\documentstyle[12pt]{article} \newcommand {\Ebar} {{\mbox{\rm$\mbox{I}\!\mbox{E}$}}} \newcommand {\Rbar} {{\mbox{\rm$\mbox{I}\!\mbox{R}$}}} \newcommand {\Nbar} {{\mbox{\rm$\mbox{I}\!\mbox{N}$}}} \newcommand {\Cbar} {\mathord{\setlength{\unitlength}{1em} \begin{picture}(0.6,0.7)(-0.1,0) \put(-0.1,0){\rm C} \thicklines \put(0.2,0.05){\line(0,1){0.55}} \end {picture}}} \newcommand {\Qbar} {\mathord{\setlength{\unitlength}{1em} \begin{picture}(0.6,0.7)(-0.1,0) \put(-0.1,0){\rm Q} \thicklines \put(0.2,0.05){\line(0,1){0.55}} \end {picture}}} \newsavebox{\zzzbar} \sbox{\zzzbar} {\setlength{\unitlength}{0.9em} \begin{picture}(0.6,0.7) \thinlines % \put(0,0){\framebox(0.6,0.7){}} \put(0,0){\line(1,0){0.6}} \put(0,0.75){\line(1,0){0.575}} \multiput(0,0)(0.0125,0.025){30}{\rule{0.3pt}{0.3pt}} \multiput(0.2,0)(0.0125,0.025){30}{\rule{0.3pt}{0.3pt}} \put(0,0.75){\line(0,-1){0.15}} \put(0.015,0.75){\line(0,-1){0.1}} \put(0.03,0.75){\line(0,-1){0.075}} \put(0.045,0.75){\line(0,-1){0.05}} \put(0.05,0.75){\line(0,-1){0.025}} \put(0.6,0){\line(0,1){0.15}} \put(0.585,0){\line(0,1){0.1}} \put(0.57,0){\line(0,1){0.075}} \put(0.555,0){\line(0,1){0.05}} \put(0.55,0){\line(0,1){0.025}} \end{picture}} \newcommand{\Zbar}{\mathord{\!{\usebox{\zzzbar}}}} \newsavebox{\uuunit} \sbox{\uuunit} {\setlength{\unitlength}{0.825em} \begin{picture}(0.6,0.7) \thinlines % \put(0,0){\framebox(0.6,0.7){}} \put(0,0){\line(1,0){0.5}} \put(0.15,0){\line(0,1){0.7}} \put(0.35,0){\line(0,1){0.8}} \multiput(0.3,0.8)(-0.04,-0.02){12}{\rule{0.5pt}{0.5pt}} \end {picture}} \newcommand {\unity}{\mathord{\!\usebox{\uuunit}}} \newcommand{\QED}{{\hspace*{\fill}\rule{2mm}{2mm}\linebreak}} \newcommand{\integ}{\Zbar} \newcommand{\reals }{\Rbar} \renewcommand{\L}{{\cal L}} \renewcommand{\theequation}{\thesection.\arabic{equation}} \setlength{\textheight}{21cm} \setlength{\textwidth}{16cm} \oddsidemargin 0.0in \evensidemargin 0.0in \topmargin 0.0in \pagestyle{plain} %\renewcommand{\baselinestretch}{2} \newtheorem{definition}{Definition} \newtheorem{prop}{Proposition} \newtheorem{lemma}{Lemma} \begin{document} \title{Relative energies for non-Gibbsian states} %\renewcommand{\baselinestretch}{1} \author{Christian Maes \thanks{ Onderzoeksleider N.F.W.O. Belgium \hfill \newline Research supported by EC grant CHRX-CT93-0411} \hspace{.14cm} and Koen Vande Velde \\ Instituut voor Theoretische Fysica \\ K.U. Leuven \\ Celestijnenlaan 200D\\ B-3001 Leuven (Belgium) \\ e--mail: Christian.Maes@fys.kuleuven.ac.be \\ e--mail: Koen.VandeVelde@fys.kuleuven.ac.be \\} \date{} \maketitle \begin{center} In memory of Roland Dobrushin \end{center} \begin{abstract} \noindent We investigate the suggestion of R.L. Dobrushin, that for many examples of non-Gibbsian measures, a well-defined interaction potential can still be derived. \\ Such an interaction cannot be absolutely summable, uniformly in all configurations. Rather there will exist a typical set of configurations on which the interaction decays sufficiently. \\ We are showing how to complete this program, using quite elementary methods, for the example of the restriction of the Ising pure phases to one layer of the lattice. We also treat the Ising pure phases under Kadanoff transformation with block-size one and a toy model. \\ We sketch a general setup, showing that there is an intimate relation between the problem treated here and the decay of correlations in certain disordered systems. \end{abstract} {\bf keywords:} Gibbsian formalism, relative energies, non-Gibbsian measures, \\ \hspace*{2cm} renormalization-group pathologies. \newpage \section{Introduction} Ever since the first appearance of natural' examples of non-Gibbsian measures \cite{GP,LM,LS} in simple models of statistical mechanics, there has been the feeling that in many cases the non-Gibbsianness is very weak. After all, all that matters is that one can make sense of relative energies somehow. These relative energies are indeed the objects most people concentrate on to parametrize or to investigate the effect of transformations of equilibrium states. This was especially emphasized by R.L. Dobrushin, who pointed out that the pathologies' of certain transformations \cite{EFS} are not so worrisome as one might think at first. He explained that, in practice one would not see much of these pathologies, if the non-Gibbsian aspect is due only to special configurations of the system, which are very untypical anyway. In \cite{D} he has shown (at least for one example \cite{SCH}) how relative energies can be defined for typical configurations. \\ In this paper, we investigate his suggestion for three examples ( including the model treated in \cite{D}), where we show by quite elementary methods how to complete his program. The simplification of our method with respect to the one in \cite{D} lies in the way we construct the interaction potential. For this we have benefitted from discussions with J. Bricmont, who together with A. Kupiainen \cite{BK} is using very similar methods to investigate the statistical mechanics of infinite dimensional dynamical systems. \\ As will become clear, an interesting relation (also observed by J. Bricmont) follows between non-Gibbsian aspects and certain singularities in disordered systems. These singularities show up if one considers atypical realizations of the disorder, not allowing a nice decay of the correlation functions uniformly in the set of all disorder realizations. \\ The paper is organized as follows. In section 2 we sketch a general setup and make the link to problems in disordered systems. Section 3 is devoted to the construction of an interaction for the projection of the Ising phases. In Section 4 we make a similar construction for the images of the Ising pure phases under Kadanoff transformation with block-size one. In Section 5 we study a toy model, which illustrates nicely the general philosophy. \section{General setup} \setcounter{equation}{0} The main problem of the paper is how to construct interactions for states that are obtained from Gibbs states via some transformation. For more details on the relationship between Gibbs states and interactions we refer to \cite{EFS,G}. \\ Let $\mu$ be a probability measure on $\Omega _1:=\{ +,- \} ^\Lambda$ for some finite volume $\Lambda \subset \integ ^d$. From $\mu$ we obtain another state $\nu$ on $\Omega _2:=\{ +,- \} ^V$ ($V\subset \Lambda$) via a stochastic transformation $T$. \\ $T$ is a stochastic matrix, i.e.: \begin{eqnarray*} && T(\xi,\sigma ) \geq 0, \hspace{1cm} \sigma \in \Omega _1,\xi \in \Omega _2 \\ && \sum_{\xi \in \Omega _2} T(\xi ,\sigma )=1 \end{eqnarray*} and $\mu T= \nu$, i.e.: $\sum_{\sigma \in \Omega _1} T(\xi ,\sigma )\mu (\sigma)=\nu (\xi )$ Let $H$ be some (energy) function on $\Lambda$ and $\beta \geq 0$. Think of $\mu$ as a Gibbs measure with respect to the Hamiltonian $H$ at inverse temperature $\beta$. Of course, for finite volumes $\Lambda$ and $V$, $\nu$ will always be Gibbsian. The problem is then to see whether and in what sense the Gibbsian character is preserved as the volumes tend to infinity. \\ We assume that for all sets $A\subset V$ there is a set $B_A\subset \Lambda$ such that \begin{equation} \sum_{\sigma \in \Omega _1} T(\xi ^A,\sigma )\mu (\sigma) =\sum_{\sigma \in \Omega _1} T(\xi ,\sigma )\mu (\sigma) \exp [-\beta \Delta _{B_A} H(\sigma )] \label{vw} \end{equation} where, for any sets $A\subset V$ and $B\subset \Lambda$ \begin{eqnarray*} && \xi ^A(x)=-\xi (x), \hspace{1cm} x\in A \\ && \xi ^A(x)=\phantom{-} \xi (x), \hspace{1cm} x\not\in A \\ && \Delta _{B} H(\sigma )=H(\sigma ^{B})-H(\sigma ) \end{eqnarray*} the relative energy with respect to a spin-flip in $B$. (\ref{vw}) is satisfied whenever \begin{eqnarray} && \frac{\mu (\sigma ^B)}{\mu (\sigma )}= \exp [-\beta \Delta _B H(\sigma )] \nonumber \\ && \mbox{and } T (\xi ^A,\sigma )=T(\xi ,\sigma ^{B_A}) \end{eqnarray} In particular this is the case when $T$ is a projection or decimation transformation, because then \begin{equation} T(\xi ,\sigma )=\prod _{x\in V} \frac{1+\xi (x)\sigma (x)}{2} \end{equation} and we can take $B_A=A$. \\ It holds also for (real) stochastic transformations like the Kadanoff transformation \cite{Ka}, where $\Lambda$ is divided into disjoint blocks $B_x,x\in V$ such that $\cup _{x\in V}B_x=\Lambda$. The transformation is \begin{equation} \label{kada} T(\xi ,\sigma )=\prod _{x\in V}\frac{\exp [p\xi (x)\sum_{y\in B_x}\sigma (y)]}{2\cosh [p\sum_{y\in B_x}\sigma (y)] } \end{equation} $00\}$. Let $\mu ^+_n$ be the Ising measure on the volume $\Lambda _n$ with $+$ boundary conditions: \begin{eqnarray} && H^+_n(\sigma ):= - \sum _{\langle xy \rangle \subset \Lambda _n }(\sigma (x)\sigma (y)-1) -\sum _{\langle xy\rangle \atop x\in \Lambda _n;y \in \Lambda _n^c} (\sigma (x)-1) \\ && \mu ^+_n(\sigma ):= \frac{1}{Z^+_n(\beta )} \exp (-\beta H^+_n(\sigma )), \end{eqnarray} where $Z^+_n(\beta )$ is the usual normalizing partition function. \\ Denote with $\nu ^+_n$ the restriction of $\mu ^+_n$ to $V_n$. \\ The relative energy for a spin-flip at the origin is (see (\ref{spinflip})) \begin{equation} h^+_n(\xi )=\log \frac{\nu ^+_n(\xi )}{\nu ^+_n(\xi ^o)} \label{rele} \end{equation} A small calculation shows that \begin{equation} h^+_n(\xi )= 2\beta \xi (0)(\xi (1)+\xi (-1))+ 2\log \frac{Z^{+,\xi }_n(\beta )}{Z^{+,\xi ^o}_n(\beta )}, \end{equation} where \begin{equation} Z^{+,\xi }_n(\beta ):=\sum _{\sigma (x)=\pm,x\in \Gamma _n} \exp (-\beta H^{+,\xi }_n(\sigma )) \end{equation} with $H_n^{+,\xi }$ the Ising Hamiltonian on the volume $\Gamma _n$ with $+$ boundary conditions on $\Gamma _n^c\backslash \integ$ and $\xi$ boundary conditions on $\integ$: \begin{eqnarray} && H^{+,\xi }_n(\sigma ):= - \! \! \! \sum _{\langle xy \rangle \subset \Gamma _n }(\sigma (x)\sigma (y)-1) -\!\! \! \! \sum _{\langle xy\rangle \atop x\in \Gamma _n;y \in \Lambda _n^c} (\sigma (x)-1) -\sum _{x \in V_n} (\xi (x)\sigma (x')-1) \end{eqnarray} where $\sigma (x')$ is the neighboring spin in $\Gamma _n$ of $\xi (x)$. The relative energy (\ref{rele}) can be expanded using the telescopic identity (as in \cite{BK}) \begin{equation} h^+_n(\xi)=\sum_{k=1}^{2n+1}(h^+_n(\xi_k)-h^+_n(\xi_{k-1})) + \Phi _0^n, \label{confituur} \end{equation} where $\xi_k(x)=\xi (x)$ if $x\in D_k$ and $+$ otherwise and $D_k, k\in \Nbar$ is defined by $D_{2l+1}=[-l,l]\cap \integ$ and $D_{2l}=[-l+1,l]\cap \integ$ except for $D_0=\emptyset$. \\ The constant $\Phi _0^n$ in (\ref{confituur}) is $h_n^+({\bf +})$ with ${\bf +}(x)=+, \forall x$. This is the vacuum or reference configuration for which we {\em a priori} know that $\lim _n h_n^+({\bf +})$ makes sense. Now define the interaction potential \begin{equation} \Phi ^n_k(\xi ):=h^+_n(\xi_k)-h^+_n(\xi_{k-1}) \end{equation} for $k\geq 1$. More explicitly, \begin{eqnarray} && \Phi^n_1(\xi )=h^+_n(\xi _1)-h^+_n(+)=(\xi (0)-1)h^+_n(+) \nonumber \\ && \Phi^n_2(\xi )+\Phi^n_3(\xi )=2\beta \xi (0)(\xi (1)+\xi (-1)) -4\beta \xi (0)+ \nonumber \\ && 2\log \frac{Z_n^{+,\xi _2}(\beta )Z_n^{+,\xi ^o_{1}}(\beta )} {Z_n^{+,\xi ^o_2}(\beta )Z_n^{+,\xi _{1}}(\beta )} +2\log \frac{Z_n^{+,\xi _3}(\beta )Z_n^{+,\xi ^o_{2}}(\beta )} {Z_n^{+,\xi ^o_3}(\beta )Z_n^{+,\xi _{2}}(\beta )} \label{if} \end{eqnarray} \begin{equation} \Phi^n_k(\xi )=2\log \frac{Z_n^{+,\xi _k}(\beta )Z_n^{+,\xi ^o_{k-1}}(\beta )}{Z_n^{+,\xi ^o_k}(\beta )Z_n^{+,\xi _{k-1}}(\beta )} \label{fi} \end{equation} for all $k>3$. Let $a_k=D_k\backslash D_{k-1}$. Observe that (\ref{fi}) equals zero whenever $\xi _k=\xi _{k-1}$, i.e. whenever $\xi (a_k)=+$. %At the same time we can freely choose $\xi (0)=-1$. So we can and %will assume while discussing $\Phi _k^n(\xi )$ that $\xi (0)=\xi (a_k)=-$ \\ As pointed out in the general setup, we can write the potential in terms of correlation functions. For any $\xi \in \Omega$ and any set $C\subset \Gamma _n$, let $\mu _C^{+,\xi}$ be the measure on $C$ with $\xi$ boundary conditions on $V_n$ and $+$ boundary conditions on the rest of $\partial C$. When $C=\Gamma _n$, we just write $\mu _n^{+,\xi }$. One readily sees that from (\ref{fi}) that for $k>3$ \begin{equation} \Phi ^n_k(\xi )=(1-\xi (a_k))\log \frac{\mu _n^{+,\xi _k}( e^{-2\beta \xi (0) \sigma (0,1)} e^{2\beta \sigma (a_k,1)})} {\mu _n^{+,\xi _k}(e^{-2\beta \xi (0) \sigma (0,1)} )\mu _n^{+,\xi _k} ( e^{2\beta \sigma (a_k,1)})} \label{corr} \end{equation} Our potential for $\nu ^+$ will then be $\{ \Phi_k \} _k$ \begin{equation} \Phi_k (\xi )=\lim _n \Phi^n_k(\xi ) \label{rroc} \end{equation} The existence of this limit can be proven for all $\xi \in \Omega$ by a monotonicity argument. We want to show that there exists a set $\Omega ^+\subset \Omega$, with $\nu ^+(\Omega ^+)=1$, such that for $\xi \in \Omega ^+$, $\Phi^n_k (\xi )$ is exponentially small in $k$, uniformly in $n$. Explicitly, we choose \begin{eqnarray} && \Omega _l^+:=\{ \xi \in \Omega | \forall k\geq l ,\,\,\,\frac{1}{k} \sum_{x \in D_k} \xi (x) \geq 3/4 \} \nonumber \\ && \Omega ^+:=\bigcup _l \Omega _l^+ \end{eqnarray} The $3/4$ is of course arbitrary. The most important thing is that \begin{lemma} If $\beta$ is large enough, then $\nu ^+(\Omega ^+)=1$ \end{lemma} Proof: \\ The proof is easy using that for $\beta$ large enough, there exists $\alpha =\alpha (\beta )>0$ such that \begin{equation} \nu^+(\{ \xi \in \Omega | \frac{1}{k} \sum_{x \in D_k} \xi (x) < 3/4 \}) l$,$\delta (\beta )\uparrow \infty $as$\beta \uparrow \infty $\item$\sum _{k=0}^{2n+1} |\Phi _k^n(\xi )| \leq c(\xi ) < \infty$for all$\xi \in \Omega ^+$; \end{enumerate} } \noindent The first statement of Theorem 1 (existence of the limit) was already observed after (\ref{corr})-(\ref{rroc}). The proof of 2,3 and 4 follows from combining (\ref{corr}) with \begin{prop} \label{p2} Take$f_m(\sigma )=\exp [-2\beta \xi (m)\sigma (m,1)]$. \\ For$\beta $large, there exists$\delta =\delta (\beta )$($\delta \uparrow \infty $as$\beta \uparrow \infty $), such that for$\xi \in \Omega _l^+$and$2n+1\geq k\geq l$\begin{equation} |\mu _n^{+,\xi _k}( f_0f_{a_k} )-\mu _n^{+,\xi _k}(f_0)\mu _n^{+,\xi _k} (f_{a_k})| \leq \mbox{const}(\beta ) e^{-\delta k} \end{equation} \end{prop} We introduce the following notation. For any two sets$A,B\subset \integ ^2$, define the event$A\rightarrow B=\{ (\sigma ,\sigma ')\in \Theta \times \Theta |$there is a path$\pi $from$A$to$B$such that$(\sigma (x),\sigma '(x)) \not= (+,+)$for all$x \in \pi \} $. A path from$A$to$B$is a sequence$x(0)\in A$,$x(1),\ldots ,x(n)\in B$of consecutive nearest neighbor sites. \\ Before proving Proposition \ref{p2} we need the following \begin{lemma} \label{mmm} If$\beta $is large enough there exists$\delta =\delta (\beta )$($\delta \uparrow \infty $as$\beta \uparrow \infty $), such that for$\xi \in \Omega _l$and$k\geq l$\begin{equation} \mu _n^{+,\xi _k} \times \mu _n^{+,\xi _k}[(0,1)\rightarrow \Lambda _k^c] \leq \mbox{const}(\beta ) e^{-\delta k} \end{equation} \end{lemma} Proof: \newline \newline First note that for$\xi _k$with$\xi \in \Omega _k^+$only one spin out of eight can be minus in$D_k$and thus \begin{eqnarray} && \mu _n^{+,\xi _k} \times \mu _n^{+,\xi _k}[(0,1)\rightarrow \Lambda _k^c] \leq \exp [4\beta \sum _{x\in D_k}(1-\xi (x))] \, \mu _n^{+,+} \times \mu _n^{+,+}[(0,1)\rightarrow \Lambda _k^c] \nonumber \\ && \phantom{\mu _n^{+,\xi _k} \times \mu _n^{+,\xi _k}[(0,1)\rightarrow \Lambda _k^c]} \leq e^{\beta k} \, \mu _n^{+,+} \times \mu _n^{+,+}[(0,1)\rightarrow \Lambda _k^c] \end{eqnarray} We can then use Proposition 2.4 in \cite{BS} (with some trivial modifications) to conclude that for$\beta $large \begin{equation} \mu _n^{+,\xi _k} \times \mu _n^{+,\xi _k}[(0,1)\rightarrow \Lambda _k^c] \leq \mbox{const}(\beta )e^{-\delta k} \end{equation} and$\delta \uparrow \infty $as$\beta \uparrow \infty $. \QED \newline Let$\mu _{k}^{+,\xi,\eta }$be the measure on$\Gamma _k$with$\xi $boundary condition on$V_k$and$\eta $boundary condition on$\partial \Lambda _k^c$. \newline \newline Proof of Proposition \ref{p2}: \newline For any configuration$\xi \in \Omega $we have that \begin{eqnarray} && |\mu _n^{+,\xi }( f_0f_{a_k} )-\mu _n^{+,\xi }(f_0)\mu _n^{+,\xi} (f_{a_k})|= \nonumber \\ && |\mu _n^{+,\xi }\left( \mu _n^{+,\xi } (f_{a_k}|\sigma =\eta \, \mbox{on} \, \partial \Lambda _k^c ) [\mu _k^{+,\xi,\eta }(f_0)- \mu _n^{+,\xi }(f_0)] \right)| \nonumber \\ && \leq e^{2\beta }\mu _n^{+,\xi }(|\mu _k^{+,\xi ,\eta }(f_0)- \mu _n^{+,\xi }(f_0)|) \nonumber \\ && = e^{2\beta }\mu _n^{+,\xi }(| \mu _k^{+,\xi ,\eta }\times \mu _n^{+,\xi }(f_0\times \unity -\unity \times f_0)|) \label{moe} \end{eqnarray} The rest of the argument is similar to the ideas in \cite{BM}. Pick a configuration$(\sigma ,\sigma ')$from the distribution$\mu _k^{+,\xi ,\eta }\times \mu _n^{+,\xi }$. If there is no path from$(0,1)$to$\Lambda _k^c$on which$(\sigma ,\sigma ')\not \equiv (+,+)$, there exists a$*$-chain around$(0,1)$separating it from$\Lambda _k^c$and on which$\sigma \equiv \sigma '$. This chain has a part in$\Gamma _k$on which$(\sigma ,\sigma ') \equiv (+,+)$and a part in$V_k$on which lives the configuration$\xi $and$(\sigma ,\sigma ')\equiv (\xi ,\xi )$. It follows that there exists a maximal$*$-chain$\Delta $with this property inside$\Lambda _k$(maximal in the sense that it is contained in no other$*$-chain). Therefore \begin{eqnarray} && \mu _k^{+,\xi ,\eta }\times \mu _n^{+,\xi } (f_0\times \unity -\unity \times f_0)= \\ && \nonumber \\ && \sum _{* \mbox{\scriptsize{-chains }} \Delta \atop \mbox{\scriptsize{around }} (0,1)} \mu _k^{+,\xi ,\eta }\times \mu _n^{+,\xi } (f_0\times \unity -\unity \times f_0|\Delta ) \, \mu _k^{+,\xi ,\eta }\times \mu _n^{+,\xi } (\Delta )+ \nonumber \\ && \mu _k^{+,\xi ,\eta }\times \mu _n^{+,\xi } (f_0\times \unity -\unity \times f_0|(0,1)\rightarrow \Lambda _k^c )\, \mu _k^{+,\xi ,\eta }\times \mu _n^{+,\xi } [(0,1)\rightarrow \Lambda _k^c] \nonumber \end{eqnarray} Now since$\sigma =\sigma '$on$\Delta $\begin{equation} \mu _k^{+,\xi ,\eta }\times \mu _n^{+,\xi } (f_0\times \unity -\unity \times f_0|\Delta ) =0 \end{equation} for every$\Delta $so \begin{equation} \label{ya} |\mu _k^{+,\xi ,\eta }\times \mu _n^{+,\xi } (f_0\times \unity -\unity \times f_0)|\leq 2e^{2\beta } \mu _k^{+,\xi ,\eta }\times \mu _n^{+,\xi } [(0,1)\rightarrow \Lambda _k^c] \end{equation} Lemma \ref{mmm} and (\ref{moe}),(\ref{ya}) now yield that \begin{eqnarray} && |\mu _n^{+,\xi _k}( f_0f_{a_k} )-\mu _n^{+,\xi _k}(f_0)\mu _n^{+,\xi_k} (f_{a_k})| \leq \nonumber \\ && e^{2\beta }\mu _n^{+,\xi _k}[| \mu _k^{+,\xi _k,\eta }\times \mu _n^{+,\xi _k}(f_0\times \unity -\unity \times f_0)|] \leq \nonumber \\ && 2e^{4\beta } \mu _n^{+,\xi _k}\times \mu _n^{+,\xi _k} [(0,1)\rightarrow \Lambda _k^c] \leq \mbox{const}(\beta ) e^{-\delta k} \end{eqnarray} \QED \newline \underline{Remarks:} \begin{itemize} \item The proof of Lemma \ref{mmm} works only for dimension$d=2$. \\ The result is however more general. It can be proven with a little bit more effort, in a way analogous to Lemma 3 for the Ising model under Kadanoff transformation (see Section 4). \item The proof of the previous Proposition does not depend essentially on specific features of the Ising model. The important thing is to be able to prove Lemma \ref{mmm}, but this is expected to hold for a much wider class of measures, see chapter 6 in \cite{G}. \\ A result that comes close to Lemma \ref{mmm} is found in \cite{GLM}. In fact, from \cite{GLM} we know that for all temperatures$T\beta _0$any Gibbs measure$\mu $for the Ising model with zero field and inverse temperature$\beta $is transformed by$T$into a non-Gibbsian measure. This holds for any dimension$d\geq 2$. We restrict ourselves here to the case$d=2$. We show that a potential can nevertheless be constructed, using the same methods as in the previous section. \\ Let$\mu _n^+$be as in the previous section and$\nu _n^+=\mu _n^+T$. The relative energy of$\nu _n^+$for a spin-flip at the origin is given by \begin{eqnarray} && h_n^+(\xi )= \log \frac{Z_n^{+,\xi }}{Z_n^{+,\xi ^o}} \\ && Z^{+,\xi }_n:=\sum _{\sigma (x)=\pm,x\in \Lambda _n} \exp (- H^{+,\xi }_n(\sigma )) \nonumber \\ && H^{+,\xi }_n(\sigma ):= - \beta \sum _{\langle xy \rangle \subset \Lambda _n }(\sigma (x)\sigma (y)-1) -\beta \sum _{\langle xy\rangle \atop x\in \Lambda _n;y \in \Lambda _n^c} (\sigma (x)-1) -p\sum _{x \in \Lambda _n} \xi (x) (\sigma (x)-1) \nonumber \\ && \hspace{2cm} -p\sum _{x \in \Lambda _n} \xi (x) \end{eqnarray} We now choose the volumes$D_k$as follows:$D_0=\emptyset $,$D_1=\{ 0\} $and$D_k$is obtained from$D_{k-1}$by adding one of the points closest to the origin and not already contained in$D_{k-1}$. (The distance of the point$(i,j)$from the origin is given by max$\{ |i|,|j| \} $.) We call$x_k$the point contained in$D_k$but not in$D_{k-1}$. Again$\xi _k$is the configuration taking the values of$\xi $inside$D_k$and$+$outside. We define$\Phi _0^n(\xi ):=h_n^+(+)$and for$k\geq 1$\begin{equation} \Phi ^n_k(\xi ):=h^+_n(\xi_k)-h^+_n(\xi_{k-1}) \end{equation} \begin{equation} \Phi ^n_1(\xi )=h^+_n(+)(\xi (o)-1) \end{equation} and for$k>1$\begin{equation} \Phi ^n_k(\xi )=\frac{1-\xi (x_k)}{2}\log \frac{\mu _n^{+,\xi _k}( e^{-2\beta \xi (o)\sigma (o)} e^{2\beta \sigma (x_k)})} {\mu _n^{+,\xi _k}(e^{-2\beta \xi (o)\sigma (o)} )\mu _n^{+,\xi _k} ( e^{2\beta \sigma (x_k)})} \end{equation} where$\mu _n^{+,\xi }$is the Gibbs state for$H_n^{+,\xi }$. \\ Following \cite{BS} we call a finite set$A\subset \integ ^2$enclosing if$A$and$A^c$are connected sets. For a connected set$B$we denote$\bar{B}$for the union of$B$and the finite components of its complement.$\bar{B}$is always enclosing. There is a one-to-one correspondence between enclosing sets and contours as they are usually defined for the Ising model (see e.g. \cite{SI}). Each contour borders an enclosing set, i.e. for each contour$\gamma $there is an enclosing set$A$such that$A=\mbox{Int} \gamma $(the interior of the contour). \\ Let \begin{eqnarray} && \Theta _l^+:=\{ \xi \in \ \Theta | \forall k\geq l \mbox{ and all$V$, connected sets with } \bar{V}\ni o , |V|\geq k : \nonumber \\ && \hspace{3cm} \frac{1}{|V|} \sum_{x \in V } \xi (x) \geq 1/2 \} \nonumber \\ && \Theta ^+:=\bigcup _l \Theta _l^+ \end{eqnarray} Again$\mu ^+(\Theta ^+)=1$for large enough$\beta $and \newline \newline {\bf Theorem 2} \newline {\em The measure$\nu _n^+$satisfies \begin{equation} \nu _n^+(\xi ^0)=\nu _n^+(\xi )\exp [-\sum _{k=0}^{(2n+1)^2} \Phi _k^n(\xi )] \end{equation} with interaction potential$\{ \Phi _k^n \} _k $satisfying: \begin{enumerate} \item$\lim _n \Phi _k^n(\xi )=\Phi _k(\xi )$exists for all$\xi \in \Theta $; \end{enumerate} and for$p$sufficiently large, uniformly as$n\uparrow \infty $and uniformly in$\beta $\begin{enumerate} \setcounter{enumi}{1} \item$|\Phi _k^n(\xi )|\leq C(\beta ,p) \, \, e^{-\lambda (p) \scriptsize{\mbox{diam}}{D_k}}$for all$\xi \in \Theta ^+ _l$whenever${\rm diam}D_k>l$({\rm diam}$D_k \sim \sqrt{k}$is the diameter of the set$D_k$); ($\lambda (p) \uparrow \infty $as$p \uparrow \infty $) \item$\sum _{k=0}^{(2n+1)^2} |\Phi _k^n(\xi )| \leq c(\xi ) < \infty$for all$\xi \in \Theta ^+$; \end{enumerate} } The proof of Theorem 2 goes along the same lines as that of Theorem 1. It follows from \begin{prop} \label{p3} Take$f_x(\sigma )=\exp [-2\beta \xi (x)\sigma (x)]$. If$p$is large enough, there exists$\lambda =\lambda (p)$($\lambda \uparrow \infty$as$p \uparrow \infty $), such that if$\xi \in \Theta _l^+$, we have that for$2n+1\geq \mbox{{\rm diam}}D_k \geq l$\begin{equation} |\mu _n^{+,\xi }( f_0f_{x_k} )-\mu _n^{+,\xi }(f_0)\mu _n^{+,\xi } (f_{x_k})| \leq \mbox{{\rm const}}(\beta ,p) e^{-\lambda \scriptsize{\mbox{diam}}D_k} \end{equation} \end{prop} We need the following \begin{lemma} \label{nnn} If$p$is large enough, there exists$\alpha =\alpha (p)$($\alpha \uparrow \infty $as$p \uparrow \infty $), such that for$\xi \in \Theta ^+_l$,$m\geq l$\begin{equation} \mu _n^{+,\xi } \times \mu _n^{+,\xi }[(0,1)\rightarrow \Lambda _m^c] \leq \mbox{{\rm const}}(p) e^{-\alpha m} \end{equation} \end{lemma} Proof: \newline \newline Denote$C(\sigma ,\sigma ')$for the maximal connected set containing$o$on which$(\sigma ,\sigma ')\not\equiv (+,+)$. Denote with${\cal E}(o)$all enclosing sets containing the origin. Let$C\in {\cal E}(o)$. \begin{eqnarray} && \mu _n^{+,\xi } \times \mu _n^{+,\xi }(\bar{C}(\sigma ,\sigma ')=C) \leq \nonumber \\ && \sum _{R\subset \partial C} \mu _n^{+,\xi } \times \mu _n^{+,\xi } (\sigma \equiv - \, \mbox{on} \, R, \sigma \equiv + \, \mbox{on} \, \partial C^c, \sigma '\equiv - \, \mbox{on} \, \partial C \backslash R, \sigma '\equiv + \, \mbox{on} \, \partial C^c) \nonumber \\ && \sum _{R\subset \partial C} \mu _n^{+,\xi } (\sigma \equiv - \, \mbox{on} \, R, \sigma \equiv + \, \mbox{on} \, \partial C^c) \mu _n^{+,\xi } ( \sigma '\equiv - \, \mbox{on} \, \partial C \backslash R, \sigma '\equiv + \, \mbox{on} \, \partial C^c) \end{eqnarray} We now need the following Lemma, which we will prove afterwards. \begin{lemma} \label{black} Let$C\in {\cal E}(o)$and$R\subset \partial C$. If$\xi \in \Theta ^+_l$and$\partial C$contains at least$m\geq l$sites, there exists$\delta =\delta (p)$($\delta \uparrow \infty $as$p \uparrow \infty $), such that \begin{equation} \mu _n^{+,\xi } (\sigma \equiv - \, \mbox{on} \, R, \sigma \equiv + \, \mbox{on} \, \partial C^c) \mu _n^{+,\xi } ( \sigma '\equiv - \, \mbox{on} \, \partial C \backslash R, \sigma '\equiv + \, \mbox{on} \, \partial C^c) \leq \mbox{const}(p) e^{-\delta |\partial C|} \end{equation} \end{lemma} If we demand that there is a path from$o$to$\Lambda _m^c$then indeed$|\partial \bar{C}(\sigma ,\sigma ')| >m$. Using Lemma \ref{black}, we have that for$\xi \in \Theta ^+_l$\begin{eqnarray} && \mu _n^{+,\xi } \times \mu _n^{+,\xi }(\bar{C}(\sigma ,\sigma ')=C) \leq \nonumber \\ && 2^{|\partial C|} \mbox{const}(p) e^{-\delta |\partial C|} \leq \mbox{const}(p) e^{-\lambda |\partial C|} \end{eqnarray} and$\lambda \uparrow \infty$when$p \uparrow \infty $. \\ We can now continue as follows. Let$W_s$denote the number of enclosing sets bordered by a contour of length$s$and containing$o$. By the isoperimetric inequality on the square lattice we have that$W_s\leq (s^2/16) 4^s $\begin{eqnarray} &&\mu _n^{+,\xi } \times \mu _n^{+,\xi }[o\rightarrow \Lambda _m^c] = \nonumber \\ && \mu _n^{+,\xi } \times \mu _n^{+,\xi }[C(\sigma ,\sigma ' )\cap \Lambda _m^c \not= \emptyset ] \leq \nonumber \\ && \sum _{s=m}^{\infty } \sum _{C \in {\cal E}(o) \atop |\partial C|=s} \mu _n^{+,\xi } \times \mu _n^{+,\xi }[\bar{C}(\sigma ,\sigma ')=C] \leq \nonumber \\ && \mbox{const}(p) \sum _{s=m}^{\infty } e^{-\lambda s} \frac{s^2}{16} 4^s \leq \mbox{const}(p) e^{-\alpha m} \end{eqnarray} for some$\alpha >0$. \QED \newline Proof of Lemma \ref{black}: \\ \\ Let for any enclosing set$V\subset \integ ^2$\begin{eqnarray} && \tilde{H_V}^{+,\xi }(\sigma ):= - \beta \sum _{\langle xy \rangle \subset V }(\sigma (x)\sigma (y)-1) -\beta \sum _{\langle xy\rangle \atop x\in V;y \in V^c } (\sigma (x)-1) -p\sum _{x \in V} \xi (x) (\sigma (x)-1) \nonumber \\ && \tilde{H_V}^{-,\xi }(\sigma ):= - \beta \sum _{\langle xy \rangle \subset V }(\sigma (x)\sigma (y)-1) -\beta \sum _{\langle xy\rangle \atop x\in V;y \in V^c } (-\sigma (x)-1) -p\sum _{x \in V} \xi (x) (\sigma (x)+1) \nonumber \\ && \tilde{Z}^{+,\xi }(V):= \sum _{\sigma (x);x\in V} \exp [ \tilde{H_V}^{+,\xi }(\sigma )] \nonumber \\ && \tilde{Z}^{-,\xi }(V):= \sum _{\sigma (x);x\in V} \exp [ \tilde{H_V}^{-,\xi }(\sigma )] \end{eqnarray} and observe that \begin{equation} \label{zetten} \frac{\tilde{Z}^{-,\xi }(V)}{\tilde{Z}^{+,\xi }(V)} \leq \exp [2\beta |\partial V^c|] \, \exp [2p \sum _{x\in V} \xi (x)] \end{equation} We have that \begin{eqnarray} && \mu _n^{+,\xi } (\sigma \equiv - \mbox{ on } R, \sigma \equiv + \mbox{ on } \partial C^c) \mu _n^{+,\xi } ( \sigma '\equiv - \mbox{ on } \partial C \backslash R, \sigma '\equiv + \mbox{ on } \partial C^c) \leq \nonumber \\ && \mu _n^{+,\xi } (\sigma \equiv - \mbox{ on } R| \sigma \equiv + \mbox{ on } \partial C^c) \mu _n^{+,\xi } ( \sigma '\equiv - \mbox{ on } \partial C \backslash R| \sigma '\equiv + \mbox{ on } \partial C^c) = \nonumber \\ && \mu _C^{+,\xi } (\sigma \equiv - \mbox{ on } R) \mu _C^{+,\xi } ( \sigma '\equiv - \mbox{ on } \partial C \backslash R) \leq \nonumber \\ && \sum _{{ \{\gamma \}, \{\gamma '\} \mbox{\scriptsize{ inside }} C \atop \{ \gamma \} \mbox{\scriptsize{ compatible with }} R} \atop \{\gamma '\} \mbox{\scriptsize{ compatible with }} \partial C \backslash R } \prod _{\gamma }w_{\xi }(\gamma ) \prod _{\gamma '}w_{\xi }(\gamma ') \label{form2} \end{eqnarray} The set$\{ \gamma \} $of contours is compatible with$R$if it is compatible with$\sigma \equiv -$on$R$,$R\subset \cup _{\gamma }\mbox{Int}\gamma $and$R\cap \mbox{Int}\gamma \not= \emptyset$for each$\gamma $. The weight$w_{\xi }(\gamma )$of a contour$\gamma $is given by \begin{equation} w_{\xi }(\gamma ):=\exp [-2\beta |\gamma |-2p\sum _{x\in \mbox{\scriptsize{Int}}\gamma } \xi (x)] \frac{\tilde{Z}^{-,\xi }(\mbox{Int}\gamma \backslash \partial \mbox{Int}\gamma )}{\tilde{Z}^{+,\xi }(\mbox{Int}\gamma \backslash \partial \mbox{Int}\gamma )} \end{equation} Using (\ref{zetten}) we see that \begin{equation} w_{\xi }(\gamma )\leq \exp [-2p\sum _{x\in \partial\mbox{\scriptsize{Int}}\gamma } \xi (x)] \end{equation} Now regard the sets$\{ \gamma \} , \{\gamma '\} $in (\ref{form2}) as a single set of contours$\{ \tilde{ \gamma } \}= \{ \gamma \} \cup \{\gamma '\} $. The contours$\tilde{\gamma }$can overlap, but a site of$\partial C$can be at most inside two contours. Moreover, every site of$\partial C$is in the interior of at least one contour of$\{ \tilde{ \gamma } \} $and$|\partial C|>m$. \\ Let$V:=\cup _{\tilde{\gamma }}\partial \mbox{Int}\tilde{\gamma }$.$V$is a connected set such that$\bar{V}\ni o$and$|V|>m$. Because$\xi \in \Theta ^+_l$,$m\geq l$, \begin{equation} \sum _{\tilde{\gamma }} \sum _{x\in \partial \mbox{\scriptsize{Int}}\tilde{\gamma }}\xi (x) \geq \frac{1}{4} \sum _{\tilde{\gamma }} |\partial \mbox{Int}\tilde{\gamma }| \end{equation} This takes into account the worst situation where all$-$spins are counted twice and all$+$spins only once. Hence, in (\ref{form2}) \begin{equation} \prod _{\gamma }w_{\xi }(\gamma ) \prod _{\gamma '}w_{\xi }(\gamma ') \leq \prod _{\gamma } e^{-\frac{1}{2}p|\partial \mbox{Int}\gamma |} \prod _{\gamma '} e^{-\frac{1}{2}p|\partial \mbox{Int}\gamma '|} \end{equation} Now we can proceed as in Lemma 2.5 of \cite{BS}. Enumerate now the set$R$as$\{ x_1, \ldots ,x_q \} $. Each$x_i,i=1,\ldots q$is enclosed by a contour of the set$\{ \gamma \} $. The set$\{ \gamma \} $contains at most$q$contours. If$\{ \gamma \}$has$s$contours, let us call them$\gamma _{i_1}, \ldots ,\gamma _{i_s}$, where$x_{i_j}$is the point of$R$with the smallest index, enclosed by$\gamma _{i_j}$. Put$L_{i_j}=|\gamma _{i_j}|$for$j=1,\ldots ,s$and$L_k=0$if$k\in \{1,\ldots ,q\} $but$k\not=L_{i_1},\ldots ,L_{i_s}$. Let$L=\sum _{\gamma }|\gamma |$. Then$L_1,\ldots, L_q$gives a partitioning of$L$into$q$nonnegative numbers. It is well known that the number of such partitions is smaller then$K \, 4^L$for some constant$K$. On the other hand, the number of contours passing a fixed point$x$and having length$l$is bounded by$l\, 4^l \leq 8^l$. Therefore \begin{eqnarray} && \sum _{{ \{\gamma \}, \{\gamma '\} \mbox{\scriptsize{ inside }} C \atop \{ \gamma \} \, \mbox{\scriptsize{ compatible with }} \, R} \atop \{\gamma '\} \, \mbox{\scriptsize{ compatible with }} \, \partial C \backslash R } \prod _{\gamma }w_{\xi }(\gamma ) \prod _{\gamma '}w_{\xi }(\gamma ') \leq \nonumber \\ && \sum _{r \geq |R|} K\, 32^r e^{-\frac{1}{2}pr} \sum _{s \geq |\partial C \backslash R|} K\, 32^s e^{-\frac{1}{2}ps} \leq \mbox{const}(p) e^{-\delta |\partial C|} \end{eqnarray} for some constant$\delta >0$. \QED The proof of Proposition \ref{p3} is now analogous to that of Proposition \ref{p2}. \section{A toy model} \setcounter{equation}{0} In this section we treat a toy model to show more clearly where problems may arise in applying this method to other examples and to illustrate in a simple example the main philosophy of the method. \\ Consider independent spins$(\sigma (x), x\in \integ )$with identical distribution$\mu $,$ \mu (\sigma (x) =+)=p \not=1/2$,$00 \end{eqnarray} and this for every configuration $\xi (-l)=\alpha _{-l}\cdots \eta (l-1)=\alpha _{l-1}$. In other words, the conditional probabilities are nowhere continuous (this observation is due to J. van den Berg). The measure $\mu _p$ is therefore strongly non-Gibbsian. \\ This phenomenon occurs, because the severe constraint ($\ref{constraint}$) makes it possible to transmit information directly over arbitrary long distances. A reconstruction of the potential as in the previous section fails. There is no decay whatsoever. Indeed, let $\nu _n$ be the restriction of $\nu$ to $[-n,n]\cap \integ$ and \begin{equation} h_n(\xi )=\log \frac{\nu _n(\xi )}{\nu _n(\xi ^0)} \end{equation} It is easily seen that for \$|x|