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Supercritical bond percolation, 
polymer expansion,  Ornstein-Zernike behavior
---------------0211271142345
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\begin{document}

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\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 $|\l|<r_0$ and for all pairs $x,y\in \Z^d$. Moreover
$$
 \t^f_{x,y}(\l)\Big|_{\l={(1-p)/p}}= \t^{\rm f}_{x,y}(p)~~~~~~
\mbox{for all $p$ such that ~~${1\over 1+r_0}<p< 1$}
\Eq(tltp)
$$
where $\t^{\rm f}_{x,y}(p)$ is defined via \equ(con) and \equ(confin).
}
\vv
\\{\it Proof}. The proof is  very similar to that presented in \cite{BPS} for the
analyticity of the percolation probability. For self consistency reasons we give here
a (sketchy) description of this proof.
Define, for $\e\ge 0$
$$
|\t^{\rm f}|_{x,y}(\e)=
\sum_{\eta\in {\cal B}\atop \eta\bigodot \{x,y\} }{k_\eta \over\|\eta\|!}
|\Phi^{T}(\eta)|
\e^{|\eta|}
\Eq(abso)
$$
Then
we get  the bound
$$
|\t^{\rm f}_{x,y}(\l)|\le |\t^{\rm f}|_{x,y}(|\l|)\le
\sum_{n=1}^{\infty}{1\over (n-1)!}
\sum\limits_{\t\in T_n}\sum_{{\eta\in {\cal B}: ~\|\eta\|=n
\atop \eta\bigodot \{x,y\}}
\atop  g_\eta\supset \t }
|\l|^{|\eta|}\Eq(III)
$$
where the factor $|\Phi^{T}(\eta)|$ has been bounded, when non zero,
via the well known tree graph domination formula by the number
of tree sub-graphs in $ g_\eta$ (see e.g. \cite{Br}) and
where $\t$ is a tree graph in $\{1, \dots ,n\}$, $T_n$ is
the set of all connected trees
in $\{1, \dots ,n\}$.
Observe now that, for a fixed tree $\t$, if
$\{i,j\}\in \t$, then we have to sum over $\eta=
(\g_1, \dots \g_n)$ in such way that $\g_i\not\approx\g_j$.
Thus we have to bound quantities of the form
$\sum_{\g\in \GG^{x,y}:~\g\not\approx\g_0}\l^{|\g|}$.
Using the definitions of the previous section we write
$$
\sum_{\g\in \GG^{x,y}\atop \g\not\approx\g_0}\l^{|\g|}\le
|\g_0|\Big[
\sum_{\g\in \G \atop\g\not\sim\D_0}\l^{|\g|}+
\sum_{\g\in \G^{x,y}}\l^{|\g|}\Big]
\doteq
|\g_0|{\sum_{\g\in \GG^{x,y}}}^*
\l^{|\g|}
\Eq(boI)
$$
where, by symmetry,  $\D_0$ is any fixed plaquette;
moreover we have used, due to definitions, 2.1 and 2.2,
that $\GG^{x,y}\backslash \G^{x,y}\subset \G$.
Using \equ(boI) it is
now a standard calculus in the cluster expansion framework (see  \cite{BPS})
to obtain
$$
|\t^{\rm f}|_{x,y}(|\l|)\le
\sum_{\g_1\in \GG^{x,y}
\atop \g_1\bigodot \{x,y\}}(e|\l|)^{|\g_1|}|
\sum_{n=1}^{\infty}{n}\left[{\sum_{\g\in \GG^{x,y}}}^*
(e|\l|)^{|\g|}\right]^{n-1}\Eq(uppr)
$$
The last sum converges if
$$
{\sum_{\g\in \GG^{x,y}}}^*
(e|\l|)^{|\g|}<1\Eq(Kp)
$$
By definition \equ(boI)
$$
{\sum_{\g\in \GG^{x,y}}}^*
(e|\l|)^{|\g|}=
\sum_{n=1}^{\infty}(|\l|e)^{n}B_n +
\sum_{n=2d}^{\infty}(|\l|e)^{n}C_n
$$

where
$B_n=\sum_{\g\in \G: \,|\g|=n,\,\g\not\sim\D_0}1$ and $C_n=\sum_{\g\in \G^{x,y}:\,|\g|=n}1$.
It is easy to check that $B_n\le B^n$ for some positive $B$ depending only on $d$
(see \cite{BPS}).
\\Concerning $C_n$, proceeding as in lemmas 4.2 and 4.3 of \cite{BPS}, we
obtain
$$
C_n
\le
\sum_{k=1}^n
~\sum_{n_1+\dots+n_k=n}~\prod_{i=1}^{k}
\left[2\sum_{\g\in \G:~|\g|=n_i
\atop {\rm either}~ \g\bigodot x,~{\rm or}~\g\bigodot y}1\right]
\le
\sum_{k=1}^n~~ 2^k~
\sum_{n_1+\dots+n_k=n}~\prod_{i=1}^{k}
[2^{d}B]^{2n_i}
=
$$
$$
=
[2^{d}B]^{2n}\sum_{k=1}^{n}2^k\sum_{n_1+\dots+n_k=n} 1\leq
[2^{d}B]^{2n}\sum_{k=1}^{n} 2^k\left({n-1\atop k-1}\right)\le
[3\cdot (2^{d}B)^2]^n\doteq C ^n
\Eq(8b)
$$
We stress that bounds above are not at all optimal.
Thus, if $|\l|<r_0$, where $r_0$ can be chosen so small
to satisfy e.g. the inequality
$$
\sum_{n=1}^{\infty}(B e r_0 )^{n} +
\sum_{n=1}^{\infty}(C e r_0)^{n}< {1\over 2}\Eq(Kopi)
$$
then \equ(uppr) converges and $|\t^{\rm f}|_{x,y}(|\l|)\le 2$.
We stress that the size  of the convergence radius $r_0$ in \equ(Kopi) is not at all optimal.
{\it A fortiori}
we can conclude that also
$\t^{\L}_{x,y}(\l)$ is analytic as a function of $\l$ for  $|\l|<r_0$
and for  all $\L$.
It is now a simple cluster expansion
exercise to show
that $\lim_{\L\to \infty} |\t^{\L}_{x,y}(\l)-\t^{\rm f}_{x,y}(\l)|=0$
whenever $|\l|<r_0$
(see e.g corollary 3.1 \cite {BPS}).
This ends the proof the  theorem 2.3.
$\qed$
\vskip.2cm
{\it Remark}. It is immediate to obtain an upper bound for the (analytic continuation of the)
finite
connectivity function. Just observe that the power
series \equ(III) starts at least with a term proportional to $|\l|^{n_{x,y}}$
where $n_{x,y}= (2d-2)(|x-y|+1)+ 2$ is the the minimal number of plaquettes
of a polymer $\g$ such that $\g\bigodot \{x,y\}$, and hence, for all $|\l|<r_0$ we have

$$
|\t^{\rm f}_{x,y}(\l)|\le|\t^{\rm f}|_{x,y}(|\l|)\le
\left({\l\over r_0}\right)^{n_{x,y}}
|\t^{\rm f}|_{x,y}(r_0)\le 2 \left({\l\over r_0}\right)^{n_{x,y}}\Eq(upper)
$$



\vv






\S 3. {\it Hyperplane decoupling and the convolutive inverse  of $\t^{\rm f}_{x,y}(\l)$}
\v
\numsec=3\numfor=1






Let $k\in\Z$ and let $H_{k}=\{(x_1, x_2, \dots ,x_d)\in \Z^d: x_1=k\}$
be the hyperplanes  perpendicular to the
$x_1$ axis in $\Z^d$. We denote by $E_k\subset \Ed$
the set of edges in $H_{k}$. The set
$E^*_k\subset \Ed^*$, formed by the  plaquettes which are dual to edges in  $E_k$,
will be called the  {\it dual hyperplane}. Moreover we put
$E^*_c= \Ed^*\backslash \left(\cup_{k\in \Z}E^*_k\right)$, i.e. $E^*_c$ is the
set of  plaquettes which  belong to no $E^*_k$.
Note that the dual hyperplane
$E^*_{k}$ is a set of
plaquettes with a ''waffle mold'' shape containing all those
plaquettes that cut edges in the hyperplane $H_{k}$.

\vv
{\bf Definition 3.1}
{\it
We associate to each hyperplane $H_k$ a complex variable $w_k\in \mathbb{C}$.
We will indicate shortly with  $w$ the set of all these  variables
$\{w_{k}\}_{k\in \Z}$. Then
define, for any pair $u,v\in \Z^d$, the function
$\t^{\rm f}_{u,v}(\l,w)$ of variables $\l,w$ as follows:
\def\z{\zeta}
$$
\t^{\rm f}_{u,v}(\l,w)=
\sum_{n=1}^{\infty}{1\over n!}
\sum_{(\g)_n\in (\GG^{u,v})^n \atop \exists \g_i:~\g_i\bigodot \{u,v\} }
k[(\g)_n] \Phi^{T}[(\g)_n]\prod_{j=1}^n\Big[ \prod_{\D\in \g_j}\l_\D\Big]
\Eq(Polyhk)
$$
where
$\l_{\D}=\l$  if $\D\in E^*_c$, and
$\l_{\D}=w_i$ if  $\D\in E^*_i$.
}
\vv
By construction,
$\t^{\rm f}_{u,v}(\l,w)$
is analytic in the polydisk  $|\l|, \, |w_{i}|<
r_0$ and it is equal to $\t^{\rm f }_{u,v}(\l)$ when
$w_{i}=\l$. If $x\in \mathbb{Z}^d$ we will use the notation
$x=(x_1, \vec x)$ with $x_1\in \mathbb{Z}$ and $\vec x\in \mathbb{Z}^{d-1}$.

We now prove the following theorem




\vv
{\bf Theorem 3.2}
{\it In the polydisk  $|\l|, \, |w_{i}|<
r_0$ the function defined in \equ(Polyhk)
satisfies
$$
\frac{\partial^{s}{\t}^{\rm f}_{x,y}(\l,w) } {\partial \l^{s}}\Bigg|_{\l=0} = 0,
~~~~~~~~~~~{\rm for}~~ s\le 2(|\vec x - \vec y|+1)\Eq(deril)
$$
Moreover, for all $k$ such that $x_1 \le  k\le y_1$
$$
\frac{\partial^{s}{\t}^{\rm f}_{x,y}(\l,w) } {\partial w_{k}^{s}}\Bigg|_{w_{k}=0} = 0,
~~~~~~~~~~~{\rm for}~~ s\le 2d-3\Eq(deri-tau)
$$
Finally, for  $k$ such that $x_1 < k < y_1$ we have
$$
{\partial^{2d-2}\t^{\rm f}_{x,y}(\l,w) \over\partial w_{k}^{2d-2}}
\Bigg|_{\l_{k}=0}={(2d-2)!\over \l^2}\sum_{z\in H_{k}}
\t^{\rm f}_{x,z-e_1}(\l,w) \Big|_{w_{k}=0}\t^{\rm f}_{z+e_1,y}(\l,w)
\Big|_{w_{k}=0}
\Eq(2d-2)
$$
while for  $k$ such that $x_1 < k =y_1$ we have
$$
{\partial^{2d-2}\t^{\rm f}_{x,y}(\l,w) \over\partial w_{k}^{2d-2}}
\Bigg|_{\l_{k}=0}={(2d-2)!\over \l^2}
\t^{\rm f}_{x,y-e_1}(\l,w) \Big|_{w_{k}=0}
\s^{\rm}_{y+e_1}(\l,w)
\Big|_{w_{k}=0}
\Eq(2d-2b)
$$
where $e_1=(1,0,\dots, 0)\in \Z^d$ and $\s^{\rm f}_{z+e_1,y}(\l,w)$
is analytic in the polydisk  $|\l|, \, |w_{i}|<
r_0$ (see (3.12)).
}
\vv


\\{\it Proof}.
Concerning  \equ(deri-tau) and \equ(deril) we just observe that
the lowest  power monomial in the expansion \equ(Polyhk) comes from the contribution
in which only one Peierls contour, say $\g$,   surrounding $\{x,y\}$ with
minimal cardinality, i.e. $|\g|= (2d-2)(|x-y|+1)+2$.
If $x_1 \le k \le y_1$, then at  least $2d-2$ plaquettes of $\g$
belong to the dual hyperplane $E^*_{k}$. Moreover, consider the projection $\g^{\bot}$
of $\g$ onto the hyperplane $H_k$. Such a projection is the union of $|\vec x- \vec y| +1$
unit hypersquares and in order to close $\g$ we need two plaquettes belonging to $E^*_c$,
(top and bottom) for each of the hypersquares forming $\g^{\bot}$. Thus $\g$ has at least
least $2(|\vec x- \vec y| +1)$ plaquettes in $E^*_c$.
So
the lowest power of $w_{k}$ in the expansion \equ(Polyhk) will be
$w_{k}^{2d-2}$ and the lowest power in $\l$ will be $\l^{2(|\vec x- \vec y| +1)}$.


Concerning \equ(2d-2), denote $E^{*+}_k=\{\D_{\{x,y\}}\in \Ed^*: y_1>k\}$ and
$E^{*-}_k=\{\D_{\{x,y\}}\in \Ed^*: y_1<k\}$, i.e. $E^{*+}_k$ is the set of all plaquettes
which are above $H_k$ while  $E^{*+}_k$ is the set of all plaquettes
which are below $H_k$.
Note that $E^{*\pm}_{k}\cap E^*_{k}=\emptyset$ and
also $E^{*+}_{k}\cap E^{*-}_{k}=\emptyset$. Let now $z\in H_k$.
Since the plaquette dual to the edge
$\{x,z-e_1\}$ belongs to $E^{*-}_{k}$, in the explicit
expansion for $\t^{\rm f}_{x,z-e_1}|_{w_{k}=0}$ given by
r.h.s. of \equ(Polyhk) with $u=x$, $v=z-e_1$, the
$n$-uples $(\g)_n$ will contribute  to $\t^{\rm f}_{x,z-e_1}|_{w_{k}=0}$ have to
fulfill the following conditions:
$(\g)_n\in \G^n$ where $\G^n$ is the $n$-times Cartesian product of $\G$
(i.e. all $\g_j$ are dual animals); for all
$\g_j$, $\g_j\cap B^*_{k}=\emptyset$; and there exists a unique
contour, say $\g_i$, such that $\g_i\bigodot \{x,z-e_1\} $. Hence
$$
\t^{\rm f}_{x,z-e_1}(\l,w)\Big|_{w_{k}=0}=
 \sum_{n=1}^{\infty}{1\over n!}
\sum_{(\g)_n\in \G^n: ~\g_j\cap E^*_{k}=\emptyset \;\forall j \atop\exists! \g_i:~\g_i\bigodot \{x,z-e_1\} }
\Phi^{T}[(\g)_n]
\prod_{j=1}^n
\Big[ \prod_{\D\in \g_j}\l_\D\Big]\Eq(tsu)
$$
Now recall that, by definition \equ(Phi),
$\Phi^{T}[(\g)_n]=0$ whenever $G[(\g)_n]$ is  a disconnected graph
(remind that $G[(\g)_n]$ is is the graph with vertex set $\{1, \dots ,n\}$ which has the edge
$\{i,j\}$ iff $\g_i\not\approx\g_j$). This means that the sum over ordered $n$-uples  $(\g)_n$ in r.h.s. of \equ(tsu)
is submitted to the further restriction that $ G[(\g)_n]$ has to be a connected graph i.e. the polymers
$\g_1, \dots \g_n$ have to be pairwise incompatible and the set $\cup_{i=1}^n\g_i$,  viewed as a subset of
$ E^*_{\mathbb{Z}^d}$, has to be connected. This condition and  can happen simultaneously
with the condition $\g_j\cap E^*_{k}$ only if all $\g_i\in  E^{*-}_{k}$.
Hence
$$
\t^{\rm f}_{x,z-e_1}(\l,w)\Big|_{w_{k}=0}=
 \sum_{n=1}^{\infty}{1\over n!}
\sum_{(\g)_n\in \G^n: ~\g_j\subset E^{*-}_{k}\;\forall j\atop\exists! \g_i:~\g_i\bigodot \{x,z-e_1\} }
\Phi^{T}[(\g)_n]
\prod_{j=1}^n
\Big[ \prod_{\D\in \g_j}\l_\D\Big]\Eq(tsu2)
$$
analogously for $\t^{\rm f}_{z+e_1,y}
|_{w_{k}=0}$ one obtains
$$
\t^{\rm f}_{z+e_1,y}(\l,w)\Big|_{w_{k}=0}=
 \sum_{n=1}^{\infty}{1\over n!}
\sum_{(\g)_n\in \G^n: \,\g_j\subset E^{*+}_{k}\;\forall j\atop\!\!\!\!\exists \g_i:\,\g_i\bigodot \{z+e_1,y\} }
\Phi^{T}[(\g)_n]
\prod_{j=1}^n
\Big[ \prod_{\D\in \g_j}\l_\D\Big]\Eq(tgiu)
$$


\\We now derive an explicit expansion for $(\partial^{2d-2}\t^{\rm f}_{x,y}/\partial w_{k}^{2d-2})
|_{w_{k}=0}$  for all $k$ such that $x_1<k<y_1$ assuming that $x_1+1<y_1$.
The derivative of order $2d-2$ with respect to
$w_{k}$ is the coefficient of
$w_{k}^{2d-2}$, i.e. the lowest power of $w_{k}$ in the expansion \equ(Polyhk),
times the factor $(2d-2)!$.
The $n$-uples $(\g)_n$ for which the power $w_{k}^{2d-2}$
appears in the r.h.s. of \equ(Polyhk)
are those for
which  all $\g_j\in \G$ (i.e. all $\g_j\in \G$ are dual animals) and there exists among the $(\g)_n$ a
unique {\it contour}, say  $\g_i$, which surrounds
$\{x,z,y\}$, where $z\in H_k$,  and has exactly $2d-2$ plaquettes
in common with $E^*_{k}$. These  $n$-uples corresponds to
configurations in $\Omega$ for
which one and only one site of $H_{k}$, namely $z$, belongs to the finite
cluster containing $x$ and $y$. Hence
$$
{\partial^{2d-2}\t^{\rm f}_{x,y}\over\partial w_{k}^{2d-2}}
\Bigg|_{w_{k}=0}=(2d-2)!\!\!
\sum_{z\in H_{k}} \sum_{n=1}^{\infty}{1\over n!}
\sum_{{(\g)_n\in \G^n: ~ \exists! \g_i:~\g_i\bigodot \{x,z,y\}\atop
|\g_i\cap E^*_{k}|=2d-2}\atop
\g_j\cap E^*_{k}=\emptyset,~\forall j\neq i }
\!\!\!\! \Phi^{T}[(\g)_n]\prod_{j=1}^n
\Big[ \prod_{\D\in \g_j\atop \D\notin E^*_k}\l_\D\Big]=
$$
$$=(2d-2)!\!\!
\sum_{z\in H_{k}}
\sum_{\g\in\G:~\g\bigodot \{x,z,y\}\atop
|\g\cap E^*_{k}|=2d-2}
\prod_{\D\in \g \atop\D\notin E^*_k}
\l_\D
\sum_{n=0}^{\infty}{1\over n!}
\sum_{(\g)_n\in \G^n\atop
\g_j\cap E^*_{k}=\emptyset,~\forall j }
 \Phi^{T}[\g,(\g)_n] \prod_{j=1}^n
\Big[ \prod_{\D\in \g_j}\l_\D\Big]
\Eq(sum)
$$

Consider now the  plaquettes $\D_z^+\in E^{*+}_k$ and $\D_z^-\in E^{*-}_k$ dual to the
edges $\{z,z+e_1\}$ and $\{z,z-e_1\}$ respectively and observe that
$\l_{\D_z^+}=\l_{\D_z^-}=\l$ since they both are plaquettes perpendicular to the $x_1$ axis.
Then, for  $\g$ contour such that $\g\bigodot\{x,z,y\}$ and
$|\g\cap E^*_{k}|=2d-2$ we can associate  uniquely to $\g$ the pair
$\{\g^+,\g^-\}$
where $\g^+ =(\g\cap E^{*+}_{k})\cup \D_z^+$ and
$\g^-= (\g\cap E^{*-}_{k})\cup \D_z^-$. Observe that
$\g^-\bigodot \{x,z-e_1\}$ and $\g^+\bigodot \{y,z+e_1\}$.
Hence the sum over
$\g$ factorizes as
$$
\sum_{\g\in\G:~\g\bigodot \{x,z,y\}\atop
|\g\cap E^*_{k}|=2d-2}
\prod_{\D\in \g \atop\D\notin E^*_k}
\l_\D=
{1\over \l^2}
\sum_{\g^-\in \G: ~\g^-\subset E^*(H^-_{k\n})\atop
\g^-\bigodot \{x,z-e_1\}}\prod_{\D\in \g^-}\l_\D
\sum_{\g^+\in \G: ~\g^+\subset E^*(H^+_{k\n})\atop
\g\bigodot \{z+e_1,y\}}\prod_{\D\in \g^+}\l_\D
\Eq(fac1)
$$
Now observe that the $n$-uple
$(\g)_n\in (\GG^{x,y})^n$ such that $\g_j\cap E^*_{k}=\emptyset,~\forall j$
can always be organized as $(\g)_n=\left((\g^+)_{n_1}, (\g^-)_{n_2}\right)$
such that $n_1+n_2=n$
and $\g^{+}_i\subset E^{*+}_{k}$ for all $i=1, \dots , n_1$ and
$\g^{-}_i\subset E^{*-}_{k}$ for all $j=1, \dots , n_2$.



By definition \equ(Phi) it is not difficult to check that
$$
\Phi^{T}[\g,(\g)_n]=   \Phi^{T}[\g^-,(\g^-)_{n_2}]\Phi^{T}[\g^+,(\g^+)_{n_1}]\Eq(fac2)
$$


Formulas \equ(fac1) and \equ(fac2) then imply that
$$
{\partial^{2d-2}\t^{\rm f}_{x,y}\over\partial w_{k}^{2d-2}}(\l,w)
\Bigg|_{w_{k}=0}= {(2d-d)!\over \l^2}
~~\times
$$
$$
\times~~
\sum_{z\in H_{k}}\Bigg[
\sum_{\g^-\in \G: ~\g^-\subset E^{*-}_{k}\atop
\g^-\bigodot \{x,z-e_\n\}}\prod_{\D\in \g^-}\l_\D
\sum_{n_2=0}^{\infty}{1\over n_2!}
\sum_{(\g^-)_n\in \G^n\atop
\g^-_j\subset E^{*-}_{k},~\forall j }
\l^{|(\g^-)_n|} \Phi^{T}[\g^-,(\g^-)_n]\Bigg]~~\times
$$
$$
\times~~\Bigg[
\sum_{\g^+\in \G: ~\g^+\subset E^{*+}_{k}\atop
\g^+\bigodot \{y,z+e_\n\}}\prod_{\D\in \g^+}\l_\D
\sum_{n_1=0}^{\infty}{1\over n_1!}
\sum_{(\g^+)_n\in \G^n\atop
\g^+_j\subset ^E_{k},~\forall j }
\l^{|(\g^+)_n|} \Phi^{T}[\g^+,(\g^+)_n]\Bigg]=
$$
$$=
{(2d-d)!\over \l^2}\sum_{z\in H_{k}}
\t^{\rm f}_{x,z-e_1}
\Big|_{w_{k}=0}
\t^{\rm
f}_{z+e_1,y}\Big|_{w_{k}=0}
$$
where in last line we used \equ(tsu2) and \equ(tgiu).

The case  $k$ such that $x_1 < k =y_1$, i.e. the proof of identity \equ(2d-2b),
can be treated analogously. We left to the reader
to check that in this case formula \equ(2d-2b) holds with
$$
\s^{\rm f}_{z+e_1,y}(\l,w)=\t^{\rm f}_{y+e_1,y+e_1}(\l,w)+\l
\sum_{n=1}^{\infty}{1\over n!}
\sum_{(\g)_n\in \G^n: ~\g_i\not\bigodot \{y+e_1\}\atop
\g_i\not\bigodot \{y+e_1\},\; \exists \g_j\ni \D^+_y}
\Phi^{T}[(\g)_n]
\prod_{j=1}^n
\Big[ \prod_{\D\in \g_j}\l_\D\Big]\Eq(sig)
$$
where $\D^+_{y}$ denotes the plaquette dual to the edge $\{y,y+e_1\}$
and the sum runs over ordered $n$-uples of contours in $\G$ such that
no contour surrounds $y+e_1$ and there exists a contour which contains the
plaquette $\D_{\{y,y+e_1\}}$.

\qed

\vskip.2cm

As a direct consequence of the convergence of the cluster expansion and
Theorem 3.2 we have the following corollary
\vv
{\bf Corollary 3.3}
{\it
Let $r_1 <r_0$. Then
for all $|w_i|, |\l|\le r_1 $ the following bound holds
$$
|\t^{\rm f}_{x,y}(\l,w)|\le 2
\left({|\l|\over r_1}\right)^{2(|\vec x- \vec y| +1)}
\prod_{i=x_1}^{y_1}\left({|w_i|\over r_1}\right)^{2d-2}
\Eq(utlw)
$$
}
\vv
{\it Proof}.
By theorem 3.2 and recalling \equ(abso) and comment below \equ(Kopi),
we have for all $|\l|, |w|\le r_1$

$$
|\t^{\rm f}_{x,y}(\l,w)|\le \sum_{m\ge 0}r_1^{m} \sum_{\eta\in {\cal B}\atop |\eta|=m}
{k_\eta \over\|\eta\|!}
|\Phi^{T}[\eta]|= |\t^{\rm f}|_{x,y}(r_1)\le 2
$$
The analyticity of $\t^{\rm f}_{x,y}(\l,w)$ in the polydisk $|\l|,|w|<r_0$
 also implies that we can differentiate
$\t^{\rm f}_{x,y}(\l,w)$  by differentiating the series term by term.
Thus, \equ(deril) and \equ(deri-tau) imply that for any $\eta\in \cal B$
the monomial
$\prod_{\D\in \eta}\l_\D$ is of the form
$\l^{2(|\vec x- \vec y| +1)}\prod_{i=x_1}^{y_1}w_i^{2d-2}M_\eta(\l,w)$ where
$M_\eta(\l,w)$ is again a monomial in the variables $\l$ and $w$ which depends on $\eta$
whose total degree is $|\eta| - [2(|\vec x- \vec y| +1) + (2d-2)(|y_1-x_1|+1)]$. Thus for
$|\l|,|w|\le r_1$ and for all
$\eta$ contributing to $\t^{\rm f}_{x,y}(\l,w)$ we get


$$
|\prod_{\D\in \eta}\l_\D|\le |\l|^{2(|\vec x- \vec y| +1)}
\prod_{i=x_1}^{y_1}|w_i|^{2d-2} |M_\eta(\l,w)|\le
\left[{|\l|\over r_1}\right]^{2(|\vec x- \vec y| +1)}\prod_{i=x_1}^{y_1}
\left[{|w_i|\over r_1}\right]^{2d-2} r_1^{|\eta|}
$$
Whence, for all $\l, w$ in the polydisk
$|\l|,|w|\le r_1$
$$
|\t^{\rm f}_{u,v}(\l,w)|\le |\t^{\rm f}|_{x,y}(r_1)
\left[{|\l|\over r_1}\right]^{2(|\vec x- \vec y| +1)}\prod_{i=x_1}^{y_1}
\left[{|w_i|\over r_1}\right]^{2d-2}
$$


\qed

\vskip.2cm
Let $l^2(\Z^d)$ the Hilbert space of the square summable functions $\phi: \Z^d\to \C$
(i.e functions $x\mapsto\phi_x$ such that $\|\phi\|=\sum_{x\in \Z^d}|\phi_x|^2<\infty$).
Define the operator $\t^{\rm f}:l^2(\Z^d) \rightarrow l^2(\Z^d)$ as follows
$$
(\t^{\rm f}\phi)_x=\sum_{y\in\Z^d}\t^{\rm f}_{x,y}(\l,w)\phi_y,\Eq(tf)
$$
Note that, by \equ(upper),
$\t^{\rm f}$ is a  bounded operator in $l^2(\Z^d)$ whenever $|\l|, |w_i| <r_0$.
We now consider the question of invertibility of the operator $\t^{\rm f}$
when $|\l|$ and $|w_i|$ are small. We have to circumvent the problem that
$\t^{\rm f}=0$ when $w_i=\l=0$. We follow the strategy  explained in \cite{Ri}.
Namely, we write
$\t^{\rm f}= P+S$ where $P$ and $S$ are operators with entries
$P_{x,y}(\l,w)= \t^{\rm f}_{x,y}(\l,w)\d_{x,y}$ and
$S_{x,y}(\l,w)= \t^{\rm f}_{x,y}(\l,w)(1-\d_{x,y})$. Then one shows that
$P$ is invertible as long as $w_k,\l\neq 0$ and thus we can rewrite
$\t^{\rm f}= P(I+P^{-1}S)$ where $I$ denotes the identity operator in  $l^2(\Z^d)$.
Finally one checks that $P^{-1}S$ has small norm
when $|w_i|, |\l|$ are sufficiently
small so that also $I+P^{-1}S$ is  invertible.
This will imply that $\t^{\rm f}$ is invertible when $|w_i|, |\l|$ are
sufficiently small and  different from $0$.
In order to develop this strategy
we will need
a lower bound for $\t^{\rm f}_{x,x}(\l,w)$ for $\l, w_i\neq 0$. We thus show the
following proposition.

\vv
{\bf Proposition 3.4}
\label{prop:lowe}
{\it There exists $r_2<r_1$ such that, for all $|\l|, |w_i|\le r_2$,
$$
|\t^{\rm f}_{x,x}(\l,w)|\geq {1\over  2}|\l|^2|w_{x_1}|^{2d-2}\Eq(upper2)
$$
and thus $P$ is invertible in the punctued polydisk
$0< |\l|,|w|\le r_1$.
}
\vv
{\it Proof}.
This is
a direct consequence of the definition \equ(Polyhk). Just observe that
the function $ f_{x,x}(\l,w)=
\t^{\rm f}_{x,x}(\l,w)/ \l^2 w_{x_1}^{2d-2}$ is analytic
and uniformly bounded by $C_{r_1}$ for all $|\l|, |w_i|\le r_1$. Moreover via
\equ(Polyhk) it is easy to check that $ f_{x,x}(\l=0,w=0)=1$ and therefore
we get $|f_{x,x}(\l,w)-1|<1/2$ if $|w_i|,|\l|\le r_2$ for a certain
$r_2<r_1$ and the proposition is proved.
\qed
\v
Let $Q=P^{-1}S$ and $R=I+Q$. We now prove the following theorem.
\vv
{\bf Theorem 3.5}
{\it One can choose $r_2$ so small such that $\|Q\|\le 1/2 $ in the polydisk  $|\l|, |w|\le r_2$
and  hence the inverse
$M=R^{-1}=(I+Q)^{-1}$ exists, it is analytic and its norm $\|M\|\le 2$.
}
\vv
{\it Proof}. The entries
$Q_{x,y}(\l,w)=(1-\d_{x,y})\t_{xy}(\l,w)/\t_{xx}(\l,w)$
are analytic for any $|w_i|,|\l|\le  r_1$ and, by \equ(utlw) and \equ(upper2),
admit the following upper bound for all $|w_i|,|\l|\le  r_2<r_1$
$$
|Q_{x,y}(\l,w)|\le
4
\left({|\l|\over r_1}\right)^{2|\vec x- \vec y|}
\prod_{i=x_1+1}^{y_1}\left({|w_i|\over r_1}\right)^{2d-2}
\Eq(uppq)
$$
To prove that $I+Q$ is an invertible operator in a certain polydisk
$|\l|, |w_i|<r_3$ around the origin, we just have to show that in this polydisk
the norm of the operator $Q$ is small, which is an easy task using the upper bound
\equ(uppq).
Put $\e= r_2/r_1$. By hypothesis $\e<1$. Hence
using  \equ(uppq), for all $|\l|, |w|\le r_2<r_1$ we get:
$$
\|Q\|^2=\sup_{y\in \Z^d}\sum_{x\in \Z^d, \,
x\not=y }|Q_{x,y}|^2
\le C(\e)\e^{2}
$$

where $\lim_{\e\to 0}C(\e)\e^2=0$. Thus choosing $ \e $ so small,
say,  such that $C(\e)\e^{2}<1/2$,  we have that for all
$|\l|, |w| \le r_2=\e \,r_1$ the operator norm $\|Q\|<1/2$ and $I+Q$ is invertible.
Note that the inverse
$$
M=R^{-1}=(I+Q)^{-1}= I +\sum_{n=1}^{\infty} (-1)^n Q^n\Eq(Mex)
$$
is analytic in the polydisk $|\l|, |w| \le r_2$ and is a bounded operator
with norm $\|M\|\le 2$.


\qed
\vskip.2cm
{\it Remark}. Clearly \equ(uppq) is not such a good bound for $|Q_{x,y}(\l)|$ (i.e. when we make
$w_i=\l$). But since we know that $Q_{x,y}(\l)=\t^{\rm f}_{x,y}(\l)/\t^{\rm f}_{x,x}(\l)$ is analytic in the disc
$|\l|<r_2$ with $|Q_{x,y}(\l)|\le 1/2$,  and its expansion as a power series of $\l$ starts at least with a
the power $\l^{(2d-2)|x-y|}$, this is enough to conclude that


$$
|Q_{x,y}(\l)|\le {1\over 2} \left( { |\l|\over r_2}\right)^{(2d-2)|x-y|}\Eq(uppql)
$$




\vskip.2cm
Our aim for the rest of this section is to prove that
$M_{x,y}$ decays faster than  $\t^{\rm f}_{x,y}$ along the principal direction $x_1$.
With this in mind we prove the following theorem.






\vv
{\bf Theorem 3.6}
{\it Let $\l,w$ such that $|\l|,|w|\le r_2$. Take $x,y\in \mathbb{Z}^d$ such that
$|x-y|\ge 2$.
Then it holds
$$
{\partial^{j}M_{x,y}(\l, w)\over\partial w_{k}^{j}}\Bigg|_{w_{k}=0}=0~~~
~~~~~~~\mbox{ for~ $0\le j\le 2d-3$~ and ~ $x_1 < k\le y_1$}\Eq(dew)
$$
and
$$
{\partial^{2d-2}M_{x,y}(\l, w)\over\partial w_{k}^{2d-2}}\Bigg|_{w_{k}=0}=0~~~
~~~\mbox{for  ~ $x_1 < k+1\le y_1$}\Eq(dew2)
$$
}
\vv
{\it Proof:}
Let $u,v\in \mathbb{Z}^d$. We have, by definition,
$$
R_{u,v}(\l,w)= {\t^{\rm f}_{u,v}(\l,w)\over \t^{\rm f}_{u,u}(\l,w)}
$$
Hence, by theorem 3.2 and proposition 3.4,
we also obtain
$$
{\partial^{\ell}R_{u,v}(\l,w)\over\partial w_{k}^{\ell}}
\Bigg|_{w_{k}=0}=0 ~~~~~~~\mbox{ for~ $0\le\ell\le 2d-3$~ and ~ $u_1 < k\le v_1$}\Eq(der0)
$$
and
$$
{\partial^{2d-2}R_{u,v}(\l,w)\over\partial w_{k}^{2d-2}}
\Bigg|_{w_{k}=0}= \left[{1\over \t^{\rm f}_{u,u}(\l,w)}  {\partial^{2d-2}\t^{\rm f}_{u,v}(\l,w)
\over\partial w_{k}^{2d-2}}\right]\Bigg|_{w_{k}=0}~~~~\mbox{ for~  ~ $k\neq u_1$}
$$
Observe that \equ(der0) does not hold when $u_1=k$
(in this case even $R_{u,v}(\l,w)|_{w_{k}=0}\neq 0$).

Hence, for $u_1 < k<v_1$ we can use identity \equ(2d-2) and get
$$
{\partial^{2d-2}R_{u,v}\over\partial w_{k}^{2d-2}}
\Big|_{w_{k}=0}=
{(2d-2)!\over  \l^2}\sum_{z\in H_{k}}
R_{u,z-e_1}(\l,w)\Bigg|_{w_{k}=0}\t^{\rm f}_{z+e_1,v}(\l,w)
\Bigg|_{w_{k}=0}\Eq(2d-2R)
$$
while, for $u_1 < k=v_1$, we can use \equ(2d-2b) to get
$$
{\partial^{2d-2}R_{u,v}\over\partial w_{k}^{2d-2}}
\Big|_{w_{k}=0}=
{(2d-2)!\over  \l^2}
R_{u,v-e_1}(\l,w)\Bigg|_{w_{k}=0}\s_{v+e_1}(\l,w)
\Bigg|_{w_{k}=0}\Eq(2d-2Rb)
$$



>From \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
$$
{\partial^{2d-2}M_{x,y}(\l,w)\over\partial w_{k}^{2d-2}}
\Bigg|_{w_{k}=0}=-\Big[\sum_{u,v\in\Z^d\atop
u_1<k\le v_1}
M_{xu}(\l,w){\partial^{2d-2}R_{u,v}(\l,w)\over\partial w_{k}^{2d-2}}
M_{vy}(\l,w)\Big]\Bigg|_{w_{k}=0}=
$$
$$
= -\Big[\sum_{u,v\in\Z^d\atop
u_1<k< v_1}
M_{xu}{\partial^{2d-2}R_{u,v}\over\partial w_{k}^{2d-2}}
M_{vy}\Big]\Bigg|_{w_{k}=0}-\Big[\sum_{u,v\in\Z^d\atop
u_1<k= v_1}
M_{xu}{\partial^{2d-2}R_{u,v}\over\partial w_{k}^{2d-2}}
M_{vy}\Big]\Bigg|_{w_{k}=0}\Eq(falsa)
$$
We now can use \equ(2d-2R) and \equ(2d-2Rb) to get

$$
{\partial^{2d-2}M_{x,y}\over\partial w_{k}^{2d-2}}\Bigg|_{w_{k}=0}
=
-{(2d-2)!\over  \l^2}\Big[\sum_{z\in H_{k}}\sum_{u\in\Z^d\atop u_1<k}
M_{xu}
R_{u,z-e_1}\sum_{v\in\Z^d\atop k< v_1}\t^{\rm f}_{z+e_1,v}
M_{vy}\Big]\Bigg|_{w_{k}=0}-
$$
$$
-{(2d-2)!\over  \l^2}\Big[\sum_{v\in\Z^d\atop v_1=k}\sum_{u\in\Z^d\atop
u_1<k}
M_{xu}R_{u,v-e_1}\s_{v+e_1}
M_{vy}\Big]\Bigg|_{w_{k}=0}\Eq(Der0)
$$
Now, since $M_{x,u}|_{w_{k}=0}=0$ if $u_1\ge k$
we have
$$
\sum_{u\in\Z^d\atop u_1<k}
M_{xu}
R_{u,z-e_1}\Big|_{w_{k}=0}=\sum_{u\in\Z^d}
M_{xu}
R_{u,z-e_1}\Big|_{w_{k}=0}=\d_{x,z-e_1}=0\Eq(zer1)
$$
and
$$
\sum_{u\in\Z^d\atop
u_1<k}
M_{xu}R_{u,v-e_1}=\sum_{u\in\Z^d\atop}
M_{xu}R_{u,v-e_1}=\d_{x,v-e_1}=0\Eq(zer2)
$$
where $\d_{x,z-e_1}=0$ and $\d_{x,v-e_1}=0$ follow from
the hypothesis that $x_1<k+1$. Inserting now \equ(zer1) and \equ(zer2) in \equ(Der0) the theorem
is proved.
\qed

\vskip.2cm



As an immediate corollary of theorem 3.6 above we have the following
\vv
{\bf Corollary 3.7}
{\it Let $\l,w$ such that
$|\l|,|w|\le r_2$. Take $x,y\in \mathbb{Z}^d$ such that
$|x-y|\ge 2$.
Then
$$
|M_{x,y}(\l,w)|\leq 2\left( {|w_{x_1+1}|\over r_2}\right)^{(2d-2)}\prod_{k=x_1+2}^{y_1}
\left( {|w_k|\over r_2}\right)^{(2d-1)} \Eq(Mxy)
$$
}
\vv
{\it Proof}.
Since by theorem 3.5 $M_{x,y}(\l,w)$ is analytic in the polydisk
$|\l|,|w|\le r_2$ and $\|M\|\le 2$, one can use identities \equ(dew) and \equ(dew2),  and
proceed as in the proof of corollary 3.3.

\qed
\v

The auxiliary variables have done their job and we can now let all variables
$\{\l,w\}$ be equal to $\l$ to get the following immediate corollary.

\vv
{\bf Corollary 3.8}
{\it Let $\l$ such that $|\l|\le r_2$. Take $x,y\in \mathbb{Z}^d$ such that
$|x-y|\ge 2$.
Then
$$
|M_{x,y}(\l)|\leq 2
\left( {|\l|\over r_2}\right)^{(2d-1)\|x-y\|-1} ~~~
\mbox{where $\|x-y\|=\max_{i=1}^d|x_i-y_i|$}\Eq(Mxyl)
$$
}
\vv
{\it Proof}. The bound \equ(Mxyl) follows immediately from the fact
that  \equ(Mxy) can be obtained for any principal direction and by making
all $w$ equal to $\l$.
\qed
\v
{\it Remark}.
The bound \equ(Mxyl) is optimal when $x$ is a vector along some principal direction,
but clearly is not a  good bound in all the other cases.
A  more efficient bound for general $x-y$  can be obtained by noting that, if
$\l^{m_{x,y}}$ is the lowest power in the expansion of $M_{x,y}(\l)$, then,
since we know that $M_{x,y}(\l)$ is analytic in the disc
$|\l|<r_2$ and that $|M_{x,y}(\l)|\le 2$ in in this disc, this is enough to conclude that
$|M_{x,y}|\le 2(|\l|/ r_2)^{m_{x,y}}$.


By \equ(Mex) the lowest power of $\l$ in the expansion of
$M_{x,y}(\l)$ is at least equal to the the lowest power in the expansion
of $Q_{x,y(\l)}=\t^{\rm f}_{x,y}(\l)/\t^{\rm f}_{x,y}(\l)$, i.e.
$\l^{(2d-2)|x-y|}$.
On the other hand, bound \equ(Mxyl) clearly shows that some cancellations
are occurring in the expansion for $M_{x,y}$. So one should reasonably expect
that the lowest power
in the expansion $M_{x,y}(\l)$ is actually bigger that $\l^{(2d-2)|x-y|}$.
As a matter of fact, looking at \equ(Mex), the lowest power $\l^{(2d-2)|x-y|}$ is contained in
the first $|x-y|$ terms of the Neuman expansion  of $M_{x,y}(\l)$, namely
$Q_{x,y}, [Q^2]_{x,y}, [Q^3]_{x,y}, \dots , [Q^{|x-y|}]_{x,y}$. We now show that
these terms cancel in the alternate sum.
Consider first for simplicity the case in which
$x=(0,\vec 0)$ and $y=(n, \vec 0)$. In this case we have that
the power $\l^{(2d-2)n}$ appears only one time in the expansion of
$Q$ since there is only one minimal contour (i.e. with $(2d-2)(n+1)+2$ plaquettes)
surrounding  $x$ and $y$.
It appears   $n$ times in the expansion of $Q^2$, corresponding to
the $n-1$ configurations in which
there is a minimal tube from $x$ to $z$ and a minimal tube
from $z$ to $y$ (where of course $z=(m,\vec 0)$ with $1\le m\le n-1$.
It appears a number of times, say $C_{m,n}$, in the expansion of
$Q^m$, each time corresponding to a configuration in which
there is a minimal contour surrounding $x$, $z_1$, then a minimal contour
surrounding $z_1$,  $z_2$ and so one finishing with a minimal contour surrounding
$z_{m-1}$, $y$ where of course $1\le z_1<z_2\dots <z_{m-1}\le n-1$.
Thus $C_{m,n}$ is exactly the number of ways in which we can choose
$m-1$ objects among $n-1$ objects.
Since in the term $Q^m$ appears the factor $(-1)^{m}$ we have that
the coefficient in front of $\l^{(2d-2)n}$ in the expansion of
$M_{x,y}(\l)$, at least in the case in which $x=(0,\vec 0)$ and $y=(n, \vec 0)$,
is
$$
\sum_{m=1}^n{n-1\choose m-1}(-1)^m=
0
$$
When $x$ and $y$ are not in the same principal direction
then there are
$|x-y|!/( |x_1-y_1|!\cdots |x_d-y_d|!)$ minimal
contours surrounding $\{x,y\}$. But we can repeat for each of these minimal contours
the same reasoning seen before (which does not depend on the shape of the minimal contour but only
on its length).
Hence we can conclude that the coefficient in front of
$\l^{(2d-2)|x-y|}$ in the expansion of $M_{x,y}(\l)$ is actually zero and
thus we have actually a bound as follows


$$
|M_{x,y}|\le 2\left[{|\l|\over r_2}\right]^{(2d-2)|x-y|+1}
~~~~~~\mbox{whenever $|x-y|\ge 2$}\Eq(caz2)
$$
We will use both estimates \equ(caz2) and \equ(Mxyl) in the sequel.
We  will need also a lower  bound for the terms $M_{x,y}(\l)|_{|x-y|=0}$
and  $M_{x,y}(\l)|_{|x-y|=1}$.
Again, by symmetry these terms are constant and independent of $x, y$. We will
put
$$
M_{0}(\l)= M_{x,y}(\l)|_{|x-y|=0} ~~~~~~
M_{1}(\l)=M_{x,y}(\l)|_{|x-y|=1}\Eq(M0M1)
$$
>From \equ(uppq) and \equ(Mex) and the polymer expansion \equ(Poly) it follows easily
that, for all $|\l|<r_2$,

$$
M_{0}(\l)= 1+ f(\l),~~~~~~~M_{1}(\l)=-\l^{2d-2}(1+f'(\l)),
\Eq(Mnon)
$$
where $ f(\l)$ and $ f'(\l)$ are analytic for $|\l|<r_2$ and $f(\l)|_{\l=0}=f'(\l)|_{\l=0}=0$.
For a shorter notation we will denote
$O(1)$ any analytic  function of the form $1+f(\l)$ and, for any
$a>0$ 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 $\{|\mbox{Im}k_1|<m_2(\l)\}$.
We will further show that when $\l$ is real positive the
Fourier transform $\hat{M}(k_1,\vec k)$ has two simples zeroes inside
this  strip  on the imaginary axis at the points
$k_1=\pm i\o(\vec k)$, where $\o(\vec k)$ is real positive.
Thus  we can state the following theorem.
\vv
{\bf Theorem 4.1}
{\it There exists $r_3<r_2$ such that
for any fixed $\vk\in {\Z}^{d-1}$ and
$0<\l<r_3$, the function $\hat{\t^{\rm f}}(k_1,
\vk)$ is meromorphic in the strip $|{\rm Im} k_1|< \tilde m_2(\l)$,
where $\tilde m_2(\l)= (2d-7/4)(|\ln\l|-|\ln r_2|)$,
and has two simple poles in this strip at the points $k_1=\pm
i\o(\vec{k})$ with $\o(\vec{k})$ being
a real positive even  analytic function of $\vec{k}\in{\R}^{d-1}$ such that
$e^{-\o(\vk)}= \l^{(2d-2)}O(1)$.
}
\vv

{\it Proof}. Equation \equ(tM)  implies that
$\hat{\t^{\rm f}}(k_1,
\vk)= \t^{\rm f}_0 \times [1/ \hat{M}(k_1,\vec k)]$. Hence
we just have to show that
$\hat{M}(k_1,\vec k)$ has two simple zeroes inside
the  strip above  on the imaginary axis at the points
$k_1=\pm i\o(\vec k)$.

Let us study in detail the Fourier transform $\hat{M}(k_1,\vec k)$.
First, recalling  that $M_{x,y}=M_{x-y, 0}$, denote
$M_{x}=M_{x, 0}$. Recall also the definition of $M_0=M_{x=0}$ and $M_1=M_{x}|_{|x|=1}$.
We also use the notation $x=(x_1, \vec x)$. With these notations the Fourier transform
of $M_{x}\equiv M_{x_1, \vx}$ is defined as
$$
\hat{M}(k_1, \vec{k}) =
\sum_{x\in \mathbb{Z}^d}e^{-ik_1x_1}e^{-i\vec k \cdot\vec x}{M}_{x_1,\vec x}
$$
Using that $M_{x_1, -\vec x}=M_{x_1, \vec x}=M_{-x_1,\vec x}$ one can rewrite
$\hat{M}(k_1, \vec{k})$ as follows.

$$
\hat{M}(k_1, \vec{k})= g_0(\vec k) + \left[ e^{ik_1} + e^{-ik_1}\right]
g_{1}(\vec k)
+ \sum_{n=2}^{\infty} \left[e^{-ik_1 n}+ e^{+ik_1n}\right]g_n(\vec k)\Eq(Mexp)
$$
where
$$
g_n(\vec k)= \sum_{\vec x\in \mathbb{Z}^{d-1}}\cos(\vec k \cdot\vec x){M}_{n,\vec x}~~~~~~
\mbox{for $n=0,1,\dots$}
\Eq(gn)
$$
Observe that the functions $g_n(\vk)$ are analytic in
$\vk$, real for $\vk$ real and even, i.e., $g(\vk)=g(-\vk)$.
Moreover by \equ(Mnon) and  \equ(Mxyl) it is easy to check that
$$
g_0(\vk)= O(1), ~~~~~g_1(\vec k)=-\l^{2d-2} O(1)\Eq(ord)
$$




Let now $z=e^{ik_1}$ and define the function of the complex variable $z$
$$
C(z, \vec k)= C_0(z, \vec k)+ C_s(z, \vec k)
$$
where
$$
C_0(z, \vec k)= g_0(\vec k) + \left[ z + {1\over z}\right]g_{1}(\vec k)
~~~~~~
C_s(z, \vec{k})=
\sum_{n=2}^{\infty} \left[z^n+ {1\over z^n}\right]g_n(\vec k)\Eq(c0cs)
$$
Since
$C(e^{ik_1}, \vec{k})=\hat{M}(k_1, \vec{k})$,
to find the zeroes of the function $\hat M(k_1, \vk)$ in the rectangle
$\{|\mbox{Im}\, k_1|<m_2(\l), -\pi < |\mbox{Re}\, k_1|\le \pi\}$
is equivalent to find the zeroes of $C(z, \vec k)$ in the
annulus $A\equiv\{e^{-m_2(\l)}<|z|<e^{m_2(\l)}\}$.
Note that the zeroes of $\hat M(k_1, \vk)$ are on the imaginary axis if
the zeroes of $C(z, \vec k)$ are real and positive.
Moreover, since  the functions $g_n(\vk)$ are even in the variable $\vec k$ then,
if $z_0(\vk)$ is a zero of  $C(z, \vec k)$, $z_0(-\vk)=z_0(\vk)$,
or equivalently, if $k_1^0(\vk)$ is a zero of $\hat M(k_1, \vk)$,  then
$k^0_1(-\vk)=k^0_1(\vk)$. Finally, due to the fact that $C(z, \vec k)= C(1/z, \vec k)$, if
$z_0(\vk)$ is a zero of $C(z, \vec k)$ then  $1/z_0(\vk)$ is also a zero,
or, equivalently,  if $k_1^0(\vk)$ is a zero of $\hat M(k_1, \vk)$, then
$-k_1^0(\vk)$ is also a zero.







We will now show that  for any  $\vk\in {\Z}^{d-1}$ and $\l\in\ \mathbb{R}^+$, $\l<r_3$,
one can find an annulus $\tilde A\subset A$ such that the function
$C(z, \vec{k})$ has only two real zeroes, say $\tilde z_1(\l, \vk)$ and
$\tilde  z_2(\l, \vk)=1/\tilde z_1(\l, \vk)$, in  $\tilde A$, and they
are real positive.

First observe that,
by \equ(ord), for fixed $\vk\in {\Z}^{d-1}$ and $\l\in\ \mathbb{R}^+$, $\l<r_3$, the function
$C_0(z, \vk)$ has two real positive zeroes at $z_1(\l,\vk)$ and $z_2(\l,\vk)$,
given by
$$
z_1(\l, \vk) = {{-2g_1(\vec k)/ g_0(\vec k)}\over 1+ \sqrt{1-
4\left[{g_1(\vec k)/ g_0(\vec k)}\right]^2}} =
\l^{2d-2}O(1)~~~;~~~z_2(\l, \vk)= {1\over z_1(\l, \vk)}= \l^{-(2d-2)}O(1)
$$
Note that the minus in the second function of \equ(ord),
which is a consequence of the expansion \equ(Mex),
is crucial to conclude that $z_1(\l, \vk)$ and $z_2(\l, \vk)$ are real positive.

On the other hand, using bound \equ(Mxy), we get easily that
$$
|g_n(\vec k)|\le C_ d \,n^d\,\l^{-1} e^{-m_2(\l)n}~~~ {\rm for}~~~~n\ge 2\Eq(bgn)
$$
hence
$$
|C_s(z, \vec{k})|\le
{C_d\over |\l|}\sum_{n=2}^{\infty} \left[|z|^n+ {1\over |z|^n}\right]n^d\, e^{-m_2(\l)n}
$$
Note that, as expected, the series in
r.h.s. above converges in the annulus $A$ given by
${1\over \e}e^{-m_2(\l)}<|z|<\e e^{+m_2(\l)}$ for any $\e<1$;
moreover, for $\e<1/2$, we also have
$$
|C_s(z, \vec{k})|\le \tilde C_d\l^{-1}\e^2\Eq(pip)
$$
where
$\tilde C_d =2C_d \sum_{n=0}^{\infty}n^d 2^{-n}$. Let us choose $\e= (\l/r_2)^{3/4}$  such
that $\e e^{+m_2(\l)}=e^{+\tilde m_2(\l)}$
where
$\tilde m_2(\l)= \left(2d-{7/4}\right)(|\ln\l|-|\ln r_2|)$
and define the annulus  $\tilde A\subset A$ as
$\tilde A = \{ e^{-\tilde m_2(\l)} < |z| <e^{+\tilde m_2(\l)}\}$.
We will denote $\g_1$ the circle $|z|=e^{-\tilde m_2(\l)}$ and
$\g_2$ the circle $|z|=e^{+\tilde m_2(\l)}$. Thus $\partial \tilde A=\g_1\cup\g_2$
is the boundary
of $\tilde A$.
Since the root $z_1(\l,\vk) $ is of order ${\l}^{2d - 2}$ while
$e^{-\tilde m_2(\l)}$ is of the order ${\l}^{2d - 7/4}$,
one can choose $r_3<r_2$ such small  that
$\{z_1(\l,\vk),1/z_1(\l,\vk)\}\subset \tilde A$ and $\e= (\l/r_2)^{3/4}<1/2$
whenever $\l< r_3$. Thus, by  \equ(pip)
we get
$$
|C_s(z, \vk)|\le {C_d\over r_2^{3/2}}\l^{1/2}~~~~
\mbox{for $z\in \tilde A\cup\partial A$ and $\l<r_3$}\Eq(Bcs)
$$
i.e. $C_s(z, \vk)$ is of order
$\l^{1/2}$ in  $\tilde A\cup \partial \tilde A$. Finally it is easy to show that
$|C_0(z, \vk)|$ is of order
$\l^{-1/4}$ for $z\in\partial A$ and  is of order
$1$ for $z$ in the real negative interval
$I^-=  (-e^{+\tilde m_2(\l)}, -e^{-\tilde m_2(\l)})$.

We can now apply Rouch\'e's theorem to find the zeroes of the
function $C(z, \vec k)= C_0(z, \vec k)+ C_s(z, \vec k)$ in the interior
$I(\g)$
of the closed curve
$\g=\g_1\cup\g_2\cup I$ (with $I^-$ traversed  twice in opposite
directions) and  conclude that $C(z, \vec{k})$ has still two
zeroes in the interior $I(\g)$ of $\g$ (hence these zeroes are the
only two zeroes in the annulus). Note that
$I(\g)=\tilde A\backslash I^-$ and it is easy to show that $C(z, \vec{k})$
is, as $C_0(z, \vec{k})$, of order $O(1)$ in the whole interval
$I^-$. Hence we conclude that $C(z, \vec{k})$ has two zeroes inside
the annulus $\tilde A$.

Finally it is easy to check that the two zeroes $\tilde z_1(\l,\vk)$
and $\tilde z_2(\l,\vk)= 1/\tilde z_1(\l,\vk)$ in the annulus $\tilde A$
lay actually in the
real positive interval $I^+=(e^{-\tilde m_2(\l)}, e^{+\tilde m_2(\l)})$
of $C(z, \vec{k})$ and
$$
\tilde z_1(\l,\vk)=e^{-\o(\vk)}=\l^{2d-2}O(1)\Eq(z12d)
$$

As a matter of fact this follows
from the fact that the function $C_s(z, \vec{k})$ (see \equ(c0cs))
is real (for $\l$ real)  and  by  \equ(Bcs) $|C_s(z, \vec{k})|\le C\l^{1/2}$
for some constant $C$ in the interval $I^+$.
This immediately implies that the function
$C_0(z, \vec k)+ C_s(z, \vec k)$ has at least two zeroes in the interval $I^+$.
More precisely, it has at least
one zero laying in the interval $I_1=(z^-_1,z^+_1)$ around $z_1$ and
at least one zero laying in the interval $I_2=(z^+_2 ,z^-_2)$ around $z_1$, where
$z^+_1$ and $z_2^+=1/z^+_1$ are the solution of the equation
$C_0(z, \vec k)= +C\l^{1/2}$ and $z^-_1$ and $z_2^-=1/z^-_1$
are the solution of the equation $C_0(z, \vec k)= -C\l^{1/2}$.
Since $z^{\pm}_1=\l^{2d-2}O(1)$, \equ(z12d) follows.










Concerning the analyticity of $\o(\vec{k})$, just observe that
$\hat{M}(k_1, \vec{k}) $ is analytic as
a function of $k_1$ and $\vk$. Since $i\o(\vk)$ is a simple zero
of $\hat{M}$, the conclusion
is a straightforward application of the implicit
function theorem. Finally we recall that  $\o(\vec{k})=  \o(-\vec{k})$ follows
from the fact that  the functions $g_n(\vk)$ defined in \equ(gn) are even.
\qed
\v

In order to proof our main theorem we just need to show that the function
$\o(\vk)$ reaches is unique minimum in the square $-\pi<k_i\le \pi$ $i=2,\dots ,d$
at the point $\vk=0$. This occurs at least if one chooses $\l$ strictly positive,
which of course is always the case.


Namely we will show the following proposition

\vv
{\bf Proposition 4.2} {\it For
all $\l\in \mathbb{R}^+$ such that
$0<\l<r_3$, the
following equalities hold.
%$$
%\left. \frac{\partial \o (\vk)}{\partial k_j} \right|_{\vk = 0} = 0\Eq(dM1)
%$$
$$
\frac{\partial \o (\vk) }{\partial k_i}  =2\sin k_i \l^{2d-2}O(1)~~~
\mbox{for $-\pi<k_i<\pi$ $i=2,\dots ,d$}\Eq(dM2)
$$
$$
\left. \frac{\partial^2 \o (\vk) }{\partial k_i \partial k_j}
\right|_{\vk = 0} =2\l^{2d-2}O(1)\d_{ij}\Eq(dM3)
$$
and hence $\vk = 0$ is a global minimum of $\o(\vk )$ for
$\vk \in [-\pi , \pi]^{d-1}$.
}
\vv
{\it Proof}. Recalling \equ(Mexp), by differentiating
the equation $\hat M (i\o(\vec{k}), \vec{k}) = 0$ with respect to
$\vk_j$, for
$j\not = 1$, we obtain the identity
$$
\frac{\partial i\o(\vec{k}) }{\partial k_j}
{\partial{\hat M}(i\o(\vec{k}), \vec{k})\over\partial k_1 } +
{\partial {\hat M(i\o(\vec{k}), \vec{k})}\over \partial k_j}=0 \Eq(ide)
$$
%Since $k_1=\pm i\o(\vec{k})$ are simple zeroes of $\hat M (k_1, \vec{k})$ for any
%fixed $\vk$,
%then $\partial{\hat M}(i\o(\vec{k}), \vec{k})/\partial k_1\neq 0$. Moreover
%since $g_n(\vk)$ defined in
%\equ(gn) is even (and hence $\partial g_n(\vk)/\partial k_j$ is odd),
%the $k_j$ derivative  of $\hat M $ is zero at
%$k_j=0$. This gives \equ(dM1).

\vskip.2cm
Observe now that,
$$
{\partial {\hat M(k_1, \vec{k})}\over \partial k_1}\Bigg|_{k_1=i\o(\vk)}
=- i \left\{ \sinh[\o(\vk)]g_1(\vk) +\sum_{n=2}^{\infty} \sinh[n\o(\vk)]g_n(\vk)\right\}
=+iO(1)\Eq(dMk)
$$
since, recalling that $\sinh[\o(\vk)]=\l^{-(2d-2)}O(1)$ and \equ(ord), we have
that
$\sinh[\o(\vk)]g_1(\vk)= -O(1)$, while, using \equ(bgn), we get
$\sum_{n=2}^{\infty} \sinh[n\o(\vk)]g_n(\vk)=O(\l)$.


On the other hand
$$
{\partial {\hat M(k_1, \vec{k})}\over \partial k_i}\Bigg|_{k_1=i\o(\vk)}
= \left\{ {{\partial g_0(\vk)\over\partial k_i}}+
\sum_{n=1}^{\infty} \cosh[n\o(\vk)]{{\partial g_n(\vk)\over\partial k_i}}\right\}
\Eq(pir)$$



Now, for $n\ge1$ that, we have
$$
{\partial g_n(\vk)\over\partial k_i}=  -
\sum_{\vx\in \mathbb{Z}^{d-1}} x_i\sin(\vk\cdot\vx)M_{n,\vx}
=-2\sum_{s=1}^{\infty}s \sin(sk_i)\sum_{\vx\in \mathbb{Z}^{d-1}\atop x_i=s}
\cos(\vk\cdot\vx-k_ix_i)M_{n,\vx}=
$$
$$
=
-2\sin k_i\sum_{s=1}^{\infty}s {\sin(s k_i)\over \sin k_i}\sum_{\vx\in \mathbb{Z}^{d-1}\atop x_i=s}
\cos(\vk\cdot\vx-k_ix_i)M_{n,\vx}=\sin k_i O(\l^{(2d-2)(n+1) +1})
$$
where in the last line
we have used the bound \equ(caz2) and the fact that $ |{\sin(sk_i)/ \sin k_i}|\le s$.
Hence the function
$F(\l)=\sum_{n=1}^{\infty} \cosh[n\o(\vk)](\partial g_n(\vk)/\partial k_i)$
which we know to be analytic in $\l$ at least in the disc $|\l|<r_3$,
starts with a power which is at least of the
order $\l^{2d-1}$ with a coefficient proportional to $\sin k_i$.
Hence $F(\l)=\sin k_iO(\l^{2d-1})$ (here the signal of $O(\l^{2d-1})$ is out of control).

On the other hand, recalling that $M_1=-\l^{2d-2}O(1)$ and using again
bound \equ(caz2), we have
$$
{\partial g_0(\vk)\over\partial k_i}= - 2\sin k_i \Bigg[M_{1} +
\sum_{\vx\in \mathbb{Z}^{d-1}\atop x_i=1, \,|\vx|>1}\cos(\vk\cdot\vx-k_ix_i)M_{0,\vx}~
+ $$
$$
+~\sum_{s=2}^{\infty}s {\sin s k_i\over \sin k_i}
\sum_{\vx\in \mathbb{Z}^{d-1}\atop x_i=s, }\cos(\vk\cdot\vx-k_ix_i)M_{0,\vx}\Bigg]~
=~2 \sin k_i\l^{2d-2}O(1)
$$



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






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