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\title{Remarks on decay of correlations
and Witten Laplacians III\\
Application to logarithmic Sobolev inequalities.}
\author{Bernard Helffer\\ UA 760 du CNRS,\\
D\'epartement de
math\'ematiques, Bat 425\\
F-91405 Orsay C\'edex, FRANCE}
\date{December 26, 1997}
%\makeindex
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\maketitle
%
%RESUME
%
\begin{abstract}
This is the continuation\footnote{A first version of these notes was
distributed at the end of October 97.} of previous notes on the subject referred as
\cite{He4} and \cite{He5}. The main application treated in Part I
was a semi-classical one. The second application was more perturbative
in spirit and gave very explicit explicit estimates for the lower bound
of the Witten Laplacian in the case of a quartic model. \\
We shall develop in this third part a remark given in the
second part concerning the possibility of relating our studies of the
Witten Laplacian with the existence of uniform logarithmic Sobolev inequalities
through a criterion of B. Zegarlinski. More precisely, we shall show
how to control the decay of correlations uniformly with respect to
various parameters. \end{abstract}
% %FIN DE RESUME %
\section{Introduction}
In the recent years, a new insight has been given in the study of the decay
of the correlation pairs, through the analysis of a Witten Laplacian on $1$-forms
(\cite{Sj2}, \cite{He4}, \cite{NaSp}, \cite{AA2}).
This gave not only a nice way to recover the Brascamp-Lieb inequality \cite{He1}
but
also permits the analysis of non-convex situations (\cite{Sj1}, \cite{BaJeSj},
\cite{He2}, \cite{He3}, \cite{He4} and \cite{He5}). In this spirit, it is
natural to analyze if this approach gives new results concerning logarithmic Sobolev
inequalities in the non convex case. The main point would be to get
an extension of the Bacry-Emery criterion \cite{BaEm} in the case of non-convex situations
by relating the logarithmic Sobolev best constant to the lowest eigenvalue of
ths Witten Laplacian. We discuss this question
in \cite{He2} and \cite{He3} in connection
with recent studies of Antoniouk-Antoniouk \cite{AA1}, \cite{AA2} but
do not obtain decisive results outside the case when the
gradient of the interaction is bounded and small.\\
Motivated by discussions with T.~Bodineau and a talk by B.~Zegarlinski
in Brisbane (July 97), we consider here another approach initiated by Strook-Zegarlinski \cite{StZe1}, \cite{StZe2}
and
continued by Zegarlinski \cite{Ze} consisting in deducing
uniform logarithmic Sobolev inequalities
from the existence of uniform decay estimates for the
correlations. This is what is obtained
here through the Witten Laplacian approach.\\
More precisely our aim is to analyze the thermodynamic properties of the measure
$\exp - \Phi^{\Lambda,\omega}(X)\;dX$
in the case when $\Phi^{\Lambda,\omega}$, which is associated with cubes
$\Lambda\subset
\zz^d$ and some $\omega\in \rz^{\zz^d}$ defing the boundary condition, has the form, for $X\in \rz^\Lambda$,
\begin{equation}\label{1.1}
\Phi^{\Lambda, \omega}(X) = \sum_{j\in \Lambda} \phi (x_j)
+ \frac{\Jg}{2} \sum_{(\{j \}\cup\{ k \}) \cap \Lambda \neq \emptyset \;,\;j\sim k} |z_j-z_{k}|^2
\end{equation} where
\begin{itemize}
\item $X=(x_j)_{j\in \Lambda}$,
\item
$\phi$ is a one particle phase with at least quadratic increase at
$\infty$,
\item
\begin{equation}\label{1.2}
\begin{array}{lll}
z_j &= x_j &\mbox{ if }j\in \Lambda\\
z_j&=\omega_j &\mbox{ if }j\not\in \Lambda\;.
\end{array}
\end{equation}
\item $j\sim k$ means that $j$ and $k$ are nearest neighbors\footnote{In our
previous studies \cite{He4}, \cite{He5}, we were mainly analyzing the case when $j$ and $k$ were nearest neighbors in
$\Lambda$
considered as a discrete torus.} in $\zz^d$.
\end{itemize}
We shall sometimes use the following decomposition
\begin{equation}\label{1.2a}
\Phi^{\Lambda,\omega} = \Phi_d^{\Lambda}+ \Jg\; \Phi_i^{\Lambda,\omega}\;,
\end{equation}
with
\begin{equation}\label{1.2b}
\Phi_d^{\Lambda}(X) = \sum_{j\in \Lambda} \phi(x_j)\;,
\end{equation}
and
\begin{equation}\label{1.2c}
\Phi_i^{\Lambda,\omega} (X) =
\sum_{(\{j \}\cup\{ k \}) \cap \Lambda \neq \emptyset \;,\;j\sim k}
|z_j-z_{k}|^2\;.
\end{equation}
We shall also meet the Dirichlet boundary condition which by definition
corresponds to the phase
\begin{equation}\label{1.2d}
\Phi^{\Lambda, D} = \Phi_d^{\Lambda}+\Jg \;\Phi_i^{\Lambda,D}\;,
\end{equation}
with
\begin{equation}\label{1.2e}
\Phi_i^{\Lambda, D} (X) =
\sum_{j,k\in \Lambda \;,\;j\sim k}
|x_j-x_{k}|^2
\end{equation}
If necessary, the dependence on $\Jg$ will be mentioned by the notation\break
$\Phi^{\Lambda, \omega}= \Phi^{\Lambda, \omega,\Jg}$ or
$\Phi^{\Lambda, D}=\Phi^{\Lambda, D, \Jg}$.\\
Let us now state the assumptions on the single-spin phase $\phi$. We assume that $\phi$
is $C^\infty$ and convex at $\infty$, so there exists $C>0$
such that
\begin{equation}\label{1.3}
\phi''(x)\geq \frac 1C\;,\;\forall x\in \rz\mbox{ s.t. } |x|\geq C\;.
\end{equation}
We add also the technical condition that, there exists $\rho>0$ and,
for all $k\in \nz$,
$C_k$ such that,
\begin{equation}\label{1.4}
|\phi^{(k+1)}(x)|\leq C_k <\phi'(x)>^{(1-\rho k)_+}\;,
\end{equation}
where, for $u\in \rz^n$, $__:=(1+|u|^2)^\frac 12$ and, for $t\in \rz$, $(t)_+:= \max (t,0)$.\\
The typical example will be
\begin{equation}\label{1.3a}
\phi(x) =\frac{1}{12} \lambda x^4 +\frac 12 \nu x^2\;.
\end{equation}
where
the parameters $\lambda$ and $\nu$ satisfy
\begin{equation}
\label{1.4a}
\lambda> 0 \;,\;
\end{equation}
We would like to analyze the possibility of having any sign for $\nu
$. The sum in (\ref{1.2e}) is over the non-oriented pairs of $\Lambda\times \Lambda$.\\
Our main problem will be to analyze the properties of the measure
\begin{equation}
\label{1.5}
d\mu_{\Lambda, \omega}:= \exp -\Phi^{\Lambda,\omega}(X) \; dX
/ \left(\int_{\rz^\Lambda} \exp - \Phi^{\Lambda,\omega}(X)\; dX \right)\;,
\end{equation}
or of the measure
\begin{equation}
\label{1.5a}
d\mu_{\Lambda}:= \exp -\Phi^{\Lambda,D}(X) \; dX
/ \left(\int_{\rz^\Lambda} \exp - \Phi^{\Lambda,D}(X)\; dX \right)\;.
\end{equation}
We shall in particular analyze the covariance associating to
$f,g \in C^{\infty}_{temp}(\rz^\Lambda)$
\begin{equation}
\label{1.6}
\cov_{\Lambda,\omega}(f,g) = \langle (f-\langle f\rangle_{\Lambda,\omega})(g-\langle g \rangle_{\Lambda,\omega})\rangle_{\Lambda,\omega}
\end{equation}
where $\langle\; \cdot\; \rangle_{\Lambda,\omega}$ denotes the mean value with respect to the
measure
$d \mu_{\Lambda,\omega}$ and $C^{\infty}_{temp}(\rz^\Lambda)$ is the space
of $C^\infty$ functions with polynomial growth.\\
Similarly, we associate with $\Phi^{\Lambda,D}$ and the measure $ d\mu_\Lambda$
the mean value $\langle\;\cdot\;\rangle_\Lambda$ and
the covariance $\cov_\Lambda$.
Our main theorem is the following
\begin{theorem}\label{Theorem1.1}.
Let $\Phi=\Phi^{\Lambda,D,\Jg} = \Phi_d^\Lambda +
\Jg \Phi_i^{\Lambda,D}$ with $\phi$ satisfying
(\ref{1.3})
and (\ref{1.4}).
Then there exists a constant $C$ and $ \Jg_0 >0$ such that,
for $\Jg \in [-\Jg_0,\Jg_0]$ and for any cube $\Lambda\subset \zz^d$, we
have:
\begin{equation}\label{1.7}
\langle f \ln f \rangle_{\Lambda} - \langle f\rangle_{\Lambda}\ln
\langle f\rangle_{\Lambda}
\leq C \langle \nabla f^\frac 12 \rangle_{\Lambda}\;.
\end{equation}
for all non-negative function $f$ for which the right hand side is
finite.
\end{theorem}
As communicated by B. Zegarlinski,
example (\ref{1.3a})
is also analyzed by N. Yoshida \cite{Yo}.\\
In this case, we say shortly that the uniform logarithmic Sobolev
inequality (ULS inequality) is
true.\\
The proof will be the conjonction of
\begin{itemize}
\item a criterion by Zegarlinski \cite{Ze} relating the existence of ``uniform''
decay estimates with the existence of ``uniform'' logarithmic Sobolev
inequalities,
\item the proof (mainly given in \cite{He5}), that the existence of ``uniform''
decay estimates may be obtained from the existence of an ``uniform''
lower bound for the lowest eigenvalue of a Witten Laplacian on
$1$-forms,
\item the proof of this ``uniform'' lower bound in the case of a weak nearest
neighbors interaction by comparison with a family of ``one-particle''
differential
operators on $\rz$.
\end{itemize}
Although the main techniques were already present in our previous
notes. We think it is worthwhile to follow in detail this problem of
the control of
the uniformity with respect to $\Lambda$ and $\omega$ and to show
that
the assumption of strict convexity at $\infty$ for the single spin
phase is a natural condition under which this technique works.
\section{Lower bound for the spectrum of the Witten Laplacian}\label{Section2}
Let us recall that the Witten Laplacian on $1$-forms attached to the phase
$\Phi=\Phi^{\Lambda,\omega}$
is defined as\footnote{ It was
denoted by $\Delta_{\Phi}^{(1)}$ in \cite{Sj2}.},
\begin{equation}
\label{2.1}
W_1^{\Phi} := \left[\sum_{j\in \Lambda}
\left(- \pa/\pa x_j +\frac 12 \pa \Phi/\pa x_j\right) \left(\pa/ \pa x_j
+\frac 12 \pa \Phi/\pa x_j \right)\right]\otimes I + \;\Hess \Phi\;.
\end{equation}
defined on the $L^2$ $1-$forms with respect to the standard Lebesgue
measure on $\rz^m$, with $m=|\Lambda|$.
The aim of this section is the proof of the following theorem
\begin{theorem}\label{Theorem2.1}:\\
For any cube $\Lambda\subset \zz^d$ and $\omega \in \rz^{\zz^d}$, let
$\Phi:=\Phi^{\Lambda,\omega,\Jg}$ be the
phase on $\rz^\Lambda$ with $\phi$ satisfying (\ref{1.3})-(\ref{1.4}).
There exists $\Jg_0$ and $\sigma_1 >0$, such that the lowest
eigenvalue $\lambda_1^{\Lambda,\omega, \Jg}$,
of the corresponding Witten Laplacian on $1$-forms $W_1^\Phi$,
satisfies, for any
cube $\Lambda$, $\omega\in \rz^{\zz^d}$ and
$ \Jg \in [- \Jg_0, \Jg_0]$,
\begin{equation}
\lambda_1^{\Lambda,\omega, \Jg}\geq \sigma_1\;.
\end{equation}
\end{theorem}
Similarly, we have also
\begin{theorem}\label{Theorem2.1a}:\\
For any cube $\Lambda\subset \zz^d$, let
$\Phi:=\Phi^{\Lambda,D,\Jg}$ be the
phase on $\rz^\Lambda$ with $\phi$ satisfying (\ref{1.3})-(\ref{1.4}).
There exists $\Jg_0$ and $\sigma_1 >0$, such that the lowest
eigenvalue $\lambda_1^{\Lambda,D, \Jg}$,
of the corresponding Witten Laplacian on $1$-forms $W_1^\Phi$,
satisfies, for any
cube $\Lambda$ and
$ \Jg \in [- \Jg_0, \Jg_0]$,
\begin{equation}
\lambda_1^{\Lambda,D, \Jg}\geq \sigma_1\;.
\end{equation}
\end{theorem}
The starting point for the proof is the basic identity
\begin{equation}\label{4.1}
\langle W^\Phi_1 u\;|\; u\rangle_{L^2} = \sum_{j,k} ||X_k\,u_j||^2
+ \sum_{j,k}\int \frac{\pa^2\Phi}{\pa x_j\,\pa x_k} u_j\;u_k \;dX\;,
\end{equation}
with
\begin{equation}
X_j= \frac{\pa}{\pa x_j} + \frac 12 \frac{\pa \Phi}{\pa x_j}\;.
\end{equation}
Let us denote by $w_j^{(0)}$ and $w_j^{(1)}$ the single-spin Witten
Laplacians (respectively on $0$- and $1$- forms) attached to the variable
$x_j$ and the phase on $\rz$
\begin{equation}
\phi_j(x_j):=\Phi^{\Lambda,\omega, \Jg} (X)\;.
\end{equation}
The differential of this phase $\phi_j$ depends actually only on the $z_\ell$ with $\ell\sim j$
and recall that $z_\ell = x_\ell$ if $\ell\in \Lambda$
and $z_\ell = \omega_\ell$ if $\ell \in \zz^d\setminus \Lambda$.
We have indeed
\begin{equation}
\phi_j(t) = \phi(t) +\Jg
\sum_{(\{\ell\}\cup \{j\} )
\cap \Lambda\neq \emptyset\;,\;\ell \sim j} |t-z_\ell|^2
+ {\hat \phi}({\hat z}_j)
\end{equation}
where the last term is independent of $t$ and
will be irrelevant in the discussion. More precisely, the operators
$w_j^{(0)}$ and $w_j^{(1)}$ depend only
on the $z_\ell$ with $\ell\sim j$. \\
It will be quite important that the estimates
which will be proved are independent of these parameters.\\
We note the relations
\begin{equation}
w_j^{(0)}= X_j^*\;X_j\;,
\end{equation}
and
\begin{equation}
w_j^{(1)} = X_j\; X_j^* = X_j^*\;X_j\; + \frac 12 \frac{\pa^2 \Phi}{\pa x_j^2}\;.
\end{equation}
According to the context, we shall see these identities as identities between
differential operators on $L^2(\rz^\Lambda)$ or on $L^2(\rz_{x_j})$ (the other variables
being considered as parameters).\\
With these conventions, we have
\begin{equation}
\langle W_1^\Phi u\;|\; v\rangle_{L^2} = \sum_{j,k\in \Lambda\;,\;j\neq k}
\langle w_k^{(0)}\,u_j\;|\; v_j\rangle
+ \sum_{j\in \Lambda} \langle w_j^{(1)} u_j\;|\;v_j\rangle + \Jg
\; \langle \Hess' \Phi_i\, u\;|\; v\rangle \;,
\end{equation}
where $\Phi_i=\Phi_i^{\Lambda,\omega}$ denotes the interaction phase
and $\Hess'$ means that we consider only the terms outside of
the diagonal of the Hessian, that is such that for $k,\ell \in \Lambda$
\begin{equation}
\begin{array}{lll}
(\Hess' \Phi_i)_{k\ell}& = - 1 &\mbox{ if } k\sim \ell\\
&=0& \mbox{ else }\;.
\end{array}
\end{equation}
Here we observe that $\Hess \Phi^i$ is independent of $z$
and that $\Jg\,\Hess \Phi^i$ corresponds to a perturbation
in $\Og (\Jg)$, where $\Og$ is uniform with respect to $\Lambda$, using the Schur's
Lemma.\\
We shall prove later the following theorem
\begin{theorem}\label{Theorem2.3}.
Let us assume that there exists $\rho_1 >0$ such that, for any $j\in \zz^d$ and any
$z\in \rz^{\zz^d\setminus\{j\}}$,
the operator\footnote{For a given $j\in \zz^d$, the effective parameters are actually
the $z_\ell$ such that $\ell\sim j$,} $w_j^{(1)}$ satisfies
\begin{equation}
w_j^{(1)} \geq \rho_1\;,
\end{equation}
then, for any $\epsilon >0$, there exists $\Jg_0 >0$ such that the Witten Laplacian
$W_1^\Phi$, with $\Phi:=\Phi^{\Lambda,\omega,\Jg}$ or with $\Phi:=\Phi^{\Lambda,D,\Jg}$ ,
satisfies
for any $\Lambda$, $\omega\in \rz^{\zz^d}$ and $\Jg \in[-\Jg_0,+\Jg_0]$,
\begin{equation}
W_1^\Phi \geq (\rho_1 - \epsilon)\;.
\end{equation}
\end{theorem}
We recall that the positivity of $w_j^{(1)}$ is immediate from the definition.
We also observe that it is sufficient to treat the case of a fixed $j_0$ all the families
being unitary equivalent (after a simple change of the names of the parameters).
We shall see that the strict positivity for a fixed value of the parameter $z$ is a consequence
of general arguments
(see \cite{Sj2} and \cite{Jo}), at least for $\Jg\in[-\Jg_0,+\Jg_0]$ with $\Jg_0$ small enough.
The other important point is to verify the condition of
uniformity. This will be done in the next section.
\section{ Uniform estimates for a family of $1$-dimensional Witten Laplacians.}
In the preceding section, we have proved that the proof of a uniform lower bound for the Witten Laplacian $W_1$
can be deduced from the study of one-dimensional Witten Laplacian. In this sense,
we are not far of the Dobrushin's approach as described for example in \cite{AA2}. We want to analyze
\begin{equation}\label{5.1}
w(t,\frac{d}{dt}):= - \frac{d^2}{d t^2} + \frac 14 \left(\phi'(t) + 2\Jg d\; t -
2 \Jg \sum_{0\sim k} z_k\right)^2 + \frac 12 (\phi''(t) + 2\Jg\; d)\;.
\end{equation}
As observed in \cite{He5}, it is enough
to analyze the following family depending on $\Jg$:
\begin{equation}\label{5.3}
w(t,\frac{d}{dt}, \alpha, \Jg):= - \frac{d^2}{d t^2} + \frac 14\left
( \phi'(t) + 2\Jg d\; t -\alpha\right )^2
+ \frac 12 (\phi''(t) + 2\Jg d)\;,
\end{equation}
with
\begin{equation}\label{relation}
\alpha= -2 \Jg \sum_{0\sim k} z_k\;.
\end{equation}
Of course, there is a singular situation when $\Jg=0$ because $\alpha$ becomes
identically $0$
in this case but, because we are interested in uniform properties with
respect with $\Jg$ and $z$, we now forget the relation (\ref{relation}) and
will analyze the family $w(t,\frac{d}{dt}, \alpha, \Jg)$ for any $(\alpha,\Jg)
\in \rz\times [-\Jg_0,+\Jg_0]$.\\
\begin{theorem}\label{Theorem3.1}.
Let $\phi$ be a phase satisfying (\ref{1.3}) and (\ref{1.4}). Then,
there exists $\Jg_0$, and $\rho_1>0$ such that, for any $(\alpha,\Jg)
\in \rz\times [-\Jg_0,+\Jg_0]$, we have
\begin{equation}
w(t,\frac{d}{dt}, \alpha, \Jg)\geq \rho_1\;.
\end{equation}
\end{theorem}
For any fixed pair $(\alpha,\Jg)$, the operator is strictly positive according
to the assumptions (\ref{1.3}) and (\ref{1.4}) and the theorem by J.~Johnsen \cite{Jo}
generalizing \cite{Sj2}. We observe indeed that (\ref{1.4}) implies their condition
through
\begin{equation}
t\phi'(t) \geq \frac 1C t^2\;,\;\mbox{ for } |t|\geq C\;.
\end{equation}
A similar inequality (with another constant $C$ which depends on $\alpha$ and $\Jg$) holds for the phase
$t\mapsto\phi_{\alpha,\Jg}(t):= \phi(t) + \Jg\,d\, t^2 - \alpha t$ under the condition that
$\Jg \geq -\Jg_0$ for some small enough strictly positive $\Jg_0$. It is also clear that
$|\phi'(t)|\geq \frac {|t|}{C}$ , and the operator is with discrete spectrum
and strictly positive for any pair $(\alpha,\Jg)$
with $|\Jg|$ small enough.\\
Let us give an explicit proof of the injectivity (admitting that we
have proved that the
kernel is in $\Sg(\rz)$) because this property is
actually quite easy in
dimension $1$.
\begin{lemma}\label{Lemma3.2}.
If $u$ is in $\Sg(\rz)$ and solution of the equation
$$ - u''(t) + \frac 14 \varphi'(t)^2 \, u(t) + \frac 12
\varphi''(t)\, u(t) =0\;,$$
for a phase $\varphi$ satisfying (\ref{1.3}) and (\ref{1.4}), then $u=0$.
\end{lemma}
For the proof, one first remarks that this solution is also a solution of
$$
u'(t) - \frac 12 \varphi'(t)\, u(t)=0\;,
$$
and consequently given by $u(t) = C \,\exp \frac 12 \varphi(t)$. But
it is then necessarily identically $0$ when it is assumed to be
rapidly decreasing.\\
\noindent This lemma is then applied to $\varphi=\phi_{\alpha,\Jg}$.\\
\noindent The second point is that the lowest eigenvalue depends continuously
on $(\alpha,\Jg)$ by rather simple argument of perturbation.\\
\noindent So the remaining point is a problem of uniformity
as $\alpha\ar \pm \infty$. The specific model
$\phi(t)= \frac {\lambda}{12} \, t^4 + \frac \nu 2 \, t^2$ was treated
directly in \cite{He5} using a semi-classical approach.
We give here an easy argument which is quite general although not
optimal for the specific model.
\\
It is indeed immediate to see from (\ref{1.4}) that
there exists $\tau>0$
such that, for $|\alpha|\geq \tau$ and $|\Jg|\leq \frac 1\tau$
and all $t\in \rz$,
\begin{equation}
\frac 14 (\phi'(t)+ 2\Jg\,d\,t -\alpha)^2 +\frac 12 \phi''(t) + \Jg\,d\geq \frac 1\tau\;.
\end{equation}
This gives immediately a lower bound for the corresponding
operator\break $w(t,\frac {d}{dt},\alpha,\Jg)$, for any $(\alpha,\Jg)$
such that $|\alpha| \geq \tau$ and $\Jg \in [-\Jg_0,+\Jg_0]$
with a possibly new smaller $\Jg_0>0$. Combining this with the locally uniform lower bound,
we have proven the theorem.
\section{
Uniform decay estimates}
We recall the argument developed in \cite{He4} (see
also \cite{Sj2}, \cite{NaSp} (Theorem B) or more recently
\cite{BaJeSj}, \cite{SW}) and
control its uniformity with respect to $\Lambda$
and $\omega$. The analysis in \cite{He4}
was only given for the periodic case and for $d=1$ and we consider here other boundary conditions. Our starting point is the formula.
\begin{equation}
\label{c1}
\cov_{\Lambda,\omega} (f,g) = \left(\int\left[( A_1^{-1}) \nabla f \right]\cdot \nabla g\;
\exp -\Phi\; dX \right) /
\left(\int \exp - \Phi\; dX\right)
\end{equation}
Here $A_1$ is unitary equivalent to $W_1^\Phi$ by conjugation by
the map $U \,\omega = \exp -\frac \Phi 2 \;\omega$, and in particular isospectral.
We have in mind to take $f=x_i$, $g=x_j$
with $i$ and $j$ in $\Lambda$. The idea is reminiscent of \cite{CT} and
consists in the introduction of weighted spaces on $\Lambda$,
associated with
strictly positive weights
satisfying
\begin{equation}\label{poids2} \exp -\kappa \leq \rho (\ell) /
\rho(k) \leq \exp \kappa
\;,\end{equation}
where $\ell \sim k$ (this means that $\ell$ and $k$ are nearest neighbors
in $\zz^d$) and
$\kappa$ will be determined later.
\\
For a given $i\in\Lambda$, the functions $\rho(\ell)= \exp - \kappa d(i,\ell)$
where $d$ is a usual distance on $\rz^d$ satisfy this condition.
For a given $\Lambda\subset \zz^d$ with $|\Lambda|=m$,
let us now associate with a given weight $\rho$ on $\Lambda$ the $m\times m$
diagonal matrix $M_\Lambda$ defined by
\begin{equation}
\label{c2}
M_{k\ell} = \delta_{k\ell}\;
\rho(\ell)\;,\;\mbox{ for }\ell, \,k\in \Lambda\;.
\end{equation}
For any slowly increasing functions $f,g$, we can rewrite (\ref{c1}) in the form
\begin{equation}
\label{c3}
\cov_{\Lambda,\omega} (f,g) =\frac{ \left(\int\left(( M^{-1}A_1^{-1}M) M^{-1} \nabla f\right)
\cdot
\left( M \nabla g \right)\exp -\Phi\; dX \right) }{
\left(\int \exp - \Phi\; dX\right)}
\end{equation}
and we deduce the estimate (we can assume that $\int \exp - \Phi\; dX
=1$ after renormalization)
\begin{equation}
|\cov_{\Lambda,\omega} (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_i $, $g(X) = x_j$ and
choose $\rho=\rho_i=\exp- \kappa d(i,\cdot)$
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 d(i,j)
\;.
\end{equation}
Everything is then reduced to the control, uniformly with respect to $\Lambda$ and $\omega$,
of $ M^{-1}\, A_1^{-1}\, M
$
in suitable $L^2$- 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
under the assumptions (\ref{1.1})-(\ref{1.4}) as proved in \cite{Jo}.
We observe that, for all $X\in \rz^\Lambda$,
\begin{equation}
\begin{array}{l}
||\Hess \Phi(X) - M^{-1} \Hess \Phi(X) M ||_{\Lg(\ell^2)} = \\
|\Jg|\;||\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 $$
\delta_M(\Hess \Phi_i)=: \Hess \Phi_i - M^{-1} \Hess \Phi_i M
$$
vanish if
$k\not\sim \ell$, it is immediate, using Schur's Lemma, to get that
\begin{equation} \label{c6}
\begin{array}{ll}
||\delta_M(\Hess \Phi_i) ||_{\Lg(\ell^2)}& \leq
2 d \,\sup_{\ell\sim k} | (1- \frac{\rho(\ell)}{\rho(k)})|\\
& \leq
2d\,\max\left( (1-\exp -\kappa), (\exp \kappa -1)\right)\\
&=2d \,\theta\;,
\end{array}
\end{equation}
with,
\begin{equation}
\theta:=(\exp \kappa -1)\;,
\end{equation}
and this is clearly uniform with respect to the lattice .\\
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} \left[ I +\Jg \delta_M (\Hess \Phi_i)
A_1^{-1}\right]^{-1}\;,
\end{equation}
where $\delta_M(\Hess \Phi_i)$ is now
considered as an operator (of order $0$) on the $L^2$ $1$-forms.
According to (\ref{c5}) and (\ref{c6}), we finally obtain that,
if $$0<2\Jg d\, \theta \leq \frac 12 \sigma_1\;,$$
then
\begin{equation}
\label{c9}
|| M^{-1} A_1^{-1} M || \leq \frac{2}{\sigma_1}
\end{equation}
We have obtained the following~:
\begin{theorem}\label{Theorem4.1}.
Under the same assumptions as in Theorem \ref{Theorem3.1}, there exists $\Jg_0>0$,
$C$ and $\kappa$, such that,
for any $\Lambda$, $\omega$ and $\Jg\in[-\Jg_0,+\Jg_0]$, the correlation pair function satisfies
\begin{equation}
|\Cov_{\Lambda,\omega}(x_i,x_j)| \leq C \exp -\kappa d(i,j)\;,\;\forall i,j\in \Lambda
\;.
\end{equation}
\end{theorem}
\begin{remark}.{\rm
The proof gives actually a decay with $\kappa$ such that $\exp
\kappa\;\cdot\;\Jg$ is small enough. This is coherent with the Dobrushin's
approach (See \cite{He3}, for a discussion about the links between the
Dobrushin's approach and the Witten Laplacian approach). The transfer
matrix approach (in the case $d=1$) or the formal
perturbative approach suggest also that $\kappa \asymp \ln \frac 1\Jg$.}
\end{remark}
\section{Proof of the logarithmic Sobolev estimates}
We can just finish the proof by recalling the following theorem by B. Zegarlinski
\cite{Ze}. B. Zegarlinski considers a Gaussian measure on $ \rz^{\zz^d}$ with mean $0$ and a covariance $G$ whose inverse
$A=G^{-1}$ satisfies that there exists $R>0$ such that
\begin{equation}\label{z1}
A_{ij}\equiv 0\mbox{ if } d(i,j)>R\;,
\end{equation}
and that
\begin{equation}
||A||:=\sup_j\sum_i |A_{ij}| < +\infty
\end{equation}
This contains our case, with $R=1$, $A_{ii}=\Jg$ and $A_{ij}=-1$ when
$i\sim j$.\\
The phase $\phi$ is a semibounded function which can be represented
as the sum
\begin{equation}\label{z2}
\phi = v + w\;,
\end{equation}
where $w$
has a bounded first and second derivative and $v$ is
with non negative second derivative.\\
It is clear that (\ref{1.3}) permits to get (\ref{z2}).
\begin{theorem}\label{Theorem5.1}.
Let us assume that (\ref{1.3}) and (\ref{1.4}) are satisfied.
Suppose, that for some $\Jg$, there are constants $C$, $\kappa \in ]0,+\infty[$ such that for any sufficiently large
cube $\Lambda_0 \subset \zz^d$ and any $\omega\in \rz^{\zz^d}$ we have
\begin{equation}
|\cov_{\Lambda_0,\omega}(x_i,x_j)|\leq C\exp - \kappa d(i,j)\;,\;\forall i,j\in \Lambda_0\;.
\end{equation}
Then there exists a constant $c\in ]0,+\infty[$ such that for any cube $\Lambda\subset
\zz^d$ we have
\begin{equation}
\langle f \ln f \rangle_{\Lambda}\leq 2c \langle |\nabla f^{\frac 12}|\rangle_\Lambda
+ \langle f \rangle_{\Lambda}\; \ln \langle f \rangle_{\Lambda}\;,
\end{equation}
for all nonnegative functions $f$ for which the right-hand side is finite.
\end{theorem}
Combining all the statements Theorems \ref{Theorem2.1},
\ref{Theorem2.3},
\ref{Theorem3.1}, \ref{Theorem4.1}
and \ref{Theorem5.1}, we get Theorem \ref{Theorem1.1}.
\begin{remark}.
{\rm The preprint \cite{Yo} considers generalizations going in other
directions. Let us mention some of them where our proof is still
relevant.
We can consider more generally, for any $h\in \zz^d$, the phases
$\Phi^{\Lambda,D,\Jg,h}(X) = \Phi^{\Lambda,D,\Jg}(X)
+ \sum_{i\in \Lambda} h_i\cdot x_i$ and get the previous result
uniformly
with respect to $h$.\\
We can also consider more generals $A_{ij}$ satisfying the assumptions
of Zegarlinski \cite{Ze}. The condition on $\Jg$ is then replaced by a
condition on $||A||$.\\
N. Yoshida mentioned also the case when, for fixed $\Jg$, one
considers weak perturbations of convex situations. Here the techniques
and results of our previous papers \cite{He4} and \cite{He5} can be
used
for verifying the assumptions of the Theorem 5.1 of \cite{Ze}.\\
On the other hand, our assumptions on the single spin phase
seem weaker than in Yoshida \cite{Yo} (see for example his Lemma 3.3). As
discussed
with him\footnote{Personal e-mail exchange in November 1997} quite recently, the use
of GHS inequality seems important. This makes reasonable extension of
the quartic model to other polynomials like
$$\phi(x)= a_{2n}x^{2n}+...+a_{2}x^{2}$$
with non negative $a_{4},\cdots,a_{2(n-1)}\geq 0$ and
$a_{2n}>0$.}
\end{remark}
\section{The case when the one particle phase is defined on $\rz^N$}\label{Section6}
Some of the methods for proving decay estimates use standard
estimates in statistical mechanics (for
example GHS) where the dimension $N$ of the space on which the one particle phase
is defined plays a role (for example the condition $N\leq 4$ is
mentioned by A.~Sokal in \cite{So}, see the discussion after his
Theorem 3). We shall show here that, in the Witten
Laplacian approach, the extension from $N=1$ to general $N$ is only a minor difficulty. Let us assume
indeed that
our phase $\phi$ is now defined on $\rz^N$ and satisfies the two
following natural
conditions.
\begin{itemize}
\item
There exists \footnote{We note
that this condition implies the superstability for the
corresponding phase $\Phi$ defined in (\ref{1.1}).} $C$ such that, $\forall
x\in \rz^N$ with $|x|\geq C$, we have
\begin{equation}\label{6.1}
\Hess \phi \geq \frac 1C\;.
\end{equation}
\item
There exists $\rho>0$ and, for all $\beta \in \nz^N$, a constant $C_\beta$ such that
\begin{equation}\label{6.2}
|\pa_x^\beta\nabla\phi (x)| \leq C_\beta
<\nabla \phi (x)>^{(1-\rho|\beta|)_+}\;.
\end{equation}\end{itemize}
The second condition is probably technical.
\begin{theorem}.
Under these two conditions (\ref{6.1}) and (\ref{6.2}), there exists $\Jg_0>0$
such that, for $|\Jg|\leq \Jg_0$, the Witten
Laplacian on the $1$-forms attached to the phase $\Phi$ on
$\rz^{N|\Lambda|}$
is uniformly strictly positive with respect to the cubes $\Lambda$,
the boundary conditions and $\Jg\in[-\Jg_0,+\Jg_0]$.\end{theorem}
The basic example satisfying these assumptions appears in the so
called
Lattice vector field theory (See \cite{Fr}). We take $N=4$, $d=4$ and
\begin{equation}\label{6.3}
\phi (x)= \frac {1}{12}\lambda |x|^4 +\frac{\nu}{2} |x|^2\;,
\end{equation}
with $\lambda>0$ and $\nu < 0$.\\
The proof is a consequence of the following
\begin{lemma}.
Let $\phi$ satisfy (\ref{6.1}) and (\ref{6.2}) and
$w (x,\pa_x,\alpha, \Jg)$ the Witten Laplacian on $1$-forms on
$\rz^N$
attached to the phase
$$\rz^N\ni x\mapsto \phi_{\alpha,\Jg}(x) = \phi(x) + 2 \Jg d|x|^2 -
\alpha\cdot x\;,
$$
where $\alpha\in \rz^N$ and $\Jg \in ]-\frac1C, +\frac 1C[$.\\
Then there exists $\Jg_0>0$ and $\varrho_1>0$ such that, for all
$(\Jg,\alpha)$
in\break $]-\Jg_0,+\Jg_0[\times \rz^N$, the lowest eigenvalue of $w
(x,\pa_x,\alpha, \Jg)$ is larger than $\varrho_1$.
\end{lemma}
\noindent {\bf Proof}.\\
The Witten Laplacian on $1$-forms on $\rz^N$ attached to the phase
$\phi_{\alpha,\Jg}$ takes now the form
$$
w(x,\pa_x,\alpha, \Jg):= (-\Delta + \frac 14 |\nabla \phi_{\alpha,\Jg}|^2 - \frac 12 \Delta
\phi_{\alpha,\Jg})\otimes I + \Hess \phi_{\alpha,\Jg}\;.
$$
We introduce a partition of unity starting of a
$C^\infty$ function $\chi (x)$ satisfying \break $0\leq \chi\leq 1$, equal to $1$ on
$B_{\rz^N} (0,1)$ and with
support in $B_{\rz^N} (0,2)$. We introduce also a (large) parameter
$R$
and define
$$
\chi_{1;R}(x)= \chi(\frac{x}{R})\;,\; \chi_{2;R}(x) = \sqrt{1 -
\chi_{1;R}(x)^2}\;.
$$
We use the standard identity associated to this partition of unity
$$
\langle w \,u\;|\; u\rangle_{L^2(\rz^N)}
= \langle {\hat w } \,(\chi_1\, u)\;|\;
(\chi_1\,u)\rangle_{L^2(\rz^N)} +
\langle
{\hat w}\,( \chi_2\,u)\;|\; (\chi_2 \,u)\rangle_{L^2(\rz^N)}\;,
$$
where ${\hat w} = w\; -\; \; (|\nabla \chi_1|^2 - |\nabla \chi_2|^2)
\otimes I $.\\
We observe also that ${\hat w}\, - \,w = \Og(\frac{1}{R^2})$. We first
observe that if
$R \geq C $ large enough such that $\langle w (\chi_2 \,u)\;|\;\chi_2
\,u\rangle \geq \frac {1}{C}||\chi_2 u||^2\;.$
We can then find $R_0\geq C$ and $0<\Jg_0<\frac 1C $ such that, for any
$\Jg\in[-\Jg_0,+\Jg_0]$, any $\alpha\in\rz^N$ and $R\geq R_0$
\begin{equation}\label{6.4}
\langle{\hat w} \,(\chi_2\, u)\;|\;(\chi_2\,
u)\rangle \geq \frac {1}{2C}||\chi_2\, u||^2\;.
\end{equation}
We fix now $R$. It is easy to get, for any $A >0$, the existence of $D$ such that
for all $(\alpha,\Jg)\in \rz^N\times[-\Jg_0,+\Jg_0]$ such that
$|\alpha|\geq D$, we have
\begin{equation}\label{6.5}
\langle{\hat w}\,( \chi_1\, u)\;|\;(\chi_1\,u)
\rangle \geq A ||\chi_1 \,u||^2\;.
\end{equation}
Putting (\ref{6.4}) and (\ref{6.5}) together, we have proved the existence of
$\varrho_0>0$, $\Jg_0>0$ and $D>0$
such that,
for all $(\alpha,\Jg)\in \rz^N\times[-\Jg_0,+\Jg_0]$ with
$|\alpha|\geq D$, we have
\begin{equation}
\langle w (x,\pa_x,\alpha, \Jg)\, u\;|
\;u\rangle \geq \varrho_0 ||u||^2\;.
\end{equation}
We now only need the local control of the lower bound with respect to
$(\alpha,\Jg)$. We observe that, for any $(\alpha,\Jg)$, the lowest
eigenvalue of $w$ is strictly positive (See \cite{Jo}) and continuous
(under the assumptions (\ref{6.1}) and (\ref{6.2}) with
respect to $(\alpha,\Jg)\in \rz^N\times[-\Jg_0,+\Jg_0]$. This proves
the lemma.\\
One deduces the theorem from the lemma along the same lines as when
$N=1$.
\begin{remark}.
{\rm Under the assumptions of the theorem, one can hope to obtain
the uniform logarithmic Sobolev inequality for $\Jg$ small enough and
any $N$. Nethertheless, there would be a need of an extension of
Zegarlinski's Theorem~5.1, for $N>1$.}
\end{remark}
{\bf Acknowledgements}.\\
Our interest for logarithmic Sobolev inequalities comes initially from
discussions with T.~Bodineau.\\
We thank also A.~Val.~Antoniouk, A.~Vict.~Antoniouk, V.~Bach, J.-D~Deuschel,
A.~Guionnet and B.~Zegarlinski for useful discussions or comments on
this subject.\\
We thank also B.~Zegarlinski for communicating the
preprint \cite{Yo} and N.~Yoshida for fruitful exchange of
correspondence.\\
Financial support of the European
Union through the TMR network
FMRX-CT 960001 is gratefully acknowledged.\\
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\end{document}
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