% Format : LaTeX \documentstyle[12pt]{article} \newcommand {\Ebar} {{\mbox{\rm$\mbox{I}\!\mbox{E}$}}} \newcommand {\Rbar} {{\mbox{\rm$\mbox{I}\!\mbox{R}$}}} \newcommand {\Hbar} {{\mbox{\rm$\mbox{I}\!\mbox{H}$}}} \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}}} \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}}}} \newcommand{\Zzbar} {\mathord{\!\setlength{\unitlength}{0.9em} \begin{picture}(0.6,0.7) \thinlines \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}}} \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}} \def\cartprod{\mathop{\!\setlength{\unitlength}{1em}\begin{picture}(1,1) \thinlines \put(0,0){\line(1,1){1}} \put(1,0){\line(-1,1){1}} \end{picture} }} \newcommand{\integ}{\Zbar} \newcommand{\reals }{\Rbar} \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} \begin{document} \title{Almost sure quasilocality in the random cluster model} \renewcommand{\baselinestretch}{1} \author{Charles-Edouard Pfister \\ {\small D\'{e}partement de Math\'{e}matiques, Ecole Polytechnique F\'{e}d\'{e}rale de Lausanne} \\ {\small CH-1015 Lausanne Switzerland}\\ \\ Koen Vande Velde\thanks{Aspirant N.F.W.O. Belgium}\\ {\small Instituut voor Theoretische Fysica, K.U. Leuven}\\ {\small Celestijnenlaan 200D, B-3001 Leuven Belgium} } \date{} \maketitle \begin{abstract} \noindent We investigate the Gibbsianness of the random cluster measures $\mu^{q,p}$ and $\tilde{\mu }^{q,p}$, obtained as the infinite-volume limit of finite volume measures with free and wired boundary conditions. For $q>1$, the measures are not Gibbs measures, but it turns out that the conditional distribution on one edge, given the configuration outside that edge is almost surely quasilocal. \end{abstract} keywords: random cluster model, non-Gibbs states, \\ \phantom{keywords:} quasilocality of conditional distributions. \newpage \section{Introduction.} In recent years, it has become apparent (see \cite{efs} and references therein ) that not all states of physical interest for statistical mechanics are Gibbs measures \cite{georgii}. Examples can be found in renormalization group theory, where applying renormalization group transformations to Gibbs states may lead out of the class of Gibbs measures \cite{efs}. Other examples come from the theory of interacting particle systems, where nonreversible processes may have non-Gibbsian stationary states \cite{lebsch}. A general theory of non-Gibbsian states is not available. All one can do for the moment is investigate particular models and try to classify the examples one has of non-Gibbsian states. One way of classification is the following. Gibbs states verify the property of quasilocality. This means that conditional expectations of local events are continuous functions of the configuration one conditions on. Non-Gibbsian states (if they are not non-Gibbsian for the reason that there are some constraints or hard-core interactions) lack this quasilocality property. A way of classification is therefore to look how large the set is on which quasilocality fails \cite{fp}. In many cases, this seems very difficult \cite{mv,fp}. \\ In this paper we investigate the free and wired random cluster measures with $q>1$ and $0
1$. For the construction of the free boundary condition state, we start with a finite set of edges $\Lambda \subset \integ ^2_*$. We define a probability measure $\mu _{\Lambda }$ by its weights: \begin{eqnarray} \mu _{\Lambda }^{q,p}(\eta ):= \cases{ \frac{1}{Z_{\Lambda }}p^{N_1(\eta_{\Lambda} )} (1-p)^{N_0(\eta_{\Lambda} )}q^{c(\eta )}& if $\eta_e=0$ outside $\Lambda$\cr &\cr 0&otherwise.\cr} \end{eqnarray} $Z_{\Lambda } $ is a normalization constant. $N_1(\eta_{\Lambda} )$ is the number of open edges of $\eta $ in $\Lambda$; $N_0(\eta_{\Lambda} )$ is the number of closed edges of $\eta $ in $\Lambda$; $c(\eta )$ is the number of clusters of $\eta $. \\ We proceed in a similar way to construct the wired state $\tilde{\mu }^{q,p}$. The finite volume measures $\tilde{\mu }_{\Lambda }^{q,p}$ are defined by \begin{eqnarray} \tilde{\mu }_{\Lambda }^{q,p}(\eta ) := \cases{\frac{1}{Z_{\Lambda }}p^{N_1(\eta_{\Lambda} )} (1-p)^{N_0(\eta_{\Lambda} )}q^{c(\eta )}& if $\eta _e=1$ outside $\Lambda$ \cr & \cr 0& otherwise. \cr} \end{eqnarray} The following results are well known \cite{aiz}. \newtheorem{Lemma}{Lemma} \begin{Lemma} \begin{enumerate} \item $\mu _{\Lambda }^{q,p}$ and $\tilde{\mu }_{\Lambda }^{q,p}$ have the FKG property, i.e. for any increasing functions $g,h$ on $\Omega $ \begin{eqnarray} && \mu _{\Lambda }^{q,p}(gh)\geq \mu _{\Lambda }^{q,p}(g) \mu _{\Lambda }^{q,p}(h) \nonumber \\ && \tilde{\mu }_{\Lambda }^{q,p}(gh) \geq \tilde{\mu }_{\Lambda }^{q,p}(g) \tilde{\mu }_{\Lambda }^{q,p}(h) \end{eqnarray} \item For any increasing function $g$ on $\Omega $, \begin{eqnarray} && \mu _{\Lambda } ^{q,p}(g) \leq \mu _{\Lambda '}^{q,p}(g) \nonumber \\ && \tilde{\mu }_{\Lambda } ^{q,p}(g) \geq \tilde{\mu }_{\Lambda '}^{q,p}(g), \end{eqnarray} if $\Lambda \subset \Lambda '$. \item The weak limits $\mu ^{q,p}:=\lim _{\Lambda }\mu ^{q,p}_{\Lambda }$ and $\tilde{\mu }^{q,p}:=\lim _{\Lambda }\tilde{\mu } ^{q,p}_{\Lambda }$ exist. \end{enumerate} \end{Lemma} It is also well known that there exists a critical value $p_c(q)$ for $p$ above which there is percolation in the state $\mu ^{q,p}$ and below which there is absence of percolation. By percolation is meant the almost sure existence of an infinite connected cluster. Off the critical point, it is believed that $\mu ^{q,p}=\tilde{\mu }^{q,p}$ and that there is thus a unique state for the model (because free and wired boundary conditions are extremal in FKG sense). It can be proven rigorously for $q=1,2,\ldots $ by the existing connection with the Potts model (see \cite{aiz} for more information on this). Thus $\mu ^{q,p}$ and $\tilde{\mu }^{q,p} $ can only differ at $p_c(q)$, and they indeed do at high integer values of $q$ where one can make the connection with a first order transition for the Potts model. For notational convenience we will drop the superscript $q,p$ unless explicitly needed. \newtheorem{Definition}{Definition} \begin{Definition} We call a function $g$ on $\Omega $ quasilocal at $\eta $ iff for any $\epsilon >0$, there exists a finite region $\Lambda _{\epsilon }$ such that \begin{equation} \sup _{\stackrel{\zeta :}{\zeta _{\Lambda _{\epsilon}}= \eta _{\Lambda _{\epsilon}}}}|g(\zeta )- g(\eta )|<\epsilon . \end{equation} \end{Definition} The following result is well known \cite{sul,koz,efs}. \begin{Lemma} Let ${\cal F}^e$ be the $\sigma $-algebra of events {\em not} depending on $\omega _e$. The measure $\rho $ on $\Omega $ is a Gibbs measure iff for every edge $e$ there exists a version $\pi _e(\chi_e|\cdot )$ of $\Ebar _{\rho }[\chi _e|{\cal F}^e]$ satisfying \begin{enumerate} \item $0<\pi _e(\chi_e|\cdot )<1 $ \item $\pi _e(\chi_e|\cdot )$ is quasilocal at every $\eta \in \Omega $. \end{enumerate} \end{Lemma} We can now state the main result of the paper.\\ \\ \newtheorem{Theorem}{Theorem} \begin{Theorem} Let $\mu$ and $\tilde{\mu }$ be the probability measures constructed above. \begin{enumerate} \item $\mu $ is not a Gibbs measure. No version of $\Ebar _{\mu }[\chi_e|{\cal F}^e]$ is quasilocal everywhere. \item There exists a version of $\Ebar _{\mu }[\chi_e|{\cal F}^e]$ which is quasilocal $\mu $-a.s.. \end{enumerate} The same is true for $\tilde{\mu }$. \end{Theorem} \section{Conditional probabilities.} \setcounter{equation}{0} Let us now construct a version $\pi _e(\chi_e|\cdot )$ of the conditional probability $\Ebar _{\mu }[\chi_e|{\cal F}^e]$. Let $\Lambda$ be a finite subset of $\integ ^2_*$ and $\Lambda^c:=\integ ^2_*\backslash \Lambda $. We denote by $0$ the configuration in which every edge is closed, and by $1$ the configuration in which every edge is open. We define $\pi _e(\chi_e|\eta)$ for all $\eta=\eta _{\Lambda }0_{\Lambda ^c}$ as the conditional expectation of $\chi_e$ with respect to $\mu_{\Lambda}$; since those configurations are dense in $\Omega$ we then extend the definition by a limiting procedure. (The proof of the existence of the limit is given after the next Lemma.) Thus, \begin{eqnarray} &&\pi _e(\chi_e|\eta _{\Lambda }0_{\Lambda ^c}):=\mu _{\Lambda } (\chi_e|\eta _{\Lambda \backslash \{ e\}}0_{\Lambda ^c} ) \\ && \pi _e(\chi_e|\eta ):=\lim _{\Lambda } \pi _e(\chi_e|\eta _{\Lambda }0_{\Lambda ^c}). \end{eqnarray} Similarly we construct a version $\tilde{\pi }_e(\chi_e|\cdot)$ of $\Ebar _{\tilde{\mu }}[\chi _e|{\cal F}^e]$. \begin{eqnarray} && \tilde{\pi }_e(\chi_e|\eta _{\Lambda }1_{\Lambda ^c}):= \tilde{\mu }_{\Lambda } (\chi_e|\eta _{\Lambda \backslash \{ e\} }1_{\Lambda ^c}) \\ && \tilde{\pi }_e(\chi_e|\eta ):= \lim _{\Lambda }\tilde{\pi }_e(\chi_e|\eta _{\Lambda } 1_{\Lambda^c}). \end{eqnarray} \begin{Lemma} Let $V\ni e$ be a finite subset of $\integ ^2_*$. For a ${\cal F}^e$-measurable function $g$ \begin{eqnarray} && \mu _V(\pi _e(\chi_e|\cdot )g)=\mu _V(\chi_eg) \\ && \tilde{\mu }_V(\tilde{\pi }_e(\chi_e|\cdot )g)= \tilde{\mu }_V(\chi_e g) \end{eqnarray} \end{Lemma} \underline{Proof:}\\ \begin{eqnarray*} \mu _V(\pi _e(\chi_e|\cdot )g) & = & \int \mu _V(d\omega )\pi _e(\chi_e|\omega )g(\omega ) \\ & = & \int \mu _V(d\omega )\pi _e(\chi_e|\omega _V0_{V^c}) g(\omega_V0_{V^c} ) \\ & = & \int \mu _V(d\omega )\Ebar _{\mu _V }[\chi_e|{\cal F}^e]g(\omega)\\ & = & \mu _V(\chi_eg ) \end{eqnarray*} The proof of the second statement is similar. \QED \\ Let us compute now more explicitly our expression for $\pi _e(\chi_e|\eta )$. Denote by $\eta ^0$ the configuration equal to $\eta $ off $e$ and equal to $0$ on $e$ and by $\eta ^1$ the configuration equal to $\eta $ off $e$ and equal to $1$ on $e$. Now \begin{equation} \pi _e(\chi_e|\eta _{\Lambda }0_{\Lambda ^c})=\frac{\mu _{\Lambda }(\eta ^1)}{ \mu _{\Lambda }(\eta ^1)+ \mu _{\Lambda }(\eta ^0)}. \end{equation} It is thus clear that \begin{eqnarray} \label{energy} \pi _e(\chi _e|\eta _{\Lambda }0_{\Lambda ^c})= \cases{ \frac{p}{p+q(1-p)} & if $c(\eta_{\Lambda}^0 0_{\Lambda ^c})= c(\eta _{\Lambda }^1 0_{\Lambda ^c})+1$\cr & \cr p & if $c(\eta _{\Lambda}^00_{\Lambda ^c})= c(\eta _{\Lambda }^1 0_{\Lambda ^c}).$\cr} \end{eqnarray} The event where $c(\eta _{\Lambda }^0 0_{\Lambda ^c})= c(\eta _{\Lambda }^1 0_{\Lambda ^c})-1$ is impossible. Indeed, since isolated lattice sites are also counted as clusters, the creation of an open edge cannot increase the number of clusters.\\ For any finite $\Lambda$ we say that two sites $x$ and $y$ are connected inside $\Lambda$ for the configuration $\eta $ if they are connected for the configuration $\eta _{\Lambda }0_{\Lambda ^c}$ . Similarly, we say that two points are connected outside $\Lambda$ for $\eta $ if they are connected for $\eta _{\Lambda }1_{\Lambda ^c}$. Let $e$ be the edge with sites $x$ and $y$ as endpoints. Define the following events: $$ E^i_{\sqcap,\Lambda }:= \{x \mbox{ is connected to $y$ inside $\Lambda$}\}, $$ and $$ E^i_{\sqcap}:= \bigcup_{\Lambda}E^i_{\sqcap,\Lambda }. $$ Similarly, define $$ E^o_{\sqcap ,\Lambda}:= \{x \mbox{ is connected to $y$ outside $\Lambda$}\}, $$ and $$ E^o_{\sqcap}:=\left(\bigcup_{\Lambda}E^i_{\sqcap,\Lambda }\right) \bigcup \left( \bigcap_{\Lambda}E^o_{\sqcap,\Lambda }\right) . $$ Let $\chi ^i_{\sqcap }$ denote the indicator function of $E^i_{\sqcap }$. Denote with $\chi ^o_{\sqcap }$ the indicator function of $E^o_{\sqcap }$. Then \begin{eqnarray*} & c(\eta _{\Lambda }^0 0_{\Lambda^c})=c(\eta _{\Lambda }^1 0_{\Lambda^c})+1 & \mbox{ iff $\eta ^0 \not\in E^i_{\sqcap ,\Lambda } $} \\ & & \\ & c(\eta _{\Lambda }^0 0_{\Lambda^c})=c(\eta _{\Lambda }^1 0_{\Lambda^c}) & \mbox{ iff $\eta ^0 \in E^i_{\sqcap ,\Lambda } $}. \end{eqnarray*} Now we want to take the limit $\Lambda \nearrow \infty$. If $\eta ^0\in E^i_{\sqcap }$ there has to exist some finite volume $\Lambda $ such that $\eta ^0 \in E^i_{\sqcap ,\Lambda }$. Therefore, for each such $\eta $ the limit over $\Lambda $ exists and \begin{eqnarray} \pi _e(\chi _e|\eta ) & = & \phantom{+} \frac{p}{p+q(1-p)} \left( 1-\chi ^i_{\sqcap }(\eta ^0) \right)+p \chi ^i_{\sqcap }(\eta ^0) \nonumber \\ & = & \frac{p}{p+q(1-p)}+\left( p-\frac{p}{p+q(1-p)} \right) \chi ^i_{\sqcap }(\eta ^0). \label{1} \end{eqnarray} It is clear that $\pi _e(\chi _e|\cdot )$ is a nonlocal, nonnegative and increasing function since $q>1$. In a similar way \begin{eqnarray} \tilde{\pi }_e(\chi _e|\eta _{\Lambda }1_{\Lambda ^c})= \cases{ \frac{p}{p+q(1-p)} & if $c(\eta_{\Lambda}^0 1_{\Lambda ^c})= c(\eta _{\Lambda }^1 1_{\Lambda ^c})+1$\cr & \cr p & if $c(\eta _{\Lambda}^01_{\Lambda ^c})= c(\eta _{\Lambda }^1 1_{\Lambda ^c}).$\cr} \end{eqnarray} It is again clear then that \begin{eqnarray*} & c(\eta _{\Lambda }^0 1_{\Lambda^c})=c(\eta _{\Lambda }^1 1_{\Lambda^c})+1 & \mbox{ iff $\eta ^0 \not\in E^o_{\sqcap ,\Lambda } $} \\ & & \\ & c(\eta _{\Lambda }^0 1_{\Lambda^c})=c(\eta _{\Lambda }^1 1_{\Lambda^c}) & \mbox{ iff $\eta ^0 \in E^o_{\sqcap ,\Lambda } $}. \end{eqnarray*} For $\eta ^0\in E^i_{\sqcap }$ we already know that we can take the limit $\Lambda \nearrow \infty$. Thus now take $\eta \in \Omega $, $\eta ^0\in E^o_{\sqcap }$ but $\eta ^0\not\in E^i_{\sqcap }$. This $\eta $ is such that $\eta ^0 \in E^o_{\sqcap, \Lambda }$ for all finite $\Lambda $. Thus, $$ \tilde{\pi }_e(\chi _e |\eta _{\Lambda }1_{\Lambda ^c})=p. $$ The limit over $\Lambda $ thus exists and equals $p$. It is then clear that \begin{equation} \label{2} \tilde{\pi }_e(\chi _e|\eta ) = \frac{p}{p+q(1-p)}+\left( p-\frac{p}{p+q(1-p)} \right) \chi ^o_{\sqcap }(\eta ^0). \end{equation} The expressions for $\pi _e(\chi _e|\cdot )$ and $\tilde{\pi _e}(\chi _e|\cdot )$ differ thus on the set of configurations $\eta $ such that in $\eta ^0$ both $x$ and $y$ are connected to infinity, but not to each other.\\ The following Lemma is an adaptation of Lemma 3.1 in \cite{fp}. \begin{Lemma} Let $g$ be a monotone bounded function. If \begin{equation} g(\omega )=\lim _{\Lambda }g(\omega _{\Lambda }0_{\Lambda ^c}) \end{equation} then \begin{equation} \label{mu} \mu (g)=\lim _{\Lambda }\mu _{\Lambda }(g). \end{equation} If \begin{equation} g(\omega )=\lim _{\Lambda }g(\omega _{\Lambda }1_{\Lambda ^c}) \end{equation} then \begin{equation} \tilde{\mu } (g)=\lim _{\Lambda }\tilde{\mu } _{\Lambda }(g). \end{equation} \end{Lemma} \underline{Proof:} \\ We just prove (\ref{mu}) for monotone increasing functions. The proof of the other statements is similar. We denote $g_{\Lambda }(\omega ):=g(\omega _{\Lambda }0_{\Lambda ^c})$. Since $g$ is increasing, $g_{\Lambda }\leq g_{\Lambda '}$ for $\Lambda \subset \Lambda '$. Let $M\subset \Lambda \subset N$, $|N|<\infty$. \begin{equation} \mu _{\Lambda } (g_M)\leq \mu _{\Lambda } (g_{\Lambda })= \mu _{\Lambda } (g)\leq \mu _{N} (g_{\Lambda }). \end{equation} The left inequality is due to the above remark, the right one is due to Lemma 1 since $g_{\Lambda}$ is monotone increasing. Since $g_{\Lambda }$ is a local function, we can take the limit over $N$, and get (Lemma 1) \begin{equation} \mu _{\Lambda } (g_M)\leq \mu _{\Lambda } (g)\leq \sup _N\mu _N(g_{\Lambda })= \mu (g_{\Lambda }). \end{equation} By hypothesis $\lim_{\Lambda}g_{\Lambda}=g$; by the monotone convergence theorem we obtain \begin{equation} \mu (g_M)\leq \lim _{\Lambda }\inf \mu _{\Lambda } (g) \leq \lim _{\Lambda } \sup \mu _{\Lambda } (g)\leq \mu (g). \end{equation} Finally taking the limit over $M$ yields \begin{equation} \mu (g)\leq \lim _{\Lambda } \mu _{\Lambda } (g)\leq \mu (g). \end{equation} \QED \\ \newtheorem{Proposition}{Proposition} \begin{Proposition} \label{ja} $\pi _e(\chi_e|\cdot )$ is a version of $\Ebar _{\mu }[\chi_e|{\cal F}^e]$. \end{Proposition} \underline{Proof:} It suffices to prove that for any nonnegative increasing local ${\cal F}^e$-measurable function $g$ \begin{equation} \mu (\chi_eg)=\mu (\pi _e(\chi _e|\cdot )g). \end{equation} Because both $\chi_e$ and $g$ are local we have from weak convergence and Lemma 3 that \begin{eqnarray} \mu (\chi_eg) & = & \lim _{\Lambda }\mu _{\Lambda }(\chi_eg) \\ & =& \lim _{\Lambda }\mu _{\Lambda }(\pi _e(\chi_e|\cdot )g). \nonumber \end{eqnarray} Now $\pi _e(\chi_e|\cdot )$ is nonnegative and increasing and so is $g$. Then so is $\pi _e(\chi_e|\cdot )g$. By definition $\pi _e(\chi_e|\omega)=\lim_{\Lambda} \pi_e(\chi_e|\omega_{\Lambda}0_{\Lambda^c})$; since $g$ is local the same is true for $\pi_e(\chi_e|\cdot )g$. We can therefore apply Lemma 4 which guarantees the convergence of $\mu _{\Lambda }(\pi _e(\chi _e|\cdot )g)$ to $\mu (\pi _e(\chi_e|\cdot )g)$. This concludes the proof. \QED \\ In the same manner we prove that $\tilde{\pi }_e(\chi_e|\cdot )$ is a version of $\Ebar _{\tilde{\mu }}[\chi_e|{\cal F}^e]$. \section{Almost sure quasilocality.} \setcounter{equation}{0} We now investigate the quasilocality properties of $\pi _e(\chi_e|\cdot )$. We follow \cite{efs} section 4.5.3. Because $\pi _e(\chi_e|\cdot )$ is only one version of $\Ebar _{\mu }[\chi_e|{\cal F}^e]$, to prove non-quasilocality of $\Ebar _{\mu }[\chi_e|{\cal F}^e]$, we have to show that no function that equals $\pi _e(\chi_e|\cdot )$ $\mu$-a.s. is quasilocal everywhere. We say then that $\pi _e(\chi_e|\cdot )$ displays an essential non-quasilocality. Because $\mu $ gives nonzero probability to any open set, it suffices to investigate the function $\pi _e(\chi_e|\cdot )$ on a neighborhood of $\eta $, to search for essential non-quasilocality of $\pi _e(\chi_e|\cdot )$ at $\eta$. A neighborhood of $\eta$ is constructed in the following way. Fix a finite set $\Lambda$, then \begin{equation} {\cal N}_{\Lambda }(\eta):=\{ \eta\in \Omega : \zeta _{\Lambda }= \eta _{\Lambda } \} \end{equation} is a neighborhood of $\eta$. \begin{Proposition} No version of $\Ebar _{\mu }[\chi_e|{\cal F}^e]$ is quasilocal everywhere. \end{Proposition} \underline{Proof:} \\ The proof is given in \cite{efs}. We repeat it here for the sake of completeness and because it gives more insight in the properties of $\pi _e(\chi _e|\cdot )$. \\ Let $\Lambda _n:=[-n,n]^2$. Let $\eta $ be the configuration which sets $\eta _f:=1 $ on parallel rays running from $x$ and $y$ to infinity, perpendicular to the edge $e$, and sets $\eta_f=0$ on all other edges. We choose two subsets of ${\cal N}_{\Lambda _n}(\eta )$: ${\cal N}_{\Lambda _n}^1(\eta )$ in which an open edge in $\Lambda _{n+1} \backslash \Lambda _n$ connects the two rays, ${\cal N}_{\Lambda _n}^0(\eta )$ in which all edges of $\Lambda _{n+1} \backslash \Lambda _n$ are closed, so the parallel rays cannot be connected no matter what the configuration ouside $\Lambda _{n+1}$ is. Now for all $\zeta ^{1,n}\in {\cal N}_{\Lambda _n}^1(\eta )$, $\zeta ^{2,n}\in {\cal N}_{\Lambda _n}^0(\eta )$ \begin{equation} \pi _e(\chi _e|\zeta ^{1,n})-\pi _e(\chi _e| \zeta^{2,n})=p-\frac{p}{p+q(1-p)}>0, \end{equation} uniformly in $n$. \\ Since ${\cal N}_{\Lambda _n}^1(\eta )$ and ${\cal N}_{\Lambda _n}^0(\eta )$ carry positive $\mu $-measure, it follows that no function that equals $\pi _e(\chi _e|\cdot )$ $\mu $-a.s. can be quasilocal at $\eta $. \QED \\ We set out now to find how large the set of configurations is, where $\pi _e(\chi_e|\cdot )$ exhibits essential non-quasilocality. It turns out that this set is rather small in measure-theoretical sense. \begin{Lemma} \label{cluster} $\mu $-a.s. there exists no more than one infinite cluster. The same holds for $\tilde{\mu }$. \end{Lemma} \underline{Proof:} \\ As a consequence of the Burton-Keane uniqueness Theorem \cite{bk}, the result follows from translation invariance of $\mu $ and the so-called finite-energy property, i.e. \begin{equation} 0<\Ebar _{\mu }[\chi_e|{\cal F}^e]<1, \mbox{ $\mu $-a.s.}. \end{equation} But this condition is easily verified from expression (\ref{energy}) for $\pi _e(\chi _e|\cdot )$ and Proposition \ref{ja}. The same reasoning applies to $\tilde{\mu }$. \QED \begin{Proposition} The function $\pi _e(\chi_e|\cdot )$ is quasilocal $\mu $-a.s.. \end{Proposition} \underline{Proof:} \\ According to Lemma \ref{cluster}, the set $\Omega _1$ of configurations where there is no infinite cluster or a unique infinite cluster carries full measure. Take thus any $\eta \in \Omega _1$. Suppose first that $\eta $ has no infinite cluster. Then there exists some finite set $\Lambda $, $e \in \Lambda $, for which no site in $\Lambda $ is connected to $\Lambda ^c$. But then $\pi _e(\chi_e|\eta )=\pi _e(\chi_e|\zeta )$ whenever $\eta $ and $\zeta $ agree inside $\Lambda $. This proves locality of $\pi _e(\chi_e|\cdot )$ at such $\eta$.\\ Now take any configuration $\eta \in \Omega _1$ that has a unique infinite cluster. If not both $x$ and $y$ are connected to infinity, then again there exists a finite set $\Lambda $ such that $\pi _e(\chi _e|\eta )=\pi _e(\chi _e|\zeta )$ whenever $\eta $ and $\zeta $ agree inside $\Lambda $. So suppose that both $x$ and $y$ are connected to infinity. Because of the uniqueness of the infinite cluster, there exists now a finite set $\Lambda $ such that $x$ and $y$ are connected by a path of open edges within $\Lambda $. In that case again $\pi _e(\chi _e|\zeta)=\pi _e(\chi _e|\eta )$ for all $\zeta $ that agree with $\eta $ inside $\Lambda $. Hence $\pi _e(\chi _e|\cdot )$ is quasilocal for all configurations $\eta \in \Omega _1$. \QED \\ Because both $\mu $ and $\tilde{\mu }$ have no more than one infinite cluster it is now clear from expressions (\ref{1}) and (\ref{2}) that $\mu $-a.s. and $\tilde{\mu }$-a.s. $\pi _e(\chi _e|\cdot )=\tilde{\pi }_e(\chi _e|\cdot )$. It follows that $\pi _e(\chi _e|\cdot )$ is a version of both $\Ebar _{\mu }[\chi _e|{\cal F}^e]$ and $\Ebar _{\tilde{\mu }}[\chi _e|{\cal F}^e]$ and that $\pi _e(\chi _e|\cdot )$ is also quasilocal $\tilde{\mu }$-a.s.. \\ \\ \underline{Acknowledgment:} KVV thanks the EPFL in Lausanne, where part of the work was done, for kind hospitality. \begin{thebibliography}{10} \bibitem{efs} A.C.D. van Enter, R. Fern\'{a}ndez and A.D. Sokal: {\em Regularity properties and pathologies of position-space renormalization transformations: scope and limitations of Gibbsian theory}, J. Stat. Phys. {\bf 72}, 879 (1993) \bibitem{georgii} H.-O. 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