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determinantal point process, fermion poin process, Gibbs point process, 
Papangelou intensity, percolation
---------------0402040517717
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\def\r{\rho}      \def\vr{\varrho}    \def\om{\omega}     \def\Om{\Omega}
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\pagestyle{myheadings} \markboth{Georgii and Yoo}{Determinantal point processes}

\begin{document}
\title{Conditional Intensity and Gibbsianness\\ of Determinantal Point Processes}
\author{ Hans-Otto Georgii\footnotemark[1] \  and Hyun Jae Yoo\footnotemark[2] }
\date{}
%\subjclass{Primary: 60K35; Secondary: 82B43 }
 \maketitle

\begin{abstract}
The Papangelou intensities of determinantal (or fermion) point processes
are investigated. These exhibit a monotonicity property expressing the
repulsive nature of the interaction, and satisfy a bound implying
stochastic domination by a Poisson point process. We also show that
determinantal point processes satisfy the so-called condition
$(\S_{\l})$ which is a general form of Gibbsianness. In the absence of
percolation, the Gibbsian conditional probabilities can be identified
explicitly.
\end{abstract}
\noindent
{\bf Keywords}. {Determinantal point process, fermion point process,
Gibbs point process, Papangelou intensity, stochastic domination,
percolation}\\
{\bf Running head}. {Determinantal point processes}

\footnotetext[1]{Mathematisches Institut der Universit\"{a}t
        M\"{u}nchen, Theresienstra{\ss}e 39, 80333 M\"{u}nchen, Germany.
        E-mail: georgii@lmu.de}
\footnotetext[2]{University College , Yonsei University, 134
Shinchon-dong, Seodaemoon-gu, Seoul 120-749, Korea. E-mail:
yoohj@yonsei.ac.kr}
%\today

\section{Introduction}

The aim of this paper is to investigate the dependence structure of
determinantal (or fermion) point processes, abbreviated DPP's. These are
point processes on $\R^d$ (or more general spaces) with a particular
repulsive dependence structure induced by the fact that their
correlation functions are given by suitable determinants. More
explicitly, the correlation function $\r(\a)$ at a finite configuration
$\a$ is the determinant of the matrix obtained by evaluating a positive
definite function at the points of $\a$; see Subsection \ref{subsec:DPP}
for details.

Since their invention by Macchi \cite{M}, DPP's have attracted much
interest from various viewpoints. Spohn \cite{Sp1, Sp2} investigated the
dynamics of the so called Dyson's model, a model of interacting Brownian
particles in one dimension with pair force $1/x$, or pair interaction
$-\log x$. Its invariant measure is a typical DPP having the sine kernel
as defining positive definite function. Rather recently, the theory of
DPP's has been developed much further. Soshnikov  \cite{So} established
the full existence theorem for DPP's and discussed many examples
occurring in various fields of physics and mathematics. Shirai and
Takahashi  \cite{ST1}  also dealt with the existence theorem and
extended the theory to some generalized point processes including boson
processes. They also established some particular properties such as
limit theorems, ergodic properties, and the Gibbs property of the
corresponding discrete model \cite{ST2}. In a series of papers, Borodin
and Olshanski studied the DPP's appearing in the representation of the
infinite symmetric group, see \cite{BO} and the references therein.
Lyons and Steif \cite{L, LS} investigated the ergodic and stochastic
domination properties of DPP's on discrete lattices. The Glauber
dynamics leaving some DPP's invariant was studied by Shirai and the
second author \cite{SY}.

In this paper we ask for the dependence properties of DPP's. Our leading
questions are the following:
\begin{itemize}
\item[--]What can be said about the repulsive nature of the interaction? 
\item[--]When are determinantal point processes Gibbsian?
\end{itemize}
A key quantity we consider is the conditional intensity in the sense of
Papangelou, which is a function $c(x,\x)$ of points $x\in\R^d$ and
configurations $\x$. Its intuitive meaning is that $c(x,\x)dx$ is the
conditional probability of having a particle in $dx$ when the
configuration $\x$ is given. We will show that, locally on bounded
regions, Papangelou intensities always exists, are given by ratios of
determinants, and are decreasing functions of $\x$
(Theorem~\ref{thm:PapMon}). This monotonicity is a natural expression of
the repulsiveness of DPP's. In particular, it implies negative
correlations of local vacuum events stating that some bounded regions
contain no particles (Corollary~\ref{cor:Monoton}). We also show that
the local Papangelou intensities are bounded from above. This implies
stochastic domination of DPP's by Poisson point processes and, in
particular, the absence of percolation in the associated Boolean model
when the underlying positive definite function is small enough
(Corollaries~\ref{cor:PoissonDom} and \ref{cor:NoPerc}).

Next we will show that the Papangelou intensities of DPP's exist not
only locally but also globally (Theorem~\ref{thm:Sigma}). This means
that all DPP's are Gibbsian in a general sense, in that one can write
down natural versions of their conditional distributions in bounded
regions when the configuration outside of this region is fixed, and
these conditional distributions are absolutely continuous with respect
to the Poisson point process. It is a more difficult question to decide
whether the associated conditional Hamiltonians can be expressed in
terms of the underlying positive definite function in the natural way
one expects. Here we only have a partial result, stating in particular
that this holds whenever the underlying positive definite function has
finite range and is small enough (Theorem~\ref{thm:Gibbs}).

In the next section we set up the stage. That is, in
Subsection~\ref{subsec:DPP} we recall the definition of DPP's together
with some basic facts, while Subsection~\ref{subsec:PI} contains a
discussion of the Papangelou intensity and its significance. Our results
are stated in Section~\ref{sec:results}. Section~\ref{sec:aux} provides
some auxiliary results on operators and determinants, and the proofs of
our results follow in Section~\ref{sec:proofs}.

\vspace{0cm plus 4cm}


\section{Preliminaries}
\subsection{Determinantal point processes}
\label{subsec:DPP}

In this section we describe the general setting and recall the
definition of determinantal point random fields (DPP). For a more
complete account of DPP's we refer to the survey \cite{So} and the
articles \cite{L, LS, ST1, ST2}. The state space of a DPP may be a quite
general separable Hausdorff space, but in this paper we simply take
$\R^d$. We write $\cB$ for the Borel $\s$-algebra on $\R^d$, $\l$ for
Lebesgue measure on $(\R^d,\cB)$, and $\cB_0$ for the system of all
\emph{bounded} Borel sets in $\R^d$.

Let $\cX$ be the space of all locally finite subsets (configurations) in
$\R^d$, i.e.,
\[
\cX:=\{\x  \sub \R^d:| \x \cap \L|<\infty \text{ for all 
}\L \in \cB_0\}\,,
\]
where $|A|$ denotes the cardinality of a set $A$. Given any $\L \sub
\R^d$, we write $\cX_\L:=\{\x \in\cX: \x \sub\L\}$ for the set of all
configurations in $\L$, and $r_\L:\x \to\x_\L:= \x \cap\L$ for the
corresponding projection from $\cX$ onto $\cX_\L$. Also let $N_\L:\x 
\to | \x \cap \L|$ be the associated counting variable on $\cX$, and
$\cF_\L$ be the smallest $\s$-algebra on $\cX$ such that $N_\D$ is
measurable for all bounded Borel subsets $\D \sub \L$. We write $\cF$
for $\cF_{\R^d}$. Each configuration $\x \in \cX$ can be identified with
the integer-valued Radon measure $\sum_{x\in  \x }\d_x$ on the Borel
$\s$-algebra on $\R^d$. The vague topology for the latter then induces a
topology on $\cX$ turning $\cX$ into a Polish space. It is well-known
that $\cF$ is the associated Borel $\s$-algebra \cite{DV, Sh}. A
\emph{point process} (PP for short) is a probability measure $\m$ on
$(\cX,\cF)$. We write $\m_\L:= \m\circ r_\L^{-1}$ for its marginal on
$\cX_\L$.

Next let $\cX_0=\{\a\in\cX: |\a|<\infty\}$ be the set of all
\emph{finite} configurations in $\R^d$. $\cX_0$ is equipped with the
trace $\s$-algebra $\cF_0=\cF|_{ \cX_0}$ and the \emph{Lebesgue-Poisson
measure} $L$ defined by the identity
\[
\int_{\cX_0} f(\a)\,L(d\a) = \sum_{m=0}^{\infty}\frac{1}{m!}\int_{(\R^d)^m}
f(\{x_1,\ldots, x_m\})\, dx_1\cdots dx_m
\]
for any measurable function $f:\cX_0\to\R_+$. For any $\L\subset\R^d$ we
let $L_\L(d\a) = 1_{\{\a\subset\L\}}L(d\a)$ be the restriction of $L$ to
$\cX_\L$. Here  we use the notation $1_A$ for the indicator function of
a set $A$.

Recall that a PP $\m$ is said to have the \emph{correlation function}
$\r:\cX_0\to \R_+$ if $\r$  is measurable and satisfies
\begin{equation}\label{eq:corrfct}
\int_\cX \  \sum_{ \a\in \cX_0:\, \a \sub \x} u( \a) \ \m(d \x )=
\int_{\cX_0}  u( \a) \r( \a)\,L(d \a)
\end{equation}
for any measurable $u:\cX_0\to \R_+$. The \emph{Poisson point process
$\p^z$ with intensity $z>0$} is the unique PP with correlation function
$\r(\a )=z^{|\a |}$. Equivalently,  $\p^z$ is the unique PP such that,
for each $\L \in\cB_0$, the projection $\p^z_\L$ has the Radon-Nikodym
density $\x\to e^{-z\l(\L)}  z^{|\x|}$ relative to $L_\L$. The
characteristic feature of the determinantal point processes to be
considered here is that their correlation function is given by suitable
determinants. Given a function $K:\R^d\times\R^d \to\C$, we write 
\begin{equation}\label{eq:Kmatrix}
K(\a,\a)=\big(K(x,y)\big)_{x,y\in\a}
\end{equation}
for the matrix obtained by evaluating $K$ at a finite configuration $\a\in\cX_0$.

%:   Def DPP
\begin{define}\label{def:DPP} Let $K(x,y)$, $x,y\in \R^d$, be the integral kernel
function of a positive Hermitian integral operator $K$ on $L^2(\R^d)$.
A PP $\m$ with correlation function
\[
\r(\a)=\det K(\a,\a)
\]
is called a \emph{determinantal point process} (abbreviated DPP) with respect to $K$.
\end{define}
The DPP's defined above are also known as \emph{fermion point
processes}; see e.g. \cite{ST1} for the boson case where the determinant
is replaced by the permanent, and more general determinantal processes.
For the existence of DPP's we state the following theorem from
\cite{So}, see also \cite{M, DV, ST1}. $I$ stands for the identity
operator, and the ordering $S\le T$ between operators means that $T-S$
is a positive operator.
%
\begin{thm}[Macchi, Soshnikov]\label{thm:frpf} 
A Hermitian locally trace class operator $K$ on $L^2(\R^d)$
defines a DPP $\m$ if and only if $0\le K\le I$, and in this case $\m$ is unique.
\end{thm}
%
Any DPP $\m$ is locally absolutely continuous with respect to the
Lebesgue-Poisson measure $L$ and admits explicit expressions for the
local densities. To be specific, for each  $\L \in \cB_0$ let $P_\L:
L^2(\R^d)\to L^2(\L)$ be the projection operator and $K_\L:=P_\L KP_\L$
the restriction of $K$ onto $L^2(\L)$. That is, $K_\L$ has the kernel
$K_\L(x,y)=1_\L(x)K(x,y)1_\L(y)$. Suppose that  $\m$ is the unique DPP
for an operator $K$ as in Theorem \ref{thm:frpf}. Then the density
function of $\m_\L$ with respect to $L_\L$ is given by \cite{ST1, So}
\begin{equation}\label{eq:density_function}
\s_\L(\x)=\det(I-K_\L)\det J_{[\L]}(\x,\x), \quad \x\in\cX_\L;
\end{equation}
here  $J_{[\L]}:=K_\L(I-K_\L)^{-1}$, and the normalization constant
$\det(I-K_\L)$ is to be understood as a Fredholm determinant \cite{Si}.
Moreover, the correlation function $\r$ is recovered from the local
densities $(\s_\L)$ by 
\[
\r(\a)=\int_{\cX_\L}\s_{\L}(\a\x)\,L_\L(d\x)
\]
for $\a\in \cX_\L$, where $\a\x$ is shorthand for $\a\cup\x$.

\subsection{Papangelou intensities}
\label{subsec:PI}

Here we summarize some facts concerning the reduced (compound) Campbell 
measure of a PP $\m$ as well as its Papangelou intensity which describes 
the local dependence of particles.
%
\begin{define}\label{def:redCampb}
(a) The \emph{reduced} (or modified) \emph{Campbell measure} of a PP
$\m$ is the measure $\CM$ on the product space $(\R^d\times\cX,
\cB\otimes\cF)$ defined by
\[
\CM(A) = \int \m(d\x ) \sum_{x\in \x } 1_A(x, \x \sm x)\,, \quad A\in \cB\otimes\cF,
\]
where $\x \sm x:=\x \sm\{ x\}$.

(b) The \emph{reduced compound Campbell measure} of $\m$ is the measure $\CCM$ 
on the product space $(\cX_0\times\cX, \cF_0\otimes\cF)$ satisfying
\[
\CCM(B) = \int \m(d\x ) \sum_{\a\in\cX_0:\,\a\subset \x } 1_B(\a, \x \sm \a)\,, 
\quad B\in \cF_0\otimes\cF.
\]
\end{define}
%
It is well-known and easy to check that the reduced Campbell measure of
the Poisson PP $\p^z$ is given by $\CM[\p^z]=z \l\otimes\p^z$, and a
classical result of Mecke \cite{Mecke} states that $\p^z$ is the only PP
with this property. This fact suggests the following concept.

\begin{define}
A PP $\m$ is said to satisfy \emph{condition} $(\S_\l)$ if $\CM \ll
\l\otimes\m$. Any Radon-Nikydom density $c$ of $\CM$ relative to
$\l\otimes\m$ is then called (a version of) the \emph{Papangelou
(conditional) intensity} (abbreviated PI) of $\m$.
\end{define}
%
More explicitly, $c$ is a PI of $\m$ if
\begin{equation}\label{eq:Papint}
\int \m(d\x ) \sum_{x\in \x } f(x, \x \sm x) = \int dx \int \m(d\x )\, c(x,\x ) f(x,\x )
\end{equation}
for all measurable functions $f:\R^d\times\cX \to \R_+$. Intuitively,
$c(x,\x )$ is the conditional probability for a particle in the
differential region $dx$ given the configuration $\x $. Also, if
\[
Gh(\x ) = \int dx\ c(x,\x ) \big[h(\x \cup x)-h(\x )\big] + 
\sum_{x\in \x }\big[h(\x \sm x)-h(\x )\big]
\]
is the formal generator of a birth-and-death process with birth rate
$c(x,\x )dx$ for a particle in $dx$ and death rate $1$ for each
particle, (\ref{eq:Papint}) is equivalent to the reversibility equation
$\int g\, Gh\, d\m = \int h\, Gg\, d\m$; to see this it is sufficient to
set $f(x,\x )=g(\x )h(\x \cup x)$.

The following remark exhibits the significance of condition $(\S_\l)$.
In particular, it shows that $(\S_\l)$ processes are Gibbsian in a
general sense.

%:  Remark
\begin{rem}\label{rem:MWM}
For any $\m$ satisfying condition $(\S_\l)$ the following conclusions hold.

\smallskip
(a) The reduced compound Campbell measure $\CCM$ is absolutely
continuous with respect to $L\otimes\mu$ with a Radon-Nikodym density
$\hat c$ satisfying $\hat c(\emptyset,\xi) =1$ and
\begin{equation}\label{eq:compPapdens}
\hat c(\a,\x ) = c(x_1,\x) \prod_{i=2}^n 
c(x_i,x_1 \ldots x_{i-1}\x) \quad\text{when }\a=\{x_1,\ldots, x_n\};
\end{equation}
here $x_1 \ldots x_{i-1}\x = \{x_1, \ldots, x_{i-1}\}\cup \x$. In
particular, the right-hand side of \eqref{eq:compPapdens} is almost
surely symmetric in $x_1,\ldots, x_n$. $\hat c$ is called the
\emph{compound Papangelou intensity} (CPI). Explicitly, the relation
$\hat c=d\CCM/d( L\otimes\mu)$   means that
\begin{equation}\label{eq:compPapint}
\int \m(d\x ) \sum_{\a\in\cX_0:\,\a\subset \x } f(\a, \x \sm \a) = 
\int L(d\a) \int \m(d\x )\, \hat c(\a,\x )\, f(\a,\x )
\end{equation}
for any measurable $f:\cX_0\times\cX \to \R_+$, and follows easily from
\eqref{eq:Papint} by induction on $|\a|$.

\smallskip
(b) For an $f$ depending only on $\a$, a comparison of
\eqref{eq:compPapint} and \eqref{eq:corrfct} shows that the correlation
function and the CPI of $\m$ are related to each other by
\[
\r(\a) = \int \m(d\x)\, \hat c(\a,\x)\quad\text{for $L$-almost all $\a$.}
\]

\smallskip
(c) Let $\L\in\cB_0$. Applying \eqref{eq:compPapint} to $f(\a,\x)=
g(\a)h(\x) 1_{\{\a\subset\L, \,N_\L(\x)=0\}}$ for any
$\cF_\L$-measurable $g:\cX\to\R_+$ and $\cF_{\L^c}$-measurable
$h:\cX\to\R_+$ and taking conditional expectations we find
\[
\begin{split}
&\int \E_\m(g|\cF_{\L^c})\,h\, d\m = \int g h\, d\m
= \int f \,d\CCM\\
&=\int \m(d\x)\,h(\x)\  \m(N_\L=0|\cF_{\L^c})(\x)  \int L_\L(d\a)\, g(\a)\,\hat c(\a,\x_{\L^c})
\end{split}
\]
Hence
\begin{equation}\label{eq:condexpect}
\E_\m(g|\cF_{\L^c})(\x) = \m(N_\L=0|\cF_{\L^c})(\x)\ \int g\; \hat c(\cdot, \x_{\L^c}) \,dL_\L
\end{equation}
for $\m$-almost all $\x$ and any $\cF_\L$-measurable $g$. In particular,
for $g\equiv 1$ we find that $\m(N_\L=0|\cF_{\L^c})>0$ almost surely for
each $\L\in\cB_0$, a property introduced by Papangelou \cite{Papangelou}
as condition $(\S)$, and by Kozlov \cite{Kozlov} as the condition of
non-degenerate vacuum. 
\end{rem}

The observations in the preceding remark are due to Matthes, Warmuth and
Mecke \cite[Section 3]{MWM} and give one part of their theorem below;
cf. also \cite[Theorem 2$'$\,]{NZ}.

\begin{thm}[Matthes, Warmuth and Mecke]\label{thm:MWM}
A point process $\m$ satisfies condition $(\S_\l)$ with PI $c$ if and
only if, for each $\L\in\cB_0$, $\m$ is absolutely continuous relative
to $L_\L\otimes\m_{\L^c}$ with a density  $\g_\L$ satisfying
$\g_\L(x\x)=0$ whenever $\g_\L(\x)=0$ and $x\in\L\sm\x$. In this case, 
for $L_\L\otimes\m_{\L^c}$-almost all $\x \in\cX$ we have
\begin{equation}\label{eq:condPapdens}
\g_\L(\x ) = Z_\L(\x _{\L^c})^{-1}\; \hat c(\x_\L,\x _{\L^c}) 
\end{equation}
with $0< Z_\L(\x _{\L^c}) = \int \hat c(\cdot,\x _{\L^c})\,dL_\L
<\infty$, and $c(x,\x ) = \g_\L(x\x )/\g_\L(\x )$ for
$\l\otimes\m$-almost all $(x,\x )\in \L\times\cX$.
\end{thm}
%
More explicitly, Equation (\ref{eq:condPapdens}) means that for each
bounded measurable function $f:\cX\to\R$, the conditional expectation
$\E_\m(f|\cF_{\L^c})(\x )$ has the version
\begin{equation}\label{eq:condProb}
G_\L(f|\x ) := Z_\L(\x _{\L^c})^{-1} \int f(\a\x _{\L^c})\, \hat c(\a,\x _{\L^c})
L_\L(d\a).
\end{equation}

\begin{rem}\label{rem:locMWM}
Theorem \ref{thm:MWM} has a counterpart for the restriction of a PP $\m$
to a bounded region $\L\in\cB_0$. Let $\l_\L$ be the restriction of $\l$
to the $\s$-algebra $ \cB_\L$ of Borel subsets of $\L$. Then $\m_\L$
satisfies condition $(\S_{\l_\L})$ with a PI $c_\L$ if and only if
$\mu_\L$ is absolutely continuous with respect to $L_\L$ with a density 
$\s_\L$ having an increasing zero-set $\{\s_\L=0\}$. In this case,
$\s_\L = \s_\L(\emptyset)\,\hat c_\L(\cdot,\emptyset)$ $L_\L$-almost
surely, and $c_\L(x,\xi) = \s_\L(x\x)/\s_\L(\x)$ for $\l_\L\otimes\m_\L$
almost all $(x,\x)\in \L\times\cX_\L$. This follows from the above by
noting that $\m_\L$ is trivial on $\cF_{\L^c}$; cf. also
\cite[Proposition 3.1]{GK}.
\end{rem}

\section{Results}
\label{sec:results}

The DPP's considered in this paper are defined via some positive
Hermitian integral operator $J$ on $L^2(\R^d)$ satisfying the following
condition of boundedness, continuity, and translation invariance.

\begin{ass}\label{ass:assumption} The kernel function $J(\cdot,\cdot)$
of $J$ is given by $J(x,y)=j(x-y)$, $x,y\in \R^d$, where
\[
j(x):=(2\p)^{-d/2}
\int_{\R^d}e^{ix\cdot t}\,\ph(t)\,dt
\] 
is the Fourier transform of a nonnegative integrable function $\ph$ on
$\R^d$. We call $z(J):=j(0)=(2\p)^{-d/2}\|\ph\|_1$ the \emph{activity}
of $J$.
\end{ass}

To any such $J$ we associate the operator
\begin{equation}\label{eq:K-operator}
K^J:=\frac{J}{I+J}.
\end{equation}
By construction, $K^J$ has the integral kernel
$K^J(x,y)=k^J(x-y)$, $x,y\in \R^d$, where $k^J$ is given by
\begin{equation}\label{eq:kJ}
k^J(x):=(2\p)^{-d/2}
\int_{\R^d}e^{ix\cdot t}\,\frac{\ph(t)}{1+\ph(t)}\,dt.
\end{equation}
$K^J$ then satisfies the conditions of Theorem \ref{thm:frpf} and hence
defines a unique DPP  $\m^J$. Our basic observations are the following;
recall the notation \eqref{eq:Kmatrix}.

%:  Thm1
\begin{thm}\label{thm:PapMon}
For each $\L\in\cB_0$, $\m^J_\L$ satisfies  condition $(\S_{\l_\L})$,
and the associated PI $c^J_\L$ exhibits the following properties:

\smallskip{\rm(a)}
For $L_\L\otimes\m_\L$-almost all $(\a,\x)\in\cX_\L\times\cX_\L$,
\begin{equation}\label{eq:cLformula}
\hat c^J_\L(\a,\x)= \det J_{[\L]}(\a\x,\a\x)/\det J_{[\L]}(\x,\x)\,,
\end{equation}
where the ratio is defined as zero if the denominator vanishes. As in 
(\ref{eq:density_function}), $J_{[\L]}$ is a continuous kernel of the operator
$J_{[\L]}=K_\L^J(I-K_\L^J)^{-1}$ on $L^2(\L)$ and $K^J_\L$ denotes the
restriction of $K^J$ to $L^2(\L)$. In particular, $c^J_\L(x,\x)= \det
J_{[\L]}(x\x,x\x)/\det J_{[\L]}(\x,\x)$ for $\l_\L\otimes\mu_\L$-almost
all $(x,\x)\in\L\times\cX_\L$.

\smallskip{\rm(b)} 
For any $\x\sub\eta\in\cX_\L$ and $x\in \L\sm\eta$, 
\[
c^J_\L(x,\x)\ge c^J_\L(x,\eta)\quad\text{and}\quad 0\le c^J_\L(x,\x)\le z(J).
\] 
\end{thm}
%
Here are some consequences of the theorem. First, let $\L,\D\in\cB_0$
with $\L\subset\D$ and $f:\cX_\L\to\R_+$ be any measurable function.
Then, combining \eqref{eq:cLformula} with Remark \ref{rem:locMWM} and
\eqref{eq:condProb} we find that
\begin{equation}\label{eq:locCondProb}
\begin{split}
\E_\m(f|\cF_{\D\sm\L})(\x) &= G^J_{\L,\D}(f|\x)\\ 
&:= Z^J_{\L,\D}(\x_{\D\sm\L})^{-1}
\int L_\L(d\a) \, f(\a)\,\hat c_\D(\a,\x_{\D\sm\L})
\end{split}
\end{equation}
for $\m$-almost all $\x\in\cX$. Note that the normalization constant
\[
Z^J_{\L,\D}(\x)= \int L_\L(d\a) \,\hat c_\D(\a,\x)
\]
is always finite due to the bound in assertion (b), and non-zero
whenever $\det J_{[\D]}(\x,\x)>0$ because $L_\L(\{\emptyset\})=1$. This
means that $G^J_{\L,\D}(f|\x)$ is well-defined for \emph{all} such $\x$.

Statement (b) of Theorem \ref{thm:PapMon},  the monotonicity of the PI's
$c^J_\L(x,\cdot)$, expresses the repulsive nature of the particle
interaction in an infinitesimal way. The following corollary provides an
integral version of this repulsiveness.

\begin{cor}\label{cor:Monoton}
For any $\L\subset\D\in\cB_0$ and any measurable function
$f:\cX_\L\to\R_+$, the ratio $G^J_{\L,\D}(f|\x )/G^J_{\L,\D}(N_\L=0|\x
)$ is decreasing in $\x\in\cX_{\D\sm\L} $. In particular,
$G^J_{\L,\D}(N_\L=0|\,\cdot)$ is increasing, and for
$\L\sub\D_1\sub\D_2\in\cB_0$ we have
\begin{equation}\label{eq:Monoton}
\m^J(N_\L=0|N_{\D_2\sm\L}=0)\le 
\m^J(N_\L=0|N_{\D_1\sm\L}=0) \le \m^J(N_\L=0)\,.
\end{equation}
\end{cor}

\begin{rem}\label{rem:Monoton}
The statement of the corollary is weaker than one may hope. In fact, one
might guess that $G^J_{\L,\D}(f|\x )$ is decreasing in $\x$ for any
increasing $\cF_{\L}$-measurable function $f$. (The corollary implies
this assertion only for $f=1_{\{N_\L\ge 1\}}$. This guess, however,
cannot be expected to be true in general. For if $f$ depends only on
some part $\L_0$ of $\L$, then an increase of $\x$ may repel some
particles from $\L\setminus\L_0$, giving a chance to some additional
particles in $\L_0$, so that $G^J_{\L,\D}(f|\x )$ will increase. So, the
situation is less satisfactory for point processes than in the discrete
case; cf. Theorem 6.5 of \cite{L}. Nevertheless, \eqref{eq:Monoton}
implies that, for disjoint $\L,\D\in\cB_0$, the events $\{N_\L=0\}$ and
$\{N_\D=0\}$ are negatively associated; see Proposition 2.7 of
\cite{ST2} for the corresponding result in the discrete case.
\end{rem}

Next we exploit the domination bound $c^J_\L\le z(J)$. Let $\m,\n$ be
two PP's. One says \emph{$\m$ is dominated by $\n$}, written
$\m\preceq\n$, if
\[
\int f\,d\m\le \int f\,d\n
\]
for every increasing measurable function $f:\cX\to\R$. We then have 
the following Poisson domination result.

\begin{cor}\label{cor:PoissonDom}
For any $J$ as above, $\m^J\preceq\p^{z(J)}$.
\end{cor}
%
The last result implies that for small activity there is no percolation
in the Boolean model associated to $\m^J$. Let $b_R(x)$ denote the
closed ball of radius $R<\infty$ in $\R^d$ centered at $x$, and for $\x
\in\cX$ let 
\[
B_R(\x )=\bigcup_{x\in \x }b_R(x)
\]
the associated Boolean set. $B_R(\x )$ splits into connected components
called \emph{clusters}. A cluster is called \emph{infinite} if it
consists of infinitely many points of $\x $, or equivalently, if its
diameter is infinite. It is well-known \cite{G, MR} that, for $d\ge 2$,
there exists a critical threshold $0<z_c\equiv z_c(d,R)<\infty$ for
Poisson percolation: 
\begin{equation*}
\p^z\big(\exists \text{ infinite cluster of }B_R(\cdot)\big)=
\begin{cases} 0 &\text{if }z<z_c\,,\\1 &\text{if }z>z_c\,.\end{cases}
\end{equation*}
For $d=1$ one has $z_c=+\infty$ \cite[Theorem 3.1]{MR}.

\begin{cor}\label{cor:NoPerc}
For $z(J)<z_c$, $\m^J(\exists \text{ infinite cluster of }B_R(\cdot))=0$.
\end{cor}

We now address the question whether DPP's are Gibbsian. 
In view of Remark \ref{rem:MWM} and Theorem \ref{thm:MWM}, the following
result implies that this is the case at least in a general sense.

%:  Thm2
\begin{thm}\label{thm:Sigma}
$\m^J$ satisfies condition $(\S_\l)$.  Its CPI is 
given by
\begin{equation}\label{eq:cJ-lim}
\hat c^J(\a,\x) = \lim_{n\to\infty} \hat c^J_{\D_n}(\a,\x_{\D_n})\quad
\text{for $L\otimes\m$-almost all $(\a,\x)$,}
\end{equation}
 where $\hat c^J_{\D_n}$ is given by \eqref{eq:cLformula} and $(\D_n)$ is any 
sequence in $\cB_0$ that increases to $\R^d$. In particular, 
$\m^J(N_\L=0|\cF_{\L^c}) >0$ $\m^J$-almost surely for each
$\L\in\cB_0$ and, for any $\cF_\L$-measurable $f:\cX\to\R_+$, the ratio 
\[
\E_{\m^J}(f|\cF_{\L^c})/\m^J(N_\L=0|\cF_{\L^c}) 
\]
of conditional expectations has a decreasing version.
\end{thm}
%
Combining  \eqref{eq:cJ-lim} with \eqref{eq:cLformula} and
\eqref{eq:condProb} we obtain at least an implicit formula for the
conditional probabilities of $\m^J$ given the events outside of a
bounded region. But the question remains of whether the CPI, and thereby
the conditional probabilities, of $\m^J$ can be identified in a more
specific way. In fact, there exists a natural candidate for $\hat c^J$,
namely
\begin{equation}\label{eq:candidate}
\hat c^J_*(\a,\x) = \lim_{\D\uparrow\R^d} \det J(\a\x_\D,\a\x_\D)/\det J(\x_\D,\x_\D)\,.
\end{equation}
As a matter of fact, this limit exists because the expression on the
righthand side is decreasing in $\D$, as follows from \eqref{eq:det3}
below. (As before, we set a ratio of determinants equal to zero if the
denominator vanishes.) In contrast to \eqref{eq:cJ-lim}, the
determinants in \eqref{eq:candidate} involve $J$ itself rather than
$J_{[\D_n]}$. We thus arrive at the following question.

%:  Question
\begin{qu}\label{question} When is it true that
$\hat c^J=\hat c^J_*$ almost surely for $L\otimes\m^J$?
\end{qu}
%
In fact, this question is closely related to the problem of whether $\m^J$ is Gibbsian
in the usual sense that its conditional probabilities are given by an interaction potential.
We will discuss this point in Remark \ref{rem:potential} below.

Unfortunately, we are unable to settle Question \ref{question} in the
same generality as this was done in the lattice case by Shirai and
Takahashi \cite[Theorem 6.2]{ST2}. Their argument exploits the symmetry
between occupied and empty lattice sites and therefore does not carry
over to our continuous setting. The following theorem gives, at least, a
partial answer. $J$ is said to have range $R<\infty$ if $J(x-y)=0$
whenever $|x-y|>R$.

%:  Thm3
\begin{thm} \label{thm:Gibbs}
{\rm(a)} In general, the inequality $\hat c^J\le\hat c^J_*$ holds 
$L\otimes\m^J$-almost surely.

{\rm(b)} If $J$ has range $R<\infty$ and 
$\m^J(\exists \text{ \rm infinite cluster of }B_R(\cdot))=0$
(which is the case at least when $d=1$ or $z(J)<z_c$) then 
\begin{equation}\label{eq:cJ-subcr}
\hat c^J(\a,\x) = \hat c^J_*(\a,\x)=\det J(\a\x^\sharp,\a\x^\sharp)/
\det J(\x^\sharp,\x^\sharp)\ 1_{\{\diam W(\a,\x)<\infty\}}
\end{equation}
for $L\otimes\m^J$-almost all $(\a,\x)$. 
Here $W(\a,\x)$ is the union of the clusters of $B_R(\a\x)$ hitting $\a$, and 
$\x^\sharp:=\x_{W(\a,\x)}$ is the restriction of $\x$ to $W(\a,\x)$. 
\end{thm}
%
Assertion (b) of this theorem has already been derived by the second
author in \cite{Y} by different methods. In the situation of this
statement, the CPI $\hat c^J_*(\a,\cdot)$ is continuous at every
configuration $\x$ for which $B_R(\x)$ contains no infinite cluster.
This implies that the conditional expectation $\E_\m(f|\cF_{\L^c})$ has
a version which is continuous at every configuration $\x$ for which
$B_R(\x)$ contains no infinite cluster hitting $\L$, and the latter
occurs with probability~1. This is Gibbsianness in the best sense one
can expect. In particular,  the conditional probabilities of $\m^J$ can
be expressed in terms of a suitable many-body potential. The following
remark shows that such a potential exists whenever Question
\ref{question} can be answered in the positive.


\begin{rem}\label{rem:potential}  For each $\L\in\cB_0$ and any
$\x\in\cX_{\L^c}$ satisfying $\det J(\z,\z)>0$ for all finite
$\z\sub\x$, we can define a Gibbs distribution $G_\L(\cdot|\x)$ by
inserting $\hat c^J_*$ into \eqref{eq:condProb}.  In fact,
$G_\L(\cdot|\x)$ is a Gibbs distribution for the Hamiltonian
$H_\L(\cdot|\x)= -\log \hat c^J_*(\cdot,\x)$ on $\cX_\L$,
$\x\in\cX_{\L^c}$. (If desired, one can express the Hamiltonian in terms
of a many-body potential $\Phi$, but this is not particularly useful.)
These Gibbs distributions altogether form a Gibbsian specification
$G=(G_\L)_{\L\in\cB_0}$ in the sense of Preston \cite[pp. 16,
17]{PrLNM}; this has been proved by Gl\"otzl \cite{Gloetzl2} in a
general setting, and in \cite{Y} for the particular case of DPP's. By
construction, $\m^J$ is a Gibbs measure for $G$ whenever it admits the
CPI $\hat c^J_*$.

Conversely, suppose $\m^J$ is a Gibbs measure for some Hamiltonian $H$.
Then $\hat c^H(\a,\x)$ $:=\exp[-H_\a(\a|\x_{\a^c}]$ is a version of its
CPI \cite{NZ} and satisfies the continuity condition $\hat c^H(\a,\x) =
\lim_{\D\ua\R^d}\hat c^H(\a,\x_\D)$. If one assumes that the function
$c^J=\lim_{n\to\infty} c^J_{\D_n}$ in  \eqref{eq:cJ-lim} has the same
continuity property, one can conclude that $c^J=c^J_*$ almost surely.
For, Proposition \ref{prop:local_convergence} implies that
$\lim_{n\to\infty} c^J_{\D_n}(\a,\x_\D) =c^J_*(a,\x_\D)$ whenever
$\a\in\cX_0$, $\D\in\cB_0$, and $\x\in\cX$ is such that $\det
J(\x_\D,\x_\D)>0$. But the last condition holds $\m^J$-almost surely.
\end{rem} 


\section{Auxiliary results on operators and determinants}
\label{sec:aux}

In this section we collect some results on the local versions of the
operators $J$ and $K^J$ defined in Assumption \ref{ass:assumption} and
Eq. (\ref{eq:K-operator}), as well as their associated matrices. As
before, let $P_\L: L^2(\R^d)\to L^2(\L)$ be the projection operator, and
$K_\L:=P_\L KP_\L$ resp. $J_\L:=P_\L JP_\L$ the restrictions of $K$ and
$J$ to $L^2(\L)$. We also consider the local $J$-operator
$J_{[\L]}:=K_\L^J(I-K_\L^J)^{-1}$ associated to $K_\L$. Notice that
$J_{[\L]}$ is different from $J_\L$.

\begin{lem}\label{lem:contkernel}
For each $\L \in\cB_0$, the operator $J_{[\L]}$ admits a continuous kernel 
function $J_{[\L]}(x,y)$, $x,y\in \L$.
\end{lem}
%
\Proof  A constructive approach to the Radon-Nikodym theorem implies that
$J_{[\L]}$ admits a (jointly) measurable kernel function $J_{[\L]}'(x,y)$, $x,y\in\L$. 
Since \[J_{[\L]}=K_\L^J+(K_\L^J)^2+K_\L^JJ_{[\L]}K_\L^J,\]
$J_{[\L]}'(x,y)$ coincides for almost all $x,y\in\L$ with
\[\begin{split}
J_{[\L]}(x,y):= & \; k^J(x-y) + \int_\L k^J(x-u)\,k^J(u-y)\,du\\ & + 
\int_\L\int_\L k^J(x-u)\,J_{[\L]}'(u,v)\,k^J(v-y)\,du\, dv\,.
\end{split}
\]
Since $k^J$ is uniformly continuous by Eq.~(\ref{eq:kJ}), it is easily
checked that $J_{[\L]}(x,y)$ is continuous, as required.\EndProof

Henceforth we fix any continuous kernel of $J_{[\L]}$. We are interested
in the matrices $J_{[\L]}(\x,\x)$ obtained by evaluating this kernel at
the points of arbitrary configurations $\x\in \cX_\L$; cf. Definition
\ref{def:DPP}. Our analysis relies on the following general fact which
appears already in \cite[p. 18]{OP}. Since no proof is given there, we
provide a proof here for the convenience of the reader.

\begin{lem}\label{lem:basic-order}
For  any invertible positive operator $T$ and any projection $P$,
\[
PT^{-1}P\ge P(PTP)^{-1}P.
\]
\end{lem}

\Proof Since $T$ is positive and invertible, any restriction of $T$ to
some subspace is invertible in this subspace. This means that the
operators $T_P^{-1}:=P(PTP)^{-1}P$ and $T_{P^\bot}^{-1}:=P^\bot(P^\bot
T^{-1}P^\bot)^{-1}P^\bot$ are well-defined, where $P^{\bot}=I-P$. The
key observation is the decomposition formula
\begin{equation}\label{eq:decomposition}
PT^{-1}P=T_P^{-1}+PT^{-1}T_{P^\bot}^{-1}T^{-1}P\,.
\end{equation}
To see this we observe that $P^\bot T^{-1}(P^\bot+P)TP=0$. Multiplying
with $T_{P^\bot}^{-1}$ from the left we find $P^\bot TP=-T_{P^\bot}^{-1}
T^{-1}PTP$. Inserting this into the identity $PT^{-1}(PTP+P^\bot TP)=P$
we get
\[
PT^{-1}\Big(I-T_{P^\bot}^{-1}T^{-1}\Big)PTP=P.
\]
Multiplying with $T_P^{-1}$ from the right and rearranging we arrive at
\eqref{eq:decomposition}. As the second operator on the right-hand side
of \eqref{eq:decomposition} is positive,
the lemma follows.\EndProof

Next we state a monotonicity result which is interesting in its own 
right; we will only need a weak form of it.

\begin{lem}\label{lem:monotonicity}
For any $\L \sub\D\in \cB_0$, $J_{[\L]}\le P_\L J_{[\D]}P_\L\le J_\L$ 
in the operator ordering. In particular,
\begin{equation}\label{eq:det-monotone}
\det J_{[\L]}(\x,\x) \le \det J_{[\D]}(\x,\x)\le \det J(\x,\x)
\end{equation}
for all $\x\in \cX_\L$.
\end{lem}

\Proof 
We prove the inequality $J_{[\L]}\le J_\L$ which implies in particular
that $P_\L J_{[\D]}P_\L\le P_\L J_\D P_\L =J_\L$. The proof of the
relation $J_{[\L]}\le P_\L J_{[\D]}P_\L$ is similar. Since
$J=K^J(I-K^J)^{-1}$ and $L(I-L)^{-1}=-I+(I-L)^{-1}$ for any $0\le L< I$,
the inequality $J_{[\L]}\le J_\L$ is equivalent to
\[
P_\L(I-K_\L^J)^{-1}P_\L\le P_\L(I-K^J)^{-1}P_\L.
\]
But this follows from Lemma \ref{lem:basic-order} as applied to $P=P_\L$ and $T=I-K^J$, 
because $P_\L(I-K_\L^J)^{-1}P_\L=P_\L(P_\L(I-K^J)P_\L)^{-1}P_\L$.

To prove inequality (\ref{eq:det-monotone}) we note that, for any two
positive definite $n\times n$ matrices $0\le A\le B$,  the decreasingly
ordered eigenvalues of $A$ resp. $B$ satisfy the inequalities 
\[
0\le \l_i^{\da}(A)\le \l_i^{\da}(B), \quad i=1,\ldots,n;
\]
see \cite[Corollary III.2.3]{B}. The determinant is therefore an
increasing function relative to the operator ordering. The  inequalities
(\ref{eq:det-monotone}) will thus be proved once we have shown that
\[
J_{[\L]}(\x,\x)\le J_{[\D]}(\x,\x) \le J(\x,\x)
\]
as operators on $\C^n$, $n=|\x|$. This, however, is a direct consequence
of the continuity of the kernels $J_{[\L]}(x,y)$ established in Lemma
\ref{lem:contkernel} together with the operator inequalities just
proved; it is enough to approximate the Dirac delta-measures  at the
points $x\in\x$ by $L^2(\L)$-functions.\EndProof

We will also need the following inequalities for determinants;
\eqref{eq:det2} is known as the Fischer inequality. 

\begin{lem}\label{lem:determinant-relation}
The following relations hold for determinants of positive definite 
Hermitian block matrices:
\begin{align}
\label{eq:det1}&\det\begin{pmatrix}A&D\\D^*&B\end{pmatrix}
=
\det(A-DB^{-1}D^*)\,\det B\quad \text{if }\det B>0,\\
\label{eq:det2}&\det\begin{pmatrix}A&D\\D^*&B\end{pmatrix} \le \det A\,\det B\,,\\ 
\label{eq:det3}&\det\begin{pmatrix}A&D&E\\D^*&B&F\\ E^*& F^*&C\end{pmatrix} \det B
\le \det\begin{pmatrix}A&D\\D^*&B\end{pmatrix}\, 
\det\begin{pmatrix}B&F\\F^*&C\end{pmatrix}\,.
\end{align}
\end{lem}

\Proof
Eqn. \eqref{eq:det1} follows from the identity
\[
\begin{pmatrix}A&D\\D^*&B\end{pmatrix} =
\begin{pmatrix}A-DB^{-1}D^*&DB^{-1}\\0&I\end{pmatrix}
\begin{pmatrix}I&0\\ D^*&B\end{pmatrix}\,,
\]
where $I$ is the identity matrix. Since $DB^{-1}D^*\ge0$ and the
determinant is monotone with respect to the operator ordering, this
implies \eqref{eq:det2}  in the invertible case. For general $B$, one
can approximate $B$ by invertible matrices. To prove \eqref{eq:det3} one
can assume that $\det B>0$ and then divide both sides by $\det^2 B$. In
view of \eqref{eq:det1}, \eqref{eq:det3} then takes the form 
\[
\det \left[\begin{pmatrix}A&E\\E^*&C\end{pmatrix}
- \begin{pmatrix}D\\F^*\end{pmatrix}B^{-1} \big(D^*F\big)\right]
\le \det(A-DB^{-1}D^*)\, \det (C-F^*B^{-1}F)
\]
which follows from \eqref{eq:det2}.
\EndProof

We conclude this section with a convergence result.

%:  Prop
\begin{prop}\label{prop:local_convergence}
As $\L\in\cB_0$ increases to $\R^d$, $J_{[\L]}(x,y)$ converges to 
$j(x-y)$ locally uniformly in the variables $x,y\in\R^d$.
\end{prop}
%
\Proof 
First we observe that $k^J\in L^2(\R^d)$. This follows from
Eq.~\eqref{eq:kJ} and Plancherel's theorem because $\ph/(1+\ph)\in
L^2(\R^d)$; see  Example (c) on p. 483 of \cite{Feller}, for example. In
particular, if we set $k_x(u):=k^J(u-x)$ for $u\in \R^d$ then
$k_x(\cdot)\in L^2(\R^d)$ for each $x\in \R^d$.

Next, for each $N\ge1$ we can write
\[
J_{[\L]}=\sum_{i=1}^{N+2}(K_\L)^i+(K_{\L})^{N+2}J_{[\L]}\text{ \
 and \ }J =\sum_{i=1}^{N+2} K^i+K^{N+2}J \,.
\]
Since $0\le K_\L\le K<I$, the operator norms
$\|(K_{\L})^{N}\|\le\|K^{N}\|$ become arbitrarily small when $N$ is
large enough. Moreover, $\|J_{\L}\|$ is bounded uniformly in $\L$.
Hence, if $N$ is sufficiently large then the kernel
\[
(K_{\L})^{N+2}J_{[\L]}(x,y)=(k_x,(K_\L)^{N}J_{[\L]}k_y)_{L^2(\R^d)}
\]
is arbitrarily small uniformly in $x,y,\L$ (including $\L=\R^d$).
It is therefore sufficient to show that, for each $i\in\N$,
\[
(K_\L)^i(x,y)\to K^i(x,y)
\]
locally uniformly in $x,y$ as $\L\ua\R^d$. The case $i=1$ is trivial.
For  $i=2$ we have  
\[
(K_{ \L})^2(x,y)=(P_\L k_x,P_\L k_y)_{L^2(\R^d)}\,,
\] 
and this converges to $(k_x, k_y)_{L^2(\R^d)}=K^2(x,y)$ locally
uniformly in $x,y$. In the case $i\ge 3$ we write
\[
(K_{\L})^i(x,y)=(k_x,(K_{\L})^{i-2}k_y)_{L^2(\R^d)}.
\]
Since $K_\L$ converges strongly to $K$ as $\L\ua \R^d$, the inner
product on the right-hand side converges to
$(k_x,(K)^{i-2}k_y)_{L^2(\R^d)}=K^i(x,y)$ locally uniformly in
$x,y$.\EndProof


\section{Proofs}
\label{sec:proofs} 

Let us start proving Theorem \ref{thm:PapMon}. 

\medskip
\Proof[ of Theorem \ref{thm:PapMon}] (a)
Fix any $\L\in\cB_0$. From Eq. (\ref{eq:density_function}) we know that
$\m_\L$ is absolutely continuous relative to $L_\L$ with density
$\s_\L(\x)\propto \det J_{[\L]}(\x,\x)$. Inequality \eqref{eq:det2}
implies that the zero set $\{\s_\L=0\}$ is increasing. Remark
\ref{rem:locMWM} thus tells us that $\m_\L$ satisfies condition
$(\S_{\l_\L})$ with CPI given by (\ref{eq:cLformula}). This proves
statement (a) of the theorem.

(b) The first inequality in (b) is an immediate consequence of
\eqref{eq:det3} which asserts that $\det J_{[\L]}(x\eta,x\eta)\det
J_{[\L]}(\x,\x) \le \det J_{[\L]}(\eta,\eta) \det J_{[\L]}(x\x,x\x)$
whenever $\x\subset\eta\in\cX_\L$.
%To prove the first inequality in (b) let $\x\subset\eta\in\cX_\L$ 
%and $x\in\L\sm\eta$ be given. We can
%assume that  $\det J_{[\L]}(x\eta,x\eta)>0$ because otherwise there is nothing
%to show. Lemma \ref{lem:determinant-relation}(a) then implies that also
%$\det J_{[\L]}(\z,\z)>0$ for any $\z\sub x\eta$. Writing $Q_{\z}$ for the 
%projection onto the subspace corresponding to the index set $\z$, we conclude 
%from Cramer's rule that
%\begin{equation}\label{eq:an-identity}
%c^J_\L(x,\x)^{-1}=Q_{\{x\}}\,J_{[\L]}(x\x,x\x)^{-1}\,Q_{\{x\}}.
%\end{equation}
%On the other hand, Lemma \ref{lem:basic-order} implies that
%\[
%Q_{x\x}\,J_{[\L]}(x\eta,x\eta)^{-1}\,Q_{x\x}\ge
%J_{[\L]}(x\x,x\x)^{-1}.
%\]
%Applying the projection $Q_{\{x\}}$ on both sides we find
%\[
%Q_{\{x\}}\,J_{[\L]}(x\eta,x\eta)^{-1}\,Q_{\{x\}}\ge
%Q_{\{x\}}\,J_{[\L]}(x\x,x\x)^{-1}\,Q_{\{x\}}.
%\]
%Together with (\ref{eq:an-identity}), the first inequality follows. 
In particular, we have $c^J_\L(x,\x)\le c^J_\L(x,\emptyset)$ for the
empty configuration $\emptyset$. But the last quantity equals $\det
J_{[\L]}(x,x)$ by definition, which in turn is not larger than $ z(J)$,
by inequality (\ref{eq:det-monotone}). This proves the second inequality
of assertion (b).\EndProof

We now turn to the proofs of Corollaries \ref{cor:Monoton} through
\ref{cor:NoPerc}.

\medskip
\Proof[ of Corollary \ref{cor:Monoton}] In view of Eq. (\ref{eq:locCondProb}), 
\[
G^J_{\L,\D}(f|\x )\,\big/\, G^J_{\L,\D}(N_\L=0|\x )= 
\int L_\L(d\a)\,  f(\a) \, 
\hat c^J_\D(\a,\x)
\]
for any $\x\in\cX_{\D\sm\L}$. Since $f\ge0$, Theorem \ref{thm:PapMon}(b)
together with equation \eqref{eq:compPapdens} implies that the integrand
on the right-hand side is a decreasing function of $\x$. In particular,
for $f\equiv1$ we find that $G^J_{\L,\D}(N_\L=0|\,\cdot)$ is increasing.
To prove \eqref{eq:Monoton} we note that 
\[
\begin{split}
&\m^J(N_\L=0|N_{\D_2\sm\L}=0)= G^J_{\L,\D_2}(N_\L=0|\emptyset )\\
&\le \E_{\m^J}\big(G^J_{\L,\D_2}(N_\L=0|\,\cdot)\big|\cF_{\D_1\sm\L}\big)(\emptyset)\\
&= \m^J(N_\L=0|\cF_{\D_1\sm\L})(\emptyset) =\m^J(N_\L=0|N_{\D_1\sm\L}=0)
\end{split}
\]
because $G^J_{\L,\D_2}(N_\L=0|\emptyset )\le G^J_{\L,\D_2}(N_\L=0|\,\cdot )$.
The final inequality follows by setting $\D_1=\L$.\EndProof


\Proof[ of Corollary \ref{cor:PoissonDom}] 
We show first that $\m^J_\L\preceq\p^{z(J)}_\L$ for  any $\L\in\cB_0$.
It is well-known and easy to check that the Poisson PP $\p^{z(J)}_\L$
has the constant PI $z(J)$ on $\L$; cf. \cite{Mecke, NZ}. The inequality
$c^J_\L(x,\x)\le z(J)$ established in Theorem \ref{thm:PapMon}(b) thus
means that the PI of $\m^J_\L$ at a configuration $\x\in\cX_\L$ is not
larger than the PI of $\p^{z(J)}_\L$ evaluated at any larger
configuration $\eta\in\cX_\L$, as long as $x\notin\eta$. This, however,
is precisely the hypothesis of the point-process counterpart of the
well-known FKG-Holley-Preston inequality for lattice systems. This
continuous counterpart was obtained first by Preston \cite{Pr}; an
alternative simplified proof can be found in \cite[Theorem 1.1]{GK}. It
asserts that, under the above condition on the PI's,
$\m^J_\L\preceq\p^{z(J)}_\L$, as required.

To get rid of the locality restriction we argue as follows. For any
compact $\L$, a celebrated theorem of Strassen \cite{Strassen} provides
us with a probability measure $m_\L$ on $\cX\times\cX$ having marginals
$\m^J_\L$ resp. $\p^{z(J)}_\L$ and being supported on the set
$D:=\{(\x,\eta)\in\cX\times\cX:\x\sub\eta\}$. Note that $D$ is closed
when $\cX\times\cX$ is equipped with the product of the vague topology
on $\cX$. By a standard compactness criterion for point processes
\cite[Proposition 9.1.V]{DV}, the measures $m_\L$ admit a weak limiting
measure $m$ as $\L$ increases to $\R^d$. By construction, $m$ has
marginals $\m^J$ resp. $\p^{z(J)}$, and $m(D)=1$ because $D$ is closed.
This implies that $\m^J\preceq\p^{z(J)}$.\EndProof

\Proof[ of Corollary \ref{cor:NoPerc}]
As the event $\{\exists \text{ infinite cluster of }B_R(\cdot))\}$ is
increasing, the result follows immediately from Corollary
\ref{cor:PoissonDom}.\EndProof

%:  Pf Thm 3.7
Finally we provide the proofs of Theorems \ref{thm:Sigma} and  \ref{thm:Gibbs}.

\medskip
\Proof[ of Theorem \ref{thm:Sigma}]
We only need to show that $\m^J$ satisfies condition $(\S_\l)$ with CPI
\eqref{eq:cJ-lim}; the remaining assertions then follow from Theorem
\ref{thm:PapMon} and Remark \ref{rem:MWM}(c). Let $(\D_n)$ be any
increasing sequence in $\cB_0$ exhausting $\R^d$, $\L\in\cB_0$ a fixed
set, and $n$ so large that $\L\sub\D_n$. Consider the product space
$\cX_\L\times\cX$, equipped with the probability measure
$\n_\L^J:=\p_\L^1\otimes\m^J$ and the $\s$-algebras $\cG_n=
(\cF|_{\cX_\L})\otimes\cF_{\D_n}$. Now, Theorem \ref{thm:PapMon}(a)
asserts that, on $\cG_n$, the restriction of $\CCM[\m^J]$ to
$\cX_\L\times\cX$ is absolutely continuous with respect to $\n_\L^J$
with Radon-Nikodym density $R_n:=e^{|\L|}\,\hat c^J_{\D_n}$. The
sequence $(R_n)$ is therefore a nonnegative martingale relative to
$\n_\L^J$. In view of Theorem \ref{thm:PapMon}(b), $R_n$ satifies the
uniform bound $R_n(\a,\x)\le S(\a):= e^{|\L|}\,z(J)^{|\a|}$, and $S$ is
integrable relative to $\p_\L^1$. This means that $R_n$ converges
$\n_\L^J$-almost surely and in $L^1(\n_\L^J)$-norm to a limit $R$. By
the norm-convergence,  $R$ is a Radon-Nikodym density of $\CCM[\m^J]$
relative to $\n_\L^J$ on the limiting $\s$-algebra $\s(\bigcup_n\cG_n)=
(\cF|_{\cX_\L})\otimes\cF$. Finally we replace $\p_\L^1$ with $L_\L$ by
dropping the constant factor $e^{|\L|}$,  and use the fact that
$\cX_\L\uparrow\cX_0$ as $\L \uparrow\R^d$. We then get the desired
result that $\CCM[\m^J]\ll L\otimes\m^J$ with density
\eqref{eq:cJ-lim}.\EndProof

%:  Pf Thm 3.9
\Proof[ of Theorem \ref{thm:Gibbs}] (a) Consider the CPI
\[
\hat c^J(\a,\x) = \lim_{n\to\infty} \hat c^J_{\D_n}(\a,\x_{\D_n})\,.
\]
that occurs in \eqref{eq:cJ-lim}; we know that this limit exists for
$L\otimes\m^J$-almost all $(\a,\x)\in\cX_0\times\cX$. The monotonicity
result of Theorem \ref{thm:PapMon}(b) implies that, for each
$\L\in\cB_0$,
\[
\hat c^J(\a,\x) \le \liminf_{n\to\infty} \hat c^J_{\D_n}(\a,\x_\L)\,.
\]
Now, Proposition \ref{prop:local_convergence} asserts that 
\[
\hat c^J_{\D_n}(\a,\x_\L)\to \hat c^J_*(\a,\x_\L):= \det J(\a\x_\L,\a\x_\L)/\det J(\x_\L,\x_\L)
\]
as $n\to\infty$, provided that $\det J(\x_\L,\x_\L)>0$. In view of 
\eqref{eq:density_function} and \eqref{eq:det-monotone}, the last
proviso holds true for $\m^J$-almost all $\x$. Hence $\hat c^J(\a,\x)
\le c^J_*(\a,\x_\L)$ for $L\otimes\m^J$-almost all $(\a,\x)$. As we have
noticed after \eqref{eq:candidate}, $c^J_*(\a,\x_\L)$ is a decreasing
function of $\L$, and its limit is equal to $c^J_*(\a,\x)$ by
definition. This completes the proof of statement (a).

(b) Fix any $\L\in\cB_0$, $\a\in\cX_\L$, and consider the function $\hat c^J_*(\a,\cdot)$. Let $W(\a,\cdot)$
be the union of all clusters of $B_R(\cdot)$ hitting $\a$. If $\x\in\cX$
is such that $W(\a,\x)$ is finite then $J(x,y)=0$ for all
$x\in\a\x^\sharp$ and $y\in\x^\flat$, where $\x^\sharp= \x_{W(\a,\x)}$
and $\x^\flat=\x_{\R^d\sm W(\a,\x)}$. Hence, for any bounded $\D\supset
W(\a,\x)$, the matrix $J(\a\x_\D,\a\x_\D)$ consists of the two diagonal
blocks $J(\a\x^\sharp,\a\x^\sharp)$ and $J(\x^\flat_\D,\x^\flat_\D)$. Thus 
\[
\det J(\a\x_\D,\a\x_\D)=
\det J(\a\x^\sharp,\a\x^\sharp) \det J(\x^\flat_\D,\x^\flat_\D)\,.
\] 
Dividing this by the analoguous equation for $\a=\emptyset$ we arrive at
the second identity in \eqref{eq:cJ-subcr}. In particular, it follows
that $c^J_*(\a,\cdot)$ is continuous on the set of configurations having
no infinite cluster, and thus is continuous almost everywhere.

Consider now the DPP's $\m^{J_\D}$ relative to $J_\D=P_\D J P_\D$,
$\L\sub\D\in\cB_0$. According to Theorem 6.17 of \cite{ST1}, $\m^{J_\D}$
converges weakly to $\m^J$ as $\D$ increases to $\R^d$. Let $f$ be any
bounded continuous function on $\cX_0\times\cX$ such that $f(\a,\x)=0$
unless $\a\sub\L$ and $|\a\x_\L|\le k$ for some number $k$. The
portmanteau theorem \cite[Proposition A2.3.V]{DV} then implies that
\[
\int \hat c^J_*\, f\, d(L_\L\otimes\m^J)
= \lim_{\D\ua\R^d}\int\hat c^J_* \, f \,d (L_\L\otimes\m^{J_\D})\,. 
\]
But $\m^{J_\D}$ is supported on $\cX_\D$, and $\hat c^J_*=\hat
c^{J_\D}_*$ on $\cX_\L\times\cX_\D$. Moreover, it is evident from the 
proof  of statement (a) that $\hat c^{J_\D}_*$ is a CPI of $\m^{J_\D}$.
Eq. \eqref{eq:compPapint} thus shows that 
\[
\int\hat c^J_* \, f \,d (L_\L\otimes\m^{J_\D}) = 
\int \m^{J_\D}(d\x ) \sum_{\a\in\cX_0:\,\a\subset \x } f(\a, \x \sm \a)\,. 
\]
The integrand on the right-hand side is still a bounded continuous
function of $\x$. Letting $\D\ua\R^d$  we thus find that
\[
\int \hat c^J_*\,f d(L\otimes\m^J)= \int  f\, d\CCM[\m^J]\,.
\] 
As $f$, $k$ and $\L$ were arbitrarily chosen, it follows that $\hat
c^J_*$ is a CPI of $\m^J$. \EndProof

\small
\vskip 0.5 true cm \noindent{\it Acknowledgments}.  H.J.Y. would like to
thank  Prof. T. Shirai and Prof. Y. Takahashi for inviting him to
Kanazawa University. Part of this work was done during this stay. We
also thank Prof. Y. M. Park for informing us about reference \cite{OP}.
H.J.Y. was supported by Korea Research Foundation Grant
(KRF-2002-015-CP0038).

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