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%% A LECTURE ON CLUSTER EXPANSIONS
%% Salvador Miracle-Sole
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\begin{document}
\def\supp{\mathop{\rm supp}}
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\def\Exp{\mathop{\rm Exp}}
\def\Log{\mathop{\rm Log}}
\title{\Large A LECTURE ON CLUSTER EXPANSIONS}
\author{Salvador Miracle-Sole}
\date{\footnotesize Centre de Physique Th\'eorique, CNRS
Luminy, Case 907,\break F-13288 Marseille Cedex 9, France.
E-mail: miracle@cpt.univ-mrs.fr}
\maketitle
\begin{abstract}
\footnotesize\noindent
A short exposition with complete proofs of the
theory of cluster expansions for an abstract polymer system
is presented.
\end{abstract}
\bigskip
Consider a countable set ${\cal P}$,
whose elements will be called {polymers}.
Let ${\cal I}$ be a subset of
${\rm P}_2({\cal P})$, the set of all subsets of ${\cal P}$
with two elements.
We say that two polymers $\gamma$ and $\gamma'$ are
{incompatible} if $\{\gamma,\gamma'\}\in{\cal I}$ or
if $\gamma=\gamma'$,
and we will also write $\gamma\not\sim\gamma'$.
If $\{\gamma,\gamma'\}\not\in{\cal I}$ we say that the two
polymers are {compatible} and we write $\gamma\sim\gamma'$.
Assume that a complex valued function $\phi(\gamma)$,
$\gamma\in{\cal P}$, is given.
We call $\phi(\gamma)$ the {weight}, or the {activity},
of the polymer $\gamma$.
For any finite subset $\Lambda\subset{\cal P}$,
the {partition function} $Z(\Lambda)$ of the
polymer system is defined by
\beq
Z(\Lambda)=
\sum_{{\scriptstyle X\subset\Lambda}
\atop{\scriptstyle {\rm compatible}}}
\prod_{\gamma\in X}\phi(\gamma)
\label{1}\eeq
The sum runs over all subsets $X$ of ${\Lambda}$ such that
$\gamma\sim\gamma'$ for any two distinct elements of $X$.
If $X$ contains only one element, $X$ is considered a
compatible subset, and if $X=\emptyset$,
the product is interpreted as the number $1$.
We introduce the following
function on ${\cal P}\times{\cal P}$
\beq
f(\gamma,\gamma')=\cases{-1 &if $\gamma\not\sim\gamma'$ \cr
0 &otherwise \cr}
\label{2}\eeq
Let ${\cal G}_n$, $n\ge2$ be the set of connected
graphs with $n$ vertices, $1,\dots,n$.
We consider undirected graphs without multiple edges,
equivalently defined by a subset of ${\rm P}_2(\{1,\dots,n\})$
which determines the edges.
Given $g\in{\cal G}_n$ we define the value of $g$ on a
sequence $(\gamma_1,\dots,\gamma_n)\in{\cal P}^n$ as
\beq
g(\gamma_1,\dots,\gamma_n)=\prod_{(i,j)\in g}f(\gamma_i,\gamma_j)
\label{3}\eeq
where $(i,j)\in g$ means that the graph $g$ has
an edge connecting $i$ with $j$.
We also define ${\cal G}_1$
as the set containing only one graph $g$ having
only one vertex (and no edges) and write
\beq
g(\gamma)=1,\quad \gamma\in{\cal P}
\label{4}\eeq
\begin{Th}[Expansion]
\label{T1}
Define
\beq
a^{\rm T}(\gamma_1,\dots,\gamma_{n})=
\sum_{g\in {\cal G}_{n}}g(\gamma_1,\dots,\gamma_{n})
\label{5}\eeq
Then, we have
\beq
\ln Z(\Lambda)=\sum_{n=1}^\infty {1\over{n !}}
\sum_{(\gamma_1,\dots,\gamma_{n})\in\Lambda^n}
a^{\rm T}(\gamma_1,\dots,\gamma_{n})
\prod_{i=1}^n\phi(\gamma_i)
\label{6}\eeq
\end{Th}
{\it Proof.}
The partition function can be written as
\beq
Z(\Lambda)=1+\sum_{\gamma\in{\Lambda}}\phi(\gamma)+
\sum_{n=2}^\infty{1\over{n!}}
\sum_{(\gamma_1,\dots,\gamma_n)\in{\Lambda}^n}
\prod_{i=1}^n\phi(\gamma_i)
\prod_{1\le iFrom the definition of $Z^*$ we see that
\beq
Z^*(\Lambda\cup\{\gamma_0\}) = Z^*(\Lambda)
-\vert\phi(\gamma_0)\vert Z^*(\Lambda_0)
\label{17}\eeq
with
$\Lambda_0=\{\gamma\in\Lambda : \gamma\sim\gamma_0\}$,
and
\beq
-\ln\,Z^*(\Lambda\cup\{\gamma_0\}) = -\ln\,Z^*(\Lambda)
-\ln \Bigg(1-{{\vert\phi(\gamma_0)\vert Z^*(\Lambda_0)}
\over{Z^*(\Lambda)}}\Bigg)
\label{18}\eeq
On the other hand, we have
\beq
Z^*(\Lambda_0)/Z^*(\Lambda)=
\exp\sum_{X\in{\rm M}(\Lambda)\backslash{\rm M}(\Lambda_0)}
\vert\phi^{\rm T}(X)\vert
\label{19}\eeq
This shows the positivity of all the terms in the expansion of
the second term in the right hand side of equation (\ref{18})
(remark that the series expansions of the functions $\exp x$
and $-\ln(1-x)$ have only positive terms for $x\ge0$).
Since, by assumption, this is also
the case for the first term, it follows that also
$-\ln\,Z^*(\Lambda\cup\{\gamma_0\})$ satisfies
the induction hypothesis.
The lemma is proved.
\bigskip
\noindent{\it Proof of theorem \ref{T2}.}
We use again an induction argument on the subsets $\Lambda$.
Assume that, for a given $\Lambda$ and any $\gamma\in\Lambda$,
the following estimate holds
\beq
\sum_{X\in{\rm M}(\Lambda),\,X(\gamma)\ge1}
\vert\phi^{\rm T}(X)\vert\le\mu(\gamma)
\label{20}\eeq
This inequality can also be written as
\beq
-\ln Z^*(\Lambda)+\ln Z^*(\Lambda\backslash\{\gamma\})
\le\mu(\gamma)
\label{21}\eeq
and, for all $\Lambda'\subset\Lambda$, it implies
\beq
-\ln Z^*(\Lambda)+\ln Z^*(\Lambda')
\le\sum_{\gamma\in\Lambda\backslash\Lambda'}\mu(\gamma)
\label{22}\eeq
and, in particular,
\beq
{Z^*(\Lambda_0)}/{Z^*(\Lambda)}
\le \exp\bigg(
\sum_{\gamma:\gamma\in\Lambda, \gamma\not\sim\gamma_0}
\mu(\gamma)\bigg)
\label{23}\eeq
because
$\Lambda\backslash\Lambda_0
=\{\gamma\in\Lambda : \gamma\not\sim\gamma_0\}$.
Since $\Lambda$ does not contain $\gamma_0$, we get
\beq
\vert\phi(\gamma_0)\vert
{{Z^*(\Lambda_0)}\over{Z^*(\Lambda)}}
\le \vert\phi(\gamma_0)\vert \exp\bigg(-\mu(\gamma_0)
+\sum_{\gamma:\gamma\in{\cal P}, \gamma\not\sim\gamma_0}
\mu(\gamma)\bigg)
\label{24}\eeq
and, taking the assumption (\ref{i1})
of the theorem into account,
\beq
\bigg\vert{{\phi(\gamma_0)Z^*(\Lambda_0)}\over{Z^*(\Lambda)}}\bigg\vert
\le e^{-\mu(\gamma_0)}(e^{\mu(\gamma_0)}-1)
= 1-e^{-\mu(\gamma_0)}
\label{25}\eeq
Then, using (\ref{18}) and the fact that
$-\ln(1-x)$ is an increasing function of $x$,
for any real $x$ in the interval $-1