\documentstyle[11pt]{article}
\textwidth 146 mm
\textheight 230 mm
\oddsidemargin 7mm \evensidemargin -1mm \topmargin -4mm
% \def\baselinestretch{1.4}
\def\R{{\bf R}}
\def\Z{{\bf Z}}
\def\P{{I\!\!P}}
\def\EE{{\cal E}}
\def\XX{{\cal X}}
\def\XXq{\XX^{(q)}}
\def\X{{\vec X}}
\def\L{\Lambda}
\def\LL{{\cal L}}
\def\Ln{L(n)}
\def\D{\Delta}
\def\LY{{\L|Y}}
\def\s{\sigma}
\def\th{\vartheta}
\def\ph{\varphi}
\def\sp{\mbox{sp}}
\def\inv{^{-1}}
\def\be{\begin{equation}\label}
\def\ee{\end{equation}}
\def\bea{\begin{eqnarray*}}
\def\eea{\end{eqnarray*}}
\begin{document}
\title{{\bf Phase Transition\\in Continuum Potts Models}}
\author{Hans-Otto Georgii\thanks{Research partially supported by the Isaac Newton Institute Cambridge}
\and Olle H\"{a}ggstr\"{o}m\thanks{Research supported by
the Swedish Natural Science Research Council and
the Deutsche For\-schungs\-gemeinschaft}
\and \\ University of Munich and Chalmers University of Technology}
\date{}
\maketitle
\begin{abstract}
We establish phase transitions for a class of continuum multi--type particle systems with finite range repulsive pair interaction between particles
of different type. This proves an old conjecture of Lebowitz and Lieb.
A phase transition still occurs when we allow a background pair interaction
(between all particles) which is superstable and has sufficiently short
range of repulsion. Our approach
involves a random--cluster representation analogous to the Fortuin--Kasteleyn
representation of the Potts model. In the course of our argument, we establish the existence of a percolation transition for Gibbsian particle systems with random edges between the particles, and also give an alternative proof for the existence of Gibbs measures with superstable interaction.
\end{abstract}
\section{Introduction}
Although the study of phase transitions for Gibbsian systems is one of the main subjects of statistical mechanics, examples of models exhibiting phase transition are mainly restricted to lattice systems. For systems of particles in the continuum the situation is still quite unsatisfactory. Besides the mere existence of phase transitions for some unknown interactions (as shown in Appendix B of Israel
\cite{Israel}) and the canonical--ensemble approach of Johansson
\cite{Jo1,Jo2} in one dimension, there exists essentially only one specific
model for which a phase transition is known to occur: the model of
Widom and Rowlinson \cite{WR}. This is a multi--type particle system in
$\R^d $, $d\geq 2$, with hard-core exclusion between particles
of different type, and no interaction between particles of the same type.
The phase transition in this model for large activities $z$ was established by
Ruelle \cite{Ru} using a version of the Peierls argument. Lebowitz and Lieb
\cite{LL} extended his result to multi--type particle systems in which the hard--core exclusion is replaced by a soft--core repulsion between
unlike particles. Such models can also be viewed as continuum versions
of the Ising or Potts model. Lebowitz and Lieb needed the condition that the
soft--core repulsion is sufficiently strong (or, equivalently, that the temperature is sufficiently low), and conjectured that a phase transition for large $z$ should still occur without this proviso.
In this paper we establish the existence of phase transitions for a large class of continuum Potts models including
the Widom--Rowlinson model and its soft--core variant. In particular, we shall verify the preceding conjecture. We also include a background interaction between all particles (independently of their types) which is superstable but not too repulsive in relation to the intertype repulsion.
Our approach is based on a
random--cluster representation analogous to the Fortuin--Kasteleyn
representation of the lattice Potts model; see \cite{FK,ES,Gr}. This gives a certain continuum percolation process,
which has previously been studied e.g\frenchspacing.
by Given and Stell \cite{GS} from a different point of view.
A phase transition for the continuum Potts model follows if we can show that
the corresponding percolation process contains an infinite cluster.
A similar program was recently carried out
by Chayes {\sl et al\frenchspacing.} \cite{CCK} for the hard--core
Widom--Rowlinson model (related ideas appear also in \cite{GLM}). In that
case, the existence of infinite clusters follows from
a stochastic comparison with the Poisson Boolean model of continuum
percolation (see \cite{MR}), while our more general framework requires somewhat
more involved arguments.
We now describe the setup. We consider the Euclidean space $\R^d$, $d\geq 2$,
and fix an integer $q\ge2$. Let $\XX$ denote the class of all locally finite subsets of $\R^d$. A configuration of particles in $\R^d$ with $q$
distinct types (without multiple occupancies)
is described by a vector $\X=(X_1,\ldots,X_q)$ of pairwise disjoint sets $X_1,\ldots, X_q\in\XX$. We write $\XXq$ for the class of all such multitype configurations. Any $\X\in\XXq$ is uniquely determined by the pair $(X,\s)$,
where $X=\sp(\X):= X_1\cup\ldots\cup X_q$ is the set of all occupied positions, and the type function $\s=\s(\X): X\to\{1,\ldots,q\}$ is defined by $\s(x)=s$ if $x\in X_s$, $1\le s\le q$. $\XX$ is equipped with the $\s$--algebra which is generated by the counting variables $\XX\ni X\to\#(X\cap
\Delta)$ for bounded measurables $\Delta\subset\R^d$, and $\XXq$ with the restriction of the product $\s$--algebra on $\XX^q$.
The set of all configurations in a measurable set $\L\subset\R^d$ will be denoted by $\XXq_\L$, and the corresponding $\s$--algebra is similarly defined.
The interaction between the particles is assumed to be given by a formal Hamiltonian of the form
\bea
H(\X)&=& H^\ph(\X)+H^\psi(X)\\
&=&
\sum_{\{x,y\}\subseteq X\atop \s(x)\neq \s(y)} \varphi(x-y)
+\sum_{\{x,y\}\subseteq X} \psi(x-y)\;.
\eea
Here $\X\in\XXq,\,X=\sp(\X),\,\s=\s(\X)$, and the pair potentials
$\varphi,\psi:\R^d\rightarrow\,]-\infty,\infty]$ are even measurable functions.
The first term $H^\ph$ is the most important
contribution to the Hamiltonian. It describes a repulsion between particles of different types. The second term $H^\psi$ corresponds to a type--independent background pair interaction. Specifically, we assume the following. There
exist constants $u>0$ and $0\le r_1\le r_20$, the associated {\sl Gibbs distribution} $Q_{\L|\vec Y}$ on $\XXq_\L$ is defined by
\begin{equation}\label{Q}
Q_{\L|\vec Y} (d\X)
= Z_{\L|\vec Y} ^{-1}\exp[-H_{\L|\vec Y}(\X)]\;
\pi_\L (dX_1)\ldots\pi_\L(dX_q) \;.
\end{equation}
In the above,
\bea
H_{\L|\vec Y}(\X)&=& H_{\L|\vec Y}^\ph(\X) +H_{\L|Y}^\psi(X)\\
&=& \sum_{\{x,y\}\subseteq X\cup Y\atop \{x,y\}\cap\L\ne\emptyset,\, \s(x)\neq \s(y)}\varphi(x-y)
+ \sum_{\{x,y\}\subseteq X\cup Y\atop \{x,y\}\cap\L\ne\emptyset}\psi(x-y)
\;
\eea
[with $\X\in\XXq_\L,\, X=\sp(\X),\,Y=\sp(\vec Y)$, and $\s$ the common extension of $\s(\X)$ and $\s(\vec Y)$] is
the Hamiltonian in $\L$ with boundary condition $\vec Y$;
$\pi_\L=\pi_\L^{z}$ is the Poisson point process on $\XX_\L$ with constant intensity $z$, i.e.
\be{Poisson}
\int f\:d\pi_\L = e^{-z|\L|} \sum_{n=0}^\infty \frac{z^n}{n!} \int_{\L^n}f(\{x_1,\ldots,x_n\})\:dx_1\ldots dx_n
\ee
for any bounded measurable function $f$ on $\XX_\L$; and $Z_{\L|\vec Y}$ is a normalizing constant usually called the {\sl partition function}. (We will always use symbols like $Z_{\L|\vec Y}$ to denote such normalizing constants, but their precise meaning will vary with the context. Also,
throughout the paper we will often suppress the dependence of some quantities on the parameters $z, q, \varphi, \psi$ which will be considered as fixed.)
Later on, we will prefer to consider $Q_{\L|\vec Y}$ as a measure on $\XXq$ which is supported on the set $\XXq_{\L|\vec Y}=\{\X\in\XXq: X_i\setminus\L=Y_i
\mbox{ for }1\le i\le q\}$.
Since $\psi$ is not necessarily of finite range, $Q_{\L|\vec Y}$ is only well-defined if we require that $\vec Y$ (resp.\ its support $Y$) is {\sl tempered} in the sense that
\[
\sup_{k\ge1} k^{-d} \sum_{i\in{\bf Z}^d:|i|\le k} N_i(\vec Y)^2 < \infty\,,
\]
where $N_i(\vec Y)= \#(\vec Y\cap\Delta(i))$ is the number of particles in the cubic cell
\be{cell}
\Delta(i)= [-\delta/2, \delta/2[^d + \delta i\:, \qquad i\in\Z^d,
\ee
of an arbitrarily fixed side length $\delta>0$. It is then easy to see from (\ref{ss}) and (\ref{lr}) that $ H_{\L|\vec Y}$ is well--defined and $\exp[-H_{\L|\vec Y}]$ is integrable with respect to $(\pi_\L)^q$.
It is evident from (\ref{Q}) that, for fixed $X\in\XX_\L$, the conditional distribution of $\s(\X)$ under the condition $\sp \X=X$ relative to
$Q_{\L|\vec Y}$
is that of a discrete Potts model on $X$ with a position--dependent interaction between the type--variables. This justifies calling our model a {\sl continuum Potts model}.
\medskip\noindent
{\bf Definition}
{\sl A probability measure $Q$ on $\XXq$ is called a tempered Gibbs measure for the continuum Potts model or, for short, a continuum Potts
measure with activity $z>0$ and interaction potentials $\varphi$, $\psi$ if
$Q$ is supported on the set of tempered configurations and, for any box $\L\subset\R^d$, the probability kernel $\vec Y\to Q_{\L|\vec Y}$ from $\XXq_{\L^c}$ to $\XXq_\L$ is a version of the conditional distribution under $Q$ of the configuration in $\L$ given the configuration outside of $\L$. }
\medskip
Continuum Potts measures always exist, as will be shown in Section 4. A phase transition is said to occur
if there exists more than one continuum Potts
measure. The following theorem,
which is our main result, shows that this happens when $z$ is
sufficiently large.
\medskip\noindent
{\bf Theorem 1.1} \quad
{\sl Suppose $\varphi$ and $\psi$ satisfy assumptions (A1) to (A5), and let $q\geq 2$.
If the
activity $z$ is sufficiently large (depending on the model parameters $d,\, u,$ and $r_1$ to $r_4$), there exist at least $q$ distinct continuum Potts measures for $z,\,q,\,\varphi,\,\psi$ which are invariant and ergodic under translations and can be distinguished by the densities of particles with a fixed type.}
\medskip
In Remark 4.3 at the end of the paper we will show that the phase transition manifests itself also thermodynamically by a non--differentiablity (``discontinuity") of the pressure. Here is a number of further remarks.
\medskip\noindent
{\bf Remark 1.2} \quad The scale assumptions (A5), in their present form, are certainly an artefact of our method. We note, however, that the condition $r_1z_c$. We are unable to settle this issue.
\medskip\noindent
{\bf Remark 1.4} \quad In the above, the inverse temperature $\beta$ was set equal to 1. This is no
loss of generality because $\beta$ can be absorbed into $\varphi$ and $\psi$.
However, one may ask for the phase transition region in the
$(z,\beta)$--plane
when $q,\,\varphi,\,\psi$ are fixed. As the proof of Theorem 1.1 will show, there exists a decreasing function $z_0:\:]0,\infty[\,\to\,]0,\infty[$ (depending on all model parameters) such that the continuum Potts measure is non--unique whenever $\beta>0$ and $z>z_0(\beta)$. In particular, for given $z>z_0(\infty)$ a phase transition occurs for sufficiently large $\beta$, but as in the
previous remark we are unable to answer the question of monotonicity
in $\beta$.
\medskip\noindent
{\bf Remark 1.5} \quad
Suppose we take $q=2$ and $\psi\equiv 0$ and consider, for each $\X\in\XX^{(2)}$, the set
$X_1\in\XX$ of particles of type 1. Let $\L\subset \R^d$ be a fixed box and $\Delta\supset\L$ so large that the distance between $\L$ and $\R^d\setminus\Delta$ exceeds the range $r_4$ of $\varphi$. It is easy to see that the conditional distribution of $X_1\cap\L$ given $X_1\setminus\L$ under $Q_{\Delta|\vec Y}$ does not depend on $\vec Y$ and has a
$\pi_\L$--density proportional to
\begin{eqnarray*}
& &\hspace{-4ex}
\int \pi_{\Delta}(dX_2)\exp\Big[-\sum_{y\in X_2} \sum_{x\in X_1} \varphi(x-y)\Big]
\\
& =&\exp\left[-z\int\Big(1-\exp[- \sum_{x\in X_1}\varphi(x-y)]\Big)\,dy \right].
\end{eqnarray*}
The image of the a continuum Potts measure under the projection $\X\to X_1$ is therefore a Gibbs measure on $\XX$ for the formal Hamiltonian
\[
H_z^{(1)}(X)=z\int\Big(1-\exp\Big[
-\sum_{x\in X}\varphi(x-y)\Big]\Big)dy.
\]
and the activity $z$. In the special case of the hard--core potential
\be{WR}
\varphi(x)=\left\{
\begin{array}{ll}
\infty & \mbox{for } | x | \leq r \\
0 & \mbox{otherwise},
\end{array} \right.
\ee
this relation between the two--type model and its
one--type marginal was discovered by Widom and Rowlinson \cite{WR}. We conclude from Theorem 1.1 that, for large $z$, the one--type particle system
with Hamiltonian $H_z^{(1)}$ and activity $z$ exhibits a phase transition, in that there exist two translation--invariant Gibbs measures with different particle densities.
\medskip
The rest of this paper is organized as follows. In Section 2, we obtain the
random--cluster representation for the continuum Potts model on a compact
set $\Lambda\subset{\bf R}^d$. Section 3 is devoted to the existence of percolation in the random cluster model and, in particular, for Gibbsian systems of particles of a single type. By `percolation' we here mean the occurence
of infinite clusters. In Section 4 we conclude the proof of Theorem 1.1
by taking the infinite--volume limit $\Lambda \uparrow\R^d$. In particular, we obtain the existence of continuum Potts measures for arbitrary $z>0$ and thus,
in the case $q=1$, the existence of tempered Gibbs measures for pair interactions $\psi$ satisfying (A3) and a condition of upper regularity.
\section{The random--cluster representation}
The purpose of this section is to obtain a joint construction of the
continuum
Potts model on a fixed finite box $\Lambda\subset{\bf R}^d$ with parameters
$z$, $\varphi$ and $q$, and a related edge
process which we call the {\sl continuum random--cluster model}.
This is analogous to the joint construction of the discrete Potts model and
its Fortuin--Kasteleyn representation, which was discovered by Edwards and
Sokal \cite{ES}. The continuum random--cluster model was introduced first by Klein \cite{Klein}; see \cite{GS} and the references there for some subsequent
work. In the particularly nice case of the Widom--Rowlinson model when $\psi\equiv0$ and $\varphi$ is given by (\ref{WR}), the random--cluster representation was rediscovered independently by Chayes {\sl et al.} \cite{CCK} and by the present authors.
In the course of the construction, we will pay particular attention to
measurability questions. One reason for doing so is that some of the
earlier work on continuum percolation involves constructions of probability
spaces in which most of the interesting events turn out not to be measurable,
as pointed out by Meester and Steif \cite{MS,MR}.
Let $Y\in\XX_{\L^c}$ be a fixed tempered boundary configuration. We identify $Y$ with the boundary condition $\vec Y = (Y,\emptyset,\ldots,\emptyset)\in\XXq_{\L^c}$ of type--1 particles. (The assumption of constant type might seem unnaturally restrictive, but it keeps things simpler
and is sufficient for our purposes.)
The basic idea of the random--cluster representation is to introduce random edges between the particle positions. We thus introduce the set
\[
E_{\R^d} =\Big\{e=\{x,y\}\subset \R^d: x\ne y\Big\}
\]
of all possible edges between pairs of points in $\R^d$. The natural $\s$--algebra on $E_{\R^d}$ consists of all sets of the form $\{\{x,y\}\in
E_{\R^d}:(x,y)\in A\}$ with measurable $A\subseteq\R^d\times\R^d$. In particular, for any measurable $\Delta\subseteq\R^d$ the set
\[
E_{\Delta}= \{e\in E_{\R^d}:e\subseteq\Delta\}
\]
of all edges within $\Delta$ is measurable. We write
\[
\EE =\{E\subset E_{\R^d}: E \mbox{ locally finite}\}
\]
for the set of all edge configurations. $\EE$ comes equipped with the $\s$--algebra which is generated by the counting variables $E\to\#(E\cap F)$ with bounded measurable $F\subset E_{\R^d}$.
We will construct a probability measure $\P=\P_\LY^{zq}$ on $\XXq\times\EE$
which models the following random mechanism. Indistinguishable
particles appear in $\Lambda$ according to a Gibbs distribution for
the background interaction $\psi$ with activity $zq$ and boundary condition $Y$. Each particle picks a type at random from
$\{1,\ldots,q\}$. Finally, each pair $e=\{x,y\}$ of particles (one of which may belong to $Y$) is connected by an edge with probability
$p(e)=1-\exp[-\ph(x-y)]$, independently of all other pairs; the particles in $Y$ are connected to each other deterministically. Note that $\varphi$ enters the
construction only via the edge probabilities $p(e)$.
Specifically, the measure $\P$ will be built from the following three constituents.
\begin{itemize}
\item {\sl The distribution of particle positions}. This is given by the (single--type particle) Gibbs distribution $P_\LY^{zq}$ in $\L$ with boundary condition $Y$, interaction $\psi$, and activity $zq$, i.e.,
\be{P}
P_\LY^{zq}(dX)=Z_\LY^{-1} \exp\Big[-\sum_{\{x,y\}\in E_{X\cup Y}\setminus E_Y}\psi(x-y)\Big]\;\pi_\L^{zq} (dX)\;,
\ee
$X\in\XX_\L$. (Note that this measure coincides with $Q_{\L|\vec Y}$ when $q=1$.) Henceforth we shall view $P_\LY^{zq}$ as a probability measure on $\XX$ which is supported on the set $\XX_\LY = \{X\in\XX: X\setminus\L=Y\}$.
\item {\sl The type picking mechanism}. For a fixed set of positions $X\in\XX_\LY$, we let $\lambda_{X,\L}$ denote the distribution of the random vector
\[
\Big(\{x\in X: \tau_x=s\}\Big)_{1\le s\le q} \in \XXq\;,
\]
where $(\tau_x)_{x\in X\cap\L}$ are independent and uniformly distributed on
$\{1,\ldots,q\}$, whereas $\tau_x=1$ for $x\in X\setminus\L=Y$.
\item {\sl The edge drawing mechanism}. For given $X\in\XX_\LY$, let $\mu_{X,\L}$ denote the distribution of the random edge configuration
$\{e\in E_{X}:\eta_e=1\}\in\EE$, where $(\eta_e)_{e\in E_{X}}$ are independent $\{0,1\}$--valued random variables with
\[
\mbox{Prob}(\eta_e=1)= p_\L(e):= \left\{ \begin{array}{cl}
1- e^{-\ph(x-y)} &\mbox{when }e=\{x,y\}\in E_{\R^d}\setminus E_{\L^c}\\
1&\mbox{when } e\in E_{\L^c} \;. \end{array}\right.
\]
The measure $\mu_{X,\L}$ is a point process on $E_{\R^d}$ (which may be viewed as a $p_\L$--thinning of the complete edge set $E_{X}$).
\end{itemize}
The desired probability measure $\P$ on $\XXq\times\EE$ is now defined by
\[
\P = \P_\LY^{zq} := \int P_\LY^{zq}(dX)\; \lambda_{X,\L} \otimes \mu_{X,\L} \;.
\]
This definition makes sense because the mappings $X\to\lambda_{X,\L}$ and $X\to\mu_{X,\L}$ are probability kernels, as we will show in Lemma 2.4 below.
$\P$ is supported on the set of all $(\X,E) \in \XXq\times \EE$ which are such that $\X\in\XXq_{\L|\vec Y}$ and $E\subseteq E_{X}$ with $X=\sp\X$.
We now consider the event $\Omega\subset\XXq\times\EE$ that the particle types are constant on each connected component of the graph $(\sp\X,E\cap E_{\mbox{\scriptsize sp}\X})$. Equivalently, $\Omega$ can be described as the event that no two vertices
with different types have an edge in common.
Formally, we can write
\[
\Omega = \Big\{(\X,E)\in\XXq\times\EE :
\sum_{\{x,y\}\in E}\, \sum_{s=1}^q 1_{X_s}(x)\: (1-1_{X_s}(y)) = 0
\Big\}\;.
\]
Since the functions $(\X,x) \to 1_{X_s}(x)$ are measurable (see 5.1.9 in \cite{KMM}) and $(\X,E)\to \sum_{e\in E}g(\X,e)$ is measurable for any measurable $g$ on $\XXq\times\EE$, this way of writing shows that $\Omega$ is measurable. We also have
\[
\P(\Omega) \ge P_\LY^{zq}(\{Y\}) = e^{-zq|\L|}/Z_\LY > 0\;.
\]
Thus we can introduce the conditional measure
\[
\P_\Omega =I\!\!P(\:\cdot\:|\Omega)\;.
\]
$\P_\Omega$ is the random--cluster representation measure with which the rest of this section is concerned.
Our first result is
that if we disregard the edges, and look only at the particle positions and
their types, then we obtain the continuum Potts model. We write
$\mbox{pr}$ for the projection from
$\XXq\times\EE$ onto $\XXq$.
\medskip\noindent
{\bf Proposition 2.1} \quad $\P_\Omega\circ \mbox{pr}^{-1} = Q_{\L|\vec Y}$.
\medskip\noindent
{\sl Proof.} \quad For $\X\in\XXq_{\L|\vec Y}$, let $\Omega_\X$ be the $\X$--section of $\Omega$. It follows straight from the definitions that, for $X=\sp\X$,
\bea
\mu_{X,\L}(\Omega_\X)
&=&\prod_{e\in E_{X}: e \not\subseteq X_s\,\mbox{\scriptsize for any }s}
\Big(1-p_\L(e)\Big)\\[1ex]
&=& \exp\left[-H_{\vec Y}^\ph(\X)\right]\; .
\eea
It is also easy to see that $\int \pi_\L^{zq}(dX)\,\lambda_{X,\L}$ is precisely the $q$--fold product measure $(\pi_\L^z)^q$ on $\XX^q_\L$ (which is in fact supported on $\XXq_\L$). Hence, for any bounded measurable function $f$ on $\XXq$ we can write
\bea
\int f\circ\mbox{pr}\:d\P_\Omega &=& c_1\int_\Omega f\circ\mbox{pr}\:d\P\\
& = & c_1 \int P_\LY^{zq}(dX)\int \lambda_{X,\L}(d\X)\:f(\X) \:\mu_{X,\L}(\Omega_\X)\\
&=& c_2 \int (\pi_\L)^q(d\X)\: f(\X\cup\vec Y)\: \exp
\left[-H^\psi_\LY(X)-H^\ph_{\L|\vec Y}(\X)\right]\\
&=&c_3 \int f\:dQ_{\L|\vec Y}\; ,
\eea
where $c_1,\,c_2,\,c_3$ are suitable constants. Since both $\P_\Omega$ and
$Q_{\L|\vec Y}$ are probability measures, we have
$c_3 =1$ and the proposition follows.
$\Box$
\medskip
Proposition 2.1 shows in particular that, under $\P_\Omega$, the conditional distribution of $\s(\X)$ under the condition $\sp\X=X$ is nothing other than the Gibbs distribution for a discrete Potts model on the finite set $X$ with constant boundary condition 1 on $Y=X\setminus\L$. It is therefore no surprise that the conditional distribution of $E$ under $\sp\X=X$ corresponds to the classical Fortuin--Kasteleyn random--cluster representation of the Potts model, as we will show next.
Consider the mapping $\sp:(\X,E)\to(\sp\X,E)$ from $\XXq\times\EE$ onto
$\XX\times\EE$. For each $(X,E)\in\XXq\times\EE$ with $E\subseteq E_X$ we let $K(X,E)$ denote the
number of connected components in the graph $(X,E)$
. We will show in Lemma 2.4 below that $K(\cdot,\cdot)$ is measurable. We define the {\sl continuum random--cluster distribution} $C_\LY$ on $\XXq\times\EE$ by
\be{C}
C_\LY(dX,dE) = Z_\LY^{-1}\;q^{K(X,E)}\; P^z_\LY(dX)\:\mu_{X,\L}(dE)\;,
\ee
where $P^z_\LY$ is as in (\ref{P}) with activity $z$ rather than $zq$, and
$Z_\LY$ is again a normalizing constant (different from the one in (\ref{P})).
Recall that, for any $X\in\XX_\LY$, the edges drawn by $\mu_{X,\L}$ include all edges in $E_Y$. This means that $C_\LY$ is defined by means of the so--called
{\sl wired boundary condition}.
Note further that the definition (\ref{C}) makes sense also for non--integer $q>0$.
\medskip\noindent
{\bf Proposition 2.2} \quad $\P_\Omega\circ\sp^{-1} = C_\LY\;$.
\medskip\noindent
{\sl Proof.} \quad For $(X,E)\in \XX\times\EE$ with $X\in \XX_\LY$ and $E_Y\subseteq E\subseteq E_X$ let
\[
\Omega_{(X,E)} = \{\X\in\XXq: \sp\X=X, (\X,E)\in\Omega\}\;.
\]
The definition of $\lambda_{X,\L}$ then implies that
\[
\lambda_{X,\L}(\Omega_{(X,E)}) = q^{K_\L(X,E)}\Big/ q^{\#(X\cap\L)}\;,
\]
where $K_\L(X,E)$ is the number of connected components of $(X,E)$
that are completely contained in $\L$. Indeed, under $\lambda_{X,\L}$ all $q^{\# (X\cap\L)}$ possible values of $\s(\X)$ have equal probability. Getting an element of $\Omega_{(X,E)}$ requires consensus of $\s(\X)$ on each connected component of $(X,E)$, and $\s(\X)$ is equal to 1 on the unique component
containing $Y$; this means that each of the $K_\L(X,E)$
connected components of $(X,E)$ not containing $Y$ has a choice
between $q$ different values.
Proceeding as in the proof of Proposition 2.1, we obtain for any measurable function $f$ on $\XX_\L\times\EE_\LY$
\bea
\int f\circ\sp\;d\P_\Omega &=& c_1 \int P_\LY^{zq}(dX) \int \mu_{X,\L}(dE)\:
f(X,E)\: \lambda_{X,\L}(\Omega_{(X,E)})\\
&=& c_2 \int P_\LY^z(dX)\int \mu_{X,\L}(dE)\: f(X,E)\: q^{K_\L(X,E)}
\eea
with suitable constants $c_1,\, c_2>0$. Here we used the fact that $P_\LY^{zq}$ is absolutely continuous relative to $P^z_\LY$ with density proportional to
$X\to q^{\# (X\cap\L)}$. Comparing this with the definition (\ref{C}) of $C_\LY$
we obtain the desired result
because $K(X,E)-K_\L(X,E)$ is constant and equal to
either 0 or 1 according to whether $Y=\emptyset$ or not.
$\Box$
\medskip
Our third result in this section
relates the influence of the
boundary condition on the type of a single particle to connectivity
probabilities in the random-cluster representation.
For measurable $\Delta \subseteq\L,\: 1\le s\le q,\: \X\in\XXq$ and $(X,E)
\in \XX\times\EE$ with $E\subseteq E_X$ we define
\[
N_{\Delta,s}(\X) = \#(X_s\cap\Delta)\;, \quad N_\Delta(\X) = \sum_{s=1}^q N_{\Delta,s}(\X) = \#(\sp\X\cap\Delta)
\]
and
\[
N_{\Delta\leftrightarrow \L^c}(X,E) = \#\Big\{x\in X\cap\Delta: \mbox{$\exists$ a path in $(X,E)$ from $x$ to some } y\in X\setminus\L \Big\}\;.
\]
The functions $N_{\Delta,s}$ and $N_\Delta$ on $\XXq$ are measurable by definition, and the measurability of $N_{\Delta\leftrightarrow \L^c}$ will be shown in Lemma 2.4 below.
\medskip\noindent
{\bf Proposition 2.3}
{\sl For any measurable $\Delta\subseteq\L$,}
\be{N}
\int (q\:N_{\Delta,1} - N_\Delta)\,dQ_{\L|\vec Y} = (q-1) \int N_{\Delta\leftrightarrow \L^c}\; dC_\LY\;.
\ee
\medskip\noindent
{\sl Proof.} By Proposition 2.1, the integral on the left--hand side is equal to
\[
\P(\Omega)^{-1} \int P^{zq}_\LY(dX) \int \mu_{X,\L}(dE) \sum_{x\in X\cap\Delta}
\int_{\Omega_{(X,E)}}\lambda_{X,\L}(d\X) \Big(q\: 1_{\{\s(\X,x)=1\}}-1\Big)\;,
\]
where $\Omega_{(X,E)}$ is as in the proof of Proposition 2.2. If $x$ is connected to $\L^c$ in $(X,E)$,
the innermost integral is equal to $(q-1) \,\lambda_{X,\L}(\Omega_{(X,E)})$ because in this case the particle type $\s(x)=1$ on $\Omega_{(X,E)}$. Otherwise, $\s(x)$ is independent of $\Omega_{(X,E)}$ under $\lambda_{X,\L}$ and takes all values in $\{1,\ldots,q\}$ with equal probability, which implies that the innermost integral vanishes. The result now follows as in the proof of Proposition 2.2. $\Box$
\medskip
We finally need to settle the measurability questions.
\medskip\noindent
{\bf Lemma 2.4}
\begin{itemize}
\item[(a)] {\sl The mappings $X\to\lambda_{X,\L}$ and $X\to\mu_{X,\L}$ are probability kernels from $\XX$ to $\XXq$ resp.\ $\EE$.}
\item[(b)] {\sl For any measurable $\Delta \subseteq\L$, the functions
$N_{\Delta\leftrightarrow \L^c}$ and
and $K(\cdot,\cdot)$ on $\XX\times\EE$ are measurable.}
\end{itemize}
\medskip\noindent
{\sl Proof.} (a) The fact that $\lambda_{X,\L}$ depends measurably on $X$ is well-known in connection with the randomization of a point process; see \cite{Kall},
p.\ 17 and Lemma 1.7. As for $\mu_{X,\L}$, we consider its Laplace transform
$L_{X,\L}$. For any measurable function $f:E_{\R^d}\to\;[0,\infty[$ we have
\bea
L_{X,\L}(f)&:=& \int \exp[-\sum_{e\in E}f(e)]\; \mu_{X,\L}(dE)\\
&=& \prod_{e\in E_{X}} [p_\L(e)\, e^{-f(e)}+1-p_\L(e)]\\
&=&\exp\Big[-\sum_{\{x,y\}\subseteq X} \tilde f(x,y)\Big]\;,
\eea
where
\[
\tilde f(x,y) = - \log\Big [ e^{-f(\{x,y\})} +1_{\big\{ \{x,y\}\cap\L\ne\emptyset\big\}}e^{-\ph(x-y)}(1- e^{-f(\{x,y\})})\Big]\;.
\]
Since $\tilde f$ is measurable, the function $X\to L_{X,\L}(f)$ is measurable. Lemma 1.7 of \cite{Kall} thus again implies that $\mu_{X,\L}$ depends measurably on $X$.
(b) Consider the set $B$ of all quadrupels $(x,y,X,E)\in \R^d\times\R^d \times \XX\times\EE$ which are such that $x,y\in X,\;x\ne y$, and $x$ is connected to $y$ in the graph $(X, E\cap E_X)$. We may write $B= \bigcup_{n\ge1}B_n$, where
\[
B_1 = \Big\{(x,y,X,E): x, y \in X,\;\{x,y\}\in E\Big\}
\]
and, for $n\ge 1$,
\[
B_{n+1} = \Big\{(x,y,X,E): \sum_{v\in X} 1_{B_1}(x,v,X,E)\: 1_{B_n}(v,y,X,E)
>0\Big\}\;.
\]
Since the functions $(x,X)\to 1_X(x)$, $(x,y,E)\to 1_E(\{x,y\})$ and $ X \to \sum_{v\in X} g(v,X)$ [for measurable $g$] are measurable (cf. 5.1.2 and 5.1.9 of \cite{KMM}), it follows by induction that $B_n$ is measurable for any $n$, whence $B$ is also measurable.
Since
\[
N_{\Delta\leftrightarrow \L^c}(X,E) = \sum_{x\in X\cap\Delta} g(x,X,E)
\]
with $g(x,X,E) = 1$ when $\sum_{y\in X\setminus\L} 1_B(x,y,X,E)>0$ and $g(x,X,E)=0$ otherwise, we now see that $N_{\Delta\leftrightarrow \L^c}$
is measurable on $\XX\times\EE$. Also, for any $k\ge1$ we have $K(X,E) \ge k$ if and only if
\[
\sum_{x_1,\ldots,x_{k}\in X} \prod_{1\le i 0\;.
\]
In view of the above, the last expression depends measurably on $(X,E)$, and this yields the measurability of $K$. $\Box$
\section{Percolation in the random--cluster model}
In this section we will establish the existence of percolation for continuum random cluster distributions $C_\LY$ when $q\ge1$ is an arbitrary real number, $z$ is sufficiently large and the boundary condition $Y$ contains sufficiently many particles.
For any box $\Delta\subset\R^d$ we set
\[
N_{\Delta\leftrightarrow\infty}(X,E) = \#\Big\{x\in X\cap\Delta: x \mbox{ belongs to an infinite cluster of } (X,E\cap E_X)\Big\}\;.
\]
It follows from Lemma 2.4 that $N_{\Delta\leftrightarrow\infty}$ is measurable.
We consider again the cells $\Delta(i)$ defined by (\ref{cell}) for the side length $\delta=r_3/\sqrt{d+3}$.
The main result of this section is the following.
\medskip\noindent
{\bf Proposition 3.1}\quad {\sl Suppose assumptions (A1) to (A5) hold and $z$ is sufficiently large (depending on the parameters of our model). Then there exists an $\varepsilon>0$ and an integer $n_*\ge1$ such that
\[
\int N_{\Delta\leftrightarrow\infty} dC_\LY \ge \varepsilon
\]
for any cell $\Delta=\Delta(i)$, any finite union $\L$ of cells, and any boundary condition $Y\in\XX_{\L^c}$ with at least $n_*$ particles in each cell.}
\medskip
The proof of this proposition consists of two steps. First we will find a
measure $\tilde C_\LY$ which is stochastically smaller than $C_\LY$ and such
that, conditionally on the particle configuration, the edges are drawn independently of each other. It is therefore sufficient to establish percolation for $\tilde C_\LY$. This will be done by discretization and comparison with a mixed site--bond percolation problem on the lattice $\Z^d$.
The definition of $\tilde C_\LY$ is based on the following two ingredients.
\begin{itemize}
\item The distribution of particle positions is given by the marginal distribution
\[
M_\LY = C_\LY(\cdot\:\times\EE)
\]
of $C_\LY$ on $\XX$.
\item For given $X\in\XX$, the distribution $\tilde\mu_X$ of edges is defined as the distribution of the random set $\{e\in E_X:\tilde\eta_e=1\}$, where
$(\tilde\eta_e)_{e\in E_X}$ are independent $\{0,1\}$--valued random variables with
\[
\mbox{Prob}(\tilde\eta_e=1)=\tilde p := \frac{1-e^{-u}}
{qe^{-u}+1-e^{-u}} \mbox{ when } e=\{x,y\} \mbox{ with } |x-y|\le r_3
\]
and $\mbox{Prob}(\tilde\eta_e=1) =0$ otherwise; the constants $r_3$ and $u$ are as in assumption (A1). It follows as in Lemma 2.4 that $X\to\tilde\mu_X$ is a probability kernel from $\XX$ to $\EE$.
\end{itemize}
We now define $\tilde C_\LY$ on $\XX\times\EE$ by
\be{CMT}
\tilde C_\LY(dX,dE) = M_\LY(dX)\,\tilde\mu_X(dE)\;.
\ee
For the sake of comparison we note from (\ref{C}) that
\be{CM}
C_\LY(dX,dE) = M_\LY(dX)\,\mu^q_{X,\L}(dE)\;,
\ee
where
\[
\mu^q_{X,\L}(dE) = q^{K(X,E)} \mu_{X,\L}(dE)\Big/\int q^{K(X,\cdot)}d\mu_{X,\L}
\]
is the discrete random--cluster distribution on $X\cap\L$ with wired boundary condition on the set $X\setminus\L$.
To state the relationship between $C_\LY$ and $\tilde C_\LY$ we recall that a probability measure $\nu_1$ on a partially ordered measurable space is said to
{\sl dominate} a second probability measure $\nu_2$
if $\int f\,d\nu_1\ge\int f\,d\nu_2$ for any increasing bounded measurable function $f$. In this case we write $\nu_1\succeq\nu_2$.
\medskip\noindent
{\bf Lemma 3.2} {\sl For all $q\ge1$ and $X\in\XX$, we have $\mu^q_{X,\L}\succeq \tilde\mu_X$ and therefore $C_\LY\succeq\tilde C_\LY$.}
\medskip\noindent
{\sl Proof.} \quad By assumption (A1),
\[
p_\L(e)/q(1-p_\L(e)) \ge \tilde p/(1-\tilde p)
\]
for all $e\in E$. The relation $\mu^q_{X,\L}\succeq\tilde\mu_X$ thus follows from a well--known comparison inequality for discrete random--cluster
distributions; see e.g.\ inequality (7.11) of \cite{Gr}
(this reference deals only with the case of equal edge probabilities, but the proof works without change in the general case). The second assertion is immediate from (\ref{CMT}) and (\ref{CM}).
$\Box$
\medskip
Since $N_{\Delta\leftrightarrow\infty}$ is increasing, Lemma 3.2 shows that in order to prove Proposition 3.1 it is sufficient to prove the analogous result for $\tilde C_\LY$. In the case $\psi\equiv0$, this can be achieved by a further
stochastic comparison. Namely, the point--process
analog of Holley's inequality \cite{P,GK} together with Lemma 3.5 below then
imply that $M_\LY\succeq \pi_\L^{\alpha z}$, so that Proposition 3.1 follows from the fact that the so--called random connection model of Poisson particles
with random edges exhibits percolation when the particle density is large enough \cite{Pe,MR}. In the general case such a stochastic comparison does not work. Nevertheless, it is possible to extend the proof of
percolation in the random connection model to our present setting. Specifically, we shall use a discretization to compare the percolation problem for $\tilde C_\LY$ with a site--bond percolation problem on the lattice $\Z^d$.
The basic idea is as follows. We consider the division of $\R^d$ into the cells $\Delta(i)$ defined by (\ref{cell}). The side length $\delta$ is chosen so small that any two points in adjacent cells have distance less than $r_3$, which means that $\delta=r_3/\sqrt{d+3}$. For a suitably chosen number $n_*$, we will call a cell {\sl good} relative to a given particle--edge configuration if there exist at least $n_*$ particles in the cell which, together with the associated edges, form a connected graph. Two neigboring cells will be said to be {\sl linked} if there exists an edge joining a particle in one cell to a particle in the other cell. An infinite cluster of particles joined by edges then certainly exists whenever there exists an infinite cluster of good cells joined by links. This is a correlated mixed site--bond percolation problem on $\Z^d$. For suitably chosen parameters, percolation in this lattice model can be established by means of a comparison with the Bernoulli site--bond percolation problem to
which we turn now.
Consider the integer lattice $\Z^d$, and let $0\le p\le1$. Declare a site $i\in\Z^d$, or a bond $\{i,j\}$ between two sites of distance 1, to be open with probability $p$, and closed otherwise. These designations are done independently for all sites and bonds. Let $\theta(p)$ denote the probability that the origin is open and belongs to an infinite component of the open graph consisting of open sites and open bonds. We shall use the following fact.
\medskip\noindent
{\bf Lemma 3.3} {\sl For $d\ge2$, there exists a threshold value $p_c<1$ (depending on $d$) such that $\theta(p)>0$ when $p>p_c$.}
\medskip\noindent
{\sl Proof.} \quad This follows either by an adaptation of the usual Peierls argument, or by an application of an inequality of McDiarmid and Hammersley
(see inequality (10.8) in \cite{Kesten}) which implies that $\theta(p)$ exceeds
the corresponding probability for the Bernoulli pure site percolation problem with parameter $p^2$. $\Box$
\medskip
We shall also take advantage of the following well--known fact on
random graphs.
\medskip\noindent
{\bf Lemma 3.4}\quad{\sl For any positive integer $n$ and $0< p<1$, let ${\cal G}_{n,p}$ be the
random graph on $n$ vertices where each pair of vertices independently
forms an edge with probability $p$. Then, for fixed $p$,
\[
\gamma(n,p)= \mbox{\rm Prob}
\Big({\cal G}_{n,p} \mbox{\rm \hspace{1 mm}is connected}\Big)\to 1 \mbox{ as }n\to\infty\;.
\]
In particular, $\gamma(p)=\inf_{n\ge1}\gamma(n,p)>0$.}
\medskip\noindent
{\sl Proof.}\quad The first statement follows from the simple observation that ${\cal G}_{n,p}$ is certainly connected if the first vertex is connected to any other vertex by a path of length 2. A straightforward estimate shows that the latter event has probability at least $ 1-(n-1)(1-p^2)^{n-2}$. (An asymptotic expansion for $\gamma(n,p)$ has been derived by Stepanov \cite{Step}, see also the references there.) This also gives the second assertion
because each $\gamma(n,p)$ is positive. $\Box$
\medskip
We finally need a third preparatory lemma which will allow us to control the measure $M_\LY$. For $X\in\XX_\LY$ let
\[
h_\L(X) = Z_{\LY}^{-1}\int q^{K(X,\cdot)}\, d\mu_{X,\L}\;,
\]
where $Z_\LY$ is as in (\ref{C}). Note that $h_\L$ is the
Radon--Nikodym derivative of $M_\LY$ with respect
to $P^z_\LY$.
\medskip\noindent
{\bf Lemma 3.5}
{\sl For $q\ge1$, there exists a constant $\alpha>0$ (depending on $q, r_4/r_3$ and $\tilde p$) such that for any box $\L\subset\R^d$, $X\in\XX_\LY$ and $x\in\L\setminus X$
\[
h_\L(X\cup\{x\})\ge \alpha\: h_\L(X)\;.
\]}
\noindent
{\sl Proof.}\quad Since $x\in\L$, we can ignore the normalizing factor $ Z_{\LY}$. Let $\mu_{x|X,\L}$ denote the distribution of the random
edge set $\{e\in E_{X\cup x}\setminus E_X:\eta_e=1\}\in\EE$, where
$(\eta_e)$ are independent $\{0,1\}$--valued with $\mbox{Prob}(\eta_e=1) = p_\L(e)$. (We omit the braces around $x$ for brevity.) Then we can write
\[
h_\L(X\cup x)\big/ h_\L(X) = \int \mu^q_{X,\L}(dE) \int \mu_{x|X,\L}(dE')\:
q^{K(X\cup x,E\cup E')-K(X,E)}
\]
with $\mu^q_{X,\L}$ as in (\ref{CM}). To estimate the last expression we note that, due to the finite range assumption (A2), all particles of $X$ joined to $x$ by $\mu_{x|X,\L}$ belong to the ball $B=\{y\in\R^d:|x-y|\le r_4\}$.
We split $B$ into a minimal number $k$ of disjoint Borel sets $B_1,\ldots,B_k$ of diameter at most $r_3$; $k$ obviously depends only on the ratio $r_4/r_3$.
Writing $X^j=X\cap B_j$ for the configuration in $B_j$, we let $A_j$ denote
the set of all $E\in\EE$ for which the graph $(X^j,E\cap E_{X^j})$ is
connected, $1\le j\le k$. For $E\in A= \bigcap_{j=1}^k A_j$, there exist
at most $k$ different clusters of $(X,E)$ which can be joined together by
the edges in $E'$ drawn by $\mu_{x|X,\L}$. Hence $K(X\cup x,E\cup E')\ge K(X,E)-k$ when $E\in A$.
This gives us the first inequality in
\bea
h_\L(X\cup x)\big/ h_\L(X) &\ge& q^{-k}\mu^q_{X,\L}(A)\\
&\ge& q^{-k} \tilde\mu_X(A)\\
&\ge& q^{-k} \gamma(\tilde p)^k =: \alpha .
\eea
The second inequality above comes from Lemma 3.2 because $A$ is increasing. The last estimate follows from the fact that $A_1,\ldots,A_k$ are independent under $\tilde\mu_X$, each $A_j$ having
probability at least $\gamma(\tilde p)$ by Lemma 3.4. $\Box$
\medskip
We now turn to the
comparison of the percolation problem for $\tilde C_\LY$ with the Bernoulli site--bond percolation problem on $\Z^d$. First of all, we introduce the number $n_*$ on which the definition of good cells is based. To this end we fix any $p_*$ with $p_c0
\]
by assumption (\ref{r1}). Here $|\cdot|$ stands for the volume of a set in $\R^d$. On the other hand, Markov's inequality yields for any
$c>0$
\[
\Big|\Big\{x\in\D_X:\sum_{y\in X}\psi(x-y)\ge c\Big\}\Big|\le (n_*-1)\, b(r_1)/c \;,
\]
where $b(r_1)$ is the integral in assumption (A4). This shows that
\[
g_{i|Y'}(X)\ge \alpha e^{-c}[v_*-(n_*-1)b(r_1)/c]
\]
when $N_i(X)0$ such that $g_{i|Y'}(X)\ge b\, n_*$ whenever $N_i(X)1$. The lemma thus follows when $z$ is so large that $1-(bz-1)^{-1}\ge\sqrt{p_*}$. $\Box$
\medskip
We now return to the picture of good cells and links between good cells.
Suppose $z$ is as large as required by Lemma 3.6, and $\L$ is a finite union of cells. By (\ref{n*}) and the choice of $z$, the conditional probability under $\tilde C_\LY$ of the event that a cell $\D(i)$ is good given the particle configuration $Y'$ outside of $\D(i)$ and all edges which are not contained in
$\D(i)$ is at least
\[
\int_{\{N_i\ge n_*\}}M_{\L,i|Y'}(dX)\: \gamma(N_i(X),\tilde p) \ge p_*\; .
\]
Condition (\ref{n*}) also guarantees that the conditional probability for two neighboring cells to be linked when they are good is not less than $\lambda(n_*,\tilde p)\ge p_*$. By a standard comparison argument, we can conclude that
\[
\int N_{\D(i)\leftrightarrow\infty} d\tilde{C}_\LY \geq n_*\, \theta(p_*)=:\varepsilon>0
\]
for such $z$ and $\L$ and all $i$. Together with Lemma 3.2, this observation completes the proof of Proposition 3.1.
\medskip
We now specialize to the case $q=1$. Let $P$ be a tempered Gibbs measure on $\XX$ with pair potential $\psi$ and activity $z$. In complete analogy to Section 1, this means that $P$ is supported on the tempered configurations in $\XX$ and, for any finite box $\L$ in $\R^d$, admits $P^{z}_\LY$ as conditional distribution under the condition
$X\setminus\L=Y$. Assumptions (A3) and (A4) imply that such a $P$ exists; see Remark 4.2. By the case $q=1$ of Lemma 3.6, the conditional probabilities of $P$ satisfy $P^z_{\D(i)|Y'}(N_i\ge n_*)\ge\sqrt{p_*}$ for all $i\in\Z^d$ and all
tempered $Y'\in\XX_{\D(i)^c}$ when $z$ is large enough. The argument above thus
gives the following corollary of independent interest.
\medskip\noindent
{\bf Corollary 3.7} {\sl Suppose $X$ is a configuration of particles chosen at random according to a tempered Gibbs measure $P$ with pair interaction $\psi$ and activity $z$. Suppose further that any two distinct particles $x,y\in X$ with $|x-y|\le r_3$ are connected by an edge with probability $\tilde p>0$, independently for all such pairs of particles. If (A3) to (A5) hold and $z$ is sufficiently large then, with probability 1, there exists an infinite cluster of particles connected by edges. Explicitly, the measure $\tilde C_P(dX,dY) =P(dX)\tilde\mu_X(dE)$ satisfies}
\[
\tilde C_P\Big(\sup_{i\in\Z^d} N_{\D(i)\leftrightarrow\infty}\ge1 \Big)=1\;.
\]
\medskip
In the case of no interaction ($\psi\equiv 0$) when $P$ is a Poisson point process, this result was obtained first by M.~Penrose \cite{Pe}. (It is then not necessary to assume that the edge probabilities are uniformly positive for short distances.)
\section{The infinite--volume limit}
In this section we will complete the proof of Theorem 1.1 by investigating the Gibbs distributions $Q_{\L|\vec Y}$ in the infinite volume limit $\L \uparrow\R^d$. Many of our arguments are standard, but we have to be careful
with various details.
We assume again that $q$ is an integer $\ge2$. As before, we consider the cells $\D(i)$ defined by (\ref{cell}) with $\delta =r_3/\sqrt{d+3}$. For notational convenience we assume without loss of generality that the units in $\R^d$ are chosen in such a way that $\delta=1$. The limit $\L \uparrow\R^d$ will be taken along the sequence
\[
\L(n)=[-n-1/2,n+1/2[^d
\]
of boxes of volume $v_n=(2n+1)^d$.
Let $z$ and $n_*$ be as large as required by Proposition 3.1, and let $Y\in\XX$ be such that $N_i(Y)=n_*$ for all $i\in\Z^d$. For each $n$, $Y$ defines the boundary condition
\[
\vec Y(n)= (Y\setminus\L(n),\emptyset,\ldots,\emptyset) \in\XXq_{\L(n)^c}
\]
of particles of type 1. We look at the projection
\[
Q_n = Q_{\L(n)|\vec Y(n)}\Big(\X\in\XXq: \X\cap\L(n)\in\cdot \Big)
\]
of $Q_{\L(n)|\vec Y(n)}$ to $\L(n)$; here $\X\cap\L(n)=(X_s\cap\L(n))_{1\le s\le q}\in\XXq_{\L(n)}$ is the restriction of $\X\in\XXq$ to $\L(n)$.
To obtain asymptotic translation invariance we take a spatial average of $Q_n$.
For $x\in\R^d$, the translation $\th_x:\XXq\to\XXq$ is defined by
\[
\th_x\X=\Big(\{y-x:y\in X_s\}\Big)_{1\le s\le q}\;.
\]
It is well--known \cite{KMM} that the mapping $(x,\X)\to \th_x\X$ is measurable.
In view of the cell structure we have imposed on $\R^d$, we will confine ourselves first to lattice translations. So we define
\[
\bar Q_n= v_n\inv\sum_{i\in \Ln} Q_n\circ\th_i\inv
\]
with $\Ln=\L(n)\cap\Z^d$. We will show that the sequence $(\bar Q_n)$ admits an accumulation point in a fairly strong topology which is defined as follows.
Let $\LL$ denote the class of all measurable functions $f:\XXq\to\R$ which are
{\sl local} and {\sl tame}, in that there exists some $\ell\ge1$ such that $f= f(\cdot\cap\L(\ell))$ and $|f|\le b(1+N_{\L(\ell)})$ for some constant $b=b(f)<\infty$. The {\sl topology $\tau_\LL$ of local convergence} on the set of all probability measures $R$ on $\XXq$ with $\int N_{\D} dR<\infty$
for each finite box $\D$ is then defined as the weak$^*$ topology induced by $\LL$, i.e., as the smallest topology for which the mappings $R\to\int f\,dR$ with
$f\in\LL$ are continuous.
The basic result of this section is the following.
\medskip\noindent
{\bf Proposition 4.1} {\sl Suppose $\varphi$ and $\psi$ satisfy (A1) to (A4)
and $z>0$ is arbitrary. Then
there exists a subsequence of $(\bar Q_n)$ which converges in $\tau_\LL$ to a limit $Q$. This $Q$ is a continuum Potts measure for $\varphi,\psi$ and $z$.
It is invariant under the lattice translations $(\th_i)_{i\in\Z^d}$ and of second order in the sense that $\int N_\D^2\:dQ<\infty$ for each finite box $\D\subset\R^d$.}
\medskip
Let us show first how Theorem 1.1 follows from the preceding proposition and the results of the foregoing sections. To achieve invariance under the full translation group $(\th_x)_{x\in\R^d}$ we first pass from $Q$ to the average
\[
Q^{(1)}=\int_{\D(0)} Q\circ\th_x\inv\:dx\;.
\]
$Q^{(1)}$ is a continuum Potts measure of second order. Also, $Q^{(1)}$ is symmetric under exchanges of the types $2,\ldots,q$ because all $Q_n$ exhibit
this symmetry.
Applying Propositions 2.3 and 3.1 we see that, for $\D=\D(0)$,
\bea
\int(qN_{\D,1}-N_\D)d\bar Q_n &\ge& (q-1)v_n\inv\sum_{i\in\Ln} \int N_{\D(i)\leftrightarrow\infty} dC_{\L(n)|Y\setminus\L(n)}\\
&\ge&(q-1)\varepsilon\;,
\eea
and also that the expression on the left--hand side is increasing in $\D$.
Since $N_{\D,1},N_\D\in\LL$, the same inequality holds for $Q$ in place of $\bar Q_n$, and using the monotonicity in $\D$ we obtain that
\[
\int(qN_{\D,1}-N_\D)d Q^{(1)} > 0
\]
for $\D=[-1,1[^d$, and by translation invariance also for $\D=\D(0)$.
Together with the symmetry of $Q^{(1)}$, this means that
\[
\rho_1(Q^{(1)})>\rho_2(Q^{(1)})=\ldots=\rho_q(Q^{(1)})\;,
\]
where $\rho_s(Q^{(1)})=\int N_{\D(0),s}\,d Q^{(1)}$ is the expected density
of particles of type $s$. For $2\le t\le q$, let $Q^{(t)}$ be the $(\th_x)_{x\in\R^d}$--invariant continuum Potts measure which is obtained from
$Q^{(1)}$ by interchanging the roles of the types 1 and $t$. It is then obvious
that the matrix $(\rho_s(Q^{(t)})_{1\le s,t\le q}$ is regular. By the ergodic
decomposition theorem (which can be derived in analogy to the lattice case considered in Chapter 14 of \cite{Geo}), this implies the existence of
$(\th_x)_{x\in\R^d}$--ergodic continuum Potts measures with the same property;
cf. Corollary (14.18) and Comment (16.19) in \cite{Geo}. The proof of Theorem 1.1 is thus complete.
To conclude, let us note that our estimates also imply that $\rho_s(Q^{(s)})/\rho(Q^{(s)})\to1$ as $z\to\infty$ for each type $s$.
\medskip
We now turn to the proof of Proposition 4.1. We might follow the lines of Ruelle
\cite{Rue1} or, under slightly stronger assumptions on $\psi$, of Dobrushin
\cite{D1,D2}; these references, however, work with other topologies than $\tau_\LL$. We prefer to sketch here an alternative ``thermodynamic"
approach in the spirit of large deviation theory. We proceed in four stages.
\medskip
{\sl Step 1. Energy and entropy estimates}. Suppose $\psi$ is superstable
and lower regular. By definition, {\sl superstability} means that there exist constants $a,b>0$ such that, for all $n$ and $X\in\XX_{\L(n)}$,
\be{ss}
H^\psi(X)\ge\sum_{i\in\Ln}\big[ a\,N_i(X)^2-b\, N_i(X)\big]\;.
\ee
{\sl Lower regularity} is defined as the existence of numbers $\Psi_i\ge0$, $i\in\Z^d$, such that $\Psi :=\sum_{i\in\Z^d}\Psi_i<\infty$ and
\be{lr}
\sum_{x\in X\cap\D(i),\, y\in X\cap\D(j)}\psi(x-y)\ge - \Psi_{i-j}N_i(X)N_j(X)
\ee
for all $X\in\XX$ and $i,j\in\Z^d$. For our fixed configuration $Y$, both
conditions together imply that
\be{ab}
H_{\L(n)|\vec Y(n)} \ge \sum_{i\in\Ln} [a\,N_i^2-b_*\,N_i] \ge -c_*\,v_n
\ee
for all $n$, where $b_*= b+\Psi n_*$ and $c_*= b_*^2/4a$.
As is well--known, the relative entropy (or Kullback--Leibler information) of
two probability measures $\mu,\nu$ on the same measurable space is defined by
$I(\mu;\nu)=\int \log f\,d\mu\ge0$ if $\mu$ has Radon--Nikodym density $f$ relative to $\nu$, and $I(\mu;\nu)=\infty$ otherwise. Using (\ref{Q}) we obtain for each $n$, writing $\pi_{n,q}=(\pi_{\L(n)})^q$,
\begin{eqnarray}\label{ee}
v_n\inv I(Q_n;\pi_{n,q}) &=& -v_n\inv\int H_{\L(n)|\vec Y(n)}dQ_n - v_n\inv \log Z_{\L(n)|\vec Y(n)}\nonumber\\
&\le& c_*+zq =: c\;.
\end{eqnarray}
The inequality follows from the energy estimate (\ref{ab}) and the fact that
$Z_{\L(n)|\vec Y(n)}\ge\pi_{\L(n)}(\{\emptyset\})^q$. In the case $\psi\ge0$, the preceding inequality (\ref{ee}) holds trivially with $c=zq$.
\medskip {\sl Step 2. The existence of an accumulation point $Q$}. We first need to introduce the relative entropy density. For any $(\th_i)_{i\in\Z^d}$--invariant probability measure $R$ on $\XXq$, this is defined as
\[
I(R)=\lim_{k\to\infty} v_k\inv I(R_{(k)};\pi_{k,q})\;,
\]
where $R_{(k)}$ is the projection of $R$ onto $\XXq_{\L(k)}$. For the existence of this limit we refer to \cite{Geo}, Theorem 15.12. The important fact we need here is that the sublevel sets $\{I\le c\}$ are sequentially compact in the topology $\tau_\LL$.. This was shown in Proposition 2.6 of \cite{GZ}.
We now consider the following particular sequence $(\tilde Q_n)$ of $(\th_i)_{i\in\Z^d}$--invariant measures. For each $n$, we write $\hat Q_n$ for the probability measure on $\XXq$ relative to which the particle configurations in the disjoint blocks $\L(n)+(2n+1)i$, $i\in\Z^d$, are independent with identical distribution $Q_n$, and we set
\[
\tilde Q_n = v_n\inv \sum_{i\in\Ln}\hat Q_n\circ\th_i\inv\;.
\]
It is not difficult to see that $I(\tilde Q_n) = v_n\inv I(Q_n;\pi_{n,q})$,
cf. the proof of Proposition (16.34) in \cite{Geo}. Together with (\ref{ee})
this shows that the sequence $(\tilde Q_n)$ has a subsequential limit $Q$ in $\tau_\LL$. Clearly, $Q$ inherits the $(\th_i)_{i\in\Z^d}$--invariance of the
$\tilde Q_n$. Now the point is that, by a slight variant of Lemma 5.7 in \cite{GZ}, the sequences $(\tilde Q_n)$ and $(\bar Q_n)$ are asymptotically equivalent in $\tau_\LL$. We thus conclude that a subsequence of $(\bar Q_n)$
converges to $Q$.
\medskip
{\sl Step 3. Bounds on the second moments.} Suppose first that $\psi$ is superstable and lower regular, and let $i\in\Z^d$ and $n\ge0$ be arbitrary.
Using the estimates (\ref{ab}) and (\ref{ee}) together with the nonnegativity of the relative entropy we find
\begin{eqnarray}\label{m}
\int N_i^2\, d\bar Q_n &=&v_n\inv\sum_{j\in\Ln} \int N_{i+j}^2\, dQ_n\nonumber\\
&\le& n_*^2 + v_n\inv \sum_{j\in\Ln}\int N_j^2\, dQ_n\nonumber\\
&\le& n_*^2+ \frac{2}{av_n}\int H_{\L(n)|\vec Y(n)}\, dQ_n +
\int \Big(\frac{2b_*}{a}N_0-N_0^2\Big)d\bar Q_n\nonumber\\
&\le& n_*^2+2zq/a+(b_*/a)^2 =: m\;.
\end{eqnarray}
In the alternative case when $\psi\ge0$, a uniform bound on $\int N_i^2\, d\bar Q_n$ can be obtained from the fact that, in this case, the density of $Q_{\D(i)|\X}$ relative to $(\pi_{\D(i)})^q$ is bounded uniformly in $\X\in\XXq_{\D(i)^c}$. Since $N_i^2$ is the supremum of functions in $\LL$, we conclude in particular that $\int N_i^2 dQ<\infty$. Hence $Q$ is of second order.
\medskip
{\sl Step 4. Verification of the Gibbs property.} To show that $Q$ is a continuum Potts measure we need to check that, for any $k\ge0$ and any local function $f$ on $\XXq$ with $0\le f\le1$,
\[
\int f\,dQ = \int f_{\L(k)}\,dQ\;.
\]
Here we use the abbreviation $f_\L(\X) = \int f\,dQ_{\L|\X}$, $\X\in\XXq_{\L^c}$.
For notational convenience we confine ourselves to the case $k=0$; for general $k$ one can argue similarly. Since $f\in\LL$, we know that $\int f\,dQ = \lim \int f\,\bar Q_{n'}$ along the subsequence $(n')$ with $\bar Q_{n'}\to Q$. The consistency and translational homogeneity of the Gibbs distributions imply that, for each $n$,
\[
\int f\,d\bar Q_{n}=v_n\inv \sum_{i\in\Ln}\int (f\circ\th_i)_{\D(i)}\:dQ_n =\int f_{\D(0)}\:d\bar Q_{n}\;.
\]
Therefore it only remains to show that
\be{lim}
\int f_{\D(0)}\:dQ = \lim f_{\D(0)}\: d\bar Q_{n'}
\ee
along the subsequence $(n')$. This is obvious when $\psi$ has finite range because then $f_{\D(0)}$ is local. In the general case, we approximate $f_{\D(0)}$ by the bounded local functions $f_{\D(0),k}= f_{\D(0)}(\cdot\cap\L(k))$, $k\ge1$. Namely, arguing as in the first lines of the proof of Lemma 7.4 of \cite{G2} we find that
\begin{eqnarray}\label{f}
\Big|f_{\D(0)}(\X)-f_{\D(0),k}(\X)\Big| &\le&
2\sum_{j\in\Z^d\setminus\L(k)} \Psi_j\:N_j(\X)\nonumber\\
&&\times\Big[\int N_0\, dQ_{\D(0)|\X}+\int N_0\, dQ_{\D(0)|\X\cap\L(k)}\Big]
\end{eqnarray}
for sufficiently large $k$. In addition to our assumptions $\varphi\ge0$ and
(\ref{lr}), this estimate requires only the {\sl upper regularity} of
$\varphi+\psi$, which means that
\be{ur}
\sum_{x\in X\cap\D(i),\, y\in X\cap\D(j)}(\varphi+\psi)(x-y)\le \Psi_{i-j}N_i(X)N_j(X)
\ee
for all $X\in\XX$ whenever $|i-j|$ exceeds some constant. This property follows
trivially from (A2) and (A4). If $\psi$ is superstable,
a variant of Lemma 7.3 in \cite{G2} shows that there exists a constant $C<\infty$ such that
\[
\int N_0\, dQ_{\D(0)|\X} \le C+ D\sum_{i\ne0}\Psi_i\: N_i(\X)
\]
with $D=2/a$. If $\psi\ge0$, we can even set $D=0$.
Inserting this into (\ref{f}) and using the Cauchy--Schwarz inequality
and (\ref{m}) we arrive at the estimate
\[
\int |f_{\D(0)}-f_{\D(0),k}|\,d\bar Q_n \le 4\big[Cm^{1/2}+D\Psi m\big]
\sum_{j\in\Z^d\setminus\L(k)} \Psi_j
\; =:\varepsilon(k)\;.
\]
The same estimate holds with $Q$ in place of $\bar Q_n$.
Since $\varepsilon(k)\to0$ as $k\to\infty$ and $f_{\D(0),k}\in\LL$
for all $k$, assertion (\ref{lim}) follows immediately. The proof of Proposition 4.1 is therefore complete. $\Box$
\medskip
We conclude with two remarks.
\medskip\noindent
{\bf Remark 4.2}\quad Proposition 4.1 evidently implies the existence of continuum Potts measures for arbitrary activities $z>0$. Of course, it is also true in the case $q=1$ and then gives us the existence of tempered Gibbs measures for pair interactions $\psi$ satisfying (A3) and the upper regularity (\ref{ur}) (with
$\varphi\equiv0$), as established first by Dobrushin \cite{D2} and Ruelle \cite{Rue1}. Compared with \cite{Rue1}, the preceding argument is less elementary but also much less involved, whereas the methods of \cite{D2} need
a non--integrable singularity of $\psi$ at the origin. Note, however, that these references provide stronger information on the moments of the limiting Gibbs measures.
\medskip\noindent
{\bf Remark 4.3}\quad Suppose each particle type $s$ is equipped with its own
activity $z_s>0$, so that the parameter $z>0$ is replaced by the vector
$\vec z=(z_1,\ldots,z_q)$. Let $Z_{\L|\vec Y}(\vec z)$ be the associated partition function, cf. (\ref{Q}). Using the techniques of \cite{G1}, one can show that the {\sl pressure}
\[
p(\vec z) = \lim_{n\to\infty}v_n\inv\log Z_{\L(n)|\vec Y(n)}(\vec z)
\]
(exists and) is equal to
\[
-\inf_R\Big[ U(R)+I_1(R)- \sum_{s=1}^q\rho_s(R)\log z_s\Big]\;.
\]
In the above, the infimum extends over all probability measures $R$ on $\XXq$,
\[
U(R)= \lim_{n\to\infty} v_n\inv\int H_{\L(n)|\vec Y(n)}\:dR_{(n)}
\]
is the energy density, and $I_1(R)$ is the relative entropy density of $R$ with the Poisson point processes of intensity 1 as reference measures. If $z$ is so large as required before and $\vec z=(z,\ldots,z)$, one can further deduce from
(\ref{ee}) with the methods of \cite{G1} that the infimum above is attained by
$\bar Q$ and thus also by $Q_1,\ldots,Q_q$. (In fact, if $\psi$ has a non--integrable singularity at the origin, an analog
of the variational principle in \cite{G2} shows that the infimum is attained precisely for all $(\th_x)_{x\in\R^d}$--invariant continuum Potts measures.) Since $\rho_s(Q_s)>\rho_s(Q_t)$ for
$1\le s\ne t\le q$, this means that the convex function $(\log z_1,\ldots,\log z_q) \to p(\vec z)$ admits $q$ tangent hyperplanes at $(\log z,\ldots,\log z)$
with distinct slopes in each direction. Consequently, the pressure $p(\vec z)$
is not differentiable with respect to $z_s$ at $(z,\ldots,z)$ for each $1\le s\le q$, provided $z$ is large enough. By the usual Legendre--Fenchel duality
formalism, this non-differentiability corresponds to a discontinuity of the pressure as a function of the particle densities $\rho_1,\ldots,\rho_q$.
\small
\begin{thebibliography}{19}
%\bibitem{ACCN} Aizenman, M., Chayes, J.T., Chayes, L. and Newman, C.M. (1988)
%Discontinuity of the magnetization in one-dimensional $1/|x-y|^2$ Ising
%and Potts models, {\sl J. Statist. Phys.} {\bf 50}, 1--40.
\bibitem{CCK} Chayes, J.T., Chayes, L. and Kotecky, R. (1995)
The analysis of the Widom-Rowlinson model by stochastic geometric methods,
{\sl Commun. Math. Phys.} {\bf 172}, 551--569.
\bibitem{D1} Dobrushin, R.L. (1970) Prescribing a system of random variables by conditional distributions,
{\sl Th. Probab. Appl.} {\bf 15}, 458--486.
\bibitem{D2} Dobrushin, R.L. (1970) Gibbsian random fields for particles without hard core, {\sl Theor. Math. Phys.} {\bf4}, 705--719.
\bibitem{DM} Dobrushin, R.L. and Minlos, R.A. (1967) Existence and continuity
of pressure in classical statistical physics, {\sl Th. Probab. Appl.} {\bf 12},
535--559.
\bibitem{ES} Edwards, R.G. and Sokal, A.D. (1988) Generalization of the
Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo
algorithm, {\sl Phys. Rev.} D {\bf 38}, 2009--2012.
\bibitem{FK} Fortuin, C.M. and Kasteleyn, P.W. (1972)
On the random-cluster model. I. Introduction and relation to other models,
{\sl Physica} {\bf 57}, 536--564.
\bibitem{Geo} Georgii, H.-O. (1988) {\sl Gibbs Measures and Phase
Transitions}, de Gruyter, New York.
\bibitem{G1} Georgii, H.-O. (1994) Large deviations and the equivalence of
ensembles for Gibbsian particle systems with superstable interaction. {\sl Probab.\ Th.\ Rel.\ Fields} {\bf99}, 171--195.
\bibitem{G2} Georgii, H.-O. (1995) The equivalence of ensembles for classical systems of particles. {\sl J.\ Statist.\ Phys.} {\bf80}, 1341--1378
\bibitem{GK} Georgii H.-O. and K\"uneth, T. (1995) Stochastic comparison of
point random fields, preprint.
\bibitem{GZ} Georgii, H.-O. and Zessin, H. (1993) Large deviations and the maximum entropy principle for marked point random fields. {\sl Probab.\ Th.\ Rel.\ Fields} {\bf96}, 177--204.
\bibitem{GLM} Giacomin, G., Lebowitz, J.L. and Maes, C. (1995) Agreement
percolation and phase coexistence in some Gibbs systems, {\sl J. Statist.
Phys.} {\bf 80}, 1379--1403.
\bibitem{GS} Given, J.A. and Stell, G. (1990) The Kirkwood-Salsburg
equations for continuum percolation, {\sl J. Statist. Phys.} {\bf 59},
981--1018.
\bibitem{Gr} Grimmett, G. (1994) Percolative problems, {\sl Probability and
Phase Transition} (ed. G. Grimmett), Kluwer, Dordrecht, 69--86.
%\bibitem{Ho} Holley, R. (1974) Remarks on the FKG inequalities,
%{\sl Commun. Math. Phys.} {\bf 36}, 227--231.
\bibitem{Israel} Israel, R.B. (1979) {\sl Convexity in the Theory of Lattice Gases}, Princeton University Press, Princeton.
%\bibitem{Ja} Janson, S. (1984) Bounds on the distribution of extreme values
%of a scanning process, {\sl Stoch. Proc. Appl.} {\bf 18},
%313--328.
\bibitem{Jo1} Johansson, K. (1991) Separation of phases at low temperatures
in a one-dimensional continuous gas, {\sl Commun. Math. Phys.} {\bf 141},
259--278.
\bibitem{Jo2} Johansson, K. (1995) On separation of phases in
one-dimensional gases, {\sl Commun. Math. Phys.} {\bf 169}, 521--561.
\bibitem{Kall} Kallenberg, O. (1983) {\sl Random measures}, 3rd edition. Akademie Verlag, Berlin.
\bibitem{Kesten} Kesten, H. (1982) {\sl Percolation Theory for Mathematicians}, Birkh\"auser, Boston etc.
\bibitem{Klein} Klein, W. (1982) Potts--model formulation of continuum percolation, {\sl Phys. Rev.} {\bf B 26}, 2677-2678.
\bibitem{LL} Lebowitz, J.L. and Lieb, E.H. (1972) Phase transition in a continuum classical system with finite interactions. {\sl Phys. Lett.} {\bf 39A}, 98--100.
%\bibitem{Lin} Lindvall, T. (1992) {\sl Lectures on the Coupling Method},
%Wiley, New York.
\bibitem{KMM} Matthes, K., Kerstan, J. and Mecke, J. (1978) {\sl Infinitely Divisible Point Processes} (Wiley, Chichester).
\bibitem{MR} Meester, R. and Roy, R. (1996) {\sl Continuum Percolation},
Cambridge University Press, to appear.
\bibitem{MS} Meester, R. and Steif, J. (1994) Personal communication.
\bibitem{Minlos} Minlos, R.A. (1967) Limiting Gibbs distribution, {\sl Funct. Anal. Appl.} {\bf 1}, 140--150\&206--217.
\bibitem{Pe} Penrose, M.D. (1991) On a continuum percolation model,
{\sl Adv. Appl. Probab.} {\bf 23}, 536--556.
\bibitem{P} Preston, C.J. (1977) Spatial birth-and-death processes,
{\sl Bull. Int. Statist. Inst.} {\bf 46}(2), 371--391.
\bibitem{Rue1}Ruelle, D. (1970) Superstable interactions in classical statistical mechanics, {\sl Commun. Math. Phys.} {\bf18}, 127--159.
\bibitem{Ru} Ruelle, D. (1971) Existence of a phase transition in a continuous
classical system, {\sl Phys. Rev. Lett.} {\bf 27}, 1040--1041.
\bibitem{Step} Stepanov, V.E. (1970) On the probability of connectedness of a random graph ${\cal G}_m(t)$, {\sl Th. Probab. Appl.} {\bf 15}, 55--67.
\bibitem{WR} Widom, B. and Rowlinson, J.S. (1970) New model for the study of
liquid-vapor phase transition, {\sl J. Chem. Phys.} {\bf 52}, 1670--1684.
\end{thebibliography}
\end{document}