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11 pages
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Cluster expansion, correlation functions
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\begin{document}


{\hfill\small Moscow Mathematical Journal {\bf 4} 511--522 (2004)}
\vspace{2mm}

\title{Cluster expansions \& correlation functions}

\author{Daniel Ueltschi}

\address{Daniel Ueltschi \hfill\newline
Department of Mathematics \hfill\newline
University of Arizona \hfill\newline
Tucson, AZ 85721, USA\hfill\newline
{\small\rm\indent http://math.arizona.edu/$\sim$ueltschi}}
\email{ueltschi@math.arizona.edu}

\maketitle

\vspace{-5mm}

\begin{centering}
{\small\it
Department of Mathematics, University of California, Davis\\
}
\end{centering}

\vspace{2mm}


\begin{quote}
{\small
{\bf Abstract.}
A cluster expansion is proposed, that applies to both continuous and discrete systems. The
assumption for its convergence involves an extension of the neat Koteck\'y-Preiss
criterion.
Expressions and estimates for correlation functions are also presented. The results are applied to
systems of interacting classical and quantum particles, and to a lattice polymer model.

\vspace{1mm}

}  % end \small

\vspace{1mm}
\noindent
{\footnotesize {\it Keywords:} Cluster expansion, correlation functions.}

\vspace{1mm}
\noindent
{\footnotesize {\it 2000 Math.\ Subj.\ Class.:} 82B05, 82B10.}
\end{quote}

\vspace{1mm}


\section{Introduction}

Cluster expansions were introduced at the dawn of Statistical Mechanics for the study of high
temperature gases of interacting particles. They constitute a powerful perturbative method that is
suitable for geometrically large systems, such as encountered in Statistical Physics. Numerous
articles have contributed to the subject; a list of relevant publications is
\cite{GK,Cam,Bry,KP,Pfi,Dob,BZ,Mir,Far} and
references therein. Cluster expansions can now be found in standard books, see Chapter 4 of Ruelle
\cite{Rue}, or Chapter V of Simon \cite{Sim}. They apply to continuous systems such as classical or
quantum models of interacting particles, and also to discrete systems such as polymer models, spin
models (with discrete or continuous spin spaces), or lattice particle models. The methods
for treating these various situations share many similarities, but a somewhat different cluster
expansion was so far required in each case.

The exposition of the cluster expansion is often intricate, and the paper of Koteck\'y and
Preiss \cite{KP} should be singled out for proposing a clear and concise theorem, that involves a
neat criterion for the convergence of the expansion. This theorem applies to discrete systems only,
and its rather difficult proof was subsequently simplified in \cite{Dob,BZ,Mir}.
Explicit expressions for the contribution of cluster terms can be found in the article of Pfister
\cite{Pfi}; this helps
clarifying the situation and allows for computations of lowest order terms. The condition for the
convergence is the same as in \cite{KP} in the case where polymers are subsets of a lattice.

The goal of this paper is to present a general theorem that applies to both continuous and discrete
systems. The condition for the convergence is given by an extension of the criterion of Koteck\'y
and Preiss, see Equation \eqref{KPcrit1} below, and the contribution of cluster terms involves explicit
expressions. Furthermore, we derive expressions and
estimates for correlation functions.

The theorems presented here are illustrated in Section \ref{secill} in three different situations,
namely classical and quantum gases of interacting particles, and lattice polymer models.



\section{Cluster expansions}

Let $(\bbA, \caA, \mu)$ be a measure space; $\mu$ is a complex measure and 
$|\mu|(\bbA)<\infty$, where $|\mu|$ is the total variation (absolute value) of $\mu$.
Let $\zeta$ be a complex measurable symmetric function on $\bbA\times\bbA$.
The {\it partition function} $Z$ is defined by
\be
\label{deffpart}
Z = \sum_{n\geq0} \frac1{n!} \int\dd\mu(A_1) \dots \int\dd\mu(A_n) \prod_{1\leq
i<j\leq n} \bigl( 1 + \zeta(A_i,A_j) \bigr).
\ee
The term $n=0$ of the sum is understood to be 1.

We denote by $\caG_n$ the set of all (unoriented)
graphs with $n$ vertices, and $\caC_n \subset \caG_n$ the set of connected graphs of $n$ vertices.
We introduce the following combinatorial function on finite sequences $(A_1,\dots,A_n)$ of $\bbA$:
\be
\varphi(A_1,\dots,A_n) = \begin{cases} 1 & \text{if } n=1 \\ \frac1{n!} \sum_{G \in \caC_n} \prod_{(i,j)
\in G} \zeta(A_i,A_j) & \text{if } n\geq2. \end{cases}
\ee
The product is over edges of $G$. A sequence
$(A_1,\dots,A_n)$ is a {\it cluster} if the graph with $n$ vertices and an edge between $i$ and $j$ whenever
$\zeta(A_i,A_j) \neq 0$, is connected.

The cluster expansion allows to express the logarithm of the partition function as a sum (or an
integral) over clusters.


\begin{theorem}[Cluster expansion]\hfill
\label{thmclexp}

\noindent
Assume that $|1+\zeta(A,A')|\leq1$ for all $A,A'\in\bbA$, and that there exists a nonnegative
function $a$ on $\bbA$ such that for all $A\in\bbA$,
\be
\label{KPcrit1}
\int\dd|\mu|(A') \, |\zeta(A,A')| \, \e{a(A')} \leq a(A),
\ee
and $\int\dd|\mu|(A) \e{a(A)} < \infty$. Then we have
$$
Z = \exp\Bigl\{ \sum_{n\geq1} \int\dd\mu(A_1) \dots \int\dd\mu(A_n) \, \varphi(A_1,\dots,A_n) \Bigr\}.
$$
Combined sum and integrals converge absolutely. Furthermore, we have for all $A_1\in\bbA$
\be
\label{b1}
1 + \sum_{n\geq2} n \int\dd|\mu|(A_2) \dots \int\dd|\mu|(A_n) \, |\varphi(A_1,\dots,A_n)| \leq
\e{a(A_1)}.
\ee
\end{theorem}


The rest of the section is devoted to the proof of this theorem; the reader interested in results only should jump
to Section \ref{seccorrfcts} that discusses correlation functions.


\begin{proof}
Another inequality turns out to be helpful. Multiplying both sides of \eqref{b1} by $|\zeta(A,A_1)|$
and integrating over $A_1$, we find using \eqref{KPcrit1}
\be
\label{b2}
\sum_{n\geq1} \int\dd|\mu|(A_1) \dots \int\dd|\mu|(A_n) \Bigl( \sum_{i=1}^n |\zeta(A,A_i)|
\Bigr) |\varphi(A_1,\dots,A_n)| \leq a(A)
\ee
for all $A\in\bbA$.

The strategy is to show inductively that \eqref{KPcrit1} implies \eqref{b1}.
Convergence of the cluster expansion follows and allows to prove Theorem \ref{thmclexp}.

We prove that the following holds for all $N$,
\be
1 + \sum_{n=2}^N n \int\dd|\mu|(A_2) \dots \int\dd|\mu|(A_n) |\varphi(A_1,\dots,A_n)| \leq
\e{a(A_1)}.
\ee
The case $N=1$ is clear and we consider now any $N$. The left side is equal to
\be
\label{uneborne}
1 + \sum_{n=2}^N \int\dd|\mu|(A_2) \dots \int\dd|\mu|(A_n) \frac1{(n-1)!} \Bigl|
\sum_{G \in \caC_n} \prod_{(i,j)\in G} \zeta(A_i,A_j) \Bigr|.
\ee
Let us focus on the sum over connected graphs $G$.
Removing all edges of $G$ with one endpoint on 1 yields a possibly disconnected graph $G'$. Let $(G_1,
\dots, G_k)$ be a sequence of connected graphs where $G_i$ has set of vertices $V_i$, $V_1 \cup \dots
\cup V_k = \{2,\dots,n\}$, and $V_i \cap V_j = \emptyset$ if $i \neq j$. Each sequence determines a graph
$G'$, and to each $G'$ corresponds $k!$ such sequences. See Fig.\ \ref{figcluster} for an illustration. Therefore
\bfig
\epsfxsize=70mm
\centerline{\epsffile{figcluster.eps}}
\figtext{
\writefig	8.2	5.5	{$G_1$}
\writefig	10.5	2.0	{$G_2$}
\writefig	3.8	1.6	{$G_3$}
}
\caption{Illustration for $G$, $G'$, and $(G_1,\dots,G_k)$.}
\label{figcluster}
\efig
\be
\Bigl| \sum_{G \in \caC_n} \prod_{(i,j)\in G} \zeta(A_i,A_j) \Bigr| \leq
\sum_{k\geq1} \frac1{k!} \Bigl| \sum_{(G_1,\dots,G_k)} \prod_{\ell=1}^k \Bigl\{ \prod_{(i,j)\in
G_\ell} \zeta(A_i,A_j) \sum_{G_\ell'} \prod_{(i,j)\in G_\ell'} \zeta(A_i,A_j) \Bigr\} \Bigr|.
\ee
The sum over $G_\ell'$ runs over nonempty sets of edges with one endpoint on
1 and one endpoint in $V_\ell$ ($i=1$ in the last product). We have
\be
\label{combi}
\sum_{G_\ell'} \prod_{(i,j)\in G_\ell'} \zeta(A_i,A_j) = \prod_{i\in V_\ell} \bigl( 1 +
\zeta(A_1,A_i) \bigr) - 1.
\ee
Using
\be
\label{combi2}
\prod_{i=1}^n (1+\alpha_i) - 1 = \Bigl[ \prod_{i=1}^{n-1} (1+\alpha_i) - 1 \Bigr] (1+\alpha_n) +
\alpha_n,
\ee
one easily proves inductively that the absolute value of \eqref{combi} is smaller than $\sum_{i\in V_\ell}
|\zeta(A_1,A_i)|$.

The sum over sequences $(G_1, \dots, G_k)$ can be done by first choosing the respective
numbers of vertices $m_1, \dots, m_k$ whose sum is $n-1$, then by summing over partitions of
$\{2,\dots,n\}$ in sets $V_1,\dots,V_k$ with $|V_i|=m_i$, and
finally by choosing connected graphs for each set of vertices. The number of partitions is
$\frac{(n-1)!}{m_1! \dots m_k!}$. Then \eqref{uneborne} can be bounded by
\bm
1 + \sum_{n=2}^N \sum_{k\geq1} \frac1{k!} \sumtwo{m_1,\dots,m_k\geq1}{m_1+\dots+m_k=n-1}
\prod_{\ell=1}^k \Bigl[ \int\dd|\mu|(A_1') \dots \int\dd|\mu|(A_{m_\ell}') \\
|\varphi(A_1',\dots,A_{m_\ell}')| \sum_{i=1}^{m_\ell} |\zeta(A_1,A_i')| \Bigr]. \nn
\end{multline}
We can sum over $n$; the constraint $m_1+\dots+m_k \leq N-1$
can be relaxed into $m_\ell \leq N-1$ for all $\ell$. Using \eqref{b2} with $n \leq N-1$,
we obtain the bound
\be
1 + \sum_{k\geq1} \frac1{k!} [a(A_1)]^k = \e{a(A_1)}.
\ee
This proves inequality \eqref{b1}. Absolute convergence of the cluster expansion follows from
\eqref{b1} and summability of $\e{a(A)}$.

The rest of the proof is standard and consists in showing that clusters are
indeed the terms of the expansion of the logarithm of the partition function. The idea is to expand
$Z$ so as to recognize an exponential.
\be
Z = 1 + \sum_{n\geq1} \frac1{n!} \int\dd\mu(A_1) \dots \int\dd\mu(A_n) \sum_{G
\in \caG_n} \prod_{(i,j)\in G} \zeta(A_i,A_j).
\ee
The sum over $n$ converges absolutely.
We proceed as above and consider sequences of connected graphs $(G_1,\dots,G_k)$ whose sets of
vertices form a partition of $\{1,\dots,n\}$. Summing first over the number of vertices of each
partition, then over partitions, we get
\ba
Z &= 1 + \sum_{n\geq1} \sum_{k\geq1} \frac1{k!}
\sumtwo{m_1,\dots,m_k\geq1}{m_1+\dots+m_k=n} \frac1{m_1! \dots m_k!} \nn\\
&\hspace{35mm} \prod_{\ell=1}^k \Bigl\{
\int\dd\mu(A_1) \dots \int\dd\mu(A_{m_\ell}) \sum_{G \in \caC_{m_\ell}} \prod_{(i,j)\in G} \zeta(A_i,A_j) \Bigr\} \\
&= 1 + \sum_{n\geq1} \sum_{k\geq1} \frac1{k!} \sumtwo{m_1,\dots,m_k\geq1}{m_1+\dots+m_k=n}
\prod_{\ell=1}^k \int\dd\mu(A_1) \dots \int\dd\mu(A_{m_\ell}) \, \varphi(A_1,\dots,A_{m_\ell}). \nn
\end{align}
Absolute convergence of the clusters allows to remove the sum over $n$, and
this completes the proof.
\end{proof}



\section{Correlation functions}
\label{seccorrfcts}

An advantage of the cluster expansion is to characterize correlation functions. The relevant
general expressions are
\ba
\label{corr}
Z(A_1,\dots,A_m) &= \sum_{n\geq m} \frac1{(n-m)!} \int\dd\mu(A_{m+1}) \dots \int\dd\mu(A_n)
\prod_{1\leq i<j\leq n} \bigl( 1 + \zeta(A_i,A_j) \bigr); \\
\hat Z(A_1,\dots,A_m) &= \sum_{n\geq m} \frac{n!}{(n-m)!}
\int\dd\mu(A_{m+1}) \dots \int\dd\mu(A_n) \, \varphi(A_1,\dots,A_n).
\label{defcorr}
\end{align}
It is understood that in both expressions, the case $n=m$ corresponds to taking the integrand
without integrating on polymers.


\begin{theorem}[Correlation functions]\hfill
\label{thmcorr}\nopagebreak

\noindent
Under the same assumptions as in Theorem \ref{thmclexp}, we have
$$
\frac{Z(A_1,\dots,A_m)}Z = \sum_{\{V_1,\dots,V_k\}} \prod_{j=1}^k \hat Z \bigl( (A_i)_{i \in V_j}
\bigr)
$$
where the sum is over partitions of $\{1,\dots,m\}$, i.e.\ $V_1 \cup \dots \cup V_k = \{1,\dots,m\}$,
and $V_i \cap V_j = \emptyset$ if $i\neq j$.
\end{theorem}


The next result deals with estimates of correlations. To exhibit a suitable decay, an efficient
strategy is to establish the criterion \eqref{KPcrit1} in a stronger form. We consider a nonnegative
function $b$ on $\bbA$, and a nonnegative symmetric function $c$ on $\bbA \times \bbA$ (both can be
identically zero, but the larger they are the better). We introduce the notation
\ba
&\mu_b(A) = \mu(A) \e{b(A)} \nn\\
&\zeta_c(A,A') = \zeta(A,A') \e{c(A,A')} \nn\\
& c(A_1,\dots,A_n) = \min_{G \in \caC_n} \sum_{(i,j) \in G} c(A_i,A_j) \quad \text{if } n\geq3 \nn\\
& \varphi_c(A_1,\dots,A_n) = \varphi(A_1,\dots,A_n) \e{c(A_1,\dots,A_n)}. \nn
\end{align}
The utility of functions $b$ and $c$ will be illustrated in Section \ref{secill}. The following
theorem contains estimates on correlations; compare with the definition \eqref{defcorr} of $\hat
Z(A_1,\dots,A_m)$.


\begin{theorem}[Decay of correlations]\hfill
\label{thmdecay}

\noindent
Assume that $|1 + \zeta_c(A,A')| \leq 1$ for all $A,A'\in\bbA$, and that there exists a nonnegative
function $a$ on $\bbA$ such that
\be
\label{KPcrit2}
\int\dd|\mu_b|(A') \, |\zeta_c(A,A')| \, \e{a(A')} \leq a(A)
\ee
for all $A\in\bbA$. Then the following estimate holds true for all $m\geq1$, and all $A_1,\dots,A_m
\in \bbA$,
\ba
\sum_{n\geq m} \frac{n!}{(n-m)!} &\int\dd|\mu_b|(A_{m+1}) \dots \int\dd|\mu_b|(A_n) \,
|\varphi_c(A_1,\dots,A_n)| \nn\\
&\leq \exp\Bigl\{ \tfrac1{m\gamma} \bigl[ (1+\gamma)^m - 1 \bigr] \sum_{i=1}^m
a(A_i) \Bigr\} \prod_{1\leq i<j\leq m} \bigl( 1 + |\zeta_c(A_i,A_j)| \bigr). \nn
\end{align}
Here, we set
\be
\label{defgamma}
\gamma = \sup_n \sup_{A_0,\dots,A_n} \Bigl| \prod_{i=1}^n \bigl( 1 + \zeta(A_0,A_i) \bigr) - 1 \Bigr|
\ee
(clearly, $0\leq\gamma\leq2$).
\end{theorem}


Notice that the case $m=1$ is
\be
\label{b3}
1 + \sum_{n\geq2} n \int\dd|\mu_b|(A_2) \dots \int\dd|\mu_b|(A_n) \, |\varphi_c(A_1,\dots,A_n)|
\leq \e{a(A_1)}
\ee
which is reminiscent of \eqref{b1}. Multiplying both sides of \eqref{b3} by $|\zeta_c(A,A_1)|$ and
integrating over $A_1$, we obtain from \eqref{KPcrit2}
\be
\label{b4}
\sum_{n\geq1} \int\dd|\mu_b|(A_1) \dots \int\dd|\mu_b|(A_n) \Bigl( \sum_{i=1}^n |\zeta_c(A,A_i)|
\Bigr) |\varphi_c(A_1,\dots,A_n)| \leq a(A)
\ee
for all $A\in\bbA$. \eqref{b3} and \eqref{b4} can be proved exactly the same way as \eqref{b1} and
\eqref{b2}.

We turn to the proofs of Theorems \ref{thmcorr} and \ref{thmdecay}. Readers interested in applications
should jump to Section \ref{secill}. We start with the proof of Theorem \ref{thmdecay} as it will
imply the convergence of the terms appearing in Theorem \ref{thmcorr}. We assume that \eqref{b4} has
been established.


\begin{proof}[Proof of Theorem \ref{thmdecay}]
A connected graph $G$ on $\{1,\dots,n\}$ can be decomposed into a graph $G'$ on $\{1,\dots,m\}$; a
partition $\{W_1,\dots,W_k\}$ of $\{m+1,\dots,n\}$; connected graphs $G_1,\dots,G_k$ where $G_i$
has set of vertices $W_i$; non-empty subsets $V_1,\dots,V_k$ of $\{1,\dots,m\}$; non-empty sets of edges
between $W_j$ and each vertex of $V_j$. This is illustrated in Fig.\ \ref{figcluster2}. Conversely,
choosing these graphs and sets of vertices yields a graph on $\{1,\dots,n\}$; it is
not necessarily connected, but we obtain an upper bound by dropping this
constraint. Therefore we get
\bfig
\epsfxsize=70mm
\centerline{\epsffile{figcluster2.eps}}
\figtext{
\writefig	3.4	3.5	{$G_1$}
\writefig	3.4	2.9	{$W_1$}
\writefig	10.7	2.9	{$G_2$}
\writefig	10.8	2.3	{$W_2$}
\writefig	7.2	1.9	{$G'$}
\writefig	6.4	1.4	{$\{1,\dots,m\}$}
}
\caption{}
\label{figcluster2}
\efig
\bm
\Bigl| \sum_{G\in\caC_n} \prod_{(i,j)\in G} \zeta(A_i,A_j) \Bigr| \e{c(A_1,\dots,A_n)} \leq
\sum_{G' \in \caG_m} \prod_{(i,j)\in G'} |\zeta_c(A_i,A_j)| \sum_{\{W_1,\dots,W_k\}} \\
\prod_{\ell=1}^k \biggl\{ \Bigl| \sum_{G_\ell \in \caC(W_\ell)} \prod_{(i,j)\in G_\ell} \zeta(A_i,A_j)
\Bigr| \e{c((A_i)_{i\in W_\ell})} \\
\sumtwo{V_\ell \subset \{1,\dots,m\}}{V_\ell \neq \emptyset} \exp\bigl\{ \min_{i\in V_\ell, j\in
W_\ell} c(A_i,A_j) \bigr\} \prod_{i \in V_\ell} \Bigl| \prod_{j\in W_\ell} \bigl( 1 + \zeta(A_i,A_j)
\bigr) - 1 \Bigr| \biggr\}.
\end{multline}

From the definition \eqref{defgamma} of $\gamma$, we have
\ba
\prod_{i\in V_\ell} \Bigl| \prod_{j\in W_\ell} \bigl( 1 + \zeta(A_i,A_j) \bigr) - 1 \Bigr| &\leq
\frac{\gamma^{|V_\ell|-1}}{|V_\ell|} \sum_{i\in V_\ell} \Bigl| \prod_{j\in W_\ell} \bigl( 1 +
\zeta(A_i,A_j) \bigr) - 1 \Bigr| \nn\\
&\leq \frac{\gamma^{|V_\ell|-1}}{|V_\ell|} \sum_{i\in V_\ell, j\in W_\ell} |\zeta(A_i,A_j)|.
\end{align}
We used \eqref{combi} and \eqref{combi2} for the second inequality. Furthermore
\be
\sumtwo{V_\ell \subset \{1,\dots,m\}}{V_\ell \neq \emptyset} \frac{\gamma^{|V_\ell|-1}}{|V_\ell|}
\sum_{i\in V_\ell, j\in W_\ell} |\zeta(A_i,A_j)| = \frac1{m\gamma} \bigl[ (1+\gamma)^m - 1 \bigr]
\sum_{i=1}^m \sum_{j\in W_\ell} |\zeta(A_i,A_j)|.
\ee
Then
\bm
\sum_{n\geq m} \frac1{(n-m)!} \int\dd|\mu_b|(A_{m+1}) \dots \int\dd|\mu_b|(A_n) \Bigl| \sum_{G\in\caC_n}
\prod_{(i,j)\in G} \zeta(A_i,A_j) \Bigr| \e{c(A_1,\dots,A_n)} \\
\leq \sum_{G'\in\caG_m} \prod_{(i,j)\in G'}
|\zeta_c(A_i,A_j)| \sum_{n\geq m} \sum_{k\geq0} \frac1{k!}
\sumtwo{n_1,\dots,n_k\geq1}{n_1+\dots+n_k=n-m} \\
\prod_{\ell=1}^k \biggl\{ \frac1{n_\ell!} \int\dd|\mu_b|(A_1') \dots
\int\dd|\mu_b|(A_{n_\ell}') \Bigl| \sum_{G_\ell \in \caC_{n_\ell}} \prod_{(i,j)\in G_\ell}
\zeta(A_i,A_j) \Bigr| \e{c(A_1',\dots,A_{n_\ell}')} \\
\frac1{m\gamma} \bigl[ (1+\gamma)^m - 1 \bigr] \sum_{i=1}^m \sum_{j=1}^{n_\ell} |\zeta_c(A_i,A_j)|
\biggr\}.
\end{multline}
The case $k=0$ of the right side should be understood as the case $n=m$ of the left side. The expression
inside brackets in
the right side converges because of \eqref{b4}; this allows to sum over $n$, and we get Theorem
\ref{thmdecay}.
\end{proof}




\begin{proof}[Proof of Theorem \ref{thmcorr}]
We expand the product in \eqref{corr} and we obtain a sum over graphs of $\caG_n$. A graph of
$\caG_n$ can be written as a sequence of graphs $(G_1,\dots,G_k,G')$, where each $G_i$ is connected
and has at least one vertex in $\{1,\dots,m\}$, and the vertices of $G'$ are in $\{m+1,\dots,n\}$.

Let $W_i$ be the set of vertices of $G_i$, $V_i = W_i \cap \{1,\dots,m\}$, and $W'$ be the set of
vertices of $G'$. $\{V_1,\dots,V_k\}$ is a partition of $\{1,\dots,m\}$ and $W' \subset
\{m+1,\dots,n\}$. Furthermore, let $n_j = |W_j \setminus V_j|$, $1\leq j\leq k$, and $p = |W'|$.

The number of partitions $\{W_1,\dots,W_k,W'\}$ of $\{1,\dots,n\}$ corresponding to given
$n_1$, \dots, $n_k$, $p$, and $\{V_1,\dots,V_k\}$, is equal to $\frac{(n-m)!}{p! \prod_i n_i!}$.
Therefore
\bm
Z(A_1,\dots,A_m) = \sum_{\{V_1,\dots,V_k\}} \sum_{n\geq m} \sumtwo{n_1\geq0, \dots,
n_k\geq0}{n_1+\dots+n_k \leq n-m} \prod_{\ell=1}^k \Bigl[ \frac{(|V_\ell|+n_\ell)!}{n_\ell!}
\int\dd\mu(A_1') \dots \int\dd\mu(A_{n_\ell}') \\
\varphi \bigl( (A_i)_{i\in V_\ell}, A_1', \dots,
A_{n_\ell}' \bigr) \Bigr] \, \frac1{p!} \int\dd\mu(A_1') \dots \int\dd\mu(A_p') \sum_{G\in\caG_p}
\prod_{(i,j) \in G} \zeta(A_i',A_j'),
\end{multline}
where $p = n - m - \sum_{i=1}^k n_i$. All terms converge absolutely and uniformly in $n$ because of
Theorems \ref{thmclexp} and \ref{thmdecay}; this allows to perform the sum
over $n$, and we obtain Theorem \ref{thmcorr}.
\end{proof}




\section{Illustrations}
\label{secill}

The first two examples are classical interacting particles and lattice polymers --- they are well-known
but nevertheless constitute nice illustrations of the results above. The last example deals with
quantum interacting particles and is more involved.


\subsection{Classical interacting gas}

We consider a gas of particles that are subject to pair interactions. Let $\bbA = \Lambda$ be a bounded
subset of $\bbR^d$ and $\dd\mu(x) = z \dd x$, where $z \in \bbR_+$ is the fugacity and $\dd x$ is the
Lebesgue measure. Let $\beta$ be the inverse temperature and $U(x-y) \geq 0$ represent the interactions between
particles at positions $x,y\in\Lambda$. Setting $\zeta(x,y) = \e{-\beta U(x-y)}-1$, the partition
function is given by \eqref{deffpart}. We choose $a(x)=1$ and we easily check that the
criterion \eqref{KPcrit1} holds whenever
\be
\label{condconv}
z \int\dd x (1-\e{-\beta U(x)}) \leq \e{-1}.
\ee
This condition for the convergence of the cluster expansion is well-known, see \cite{Rue} Chapter 4.
One then obtains an expression for the thermodynamic pressure $p(\beta,z)$, namely
\be
\beta \, p(\beta,z) = z + \sum_{n\geq1} z^{n+1} \int\dd x_1 \dots \int\dd x_n \varphi(0,x_1,\dots,x_n)
\ee
where integrals are over $\bbR^d$. This expression is absolutely convergent because of \eqref{b1}.
Furthermore the function $\varphi(\cdot)$ is analytic in $\beta$; by Vitali convergence theorem (see e.g.\
\cite{Sim} Theorem V.2.7 in the context of statistical physics), $p(\beta,z)$ is analytic in $\beta,z$
when the condition \eqref{condconv} is satisfied.

We study now the correlations. For bounded $\Lambda$, we consider the functions
\ba
&\rho_1(x_1) = \frac1Z \sum_{n\geq1} \frac{z^n}{(n-1)!} \int\dd x_2 \dots \int\dd x_n
\prod_{1\leq i<j\leq n} \bigl( 1 + \zeta(x_i,x_j) \bigr), \nn\\
&\rho_2(x_1,x_2) = \frac1Z \sum_{n\geq2} \frac{z^n}{(n-2)!} \int\dd x_3 \dots \int\dd x_n
\prod_{1\leq i<j\leq n} \bigl( 1 + \zeta(x_i,x_j) \bigr). \nn
\end{align}
By Theorem \ref{thmcorr}, the truncated two-point correlation function $\rho_2^{\text t}(x_1,x_2)$ is
given by
\ba
\rho_2^{\text t}(x_1,x_2) &= \rho_2(x_1,x_2) - \rho_1(x_1) \rho_1(x_2) \nn\\
&= \sum_{n\geq2} n(n-1) z^n \int\dd x_3 \dots \int\dd x_n \; \varphi(x_1,\dots,x_n).
\end{align}
We use Theorem \ref{thmdecay} (with $\gamma=1$) to get a bound for the decay of correlations. Let
$b\equiv0$ and $c(x)\geq0$ satisfy the triangle inequality, such that
\be
\label{condconv2}
z \int\dd x \bigl( 1 - \e{-\beta U(x)} \bigr) \e{c(x)} \leq \e{-1}.
\ee
We have
\ba
\e{c(x_1-x_2)} |\rho_2^{\text t}(x_1,x_2)| &\leq \sum_{n\geq2} n(n-1) z^n \int\dd x_3 \dots \int\dd x_n
|\varphi(x_1,\dots,x_n)| \e{c(x_1,\dots,x_n)} \nn\\
&\leq \e3 \bigl\{ 1 + \bigl( 1 - \e{-\beta U(x_1-x_2)} \bigr) \e{c(x_1-x_2)} \bigr\}.
\end{align}
The right side converges to $\e3$ as $|x_1-x_2| \to \infty$. This shows that
\be
|\rho_2^{\text t}(x_1,x_2)| \leq \const \e{-c(x_1-x_2)}
\ee
for all functions $c$ satisfying \eqref{condconv2}.



\subsection{Polymer models}

A polymer is a connected subset of $\bbZ^d$. Let $\bbA$ be the set of polymers in a finite set $\Lambda
\subset \bbZ^d$. The measure $\mu$ is taken to be the counting measure multiplied by a weight $w(A)$
satisfying $|w(A)| \leq \e{-\eta |A|}$ with $\eta = 2\log(2d\phi) + \phi^{-1}$. Here
$\phi=\frac{\sqrt5+1}2$ is the Golden Ratio. Polymers interact through a condition
of non-intersection, that is, $\zeta(A,A')$ is $-1$ if $A\cap A' \neq \emptyset$, and is 0 otherwise.

To check the criterion \eqref{KPcrit1}, we choose $a(A) = \phi^{-1} |A|$. It is enough to consider
the case where $A=\{0\}$. If $A$ is a connected set, there exists a closed walk with nearest-neighbor
jumps whose support is $A$, and whose length is at most $2|A|$. This can be seen by induction: knowing
the walk for $A$, it is easy to construct one for $A\cup\{x\}$. The number of connected sets of
cardinality $n$ that contain the origin is therefore smaller than the number of walks of length $2n$
starting at the origin, which is equal to $(2d)^{2n}$. The left side of \eqref{KPcrit1} is
bounded by $\sum_{n\geq1} (2d)^{2n} \e{-(\eta-\phi^{-1}) n}$ and this is equal to $\phi^{-1}$.

The cluster expansion provides absolutely convergent series for thermodynamic quantities.
Many physical models can be mapped onto a polymer model, and Theorem \ref{thmdecay} typically
provides informations on correlation functions.



\subsection{Quantum interacting gas}

The description of a gas of quantum particles in a bounded domain $\Lambda \subset \bbR^d$ should start
with the state space of the system, which is the Fock space $\caF_\pm = \bigoplus_{N\geq0} P_\pm(L^2(\Lambda))^{\otimes N}$, where $P_+$ (resp.\
$P_-$) is the projector onto symmetric (resp.\ antisymmetric) functions. Next one should introduce the Hamiltonian (Laplacian for the kinetic energy of the
particles, and another operator for the interactions). One could then write down the partition
function $Z = \Tr \e{-\beta (H-\mu N)}$.

It serves better our purpose to define the model in the Feynman-Kac representation. See \cite{Gin} for
a complete description of this representation, and for the definition of the Wiener measure, to be used below. Results of this section can actually be found in \cite{Gin},
but the cluster expansion used there is very intricate.

We start with the partition function
\bm
\label{fpartquant}
Z = \sum_{N\geq0} \frac{z^N}{N!} \int_\Lambda \dd x_1 \dots \int_\Lambda \dd x_N \sum_{\pi\in S_N}
\varepsilon^{|\pi|} \int\dd W_{x_1x_{\pi(1)}}^\beta(\omega_1) \dots \int\dd
W_{x_Nx_{\pi(N)}}^\beta(\omega_N) \\
\prod_{i=1}^N \upchi_\Lambda(\omega_i) \prod_{1\leq i<j\leq N}
\exp\Bigl\{ -\int_0^\beta U \bigl( \omega_i(t) - \omega_j(t) \bigr) \dd t \Bigr\}.
\end{multline}
Here, $S_N$ is the permutation group of $N$ elements, $\varepsilon=1$ for bosons and $\varepsilon=-1$
for fermions, $|\pi|$ is the number of transpositions of $\pi$, $\upchi_\Lambda(\omega)$ is 1 if
$\omega(t) \in \Lambda$ for all $0\leq t\leq\beta$ and is 0 otherwise, and $U(x)$ represents a
pair interaction potential. For simplicity we assume that $U$ is nonnegative and summable. Recall
that the Wiener measure $\dd W_{xy}^\beta(\omega)$ satisfies the following property; if
$0<t_1<\dots<t_n<\beta$ and if $f$ is a function $\bbR^{nd} \to \bbR$, we have
\bm
\int\dd W_{xy}^\beta(\omega) f \bigl( \omega(t_1), \dots, \omega(t_n) \bigr) \\
= \int_{\bbR^d} \dd x_1 \dots
\int_{\bbR^d} \dd x_n \psi_{t_1}(x_1-x) \psi_{t_2-t_1}(x_2-x_1) \dots \psi_{\beta-t_n}(y-x_n)
f(x_1,\dots,x_n),
\end{multline}
where $\psi_t(x)$ is the Gaussian
\be
\psi_t(x) = (2\pi t)^{-d/2} \e{-x^2/2t}.
\ee

The first step is to derive an expression for the partition function that is of the form
\eqref{deffpart}.
A permutation $\pi\in S_N$ can be decomposed in $k$ cycles of lengths $\ell_1,\dots\ell_k$ whose sum is
$N$. Given $\ell_1,\dots,\ell_k$, there are
$$
\frac1{k!} \frac{N!}{\prod_{i=1}^k \ell_i}
$$
corresponding permutations. Furthermore, the integrations over $x_1,\dots,x_\ell$ and over
$\omega_1$, \dots, $\omega_\ell$, where $\omega_j$ is a path from $x_j$ to $x_{j+1}$ (with $x_{\ell+1}
\equiv x_1$), can be performed by a single integration over $x$, followed by an integration over a path
$\omega$ with the measure $W_{xx}^{\ell\beta}$. We denote $\bsomega = (\ell,x,\omega)$ where $\ell$ is
a positive integer, $x\in\Lambda$, and $\omega$ is a path $[0,\ell\beta] \mapsto \bbR^d$ with
$\omega(0) = \omega(\ell\beta) = x$. We consider the following measure $\mu$
\bm
\dd\mu(\bsomega) = \frac{z^\ell \varepsilon^{\ell+1}}\ell \, \dd x \; \dd W_{xx}^{\ell\beta}(\omega)
\;\upchi_\Lambda(\omega) \\
\prod_{0\leq m<n\leq \ell-1} \exp\Bigl\{
-\int_0^\beta U \bigl( \omega(m\beta+t) - \omega(n\beta+t) \bigr) \dd t \Bigr\}.
\end{multline}
The partition function \eqref{fpartquant} can be written as
\bm
Z = \sum_{N\geq0} \frac1{N!} \int\dd\mu(\bsomega_1) \dots \int\dd\mu(\bsomega_N) \\
\prod_{1\leq i<j\leq
N} \exp\Bigl\{ -\sum_{m=0}^{\ell_i-1} \sum_{n=0}^{\ell_j-1} \int_0^\beta U \bigl( \omega_i(m\beta+t)
- \omega_j(n\beta+t) \bigr) \dd t \Bigr\}.
\end{multline}
We obtain \eqref{deffpart} by setting
\be
\zeta(\bsomega,\bsomega') = \exp\Bigl\{ -\sum_{m=0}^{\ell-1} \sum_{n=0}^{\ell'-1} \int_0^\beta U
\bigl( \omega(m\beta+t) - \omega'(n\beta+t) \bigr) \dd t \Bigr\} - 1.
\ee

We now establish the criterion \eqref{KPcrit1}. The condition below involves $z$, $\beta$, and $U$,
see \eqref{condconvquant}, and is not the most general that can be achieved. It is enough for the
purpose of an illustration of the use of cluster expansions, however.

We take $a(\bsomega)=(-\log z)\ell$. Since $1-\e{-t} \leq t$, we have
\bm
\int\dd|\mu|(\bsomega') |\zeta(\bsomega,\bsomega')| z^{-\ell'} \leq \sum_{\ell'\geq1}
\frac1{\ell'} \int\dd x' \int\dd W_{x'x'}^{\ell'\beta}(\omega') \\
\sum_{m=0}^{\ell-1}
\sum_{n=0}^{\ell'-1} \int_0^\beta U \bigl( \omega(m\beta+t) - \omega'(n\beta+t) \bigr) \dd t.
\end{multline}
Now for all $y$ and all $0<t'<\ell'\beta$, it is not hard to see that
\be
\int\dd x' \int\dd W_{x'x'}^{\ell'\beta}(\omega') U \bigl( y - \omega'(t') \bigr) =
(2\pi\ell'\beta)^{-d/2} \int U(x) \dd x.
\ee
We get then
\be
\int\dd|\mu|(\bsomega') |\zeta(\bsomega,\bsomega')| z^{-\ell'} \leq \frac{\ell\beta}{(2\pi\beta)^{d/2}} \int U(x) \dd
x \sum_{\ell'\geq1} {\ell'}^{-d/2}.
\ee
The criterion is fulfilled if the right side of the expression above is smaller than
$(-\log z)\ell$, that is,
if
\be
\label{condconvquant}
\frac\beta{(2\pi\beta)^{d/2}} \int U(x) \dd x \sum_{\ell\geq1} \ell^{-d/2} \leq -\log z.
\ee
This assumes in particular that $z \leq 1$. The thermodynamic pressure is analytic in $z,\beta$
in the range of parameters where \eqref{condconvquant} holds, and no condensation takes place, whether
classical or of the Bose-Einstein type.

Correlations are given by `reduced density matrices' and require more efforts.



\begin{thebibliography}{99}
\vspace{-1mm}

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\end{thebibliography}

\end{document}

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 1689 2816 l 1688 2862 l 1688 2863 l 1687 2907 l 1688 2951 l
 1689 2996 l 1690 3040 l 1693 3085 l 1696 3129 l 1700 3174 l
 1705 3218 l 1711 3263 l 1717 3307 l 1724 3351 l 1733 3396 l
 1742 3439 l 1752 3483 l 1763 3526 l 1775 3568 l 1787 3610 l
 1801 3651 l 1816 3692 l 1832 3731 l 1849 3770 l 1866 3807 l
 1885 3844 l 1905 3879 l 1925 3913 l 1947 3945 l 1969 3976 l
 1992 4005 l 2017 4032 l 2042 4058 l 2068 4082 l 2094 4104 l
 2122 4124 l 2150 4142 l 2179 4158 l 2209 4172 l 2240 4184 l
 2271 4194 l 2303 4201 l 2335 4206 l 2368 4209 l 2402 4210 l
 2436 4208 l 2471 4204 l 2507 4198 l 2543 4189 l 2580 4179 l
 2618 4165 l 2656 4149 l 2695 4131 l 2734 4111 l
 cp gs col7 0.85 shd ef gr  [] 0 sd
% Polyline
 [30] 0 sd
n 9638 4850 m 9666 4861 l 9694 4870 l 9720 4875 l 9745 4878 l 9769 4878 l
 9791 4875 l 9813 4869 l 9833 4861 l 9852 4850 l 9870 4836 l
 9887 4820 l 9902 4801 l 9916 4780 l 9929 4756 l 9940 4730 l
 9950 4702 l 9959 4671 l 9966 4638 l 9972 4603 l 9976 4566 l
 9979 4527 l 9981 4486 l 9981 4444 l 9979 4401 l 9977 4356 l
 9972 4309 l 9967 4262 l 9960 4214 l 9951 4165 l 9942 4115 l
 9930 4065 l 9918 4014 l 9904 3963 l 9889 3912 l 9873 3861 l
 9856 3810 l 9837 3760 l 9818 3709 l 9797 3660 l 9775 3610 l
 9752 3561 l 9728 3513 l 9703 3465 l 9677 3418 l 9650 3372 l
 9622 3326 l 9593 3282 l 9563 3238 l 9530 3192 l 9496 3148 l
 9461 3104 l 9424 3061 l 9387 3019 l 9348 2977 l 9308 2936 l
 9267 2896 l 9225 2857 l 9183 2818 l 9139 2781 l 9095 2744 l
 9049 2709 l 9004 2674 l 8958 2641 l 8912 2609 l 8865 2579 l
 8819 2550 l 8772 2522 l 8726 2497 l 8680 2473 l 8635 2451 l
 8590 2430 l 8547 2412 l 8504 2395 l 8462 2381 l 8422 2369 l
 8383 2359 l 8346 2351 l 8310 2345 l 8276 2342 l 8244 2341 l
 8214 2342 l 8185 2345 l 8159 2351 l 8135 2358 l 8113 2369 l
 8094 2381 l 8077 2396 l 8062 2413 l 8050 2432 l 8040 2453 l
 8032 2477 l 8027 2503 l 8025 2532 l 8025 2563 l 8027 2592 l
 8032 2623 l 8038 2656 l 8047 2692 l 8057 2729 l 8069 2767 l
 8083 2808 l 8099 2850 l 8117 2894 l 8137 2940 l 8158 2987 l
 8181 3036 l 8206 3086 l 8233 3138 l 8261 3191 l 8291 3245 l
 8322 3300 l 8354 3355 l 8388 3412 l 8423 3469 l 8460 3527 l
 8497 3585 l 8535 3644 l 8574 3702 l 8614 3761 l 8655 3819 l
 8696 3877 l 8738 3934 l 8780 3991 l 8822 4047 l 8865 4102 l
 8907 4156 l 8950 4208 l 8992 4260 l 9034 4310 l 9075 4358 l
 9117 4404 l 9157 4449 l 9197 4492 l 9237 4533 l 9275 4572 l
 9313 4609 l 9350 4643 l 9386 4676 l 9421 4706 l 9455 4734 l
 9488 4759 l 9520 4782 l 9551 4803 l 9581 4821 l 9610 4837 l

 cp gs col7 0.85 shd ef gr  [] 0 sd
% Polyline
 [30] 0 sd
n 6938 3525 m 6982 3547 l 7026 3571 l 7071 3595 l 7115 3621 l 7159 3648 l
 7202 3675 l 7246 3704 l 7289 3734 l 7332 3764 l 7375 3796 l
 7417 3829 l 7459 3863 l 7501 3898 l 7542 3935 l 7582 3972 l
 7622 4010 l 7661 4048 l 7700 4088 l 7737 4129 l 7773 4170 l
 7809 4212 l 7843 4254 l 7875 4297 l 7907 4341 l 7937 4385 l
 7966 4429 l 7993 4473 l 8018 4518 l 8042 4563 l 8064 4607 l
 8084 4652 l 8103 4697 l 8119 4741 l 8134 4785 l 8146 4829 l
 8157 4872 l 8165 4915 l 8172 4958 l 8176 5000 l 8179 5041 l
 8179 5082 l 8177 5122 l 8174 5162 l 8168 5201 l 8160 5240 l
 8150 5278 l 8137 5316 l 8123 5353 l 8106 5389 l 8088 5425 l
 8068 5458 l 8047 5491 l 8023 5523 l 7998 5555 l 7971 5587 l
 7942 5618 l 7910 5649 l 7877 5680 l 7842 5711 l 7805 5742 l
 7766 5772 l 7725 5802 l 7683 5832 l 7638 5861 l 7592 5890 l
 7544 5919 l 7494 5947 l 7443 5975 l 7390 6002 l 7335 6029 l
 7279 6055 l 7222 6081 l 7163 6106 l 7104 6130 l 7043 6154 l
 6981 6177 l 6919 6200 l 6856 6221 l 6792 6242 l 6728 6262 l
 6663 6281 l 6598 6299 l 6533 6316 l 6467 6332 l 6402 6347 l
 6337 6361 l 6272 6375 l 6207 6387 l 6143 6398 l 6079 6409 l
 6016 6418 l 5954 6426 l 5892 6434 l 5831 6440 l 5770 6446 l
 5710 6450 l 5652 6453 l 5593 6456 l 5536 6457 l 5480 6458 l
 5424 6457 l 5370 6456 l 5316 6453 l 5263 6450 l 5203 6445 l
 5145 6439 l 5088 6432 l 5031 6423 l 4975 6413 l 4919 6402 l
 4864 6390 l 4809 6376 l 4755 6362 l 4702 6345 l 4649 6328 l
 4596 6310 l 4545 6290 l 4493 6269 l 4443 6247 l 4393 6223 l
 4344 6199 l 4296 6173 l 4249 6147 l 4203 6119 l 4158 6090 l
 4114 6061 l 4071 6030 l 4030 5999 l 3990 5967 l 3951 5934 l
 3913 5901 l 3877 5867 l 3843 5832 l 3809 5797 l 3778 5762 l
 3748 5727 l 3719 5691 l 3692 5655 l 3667 5619 l 3643 5582 l
 3621 5546 l 3601 5510 l 3581 5474 l 3564 5437 l 3548 5401 l
 3533 5365 l 3520 5329 l 3508 5293 l 3498 5257 l 3489 5222 l
 3481 5186 l 3475 5150 l 3470 5111 l 3466 5072 l 3463 5032 l
 3462 4992 l 3463 4952 l 3465 4912 l 3468 4872 l 3473 4831 l
 3480 4789 l 3488 4748 l 3498 4706 l 3510 4663 l 3523 4621 l
 3537 4578 l 3554 4536 l 3571 4493 l 3591 4450 l 3612 4407 l
 3634 4364 l 3658 4322 l 3683 4280 l 3709 4238 l 3737 4197 l
 3766 4156 l 3796 4116 l 3827 4076 l 3859 4038 l 3893 4000 l
 3927 3962 l 3962 3926 l 3998 3891 l 4034 3857 l 4071 3823 l
 4109 3791 l 4147 3760 l 4186 3730 l 4225 3701 l 4264 3673 l
 4305 3646 l 4345 3620 l 4386 3595 l 4428 3570 l 4470 3547 l
 4513 3525 l 4552 3505 l 4592 3486 l 4633 3468 l 4675 3450 l
 4718 3433 l 4761 3417 l 4805 3401 l 4850 3386 l 4896 3371 l
 4943 3358 l 4991 3345 l 5040 3333 l 5089 3321 l 5140 3311 l
 5191 3301 l 5243 3292 l 5295 3284 l 5349 3277 l 5403 3270 l
 5457 3265 l 5513 3261 l 5568 3258 l 5624 3255 l 5680 3254 l
 5736 3254 l 5792 3254 l 5849 3256 l 5905 3259 l 5961 3263 l
 6017 3268 l 6073 3274 l 6128 3281 l 6183 3289 l 6237 3298 l
 6291 3308 l 6344 3319 l 6397 3331 l 6449 3344 l 6500 3358 l
 6551 3373 l 6601 3389 l 6651 3405 l 6700 3423 l 6748 3441 l
 6796 3461 l 6844 3481 l 6891 3503 l
 cp gs col7 0.90 shd ef gr  [] 0 sd
7.500 slw
% Ellipse
n 3900 4800 75 75 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 4800 6000 75 75 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 4500 3900 75 75 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 7800 4800 75 75 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 6900 5700 75 75 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 9600 4500 75 75 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 9300 3600 75 75 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 8400 2700 75 75 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 2400 3900 75 75 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 2100 2400 75 75 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 3300 2400 75 75 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 2700 1200 75 75 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 4500 900 75 75 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 6300 3900 75 75 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Polyline
30.000 slw
n 3300 2400 m
 2700 1200 l gs col7 0.00 shd ef gr gs col0 s gr 
% Polyline
n 3300 2400 m
 4500 900 l gs col7 0.00 shd ef gr gs col0 s gr 
% Polyline
n 2100 2400 m
 3300 2400 l gs col7 0.00 shd ef gr gs col0 s gr 
% Polyline
n 3300 2400 m
 2400 3900 l gs col7 0.00 shd ef gr gs col0 s gr 
% Polyline
n 2400 3900 m
 2100 2400 l gs col7 0.00 shd ef gr gs col0 s gr 
% Polyline
n 3900 4800 m
 4500 3900 l gs col7 0.00 shd ef gr gs col0 s gr 
% Polyline
n 3900 4800 m
 6300 3900 l gs col7 0.00 shd ef gr gs col0 s gr 
% Polyline
n 4500 3900 m
 6300 3900 l gs col7 0.00 shd ef gr gs col0 s gr 
% Polyline
n 3900 4800 m
 4800 6000 l gs col7 0.00 shd ef gr gs col0 s gr 
% Polyline
n 6900 5700 m
 7800 4800 l gs col7 0.00 shd ef gr gs col0 s gr 
% Polyline
n 8400 2700 m
 9300 3600 l gs col7 0.00 shd ef gr gs col0 s gr 
% Polyline
n 9300 3600 m
 9600 4500 l gs col7 0.00 shd ef gr gs col0 s gr 
% Polyline
 [120] 0 sd
n 3300 2400 m
 4500 3900 l gs col0 s gr  [] 0 sd
% Polyline
 [120] 0 sd
n 3300 2400 m
 6300 3900 l gs col0 s gr  [] 0 sd
% Polyline
 [120] 0 sd
n 2400 3900 m
 3900 4800 l gs col0 s gr  [] 0 sd
% Polyline
 [120] 0 sd
n 6300 3900 m
 8400 2700 l gs col0 s gr  [] 0 sd
% Polyline
 [120] 0 sd
n 7800 4800 m
 9600 4500 l gs col0 s gr  [] 0 sd
$F2psEnd
rs

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