z\over3} \b j } \,. $$ In a similar way one finds, for $j>0$ $$ \ms\{ \ph(x) \le k-j \} \le \nep{ -{\z\over3} \b j } \,. $$ Therefore, for all $x\in\L$, $$ \mean\vv |\ph(x) - k | \le 1 + 2 \sum_{j=1}^\infty \nep{ -{\z\over3} \b j } + \le 1.1 \QED $$ \endgroup %------------------------------ \fine \expandafter\ifx\csname sezioniseparate\endcsname\relax% \input macro \fi %---------------------------------------------------------- \numsec=4 \numfor=1 \numtheo=1 \pgn=1 \beginsection 4. Lower bound on the gap at $\hkst$ In this section we complete the proof of Theorem \thm[1.2] by proving a lower bound on the gap of the generator of the dynamics for $h=\hkst$ and arbitrary boundary conditions, which is of the same order of the {\it upper\/} bound discussed in the previous section, namely a negative exponential of the {\it boundary\/} of the square $Q_L$. \smallno As is \papI\ we define $N_0(h) = (\z^3 h)^{-1}$, where $\z = 1000^{-1}$. We prove that (remember that we always assume $0\le J(x,y)\le 1$ and that $J\in\d Q$ has been defined in Section 1.2) \nproclaim Theorem [8.1a]. If $\b$ is large enough, then, for each $1 \le k\le\kmax=\kmx$, there exists a positive constant $G(\b,\hkst)$ such that $$ \inf_{J\in\d Q} \inf_{\psi\in\O} \gap\bigl( L^{J,\hkst,\psi}_Q \bigr) \ge G(\b,\hkst)^{\log L} \, \nep{- 80\b k L } $$ for any square $Q$ of side $L \ge 20 N_0(\hkst)$. \noindent {\it Remarks.} \textindent{(1)} Free boundary conditions are recovered by simply setting $J(x,y)=1$ everywhere but $J(x,y)=0$ for all $x,y$ such that $[x,y]^*\in\d Q$. Moreover, using the fact that the gap is positive uniformly in the boundary conditions for each $L\ge \inte{8/h+1}$ (Proposition \thm[6.3b]), we can easily find a constant $C_1(\b,h)>0$ such that statement $(ii)$ of Theorem \thm[1.2] holds. \textindent{(2)} By using a more sophisticated approach which requires the so called ``surgery'' technique one can replace the factor $80 \b k$ with $80 \b$. \medno \Pro\ Let $\hst=\hkst$, $N_0 = N_0(\hst)$ and $\Nh_0 = 10 N_0$. The proof is obtained by recursion as follows. For $L\in \Zp$, let $$ g_L(\b,h) = \inf_{J\in\d Q_L} \inf_{\psi\in\O} \gap(L^{J, h,\psi }_{Q_L}) $$ Proposition \thm[6.3b] says that $g_L>0$ if $L\ge\inte{8/h+1}$. Given now $L\ge 2 \Nh_0$, it is possible to find positive integers $s$, $\{ L_i \}_{i=1}^s$ such that $$ \eqalign{ & L= L_0 > L_1 > \cdots > L_s = \Nh_0 \cr & {5\over4} L_i \le L_{i-1} \le {7\over4} L_i \qquad i=1,\ldots, s\cr } $$ (a proof of this is given in the Appendix A2). We clearly have $$ s\le { \log L -\log \Nh_0 \over \log(5/4) } \,. $$ Then we prove the following recursive inequality $$ g_{L_{i-1}}(\b,\hst) \ge g_{L_i}(\b,\hst) \, G_0(\b,\hst) \, \nep{ -20 \b k L_{i} } \qquad \forall 1\le i \le s \Eq(7.2) $$ for a suitable $G_0(\b,\hst)>0$. Iterating \equ(7.2) and observing that $$ \sum_{i=1}^s L_i \le L \sum_{i=1}^s (4/5)^i \le 4 L $$ we get $$ g_L(\b,\hst) \ge g_{\Nh_0}(\b,\hst) \, \prod_{i=1}^s \Bigl[ G_0(\b,\hst) \, \nep{ -20 \b k L_i } \Bigr] \ge G(\b,\hst)^{\log L} \nep{ -80\b k L } $$ for some $G(\b,\hst)>0$, and so the theorem is proven. In order to establish \equ(7.2), we actually prove the slightly more general inequality $$ \inf_{{5\over4} N\le L\le {7\over4} N} g_L(\b,\hst) \geq g_N(\b,\hst) \, G_0(\b,\hst) \, \nep{ -20 \b k N } \qquad \forall N \geq \Nh_0 \,. $$ We proceed as follows. Let $N\ge \Nh_0$, $5N/4< L \le 7N/4$ and $\L=Q_L$. $\L$ can be obtained in a unique way (apart from permutations) as a union of four overlapping squares of side $N$ that we denote by $\L_i$, $i=1,\dots, 4$ (see Fig.~1). Let $\L_1$ be the northeast square in $\L$, $\L_2$ the northwest one and so on, proceeding counterclockwise. Let $\calD=\{\L_1,\L_2,\L_3,\L_4\}$, and consider the block--dynamics defined in Appendix A1, determined by $\b,\hst,J$, the b.c. $\psi$ and the collection $\calD$, through the Markov generator $L_\calD^{J,\hst,\psi}$ given by \eqv(aa3). Thanks to Proposition \thf[aa1] we know that $$ \gap(L_\L^{J,\hst,\psi}) \geq \gap( L_\calD^{J,\hst,\psi}) \, g_N(\b,\hst) \Eq(7.4) $$ Thus, in order to prove the theorem, it remains to show that there exists $G_0(\b,\hst)>0$ such that $$ \inf_{J\in\d \L} \inf_{\psi\in\O} \gap( L_\calD^{J,\hst,\psi}) \geq G_0(\b,\hst) \, \nep{ -20\b k N } \Eq(7.9) $$ By Proposition \thf[aa2], this in turn follows, if we can prove that $$ \PP^{J,\hst,\psi,(\ph,\phb)}_\L \{ (\ph_t,\phb_t) :\ph_1 = \phb_1 \} \ge G_0(\b,\hst) \, \nep{ -20\b k N } \quad {\rm for\ all\ } \ph,\phb {\rm\ such\ that\ } \ph\le\phb \Eq(7.20) $$ where $(\ph_t,\phb_t)$ is the coupled process defined in the Appendix A1. The idea at this point is that in order to have $\ph_t = \phb_t$ on $\L$ it is \oqq almost'' sufficient to impose that the two configurations agree on a subset of $\L$ whose cardinality is of the order of the boundary. Let in fact (see Fig.~1) $$ A_i = \L_i \cap \Bigl( \bigcup_{j\ne i} \de^+ \L_j \Bigr) \qquad i=1,\ldots,4 \,. $$ Using the explicit construction and notation for this coupled process given in the Appendix A1, we have $$ \PP^{J,\hst,\psi,(\ph,\phb)}_\L \{ \ph_1 = \phb_1 \} \ge p_1 \, p_2\, p_3 \Eq(7.21) $$ where $p_1$ is the probability of having $\t(6)\le 1$, $p_2$ is the probability that the order in which the first six blocks are updated is given by $$ l(1),\ldots,l(6) = 1,3,2,4,1,3 J \Eq(l16) $$ and $p_3$ is the probability of having $\ph_{\t(6)} = \phb_{\t(6)}$, given \equ(l16). Define now the events (the random variables $(\h,\hb)$ have been introduced in the Appendix A1) $$ F^{(\ph',\ph'')}_{i,j} = \{ (\h(x),\hb(x))^{(\ph',\ph'')}_{i,j} = (1,1)\ \forall x\in A_{i} \} $$ Then we observe, and this is the key remark, that, since the interaction is between nearest neighbors, if the event $F_{1,1}^{(\ph_0,\phb_0)}$ and $F_{3,2}^{(\ph_{\t(1)},\phb_{\t(1)})}$ happen, then necessarily (given \equ(l16)) $\ph_{\t(6)} = \phb_{\t(6)}$. Thus, taking into account \item{$(i)$} that $(\h(x),\hb(x))=(1,1)$ if and only if $\hb(x)=1$ \item{$(ii)$} the properties of the standard coupling $\nu_{\L_i}^{\ph',\ph''}(\h,\hb)$ given in Section 1.3 \smallno we get $$ \eqalign{ & p_3 \ge \inf_{\ph' \le \ph''\,,\,\pht'\le\pht''} \prob( F^{(\ph',\ph'')}_{1,1} \cap F^{(\pht',\pht'')}_{3,2} ) \ge \cr & \ge \inf_{J\in\d \L_1} \inf_{\ph'\in\O} \bigl( \, \mu_{\L_1}^{J,\hst,\ph'} \{ \ph\in\O_{\L_1} : \ph(x) = 1\,, \ \forall x\in A_1 \} \, \bigr)^2 \,. \cr } \Eq(7.22) $$ Clearly the event $\{\ph (x) = 1\ \forall \,x\in A_i\}$ is a decreasing event so that, using the FKG property we get $$ \mu_{\L_1}^{J,\hst,\ph'} \{\ph (x) = 1\ \forall \,x\in A_1\} \ge \prod_{x\in A_1} \mu^{J,\hst,\ph'}_{\L_1} \{ \ph(x)=1 \} \Eq(7.11) $$ Now we subdivide the $x$'s in $A_1$ into two groups, depending on whether they are close to the boundary or not. Let then $$ \eqalign{ A'_1 &= \{ x\in A_1 : d(x,\de\L_1) \le N_0 \} \cr A''_1 &= A_1 \setm A'_1 \cr } $$ We claim that \item{$(a)$} $|A'_1| \le 4 N_0$ and $|A''_1| \le 2 N$ \item{$(b)$} if $x\in A'_1$ then for all $J\in\d\L_1$, $\ph'\in\O$, $\mu^{J,\hst,\ph'}_{\L_1} \{\ph(x)=1 \} \ge b_2(b,\hst) >0$ \item{$(c)$} if $x\in A''_1$ then for all $J\in\d\L_1$, $\ph'\in\O$, $\mu^{J,\hst,\ph'}_{\L_1} \{ \ph(x)=1 \} \ge \ov8 \nep{-4\b k}$ \medno $(a)$, $(b)$ and $(c)$ together with the obvious bound $$ p_1 \, p_2 \ge \nep{-\b} $$ clearly imply \equ(7.9) with $$ G_0(\b,\hst) = \nep{-\b} \, b_2(\b,\hst)^{8 N_0} $$ and so the theorem would be proven. \smallno \textindent{$\bullet$} To prove $(a)$ it is sufficient to observe that $A_1= T_v \cup T_h$ (see Fig.~1), where $T_v$ is a vertical line of length $N$ whose distance from the vertical sides of $\L_1$ is greater than $$ ((L-N) \mmin (2N-L)) -2 \ge \ov5 N \ge 2 N_0 $$ since $5N/4 < L \le 7N/4$. An analogous statement holds for $T_h$. \textindent{$\bullet$} $(b)$ is proven in Proposition \teo[3.2] of \papI. \textindent{$\bullet$} The proof of $(c)$ is a little more involved and we have to use the fact the at distances greater than $N_0$ from the boundary is very likely to have $k\le \ph \le k+1$. Let in fact $x\in A''_1$. Let $V_0$ be a square of side $N_0$ (or $N_0+1$ if $N_0$ is even) centered at $x$ and let $V_1$ be the square still centered at $x$ but with side $N_0 - 2 \inte{N_0/4} - 2$. By \equ(DLR), taking into account that if $J\in\d\L_1$ then $J(x,y)=1$ for all $\{x,y\}\sset\L_1$, we obtain $$ \inf_{J\in\d \L_1} \inf_{\ph'\in\O} \mu^{J,\hst,\ph'}_{\L_1} \{ \ph(x)=1 \} \ge \inf_{\ph'\in\O} \mu^{\hst,\ph'}_{V_0} \{ \ph(x)=1 \} \,. \Eq(7.30) $$ Take now $\hba = \ov{4\b} \nep{ -4\b k} \le \hst$. Because $\{ \ph(x) = 1 \}$ is a negative event, by the FKG property, $$ \mu^{\hst,\ph'}_{V_0} \{ \ph(x)=1 \} \ge \mu^{\hba,\ph'}_{V_0} \{ \ph(x)=1 \} \,. \Eq(7.31) $$ Furthermore, if we let $$ A = S_+(V_0,k+2, \de^+ V_1) $$ (the event $S_+(\L,j,V)$ has been defined in Proposition \thm[8.6]), by \teo[5.1] of \papI\ and \teo[A1.1] of \papI\ we have $$ \eqalign{ & \mu^{\hba,\ph'}_{V_0} \{ \ph(x)=1 \} \ge \mu^{\hba,\ph'}_{V_0} ( \ph(x)=1 \tc A^c ) \, \mu^{\hba,\ph'}_{V_0} ( A^c ) \ge \cr & \ge \ov2 \sup_{\st V : \, V_1\sset V \sset V_0 \atop {V \rm\ connected\ and \atop \rm simply\ connected } } \mu^{\hba,k+1}_V \{ \ph(x)=1 \} \cr } \Eq(7.32) $$ On the other side, letting $$ F = \{ \ph\in\O_V : \ph(y) = k+1\ \forall y\in V, \ {\rm such\ that\ } |x-y|=1 \} $$ and using Proposition \teo[3.5] of \papI\ , we find, for each simply connected $V$ containing $V_1$, $$ \eqalign{ & \mu^{\hba,k+1}_V \{ \ph(x)=1 \} \ge \mu^{\hba,k+1}_V (\ph(x)=1 \tc F ) \, \mu^{\hba,k+1}_V (F) \ge \cr & \ge \mu^{\hba,k+1}_V (\ph(x)=1 \tc F ) \, \Bigl( 1 - 4 \sup_{y\in V} \mu^{\hba,k+1}_V \{ \ph(y) \ne k+1 \} \Bigr) \ge \cr & \ov2 \, \mu^{\hba,k+1}_\xx \{ \ph(x)=1 \} \cr } \Eq(7.33) $$ But an explicit computation, gives $$ \mu^{\hba,k+1}_\xx \{ \ph(x)=1 \} \ge \ov2 \nep{ -4\b k} \,. $$ In this way we have found $$ \inf_{J\in\d\L_1} \inf_{\st \ph'\in\O} \mu^{J,\hst,\ph'}_{\L_1} \{ \ph(x)=1 \} \ge \ov8 \nep{-4\b k} \qquad \forall x\in A''_1 \QED $$ \fine \expandafter\ifx\csname sezioniseparate\endcsname\relax% \input macro \fi %---------------------------------------------------------- \numsec=-1 \numfor=1 \numtheo=1 \pgn=1 \beginsection Appendix A1. The block dynamics Given \item{$(i)$} the hamiltonian \equ(H), with parameters $(\b, h, J)$ \item{$(ii)$} a boundary condition $\psi\in\O$ \item{$(iii)$} a collection $\calD = \{ \L_1,\ldots,\L_n\}$ of finite (possibly overlapping) subsets of $\ZZ$, \smallno and letting $$ \L=\cup_{i=1}^n \L_i $$ we define as the standard block dynamics (with respect to $\b,h,J,\psi,\calD$) the Markov process on $\O_\L$ whose generator is $$ (L_\calD^\psi f)(\ph) = \sum_{i=1}^n \sum_{\h\in\O} \mu_{\L_i}^\ph(\h) [ f(\h) -f(\ph) ] \qquad {\rm for\ all\ }\ph\ {\rm such\ that\ } \ph_{\L^c} = \psi_{\L^c} \,. \Eq(aa3) $$ The sum over $\h$ is actually restricted to those configuration that agree with $\ph$ outside $\L_i$ because of \equ(finvolmea). A straightforward computation shows that $L_\calD^\psi$ is still a bounded self--adjoint operator on $L^2( \O_\L, d\mu_\L^\psi)$. We assume to have chosen $\b,h,J$ and won't mention them explicitly in the following. Then \nproclaim Proposition [aa1]. %------------------------------- For all $\b, h >0$, all $J$, $\psi$, let $L_\L^\psi$ be given by \equ(gnrt) with transition rates satisfying $(H_1)-(H_5)$. Then $$ \gap( L_\L^\psi ) \ge \gap( L_\calD^\psi ) \, \inf_i \inf_{\ph\in\O} \gap( L_{\L_i}^\ph ) \, \Bigl( \sup_{x\in\L} \#\{ i : \L_i\ni x \} \Bigr)^{-1} \Eq(aa0) $$ \Pro\ Let $$ g = \inf_i \inf_{\ph\in\O} \gap( L_{\L_i}^\ph ) $$ Thanks to \equ(gap), \equ(aa0) is proven if we can show that $$ \Dir_\calD^\psi(f,f) \le g^{-1} \, \sup_{x\in\L} \#\{ i : \L_i\ni x \} \, \Dir_\L^\psi(f,f) \qquad {\rm for\ all\ } f\in L^2(\O_\L, d \mu_\L^\psi) \,. \Eq(aa1) $$ But, using \equ(DLR), we find $$ \eqalign{ & \Dir_\calD^\psi(f,f) \le \ov2 \sum_{\ph\in\O} \mu_\L^\psi(\ph) \sum_i \sum_{\h\in\O} \mu_{\L_i}^\ph(\h) \, [ f(\h) - f(\ph) ]^2 = \cr & = \ov2 \sum_{\ph\in\O} \sum_{\ph'\in\O} \mu_\L^\psi(\ph') \sum_i \mu_{\L_i}^{\ph'}(\ph) \sum_{\h\in\O} \mu_{\L_i}^\ph(\h) \, [ f(\h) - f(\ph) ]^2 = \cr & = \sum_{\ph'\in\O} \mu_\L^\psi(\ph') \sum_i \Var_{\L_i}^{\ph'}(f) \le g^{-1} \, \sum_{\ph'\in\O} \mu_\L^\psi(\ph') \sum_i \Dir_{\L_i}^{\ph'}(f,f) \cr} \Eq(aa2) $$ On the other side, by \equ(DLR) again, $$ \eqalign{ & \sum_{\ph'\in\O} \mu_\L^\psi(\ph') \sum_i \Dir_{\L_i}^{\ph'}(f,f) = \cr & = \ov2 \sum_{\ph'\in\O} \mu_\L^\psi(\ph') \sum_i \sum_{\ph\in\O} \mu_{\L_i}^{\ph'}(\ph) \sum_{x\in\L_i,\, s=\pm1} c(x,\ph,s) \, [ f(\ph^{x,s}) - f(\ph) ]^2 \le \cr & \le \sup_{x\in\L} \#\{ i : \L_i\ni x \} \, \Dir_\L^\psi(f,f) \cr } $$ \equ(aa1) and the proposition are thus proven. \QED \medno We want to introduce now a coupling between two copies of the block dynamic process in such a way to preserve the order given by the initial conditions. Given $\psi$, $\ph$, $\phb$ such that $\psi_{\L^c}=\ph_{\L^c}=\phb_{\L^c}$, consider the the process $(\ph_t,\phb_t)$ which starts from $(\ph,\phb)$ at time zero and evolves according to the generator $$ (\hat L_\calD^{\psi} f)(\ph,\phb) = \sum_i \sum_{\h,\hb\in\O} \nu_{\L_i}^{\ph,\phb}(\h,\hb) \, [ f(\h,\hb) - f(\ph,\phb) ] $$ where $\nu_{\L_i}^{\ph,\phb}$ is the standard coupling defined in Section 1. \smallno Given the starting point $(\ph,\phb) \in \O^2$, the process $(\ph_t,\phb_t)$ can be constructed explicitly as follows. Let \item{$(a)$} $\t(j)$, $j=1,2,\ldots$, be the jump times of a Poisson process with rate $n=$ number of blocks. \item{$(b)$} $(\h,\hb)^{(\ph,\phb)}_{i,j}$, for $(\ph,\phb)\in\O^2$, $i=1,\ldots,n$ and $j=1,2,\ldots$ be a collection of $\O^2$ valued independent random variables (and independent of the $\t_j$'s) distributed according to $\nu_{\L_i}^{\ph,\phb}(\h,\hb)$. \item{$(c)$} $l(j)$ for $j=1,2,\ldots$ be a collection of $i.i.d.$ (and independent of what has been defined before) random variables with uniform distribution on $\{1,\ldots,n\}$ \smallno Define now \item{(1)} $(\ph_0,\phb_0) = (\ph, \phb)$ \item{(2)} $(\ph_t,\phb_t) = (\ph_{\t(j)},\phb_{\t(j)})$, for all $\t(j)\le t < \t(j+1)$ \item{(2)} $(\ph_{\t(j+1)},\phb_{\t(j+1)}) = (\h,\hb)^{(\ph_{\t(j)}, \phb_{\t(j)} ) }_{l(j+1), \, j+1}$ \smallno Then we have \nproclaim Proposition [aa2]. With respect to the quantities previously defined \item{$(i)$} both $\ph_t$ and $\phb_t$ have the same distribution as the process associated with the generator \equ(aa3) \item{$(ii)$} if $\ph\le\phb$, then $$ \prob \{ \ph_t \le \phb_t \} =1 \qquad {\rm for\ all\ } t\ge 0 $$ \item{$(iii)$} If $\prob \{ \ph_1 = \phb_1 \} \ge p$ for all starting points $\ph,\phb$ such that $\ph\le\phb$ (and such that they agree with $\psi$ on $\L^c$), then $$ \gap( L^\psi_\calD ) \ge p $$ \noindent $(i)$ is a standard computation. $(ii)$ follows from the definition of $(\ph_t,\phb_t)$ and from the properties of the coupling measure $\nu_\L^{\ph,\phb}$. To get $(iii)$ we use the (time) Markov property which yieds $$ \prob \{ \ph_t \ne \phb_t \} \le (1-p)^{\inte{t}} \le \nep{ -(t-1) p } \QED $$ \vfill\eject %-------------------------------------------------------------- \numsec=-2 \numfor=1 \numtheo=1 \pgn=1 \beginsection Appendix A2. \nproclaim Proposition [aa3]. Let $N\ge 100$, $L\ge 2N$. Then there exist integers $s$, $L_i$, such that $$ \eqalign{ & L= L_0 > L_1 > \cdots > L_s = N \cr & {5\over4} L_i \le L_{i-1} \le {7\over4} L_i \qquad i=1,\ldots, s\cr } $$ \Pro\ Let also $X$ be the set of $L$'s such that the proposition holds. We want to show that $X$ contains all integers greater than or equal to $2N$. Let $N_0^+=N_0^-=N$ and, for each $i\in\Zp$ we set $K_i = [N_i^-, N_i^+]$, where $$ \txt N_i^+ = \inte{ {7\over4} N_{i-1}^+ } \quad {\rm and} \quad N_i^- = \ceil{ {5\over4} N_{i-1}^- } $$ ($\ceil x$ is the smallest integer not less than $x$). Obviously $$ K_i \supset \bigcup_{n\in K_{i-1}} \bigl[ \, \ceil{5n/4} , \inte{7n/4} \, \bigr] $$ and, since for $n\ge 100$, $$ \inte{7n/4} \ge \ceil{5(n+1)/4} $$ we have in fact $$ K_i = \bigcup_{n\in K_{i-1}} \bigl[ \, \ceil{5n/4} , \inte{7n/4} \, \bigr] \,. $$ So, for each $m\in K_i$ there exists $n\in K_{i-1}$ such that $$ {5\over4} \le {m\over n} \le {7\over4} \,. $$ This implies $$ X \supset \bigcup_{i=1}^\infty K_i \,. $$ Furthermore, since $N\ge 100$, we have $$ N_2^- \le (5/4)^2 N + 5/4 + 1 \le (7/4) N - 1 \le N_1^+ $$ and it is easy to check, by induction, that $$ N_i^- \le N_{i-1}^+ \qquad \forall i\ge 2 \,. $$ Thus we get $$ \bigcup_{i=1}^\infty K_i \supset \bigcup_{i=2}^\infty [ N_{i-1}^+, N_i^+ ] = [N_1^+, \infty ) \supset [2N, \infty) \,. \QED $$ \fine \vfill\eject %---------------------------------------------------------- \beginsection References \frenchspacing \item{[CM]} F. Cesi and F. 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Phys. {\bf 165}, 33 (1994) \item{[S]} R.~H.~Schonmann: Second order large deviation estimates for ferromagnetic systems in the phase coexistence region. Commun. Math. Phys. {\bf 112}, 409 (1987) \fine \beginsection Figure captions \item{(1)} Proof of Proposition \thm[8.1a] \fine \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} \bye