Content-Type: multipart/mixed; boundary="-------------0307160855945" This is a multi-part message in MIME format. ---------------0307160855945 Content-Type: text/plain; name="03-334.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-334.keywords" Birth and death process; Continuous system; Gibbs measure; Glauber dynamics; Spectral gap ---------------0307160855945 Content-Type: application/x-tex; name="Glauber_preprint.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Glauber_preprint.tex" \documentstyle[12pt,amsmath,amssymb]{article} \textwidth15.5cm \textheight21cm \oddsidemargin 5mm %\renewcommand{\baselinestretch}{2} \begin{document} \newtheorem{lem}{Lemma}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{prop}{Proposition}[section] \newtheorem{rem}{Remark}[section] \newtheorem{define}{Definition}[section] \newtheorem{cor}{Corollary}[section] \allowdisplaybreaks \makeatletter\@addtoreset{equation}{section}\makeatother \def\theequation{\arabic{section}.\arabic{equation}} \newcommand{\D}{{\cal D}} \newcommand{\N}{{\Bbb N}} \newcommand{\C}{{\Bbb C}} \newcommand{\Z}{{\Bbb Z}} \newcommand{\R}{{\Bbb R}} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \newcommand{\rom}[1]{{\rm #1}} \newcommand{\FC}{{\cal F}C_{\mathrm b}^\infty({\cal D},\Gamma)} \newcommand{\eps}{\varepsilon} \newcommand{\dd}{\overset{{.}{.}}} \newcommand{\fii}{\varphi} \def\stackunder#1#2{\mathrel{\mathop{#2}\limits_{#1}}} \newcommand{\FCo}{{\cal F}C_{\mathrm b}^\infty({\cal D},\dd\Gamma)} \renewcommand{\author}[1]{\medskip{\large #1}\par\medskip} \begin{center}{\Large \bf Glauber dynamics of continuous particle systems }\end{center} {\large Yuri Kondratiev}\\ Fakult\"at f\"ur Mathematik, Universit\"at Bielefeld, Postfach 10 01 31, D-33501 Bielefeld, Germany; Institute of Mathematics, Kiev, Ukraine; BiBoS, Univ.\ Bielefeld, Germany. e-mail: \texttt{kondrat@mathematik.uni-bielefeld.de}\vspace{2mm} {\large Eugene Lytvynov}\\ Institut f\"{u}r Angewandte Mathematik, Universit\"{a}t Bonn, Wegelerstr.~6, D-53115 Bonn, Germany; BiBoS, Univ.\ Bielefeld, Germany; SFB 611, Univ.~Bonn, Germany. e-mail: \texttt{lytvynov@wiener.iam.uni-bonn.de} {\small \begin{center} {\bf Abstract} \end{center} \noindent This paper is devoted to the construction and study of an equilibrium Glauber-type dynamics of infinite continuous particle systems. This dynamics is a special case of a spatial birth and death process. On the space $\Gamma$ of all locally finite subsets (configurations) in $\R^d$, we fix a Gibbs measure $\mu$ corresponding to a general pair potential $\phi$ and activity $z>0$. We consider a Dirichlet form $ \cal E$ on $L^2(\Gamma,\mu)$ which corresponds to the generator $H$ of the Glauber dynamics. We prove the existence of a Markov process $\bf M$ on $\Gamma$ that is properly associated with $\cal E$. In the case of a positive potential $\phi$ which satisfies $\delta{:=}\int_{\R^d}(1-e^{-\phi(x)})\, z\, dx<1$, we also prove that the generator $H$ has a spectral gap $\ge1-\delta$. Furthermore, for any pure Gibbs state $\mu$, we derive a Poincar\'e inequality. The results about the spectral gap and the Poincar\'e inequality are a generalization and a refinement of a recent result from \cite{BCC}. \noindent %2000 {\it AMS Mathematics Subject Classification:} {\it MSC:} 60K35, 60J75, 60J80, 82C21, 82C22 \vspace{1.5mm} \noindent{\it Keywords:} Birth and death process; Continuous system; Gibbs measure; Glauber dynamics; Spectral gap\vspace{1.5mm} %\noindent{\it Running head:} Glauber dynamics of continuous %systems \section{Introduction} This paper is devoted to the construction and study of an equilibrium Glauber-type dynamics (GD) of infinite continuous particle systems. This dynamics is a special case of a spatial birth and death process on $\R^d$. For a system of particles in a bounded volume, such processes were introduced and studied by C.~Preston in \cite{P}. In the latter case, the total number of particles is finite at any moment of time. In the recent paper by Bertini {\it et al.}, \cite{BCC}, the generator of the GD in a finite volume was studied. This generator corresponds to a special case of birth and death coefficients in Preston's dynamics. A positive, finite range, pair potential $\phi$ and an activity $z>0$ were fixed which satisfy the condition of the low activity-high temperature regime. Then, with any finite volume $\Lambda\subset\R^d$ and a boundary condition $\eta$ outside $\Lambda$, one may associate a finite volume Gibbs measure $\mu_{\Lambda,\eta}$. A non-local Dirichlet form ${\cal E}_{\Lambda,\eta}$ on $L^2(\mu_{\Lambda,\eta})$ was considered which corresponds to the generator of the GD on $\Lambda$. It was shown that the generator $H_{\Lambda,\eta}$ of ${\cal E}_{\Lambda,\eta}$ has a spectral gap which is uniformly positive with respect to all finite volumes $\Lambda$ and boundary conditions $\eta$. In this paper, we discuss the GD in the infinite volume. The problem of construction of a spatial birth and death process in the infinite volume was initiated in paper \cite{HS}, where it was solved in a very special case of nearest neighbor birth and death processes on the real line. So, we consider the space $\Gamma$ of all locally finite subsets (configurations) in $\R^d$, and a grand canonical Gibbs measure $\mu$ on $\Gamma$ which corresponds to a pair potential $\phi$ and activity $z>0$. The measure $\mu$ is supposed to be either of the Ruelle type or corresponding to a positive potential $\phi$ satisfying the integrability condition. In Section~\ref{Section2}, we shortly recall some facts about Gibbs measures which we use later on. In Section~\ref{jnhbh}, we consider the following bilinear form on $L^2(\Gamma,\mu)$ which is defined on a proper set of cylinder functions: \begin{equation} {\cal E}(F,G)=\int_\Gamma\sum_{x\in\gamma}(F(\gamma\setminus x)-F(\gamma)) \,(G(\gamma\setminus x)-G(\gamma))\,\mu(d\gamma)\label{dfhibhvi}\end{equation} (here and below, for simplicity of notations we will just write $x$ instead of $\{x\}$ for any $x\in\R^d$). We prove that this form is closable and its closure is a Dirichlet form. By using the general theory of Dirichlet forms (cf.\ \cite{MR}), we prove that there exists a Hunt process $\bf M$ on $\Gamma$ properly associated with $\cal E$. In particular, $\bf M$ is a conservative Markov process on $\Gamma$ with {\it cadlag\/} paths. By construction, $\bf M$ is an equilibrium GD on $\Gamma$ with the stationary measure $\mu$. Let us mention that the birth and death coefficients were supposed to be bounded in \cite{HS}, which is not the case for the GD, provided the potential $\phi$ has a negative part. In the case where the interaction between the particles is absent (i.e., $\phi=0$ and, therefore, $\mu$ is the Poisson measure $\pi_z$ with intensity $z$), the Markov process corresponding to the Dirichlet form \eqref{dfhibhvi} was explicitly constructed and studied by D.~Surgailis \cite{S1,S2}. In Sections~\ref{gzguz4545} and \ref{hucddhs}, we only consider the case of a positive potential $\phi$ and study the problem of the spectral gap for the generator $H$ of the Dirichlet form $\cal E$. Let us recall that the Poisson measure $\pi_z$ possesses the chaos decomposition property, and hence the space $L^2(\Gamma,\pi_z)$ is unitarily isomorphic to the symmetric Fock space over $L^2(\R^d,z\,dx)$, see e.g.\ \cite{S1}. Under this isomorphism, the operator $H$ goes over into the number operator $N$ in the Fock space, see \cite[Theorem~5.1]{AKR1}. Evidently, $N$ (and thus $H$) has spectral gap 1. Therefore, one may expect that, at least in the case of a ``small perturbation'' of the Poisson measure, the operator $H$ still has a spectral gap. One way to prove the existence of a spectral gap of a generator $H_E$ of a Dirichlet form $ E$ is to derive a coercivity identity for $ H_E$ on a class $\cal C$ of ``nice functions,'' and using it, to show that, for each $F\in\cal C$, $\|H_EF\|^2\ge G (H_E F,F)$ with $G>0$. If one additionally knows that the operator $ H_E$ is essentially selfadjoint on $\cal C$, the latter estimate implies that $H_E$ has a spectral gap $\ge G$. In the case of a probability measure defined on a Hilbert space, this approach was developed in \cite{K}, see also \cite[Ch.~6, Sect.~4]{BK}. So, having in mind this idea, we first prove in Section~\ref{gzguz4545} that the operator $H$ is essentially selfadjoint. This is technically the most difficult part of the paper. Then, in Section~\ref{hucddhs}, we prove a coercivity identity for the operator $H$ on cylinder functions, and using it and the essential selfadjointness of $H$, we show that the set $(0,1-\delta)$ does not belong to the spectrum of $H$, provided that $\delta{:=}\int_{\R^d}(1-e^{-\phi(x)})\, z\, dx<1$. This statement leads us to the Poincar\'e inequality if we are able to show that zero is a nondegenerate eigenvalue of $H$. We prove the latter statement for any $\mu$ that is an extreme point in the set of all Gibbs measures corresponding to $\phi$ and $z$. In the low activity-high temperature regime, the latter set consists of exactly one point, which is therefore extreme. Thus, compared with the result of \cite{BCC}, the progress achieved in the study of the spectral gap is as follows: \begin{enumerate} \item We work in the whole space $\R^d$, instead of taking finite volumes $\Lambda$ in $\R^d$ and boundary conditions $\eta$; \item We do not suppose that the potential $\phi$ has a finite range; \item The essential selfadjointness of $H$ is proven; \item For $\delta<1$, an explicit estimate for the value of the spectral gap of $H$ is found, and a Poincar\'e inequality is proven for any pure Gibbs state. \end{enumerate} In a forthcoming paper, we are going to discuss the existence problem for general birth and death processes on configuration spaces and study a scaling limit of these processes. \section{Gibbs measures on configuration spaces}\label{Section2} The configuration space $\Gamma:=\Gamma_{\R^d}$ over $\R^d$, $d\in\N$, is defined as the set of all subsets of $\R^d$ which are locally finite: $$\Gamma:=\big\{\,\gamma\subset \R^d\mid |\gamma_\Lambda|<\infty\text{ for each compact }\Lambda\subset \R^d\,\big\},$$ where $|\cdot|$ denotes the cardinality of a set and $\gamma_\Lambda:= \gamma\cap\Lambda$. One can identify any $\gamma\in\Gamma$ with the positive Radon measure $\sum_{x\in\gamma}\eps_x\in{\cal M}(\R^d)$, where $\eps_x$ is the Dirac measure with mass at $x$, $\sum_{x\in\varnothing}\varepsilon_x{:=}$zero measure, and ${\cal M}(\R^d)$ stands for the set of all positive Radon measures on the Borel $\sigma$-algebra ${\cal B}(\R^d)$. The space $\Gamma$ can be endowed with the relative topology as a subset of the space ${\cal M}(\R^d)$ with the vague topology, i.e., the weakest topology on $\Gamma$ with respect to which all maps $\Gamma\ni\gamma\mapsto\la f,\gamma\ra:=\int_{\R^d} f(x)\,\gamma(dx) =\sum_{x\in\gamma}f(x)$, $f\in{\cal D}$, are continuous. Here, $\D:=C_0(\R^d)$ is the space of all continuous real-valued functions on $\R^d$ with compact support. We will denote by ${\cal B}(\Gamma)$ the Borel $\sigma$-algebra on $\Gamma$. Now, we proceed to consider Gibbs measures on $\Gamma$. A pair potential is a Borel measurable function $\phi\colon \R^d\to {\Bbb R}\cup\{+\infty\}$ such that $\phi(-x)=\phi(x)\in\R$ for all $x\in\R^d\setminus\{0\}$. %Throughout this paper, we will suppose that the following %condition is satisfied: %\begin{description} %\item[(P)] ({\it Positivity}) %We have $\phi(x)\ge0$ for all $x\in\R^d$. %\end{description} A grand canonical Gibbs measure $\mu $ (or just Gibbs measure for short) corresponding to the pair potential $\phi$ and activity $z>0$ is usually defined through the Dobrushin--Lanford--Ruelle equation, see e.g.\ \cite{Ru69}. However, it is convenient for us to give an equivalent definition through the Georgii--Nguyen--Zessin identity (\cite[Theorem~2]{NZ}, see also \cite[Theorem~2.2.4]{Kuna}). For $\gamma\in\Gamma$ and $x\in\R^d\setminus\gamma$, we define a relative energy of interaction between a particle located at $x$ and the configuration $\gamma$ as follows: $$ E(x,\gamma){:=}\begin{cases}\sum_{y\in\gamma}\phi(x-y),&\text{if }\sum_{y\in\gamma}|\phi(x-y)|<\infty,\\ +\infty,&\text{otherwise}.\end{cases}$$ A probability measure $\mu$ on $(\Gamma,{\cal B}(\Gamma))$ is called a Gibbs measure if it satisfies \begin{equation}\int_\Gamma \mu(d\gamma)\int_{\R^d} \gamma(dx) \, F(\gamma,x) =\int_\Gamma \mu(d\gamma)\int_{\R^d} z\,dx\,\exp\left[-E(x,\gamma)\right] F(\gamma\cup x,x)\label{fdrtsdrt}\end{equation} for any measurable function $F:\Gamma\times\R^d\to[0,+\infty]$. (Notice that any fixed set $\gamma\in\Gamma$ has zero Lebesgue measure, so that the expression $E(x,\gamma)$ on the right hand side of \eqref{fdrtsdrt} is a.s.\ well-defined.) Let ${\cal G}(z,\phi)$ denote the set of all Gibbs measures corresponding to $z$ and $\phi$. In particular, if $\phi\equiv0$, then \eqref{fdrtsdrt} is the Mecke identity, which holds if and only if $\mu$ is the Poisson measure $\pi_z$ with intensity measure $z\, dx$. %%%%%%%%%%%%%%%%% Let us now describe some classes of Gibbs measures which appear in classical statistical mechanics of continuous systems. For every $r=(r^1,\dots,r^d)\in\Z^d$, we define a cube $$Q_r:=\left\{\, x\in\R^d\mid r^i-\frac 12\le x^i0$, $B\ge0$ such that, if $\gamma\in\Gamma_ {\Lambda_N}$ for some $N$, then $$\sum_{\{x,y\}\subset\gamma}\phi(x-y)\ge\sum_{r\in\Z^d}\big(A|\gamma_r| ^2-B|\gamma_r|\big).$$ \end{description} Notice that the superstability condition automatically implies that the potential $\phi$ is semi-bounded from below. %This condition is %evidently stronger than (S). \begin{description} \item[(LR)] ({\it Lower regularity}) There exists a decreasing positive function $a\colon\N\to{\Bbb R}_+$ such that $$\sum_{r\in\Z^d}a(\|r\|)<\infty$$ and for any $\Lambda',\Lambda''$ which are finite unions of cubes $Q_r$ and disjoint, with $\gamma'\in\Gamma_{\Lambda'}$, $\gamma''\in\Gamma_{\Lambda''}$, $$\sum_{x\in\gamma',\, y\in\gamma'' }\phi(x-y)\ge-\sum_{r',r''\in\Z^d}a(\|r'-r''\|) |\gamma_{r'}'|\,|\gamma_{r''}''|.$$ Here, $\|\cdot\|$ denotes the maximum norm on $\R^d$. \item[(I)] ({\it Integrability}) $$\int_{\R^d}|1-\exp[-\phi(x)]|\,dx<+\infty.$$ \end{description} For results related to spectral properties of the generator of the GD, we will need the following condition. \begin{description} %\item[(LAHT)] ({\it Low activity--high temperature regime}) We have %$$\int_\R|1-e^{-\phi(x)}|\,dx< z^{-1}\exp(-1-2B), $$ where $B$ is %as in (S). \item[(P)] ({\it Positivity}) $\phi(x)\ge0$ for all $x\in\R^d$. \end{description} A probability measure $\mu$ on $(\Gamma,{\cal B}(\Gamma))$ is called tempered if $\mu$ is supported by $$S_\infty{:=}\bigcup_{n=1}^\infty S_n,$$ where $$S_n:=\left\{\, \gamma\in\Gamma\mid \forall N\in\N\ \sum_{r\in\Lambda_N\cap\Z^d} |\gamma_r|^2\le n^2|\Lambda_N \cap\Z^d| \,\right\}.$$ By ${\cal G}^t(z,\phi)\subset{\cal G}(z,\phi)$ we denote the set of all tempered grand canonical Gibbs measures (Ruelle measures for short). Due to \cite{Ru70} the set ${\cal G}^t(z,\phi)$ is non-empty for all $z>0$ and any potential $\phi$ satisfying conditions (SS), (LR), and (I). Furthermore, the set ${\cal G}(z,\phi)$ is non-empty for all $z>0$ and any potential $\phi$ satisfying (P) and (I), see \cite[Proposition~2.7.15]{Kuna}. %%%%%%%%%%%%%%%%%%%%%% Let us now recall the so-called Ruelle bound (cf.\ \cite{Ru70}). \begin{prop}\label{waessedf} Suppose that either conditions \rom{(I), (SS), (LR)} are satisfied and $\mu\in{\cal G}^t(z,\phi)$\rom, $z>0$\rom, or conditions \rom{(P), (I)} are satisfied and $\mu\in{\cal G}(z,\phi)$\rom, $z>0$\rom. Then\rom, for any $n\in\N$\rom, there exists a non-negative measurable symmetric function $k_\mu^{(n)}$ on $({\Bbb R}^d)^n$ such that, for any measurable symmetric function $f^{(n)}:(\R^d)^n\to[0,\infty]$, \begin{align} &\int_\Gamma \sum_{\{x_1,\dots,x_n\}\subset\gamma} f^{(n)}(x_1,\dots,x_n)\,\mu(d\gamma)\notag\\&\qquad =\frac1{n!}\, \int_{(\R^d)^n} f^{(n)}(x_1,\dots,x_n) k_\mu^{(n)}(x_1,\dots,x_n)\,dx_1\dotsm dx_n,\label{6t565r7}\end{align} and \begin{equation}\label{swaswea957}\forall (x_1,\dots,x_n)\in(\R^d)^n:\quad k_\mu^{(n)}(x_1,\dots,x_n)\le \xi^n,\end{equation} where $\xi> 0$ is independent of $n$. \end{prop} The functions $k_\mu^{(n)}$, $n\in\N$, are called correlation functions of the measure $\mu$, while \eqref{swaswea957} is called the Ruelle bound. Notice that any measure $\mu\in{\cal G}(z,\phi)$ as in Proposition~\ref{waessedf} satisfies \begin{equation}\label{sweqaw}\int_\Gamma \la \fii,\gamma\ra^n\,\mu(d\gamma)<\infty,\qquad \fii\in{\cal D},\ \fii\ge0, n\in\N.\end{equation} that is, $\mu$ has all local moments finite. \section{The Dirichlet form $\cal E$ and associated Markov process}\label{jnhbh} We introduce a set $\FC$ of all functions on $\Gamma$ of the form \begin{equation}\label{hbdv} F(\gamma)=g_F(\la\fii_1,\gamma\ra,\dots,\la\fii_N,\gamma\ra),\end{equation} where $N\in\N$, $\fii_1,\dots,\fii_N\in\D$, and $g_F\in C^\infty_{\mathrm b}({\Bbb R}^N)$. Here, $C^\infty_{\mathrm b}({\Bbb R}^N)$ denotes the set of all infinitely differentiable functions on $\R^N$ which are bounded together with all their derivatives. For any $\gamma\in\Gamma$, we consider $T_\gamma{:=}L^2(\R^d,\gamma)$ as a ``tangent'' space to $\Gamma$ at the point $\gamma$, and for any $F\in\FC$ we define the ``gradient'' of $F$ at $\gamma$ as the element of $T_\gamma$ given by $D^-F(\gamma,x){:=}D^-_xF(\gamma){:=}F(\gamma\setminus x)-F(\gamma)$, $x\in\R^d$. (Evidently, $D^-F(\gamma)$ indeed belongs to $T_\gamma$.) Let $\mu$ be a Gibbs measure as in Proposition~\ref{waessedf}. We will preserve the notation $\FC$ for the set of all $\mu$-classes of functions from $\FC$. The set $\FC$ is dense in $L^2(\Gamma,\mu)$. We now define \begin{align} {\cal E}(F,G){:=}& \int_\Gamma (D^-F(\gamma),D^-G(\gamma))_{T_\gamma}\,\mu(d\gamma)\notag \\=&\int_\Gamma \mu(d\gamma)\int_{\R^d}\gamma(dx)\, D^-_x F(\gamma)\, D^-_x G(\gamma), \qquad F,G\in\FC.\label{gcdtuf}\end{align} Notice that, for any $F\in\FC$, there exists $f\in\D$ such that $|D^-_xF(\gamma)|\le f(x)$ for all $\gamma\in\Gamma$ and $x\in\gamma$. Hence, by \eqref{sweqaw}, the right hand side of \eqref{gcdtuf} is well defined. By \eqref{fdrtsdrt}, we also get, for $F,G\in\FC$, \begin{equation}\label{iud7u}{\cal E}(F,G)=\int_\Gamma\mu(d\gamma)\int_{\R^d} z\, dx\, \exp\left[-E(x,\gamma)\right]D_x^+ F(\gamma)\, D_x^+ G(\gamma),\end{equation} where $D_x^+F(\gamma){:=}F(\gamma)-F(\gamma\cup x)$. \begin{lem} %Let the conditions of Proposition~\rom{\ref{waessedf}} be %satisfied\rom. Then\rom, We have\rom: ${\cal E}(F,G)=0$ for all $F,G\in\FC$ such that $F=0$ $\mu$-a\rom.e\rom. \end{lem} \noindent{\it Proof}. Let $F\in\FC$, $F=0$ $\mu$-a.e. Denote $B(r){:=}\{x\in\R^d:|x|0$\rom. Then\rom, We have\rom: \begin{equation}\label{hvgzfc} {\cal E}(F,G)=\int_\Gamma HF(\gamma)G(\gamma)\, \mu(d\gamma),\qquad F,G\in\FC,\end{equation} where \begin{equation}\label{gen} HF(\gamma)=\int_{\R^d}īz\, dx\,\exp\left[-E(x,\gamma)\right]D^+_xF(\gamma)-\int_{\R^d}\gamma(dx)\,D^-_xF(\gamma) \end{equation} and $HF\in L^2(\Gamma,\mu)$\rom. The bilinear form $({\cal E},\FC)$ is closable on $L^2(\Gamma,\mu)$ and its closure will be denoted by $({\cal E },D({\cal E}))$\rom. The operator $(H,\FC)$ in $L^2(\Gamma,\mu)$ has Friedrichs' extension\rom, which we denote by $(H,D(H))$\rom. \end{prop} \noindent{\it Proof}. Equations \eqref{hvgzfc}, \eqref{gen} easily follow from \eqref{fdrtsdrt} and \eqref{gcdtuf}. Let us show that $HF\in L^2(\Gamma,\mu)$. By \eqref{sweqaw}, the inclusion $\int_{\R^d}\gamma(dx)\,D^-_xF(\gamma)\in L^2(\Gamma,\mu)$ is trivial. Next, we can find a compact $\Lambda\subset \R^d$ and $C_1>0$ such that $|D_x^+F(\gamma)|\le C_1\chi_\Lambda(x)$ for all $\gamma\in\Gamma$ and $x\in\R^d$. Here, $\chi_\Lambda$ denotes the indicator of $\Lambda$. Hence, by \eqref{fdrtsdrt} and \eqref{6t565r7}, \begin{align}& \int_\Gamma\left(\int_{\R^d}īz\, dx\,\exp\left[-E(x,\gamma)\right]D^+_xF(\gamma)\right)^2\,\mu(d\gamma)\notag\\ &\quad \le C_1^2\int_\Gamma \mu(d\gamma)\int_\Lambda z\, dx\int_\Lambda z\, dy\,%\notag\\&\qquad\times \exp\left[-E(x,\gamma)-E(y,\gamma)-\phi(x-y)\right]\exp[\phi(x-y)]\notag\\ &\quad=2C_1^2 \int_\Gamma \sum_{\{x,y\}\in\gamma_\Lambda}\exp[\phi(x-y)]\,\mu(d\gamma)\notag\\ &\quad= C_1^2\int_{\Lambda^2}\exp[\phi(x-y)]k_\mu^{(2)}(x,y)\, dx\, dy<\infty,\label{zgussguz} \end{align} since $k_\mu^{(2)}(x,y)\le C_2\exp[-\phi(x-y)]$ for all $x,y\in\R^d$, $C_2>0$, cf.\ \cite[Eq.~(4.29)]{AKR4}. Therefore, $H$ is the $L^2(\Gamma,\mu)$-generator of the bilinear form ${\cal E }$. The rest of the proposition now follows from e.g.\ \cite[Theorem~X.23]{RS}.\quad $\square$ For the notion of a ``Dirichlet form,'' appearing in the following lemma, we refer to e.g.\ \cite[Chap.~I, Sect.~4]{MR}. \begin{lem}\label{guzazagus}% Let the conditions of Proposition~\rom{\ref{ghvv}} be %satisfied\rom. Then\rom, $({\cal E},D({\cal E}))$ is a Dirichlet form on $L^2(\Gamma,\mu)$. \end{lem} \noindent{\it Proof}. On $D({\cal E})$ consider the norm $\|F\|_{D({\cal E})}{:=}\big(\|F\|^2_{L^2(\mu)}+{\cal E}(F)\big)^{1/2}$, $F\in D({\cal E})$. Here, we denoted ${\cal E }(F){:=}{\cal E}(F,F)$. For any $F,G\in\FC$, we define $$ { S}(F,G)(x,\gamma){:=}D_x^-F(\gamma)\, D_x^-G(\gamma),\qquad x\in\R^d,\ \gamma\in\Gamma.$$ Using the Cauchy--Schwarz inequality, we conclude that $ S$ extends to a bilinear continuous map from $(D({\cal E}),\|\cdot\|_{D({\cal E})})\times (D({\cal E}),\|\cdot\|_{D({\cal E})})$ into $L^1(\R^d\times\Gamma,\tilde\mu)$, where $\tilde\mu(dx,d\gamma){:=}\linebreak\gamma(dx) \mu(d\gamma)$. Let $F\in D({\cal E})$ and consider any sequence $F_n\in\FC$, $n\in\N$, such that $F_n\to F$ in $(D({\cal E}),\|\cdot\|_{D({\cal E})})$ as $n\to\infty$ . Then, $F_n(\gamma)\to F(\gamma)$ as $n\to\infty$ for $\mu$-a.e.\ $\gamma\in\Gamma$ (if necessary, take a subsequence of $(F_n)_{n\in\N}$ with this property). Furthermore, for any $r>0$, we have, analogously to \eqref{zgussguz}: \begin{align}& \int_{B(r)\times\Gamma }|F_n(\gamma\setminus x)-F(\gamma\setminus x)|\, \tilde\mu(dx,d\gamma)\notag\\&\qquad = \int_{\Gamma}\mu(d\gamma)\int_{B(r)} z\, dx\,\exp\left[-E(x,\gamma) \right] |F_n(\gamma)-F(\gamma)|\notag\\&\qquad\le \left(\int_{\Gamma}|F_n(\gamma)-F(\gamma)|^2\,\mu(d\gamma)\right)^{1/2}\notag\\&\quad\qquad\times\left(\int_{\Gamma} \left(\int_{B(r)}z\, dx\,\exp\left[-E(x,\gamma)\right]\right)^2\,\mu(d\gamma)\right)^{1/2}\to 0\label{12345} \end{align} as $n\to\infty$. Therefore, $F_n(\gamma\setminus x)\to F(\gamma\setminus x)$ for $\tilde\mu$-a.e.\ $(x,\gamma)\in\R^d\times\Gamma$. Thus, $D_x^- F_n(\gamma)\to D_x^-F(\gamma)$ as $n\to\infty$ for $\tilde\mu$-a.e.\ $(x,\gamma)\in\R^d\times\Gamma$, which yields: \begin{equation}\label{zguzg} { S}(F,G)(x,\gamma)=D_x^-F(\gamma)\,D^-_x G(\gamma),\qquad\text{$\tilde\mu$-a.e.\ }(x,\gamma)\in\R^d\times \Gamma,\ F,G\in D({\cal E}).\end{equation} Hence, \begin{equation}\label{hh}{\cal E}(F,G)=\int_{\R^d\times\Gamma}D^-_x F(\gamma) D_x^-G(\gamma) \,\tilde\mu(dx,d\gamma),\qquad F,G\in D({\cal E}) .\end{equation} Define $\R\ni x\mapsto g(x){:=}(0\vee x)\wedge 1$. Let $(g_n)_{n\in\N}$ be a sequence of functions from $C_{\mathrm b}^\infty (\R)$ such that, for all $x\in\R$: $0\le g_n(x)\le1$, $0\le g_n'(x)\le 2$, $g_n(x)\to g(x)$ as $n\to\infty$. We again fix any $F\in D({\cal E })$ and and let $(F_n)_{n\in \N}$ be a sequence of functions from $\FC$ such that $F_n\to F$ in $(D({\cal E}),\|\cdot\|_{D({\cal E })})$. Consider the sequence $(g_n(F_n))_{n\in\N}$. We evidently have: $g_n(F_n)\in\FC$ for each $n\in\N$ and $g_n(F_n)\to g(F)$ as $n\to\infty$ in $L^2(\Gamma,\mu)$. Next, by the above argument, we have: $D_x^-g_n(F_n(\gamma))\to D_x ^-g(F(\gamma))$ as $n\to\infty$ for $\tilde\mu$-a.e.\ $(x,\gamma)$. Furthermore, the sequence $(D^-_x g_n(F_n(\gamma)))_{n\in\N}$ is $\tilde\mu$-uniformly square-integrable, since so is the sequence $(D^-_xF_n(\gamma))_{n\in\N}$. Therefore, the sequence $(D_x^-g_n(F_n(\gamma)))_{n\in\N}$ converges to $D_x ^-g(F(\gamma))$ in $L^2(\R^d\times\Gamma,\tilde\mu)$. This yields: $g(F)\in D({\cal E})$. For any $x,y\in\R$, we evidently have $|g(x)-g(y)|\le|x-y|$. By \eqref{hh}, we then finally have: ${\cal E}(g(F))\le {\cal E}(F)$, which means that $({\cal E},D({\cal E}))$ is a Dirichlet form.\quad $\square$ We will now need the bigger space $\dd\Gamma$ consisting of all $\Z_+$-valued Radon measures on $\R^d$ (which is Polish, see e.g.\ \cite{Ka75}). Since $\Gamma\subset\dd\Gamma$ and ${\cal B}(\dd\Gamma)\cap\Gamma={\cal B}(\Gamma)$, we can consider $\mu $ as a measure on $(\dd\Gamma,{\cal B}(\dd\Gamma))$ and correspondingly $({\cal E}, D({\cal E}))$ as a Dirichlet form on $L^2(\dd\Gamma,\mu)$. For the notion of a``quasi-regular Dirichlet form,'' appearing in the following lemma, we refer to \cite[Chap.~IV, Sect.~3]{MR}. \begin{prop}\label{scfxgus} %Let the conditions of Proposition~\rom{\ref{ghvv}} be %satisfied\rom. Then,\linebreak $({\cal E},D({\cal E}))$ is a quasi-regular Dirichlet form on $L^2(\dd\Gamma,\mu)$. \end{prop} \noindent {\it Proof}. By \cite[Proposition~4.1]{MR98}, it suffices to show that there exists a bounded, complete metric $\rho$ on $\dd\Gamma$ generating the vague topology such that, for all $\gamma_0\in\dd\Gamma$, $\rho(\cdot,\gamma_0) \in D({\cal E})$ and $\int_{\R^d} S(\rho(\cdot,\gamma_0))(x,\gamma)\,\gamma(dx)\le \eta(\gamma)$ $\mu$-a.e.\ for some $\eta\in L^1(\dd\Gamma,\mu)$ (independent of $\gamma_0$). Here, $S(F){:=}S(F,F)$. Let us recall some well-known facts about cylinder functions on the configuration space (see e.g.\ \cite{KK} for details). Let ${\cal O}_{\mathrm c}(\R^d)$ denote the set of all open, relatively compact sets in $\R^d$. For $\Lambda\in {\cal O}_{\mathrm c}(\R^d)$, we have: $\Gamma_\Lambda=\bigsqcup_{n=0}^\infty \Gamma_\Lambda^{(n)}$, where $\Gamma_\Lambda^{(n)}$ denotes the set of all $n$-point subsets of $\Lambda$, $n\in\N$, and $\Gamma_\Lambda^{(0)}=\{\varnothing\}$. For $n\in\N$, we can naturally identify $\Gamma_\Lambda^{(n)}$ with $\tilde\Lambda^n/S_n$, where $\tilde\Lambda^n{:=}\{(x_1,\dots,x_n)\in\Lambda^ n: x_i\ne x_j\text{ if }i\ne j\}$ and $S_n$ denotes the group of permutations of $\{1,\dots,n\}$ that acts on $\tilde\Lambda^n$ by permuting the numbers of the coordinates. Furthermore, the trace $\sigma$-algebra of ${\cal B}(\Gamma)$ on $\Gamma_\Lambda^{(n)}$ coincides with the $\sigma$-algebra ${\cal B}_{\mathrm sym}(\tilde \Lambda^n )$ of all symmetric Borel subsets of $\tilde\Lambda^n$ (again under a natural isomorphism). Finally, any measurable function $F_\Lambda$ on $\Gamma_\Lambda$ may be identified with a measurable cylinder function $F$ on $\Gamma$ by setting $\Gamma\ni\gamma\mapsto F(\gamma){:=}F_\Lambda(\gamma_\Lambda)$. \begin{lem}\label{lemm1} Let $\Lambda\in {\cal O}_{\mathrm c}(\R^d)$\rom. Any measurable bounded function $F$ on $\Gamma_\Lambda$ such that $F\restriction \Gamma^{(n)}_\Lambda\equiv0$ for all $n\ge N$\rom, $N\in\N$\rom, belongs to $D({\cal E})$\rom. \end{lem} \noindent {\it Proof}. We take arbitrary, open, disjoint subsets $O_1,\dots,O_n$ of $\Lambda$. Consider functions $g_1,g_2\in C_{\mathrm b}^\infty(\R)$ such that $g_1(1)=1$, $g_2(0)=1$ and $g_1(x)=0$ if $|x-1|>1/2$, $g_2(x)=0$ if $|x|>1/2$. Approximating the indicator functions $\chi_{O_i}$, $i=1,\dots,n$, and $\chi_{\Lambda\setminus (O_1\cup\dotsm\cup O_n)}$ by functions from $\D$, we easily conclude that the statement of the lemma holds for the function \begin{align*}F(\gamma)&=g_1(\la\chi_{O_1},\gamma\ra)\dotsm g_1(\la\chi_{O_n},\gamma\ra) g_2(\la \chi_{\Lambda\setminus (O_1\cup\dotsm\cup O_n)},\gamma\ra)\\ &=\chi_{\{\,|\gamma_{O_1}|=1,\dots,|\gamma_{O_n}|=1,\, |\gamma_{\Lambda\setminus (O_1\cup\dotsm\cup O_n)}|=0\,\}}(\gamma). \end{align*} Hence, $F(\gamma)$ may be identified with the indicator function $\chi_{S_n(O_1\times\dotsm\times O_n)}(x_1,\dots,\linebreak x_n)$ on $\tilde\Lambda^n/S_n$. Using a monotone class argument, we get the statement for any indicator function, and then for any measurable bounded function on $\Gamma_\Lambda^{(n)}$.\quad $\square$ \begin{lem}\label{jjy} Let $f:\R^d\to\R$ be measurable\rom, bounded\rom, and with compact support\rom. Let $\zeta\in C_{\mathrm b}^\infty(\R)$ and $a\in\R$\rom. Then\rom, $\zeta(|\la f,\cdot\ra-a|)\in D({\cal E})$\rom. \end{lem} \noindent {\it Proof}. Let $\Lambda\in{\cal O}_c(\R^d)$ be such that $\operatorname{supp}f\subset\Lambda$. Define $$F_n(\gamma){:=}\zeta(|\la f,\gamma\ra-a|)\, \chi_{\{|\gamma_\Lambda|\le n\}}(\gamma),\qquad \gamma\in\Gamma,\ n\in\N.$$ By Lemma~\ref{lemm1}, $F_n\in D({\cal E})$, $n\in\N$. We evidently have: $F_n\to F$ in $L^2(\dd\Gamma,\mu)$. Furthermore, using Proposition~\ref{waessedf} and the majorized convergence theorem, we get: ${\cal E}(F_n-F_m)\to0$ as $n,m\to\infty$. From here the statement follows. \quad $\square$ The rest of the proof of Proposition~\ref{scfxgus} is quite analogous to the proof of \cite[Proposition~4.8]{MR98}. So, we only outline the main changes needed. Let $E_k{:=}B(k)\subset\R^d$, $\delta_k=1/2$, $\gamma_0\in\dd\Gamma$ and let $g_{E_k,\delta_k}$, $\phi_k$, and $\zeta$ be defined as in \cite{MR98}, and we additionally demand that $\zeta'(x)\in[0,1]$ on $[0,\infty)$. Since $\phi_k g_{E_k,\delta_k}$ is bounded and has a compact support, we have by Lemma~\ref{jjy}: $$\zeta(|\la \phi_k g_{E_k,\delta_k},\cdot \ra-\la\phi_k g_{E_k,\delta_k},\gamma_0\ra|)\in D({\cal E}).$$ (Notice that $\la\phi_k g_{E_k,\delta_k},\gamma_0\ra$ is a constant.) Furthermore, taking to notice that $\zeta'(x)\in[0,1]$, we get from \eqref{zguzg} and the mean value theorem: \begin{equation} S(\zeta(|\la \phi_k g_{E_k,\delta_k},\cdot \ra-\la\phi_k g_{E_k,\delta_k},\gamma_0\ra|)(x,\gamma)\le (\phi_k g_{E_k,\delta_k})^2(x)\le \chi_{B(k+1/2)}(x).\label{hahgfi}\end{equation} Set $$c_k{:=}\left(1+\int_{B(k+1/2)}k_\mu^{(1)}(x)\, dx \right)^{-1/2}2^{-k/2}.$$ Using estimate \eqref{hahgfi} and the numbers $c_k$, we now easily obtain the statement of the proposition absolutely analogously to the proof of \cite[Lemma~4.11 and Proposition~4.8]{MR98}.\quad $\square$ For the notion of an ``${\cal E}$-exceptional set,'' appearing in the next proposition, we refer to e.g.\ \cite[Chap.~III, Sect.~2]{MR}. \begin{prop}\label{fdj} The set $\dd\Gamma\setminus\Gamma$ is ${\cal E}$-exceptional\rom. \end{prop} \noindent {\it Proof}. We modify the proof of \cite[Proposition~1 and Corollary~1]{RS98} according to our situation. It suffices to prove the result locally, that is, to show that, for every fixed $a\in\N$, the set $$N{:=}\big\{\,\gamma\in\dd\Gamma: \sup(\gamma(\{x\}): x\in [-a,a]^d)\ge 2\,\big\}$$ is ${\cal E}$-exceptional. By \cite[Lemma~1]{RS98}, we need to prove that there exists a sequence $u_n\in D({\cal E})$, $n\in\N$, such that each $u_n$ is a continuous function on $\dd\Gamma$, $u_n\to {\bf 1} _{N}$ pointwise as $n\to\infty$, and $\sup_{n\in\N}{\cal E} (u_n)<\infty$. Let $f\in C_0({\Bbb R})$ be such that ${\bf 1}_{[0,1]}\le f\le {\bf 1}_{[-1/2,3/2)}$. For any $n\in\N$ and $i=(i_1,\dots,i_d)\in\Z^d$, define a function $f_i^{(n)} \in{\cal D}$ by $$ f_i^{(n)}(x){:=}\prod_{k=1}^d f(n x_k- i_k),\qquad x\in\R^d.$$ Let also $$I_i^{(n)}(x){:=}\prod_{k=1}^d {\bf 1} _{[-1/2,3/2)}(nx_k-i_k),\qquad x\in\R^d,$$ and note that $f_i^{(n)}\le I_i^{(n)}$. Let $\psi\in C_{\mathrm b}^\infty ({\Bbb R})$ be such that ${\bf 1} _{[2,\infty)}\le\psi\le{\bf 1}_{[1,\infty)}$ and $0\le\psi'\le 2\, {\bf 1}_{ (1,\infty)}$. Set ${\cal A}_n{:=}\Z^d\cap[-na,na]^d$ and define continuous functions $$\dd\Gamma\ni\gamma\mapsto u_n(\gamma){:=}\psi \left( \sup_{i\in{\cal A}_n}\la f_i^{(n)},\gamma\ra\right),\qquad n\in\N.$$ Evidently, $u_n\to {\bf 1}_N$ pointwise as $n\to\infty$. Furthermore, by an appropriate approximation of the function ${\Bbb R}^{|{\cal A}_n|} \ni (y_1,\dots,y_{|{\cal A}_n|})\mapsto \sup_{i=1,\dots, |{\cal A}_n|}y_i$ by $C_{\mathrm b}^\infty ({\Bbb R}^{|{\cal A}_n|})$ functions, we conclude that, for each $n\in\N$, $u_n\in{D}({\cal E})$. We now have: $$ S(u_n)(x,\gamma)=\left(\psi\left(\sup_{i\in{\cal A }_n}\la f_i^{(n)},\gamma-\varepsilon_x\ra \right)-\psi\left(\sup_{i\in{\cal A }_n}\la f_i^{(n)},\gamma\ra \right)\right)^2\quad \text{$\tilde\mu$-a.e.}$$ By the mean value theorem, we get, for $\tilde\mu$-a.e.\ $(x,\gamma)\in\R^d\times\dd\Gamma $,: \begin{equation}\label{gug}S(u_n)(x,\gamma) =\psi'(T_n(\gamma,x))^2 \left(\sup_{i\in{\cal A }_n}\la f_i^{(n)},\gamma-\varepsilon_x\ra-\sup_{i\in{\cal A }_n}\la f_i^{(n)},\gamma\ra\right)^2,\end{equation} where $T_n(\gamma,x)\in\R$ is a point between $\sup_{i\in{\cal A }_n}\la f_i^{(n)},\gamma-\varepsilon_x\ra$ and $\sup_{i\in{\cal A }_n}\la f_i^{(n)},\gamma\ra$. It is easy to see that the following estimate holds, for any $\gamma\in\dd\Gamma$ and $x\in\R^d$: \begin{align} \left|\sup_{i\in{\cal A }_n}\la f_i^{(n)},\gamma-\varepsilon_x\ra-\sup_{i\in{\cal A }_n}\la f_i^{(n)},\gamma\ra\right|&\le \sup_{i\in{\cal A }_n} |\la f_i^{(n)},\gamma-\varepsilon_x\ra-\la f_i^{(n)},\gamma\ra|\notag \\ &= \sup_{i\in{\cal A }_n} f_i^{(n)}(x)\notag\\ &\le \sup_{i\in{\cal A }_n} I_i^{(n)}(x)\notag \\ &\le \pmb 1_{[-a-1,a+1]^d}(x). \label{hfh}\end{align} We evidently have, for each $\gamma\in\dd\Gamma$ and $x\in\operatorname{supp}(\gamma)$: $$ \sup_{i\in{\cal A }_n}\la f_i^{(n)},\gamma-\varepsilon_x\ra\le T_n(\gamma,x)\le\sup_{i\in{\cal A }_n}\la f_i^{(n)},\gamma\ra.$$ Hence, \begin{align} \psi'(T_n(\gamma,x))^2&\le 4\,\pmb 1_{ \{\sup_{i\in{\cal A}_n}\la f_i^{(n)},\cdot\ra>1\} }(\gamma)\notag\\ & \le 4\,\pmb 1_{ \{\sup_{i\in{\cal A}_n}\la I_i^{(n)},\cdot\ra\ge2\} }(\gamma)\notag\\ &\le4\sum_{i\in{\cal A}_n}\pmb1_{\{\la I_i^{(n)},\cdot\ra\ge2\}}(\gamma),\label{jja} \end{align} where we used the fact that $I_i^{(n)}$ is integer-valued. By \eqref{gug}--\eqref{jja}, we have, for $\tilde\mu$-a.e.\ $(x,\gamma)\in\R^d\times\dd\Gamma$: $$ S(u_n)(x,\gamma)\le 4\sum_{i\in{\cal A}_n}\pmb1_{\{\la I_i^{(n)},\cdot\ra\ge2\}}(\gamma)\, \pmb 1_{[-a-1,a+1]^d}(x).$$ Therefore, by the Cauchy--Schwarz inequality and \eqref{hh}, \begin{equation} {\cal E}(u_n)\le 4\sum_{i\in{\cal A}_n} (\mu(\{\la I_i^{(n)},\cdot\ra\ge2\}))^{1/2}\left(\int_{\dd\Gamma}\la\gamma,\pmb1 _{[-a-1,a+1]^d}\ra\,\mu(d\gamma)\right)^{1/2}.\label{bsddbh}\end{equation} By using \cite[Theorem~5.5]{Ru70}, we easily conclude that there exists a constant $C_3>0 $, independent of $i$ and $n$, such that, for all $i\in{\cal A}_n$ and $n\in\N$, \begin{equation}\mu(\{\la I_i^{(n)},\cdot\ra\ge2\})\le C_3 \left(\int_{\R^d}I_i^{(n)}(x)\,dx\right)^2= C_3\left(\frac2n\right)^{2d} .\label{five} \end{equation} Since $|{\cal A}_n|=(2na+1)^d$, we get from \eqref{6t565r7}, \eqref{bsddbh}, and \eqref{five}: $$ {\cal E}(u_n)\le 4C_3^{1/2}(2na+1)^d \left(\frac2n\right)^d\left(\int _ {[-a-1,a+1]^d} k_\mu^{(1)}(x)\,dx\right)^{1/2},\qquad n\in\N.$$ Therefore, there exists a constant $C_4>0$, independent of $n$, such that ${\cal E}(u_n)\le C_4$ for all $n\in\N$.\quad $\square$ We now have the main result of this section. \begin{theorem}\label{8435476} \rom{1)} Suppose that the conditions of Proposition~\rom{\ref{waessedf}} are satisfied\rom. Then\rom, there exists a Hunt process %\rom(i\rom.e\rom{.,} a conservative %strong Markov process with continuous sample paths\rom) $${\bf M}=({\pmb{ \Omega}},{\bf F},({\bf F}_t)_{t\ge0},({\pmb \Theta}_t)_{t\ge0}, ({\bf X}(t))_{t\ge 0},({\bf P }_\gamma)_{\gamma\in\Gamma})$$ on $\Gamma$ \rom(see e\rom.g\rom.\ \rom{\cite[p.~92]{MR})} which is properly associated with $({\cal E},D({\cal E}))$\rom, i\rom.e\rom{.,} for all \rom($\mu$-versions of\/\rom) $F\in L^2(\Gamma,\mu)$ and all $t>0$ the function \begin{equation}\label{zrd9665} \Gamma\ni\gamma\mapsto p_tF(\gamma){:=}\int_{\pmb\Omega} F({\bf X}(t))\, d{\bf P}_\gamma\end{equation} is an ${\cal E}$-quasi-continuous version of $\exp(-t{H})F$\rom, where $H$ is the generator of $({\cal E},D({\cal E}))$\rom. $\bf M$ is up to $\mu$-equivalence unique \rom(cf\rom.\ \rom{\cite[Chap.~IV, Sect.~6]{MR}).} In particular\rom, $\bf M$ is $\mu$-symmetric \rom(i\rom.e\rom{.,} $\int G\, p_tF\, d\mu=\int F \, p_t G\, d\mu$ for all $F,G:\Gamma\to{\Bbb R}_+$\rom, ${\cal B}(\Gamma)$-measurable\rom) and has $\mu$ as an invariant measure\rom. \rom{2)} $\bf M$ from \rom{1)} is up to $\mu$-equivalence \rom(cf\rom.\ \rom{\cite[Definition~6.3]{MR}}\rom) unique between all Hunt processes ${\bf M}'=({\pmb{ \Omega}}',{\bf F}',({\bf F}'_t)_{t\ge0},({\pmb \Theta}'_t)_{t\ge0}, ({\bf X}'(t))_{t\ge 0},({\bf P }'_\gamma)_{\gamma\in\Gamma})$ on $\Gamma$ having $\mu$ as an invariant measure and solving the martingale problem for $(-H, D(H))$\rom, i\rom.e\rom.\rom, for all $G\in D(H)$ $$\widetilde G({\bf X}'(t))-\widetilde G({\bf X}'(0))+\int_0^t (H G)({\bf X}'(s))\,ds,\qquad t\ge0,$$ is an $({\bf F}_t')$-martingale under ${\bf P}_\gamma'$ for ${\cal E}$-q\rom.e\rom.\ $\gamma\in\Gamma$\rom. \rom(Here\rom, $\widetilde G$ denotes a quasi-continuous version of $G$\rom, cf\rom. \rom{\cite[Ch.~IV, Proposition~3.3]{MR}.)} \end{theorem} \begin{rem}\rom{In fact, the statement of Theorem~\ref{8435476} remains true for any Gibbs measure $\mu\in{\cal G}(z,\phi)$ whose correlation functions satisfy the Ruelle bound\rom. }\end{rem} \noindent {\it Proof of Theorem\/}~\ref{8435476}. The first part of the theorem follows from Propositions~\ref{scfxgus}, \ref{fdj} and \cite[Chap.~IV, Theorem~3.5 and Chap.~V, Proposition~2.15]{MR}. The second part follows directly from (the proof of) \cite[Theorem~3.5]{AR}.\quad $\square$ In the above theorem, $\bf M$ is canonical, i.e., $\pmb\Omega$ is the set of all {\it cadlag} functions $\omega:[0,\infty)\to \Gamma$ (i.e., $\omega$ is right continuous on $[0,\infty)$ and has left limits on $(0,\infty)$), ${\bf X}(t)(\omega){:=}\omega(t)$, $t\ge 0$, $\omega\in\pmb\Omega$, $({\bf F}_t)_{t\ge 0}$ together with $\bf F$ is the corresponding minimum completed admissible family (cf.\ \cite[Section~4.1]{Fu80}) and ${\pmb \Theta}_t$, $t\ge0$, are the corresponding natural time shifts. \section{Selfadjointness of the generator}\label{gzguz4545} In what follows, we will always suppose that the potential $\phi$ is positive. \begin{theorem}\label{ufhhu} Suppose that conditions \rom{(P), (I)} are satisfied and $\mu\in{\cal G}(z,\phi)$\rom, $z>0$\rom. Then\rom, the operator $(H,\FC)$ is essentially selfadjoint in $L^2(\Gamma,\mu)$\rom. In particular, Friedrichs' extension of $(H,\FC)$ coincides with its closure\rom. \end{theorem} \noindent {\it Proof}. Let $(\tilde H, D(\tilde H))$ denote the closure of $(H,\FC)$, which exists since the latter operator is a Hermitian one. We have to show that $(\tilde H, D(\tilde H))$ is selfadjoint. Since $\tilde H\ge0$, by the Nussbaum theorem, it is enough to show that there exists a set ${\cal S}\subset\bigcap_{n=1}^\infty D(\tilde H ^n)$ which is total in $L^2(\Gamma,\mu)$ and each $F\in {\cal S}$ satisfies: $$\sum_{n=0}^\infty \frac{\|\tilde H^n F \|_{L^2(\mu)}}{(2n)!}\, t^n<\infty$$ for some $t>0$, see e.g.\ \cite[Theorem~X.40]{RS}. We have the following lemma, whose proof is completely analogous to that of Lemma~\ref{lemm1}. \begin{lem}\label{lem1} Suppose that the conditions of Theorem~\rom{\ref{ufhhu}} are satisfied\rom. Let a function $F$ be as in the formulation of Lemma~\rom{\ref{lemm1}}. Then\rom, $F\in D(\tilde H)$ and the action of $\tilde H$ on $F$ is given by the right hand side of \eqref{gen}\rom. \end{lem} We denote by ${\cal P}$ the set of all continuous polynomials on $\Gamma$, i.e., the set of all finite sums of functions of the form $ F(\gamma)=\prod_{i=1}^n\la \gamma,\varphi_i\ra$, $ \varphi_i\in \D$, $i=1,\dots,n$, $n\in\N$, and constants. We preserve the notation $\cal P$ for the set of all $\mu$-classes of functions from $\cal P$. Using \eqref{swaswea957} and e.g.\ \cite{BKKL}, we see that ${\cal P}$ is a dense subset in $L^2(\Gamma,\mu)$. Furthermore, any function from $\cal P$ is cylinder, and we can easily conclude from Lemma~\ref{lem1} that its statement remains true for any function $F\in{\cal P}$. We will now show that ${\cal P}\subset\bigcap_{n=1}^\infty D(\tilde H^n )$. We first make some informal calculations. So, we define: \begin{align*} H_1F(\gamma)&{:=}\int z\, dx \exp\left[-E(x,\gamma)\right] D^+_x F(\gamma),\\ H_2F(\gamma)&{:=}\int \gamma(dx)\, D^-_x F(\gamma),\end{align*} so that $H=H_1-H_2$. Then, \begin{equation}\label{fctukaf}H^n=(H_1-H_2)^n=\sum_{I\subset\{1,\dots,n\}}(-1)^{n-|I|}{\cal H}_I^{(n)},\end{equation} where \begin{align} {\cal H}_I^{(n)}&{:=}{\cal H}_{I,1}{\cal H}_{I,2}\dotsm {\cal H}_{I,n},\notag \\ {\cal H}_{I,i}&{:=}\begin{cases}H_1,&i\in I,\\ H_2,&i\not\in I,\end{cases}\quad i=1,\dots,n. \label{sdgioa}\end{align} Furthermore, by induction, we conclude: \begin{equation}\label{whuwiuiw} {\cal H}_I^{(n)}=\sum_{J\subset \big\{1,\dots,\, \max\{i:\, i\in I\}-1\big\}} {\cal H}_{I,J}^{(n)},\end{equation} where \begin{align} &{\cal H}_{I,J}^{(n)}F(\gamma)=\bigg(\int m_{I,n}(dx_n)\, U_{I,J,n,x_n}\int m_{I,n-1}(dx_{n-1})U_{I,J,n-1,x_{n-1}}\notag \\&\qquad \dotsm \int m_{I,1}(dx_1)\, U_{I,J,1,x_1} \prod_{i\in I}\exp\left[-\sum_{u\in\eta\setminus\{x_s:\, s\in I^c,\, s>i\}} \phi(x_i-u)\right] \notag \\ &\qquad \times \prod_{j=1}^n G_{I,J,j}(x_1,\dots,x_n)F(\gamma)\bigg){\bigg|}_{\eta=\gamma},\notag\\ & I^c{:=}\{1,\dots,n\}\setminus I,\notag\\ & m_{I,i}(dx_i){:=}\begin{cases}z\, dx_i,&i\in I,\\ \gamma(dx_i),& i\in I^c,\end{cases}\notag\\ & U_{I,J,i,x_i}{:=}\begin{cases}D^+_{x_i},& i\in I,\ i\in J^c,\\ D^-_{x_i},& i\in I^c,\ i\in J^c,\\ \operatorname{id},& i\in J,\end{cases}\notag \\ & G_{I,J,j}^{(n)}(x_1,\dots,x_n){:=}\begin{cases}\exp\left[-\sum_{r\in I,\, r0$ such that $\sum_{n=0}^\infty \|\tilde H^n F \|_{L^2(\mu)}t^n/(2n)!<\infty$\rom. \end{lem} \noindent {\it Proof}. We first derive some estimates. Let $f_1,\dots,f_k$ be bounded integrable functions on $\R^d$. Consider $G(\gamma){:=}\linebreak\prod_{i=1}^k\la \gamma, f_i\ra$. From \eqref{6t565r7}, we conclude that \begin{align*}&\int_\Gamma G(\gamma)\,\mu(d\gamma)= \sum_{i=1}^k \sum_{\stackunder{\text{$A_j$'s disjoint},\, A_1\cup\dotsm\cup A_i=\{1,\dots,k\} }{(A_1,\dots,A_i):\, \varnothing \ne A_j\subset\{1,\dots,k\},\, j=1,\dots,i}}\\&\qquad\times \int_{(\R^d)^i} g^{(k)}(\underbrace{x_1,\dots,x_1}_{\text{$|A_1|$ times}},\underbrace{x_2,\dots,x_2}_{\text{$|A_2|$ times}},\dots, \underbrace{x_i,\dots,x_i}_{\text{$|A_i|$ times}}) k_\mu^{(i)}(x_1,\dots,x_i)\, dx_1\dotsm dx_i, \end{align*} where $g^{(k)}(x_1,\dots,x_k)=(1/k!)\sum_{\sigma\in S_k}f_1(x_{\sigma(1)})\dotsm f_k(x_{\sigma(k)})$. By induction, we prove $$ \sum_{i=1}^k \sum_{\stackunder{\text{$A_j$'s disjoint},\, A_1\cup\dotsm\cup A_i=\{1,\dots,k\} }{(A_1,\dots,A_i):\, \varnothing \ne A_j\subset\{1,\dots,k\},\, j=1,\dots,i}} 1\le 2^{k-1}k!,\qquad k\in\N.$$ Therefore, by \eqref{swaswea957}, \begin{equation}\label{a}\int_\Gamma |G(\gamma)|\,\mu(d\gamma)\le 2^ {k-1}\max\{1,\xi\}^k\, k!\, \prod_{i=1}^k\max\{\|f_i\|_{L^1},\|f_i\|_{L^\infty}\}.\end{equation} Note that, for $\mu$-a.e.\ $\gamma\in\Gamma$, \begin{align}D_x^+ G(\gamma)&=-\sum_{i=0}^{k-1} \binom i k \int \gamma (dy_1)\dotsm \gamma(dy_{i})\,g^{(k)}(y_1,\dots,y_{i},\underbrace{x,\dots,x}_{\text{$(k-i)$ times }}),\notag\\ D^-_x G(\gamma)&=\sum_{i=0}^{k-1} \binom i k (-1)^{k-i}\int \gamma (dy_1)\dotsm \gamma(dy_{i})\,g^{(k)}(y_1,\dots,y_{i},\underbrace{x,\dots,x}_{\text{$(k-i)$ times }}).\label{c} \end{align} We next easily get the following identity: \begin{equation}\label{d} \sum_{k_1=0}^n\binom {k_1}n\sum_{k_2=0}^{k_1}\binom{k_2}{k_1}\sum_{k_3=0}^{k_2}\binom{k_3}{k_2}\dotsm \sum_{k_m=0}^{k_{m-1}}\binom{k_m}{k_{m-1}}=m^n,\qquad m,n\in\N. \end{equation} Recall also the standard estimate \begin{equation}\label{e}(a_1+\dots+a_n)^2\le n(a_1^2+\dots+a_n^2),\qquad a_1,\dots,a_n\in\R,\ n\in\N.\end{equation} Finally, using condition (P), we get, for any $x,y_1,\dots,y_k\in\R^d$ and $k\in\N$, \begin{equation}\label{f} 1-\exp\left[-\sum_{i=1}^k\phi(x-y_i)\right]\le\sum_{i=1}^k(1-\exp[-\phi(x-y_i)]). \end{equation} Let now $F(\gamma)$ be as in the formulation of the lemma. Fix $\Lambda\in {\cal O}_{\mathrm c}(\R^d)$ and $C_5>0$ such that \begin{equation}\label{g} |f_1(x_1)\dotsm f_l(x_l)|\le C_5\chi_{\Lambda^l}(x_1,\dots,x_l),\qquad x_1,\dots,x_l\in\R^d. \end{equation} For any sets $I,J\subset\{1,\dots,n\}$ as in \eqref{whuwiuiw}, we define $$ n_1{:=}|I\cap J|,\quad n_2{:=}|I^c\cap J|,\quad n_3{:=}|I\cap J^c|,\quad n_4{:=}|I^c\cap J^c|.$$ Notice that $n_1+n_2+n_3+n_4=n$. Estimating all the multipliers of the form $e^{-\phi(\cdot)}$ by 1, and using \eqref{a}--\eqref{g}, we get from \eqref{sigidz} \begin{align}& \|{\cal H}_{I,J}^{(n)}F\|_{L^2(\mu)}^2\le \max\{1,z\}^{2n}C_5^2 \max\left\{1,\int_\Lambda dx\right\}^{2(n+l)}\max\{1,\xi\}^{2(n+l)}\notag\\ &\qquad\times \max\left\{1,\int_{\R^d}(1-\exp[-\phi(x)])\right\}^{2n}\notag\\ &\qquad\times 2^{2(l+n_2)-1}(2(l+n_2))!\,(n_1+n_3)!\, (n_1+n_3)^{n_2}(n_3+n_4)^{l+n_2},\label{vydvsj} \end{align} where the factor $2^{2(l+n_2)-1}(2(l+n_2))!$ is connected with estimate \eqref{a} and the fact that we get monomials of order $\le l+n_2$, the factor $(n_1+n_3)!$ is connected with the application of \eqref{f} to the terms connected with the set $I$, the factor $(n_1+n_3)^{n_2}$ is connected with the application of \eqref{f} to the terms connected with $I^c\cap J$, and the factor $(n_3+n_4)^{l+n_2}$ is connected with the application of \eqref c, \eqref d to a monomial of order $\le(l+n_2)$. Using the estimate $(2k)!\le 4^k(k!)^2$, $k\in\N$, we conclude from \eqref{vydvsj} that there exists a constant $C_6>0$, independent of $n,I,J$ and thus depending only on $F$, such that $$ \|{\cal H}_{I,J}^{(n)}F\|_{L^2(\mu)}^2\le C_6^n (n!)^2 n^{2n}.$$ Hence, by \eqref{fctukaf} \begin{equation}\label{dcgh} \|\tilde H^n f\|_{L^2(\mu)}\le (2C_6)^{n/2}n!\, n^n.\end{equation} Estimate \eqref{dcgh}, together with Stirling's formula, easily implies the statement of the lemma, and hence the statement of the theorem.\quad$\square$ \section{Spectral gap of the generator}\label{hucddhs} We first prove a coercivity identity for the gradient $D^-$. We note that, for any $\gamma\in\Gamma$ and $F\in\FC$, $(D^-)^2 F(\gamma)$ is the element of the Hilbert space $T_\gamma^{\otimes 2}=L^2((\R^d)^2,\gamma^{\otimes 2})$ given by $(D^-)^2F(\gamma,x,y)=D^-_xD^-_yF(\gamma)$, $x,y\in\R^d$. Furthermore, for any $x,y\in\gamma$: \begin{equation}\label{hsg} D^-_xD^-_y F(\gamma)=\begin{cases} F(\gamma\setminus\{x,y\})-F(\gamma\setminus x)-F(\gamma\setminus y)+F(\gamma),&x\ne y,\\ F(\gamma)-F(\gamma\setminus x)=-D^-_xF(\gamma),& x=y.\end{cases}\end{equation} Through the natural identification of the elements of $T_\gamma^{\otimes 2}$ with linear continuous operators in $T_\gamma$, we get: \begin{equation}\label{uisdz}\operatorname{Tr} (D^-)^2F(\gamma)((D^-)^2F(\gamma))^*=\sum_{x,y\in\gamma}(D^-_xD^-_yF(\gamma))^2.\end{equation} Here, $\operatorname{Tr}$ denotes the trace of an operator and $((D^-)^2F(\gamma))^*$ is the adjoint operator of $(D^-)^2F(\gamma)$. \begin{lem}[Coercivity identity]\label{zsfkllk} Suppose that the conditions of Theorem~\rom{\ref{ufhhu}} are satisfied\rom. Then\rom, for any $F\in\FC$\rom, we have \begin{align*}&\int_\Gamma (HF(\gamma))^2\,\mu(d\gamma)=\int_\Gamma\bigg[ \operatorname{Tr}(D^-)^2F(\gamma)((D^-)^2F(\gamma))^* +\sum_{x,y\in\gamma,\, x\ne y}(\exp[\phi(x-y)]-1)\\&\qquad\times(F(\gamma\setminus\{x,y\})-F(\gamma\setminus x)) (F(\gamma\setminus\{x,y\})-F(\gamma\setminus y))\bigg]\,\mu(d\gamma).\end{align*} \end{lem} \noindent{\it Proof}. Analogously to \eqref{zgussguz}, we get from \eqref{fdrtsdrt}: \begin{align}&\int_\Gamma \mu(d\gamma)\,\left(\int_{\R^d}z \, dx\, \exp\left[- E(x,\gamma)\right]D_x^+F(\gamma)\right)^2\notag\\&\qquad = \int_\Gamma \mu(d\gamma)\sum_{x,y\in\gamma,\,x\ne y}\exp[\phi(x-y)](F(\gamma\setminus\{x,y\})-F(\gamma\setminus x))\notag\\&\qquad\quad\times (F(\gamma\setminus\{x,y\})-F(\gamma\setminus y)).\label{gaft}\end{align} Next, \begin{align}& -2\int_\Gamma\mu(d\gamma)\int_{\R^d}z\, dx\, \exp\left[-E(x,\gamma)\right]D_x^+ F(\gamma)\int_{\R^d}\gamma(dy)\,D^-_y F(\gamma)\notag\\ &\qquad= -2\int_\Gamma\mu(d\gamma)\int_{\R^d}\gamma(dx)\,(F(\gamma\setminus x)-F(\gamma))\notag\\&\qquad\quad\times\int_{\R^d} (\gamma\setminus x)(dy)\,(F(\gamma\setminus\{x,y\})-F(\gamma\setminus x))\notag\\&\qquad=- \int_\Gamma\mu(d\gamma)\sum_{x,y\in\gamma,\,x\ne y}\big[(F(\gamma\setminus x)-F(\gamma))(F(\gamma\setminus\{x,y\})-F(\gamma\setminus x))\notag\\ &\qquad\quad- (F(\gamma\setminus y)-F(\gamma))(F(\gamma\setminus\{x,y\})-F(\gamma\setminus y))\big].\label{hdh} \end{align} Finally, \begin{align}&\int_\Gamma\mu(d\gamma)\left(\int_{\R^d}\gamma(dx)\, D^-_xF(\gamma)\right)^2\notag\\&\qquad=\int_\Gamma \mu(d\gamma)\sum_{x\in\gamma}(F(\gamma\setminus x)-F(\gamma))^2\notag\\&\qquad\quad +\int_\Gamma \mu(d\gamma)\sum_{x,y\in\gamma,\,x\ne y}(F(\gamma\setminus x)-F(\gamma))(F(\gamma\setminus y)-F(\gamma))\label{ghjs}\end{align} By \eqref{gen} and \eqref{hsg}--\eqref{ghjs}, the lemma follows.\quad $\square$ \begin{theorem}\label{ewsrth080} Suppose that \rom{(P)} holds\rom, $z>0$\rom, and \begin{equation} \delta{:=}\int_{\R^d}(1-\exp[-\phi(x)])\, z\, dx<1.\label{guzggz} \end{equation} Let $\mu\in{\cal G}(z,\phi)$\rom. Then\rom, the set $(0,1-\delta)$ does not belong to the spectrum of $H$\rom. \end{theorem} \noindent {\it Proof}. We fix any $F\in\FC$. By \eqref{hsg} and \eqref{uisdz}, we have: \begin{equation}\label{jvh}\operatorname{Tr} (D^-)^2F(\gamma)((D^-)^2F(\gamma))^* \ge\sum_{x\in\gamma}(D^-_xD^-_xF(\gamma))^2=\sum_{x\in\gamma}(D^-_xF(\gamma))^2,\qquad \gamma\in\Gamma.\end{equation} Using (P), \eqref{fdrtsdrt}, \eqref{guzggz}, and the Cauchy--Schwarz inequality, we next have \begin{align} &\bigg|\int_\Gamma \sum_{x,y\in\gamma,\, x\ne y}(\exp[\phi(x-y)]-1)(F(\gamma\setminus\{x,y\})-F(\gamma\setminus x))\notag\\ &\qquad\quad\times (F(\gamma\setminus\{x,y\})-F(\gamma\setminus y))\,\mu(d\gamma)\bigg|\notag\\ &\qquad \le \int_\Gamma \sum_{x,y\in\gamma,\, x\ne y}(\exp[\phi(x-y)]-1)(F(\gamma\setminus\{x,y\})-F(\gamma\setminus y))^2 \,\mu(d\gamma)\notag\\ &\qquad= \int_\Gamma\mu(d\gamma)\int_{\R^d}\gamma(dy)\int_{\R^d}(\gamma\setminus y)(dx)\, (\exp[\phi(x-y)]-1)\notag\\ &\qquad\quad\times (F(\gamma\setminus\{x,y\})-F(\gamma\setminus y))^2\notag\\ &\qquad=\int_\Gamma\mu(d\gamma)\int_{\R^d}z\, dy\,\exp\left[-E(y,\gamma)\right]\int_{\R^d}\gamma(dx)\,(\exp[\phi(x-y)]-1)\notag\\ &\qquad\quad\times (F(\gamma\setminus x)-F(\gamma))^2\notag\\ &\qquad= \int_\Gamma\mu(d\gamma)\int_{\R^d}z\, dy\int_{\R^d}\gamma(dx)\, \exp\left[-E(y,\gamma\setminus x )\right]\notag\\ &\qquad\quad\times(1-\exp[-\phi(x-y)])(F(\gamma\setminus x)-F(\gamma))^2\notag\\ &\qquad\le \int_\Gamma\mu(d\gamma)\int_{\R^d}z\, dy\int_{\R^d}\gamma(dx)\, (1-\exp[-\phi(x-y)])(F(\gamma\setminus x)-F(\gamma))^2\notag\\&\qquad= \int_{\R^d}(1-\exp[-\phi(y)])\, z\,dy\times \int_\Gamma\mu(d\gamma)\int_{\R^d}\gamma(dx)\,(D^-_xF(\gamma))^2\notag\\&\qquad= \delta\, (HF,F)_{L^2(\mu)}.\label{huui} \end{align} Using Lemma~\ref{zsfkllk}, \eqref{jvh}, and \eqref{huui}, we get, for each $F\in\FC$: \begin{equation} (HF,HF)_{L^2(\mu)}\ge(1-\delta)(HF,F)_{L^2(\mu)}.\label{shjn}\end{equation} From Theorem~\ref{ufhhu}, we then conclude that \eqref{shjn} holds true for each $F\in D(H)$. Therefore, denoting by $(E_\lambda)_{\lambda\ge0}$ the resolution of the identity of the operator $H$, we have: $$\int_{[0,\infty)}\lambda(\lambda-(1-\delta))\, d(E_\lambda F,F)_{L^2(\mu)}\ge0,\qquad F\in D(H). $$ From here the statement of the theorem trivially follows.\quad $\square$ \begin{cor}[Poincar\'e inequality]\label{jndfjdj4} Suppose \rom{(P)} and \rom{\eqref{guzggz}} hold and suppose $\mu$ is an extreme point of the convex set ${\cal G}(z,\phi)$\rom. Then\rom, \begin{equation} {\cal E}(F,F)\ge(1-\delta)\int_\Gamma(F(\gamma)-\la F\ra_\mu)^2\,\mu(d\gamma),\qquad F\in D({\cal E}), \label{chdhbbh}\end{equation} where $\la F\ra_\mu{:=}\int_\Gamma F(\gamma)\,\mu(d\gamma)$. \end{cor} \begin{rem} \rom{The Poincar\'e inequality \eqref{chdhbbh} means that, in addition to the fact that the set $(0,1-\delta)$ does not belong to the spectrum of $H$, we also have that the kernel of $H$ consists only of the constants. } \end{rem} \noindent {\it Proof of Corrolary~\rom{\ref{jndfjdj4}}}. Since $\mu$ is extreme in ${\cal G}(z,\phi)$, analogously to proof of the part (i)$\Rightarrow$(ii) of \cite[Theorem~6.2]{AKR4}, we conclude: \begin{align}& \{ \nu\in {\cal G}(z,\phi) \mid \nu=\rho\cdot \mu \text{\rom{ for some bounded, ${\cal B}(\Gamma)$-measurable function $\rho:\Gamma\to\R_+$}}\}\notag \\&\qquad =\{\mu\}.\label{hjdvhj}\end{align} Let $G\in D({\cal E})$ be such that ${\cal E}(G)=0$. It suffices to prove that $G=\operatorname{const}$. By the proof of \cite[Lemma~6.1]{AKR4}, without loss of generality, we can suppose that the function $G$ is bounded. Now, we modify the proof of the part (ii)$\Rightarrow$(iii) of \cite[Theorem~6.2]{AKR4}. Since $1\in\FC$ and $D^-1=0$, replacing $G$ by $G-\operatorname{ess\,inf}G$, we may suppose that $G\ge0$, and, in addition, that $\int G\, d\mu=1$. Define $\nu{:=}G\cdot \mu$. Since ${\cal E}(G)=0$, by \eqref{hh}, we have that $G(\gamma\setminus x)-G(\gamma)=0$ $\tilde \mu$-a.e. Since $\mu\in{\cal G}(z,\phi)$, we have, for any measurable function $F:\Gamma\times\R^d\to [0,+\infty]$, \begin{align*} \int_\Gamma \nu(d\gamma) \int_{\R^d}\gamma(dx)\, F(\gamma,x)& =\int_\Gamma \mu(d\gamma)\int_{\R^d}\gamma(dx)\, G(\gamma) F(\gamma,x)\\ &= \int_\Gamma\mu(d\gamma)\int_{\R^d}\gamma(dx)\, G(\gamma\setminus x)F(\gamma,x)\\&= \int_\Gamma \mu(d\gamma)\int_{\R^d}z\, dx\, \exp[-E(x,\gamma)]G(\gamma)F(\gamma\cup x,x)\\&= \int_\Gamma \nu(d\gamma)\int_{\R^d}z\,dx\, \exp[-E(x,\gamma)]F(\gamma\cup x,x). \end{align*} Hence, $\nu\in{\cal G}(z,\phi)$ and, by \eqref{hjdvhj}, $G=1$ $\mu$-a.e.\quad $\square$ Let us suppose that the potential $\phi$ satisfies the following condition: \begin{description} \item[(LAHT)] ({\it Low activity-high temperature regime}) $$\delta =\int_{\R^d}(1-\exp[-\phi(x)])\, z\,dx <\exp(-1).$$ \end{description} Under (P) and (LAHT), there exists a unique Gibbs measure $\mu\in{\cal G}(z,\phi)$, see \cite{Ru69} and \cite[Theorem~6.2]{Kuna2} (notice that \eqref{fdrtsdrt} and (P) imply that the correlation functions of any $\mu\in{\cal G}(z,\phi)$ satisfy the Ruelle bound \eqref{swaswea957} with $\xi=1$, and hence we can take the constant $C_R$ in \cite[Theorem~6.2]{Kuna2} to be equal to $e$). 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