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\def\Cor{{\rm Cor}\,}%correlation
\def\cor{{\rm Cor}\,}%correlation
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\def\Cov{{\rm Cov}\,}%correlation
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\title{Remarks on decay of correlations
and Witten Laplacians\\
-- Brascamp-Lieb inequalities and semiclassical limit --}
\author{Bernard Helffer\\ UA 760 du CNRS, D\'epartement de
math\'ematiques\\
Bat 425\\
F-91405 Orsay C\'edex FRANCE}
\date{November 18, 1997}
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\maketitle
%
%RESUME
%
\begin{abstract}
As it appears in recent articles by
Helffer or Sj\"ostrand and Naddaf-Spencer, the
analysis, in the context of the statistical mechanics,
of measures of the type $\exp - \Phi (x) \; dx$ is connected with
the analysis of suitable Witten Laplacians on
$1$-forms. For illustrating this point of view,
we present here remarks about the Brascamp-Lieb inequalities and its
extensions and prove the decay of the
correlation
in some cases when $\Phi$ is weakly non convex. \end{abstract}
% %FIN DE RESUME %
\section{Introduction}
Our aim\footnote{ A first version of these remarks was diffused in June 1996.} is to analyze Laplace integrals associated to a measure whose density with respect to the Lebesgue measure
takes the form $\exp -\frac 1h \Phi$, up to a multiplicative normalization constant, in the case when the potential $\Phi$
is weakly convex or weakly non convex. We analyze as a starting
point the Brascamp-Lieb inequality or the Poincar\'e inequality
in connection with the lowest eigenvalue of a suitable Witten
Laplacian on one-forms. The role of this Laplacian which appears
implicitly in \cite{Sj1990}, \cite{Sj1991}, \cite{HeSj1993} and \cite{He1993b} was emphasized in
\cite{Sj1994b} with new applications. This was then used in \cite{NaSp} and \cite{He1994e} in
connection with the Brascamp-Lieb inequality. We have in mind applications to a
potential of the form
\begin{equation}\label{a1}
\Phi(x) = \sum_{j=1}^m v (x_j) + \frac{\Jg}{2} \sum_{j=1}^m |x_j-x_{j+1}|^2
\end{equation}
with the convention that $x_{m+1} = x_1$ and where the one particle phase
$v$ takes the form $v(x) = \lambda x^4 + \nu x^2$.
The parameters $\lambda$ and $\nu$ (possibly depending on $h$) satisfy for the moment
\begin{equation}
\label{i1}
\lambda> 0 \;,
\end{equation}
and \begin{equation}
\label{a2}
\nu\geq 0\;,
\end{equation}
but the case $\nu<0$ will also be analyzed under a condition of the type\break $\nu \geq -\epsilon(\lambda)$.
More generally we will be interested in the similar problem attached
to a $d$-dimensional (periodic) lattice $\Lambda$ (identified (modulo translation) with a subset of $\zz^d$)
\begin{equation}\label{a1a}
\Phi(x)= \sum_{\ell\in \Lambda} v (x_\ell) + \frac{\Jg}{2}
\sum_{j\sim k} |x_j-x_{k}|^2\;,
\end{equation}
where $j\sim k$ means that $j$ and $k$ are nearest neighbors in
$\Lambda$
considered as living on a torus.\\
With this technique we can also consider examples like
\begin{equation}\label{a1b}
\Phi(x)= \sum_{\ell\in \Lambda} v_\ell (x_\ell) + \Phi_i (x) \;,
\end{equation}
where $\Phi_i(x)$ is a convex interaction potential with uniformly
bounded
second derivatives and $t\mapsto v_\ell(t)$ is a family of potentials
whose dependence with respect to $\ell$ is controlled uniformly.\\
Our main problem will be to analyze the properties of the measure
\begin{equation}
\label{i3}
d\mu:=\exp - \frac{\Phi(x)}{h} dx / \left(\int \exp - \frac{\Phi(x)}{h} dx \right)
\end{equation}
and more precisely the covariance associating to $(f,g)$
\begin{equation}
\label{i4}
\cov(f,g) = \langle (f-\langle f\rangle)(g-\langle g \rangle)\rangle
\end{equation}
where $\langle\; \cdot\; \rangle$ denotes the mean value with respect to the
measure
$d \mu$.\\As usual we denote the variance by
\begin{equation}
\label{i5}
\var g := \cov (g,g)\;.
\end{equation}
We shall sometimes use the notation $\var^{(m)} g$ and $\Cov^{(m)}(f,g)$ if we want to refer to the family
of phases $\Phi=\Phi^\Lambda$ with $\Lambda=\{1,\cdots,m\}$ and follow the uniformity with respect to $m$.
\section{About Brascamp-Lieb}
We follow partially \cite{Sj1994b} or \cite{He1994e}.
H.J.Brascamp and E.Lieb \cite{BraLi1976} have proved the following celebrated inequality
which plays an important role in different contexts in the study of
the Schr\"odinger equation.
\begin{theorem} \label{Theoremli6.1}:\\
Let $F(x)=\exp(-\Phi(x)), x\in \rz^m$, with $\Phi$ in $C^2 $
and strictly convex. We assume that $\Phi$ has a minimum and consequently
$F$ decays exponentially in all directions.
Let $g \in C^1(\rz^m)$, and let us assume that $ \var\; g < \infty $. Then
\begin{equation} \label{li6.1}
\var g \leq \langle~\nabla g \cdot ((\Hess \Phi\,)^{-1}\nabla g)~\rangle
\end{equation}
where $\nabla g$ is the gradient of $g$.
\end{theorem}
\begin{remark}:\\
In the semi-classical case, we obtain (for the normalized measure associated with $\exp -
\frac{\Phi}{h} dx$),
\begin{equation} \label{li6.1a}
\var g \leq h \; \langle~\nabla g \cdot \left( (\Hess \Phi\,)^{-1}\nabla
g \right) ~\rangle
\end{equation}
\end{remark}
We first recall a proof of this theorem inspired by
\cite{HeSj1993}, \cite{He1993b}, \cite{Sj1994b} and \cite{He1994e}
which was only given for $\Phi$ uniformly strictly convex and with
bounded second derivatives. We will remain rather sketchy and refer to
the complete study given by J. Johnsen in \cite{Jo} for a general
justification. Note
nevertheless that the proof is easier to verify in the case of our
main example (\ref{a1}). For $g$
in $C^1$, such that $\nabla g$ is bounded, we have seen that there exists
$f$ such
that $\langle f\rangle =0$ and
\begin{equation} \label{li6.2a}
g - \langle g \rangle = \nabla \Phi \cdot \nabla f - \Delta f =: A_0 f\;.
\end{equation}
The operator $A_0$ is selfadjoint\footnote
{We do not discuss in detail the problem of the essential selfadjointness
of $A_0$ and later of $A_1$. These problems are interesting in themselves. Let
us just observe
that we consider here the Friedrichs extension and that, for our particular examples,
the essential selfadjointness can be easily analyzed because
the operators are globally quasielliptic
(see Helffer-Robert \cite{HeRo}, Helffer \cite{He}). A more general study is developed
in Johnsen \cite{Jo}.}
on $L^2(\rz^m;\exp - \Phi)$.\\
If we take the formula giving the variance, we get
\begin{equation} \label{BraLi3}
\begin{array}{ll}
\var (g) &\stackrel{\rm def}{=} \int (g-\langle g \rangle )^2 \exp -\Phi\; dx /
\int \exp -\Phi \; dx \quad \quad \quad \\
&= \int (g-\langle g \rangle )(\nabla f \cdot \nabla \Phi - \div \nabla f) \exp -\Phi \;dx/
\int \exp -\Phi \; dx \\
& = ( \int \nabla f \cdot \nabla g \; \exp -\Phi\;dx) / \int \exp -\Phi \; dx
\end{array}
\end{equation}
As in \cite{HeSj1993}, we get by differentiation of (\ref{li6.2a}) and
with $v:=\nabla f$,
\begin{equation}\label{li6.5}
\nabla g = A_1 v\;,
\end{equation}
where $A_1$ is the operator
$$ v\mapsto A_1 v = (\nabla \Phi \cdot \nabla -\Delta) v + \Hess \Phi \cdot v \;, $$
which can be interpreted as a Witten Laplacian.
This operator is positive and actually strictly positive
(as first observed in \cite{Sj1994b}) under rather weak
assumptions\footnote{ The main assumption is the existence of
$\delta >0$ such that $x\cdot \nabla \Phi
(x)\geq \delta^{1+\delta} - \frac{1}{\delta}$.} but {\bf without any assumption
of strict convexity}. The proof given in \cite{Sj1994b} has been extended in \cite{Jo}
where technical conditions previously imposed by J. Sj\"ostrand are eliminated.
In this paper, we shall actually get
the strict positivity by explicit lower bounds.\\
We can then rewrite (\ref{BraLi3}) as
\begin{equation} \label{li6.12}
\var (g) = \left(\int A_1^{-1} \nabla g\cdot \nabla g\, \exp -\Phi\;dx \right) / \left( \int \exp -\Phi \; dx\right)
\end{equation}
{\bf In the case} when $\Phi$ is uniformly strictly convex, we observe
the following inequality between selfadjoint operators
\begin{equation}\label{bra1}
A_1 \geq \Hess\Phi\geq \sigma>0\;,
\end{equation} and, using abstract analysis (extending the result
mentioned in \cite{Ru1969}),
we obtain
\begin{equation}\label{bra2}
A_1^{-1} \leq (\Hess \Phi)^{-1}\;.
\end{equation}
The Brascamp-Lieb inequality is then an immediate consequence of (\ref{li6.12}).
\\
{\bf In the general case}, let us denote by
$\rho_1$ the lowest eigenvalue of $A_1$ which satisfies (as accounted
for above)
\begin{equation}\label{li6.11}
\rho_1>0\;.
\end{equation}
We now deduce from (\ref{li6.12}) the upper bound
\begin{equation} \label{li6.13}
\var (g) \leq \rho_1^{-1} \left( \int \nabla g \cdot \nabla g \,\exp
-\Phi\;dx \right) / \left( \int \exp -\Phi \; dx \right)\;.
\end{equation}
This is of course a stronger result than the following consequence of the
Brascamp-Lieb inequality in the convex case~:
\begin{equation}\label{li6.16}
\var (g) \leq \left(\inf_x \lambda_{min} ( \Hess \Phi (x)) \right)^{-1}
\left(\int \nabla g \cdot \nabla g \,\exp -\Phi\;dx\right)
/ \left( \int \exp -\Phi \; dx\right)\;.
\end{equation}
We have indeed in this case
\begin{equation}\label{li6.17}
\rho_1 \geq \inf_x \lambda_{min} ( \Hess \Phi (x))
\end{equation}
as a consequence of (\ref{bra1}).
But what is the most interesting here is that the proof of (\ref{li6.13})
is independent of the convexity!
\\
Coming back to the semi-classical situation, J. Sj\"ostrand (\cite{Sj1994b}) observed
that, after conjugation by the operator of multiplication by $\exp -
\frac{\phi}{2h}$, the operator $h^2 A_1$ (defined by starting from
the potential $\Phi/h$) becomes the following
more standard Witten Laplacian $W_1$,
\begin{equation}
\label{w1}
W_1 := \left[\sum_j
\left(-h \pa/\pa x_j +\frac 12 \pa \Phi/\pa x_j\right) \left(h\pa/ \pa x_j
+\frac 12 \pa \Phi/\pa x_j \right)\right]\otimes I + h \;\Hess \Phi\;,
\end{equation}
defined on the $L^2$ $1-$forms with respect to the standard Lebesgue
measure on $\rz^m$, with $m=|\Lambda|$.
Let us recall also that $W_1$ is related to the Witten Laplacian $W_0$ on the $0$-forms
by
\begin{equation}
W_0:= \left[\sum_j
\left(-h \pa/\pa x_j +\frac 12 \pa \Phi/\pa x_j\right) \left(h\pa/ \pa x_j
+\frac 12 \pa \Phi/\pa x_j \right)\right]
\;,
\end{equation}
through the identity
\begin{equation}
W_1= W_0\otimes I + h\; \Hess \Phi\;.
\end{equation}
The basic philosophy that we want to develop is that in many cases occuring in statistical
mechanics, the results obtained in the uniformly
strictly convex situation by use of the
strictly positive constant
\begin{equation}
\rho_0:=\inf_{x}\lambda_{min}(\Hess\Phi (x))\;,
\end{equation}
will also be true in non-convex situations with $\rho_0$ replaced by $\rho_1$. This will
be particularly important in the case when one can find
strictly positive lower bounds of $\rho_0$ or $\rho_1$ which are suitably controlled
with respect to $\Lambda$ or the parameter $h$.
\section{Lower bound for the spectrum of the Witten Laplacian in the
semi-classical case}
The aim of this section is to show that the approach developed in
the preceding section is performant. As a typical example, we prove the following
\begin{theorem}\label{ThWitten}:\\
Let $m\in \nz$ and $\Phi^{(m)}=\Phi$ the phase on $\rz^m$
\begin{equation}\label{3.1}
\Phi(x) = \sum_{j=1}^m \lambda_j x_j^4 + \sum_j \nu_j x_j^2+ \frac{\Jg}{2} \sum_j|x_j-x_{j+1}|^2\;,
\end{equation}
where the $\lambda_j$ satisfy
\begin{equation}\label{general}
0<{\underline \lambda}\leq \lambda_j\;.
\end{equation}
and $\nu_j$ satisfies for some
$j$\- and $m$\- independent sufficiently small $\epsilon_0 >0$
\begin{equation}
\nu_j\geq - \epsilon_0 h\;.
\end{equation}
Then there exists
$c>0$ and $h_0$ such that, for all
$m$ and all $h$ such that $00$ and $h_0$ such that, for all
$m$ and all $h$, such that \break $01$.
\item In the case of a one dimensional lattice ($d=1$), these problems can
also be analyzed through
the
technique of the transfer matrix, that is by the analysis of a spectral
problem attached to the operator
$$K_v=\exp -\frac{v}{2h} \;\cdot\;\exp h\frac{d^2}{dx^2}\;\cdot\;
\exp -\frac{v}{2h}\;.$$
\item This problem can also be analyzed through Sokal's approach \cite{Sok}.
\end{itemize}
\end{remark}
\noindent {\bf Proof of Theorem \ref{ThWitten}}:\\
Letting $X_j = h \pa_j + \frac 12 \pa_j\Phi $, we start from
\begin{equation}
\label{w5}
\langle W_1 u\;|\; u\rangle_{L^2} = \sum_{j,k} ||X_ku_j||^2
+ h \sum_{j,k}\int \frac{\pa^2\Phi}{\pa x_j\,\pa x_k} u_j\;u_k \;dx\;.
\end{equation}
We first ``omit'' the terms $||X_k\;u_j||^2$ with $k\neq j$
\begin{equation}
\label{w6}
\begin{array}{ll}
\langle W_1 u\;|\; u\rangle_{L^2} &\geq \sum_{j}\left( ||X_j u_j||^2 +
h \int \frac{\pa^2\Phi}{\pa x_j\,\pa x_j} u_j^2 dx\right)\\
&
+ h \sum_{j\neq k}\int \frac{\pa^2\Phi}{\pa x_j\,\pa x_k} u_j\;u_k\;
dx\;.
\end{array}
\end{equation}
We then analyze for fixed $j$ the term
$$
\langle w_j u_j \;|\; u_j\rangle := ||X_j u_j||^2 +
h \int \frac{\pa^2\Phi}{\pa x_j\,\pa x_j} u_j^2 \;dx\;.
$$
Easy computations give
\begin{equation}\label{3.9}
\label{w10}
\langle w_j u_j \;|\; u_j\rangle = || h\pa_{x_j} u_j||^2
+ \frac 14 ||(\frac{\pa\Phi}{\pa x_j})\; u_j||^2 + \frac h2 \int \frac{\pa^2\Phi}{\pa x_j\,\pa x_j} u_j^2\; dx\;.
\end{equation}
Of course we have the lower bound
\begin{equation}\label{3.10}
\label{w11}
|| h\pa_{x_j} u_j||^2
+ \frac 14 ||(\frac{\pa\Phi}{\pa x_j})\; u_j||^2\geq \frac h2 \int \frac{\pa^2\Phi}{\pa x_j\,\pa x_j} u_j^2 dx\;,
\end{equation}
using the standard commutator argument but this is of no interest because
this does not give any new inequality. In order to go further, we
introduce a possibly $h$-dependent $\epsilon$ with $0<\epsilon\leq 1$ and get first, using (\ref{3.9}) and (\ref{3.10}),
\begin{equation}
\label{w12}
\begin{array}{ll}
\langle w_j u_j \;|\; u_j\rangle&= (1-\epsilon)\langle w_j u_j \;|\; u_j\rangle+ \epsilon \langle w_j u_j \;|\; u_j\rangle\\
& \geq \epsilon || h\pa_{x_j} u_j||^2
+ (2-\epsilon) \frac h2 \int \frac{\pa^2\Phi}{\pa x_j\,\pa x_j} u_j^2 \;dx\;.
\end{array}
\end{equation}
Let us treat the case $\nu_j=0$. We introduce the following decomposition of the phase $\Phi$
\begin{equation}
\label{w14}
\Phi (x) = \sum_{j=1}^m \lambda_j x_j^4 + \frac J2 \sum_{j\sim k}|x_j-x_k|^2
=: \Phi_d + \Phi_i\;,
\end{equation}
where $\Phi_d$ is the sum of the single spin potentials $v_j$ defined by $v_j(t)=\lambda_j t^4$,
$$
\Phi_d(x) =\sum_{j=1}^m v_j(x_j)\;.
$$
We rewrite (\ref{w12}) in the form
\begin{equation}
\label{w15}
\begin{array}{l}
\langle w_j u_j \;|\; u_j\rangle \\
\quad \geq \epsilon h \left(|| h^\frac 12 \pa_{x_j} u_j||^2
+\frac{2-\epsilon}{2\epsilon} \int \frac{\pa^2\Phi_d}{\pa x_j\,\pa x_j} u_j^2 dx\right) +
(1-\frac \epsilon 2)h \int \frac{\pa^2\Phi_i}{\pa x_j\,\pa x_j} u_j^2
dx.
\end{array}
\end{equation}
We observe now the property that $$|| h^\frac 12 \pa_{x_j} u_j||^2
+\frac{2-\epsilon}{2\epsilon} \int \frac{\pa^2\Phi_d}{\pa x_j^2}
u_j^2 dx - \frac 12 \int \frac{\pa^2\Phi_i}{\pa x_j^2} u_j^2
dx$$
is the quadratic form attached to a new ``harmonic oscillator'' (in the $x_j$ variable) and a new commutator argument, using for the first time the structure of $\Phi_d$, gives
\begin{equation}
\label{w16}
|| h^\frac 12 \pa_{x_j} u_j||^2
+ \frac{2-\epsilon}{2\epsilon}\int \frac{\pa^2\Phi_d}{\pa x_j\,\pa x_j}
u_j^2 dx
\geq h^\frac 12 (12 {\underline \lambda}\frac{2-\epsilon}{2\epsilon})^\frac 12
\;\int u_j^2 \;dx\;.
\end{equation}
We realize that $\frac{h}{\epsilon}$ has to be chosen sufficiently large in order to control $- \frac 12 \int \frac{\pa^2\Phi_i}{\pa x_j^2} u_j^2
dx$.\\
We consequently look for $0<\epsilon<1$ in the form $\epsilon = \frac{h}{C_1}$ with $C_1$ to be determined
large enough and get
\begin{equation}
\label{w17}
\langle w_j u_j \;|\; u_j\rangle \geq\frac{1}{C_1} h^2 \left((6{\underline \lambda}
C_1)^\frac 12 - J \right) ||u_j||^2 + h \int \frac{\pa^2\Phi_i}{\pa x_j\,\pa x_j} u_j^2 dx\;.
\end{equation}
The constant $C_1$ is now chosen in order to get
\begin{equation}
\label{18}
(6{\underline \lambda}
C_1)^\frac 12 - J >0\;.
\end{equation}
\noindent Returning to (\ref{w6}), we obtain the existence of $C$
for which
\begin{equation}
\label{w18}
\begin{array}{ll}
\langle W_1 u\;|\; u\rangle_{L^2} &\geq \frac{h^2}{C}\sum_{j} ||u_j||^2 \\
&
+ h \sum_{j, k}\int \frac{\pa^2\Phi_i}{\pa x_j\,\pa x_k} u_j\;u_k\;
dx\;.
\end{array}
\end{equation}
But $\Phi_i$ is convex and we have finally the existence of $C$ such
that
\begin{equation}
\label{w19}
\langle W_1 u\;|\; u\rangle_{L^2}\geq \frac{h^2}{C} ||u||^2\;,
\end{equation}
as announced in the theorem.\\
The case when $\nu_j\neq 0$ is then easily obtained by a variant of
the argument leading to (\ref{w16}).
\begin{remark}:\\
As observed by V. Bach, T. Jecko and J. Sj\"ostrand in recent
discussions (see also \cite{BaJeSj}),
the omission in the proof, of the positive terms $\sum_{j\neq k}
||X_k u_j||^2$, when going from (\ref{w5}) to
(\ref{w6}) will surely limit the class
of interactions in consideration. We hope to come back to this point
elsewhere \cite{He10}.
\end{remark}
\begin{remark}:\\
The condition ``~$h$ small enough~'' appears when we assume that $\epsilon
<1$ in our estimates.
\end{remark}
\begin{remark}
\label{rem12}:\\
We have only used in the proof the property that the interaction phase $\Phi_i$ is convex and that
$|\pa^2\Phi_i/\pa x_j^2|$ is uniformly bounded on $\rz^m$.
\end{remark}
\begin{remark}
\label{rem13}:\\
The paper by Sokal \cite{Sok} treats similar models but an important assumption
in the argument seems, when the interaction potential is given by
$$\Phi_i(x) = \sum_{jk} J_{jk} x_j \, x_k \;,$$ the condition that
$J_{jk}\leq 0$ for $j\neq k$. Our assumption is simply a
"weak" convexity assumption. This convexity of the interaction appears also in
a recent contribution by A\&A. Antoniouk \cite{AA}.\\ The other point that we
have to explore is when
$x_j\in
\rz^n$ ($n>1$).
The use of the GHS and zero-field Lebowitz inequalities is only possible
for $n\leq 4$. Our approach apparently does not meet
such a restriction but this will probably give weaker results.
\end{remark}
\section{A proof of the correlation decay without the Maximum
Principle}
Inspired by a recent
paper
by J. Sj\"ostrand \cite{Sj1994b}, we use only an $L^2$ theory and
avoid the use of
the Maximum Principle which was playing an important role in
\cite{Sj1994a} or \cite{HeSj1993}. This was also used in a somewhat different
context by A. Naddaf and T. Spencer \cite{NaSp} (Theorem B).\\
This will permit us to weaken the assumption
of convexity. We consider only the case when the lattice is of
dimension $1$ but this is only for simplification and we could also
analyze correlations attached to periodic lattices in $\zz^d$ ($d>1$).\\
The starting point (we take for simplification $h=1$ and assume,
after renormalization, $\int
\exp - \Phi\; dx=1$) is the formula for the correlation
\begin{equation}
\label{c1}
\cov (f,g) = \left(\int\left[( A_1^{-1}) \nabla f \right]\cdot \nabla g\;
\exp -\Phi\; dx \right)
\end{equation}
We have in mind to take $f=x_i$, $g=x_j$ with $|i-j|$ large but much smaller than the size of the lattice $|\Lambda|$. We recall that we first
consider the thermodynamic limit $|\Lambda|\ar +\infty $ and then the
behavior $|i-j|$ large. We consider for
simplicity $\Lambda := \zz/m\,\zz$ that we identify with $\{1,\cdots,m\}$.\\
The idea\footnote{
This is actually not different in spirit from the much older
techniques by Combes-Thomas introduced for the study of the decay of
eigenfunctions \cite{CT}.}, which was already present in \cite{Sj1991b},
\cite{HeSj1992a}, \cite{HeSj1993} and also in the more recent \cite{NaSp}
or
\cite{BaJeSj}, is to introduce weighted spaces
$\ell^2_\rho (\zz/ m \,\zz)$, for suitable strictly positive weights
satisfying
\begin{equation}\label{poids2} \exp -\kappa \leq \rho (\ell) / \rho(\ell + 1) \leq \exp \kappa
\;,\end{equation}
with $\kappa$ to be determined later.\\
For a given $j$ satisfying
$$ 1\leq j\leq \frac m2\;,$$
we are mainly thinking of weights of the form $$\rho_{j}(\ell) = \exp
\left(\kappa \sup [0, (\inf (\ell -1, 2 j-\ell)]\right)$$ or $$\rho_{j}(\ell) = \exp -
\left(\kappa \sup [0, (\inf (\ell -1, 2 j-\ell)]\right)\;.$$
Let us now associate with a given weight $\rho$ the $m\times m$
diagonal matrix $M$ defined by
\begin{equation}
\label{c2}
M_{k\ell} = \delta_{k\ell}\; \rho(\ell)\;.
\end{equation}
For arbitrary slowly increasing functions $f,g$, we can rewrite (\ref{c1}) in the form
\begin{equation}
\label{c3}
\cov (f,g) = \left(\int\left(( M^{-1}A_1^{-1}M) M^{-1} \nabla f\right)
\cdot
\left( M \nabla g \right)\exp -\Phi\; dx \right)
\end{equation}
and we deduce the estimate
\begin{equation}
|\cov (f,g)| \leq || M^{-1}\, A_1^{-1}\, M ||\cdot || M^{-1} \nabla f
||\cdot ||M \nabla g||\;.
\end{equation}
We now take $f(x) = x_1 $, $g(x) = x_j$ and $j<\frac m2$ and choose $\rho_j$ as above so that (\ref{poids2}) is satisfied. We immediately observe that for this choice
\begin{equation}
\label{c5}
|| M^{-1} \nabla f||=1\;,\; || M \nabla g|| = \exp - \kappa (j-1)\;.
\end{equation}
Everything is then reduced to the control of $ M^{-1}\, A_1^{-1}\, M
$
in weighted $L^2$- norms. We have only here to analyze the effect of
the ``~distorsion~'' by $M$. This will be done by a simple perturbation
argument, once we have characterized the domain of the selfadjoint operator $A_1$
and verified that the domain is conserved in the distorsion. This is easily done
in the case of our example (see \cite{Jo} for more general situations).
We observe that for this example (cf (\ref{3.1}))
\begin{equation}\label{c5a}
\begin{array}{l}
||\Hess \Phi(x) - M^{-1} \Hess \Phi(x) M ||_{\Lg(\ell^2)} = \\
||\Hess \Phi_i(x) - M^{-1} \Hess \Phi_i(x) M ||_{\Lg(\ell^2)} \;.
\end{array}
\end{equation}
In this example, observing that the coefficients of $$
\delta_M(\Hess \Phi_i):= \Hess \Phi_i(x) - M^{-1} \Hess \Phi_i(x) M
$$
vanish if
$k\not\sim \ell$, it is immediate to get, uniformly with respect to $m$, that
\begin{equation} \label{c6}
||\delta_M(\Hess \Phi_i) ||_{\Lg(\ell^2)} \leq \Jg \sup_{\ell\sim k} |
(1- \frac{\rho(\ell)}{\rho(k)})| = \Og (\kappa)\;.
\end{equation}
We now estimate the operator $ M^{-1}\, A_1^{-1}\, M$. An
immediate computation gives
\begin{equation}
\label{c7}
M^{-1} A_1^{-1} M = A_1^{-1} [ I + \delta_M (\Hess \Phi) A_1^{-1}]^{-1}\;,
\end{equation}
where $\delta_M(\Hess \Phi)=\delta_M(\Hess \Phi_i)$ is now
considered as an operator (of order $0$) on the $L^2$ $1$-forms. But the
norm of this operator is $\Og (\kappa)$
according to (\ref{c5a}) and (\ref{c6}). We finally obtain the existence
of $C$ such that,
if $0<\kappa < \frac 1C \rho_1$, then
\begin{equation}
\label{c9}
|| M^{-1} A_1^{-1} M || \leq \frac{1}{\rho_1}
[ 1 - C \frac{\kappa}{\rho_1}]^{-1}
\end{equation}
We are done and this gives more generally, each time that
some lattice-independent lower bound
of $\rho_1$ (the bottom of the spectrum of $A_1$) is
available, a general scheme to get the decay for the correlation
(without use of the Maximum Principle).\\
Returning to the semiclassical situation, and following the proof with
respect to $h$, we obtain the following
\begin{theorem}:\\
Under the same assumptions as in Theorem \ref{ThWitten}, there exists
$D$
and $h_0$, such that the correlation pair function
$\Cov^{(m)}(x_1,x_j)$, for any pair $(j,m)$ s.t.
$$
1\leq j\leq \frac m2\;,
$$
and any $h$ s.t. $0\mu_2$ are the two largest eigenvalues of
this compact operator, which is called in statistical mechnics ``~transfer operator~''.
Let us
analyze the case when
\begin{equation}\label{t1}
\Phi(x) =\lambda \sum_{j=1}^m x_j^4 + \nu \sum_{j=1}^m x_j^2 + \frac14
\sum_{j=1}^m\;
|x_j - x_{j+1}|^2\;.
\end{equation}
Here $\lambda$ is strictly positive and $ \nu$ may be of indefinite sign and
$h$-dependent.\\
We are interested in the correlation pair function that is
$$
\Cov ^{(\infty)} (x_1,x_j) =: \lim_{m\ar +\infty} \left(\int x_1 x_j \exp -\frac{\Phi(x)}{h} \;dx \right)/\left(\int \exp -\frac{\Phi(x)}{h} \;dx \right)\;.
$$
It is easy to prove that this correlation pair function behaves like $(\mu_2/\mu_1)^j$.
This operator $K_v$ takes the form
\begin{equation}\label{t2}
c(h)\exp - \frac{\lambda x^4+ \nu x^2}{2h}\;\cdot\;
\exp h \frac{d^2}{dx^2}\;\cdot\;\exp - \frac{\lambda x^4+ \nu x^2}{2h}\;.
\end{equation}
We shall analyze the ``splitting'' $\mu_2/\mu_1$ between the two
first largest eigenvalues of the transfer operator. We will
be rather sketchy and leave the details
to the reader. We recall (cf for example \cite{He1995b}) that
in the case $\lambda =0$, $\nu > 0$, a dilation $x= h^\frac 12 y$ reduces the problem
to an $h$-independent ``Kac'' harmonic oscillator for which the
splitting is explicitely computable. In the case when $\lambda >0$, we
use another dilation and introduce the change $x = h^\frac 13 y$ which
leads
to the new (unitarily equivalent) operator
\begin{equation}\label{t3}
c(h)\exp -\frac 12\; h^\frac 13 \left(\lambda y^4+ \nu\,h^{-\frac 23}y^2\right)\;\cdot\;
\exp h^\frac 13 \frac{d^2}{dy^2}\;\cdot\;\exp -\frac 12\; h^\frac 13 \left(\lambda y^4+ \nu\,h^{-\frac 23}y^2\right)\;.
\end{equation}
If $\nu$ satisfies for some constant $C$ the condition
\begin{equation}
\label{t4}
\nu h^{-\frac 23}\geq C\;,
\end{equation}
then the possible non-convexity due to the presence of $\nu$ has no
effect. One finds an estimate of the splitting in the form
\begin{equation}
\label{t5}
\frac{\mu_2}{\mu_1} \sim \exp - D h^\frac 13
\end{equation}
where $D$ is a smooth function of $\lambda$ and
$\tilde \nu = \nu h^{-\frac{2}{3}}$.\\
The comparison between the Kac operator and the Schr\"odinger
operator can be done by the Trotter-Kato formula (See for example \cite{He1994d}). Modulo an error of $o(h^\frac 13)$ (cf in the quartic case \cite{I} or \cite{DS}), this comparison leads to the study of the operator
$ \exp - h^\frac 13 S$ where
$$ S:= - \frac{d^2}{dy^2}+ \lambda y^4+ \nu\,h^{-\frac 23}y^2\;.
$$
The study of the Schr\"odinger operator $S$ is relatively standard and was analyzed for example in
\cite{He92}. \\
This gives eventually the following decay for the correlation
\begin{proposition}:\\
Under the condition that $\nu$ satisfies (\ref{t4}), there exists $D$, $C_1$ and $h_0$ such that the correlation decays, for $h