Content-Type: multipart/mixed; boundary="-------------0605120931144" This is a multi-part message in MIME format. ---------------0605120931144 Content-Type: text/plain; name="06-154.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-154.keywords" Cluster expansions, polymer systems, domino systems, random geometrical objects ---------------0605120931144 Content-Type: application/x-tex; name="aldorob-04-06.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="aldorob-04-06.tex" %\input colordvi \documentclass[11pt]{article} \usepackage{amsfonts,amssymb, graphicx,a4} \def\theequation{\thesection.\arabic{equation}} \newcommand{\zeq}{\setcounter{equation}{0}} \newcommand{\qed}{\hfill\rule{3mm}{3mm}} \topmargin 0cm \textheight 22.5cm \textwidth 16cm \oddsidemargin 0.5cm \renewcommand{\baselinestretch}{1.0} \newtheorem{teorema}{Theorem}%[section] \newtheorem{lema}[teorema]{Lemma}%[section] \newtheorem{proposition}[teorema]{Proposition}%[section] \newtheorem{corollary}[teorema]{Corollary}%[section] \newtheorem{Remark}[teorema]{Remark}%[section] \newenvironment{remark}{\begin{Remark}\rm}{\end{Remark}} \begin{document} %\pagestyle{headings} %%%%%%%%%%%%%%%%% FORMATO %\magnification=\magstep1\hoffset=0.cm \voffset=-1.5truecm\hsize=16.5truecm \vsize=24.truecm \baselineskip=14pt plus0.1pt minus0.1pt \parindent=12pt \lineskip=4pt\lineskiplimit=0.1pt \parskip=0.1pt plus1pt \def\ds{\displaystyle}\def\st{\scriptstyle}\def\sst{\scriptscriptstyle} %%%%%%%%%%%%%%%% GRECO \let\a=\alpha \let\b=\beta %\let\c=\chi \let\d=\delta \let\e=\varepsilon \let\f=\varphi \let\g=\gamma \let\h=\eta \let\k=\kappa \let\l=\lambda \let\m=\mu \let\n=\nu \let\o=\omega \let\p=\pi \let\ph=\varphi \let\r=\rho \let\s=\sigma \let\t=\tau \let\th=\vartheta \let\y=\upsilon \let\x=\xi \let\z=\zeta \let\D=\Delta \let\F=\Phi \let\G=\Gamma \let\L=\Lambda \let\Th=\Theta \let\O=\Omega \let\P=\Pi \let\Ps=\Psi \let\Si=\Sigma \let\X=\Xi \let\Y=\Upsilon %%%%%%%%%%%%%%%% EQUAZIONI CON NOMI SIMBOLICI %%% per assegnare un nome simbolico ad una equazione basta %%% scrivere \Eq(...) o, in \eqalignno, \eq(...) o, %%% nelle appendici, \Eqa(...) o \eqa(...): %%% dentro le parentesi e al posto dei ... %%% si puo' scrivere qualsiasi commento; per avere i nomi %%% simbolici segnati a sinistra delle formule si deve %%% dichiarare il documento come bozza, iniziando il testo con %%% \BOZZA. Sinonimi \Eq,\EQ. %%% All' inizio di ogni paragrafo si devono definire il %%% numero del paragrafo e della prima formula dichiarando %%% \numsec=... \numfor=... (brevetto Eckmannn). \global\newcount\numsec\global\newcount\numfor \gdef\profonditastruttura{\dp\strutbox} \def\senondefinito#1{\expandafter\ifx\csname#1\endcsname\relax} \def\SIA #1,#2,#3 {\senondefinito{#1#2} \expandafter\xdef\csname #1#2\endcsname{#3} \else \write16{???? il simbolo #2 e' gia' stato definito !!!!} \fi} \def\etichetta(#1){(\veroparagrafo.\veraformula) \SIA e,#1,(\veroparagrafo.\veraformula) \global\advance\numfor by 1 % \write15{@def@equ(#1){\equ(#1)} \%:: ha simbolo= #1 } \write16{ EQ \equ(#1) ha simbolo #1 }} \def\etichettaa(#1){(A\veroparagrafo.\veraformula) \SIA e,#1,(A\veroparagrafo.\veraformula) \global\advance\numfor by 1\write16{ EQ \equ(#1) ha simbolo #1 }} \def\BOZZA{\def\alato(##1){ {\vtop to \profonditastruttura{\baselineskip \profonditastruttura\vss \rlap{\kern-\hsize\kern-1.2truecm{$\scriptstyle##1$}}}}}} \def\alato(#1){} \def\veroparagrafo{\number\numsec}\def\veraformula{\number\numfor} \def\Eq(#1){\eqno{\etichetta(#1)\alato(#1)}} \def\eq(#1){\etichetta(#1)\alato(#1)} \def\Eqa(#1){\eqno{\etichettaa(#1)\alato(#1)}} \def\eqa(#1){\etichettaa(#1)\alato(#1)} \def\equ(#1){\senondefinito{e#1}$\clubsuit$#1\else\csname e#1\endcsname\fi} \let\EQ=\Eq %%%%%%%%%%%%%%% DEFINIZIONI LOCALI \def\\{\noindent} \let\io=\infty \def\VU{{\mathbb{V}}} \def\EE{{\mathbb{E}}} \def\GI{{\mathbb{G}}} \def\TT{{\mathbb{T}}} \def\C{\mathbb{C}} \def\LL{{\cal L}} \def\RR{{\cal R}} \def\SS{{\cal S}} \def\NN{{\cal N}} \def\HH{{\cal H}} \def\GG{{\cal G}} \def\PP{{\cal P}} \def\AA{{\cal A}} \def\BB{{\cal B}} \def\FF{{\cal F}} \def\v{\vskip.1cm} \def\vv{\vskip.2cm} \def\gt{{\tilde\g}} \def\E{{\mathcal E} } \def\I{{\rm I}} \def\tende#1{\vtop{\ialign{##\crcr\rightarrowfill\crcr \noalign{\kern-1pt\nointerlineskip} \hskip3.pt${\scriptstyle #1}$\hskip3.pt\crcr}}} \def\otto{{\kern-1.truept\leftarrow\kern-5.truept\to\kern-1.truept}} \def\arm{{}} \font\bigfnt=cmbx10 scaled\magstep1 %%%%%%%%%%%%%%% DEFINIZIONI ROBERTO \newcommand{\card}[1]{\left|#1\right|} \newcommand{\und}[1]{\underline{#1}} \def\1{\rlap{\mbox{\small\rm 1}}\kern.15em 1} \def\ind#1{\1_{\{#1\}}} \def\bydef{:=} \def\defby{=:} \def\buildd#1#2{\mathrel{\mathop{\kern 0pt#1}\limits_{#2}}} \def\card#1{\left|#1\right|} \def\proof{\noindent{\bf Proof. }} \def\qed{ \square} \def\trp{\mathbb{T}} \def\trt{\mathcal{T}} %\BOZZA \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