Content-Type: multipart/mixed; boundary="-------------0211271142345" This is a multi-part message in MIME format. ---------------0211271142345 Content-Type: text/plain; name="02-494.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-494.keywords" Supercritical bond percolation, polymer expansion, Ornstein-Zernike behavior ---------------0211271142345 Content-Type: application/x-tex; name="oz.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="oz.tex" \documentclass[11pt]{article} %\documentstyle[12pt]{article} %%%\usepackage[brazil]{babel} %%%\usepackage[latin1]{inputenc} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsfonts,amssymb, graphicx} \def\theequation{\thesection.\arabic{equation}} %\renewcommand{\baselinestretch}{1.0} \topmargin 0cm \textheight 22.5cm \textwidth 16cm \oddsidemargin 0.5cm \setlength{\parindent}{0cm} \evensidemargin 0 in \oddsidemargin 0.25 in \setlength{\headheight}{2.5 cm} \setlength{\headsep}{0cm} \setlength{\parindent}{0cm} \setlength{\textwidth}{6 in} \setlength{\textheight}{20 cm} \setlength{\topmargin}{2cm} \setlength{\topmargin}{-2 cm} \setlength{\topskip}{0cm} \renewcommand{\v}{\vskip 0.2cm} \newcommand{\vv}{\vskip.4cm} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \renewcommand{\a}{\alpha} \newcommand{\G}{\Gamma} \renewcommand{\b}{\beta} \newcommand{\g}{\gamma} \newcommand{\e}{\epsilon} \renewcommand{\l}{\lambda} \renewcommand{\th}{\theta} \renewcommand{\t}{\tau} \renewcommand{\L}{\Lambda} \newcommand{\D}{\Delta} \renewcommand{\d}{\delta} \renewcommand{\o}{\omega} \renewcommand{\O}{\Omega} \newcommand{\s}{\sigma} \renewcommand{\r}{\rho} \newcommand{\X}{\Xi} \newcommand{\F}{\Phi} \newcommand{\n}{\nu} \newcommand{\la}{\leftarrow} %\renewcommand{\ra}{\rightarrow} \newtheorem{teorema}{Theorem}[section] \newtheorem{Proposition}{Proposition}[section] \newtheorem{definicao}{Definition}[section] \newtheorem{lema}{Lemma}[section] \newtheorem{proposicao}{Proposition}[section] \newtheorem{corolario}{Corollary}[section] \newtheorem{conjectura}{Conjecture}[section] %\renewcommand{\baselinestretch}{2.0} \newcommand{\qed}{\nopagebreak\hfill\fbox{ }} \newcommand{\hR}{\widehat{R}} \newcommand{\hG}{\widehat{\Gamma}} \newcommand{\vk}{\vec{k}} \newcommand{\vx}{\vec{x}} \newcommand{\hta}{\widehat{\tau}} \newcommand{\bc}{\begin{center}} \newcommand{\ec}{\end{center}} \newcommand{\be}{\begin{equation}} \newcommand{\ee}{\end{equation}} \begin{document} \def\<{\langle} \def\>{\rangle} %%%%%%%%%%%%%%%% 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 \def\\{\noindent} %\BOZZA \def\Zd{ \mathbb{Z}^d} \title{Ornstein-Zernike behavior for the Bernoulli bond percolation on $\mathbb{Z}^d$ in the supercritical regime} \author{G. A. Braga, A. Procacci\footnote{Partially supported by Conselho Nacional de Desenvolvimento Cient\'{\i}fico e Tecnol\'ogico - CNPq (Brazil)}, R. Sanchis\footnote{Supported by a graduate scholarship from Coordenadoria de Aperfei\c{c}oamento de Pessoal de N\'\i vel Superior - CAPES (Brazil)} \\ \small{Departamento de Matem\'atica - UFMG}\\ \small{Caixa Postal 1621} - \small{30161-970 - Belo Horizonte - MG - Brazil} } \maketitle \begin{abstract} \noindent We derive an Ornstein-Zernike asymptotic formula for the decay of the two point finite connectivity function $\t^{\rm f}_{x,y}(p)$ of the Bernoulli bond percolation process on $\Z^d$, along the principal directions, for $d\ge 3$, and for supercritical values of $p$ sufficiently near to $p=1$. \end{abstract} \vv{ \small{\bf Key Words:} \small{Supercritical bond percolation, polymer expansion, Ornstein-Zernike behavior}.} {\small {\bf Running title:} Ornstein-Zernike behavior for supercritical bond percolation on $\mathbb{Z}^d$.} \vv\vv \def\Lad{{\mathbb{L}^d}} \def\Ed{{\mathbb{E}^d}} \maketitle \numsec=1\numfor=1 \S 1. {\it Introduction and results} \v Consider the $d$-dimensional cubic lattice, i.e., the infinite graph $\Lad=(\Z^d,\Ed)$ with vertex set $\Z^d=\{x=(x_1, \dots ,x_d): ~x_i\in \mathbb{Z}\}$ and edge set $\Ed=\{\{x,y\}\subset \Z^d : ~|x-y|=1\}$ (nearest neighbors) where $\mathbb{Z}$ is the set of all integers and $|x-y|$ is the usual graph distance, i.e. $|x-y|=\sum_{i=1}^d|x_i-y_i|$. In the translation invariant bond percolation process on $\Lad$ an edge $e\in \Ed$ is open with probability $p$ and closed with probability $1-p$ independently of all other edges. The product measure on the configurations of edges, denoted by $P_p$, is the Bernoulli probability on $\Ed$. Let $C_x$ denote the open cluster at $x$ (i.e., the set of endpoints of the open edges that are connected to $x$) and let $\{x\leftrightarrow y\}$ denote the event $x$ connected to $y$ through open edges. The finite connectivity function is defined as $$ \t^{\rm f}_{x,y}(p)=P_p(x\leftrightarrow y, |C_x|<\infty). \Eq(tau) $$ In this paper we study the behavior of $\t^{\rm f}_{xy}(p)$ in the highly supercritical phase, i.e. for $p$ sufficiently near 1. Our main result can be stated in terms of the following theorem \vv {\bf Theorem 1.1} {\it Let $d\geq 3$ and let $x, y\in\Z^d$ such that $x-y = (0, \cdots , x_\n - y_\n , \cdots , 0)$ for some $\n = 1, 2, \cdots, d$. Then there exists a positive number $r$ such that the two point finite connectivity function can be written as $$ \t^{\rm f}_{x,y}(p)= C(p,x-y) {e^{-m(p)|x-y|}\over |x-y|^{d-1\over 2}} ~~~~\mbox{whenever ~~ ${1\over 1+r}< p <1$} \Eq(oz) $$ where $e^{-m(p)}=(1-p)^{2d-2}[1+f(p)]$ and $C(p,x-y)$ is such that $$ \lim_{x_\n-y_\n\to\infty} C(p,x-y) ={(1-p)^{4d-d^2-1}\over (2\pi)^{d-1\over 2}}[1+g(p)]\Eq(limit) $$ and $f(p)$, $g(p)$ are analytic functions of $p$, ${1\over 1+r}< p <1$, such that $\lim_{p\to 1}f(p)=0=\lim_{p\to 1}g(p)$. } \vv Asymptotic behavior of the form \equ(oz) is known as the {\em Ornstein-Zernike behavior}, after the work of L. Ornstein and F. Zernike \cite{OZ}. In statistical mechanics of spin systems, Paes Leme \cite{PJ} obtained the OZ behavior for the spin-spin correlation function of the $d$-dimensional Ising model at very high temperature, $d\geq 2$, while Schor \cite{Ri} obtained the same result for the truncated spin-spin correlation function at very low temperature and $d\geq 3$. Since then, the OZ behavior has been rigorously established for a variety of models in statistical mechanics and field theory: classical fluids at low density \cite{AK}, strongly coupled lattice gauge theories \cite{Sc,SO1,OBS,BF2}, random surfaces \cite{ACC,BF2,J}, $O(N)$ model at high temperature \cite{BF2}, self-avoiding walks at all noncritical temperatures \cite{CC,M,I}, Ising type models at high temperatures \cite{MZ2, MZ1}, finite range Ising type models for all temperatures above $T_c$ \cite{CIV}, quantum Ising model in a transverse field \cite{K} and surface tension for the low temperature 2D Blume-Capel model \cite{HK}. Concerning in particular the Bernoulli bond percolation process, the Ornstein-Zernike behavior of the two point connectivity function has been proved in the whole subcritical regime \cite{bib:camp-chay-chay} and for all directions \cite{CI}. On the other hand, a rigorous proof of the the Ornstein-Zernike behavior for the {\it finite} connectivity function in the supercritical phase was still lacking. Such a behavior was conjectured to occur at least for $d\ge 3$ and for $p$ near 1, on the basis of the analogy between bond percolation in the highly supercritical regime and the Ising model in the very low temperature regime. %It is thus rather surprising that till today no rigorous %proof of this conjecture has been given. Probably one of reasons is due to the fact %that the main technical tool used in \cite{Ri} to prove the Ornstein-Zernike behavior %in the low temperature $d\ge 3$ Ising model, namely the polymer expansion, %was not available for percolation connectivity in the supercritical %regime till recently (see \cite{BPS} and \cite{PSBS}). \noindent In this paper we proceed with the analysis started in \cite{BPS,PSBS} and we rewrite the finite connectivity in terms of an absolutely convergent polymer type expansion, a key ingredient in order to prove theorem 1.1. We then follow the strategy illustrated in \cite{Ri,Sc,SO1,OBS}, using analyticity of the finite connectivity and Spencer's hyperplane decoupling technique \cite{Sp, SZ} to prove that the convolutive inverse of $\t^{\rm f}_{x,y}(p)$ decay faster than $\t^{\rm f}_{x,y}(p)$. Then, proceeding as in \cite{Si}, we use Rouch\'e's theorem and careful bounds on the first terms of the Newman series for the inverse of $\t^{\rm f}_{x,y}(p)$ to check that such faster decay implies that the Fourier transform of the connectivity is meromorphic in a strip containing the real axis. Moreover, there are two simple poles in this strip localized on the imaginary axis in symmetric positions with respect to the real axis. Finally, the OZ decay follows by a straightforward application of residue theorem. We would like to comment that, once a polymer expansion is available, the strategy described above is not the only possibility, see e.g. \cite{BF2,K,ACC} for possible alternative methods. %The paper is organized as follows. %In section 2 we give the polymer expansion representation %for the finite connectivity. In section 3 we introduce the %hyperplane decoupling technique and we show that, for $p$ close to $1$, %$\tau^{\rm f}_x$ is the kernel of an invertible operator %$\tau^{\rm f}$. Moreover we show that the kernel $\Gamma_x$ of its inverse decays faster %than $\tau^{\rm f}_x$, implying %that $\hat{\G}(k_1,\vk)$, as a function of $k_1$, is analytic in a %strip larger than the one for $\hat{\t^{\rm f}}(k_1, \vk)$. In section %4 we prove that $\hat{\G}(k_1,\vk)$ has two simple zeroes %in this strip, at $k_1=\pm i\o(\vk)$, and that they are the poles of %$\hat{\tau^{\rm f}}(k_1,\vk)$ and obtain some key %properties of $\o(\vk)$. Finally, in section 5 we prove %theorem 1.1. \vv \S 2. {\it Analyticity of the finite connectivity.} \numsec=2\numfor=1 \v In this section we obtain an explicit polymer expansion formula for the finite connectivity following essentially the construction given in \cite{BPS,PSBS}. To make this section self-contained, we make a brief review of some definitions and results we have obtained earlier. If $R$ denotes a finite set we denote by $|R|$ the number of its elements. Consider a set $\L\subset {\mathbb{Z}^d}$ and let $E_{\L}$ be the set of all {edges} $e\in \Ed$ such that $e\subset\L$. Let $\partial \L=\{x\in\L: \exists y\not\in\L~{\rm such~that}~|y-x|=1\}$. Let $\Omega$ be the set of all functions from $\Ed$ to $\{0,1\}$. An element $\omega\in \Omega$, i.e. a {\it configuration} of the system, is a function that assigns to each edge $e\in \Ed$ either the value $1$ (open bond) or $0$ (closed bond). $\Omega_{\L}$ will denote the set of all functions from $E_{\L}$ to $\{0,1\}$. From now on we will assume that $\L$ is finite, e.g., the cube $[-N,N]^d$ in $\Z^d$. So, ${\L\to\infty}$ means simply ${N\to\infty}$. For fixed $p\in (0,1)$, in $\Omega_{\L}$ is naturally defined a probability measure $P_p^\L$, which is simply the restriction to $E_{\L}$ of the Bernoulli probability $P_p$ on $\Ed$. Given a configuration $\omega $ in $\O_{E_{\L}}$ we denote by $O_{\omega }$ the set of all open bonds of $\omega $ and with $C_{\omega }$ the set of all closed bonds in $\omega $. Hence the probability assigned to $\omega $ is simply $P_p^\L(\omega )=p^{|O_{\omega }|}(1-p)^{|C_{\omega }|}$. This probability is normalized, thus $$ \sum_{\omega \in \Omega_{\L}}P^\L_p(\omega )= \sum_{\omega \in \Omega_{\L}}p^{|O_{\omega }|}(1-p)^{|C_{\omega }|}= p^{|E_{\L}|}\sum_{\omega \in \Omega_{\L}} \left({1-p\over p}\right)^{|C_{\omega }|}=1\Eq(1) $$ Let us now define, for $\l\in \C$, the ``partition function'' $$ Z_{\L}(\l)=\sum_{\omega \in \,\Omega_{\L}}\l^{|C_{\omega }|} \Eq(ZL) $$ then, by \equ(1) \def\supg{{{\rm supp}\,\g}} \def\supa{{{\rm supp}\,\a}} \def\supb{{{\rm supp}\,\b}} $$ Z_\L(\l)\Big|_{\l={1-p\over p}} = p^{-|E_{\L}|}\Eq(lambda) $$ Draw now for each {\it closed} edge $e$ of the configuration $\omega $ a $(d-1)$-dimensional unit hyper-square $\D_e$ perpendicular to $e$, with edges parallel to the axis of $\Z^d$, and in such way that $\D_e\cap e$ is a point which coincides with the center of hyper-square $\D$ and the middle point of the segment $e$. We call such hyper-square the {\it dual plaquette to the edge $e$}. The vertices of a plaquette lie in the so called dual lattice shifted by the vector $(1/2, \dots, 1/2)$ with respect to $ \mathbb{Z}^d$. We denote by $\Ed^*$ the set of all plaquettes dual of edges in $\Ed$. In general, given a subset $S\subset \Ed$, we denote by $S^*$ its dual, i.e. $S^*=\{\D_e\in \Ed^*: ~ e\in S\}$. Given two sets $\g\subset \Ed^*$ and $\g' \subset \Ed^*$ we denote $\g\cap \g' =\{\D\in \Ed^*: \D\in \g~{\rm and }~\D\in \g'\}$. Two plaquettes in $ \Ed^*$ are {\it connected} if they share a $(d-2)$-dimensional edge. A set $\g\subset \Ed^*$ is {\it connected} if it is formed by pairwise connected plaquettes. A {\it finite} connected set $\g\subset \Ed^*$ will be called a {\it dual animal}. We will denote by $\G$ the set of all dual animals in $\Ed^*$ and by $\G_\L$ in the set of all dual animals in $E^*_\L$. We recall that a connected subgraph of $\mathbb{L}^d$ is called an {\it animal} $\cal A$ and we denote by $V_{\cal A}$ ($E_{\cal A}$) the set of vertices (edges) of $\cal A$. A dual animal $\g$ with the property that each $d-2$ dimensional edge is shared between two and only two plaquettes is called a {\it Peierls contour}. Note that a Peierls contour is always a closed polyedron, i.e., a $d-1$ dimensional closed surface which has the property that $ V_{\mathbb{Z}^d}$ is uniquely partitioned in two disjoint components, one finite and the other infinite; the finite component is the {\it interior of $\g$} and is denoted by $I(\g)$. Hence $I(\g)\subset V_{\mathbb{Z}^d} $ and $|I(\g)|<\infty$. A {\it contour} is a dual animal $\g$ such that there exists at least one $\g'\subset \g$ which is a Peierls contour. Let $\g\in \G$ be a contour and let $\g_1, \dots, \g_n$ be the set of all the Peierls contours contained in $\g$. Then we denote by $I(\g)=\cup_{i=1}^n I(\g_i)$. The set $I(\g)$, which is a finite subset of $\mathbb{Z}^d$, will be called {\it the interior of the contour $\g$}. Given a contour $\g \in \G$ and $k$ vertices $x_1,\dots,x_k$ we say that {\it $\g$ surrounds $x_1,\dots,x_k$} and write $\g\bigodot \{x_1,\dots,x_k\}$ if there exists a Peierls contour $\g'\subset \g$ and an animal ${\cal A}\subset {\mathbb{L}^d} $ such that $\{x_1,\dots,x_k\}\subset V_{\cal A}\subset I(\g')$, and $E_{\cal A}^*\cap\g=\emptyset$. \\We will say that two dual animals $\g,\g'\in \G$ are {\it compatible}, and we write $\g\sim\g'$, if $\g\cup\g'\notin \G$. \\We will denote by $|\g|$ the cardinality of $\g$, i.e. the number of plaquettes which form the dual animal $\g$. We associate to any dual animal $\g$ an activity $\l^{|\g|}$. With these notations, the partition function \equ(ZL) can be rewritten as $$ Z_{\L}(\l)=1+\sum_{n\ge 1} \sum_{\{\g\}_n:~\g_i\in \G_\L\atop \g_i\sim\g_j} \l^{|\{\g\}_n|}\Eq(OKK) $$ where we denoted shortly $\{\g\}_n =\{\g_1, \dots ,\g_n\}$, $|\{\g\}_n|=\sum_{i=1}^n|\g_i|$. Note that the term 1 is the contribution of the configuration in which all bonds are open. \def\gt{{\tilde\g}} \\We define {\it the finite volume finite connectivity function} as $$ \t^\L_{x,y}(p)= {\rm Prob}(x\leftrightarrow y: C_x\cap\partial\L=\emptyset)= p^{|E_{\L}|}\sum_{{\omega \in \Omega_{\L}\atop \exists C_x\subset O_{\omega}: ~ x\leftrightarrow y} \atop C_x\cap \partial\L=\emptyset} \left({1-p\over p}\right)^{|C_{\omega }|}\Eq(confin) $$ i.e. $\t^\L_{x,y}(p)$ is the probability that $x$ and $y$ are connected through a path of open bonds which does not intersect the boundary of $\L$. Then, $$ \lim_{\L\to \infty}\t^\L_{x,y}(p)= \t^{\rm f}_{x,y}(p)\Eq(con) $$ where $\t^{\rm f}_{x,y}(p)$ is the {\it finite connectivity function} which is the probability that $x$ and $y$ belong to the same {\it finite} open cluster. In term of dual animals the condition that $\omega $ contain a path $C_x$ of open bonds which connects $x$ to $y$ and does not intersect the boundary, corresponds to a configuration of dual animals $\{\g\}_n$ such that there is at least one {\it contour}, say $\g_i$, which surrounds the set $\{x,y\}$, i.e. $\g_i\bigodot \{x,y\}$. This means that in allowed configurations $\{\g\}_n$ no dual animal can be a contour surrounding $x$ but not $y$ and viceversa. It is important to stress that this is a condition {\it on the whole configuration} of dual animals. If an unordered $n$-uple $\{\g\}_n$ of contours satisfies such a condition we also denote it $\{\g\}_n\bigodot \{x,y\}$. Thus, in the contour notation, recalling \equ(lambda), \equ(OKK) and \equ(confin), we can rewrite the finite connectivity as $$ \t^\L_{x,y}(p)=\t^{\L}_{x,y}(\l)\Big|_{\l={1-p\over p}}\Eq(fracc) $$ \\where $\t^{\L}_{x,y}(\l)$ is a function of the complex variable $\l$ (where defined) given by $$ \t^{\L}_{x,y}(\l)={\sum_{n\ge 1} \sum_{\{\g\}_n:~\g_i\in \G_\L,~ \g_i\sim\g_j \atop \{\g\}_n\bigodot \{x,y\}} \l^{|\{\g\}_n|}\over 1+\sum_{n\ge 1} \sum_{\{\g\}_n:~\g_i\in \G_\L\atop \g_i\sim\g_j} \l^{|\{\g\}_n|}} \Eq(frac) $$ Our first aim is to show that in some small disk around $\l=0$ the function $\t^{\L}_{x,y}(\l)$ and its $\L\to \infty$ limit are analytic. To this goal we rewrite the ratio \equ(frac) as a power series in $\l$. In order to do this we will follow the ideas in \cite{BPS} defining a new set of ``connected'' objects called {\it polymers}. \vv {\bf Definition 2.1} {\it A set $\g\in \Ed^*$ is said to be $\{x,y\}$-connected if $\g=\cup_{i=1}^k\g_i$ with $k\ge 1$ such that, for all $i=1,2,\dots ,k$: $\g_i\in \G$; $\g_i$ is a contour; either $\g_i\bigodot x$, or $\g_i\bigodot y$; and $\g_i\sim\g_j$ for all $j\neq i$. } \vv \def\GG{{\mathcal G}} \\We will denote by $\G^{x,y}$ the set of all $\{x,y\}$-connected sets in $\Ed^*$ and by $\G^{x,y}_\L$ in the set of all $\{x,y\}$-connected sets in $E^*_\L$. We will also put $\GG^{x,y}=\G\cup\G^{x,y}$ and $\GG^{x,y}_\L=\G_\L\cup\G^{x,y}_\L$. \vv {\bf Definition 2.2} {\it A set $\g\in \GG^{x,y}$ will be called a $xy$-polymer (or simply polymer when it is clear from the contest). We will say that two polymers $\g_i\in \GG^{x,y}$ and $\g_j\in \GG^{x,y}$ are {\it compatible}, and we write $\g_i\approx\g_j$, if $\g_i\cup\g_j\notin \GG^{x,y}$; viceversa, $\g_i\in \GG^{x,y}$ and $\g_j\in \GG^{x,y}$ are {\it incompatible}, and we write $\g_i\not\approx\g_j$, if $\g_i\cup\g_j\in \GG^{x,y}$. } \vv \def\P{\Pi} Note that if $\g\in \G^{x,y}$ and $\g'\in \G^{x,y}$ then necessarily $\g\not\approx\g'$. To any $\g\in \GG^{x,y}$ we associate an activity $\l^{|\g|}$ where $|\g|$ is the cardinality of $\g$, i.e. the number of plaquettes which form the polymer $\g$. We say that a polymer $\g\in \GG^{x,y}$ {\it surrounds} $\{x,y\}$ if either $\g\in \G$ is a contour such that $\g\bigodot \{x,y\}$, or if $\g=\cup_{i=1}^k\g_i\in \G^{x,y}$ and $\{\g\}_k\bigodot \{x,y\}$. Note that if two polymers $\g$ and $\g'$ are such that $\g\bigodot\{x,y\}$ and $\g'\bigodot\{x,y\}$, then necessarily $\g\not\approx\g'$. \\With this definitions, the r.h.s. of \equ(frac) can be rewritten as $$ \t^{\L}_{x,y}(\l)={\sum_{n\ge 1} \sum_{\{\g\}_n:~\g_i\in \GG^{x,y}_\L,~ \g_i\approx\g_j \atop \exists! \g_i: ~\g_i\bigodot \{x,y\}} \l^{|\{\g\}_n|}\over 1+ \sum_{n\ge 1} \sum_{\{\g\}_n:~\g_i\in \GG^{x,y}_\L\atop \g_i\approx\g_j} \l^{|\{\g\}_n|}} $$ \\Define now, for $\a\in \mathbb{R}$ and $\g\in \GG^{x,y}$ $$ \r_\a( \g)=\cases {(1+\a)\l^{|\g|} &if $\g\bigodot \{x,y\}$\cr\cr \l^{|\g|} &otherwise} $$ then $$ \t^\L_{x,y}(p)= \left. {d\over d\a}\ln \Xi_\L(\a)\right|_{\a=0}\Eq(t) $$ where $$ \Xi_\L(\a)=1+\sum_{n\ge 1} \sum\limits_{\{\g\}_n:~\g_i\in \GG^{x,y}_\L\atop \g_i\approx\g_j} \prod_{i=1}^n\r_\a(\g_i)\Eq(Xi) $$ The function $\Xi_\L(\a)$ is the grand-canonical partition function of a standard hard-core polymer gas in which the polymers $\g$ are elements of $\GG^{x,y}_\L$, they have activity $\r_\a(\g)$ and the hard core condition is expressed by the fact that any two pairs $\g_i, \g_j$ in $\{\g\}_n$ must be compatible with the notion of compatibility being $\g_i\approx\g_j$. As it is well known, its logarithm can be expressed in term of a formal power series (see e.g. \cite{Br}, \cite{Si}). To simplify the expression of this formal power series let us introduce some further notations. Let's thus denote ${\cal B}_\L=\cup_{n\in \mathbb{N}} (\GG^{x,y}_\L)^n $ (${\cal B}=\cup_{n\in \mathbb{N}} (\GG^{x,y})^n $), where $(\GG^{x,y}_\L)^n$ ($(\GG^{x,y})^n $)) is the $n$ times cartesian product of $\GG^{x,y}_\L$ ($\GG^{x,y}$), and let's denote by $\eta$ a generic element of $\cal B$. Thus an element $\eta\in \cal B$ is an the ordered $n$-uple $(\g_1, \dots , \g_n)$ in $(\GG^{x,y})^n$ for some $n\in \mathbb{N}$. If $\eta=(\g_1, \dots , \g_n)$, we denote $|\eta|=\sum_{i=1}^n |\g_i|$ and $\|\eta\|=n$. The formal power series for $\ln \Xi_\L(\a)$ can thus be written as $$ \ln \Xi_\L(\a)= \sum_{\eta\in {\cal B}_\L} {1\over \|\eta\|!}\Phi^{T}(\eta) \r_a(\eta)\Eq(lnxi) $$ where, if $\eta=(\g_1, \dots ,\g_n)$, then $\r_a(\eta)=\prod_{i=1}^n\r_a(\g_i)$ and where {\it the Ursell factor} $\Phi^{T}(\eta)$ is defined as follows: $$ \Phi^{T}(\eta)= \cases{1&if $\|\eta\|=1$\cr\cr 0 &if $\|\eta\|\ge 2$ ~and~ $ g_\eta\notin G_{\|\eta\|}$\cr\cr \sum\limits_{f\in G_{\|\eta\|}\atop f\subset g_\eta}(-1)^{|f|} &if $\|\eta\|\ge 2$ ~and ~$ g_\eta\in G_{\|\eta\|}$ }\Eq(Phi) $$ with $G_{\|\eta\|}$ being the set of all connected graphs with vertex set $\{1, 2,\dots ,\|\eta\|\}$ and $g_\eta$ is the graph with vertex set $V=\{1, \dots ,\|\eta\|\}$ and with edge set $E=\{\{i,j\}:~\g_i\not\approx\g_j\}$. \\Thus, inserting \equ(lnxi) in \equ(t) we have an explicit (formal) expansion for $\t^{\L}_{x,y}(\l)$ given by $$ \t^{\L}_{x,y}(\l)= \sum_{\eta\in {\cal B}_\L\atop \eta\bigodot \{x,y\} }{k_\eta\over\|\eta\|!} \Phi^{T}(\eta) \l^{|\eta|} \Eq(Okei) $$ where, for $\eta=(\g_1,\dots , \g_n)$, the notation $\eta\bigodot\{x,y\}$ means that at least for one $i=1,\dots ,n$ it occurs that $\g_i:~\g_i\bigodot \{x,y\}$. Moreover $$ k_\eta= \#~{\rm of }~\g_i~{\rm such ~that}~\mbox{$\g_i\bigodot \{x,y\}$}\Eq(OK2) $$ By taking the limit $\L\to \infty$ in \equ(Okei) we can also define $$ \t^{\rm f}_{x,y}(\l)= \sum_{\eta\in {\cal B}\atop \eta\bigodot \{x,y\} }{k_\eta \over\|\eta\|!} \Phi^{T}(\eta) \l^{|\eta|} \Eq(Poly) $$ which represents an expansion for the finite connectivity function. It is now a rather standard application of cluster expansion methods to show that \equ(Poly) (and hence {\it a fortiori} \equ(Okei)) is actually an absolutely convergent power series in $\l$ if $|\l|$ is sufficiently small. Therefore the r.h.s. of \equ(Poly) provides an explicit polymer expansion representation of the finite connectivity function for $p$ is sufficiently near to 1, via the following theorem. \vv {\bf Theorem 2.3} {\it Let $\l\in \C$ and let $\t^{\rm f}_{x,y}(\l)$ as in \equ(Poly). Then there exists a positive number $r_0$ such that the function $\t^{\rm f}_{x,y}(\l)$ is an absolutely convergent power series of $\l$ and hence an analytic function for all complex $\l$ in the disk $|\l|k\}$ and $E^{*-}_k=\{\D_{\{x,y\}}\in \Ed^*: y_1From \equ(der0) (in the case $\ell=0$) and representation \equ(Mex) for $M_{x, y}$ in the case $x\neq y$ one can easily check that $$ M_{x,y}(\l,w) \Big|_{w_{k}=0}=0 ~~~~~~~~~~{\rm for} ~~~~x_1 < k\le y_1\Eq(der2) $$ Now we note that since $I=MR$, by the product rule we have, $$ {\partial^{m}M\over\partial w_{k}^{m}}\Bigg|_{w_{k}=0}=- \left[\left(\sum_{i=0}^{m-1} {m\choose i} {\partial^i M\over \partial w_{k}^i} \cdot{\partial^{m-i}R\over\partial w_{k}^{m-i}} \right)M\right]\Bigg|_{w_{k}=0} \Eq(prod) $$ whence, by \equ(der0) and \equ(der2) above, we get immediately the identity \equ(dew). Concerning the proof of identity \equ(dew2) we have $$ {\partial^{2d-2}M_{x,y}(\l,w)\over\partial w_{k}^{2d-2}} \Bigg|_{w_{k}=0}=-\Big[\sum_{u,v\in\Z^d\atop u_1From \equ(uppq) and \equ(Mex) and the polymer expansion \equ(Poly) it follows easily that, for all $|\l|0$ we denote $O(\l^a)$ any analytic function of the form $C\l^a[1+f(\l)]$ with $f(\l)|_{\l=0}=0$ and $C$ constant. In what follow $O(1)$ and $O(\l^a)$ may refer to different functions. Moreover, since $\t^{\rm f}_{x,x}(\l,w=\l)$ does not depend on $x$, we have that $$ [\t^{\rm f}(\l)]^{-1}= {1\over \t^{\rm f}_0(\l)} M(\l)\Eq(tM) $$ where $\t^{\rm f}_0(\l)=\t^{\rm f}_{0,0}(\l)$. \vv \S 4. {\it Location of zeroes of $\hat{M}(k_1,\vk)$} \numsec=4\numfor=1 \v Put $m_2(\l)= (2d-1)(|\ln\l|-|\ln r_2|)$. By \equ(Mxyl) the Fourier transform $\hat{M}(k_1,\vec k)$ of $M_{x,y}$ is analytic for fixed $\vec{k}\in{\R}^{d-1}$ and $|\l|< r_2$, as a function of $k_1\in \mathbb{C} $, at least in the strip $\{|\mbox{Im}k_1|1}\cos(\vk\cdot\vx-k_ix_i)M_{0,\vx}~ + $$ $$ +~\sum_{s=2}^{\infty}s {\sin s k_i\over \sin k_i} \sum_{\vx\in \mathbb{Z}^{d-1}\atop x_i=s, }\cos(\vk\cdot\vx-k_ix_i)M_{0,\vx}\Bigg]~ =~2 \sin k_i\l^{2d-2}O(1) $$ This means that $$ {\partial {\hat M(k_1, \vec{k})}\over \partial k_i}\Bigg|_{k_1=i\o(\vk)} = 2\sin k_i \l^{2d-2}O(1) $$ and thus using \equ(ide) $$ \frac{\partial \o(\vec{k}) }{\partial k_j} = 2\sin k_i \l^{2d-2}O(1) $$ and \equ(dM2) is proved. Concerning the identity \equ(dM3), since $\o(\vk)$ is even, then $ (\partial^2 \o (\vk) /\partial k_i \partial k_j)|_{\vk=0}=0$ if $i\neq j$. In the case $i=j$, by \equ(ide) we have that $(\partial\o(\vk)/\partial k_j)|_{\vk = 0} = 0$, and thus $$ \frac{\partial^2 i\o(\vec{k})}{\partial^2 k_i}\Bigg|_{\vk=0} \frac{\partial \hat M (i\o(\vec{k}), \vec{k})}{\partial k_1}\Bigg|_{\vk=0} + \frac{\partial^2 \hat M (i\o(\vec{k}), \vec{k})}{\partial^2 k_i}\Bigg|_{\vk=0} = 0. \Eq(segunda) $$ and calculating $({\partial^2 \hat M (i\o(\vec{k}), \vec{k})}/{\partial^2 k_i})|_{\vk=0}$ $$ \frac{\partial^2 \hat M (i\o(\vec{k}), \vec{k})}{\partial^2 k_i}\Bigg|_{\vk=0}= {\partial^2 g_0(\vk)\over\partial^2 k_i}\Bigg|_{\vk=0}+ \sum_{n=1}^{\infty} \cosh[n\o(\vk)]{{\partial^2 g_n(\vk)\over\partial^2 k_i}}\Bigg|_{\vk=0} $$ By the same reasoning seen before it is easy to check that ${\partial g_n(\vk)/\partial k_i}$ is at least of the order $\l^{(2d-2)(n+2) +1}$ and hence that $G(\l)= \sum_{n=1}^{\infty} \cosh[n\o(\vk)] ({{\partial^2 g_n(\vk)/\partial^2 k_i}})|_{\vk=0}$ is at least of the order $\l^{2d}$, while $({\partial^2 g_0(\vk)/\partial^2 k_i})|_{\vk=0}= + 2\l^{2d-2}O(1)$ hence $$ \frac{\partial^2 \hat M (i\o(\vec{k}), \vec{k})}{\partial^2 k_i}\Bigg|_{\vk=0}= 2\l^{2d-2}O(1)\Eq(ddMk) $$ Thus, putting \equ(dMk) and \equ(ddMk) in \equ(segunda) we obtain \equ(dM3). \qed \v \vv \S \,5. {\it Proof of theorem 1.1}. \v \numsec=5\numfor=1 >From the definition of the Fourier transform, we have $$ \t^{\rm f}_{(x_1, 0)} = {1\over (2\pi)^d} \int_{V_\pi} d\vk\int_{-\pi}^{+\pi}dk_1e^{ik_1x_1} \hta(k_1, \vk) $$ where $V_\pi=\{\vk\in \mathbb{Z}^{d-1}: -\pi\le k_i\le \pi, ~{\rm for}~i=2, \dots ,d\}$. Suppose, without loss of generality, that $x_1>0$. Let $\e>0$ such that $ \tilde m_2(\l) >\o(\vk)+\e$ (by theorem 4.1 this $\e$ always exists) and consider the rectangle in the complex plane $R= \{k_1\in \mathbb{C}:~ -\pi {\le \rm Re}k_1 \le \pi, ~0\le {\rm Im}k_1\le \o(\vk)+\e \}$ with boundary $\partial R$ (if $x_1$ is negative consider the symmetric of $R$ with respect to the real axis). By theorem 4.1 and due to the choice of $\e$, the function $\hta(k_1, \vk)$ is meromorphic in $R$ with a unique pole at $k_1=i\o(\vk)$. Thus, applying the residue theorem and using that $\hta(k_1, \vk)=\t^{\rm f}_0(\l)/\hat M(k_1,\vk))$ and \equ(dMk) we get $$ \int_{\partial R}e^{ik_1x_1} \hta(k_1, \vk) dk_1 %=2\pi i \lim_{k_1\to i\o(\vk)}(k_1-\o(\vk))e^{ik_1x_1}\hta(k_1, \vk) = 2\pi i\t^{\rm f}_0(\l) e^{-\o(\vk)x_1}r(\vk) $$ where, by equation \equ(dMk) $$ r(\vk)=\left[{\partial \hat M(k_1, \vec{k})\over \partial k_1} \Bigg|_{k_1=i\o(\vk)}\right]^{-1}= {1\over i}O(1) $$ On the other hand, by direct calculation $$ \int_{\partial R}e^{ik_1x_1} \hta(k_1, \vk) dk_1 = \int_{-\pi}^{\pi}e^{ik_1x_1} \hta(k_1, \vk)dk_1 - C_\e(x_1,\vk) e^{-(\tilde \o(\vk)+\e)x_1} $$ where $C_\e(x_1,\vk)=\int_{-\pi}^{\pi}e^{ik_1x_1} \hta(k_1 +i( \o(\vk)+\e), \vk) dk_1$ and note that $|C_\e(x_1,\vk)|\le C_\e$ where $$ C_\e=\sup_{\vk\in \mathbb{R}^{d-1}}\int_{\pi}^{\pi} |\hta(k_1 +i(\o(\vk)+\e), \vk)| dk_1 $$ Thus, using also that $\o(\vk)\ge \o(0)$ (by proposition 4.2) we obtain $$ \int_{-\pi}^{\pi}e^{ik_1x_1} \hta(k_1, \vk)= 2\pi\t^{\rm f}_0(\l) e^{-\o(\vk)}r(\vk) + O(e^{-[\o(0)+\e]x_1}) $$ Then we can write $$ \t(x_1, 0) ={\t^{\rm f}_0(\l)\over {( 2\pi)^{d-1}}}{e^{-\o(0)x_1}\over x_1^{{d-1\over 2}}} \left[x_1^{{d-1\over 2}} \int_{V_\pi}r(\vk) e^{-\D\o(\vk)x_1} {d\vk}\right] + O(e^{-[\o(0)+\e]x_1}). $$ where $\D\o(\vk)=\o(\vk)-\o(0)$. Note that the second term in the above identity decays to zero faster than $e^{-\o(0)|x_1|}$. Now, changing the variable $\vk=|x_1|^{1/2}\vec q$, we get $$ |x_1|^{{d-1\over 2}}\int_{V_\pi}r(\vk) e^{-\D\o(\vk)|x_1|} {d\vk}= \int_{V_{\pi|x_1|^{1/2}}} r(\vec{q}|x_1|^{-1/2}) e^{-\Delta\o(\frac{\vec{q}}{\sqrt{|x_1|}})|x_1|} d\vec{q} $$ By \equ(dM3) we have that $\lim_{\l\to 0}\D\o(\l\vec q)/\l^2=2O(1)\vec q^{\,2}$, hence $$ \lim_{x_1\to\infty} \int_{V_{\pi|x_1|^{1\over 2}}} r\left({\vec{q}\over |x_1|^{1\over 2}}\right) e^{-\Delta\o(\frac{\vec{q}}{\sqrt{|x_1|}})|x_1|} d\vec{q} =r(0) \int_{\mathbb{R}^{d-1}} e^{-2\l^{2d-2}O(1) \vec q^{\,2}}{d\vec{q}}={(2\pi)^{d-1\over 2}\over \l^{(d-1)^2}}O(1) \Eq(limi)$$ which is non-zero. Note that the function $C(\l,x_1)$ appearing in theorem 1.1 is given explicitly by $$ C(\l,x_1)= {\t^{\rm f}_0(\l)\over {( 2\pi)^{d-1}}}\int_{V_{\pi|x_1|^{1/2}}} r(\vec{q}|x_1|^{-1/2}) e^{-\Delta\o(\frac{\vec{q}}{\sqrt{|x_1|}})|x_1|} d\vec{q} + x_1^{{d-1\over 2}} O(e^{-\e x_1}) $$ and, by \equ(limi) and recalling that $\t^{\rm f}_0(\l)=\l^{2d}O(1)$, equation \equ(limit) follows. \qed \v {\it Remark}. Note that the leading exponential decay of $\t^{\rm f}_{x_1, 0}(\l)$ occurs with a rate $m(\l) = \o(0)$. In particular, since $\exp\{-\o(0)\}=\l^{2d-2}O(1)$ we recover a result stated in \cite{PSBS}, saying that $m(\l) = 2(d-2)|\ln \l| + O(\l)$. \vskip.5cm \noindent{\bf Acknowledgements} \noindent We thank Michael O'Carroll and Benedetto Scoppola for useful discussions. \appendix \begin{thebibliography}{99} \bibitem{ACC} D. B. Abraham, J. T. Chayes, L. Chayes: {\it Random surface correlation functions}. Comm. Math. Phys. {\bf 96}, no. 4, 439-471 (1984). \bibitem{AK} D. B. Abraham, H. 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