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Cluster expansions, polymer systems, domino systems, random geometrical objects
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\begin{center}
{\LARGE Cluster expansion for abstract polymer models.\\
New bounds from an old approach}
\vskip1.0cm
{\Large Roberto Fern\'andez$^{1}$ and Aldo Procacci$^{1,2}$}

\vskip.5cm

$^{1}$Labo. de Maths Raphael SALEM,
UMR 6085 CNRS-Univ. de Rouen, Avenue de l'Universit\'e, BP.12,
76801 Saint Etienne du Rouvray,  France\\
\vskip.1cm
 $^{2}$
Dep. Matem\'atica-ICEx, Universidade Federal de Minas Gerais, CP 702,
Belo Horizonte 
MG 30.161-970, Brazil

\vskip.1cm
email: $^1${\tt Roberto.Fernandez@univ-rouen.fr};
~ $^2${\tt aldo@mat.ufmg.br}

\end{center}

\def\be{\begin{equation}}
\def\ee{\end{equation}}


\vskip1.0cm
\begin{abstract}
  
  We revisit the classical approach to cluster expansions, based on
  tree graphs, and establish a new convergence
  condition that improves those by Koteck\'y-Preiss and
  Dobrushin, as we show in some examples.  The strategy is to
  better exploit a well known tree-graph expression, due to Penrose.

\end{abstract}
\vskip1.0cm
\numsec=2\numfor=1
\\{\bf 1. Introduction}
\vv
%\vskip6.0cm
\noindent
Cluster expansions, originally developed to express thermodynamic
potentials as power series in activities, are at the heart of
important perturbative arguments in statistical mechanics and other
branches of mathematical physics.  The classical approach to obtain
convergence conditions was based on combinatorial
considerations~\cite{mal80, sei82}, which were greatly simplified
through the use of tree-graph bounds~\cite{cam82, bry84}.  A
completely new inductive approach originated in the work of Koteck\'y
and Preiss \cite{kotpre86}, later refined by Dobrushin
\cite{dob96,dob96a} and many
others~\cite{narolizah99,bovzah00,mir00,uel04,scosok05}.  This later
approach is mathematically very appealing and, in its more elegant
version~\cite{dob96,scosok05}, it even disposes of any reference to power
series, becoming, in Dobrushin's words,  a ``no-cluster-expansion''
approach.  The combinatorial approach, however, kept its adepts who
reformulated it in a very clear and compact way~\cite{pfi91} and showed
how it can lead to bounds at least as good as those given by Koteck\'y
and Preiss~\cite{PS}.

In this paper, we revisit the classical combinatorial approach and
point out that it can be used, in a rather simple and natural way, to
produce improved bounds on the convergence region (and the
sum of the expansion).  Our main tool is an \emph{identity}, 
due to Oliver Penrose~\cite{pen67}, relating the coefficients of the 
expansion to a family of trees determined by compatibility constraints. 
(In fact, we learnt this identity from the nice exposition in \cite[Section 3]{pfi91}.) 
Succesive approximations are obtained by
considering larger families of trees that neglect some of the
constraints. If only the very basic constraint is kept (links in the
tree must relate incompatible objects), the Kotecky-Preiss
condition emerges.  To the next order of precision (branches must end
in different objects) Dobrushin's condition is found.  By refining this
last constraint (branches' ends must be mutually compatible rather than
just different) we obtain a new convergence condition which 
leads to improvements in several well-studied cases.  All these
conditions are obtained for general systems under minimal hypotheses.
The same approach should yield further improvements for more specific
systems through a better description of the resulting constraints on
tree diagrams.


\numsec=2\numfor=1
\vv\vv
\\{\bf 2. The set up and the problem}
\vv
\noindent
We adopt the abstract polymer setting of \cite{kotpre86}.  Its
starting point is an unoriented graph $\GG=(\PP,\E)$ ---the
\emph{interaction graph}--- on a countable vertex set.  The vertices
$\g\in\PP$ are called \emph{polymers} for historical
reasons~\cite{grukun71} (the name is misleading;
Dobrushin~\cite{dob96a} proposes to call them \emph{animals}, but the
traditional name holds on).  The edge set corresponds to an
\emph{incompatibility relation}: Two polymers $\g,\g'$ are
incompatible if $\{\g,\g'\}\in\E$, in which case we write
$\g\nsim\g'$.  Otherwise they are \emph{compatible} and we write
$\g\sim\g'$.  The set of edges is arbitrary, except for the assumption
that it contains all pairs of the form $\{\g,\g\}$, that is, \emph{
every polymer is assumed to be incompatible with itself}.  In
in particular vertices can be of infinite degree (each polymer can be
incompatible with infinitely many other polymers).  This happens, for
instance, for graphs associated to gases of low-temperature contours
or ``defects".


The physical information of each polymer model is given by the
incompatibility relation and a family of \emph{activities} $\und
z=\{z_\g\}_{\g\in \PP}\in\C^{\PP}$.  For each \emph{finite} family
$\L\subset \PP$, these ingredients define probability weights on the
set of subsets of $\Lambda$:
$$ {\rm Prob}_\Lambda\bigl(\{\g_{1},\g_{2},\ldots,\g_{n}\}\bigr) \;=\;
\frac{1}{\Xi_{\Lambda}(\und z)}\, z_{\g_{1}} z_{\g_{2}}\cdots z_{\g_{n}}
\prod_{j<k} \ind{\g_{j}\sim \g_{k}}
\Eq(r.1)
$$
for $n\ge 1$ and $ {\rm Prob}_\Lambda(\emptyset)=1/\Xi_{\L}$,
where 
$$
\Xi_{\L}(\und z)=1+\sum_{n\geq 1}{1\over n!}
\sum_{(\g_{1},\dots ,\g_{n})\in\L^n}
{z_{\g_1}}{z_{\g_2}}\dots{z_{\g_n}}
\prod_{j<k} \ind{\g_{j}\sim \g_{k}}\;.
\Eq(2)
$$

In physical terms, the measure \equ(r.1) corresponds to the
grand-canonical ensemble of a polymer gas with activities $\und z$ and
hard-core interaction defined by the incompatibility relation.  The abstract
formalism makes it equivalent to a lattice gas on the graph $\GG$
with self- and nearest-neighbor hard-core repulsion. The
normalization constant \equ(2) is the grand-canonical partition function in the
``volume" $\L$.  Cluster expansions allow the control of the measures
\equ(r.1) uniformly in $\L$ and absolutely in the activities.  [Thus,
the control extends to the unphysical region of non-positive (complex)
activities, where the expressions on the right-hand side of \equ(r.1) 
do not define probability
measures.]  This is achieved by determining the radius of convergence
of the logarithm of the partition function, whose  well-known formal power series
(Mayer expansion, see e.g.~\cite{R}) is 
$$
\log \Xi_{\L}(\und z)\;=\; \sum_{n=1}^{\infty}{1\over n!}
\sum_{(\g_{1},\dots ,\g_{n})\subset\L^n}
\phi^{T}(\g_1 ,\dots , \g_n)\,z_{\g_1}\dots z_{\g_n}\Eq(6)
$$
with
$$
\phi^{T}(\g_{1},\dots ,\g_{n})\;=\;\left\{
\begin{array}{ll}
1&n=1\\[8pt]
0 & n\ge 2\,,\; \GG_{\{\g_{1},\dots ,\g_{n}\}} \mbox{ not connected}\\[8pt]
\sum\limits_{G\subset \GG_{\{\g_{1},\dots ,\g_{n}\}}\atop G\ {\rm connected}}
 (-1)^{\card{E(G)}}&
n\ge 2\,,\;\GG_{\{\g_{1},\dots ,\g_{n}\}} \mbox{ connected}
\end{array}\right.
\Eq(7)
$$
where $\GG_{\{\g_{1},\dots ,\g_{n}\}}$ is the 
restriction of $\GG$ to the vertex set $\{\g_{1},\dots ,\g_{n}\}$ and $G$
ranges over all its connected subgraphs, each of which has edge set
$E(G)$.  The functions $\phi^{T}$ are the \emph{truncated functions}
of order $n$ (also called \emph{Ursell functions}).

Simple arguments show that the uniform convergence of \equ(6) 
is determined by that of the ``pinned'' (formal) expansions
$$
\Si_{\g_0}(\und\r)\;\bydef\;
\sum_{n=1}^{\infty}{1\over n!}
\sum_{(\g_{1},\dots ,\g_{n})\subset\PP^n\atop \exists i:\,\g_i=\g_0}
\card{\phi^{T}(\g_1 ,\dots , \g_n)}\,\r_{\g_1}\dots \r_{\g_n}
\Eq(6b)
$$
for $\und\r\in[0,\infty)^\PP$.  The convergence of this
positive-term series for a certain $\und\r$ implies the absolute
convergence of \equ(6) ---and the
control of the measures \equ(r.1)~\cite{dob96a}---
 uniformly for $\card{z_\g}\le\r_\g$.

A slightly more efficient alternative is to consider, instead of
\equ(6b), the formal series
$$
\Pi_{\g_0}(\und\r)\;\bydef\; 1+
\sum_{n=1}^{\infty}{1\over n!}
\sum_{(\g_{1},\dots ,\g_{n})\subset\PP^n}
\card{\phi^{T}(\g_0,\g_1 ,\dots , \g_n)}\,\r_{\g_1}\dots \r_{\g_n}\;.
\Eq(TP)
$$
The convergence of \equ(TP) implies that of \equ(6b) via the relation
$$
\Si_{\g_0}(\und\r)\;=\; \r_{\g_0}\int_0^1 \Pi_{\g_0}(\und\r^\a)\,d\a
\Eq(relspm)
$$
with
$$
\r_\g^\a\;=\;\cases{\r_{\g} &if $\g\neq \g_0$\cr\cr
\a \r_{\g} &if $\g=\g_0$
}
$$
and, thus,
$$
\Si_{\g_0}(\und\r)\;\le\; \r_{\g_0}\,\Pi_{\g_0}(\und\r)\;.
\Eq(PminTP)
$$

In the sequel we focus on the convergence of the series \equ(TP)
for positive activities.


\numsec=3\numfor=1
\vv\vv
\\{\bf 3. New convergence criterion}
\vv
\noindent
Our new criterion involves the grand-canonical partition functions
$\Xi_{\PP_{\g_0}}$, associated to the polymer families
$\PP_{\g_0}=\{\g\in\PP:\g\nsim\g_0\}$, $\g_0\in\PP$.  These functions,
defined in \equ(2), can also be written in the form
$$
\Xi_{\PP_{\g_0}}(\und \mu)=1+\sum_{n\geq 1} \,\frac{1}{n!}
\sum_{(\g_{1},\dots ,\g_{n})\in\PP^n\atop
\g_0\nsim\g_i\,,\, \g_i\sim\g_j\,,\, 1\le i ,j\le n}
{\mu_{\g_1}}{\mu_{\g_2}}\dots{\mu_{\g_n}}
\Eq(r.2)
$$
because compatible polymers are different.
\begin{teorema}\label{th:1}
Let   $\und\r\in [0,\infty)^{\PP}$.  If there exists a $\und\mu\in
  [0,\infty)^{\PP}$ such that
$$
\r_{\g_0}\,\Xi_{\PP_{\g_0}}(\und\mu) \;\le\; \mu_{\g_0} \;,\quad
\forall \g_0\in\PP\;,
\Eq(r.3)
$$
then the series $\Pi_{\g_0}(\r)$ [defined in \equ(TP)] converges for
such $\und\rho$ and satisfies $\r_{\g_0}\,\Pi_{\g_0}(\r)\le\mu_{\g_0}$.
\end{teorema}

\begin{remark}
Koteck\'y-Preiss and Dobrushin criteria produce similar results,
but starting from the conditions
$$
\r_{\g_0}\; \exp\Bigl[\sum_{\g\nsim\g_0} \mu_{\g}\Bigr] 
\;\le\; \mu_{\g_0} \ \mbox{ (Koteck\'y-Preiss)}
\Eq(r.kp)
$$
and
$$
\r_{\g_0}\,\prod_{\g\nsim\g_0} \bigl(1 + \mu_\g\bigr) 
\;\le\; \mu_{\g_0} \ \mbox{ (Dobrushin)}\;.
\Eq(r.dob)
$$
The inequalities 
$$
\Xi_{\PP_{\g_0}}(\und\mu) \;\le\; \prod_{\g\nsim\g_0} \bigl(1 + \mu_\g\bigr)
\;\le\; \exp\Bigl[\sum_{\g\nsim\g_0} \mu_{\g}\Bigr]
\Eq(r.3b)
$$
show that condition \equ(r.3) is an improvement over Dobrushin's condition
which in turns is an improvement over Koteck\'y-Preiss' condition.  The examples below
show that the successive improvements can be noticeable.
\end{remark}

\begin{remark}
  Condition \equ(r.3) can be exploited in two complementary
  directions: (i) To find the radius of convergence, that is the
  largest values $\rho_\g$ for which there is a $\und\mu$ satisfying
  \equ(r.3).  (ii) To find the sharpest bound on $\Pi_{\g_0}$, that
  is, the smallest values $\mu_\g$ yielding \equ(r.3) for a fixed
  $\und\r$.  In the following examples we focus in direction (i).
  Regarding direction (ii), Corollary \ref{cor:1} below implies that
  bounds on $\Pi_{\g_0}$ can be improved up to limit functions
  $\Pi^*(\und\r,\und\mu)$ and $\Pi^*(\und\r)$ constructed as follows.  Let
  $T:[0,\infty)^\PP\to[0,\infty]^\PP$ defined by
$$ T_{\g_0}(\und\mu) \;=\; \und\r\; \Xi_{\PP_{\g_0}}(\und\mu)\Eq(r.4) $$
for each $\g_0\in\PP$.  Then, 
$$ \mu_{\g_0}\ge T_{\g_0}(\und\mu) \ge \cdots
\ge T^n_{\g_0}(\und\mu) \buildd{\searrow}{n\to\infty} 
T^\infty_{\g_0}(\und\mu) \defby \r_{\g_0}\,\Pi_{\g_0}^*(\und\r,\und\mu) 
\ge T^\infty_{\g_0}(\und\r) \defby \r_{\g_0}\,\Pi_{\g_0}^*(\und\r)
\ge \r_{\g_0}\,\Pi_{\g_0}(\und\r)\;.
\Eq(r.5)
$$
Similar chains of inequalities hold within the Koteck\'y-Preiss and
Dobrushin criteria, for operators $T$ defined as in \equ(r.4)
but with $\Xi_{\PP_{\g_0}}(\und\mu)$ substituted, respectively,
by $\exp\bigl[\sum_{\g\nsim\g_0} \mu_{\g}\bigr] $ and
$\prod_{\g\nsim\g_0} \bigl(1 + \mu_\g\bigr)$.  Of course, 
$\Pi^{*{\rm KP}}\ge \Pi^{*{\rm Dob}}\ge \Pi^*$.
\end{remark}

To test our criterion we compare the criteria \equ(r.3)--\equ(r.dob)
for two families of benchmark examples.

\paragraph{Polymers with finitely many incompatibilities}
These are examples where $\GG$ has finite coordination numbers.  We
shall suppose that all polymers have equal activity $\r_\g\equiv\r$
for all $\g\in\GG$, and therefore we search for equally constant
functions $\mu_\g\equiv\mu$.  The preceding criteria take the form
$\r\le \mu/\varphi(\mu)$ for appropriate functions $\varphi$, and the
maximization of the right-hand side with respect to $\mu$ yields the
best lower bounds of the radius of convergence of \equ(TP) [and hence
of \equ(6)].  We study two systems.  First, the domino model in
$\mathbb{Z}^2$, in which polymers are pairs of nearest-neighbor sites
and incompatibility is just overlapping.  Second, the hard-core
lattice gas on graphs of bounded degree (=number of neighbors).  To
highlight differences in criteria, we consider two graphs, which we
denote $\GG_{\Delta}$ and $\GG_{\rm Triang}$.  The first one is any
graph of maximum degree $\Delta$ where \emph{all neighbors to a vertex
  are compatible between themselves} (e.g.~$\mathbb{Z}^d$, trees).
The second graph is the bidimensional triangular lattice ($\Delta=6$).
Calculations and bounds on the radius of convergence are summarized in
Tables \ref{tab:1} and \ref{tab:2}.

\begin{table}[h]
\renewcommand{\arraystretch}{1.7}
\begin{center}
\begin{tabular}{|c||c|c|c|}\hline
Model & Koteck\'y-Preiss & Dobrushin & \protect\equ(r.3)\\
\hline\hline
\rule[-5mm]{0mm}{11mm}Hard-core in $\GG_\Delta$
& $\displaystyle \r\le \mu\,e^{-(\Delta+1)\mu}$
& $\displaystyle \r\le \frac{\mu}{(1+\mu)^{\Delta+1}}$
& $\displaystyle \r\le \frac{\mu}{\mu+(1+\mu)^\Delta}$\\
\hline
\rule[-5mm]{0mm}{11mm}Hard-core in $\GG_{\rm Triang}$
& $\displaystyle\r\le\mu\,e^{-7\mu}$ 
& $\displaystyle\r\le \frac{\mu}{(1+\mu)^7}$
& $\displaystyle \r\le \frac{\mu}{1+7\mu+8\mu^2+2\mu^3}$\\
\hline
\rule[-5mm]{0mm}{11mm}Domino in $\mathbb{Z}^2$
& Same as above & Same as above
& $\displaystyle\r\le \frac{\mu}{1+7\mu+9\mu^2}$\\
\hline
\end{tabular}
\end{center}
\caption{Comparison of convergence conditions for
some polymer systems with finitely many incompatibilities}
\label{tab:1}
\end{table}

\begin{table}[h]
\renewcommand{\arraystretch}{1.7}
\begin{center}
\begin{tabular}{|c||c|c|c|}\hline
Model & Koteck\'y-Preiss & Dobrushin & \protect\equ(r.3)\\
\hline\hline
\rule[-6.5mm]{0mm}{13mm}Hard-core in $\GG_\Delta$
& $\displaystyle {1\over (\D+1)\,e}$
& \raisebox{-1mm}{$\displaystyle {\D^\D\over (\D+1)^{\D+1}}$}
& $\displaystyle{1\over 1+{\D^\D\over (\D-1)^{\D-1}}}$\\
\hline
\rule[-5.5mm]{0mm}{13mm}Hard-core in $\GG_{\rm Triang}$
& $\displaystyle{1\over 7e}\approx 0.0525$ 
& $\displaystyle{{1\over 6}\over (1+{1\over 6})^7}\approx 0.0566$
& $\displaystyle 4R^3+8R^2=1\;,\; R\approx 0,078$\\
\hline
\rule[-5.5mm]{0mm}{13mm}Domino in $\mathbb{Z}^2$
& Same as above & Same as above
& $\displaystyle{1\over 13}\approx 0.0769$\\
\hline
\end{tabular}
\end{center}
\caption{Lower bounds of the radius of convergence for cluster
expansions of some polymer systems with finitely many incompatibilities}
\label{tab:2}
\end{table}

We see that Dobrushin and Koteck\'y-Preiss criteria are only sensitive
to the coordination number, that is, the number of polymers that are
incompatible to a given one.  In the new condition \equ(r.3),
incompatibilities between these neighbors are also accounted for.
This is not the only difference, because there is also a (less
important) improvement for the graph $\GG_\Delta$, where neighbors are
mutually compatible.  The best bound available for this last graph,
however, is due to Scott and Sokal~\cite{scosok05}.  Their estimation
of the radius of convergence is
$$
R\;=\; {(\D-1)^{(\D-1)}\over \D^\D}\;,
\Eq(treeb)
$$
which, as they show, is optimal for trees.  While our
estimate approaches \equ(treeb) for large $\Delta$, the differences
for small $\Delta$ are important (1/4 vs.\ 1/5 for $\Delta=2$).  On
the other hand, our estimate for the triangular graph is better than
that coming from \equ(treeb) (0.078 vs.\ 0.067), precisely because of
the presence of incompatibilities between neighbors of a site.  The
estimate \equ(treeb) was derived through a sequence of
volume-dependent Dobrushin conditions.  It would be interesting to see
whether a similar strategy could be developed within our approach.

 
\paragraph{Random sets on a graph}
This is the general example of cluster expansions for
graphs with vertices of infinite degree.  Applications include contour ensembles of
low-temperature phases, geometrical objects of high-temperature
expansions, random sets of the Fortuin-Kasteleyn representation of the
Potts model, \dots The general setup for these models is a polymer
family formed by the finite parts of a given set $\VU$  with incompatibility
defined by overlapping.  (Usually, $\VU$ is formed by the vertices
of a graph with respect to which polymers form connected sets.)

For these systems it is useful and traditional to pass to exponential 
weight functions $a(\g)$ defined by $\m_\g=\r_\g\,e^{a(\g)}$.  
Condition \equ(r.3) becomes
$$
1+\sum_{n\geq 1}
\sum_{\{\g_{1},\dots ,\g_{n}\}\subset\PP\atop
\g_0\cap\g_i\neq\emptyset\,,\, 
\g_i\cap\g_j=\emptyset\,,\, 1\le i ,j\le n}
\prod_{i=1}^n\r_{\g_i}\, e^{a(\g_i)}
\;\le\; e^{a(\g_0)}\;
\Eq(r.6)
$$
From the constraint in the sum we only keep the fact that each of the 
polymers $\g_1,\ldots,\g_n$ must intersect \emph{different} points in $\g_0$
(otherwise they would overlap).  This implies: (i) $n\le \card{\g_0}$, and (ii) 
there are $n$ different points in $\g_0$ touched by $\g_1\cup\cdots\cup\g_n$.
These points can be chosen in ${\card{\g_0}\choose n}$ ways.  Hence,
the left-hand side of \equ(r.6) is less or equal than
$$
1+ \sum_{n=1}^{\card{\g_0}}
{\card{\g_0}\choose n}\Bigg[\sup_{x\in \g_0}\,\sum_{\g\in  \PP\atop \g\ni x}
{\r_{\g}}\,e^{a(\g)}\Bigg]^n\;=\;
\Bigg[1+\sup_{x\in \g_0}\,\sum_{\g\in  \PP\atop \g\ni x}
{\r_{\g}}\,e^{a(\g)}\Bigg]^{\card{\g_0}}\;,
$$
which leads us to the following sufficient condition for \equ(r.6):
$$
\sup_{x\in \g_0}\,\sum_{\g\in  \PP\atop \g\ni x}
{\r_{\g}}\,e^{a(\g)}\;\le\; e^{a(\g_0)/\card{\g_0}}-1
\Eq(usht1)
$$

The corresponding convergence conditions obtained from our
reference criteria are:
$$
\sup_{x\in \g_0}\,\sum_{\g\in  \PP\atop \g\ni x}
{\r_{\g}}\,e^{a(\g)}\;\le\;  \frac{a(\g_0)}{\card{\g_0}}
\ \mbox{ (Koteck\'y-Preiss)}
\Eq(r.7)
$$
and
$$
\sup_{x\in \g_0}\,\prod_{\g\in  \PP\atop \g\ni x}
\Bigl[1+ {\r_{\g}}\,e^{a(\g)}\Bigr] \;\le\;  e^{a(\g_0)/\card{\g_0}}
\ \mbox{ (Dobrushin)}\;.
\Eq(r.8)
$$
In all practical cases the function $a(\g)$ has been chosen of the form
$a(\g)=a\card\g$, with $a$ a positive constant.  Also, the criteria are
usually presented in the slightly weaker form obtained by replacing
the supremum over $x\in\g_0$ by a supremum over $x\in\VU$.  In this form,
a condition like \equ(usht1) is, in fact, present in the seminal paper
by Gruber and Kunz~\cite{grukun71} [formula (42) with 
normalization $\phi(x)=1$ and parametrization $\xi_0=e^a$.]

\numsec=4\numfor=1
\vv\vv
\\{\bf 4. The proof and the approach}
\vv
\\
\noindent
The argument has two distinct parts.  The first one is a rather elementary
convergence argument for positive series; the second one is the
Penrose tree identity for the truncated functions.

\vv\vv
\\{\bf 4.1 Trees and convergence}
\vv
\\
\noindent
Let us fix some $\und\r\in[0,\infty)^\PP$ and consider maps of the form
$$
\begin{array}{rcl}
T:[0,\infty)^\PP&\longrightarrow&[0,\infty]^\PP\\
\und\mu& \longmapsto & T_{\g}(\und\mu)= \r_\g\,\varphi_{\g}(\und\mu)
\end{array}
\Eq(r.7.0)
$$
with 
$$
\varphi _{\g_0} (\und\mu)\;=\; 1+\sum_{n\geq 1}
\sum_{(\g_{1},\dots ,\g_{n})\in\PP^n} b_n(\g_0,\g_1,\ldots,\g_n)\,
{\mu_{\g_1}}\dots{\mu_{\g_n}}
\Eq(r.7.1)
$$
for certain functions $b_n:\PP^{n+1}\to [0,\infty)$.  We find it useful to visualize
these maps in the diagrammatic form
\setlength{\unitlength}{1cm}
$$
\begin{picture}(12,2.5)
\thicklines
\put(0.5,1.5){$T_{\g_0}(\und\mu)$}
\put(2,1.5){=}
\put(2.7,1.5){$\circ$}
\put(2.7,1.3){$\scriptstyle\g_0$}
\put(3.15,1.5){$+$}
\put(3.7,1.5){$\circ$}
\put(3.7,1.3){$\scriptstyle\g_0$}
\put(3.85,1.58){\line(1,0){1}}
\put(4.8,1.5){$\bullet{\scriptstyle 1}$}
\put(5.3,1.5){$+$}
\put(5.8,1.5){$\circ$}
\put(5.8,1.3){$\scriptstyle\g_0$}
\put(5.97,1.58){\line(2,1){0.8}}
\put(6.67,1.89){$\bullet{\scriptstyle 1}$}
\put(5.97,1.58){\line(2,-1){0.8}}
\put(6.67,1.08){$\bullet{\scriptstyle 2}$}
\put(7.12,1.5){$+$}
\put(7.6,1.5){$\cdots$}
\put(8.25,1.5){$+$}
\put(8.8,1.5){$\circ$}
\put(8.8,1.3){$\scriptstyle\g_0$}
\put(8.97,1.58){\line(1,1){0.7}}
\put(9.64,2.23){$\bullet{\scriptstyle 1}$}
\put(8.97,1.58){\line(2,1){0.8}}
\put(9.64,1.86){$\bullet{\scriptstyle 2}$}
\put(9.64,1.3){\vdots}
\put(8.97,1.58){\line(1,-1){0.7}}
\put(9.64,0.74){$\bullet{\scriptstyle n}$}
\put(10.09,1.5){$+$}
\put(10.57,1.5){$\cdots$}
\end{picture}
$$
The sum is over all single-generation rooted trees.  In each tree,
open circles represents a factor $\r$, bullets a factor $\mu$ and
vertices other than the root must be summed over all possible polymers
$\g$.  At each vertex with $n$ descendants, a ``vertex function'' $b_n$
acts, having as arguments the ordered $n+1$-tuple formed by the polymer
at the vertex, the polymer at the top offspring, the polymer at the
next offspring from the top,\ldots, in that order.
With this representation, the iteration $T^2(\und\mu)$ corresponds to
replacing each of the bullets by each one of the diagrams of the
expansion for $T$.  This leads to rooted trees of up to two
generations, with open circles at first-generation vertices and bullets
at second-generation ones.  In particular, all single-generation trees
have only open circles.  Notice that the two drawings of Figure
\ref{fig:1} appear in two different terms of the expansion, and hence
should be counted as \emph{different} diagrams.  More generally, the
$k$-th iteration of $T$ involves all possible rooted tree diagrams,
counting as different those obtained by permutations of non-identical
branches.  We shall call these diagrams \emph{planar rooted trees}.
In each term of the expansion, vertices of generation $k$ are occupied
by bullets and all the others by open circles.

Formally, the definition of planar rooted trees is determined by a
labeling choice which we fix as follows.  There is a special vertex,
labeled $0$ (the root), placed, say, at the leftmost position of the
drawing.  From it there emerge $s_{0}$ branches ending at the
first-generation vertices.  The value $s_{0}=0$ describes the trivial
tree with the root as its only vertex.  Otherwise these vertices are
drawn along a vertical line at the right of the root and labeled
$(0,1),\dots (0,s_{0})$ with the second subscript increasing from the
top to the bottom of the line.  The construction continues rightwards:
Each of the vertices $(0,i)$, gives rise to a family of
second-generation vertices $(0,i,1),\dots (0,i,s_{(0,i)})$ and so on.
The vertex $v$ is of generation $\ell$ if its label has the form
$v=(0,i_1,\ldots,i_\ell)$ with $1\le i_j\le s_{(0,i_1,\ldots,i_{j-1})}$,
$1\le j\le \ell$ ($i_0\equiv0$).  A planar rooted tree of maximal
generation number $k$ is defined by a sequence
$$ \Bigl(s_{(0,i_1,\ldots,i_\ell)}\in\mathbb{N}\cup\{0\} : 
1\le i_j\le s_{(0,i_1,\ldots,i_{j-1})}\,,\,
1\le j\le \ell\Bigr)_{1\le \ell\le k}
\Eq(r.8b)
$$
with $s_{(0,i_1,\ldots,i_k)}=0$.  Let us denote $\trp^{0,k}$ the
set of trees with maximal generation number $k$; $\trp^{0,0}$ being
the trivial tree.  Figure \ref{fig:1} shows two different trees of
$\trp^{0,2}$.  The labels define a natural total order among the
vertices, with $(0,i_1,\ldots,i_\ell)<(0,i'_1,\ldots,i'_{\ell'})$ if
$\ell< \ell'$ or $\ell= \ell'$ and $i_j\le i'_{j}$ for all
$j=1,\cdots, \ell$ with some $i_j< i'_j$.  We use this order to
enumerate the set $V_t=\{v_1,v_2,\ldots,v_{\card{V_t}}\}$ formed by
the vertex, other than the root, of a planar tree $t$; $v_i< v_j$ if
$i< j$.

\begin{figure}[h]
\begin{center}
\begin{picture}(3,3)
\thicklines
\put(0,1.5){$\circ$}
\put(0,1.25){$\scriptscriptstyle 0$}
\put(0.17,1.59){\line(2,1){0.8}}
\put(0.93,1.93){$\circ$}
\put(0.77,1.68){$\scriptscriptstyle (0,1)$}
\put(0.17,1.59){\line(2,-1){0.8}}
\put(0.93,1.07){$\circ$}
\put(0.77,0.82){$\scriptscriptstyle (0,2)$}
\put(1.12,2.05){\line(2,1){0.8}}
\put(1.88,2.39){$\bullet$}
\put(1.63,2.15){$\scriptscriptstyle (0,1,1)$}
\put(1.12,2.05){\line(2,-1){0.8}}
\put(1.88,1.53){$\bullet$}
\put(1.63,1.29){$\scriptscriptstyle (0,1,2)$}
\put(0.93,0){(a)}
\end{picture}
\hspace{2cm}
\begin{picture}(3,3)
\thicklines
\put(0,1.5){$\circ$}
\put(0,1.25){$\scriptscriptstyle 0$}
\put(0.17,1.59){\line(2,1){0.8}}
\put(0.93,1.93){$\circ$}
\put(0.77,1.68){$\scriptscriptstyle (0,1)$}
\put(0.17,1.59){\line(2,-1){0.8}}
\put(0.93,1.07){$\circ$}
\put(0.77,0.82){$\scriptscriptstyle (0,2)$}
\put(1.12,1.16){\line(2,1){0.8}}
\put(1.88,1.5){$\bullet$}
\put(1.63,1.25){$\scriptscriptstyle (0,2,1)$}
\put(1.12,1.16){\line(2,-1){0.8}}
\put(1.88,0.64){$\bullet$}
\put(1.63,0.39){$\scriptscriptstyle (0,2,2)$}
\put(0.93,0){(b)}
\end{picture}
\end{center}
\caption{Planar rooted trees defined by 
(a) $s_0= s_{(0,1)}=2$ and $s_{(0,2)}=s_{(0,1,1)}=s_{(0,1,2)}=0$; 
(b) $s_0= s_{(0,2)}=2$ and $s_{(0,1)}=s_{(0,2,1)}=s_{(0,2,2)}=0$}
\label{fig:1}
\end{figure}

A straightforward inductive argument shows that
$$
T^k_{\g_0}(\und\mu) \;=\; \r_{\g_0} \Bigl[\sum_{\ell=0}^{k-1}
R^{(\ell)}_{\g_0}(\und\r) + R^{(k)}_{\g_0}(\und\r,\und\mu)\Bigr]
\Eq(r.8.1)
$$
with
$$
R^{(\ell)}_{\g_0}(\und\r) \;=\; \,\sum_{t \in\trp^{0,\ell}}\,
\,\sum_{(\g_{v_1},\ldots ,\g_{v_{\card{V_t }}})\in\PP^{\card{V_t }}}\,
\prod_{i=0}^{\card{V_t}} b_{s_{v_i}}(\g_{v_i},\g_{({v_i},1)},\ldots,\g_{({v_i},s_{v_i})})\,
\r_{\g_{({v_i},1)}}\dots\r_{\g_{({v_i},s_{v_i})}} 
\Eq(r.9)
$$
and $R^{(k)}_{\g_0}(\und\r,\und\mu)$ has a similar expression but 
with the activities of the vertex of the $k$-th generation 
weighted by $\und\mu$.  
Here we agree that $b_0(\g_v)\equiv 1$
and $\prod_\emptyset \equiv 1$.
We are interested in the $k\to\infty$ limit
of \equ(r.8.1).  Let us denote $\TT^0=\cup_\ell\TT^{0,\ell}$.

\begin{proposition}\label{prop:1}
Let $T$ be a map of the form \equ(r.7.0)--\equ(r.7.1) and assume there
exists $\und\mu\in[0,\infty)^\PP$ such that  
$$T_\g(\und\mu) \;\le\; \mu_\g \quad \forall \g\in\PP\;.
\Eq(r.10)
$$
Then the series
$$
\r^*_{\g_0} \;\bydef\; \r_{\g_0}
\,\sum_{t \in\trp^0} 
\,\sum_{(\g_{v_1},\ldots ,\g_{v_{\card{V_t }}})\in\PP^{\card{V_t }}}\,
\prod_{i=0}^{\card{V_t}} b_{s_{v_i}}(\g_{v_i},\g_{({v_i},1)},\ldots,\g_{({v_i},s_{v_i})})\,
\r_{\g_{({v_i},1)}}\dots\r_{\g_{({v_i},s_{v_i})}} 
\Eq(r.11)
$$
is finite for each $\g_0\in\PP$.  Furthermore, the limits
$T^\infty(\und\mu)=\lim_n T^n (\und\mu)$ and
$T^\infty(\und\r)=\lim_n T^n (\und\r)$ belong to
$[0,\infty)^\PP$ and are such that
$$ \mu_{\g_0}\ge T_{\g_0}(\und\mu) \ge \cdots
\ge T^n_{\g_0}(\und\mu) \buildd{\searrow}{n\to\infty} 
T^\infty_{\g_0}(\und\mu)\ge T^\infty_{\g_0}(\und\r)=\r^*_{\g_0}
\buildd{\nwarrow}{n\to\infty} T^n_{\g_0}(\und\r)
\ge \cdots \ge T_{\g_0}(\und\r)\ge\r_{\g_0}
\Eq(r.12)
$$
for each $\g_0\in\PP$.
\end{proposition}

\proof   Let us consider the
coordinatewise partial order of $[0,\infty]^\PP$ defined by
$\und\r\le\und\mu \Longleftrightarrow \r_\g\le\mu_\g \ \forall\g\in\PP$.
With respect to it $T$ is an increasing map 
and $\und\mu\ge T(\und\mu)\ge\und\r$. By induction
$$
\und\mu\;\ge\; T^{n}(\und\mu)\;\ge\; T^{n+1}(\und\mu)
\;\ge\;T^{n+\ell+1}(\und\mu)\;\ge\;
T^{n+\ell+1}(\und\r)\;\ge\;T^{\ell+1}(\und\r)
\;\ge\;T^\ell(\und\r)\;\ge\;\und\r
\Eq(r.13)
$$
for all $\ell,n\in\mathbb{N}$.  This shows that, for each
$\g\in\GG$, the series $[T_\g^\ell(\und\r)]_\ell$ is increasing and bounded
above while $[T_\g^n(\und\mu)]_n$ is decreasing and bounded below.
Thus, the limits 
$T^\infty(\und\r)\bydef\sup_\ell T^\ell(\und\r)$ and
$T^\infty(\und\mu)\bydef\inf_\ell T^\ell(\und\mu)$ exist and are 
finite and, by letting alternatingly $\ell\to\infty$ 
and $n\to\infty$ in \equ(r.12), we obtain the inequalities 
$$
\und\mu\;\ge\; T^{n}(\und\mu)\;\ge\; T^{n+1}(\und\mu)
\;\ge\;T^{\infty}(\und\mu)\;\ge\;
T^{\infty}(\und\r)\;\ge\; T^{\ell+1}(\und\r)\;\ge\;T^\ell(\und\r)
\;\ge\;\und\r\;.
\Eq(r.14)
$$
The fact that $T^\infty(\und\r)=\und\r^*$ is immediate from expression
\equ(r.8.1). $\qed$
\bigskip

We notice that 
$T^{\infty}_{\g_0}(\und\mu)=\r^*_{\g_0} + \lim_k R^{(k)}_{\g_0}(\und\mu)$.
The last limit is in fact an infimum because
$R^{(k)}(\und\r,\und\mu)\le R^{(k-1)}\bigl(\und\r,T(\und\mu)\bigr)\le 
R^{(k-1)}(\und\r,\und\mu)$.

To apply this proposition to the series \equ(TP), it is convenient
to permute the sums in \equ(r.11) (this is allowed because it is a series
of positive terms).  For each $n\in\mathbb{N}$ let us denote
$\trp^0_n$ the set of planar rooted trees with $n$ vertices besides the root.
We have
$$
\r^*_{\g_0} \;=\; \r_{\g_0}\,\sum_{n\ge 0} 
\,\sum_{(\g_{1},\ldots ,\g_n)\in\PP^n}\,
\Bigl[\sum_{t \in\trp^0_n} \;w_t
\prod_{i=1}^n \,\ind{\g_{v_i}=\g_i}\Bigr]
\; \r_{\g_1} \cdots\r_{\g_n} 
\Eq(r.15)
$$
with
$$
w_t \;=\; \prod_{i=0}^n \,
b_{s_{v_i}}(\g_{v_i},\g_{({v_i},1)},\ldots,\g_{({v_i},s_{v_i})})
\;.
\Eq(r.15.1)
$$

We are particularly interested in vertex functions
$b_n(\g_0,\g_1,\ldots,g_n)$ that are invariant under permutations of
the last $n$ arguments.  They are associated to weights $w_t$ that are
invariant under permutations of the branches of the planar tree $t$.
That is, they depend only on the underlying tree $\tau$ obtained by
neglecting the order of the vertices.  Formally, if $\trt^0_n$ is the
set of rooted trees 
on $\{0,1,\ldots,n\}$ (=labelled trees of $n+1$
vertices), there is a map $\trt^0_n\ni \tau\mapsto t_\tau\in\trp^0_n$
where $t_\tau$ is the planar tree obtained by drawing branches
starting on the root according to the order given by the labels of the
first offspring, and continuing in this way for branches within
branches.  This map is many-to-one, in fact, the cardinality of the 
preimage of a tree
$t$ (=number of ways of labelling the $n$ non-root vertices of a
planar rooted tree with $n$ distinct labels
consistently with the rule ``from high to low") is given by
$$
\b_n(t)\;=\; {n!\over\prod_{i=0}^n s_{v_i}!}\Eq(rel1)
$$
(see e.g. theorem 145B in \cite{but87}).
Therefore, for permutation-invariant functions $b_n$,
we can pass to a sum over trees $\tau\in\trt^0_n$ by changing
$w_t \to w_{t_\tau}/\b_n(t_\tau)$ and we obtain
$$
\r^*_{\g_0} \;=\; \r_{\g_0}\,\sum_{n\ge 0} \;\frac{1}{n!}
\,\sum_{(\g_{1},\ldots ,\g_n)\in\PP^n}\,
\Bigl[\sum_{\tau \in\trt^0_n} \,
\prod_{i=0}^n \,s_i!\;
b_{s_i}(\g_i,\g_{i_1},\ldots,\g_{i_{s_i}})\Bigr]
\; \r_{\g_1} \cdots\r_{\g_n}
\Eq(r.15.1b)
$$
where $i_j$ denotes the descendants of the vertex $i$ and $s_i$ its
coordination number.

Comparing \equ(r.15.1b) with \equ(TP) we immediately
see that Proposition \ref{prop:1} yields
the following criterion for the convergence of cluster expansions.
\begin{corollary}\label{cor:1}
Let $\GG=(\PP,\E)$ be a polymer system and assume
there exist functions $c_n:\PP^{n+1}\to [0,\infty)$, for
$n\in\mathbb{N}$, invariant under permutations of
the last $n$ arguments and satisfying
$$
\card{\phi^{T}(\g_0,\g_1 ,\dots , \g_n)}\;\le\;
\sum_{\tau \in\trt^0_n} 
\,\prod_{i=0}^n 
c_{s_i}(\g_i,\g_{i_1},\ldots,\g_{i_{s_i}})\;.
\Eq(r.16)
$$
Let $\und\r,\und\mu\in[0,\infty)$ and 
$$
\varphi _{\g_0} (\und\mu)\;=\; 1+\sum_{n\geq 1} \frac{1}{n!}\,
\sum_{(\g_{1},\dots ,\g_{n})\in\PP^n} c_n(\g_0,\g_1,\ldots,\g_n)\,
{\mu_{\g_1}}\dots{\mu_{\g_n}}\;,
\Eq(r.15.2)
$$
$\g_0\in\PP$.
Then, the existence of $\und\mu\in[0,\infty)^\PP$ such that
$$
\r_{\g_0}\,\varphi_{\g_0}(\und\mu)\;\le\;\ \mu_{\g_0}
\Eq(r.17)
$$
for each $\g_0\in\PP$, entails the absolute
convergence of the cluster expansion \equ(6)
for the system $\GG$, uniformly in $\Lambda$
and in the activities $\und z$ with $\card{z_\g} \le\r_\g$,
$\g\in\PP$.  Furthermore, if $T(\und\mu)$ and
$T^\infty(\und\mu)$ are defined as in the previous proposition for
$\varphi$ defined in \equ(r.15.2),
$$ \mu_{\g_0}\;\ge\; T_{\g_0}(\und\mu) \;\ge\; \cdots
\;\ge\; T^n_{\g_0}(\und\mu) \buildd{\searrow}{n\to\infty} 
T^\infty_{\g_0}(\und\mu)\;\ge\; \r_{\g_0}\,\Pi_{\g_0}(\und\r)
\Eq(r.18)
$$
for each $\g_0\in\PP$.
\end{corollary}




\vv\vv
\\{\bf 4.2 Penrose bound and the proof of Theorem \ref{th:1}}
\vv
\\
\noindent
The core of our approach is the deep tree identity obtained
by Oliver  Penrose~\cite{pen67} in 1967.
It relies on a map established between graphs and trees defined
in such a way that each connected graph has a unique associated tree, and
each tree defines a unique maximal graph among those mapped into it.
The latter is the direction needed for our purposes.  Consider a finite
graph $G=\bigl(\{v_0,v_1,\ldots,v_n\},E_G\bigr)$ and let $\TT_G$
be the set of subgraphs that are trees (thought of as trees rooted in $v_0$).  
Given a tree $\tau \in\TT_G$ and $i=1,\ldots,n$,
let $d(i)$ denote the tree distance of the vertex $v_i$ to $v_0$.
Let $p$ be the map that to each tree $\tau$ associates the graph 
$p(\tau)\subset G$ formed by adding (only once) to $\tau$
all edges $\{v_i,v_j\}\in E_{G}\setminus E_\tau$ such that either:
\begin{itemize}
\item[(p1)]  $d(i)=d(j)$ (edges between vertices of the same generation), or
\item[(p2)]  $d(i)=d(j)-1$ and $i<j$ (edges connecting to predecessors with smaller index).
\end{itemize}
The set of \emph{Penrose trees} of $G$ is
$$
\mathbb{P}_G \;=\; \bigl\{ \tau\in\TT_G : p(\tau)=\tau \bigr\}\;.
\Eq(r.19)
$$
Thus, a Penrose tree must be such that if two vertices $v_i$ and $v_j$
\begin{itemize}
\item[(t1)]  are descendants of the same predecessor, or
\item[(t2)]  have the same generation number, or
\item[(t3)]  their generation number differs by one and the smaller number
corresponds to $v_{i\wedge j}$, 
\end{itemize}
\\then $\{v_i,v_j\}\not\in E_G$.
\medskip

The remarkable \emph{Penrose identity} says that
$$
\phi^{T}(\g_0,\g_1 ,\dots , \g_n)\;=\; (-1)^n\,
\card{\mathbb{P}_{\GG_{(\g_0,\g_{1},\dots ,\g_{n})}}}
\Eq(r.20)
$$
where $\GG_{(\g_0,\g_{1},\dots ,\g_{n})}$ is the graph of vertices
$\{0,1,\ldots,n\}$ and edges 
$\bigl\{\{i.j\}: \g_i\nsim\g_j, 0\le i,j \le n\bigr\}$.
 To obtain our new criterion, we keep only condition (t1),
 which refers to individual vertices.  That is,
 rather than Penrose trees we consider the larger family of trees $\tau$
 such that 
 $$ \mbox{If $\{i,i_1\}$ and $\{i,i_2\}$ are edges of $\tau$, then $\g_{i_1}\sim\g_{i_2}$}
 \Eq(r.20b)
 $$

In this way we obtain bounds of the form \equ(r.16) with
$$
c_n(\g_0,\g_1,\ldots,\g_n) \;=\;
\prod_{i=1}^n \ind{\g_0\nsim\g_i} \prod_{j=1}^n \ind{\g_i\sim\g_j}\;,
\Eq(r.21)
$$
and Corollary \ref{cor:1} applies with
$$
\varphi _{\g_0} (\und\mu)\;=\; 1+\sum_{n\geq 1} \,\frac{1}{n!}
\sum_{(\g_{1},\dots ,\g_{n})\in\PP^n\atop
\g_0\nsim\g_i\,,\, \g_i\sim\g_j\,,\, 1\le i ,j\le n}
{\mu_{\g_1}}\dots{\mu_{\g_n}}\;,=\; \Xi_{\PP_{\g_0}}(\und \mu)\;.\;\qed
\Eq(r.22)
$$

If we replace in \equ(r.20b) the condition $\g_i\nsim\g_j$ by the
weaker requirement $\g_i\neq\g_j$ we obtain
$$
c^{\rm Dob}_n(\g_0,\g_1,\ldots,\g_n) \;=\;
\prod_{i=1}^n \ind{\g_0\nsim\g_i} \prod_{j=1}^n \ind{\g_i\neq\g_j}\;,
\Eq(r.24)
$$
and
$$
\varphi^{\rm Dob} _{\g_0} (\und\mu)\;=\; 1+\sum_{n\geq 1} 
\frac{1}{n!}\,\sum_{(\g_{1},\dots ,\g_{n})\in\PP^n\atop
\g_0\nsim\g_i\,,\, \g_i\neq\g_j\,,\, 1\le i ,j\le n}
{\mu_{\g_1}}\dots{\mu_{\g_n}}\;=\; 
\prod_{\g\nsim\g_0} (1+\mu_\g)\;,
\Eq(r.25)
$$
which corresponds to Dobrushin condition.   Finally,
if requirement \equ(r.20b) is ignored altogether, 
$$
c^{\rm KP}_n(\g_0,\g_1,\ldots,\g_n) \;=\;
\prod_{i=1}^n \ind{\g_0\nsim\g_i} \;,
\Eq(r.26)
$$
and
$$
\varphi^{\rm KP} _{\g_0} (\und\mu)\;=\; 1+\sum_{n\geq 1} \,\frac{1}{n!}
\sum_{(\g_{1},\dots ,\g_{n})\in\PP^n\atop
\g_0\nsim\g_i\,,\,  1\le i \le n}
{\mu_{\g_1}}\dots{\mu_{\g_n}}\;=\; 
\exp\Bigl[\sum_{\g\nsim\g_0} \mu_{\g}\Bigr] 
\Eq(r.27)
$$
yields the criterion of Koteck\'y and Preiss.


\section*{Acknowledgements}  It is a pleasure to thank Aernout van
Enter and Alan Sokal for useful comments and for a critical reading of
the manuscript (AvE), and Charles-Edouard Pfister and Daniel Ueltschi
for encouragement.  The work of AP was supported by a visitor grant of
CAPES (Coordena\c{c}\~ao de Aperfei\c{c}oamento de Pessoal de
N\'{\i}vel Superior, Brasil).  He also thanks the Mathematics
Laboratory Raphael Salem of the University of Rouen for the invitation
that started the project and for hospitality during its realization. 

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