%% ------------------------------------------------------------------------- %% %% Macro 1.1 - agg: 11-5-95 %% %% %% ------------------------------------------------------------------------- %% %% 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. Se non sono %% disponobili commentare lo statement \fnts=1 %% ------------------------------------------------------------------------- %%%%%%%%%%%%%%% FORMATO \magnification=\magstep1 \tolerance=10000 %\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 %%%%%%%%%%%%%%%%%%%%%%%%% 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 $} \newcount\pgn \pgn=1 \def\foglio{\veroparagrafo:\number\pgn \global\advance\pgn by 1} %%%%%%%%%%%%%%%%% EQUAZIONI E TEOREMI CON NOMI SIMBOLICI \global\newcount\numsec \global\newcount\numfor \global\newcount\numfig \global\newcount\numtheo \gdef\profonditastruttura{\dp\strutbox} \def\senondefinito#1{\expandafter\ifx\csname#1\endcsname\relax} \def\SIA #1,#2,#3 {\senondefinito{#1#2}% \expandafter\xdef\csname #1#2\endcsname{#3}\else \write16{???? ma #1,#2 e' gia' stato definito !!!!}\fi} \def\etichetta(#1){(\veroparagrafo.\veraformula) \SIA e,#1,(\veroparagrafo.\veraformula) \global\advance\numfor by 1 \write15{\string\FU (#1){\equ(#1)}} \write16{ EQ \equ(#1) == #1 }} \def\oldetichetta(#1){ \senondefinito{fu#1}\clubsuit(#1)\else \csname fu#1\endcsname\fi} \def\FU(#1)#2{\SIA fu,#1,#2 } \def\tetichetta(#1){{\veroparagrafo.\verotheo}% \SIA theo,#1,{\veroparagrafo.\verotheo} \global\advance\numtheo by 1% \write15{\string\FUth (#1){\thm[#1]}}% \write16{ TH \thm[#1] == #1 }} \def\oldtetichetta(#1){%----------------------- mnemonic label \senondefinito{futh#1}\clubsuit(#1)\else \csname futh#1\endcsname\fi} \def\FUth(#1)#2{\SIA futh,#1,#2 } \def\getichetta(#1){Fig. \verafigura \SIA e,#1,{\verafigura} \global\advance\numfig by 1 \write15{\string\FU (#1){\equ(#1)}} \write16{ Fig. \equ(#1) ha simbolo #1 }} \newdimen\gwidth \def\BOZZA{ \def\alato(##1){ {\vtop to \profonditastruttura{\baselineskip \profonditastruttura\vss \rlap{\kern-\hsize\kern-1.3truecm{$\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$}}}}} \def\talato(##1){\rlap{\sixrm\kern -1.3truecm ##1}} } \def\alato(#1){} \def\galato(#1){} \def\talato(#1){} \def\veroparagrafo{\ifnum\numsec<0 A\number-\numsec\else \number\numsec\fi} \def\veraformula{\number\numfor} \def\verotheo{\number\numtheo} \def\verafigura{\number\numfig} %\def\geq(#1){\getichetta(#1)\galato(#1)} \def\Thm[#1]{\tetichetta(#1)} \def\thf[#1]{\senondefinito{futh#1}$\clubsuit$[#1]\else \csname futh#1\endcsname\fi} \def\thm[#1]{\senondefinito{theo#1}$\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} \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 fonts msam, msbm, eufm are not available %------------------------- 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 % \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 -------- % \ifnum\fnts=0% ------------Se non ci sono le fonti % \def\bZ{ { {\rm Z} \mskip -6.6mu {\rm Z} } } \def\bR{{\rm I\!R}} \def\bb{ \vrule height 6.7pt width 0.5pt depth 0pt } \def\bC{ { {\rm C} \mskip -8mu \bb \mskip 8mu } } \def\bE{{\rm I\!E}} \def\bP{{{\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 % \else% ------ o se ci sono % \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 % \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}} \fi % %------------------------------------------------------------------- % % ------- 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_*} % \let\neper=e \let\ii=i \let\mmin=\wedge \let\mmax=\vee \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\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\tto#1{\buildrel #1 \over \longrightarrow} % \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 %\BOZZA\def\thm{\teo}\def\thf{\teo} %non numera i teoremi \def\JL{\|J\|_\L} \def\JV{\|J\|_V} \def\Jl{J^{(\l)}} \def\dd{ {d-1\over d} } \def\emd#1#2{\nep{ - \b \nabla_{#1} H_{#2}}} \def\bw{{\bar w}} \def\hA{{\widehat A}} \def\hB{{\widehat B}} \def\Ab{{\bar A}} \def\hm{{\widehat m}} \def\dB{{\dep_r B}} \def\tL{{\ntilde L}} \def\tB{{\ntilde B}} \def\Thb{{\bar \Th}} \def\Ld{{L^2(\mu^J)}} \def\LdF{{L^2(\mu^J( \cdot \tc F_\L))}} \def\LuF{{L^1(\mu^J( \cdot \tc F_\L))}} \def\JJ{{J_0}} \def\Zar{Zahradn\'\i k} %--------------------------------- INIZIO \expandafter\ifx\csname sezioniseparate\endcsname%--- non toccare \relax\input macro\input xmac \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 Relaxation to equilibrium for two dimensional } \centerline{\ttlfnt disordered Ising systems in the Griffiths phase} \vskip 0.5truecm \author{F. Cesi $^{1}$, C. Maes $^{2}$, F. Martinelli $^{3}$ } % \address{\ninerm $^{1}$ Dipartimento di Fisica, Universit\`a \lq\lq La Sapienza", P.le A. Moro 2, 00185 Roma, Italy. \hfill\break $\phantom{^1}$ e-mail: cesi@vaxrom.roma1.infn.it \hfill\break %%%%%%%%%%%% $^2$ Instituut voor Theoretische Fysika, K.U. Leuven, Celestijnenlaan 200D B-3001 Leuven and \hfill\break $\phantom{^1}$ Onderzoeksleider N.F.W.O., Belgium. \hfill\break $\phantom{^1}$ e-mail: Christian.Maes@fys.kuleuven.ac.be \hfill\break %%%%%%%%%%%% $^3$ Dipartimento di Energetica, Universit\`a dell' Aquila, Italy. \hfill\break $\phantom{^1}$ e-mail: martin@mat.uniroma3.it } %%%%%%%%%%%% \bigno {\it Dedicated to the memory of Roland Dobrushin} \bigno \abstract{ We consider Glauber--type dynamics for two dimensional disordered magnets of Ising type. We prove that, if in equilibrium the disorder--averaged influence of the boundary condition is sufficiently small, then the corresponding Glauber dynamics is ergodic with probability one and the disorder--averaged of time--autocorrelations decays like $\nep{-m (\log t)^{2}}$. For the standard dilute Ising ferromagnet with i.i.d. random nearest neighbor couplings taking the values $0$ or $J>0$, our results apply even if the active bonds percolate and $J$ is larger than the critical value for the corresponding pure Ising model. For this model we also rigorously prove the existence of a dynamical phase transition when $J$ crosses the critical value $J_c$ for the standard two dimensional Ising model. } % \vskip 1cm \noindent {\bf Key Words:} Dilute Ising model, Griffiths phase, Glauber dynamics. {\parindent=0pt \footnote{}{\ninerm Work partially supported by grant CHRX-CT93-0411 of the EEC} \footnote{}{\ninerm Mathematics Subject Classification: 82B44, 82C22, 82C44, 60K35} } \vfill\eject \endgroup %--------------------------------- INIZIO \expandafter\ifx\csname sezioniseparate\endcsname\relax% \input macro \input xmac\fi %@E1 \numsec=1 \numfor=1 \numtheo=1 \pgn=1 \beginsection 1. Introduction Roland Dobrushin certainly was one of the pioneers in the theory of interacting particle systems and in the study of stochastic dynamics for Gibbs fields in particular. From the start (at the end of the 60's) his insights related important concepts in probability theory to those of statistical physics. The mathematical tools he developed have turned out to be extremely useful in several physically relevant problems. The present paper concerns a topic he was very much interested in and to which he contributed many ideas. The general question is the study of Gibbs fields with random interactions. In particular we want to study (see \ref[GM1], \ref[GZ1], \ref[GZ2] and \ref[CMM] for related results) the speed of convergence to equilibrium in a single spin flip, Glauber--type, dynamics, which has a reversible (random) Gibbs measure in the so--called {\it Griffiths' phase}, namely that region of the phase diagram between the paramagnetic (high--temperature) phase and the phase with long--range order (at least for ferromagnetic systems). The simplest example of such a system is the dilute Ising ferromagnet below the percolation threshold. In this case the random couplings $J_{xy}$ between nearest neighbor spins take only two values, $J_{xy}=0 $ and $J_{xy}=J$ with probability $1-p$ and $p$ respectively, with $J>J_c$ and $pJ_c$ and $p>p_c$, since now there is, with probability one, an infinite cluster of low-temperature bonds. It is remarkable that, using the random--cluster representation for the Ising model (see \ref[ACCN] and \ref[N]) and some older results relating averaged correlations to the usual type of correlations for a translation--invariant system (see \ref[OPG] ), one can still find, for all $p\ge p_c$, an explicit value $J(p) >J_c$ (see Corollary \thf[Dilu] below) such that for all $J_c \le J 0$ and $h_\infty\geq 0$ such that $\bP\{|J_b| > J_\infty \hbox{ or } |h_x|> h_\infty\} = 0$. \smallno We denote the expectation with respect to $\bP$ by $\bE(\cdot)$. For each $V \ssset \Z^d$ we define the Hamiltonian $H_V : \O \mapsto \bR$ by $$ H_V^{J,h}(\s) = - \sum_{b\in \hat V}J_b\s_b - \sum_{x\in V}h_x\s(x) $$ We also set, for $\s, \t \in \O$ $$ H_V^{J,h,\t}(\s) = H_V^{J,h} (\s_V \t_{V^c} ) \Eq(finvolH) $$ and $\t$ is called the {\it boundary condition}. For each $V\ssset \Z^d$ and $\t\in \O$ we define the (finite volume) conditional Gibbs measure by $$ \mu^{J,h\t}_V(\s) = \cases{ \bigl(Z^{J,h,\t}_V\bigr)^{-1} \exp[ \,- H^{J,h,\t}_V(\s) \,] & if $\s(x) = \t(x)$ for all $x\in V^c$ \cr \vphantom{\Bigl(} 0 & otherwise. \cr } \Eq(finvolmea) $$ where $Z^{J,h,\t}_V$ is the proper normalization factor called partition function. Notice that we have absorbed the inverse temperature $\beta$ in the Hamiltonian. We will sometimes drop the superscripts $J,h$ if that does not generate confusion. Given a measurable bounded function $f$ on $\O$, $\mu_V^\t (f)$ denotes its average and $\mu_V^\t(f,g)$ stands for the covariance of $f$ and $g$. Finally, given $\D\sset \L \sset \Z^d$ we denote by $\mu_{\L,\D}^{\t}$ the projection of the Gibbs measure $\mu_\L^\t$ over the set $\D$ that is $$ \mu_{\L,\D}^{\t}(\s_\D) \equiv \mu_\L^\t(\h: \, \h_\D=\s_\D) $$ \medno {\it The dynamics.} The stochastic dynamics we want to study is determined by the Markov generators $L_V^{J,h}$, $V\ssset \Z^d$, defined by $$ (L_V^{J,h} f)(\s) = \sum_{x\in V} c_{J,h}(x,\s) (\nabla_x f)(\s) \Eq(gnrt) $$ The nonnegative real quantities $c_{J,h}(x,\s ),\; x\in\Z^d\,,\; \s\in\O\,$, are the {\it transition rates\/} for the process. \smallno The general assumptions on the transition rates are \item{(1)} {\it Finite range.} If $\s(y)=\s'(y)$ for all $y$ such that $|x-y|\le 1$, then $c_{J,h}(x,\s) = c_{J,h}(x,\s')$ \item{(2)} {\it Detailed balance.} For all $\s\in\O$ and $x\in \Z^d$, $$ \exp\bigl[ - H^{J,h}_\xx(\s) \bigr] c_{J,h}(x,\s) = \exp\bigl[ - H^{J,h}_\xx(\s^x) \bigr] c_{J,h}(x,\s^x) \Eq(dbal) $$ \item{(3)} {\it Positivity and boundedness.} There exist non--negative real numbers $c_m$ and $c_M$ such that for all $J$ $$ c_m \le \inf_{x,\s,J,h} c_{J,h}(x,\s) \quad \hbox{ and }\quad \sup_{x,\s} c_{J,h}(x,\s) \le c_M \Eq(bounded) $$ \smallno Two cases one may want to keep in mind are $$ \eqalignno{ & c_{J,h}(x,\s) = \min\bigl\{ \nep{ - (\nabla_{x} H_\xx)(\s) } \,,\, 1 \bigr\} &\eq(MET) \cr & c_{J,h}(x,\s) = \mu^{J, \s}_\xx ( \s^x ) = \left[ \, 1 + \nep{ (\nabla_{x} H_\xx)(\s) } \, \right]^{-1} &\eq(HB) \cr } $$ corresponding to the Metropolis and heat--bath dynamics respectively. We denote by $L_V^{J,h,\t}$ the operator $L_V^{J,h}$ acting on $L^2(\O,d\mu_V^{J,h,\t})$ (this amounts to choose $\t$ as the boundary condition). Assumptions (1), (2) and (3) guarantee that there exists a unique Markov process whose generator is $L^{J,h,\t}_V$, and whose semigroup we denote by $\{T^{J,h,\t}_V(t)\}_{t\ge 0}$. $L^{J,h,\t}_V$ is actually a bounded selfadjoint operator on $L^2(\O,d\mu_V^{J,h,\t})$. The process has a unique invariant measure given by $\mu_V^{J,h,\t}$. Moreover $\mu_V^{J,h,\t}$ is {\it reversible} with respect to the process, \ie $L_V^{J,h,\t}$ is self--adjoint on $L^2(\O,d\mu_V^{J,h,\t})$. \smallno {\it Infinite volume dynamics}. Let $\mu$ be a Gibbs measure for the interaction $J,h$. Since the transition rates are bounded, then the infinite volume generator $L^{J,h}$ obtained by choosing $V= \Z^d$ in \equ(gnrt) is well defined on the set of functions $f$ in $L^2(\O, d\mu)$ (or $C(\O)$) such that $\tpl f \tpl\equiv \sum_{x\in \Z^d}\|\nabla_x f\|_\infty$ is finite. The closure of $L^{J,h}$ in $L^2(\O, d\mu)$ ($C(\O)$) is a Markov generator (see, for instance Theorems \teo[3.9] in Chapter I and \teo[4.1] in Chapter IV of \ref[L]), which defines a Markov semigroup denoted by $T(t)$. Again $L^{J,h}$ is self--adjoint on $L^2(\O, d\mu)$. \beginsubsection 2.2. Main results on the dynamics %@I In \ref[CMM] one main result on the speed of relaxation of the dynamics to its equilibrium Gibbs measure has been proved under a key probabilistic assumption on the Gibbs measure with random interactions. In order to state this hypothesis we need first the following definition. Given $V\ssset \Z^d$, $n \in \Zp$ and $\a >0$, we say that the condition $SMT(V, n, \a)$ holds if for all local functions $f$ and $g$ on $\O$ such that $d(\L_f, \L_g) \ge n$ we have $$ \sup_{\t\in\O} |\mu_V^\t(f,g)| \le |\L_f| |\L_g| \|f\|_\infty \|g\|_\infty \exp( - \a d( \L_f, \L_g ) ) $$ Then the main hypothesis of \ref[CMM] can be formulated as follows: \smallno (H) There exist $L_0 \in \Zp$, $\a>0$, $\th>0$ such that for all $L \ge L_0$ $$ \bP\{ \, SMT( Q_L, L/2, \a) \, \} \ge 1 - \nep{ - \th L } $$ \bigno {\it Remark 1.} It is not difficult to check that the main result of \ref[CMM] (see theorem \thf[UB] below) follows even if in hypothesis (H) we replace \lq\lq for all $L \ge L_0$ " with the slightly weaker \lq\lq for all $L $ multiple of $L_0$ ". It is actually this (apparently) weaker form of (H) that we will show to be implied by a simpler assumption (see theorem \thf[WME] below). \bigno {\it Remark 2.} Some of the results of \ref[CMM] were obtained under a more general assumption on the distribution of $J$ and $h$ than boundedness, namely $\bE\{\nep{|J_b|^{1+\d}}\} < \infty$ and $\bE\{\nep{|h_x|^{1+\d}}\} < \infty$ for some positive $\d$. Here we prefer to work directly in the bounded case since the latter is more relevant from the physical point of view and the results on the dynamics are more transparent. \bigno The main theorem of \ref[CMM] about the speed of relaxation to equilibrium for the model described above is as follows. \nproclaim Theorem [UB]. Assume (H). Then \item{(a)} If $d \ge 1$ there exists a set $\bar \Th \sset \Th$ of full measure such that for each $J,h\in \bar \Th$ there exists a unique infinite volume Gibbs measure $\mu^{J,h}$. Moreover there exists a constant $k$ and, for each $J,h\in \bar \Th$ and for any local function $f$ there exists $t_0({J,h},f)< \infty$ such that for all $t\ge t_0$ $$ \|T^{J,h}(t)f-\mu^{J,h}(f)\|_\infty\le \exp\Bigl[ \, - t \, \exp\bigl[ \, - k \, ( \log t)^{1 - {1 \over d} } \, (\log\log t)^{d-1} \, \bigr] \, \Bigr] \Eq(UB2) $$ \item{(b)} Let $d\ge 2$. Then there exists a constant $k$ and for any local function $f$ there exists $t_0(f)< \infty$ such that, if $t \ge t_0(f)$ then $$ \bE \, \|T^{J,h}(t)f-\mu^{J,h}(f)\|_\infty \le \exp\bigl[ \, -k \, (\log t)^{d \over d-1} \, (\log\log t )^{-d} \, \bigr] \Eq(UBb) $$ \item{(c)} Let $h\equiv 0$ and let $\pi_0(\s) = \s(0)$. For each $d\ge 2$ there exist $J_1(d) > 0$ such that if $p_1 \equiv \bP\{J_{xy}= J_1\}>0$ and $p_2 \equiv \bP\{|J_{xy}|\le 1/4 \}>0$ then we have, for all large enough $t$, $$ \bE \, \| T^J (t) \pi_0 - \mu^J( \pi_0) \|_\Ld \ge \exp\bigl[ \, -k \, (\log t)^{d\over d-1} \, \bigr] \Eq(LB) $$ for some $k$ which depends on $d$, $p_1$ and $p_2$. %@E9 %@I1 %--------------------------------- INIZIO \expandafter\ifx\csname sezioniseparate\endcsname\relax% \input macro \input xmac\fi %@E1 \numsec=3 \numfor=1 \numtheo=1 \pgn=1 \beginsection 3. An alternative assumption to hypothesis (H) in two dimensions. We present here, for {\it two dimensional} systems, an alternative assumption to the basic hypothesis (H) of \ref[CMM]. The advantage of this new assumption is that in some concrete case, like the dilute Ising ferromagnet, it can be explicitly verified in interesting region of the phase diagram. The idea behind what follows is essentially based on a result obtained in \ref[MOS] for two dimensional non--random systems and recently extended in \ref[Be] to random ones. In order to formulate the main result of this section we first need some additional definitions. \nproclaim Definition [1]. Given $C>0$, $m>0$ and $\L\sset \Z^d$, we say that $WME(\L,C,m)$ holds if for all $\D\sset \L $ $$ \bE\bigl\{\sup_{\t,\t'}\Var(\mu_{\L,\D}^{\t}, \mu_{\L,\D}^{\t'})\bigr\} \le C\sum_{x\in \D}\nep{-md(x,\L^c)} $$ where $\Var$ denotes the variation distance. We say that $WME(C,m)$ holds if $WME(\L,C,m)$ holds for all $\L\sset\Z^d$. \noindent \nproclaim Definition [2]. Given $C>0$, $m>0$ and $\L\sset \Z^d$, we say that $SME(\L,C,m)$ holds if for all $\D\sset \L $ and all $y\in \L^c$ $$ \bE\bigl\{\sup_{\t,\t^y}\Var(\mu_{\L,\D}^{\t}, \mu_{\L,\D}^{\t^y})\bigr\} \le C\sum_{x\in \D}\nep{-m|x-y|} $$ \noindent Using these notions we have the following key result \ref[Be] (see also \ref[MOS] for a similar statement in case of non--random systems). \bigno \nproclaim Theorem [V.d.Berg]. Let $d=2$. Then $WME(C,m)$ implies that there exists $L_0\in \Z_+$, $m' >0$ and $C' > 0$ such that $SME(\L,C',m')$ holds for all sets $\L$ multiple of $Q_{L_0}$. \bigno {\it Remark.} Actually in \ref[Be] the above result is proved under the hypothesis that $WME(\L,C,m)$ holds for all rectangles $\L$. \bigno We are finally in a position to give our alternative to assumption (H). \bigno \nproclaim Lemma [WME]. Let $d=2$ and let $\e\in (0,1)$. Then $WME(C,m)$ implies that there exists $L_0\in \Z_+$, $\a >0$ and $\th > 0$ such that for all integers $L$ multiple of $L_0$ $$ \bP\bigl\{ SMT(Q_L, \e L, \a)\bigr\} \ge 1 - \nep{-\th L} $$ \noindent \Pro\ Let $l_0 0$, we define $\Th(\a,\e)$ as the set of those couplings such that for all pairs $( I_1, I_2)\in {{\cal I}_\e}$ one has $$ \sup_{y\in \L\setminus \L_{I_2}} \sup_\t \Var(\mu_{\L_{I_2},\L_{I_1}}^\t, \mu_{\L_{I_2},\L_{I_1}}^{\t^y}) \le \nep{-2\a L} \Eq(WME.1) $$ We claim that for all $J\in \Th(\a,\e)$ property $SMT(Q_L,\e L, \a)$ holds, provided that $L$ is large enough depending only on $\a$. Given in fact two arbitrary functions $f$ and $g$, with $d(\L_f,\L_g)\ge \e L$, let us set $$ I_f = \{i\in \{1\dots n\}: \; B_i\cap \L_f \neq \emptyset\} \qquad I_f^c = \{1\dots n\}\setminus I_f $$ and similarly for $g$. Notice that, because of our choice of $l_1$, we have $$ d(\L_{I_f},\L_{I_g}) \ge \e L - 2l_1 \ge \e L/2 \Eq(WME.1bis) $$ Now we write, using the DLR equations $$ \mu_\L^\t(f,g) = \sum_{(\s,\s')} \mu_\L^\t(\s)\mu_\L^\t(\s') f(\s)[\mu_{\L_{I_f^c}}^\s(g) - \mu_{\L_{I_f^c}}^{\s'}(g) ] \Eq(WME.2) $$ so that, using a standard interpolation argument (see e.g. the proof of proposition 4.3 in \ref[CMM]), $$ \sup_\t | \mu_\L^\t(f,g) | \le \|f\|_\infty \|g\|_\infty \sum_{y\in \L_{I_f}} \sup_\t \Var(\mu_{\L_{I_f^c}, \L_{I_g}}^\t, \mu_{\L_{I_f^c}, \L_{I_g}}^{\t^y}) \Eq(WME.3) $$ Notice that, because of \equ(WME.1bis), the pair $(I_f^c, I_g)$ belongs to ${\cal I}_\e$. Thus, thanks to \equ(WME.1), we get $$ \hbox{RHS of \equ(WME.3) } \le \|f\|_\infty \|g\|_\infty L^2 \nep{-2\a L} \le \|f\|_\infty \|g\|_\infty |\L_f||\L_g|\nep{-\a d(\L_f,\L_g)} \Eq(WME.4) $$ namely $SMT(\L,\e L,\a)$, for any $J\in \Th(\a,\e)$ and any $L$ large enough. In conclusion we have proved that, for any $L$ large enough, $$ \bP\bigl\{ SMT(Q_L, \e L, \a) \hbox{ holds }\bigr\} \ge \bP\bigl\{\Th(\a,\e)\bigr\} $$ Let us estimate from above $\bP\bigl\{\Th(\a,\e)^c\bigr\}$. We write $$ \bP\bigl\{\Th(\a,\e)^c\bigr\} \le |{{\cal I}_\e}|^2\sup_{(I_1,I_2)\in {{\cal I}_\e}}|\L\setminus \L_{I_2}| \sup_{y\in \L\setminus \L_{I_2}} \bP\bigl\{\sup_\t \Var(\mu_{\L_{I_2},\L_{I_1}}^\t, \mu_{\L_{I_2},\L_{ I_1}}^{\t^y}) \ge \nep{-2\a L}\bigr\} \Eq(WME.5) $$ Using the standard Chebyshev inequality we can bound from above the probability appearing in the r.h.s of \equ(WME.5) by $$ \bP\bigl\{\sup_\t \Var(\mu_{\L_{I_2},\L_{I_1}}^\t, \mu_{\L_{I_2},\L_{ I_1}}^{\t^y}) \ge \nep{-2\a L}\bigr\} \le \nep{2\a L}\bE\bigl\{\sup_\t \Var(\mu_{\L_{I_2},\L_{I_1}}^\t, \mu_{\L_{I_2},\L_{ I_1}}^{\t^y}) \bigr\} \Eq(WME.6) $$ We are finally in a position to use our key hypothesis $WME(C,m)$. We know in fact, thanks to theorem \thm[V.d.Berg], that $WME(C,m)$ implies that there exist $L_0$, $C'$ and $m'>0$ such that $SME(C',m')$ holds for all sets that are multiples of the square $Q_{L_0}(0)$. Thus, if we choose $l_0=L_0$, we get that all sets of the form $\L_I$ for some $I\sset \{1\dots n\}$ are \lq\lq multiple " of $Q_{L_0}(0)$ since $l_1$ and $L$ are both multiple of $l_0$. Thus, using the definition of $SME(C',m')$, we get that $$ \hbox{RHS of \equ(WME.6) }\le L^2\nep{ 2\a L}C'\nep{-m' {\e L/2}} $$ If we finally plug the above bound into the r.h.s of \equ(WME.5) and use the trivial bounds $$ |{{\cal I}_\e}| \le 2^{(L/l_1)^2}\le 2^{L^{2/3}}\, ,\qquad \sup_I |\L\setminus \L_{I}| \le L^2 $$ we obtain $$ \bP\bigl\{\Th(\a,\e)^c\bigr\} \le 2^{L^{2/3}}L^4 \nep{ 2\a L}C'\nep{-m' {\e L/2}} \Eq(WME.7) $$ The theorem now follows with e.g. $\th = \a$ and $4\a = \e m'$ . \QED \bigno \bigno %@I1 %--------------------------------- INIZIO \expandafter\ifx\csname sezioniseparate\endcsname\relax% \input macro \input xmac\fi %@E1 \numsec=4 \numfor=1 \numtheo=1 \pgn=1 \beginsection 4. Applications We have shown in lemma \thf[WME] that in $d=2$ the hypothesis (H) of theorem \thf[UB] can be replaced by $WME(C,m)$. In this final section we will check condition $WME(C,m)$ for the dilute Ising model (ferromagnetic or not). In particular we show that, for the standard ferromagnetic case without external field $h$ in which the random couplings $J_b$ assume only two values, $J_b= \JJ$ with probability $p$ and $J_b=0$ with probability $1-p$, the results on the dynamics explained in section 2.2 apply even when $p>p_c$ and $\JJ$ is larger (but not too large) than the critical value $J_c$ for the corresponding homogeneous system. Here $p_c=1/2$ is the critical value for Bernoulli bond percolation in two dimensions. \bigno \nproclaim Theorem [Corr]. Assume \smallno \item{(a)} $h_x=0$ for all $x\in \Z^d$ \item{or} \item{(b)} $h_x\ge 0$ and $J_b\ge 0$ for all $x\in \Z^d$ and all $b\in \hat \Z^d$. \smallno Then any of the following two conditions implies that there exist two positive constants $C$ and $m$ such that $WME(C,m)$ holds. \smallno \item{(i)} $\bE\bigl\{|J_{xy}|\bigr\} < J_c$ \item{(ii)} $\bE\bigl\{(1-\nep{-|J_{xy}|})\bigr\} < p_c$ \smallno where $J_c$ and $p_c$ are the critical inverse temperature of the ordinary Ising model ($h_x\equiv 0$ and $J_b=J>0$ in \equ(finvolH)) and the critical bond percolation probability in d--dimensions respectively. \bigno {\it Proof of Theorem \thf[Corr]}. \acapo {\it (a)} Using the Fortuin-Kasteleyn representation for the Ising model in absence of external field, one can prove the following very nice inequality (see \ref[N]) $$ \sup_{\t,\t'} \Var(\mu_{\L,\D}^{J,\t}, \mu_{\L,\D}^{J,\t'}) \le 2\sum_{x\in \D} \mu_{\L}^{|J|,+}(\s(x)) \Eq(Corr.1) $$ where $\mu_{\L}^{|J|,+}$ denotes the Gibbs measure in $\L$ with plus boundary conditions and couplings $\{|J_b|\}_{b\in \hat \L}$. Since the Gibbs measure $\mu_{\L}^{|J|,+}$ is ferromagnetic, we can use at this point an inequality in Section 2 of \ref[Hi] which states that $$ \mu_{\L}^{|J|,+}(\s(x)) \le \sum_{y\notin \L}\sum_{z\in \L \atop |z-y| =1}\mu_{\L}^{|J|,f}(\s(x)\s(z)) \Eq(Corr.2) $$ where the superscript $f$ in $\mu_{\L}^{|J|,f}$ denotes free boundary conditions. Thus, in order to prove the theorem, it is sufficient to show that both (i) or (ii) imply that there exist two positive constants $C$ and $m$ such that $$ \bE\bigl\{ \mu_{\L}^{|J|,f}(\s(x)\s(z))\bigr\} \le C\nep{-m|x-z|} \Eq(Corr.2bis) $$ Let us start with (i). In this case we can use an old result \ref[OPG] that states that $$ \bE\bigl\{ \mu_{\L}^{|J|,f}(\s(x)\s(z))\bigr\} \le \mu_{\L}^{\bE\{|J|\},f}(\s(x)\s(z)) \Eq(Corr.2tris) $$ where $\mu^{\bE\{|J|\},f}$ denotes standard Ising Gibbs measure in $\L$ with zero external field, free boundary conditions and coupling $\bE\{|J_b|\}$ . Thanks to the second Griffiths inequality $$ \hbox{R.H.S. of \equ(Corr.2tris)} \le \mu^{\bE\{|J|\}}(\s(x)\s(z)) \le C\nep{-m|x-z|} \Eq(Corr.3) $$ for suitable $C$ and $m$. Here $\mu^{\bE\{|J|\}}$ denotes the infinite volume limit as $ \L\to \Z^d$ of $\mu_\L^{\bE\{|J|\},f}$ and in the last inequality in the r.h.s of \equ(Corr.3) we used the fact that $\bE\{|J|\} < J_c$ implies exponential decay of infinite volume correlations (see \ref[ABF]). \smallno Let us now consider case (ii). Here we can use another result on the FK representation for the random Ising model (see \ref[ACCN] and also \ref[N]) which states that $$ \bE\bigl\{ \mu_{\L}^{|J|,f}(\s(x)\s(z))\bigr\} \le \hbox{Prob}_p(x \to z) \Eq(Corr.4) $$ where $p\equiv \bE\bigl\{(1-\nep{-\beta |J_{xy}|})\bigr\}$ and $\hbox{Prob}_p(x \to z)$ denotes the probability for independent bond percolation with occupation density $p$ that there exists a path of occupied bonds starting in $x$ and ending in $z$. Since by assumption $p< p_c$ this last probability is exponentially small in $|x-y|$ (see e.g. \ref[Gri] and references therein) and \equ(Corr.2bis) follows. \smallno {\it (b)} In the ferromagnetic case $J_b\ge 0$ with non--negative external field $h_x$, it has been proved in section 2 of \ref[Hi] that $$ \sup_{\t,\t'} \Var(\mu_{\L,\D}^{J,\t}, \mu_{\L,\D}^{J,\t'}) \le 2\sum_{x\in \D} \mu_{\L}^{J,h=0,+}(\s(x)) \Eq(Corr.5) $$ At this point we proceed exactly as before. \QED \bigno As a corollary to the above theorem we get the following result for the two dimensional dilute Ising ferromagnet. \bigno \nproclaim Corollary [Dilu]. In the same setting of theorem \thf[Corr], let us consider the two dimensional case and let us assume that the couplings $J_b$ take only the values $J_b=0$ and $J_b=\JJ$ with probability $1-p$ and $p$ respectively (dilute Ising model). Then if $$ \JJ < \cases{{J_c\over p}\mmax \log ({p\over 2p-1}) & if $p> 1/2$ \cr \infty & if $p\le 1/2$} $$ hypothesis (H) holds and theorem \thf[UB] applies. \noindent {\it Proof of the Corollary}. If $\JJ$ is smaller than the stated value then either (i) or (ii) of theorem \thf[Corr] holds since $p_c=1/2$ for $d=2$. Thus in two dimensions we can apply theorem \thf[WME] above and get the result. \QED \bigno {\it Remark}. Using the Fortuin--Kesteleyn representation, it has also been shown in \ref[ACCN] that the dilute Ising magnet exihibits the phenomenon of spontaneous magnetization provided that $$ \bE\, \Bigl( {1-\nep{-J_b}\over 1+\nep{-J_b}} \Bigr) > p_c $$ so that the infinite volume Gibbs state is non--unique and the associated dynamics is no longer ergodic. \bigno We conclude by observing that if $\JJ$ is smaller than the critical value $J_c$, then, thanks to the Griffiths inequality, $$ \mu_{\L}^{J,h=0,f}(\s(x)\s(z)) \le C\nep{-m|x-z|} $$ for {\it any} configuration of $J$. Thus, in this case, we can apply theorem 3.1 of \ref[MO1] and get that the relaxation to equilibrium is purely exponential for {\it any} configuration of $J$ and any value of $p$. In order to show that a dynamical phase transition occurs when $\JJ$ crosses the critical value $J_c$, we need a lower bound like the one given in theorem \thf[UB] for {\it any } $\JJ > J_c$. This is actually our last result. \bigno \nproclaim Proposition [Dilu-2]. Under the same hypotheses of corollary \thf[Dilu] let us also suppose that $\JJ > J_c$. Let $f(\s)=\s(0)$. Then for any $\d>0$ there exists $t_0 < \infty$ such that for all $t\ge t_0$ $$ \bE \, \|T^{J}(t)f-\mu^{J}(f)\|_{\Ld} \geq \nep{- k(\d, p,\JJ)(log t)^{2}} $$ where $k(\d, p,\JJ) = \log (1/p)({\JJ}\t(\JJ)(1-3\d))^{-2}$ with $\t(\JJ)$ the surface tension at inverse temperature $\JJ$ in the direction of the coordinate axes for the standard ferromagnetic two dimensional Ising model. \bigno \Pro\ We follow closely the proof of the lower bound of theorem \teo[3.3] given in \ref[CMM]. Since $\JJ$ satisfies the hypothesis of corollary \thf[Dilu], there exists a unique Gibbs state $\mu^J$ with zero spontaneous magnetization. Therefore, in order to prove the result, it is sufficient to prove the required lower bound only for $\bE\, \|T^{J}(t)f\|_{\Ld}$. For this purpose let, for $\L=Q_L$, $\Thb$ be the set of all interactions $J\in \Th$ such that \smallno \item{(a)} $J_{xy} = \JJ$ for all $\xy$ such that $\xy \sset \L$ \item{(b)} $J_{xy}=0$ for all $\xy$ which intersect both $\L$ and $\L^c$ (the boundary edges) \smallno and let $$ \txt F_\L = \{ \s\in\O : \, m_\L(\s) > \ov2 \} $$ where $m_{\L}(\s )$ denotes the normalized magnetization in $\L$ of the configuration $\s$. We can then use the following estimate that was established in \ref[CMM] (see formula (6.25)$\dots$ (6.28) there) $$ \eqalign{ & \bE\, \|T^{J}(t)f\|_{\Ld} \ge \cr & \ge \bP( \Thb ) \, \Bigl[ \, \ov2 - {3\over 2} \mu_{\L}^{\JJ,f}(F_\L)^{-1} \, \Bigl( \, k \, |\L|\, t \; \mu_{\L}^{\JJ,f} \{ |m_{\L}(\s )-1/2|\le 1/(100) \} + \nep{-k' |\L| t} \, \Bigr) \, \Bigr] \cr } \Eq(dilu2.1) $$ for suitable positive constants $k,k'$. Here $\mu_{\L}^{\JJ,f}$ denotes the ordinary Ising Gibbs measure in $\L$ with free boundary conditions and coupling $\JJ$. We need at this point the following large deviation result (see \ref[CGMS]) \bigno \nproclaim Theorem [CGMS]. Let $\JJ > J_c$. Then for any $\d>0$ there exists $L_0(\d)$ such that for all $L\ge L_0(\d)$ $$ \mu_{\L}^{\JJ,f} \{ \, |m_{\L}(\s )-1/2|\le 1 \, \} \le \nep{ -\JJ \t(\JJ)(1-\d)L } \quad ;\quad \mu_{\L}^{\JJ,f} \{\, m_{\L}(\s )\ge 1/2 \, \} \ge 1/3 $$ where $\t(\JJ)$ denotes the surface tension at inverse temperature $\JJ$ in the direction of the coordinate axes for the ordinary two dimensional Ising model. \bigno Choose now $\d<1/4$. Using theorem \thf[CGMS], we see that if $t$ is large enough and $L$ is larger than \hbox{$\JJ \t(\JJ)(1-2\d)^{-1}\log t$} then $$ \hbox{RHS of \equ(dilu2.1)} \ge {1\over 3} \, p^{L^2}(1-p)^{4L} \ge \nep{- k(\d, p,\JJ)(\log t)^2} \QED $$ \bigno \beginsection References \frenchspacing \item{[AAPS]} %% R.~G.~Palmer, D.~L.~Stein, E.~Abrahams and P.~W.~Anderson: {\it Models of hierarchically constrained dynamics for glassy relaxation}. Phys. Rev. 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Lett. {\bf 54}, No. 12, 1321, (1985) \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 \end